THE C O N T R O L O F NON-LOCALIZED EXTERNALITIES W I T H A S Y M M E T R I C INFORMATION by HENRY JOHN BERNARD VAN EGTEREN B.A.(Hon.), The University of Alberta, 1981. MA, The University of Alberta. 1983. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS OF THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Economics We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA December 1989 © HENRY JOHN BERNARD VAN EGTEREN. 1989 In presenting degree freely at this the thesis in University of partial of department this or publication of thesis by his for and study. or this thesis for of her DE-6 (2/88) ^<2£&JbrV Hcotoo tA t c s \l Itfel l I further requirements representatives. that the agree be It financial gain shall not The University of British Columbia Vancouver, Canada Date the scholarly purposes may permission. Department of British Columbia, I agree available for reference copying fulfilment is for an Library shall make that permission for granted by the understood be allowed advanced that it extensive head of copying my or without my written Abstract This thesis presents a model in which there Is a single agent and two principals. The agent is a monopoly firm, while the two principals are regulators who may cooperate when selecting their optimal policies or act as rivals. The specific regulatory problem in which all are involved is the control of acid rain. A n environmental regulator chooses a design standard and a public utility regulator chooses a two-part pricing scheme. These choices are made within an environment of limited and asymmetric information. Specifically, we assume the firm knows more about its fixed abatement costs than does either regulator. The firm is able to act strategically when revealing this information. Within the context of this regulation problem, we characterize the equilibrium solutions when the regulators cooperate and when they act as rivals. The non- cooperative game endows the environmental regulator with the status of leader. We also give characterizations for two different kinds of rivalry, extreme rivalry and mild rivalry. In addition, this thesis presents some different results on the bunching properties of these models. ii TABLE O F CONTENTS ABSTRACT 11 T A B L E O F CONTENTS ill LIST O F FIGURES v ACKNOWLEDGEMENT I H INTRODUCTION 1.1 Introduction 1 1.2 Purpose: 3 W h y study asymmetric information problems? LITERATURE S U R V E Y 2.1 Introduction 6 2.2 Bilateral Principal-Agent Problem 7 2.2.1 Moral Hazard 8 2.2.2 Hidden Action Problems 2.3 Regulation 2.4 Regulating Pollution i n the Presence of Asymmetric 2.5 m vl 15 19 Information 23 Conclusion 37 T H E M O D E L A N D COOPERATIVE SOLUTION 3.1 Introduction 38 3.2 Objective Functions 39 3.3 Incentive Compatibility 51 3.4 Individual Rationality 52 3.5 Two-Type Full information Solution: 3.6 Cooperative Assumptions 52 3.5.1 Introduction 53 3.5.2 Full Information: 3.5.3 Limited Information: Cooperative Assumptions Cooperative Assumptions Monotonicity and Bunching Properties ill . . 53 58 64 3.7 IV V Summary and Conclusions THE NON-COOPERATIVE MODEL 4.1 Introduction 73 4.2 Extreme Rivalry 83 4.3 Mild Rivalry 85 4.4 Monotonicity and Bunching Properties 94 4.5 Conclusion . 97 A GENERALIZATION TO N TYPES 5.1 Introduction 99 5.2 The Full Information Solution 5.3 The Limited Information Solution: Cooperative Assumptions 5.4 5.5 VI VH 71 100 102 The Limited Information Solution: Non-cooperative Assumptions 131 5.4.1 137 Focus on Mild Rivalry Conclusions 138 6.1 Summary of Contributions 139 6.2 Directions of Future Research 141 CONCLUSION BIBLIOGRAPHY 143 APPENDIX I - CHAPTER THREE 148 APPENDIX II - CHAPTER FOUR 160 APPENDIX DJ - CHAPTER FIVE 173 iv LIST O F F I G U R E S FIGURE 1 31 FIGURE 2 70 FIGURE 3 105 FIGURE 4 108 FIGURE 5 110 FIGURE 6 115 FIGURE 7 118 FIGURE 8 120 FIGURE 9 128 FIGURE 10 . 129 v ACKNOWLEDGEMENT A large number of very talented people have contributed to this thesis and to my understanding of the general research area. Margaret Slade, Chris Archibald and Hugh Neary each contributed insightful comments which are very much appreciated. Ken Hendricks was especially helpful in forcing me to re-think many of my ideas. I would like to thank A.D. Scott for many hours of interesting conversation about the problem of acid rain, and for his support, both financial and intellectual. Special thanks are due to Chuck Blackorby for his comments and for his friendship. John Weymark, my principal advisor, has directed me to hew challenges and because of his excellent tutorship, meticulous editing and insightful comments, I was able to complete this thesis. To him, many thanks. Veronica's love and support and her constant joy have provided an immeasurable contribution to this thesis. To her I owe the largest debt of gratitude. vi CHAPTER I INTRODUCTION 1.1 Introduction Researching the many difficulties associated with environmental pollution demands a consideration of several different disciplines. Indeed, even within a particular discipline like economics, several fields of study are required to Interact in order to achieve viable solutions for the control of the pollution. This thesis uses and synthesizes several different areas within economics to characterize solutions to a nonlocalized externality problem. Our analysis promotes the understanding of regulation which is constrained by the existence of incomplete and asymmetric information regarding the regulatory environment. Non-localized externalities present particularly acute problems for regulators because the pollutants are transported to impact areas which are different from the areas in which the pollutants are produced. For us, acid rain Is the archetypal example of the non-localized externality. Having impact areas which are distinct from generation areas presents the possibility