Three essays on risk and uncertainty in agriculture

Transcription

1 Retrospetive Theses and Dissertations 2007 Three essays on risk and unertainty in agriulture Niholas David Paulson Iowa State University Follow this and additional works at: Part of the Agriultural and Resoure Eonomis Commons, and the Agriultural Eonomis Commons Reommended Citation Paulson, Niholas David, "Three essays on risk and unertainty in agriulture" (2007). Retrospetive Theses and Dissertations This Dissertation is brought to you for free and open aess by Iowa State University Digital Repository. It has been aepted for inlusion in Retrospetive Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please ontat

2 Three essays on risk and unertainty in agriulture by Niholas David Paulson A dissertation submitted to the graduate faulty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Eonomis Program of Study Committee: Brue A. Babok, Co-major Professor Dermot J. Hayes, Co-major Professor David A. Hennessy Sergio H. Lene Aliia L. Carriquiry Chad E. Hart Iowa State University Ames, Iowa 2007 Copyright Niholas David Paulson, All rights reserved.

3 UMI Number: UMI Miroform Copyright 2007 by ProQuest Information and Learning Company. All rights reserved. This miroform edition is proteted against unauthorized opying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI

4 ii ABSTRACT ACKNOWLEDGEMENTS TABLE OF CONTENTS CHAPTER 1. GENERAL INTRODUCTION 1 Introdution 1 Organization of the Dissertation 4 CHAPTER 2. A COMPARISON OF ACREAGE AND BUSHEL CONTRACTS IN SPECIALTY GRAIN MARKETS 7 Abstrat 7 Introdution 8 Literature Review 11 Survey Data 14 Contrat Model 16 Overview 16 The Proessor 18 The Produers 20 Contrat Supply 22 Areage Contrats 23 Bushel Contrats 26 Analyti Results 32 Numerial Example 36 Conlusions 45 Referenes 48 Appendix 51 CHAPTER 3. READDRESSING THE FERTILIZER PROBLEM: RECONCILING THE PARADOX 54 Abstrat 54 Introdution 55 Literature Review 60 Model 64 iv vi

5 iii Data and Estimation 69 Estimation Results 73 Optimal Fertilizer Rates 76 Farmer Survey Results 81 Conlusions 85 Referenes 88 Appendix 91 CHAPTER 4. A BAYESIAN AND SPATIAL APPROACH TO WEATHER DERIVATIVES: A FRAMEWORK FOR DEVELOPING REGIONS 93 Abstrat 93 Introdution 94 Literature Review 96 Demand for Weather Derivatives in Agriulture 96 Weather Based Insurane 98 Rainfall Interpolation 99 Markov Chain Monte Carlo Methods 100 The Rainfall Model 102 Implementation Example 104 Data 106 Poliy Struture 108 Results 110 Kriging 110 Inverse Distane Weighting 113 Insurane Rates 114 Historial Analysis 116 Conlusions 118 Referenes 119 CHAPTER 5. GENERAL CONCLUSIONS 122 Conlusions 122

6 iv ABSTRACT The general theme of this dissertation is risk and unertainty in agriulture, with eah hapter addressing a speifi topi related to agriultural risk and unertainty. Chapter 2 examines the effets of prodution unertainty on the types of ontrat strutures used in speialty grain markets. A theoretial model of a ontratual relationship between a monopsonisti proessor and risk-neutral produers is presented. Two ommon ontrat strutures, and their resulting effets on the sharing of prodution risk between buyer and seller, are ompared. The spatial struture of yields and farm-level yield volatility are shown to have signifiant impats on the proessor s preferred hoie of ontrat struture and expeted profits of both the proessor and farmers in the resulting equilibrium. Chapter 3 provides a ritial look at a lassi definition regarding the relationship between input use and risk, and attempts to reonile an apparent paradox in the prodution literature. Experimental orn yield response data is used to estimate a stohasti prodution relationship between applied fertilizer, soil nutrient availability, and orn output. Optimal fertilizer appliation rates for risk-averse and risk-neutral produers are found using numerial methods. In addition to the empirial analysis, primary data olleted through a farmer survey instrument, designed to eliit information from farmers regarding their risk attitudes and subjetive beliefs regarding the relationship between risk and fertilizer use, is presented and ompared with the results of the empirial analysis. Chapter 4 turns to the opportunities for managing weather risk using weather derivative markets. Developing regions are areas in whih weather based risk management tools show signifiant potential. However, the suess and long-term viability of insurane programs depends heavily on the availability of aurate and reliable historial data. The lak of this type of historial data for developing

7 v regions is one of the largest obstales to insurane program development in these regions. A framework whih utilizes statistial methods to estimate unbiased rainfall histories from sparse data is developed. To validate the methodology s usefulness, a drought insurane example is presented using a rih data set of historial rainfall at weather stations aross the state of Iowa.

8 vi ACKNOWLEDGEMENTS First, I would like to thank my parents Dave and Paulette, my sister Betsy, Grandma Min and Grandpa Paul, Grandma Beat and Grandpa Bob (who reently passed but will be wathing when I graduate this spring), and the rest of my family and friends for their love and support. None of my ahievements would have been possible without them. My mother and father serve as models of the type of person I ontinually aspire to be. While they may not have always understood exatly what I was doing during my undergraduate and graduate studies (nor why I was doing it for the better part of a deade), they never questioned what I was trying to aomplish and fully realized its importane. Seond, I thank my o-major professors, Brue Babok and Dermot Hayes, for the time, patiene, and energy they so willingly provided to me while writing this dissertation. Their helpful omments and advie greatly ontributed to my work. I would also like to thank my ommittee members David Hennessy, Sergio Lene, Aliia Carriquiry, and Chad Hart for their support in helping me omplete this dissertation. Finally, I thank Emily Kinser. Although she has been over 1,000 miles away while I have been finishing my PhD, her love and support has been mathed only by that of my family. She has demanded exellene without being demanding, and has served as a soure of inspiration over the past four years. Moreover, she willingly listened when I needed to vent frustrations and always helped to rebuild my onfidene whenever I began to doubt myself. I love you and annot thank you enough; nor an I put into words exatly how muh you truly mean to me.

9 1 CHAPTER 1. GENERAL INTRODUCTION Introdution It would be diffiult to imagine an industry where risk and unertainty are more important than in agriulture. Throughout the supply hain, agents are faed with soures of unertainty whih may be exogenous, endogenous, or both. At the farm level, livestok and rop produers are fored to make input hoies and asset alloation deisions in a omplex environment of volatile pries, perishable outputs, prodution lags, seasonality effets, and inreasing onentration at all levels of the supply hain. Additionally, produers are onstantly at the mery of extreme weather variability. Similarly, downstream partiipants must effiiently oordinate with produers to ensure a steady supply of inputs for further proessing and eventual sale to end users. Therefore, the development of arefully planned and well defined risk management strategies is ruial to the ontinued suess of all agents in the agriulture industry. As agriulture ontinues to evolve, new soures of risk ontinue to arise presenting new hallenges for both buyers and sellers of agriultural ommodities. The use of ontrats as a oordination mehanism in agriulture is an area that has seen onsiderable reent growth. Inreasing real inomes and onsumers subsequent demand for more highly differentiated produts is often ited as one of the main fores in this area. While ontrats an provide a oordination mehanism to improve effiieny in emerging marketplaes, they also introdue new hallenges. Contrats must be strutured to effiiently alloate the added value, risk, and deision making rights between the buyer and seller. Additionally, the inentives of buyers and sellers are often misaligned, adding yet another layer of omplexity to ontrat design. Buyers would like to eliminate prodution

10 2 and prie risk while prouring speialty produts at the lowest possible prie. On the other hand, the goal of produers is to eliminate prodution and prie risk while trying to sell their prodution at the highest possible prie. For a long-run equilibrium to emerge, ontrat strutures must evolve to reflet an effiient tradeoff between the added-value reated by speialty produts and the levels of prie and prodution risk borne by buyers and sellers. Thus far, the aademi researh has foused more on ontrating within the livestok industry. The o-existene of spot and ontrat markets, and the resulting effet on the fores of ompetition, has also garnered a signifiant amount of attention. Researh on ontrat strutures in rop prodution has been given muh less attention despite the inreasing use of prodution ontrats in the growing markets for speialty rops. Further researh on ontrat strutures would provide additional insight on the impliations of ontrat struture on the sharing of prie and prodution risk between buyers and sellers of agriultural ommodities. This topi is the fous of one of the following hapters. Beyond marketing alternatives, one of the most fundamental risk management strategies available to produers is that of the hoie of input mixes. A produer s optimal input mix will depend on many fators, inluding his risk preferenes and attitudes. Produers want to avoid exessive use of inputs to ontrol prodution osts while also ensuring that the input mix does not limit prodution when pries and onditions in the prodution environment are suh that profits an be made. The relationship between risk and input use has been extensively studied in the aademi literature. In the speifi ase of fertilizer use, there exists somewhat of a paradox. Empirial evidene implies that the variability of prodution is inreasing in the amount of fertilizer applied by the produer. Therefore risk-averse agents are predited to use lower amounts of fertilizer than risk-neutral

11 3 farmers. However, there is also a signifiant amount of empirial evidene that fertilizers are over-applied by farmers as an at of self-protetion. A lear reoniliation of the seemingly opposing viewpoints would provide a ontribution to the prodution literature. A hapter of this dissertation is devoted to this very topi. While many soures of risk and unertainty are at least partially endogenous, weather variability is most ertainly out of the ontrol of partiipants in the agriulture industry. Mother Nature an turn an otherwise bumper rop into a total failure with one hard rain or hailstorm regardless of the input mix hosen by the produer. Similarly, exessively hot or old weather an have dramatially negative impats on livestok produtivity in even the most effiient operations. Reent growth in the weather derivative industry has reated new opportunities for weather risk management in agriulture. Options and insurane ontrats based on weather an provide inome transfers when weather onditions adversely affet farm profits. While the demand for insurane based on weather events may be limited in areas with well developed rop and livestok insurane programs, suh as the United States and Canada, a large body of literature is devoted to exploring their potential for developing regions. However, the lak of reliable historial data needed for proper program development is a major obstale to weather derivative growth in developing regions. Despite the hallenges, the potential for weather based produts to enhane inomes and provide inreased stability in developing regions warrants further researh. In this dissertation, I provide a potential solution to the problem of unavailable historial weather data for developing areas and illustrate an implementation of the framework.

12 4 Organization of the Dissertation This dissertation is organized into five hapters. The urrent hapter inludes a general introdution to the hapters that follow and, in this setion, provides an outline for the doument s organization. While the general theme of this dissertation is risk and unertainty in agriulture, eah hapter is meant to stand alone by addressing a speifi omponent of the issues disussed in the previous setion. I begin in Chapter 2 by taking a look at the effets of prodution unertainty on the types of ontrat strutures used in speialty grain markets. While ontrats themselves an redue risk by mathing buyers and sellers in speialized markets, the struture of a ontrat determines how value, risk, and deision-making rights are alloated between the buyer and seller. A theoretial model of a ontratual relationship between a monopsonisti proessor and risk-neutral produers is presented. Two ommon ontrat strutures are ompared: areage ontrats and bushel ontrats. Areage ontrats plae a larger share of prodution risk on the buyer, while bushel ontrats shift a greater portion of this risk to the produer. The effets of the spatial orrelation of yields aross multiple produers, and the volatility of yields at the farm level are shown to have a signifiant impat on the proessor s preferred hoie of ontrat struture and expeted profits in the resulting equilibrium. Chapter 3 provides a ritial look at a lassi definition of the relationship between input use and risk and attempts to reonile an apparent paradox in the prodution literature. Speifially, I examine how input deisions may affet the level of prodution variability a produer faes (risk endogeneity). Experimental orn yield response data is used to estimate a stohasti prodution relationship between applied fertilizer (the farmer s hoie variable), soil nutrient availability (a stohasti funtion of applied fertilizer), and orn output (a

13 5 stohasti funtion of available soil nutrients). Optimal fertilizer appliation rates for riskaverse and risk-neutral produers are found using numerial methods. The optimal rates are ompared for agents with differing risk preferenes. In addition to the empirial analysis, primary data olleted through a farmer survey instrument is presented. The survey was designed to eliit information from farmers regarding their risk attitudes and their subjetive beliefs regarding the relationship between risk and nitrogen fertilizer use. The survey results are ompared and ontrasted with those obtained from the empirial analysis. Chapter 4 turns to the hallenges reated by weather variability in agriulture and the opportunities that are arising for managing weather risk. Speifially, the relatively new but inreasingly popular market for weather derivatives is examined. While a wide variety of well developed insurane programs are available in the United States and Canada, agriultural insurane programs are not as widely available in other areas. Developing regions are areas in whih weather based risk management tools show signifiant potential for further development. However, the suess and long-term viability of insurane programs depends heavily on the availability of aurate and reliable historial data. The lak of this type of historial data for developing regions is one of the largest obstales to insurane program development in these regions. The fous of Chapter 4 is on the development of a framework whih utilizes a statistial method to estimate unbiased rainfall histories from sparse data. Speifially, a Bayesian spatial kriging model is used in onjuntion with Markov Chain Monte Carlo methods to estimate historial rainfall for outof-sample loations. These estimated rainfall histories an be used in plae of atual historial data to develop an atuarially sound insurane program based on weather. To

14 6 validate the methodology s usefulness, a drought insurane example is presented using a rih data set of historial rainfall at weather stations aross the state of Iowa. Finally in Chapter 5, I provide a summary and some general onlusions.

15 7 CHAPTER 2. A COMPARISON OF ACREAGE AND BUSHEL CONTRACTS IN SPECIALTY GRAIN MARKETS A paper to be submitted to the Amerian Journal of Agriultural Eonomis Niholas D. Paulson Abstrat The inrease in vertial integration in agriulture has been motivated by many fators inluding the evolving demand of onsumers as well as fators speifi to agriultural markets (i.e. prodution and prie unertainty and farm poliy). The literature on agriultural ontrats has foused more on ontrating in the livestok setor relative to rop prodution under ontrat, most likely due to the fat that ontrating in livestok prodution has been historially more prevalent. However, rop prodution under ontrat has also realized extensive growth, espeially in the markets for rops with speialty traits. This paper provides a theoretial model of a ontrating relationship between a risk-neutral monopsonisti proessor of a speialty rop and risk-neutral produers. The proessor indues farmers to aept prodution ontrats, based on areage or total bushels, to grow the speialty rop by offering a premium above the ommodity prie for ontrated prodution. It is ommonly assumed in the literature that areage ontrats will be the preferred struture due to produers being relatively more risk average than proessors. However, this paper presents a market environment in whih the opposite result is found. It is shown that a bushel ontrat exists whih Pareto dominates the optimal areage ontrat, although that bushel ontrat may not be the proessor s optimal bushel ontrat, and that both the

16 8 proessor and produers will be able to ahieve higher expeted profits by using a bushel ontrat struture. While expliit analyti solutions for the model do not exist, a numerial example is provided to illustrate effets of farm level yield volatility and the spatial orrelation of farm yields on expeted profits, premium levels, and farmer partiipation for the areage and bushel ontrat equilibriums. Introdution The level of vertial integration in agriultural markets has seen onsiderable growth over the last deade. Authors have outlined many motivations for this phenomenon inluding supply-hain organization (Tsoulouhas and Vukina 1999), more disriminating onsumers (Barkema 1993), more effiient relationships between buyers and sellers (Drabenstott 1993), information asymmetries (Hennessy 1996), quality ontrol (Hueth and Ligon 1999; Hennessy and Lawrene 1999), prourement onsiderations speifi to the dynamis of agriultural deision making (NASS 2003; Sexton and Zhang 1996), delining ommodity pries (Fulton, Prithett, and Pederson 2003), and the deoupling of farm support outlined in the 1996 farm bill (Coaldrake and Sonka 1993). While a large proportion of grains and oilseeds are produed under marketing ontrats in the United States (NASS 2003), prodution ontrats have been relatively more prevalent in livestok (Goodhue 2000; Lawrene et al. 1997; Johnson and Foster 1994) and speialty grains markets (Ginder et al. 2000; Good, Bender, and Hill 2000; Fulton, Prithett, and Pederson 2003; Sykuta and Parell 2003). Speialty rop markets are generally smaller and more entralized than those for ommodity rops. Risk management options for speialty grains are also imperfet as rop insurane, and futures and options markets are

17 9 generally only available for ommodity rops. These harateristis make prodution and prourement in spot markets riskier for farmers and proessors, respetively. The risk assoiated with spot market prodution and prourement is one of the main reasons given for produers entering into speialty grain ontrats in Indiana (Fulton, Prithett, and Pederson 2003). Moreover, speialty rops are assoiated with higher prodution osts than ommodity rops. Higher osts are attributed to fators suh as inreased labor intensity, storage issues stemming from segregation and identity preservation requirements, and speifi or additional input requirements and field operations (Fulton, Prithett, and Pederson 2003; Ginder et al. 2000). Prodution of speialty rops may also require the use of speifi assets (Lajili et al. 1997; Sporleder 1992), whih may be assoiated with a higher level of equity finaning relative to less speifi assets (Williamson 1996). Thus, proessors must properly struture ontrat terms to reflet the additional osts and risks assoiated with the prodution of speialty rops to indue farmer aeptane of prodution ontrats. While onsiderable attention has been given to the effets of ontrat strutures in the livestok industry, less attention has been giving to prodution ontrats for rop prodution. This paper presents a theoretial model of a monopsonisti proessor who proures rop prodution under ontrat from a set of produers. The proessor sets the struture and terms of the ontrat as well as the number of produers to whom the ontrat is offered to maximize expeted profits subjet to a apaity onstraint and produers deision rules. Risk-neutral produers who are offered the ontrat then deide to either aept the ontrat or produe a ommodity rop for sale on the ommodity market to maximize expeted profits. Produer heterogeneity is introdued with respet to prodution osts for the

18 10 speialty rop. While the situation is desribed as one where the proessor ontrats with produers to grow a speialty rop, the ontrating relationship in the model ould easily be generalized to other markets. The ontributions of this paper are two-fold. First, the model ompares the equilibrium outomes when ontrats are based on areage (areage ontrats) to when ontrats are based on a speified prodution level (bushel ontrats). Authors generally assume that areage (bushel) ontrats will be preferred by produers (proessors), beause they shift prodution risk from the produer (proessor) to the proessor (produer) (Sykuta and Parell 2003; Lajili et al. 1997). However, the set of assumptions whih define the market environment in this analysis lead to a different result. It is shown that under ertain onditions, there exists a bushel ontrat that Pareto dominates the optimal areage ontrat. Expeted profits for both the proessor and produers may be greater when the bushel ontrat struture is implemented rather than an areage ontrat. However, that bushel ontrat may not be optimal in that it is not the proessor s expeted profit maximizing bushel ontrat. Seond, the model reognizes that the proessor s total prourement in any period is the sum of random yield realizations on all ontrated ares. The spatial orrelation struture of yields will affet the proessor s ability (or inability) to pool farm level prodution risk over a large number of ontrated produers. This is expeted to have an impat on the proessor s preferred hoie of ontrat struture (areage vs. bushel ontrat). A numerial alibration to the model provides some insight into how the spatial orrelation of produer yields may affet the proessor s hoie of ontrat type and terms. Equilibrium outomes with respet to expeted proessor and produer profits are ompared for a range of

19 11 parameterizations, finding that the proessor will, in general, be able to ahieve greater expeted profits using bushel ontrats. Expeted produer profits are also shown to be greater under bushel ontrats in all senarios analyzed. The assumption of a apaity onstraint, produer risk-neutrality, and the effets of an ex post spot market for the speialty rop under bushel ontrats are the main drivers of this result. The relative advantage of bushel ontrats to areage ontrats is shown to inrease as the level of orrelation between farm level yields is inreased. Intuitively, as yield risk beomes inreasingly systemi in nature, the proessor s benefits of plaing a larger share of prodution risk on the produers inrease. This result onfirms that the hoie of ontrat struture will hinge heavily on the poolability of prodution risk for speialty rops. The next two setions provide a brief review of the literature on prodution ontrats in agriulture and a summary of reent survey results from speialty grains markets in the Midwest, respetively. Then the ontrating model is presented followed by a setion providing some analytial results. A numerial alibration to the theoretial model is then provided with results for a range of parameterizations. The final setion onludes and disusses possible extensions to the model and diretions for further researh. Literature Review Considerable attention has been given to the effets of ontrat struture on the sharing of value and risk in agriultural markets. Goodhue (2000) used an ageny theory approah to model prodution ontrats in the broiler industry, finding that ontrats outlining relative ompensation shemes and strit input ontrol by the proessor were optimal responses to grower heterogeneity and risk aversion. Weleshuk and Kerr (1995)

20 12 used a transations ost approah to examine ontrats for speialty rops in Canada, finding market power on the part of buyers led to redued ompetition with respet to the ompensation terms of ontrats. Goodhue and Hoffman (2006) disuss the tehnial aspets of ontrats, referred to as boilerplate in the ontrating industry, that are often ignored in theoretial studies but play large roles in atual settings with regards to ontrat enforement and liability issues. Empirial approahes inlude those of Purell and Hudson (2003) onerning vertial allianes in the beef industry, and Fraser s (2005) examination of ontrats in the Australian wine grape industry. Purell and Hudson (2003) use a simulation model of attle produers, ooperative feedlots, and beef pakers to examine the effets of different ompensation strutures on risk sharing within a vertial beef alliane. Fraser (2005) applies regression analysis to atual ontrat data to identify the effets of grower and regional harateristis on ontrat strutures. Fraser s results are onsistent with Sykuta and Cook s (2001) assertion that produer harateristis have a limited effet in ontrat design, implying buyer harateristis will more often determine the speifi ontrat terms. Other authors have used experimental methods to eliit produer preferenes for marketing ontrat attributes in both livestok (Roe, Sporleder, and Belleville 2004) and rop (Lajili et al. 1997) prodution. Lajili et al. (1997), using survey data, related the levels of asset speifiity, prodution unertainty, and produer risk aversion to the preferred levels of vertial integration with respet to the sharing of prodution risk and osts. Roe, Sporleder, and Belleville (2004) used an experimental survey design for marketing ontrats in the hog industry, finding that produers strongly preferred ontrats offered by ooperative firms, validating a hypothesis by Sykuta and Cook (2001) regarding institutional onsiderations in

21 13 agriultural ontrating. Wu and Roe (2007) is another example of an experimental approah whih illustrates the importane of third-party ontrat enforement in ontratual relationships when there is buyer onentration (market power). Considerable attention has also been paid to the interation between and the oexistene of ontrat and spot markets in agriulture. Xia and Sexton (2004) present a model of attle prodution where buyer onentration leads to redued ex post spot market (prie) ompetition when ontrat premiums are based on ash market pries. Zhang and Sexton (2000) use a spatial model with high transportation osts to show how proessors an use exlusive ontrats to reate aptive supplies to gain monopsony power and redue prie ompetition on the spot market. However, Carriquiry and Babok s (2004) model of oligopsony in speialty grain markets shows that if the ontrat premium is based on a fixed ash market prie there will be inreased ompetition on the ex post spot market. The reent work by Wang and Jaenike (2006) uses a prinipal-agent framework within a market equilibrium model to show that the introdution of ontrat markets may ause spot pries and spot prie volatility to rise or fall depending on the relative sizes of the ontrat and spot markets, and the ompensation strutures outlined in the ontrats. While there have been a number of exellent studies into the effets of ontrat strutures on market equilibriums, authors have mainly foused on livestok markets and the ompensation shemes in agriultural ontrats. This is most likely due to the importane of prie unertainty relative to prodution unertainty in the livestok setor. However, prodution unertainty plays a major role in rop prodution. Therefore, one would expet prodution unertainty to play a ruial role in shaping the ontrat strutures in speialty grains markets. This is the main fous of this paper.

