ENVIRONMENTAL POLICIES AND INSTITUTIONS IN FOOD-ENERGY-WATER NEXUS:

STRUCTURAL AND REDUCED FORM APPROACHES

By

Haoyang Li

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Economics – Doctor of Philosophy

Environmental Science & Policy – Dual Major

2019

ABSTRACT

ENVIRONMENTAL POLICIES AND INSTITUTIONS IN FOOD-ENERGY-WATER NEXUS:

STRUCTURAL AND REDUCED FORM APPROACHES

By

Haoyang Li

Food, energy and water sectors are heavily affected by climate change. Multiple new technologies
have been introduced to these sectors as mitigation strategies for climate change. This dissertation
aims to study whether the use of these technologies in these sectors leads to unexpected environ-
mental consequences and whether these consequences could be alleviated by properly-designed
environmental policies and institutions. Different economic analysis tools, such as reduced-form
econometrics, structural econometrics, and structural microeconomics simulation, are applied in
the analysis.

I first study the interaction of environmental policies and technology improvements in food and
water sectors. Irrigation technology is used as an example due to its ability to bridge the two sectors.
Specifically, I study how institutions such as water rights complement new irrigation technologies in
promoting the sustainability of US agriculture. Using data from the Ogallala-High Plains Aquifer
(HPA) region of Kansas, I find that water extraction moderately increases after adoption of the
more efficient LEPA irrigation. About half of the LEPA’s rebound effects arise because adopters
tend to irrigate more land and grow more water intensive crops, with the remaining half attributable
to more intensive irrigation. More importantly, this rebound effect is in general higher for farmers
with larger water rights. A 10% reduction of water rights will reduce water use by 6% in the long
run, and if the reduction targets the majority of the water rights, which lie between 100 and 500
AF, LEPA’s rebound effects decrease by 15.4%. Finally, farmers also have incentive to apply a
small amount of water in order to preserve their water rights, but the associated water waste is
insignificant.

A related policy question is how to effectively promote the adoption of more efficient irrigation
technologies. Complementing the first study in the dissertation, a structural dynamic discrete

choice model is developed next to estimate adoption determinants and aggregate diffusion paths of
LEPA technology in the Kansas HPA. The estimation results confirm that farmers value the option
to delay adoption in the presence of uncertainties and sunk costs. Besides, behavioral biases such as
inertia to change farming practices also significantly delay LEPA adoption by increasing perceived
adoption costs. This inertia is reduced when more neighbors have adopted LEPA. Finally, under the
same level of expected government budget, increase in average adoption probability under profit
gain insurance policies is 10% higher than that under cost share subsidies.

The last study turns to the energy sector, which is critical in mitigating climate change. Specif-
ically, it looks at how the projected fast decline in solar energy capacity costs affects renewable
penetration and CO2 emissions. Using both an analytical model and a dynamic structural simu-
lation model, I show that wind energy capacity investment first increases and then decreases as
solar costs decrease due to the complex interaction of wind and solar capacities. Results indicate
that when solar costs drop from the 2030 projected level to 2050 projected level, CO2 emissions
increase by 12.9% because of the decline in wind energy investment, which indicates that cheaper
solar cost does not necessarily imply a cleaner grid without any carbon policy interventions. The
use of energy storage to increase renewable penetration in order to reduce carbon emissions, as is
suggested in the literature, is economically infeasible as the high storage capacity costs currently
prevent any storage investment. However, if the regulator requires that renewable penetration rate
could not drop below 36%, the maximum renewable penetration obtained under all levels of pos-
sible future solar capacity costs, social welfare from renewable investment in 2050 would increase
by $0.77 billion/year due to a decrease in CO2 emissions. This study illustrates the importance
of a second-best-type policy in the energy sector facing future decline in solar capacity costs if a
carbon tax is not politically available.

ACKNOWLEDGEMENTS

First and foremost, I would like to express my special thanks of gratitude to my major professor,
Dr.
Jinhua Zhao, for his guidance, support and patience through my journey in the graduate
school. He encourages me to be an independent researcher and he is always ready to help when
I encounter difficulties in my study and research. His insightful thoughts and comments are
profoundly important in finishing this dissertation. I would also like to thank the other members
of my guidance committee for their mentorship and inspiration: I thank Dr. Joseph Herriges,
Dr. Jeffrey Wooldridge, and Dr. Kyoo il Kim for their efforts to improve the econometrics in the
dissertation, and I thank Dr. David Hyndman for providing interdisciplinary insights to my work.
I am also grateful for ECON and ESPP faculties, staffs and fellow graduate students. Their
extensive comments and suggestions on the dissertation chapters make my graduate life much
easier. I also thank all my friends at MSU for all the great time we have spent together.

I am deeply thankful to my family for their love, understanding, and sacrifices. Thank you my
parents for encouraging me to follow my dreams and your support during not only my graduate
studies, but also my whole life. Thank you my wife, Wen Lin, for taking care of my life in every
possible way and for cheering me up when I am down. I am forever in their debt.

Finally, I would like to thank the department of Economics and Environmental Science and
Policy Program. I could never make it without the excellent platforms and financial support they
provide.

iv

TABLE OF CONTENTS

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Introduction . . . . . . . . .

Separate Estimation of (2.1) and (2.2)

2.4 Data . . . . . . . . . . . . .
2.5 Estimation Results . . . . . .

Issues of Endogeneity .
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LIST OF TABLES . . . . . . . . . .
LIST OF FIGURES . . . . . . . . . .
INTRODUCTION . .
CHAPTER 1

ROLE OF WATER RIGHTS (WITH JINHUA ZHAO) .
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2.1
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2.2 High Plains Aquifer and Water Rights in Kansas .
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2.3 Model Specifications
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2.3.1 A Dynamic Joint Estimation Framework .
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2.3.3 Mediators of the Effects of LEPA and Water Rights .
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2.5.1 Water Extraction and the Rebound Effects of LEPA .
2.5.2 Mediators of LEPA and Water Rights
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2.5.3 Tests of the Exogeneity of LEPA .
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2.5.4 Decision to Preserve Water Rights .
2.5.5 Robustness Checks . .
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2.6 Policy Simulations of Changing Water Governing Institutions .
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2.7 Conclusions . . . . . . . . .
APPENDICES . . . . . . . . . . . .
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CHAPTER 2 REBOUND EFFECTS OF NEW IRRIGATION TECHNOLOGIES: THE
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APPENDIX A MAXIMUM LIKELIHOOD ESTIMATION OF THE DYNAMIC
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BIBLIOGRAPHY . . . . . . . . . . .
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CHAPTER 3WHAT DRIVES (NO) ADOPTION OF NEW IRRIGATION TECHNOLO-
GIES: A STRUCTURAL DYNAMIC ESTIMATION APPROACH (WITH
JINHUA ZHAO)
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MODEL . . . . .
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APPENDIX B DATA ISSUES .
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APPENDIX C RESULTS OF ROBUSTNESS CHECKS .
APPENDIX D COPYRIGHT PERMISSION .
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3.3.1 LEPA Adoption Profit Gain and Learning .
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3.3.2 LEPA Adoption Costs .

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3.6 Estimation Strategies

3.7 Empirical Results . . . . . .

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3.6.3 Dynamic Discrete Choice Estimation .
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Profit Gain Following LEPA Adoption .
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3.5.1.2 Optimal Water Extraction .

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3.5 Empirical Adoption Model
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DISCUSSION: ENDOGENEITY IN NUMBER OF NEIGHBOR
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APPENDIX E DYNAMIC DISCRETE CHOICE ESTIMATION DETAILS .
APPENDIX F

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APPENDICES . . . . . . . . . . . .

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Introduction . . . . . . . . .

BIBLIOGRAPHY . . . . . . . . . . .
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CHAPTER 4CHEAPER SOLAR, CLEANER GRID? .
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APPENDIX G OTHER POSSIBLE LEPA ADOPTION MODEL SPECIFICATIONS 110
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APPENDIX H PROFIT GAIN TRANSITION RULE SELECTION .
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4.2.3.2 Relationship Between Wind and Solar Energy Capacities .
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4.2.1 Daily Operation Model Setup .
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Under Different Scenarios .

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4.3.1
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4.3 Empirical Simulation Model

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Social Planner’s Operation Problem .
Social Planner’s Investment Problem .
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4.4 ERCOT Data and Parameter Calibration .
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4.6 Results And Discussions
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AGGREGATE PRODUCTION COST CURVE FOR EACH GEN-
ERATION TECHNOLOGY .
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APPENDIX I
APPENDIX J

Capacities . . . . . .

4.7 Conclusions and Future Directions .
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APPENDICES . . . . . . . . . . . .

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BIBLIOGRAPHY . . . . . . . . . . .
CHAPTER 5CONCLUSIONS . . . .

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vii

LIST OF TABLES

Table 2.1: Summary Statistics .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Table 2.2: Dynamic Joint Estimation and Separate Estimation Results . . . . . . . . . . . . 22

Table 2.3: Mediation Test: Marginal Effect of LEPA and Water Rights on Acres Irrigated

(Panel Tobit)

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Table 2.4: Mediation Test: Effect of LEPA and Water Rights on Crop Acres (Linear RE) . . 29

Table 2.5: Mediation Test: Comparison of Estimates with and without Mediators (Marginal

Effects of Panel Tobit)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Table 2.6: Effects of Water Rights on LEPA Adoption . . . . . . . . . . . . . . . . . . . . 35

Table B.1: Sample Mean Comparison for Unbalanced and Balanced Samples . . . . . . . . 43

Table B.2: Estimations for Unbalanced and Balanced Samples . . . . . . . . . . . . . . . . 44
Table C.1: Robustness: Estimating (2.1) for Different Values of ¯S and Samples (Panel Tobit) 48
. 71
Table 3.1: Summary Statistics (1997-2010) . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3.2: Production Function Estimation Results . . . . . . . . . . . . . . . . . . . . . . 84

Table 3.3: Fixed Effect Instrumental Variable Production Function Estimation – First

Stage .

.

.

.

.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Table 3.4: Profit Gain Transition Rule Estimation Results

. . . . . . . . . . . . . . . . . . 87

Table 3.5: Dynamic Parameter Estimation Results

. . . . . . . . . . . . . . . . . . . . .

. 88

Table 3.6: Adoption Probability Elasticities

. . . . . . . . . . . . . . . . . . . . . . . . . 93

Table G.1: Ignoring Adoption Cost Heterogeneity in LEPA Adoption Model . . . . . . . . . 110

Table H.1: Reduced-Form Evidence on Profit Gain Transition Rule Selection . . . . . . .

. 112

Table H.2: Estimation Results with AR(1), AR(2) and AR(3) Transition Rules . . . . . . . . 113

Table 4.1: Effects of Increasing Wind/Solar/Storage Capacity on Et Under Each Scenario . 130

viii

Table 4.2: Analytical Results Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 140

Table 4.3: ERCOT Generation Technology Parameters (2015) . . . . . . . . . . . . . . . . 147

Table I.1: Optimal Energy Storage Production Under Each Scenario . . . . . . . . . . . . . 164

Table I.2: Effect of Changing Exogenous Energy Supply in One Period On Electricity

Output Changes in All Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

ix

LIST OF FIGURES

Figure 2.1: Distribution of Type 1-3 Water Rights

. . . . . . . . . . . . . . . . . . . . . .

Figure 2.2: Annual percentage type 1-3 water rights experiencing over-extraction (1991-2010)

Figure 2.3: Histogram of water extraction of type 1-3 wells (1991-2010)

. . . . . . . . .

.

8

8

9

Figure 2.4: Marginal Effects of LEPA and Water Rights in Mediation Test . . . . . . . . . . 27

Figure B.1: LEPA Rebound Effect for Different Water Right Bins (Discrete) . . . . . . . . . 46

Figure B.2: LEPA Rebound Effect for Different Water Right Levels (Smooth) . . . . . . . . 46

Figure C.1: LEPA Rebound Effect for Different Water Right Bins (Prior 1980, Discrete)

. . 51

Figure 3.1: LEPA Diffusion Path in Kansas HPA . . . . . . . . . . . . . . . . . . . . . . . 72

Figure 3.2: Percent of New LEPA Adopters Who Received Cost-Share Subsidies Among

All New LEPA Adopters by Year

. . . . . . . . . . . . . . . . . . . . . . . . . 73

Figure 3.3: Histogram of ˆα1it in Production Function (upper) and Per-period Profit Dif-

ference Between LEPA and CP across Farmers and Years (lower) . . . . . . . . 86

Figure 3.4: Simulated Average Adoption Year v.s. Observed Adoption Year . . . . . . . . . 91

Figure 3.5: Simulated v.s. Observed Number of Cumulative Adopters (upper) and New

Adopters (lower) .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Figure 3.6: NPV and Real Option Decision Rule Critical Values . . . . . . . . . . . . . . . 98

Figure 3.7: Simulated Adoption Path – NPV and Real Option Decision Rule

. . . . . . .

. 99

Figure 3.8: Effect of Insurance and Cost-Share Payments on Adoption Rate . . . . . . . . . 100

Figure 3.9: Effect of Insurance and Cost-Share Payments on Adoption Rate . . . . . . . . . 101

Figure F.1: Simulated v.s. Observed Number of Cumulative Adopters (upper) and New

Adopters (lower) .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure F.2: Simulated Average Adoption Year v.s. Observed Adoption Year . . . . . . . . . 109

Figure 4.1: Quantity-Price Orderings in the Analytical Model

. . . . . . . . . . . . . . . . 128

x

Figure 4.2: Scenario Boundaries .

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Figure 4.3: Average Renewable Energy Generation Pattern and Average Demand Pattern . . 149

Figure 4.4: Observed and Simulated Hourly Average prices for Four Seasons . . . . . . . . 150

Figure 4.5: Effect of Solar Energy On the Value of Wind Energy . . . . . . . . . . . . . . . 153

Figure 4.6: Effect of Solar Energy On the Value of Energy Storage

. . . . . . . . . . . . . 154

Figure 4.7: Optimal Wind and Solar Capacities Under Different Solar Capacity Cost

. . . . 155

Figure 4.8: Response of Renewable Energy Penetration And Annual Carbon Emissions

to Solar Capacity Cost Decline . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Figure 4.9: Changes in Social Welfare Gain From Renewable Energy Investment Due to

Solar Capacity Cost Decline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Figure 4.10: Response of Energy Storage’s Value With Respect To Solar Capacity Cost Decline159

Figure J.1: Aggregate Production Cost in the Simple Example . . . . . . . . . . . . . . . . 170

xi

CHAPTER 1

INTRODUCTION

The sectors of food, energy and water are related to each other in multiple ways. Greenhouse gas
emissions from the energy sector contribute significantly to climate change, which reduces rainfall
and increases drought occurrence. Not surprisingly, irrigation becomes increasingly important to
raise crop yield. However, as groundwater levels of major aquifers continue declining, energy
expenditures associated with irrigation water pumping go up.

Multiple new technologies have been introduced to these sectors to alleviate the threat of climate
change. In the sectors of food and water, the use of more efficient and energy-saving irrigation
technologies to water crops is encouraged, in hopes of saving the drying aquifers. In the energy
sector, renewable energy technologies such as wind turbine and solar photovoltaic (PV) panels are
gradually being used to generate CO2-free energies in major electricity grids.

However, the use of these new technologies may lead to counterproductive results. Economists
have long recognized the limitations of purely technological solutions to environmental problems,
and argued that resilient and effective institutions and policies can be effective adaptation strategies.
Given this observation, a study on the benefits of combining environmental policies and institutions
with technological improvements in these sectors becomes necessary for effective policy design.
Due to the significant role of renewable energies in preventing climate change and the ability of
irrigation to connect food, energy and water sectors, the analyses in the dissertation are rooted in
the context of irrigation agriculture and renewable energy development, in hopes of extending the
lessons learned to other sectors in the food-energy-water nexus.

Using reduced-form panel data econometrics tools, the second chapter of this dissertation shows
that the use of a more efficient irrigation technology, Low Energy Precision Application (LEPA),
has increased, rather than decreased, groundwater extraction in the Kansas part of the High Plain
Aquifer Area (HPA). The increased water use comes from irrigating more acreages and switching
to more water-intensive crops such as corn, implying that policies reducing water use by corn

1

growers could help ameliorate the rebound effects of LEPA. More importantly, this rebound effect
is higher for farmers with larger water rights.1 Thus, the unexpected water use rebound following
LEPA adoption could also be alleviated by reducing farmers’ water rights. Specifically, a 10%
reduction of water rights will reduce LEPA’s rebound effect by 15.4%. Recently, many areas inside
Kansas HPA are considering to establish Local Enhanced Management Areas (LEMA) to reduce
farmers’ water rights inside the area for a number of years. The results in this chapter provide
empirical support for this policy from the perspective of LEMA’s impact on the social performance
of efficient irrigation technologies.

The third chapter extends the second chapter by comparing the cost-effectiveness of different
adoption promotion policies. Through estimating a structural dynamic discrete choice model,
I show that delaying adoption is valuable in the presence of profit uncertainty and sunk costs,
where profit uncertainty comes from farmers’ uncertainty towards factors that determine LEPA’s
profit gains, such as energy price, crop price and rainfall. Across the U.S., two most commonly
used policies to encourage agricultural technology adoption are cost share subsidies and insurance
policies. While cost share policies direct reduce adoption costs, insurance policies promote adoption
by reducing farmers’ uncertainty towards the new technology’s profit gain. A dynamic simulation
using the estimated structural parameters show that the increase in average adoption probability
under profit gain insurance policies is 10% higher than that under cost share subsidies, given
the same level of expected government budget. Although LEPA diffusion in Kansas has already
completed, the results obtained in the first two chapters are still useful for other regions that are
considering promoting efficient irrigation technology adoption. Specifically, a profit insurance
policy combined with policies that reduce water-intensive crops’ water use or policies that reduce
farmers’ water rights would speed up diffusion while limiting water use rebound.

While the previous two chapters focus on technological and institutional (policy) adaptations
to climate change, the fourth chapter focuses on preventing climate change in the energy sector
through the use of renewable energies. It is widely expected that renewable energy penetration

1A water right is defined as a property right that specifies the maximum annual water extraction allowed for each

irrigation well.

2

(CO2 emissions) will keep increasing (decreasing) facing the projected rapid decline in solar
capacity costs. However, this is not guaranteed if the two emission-free renewable technologies,
wind and solar, interact with each other. Although energy storage could also affect renewable
penetration through affecting the values of wind and solar, investing in storage is not supported
under a wide range of solar capacity costs as storage capacity is too expensive relative to its value.
With both an analytical model and a microeconomics simulation model, I show that the value of
wind energy decreases as solar capacity increases (i.e. they are substitutes) when the level of solar
capacity is already high. Compared with the second chapter where the unexpected consequence
comes from the use of a single technology, the interaction of multiple technologies leads to the
unexpected consequence in this context. As a result, wind capacity investment decreases when
solar capacity cost drops from its 2030 projected level to 2050 projected level if no carbon policies
exist. Correspondingly, renewable penetration reduces by 6% and CO2 emissions go up by 12.9%.
However, under a renewable energy policy that requires a minimum level of 36% renewable energy
penetration, social welfare from renewable energy capacity investment in 2050 would increase by
$0.77 billion/year due to an increase in renewable penetration and a decrease in CO2 emissions.

Taken together, this dissertation demonstrates that purely technological solutions for environ-
mental problems emerging in the food-energy-water nexus might be inefficient. These inefficiencies
may come from the use of a single technology alone, or come from the interactions of multiple
different technologies. However, the reduction in these inefficiencies could be economically sig-
nificant when these technological solutions are combined with properly-designed environmental
polices and institutions.

3

CHAPTER 2

REBOUND EFFECTS OF NEW IRRIGATION TECHNOLOGIES: THE ROLE OF

WATER RIGHTS (WITH JINHUA ZHAO)

2.1 Introduction

Irrigation is a major factor influencing modern agriculture’s capacity to adapt to global climate
change. Facing higher average temperature, greater variability in rainfall and increased likelihood
and severity of droughts, production agriculture will be more dependent on irrigation in order to
meet the increasing global food demand (Zilberman, Zhao, and Heiman, 2012). However, over
60% of the world’s consumption of water is already used for irrigation (FAO, 2016; Wada and
Bierkens, 2014), and there is worldwide depletion of surface and groundwater resources. For
example, the recent California drought has resulted in increased irrigation using groundwater,
leading to substantial groundwater depletion in Central Valley (Howitt et al., 2014). In the High
Plains region, irrigation water use since 1930s has depleted over half of the aquifer capacity
in the southern part of the Ogallala-High Plains aquifer (henceforth HPA) (Haacker, Kendall,
and Hyndman, 2015; McGuire, 2009). Consequently, new irrigation technologies such as Low
Energy Precise Application (LEPA) irrigation and drip irrigation have been called upon to improve
irrigation efficiency, thereby reducing water needs while maintaining or raising agricultural outputs
(Schoengold and Zilberman, 2007; Zilberman, Zhao, and Heiman, 2012).

However, more efficient irrigation technologies do not always reduce water use. Ward and
Pulido-Velazquez (2008) finds that adoption of water conservation technologies may raise water
use due to the conversion of more land into production agriculture. Pfeiffer and Lin (2014a) finds
statistically significant rebound effects of LEPA in HPA: water use increased after farmers switched
from center pivot to LEPA. Such rebound effects are inefficient because of the spatial externalities of
groundwater extraction: when one farmer pumps water, the resulting groundwater depletion raises
the pumping costs of other farmers. The rebound effects together with imperfect property rights

4

governing groundwater use will limit the potential of new technologies in promoting agricultural
adaptation and sustainability.

Economists have long recognized the limitations of purely technological responses to external
changes, and argued that resilient and effective institutions can be effective adaptation strategies
(Zilberman, Zhao, and Heiman, 2012). Examples of such institutions include property rights
over both surface water and groundwater, water markets and trading, and water use organizations
that help regulate water use. At least theoretically, effective institutions can not only promote the
adoption and diffusion of efficient technologies but also limit or even eliminate undesirable rebound
effects of these technologies.

In this article, we use a panel data set in the HPA region of Kansas that includes well level water
use and water rights during 1991-2010 to study the interacting impacts of institutions and new
technologies, focusing on the role of water rights in reducing water use, promoting the adoption
of LEPA, and limiting the rebound effects of LEPA. First, we show that LEPA’s effect of raising
water use is heterogeneous across water rights: higher rebound effects tend to occur for wells
with larger water rights. This result indicates that policies that reduce water right levels could
help ameliorate LEPA’s rebound effects in the long run. Second, we compare and quantify the
channels through which water rights and LEPA adoption affect water use. We show that about
two thirds of water rights’ effect on water use is mediated by irrigated acreages: larger water right
holders tend to irrigate larger areas of land. In contrast, around half of LEPA’s effect on water use
is mediated by both crop choices (i.e. LEPA adopters switching to more water-intensive crops)
and irrigated acreages, and the other half is attributable to LEPA adopters simply irrigating more
intensively. Moreover, the mediating patterns of crop choice and irrigated acreage vary as the water
right level changes. Estimates of the mediation channels not only identify the behavioral changes
that contribute to the rebound effects, but also shed light on other impacts of such behavioral
changes and allow for more effective targeting in formulating policy responses. For example, our
results show that the adoption of LEPA helped raise corn acreage in Kansas, which has important
implications for fertilizer use and associated environmental impacts. Policies that reduce water use

5

by corn growers will also help ameliorate the rebound effects of LEPA.

Finally, Kansas follows a prior appropriation water right system with a three-year use-it-or-lose-
it (UIOLI) clause. We find that farmers have incentive to apply a small amount of water in order
to preserve their water rights even when irrigation is not profit maximizing, and this incentive is
slightly higher after LEPA is adopted. The potential water savings if UIOLI is removed, however,
are rather small.

There is a sizable literature on the effects of new irrigation technologies on water use in
general (Ellis, Lacewell, and Reneau, 1985; Hanak et al., 2010; Khanna, Isik, and Zilberman, 2002;
Scheierling, Young, and Cardon, 2006; Ward and Pulido-Velazquez, 2008; Whittlesey and Huffaker,
1995) and in the HPA region of Kansas in particular (Hendricks and Peterson, 2012; Peterson and
Ding, 2005; Pfeiffer and Lin, 2012; Pfeiffer and Lin, 2013; Pfeiffer and Lin, 2014a; Pfeiffer and
Lin, 2014b). Our work is closest to Pfeiffer and Lin (2014a), which identifies the rebound effects of
LEPA. Our innovations include explicitly modeling water rights and farmer incentives to preserve
water rights, demonstrating the heterogeneity of LEPA’s rebound effects in water right levels, and
quantifying the channels through which the rebound effects arise by estimating the role of such
mediators as crop choices and irrigated acreages. Further, our data span a longer time period (1991
– 2010 in our article vs. 1996 – 2005 in Pfeiffer and Lin (2014a)).

2.2 High Plains Aquifer and Water Rights in Kansas

Our study area is the HPA region of Kansas, where production agriculture relies heavily on
irrigation water withdrawal from HPA. HPA is the largest freshwater aquifer in the world (Miller
and Appel, 1997) and has variable recharge rates from the north to the south. In most of the Kansas
HPA, the recharge rate is low and irrigation is similar to mining a nonrenewable resource.

Kansas has a prior appropriation water right system (Peck, 1995; Peck, 2002; Peck, 2007), with
each water right specifying the maximum amount of water that can be extracted from the HPA
in a given year. The relationship between water rights and irrigation wells can be classified into
four types: a single well assigned with a single water right (Type 1), a single well assigned with

6

multiple water rights (Type 2), multiple wells sharing a single water right (Type 3), and multiple
wells sharing multiple water rights (Type 4). We focus on Type 1-3 wells: they account for about
72% of the total number of wells and the associated water rights during our study period. Type
4 wells are not included in our analysis due to the complex relationship between water rights and
wells. Section 4 and appendix B contain detailed descriptions of the linking procedure and the
associated sample attrition.

The seniority of a water right is determined by the time when the water right application was
recorded. Among the Type 1-3 water rights in our sample, most of the application dates are in
1970s. There is effectively a moratorium on new water right applications after 1990s (Peck, 1995).
Figure 1.1(a) shows the distribution of Type 1-3 water rights by application dates. Since most
applications for water rights occurred in 1970s and almost all water rights in our sample have
application dates that are at least five years earlier than the beginning of our sample period, we
take water rights as exogenous in our empirical estimations. In principle, when water availability
cannot satisfy the demand, a well with a more senior water right can block wells with junior water
rights from extracting water from the HPA. So far this type of blocking has never occurred so that
a farmer’s water use decisions are affected by his own water rights but not by the more senior water
rights of other farmers.

Water rights in Kansas are not always strictly binding. First, water use is self-reported by
farmers. Although water pumping is metered, it is possible that farmers misreport their actual
water use. Second, farmers are allowed to go beyond the water right limits in drought years. Each
year in our sample, about 10% - 30% of the wells reported water use above their water rights.
However, as shown in figure 1.2, this percentage has generally been going down from 1991-2010.
The water right system in Kansas has a three-year use-it-or-lose-it (UIOLI) clause, implying
that a water right is subject to cancellation if no water is used by the associated well(s) for three
consecutive years. Under the Kansas Administrative Procedure Act, farmers can appeal to the
Kansas Department of Water Resources to keep the water rights if they can establish a “due and
sufficient cause for non-use” (Peck, 1995). Such cause includes sufficient rainfall, enough surface

7

Figure 2.1: Distribution of Type 1-3 Water Rights

Figure 2.2: Annual percentage type 1-3 water rights experiencing over-extraction (1991-2010)

water as diversion sources, soil and water conservation such as CRP, etc. The burden of proof for
establishing the due and sufficient cause for non-use lies with the farmers. Given the transaction
costs involved, farmers might have incentive to use a small amount of water even when irrigation
is not needed in order to preserve their water rights. Figure 1.3 shows the histogram of annual
water extraction of Type 1-3 wells in our sample. In addition to a spike in zero water use, there is

8

also a dis-proportionally large share of observations with a positive but small amount of water use.
We will estimate farmer incentives to use a small amount of water in order to preserve their water
rights, and evaluate their impacts on water use.

Figure 2.3: Histogram of water extraction of type 1-3 wells (1991-2010)

2.3 Model Specifications

The irrigation scheduling literature recognizes that a farmer typically irrigates a field multiple
times during a single growing season, and the “optimal” intraseason scheduling depends on the
stages of crop growth, soil characteristics, and irrigation technology (Scheierling, Cardon, and
Young, 1997; Shani et al., 2009; Yaron and Dinar, 1982). During early stages of the growing
season, the farmer might not know for sure how much irrigation water is needed for the reminder
of the season due to weather uncertainties and future crop and energy prices. Constrained by his
water rights, the farmer may decide to limit the amount of water applied during the early season
and leave more options open in order to meet possible high water needs during the later season.
Therefore, water rights can affect total irrigation water use of the entire growing season even when
ex post the farmer ends up using less water than what the water rights allow.

There is a fixed cost associated with each irrigation application, arising from the minimum

9

pressure required in order for the irrigation equipment to start. LEPA reduces the required water
pressure – a reduction from 75 PSI for Center Pivot to 18 PSI for LEPA (Delano, Williams,
and O’Brien, 1997), leading adopters to irrigate more frequently.1 LEPA also greatly reduces
evaporation loss, thereby raising irrigation efficiency and reducing the marginal cost of irrigation,
leading farmers to irrigate more intensively under LEPA than under center pivot. The increased
frequency and intensity of irrigation contribute to LEPA’s rebound effects. These considerations
and the existence of UIOLI clause lead to two hypotheses that we will test:

Hypothesis 1 (H1): The rebound effects of LEPA on water use are partly attributable to increased
irrigation intensity, and are lower for farmers with smaller water rights.

Hypothesis 2 (H2): Farmers have incentives to preserve water rights under UIOLI, and the in-
centives are stronger as future irrigation profitability rises and the associated water rights are
larger.

2.3.1 A Dynamic Joint Estimation Framework

We first develop and estimate a joint model of water use and water right preservation, focusing on
LEPA replacing center pivot, i.e., using wells that have only used one or both technologies during
our sample period. Our unit of analysis is wells, with each well treated as representing a single
farmer. Let yit be the observed water extraction of well i in year t. Farmers first decide whether to
use water from the well in the current year and then decide how much water to use. Let y∗
be the
it
latent variable representing the profit maximizing water extraction level without considering the
fixed cost of irrigation. In reduced form, y∗
depends on the field characteristics, weather conditions,
it
management practices, irrigation technology and water rights. To account for the nonlinear effects
of water rights on water extraction, we divide the water right levels into eight bins, indexed by
j = 1, . . . ,8 : (0,100],(100,200], . . . ,(600,700], and greater than 700 acre-feet (AF), with the first
1Since farmers irrigate multiple times during a growing season, the reduction in the minimum water pressure of
each irrigation run implies that, viewed from the perspective of the entire growing season, the variable cost (or marginal
cost) of irrigation is also lower with LEPA.

10

j = 1) set to be the base/default level. Figure 1.1(b) shows that over 90% of wells have

bin (i.e.
water rights between 100 and 500 AF. Under a linear specification, we have

8


8


j=2

∗
it =
y

R j
i ρj +

LEPAit ∗ R j

i ωj +ACit β1 + si β2 + wit β3 + dtwit β4 + Gg + ϕt + ci + εit, (2.1)

In (2.1), R j
i

j=1
= 1 if well i’s water right falls into the jth water right bin (j = 2,3. . . ,8), and
= 0 otherwise. LEPAit is a dummy variable that equals 1 if the irrigation technology is LEPA
and 0 if the technology is center pivot. We include the interaction of LEPA and R j
to test part of
i
Hypothesis 1, and the coefficient ωj captures the moderating effect of a specific water right bin on
LEPA’s rebound effect. A larger ωj means that LEPA’s rebound effect is higher for wells in the
jth bin. This specification allows us to estimate heterogeneous rebound effects for farmers with
different amount of water right allocations. ACit is a vector of acres of various crops irrigated by
well i in year t. si is a time invariant vector of soil characteristics of the field irrigated by well i.
dtwit is the well’s depth to groundwater, which affects the energy needed to extract the groundwater
for irrigation.2 wit is a vector of weather conditions. Gg and ϕt capture the fixed effects of the
groundwater management district (GMD) well i belongs to and the time fixed effect, respectively.
ci is a well-specific unobserved heterogeneity and εit is an iid error term.

The dynamics of water use arises from the incentive to preserve one’s water rights, which
be the latent variable describing the

depends on the irrigation needs in future periods. Let z∗
it
farmer’s incentive to use water from well i in year t. In reduced form, z∗
it

is specified as:

z∗
it = rci + 1(yit+1 > 0) α1 + x y

it α2 + ηit

(2.2)

Two sets of factors affect this incentive: the need to irrigate in the current period, and the need
includes all variables used in

to preserve the water right due to the UIOLI clause. In (2.2), x y
it

2The depth to water measures the distance from soil surface to the top of the underground aquifer. Note that dtw
does not change even if a farmer deepens his well, and under perfect hydroconductivity (i.e., if water flows freely in
the aquifer), the energy needed to extract water to the soil surface is determined by dtw rather than by how deep the
well is. dtw is related to a well’s saturatedthickness, which measures the depth or thickness of water in the aquifer,
i.e., the distance between the top and bottom of the aquifer. Saturated thickness affects the pump’s water flow – more
water is pumped per unit of time when there is more water underground, i.e., when the saturated thickness is larger.
The sum of depth to water and saturated thickness equals the distance from the soil surface to the bottom of the aquifer,
and remains unchanged through time. Since water right Ri is also time invariant, only one of the depth to water and
saturated thickness can be included in (2.1) to avoid perfect collinearity.

11

differently.

it

the extraction equation (2.1) except for crop acres and ci, and captures the current period need to
irrigate. The response to the UIOLI clause is captured by the need to irrigate next year y(it + 1):
if the farmer expects that he will use water next year, his incentive to use water in the current year
would be higher because doing so helps preserve his water rights.3 Following Wooldridge (2000)
and Biewen (2009), the coefficient r on ci in (2) allows the unobserved heterogeneity to affect y∗
and z∗
it

In (2.2), we make the assumption of conditional “sequential exogeneity” of 1(yit+1 > 0): ηit is
uncorrelated with yit, yit+1, . . . , yiT conditional on control variables x y
and individual heterogeneity
it
ci.4 This is a common assumption in nonlinear reduced-form dynamic estimation models (Akay,
2012; Contoyannis, Jones, and Rice, 2004; Deininger, Ali, and Alemu, 2011; Robone, Jones,
and Rice, 2011; Stewart, 2007; Wooldridge, 2005), allowing for conditional maximum likelihood
estimation for dynamic panels proposed by Wooldridge (2005). This assumption is plausible in our
model for two reasons. First, we already control for a rich set of time-invariant variables, such as
soil conditions, water rights, GMD fixed effects and individual heterogeneity ci. Thus correlations
of yit+1 with unobserved time-invariant variables are controlled to a great extent. Second, the
inclusion of weather conditions and year fixed effects in x y
helps control for correlations of yit+1
it
with time-varying unobserved variables.5

With data on annual water extractions from each well, we need to distinguish between water
use for irrigation and water use for preserving water rights in order to connect the latent variables
and z∗
y∗
to the observables. Let ¯S be a water extraction threshold such that, because of the fixed
it
it
3Presumably the incentive to preserve water rights depends on water use in all future years beyond yt+1. In (2.2),
we do not include yiτ, τ ≥ t + 2, for two reasons. First, the impact of yt+1 is likely to be much larger than that of
yiτ, τ ≥ t + 2, since the water right in year τ can be preserved by using water in τ − 1 instead of using water in year t.
Second, yit+1 and yiτ,(τ ≥ t + 2) tend to be serially correlated, and including only yit+1 means that the coefficient of
yit+1 captures the overall effect of all future water uses. In other words, we use 1(yit+1) as a sufficient statistics for
all future water uses.
4Due to the forward looking nature of (2), our assumption is in fact an “inverse” sequential exogeneity. The typical

sequential exogeneity assumption is that ηit is uncorrelated with yit, yit−1, yit−2, . . . , yi1
5One can imagine situations where yit+1 might be endogenous without the inclusion of a rich set of controls.
For example, a prolonged drought might raise the incentive to irrigate in year t as well as water extraction in the next
year. Conversely, low output prices over several years might reduce the current incentive to irrigate as well as future
water use. In both cases, the inclusion of year fixed effects will help capture these time-varying effects, addressing the
concern that yit+1 might be endogenous.

12

cost of irrigation, it is profitable to irrigate if and only if yit∗ > ¯S. Otherwise, if y∗
is lower than
¯S, the additional profit from irrigation cannot compensate for the fixed cost, and the farmer should
not irrigate for the purpose of maximizing the current period profit. The observed water extraction
yit then satisfies

it



= y∗
∈ (0, ¯S]
= 0

yit =

if

it > ¯S, z∗
y∗
it > 0 (c1)
y∗
it ≤ ¯S, z∗
it > 0 (c2)
z∗
it ≤ 0
(c3)

(2.3)

If z∗

it

it ≤ 0, the farmer does not have any incentive to use water from well i and yit = 0 (case
(c3)). If z∗
it > 0, the farmer has incentive to use water either because it is profitable to irrigate so
that yit = y∗
(case (c1)), and/or because he wishes to preserve his water rights (case (c2)). The
former scenario occurs when the optimal water use y∗
exceeds ¯S, while the latter occurs when
it
yit∗ ≤ ¯S. In case (c2), the farmer would apply a small amount of water in order to preserve the
water rights, and the observed water use yit ∈ (0, ¯S]. In our estimation, we set ¯S to be 5 AF, in
contrast to the mean extraction of 176 AF. We also conduct a series of robustness checks by varying
¯S to show that our estimation results are not sensitive to the specific levels of ¯S.