of jurisdictional conflict. When each jurisdiction has a regulator representing its Interests, then this potential for, conflict manifests itself in conflict between the regulators. In its most extreme form, this is what we witness in transboundary disputes, like that existing between the United States and Canada over acid rain. The analysis in this thesis, however, focuses on the problems involved in regulating domestic non-localized externalities, a problem first formulated in two papers by Baron (1985a,b). Broadly speaking, the contributions made by this thesis are two-fold. This thesis presents a model which analyses the impact that a firm's private knowledge about its fixed abatement costs has on the regulatory policies of two. distinct regulatory bodies, when the institutional relationship between the two regulators is allowed to vary. So 1 far, such a characterization has not appeared i n the literature. Secondly, we make a modest contribution to the theoretical literature on multi-dimensional screening by presenting some new results on the bunching, or pooling properties of the model. Since the economics literature does not contain many examples of screening problems i n which more than two choice variables are employed, so-called multidimensional screening problems, this thesis also contributes to the body of economic knowledge by presenting a simple, intuitively appealing example from this class of problems. Following the original insights of Baron (1985a,b), this thesis analyzes how the optimal regulatory policies imposed on a monopoly firm will change as the institutional arrangement between a public utility regulator and a n environmental regulator is allowed to vary. The institutional arrangement changes as we vary our assumptions regarding cooperative versus non-cooperative behaviour between the regulators. regulatory process The for either controlling authority is complicated further by the existence of limited and asymmetric information. Specifically, we model the firm as having a n information advantage over either regulator because the firm is assumed to know more about its own fixed abatement costs than either regulator. The model i n this thesis differs from Baron (1985a) i n two main areas. we focus on only one emission control variable, the design standard. First, Secondly, we include idiosyncratic information i n the firm's abatement cost function. Somewhat surprisingly, this small change in the model significantly alters the roles played by each of the regulatory instruments i n equilibrium. The central questions influences the of this thesis focus traditional monopoly regulation o n how asymmetric information problem and on how changing institutional relationships impacts on the behaviour of the three economic 2 actors. Clearly, assumptions regarding the institutional structure profoundly influence the optimal regulatory mechanism, as well as the welfare of the three economic actors. We characterize these differences and show which type of institutional structure is preferred by each economic actor. Whereas any claim to generality at this point is premature, we show also that many of our results are consistent with a fairly broad class of asymmetric information problems. We feel that this fact lends credence to some of our new results. 1.2 Purpose: Why study asymmetric information problems? Regulatory policies based either explicitly or unknowingly on full information, or the naive assumption that privately held information will be released tmthfully upon request, are not feasible. For us the converse of tjnithful information is not lying, so we do not have to assume that firms are lying explicitly when they release information, rather it is the judicial release of only certain types, or less than complete, information which makes the regulatory policies based on this information infeasible. The question this observation begs is: Should we care about such distortions? Do they warrant special attention, given the costs which may accompany the extraction of tmthful information? Our view is that, even though individual distortions may be small; i.e. small on a firm-specific basis, the total of all these distortions is significant, and should not be ignored. Perhaps this Is the most Important reason for examining asymmetric information problems within the context of pollution control. The thesis begins with a survey of the literature. It is quite extensive because we have attempted to cover several different areas of economics which are included in this thesis. The model and the characterization of the solution in which we assume that the regulators cooperate fully when setting their regulatory decisions is presented in chapter HI. While assuming complete cooperation between the two regulators, we present a full information solution and a limited information solution. Chapter IV 3 presents the non-cooperative m o d e l w h i c h c o n t a i n s two n o t i o n s b e h a v i o u r b e t w e e n t h e two r e g u l a t o r s . T h e first is extreme r e g u l a t o r s p u r s u e separate a n d conflicting m a n d a t e s . of non-cooperative rivalry i n w h i c h both T h e second is m i l d rivalry w h i c h , a l t h o u g h it Is a n o n - c o o p e r a t i v e g a m e , h a s e m b e d d e d i n it e l e m e n t s o f c o o p e r a t i o n . O n c e s o m e sort of I n s t i t u t i o n a l regime i s i m p o s e d o n the r e g u l a t o r s , t h e y interact to d e t e r m i n e a r e g u l a t o r y p o l i c y w h i c h i s t h e n i m p o s e d o n the f i r m . p o l i c y reflects t h e fact t h a t t h e firm i s a m o n o p o l i s t i n p r o d u c t i o n a n d t h a t it a l s o p r o d u c e s t h e n o n - l o c a l i z e d externality. scheme, T h e regulatory T h a t is, it u l t i m a