22 14 Survey Data In addition to the theoretial and empirial studies on agriultural prodution ontrats, there also exists a olletion of reent survey data on speialty grain markets in the Midwest. The survey data reported in Good, Bender, and Hill (2000) for Illinois speialty grain handlers shows that the vast majority of speialty orn (waxy, high oil, white, yellow food grade) and soybeans (tofu, non-gmo) in Illinois are produed and proured under ontratual arrangements. Good, Bender, and Hill (2000) report that many firms at as intermediaries between produers and proessors by forming ontratual relations with both parties. Moreover, the intermediate firms, omprised mainly of ountry elevators, ontrat with produers based on areage but ontrat with proessors based on bushels. This implies that the intermediate firms may be able to pool prodution risk aross produers. Sykuta and Parell (2003) provide a survey on ontrat strutures offered by DuPont in their speialty soybean programs. While the ontrat premiums are generally based on the total bushels delivered, the atual ontrats are based on areage, shifting a portion of the prodution risk to the proessor. Areage ontrats differ from bushel ontrats in that the produer does not have to make up yield shortages in poor years or sell surpluses at a potentially lower prie on the spot market in good years. The prodution risk shifted to buyers under areage ontrats reates variability with respet to the amount of premium paid out to ontrated produers, as well as serious impliations for apaity onsiderations for proessors. Sykuta and Parell (2003) note that buyers may be able to redue prodution risk by pooling aross large areas (attempting to eliminate some systemi yield risk), or offer ontrats in seletive areas or to speifi produers with low historial yield variability.

23 15 The survey data from over 2,800 produers in Ginder et al. (2000) reports that the majority (60%) of speialty orn ontrats in Iowa are strutured to pay a premium over a referene prie, referred to as market prie plus a premium. The referene prie an either be a fixed prie or pegged to a speified ash market prie, suh as a loal spot market or a futures prie. The surveys onduted by Good, Bender, and Hill (2000) and Fulton, Prithett, and Pederson (2003) imply the same type of ompensation strutures for speialty grain ontrats in Illinois and Indiana, respetively. Sykuta and Parell s (2003) survey also reported the use of a premium above a market prie in DuPont s speialty soybean programs. The premiums are motivated by higher prodution osts, segregation and identity preservation, possible yield drags, and doumentation and ertifiations osts (Ginder et al. 2000). The magnitude of these additional osts will depend on the harateristis of the speialty rop as well as harateristis of the individual farm operation. Storage apaity, farm size, finaning onstraints, and the management ability of the produer are just a few examples of heterogeneity with respet to speialty rop prodution osts aross farming operations. The survey onduted by Fulton, Prithett, and Pederson (2003) reports that higher variable prodution osts, higher investment osts, and managerial time requirements were three of the top four reasons why Indiana produers hose not to grow any speialty rops. The number one reason for not produing speialty rops was the lak of a market for sale of the speialty rop. For surveyed produers who did produe speialty rops under ontrat, revenue enhanement and market aess were the top two reasons reported for ontrating.

24 16 Contrat Model The terms speialty and ommodity are used to differentiate the ontrated rop and the alternative ash market rop in the model. Examples would inlude high oil, white, or waxy orn (speialty) and #2 yellow orn (ommodity). Soybean prodution would be another example in ommodity form, with the speialty rop being DuPont s STS or tofu soybeans. Moreover, the non-gmo or organi forms of any general ommodity rop would be another example of what is referred to as a speialty rop. For simpliity, bushels will be used as the prodution unit measure for the speialty and ommodity rop. However, the model ould easily be generalized to other types of rops whose prodution is measured in other units (i.e. tons or lbs.). In the setions that follow, subsripts denote differentiation unless otherwise noted. Overview The modeled ontrating senario is a familiar one in agriulture. Consider a profitmaximizing (risk-neutral) monopsonisti proessor who proures speialty rop prodution from a group of produers by way of a prodution ontrat 1. In stage I, the proessor offers a ontrat for the speialty rop to produers. The produers then deide to either aept the ontrat to produe the speialty rop, or to rejet the ontrat and produe the ommodity rop. Produers who aept the ontrat then move into stage II where prodution of the speialty rop takes plae per speifi management guidelines outlined in the ontrat 2. If the ontrat guidelines are followed the proessor verifies the speialty rop for eah ontrated farmer through ostly monitoring, aknowledging that his prodution arries the 1 The phrase prodution ontrat is used as a general term enompassing the more speifi areage and bushel ontrat types. 2 The model abstrats from the speifis of the ontrat regarding management requirements. These ould inlude speifi input requirements as well as guidelines for the timing of field operations.

25 17 speialty trait. Thus, there is no quality or trait unertainty in the model. Furthermore, it is assumed that the proessor will not purhase any amount of unverified speialty rop prodution. This assumption preludes the possibility of produers hoosing to grow the speialty rop speulatively (i.e. there is no ex ante spot market for the speialty rop) 3. Moreover, given the management guidelines outlined in the ontrat and the intensive monitoring done by the proessor, produers do not have the ability to shirk and attempt to pass off the ommodity rop as having the speialty trait. In stage III the farmers harvest the speialty rop and yields are realized on eah farm. Eah farmer s atual yield is private information (informational asymmetry). In stage IV ex post spot market transations for the speialty rop may take plae between produers, depending on the type of ontrat offered (areage vs. bushel) and yield realizations (supply and demand). There will not be an ex post spot market under areage ontrats beause produers deliver all speialty bushels produed to the proessor. However, if the proessor offers a bushel ontrat an ex post spot market for the speialty rop may exist as ontrated farmers realizing low yields, in an effort to fulfill their bushel ontrats, may purhase bushels from ontrated farmers who realized high yields. The proessor may also enter the ex post spot market for the speialty rop to purhase additional speialty rop prodution up to his apaity onstraint. The prevailing prie on the speialty rop spot market will depend 3 Note that this assumption is not ritial under areage ontrats beause the monopsonist would never offer a prie above the salvage value for any spot market prodution in years when the aggregate prodution on ontrated ares is below his operating apaity. This implies no produer would ever hoose to produe the speialty rop without a ontrat. However, under bushel ontrats there exists a positive probability that a spot market for the speialty rop will exist ex post, and that the prie on that spot market may be higher than the premium offered by the proessor in the ontrat beause of exess demand. The assumption that the proessor will only purhase speialty rop prodution from verified ares is ritial to the results of the paper, as it prevents any produer from hoosing to speulatively produe the speialty rop for the spot market (referred to in the industry as wildatting ).

26 18 on the aggregate yield over ontrated ares. This will be explained further in the following setions. In stage V the ontrats are settled by the farmers delivering ontrated prodution to the proessor and reeiving the ompensation outlined in the ontrat. Finally in stage VI, proessing takes plae and the proessor sells output to a downstream user earning a proessing return on eah bushel proessed up to the plant apaity. The proessor sells any exess (above his apaity onstraint) ontrated speialty rop at its salvage value on the ommodity market. Figure 1 summarizes the timing of all deisions and ations in the modeled ontrating senario. Stage I: Contrat Offering Stage II: Prodution Stage III: Harvest Stage IV: Speialty Crop Spot Market Stage V: Contrat Settlement Stage VI: Proessing Proessor offers ontrat premium p and underage penalty p u to N produers N produers aept the ontrat to grow the speialty rop Farmers produe the speialty rop per the ontrat terms, proessor verifies speialty rop for eah ontrated produer Farmers harvest the speialty rop and realize their yields Farmers buy/sell bushels on the speialty rop spot market at the spot prie p s Farmers deliver bushels produed and purhased on the spot market to the proessor, pay any underage penalties, and reeive premium payment Proessor proesses ontrated prodution for sale upstream up to his apaity Q, and dumps any exess at the salvage value Figure 1: Timeline of the ontrat proess The Proessor The proessor earns a net proessing return R for eah unit of the speialty rop proessed Y, up to an exogenous apaity onstraint Q 4. Thus, proessing is modeled as a fixed proportions tehnology where eah unit proessed results in one unit of output (by appropriately speifying the unit of measure for the proessed rop) (Carriquiry and Babok 2004). 4 The problem outlined in this paper an be thought of as a short-term optimization problem for the proessor. A long-term optimization problem would inlude the hoie of the optimal plant apaity. Disussions with industry representatives in Iowa justified the assumption of an exogenous apaity onstraint in a given period.

27 19 The prie reeived by the proessor (R) is assumed to be net of any variable prodution and operating osts for the proessing plant. The apaity onstraint Q ould be thought of as a physial onstraint on the tehnology, or due to ontratual arrangements between the proessor and downstream buyers (Good, Bender, and Hill 2000). Eah unit of the speialty rop proured above apaity is dumped on the ommodity market at its salvage value 5 whih is assumed to equal the prie on the ommodity market r less a perentage handling harge δ 6. The handling harge inludes storage and transportation osts inurred by the proessor on ontrated speialty rop prodution that annot be proessed due to his apaity onstraint. (1.1) RY ( ; Q ) = RY for Y Q RQ+ ( r δ )( Y Q) for Y > Q where Y = y for all ontrated farmers i i i δ [0, r] To indue prodution of the speialty rop the proessor offers a ontrat to produers either based on areage or on total bushels to be delivered. The areage ontrat is defined by a premium level p, above a referene prie, that the proessor will pay the produer for eah bushel grown on ontrated ares. The bushel ontrat is defined by a premium p above the referene prie, the size of the ontrat in bushels y B, and an underage penalty p u that the farmer must pay to the proessor if he annot fulfill his ontrat in bushels of the speialty rop (beause his yield fell below the ontrated amount 5 Note that this assumption may not hold for ertain speialty varieties. As an example, waxy orn annot be sold on the #2 yellow orn market. 6 Net proessing returns R are assumed to be net of the handling harge δ for proessed bushels below the apaity onstraint.

28 20 and there was not any additional prodution from other ontrated produers available on the spot market). In addition to hoosing the terms (premium and state ontingent penalty) and struture (areage or bushel) of the ontrat offered, the proessor hooses the size of the ontrating region, defined by the number of produers N within the region, where he will offer the ontrat 7. In the ase of either ontrat type, the ontrat also outlines speifi guidelines for the management praties that the produer must follow. These guidelines may inlude requirements for input use (i.e. a speifi seed, fertilizers, or hemials), storage and segregation, and timing of any prodution tasks. The proessor enfores the management terms of the ontrat through ostly monitoring. The osts of monitoring m( N ) are assumed to be an inreasing onvex funtion of the size of the ontrating region, with m, m > 0. It is assumed that as ontrated farmers beome spread out over a larger region it beomes inreasingly diffiult (ostly) for the proessor to monitor their management ations. The Produers Upon reeiving the ontrat offer, the set of N profit maximizing (risk-neutral) produers within the ontrating region hoose to either aept the ontrat and produe the speialty rop, or produe the ommodity rop for sale in the ommodity market. Eah produer is assumed to farm one are and fae a farm-level yield distribution for both the speialty and ommodity rops, [ ] no yield drag for the speialty rop) where E[ y i ] y ~ f( y) 0, y for i N (i.e. it is assumed there is i max y = and [ ] 2 Var y i = σ for i. Furthermore, the joint distribution of farm level yields is haraterized by a spatial y N NN 7 N ould also be loosely interpreted as the number of ounties in whih the proessor offers the ontrat. In reality, N may be a disrete variable, but for modeling purposes its assumed to be ontinuous.

29 21 orrelation struture where the orrelation between yields on any two farms i and j is equal to ρi, j 0 for i j. Commodity rop prodution osts are also assumed to be homogeneous aross all produers within any ontrating region. Prodution osts for the ommodity rop are equal to the sum of J variable and (annualized) fixed prodution osts that inlude labor, fertilizers, hemials, land, mahinery, fuels, storage et. Expeted profits for ommodity rop C prodution π i are equal to the expeted yield times the ommodity prie r minus prodution osts. The ommodity prie is assumed to be onstant so that the analysis is foused on the effets of prodution unertainty 8. (1.2) E π C i = ry for i, where = xj for i j Given the management guidelines outlined in the ontrat, prodution of the speialty rop will result in higher prodution osts (i.e. seed with speial genetis, more intensive labor and management, the use of additional inputs or speifi assets) than the ommodity rop. Furthermore, it is assumed that these osts may vary by farm so that produers are heterogeneous with respet to speialty rop prodution osts under ontrat. The additional osts above those for the ommodity rop are denoted by a non-negative additive term τ ji,. s (1.3) i = ( xj+ τ ji, ), where τ ji, 0 for i, j j Thus, speialty rop prodution osts are higher than for the ommodity rop = for all farmers. It is also assumed that transportation and marketing osts are suh s i i 8 Alternatively, r ould be interpreted as the expeted ommodity prie at harvest and that speialty rop yields within the ontrat region are unorrelated with the ommodity market prie. This interpretation would not affet the results given the risk-neutrality assumptions for both the proessor and produers.

30 22 that produers, given equal pries, are indifferent between delivering bushels to the proessor, selling on the speialty rop spot market, or selling on the ommodity market 9. While eah produer knows his speialty rop prodution ost with ertainty, the proessor views prodution osts for the speialty rop as being randomly distributed, on a non-negative and bounded support, aross produers within the ontrating region s ~ ˆ s u h( ) [, + ], i N for N. The distribution of speialty rop prodution osts i aross produers in any region is also ommon knowledge to eah of the produers. Contrat Supply Given the assumptions on the speialty grain prodution osts, the proessor must offer premiums above the ommodity prie to indue farmers to aept speialty rop prodution ontrats. By making the ompensation terms of the ontrat more favorable the proessor inreases the total number of farmers in a given ontrating region who will aept the ontrat N, a shift up his ontrat supply urve for a given ontrat region. By inreasing the size of the ontrating region the proessor indues an outward shift to his ontrat supply urve inreasing the number of farmers who will aept the ontrat for any ompensation struture. The effets of the ompensation terms of the ontrat and the size of the region on the number of ontrats aepted is given in figure 2. The left panel provides a spatial interpretation 10 of the effets of inreasing the size of the ontrat region from N 1 to N 2 (N 2 > N 1 ) on the number of aepted ontrats. The right panel in figure 1 plots two onditional ontrat supply urves for the two ontrating region sizes. The ontrat 9 These osts are the produer analogue of the proessor s handling harge δ. 10 The term spatial here is used loosely. The only spatial aspet in the model is the spatial orrelation of yields aross farmers. Transportation osts are assumed equal aross produers.

31 23 premium is plotted on the vertial axis. Inreasing the premium for a given ontrat size indues a move up the supply urve. Inreasing the size of the ontrat region from N 1 to N 2 auses a rightward rotation in the proessor s ontrat supply urve. N 2 $0.60 N 1 $0.50 N (p N 1 ) $0.40 P Premium ($/bu.) $0.30 $0.20 N (p N 2 ) $0.10 Farmers who rejet ontrat (N-N ) Farmers who aept ontrat (N ) $ Contrated Farmers Figure 2. Effets of the premium and ontrat region size on the proessor s ontrat supply urve. In equilibrium, the proessor optimally hooses the ontrat terms to maximize expeted profits given knowledge about the profit-maximizing behavior of produers. The following subsetions outline the proessor and produers problems and market equilibrium in detail under areage and bushel ontrats, respetively. Areage Contrats A Under areage ontrats, farmer i s profit π i is equal to his yield times the sum of the referene (ommodity) prie and the ontrat premium p less speialty rop prodution osts. A (1.4) π = ( r+ p) y s i i i Farmer i aepts the ontrat if expeted profits from ontrating are greater than expeted profits from produing the ommodity rop.

32 24 (1.5) E π A C ( ) s s ˆ i E πi r+ p y i ry i = i py premium ˆ A Equation (1.5) defines the marginal produer with speialty prodution ost = py, where all farmers with a speialty rop prodution ost premium below (above) ˆ A will aept (rejet) the ontrat. Using a hange of measure, the distribution of speialty rop prodution ost premiums within any ontrating region N is given by u s ˆ ~ ( ˆ ) 0,, where ( ˆ ) ( ) ˆ s h h = h = h( ). Also, let H( ) = h( x) dx = Pr[ ˆ ] i i i i i denote the umulative distribution funtion of the speialty rop prodution ost premium. The proessor offers a premium p above the ommodity referene prie, that will be paid for eah bushel of the speialty rop grown on the N ontrated ares, within the hosen ontrating region N. Under areage ontrats there will be no ex post spot market for the speialty rop beause eah farmer delivers all bushels produed on ontrated ares to the proessor. The proessor s profit is equal to proessing returns less prourement and diversifiation osts. For eah bushel of the speialty rop dumped on the ommodity spot market the proessor inurs a handling fee δ that is expressed as a perentage of the spot prie. The proessor hooses the premium and the size of the ontrating region to maximize expeted profits, subjet to eah produer s deision rule 11 and given his information on the farm-level yield and speialty rop prodution ost premium distributions. 0 (1.6) max E A ( p, N) pn, Π = Q 0 0 N y ( δ )( ) max RYdG( Y ) + RQ + r - Y -Q dg(y) N y max ( r+ p) YdG( Y) m( N) Q 11 While not expliitly motivating the model as a prinipal-agent problem, the proessor s onsideration of produer s ations is analogous to rationality onstraints within a prinipal-agent framework.

33 25 subjet to p 0, N 0, and A C Y = ( yi E πi E πi )~ g( Y; p, N) 0, N ymax where ˆ ~ h( ˆ), y ~ f( y) i i i Total prourement for the proessor Y is equal to the sum of the yield realizations for eah ontrated produer. Therefore, the distribution of total prourement g, with umulative distribution funtion G, is a funtion of the farm level yield distribution f and the number of farmers who aept the ontrat, whih in turn is a funtion of the premium offered and the size of the ontrating region 12. Formally, N = H( py) N with N = h( py) Ny 0 and NN = H( py) 0. Thus, GY ( ; p+ dpn, ) GY ( ; pn, ) and GY ( ; pn, + dn) GY ( ; pn, ) for dp, dn 0. This implies that inreasing the premium or size of the ontrating region indues a shift of first-order stohasti dominane in G. p Using Leibniz rule and noting that N y max YdG( Y ) = E( Y ) = H( py) Ny, where H ( ) is 0 the umulative distribution funtion for the speialty rop prodution ost premium, the A* A* solution to the proessor s problem ( p, N ) satisfies the following first order onditions. All funtions are evaluated at the optimum unless otherwise noted. (1.7) A E Π Q N y max = RYdGp + RQ + ( r δ )( Y Q) dg p 0 Q ( δ )( ) ( ) + RQ+ r N y Q g N y N y max max p max A A ( r+ p ) N y N y 0, p 0 p p 12 If yields at the farm level were assumed independent, normality ould be assumed as an approximation for the distribution of total prourement using the Central Limit Theorem. However, it is widely aepted that rop yields exhibit positive spatial orrelation. Therefore, no assumptions are made on the funtional form of g, or the distribution of yields at the farm level f. The ost of this generality, as usual, is the inability to derive expliit analyti solutions for the ontrat equilibriums in either the areage or bushel ontrat ases.

34 26 (1.8) A E Π Q N y max = RYdGN + RQ + ( r δ )( Y Q) dg N 0 Q ( ) + RQ ( r δ ) N y Q + g( N y ) N y A A ( r+ p ) N y m 0, N 0 N N max max N max N Note that both non-negativity onstraints must be non-binding in any equilibrium with ontrating so that the first order onditions will be stritly equal to zero. Equations (1.7) and (1.8) equate the marginal benefits to the marginal osts of inreasing the premium p and the size of the ontrating region N, respetively. The marginal benefits of inreasing the premium are equal to the inrease in proessing returns, given by the first three terms in equation (1.7). Similarly, the net benefits of inreasing N are given in the first three terms of equation (1.8). The third term, in both ases, reflets the fat that the upper bound of the aggregate yield distribution is inreasing in the number of ares ontrated by the proessor (i.e. Y max = ). N ymax The marginal osts of inreasing the premium in (1.7) are equal to the inreased ost on eah ontrated are plus the additional ost of inreasing the amount of ontrat ares from inreasing the premium. The marginal osts of inreasing the size of the ontrat region in (1.8) are equal to the inrease in prourement osts as more ontrats will be aepted for any given premium, plus the inrease in monitoring osts from expanding the size of the ontrating region. The seond order suffiient onditions for a maximum are given in the Appendix and are assumed to hold. Bushel Contrats B Given a bushel ontrat offer, farmer i s profit π i is equal to the referene prie plus the premium times the size of the ontrat y B, less prodution osts for the speialty rop.

