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dissertation entitled

ESSAYS ON LAND USE DECISIONS FOR ENERGY CROP
PRODUCTION AND THE EFFECTS OF SUBSIDIES
UNDER UNCERTAINTY AND COSTLY REVERSIBILITY

presented by

FENG SONG

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ESSAYS ON LAND USE DECISIONS FOR ENERGY CROP PRODUCTION AND
THE EFFECTS OF SUBSIDIES UNDER UNCERTAINTY AND COSTLY
REVERSIBILITY
By

Feng Song

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Agricultural, Food and Resource Economics

2010

ABSTRACT

ESSAYS ON LAND USE DECISIONS FOR ENERGY CROP PRODUCTION AND
THE EFFECTS OF SUBSIDIES UNDER UNCERTAINTY AND COSTLY
REVERSIBILITY
By
F eng Song

The US. Energy Independence and Security Act of 2007 mandates blending into
transportation fuels of 16 billion gallons of cellulosic fuels annually by 2022. Dedicated
energy crops are being explored to provide more efficient and environmental fiiendly
feedstocks for cellulosic biofuel production. Devoting land to energy crops represents a
long-term commitment, involves adjustment costs and great uncertainties. This research
develops a dynamic land conversion model to take into account these factors. The model
is applied to address two separate but related questions: under what conditions farmers

are willing to convert production land to energy crops? Which subsidy policies encourage

energy crop production most cost effectively?

This dissertation is divided into two essays. The first essay studies a farmer’s decision
to convert a unit of traditional crop land into dedicated energy crops, taking into account
sunk conversion costs, uncertainties of traditional and energy crop returns, and learning.
The optimal decision rules differ significantly from the expected net present value rule,
which ignores learning, and from real option models that allow only one way conversions
into energy crops. These models also predict drastically different patterns of land
conversions into and out of energy crops over time. Using com-soybean rotation and
switchgrass as examples, we show that the model predictions are sensitive to assumptions

about stochastic processes of the returns.

The second essay evaluates the cost-effectiveness of four types of governmental
subsidies in encouraging energy crop production. We first present a land conversion
model to Show how the subsidies that are expected net present value (ENPV) equivalent
can change a representative farmer’s optimal land conversion rules differently for
converting land into an alternative use as well as converting out of it. This is because
these subsidies affect the land conversion costs, land return level and uncertainty
differently. Then in the context of encouraging switchgrass production, we compare the
probabilities of inducing the representative farmer to convert land from com-soybean to
switchgrass across four subsidies for the same, fixed 30-year expected government
budget. Results of Monte Carlo simulations show that the insurance subsidy results in the
highest probability of land being converted to the energy crop, followed by the constant
subsidy. Although the cost-sharing subsidy and the variable subsidy encourage land
conversion to the energy crop, they also discourage land from staying in it. Over time,
these two subsidies have little effect on the land area in energy crops compared to the no-
subsidy baseline. Combining the establishment cost-sharing subsidy with other annual

subsidies has no added effect over single subsidies in inducing land conversion to the

energy crop.

Copyright by
Feng Song

2010

ACKNOWLEDGMENTS

I would like to thank my major professor, Scott M. Swinton, for being such a
wonderful supervisor. He guides me through the whole Ph.D program and makes sure
everything on the right track. He encourages me to think like an independent researcher.
He cares about the career development of students and is willing to share his experience

and help me out when I feel confused.

I would like to thank my dissertation supervisor, J inhua Zhao, for playing an
indispensible role during my dissertation completion process. He is an inspiring and
knowledgeable researcher. He gives me constructive comments every time I approached
him with questions. He helps me to develop as a researcher and has provided an
excellent model of professional success and balanced life, for which I am deeply grateful.

I would like to express my appreciation to the members of my committee, Richard
Horan and Robert Myers for inspiring discussions and valuable comments that help
improve this research. Other faculty members, including Zhengfei Guan, Frank Lupi,
Songqing Jin, Jeffery Wooldridge, Scott Loveridge, and Eric Crawford deserve my
special thanks too. Thank you goes to Debbie Conway, Nancy Creed and Ann Robinson
too for their help with administrative issues. Ialso appreciate the financial support

provided by the DOE Great Lakes Bioenergy Research Center (DOE BER Office of

Science DE-FCO2-O7ER64494).

I would like to thank my dear friends and colleagues who shared my laughter,

excitement, depression and many other things together in the past five years at East

Lansing. It is such a long list that there is no way to have all your names here. I believe
you know who I am talking about. It is because of you that my life in East Lansing was
very enjoyable. I already began to miss the days we were together. I wish you good luck
for the rest of the journey.

Finally and most importantly, I want to thank my family for their unconditional love
and constant support. I owe too much to my parents for making them worry about me and
miss me so much, for not being able to take care of them when I should, for my
occasionally impatience. I thank my sister and brother-in-law for taking care of my
parents when I was absent so that I could focus on my intensive work in the graduate
school. My last words are reserved for my husband. We met in graduate school at
Michigan State University. I feel fortunate to find a guy who is funny and full of
enthusiasm for life. His support, encouragement and humor have greatly comforted me
during difficult times. Life is full of ups and downs. With his company, I have confidence

to meet any challenge ahead.

vi

TABLE OF CONTENTS

LIST OF TABLES .............................................................................. ix
LIST OF FIGURES ............................................................................. x
INTRODUCTION ............................................................................... 1
References ...................................................................................... 4
ESSAY l
SWITCHING TO PERRENIAL ENERGY CROPS UNDER
UNCERTAIN TY AND COST REVERSIBILITY .......................................... 5
Introduction .................................................................................... 5
A General Land Conversion Model .......................................................... 9
Data and Parameter Estimation ............................................................. 14
Construction of Returns to Land Uses .................................................. 15
Parameter Estimation ...................................................................... 16
Land Conversion Costs ................................................................... 20
Results and Sensitivity Analysis ............................................................. 21
Comparison with NPV and One-way Conversion Rules .............................. 23
Effects of Different Stochastic Processes ................................................ 25
Effects of Conversion Costs ............................................................... 28
Effects of Uncertainties ..................................................................... 29
Effect of Positive Correlated Crop Returns ............................................... 30
Effect of Changing Growth Rate of Switchgrass
Under GBM Assumption ................................................................... 31
Conclusions and Discussions .................................................................. 31
Appendix A: A Homogenous Land Conversion Model ................................... 52
Appendix B: Numerical Solution for the Optimal Land Use Decision Model ......... 65
Appendix C: Specification Tests .............................................................. 70
References ....................................................................................... 77

ESSAY 2

ALTERNATIVE LAND USE POLICIES:

REAL OPTIONS WITH COSTLY REVERSIBILITY ................................... ....80
Introduction ........................................................................................ 80
Land Conversion Decision Model ................................................................................ 84

Decision without Governmental Intervention ............................................. 84

Decision under Different Subsidies ........................................................ 9O

Simulation of Land Use Choice ............................................................... 92

Results ............................................................................................. 95
vii

Critical Land Conversion Boundaries ...................................................... 95

The Proportion of Land in Switchgrass .................................................. 96

The Change and Variation of Governmental Costs over Years ....................... .98

The Effects of Combining Cost-sharing Subsidy with Other Types of Subsides. . .99
Conclusions and Discussions ................................................................. 101
References ...................................................................................... 1 19
CONCLUSDING REMARKS ................................................................... 122
References ...................................................................................... 1 25

viii

LIST OF TABLES

Table 1.1 Baseline Parameters for Numerically Solving

the Dynamic Optimal Land Conversion Rule ...................................... 34
Table C. 1. Simple Dicky-Fuller Test Results ................................................... 72
Table C.2 Phillips-Perron Test Results ........................................................... 73
Table C.3 KPSS Test Results ...................................................................... 74
Table C.4 Serial Correlation AR(1) Test ........................................................ 75
Table C.5 Heteroskedasticity ARCH(1) Test ................................................... 76
Table 2.1 Parameters for Solving Optimal Land Conversion Rule ........................ 103
Table 2.2 Land Conversion Optimality Conditions under Different Subsidies... . . . . . . . .. 104
Table 2.3 Parameters for Policy Simulation: Single Subsidy ................................ 105

Table 2.4 Parameters for Policy Simulation: Single Subsidy vs. Combined Subsidy. . ..106

ix

LIST OF FIGURES

Figure 1.1 Conversion boundaries ............................................................ 35

Figure 1.2 Average returns to com—soybean and switchgrass in
North-central US. (1982 -2008, in 1982 dollars) .............................. 36

Figure 1.3a NPV vs. Dynamic optimal (under GBM): conversion boundaries ......... 37

Figure 1.3b NPV vs. Dynamic optimal rule (under GBM):
proportion of land in switchgrass ................................................ 38

Figure 1.4a Two-way vs. one-way conversion (under GBM):
conversion boundaries ............................................................ 39

Figure 1.4b Two-way vs. one-way conversion (under GBM):

proportion of land in switchgrass ................................................ 40
Figure 1.5a GBM vs. MR: conversion boundaries ........................................... 41
Figure 1.5b GBM vs. MR: proportion of land in switchgrass .............................. 42
Figure 1.6a Reducing conversion costs (under GBM): conversion boundary ............ 43

Figure 1.6b Reducing conversion costs (under GBM):
proportion of land in switchgrass ................................................. 44

Figure 1.7a Return volatality (under GBM): conversion boundary ........................ 45

Figure 1.7b Return volatality (under GBM):

proportion of land in switchgrass ................................................. 46
Figure 1.8a Return correlation (under GBM): conversion bounary ......................... 47
Figure 1.8b Return correlation: proportion of land in switchgrass .......................... 48

Figure 1.9a Increasing growth rate to 0.07 (under GBM): V
conversion boundary ................................................................ 49

Figure 1.9b Decreasing growth rate to 0.01 (under GBM):
conversion boundary ................................................................ 50

 

Figure 1.9c Varying growth rate (under GBM):
proportion of land in switchgrass ................................................ 51

Figure A] The critical land conversion boundaries .......................................... 60

Figure A.2 Sensitivity of the critical return ratios to
the variance of switchgrass return ................................................ 61

Figure A.3 Sensitivity of the critical return ratios to
the drift rate of switchgrass return ................................................ 62

Figure A.4 Sensitivity of the critical return ratios to conversion costs ..................... 63

Figure A.5 Sensitivity of the critical return ratios to

correlation coefficient between two return series ................................. 64
Figure 2.1 Optimal land conversion rule: no subsidy ........................................ 107
Figure 2.2 Land conversion simulation steps ................................................. 108
Figure 2.3a Optimal land conversion rule: constant subsidy ................................ 109
Figure 2.3b Optimal land conversion rule: variable subsidy ................................ 110
Figure 2.3c Optimal land conversion rule: insurance subsidy ............................... 111
Figure 2.3d Optimal land conversion rule: cost-sharing subsidy ........................... 112
Figure 2.4 Probability of land in switchgrass: comparison of single subsidy

(expected NPV of governmental costs=$30/acre) ............................... 113
Figure 2.5a Mean NPVs of governmental costs over years .................................. 114
Figure 2.5b Standard errors of NPV of governmental costs over years .................... 115

Figure 2.6a Effect of combining a cost-sharing subsidy with
an insurance subsidy (expected NPV of governmental costs is $80/acre). . .1 16

Figure 2.6b Effect of combining a cost-sharing subsidy with
a constant subsidy (expected NPV of governmental costs is $80/acre)......117

Figure 2.6c Effect of a combining cost-sharing subsidy with a
variable subsidy (expected NPV of governmental costs is $80/acre) ........ 118

xi

Introduction

The soaring oil price between 2005 and 2008 coupled with pressures of energy security
and climate change crisis have triggered another round of efforts to search for renewable
energy sources. Included in the solution portfolio is biomass energy, which is especially
attractive in countries with ample agricultural land, such as United State. In addition to
co-firing to generate electricity, special interest is given to convert biomass to
transportation fuel. Although currently most US. liquid biofuels are derived from corn
grain, cellulosic biofuels are being strongly advocated because of their better
environmental performance ( Schemer et a1. 2008; Paine et al. 1996). The US. Energy
Independence and Security Act (EISA) of 2007 mandates the use of cellulosic biofuel,
increasing from 0.1 billion gallons annually in 2010 to 16 billion gallons in 2022.
Dedicated energy crops, such as switchgrass, have been identified as the one of the major

feedstocks for cellulosic biofuel production in the United States (Perlack et a1. 2005).

As pointed by Khanna (2008), to be economically viable, energy crops must compete
successfully both as crops and as fuels. Cellulosic biofuel produced from the energy
crops need to compete with fossil fuels and com—based ethanol. Farmers need to make
land allocation decisions for energy crops. Besides expected profitability considerations,
which underpin the net present value (NPV) decision rule, uncertainty plays an important
role in famers’ adoption decision of energy crops since growing them is fraught with
risks. Farmers face at least two categories of risks: policy risks and market risks. Given
current technology, the potential demand for energy crops as cellulosic feedstock is
largely from ethanol blending mandate of EISA. There are concerns about how the

mandate is going to be implemented (Taheripour and Tyner 2008). More importantly,

I

although the price of energy crops is largely undetermined, it is likely to exhibit different
volatility patterns from traditional crops prices. As ethanol feedstocks, energy crops will
have prices determined in large part by the ethanol market, which is linked to the gasoline
market (Tyner 2008). Energy crop price volatility is likely to be aggravated as ethanol
shifts in and out of status as a cost-effective fuel substitute for gasoline, based on the
relative prices of petroleum and com-based ethanol. In short, the uncertainties should be
taken into account in modeling farmers’ adoption decision of the energy crops. This

dissertation focuses on the market risks.

Studying fanners’ land use decision mechanism also has important implications on
design of effective governmental policies to support energy crop production. Coupled
with energy policies that induce energy crop production by creating new markets for
them, many countries also use agricultural subsidies to provide direct production
incentives. These range from cost—sharing at start-up to price subsidy at final sale. The
effectiveness of these subsidies should be systematically analyzed and can only be

correctly evaluated if we can understand how the farmers make decisions.

The real options framework has been used to investigate the effects of uncertainty on
the timing of land conversion to alternative uses. The general finding is that combined
effects of uncertainty and sunk costs involved in the conversion generate substantial
“hurdle rates”, which is not accounted for in present value analyses that are based on
expected values. However it is often assumed the land conversion decision is irreversible.
This may be a reasonable assumption for urban land conversion, but for agricultural land,
a farmer can switch between different uses at some cost. Therefore we adopt an

innovative approach by allowing the costly conversion of land back to original use. With

2

this improved model, I address two related questions about land use decisions for energy
crop production. In the first essay I derive the dynamically optimal land conversion rule
between a dedicated energy crop and a traditional food crop rotation. The resulting
decision rule is contrasted with alternative decision rules, including the NPV rule and the
one-way dynamically optimal conversion rule. We also compare two stochastic processes
for payoffs from the two land uses using market and agronomic yield data. These
processes go beyond the typical geometric Brownian motion assumptions in the literature.
The second essay examines four types of ENPV-equivalent subsidies, Showing how they
can affect a representative farmer’s optimal land conversion rule differently, depending
on their effects on the conversion costs, level of returns and variability of returns. For a
given governmental budget, the subsidies are compared by their ability to attract land

conversion to energy crops.

References

Khanna, M. 2008. "Cellulosic Biofuels: Are They Economically Viable and
Environmentally Sustainable?" Choices 23(3): 16-21 .

Paine, L. K., T. L. Peterson, D. J. Undersander, K. C. Rineer, G. A. Bartelt, S. A. Temple,
D. W. Sample, and R. M. Klemme. 1996. "Some Ecological and Socio-economic
Considerations for Biomass Energy Crop Production." Biomass and Bioenergy
10(4):231-242.

Perlack, R D.,L. L. Wright, A. F. Turhollow, R. L. Graham, B. Stokes, and D. C. Erbach.
2005. "Biomass as Feedstock for a Bioenergy and Bioproducts Industry: The
Technical Feasibility of a Billion-Ton Annual Supply. " Department of Energy.

Schmer, M. R., K. P. Vogel, R. B. Mitchell, and R. K. Perrin. 2008. "Net Energy of
C ellulosic Ethanol from Switchgrass." Proceedings of the National Academy of
Sciences, 105(2): 464-69.

Taheripour, F., and WE. Tyner. 2008. "Ethanol Policy Analysis -- What Have We
Learned So Far?" Choices 23(3): 6-11.

Tyner, W. E. 2008. "The US. Ethanol and Biofuels Boom: Its Origins, Current Status,
and Future Prospects." BioScience 58:646-653.