t e l y c o n t a i n s a t w o - p a r t p r i c i n g w h i c h reflects m a n y u t i l i t y p r i c i n g a r r a n g e m e n t s ; a n d , a d e s i g n s t a n d a r d , w h i c h regulates t h e a b a t e m e n t t e c h n o l o g y of the firm. C h a p t e r s three a n d four deal with a n asymmetric information p r o b l e m i n w h i c h t h e f i r m i s a s s u m e d to b e one o f o n l y two p o s s i b l e cost types, either a h i g h cost o r low cost f i r m . S i n c e t h i s is s o m e w h a t restrictive, c h a p t e r five generalizes o u r r e s u l t s to t h e c a s e w h e r e t h e f i r m c a n b e a n y one of a p o s s i b l e 'n' different c o s t types. Also i n c h a p t e r five, we p r e s e n t s o m e different r e s u l t s o n the b u n c h i n g properties a s s o c i a t e d with multi-dimensional screening. values Basically this m e a n s for t h e r e g u l a t o r y v a r i a b l e s c h a n g e t h a t we f o c u s o n h o w a s the firm's cost type c h a n g e s . the These r e s u l t s are different b e c a u s e t h e y s h o w h o w a n a d d i t i o n a l r e g u l a t o r y v a r i a b l e c a n create some regulatory independence. T h a t is, t h e r e l a t i o n s h i p s b e t w e e n t h e r e g u l a t o r y choice v a r i a b l e s are n o t a s r i g i d a s i n t r a d i t i o n a l s c r e e n i n g m o d e l s . T h e final c h a p t e r s u m m a r i z e s o u r c o n t r i b u t i o n s a n d t h e n p i c k s u p o n several t h e m e s i n t r o d u c e d i n t h e literature s u r v e y w h i c h are categorized i n the f i n a l c h a p t e r a s d i r e c t i o n s for f u t u r e r e s e a r c h . at t h e e n d o f t h e t h e s i s . is contained. T h e r e a d e r also i s d i r e c t e d to the three appendices T h e s e a p p e n d i c e s are w h e r e m o s t of t h e c o m p u t a t i o n a l w o r k F o r e x a m p l e , t h e y c o n t a i n t h e l o n g - h a n d v e r s i o n s of t h e m a x i m i z a t i o n p r o b l e m s , f i r s t - o r d e r c o n d i t i o n s , d e r i v a t i o n s of e q u a t i o n s a n d m o s t of t h e proofs of t h e 4 propositions. 5 CHAPTER H LITERATURE SURVEY 2.1 Introduction The primary objective of this literature survey is to Indicate the different sections of the literature to which this thesis makes a contribution. In addition to this primary objective, there are several more subtle objectives of which the reader should be aware. First of all, some of the material presented in this survey is included because it places key contributions to the literature within the proper historical context. This is particularly true of section 2.4 which develops a series of papers that deal with pollution control and asymmetric Information. The most important papers in this section are those by Baron (1985a,b) and Spulber (1988), since they are most closely associated with this thesis. Another objective is to introduce material which develops results from the literature that we use to solve the models presented in this thesis. Certainly this is true of the material presented on the revelation principle (section 2.2.3) as well as the intuition gathered from Baron and Myerson (1982) - section 2.3. Still other material is presented because of its importance to models of asymmetric information. Even though we do not exploit this material In the core chapters of the thesis, nor make a contribution to this material, we do recognize its importance. In fact, these sections of the literature survey form the basis for many of the criticisms we level against the thesis, and several of the themes introduced in these sections are addressed again in directions for further research (section 6.2). The material on moral hazard presented in section 2.3 and 2.3.1 fall into this category. So the objective here is to provide some background for the assumptions we make in the model and to 6 suggest that we are aware of some of the problems this creates. 2.2 Bilateral principal-agent problem At its most fundamental level, this thesis presents a model which falls into the general class of incentive problems. Whereas the incentive problems are intimately related to information problems in general, this latter class of problems has a wider focus. Laffont and Maskin (1985) identify an incentive problem as the problem that a planner (alternatively called a designer, principal, or government, depending on context) faces when his own objectives do not coincide with those of the members of society (whom we shall call agents). 1 Refining this definition of an incentive problem even further allows for a clearer separation of incentive problems from information problems. First, the incentives literature differs from the social choice literature in that the planner or principal already is endowed with an objective function (which even may be a social objective function). The social choice literature on the other hand, attempts to derive relevant social objective functions from the preferences of the members of a society. Second, the planner or principal in the incentive problem is in some way influenced by the other members of society or the agents, so there is some dependency between the principal and the agent. The main two ways in which this dependency can manifest itself is by the agent taking some action which influences, for example, the principal's welfare, or by the agent possessing some private information, about something which the principal wants/needs to know, so that knowing this Information Influences the principal's welfare. The first type of dependence, termed a pure behaviourial dependence by La£font and Maskin (1985). pg. 31 1 7 Laffont and Maskin (1985), Is often referred to as a hidden action or moral hazard problem. The second dependence Is termed pure informational dependence and is often referred to in the literature as a hidden characteristic or adverse selection problem. 