The remaining issue is to model individual heterogeneity ci. Since water rights as well as
soil characteristics are time invariant for a specific well, we use a random effects rather than a
fixed effects specification at the well level in order to identify the coefficients of the time-invariant
variables. We adopt the correlated random effect (CRE) framework of the Chamberlain-Mundlak
Device suggested by Wooldridge (2002) that makes the following distributional assumption for ci:

LEPAi ∗ R j

t γ1j + ACiγ2 + wiγ3 + dtwiγ4 + 1(yT > 0) γ5 + ai

(2.4)

ci = γ0 +


j

where ai ∼ N(0, σ

a) and is independent of the explanatory variables. The bars over variables
2
indicate time averages of time-varying variables across all sample periods for a well. By including
these averages, we allow for arbitrary correlations between ci and the time-varying variables in
(2.1) and (2.2), which is analogous to that of a fixed effects specification.6 Because the joint
6In fact, the CRE specification reduces to a fixed effects specification if (2.1) - (2.2) are estimated through linear

models.

13

estimation model is dynamic, the “end condition" 1(yT > 0) is included to capture the potential
correlation between yT and ci. This “end condition" is analogous to the “initial condition" used in
the dynamic panel estimation approach proposed by Wooldridge (2002) and Wooldridge (2005).
The only difference is that the dynamics in our model “goes backwards” (i.e. yit is influenced by
yit+1 but not yit−1) so that an “end condition” rather than an “initial condition” is included. The
conditional maximum likelihood estimation of the CRE specification in a dynamic setting gives
consistent estimates of the model parameters, as shown in Wooldridge (2002) and Wooldridge
(2005).

and ¯S for y∗
it

Equations (2.1) - (2.4) describe a forward-looking dynamic system in which decisions are
characterized by double hurdles (0 for z∗
) with two latent variables. The likelihood
it
function is derived in appendix A. Our model structure goes beyond the commonly used dynamic
panel models in Wooldridge (2005), which are either linear or Probit/Tobit models. Most double
hurdle models are applied to the analysis of cross-sectional data (Newman, Henchion, and Matthews,
2003; Yen, 1993), and the few applications of double hurdle models to panel data, such as Dong and
Kaiser (2008), Dong, Chung, and Kaiser (2001), and Ricker-Gilbert, Jayne, and Chirwa (2011),
have much simpler dynamic structures. For example, Dong and Kaiser (2008) assumes that each
agent only makes participation decision once over the study period, though purchasing quantity
decisions are made every year.
In Dong, Chung, and Kaiser (2001), the dynamic structure is
represented by autocorrelations in one of the error terms, whereas in our model the influence of
yit+1 on yit is introduced explicitly. There are no dynamic decisions in Ricker-Gilbert, Jayne, and
Chirwa (2011).

2.3.2 Separate Estimation of (2.1) and (2.2)

The dynamic joint estimation of (2.1) and (2.2) is consistent and efficient, but suffers from two
limitations. First, we need to establish causality of LEPA on water use but the econometric literature
has not developed an effective methodology to test and correct for the endogeneity of explanatory
variables in a nonlinear dynamic model with unobserved heterogeneities. Second, it is difficult to

14

calculate the marginal effects of key variables. In (2.1) - (2.4), as z∗
is a function of yit+1 and
it
the value of yit+1 depends on z∗
it+1, E(yit|ci) is a complex function of yit+1. Specifically, when
and y∗
we take the derivative of E(yit|ci) with respect to a particular xit that affects both y∗
it+1
(e.g. time-invariant variables such as water rights), we also need to calculate the derivative of z∗
it
with respect to yit+1 in (2.2). However, yit+1 again depends nonlinearly on y∗
it+1. The
marginal effects thus involve evaluating a nonlinear chain of derivatives. The computation burden
is further increased since we have to integrate ci out of the marginal effects. Finally, since the
marginal effects are highly nonlinear, we will have to rely on bootstrap methods instead of the
delta method to calculate the standard errors of the marginal effects. Bootstrapping is essentially
impossible given the computation burdens involved.

it+1 and z∗

it

To address these limitations, we also estimate the two decisions in (2.1) and (2.2) separately,
and use the separate models to test the exogeneity of LEPA adoption and to evaluate the marginal
effects of key variables. Specifically, we use the correlated random effect (CRE) Tobit method to
estimate (2.1), with (2.4) replaced by

LEPAit ∗ R j

i θ1j + ACiθ2 + wiθ3 + dtwiθ4 + au
i ,

(2.5)

ci = θ0 +


j

which excludes yT because (2.1) by itself is a static model so that the ending period condition is
not needed. In separately estimating (2.2), we use di to represent the unobserved heterogeneity:

with di = ϕ0 +
j LEPAi ∗ R j

z∗
it = di + 1(yit+1 > 0) α1 + x y
i ϕ1j + dtwit ϕ2 + wi ϕ3 + zT ϕ4 + az
i

it α2 + ηit,

(2.6)

. Define zit = 1 if z∗

i > 0 (or

equivalently yit > 0), and zit = 0 otherwise, we know that (2.6) is equivalent to

z∗
it = rci + zit+1α1 + x y

itα2 + ηit,

(2.7)

which reduces to a standard CRE dynamic Probit model.

Since separate estimations are biased and less efficient, we compare the estimation results of
the joint dynamic model with those of the separate models to assess the magnitude of the bias and
the efficiency loss. We find that the estimates of the key variables are close across the two models.

15

2.3.3 Mediators of the Effects of LEPA and Water Rights

Pfeiffer and Lin (2014a) shows that LEPA adoption could raise water extraction through adopters
expanding the irrigated acreage and switching to more water intensive crops. To more formally
study the channels through which LEPA and water rights affect water use, we adopt the three-step
procedure of a mediation effect model with two mediators, crop choices and irrigated acreage.
Specifically, in Step 1, we regress water use on LEPA and water rights without the mediators, i.e.,
we estimate the model in (2.1) and (2.5) without the crop acreages.
In Step 2, we regress the
mediators on LEPA and water rights. Finally in Step 3, we estimate the model in (2.1) and (2.5)
but with the mediators included.

Since the estimation model in (2.1) and (2.5) is nonlinear, we cannot rely on the Sobel test to
estimate the magnitude of the mediation effects as in, for example, Hendricks and Peterson (2012),
which conducts a Sobel test to evaluate the effects of water price on water demand. Instead, we
follow the general rule that mediation effects are present if the marginal effects of LEPA and water
rights are significant in Step 1, but their magnitudes and/or significance levels are reduced in Step
3 (relative to Step 1), and they are significant in Step 2. The logic is that the independent variables
(LEPA and Water Rights) affect the mediators (crop choices and irrigated acreage), which in turn
affect the dependent variable (water use). We compare the changes in the marginal effects on water
extraction of LEPA and water rights, with and without the mediators (i.e., in Steps 1 and 3). The
reduction in their marginal effects when the mediating variables are included show how much of
their effects on water use manifest through the mediators.

2.3.4

Issues of Endogeneity

One might be concerned that dtwit is endogenous in (2.1) as the depth to water is greater in
areas where more water was extracted in the past. However, our regression analysis shows that
an individual well’s water extraction has little impact on annual changes in dtwit – the larger
influence comes from the aggregate water extraction of all wells above the aquifer. This spatial
externality of water extraction, however, might in itself lead to endogeneity of dtwit. Well i’s water

16

extraction might be influenced by its neighbor wells in two ways. First, farmers might engage
in a simultaneous move game of water extraction so that their water extraction levels in a Nash
equilibrium are mutually dependent. Second, one well’s extraction affect the depth to water of
other wells. Explicitly including dtwit in (2.1) addresses the second reason but not the first. Due
to the large number of wells above the aquifer, we expect the strategic incentives in the first reason
to be moderate. Thus we treat dtwit as exogenous, similar to Pfeiffer and Lin (2014a), Pfeiffer and
Lin (2014b) and Hendricks and Peterson (2012).

The adoption of LEPA might also be endogenous in (2.1) – it is possible that unobserved factors
raise water use as well as promote LEPA adoption. It is difficult to come up with valid instrument
variables.7 Instead, we take an indirect approach that combines several methodologies developed
in the literature for linear models, to establish that LEPA adoption is exogenous in a linear model
of (2.1). Once we establish the exogeneity in the linear model, we then argue that the exogeneity
will carry over to nonlinear CRE Tobit estimation since the latter includes more controls such as
individual heterogeneity.

Our methodology for the linear model is based on the following identity:

Cov(∆LEPAit, ∆εit) = Cov(LEPAit, εit) − Cov(LEPAit−1, εit)

− Cov(LEPAit, εit−1) + Cov(LEPAit−1, εit−1)
= 2 Cov(LEPAit, εit) − Cov(LEPAit−1, εit) − Cov(LEPAit, εit−1)

, (2.8)

where ∆LEPAit = LEPAit−LEPAit−1 and ∆εit = εit−εit−1. The second equality in (2.8) follows
from the stationarity of εit (which implies that Cov(LEPAit, εit) = Cov(LEPAit−1, εit−1)). The
exogeneity of LEPAit in (2.1), i.e., Cov(LEPAit, εit) = 0, is equivalent to Cov(∆LEPAit, ∆εit) +
7Pfeiffer and Lin (2014a) uses county level subsidy expenditures for LEPA adoption as the instrument in a fixed
effects model and shows that correcting for the endogeneity of LEPA makes a small but statistically significant difference.
One potential issue about this IV is that the subsidy expenditures of a county are partly driven by that county’s adoption
rate, which in turn might be correlated with individual adoption decisions within the county. Another possible IV is the
adoption decisions of one’s neighbors in previous years: the technology diffusion literature provides rich evidence about
neighbor influences (Zhao, 2007). However, the validity of this IV requires controlling for correlated unobservables
among neighbors and location-specific time trends, which is difficult to implement. In fact, regression analysis of
our data provides evidence against the validity of this IV. (Details of the analysis is available from the authors upon
request.)

17

Cov(LEPAit−1, εit) + Cov(LEPAit, εit−1) = 0, according to (2.8). We will test a stronger version
of this condition:

Hypothesis 3 (H3): Cov(LEPAit−1, εit) = 0,Cov(LEPAit, εit−1) = 0 and Cov(∆LEPAit, ∆εit) =
0.

Compared with testing the sum of the covariances being zero, Hypothesis 3 has a higher
probability of being rejected, resulting in a test with a higher statistical power. That is, if we
fail to reject Hypothesis 3, we will also fail to reject the hypothesis of Cov(∆LEPAit, ∆εit) +
Cov(LEPAit−1, εit) +Cov(LEPAit, εit−1) = 0. We will test each of the three conditions separately
and obtain the associated p-values, and then draw conclusions about the joint hypothesis.

Specifically, to test the first condition in Hypothesis 3, we follow the strategy suggested by
Wooldridge (2002) and Wooldridge (2005): we add LEPAit−1 ∗ R j
i , j = 1, . . . ,8 to equation (2.1)
and run a fixed effect regression. We fail to reject the null of Cov(LEPAit−1, εit) = 0 if the F-test of
the coefficients of LEPAit−1 ∗ R j
i , j = 1, . . . ,8 being zero is not statistically significant.8 Similarly,
to test the second condition in Hypothesis 3, which is equivalent to Cov(LEPAit+1, εit) = 0, we
run the same FE regression but with LEPAit−1 replaced by LEPAit+1.

To test the third condition in Hypothesis 3, we follow another strategy suggested by Wooldridge
(2002, p308-309) and adopted by Hendricks and Peterson (2012), which applies an IV strategy to
linear first difference (FD) models. Specifically, consider the linear first differencing of (2.1).

8


j=1

∆yit =

∆LEPA∗

it R j

i ωj + ∆ACπ β1 + ∆wit β3 + ∆dtwit β4 + ∆ϕt + ∆εit,

(2.9)

where ∆ means first differencing. A Hausman test can be performed to test the exogeneity of
∆LEPAit in (2.9), i.e., Cov(∆LEPAit, ∆εit) = 0, if we can find a valid IV for ∆LEPAit ∗ R j
. One
IV that is often used is the lagged explanatory variable, i.e., LEPAit−2 ∗ R j
. The argument is that

i

i

8The setup in Wooldridge (2002) is to test whether εit is correlated with the lead of the independent variable (xit),
not the lag, in order to test the strict exogeneity of xit. The rationale is the same in our model where we perform the
test using lagged independent variables. This method has been extended by Su, Zhang, and Wei (2016) to allow for
more periods of lags and leads. The test also requires that LEPAit be exogenous. This condition is satisfied under the
null of Hypothesis 3: when all three covariance terms are equal to zero, (2.8) implies that Cov(LEPAit, εit) = 0.

18

a farmer’s LEPA adoption decision in previous periods, although possibly correlated with yit−1
and yit, should have no impact on the cross period changes in water use shocks in future periods
(i.e. Cov(LEPAit−2, ∆εit) = 0), especially after a number of covariates have been controlled for
in (2.1), Thus, we use LEPAit−2 ∗ R j
as an IV to conduct a Hausman test of the endogeneity of
in the FD equation of (2.1). We fail to reject the null that Cov(∆LEPAit, ∆εit) = 0
∆LEPAit ∗ R j
if the Hausman test statistics is not statistically significant.

i

i

Moving from the linear model to the CRE Tobit estimation, we note that in the CRE specification
of the unobserved heterogeneity ci in equation (2.5), LEPAit is uncorrelated with the projection
error au
by construction (Chamberlain, 1982; Mundlak, 1978; Wooldridge, 2002).9 Therefore, as
i
long as the tests above show that LEPA is exogenous to the idiosyncratic error εit, it will also be
exogenous to au
i

+ εit.

2.4 Data

The data used for our estimation are drawn from multiple sources. Information about wells and
water rights is drawn from the Water Information Management and Analysis System (WIMAS)
maintained by the Kansas Water Office. Among the four types of wells, we focus on Type 1-3
wells for two reasons: (i) the aggregation of information about wells and water rights is relatively
straightforward, and (ii) little information will be lost due to the high proportion of Type 1-3 wells
in our sample: among 24961 wells in total, 17903 wells are of Types 1-3, accounting for 72%
of all wells. For Type 2 and 3 wells, we aggregate water rights and well characteristics to form
one-to-one well-water right pairs. The detailed aggregation procedure is described in appendix
B.). In contrast, multiple wells are associated with multiple water rights for Type 4 wells, making
it hard to aggregate relevant information. The appendix also describes the subsequent steps of data
cleaning towards the unbalanced sample of 11,099 wells and 175,373 observations as well as the
final balanced sample of 4,663 wells and 93,260 observations. It shows the summary statistics and
regression results used to ensure representation and consistency as data points are removed.

9A well-known implication of this fact is that estimates for time-varying variables (both the estimated coefficients

and standard errors) in linear CRE models are the same as those in FE models.

19

Spatially explicit soil characteristics are obtained from SSURGO soil survey on the website
of USDA Natural Resources Conservation Service. These characteristics include detailed soil
information about surface soil slope, available water capacity, irrigated capacity class and saturated
hydraulic conductivity. Data on depth to water are obtained from the output files of Haacker,
Kendall, and Hyndman (2015). Weather data are obtained from the North America Land Data
Assimilation System (NLDAS) maintained by NASA that reports geo-referenced information about
annual precipitation and potential evapotranspiration.

Finally, the well and water right data obtained from WIMAS are matched to soil and weather
data using the spatial join function in ArcGIS. Table 2.1 presents the summary statistics of the
variables used in our estimation based on the balanced panel.

2.5 Estimation Results

Table 2.2 presents the dynamic joint estimation and separate estimation results for water extrac-
tion in (2.1) and incentive to use water in (2.2). Comparing the first two columns of table 2.2, we
find that the two estimation approaches lead to similar estimates and significance levels for most
variables. Specifically, (i) the estimated coefficient of LEPA interacted with the water right bins
follow the same pattern of first increasing and then decreasing, peaking at the (300,400] bin. (ii)
The estimated coefficients of the water right bins follow a similar pattern as the water right levels
increase. (iii) The estimated coefficients of the weather and soil variables are similar. Relatively
large differences between the two approaches arise for two sets of coefficients. (i) The coefficients
of crop acres in the separate estimation roughly equal those in the joint dynamic estimation plus
0.30 for all crops. One possible reason is that the total acreage of planted crops is almost perfectly
correlated with the incentive to use water, and in joint estimation the indicator variable of using
water 1(yit+1 > 0) partly captures the crop acreage effect. (ii) The coefficients of the time-invariant
water right bins differ to some extent across the two approaches, possibly because the individual
heterogeneities in (2.1) and (2.2) are time-invariant and are allowed to be correlated with each other
in the joint estimation but not in the separate estimations, making time-invariant variables more

20

Table 2.1: Summary Statistics

Balanced Panel 1991-2010
Definition
Annual groundwater ex-
traction

N

Annual water use cap
Annual acres irrigated

Water Extraction\Acres
Irrigated

=1 if LEPA is used; 0 oth-
erwise
Annual rainfall
Distance from land sur-
face to top of aquifer

Variables (units)

Water
(AF)

Extraction

Center Pivot
LEPA

Water Right (AF)
Acres
(Acres)

Irrigated

Center Pivot
LEPA

Extraction per acre
(AF/Acre)

Center Pivot
LEPA

LEPA

Precipitation (in)
Depth to Water (ft)

Slope (% of distance) Ground surface gradient
Crop water evaporation
Potential Evapotran-
spiration (in)
and transpiration
Rate of water moving in
Saturated Hydrocon-
saturated soil
ductivity
Soil’s ability to retain wa-
Available Water Ca-
pacity (cm/cm)
ter
=1 if soil is suitable for
Irrigated Capability
Class (dummy)
Irrigation; 0 otherwise
Percent of wells planting certain crops
Corn
Soybeans
Alfalfa
Corn and Wheat
Corn and Soy
Wheat
Sorghum
Fallow/Dryland Crop
Other Crops
# Observations with
water extraction = 0
(AF)
# with extraction in
(0,5]
# with extraction in
(0,10]
# with extraction in
(0,20]

Mean

Std.D

Min

Max

142.24

142.05
142.24
182.51
84.69

82.98
86.08
0.51

0.55
0.47
0.5

5.94
115.79

3.3
3.92

35.66

0.045

0.46

0

0
0
0.1
0

0
0
0

0
0
0

10.86
0

0
40.51

0

0

0

1720.66

1720.66
1349
2074
640

640
640
33.03

33.03
6.71
1

50.06
3151.42

20
61.38

92

0.23

1

176.03

170.74
180.92
274.72
150.61

146.6
154.32
1.15

1.14
1.15
0.52

25.28
94.48

2.93
52.23

36.54

0.16

0.31

42.47%
8.48
9.17
3.22
3.09
3.46
2.01
2.58
25.53

93,260

44,845
48,415
93,260
93,260

44,845
48,415
90,304

43,119
47,185
93,260

93,260
93,260

93,260
93,260

93,260

93,260

93260

93,260
93,260
93,260
93,260
93,260
93,260
93,260
93,260
93,260
2956

393

851

2076

21

prone to differ across the two approaches.

Table 2.2: Dynamic Joint Estimation and Separate Estimation Results

Water Extraction
Dynamic Joint Separate

Variables
1(yit+1 > 0)
LEPA ∗ 1[0 < R ≤ 100]
-4.92*
(2.73)
LEPA ∗ 1[100 < R ≤ 200]
-0.72
(0.84)
LEPA ∗ 1[200 < R ≤ 300] 2.38**
(0.95)
LEPA ∗ 1[300 < R ≤ 400] 7.32***
(1.59)
LEPA ∗ 1[400 < R ≤ 500] 6.09**
(2.40)
LEPA ∗ 1[500 < R ≤ 600]
-1.72
(2.05)
LEPA ∗ 1[600 < R ≤ 700]
-5.62
(3.93)
LEPA ∗ 1[R > 700]
-2.38
(2.40)
1[100 < R ≤ 200]
10.70**
(4.93)
1[200 < R ≤ 300]
15.38***
(5.19)
1[300 < R ≤ 400]
29.54***
(7.26)
1[400 < R ≤ 500]
26.30***
(7.42)
1[500 < R ≤ 600]
47.74***
(9.76)
1[600 < R ≤ 700]
18.12**
(9.11)
1[R > 700]
144.74***
(8.46)
-1.86***
(0.11)
-0.62***
(0.03)
4.21***

Depth to Water

Precipitation

Potential Evapotranspiration

Incentive to Use Water
Dynamic Joint
2.26***
-0.034
0.14*
(0.08)
0.18***
(0.04)
0.23***
(0.05)
0.31***
(0.08)
0.22*
(0.12)
-0.076
(0.11)
0.34
(0.20)
0.33**
(0.15)
0.36***
(0.06)
0.40***
(0.07)
0.36***
(0.09)
0.25**
(0.11)
0.26**
(0.12)
-0.09
(0.15)
0.42***
(0.13)
-0.0014
(0.00)
-0.0044***
0.00
0.010***

Separate
1.73***
-0.057
0.19*
(0.11)
0.24***
(0.06)
0.29***
(0.66)
0.33***
(0.11)
0.16
(0.14)
0.0062
(0.16)
0.24
(0.20)
0.33*
(0.18)
0.55***
(0.12)
0.70***
(0.14)
0.65***
(0.16)
0.65***
(0.22)
0.83***
(0.22)
0.46*
(0.27)
0.91***
(0.29)
-0.0047
(0.01)
-0.0032*
(0.00)
0.049***

-4.29
(2.62)
0.15
(0.82)
2.19**
(0.93)
5.90***
(1.55)
4.40*
(2.35)
-4.51**
(2.00)
-6.83*
(3.85)
-2.99
(2.34)
18.92***
(6.55)
24.17***
(6.85)
26.00***
(8.23)
18.39*
(10.60)
41.71***
(9.91)
-13.02
(13.99)
76.33***
(11.16)
-1.82***
(0.10)
-0.57***
(0.03)
4.21***

22

Table 2.2 (cont’d)

(0.26)
-0.41
(1.56)
3.03***
(0.25)
-152.97***
(27.48)
0.036
(0.04)
0.62***
(0.01)
0.57***
(0.01)
0.63***
(0.01)
0.55***
(0.01)
0.61***
(0.01)
0.12***
(0.01)
0.37***
(0.01)
0.50***
(0.01)
38.02
(66.01)

Irrigated capacity class

Slope

Available water capacity

Saturated hydroconductivity

Corn Acres

Soybeans Acres

Alfalfa Acres

Corn-Wheat Acres

Corn-Soybeans Acre

Wheat Acres

Sorghum Acres

Other Crop Acres

Constant

r

Marginal Effects
1[100 < R ≤ 200]
1[200 < R ≤ 300]
1[300 < R ≤ 400]
1[400 < R ≤ 500]
1[500 < R ≤ 600]
1[600 < R ≤ 700]

(0.25)
-2.17
(1.65)
2.59***
(0.27)
-162.51***
(28.32)
0.058*
(0.04)
0.97***
(0.01)
0.93***
(0.01)
0.98***
(0.01)
0.88***
(0.01)
0.95***
(0.01)
0.48***
(0.01)
0.73***
(0.01)
0.85***
(0.01)
81.03
(60.97)

19.36***
(6.01)
25.25***
(6.32)
28.79***
(7.65)
20.95**
(9.86)
38.48***
(9.34)
-12.62
(12.36)

23

(0.00)
0.035
(0.03)
0.0035
(0.01)
-0.64
(0.54)
0.00027
(0.00)

(0.02)
0.045
(0.04)
0.0062
(0.01)
-1.04
(0.70)
0.00062
(0.00)

0.68***
-0.031

-2.76*
(1.57)

0.028**
(0.01)
0.034**
(0.02)
0.033**
(0.01)
0.029*
(0.02)
0.030*
(0.02)
0.026
(0.02)

1[R > 700]

LEPA (overall)

@0 < R ≤ 100
@100 < R ≤ 200
@200 < R ≤ 300
@300 < R ≤ 400
@400 < R ≤ 500
@500 < R ≤ 600
@600 < R ≤ 700
@R > 700

1(yit+1 > 0)
Time Fixed Effect
GMD Fixed Effect
N

Table 2.2 (cont’d)

72.64***
(10.76)
0.81
(0.59)
-3.84*
(2.33)
0.14
(0.76)
2.05**
(0.87)
5.55***
(1.46)
4.09*
(2.19)
-4.29**
(1.90)
-5.91*
(3.31)
-2.92
(2.29)

Yes
Yes
88597

Yes
Yes
93260

Yes
Yes
88597

0.038**
(0.02)

0.0064
(0.00)
0.0078***
(0.00)
0.0092***
(0.00)
0.010***
(0.00)
0.0055
(0.00)
2.36E-03
(0.00)
0.008
(0.01)
0.010**
(0.00)
0.206***
(0.02)
Yes
Yes
88597

Note: Standard errors are reported in parentheses in the first three columns. The last column reports cluster
robust standard errors (clustered at the well level). Significance levels: ∗ ∗ ∗ = 1%,∗∗ = 5%, and ∗ = 10%
Comparing the last two columns of table 2.2, we find that the estimated coefficients of LEPA
interacted with the water right bins again follow similar patterns across the two approaches, and the
same holds true for the estimated coefficients of water right bins themselves. Further, the estimated
coefficients of LEPA interacted with the water right bins have similar magnitudes across the two
approaches. The estimated coefficients of the water right bins do differ across the two approaches,
again due to the correlation of individual heterogeneities in (2.1) and (2.2).

Overall, the coefficients of key variables (e.g., LEPA interacted with different water right bins)
are similar in magnitude and trend between the joint and separate estimation approaches, showing

24

that the bias of separate estimation is limited. We will thus rely on the separate estimation results
in subsequent discussions and analyses.

2.5.1 Water Extraction and the Rebound Effects of LEPA

The CRE Tobit estimates of the water extraction decisions in (2.1) and (2.5) are reported in the
second column of table 2.2. The estimated coefficients of most variables have expected signs.
Water extraction goes up as the irrigated crop acreages rise, and this effect is larger for water
intensive crops including corn, soybeans, alfalfa and corn/soybeans rotation. Water extraction also
rises in potential evapotranspiration (a higher value of which represents a higher demand for water
by crops), in saturated hydraulic conductivity (which measures how easy water flows through the
aquifer and thus out of pumps), and in slope of the land (since more sloped land requires more water
to irrigate due to more runoff). On the other hand, water extraction decreases in precipitation, in
depth to water (since more depth means more energy expenditure needed to extract water to soil
surface), and in available water capacity (a higher value of which indicates more water is stored in
a crop’s root zone for crop needs).

Regarding the rebound effects of LEPA, the second column of table 2 reports the estimated
coefficients of LEPA interacted with the water right bins, as well as the estimated marginal effects
of LEPA, both overall and at different water right bins.10 The rebound effect is most pronounced
for water rights between 200 and 500 AF, but is negative for extremely small (below 100 AF) and
large water rights (above 500 AF). Thus, the adoption of LEPA raises water extraction for farmers
with “moderate” water rights (between 100 and 500 AF) but reduces water extraction for those
with extremely small and large water rights (below 100 AF and over 500 AF). As discussed earlier,
a farmer’s intraseason irrigation decisions are inherently dynamic: facing possibly binding water
rights, the farmer will likely restrict water extraction during the early growing season in order to

10Since the Tobit model is nonlinear, the coefficients of LEPA*water right bins in (2.1) and the corresponding
marginal effects can differ in both their estimated magnitudes and statistical significance. For example, while the
coefficient to of LEPA ∗ 1(0 < R ≤ 100] is -4.29 and not significant, the marginal effect of LEPA at water right of
(0,100] is -3.84 and significant. This kind of difference often occurs when calculating marginal effects in nonlinear
models, see, for example, Dowd, Greene, and Norton (2014).

25

leave more options open to increase water use during the later season. For those with moderate
water rights, the higher irrigation efficiency of LEPA means that farmers have access to more
“effective water,” i.e., water that is available to be used by crops, so that he has more flexibility
of increasing the effective water to crops when given the same amount of available water. Thus,
with the adoption of LEPA, he has incentive to use more water during the early season since doing
so will be less costly in restricting the options during the later season. This intuition provides a
dynamic explanation for the rebound effects of new technologies, over and above the static reasons
discussed in, for example, Ward and Pulido-Velazquez (2008).11 For those with extremely large
water rights, it is likely that they already apply too much water, e.g., to maximize crop yields instead
of profit. LEPA, with its higher irrigation efficiency, can reduce the extracted water that is needed
to maximize the yields. Conversely, those with extremely small water rights may have already
exceeded their water rights, and the adoption of LEPA can help mitigate violations of water rights
without reducing the effective water received by the crops.12 In these two cases, LEPA reduces
water use and has no rebound effects.13

The rebound effect of LEPA follows a nonlinear pattern as water rights change: it first increases
as water rights rise up to about 500 AF, and then decreases as water rights further rise. This pattern
is also graphed in figure 2.4 (the solid lines). Since almost 90% of the wells in our sample have
water rights below 500 AF, the overall pattern depicts the rebound effects of LEPA increasing
in water rights. Again, this pattern is consistent with the framework of intraseasonal dynamic
irrigation decisions. For those with smaller water rights, there is a higher probability of the water
rights being binding and the incentive to restrict water use in the early season is higher, leading to
lower total water use even if ex post the water rights are not binding. The larger the water rights, the

11This dynamic explanation is speculative as testing it requires intraseasonal irrigation scheduling data, which are
not available. Nevertheless, the agronomy literature shows that for some crops such as corn, the yield is more sensitive
to water deficits later in the season, especially during tasseling and grain filling periods, than early in the season
(NeSmith and Ritchie, 1992a; NeSmith and Ritchie, 1992b).

12About 62% of wells with water rights in (0, 100] AF have exceeded their water right allocations in at least one
year during the periods when they use center pivot, but this percentage drops to 48% during the periods when they use
LEPA. In contrast, for wells with significant rebound effects, the percentages are respectively 43% and 38% for those
in (200,300], and 24% and 20% for those in (300,400].
beyond the scope of this article.

13Formal tests of these explanations require dynamic and structural estimation of the irrigation decisions, and are

26

Figure 2.4: Marginal Effects of LEPA and Water Rights in Mediation Test

more room the farmer has in adjusting water use later in the season, raising the farmer’s incentive
to use more water early in the season. These results confirm part of Hypothesis 1, and highlight the
importance of institutional approaches such as restricting water rights in ameliorating the rebound
effects of LEPA. We will conduct a series of counterfactual analysis of reducing water rights later
in this article.

Overall, conditional on crop choices, the rebound effect of LEPA is positive (at 0.81) but not
statistically significant – we will come back to this issue below when we discuss the mediation
results.

27

2.5.2 Mediators of LEPA and Water Rights

To investigate the channels through which LEPA and water rights affect water extraction, we
conduct the three step regressions used in formal mediation tests with two possible mediators, acres
irrigated and crop choices (tables 2.3 - 2.5). Tables 2.3 and 2.4 show that the two independent
variables, LEPA and water rights, do affect the two mediators. In table 2.3, the marginal effects
of water rights and LEPA on acres irrigated are positive and significant. In table 2.4, LEPA raises
the acreage of corn, soybeans and alfalfa, three water intensive crops. Compared to farmers with
water rights between 0 and 100 AF (the base water right bin), acreages of water intensive crops
(corn, soybean, alfalfa, and mixture of corn with other crops) are larger for wells in higher water
right bins.

Table 2.3: Mediation Test: Marginal Effect of LEPA and Water

Rights on Acres Irrigated (Panel Tobit)

Variables
R1

R2

R3

R4

R5

R6

R7

LEPA

Marginal Effects
80.29***
(4.49)
97.11***
(4.80)
135.35***
(6.59)
183.82***
(9.36)
224.84***
(8.24)
210.17***
(13.18)
365.31***
(8.89)
2.39***
(0.32)

Note: Standard errors are reported in parentheses. Significance levels:
∗ ∗ ∗ = 1%,∗∗ = 5%, and ∗ = 10%.

28

Table 2.4: Mediation Test: Effect of LEPA and Water Rights on Crop Acres (Linear RE)

LEPA Water Right Bins (lower bound upper bound)

Marginal
Effects

100

200

200

300

300

400

400

500

500

600

600

700

>700

Corn

(3.83)

(6.99)

Alfalfa

(3.21)
(11.66)
12.67*** 10.72*** 9.11*** 5.74**
(2.67)
(1.77)
2.36
7.06***
(2.46)
(4.14)
0.59
(1.14)
2.71***
(1.02)

3.82*** 41.83*** 61.66*** 51.40*** 43.08*** 64.78*** 32.81** 65.16***
(0.93)
Soybeans 0.77**
(0.38)
1.17*
(0.66)
-0.55
(0.56)
-0.35
(0.31)
-0.83** 4.02***
(0.33)
Sorghum 0.022
(0.25)
-1.95*
(1.03)

(14.76)
(16.60)
(10.12)
6.32*** 8.75**
8.31***
(3.83)
(2.03)
(2.18)
(2.24)
65.92***
36.55*** 5.55
5.83*
(19.58)
(10.64)
(7.11)
(3.01)
26.72*** -0.98
14.02*
13.81*** 23.21**
(7.62)
(4.90)
(10.04)
(10.30)
7.41***
9.63*** 10.48*** 4.01*
(2.63)
(2.08)
(2.60)
0.03
1.72
3.26*
(2.41)
(2.41)
(1.78)
1.43
0.68
1.14
(1.35)
(1.20)
(1.53)

(1.75)
5.05*
(2.83)
1.96
(1.99)
5.22***
(1.12)
1.75
(1.30)
1.33
(1.15)

(10.09)
3.56**
(1.39)
6.96
(8.61)
3.21
(3.64)

Corn
& Wheat
Corn
& Soy
Wheat

(3.27)
-1.7
(2.95)
1.53
(1.52)

(1.11)
2.35**
(0.98)
12.19*** 12.87*** 45.58*** 104.05***84.61*** 159.03***205.23***
(3.74)

(15.17)

(29.67)

(19.52)

(43.96)

(8.42)

(4.20)

Others

Note: Clustered robust standard errors are reported in parentheses (clustered at well level). Significance levels:
∗ ∗ ∗ = 1%,∗∗ = 5%, and ∗ = 10%.
In table 2.5, the first column reports the marginal effects of various water right bins (relative to
the default bin) as well as those of LEPA at each of these bins when both mediators are excluded
from the regression; the second column contains these marginal effects when one of the mediators,
acres irrigated per well, is added as an explanatory variable; and the third column reproduces the
marginal effects reported in column two of table 2.2, which contain all the mediators.14 These
marginal effects are also plotted in figure 2.4, with panel (a) showing the marginal effects of LEPA
and panel (b) showing those of water right bins. The effects without the mediators, i.e., those
corresponding to the first column of table 2.5, are plotted as broken lines, those with only the total
irrigated acres included (corresponding to the second column of table 2.5) are plotted as dashed

14Instead of adding the irrigated acreage and the crop choices (excluding a default crop) as separate variables, the

third column simply contains the irrigated acreage of all crops. The results of the two approaches are equivalent.

29

lines, and those with crop acreages included (corresponding to the last column of table 2.5) are
plotted as solid lines.

Table 2.5: Mediation Test: Comparison of Estimates with and without Mediators (Marginal

Effects of Panel Tobit)

Water Extraction Estimation With/Without Mediators

Variables

1[100 < R ≤ 200]
1[200 < R ≤ 300]
1[300 < R ≤ 400]
1[400 < R ≤ 500]
1[500 < R ≤ 600]
1[600 < R ≤ 700]
1[R > 700]
LEPA (overall)

@0 < R ≤ 100
@100 < R ≤ 200
@200 < R ≤ 300
@300 < R ≤ 400
@400 < R ≤ 500
@500 < R ≤ 600
@600 < R ≤ 700
@R > 700

No Mediators
92.38***
(5.50)
122.09***
(6.17)
159.89***
(9.24)
201.16***
(13.86)
281.11***
(12.25)
192.57***
(19.94)
492.76***
(13.29)
3.09***
(0.63)
-1.13
(1.34)
1.78**
(0.80)
4.76***
(0.95)
10.23***
(1.66)
7.50***
(2.54)
-6.22***
(2.18)
-7.09*
(4.16)
5.05**
(2.57)

With Acres Irrigated With Crop Acres
32.44***
(6.68)
45.22***
(7.05)
39.56***
(8.65)
25.05**
(11.19)
61.73***
(10.73)
-11.11
(13.74)
115.96***
(12.44)
1.69***
(0.60)
-2.59
(2.30)
0.83
(0.78)
3.44***
(0.90)
6.11***
(1.49)
6.47***
(2.22)
-4.66**
(1.96)
-5.84*
(3.23)
-2.36
(2.38)

19.36***
(6.01)
25.25***
(6.32)
28.79***
(7.65)
20.95**
(9.86)
38.48***
(9.34)
-12.62
(12.36)
72.64***
(10.76)
0.81
(0.59)
-3.84*
(2.33)
0.14
(0.76)
2.05**
(0.87)
5.55***
(1.46)
4.09*
(2.19)
-4.29**
(1.90)
-5.91*
(3.31)
-2.92
(2.29)

Note: Standard errors are reported in parentheses. Significance levels: ∗ ∗ ∗ = 1%,∗∗ = 5%, and ∗ = 10%.

