发展中国家农业生产者对商品保险的需求：理论分析和对于加纳可可业的应用研究

tvPS POLICY RESEARCH WORKING PAPER 28 8 7 The Demand for Commodity Insurance by Developing Country Agricultural Producers Theory and an Application to Cocoa in Ghana Alexander Sarris The World Bank PPh Development Research Group Rural Development September 2002  PouIcY RISEARCH WOkKING PA\PE 2887 Abstract Sarris considers the benefit to agricultural producers of Living Standards Survev data to specify variouS classes of commodity price insuranlcc that provides in every ycar- cocoa-producing houselholds and moithly price data for but in advance of the resolutioll of productioni and price both domestic and international prices, to formulate uncertainity-a niinimuli- price for a fixed or variablc appropriate models for ascertaining price risks faced by portionl of production. Under the assuLIMption that producers. The author gives empirical estimates of the producers do not chanigc their long term productioll and actuarially fair premium, and shiows that they arc smaller incomic cliversification patteril, the author- suggcsts a than markct-based put option prices from organized theoretical framework that leads til explicit fornitilas of exchianiges. rhe overall bencfit in providing minimum the benefit in providing this type of insuraice. He shows price insurance to houselholds, however, turnis out to be that this beniefit depends not only on the actuarially fair sUbstantially highier thanl the acttiarially fair premitinis instirance premium, but also oln hoIselhold-specific and the market-based ptIt option priccs. This is due to factors that depcnd on the attitudes to risk, the both the magnitude of the uncertainities facing the consumption smoothinig parameters, and the liouselhold- households, as well as their risk and consumilption specific cxposures to income risks. The author applies smoothinig behavior. the theoretical framework for GChana, usinig the GhalLaL This paper-a product of Rural Developimienit, Development Researcih Group-is part of a larger cffort in the group to analyze mcchianiisilis for risk mitigation in agriculture. Copies of the paper are available free from the World Bank, 18 18 H Street NW, Washinigtoni, DC 20433. I'lease contact Maria Fernanidez, room MC3-305, telephone 202-473-3766, fax 202-522-1 151, email addrcss mferinanidez2@iwvorldbank.org. IPolicy Research\ Workinig lPapers are also posted on the Web at http://econn.worldbanik.org. The authlor may bc conitactcd at asarris Chol.gr. September 2002. (48 pages) Tim^e l'olcr b Research dorving Pape Seres dissi-Pinates tt' lindiiigs of work io progress taf encourage the exrhanSge of ideas abofit deveJlopmllen t iss ues. A11 objestir e o/ thle series is to get tl) finidinigs ou lt qt fickli. evJen if tbe presenltationis are less tl7al fiel iy polisb7ed. 7117e pape rs c l arn the niam7es ol/-the vaiabors and sbouild be cited aIccordinlgly. Th1e linzdinzgs interpretationis anld coiici(siopis expressed inl this pI7per arif eiztirely, tbo)se f)/ the auItlors. '//)C (lo IIOt liecess7irdi relpresenit tb)e vwiew lz0/the W/orld BLInIk, its Execuftivoe Directors, or the cl)lllltrics the), represeitt. 17roduceci bv the Research Advisorv Staff  The Demand for Commodity Insurance by Developing Country Agricultural Producers: Theory and an Application to Cocoa in Ghana By Alexander Sarris1 Keywords: Commodity price insurance, risk management, cocoa, Ghana. Professor of Econornics, University of Athens, Greece, and Visiting Research Fellow, World Bank. The author would like to thank Hua Di for excellent research assistance, Awudu Abdulai, Yiannis Baffes, Patience Mensah, and Gerald Shively for help with data acquisition, and Yiannis Baffes, Gershon Feder, Don Larson, and Panos Varangis for helpful discussions at various stages of the work. Conmnents received at two seminars presented at the World Bank are also acknowledged. The usual disclaimer applies. E-mail asarris(a)worldbank.org and asarris(ahol.r .   1. Introduction Agricultural producers around the world are exposed to a variety of income uncertainties, both market related, such as price variations, as well as non-market related, such as unstable weather patterns. It is well known that such uncertainties induce substantial income risks, and these can be particularly detrimental to small and/or poor producers in developing countries. It is also well known that farmers have developed several ways for dealing with the various risks they face. These involve risk management strategies, namely actions taken ahead of the resolution of any uncertainty to improve the ex-ante exposure of the producer's household to various risks, as well as risk coping strategies, namely rules adopted ex-ante to help the household to deal ex-post with any undesirable consequences. Risk management strategies include among others crop diversification, income diversification through off-farm work, sharecropping, etc. Such ex-ante strategies are usually designed to sacrifice higher expected income for a more stable income stream Risk coping strategies may include the availability of short term consumption