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The Economic Effects of Alternative
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THE ECONOMIC EFFECTS OF ALTERNATIVE INSTITUTIONAL DESIGNS FOR U.S.
CROP INSURANCE.

by

John Duncan

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

DOCTOR OF PHILOSOPHY

Department of Agricultural Economics

1994

ABSTRACT

THE ECONOMIC EFFECTS OF ALTERNATIVE INSTITUTIONAL DESIGNS FOR U.S.
CROP INSURANCE

By
John Duncan

This study compares the performance of alternative institutional designs for crop insurance
delivery. The aim is to investigate institutional designs that incorporate a role for the government
and competitive firms in meeting the goals of actuarial soundness and high coverage levels (low

cost) for farmers. In particular, four alternative designs are investigated:

1. Competitive firms deliver all crop insurance without any government reinsurance.

II. The government provides insurance to all farmers with no role for competitive firms.

III. Both the government and competitive firms deliver insurance in a mix detennined by
the current federal crop insurance program.

IV. Competitive firms deliver all crop insurance with the government providing only

reinsurance.

These four designs are investigated with simple stylized models where the government is portrayed
as risk neutral with no ability to distinguish between farm risk types, while competitive firms are
portrayed as risk averse with full information to distinguish between farm risk types.

Results indicate that, with more than one farmer risk type, a self sustaining federal crop
insurance program could provide maximum coverage to all farmers if low risk policies are taxed
and high risk policies are subsidized. This does not occur in the current federal crop insurance

design. Also, if the government continues to deliver crop insurance under the current design, it

could cut costs if all policies were delivered through private firms without requiring these firms to
share in the risk.

For competitive firms, results indicate that high levels of systematic risk are a major
reason for the lack of competitive markets for multi-peril crop insurance. The analysis shows that
if there were no such markets (due to high levels of systematic risk) then proportional reinsurance
schemes without subsidies would not facilitate an equilibrium. The Area-Yield Insurance Plan
(currently available only to farmers) has the potential to serve as a reinsurance scheme that
facilitates equilibrium without undue burden on taxpayers. Once a competitive crop insurance

market is established, proportional reinsurance may be introduced to increase coverage to farmers.

Rebecca, Timothy and Annette

iv

ACIG‘IOWLEDGMENTS

I would like to thank Robert Myers and Jack Meyer for carefully reading each chapter of
my dissertation and advising me in the course of the research. Other members of my dissertation
committee were Roy Black and Steve Hanson. I would also like to acknowledge John Hoehn and
J irn Oehmke who have influenced my graduate studies, although they did not directly contribute to

the dissertation.

Funding for my graduate training at Michigan State University came through various
assistantships and fellowships. Majority of my financial support while writing my dissertation
came from the Food Security Project in the form of assistantships. I thank Valerie Kelly for
encouraging me and letting me do assistantship work around my dissertation writing as it was not
easy juggling between two very distinct areas. I am also grateful to the Department of Agricultural
Economics, the Federal Crop Insurance Corporation, and the Graduate School at Michigan State
University for partial funding. I thank Eric Crawford, our graduate coordinator, for advice and for
facilitating financial support in the last legs of my writing.

I thank my friends for their prayers and support. Finally, my children and wife have
sufl‘ered many hardships during the course of my graduate studies. Their love has been a source of

my strength. To Rebecca, Timothy and Annette, thank you for your smiles and thank you for your

faith to walk with me.

TABLE OF CONTENTS

 

 

ACKNOWLEDGMENTS ........................................................................................................... v
LIST OF TABLES ...................................................................................................................... x
LIST OF FIGURES ................................................................................................................... xi
1. INTRODUCTION -- -- -- -- ........... - - ..... 1
1.1 THE PROBLEM ....................................................................................................................... 3
1.2 OBJECTIVES OF THE STUDY ................................................................................................... 5
1.3 WORKING HYPOTHESES FOR THE STUDY ............................................................................... 7
1.4 OUIIJNE OF THE DISSERTATION ............................................................................................ 9
2. AN OVERVIEW OF CROP INSURANCE IN THE U.S. - - . .......... 10
2.1 HISTORICAL OVERVIEW ....................................................................................................... 10
2.2 THE 1980 CROP INSURANCE ACT ........................................................................................ 14
2.3 DESIGN OF THE F CIC PROGRAM .......................................................................................... 15
2.4 PERFORMANCE OF CROP INSURANCE SINCE THE 1980 CROP INSURANCE ACT ...................... 16
2.5 PROBLEMS OF THE FCIC PROGRAM ..................................................................................... 20

2.5.1 Actuarial Soundness ..................................................................................................... 20

2.5.2 Crop Price Forecasts ..................................................................................................... 21

2.5.3 Lack of Risk Sharing by Reinsurance Companies .......................................................... 22
3. SYSTEMATIC RISK AND THE DELIVERY OF CROP INSURANCE ........................ 24

3.1 STYLIZED RISK MARKET MODEL AND A FRAMEWORK FOR ANALYSIS .................................. 24

3.1.1The RiskMarket ........................................................................................................... 24
3.1.2 The Mean-Variance Framework .................................................................................... 26
3 .2 THE DEMAND FOR CROP INSURANCE ................................................................................... 23
3 .3 COMPETITIVE EQUILIBRIUM IN THE MARKET FOR CROP INSURANCE .................................... 32
3.3.1 Competitive Supply of Crop Insurance .......................................................................... 32
3.3.2 A Definition of Equilibrium .......................................................................................... 36
3.3.3 The Competitive Equilibrium ........................................................................................ 38
3.3.4 Properties of the Competitive Equilibrium ..................................................................... 38
3.4 FCIC DELIVERY OF CROP INSURANCE ................................................................................. 45
3.4.1 The FCIC equilibrium ................................................................................................... 45
3.4.2 Properties of the Equilibrium ........................................................................................ 46
3 .5 COMPARISON OF THE COMPETITIVE EQUILIBRIUM AND FCIC DELIVERY .............................. 47

4. CROP INSURANCE DELIVERY UNDER MULTIPLE FARMER RISK CLASSES ...50

4.1 THE RISK MARKET .............................................................................................................. 51
4.2 THE FCIC EQUILIBRIUM ...................................................................................................... 52
4.2.1 Definition of an FCIC equilibrium ................................................................................. 52
4.2.2 Separating Equilibrium With No Cross-Subsidization Across Risk Pools ....................... 54
4.2.3 FCIC Separating Equilibrium With Cross-Subsidization Across Risk Pools ................... 64
4.2.4 The FCIC Pooling Equilibrium ..................................................................................... 72
4.2.5 Implications for Current FCIC Policies ......................................................................... 76
4.3 THE COMPETITIVE DELIVERY OF CROP INSURANCE WITH TWO RISK TYPES ........................ 78
4.3.1 The Competitive Equilibrium ........................................................................................ 79

vii

4.4 COMPARISON OF THE COMPETITIVE EQUILIBRIUM AND FCIC DELIVERY .............................. 82

 

4.4.1 The Case of a Separating Equilibrium With No Cross Subsidies .................................... 83
4.4.2 The Case of a Separating Equilibrirun With Cross Subsidies ......................................... 84
4.5 CONCLUSIONS ..................................................................................................................... 85
5. FCIC REINSURANCE UNDER THE 1980 CROP INSURANCE ACT -- -_ 87
5.1 THE FEDERAL CROP INSURANCE PROGRAM ......................................................................... 88
5.1.1 The FCIC Reinsurance Program ................................................................................... 88
5.2 FCIC USE OF THE INFORMATION AND DELIVERY COST POSITION OF PRIVATE FIRMS ........... 90
5.3 FCIC REINSURANCE OF PRIVATE FIRMS WITH ONE RISK CLASS OF FARMERS ..................... 92
5.3.1 FCIC Supply of Insurance to Farmers ........................................................................... 92
5.3.2 The Reinsurance Decisions of Private Firms .................................................................. 93
5.4 F CIC REINSURANCE WITH MORE THAN ONE RISK CLASS OF FARMERS ............................. 104
5.4.1 FCIC Reinsurance in a Separating Equilibrium ........................................................... 106
5.5 IMPLICATIONS FOR THE FCIC REINSURANCE PROGRAMS ................................................... l 10
6. REINSURANCE FOR A COMPETITIVE CROP INSURANCE MARKET. ............... 114
6.1 CONSTRAINTS ON EXISTENCE OF COMPETITIVE INSURANCE MARKETS .............................. I 15
6.2 PROPORTIONAL REINSURANCE ........................................................................................... 120
6.2.1 Proportional Reinsurance Described ............................................................................ 121
6.2.2 The Model .................................................................................................................. 122
6.2.3 Can Proportional Reinsurance Facilitate Equilibrium? ................................................. 125
6.2.4 Can Proportional Reinsurance Increase Coverage to Farmers? ..................................... 128
6.3 PROPORTIONAL REINSURANCE WITH GOVERNMENT SUBSIDIES .......................................... 132
6.4 IMPLICATIONS FOR CONVENTIONAL REINSURANCE SCHEMES ............................................ 135

viii

6.5 AREA-YIELD CROP INSURANCE AS “REINSURANCE” .......................................................... 136

6.5.1 Implications of Area-Yield Insurance as “Reinsurance” ............................................... 139
7. SUMMARY, CONCLUSIONS AND POLICY IMPLICATIONS .................................. 141
8. BIBLIOGRAPHY .................... - - -- - - - - -- 147

 

ix

LIST OF TABLES

TABLE 2-1: GOVERNMENT AGRICULTURAL DISASTER ASSISTANCE (‘000 8), 1980-90. ......... 17
TABLE 2-2: PARTICIPATION RATES IN THE FCIC PROGRAM, 1980-90. ...................................... 18
TABLE 3-1: COMPETITIVE EQUILIBRIUM COVERAGE LEVELS: w=0.01, A=0.01, B=lO. ............ 42
TABLE 3-2: COMPETITIVE EQUILIBRIUM COVERAGE LEVELS: \y=0.01, 7L=0.0l, B=20. ............ 42
TABLE 3-3: COMPETITIVE EQUILIBRIUM COVERAGE LEVELS: w=0.001, A=0.0l, B=10. .......... 43
TABLE 4-1 FCIC EQUILIBRIUM COVERAGE LEVELS WITH TAXES AND SUBSIDIES: R=l ...... 69
TABLE 4-2 FCIC EQUILIBRIUM COVERAGE LEVELS WITI-I TAXES AND SUBSIDIES: R=2 ...... 69

TABLE 4-3 FCIC EQUILIBRIUM COVERAGE LEVELS WITH TAXES AND SUBSIDIES: R=O.5 70
TABLE 6-1: COMPETITIVE COVERAGE LEVELS WITH FCIC PROPORTIONAL REINSURANCE:
K=l; p=0.05 ................................................................................................................................. 130
TABLE 6-2: COMPETITIVE COVERAGE LEVELS WITH FCIC PROPORTIONAL REINSURANCE:
K=l; p=0. 1. ................................................................................................................................. 130
TABLE 6-3: COMPETITIVE COVERAGE LEVELS WITH FCIC PROPORTIONAL REINSURANCE:
K=0.5; p=0.l ................................................................................................................................ 130
TABLE 64: COMPETITIVE COVERAGE LEVELS WITH FCIC PROPORTIONAL REINSURANCE:

K=0; p=0.l. ................................................................................................................................. 131

LIST OF FIGURES

FIGURE 6-1: RESERVATION UTILITY AND COMPETITIVE EQUILIBRIUM ................................ 118
FIGURE 6-2: COMPETITIVE EQUILIBRIUM AND PROPORTIONAL REINSURANCE .................. 127
FIGURE 6-3: COMPETITIVE EQUILIBRIUM AND SUBSIDIZED PROPORTIONAL

REINSURANCE. ......................................................................................................................... 135

1. Introduction

The government is generally considered to be in a better position than private companies to
undertake very risky investments. This is because of the government's ability to pool risk across a
large and diverse portfolio of activities, and also its ability to spread risk across a large number Of
taxpayers such that the risk borne by an individual taxpayer is negligible. Both these risk pooling
and risk spreading aspects of the government put it in a unique position to develop programs which
private companies will not undertake because they require a higher rate of return to compensate for
the higher risk (Arrow and Lind). One example where the U.S. government uses its capacity to

pool and spread risks is in the delivery of multi-peril crop insurance.

The U.S. government provides multi—peril crop insurance through the Federal Crop
Insurance Corporation (F CIC). NO multi-peril insurance exists in the private sector although
insurance for individual perils such as hail and fire exist for many crops. Multi-peril crop
insurance protects participating farmers against all unavoidable yield risks such as droughts,
floods, insect infestations, and other natural disasters. Government involvement in crop insurance
is driven not only by the desire to compensate for negative income shocks resulting from farm yield
losses, but also the desire to make crop insurance the primary provider of disaster assistance to
farmers. This was done because other disaster assistance programs, such as direct cash payments,
were considered too expensive and involved the farmer carrying too little of the production risk
(GAO/RCED-92-25). Because crop insurance is based on actuarial data, it is argued that an
insurance system provides for a more stable funding of disaster losses. Further, with crop

insurance there is a pre-commitment of government funds with a contingent plan in place to deal

2
with disasters while other forms of disaster assistance are highly discretionary (GAO/RECD-88-

leBR).

The 1980 Crop Insurance Act (Public Law 96-365, Sept. 26, 1980) legislated the FCIC to
perform the role of primary disaster assistance provider. The FCIC was directed to: (i) increase
crop insurance participation rates in order to abolish other government funded disaster assistance
programs; and (ii) reduce the amount of federal costs the government was bearing for unavoidable
crop failure. The objective of the government was to convert uncertain disaster payments to a
system where the farmers recognized the risks they faced and partially paid to insure such risks.
The program was to operate within a budget although federal subsidies were to be provided to
reduce insurance costs to farmers while maintaining adequate coverage levels. To increase the
efficiency of the program and achieve the above Objectives, the FCIC was directed to use the

expertise of the private sector in providing federal crop insurance.

A key feature of the 1980 Crop Insurance Act was for the FCIC to involve the private
sector in the federal crop insurance program. It was envisioned that the expertise of private
insurance firms would maintain and/or improve services to farmers and save money by increasing
the “efliciency” of the program. The FCIC did not single out any particular expertise of the private
companies that would help the crop insurance program to gain this “eficiency”. The path that the
FCIC chose was to involve private insurance firms by having them deliver the major share of crop
insurance to farmers. In fact, all the crop insurance delivery would take place through two
channels: Master Marketers and reinsured companies. Direct crop insurance sales by FCIC

employees were terminated in 1983 (GAO/RCED-87-77).

Both Master Marketers and reinsured companies are private insurance firms but there are

differences in the ways each participates in the FCIC program. Master Marketers are private

3
insurance firms that sell crop insurance as agents for the FCIC. These firms bear no risk on the
policies they sell and do not adjust claims. The federal government retains all premiums and pays
all indernnities on policies sold by the Master Marketers. The Master Marketers get approximately
15% of the premiums as fees. Unlike Master Marketers, reinsured firms bear a portion of the risk
on the policies they sell. The firms are called “reinsured” because they buy reinsurance fiom the

FCIC through proportional and/or stop-loss reinsurance measures.

In proportional reinsurance, the FCIC and the private insurance companies share specified
portions of both the policy premiums and the indemnities, while in stop-loss reinsurance the FCIC
agrees to reimburse the private insurance company for all indemnity payments above a
predetermined level (see Chapter 6 for a detailed discussion of the reinsurance schemes). Although
the reinsured fimis sell, service and settle claims on their own crop insurance policies, the policy
features and the premiums farmers pay are determined and set by the FCIC. The reinsured firms
are allowed to differentiate between high and low risk farmers for proportional reinsurance
purposes and have the option to retain less of the high risk policies. The F CIC firrther reimburses
reinsured firms for administrative costs. By 1990, reinsured companies accounted for 89% of the
total number of crop insurance policies sold to farmers. Reinsurance, therefore, is currently the

primary mode of crop insurance delivery used by the F CIC.

1.1 The Problem

Despite federal payments for all administrative expenses and federal subsidies for crop
insurance premiums, the FCIC bore losses every year during the 1980-90 decade without any

establishment of reserves against catastrophic losses or obtaining the desired level of participation

4
rates in the program. This performance generated serious criticisms of the workings of the federal
crop insurance program. Various General Accounting Oflice reports (GAO/RCED-88-7;
GAO/RCED-89-10; GAO/RCED-90-32; etc.), attributed this poor performance mainly to (i) the
lack of actuarial soundness of crop insurance premiums and (ii) overgenerous reinsurance
agreements with the private reinsured companies]. While many studies have addressed the actuarial
soundness question (Chambers; Nelson; Nelson and Preckel; Skees and Reed; etc.), there is
currently no research addressing the performance of alternative designs of the FCIC reinsurance
program, and the way it includes private firms. Preliminary General Accounting Office studies
(various issues) claim that the historical involvement of private insurance firms has aggravated the

performance of the FCIC rather than helped it.

Related to the problem of insurance provision by private companies is the problem of
systematic or catastrophic risk in crop production. The systematic nature of yield risks means that
when disaster strikes it generally affects a large number of farmers. Since the proportion of
systematic risks is considered extensive in the agricultural sector, an insurance firm could face
large indemnity outlays within a short period of time, such that it drives the insurance firm to ruin.
In such cases insurance firms may demand a high “risk premium” to provide crop insurance to
farmers (Hara). This role of systematic risk has received some attention in the literature (Miranda;
Chambers) but not in the context of risk averse firms delivering crop insurance. Given that
reinsured firms are now the primary mode of crop insurance delivery in the U.S., it seems
particularly important to examine the role of such firms in providing insurance services in the

presence of systematic risks.

 

lOther criticisms include too rapid an expansion of the program, and the continued existence of other
disaster assistance programs such as Direct Cash Payments and Emergency Loan Programs.

1.2 tiv of heS d

The purpose of this study is to compare the performance of alternative institutional designs
for crop insurance delivery in the context of simple stylized models. The aim is to study
institutional designs that incorporate a role for the FCIC and competitive firms in meeting the
federal crop insurance policy goals of actuarial soundness and high participation rates (low cost).

In particular, four alternative stylized designs are investigated.

1. Competitive firms deliver all crop insurance to farmers without any reinsurance role for
the government. Such a design would involve risk averse competitive firms who face a
trade off between profits and risk in their decision to provide insurance. The evaluation of
this design would include studying the “risk premium” charged by these firms given their
degree of risk aversion. It is of interest to see the conditions under which this market is
viable and, if it is viable, how the resulting coverage meets the government’s objective of
providing adequate coverage to all farmers. In the absence of any subsidization, this design

has no cost to taxpayers.

’ II. The government (through the FCIC) provides insurance to all farmers directly with no
role for competitive firms. Here, the F CIC acts as if it is a giant insurance firm that sells
and services crop insurance for the whole country but is mandated to operate at zero
expected profits. Unlike competitive firms, the FCIC is assumed to be a risk neutral
insurance provider (there is no “risk premium”) but it is expected to face higher costs of

providing insurance and have less information on farmer risk classification.

6
III. Both the government and competitive firms deliver insurance in a mix determined by
a reinsurance agreement between the two institutions. The purpose of this design is to
study how well the current F CIC reinsurance program meets the stated objectives of the
government. The design embodies the FCIC setting major policy variables, such as pricing
and coverage levels, while the competitive firms sell and adjust the policies sold. The
competitive firms bear some risk on the policies they sell and transfer the rest to the F CIC

through proportional and stop-loss reinsurance.

IV. Competitive firms deliver all crop insurance with the government providing
reinsurance. Unlike Design III, all primary insurance decisions are now made by the
competitive insurance firms who also decide whether or not to buy FCIC reinsurance.
Since multi-peril private crop insurance does not exist, this design evaluates under what
conditions such a market breaks down. This information helps to investigate whether
alternative federal reinsurance programs would facilitate such a market to evolve and, if it

does, whether this market would meet the stated crop insurance goals of the government.

Designs I and H are polar cases designed to identify the performance of the two main
alternative institutions for meeting the government’s objectives for crop insurance. Designs III and
IV are alternative formulations to investigate the effects of combining the two insurance institutions

to deliver crop insurance in the presence of non-diversifiable risk.

1.3 Working Hypotheses for the Study

Several working hypotheses are employed in the models of this study. These working
hypotheses simplify the models but they do so in a way that allows important insights to be gained
into the role of private firms and the government in delivering crop insurance. These working

hypotheses are:

(i) Competitive firms can deliver crop insurance at a lower cost than the FCIC.

This hypothesis is supported by the arguments of Hazel], who suggests that government
agencies have little incentive to be cost-effective when the insurer has a government guarantee.
There is no direct evidence to compare the delivery cost of the government to that of a competitive
firm since a competitive crop insurance market does not exist. However, if the health industry is to
be used as a proxy for competitive firms, administrative costs (as a ratio of total premiums) is

placed at about 5 percent (Diamond) compared to the FCIC’s 51 percent (Hazell).

(ii) When farmers belong to different risk classes then competitive firms have perfect information
regarding each farm ’s risk classification but the FCIC cannot distinguish between risk classes.

This working hypothesis is a simple stylized way of incorporating the general idea that
competitive firms have more information on farmer risk types than the government. The idea comes
from Hazell’s assertions that government agencies have little incentive to be cost-effective when the
insurer has a government guarantee. Hazell’s arguments suggest that the government would have
higher costs of getting the same information on farmer characteristics as competitive firms, which
implies that they have less incentive to collect such information compared to firms. An extreme
case would be when the competitive firms collect full information on farmer risk types but the

FCIC collects no information. Furthermore, there is evidence to suggest that political factors

8
constrain the government fi'om collecting information that would help better classify farmer risk
types. For example, attempts to include farm size in premium calculations have been seen as
government discrimination against small farmersz. This would not be a factor in the case of

competitive fimrs.

(iii) Competitive firms delivering crop insurance are risk averse while the F CIC is risk neutral.
Competitive insurance firms are often hypothesized to be driven by safety designed to
avoid bankruptcy. This is empirically supported by the fact that private insurance firms purchase
reinsurance on their portfolio. This aversion to risk is modeled using a mean-variance framework in
this study. The government, on the other hand, is backed by the federal treasury and following
arguments forwarded by Arrow and Lind (that the government is able to spread risk across a large
number of taxpayers and diversify over a large portfolio), the government’s preferences are

portrayed as risk neutral.

(iv) Competitive firms are assumed to operate at reservation utility levels in long-run
equilibrium.

This concept was first introduced by Appelbaum and Katz whereby a competitive firm
enters or exits an industry depending on Whether their expected utility from participating in the
industry is higher or lower than a reservation utility level. The reservation utility indicates
preferences for alternative investments opportunities available to the firm. By defining long run
equilibrium in this manner, the risk aversion of competitive firms is modeled to investigate the

effects of systematic risk on coverage to farmers. This approach difi’ers from the usual

 

2 Personal communication with Roy Black, Michigan State University.

9
assumptions of risk neutrality and zero profits as conditions for long-run equilibrium and is an
important innovation in this study.
(v) Agricultural yield risks are highly correlated across farms (systematic risk).
Systematic risks occur in the agricultural sector mainly because the sector is prone to
unpredictable, widespread natural disasters. Empirical support for this assumption is provided by

Miranda, Hazell, and Hara.

(vi) Yield risk is the only source of risk to farm income.
Other risks, such as price risk, are assumed away SO as to concentrate on risks that are
purely a result of the technical production process. This also facilitates isolation of the effects of

systematic risks on coverage levels of farmers.

1.4 utline of the Dissertation

The rest of the study is organized as follows. Chapter 2 provides an historical overview of
the U.S. Federal Crop Insurance Program. Chapter 3 presents a basic model and studies
competitive firms and FCIC delivery of crop insurance in the presence of high positive correlation
between farm risks (but with only one farmer risk class). Chapter 4 extends the analysis of Chapter
3 by including multiple farmer risk classes and assuming that the FCIC cannot distinguish which
farmer belongs to which class. Chapter 5 studies a stylized version of the current FCIC reinsurance
program where the F CIC makes all of the primary insurance decisions, while Chapter 6
investigates reinsurance programs where the FCIC provides reinsurance only and does not make all
primary insurance decisions. Chapter 7 provides a summary and conclusions and highlights the

main implications of the study.

2. An Overview of Crop Insurance in the U.S.

The congress created federal crop insurance in 1938 after private insurance companies
were unable to establish a financially viable multi-peril crop insurance business. The 1980 Crop
Insurance Act extended the federal crop insurance program to be the primary disaster assistance
provider in the United States. However, since its inception in 1938, and throughout the period after
the 1980 Crop Insurance Act, critics have rated the performance of the program as poor. This
section gives a brief historical overview of the program before 1980 and elaborates on the
workings and performance of the program since 1980. The primary source of information in
compiling this information has been various General Accounting Office (GAO) reports while the

historical overview has been heavily drawn fi'om Gardner and Kramer.

2.1 Historical Overview

Crop insurance was offered on a limited basis for bail and fire before the 20th century
(Gardner and Kramer). The first multiple peril crop insurance sold was recorded in 1899 by
Realty Revenue Guaranty of Minneapolis but none of the programs were sustained over time. In
1922 crop insurance was approached as a national problem for the first time when the U.S. Senate
passed a resolution calling for the investigation of crop insurance (U .8. Congress 1923). The
resolution of 1922 provided for the appointment of a select Senate Committee to investigate (i) the
kinds and costs of insurance available; (ii) the adequacy of protection; (iii) the desirability of any
practical methods for extending the scope of such insurance; and (iv) the availability and
sufliciency of statistics to properly and safely issue additional insurance. However, by the summer

of 1924 no action on crop insurance was undertaken by Congress.

