Implementable Ramsey-Boiteux pricing in agricultural and environmental policy.

Agricultural producers routinely face price discrimination that is based either on characteristics of the product that they produce, or on the physical manner in which the product is produced. For example, within the U.S. federal milk marketing order system, producers traditionally faced different administered prices for butterfat and skim milk (Buccola and Iizuka, 1997). Wheat marketed by producers at local elevators in the United States is priced according to its measurable characteristics including test weight, moisture content, percentage of foreign matter ("dockage"), and protein content (Barkley and Porter 1996; Lavoie 2005). In the United States, hog and other livestock producers receive different per unit prices depending on the lean/fat content of the animals that they deliver (Wang and Jaenicke 2006). In Europe, wine cooperatives reward producers of the same varietal grape differentially depending upon the average sugar content of their delivered grapes (Touzard et al. 2001; Zago 2006). In a policy context, receipt of U.S. farm program benefits, including subsidized prices, is often contingent upon participating farmers complying with environmental constraints on production practices (e.g., "swamp-buster" and "sodbuster" provisions).

This article considers how to set such discriminatory prices, which differentiate optimally in terms of measurable and contractible quality characteristics or measurable and contractible production characteristics, in the presence of hidden knowledge between the economic actor setting the discriminatory prices and the producers receiving the discriminatory price. Our analysis specifically treats the case where the hidden knowledge is about the producer's physical cost structure. Each producer knows his or her cost structure exactly, but the economic actor setting the price can only observe, and therefore contract upon, the quantity and the quality of the product (or other production characteristic) delivered by the producer.

For the sake of a concrete example in discussing our results, we adopt the economic metaphor of a government or regulator trying to simultaneously subsidize farmers while controlling for the adverse environmental consequences of farming indirectly through output-price subsidies that are coupled with acreage retirement provisions. In this setting, the observables and contractibles are the farmer's output and his or her retirement of acreage, and the hidden knowledge is about the farmer's cost structure. Thus, we extend the results of Chambers (1992, 2002), Smith (1995), Bourgeon and Chambers (2000), and Innes (2003) on optimal policy formulation based on a single observable to the case where there are two observables and contractibles.

However, while the results are stated and interpreted in this framework, it is obvious that they also apply to other discriminatory settings with only minor changes. For example, take the problem of a wine cooperative determining the optimal strategy for rewarding its producers according to the quality (as measured by sugar content) of the grapes that they deliver. Then, the observables would be the quantity of the grapes delivered and the average sugar content of delivered grapes. Another example comes from the pricing policy of poultry integrators who pay growers on the basis of the quantity of product produced and upon their use of specific contractually specified inputs. (1)

Conventional wisdom is that the presence of hidden knowledge on the part of farmers in such a setting would prevent the government or regulator from achieving its most preferred policy (Guesnerie and Laffont 1984; Laffont 1988; Chambers 1992, 2002; Smith 1995; Bourgeon and Chambers 2000; Innes 2003). We reach the surprising, at least to us, conclusion that for a very broad range of production technologies, the perfect-information voluntary policy, which involves a modified version of Ramsey-Boiteux pricing for the agricultural commodity, is implementable even in the presence of hidden knowledge by farmers about their types.

In what follows, we first detail the basic model. Then we consider the best perfect-information voluntary policy and show that it involves a modified form of Ramsey-Boiteux pricing. After that, we show that if only the first-order necessary conditions for truthful implementation are considered, the modified Ramsey-Boiteux pricing rule is implementable, and we identify a class of technologies for which optimality in the presence of hidden knowledge always involves the modified Ramsey-Boiteux pricing rule. Then we turn to a brief analysis of the satisfaction of the second-order conditions for truthful implementation, and the article then concludes.

The Model and Notation

Each farm's technology is given by the restricted profit function, [pi](p, a, [theta]), where p is the per unit price received by the farm for the product produced, a is either an input such as land, or a nonpriced output, such as pollution, and [theta] represents an efficiency parameter. The efficiency parameter has a number of potential interpretations. Perhaps the most intuitive, however, is that it indexes the imperfectly measurable human capital of the farm operator that we typically think of as the farmer's ability. In this context, the interpretation of [pi](p, a, [theta]) is the maximum variable profit available to a farmer of ability [theta] given that he or she farms a acres and faces a market price of p for his or her product.

For the sake of simplicity, we assume that [pi] is sufficiently smooth to admit any derivatives that we wish to take. In what follows, our intuitive focus is on land retirement for either conservation or environmental purposes, and so we will refer to a mnemonically as land. Without any true loss of generality, farms are ranked so that the efficiency parameter indexes them positively and, thus, [[pi].sub.[theta]] (p, a, [theta]) > 0. Higher ability means higher profits. We also assume that the efficiency parameter positively indexes both production and the shadow price of land so that [[pi].sub.p[theta]](p, a, [theta]) > 0, [[pi].sub.a[theta]](p, a, [theta]) > 0. In the context of our farmer ability interpretation of [theta], assuming that [[pi].sub.p[theta]] (p, a, [theta]) > 0 implies, by Hotelling's lemma, that farmers with higher ability optimally produce more output from a given acreage and for a given price than farmers with lower ability. Assuming that [[pi].sub.a[theta]] (p, a, [theta]) > 0 implies that the shadow price of land to farmers of higher ability is greater than the shadow price to farmers of lower ability. Without loss of generality, [theta] is assumed to be distributed according to G([theta]), which is strictly increasing and smooth on its support [THETA]. Both the government and the farmer know [pi] and G.

There exists a perfectly competitive market (with perfectly elastic demand) to which all farmers have access. In that market, the prevailing price is [p.sub.m], which is independent of the actions taken by any farmer or the government. Following Chambers (1992) and Bourgeon and Chambers (2000), we therefore assume, for the sake of simplicity, that the country in question is small relative to the market. Following arguments developed in Chambers (2002), we can extend our argument to the case where the country is large. When faced with a price of [p.sub.m], farmers choose a according to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

First-order conditions for an interior solution require

[[pi].sub.a]([p.sub.m], [a.sup.*]([theta]), [theta]) = 0

while the farmer's output choice [y.sup.*]([theta]) by Hotelling's lemma is governed by

[y.sup.*]([theta]) = [[pi].sub.p]([p.sub.m], [a.sup.*]([theta]), [theta])

so that

[theta]' > [theta] [??] [y.sup.*]([theta]) > [y.sup.*]([theta]).

Here [y.sup.*]([theta]) and [a.sup.*]([theta]) represent the amount produced and the acreage farmed by a farmer of type [theta] when he or she relies exclusively on the competitive market, and [THETA]([theta]) represents the farmer's reservation profit in the absence of a government program.

The government makes available to farmers a subsidy scheme in return for them (the farmers) taking some action on a. For analytic simplicity, we assume that all such restrictions can be modeled as direct controls on the level of a. Specifically, the government will offer farmers a price for y that is contingent on the actions that they take with regard to a. We assume that a is both observable and fully contractible.

One could model such price discrimination as a nonlinear pricing problem, that is, as one of the government choosing a payment function, [??], relating the farmer's per unit price to a as, for example, [??](a). However, because the farmer's optimal choice of a will depend upon his efficiency parameter, [theta], we follow Guesnerie and Laffont (1984) and recognize that price is also implicitly a function of the efficiency parameter because one can always define (2)

p([theta]) = [??] (a([theta])).