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])).

Thus, the government is considering offering a set of type-specific contracts {p([theta]), a([theta]) : [theta] [member of] [THETA]} [equivalent to] {(p([THETA]), a([THETA])}, which the farmers can take or leave as they see fit. If they agree to participate, they receive an enhanced per unit output price p([theta]) compensating for a restricted use of land. If they do not take the contract, then they must rely on the free market, which means that they implicitly choose the alternative contract {[p.sub.m], [a.sup.*]([theta])}, where [a.sup.*] ([theta]) is the amount that the producer would farm for the competitive market. A [theta]-type individual will voluntarily participate in such a subsidy program if and only if it satisfies the so-called "individual rationality constraint" given by

(IR) [pi](p([theta]), a([theta]), [theta]) [greater than or equal to] [PI]([theta]).

The cost to the government for a [theta]-type farmer is thus

B([theta]) = (p([theta]) - [p.sub.m]) [[pi].sub.p] (p([theta]), a([theta]),[theta]).

We assume that the usage of land by a [theta]-farmer incurs a constant marginal social cost of v([theta]), which is type specific. Allowing social damage to be type specific encompasses the possibility that overall damage is nonlinear in a tractable fashion. An example illustrates. Suppose that the damage imposed upon the environment of farmed acreage were of the nonlinear form d(a), which is not directly tied to the farmer's efficiency parameter. Then, in equilibrium, a [theta]-type farmer who farmed a([theta]) would impose damage of d(a([theta])) upon the environment. Our specification approximates this damage with the linear approximation

v([theta])a([theta]) = d(a([theta]))

so that v([theta]) [approximately equal to] d(a([theta]))/a([theta]). More generally, however, it is not unreasonable to believe (and thus generality requires us to account for) that the damage imposed upon the environment of farmed acreage may depend not only upon acreage but upon the farm's efficiency parameter, that is, damage is of the form d(a, [theta]). For example, if [theta] indexes human capital, it is not unreasonable to suppose in many instances that environmental damage imposed in the farming process may be directly linked to the farmer's overall ability. (3)

The government's overall objective in designing the program is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [mu], > 1 represents the exogenous cost of public funds.

First-Best Policy and Perfect Information Policy

Intuitively, it may seem that the first-best policy is the simple Pigovian policy, where the government simply confronts each farm with its social marginal cost and lets each farm choose the socially optimal level of a at the prevailing market price. It is straightforward, however, that this policy, which is characterized by

[[pi].sub.a] ([p.sub.m],[ a.sup.P] ([theta]), [theta]) = v([theta])

does not allow the farmers to reach [PI]([theta]). Hence, the farmer is better off relying on the competitive market, and he or she will not voluntarily participate. If the government, instead, tries to implement a policy where each farmer receives Pm but farms [a.sup.P]([theta]), it still follows by optimality that

[pi]([P.sub.m], [a.sup.P] ([theta]), [theta]) [less than or equal to] [PI] ([theta]).

Thus, the first-best cannot be achieved by relying on the market price if the program is to be voluntary even if the government can observe the farmer's type. Accordingly, any optimal policy must accommodate (IR).

If the government could observe each farmer's type (which is contrary to our assumptions), the best voluntary policy, therefore, is derived as the solution to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This policy is separable across types so that we may conveniently derive the optimal policy pointwise as the one that solves

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each [theta] and then integrate over the support to obtain the maximal value. It is natural to derive the optimum here by using a Lagrange multiplier for the constraint and then taking first-order conditions. However, a variational argument (which can be made precise by the Lagrange method) proves more convenient in explaining our central result.

When (IR) binds, differentiation with respect to the choice variables gives

[[pi].sub.p](p([theta]), a([theta]), [theta]) dp([theta]) + [[pi].sub.a](p([theta]), a([theta]), [theta]) da([theta]) = 0

so that

(1) dp([theta])/da([theta]) = - [[pi].sub.a](p([theta]), a ([theta]), [theta])/[[pi].sub.p](p([theta]), a([theta]), [theta]).