Essay 1
Switching to Perennial Energy Crops under Uncertainty and Costly Reversibilityl

Introduction

Replacing fossil fuel with renewable fuels, including biofuels such as ethanol, has been
advocated for contributing to energy independence and mitigating climate change.
Currently most of the ethanol produced in the United States comes from corn grain,
raising concerns about the negative environmental impacts associated with corn
production, upward pressure on food prices, and greenhouse gas emissions due to indirect
land use changes as rising food prices induce cultivation of new lands (Searchinger et a1.
2008). A promising alternative is cellulosic ethanol, which relies on nonfood feedstocks.
The US. Energy Independence and Security Act (EISA) of 2007 mandates blending into
transportation fuels of 36 billion gallons of renewable liquid fiJClS annually by 2022, out
of which at least 16 billion gallons must be cellulosic ethanol. Significant expansion of
cellulosic ethanol production will require more land to grow dedicated energy crops. A
recent simulation of potential US switchgrass production implies a need for 71 million
acres of crop land to meet the 2007 EISA mandate (Thomson et a1. 2008). Yet idle land
in the United States, including CRP land, is only about 40 million acres (Lubowski et al.
2006). This implies that current production land will need to be converted to grow

cellulosic energy crops.

Although large scale production of cellulosic energy crops is not commercially viable

at present, its advent could have dramatic effects on land use change and associated

 

A manuscript Song, Zhao and Swinton (2010), which is based on the content of this essay, has been
submitted for consideration for publication.

economic and environmental impacts. Forecasting the conditions under which such
change would occur is an important first step toward evaluating likely outcomes and
relevant policy interventions. Forecasting land use change depends critically on farmers’
land use decisions, which in turn are driven by several salient features of dedicated

energy crops.

First, all of the major cellulosic energy crop contenders are perennial. They need
several years to establish before achieving full yield potential (Powlson, Riche, and
Shield 2005). Devoting land to energy crops represents a long-term commitment by the
farmer and incurs sunk costs. Moreover, converting land back to traditional annual crops
also incurs (possibly substantial) costs (e. g., costs of killing persistent perennial

rootstocks).

Second, farmers growing cellulosic energy crops face great revenue uncertainty due to
both production and price uncertainties. . The production uncertainties are inherent in
any agricultural production. More importantly, energy crop prices are largely
undetermined but likely to exhibit different volatility patterns from traditional crops.
Crops destined for conversion into ethanol will have prices determined in large part by
the ethanol market, which is linked to the gasoline market (Tyner 2008).2 Energy crop
price volatility is likely to be aggravated as ethanol shifts in and out of status as a cost-
effective fuel substitute for gasoline, based on the relative prices of petroleum and corn
grain, the leading current ethanol feedstock in the United States. Although mandated

growth in cellulosic ethanol demand under EISA may mitigate one policy related source

 

Another large demand for energy crops is co-firing to provide electricity.

of price uncertainty, there remain important uncertainties regarding federal climate-

change policy and state-level renewable energy policies.

Finally, although real options methods exist for modeling stochastic revenue streams
and uncovering optimal decision rules, the real options literature typically relies upon
Brownian motion that is mathematically convenient but not necessarily consistent with
observed variability of energy returns. For instance, Pindyck (1999) examined the long-
run evolution of oil, coal and natural gas prices using the US. data from year 1870 to
1996 and found that these energy prices are mean reverting, in spite that the rate of
reversion is slow. In capital investment literature, the effect of different assumption on
the underlying stochastic process has been investigated by Metcalf and Hassett (1995)
and Sarkar (2003), etc. Some found that the effect is negligible while others gave
opposite conclusion. It is of interest to examine how the assumption of different

stochastic processes will affect the agricultural land conversion decision.

In this paper, we study land conversions between traditional crops and energy crops,
incorporating these three features of energy crops. We make several contributions to the
literature. First, we extend studies based on the net present value (NPV) approaches to
allow for uncertainty, sunk costs and learning.3 In the NPV approach, a farmer will
convert land to energy crops if the expected NPV of the returns from energy crOps exceed
those from current (traditional) crops. But under uncertainty and sunk costs, the farmer
may be more reluctant to convert land into and out of the two uses, similar to the

predictions of real option theory (Dixit and Pindyck 1994).

—‘

Here we model an autonomous learning process. The uncertainties of the crop returns change over time.
By observing the historical realizations of crop retums, the farmer can update his information set and obtain
the new marginal distribution of the returns in the future.

7

Second, we extend the real options studies and allow for land use conversion in two
directions, so a farmer deciding on converting to energy crops is allowed to take into
consideration the future possibility of converting the land back to traditional crops under
plausible market conditions. Real option theory has been widely applied in urban land use
decisions since Titman (1985) (e.g., Capozza and Li 1994; Abebayehu, Keith, and
Betsey 1999). A common assumption in this literature is that land conversion is
irreversible. This assumption might be reasonable for urban development, but for
agricultural land, a farmer can switch between different uses with costs. Allowing two-
way land conversion is important to capture the flexibility of farmer’s land use decisions,
and is particularly important for energy crops given the high degrees of uncertainties

involved.

Third, we contrast the effects of two alternative stochastic processes for returns from
the two competing crop choices, geometric Brownian motion (GBM) and mean reverting
(MR). We use both historical market data and simulated agronomic yield data to
parameterize and test the stochastic processes and Show how optimizing farmer behavior

responds to alternative assumptions about the stochastic processes.

Our paper is closely related to the broader real Options literature (Dixit and Pindyck
1994), especially those allowing for two-way decisions. Dixit (1989) studies a firm’s
entry and exit decisions assuming that the output price is the only state variable and
evolves according to GBM. Mason (2001) extends this work to examine a mine’s
decision to start and to shut down, for which not only the output price is uncertain'but the
resource reserve is limited. This results in the optimal decision rule depending on both

the reserve stock level and the price. Dixit (1989) and Mason (2001) both assume that the

8

decision maker only obtains a return in one state (entry or active) and thus only one
stochastic state variable (price) governs the optimal action. Our model allows two
separate but possibly correlated returns, those fi'om traditional crops and from energy
crops. Kassar and Lasserre (2004) examine the optimal abandonment rule between two
species, both of which have stochastic values. However, in their study, a species cannot
be recovered once it is lost, which is equivalent to switching in one direction only.

A General Land Conversion Model

Consider a risk-neutral farmer4 with a unit of land facing two land use alternatives.

is S = {c,s}, who can convert from alternative 1' to j with a lump-sum sunk cost C ij .

For concreteness, we use com-soybean rotation (denoted by c) and switchgrass (denoted

by s) as representative of a traditional crop and an energy crop. The return to alternative
iin period t is denoted by 721- (t) , which is assumed to evolve according to a known
stochastic process of the general form:

(I) dig-(t)=ai(7ri,t)dt+O',-(7rl-,t)dzl~ i=C,S

where the drift term (ll-(72,30 and the variance term 0305,31) are known nonrandom

fimctions, and (12,- is the increment of a Wiener process. Thus, new information about

future returns of the two crops comes in the form of newly observed return levels, which
become starting points for the distributions of future returns. Note that we model the
returns directly, instead of modeling the price and yield uncertainties separately. This

approach simplifies our analysis, and in the empirical section we derive the return

 

We assume risk neutrality for simplicity; similar results can be obtained when the farmer is risk averse.

processes from the underlying price, yield and cost processes. The correlation coefficient

of the two return processes is p , i.e., E (dzcdzs) = pdt. Traditional crop and energy

crop returns could be correlated for a variety of reasons, e. g. both are linked with energy
prices and are subject to macro-economic shocks. Moreover, farmers switching land
between the two crops could introduce a long-term relationship between their return

series. Finally, let r be the farmer’s discount rate.

A key insight of the real options approach is that when the land is in use i, say in
traditional crops, the farmer has the option of converting it into energy crops when
market conditions are “favorable.” Once converted, it is costly to revert it back to
traditional crops if the market conditions turn out to be less favorable. Thus, sticking to
the current land use (in traditional mom) has an additional value, called option value,
derived from the option of converting it into the alternative use (in energy crOps). But
since the land in energy crops can be further converted back to traditional crops (albeit at
a cost), this option value of converting from traditional to energy crops further depends
on the option value associated with converting from energy to traditional crops. The
mutual dependence of the two option values significantly complicates the solution
algorithm. To make the analysis traceable and rigorous, we solve a simplified model in
Appendix A, which admits an analytical solution. Thus we can reveal some useful

information through comparative statics.

Let Vi(7rc (t), zrs (t )) be the farmer’s period t expected present-value payoff starting

with land use i and following optimal land conversion rules. Due to the option of

converting into use j 1: i, the payoff depends on the distribution of future returns of both

10

land uses, the information for which is contained in the two current returns, 7Q. (t) and
”s (t) . At time t, the farmer chooses between keeping the land in use i and converting it

into alternative use j :

rri(t)dt + e‘m” EVi (7:, (t + dt),

(2) Vi(7rc(t),7rs (t)) = max .
”s0 + a’0), VJ (flc(t),7rs(t)) .. Ci].

The first term on the right-hand side describes the payoffs if the land is kept in use i: in

the infinitesimal period [t,t + dt] , the farmer receives profit from land use i at rate 7r,- (t) ,

and at the end of the period, receives the new discounted expected payoff

6—,.th Vi(t + dt). The second term on the right-hand side describes the payoff if the

land is converted into use j: the farmer receives the expected payoff of use j, Vj (t) , but

incurs the conversion cost Cij .

As shown by Brekke and Oksendal (1994), the decision problem in (2) can be

equivalently expressed by a set of complementary slackness conditions, as long as the

value functions Vc(0) and V 3(0) are stochastically C2 .5 First, (2) implies that

(3) V‘pzc (z), 7:, (t)) 2. 7r,(t)dt + e_rdt EV‘ (a, (t + (10,”, (t + dt))

which, after applying Ito’s lemma, can be expressed as L Vi(7rc (t), 72's (t)) .>_ 0, where

 

5 i . . . . . . . . . . .
V (ED rs stochastrcally C2 if it rs continuously twrce differentiable rn ”i and satisfies the generalized

Dynkin formula, which gives the expected value of any suitably smooth statistic of an [to diffusion process
at a stopping time. See more details in Brekke and Oksendal (1994). This condition is trivially satisfied in
our applications.

11

(4)
LViUchIS) E rV’(7rc,7rs) _ ”1.0) .. ac(7rc,t)V7: — as(7rs,t)V7:
c S

2 i 2 i i
..1/20'c (7rc,t)Vflcflc -— l/20'S (”pm/”Ms -— poc (flc,t)0's (173,056,!

and the subscripts of V’ denote partial derivatives.6 Equation (2) also implies that
Vi(7rc (t), yrs (1)) 2 Vj(7rc (t), 7:5 (t)) - CU" Then we know the value functions Vi and

Vj have to satisfy:
(i) LVi(7rc,7rs)_>_0, i=c,s

(5) (ii) Vi(7rc,7rs)2Vj(7rc,7rS)—Cij i,je{c,s}andi¢j

(iii) either (i) or (ii) has to hold as a strict equality.

If (i) is an equality, the farmer should keep his land in current use i, and if (ii) is an
equality, the farmer should switch the land use to j. If both are equalities (a nongeneric
case), the farmer is indifferent between converting and not converting. The optimal
conversion decisions (i.e., solutions to (5)) are represented by two conversion boundaries

in the fl'c — as space, one for each type of current land use, as shown in figure 1.1. If the

 

6 e—rthViUl'c (t + (If), its (1 + d1» can be rewritten as
l

1+r*dt

 

[Vi (”c (t), ”s (t)) + EdVi ] , which, by applying Ito’s lemma, can be written

1 - . i . .
1 H *dt[V’(Itc(t),itsft))+(ac(7rc,t)V;rc +as(”59t)V,,s +1/2°%(”C”)Virczrc +1/2"~i(”s”)Vi,

 

”s
+pO'CO'SV; 7r )* dt] . After multiply both sides of equation by (l + r * dt) , rearrange the terms and

c s
divide both sides by dr, we can obtain equation (4).

12

current land use is in traditional crops (i =c), the conversion boundary, yrs = b“ (”C ) ,

denotes the returns from traditional and energy crops that the farmer is indifferent

between converting to energy crops and sticking to traditional crops. Above this
boundary, i.e., when 7:5 > 1)“ (ac) , the returns from energy crOps are sufficiently high
that it is optimal for the farmer to convert to energy crops. Conversely, if as < I)“ (756),
the farmer should stick to growing traditional crops. Intuitively, as shown in figure 1.],
boundary bcs lies above the 45" line: given the sunk costs and uncertainties, the farmer
is reluctant to convert to energy crops even when its return E; (t) slightly exceeds

”c (t) , the return from traditional crops.

Similarly, if the current land use is in energy crops, the boundary for converting to
traditional crops is given by 71's = bSCUIC). If the return from energy crops is too low
compared with traditional crops so that its < bsc (ac) , the farmer should convert to
traditional crops. Otherwise, it is Optimal for the farmer to stick to the current use (in

energy crops). Again, due to uncertainty and sunk costs, b3C lies below the 45" line in
1.1: once the land is already in energy crops, the farmer is reluctant to convert it to

traditional crops unless the return from traditional crops is sufficiently high. Thus, the

two boundaries divide the ac - 71's space into three regions: above the boundary bcs , it

is optimal to convert from traditional crops to energy crops; below the boundary bsc , it is
optimal to convert from energy crops to traditional crops; and between the two

boundaries, it is optimal to keep land in its current use.

13

Since the two value functions V 6(0) and V 5(0) are interdependent, there are no

analytical solutions to (5).7 We follow Miranda and F ackler (2002) and employ the

collocation method, which approximates the unknown value functions

V (71' c , ”s ) usrng a lrnear comblnatron of n known basrs functions:

. ”c ”s
‘1 — o O o I
(6) V (7Z'c,7Z'S)- Z Z CJCJS¢JCJS (”c’fl's)
jc==ljs=l

Where V1.0) represents the numerical approximation of V 1.(0) . Compared to the
shooting algorithm and finite difference method that are often used to numerically solve
the value functions in stochastic dynamic optimization problems, the collocation method
is a fast and robust alternative (Dangl and Wirl 2004). The coefficients cjc js are found
by requiring the approximant to satisfy the optimality condition (5) at a set of
interpolation nodal points. The threshold returns that induce the farmer to convert land
will also be solved at the nodal points. The details about the numerical solution are

documented in Appendix B.

Data and Parameter Estimation

 

7 . . . . .
If we allow land conversion in one direction only, then analytical solutions exrst for specral stochastic
processes of £6 (t) and 71's (I) , e.g., geometric Brownian motions. Numerical methods must be

employed for more general processes even for one-way conversion models. For instance, lnsley and Rollins
(2005) and Conrad and Kotani (2005) formulate (4) as a linear complementarity problem and then use the
finite difference approach to solve it numerically.

14

The empirical analysis focuses on a representative farmer’s optimal land conversion
decision in the North-central United States, 8 where com and soybean are two major crops
frequently grown in rotation. We assume that the farmer currently grows corn and
soybean in a balanced rotation (with half of his land in each crop). The alternative land

use is to grow switchgrass. The model can be easily adapted to other locations and crops.

Construction of Returns to Land Uses

We first estimate the drift and variance parameter fimctions a, (If, ,t) and o;(7r,, t) for the

two land uses, and the correlation coefficient of the two stochastic processes ,0 .

Typically, these parameters are estimated using historical time series on returns. We use
US. Department of Agriculture’s North-central regional data for 1982 to 2008 (USDA
2008) to calculate the annual average return to the com-soybean rotation, which equal the
value of production minus the Operating cost. The returns are deflated by the Consumer

Price Index (CPI, 1982=100). The data series is plotted in figure 1.2.

Since switchgrass has not been grown commercially as a biofuel feedstock, we do not
have historical data for switchgrass returns. We instead construct a hypothetical series of
returns for switchgrass grown as an energy crop. The return equals the farmgate price of
switchgrass times yield minus the production costs. The farmgate price is determined by
ethanol producers’ willingness to pay (WTP) as well as government subsidies.9 Ethanol

producers’ WTP is equal to the ethanol price minus the conversion, costs from

 

8
The North-central area includes Illinois, Indiana, Iowa, Michigan, Minnesota, Missouri, Ohio, and
Wisconsin.

Governmental subsidies are important in the competitiveness of switchgrass. With our baseline
parameters assumptions, ethanol prices in some years were so low that the switchgrass revenue was not
able to cover the production costs if there had been no governmental subsidy.