35 27 For simpliity, the analysis is limited to bushel ontrats where B y = y. This of ourse implies that aggregate ontrated bushels will equal N y = Y. When atual yield is less than the size of the ontrat the farmer must either pay a fixed underage penalty (speified in the ontrat offer) on eah unit below the ontrated amount, or enter the speialty rop spot market (if there is positive supply) to purhase exess prodution from other ontrated farmers to fulfill his ontrat obligation. When farmer i s yield is greater than the ontrat size he an sell the exess speialty rop prodution at its salvage value on the ommodity market or sell it on the speialty rop spot market (if demand exists). (1.9) ( ) ( )( ) B s u π = ( r+ p) y + I y p Y, p, r y > y y y i i i i s u ( ) ( )( ) + 1 I yi p Y, p, r yi y yi y i where 1 if yi > y I( yi ) = 0 otherwise The prevailing prie on the speialty rop spot market p s will depend on the underage penalty p u set by the proessor in the bushel ontrat, the salvage value r, and the aggregate speialty rop prodution Y aross all ontrated farmers. At the time of ontrat signing, produers use their knowledge of the distribution of prodution ost premiums for the speialty rop and their information on the joint distribution of yields within the ontrating region to formulate an expetation for the ex post speialty rop spot market prie. There are two possible senarios. The aggregate yield realization will either be equal to or below the total ontrated by the proessor Y Y Y Y, or greater than the ontrated amount Y > Y. If, exess demand on the speialty rop spot market will bid the spot prie up to the

36 28 underage penalty 13, p s u = p. If Y Y > there will be exess supply on the speialty rop spot market and the prevailing spot prie will equal the salvage value of the ommodity market prie, s p = r. In either aggregate yield ase, farmer i s yield ould be greater than or less than the individual ontrat size, y > y or y y, reating four possible ex post spot market senarios i i under bushel ontrats. Farmer i s expetations for the prevailing speialty rop spot prie onditional on his yield falling above or below the ontrat size are given below. (1.10) (1.11) E p y y p p r p s su, u u i = = θ1 + (1 θ1), E p s y s o u i > y = p = θ2p + (1 θ2) r r where θ1 = Pr Y Y yi y θ2 = Pr Y Y yi > y The onditional probabilities θ 1 and θ 2 will depend on how farmer yields are jointly distributed aross the ontrating region (i.e. the spatial orrelation), with θ1 ( ) θ2 when yields are positively (negatively) orrelated. If yields are perfetly orrelated there will not be an ex post spot market beause farmers will either pay the underage penalty p u to the proessor when (all) yields are below the mean, or sell the exess speialty rop on the ommodity market at the salvage value r when (all) yields are above the mean (i.e. θ 1 = 1, θ 2 = 0 ). If yields are independent, θ1 θ2 = = and the expeted spot prie is 1 u 2 ( p r) in 13 Note that if the aggregate yield realization is suh that only one farmer would not be able to fulfill his ontrat obligation with purhases on the spot market the prevailing spot market prie would not be p u. There is no solution in this speifi ase where there are a limited number of buyers bidding for a fixed supply. This is the same problem faed by Carriquiry and Babok (2004). For simpliity, it is assumed that farmers expet the prie on the speialty rop spot market to be bid up to p u for all exess demand senarios.

37 29 both the underage and overage senarios. This implies that if yields are independent, the magnitude of the underage penalty has no effet on farmer partiipation beause the expeted ost of the penalty when farmer i s yield is below the mean is exatly equal to the benefits of the penalty when his atual yield is above average. This is purely a result of farmer expetations for the ex post speialty rop spot prie. Given farmer i s expetation for the ex post speialty rop spot market prie, expeted profit for the bushel ontrat is given below in equation (1.12). (1.12) ( ) ( ) y y max,, 0 y { θ1 θ1 } { θ2 θ2 } B s s u s o E π i = ( r + p) y i p y y df( y) + p y y df( y) s u u = ( r+ p) y p + (1 ) r Δ y+ p + (1 ) r Δy i u ( )( θ1 θ2) s = ( r+ p) y + r p Δy i, y y max y y y df( y) y y df( y) by the definition of y. Thus, if yields are where Δ = ( ) = ( ) 0 y positively orrelated, the farmer will pay less (in expetation) than u p in the ase of an individual underage and will reeive a prie greater (in expetation) than the salvage value r in the ase of an individual overage 14. Equation (1.12) defines the marginal prodution ost premium for the speialty rop ˆB, where all produers with ˆ i B ( > ) ˆ will aept (rejet) the bushel ontrat offered by the proessor. Comparing this to the marginal produer under areage ontrats, defined by ˆ A, any underage penalty that is greater than the ommodity prie will require a greater bushel ontrat premium to indue the same level of farmer ontrat aeptane relative to the areage ontrat (for a given N). 14 This is assuming aross spae. u p r in equilibrium, whih is shown to hold, and that yields are positively orrelated

38 30 (1.13) ˆB ( u )( θ θ ) = py + r p Δ y 1 2 Given the profit-maximizing deision rule of the produers, the proessor hooses the premium level p and the underage penalty u p to maximize expeted profits. In equilibrium there must be some onstraints on the values of the underage penalty. If the proessor sets the underage penalty to a value below the ommodity prie, eah ontrated produer has an inentive to report zero yield (private information) to the proessor and pay the underage fee while selling the speialty rop on the ommodity market. This nets eah produer the referene prie plus premium times the ontrat size (a sort of lump sum payment), plus the differene between the underage penalty and referene prie for eah bushel produed up to the ontrat size (or atual yield if it is below the mean), plus the referene prie for eah unit produed above the mean. The maximum underage penalty that the proessor an harge is assumed to be equal to his net proessing return R 15. For any underage penalty greater than the net proessing margin, the proessor would be better off olleting the underage than aepting delivery of ontrated bushels for proessing. As an arbitrage ondition, it is assumed produers would simply not aept suh a ontrat. The proessor s maximization problem for bushel ontrats is presented formally below in (1.14). The first order onditions are obtained by differentiating the proessor s onstrained objetive funtion using Leibniz rule, and are presented in (1.15)-(1.19) where λ and μ are the Lagrange (L) multipliers for the inequality onstraints on the underage penalty. The seond order onditions for the proessor s problem with bushel ontrats, whih are assumed to hold, are provided in the Appendix. 15 Again, from a prinipal-agent perspetive, the onstraints on the underage penalty would be analogous to inentive ompatibility onstraints.

39 31 (1.14) B u max E[ Π ( p, N, p )] = u pn,, p Q 0 N y max ( ) ( ) 0 R r YdG( Y ) + R r QdG( Y ) pn y N y Q ( ) ( ) ( ) + u p N y Y dg Y m N u subjet to p, N 0, r p R, and B C Y = ( yi E πi E πi )~ g( Y; p, N) where ˆ ~ h( ˆ), y ~ f( y) i i i (1.15) (1.16) (1.17) N y max ( ) ( ) ( ) Q L = R rydg + R r QdG + R r Qg( N y ) N y p 0 Q N y N y { ( ) 0 0 } p p max p max B u B p N y N y + p N y Y dg + N ydg 0, p 0 p p p N y max ( ) ( ) ( ) Q L = R rydg + R r QdG + R r Qg( N y ) N y N 0 Q N y N y { ( ) 0 0 } N N max N max B u B p N y m + p N y Y dg + N ydg 0, N 0 N N N N Q N y L max u ( ) u ( ) u R r YdG 0 p R r QdG Q p = + p N y B ( ) ( max) u u p max p ( ) 0 N y N y u p { ( ) u u N y Y dg N ydg 0 0 } λ μ 0 p p + R r Qg N y N y p N y + N y Y dg = (1.18) L R p u u 0, λ 0, and λ R p = = 0 λ (1.19) L u u = p r 0, μ 0, and μ p r = 0 μ As in the areage ontrat solution, both of the non-negativity onstraints on p and N will not bind for any interior solution with ontrating. Conditions (1.15)-(1.17) equate the marginal benefits of inreasing the premium, size of the ontrating region, and the underage penalty to their marginal osts. The marginal benefits of inreasing the premium and size of

40 32 the ontrating region are equal to the expeted marginal inreases in revenues from proessing returns, while the marginal osts are equal to the expeted inreases in prourement and monitoring osts. The marginal benefits of inreasing the underage penalty are equal to the expeted inrease in underage penalties and the redution in prourement osts due to lower farmer aeptane. The marginal osts of inreasing the underage penalty are equal to the expeted redution in proessing returns due to lower farmer aeptane of the ontrat. Conditions (1.18) and (1.19) ensure that the onstraints are satisfied at the optimal solution. Analyti Results While the generality of the model preludes the derivation of expliit analytial solutions for the optimal areage and bushel ontrat equilibriums, some laims an still be made regarding the two ontrat strutures. The first result shows that bushel ontrats an Pareto dominate areage ontrats. Proof: Proposition 1: There exists a bushel ontrat whih Pareto dominates the optimal areage ontrat, although that bushel ontrat may not be the proessor s optimal bushel ontrat. Consider a bushel ontrat where the premium and size of the ontrat region are set equal to the values of the optimal solution for the areage ontrat, and the underage penalty is set equal to the salvage value r. Note that farmer aeptane of the two ontrats will be equal under these onditions, so that expeted farmer profits are the same for the optimal areage

41 33 and proposed bushel ontrats (i.e. farmers are no worse off) 16. Given equal ontrat aeptane, the aggregate distribution of ontrated prodution will be the same so that the proessor s expeted profits an be ompared under the two ontrat speifiations. Subtrating the proessor s expeted profit under the optimal areage ontrat from expeted profits under the proposed bushel ontrat yields the desired result that the proessor is at least as well off under the proposed bushel ontrat than the optimal areage ontrat. Formally, N * * * * ( ) ( ) y max δ [ ] B A A u A A A E p, N, p r E p, N Π = Π = Y -Q dg 0. Q Moreover, if the handling harge δ is stritly greater than zero, the proessor s expeted profits under the bushel ontrat are stritly greater than expeted profits under the optimal areage ontrat. Therefore, the proposed bushel ontrat Pareto dominates the optimal B B* B* u* B A* A* u areage ontrat. Finally, E Π ( p, N, p ) E Π ( p, N, p = r) by the definition of a maximum. This trivially proves that the proposed bushel ontrat may not be optimal. The intuition behind Proposition 1 is that by offering an equivalent (as far as farmers are onerned) bushel ontrat, the proessor is able to avoid handling exess speialty rop prodution in years with above average yield realizations. This result relies heavily on the assumptions that 1) farmers are indifferent between delivering bushels to the proessor and selling them on the ommodity market, and 2) the salvage value is equal to the ommodity 16 Note that this result would also hold for risk-averse produers, at least in a mean-variane framework. By definition, expeted produer profits are equal under the proposed bushel and optimal areage ontrats, while the variane of expeted profits under the proposed bushel ontrat are atually smaller than the optimal areage A 2 2 for A r+ p σ rσ p 0. ( ) ontrat (( ) y) ( y)

42 34 prie r. Note that under the proposed bushel ontrat, the produer is guaranteed the premium p on eah bushel ontrated. For aggregate yield realizations below the ontrated level the proessor realizes higher prourement osts than those in the optimal areage ontrat beause he is paying the premium for the total ontrat size, but only being reimbursed r for eah short bushel in the ase of an aggregate yield shortage. This represents the prie the proessor must pay for the farmer to take on the risk of yield realizations below the bushel ontrat size. A orollary to Proposition 1 is that produers will prefer the bushel ontrat proposed in Proposition 1 to the optimal areage ontrat with δ 0. This an be formally stated as: B A* A* u A A* A* (1.20) πi ( p, N, p r) πi ( p, N ; δ 0) = for i The proof of this laim is straightforward. Total differentiation of the first order onditions for the areage ontrat with respet to δ yields the following omparative stati results. (1.21) (1.22) p N A* N y ( ) max Y-Q dgp Q = 0 δ 2 A Π 2 p N y max ( ) Y-Q dg A* N Q = 2 A δ Π N 2 0 The denominators of (1.21) and (1.22) are negative by the seond order onditions, while the numerators are both non-negative given inreases in p and N indue shifts of firstorder stohasti dominane in the distribution funtion G. Thus, both the premium and size of the ontrating region are larger under areage ontrats when δ = 0 implying farmer partiipation and expeted produer profits are also greater when δ = 0.

43 35 Noting that the proposed bushel ontrat is equivalent to the optimal areage ontrat when δ = 0, expeted produer profits are greater under the proposed bushel ontrat than under the optimal areage ontrat when δ > 0, or * * * * * * ( p, N, p 1 ) ( p, N ; 0 ) ( p, N ; 0) π = = π δ = π δ. B A A u A A A A A A i i i Finally, a laim an be made on the optimal bushel ontrat underage penalty. Exept for very speifi onditions, one of the onstraints on the optimal underage penalty will bind. Proof: Proposition 2: The optimal underage penalty equals r (R) if ( Y ) Y Y dg 0 1 <(>) N Δy ( θ θ ) Suppose at the optimum that the onstraints on the optimal underage penalty do not bind, p u* ( rr, ). Define a simultaneous hange in the bushel ontrat premium and the underage penalty suh that farmer aeptane is held onstant (i.e. the marginal speialty rop prodution ost premium is held onstant). (1.23) [ ] ( θ θ ) ( θ θ ) Δy dˆ = y dp Δ y dp = 0 dp = dp y B* u 1 2 u 1 2 Then the hange in the proessor s expeted profits for the simultaneous hange in the Y premium and underage penalty is given by ( ) ( θ1 θ2) Y 0 ( Y ) ( θ θ ) Δ u Y Y dg N y dp. Thus, if Y dg 1 > 0 and the underage penalty is less than R, the proessor an inrease N Δy expeted profits by inreasing the underage penalty to R while simultaneously inreasing the u* premium aording to (1.23), whih violates an optimum for p ( rr, ). The same logi

44 36 holds when Y ( Y ) 0 ( θ θ ) N Y dg 1 < 0, in that expeted profits will inrease if the underage 1 2 Δy penalty is redued to r while the premium is simultaneously redued aording to (1.23), u* again a violation of the supposition of a maximum with p ( rr, ). The intuition behind Proposition 2 lies in what the ratio Y ( Y ) 0 ( θ θ ) N Y dg represents. Δy 1 2 The numerator reflets the marginal valuation of the underage penalty to the proessor. Similarly, the denominator reflets the marginal ost of the underage penalty for the produers. If the marginal value to the proessor is greater than the aggregate marginal ost to produers, the optimum is attained at the maximum underage penalty. An analogous argument holds when the marginal valuation of the underage penalty to the proessor is less than the aggregate marginal ost to produers. Numerial Example Sine an expliit analytial solution does not exist for the model, a numerial approah was used to solve the model given funtional form assumptions for the farm level yield and speialty rop prodution ost premium distributions, and the monitoring ost funtion. Speialty rop prodution ost premiums were assumed to follow a uniform u distribution, ˆ i ~ U 0,. A simple quadrati form was assumed for the monitoring ost funtion. (1.24) β mn ( ) = N 2 2

45 37 The three parameter beta distribution, desribed by (1.25), was hosen for the yield distribution beause it an easily be parameterized to have finite bounds and to be either symmetri around the mean, left skewed, or right skewed. Moreover, the three parameter beta distribution has previously been used to approximate rop yield distributions (Babok and Hennessy 1996). The beta distribution was alibrated so that farm level yields had a mean of 100 and a oeffiient of variation of 20%. This level of yield volatility is onsistent with federal rop insurane rates for orn and soybeans in many ounties throughout the Midwest. (1.25) [ ] [ ] Γ[ ] Γ a+ b ( y) ( y y) f ( y) =, 0 y Γ a b y a 1 b 1 max a+b-1 max y max A baseline parameterization for an areage ontrat with an optimal premium and ontrating region size of 0.2 and 750, respetively, was ahieved by solving the model with yields fixed at their means. The ommodity prie was set equal to one, the average yield level was set to 100, and the upper bound on the speialty rop prodution ost premium was set equal to 150. With fixed yields, the first order onditions (1.7) and (1.8) for the optimal areage ontrat redue to equations (1.26) and (1.27), whih solve for the net proessing return R and the diversifiation ost funtion parameter β in terms of the other model parameters and the desired optimal ontrat premium and region size. The plant apaity Q was then set equal to the optimal aggregate prourement level with yields fixed at their means. A summary of the alibrated parameters for the baseline ase is given in table 1. (1.26) R = 1+ 2 p (1.27) β = N u ( py ) 2

46 38 Table 1. Baseline Parameter Values Parameter Value Parameter Value y 100 R 1.4 u 150 r 1 β a 12 Q b 12 δ 0.00 y max 200 The solution to the proessor s profit maximization problem was then solved under yield unertainty using numerial methods. Four different yield orrelation strutures were examined, ranging from independent to perfetly orrelated yields 17. A resorting method based on rank orrelations was used to impose the desired level of orrelation between individual farmer yield draws (Iman and Conover 1982). Additionally, model results were also alulated assuming the volatility of farm level yields was 40% to examine the effets of inreasing yield volatility at the individual farm level. Table 2 reports the optimal areage ontrat terms for the baseline ase. Compared to when yields are fixed at their mean (reported in the seond olumn), the proessor offers an areage ontrat with a lower premium to fewer produers when yield unertainty is introdued. The optimal premium dereases from $0.197 to $0.182 as yields beome more orrelated, while the number of ontrated produers falls by more than 20% from nearly 96 to A (possibly) more realisti assumption would be that the orrelation struture was a funtion of the size of ontrating region N and the distane between farming operations. However, this would have required estimation (and possible misspeifiation) of a relationship between distane and orrelation of yields as well as signifiantly inreased the omputing time needed for solution onvergene. The simpler approah was adopted beause of a lak of farm-level data for speialty rop yields.

47 39 The introdution of yield unertainty exposes the proessor to the risk of above average yield realizations and having to handle grain in exess of his proessing apaity. This effet inreases as the individual yields beome more positively orrelated. This is due to the diret relationship between the volatility of total prourement and the spatial orrelation of yields. When yields are independent, low yield realizations are balaned by above average yields on other farms. In short, the proessor is able to pool the prodution risk. At the other extreme, when yields are perfetly orrelated, the aggregate prourement of the proessor is extremely volatile and the probability of the proessor being obligated to purhase prodution in exess of plant apaity inreases. In this extreme ase, prodution risk is purely systemi and the proessor is unable to pool any of the prodution risk aross ontrated farmers. With areage ontrats the proessor earns a negative profit margin, equal to the areage ontrat premium plus the handling harge, on every bushel proured above apaity. To insure against these losses, the proessor redues the premium and size of the ontrating region to redue the hane of having to operate above apaity. This results in fewer farmers aepting the ontrat, implying a redution in farmer profits when yield unertainty is introdued. This effet is magnified as yield risk beomes inreasingly systemi. The last row of table 2 illustrates this effet, showing that the additional profits earned by produers delines as the level of orrelation between farm level yields inreases. Table 3 reports the optimal areage ontrat parameters when the proessor s handling fee δ for prourement above his plant apaity is equal to The 10% handling harge implies even larger losses on every bushel proured above apaity ompared to the situation in table 2. The resulting optimal areage ontrat is haraterized by a slightly

48 40 lower premium, and is offered to fewer total farmers resulting in a lower level of farmer partiipation and a redution in expeted profits for both the proessor and the produer for all levels of yield orrelation. Again, the last row of table 3 illustrates the deline in additional profits earned by the produers through ontrating as the poolability of yield risk delines. Table 2. Optimal Areage Contrats, σ y = 20, δ = 0 yi = y y ~ f( y) = Beta( a, b, y ) ρ i, j p A * N A * N E[ Π ] A C E ( πi πi ) i i max Table 3. Optimal Areage Contrats, σ y = 20, δ = 0.10 yi = y y ~ f( y) = Beta( a, b, y ) ρ i, j p A * N A * N E[ Π ] A C E ( πi πi ) i i max The optimal bushel ontrat parameters, when farm level yields have a 20% oeffiient of variation, are reported in table 4 over a range of yield orrelation levels. Using the bushel ontrat struture, the proessor is able to eliminate the risk of prouring a prodution level above his plant apaity. Moreover, the proessor an set an underage

49 41 penalty to reover profits when aggregate prodution is less than the total ontrated. When aggregate prodution is greater than the total ontrated, the proessor an realize even greater profits by purhasing any exess (up to his apaity onstraint) at the salvage value (ommodity prie) from ontrated produers on the ex post spot market. This effetively shifts the majority of prodution risk on to the produers. However, produers are ompensated for taking on a greater portion of the yield risk through higher ontrat premiums relative to the areage ontrat struture for any given yield orrelation struture. For example, when the orrelation between farm yields is 0.8 and there is no handling harge, the areage ontrat equilibrium results in ontrated farmers with a premium of The bushel ontrat equilibrium under the same onditions results in farmers under ontrat at a premium of Table 4. Optimal Bushel Contrats, σ y = 20 yi = y y ~ f( y) = Beta( a, b, y ) i ρ i, j p B * N B * N E[ Π ] p u * p s.u p s,o B C E ( πi πi ) i max When yields are independent so that prodution risk is poolable, the inrease in the proessor s expeted profits from using bushel ontrats is minimal. Expeted profits under

50 42 bushel ontrats and independent yields was estimated to be $ , an inrease of only $30 ompared to both areage ontrat senarios reported. However as the level of orrelation between yields is inreased, and prodution risk beomes inreasingly systemi, the proessor is able to extrat even greater relative gains from using bushel ontrats. In fat, the proessor is able to earn greater expeted profits using bushel ontrats when yields are unertain ompared to when yields are fixed at the mean. For example, expeted proessor profits for bushel ontrats were estimated to be over $1,200 when yields are unertain and positively orrelated ompared to only $1,000 in the ase of ertain yields. This is beause of the ex post spot market that is reated by bushel ontrats. When aggregate prodution exeeds the total ontrated, the proessor is able to enter the spot market and purhase the exess at the ommodity prie, earning an even greater profit margin on eah bushel in exess of the total ontrated (up to his apaity onstraint). These results imply that bushel ontrats may be more prevalent when proessors are unable to pool prodution risk aross ontrated produers. This may be the ase for ertain rops or geographi regions. Comparing produer profits to the orresponding values in table 2 shows that the produers also earn greater profits, in expetation, under bushel ontrats. Table 4 also reports the pries on the speialty rop spot market expeted by produers when farm-level yield is below or above the ontrated amount (mean yield). In the ase of an individual underage (overage), the produer s expetation for the spot prie p s.u (p s.o )ranges from 1.20 (1.20) when yields are independent to 1.34 (1.058) when the orrelation between farm level yields is equal to The last row of table 4 reports additional profits earned by the produers through ontrating.