2.2.1 Moral Hazard An understanding of pure behavioural dependence or moral hazard has its intellectual roots in the study of teams, Marshack and Radner (1972). The team, which may be something concrete like an organization, must decide upon an allocation mechanism (a rule or even a system of rules) which will translate observations on the state of the world and any communications or messages between team members into some sort of decision or action. Each member of the team is distinct, each possessing different decision variables but pursuing a common team objective. The idea here is that different members of the team operate in a restricted environment in which communication between team members is imperfect. Agents do not act strategically to take advantage of this imperfect information because there is a common team objective. The allocation mechanism must ensure an effective outcome when faced with less than perfect communication between the members of the same team. Groves (1973) takes one step back from this analysis of the team and examines a situation ln which Incentives are structured so as to ensure that the decisions made by the members of the team conform to team objectives. In the theory of teams, there was no incentive problem because all members of the team had the same objective. Now, team members have conflicting objectives. The trick is to get agents to conform to the team objective; ie. how can we formulate a team in the presence of uncertain payoffs and unknown decisions? The compensation rule established by Groves (1973) ensures that the messages sent by the agent about its decisions are in fact accurate. The model discussed by Groves (1973) has elements of both hidden action and hidden 8 characteristic problems. A major technical breakthrough in the work on principal-agent problems Is associated with Mirrlees. In Mlrrlees (1976). the author examines a problem similar to the one set out in Groves (1973); Mirrlees sets out to explain three stylized facts, or observed characteristics: 1. The distribution of incomes within the firm. 2. The existence of authoritative relationships. 3. The existence of hierarchical structure and its impact on production. Since the model In Mirrlees (1976) has elements of both moral hazard and adverse selection, we will focus on only a portion of the model: that part dealing with moral hazard. Mirrlees states that there Is considerable uncertainty associated with "the tasks (or the value of the tasks) that any member of the organization will be undertaking, about his capabilities to undertake these tasks, and about the way in which he will or has undertaken these tasks." 2 If we focus on the undertaking of tasks, or broadly speaking, the agent's actions, then Mirrlees identifies two distinct types of constraints on the principal's choices which must be considered when resolving this type of uncertainty. The first of these constraints is the participation constraint. If the agent is going to continue to be part of the organization and participate in any proposed compensation scheme, then the scheme must ensure that the agent receives a payment which Is at least equivalent to what the agent could receive in other opportunities. The second constraint directly deals with the agent's incentives. Given 2 Mirriees (1976). pg. 107 9 the compensation scheme, which is based on the agent's action, the planner must ensure that the agent's action perceived and used by the planner is actually the action the agent would choose when faced with the particular compensation scheme. Since the agent's actions are not directly observable, this becomes problematic. The incentive problem is complicated further by the existence of a random state of nature, which does not allow the principal to infer the agent's action by observing some other variable that is the direct result of the agent's action. Typically, the dimensionality of the incentive constraint presented a severe computational problem for modellers. To overcome this, the first-order condition associated with the agent's objective replaced the incentive constraint. The idea here was that, if the action used by the principal in the compensation scheme was actually to be chosen by the agent, then this action should satisfy the first-order necessary condition of the agent's maximization problem. This reduced the dimensionality problem down to manageable levels, but ignored second-order conditions. The first-order approach as set forth in Mirrlees (1976) was an important conceptual breakthrough because it introduced a new method for examining the principal-agent problem. In all previous examples of the principal-agent problem, including Spence and Zeckhauser (1971) and Ross (1973), the authors examined the incentive problem from a specific viewpoint. Suppose that the principal wished to maximize the value of some output produced by the agent. The level of output is a function of the agent's effort and a random state of nature. The dependence of output on effort and the random state of nature was modelled directly. This meant that when the principal's expected value function was set out, the expectation was taken over the distribution of random states. Neither effort nor the particular state of nature were perfectly observable. Since effort provided disutility to the agent, the principal's choice of sharing rule was problematic; i.e., there was an incentive problem to solve. Most authors solved the large number of incentive constraints by using a first-order approach which involved differentiating the agent's objective function with respect to 10 effort. But since the proposed sharing rule must be a function of only