30

When Acres Irrigated is added to the explanatory variables (comparing between the first and
second columns of table 2.5, and between the broken and dashed lines in figure 2.4(b)), the marginal
effects of water rights on water extraction decrease by 65%-76% for different water right bins. In
contrast, when crop choices are added (comparing between the second and third columns, or dashed
and solid lines), the marginal effects of water rights decrease slightly, by about 10% on average.
These results indicate that about two thirds of the effects of water right on water extraction operate
through irrigated acreage. That is, farmers with larger water rights tend to use more irrigation
water mainly because they irrigate larger areas of land.

For the rebound effect of LEPA, both the irrigated acreage and crop choices are significant
mediators over the range of water rights between 100 and 500 AF, and the mediators’ effects are
heterogeneous across the water right bins (table 2.5 and figure 2.4(a)). For wells with small water
rights (in (100,200]), the mediators together explain over 90% of LEPA’s effects on water use,
while for those with water rights in (200,500], the mediators together explain about 50% of LEPA’s
effects. The results confirm part of Hypothesis 1: even after expanded irrigation acreage and more
water intensive crop choices are accounted for, about half of LEPA’s rebound effect is attributable
to farmers simply irrigating more intensively under LEPA than under center pivot for the majority
of farmers (those with water rights between 200 and 500 AF). Further, comparing the marginal
effects of LEPA across the three columns of table 2.5, we find that the effects of each mediator vary
across water right bins: the effects of irrigated acreage seems to dominate those of the crop choices
for water rights in (100,400], while the opposite is true for water rights in (400,500].

The first column of table 2.5 shows that when crop acreages are not included as explanatory
variables, the estimated overall marginal effect of LEPA on water use is 3.09 and is statistically
significant. This estimate is close to those obtained in Pfeiffer and Lin (2014a), which does not
include crop acreages: 3.95 and 3.60 in fixed effects models with and without instrumenting for
LEPA adoption. Our results show that a large share of the marginal effects can be attributed to
the mediators including irrigated acreage and crop choices, and that the effects vary greatly across
water right levels.

31

2.5.3 Tests of the Exogeneity of LEPA

The p-values of the three regressions used to test the three conditions in Hypothesis 3 are respectively
0.62 (F-test of the coefficients of LEPAit−1 ∗ R j
i , j = 1, . . . ,8), 0.35 (F-test of the coefficient of
LEPAit+1 ∗ R j
i , j = 1, . . . ,8) and 0.65 (Hausman test with LEPAit−2 ad the IV). Thus, we fail
to reject each of the three individual hypotheses. The p-value for the joint Hypothesis 3 depends
on how the test statistics of the three individual hypotheses are correlated with each other. For
example, if the three test statistics are independent, then the p-value of the joint hypothesis equals
1−(1 − p1)(1 − p2)(1 − p3) = 0.91.15 But in general the test statistics of the individual hypotheses
are correlated, and the classical approach to control for this is the Bonferroni method (Shaffer, 1995).
Specifically, let α be the significance level of the joint hypothesis (i.e., rejecting Hypothesis 3 if
and only if the p-value is less than α). Then the joint hypothesis is rejected if the p-value of each
of the three separate tests is less than α/3. We fail to reject Hypothesis 3 even if we choose a
high significant level, e.g., α = 10%, since the p-value of each individual hypothesis far exceeds
0.10/3=0.033. There are different variations of the Bonferroni method (e.g., with different weights
and orders attached to the three individual tests) but the same conclusion holds: we fail to reject
the null hypothesis that LEPA is exogenous in (2.1). This finding is largely consistent with Pfeiffer
and Lin (2014a), which controls for the potential endogeneity of LEPA adoption using fixed effects
2SLS: the estimated coefficients of LEPA with and without the instrument variable are close to
each other.

2.5.4 Decision to Preserve Water Rights

The last column of table 2.2 reports the panel Probit regression results of the decision to use
water (equation (2.2)). Confirming part of Hypothesis 2 that farmers’ decisions are influenced by
expectations about future water use, the coefficient of the indicator variable of water use in the next

15Let H0 denote the joint hull hypothesis, i.e., Hypothesis 3, and H0i be the ith individual hypothesis. Then the
p-value of the joint hypothesis = Prob (reject H0 | H0 is true) = Prob (reject H01 OR reject H02 OR reject H03 | H0i
is true, i = 1,2,3) = 1 – Prob (fail to reject H01 AND fail to reject H02 AND fail to reject H03 | H0i is true, i = 1,2,3)
= 1 − (1 − p1)(1 − p2)(1 − p3).

32

period (yit+1 > 0) is positive and significant. That is, farmers have more incentive to preserve their
water rights in the current period if they expect that they will again irrigate the next year.

The bottom part of the column reports the marginal effects of water rights and LEPA on the
incentive to use water. LEPA has a positive influence on the incentive for all levels of water rights
and this effect is significant for water rights between 100 and 400 AF, although the magnitudes
of the effects are small. For example, the marginal effect of LEPA is 0.0078 for water rights in
(100,200], implying that on average a farmer with these levels of water right is 0.78% more likely
to preserve his water rights if LEPA has been adopted than if center pivot is still used.

Relative to the base water right bin of (0,100], the marginal effects of other water right bins are
positive and generally significant. Again their magnitudes are modest, possibly because we have
controlled for a large number of covariates in the estimation and because farmers view water rights
as providing irrigation services with limited values over and above the irrigation services. In other
words, once future irrigation water use is controlled for, the water rights only represent a modest
value to the farmer and thus do not significantly affect their incentives to preserve the water rights.

2.5.5 Robustness Checks

In estimating the water extraction decisions in (2.1) and (2.5), we introduced the water extraction
threshold ¯S to interpret whether observed water use is for irrigation in the current period or for
preserving water rights (cf. (2.3)). We set ¯S = 5 AF in our estimation, but this threshold could be
at other values. We re-estimate (2.1) and (2.5) using different values of ¯S (at 5, 10, and 20 AF) and
report the estimation results in appendix C (table C.1 and figure C.1). The results show that the
estimated coefficients and marginal effects of key variables (e.g., LEPA interacted with water right
bins) remain largely unchanged, establishing that our estimates are robust to different values of ¯S.
Since water right applications occurred much earlier than our study period (cf. figure 2.1(a)),
water rights are treated as being exogenous in our regression analysis. One might argue that farmers
with higher water extraction applied for larger water rights, and serial correlation in water extraction
could cause water rights to be endogenous. We are less concerned about this potential endogeneity

33

because we control for a wide range of factors affecting water extraction, including soil and weather
characteristics, crops and depth to water, as well as time and groundwater management district
fixed effects. Nevertheless, in case there are unobserved variables correlated with water rights, we
conduct a robustness check in which only wells with water right application dates prior to 1980 are
included. The estimation results are also reported in appendix C (table C.1 and figure C.1). Again,
our estimation results are robust to this restriction.

2.6 Policy Simulations of Changing Water Governing Institutions

While it is important to recognize the existence of the rebound effects of more efficient technolo-
gies such as LEPA, it is perhaps even more important to investigate what can be done in response
to such rebound effects. The previous discussions highlight the importance of water governing
institutions: the rebound effects of LEPA tend to be lower for farmers with smaller water rights.
Further, farmers do respond to the use-it-or-lose-it clause by using water in order to preserve their
water rights. If water is used only to preserve the water rights instead of for profitable irrigation,
removing the UIOLI clause might help save water. In this section, we conduct a series of counter-
factual simulations to evaluate how water use will be affected as the water rights are reduced and
as the UIOLI clause is removed.

Since the water rights are fixed during our sample period, the estimated effects of the water
rights are due to cross-sectional variation and thus represent the long-term impacts of differing
water rights that would have incorporated such adjustments as growing different crops and adopting
LEPA. To simulate such long-term impacts, we rely on the CRE Tobit estimates of (2.1) without
the mediators, i.e., we use the coefficients underlying the marginal effects reported in the first
column of table 2.5. As established in the mediation tests (tables 2.3 and 2.4), long-term changes
in water rights will indeed influence irrigated acreage and crop choices, which in turn affect water
extraction. Excluding the mediators allows us to capture the long-run effects of water rights, after
farmers have adjusted their irrigated acreage and planted crops.

To account for possible adjustments through the adoption of LEPA, we could re-estimate (2.1)

34

Table 2.6: Effects of Water Rights on LEPA Adoption

Water Right Bins
1[100 < R ≤ 200]
1[200 < R ≤ 300]
1[300 < R ≤ 400]
1[400 < R ≤ 500]
1[500 < R ≤ 600]
1[600 < R ≤ 700]
1[R > 700]
Precipitation

Depth to Groundwater

Potential Evapotranspiration

Irrigated capacity class

Slope

Available water capacity

Saturated hydroconductivity

Marginal Effects
0.097***
(0.018)
0.098***
(0.019)
0.097***
(0.021)
0.111***
(0.025)
0.111***
(0.024)
0.083**
(0.033)
0.124***
(0.025)
-0.0017***
(0.001)
0.00022*
(0.000)
-0.0036***
(0.001)
-0.0064
(0.009)
-0.00028
(0.001)
0.081
(0.150)
0.00032*
(0.000)

Note: Standard errors are reported in parentheses. Significance levels: ∗ ∗ ∗ = 1%,∗∗ = 5%, and ∗ = 10%.

35

without LEPA, or capture any effects that water rights might have on LEPA adoption. We take
the latter approach since LEPA is a primary interest of our study. Specifically, we regress LEPA
adoption on water rights as well as the weather and soil characteristics, using a CRE panel linear
probability model, a CRE panel Probit model, and a CRE pooled Probit model. These models
lead to similar estimation results, and table 2.6 reports those from the CRE panel linear probability
model. The water right bins over 100 AF all have positive and significant coefficients, and their
coefficients are of similar magnitudes. That is, although farmers with water rights exceeding 100
AF are more likely to adopt LEPA than those with water rights below 100 AF, the incentive to
adopt LEPA is similar among the former so that when water rights are reduced within this group,
the incentive to adopt LEPA will not change. This observation will likely carry over to the entire
sample given the small percentage of farmers with water rights lower than 100 AF, making it
feasible to condition the long-term effects of water rights on irrigation technologies.

We estimate the expected reductions in water extraction when water rights for all wells are
reduced by 5%, 10% and 15% respectively for three technology scenarios of all wells using center
pivot in all sample years, all wells using LEPA, and wells using technologies given in our data. We
first calculate each observation’s expected water extraction given the existing water rights using the
coefficients underlying the first column of table 2.5, and recalculate the expected water extraction
when each well’s water right is reduced. A well’s expected water extraction will change if and only
if its water right falls in a different water right bin after the reduction.16 We find that that a 10%
reduction in water rights leads to about 6% reduction in water extraction, and this relationship
is about linear within the range of water right reduction levels. The reduction in water extraction
is slightly more pronounced if all wells use LEPA than if all use center pivot, consistent with the
earlier finding that reducing water rights helps limit the rebound effects of LEPA. If the government
specifically targets the water rights with positive rebound effects, e.g., reducing by 10% of the

16Since we divide the water right levels into eight water right bins, a well’s water extraction varies only if its water
right level moves across bins as it is reduced. This is of course an artifact of the discretization of water right levels.
While for each individual well the effects are under- or over-estimated, the estimated average effects tend to be accurate;
in fact, the marginal effects of water right reductions for all wells remain similar across the three water right reduction
magnitudes of 5%, 10%, and 15%. Note that water right reductions affect water extraction even if the water right is
not binding, due to the dynamic nature of intraseasonal irrigation scheduling.

36

water rights between 100 and 500 AF, then their water extraction will decrease by about 4.6%, and
LEPA’s rebound effect will decrease by about 15.4%.

Water is wasted when it is not profitable to irrigate but water is nevertheless extracted in order
to preserve the water rights, i.e., when y∗
it > 0. The actual waste equals a small amount
not higher than ¯S. If we assume that the waste is at the upper bound of ¯S, then the expected water
waste per well equals:

it < ¯S and z∗

Prob(y

it < ¯S, z∗
∗
(i.e. εit + au
i

it > 0) × ¯S
and ηit + az
i

and z∗
it

(2.10)

Since the error terms of y∗
it

) specified in equations (2.1) and
(2.2) are both normally distributed, the joint probability in (2.10) could be calculated as long as
we know the correlation coefficient of εit + au
.17 We calculate the expected waste in
i
(2.10) and bootstrap its standard errors using -1, 0.5, 0, 0.5 and 1 as the correlation coefficients
between εit + au
i

and ηit + aZ
i

and ηit + au
i

.

We find that the annual expected water waste per well is rather small, at about 0.25-0.35 AF
per year for each well, and this result is not sensitive to the correlation coefficients. Although
this amount is significantly different from zero, it is insignificant in magnitude compared with the
average annual water extraction of 176 AF per well. Even if the amount applied each time is higher,
say, at 10 (or 20) AF, the resulting annual waste of 0.5-0.7 (or 1-1.4) AF is still small. We therefore
conclude that the UIOLI clause has not resulted in significant water waste.

2.7 Conclusions

In this article we investigate how institutions can complement new technologies in promoting
the sustainability of US agriculture using irrigation as an example. Specifically, we study the role
of water rights in limiting the rebound effects of LEPA and affecting farmer incentives to preserve
their water rights in the HPA region of Kansas. We find that water extraction moderately increases
after LEPA adoption, and the rebound effect is heterogeneous across farmers with different levels

17The correlation coefficient could be estimated through joint dynamic estimation of the water extraction and water
use incentive equations while allowing for the error terms to be correlated. However, since the standard errors can only
be estimated through bootstrapping, the computation time involved is prohibitively long.

37

of water rights. It is negative for the few farmers with very small (<100 AF) and very large water
rights (>500 AF), but is positive for the majority of farmers, i.e., those with water rights between
200 and 500 AF. For these farmers, LEPA’s rebound effects are larger for those with larger water
rights, implying that policies restricting water rights can help ameliorate LEPA’s rebound effects
in the long run. Specifically, a 10% reduction of water rights for all wells will in the long run
reduce water use by about 6%. If the 10% reduction is targeted at the medium water rights between
100 and 500 AF, their expected water use will decrease by 4.6% and LEPA’s rebound effect will
decrease by 15.4%. Further, restricting water rights will not affect the incentives to adopt LEPA
among farmers with water rights exceeding 100 AF. Thus, institutional changes such as reducing
water rights can limit the undesirable rebound effects of new technologies such as LEPA without
hurting incentives to adopt them.

We also find that on average about a half of the effects of LEPA in raising water extraction across
a wide range of water right levels arises from the channel of crop choices and expansion in irrigated
acreage: farmers using LEPA are more likely to grow water intensive crops such as corn, soybean
and alfalfa as well as irrigate more lands compared with those using center pivot. The remaining
half of LEPA’s rebound effect is attributable to more intensive irrigation. In contrast, about two
thirds of the effects of water rights on water extraction operate through irrigated acreage: famers
with larger water rights will irrigate larger areas of their fields. Crop choices explain about 10%
of the effects of water rights: farmers with larger water rights tend to plant more water intensive
crops. The remaining quarter of the effects is attributable to more intensive irrigation by farmers
with larger water rights.

We find that farmers have incentive to preserve their water rights even when no irrigation is
needed, in response to the use-it-or-lose-it clause of the water right system. This incentive is
higher if the farmer expects to irrigate in the next period. Nevertheless, we find that the associated
water waste is insignificant, at about 0.25-0.35 AF per well each year, relative to the annual water
extraction of 176 AF.

38

APPENDICES

39

APPENDIX A

MAXIMUM LIKELIHOOD ESTIMATION OF THE DYNAMIC MODEL

Assuming that εit ∼ iidN
density function for yit has three possible cases. Under condition (c1),

, ηit ∼,iidN(0,1), and cov(εit, ηit) = 0,1 we know the period t

2
ε

(cid:16)0, σ

(cid:17)

1
σε

φ

=

(cid:34)

(cid:17)

(cid:17)

(cid:16)

(cid:16)

fc1

yit|yit+1, ci, x y

it

Under condition (c2),

fc2

yit|yit+1, ci, x y

it

1 − Φ

=

Finally, under condition (c3),

fc3

yit|yit+1, ci, x y

it

(cid:16)

f

yit|yit+1, ci, x y

it

(cid:33)

(cid:33)

itα2

itα2

σε

(cid:32) yit − ci − x y
(cid:32) ci + x y
(cid:17)

= 1 − Φ

σε

it β − S

Φ

Φ

ση

it β

(cid:32)rci + 1(yit+1 > 0) α1 + x y
(cid:32)rci + 1(yit+1 > 0) α1 + x y
(cid:33)

(cid:33)
(cid:33)(cid:35)
(cid:32)rci + 1(yit+1 > 0) α1 + x y
(cid:17)
(cid:17)(cid:105)1(cid:16)
(cid:17)(cid:105)1(cid:16)0<yit≤S

fc3 (yit+1) yit+1, x y

∗(cid:104)

yit >S

itα2

ση

ση

(cid:17)

it

(cid:17)]1(yn=0)

(A.1)

(A.2)

(A.3)

(A.4)

fc1

fc2

yit|yit+1, ci, x y
yit|yit+1, ci, x y

it

it

(cid:16)

(cid:17)

(cid:104)
∗(cid:104)

=

Together, the likelihood function for a particular well i in specific year t is:

(cid:16)
(cid:16)

(cid:16)

Note that, due to the dynamic structure in equation (2.2) in the main text, the likelihood
function in period t depends on yit+1. Utilizing the property of conditional density functions,
the full likelihood function (conditional on ci) of well i over all years is given by the product of
per-period likelihood functions:

f

yi|yiT, ci, x y

it

=

f

yit|yit+1, ci, x y

it

(A.5)

Using the distribution of ci in equation (2.3) specified in the main text, the unconditional

likelihood contribution becomes:

(cid:16)

(cid:17)

Þ

(cid:17)

(cid:16)

f

yi|yiT, x y

it

=

f

yi|yiT, ci, x y

it

h

ci|xi, yTi, θ

dci

(A.6)

(cid:16)

Ti−1


t=1

(cid:17)

(cid:16)

(cid:17)

(cid:17)

1We also make the typical assumption that σ

2
η = 1 in order to identify the model.

40

APPENDIX B

DATA ISSUES

In this appendix, we describe how we aggregate information for Types 2-3 wells, and the data
cleaning procedures.

B.1 Well-Water Right Data Aggregation

For Type 2 wells, i.e., cases where a single well is associated with multiple water rights, we sum
the amount of water extraction caps specified by all associated water rights and treat the sum as the
“single” water right for the well. For Type 3 wells, i.e., cases where multiple wells share the same
water rights, their soil and weather characteristics (e.g., precipitation and dtw) are similar as the
wells are located close to each other. The averages of these characteristics are used as those of the
“aggregate well.” For water extraction, acres irrigated and crops planted, we sum up the reported
levels of individual wells to obtain the corresponding levels of the aggregate well. In cases where
different crops are planted for the individual wells (constituting about 9% of Type 3 wells), we
represent the planted crop of the aggregate well as “other crops.” For the irrigation technology of
the aggregate well, we take the most advanced technology of the individual wells.1

B.2 Data Cleaning Procedures

We drop wells that do not report any irrigation technology during the study period and those that
do not have annual extraction caps on record,2 resulting in an unbalanced panel of 12459 wells and
233206 observations in total. We also drop a small number of wells for which the reported water

1We also tried to use the most frequently used technology among the individual wells, and the regression results

are almost the same.

2Typically, when water rights are abandoned, the whole histories of their relevant information (except for water
right IDs) are dropped from the WIMAS system. WIMAS records the year when the water rights were abandoned, but
this information could not be downloaded. Instead, one needs to go into the WIMAS website and manually check this
information from the “Action Trail” menu for each selected water right. We manually checked 1000 randomly selected
abandoned water rights and found that about 80% of these water rights were abandoned before or during 1980s. Since
our study period spans from 1991 to 2010, dropping abandoned water rights from our sample does not affect our
results.

41

extraction is positive but no planted crops are reported (1,610 observations in total). It is required
that farmers must report the crop they are irrigating if they report a positive water use.3 Thus, these
observations are treated as having missing data, rather than as farmers simply using some water to
preserve their water rights. Similar to Pfeiffer and Lin (2014a), we exclude observations that use
flood irrigation during our study period, resulting in an unbalanced sample with 11,220 wells and
177,195 observations. The sample means of key variables up to this data-cleaning step is presented
in column 1 of table B.1.

Similar to Pfeiffer and Lin (2014a), we remove outlier wells by excluding those with the
maximum irrigated acres in the study period exceeding 640. Doing so reduces the sample size
to 11,099 wells and 175,373 observations. The sample mean of key variables up to this step is
listed in the second column of table B.1. A quick comparison of the first two columns of table B.1
shows that the number of observations drops only by 1.03% after excluding the outlier wells, and
the sample means remain largely unchanged for most key variables. The sample means of water
extraction and water rights slightly decrease after outliers are dropped. The exclusion of the outliers
helps prevent the estimation results from being skewed by a small amount of extreme observations.
Our analysis that uses pooled data will be based on the unbalanced sample of 11,099 wells
and 175,373 observations. However, our correlated random effects model specified in Section 3
can only be applied to balanced samples. For these models, we drop those wells with missing
data for some of the sample years, resulting in a balanced sample of 4,663 wells and 93,260
observations. The associated sample means of the balanced sample are shown in the last column
of table B.1. Comparing the last two columns of table B.1 shows that the average water rights and
dtw are relatively lower in the balanced sample, while the other attributes are similar across the two
samples.

Since the sample size drops significantly going from the unbalanced to the balanced sample, we
run a linear fixed effects variation of equation (2.1) specified in the main text using the two samples
to check whether the two samples lead to significantly different estimates of LEPA’s rebound effects.

3Personal communication with Brownie Wilson, the GIS/Support Services Manager for Kansas WIMAS database

and Kansas Geological Survey.

42

Table B.1: Sample Mean Comparison for Unbalanced and Balanced Samples

Samples
Max Irrigated Acres Outliers
Water Extraction (AF)

Center Pivot Irrigation
LEPA Irrigation
Water Right (AF)
Acres Irrigated (Acres)

Center Pivot Irrigation
LEPA Irrigation

Extraction per acre (AF/Acre)

Center Pivot Irrigation
LEPA Irrigation

LEPA
Precipitation (in)
Depth to Water (ft)
Slope (% of distance)
Potential Evapotranspiration (in)
Saturated Hydroconductivity
Available Water Capacity (cm/cm)
Irrigated Capability Class (dummy)
Percent of Wells planting Certain Crops
Corn
Soybeans
Alfalfa
Corn and Wheat
Corn and Soy
Wheat
Sorghum
Fallow/Dryland Crop
Other Crops
N
# Wells
# Observations with water use = 0
# Observations with water use ∈ (0,5]

Sample Mean
Unbalanced Unbalanced Balanced
Include
Exclude
176.03
180.04
170.74
172.51
185.46
180.92
274.72
306.35
150.61
155.86
149.7
146.6
154.32
160.3
1.15
1.14
1.14
1.14
1.15
1.15
0.52
0.58
25.28
24.49
116.1
94.48
2.93
2.46
52.23
52.6
36.54
29.21
0.17
0.16
0.31
0.39

Exclude
175.37
168.26
180.49
299.07
151.97
146.12
156.19
1.14
1.14
1.15
0.58
24.52
115.66
2.44
52.58
29.13
0.17
0.39

40.46%
7.22
7.6
5.32
3
3.66
1.98
3.26
27.5
177,195
11220
7194
1303

40.70%
7.28
7.61
5.29
3.02
3.67
1.99
3.28
27.15
175,373
11099
7159
1303

42.47%
8.48
9.17
3.22
3.09
3.46
2.01
2.58
25.53
93,260
4663
2956
393

43

Unlike the CRE Tobit model, the linear FE model does not require a balanced panel. The first
two columns of table B.2 show the regression results using the unbalanced and balanced panels
respectively. Figure B.1 graphs the estimated LEPA’s rebound effects for different water right bins,
with the solid line representing the effects estimated using the unbalanced panel and the dashed line
representing those using the balanced panel. To further compare the estimated rebound effects of
LEPA between the two samples, we re-run the linear FE model using cubic splines of water rights
instead of the water right bins, in order to “smooth out” the nonlinear effects.4 Figure B.2 plots the
estimated rebound effects of LEPA for the two samples.5

Table B.2: Estimations for Unbalanced and Balanced Samples

Method
N
Number of Wells
LEPA ∗ 1[0 < R ≤ 100]
LEPA ∗ 1[100 < R ≤ 200]
LEPA ∗ 1[200 < R ≤ 300]
LEPA ∗ 1[300 < R ≤ 400]
LEPA ∗ 1[400 < R ≤ 500]
LEPA ∗ 1[500 < R ≤ 600]
LEPA ∗ 1[600 < R ≤ 700]
LEPA ∗ 1[R > 700]
1[100 < R ≤ 200]
1[200 < R ≤ 300]

Unbalanced
Linear FE
175373
11099
-3.47*
(1.81)
1.83***
(0.68)
2.02***
(0.73)
4.81***
(1.07)
4.30***
(1.64)
-5.58***
(1.54)
-6.21***
(2.12)
-2.47
(1.73)

Balanced
Linear FE
93260
4663
-5.22**
(2.45)
-0.53
(0.80)
1.49*
(0.91)
5.38***
(1.52)
4.70**
(2.29)
-5.01***
(1.96)
-7.62**
(3.78)
-2.78
(2.30)

Balanced
CRE Tobit
93260
4663
-4.29
(2.62)
0.15
(0.82)
2.19**
(0.93)
5.90***
(1.55)
4.40*
(2.35)
-4.51**
(2.00)
-6.83*
(3.85)
-2.99
(2.34)
18.92***
(6.55)
24.17***

(cid:1), where θk(·) is the
4Specifically, we replace
8
to be estimated. Therefore,
 ˆαkθk(r) is the estimate of the rebound effect of LEPA for a well with water right Ri = r.

kth basis function defined over the sample range of water right levels and αk is the coefficient of the kth basis function
In our estimation, we set the number of basis functions to be K = 10.

j=2 LEPAit ∗ R j
i

in equation (2.1) by LEPAit ∗
K

k=1 αkθk(cid:0)Ri

5We do not use the cubic spline method in our main analysis due to the difficulty in calculating the marginal effects

of water rights and the associated standard errors in the CRE Tobit model.

44

1[300 < R ≤ 400]
1[400 < R ≤ 500]
1[500 < R ≤ 600]
1[600 < R ≤ 700]
1[R > 700]
Precipitation

Depth to Water

Potential Evapotranspiration

Irrigated capacity class

Slope

Available water capacity

Saturated hydroconductivity

Corn Acres

Soybeans Acres

Alfalfa Acres

Corn-Wheat Acres

Corn-Soybeans Acre

Wheat Acres

Sorghum Acres

Other Crop Acres

Marginal Effects

Table B.2 (cont’d)

-1.46***
(0.077)
-0.52***
(0.020)
4.29***
(0.190)

-1.76***
(0.100)
-0.54***
(0.025)
4.04***
(0.250)

0.83***
(0.007)
0.78***
(0.009)
0.84***
(0.008)
0.74***
(0.008)
0.80***
(0.011)
0.32***
(0.010)
0.57***
(0.013)
0.71***
(0.007)

0.78***
(0.005)
0.73***
(0.007)
0.81***
(0.006)
0.69***
(0.005)
0.76***
(0.008)
0.30***
(0.007)
0.53***
(0.009)
0.68***
(0.005)

45

(6.85)
26.00***
(8.23)
18.39*
(10.60)
41.71***
(9.91)
-13.02
(13.99)
76.33***
(11.16)
-1.82***
(0.100)
-0.57***
(0.026)
4.21***
(0.250)
-2.17
(1.650)
2.59***
(0.270)
-162.51***
(28.320)
0.058*
(0.035)
0.97***
(0.008)
0.93***
(0.010)
0.98***
(0.009)
0.88***
(0.009)
0.95***
(0.011)
0.48***
(0.011)
0.73***
(0.013)
0.85***
(0.008)

Table B.2 (cont’d)

LEPA

1.21**
(0.480)

0.193
(0.620)

0.81
(0.590)

Note: Standard errors are reported in parentheses. Significance levels: ∗∗∗ = 1%,∗∗ = 5%,
and ∗ = 10%

Figure B.1: LEPA Rebound Effect for Different Water Right Bins (Discrete)

Figure B.2: LEPA Rebound Effect for Different Water Right Levels (Smooth)

46

Table B.2, figures B.1 and B.2 show that the estimates are similar across the unbalanced
and balanced samples. The general pattern of LEPA’s rebound effects first increasing and then
decreasing in water rights is the same, and the magnitudes of LEPA’s rebound effects obtained from
the two samples are close for water rights below 600 AF, which constitute the majority of our data
(cf. figure 2.1(b) in the main text).6 Finally, we also compare the estimates using the balanced
sample of both the linear and CRE Tobit models (the last two columns of table B.2), and find that
the key estimates and the general pattern of LEPA’s rebound effects are similar.

6The rebound effects do differ significantly across the balanced and unbalanced samples for water rights exceeding
600 AF. This difference underlies the difference in the estimated marginal effects of LEPA in the last row of table B.2.

47

APPENDIX C

RESULTS OF ROBUSTNESS CHECKS

In this section, we report the estimation results of the robustness checks with respect to the threshold
value ¯S and the sample restriction to water right application dates prior to 1980. The first three
columns of table C.1 show that the estimated coefficients and marginal effects of key variables (e.g.,
LEPA interacted with water right bins) remain largely unchanged, establishing that our estimates
are robust to different values ¯S. The estimated marginal effects of LEPA across different water right
bins for the three specifications are plotted in figure C.1.
Table C.1: Robustness: Estimating (2.1) for Different Values of ¯S and Samples (Panel Tobit)

Variables
LEPA ∗ 1[0 < R ≤ 100]
LEPA ∗ 1[100 < R ≤ 200]
LEPA ∗ 1[200 < R ≤ 300]
LEPA ∗ 1[300 < R ≤ 400]
LEPA ∗ 1[400 < R ≤ 500]
LEPA ∗ 1[500 < R ≤ 600]
LEPA ∗ 1[600 < R ≤ 700]
LEPA ∗ 1[R > 700]
1[100 < R ≤ 200]
1[200 < R ≤ 300]
1[300 < R ≤ 400]
1[400 < R ≤ 500]

Water Extraction
All years
-2.14
(2.770)
0.23
(0.830)
2.13**
(0.940)
5.87***
(1.570)
4.42*
(2.370)
-4.52**
(2.020)
-6.96*
(3.880)
-2.88
(2.360)
40.12***
(6.810)
45.54***
(7.100)
48.42***
(8.440)
40.84***
(10.790)

All years
-3.47
(2.660)
0.17
(0.830)
2.11**
(0.930)
5.84***
(1.560)
4.33*
(2.360)
-4.52**
(2.000)
-6.90*
(3.860)
-2.98
(2.350)
28.46***
(6.630)
33.78***
(6.930)
36.01***
(8.290)
28.58***
(10.650)

Before 1980
-6.54**
(3.150)
0.3
(0.920)
3.15***
(0.990)
6.66***
(1.630)
4
(2.500)
-5.25**
(2.090)
-4.77
(4.170)
-3.57
(2.540)
22.57***
(8.610)
28.45***
(8.840)
30.88***
(10.090)
23.43*
(12.350)

All years
-4.29
(2.620)
0.15
(0.820)
2.19**
(0.930)
5.90***
(1.550)
4.40*
(2.350)
-4.51**
(2.000)
-6.83*
(3.850)
-2.99
(2.340)
18.92***
(6.550)
24.17***
(6.850)
26.00***
(8.230)
18.39*
(10.600)

48

1[500 < R ≤ 600]
1[600 < R ≤ 700]
1[R > 700]
Precipitation

Depth to Water

Potential Evapotranspiration

Irrigated capacity class

Slope

Available water capacity

Saturated hydroconductivity

Corn Acres

Soybeans Acres

Alfalfa Acres

Corn-Wheat Acres

Corn-Soybeans Acre

Wheat Acres

Sorghum Acres

Other Crop Acres

Constant

Marginal Effects
1[100 < R ≤ 200]
1[200 < R ≤ 300]

Table C.1 (cont’d)

51.29***
(9.960)
-2.62
(14.040)
85.88***
(11.210)
-1.83***
(0.100)
-0.57***
(0.026)
4.23***
(0.250)
-2.24
(1.650)
2.58***
(0.270)

41.30***
(11.670)
-18.84
(16.030)
80.53***
(13.030)
-1.87***
(0.110)
-0.60***
(0.028)
4.22***
(0.280)
-2.63
(1.820)
2.45***
(0.300)

41.71***
(9.910)
-13.02
(13.990)
76.33***
(11.160)
-1.82***
(0.100)
-0.57***
(0.026)
4.21***
(0.250)
-2.17
(1.650)
2.59***
(0.270)
-162.51*** -164.52*** -167.21*** -164.14***
(28.320)
0.058*
(0.035)
0.97***
(0.008)
0.93***
(0.010)
0.98***
(0.009)
0.88***
(0.009)
0.95***
(0.011)
0.48***
(0.011)
0.73***
(0.013)
0.85***
(0.008)
81.03
(60.970)

(28.350)
0.057
(0.035)
0.96***
(0.008)
0.92***
(0.010)
0.97***
(0.009)
0.87***
(0.009)
0.94***
(0.012)
0.47***
(0.011)
0.72***
(0.013)
0.84***
(0.008)
69.02
(61.020)

(31.450)
0.087**
(0.039)
0.97***
(0.009)
0.93***
(0.010)
0.98***
(0.009)
0.87***
(0.010)
0.94***
(0.013)
0.48***
(0.011)
0.72***
(0.014)
0.84***
(0.008)
98.84
(67.820)

63.35***
(10.080)
10.63
(14.150)
98.03***
(11.320)
-1.86***
(0.110)
-0.57***
(0.026)
4.29***
(0.260)
-2.33
(1.650)
2.58***
(0.270)

(28.430)
0.054
(0.035)
0.95***
(0.008)
0.90***
(0.010)
0.96***
(0.009)
0.86***
(0.009)
0.92***
(0.012)
0.44***
(0.011)
0.70***
(0.013)
0.82***
(0.008)
53.36
(61.200)

19.36***
(6.010)
25.25***

27.44***
(5.980)
33.34***

36.93***
(5.990)
42.91***

23.76***
(7.800)
30.65***

49

1[300 < R ≤ 400]
1[400 < R ≤ 500]
1[500 < R ≤ 600]
1[600 < R ≤ 700]
1[R > 700]
LEPA (overall)

@0 < R ≤ 100
@100 < R ≤ 200
@200 < R ≤ 300
@300 < R ≤ 400
@400 < R ≤ 500
@500 < R ≤ 600
@600 < R ≤ 700
@R > 700

Table C.1 (cont’d)

(6.320)
28.79***
(7.650)
20.95**
(9.860)
38.48***
(9.340)
-12.62
(12.360)
72.64***
(10.760)
0.81
(0.590)
-3.84*
(2.330)
0.14
(0.760)
2.05**
(0.870)
5.55***
(1.460)
4.09*
(2.190)
-4.29**
(1.900)
-5.91*
(3.310)
-2.92
(2.290)

(6.290)
37.28***
(7.630)
29.58***
(9.860)
46.58***
(9.320)
-3.85
(12.380)
80.75***
(10.740)
0.8
(0.590)
-3.04
(2.320)
0.16
(0.770)
1.97**
(0.870)
5.49***
(1.470)
4.03*
(2.200)
-4.30**
(1.900)
-5.98*
(3.320)
-2.91
(2.290)

(6.290)
47.45***
(7.640)
39.64***
(9.890)
56.41***
(9.330)
6.97
(12.440)
90.73***
(10.750)
0.87
(0.600)
-1.82
(2.350)
0.21
(0.770)
1.99**
(0.880)
5.52***
(1.480)
4.11*
(2.210)
-4.29**
(1.910)
-6.05*
(3.350)
-2.81
(2.310)

(8.030)
34.69***
(9.270)
26.39**
(11.410)
38.54***
(10.840)
-15.5
(13.930)
77.17***
(12.410)
1.20*
(0.650)
-5.79**
(2.770)
0.28
(0.850)
2.95***
(0.930)
6.26***
(1.540)
3.72
(2.330)
-4.97**
(1.970)
-4.04
(3.510)
-3.48
(2.470)

Note: Standard errors are reported in parentheses. Significance levels: ∗ ∗ ∗ = 1%,∗∗ = 5%, and
∗ = 10%

The estimation results for the subsample with water right application dates prior to 1980 are
reported in the last column of table C.1, and the marginal effects of LEPA are plotted in figure C.1.
The trends of the marginal effects of LEPA across different water right bins are the same between
the full sample and the prior to 1980 sample. The magnitudes of the rebound effects are also rather
close to each other over a wide range of water right bins, so are the magnitudes of the water right
bins themselves.

50

Figure C.1: LEPA Rebound Effect for Different Water Right Bins (Prior 1980, Discrete)

51

APPENDIX D

COPYRIGHT PERMISSION

52

53

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54

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58

CHAPTER 3

WHAT DRIVES (NO) ADOPTION OF NEW IRRIGATION TECHNOLOGIES: A
STRUCTURAL DYNAMIC ESTIMATION APPROACH (WITH JINHUA ZHAO)

3.1 Introduction

Agriculture is among the most vulnerable sectors to global climate change. With the prospect
of greater variabilities in rainfall and higher frequencies of serious droughts, irrigation is becoming
increasingly important as an adaptation strategy to climate change (Zilberman, Zhao, and Heiman,
2012). However, throughout the world, irrigation is facing the major challenge of declining
groundwater, and this is the most serious in semi-arid agricultural zones such as California and the
High Plain Aquifer (HPA) area in the US. For example, half of groundwater storage in the southern
Ogallala aquifer underlying the HPA has been depleted, threatening agricultural sustainability in
this important production area (Haacker, Kendall, and Hyndman, 2015). As a result, more efficient
irrigation technologies are needed in many parts of the world in order to preserve water resources
while maintaining and even raising agricultural outputs and profits.