51 43 When yields are perfetly orrelated, the expeted spot prie in the ase of a farmlevel underage equals the underage penalty of 1.4, and is equal to the ommodity prie of one in the ase of an above average yield realization at the farm level. Farmers expetations of the ex post spot prie depend only on their own yield distribution, whih is the aggregate distribution when yields are perfetly orrelated. The farmer s expetations do not dampen the underage penalty. The proessor then hooses a lower underage penalty ompared to the ases of positively, but not perfetly orrelated, yields and expeted profits fall. Tables 5-7 report the numerial results when farm yield volatility is doubled to 40%. Inreasing volatility at the farm level inreases the volatility of aggregate ontrated prodution for all levels of yield orrelation. The proessor offers areage ontrats with a lower premium to an even smaller group of produers when the farm level yield volatility is inreased. Similar to the baseline volatility ase, when a positive handling fee is imposed the proessor further redues the premium and ontrating region size to ontrat with a smaller group of produers to redue the magnitude an probability of losses when yields are above average, although the effets are relatively small. Additionally, inreased farm-level yield volatility inreases the expeted osts of an underage to the produer, requiring either a higher premium or lower underage penalty to keep partiipation onstant. However, inreased farm level yield volatility atually inreases expeted profits, relative to the baseline ase, when the proessor uses bushel ontrats. Table 7 shows that expeted profits inrease over $100 relative to the 20% yield volatility solution. Again, this is illustrating the benefit of the ex post speialty rop spot market to the proessor. When yields are more volatile at the farm level, aggregate ontrated prodution will also be more volatile. The proessor optimally ontrats with fewer farmers by reduing the premium and

52 44 size of the ontrating region, and is able to take advantage of above average aggregate prodution by purhasing the speialty rop on the spot market at the ommodity prie up to his apaity onstraint. Table 5. Optimal Areage Contrats, σ y = 40, δ = 0 yi = y y ~ f( y) = Beta( a, b, y ) ρ i, j i p A * N A * N E[ Π ] A C ( i i ) E π π i Table 6. Optimal Areage Contrats, σ y = 40, δ = 0.10 yi max = y y ~ f( y) = Beta( a, b, y ) ρ i, j i p A * N A * N E[ Π ] A C ( i i ) E π π i max When yields are perfetly orrelated, farmers expet ex post spot pries to be equal to the underage penalty or the salvage value depending on aggregate yields, whih are equal to farm level yields. There is no dampening effet on the spot prie expetations of farmers so the proessor must lower the underage penalty and realizes smaller expeted profits ompared to the ases of positively, but not perfetly, orrelated yields. Again, omparing

53 45 the last row in tables 5-7 shows that produers also prefer the bushel ontrat struture beause they earn greater expeted profits ompared to either areage ontrat senario. Table 7. Optimal Bushel Contrats, σ y = 40 yi = y y ~ f( y) = Beta( a, b, y ) ρ i, j i p B * N B * N E[ Π ] p u * p s,u p s,o B C ( i i ) E π π i max Conlusions The fat that vertial oordination through ontratual relationships in agriulture is beoming inreasingly important is well doumented. The rise of vertial oordination in agriulture has been more apparent in livestok markets, whih is refleted in the aademi literature. Many authors have explored the effets of varying ontrat strutures using theoretial, empirial, and experimental approahes. Speial onsideration has been given to the effet of ompensation strutures on the effiieny of ontrat market equilibriums and the o-existene of ontrat and spot markets for the same ommodity. However, previous studies have foused more heavily on ontrating in livestok markets, while prodution ontrats in rop prodution have been given muh less attention.

54 46 This paper makes a ontribution in this area by presenting a omparison of ontrat strutures within a theoretial model of a ontrating relationship between a risk-neutral monopsonisti proessor and risk-neutral produers. The main analytial result is that there exists a bushel ontrat struture whih Pareto dominates the optimal areage ontrat. However, this bushel ontrat may not be optimal. This result departs from the onventional thinking in the ontrat literature that areage ontrats will be the preferred hoie of ontrat struture. Furthermore, it is shown that the magnitude of the optimal underage penalty for bushel ontrats depends on the relative marginal valuations of low yield realizations for the proessor and produers, whih are themselves funtions of the spatial orrelation of yields, or the poolability of prodution risk. Moreover, the (expeted) magnitude of the underage penalty is dampened at the farmer level beause of the produers expetations of ex post spot market pries for the speialty rop under bushel ontrats, whih are onditional on the aggregate prodution of the speialty rop. A alibrated numerial example shows that areage ontrat premiums, farmer partiipation, and expeted profits for both the proessor and produers deline as the orrelation between farm level yields aross spae inreases. Inreasing the level of yield volatility at the farm level results in the same type of effet. Numerial solutions for the optimal bushel ontrat under different assumptions for the spatial orrelation of yields and farm level yield volatility illustrate the analyti result that the proessor will prefer bushel ontrats (greater expeted profits) and that they may Pareto dominate areage ontrats. The bushel ontrat struture allows the proessor to ontrat with a greater number of produers at higher premium rate, inreasing farmer profits relative to the areage ontrat

55 47 equilibrium. As prodution risk beomes more systemi (larger orrelation between yields), the proessor benefits relatively more from using bushel ontrats. When risk is largely poolable, the two ontrat strutures are nearly equivalent with respet to expeted profits and farmer partiipation in the resulting equilibriums. These results imply that bushel ontrats may be more prevalent for rops and regions where the nature of prodution risk is highly systemi. When prodution risk an be pooled, the hoie of ontrat struture may be less important. These results must be interpreted with are. The model inludes a set of restritive assumptions inluding produer risk-neutrality, an exogenous ommodity market with no prie unertainty, and the ability to dump the speialty rop on the ommodity market at the ommodity prie. Thus, the model an be thought of as a starting point, providing for a multitude of possible extensions for future researh. Obviously, produer risk aversion and ommodity market prie unertainty are two potential areas for further analysis. Risk averse produers will, in general, require more ompensation for taking on a larger share of prodution risk, potentially eroding away the gains from bushel ontrats. The addition of prie unertainty may also greatly affet the results, espeially if the ommodity prie is assumed to be orrelated with aggregate or farm-level yields. Furthermore, extending the model to an oligopsony setting where multiple proessors ompete in prodution ontrats would redue the ability of the proessor to inrease profits using the ex post spot market when bushel ontrats are offered. While the limitations of the urrent model are well reognized, the results of this analysis do provide irumstanes where bushel ontrats are stritly preferred by all agents.

56 48 Empirial testing of the validity of these assumptions in real-world ontrat markets is another area for further researh. Referenes Babok, B.A. and D.A. Hennessy Input Demand Under Yield and Revenue Insurane. Amerian Journal of Agriultural Eonomis 78(2): Barkema, A Reahing Consumers in the 21 st Century: The Short Way Around the Barn. Amerian Journal of Agriultural Eonomis 75: Carriquiry, M. and B.A. Babok Can Spot and Contrat Markets Co-Exist in Agriulture? Center for Agriultural and Rural Development, Working Paper 02-WP 311, Iowa State University. Coaldrake, K. and S.T. Sonka Contratual Arrangements in the Prodution of High Values Crops. Journal of the Amerian Soiety of Farm Managers and Rural Appraisers 57: Drabenstott, M Consolidation in US Agriulture: The New Rural Landsape and Publi Poliy. Eonomi Review 84: Fraser, I Miroeonometri Analysis of Wine Grape Supply Contrats in Australia. Australian Journal of Agriultural Eonomis 49: Fulton, J., J. Prithett, and R. Pederson Contrat Prodution and Market Coordination for Speialty Crops: The Case of Indiana. Paper presented at Produt Differentiation and Market Segmentation in Grains and Oilseeds: Impliations for Industry in Transition Symposium, ERS USDA and Farm Foundation In., Washington DC, January Ginder, R., G. Artz, D. Jarboe, H. Hommes, J. Cashman, and H. Holden Output Trait Speialty Corn Prodution in Iowa. Iowa State University Extension Publiation. Viewed Marh Good, D., K. Bender, and L. Hill Marketing of Speialty Corn and Soybean Crops. Department of Agriultural and Consumer Eonomis, Report AE University of Illinois, Urbana-Champaign. Goodhue, R.E Broiler Prodution Contrats as a Multi-Agent Problem: Common Risk, Inentives and Heterogeneity. Amerian Journal of Agriultural Eonomis 82:

57 49 Goodhue, R.E. and S. Hoffman Reading the Fine Print in Agriultural Contrats: Conventional Contrat Clauses, Risks and Returns. Amerian Journal of Agriultural Eonomis 88(5): Hennessy, D.A Information Asymmetry as a Reason for Food Industry Vertial Integration. Amerian Journal of Agriultural Eonomis 78: Hennessy, D.A., and J.D. Lawrene Contratual Relations, Control, and Quality in the Hog Setor. Review of Agriultural Eonomis 21: Hueth B. and E. Ligon Produer Prie Risk and Quality Measurement. Amerian Journal of Agriultural Eonomis 81: Iman, R. and W. Conover A Distribution-Free Approah to Induing Rank Correlation Among Input Variables. Communiations in Statistis B11 3: Johnson, C.S. and K.A. Foster Risk Preferenes and Contrating in the US Hog Industry. Journal of Agriultural and Applied Eonomis 26: Lajili, K., P.J. Barry, S.T. Sonka, and J.T. Mahoney Farmers Preferenes for Crop Contrats. Journal of Agriultural and Resoure Eonomis 22(2): Lawrene, J.D., V.J. Rhodes, G.A. Grimes, and M.L. Hayenga Vertial Coordination in the US Pork Industry: Status, Motivations, and Expetations. Agribusiness 33: National Agriultural Statistis Servie (NASS) Corn, Soybeans, and Wheat Sold Through Marketing Contrats: 2001 Summary. U.S. Department of Agriulture, Washington, DC. Purell, W.D. and W.T. Hudson Risk Sharing and Compensation Guides for Managers and Members of Vertial Beef Allianes. Review of Agriultural Eonomis 25(1): Roe, B., T.L. Sporleder, and B. Belleville Hog Produer Preferenes for Marketing Contrat Attributes. Amerian Journal of Agriultural Eonomis 86(1): Sexton, R.J. and M. Zhang A Model of Prie Determination for Fresh Produe with Appliation to California Ieberg Lettue. Amerian Journal of Agriultural Eonomis 78: Sporleder, T.L Managerial Eonomis of Vertially Coordinated Agriultural Firms. Amerian Journal of Agriultural Eonomis 74:

58 50 Sykuta, M.E. and M.L. Cook A New Institutional Eonomis Approah to Contrats and Cooperatives. Amerian Journal of Agriultural Eonomis 83(5): Sykuta, M. and J. Parell Contrat Struture and Design in Identity-Preserved Soybean Prodution. Review of Agriultural Eonomis 25(2): Tsoulouhas, T. and T. Vukina Integrator Contrats with Many Agents and Bankrupty. Amerian Journal of Agriultural Eonomis 81: Wang, W. and E.C. Jaenike Simulating the Impats of Contrat Supplies in a Spot Market-Contrat Market Equilibrium. Amerian Journal of Agriultural Eonomis 88(4): Weleshuk, I.T. and W.A. Kerr The Sharing of Risks and Returns in Prairie Speial Crops: A Transation Cost Approah. Canadian Journal of Agriultural Eonomis 43: Williamson, O.E The Mehanisms of Governane. New York: The Free Press. Wu, S.Y. and B. Roe Contrat Enforement, Soial Effiieny, and Distribution: Some Experimental Evidene. Amerian Journal of Agriultural Eonomis 89(1): Xia, T. and R.J. Sexton The Competitive Impliations of Top-of-the-Market and Related Contrat-Priing Clauses. Amerian Journal of Agriultural Eonomis 86(1): Zhang, M. and R.J. Sexton Captive Supplies and the Cash Market Prie: A Spatial Markets Approah. Journal of Agriultural and Resoure Eonomis 25:

59 51 Appendix The seond order suffiient onditions for the proessor s problem with areage ontrats. 2 A 2 A Π Π 2 A p pn Η = 0 2 A 2 A, where Π Π 2 N p N 2 A Q N y Π max 2 = pp + + ( δ ) 0 Q p ( δ ) ( ) ( ) RYdG RQ r Y Q dg p( max) p max ( )( ) ( max ) pp max g N y N y 2 + RQ ( r δ )( N ymax Q) + g p N ymax N p ymax g N y N y 2 A* max p max pp + r g N y N y ( r+ p ) N y 2Np y 2 A Q N y Π max = pn + + ( δ ) 0 Q pn ( δ ) ( ) 2 max N p max ( ) gp( N ymax ) NN y max + RQ ( r δ )( N ymax Q) + + g( N ymax) N pn ymax 2 + gn( N ymax) NNNpymax + RYdG RQ r Y Q dg r g N y N N y ( r+ p) N y N y pn 2 A Q N y Π max = NN + + ( δ ) 0 Q pn N ( ) 2 * ( δ ) ( )( ) ( ) pp pn N ( max ) N max ( )( ) ( max ) NN max g N y N y 2 + RQ ( r δ )( N ymax Q) + gn N ymax N N ymax g N y N y + RYdG RQ r Y Q dg r g N y N y r+ p N y m A max N max NN NN NN The seond order suffiient onditions for the proessor s problem with bushel ontrats.

60 52 2 B 2 B 2 B Π Π Π 2 u p pn p p B 2 B 2 B 2 B Η = Π Π Π 2 u 0, where N p N p N 2 B 2 B 2 B Π Π Π u u u2 pp p N p 2 Q Π = + ( ) N y max ( ) ( ) B B* 2 R r YdGpp R r QdGpp p Npp y 2Np y p 0 Q gp ( N ymax ) Np y max 2 p ( max )( p max ) g( N ymax ) Npp ymax N y N y N y 2 { 2 p p ( ) pp pp ( )( p ) } + R r Q g N y N y u p N ydg N y Y dg N ydg g N y N y Q Π = + ( ) N y max ( ) ( ) B B* 2 R r YdGNN R r QdGNN p NNN y mnn N 0 Q gn ( N ymax) NNy max 2 N ( max )( N max ) g( N ymax ) NNN ymax N y N y N y 2 { 2 N N ( ) NN NN ( )( N ) } + R r Q g N y N y u p N ydg N y Y dg N ydg g N y N y B Q N y Π max B* 2 = ( ) u u ( ) u u u u + 0 p p Q p p p p u ( p ) ( max ) g( N ymax ) N u ymax p N y N y u { ( ) u N ydg 0 p N y Y dg 0 p } N y N y N y 2 u* 2 u u u ( ) u u u u { 0 p p 0 p p 0 p p ( )( u p p ) } g N y N y R r YdG R r QdG p N y g u N y N uy p p max 2 + ( R r) Q g u ( N ymax )( N uy p p max ) + p N ydg + N ydg+ N y Y dg +

61 53 2 B Q Π = + ( ) N y max ( ) ( ) B* R r YdGpN R r QdGpN p NpN y NN y pn 0 Q gp ( N ymax ) NN y max 2 N ( max) N p max g( N ymax ) NpN ymax N y N y N y N y { N p p N ( ) pn pn ( ) N p } + R r Q g N y N N y + p N ydg + N ydg + N y Y dg + N ydg + g N y N N y u* 2 2 B Q N y Π max B* u = ( ) u + ( ) u u u p p Q p p p p p N y p p 0 u p ( ) ( ) ( max ) N y NpydG ( N y Y) dgp N y N y N y N y 2 u p p u ( ) u { 0 p 0 p + u 0 p p ( ) u p 0 p p p } N ydg + g N y N N y 0 0 R r YdG R r QdG p N y N y g N ymax Np y max 2 + ( R r) Q gp N ymax N u N p pymax g N y N u y p p max + + u* p N ydg N ydg N y Y dg B Q N y Π max B* u = ( ) u + ( ) u u p N Q p N p N N y p N 0 u p ( ) ( ) ( max ) N y NNydG ( N y Y) dgn N y N y N y N y 2 u N N u ( ) u u { 0 p 0 p 0 p N p N ( ) u ydg + g N y N N N y 0 p } 0 0 R r YdG R r QdG p N y g N ymax NN y max 2 + ( R r) Q gn N ymax N u N p N ymax g N y N u y p N max p N ydg + N ydg + N y Y dg + N u*

62 54 CHAPTER 3. READDRESSING THE FERTILIZER PROBLEM: RECONCILING THE PARADOX A paper to be submitted to the Amerian Journal of Agriultural Eonomis Niholas D. Paulson Abstrat Pope and Kramer (1979) defined an input to be marginally risk-inreasing if the riskaverse agent s marginal risk premium, the wedge between an input s expeted marginal produt and its prie, was positive at the optimum. This implies that eterus paribus, a riskaverse agent will use less of a risk-inreasing input than a risk-neutral agent. Empirial work has shown that many inputs to rop prodution, inluding fertilizer, are risk-inreasing (Roumasset et al. 1989). However, there exists a large body of literature whih ontends that farmers apply nitrogen in exess of profit-maximizing levels (over-apply) to self protet (Ehrlih and Beker 1972) against fertilizer being a limiting input in years of optimal growing onditions (Babok 1992; Below and Brandau 2001). This presents somewhat of a paradox with respet to the way in whih farmer s use fertilizer under unertainty. This paper presents a model of optimal input appliation under output and input unertainty. Using experimental yield response data, a stohasti prodution relationship between yield and available soil nitrate is estimated. Input unertainty is introdued by assuming that available soil nitrate is a stohasti funtion of fertilizer applied by the farmer. Numerial results imply that while risk-averse farmers may use less fertilizer (i.e. fertilizer is riskinreasing), farmers with both types of risk preferene may over-apply nitrogen beause of

63 55 input unertainty. In addition to the empirial analysis, primary data from a survey on farmers risk preferenes and subjetive beliefs about the relationship between fertilizer and yield variability are presented. The survey data suggest that while farmers do exhibit risk aversion in that ertain outomes are preferred to gambles, farmers show preferene towards yield gambles with small hanes of very large gains (high yields) to gambles with small hanes of very low yields (rop failures). Introdution Two general approahes are used to explore the use of prodution inputs under unertainty, often leading to very different onlusions. Pope and Kramer (1979) defined an input to be marginally risk-inreasing (dereasing) 1 if, under risk aversion, the expeted marginal produt is greater (less) than the fator prie at the expeted utility maximizing level (i.e. the marginal risk premium is positive at the optimum). In a eterus paribus framework, this definition an be restated as an input is risk-inreasing (dereasing) if the risk averse firm s optimal demand for the input is less (more) than that of the risk neutral firm. A suffiient ondition for an input to be risk-inreasing is that the variability of output be inreasing in the level of input use 2. Empirial evidene has shown this to be the ase for many agriultural inputs, inluding fertilizer (Roumasset et al. 1989). Alternatively, an approah whih generally relies on the agronomi theory of a limiting input tehnology shows that the over-appliation of inputs may be viewed as an 1 The terms marginally risk-inreasing (dereasing) and risk-inreasing (dereasing) will be used interhangeably throughout this paper. 2 Furthermore, Pope and Kramer (1979) showed that the (then) ommon multipliative prodution funtion speifiation implies that all marginally produtive inputs are risk-inreasing. Moreover, the ommonly estimated log-linear prodution speifiation is simply a linear representation of the multipliative form. Thus log-linearity also implies that all marginally produtive inputs are risk-inreasing.

64 56 ativity of what Ehrlih and Beker (1972) define as self-protetion, or an at to redue the size of losses. The limiting input theory, first proposed by Von Liebig (1840), implies agriultural prodution an be defined by a fixed proportions tehnology where rop yield is determined by the most limiting input. This speifiation an be haraterized by a simple linear response and plateau (LRP) prodution funtion (Cate and Nelson 1971). Complementarity between all inputs is imposed by the LRP speifiation, so that farmers are thought to use inputs in exess to ensure that none of the elements under their ontrol will be limiting. For example, a farmer may over-apply 3 fertilizer to ensure that nutrients are not limiting when exogenous and/or stohasti inputs, suh as weather, have optimal realizations (Babok 1992). As another example, farmers may over-apply inputs to redue the probability of low yield realizations and inrease the probability of high yields. This result implies a distributional effet on yields onditioned on input levels illustrated by Babok and Hennessy (1996). The traditional Pope and Kramer definition relies on the global onavity of the produer s utility funtion. Both risk-averse and risk-neutral farmers value the mean effets of produtive inputs equally. However, the risk averse farmer disounts (values) the onsequential inrease (derease) in output variability from additional input use. However, beause random yield enters linearly into the objetive funtion (profit) of the risk-neutral produer, yield variability has no effet on input hoies 4. Thus, an input whih has an 3 For the purposes of this study, over-applying an input is defined as applying more than the profit maximizing level when the input available to the orn plant is a deterministi funtion of the appliation rate (i.e. the nitrogen available to the plant is exatly equal to some mean level given the rate of appliation). If nitrogen fertilizer is applied above this mean optimal rate, then on average nitrogen availability will be in exess of the optimal level. This is onsistent with the appliation rate used by agronomists and rop sientists to formulate appliation rate reommendations, defined as the eonomi optimum in Sawyer et al. (2006). 4 Ignoring the relationship between random output pries and yields.