credit, mutual family or village based reciprocal giving arrangements, etc. For a recent survey of these practices see Dercon (2000). The acknowledged precarious situation of many poor rural residents in developing countries has led to calls for the adoption of various additional safety nets (World Bank, 2001). Apart from publicly based such safety nets for rural residents, some proposals have advocated market based insurance systems. For instance, the recent initiative of the International Task Force (ITF) on Commodity Risk Management, has proposed using market based derivative instruments to provide price insurance for internationally traded commodities (ITF, 1999), while other proposals have suggested using market based weather insurance to cover yield risks (Skees, Hazell and Miranda, 1999). Varangis, Larson and Anderson (2002) have suggested using combinations of the above instruments to manage agricultural market risks in developing countries. The various proposals, however, have not considered the demand for such safety nets, by the beneficiaries. The issue in the context of agricultural income insurance is the following. Under the structural conditions, exposure to risk, and risk mitigations strategies agricultural producers have adopted, how much yield and price insurance for the commodities produced would they wish to obtain, and how much would they be willing to pay for it? These are crucial questions that must be answered if a system of providing rural safety nets and in particular various types of commodity insurance (quantity or price based) are to be promoted in developing countries. The purpose of this paper is to explore the issue of the demand for commodity insurance theoretically as well as empirically for the case of price insurance, and in the context of a poor agrarian economy, with rural households significantly dependent on agricultural commodity risks. A significant share of the income variations of rural producers in developing countries seem to be due to idiosyncratic shocks, namely shocks particular to a household (such as sickness) (Morduch, 1995, Townsend, 1995, Carter, 1997). Such risks can be insured through formal or informal pooling of a large number of such shocks, such as through village reciprocity relations, that exist in many developing countries, or formal private or public insurance schemes that exist in many developed countries. Covariate shocks, however, namely those that affect all households in a given community or region, such as weather or price shocks, cannot be insured by pooling them within a small region, and can be insured only if pooled over a much wider range of potentially affected households. It is the need to insure farmers, against such covariate shocks that have induced the governments of most developed countries to institute various price or  2 income support schemes, under the perception that the private insurance industry would not be able to provide adequate coverage at reasonable cost. The non-existence of such arrangements in developing countries is what induces rural households to develop self insurance, or what has been termed "consumption smoothing strategies" to deal with covariate shocks. These strategies basically involve building "precautionary savings", in the form of liquid or near liquid assets (cash, grain stocks, livestock, jewelry, etc.) in good years, and depleting them in years of adverse covariate shocks (Deaton, 1991). There is conflicting evidence, however, on whether such strategies are effective at smoothing consumption (Rosenzweig and Binswanger (1993), Rosenzweig and Wolpin, 1993, Fafchamps, et. al, 1998, Dercon, 2000). The consensus, nevertheless, appears to be that despite the variety of smoothing strategies adopted by poor households in developing countries there is substantial residual consumption risk (Jalan and Ravallion, 1999). There is also evidence that these practices are costly at the micro level in terms of current income and consumption foregone, as well as the types of investments undertaken (Fafchamps and Pender, 1997). Finally there is evidence that commodity price instability is detrimental to overall macroeconomic growth (Collier and Dehn, 2001). For all these reasons the additional provisions of safety nets or insurance mechanisms in rural areas is crucial to poverty alleviation, as well as growth. Under the above circumstances, how is one to assess the benefit, and hence the potential Willingness of Producers to Pay (WTP), namely their underlying demand, for additional income insurance, and in particular insurance against downward commodity price risks? The question cannot be judged in abstract, but only in the context of the already existing self or community based insurance systems available to households, and after taking into account the degree to which they have altered their economic behavior to take into account such possibilities. If, for instance, households are already using precautionary savings to smooth out covariate price and quantity variations to income, they are already paying for such insurance through the opportunity