Better technologies are not always adopted; there is a rich economics literature documenting
the slow diffusion of efficient new technologies, including irrigation technologies (Sunding and
Zilberman, 2001; Zilberman, Zhao, and Heiman, 2012). One example, which is the subject of this
paper, is the adoption and diffusion of Low Energy Precision Application (LEPA) in the Kansas
portion of HPA: it took more than 20 years before LEPA diffusion was completed. A major reason
for the slow diffusion, especially at the early stages, is the combination of sunk adoption costs,
which include the upfront purchasing, installing and learning costs, and uncertainties regarding
the new technology’s profitability. A forward-looking farmer has incentive to wait and obtain
more information about the new technology’s profit gains before incurring the irreversible (sunk)
adoption costs (Carey and Zilberman, 2002; Zhao, 2007). In other words, farmers do not simply
choose whether or not to adopt. Instead, they also have the option to delay the adoption decision

59

until more information is available so that they are more certain about the technology’s profitability
(Taylor and Zilberman, 2015). Thus, waiting is treated as a "real option" for farmers. This real
options framework has been applied to a wide range of investment and policy decisions beyond
technology adoption (Dixit and Pindyck, 1994). Besides these rational considerations, certain
irrational behavioral biases could also lead to slow technology diffusion. For example, farmers
might have psychological inertia in choosing alternative farming practices. Such inertia raises
switching cost and significantly reduces the incentive to adopt new technologies (Handel, 2011;
Hortaccsu, Madanizadeh, and Puller, 2017; Ito, Ida, and Tanaka, 2016).

In this paper, we study the underlying farm-level incentives that drive LEPA diffusion in
the Kansas part of HPA by structurally estimating a dynamic technology adoption model that
incorporates uncertainty, learning, irreversibility, inertia and delay. We are able to provide an
in-depth understanding of irrigation technology adoption behavior by incorporating a number of
potential adoption determinants in a theoretical-consistent framework. We first develop a real
options model of LEPA adoption, and then structurally estimate the underlying parameters using
observed LEPA diffusion data. The structural modeling framework also allows us to make out-of-
sample predictions and conduct counterfactual policy simulations.

We find empirical evidence that LEPA profit gain level, profit gain uncertainty and upfront
adoption costs are all important determinants of LEPA adoption behaviors. Specifically, farmers do
value the option to delay adoption when they are making an irreversible adoption decision and when
they are uncertain about LEPA’s profit gain. Consistent with the predictions of the real options
adoption model, this option significantly decreases LEPA diffusion speed. If farmers simply choose
whether or not to adopt LEPA in each year but do not consider the option to delay adoption, average
LEPA adoption rate would increase by as much as 42%, and the time it takes to reach the long-run
diffusion rate observed in the sample would shrink by over 50%.

We also find evidence that peer effects positively affect LEPA adoption, which is consistent with
the findings in Sampson and Perry (2018). The interpretation of peer effects in our model is fairly
flexible due to our flexible model specification regarding peer effects. One such interpretation is

60

that the presence of more neighbor LEPA adopters significantly reduces farmers’ inertia to choose
alternative farming practices. Unlike the rational incentive to delay adoption created by profit
gain uncertainty and adoption irreversibility, inertia represents a form of behavioral bias that raises
farmers’ subjective adoption costs. By reducing inertia, peer effects reduce perceived adoption
costs and increase LEPA adoption probability.

Many important agricultural districts across U.S and across the world are subsidizing or are
considering subsidizing water-saving irrigation technology adoptions. For example, Arizona,
California, Colorado, Texas and Utah have received nearly half of Environmental Quality Incentives
Program’s (EQIP) funding to promote irrigation efficiency improvement (Nixon, 2013). As another
example, the Comprehensive Water Management Plan (CWMP) issued by Chinese government in
2006 also requires the central government to subsidize a major part of water management equipment
construction (Yao, Zhao, and Xu, 2017). We study the relative cost-effectiveness of two frequently
used subsidy policies through counterfactual simulations: a LEPA profit gain insurance policy
that guarantees a minimum annual per-acre profit gain level, and a cost share policy that directly
reduces LEPA adoption costs. Simulation results show that insurance policies are always more
cost-effective than the cost share policies in promoting LEPA adoption, given the same government
budget.

To our knowledge, this is the first paper that decomposes the effects of profit gain level, profit gain
uncertainty and adoption costs on irrigation technology adoption. Most empirical work utilizes
reduced-form techniques to study irrigation technology adoption (Alcon, Miguel, and Burton,
2011; Caswell and Zilberman, 1986; Dinar and Yaron, 1992; Foltz, 2003; Koundouri, Nauges,
and Tzouvelekas, 2006; Kulecho and Weatherhead, 2006; Salazar and Rand, 2016; Shrestha and
Gopalakrishnan, 1993). While reduced-form studies can estimate the overall effects of a certain
factor on adoption decisions, the pathways through which the causality relationship is established
remains unclear due to the lack of variation in uncertainty levels across farmers and lack of true
adoption cost data. Our study, on the other hand, employs a structural dynamic estimation approach
to overcome these difficulties.

61

Despite their advantages, structural dynamic models are less often used in the literature than
reduced-form models for two main reasons. The first is the curse of dimensionality (Rust, 1987):
computational cost increases exponentially as the number of state variables increases. Technology
adoption decisions are affected by multi-dimensional uncertainties such as input and output price
uncertainties, and incorporating all of them (as state variables) typically leads to a Bellman equation
that cannot be evaluated numerically. The second challenge facing structural dynamic models is
the lack of robustness (Arcidiacono and Miller, 2011): the estimation results are typically more
sensitive to model specification than reduced form models. A main reason is that structural dynamic
models typically assume that a parsimonious set of parameters can capture all underlying incentives,
while reduced form models can resort to such tools as fixed effects models using panel data to deal
with the many individual characteristics not captured by available data.

In this paper, we adapt the novel methodology proposed by Arcidiacono and Miller (2011), which
simultaneously addresses both issues of dimensionality and unobserved individual heterogeneity.
The importance of accounting for individual heterogeneity in technology adoption studies has been
documented by Suri (2011), which shows that individual heterogeneity explains large variations
in the adoption of hybrid maize in Kenya.
In addition, unobserved heterogeneity introduces a
special pattern of serial correlation in error terms and represents a form of dynamic selection,
which could bias the estimation of other parameters if not properly accounted for. The difficulty
of incorporating individual heterogeneity in structural dynamic discrete choice models, similar to
that of incorporating multiple state variables, mainly arise from the computational burden, which is
alleviated in Arcidiacono and Miller (2011) by combining reduced form techniques with structural
estimation in a novel fashion. Our paper represents the first application of their methodology to
irrigation technology adoption. Other applications include Chung, Steenburgh, and Sudhir (2013),
which studies sale force responses to a bonus-based compensation plan and Scott (2013), which
studies agricultural land use dynamics in the U.S.

We construct a large well-level panel dataset of LEPA adoption for the estimation.

In the
literature, cross sectional (Caswell and Zilberman, 1986; Foltz, 2003; Koundouri, Nauges, and

62

Tzouvelekas, 2006) and aggregate time-series datasets (Alcon, Miguel, and Burton, 2011) have
been used to study irrigation technology adoption decisions. Cross sectional data, being a snapshot
of the diffusion process, provide little information on the diffusion process. Time series data, on
the other hand, depicts the aggregate diffusion process without information on individual decisions
and individual heterogeneity. The panel dataset enables a more comprehensive investigation of
individual adoption decisions and aggregate diffusion processes. Only a few studies in the irrigation
adoption literature use panel datasets, e.g., Shrestha and Gopalakrishnan (1993), and Genius et al.
(2014), but even with these studies, the datasets contain either too few study periods or too few
cross-sectional observations (i.e. farmers).

The remainder of the paper proceeds as follows. We review the relevant literature in section 3.2
and construct a conceptual real options adoption model in section 3.3. We introduce the study
area and the dataset we use in section 3.4. Section 3.5 enriches the conceptual model to make
it feasible for empirical implementation, given data available to us. Section 3.6 explains our
estimation strategies. We present and discuss estimation results in section 3.7 and conduct a set of
counterfactual simulations in section 3.8. The paper concludes in section 3.9.

3.2 Literature Review

This paper brings together two branches of literature, namely the literature on technology

adoption and that on the estimation of structural dynamic discrete choice models.

Since the seminal work of Griliches (1957), technology adoption has been characterized as
a rational behavior that is driven by the new technology’s profitability. The S-shaped diffusion
pattern of many new technologies is shown to be a result of profit gain heterogeneity among
potential adopters (David, 1966; Mansfield, 1961; Rogers, 1995; Sunding and Zilberman, 2001;
Weir, Knight, et al., 2000). In a study of hybrid maize adoption in Kenya, Suri (2011) finds strong
evidence that the adoption decisions are rational and could be explained well by both observed
and unobserved heterogeneous net benefits of the technology. With a structural dynamic model,
Ryan and Tucker (2012) also finds that observable heterogeneities in benefits and adoption costs

63

are important determinants of intra-firm video-call technology diffusion.

When technology adoption is treated as a dynamic decision, not only a new technology’s prof-
itability, but also decision makers’ uncertainty towards its profitability would affect its adoption
pattern. As explained by Dixit and Pindyck (1994), decision makers not only choose between adopt-
ing or not adopting in a given period, but also choose whether or not to delay the adoption/investment
decision to the future until more information becomes available (Taylor and Zilberman, 2015). In
one setting, potential adopters are uncertain about the technology itself. For example, how to best
operate the technology. Consequently, economic agents have incentive to obtain more information,
e.g., through other agents’ experience of using the technology (Besley and Case, 1994; Conley and
Udry, 2010; Foster and Rosenzweig, 1995; Goolsbee and Klenow, 2002). If the agent expects that
many of his neighbors will adopt soon, he might want to delay adoption until his neighbors adopt
first(Foster and Rosenzweig, 1995; Zhao, 2003). This delaying behavior is known as "strategic
delay". In another setting, decision makers have perfect information about the technology itself but
are uncertain about future realizations of exogenous factors (e.g. output prices) that determine the
net profit of the new technology (Farzin, Huisman, and Kort, 1998). Thus, they would not adopt
the technology until its profit gain raises to a fairly high level such that they are more or less sure
that it would not drop to very low levels in the future. Examples of this case includes but are not
limited to studies on electricity generator investment (Reedman, Graham, and Coombes, 2006), oil
drilling (Kellogg, 2014), and choice among competing new technologies (Kauffman and Li, 2005).
Irrigation technology adoption is among one of the most frequently studied technology adoption
topics in the literature. Most empirical studies on irrigation technology adoption take reduced-form
approaches (e.g. Probit/Logit and duration models). Exogenous factors that affect new irrigation
technology profits such as well depth, water price, land quality, crop price, farmer characteristics,
credit availability and peer effects are found to be important drivers of irrigation technology adoption
(Alcon, Miguel, and Burton, 2011; Caswell and Zilberman, 1986; Dinar and Yaron, 1992; Foltz,
2003; Genius et al., 2014; Koundouri, Nauges, and Tzouvelekas, 2006; Kulecho and Weatherhead,
2006; Salazar and Rand, 2016; Sampson and Perry, 2018; Sampson and Perry, 2019; Shrestha

64

and Gopalakrishnan, 1993). However, we are not aware of structure estimation papers that fully
account for the dynamic nature of irrigation technology adoption.

Traditionally, the Nested Fixed Point Algorithm (NFXP) proposed by Rust (1987) has been used
to estimate structural dynamic discrete choice models. A significant drawback of this approach is the
high computational burden involved, since the highly nonlinear dynamic programming problem has
to be solved for every observation for each iteration of parameter search. With many observations,
the execution speed is extremely slow. More importantly, accounting for unobserved heterogeneity
could make NFXP computationally infeasible (Arcidiacono and Miller, 2011). Alternatively, two-
step procedures in Hotz and Miller (1993) and Bajari, Benkard, and Levin (2007) have been
developed to reduce computational burden associated with NFXP. The idea is to estimate reduced-
form conditional choice probabilities (CCPs) of choosing each option as a flexible function of
state variables in the first step. These estimated probabilities are subsequently used as inputs in a
second step estimation to recover structural parameters. In particular, the value functions could be
re-formatted as functions of state variables and CCPs in the second step, eliminating the need to
solve the dynamic programming problem for each iteration. This approach is used frequently in
industrial organization studies. For example, Ryan (2012) uses the model to study the effects of
environmental regulation on cement firm competition and their entry/exit decisions. Bajari et al.
(2013) studies the impacts of income and credit constraints on housing demand. However, this
approach cannot deal with unobserved heterogeneity, which is critically important for understanding
technology adoption.

Unobserved heterogeneities in dynamic discrete choices are successfully controlled for in Ar-
cidiacono and Miller (2011). This approach, through a combination of Expectation-Maximization
(E-M) algorithm and the two step-procedures similar to Hotz and Miller (1993) and Bajari, Benkard,
and Levin (2007), incorporates unobserved individual heterogeneities without substantially increas-
ing computational burden. We adopt this approach in this study, offering one of the first applications
of this estimation method to technology adoption.

65

3.3 Theoretical Framework: A Real Options Approach

This section outlines a model of irrigation technology adoption for farmers who follow real
options decision rule. Specifically, each farmer decides the optimal time to replace his current CP
(P) irrigation system with a more efficient substitute, LEPA (L). While this simple model abstracts
away from many details of the real decision problem, we extend it in section 3.5 to make it more
realistic for empirical implementation.

At the beginning of each year (t), a farmer (i) who has not adopted LEPA considers whether
or not to adopt LEPA in the current year. If he adopts LEPA, he starts to use LEPA immediately
after paying the upfront adoption costs. LEPA adoption is assumed to be irreversible – farmers
never switch back to CP once LEPA has been adopted.1 Therefore, farmers who adopted LEPA
in previous years do not make adoption decisions again. On the other hand, if he chooses not to
adopt LEPA in year t, he continues to make adoption decision in the next year. The farmer solves
a single-agent dynamic programing problem to maximize the sum of discounted expected lifetime
profits by determining the optimal time to adopt LEPA.

3.3.1 LEPA Adoption Profit Gain and Learning

Farmer i’s profit gain from using LEPA (compared to using CP) in year t is πD
it
farmers and years. Additionally, πD
it

follows a transition process that is known to the farmer:

, which varies across

πD
it+1 = γ0 + γ1πD

it + γ2πD

it−1 + γ3t + ηit+1.

(3.1)

where ηit ∼ i.i.dN(0, ση). In addition, we assume that future profit gain realizations are positively
correlated with previous years’ profit gain realizations. That is, γ1 > 0 and γ2 > 0. Thus, current
and past profit gain realizations are informative about its realizations in the future. This represents
a form of learning that is used frequently in real options models. This transition structure has three
important features:

1In the data, farmers seldom switch back from LEPA to CP because LEPA is almost always more profitable than
CP due to the increased efficiency and reduced energy expenditures. Once adopted and the sunk adoption costs have
been incurred, there is no reason for farmers to switch back to CP.

66

First, although a farmer could observe his profit gain realizations in the current year (πD
it

) and in
it−1) , he could not predict perfectly those in the future. The random noise ηit+1 captures

the past (πD
both profit gain volatility and the farmer’s uncertainty toward future profit gain realizations.

Second, due to this volatility, it is always possible for πD

it+1 to drop to very low levels, even

though πD
it

and/or πD

it−1 are relatively high.

Third, as πD

it+1 is positively correlated with πD
it

and πD

it+1, the probability of πD

it+1 dropping to

a very low level is lower when πD
it

and πD

it−1 are higher.

These three features have important implications on farmers’ adoption behaviors, as will be

discussed later in this section.

3.3.2 LEPA Adoption Costs

Converting from CP irrigation to LEPA requires purchasing water-delivering nozzles and attaching
the nozzles to the CP frame. The amount and/or type of nozzles to purchase depend on many
factors such as the initial CP frame’s size and age. Besides, the conversion also requires a change in
farming practices such as constructing round furrows and performing conventional tillage. Again,
the costs of such changes are different for different farmers depending on their initial farming
practices. We assume that the various cost components above are time-constant and denote these
heterogeneous upfront adoption costs by Ci.

Moreover, farmers may have psychological inertia in choosing alternative irrigation systems
and farming practices, which is a form of perceived adoption cost. The role of inertia as an
"implicit" switching cost has been well-documented (Handel, 2011; Hortaccsu, Madanizadeh,
and Puller, 2017; Ito, Ida, and Tanaka, 2016). This inertia, however, is lower when farmers are
surrounded by more neighbor LEPA adopters. The effects of neighbor adopters on inertia (peer
effects) is represented by f(nit), where nit is the number of neighbor LEPA adopters for farmer i
at the beginning of year t (or equivalently, at the end of year t − 1). Consequently, the complete
characterization of adoption costs is:

Cit = Ci + f(nit).

67

(3.2)

The interpretation of neighbor LEPA adopters’ effects on LEPA adoption is flexible. Besides
reducing perceived adoption costs, peer effects could take multiple other forms. For example,
farmers could derive utility not only from profit gains of using LEPA, but also from their neighbor
It is possible that the appearance of neighbor adopters simply makes farmers feel
adopters.
comfortable about adopting LEPA, even though it does not really affect profit gains.
It is also
possible that farmers are not familiar with the performance of the technology and learn from his
neighbor adopters about this information. Even though they may not be able to observe their
neighbors’ profit gain realizations, number of neighbor adopters could also provide them with
additional information on the merits of the new technology. Under these alternative settings,
+ f(nit), where now f(nit)
Cit = Ci, but the annual profit gain function could be modified as πD
it
translates utility gains from witnessing more neighbor adopters into a monetary value. Although
we incorporate f(nit) in the adoption cost function rather than the profit function in our model, this
choice does not affect parameter estimations in the empirical model, a point that will be made more
clear in section 3.6.

3.3.3 Value of Immediate LEPA Adoption

If a farmer adopts LEPA in the current year (i.e. year t), he will accumulate profit gain from LEPA
adoption for all periods τ ≥ t (since LEPA adoption is irreversible). Without considering adoption
costs, the value of adopting LEPA in year t is simply the summation of discounted expected LEPA
profit gain from the current year on:

VL(Γit) = Et

βτπD
iτ

= πD

it + βE {VL(Γit+1)|Γit},

(3.3)

(cid:40) ∞


(cid:41)

τ=t
where β is the annual discount factor.
expectation is taken with respect to πD

In addition, Γit = {πD
and πD

it+1 conditional on πD
it

it−1}, which means that the

it , πD
it−1, according to (3.1).

If the farmer follows Net Present Value (NPV) rule in making adoption decision, he would
adopt LEPA in year t as long as the ex-ante expected NPV of adoption in year t is positive:
VL(Γit) − Cit > 0. By doing so, the farmer only compares the value of adopting LEPA now (i.e.

68

VL(Γit) − Cit) and the value of never adopting (i.e. 0). However, he ignores the value of the option
to remake adoption decision in the future if he gives up adopting LEPA in this year. In this sense,
the farmer is myopic.

Delaying adoption decision is valuable when farmers are uncertain about future LEPA profit
gain realizations and when LEPA adoption is irreversible. As discussed in section 3.3.1, even if
current year’s ex-ante expected NPV of adoption is positive, there is always a chance that ex-post
profit gain in the future drops to a fairly low level due to its inherent volatility, and the adoption costs
might not be recouped if adoption has already occurred. If this situation happens, the farmer would
regret of having adopted LEPA previously, even though the ex-ante expected NPV of adoption at
that time used to be positive. The irreversibility of LEPA adoption and uncertainty about future
profit gain realizations together render a positive value of delaying adoption until more positive
information about future profit gain realizations arrives.

3.3.4 Value of Delay and The Adoption Decision

Formally, the value of the option to delay adoption decision to the next year could be expressed as:

VP(Γit,Cit) = βE(cid:8)max[VP(Γit+1,Cit),VL(Γit+1) − Cit]|Γit

(cid:9).

(3.4)

In other words, a farmer will adopt LEPA in the next year if the realized conditions next year

favor LEPA, but he could choose to continue waiting if otherwise.

Note that Cit, rather than Cit+1, appears in the right hand side of (3.4). Since the only time-
varying component in Cit is inertia, we implicitly assume that farmers do not think/realize that
their inertia levels would change in the next year if more neighbors adopt LEPA this year.2 In other
words, we do not model strategic delay. At the beginning of each year, inertia levels are set after
farmers observe how many of their neighbors have already adopted LEPA, but they do not consider
the possibility that their inertia levels could be further reduced if they delay adoption until more
neighbors adopt.

2This assumption does not rule out that farmers are forward-looking in other aspects. For example, the farmer is

still projecting πD

it+1 based on their current and past profit gain realizations.

69

The model would be extended to a dynamic game model among farmers if strategic delay is
considered. When making his adoption decision in one year, a farmer needs to predict how many
neighbors are going to adopt LEPA in that year. However, his neighbors’ adoption behaviors are
also affected by many other farmers (i.e. his neighbor’s neighbors), whose behaviors again are
affected by another set of farmers. As there is a large set of farmers in the region, each farmer
would take a very large number of other farmers’ adoption behaviors into account, which makes
empirical implementation infeasible. Therefore, we assume that farmers do not practice strategic
delay in reality and leave the test of this assumption in future research.

3.3.5 Adoption Decision

Given the model structure above, farmer i would adopt LEPA in year t if and only if:

VL(Γit) − Cit > VP(Γit,Cit).

(3.5)

That is, he will only adopt LEPA if the current year’s expected NPV from LEPA adoption is
higher than that from delaying adoption decision to the future. Another interpretation of (3.5)
comes from rewriting it as VL(Γit) > Cit + VP(Γit,Cit). That is, the sum of discounted expected
profit gains from LEPA adoption must be high enough to compensate both adoption costs and the
option value. This situation will happen if year t profit gain is high enough such that the farmer is
more or less certain that it would not drop to a fairly low level in the future.

3.4 Study Region and Data

Our study region is the Kansas part of HPA, which spans more than 30,000 square miles across
western Kansas.
Irrigation agriculture accounts for more than 90 percent of total agricultural
production in this area and groundwater is almost the only irrigation water source. Information on
well locations and irrigation technologies is drawn from the Water Information Management and
Analysis System (WIMAS) maintained by the Kansas Water Office. Our sample contains a large
number of irrigation wells in this area – 7251 in total. The data also spans a long time period –
from 1997 to 2010.

70

Spatially explicit data on depth to groundwater are obtained from the output files of Haacker,
Kendall, and Hyndman (2015). Geo-referenced precipitation and temperature data are obtained
from the North America Land Data Assimilation System (NLDAS) maintained by NASA. We
match the well-level data with these spatially explicit data in ArcGIS according to each well’s
geographical coordinates. Finally, information on crop price comes from NASS quick stats and
information on natural gas price is obtained from EIA. All price data are converted to 2010 dollar
values, adjusting for inflation rate.

Table 3.1: Summary Statistics (1997-2010)

N

Mean

Std. Dev. Min. Max.

1.05
1.13
1.02
2.86
5.89
113.36
1.01
15.88

19.32
24.56
28.19
27.08
22.45
14.49
7.08

35,332
6,533
28,799
35,332
35,332
35,332
35,332
28081

35,332
35,332
35,332
35,332
35,332
35,332
35,332
7,251
6,533

0.59
0.50
0.59
0.59
2.53
89.65
0.19
7.45

1.49
1.39
1.24
1.61
2.18
1.49
2.36

0
0
0
2.27
3.59
0
0.43
0

15.36
21.04
24.52
22.71
17.68
7.78
-0.86

17.42
5.10
17.42
4.95
13.65
386.51
1.54
30

23.02
28.75
31.23
30.65
27.38
17.83
11.58

6,533
4,658

1999.85
2001.02

3.14
3.04

1997
1998

2010
2010

Variables
Water Use (feet/acre)

LEPA
CP

Real Corn Price ($/Bu)
Real Gas Price ($/Mcf)
Depth to Water (ft)
Effective Precipitation (feet/year)
Number of Neighbor LEPA Adopters
Monthly Average Temperature (°C)

May
Jun
Jul
Aug
Sep
Oct
Nov
# Wells
# Wells that Adopted LEPA
LEPA Adoption Year

Full Sample
Post-1997 Adopters

Table 3.1 presents the summary statistics of the variables. The major crop in this region is
corn. Western Kansas has a semi-arid climate with average annual precipitation around 1 foot/year,
which could not meet water demand for most crops. Therefore, irrigation is an important factor
determining crop yields. Prior to the introduction of LEPA in 1991, Center Pivot (CP) was the

71

dominant irrigation technology in the study region. Compared to CP, LEPA is more efficient3 and
requires less energy to lift groundwater. Therefore, LEPA is in general more profitable.

Figure 3.1 depicts LEPA’s diffusion path. A sudden jump in total number of adopters is observed
in 1997, which is mainly caused by inconsistent definition of LEPA before and after 1997.4 We
only use data after 1997 in our empirical exercise for internal consistency.

Figure 3.1: LEPA Diffusion Path in Kansas HPA

Another potential reason for the sudden jump might be that the state of Kansas started to provide
cost shares to a small number of adopters after 1996. We do not have information on exactly who
received the subsidy, but we do know the total number of farmers who received the subsidy in each

3efficiency =

amount of water utilized by crop

total water extraction

(Pfeiffer and Lin, 2014). The efficiency level is always less than 1

because part of irrigation water is lost due to evaporation, soil water percolation and wind drift.

4Before 1997, only Center Pivot with LEPA nozzles was classified as LEPA in the dataset. However, all Center
Pivot systems equipped with dropped nozzles were classified as LEPA after 1997. The different types of nozzles
have similar efficiency levels, with the most significant difference between them being the ability to be adapted to
local farming conditions (personal communication with Dr. Danny H. Rogers, Kansas State University Research and
Extension, Department of Biological and Agricultural Engineering).

72

year. Across all sample years, only around 12% of LEPA adopters received the subsidy from 1996
to 2010 and the average cost share per farmer is $16/acre. Figure (3.2) plots the annual percent of
new LEPA adopters who received cost share subsidies. In each year during 1997-2006, a period
when most LEPA adoptions happen, the share of LEPA adopters who received cost share subsidies
is relatively low, varying between 7% to 20%. This share drops to 0 after 2007. Given this low cost
share subsidy reception rate and low average cost share amount, we ignore the cost share program
in our empirical model.

Figure 3.2: Percent of New LEPA Adopters Who Received Cost-Share Subsidies Among All New
LEPA Adopters by Year

3.5 Empirical Adoption Model

This section enriches the theoretical model outlined in section 3.3 to make it feasible for
empirical implementation. We first derive the expression of πD
from the underlying production
it
and cost functions, whose parameters are empirically estimated in section 3.6. We also distinguish
variables that are observed by both farmers and econometricians from variables that are only
observed by farmers. Finally, we show a transformation of Bellman equations associated with the
adoption decision, which forms the basis of the dynamic discrete choice estimation strategy in
section 3.6.

73

3.5.1 Profit Gain Following LEPA Adoption

3.5.1.1 Production Function Specification

The first step in calculating profit gain following LEPA adoption is to specify a crop production
function. Since our focus is on irrigation technologies, we model water as the single variable input,
with other inputs optimally chosen given water application rates. In order to maintain flexibility,
we specify a concave per-acre crop-water production function, following many agronomic studies
(Martin, Watts, and Gilley, 1984; Tong and Guo, 2013):

qi jt(wi jt) = α0it + α1it ∗ (θ j ∗ wi jt + rit) + α2(θ j ∗ wi jt + rit)2

,

for j ∈ {P, L},

(3.6)

where wi jt is annual water extraction, rit is annual rainfall, and θ j is water use efficiency of
technology j ∈ {P, L}, which equals the percentage of total water extraction that is utilized by the
crop. Therefore, θ j ∗ wi jt + rit is the total effective water from irrigation and rainfall that is utilized
by the crop. LEPA is more efficient than CP: the same level of effective water is reached with less
water extraction when LEPA is used. We set θP = 0.85 and θL = 0.95 following Amosson et al.
(2001) and Irmak et al. (2011). We assume that the concavity parameter α2 is constant but allow
α0it and α1it to be heterogeneous across farmers or years to capture the effects of unobservables on
crop yields.5

Revenue per acreage equals the output in (3.6) multiplied by crop price. Although farmers plant
a variety of crops in Kansas HPA, corn is the dominant crop. According to USDA NASS statistics,
corn production ranks top1 or top2 in each year during 1996-2010 in the study area. Although
we have data on self-reported crop code in WIMAS database, the information is noisy. Reported
crop type often represents a combination of multiple crops (e.g. corn, wheat and barley), and the
areas planted to different crops in the combination are not reported, making it difficult to utilize
this information. We treat the production function in (3.6) as a generic production function without
distinguishing among different crops and use corn price as crop price. The cost of this simplification

5 α0it will finally be canceled out when calculating profit difference between LEPA and CP.

74

is low: corn dominates the production region, and heterogeneity of α1it across farmers and years
can also capture farm-specific and year-specific crop choices.

We next specify the production cost function. Following Rogers and Alam (2006) and Pfeiffer
and Lin (2014), the variable cost to pump one acre-foot of groundwater is specified as a linear
function of total water extraction:

ci jt(wi jt) = pe

i jt ∗ wi jt = η ∗ pgas

t

∗ (dtwi + 2.31 ∗ PSIj) ∗ wi jt,

j ∈ {P, L},

(3.7)

being energy cost to lift one acre-foot of groundwater for farmer i using technology j in
with pe
i jt
year t. Energy cost is calculated from the engineering relationship between the amount of natural
gas needed to lift one acre-foot groundwater up for one foot (η = 0.022), natural gas price (pgas
),
irrigation well’s depth to groundwater (dtwi) and water pressure required to operate irrigation
system (PSIj). LEPA requires less energy to operate than CP – the required water pressure to
operate the system decreases from around 75 PSI for CP to 18 PSI for LEPA (Delano, Williams,
and O’Brien, 1997). Similar to Hendricks and Peterson (2012), we assume that all farmers use
natural gas to power their irrigation equipment.6 Even though farmers face the same natural gas
price in a given year, different wells have different levels of depth to water and uses different
irrigation systems, so the energy cost per unit water extraction is farm-and-technology specific.
Finally, the cost of water pumping is known to both farmers and econometricians.

t

Combining (3.6) and (3.7), farmer i’s per-acre profit from using irrigation system j is

j

it(wi jt) = pt ∗ qi jt(wi jt) − ci jt(wi jt),

π

(3.8)

where pt is crop price in year t.

3.5.1.2 Optimal Water Extraction

In each year, water extraction decisions are made after farmers haven selected irrigation technolo-
gies. Groundwater is a common pool resource, shared by multiple farmers over the same aquifer. A

6The public available database in WIMAS does not contain information on irrigation well level energy source.

75

single farmer’s water use im a given year has insignificant effects on the aquifer level, and even less
effect on his future water use and profits. Therefore, we assume that a farmer’s water use does not
influence his well’s depth to groundwater in the future (i.e, dtwi is time-constant).7 Consequently,
the water extraction decision is static. Maximize (3.8) with respect to wi jt, the first order condition
(FOC) gives optimal water extraction per acre for farmer i in year t with irrigation technology j:

w

∗
i jt = − α1it
2α2θ j

+

1
2α2

pe
i jt
2
j pt

θ

− rit
θ j

.

(3.9)

into equation (3.8) gives the optimal per-period profit function π

Plugging w∗
it(xit), where xit =
i jt
{pe
i jt, pt,rit} is the set of state variables that affect optimal water extraction. Finally, annual profit
gain from using LEPA (compared to using CP) is expressed as:

j

πD
it = πL

it(xit) − πP

it(xit).

(3.10)

3.5.2 Adoption Decision in the Empirical Model

The adoption decision in the empirical model is the same as that in the theoretical model specified
in section 3.3 except that we add another component, εit, to the per-period profit gain function:

it (xit, εit) = πD
πD

it + εit

(3.11)

Unlike πD
it

which is observable to both farmers and econometricians, εit is only observed by
farmers. Following Rust (1987), we interpret this term as a per-period profit gain shock. If all
state variables observed by farmers could also be observed by econometricians, then two farmers’
adoption decisions should be exactly the same if all their state variables coincide. This is rarely
the case observed in the data. The goal of adding an additional state variable that is unobservable
to econometricians is to rationalize different adoption decisions observed when observable state
variables are the same. We assume εit conforms to an i.i.d logistic distribution with location
parameter zero and scale parameter σε.

7During our study period, the average depth to groundwater dtw across the study region only increased by 5 feet,

compared to the average depth to groundwater level of 108 feet.

76

Under this enriched setup, the choice-specific value functions 8 corresponding to (3.3) and (3.4)

are:

and

Þ

vL(Γit) = πD
= πD

it + βE [vL(Γit+1) + εit+1|Γit, εit]
vL(Γit+1) f(Γit+1|Γit)d(Γit+1),
it + β
(cid:9)
vP(Γit,Cit) = βE(cid:8)max[vP(Γit+1,Cit), vL(Γit+1) − Cit + εit+1]|Γit, εit
(cid:18) vP(Γit+1,Cit)
(cid:19)(cid:27)
(cid:19)

(cid:18) vL(Γit+1) − Cit

Þ

(cid:26)

= βσε

ln

exp

σε

+ exp

σε

f(Γit+1|Γit)dΓit+1

where the second equality follows from the i.i.d logistic distribution assumption of εit (Arcidiacono
and Miller, 2011). Moreover, we parameterize LEPA adoption costs Cit as:

Cit = Ci + f(nit) = Ci + γNnit .

(3.14)

(3.12)

(3.13)

,

Further, each farmer knows his own Ci, but this information is not observed by econometricians.
We assume there are two levels of Ci – high (CH) and low (CL), and each farmer has non-negative
probabilities of belonging to each cost group (τH
).9 We assume, however, that there is no
i
correlation between Ci for different farmers after conditioning on a bunch of observable variables.
A farmer adopts LEPA in year t if and only if the value of adopting LEPA net of adoption costs
is higher than the value of delaying adoption. Let diLt ∈ {0,1} denote whether farmer i adopts
LEPA in year t. The adoption decision in the empirical model becomes:

and τL
i

vL(Γit) − Cit + εit ≥ vP(Γit,Cit) ⇔ diLt = 1

vL(Γit) − Cit + εit < vP(Γit,Cit) ⇔ diLt = 0

.

(3.15)

3.6 Estimation Strategies

There are three sets of parameters in the model specified in Section 3.5: Ω1 = {α1it, α2i},
parameters in the production function in (3.6); Ω2 = {γ0, γ1, γ2, γ3}, parameters in f(Γit+1|Γit), the
8Following Rust (1987), the choice-specific value functions are the lifetime expected profits without considering

εit.
9Assuming discrete supports for unobserved heterogeneities (a.k.a “finite mixture") is a common practice in the
dynamic discrete choice estimation literature, which provides a good approximation to the true distributions. See for
example Mroz (1999), Arcidiacono and Miller (2011) and Chung, Steenburgh, and Sudhir (2013).

77

transition function of profit gain; and Ω3 = {CH,CL, τH
, σε, γN}, the collection of Ci distribution
i
parameters, the standard deviation of per-period profit shocks, and the effect of neighbor LEPA
adopters on farmers’ inertia.
In this section, we discuss the estimation strategies employed to
estimate these three sets of parameters.

3.6.1 Production Function Estimation

Due to the lack of individual farmer level crop yield data, we could not estimate the production
function in (3.6) directly. Instead, we rely on (3.9), the optimality condition associated with annual
water extraction, to identify parameters in production function (3.6). Rearranging (3.9), we have

θ j w

∗
i jt + rit = − α1it
2α2

+

1
2α2

pe
i jt
θ j pt

.

α1it is further parameterized as follows:

α1it = α1i + α1gt Dt ∗ gi + α1temptempit + µit,

(3.16)

(3.17)

where α1i is a farmer-level fixed effect, Dt is a series of year dummies, gt is a series of groundwater
management district (GMD) dummies,10 tempit is a vector of monthly average temperatures in
the growing season, and µit is an idiosyncratic unobserved term. The estimation equations in
(3.16) and (3.17) are broadly consistent with the reduced-form water use estimation equations in
Hendricks and Peterson (2012), Pfeiffer and Lin (2014), and Li and Zhao (2018).

Ideally, equation (3.16) could be estimated with a fixed effects model. However, crop price is
only available at the whole state level, rather than at individual farmer level, leading to possible
measurement errors associated with crop prices. For identification, we use the annual log of ethanol
demand in Kansas as the IV for crop price. This choice of IV is reasonable because large share of
corn is used in ethanol production. The output of this estimation step is ˆΩ1 = { ˆα1it, ˆα2}.

10The Kansas HPA is divided into five GMDs for groundwater monitoring and regulation purpose. Areas within

the same GMD share similar hydrological and climatically conditions.