10

11

In 193 6, President Roosevelt appointed a committee to make recommendations for
legislation providing for government sponsored crop insurance. The legislation established the
Federal Crop Insurance Corporation (F CIC), an agency within the United States Department of
Agriculture (The Federal Crop Insurance Act of 193 8). The management of the corporation was
vested in a board of directors appointed by the secretary of Agriculture and the FCIC was backed
by the resources of the treasury. The initial insurance was only for wheat and then it was extended
to cotton. Local committees of the Agriculture Adjustment Administration administered the crop
insurance program.

The initial experience of the crop insurance program was not very encouraging as the
program’s indemnities exceeded premiums. The program was hampered by high costs, low
participation, and the inability to accumulate adequate reserves for catastrophic losses. It seems
that inexperienced estimators relied too heavily on county averages for setting premiums (Gardner
and Kramer). Added to these problems, many states had also experienced drought during this time
(Clendenin). Private companies continued to provide coverage for hail and fire damage, which

generally are not prone to systematic risk.

After the 1939 experience of appraising yields too high, the F CIC established a key farm
system for future years (F CIC Annual Report 1939). Under this system, fifty to a hundred farms
with good yield data were selected from each county. To appraise yields, a comparable firm from
the key firms was appraised to calculate premiums. Participation in the program increased steadily
with the number of firms insured increasing from 165,775 in 1939 to 371,392 in 1941. Despite the
growth in participation, the F CIC continued to pay indemnities in excess of premiums received
(F CIC Annual Report, 1943). The program for wheat and cotton was highly criticized in the 1943

congressional hearings because of large underwriting losses and the low level of participation. The

12
congress expected to subsidize administrative costs but not cover indemnities for normal years
(Agriculture Finance Review, 1943). It became clear that the program would not be self supporting
and the annual underwriting loss of $4 million each year was troublesome because losses occurred

even during relatively good crop years.

In 1946 several new features were added to the program. A three year contract for wheat
was introduced to overcome adverse selection (as many wheat firrners bought insurance after soil
moisture indicated a poor crop). The three year contract was designed to eliminate adverse
selection, at least for the second and third year of the program (Gardner and Kramer). Also, since
there was not enough data available to determine individual firm yield variability, all crops were
insured under countywide rates. Further, partial coverage was Ofi‘ered for the first time (i.e.,
receive a lower level of protection for a lower premium) but this did not prove popular with

firmers.

In 1947, the FCIC collected premiums in excess of its indemnities for the first time in its
history. However, the legislation passed that year severely curtailed the Operations of the
corporation. Although the scope of the program was reduced, the FCIC was given greater latitude
in experimenting with alternate forms of insurance (F CIC Annual Report 1948). Over the
experimental phase (1948-52) the surplus of premiums over indemnities was $2.25 million.
Beginning in 1956, FCIC announced that it would no longer be selling insurance to counties in
Colorado, New Mexico, and Texas (these counties were considered high risk farming areas and not
selling insurance to these areas was expected to result in a surplus for the national program). Other
factors, such as improved operation methods and advancement of closing date for application to

reduce adverse selection, also contributed to the improved performance.

1 3

During the 1960s the FCIC concentrated on increasing its coverage reaching $920 million
in 1969 (versus $470 million in 1947 and $271 million in 1959). However, this rapid increase
came at a cost - in the last three years ofthe 1960s the corporation paid 29% ofthe total
indemnities paid during the entire 1948-69 period (Gardner and Kramer). The secretary of
agriculture appointed new management who concluded that, for many crops, risks had been
miscalculated and premium rates set too low. The cotton program was identified as one ofthe
corporations major problems. At this time, premiums for cotton were increased and potatoes were

dropped from the program altogether.

In 1970, the secretary of agriculture appointed a new task force on non-governmental
insurance to study the F CIC. The task force criticized the practice of establishing premiums on a
county-wide basis, and concluded that the most urgent needed change was to base the program on
individual firm risks (F CIC Task Force, 1970). It was concluded that low participation prevented
F CIC from operating an effective protection program. To increase farmers participation in the crop

insurance program, individualized firm protection was proposed.

The federal government also offers disaster assistance programs and various emergency
loan programs to cover gaps left by crop insurance. For example, beginning in the mid 1970s the
disaster assistance program paid an average of $436 million annually (mainly cash) to firrners
between 1974 and 1980. USDA made an average of $965 million in emergency loans between

1970 and 1979.

14

2.2 Th 1980 In uran eA

By 1980, direct cash payments and emergency loans were being criticized for being too
expensive and allowing finners carry too little of their production risks. This encouraged farmers
to plant crops in marginal lands susceptible to natural disasters (GAO/RCED-92-25). These
programs were also considered inequitable since they only provided payments to finners of the six
primary crops (wheat, corn, sorghum, barley, upland cotton and rice). The crop insurance program
was seen as an alternative way to deliver disaster assistance to firmers. It was considered the best
disaster assistance method because (GAO/RECD-88-211BR): (i) insurance rates are based on
actuarial data and such a system provides for more stable funding of disaster losses; and (ii) with
crop insurance there is only one value judgment to be made - the level of subsidization of the
insurance premium (while with loans and direct payments several value judgments, such as the
terms and the timing of assistance, which are not related to actual severity of the disaster have to
be made). To make crop insurance the primary disaster assistance provider the congress enacted
the Crop Insurance Act of 1980 (PL 96-365, September 26, 1980) and greatly expanded the FCIC
program. Specifically, the objectives of the 1980 Crop Insurance Act for the FCIC may be
summarized as:

(i) The program is to work on an actuarially sound basis;

(ii) The program is to provide maximum coverage feasible to all farmers such that the

program serves as the primary provider of disaster assistance;

(iii) The program is to include the private sector to make it more efficient and avoid undue

burden on taxpayers.

15

2,3 Dgign of the FCIC Progr_am

The crop insurance delivered by the FCIC is designed to protect participating farmers
against unavoidable risks, such as droughts, floods, insect infestations, and other natural disasters.
All firmers are eligible to participate if FCIC ofi‘ers an insurance program in their county.
Participants can elect coverage of 50, 65, or 75 percent of their normal yield which also includes
damage on quality for most crops. This normal yield is calculated as Average Production History
(APH) using the firmer’s own records for yields in the years available, and county averages for the
years when yield records are not available. The firrners can choose any price selection ranging
from 30 percent to 100 percent of the crop's expected market price and, accordingly, the insurance
premium rates depend on the dollar amount of insurance protection that one chooses to purchase.
FCIC subsidizes 30 percent of the premium costs for all policies up to the 65 percent coverage
level, and pays for the program’s administrative costs. The rates that the firmer fices are net rates
after government subsidies and all crop insurance premiums are tax deductible. If the firmer's
individual yields are above county yields, and records are available, then premiruns for that farmer

may be reduced. The premium is also reduced if the firmer buys separate hail and fire insurance.

The deadline for purchasing insurance is one month prior to usual planting. The bill for the
premium is not sent out until harvest time, which implies that premiums do not add to input cost at
planting time. In a year of loss, the premium is deducted and the indemnity is based on an estimate
of actual production by an independent loss adjuster. The firmer can choose the mix of crops to

insure but the firmer must insure all planted acreage owned in the county.

To satisfy the legislative mandate of the 1980 crop insurance act for greater private sector

involvement, FCIC developed two systems of delivering crop insurance - Masters Marketers and

l6
reinsured companies. Master Marketers are private insurance companies that sell crop insurance as
agents for the F CIC. These private companies bear no risk on the policies they sell and do not
adjust claims. The federal govermnent retains all premiums and pays all indemnities. The Master

Marketers get approximately 15% of the premiums as fees.

Reinsured companies sell, service and settle claims on their own crop insurance policies,
although the premium and other policies are determined by FCIC. Unlike Master Marketers,
reinsured companies bear risk on the policies they sell but obtain reinsurance from the FCIC.
Consequently, reinsured companies and the F CIC share gains and losses from policies. In addition,
the F CIC reimburses reinsured companies for administrative costs amounting to 32-33 percent Of
the premiums (premiums include government subsidy at the 50 and 65 percent coverage level). By

1990, reinsured companies accounted for 89% of total crop insurance policies sold to firmers.

2.4 Performance of Crop Insurance since the 1980 Crop Insurance Act

Various GAO reports have indicated that FCIC performance has fillen fir short of that
envisioned by Congress in the 1980 Crop Insurance Act. Despite federal payments for all
administrative expenses, and federal subsidies on crop insurance premiums, federal crop insurance
has continuously borne losses every year in the 1980-90 decade, without any establishment of
reserves against catastrophic losses. Participation rates (gross insured area/planted area) has been
less than the 50 percent hoped for and has stayed below 25 percent of the eligible acres for most of
the decade (GAO/RCED-92-25). Moreover afier a decade of operation, there is no indication that
crop insurance has replaced other disaster assistance programs. This is, in part, because of the

government's provision of ad hoc disaster payments and emergency loans. In 1980-90, the USDA

l7
spent $25 billion on insurance loans and direct payments, 76 percent or $19 billion of which was
spent for disaster payments and emergency loan programs. Government outlays over 1980—90 for

the three major disaster assistance programs are given in Table 2-1.

Table 2-1: Government Agricultural Disaster Assistance (‘000 $), 1980-90.

 

 

Program Crop Disas. Emergency Total
Insurance Payment Loan
1980 28,015 303,352 245,261 576,628
1981 426,923 1,422,363 402,171 2,251,459
1982 480,724 337,390 440,681 1,258,795
1983 345,865 127,897 436,225 909,987
1984 325,956 26,979 438,673 791,608
1985 462,696 17,795 730,337 1,210,828
1986 733,136 16,610 865,598 1,615,344
1987 562,470 824,193 1,180,047 2,566,710
1988 1,223,054 1 14,203 1,647,491 2,984,748
1989 811,292 4,012,104 2,242,010 7,065,406
1990 751,864 1,659,607 1,461,491 3,872,962
Total 6,151,997 8,862,493 10,089,985 25,104,475

 

Source. GAO/RCED-92-25 (GAO analysis of USDA data).

One of the reasons given by the congress for continuing support for all three forms of
disaster assistance programs is that the participation rates in the FCIC program has been too low.
The question of participation rates in the F CIC program was important enough to warrant a
General Accounting Ofice investigation which is documented in GAO/RCED-88-l7lBR. The
study indicates that often states surveyed in 1987, participation rates varied from 2.9 percent to

44.9 percent against a national average of 24.9 percent (since then the figure has been revised to

18
20.1 percent). Some crops were not insured at all while others had over 60 percent participation

rates. Table 22 reports the FCIC participation trends for the years 1980-91.

Table 2-2: Participation Rates in the FCIC Program, 1980-90.

 

 

 

Category # of County # of Crops Acres Acres Participation
programs Insured Eligible“ Insureda Rate (%)

1980 4,632 30 273,889 26,272 9.6
1981 5,969 30 282,333 44,996 15 .9
1982 14,498 29 280,046 42,721 15 .3
1983 15,415 32 240,103 27,935 11.6
1984 17,868 37 276,073 42,668 15.5
1985 18,892 39 265,967 48,537 18.2
1986 19,053 41 247,987 48,632 19.6
1987 19,263 42 244,807 49,134 20. l
1988 19,611 44 243,114 55,589 22.9
1989 20,507 49 253,795 101,502 40c
1990 21,354 51 254,047 101,126 39.8c
1991 21,373 51 b b b
Source: GAO/RCED-92-25 (GAO analysis of FCIC data).
1’in thousands.

t’data not available at report time.

cmany producers who participated in the 1988 or 1989 disaster assistance programs were required
to purchase crop insurance the following year.

According to the GAO/RCED-88-l7 1BR report, reasons for low participation rates are:
(i) Climate conditions - some areas are less prone to adverse climatic conditions than others; (ii)
Complex record keeping - farmers feel that the complex record keeping is more trouble than it is
worth; (iii) Condition of the farm economy - with decreasing profits in the farming sector, farmers

see insurance as an additional cost in operations; (iv) Costly premiums - premium rates are

19
considered actuarially out of line and the costs outweigh the benefits from insurance; (v) Crop
diversification - crop diversification is seen as adequate insurance by many farmers; (vi) Crop
tolerance - corn is more tolerant to hail than soybeans, but soybeans are more tolerant to drought
and this affects insurance purchasing; (vii) Delivery system - many finners believe that crop
insurance is becoming more concerned with higher commissions to agents and less concerned with
servicing firmers; (viii) Distrust of federal programs - conflicts about insurance claims have
created negative attitudes towards FCIC and the Federal Program; (ix) Farmer optimism - many
farmers have an optimistic outlook for a good crop and, hence, a decreased demand for insurance;
(x) Other federal programs - other firm programs provide income and price support with direct
cash payments at no cost; (xi) Frequent program changes - frequent program changes gives
insurance agents little time to develop efi‘ective marketing plans to sell crop insurance; (xii)
Education - the FCIC is seen to give insufficient education to farmers on the benefits of purchasing
insurance; (xiii) Insurance agent's knowledge - many insurance agents do not have the knowledge
to explain the program accurately to firmers; (xiv) Limited coverage - the insurance program is
seen to have limited coverage, so that buying this insurance is not beneficial; (xv) No guaranteed
payments - unlike other government programs, the crop insurance does not guarantee annual
payments to firmers; and (xvi) Actuarial fiimess - the FCIC’s rates may not reflect the actual risk

level of firmers and price elections may not always reflect the current year’s crop prices.

The report also gives the following reasons for different participation rates among states:
(i) Agent incentives - high risk areas have high premiums implying agent commissions are high and
therefore agents are more active in selling insurance. In the North East firms are small and there
are fewer crops, therefore less profitability; (ii) Crop diversification - reduces the expected return

from crop insurance; (iii) Crop value - high value crops, such as peanuts, sugar beets, and tobacco,

20
are more likely to be insured; (iv) Insurance education - the way in which the FCIC promote
insurance education is not adequate; and (v) Weather patterns -stable weather patterns, such as in

Arizona, cause farmers to be less likely to buy insurance.

2.5 Problems of the FCIC Program

GAO reports identify some key problem areas for the FCIC program. These areas are
actuarial soundness of the program, crop price forecasts, and inadequate risk sharing by the

insurance delivering private sector.

2.5.1 Actuarial Soundness

The Comptroller General’s report (GAO/AFMD-87-36) says that the premiums rates set
by the FCIC are not adequate to cover losses on insured crops and establishment of reasonable
reserves against unforeseen losses. The FCIC'S response to this allegation is that higher premiums
rates would further aggravate producer participation in the program (one of the goals Of the 1980
crop insurance act). Some of the specific reasons for this lack of actuarial soundness are

documented below:

2.5.1.1 LflofFarm Level Data

One clear problem is the lack of individual firm level data. This lack of data has resulted

in classification problems which has led to adverse selection, compromising the actuarial soundness

Of the program.

21

2.5.1.2 Political Pressure

Political pressure also compromises the actuarial soundness of the program. The GAO
report notes "we advised the FCIC manager that rapid expansion of the program might result in
increased exposure to loss because insurance rates might be based on questionable actuarial
assumptions and methodologies" (p.24, GAO/RECD-92-25). An example of expansion of the
program at the cost of actuarial soundness is given by the sale of insurance for non-irrigated
safilower in California counties that had sufi‘ered four straight years of drought. The policies
ofl‘ered had too high a yield guarantee to work on an actuarially sound basis. FCIC staff involved
in the California program said losses occurred for two reasons: (i) weak internal controls for
expansion of county programs and (ii) inability to establish an actuarially sound program because

of political pressure (GAO/PEMD-9l-27).

2.5.2 Crop Price Forecasts

Crop price forecasts have been the other major influence on the actuarial soundness of the
program. FCIC'S forecast are intended to reflect actual seasonal average market prices for the crop
year, and provide the basis for the different program price options. Price elections are important
because they directly affect the amount of the premium a firmer pays and the amount of indemnity
the FCIC pays. Price elections also afi‘ect private companies selling insurance because it alters
commissions. If prices are low, participation rates fill and if prices are high, indemnities are high
and moral hamrd problems increase.

Some of the problems identified with price forecasts are: (i) forecasts being made earlier in

the crop year than necessary; (ii) lack of an effective management process to evaluate accuracy and

methods; and (iii) the failure to deduct harvest costs when total crop losses occur.

22
The GAO/PEMD-92-4 report indicates that F CIC's corn, wheat, and soybean price
forecasts exhibit large bias errors. For example, in 1983-89 the FCIC would have spent $194
million less if it had used the World Agriculture Outlook Board (WAOB) price forecast for crops.
For the same period, if the A&US (Actuarial and Underwriting Service) forecasts had been used it
would have paid $167 million less. However, the WAOB is restricted by administration regulations
from publicly releasing supply and demand forecasts before the President's annual budget

submission.

F CIC's forecasts are intended to reflect actual seasonal average market prices for the crop
year, and provide the basis for the different program price options. Currently FCIC makes price
forecasts nine months prior to the closing date on insurance sales, while firmers generally delay
insurance purchases as long as possible. This puts inflexibility into the system. Also, the FCIC
forecasts national average prices rather than for regions. The Agricultural Stabilization and
Conservation Service (ASCS) has established differentials for many of the program crops on
account of local difl‘erences in grain prices. ASCS data Show that regional differences may vary
above or below average by as much as 20 percent. The GAO/PEMD-92-4 report suggests that
FCIC could improve forecasting accuracy by using the available price differentials. FCIC started
using the commodity futures market to forecast the price for soybeans in crop year 1989 and added
wheat and corn in crop year 1990. Otherwise actuarial staff prepares forecasts by discussion with,

and input from, USDA analysts and non government commodity analysts.

2.5.3 Lack of Risk Sharing by Reinsurance Companies

Under the reinsurance program of the FCIC, private insurance companies sell policies

whose premiums and yield loss adjustment policies are set by the FCIC. The private firms service

23
and adjust the losses on FCIC policies sold under their names. The reinsurance companies are
compensated for administrative, Operating and claim adjustment expenses. Through the reinsurance
agreements, the F CIC shares any gains or losses that are incurred with reinsurance companies. The

details of the reinsurance agreement are given in Chapter 5 of this study.

During the 1980s, the government has borne most of the risk for excess program losses to
ensure that companies would participate in the program, "In fict, the reinsured companies
collectively realized small profits in the years when the policies they sold had large losses" (p.3
GAO/PEMD-92-4). The report also indicates that in 1981-90 FCIC sustained over $2.3 billion in

excess losses while reinsurance companies made $101 million of profits.

3. Systematic Risk and the Delivery of Crop Insurance

This chapter compares the effects of crop insurance when insurance is delivered by a
competitive firm and when it is delivered by the federal government (also referred to as the FCIC).
Both the institutions are modeled in a highly stylized fashion that incorporates two of the basic
features outlined in the working hypotheses (see Chapter 1): (i) that competitive firms have a lower
cost of delivery than the government; and (ii) that competitive firms are risk averse while the
government is risk neutral. The analysis is done in a full information setting by assuming only one
risk type of firmers. The long-run competitive and F CIC equilibria are both developed assuming

positively correlated agricultural risks.

The rest of the chapter is organized as follows: section 3.1 describes the stylized risk
market model and the framework for analysis in this study; section 3.2 develops the model for crop
insurance demand by firmers; section 3.3 presents the competitive equilibrium for crop insurance;
section 3.4 presents the FCIC equilibrium of crop insurance; and section 3.5 compares the two

equilibria.

3.1 Stylized Risk Market Model and a Framework for Analysis

3.1.1 The Risk Market

The agricultural risk market is modeled in a simple binary loss/no-loss framework as it
provides a straightforward way to generate useful results. There is one risk class of firmers
producing the same crop. The maximum yield for each farmer is M and the yield loss when disaster

occurs is l. The loss I is binary and takes the value L with probability P and zero otherwise. The

24

marginal distribution of loss is identical for all farmers. The correlation of loss between any two

firmers is identical and indicated by the correlation coefficient p. This correlation coefficient is

taken to be a measure of systematic risk in the system since an increase in p would increase

correlation of loss between all firmers in the group. To understand the probability distribution and

the correlation of loss between firmers, consider the loss distributions of any two firmers, say 1

and 2, in the bivariate distribution below:

 

 

 

 

 

Loss Farm 2 No Loss Farm 2 Mar. Dist. Farm 1
Loss Farm 1 P11 P10 P
No Loss Farm 1 Pm Poo (1-P)
Mar. Dist. Farm 2 P (l-P) l

 

 

 

 

The bivariate distribution is for crop losses on two firms. It indicates that the probability

that firm 1 suffers loss L is P (given as the marginal distribution in column 4). There are two

scenarios when farm 1 makes a loss: (i) farm 1 loses L and firm 2 loses L (given by probability

P“) or (ii) firm 1 loses L and firm 2 makes no loss (given by probability P10). The marginal

distribution of both the firms are equal since they are from a single risk group. Although the

marginal distributions are identical, the degree of covariance between the firm losses is determined

by the probabilities of the bivariate distribution. The correlation coefficient for the two losses is

 

26
larger for higher values of P“ and P00. The expected value, variance’, and correlation4 of loss are

denoted respectively by
EU) = Z = PL ;
Var(l) = a: = P(1— P)L’;

Plo

COI'7'(I-, 1:) p_1_P—(1_j;i

In the crop insurance market only zero or positive correlation between the losses are

assumed since the agricultural market displays high levels of positive cO-variability.

3.1.2 The Mean-Variance Framework

The framework selected to analyze the agricultural insurance market is the mean-variance
(MV) decision rule for firmers and insurance firms. This approach is chosen because the analysis
involves insurance firms making coverage decisions on more than one source of risk in their
portfolios. For such an analysis, approaches such as the Expected Utility (EU) framework soon

become very complicated to analyze, even for simple models.

 

3 It should be noted that in such a bivariate loss distribution while the expected value of loss increases
with higher disaster probability, the variance reaches its maximum at probability of loss at P=0.5 and then
decreases. Ifthe probability of loss remains below 0.5 (this should be the case for all practical purposes),
any increase in probability of disaster would also increase the variance of output.

4 The Correlation coeflicient is calculated from the expression

Cov(l ,, 1)].

”WV ar<I.)Var ara)
where Cov(l ,, 1,.) = E(l,.,IJ)— E(l )E(I )

Corr(ll ,, I) i j

 

27

Although choosing the MV framework may at first appear restrictive, Meyer has shown
that if there is only one source of risk then location scale transformation of the risk would result in
the MV analysis being consistent with the EU fiamework. For example, consider output of a firm
selected so as to maximize expected utility fi'om profits which are given by n=px-c(x) (where p is
the random output price and c(x) is the non-random variable cost). Then, Since all profit
alternatives are a linear transformation of the random price p and hence related to each other only
by location and scale parameters, the expected utility-maximizing choice of x can instead be
represented as one which maximizes preferences over only mean and variance of profits. This is the
case for farmers in the model of this study because profits from firming is a linear transformation
of random crop yield. However, similar arguments cannot be extended to competitive insurance

firms since the portfolio of these agents consist of more than one source of risk.

Empirically there is very little work showing preference of the EU framework over MV
framework. Of the little work there is, Meyer and Rasche have analyzed common stocks and shown
that, within measurement errors, “one cannot reject the hypothesis that the EU efficient set of
portfolios for risk averse investors are contained in the MS [MV] efiicient set” (p. 92). If
consistency with the EU framework is desired, the MV analysis may be thought of as a two
moment approximation of the EU fiarnework. Robison and Barry write that “we justify the use of
the [MV] model even when it is not consistent with the EU models because it is interesting in its
own right. Moreover the proper test of the [MV] model is not its absolute consistency with the EU

models but its ability to describe and predict decision maker behavior under risk.” (p.84).

In this study the linear functional form

Utility(y7) = MeanCy) — g-varianceG)

28
is used where the expected value and variance are the objects of utility. Such a function allows for
relative ease in deriving optimal solutions and conducting equilibrium analysis. Holding A. constant
and maximizing the utility function gives a solution on the MV frontier that is consistent with
arriving at the frontier with any other criteria. For comparative statics analysis, holding A constant
approximates substitution efl‘ects but not income effects (Robison). A drawback of the MV
approach is that if the loss distributions are highly skewed, the decision rule does not explicitly

account for moments above the second.

For the remainder of the study the linear MV objective function is referred to as utility of

the agents since it provides a consistent ranking for the preferences of the agents.

3.2 The Demand for Crop Insurance

Let there be N farmers in the risk group Where the probability of disaster P is known to
each of the firmers. The firmers are assumed to be risk averse agents who maximize their end of
the period utility of profits. Ifan insurance market exists, it is fully described by the pair (w(cb), II),
where MO) is the premium as a function of coverage 4). The decision problem of the farmer is to
maximize his or her utility of profits by choosing the Optimal coverage level. For example, let a
farmer's maximum average yield per acre be M=140 bushels. Ifdisaster strikes the loss in yield is
L=70 bushels. The farmer may decide to purchase 50 percent coverage at w(5 O)=5 bushels. If
disaster strikes, the firmer has (l40-70+35-5)= 100 bushels of corn per acre; and if disaster does
not strike, the firmer has (140-5): 135 bushels of corn rather than in the uninsured state of 70 and

140 bushels per acre respectively.