If (p, a) have been chosen optimally then any variation in the objective function must also satisfy (for an interior solution)

[[pi].sub.p]dp([theta]) + [[pi].sub.a]da([theta]) - v([theta])da([theta]) -[mu][[[pi].sub.p] + (p - [p.sub.m])[[pi].sub.pp]]dp([theta]) -[mu](p - [p.sub.m])[[pi].sub.pa]da([theta]) = 0

where function arguments have been dropped for notational convenience. Together these two equations and (IR) imply that at the optimum

(2) [[pi].sub.p] - [mu][[[pi].sub.p] + (p - [p.sub.m]) [[pi].sub.pp]]/[[pi].sub.p] = [[pi].sub.a] - v([theta]) - [mu](p - [p.sub.m])[[pi].sub.pa], [pi](p([theta]), a ([theta]), [theta]) = [PI]([theta]).

The interpretation of these conditions is straightforward. In the absence of a program, the farmer is initially at [pi]([p.sub.m], [a.sup.*]([theta]), [theta]) = [PI] ([theta]). Suppose that the government wishes to retire an acre of farmland. Doing so lowers the farmer's rent by [[pi].sub.a]([p.sub.m], a([theta]), [theta]), which leaves him or her below the reservation utility. If the farmer is to be induced to participate in the program, he or she must be compensated. Thus, the government must offer a higher price, and the price change that just exactly balances the acreage change at the margin is given by the right-hand side of (1). Thus, the range of price and acreage variation is limited by the fact that once the farmer has been raised to a level of compensation that just equals his or her market return, every unit that the acreage falls must be matched by a corresponding increase in the price received implied by (1).

Increases in the price received by the farmer have two effects. The first is an increase in farmer welfare that is measured locally by the amount that he or she supplies, [[pi].sub.p](p([theta]), a([theta]), [theta]). The second is the budgetary effect of that price rise, which consists of two components. The first is the increased expenditure on the farmer's preexisting supply, measured by [[pi].sub.p](p([theta]), a([theta]), [theta]), times the budget weight. The second is the expenditure on new supply that is called forth by the price increase, measured by (p - [p.sub.m][[pi].sub.pp]. If price were freely variable, optimality would require that these two effects exactly offset one another. This would lead to the classic Ramsey-Boiteux inverse elasticity formula (Atkinson and Stiglitz)

p([theta]) - [p.sub.m]/p([theta]) = - [mu] - 1/[mu] [[pi].sub.p](p([theta]), a([theta]), [theta])/[[pi].sub.pp](p([theta]), a([theta]), [theta])p([theta])

relating the divergence in the subsidized price from the market price to the reciprocal of the supply elasticity.

Observe, however, that the classic inverse-elasticity rule results in a price to farmers that is lower than the market price. Indeed, if (IR) is neglected (i.e., the program is not voluntary), the government would be tempted to raise money from farmers because the shadow cost of public funds is greater than one. This reflects the implicit assumption that the government incurs a deadweight loss in raising government revenues. (Empirical estimates of [mu] for the United States range as high as 1.3.) Thus, the inverse-elasticity formula must be adjusted for the voluntary nature of the program.

Moreover, there are two, not just one, policy variables that are being manipulated, and the second, acreage, also has direct and budgetary effects. The direct effect is measured by the marginal social benefit of an acre of land farmed, [[pi].sub.a] (p([theta]), a([theta]), [theta]) - v([theta]), which initially is negative. The budgetary effect is measured by the change in budgetary cost caused by the supply response associated with the acreage change, [mu] (p - [p.sub.m]) [[pi].sub.pa]. If acreage were freely variable, optimality would require that these effects exactly offset one another leading to a condition consistent with the Ramsey-Boiteux inverse elasticity rule

[[pi].sub.a](p([theta]), a ([theta]), [theta]) - v([theta]) = [mu](p - [p.sub.m]) [[pi].sub.pa] = - ([mu] - 1) [[pi].sub.p]/[[pi].sub.pp] [[pi].sub.pa]

so that the divergence from first-best pricing would take the sign opposite of [[pi].sub.pa] and would only converge to first-best pricing as either [[pi].sub.pa] [right arrow] 0 or [[pi].sub.pp]/[[pi].sub.p] [right arrow] [infinity].

Neither acreage or price, however, are freely variable because of the need to ensure that the farmer is left no worse off than he or she would be in the free market. As noted, the range of allowable price-acreage variability is given by (1). Accommodating this restriction on the variability of price and acreage generally prevents achievement of the Ramsey-Boiteux rules. Thus, the left-hand side in the first expression in (2) measures the divergence in output price from the Ramsey-Boiteux rule and the right-hand side measures the divergence in the social marginal benefit of acreage from the Ramsey-Boiteux rule.