15

switchgrass to ethanol and the transportation costs from field to a processing facility. The
ethanol price (in $ per gallon) is obtained from the Nebraska Energy Office from 1982 to
2008. The estimated conversion cost is assumed to be S 0.91 per gallon (DiPardo 2004).
Assuming that one ton of switchgrass yields 91 gallons of ethanol (Schmer et a1. 2008)
and multiplying by this conversion rate, we convert the ethanol price and conversion cost
from a per gallon basis into per ton basis. The transportation cost is assumed to be $8 per
ton (Babcock et al. 2007). For government subsidies, we use the $45/ton matching
payment currently provided by USDA to biofuel producers for their costs of collection,
harvest, transport and storage of biomass. Switchgrass yield for 1982-2008 in the same
North-central states as the com-soybean returns data is obtained from Thomson et a1.
(2008), who used the EPIC Model at the 8-digit watershed level that can be aggregated to
the state level. The average switchgrass yield is 3.10 tons/acre and the standard deviation
is 0.20 tons/acre. The production cost is assumed to be $190/acre (Duffy and Nanhou
2001). This is the operation cost in maintenance year 2-10, which does not include the
establishment cost since it will be accounted in the land conversion cost discussed below.
Finally, the nominal returns are deflated to a 1982 base using the CPI. These returns are

plotted in figure 1.2.
Parameter Estimation

We consider two commonly used return processes, geometric Brownian motion (GBM)
and geometric mean reversion (MR). The GBM process is widely used in real option

studies for its analytical tractability, and is represented as
(7) d3} =aiflidt+oi7ridzi, i=C,S

l6

where a, and o; are drift and variance parameters respectively. If a return follows a GBM

process, the conditional mean and variance of the return rise over time without boundary.
Thus it is a nonstationary series. The MR process, on the other hand, assumes that the

random variable will revert to a long-term average, is stationary and is described by
(8) drrl- = 77,-(76 -— 7r,)7r,-dt + airtidzi, i 2 0,5
where 7f" is the long- term average return of land use i, 771' is the speed of reversion

and 0',- is the variance rate. The returns revert to a long-run equilibrium 751' at a speed Of

771'71'1' . The further the returns divert from the if,- , the quicker the reversion will be.

Theoretically, both processes can be justified in describing how agricultural returns
evolve over time. GBM can better reflect a trend which could be positive due to
technological advances that boost productivity while a MR process can better reflect the
long-term equilibrium conditions when technology is unchanging. Statistical tests of the
stationarity of com-soybean and switchgrass returns generate mixed signals. For instance,
for the logarithmic com-soybean returns, the null hypothesis of unit root can not be
rejected at 10% level based on the Dickey-Fuller test and Phillip-Perron test, but the null
hypothesis of stationarity can not be rejected at 10% level based on KPSS test, either.
Similar results are found for logarithmic switchgrass returns. The detailed test results can
be found in Appendix C. Given the nascent state of biofuel crop technology and markets,
the dynamic growth implicit in a GBM process seems the more justified of the two. Thus,
we use GBM processes in our baseline model, and study how the results change if MR

processes are used instead.

17

To estimate the parameters of the two processes for comasoybean and switchgrass

returns, we first discretize the two continuous time processes. If it; follows a GBM, then
In it,- follows a simple Brownian motion with drift, which is the limit of a random walk.

In particular,
(9) Inna“ -ln7r,-,, =(ai-l/2of7')+O',-e,~, i=c,s

where 6,- ~ N (0, 1). The maximum likelihood estimates of the drift al- and the standard

deviation 0',- are thus (it,- = m,- + 0.5.5}2 and 6',- = si, where m,- and Si are respectively
the mean and standard deviation of the series In 7:, t — In ”1' t—l . The estimate of the
correlation coefficient p is the correlation between series In ac t — 1n ac t—l and

In 71's, t — In as ,__1 . The parameter estimates for the GBM representation of the com-

soybean and switchgrass returns are presented in table 1.10

The discrete time approximation to the MR process in (8) is as follows:

(10) ”in “ ”Lt—1 = ”iv-ii ‘ ”i,t—l)”i,t—l + “Win—lei: 1': CBS
where again el- ~ N (0, 1) . Dividing both sides by 7:,- t_1, we obtain

”Lt " ”Lt—l

 

(11)

= az- + bi”i,t—1 + diet, 1 = c,s
”Lt—l

 

10 The estimated growth rate of the switchgrass return is 0.17, which is higher than the assumed discount
rate of 0.08. For the dynamic optimization problem to have a solution, the expected growth rate cannot
exceed the discount rate (otherwise, the expected payoff from switchgrass is infinite). We thus assume that
the switchgrass return grows at the same rate as com-soybean. We later tested the results sensitivity to this
assumption by varying the grth rate of switchgrass return from 0.01 to 0.07.

18

Since the stochastic processes of the returns to com-soybean and switchgrass have

correlated residuals (i.e., cc and es are correlated), we use a seemingly unrelated

regression model to estimate the parameters 61,-, bi, 0',- and p. Consistent estimates of

A
A

A a'
_. . A -— I . .
77,- and It,- are then obtained as 17,- = -b,- , and fl,- = --:-. The regressron estimates are as
i

A

follows: [to = 0.48, be = -—0.0046 with standard errors 0.20 and 0.0019, respectively,

and (is = 0.72 ’55 = —0.0038 with standard errors 0.35 and 0.0022, respectively. The

standard error of the regression is 0.31 and 1.02 for com-soybean and switchgrass

respectively. The calculated results for m and Triare reported in table 1. Consistent

with the observation in figure 1.2, the return to switchgrass has higher volatility than that

to com-soybean, no matter which type of stochastic processes they follow.

Land conversion between traditional crops and energy crops will introduce an
equilibrium relationship between the two markets. Thus these two crop return series
could be cointegrated. However, using the constructed historical data may fail to capture
this relationship since the agricultural market and energy market were not integrated
historically (Tyner and Taheripour 2008; Tyner 2009). Historical returns to com-soybean
and switchgrass could be negatively correlated as indicated by our estimation results (-
024 under GBM assumption and -0.23 under MR assumption). Part of the reason is that
switchgrass revenue is simulated as a function of petroleum price and thus highly
positively correlated with petroleum, whereas until 2005, com-soybean returns were
negatively correlated with petroleum prices due to petroleum used as transportation and

fertilizer inputs. However, this pattern of negative correlation could change as more com

19

and soybean are used to produce biofuels, and as agricultural and petroleum markets
become more integrated. Then high petroleum prices may push up corn and soybean
prices, increasing their returns. A supporting evidence is that the correlation between the
annual ethanol price and com-soybean return for year 1982-2005 is -0.07, and it changes
to 0.28 for year 2006-2008. Tyner (2009) shows similar result that the price correlation
between crude oil and corn change from -0.29 during period 1988-2005 to 0.8 during
period 2006-2008. Furthermore, the positive correlation may become stronger as
switchgrass or other energy crops expand production and compete with corn-soybean for
limited land. In response to this possibility, we test the sensitivity of our results to the

effect of positive correlation between the two crops.
Land Conversion Costs

As a perennial crop, switchgrass needs to become established and will not achieve firll
yield until the third or fourth year after seeding. It needs to be replanted every ten years.
We use the NPV of the (infinite) sequence of first-year switchgrass establishment costs as

an estimate of land conversion costs from com-soybean to switchgrass, Ccs .”

Switchgrass establishment costs include seed, chemicals, machinery and labor (e. g.
Hallam, Anderson, and Buxton 2001; Duffy and Nanhou 2001; Khanna, Dhungana, and
Clifton-Brown 2008; Perrin et al. 2008). The estimated costs vary widely across studies
because of different assumptions, methods employed, and production locations. We use

$109/acre estimated by Khanna, Dhungana, and Clifton-Brown (2008) because they

 

11
The NPV of the decennial establishment cost for switchgrass ($136/acre) overestimates Cos if the

farmer does not permanently stay with switchgrass. In this case, we overestimate the farmer’s reluctance to
convert to switchgrass but not to a large extent (one time establishment cost is $109/acre).

20

report the detailed costs categories year by year, which facilitates our calculations. The
NPV of the cost sequence is $136/acre at a discount rate of 8%.

Conversion of land from switchgrass to com-soybean production requires clearing
existing vegetation residue by tillage or herbicides. We use the costs of converting land
in Conservation Reserve Program back to crop production as an approximate of the
conversion costs from switchgrass to com-soybean. Higher than normal fertilizer rates
may be required for two years after conversion (Blocksome et al. 2008). We assume
$47/acre conversion cost from switchgrass to com—soybean production, which includes
$17/acre disking operation costs and $30/acre total additional fertilizer costs for the first

two years (Williams et al. 2009).

Results and Sensitivity Analysis

Given the baseline parameter values, we solve the optimality condition in (5) using
OSSOLVER (Fackler 2004), implemented with CompEcon Toolbox in Matlab (Miranda
and F ackler 2002). The same solver was employed by Netbakken (2006) to solve her
model of a fleet’s optimal decision to enter or to leave a fishery. The family basis

function we use is a piecewise linear spline. For each state variable (i.e., 7:6 and as ), the

nodal points are evenly spaced over the revenue interval [0, 5] (in hundred dollars) with

an increment of 0.1.

Figure 1.3a shows the two boundaries (the solid lines) for conversions from com-

bCS

soybean to switchgrass ( ) and from switchgrass to com-soybean (1)“) assuming that

both returns follow GBM. The boundaries indicate significant hysteresis in land

21

conversion decisions. For instance, the real average annual returns based on 2008 prices
in 1982 dollars are fl'c =$92/acre for com-soybean and 71's 2 $135/acre for switchgrass.

If the land is currently in corn-soybean, the minimum switchgrass return for converting

the land to switchgrass is bcs (92) = $295/acre, which is significantly higher than

$135/acre. Thus, the land will be kept in com-soybean rotation even though 72's > 7:6.

Conversely, if the land is already in switchgrass, the required minimum com-soybean
return for converting into com-soybean is about $280/acre. Thus, the land will not be

converted either.

Given the two boundaries, we calculate the expected probabilities that a piece of land in

com—soybean will be in switchgrass for each year during a 30 year period, with the 2008
returns as the initial (time zero) returns: rte (0) == $92/acre and 7’s (0) = $135/acre. Given

this starting point, we draw N (=5000) sample paths of the joint return processes for 30

1.12

years according to (7) and the parameter values in table Each sample path of the two

returns, {(fl'c (t), ”s (t)), t = l,..., 30} , is then compared with the conversion boundaries,

( has (0), bsc (0)) , to decide whether the land is kept in its current use or should be
converted to the alternative use. For instance, in year 1, when the land is still in corn-

soybean, the realized returns on a particular sample path, (276(1), 7rs (1)) , are compared

with boundary b“. If the realized returns are in the “no action zone” (e. g., if

7219(1) < bcs (72": (1)) according to the optimal decision rule), the land is kept in cOrn-

 

12
The two correlated stochastic processes {(fl'c (t), JIS (t)),t E [0,30]} are approximated by the
Euler method and implemented using Matlab’s Econometric toolbox.

22

soybean, and similar comparisons are made in year 2. If, on the other hand, the realized
returns are in the “conversion zone” (i.e., if 7’s (l) 2 b“ (ac (1)) ), the land is converted to
switchgrass, and in year two, the realized returns (ac (2), as (2)) will be compared with
boundary b“ to decide whether the land should be converted into com-soybean. Finally,
for each period we count the number of sample paths on which the land is in switchgrass.

Dividing this number by N, we obtain the proportion of land in switchgrass for each

period.

Figure 1.3b illustrates the proportion of land in switchgrass for the 30 year period (solid
line). Since the starting level of switchgrass return (at $135/acre) is much higher than that
of com-soybean (at $92/acre), more land is gradually converted into switchgrass, peaking
at 29% of the total land area. However, the switchgrass return also has a higher level of
uncertainty, and eventually some land in switchgrass is converted back to com-soybean,

stabilizing at about 16% of the land area.
Comparison with NPVand One- Way Conversion Rules

We next compare the two conversion boundaries found above with those based on the
NPV rule. According to the NPV decision rule, the farmer will switch from com-soybean

to switchgrass when the expected NPV of switching is higher than staying in com-

00 _ 00 _
soybean, i.e., when E Ji 0 as (t)e rt dt __ cos 2 E l0 ”c (t) e rt dt . Similarly, the
farmer will convert from switchgrass to com-soybean when

00 _ 00 _ .
E I 0 ac(t)e rtdt—CSC 2 E I 0 aS(t)e rtdt. Grven that both ac(t) and as(t)

23

follow GBM, we use (7) and obtain two NPV conversion boundaries:

r a .
bfi‘ipy(ac) 2 ac S + (r -— as )Ccs for conversron from com-soybean to
T c

 

r a
switchgrass, and bifpy(ac) = as S -— (r - as )Csc for conversion from
T c

 

switchgrass to com-soybean. The two NPV boundaries (in dash lines, based on GBM

process) are shown in figure 1.3a.

As illustrated in figure 1.3a, the NPV rule predicts that the farmer will convert between
land uses far more readily than under the dynamically optimal real options rule. For
instance, if the com-soybean return is $98/acre, the average historical return during 1975
- 2007, the farmer who grows com—soybean will convert to switchgrass if the switchgrass
return exceeds SIDS/acre, and the farmer who grows switchgrass will convert to com-

soybean if the switchgrass return is less than $100/acre. But the real option rule, given by

bcs (as) and bsc (as) , indicates that the com-soybean farmer will convert to

switchgrass only if the switchgrass return exceeds $315/acre, which is 3 times the NPV
threshold. The switchgrass farmer will convert to com-soybean only if the switchgrass

return is lower than $40/acre, 60% lower than the corresponding NPV threshold.

The different conversion boundaries under these two decision rules,

( bcs (”c ), bsc (7%)) and (bxfpV(flc ), bigchVU’c) ), imply different amounts of land

converted between the two uses. Figure 1.3b compares the proportions of land in .

switchgrass under the real option and NPV rules. The NPV rule predicts that land will be

quickly converted into switchgrass (peaking at 73% of total land area), followed by a

24

gradual decline, and eventually stabilizing at about 57%. The predicted proportion of land
in switchgrass is consistently higher than the predictions of the dynamically optimal

model.

A real options model that only allows one-way conversion will predict significantly
greater farmer reluctance to convert than a two-way conversion model, as shown in figure

1.4a.13 For instance, the threshold return for converting from com-soybean to switchgrass,

bgsW , doubles bcs , the threshold boundary when two-way conversion is accounted for.

Similarly, the com—soybean return threshold for a farmer to convert from switchgrass to
corn-soybean is twice as high under the one-way conversion model compared to the two-
way conversion model. Because of the increased hysteresis, the one-way real options

model predicts much lower proportions of land in switchgrass, as shown in figure 1.4b.

Effects ofDifl'erent Stochastic Processes

We next investigate the effects of assuming different stochastic processes by comparing
the conversion boundaries and switchgrass proportion under GBM and MR processes.

We first “anchor” the two processes so that they are comparable by estimating the
parameter values of the two processes using the same time series data for ac and as.
Estimates for the two processes are presented in table 1.1. This anchoring approach
implies that the parameter values may not be completely comparable. For instance,

although the variance rate for the com-soybean return under the GBM assumption (at

0.29) is roughly the same as that under the MR assumption (at 0.31), the variance rate for

 

'3 The two corresponding conversion boundaries, bgsw and bch , are obtained by imposing a
prohibitively large cost of reverting back the earlier conversion.

25

switchgrass return under the GBM assumption (at 0.64) is estimated to be much smaller

than that under the MR assumption (at 1.0).

The two-way conversion boundaries for com-soybean and switchgrass returns follow
distinct patterns according to whether the underlying stochastic processes follow GBM or
MR parameters, as illustrated in figure 1.5a. The solid lines define the optimal land
conversion boundaries assuming GBM. Under a GBM process, the conversion pattern
mainly depends on the relative volatility and conversion costs. This yields a fairly
symmetric boundary pair, albeit with a lower threshold for conversion from switchgrass
to com-soybean at low returns than vice-versa. Under a MR process, the pattern is
asymmetric, with a lower threshold for conversion to switchgrass and a much higher
threshold to com-soybean for low return rate than under GBM. But this pattern at low
return levels is reversed at higher returns, with declining tendency to convert to
switchgrass and rising tendency to convert to com-soybean. The difference in conversion
boundary patterns arises from three distinct effects of MR as compared to GBM

processes: the relative effects of uncertainty, distant time horizon and mean reversion.

The uncertainty effect follows from the higher relative volatility of the switchgrass
return in MR process than in GBM case (3 vs. 2 times the corresponding com-soybean
uncertainty). Hence, the optimizing farmer is more reluctant to convert land into

switchgrass as well as out of it.14 Due to the higher relative volatility for MR, the

 

‘4 The reluctance to convert out of switchgrass as the switchgrass return becomes more uncertain is a
feature of the real option argument: as 0' 8 increases, it is more likely that future return if S is high, which
implies that the farmer should not convert out of switchgrass. In response, the farmer has more incentive to
wait until as is low relative to 7Q. before converting out. This prediction is opposite to that of standard

risk aversion arguments, and has been used by Schatzki (2003) to test real option versus risk aversion
assumptions.