29
To determine the demand for crop insurance let profits fi'om firming be the only argument
entering the utility of the firmer. Let the premium be quoted on a per acre basis and it is assumed
that w(¢)=w¢, so that the premium is linear in coverage. Further the coverage is restricted to be
positive and not exceed the total loss, i.e., ¢e[0,1]. The profits5 of a firmer are:
(3'1) 72",. = M—w¢—(1— ¢)l

where [takes the values L or 0 with probabilities P and (l-P) respectively. If positive coverage is
chosen, the associated premium W4) is paid irrespective of the state of nature. If there is a disaster,

the firmer loses (l-¢)L of his output, the rest is covered by insurance.
The relevant moments needed for the MV model are
as M- w ¢—(1—¢)Z
and
oi. =(1-¢)’oi
It is assumed that the firmer’s preferences are a positive firnction of expected end-of-period profits

and a negative finction of the variance of end-of-period profits from firming. The firmer’s

insurance problem is to choose coverage to maximize his or her preference function.

To obtain a specific crop insurance demand for the firmer the linear preference function

_ A
U(.) = IIF _26’2"

 

5 This satisfies Meyer’s location scale conditions Since it is linear in I

3O

2
is assumed where 2— equals the equilibrium slope at the tangency between an iso-expected utility

line and the mean-variance set (Robison and Barry). The slope reflects the degree of risk aversion
of the agent with higher values indicating higher degrees of risk aversion. The first-order condition6

(FOC) to the maximization problem of the firmer then becomes
(-w +Z)+,1(1—¢)a§ = o

and the demand for insurance at premium w is

= _ WI
(3-2) ¢ 1 [’10:].

 

Equation (3-2) helps to identify constraints on the premium (w) that must hold if an

interior solution to the problem exists. Firstly, for positive cO-insurance it must be the case that

 

w -Z —
l-[ ’10: ) >0 or (w -L)</IO'i.

In words, the difference between the premiums and expected indemnities cannot exceed the product
of the firmer’s risk preferences and the variance of the loss. The possibility of an interior solution
increases if a firmer is highly risk averse (indicated by larger values of A) or the loss displays high
variability.

Secondly an interior solution also requires that the coinsurance be less than 100 percent.

For this to hold

 

5 The second order condition is - Ila: < O and therefore the solution gives a maximum.

31

 

l-[W -L)<1 or w -Z—>0.

2
L
This implies that the prenrium must exceed the expected loss for the firmer. Further implications of

firmers purchasing insurance are summarized in the following propositions (the results of these

propositions are common knowledge and may be found in various text books’).

Proposition (3-1): If the insurance premium is actuarially fair, a risk averse farmer will always

seek fill] coverage

Proof.

If the insurance is actuarially fair, i.e., premium equals indemnity then w = I—. in which case

Equation (3-2) indicate that ¢=l.

Proposition (3-2): A more risk averse farmer will seek higher coverage than a less risk averse

farmer, provided the premium is above actuarially fair.

Proof.

From Equation (3-2), increases in A will increase 4) given that w — I—. > 0 .

Proposition (3-3): A risk averse farmer will buy less coverage if the premium per acre increases.

Proof

 

7 All these results also hold for an Expected Utility model with general distribution for l.

32

From Equation (3-2) any increase in premium (w) will lead to a decrease in coverage.

Proposition (3-4): For probabilities of loss below 0.5, a farmer with greater probability of loss

will seek to buy more coverage than a farmer with a lower probability of loss.

Proof

Note that the probability of loss affects both the mean and the variance. Consider the relevant

“primal-dual”8 element to study the response of (b to changes in P

[1+2n2L(1— ¢)P(1— er— 2P)] 3;; > o

.. 5¢
Thr fPSO.5th ——->o.
SI 6 W

3.3 Competitive Equilibrium in the Market for Crop Insurance

3.3.1 Competitive Supply of Crop Insurance

The competitive crop insurance industry is assumed to be comprised of risk averse9 firms
selling only crop insurance because the primary business of many of these firms actually engaged

in the industry is selling crop insurance. Since there iS only one risk group there are no farmer

 

8 see Silberberg

9 Although Rothschild and Stiglitz assume risk neutral behavior for private firms they indicate that
justification for such an assumption cannot be supported from the literature of firm behavior under
uncertainty since it is one of the more unsettled areas of economics. This study appeals to the empirical
evidence that insurance firms buy reinsurance on the portfolio they insure and therefore this reflects risk
averse behavior.

33
classification problems and all the properties discussed in Section 3.2 about the farmer are
assumed to hold. For convenience the firms are assumed to incur administrative costs of e per acre,

which is assumed to be linear in coverage, and there are no fixed costs.

The end of period profit of a firm from selling crop insurance to n finners is
7’s : "(Ws¢s " ¢scl " ¢SZII
i=1

The subscript s refers to the competitive insurance firm. The profits are a function of premiums
less administrative costs and indemnities paid to the n firmers. The mean and the variance'0 of the

portfolio of the competitive firm are
77s : n¢s(ws _ Z‘ C)

and

 

'0 The variance of a sum of random variables is

Varé: X,)= :Var(X,)+::Cov(X,,Xj).

Since in this study, all the random variables (yield losses) have the same marginal distribution and the
covariance between any two random variable (yield losses) is positive and identical, the correlation
coefficient between any two of the random variable is

_ Cov(X,.,Xj)

0,2

 

Substituting the identical variance and correlation coefficient values in the variance expression

Var(z X,) = 202 + 221,002
i=1 1:1 l=l j:
jti

01'

Vadi X, ) = Ozn[l + (n —1)p].

i=1

34
ai, = n¢§cifl + (n -1)p]

where p is the correlation coefficient between any two firms. Recall that p is assumed to be non-
negative. The higher the value of p, the higher the co-variability of yield loss between firms. As the
value of p increases the correlation between all the firmers increases, and therefore p also indicates
the level of non-diversifiable risk in the portfolio of the insurance firms. Therefore, with each
increase in risk, the degree of increase in the variance of the portfolio is also a function of the
degree of correlation between the risks. Ifthe risks are independent (p=0) then the portfolio
variance does not increase as much as if there is some degree of positive correlation between the

risks (p>0).

As in the case of the firmer, it is assumed that the preferences of the insurance firm are a
positive function of the end-Of-period profits and a negative function of the variance of the

end-Of-period profits, i.e.,
V(ES ’ 0:3) '

The insurance firm’s problem is to maximize its preferences by choosing the coverage it would
provide to firmers given the premium per acre. The number of policies is determined by
competition in the insurance market and at this point it is assumed to be exogenously given. To

determine the supply of insurance, as in the case of the firmer, a specific utility form of the type

V(.)=r_rs —%ofis

35
is assumed where \y reflects the degree of risk aversion of the competitive firm. The FOCll to the

firrn’s maximization problem is
(3-3) n(ws — I: — c) — gringo: [1 + (n - l)p] = O
The coverage that would be offered by the insurance firm is

¢ : (WS‘Z-C)
S Wil1+(n-l)p]°

 

This indicates that, given a premium, the coverage level decreases if either the variance or the
degree of correlation between the risks increases. Similarly if the firm becomes more risk averse.
Since the coverage must lie between 0 and 1, there are some restrictions put on the insurance

premium. Firstly, the coverage must be positive. This implies that
(w. — Z — c) > 0 .

Since the right hand side (RHS) is always positive, for the competitive firm to offer coverage, the
premium must exceed expected indemnities plus cost. In other words, the competitive firm needs a
positive “risk premium” to insure the farmer risk. However, by requiring the coverage to be less

than one, an interior solution also requires the restriction that
(“’5 - Z - C) < will + (n -1)p]

or the “risk premium” must not exceed the RHS. We know that the value of the RHS is positive by
definition. Higher values of any parameters in the RHS would decrease the likelihood of coverage

exceeding unity.

 

” Second order condition: -t//nO': [l + (n - l)p] < 0 therefore the solution gives a maximum.

36

3.3.2 A Definition of Equilibrium

In a competitive equilibrium without risk, the long-run equilibrium is characterized by zero
profit brought about by entry and exit of firms. When risk is introduced, the competitive
equilibrium is described by zero expected profit if the producers are risk neutral (Rothschild and
Stiglitz). Ifthe producers are risk averse then Sandmo’s work provides the beginnings of a

framework to analyze such a problem. The basic model of the firm in Sandmo’s paper is

(3-4) Mqax Ell/(%)] = Ell/(P61 - C((I) - B]

where p is the random output price, q is the quantity to be produced, c(q) is the variable cost and B

is the fixed cost. The maximization process involves solving the FOC
(3-5) ElU' (70(1) - 0' (61)] = 0-

Given that the second order condition is satisfied, comparative statics are drawn from this

framework.

The equilibrium described above is considered to be a Short run analysis because output
prices are treated as exogenously given with a known distribution. Appelbaum and Katz extend
Sandmo’s article to incorporate the behavior of expected utility maximizing firms in long-run
equilibrium. This is achieved by (i) letting stochastic output price be a function of industry output;
and (ii) letting the expected utility of the firm be equal to a reservation utility b Of some benchmark
activity. Entry and exit of firms require that the expected utility of profits equal this benchmark

utility level, i.e.,

(345) BMW - C(q) - Bl] - b = 0

37
Thus the industry equilibrium is determined by the intersections of equations (3 -5) and (3 -6) and
the number of firms in the industry are determined endogenously. In this model firm output adjusts
with free entry and exit of firms, which in turn efi‘ects output price and keeps the expected utility of
the firm at the reservation utility level b. Appelbaum and Katz find that comparative statics for the
firm are altered in the industry equilibrium as opposed to the short run equilibrium studied in
Sandmo’s framework. An application of the principle of reservation utility is found in Meyer and
Robison which look at land (an input) prices adjusting (affecting profitability through cost Of

production) to maintain industry equilibrium in the agricultural land market.

A long-run equilibrium concept Similar to that suggested by Appelbaum and Katz is
adopted here to study the competitive equilibrium in crop insurance. Appelbaum and Katz’s
framework is modified such that, in the long-run, competition adjusts the number of insurance
policies sold to maintain a reservation utility level. The number of firms in the industry is
determined endogenously and is given by N/n where N is the total number of firmers. All
information is assumed to be common knowledge. All the firms in the industry are identical in all
respects and therefore it is suficient to determine the equilibrium by analyzing one firm. For
convenience, it is assumed that when the value of n is not a positive integer, the firms collude to
provide insurance on the fraction such that the farmer is provided insurance as if he were receiving

it fi'om one firm.

38

3.3.3 The Competitive Equilibrium

Definition (3-1): A competitive equilibrium in this model of crop insurance delivery is a premium

level wo, coverage level pic and number of policies no for each firm such that:

(3-7) ( -w° + Z)+ 2.(1-¢°)a: = o
(3-8) n°(w° — Z — c) — ym°¢°ai[1+(n° —1)p]= 0
(3-9) n°¢°(w° — Z — c) — %-n°(¢°)zoi[1 + (n0 —1)p]— b = O.

The first expression is the demand for insurance by the firmer, the second expression is the short-
run supply of insurance by a competitive firm, and the third expression is the long-run equilibrium
condition that the utility of the firm is at the reservation level b. The simultaneous solution of the

three non-linear system of equations determines the long-run equilibrium premium wo, coverage tho

and the number of policies sold It0 by each insurance firm.

3.3.4 Properties of the Competitive Equilibrium

The properties of the equilibrium are discussed in a series of propositions below. In stating
the propositions it is assumed that a solution exists (the alternative being no insurance bought or

sold at equilibrium).

Proposition (3-5): With independent risks (p=0) the equilibrium coverage (¢0) to the farmer is

determined independently of the reservation utility (b) of the firm.

39

Proof

At equilibrium, the FOC of the firm may be rewritten as

_ w°—L—c .
W012. [1 + (n° - 1)p] ’

 

¢0

with p = 0, the FOC reduces to

wo—L—c

0
¢— WI

and therefore premium and coverage are determined by equations (3 -7) and (3 -8) which are

independent of n°.

Proposition (3-6): With independent risks (p=0) reservation utility (b) determines the number of

policies held by the firm at equilibrium, and it is increasing in b.
Proof:
Substituting equilibrium coverage from Equation (3-8) into (3-9) to solve for n0

0

n _ th/IO’: .
[1+(n° —1)p] (w° —I-,—c)2 ’

 

(3-10)

with p = 0 this expression reduces to

o_ ZbV’UI
n — 0_—— 2.
(w L c)

 

Since w is determined independently of this equation (see proposition (3 -5)), and given that all the

parameters are known, any increase in b will increase n°.

40
Proposition (3-7): With identical risks (p=1), reservation utility (b) determines the premium over
and above expected indemnities and cost charged to farmers at equilibrium, and it is increasing

in b.
Proof'

with p=l, expression (3-10) reduces to

(w° —Z—c)= ‘Ith/IO': .

Any increase in b will increase the equilibrium premium charged to farmers.

Proposition (3-8): With independent risks (p=0) , the equilibrium coverage decreases with
increases in cost of delivery (c) and/or increases in risk aversion of the insurance provider (w);
and the equilibrium coverage increases with increase in variance (of) and/or increases in the

risk aversion (2) of the farmer .
Proof:

Adi—c

uilibriumcove eis °=-—————.
Eq rag ¢ OWN/I)

From this equation it is clear that an increase in c or u: would decrease equilibrium coverage.

Further

o
§¢2 = 2 20 >0
501. (UL) (II/+4)

 

 

d¢° _ Oil/1+0

— 2 2>0
621 CLO/1+2)

 

41
Both the derivatives are positive indicating increases in variance and risk aversion of the farmer

would both increase equilibrium coverage.

Proposition (3-9): With identical risks (p=1), the equilibrium coverage decreases with increases
in cost of delivery (0) and/or increases in risk aversion of the insurance provider (w) and/or
increases in the reservation utility of the firm (b); and the equilibrium coverage increases with

increases in variance (of) and/or increases in the risk aversion of the farmer ()1).

Proof

(fibula: +c

2
2.0L

 

With (p=1) equilibrium coverage is ¢° = 1— and it is clear from this equation that

increases in b, \y, or c would decrease coverage and increase in A would increase coverage. Also,
Since

wo — 1(‘12by/0'2 +6)

602 ' wait

 

>0

increases in variance will increase coverage.

So fir, the comparative statics in the propositions above were limited to polar cases of p=0
and p=1. This is because when positive but less than perfect correlation exists, Obtaining
equilibrium values requires solving a system of non-linear equations in three unknowns. Because of
this dificulty numerical examples are explored and results are reported in Table 3-1 through Table
3-3. The parameters chosen for the numerical example are P=0.3 and 0.4; c=5 bushels; M=l40
bushels; L=70 bushels; \y=0.01 and 0.001;7.=0.01; and b=10 or 20. The parameters chosen do not

reflect any real world application and are only for expository purposes.

42

Table 3-1: Competitive Equilibrium Coverage Levels: \y=0.01, 1:001, b=10.

 

 

 

 

 

 

 

E P=0.30 i P=0.40
p i wl (p1 ul n1 I w2 b2 u2 n2

i 0 0 113.855 0 i 0 0 106.120 0
0.0 : 27.645 0.354 114.501 15.490 : 35.380 0.372 106.936 12.260
0.1 : 29.443 0.179 114.021 20.472 : 37.159 0.221 106.408 14.683
0.2 : 30.669 0.060 113.874 49.683 : 38.417 0.114 106.197 23.610
0.3 : a a a a : 39.494 0.023 103.846 103.846
0.4 l a a a a l a a a a
a = interior solution did not exist.
Table 3-2: Competitive Equilibrium Coverage Levels: \y=0.01, 1:001, b=20.

I I

1 P=O.30 1 P=0.40
p I WI (1)] ul n1 E w2 (1)2 u2 n2
0.0 i 27.645 0.354 114.501 30.980 E 35.380 0.372 106.936 24.520
0.1 : 30.691 0.058 113.872 102.677 : 38.466 0.110 106.191 48.681
0.14 : a a a a : 39.311 0.038 106.129 125.972
0.15 1 a a a a l a a a a

 

a = interior solution did not exist.

43

Table 3-3: Competitive Equilibrium Coverage Levels: w=0.001, k=0.01, b=10.

 

 

 

I P=0.30 I P=0.40
p E wl «bl u1 n1 I w2 <|>2 u2 n2
0.0 i 24.663 0.644 115.989 46.857 i 31.796 0.677 108.816 37.086
0.1 : 25.708 0.543 115.369 21.589 : 32.874 0.586 108.136 18.223
0.2 : 26.241 0.491 115.094 18.190 : 33.437 0.538 107.820 15.266
0.3 : 26.652 0.451 114.900 16.731 : 33.870 0.501 107.596 13.913
0.4 : 27.001 0.417 114.749 15.990 : 34.237 0.470 107.417 13.155
0.5 : 27.309 0.387 114.625 15.623 : 34.562 0.442 107.269 12.703
0.6 : 27.589 0.360 114.521 15.494 : 34.856 0.417 107.143 12.438
0.7 : 27.848 0.335 114.431 15.538 : 35.128 0.394 107.032 12.301
0.8 : 28.090 0.311 114.353 15.724 : 35.382 0.372 106.935 12.260
0.9 : 28.319 0.289 114.284 16.038 : 35.622 0.352 106.848 12.297
1.0 1 28.537 0.268 114.223 16.476 1 35.850 0.333 106.770 12.403

 

In all of the tables, results are presented for two probabilities of disaster (P=0.3 and

=0.4). This is done to see how the results are affected by an increase in the probability of disaster.

Table 3-1 and Table 3-2 are presented to illustrate the effects of increases in reservation utility Of

the firm from b=10 to b=20. And Table 3-1 and Table 3-3 are presented to see the effects of

decreasing the degree of risk aversion of the firm from w = 0.01 to \y = 0.001 (the degree of risk

aversion of the firmer A is kept constant in all the examples) and ul and u2 are the equilibrium

utility levels of the firmers when P=0.3 and P=0.4 respectively. Similarly n1 and n2 are

equilibrium number of policies held by the insurance firms when P=0.3 and P=0.4 respectively.

The first row of Table 3-1 Shows the utility of the farmer when no insurance is bought. AS

expected, if positive levels of coverage are bought in equilibrium, the utility of the firmer is greater

44
than without insurance. Also as expected, the coverage to finners increases as the degree of risk

aversion of the firm decreases (compare Table 3-1 and Table 3-3).

fluilibrium Effects of Correlation (9) Between LOSS_e_S_

The numerical results Show that, in all cases, as the degree of correlation among risks
increases, the insurance premium increases and the coverage decreases in equilibrium. The
equilibrium utility of the firmers is always at its highest when risks are independent (p=0). Results
in Table 3-1 and Table 3-2 indicate that, as risks become more non-diversifiable, competitive
equilibrium coverage decreases. Given a set of parameters and a cost structure, correlation between

firm losses could reach a point such that no competitive crop insurance market exists.

There appears to be no pattern to the number of policies the firm would hold as the

correlation between losses increases. The results indicate that the number of policies held by a firm

may increase or decrease as p increases, depending on the parameters of the system.

fiuilibrium Effects of Reservation Utility (b)

As indicated by proposition (3 -6) the reservation utility of the firm has no effect on the
equilibrium coverage if risks are un-correlated (see results in Table 3-1 and Table 3-2 for p = 0).
For all other cases the equilibrium premium increases and the coverage decreases with increases in
reservation utility of the firm. Therefore, with correlated risks the competitive equilibrium coverage
depends on the reservation utility of the firm. As stated earlier the only exception to this rule is
when the degree of correlation among the firms is zero and the level of reservation utility decides

only the number of policies (It) the firm holds at equilibrium. Theoretically, there is a possibility of

45
only one firm supplying crop insurance in a competitive equilibrium, depending on the level of

reservation utility.

3.4 FCIC Deliveg of Crop Insurance

Now we turn to an alternate institutional arrangement, the FCIC, for delivering crop
insurance to the same firmers. The FCIC crop insurance delivery system is modeled in a stylized
way according to the mandates Of the 1980 Crop Insurance Act, i.e., provide crop insurance to all
firmers at actuarially fair premiums plus delivery cost. Such a program would be self sustaining in
the long-run (as desired by Congress) and therefore the current program’s subsidies are not
included in the design. Since there is only one risk type, there are no firmer classification problems

faced by the FCIC and all information is assumed to be common knowledge.

Delivery cost is g which is assumed to be linear in coverage and there are no fixed costs.

The FCIC premium per acre is
ws 2 L + g

where g is the cost of delivery of the FCIC. Farmer behavior remains exactly the same as in the

competitive equilibrium developed earlier. The F CIC equilibrium may now be introduced.

3.4.1 The FCIC equilibrium

Definition (3-2): An FC'IC equilibrium in this model of crop insurance delivery is a premium w‘

and a coverage level 45. such that:

46
(3-11) w‘ =Z+g

. _

w-L

2
L

(3-12) ¢‘ =1-

 

Since there are no informational problems the FCIC breaks even in the long-run. Given the
premiums, the firmers choose coverage 4). at equilibrium. Note that for an interior solution to

exists premiums must exceed expected indemnities in this model.

3.4.2 Properties of the Equilibrium

The equilibrium coverage under the F CIC program is

 

r g
:1-
¢ 20':

Depending on the delivery cost (g), the coverage to the firmer could be minimal (for high values of
g) or close to unity (for very low values of g). This equilibrium coverage is an increasing function

of the degree of risk aversion of the firmer and the variance of the risk.

Note that because the FCIC sets premiums at actuarially fiir levels plus delivery costs, the
degree of correlation between the firm losses does not enter the FCIC decision making in crop
insurance delivery. This suggests risk neutral behavior of the FCIC for reasons such as the
government’s ability to spread risk among a large number of taxpayers and also diversify its

portfolio over a large number of activities (Arrow and Lind).

47

3.5 Comnarigon of {he Competigive Equilibrium and FCIC Delivery

It has been a working hypothesis in this study that the cost Of delivery associated with the
F CIC is higher than the competitive firms. However, whether the firmer benefits more fi'om the
FCIC or the competitive equilibrium is also a function of risk preferences of the competitive firms,
the level of non-diversifiable risk in the system and the level of reservation utility of the competitive

firms.

Proposition (3-1 0) : If premium and coverage of both the F CIC and the competitive equilibrium
are identical (w0=w‘, qo=q') then it must be the case that delivery cost associated with the FCIC

is higher than that of the competitive firm (g>c).
Proof:

If equilibrium coverage levels are identical then

> 20': - c _ 2.0: — g
Mi +aiw(l+(n° -1)p) Mi

 

¢°=¢‘=

From the expression above, it can be seen that the denominator associated with the competitive
firm coverage is larger than that of the FCIC coverage because OSpSI and given that the firm

holds at least one policy (n21). Therefore, for equality to hold c<g.

48
Proposition (3-11): The farmer has greater (equal or less) coverage in an FCIC equilibrium
than in a competitive equilibrium if the cost of delivery (g) associated with the FCIC is
Adi—c

1+%(l+(n°—l)p) .

 

g < (2)2102 -

Proof:

Follows directly from proposition (3-10) above.

The propositions indicate that the cost of delivery of the FCIC can always be made high
enough such that competitive equilibrium could be made more desirable to firmers. Proposition (3-
11) indicates the relationship between FCIC cost of delivery and the parameters of the competitive
firm under which the farmer is indifferent between the two institutions delivering crop insurance.
This relationship shows that cost of delivery alone does not drive the equilibrium coverage to
firmers. As can be seen from the results of Section 3.3.4, the amount of non-diversifiable risk in
the system and the reservation utility of the competitive firm are important fictors in detemrining

the competitive equilibrium coverage along with degree of risk aversion Of the firm.

Depending on the value of the parameters, increases in the level of non-diversifiable risk in
the system could substantially reduce equilibrium coverage to the extent that no competitive
equilibrium exits. Since the level of non-diversifiable risk is high in the agricultural sector, these
results indicate that competitive equilibrium coverage could be substantially reduced in this market.
In practice one observes private insurance for hail and fire (which are fairly independent across
firms) but no private insurance for multi-peril disasters as floods, pests, etc. which are highly

correlated across firms (for further discussion on systematic risk and equilibrium, see Chapter 6).

49

Furthermore, whether the competitive or the FCIC equilibrium provides higher coverage to
farmers is a function of the reservation utility of the firms. The analysis indicates that if risks are

dent then the reservation utility does not affect the coverage to farmers and equilibrium is
d rmined by the degrees of risk aversion of the insurance seller and buyer, as well as the cost of
de 'very. This is the general scenario of most studies that assume independent risks. However with
non-diversifiable risks, the reservation utility affects the coverage to firmers such that coverage
decreases with increases in the reservation utility. Ifthe reservation utility is very high then
equilibrium coverage to firmers could be minimal or even non-existent. The non-independence of

risk, therefore, affects coverage indirectly through reservation utility.

The results in this chapter suggest that if there are high levels of systematic risks in the
crop insurance market the only recourse may be for the government to deliver crop insurance
directly if it is politically desirable to do so. However, the effectiveness of federal crop insurance
delivery may be reduced if the FCIC fices high information costs such that classification of
firmers into risk groups is a problem. This could compromise the actuarial soundness of the
program and pose serious budgetary outlays to the taxpayers. The next chapter studies this aspect

of the problem in more detail.

4. Crop Insurance Delivery Under Multiple Farmer Risk Classes

The purpose of this chapter is to build on the stylized model of the previous chapter by
including multiple firmer risk types. It is assumed that there are heterogeneous firm risk types and,
as discussed in the working hypotheses (see Chapter 1), the government (or the FCIC) is assumed
to be unable to obtain information to distinguish firmers into risk classes for insurance selling
purposes. On the other hand competitive firms providing crop insurance are assumed to be able to
obtain this information costlessly. These polar cases are used to highlight the issues resulting from
information differences between the government and competitive firms. All of the other

assumptions outlined in Chapter 3 for the crop insurance model also hold in this chapter.