observable variables, namely output in this case, differentiating with respect to effort meant that the sharing rule also must be differentiated. The problem with this is that Mirrlees had shown that relevant sharing rules existed which were not differentiable. Mirrlees' answer to this was to reformulate the problem noting that for a given effort level, the random states of nature induced a distribution on output. Therefore, the problem could be restated so that the expectation was taken over output rather than the state of nature. This meant that the first-order approach could be used because the sharing rule need no longer be differentiated with respect to effort. Apart from Mirrlees* interesting result, there is another reason why his solution technique is valuable. Because of this technique, others began solving very general principal agent problems, which ultimately uncovered the basic structure of moral hazard problems. The analysis of the principal-agent framework then became less problem specific. In both Holmstrom (1979) and Shavell (1979) we get a very clear presentation of the generic moral hazard problem. Holmstrdm (1979) and Shavell (1979) examine models in which an agent's action is unobservable and affects the probability distribution of an outcome like output. The output Influences both the principal's and the agent's well-being. In addition, output is subject to stochastic disturbances so that if low output occurs, for example, the unobservability of the agent's action means that the principal cannot determine if the low output is the result of the agent's action or because of a bad draw from mother nature. The principal's task Is to construct a payments scheme as a function of observable variables and an action for the agent so as to maximize expected welfare subject to the participation constraint and the incentive constraint. This is the classic principalagent problem. Clearly it is often the case that the results of theoretical models are influenced 11 by technique. This is certainly true in the case of the principal-agent model. Since the Incentive problems associated with moral hazard create distortions away from firstbest results (more appropriate would be to term these results as full information solutions), a natural question which arose in the literature focused on how first-best results might be obtainable in the presence of moral hazard. In Harris and Ravlv (1977). Mirrlees (1976) and Gjesdal (1982), first-best results are obtained by using forcing contracts. In order to use such contracts, which impose enormous penalties on the agent when the desired action is not taken, the principal must either observe the agent's action directly or the principal must be able to infer the agent's action from some observed variable. The latter may happen when the support of output Is dependent upon the agent's action for a given value of the random variable. That is, the maximum amount that can be produced for a given distribution of the random variable is dependent upon the agent's action. So the technique here, a moving support, greatly influenced the types of results obtained. Lewis (1980) and Sapplngton (1983) discuss the Implications of attaining firstbest results when a forcing contract is not legally feasible. That is when there is limited liability. In Holmstrom (1979), we find that the technique changes in that HolmstrSm assumes that there is a non-moving support for output. He uses the first-order approach, in the spirit of Mirrlees (1976), to address the question of whether imperfect monitoring of the agent's action, which is costly, can improve the welfare of those participating in the scheme. Holmstrom answers this question in the affirmative. Shavell (1979), like Holmstr6m (1979), uncovers the basic intuition associated with solving the principal-agent problem when there are risk-averse agents. Shavell shows that the optimal sharing rule always depends upon the random level of output, 12 but that it would never leave the agent bearing all of the risk. When a monitor or signal is used, and both principal and agent are risk averse, the information on the agent's action is clearly imperfect. Thus, there is the possibility that this monitor may lead to the wrong conclusion regarding the true level of effort. This new risk has the possibility of decreasing welfare when both participant's are risk averse. Thus, it is unclear that this partial information Is valuable. Both Shavell (1979) and HolmstrSm (1979) show that this monitoring information leads to improvements in welfare. Grossman and Hart (1983) develop a technique for solving the principal-agent problem which does not rely on the first-order approach. This is important for us because the recursive nature of the solution technique in a moral hazard setting is very similar to ours in an adverse selection setting. The first-order approach involves relaxing the constraint set to examine all stationary points for the agent's maximization problem. However, these are only necessary conditions so that the second-order conditions may not be satisfied and the resulting solution using the first-order condition approach may not be a global optimum. To get around this problem, Grossman and Hart (1983) assume that the agent's objective function is additively separable in the action and the reward. This assumption means that the principal's problem can be solved by first maximizing with respect to an observed variable for a given value of an unobserved variable. This maximization proceeds subject to the incentive compatibility and participation constraints. After this is complete, the resulting value function is maximized with respect to the unobserved variable. This was an interesting approach but concrete results were difficult to obtain because closed form solutions