Optimal state-contingent regulation under limited liability.

We consider an optimal regulation model in which the regulated firm's production cost is subject to random, publicly observable shocks. The distribution of these shocks is correlated with the firm's cost type, which is private information. The regulator designs an incentive-compatible regulatory scheme, which adjusts itself automatically ex post given the realization of the cost shock. We derive the optimal scheme, assuming that there is an upper bound on the financial losses that the firm can sustain in any given state. We first consider a two-type, two-state case, and then extend the results to the case of a continuum of firm types and an arbitrary finite number of states. We show that the first-best allocation can be implemented if the state of nature conveys enough information about the firm's type and/or the maximal loss that the firm can sustain is sufficiently large. Otherwise, the solution is characterized by classical second-best features.

1. Introduction

* The optimal regulation literature has seen major developments in the past twenty-five years. This literature has mainly focused on the issue of how to regulate a firm when it has private information about its demand and cost functions. In practice, however, regulated rates are set for an extended period, typically a few years. During this period, the demand and cost conditions may be subject to random shocks. Therefore, it is important to design flexible regulatory mechanisms that can respond to these shocks. In particular, when regulated rates are not appropriately adjusted following negative external shocks, the firm may incur a large deficit that, due to limited liability, it may be unable to sustain. In this article we consider the design of an optimal regulatory mechanism that responds to ex post cost shocks and takes explicit account of the firm's limited ability to sustain losses.

Our analysis differs from classical optimal regulation theory (e.g., Baron and Myerson, 1982; Laffont and Tirole, 1986; and Lewis and Sappington, 1988) in three respects. First, we explicitly consider limited liability constraints: the ex post profit of the firm cannot fall below some given (possibly negative) level. (1) Second, a publicly observable signal conveys information about the firm's hidden cost type and is used by the regulator to set the regulated rate. Finally, the public signal is a real cost shock that affects the firm's cost directly; consequently, the optimal production level of the firm is signal dependent. In this setting, the regulator designs an incentive-compatible regulatory scheme that adjusts itself automatically ex post given the realization of the publicly observed cost shock, before the firm produces. This regulatory scheme can be thought of as an "indexed" or state-contingent incentive scheme: the regulator does not have to redesign it after the realization of each cost shock.

Examples for the type of public cost shocks that we have in mind include equipment failures and fluctuations in input prices (e.g., fuel prices in the case of electric utilities). A large number of equipment failures may indicate that the firm's technology, and hence its unobserved cost type, is inefficient. The correlation between observed input prices and the firm's unobservable cost type may reflect the fact that the firm chooses its technology ex ante on the basis of its input price forecasts. If ex ante technological choices are relatively inflexible, the ex post realization of input prices will be correlated with the firm's technology, although due to forecasting errors, the correlation is bound to be imperfect. (2)

The optimal regulatory mechanism depends in our setting on the degree of correlation between the public signal and the firm's hidden cost type, as well as on the maximal deficit that the regulated firm can sustain. We find that whenever the realization of the random cost shocks conveys enough information about the firm's type and/or the maximal deficit that the firm can sustain in any given state is sufficiently large, the optimal regulatory scheme implements the first-best allocation, despite the fact that the firm's type, which determines the distribution of its costs in the various states of nature, is private information. By contrast, if the first-best allocation cannot be implemented, the solution is characterized by classical second-best features, i.e., the production levels of inefficient types are distorted downward to reduce the expected cost of informational rents, there is no distortion "at the top," and there is no (expected) rent "at the bottom." We obtain these results first in a simple two-type, two-state case, and then extend them to the case of a continuum of firm types and an arbitrarily large, but finite, number of states of nature.

The idea that regulators can exploit the correlation between ex post public signals and the firm's type and design signal-dependent transfers that implement the first-best allocation was first explored by Riordan and Sappington (1988). Their results are analogous to those of Cremer and McLean (1985, 1988) in the context of auction theory--the main difference being that in the context of auctions, the reports of other bidders play the role that the ex post public signals play in the Riordan and Sappington model. Our article differs from Riordan and Sappington in that the ex post signals are purely informational in their model, whereas in our model they are real cost shocks that affect not only the firm's transfers but also its cost and, hence, its output. More important, Riordan and Sappington's methodology allows them to prove that the first-best solution can be implemented under certain conditions, but it does not provide a characterization of the optimal regulatory scheme. In contrast, we fully characterize the optimal regulatory scheme, both when the first-best solution can be implemented and when it cannot.

It has long been recognized that limited liability constraints are important to assess the robustness of the first-best implementation results of Cremer and McLean (1985, 1988) and Riordan and Sappington (1988). Robert (1991) considers an auction problem in which each bidder can have finitely many possible types, but the types of different bidders are correlated. He shows that if there are upper bounds on the payments that the bidders can make, then the auctioneer may not be able to extract the full surplus from each bidder, as in Cremer and McLean (1988). Kosmopoulou and Williams (1998) consider a related model of group decision making with a continuum of agents' types. They show that it is impossible to implement the first-best allocation if agents' types are approximately independent and either the monetary transfers among agents, or their ex post payoffs, are subject to limited liability constraints. (3)

Closer to our article, Demougin and Garvie (1991) were the first to study optimal regulation with a continuum of firm types, correlated information, and limited liability constraints. We extend their analysis in several ways. First, as in Riordan and Sappington (1988), the signals in Demougin and Garvie are purely informational. Hence, in their article the firm's output is independent of the signals, whereas in ours it is state-contingent. Second, Demougin and Garvie consider either the case where the firm must earn nonnegative profits in each state of nature, or the case in which the regulator must use nonnegative transfers in every state. By contrast, in our model, the maximal loss that the firm can sustain in each state of nature is a parameter. In particular, we characterize the solution to the regulator's problem for various levels of this parameter. Third, the signal in Demougin and Garvie is binary, whereas we consider an arbitrarily large (but finite) number of states of nature. One of our contributions is to show that in order to implement the first-best solution, the regulator should use transfers that reward the firm in exactly one state and impose the same (minimal) punishment on the firm in all other states. Finally, while Demougin and Garvie rely on constrained calculus of variations techniques, our approach has the advantage of building on the by-now familiar and relatively simple methodology of Baron and Myerson (1982), which we adapt to the case of ex post cost shocks and limited liability constraints.

The rest of the article is organized as follows. In Section 2 we present the simple case of two types and two states. In Section 3 we study the case of a continuum of types and multiple signals. Section 4 concludes. The proofs are relegated to the Appendix.

2. The two-type, two-state case

* Consider a regulated firm that produces a single product. The consumers' utility is

U(q, t) = S(q) - t, (1)

where q is the firm's output and t is the total transfer made to the firm. We assume that S(*) is twice continuously differentiable, strictly increasing, and concave. The function S(*) and the transfer t can have at least two interpretations. If the regulator procures a public good from the firm, then q is simply the size or the quality of the public good, S(q) is the gross aggregate utility that consumers derive from the public good, and t is the amount paid to the firm out of the state's budget. If the firm is a regulated monopoly producing a private good, then S(q) = [[integral].sup.q.sub.0] P([xi])d[xi] is the gross consumers' surplus, and P(*) is the inverse-demand function for the good. In that case, t = P(q)q + A is the regulated firm's revenue, where P(q)q is the aggregate sum of the usage fees that consumers pay, and A is either a subsidy paid to the firm out of the state's budget, or the aggregate sum of fixed fees paid by consumers.