26

uncertainty effect raises the conversion boundary from com-soybean to switchgrass and

lowers the boundaries in both directions.

The distant time horizon effect arises from the higher projected long-term average
return from switchgrass as compared to com-soybean. This effect lowers the conversion
boundary from corn-soybean to switchgrass and raises the conversion boundary from

switchgrass to com-soybean compared to GBM case.

While the previous two effects hold true for low as well as high return levels, the mean
reversion effect on the conversion boundaries behaves differently at low return levels
than at high ones. When both com-soybean and switchgrass returns are high, mean
reversion pulls them downward towards the long-term average. However, the
switchgrass return reverts to its mean more slowly than the com-soybean return because
it has both a smaller reversion speed parameter and a smaller absolute difference between
the current return and the long-term average. At high return levels, these effects lower
the conversion boundary from com-soybean to switchgrass and raise the boundary from
switchgrass to com-soybean. By contrast, when the corn-soybean and switchgrass returns
are low, mean reversion causes them to rise. If the switchgrass return reverts more
slowly than the com-soybean return, it raises the boundary from com-soybean to
switchgrass and lowers the boundary from switchgrass to com-soybean. When the order
of the reversion speed is reversed, the net effect is ambiguous because the switchgrass
return has a lower reversion speed parameter but a higher difference between current

return and long-term average.

27

The asymmetric pattern of the MR returns in figure 1.5a arises from the interaction of

the three effects. For conversion boundary bcs , when com-soybean returns are low the
distant time horizon effect dominates the other two effects, lowering the boundary. But

when corn-soybean returns are high, the uncertainty effect dominates and the boundary is

raised. For boundary bCS , the three effects work together to lower the boundary when the
com-soybean returns are low, and the mean reversion effect dominates at high com-

soybean return levels.

Consistent with the conversion boundaries in figure 1.5a, figure 1.5b shows that the
predicted proportion of land in switchgrass is higher under MR for the first 4 years, since
the conversion boundary into switchgrass is low initially. As the returns grow over time,
the proportion of land in switchgrass declines and becomes lower than under GBM

processes. Eventually the switchgrass land proportion under both processes stabilizes

around 16%.
Eflects of Conversion Costs

Figure 1.6a shows how halving the cost of conversion costs affects the optimal

conversion rule under the GBM assumption. In the top panel, reducing the conversion
costs from com—soybean to switchgrass (C c s ) creates the desired incentive by making

the corn-soybean grower less reluctant to make the conversion. However, it also has the
indirect effect of making the switchgrass grower more prone to convert (back) to com-
soybean, because although the farmer currently growing switchgrass will not directly
benefit from the subsidy for conversion to switchgrass, its existence reduces the expected

cost of converting from com-soybean back to switchgrass, thereby reducing the implied

28

cost of switching back to com-soybean. Thus it indirectly increases his incentive to
convert land to com-soybean. The direct effect of lowering C cs is greater than the
indirect effect. The reduction in C cs lowers the boundary from com-soybean to
switchgrass more than the boundary from switchgrass to com—soybean. Similarly, the
reduction in C so lowers both of the conversion boundaries but lowers the boundary

from switchgrass to com—soybean more than the boundary from com-soybean to

switchgrass. These results also hold under the MR assumption.

An important insight from this two-way model is that a policy to subsidize conversion
to dedicated energy crops, such as the USDA Biomass Crop Assistance Program, can
have the joint effect of encouraging conversion both into and away from the biomass crop.

Figure 1.6b illustrates how the two effects interact over time. Reducing the conversion

cost from com-soybean to switchgrass (C cs ) leads to higher proportions of land in

switchgrass. But after 5 years, more switchgrass land is converted back to com—soybean,
due to the higher incentive to switch back to com—soybean. As time approaching end of

30 years , the predicted portion switchgrass land is only 1% higher than baseline. Further,

lowering conversion cost into com-soybean, C re , in fact promotes conversion into
swrtchgrass for the first seven years. Finally, in the long run, lowering C es and lowering

C sc have almost the same effects on land in switchgrass.

Effects of Uncertainties

As discussed earlier, higher uncertainties in either com-soybean or switchgrass returns

will cause the farmer to be more reluctant to take any conversion action. Figure 1.7a

29

shows that doubling the variance parameter of com-soybean return 0's (or switchgrass

return as ) significantly raises the conversion boundary from switchgrass (or com—

soybean) to com-soybean (or switchgrass), I)“ (or bcs ), and slightly raise the

conversion boundary from com-soybean to switchgrass I)“ (or bsC ). As argued by
Sarkar (2003), high uncertainties do not automatically translate into fewer conversions:
although conversion is undertaken only with “more extreme” returns with the higher

boundaries, higher uncertainty levels also mean that “extreme returns” occur more

frequently. Figure 1.7b shows that doubling 0's and doubling as have strikingly
different impacts: the proportion of land in switchgrass increases significantly as 0c

doubles, but decreases to nearly zero (at 0.06) in the long run as as doubles.

Effect of Positive Correlated Crop Returns

Here we examine how the land conversion will change if the com-soybean return and
switchgrass return are positively correlated instead of negative correlated. Positive
correlation implies the returns will more likely move to the same direction. The change of
relative values will be smaller than when the correlation is negative. Thus the farmer will
have less incentive to wait and option value of waiting is reduced. With perfect
correlation, the current land use will remain preferred forever so that the option value of
the alternative use will disappear. As shown in figure 1.8a, correlation between the two
crop returns changing from -0.23 to 0.3 moves the conversion boundaries towards’each

other and reduces the inaction zone. However, this change does not translate to much

30

difference in the proportion of land in switchgrassas( figure 1.8b) as the incentives for

converting into switchgrass and converting out both increase.
Eflect of C hanging Growth Rate of Switchgrass Return under GBM Assumption

The growth rate of the two return series are imposed the same. We tested the results
sensitivity to this assumption by varying the growth rate of switchgrass return from 0.01
to 0.07. In general, increasing in the growth rate of switchgrass return will lower the
conversion boundary from com-soybean to switchgrass and raise the conversion

boundary from switchgrass to com-soybean while decreasing in it will have the opposite

effect. Increasing 0tS from 0.03 to 0.07 will make the proportional land in switchgrass

peak at 0.44 (1.5 times baseline) and stabilize at 0.22 (1.38 times baseline). Decreasing as

from 0.03 to 0.01 will make the proportional land in switchgrass peak at 0.24 (84% of

baseline) and stabilize at 0.13 (81% of baseline). We report here two selective cases,

Og=0.07 and GIST—0.01 in figure 1.9a, 1.9b and 1.9c.

Conclusions and Discussions

This study develops a real options framework to analyze the farmer’s land use decision
between traditional annual crops and perennial energy crOps. The study innovates fiom
existing models of optimal conversion under the assumption of irreversible decisions by
introducing a model for two-way conversion. The possibility of costly reversibility in
crop production is illustrated using an annual com—soybean crop rotation and perennial
switchgrass representative alternative crop systems. Consistent with real options theory,
the option value of sticking to the current land use delays converting land into

switchgrass as well as converting out of it. By comparison with the real options results,

31

an NPV model predicts that an optimizing farmer would be much more prone to convert
land to switchgrass. A one-way real option model characterizing the land conversion
decision as irreversible predicts much greater reluctance to convert land from annual
com-soybean to a perennial switchgrass energy crop, also implying lower accumulated
land under energy crops over a 30-year time horizon. We further show how two
alternative stochastic process assumptions affect the optimal conversion rule and the

proportion of land devoted to the dedicated energy crop.

From a policy perspective, this model offers two important insights. First, compared to
deterministic break-even analyses (e. g., Tyner 2008; James, Swinton, and Thelen 2010),
it highlights the significant option value of delaying land conversion even when a static
net present value threshold is passed. The illustrative case here suggests that returns from
dedicated energy crops may have to exceed double the breakeven NPV level before

becoming a dynamically optimized choice.

Second, compared with past real options models that assume complete irreversibility of
decisions, this two-way model reveals that conversion subsidies to encourage biofuel
crop planting have a two-edged impact. The effect of reducing conversion costs from
com-soybean to an energy crop (switchgrass) is to lower the conversion threshold
revenue levels in both directions, meaning that not only is it easier to convert from com-
soybean into switchgrass, but it also is easier to convert the other way. Compared to the
case of no subsidy, the predicted proportion of land planted to the switchgrass energy
crop is higher with the subsidy in the intermediate period but difference becomes

negligible toward the latter part of a 30-year time horizon.

32

One limitation of this model is that we did not incorporate the switchgrass age into land
conversion model. Several issues relate to the age of switchgrass (or more generally, an
energy crop) in determining farmer’s decision. The first is that farmer may face liquidity
constraint problem in growing energy crops since costs for establishment and reseeding
incur in the first several years after planting while the yields are lower at the same time.
The net returns in the beginning of the production cycle may be even negative. Second,
although we use ten years as a production cycle in our paper because it is recommended
by the agronomists and is used in most literature to calculate the switchgrass production
costs, ten years may not be the optimal replanting time. The energy crop grower can
choose the optimal replanting time subject to the uncertainties and has the option to
convert land to an alternative use. Third, while the conversion costs from switchgrass to
other crop stay fairly constant over years since its roots do not change much, this is not
true for some other energy crops, such as miscanthus. As the roots grow over time, the
conversion costs from miscanthus to other crops grow over time too. Introducing the age
of energy crops into the model can help (at least partially) address the above issues and

will be an extension of our model.

33

Table 1.1 Baseline Parameters for N umerically Solving the Dynamic Optimal Land

Conversion Rule

 

Parameters of the stochastic processes of the returns to corn-soy and switchgrass

Returns to com-soy

Returns to switchgrass

 

A

A

 

Drift parameter as 0_03 as 0.03
GBM

Variance parameter 6's 0.29 6's 054

Correlation parameter ,5 -024
Lon g-run production profit iris 104 72's 191

MR Reverting speed tic 0.005 is 0.004

Variance parameter 6's 0. 31 5's 1.0
Correlation parameter [7 -023

 

Land conversion costs

Com-soy to switchgrass Ccs: 136 $/acre Switchgrass to com-soy C SC: 47$/acre

 

Discount factor r

0.08

 

34

 

bCS(fl,C)

Convert from com-
soybean to switchgrass

 

O
,I 45
’/ Stick to current use
x' bSC (72' )
c
x’ Convert from switchgrass
x" to com-soybean
71' C

Figure 1.1 Conversion boundaries

35

 

 

 

Corn-soybean

s
u
r.
g
h
c
..oh
w
S
n
.
.
.
.

 

 

400 -

u - # d u
0 0 0 0 0
0 5 0 5 0
3 2 2 1 I

sing: 22;:
9:53— 3:53. :3“

 

accn
eacu
vccn
Nccn
cch
waa~
eaaw
vaa~
Nam—
eaa—
«mam
ena—
vam—
awa—

Year

 

 

Figure 1.2 Average returns to com-soybean and switchgrass in North-central US.

(1982 -2008, in 1982 dollars)

36

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Return to Switchgrass (in $1 OO/acre)

,/ —— Dynamic Optima] Rule(GBM)
0 £55 L A_‘* ‘NPV Rule
0 1 2 3 4 5

 

 

 

Return to Com-soybean (in $ 1 OO/acre)

Figure 1.3a NPV vs. Dynamic optimal (under GBM): conversion boundaries

37

 

 

 

 

 

 

 

 

 

\
\\
\\
\\

\\
\\
\\
\

4:.
\
‘\
\
1

(a)
i
\x
J—

 

Retum to Switchgrass (in $1 OO/acre)

 

 

 

“—‘ Dynamic Optimal Rule(GBM)
O ' — ~ * NPV Rule A
O 1 2 3 4 5

Return to Com-soybean (in $ 1 OO/acre)

Figure 1.33 NPV vs. Dynamic optimal (under GBM): conversion boundaries

37

Proportion of Land in Switchgra

 

99°
UIQQ
11]

0.4 -

 

PP?
HNU)

 

O

0 2 4 6 81012141618202224262830

Year

ss
. P
on

I

— NPV Rule

----- Dynamic Optimal

Rule

 

38

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/
/ bcs // ——*— Two Way Optimal Conversion Rule
“ _ “ One Way Optimal Conversion Rule 1

.3;
Q
Q

U)
BF:
Ca
4

 

N

\
1_

Return to Sw1tchgrass (1n $100/acre)
.__ ___ _.__R \H
\
\
i
1 \

fl

 

    

w

 

09
R
'
r

1 2 3 4 5
Return to Corn-soybean (in $100/acre)

Figure 1.4a Two-way vs. one-way conversion (under GBM): conversion boundaries

39

Proportion of Land in Switchgrass

 

0.4

0.3

0.2

0.1

— Two Way Conversion

----- One Way Conversion

 

0 2 4 6 8 1012141618202224262830

Year

 

 

4O

 

 

 

 

 

 

 

h.
T
L

w
E
k
\

Return to Swntchgrass (m $100/acre)
N
k K
\ M
\
\
\ \\
\
\\

/ " sc
/ bGBM

 

 

”‘ ‘ “ Returns Follow MR

 

0‘ ‘ fl 1 2 3 4
Return to Com-soybean (in S 1 00/ acre)

”w Returns Follow GBM T!
J
5

Figure 1.5a GBM vs. MR: conversion boundaries

41

 

 

 

Proportion of Land in Switchgrass

P
u-

9
A

a
i»

P
N

P
_

=5

 

‘ ’1' ‘~.. ..... Returns Follow GBM

I ‘~-_

.' '~~--- ----- Returns Follow MR
0
l

 

 

0 2 4 6 8 1012141618202224262830

Year

 

Figure 1.5b GBM vs. MR: proportion of land in switchgrass

42

 

 

 

Return to Switchgrass (in $100/acre)

N Baseline
~ ~ — Reducing C12 by Half

 

O 1 2 3 4 5
Return to Corn-soybean (in $ 1 OO/acre)

 

 

”‘1'; 1' /
s 4 ,//
S /,/
m l/
.E bCS f
t: 3t ,5”
m /" /
g) i // / ”’ / 1,
e w” ’
r: / i
U3) / bSC
o / f
E /,///
.3 / / // ‘ Baseline
‘35 / '/ — — — Reducing C2] by Half
1 2 3 4 5

 

Return to Corn-soybean (in $100/acre)

Figure 1.6a Reducing conversion costs (under GBM): conversion boundary

43

 

 

 

P
(II
J

 

P
i—s

 

§

53

g 0.4 ~

3; --------- Reducing Ccs by
.5 0.3 Half

E Baseline

:51 0.2

g - - - Reducing Csc by
E Half

Q

a.

2

A.

 

 

G

0 2 4 6 81012141618202224262830
Year

 

 

Figure 1.6b Reducing conversion costs (under GBM): proportion of land in

switchgrass

44

 

£11

Return to Switchgrass (in $100/acre)

4:-

DJ

M

fl

 

 

 

—* Baseline

“ ~ — Double 0

 

 

 

 

 

 

 

 

 

 

0 LL’Z'_/ i
O 1 2 3 4 5
Return to Corn-soybean (in $100/acre)

3 5 /' ‘
/
g 4 bcs / / ——~ Baseline
.— ” / ,
99 / / _ — — Double as
.E / /
v .r
§ 3 i / // i
3‘0 X // f/ l
f.) 2 J / // i
g I // ,4’// f
m ' / / SC
8 y , ;’/ b 4
e 1 f -/
§ /
a 0 if/ 1 9f 1 _L_.___,_J

0 l 2 3 4 5

Return to Com-soybean (in $100/acre)

Figure 1.7a Return volatality (under GBM): conversion boundary

45

 

 

 

9
UI
l

 

 

 

::
.g 0.4 ‘
.3 § . ................................
,_ .0 3 .............................
a DD
3 g ......... Double ac
0‘0 2 _ ... .
g at: Baseline
a. _ ~
8 0.1 ~ ' "" ~. g - - - Double as
flu ~ ~ ~
0 ‘ ------- ——-_-_- --

0 2 4 6 8 1012141618202224262830

Year

 

Figure 1.7b Return volatality (under GBM): proportion of land in switchgrass

46

 

 

 

 

 

P
(II
J

 

 

:5 0.4 -

fl

3 g0 3 0 ......................................

g a .............................

5 g -------------- Double cc

° ' 0 2 4 ,.