at the first stage were not obtainable. The final contribution to the principal-agent literature which is of relevance to us was made recently by Bernheim and Whinston (1986). This article expands the bilateral principal-agent model by considering multiple principals. Examples of common agency as described by Bernheim and Whinston (1986) include wholesale 13 trade (which they term delegated common agency) In which several principals endow a common agent (broker) with the power to make decisions influencing all participants. Intrinsic common agency describes a situation ln which a single agent is naturally endowed with the power to influence the well-being of several principals. An example of this would be more than one regulator being influenced, and in turn influencing, the actions of a single firm. This is similar to the situation analyzed ln this thesis. Our analysis differs because we assume away all moral hazard problems. So Instead of the regulators being worried about designing mechanisms to handle hidden action problems, they will be worried about hidden characteristic problems. As we alluded to earlier, the problem we discuss in this thesis clearly contains elements of moral hazard. These are Important problems in any practical setting of environmental regulation. However, we shall simply suppress them and focus exclusively on the hidden characteristic problems, to be discussed in the next section. 2.2.2 Hidden Characteristic Problems Hidden characteristic information problems capture the difficulties created when the agent possesses better Information than the principal regarding some idiosyncratic parameter. This parameter may take the form of preferences, endowments or technologies, for example. Obviously this is problematic for the principal because it usually needs to know the true value of this parameter in order to make effective 3 choices. The problem of course is to motivate the agent to release this privately-held information truthfully. As in the hidden-action problem, one consideration which must be incorporated information which the agent knows better than the principal is usually modelled as a one-dimensional parameter. Simplicity being the motivating factor behind this decision. Only recently. Quinzii and Rochet (1985), for example, have there been attempts at expanding the information space. 14 directly into the optimal mechanism (contract) is the fact that the agent always has the option of non-participation. Thus, any attempt to get an agent to reveal its information must at least ensure participation (These constraints are usually referred to as individual rationality constraints). This must happen for any realized value of the privately-held parameter. Thus, if the agent reports tmthfully, it must be the case that the compensation it receives from doing so is at least as good as the agent could receive by not participating in the scheme. In addition to the problem of participation, the principal must handle the agent's incentive to lie. The principal must construct a mechanism which ensures that the strategy of truth-telling on the part of the agent is better for the agent than reporting a he. Stated more generally, the incentive constraint simply ensures that the mechanism chosen Incorporates the agent choosing its best strategy. When the mechanism is a direct mechanism, so that the firm's type is reported, then the incentive constraints in the hidden characteristic problem problem are very similar in character to those encountered in the hidden-action problem. The difference is that. Instead of choosing an optimal action, the agent now is involved in choosing an optimal report. And like the constraints encountered in the moral hazard problem, the incentive constraints in the adverse selection problem are plagued by dimensionality problems when an actual solution is attempted. This is because there may be a very large number of possible values for the agent's idiosyncratic parameter to take. In order to ensure that the agent reveals this information truthfully, there must be a constraint associated with every possible kind of he - thus the dimensionahty problems. Fortunately, several solution techniques have evolved which allow researchers to address some of these problems. The first, and most important, result is called the revelation principle. This result allows the principal to restrict attention to direct mechanisms in which the agent reports its type truthfully. 15 The Revelation Principle. Any equilibrium allocation of any mechanism can be achieved by a truthful direct mechanism... By a direct mechanism, we mean a game ln which all agents first simultaneously declare values for whatever parameters they have observed, for example, parameters describing their own tastes, etc. After these "messages' or 'signals' are sent, some allocation is affected as a function of the declaration of all the players. This allocation rule is specified in advance. Players in a direct mechanism need not tell the truth about their observed parameters. In a truthful, direct mechanism, however, there is an equilibrium in which all players do tell the truth: a truthful mechanism, then. Is one in which each player is given an incentive (by the allocation rule) not to lie, provided that he expects all other players to tell the truth. 