Optimum Crop Insurance under Joint Yield and Price Risk

Abstract

ABSTRACT This article examines the design of an optimal policy for an insurable risk in the presence of another uninsurable risk. In the context of incomplete markets, indemnity payments are made whenever the insurable realized index is lower or higher than a critical level, depending on whether the policyholder's revenue increases or decreases with this index. These results are illustrated in an agricultural context in which crop insurance is available and price risk is uninsurable. Under the principle of increasing uncertainty, price risk induces the prudent producer to reduce his or her optimal price selection. The effect of direct subsidies on the optimal price selection is shown to depend on the producer's temperate behavior. INTRODUCTION Early literature on the theory of insurance, such as Borch (1962), Arrow (1971), and Raviv (1979), focused on optimal insurance purchasing in the context of a complete set of insurance markets for future risks. In recent years, the insurance-buying decision in incomplete markets has been examined when the insurable risk interacts in an additive manner with the uninsurable background risk, which can be correlated with the former (see Doherty and Schlesinger, 1983; Gollier, 1996). The purpose of this article is to investigate the optimal insurance policy when the indemnity is contingent only on an imperfect signal of the policyholder's revenue. More specifically, his or her random revenue is a function of an insurable random index and an independent background risk, which is assumed uninsurable. This source of uninsurability can stem from asymmetric information between insureds and insurers. In this context of incomplete markets, the design of an optimal insurance contract is derived. The optimal form depends on the effect of an increase in the insurable index on the policyholder's revenue. If the principle of increasing uncertainty (Leland, 1972) is satisfied and if the policyholder exhibits prudence (Kimball, 1990), an upper or lower bound associated with the slope of the optimal marginal coverage can be defined. Full insurance against the index is shown to be optimal if and only if the insurance premium is actuarially fair. These results are illustrated in an agricultural context in which the producer faces both yield and price uncertainty and in which a crop insurance market is available. Under a multiple peril crop insurance program, the producer selects a yield guarantee and a price selection at which a unit loss of output is compensated. The article first examines the effect of demand uncertainty on the optimal price selection and then analyzes how the payment of direct subsidies affects this price selection. To sign this effect, the concept of temperance introduced by Kimball (1992) is used. After the model has been presented in the next section, the design of an optimal insurance contract in the presence of background risk is derived. In the fourth section, these results are used to examine the producers' rationale for purchasing crop insurance when output price risk cannot be hedged. THE MODEL The problem of the insurance-purchasing decision of a risk-averse producer is considered in a static model. His or her revenue is based on an index, or a proxy, y, which is assumed insurable, and on a background risk [epsilon], which is assumed uninsurable. [1] The stochastic dependence between revenue risk and index risk is formalized by the deterministic revenue function R(y, [epsilon]), where [epsilon], where is independent of y and E([epsilon]) = 0, E denoting the expectation operator. The support of the cumulative distribution function associated with random variables y or [epsilon] is contained in [0, [y.sub.max]] or [[[epsilon].sub.min], [[epsilon].sub.max]] with [y.sub.max] [less than] 0 and [[epsilon].sub.min] [less than or equal to] 0 [less than or equal to] [[epsilon].sub.max], respectively. The revenue function is assumed strictly monotonic in the background risk and, without a loss of generality, a larger value of the [epsilon] risk generates a larger revenue; i. …