Valuing agricultural insurance.

Agricultural insurance, in its various guises, has a spotty history in the United States. For the most part, multiple-peril insurance products are not commercially viable without government subsidies. A variety of explanations have been offered for their poor commercial performance (Sanderson 1943; Lee 1953; Gardner and Kramer 1986; Nelson and Loehman 1987; Chambers 1989; Miranda 1991). Roughly put, each revolves around some form of market incompletion, be it information asymmetries (adverse selection, moral hazard) or incomplete contingent claims markets. While this literature enhances our understanding of agricultural insurance markets, it leaves unanswered a central empirical question for publicly supported insurance. What is its value to producers?

This question has not been ignored (Turvey and Amanor-Boadu 1992; Turvey 1992; Skees and Reed 1986; Skees, Black, and Barnett 1997; Stokes, Nayda, and English 1997; Goodwin and Ker 1998; Ker and Goodwin 2000; Yin and Turvey 2003; Babcock, Hart, and Hayes 2004). And, although the technical details differ, attempts to price insurance products all follow the basic principles of asset pricing. An asset's price should equal the expected value of the product of its stochastic payout and an appropriate stochastic discount factor (pricing kernel) (Ross 1976; Harrison and Kreps 1979; Hansen and Singleton 1982; Clark 1993; Cochrane 2001; Campbell 2003; Duffle 2001). More concretely, if the asset's stochastic payout is {??], and the proper stochastic discount factor is [??], then its price should be E [[??] [??]], where E is the expectation operator.

However, no consensus has emerged on what to use as [??] for agricultural-insurance products. Recently, Myers, Liu, and Hanson (2005) have classified agricultural-insurance valuation models into four basic types: present-value models; Black Scholes option-pricing models; arbitrage-based pricing models; and general-equilibrium, representative-agent consumer pricing models.

The present-value method interprets [??] as the product of an intertemporal discount factor and the probability measure underlying the states of Nature; the Black-Scholes option-pricing method assumes complete markets, continuous and frictionless trading, and the law of one price to derive [??] as the market-based pricing kernel; the arbitrage-based pricing model, which can be traced to Ross (1976), replaces the Black-Scholes complete-markets assumption with an assumption of either spanning assets or spanning factors to construct [??]; and the representative-agent, consumption-based model takes [??] (in an expected-utility framework) as a representative consumer's stochastic intertemporal marginal rate of substitution (see especially Cochrane 2001; Campbell 2003; and Duffle 2001).

Myers, Liu, and Hanson (2005) criticize these approaches as being, in various degrees, either unrealistic or irrelevant to an agricultural insurance context. In their place, they offer as a fifth alternative an extension of the general equilibrium, representative-agent, consumption-based model that allows for market incompletion.

This article makes a simple point. These approaches ignore or trivialize the most salient characteristic of agricultural insurance. It is insurance for agricultural producers. Inevitably, that implies that an empirically tractable method for estimating [??] is ignored. The method hinges not upon stringent market-spanning assumptions or on the farmer's role as a "representative consumer." Instead, it concentrates on his or her role as a producer, perhaps the most fundamental characteristic of farming. The essence of the approach is that rational producers, regardless of their risk preferences, never forego opportunities to risklessly raise profit.

In what follows, the model is first presented. Equilibrium production behavior is then characterized for a farmer facing a stochastic technology and stochastic markets. This characterization holds regardless of the farmer's attitudes towards risk. For the theory to be valid, farmers need only prefer more to less. When the stochastic technology is suitably smooth (Gateaux differentiable), the characterization reveals that the farmer's production cost structure defines a proper stochastic discount factor for the farmer. That, in turn, implies that the farmer's willingness to pay for an agricultural insurance product can be derived as the expected value of the product of the stochastic insurance payout and that stochastic discount factor is derived from the farmer's production cost structure. An empirical illustration of the method for estimating [??], which uses aggregate data, is presented, and a brief discussion of the empirical results follows.

