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...their costs. The second property is separability of less efficient types. The third property is full bunching of types when the available fund is small enough. The fourth property of the mechanism is that it is a third best one, that is, the output under this regulatory mechanism is strictly lower than the second best output for any given type.
Keywords and Phrases: Regulation, Asymmetric information, Limited funds.
JEL Classification Numbers: D82, H42, L51.
1 Introduction
We analyze the problem of regulating a monopolist with unknown cost when the regulator has limited funds. Baron and Myerson (1982) and Laffont and Tirole (1993) developed a procedure to regulate a monopolist with unknown cost in the absence of any fund constraints. The main property of the optimal (or second best) mechanism is full separability of types, that is, if the monopolist is a high (low) cost type then she produces lower (higher) quantity and recieve a lower (higher) transfer. The cost of separation, induced by the optimal mechanism, is the information rent enjoyed by the lower cost or more efficient types. The regulator uses its fund to pay this information rent. However, if there are numerous projects, public funds are usually scarce. It is reasonable to imagine that the fund provider may be unable to finance the monopolist at the level prescribed by the optimal mechanism.
When funds are limited, the regulator has two instruments to limit the transfer: (a) bunching the more efficient types and (b) under-production. The optimal regulatory mechanism, which we call the constrained optimal mechanism, prescribes that the monopolist supplies a good of lower quantity (compared to the second best quantity) and that the more efficient types produce the same quantity. These two distortions reduce the information rent and the quantity produced by the more efficient types. However, if the fund crisis is "too" strong, the constrained optimal mechanism prescribes full bunching. We also highlight the difference between the optimal and the constrained optimal mechanism. Our comparative static result show that a reduction in available fund reduces the quantities produced by all types and increases the interval in which there is bunching.
There are several papers dealing with mechanism design problems under asymmetric information when there exists budget constraints. Laffont and Robert (1996) describe the optimal auction when all the bidders have a financial constraint which is common knowledge. Like in our constrained optimal mechanism, the financial constraint in Laffont and Roberts (1996) reduces the bids of all participants (even those with a low valuation for the good). Che and Gale (2000) extends the result in Laffont and Roberts (1996) by relaxing the assumption that financial constraints are common knowledge. Monteiro and Page Jr. (1998) describe the optimal selling mechanisms for multiproduct monopolists in the presence of budget constrained buyers. To construct the constrained optimal mechanism, we extend the methodology of Thomas (2002). Thomas (2002) considers the incentive problem of a monopolist who faces financially constrained buyers. Finally, Gautier (2002) considers the regulator's mechanism design problem under financial constraint when there are two types of firm. In Gautier (2002), bunching is an issue only if the financial constraint is sufficiently strong. We develop and analyze our model in Sections 2-4. All proofs are relegated in the Appendix.
2 The model
The utility of the monopolist is [U.sub.m] = t - [theta]q where t is the transfer that she receives from the regulator and [theta] is her marginal cost and q is the quantity of the public good she produces. The utility function of the regulator is [U.sub.r] = S(q) - t where S(q) is the consumer's surplus when a quantity q of public good is supplied and t is the transfer to the monopolist. S(q) is assumed to be twice differentiable with S'(q) > 0, S"(q) is the first best outcome. (3)
We assume that the marginal cost of the monopolist is private information. In this context, we assume that the marginal cost of the monopolist [theta] belongs to the interval [[[theta].bar], [bar.[theta]]] where for all [theta] [member of] [[[theta].bar], [bar.[theta]]]. The regulator's objective is to maximize [[integral].sub.[[theta].bar].sup.[bar.[theta]]{S(q([theta])) - t([theta])} f([theta])d[theta] subject to incentive compatibility constraint (or IC) and participation constraint (or PC). A direct mechanism M = , in this context, simply specifies a type contingent quantity-transfer pair. Here q : [[[theta].bar], [bar.[theta]]] [right arrow] [R.sub.+] and t : [[[theta].bar], [bar.[theta]]] [right arrow] [R.sub.+]. For simplicity we restrict attention to continuous mechanisms. Let [U.sub.m]([theta]; [theta]') = t([theta]') - [theta]q([theta]') be the utility of the monopolist under the mechanism M if her true type is [theta] and if she announces [theta]' [member of] [[[theta].bar], [bar.[theta]]]. With slight abuse of notation, let us define [U.sub.m]([theta]) [equivalent to] [U.sub.m]([theta]; [theta]), for all [theta] [member of] [[[theta].bar], [bar.[theta]]]. Incentive compatibility requires that [U.sub.m]([theta]) [greater than or equal to] [U.sub.m]([theta]; [theta]'), for all [theta], [theta]' [member of] [[[theta].bar], [bar.[theta]]] and participation constraint states that [U.sub.m]([theta]) [greater than or equal to] 0, for all [theta] [member of] [[[theta].bar], [bar.[theta]]]. It is well known in the literature that the optimal mechanism M satisfies both the incentive compatibility constraint and the participation constraint if and only if the utility of any type [theta] [member of] [[[theta].bar], [bar.[theta]]] is given by [U.sub.m]([theta]) = [[integral].sub.[theta].sup.[bar.[theta]]] q([tau])d[tau] and the optimal type-contingent...
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