cost of any consumption that is put into saving. Unless the provision of additional income insurance in such circumstances costs less, or provides for more reliable insurance at similar cost, households may not be willing to pay for it. There are two consequences of providing in every period some type of income insurance to a household, including special types, such as commodity price insurance. The first involves a ceteris paribus increase in overall welfare, assuming that nothing else in the household structure changes. This will be the object of this paper, and can be considered as the minimum possible benefit from the insurance. The second consequence involves changes in the overall income and production pattern. These changes will occur if the household believes that the insurance provided is permanent, namely will be provided in every period. Both common sense as well as theory suggest that if a household is covered through adequate safety nets, then it may adopt a production and income pattern that is more risky, in the sense that it includes larger amounts of activities that have uncertain returns (Newberry and Stiglitz, 1981, Fraser, 1988, Finkelshtain and Chalfant, 1991, Moschini and Lapan, 1995, Gollier, 1995). Empirical verification of these theoretical predictions, however, is difficult (for a review of relevant studies and problems see Moschini and Hennessy, 2001). This paper outlines a theory of the benefit from commodity insurance, under fixed production structure, and a methodology for empirical assessment of this benefit. The theory developed is applied to Ghana, and for the case of price insurance for cocoa. Ghana is a poor country (per capita income around 400 USD), with a large rural population (68 percent of the 18.9 million  3 inhabitants) that depends substantially on agriculture (35 percent of GDP). About 42.6 percent of the total population and 51.6 percent of the rural population live below the poverty line. Ghana is heavily dependent on three commodities (gold, timber and cocoa) for its exports. In year 2000 cocoa accounted for 20 percent of export earnings, down from about 40 percent in 1990 and more in the 1980s. Cocoa is the most important cash crop for farmers in the southern "Rural Forest" agroecological zone, and it accounts for 13 percent of national agricultural GDP. There are about 500 000 cocoa producing households (11 percent of all Ghana households or 16 percent of all households producing some agricultural output). Cocoa is mostly exported, and a government parastatal marketing agency "Cocobod" has monopolized cocoa trade until recently when liberalization allowed the participation of private marketing agents. Domestic cocoa prices have been stable and under government control, and the government through Cocobod has absorbed most of the international price fluctuations, in effect through variations in its export tax revenue from exports. Cocoa export tax revenues have fluctuated widely and have averaged between 4 and 15 percent of total tax revenues during the decade of the 1990s (for a review of the history and consequences of the various cocoa related policies of the government of Ghana (GOG) see Varangis and Schreiber, 2001). A more open trading regime, however, will expose domestic producers directly to international price fluctuations. Even if, however, Cocobod maintains its price guarantee function, it may consider insuring its minimum price offered to farmers in international markets, and charge farmers an insurance premium for it (which could be implicit in the prices offered). Thus the estimation of the WTP of farmers for insurance is relevant to this case as well. The availability of organized international futures and options markets, make the study of the demand for commodity price insurance for cocoa in Ghana particularly interesting, as the possibility arises of providing insurance through commercially available derivative instruments like options The plan of the paper is the following. In section 2 a review of previous relevant studies is presented. Section 3 outlines the theoretical framework. Section 4 explains the application of the theory to an empirical setting. Section 5 presents the data and estimations of the various model parameters for Ghana. Section 6 presents the empirical results concerning WTP, while section 8 compares the WTP with actual market based put option prices. The final section summarizes the conclusions and implications. 