78

3.6.2 Profit Gain Transition Rule Estimation
Using ˆΩ1 = { ˆα1it, ˆα2}, we predict annual water extraction if farmer i uses CP ( ˆw∗
( ˆw∗
iLt
and ˆπL
profits, ˆπP
it
it
farmer i in year t:

) or LEPA
) into (3.8), we obtain the predicted
, as well as the predicted per-period profit difference between LEPA and CP for

) in year t according to (3.9). Plugging ( ˆw∗
iPt

) and ( ˆw∗
iLt

iPt

it−1, as well as a time trend. We enrich
According to (3.1), farmers forecast πD
(3.1) by adding the predicted fixed effects component of α1it in (3.17) to the prediction equation:

it = ˆπL
ˆπD

it − ˆπP
it .
, πD
it+1 based on πD
it

it+1 = γ0 + γ1 ˆπD
ˆπD

it + γ2 ˆπD

it−1 + γ3 ˆα1i + γ4t + ηit .

(3.18)

Equation (3.18) is estimated via OLS. The output of this step is ˆΩ2 = { ˆγ0 − ˆγ4, ˆση}. Equation
(3.18) assumes a transition function analogous to that of an AR(2) process, and transition rule
functional form selection issues are discussed in appendix H. Note that instead of estimating the
transition function for xit (underlying variables that determines πD
), we estimate the transition
it
directly. This approach has two advantages. First, the dimension of the state space
function for πD
it
is reduced from nine (i.e. current and past values of the crop price, gas price and precipitation,
year dummies, depth to groundwater as well as α1i) to only four (i.e. the current and past profit
gains, a time trend, and α1i), greatly reducing computational burden in estimating and simulating
the dynamic discrete choice model. Second, it is more realistic to assume that farmers forecast
future profit gains based on current and past profit gains instead of all underlying variables that
affect profit gain, because the latter approach would require farmers to possess fairly high levels
of computational power and cognitive ability (Hendel and Nevo, 2006). See Gowrisankaran and
Rysman (2012) and Murphy (2018) for similar approaches.

3.6.3 Dynamic Discrete Choice Estimation
The remaining set of parameters Ω3 = {CH,CL, τH
, σε, γN} are estimated through the dynamic
i
LEPA adoption model (3.12) - (3.15). Rust (1987) pioneered the Nested Fixed Point Algorithm

79

(NFXP) to structurally estimate dynamic stochastic discrete choice models. For each iteration of
model parameter search, the NFXP solves the dynamic programming problem defined by (3.12)-
(3.15) and computes choice-specific value functions. However, the computational burden of NFXP
is relatively high due to this repeated solution of dynamic programming problems, especially for
problems with multiple state variables. More Importantly, unobserved heterogeneity, which is
represented in our model by individual-specific part of adoption costs (i.e. Ci), could make NFXP
computationally infeasible (Arcidiacono and Miller, 2011). Instead, we follow Hotz and Miller
(1993), Bajari, Benkard, and Levin (2007) and Arcidiacono and Miller (2011) to estimate (3.12)
- (3.15) without explicitly solving the dynamic programming model. To proceed, we rearrange
equation (3.13) as:

vP(Γit,Ci, nit) = β

[vL(Γit+1) − Ci − γN nit − σεlnPrL(Γit+1,Ci, nit)] f(Γit+1|Γit)dΓit+1,
where PrL(Γit+1,Ci, nit) is the probability of choosing LEPA in year t + 1 given Γit+1:

(3.19)

PrL(Γit+1,Ci, nit) =

=

exp[ vP(Γit+1,Ci,nit)
1

σε

exp[ vL(Γit+1)−Ci−γN nit

]

σε

] + exp[ vL(Γit+1)−Ci−γN nit

σε

]

(3.20)

1 + exp[ vPt+1−(vLt+1−Ci−γN nit)

] .

σε

Þ

Subtracting (3.19) from the value of LEPA in (3.12) and adoption cost Cit, we get

vL − Ci − γNnit − vP = πD

[lnPrL(Γit+1,Ci, nit)] f(Γit+1|Γit)dΓit+1

Þ

it + βσε
− (1 − β)(Ci + γNnit),

(3.21)

which is a function of the per-period profit gain of using LEPA (πD
) and the next
it
period’s probability of adopting LEPA (if LEPA is not adopted in the current period). Details of
the derivation are presented in appendix E.1.

= πL

it − πP

it

Since πD
it

and nit enter (3.21) linearly,11 the interpretation of peer effects becomes very flex-
ible, as discussed in section 3.3. For example, if peer effects are interpreted as utility gains
and nit enter (3.21) nonlinearly in the second term through PrL(Γit+1,Ci, nit). However, as will be shown

in appendix E.2, this term is treated as known when estimating (3.21).

11πD
it

80

Þ [lnPrL(Γit+1,Ci, nit)] f(Γit+1|Γit)dΓit+1 −(1− β)Ci, where πD

+ ˜γNnit] +
from witnessing more neighbor LEPA adopters, (3.21) becomes vL − Ci − vP = [πD
+ ˜γNnit is annual utility gain
βσε
from LEPA adoption. Given the same structure of this equation and (3.21), estimates of parameters
other than γN and ˜γN are not affected. ˜γN is also obtained by a simple constant-scale transforma-
tion of γN: ˜γN = (β − 1)γN. Obviously, peer effects could also be easily interpreted as neighbor
adopters’ effect on utility gains, even though it is currently interpreted as neighbor adopters’ effect
on adoption costs in our model (through its effect on inertia).

it

it

According to (3.15), the likelihood of observing farmer i whose unobserved portion of adoption

costs being Ci ∈ {CH,CL} choosing irrigation technology j ∈ {P, L} in year t is:

Pr[vL(Γit) − Ci + εit > vP(Γit,Ci, nit)], j = L

Pr[vL(Γit) − Ci + εit < vP(Γit,Ci, nit)], j = P

.

(3.22)

li jt(Γit; Ci, nit, σε) ≡ Prj(Γit; Ci, nit) =

Ti


t=1

Ti


t=1

i

Therefore, the likelihood contribution of this farmer in year t is:

lit(dit, Γit+1|Γit,Ci, nit; σε) = 


j∈{P,L}

[li jt(Γit,Ci, nit; σε)]di jt f(Γit+1|Γit).

(3.23)

Integrating out Ci in (3.23), the full likelihood contribution of farmer i is:

i

li(di, Γi|Γi1, ni, σε) = τH

lit(dit, Γit+1|Γit,CH, nit, σε) + τL

lit(dit, Γit+1|Γit,CL, nit, σε),
(3.24)
where Ti is the number of years that farmer i waited until he adopted LEPA. In addition, di =
{di1, di2 . . . diTi}, ΓD
} and ni = {ni1, ni2 . . . niTi}. To complete the model, we
parameterize the probability of farmer i falling into adoption class H as:

i1, ΓD

i2 . . . ΓD
iTi

= {ΓD

i

i = τH(Γi1; δ) =
τH

exp(δ0 + δ1Γi1)
1 + exp(δ0 + δ1Γi1) .

(3.25)

The E-M CCP algorithm in Arcidiacono and Miller (2011) is taken to estimate the model
parameters. Details of the algorithm are presented in at appendix E.2. Standard errors of parameters

81

estimated by the dynamic discrete choice model are obtained via nonparametric panel bootstrapping
with 380 bootstrap runs.12

One problem in the empirical model above is that the number of neighbor adopters (nit) might be
endogenous due to correlated unobservables between a farmer and his neighbors. One solution in the
context of identifying peer effects on groundwater appropriation and irrigation technology adoption
with reduced-form techniques is to include a detailed list of control variables in the estimation to
capture as much correlation among neighbors as possible (Sampson and Perry, 2018; Sampson and
Perry, 2019). In our model, annual profit gains πD
is a function of various factors such as crop price,
it
gas price, depth to groundwater and precipitation. We also allow marginal productivity of water’s
slope to be farmer(well)-specific and time-specific to control for time-constant and time-varying
correlated unobservables among farmers and their neighbors. Therefore, we are controlling for
a large set of observed and unobserved variables that might be correlated with farmers and their
neighbors. Consequently, we believe that the problem of endogeneity in estimation results are
minimized even though we could not completely rule out all potential sources of endogeneity in
nit. We provide more discussion on this issue in section 3.7.
3.7 Empirical Results

In this section, we present the estimation results of production function parameters ( ˆΩ1) from
(3.16), state variable transition function parameters ( ˆΩ2) from (3.18), and dynamic parameters ( ˆΩ3)
from (3.24) and (3.25). Each step relies on the estimation results of the previous steps. We conduct
robustness checks at each step to ensure that robust estimation results are used in the subsequent
steps.

12Due to the randomness in selecting wells in each bootstrap sample, it is possible to resemble a sample with all
(or almost all) farmers/wells falling into the low adoption cost or high adoption cost class. With little adoption class
heterogeneity in the drawn bootstrap sample, the estimated CL, CH, and corresponding class probabilities will be very
imprecise. To avoid this situation, we initially draw 500 bootstrap samples. After all bootstrap runs, we delete 120 runs
whose estimated τH
) are very close to 1 (or that give extremely large δ0 and δ1 estimates) when calculating
i
standard errors, leaving us 380 effective bootstrap runs.

(or τL
i

82

3.7.1 Production Function Estimation Results

Production function estimation results are reported in Table 3.2. The results in column (I) and
(III) are obtained by applying a farm-level fixed effects model to the first order condition of profit
maximization (3.16), using ethanol production as the instrument variable for corn price. As a
robustness check, columns (II) and (IV) reports direct estimates of (3.6) using county level yield
data.

From (3.17), the intercept of effective water’s marginal product (α1it) depends on farmer fixed
effects and year-by-GMD fixed effects, and would also depend on the monthly temperature in the
growing season. Column (I) reports the second stage estimation results of the 2SLS estimation
of (3.16), excluding monthly temperature in (3.17), while column (III) reports the results with the
temperatures included. The first stage estimates are reported in Table 3.3. The first column in
Table 3.3 corresponds to specification (I) in Table 3.2, while the second column corresponds to
specification (III). The instrument passes both under-identification test and weak instrument test.
The slope of water’s marginal product (α2) is around -33 in both specifications, which is negative
and statistically significant, confirming that the production function is concave in effective water.
Column (III) indicates that while a higher temperature improves yields in May and June, it
reduces the yield during the hot summer months of July - September. Between columns (I) and
(III), the average α1it across farmers and years are very close, at around 143.

We estimated the production function in (3.6) indirectly through (3.16) due to the lack of farm
level yield data. As a robustness check, we also provide a direct production function estimation
using county-level crop yield data. Specifically, we estimate the following adaptation of (3.6) to
county level data:

qct(wct) = α0ct + α1ctW e

it + α2ctW e2

it + µct,

(3.26)

where subscript c indexes county in the study region and W e is effective water calculated from
individual farmer-level water use averaged to county level.13 We use a county level fixed effects

13Specifically, we calculate annual effective water for farmer i, θit wit +rit, for each farmer in our dataset and obtain

effective water in county c in year t by averaging effective water for farmers in that county.

83

Table 3.2: Production Function Estimation Results

Without Temperature
Water Use County-Level Water Use County-Level

With Temperature

FOC
(I)

-33.03***

(0.87)

Yield
(II)
-29.9*
(17.81)

FOC
(III)

-32.91***

(0.88)

Yield
(IV)

-37.04*
(22.07)

Identification
Strategy

α2

α1it

α1temp5

α1temp6

α1temp7

α1temp8

α1temp9

α1temp10

α1temp11

2.37***
(0.49)
1.83***
(0.45)
-2.95***
(0.42)
-0.91*
(0.51)
-1.76***
(0.48)
-8.52***
(0.59)
2.64***
(0.41)

Y

Y

142.71
0.132
35332

-0.05
(2.38)
-2.60
(2.48)
-0.44
(2.09)
-0.99
(2.31)
2.02
(1.97)
9.32***
(2.53)
-0.58
(1.84)

Y
Y

Y
Y

152.62
0.823
333

Farmer FE
County FE
Year-by-GMD FE

Y

Y

αoit

County FE
Year FE
E(α1it)
R2
Obs

143.20
0.130
35332

Y
Y

Y
Y

120.64
0.804
333

Notes: Standard errors are reported in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01

84

Table 3.3: Fixed Effect Instrumental Variable Production Function Estimation – First Stage
Dependent Variable: pE/(θ j ∗ pC)
(III)
pE/(θ j ∗ log(ethanol))

(I)

2.81***
(0.0012)

2.84***
(0.0052)

Monthly Average Temperature
Farmer FE
Year-by-GMD FE
F-statistic
Under-identification test (Anderson canon. corr. LM P-value)
Weak-instrument-robust inference (Anderson-Rubin Wald test p-value)

3.0E+05
0.000
0.000
Notes: Standard errors are reported in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01

2.8e+05
0.000
0.000

N
Y
Y

Y
Y
Y

model to estimate (3.26) and the results are shown in columns (II) and (IV) in Table 3.2. Most of the
temperature variables in (IV) are insignificant, possibly due to the small number of observations.
The estimated slope of marginal product (α2) and the average of the intercept of marginal product
(α1) across farmers (or counties) and time are similar to each other across specifications (I) - (IV),
providing strong support that our identification strategy based on individual water use data works
well in identifying production function parameters. We rely on the results of specification (I) in
subsequent estimations. We do not use the direct yield function estimation results in specification
(II) and (IV) because farmer-level fixed effects estimation allows us to estimate farmer-specific
intercept of effective water’s marginal product (i.e. α1i).

The importance of accounting for individual level heterogeneity has been well documented by
Suri (2011). One such farmer-level heterogeneities comes from heterogeneity in farmers’ profit
gains from using LEPA, which in turn comes from farmers’ heterogeneous production functions.
Indeed, the histogram of estimated ˆα1it across farmers and years in the upper panel of Figure 3.3
exhibits strong heterogeneity in α1. This heterogeneity is then passed on to the heterogeneity of
annual profit gain in the lower panel of figure 3.3. The mean and standard deviation of annual profit
gain are $24.07/acre and $14.25/acre, respectively.

As a final robustness check, we calculate the average elasticity of water extraction per acre with

85

Figure 3.3: Histogram of ˆα1it in Production Function (upper) and Per-period Profit Difference
Between LEPA and CP across Farmers and Years (lower)

respect to energy cost. The elasticity is -0.562 for farmers who uses CP and -0.279 for those using
LEPA. Using a similar dataset in the same study region, Hendricks and Peterson (2012) estimates
the average elasticity across all farmers and all years in their sample to be -0.09. However, they do
not take the energy requirement to start irrigation equipment into account (i.e. PSIj in (3.7)) when
calculating the energy costs. After such factor is incorporated into energy cost calculation, Pfeiffer
and Lin (2014) shows that the estimated elasticity for CP becomes -0.577 (Pfeiffer and Lin, 2014),
which is about the same as our estimate of -0.562.

3.7.2 Profit Gain Transition Rule Estimates

The estimated AR(2)-type transition process for profit gains is presented in table 3.4. Most
variables used to predict future profit gain realizations are statistically significant. In particular,
it−1) are positively correlated with future profit
current and past profit gain realizations (πD
it

and πD

86

gain realizations. The uncertainty of profit gain transition is captured by σε. The more uncertain
farmers are about the future realizations of these variables, the more dispersed is the distribution
represented by the probability density function f(πit+1|πit, πit−1, α1i, t).

Table 3.4: Profit Gain Transition Rule Estimation Results

Dependent Variable: ˆπD
πD
it

it+1 Estimates
0.21***
(0.0059)
0.55***
(0.0069)
6.04***
(0.1789)
0.031
(0.025)
-5.25**
(.4715)
8.82

πD
it−1

α1i

t

constant

ση

Notes: Standard errors are reported in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01

3.7.3 Dynamic Parameter Estimation Results

3.7.3.1 Parameter Estimates

The estimation results for adoption cost distribution parameters and per-period profit gain shock’s
scale parameter are shown in table 3.5. All estimated parameters have the expected signs and are
statistically significant. We discuss several important lessons learned from the estimation results.
First, 66% of farmers fall into the low adoption cost class and 34% of farmers fall into the
high cost class on average. The large difference between the two cost classes and the nontrivial
probability of each class demonstrate the importance of accounting for cost heterogeneities in
adoption models.

Second, the unobserved adoption costs for farmers with low adoption costs are estimated to be
$74.15/acre, which falls in the $40-$77.37/acre equipment and installation cost range (converted
to 2010 dollar) used in agronomic studies (Delano, Williams, and O’Brien, 1997; Hutton et al.,

87

Table 3.5: Dynamic Parameter Estimation Results

Adoption Cost Cit = Ci + γNnit ($/acre)

Parameter Estimates

CL

CH

γN

Cost Class Probability
C = CL
C = CH

Profit Shock Scale Parameter
σε

Obs

74.15***
(28.77)
277.70***
(88.72)
-2.7**
(1.3)

64%
36%

33.49***
(5.28)
28081

Notes: Bootstrapped standard errors are reported in parentheses. * p < 0.1, ** p < 0.05, ***
p < 0.01

1989; Peters, Neibling, and Stroh, 2016). However, estimated adoption costs for farmers with high
adoption costs are $277.7/acre, which is much higher than the upper bound of the range above.14
Adoption costs in our model include not only equipment and installation cost, but also include
farmers’ inertia to change their current farming systems. Shown by the cost range in the agronomic
literature, equipment and installation costs do not vary too much. Therefore, farmers with high
adoption costs are likely to have higher psychological inertia towards changing their current farming
practices than other farmers. Thus, allowing adoption costs to be heterogeneous across farmers
captures this irrational behavioral bias.15

Third, similar to Sampson and Perry (2018), which studies peer effects on LEPA adoption using
reduced-form estimation with the same data source, we also find statistically significant evidence
of peer effects. As the number of neighbor LEPA adopters increases, inertia level decreases, and

14We also tried to increase the number of classes to three and four, but the estimated population level probabilities

of the third and forth classes turn out to be close to zero.

15We interpret inertia as a behavioral bias because inertia itself does not bring any profit/utility gains to farmers.

88

perceived adoption costs decrease. As discussed in section 3.3 and section 3.6, we can also interpret
peer effects as a farmer’s utility gain from observing more neighbor adopters: if more neighbors
adopt LEPA, I am more comfortable of adopting LEPA myself. Compared to Sampson and Perry
(2018), our study utilizes a structural estimation model that not only quantifies peer effects, but
also the effects of profit gain level, profit gain uncertainty and adoption costs on LEPA diffusion.
Lastly, as the deterministic part of the per-period profit difference is estimated in the previous
step, the per-period profit shock scale parameter σε can be identified. The estimated scale parameter
is relatively large – around $33.49/acre, higher than the mean of the per-period per-acre profit gain
from using LEPA ($24.07/acre). This finding is consistent with studies that use the same dataset
to study LEPA adoption decisions with reduced-form estimation techniques. For example, after
controlling for a rich set of control variables, the R2 in a fixed effects linear probability model is
still only around 0.1 in Sampson and Perry (2018). The relatively large scale parameter estimate
indicates that there are possibly many other factors that affect per-period LEPA profit gains that are
not captured by the model. As we have controlled for a rich set of climate factors and economic
factors in the profit function, one possible source of the missing information is the deviation of
farmers’ perceived profit gain from the objective profit gain. Another possible explanation is that
farmers may not be extremely rational when making adoption decisions and a rational dynamic
programming model here is not able to capture all irrational behaviors. Although we introduce
behavioral bias in adoption costs through inertia, there might still be other forms of irrational
behaviors. For example, farmers might be yield-maximizing rather than profit-maximizing. Such
behaviors might be related to factors such as farmers’ socioeconomic characteristics. Unfortunately,
we could not identify such behavioral biases given the information in the dataset, but a future
extension that incorporates more irrational behaviors into the modeling framework here is valuable.

3.7.3.2 Model Validity

To check how well the structural adoption model works, we test model validity by comparing the
observed LEPA diffusion path to the path predicted by our model. To do so, we parameterize the

89

model in (3.12) to (3.15) with the estimated model parameters and solve explicitly the dynamic
programming model faced by each farmer in each year using collocation method described by
Miranda and Fackler (2004). In a nutshell, the simulation follows the procedure below:

• Decide each farmer’s adoption cost (high or low) according to the estimated individual
). We set a farmer to be a

farmer’s probability of belonging to each adoption cost class (πH
i
high-adoption-cost farmer if πH

i > 0.5 and to be a low-adoption-cost farmer otherwise.

• Randomly draw the per-period profit shock for each farmer in each year according to the

estimated shock distribution.

• Calculate the deterministic part of the per-period profit gain based on observed variables such

as crop price, energy price and precipitation.

• Solve vP and vL defined in (3.12) and (3.13) by collocation method.
• Decide whether to adopt LEPA for each farmer in each year according to (3.15). If adopt,
record adoption year and this farmer would not make adoption decisions again in the future.
If not, continue to the next year.

• Repeat the previous procedure for each year in the sample.

• Iterate the procedure above over different draws of per-period profit shocks.

• Across all the draws, calculate the average of each farmer’s adoption years and obtain the

average new and cumulative adopters in each year.

Figure 3.4 plots each irrigation well’s simulated LEPA adoption year against its observed
adoption year. A linear regression line is also fitted in the figure. All the points would stay on the
45°line for a perfectly fit model. A clear upward sloping trend is observed in the figure, meaning
that the model predicts individual-level diffusion patterns relatively well.

Figure 3.5 presents the comparison between the simulated and observed number of total adopters
(upper panel) and the number of new adopters (lower panel) in each year. The solid lines represent
observed paths and the dashed lines represent simulated paths. In general, the predicted trends of

90

Note: each dot in the figure represents an observed-simulated adoption year combination for an
irrigation well. The line in the figure is the line fit of wells’ simulated adoption years on observed
adoption years.

Figure 3.4: Simulated Average Adoption Year v.s. Observed Adoption Year

both total adopters and new adopters over time are similar to the trends observed in the data. The
combined evidence from figures 3.4 and 3.5 implies that the model does a relatively good job in
terms of predicting both average adoption pattern and individual well-level LEPA diffusion pattern.

Figure 3.5: Simulated v.s. Observed Number of Cumulative Adopters (upper) and New Adopters
(lower)

91

One possible reason that could bias the estimation results and harm model validity is the
potential endogeneity in the number of neighbor LEPA adopters. As we have discussed at the
end of section 3.6, we could not completely rule out the possibility that the number of neighbor
adopters is endogenous due to correlation in unobservables, even though we have included a rich
set of control variables in both the main regression equation (3.21) and adoption class distribution
equation (3.25). However, we still use the estimation results reported above in our subsequent
analysis because: (1) the overall adoption patterns predicted by the model are relatively reasonable
shown by figures 3.4 and 3.5, and (2) it is difficult to find a suitable IV for neighbor adopters in this
context and the approach we take (i.e. including a detailed list of control variables to identify peer
effects) is consistent with the approach commonly used in the literature (Sampson and Perry, 2018;
Sampson and Perry, 2019). We check whether our estimation results are sensitive to our definition
of neighbors in appendix F.

3.7.3.3 Effects of Uncertainty and Adoption Costs on Adoption Rate

In this section, we consider the effects of adoption costs, number of neighbor LEPA adopters,
profit gain level and profit gain uncertainty on LEPA adoption rate. Specifically, we calculate the
elasticities of adoption probability with respect to adoption costs and state variable uncertainties.
Elasticity with respect to adoption costs is relatively straightforward to calculate. In practice,
we increase estimated unobserved adoption costs by one percent and calculate the percent change
in adoption probability for each individual in each year through simulating the model defined by
(3.10)-(3.15). The mean percent change in adoption probability across all farmers and all years is
reported in the first row in table 3.6 as an estimate of the elasticity.

Ideally, standard errors of the elasticity estimates could also be obtained if we bootstrap the
simulation, which is computationally costly.
Instead, we report the standard deviation of this
percent change across farmers and years in parenthesis in table 3.6. As this standard deviation is
not the standard error of the estimated elasticity, we could not calculate p-value and show statistical
significance. However, it gives a broad idea about how disperse the estimated elasticities are for

92

Table 3.6: Adoption Probability Elasticities

Adoption Cost
Overall

High Adoption Cost Class

Low Adoption Cost Class

# of Neighbor LEPA Adopters
Level

Natural Gas Price
Level

Uncertainty

Crop Price
Level

Uncertainty

Growing Season Precipitation
Level

Uncertainty

Profit Gain
Level

Uncertainty

Elasticities

-1.75
(1.44)
-2.53
(1.23)
-0.49
(0.66)

0.42
(0.26)

0.48
(0.33)
-0.034
(0.14)

0.17
(0.18)
-0.1
(0.29)

-0.73
(0.42)
0.027
(0.055)

0.68
(0.34)
-0.10
(0.033)

Notes: elasticities are calculated as the percent change in adoption rate (i.e. adoption probability)
with respect to 1 percent change in stata variable level or uncertainty. For example, if initial
probability of adoption is 40%, an elasticity of 1 means one hundred percent increase in the
variable being considered leads to 100% increase in adoption probability, resulting in an adoption
probability of 80%.

93

different farmers in different years. A 1% increase in adoption cost will lead to a 1.75% reduction
in adoption rate on average.16 The mean of this elasticity across farmers and years is not too
much larger than its standard deviation, meaning that the effect of increasing adoption costs on
adoption probability is relatively dispersed across farmers. To investigate the source of dispersion,
we also report separately the elasticity estimates (and their standard deviations) for farmers with
high adoption costs and with low adoption costs.17 Shown by the results, the effect of increasing
adoption costs on adoption probability is much higher for farmers with high adoption costs than
those with low adoption costs. This is partly because the level of high adoption costs is much higher
than that of low adoption costs. Therefore, one percentage change in the level of high adoption
costs is also higher than that of low adoption costs, leading to more changes in adoption probability.
The procedure of calculating adoption probability elasticity with respect to the number of
neighbor adopters is the same as the one above. We increase the amount of neighbor adopters for
each farmer in each year by one percent and calculate percent change in adoption probability. The
mean and standard deviation of this elasticity are also reported in table 3.6. As expected, increase
in the number of neighbor LEPA adopters increases adoption rate because it lowers down farmers’
perceived adoption costs through reducing inertia.

To calculate elasticities with respect to profit gain uncertainties and profit gain levels, we first
decompose overall profit gain uncertainty into uncertainties in the underlying factors (i.e. gas price,
crop price, precipitation, and profit shocks). We estimate the following AR(2) transition rule for
each underlying factor:18

xit+1 = φ0 + φ1xit + φ2xit−1 + νit .

We define the elasticity of adoption probability with respect to an underlying variable’s un-
certainty as the percentage change in adoption rate with respect to one percentage change in the

16Note that this is a percentage reduction in percentage. For example, if the initial adoption probability is 50%,

then a 1.75% reduction in adoption rate gives an adoption probability of 50% ∗ (1 − 0.0175) = 49.13%.
high adoption costs.

17We assign farmers whose probability of falling into high adoption cost class is greater than 0.5 to be farmers with

18Our data shows that precipitation does not demonstrate strong serial correlation. Therefore, precipitation is

assumed to be normally distributed across farmers and years.

94

standard deviation of νit (σν).19 Similarly, the adoption probability elasticities with respect to
underlying variables’ levels are calculated as the percentage changes in adoption rate with respect
to one percentage change in the levels of xit.

The estimated elasticities with respect to underlying variables’ levels and uncertainties are
reported in table 3.6. The signs of the elasticities with respect to these variables’ levels are as
expected: increases in natural gas price and crop price lead to increases in adoption rate because
such increases make LEPA more profitable compared to CP. On the other hand, an increase in
precipitation reduces adoption rate because irrigation demand is lower with more precipitation, so
there is less need to adopt more efficient irrigation technologies.

The effects of natural gas price, crop price and precipitation uncertainties are relatively smaller
than the effects of their levels. Holding profit gain level constant, higher uncertainty in profit
gain is expected to reduce adoption probability because it raises the threshold of profit gain above
which adoption happens. However, increasing precipitation uncertainty increases, rather than
decreases, adoption probability. Moreover, the mean of each elasticity across farmers and years is
smaller than the corresponding standard deviation (reported in parentheses). Indeed, increases in
levels/uncertainties of gas price, crop price and precipitation do not necessarily translate in to the
same change in profit gain uncertainty due to the nonlinear relationship between profit gain and
underlying variables that affect profit gains. Therefore, we also directly calculate the elasticities of
adoption probability with respect to profit gain’s level and uncertainty (refer to (3.18)). The results
are shown in the last two rows of table 3.6. As expected, increase in profit gain level increases
adoption probability, while increase in profit gain uncertainty decreases adoption probability.
The means of these elasticities are also much higher than their standard deviations (reported in
parentheses).

19Specifically, we simulate the paths for all state variables in each year based on the new transition rule and calculate
LEPA profit gains. To be consistent with the estimation procedure, we then use the calculated profit gains to fit a profit
gain transition rule. With this transition rule, we numerically solve the dynamic programming problem and calculate
the probability of adoption under different magnitudes of σν, holding the levels of profit gain constant at the observed
levels.

95

3.8 Counterfactual Simulations

In section 3.7.3.3, we calculate the elasticity of LEPA adoption probability with respect to profit
gain level, profit gain uncertainty, neighbor adopters and adoption costs. These elasticity estimates
could be regarded as "marginal effects".

In this section, we ask two questions: (1) what is the effect of the incentive to delay adoption
created by profit gain uncertainty and adoption irreversibility on adoption rate? (2) what kind of
policy is the most cost-effective in promoting efficient irrigation technology adoption? Answering
these two questions requires changing certain model parameters "non-marginally". One merit of
structural models is the ability to evaluate the effects of non-marginal changes in key variables on
adoption decisions through simulation. We lay out the details of the simulations used to answer
theses two questions and discuss the simulation results in the following two subsections.

3.8.1 Option to Delay and Adoption Rate

According to the Net Present Value (NPV) rule, a farmer should adopt LEPA as long as the expected
sum of discounted future profit gains from LEPA is higher than the adoption costs. To put it in
another way, NPV rule only considers whether or not to adopt now, but not the option to delay
adoption decision to the future.

However, this other option (i.e. option to delay) is important when farmers are uncertain about
future profit gain realizations and when the adoption procedure is irreversible. Even if the NPV of
adoption is positive now, there is always a chance that profit gain drops to a fairly low level in the
future due to its inherent volatility, and adoption costs might not be recouped if this happens but
adoption has already occurred.

By real options theory, farmers do not simply choose whether or not to adopt now, but also
consider the option to delay the decision until better information is available (Taylor and Zilberman,
2015). If he does not adopt this year, he still has the option to make adoption decision in the future.
But if he adopts now, he would not have the chance to recoup the upfront adoption costs if profit gains

96

become very low in the future. This "option value" creates incentive for forward looking farmers
to delay adoption even though adoption is profitable through the lens of NPV rule. Therefore, it
is possible that a farmer should adopt LEPA in a given year by NPV rule, but should not adoption
by real options rule. A rational farmer tends to delay adoption until profit gain realization in the
current year is high enough such that the probability that it drops to a low level later is small.

Formally, the NPV decision rule is to adopt LEPA as long as:

(cid:41)

(cid:40) ∞


τ=t

Et

βτπD
iτ

− Cit > 0.

(3.27)

On the other hand, the real-option decision rule is to adopt if:

Obviously, VL(Tit) = Et

τ=t βτπD
iτ

(cid:110)
∞

vL(Γit) − vP(Γit,Cit) − Cit > 0.
For each observation in the dataset, we simulate VL(Γit) = Et

and Cit = Ci + γNnit using observed state variables and estimated structural parameters in table
3.4 and table 3.5. Figure 3.6 plots average of E
average of Cit in two axes, one for farmers with low adoption costs and the other for farmers with
high adoption costs. All averages above are taken across farmers and years in the sample.

τ=t βτπD
iτ

(3.28)

τ=t βτπD
iτ

(cid:110)
∞
(cid:111), vL(Γit)−vP(Γit,Cit)
(cid:110)
∞
(cid:111), average of vL(Γit)− vP(Γit,Cit), and
(cid:111) is always greater than vL(Γit) − vP(Γit,Cit) because
(cid:110)
∞
(cid:111), which
(cid:111). Therefore, they are also

(cid:110)
∞

τ=t βτπD
iτ

vP(Γit,Cit) represents the value of the option to make adoption decisions in the future, which is
positive and is killed once LEPA is adopted. For farmers with low adoption costs (upper panel),
adoption costs are higher than vL(Γit)− vP(Γit,Cit) but is much lower than Et
means that farmers should adopt LEPA if they follow NPV rule, but should not adopt if they follow
real options rule instead. For those with high adoption costs (lower panel), adoption costs are
higher than vL(Γit)− vP(Γit,Cit) and is about the same as E
unlikely to adopt under real options decision rule unless a fairly high profit realization is observed,
and they are on the margin of adoption under NPV rule. Taking these two cases together, if all
farmers follow NPV rule, average adoption probability across farmers and years would be higher.
Figure 3.7 shows that if all farmers had followed NPV rule, average LEPA adoption rate across
all farmers and all sample periods would have increased by 13%. More specifically, the increase in

τ=t βτπD
iτ

97

average adoption rate in the first two years (i.e. 1998 and 1999) would have been as high as 42%
and the diffusion process would have been completed by as early as 1999. The huge difference
in the two adoption paths shows that the option to delay adoption decision until more information
arrives significantly reduces adoption rate.

Figure 3.6: NPV and Real Option Decision Rule Critical Values

If farmers are not uncertain towards LEPA’s profit gain (i.e.

they know perfectly profit gain
realizations in the future), there will be no incentive to delay adoption. Thus, adoption paths
produced by NPV rule and real option rule would be the same. Given the large difference between
the two predicted adoption patterns, profit gain uncertainty is an important factor that lowers down
adoption rate (through creating an option value to delay). Even though the marginal effect of
profit gain uncertainty is relatively small as shown in table 3.6, the total effects of uncertainty are
important given the highly nonlinear nature of Bellman equations.

3.8.2

Insurance, Cost-Share Programs and Adoption

Two commonly adopted approaches to promote technology adoption are to provide (i) cost share
payments to reduce initial adoption costs (Isik, 2004; Seo et al., 2008), and (ii) insurance to reduce
profit uncertainties (Freudenreich and Mußhoff, 2018; Salazar et al., 2017). Utilizing a dynamic
simulation framework, Song (2018) shows that profit insurance program is more cost-effective than
adoption cost subsidies in promoting the adoption of energy crops such as switchgrass. Profit

98

Figure 3.7: Simulated Adoption Path – NPV and Real Option Decision Rule

insurance, while reducing profit uncertainty, also increases profit levels when the actual profit level
drops below the insured level. Thus, the overall effects of profit insurance tend to be larger than
that of cost-share in their application.

However, performances of profit insurances and cost-share subsidies in terms of efficient irri-
gation technology promotions have not been compared in the literature. In this section, we evaluate
the cost-effectiveness of a LEPA profit gain insurance policy and a cost share subsidy program.
Under the insurance policy, LEPA adopters will receive the insured LEPA profit gain once the level
of the observed profit gain drops below this insured level. Under the cost share policy, farmers
receive a one-time reimbursement of a fraction of adoption costs after they adopt LEPA.

For the cost share counterfactual policy, we set cost share rate at $20/acre.20 The simulation
procedure is the same as the one described in section 3.7.3.2, except that Cit in (3.13) is replaced
by Cit − 20. For the counterfactual insurance policy simulation, the choice of insurance rate
($R/acre) is a bit trickier. To make the comparisons between cost share policy and insurance policy
meaningful, the two policies should have the same level of expected government budget. Therefore,
we first run the cost share policy counterfactual simulation and record the expected government
expenditure under this policy. Then, we pick an arbitrary insurance rate, run the simulation, and

20The average cost-share rates among farmers who received the payments under the real Kansas program is around

$16/acre.

99

calculate government expenditure under this rate. If this expenditure is lower (higher) than the
one under the cost share policy, we adjust up (down) the insurance rate until the level of expected
government budget under the insurance policy equals to that under the cost share policy. Following
this procedure, the selected insurance rate is R = $16/acre. Again, the simulation procedure is the
it , R}.
same as the one described in section 3.7.3.2, except that πD
it

in (3.12) is replaced by max{πD

Note: The expected net govern present payments to farmers under the insurance and cost-share
policy are both $11.4 million.

Figure 3.8: Effect of Insurance and Cost-Share Payments on Adoption Rate

The results are plotted in figure 3.8. The solid line represents the simulated adoption path
without policy interventions, the dashed line is the simulated adoption path under the $20/acre
cost-share policy, and the dash-dotted line is the simulated adoption path under the $16/acre
insurance program. Both policies cost the government around $11.4 million in expectation. It is
straightforward to tell from the figure that the insurance policy is more efficient than the cost-share
subsidy. More precisely, the average adoption probability over all sampling periods increases by
11.9% under the insurance program, but it only increases by 1.8% under the cost-share program.
There are two possible explanations for why insurance program works more efficiently. First, as
discussed above, profit insurance affects adoption probability by both reducing profit uncertainty
and increasing profit levels when the actual profit level drops below the insured level. Second, even

100

Note: The expected net govern present payments to farmers under the insurance and cost-share
policy are both $20.9 million.

Figure 3.9: Effect of Insurance and Cost-Share Payments on Adoption Rate

though the marginal effects of profit gain level and profit gain uncertainty are smaller than that of
adoption costs shown in table 3.6, the total effects could be reversed given the highly nonlinear
nature of bellman equation.

To make sure that this result does not change when the level of government budget changes,
we perform the same exercise except that we set a new cost share rate to $36/acre21 and a new
insurance rate to $20/acre. The levels of expected government expenditure for the two policies are
both $20.9 million. The adoption path comparison is shown in figure 3.9. Again, the insurance
program is the more cost-effective option.