The initial sections of this chapter are devoted to developing an F CIC equilibrium while
imposing the assumptions that (i) the program is actuarially sound”; and (ii) the program provides
maximum coverage to all farmers. Conditions under which the inability to classify finners into risk
classes becomes a problem are developed. The later sections of the chapter develop a competitive
model of insurance delivery with more than one risk type in the market (and positive correlation
between firm risks). The chapter states the conditions under which a competitive insurance firm
would prefer to hold contracts for both risk types in its portfolio, and the resulting implications for

firmer coverage. The chapter then compares the FCIC equilibrium to the competitive equilibrium.

 

‘2 Actuarially sound is taken to mean that the program is self sustaining.
50

51

4.1 The Risk Market

To introduce imperfect information into the FCIC equilibrium, firmers of two risk types, 1
and 2, are assumed. Type 2 firmers are the high risk group. The maximum yield for each firmer in
each risk class is M and the random yield loss is l,- for type i=1,2. The loss I,- is binary and takes the
value L with probability P, for risk type i and zero otherwise. There are N of type 1 firmers and
N2 of type 2 firmers. The ratio of type 1 to type 2 firmers is R. For example, R=1 implies that

there are equal numbers of both types of farmers and R=2 implies that there are twice as many
type 1 than type 2.
To understand the joint probability distribution of any two firms, consider the loss

distributions of the firms, say A and B, in the bivariate distribution below:

 

 

 

 

 

Loss Farm B No Loss Farm B Mar. Dist. Farm A
Loss Farm A PAB PAo PA
No Loss Farm A P03 P00 (l-PA)
Mar. Dist. Farm B PB (l-PB) l

 

 

 

 

The bivariate distribution is for crop losses on two firms. It indicates that the probability

that firm A sufi’ers loss L is PA (given as the marginal distribution in column 4). There are two

scenarios when firm A makes a loss: (i) firm A loses L and farm B loses L (given by probability

PAB) or (ii) firm A loses L and farm B has no loss (given by probability PM). Similarly for firm

 

52
B’s marginal probability distribution. Ifthe two farmers are of the same risk type then PA=PB and

the marginal distributions of the two firms are equal.

The expected value, variance and correlation coeflicient of loss for the two firms are:
E(l,.)=I—.,.=P,.L i=A,B

Var(l,.) = of = B(l—R)L’ i: M

P.. —P.P.
Corr(l,,lj) 2 pi}. = " ' ’
Jen—mead»

i=A,B

 

Since the study refers to risk types as type 1 and type 2, for notational purposes correlation
of losses between two finners within a risk type are p. and p2 for types 1 and 2 respectively and

correlation of losses between a type 1 farmer and a type 2 firmer is p12.

4.2 The FCIC Eguilibrium

4.2.1 Definition of an FCIC equilibrium

The FCIC equilibrium is modeled to be consistent with the mandates of the 1980 Crop
Insurance Act, i.e., (i) the program should operate on an actuarially sound basis with premium
income suficient to cover losses; and (ii) the program must make maximum crop insurance
coverage available to all farmers such that the F CIC program serves as the primary provider of

disaster assistance.

Unlike the previous chapter, the FCIC is now assumed to Operate with imperfect

information about firmer risk types. The informational problem of the FCIC is such that it lacks

53
information to classify farmers into risk types. It is assumed that the FCIC has knowledge about
the amount of loss, the probability of loss of high and low risk types, the ratio of low to high risk
firmers, and the preferences of the farmers. In other words the only information that is lacking is

whether a particular farmer is oftype l or 2.

Given that the FCIC is mandated to operate with zero expected profits and maximum
coverage to all firmers, an ideal program with full information would operate by selling insurance
to each firm at an actuarially fiir price for that firm, plus cost of delivery, and the farmers would
choose their optimum coverage. However, with the information problems described this may not be
feasible since a high risk firmer cannot be distinguished from a low risk firmer and the high risk
farmer does not have an incentive to reveal that he is high risk, thereby paying higher premiums.
Therefore if type 2 firmers have an incentive to buy type 1 contracts, the contracts must be

designed to correct for the classification problems.

To arrive at an FCIC equilibrium it is assruned that the FCIC is the sole provider of crop
insurance, and that it can offer contracts that specify both premium and coverage levels. Since the
second mandate requires that maximum coverage be provided to all risk types, the FCIC provides
only those contracts that are emcient. In other words, if the coverage (and utility) of a firmer can
be increased while keeping the other firmers unaffected , such a contract will be ofl’ered. Because
there are two risk types (and given all have identical preferences), more than two contracts are not

considered since a third contract would be redundant.

Ideally, in this model the F CIC would like to classify firmers into two risk pools: a high

risk pool which has all the high risk firmers and a low risk pool that has all the low risk farmers.

54
With two risk pools there are three situations envisioned where an FCIC equilibrium can occur
under firmer classification problems: (i) a separating equilibrium with no cross subsidies" across
risk pools; (ii) a separating equilibrium with cross subsidies across risk pools; and (iii) a pooling
equilibrium with average pricing of the premium. Equilibrium in situations (i) and (ii) require that
when two contracts, high and low, are offered by the FCIC the high risk types have no incentive to
buy the low risk contracts and vice versa (there is self selection of contracts). In Situation (iii)
everyone that buys insurance buys the same contract since only one premium and coverage is
offered by the F CIC to all firmers irrespective of risk type. It is self evident that in a pooling
equilibrium there are cross subsidies across risk types. The three situations are discussed in detail

in the sections below.

4.2.2 Separating Equilibrium With N o Cross-Subsidization Across Risk Pools

As mentioned earlier, a separating equilibrium requires that the high risk types buy the
high risk policies and the low risk types buy the low risk policies offered by the FCIC. The
premium and coverage must be such the one risk type has no incentive to buy the contract meant
for the other risk type. The other requirement in this equilibrium is that there are to be no cross
subsidies across risk types. Therefore the FCIC makes zero expected profits from each risk pool.
In other words the premiums in each risk pool must be such that they equal expected indemnities

plus cost of delivery or

 

'3 Subsidies or taxes are referred to in the context of subsidizing or taxing a coverage that would be
ofl‘ered in full information. For example if 50 percent coverage is priced at 2 bushels in a full information
environment then selling the same coverage for 3 bushels implies that there is a tax of 1 bushel. Similarly
for subsidies.

55
(4-1) ¢wa’ -- ¢Z(IT.- +g) I-—— 1.2
where ¢: is the separating equilibrium coverage to risk types 1 and 2 and wf’ is the full

information premium per acre (which equals expected loss I: and cost of delivery g). Note that as

l
in Chapter 3 the premium and cost of delivery are assumed to be linear in coverage. Given that this

coverage leads to a separating equilibrium, the following conditions hold

(4-2) NAM“ -Z -g)¢.’ = 0 and N205" 43 -g)¢2 = 0.

The separating equilibrium without cross subsidies is identical in most respects the model
developed by Rothschild and Stiglitz (R-S) for a risk neutral competitive insurance firm. The only
difference is that, while in the R-S model the zero profit condition on each risk pool is arrived at by
competition, here it is arrived at by the mandates of the FCIC and the assumption of no cross
subsidies across risk types. The other major difference to be noted is that here farm risks are
allowed to be positively correlated while R-S assume no correlation. However, this has no effect
because both the R-S model and the FCIC model assume insurance providers are risk neutral so

higher moments do not come into play.

Most of the results arrived at by the R-S model hold for the current FCIC separating
equilibrium without cross subsidies. While the R-S model is mainly a graphical exposition of the
separating equilibrium, the FCIC equilibrium is developed here with a more mathematical
approach using the mean-variance framework. Certain specific results which are more relevant to

the study of crop insurance are highlighted in the ensuing paragraphs.

56

4.2.2.1 The Model

Since the FCIC is unable to distinguish between high and low risk firmers, and premiums
must satisfy expression (4-1), there must be restrictions on coverage to achieve a separating
equilibrium. The restrictions on coverage must be such that the high risk firmers are indifi'erent to
or less prefer the coverage associated with the premium for the low risk types and vice versa". To
arrive at the restrictions on coverage in this equilibrium, the second mandate of providing
maximum coverage to all risk types is used. The FCIC offers only two contracts specifying both
coverage level and premium for both the contracts. Coverage levels are obtained by providing full
information coverage to the high risk type and, keeping the utility of the high risk types constant at
this level, determining the maximum coverage to the lower risk group such that the high risk types
are indifl’erent between the two contracts". It is shown later in this chapter that the separating
equilibrium arrived at in this fashion leads to the greatest utility for both risk types under F CIC

classification problems.

As in Chapter 3 assume the preference function

U(fi,oz)=fi—::—oz

 

” There is an important implicit restriction in the separating equilibrium without cross subsidies
regarding the number of high risk farmers. It is obvious from the equation above that if there were a low
proportion of high risk types, then premiums set at the actuarially fair level plus cost is not a preferred
equilibrium. For example if there was only one high risk farmer and all the others were low risk types,
then by making a loss on the one high risk farmer and spreading this loss across all low risk farmers
potentially could increase the utility of the low risk farmers as they would be paying only slightly higher
prices for coverage than the case when there is full information. Such an equilibrium that allows losses on
some risk types and profits on other risk types, but overall zero profits can be superior to the equilibrium
studied here. Such situations are studied later in the chapter.

1’ The probability of disaster of the risk types and preferences of firmers are assumed known by the FCIC.

57
f0? both types of firmers, so that the preference function is increasing in expected value and
decreasing in the variance of profits from firming. The value of A indicates the risk preferences of
l

the firmer, with higher values indicating greater degrees of risk aversion. Let the optimum
coVerage associated with the high risk type at full information be 45:7 ; superscript F7 indicates firll
information.
In this separating equilibrium the contract offered to type 2 farmers is the same contract as
F]

in the full information situation, i.e., (a52 w? , ¢§7 ). The preference function of a type 2 farmer

when he buys this contract is16
— 2.
U"(.|wf’,¢§”)= M— ¢§’W§’ —(1— SOL. ~30- 5%:

where UH is the value of the preference function of the higher risk type. Now let the FCIC offer a
contract for the low risk type given by (¢,w1” , ¢1), where ¢1 is some positive coverage level meant

for a firmer fi'om risk type 1. The high risk firmer may opt to buy this since the FCIC cannot
distinguish him from a type 1 firmer. If the firmer buys the low risk contract, the utility of the

farmer is
U”<.le,¢.>=M—¢.wf’ 41402—914020:

If U "(.lwf I , ¢l ) > U H (.lw:7 , ¢§7 ) then the coverage and premium meant for the type 1

firmer is preferred by the high risk firmer. A pooling situation occurs if both groups buy the same

contract and the program would lose money on the high risk farmers. Therefore, for a separating

 

'6 See the previous chapter for a derivation of the expected value and variance of farm profits.

58
equilibrium to exist, it must be the case that UH(.|WIFI,¢1)S U” (.lwf’, :7 ); it is assumed that
when the preferences from the two contracts are equal the firmer purchases the contract meant for
the high risk type. However, contracts of the form U "(. lw,“ , ¢l ) < U H (.lwf’ , ¢§7 ) would imply

decreasing coverage of the lower risk group, without any apparent gain to the higher risk group
(goes against the FCIC mandate of maximum coverage feasible to all risk types) and as such these
contracts are not considered. The coverage to the lower risk type at which the higher risk type is

indifferent between buying the two FCIC contracts is the equilibrium coverage for the lower risk

type.

Definition (4-1): AN FCIC separating equilibrium without cross subsidies between risk pools in

this model of crop insurance delivery is a premium w,” and coverages ¢I and ¢2Fl such that:

(4-3) w.” = Z. + g t: 1,2

(44) §’=1- g

 

(4-5) ¢I(—wf’+fz)——§-(¢I-l)’a§= §’(—w§’+Z.)—-§-(¢§’-l)’o§-

The equilibrium values are obtained by solving the four equations sequentially. With known
probability of disasters, the premiums to the two risk types are set at expected indemnity plus cost
(full information premiums). Only two coverage levels are marketed by the F CIC: (i) the full
information coverage for the high risk types arrived at by the second expression above; and (ii)
separating equilibrium coverage for the low risk types detennined by the point where the high risk

type is indifi‘erent between the two contracts.

59

At equilibrium with positive g, an interior solution would be ensured for risk type 2 as long
as g < 110;. All the conditions discussed for an interior solution in Chapter 3 hold for both types

of farmers. However, with classification problems there are some additional concerns associated
with this equilibrium that must be addressed. Most of these concerns effect only risk type 1 since

the higher risk type enjoys the same conditions as a full information situation. For risk type 1 there
is a possibility that the coverage (45;) arrived at from expression (4-5) is higher than what it would

be under full information, i.e., the parameters of the model are such that

‘> Fl:1_ g
¢I l 1012

 

In such a case the utility of type 1 firmers could be increased by setting the coverage less than
(¢: )at the full information level (515:7 ) with premium #7wa . Ifsuch a situation exists, then the
FCIC dificulty with firmer classification is not a problem. The FCIC could simply restrict

coverage with each premium at the optimum level and each group would buy contracts meant for

them.

Consider the following discussion to see the conditions under which information
asymmetry is always a problem with positive delivery cost. Start with a situation in which full

information coverage is dictated by the FCIC for both risk pools. The two contracts offered are (i)

coverage ¢f7 and premium ¢flwlm and (ii) coverage ¢§I with premium ¢§7wfl . Note that both

premium and coverage are restricted to the ones specified in the contracts. Given that both the

contracts are marketed by the FCIC, the firmer may buy any contract he chooses. Also note that

My < 4‘? always for positive g.

60

2 2
Proposition (4-1): If g 5 (2;, then FCIC premium and coverage to farmers cannot be
02 01

delivered at fill] information levels if the program is to remain actuarially sound.

Proof

The situation where the low risk contract is more attractive (>), indifferent (=), or less

attractive (<) to the high risk contract for a type 2 firmer is given by the condition
— 2 — 2
¢f"(—w.” +L.)— 5w?" -1)Za§ (>,=,<> ¢f’<—w§’ + Lajos? — Ito:

where the left hand side (LHS) is the utility of the firmer buying low risk contract and the right
hand side (RHS) is the utility of the firmer buying high risk contracts (the common terms are
canceled out). By substituting in the premium and the full information values for 4), the above

expressions reduces to

- — (101’ - g) g [
._ *_ ___
(4.6) (L2 L1) 2 l2 (>a s <) a 120:

8((0§)2 -(012 )2)

20,2

 

—<a:—ai>].

As indicated earlier, as long as the LHS < RHS, the high risk firmer would always prefer the
contracts written for the high risk types and farmer classification is not a problem in this model. If
LHS > RHS then lack of information on firmer classification is a problem and the coverage to the
low risk types is determined in a separating equilibrium stated above (coverage to type 1 finners
will be always less than coverage at full information). For lack of information on firmer
classification to be always a problem in this model (with positive cost of delivery insurance), it is
sumcient that the terms in the brackets in the RHS is non-positive or

8((Ui)2 -(0.2)2)

20',2

 

-(0’§-0'i’)50

61

01'

202
4-7 s ‘
( ’ g (as?)

It must be remembered that imperfect information may still cause a problem even if g does
not satisfies the condition. This would depend on expression (4-6). Also, as can be seen from the
above proposition at g=0, imperfect information would never allow F CIC equilibrium to deliver
crop insurance at full information levels. For the remainder of the study it is assumed that (4-1)

holds such that the separating equilibrium coverage to type 1 firmers is less than full information
coverage (if < ¢f’ ).
So fir in the discussion it has been implicitly assumed that when a separating equilibrium

exists, the low risk types will always purchase the contract meant for the low risk firmer. Here it is

shown that this is always true in equilibrium.

Proposition (4-2): In a separating equilibrium, the low risk type always prefers the low risk

contract to the high risk contract.
Proof:
Suppose that both the risk types buy the high risk contract and preferences are
U H (.lwf7 ,¢f’ ) and U L (.wa’ , a5? ) respectively for risk types 2 and 1. Now decrease the
premium to w,” and adjust coverage such that the high risk type is indifferent between the

contracts. This is the separating equilibrium point U H (.lwlm , ¢I) = U ”(.waI , W?) . Recall from

62

Chapter 3 that for the same premium a high risk firmer would always seek higher coverage than a

low risk firmer. Thus for the low risk firmer to be indifferent to UL (.lwfI , a}? ), the coverage will

be lower than it". Since it is assumed that Proposition (4-1) holds then

U‘(.|wf',¢:)>U‘(.|w:",¢§’).

Having established the conditions under which the separating equilibrium is feasible, the

properties of the equilibrium with no cross subsidies are now discussed.

Proposition (4-3): Any increase in the cost of delivery (g) will decrease coverage to both the high

and low risk farmers.

Proof'

Increase in cost of delivery would increase premiums by the same amount to both risk
groups, i.e.,

W?" = A + g + Ag

W?“ = L2 + g + Ag

Without an adjustment in equilibrium coverage, then the utility associated with the low risk
contract will be higher, since there will be a greater decrease in the expected value of loss

associated with the high risk type, i.e.,

48¢? > 48¢;-

63
After the equilibrium coverage to the high risk group adjusts to a lower level, the contract of the
lower risk type gets even more attractive if no adjustment in coverage to this contract takes place.

To restore separating equilibrium, the coverage to the lower risk group must decrease.

Proposition (4-4): With a decrease in the probability of disaster for the low risk types (Pl),
coverage provided to the low risk types must decrease if a new separating equilibrium is to exist.
The level of decrease in the coverage is moderated by the cost of delivery (g) - the higher the

cost the lower the decrease.
Proof

Rewrite equilibrium expression (4-5) as
‘ I ‘ A " 2 F1 2 2
mm — BL) + (I: — «I. )g = 3((¢. — I) — (¢2 — 1) )0.

Assume that equilibrium exists and at equilibrium there is a decrease in P1; this implies LHS >
RHS. Since 45:7 is fixed (there is no change in P2), the only variable that can change is ¢:.
Consider the case where there is no cost of delivery (g=0). Then any decrease in P, would lead a to

decrease in ¢I . However with positive cost of delivery there must still be a decrease in ¢I for the

equilibrium to hold, but the decrease in coverage must be less since any drop in ¢I would tend to

increase the LHS via the effect of g — the higher the value of g the less the coverage to type 1

falls.

64
Proposition (4-5): The coverage (and utility) to the low risk type is at a maximum in this model

of FCIC separating equilibrium if the coverage to the higher risk type equals the coverage at fit”
information (£7).
Proof'

Let the initial equilibrium be at the unconstrained Optimum for the higher risk type at a};
and the corresponding separating equilibrium for the lower risk type at ¢f , such that the higher
risk group is indifferent between the two contracts. Now if the coverage level for type 2 is varied

above or below ¢§ the utility of the firmer would decrease and for any such situation the contract

of the lower risk type is now more attractive and a pooling situation occurs. To regain separating
equilibrium, the coverage to the lower risk type must be lowered thus decreasing the utility of these
firmers. In other words the highest level of utility to the lowest risk type is attained when the

higher risk type is given the optimum coverage, which is the same as coverage under firll

information (til; = elf] ).

4.2.3 FCIC Separating Equilibrium With Cross-Subsidization Across Risk Pools

The equilibrium with no cross subsidies across risk pools may not be the most desirable
one to all firmers. Cross subsidies across risk pools would imply that the FCIC makes expected
positive profits on one risk pool and expected losses on other risk pool but overall it make zero
expected profits. Such a situation does not violate the mandates of the F CIC stated earlier as it

keeps the program at actuarially sound levels. This section develops a separating equilibrium with

65
cross subsidies across risk pools and states the conditions under which the separating equilibrium

with cross subsidies provides higher utility to both types of firmers who buy insurance.

4.2.3.1 The Model

In a separating equilibrium with cross subsidiesl7 the premiums need no longer be at
expected indemnity plus costs. If w,’ and qt: are the cross subsidy premiums and coverage offered
by the FCIC in a separating equilibrium to type i firms, the expected profits of the FCIC are
(4-8) N109? -Z - g)¢f +N2(w;' - Z. - g)¢§ = 0
where overall there are zero expected profits, but there may be expected profits and losses in each
risk pool.

As before, the sole provider position of the FCIC allows it to ofi‘er only two contracts such
that both premium and coverage are specified for each contract. It is assumed that the conditions

specified in Proposition (4-1) hold and full information premium and coverage cannot be ofi‘ered to

firmers when there are classification problems.

Results from the previous section illustrate that taxing high risk types would further reduce
the coverage to the low risk types to maintain a separating equilibrium. Thus, if there is to be any
cross subsidization it must be in the form of taxing the low risk premiums, subsidizing the high risk

premiums, and adjusting coverage such that a separating equilibrium is achieved.

 

'7 A separating equilibrium with cross subsidization is not feasible in a competitive market as described by
the Rothschild and Stiglitz model. The only reason it is feasible here is because the FCIC is assumed be
the sole provider of crop insurance in the model and as such does not face competitive forces.

66

Definition (4-2): AN FCIC separating equilibrium with cross subsidization in this model are the

premiums w,’ and coverage ¢:.' for risk types i (i=1 , 2) such that:
(4-9) wf’ = w,”7 + d
(4-10) w; = wf’ — Rd

+ __

”’2 -142
2
20',

 

(4-11) a5; =1—

(4-12) «Ix—w." +£>~§<¢r — Ito: = ¢;(—w; +£.)-12'-(¢; —1)2o-§

The first expression in the equilibrium conditions is the premium per acre for the low risk
types and it is arrived at by adding a tax (1 to the full information premium" for the low risk types.
The second expression is the premitun for the low risk types and it is arrived at by subsidizing the
full information premium by subtracting Rd from the full information premium per acre. R reflects
the ratio of type 1 to type 2 firmers. The expression Rd indicates that if R=1 then the tax on the
low risk firmer premium equals the subsidy on the high risk premium. If R=2, indicating that twice
as many low risk types to high risk types, then the subsidies go the higher risk type goes even
further. The third expression is the optimal coverage that the high risk type would receive atw,+

(the choice of this coverage follows from a similar proof to that of Proposition (4-5)). And finally,
the last expression determines the coverage to the low risk types required to generate a separating

equilibrium at the given premium levels.

 

'8 Recall that premium is linear in coverage and therefore tax also enters as linear in coverage.

67
The equilibrium above shows that the a high risk type always enjoys higher preference
level from the separating equilibrium with cross subsidies than the separating equilibrium without

cross subsidies. This is because the premium per acre with cross subsidy to the type 2 firmer is

lower that the with no cross subsidy (w; < w? ) and coverage is higher (¢; > a}? ). On the other
hand, the type 2 firmer fices higher premiums (w,+ > w,” ) but also higher coverage (45; > g6; ).
Whether the preferences of type 1 finners is higher or lower is yet to be seen.

With tax d on the premium per acre, the coverage ¢f at which the low risk farmer is
indifferent to a no tax Situation is U L (le , all.) = U L (wa7 + d, M); ¢1d is the critical coverage
that a separating equilibrium must ensure for the cross subsidies equilibrium to be more desirable

w:,¢:)2UL(.w:,¢t).

 

 

than the equilibrium with no cross subsidies, i.e., U L (.

Proposition (4-6): A separating equilibrium with cross subsidies ( ¢1+w1’ , ¢,’ ) is superior to an
equilibrium without cross subsidies if the tax d on the lower risk type satisfies:

a,<t.'(¢.*-¢I)+ II

A. 2¢r of((¢: -1>2 —(¢r -1>2)

 

Proof

Consider the utility of the low risk type in a separating equilibrium with no cross subsidies
— ‘ Fl — ’1 " 2 2
(4'13) M_Ifi+¢i(-wi +Ifi)—§(¢l_l) 0', :A

where A is the utility level of the firmer. Now if coverage is allowed to increase to ¢f , the utility of

the firmer would increase and the farmer would desire this new contract. This increase in utility of

the firmer would be less if the premium is allowed to increase simultaneously since increases in

68

premium decrease utility. Let the increase in premium per acre be d (the tax level) such that it is at

w,+ (= w,“Y + d). The utility of the farmer is reduced but it is still higher than the level A. Then
subtracting expression (4-13) from the utility with coverage ¢f and premium ¢l+wf will yield a

positive value or
III—w“ + Z)— int -1)202_[¢‘(_wn + Z)— 1w — 1w) > o
l 1 2 l l l l 2 l l
Rewriting this expression to isolate w gives
+ + "‘ F] + ‘ _ A" 2 ‘ 2 + 2
1w. -¢.w. <<¢, —¢.>I.+;o.(<¢. -I> —<¢, —1>)

The RHS of this expression is positive. Since w,+ = w,” + d (where w,” = I: + g). Rewriting the

expression above and isolating d gives

d< g(¢i -¢I)+ 21
¢I 2¢I

 

of((¢: —1)2 — (¢I' —1)2).

The condition in Proposition (4-6),is more likely to hold if the tax d on the low risk type
goes further in subsidizing the high risk types. From the equilibrium condition it is evident that this
happens as the value of R (the ratio of low to high risk farmers) increases. Therefore, if the market
has higher ratios of low to high risk firmers, then a cross subsidizing separating equilibrium would
increase utility (and coverage) to all firmers than in a separating equilibrium with no cross

subsidies.

To highlight discussions of this section further consider a simple numerical example

below. The same parameters as in Chapter 3 are used i.e., P1=0.3; P2=0.4; g=3 bushels; M=14O

69

bushels; L=70 bushels; 1:0.01; R=ratio of low to high risk types. The reader is reminded that the

parameters chosen do not reflect any real world application and are only for expository purposes.