Optimality with Hidden Knowledge

More generally, we are interested in regulation where there exists asymmetric information between the regulator and the farmer. In the preceding section, we assumed implicitly that the regulator could costlessly design type-specific contracts. If the farmer's type is his or her own private knowledge, then it is well known that the government faces constraints in the type of contracts that it can implement (Laffont 1988; Chambers 2002). To distinguish the optimal contract defined by (2), we will refer to it as the perfect information policy or contract.

Viewing the government's problem in the presence of asymmetric information as one of mechanism design implies that these informational constraints can be framed in terms of the farmer's truthful revelation of his or her [theta] (Myerson 1981; Guesnerie and Laffont 1984). Truthful revelation requires that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which, in words, requires that if asked to report his or her [theta] in order to receive access to {p([THETA]), a([THETA])}, the farmer would find it optimal to report his or her true [theta]. The first-order condition for truthtelling is that almost everywhere

(3) [[pi].sub.p](p([theta]), a ([theta]), [theta]) p'([theta]) + [[pi].sub.a](p([theta]), a ([theta]), [theta])a' ([theta]) = 0

where primes on functions denote derivatives. Using standard methods (Guesnerie and Laffont 1984), it is easy to ascertain that the second-order conditions for truthful revelation require

(4) [[pi].sub.[theta]p](p([theta]), a([theta]), [theta])p'([theta]) + [[pi].sub.[theta]a](p([theta]), a([theta]), [theta])a'([theta]) [greater than or equal to] 0.

Thus, the first-and second-order conditions for truthtelling are satisfied only if

(5) a'([theta]) [[[pi].sub.[theta]a]]/[[pi].sub.a] - [[pi].sub.[theta]p]/[[pi].sub.p]] [greater than or equal to] 0.

There are several observations to make. Note, first, the similarity between expression (3) and (1). This suggests (which we verify below) that if the regulator can achieve the farmer's reservation utility via the modified Ramsey-Boiteux pricing rule, then he or she can use the same variational rule on price and acreage to ensure consistency with the first-order condition for truthtelling.

One should also observe that the second-order conditions for truthtelling involve both price variation and acreage variation. It is usually assumed in the canonical adverse selection framework that (4)

[partially derivative]/[partially derivative][theta][[[pi].sub.a]/[[pi].sub.p]]] = [[pi].sub.a/[[pi].sub.p] [[[pi].sub.[theta]a]/[[pi].sub.a] - [[pi].sub.[theta]p/[[pi].sub.p]]]

is either positive or negative for all a(x) and p(x), i.e., that the marginal rate of substitution between land and output price is monotonic in [theta]. In that case, a simple monotonicity restriction on the motion of price and/or acreage is sufficient to guarantee that the second-order conditions for truthtelling are met, as stated formally in the next lemma:

LEMMA 1. If [[pi].sub.[theta]a]/[[pi].sub.a] - [[pi].sub.a] - [[pi].sub.[theta]p] > 0 (resp. <0) everywhere, then any schedule {a([THETA]), p([THETA])} satisfying the first-order condition for truthtelling and a'([theta]) [greater than or equal to] 0 (resp. [less than or equal to] 0) for all [theta] [member of] [THETA] also satisfies the second-order solution for truthtelling. If [[pi].sub.[theta]a]/[[pi].sub.a] - [[pi].sub.[theta]p]/[[pi].sub.p] = 0 everywhere, so that [pi](p, a, [theta]) = [??](m(p, a), [theta]), then any schedule {a([THETA]), p([THETA])} satisfying the first-order condition for truthtelling for all [theta] [member of] [THETA] also satisfies the second-order condition for truthtelling.

To understand why we cannot expect such a regularity in our context, notice that the bracketed term in (5) is of an ambiguous sign under our assumptions. The two expressions therein measure the marginal effect of the efficiency parameter on the shadow price (to the farmer) of land and the marginal effect of the efficiency parameter on supply. If efficiency is interpreted in its usual sense, we expect (and thus we have imposed by assumption) that both of these terms are positive. Thus, their difference can be either positive or negative, which means that standard procedures for ensuring consistency with the second-order conditions may not be available.

To proceed, it is convenient to introduce some further notation. Denote the difference between the farmer's income and his reservation utility as

(6) R([theta]) = [pi](p([theta]), a([theta]), [theta]) - [PI]([theta]).