3.9 Conclusions and Future Directions

Climate change and the continuing exhaustion of groundwater resources in many important
agricultural zones impose significant challenges to modern agriculture. Adopting more efficient
irrigation technologies is an effective adaptation strategy facing the challenges. However, the

21This is the 95th percentile of cost-share payments received by farmers under the real Kansas program during

1996-2014

101

diffusions of these technologies are typically very slow. The effectiveness of various policies that
promote adoption depend on the underlying incentives of potential adopters. Being able to identify
the most important adoption drivers helps promote the diffusions of these technologies, but it is
difficult to achieve this goal with traditional reduced-form estimation approaches.

This paper develops and estimates a structural dynamic discrete choice model that explicitly
accounts for LEPA’s profit gains, profit gain uncertainties, individual-specific adoption costs and
peer effects on adoption decisions. Our estimation results confirm that farmers value the option to
delay adoption when they face uncertainties and sunk costs. Profit gain uncertainty and irreversibil-
ity of LEPA’s adoption together make it more attractive to delay adoption until more information
about future profit gain arrives. Had farmers ignored the option to delay adoption and used NPV
decision rule instead of real options rule when making LEPA adoption decisions, average adoption
probability would have increased by 13% to 42% in different sampling years.

Besides the rational incentive to delay adoption created by profit gain uncertainty, we also find
evidence that LEPA adoption rate has been driven down by farmers’ behavioral biases, namely,
farmers’ inertia to change farming practices. Inertia adds up significantly to some farmers’ perceived
adoption costs, making them less likely to adopt LEPA, ceteris paribus. However, such inertia could
be reduced when more neighbors of the farmer adopt LEPA, which is consistent with the literature
on peer effects.

Following many local governments’ increasing interest to promote efficient irrigation technology
adoption through subsidies, we perform counterfactual policy simulations based on our model to
test the relative cost-effectiveness of two policies that are commonly used: a cost-share policy and a
profit gain insurance policy. Simulation results show that insurance policies are more cost-effective
than cost-share programs under the same levels of expected government budget.
In particular,
increase in adoption probability following the implementation of the insurance policy is around
10% higher than that of the cost-share program.

There are several directions to extend the analyses in this study. First, we ignore strategic
delay when modeling the effects of neighbor adopters on farmers’ adoption decisions. As stated

102

in section 3.3, the main reason of ignoring strategic delay is that farmers need to predict a large
number of other farmers’ adoption decisions when making their own adoption decisions. It might
help to justify this simplification if a model with strategic delay is estimated using a small subset of
farmers (e.g. farmers in a township) and we test the strength of strategic delay based on this model.
However, with much fewer observations, the power of statistical test might drop significantly, unless
a dataset with much more detailed farmer-level information such as the one used in Besley and
Case (1994) is obtained. Second, farmers use either natural gas or electricity in this region to power
their irrigation system. As we do not have information on individual farmers’ energy sources, we
assume that natural gas is the only energy source in the region following the literature. A natural
extension of this paper is to get nonpublic energy source data from Kansas government and examine
the differences in adoption behaviors between farmers who rely on natural gas and who rely on
electricity to operate their irrigation systems. Finally, we have illustrated that LEPA adoption
behaviors might also be affected by certain irrational behavioral biases that are not captured in
the model. A fruitful future research direction is to identify these irrational components, which
probably requires obtaining information on farmers’ socioeconomic characteristics.

103

APPENDICES

104

Þ
Þ

(cid:26)
(cid:26)

ln

ln

exp

exp

σε

(cid:19)
(cid:18) vL(Γit+1) − Ci − γN nit
(cid:18) vP(Γit+1,Ci, nit)
(cid:18) vLt+1 − Ci − γN nit
(cid:19)
(cid:18) vPt+1 − [vLt+1 − Ci − γN nit]
(cid:20)

(cid:19)(cid:27)

+exp

σε

σε

1 + exp

σε

f(Γit+1|Γit)dΓit+1

(cid:19)(cid:21)(cid:27)

f(Γit+1|Γit)dΓit+1

= βσε

= βσε

Þ

= β

APPENDIX E

DYNAMIC DISCRETE CHOICE ESTIMATION DETAILS

This section shows how to derive alternative representations of choice specific value functions and
the details of the E-M CCP algorithm proposed by Arcidiacono and Miller (2011), tailored to our
technology adoption model.

E.1 Alternative Choice-Specific Value Function Representation

This section shows how to derive equation (3.19) from equation (3.13):

vP(Γit,Ci, nit) = βE(cid:8)max[vP(Γit+1,Ci, nit), vL(Γit+1) − Ci − γN nit + εit+1]|Γit, εit

(cid:9)

[vL(Γit+1) − Ci − γN nit − σεlnPrL(Γit+1,Ci, nit)] f(Γit+1|Γit)dΓit+1

where the last line follows from the fact that:

PrL(Γit+1,Ci, nit) =

=

exp[ vL(Γit+1)−Ci−γN nit

]

σε

exp[ vP(Γit+1,Ci,nit)
1

σε

] + exp[ vL(Γit+1)−Ci−γN nit

σε

1 + exp[ vPt+1−(vLt+1−Ci−γN nit)

] .

σε

(E.1)

]

E.2 E-M CCP Estimation Algorithm

If Ci is observable, a simple two-step CCP approach is feasible. In a first step, one perform a
reduced-form flexible logit regression of LEPA adoption decision on state variables in Γit and nit
to get predicted ˆPr L(Γit+1, nit). In the second step, plug ˆPr L(Γit+1, nit) into equation (3.21) and

105

apply Maximum Likelihood Estimation (MLE) to estimate σε and γN. However, as adoption costs
are unobserved by econometricians, the two steps needs to be modified following the EM-CCP
approach in Arcidiacono and Miller, 2011.

To initiate the estimation procedure, recall that we assume that their are two possible realizations
of the unobserved part of adoption costs (i.e. Ci ∈ {CH,CL}), so we replicate the dataset twice. The
two replications share exactly the same variable values except that they have different unobserved Ci
– one group with CH and another with CL. Next, we give arbitrary starting values to parameters in
(0)
N }, where the(0) superscripts
Ω3. Denote these starting values as Ω
stand for iteration (0). In addition, we also give arbitrary starting values to q(0)
, the probability of
iCk
farmer i falling in adoption cost category Ck ∈ {CH,CL}. We then perform parameter iterations
until the algorithm converges. The steps in each iteration and the convergence criterion are defined
as follows. In iteration (m+1):

(0)
3 = {CH(0),CL(0), δ(0), σ

(0)



, γ

Step 1: perform a reduced-form flexible logit regression of LEPA adoption decision on πD
,
it
it−1, α1i, nit, time trend t, CH(m) and CL(m), using q(m)
as sampling weights. Variables other than
πD
and nit are also included in the regression because they are used by farmers to predict future
πD
it
profit gains. Using the estimated parameters, we predict ˆPr(m+1)
(Γit+1,Ck(m), nit), the probability
that farmer i whose unobserved adoption costs are Ck ∈ {CH,CL} with nit neighbor LEPA adopters
at the beginning of year t will adopt LEPA in year t + 1.

iC

L

L

L

Step 2:

. Plug Pr(m+1)

(Γit+1,Ck(m), nit) and q(m)
iCk

this step of CCP estimation is to update the structural parameters that maximize
the likelihood function derived from the dynamic programming problem, given the predicted
Pr(m+1)
(Γit+1,Ck(m), nit) into equation (3.21), then plug
equation (3.21) into equation (3.20), we obtain the per-period likelihood of observed choices for
farmer i with unobserved adoption costs being Ck as l(m)
(dit|Γit, nit, ˆPr(m+1)
Lit+1 ; Ci = Ck, σε, γN),
which is analogues to equation (3.23) except it is augmented with ˆPr(m+1)
Lit+1 and f(Γit+1|Γit) is
dropped.1 The theoretical-consistent likelihood contribution for farmer i with unobserved adoption
1Take log with respect to equation (3.23), it is straightforward to see that f(Γit+1|Γit) is additively separable
from the other parts of log likelihood function. Therefore, it does not influence the final MLE estimation. Thus, the
transition rule is separately estimated in section 3.6.2.

it

106

costs being Ck across all years becomes:

l(m)

i

(di|Γi, ni, ˆPr(m+1)

Li

; Ci = Ck, σε, γN) =

Ti


t=1

l(m)

i

(dit|Γit, nit, ˆPr(m+1)

(E.2)

Lit+1 ; Ci = Ck, σε, γN)
(m+1)
ε

According to the E−M algorithm, the structural parameters {CH(m+1),CL(m+1), σ

}
that maximize the likelihood function derived from the dynamic programming problem are found

γ

(m+1)
N

by maximizing: 





Ck∈{CH,CL}
with respect to {CH,CL, σε, γN}.

i

q(m)
iCk lnl(m)

i

(di|Γi, ni, ˆPr(m+1)

Lit+1 ; Ci = Ck, σε, γN)

(E.3)

Step 3: this step updates the parameters of population-level unobserved adoption cost distribu-
tion, δ(m+1), and update the probability of farmer i falling in unobserved cost category Ck, q(m+1)
.
iCk
Given {CH(m+1),CL(m+1), σ
(m)
(di,Ci = Ck(m+1)|Γi, ni, ˆPr
l
i

}, calculate:
(m+1)
N

(m+1)
ε
(m+1)
Lit+1 ; δ, σ

(m+1)
N

(m+1)
ε

, γ

, γ

Ti


) = τ(Ck|Γi1, δ)
∗

(m)
l
i

(di|Γi, ˆPr

t=1

(m+1)
Lit+1 ; Ci = C(m+1), σ

(m+1)
ε

)

(E.4)

Maximize the equation above with respect to δ gives δ(m+1). Finally, q(m+1)
iCk
(m+1)
N
(m+1)
ε

(m+1)
N
The iterative inner loop and outer loop continues until function (E.3) converges.2



Ck∈{CH,CL} l(m+1)

(m+1)
ε
Lit+1; δ(m+1), σ

(di,Ci = Ck(m+1)|Γi, ni, ˆPr(m)

(di,Ci = Ck(m+1)|Γi, ni, ˆPr(m)

Lit+1; δ(m+1), σ

l(m+1)

q(m+1)
iCk

, γ

, γ

)

=

i

i

is updated as:

(E.5)

)

2Convergence is obtained if the difference between the value of function (E.3) in two consecutive iterations is less

than 10−7.

107

DISCUSSION: ENDOGENEITY IN NUMBER OF NEIGHBOR LEPA ADOPTERS

APPENDIX F

In our main specification, we define neighbors are the nearest 30 wells around each well. We test
whether the estimation results are sensitive to our definition of neighbors in this section. In an
alternative specification, we define neighbors as the 21st-30th nearest wells around each well. In a
third specification, we also define neighbors to be the nearest 30 wells around each well, but restrict
that only the new adopters last year influence farmers’ adoption decisions.

Total number of adopters in each year and annual new adopters predicted by the initial model
and the two alternative specifications are plotted in figure F.1 (parameter estimation results are
omitted here).

Figure F.1: Simulated v.s. Observed Number of Cumulative Adopters (upper) and New Adopters
(lower)

The predicted patterns for the initial specification and the first alternative (21st-30th nearest

108

Figure F.2: Simulated Average Adoption Year v.s. Observed Adoption Year

wells as neighbors) are very similar. When we restrict that only new adopters last year influence
adoption decisions, the predicted aggregate adoption path moves closer to the observed path, but
the difference between this path and the previous two paths are not very large. However, shown
by figure F.2, the individual farmer (or well) level adoption year predictions are much worse when
only last year’s new neighbor adopters are allowed to affect farmers’ adoption decisions.

The comparisons above reveal that the predicted aggregate adoption pattern is insensitive
to neighbor definition. However, the prediction power with respect to individual level adoption
decisions is stronger when we allow all past neighbor adopters to affect farmers’ adoption decisions.
The results of this exercise lend support to our inclusion of total neighbor adopters in the adoption
model.

109

APPENDIX G

OTHER POSSIBLE LEPA ADOPTION MODEL SPECIFICATIONS

G.1 No Adoption Cost Heterogeneity

This section contains LEPA adoption model without considering individual heterogeneity in
adoption costs. The estimation of this model is straightforward, which follows the procedure
described in the first paragraph in appendix E.2. The estimation results are shown in the first
column of table G.1. The estimation results of the preferred model in table 3.5 are also duplicated
in the second column of table G.1 to facilitate comparison.

Table G.1: Ignoring Adoption Cost Heterogeneity in LEPA Adoption Model

No Heterogeneity Preferred
in Adoption Cost Model

Adoption Cost Cit = Ci + γNnit ($/acre)

CL

CH

γN

Cost Class Probability
C = CL
C = CH

Profit Shock Scale Parameter
σε

Obs

503.7***
(56.2)

20.69***
(4.77)

66.14***
(6.12)
28081

74.15***
(28.77)
277.70***
(88.72)
-2.7**
(1.3)

64%
36%

33.49***
(5.28)
28081

Notes: Bootstrapped standard errors are reported. * p < 0.1, ** p < 0.05, *** p < 0.01

Without modeling adoption cost heterogeneity, the estimated adoption costs goes to $503.7/acre,
which is unreasonably high. The scale parameter of profit shock ($66.14/acre) is also much higher
than the one estimated when adoption cost heterogeneity is modeled. We also estimate the adoption

110

model by the BBL method in Bajari, Benkard, and Levin (2007) without accounting for adoption
cost heterogeneity. The results (not shown here, available upon request) are consistent with those
shown in column 1 here. Intuitively, If farmers all have the same adoption costs, the observable
LEPA profit gain at which farmers adopt LEPA should be similar, ceteris paribus. If adoption cost
heterogeneity presents, the observable profit gain level above which farmers adopt LEPA will be
higher for farmers with high adoption costs and lower for those with low adoption costs. When
such heterogeneity is not properly accounted for, a large profit gain shock dispersion is needed to
justify the large dispersion of observed profit gains at which different farmers adopt LEPA. The
unreasonably high adoption costs and profit shock scale parameter estimates shown in column 1
further illustrate the importance of modeling adoption cost heterogeneity.

111

APPENDIX H

PROFIT GAIN TRANSITION RULE SELECTION

Selecting a correct transition rule for profit difference is a critical step in dynamic structural
estimation. This section discusses the procedures we follow to make such selection.

We fit a fixed effects linear probability model in table (H.1), where the dependent variable is
whether a farmer adopts LEPA in each year, whereas the independent variables are LEPA profit
gains in the current and past years. The results show that profit gains in year t, t − 1 and t − 2 all
have significant impacts on LEPA adoption probability, while the influence of that in year t − 3
becomes insignificant. According to the dynamic discrete choice problem defined by equation
(3.12) to (3.15), past realizations of LEPA profit gain (i.e. πD
it−3, etc) could only affect
LEPA adoption probability in year t through their influence on LEPA profit gain’s transition rule.
Therefore, possible LEPA profit gain realization in the next year should be a function of its current
realization and its realizations in the past two years, which gives an AR(3) type of transition rule.

it−2, πD

it−1, πD

Table H.1: Reduced-Form Evidence on Profit Gain Transition Rule Selection

(1)

(2)

(3)

(4)

πD
it

πD
it−1

πD
it−2

πD
it−3

Farmer FE
Year FE
R2

0.0032***
(0.000)

0.0027***
(0.000)
0.0016***
(0.001)

0.0015***
(0.001)
0.0016***
(0.001)
0.0018***
(0.001)

0.0015***
(0.001)
0.0012**
(0.001)
0.0023***
(0.001)
-0.0003
(0.0007)

Y
Y

0.1650

Y
Y

0.1623

Y
Y

0.1452

Y
Y

0.1394

Notes: Standard errors are reported in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01

However, we finally select an AR(2) transition rule specified by equation (3.18) because it gives

112

us an additional year in the estimation compared to if we choose and AR(3) rule. The first year
in the dynamic estimation is 1998 if an AR(2) transition rule is selected and is 1999 if an AR(3)
transition rule is selected instead. Additionally, it also reduces the dimension of the discrete choice
model by one. Finally, shown by table H.2, the difference between dynamic estimation results with
AR(1) and AR(2) transition rules is very large, while the difference between results with AR(2)
and AR(3) transition rules is small. Therefore, the results from using the AR(2) type transition rule
is reported in the main text and all the subsequent analyses are based on these results.

Table H.2: Estimation Results with AR(1), AR(2) and AR(3) Transition Rules

Adoption Cost Cit = Ci + γNnit ($/acre)

AR(1)

AR(2)

AR(3)

CL

CH

γN

Cost Class Probability
C = CL
C = CH

Profit Shock Scale Parameter
σε

-101.93***

(14.97)
431.65***
(61.44)
-20.36***

(7.34)

74.15***
(20.13)
277.70***
(17.77)
-2.7***
(1.3)

80.12***
(24.13)
285.78***
(14.76)
-2.4***
(1.3)

68%
32%

64%
36%

61%
39%

241.55***
(97.28)

33.49***
(5.28)

31.45***
(5.37)

Notes: Bootstrapped standard errors are reported in parentheses. * p < 0.1, ** p < 0.05, ***
p < 0.01

113

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114

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CHAPTER 4

CHEAPER SOLAR, CLEANER GRID?

4.1 Introduction

Thanks to the declining renewable energy costs and various federal support programs, renewable
energies’ share of electricity generation across the U.S has more than doubled from 2007 to 2017.
Texas, for example, has already reached its 2025 renewable capacity goal specified by the state’s
Renewable Portfolio Standard (RPS) in 2009. Although wind capacity costs are not expected to
experience large changes in the near future, the decline in solar capacity costs is still strong, with a
62% decline expected in only thirty years (Augustin et al., 2018). As solar capacity costs continue
declining, it is widely expected that renewable energy penetration1 would further increase, even
without stricter RPS requirements.

However, although solar capacity is expected to grow with the fast decline in its capacity costs,
the overall change in renewable energy penetration also depends on how the capacity levels of
other renewable resources, such as wind and energy storage, are adjusted correspondingly. If the
expansion of solar capacity increases their values (i.e. solar complements wind and storage), the
capacities of all renewable resources and renewable energy penetration will increase. If they are
substitutes instead, increase in solar capacity will push the capacities of other renewable resources
down. Consequently, the change in renewable energy penetration becomes ambiguous.

This chapter studies the relationships between solar, wind and energy storage, both analytically
and empirically. I then investigate how renewable energy penetration, carbon emissions and social
welfare are affected by the projected decline in solar capacity costs. I first build a three-period
analytical model where wind, solar, storage and fossil-fuel energy generators compete to supply
electricity (energy).2 On the one hand, different renewable energy resources substitute each other

1Renewable energy penetration is defined as the share of electricity generated from renewable energy sources.
2Throughout the chapter, "electricity" and "energy" are used interchangeably.

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because the energy supply from one resource is a perfect substitute for that from another.3 On
the other hand, they can also complement each other through influencing outputs of fossil-fuel
energies. Wind and solar energies are intermittent – they are only produced when wind blows
and the sun is shining. Facing almost perfectly inelastic real-time electricity demand, fossil-
fuel power generators must be changing output frequently to meet the increasingly fluctuating
residual demand due to renewable expansion, which imposes high output adjustment costs (i.e.
ramping costs and startup costs) on these generators. Fortunately, wind and solar energy resources
share complementary production patterns (Slusarewicz and Cohan, 2018). Mixing wind and
solar capacities helps smoothing out aggregate renewable energy production and reducing fossil
fuel generation capacities’ output adjustment costs (Shaner et al., 2018; Slusarewicz and Cohan,
2018; Solomon, Kammen, and Callaway, 2016; Solomon, Faiman, and Meron, 2010). Due to
the reduction in ramping costs, increasing the capacity of one renewable resource increases the
value of the other. In equilibrium, whether renewable energy resources complement or substitute
each other depends on the relative strengths of these two countering forces. The strength of
the complementarity effect, in particular, depends on the capacity levels of different renewable
energy resources, which affect fossil-fuel energy capacities’ output ramping patterns. Finally, the
analytical model predicts that solar complements (substitutes) wind when the level of solar capacity
is relatively low (high). While storage never substitutes wind, it only complements solar when the
level of solar capacity is relatively high.

A dynamic simulation model is then developed to simulate daily operation decisions of elec-
tricity generation capacities in the ERCOT electricity market. The relationships between wind,
solar and storage predicted by the analytical model are all confirmed by the simulation results. The
results from the operation simulation model are then combined with a stylized investment model to
decide optimal wind/solar/storage capacities as well as renewable energy penetrations at different
levels of solar capacity costs. Solar capacity level is low when solar capacity costs are high. At low
solar levels, solar and wind are complements and solar cost reduction increases both solar capacity

3An exception is energy storage, it complements other resource in the period when it is charging because it

demands, rather than supplies, energy.

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and wind capacity, driving up renewable energy penetration. However, solar and wind become
substitutes as solar costs gradually go down (and solar capacity becomes high). A decline in solar
costs then increases solar capacity but decreases wind capacity. Fixing the capacity costs of wind
at its long-term average,4 renewable energy penetration first goes up from 22% to 36% when solar
costs decline from the 2016 projection to the 2030 projection, and then goes down to 30% as solar
costs keep declining to the 2050 projection. Correspondingly, carbon emissions first decrease by
22%, but then increase by 12.9% from the minimum point. Due to the lack of carbon policies, social
welfare from renewable energy investment (i.e. benefits of energy consumption net of fossil-fuel
energy generators’ operation costs, renewable energy investment costs and social damage of carbon
emissions) reduces by $0.57 billion/year as solar costs decrease from the 2030 projection to the
2050 projection. If, on the other hand, renewable energy penetration is not allowed to drop below
36%, the highest penetration level achieved over the range of solar capacity costs being studied,
social welfare would increase by $0.77 billion/year when the capacity costs of solar drop to the
2050 projection. Thus, solar cost reduction alone does not guarantee a cleaner grid and a better
off electricity market. Environmental policies are needed to reduce the unexpected consequence of
solar cost decline and improve social welfare.

The simulation model also shows that it is not optimal to invest in energy storage if solar
capacity costs fall in the range used in this study. However, As solar capacity costs decline from the
2016 projection to the 2050 projection, a 92.5% increase in the value of storage is observed because
solar complements storage at high solar capacity levels. It is reasonable to expect that more storage
capacity would be added to the grid if solar capacity costs continuous to decline after 2050, and if
storage capacity costs could be driven down due to future technological improvements. As storage
complements solar at high solar capacity levels, the presence of storage capacity by the time when
solar capacity costs are very low would further increase renewable penetration and is likely to drive
carbon emissions down.

This study is related to the literature in several ways. First, by investigating how the values of

4Unlike solar, the projected average capacity costs of wind is almost constant from 2016 to 2050.

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wind, solar and energy storage depend on each other’s capacity level, this study looks into how
the expansion of one renewable resource affects optimal capacities of other renewable resources
and renewable energy penetration. This is in contrast with Cullen and Reynolds (2017) and
other studies that investigate how fossil-fuel energy capacities are affected by renewable energy
expansions (Bushnell, 2010; Gowrisankaran, Reynolds, and Samano, 2016; Verdolini, Vona, and
Popp, 2018). There is also a large literature in quantifying the marginal values of renewable
energy and energy storage (emission offsets and/or production cost savings) using reduced-form
econometrics techniques (Abrell, Kosch, and Rausch, 2019; Callaway, Fowlie, and McCormick,
2018; Carson and Novan, 2013; Cullen, 2013; Graff-Zivin, Kotchen, and Mansur, 2014; Kaffine,
McBee, and Lieskovsky, 2013; Novan, 2015). However, none of these studies investigate the
interactions of wind energy, solar energy, and energy storage, which directly affect equilibrium
capacity of each renewable resource.

Second, it combines two literatures that study the effects of mixing different renewable resources
on renewable energy penetration. In one literature, the share of each renewable resource in the
mixture is arbitrarily chosen (Lew et al., 2013). A detailed maximization problem that determines
fossil-fuel energy capacity’s output is solved, taking fossil-fuel capacities’ operation costs (i.e.,
production costs and startup costs) into consideration. The share of renewable energy output in
total energy generation is then calculated. Due to the ignorance of renewable resources’ investment
costs, the mixture arbitrarily chosen may not be supported in an economic equilibrium. In the other
literature, the share of each resource in the mixture is decided by maximizing renewable energy
output while satisfying a "limited curtailment" constraint and keeping investment costs as low as
possible (Shaner et al., 2018; Solomon, Kammen, and Callaway, 2016; Solomon, Faiman, and
Meron, 2010). This approach, however, ignores the impact of renewable energy production on
fossil-fuel energy generators’ operations, which in turn affect the values of renewable energies and
optimal investment. Moreover, this literature does not study how renewable penetration changes
as investment costs decline. This study combines the two literatures by adopting an approach
similar to that used in Cullen and Reynolds (2017), where optimal investment decisions are solved

123

after optimal daily operation decisions for each possible renewable capacity combination are first
decided based on fossil-fuel energy generators’ production costs and startup costs. This exercise is
then performed multiple times to decide the changes in renewable energy penetration and carbon
emissions as the capacity costs of solar keep declining.

Finally, this study also fits into a growing literature that emphasizes the effects of dynamic
frictions on renewable energy valuation. Comparing simulation results obtained by a dynamic
simulation model that accounts for fossil-fuel energy generators’ startup costs (i.e. costs incurred
when a generator is started up from zero production) and a static model that does not, Cullen (2015)
and Cullen and Reynolds (2017) show that ignoring startup costs significantly overestimates the
value of wind energy because the intermittency of wind increases fossil-fuel energy generators’
total startup costs. Similar arguments have also been made in several other studies (Eser et al., 2016;
Kopiske, Spieker, and Tsatsaronis, 2017; Milligan and Kirby, 2009). I extend this literature by
showing both analytically and empirically that startup costs not only affect the values of renewable
resources themselves, but also influence the effect of one renewable resource on the value of
another. In a similar study, Castro (2018) looks into the interaction of wind energy and energy
storage with a simulation model, but ignores startup costs in the analysis. Without startup costs, the
simulated relationships between wind, solar and storage capacities would be biased. For example,
aggregate renewable energy output smoothing through mixing wind and solar energies is not
valuable anymore because fossil-fuel energy capacities’ output changes are costless. Consequently,
the degree of substitution between wind and solar would be overestimated due to the loss of this
complementarity effect.

The rest of the paper is organized as follows. A three-period model is analyzed in section
4.2 to show why and how the values of wind energy, solar energy and energy storage depend on
each other’s capacity when fossil-fuel energy generators’ output adjustment costs are considered.
Section 4.3 presents the details of the daily dynamic simulation model used to empirically quantity
the interactions between different renewable resources and energy storage. The stylized investment
model that determines renewable energy penetrations under different solar capacity costs is also

124

presented. Section 4.4 shows the data and procedure used to calibrate the simulation model. Model
validation results are shown in chapter 4.5. Section 4.6 shows and discusses the implications of the
simulation results. Finally, section 4.7 concludes the paper and proposes several future research
directions.

4.2 Analytical Model: Value Dependence

In this section, I analyze why and how do the values of wind, solar and energy storage depend
on each other in a multi-period daily operation model. I first lay out demand and supply in the
electricity market. Then, social planner’s problem regarding electricity generation in each period,
facing given levels wind, solar, storage and fossil-fuel energy capacities, is presented. Values of
wind energy, solar energy and energy storage are then derived from the value function associated
with the social planner’s optimal operation decisions. Based on the previous two steps, I analyze
complementarity/substitution relationships among wind energy, solar energy and energy storage.

4.2.1 Daily Operation Model Setup

In the market, there is one wind turbine with capacity kw, one solar PV with capacity kso, and a
battery storage with capacity ks. For simplicity, I assume that there is only one type of fossil-fuel
power generation technology with unlimited capacity.

Given these capacity levels, a social planner maximize the payoff from electricity production
and consumption in a representative day, which is divided into three periods (t = 1,2,3). For
each t ∈ {1,2,3}, benefit from electricity consumption is a concave function of total electricity
production, Bt(Qt). Demand is given by a downward-sloping linear function pt(Qt) = B(cid:48)
t(Qt) =
at − bQt, where b > 0. Wind and solar energy outputs in period t are αw
t kso, where αw
t
are wind and solar energy capacity factors that are exogenously given. In contrast, energy
and αso
t
storage is charged during period(s) when price(s) is(are) low, and is discharged during period(s)
when price(s) is(are) high.

t kw and αso

In the representative day, period 1 corresponds to early morning period (4:00am to 9:00am),

125

where electricity demand is low, wind energy production is high, and solar energy is not being pro-
duced. Period 2 corresponds to mainly daytime hours (9:00am to 18:00pm), when both electricity
demand and solar energy production are high, but wind energy production is low. Finally, period 3
corresponds to late afternoon and early night hours (18:00pm to 22:00pm), when solar energy gen-
eration quickly drops to zero, but wind energy production and demand remain largely unchanged.
Given this, it is assumed that a1 < a2 = a3, αw1 > αw2 = αw3 > 0 and αso
3 = 0,
respectively.

2 > αso

1 = αso

Renewable energy generation does not incur any costs, but fossil-fuel energy production is

costly. The total fossil-fuel energy generation costs summed across all periods is:

C(q1, q2, . . . , qT) =

CP(qt) +

˜Cr(qt − qt−1),

(4.1)

where CP(qt) represents production costs and ˜Cr(qt − qt−1) represents intertemporal output adjust-
ment costs (or ramping costs), which has the following form:

T


t=2

t=1

T



˜C(x) =

0,
x < 0
Cr(x), x ≥ 0

,

(4.2)

P

r > 0, C(cid:48)(cid:48)

= constant > 0 and C(cid:48)(cid:48)

where CP(.) and Cr(.) are assumed to be convex, increasing and quadratic functions (i.e. C(cid:48)
P >
0,C(cid:48)
r = constant > 0). The form of the outcome adjustment
costs implies that the production technology favors small positive (or no/negative) intertemporal
output changes than large ones. Further, I assume C(cid:48)
r(0) > 0, which implies that increasing output
from the previous period is not always profitable for fossil-fuel power generators, even if price in
the current period is relatively high.

4.2.2 Social Planner’s Operation Decision and Values of Wind, Solar and Storage

The social planner’s daily operation problem is to choose energy production from fossil-fuel energy
capacities and energy storage to maximize the surplus from electricity production and consumption:

maxq1,q2,q3,s1,s2,s3

Bt(Qt) − C(q1, q2, q3).

(4.3)

3


t=1

126

subject to:

Qt = qt + Et,

(4.4)

where st is energy supply from storage in period t and Et = αw
t kso + st is the aggregate
energy output from non-fossil energy sources. For the ease of explanation, define energy storage’s
. Finally, carbon emission costs are not included in (4.3)
capacity factor in each period as αs
because no carbon taxes (or other carbon polices) exit in ERCOT, the study region of the empirical
model presented later.

t kw + αso

∂s∗
t
∂ks

t =

and the effect of type j capacity on the value of type i is Vki k j

The optimized value of (4.3) is the value function associated with the social planner’s optimal
operation decisions. Denote this value by V(kw, kso, ks). The value of type i ∈ {w, so, s} capacity
is Vki
.5 The focus of the analytical
analysis is to sign Vki k j
and Vki k j

, for i, j ∈ {w, so, s} and i (cid:44) j.
could be obtained after the social planner’s problem defined by (4.3) and
(4.4) is solved. Analytically solving this maximization problem is complicated by the kink at
qt = qt−1 in C(q1, q2, q3) specified in (4.1) and (4.2). Thus, solving the social planner’s problem
requires an ordering of q∗
3.6 This ordering, in particular, changes with varying levels of
renewable energy/energy storage capacity. Many possible orderings exist, at least in theory.

2 and q∗

Both Vki

1, q∗

Fortunately, q∗

1 < q∗

3 and q∗

2 < q∗

3 hold in almost all situations according to the later empirical
simulation results. Therefore, I only focus on the cases where these two inequalities are satisfied in
the following analytical analysis. So, we are left with four quantity orderings: (1) q∗
2 < q∗
3,
1 = q∗
(2) q∗
3. As solar energy produces only in
period 2, the magnitude of q∗

2 < q∗
2 is largely driven by the amount of solar capacity.

3 and (4) q∗

3, (3) q∗

1 < q∗

1 < q∗

1 > q∗

1 < q∗

2 = q∗

2, q∗

As discussed earlier, the operation of energy storage depends critically on the electricity price
in each period. Similar to the quantity orderings, there are also many possible price orderings.
However, p3 > p1 happens in almost all situations according to the later empirical simulation
results. Therefore, I only focus on the cases where this inequality is satisfied. Now, price ordering

2V
5Notationally, Vki
∂ki ∂k j
6The "*" superscript means "optimal choice".

and Vki k j

= ∂V
∂ki

= ∂

.

127

depends only on the magnitude of p2 relative to p1 and p3. Consequently, we are left with the
same set of five price orderings for each quantity ordering. Namely, p1 < p2 > p3, p1 < p2 = p3,
p1 < p2 < p3, p1 = p2 < p3 and p1 > p2 < p3.

However, only the quantity-price orderings shown in figure 4.1 happen frequently in the em-
pirical simulation defined later in section 4.3 using model parameters that are calibrated to fit the
real ERCOT power system. Since demand in period two is higher than that in period one and solar
energy produces only in period two, a relatively large solar capacity is needed to drive p2 drop
below p1. But this also tends to push q∗
1. Therefore, p1 ≥ p2 is very unlikely to happen if
q∗
1 < q∗
2. Similarly, to keep p2 above p1, the amount of solar capacity must not be too high, which
1 > q∗
makes q∗

2 difficult to achieve. Therefore, p1 < p2 is also unlikely to happen if q∗

2 below q∗

1 ≥ q∗
2.

Note: In each possible price ordering, the first, second and third dots represent prices in period
one, two and three, respectively. The height of the dots represents corresponding price levels.

Figure 4.1: Quantity-Price Orderings in the Analytical Model

In the subsequent analytical analysis, I only focus on quantity-price ordering combinations
denoted as L1, L2, L3 and H in figure 4.1. As solar capacity is an important determinant of
which combination happens in equilibrium, I refer to these combinations as the low-low solar,
low-medium solar, low-high solar, and high solar scenarios, respectively.7 Since scenarios L1, L2
and L3 share the same quantity ordering q∗
3, I denote the union of these three scenarios
as L = {L1, L2, L3}, and refer to it as low solar scenarios.

1 < q∗

2 = q∗

The mechanism that drives the relationships between wind, solar and storage under scenario

7Alternatively, I can also analytically solve out the conditions under which each of the twenty cases happen (five
price orderings for four quantity orderings), which is straightforward but tedious because they depend on the (nonlinear)
relationships between all model parameters.

128

the medium solar scenario) is similar to that under scenario L2 (i.e.

M (i.e.
the low-medium
solar scenario). Therefore, I will simply mention the results under this scenario in the analytical
analysis, without providing detailed discussions. Ignoring other quantity-price orderings would
barely affect the analysis results because they are unlikely to happen under the model parameters
that are calibrated to fit the real ERCOT power system.

4.2.3 Value Dependence: Solar Capacity, Wind Capacity and Energy Storage
Under all five scenarios, the value of type i ∈ {w, so, st} capacity is (derivations are in appendix
I.1):

=

Vki

∂V
∂ki

= p1(Q∗

1)αi

1 + p2(Q∗

2)αi

2 + p3(Q∗

3)αi
3

(4.5)

The value of an additional unit of type i capacity equals to the summation of its marginal
revenue products across all periods, which is consistent with the results obtained in Carson and
Novan (2013) and Castro (2018).

Given (4.5), the effect of capacity j on the value of capacity i depends on capacity i’s production
pattern as well as the effect of capacity j expansion on each period’s equilibrium electricity price
through its influence on total energy output in each period:

Vki k j

3
∂k j

1(.) ∂Q∗
1p(cid:48)
1
= αi
∂k j
1( ∂Q∗
= −bαi
1
e j
1 +
∂E1
2( ∂Q∗
2
∂E1
3( ∂Q∗
3
∂E1

3(.) ∂Q∗
2(.) ∂Q∗
3p(cid:48)
2p(cid:48)
2
+ αi
+ αi
∂k j
∂Q∗
∂Q∗
3)
1
1
e j
e j
2 +
∂E2
∂E3
∂Q∗
∂Q∗
3)
2
2
e j
∂E2
∂E3
∂Q∗
∂Q∗
3)
3
3
e j
∂E3
∂E2

− bαi

− bαi

e j
1 +

e j
1 +

e j
2 +

e j
2 +

,

(4.6)

where the terms in the parentheses in (4.6) represent changes in period t total electricity output as a
response of changes in non-fossil fuel energy output (e j
) in all four periods due to type j capacity
t
expansion. In other words, the effect of type j capacity on the value of type i capacity is simply the
weighted average of each period’s price level change brought by type j capacity expansion, where
the weights are type i capacity’s capacity factors in different periods.

129

From (4.6), the relationship between type i and type j capacity depends critically on (i) how
non-fossil fuel energy output in each period is affected by wind/solar/storage capacity expansion
(i.e. e j
), and (ii) how total energy output in each period is affected by changes in non-fossil fuel
t
energy supply in that period and other periods (i.e. ∂Q∗
, for t, τ ∈ {1,2,3}). Detailed discussions
regarding these two key factors are provided next.

t
∂Eτ

4.2.3.1

t and ∂Q∗
e j

t
∂Eτ

Under Different Scenarios

Define the effect of increasing wind/solar/storage capacity on non-fossil fuel energy output in each
period as:

e j
t =

∂E∗
t
∂k j

∂αw

=

t kso + ∂s∗
t kw + ∂αso
∂k j

t

.