65 57 inreasing (dereasing) effet on yield variability will result in lower (higher) levels of use for the risk-averse produer relative to a farmer who is risk-neutral. Under the limiting input view, farmers may over-apply nitrogen as self-protetion (Ehrlih and Beker 1972) even though the theoretial foundations for this argument imply that yield variability is non-dereasing in the level of inputs used 5. As an example, Babok (1992) notes that US farmers tend to over-apply fertilizer based on expetations of ex post realizations of other unertain fators despite the fat that yield data implies that yield variability is inreasing in the level of fertilizer applied. Further evidene for the limiting input argument was exhibited by Gallagher (1987), where soybean yield variability was shown to be inreasing over time. This is attributed to the fat that while yield apaities have inreased, the possibility of extremely low or zero yield levels may always be present due to fores suh as severe weather variability. Moreover, empirial evidene implies that farmers onsistently over-apply fertilizer (SriRamaratnam et al. 1987; Below and Brandau 2001; Sawyer et al. 2006). The National Researh Counil (1993) estimated that there are up to 8 billion pounds of exess nitrogen left in the soil eah year. Yadav, Peterson, and Easter (1997) showed that farmers in southeastern Minnesota applied nitrogen at rates exeeding both reommended rates and estimated profit-maximizing levels. Below and Brandau (2001) state that Nitrogen fertilizer over-appliation has been viewed by many as a heap form of insurane to insure against the possibility of N losses and to make ertain that suffiient N is available in ase the 5 As the level of use for any single input inrease from a binding state to a non-binding state the set of possible yield realizations beomes larger, thus inreasing the variability of output by definition.

66 58 environment is supportive of high yields., further justifying the limiting input and selfprotetion arguments. This apparent paradox is what is defined as the fertilizer problem. The obvious question is whih view is orret? Should the relationship between risk and input use be haraterized aording to the definition provided by Pope and Kramer (1979), or are inputs used by agriultural produers as self-protetion against the fores whih are out of their ontrol (i.e. weather, soil harateristis, nutrient availability)? More importantly, is it possible to speify a model whih implies that nitrogen is both risk-inreasing and overapplied by both risk-averse and risk-neutral agents? This paper attempts to reonile this paradox by examining these questions using experimental data on orn prodution in Iowa (Binford, Blakmer, and Cerrato 1992; Blakmer et al. 1989). The experimental nature of the data allows us to examine the effets of nitrogen fertilizer on orn yields eterus paribus. A flexible funtional form (translog) is used to model orn yield response to available soil nitrogen after plant emergene in the spring. An estimation proedure proposed by Just and Pope (1979), allowing for flexibility with regards to the effet on input use on output variability, is used to estimate the prodution funtion. Consistent with previous empirial studies, it is shown that yield variability is indeed inreasing in the amount of soil nitrogen available after emergene in the spring. Then, following Babok and Blakmer (1992), available soil nitrogen is speified as a stohasti funtion of applied nitrogen fertilizer (the farmer s hoie variable). This approah embodies the notion that applied nitrogen fertilizer is not neessarily the ritial input; rather the atual nitrogen available to the plant in the soil determines rop yields. Using numerial tehniques, the optimal nitrogen appliation rates are alulated and

67 59 ompared for both a risk-neutral and risk-averse produer. While the effets of nitrogen on yield variability do indeed ause the optimal appliation rates for the risk averse farmer to be lower than those for the risk neutral produer, the effets of unertainty with respet to available soil nitrogen are shown to be able to inrease the optimal appliation rates above the passive profit-maximizing optimum 6 for produers with both types of risk preferenes (i.e. fertilizer is over-applied). Thus, nitrogen fertilizer would be haraterized in this setting as a marginally risk-inreasing input aording to the Pope and Kramer definition, while it is also over-applied relative to the passive optimal appliation rate due to the unertainty assoiated with atual soil nitrogen availability. Both effets are shown to be due to the urvature properties of the (assumed) utility and (estimated) prodution funtions. Beause assumed funtional forms may not perfetly represent real-world deision making and estimated relationships are subjet to speifiation error, primary data from atual eonomi agents (farmers) was also olleted by way of a survey instrument. In addition to the analysis of the Blakmer data, results to the survey are provided. The survey, designed following the famous work by Kahneman and Tversky (1979) on prospet theory, shows that the majority of farmers preferenes over their yield distributions violate some of the lassi axioms of expeted utility theory. Moreover, the survey results show that while about 50% of the surveyed farmers believe that inreased fertilizer use inreases yield variability, a muh smaller proportion of the sample population feel that inreased nitrogen use inreases risk. While farmers do show preferene for senarios with sure payouts to gambles of equal expeted value, it is shown that farmers do not neessarily equate yield variability with yield risk as is impliitly assumed in ommon mean-variane frameworks. In 6 The passive profit-maximizing optimal appliation rate is defined in a later setion of the paper.

68 60 fat, yield gambles with small hanes of realizations well above the mean (i.e. bumper rops) are preferred to gambles with small hanes of yields well below the mean (i.e. rop failures), when the expeted yield levels are equal. This result implies that risk-averse farmers may not onsider large yield realizations as risky outomes. However, a globally onave utility funtion disounts yield variability above and below the mean equally. The remainder of this paper is organized as follows. The next setion provides some bakground on previous work related to this study. The third setion speifies the yield response model. The fourth setion disusses the data and outlines the estimation tehniques employed. Setion five reports and disusses the estimation results. Setion six provides numerial examples of optimal nitrogen appliation rates under various irumstanes. Setion seven outlines the farmer survey and the olletion methods, as well as a detailed look into the survey results. The final setion provides some onluding remarks and disusses areas for further researh. Literature Review The literature on the relationship between risk, prodution levels, and input use in agriulture is vast. Ratti and Ullah (1976) examined the effets of prodution unertainty on input use in a two fator model. They showed that input demands under unertainty are less for the risk averse firm, ompared to the risk neutral firm, given assumptions on the elastiities of the marginal produt urves and the level of omplementarity between the two fators. Furthermore, they show that the required assumptions generally hold for many of the (then) ommon prodution funtion speifiations, inluding the Cobb-Douglass, CES, and Transendental models. MaMinn and Holtmann (1983) analyze input hoie under a very

69 61 general form of tehnologial unertainty, finding that risk-averse agents may inrease or derease input use as the level of prodution unertainty inreases. Rothshild and Stiglitz (1970 and 1971) showed that the effet of inreases in risk, defined by a mean preserving spread, on optimal input levels depends on the urvature of the marginal utility or produt urve with respet to the stohasti shok 7. The effets of prie unertainty have also been explored extensively (Sandmo 1971; Ishii 1977; Hartman 1976). Examples of agriultural inputs with both risk-inreasing and risk-dereasing harateristis have been reported in the literature. Pest ontrol inputs have been shown to provide some level of protetion against prodution unertainty implying a risk-dereasing effet (Feder 1979). However, Hurley and Babok (2003) find that pestiides would be defined as risk-inreasing by the Pope and Kramer definition. As another example, inreases in the apital-labor ratio have been shown to redue the effets of weather variability on prodution in agriulture (see Pope and Kramer 1979 for examples). Hurley, Mithell, and Rie (2004) emphasize the endogeneity of risk aused by input hoies. They apply their oneptual model of the adoption of Bt orn hybrid tehnology to two Midwestern ounties and find that Bt orn, while generally though to be an input used as self-protetion against orn borer infestation, may be defined as a risk-inreasing or riskdereasing input depending on the prie of Bt seed. However, their model implies an interesting result in that When planting orn is optimal, Bt orn is risk inreasing if the expeted loss it eliminates exeeds its prie... (Proposition 1, p. 347 of Hurley, Mithell, and Rie 2004). In other words, Bt orn is risk inreasing when the expeted benefit of its 7 Rothshild and Stiglitz (1971) assumed risk-aversion for the optimizing agent. For a risk-neutral agent under prodution unertainty, their urvature result applies to the marginal produt urve.

70 62 adoption is greater than the ost of adoption (i.e. when Bt orn is expeted to inrease expeted profits). SriRamaratnam et al. (1987) used an experimental approah to eliit farmer s subjetive beliefs about the responsiveness of sorghum yields to nitrogen fertilizer in Texas. They found that farmers generally overestimate the response of yields to fertilizer ausing them to over-apply. More reent studies mentioned in the previous setion have also found evidene that farmers apply fertilizer at rates whih exeed the profit-maximizing level (NRC 1993; Yadav, Peterson, and Easter 1997; Below and Brandau 2001). Just and Pope (1979) use a three-stage estimation proess to separate the effets of nitrogen fertilizer on average yield and yield variability. They find that both orn and oat yield variability are indeed inreasing funtions of nitrogen fertilizer appliation rates. Additionally, Just and Pope (1979) provide referenes to previous studies in whih yield variability was found to be inreasing in fertilizer appliation rates, although the goal of their paper was to show that the multipliative, or log-linear, forms used in the estimation of these studies imposed this ondition. More reently, Ramaswami (1992) provided more generalized onditions under whih input ould be haraterized aording to the Pope and Kramer definition 8. Using the same experimental data, Ramaswami (1992) showed that, using his relaxed definition, nitrogen fertilizer was risk-inreasing at low (high) appliation rates for otton (orn). However, assumptions on preferenes were required to sign the marginal risk premium for high (low) appliation rates for otton (orn). 8 Ramaswami (1992) derived onditions under whih an input ould be haraterized as risk inreasing (dereasing) based purely on tehnologial assumptions, noting that Pope and Kramer s original definition inludes assumptions on both preferenes and tehnology. His result yields the weakest ondition neessary to define an input as risk inreasing (dereasing).

71 63 Paris and Knapp (1989) outlined multiple methods for estimation of LRP funtions from input and output data. Using these methods with experimental data on orn response to nitrogen and phosphorus fertilizers, Paris (1992) shows that the LRP speifiation provided the best interpretation of the prodution relationship. Lanzer and Paris (1981) estimated LRP prodution funtions for wheat and soybean prodution in Brazil. By inorporating nutrient arry-over, they demonstrate the ineffiieny of fertilizer appliation rate reommendations whih are based on the traditional polynomial fertilizer response funtions. Examples of other appliations of the LRP prodution funtion to agriultural studies inlude Babok and Blakmer (1992, 1994) and Babok (1992). Babok and Blakmer (1992) use the LRP speifiation to model orn yield response as a funtion of available soil nitrogen. They find that nitrogen appliation rates ould be redued by up to 38% through the use of late spring soil tests followed by side-dressed nitrogen appliations if the soil tests indiate low levels of available soil nitrogen. Their results imply that farmers may be over-applying nitrogen to offset the unertainty with respet to the nitrogen that will be available to the plants during growth. In another study, Babok and Blakmer (1994) use the LRP speifiation with experimental data on Iowa orn prodution to show that there may be a positive relationship between optimal nitrogen fertilizer appliation rates and growing onditions, indiating that the notion of over-applying fertilizer inputs to take advantage of the good years may have some validity. In another appliation of the LRP funtion to agriultural data, Babok (1992) shows that unertainty indues the over-appliation of nitrogen fertilizer as long as the slope of the LRP funtion, in the range where nitrogen is limiting, is more than double the prie of nitrogen fertilizer. While this result relies on a distributional assumption, it is quite intuitive.

72 64 Given the expeted value for growing onditions, the LRP speifiation implies an optimal appliation rate equal to the point where nitrogen fertilizer beomes non-binding (i.e. the kink in the LRP funtion where yield response reahes the plateau) 9. When unertainty with respet to growing onditions is introdued there is a 50% hane of either a better or worse realization for growing onditions. Thus, there is a 50% hane that nitrogen ould beome limiting, whih implies Babok s result. However, the LRP approah to modeling prodution in agriulture has not been adopted extensively in the literature beause eonomists prefer to work with smooth differentiable funtions (Lanzer and Paris 1981). Moreover, the notion of a limiting input is based on soil siene theory at the plant level, while eonomists tend to examine problems from a more aggregated viewpoint at a field or whole farm level. Berk and Helfand (1990) reoniled the opposing views of the LRP and differentiable polynomial prodution funtion speifiations by showing that the traditional funtional forms an be derived as an aggregation of the LRP model aross many heterogeneous inputs. The deision to estimate a smooth, differentiable translog prodution funtion for this study rather than using the LRP model was made based on their results. Model Consider a risk-neutral farmer who hooses an amount of fertilizer to apply x, at unit prie w, to produe stohasti output q to maximize expeted profits E [ π ]. Furthermore, assume that the amount of fertilizer relevant to prodution is that whih is available in the 9 This is assuming the slope of the LRP funtion over the range where nitrogen would be the binding input is less than the unit prie of nitrogen fertilizer. If the slope, or the marginal response, is less than the unit prie the optimal appliation rate is zero.

73 65 soil x, whih is assumed to be a stohasti funtion of applied fertilizer. The relationship between applied fertilizer and available fertilizer is given by x = x+ x+ φμ, where μ = x E x and x, φ 0. (1.1) [ ] The error term μ is mean zero by onstrution so that the expeted level of available nutrients in the soil is equal to the amount in the soil prior to fertilizer appliation x 10 plus the amount of fertilizer applied x. The speial ase where φ = 0 is that of input ertainty with respet to applied fertilizer. Inreasing the value of φ is a measure of inreasing input unertainty by way of a mean preserving spread. This is onsistent with the definitions of inreasing risk proposed by Rothshild and Stiglitz (1970) in that φ > 0 implies available fertilizer is equal to applied fertilizer plus noise. Available fertilizer may differ from what is applied for a variety of reasons suh as nutrient losses (or gains) from weather events 11 and differenes in natural levels of nutrients due to heterogeneity with respet to soil types and previous praties (i.e. rop rotations), as well as unertainty with respet to appliation tehnologies. Output is assumed to be an inreasing onave funtion in the amount of available fertilizer x, q 0, q 0 for x 0. All other prodution inputs, denoted by vetor z with x xx unit prie vetor r, are taken as given. Thus, the analysis is limited to the maximization of expeted profits onditional on all inputs (other than fertilizer) being exogenous. (1.2) π q( x x φ ε z) = (, ), ; wx r' z. 10 The value of x will, in general, be a funtion of a number of variable inluding the previous year s rop (rotation effets) and the speifi soil type. 11 Farmers in the Midwest generally apply nitrogen fertilizer for orn prodution in the fall after harvest or in the spring prior to planting. In either ase there is generally a period of time between appliation and when nutrient uptake by the plant ours. Therefore, (stohasti) weather events suh as rainfall that our between appliation and plant growth may affet available nutrient levels in the soil.

74 66 The stohasti omponent of output ε is assumed to be a mean-zero disturbane whose variane may or may not depend on inputs x, x, and z. Without loss of generality, it is also assumed that high draws of ε are assoiated with high draws of output q. The stohasti disturbane ould be thought of as a proxy for growing onditions throughout the prodution period, where better growing onditions are given by larger draws of ε. The farmer s profit maximization problem is given below, where it is assumed that all inputs must be non-negative. (1.3) [ π ] ( φ ε ) max E z = E q x( x, ), ; z wx r ' z x subjet to x 0 and equation (1.1) (1.4) [ π ] x x ( ( ) ε z) * E = E q x x, ; w 0, x 0 Denoting the expeted profit-maximizing level of applied fertilizer by x ( w, φ) first order ondition for a maximum is given by equation (1.4), where derivatives are denoted with a subsript and the first order ondition holds with equality if xˆ > 0. The seond order ondition requires that the expeted value of the seond derivative of output with respet to fertilizer be non-positive E qxx ( x( x ) ε ), 0, whih is satisfied given the urvature assumptions on q. The optimal appliation rate given in (1.4) when φ = 0 will be defined as the passive profit-maximizing optimum beause it ignores the unertainty assoiated with soil nutrient levels. This leads to the following definiton Definition 1: An input is said to be over(under) applied if the appliation rate is x w, φ = 0. greater (less) than the passive risk-neutral optimum ( ), the

75 67 Now onsider a risk-averse farmer who hooses the amount of fertilizer to apply to maximize the expeted utility of profits given in (1.2). The risk-averse farmer s problem is given by (1.5) ( ) ( ) ( ( ) ) max E U π z = E U q x x, φ, ε; wx ' z r z x subjet to x 0 and equation (1.1) where the utility funtion U is assumed to be inreasing and onave, U 0, U 0 for π. Let the expeted utility maximizing level of fertilizer be denoted by x ( w, φ). The risk-averse farmer s first and seond order onditions are given by ( ( ε z) ) (1.6) [ ] x π x ( ), ; 0, EU = E U q x x w x 0 and (1.7) [ ] ππ ( ( ) ε z) 2 ( ) π ( ( ) ε z) EUxx = EU q x x x, ; w + Uq xx x x, ; 0. Note the seond order ondition in (1.7) is satisfied given the urvature assumptions on U and q. Suppose that interior solutions exist for both the risk-neutral and risk averse farmer so that (1.4) and (1.6) both hold with equality. Then, using Cov ( x, y ) to denote the ovariane between x and y, (1.6) an be rewritten as (, ε; z) (1.8) x ( ) (, q π x ) [ ] Cov U E q x x w=, EU π π ππ where Cov U EU (, q π x ) [ ] π is what Pope and Kramer (1979) defined as the marginal risk premium (MRP). Pope and Kramer (1979) defined the input x to be risk inreasing (dereasing) if the MRP is positive (negative). Or, x ( > ) x if the MRP is positive (negative) at the

76 68 optimum. Given positive marginal utility, the denominator is positive so that the sign of the MRP is the opposite of the sign of the ovariane between marginal utility and marginal produt. Differentiating the first order onditions (1.4) and (1.6) with respet to φ yields the following omparative stati results. (1.9) (1.10) x Cov q = φ E q (, xx μ) [ ] xx ( ππ x ( ( ), ε; ) z, μ ) 2 ( ( ), ε; z) + π xx (, ε; z) Cov U q x x w x = φ E U ( q ππ x x x w) U q x( x ) The signs of (1.9) and (1.10) are equal to the signs of the numerators, beause the denominator of eah term is negative by the seond order onditions of the risk-neutral and risk-averse farmer s maximization problems, respetively. The sign of equation (1.9) depends on the sign of qxxx, with x > ( < ) 0 φ as q > ( < ) 0. Similarly, a suffiient xxx ondition for x > ( < ) 0 φ is that q and U πππ > ( < ) 0. These are analogous to the results xxx shown by Rothshild and Stiglitz (1971). Assuming onvex (onave) marginal utility and onvex (onave) marginal produt implies that the optimal level of input use when φ > 0 is greater (less) than when φ = 0. However, one of the purposes of this analysis is to ompare the risk-averse farmer s optimal input use under input unertainty x ( w, φ 0) appliation without input unertainty x ( w, φ 0) > to the risk-neutral farmer s optimal fertilizer = (i.e. the passive optimum). To

77 69 aomplish this task experimental data on yield response to nitrogen is used to estimate a yield response funtion for orn. Data and Estimation Corn prodution or yield, q, is modeled as a funtion of soil nitrogen x and a stohasti error term ε. Following Just and Pope (1979), the prodution funtion is the sum of a mean and variane omponent whih are both funtions of the input x. The error term enters multipliatively with the variane omponent and is assumed to be distributed aording to the standard normal distribution. (1.11) q= f ( x) + h 2 ( x) ε, ε ~ N( 0,1 ) 1 This of ourse implies that the variane of output is equal to h( x ) and the effet of input use on output variability is given by ( ) Var q = h x x. The error term in the prodution funtion is assumed to apture the effets of exogenous fores, suh as weather, while the sale of this variability is determined endogenously by input levels through the variane funtion omponent. Thus, this speifiation aptures the endogeneity of risk emphasized by Hurley, Mithell, and Rie (2004), where risk is loosely defined as yield variability around the mean. Furthermore, the marginal produtivity of an input, aptured by f, does not impose any a priori restritions on that input s effet on yield variability. The effet of input levels on yield variability depends on the sign of h 12 x. h > is a suffiient ondition for the MRP 0 12 Note that x ; Ramaswami 1992). assuming onave utility (Pope and Kramer

78 70 The data used for this study omes from the earlier works of Binford, Blakmer, and Cerrato (1992) and Blakmer et al. (1989). Subsets of the data have also been used by Babok and Hennessy (1996) and Babok and Blakmer (1994, 1992). The data set ontains information on orn yields, nitrogen fertilizer appliation rates, and results from a late spring soil nitrogen test for 15 experiment stations aross the state of Iowa olleted from 1985 to A signifiant amount of weather variability is inluded in the data with years of exellent growing onditions and high yield levels (1987, 1990) and years of extremely poor growing onditions and low yield levels (1988 drought). All input levels, other than applied nitrogen fertilizer, were held onstant and at non-limiting levels aross years and sites to isolate the effets of nitrogen fertilizer on orn yields. Additionally, both ontinuous orn (orn-orn) and orn following soybeans (orn-soybean) rotations were examined. The orn-soybean rotation data onsisted of a total of 750 observations, while the orn-orn data inluded 1248 observations. Data on ontinuous orn overed all 6 years and all 15 experiment station loations in the full data set. Data for the orn-soybean rotation was only available for 8 experiment station sites over a 4 year period ( ). Nitrogen fertilizer rates ranged from zero to 300 pounds per are of nitrogen fertilizer in lb. inrements, with three repetitions of eah appliation rate performed annually at eah experiment station site. A late spring soil nitrate test was also onduted and reorded to determine the level of nitrogen, in parts per million (ppm), available in the soil for plant growth. The available soil nitrate levels were highly (but not perfetly) orrelated with fertilizer appliation rates (0.70 orrelation oeffiient). Soil nitrate levels ranged from 3.8 to