2. Previous literature related to commodity insurance There are three ways that have been utilized to assess the WTP of farmers for price or income insurance. The first involves direct questioning of producers, and is related to the literature on contingent valuation (we shall term this the CV method). The second method involves the use of theory along with the combination of microeconomic household information, and market information to estimate indirectly the appropriate premiums (this will be termed the indirect method). The third involves inference of the willingness to pay from analysis of the patterns of production and other behavior of producers (we shall term this the revealed preference method). The CV methods are based on direct questioning of agents (producers, households, etc.) on how much they are willing to pay for avoiding an undesirable event, or for a given amount of an insurance contract. The major problems with this approach have largely to do with the specification of the "scenario" or the "benchmark" against which the agent is supposed to compare the current situation, and express a monetary value for what it is worth to him/her to move to the new situation, or avoid a bad one. It is not always easy to specify well this scenario,  4 especially if it involves a rather improbable event, and this lies at the heart of most criticisms of this approach (see e.g. the papers in Hausman (1993)). However, in the case of well specified risks, such as price or yield variations, it is likely that farm households are familiar not only with their normal values, but also with their variability over time, and hence the above criticism may not be valid. Another problem with direct WTP studies involves the fact that reported values are likely to be influenced by recent experiences, For instance farmers are more likely to express high demand for price insurance if prices in recent periods have been low. There are also several issues concerning the method of deriving the WTP from either direct expression of values, or contingent rankings of alternative choices, but these seem to have been largely resolved. The literature on CV based methods has recently been surveyed by McCarthy (2002), who provides more discussion on both conceptual as well as estimation issues, and more references. There are very few studies relevant to agricultural insurance, that use the CV approach. Patrick (1988) analyses producers' demand for a multiple peril crop insurance (MPCI) program with indemnities based on actual yields, and a rainfall insurance program with indemnities based on area rainfall, and uses tobit procedures to analyze factors influencing farmers' WTP for the alternative programs. Vandeveer and Loehman (1994) applied both dichotomous choice and ranking of activities in a study of farmer response to modifications in crop insurance. The ranked responses were used in a ranked logit model to derive WTP. The indirect methods of estimating WTP involve first the specification of a model of the random income or other variable of direct relevance to the farmer's welfare (e.g. consumption), and expressing the WTP as the amount of money that would equate the expected utilities of the relevant variable with and without the insurance. This amount of money (the premium) is then estimated for objectively estimated values of the risks with and without the insurance, and for a range of relevant utilities, or relevant parameters (such as degrees of risk aversion) from a given class of utilities. There are also very few studies attempting to estimate WTP for agricultural insurance by the indirect approach. Hazell, Bassoco and Arcia (1986) applied a programming model to infer the demand for crop yield insurance by the representative farmer in Mexico. Fraser (1992) uses an indirect method to estimate WTP for crop insurance. He does this by estimating and comparing certainty equivalents, in the presence and absence of insurance, of expected utility, based on the mean-variance framework and constant relative risk aversion. Bardsley, Abey and Davenport (1984), use a simulation model to estimate the amount of insurance at a given minimum price that will be purchased, per unit of insured quantity. All the indirect methods have to use market data to infer the various parameters of the models, such as price and yield variabilities, as well as estimates of risk aversion parameters. While estimates of market parameters can be estimated readily if the appropriate data is available, risk parameters are not easy to estimate (for available empirical methodologies see Moscardi and deJanvry 1977, Binswanger 1980, Antle (1987,89), and Bardsley and Harris 1987). This suggests that the relevant measures of the WTP must be estimated using a set of risk aversion parameters that are considered as spanning the appropriate true values in the relevant region. An additional restriction of these methods is that they must assume some parametric form of utility that can subsequently be simulated. Nevertheless, the methods avoid many of the subjectivity issues, as well as the scenario design problems that plague the CV based approaches.  