The results of the numerical analysis is reported in the Table 4-1 through Table 4-1 below.

Table 4-1 FCIC Equilibrium Coverage Levels With Taxes and Subsidies: R=1

 

 

 

 

 

 

 

I P=0.30 I P=0.40
Tax (d) I w: I: III E w; I; u2
0 :r 26.0 0.151 114.530 i 33.0 0.575 108.060
0.1 : 26.1 0.157 114.540 : 32.9 0.583 108.120
0.7 i 26.7 0.199 114.565 : 32.3 0.634 108.486
0.9 1 26.9 0.215 114.5614I 32.1 0.651 108.615
Table 4-2 FCIC Equilibrium Coverage Levels With Taxes and Subsidies: R=2

E P=0.30 j P=0.40
Tax (.1) i w: I: uI:r w; A u2
0.1 i 26.1 0.162 114.561 i 32.8 0.592 108.180
0.7 : 26.7 0.243 114.668 : 31.6 0.694 108.951
0.9 : 26.9 0.277 114.677 : 31.2 0.728 109.235
1.0 1 27.0 0.296 114.674 I 31.0 0.745 109.382

 

70

Table 4-3 FCIC Equilibrium Coverage Levels With Taxes and Subsidies: R=0.5

 

 

 

E P=0.30 j P=0.40
Tax (d) E w: I: ul E w; 4; u2
0.04 E 26.04 0.152 114.536 E 32.93 0.577 103.074
0.1 g 26.10 0.155 114.535 I 32.95 0.579 108.092

 

The first column of each table indicates the tax imposed on the lower risk group and the
remaining columns indicate premium (w), coverage (4)) and utility (It) for risk type 1 and 2
respectively. Table 4-1 shows the results where there are equal numbers of low to high risk
firmers; Table 4-2 shows the results when the number of low risk types are twice as many as the
high risk types; and Table 4-3 indicates the case when there are twice as many high risk types to

low risk types. The results are reported for selective tax levels.

Results in Table 4-1 show that subsidizing the high risk type (type 2) leads to decreased
premium and increased coverage (and higher utility to risk type 2). Since in this equilibrium the
subsidies to the higher risk type is financed by taxing low risk firms, the premiums for these firms
increase. However, increases in premium to the lower risk type (by taxation) does not lead to
lowering of the preference value of the lower risk type in a separating equilibritun. As the higher
risk class is made increasingly better Ofi‘ with subsidies the separating equilibrium allows an
increase in the coverage ofi’ered to the lower risk class. In this example both the equilibrium
premium and coverage to the lower risk type increases in the separating equilibrium. The
preferences of the lower risk group (indicated by III in Table 4-1) increases as the tax increases
and reaches a maximum at an approximate tax of 0.7 bushels, after which it starts to decline. This

implies that given a choice of insurance contracts, the low risk group would choose a contract at

7 1
which the premium is higher than in an FCIC separating equilibrium contract (premium at 26.7
bushels) with a higher coverage (0.199 percent) as opposed to a premium of 26 bushels and
coverage of 0.151 percent, without any taxes. Taxing of the low risk types also makes it possible

to increase the coverage (and utility) of the high risk types from 0.575 percent 0.651 percent.

However, not all situations permit taxing and subsidizing to increase the preferences of all
farmers. The results depend on the proportion of low to high risk farmers in the market. Table 4-2
and Table 4-3 report the results for the same example but with the proportion of high to low risk
firmers changed. Results of Table 4-2 indicate that, when there are twice as many low to high risk
‘ farmers, the same level of tax to the low risk type increases the preferences of both risk groups
compared to the case when there was an equal number of high and low risk groups. For example, a
tax of 0.1 bushels now has coverage of0. 162 bushels (u1=1 14.561) compared to 0.157 bushels
(u1=114.540) in Table 4-1. While earlier the preferences of the low risk firmers were maximized
at a tax of 0.7 bushels, it is now maximized at 0.9 bushels, indicating that as the ratio of low to
high risk types increase, higher levels of tax are preferred. Furthermore, the high risk types are
increasingly better off (compare u2 of Table 4-2 to that of Table 4-1 for each level of tax) with the
tax and subsidy plan as the proportion of low to high risk firms increased. This result is driven by

the fict that the taxes go a lot further as there are more type 1 firmers than type 2 farmers.

A Situation where tax and subsidy schemes are less effective is indicated in Table 4-3
where there are twice as many high risk types as low risk types. These results show that at taxes
any higher than 0.04 bushels the utility of the low risk types decreases (at 3 decimal places the
utility does not change from the case when there are no taxes). With this amount of tax the

preference function of the high risk types is increased only slightly.

72

4.2.4 The FCIC Pooling Equilibrium

The third and final possibility considered in this chapter is a pooling equilibrium. In this
case, only one contract is offered by the FCIC and every farmer buys the same coverage and pays
the same premium irrespective of their risk classes. In this section it is also assumed that

proposition (4-1) holds and that the FCIC cannot classify finners into risk classes.

4.2.4.1 The Model

A pooling equilibrium occurs where the premium per acre (WP ) is an average premium,

i.e.,

 

A A
= lel -t-N2w2

(4-14) w"
Nl +N2

where wf is the actuarially fiir premium for risk class i and g is the cost Of delivery per acre. As

before, the premium is assumed to be linear in coverage. The FCIC is able to calculate this
premium because all the information in the expression is known. The pooling premium depends on
the number of high to low risk types. Ifthere are no high risk types (N 2:0) than the premium

reduces to the premium for the low risk types with firll information and vice versa.

A A
lel + Nzw2

N,+N2

 

Proposition (4- 7) : If the premium to all risk types is w” = + g there can be only

one coverage level in equilibrium.

Proof:

73
Let the coverage level to low and high risk types be (I); and 4»; respectively. The expected
profits of the FCIC is
E05): Ni¢i(W’ - I; - g)+ N2¢2(W’l- 2:2 - g)
The first term in the RHS is the expected profits on the low risk pool and the second term is the

expected profits on the high risk pool. Substituting the value w" from expression (4-14) and

 

simplifying gives
._ Ner _ _ _ _
E(”,)- N] +N2 (L2 L1)(¢l ¢2)

It is clear from the expression above that for the FCIC to break even, E(7l'g) = 0 , the coverage

offered to both firmer types must be the same (4); = (1)1).

From Proposition (4-7) it appears that there are no limits to the coverage for equilibritun
as long as the coverage to all firmers is the same. However, from the demand side there are

boundaries to the coverage that the FCIC can sell.

Proposition (4-8): In the pooling equilibrium of this model, the pooling coverage ¢” must be

such that ¢p _<_ 2(1- M] .

’10"2
Proof:

For the given premium w” , the farmer would not buy F CIC insurance if the utility of
buying insurance is lower than the utility from no insurance, i.e. U (.lw” , ¢p ) > U (.|0,0). For the

specific utility function in this study the condition to buy insurance is

74

M—Z+¢(—w+f)—g—(¢—l)zai ZM—Z—gai.

Simplifying and rearranging the expression gives

“2011—2),

2
AOL

The expression in the brackets in the RHS is the expression for optimum coverage that the firmer

would choose given the premiumw’ . Therefore, in this linear mean-variance framework, coverage
above twice the optimum level would not be bought by the firmer. Since the FCIC program is to
serve as the primary disaster provider, it must ensure that both risk types buy insurance or the

pooling coverage must satisfy

_ p _(wp_l—1)]
(415) ¢ S2(1 ———-’10_‘2 .

For the rest of the analysis, it is assumed that FCIC sets coverage such that both risk types
buy insurance. Since coverage could essentially be set anywhere within the stated bounds, selection

of the coverage level becomes difi'rcult without further information. To see how different coverage

levels effect the utility of the firmers let ¢f and ¢§ be the optimum coverage for risk types 1 and
2 with premiums at WW,” and MW; respectively. Also note that ¢§ > 051’ Let W be the pooling
equilibrium coverage chosen by the F CIC. If V = { then the utility of the low risk type is

maximized while that of the high risk type is minimized under the pooling premium. Note that the

full information utility cannot be achieved for the low risk types because w” > w,” and all” > ¢f .

Ifcoverage is set such that W = if; , then the coverage of the high risk type is at a maximum (and

75

that of the low risk type is minimized). Here the high risk type is better OE than in the case of full

information because both coverage ¢f7 < ¢§ and premium w” < wf’ are better.

To compare the pooling equilibrium with the separating equilibrium, consider setting the
pooling coverage at level a5", such that the utility of the high risk type is maintained at a
separating equilibrium level with some tax d. To ficilitate the comparison, consider a separating

equilibrium with tax 61’ such that w,+ : w” = w; . The level at which d ’ must be set is

(4-16) d’=fi;(wf’—wf” -

At this level of tax a separating equilibrium breaks down and a pooling equilibrium occurs because
if taxes are at Equation (4-16), then Proposition (4-7) shows that there can only be one coverage
for both risk types. This is because the premium is the same and there can no longer be a
separating effect through coverage, and the FCIC still maintain zero profits. From Proposition (4-

6) we know that the separating equilibrium provides higher utility to the low risk type only if

 

MM; "’+ 2:. 012((¢I-1)2 —(¢: -1)2).

So in a pooling equilibrium this condition is more likely to hold if R is as large as possible so that
d p is as small as possible, or a pooling equilibrium is approached by a separating equilibrium
when the ratio of R is very high.

We have seen in this section that a separating equilibrium with cross subsidies can lead to
a pooling equilibrium with the right amount of tax on the low risk types. Calculations in the

numerical examples (presented in the previous section) support a separating equilibrium with cross

subsidies rather than a pooling equilibrium where the utility of the low risk types are maximized.

76
This is because, in these examples the ratio of low to high firmers is not large enough to justify a
pooling equilibrium. Since a pooling equilibrium can always be approached fi'om a separating
equilibrium with cross subsidies, for the remainder of this study, only separating equilibria are

considered.

4.2.5 Implications for Current FCIC Policies

So fir the study has shown what an FCIC equilibrium might look like with lack of
information to classifying finners into risk types. This section briefly connects the results of this

model with the current F CIC program policies.

The current FCIC program Operates as if it has firll information on firmer risk types since
measures to addresses classification problems are not present. All the firmers have a choice of 50,
65 or 75 percent coverage level with rates based on their past 10 years average yield. Skees and
Reed have indicated that using only average yield is not adequate for proper firmer classification.
The federal crop insurance figures available for the 1980-90 decade indicate that close to a billion
dollars have been transferred fiom the FCIC to the firmers — this has been over and above any
predetermined subsidies. Therefore, among other problems such as moral hazard, timing of sales,

etc., the classification issue appears to be a problem to the crop insurance program.

Results from this study indicate that if the FCIC is to operate on an actuarially sound
basis, attempts must be made to price policies such that high risk farmers self select high risk
policies (if information on classification cannot be obtained inexpensively). One way to do this is
subsidize high coverage levels and tax low coverage levels. Results Show that a pooling equilibrium

may severely penalize the low risk types unless the proportion of low to high risk finners is large.

77
The results of the separating equilibrium gives an idea of how the contracts would look. The

implications of some of the current government practice of providing subsidies is summarized

below.

Proposition (4-9): The current F CIC practice of providing subsidies to lower coverages and not
to the higher coverage will either (i) lower the coverage to the lower risk type in a separating

equilibrium or (ii) further increase budgetary outlays through adverse selection
Proof'

Starting in a separating equilibrium, provision of subsidies to the lower risk group makes
these policies more attractive to the higher risk group then they otherwise would be. To restore a

separating equilibrium expression (4-12) indicates that there must be a decrease in the coverage of

the low risk group or the government will fice losses due to adverse selection.

Proposition (4-10): If the maximum coverage to the high risk types is limited to below the fitll
information optimum, then either (i) the coverage to the lower risk type is reduced in a
separating equilibrium or (ii) there are fitrther increases in budgetary outlays through adverse

selection. .
Proof'

Follows fi'om Proposition (4-5).

Proposition (4-11): Subsidization of administrative costs (g) is beneficial to all risk groups in an

FCIC separating equilibrium and in an FCIC pooling equilibrium.

78

Proof:

From Proposition (4-3) it follows that lowering administrative costs would increase the utility of
both the high and low risk types while still maintaining a separating equilibrium. For a pooling

equilibrium, given the same coverage, lower premiums would be ficed by both risk types.

For a separating equilibrium with or without cross subsidies, policies that subsidize all risk
types will increase the utility of both risk types (the current program pays all administrative cosis).
Even policies that only subsidize the higher risk types are desirable in a separating equilibrium but
policies that subsidize only the lower risk types have the effect of reducing coverage (and utility) to

these risk types in equilibrium.

4.3 The Competitive Deliver_'y of Crop Insurance with Two Risk Types

Recall the working hypothesis that competitive firms are risk averse with very low cost of
gathering information such that they are assumed to operate with full information. This, of course
is a very strong assumption, but it is used here to highlight the difierences between the FCIC and
private competitive firms. Also recall that these firms are also assumed to face insurance delivery
costs lower than the FCIC. This section of the chapter investigates how the competitive equilibrium
will behave when there are two risk types in the market, and under what conditions a firm would

hold only one risk type in its portfolio.

79

4.3.1 The Competitive Equilibrium

All the characteristics of the competitive firm specified in Chapter 3 hold. This implies
that, in the long-run, the competitive firm is operating at reservation utility b such that no entry
’ occurs. Competition determines the number of policies held by each firm. As before, assume the

specific preference function for the firm

V(.) = ES — 22:02

‘3'

Definition (4-3): A competitive equilibrium in this model of crop insurance delivery with two risk
types is a premium level w? , coverage level 915? and number of policies n? for each firm such

that:

(4-17) —w,° + Z, + 2(1- ¢f )0,2 = 0

(4-18) —w;’ +Z, +2(1— ¢§)o§ = 0

(4-19) n,°(wl°I—., — c—)- I;/[¢1nl °[1+(n, 1pl )]cr1 +nl °n2¢2012]= 0
(420) n30»? — z, — c)— PM"? [1 + 023— —1>p.ia§ + nfné’ $01.] = o

nf’(w.° - Z — c)+n§(w§ 4? — c)
(421) —-'4’-[(¢?>2n.° [1 +02? — 1);).103 +(¢‘: )2 n: [1 +023 — 0122103]

__nln2¢l¢2612b:

(4-22) n,° = Rn:

80
The first two equations are the first-order conditions (FOC) for maximizing the preference
functions of the two risk type farmers as they choose coverage given the insurance premiums. The
next two equations are the F CC for maximizing the preference of the competitive by choosing the
coverage it would provide to risk type 1 and 2 respectively. The fifth equation states the long-run
condition whereby the preferences of the competitive firm is at reservation level b and the final
equation gives the proportion of low risk to high risk firmers. The equilibrium values are attained

by solving the non-linear system of equations for the six unknowns.

Although it appears that in equilibrium firms hold both types of policies in their portfolio,
this is not necessarily the case. A competitive firm will only hold both risk types in its portfolio if it
can ofi‘er better coverage to a firmer than a firm which is holding only that firmer risk type in its
portfolio. If this were not the case, then some firms would hold only one risk type in its portfolio
and these firms would be able to take business away from the firms that hold both risk types. The

condition under which the company would hold both risk groups in its portfolio is outlined below.

Consider the coverage to farmer type 1 (type 1 is chosen arbitrarily) and let this be (bu,
when the firm is holding both farmer types in its portfolio and 4)], when it is holding only that

firmer type in its portfolio. The respective coverages at a competitive equilibrium are

20,2 —c
012(1' + WNla)

: 1012 —c" WRnlb¢2012
01201 + WNlb)

 

 

¢lb ¢la =

where N,=[l+(ni-l)p] and 012 = pIZO’lO'z.

For the firm to hold both risk groups in its portfolio, it must be the case that in, _>. 4),

otherwise the firmers would not buy insurance from a firm that has both firmer types in its

81
portfolio. After some algebraic manipulation it is seen that the company will hold both types in its
portfolio if

(4012 — CX’A. — nib).
Rnlb¢2

 

012 -<- PI
Although this condition is alone suficient, a similaf argument may be developed for the high risk
I
firmer types, i.e.,

(40227 " C)("2.. — an)
Rn2b¢l

 

012 5 P2

again this alone is sufficient. Note: n,.-n,b (i=1,2) is positive19 since negative covariance between

any two firmers is not permitted in this model.

Proposition (4-12): A competitive firm would hold only one risk type of farmers in its portfolio if

(’10-; T C )(nZa - an)
Rn20¢1

(1012 - c)(nla - nlb)
Rnlb¢2

 

 

either an 5 p1 or 012 S p2 is satisfied.

It is then clear that if risks between two finners in any risk class are independent but not

independent across risk classes ( p, = 0, p,2 :13 0 ), the firms in this model will hold only one risk

 

'9 Alternatively it may be proved as follows: consider the case when the company is holding only one risk
type in its portfolio. Now with the addition of the second risk type the number of policies it holds of the
first risk type will not change if

n2¢2(W2 - Z2 - 6)“ glfinz [1 '1’ (n2 _ llhipz + n1n2¢1¢2012]= O

This is not feasible if the firm’s FOC is to hold (this expression is positive). Since the reservation utility
remains at b an addition of a positive value is tantamount to reducing the reservation utility to a lower
level than b. Results from Chapter 3 Show that with reduction of reservation utility, the number of policies
must decrease.

82
type in its portfolio. Onthe otherth if p, > O and p12 2 0, they will hold both risk types in
their portfolio. More generally, the intuitive reasoning of the proposition is that firms would hold

both risk types whenever there are opportunities for diversification.

4.4 Comparison of the Competitive Equilibrium and FCIC Deliveg

Chapter 3 highlighted the difi‘erences between an FCIC equilibrium and a competitive
equilibrium under full information for both agents. The results on coverage to firmers were driven
by the working hypothesis that the F CIC has a high cost of delivery but behaves as risk neutral
while competitive firms have lower cost of delivery but are risk averse and operate at reservation
utility. While the cost of delivery reduces the FCIC coverage, high levels of positive systematic risk
reduce the competitive coverage. The comparison between the two equilibria in this chapter
complement the results of chapter 3 by adding classification problems to the F CIC equilibrium,

and adding higher diversification possibilities for the competitive firm.

For reasons mentioned earlier, only the separating FCIC equilibrium is considered and for
exposition purposes the FCIC separating equilibrium with no cross subsidies is discussed first
followed by the separating equilibrium with cross subsidies. The competitive firm may have one or

more risk types in its portfolio, depending on the correlation of loss between firms.

83

4.4.1 The Case of a Separating Equilibrium With No Cross Subsidies

High Risk Farmers

The comparison of results between the FCIC and the competitive equilibrium for the high
risk firmers does not change from those presented in Chapter 3 if Proposition (4-12) holds and
only one risk type is held by a competitive firm. The conditions assumed are such that the
competitive firms supplying insurance to the high risk types hold only high risk policies in their
portfolios while in an FCIC separating equilibrium the coverage offered to the firmers is the same
as in the FCIC full information case. It may be concluded that classification problems do not
change the position of the high risk farmer away fi'om a full information FCIC equilibrium. If
Proposition (4-12) does not hold then the competitive firm is able to provide higher coverage to
high risk types because of diversification possibilities. This implies that if we start at a situation
where the high risk type is indifi'erent between the FCIC and competitive coverage at full

information, the firmer now prefers to buy insurance from a competitive firm.

Low Risk Farmers

The coverage under a competitive equilibrium is as described in Chapter 3 if Proposition
(4-12) holds. If it does not hold, then diversification possibilities ensure that higher coverage is
attained now under the competitive equilibrium. If, in the fill information F CIC equilibrium, the
firmer preferred purchasing insurance from a competitive firm, the situation now is more in fivor
for doing so from a competitive firm because F CIC coverage is now lower because of classification
problems and competitive coverage is higher because of diversification. A Situation which makes

the FCIC separating equilibrium coverage decrease compared to an FCIC full information

84
equilibrium is if there are large difl‘erence in the probabilities of disaster of the two risk types.
Proposition (44) shows that as the difference in the probability of disaster increases, the coverage

to the lower risk group is reduced.

4.4.2 The Case of a Separating Equilibrium With Cross Subsidies

High Risk Farmers

With positive cross subsidies from the low risk types, the higher risk farmers are made
better Off in this separating equilibrium than under the FCIC equilibrium with full information.
Similarly if there are diversification possibilities under the competitive equilibrium. An additional
fictor that affects the coverage level in the FCIC equilibrium is the proportion of low to high risk
farmers — as the proportion increases the subsidies increase and the high risk types are
increasingly better off. Whether this coverage is higher than the competitive equilibrium still
remains a function of cost of delivery of the FCIC and the degree of risk aversion of a competitive

firm and the degree of correlation between firm risks.

Low Risk Farmers

The low risk finners’ utility still remain below the full information level although the
utility is higher than in the case of no cross subsidies. How much higher is a function of the ratio of
low to high risk firmers. As the proportion of low to high risk increases, then firmers have higher
coverage (and utility) under the separating equilibrium. Thus with classification problems, a
separating equilibrium is able to achieve more for both risk types as the proportion of low to high

risk firmers increases.

85

4.5 Conclusions

The FCIC equilibrium was characterized by informational problems in differentiating
between firmer risk classes. Conditions were developed and stated when this lack of information
on firmer risk classification was a problem in the FCIC equilibrium. Three specific FCIC
equilibria with classification problems were considered (i) a separating equilibrium with no cross
subsidies between risk pools; (ii) a separating equilibrium with cross subsidies across risk pools;
and (iii) a pooling equilibrium. Conditions were developed under which each of these equilibria
were preferable to firmers. Of the three equilibria, it is seen that (i) and (iii) are special cases of
equilibrium (ii). To low risk firmers, a separating equilibrium with no cross subsidies provides the
lowest coverage when there is a large proportion of low to high risk types and the pooling
equilibrium provides the lowest coverage when there is small proportion of low to high risk

firmers.

When lack of information on firmer classification is a problem, contracts require that high
risk firmers self select high risk contracts. The basic nature of the separating equilibrium implies
that low risk firmers are penalized in the presence of high risk firmers. If cross subsidization
between risk classes is allowed, then the penalty on the low risk type is reduced as the ratio of low

to high risk firmers increases.

FCIC separating equilibrium has implications for government practices. If there are to be
any subsidies fi'om taxpayers, then subsidization of all risk types is desirable to better meet the
mandates of the FCIC. The practice of subsidizing lower coverage levels has adverse effects on the
coverage of the lower risk types and, in the long-run, such subsidies penalize these risk types

instead of helping in a separating equilibrium.

86

When there is more than one risk type, there are also implications for the coverage offered
by competitive firms. Conditions are stated on correlation between risks under which a competitive
firm would sell both types of policies and diversify to provide higher coverage to all firmers. The
presence of two risk classes of finners impacts the comparisons made in Chapter 3 between a full
information FCIC equilibrium and a competitive equilibrium. In all cases, in an FCIC equilibrium,
the high risk types stand to gain and the low risk types lose to a varying degree. Which equilibrium
provides higher coverage to the low risk farmers is a firnction of the difference in the probability of
disaster between low and high risk firmers, the ratio of low to high firmers, the cost of delivery of

the FCIC and the competitive firm, the degree of correlation between firm losses, and the degree of

risk aversion of competitive firms.

5. FCIC Reinsurance Under the 1980 Crop Insurance Act

One of the mandates of the 1980 Crop Insurance Act was to include the private sector in
the federal crop insurance program to make it “more efficient”. The purpose of this chapter is to
introduce the private sector into the FCIC model through a reinsurance contract in which
competitive firms administer contracts and bear some risks but the FCIC sets insurance contract
characteristics and provides reinsurance to these firms. The expected performance of such a
program is evaluated in terms of its ability to meet two main objectives of the 1980 Crop Insurance
Act: (i) the program should work on an actuarially sound basis and (ii) the program should provide
maximum coverage feasible to all finners, such that the program serves as the primary provider of

disaster assistance.

According to the working hypothesis of the study and the results from Chapters 3 and 4,
the F CIC could increase coverage to all firmers and still function on an actuarially sound basis, if
it had more information on classifying farmers into risk types and lower costs of delivering
insurance. However, unlike the FCIC, private insurance firms are risk averse profit making agents
that would presumably charge a premium to compensate for bearing risk. This premium is likely to
be high in a market where the losses are highly correlated, as in the agricultural sector. This
chapter develops a model to incorporate competitive private firms into the FCIC model provided
earlier in the study. The competitive firms participate in the program as long as the utility from

participating is higher than a reservation utility.

This chapter’s proceeds as follows. First the current FCIC reinsurance program is outlined
in detail. Then the program is studied when there is only one firmer risk type. Most of the major

results are derived in this section. Next the reinsurance program is examined under multiple firmer

87

88
risk types and classification problems by the FCIC. The chapter concludes by looking at the

implications of the model for the current objectives of the FCIC program.

5.1 Th F de l ro Insurance Pro ram

Under the current program the FCIC determines the premiums and coverage for all policies
sold to firmers including loss adjustment standards and procedures. The F CIC is mandated by the
1980 Crop Insurance Act to set premiums at actuarially sound levels and ofi‘er coverage at 50, 65
and 75 percent of 10 year average yields of the farm. To encourage widespread adoption of FCIC
crop insurance, the FCIC decided to subsidize all delivery costs of the program. The private firm
delivering F CIC crop insurance is obligated to sell insurance to any firmer that seeks insurance if
federal crop insurance is offered in that county for the particular crop. For expenses incurred in
selling and servicing the FCIC policies, the FCIC pays between 32-33 percent of the premirun of
every policy sold to the participating private firm. This percentage is decided annually by the

Standard Reinsurance Act20 between the FCIC and the private firms.