Expression (3), therefore, implies

(7) R'([theta]) = [[pi].sub.[theta]](p([theta]), a([theta]), [theta]) - [PI]'([theta]).

Differentiation establishes that

R"([theta]) = [[pi].sub. [theta] [theta]](p ([theta]), a([theta]), [theta]) - [PI]"([theta]) + [[pi].sub. [theta]p] (p([theta]), a([theta]), [theta])p'([theta]) + [[pi].sub. [theta]p](p([theta]), a([theta]), [theta])a'([theta])

so that invoking (4) shows that for the second-order condition to be satisfied

(8) R"([theta]) - [[pi].sub. [theta] [theta]](p([theta]), a([theta]), [theta]) + [PI]"( [theta]) [greater than or equal to] 0.

By these arguments, the regulator's problem in the presence of asymmetric information between itself and the farmer can thus be recognized as choosing acreage and price according to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with R([theta]) > 0 for all [theta].

Initially, we disregard the second-order conditions for truthful revelation, (8), and only impose (6) and (7). Denote by [tau] ([theta]), [lambda] ([theta]) the Lagrange multipliers corresponding to (6) and (7) to obtain the following Lagrangian expression for this program

L = [[integral].sub.[THETA]]{[R + [PI] -va - [mu](p - [p.sub.m])[[pi].sub.p]]g +[tau][R + [PI] - [pi]] + [lambda][R' - [[pi.sub.[THETA]] + [PI]']}d[theta]

where g([theta]) = G'([theta]) > 0. Integrating by parts gives

[[integral].sub.[THETA]] [lambda]([theta])R'([theta])d[theta] = [lambda]([bar.[theta]])R([bar.[theta]])-[lambda]([[theta].bar])R([[theta].bar) - [[integral].sub.[THETA]][lambda]'([theta])R([theta])d[theta].

Substituting this expression into the Lagrangian gives us the following Hamiltonian expression for the regulator's objective function

L = [[integral].sub.[THETA]]H(R,p,a,[theta])d[theta] + [lambda]([bar.[theta]]) R([bar.[theta]]) - [lambda]([[theta].bar])R([[theta].bar])

where

H(R, p, a, [theta]) = [R + [PI] - va - [mu](p - [p.sub.m]) [[pi].sub.p])]g + [tau][R + [PI] - [pi]] - [lambda]'R - [lambda][[[pi].sub. [theta] - [PI]'].

This problem is separable across types, and in principle, we could pursue a variational argument similar to that used in deriving and discussing the optimality conditions for the first best. Instead, we rely here on simpler first-order arguments. By the requirement for pointwise optimization assuming an interior solution for both p and a, we have the following necessary conditions for a solution:

(9) [partially derivative]H/[partially derivative]R = g([theta]) + [tau]( [theta]) - [lambda]'([theta]) [less than or equal to] 0, R([theta]) [greater than or equal to] 0

(10) [partially derivative]H/partially derivative]p] = [mu][[[pi].sub.p] + (p - [p.sub.m]) [[pi].sub.pp]]g - [tau] [[pi].sub.p] - [lambda]( [theta]) [[pi].sub. [theta]p] = 0

(11) [partially derivative]H/[partially derivative]a = [v([theta]) + [mu](p - [p.sub.m]) [[pi].sub.pa]]g - [tau] [[pi].sub.a] - [lambda] [[pi].sub. [theta]a] = 0

and the transversality conditions

[partially derivative]L/[partially derivative]R([[theta].bar]] = - [lambda]([theta]) [less than or equal to] 0 (R([[theta].bar]]) greater than or equal to] 0)

where [[theta].bar] and [bar.[theta]] define the upper and lower supports, respectively, of [THETA].

Is the Perfect Information Policy Implementable?

We now show that the perfect information policy described by (2) can satisfy expressions (9)-(11). In the perfect information case, R([theta]) = 0. Set [lambda] ([theta]) = 0 for all [theta]. This ensures that the tranversality conditions are satisfied. Moreover, use of (10) yields

[tau] = - [mu][1 + (p - [p.sub.m]) [[pi].sub.pp]/[[pi].sub.p]]g

which allows us to simplify (9) to

1 - [mu][1 + (p - [p.sub.m]) [[pi].sub.pp]/[[pi].sub.p]] < 0

which is thus satisfied so long as p([theta]) [greater than or equal to] [p.sub.m]. Using (11), we also get

[tau] = - [v/[[pi].sub.a] + [mu](p - [p.sub.m]) [[pi].sub.pa]/[[pi].sub.a]]g.