5 The revealed preference (RP) method relies on the idea that the producers are behaving with respect to their production and saving-investment decisions in a way that is compatible with their attitudes toward risk. Their desire and WTP for insurance is expressed implicitly in these decisions. If a model can be constructed that takes all these decisions into account, then observable behavioral patterns can be deduced, from which risk attitudes as well as WTP measures can be estimated. The problem, of course, if to specify a general enough model that allows for the derivation of risk attitudes and WTP measures. An early paper by Binswanger and Sillers (1983) utilized this methodology to estimate the implied risk attitude parameters of farmers, but did not consider explicitly insurance. The first paper using a methodology of this type to estimate risk premiums for insurance is the one by Gautam, Hazell and Alderman (1994). In that paper the farm household's behavior is assumed to be described by the maximization of the expected value of an intertemporal utility function. The production, saving, labor allocation, diversification, borrowing, and insurance decisions are assumed to be endogenous. The equilibrium conditions of the optimization problem are manipulated to infer the production and diversification decisions of the household as fumctions of both standard variables as well as a variable that measures the relative preference of the household for risky versus non-risky income. Under the assumption that the household is already well diversified and insured, implying that there is no unmet need for further insurance, the value of this parameter should take a value that can be inferred from non-experimental data. The authors use panel data to estimate the value of this parameter implied by the actual behavior of farmers, and deduce that there seems to be considerable latent demand for crop insurance, and furthermore, that the implied WTP is in the neighborhood of 13-17 percent of the indemnity value. These numbers (supplemented by estimated transactions cost) suggested in that case, that the WTP of farmers for drought insurance is above the cost of actuarially fair drought insurance, and hence that the provision of such insurance would be commercially viable. The strength of this methodology lies in the fact that it can estimate the "latent demand" for drought insurance, namely the additional and as yet unmet demand for insurance, given that the households already have some self insurance mechanism. The underlying assumption is that the way the households have adjusted to the recurring weather risks is by diversifying, as well as adopting different production patterns than what would be dictated through simple expected income calculations. As such, the empirical estimates involve the long run or steady state production pattern of the farm household, given the household's perceptions of drought risks. This approach seems suitable for the issue of assessing how farm households who are exposed to price risk adjust their long tern production structures (for instance through diversification), and what implicit risk attitudes dictate the observed production patterns. The method may also be suitable for assessing the WTP for price insurance, but, the data requirements are quite heavy, as they invariably involve panel survey data. The same approach is essentially followed by Sakurai and Reardon (1997) who utilized panel data for Burkina Faso. The additional feature of this study is that the researchers regress their estimates of farm level demands for drought insurance on a set of variables, so as to identify variables that increase or decrease such demand. They find, as expected, that the demand for drought insurance depends on the perceived probabilities of droughts, and is higher for regions with higher such probabilities. They also find that variables such as the size of cultivated area,  6 and the age of household significantly affect positively the demand for insurance, while the amount of off-farm income, the availability of public aid and private gifts, and the size of household significantly affect negatively the demand for insurance. These are reasonable and expected findings. There are finally few studies who utilize panel data to infer simultaneously the risk attitudes and consumption smoothing parameters of rural households. All of these studies use the RP methodology to assess risk attitudes and consumption smoothing as well as diversifications patterns and savings parameters for rural households, using panel data, but do not consider the demand for insurance. Examples of such studies are the ones by Kurosaki (1998), Kurosaki and Fafchamps (2002), Fafcharnps, Udry and Czukas (1998), and Dercon (1996, 1998). 3. A model for the demand for commodity price insurance Commodity price insurance for an agricultural producer is like a put option, or a minimum price guarantee. In other words it guarantees for the amount of contracts purchased or quantity covered, and over a period stated in the contract, a minimum price (the strike price of the option like contract), but allows the producer to obtain a higher price. This similarity is the basic reason that renders price insurance schemes based on derivative instruments traded in organized markets possible (Duncan, 1997, Sarris, 1997, 2000, 2002, Varangis, Larson and Anderson, 2002). Commodity yield insurance is similar to price insurance except that the role of price and quantity are reversed. In other words rather than guaranteeing a minimum price for a given quantity, yield insurance guarantees to farmers a given price for a minimum insured quantity. Thus the put-like