E g Baseline

E 0.1- ’-“~\ ---Doubleos
o ‘ -------- _:_._. --

 

0 2 4 6 81012141618202224262830

Year

 

Figure 1.7b Return volatality (under GBM): proportion of land in switchgrass

46

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

//
//
70‘ //
H cs
5: 4 ~ b / -
O _//
S I
66 {K
.5 4
7,: 3 . /
(I)
‘3 /
DD
5 ,4
+0 /
E 2 * /
(I) I
o /
H ’6,
g 4
*5 1 7
a: f _
/, —— Returns Negatively Correlated
0 g/ i * _ '“ Returns Positively Correlated
0 1 2 3 4 5

Return to Com-soybean (in $ 1 OO/acre)

Figure 1.8a Return correlation (under GBM): conversion bounary

47

 

 

 

Proportion of Land in

Switchgrass

0.5 4

 

 

0.4 -
0.3 -
x" """ - ..... Returns Negatively
0.2 a ’1’ “kn“ Correlated
0 1 ‘ I” ............ Returns Positively
' Correlated
0

 

0 2 4 6 8 1012141618202224262830

Year

 

Figure 1.8b. Return correlation: proportion of land in switchgrass

48

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 r r
! /////
/////
4 L CS // A
b / /
/
//
f l
/ 4

 

L»)
\

 

 

Return to Switchgrass (in $100/acre)

 

 

 

 

 

//,
////
2 L l//
/’//
//
7
1 4
l / ' —
/ //////// ~——— Baseline, ens—0.03
4 ’fl/ — — ~ (15:0.05
0 ‘Jififl / d i i l
0 1 2 3 4 5

Return to Com-soybean (in $100/acre)

Figure 1.9b Decreasing growth rate to 0.01 (under GBM): conversion boundary

50

 

----- as=0.07
- ° - as=0.05
Baseline: as=0.03
......... (18:0.01

 

 

Proportion of Land in
Swntchgrass

 

 

0 2 4 6 8 1012141618202224262830

Year

 

 

Figure 1.9c Varying growth rate (under GBM): proportion of land in switchgrass

51

 

Appendix A
Homogenous Two-way Land Conversion Model

Model Solution

In this appendix, we solve a two-way land conversion model, of which the value
functions are homogenous of degree one in the two return variables. The key assumption
to make the value functions homogenous is that conversion costs are proportional to the

return to current use, C ij = kl-jrri. The other assumptions are maintained the same as in

table 1.1.

At time t, if the farmer chooses between keeping the land in use i, the value function

has to satisfy the Bellman equation A(l). Left hand side (LHS) is the market rate of

return of investing Vi dollars while right hand side (RHS) is the rate of return generated

from the land use i, which equals the instantaneous retum 7r,-(t)p1us the change of value

function. The no arbitrage condition requires the RHS equals LHS.

EdV’(7zc,zzS)
dt

 

A(1) rV'(7rc,7rs)=7r,-(t)+
By Ito’s Lemma, it can also be written as a partial differential equation

rV'(7rC,7rS) = 7r,(t) + aCIZ'CVfic + aserVfls
A(2) /2 2 Vi +1/2027z' Vi +p0'7r0'7rVi
+1 O-Cfl'c flcflc S 3 ”571.5 C C S S ”cfls

52

where the subscripts denote the partial derivatives. Equation A(2) captures the

relationship between the two state variables 7:6 and 71's. It must be satisfied by

i . . . . .
V (fl'c,7Z'S)1f the land 18 currently in use 1. We can also observe that value functions

should be homogenous of degree one in 7rc and 7:3 .Thus the optimal land conversion

7!-
rule depends only on the ratio of the two land returns. We define dz'j — ——1, then the
”i

value functions can be written as V l = 71'1- * gl(dI-j). Successive differentiation gives

”i=g i(dij) dijgdij (di 'j) Vflly zgzlg<dij)
_zd‘2 . .
A3 Vi ’° (dz-j) V’ = ’ (d--)/7r-
U 70-70 _ #ng 'ijd ”1’71 gdo'dzy' ’1 ’
”I
" =—g" (door-Hr?)

Substituting these derivatives into equation A(2) and grouping terms, we obtain two

ordinary differential equations for each unknown function g1 (d 1]) .

A(4)

1/2}3l_CSo-.2 (1057(1wa j-—)+(uj ul-d)l-jgdij(dl- )+(u,.—r)gi(d,-j)+1=0

53

general solution and a particular solution:
A(5) gl(dl]')=Aildyflil +Al'2di'fi2 +1/(r—ai)

Where Ail and AiZ are constants to be determined and [Bil and [fl-2 are the positive and

negative roots of the following fundamental quadratic equation

A(6) 1/2(0-3 +03 -pO-Co's)fli(fli —l)+(aj —al)fli +(ai —r)=O

Corresponding] y the value function V i can be expressed as
i _ , , 51'] A- dfliz ./ _ .
A(7) V (zc,zS)—7r,(A,1dI-j + ,2 ij )+7r, (r at)

There is a nice economic interpretation about A(7): the second term is the expected NPV

if the status quo is maintained forever while the first term is the value to have the option

to convert land to use j.

Up to now we have characterized the conditions that the value functions need to satisfy

to stay in the current use. To fully solve the optimal land conversion rule, we need to find

a:
the constants Acl , AcZ’ AS], and A52 as well as the critical land conversmn ratio dcs

and (1:6. This requires the value functions to satisfy three additional boundary

conditions.

54

 

A(8) V‘(o,zrj)=0 <6)

A(9) Value-matching condition V i(7rc, 72's) = V j (”ca ”5) — kl-J-Jrl- when
* ,
7! j = j(7fi)

V]; (£6,713) 2 V7; (news) — kij
A( 10) Smooth-pasting conditions it I_ when
Vflj (news) = ng ow.)

*
”j: jUTi)

The first boundary condition A(8) is derived from the fact that V1.0) will be zero if the

75- goes to zero since VI = 7r,- *gl(dl-j). This implies A12 2 0. The other two

conditions come from consideration of optional conversion (see details in Dixit and

Pindyck (1994)), chapter 4). When 7! j > 7Z'i, converting land from use i to j is optimal if

' . . * ..
7! j 18 greater than the critical boundary 7! j (7Q) (denoted by by )- AS we noted above,

*
7Tj(7’i) . .
, whlch is the slope of the

 

*
the decision depends only on the ratio dij =
7:,-

Conversion boundary. On the conversion boundary, the value functions need to satisfy the

value matching conditions and smooth pasting condition. Value matching conditions says

that on conversion boundary the farrner is indifferent between staying in current land use

55

and converting to alternative use. Smooth pasting condition impose the equality of

a:
marginal changes in the continuation value and the stopping value when fl' j = 7: j (Iii) .

Now substituting the A(5) into equation A(9) and A( 10) for i = c,s respectively and

grouping terms, we can obtain a system of four equations with four unknowns, Acl ,

* * .
A31, dcs and dsc. Since the term Aizdf’z is eliminated, we depress the notation 1 in

the Ail and flu and simply use A,- and ,6, in the following.

AM; We — Asd:s(“‘/”s) + l/(r — aC)- dc, /(r — as ) + kc, = 0
flcAc(dZs>‘/”c“” — (1 — fismsdl'fl’S — dcs /<r — as) = 0

AU 1) Ac(d:c)(1-flc) - A3424 + d; /(r — ac) — l/(r — as) + ksc = 0

(1" 18c)Ac(d.:c )‘flc " flsAsd:c(IBS -1) + l/(r ‘“ ac) : O

The critical ratio of land conversion can be solved after parameterize the system of

equations A(1 1). We use the same parameters as in Essay 1 table 1.1 except that the

conversion costs parameters. The kcs and ksc are assumed to be 0.9 and 0.3 respectively.
9 . . l * 0
Given these parameters, the critical return who of dcs to lnduce farmer to convert land

*
from com—soy to switchgrass is 2.39. The critical return ratio of dsc for farmer to make

the opposite conversion is 2.12. As expected, the critical ratios are higher than 1 due to

uncertainty and costs. The farmer with an opportunity to convert land to an alternative

56

 

 

use (switchgrass) is like holding an “option”, which enables him to wait for more
information to arrive and make more informed decision. Once he converts, it is costly to
convert back to original land use (com-soybean) if the market conditions turn out to be
less favorable. Thus, sticking to current land se has an additional value, called “option
value”, derived from converting to alternative land use (switchgrass). Once the farmer
converts land to switchgrass, there is a similar option value associated with converting
back to corn-soybean. These two option values are mutually dependent and need to be

solved simultaneously as shown in the conditions A(9) and A(lO).

The result is shown in figure A1 where the switchgrass return increases from zero to
$500/acre. Each critical boundary separates a waiting region (staying in current use) from

an exercise region (converting to alternative use). As shown in figure Al , the two

conversion boundaries starting from origin divide the ”c —- 7:5 region into three parts:

above [JCS , it is optimal to convert land from com-soybean to switchgrass; below I)“ , it
is optimal to convert land from switchgrass to com-soybean; between them, it is optimal
to stay in current use. Once the land is already in one use, the farmer is reluctant to
convert to the alternative use unless the retum from the latter is sufficiently high. There is

a large inaction zone between the conversion boundaries.

Comparative Statics

a: a:
We can examine the sensitivity of the critical return ratios (1 cs and d sc to the change of

the key parameters. In essay 2 we will analyze the effects of the subsidies, which could
change the dynamics of the switchgrass return. These comparative statics can help

explain the later simulation results.

57

Different from a one-way land conversion model, any change associated with one land

use return in a two-way land conversion model, such as the variance or conversion costs,
will affect the land use decision rules in both directions. First we vary variance as from
O. l to 1, holding other parameters unchanged. A rise in the uncertainty over switchgrass

return will increase the option value, not only to convert land into the switchgrass but

a:
also to convert out. Therefore it requires higher current thresholds of return ratios, d cs

1:
and dsc , to make both direction conversions. The results are shown in figure A2.

Second, we vary drift rate of switchgrass return, as , from 0.01 to 0.07. The drift rate

in the GBM represents how fast the stochastic variable is expected to grow. Anticipating
a faster growth rate of switchgrass return the farmer growing com—soy is more willing to

convert land to switchgrass while farmer growing switchgrass is less willing to withdraw

*
land from it. Therefore, critical ratio of converting into switchgrass, dc , is raised while

2|:
the critical ratio of converting out of switchgrass, dsc , is lowered (figure A3).

Third, we examine the effect of conversion costs on optimal conversion rules by

varying the proportional parameter of conversion costs from corn—soybean to switchgrass,
kcs , from 0.1 to 1. Clearly, the higher the conversion cost from com-soybean to
switchgrass is, the more costly to falsely make the conversion. The com-soybean grower
will therefore be more reluctant to convert to switchgrass. Moreover, higher kcs _
increases the expected cost of converting from com-soybean back to switchgrass, thereby

increasing the implied cost of switching to com-soybean. Thus it indirectly depresses the

switchgrass grower’s incentive to convert land to com—soybean. The direct effect of

58

increasing kcs on critical ratio of converting into switchgrass is larger than the effect of

the indirect effect on critical ratio of converting back to com-soybean. As shown in figure

* *
A4, dcs rises faster than dsc.

Last, we examine the effect of return correlation. In the absence of uncertainty, the
coefficient of correlation is meaningless since land returns do not change. Under
uncertainty, a positive correlation implies the returns will more likely move to the same
direction. The change of relative values will be smaller than when the correlation is
negative. Thus the farmer will have less incentive to wait and option value of waiting is
reduced. With perfect correlation, the current land use will remain preferred forever so
that the option value of the alternative use will disappear. As shown in figure A5,
correlation between the two crop returns changing from -O.5 to 0.5 reduces the critical
return ratio of both directions. This moves the conversion boundaries towards each other

and reduces the inaction zone.

59

 

 

Switchgrass Return (S/acre)

1400

1200

1000

800

600

400

200

 

 

 

_ bCS
Convert Land to Corn-soybean
- dcs*
Stay in Current Use
000000000 dsc*
.. bsc
.................................... Convert Land to Switchgrass
‘T...:";“.r I T 1 T T 1— l 7 T r T l 1 T T T I— I—j—Tl W

0 40 80 120 160 200 240 280 320 360 400 440 480
Com-soybean Return (Slacre)

 

Figure A.l The critical land conversion boundaries

60

 

 

 

 

 

 

3.5 ~
3 ..
2.5 a
G
‘3
n: 2 ‘
5 15
0: 0 fl *
U dcs
l _ 000000000 dsc*
0.5 -
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
GS

 

 

 

Figure A.2 Sensitivity of the critical return ratios to the variance of switchgrass
return

61

 

2.5 -

2.4 “ \

2.3 *

 

dcs*

2.2 "
......... dsc*

Critical Ratio

2.1 ..

 

 

1.9

0.01 0.02 0.03 0.04 0.05 0.06 0.07
as

 

 

 

Figure A.3 Sensitivity of the critical return ratios to the drift rate of switchgrass
return

62

 

 

 

 

 

2.5 r
E
a 2 q
a:
"g l 5 -
° dcs*
"J
U 1 fl ......... dscir
0.5 r
0

 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Conversion Cost Ks

 

Figure A.4 Sensitivity of the critical return ratios to conversion costs

63

 

 

 

DJ
J

 

 

 

2.5 a
3
a 2 d
M
'5 l 5 ‘
IE 1 - dcs*
U ......... dsc"
0.5 a
0

 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Conversion Cost Ks

 

Figure A.4 Sensitivity of the critical return ratios to conversion costs

63

 

 

2.5“ —

 

 

dcs*

......... dsc*

Critical Ratio
g—A
UI

0.5

1

 

 

-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5

Correlation Coefficient p

 

 

 

Figure A.5 Sensitivity of the critical return ratios to correlation coefficient between
two return series

64

Appendix B
Numerical Solution for the Optimal Land Use Decision Model

The numerical solution is implemented in Matlab using OSSOLVE solver (F ackler 2004)
and CompEcon toolbox Miranda and Fackler, 2002). More reference can be found in
Brekke and Brent Oksendal (1994) and Qi and Liao (1999). Our multi-states and
dimensions optimal land conversion problem (3) can be reduced to a set of optimal
conditions that value functions satisfy Bellman differential equations and boundary

conditions respective to different states:

. . . 2 .
(-) er(7Tc,7Ts)_”i(t)—ac(7’c’t)V,l,c —as(n's,t)V7'Is —1/206(”C’t)Vzlrczrc
l

2 . . .—
_1/205(7[S’t)V71zsrr5 — pO'C(7rc,t)0'S(7rS,t)V7’rcflS Z O z— c,s

3(1) (ii) Vi(7rc,7rs)2Vj(7rc,7rS)-C,-j t.jeSandi¢j

(iii) either of the (i) or (ii) has to hold as strict equality.

We cannot obtain the explicit form of value functions and have to rely on the
numerical solution methods. This problem falls into a more general category of so called

functional equation problem, e. g., we need to find a function f that satisfies:

13(2) Tf = 0

65

method is a function approximation method widely used to solve functional equations

and will be employed here (Judd, 1998; Miranda and F ackler, 2002).

Collocation method is to approximate unknown function f by using a linear

combination of n pre-defined basis functions drawn from a family of approximating
.. n
functions: f (x) = 20 j¢ j (x). The n coefficients Cl ,62 - - -C” can be found by solving
1

a system of n equations resulting from evaluating the approximants at n nodal points

x1 9x2 ~ ° -x” and requiring them to satisfy the functional equations:

11
3(3) g(x,-,ch¢j(x))=0 V1'=l,2---n
l

The collocation method thus can approximate the function indirectly by using the

rootfinding techniques for non-linear equations, for example, Newton’s method or quasi-

Newton methods.

Specifically, the value function V l(JZ'C,7Z’S)can be approximated by a linear

combination of n (”c * n3 ) known basis function.

, ”c ”s
8(4) V’(7rc.rr.)= Z Z Cjcjs¢jcjs(7rc’”5)
Jb=1j5=1

66

and (91 is a vector of coefficients for the value functions associated with the ith state. The

family basis function we use is piecewise linear spline with a 101-degree on the
interval {(fl'c, 71's ) l 0 S 7Z'C S 10,0 S ”s S 10} . More details about how to define basis

function can be found in Miranda and Fackler (2002, page 136-149) .

Define the approximation differential operator:

Miran.) = mew.) — err-(70.0%,- (news)

3(5) — 1/2012¢”i”i (717,1) ‘ pacas¢7ri7ri (”1'30

Now the optimality conditions B(l) can be written as

(i) fli(il'c,7[s)6i —72'- 2 0 1' = 1,2
B(6) (ii) ¢(7rc,7rs)6l,- — ¢(fl'c,fl's)6j — Ci]- 2 0 w ¢ j
(iii) either ofthe (i) or (ii) has to hold as strict equality.