79 ppm, with an average of 27.5 (30.7) ppm and standard deviation of 18.1 (16.3) ppm in the orn-orn (orn-soybean) data. Yields ranged from 4 to 218 bushels per are, with an average of (143.2) bushels per are with a standard deviation of 45.9 (39.7) bushels per are in the orn-orn (orn-soybean) data. Overall, average soil nitrate and yield levels were higher and less variable for the orn-soybean rotation data whih is onsistent with previous findings regarding the benefits of rop rotation. Using a three stage approah outlined by Just and Pope (1979), translog funtional forms were estimated for the mean and variane omponents in the prodution funtion. The translog form was ompared to alternative speifiations, inluding the Cobb-Douglas, linear, and quadrati speifiations. A likelihood ratio test rejeted the Cobb-Douglass form, while the linear and quadrati speifiations were found to provide an inferior fit to the translog form when plotted against the data. While basi soil harateristis may have differed between experiment sites, it is noted in Blakmer et al. (1989) and Binford, Blakmer, and Cerrato (1992) that muh are was taken to make eah of the observations as omparable to eah other as possible. Tillage, planting, and harvest praties were oordinated to be nearly idential aross the experiment sites with regard to both methods and timing. Of ourse, heterogeneity due to weather variability and site speifis suh as soil type were not able to be ontrolled. Dummy variables were inluded to apture site (d s ) effets 13, In an earlier version of this paper dummies were also inluded to apture year effets. However, this had the effet of ontrolling for some of the exogenous risk farmers fae when they hoose input levels. Therefore, the year dummies were removed. Site dummies were left in the analysis, assuming site effets represent farmspeifi measures that would be known by the produer. Thanks to Sergio Lene and Brue Babok of Iowa State University for their omments on this issue. 14 Just and Pope (1979) use a variane omponents tehnique to apture time and site effets in their experimental data. They note that Maddala, and Wallae and Hussein disuss the advantages to variane

80 72 The first stage of the estimation proedure provides onsistent estimates of the mean yield omponent parameters by estimating the following equation using a non-linear least squares (NLS) estimator qis, = f xis, + εis, * = α0 xis, exp αxxlog xis. + αs' ds+ εis, *, 2 αx (1.12) ( ) ( ) 2 where ε * = h ( x ) 1 ε is, is, is, E εis, = E εis, x = 0 Given the assumptions on ε, the omposed error term ε* is normally distributed with a zero mean, but is heteroskedasti. Thus while the first-stage parameter estimates are unbiased and onsistent, they are not effiient. Moreover the standard errors annot be used for hypothesis testing due to the heteroskedasti nature of the error term (Greene 2003). 2 2 Noting that E ( ε, *) is = E h( xis, ) ε is, = h( xis, ) and (, *) E (, *) ε 2 2 is = ε is ξ i, where ξ i is suh that E [ ξ i ] = 1, the onsistent parameter estimates for the mean omponent from the first stage an be used to obtain onsistent estimates of the first stage residuals, ˆ ε is, *. The squared residuals an then be regressed, in a non-linear framework, on the nitrogen levels in a translog funtional form to obtain onsistent estimates of the parameters for the variane omponent of the prodution funtion. omponents versus the dummy variable approah. Wallae and Hussein note that asymptoti effiieny is only attained through the variane omponents approah with data where regressors repeat from year to year, whih is the ase with the applied nitrogen rates. However, we adopt available soil nitrogen as an independent variable, whih is ontinuous. Thus, dummy variables are assumed to be adequate to apture site effets. 15 The non-linear least squares estimation was arried out in MatLab using ode written by the author. The model was estimated using the linearized approah and onvergene riterion disussed in Greene (2003).

81 73 Finally, the third stage of estimation is done within a generalized NLS estimation proedure, where the ovariane matrix is estimated using the seond stage parameter estimates for the variane omponent. The third stage of the proedure re-estimates the parameters of the mean yield omponent, providing unbiased, onsistent, and effiient parameter estimates. The use of dummy variables in the model implies a simple heteroskedasti ovariane struture where the non-diagonal elements of the estimated ovariane matrix are equal to zero. Prodution funtions for the orn-orn and ornsoybean rotations were estimated to separate the rotational effets. The estimation results are reported in the next setion. Estimation Results Table 1 reports the parameter estimates (t-statistis) for the orn-soybean rotation data. Table 2 reports the parameter estimates (t-statistis) for the ontinuous orn data. The first and third set of olumns in eah table report the first-stage and third-stage parameter estimates, respetively, for the mean omponent of the prodution funtion. The seond set of olumns in the tables report the oeffiient estimates for the variane omponent of the prodution funtion. For eah rotation, an unrestrited model was estimated inluding a full set of site dummies. A restrited model was then estimated, eliminating the site dummies whih were not statistially signifiant at a 5% signifiane level in the unrestrited model. Comparisons of yield plots aross years and sites were onsistent with the statistial signifiane of the dummy estimates. Site 9 was arbitrarily hosen as the baseline site. Negative effets for site

82 74 13 were shown for orn following soybeans, while negative effets at sites 13 and 4 were signifiant for the ontinuous orn data 16. Table 1. Prodution Funtion Estimates, Corn-Soybean Rotation 1 st Stage 2 nd Stage 3 rd Stage Unrestrited Restrited Unrestrited Restrited Unrestrited Restrited α 0 α x α xx Site dummies (0.30) (23.71) (-69.23) (0.26) (19.90) (-56.35) (0.002) (0.578) (-1.003) (0.003) (0.902) (-2.412) (0.28) (21.43) (-61.91) (0.24) (17.94) (-49.79) d (0.445) (0.546) d (1.282) (1.247) d (-0.391) (-0.436) d (-0.393) (-0.349) d (-1.689) (-2.275) (-1.669) (-2.288) d (0.777) (0.779) d (-0.081) (-0.037) Average yield levels were found to be inreasing and onave funtions of available soil nitrate for both rotations. Yield variability was also found to be an inreasing funtion of available soil nitrate for both rotations. Figures 1 plots the yield and variane funtions for ontinuous orn and orn-soybean rotations. The yield funtions for soil nitrate tend to flatten out as the level of soil nitrate inreases beyond 60 ppm. For the ontinuous orn data, yield variability inreases almost linearly with soil nitrate (above 10 ppm), while yield variability 16 The estimation results should not be interpreted as any type of best-fit yield response funtion for fertilizer reommendations. This was not the goal of the analysis. Moreover, the experimental data is up to twenty years old and is not expeted to reflet yield response to nitrogen fertilizer for more modern hybrid genetis.

83 75 is estimated to be inreasing but onave in soil nitrate. The estimation results onfirm that yield variability does indeed inrease with the level of available soil nitrate (i.e. h x > 0 ) for both rop rotations. Therefore, by the Pope and Kramer definition, nitrogen would be onsidered a risk-inreasing input to orn prodution. The next setion examines the effets on input on unertainty on farmers optimal appliation rates for nitrogen fertilizer Mean Yield Yield Variane Soil Nitrate (ppm) Yield, orn-orn Variane, orn-orn 0 Yield, orn-soy Variane, orn-soy Figure1. Estimated yield and variane funtions, orn-orn and orn-soy rotations Table 2. Prodution Funtion Estimates, Corn-Corn Rotation 1 st Stage 2 nd Stage 3 rd Stage Unrestrited Restrited Unrestrited Restrited Unrestrited Restrited α 0 α x α xx Site dummies d 1 d 3 d (1.094) (61.85) ( ) (-0.598) (0.551) (-2.122) (0.953) (53.28) ( ) (-2.051) (0.003) (0.747) (2.41) (0.004) (1.668) (-0.554) (1.071) (61.16) ( ) (-0.565) (0.689) (-2.063) (0.911) (50.66) ( ) (-2.009)

84 76 Table 2. Prodution Funtion Estimates, Corn-Corn Rotation (ontinued) 1 st Stage 2 nd Stage 3 rd Stage Unrestrited Restrited Unrestrited Restrited Unrestrited Restrited d 5 d 6 d 8 d 10 d 11 d 12 d 13 d 14 d 15 d 16 d (0.767) (0.440) 2.41 (0.097) (-0.094) (0.484) (-0.365) (-2.788) (-0.142) (-0.101) (0.790) (-0.015) Optimal Fertilizer Rates (-3.255) (0.738) 8.52 (0.333) 1.39 (0.056) (-0.065) 8.80 (0.395) (-0.243) (-2.604) (-0.085) (-0.098) (0.804) (0.0009) (-3.097) For any fertilizer appliation rate, the level of available soil nitrate in late spring is stohasti. Using the same experimental data, Babok and Blakmer (1992) estimated three-parameter gamma distributions for the distribution of available soil nitrate x onditional on fertilizer appliation rates. They speified the gamma distribution s parameters as linear funtions of the appliation rates x. (1.13) p( x ) = θ 1 ( x ) ( x ) θ λ Γ( θ) γ exp γ / λ where θ > 0, λ > 0, γ > 0

85 77 θ = θ + θ x 0 1 λ = λ + λ x 0 1 γ = γ + γ x 0 1 Maximum likelihood estimates for the parameter funtions were omputed to onfirm and repliate the results of Babok and Blakmer (1992). Alternative speifiations were also examined, but likelihood ratio tests showed that their linear speifiations best fit the experimental data. The parameter estimates (t-statistis) are equivalent to those reported in Babok and Blakmer (2002) (Table 1) and are reported in table 3. As in their study, the value of γ 0 was restrited to equal to zero, implying a lower bound of zero ppm for soil nitrate onentration when no fertilizer is applied. Both the mean and variane of soil nitrate levels are inreasing in the level of fertilizer applied for both rotations. The skewness exhibited in the soil nitrate distributions also inreases with the amount of nitrogen fertilizer applied. See Babok and Blakmer (1992) for plots of the distributions at varying nitrogen fertilizer appliation rates. Table 3. Parameter Estimates (t-statistis) for Soil Nitrate Distributions Rotation Continuous Corn Corn-Soybean θ 0 θ 1 λ 0 λ 1 γ (16.4) (4.78) (14.1) (17.4) (5.7) 5.94 (10.1) (2.23) (11.6) (10.4) (7.9) Using the soil nitrate distribution parameter estimates, soil nitrate draws were generated over a grid of nitrogen appliation rates. For eah soil nitrate draw, expeted profit

86 78 and utility were omputed assuming a onstant absolute risk aversion (CARA) utility funtion for the risk-averse farmer and using a random draw of standard normal deviates for ε with the estimated prodution funtion 17. This defined distributions of expeted profits and utilities for eah nitrogen appliation rate. (1.14) U ( π ) = exp[ βπ ] The nitrogen appliation rates whih maximized expeted profits and expeted utility were found for both rotation types and over a range of risk aversion levels. The relative prie of nitrogen fertilizer was set to alibrate optimal appliation rates lose to urrent reommendations under risk-neutrality and ertainty with respet to soil nitrate levels for both ontinuous orn and orn-soybean rotations 18. These results were ompared to the expeted profit and utility maximizing appliation rates under ertainty with respet to soil nitrate availability 19. Following Babok, Choi, and Feinerman (1993) the oeffiient of absolute risk aversion was alibrated to to yield a risk premium ratio of 25% for both the ornsoybean and ontinuous orn rotations. Risk aversion levels above and below this alibrated level were examined. The effet of unertainty with respet to soil nitrate levels on optimal fertilization rates will depend on the urvature implied by the prodution and utility funtions (Rothshild and Stiglitz 1971). The estimated prodution funtion implies onvex marginal produt, although no a priori restritions were made to ensure this. Additionally, CARA utility funtions belong to the family of non-dereasing absolute risk aversion funtions, 17 For eah draw of soil nitrate, the yield distribution is determined by the variane omponent of the yield funtion and the draw for the exogenous risk omponent, ε. 18 Current reommendations for appliation rates in southern Minnesota are around 130 (170) lbs/are for orn following soybeans (ontinuous orn). 19 For the ase of input ertainty, the level of soil nitrate was set equal to the mean of the gamma distribution implied by the fertilizer appliation rate.

87 79 implying onvex marginal utility. Based on equations (1.9) and (1.10), the optimal nitrogen appliation rates were expeted to be larger under soil nitrate onentration unertainty. Table 4 reports the optimal nitrogen appliation rates over varying risk attitudes for the orn-soybean rotation. The third olumn reports optimal appliation rates when soil nitrate is a deterministi funtion of applied nitrogen ( φ = 0 ). The fourth olumn reports optimal appliation rates when soil nitrate is a stohasti variable distributed aording to the gamma distribution onditional on the fertilizer appliation rate aording to relation (1.13) and the parameter estimates reported in table 3. The input unertainty ases reported in the fourth olumn orrespond to φ > For all levels of risk aversion the optimal appliation rate for the risk averse farmer ˆx is less than that of the risk-neutral farmer ˆx. These results apply for both ases of input ertainty ( φ = 0 ) and unertainty ( φ > 0 ), and illustrate the fat that nitrogen fertilizer is risk-inreasing. However, omparing the optimal appliation rates under input ertainty and input unertainty shows that farmers with all risk preferenes apply more fertilizer when soil nitrate is stohasti ( φ > 0 ). Optimal appliation rates under input unertainty are 8-17% greater than when soil nitrate is ertain given applied nitrogen fertilizer. Note also that for farmers with risk aversion oeffiients less than 0.001, the optimal appliation rate under input unertainty is greater than the optimal rate for the risk-neutral farmer when soil nitrate is ertain (the passive optimum). Given unertainty with respet to soil nitrate onentrations, some risk-averse farmers may optimally over-apply nitrogen fertilizer. The 20 The fourth olumn in tables 4 and 5 report optimal appliation rates when φ > 0. However, eah row in tables 4 and 5 represent different values of φ onditional on the amount of fertilizer applied by the relation (1.13) and the parameter estimates in table 3.

88 80 result that a farmer would over-apply nitrogen fertilizer under soil nitrate unertainty depends on the urvature of the yield response and utility funtions (Rothshild and Stiglitz 1971). A onvex marginal produt urve implies that, on average, the gains from an additional unit of fertilizer are greater than the losses or the prie of the additional unit, whih is the intuition behind the result of Rothshild and Stiglitz (1971). x Table 4. Optimal Nitrogen Appliation Rates (lbs/are), Corn-Soybean Rotation β φ = 0 φ > 0 x Risk Neutrality The optimal nitrogen appliation rates for ontinuous orn are reported in Table 5. Again, as the level of risk aversion inreases the optimal appliation rates deline beause fertilizer is a risk-inreasing input. However, appliation rates for farmers with all types of risk preferenes inrease by 2-6% when input unertainty is introdued. As with the ornsoybean rotation results, farmers with risk aversion oeffiients less than have optimal appliation rates greater than risk-neutral farmers under input ertainty. x Table 5. Optimal Nitrogen Appliation Rates (lbs/are), Corn-Corn Rotation β φ = 0 φ > 0 x Risk Neutrality

89 81 Farmer Survey Results In addition to the empirial analysis of the Blakmer nitrogen response data, a survey was distributed to Midwestern orn farmers to analyze their risk preferenes and subjetive beliefs about the relationship between fertilizer use and yield risk. The results thus far have been based on assumed preferene relationships and an estimated prodution funtion. The olletion of primary data, by way of the survey, was done in an attempt to gain additional insight into whether the assumptions made on farmer preferenes are onsistent with the ations and beliefs of atual farmers. The survey results were intended to be used to try and validate or disredit the empirial and theoretial findings of this and previous studies. The survey was distributed to farmers in Illinois, Iowa, Missouri, Minnesota and North Dakota during August and September of The majority of the surveys were ompleted voluntarily by orn produers who attended informational meetings held by Deision Commodities 21 of Ames, IA. Additionally, a small portion of the surveys were personally administered to farmers who were delivering grain to loal grain elevators in southeastern Minnesota. Farmers were paid $5 to omplete the survey, whih was omprised of seven questions pertaining to the produer s risk preferenes and subjetive beliefs about the relationship between fertilizer use and orn yields. A sample survey is provided in the Appendix. A total of 130 responses were obtained, with all respondents filling out the entire survey. Survey results were also ompiled at the state level. The state level results did not differ from those of the entire population at ommon levels of statistial signifiane and are not reported. 21 Deision Commodities is a ompany based in Ames, IA. Deision Commodities offers market-based index ontrats as risk management and marketing tools for farmers throughout Iowa, Illinois, Missouri, and North Dakota.

90 82 The survey design was modeled after that of Kahneman and Tversky s (1979) famous ritique of expeted utility theory. The first two questions of the survey were onerned with the farmer s preferenes over monetary gambles with expeted outomes of equal value, and were inluded as warm-up questions for the farmer. The perentage of farmers from the total sample who hose eah option is provided in parentheses following eah hoie. An asterisk denotes preferene at a 5% signifiane level. Questions 1 and 2 asked the produer to hoose between two monetary gambles. Question 1: A. 50% hane of winning $1000, 50% hane of winning nothing (20%) B. $450 with ertainty (80%*) Question 2: A. 4% hane of winning $12,000 and a 96% hane of losing $500 (40%) B. 95% hane of winning $500 and a 5% hane of losing $9,500 (60%*) The results of question 1 imply a strong preferene for the ertain amount versus a 50/50 gamble with a greater expeted value ($500 vs. $450), implying risk aversion among the surveyed orn farmers. In question 2 there is a statistially signifiant preferene for gamble B. Both gambles A and B in question two have expeted values of zero. Gamble A provides a small hane of a large gain and a large hane of relatively small loss, whereas gamble B has a large hane of a small gain and small hane of a large loss. Noting that the variane of gamble A is greater than the variane of gamble B, farmer preferenes are onsistent with expeted utility from a mean-variane standpoint. Questions 3 through 5 asked the farmer to ompare yield gambles rather than monetary gambles. The survey asked respondents to answer the questions assuming that they

91 83 did not have aess to any type of government support programs, suh as federal rop insurane. If the farmer answered the questions assuming he would be insured, low yield senarios may not be valued at the full loss level. Question 3 gives the farmer the options of a yield senario with a large hane of yields slightly above the mean (180 bu/are) and a small hane of very low yields (i.e. a rop failure), and a senario with a small hane of a very large yield realization (i.e. a bumper rop) and relatively large hane of a yield realization slightly below the mean. Farmers had a statistially signifiant preferene for the small hane of a very large yield gamble (B) over the small hane of a very low yield gamble (A). This would violate riskaversion, in mean-variane terms, as the expeted yield in both gambles is equal (180 bu/are) with gamble B having a higher variane. The majority of surveyed farmers prefer yield risk above the mean relative to yield risk below the mean. Question 3: A. 95% hane of 185 bu/are and a 5% hane of 85 bu/are (41%) B. 4% hane of 300 bu/are and a 96% hane of 175 bu/are (59%*) Question 4: A. 95% hane of 185 bu/are and a 5% hane of 85 bu/are (18%) B. 180 bu/are (82%*) Question 5: A. 4% hane of 300 bu/are and a 96% hane of 175 bu/are (48%) B. 180 bu/are (52%) Questions 4 and 5 asked the farmers to ompare the yield gambles from question 3 to ertain yields at the mean level. In question 4 there is a signifiant preferene for the ertain

92 84 mean yield over the small hane of a very low yield gamble. However, the results for question 5 show no strong signifiant preferene for the ertain mean yield and the small hane of a very large yield gamble. Expeted utility-maximizers would stritly prefer the ertain mean yield over either gamble. While the farmers have a very strong preferene for the ertain mean yield over the small hane of a very low yield gamble, they are statistially indifferent between the ertain mean yield and a small hane of very high yields gamble. This provides further evidene to the hypothesis that farmers do not onsider above average yield realizations as risky. Questions 3 through 5 also allowed for testing of the transitivity axiom of expeted utility theory. Respondents who answered A-B-A or B-A-B for questions three through five violate transitivity. For example, denoting the utility of a gamble where the agent reeives payout x (y) with probability π 1 ( π 2 ) by U( π1, x; π 2, y), hoosing option A for question 3 implies U( 0.95,185;0.05,85) U( 0.04,300;0.96,175) question 4 implies that U( 1,180) U( 0.95,185;0.05,85) question 5 implies U( 0.04,300;0.96,175) U( 1,180) U ( 0.04,300;0.96,175) U( 0.95,185;0.05,85) >. Similarly the hoie of senario B in > and the hoie of senario A in >. The latter two relations imply >, violating the respondent s preferene ranking implied by hoosing senario A in question The A-B-A type of transitivity violation was done by 14 (10.8%) of the respondents. Only one (<1%) farmer s response qualified as a B- A-B type of transitivity violation. Finally, the survey asked two questions regarding the farmer s subjetive beliefs of the effet of fertilizer appliation on yield risk and variability. While a slight majority of the 22 Indifferene over all three gambles is also a possibility and would not imply a violation of transitivity.