option is on quantity produced rather than price. In the sequel the discussion will refer to price insurance, with the understanding that all theoretical analysis can be easily transposed to the yield insurance problem. There are several points of clarification worth mentioning in the context of commodity price insurance. First, the price insurance may not affect all of the production of a producer, but only the amount of production covered, and hence it is not strictly similar to a minimum price guarantee scheme for whatever output is produced, of the type that have been adopted by many governments. Nevertheless, an agricultural producer is probably more interested in a minimum price guarantee for whatever amount of product he/she2 decides to sell. The second issue concerns the type of price and market that is relevant for a producer, and the ones that must be considered in estimating WTP for price insurance. It is clear that a producer is mainly interested in the price he receives for his commodity locally. If this price happens to be the same or partially correlated with some international price or price in some organized exchange, then such price offers good signals to the producer. However, as far as his WTP is concerned, price information must relate to local prices. Third, the WTP for commodity price insurance depends at what point in the production cycle the producer is faced with the possibility to buy insurance. For instance, the insurance can be provided at a point in the year, after all production inputs are committed, and the only uncertainties facing the producer are environment related and price related. Alternatively, the insurance can be offered before the annual production decision is made. The WTP under each of these alternatives should be different, and in fact is expected to be larger on average in the 2 In the sequel the male gender is used to refer to a producer, without implying any prejudice concerning the type of agricultural household head.  7 second situation above, as in that case the producer has larger flexibility to adjust, and hence can achieve larger expected utility with the insurance. Fourth, it makes a difference whether insurance, under either of the two types indicated above (before or after annual production decisions) is temporary, namely a one shot affair, or is offered every year. In the former case, the producer is not expected to alter long term behavior, while in the latter case he is. The theory outlined below pertains to the case of insurance offered within one crop year, and after the major short term production decisions, such as land and fertilizer allocations, have been made. It thus assumes that the long term diversification pattern of the producer stays unaffected. In this sense, the estimated benefit, and WTP can be considered as the minimum demand for price insurance. Any changes in production structure will provide an additional benefit, but will not be considered here. Assume that for a farm household time is measured in crop years, indexed by an integer T. Each crop year is divided into two, not necessarily equal, periods 1 and 2, indexed by j. The first period within each crop year is meant to represent the period after planting, but before the resolution of production and price uncertainty, while the second period is meant to represent the resolution of production and price uncertainty, and the realization of annual crop income. In the first period the household income consists of sources other than agriculture, while all agricultural income is assumed to be realized in the second period (in addition to other possible sources of income). Time is indexed by an integer variable t=2T+j, where j=1 or 2. Hence, odd values of t denote the first part of any crop year, while even values the second part. Denote the vector of consumed goods (it may include leisure) of the farm household in period t by Ct, the vector of quantities of assets in the beginning of period t by At, the vector of decision variables (such as inputs, land allocation, amount of insurance instruments to buy, savings and investment decisions, etc.) that are determined in period t by xt , the information available to the decision maker at the beginning of period t by It (such as values of all realized economic variables as well as states of nature in previous years), and the state of nature that is revealed in the beginning of period t by St (this may include uncertainty about income affecting variables such as weather, prices, sickness, etc.). Also denote by PAt , Pct and pt , the vectors of prices of assets, consumption goods, and income eaming activities (including labor) respectively at time t. Denote by U(C,) the instantaneous household utility in period t. The household will be postulated to maximize the ex-ante expected value of the discounted sum of instantaneous utilities, over n crop years. W = E{[ E U(CO)]/I1} (1) where o denotes an appropriate discount factor. The expectation in (1) is taken over all states of nature St (t=1,2,,.,2n), based on information at the beginning of the relevant horizon