Hence, conditions

[[pi].sub.p] - [mu][[[pi].sub.p] + (p - [p.sub.m]) [[pi].sub.pp]/[[pi].sub.p]]

= [[pi].sub.a] - v([theta]) - [mu](p - [p.sub.m]) [[pi].sub.pa]/[[pi].sub.a]

R([theta]) = 0

solve the regulator's problem.

These conditions correspond exactly to (2). Thus, we have established

PROPOSITION 1. The solution to (2) satisfies (9)-(11).

Proposition 1 can be explained as follows. In designing a contract structure {p([THETA]), a([THETA])} that is consistent with (IR), the government implicitly chooses a nonlinear price schedule [??](a) whose slope in (p, a) space is given by the right-hand side of (1). For a contract structure to be incentive compatible in the presence of asymmetric information, the revelation principle implies that the farmer's true [theta] must be the solution to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first-order condition for this problem, which is given by (3), also implicitly defines a nonlinear price structure [??](a) whose slope in (p, a) space to the first-order elicits truthful revelation. By (3), that nonlinear price schedule has exactly the same slope in (p, a) space as (1). Hence, to the first-order, any individually rational contract structure automatically elicits truthful revelation. Thus, if a scheme can be made individually rational for all [theta], expression (7), which is derived from (3), is redundant in formulating the government's optimal program in the presence of asymmetric information.

Together with Lemma 1, Proposition 1 gives conditions under which the government can attain the perfect information (symmetric information) policy even in the presence of asymmetric information between itself and farmers. If the farmer's technology satisfies either of the conditions specified in Lemma 1, the perfect information policy is incentive compatible and can be achieved. Because it is generally believed that the perfect information policy is not attainable in the presence of asymmetric information between the principal and the agent, this conclusion is somewhat startling. In fact, in the canonical adverse selection model, conditions analogous to those specified in Lemma 1 only ensure the existence of an implementable monotonic policy in the presence of asymmetric information. Therefore, imposing them only ensures that one can ignore (8) in examining the design of the optimal government policy in the presence of asymmetric information. Imposing them, however, would not imply that the principal could implement the perfect information policy as in the case in our model. Hence, taken together Proposition 1 and Lemma 1 describe a situation where the government pays no informational rents to farmers (R([theta]) = 0 for all [theta]), and truthful implementation is ensured. Hence, this result bears closer examination.

The conditions in Lemma 1 are derived from expression (5) and are expressed in terms of partial derivatives of the profit function. The second part of Lemma 1 shows that if (p, a) are separable from 0 in the farmer's profit function, then any schedule {a([THETA]), p([THETA])} satisfying the first-order condition for truthtelling for all [theta] [member of] [THETA] also satisfies the second-order condition for truthtelling. Hence, when the profit function is separable in this manner, then by Proposition 1 the perfect information policy is always achievable. In the following proposition, we identify a nonseparable technology for which the first-best, perfect information policy is implementable. (The separable case is the special case of the technology in Proposition 2 where k(a) = 0 for all a.)

PROPOSITION 2. If [pi](p, a, [theta]) = [??](m(p, a), [theta])-k(a), where k(a) [greater than or equal to] 0, k'(a) > 0 and k"(a) [greater than or equal to] 0, then the perfect information policy is implementable provided a(.) is increasing.

Proof: We have [[pi].sub.a] = [[pi].sub.a] = [[??].sub.m] [m.sub.a] - k'(a); [[pi].sub.a[theta]] = [[??].sub.m[theta]][m.sub.a]; [[pi].sub.p] = [[??].sub.m][m.sub.p]; and [[pi].sub.p[theta]] = [[??].sub.m[theta]] [m.sub.p], which give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[??].sub.m] [m.sub.a] - k'(a) > 0 since we have a([theta]) < [a.sup.*]([theta]) for all [theta].

Proposition 2 demonstrates that the best voluntary policy is implementable even in the presence of asymmetric information for an important class of structural models, provided that the corresponding acreage schedule is increasing. Thus, for these models the perfect information policy is implementable. As discussed above, the perfect information policy corresponds to a policy that essentially perturbs the Ramsey-Boiteux pricing rule in a manner that is consistent with ensuring that R([theta]) [greater than or equal to] 0. More generally, however, one expects to see the largest divergences between p([theta]) and [p.sub.m] associated with the individuals having the least elastic supply, and the smallest divergences associated with individuals having the most elastic supply.