93 85 surveyed farmers responded that they believe fertilizer inreased yield variability, the signifiant majority of produers believe that fertilizer does not inrease yield risk. The survey results imply that farmers do not equate yield risk with yield variability. Question 6: Do you think applying more nitrogen fertilizer inreases your yield risk? Yes (28%) No (72%*) Question 7: Do you think applying more nitrogen fertilizer inreases your yield variability? Yes (56%) No (44%) Conlusions The prodution literature is rih with studies examining the relationship between prodution unertainty and optimal input use. Many authors have onluded that fertilizer is a risk-inreasing input aording to the original definition of Pope and Kramer (1979) using experimental yield response data (Ramaswami 1992; Just and Pope 1979). Empirial evidene tends to support these laims in that yield variability is generally found to be inreasing in the amount of fertilizer applied (Roumasset et al. 1989). The Pope and Kramer definition is also applied in theoretial analyses where produtive inputs are generally found to be risk inreasing (Hurley et al. 2004; Hurley and Babok 2003). However, there are a number of empirial studies whih illustrate that farmers onsistently over-apply fertilizer (Yadav, Peterson, and Easter 1997; NRC 1993). The overappliation of fertilizer is generally motivated as an at of self-protetion (Ehrlih and Beker 1972) in response to unertainty with respet to growing onditions (Below and Brandau 2001; Babok 1992) and/or input availability (Babok and Blakmer 1992, 1994),

94 86 or as the result of (inorret) subjetive beliefs regarding yield response (SriRamaratnam et al. 1989). Obviously these views present somewhat of a paradox. This paper has shown that an input, nitrogen fertilizer, an be simultaneously defined as risk inreasing and over-applied by both risk-neutral and risk-averse produers. All else equal, the risk-averse farmer will hoose appliation rates below those of the risk-neutral farmer. This effet is attributed to the fat that the variane of yields is inreasing in the level of fertilizer applied and the onavity of the utility funtion, the ombination of whih implies a positive marginal risk premium. However, if applied fertilizer is assumed to be an imperfet proxy for stohasti soil nutrients both risk-averse and risk-neutral farmers may apply fertilizer at rates exeeding the passive optimum, where the passive optimum is defined as the profit-maximizing appliation rate when soil nutrients are assumed to be a deterministi funtion of applied fertilizer. This response is due to the onvexity of the (estimated) marginal produt and (assumed) utility funtions (Rothshild and Stiglitz 1971). In either ase, the optimal response of the agent (farmer) to unertainty relies on the urvature of the prodution tehnology and preferene relation, both of whih an be easily manipulated by the researher through estimation speifiation (tehnology) or assumption (tehnology and preferenes). Furthermore, the use of various risk-inreasing definitions by authors an lead to onfusing and onfliting results in the literature. This paper has learly shown the differene between the Pope and Kramer (1979) and Rothshild and Stiglitz (1971) approahes to defining the relationship between risk and input use. Pope and Kramer s definition ompares optimal input use aross risk preferenes while the Rothshild and Stiglitz approah is onerned with how the introdution of unertainty may effet

95 87 optimal hoies. The empirial results from this analysis show that fertilizer an be defined as risk-inreasing by the Pope and Kramer definition and also be over-applied by both riskaverse and risk-neutral farmers due to the urvature properties of the prodution and utility relations first derived by Rothshild and Stiglitz (1971). In addition to the empirial analysis, the results from a farmer survey were presented. The survey was designed to eliit information on farmer preferenes over yield outomes as well as their subjetive beliefs about the relationship between risk and input use. The survey results imply that while farmers do prefer ertain outomes to gambles, they disount risk above the mean less than risk below the mean. Yield gambles with small hanes of very large yield realizations are preferred to those with small hanes of yields that would be defined as rop failures (where the expeted yield is held onstant). Moreover, while roughly 50% of the farmer respondents reognize that inreased fertilizer use may inrease yield variability, a muh smaller perentage (28%) of respondents feel that additional fertilizer inreases yield risk. The survey results imply that the appliation of theoretial frameworks other than expeted utility theory to deision making in agriulture are warranted. The appliation of behavioral methods in the general eonomis literature is rapidly growing. Similarly, there have been a signifiant number of reent studies applying experimental based behavioral methods to agriultural eonomis. Using these results as the basis for future models of hoie under unertainty in agriulture provides signifiant potential for further researh and the re-examination of results obtained from previous theoretial frameworks (i.e. expeted utility theory).

96 88 Referenes Babok, B.A The Effets of Unertainty on Optimal Nitrogen Appliations. Review of Agriultural Eonomis 14(2): Babok, B.A. and A.M. Blakmer The Ex-Post Relationship Between Growing Conditions and Optimal Fertilizer Levels. Review of Agriultural Eonomis 16(3): The Value of Reduing Temporal Input Nonuniformities. Journal of Agriultural and Resoure Eonomis 17(2): Babok, B.A. and D.A. Hennessy Input Demand Under Yield and Revenue Insurane. Amerian Journal of Agriultural Eonomis 78(2): Babok, B.A., E.K. Choi, and E. Feinerman Risk and Probability Premiums for CARA Utility Funtions. Journal of Agriultural and Resoure Eonomis 18(1): Below, F. and P. Brandau How Muh Nitrogen Does Corn Need? 2001 Agronomy Field Day Presentation, University of Illinois Extension, University of Illinois Urbana- Champaign, Urbana, IL. Berk, P. and G. Helfand Reoniling the Von Liebig and Differentiable Crop Prodution Funtions. Amerian Journal of Agriultural Eonomis 72: Binford, G.D., A.M. Blakmer, and M.E. Cerrato Relationships Between Corn Yields and Soil Nitrate in Late Spring. Agronomy Journal 84: Blakmer, A.M., D. Pottker, M.E. Cerrato, and J. Webb Correlations Between Soil Nitrate Conentrations in Late Spring and Corn Yields in Iowa. Journal of Prodution Agriulture 2: Cate, R.B. and L.A. Nelson A Simple Statistial Proedure for Partitioning Soil Test Correlation Data into Two Classes. Soil Si. So. Amer. Pro. 35: Ehrlih, I. and G.S. Beker Market Insurane, Self-Insurane, and Self-Protetion. The Journal of Politial Eonomy 80(4): Feder, G Pestiides, Information, and Pest Management Under Unertainty. Amerian Journal of Agriultural Eonomis 61(1): Greene, W.H Eonometri Analysis, 5 th Edition. Prentie Hall, Upper Saddle River, New Jersey. Gallagher, P US Soybean Yields: Estimation and Foreasting with Nonsymmetri Disturbanes. Amerian Journal of Agriultural Eonomis 69(4):

97 89 Hartman, R "Fator Demand with Output Prie Unertainty." Amerian Eonomi Review 66(4): Hurley, T.M. and B.A. Babok Valuing Pest Control: How Muh is Due to Risk Aversion? Risk Management and the Environment: Agriulture in Perspetive. Kluwer Aademi Publishers, Dordreht, Netherlands Hurley, T.M., P.D. Mithell, and M.E. Rie Risk and the Value of Bt Corn. Amerian Journal of Agriultural Eonomis 86(2): Ishii, Y On the Theory of the Competitive Firm Under Prie Unertainty: Note. The Amerian Eonomi Review 67(4): Just, R.E. and R.D. Pope Prodution Funtion Estimation and Related Risk Considerations. Amerian Journal of Agriultural Eonomis 61(2): Kahneman, D. and A. Tversky Prospet Theory: An Analysis of Deision Under Risk. Eonometria 47(2): Lanzer, E.A. and Q. Paris A New Analytial Framework for the Fertilization Problem. Amerian Journal of Agriultural Eonomis 63(1): MaMinn, R.D. and A.G. Holtmann Tehnologial Unertainty and the Theory of the Firm. Southern Eonomi Journal 50: Maddala, G.S The Likelihood Approah to Pooling Cross-Setion and Time-Series Data. Eonometria 39(6): Moshini, G Nonparametri and Semiparametri Estimation: An Analysis of Multiprodut Returns to Sale. Amerian Journal of Agriultural Eonomis 72: National Researh Counil (NRC) Board on Agriulture. Soil and Water Quality: An Agenda For Agriulture. National Aademy Press. Washington, DC. NRDC Natural Resoures Defense Counil. Think Before You Drink: Update. Washington, DC. Paris, Q The Von Liebig Hypothesis. Amerian Journal of Agriultural Eonomis 74(4): Paris, Q. and K. Knapp Estimation of Von Liebig Response Funtions. Amerian Journal of Agriultural Eonomis 71(1): Pope, R.D. and R.A. Kramer Prodution Unertainty and Fator Demands for the Competitive Firm. Southern Eonomi Journal 46(2):

98 90 Ramaswami, B Prodution Risk and Input Deisions. Amerian Journal of Agriultural Eonomis 74(4): Ratti, R.A. and A. Ullah Unertainty in Prodution and the Competitive Firm. Southern Eonomi Journal 42: Rothshild, M. and J.E. Stiglitz Inreasing Risk I: A Definition. Journal of Eonomi Theory 2: Inreasing Risk II: Its Eonomi Consequenes. Journal of Eonomi Theory 3: Roumasset, J., M. Rosegrant, U. Chakravorty, and J. Anderson Fertilizer and Crop Yield Variability: A Review. Variability in Grain Yields, ed. J. Anderson and P. Hazell. Baltimore and London: The Johns Hopkins University Press. Sandmo, A On the Theory of the Competitive Firm Under Prie Unertainty. The Amerian Eonomi Review 61(1): Sawyer, J., E. Hafziger, G. Randall, L. Bundy, G. Rehm, and B. Joern Conepts and Rationale for Regional Nitrogen Rate Guidelines for Corn. Iowa State University Extension Publiation PM 2015 (April 2006), Ames, IA. SriRamaratnam, S., D.A. Bessler, M.E. Rister, J.E. Matoha, and J. Novak Fertilization Under Unertainty: An Analysis Based on Produer Yield Expetations. Amerian Journal of Agriultural Eonomis 69(2): Von Liebig, J Organi Chemistry in its Appliation to Agriulture and Physiology. Playfair, London. Wallae, T.D. and A. Hussein The Use of Error Components Models in Combining Cross Setion with Time Series Data. Eonometria 37(1): Yadav, S.N., W. Peterson, and K.W. Easter Do Farmers Overuse Nitrogen Fertilizer to the Detriment of the Environment? Environmental and Resoure Eonomis 9(3):

99 91 Appendix Sample Farmer Survey The first two survey questions ask you to ompare different situations involving monetary outomes. 1. Whih situation would you prefer? A: 50% hane of winning $1,000 and a 50% hane of winning nothing B: Winning $450 with ertainty 2. Whih situation would you prefer? A: 4% hane of winning $12,000 and a 96% hane of losing $500 B: 95% hane of winning $500 and a 5% hane of losing $9,500 The following three survey questions ask you to ompare different senarios for the orn yields on your farm for a given year. Assume you do NOT have aess to government support programs, suh as Federal rop insurane, when answering these questions. 3. Whih situation would you prefer for your average orn yields? A: 95% hane of 185 bu/are and a 5% hane of 85 bu/are B: 4% hane of 300 bu/are and a 96% hane of 175 bu/are 4. Whih situation would you prefer for your average orn yields? A: 95% hane of 185 bu/are and a 5% hane of 85 bu/are B: 180 bu/are 5. Whih situation would you prefer for your average orn yields? A: 4% hane of 300 bu/are and a 96% hane of 175 bu/are B: 180 bu/are The final two questions of this survey ask you about the relationship between nitrogen fertilizer and your orn yields.

100 92 6. Do you think applying more nitrogen fertilizer inreases your yield risk? Yes No 7. Do you think applying more nitrogen fertilizer inreases your yield variability? Yes No Note: The results of this ISU study are for researh purposes only and your identity will remain onfidential. The results of this survey will not be used for any type of ommerial purpose suh as the sales or promotion of any produt or tehnology

101 93 CHAPTER 4. A BAYESIAN AND SPATIAL APPROACH TO WEATHER DERIVATIVES: A FRAMEWORK FOR DEVELOPING REGIONS A paper submitted to the Amerian Journal of Agriultural Eonomis Niholas D. Paulson 1, Chad E. Hart, and Dermot J. Hayes Abstrat There are a wide variety of farm and ounty level insurane programs available to livestok and rop produers in the United States and Canada. These programs rely on reliable long-term data for atuarial soundness. However, the expansion of rop insurane programs in other areas has been limited. Reently, the use of weather indexes as risk management tools has seen inredible growth. While the demand for weather based agriultural insurane in developing regions is limited, there exists signifiant potential for the use of weather indexes in developing regions (Varangis, Skees, and Barnett 2002). This paper proposes a Bayesian rainfall model whih uses spatial kriging and Markov Chain Monte Carlo tehniques to estimate unbiased rainfall histories from sparse historial data. The estimated history an then be used to develop atuarially sound weather based insurane. The method is validated using a rih data set of historial rainfall in Iowa. An example drought insurane poliy is presented. The fair rates are alulated using Monte Carlo analysis and a historial analysis is arried out to assess potential poliy performane. While the appliation is speifi to forage prodution in Iowa, our method provides a framework whih ould easily be applied to other regions, suh as developing areas, and for other rops. 1 Lead researher and primary author.

102 94 Introdution Crop and livestok produers in the United States have aess to a rih variety of yield and revenue insurane programs. Some of these produts insure farm-level yields or revenues and others insure against delines in ounty-level yields and revenues. The atuarial suess of these produts depends on the availability of aurate yield histories at the farm and/or ounty levels and on effiient futures markets. Yield histories are needed to provide a yield guarantee, and futures pries are used to provide the prie omponent of a revenue guarantee. These produts are subsidized, and the federal government provides reinsurane. The provision of federal reinsurane is neessary in part beause private setor reinsurers are wary of reinsuring the kind of systemi risks that an exist in agriulture (Miranda and Glauber 1997). An equally rih range of produts is available in Canada, with provinial governments rather than federal governments providing institutional support. Crop insurane (other than hail) has not developed at a similar rate outside of the United States and Canada. Possible reasons inlude a lak of government provision of reinsurane, a lak of aurate and long-term yield data at the farm or regional level, and a lak of interest among produers due to the availability of other revenue support programs. An alternative to traditional insurane based on farm or area yields is that of agriultural insurane based on weather events. Varangis, Skees, and Barnett (2002) note that while demand for weather risk management tools for agriulture in developed nations is limited beause of the availability of subsidized insurane programs, there is onsiderable potential for their use in developing nations. This is attributable to many fators, inluding the lak of subsidized insurane programs and greater relative dependene on agriulture in developing areas, as well as the fat that weather-related disasters have a muh larger adverse effet on

103 95 eonomies in developing regions. Moreover, sine weather derivatives fall into the ategory of index produts, the osts assoiated with administering their use are relatively low and adverse seletion and moral hazard problems are virtually eliminated. Our working hypothesis is that the provision of rop or revenue insurane programs would benefit agriultural produers in ountries that do not already have aess to these programs (i.e. developing areas). This is true beause insurane programs typially inrease the ertainty equivalent returns (CER) by a multiple of the fair premium value (Hart, Babok, and Hayes 2001) 2. It is also true beause ongoing multilateral and bilateral trade agreements are having the effet of reduing other forms of inome stabilization, and beause the provision of inome insurane programs an be viewed as green box support (nontrade-distorting) by the World Trade Organization (WTO 1994). This artile addresses two of the prinipal barriers to the international expansion of rop insurane programs previously desribed. The first barrier is the lak of high-quality, long-term data that an be used for insurane program development. We diretly address this by developing a method to interpolate among available weather stations to measure atual rainfall at a partiular site. We propose a Bayesian rainfall model that uses reently developed spatial kriging and Markov Chain Monte Carlo tehniques. Using the proposed method, dense unbiased rainfall histories an be estimated from a sparse grid of historial data. The rainfall distributions that we generate an then be used to find the atuarially fair rates for a weather derivative that is designed to indemnify produers against drought. 2. Note that this result was shown for the speifi ase of insurane for livestok and was not generalized to all insurane programs.

104 96 The seond problem is finding a private setor substitute for the reinsurane that is urrently provided by the government in the United States and Canada. While we do not expliitly outline a reinsurane sheme, the method we propose generates rainfall distributions for non-sample sites with the spatial orrelation struture between observed loations. The availability of this spatial orrelation struture is key beause it provides the information needed by a reinsurer to separate systemi and non-systemi risks. In order to validate the proposed method, we apply it to a problem in whih the answer is already known. Iowa has a rih series of rainfall data from numerous weather stations, and this allows us to first predit the atual rainfall at eah station and then ompare the predited and atual values. We show that the method is aurate and unbiased and that we an suessfully unover the spatial orrelation struture aross sites. We also show that it is possible to develop and rate a pratial drought insurane poliy using the Iowa rainfall data. While the appliation is speifi to forage prodution in Iowa, the methods used an easily be generalized to rate a weather derivative for a variety of rops and regions, inluding developing areas. More importantly, spatial kriging an generate unbiased rainfall histories in areas for whih the density of historial data may be quite low. This would allow weather risk produts to be aurately pried for many developing areas for whih historial information may be sare. Literature Review Demand for Weather Derivatives in Agriulture The largest obstales faing development of weather derivative produts in developing areas are basis risk and the lak of historial weather data. Basis risk, in the speifi ase of

105 97 rainfall and its effet on agriulture, refers to the relationship between the preipitation measured at the weather station and the prodution or revenue on the farm. Basis risk is more problemati for individual purhasers whose risk exposure is more entralized (Varangis, Skees, and Barnett 2002). The users, or purhasers, of weather derivatives would like to minimize the basis risk involved with the use of weather data olleted at a site that does not neessarily orrespond with their exposure loation (Dishel 2000). Martin, Barnett, and Coble (2001) propose that weather derivative basis risk may be redued onsiderably through a portfolio holding of various weather derivatives based on several surrounding weather stations. Dishel (2002) notes that Farmers, growers and hydroeletri generators would like to have ontrats written on rain falling on their fields, in their groves or over their watersheds. This is generally impossible beause the market needs long and aurate measurement reords to assess the value of a weather derivative, and unaffiliated parties do not generally ompile measurement reords at these loations. Other studies have explored the potential demand for agriultural insurane based on preipitation. Sakurai and Reardon (1997) and Gautam, Hazell, and Alderman (1994) use household survey data to estimate latent demand for drought insurane in West Afria and southern India, respetively. Using a set of redued-form equations resulting from the optimality onditions of a dynami household optimization problem, both studies estimate a positive latent demand for drought insurane. Additionally, it is estimated that the insurane would be implementable on a full-ost basis. MCarthy (2003) estimates the demand for rainfall-based insurane ontrats for four regions in Moroo, finding that the median willingness to pay for rainfall-based insurane was 12% 20% above the fair value of the ontrats.

106 98 Weather Based Insurane Weather patterns tend to exhibit positive spatial orrelation, making losses more volatile from the perspetive of the insurer, inreasing the ost of maintaining adequate reserves to over potential losses from systemi events. Thus, insurane may not be the optimal mehanism for providing effiient risk sharing (Skees and Barnett 1999). However, if the insurer an over an area large enough to diversify the systemi risk of weather events or has aess to an adequate reinsurane program, an insurane mehanism should be feasible and implementable (Dunan and Myers 2000). Despite the largely systemi omponent of weather risk, there have been many reent studies examining the feasibility of developing agriultural insurane based on weather indexes. Martin, Barnett, and Coble (2001) outline various option strutures for preipitation insurane and provide a rating method appliation for otton in Mississippi. Skees et al. (2001) investigate the development of drought insurane based on a rainfall index in Moroo and find that the produt would be both feasible and of signifiant benefit to Moroan farmers. Turvey (1999, 2001) also disusses the appliation of weather derivatives in agriulture by rating various examples of rainfall and temperature options for various loations in Canada. To relate rop yields to weather events, Turvey (2001) examines the orrelation of orn, soybean, and hay yields with measures of both rainfall and temperature. Temperature was found to be highly orrelated with orn and soybean yields, while preipitation showed more orrelation with hay yields. Preipitation insurane poliies have also been explored and utilized in other ountries. Argentina, Ethiopia, Mexio, Moroo, Niaragua, and Tunisia have all tested the feasibility of weather-based insurane produts for agriulture (Varangis 2001), while

107 99 Australia is urrently exploring the possibility of developing rainfall insurane (Plate 2004). Two Canadian provines, Ontario and Saskathewan, have preipitation insurane produts on the market. The use of preipitation-based insurane in the Canadian provines is attributed to the high orrelation between attle pasture produtivity and rainfall (Varangis 2001). Rainfall Interpolation There is an extensive literature foused on rainfall interpolation tehniques. The simplest method sets the value of rainfall at out-of-sample loations equal to the rainfall reorded at the nearest observed site (Thiessen 1911). In 1972 the National Weather Servie adopted another method, with rainfall estimated as a weighted average of surrounding observed values, in whih the weights were inversely proportional to the squared distanes from the unobserved site (Bedient and Huber 1992). This method is not useful for our purposes beause eah site is treated as an independent observation and provides no information on the spatial orrelation struture of rainfall. More reently, advanes in the area of geostatistis have reated more statistially sophistiated interpolation methods through the use of kriging. Kriging, or optimal predition, refers to the pratie of making inferenes on unobserved values of a random proess given data generated from the same proess (Cressie 1993). In pratie, kriging tehniques form a preditor that is equal to a weighted average of the data in the sample. The weights used in the average are determined from the orrelation struture of the proess, whih may be given, assumed, or estimated from the data. Kriging tehniques have been shown to provide preditors that are both statistially unbiased and effiient.