for the household. The maximization will be assumed to be over all sets of decision vectors xt The restrictions relating the various variables are the folfowing. PAeA1+I = PAtA, + p,yj (A, X, S,) - pCC, =- R, - pc C (2) x, E= X, (3)  8 The equation in (2) defines the value of end of period assets at period t prices. The variable Rt denotes the value of resources available to the household at the beginning of period t, namely previous period assets valued at current period prices, plus current income from these assets. Xt is an appropriate constraint set for the decision variables, and yj (.) denotes the vector of quantity of netput activities (positive if outputs, negative if inputs) affecting the income of the household in period t3. The subscript j in the income function denotes the possibility that income sources may be different in the two periods of each crop year. However, note that the nature of the income function y is time invariant. In other words it is assumed that during the planning horizon of the household, the nature of the income generating activities, stays unchanged. This implies, for instance, that unknown future technological improvements are not taken into account in the household's planning problem. Notice that the budget constraint (2) takes into account appreciation of assets, through the revaluation of assets carried over from last period (At ) at current period asset prices. Notice that no restriction is placed on the sign of assets. Hence negative assets (namely liabilities such as borrowing) are allowed in this general formulation. If the household is liquidity constrained, then the restriction that some or all assets should be non- negative must be imposed (Deaton, 1991). The nature of the solution to such a problem is theoretically well known, and involves the application of Kuhn-Tucker first order conditions (if there are non-negativity constraints) to the standard Bellman equation (for illustrations see Deaton, 1992a, Zeldes, 1989). If the utility and income generating functions are time invariant, as has been assumed here, and if the stochastic processes determining prices as well as the other uncertainties affecting household incomes are stationary, the general solution for the consumption in each period is a time invariant function of the "state variables" in period t, namely variables that summarize the information available to the household in the beginning of period t. Such information generally include the volumes of assets at the start of period t, the state of nature in period t (such as uncertain types of income), and the prices of the various assets and products that enter production and consumption. Under some restrictive assumptions such as equality of all prices, and simple linear income generating rules, the solution can be obtained numerically (e.g. Deaton, 1991). In general the solution is not analytically tractable, and can be written as follows. C, = f (I,) = f (A,, yj (A,, S,), P, I PA, I Pc) (4) If an equation like (4) is the solution to the overall optimization problem (1)-(3), then the utility function in (1) can be rewritten as follows. W = EI 5T [U(C2+I) +5E(C2T+2 I2T+I)iIO i = } E{Z5ITV(C2T+I, I2T+1)1V0 } (5) In (5) 51 = ,2, the consumption within the various parentheses and brackets has a form like (4), and the function V just defines the quantity inside the bracket in the left had side of (4). The expectation inside the brackets are taken conditional on information available in the first period 3 The returns to any financial assets, such as interest on deposits or loans, are included in the income terms. Similarlv the depreciation of physical assets can also be considered as included in y in this general notation.  9 of a given crop year T, while the unconditional expectations outside the brackets are taken with information available in the beginning of the planning horizon, namely year 04. Consider now the provision of an insurance contract to the farmer in the first period of the crop year, whose outcome depends on events of the second period. The contract considered is in the form of an option to sell all or a portion of a produced crop at a minimum "strike" price. Denote the amount of the crop that is insured as q (can be fixed or variable), and the return to the insurance contract per unit of the insured crop as r. The insurance contract is similar to the minimum price guarantee schemes that have been popular in developed countries (such as the loan rate system for cereals in the US), and hence the theory applies to these settings as well. If we assume that the nature of the function f in (4) is not affected by the provision of this contract, then we can define the benefit of this contract as the amount that must be subtracted from income of the first period in the crop year, so that the two-period utility with the contract is equal to the utility without it. Analytically we define the benefit in year T to be the solution B to the following implicit equation. U(C2T+I (Y, - B)) + SE[U(C2T+2 (Y2 + rq))jI2T+l ] = U(C2T+. (YO)) + 5E[U(C2T+2 (Y2 ))JI2T+1| (6) The key assumption