We close this section with an example of a technology for which the optimal Ramsey-Boiteux pricing rule is implementable in the presence of asymmetric information between the government and farmers. It is worth noting that this class of technologies is routinely imposed in many analyses of hidden-knowledge problems.(see, e.g., Laffont and Tirole 1993; Lewis and Sappington 1989).

EXAMPLE 1. Consider the class of restricted profit functions given by

[[pi](p, a, [theta]) = - [phi]([theta]) + h([theta])m(p, a) - ka

where [phi] ([theta]) [greater than or equal to] 0 corresponds to a fixed cost for the [theta]-type farm. We assume that [phi]' ([theta]) > 0, h([theta]) > 0, h'([theta]) [greater than or equal to] 0, that m(p, a) = [p.sup.2]a, and that there is a maximum amount of land {[bar.a] that each farmer can use for producing. We have [[pi].sub.a] = h([theta])[p.sup.2] - k, which yields [a.sup.*]([theta]) = [bar.a] for all [theta] verifying h([theta]) > k/[p.sup.2.sub.m] and [phi]([theta]) < [h([theta])[p.sup.2.sub.m] - k][bar.a]. The latter condition is always satisfied provided that [bar.a] is large enough, while the former holds for all [theta] if it holds for [[theta].bar]. We assume that both conditions are satisfied in the following. Hence, without regulation, farmers use all their land and earn a profit given by

[PI]([theta]) = - [phi]([theta]) + [h([theta]) [p.sup.2.sub.m] - k][bar.a]

for all [theta] [member of] [[theta].bar], [bar.[theta]]. The first condition in (2) simplifies to

[p.sub.m]/p([theta]) = v([theta]) + 2 [micro]k/[mu](h([theta])[p.sup.2] + k)

[FIGURE 1 OMITTED]

which implicitly defines price p([theta]) > [p.sub.m] for given [theta] if the corresponding marginal damage v([theta]) is large enough, and more precisely if

v([theta]) > 2 [mu][[square root of h([theta])k][p.sub.m] - k].

For all [theta] satisfying this condition, the optimal price is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a decreasing function of 0 provided that v'([theta]) is small. Imposing (IR) as a binding constraint gives the nonlinear, land-price relationship

A(p) = [bar.a] h([theta])[p.sup.2.sub.m] - k/h([theta])[p.sup.2] - k.

Observe that A(p) is decreasing for all p [member of] [p([bar.[theta]]), p([[theta].bar] and convex, as depicted in Figure 1.

Ensuring Satisfaction of the Second-Order Conditions

In general verifying whether the second-order conditions are satisfied or not is quite difficult. Because there is no reason to expect that

[[pi].sub.[theta]a]/[[pi].sub.a] - [[pi].sub.[theta]p]/[[pi].sub.p]

takes a particular sign, it is difficult to ascertain even at the crudest levels just what one expects the conditions for satisfaction of the second-order condition for truthtelling to be intuitively.

Suppose that the perfect information policy does not satisfy (5) over [[THETA].sub.1] [subset] [THETA]. Hence, over a nonnegligible subset [GAMMA] [subset or equal to] [THETA] (with [[THETA].sub.1] [subset or equal to] [GAMMA]) condition (4) is binding. Using (3), [??]([theta]), [??]([theta]) must thus satisfy

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [theta] [member of] [GAMMA]. There are two general solutions to this problem: either [??]'([theta]) = [??]'([theta]) = 0 or

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [theta] [member of] [GAMMA]. We already investigated the special case [pi](p, a, [theta]) = [??](m(p, a), [theta]). Apart from this case, (13) implicitly defines functions [??](x) and [??](x) over [GAMMA] (up to a constant). However, to be implementable, these functions must also satisfy (3). A total differentiation of (13) will rapidly convince the reader that this is only true under the most stringent conditions. Consequently, as a general rule, one expects that the solution when (4) is binding involves bunching over the interval [GAMMA]. More precisely, one expects that [??]([theta]) = [bar.x] and [??]([theta]) = [bar.a] over [GAMMA].