. . 1'
Define the matrix (D and B1 as the function ¢(flc,7rs) and ,6 (HCJZ'S) evaluated at
the state variables nodal values. The optimality conditions (1) can be reformulated as an

Extended Vertical Linear Complementarity Problem:
B(7) G(z)=min(Mcz+qc,Msz+qS)=O
Where 2 is the state associated coefficients 6:. stacked vertically. The M, and q is the

given by

67

3(8) Mczl—B; glq =[C:clcs..lM =[o Bs qs=[C::I15n]

where 1n is a column vector composed of n ones and the min operator is applied element
wise. The collocation method can be used to solve for 2. Notice that the function (3(2) in
EVLCP problem is not continuously differentiable, thus standard Newton method is not

applicable. Qi and Liao (1999) proposed a smoothing Newton method by using the
aggregation technique. Rewrite G(z) as = —max {—(M CZ + qc ), —(Msz + ‘15)} and

define an aggregation function G(t, Z):

—|t|ln( Z exp(—sz + q 1),/M) if t > 0
j=c,s

G(z) if t: 0

B(9) (G(t,Z)),- = i

The smoothing parameter tis defined as the first component of the following equation:

B(10) H(t,Z) = (Got 2)]

The equation H (t, Z) can be solved by invoking Newton method and enforcing the

parameter t positively converging to zero. The point is to approximate the nonsmooth
function G(z) by a strongly semismooth function H (t, Z) by using the aggregation
technique. The solution of the G(1,Z) is the same as the function H (t,Z ) when tis equal
to O. The function H (t, Z) can be approximated by the Newton method for every
parameter (>0 and gradually reduce t to zero. Qi and Liao proved that when t

approaches zero, the solution to H (t, Z) approaches to the solution to 0(2) 2 O . Based on

68

the algorithm proposed by Qi and Liao, Falker (2004) implement the algorithm in Matlab

coded a procedure OSSOLVE.

69

Appendix C
Specification Tests

Testing Unit Root: Dicky-Fuller Test and Phillips-Perron Test

Dicky-Fuller (DF) test requires to run. the regression
A111 717,; = ,50 + ,8] In ”Lt + 31',“ i: CBS-The null hypothesis is the infra has

a unit root or ,6, = 0. There is no evidence of AR( 1) serial correlation in the logarithms of

both crop returns. To address other type of potential serial correlations, we perform

Philips-Perron (PP) test (Phillips and Perrom 1988), which requires to run the regression
on in in", = 70 +7111] ”Lt—l +1113), i: C, S. The null is In 7Q, has a unit root or 71 =1.

For logarithm of return to com-soybean, both DF test and PP test can not reject the unit

root. Similar results are found for the logarithm of return to switchgrass.

Testing Stationarity: KPSS Test

The KPSS test (Kwiatkowski et al. 1992) can have a null hypothesis of either level
stationary or trend stationary. We choose the level stationary as our null hypothesis. For

both logarithms of returns to com-soybean and switchgrass, the null can not be rejected at

10% level.

Testing Serial Correlations and Heteroskedasticity

To estimate the parameters in GBM and MR processes, we discretize them into equations
(9) and ( l l), in which conditional variance term is assumed to be serial uncorrelated and

constant. Here we test whether the error terms show AR( 1) type serial correlation and

70

ARCH type of heteroskedasticity. Following Wooldridge (2000), the steps of testing for

AR(l) serial correlation in GBM are as following:

1) Run the regression (9) save the residues £13! for all t.

2) Run the regression of éi’, on éi,,_1 and explanatory variables in (9).

3) If the coefficient of gig—1 is statistically different from zero, then there is sign of

serial correlation.
The steps of testing for heteroskedasticity are as following:

1) Run the regression (9) save the residues 5),, for all t.

2) Run the regression of 53, on 53,4 (including constant).

3) If the coefficient of 53,4 is statistically different from. zero, then there is sign of

ARCH( 1).

The same steps apply to MR process (equation 1 1). Table A4 and A5 show the testing

results. In summary, there is no evidence of serial correlations and ARCH(1) for the two

returns under both GBM and MR assumptions.

7]

Table C.1 Simple Dicky-Fuller Test Results

 

 

 

 

Coefficient ,6] and t-statistics

Lagged logarithm of return to com-soybean -0.31 (-2.16)

Lagged logarithm of return to switchgrass -0.37 (-2.53)

 

 

 

   

Dickey-Fuller critical values: 1%, -3.45; 5%, -2.87; 10%, -2.57.

 

72

Table C.2 Phillips-Perron Test Results

 

Test statistics for 71 = 1

 

Lagged logarithm of return to com-soybean -2.07

Lagged logarithm of return to switchgrass -2.39

 

Critical values: 1%, -3.743; 5%, -2.997; 10%, -2.629.

73

Table C.3 KPSS Test Results

 

Test statistics on 8th lag

 

Logarithm of return to com-soybean 0, 34

Logarithm of return to switchgrass 0.26

 

Maxlag = 8 chosen by Schwert criterion. Critical values for Ho: log III-,1 is trend stationary: 1% , 0.74; 5% ,

0.46 ; 10%: 0.35 .

74

Table C.4 Serial Correlation AR(l) Test

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Coefficients and t-statistics on Lagged Errors

 

 

 

 

 

 

 

 

 

 

 

 

GBM MR
Com-soybean -0.3 7(-1.36) -0.38(-1.67)
Switchgrass -O.20 (-0.64) -0.22 (-1.02)

 

 

 

 

 

 

 

 

 

 

 

 

75

Table C.5 Heteroskedasticity ARCH(1) Test

 

Coefficient and t-statistics on Lagged Squared Errors

 

GBM MR
Com-soybean 0.36 (1.91) O.38(2.02)
Switchgrass 0.19 (0.94) 0.05 (0.25)

 

76

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79

Essay 2
Alternative Land Use Policies: Real Options with Costly Reversibility

Introduction

Agricultural subsidies have been used to induce socially desirable land uses for a long
time. An example is the US. Conservation Reserve Program, in which farmers set aside
production land to provide environmental benefits and receive payments from
government in return. One strategy to mitigate climate change proposed in the United
States and Canada is subsidizing farmers to convert the marginal agricultural land to
forest for more carbon sequestration (Stavins 1999; McKenney et al. 2004; Lubowski,
Plantinga, and Stavins 2006). A body of literature has analyzed the effects of subsidies
on the land use change, such as Stavins and Jaffe (1990) and Plantinga, Mauldin and
Miller (1999). A common assumption in these studies is that a farmer will compare the
expected net present value (ENPV) of returns to different land uses and choose the one
with the highest ENPV. Thus subsidizing a desirable land use will raise its return and
induce land converted to it. ENPV decision rule implies that the form of the subsidy, such
as lump sum or continuous, constant or variable, does not matter. Subsidies that are equal
under the ENPV rule are implicitly assumed to give farmers the same incentive to convert
land to the desirable use.

It has been observed that farmers often do not convert land even it is profitable to do
so under the ENPV rule (Isik and Yang 2004; Plantinga et al. 2002). Parks (1995)
CXplained land conversion hysteresis as a consequence of risk aversion and expected
caPita] gains. He also explored the effects of some types of conversion subsidies.

Although not explicitly clear, in his model a cost-sharing subsidy and a constant annual

80

rental payment, which do not change the uncertainties of land returns, can give a farmer
the same conversion incentive if their annualized values are equal. In contrast, the real
options framework shows that the interaction among irreversible sunk cost, uncertainty,
and learning can cause even risk-neutral farmers to be more reluctant to change land uses
than the ENPV rule predicts (e. g., Titman 1985). More importantly, subsidies taking
various forms can affect the conversion costs, the level and uncertainty of land use
returns differently. Subsequently they will affect the farmers’ land conversion decision
differently even though they are ENPV-equivalent.

The purpose of this paper is to compare the long term effectiveness of different forms
of agricultural subsidies in achieving an increase in a desired land use. We adopt an
innovative real options approach by relaxing the absolute irreversibility assumption in
previous literature and allowing for land use conversion in two directions. A farmer
deciding on converting to another land use is allowed to take into consideration the future
possibility of converting the land back to its original use under plausible market
conditions. The absolute irreversibility assumption might be reasonable for urban
development (Capozza and Li 1994; Abebayehu, Keith, and Betsey 1999). However for
agricultural land, a farmer can switch between different uses with costs. Allowing two-
way land conversion can help capture the flexibility of farmer’s land use decisions.
Moreover, it has important implications in designing subsidy programs since the
subsidies not only change the farmer’s willingness to convert land into the desirable use
but also the willingness to convert it out.

To make our ideas concrete, we evaluate land conversion subsidies in the context of

encouraging production of energy crops, which can be directly combusted to provide

81

electricity or converted to transportation fuel. Globally the market demand for energy
crops is largely driven by various renewable energy policies. For example, in the United
States, more than 20 states mandate Renewable Portfolio Standards (RPS), which require
a certain minimum quantity of electricity produced from eligible renewable energy
sources. Biomass is an eligible energy source in some states. But more (potential)
demand for energy crops may come fi'om cellulosic biofuel production. Currently liquid
biofuels are strongly advocated in many countries, including the United States, due to
political concerns related to energy security, climate change and rural development
(Khanna 2008 ; Raj agopal and Zilberman 2007). Although grain-based biofuel currently
dominates the market, cellulosic biofuel is believed to have superior environmental
performance, such as higher net energy, higher carbon credit and more-environmental
friendly-feedstocks (Schmer et al. 2008; Paine et al. 1996). For this reason, the US.
Energy Independence and Security Act (EISA) of 2007 mandates the use of cellulosic
ethanol, increasing from 0.1 billion gallons annually in 2010 to 16 billion gallons in 2022.
To meet this mandate, significant expansion of energy mom is expected to occur on
agricultural land and compete with traditional crops for the limited acres (Thomson et al.
2008; Walsh et al. 2003).

Coupled with energy policies that induce energy crop production through creating new
markets for them, many countries also use agricultural subsidies to provide direct
production incentives. The perennial nature of most energy crops involves sunk costs to
establish the plants, which may become prohibitively high for some woody crops. To
overcome this barrier, a lump-sum payment is often provided to cover the establishment

costs in full or partial. In the 19905, Sweden offered 10,000SEK/ha (roughly $573/acre)

82

establishment subsidy for planting willow (Helby, Rosenqvist, and Roos 2006). In early
2007, the Irish government announced it would subsidize half of the establishment costs
for willow and miscanthus (Styles, Thome, and Jones 2008). In the United States, the
Food, Conservation and Energy Act (FCEA) of 2008 introduced direct payments for up
to 75% of establishment costs for eligible energy crOps. In addition to a cost-sharing
subsidy to help start-up, annual payment is also provided by governments to support
production, collection, harvest, storage and transportation of energy crops. For example,
European Union (EU) farmers can receive an annual payment of €45/ha (roughly
$25/acre) for growing energy crOps on production land (Raj agOpal and Ziberman 2007).
The Irish government subsidizes additional €85/ha (about $45/acre) for growing willow
and miscanthus (Styles, Thome, and Jones 2008). In contrast, the US. farmers can
receive a payment to cover costs of harvest, storage and transportation that is equal to
what they obtain from biorefiners for 2 years (up to $45/ton). This type of subsidy will
vary with the market price and yield of biomass. F CEA also required the Federal Crop
Insurance Cooperation to study the insurance policies for energy crops, providing the
future possibility of subsidizing the energy crops insurance. Given that large subsidy
amounts are spent and take different forms, the effectiveness of these subsidies should be
systematically evaluated.

Our paper contributes to the literature in several aspects. First, we examine a range of
ENPV-equivalent subsidies, showing how they can affect a representative farmer’s
optimal land conversion rule differently, depending on their effects on the conversion
costs, returns level and variability of returns. Second, we examine how optimal land

conversion strategies differ between a real options framework assuming irreversible land

83

use decisions and our framework, which allows reversion to a prior land use. In this
framework, it turns out that subsidies not only change the farmer’s willingness to convert
land into energy crops but also the willingness to convert back out. Third, based on this
improved model, we compare the effectiveness of subsidy programs for encouraging the

production of energy crops.

The remainder of the paper is organized as follows. In the next section, we present a
general land conversion decision model without governmental intervention to better
expose the idea of uncertainty and sunk costs causing hysteresis in land conversion. Next,
we examine how various forms of subsidies for energy crops can change a representative
farmer’s land conversion decision rule differently even though they are ENPV-equivalent.
Then we perform a Monte Carlo simulation on the farmer’s annual land use choice under
each type of subsidy over a period of 30 years. The subsidy levels are calibrated so that
they have the same expected cost to the governmental and their long-term performances
are compared according to the increased expected conversion rate into energy crops than

no subsidy support. Finally we give results and conclusions.
Land Conversion Decision Model
Decision without Governmental Intervention

Consider a representative, risk neutral farmer with a unit of land facing two competing
crop production alternatives: a corn-soybean rotation and switchgrass, which are selected

as representative of a traditional crop and an energy crop. The returns to com-soybean

and switchgrass at period t are denoted by ac (t) and as (t) , respectively. The farmer

can convert land from com-soybean to switchgrass with a lump-sum cost Ccs or vice

84

versa with a lump-sum cost C 30’ The farmer seeks to maximize the net present value of

current and future returns at a discount rate r over an infinite time horizon. The future
returns to com-soybean and switchgrass are assumed to evolve according to geometric

Brownian motion (GBM) 1:
(1) dfl'i =a,-7r,~dt+0',-7ridzl- ie{c,s}
where dz, is the increment of a Wiener process. The correlation coefficient of the two

return processes is p , i.e., E (dzcdzs) = pdt . Traditional crop and energy crop returns
could be correlated for a variety of reasons, e. g. both are linked with energy prices and
are subject to macro-economic shocks.2

According to the ENPV decision rule, the farmer will switch from one crop to another

when the ENPV of switching is higher than staying, i.e.,

(2) Convert if Efflj(t)e—rtdt—CU _>_ EfflI-(tk—rtdt i,jE{C,S}, andiij

The real options literature has pointed out that the ENPV approach ignores that the agent
can optimally postpone their irreversible actions. In this case, the farmer with an
opportunity to switch crop is like holding an “option”. When he exercises this option
(convert to another crop), he gives up the opportunity of waiting for new information that
enables him to make more informed decision. It is costly to revert it back to original crop
if the market conditions turn out to be undesirable. The lost option value is an opportunity

costs that must be include as part of cost of land conversion.

 

l . .
We drop off the time t to simplify the notation whenever it does not cause confusron.

Land switching between the two crops could make the crop prices endogenous. However, explicitly
modeling this relationship is beyond the scope of our paper.

85

Next we derive the optimal decision using real options approach. Let V1 (ac, as) be

the value function of currently being in land use i, which is defined as the expected net
present value of all future returns starting from com-soybean and then following optimal

policies. Due to the option of converting into use j ¢ i, the payoff depends on the
distribution of future returns of both land uses, the information for which is contained in
the two current returns, aC (t) and as (t). At time t, the farmer chooses between
keeping the land in use i and converting it into alternative use j:

7;,(t)dt + e""d‘EV" (a, (t + dt), 7:, (t + dt)),

(3) Vi(7tc,7rs) = max ,
VJ (”c(t)a7[s (t)) ‘- Cij

The first term on the right hand side describes the payoffs if the land is kept in use i: in
the infinitesimal period [t,t + dt] , the farmer receives profit from land use i at rate 7!,- (t) ,

and at the end of the period, receives the new discounted expected payoff

6".th Vi(t + dt). The second term on the right hand side describes the payoff if the

land is converted into use j: the farmer receives the expected payoff of use j, VJ (t), but
incurs the conversion cost Cij .