5.1.1 The FCIC Reinsurance Program

Reinsurance is insurance for firms selling insurance. In the private sector reinsurance is
generally sold by companies that specialize in this business, such as Lloyds of London. In most

cases the decisions of pricing, coverage, etc. are made by the primary insuring firm and if

 

2° This is the reinsurance agreement between the FCIC and the private firms and it is revised annually.
The contracts reported here are taken from 1991 and 1993 Standard Reinsurance Agreements.

89
reinsurance is sought, the reinsurer and the insurance company decide on the premium paid for the

portion of policies to be reinsured.

The FCIC chose to use reinsurance of private insurance firms as the primary means to
deliver federal crop insurance to firmers. Unlike traditional private reinsurance agreements, the
FCIC, who is also the reinsurer, sets the premiums and coverage on the policies and the private
firms are passive deliverers. Once the policies are delivered, the private firms determine how much
they will retain under the FCIC reinsurance contracts available to them. Reinsurance contracts
between the FCIC and the private companies take two forms: (i) proportional reinsurance; and (ii)

stop-loss (or non—proportional) reinsurance.

5. 1. 1. 1 Proportional Reinsurance

1n proportional reinsurance, FCIC and the private insurance companies share specified
portions of both the policy premiums collected and the indemnities paid. Under the Standard
Reinsurance Agreement between the FCIC and private firms, the firms categorize their business
into three risk pools: (a) Assigned Risk Fund, (b) Developmental Fund, and (c) Commercial Fund.
The Assigned risk pool contains the high risk policies where the FCIC assumes responsibility for
80 percent of the of the liabilities in exchange for 80 percent of the premium. The amount of
business that a company may put in this pool is limited by states and is determined by the F CE”.
The firm may choose to place some or all of its crop insurance business in it the Developmental
risk pool but the firm must retain a minimum 35 percent of premium and associated liabilities. The

company may place any business not included in the above two pools in the Commercial pool.

 

2' It ranges from 20 percent in some states to 75 percent in others.

90
Here the private company must retain at least 50 percent of the premium and the associated

liabilities.

5. 1. 1.2 Stop-Loss Reinsurance

That part of business that the reinsured companies do not cede to FCIC through
proportional reinsurance is then eligible for stop-loss reinsurance. For example if 60 percent of the
business was reinsured through proportional reinsurance the retained business (40 percent) is then
eligible for stop-loss reinsurance. In stop-loss reinsurance the F CIC agrees to reimburse the firms
for losses over a certain predetermined loss ratio of the crop insurance portfolio (loss ratio
expresses the amount of indemnities as a percentage of premiums which include government
subsidies). In other words there is a cap on the losses a firm can incur by participating in the
program. To pay for this stop-loss protection, the private firms enters into a gain sharing rule with
the F CIC such that if positive profits are made by the firm, the FCIC takes a percentage of these
profits. The gain sharing rule is exogenously given to the firms through the Standard Reinsurance
Agreement (revised annually). In the period 1986 through 1990, FCIC bore 100 percent of all

losses above a 156.5 loss ratio.

5.2 FCIC use of the Information and Deliver_'y Cost Position of Private Firms

There are two characteristics of private firms that are desirable to incorporate into the
model: (i) information to classify finners into risk pools; and (ii) lower delivery cost structure. The
question therefore is whether the current F CIC program described above is able to do this and what

are the implications for the program?

91

In the current program the FCIC sets all premiums and coverages to firmers. Risk classes
are determined according to the average production history (APH) of the last 10 years of yield
data. Ifdata are not available, then county level data are substituted to calculate the premium for
the particular firmer and no additional information is used to classify the firmer”. The information
available to private firms is not used at this stage of the program. Where the information position
of the private firms is used is at the time when the firms make reinsurance decisions since they are
able to assign policies to different risk pools for proportional reinsurance purposes. However, this
information is not used by the F CIC in subsequent years to classify firmers. Thus, it is concluded
that the current program does not use the information available to private firms to classify firmers
into risk pools. There is no advantage gained to improve the information position of the FCIC by

involving private firms under the current F CIC reinsurance program.

Under the reinsurance agreement, FCIC reimburses delivery (or administrative) costs to
the private firms for each policy sold irrespective of the amount of business retained by the private
firms. Whether the current F CIC program is more efficient in paying for delivery costs in such a

manner as opposed to using its own delivery cost remains to be studied.

Since the information position of the private firms are not used to classify firmers, much
of the analysis in this chapter is done with only one risk type and the focus is more on the delivery
cost aspect of the F CIC program and the implication of using risk averse competitive firms. The

efi‘ects of information problems are discussed in later sections.

 

’1 The study by Skees and Reed has shown that although the use of APH is better than no information at
all there are still adverse selection problems.

92

5.3 FCIC Reinsurance of Private Firms With One Risk Class of Farmers

This section develops a model with firmers of one risk type to study the private firm’s
responses to the FCIC reinsurance program. The model has three agents, the farmer, the FCIC and
the competitive firm; each of which are discussed in turn. Since the decision making of the firmer
and FCIC are in line with the analyses in Chapter 3, only brief comments are made and the main

results are stated. Much of the discussion focuses on the competitive firm’s decision to reinsure in

this program.

5.3.1 FCIC Supply of Insurance to Farmers

The model for the agricultural risk market is identical in most respects to the one described
in chapter 3. To recapitulate, there are a total of N finners and each firmer owns one acre of land
that yields M bushels of output. Loss L occurs with probability P and zero otherwise. The mean

and the variance of this loss are identical for all firmers and given by

Mean=L
Varzai.

The correlation of losses between two firms is p and it is the same for any two firms in the area.

As in chapter 3, the utility function
_ 2
U(.)Zfl'F ——2—O':'

is assumed for all finners and utility it is increasing in the mean and decreasing in the variance of

profits. The mean and the variance of profits of a firmer with coverage (I) and prenrium 4>w are

93

ER = M ‘ w ¢-(l-¢)Z
and

0i, = (1 -¢)201-

Since there is only one risk type, there are no classification problems and premiums are set
at actuarially fiir levels (w=w") as all delivery costs are subsidized by the FCIC. The competitive
firms deliver these policies to the firmers at coverage levels set by the FCIC. Results from Chapter
3 indicate that at actuarially fiir premiums, a risk averse firmer would seek full coverage (0:1).
However, the F CIC limits the maximum coverage to levels below one (¢=0.75 is the maximum

coverage under the current F CIC program) and therefore the risk averse firmer would prefer this

coverage to any coverage below this at actuarially fiir premiums.

5.3.2 The Reinsurance Decisions of Private Firms

Now that we have established the premium and coverage to the firmer at the FCIC
equilibrium, we turn to the competitive firm reinsurance decisions once the policies are sold. Since
everything else is determined by the F CIC the only aspect that the competitive firm has control
over is the amount of reinsurance it would seek from the F CIC. For the sake of clarity the
reinsurance decisions of the firm are first developed with only FCIC proportional reinsurance, and

then stop-loss reinsurance is introduced.

5.3 ,2.1 Proportional Reinsurance

With only one risk type of firmers, only one risk pool is considered for proportional
reinsurance purposes. If the firm participates in the FCIC program, then it is required that it

maintain a minimum proportion of each of the policies sold. The firm keeps the corresponding

94
premium and is responsible for indemnifying that portion of the policies. Let t=t° be this minimum
percent retention required of the competitive firm on each policy sold. The upper limit is t=l where
the full policy is retained by the firm. The profits of a competitive firm fi'om selling FCIC
insurance is

(5-1) 71‘, = t¢(nw” —Zn:l,)+6rw"¢ r

i=1

|/\
N
IA
..—0

The first term on the right hand side (RHS) is the net profits from selling crop insurance to
n firmers. Since the firm must sell policies to any firmer that seeks insurance, 11 is beyond the
control of the firm. At this point assume n to be exogenously given. 4w“ is the actuarially fair
premium collected from the firmer and 4)], is the indemnity paid to firmer i from the FCIC
insurance. The indemnity payment to firmer i takes the values 4)L or 0 with probability P and l-P
respectively. It has already been established that 4) will be the maximum coverage allowed by the

FCIC.

The second term on the RHS is an expression to capture any transfer that might take place
from the FCIC to a competitive firm over and above any delivery cost. Since premiums are
assumed to be at actuarially fiir levels, the only way this transfer can take place in the model is
through administrative cost remuneration. Let 0 be the percent of premiums transferred to
competitive firms for each policy sold. For example, the current reinsurance agreement pays
approximately 33 percent of premiums as administrative costs. If the actual cost for delivering
crop insurance is only 30 percent then 3 percent of premiums is a direct transfer from the FCIC to
the competitive firm. If the administrative costs equals the actual cost incurred then 0:0 and there

are no positive transfers from the FCIC to the competitive firm. Negative transfers are not

95
considered. Since the firm receives this transfer irrespective of the percent of policies it retains

through proportional reinsurance, it is not a function of reinsurance (l-t) bought by the firm.
The mean and the variance of the portfolio of the competitive firm are
7r, = atwӢ
and
of, = t2¢’II[1+(n — 1)piai
Note that the expected value is not a function of the reinsurance bought by the firm since policies
are actuarially fiir.

As in Chapter 3, assume that the utility of the firm is given by
V(.) = a. — 3250:,

which is increasing in the mean and decreasing in the variance. The problem of the competitive
firm is to maximize its utility by choosing the proportion (t) of the policies it would retain under the

current F CIC program. The first-order condition (F OC) to this maximization problem is23

(5-2) —wt¢2n[1+(n— 1)p]ai = 0.

This expression is solved only at t=0 (all the other parameters are positive by assumption) and it is

independent of positive transfers from the FCIC.

 

2’ The second-order condition —I//¢2n[1+(n - 1)p]0'i < O for maximmn is satisfied.

96
Proposition (5-1): In this model of FCIC reinsurance with one risk type and actuarially fair
premiums, the risk averse competitive firm prefers not to hold any of the FCIC policies it sells,

even with positive transfers from the FC'IC.

The proposition above does not mean that the competitive firm will not participate in the
program, only that it prefers not to hold any of the actuarially fiir premiums. If there are no
positive transfers fi'om the FCIC, it is not likely that a risk averse firm would participate in the
program since the firm’s utility from participating in the program is lower than doing nothing (in

this particular utility function, utility of doing nothing is normalized at zero).

Proposition (5-2): In this model, a participating competitive firm would never hold more than the
minimum portion of policies I‘ required of it by the FCIC.

Proof:

Any increase in t would decrease utility since it does not effect the mean and it increases

the variance.

Now introduce competition among firms participating in the F CIC program and as in
Chapter 3 assume that there is a reservation utility b such that the firm would only participate if

the utility of participation exceeds this reservation utility, i.e.,
aw -—"21<I‘)2¢2nil+(n— 0pr 2 b.

With competition among firms, the only fictor that can adjust is the number of policies (n)

held by each firm (everything else is kept fixed by the FCIC). If the transfers from the FCIC are

97
such that by participating in the FCIC program, the reservation utility of the firm is above b, then
in a competitive market there would be entry or exit of firms as n adjusts to regain reservation

utility. Equilibrium is reached when the utility of participating in the F CIC program is at b
(5-3) am) - g0): ¢2nI1 + (n —1)piai = b.

The changes in n with respect to other parameters is not clear because it appears as a quadratic
term in the expression. Another way to look at this problem is to see how utility changes with
changes in n.

Increases in It would increase utility if the expression

(54) 129 = av‘¢—K(t‘)2¢2i1+(n—I)piai la‘wainp
n 2 2

is positive and vice versa for decreases in It. To Simplify the expression, multiply throughout by n
(it is positive and this will not change the Sign of the expression). Note that the first two terms
when multiplied by It gives the expression b in equilibrium or at equilibrium signing the expression

is equivalent to determining the sign of
V c 2 2 2 2
(55) 1270 ) is as" p

This expression can take positive or negative values depending on the values of the other
parameters and the initial value of n. For example consider a small increase in t at equilibrium such
that there is a decrease in utility. Ifexpression (5-5) is positive, the number of policies It held by
each firm would increase to bring the reservation utility back to equilibrium. This would mean that

there would be exit of some firms.

98
This suggests that within each transfer level 0, the FCIC has some flexibility to adjust t as
changes in t will simply reshuflle the number of policies held by a competitive firm. Another way
to state this is that there is a minimum transfer 0., such that it would ensure t‘ is being held by a
competitive firm. Any higher transfer level would not result in the competitive firms holding more

of the policies but rather entry and exit of firms.

Consider the transfer level 06 such that for transfer levels below this competition cannot
adjust n and bring the firms utility back to reservation levels. This then is the minimum transfer
required for competitive firms to participate in the FCIC program and retain t" of the policies it
sells“. Let the number of policies associated with 0c and t° be nc. Now lets say that FCIC decreases
its transfer level below 0c such that there is a decrease in utility of the firm (the mean of profits
drop). To increase utility, the number Of policies held by each firm would decrease or increase
depending on the Sign of expression (5-5). Ifthe expression is positive then It would increase to
restore equilibrium and firms would exit the industry. This contradicts the fict that 06 is the
minimum transfer level for t°. Similarly if the expression (5-5) is negative then u would decrease,
or fimrs would enter to bring utility back to reservation levels. This too contradicts the fict that 0c
is the minimum transfer level. Therefore it must be the case that the expression (5-5) equals zero,

or at minimum transfer

(54) 0.II.W"¢--"-2’-(I‘)’¢’n.[1+(n. —1)plo-i = b

and b - Ig—(tcftizaipn: = 0 must hold.

 

2" Note that minimum transfer level would change with the minimum t value.

99

Proposition (5-3): The minimum transfer ( OJrequired for the competitive firms to participate in
the FCIC program increases with minimum retention (t‘) required of the firms.

Proof:

Follows from expression (5 -6) above as any increase in t must be followed by increase in 0

such that reservation utility b is maintained.

In other words the proposition states that if the F CIC writes reinsurance contracts such
that competitive firms retain higher levels of the policies, then the minimum transfer to the firms

must increase to compensate for the firms holding larger portions of actuarially fiir policies.

Proposition (5-4): If the correlation between farm losses is zero, then the number of firms in the

program increases directly with increases in transfer level a
Proof‘

The equilibrium expression with p=0 is
(5-7) owes — 321$)2 ¢2no§ = b

and n is exactly solved by

 

n = b .
W‘¢--';5(I’)’¢’ai

The higher the level of transfers the smaller the number of policies held by each firm at

equilibrium, and the larger the number of firms participating in the FCIC program.

100

5.3.2.2 Stop-Loss Reinsurance

So fir the discussion has concentrated only on proportional reinsurance to the competitive
firms. This section introduces stop-loss reinsurance provided by the F CIC. The stop-loss
reinsurance covers indemnity payments of a firm if the loss ratio is over a certain predetermined
level on the total retained business of the company. Stop-loss reinsurance is not on a policy by
policy basis as in proportional reinsurance, rather it is on the whole portfolio of the competitive
firm delivering FCIC crop insurance (the firms are assumed to hold only FCIC crop insurance in
their portfolio).

Assume that n is exogenously given to simplify the discussion. Stop-loss is introduced in
the model such that if the total loss of the finners appearing in the portfolio of the insurance firm
exceeds k, the FCIC is responsible for paying for all the excessive indemnity of the firm (i.e.,
losses exceeding t4)k). Note that the FCIC is already responsible for all such losses in the portion Of
the policies that it holds through proportional reinsurance, i.e., (l-t)4)k, which makes the FCIC
responsible for all losses excwding 4k. In return for providing stop-loss reinsurance to the firms,
the F CIC engages in gain sharing with these fimrs. This gain sharing is such that if the competitive
firm delivering FCIC insurance makes positive profits on its crop insurance portfolio, the
competitive firm keeps y(k) percent of the profits and the FCIC takes the rest. The gain sharing
rule is portrayed as a function of k since it would vary according to how much stop-loss coverage k
is provided. The higher the coverage, the higher the premium collected through the gain sharing
rule. Both the stop-loss level k and gain sharing are exogenously given to the firm participating in

the F CIC program.

l 0 1
With the availability of stop-loss the profits of the competitive company are the same as

expression (5-1), but now stop-loss is introduced.

r<k>t¢i<w —I.-) if 2wa —I.)>o

(5-8) a, = anv‘¢+4 , ..
maX(t¢Z(W" -1l)s-t¢k) if 204’" -1.-)S 0

 

As before the first term on the RHS is the direct transfer of funds from the FCIC to the

competitive firm. The second term now includes both the proportional and stop-loss reinsurance. If

there are positive profits (2 (WA — l, ) > O) , the firm gets to keep y(k) percent Of the profits of the

retained business and if there are non-positive profits (2 (w‘ — 1,.) S 0) , the maximum loss the

firm bears is t¢k , the rest being covered by stop-loss reinsurance25 .

Recall that stop-loss reinsurance is given exogenously to the firm and the only decision the
firm makes is the amount of business it will retain through proportional reinsurance. To see how

stop-loss reinsurance effects the competitive firm’s decisions on proportional reinsurance define

r = 2(w” - 1,.) to be the random profits with distribution F (r) and supports [nwA,-nL].

i =1
Without stop-loss, the maximum loss the policies can make is nbL (all the firmers in the portfolio
collect indemnity for that period), and the maximum gain the policies can make is n4)W’ (no
firmers collect indemnity for that period). For the moment let the direct transfers from the
government to the competitive firm be zero (0:0). Since the premiums are actuarially fair the

expected value of profits is

 

25 Note that k is actually a function of u but since u is assumed to be exogenously given, we simply state k
which makes the notation and the discussion much clearer.

102

M
E(r)= [drop-0

—nL
which may be rewritten as

t TrF(r)+ 32F(r)+mfrF(r)) = 0.

0

Now if the gain sharing rule is actuarially fiir such that the expected value of stop-loss payments

to the competitive firm equals the expected value of gain collected by the FCIC then
_,, ”4
-r¢ j tar): tar—rte» I Ire)
_. nL 0
or

514%)
Item—7%—
Ithr)

Where 7 A(k) is the value of gain sharing rule which is actuarially fiir at stOp-loss reinsurance k.
At this value the government would break even in the long-run by providing stop-loss reinsurance.
Ifthe gain sharing rule is such that y>y “, the stop-loss premium is priced above actuarially fair
and the FCIC would make profits fiom providing stop—loss and if y<y A, the stop-loss is below

actuarially fiir and the F CIC would makes losses.

103
Proposition (5-5): The minimum transfer (6..) required for competitive firms to participate in the
FCIC program is reduced with the introduction of actuarially fair stop-loss reinsurance.

Proof:

Assume that the stop-loss is provided at an actuarially fiir gain sharing rule (7“) such that
the expected profits of the competitive firm is zero. The variance of loss is now decreased
compared to the case of only proportional reinsurance because the loss is truncated at k and y<1.
Now reintroduce a positive transfer level (0 > 0) such that the competitive firm participating in the
program retains t percent of the policies (for reasons cited in the previous section). Unlike the case
where no stop-loss was available, now with stop-loss the variance is lower. Expression (5-6) Shows
that the reduction in variance could mean that the minimum transfer required of the competitive

firms to participate in the program is reduced.

Therefore stop-loss reinsurance provides another avenue for the government to transfer
risk from the competitive firms to the F CIC. If the gain sharing rule is actuarially fiir, then it
reduces the variance of the portfolio by keeping expected profits (excluding positive transfers) at
zero while reducing the variance of the portfolio. At the extreme when k=0, all losses are paid by

the FCIC and the gain sharing rule is Y’(0)=0, or all the gain is taken by the FCIC.

Thus proportional and stop-loss reinsurance both serve very similar purposes and one can
be substituted for the other to transfer risk from the competitive firm to the FCIC. For example
with the addition of stop-loss reinsurance there is a further reduction in variance and now the FCIC
can increase the minimum proportion of policies (t) to be retained by the competitive firms. One
example could be that the increase in t can be such that it compensates for the decrease in variance

and the reservation utility of the firms is maintained in b.

104

However, unlike proportional reinsurance where the premium charged and indemnities paid
are clearly defined by t, it is more difficult to come up with actuarially fiir gain sharing rule in
stop-loss reinsurance. This is because the probability distribution of profits is required which may
not be known with high degree of certainty”. There is the possibility that the premium collected
through gain sharing for stop—loss is not actuarially fiir. Consider the case where the gain sharing
rule is below actuarially fiir (premium does not cover expected payments). For the discussion here
let the direct transfers be zero (0:0) so that if stop-loss is actuarially fiir, firms will not participate
in the program. If stop-loss is provided at below actuarially fair prices then the premiums wA
collected fiom the firmer become above actuarially fiir for the firm. This is because the
probability of making positive profits is now greater than making losses. Results from Chapter 3
Show that the firm would now voluntarily opt to retain some of the policies (t > 0) through
proportional reinsurance (recall that in the case of proportional reinsurance, given the choice, the
firms would always prefer to hold t==0). Depending on how much the stop-loss premiums are below
actuarially fiir, the firms may even opt to retain more than the minimum amount required of them
or we now may observe t>t'. In such a case the competitive firms would hold higher portions of the

policies it sells. The FCIC would lose money in such a stop-loss reinsurance program.

5.4 FCIC Reinsurance With More than One Risk Class of Farmers

As indicated earlier in this chapter, the current F CIC program does not use the

informational position of the competitive firms to classify finners for insurance delivery. This

 

2" The same may be said about arriving at actuarially fair premiums but with only one risk type, this
problem is assumed away at present.

105
implies that all the informational problems of the F CIC discussed in Chapter 4 are retained in the
program. The only difi‘erence in this chapter is that now all administrative costs are subsidized by

the FCIC and the FCIC insures indirectly through reinsurance.

Ifthe firmer classifications problems are not addressed by the FCIC and attempts are
made to price premiums at actuarially fiir levels, then all firmers would signal that they are low
risk farmers and pay the same premium to buy the same maximum coverage in this model. Clearly,
there are adverse selection problems as the high risk types are paying premiums less than expected
indemnities. It is not difficult to extend the analyses of the previous section to see that increasing
amounts of positive transfers would be required for the competitive firms to participate with
adverse selection problems. The FCIC would be ficing additional expenditures in two areas: (1)
paying indemnities greater than premiums directly to the firmers on the portion of the business
they hold and (ii) paying higher transfer levels to competitive firms in order for these firms to hold
policies priced at below actuarially fiir levels. The program would not Operate at an actuarially

sound basis and would lose money.

The purpose of this section is not to study a situation as described above but to look at a
situation described in the previous chapter; a separating equilibrium such that the program is
actuarially sound on the portion of the business directly associated with firmers. In other words,
this section studies scenarios where the program’s expected indemnities to firmers equal premiums
collected fiom firmers. Given such a situation, the effects of the current reinsurance agreements on

such a program is studied.

106

5.4.1 FCIC Reinsurance in a Separating Equilibrium

The section first presents a modified version of the separating equilibrium with cross
subsidies between firmer risk types as presented in Chapter 4. The modification from the model in
Chapter 4 is that all delivery cost is removed. Recall that the F CIC is a dictator who follows the
FCIC mandates and is able to provide and sustain any contract it desires. The separating

equilibrium model of chapter 4 is reproduced below (minus delivery cost).
(5-9) w,’ = w,‘ +d
(5-10) w; = w; — Rd

(511) ¢; =¢gm :“a
+ + _ 2' + 2 2 max + _ A max 2 2
(5'12) ¢1(-Wl +L2)—E(¢l ‘1) ‘72 :¢2 (“W2 +L2)“2_(¢2 ‘1) 02

The first two expressions are the premiums to risk type 1 and 2 respectively. The
premiums are at actuarially fair levels plus there is a tax d on risk type 1 and a subsidy Rd on risk
type 2. Recall that R is the ratio of low to high risk firmers. Subsidy on risk type 2 implies that
premiums are below actuarially fiir and this means that risk type 2 would seek coverage above 100

percent (firmers seek 100 percent coverage at actuarially fair premiums). Following the current
FCIC program it is assumed that the maximum coverage is less than 100 percent (¢§"" <1). This

is represented in the third expression. The fourth and final expression presents the coverage
restrictions on type 1 firmers such that a separating equilibrium is achieved. This coverage is

determined at the point where the high risk type is indifi’erent between the low and high risk

107

contracts. The two contracts offered by the FCIC are (45:45fo ) and (45'2”, gmw; ) for low and
high risk types respectively.

The basic features of the equilibrium are that type 1 firmers fice premiums higher than
actuarially fair and type 2 finners face premiums below actuarially fiir levels. Results fiom the
previous chapter show that when delivery costs are subsidized, as in the present model, farmer
classification is always a problem. The other result to note is that truncating maximum coverage to
below the level desired by risk type 2 also lowers the coverage feasible to the low risk type. This is
because the coverage to low risk types can be increased if the utility of the high risk type is
increased. All the results regarding the tax d and the ratio of low to high risk firmers R discussed

in the previous chapter also hold here.

The reinsurance decisions of the competitive firms are now discussed. It is assumed that
parameters are such that positive coverage is feasible within the separating equilibrium to both risk
types (otherwise the problem reduces to the case of only one risk type — i.e., insurance to only
type 2 firmers). All reinsurance considered is proportional reinsurance. Stop-loss reinsurance is
removed to keep the analyses simple and, as Shown in the previous section, the effects of stop-loss

is very similar to proportional reinsurance.