The Model

I study competitive farmers facing stochastic production and stochastic markets. Formally, there are two periods. The first period, t, is nonstochastic, and the second, t + 1, is stochastic. The stochastic setting is modeled as a probability space (S, [OMEGA], [pi]) where S represents the set of states of "Nature," [pi] is a subjective probability measure, and [OMEGA] represents the measurable events (subsets of S). Random variables are defined as bounded maps from S to the reals (Savage 1954; Duffle 2001). Hence, random variable, [??], is the element of [R.sup.s] defined by

[??] = {f(s) ; s [member of] S}

where f : S [right arrow] R is the (measurable) map defining the random variable. Random variables will always be distinguished from their ex post values by a tilde (~). Notationally, [??] represents the random variable, and f(s) denotes the ex post (observed) outcome associated with realization s.

The stochastic production technology is represented by a single-product, input correspondence that maps a stochastic output, [??] [member of] [R.sup.s.sub.+], into sets of inputs that are capable of producing it. (1) Inputs are chosen in period t and are nonstochastic. Denote those inputs by x [member of] [R.sup.N.sub.+] and their prices, which are nonstochastic, by w [member of] [R.sup.N.sub.+]. The stochastic output is also chosen in period t but realized or observed in period t + 1. The period t + 1 price of the output is stochastic and denoted by [??] [member of] [R.sup.S.sub.++]. Notationally, therefore, if [??] is chosen by the producer in period t and s [member of] S is realized, the ex post or observed output in period t + 1 is z(s) and the ex post (spot) output price is p(s).

The input correspondence describing the technology, X : [R.sup.S.sub.+] [right arrow] [R.sup.N.sub.+] , maps stochastic output into variable input sets according to

X(??) = {x [member of] ] [R.sup.N.sub.+] x can produce [??]}.

The only technical requirement is that X([??]) be closed. No curvature or disposability assumptions are imposed.

The (period t) minimal cost of producing the stochastic output, [??], is given by the production cost function

c (w, [??]) = min {w' x : x [member of] X ([??])}

if X (??) is nonempty and [infinity] otherwise. As usual, c(w, [??]) is nondecreasing and superlinear (positively linearly homogeneous and concave) in w. The proof of these properties is standard and, therefore, omitted.

The only restriction on the farmer's ex ante (period t) preferences is that he or she strictly prefers more period t consumption to less and at least weakly prefers more period t + 1 consumption to less. More formally, if we denote the farmer's ex ante preferences over period t consumption, [q.sub.t], and period t + 1 stochastic consumption, [[??].sub.t+1], by W : [R.sub.+] x [R.sup.S.sub.+] [right arrow] R, then [q.sup.*.sub.t] > [q.sub.t] [??] W ([q.sup.*.sub.t], [[??].sub.t+1] > W ([q.sub.t], [??].sub.t+1])

and

[[??].sup.*.sub.t] [greater than or equal to] [[??].sub.t] [??] W ([q.sub.t], [[??].sub.t+1] [greater than or equal to] W ([q.sub.t], [??].sub.t+1])

The farmer can also transform period t income into period t + 1 consumption by investing in financial markets. These markets include all financial assets available to farmers. These markets are frictionless but stochastic, and the ex ante financial security payoffs are given by the S x J matrix A (a matrix of J random variables). The stochastic payout on the jth financial asset is denoted by [[??].sub.j] [member of] [R.sup.S.sub.+], and its period t price is denoted by [[upsilon].sub.j]. The firm's portfolio vector, corresponding to its period t holdings of the financial assets, is denoted by h [member of] [R.sup.J]. Denote the jth stochastic return by [[??].sub.j] = [[??].sub.j]/[[upsilon].sub.j]. In what follows, we will refer to the farmer's choice of h as his or her hedge.

Equilibrium Behavior

The farmer's problem in period t is to choose x, [??] [[??].sub.t+1] and h according to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[omega].sub.t] is period t wealth, v' = ([[upsilon].sub.1],..., [[upsilon].sub.j],), and [??] [??] denotes the random variable whose ex post realization in state s is p(s) z(s). We start by demonstrating a basic result, originally due to Chambers and Quiggin (2002, 2004), which characterizes equilibrium production and hedging behavior regardless of the farmers' attitudes towards risk. (2)

PROPOSITION 1. Given any stochastic consumption, ([[??].sub.t+1]) the farmer solves

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof: Suppose, to the contrary, that the farmer chooses [??] and h that are not cost minimizing as claimed, but that yield [[??].sub.t+1]. Denote these choices by [[??].sup.0] and [h.sup.0] . This cannot be optimal because by choosing

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the farmer saves