Intuitively, the conversion decision will depend on the relative returns of the two

competing crops. For example, for any return level of ac , there will be a critical value

* *
as With which continuing in corn-soybean is optimal if as < as and conversion is

- . * * u l o o -
optimal if as > as . The as (ac) wrll form a critical conversron boundary in the
ac — as space. Similarly, there is another conversion boundary from switchgrass to

86

com-soybean a; (a sw) . Following the standard procedures of solving the real options
problems, we can characterize the optimality conditions of our land conversion decision.
In the continuation region (where the agent continues in current land use), the value
functions need to satisfy the following the equation:

rV'(ac,7rs) == tri + actrcV;r + asaszis
C

(4)

+1/2o'37rcszicfl +1/20s2a3V7:S”S + pacasacasV;

C cfls

This is a no-arbitrage condition expanded by Ito’s lemma, implying that the rate of return

of investing V i dollars (left-hand side) should equal the rate of return generated by land
use i (right-hand side). On the boundaries of conversion, the payoffs of continuing in the
current use should equal the payoffs of converting minus the conversion costs, along with

their derivatives. These are the value-matching and smooth-pasting conditions:

. . *
(5) Vl(]z'c’]z's):VJ(f[c,f[s)—CU when ”i :fl'i (Hf) i,j€{C,S}, alldiij

6V’(7rc,7rs) : 8Vj(7rc,7rs)
(6) (MC arc

and

 

 

 

 

awn.- ,7rs) = aVl(7rc.7ts) when 7,1,: ”l(flj) i,j€ {6,5},andi¢j
air, an,
The system of equations (4)-(6) subject to (1) implicitly defines the unknown value
functions and two conversion boundaries. Since the land in the alternative use
(switchgrass) can be further converted back to original use (com-soybean), theoption

value of converting from corn-soybean to switchgrass further depends on the option value

associated with converting in the other direction, from switchgrass to com-soybean. The

87

mutual dependence of the two option values significantly complicates the solution
algorithm. Except in special cases, such as when value functions are homogeneous of
degree one, there is no analytical solution to (4)-(6). Instead, we solve the model
numerically using the collocation method (Miranda and F ackler 2002; F ackler 2004).
This method approximates the unknown value functions using a linear combination of it
known basis functions and fixes the coefficients by solving a system of n equations that
are derived from the optimality conditions (4)-(6). Appendix B provides more details.
Table 2.1 presents the parameters we use to solve the model. More details about the
parameter estimation are documented in the first essay (Song, Zhao, and Swinton 2010).
In summary, historical data on corn and soybean returns were obtained from the US.
Department of Agriculture (USDA), while data on switchgrass returns were constructed
from historical ethanol prices and production cost that are taken from various sources.
The drift parameters and variance parameters of the two crop return series were
econometrically estimated. The land conversion costs were taken from literature (Khanna,
Dhungana, and Clifton-Brown 2008). We assume that the two returns have a correlation
coefficient of 0.3, instead of -0.24 as estimated in essay 1. Historical returns to com-
soybean and switchgrass could be negatively correlated as indicated by our estimation
results. Part of the reason is that switchgrass revenue is simulated as a function of
petroleum price and thus highly positively correlated with petroleum, whereas until 2005,
com-soybean returns were negatively correlated with petroleum prices due to petroleum
used as transportation and fertilizer inputs. However, this pattern of negative correlation
could change as more com and soybean are used to produce biofuels, and as agricultural

and petroleum markets become more integrated. Then high petroleum prices may push up

88

corn and soybean prices, increasing their returns. A supporting evidence is that the
correlation between the annual ethanol price and com-soybean return for year 1982-2005
is -0.07, and it changes to 0.28 for year 2006-2008. Tyner (2009) shows similar result
that the price correlation between crude oil and corn change from -0.29 during period
1988-2005 to 0.8 during period 2006-2008. Furthermore, the positive correlation may
become stronger as switchgrass or other energy crops expand production and compete

with com-soybean for limited land.

Figure 2.1 shows the two boundaries for conversions from com-soybean to
switchgrass ([965) and from switchgrass to com-soybean (bSC ). The two boundaries

divide the ac — as space into three regions. Above the boundary beS , it is optimal to

convert from corn-soybean to switchgrass. Below the boundarybsc , it is optimal to
convert from switchgrass to com-soybean. Between the two boundaries, it is optimal to
keep land in its current use. The large inaction zone indicates significant hysteresis in
land conversion decisions. For instance, the calculated switchgrass returns based on 2008
prices is $135/acre while the com-soybean return in 2008 is $92/acre (both in 1982

dollars).3 If the land is currently in com-soybean, the minimum switchgrass return for

converting the land to switchgrass is bcs (92) = $295/acre, which is significantly higher

2009 corn and soybean returns are not available yet from USDA.

89

than the $ 99/acre threshold value under ENPV rule.4 Thus, the land will be kept in a
com-soybean rotation even though 72'sW > acs . Conversely, if the land is already in

switchgrass, the required minimum com-soybean return for converting into com-soybean

is about $280/acre. Thus, the land currently in switchgrass will not be converted either.

Decision under Diflerent Subsidies

Above we have described various subsidies for supporting energy cr0p production
currently used or proposed in many countries. They can be categorized into four types: (a)
a constant annual subsidy, denoted by f; (b) a variable annual subsidy, which is a

percentage of return, denoted by 77; (c) an insurance policy, which guarantees a minimum
annual return of 7L; from energy crops; and (d) a lump-sum payment made to the

switchgrass grower either for the first year of growing switchgrass or for the
reestablishment after a 10-year rotation, denoted by s. The constant subsidy and variable
subsidy are abstracted from annual payments used in European countries and the United
States, respectively. The insurance subsidy is a mimic of the proposed revenue-based
commodity support program in F CEA, which provides payment to farmers when the
market revenue falls below an expected or target revenue (more details can be referred to
Cooper 2009).

If farmers are risk-neutral and make decisions according to the ENPV rule, different

forms of subsidies can give them the same incentive to convert land to energy crOps as

 

 

 

4 r - a v
The conversion boundaries under ENPV are: bfip V (a C) = as S + (r - as )Ccs for
_ c,
. . sc r - ac
conversron from corn-soybean to swrtchgrass, and pr V (as) = as - (r — ac )Csc for
" s

conversion from switchgrass to com-soybean.

90

long as they have the same ENPV by equation (2). This implies that for a given
governmental budget, these subsidies will perform the same in terms of attracting the
land to grow energy crops. However, using the dynamic land conversion decision model
developed above, we will show that ENPV-equivalent subsidies can affect the land
conversion costs, instantaneous returns and variability of energy crops differently,
causing the optimal land conversion rules to differ.

For each type of subsidy, the value functions need to satisfy the corresponding
Bellman equations in the continuation region and the value matching and smooth pasting
conditions on the boundaries of conversion. These conditions are summarized in table 2.2.

Constant and variable subsidies will be added to the instantaneous return to switchgrass,

which are as + f and as + 77as , respectively. Under an insurance subsidy, the

. . . S . .
mstantaneous payment 1n Bellman equation of V rs max (as,as). The value-matching

conditions and smooth pasting conditions for these subsidies are the same as (5) and (6).

. . . . i
For a one-time cost-sharmg subsrdy, the Bellman equations for V are the same as (4),

but the conversion cost Ccs is reduced by s in the value-matching condition for

converting from com-soybean to switchgrass. The smooth pasting condition is the same

as (6).

The farmer’s Optimal land conversion rule under different forms of subsidy will be
solved using the same projection method described in Appendix B. The subsidy levels
need to be determined before the optimal land conversion model is solved. To make a

meaningful comparison, we need to calibrate the subsidy parameters such that the ENPVs

91

of governmental payments over a period are the same. The details about the calibration

are presented in the next section.

Simulation of Land Use Choice

Given the optimal land conversion rule, a representative com-soybean grower will
convert land to switchgrass when beS is reached while a representative switchgrass

grower will convert to com-soybean when I)“ is reached. With stochastic returns, we
can compute the ex ante expected probability of a unit of com-soybean land converting to
switchgrass within a period of time. Previous real option literature (e. g. Leahy 1993;
Pyndick and Dixit 1994) has show that in a competitive industry the Optimal investment
policy derived in a single-firm partial equilibrium setting happens to coincide with the
optimal policy rule in a general equilibrium if all firms share the same risky return
process. Given a large number of firms in that industry, the ex ante probability of
investment will also measure the fraction of available investment we can expect to be
implemented. Following Metcalf and Hasett (1995) and Sarkar (2003), we assume that
the farmers are homogenous and subject to the same stochastic processes of crop returns.
We also assume that governmental has a subsidy program whose goal is to cost
effectively attract more land to grow energy crops. Given the same governmental budget,
the higher proportion of land that a subsidy program can convert into switchgrass, the

more effective it is.

Given the optimal land conversion rules, we can simulate how the farmer responds to

the changes of land use returns and calculate the governmental costs under different

92

forms of subsidy. The simulation steps are illustrated by figure 2.2 and summarized in the
following. Note that the steps in dotted rectangular are repeated.
First, we simulate N (=5000) sample paths of com-soybean and switchgrass returns

over 30 years according to the joint stochastic processes parameterized by values in table
2.1, denoted by (ac’n (t), ”s,n (t)) for n=1,2,. . .,5000 and t=0,1,2,. . .30. This is done
with the Econometric Toolbox in Matlab. The initial returns are assumed to be at 2008
level, ac (0) = $92/ acre and a s(0) = $135/ acre , respectively. The initial land use

is com-soybean production.
Second, for each type of subsidy, we initially select a subsidy level and solve the land

conversion decision rules.

Third, for each simulated path of com-soybean and switchgrass returns, given critical
land conversion boundaries under different types of subsidies, we can predict the land use

assuming that the farmer acts according to the Optimal land use decision rule. Each

sample path of the two returns, {(nc’n (0,775,120 )),t =1,...,30} Vn , is compared with

the conversion boundaries, ( beS (O), 1956(0) ) , to decide whether the land is kept in its

current crOp or should be converted to the alternative crOp. For instance, in year 1, when

the land is still in com-soybean, the realized returns on a particular sample path,

(75c, n(t)17rs, n (t )) , are compared with boundary bcs. If the realized returns are in the

“no action zone” (e.g., if as "(1) S bcs (”an (1)) according to the Optimal decision

rule), the land is kept in com-soybean, and similar comparisons are made in year 2. If, on

the other hand, the realized returns are in the “conversion zone” (i.e.,

93

if as n (1) > beS (”cm (1)) ), the land is converted to switchgrass, and in year 2, the

realized returns (ass, (2),as,,, (2)) will be compared with boundary bsc to decide

whether the land should be converted into com-soybean. We can also predict
governmental payments based on the farmer’s land use choice. Under constant subsidy,

variable subsidy and insurance subsidy, the government pays the farmer f lacre,
77as /acre and max (0, (as - a s) )/ acre per year, respectively when the farmer is in

switchgrass production. Under the cost-sharing subsidy, once the farmer converts land
from com-soybean to switchgrasss or reestablishes after ten years of being in switchgrass,
the government will pay s /acre to the farmer. For each simulated path of com-soybean
and switchgrass returns, we calculate the NPVs of total govemmental payments over 30
years for each type Of subsidy. Then the means (ENPV) and standard errors of the
discounted governmental costs over the N simulated paths of the joint returns can be

obtained for each type of subsidy during a 30 year period.

Fourth, we calibrate the subsidies by repeating steps 1-3 so that the ENPVs of
governmental costs at the end of 30 years under different subsidies are equalized, at a
level of $30/acre (2+: $1 simulation error). The calibrated subsidy parameters are presented
in table 2.3. For each period we count the number of sample paths on which the land is in
switchgrass. Dividing this number by N, we obtain the proportion of land in switchgrass

for each form of subsidy during a 30 year period.

The government can require that a farmer has to stay in switchgrass for some minimum number of years

to receive the cost-sharing subsidy; otherwise he has to pay a penalty. In the simulation, we impose that the
farmer can receive the subsidy only if he did not convert from switchgrass to com-soybean in the past five
years.

94

Results
Critical land conversion boundaries under diflerent forms of subsidies

In this section we present the effects of different subsidies on a representative farmer’s

optimal land conversion rule. In f1 gure 2.3a-d, the solid curves are the critical boundaries

bcS and bsc under the no subsidy base case. The dashed curves are conversion

boundaries under the four different subsidies.

A constant subsidy increases the instantaneous return to switchgrass. As expected, we
can see from figure 2.3a that it lowers the conversion boundary from com-soybean to
switchgrass and raises the conversion boundary from switchgrass to com-soybean. So it

encourages farmers to convert to energy mom and discourages them from withdrawing.

Compared with a constant subsidy, a variable subsidy not only increases the
switchgrass return but also its variability. This implies two opposite effects on the
optimal land conversion decision: a higher return gives incentive to convert to
switchgrass and a disincentive tO withdraw land out Of it, while more uncertainties will
hold back converting to switchgrass and encourage converting out. Figure 2.3b shows

that the return effect dominates the uncertainty effect on converting to switchgrass but the
uncertainty effect dominates the return effect on converting out so that both I)“ and

bSC are lowered compared with no subsidy case. However, we should be aware that this

is not always the case.

Farmers are assumed to receive the insurance subsidy only when the switchgrass return

is lower than 7_Z's , which is $60/acre. The insurance subsidy generally lowers the

95

conversion boundary from corn-soybean to switchgrass, but the effect is more dramatic
when the com-soybean return is lower than $25/acre: farmers will convert to switchgrass
even if its market return is zero since the subsidy can increase it to $60/acre. Similarly,
the subsidy raises the conversion boundary from switchgrass to corn-soybean much more
when the switchgrass return is lower: for a switchgrass market return lower than $45/acre
the farmers will not convert to com-soybean until the latter reaches at least $120/acre.

These effects gradually vanish when the switchgrass return goes beyond the insured level.

Different from the annual subsidies, a cost-sharing subsidy for switchgrass always

lowers both direction land conversion boundaries (figure 2.3d). While reducing the
conversion costs from com-soybean to switchgrass (Ccs) makes the com-soybean

grower less reluctant to covert the land, it also has the indirect effect of making the
switchgrass grower more prone to convert back to com-soybean. This is because
although the farmer currently growing switchgrass will not directly benefit from the
subsidy for conversion to switchgrass, its existence reduces the expected cost of
converting from com-soybean back to switchgrass, thereby reducing the implied cost of

switching to com-soybean. Thus it indirectly increases his incentive to convert land to

com-soybean. The direct effect of lowering Ccs is greater than the indirect effect. The

reduction in Ccs lowers the boundary from com—soybean to switchgrass more than the

boundary from switchgrass to com-soybean.
The Proportion of Land in Switchgrass

The effects of a subsidy program on encouraging energy crop production can be

illustrated more clearly using the proportion of land in switchgrass over a 30 year period.

96

By changing the optimal land conversion decision rule, the subsidy program will change
the probability of land converted into energy crops as well as converting out. The lower
the conversion boundary from com-soybean to switchgrass, the more likely the realized
returns can reach the boundary, so that the farmer will convert to switchgrass. Conversely,
the lower the conversion boundary from switch grass to com-soybean, the more likely the
realizations of the returns can reach the boundary, so that the farmer will convert out of

switchgrass.

Figure 2.4 shows the expected proportion of a unit com-soybean land converting into
switchgrass over a 30 year period under the no subsidy case and the four single subsidy
program, given that the expected governmental cost is uniformly $30/acre at the end of
30 years. The cumulative proportion is not monotonically increasing over the years

because the farmer can optimally convert back to com-soybean when its return is high

enough and reach the conversion boundary b“. We first examine the case without
subsidy, indicated by the solid curve in figure 2.4. At the beginning, the prOportion of
land converted into switchgrass increases over years and peaks at 0.26 in year 9.
However, the switchgrass return also has a higher level of uncertainty, and eventually
land in switchgrass is likely to be converted back to com-soybean. At the end of the 30
years, the proportion of land in switchgrass is about 0.13. The average probability of land

in switchgrass over 30 years is 0.18.

A constant subsidy lowers beS and raises bSC , implying that it is easier to convert
into switchgrass and harder to convert out. The conversion pattern over time is similar to

the no subsidy case but the proportion of land in switchgrass peaks at 0.28 and stabilizes

97

at 0.19, increased by 0.03 and 0.06 compared with the no subsidy case. The average

proportion over all 30 years is also increased from 0.18 to 0.22. In contrast, the variable

subsidy and cost-sharing subsidy lower both bcs and bsc (although for different reasons
as we discussed above), implying that it is easier to convert into switchgrass as well as to
convert out. These two subsidies raise the peak probability of land in switchgrass to 0.28
and 0.32, respectively, but barely change where it stabilizes. The average proportion of

land in switchgrass over years is increased from 0.18 to 0.21 under both cases.

When both of the com—soybean and switchgrass returns are low, the insurance subsidy
effectively makes switchgrass the dominant choice. Once the return of com-soybean falls
below $25/acre, the land will be converted to switchgrass and will not be converted out
until the com-soybean return bounces back to at least $105/acre. So the insurance subsidy
increases the probability of land in switchgrass the most, peaking at 0.31 and stabilizing

at 0.25. The average proportion over all 30 years is 0.2.
The Change and Variation of Governmental Costs over Years

Each subsidy program is calibrated such that it will incur the same expected
governmental cost at the end of 30 years. However, these costs may change from year to
year. This information is interesting because the government may prefer a policy program
that has a stable stream of expenditures. Figure 2.5a shows the mean NPV of
governmental costs under each subsidy over 30 years. Since the cost-sharing subsidy is a
relative large one-time payment and more conversion to switchgrass happen in the first

several years, its expenditure grows faster in the first three years, slows down, until

98

intermediate period and stabilizes after year 15. The other three subsidies are annual

payments, which increase steadily over the years.