(4.7)

Energies produced from wind and solar energy capacities are exogenous, so the key determinant
is how storage operation is changed facing wind/solar/storage capacity expansion. Table 4.1

of e j
t
shows the values of e j
t

under different scenarios.

Table 4.1: Effects of Increasing Wind/Solar/Storage Capacity on Et Under Each Scenario

e j
1
¯αw
0
-1
¯αw
0
-1
¯αw
0
-1
¯αw
0
0

e j
2
αw
αso
2
1

e j
3
αw
0
0

αw
αw
2 /2 αso
2 /2
αso
1/2
1/2

αw
αso
2
0

αw
αso
2
-1

αw
0
1

αw
0
1

L1

L2

L3

H

j = w
j = so
j = s
j = w
j = so
j = s
j = w
j = so
j = s
j = w
j = so
j = s

Under scenario L1, p1 < p2 and p2 > p3. The strictly positive price gap between period two and
one implies that storage is fully charged in the first period and fully discharged in the second period.

130

When wind/solar/storage capacity is marginally increased, this price gap persists and storage still
fully charges in period one and fully discharges in period two. Thus, energy supply from storage is
not affected by changes in wind and solar capacities. Consequently, one unit increase in wind/solar
capacity increases non-fossil fuel energy supply by its capacity factor in each period. Similarly, one
unit increase in storage capacity decreases non-fossil fuel energy supply by one unit in the period
when it is charging and increases that by one unit in the period when it is discharging. Similar
arguments apply to scenarios L2 and H, where storage is fully charged in one period and fully
discharged in another.

a2−a3−bαso2 kso+bks

2b

2 =

However, p1 < p2 = p3 under scenario L2. Thus, storage is fully charge in period one. However,
storage must be discharging in both period two and three as p2 = p3. If storage discharges in only
one of these two periods, marginal profit of discharging (i.e. price) in that period must be higher
than that in the other period. The non-arbitrage condition p2 = p3 solves optimal discharge amount
in both periods s∗
2. The values in table 4.1 under scenario
L2 then follows.
Note that s∗

2 does not depend on wind capacity because wind production levels are the same in
period two and three. Thus, a change in wind capacity would not lead to price imbalance between
these two periods. Similarly, s∗
2 = q∗
3
under all low solar scenarios.

2 is also independent of fossil-fuel production8 because q∗

and s∗

3 = ks − s∗

The next building block of this study is presented in theorem 1, which establishes the effects of
increasing Et on electricity outputs in all three periods. Increasing non-fossil energy supply in one
period substitutes (i.e decreases) fossil-fuel energy supply in that period. Without ramping costs,
there are no dynamic linkages between any two consecutive periods. Therefore, total energy output
in any other period is not affected when Et changes. However, when ramping costs present, energy
output in other periods are also affected through the changes in fossil-fuel energy capacity’s output.

Theorem 1: Intertemporal Electricity Generation Dependence
(I) Own-Period Effects: under both low and high solar scenarios, increase in non-fossil fuel

8Judged by the fact that the expression of s∗

2 does not include any fossil-fuel energy capacity’s cost parameters

131

energy supply in any period decreases equilibrium fossil-fuel energy generation in that period, but
increases total energy generation in that period, i.e., ∂q∗
(II) Cross-Period Effects:

∂Q∗
t
∂Et

> 0.

< 0,

t
∂Et

(a) Low Solar Scenarios: Increase in non-fossil fuel energy supply in any period decreases
< 0 and

equilibrium fossil-fuel and total energy generation in all other periods, i.e., ∂q∗
∂Q∗
t
∂Eτ

< 0, for t, τ ∈ {1,2,3},& t (cid:44) τ.

t
∂Eτ

(b) High Solar Scenario: Increase in E2 (E3) decreases equilibrium fossil-fuel and total energy
< 0

supply in period three (two), but has no effect on period one, i.e., ∂q∗
and ∂Q∗

< 0 for t, τ ∈ {2,3}& t (cid:44) τ.

< 0, ∂q∗

< 0, ∂Q∗

1
∂Eτ

t
∂Eτ

t
∂Eτ

1
∂Eτ

| · · · > | ∂Q∗

| under all

| ∂Q∗
t−1
∂Et

| · · · > | ∂Q∗
t−m
∂Et

|, and | ∂Q∗

t+1
∂Et

t
∂Eτ

t+m
∂Et

= ∂Q∗
τ
∂Et

for t, τ ∈ {1,2,3} under all scenarios.

(III) Cross-Period Attenuation:
scenarios.
(IV) Symmetry: ∂Q∗
(V) Total Generation Neutrality: Under both low and high solar scenarios, the effect of an increase
in non-fossil fuel energy output in any period on total electricity generations summed over all three
periods are the same, i.e., ∂[Q∗
(VI) Middle-Period Effects Comparison:

≡ M for t ∈ {1,2,3}, where M =

2+Q∗
1+Q∗
3]
∂Et

C(cid:48)(cid:48)
P
b+C(cid:48)(cid:48)
P

> 0.

(a) The own-period effect for period two is higher under low solar scenarios than under the high

solar scenario, i.e., ∂Q∗L2
∂E2 >

∂Q∗H2
∂E2 .
∂E2 | < | ∂Q∗L3
(b) Under low solar scenarios, | ∂Q∗L1
∂E2 |.

In what follows, I illustrate the intuition of the theorem. Formal derivations of the quantities

shown in the theorem is moved to appendix I.2.

First, an increase in Et reduces residual demand facing fossil-fuel energy generators in period
t is increased because increase in Et raises total

t downward. But total energy output Q∗

t, pushing q∗
period t supply.

132

t , q∗

t−1, q∗

Second, upward ramping is costly but downward ramping is costless. When q∗

t > q∗
t−1,
decreasing qt−1 raises the marginal cost of production in period t through increased ramping. As
Et−1 expansion reduces q∗
t is also reduced. Similarly, decreasing qt increases the opportunity
cost to keep period t − 1 power generation high to avoid upward ramping cost. As Et expansion
reduces q∗
t−1 is also reduced. This argument only applies to the transition from period two
to period three under the high solar scenario because only this transition represents an upward
ramping process. Under low solar scenarios, a upward ramping from period one to period two
also presents, so the argument above also follows for this transition. Although q∗L2 = q∗L3
(i.e. no
upward ramping presents), this equality would not be would not be violated by marginally changing
q∗
2.9 As increasing E2(E3) decreases q∗
Going from fossil-fuel electricity generation (qt) to total electricity generation (Qt) is straight-
t for t (cid:44) τ through affecting fossil-fuel electricity generation in

2) would also be decreased.

forward since Eτ only affects Q∗
period t.

2(q∗

3), q∗

3(q∗

Third, "cross-period attenuation" emerges because Et only has direct effect on its own period
and the periods next to/prior to period t. Intuitively, each period is a "buffer" for the next/previous
period with respect to a change in Et.

Fourth, "symmetry" and "total generation neutrality" emerges due to the assumption that the
marginal ramping cost function C(cid:48)
r(·) depends solely on fossil-fuel energy generation difference in
consecutive periods (i.e. qt+1 − qt). One implication of the "total generation neutrality" property
is that the positive own-period effect dominates the summation of all (or every single) negative
cross-period effects.

Fifth, decrease in q2 (i.e. E2 expansion) only increases the marginal cost of increasing q3 under
the high solar scenario. However, under low solar scenarios, this will increase explicit/implicit cost
of production in both period one and three: q2 decline increases the opportunity cost to keep period
one power generation high. Meanwhile, since q∗
2 is marginally changed,
decreasing q∗
3, which decreases period three profit.10 This is an implicit cost

2 will also decrease q∗

3 still holds when q∗

2 = q∗

9For more details on this point, please refer to the discussion of (I.2) in appendix I.1
10This argument is a direct implication of I.2 in appendix I.1

133

of reducing q2. Since reduction in q2 affects more periods under the low solar scenarios, one unit
increase in E2 would lead to less q2 reduction under these scenarios, which means the increase in
total electricity output in period two is larger.

Finally, reduction in q∗L2 leads to an increase in the opportunity cost of keeping qL1 high, which
shifts period one fossil-fuel energy capacity’s marginal cost curve to the left. As the cost function
(i.e. production cost+ramping cost) is concave, one unit increase in marginal cost function results
in less than one unit decrease in q∗L1 . In contrast, q∗L2 = q∗L3 still holds by marginally changing E2,
results in one unit reduction in q∗L3 . Part (b)
as discussed earlier. Thus, one unit reduction in q∗L2
in theorem 1 (IV) then follows.

Given theorem 1, I analyze wind-solar, solar-storage, and wind-storage relationships in se-
quence. Specifically, I show that the these relationships are different under the low solar and high
solar scenarios.

4.2.3.2 Relationship Between Wind and Solar Energy Capacities

The relationship between wind and solar energy capacities is characterized by Vkw kso
effect of increasing solar energy capacity on the value of wind energy capacity. Given eso
t
4.1, Vkw kso

, i.e., the
in table

Vn
kw kso

2
∂E2

αso

= −b ¯αw( ∂Q∗n
1
αso
∂E2
∂Q∗n
= −b ¯αwαso
1
2
∂E2
∂Q∗n
1

under scenarios L1, L3 and H becomes:
2 ) − bαw( ∂Q∗n
2 ) − bαw( ∂Q∗n
(cid:34) ∂Q∗n
(cid:35)
(cid:35)
(cid:34)
− bαwαso
2
M − ∂Q∗n
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

∂E2(cid:124)(cid:123)(cid:122)(cid:125)

= −b ¯αwαso
2

−bαwαso
2

∂Q∗n
3
∂E2

1
∂E2

2
∂E2

+

L:(−);H:(0)

(+)

3
∂E2

2 )
αso

(4.8)

for n ∈ {L1, L3, H},

,

where αw1 = ¯αw > αw = αw2 = αw3 .

Increasing solar capacity increases non-fossil fuel energy supply in period two. Thus, total
electricity output in period two increases due to the "own-period effect". Meanwhile, total elec-

134

tricity output in period three is reduced due to the "cross-period effect" under all three scenarios.
Consequently, period two price decreases and period three price increases.

Since the positive own-period effects dominates the summation of all (or every single) negative
cross-period effects (an implication of "total generation neutrality" property), the overall price
(output) change in period two and three is negative (positive). As wind energy output are the same
in these period two and three, the net effect of solar expansion on wind energy production in these
two periods is negative.

Under the high solar scenario (H), solar expansion in period two has no effect on output(price)
= 0). Thus, the overall effect of solar capacity on the value of wind

in period one (i.e. ∂Q∗s1
capacity is negative, which means solar and wind energy capacities are substitutes.

∂E2

However, solar expansion raises period one price and thus the value of wind energy output in that
period under scenarios L1 and L3 due to the "cross-period effect". Although the summation of price
(output) changes across all three periods is negative (positive), the fact that wind energy production
is higher in period one than in the other two periods makes the sign of (4.8) undetermined.

The value of Vkw kso

is exactly the same as Vksokw

under scenario L2 is exactly the same as those under scenarios L1 and L3.
Note that Vkw kso
, the influence of wind capacity on the value of solar.
According to table 4.1, increasing wind capacity would not affect storage energy supply under all
low solar scenarios (i.e. L1, L2, L3). Thus, the effects of wind energy capacity on the value of solar
energy capacity are the same under all three low solar scenarios.

Finally, it could also be shown that wind and solar energy capacities substitute each other under
the medium solar scenario (M). The derivation is similar to the derivations above, so it is omitted
in this chapter to save space. The relationship between wind and solar capacities in summarized in
proposition 1:

Proposition 1: Wind energy capacity and solar energy capacity are likely to be complements
under low solar scenarios, but they become substitutes for sure under medium solar and high solar
scenarios.

135

Intuitively, wind energy and solar energy production overlaps in certain periods across a day.
In this sense, wind and solar are substitutes. One the other hand, wind energy and solar energy
production are volatile (i.e., intermittent), but their production patterns always complement each
other. Therefore, combining these two energy sources in the electricity grid would smooth out
renewable energy production across a day, which reduces fossil-fuel energy sources’ ramping (i.e.,
output changes). This smoothing effect is valuable when fossil-fuel energy capacity is ramping up
because upward-ramping is costly. In this sense, wind and solar are complements. The degree
of complementarity depends on the duration of upward-ramping. When solar capacity is low, the
duration of upward-ramping is three, which drops to two when solar energy is high. Thus, as
solar energy increases, the degree of complementarity drops but the degree of substitution keeps
constant. Finally, wind and solar energies become absolute substitutes at high levels of solar energy
capacity.

Also note that if ramping is costless, the value of renewable energy smoothing disappears.
In such case, wind and solar energies are always substitutes under all possible scenarios. The
degree of substitution between wind and solar would also be overestimated due to the loss of the
complementarity effect.

4.2.3.3 Relationship Between Solar Energy Capacity and Energy Storage

The relationship between solar energy and energy storage capacities is characterized by Vksoks
, i.e.,
the effect of increasing storage capacity on the value of solar energy capacity. Under the low-low
solar scenario, storage is fully charged in period one and fully discharged in period two. With this
operation pattern, Vksoks

under the low-low solar scenario becomes:

V L1
ksoks

= −bαso
2

< 0,

(4.9)

(cid:35)
(cid:34)
− ∂Q∗L2
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

∂Q∗L2
∂E2

∂E1

+

(+)

Storage capacity expansion increases storage discharging (thus non-fossil fuel energy supply)
in period two, which increases period two total energy output through the "own-period effect".

136

Storage charging in period one is also increased, which equivalent to a reduction in period one non-
fossil fuel energy supply, which increases period two total energy output through the "cross-period
effect". Since both forces raise period two total energy output, period two price decreases and the
value of solar energy capacity is reduced as solar energy is only produced in period two. Thus,
solar and storage are substitutes under the low-low solar scenario.

Under the low-medium solar scenario, storage charges fully in period one, but discharge a
becomes:

portion of stored energy in period two and the remaining energy in period three. Vksoks

− ∂Q∗L2
(cid:124)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:125)(+)

∂E1

− ∂Q∗L2

(cid:34)

∂E1

1
2

+

1
2

+

(cid:34)

∂Q∗L2

 = −bαso
(cid:124)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:125)
∂Q∗L2
∂E2(cid:124)(cid:123)(cid:122)(cid:125)(+)
(cid:32)
(cid:33)(cid:35)
∂E3
(−)
M − ∂Q∗L1

< 0,

1
2

+

2

∂E2

V L2
ksoks

= −bαso
2

= −bαso
2

(cid:32) ∂Q∗L2

∂E2

∂Q∗L3
∂E2

+

(cid:33)(cid:35)

− ∂Q∗L2

∂E1

1
2

+

(4.10)

where the second equality follows the "symmetry" property (theorem 1 (IV)). Similar to scenario
L1, increased charging in period one increases (decreases) period two total energy output (price).
However, the additional one unit charging is evenly spread to period two and three, which increases
and decreases period two total energy output, respectively. But own-period effects dominate
the cross-period effect according to theorem 1 (V). Thus, increased discharging brought by storage
expansion increases period two total energy output. Combining the influence of additional charging
and discharging, period two electricity price falls and the value of solar energy capacity declines.
Under the low-high solar scenario, storage charges fully in period one, and is fully discharged

in period three. Vksoks

becomes:

(cid:34)

(cid:35)

V L3
ksoks

= −bαso
2

− ∂Q∗L2

∂E1

∂Q∗L2
∂E3

+

= −bαso
2

(cid:35)
(cid:34)
− ∂Q∗L2
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

∂Q∗L3
∂E2

∂E1

+

(−)

> 0,

(4.11)

The additional discharging in period three now decreases period two total electricity output.
According to theorem 1 (IV) and (VI) (b), this decrease dominates the output increase brought by

137

additional charging in period one. Therefore, price in period two increases and storage complements
solar capacity under this scenario.

Under the high solar scenario, storage is fully charged in period two and fully discharged in

period three. With this operation pattern, Vksoks

(cid:34)
(cid:35)
under the high solar scenario becomes:
− ∂Q∗H2
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

∂Q∗H2
∂E3

> 0

(4.12)

V H
ksoks

= −bαso
2

∂E2

+

(−)

Both the additional charging in period two and additional discharging in period three decrease
price level in period two. The resulting increase in period two electricity price raises the value of
solar energy under this scenario.

Finally, solar energy and energy storage capacities also complement each other under the
medium solar scenario (M). Again, the derivation is similar to the derivations above and is omitted
to save space.

The relationship between solar energy and energy storage is summarized in proposition 2:

Proposition 2: Solar energy and energy storage are substitutes when the amount of solar capacity
is low and become complements as solar capacity keeps increasing.

The argument in the last section also applies here to help understand the intuition behind the
change in the relationship between solar energy and storage. When solar capacity is very low,
energy supply patterns of storage and solar energy are very similar, are overlapped and do not
smooth each other. Thus, solar and storage are substitutes. As solar capacity increases, solar and
storage energy supplies become less overlapped and starts to smooth each other’s variation. Thus,
the two energy sources gradually become complements.

4.2.3.4 Relationship Between Wind Energy and Storage

The relationship between wind energy and energy storage capacities is characterized by Vkw ks
the effect of increasing storage capacity on the value of wind energy capacity.

, i.e.,

138

Increasing storage capacity by one unit results in both a one-unit decrease and a one-unit increase
in non-fossil fuel energy supply through the charging and discharging cycle, although the increase
and decrease happen in different periods. Due to the "total generation neutrality" property, storage
expansion would not change the summation of total electricity output across all three periods.

Under all low solar scenarios, storage is fully charged in period one. The only difference among
the three low solar scenarios is the period in which the storage discharges, which then affects es
.
t
In general, Vkw ks

(cid:35)

∗ esn
2 +

∂Q∗L2
∂E3

∗ esn
3

Vn
kw ks

= −b ¯αw

− bαw

= −b ¯αw

(cid:35)

∂E1

∗ esn
1 +

(cid:34)
under all low solar scenarios shares the same form:
∂Q∗L2
− ∂Q∗L1
(cid:34)
∂E2
− ∂Q∗L3
(cid:34)
(cid:35)
− ∂Q∗L1
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

(cid:34)
− ∂Q∗L2
∂E1
∂Q∗L3
(cid:34) ∂Q∗L1
∗ esn
3
∂E3
− ∂Q∗L1

∂Q∗L1
∂E2
∗ esn
1 +
∂Q∗L1
∂E2

− bαw
∂Q∗L3
∂E2
−bαw

∗ esn
2 +

∂E1

∂E1

∂E2

∂E1

(cid:35)

(cid:35)

+

+

(−)

(4.13)

,

> 0, n ∈ {L1, L2, L3}

Both the additional charging in period one and additional discharging in the other periods

decreases period one total electricity output. Thus, price level in period one increases.

As the summation of total electricity output across all three periods is unchanged, the summation
of total electricity output in period two and three increases by the same amount as the decrease
in period one output. Thus, price summation across period two and three decreases by the same
amount as the increase in period one price. Because wind energy output is higher in period one than
in the other periods, the overall value of wind energy capacity increases and storage complements
wind under low solar scenarios.

In contrast, the effects of energy storage on the value of wind energy under the high solar

scenario is:

V H
kw ks

= −bαw

= −bαw

(cid:34)
(cid:34)

− ∂Q∗H2
− ∂Q∗H2

∂E2

∂E2

∂Q∗H2
∂E3
∂Q∗H2
∂E3

+

+

− bαw

− bαw

(cid:35)

(cid:34)
− ∂Q∗H3
(cid:34) ∂Q∗H2

∂E2

∂E2

+

∂Q∗H3
∂E3
− ∂Q∗H2

(cid:35)

∂E3

(4.14)

,
= 0

(cid:35)
(cid:35)

139

Storage now fully charges in period two and fully discharges in period three. Neither the charging
nor discharging affect prices in period one because the fossil-fuel energy output is ramping down
from period one to period two.

Again, storage expansion would not change the summation of total energy output in period two
and three, which implies that the increase in period two price is the same as the decrease in period
three price. As wind energy outputs are the same in period two and three, the overall value of wind
energy capacity would not change as storage capacity changes.

Finally, wind energy and energy storage capacities also complement each other under the
medium solar scenario (M). Again, the derivation is similar to the derivations above and is
omitted to save space. The relationship between wind energy and energy storage is summarized in
proposition 3:

Proposition 3: Wind energy and energy storage complement each other under a wide range of
renewable energy capacity levels. They only become independent after solar energy capacity
increases to a very high level.

4.2.3.5 The Critical Turning Points

In the previous section, I show that the relationships between the values of wind energy, solar
energy and energy storage vary depending on which scenario happens in equilibrium, which is
heavily affected by the level of solar capacity. The varying relationships are summarized in table
4.2.

Table 4.2: Analytical Results Summary

Wind-Solar
Solar-Storage
Wind-Storage

L1
(+)
(-)
(+)

L2
(+)
(-)
(+)

L3 M H
(+)
(-)
(+)
(+)
(+)
0

(-)
(+)
(+)

Indeed, not only solar capacity, but also wind and storage capacities, affect which scenario

140

happens in equilibrium. Figure 4.2 shows two critical solar capacity levels as functions of wind
capacity and storage capacity.

Figure 4.2: Scenario Boundaries

The first critical level, K A

so, separates scenarios L1/L2 from scenarios L3/M/H.

It also
represents the minimum level of solar capacity required for solar capacity to complement storage
so(ks) increases as storage capacity goes up. With increased
capacity. I show in appendix I.3 that k A
discharging in period three under L3, p3 tends to be lower. Thus, more solar capacity is needed to
push p2 below p3 if we still want to stay under at least the low-high solar scenario (L3) rather than
switching to the other two low solar scenarios (i.e. L1 and L2).

The second critical level, K B

so, separates scenarios L1/L2/L3 from scenarios M/H. This is
also the maximum level of solar capacity required for solar capacity to complement wind capacity.
so(kw, ks) is increasing in the capacity of wind, but is decreasing in the capacity of
Specifically, k B
storage. q∗
1 tends to be lower when wind capacity is higher because wind energy production is more
active in period one. Even with more solar capacity, q∗
1 still holds and we are still under low
solar scenarios (i.e. L1, L2, L3). Increasing storage capacity increases charging in period one under
2 > q∗
scenario L3, which increases q∗
1
if we still want to stay in scenario L3 rather than switching to medium and high solar scenarios.

1. Thus, less solar capacity could be accommodated to keep q∗

2 > q∗

Proposition 4: Wind capacity, solar capacity and storage capacity all affect which scenario occurs
in equilibrium, which in turn affects the equilibrium relationships between wind, solar and energy
storage. Specifically,

(I) Low solar scenarios, under which wind and solar energies are complements, are more likely

to happen when the level of wind capacity is high and the level of storage capacity is low.

141

(II) Among all low solar scenarios, the low-low and low-medium solar scenarios, where solar
energy and storage are substitutes, are more likely to happen when the level of storage
capacity is low.

4.3 Empirical Simulation Model

In this section, I first describe the simulation model that is used to obtain the value function
associated with the social planner’s optimal operation decisions, under given capacity levels of
wind, solar and storage. This value function serves as the basis of empirically quantify the value
dependence between wind, solar and storage capacities. It also serves as the building block of the
social planner’s optimal capacity choice problem, which is used to empirically quantify the effect
of solar capacity cost decline on renewable energy penetration. The investment simulation model
is presented following the operation model.

The setup of the simulation model is similar to those in Cullen and Reynolds (2017) and
Reynolds (2018), with certain modifications. Compared to the analytical model, the simulation
model includes more fossil-fuel power generators. In addition, the abstract output adjustment costs
is represented by startup costs and minimum generation levels.

Startup costs, as the name suggests, are incurred when a fossil-fuel power generator is started
up from inactive status. Once started up, it must be producing at a level higher than its minimum
generation. The existence of startup costs and minimum generation levels makes the intertemporal
cost structure of a fossil-fuel generator similar to the one specified in the analytical model. Therefore,
insights obtained in the analytical model also apply to analyze results obtained from the simulation
model.

4.3.1 Social Planner’s Operation Problem

Consistent with the notations used in the analytical model, wind and solar capacity levels are
, respectively. Moreover,
denoted as kw and kso, with capacity factors in hour t being αw
t
let ks still be the total capacity of energy storage, whose energy supply in hour t is ht.

and αso
t

142

becomes Qt =
J

Unlike in the analytical model, there are four types (J = 4) of fossil-fuel power generation
technologies in the grid: coal, combined cycle, gas steam and gas turbine. Total amount of type
j generation capacity summed over all type j firms is k j. Thus, total electricity supply in hour t
t kso + ht. Electricity suppliers are atomistic price taking
firms. Each firm operates one unit (arbitrarily small) of energy generation capacity (i.e. generator).
Therefore, I will use "firm" and "generator" interchangeably the in rest of this paper.

j=1 qjt + αw

t kw + αso

Fossil-fuel power generators are subject to start-up costs and minimum generation. Startup cost
gj is incurred if a type j generator switches from "off" status (i.e. not producing anything) to "on"
status in a given hour. Once it is turned on, it must be producing at a level in [mj,1]. It is assumed
that gj and mj are homogeneous within each technology type. Moreover, startups takes one hour
to be effective. That is, the amount of new capacities turned on in hour t could only be utilized to
produce electricity from hour t + 1.

Finally, production cost needs to be defined. Each fossil-fuel energy generator has a con-
stant marginal production cost over its production range, but different generators with the same
technology type are allowed to have different marginal production costs, following Reynolds
I assume that marginal production costs of type j generators fall into nj classes, with
(2018).
. Let xi jt ∈ [0, ki j] be the amount of type j capacities with production cost ci j
c1 < c2 < · · · < cnj
i qi jt

that are kept on at the beginning of hour t. For a given level of aggregate production qjt =
nj

for type j generators (where qi jt ∈ [mi j xi jt, xi jt]), the cost function is defined as:

nj


i

C(qjt; xjt) = min
qi jt

ci jqi jt

(4.15)

As firms’ startup costs and minimum generation levels within each technology type are ho-
mogeneous, only generators with the lowest production costs will be kept "on" given a feasible
production level qj. A direct implication of this result is that the aggregate production cost of
type j firms, C(qjt; xjt), is continuous and convex in qjt for given xjt (Reynolds, 2018). For more
discussion on the properties of C(qjt; xjt), please refer to Reynolds (2018). The construction of
this total production function is shown in appendix J. Due to a lack of carbon policy in the study

143

region, carbon emission costs are not included to construct C(qjt; xjt). This conforms to the current
reality that carbon costs are not taken into consideration when the grid manager arranges hourly
electricity generation everyday.

The existence of minimum generation level and discrete startup decision (i.e. a generator is
either "on" or "off") render non-convexity to generator-level production possibility sets. However,
by assuming that firms are measure zero, Cullen and Reynolds (2017) and Reynolds (2018) show
that the aggregate production possibility set is convex for each technology type. I modify their
aggregate production possibility sets by adding storage:
PT(k) = {0 ≤ yj ≤ xj,0 ≤ zj ≤ k j − xj, mj xj ≤ qj ≤ xj, ˜ks − ks ≤ h ≤ ˜ks, ˜ks ≤ ks; j = 1, . . . , J}.
(4.16)
where x = {x1, . . . , xJ} is a vector of the amount of "on" capacities at the beginning of each hour.
yj is the amount of "on" capacities that type j firms decide to carry on to the next hour. zj is
˜ks is the amount of electricity stored
the amount of new capacities that firms decide to turn on.
in the energy storage equipment. PT(k) is compact for any vector of k = {ks, kw, kso, k f }, where
k f = {k1, k2, k3, k4}. Finally, the transitions of state variables satisfy:

xt+1 = yt + zt

˜kst+1 = ˜kst − ht

.

(4.17)

The first equation in (4.17) states that the amount of "on" fossil-fuel power capacities in the
next hour is the amount of current "on" capacities that is kept on to the next hour plus the amount
of new capacity started. This specification allows one hour lag between the startup decision and
when the new start-up capacities could produce energy. The second equation states that the amount
of power stored in the next hour equals to the amount currently stored minus the amount of energy
discharged (sold) in the current hour.

Similar to Reynolds (2018), the aggregate (aggregated over each technology type) electricity
production technology is convex due to the compactness of PT(x) and linearity in the transition
rules11. Given market demand, market supply and capacity vector k, hourly payoff for the whole
11I do not introduce energy storage’s round-trip efficiency because it will render non-linearity in ˜ks’s transition rule

144

electricity grid is:

H(qt, zt, xt, ht; k) = B( J


j=1

t kso + ht) − J


j=1

qjt + αw

t kw + αso

C(qjt; xjt) − J


j=1

gj zjt,

(4.18)

where qt = {q1t, q2t, q3t, q4t}, zt = {z1t, z2t, z3t, z4t} and xt = {x1t, x2t, x3t, x4t}. The social
planner’s operation problem is to choose the sequence of {qt, zt, xt} to maximize the sum of hourly
payoffs (4.18) across the planning horizon T, subject to the aggregate production possibility set
PT(x) specified in (4.16) and state transition rules specified in (4.17). The value function associated
with the social planner’s optimal operation decisions is denoted as:

T


t=1

V(k) = maxq,z,x,h∈PT(k)

H(qt, zt, xt; k),

(4.19)

Planning horizon T in (4.19) needs to be selected to solve the social planner’s operation problem.
As quantifying Vki k j ,i, j ∈ {w, so, s} for different renewable and storage capacity combinations is
one of this study’s focus, the social planner’s operation problem needs to be solved for many times.
Setting T to be very large or infinity, as in Cullen and Reynolds (2017) and Reynolds (2018),
is infeasible for this purpose due to the heavy computational burden caused by the large set of
choice/state variables and the extremely small time step (1 hour).

Instead, I choose T to be 48 hours in two days. The two days are exactly the same in terms of
hourly energy demand and hourly renewable energy output. That is, I construct a representative
day and replicate it twice. The construction of the representative day is detailed in section 4.4.
The initial "on" capacities for all fossil-fuel technology types at the beginning of the first day are
set to 0 (i.e. xj = 0, j ∈ {1,2,3,4}). I then use only the second day to calculate V(k) because the
second day is less affected by the initial adjustment of "on" capacities from 0. In a model with long
planning horizon, the initial adjustment periods’ effect on the final simulation results is very small
compared to the sum of the rest periods. As the model is only simulated for two days, incorporating
adjustment periods in calculating V(k) will overshoot the importance of this period. Indeed, the
duration of the initial adjustment periods is relatively short – hourly fossil-fuel energy generation
looks very similar on the two days after 4:00am. As an robustness check, I also extend T to be 72

145

hours in three consecutive representative days. Hourly operations in the second and third days are
almost the same.

Note that the formulation of social planner’s problem assumes away uncertainty – the social
planner knows perfectly wind generation, solar generation and demand shocks in each hour in the
future. This assumption is not too restrictive because the chosen T is relatively small.

4.3.2 Social Planner’s Investment Problem
Knowing V(kw, kso, ks, k f ) for all k, the social planner’s investment problem is to maximize:

30


τ=0

max

kw,kso,ks

W(kw, kso, ks, k f ) =

βτ365 ∗ V(kw, kso, ks, k f ) − θwkw − θsokso − θsks

(4.20)

where θw, θso and θs are the per megawatt capacity costs of wind turbine, solar PV and energy
storage, respectively. Since V(kw, kso, ks, k f ) is the value function from optimal capacity operation
in a single representative day, it is multiplied by 365 to obtain the annual value function. On
average, wind turbine and solar PV last for thirty years.12 Thus, the present value of the operation
value function in thirty years is calculated to be the benefit from capacity investment.

Through varying θso, the maximization problem (4.20) delivers optimal kw, kso and ks under
each level of solar capacity cost, based on which renewable energy penetration is calculated. Since
the focus of this study is the effects of solar cost decline on wind/storage investment and renewable
energy penetration, the levels of different fossil-fuel energy capacities are set to be fixed.

Given the optimal capacities k∗

w, k∗

so and k∗

s, the private surplus from renewable energy capacity

investment is defined as:

W(k∗

w, k∗

so, k∗

s, k f ) − W(0,0, k f )

(4.21)

Note that the surplus is private surplus, not social welfare, as the social damage caused by
carbon emissions are not taken into consideration when making investment decisions due to the
lack of carbon policies in the study region. As will be shown later in section 4.6, this is exactly the
reason why declining solar capacity costs might lead to social welfare loss.

12Data Source: NREL Energy Analysis: https://www.nrel.gov/analysis/tech-footprint.html

146

4.4 ERCOT Data and Parameter Calibration

The Electric Reliability Council of Texas (ERCOT) is an independent system operator for an
electricity grid that covers most of Texas. ERCOT serves about 90 percent of the state’s electricity
demand and is largely unconnected with the other U.S. electricity grids. Therefore, electricity
demand in ERCOT region is almost solely supplied by energy generation capacities located inside
its serving area.

Historically, fossil-fuel generation capacities such as natural gas and coal power generators
dominate energy supply in ERCOT. Up to 2015, total fossil-fuel generation capacity reaches
70,791 MW. All coal, combined cycle, natural gas steam and natural gas turbine energy generators
located in ERCOT with a nameplate capacity greater than 10MW are utilized to calibrate fossil-
fuel capacity cost parameters. The total capacities of these generators adds up to 65,622MW,
which accounts for 93% of total generation capacities in ERCOT in 2015. Table 4.3 summarizes
information on different fossil-fuel technology types used in the study.

Table 4.3: ERCOT Generation Technology Parameters (2015)

Generation Technology Capacity Minimum Generation mj Startup Cost gj
($/MW Cap)
Coal
Combined Cycle
Gas Steam
Gas Turbine
Total

(MW)
18,771
30,648
11,923
4,340
65,622

105
79
75
32

(% of Cap)

0.53
0.51
0.33
0.64

Startup costs of different generation technologies are calibrated from Kumar et al. (2012). I
obtain minimum generation level per unit of capacity for all generators in ERCOT market from
EIA form 860. Minimum generation for each type of generation technology is calculated as the
capacity-weighted average minimum generation level of power generators within this type.
In
general, coal power generator is the most inflexible technology characterized by a high startup cost
and relatively high minimum generation level, while generators using gas as fuel are more flexible.
Information on generation capacities are drawn from EIA form 860. I use summer capacity

147

instead of nameplate capacity to calculate total capacities as the former is a more realistic measure
of the maximum hourly generation rate a power generator could achieve. The grid is dominated
primarily by combined cycle and coal generation capacities, with a small amount of natural gas
turbine capacities primarily utilized during peak hours.

Construction of the aggregate production cost function C(qjt; xjt) for each type of generation
technology is more complex. The procedure is described in appendix J. In general, production
cost is lowest for coal power generators, followed by combined cycle generators, then by gas steam
generators and finally by gas turbine generators.

A representative day is constructed using hourly renewable generation data and electricity
quantity demanded (load) data in 2015. Hourly electricity demanded and electricity price in the
representative day are also constructed by averaging the observed hourly demand and prices across
all days in 2015. Given the linear demand function P(Qt, θt) = θt − bQt, the slope parameter
b is calibrated by setting elasticity of demand calculated at the pair of daily average price and
daily average load to be -0.05, following Cullen and Reynolds (2017). Subsequently, each θt,
for t ∈ {1,2, . . . ,24}, is fixed by solving the inverse demand function using the calibrated slope
parameter. Wind and solar energy output in each hour in the representative day are obtained in the
same fashion. Starting from the early 2000s, renewable energy capacities, especial wind capacity,
keep increasing rapidly. By the end of 2015, the amount of wind capacity reaches 14,683 MW.
Although the level of solar capacity is relatively low in 2015 (1,101 MW), it is expected to increase
rapidly in the future due to the constantly declining capacity costs.

Figure 4.3 plots ERCOT’s hourly average wind and solar generation capacity factors as well
as hourly average realized electricity demand in 2015. Consistent with the analytical model, wind
energy production peaks in the morning and late night but decreases during daytime.

Finally, parameters regarding capacity costs are calibrated. According to NREL renewable
energy capacity cost projection (Augustin et al., 2018), average capacity costs of wind stays
relatively unchanged from 2016 to 2050, at roughly about $1600/kW. However, average solar
capacity costs will drop rapidly from $1778/kW in 2016 to around $700/kw in 2050. Consistent

148

Figure 4.3: Average Renewable Energy Generation Pattern and Average Demand Pattern

with this projection I set θw = $1600/kW and θso ∈ $(700,1200)/kW, with increments of $10/kW,13
Currently, both wind and solar capacity expansions are subsidized in the U.S. The Renewable
Electricity Production Tax Credit (PTC) offers $24/MWh for wind energy production in the first
ten years after initial investment, while solar investment is subject to a 30% Investment Tax Credit
(ITC). I adjust the capacity costs accordingly by deducing these tax credits from the initial capacity
costs in the simulation. Finally, capacity cost of storage (θs) is fixed at $1500/kW, its 2018 level
(U.S. EIA, 2018).

4.5 Simulation Implementation and Model Validation

Before turning to the formal analysis, I test model validity by replicating observed hourly average
price patterns. For validation purpose, I obtain hourly average demand and renewable generation
for each season (i.e. spring, summer, fall and winter). This exercise gives four representative days
and the model is executed for four times, one for each season. The simulation is performed using
Artelys Knitro nonlinear optimization toolbox in Matlab.