that allows the definition in (6) is that the nature of the income generating function yj (.) as well as the consumption function (4) are not altered by the provision of insurance. This, of course, is not strictly correct, as the household may adjust its long tern exposure to risk as is implied by theory, but as the nature of the changes in the income functions as well as the consumption function under insurance are quite intractable, the assumption can be considered as a first approximation, and one that can facilitate the estimation of the "minimum value" of WTP, for such insurance contracts. To utilize (6) for empirical analysis we first assume that total household consumption is composed of one aggregate commodity. This is done for convenience, so as to neglect commodity composition consumption effects. Then we approximate (4) by the following aggregate consumption function. C C- +A(R )= C; + p(R, - R (7) Pct where Rt has been defined in (2), and where we have normalized all nominal values by the price of aggregate consumption (namely a suitable consumer price index). The formulation in (7) is the one that has been utilized as an approximation to the optimal rule (4) in the literature of the general lifetime optimization problem under uncertainty as well as under liquidity constraints (there is a large literature on consumption under uncertainty and liquidity constraints, and consumption smoothing. For useful surveys see Deaton, 1992b, Browning and Lusardi, 1996, and Morduch, 1995). In (7) the value of "trend" real consumption C is assumed not to depend on current period random variables, albeit it may include time varying components due to seasonal or lifetime effects. The current (real) value of resources Rt includes the current real income of the household, as well as the current valuation (deflated by the consumer price index) of the 4 If the two periods within the crop year are different in duration, the discount rate within the bracket in the left hand side of (5) will be different than the discount rate outside the same bracket.  10 household assets. As such it includes both covariate risks, such as price variations, as well as idiosyncratic risks. The starred value of R is the trend or expected value of these resources (income and assets). The parameter ,B denotes the amount of smoothing that the household does in each period, and is a function of household characteristics. If P is equal to 0, then there is perfect smoothing, and current consumption is independent of current income, or the value of current assets. If 1 is equal to 1, there is no smoothing at all, and current consumption moves exactly as current resources. Notice that perfect smoothing may involve negative values of assets in some periods (namely debts). If this is impossible due to liquidity constraints, then consumption smoothing will not be perfect and the relevant value for P will be larger than zero. Denote by z the term that include the total (real) return to the insurance contract. z = rq (8) where by r we now denote the return to the insurance contact, deflated by the CPI in the relevant period. We can then write the consumption with the insurance in each of the two periods of crop year T as follows (the year specific variable T is suppressed for ease of notation, and because it does not affect the subsequent analysis which depends only on the seasonal variables). C, = C, + 9(Rj - B - R1) = C,* + (R, - R>) -iB -C> + PAR, -,B= C, -fiB (9) c2 =C; +/(R2 +z-R;)=C; +/(R2 -R;)+/z C; +AR2 +fz=C2 +/Z (10) In (9) and (10) the consumption variables with hats denote consumption with the insurance contract, while the ones without hats denote consumption without insurance. We can now expand the utilities in both the left and right hand sides of (6) about C; using Taylor's theorem. Neglecting the Taylor expansion terms higher than second order, and canceling similar terms from the left and right hand sides of (6), results in the following equation (primes denote differentiation). 0 = -U'(C )B+ I 8U(Cl) (B2 - 2B AR,) + U'(C)E(z) + 2lU(c2)* E(Z2) + 2E(zAR2 (2 ) In (11) E(.) denotes conditional expectation, given information in period I of the crop year. To proceed, assume that the trend real consumption is the same in each of the two sub-periods of the crop year. Denote this common value (which may be different in each crop year) by C*. Furthermore, define the following normalized variables. r - r (12) e Pi2 _ B (13) C( qrp2q (14) C*  z _=r q =C (15) C' rR. Rjr - 0(j=1,2) (16) c. R;r R; (j=1,2) (17) SC P C' U' (18) In (12) the price in the denominator is the expected or normal price of the insured commodity in period 2. In (13)-(17) all variables are defined as shares of trend real expenditures, and (18) just defines the coefficient of relative risk aversion. With these definitions, equation (11) can be rewritten as a quadratic equation in the normalized benefit, as follows. 1 (Br)2 + (Br)(1 _ARr )+6 E(Zr') ++E(Z ) +2E( rARr) 0 (19) 2 4- +I{2 Z 2= 0 (19 where 0 is the product of the coefficient of relative risk aversion and the consumption smoothing parameter.