A competitive firm participating in the F CIC program must sell policies to any firmer that
seeks it. In a separating equilibrium, the high risk firmer would buy the high risk policies and the
low risk firmer would buy the low risk policies. Once the policies are sold, the competitive firm
has to make decisions on how much of the policies would be retained under the proportional
reinsurance agreement and how much will be passed on to the F CIC. Assume that the F CIC

provides two risk pools for reinsurance purposes. For the high risk pool minimum retention rate is

108
t; and for the low risk pool the minimum retention rate is if such that if > t; . As in the FCIC

program there is a maximrun amount of business that may be put in the high risk pool. To simplify
the analysis it is assumed that the companies do not have a higher percent of type 2 farmers than
the rmximum allowed in the high risk pool. For example, if 70 percent is the maximum of the
portfolio allowed in the high risk pool, the competitive firms have less than 70 percent high risk
types in their portfolio.

By design of the separating equilibrium, the total portfolio that a competitive firm holds
(before reinsurance) has zero expected value. Consider the high risk types for reinsurance
purposes. It has be established earlier in the chapter that positive transfers to the competitive firms
must take place for the firms to retain a proportion of these policies (since these policies are below
actuarially fair prices). The positive transfer would now be greater for the same minimum retention
level than the case when there is only one risk type in the portfolio of the firm. The case is the
opposite for reinsurance of the low risk types. In the separating equilibrium, the low risk types are
paying above actuarially fiir prices for the coverage they receive. The competitive firms would
now voluntarily choose to hold positive proportions of these policies. Whether the fraction chosen
is above or below the minimum retention required for these policies is a function of how much

higher or lower the premium is from actuarially fiir levels.

Thus we see that a proportion of risk type 1 policies are desirable (given tax d>0) to a
competitive firm while no proportion of risk type 2 are desirable. Let n; be the number of policies
of risk type 1 and n; be the number of policies of risk type 2 in the portfolio of a firm. These
numbers are assumed to be exogenously given for simplicity. The profits of the firm from

participating in the F CIC program are

109

"I n:
(5 13) 7r. = 64m: + Witt?“ + t.¢?(nlwr — 219+ t,¢;m(n,w; — 21,.)
- i=1 i=1

The first two terms on the RHS are the direct transfers from the FCIC to the competitive firms. 0
remains the same for both types of policies Since the cost of delivery is assumed to be identical for

both policies. The third and the fourth terms are the premiums collected and indemnities paid to

risk type 1 and 2 respectively for the retained business of the firm.

The mean and variance of the portfolio of the firm are

i. = WWW? + WM?“ + 114591109? - 1—1)+t2¢'2"“"2(W§ - 1:2)

ai, = tf(¢f)’nl[1+(nl —1)p.ia: + Ito?“ )2 n. [1 + (n. —1)p.la§
+ II t.n.n2¢;‘¢?“peal’0§
where p1 is correlation between two type 1 losses, p; is correlation between two type 2 losses, and
p12 is the correlation between a type 1 and a type 2 loss. All correlations are assumed to be
positive.

Unlike the case with only one risk type, the mean of the portfolio is now affected by the
retention levels t1 and t;. The mean is increased by higher retention levels of type 1 finners and
decreased by higher retention levels of type 2 firmers. The variance is increased by higher retention
levels of any type firmer. Also note that the variance is a function of correlation between risks.
Since only positive correlation is considered, the higher the number of policies in its portfolio the

greater would be the increase in variance.

Notice that given the right demand conditions, correlation coefiicients, and coverage

restrictions, there is a combination of t, and t; such that competitive firms would hold both types of

l 10
policies without nwding any subsidy from the FCIC. This is possible because the gains fi'om risk
type 1 could compensates for the losses fi'om risk type 2 and still maintain reservation utility. Since
positive expected profits must be made for this to happen, it is the case that the expected value of
the portion of the policies transferred to the FCIC has expected negative value. The FCIC program
is then no longer sound. The situation is further aggravated if direct transfers are required to ensure

that the competitive firms participate.

Therefore the efi’ects of FCIC reinsurance with classification problems is that now the low
risk firmers also bear the burden for classification problems. While in the one risk case, FCIC was
hearing all the burden of transfer to the competitive firms, now this transfer to the competitive
firms plus transfer to the type 2 firmers are borne by the low risk finners and the FCIC. In the
separating equilibrium, the program fills short of providing maximum coverage to all finners to

serving as the primary provider of disaster assistance.

5.5 Implications for the FCIC Reinsurance Programs

The model in this chapter suggests that without firmer classification problems (only one
risk type), the government is able to meet its objective of providing the maximum possible coverage
under the federal crop insurance program. Whether this objective is met with minimum cost to the
taxpayer under the current FCIC reinsurance program is not clear. If stop-loss reinsurance is at
actuarially fiir premiums then the short answer to this question is yes, if the F CIC cannot deliver
crop insurance at costs below 33-34 percent of premium. The answer is no if its delivery costs are

higher.

l 1 l

Stop-loss reinsurance is essentially a substitute for proportional reinsurance and it is a
similar means of transferring risk from the competitive firms to the F CIC. However, unlike
proportional reinsurance the pricing of stop-loss reinsurance is not easily verified to be at
actuarially fiir levels via the gain sharing rule. If stop-loss reinsurance has very fivorable terms
(i.e., it is priced at below actuarially fair levels), then it is another way to transfer funds from the
FCIC to private competitive firms. Unlike the transfer of funds through administrative costs, this
transfer directly efi’ects the retention of policies by the private competitive firms. Ifthe stop-loss is
at very fivorable terms then the competitive firms could hold more than the minimum proportion of
policies required through proportional reinsurance. If the stop-loss reinsurance is priced at below
actuarially fiir (as various GAO reports indicate), then the delivery cost of the program is above
the 33-34 percent of premium. However, this does not suggest that F CIC can deliver it at a cheaper
rate if its own cost of delivery is very high.

The long-run efl’ects of competition among private firms suggest that associated with every
t (retention level) there is a minimum transfer such that competitive firms participate in the FCIC
program. Among other things this minimum transfer is a function of the level of reservation utility
and the degree of correlation between firm losses. If the current transfer level of the F CIC is above
this level then higher levels of transfer are being dissipated by entry of firms through competition.
The implication of this model then is that the transfer can be reduced without adversely afl‘ecting

firmers while ensuring that competitive firms still participate in the program.

With the introduction of classification problems, the inclusion of private competitive fimrs
with reinsurance would further aggravate the problem since these firms would now demand a
higher “risk premium” to hold policies that are priced below actuarially fair. If no differentiation

between the high and the low risk firmers is made, and the premiums are priced at actuarially fiir

a. A».
112 "I ‘

prices, then the program succeeds in providing maximum coverage to all firmers but would face
severe expenditures. This may partially explain why the program has paid close to $2 billion to
firmers in the 1980-90 decade over and above predetermined subsidies. prrivate competitive
firms are required to hold these policies, then high transfer rates, via administrative costs and very

fivorable stop-loss features has to be in place.

If reinsurance is applied to a separating equilibrium situation then the program may no
longer be able to provide maximum coverage desired to all firmers, as the low risk finners are
now taxed. Since private firms are still required to hold a minimum of high risk policies then

transfer of funds, either from low risk firms and/or the FCIC, will still be required.

The findings of the study on the stylized F CIC reinsurance program explaining the idea
that reinsurance to private firms is provided on better terms than commercial reinsurance
companies would provide. The basic reasons given by the F CIC are that: (i) the farming sector is
more prone to widespread natural hazards as droughts that affect a large number of finners; and
(ii) the private companies are obligated to sell FCIC crop insurance to all finners (at FCIC set
premiums) who demand it if crop insurance is offered for the particular crop (unless specifically
excluded by the F CIC). The results of this chapter provide explanations why generous reinsurance

agreements are needed.

The overall conclusion from this chapter is that the information position of private firms
are not used to classify farmers. Rather, it is used for reinsurance purposes. Whether the cost of
involving the private firms to deliver FCIC insurance is lower than the FCIC delivering insurance
itself is not clear. The model suggests that the current reinsurance program of requiring private
firms to retain a part of the policies it sells is not the most cost effective way Of delivering crop

insurance. If the cost structure of the private firms is to be used then it is best not to require private

113
firms to retain any of the business they deliver, as this appears to serve no purpose in achieving the
FCIC’S objectives. Under the current F CIC program, the model suggests that the most efficient
way is to hire these private firms to simply deliver and adjust the policies sold for a fee while the
FCIC bears all the risks. This finding is consistent with the recommendations of various General

Accounting Ofiice reports.

6. Reinsurance for a Competitive Crop Insurance Market.

This chapter focuses on the FCIC’S ability to involve the private sector through
reinsurance. The reinsurance schemes investigated are self sustaining and therefore preclude
negative expected profits. Historically, private multi-peril crop insurance has not existed in the
United States for any sustained period of time“. Therefore the primary question here is: if no
private crop insurance market existed without F CIC reinsurance then could the availability of
FCIC reinsurance ficilitate such markets? This question gains further importance because one of
the goals of the FCIC is to involve the private sector primarily through reinsurance. The second
question here is: under what conditions can FCIC reinsurance increase coverage to firmers buying

private competitive crop insurance?

The main difl‘erence between the stylized model of this chapter and that of Chapter 5 is
that here the FCIC involves itself only by making reinsurance available to competitive firms.
Unlike the reinsurance program discussed in chapter 5 (and the existing FCIC program) where all
premium, coverage, loss adjustments, etc. decisions were made by the F CIC; such decisions are
now made by competitive insurance firms. The purchase of reinsurance is not mandatory and if a

firm buys F CIC reinsurance it is only because reinsurance benefits the firm.

This chapter is organized as follows: First, the competitive insurance market model of
chapter 3 is used to investigate the conditions under which equilibrium breaks down in the
competitive market. Both a mathematical and a graphical approach are used. This is done to help
answer the first question of this chapter and study reasons why private multi-peril crop insurance

does not exist in the agricultural sector. The chapter then introduces F CIC proportional reinsurance

 

2' The only time such insurance existed was around the turn of the Century (see chapter 2).
l 14

l 15
where the FCIC pays a proportion of indemnities for a share of the same proportion of premiums
(see below for details). This section analyzes how this reinsurance scheme might bring about
insurance markets where none existed before, and its ability to increase coverage to firmers. The
chapter then looks to answer the same questions in a reinsurance scheme where the premiums for
reinsurance are actuarially fiir. Next is a discussion which summarizes results on the ability of
F CIC to provide conventional reinsurance schemes when it lacks information on firmer risk
classes. The final section provides discussion on an alternative “reinsurance” scheme of using an
Area-Yield Insurance Plan (explained below) to overcome some of the problems ficed in the more

conventional reinsurance schemes.

6.1 Constraints on Existence of Competitive Insurance Markets

Let there be one risk type of farmer with properties as described in Chapter 3. To
recapitulate, each acre yields M bushels of output. Ifdisaster occurs on a firm, output L is lost
with probability P and 0 otherwise. The correlation of loss between any two firms is the same and
it is denoted by p which is assumed to be non-negative for agricultural markets. The competitive
market is such that the portfolio of the insurance firms consists of only crop insurance policies and,
in the long-run, competition ensures that firms operate at reservation utility levels. All the
assumptions described in Chapter 3 regarding agent’s preferences, functional forms, etc. hold here.

The notation also remains identical unless otherwise stated.

To analyze the conditions under which equilibrium breaks down, the definition of a

competitive equilibrimn fi'om Chapter 3 is reproduced below.

116
Definition (3-1): A competitive equilibrium in this model of crop insurance delivery is a premium

level w”, coverage level as” and number ofpolicies n" for each firm such that:
(3-3) (-w° + Z)+/t(1— o)°)a2 = 0

(3-9) n°(w° - L — c) — wn°¢°az[l + (n° — 1)p] = 0

(3-10) n°¢°(w° — Z — c) — %n°(¢°)202[1+(n° -1)p]—b = 0.

The simultaneous solution of the three non-linear equations in coverage, premium, and number of
policies, gives the equilibrium values. To study the equilibrium analytically the first two

equilibrium expressions are solved first and then the third expression is included.
Solving the first two equations gives the equilibrium coverage

2 110': — c
(4 + w(1 + (n° -1)p))ai

 

(6-1) ¢°

This equilibrium coverage is a function of degrees of risk aversion of finners and firms, the
variance of the loss, the degree of correlation between losses, the number of policies held by the
firm, and the cost of insurance delivery. This expression shows an obvious case where equilibrium
does not exist; when cost of delivery is high enough that the numerator is non-positive.
Alternatively, one may think of the numerator being non-positive if either the degree of risk
aversion of the firmer is low (tending towards risk neutrality) or the variance of the loss is low. In
these situations an insurance market will not exist because benefits from insurance do not justify
the cost. For the rest of this chapter it is assumed that the numerator takes positive values. This
allows us to concentrate on the less obvious aspects of when equilibrium breaks down in this

model.

l 17
Exactly what coverage level would transpire (if an equilibrium exists) is a function of n0
which is to be determined at equilibrium. This is done by including the third expression for
equilibrium that states that the firm must operate at reservation utility levels”. Therefore,
expression (3-10) maps the equilibrium relation between reservation utility b and number of
policies n°. Theoretically the values of no can range from 0 to infinity. At n°=0 the utility of the
firm is zero29 (equivalent to the utility from the firm doing nothing). As It —) co , the equilibrium

utility (V) of the firm approaches
(110': - c)2
Lim V —-) -———2-—
""°° ZWPGL

These limits give us two reference points between which reservation utility takes values. If it can be
shown that the limit at which It tends to infinity is the upper limit then we can determine the values
of b which are not in the equilibrium set. One way to do this is to show that in expression (3-10), b
is monotonic for positive values of n. To check whether this expression is monotonic, differentiate

the preference function with respect to n using the envelope theorem to Obtain
0"V _
51— : ¢(w — L - c)— g-Wafll +(2n - l)p).

If this expression is monotonic in n then there Should be no turning points. Let us assume that there

exists a turning point, in which case the expression takes the value

¢(w —Z— c)— %¢’02(I +(2n — l)p) = 0.

 

23 One must note that when p=0, the number of policies does not appear in the coverage level and
therefore appear to be independent of it or reservation utility. As seen in chapter 3, this is not the case and
reservation utility still affects the premium and coverage to farmers.

29 This is a result of M-V preference function normalization.

118

At this point the coverage to firmers is solved as

lat—c

(2 + %(l + (2n — 1)p))ci '

 

¢=

If this is an equilibrium expression it must satisfy Equation (6-1). The only point where the two
expressions are identical are at p=l or perfect correlation between all risks. One is not likely to
find this extreme case in an insurance market and it may be safe to assume that the reservation

utility is monotonic in the range stated for n.
Equilibrium reservation utility is therefore bounded by

(20: — c)2

be 0 2
ZapoL

,

and it can be represented graphically as

 

<7 I

 

b0

 

 

 

110 11

Figure 6-1: Reservation Utility and Competitive Equilibrium

l 19
Equilibrium utility varies in the range [0, b] where b is the upper limit. Figure 6-1 contains all
the information necessary to solve for equilibrium coverage and premium. For example given a
reservation utility, say b0, the corresponding equilibrium number of policies is no. This value of no
can then be used to Obtain coverage in Equation (6-1), and then the premium can be determined

from the first-order condition (F OC) of the firmer (Equation 3-1).

Figure 6-1 may now be used to gain insights into how equilibrium breaks down in a crop

insurance market with positive correlation between risks. The equilibrium relation between n and b
shows that values of It cannot be determined for reservation utility values above 6 (no equilibrium
is achieved for such utility levels). The reasons why equilibrium breaks down are therefore
contained in the upper limit expression for reservation utility b, i.e.,

_ _ (20': — c)2
ZWPUI’.

To ensure a large range of reservation utility for equilibrium points this expression must be
made as large as possible. The factor that affects this range the most is the value of p, or the degree
of the correlation between risks (also a measure of systematic risk in this study). If the correlation
p between firm losses tends to zero, the upper limit to equilibrium increases to infinity. This means
that as the risks gets closer to being independent, equilibrium points are almost always ensured
(although high reservation utility would provide minimal coverage to finners). However, as p
moves away from zero towards unity, the upper limit to equilibrium reservation utility is brought
down very sharply. This Shows that even with low levels of correlation between firm losses, there
could be breakdown of competitive equilibrium. This breakdown in equilibrium is further

aggravated if the competitive firms are highly risk averse (high values of w).

120
The findings of this model provide an explanation for the observed non-existence of multi-
peril crop insurance in the private sector because of the catastrophic nature of many disasters.
Also note that private insurance markets for fire and hail exist and these risks are fiirly across
firms. Further for some perils, insurance markets may not exist (even if risks are independent)
because, as indicated earlier, the benefits from insurance may not justify the costs or because of

moral hazard and adverse selection.

6.2 Proportional Reinsurance

Following the mandates of the 1980 Crop Insurance Act, the FCIC chose to use
reinsurance of private insurance firms as the primary mode of delivering crop insurance to firmers.
This was done to make the program more “efficient” by increasing coverage to firmers and
running it on a self sustaining basis (although still eligible for some predetermined subsidies). The
working hypotheses in Chapter 1 provides reasons why private competitive firms were more suited
than F CIC to gain this “efficiency”, viz., (i) a higher level of information for firmer risk
classification and (ii) a lower delivery cost. Chapter 5 explained reasons why the current FCIC
program has fillen short of using the information position of the private firms to classify firmer
risk types. One of the primary reasons for this assertion is that the private firms participate in a
very passive framework where all coverage, premium, adjustment, etc. decisions are made by the
F CIC while the private firms only deliver crop insurance and then decide how much of the policies
to retain through reinsurance. The focus of this section is to introduce FCIC proportional
reinsurance, but let the private competitive firms make all the insurance decisions. This is to

overcome the major criticism of the current program which is that it does not use the information

12 1
position of the private competitive firms. In doing so the study analyzes the effectiveness of such a

reinsurance scheme in Opening markets where none existed before, and in increasing the coverage

to firmers.

6.2.1 Proportional Reinsurance Described

Proportional reinsurance contracts are where a firm reinsures a proportion of its portfolio
and the FCIC is responsible for indemnities on that proportion of every policy. In exchange, the
FCIC receives the same proportion of premiums charged by the competitive firms to firmers. This
proportional reinsurance structure is similar to the one existing in the current FCIC program, with
one major difi‘erence; the competitive firms now make all the coverage, premium, adjustment, etc.

decisions while the F CIC only makes reinsurance available to competitive firms.

The reinsurance structure described above makes positive expected profits for the F CIC
since it shares in the premiums set by the competitive firms. This is because the competitive firms
are risk averse and the premiums set by such firms would be priced above expected indemnity plus
cost of delivery. According to the working hypothesis of this study (the FCIC being assumed to be
risk neutral) the F CIC would be willing to sell the same reinsurance to firms at premiums that
make zero expected profits. However the proportional reinsurance described is extremely easy to
operate since it requires no actuarial calculations on the part of the FCIC. Actuarial calculations
would be dificult for the FCIC because it is assumed to lack information on firmer risk
classification. For this reason the FCIC is assumed to sell reinsurance in the manner described and
make positive profits. The profits generated from such a program may be placed back into the

program by subsidizing the cost of obtaining reinsurance to fimls, or transferring it to firmers

122
purchasing crop insurance (through tax breaks, etc.). These issues are abstracted from in this study

and it is assumed that the FCIC provides proportional reinsurance which generates positive

expected profits.

6.2.2 The Model

Since the firms are assumed to have full information, only one risk type of firmers is
assumed (this also simplifies the analyses considerably). Further, it is assumed that the insurance
firms sell crop insurance, and if a firm buys reinsurance, it must buy reinsurance for its entire crop

insurance portfolio. The profits of a competitive firm with FCIC reinsurance is

71's 2 tn¢(w — c)— «6:1,. — n¢(1—— t)k

The first term in the right hand side (RHS) is the premium w less cost c (both quoted in per acre
terms) collected from n farmers for 4) percent coverage. With proportional reinsurance, the firm
retains t percent of this net premium and passes on 1-t percent to the FCIC as premium (for H
percent proportional reinsurance). The second term indicates the total losses incurred by firmers,
which 4) percent is covered through crop insurance. Of this 4) percent paid to firmers the FCIC
pays l-t percent through a reinsurance contract and t percent is borne by the competitive firm. The
last term is the transaction cost incurred by the competitive firm to obtain FCIC proportional
reinsurance. This transaction cost k is assumed to be linear in the amount of reinsurance bought. If
(H) is the amount of reinsurance bought, then the total indemnity reinsured is n4)(1-t). The total

fee that must be paid for this reinsurance is n¢(1-t)k.

The mean and the variance of profits for the competitive firm are

123

a, = tn¢(w—L—c)—n¢(l—t)k

02,, = 12¢2n[1+ (n —1)p]a: .

Reinsurance has the effect of reducing both the mean (by reducing retained premiums and

transaction costs) and the variance (by transferring a portion of the indemnity to the FCIC).

As in previous chapters, the competitive firm is assumed to have a preference function that
is increasing in the expected value of profits and decreasing in variance of profits of its portfolio.

The specific functional form
V(.) = e, — go;

is used to represent the utility of the firm.

The decision problem of a competitive firm is to maximize its preference function by
deciding on the amount of insurance it would provide to finners given that federal reinsurance is

available in the market. The utility of the firm selling insurance is
(6-2) V = tn¢(w - Z — c)— 12’-¢212n[1+(n —1)p]c§ - n¢(1—t)k

The F OC"0 to maximize this function with respect to 4) is

(6-3) tn(w — L — c) — n(1- t)k — w¢t2n[1+(n — 1)p]O'i = 0

Solving for coverage yields

 

3° The Second Order Condition is satisfied (—ut2n[l + (n — 1)p]ai < 0)

124

(w—L-c)—(l—;t—)k
W11 + (n - 0.0103.

 

(64) ¢=

To attain equilibrium values the firmer must be introduced. The decision problem of
firmers is exactly identical to that given in Chapter 3, since F CIC reinsurance does not efi’ect the

firmer directly. Recapitulating, the firmer maximizes his preference function
- - ll 2 2
U=M-¢w—L+¢L—E(¢—l) O’L.

The F OC to this problem is

(6-5) —w +Z — 2(4) —1)a§ = o

 

and the coverage sought is
w — Z
6-6 = l— .

The equilibrium conditions may now be expressed in the presence of FCIC proportional

reinsurance.

Definition (6-1): A competitive equilibrium in this model of crop insurance delivery with FCIC
proportional reinsurance t' is a premium level w', coverage level ¢', number of policies n’ for

each firm such that:

(6-7) (—w' + L)— t(¢' —-1)a2 = 0

(6-8) t’n'(w' -— L — c) — w¢’t'2n'[l + (n' —1)p]ai - n'(1— t')k = O

125

(6-9) t'n'¢’(w' — L — c)-— gqi'zait'zn' [1 + (n' — l)p]— n'(1— t' )¢k — b = 0

(6-10) —¢'w' + ¢'L — gar -1)zc§ 2 —¢°w° + ¢°Z — §<¢° -1)2 a:

The first two expressions in the definition of a competitive equilibrium are the first-order
conditions for maximize the preferences of the firmer and the competitive firm, respectively. The
third expression is the condition that the utility of the competitive firm is at reservation level b. The
fourth expression is the condition that the utility of the firmer must be higher when the firm buys
reinsurance than without it. This last expression assures that reinsurance level t’ is in the
equilibrium set since if the firm buying reinsurance cannot provide higher coverage (and utility) to
firmers, then a firm not buying this insurance would be able to take policies away from the firm
buying insurance. If this expression does not hold then l-t proportional reinsurance is not in the
equilibrium set and such reinsurance ofl’ered by the FCIC would not be bought by a competitive

firm.

The proportional reinsurance model may now be explored to see if (i) it brings about
equilibrium points where non existed before; and (ii) it increases coverage to firmers. Each of these

aspects are studied in turn.

6.2.3 Can Proportional Reinsurance Facilitate Equilibrium?

Suppose the reservation utility is set such that equilibrium does not exist without
reinsurance. Then the question is: can proportional reinsurance bring about equilibrium? To study

this one needs to analyze only the first three expressions for equilibrium in Definition (6-1). If

126
equilibrium exists, then expression (6-10) may be used to check if such reinsurance levels would

actually be bought by competitive firms. This line of argument is clearer in the discussion below.

The presence of FCIC reinsurance (t<1) would bring new equilibrium points if it is able to
shift the upper equilibrium limit to above t=1. To solve for the equilibrium relation between
reservation utility b and number of policies n, steps identical to those in section 6.1 are followed.
First the equilibrium coverage is solved for from expressions (6-3) and (6-5) to arrive at

“ppm

6-11 = t .
( ’ ¢ (I+tv(1+(n—1>p»oi

 

As before, this coverage is a function of exogenous parameters, 11 but now also t. Assume that t is
exogenously given by the FCIC. The value of n is determined at equilibrium as the utility of the
firm settles to reservation utility level b. As before, It takes ranging from 0 to infinity. At n=0 the

utility remains at zero, while the limit as It tends to infinity is

(lag—c-(—1'—’—)lc)2
erV-+ ’

n—rao ZWPO-i

 

Further it can be proved that equilibrium reservation utility is monotonic in n (the proof is identical

to the previous section) such that reservation utility lies in the range

(Mi-c-(l—thjf

t

 

 

beO

9

2mm:

 

 

for values of p<l.

127
In the limits of b, we see that as t increases, the upper limit decreases (with positive k)

because

Lim (l—t)—)oo

t—)0 t

 

In other words, if equilibrium did not exist when t=1 (i.e., because reservation utility was above

5 ) then for values of K], the reservation utility levels in the equilibrium set cannot be increased.