13Shown by the simulation results latter, solar capacity is never developed when θso > $1090/kW. Thus, I adjust the
upper bound of θso down from $1244/kW to $1200/kW for the neatness of figures in the next section. This adjustment
will not affect the conclusions from the simulation exercises because optimal investment decisions are the same for
θso between $1200/kW and $1788/kW.

149

Note: average daily startups are 335MW in spring, 1739MW in summer, 641MW in Fall, and 19
MW in winter.

Figure 4.4: Observed and Simulated Hourly Average prices for Four Seasons

Figure 4.4 plots these validation results. Daily capacities started up is also reported for each
season below the figure. The simulation model predicts the observed hourly average price patterns
for the four seasons quite well. In particular, the summer price spike is caused by the relatively
large amount of peak hour startups during peak hours, showing the importance of modeling startup
costs explicitly.

For the ease of explaining results later in the paper, I still use the annual representative day
constructed in section 4.4, rather than the four seasonal representative days, in the formal simulation.
Results obtained by using the annual representative day and the seasonal representative days are
very similar.

4.6 Results And Discussions

This section presents the simulation results and discusses their implications. First, the empir-
ical simulation model in section 4.3 is executed with different values of wind, solar and energy
storage capacities in order to determine whether wind energy, solar energy and energy storage are
complements or substitutes in ERCOT. Then, based on the results in the first step, the effects of

150

solar capacity cost decline on the investment of other renewable resources and renewable energy
penetration are presented.

Specifically, 10000 combinations of wind, solar and energy storage capacities are randomly
drawn to run the dynamic operation simulation model.
I limit the range of wind capacity to
be between 15000MW and 75000MW, the range of solar capacity to be between 300MW and
35000MW, and the range of energy storage capacity to be between 300MW and 15000MW. The
lower bounds of wind and solar energy capacities are set according to the 2015 installed capacities
of wind and solar. The lower bound of storage capacity is set according to the 2020 energy storage
capacity projection.14 The upper bounds of the capacities are set high enough to enclose a variety
of possible renewable energy penetration levels in the future. With 75000MW of wind capacity,
35000MW of solar capacity and 15000MW of energy storage, renewable energy could serve up to
85% of ERCOT electricity demand on average across a whole day.15

A third order polynomial regression of the optimized V(kw, kso, ks) is fitted with respect to
wind capacity, solar capacity and energy storage. The partial derivative of the fitted V(kw, kso, ks)
with respect to wind (solar/storage) capacity represents the value of wind (solar/storage) energy
(Vki,i ∈ {w, so, s}).16 The regression results are omitted here but the overall fitting of the regression
is exceptional, with R2 = 0.9999.

After each Vki

for i ∈ {w, so, s} is obtained, the relationships between one type of capacity and
at different levels of
increases as k j increases, type i and type j capacities are complements. Otherwise,

the other types of capacities are characterized by comparing the value of Vki
k j, j (cid:44) i. If Vki
they are substitutes.

14There were almost no storage capacity in ERCOT by the end of 2015.
15With theses levels of renewable/storage capacities, the hourly percentage of electricity demand met by renewable

energies and energy storage ranges from 70% to 100%.

16Ideally, for each of the 10000 capacity combinations, I could calculate the value of one specific renewable energy
through increasing its capacity level by a small amount. The difference between the initial optimized social value and
the new value is the value of that energy source. However, this will greatly increase the computational burden. As
there are three renewable sources, 3 × 2 × 10000 = 60000 simulations are needed, comparing to only 10000 runs for
the selected method.

151

4.6.1 Values of Wind, Solar and Energy Storage

Before showing the results related to value dependence, the simulated values of wind, solar and
storage are compared with the estimates found in the literature as another round of robustness
check. Based on ERCOT energy generation capacity levels in 2015, the simulated values of wind,
solar and storage are $23.7/MWh, $25.4/MWh, and $20/MWh, respectively.17

Callaway, Fowlie, and McCormick (2018) estimates that the avoided operation cost associated
with wind energy production is $30/MWh in ERCOT.18 In a simulation model that does not con-
sider startup cost, Castro (2018) estimates that the value of wind is $59/MWh in ERCOT in 2015,
considering both avoided operation cost and the benefit of offsetting carbon emissions. In the liter-
ature, the value of carbon offsetting associated with wind energy production in ERCOT ranges from
$19/MWh-$25/MWh (Callaway, Fowlie, and McCormick, 2018; Cullen, 2013). Subtracting the
value of carbon offset, the value of avoided operation cost associated with wind energy production
becomes $35/MWh-$40/MWh in Castro (2018).

Wind energy’s value obtained in this study is lower than those obtained in Callaway, Fowlie,
and McCormick (2018) and Castro (2018). Part of the reason is that startup costs are ignored in
these studies. Because fossil-fuel energy capacities’ output adjustments incurs high startup costs,
ignoring startup costs tends to overestimate the value of wind energy.

Callaway, Fowlie, and McCormick (2018) estimates that the value of solar energy is $41/MWh
during 2010-2012. Solar energy is only produced during daytime. Under current solar capacity
levels, daytime hours are periods when more expensive natural gas generators set price. Dur-
ing 2010-2012, average natural gas price is 1.5 times that in 2015. Thus, the value of solar
energy estimated in Callaway, Fowlie, and McCormick (2018) is approximately equivalent to
$41MW h/1.5 = $27.3/MWh in 2015, which is similar to the estimate in this study.

production during the representative day.

by resource i’s total energy

Finally, the value of storage (without considering social cost of carbon) is estimated to be
17The initial unit of Vki
18Only the values associated with avoided operation cost in other studies are considered here because these are the
values that are directly related to investment decisions without any carbon policies. The values associated with carbon
emission offsets are re-added later in this section to calculate the full social surplus from renewable energy investment.

is $/MW-day. It is converted to $/MWh though dividing Vki

152

$42/MWh in ERCOT during 2007 to 2009 in Carson and Novan (2013). The value of storage
largely depends on arbitraging price difference between low-price and high-price periods, which
is again largely driven by natural gas price. During 2007-2009, average natural gas price is about
2.4 times higher than that in 2015. Thus, the private value of energy storage estimated in Carson
and Novan (2013) is approximately equivalent to $42MWh/2.4=$17.5/MWh in 2015, which is also
similar to the estimate in this study.

4.6.2 Relationships Between Wind Energy, Solar Energy and Energy Storage Capacities

Figure 4.5 plots the value of wind energy capacity against different levels of solar energy capacities.
For each level of solar capacity, wind capacity is set to be either 20000MW (low) or 50000MW
(high). Similarly, storage capacity is set to be either 300MW (low) or 10000MW (high). As solar

Figure 4.5: Effect of Solar Energy On the Value of Wind Energy

energy capacity increases, the value of wind energy capacity first increases and then decreases
under all combinations of wind and storage capacities. Thus, wind energy and solar energy are
complements when the level of solar capacity is relatively low, and gradually become substitutes
as the level of solar capacity increases. This result is consistent with proposition 1. Moreover,
wind and solar complement each over over a wider range of solar capacity levels when the level

153

of wind capacity is high. This observation is consistent with proposition 4, which predicts that
the threshold of solar capacity above with solar starts to substitute wind is higher when more wind
capacity presents in the grid.

The complementary/substitution relationship between wind energy and energy storage is also
illustrated by figure 4.5 through comparing the two solid lines (or the two dashed lines). Holding
wind and solar capacities constant, the value of wind is always higher when more storage capacity
presents, which is consistent with the prediction in proposition 3 that wind energy and storage are
never substitutes.

Figure 4.6 plots the value of energy storage capacity against different levels of solar energy
capacities. Again, wind capacity is set to be either 20000MW (low) or 50000MW (high), and
storage capacity is set to be either 500MW (low) or 10000MW (high).

Figure 4.6: Effect of Solar Energy On the Value of Energy Storage

Consistent with proposition 2, the value of energy storage first decreases and then increases
as solar capacity continues increasing. This relationship holds under all wind/storage capacity
combinations being considered.

154

4.6.3 Declining Solar Capacity Costs, Increasing Renewable Penetration?

Given the social planner’s investment problem defined in (4.20) as well as investment cost param-
eters assumed in section 4.4, optimal wind and solar capacity levels are solved out over the range
of solar capacity costs. The results are plotted in figure 4.7.

Note: when solving the investment problem, wind capacity is constrained to be no lower than
15000MW, its observed 2015 level. Wind turbines could last for a long time period (30 years).
Once built, it is almost irrational for wind capacity to exit due to the extremely low production cost.
Similarly, solar capacity is constraint to be no lower than its observed 2015 level (1101MW).

Figure 4.7: Optimal Wind and Solar Capacities Under Different Solar Capacity Cost

In the figure, each star marker represents a combination of optimal wind capacity and optimal
solar capacity under a specific level of solar capacity cost. The solid long arrow signifies the
direction of solar cost change. The figure also includes a dashed line that represents the boundary
above which wind and solar capacities become substitutes. Consistent with proposition 4, the
boundary line is higher when the level of wind capacity is higher.

When the capacity costs of solar are very high, it is not optimal for investors to develop solar

155

capacity, representing an "all wind" solution. As solar costs gradually decrease, more solar is added.
At first, the increase in solar capacity also drives up wind capacity because solar complements wind
when solar capacity is relatively low. However, as solar costs keep decreasing, optimal wind and
solar capacity combination passes the dashed boundary line and solar starts to substitute wind. This
happens when pre-subsidy solar capacity costs drop roughly to $1040/kW, about $200/kW higher
than the 2030 projection. As a result, solar cost decline not only pushes up optimal solar capacity,
but also drives down optimal wind capacity. At the projected 2050 pre-subsidy solar capacity costs,
it is optimal to invest only in solar capacity, reaching an "all solar" solution. Further, due to the
high capacity costs and relatively low per megawatt value, storage capacity is never deployed in
equilibrium under all solar cost scenarios.

Figure 4.8 plots the effects of such wind/solar mixture changes on renewable energy penetration
and total carbon emissions. The star shaped markers represent renewable energy penetrations
under different solar cost levels, and the triangle shaped markers represent the corresponding
annual carbon emissions.

As solar capacity costs gradually decrease (from right to the left in the figure), both solar capacity
and wind capacity increase first, driving up renewable energy penetration. Since renewable energy
production is carbon-free, carbon emissions also drop correspondingly. When solar capacity costs
drop below the point where wind and solar become substitutes, renewable energy penetration still
increases for a while because the drop in wind energies is more than offset by the increase in solar
energies (see figure 4.7). However, when solar costs drop below $880/kW, just a little bit higher
than the 2030 projection, renewable energy penetration starts to decrease and carbon emissions
start to increase due to the fast decline in wind capacity (again, see figure 4.7). After solar costs
drop to low enough and "all solar" solution emerges, further decline in solar costs then drives up
renewable penetration through increasing solar capacity, but the range of solar costs when such
changes happen is small.

To summarize, renewable energy penetration first goes up from 22% to 36% when solar costs
decline from the 2016 projection to the 2030 projection, and then goes down to 30% as solar costs

156

Figure 4.8: Response of Renewable Energy Penetration And Annual Carbon Emissions to Solar
Capacity Cost Decline

keep declining to its the 2050 projection. Correspondingly, carbon emissions first decrease by 22%,
but then increase by 12.9% from the minimum point.

Changes in renewable energy penetration and carbon emissions due to solar capacity cost
decline translate directly into changes in social welfare from renewable energy investment, which
is the private surplus from renewable investment defined by (4.21), net of social damage caused by
carbon emissions. Figure 4.9 plots this change, assuming a social cost of $38/ton of CO2 (Callaway,
Fowlie, and McCormick, 2018; U.S. Interagency Working Group on Social Cost of Carbon, 2014).
Consistent with the changes in carbon emissions (figure 4.8), social welfare from renewable
energy investment first increases, then decreases, and then increases as the capacity costs of solar
decline. The highest social welfare is reached when solar capacity costs decline to around $820/kW,
the 2030 projection. Social welfare from renewable energy investment drops by as high as 17.2%,
or $0.57 billion/year in dollars, when solar capacity costs decrease from the 2030 projection to the
2050 projection. This represents an unexpected consequence associated with solar capacity cost

157

Figure 4.9: Changes in Social Welfare Gain From Renewable Energy Investment Due to Solar
Capacity Cost Decline

decline due to the interaction of solar capacity and wind capacity.

An ideal policy to counter this bad unexpected consequence is to impose a carbon tax that
is equal to the social cost of carbon to internalize the environmental externalities when making
renewable energy investment decisions. Without a carbon tax, setting stricter RPS renewable energy
penetration thresholds could serve as a second-best policy. For example, when it is required that
renewable energy penetration could not drop below 36%, the highest penetration level achievable
shown in figure 4.8, social welfare from renewable energy investment could increase by 27%, or
$0.77 billion/year in dollars when the capacity costs of solar drop to the 2050 projection.

4.6.4 Effect Of Solar Capacity Cost Decline on The Value of Energy Storage

Although it is not optimal to invest in energy storage due to its high investment costs, the value of
energy storage does increase at a fast rate in general as solar capacity costs decline (figure 4.10).

158

Figure 4.10: Response of Energy Storage’s Value With Respect To Solar Capacity Cost Decline

When solar costs decline, there is a slight decrease in the value of energy storage when the
costs are still high. According to proposition 2 and figure 4.6, solar substitutes storage at low solar
capacity levels. Thus, an increase in solar capacity caused by solar cost decline decreases the value
of storage when solar costs are high. However, solar starts to complement storage and the value
of storage continues going up as more and more solar capacity is added due to solar cost decline.
When pre-subsidy solar capacity costs decrease from $1200/kW to around $700/kW (the 2050
projection), the private value of energy storage (evaluated at 324MW, the projected level in 2020)
increases by $23.9/MWh, representing a 92.5% increase.

Therefore, it is reasonable to expect that more storage capacity would be added to the grid if
solar capacity cost decline continuous after 2050, and if storage capacity costs could be driven down
due to future technological improvements. As storage complements solar at high solar capacity
levels, the presence of storage capacity by the time when solar capacity costs are very low is very
likely to increase renewable penetration and to drive carbon emissions down.

159

4.7 Conclusions and Future Directions

Development of clean renewable energies such as wind and solar is partly limited by their high
capacity costs. A recent report by the National Renewable Energy Laboratory (NREL), however,
projects that solar capacity costs will drop dramatically in the next thirty years. This chapter
investigates how this cost decline affects renewable energy penetration and social welfare.

Changes in renewable energy penetration depends on how the capacities of wind, solar and
storage are adjusted facing sharp solar cost decline, which in turn depends on whether these
resources complement or substitute each other. I show that wind and solar substitute each other
on the one hand because energies produced from different resources are perfect substitutes and so
compete with each other. On the other hand, mixing wind and solar resources helps smoothing
aggregate renewable energy output and reducing fossil-fuel energy generators’ startup costs. I show
in an analytical model that the complementarity effect is stronger than the substitution effect if the
level of solar capacity is relatively low, but the strengths of the two effects reverse and solar starts
to substitute wind as solar capacity gradually increases. Moreover, while storage never substitutes
wind, it only complements solar at high solar capacity levels.

Two simulation models are then combined to empirically determine the effects of solar cost
decline on renewable energy penetration and social welfare in the ERCOT electricity grid. The
first is a dynamic grid-level power plant operation simulation model that takes fossil-fuel energy
capacities’ startup costs into consideration, and the second is a stylized renewable energy investment
model. Because wind and solar switches from being complements to being substitutes as solar
capacity increases, simulation results show that the decline in solar capacity costs first increases
renewable energy penetration through increasing both solar and wind capacities, but then decreases
the penetration through decreasing wind capacities when solar capacity increases. Specifically,
renewable energy penetration first goes up from 22% to 36% when solar costs decline from the
2016 level to the 2030 projection, and then goes down to 30% as solar costs keep declining to the
2050 projection. Correspondingly, carbon emissions first decrease by 22% but then increase by
12.9% from the minimum point.

160

Due to the lack of carbon policies, social welfare from renewable energy investment drops by
$0.57 billion/year when solar costs decrease from the 2030 projection to the 2050 projection. Thus,
a decline in solar cost does not necessarily imply an increase in renewable penetration and social
welfare. I show that a policy that requires a minimum renewable energy penetration level would
lead to large increases in social welfare when solar costs are low.

As storage complements solar at high solar capacity levels, the presence of storage capacity
by the time when solar capacity costs are very low has the potential to further increase renewable
penetration and is likely to drive carbon emissions down. Even though it is not optimal to invest in
storage currently, the value of storage increases at a fast rate as solar costs decline. A 93% increase
in the value of storage is observed as solar costs decrease from the 2016 level to the 2050 projection.
Therefore, it is reasonable to expect storage to become competitive if solar capacity cost decline
continuous after 2050, and if storage capacity costs could be driven down in the future.

There are several directions to extend the analyses in this chapter. First, fossil-fuel capacity
would also vary in the long run in response to renewable energy capacities’ relative cost advantages.
A model that also takes fossil-fuel energy generators’ enter/exit decisions into consideration will
be a useful step in the future. Second, the investment decisions are also dynamic in nature.
Real-world renewable resource developments are characterized by slow year-by-year adjustments:
both project approval and site construction take considerable time. Further, investors update their
expectations of future capacity costs by observing current and past cost realizations. Therefore,
a dynamic investment model, rather than the static one used here, that takes full consideration of
these dynamic and uncertainty factors would provide more insights into how different renewable
energy resources gradually adjust their capacities (as well as how renewable penetration gradually
changes) overtime facing solar cost decline.

161

APPENDICES

162

APPENDIX I

DERIVATION DETAILS IN THE ANALYTICAL MODEL

I.1 Derivation – Value of Renewable Energies and Energy Storage

Low Solar Scenarios (q∗

1 < q∗

2 = q∗
3)

Given the ordering of optimal fossil-fuel energy generation q∗

3, as well as social
planner’s problem defined in (4.3) and (4.4), the first order conditions (FOCs) with respect to q1 is:

1 < q∗

2 = q∗

p1(Q∗

1) − C(cid:48)

P(q∗

1) + C(cid:48)

r(q∗

2 − q∗

1) = 0

(I.1)

The necessary conditions for q2 and q3 are different from that for q1 due to the kink in output
adjustment function ˜Cr(q3 − q2) at q3 − q2 = 0. Specifically, the necessary condition for period 3
fossil-fuel energy generation is:

0 < p3(Q3|q3 = q∗

2) − C(cid:48)

P(q3 = q∗

2) < C(cid:48)

r(0).

(I.2)

(I.2) has two implications: (i) it is profitable to increase q3 when there is no ramping cost.
Otherwise, downward ramping of fossil-fuel energy production would be observed since downward-
ramping is costless. (ii) marginal cost of ramping up fossil-fuel power production in period 3 must
be higher than its marginal benefit, evaluated at the point where no ramping happens.
P(·), the inequalities above still hold if q∗
2. Thus, for all q2 in the neighborhood of q∗

Due to the continuity of pt(·) and C(cid:48)
changed and q3 is set equal to the new q∗
is always equal to q2 (i.e.
fossil-fuel power production in period 2 is set to q2). Therefore, ˜Cr( ˆq∗
neighborhood of q∗
FOC associated with q2 is:

2 is marginally
2, optimal q3
3 is optimal period 3 fossil-fuel power production if
3 − q2) = 0 for all q2 in the
2) = 0 in this neighborhood. Consequently, the

2, which implies that ˜C(cid:48)

ˆq∗
3 = q2, where ˆq∗

3 − q∗

r( ˆq∗

p2(Q∗

2) + p3(Q3|q∗

3 = q∗

2) − 2C(cid:48)

P(q∗

2) − C(cid:48)

r(q∗

2 − q∗

1) = 0

(I.3)

163

Plug q∗

1, q∗

2 and q∗

3 into (4.3), we get the optimized value function V(kw, kso, ks). Take partial
, the

∂αw

=

t kso+∂s∗
t kw+∂αso
∂ki

t

derivative of V(kw, kso, ks) with respect to ki and denote ei
value of type i ∈ {w, so, s} capacity is:

t =

∂E∗
t
∂ki

t=1

3

3

3


t=1

=

Vki

=

=

[pt(Q∗

[pt(Q∗

[pt(Q∗

t )(ei
t +

t )(ei
t +

∂q∗
t
∂ki
∂q∗
t
∂ki
∂q∗
t )(ei
t
t +
∂ki
1 + p2(Q∗
2)ei

t
∂ki

) − C(cid:48)

) − C(cid:48)

t ) ∂q∗
P(q∗
t ) ∂q∗
P(q∗
t ) ∂q∗
P(q∗
t
∂ki
2 + p3(Q∗
3)ei
3

) − C(cid:48)

t
∂ki

t=1
= p1(Q∗

1)ei

] − ∂ ˜Cr(q∗
] − ∂Cr(q∗

2 − q∗
1)
∂ki
2 − q∗
1)
∂ki
1) ∂q∗
r(q∗
2 − q∗

] + C(cid:48)

1
∂ki

− ∂ ˜Cr(q∗

3 − q∗
2)
∂ki

.

(I.4)

− C(cid:48)

r(q∗

2 − q∗

1) ∂q∗

2
∂ki

Again, marginally changing ki will only deliver a new optimal q2 in the neighborhood of q∗
2.
3 − q2) = 0 still holds for each q2
= 0 in the second line. Finally, the last line is

Thus, the inequalities in (I.2) would not be violated and ˜Cr( ˆq∗
3−q∗
2. Therefore, ∂ ˜Cr(q∗
2)
in the neighborhood of q∗
∂ki
obtained by plugging (I.1) and (I.3) into Vki
.

The values of ei
t

depends critically on how storage operation changes when ki is increased. Since
are constants. However,

and ∂αso
t kso
∂ki

wind and solar energy production are exogenous, ∂αw
t kw
∂ki
∂s∗
t
∂ki
social planner. The optimal st is shown in table I.1.

may vary across different scenarios because storage production is also a control variable of the

Table I.1: Optimal Energy Storage Production Under Each Scenario

s∗
1
L1 −ks
L2 −ks
L3 −ks
0
H

a2−a3−bαso2 kso+bks

s∗
2
ks

2b
0
−ks

a3−a2+bαso2 kso+bks

s∗
3
0

2b
ks
ks

Under scenario L2, storage discharge spreads to two periods (period two and three) since
p2 = p3, which translates into the following optimality condition for storage production: a2 −

164

2 + αw2 kw + αsokso + s∗

b(q∗
table I.1 by imposing αw2 = αw3 .

2) = a3 − b(q∗

3 + αw3 kw − s∗

2), which delivers the value of αs∗

2 shown in

Finally, plug s∗

t in table I.1 into (I.4) and impose p2(Q∗

2) = p3(Q∗

3) for scenario L2, the expression

of Vki

for i ∈ {k, so, s} under all low solar scenarios could be re-written as:

= p1(Q∗

1)αi

1 + p2(Q∗

2)αi

2 + p3(Q∗

3)αi
3

Vki

(I.5)

is defined as the capacity factor of storage in period t.

where αs

t =

∂s∗
t
∂ks

The derivation under the medium solar scenario (omitted here) is similar to that under the

low-medium solar scenario (i.e. L2). Vki
High Solar Scenario (q∗

2 < q∗
3)

1 > q∗

under this scenario is exactly the same as that in (I.4).

Given this ordering of optimal fossil-fuel energy generation (q∗

1 > q∗

2 < q∗

3), the first order

conditions (FOCs) with respect to q1, q2 and q3 are:



p1(Q∗
p2(Q∗
p3(Q∗

1) − C(cid:48)
2) − C(cid:48)
3) − C(cid:48)

P(q∗
P(q∗
P(q∗

1) = 0
2) + C(cid:48)
3) − C(cid:48)

r(q∗
r(q∗

3 − q∗
3 − q∗

2) = 0
2) = 0

(I.6)

Similar to the low solar scenarios above, the value of type i ∈ {w, so, s} capacity is:

] − ∂Cr(q∗

3 − q∗
2)
∂ki
2) ∂q∗
3 − q∗
r(q∗

2
∂ki

] + C(cid:48)

− C(cid:48)

r(q∗

3 − q∗

2) ∂q∗

3
∂ki

.

(I.7)

3

3


t=1

=

Vki

=

[pt(Q∗

[pt(Q∗

t )(αi
t +

∂q∗
t
∂ki
∂q∗
t )(αi
t
t +
∂ki
1 + p2(Q∗
2)αi

t
∂ki

) − C(cid:48)

t ) ∂q∗
P(q∗
t ) ∂q∗
P(q∗
t
∂ki
2 + p3(Q∗
3)αi
3

) − C(cid:48)

t=1
= p1(Q∗

1)αi

I.2 Derivation: Theorem 1
2 = q∗
3)

Low Solar Scenarios (q∗

1 < q∗

165

−b − C(cid:48)(cid:48)





 = eL

t

dq∗L1
dEt
dq∗L2
dEt

Totally differentiating the FOCs specified by (I.1) and (I.3) with respect to q∗L1 , q∗L2 and Et, we

get:

r

P − C(cid:48)(cid:48)
C(cid:48)(cid:48)
r

C(cid:48)(cid:48)
r
P − C(cid:48)(cid:48)
−2b − 2C(cid:48)(cid:48)
where eL1 = [b 0](cid:48), eL2 = [0
b](cid:48) and eL3 = [0
b](cid:48). Solving the equation system above by
Cramer’s Rule for each t = 1,2,3 delivers the upper panel of table I.2. Note that due to the property
of Case I discussed in section I.1, dq∗L2

(I.8)

for all t = 1,2,3.

=

r

dq∗L3
dEt

dEt

Table I.2: Effect of Changing Exogenous Energy Supply in One Period On Electricity Output
Changes in All Periods

(a) Low Solar Scenarios

dE1

dE2

dE3

dq∗L1
dq∗L2
−b(2b+2C(cid:48)(cid:48)
P+C(cid:48)(cid:48)
−bC(cid:48)(cid:48)
r )
r
P+3C(cid:48)(cid:48)
P)(2b+2C(cid:48)(cid:48)
P+3C(cid:48)(cid:48)
P)(2b+2C(cid:48)(cid:48)
r )
r )
P+C(cid:48)(cid:48)
−b(b+C(cid:48)(cid:48)
−bC(cid:48)(cid:48)
r )
r
P+3C(cid:48)(cid:48)
P)(2b+2C(cid:48)(cid:48)
P+3C(cid:48)(cid:48)
P)(2b+2C(cid:48)(cid:48)
r )
r )
−b(b+C(cid:48)(cid:48)
P+C(cid:48)(cid:48)
−bC(cid:48)(cid:48)
r )
r
P)(2b+2C(cid:48)(cid:48)
P+3C(cid:48)(cid:48)
P)(2b+2C(cid:48)(cid:48)
P+3C(cid:48)(cid:48)
r )
r )
(b) High Solar Scenario

(b+C(cid:48)(cid:48)
(b+C(cid:48)(cid:48)
(b+C(cid:48)(cid:48)

(b+C(cid:48)(cid:48)
(b+C(cid:48)(cid:48)
(b+C(cid:48)(cid:48)

dq∗L3
−bC(cid:48)(cid:48)
r
P+3C(cid:48)(cid:48)
P)(2b+2C(cid:48)(cid:48)
r )
P+C(cid:48)(cid:48)
−b(b+C(cid:48)(cid:48)
r )
P+3C(cid:48)(cid:48)
P)(2b+2C(cid:48)(cid:48)
r )
−b(b+C(cid:48)(cid:48)
P+C(cid:48)(cid:48)
r )
P)(2b+2C(cid:48)(cid:48)
P+3C(cid:48)(cid:48)
r )

(b+C(cid:48)(cid:48)
(b+C(cid:48)(cid:48)
(b+C(cid:48)(cid:48)

dq∗H1
C(cid:48)(cid:48)
P
b+C(cid:48)(cid:48)
P
0

0

dE1

dE2

dE3

dq∗H2
0
P+C(cid:48)(cid:48)
−b(b+C(cid:48)(cid:48)
r )
P)(b+C(cid:48)(cid:48)
P+2C(cid:48)(cid:48)
r )
−bC(cid:48)(cid:48)
r
P)(b+C(cid:48)(cid:48)
P+2C(cid:48)(cid:48)
r )

(b+C(cid:48)(cid:48)
(b+C(cid:48)(cid:48)

dq∗H3
0
−bC(cid:48)(cid:48)
r
P)(b+C(cid:48)(cid:48)
P+2C(cid:48)(cid:48)
r )
P+C(cid:48)(cid:48)
−b(b+C(cid:48)(cid:48)
r )
P)(b+C(cid:48)(cid:48)
P+2C(cid:48)(cid:48)
r )

(b+C(cid:48)(cid:48)
(b+C(cid:48)(cid:48)

Note: the "L" and "H" subscripts index low solar and high solar scenarios.

High Solar Scenario (q∗

1 > q∗

2 < q∗
3)

Totally differentiating the FOCs specified by (I.6) with respect to q∗H1 , q∗H2 , q∗H3

and Et,

166



t ∈ {1,2,3}:

−b − C(cid:48)(cid:48)
P

r

0
0

0
0
C(cid:48)(cid:48)
−b − C(cid:48)(cid:48)
P − C(cid:48)(cid:48)
r
C(cid:48)(cid:48)
P − C(cid:48)(cid:48)
−b − C(cid:48)(cid:48)
r
b 0](cid:48) and eH3 = [0
where eH1 = [b 0
0
above for each t ∈ {1,2,3} delivers the lower panel of table I.2.
I.3 Derivation – Boundaries of Scenarios L2/L3 and L3/M

0](cid:48), eH2 = [0

r





 = eH

t

dq∗H1
dEt
dq∗H2
dEt
dq∗H3
dEt
b](cid:48). Solving the equation system

(I.9)

From table 4.2, it is clear that the left boundary of scenario L3 defines the boundary between
scenarios L2 and L3, and its right boundary defines the boundary between scenarios L3 and M.
Therefore, finding out the boundaries defining scenario L3 is enough to define the two turning
points that we are interested in. The inequalities that defines scenario L3 are:



∗L3
2
∗L3
1

)
)

p3(Q
p2(Q
∗L3
q
2

∗L3
3
∗L3
2

) > p2(Q
) > p1(Q
∗L3
> q
1

(I.10)

∗L3
From the FOCs defined in (I.1) and (I.3), q
1

∗L3
and q
2

∗L3
wind/solar/storage capacities, cost parameters as well as demand parameters. Plug q
1
into the inequalities above, (I.10) is reformulated as:

could be solved out as functions of
∗L3
and q
2



kso >

kso <

kso <

a2−a3
(a2−a3)b+(a2−a1)(4c2+6γ2)+3bγ1

+ ks = k A
so

b

a2+a3−2a1−3γ1

b

4bc2+6bγ2+b2

− 4bc2+6bγ2−b2
4bc2+6bγ2+b2 ks +
− 3ks + 2kw( ¯αw − αw) = k B1
so ). Scenario L1 or L2 emerges if kso ∈ [0, k A
so, ∞].

so

so]. Finally, scenario M or H emerges if kso ∈ [k B

Define k B
kso ∈ [k A

so = min(k B1
so, k B

so , k B2

( ¯αw−αw)
4bc2+6bγ2+b2 kw = k B2

so

(I.11)

so]. Scenario L3 emerges if

167

Further, it is straightforward to show that k A

so and k B2

both k B1
and decreasing in ks.

so are increasing in kw and decreasing in ks. Thus, K B

so is increasing in ks. Assuming 4c2 + 6γ2 > b,
so is also increasing in kw

168

APPENDIX J

AGGREGATE PRODUCTION COST CURVE FOR EACH GENERATION

TECHNOLOGY

In this section, I show how to construct C(qjt; xjt) for type j generation capacities from each
generator’s marginal production cost. I calibrate generator level marginal production cost as:

Ci j = P f i j ∗ HRi j + PSO2 ∗ SO2Ratei j + PNOX ∗ NOX Ratei j + VOMj,

where HRi is heat rate of generator i, which is the amount of heat input (measured in mmbtu)
required to generate 1 MW of electricity. I obtain generator level heat rate from EGRID database.
P f i j is fuel price measured in $/mmbtu, which is obtained from EIA. SO2Ratei j and NOX Ratei j
SO2 and NOX emission per MW of electricity generated, measured in tons. This information is also
obtained from EGRID. Finally, VOMj is technology-specific Variable Operations and Maintenance
cost calibrated from Cullen and Reynolds (2017).

The difficulty of constructing aggregate production cost function for each generation technology
comes from its dependence on the amount of capacities being kept on at the beginning of year t,
xjt. It is straightforward to show that the aggregate production cost when aggregate output is at the
minimum generation level of “on" capacities, C(mj xjt; xjt), is piecewise linear and convex in mj xjt.
The slope of this curve increases at the points of mj x1, mj(x1 + x2), . . . , mj(x1 + x2 +· · · + xnj). This
is a direct implication of the result mentioned in the main text that only the least-cost generators
within a generation type would be kept on.

For a given level of xjt, C(qjt > mj xjt; xjt) is also piecewise linear and convex in qjt. The slope
of this curve increases at points where the closest lower-cost generation capacities are exhausted.
To make the argument concrete, consider a simple case where the marginal production costs for
type j generation technology fall into three classes (nj=3): c1 < c2 < c3. The total capacities of
generators within each class are k1 = 200, k2 = 60 and k3 = 100. Further, the minimum generation
level per unit of capacity is mj = 0.5.

169

Figure J.1 plots the aggregate production cost function with different levels of “on" capacities.
The line with arrows represents production cost when producing at the aggregate minimum gener-
ation level of “on" capacities. The bold solid lines are production cost when producing above the
aggregate minimum generation level.

Figure J.1: Aggregate Production Cost in the Simple Example

In practice, I fit quadratic regression lines to C(qjt > mj xjt; xjt = k j) curve (i.e. the uppermost
curve in the figure) and C(mj xjt; xjt) curve. To adjust “on" capacity level, I adjust estimated
C(qjt > mj xi jt; xjt = k j) along the esimated C(mj xi jt; xi jt) curve. The resulting aggregate
production cost curve C(qjt; xi jt) is convex in xjt and qjt.

170

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CHAPTER 5

CONCLUSIONS

New technologies in the sectors of food, water and energy are cited as mitigation and adaptation
strategies facing climate change. However, the use of these new technologies may not lead to the
expected consequences that these technologies are designed to deliver. In food and water sectors, I
find that the use of more efficient irrigation technologies in Kansas HPA increases, rather decreases,
groundwater extraction. In the energy sector, further decline in solar capacity costs decreases wind
energy capacity investment when solar cost is already low due to the complex value dependence of
wind and solar capacities. As a result, renewable energy penetration decreases and CO2 emissions
increases. Somewhat surprisingly, cheaper renewable energy costs not always lead to a cleaner
grid.

Economists have long recognized the limitations of purely technological solutions to environ-
mental problems, and argued that resilient and effective institutions and policies can be effective
adaptation strategies. Combining the use of new technologies with properly-designed environmen-
tal policies and institutions, thereby, might lead to significant social welfare increase. For example,
the rebound effects following efficient irrigation technology adoption are found to be greater for
farmers with larger water rights. Reducing water rights by 10% is found to be able to reduce
the rebound effects by 15.4%. After rebound effects are controlled, an insurance policy which
subsidizes farmers when the new technology’ annual profit gain is lower than a minimum level
would reduce farmers’ uncertainty and increase adoption rate. Such an insurance policy is found
to be more cost-effective than the more commonly used cost share subsidies.

As for the renewable energy example, a policy that mandates a 36% renewable energy pene-
tration when solar capacity costs are low increases wind capacity investment, compared with the
no-policy scenario. Consequently, social welfare increases by $0.77 billion/year due to an increase
in renewable energy penetration and a corresponding decrease in CO2 emissions.

Most analyses in this dissertation take partial-equilibrium approaches such that policy/technology

174

interactions are studied separately for each sector (except in chapter two). Therefore, the results in
this dissertation are best interpreted as "own-sector" welfare effects of policy/technology interac-
tions in the food-energy-water nexus. Indeed, food, water and energy sectors are closely related to
each other. Therefore, I also provide a brief discussion below on the connections among different
sectors to guide future researches on food-energy-water nexus.

Shown in chapter two, a part of LEPA irrigation’s rebound effects comes from switching to
more water-intensive crops such as corn. From the perspective of food-water nexus, policies or
technology adoptions in food and agricultural sectors that reduce water use by corn growers could
also help ameliorate the rebound effects of LEPA. Moreover, LEPA adoption might also reduce
energy usage because the energy requirements of LEPA irrigation are much lower than those of
traditional irrigation technologies. Therefore, policies that promote LEPA adoption are also likely
to reduce energy use.

Renewable energy development is another bridge between energy and water sectors. Unlike
fossil-fuel energy generators, renewable energy resources such as wind and solar energy generators
not only emit zero air pollutants, but also requires much less water for cooling. As renewable energy
capacity costs continue declining, energies produced from renewable resources would gradually
replace those from fossil-fuel energy generators, which reduces water use in the energy sector.
Besides bridging energy and water sectors, renewable energy development also connects energy
and food sectors. Land requirements for wind and solar power plants are much larger than those
for fossil-fuel energy generators. A large share of these land requirements is met by converting
farmland into solar/wind farms, which reduces food production. Policies aiming at increasing
renewable energy penetration would thus also influence both regional water balance and food
production structures.

Future researches should aim to quantify these cross-sector effects of environmental policies
and institutions. These results, combined with the own-sector effects obtained in this dissertation,
would be able to provide comprehensive empirical support for the design of effective environmental
policies and institutions that complement technology improvements in the food-energy-water nexus.

175