108 100 While kriging methods provide statistially attrative properties, they an also require a signifiant amount of omputing time and effort. Thus, many studies have foused on the omparison of point estimates obtained from kriging to the estimates based on simpler interpolation approahes. While many authors have shown that kriging tehniques provide better estimates than do simpler methods (Tabios and Salas 1985), others have found that the results depend ritially on the density of the sampled loations. In general, studies have shown that kriging dominates the simpler interpolation methods for areas with smaller sampling densities while the methods are fairly equivalent for areas with sampling grids of higher density. Cressie disusses various types of kriging, whih differ with respet to the underlying assumptions for the stohasti proess. In general, the spatial proess is modeled as the sum of a mean and a spatially orrelated error omponent. Bayesian kriging assumes that the mean and error omponents are random and independent while reognizing that the model parameters are themselves stohasti. Given appropriate priors for the parameters of the mean and error struture omponents, the optimal Bayesian preditor for out-of-sample loations an be found and has been shown to be superior to other kriging methods (Cressie 1993). Markov Chain Monte Carlo Methods While point estimates for the onditional means and varianes of a stohasti spatial proess in a Bayesian model an be derived expliitly given appropriate distributional assumptions (see, e.g., Kitanidis 1986), an alternative approah is to sample diretly from the posterior distribution using Markov Chain Monte Carlo (MCMC) tehniques. MCMC methods are often employed when expliit evaluation of omplex and high dimensional

109 101 integrals is not possible. Under these irumstanes, MCMC tehniques provide an alternative to more traditional numerial or analyti methods of integration. MCMC methods differ from traditional Markov hain theory in that the proess s stationary distribution is used to identify the transition distribution rather than the reverse problem (Brooks 1998). The theorem on whih MCMC methods are based states that any hain that is both irreduible and aperiodi will have a unique stationary distribution to whih the t-step transition kernel will onverge as t approahes infinity (Brooks 1998). In pratie, hains are generated either using a single transition kernel or, in many ases, using a ombination of multiple sampling algorithms. The two most ommon transition kernels are the Gibbs sampler and the Metropolis-Hastings algorithm. The Gibbs sampler operates by splitting the urrent state vetor into a number of omponents while updating eah omponent separately in turn. The Metropolis-Hastings algorithm differs from the Gibbs sampler in that it is a generalized rejetion sampler in whih drawn values are orreted to math asymptoti properties of the stationary distribution. Implementation is ahieved by speifying starting values and non-informative priors for eah proess variable. There are several implementation issues involved with MCMC tehniques. The hains are naturally autoorrelated beause of the sampling algorithms dependene on the previous step. This is generally addressed by thinning the hains to save only a portion of the realizations. Atual onvergene is also an issue and is usually addressed by running multiple hains from different starting values and omparing running plots of the realizations to ensure onvergene to the same distribution. These plots are also used to determine the number of initial values to disard, referred to as burn-in values, so that the portion of the hains used for analysis are an aurate representation of the stationary

110 102 distribution. Suffiient hain length is also under debate. In general, the minimum hain length depends on the problem at hand and is inreasing in the standard deviation of the sample mean of some funtion of the iterations and dereasing in the level of autoorrelation between onseutive realizations. One onvergene and stationarity have been determined, point estimates an be omputed as sample moments from the sampling distributions. See Brooks (1998) and Gilks, Rihardson, and Spiegelhalter (1996) for more detailed desriptions of the theory behind MCMC methods and impliations for empirial implementation. The Rainfall Model Following Cressie, and Kitanidis (1986), in order to derive an empirial Bayes preditor for rainfall, let y i denote observed rainfall at loation i and assume that the atual rainfall at a given site is determined by the sum of a mean or drift proess, μ, and a spatially orrelated error proess, ε, whih are both funtions of site-speifi measures X and K model parameters θ. (1.1) y μ ( X θ ) + ε ( X, θ ) i =. i, i Applying Bayes theorem, the posterior distribution for the stohasti model parameters θ onditional on observed rainfall y is the produt of the likelihood funtion and the prior distribution normalized by an appropriate onstant. (1.2) p( θ y) = θ 1... θ K p( y θ ) p( θ ) p( y θ ) p( θ ) dθ... dθ K 1 For any unobserved loation j, the distribution of rainfall at a sample of N loations, y = y 1.y N, is given by y~ j onditional on observed rainfall

111 103 (1.3) p( y y)... p( y, θ y) dθ... dθ = j θ j K 1 θk 1 = θ 1... p( y θ, y) p( θ y) dθ... dθ θ K = E ( ~ p( θ y)[ p y j θ, y)]. Thus, the posterior distribution for any j K 1 y~ j given y is taken as the expeted value of the posterior onditioned on y and θ with respet to the posterior distribution of θ onditioned on y. If the only variable of interest is rainfall at an unobserved site y~ j, the posterior distribution given in equation (1.3) is all that is required. The information ontained in the posterior defined by equation (1.3) ould be used to aurately prie a weather derivative for any unobserved site j. For example, the mean rainfall E ~ y ], whih may be of interest as an [ j option strike or insurane guarantee, would be given by (1.4) E [ ~ y j ] = ~ y j p( ~ y j y) d~ y ~ j. y j However, for reinsurane purposes it is ritial to have information on how rainfall is jointly distributed aross spae. The intratemporal spatial orrelation struture aptured by the joint posterior distribution of unobserved sites provides the information needed by a reinsurer who owns a portfolio of poliies rated for individual loations. The joint posterior for a olletion of J unobserved loations, ~ y = ~ y... ~ y, is given by 1 J (1.5) p( y y)... p( y, θ y) dθ... dθ = θ 1 θ K K 1 = θ 1... p( y θ, y) p( θ y) dθ... dθ θ K K 1 = E ( ~ p ( θ y )[ p y θ, y)]. The marginal distribution for any unobserved site j is then given by

112 104 (1.6) ( ~ y y) = p j p( ~ y, θ y) dθ... ~ ~ ~ ~ K dθ1dy J.. dy j+ 1dy j 1... dy1 y1 yj 1 yj+ 1 yj θ1 θk = p( ~ y θ, y) p( θ y) dθ... dθ dy ~.. dy ~ dy ~... dy ~. ~ ~ ~ ~ K 1 y1 yj 1 yj+ 1 yj θ1 θk ~ J ~ ~ j+ 1 j 1 ~ 1 The expressions given in equations (1.5) and (1.6) are Bayesian distributions of rainfall at unobserved loations given the rainfall data from the observed sites, y. These distributions aount for parameter unertainty and differ from a non-bayesian approah in whih point estimates for the parameters might be treated as known (Kitanidis 1986). The information on the spatial orrelation of rainfall aross spae is given in both the joint posterior distribution of rainfall at the unobserved sites outlined in equation (1.5) and the posterior distribution for the model parameters given in equation (1.2). Given the potential size of the integral in equation (1.6), many ases may arise in whih expliit evaluation would be impossible. As an alternative, MCMC methods an be used to simultaneously generate Markov Chains of both the model parameters from the posterior distribution in equation (1.2) and rainfall estimates for any number of unobserved loations from the posteriors given in equations (1.5) or (1.6). Implementation Example To estimate the model, the struture of the mean and error proesses must be speified. Appropriate starting values and priors for the model parameters and rainfall at the unobserved sites are also needed. To provide an example of how the rainfall model ould be implemented, suppose that the mean omponent of rainfall at any given site is a linear

113 105 funtion of its geographi oordinates. Furthermore, assume that the spatial orrelation struture for a group of loations an be summarized by an exponential orrelogram 3. lat long (1.7) μ ( X, θ ) θ + θ X + θ X i = 0 lat i long κ ( ) (1.8) Σ = f ( d ϕ, κ ) = exp ( ϕd ) ij ij,. ij i Under the exponential speifiation, the orrelation between rainfall at loations i and j, Σ ij, is a funtion of the Eulidean distane d ij between the two loations. The exponential orrelogram assumes that the orrelation between observations delines with the distane between the observations. This property makes it a natural hoie for modeling the spatial orrelation struture of weather events aross a region. The parameters κ and φ are measures of spatial smoothing and deay, respetively. The smoothing parameter, κ, is bounded between zero and two, with larger values indiating higher levels of spatial smoothing. A value of κ equal to two implies the Gaussian orrelation funtion. The deay parameter, φ, is bounded below at zero and indiates the degree of deline in orrelation between two loations with distane. A larger (smaller) value of φ indiates a faster (slower) deline in orrelation as distane inreases (Thomas et al. 2004). Thus, larger estimates for φ indiate a smaller degree of similarity between nearby stations. Given observed rainfall for a set of loations and their geographi oordinates, ordinary least squares estimates of the mean proess parameters would provide appropriate starting values. Starting values for the orrelogram parameters an be obtained using maximum likelihood estimates if multiple periods of data are available. Alternatively, multiple hains an be generated from a wide range of starting values to ensure model 3. To larify notation, the parameterization given in equations (1.7) and (1.8) imply that the model parameters are given by θ { θ, θ, θ, ϕ, κ} =. 0 lat long

114 106 onvergene of the model to the same stationary distribution regardless of starting values. Prior seletion should be limited to diffuse distributions, suh as the uniform, to prevent any effet of prior speifiation on the results of the model. We now provide an appliation of our proposed method using rainfall data from Iowa. Data State-level monthly preipitation totals for Iowa were obtained from the National Oeani and Atmospheri Administration s National Climati Data Center (NCDC). The historial series of preipitation totals for all sequential ombinations of months were ompared to historial per-are hay yields as reported by the National Agriultural Statistis Servie (NASS) for the state of Iowa. The April through Deember time period showed the highest orrelation between umulative preipitation and hay yields for Iowa and was adopted as the overage period for the weather derivative example. In addition to aggregated state-level data, the NCDC reports data from thousands of individual weather stations loated throughout the ountry. The full data set of Iowa weather stations was ondensed to exlude those weather stations that did not have omplete preipitation reords for the months inluded in the overage period (April Deember) for the entire thirty-year period from 1973 to At the time of data olletion, the last monthly reording was for August 2003, hene the use of data to alulate the thirty-year average preipitation levels guaranteed by the poliy. Given the data requirements, the number of usable weather stations was redued to sixty-seven in the state of Iowa. The grid of sixty-seven weather stations provides a relatively dense sampling grid in omparison to previous studies (Tabios

115 107 and Salas 1985; Dirks et al. 1998). The distane between adjaent weather stations averages 20 miles, with a maximum (minimum) distane between weather stations of 50 (7) miles. Figure 1. Average and standard deviation of reported preipitation at Iowa weather stations, Figure 1 shows the means and standard deviations of reported preipitation levels, in inhes, for the ounties in whih the weather stations are loated. The weather station data show that the northwest setion of Iowa tends to be the driest, with more preipitation, on average, being reported as one moves toward the southeast setion of the state. Preipitation variability, as measured by the standard deviation of reported preipitation, follows a similar

116 108 pattern aross the state, with lower variability in the northern setion of the state and higher variability in the entral and southern regions. Two additional issues arose with the weather station data. First, for some stations and months, only estimated preipitation values were available. These estimated values were assumed to be unbiased and were left unhanged. Seond, for some other stations and months, the preipitation values were reported as inomplete. For these inomplete months, the NCDC indiated that somewhere between one and nine days of information were missing from the reported preipitation value. In order to onserve these data points, it was assumed that the inomplete months were missing the average of five days of preipitation information and that the preipitation amount during those five days was equal to the fiveday average preipitation amount for the month based on the reported total. The oordinates of the geographial enters of eah ounty in Iowa, measured in degrees of latitude and longitude, were alulated from a data file reated by Giglierano and Madhukar (1990). This yielded ninety-nine ounty referene points, or sample farms, where rainfall ould be interpolated to rate the weather derivative. The geographi oordinates of eah of the sixty-seven weather stations in Iowa were obtained from the NCDC. The Eulidean distanes between the weather stations and referene points were alulated using the oordinate data. Poliy Struture The rainfall guaranteed under the poliy was taken as the thirty-year average of reorded preipitation for the area over the insurane period, whih is patterned after the thirty-year limate normals used by the NCDC. The indemnity (I) takes the form of an

117 109 exoti put option on the 30-year average rainfall guarantee. The indemnity struture is similar to an example outlined by Martin, Barnett, and Coble (2001). (1.9) I R A = Max 0, Min L * F * C, L R30 where L = liability value F = indemnity fator C = overage level ( C [ 0,1] ) R A = atual rainfall R 30 = the rainfall guarantee Indemnities are triggered when atual preipitation is less than a seleted perentage (the overage level, C) of the historial average preipitation. The perentage shortfall in preipitation is translated into a shortfall in liability value, with the indemnity paid equal to the liability shortfall. The indemnity fator F was reated to translate preipitation shortfalls into liability losses. A regression relating NCDC preipitation levels to NASS hay yields was estimated for Iowa. To put all variables on a perentage basis, ratios were reated for eah variable. The preipitation ratio (RR) is the ratio of the urrent year s preipitation to the thirty-year average. The hay yield ratio (YR) is the ratio of the urrent year s reported hay yield to the ten-year average hay yield. It is assumed that exessive amounts of rainfall an also ause rop losses. Sine the intent of the appliation was to provide overage against drought risk, only years in whih rainfall was below the thirty-year average were used in estimating the regression relationship. Table 1 reports the regression estimates. YR RR RR RR (1.10) ( t t 1) = α + β * ( t t 1) + ε t

118 110 Table 1. Hay Yield Preipitation Regression Coeffiient Estimates (Standard Errors) αˆ βˆ R (0.20) 1.52 (0.22) 0.85 The sign of the estimated slope oeffiient was as expeted, with preipitation shortfalls leading to a redution in hay yields below the average level. The results exhibit fairly strong yield movements in Iowa, with a 1% drop in preipitation from the thirty-year average resulting in a 1.52% drop in hay yields below the ten-year average hay yield. The indemnity fator F was taken as the slope oeffiient estimate. Thus, the poliy pays 1.52% of the liability for every 1% drop in preipitation below the guaranteed historial average. Results Kriging For eah year in the data a sample from the posterior distributions of eah model parameter and rainfall for eah of the ninety-nine sample farms were generated. The program was set to estimate rainfall for eah sample site individually, signifiantly reduing the order of integration. 4 The latitude and longitude oordinates for eah of the weather stations and referene points were normalized to make the southwest orner of Iowa the grid origin. Sample autoorrelation plots from initial sample iterations exhibited autoorrelation through ten lags in the hains. To obtain a loser approximation to an independent sample, 4. Although this approah saved onsiderable time, the orrelation struture of the Markov hains for unobserved rainfall aross spae for any given year was lost. While this information was not ritial to the speifi appliation of poliy rating, the spatial struture of the rainfall distributions would be of onsiderable interest for reinsurane purposes. Information on the orrelation struture of rainfall aross spae is still provided by the orrelogram parameter estimates.

119 111 the hains were thinned to save every tenth iteration. To assess onvergene, three hains of 55,000 iterations were run from different starting values. The first 5,000 iterations of eah hain were disarded to minimize the impat of the starting values. As a diagnosti for suffiient hain length, we onfirmed that the Monte Carlo error for the samples was less than 5% of the sample standard deviation. 5 The estimation proess yielded Markov hains of 5,000 rainfall and parameter samples for eah year in the data. The point estimates for the model parameters and unobserved rainfall at the referene points were taken as the sample means from the Markov hains. The estimated thirty-year means and standard deviations of preipitation are illustrated in figure 2, and are very similar to those in the atual weather station data illustrated in figure 1. Furthermore, ross-validation onfirmed that the kriging results were statistially unbiased estimates of atual rainfall, while the average standard deviation of the bias estimates 6 was 3.01 inhes of rainfall. These results an be interpreted as upper bounds on the performane of the model, as the ross-validation results ome from a sampling grid of lower density. Summary statistis of the parameter-point estimates for the mean proess and the orrelogram are given in table 2. The point estimate for Θ 0 an be interpreted as a rainfall estimate for the southwest orner of the Iowa grid and averaged just under 29 inhes of rainfall, whih is onsistent with the true thirty-year means from weather stations in that region. The point estimates for Θ lat and Θ long indiate that, on average, preipitation delines by 1.46 inhes for every degree of latitude as you move north and inreases by 0.85 inhes 5. The Monte Carlo error is a measure of the deviation of the sampled mean from the mean of the true posterior distribution. See Gilks, Rihardson, and Spiegelhalter 1996 and Brooks 1998 for further disussion on diagnostis in MCMC appliations. 6. Cross-validation refers to estimating the model sequentially for eah observed site, or weather station, based on the data for the N-1 remaining stations. The bias estimates refer to the differene between the rainfall estimates from the model and the atual rainfall reorded at that weather station for the given year.

120 112 for every degree of longitude as you move east. These results are also onsistent with the relationship between average rainfall amounts and loation in the state of Iowa, as depited in figure 1. Figure 2. Average and standard deviation of estimated preipitation for Iowa ounties, The point estimate of the smoothing parameter, κ, ranged from 0.56 to 1.66, with an average value of Point estimates for the deay parameter, φ, varied within a onsiderable range from 0.48 to 11.34, with an average value of A larger point estimate for φ indiates a weaker spatial orrelation struture in the rainfall data for the given

121 113 year, implying the ourrene of loalized storms and volatile rainfall amounts aross the grid over the orresponding time period. Table 2. Summary Statistis of Rainfall Model Parameter Point Estimates θ 0 θ lat θ long κ φ Mean Median Standard Deviation Minimum Maximum Inverse Distane Weighting For omparison purposes, a simple inverse distane weighting (IDW) sheme was also used to interpolate preipitation in the Iowa ounties using the historial data. For eah ounty referene point, the weights assigned to the weather stations were equal to the inverse of the Eulidean distane between the referene point and the weather station normalized by the sum of weights for all weather stations. Using ross-validation, the number of surrounding weather stations to use in the interpolation was varied from the nearest station to the entire set of surrounding weather stations. All IDW estimators were found to be statistially unbiased with gains in effiieny up to four surrounding stations. Using the four nearest stations, rainfall at eah of the ounty referene points was interpolated over the 30 years of data. The results were nearly idential to those from the kriging model with the 30- year averages differing by less than 0.7 inhes for all ounty referene points. Coeffiients of variation were also nearly idential for the two methods, implying that the insurane rates generated from either method would also be nearly idential.

122 114 The equivalene of the two methods should be interpreted with are. The IDW results only provide point estimates for unobserved rainfall and are not guaranteed to be unbiased or effiient estimators. Moreover, the kriging model provides a riher set of results inluding empirial distributions of rainfall at eah of the out-of-sample loations and parameter estimates whih define the orrelation struture of rainfall aross spae. This information has signifiant value for estimating the magnitude of basis risk as well as impliations for aurately rating a reinsurane program. Insurane Rates To rate the insurane poliy, Monte Carlo analysis was used, assuming that rainfall over the overage period follows the gamma distribution defined by the historial rainfall parameters. Using a method-of-moments approah, gamma distributions were fit to the historial rainfall means and standard deviations implied by the kriging estimates for the ninety-nine referene points. For eah method, 5,000 random draws were taken from eah of the speified gamma distributions. The poliy was then rated by taking the average indemnity value over the 5,000 rainfall draws for eah of the ninety-nine referene points. Note that this risk-neutral priing approah does not inorporate a market prie of risk into the fair premium rates. The hoie of gamma distributions was based on the prevalene of this distributional hoie for preipitation in the sientifi and agriultural literature (Barger and Thom 1949; Ison, Feyerherm, and Bark 1971; Martin, Barnett, and Coble 2001). Using the method proposed by Moshini (1990), nonparametri kernel densities were fit to eah of the 30 year preipitation histories for the weather stations and ompared with the gamma distributions implied by the sample moments. The gamma density plots were very similar to the

123 115 nonparametri estimates and were determined to provide an exellent fit to the data. As an example, the gamma distribution and the nonparametri density for the Chariton weather station are illustrated in figure 3. Figure 3. Gamma and Nonparametri Rainfall Densities, Chariton Weather Station Given the rainfall insurane struture, the MCMC simulation results, and the gamma distributional assumption, Iowa premium rates average 12.4% under full overage. The average premium rate aross the Iowa referene points is equal to 1.2% for 75% overage. At 75% overage, the highest rate is 2.35% in Southeast Iowa at the Taylor ounty referene point, while the lowest premium is 0.3% in Northeast Iowa at the Clayton ounty referene point. These results are expeted, as the lowest implied preipitation oeffiient of variation (15.8%) is at the Clayton County referene point, while the largest implied oeffiient of variation (24.6%) is at the Taylor County referene point. Figure 4 maps the premium rates aross Iowa at a 75% overage level. In general, premium levels are the lowest in the Northeast setion of the state, with areas of relatively larger premium levels loated in various loations throughout the rest of Iowa.

124 116 Figure 4. Premium rates at 75% overage Historial Analysis A historial analysis of the insurane poliy was onstruted for the thirty years of available data. Preipitation estimates were taken from the kriging results and used to alulate the indemnity level for eah year. The top panel of figure 5 maps the poliy s lossost, the ratio of indemnity to liability, at 75% overage for Preipitation was below 75% of the thirty-year average for a poket of ounties in southwestern Iowa, triggering indemnity payments. Up to 26% of the total liability overed under the poliy would have been paid out in indemnities in Loss ratios, the ratio of indemnities to premiums, would have been as high as 18 in some ounties, with an average loss ratio of one aross the state. Thus, in 2000, the variability of rainfall was diversifiable aross the entire state. The bottom panel of figure 5 maps loss-ost values for eah of the ounties for 1988, a drought year throughout the Midwest. Indemnity payments would have been triggered in all but the Northwest quadrant of the state, with loss-ost exeeding 40% of the liability insured in some areas. Loss ratios in 1988 would have exeeded 50 in selet regions with an average of aross the entire state. In general, the poliy tends to pay indemnities in

125 117 onentrated areas and at fairly high loss ratios. At higher overage levels the loss regions expand to over larger areas aross the state. These results are expeted given the spatial nature of weather events. Figure 5. Loss-osts in Iowa ounties for 2000 and 1988 While the poliy is theoretially rated to yield a loss ratio of one over time for any given loation, the systemi nature of weather risk requires a large geographi area of overage to provide proper risk pooling and insurability for any given year. These results suggest that any party offering this type of overage should either hold a diversified portfolio of poliies written aross a spatially diverse area or hold suffiient reserves (or reinsurane) to over years when rainfall is well below the level of the guarantee.