We have assumed the risk-neutrality of government and an ex ante budget constraint
and compared the cost-effectiveness of different subsidies. However, the performance of
different subsidies could change if the government is risk averse, or has an ex post budget
or both. Then less variability of the governmental expenditures will be more desirable.
Figure 2.5b shows the simulated standard errors of the NPV of the governmental costs for
each subsidy program over the 30 years. The standard error of the cost-sharing subsidy
payment rises rapidly in the first three years and becomes steady at $36/acre since then.
The standard error of the constant subsidy payment keeps rising steadily to 44/acre at the
end of 30 years. The distributions of other two subsidy payments are much more heavy-
tailed. The simulated standard error is $143/acre and $72/acre, or 4.8 and 2.4 times of the
mean under the variable subsidy payment and the insurance subsidy payment,

respectively, at the end of 30 years.
The Effects of Combining Cost-sharing Subsidy with Other Types of Subsidies

In addition to considering the program implementing a single subsidy, we also evaluate
the effectiveness of combining the lump-sum cost-sharing subsidy with other three types
of annual subsidy, as often occurs in practice. For example, as we discussed above, Irish
farmers can receive a subsidy up to half of establishment costs as well as a constant
subsidy of $70/acre for planting willow and miscanthus, while US. farmers can receive
a subsidy up to 75% of the establishment costs and a 2-year variable subsidy that matches

the biorefiner’s payment for any eligible energy crop.

99

Again we compare the three forms of combined subsidies among themselves and with
their single form subsidy counterpart by how they change the expected proportion of a
unit of corn-soybean land converting to switchgrass. The simulation is performed given
an expected governmental cost at $80/acre. The calibrated subsidy levels are presented in
table 2.4. First we can examine the relative performance of the three combined subsidies.
Figures 2.6 a-c show that consistent with the relative performance Of the single subsidy
forms, subsidizing the establishment costs and insuring a minimum return will result in
the highest probability of land in switchgrass, which peaks at 0.4 and stabilizes at 0.3 at
the end of 30 years and averages at 0.33. A constant subsidy together with a cost-sharing
subsidy has 0.4 probability of land in switchgrass at the peak and 0.22 at the end of 30
years, averaging at 0.29. A variable subsidy together with a cost—sharing subsidy will
rank lowest, having probability of 0.3 for land growing switchgrass at peak and 0.13 at

the end of 30 years and averaging 0.21.

Compared with their single subsidy counterpart, the combined forms increase the
peaking proportion of land in switchgrass in the intermediate period but reduce the
proportion toward the latter part of a 30 year time horizon (except the variable subsidy
together with cost-sharing subsidy, whose proportion of land in switchgrass is slight
higher than its counterpart). This can be explained by the dual effects of the cost-sharing

subsidy on land conversion: it has a positive effect on the expected rate Of converting

land to switchgrass by lowering the conversion boundary bcS more than the annual

subsidies but also has a negative effect by inducing land converting out later since it

lowers the conversion boundary bsc . The average probability over years change little

compared to the single forms, but single form subsidies have smaller variances.

100

Conclusions and Discussions

This study examines the design of agricultural subsidy programs that aim to encourage
desirable land use using a real Options framework that reflects the following features: (a)
the dynamic characteristics of land conversion; (b) the sunk costs and future return
uncertainties associated with land conversion; and (c) flexibility in an optimizing,
representative farmer’s land use decisions. Results show that failure to consider these
factors can lead to misleading conclusions. Although the levels of different subsidy

forms were selected to be ENPV-equivalent, they are not equally cost-effective.

Using energy crop production as an example, we compare three annual subsidies and
one lump-sum subsidy that have the same expected governmental costs. The insurance
subsidy results in the highest expected proportion of land being converted to energy crops
(switchgrass), followed by the constant subsidy. Although the cost-sharing subsidy and
the variable subsidy have the positive effect of encouraging land conversion to
switchgrass, they also have a negative effect of discouraging land from staying in that
switchgrass. The two effects cancel each other out and result in an increase in the
predicted proportion of land in switchgrass in the intermediate period but a drop back to
the no-subsidy level at the end of 30 years. The relative performance of combing cost-
sharing subsidy with other annual subsidies is consistent with comparison of single

subsidies.

The results presented in this paper suggest that the existing US. energy crop subsidy
system, which is a variable subsidy combined with a cost-sharing subsidy, may not be the

most cost-effective. Greater cost-effectiveness of the insurance subsidy highlights the

101

research needs for how to reduce the uncertainties of the returns to energy crops.
Taheripour and Tyner (2008) propose a subsidy that is inversely related with the oil
price6 in order to reduce the volatility of energy crop prices. Compared with the
government providing an insurance policy, the long-term contract between energy crop
growers and biorefiners may serve as a better mechanism considering the possible

transaction costs involved in the former.

There is a caveat in evaluating the performance of cost-sharing subsidy based on our
results. We only consider the cost-effectiveness of a subsidy, i.e., the ability to convert
land to switchgrass given the same governmental expenditures. But there are other factors
that justify the cost-sharing subsidy, one of which is the farmer’s liquidity constraint.
Numerous studies show farmers are concerned about the large up-front costs of
establishing the energy crops (e.g. Sherrington, Bartley and Moran 2008; Bocqueho and
J acquet 2010). A cost-sharing subsidy can relax this constraint and thus reduce the

adoption barriers.

 

6
They call it the variable subsidy, which clearly is different from the one in our paper.

102

Table 2.1. Parameters for Solving Optimal Land Conversion Rule without Subsidies

 

Land Conversion Model Parameters

Notation and Value

 

Discount Factor

Drift Rate Of Com-soy Return

Drift Rate of Switchgrass Return
Variance Parameter Of Com-soy Return

Variance Parameter of Switchgrass Return

Correlation Coefficient between Two Returns

Land Conversion Cost from Com-soy to Switchgrass

Land Conversion Cost from Switchgrass to Com-soy

0.08

0.03

0.03

0.29

0.64

0.3

$136/acre

$47/acre

 

103

 

 

 

 

 

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104

Table 2.3 Parameters for Policy Simulation: Single Subsidy

 

 

Subsidy Form Subsidy Level
Constant Subsidy $ 12/acre
Variable Subsidy 2%
Insurance Subsidy $ 60/acre
Cost-sharing Subsidy $ 90/acre
Governmental Costs $3 0/acre

 

Note: The simulation errors of expected governmental costs across different subsidies are within S l/acre.

105

Table 2.4 Parameters for Policy Simulation: Single Subsidy vs. Combined Subsidy

 

 

Combined Form Single Form
Cost-sharing Subsidy 68/acre (Half of Baseline Conversion Costs)
Constant Subsidy S l4/acre $24/acre
Variable Subsidy 3.6% 5.5%
Insurance Subsidy $ 6l/acre SSS/acre
Governmental C osts $80/acre

 

Note: The simulation errors of expected governmental costs across different subsidies are within $2/acre.

106

5 1 /v ‘ Tw—TT‘TOI

Convert from Com-soy/

4 to Switchgrass /

3 l .
Stick to Current Use /

.l /

 

 

 

 

 

 

)

/ bsc

Return to Switchgrass (in $100/acre

 

/
/
1 ¥/ Convert from Switchgrass to
{11’ Com-soy
0 tr/ 4 1 a 4+
0 l 2 3 4 5

Return to Com-soybean (in $100/acre)

Figure 2.1 Optimal land conversion rule: no subsidy

107

 

 

l. Simulate the realized retums of com- 2. Select a subsidy level and solve the

soybean and switchgrass for 5000 times land conversion model.

using known parameters of the stochastic

processes and 2008 return values Over 30

 

 

 

 

years.

 

 

' Obtain 5000 paths of return pairs over 30

 

 

3. Predict the land use in each period
and the governmental costs.

Outputs are state of land use, the means

 

 

 

and standard errors of expected

governmental costs in each period.

I
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S

Q:

4. Repeat steps l-3 for each type of

i subsidy until their expected governmental

costs at end of 30 years are equal.

 

 

Obtain the proportion of land in

switchgrass, means and standard errors of

governmental costs over the 30 years.

Figure 2.2 Land conversion simulation steps

108

 

 

 

Return to Switchgrass (in $100/acre)

 

 

4 5
Return to Com-soy (in $100/acre)

Figure 2.3a Optimal land conversion rule: constant subsidy

109

 

 

 

Return to Switchgrass (in $100/acre)

 

1 2 3 4 5
Return to Com-soybean (in $100/acre)

Figure 2.3b Optimal land conversion rule: variable subsidy

110

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

, r" l
2? /
a... ,1
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o /.
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If? M,
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a l / /
a / /
co 1 /
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a—o / #9
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.9 t/ '1 ,/
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g 1 ~/ // / //
Q) /
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0 a" L 1 1 1 1
O l 2 3 4 5

Return to Com-soy (in $100/acre)

Figure 2.3c Optimal land conversion rule: insurance subsidy

lll

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Return to Switchgrass (in $100/acre)

 

 

 

2 3 4 5
Return to Com-soy (in $100/acre)

Figure 2.3d Optimal land conversion rule: cost-sharing subsidy

112

 

0.5

I

.6
A
l

— - Insurance Subsidy

9
cu
I
‘s
I

I

f \ - - - Constant Subsidy

Variable Subsidy

P
N

- - - Cost-sharing

Subsidy
—No Subsidy

Proportion of Land in Switchgrass
9
fl

 

 

 

0 2 4 6 8 1012141618202224262830

Year

 

 

 

Figure 2.4 Probability of land in switchgrass: comparison of single subsidy
(expected NPV of governmental costs=$30lacre)

Note: The average propOrtion of land in switchgrass over 30 years: 0.25 for the insurance subsidy, 0.22 for
the constant subsidy, 0.21 for the cost-sharing subsidy, 0.20 for the variable subsidy, and 0.18 for the no

subsidy case.

113

 

 

Expected N PV of
Governmental Costs

F‘HNNMMA
Q MGM:

M:

GM

 

' , I - "' ' "' ° — ' —; T..:--':"°5- ° 'Cost-sharing
- /./ ,v’v ........ / / subsidy
I ’ ........ / — - - °

. / I I , ’ ............ / / Constant Subsrdy
- ....... /

I ’1’ ........ / --------- Variable Subsidy
I I 1’ ....... / /
q ' a: ........ / / — - Insurance Subsidy
J! I. /

 

0 2 4 6 81012141618202224262830

Year

 

Figure 2.5a Mean NPVs of governmental costs over years

114

 

 

 

Simulated Standard Error of

Governmental Costs

160 T
140 __ .........
120 -~
100 ~-
--------- Variable Subsidy
80 " ’ — - Insurance Subsidy
_ I
60 .. ....... / I ’ ----- Constant Subsidy
.......... / / -_------_ -Cost-sharing Subsidy
40 '7 .’ _ — o —..:.L.. J#1£T:.: —————
20 --I If"?

 

 

0 2 4 6 8 1012141618202224262830

Year

 

Figure 2.5b Standard errors of NPV of governmental costs over years

115

 

 

 

Simulated Standard Error of

Governmental Costs

160 T
140 __ .........
120 --
100 '-
--------- Variable Subsidy
80 “ ’ — —Insurance Subsidy
’
60 __ ...... / I ’ ----- Constant Subsidy
........ / / __.— -Cost-sharing Subsidy
4o ,. _____ ._ . Leg-r75? 21': . _
0’ ...... o" J
20 __I ’.¢°5"

 

 

0 2 4 6 8 1012141618202224262830

Year

 

Figure 2.5b Standard errors of NPV of governmental costs over years

115

 

 

 

Simulated Standard Error of

Governmental Costs

 

160 ‘r
140 “ °°°°°°°°°
120 -- '°
100 r“
--------- Variable Subsidy
80 "' . ’ — —Insurance Subsidy
60 -- ...... / I ’ ’ ----- Constant Subsidy
........ / / ---._ - -Cost-sharing Subsidy
40 ‘7 ’ ...... . a; . 7a,..1'5'1’2. ——————
° ...:;O"7
20 --I w /
U ”

0 2 4 6 81012141618202224262830

Year

 

Figure 2.5b Standard errors of NPV of governmental costs over years

115

 

 

 

 

 

 

a 0.5 -
E
as
g 0.4 -
'E
m
.5 0.3 a
11
E, °°°°°°°°° Cost-sharing and
L: 0.2 _ Insurance Subsidy
3 Insurance Subsidy
“E 0.1 -
e
a.
E
9" 0

0 2 4 6 8 1012141618202224262830

Year

 

 

 

Figure 2.6a Effect of combining a cost-sharing subsidy with an insurance subsidy
(expected NPV of governmental costs is $80/acre)

116

 

 

0.5 -

0.4

0.3

0.2

0.1

 

Proportion of Land in Switchgrass

.......

O O.
O
I

 

--------- Cost-sharing and
Constant Subsidy

Constant Subsidy

 

 

0

 

0 3 6 9 12151821242730

Year

Figure 2.6b Effect of combining a cost-sharing subsidy with a constant subsidy
(expected NPV of governmental costs is $80/acre)

117

 

 

0.5 -

--------- Cost-sharing and
Variable Subsidy

Variable Subsidy

 

 

Proportion of Land in Switchgrass

 

 

0 3 6 912151821242730

Year

Figure 2.6c Effect of a combining cost-sharing subsidy with a variable subsidy

(expected NPV of governmental costs is $80/acre)

118

 

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121

Concluding Remarks

Large scale production Of dedicated energy crops could have dramatic effects on land
use change and associated economic and environmental impacts. However, whether
farmers are willing to adopt these mom is still an open question. This dissertation, which
consists of two essays, looks at a representative farmer’s optimal land use decisions for
energy crops and the cost-effectiveness of various governmental subsidies that occur in
practice.

This dissertation applies a new real options framework to model a farmer’s land
conversion decision, which eliminates the absolute irreversibility assumption and allows
costly reversion of land use. By comparison with the real options results, an NPV model
predicts that an optimizing farmer would be much more prone to convert land to energy
crOps. A one-way real option model characterizing the land conversion decision as
irreversible predicts much greater reluctance to convert land from traditional crops to

energy crops.

The results from the first essay suggest that sunk costs and risk tend to deter farmers’
willingness to invest in dedicated energy crops, causing them to require more than double
the estimated break—even revenue based on average yields and prices that ignore
variability. Using rotated com-soybean and switchgrass as two representative crops, the
simulation shows that the predicted probability of land converting from com-soybean to
switchgrass over a 30 year period is low. Governmental subsidies can increase the
predicted probability, as shown in essay 2. Subsidies that reduce the uncertainties of
return to energy crop perform better than the ones that increase the uncertainties.

However, the effectiveness Of cost-sharing subsidy is not as good as expected because it

122

has a counter-intuitive side-effect of encouraging land converted out of the energy crop at
a later period. The results presented in this paper suggest that the existing US. energy
crop subsidy system, which is a variable subsidy combined with a cost-sharing subsidy,
may not be the most cost-effective. Greater cost-effectiveness of the insurance subsidy
highlights the research needs for how to reduce the uncertainties of the returns to energy

crops.

This study can be extended in several aspects. First, the representative farmer is
assumed to be risk-neutral. To maximize the NPV of current and future returns, it is
optimal to convert all his land to the alternative use if the retum of alternative land use is
high enough compared tO the return of current land use. Therefore, crop diversification is
never optimal. If the risk-aversion is introduced, the crop diversification may become
optimal since it is often used as a risk reduction strategy. The model will be modified to
maximize an expected utility function that includes the risk-aversion. The control
variable could be the proportional land in each use. This modification may further

complicate the solution algorithm.

Second, our optimal land conversion decision is derived in a representative farmer
partial-equilibrium setting, in which the individual’s ex ante probability of land
conversion can be simulated over a certain period. Although the predicted ex ante
probability could measure the fraction of land in switchgrass under the homogeneity
assumption, two more factors should be accounted for a better measurement on the
aggregate level. One is that reallocation of land into different crops may have feedback
effects, which could make the stochastic processes of the crop returns endogenous.

Another is to model. the farmers’ heterogeneity. Incorporating these features may need a

123

general stochastic dynamic model. There are works that employ the computational
general equilibrium models or mathematical programming models to predict the land use
change on a regional or national level (e. g. Keeney and Hertel 2009; Walsh et. al. 2003).
However, most of these models fail to account for uncertainties. How to integrate the
micro-level model that account for the adjustment costs and the uncertainties in the land
conversion with the macro-level model that account for the endogenous evolution of crop

returns and farmers’ heterogeneities is an interesting topic.

124

References

Keeney, R. and T.W. Hertel. 2009. "The Indirect Land Use Impacts of United States
Biofuel Policies: The Importance of Acreage, Yield, and Bilateral Trade
Responses." American Journal of Agricultural Economics 91(4):895-909.

Walsh, M., D. de la Torre Ugarte, H. Shapouri, and S. Slinsky. 2003. "Bioenergy Crop
Production in the United States: Potential Quantities, Land Use Changes, and
Economic Impacts on the Agricultural Sector." Environmental and Resource
Economics 24(4):313-333.

125

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