Concluding Remarks

We have considered the optimal design of output-subsidy policies in tandem with environmentally motivated acreage controls. For a broad class of models, familiar in the literature, optimality has been shown to entail a modified version of Ramsey-Boiteux pricing. An obvious question to raise about these results is whether one can do better with policies that link acreage retirement to direct payments for acreage. It turns out that such policies cannot typically remove the informational rents that are associated with hidden knowledge on the part of farmers (see, for example, Smith 1995). Preliminary calculations, which for brevity's sake we do not report here, reveal that direct payment schemes can be worse than output-subsidy schemes at least for policies that induce "small" changes in farmers' habits. The intuitive reason is that over a limited range, output-subsidy schemes allow the regulator to use two instruments, acreage control and supply adjustment (indirectly through the price subsidy), to attack the environmental and hidden-knowledge problems.

[Received September 2006; accepted September 2007.]

References

Atkinson, A.B., and J.E. Stiglitz. 1980. Lectures on Public Economics. London: McGraw-Hill Book Co.

Barkley, A., and L. Porter. 1996. "The Determinants of Wheat Variety Selection in Kansas, 1974 to 1993." American Journal of Agricultural Economics 78(1):202-11.

Bourgeon, J.-M., and R.G. Chambers. 2000. "Stop-and-Go Agricultural Policies." American Journal of Agricultural Economics 82(1):1-13.

Buccola, S., and Y. Iizuka. 1997. "Hedonic Cost Models and the Pricing of Milk Components." American Journal of Agricultural Economics 79(2):452-62.

Chambers, R.G. 1992. "On the Design of Agricultural Policy Mechanisms." American Journal of Agricultural Economics 74(3):646-54.

--. 2002. "Information, Incentives and the Design of Agricultural Policies." In B. L. Gardner, and G. C. Rausser, eds. Handbook of Agricultural Economics. New York, NorthHolland/Elseveir.

Guesnerie, R., and J.-J. Laffont. 1984. "A Complete Solution of a Class of Principal-Agent Problems With an Application to the Control of the Self-Managed Firms." Journal of Public Economics 25:329-70.

Innes, R. 2003. "Stop and Go Agricultural Policies with a Land Market." American Journal of Agricultural Economics 85(1):198-215.

Laffont, J.-J. 1988. Fundamentals of Public Economics. Cambridge: MIT Press.

Laffont, J.-J., and J. Tirole. 1993. A Theory of Incentives in Procurement and Regulation. Cambridge: MIT Press.

Lavoie, N. 2005. "Price Discrimination in the Context of Vertical Differentiation: An Application to Canadian Wheat Exports." American Journal of Agricultural Economics 87(4):835-54.

Lewis, T.R., and D.E.M. Sappington. 1989. "Countervailing Incentives in Agency Problems." Journal of Economic Theory 49:294-313.

Myerson, R.B. 1981. "Optimal Auction Design." Mathematics of Operations Research 6:58-73.

Smith, R.B.W. 1995. "The Conservation Acreage Reserve Program As a Least-Cost Land-Retirement Program." American Journal of Agricultural Economics 77:93-105.

Touzard, J.-M., F. Jarrige, and C. Gaullier. 2001. Qualite du vin et prix du raisin : trois lectures du changement dans les cooperatives vinicoles du Languedoc. Etudes et Recherches sur les Systemes Agraires et le Developpement, INRASAD. 20 p.

Wang, Y., and E.C. Jaenicke. 2006. "Simulating the Impacts of Contract Supplies in a Spot Market-Contract Market Equilibrium Setting." American Journal of Agricultural Economics 88(4):1062-77.

Zago, A. 2006. "The Design of Quality Pricing in Procurement Settings via Technology Estimation." Working Paper, University of Verona, Italy.

(1) The authors thank the Editor for pointing this example out to us.

(2) Chambers (2002) contains a specific discussion of this issue in the context of optimal agricultural policy formulation.

(3) A reviewer asks whether taking v([theta]) to be constant across types changes the results significantly. While it certainly simplifies the analysis in some situations, it does not change the basic results that we develop below.

(4) This is the so-called "sorting," "single crossing," or "Spence-Mirrless" condition.

Jean-Marc Bourgeon is director of research, INRA, and professor of economics, Ecole Polytechnique, Palaiseau, France.

Robert G. Chambers is professor, Department of Agricultural and Resource Economics, University of Maryland, College Park, MD.