This may be illustrated graphically as

 

 

 

Figure 6-2: Competitive Equilibrium and Proportional Reinsurance

If equilibrium did not exist at t=1, the presence of proportional reinsurance cannot
ficilitate equilibrium. Making proportional reinsurance available to competitive firms will not

entice competitive firms to Ofier crop insurance to firmers where none existed before.

This result is occurs because, at the limit, with or without reinsurance the firms hold very
small portions of each risk in equilibrium. These two limits are equivalent if transaction costs are

zero. However, even though a very small portion of each risk is held by the fimi, the presence of

128
correlation between risks caps the upper limit. The higher the correlation the lower the limit. In the

presence of positive transaction costs, this limit is lowered as shown in Figure 6-2.

6.2.4 Can Proportional Reinsurance Increase Coverage to Farmers?

This second question of whether proportional reinsurance would increase coverage to
firmers becomes irrelevant if, in the first place, a private insurance market did not exist. Asking
such a question is more appropriate for private insurance markets, such as hail or fire which

already exist.

Recall fiom the preceding analysis (and from Figure 6-2 above) that with proportional
reinsurance there is a clockwise shift of the equilibrium curve. This implies that for a given
reservation utility level, the number of policies held by each firm at equilibrium is increased with
proportional reinsurance. If we look at expression (6-11), we see that there is a simultaneous but
opposite movement of n and t in the denominator of the expression. Further increases in t would

also decrease the numerator for positive k.

Assume that k=0 for the moment and concentrate on the denominator Of expression (6-11).
If the net effect (of movements in n and t) on the denominator is to increase its value then such
movements cannot be consistent with equilibrium since a firm without reinsurance can provide

higher coverage to firmers. For coverage to increase under reinsurance the expression
“/10 + (n — 1)p)0‘i must decrease. Let there be two reinsurance levels t1 and t;, such that t1>t2.

This means that t; is the higher reinsurance level. Now coverage would increase with the higher

reinsurance level only if

129
t2W(1+(”2 - 1)p)ai < tit/(1 + ("r - 1)p)0r’.

To determine if reinsurance increases coverage consider expression (6-11) reproduced below with

k=0.

= 110': — c
(2 + tlI/(l + (n - 1)p))0i

 

4’

This expression shows that coverage can be increased by increasing reinsurance (decrease in t) as
it has the effect of reducing the variance of the portfolio. At the extreme as t ——> 0 , the coverage

tends to

c
2
AOL

 

¢=1—

which is the expression for coverage provided by a risk neutral insurance provider. Ifthe
government constrains the lower limit on the amount of reinsurance that may be bought by a

competitive firm, then this lower limit would determine the coverage to firmers.

In the presence of transaction cost, higher values of k results in lower benefits (as
increased coverage) to farmers from FCIC proportional reinsurance. Consider the numerical
examples below to illustrate this. The parameters for the example are P=0.3; b=10; 2:001;

w=0.01; c=3.

130

Table 6-1: Competitive Coverage Levels With F CIC Proportional Reinsurance: k=1; p=0.05.

 

 

t E w of U n
1.0 E 28.679 0.253 114.186 16.846
0.9 : 28.643 0.257 114.195 19.064
0.8 : 28.626 0.258 114.199 22.067
0.7 : 28.637 0.257 114.196 26.333
0.6 1 28.697 0.251 114.181 32.822

 

Table 6—2: Competitive Coverage Levels With F CIC Proportional Reinsurance: k=1; p=0.1.

 

 

t E w 4) U n
1.00 E 29.443 0.179 114.020 20.471
0.95 : 29.445 0.179 114.020 20.779
0.90 1 29.452 0.178 114.019 23.297

 

Table 6-3: Competitive Coverage Levels With FCIC Proportional Reinsurance: k=0.5; p=0. 1.

 

 

t E w 4) U n
1.0 i 29.443 0.179 114.020 20.471
0.9 : 29.429 0.180 114.023 22.968
0.8 : 29.430 0.180 114.023 26.320
0.7 I 29.452 0.176 114.019 31.049

 

131

Table 6-4: Competitive Coverage Levels With FCIC Proportional Reinsurance: k=0; p=0. 1.

 

 

t E w 4) U n
1.00 r 29.443 0.179 114.020 20.471
0.90 E 29.376 0.185 114.032 22.230
0.50 E 29.064 0.216 114.095 36.520

 

Table 1 shows that, with proportional reinsurance, the coverage to firmers initially
increases, reaches a maximum at approximately t=0.8 and then starts to decline. Note that at t=0.7
coverage to firmers is still better than t=l although higher coverage is achieved nearer t=0.8. The
coverage to firmers is less at t=0.6 than when there is no proportional reinsurance. In this example,
if reinsurance is made available to competitive firms it will increase coverage to firmers because of
competition among insurance selling firms. As expected, the presence of reinsurance has increased
the number of policies held by each firm, or with reinsurance there are fewer firms in the crop

insurance market.

However, reinsurance does not increase coverage to firmers in all cases. Consider Table 6-
2, where all the parameters are identical except that the correlation between firms is increased
from p=0.05 to p=0.1. This has the effect of rotating the equilibrium curve clockwise in Figure 6-2
above. As expected, the increase in correlation decreases coverage to firmers in the absence of
proportional reinsurance. Now if F CIC proportional reinsurance is made available, the results
show that even at very low levels of reinsurance t=0.95, there is no increase in coverage to firmers.
In such situations, the presence of high correlation between firms results in even proportional

reinsurance having no efi’ect on the coverage to firmers.

132
The situation where proportional reinsurance has no effect can be partially remedied if the
transaction cost of procuring proportional reinsurance is reduced. Table 6-3 shows the case where
k is reduced from k=1 to k=0.5. In this case proportional reinsurance does increase the coverage to
firmers at t values above t=0.7. The optimum coverage is achieved at approximately F09. Thus,

as expected, a reduction in k has the effect of increasing coverage to firmers.

Table 6-4 is presented to show how coverage increases if transaction cost is totally
subsidized by the FCIC such that k=0. In this case the coverage to firmers is constrained by where
the FCIC sets a reinsurance upper limit. For example, if the upper limit on reinsurance is 10
percent (t=90), then the coverage to firmers is at 0.185. Note that this is higher than when there
was no reinsurance and, as expected, the number of policies held by each firm increases. If the
F CIC had set the upper limit to reinsurance 0.5 (t=0.5), then the coverage to firmers would

increase to 0.216 and so on.

6.3 Proportional Reinsurance with Government Subsidies

The previous section showed that proportional reinsurance cannot bring about equilibrium
where none existed previously. The major reason is that proportional reinsurance takes the same
proportion of the premium as the proportion of indemnities it covers. Since the FCIC is risk

neutral, it could settle for less premium than those associated with proportional reinsurance.

This section looks at a situation where the government subsidizes the premiums charged
for the proportional reinsurance. To highlight the results, it is assumed that the amount of subsidies
provided by the government are such that the premiums paid by the firms are actuarially fiir plus

transaction costs. This means that if the FCIC provides l-t percent reinsurance it does not charge

133
l-t percent of the premium collected from firmers. Rather, the FCIC provides the reinsurance at

expected indemnities plus cost, i.e.,

L+c

quoted in per acre terms. If the competitive firm sells 4 percent coverage to firmers and decides to

reinsure t percent of the portfolio it holds, the premium charged by the F CIC would be
n(l - t)¢(L + c) ,

i.e., the premium quoted at per acre terms is w = L + c for n firms with the FCIC covering t4 of

the loss. In such a reinsurance program, the end of the period profits of the competitive firm are
71’s 2 n¢(w — c)- MEI, — n(l - t)¢(I_. + c)— n¢(1— t)k
i=1

The first term in the RHS is the premium collected from 11 firms to provide 4 coverage per acre.
The second term is the indemnities paid after having bought (l-t) percent FCIC reinsurance. The
third term is the actuarially fiir premium paid by the competitive firm to the FCIC to obtain l-t
percent reinsurance. The last term, as before, is the transaction cost ficed by the competitive firm

for reinsurance on its portfolio. The mean and the variance of profits are
as = n¢(w—c— tL-)—n(1—t)¢(L+c)—n¢(l—t)k
0,2,8 = t2¢2n(1+(n — l)p)O,E .

The utility of the firm fiom selling crop insurance is

V(.): n¢(w— c— rZ)- n(1- l)¢(Z' +c)—n¢(1— t)k —%t2¢2n(1+(n —1)p)c:

The F OC to maximize this utility function with respect to coverage is

134
(W—C—tZ)-(1-lXZ+C+k)-—W2¢(I+(n—1)p)0’i =0

which solves to give coverage

 

___ (w-c—tL)—(1—t)(L+c+k)

¢ w’(1+(n-1)p)ai

Substituting in the value for w from expression (6—5), the equilibrium coverage is solved to obtain

20: —c—(1—t)(c+k)
(4+ W20 +01“ 1)P))012. E

 

(6-12) ¢=

The exact coverage level is determined by equilibrium values of n determined via reservation utility
b. As in the previous analysis, equilibrium relationship between b and n is mapped. To do this first

Obtain the upper limit of the utility of the firm in equilibrium as n —) 00 which is

 

LEM: —c—(l-t)(c+k)]
ZW’Wi
Note that here again, the constraint to non-existence of equilibrium (for a given t) is the presence of

correlation between firms. However, unlike the case in proportional reinsurance, the denominator

can be reduced with increases in reinsurance (l-t). At the limit

Lim b—)oo.

t-+0

The upper limit is constrained by how much reinsurance is being made available by the FCIC to
competitive firms. Graphically, this may be represented as an anti-clockwise Shift in the

equilibrium relationship between b and n

135

b K]

1:1

 

 

I)

Figure 6-3: Competitive Equilibrium and Subsidized Proportional Reinsurance.

As the reinsurance level increases, the upper limit moves upward. This implies that if there
was no feasible solution previously (because Of too high reservation utility or too high a degree of
correlation between risks), some additional points would now be in the feasible set with the
appropriate reinsurance level. This is because the upper limit is constrained only by the reinsurance
level l-t (provided that delivery costs and transaction costs are such that there is positive coverage
to start with). Moreover, since for each level of reservation utility 11 drops (see Figure 6-3) as l-t

increases, it means that coverage to finners increases (seen through expression (6-12)).

6.4 Implications for Conventional Reinsurance Schemes

The analysis in this chapter shows that the presence of correlation between firm losses can
be a major reason for the non-existence of competitive insurance for such risks. Proportional
reinsurance cannot bring about equilibrium if none existed before (even if all transaction costs were
subsidized by the FCIC). Therefore F CIC proportional reinsurance will not create a competitive

market for insurance if there was no such market to begin with. A primary reason for this is that

136
proportional reinsurance charges the same premium as the competitive firm for the coverage it
provides to firmers. Ifthe FCIC is able to price reinsurance such that it charges actuarially fiir
premiums, then the results Show that such reinsurance schemes may bring about insurance markets
where none existed before. This is a situation where the risk neutral position of the FCIC is

combined with the higher information and lower cost position of the competitive private sector.

The main drawback of pricing reinsurance at actuarially fiir levels would be the lack of
information for the FCIC to classify firmers into risk classes. In the model of this section it was
easy for the FCIC to calculate actuarially fiir premiums given that there was only one risk type
considered. Ifthere were more than one risk type, then the FCIC would not be able to calculate
actuarially fiir premiums for each risk class because it would not have the information to do so. It
is highly unlikely that the competitive firms would divulge this information to the FCIC if they
could obtain reinsurance for all risk types they hold at the lower reinsurance premiums. Therefore,
the same reasons that make it difficult for the FCIC to provide insurance directly to firmers also
make it difficult to provide reinsurance to competitive firms, such that private competitive
insurance markets emerge. Because the information problems of the FCIC are retained in providing
conventional reinsurance, a non-conventional “reinsurance” scheme is proposed below to

incorporate the risk neutral position of the F CIC with competitive risk averse firms.

6.5 Area-Yield Crop Insurance as “Reinsurance”

One of the crop insurance plans ofi‘ered by the F CIC on a pilot scale to firmers is the
Area-Yield Crop Insurance first suggested by Halcrow in 1949. Miranda has analyzed some of the

benefits of such a scheme when offered directly to firmers. In this crop insurance scheme, both

l 3 7
indemnities and premiums are not based on a producer’s individual yield but rather on the
aggregate yield of the area. In any given year every participant would receive the same indemnity
irrespective of his own yield. This scheme is attractive because the information on area yield is
easily available and there are no delivery costs (except administrative) incurred by the FCIC

because monitoring etc. is at its minimum.

It is of interest to explore if such a program could generate equilibrium where none existed
before if it was ofl‘ered to insurance firms as well as firmers. To see how such a plan would work,

consider the simplest possible case of a firm insuring {6 percent of random loss x1 for prenrium

wl (quoted in per acre terms) and incurring cost c. The end of period profit of the insurance firm is
”A : ¢(Wl —C)—¢xl-
Let mean and the variance of this random loss be 1?, and 0,2 respectively. The mean and the
variance of profits are
fix = “W1 —c—fl)
c3, = ref.
Now let this firm have access to purchasing t percent of an actuarially fair insurance
contract from the FCIC on a random loss x2 with mean 272 and variance 0': . Let the actuarially

fiir premium quoted in per acre term be w2 . If the firm buys into this contract the end of the

period profit of the firm is
”B =¢(W1—C)—¢xl —tW2 +txz-

The mean and the variance of this profit is now

138

E3 = “WI "c-fl)
of, = 420,2 + tzo': — 2t¢p 0120‘:

where p is the degree of correlation between loss xl and loss x2 . Because the insurance that the

firm buys into is actuarially fiir, the mean of end of period profits do not change. What does
change is the variance of profits. Since the preference function of a risk averse firm is increasing in
mean and decreasing in variance of profits, a decrease in the variance of the portfolio would
increase utility of the firm. Or in other words, a firm would choose to buy such an actuarially fiir
contract only if

of, < (If, .
The condition under which this holds is
a520,: +120: — 2t¢p 0,203 < 202

01'

2
to,

>— —.
p 24 a?

The RHS of the above expression is positive, so as long p is positive, there can be found a t where
this inequality holds. In fict the higher the value of p, the more the decrease in variance for a given

tand4.

139

6.5.1 Implications of Area-Yield Insurance as “Reinsurance”

The Area-Yield Insurance Plan may be used by an insurance firm in lieu of conventional
reinsurance schemes to arrive at an equilibrium where none existed before (because of high levels
of systematic risk in the system). Ifa firm selling crop insurance is given the option to buy into the
plan, then the variance of the portfolio of the competitive firm can be reduced, especially in the
agricultural sector where firm losses are highly correlated. The attractiveness of such a plan is that
the FCIC, being risk neutral, can provide Area-Yield insurance at actuarially fiir premiums (such
an scheme would not be forthcoming from the private sector). The information on area yields are
readily available and therefore actuarially fiir premiums are easier to construct. Moreover the
FCIC does not have to deal with any asymmetric information problems in such a scheme (the FCIC
does not need to have information on risk classes that a competitive firm may hold) and delivery

costs are minimal as it does not have major monitoring activities.

The option of buying into the Area-Yield Insurance Plan would have the effect of shifting
the curve in Figure 6-3 counter-clockwise because of the reduction of variance. Since the reduction
of variance is a function of correlation between risks, the greatest reduction takes place when this
correlation is highest between the portfolio of the firm and the area yield. Such a hedging scheme
directly counteracts the breakdown of competitive equilibrium because of high systematic risk in
the market. This scheme also overcomes one of the main drawbacks of the Area-Yield Insurance
Plan when ofi‘ered directly to firmers — that this Plan was not an individual insurance scheme. If
the Area-Yield Plan when bought directly by a firmer did not adequately cover the firmer, the
firmer had to seek out additional insurance to protect himself (and if no such private insurance
scheme existed the firmer had no recourse). By making the Area-Yield plan available to private

insurance firms, this drawback is removed since private firms offer individual insurance schemes.

140
The farmer now only has to deal with one insurance seller and potentially cut down on search and

transaction costs.

Once competitive equilibrium is established, the FCIC can also make proportional
reinsurance (as described in Section 6.2.4) available to competitive firms to increase coverage to
firmers. Again, such a reinsurance scheme does not require information on the part Of the FCIC
and the program would not make expected negative profits. Thus a combination of Area-Yield plan
and Proportional Reinsurance provides a “reinsurance” plan to bring into existence competitive
insurance where none existed before and use the expertise of the competitive sector to deliver crop
insurance to firmers. These plans have minimal expenditure from taxpayers while having the
potential to achieve the objectives of the FCIC to make reinsurance the primary mode of crop

insurance delivery.

7. Summary, Conclusions and Policy Implications

This study was motivated by concerns that the U.S. federal crop insurance program has
not served as the primary provider of federal disaster assistance and has not been self sustaining
(as was envisioned by the 1980 Crop Insurance Act). The key problem areas identified by various
General Accounting Office reports are crop price forecasts, actuarial soundness, and inadequate

risk sharing by the private sector. The focus of this study was on the latter two problems.

This study had three broad objectives. The first was to compare competitive and public
provision of crop insurance given the working hypotheses that (i) competitive firms can deliver
crop insurance at a lower cost than the FCIC; (ii) when farmers belong to different risk classes
then competitive fimls have perfect information regarding each farm’s risk classification but the
F CIC cannot distinguish between risk classes; (iii) competitive firms delivering crop insurance are
risk averse while the F CIC is risk neutral; (iv) competitive firms are assumed to operate at
reservation utility levels in long-run equilibrium; (v) agricultural yield risks are highly correlated
across farms (systematic risk); and (vi) yield risk is the only source of risk to farm income. The
second objective was to study the performance of the current institutional arrangement where the
Federal Crop Insurance Corporation (F CIC) involves private firms in crop insurance delivery
through a reinsurance scheme. The third and final objective was to study alternative reinsurance
schemes that may be provided by the F CIC to competitive firms, and to assess how such schemes

might meet the goals of government policy on crop insurance.

The analytical framework for the basic model was developed in Chapters 3 and 4. In the
case of competitive firms, the equilibrium concept of films operating at reservation utility in the

long-run (first introduced by Appelbaum and Katz) was incorporated and developed. This

141

142
approach difi‘ers from the usual assumptions of risk neutrality and zero profits as conditions in the
long-run equilibrium. In the case of the FCIC, separating equilibrium concepts (first introduced by
Rothscth and Stiglitz) with cross subsidies across risk types to overcome risk classification
problems were also incorporated into the models. To highlight information differences (in
identifying farmer risk classes) between competitive firms and the F CIC, competitive firms were

assumed to have full information while the FCIC was assumed to have no information.

Polar cases of crop insurance delivery by the FCIC only and by competitive firms only
were analyzed in Chapter 3. This analysis abstracted from information difl'erences between the two
institutions to concentrate on the differences solely due to cost of delivery and the degree of risk
aversion. The results of this section show that coverage to farmers is a tradeofl‘ between high cost
of delivery by the F CIC versus an additional loading on insurance premiums by risk averse
competitive firms. The loading on the insurance premium that a competitive firm seeks is directly
proportional to (i) the level of non-diversifiable risk in the portfolio of the films; and (ii) the level
of reservation utility of the competitive firms. This latter factor determines entry and exit of firms
from the crop insurance industry because it is an indicator of alternative investment opportunities

available to the firms.

Multiple risk classes were added in Chapter 4 and the FCIC was assumed to have no
information on farmer risk classes. The results indicate that if the FCIC is to overcome problems
due to this lack of information and operate without making expected losses, then it must ofl‘er
policies such that the program achieves a separating equilibrium. A separating equilibrium is
achieved when there is self selection of policies by the difi'erent risk types. In a separating
equilibrium the premiums to low risk types (identified by low coverage level policy holders) are

taxed while the premiums to high risk types (identified by high coverage level policy holders) are

 

143
subsidized. In other words the low coverage policies subsidize the high coverage policies. The size
of the cross subsidies is a function of the ratio of low to high risk farmers. While such a program
induces cross-subsidies from the low to the high risk farmers, it still provides higher coverage (and
utility) to the low risk types than if there were no such cross-subsidies. The results also show that a
pooling situation (where all risk types buy the same premium and coverage) is an equilibrium only

if there is a very high ratio of low to high risk farmers.

Chapter 4 has major implications for the current FCIC practice of subsidizing low
coverage policies. This practice is detrimental to the FCIC achieving a separating equilibrium when
information on farmer risk classes is lacking. In such a case, if federal subsidies are to be given
then they should either be given proportionally to all risk types (all coverage levels), or only to high
risk types. Subsidizing low risk types hinders separating equilibrium so either the coverage to low
risk types must be fiirther reduced to regain equilibrium or the F CIC must absorb excessive losses

through adverse selection.

Using the fi'arnework developed in Chapters 3 and 4, a stylized model of the current FCIC
reinsurance contract with private firms was analyzed in Chapter 5. The main conclusions drawn
from this analysis are that (i) the FCIC does not use the improved information position of the
private firms; and (ii) the program can save on delivery costs if the private firms are not required to
share in the risks. By requiring the private firms to keep a portion of the policies they sell (through
proportional reinsurance), there must be a transfer of funds from the FCIC to the private firms to
meet the additional premium loading that is required by the risk averse insurance firms to retain
these risks. Two areas are identified in the current FCIC program where this transfer can take

place: (i) through generous administrative cost compensation; and (ii) through generous stop-loss

144
reinsurance agreements. These findings are consistent with various General Accounting Office
allegations concerning the excessive costs of the FCIC program.

Further, for private competitive firms to participate in the FCIC program the minimum
transfer required from the FCIC to these firms increases with increases in the level of systematic
risk in the system. A higher level of transfer to firms does not convert into higher coverage to
farmers. In other words, the model suggests that the F CIC can cut back on this transfer level
without afi‘ecting the coverage to farmers. This provides a potential area to cut excessive F CIC
expenditure on the program. Furthermore, with more than one risk type among farmers, the current
practice of letting the private firms hold less of the more risky policies would aggravate the FCIC

actuarial position as the FCIC holds higher proportions of the more adversely selected policies.

Regarding delivery costs, the results of the model indicate that the current program could
reduce its cost if the FCIC used the private sector to deliver and adjust the policies for a fee, but
did not require them to retain any of the policies. This would ensure that the delivery cost position
of the private firms was used with no additional transfer by the FCIC to private firms. Ifthe
current institutional structure is to be continued, the results suggest that the FCIC should
investigate the possibility of delivering all their policies through private firms but not require the
firms to share in the risk. This suggestion stands counter to the FCIC’S stated objective of
delivering 100 percent of policies through partial reinsurance of private firms under the current
program.

Since the current F CIC reinsurance program does not appear to use the information
position of the private firms, 3 more conventional proportional reinsurance scheme was studied in
Chapter 6. Here the FCIC is responsible for paying a certain percentage of the indemnities in

return for the same percentage of premiums. In such a situation, information on farmer risk classes

145
is not required as the F CIC depends on competitive firms to make all insurance decisions including
premiums, coverage, adjustment of policies, etc. However, one of the problems of such a scheme is
that private multi-peril crop insurance does not exist in the U.S. The results of this model show that
a major reason there are no private competitive multi-peril crop insurance markets is the existence
of high levels of systematic risk in the agricultural sector. Results also indicate that if no such
market exists, then conventional proportional reinsurance would not facilitate new markets without

subsidies from the government.

Chapter 6 also presents a non-conventional “reinsurance” scheme by making a simplified
model of the Area-Yield Plan available to competitive insurance firms (this plan is currently only
available to farmers). The analysis shows that such a scheme has the potential to encourage
emergence of private markets for multi-peril crop insurance. This plan offers a hedge against non-
diversifiable risk in the portfolio of the private firms. Once a competitive market for multi-peril
crop insurance emerges, conventional proportional reinsurance schemes of the F CIC may be used
to increase coverage to farmers. This scheme has the potential to provide individualized insurance
to farmers by private firms if the scheme is offered to these firms rather than the current practice of

offering it only to farmers.

This study has a number of limitations and therefore the results must be taken as
preliminary. Firstly, the study portrayed the FCIC and the competitive private firms on polar ends
of the information scale to classify farmers. This was done to highlight the differences in
information positions of the two institutions and make the analyses more tractable. A study that
allows for more complicated informational assumptions may allow us to see a comprehensive
tradeofi‘ among information availability, delivery costs and degree of risk aversion on coverage to

farmers. Secondly, for ease of analysis, the study used a linear mean-variance model. A study that

146
uses a more generalized mean-variance or expected utility framework would also account for

income efi‘ects in addition to substitution effects of the linear model.

Future research in this area could incorporate moral hazard issues and uncertain crop
prices into the model. This study has totally abstracted from such issues to focus on concepts of
systematic risks and firmer classification problems. Also, there was only one source of risk for the
firmer and as such the firmer had no diversification opportunities. General Accounting Office
studies have identified crop diversification to be an important alternative to buying crop insurance
for firmers in certain areas and therefore should be considered in crop insurance designs for such
areas. Further, this study ignored other government firm income stabilization programs that exist
simultaneously with federal crop insurance. Information on interaction between these programs and
crop insurance program would help policy makers coordinate the programs better in their attempts
to stabilize firm incomes.

Despite the limitations of the study and suggestions for further research, this study has
taken steps to model insurance markets with systematic risk and risk averse competitive firms. The
equilibrium concepts and results provide insights into the role of government and competitive firms
in such risk markets. It is hoped that the study will lead to more informed government policies as

the search for a better institutional design for U.S. crop insurance continues.

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