4 This is an extremely useful result for it allows the principal to search among direct mechanisms which seem to be relatively simple to implement. And further, from the class of direct mechanisms, the principal can search among those that Involve truth-telling as an equilibrium strategy. And In all of this, there is no loss of generality (as long as we have a single agent and a unique equilibrium). One part of the revelation principle manifests itself in the incentive compatibility constraints. This set of constraints ensures that the mechanism or contract chosen by the regulator also involves the agent truthfully revealing its privately-held information. Of course, it is this set of constraints which creates the computational or dimensionality problems. The local approach, termed such by Matthews and Moore (1987). is a method developed in the literature which reduces the dimensionality manageable levels. problem down to Stated very cryptically, this method involves placing enough structure on the problem so that, if local incentive compatibility constraints are satisfied, then the global incentive compatibility constraints also are satisfied. 4 Harris and Townsend (1985). pg. 380. 16 Local constraints in the finite type model involve incentive compatibility between adjacent types only. involve two An upward adjacent incentive compatibility constraint would distinct types, i and 1+1. Similarly, a downward adjacent incentive compatibility constraint would involve the types i and i-1. Adjacent Incentive compatibility means that a type i firm's best strategy would be to reveal its type as i rather than as i+1 or i-1. The analogue to adjacent incentive compatibility in the case of a continuum of types is that the set of first-order conditions for the agent's optimization problem are satisfied, but the second-order conditions are ignored. If the dimensionality of the finite type model is reduced in this manner; Le., using adjacent incentive compatibility constraints, and further structure in the form of the single-crossing property. Cooper (1984), is placed on the model, then satisfying these constraints is equivalent to satisfying the global incentive constraints. Matthews and Moore (1987) show that the single-crossing condition can be ensured by ordering the marginal rates of substitution between choice variables in the agent's objective function according to type. We shall delay any further discussion of the local approach until chapter in. The main point to note about this approach is that computational ease came at the expense of a large amount of structure and its corresponding loss In generality. 5 Further insight into screening problems was gained by Mussa and Rosen (1978). They develop a model which examines a monopoly pricing problem in which the commodities being sold may be characterized according to quality. Consumers may be differentiated according to their preferences for quality versus price. The monopoly seller understands that buyers are differentiated but cannot distinguish between different types. The monopolist knows only the distribution of types. The objective of the seller is to separate out the various buyers with a pricing scheme. This allows at 5 As a counter to this point, see Repullo (1986). 17 least some partial price discrimination. The authors compare this limited information solution with the complete information solution and discuss optimal bunching patterns. An important result, common to the adverse selection literature is that when satisfying the self-selection constraints (they employ a local approach) the monopolist is required to distort both price and quality (the components of the optimal mechanism or contract) away from their full Information counterparts for all possible types but one. Maskin and Riley (1984) owes a large intellectual debt to Mussa and Rosen (1978). Maskin and Riley (1984) is important because it refines the local approach. In this article, which studies a variety of adverse selection problems in a general framework, the marginal rate of substitution ordering property manifests itself as an assumption on the marginal rate of substitution between total payment and quantity; it is assumed to be Increasing in type. This insures that indifference curves cross only once, which insures that satisfying local constraints implies that the global constraints also are satisfied. Whereas this does not begin to cover all of the contributions made to the adverse selection literature, we have introduced some key features common to the literature which will facilitate a deeper understanding of our results. 2.3 Regulation The contributions to the literature on regulation under conditions of limited and asymmetric information are indeed large. The most recent and comprehensive survey of this literature may be found in Besanko and Sappington (1987). This particular portion of the literature survey shall focus on only a very small subset of this literature. We begin by presenting the intuition behind the single-period regulatory problem 18 In the presence of adverse selection since this closely reflects the model in this thesis. The problem was analyzed for a continuum of types first by Baron and Myerson (1982). To motivate their analysis. Baron and Myerson hypothesize the regulation of a project whereby the regulator wants to enure that the maximum social surplus is obtained. The usual form of social surplus is simply some weighted sum of producer and consumer surplus. The problem of regulating the project, which may be simply the production of some socially desirable good, is that the regulator is faced with limited knowledge of the costs of production. This is important because in setting a price for the good, these costs must be known. Loeb and Magat (1979) suggest that an efficient outcome to the pricing problem can be achieved if the firm involved with the project, or producing the good, is given title to all the social surplus. The regulator would simply offer a subsidy on each unit of output equal to the consumer surplus. The firm would then set price equal to marginal cost, and maximize the total surplus available given a demand curve. The problem with this Is that now all the surplus Is in the hands of the firm. If the weights on producer and consumer surplus are not equal (the objective function is linear), then there are severe distributional problems associated with this solution. Loeb and Magat (1979) counter this by suggesting that the right to produce the good, or undertake the project, may be auctioned off with the proceeds from the sale being rebated back to consumers. Baron and Myerson (1982) note that. If the firm is a monopolist, the auction scheme cannot work. Baron and Myerson (1982) attempt to determine the optimal regulatory policy to impose on a monopolist when the regulator does not know the firm's costs. In choosing a regulatory policy, the regulator attempts to maximize a weighted sum of expected producer and consumer surplus, where the expectation is taken over the 19 firm's type. The regulator (principal) is endowed with a prior distribution of possible types. All information except the firm's type (which identifies it as a particular cost type) is common knowledge. Invoking the revelation principle simplifies the problem considerably and allows the maximization of expected social welfare to be constrained by only the incentive compatibility and individual rationality constraints. The firm is able to earn rents on its private information. In the full information solution, when producer surplus is weighted less than consumer surplus, the optimal solution has the firm earning zero rents. This solution is not implementable because the incentive compatibility constraints are violated. Because the firm must be induced to reveal its privatelyheld information, lower cost types are more valuable to the regulator than higher cost types. As a result, the rents which need to be paid to the firm are monotone in type. In this one-shot regulatory policy, the regulator receives a message from the firm in the form of a report. The report on costs then influences a regulatory policy which is then offered to the firm as a schedule composed of different four-tuples. That is, there is a contract containing four different components for each possible type. The game ends with the firm picking a single contract from the contingency list. The revelation principle and the incentive constraints ensure that the firm picks the contract which is actually designed for its type. practice, the game often does not end there. The problem with this is that. In Auditing of cost information may be undertaken to determine if the report issued to the regulator was accurate. However, if the regulator knows that the revelation principle holds, and that firms actually will report their true parameter, why would a regulatory agency undertake auditing at significant cost? This is the problem addressed in Baron and Besanko (1984). The authors discuss a model very similar to Baron and Myerson (1982) in which the regulator can commit to an audit and Impose a penalty if the ex post cost 20 parameter turns out to be different from the firm's report. The commitment to audit, and the threat of a penalty, effectively reduce the cost to the regulator of Inducing truth-telling. Even though the contracts are designed to satisfy incentive compatibility, the ex ante commitment to audit for certain values of reports reduces the rents to be paid because the value of the firm's idiosyncratic Information Is reduced in the presence of an ex post observable. This is because the ex ante value of the information is determined, in part, by the difference between a firm's type and the highest cost type. The intuition for this can be strengthened by examining the nonincentive compatible full information solution. If the regulator tried to implement this solution, the firm would always report the highest cost type. This would maximize the rents to the firm. In the limited Information solution, the regulator compensates the firm according to these rents. By imposing an audit with the threat of penalty, the ability of the firm to make the big lie is reduced. It must temper its ability to lie with the fact that it could face a large penalty if audited. Thus, the value of its information is reduced and with it the cost of truth-telling also Is reduced. This suggests that there will be a demand for auditing even when the firm always ends up making a truthful report. There are many other issues of importance to be considered in the regulation literature. Some of them we shall defer to the main chapters of the thesis, and some are the subject of future research projects. As indicated, the model of Baron and Myerson (1982). which is very close to the model in this thesis, is a one-shot regulatory game. Since most regulatory relationships are long term, this assumption is unrealistic. Recent contributions to the growing literature on multi-period contracts include Lewis and Sappington (1988a,b). Another problem encountered in adverse selection models is that implementation of the prescribed policy requires a high degree of commitment on the part of the regulator. This must continue even after the revelation of type. 21 Since this often Involves inefficient behaviour, which was required to induce revelation, it is not clear that henceforth, after revelation, this is the best policy for either the principal or the agent. We will simply acknowledge this as a shortcoming of our model. 2.4 Regulating Pollution in the Presence of Limited Information Models which relate specifically to pollution control within the context of asymmetric Information are relatively few so it is easy and convenient to group them into a single section. They provide an elegant progression of thought and technique which is mirrored In the rest of the literature. In addition, this group of papers is clearly a direct precursor to the work presented in this thesis, so it is appropriate that we focus on these works. Most of the work on asymmetric information and pollution control was addressing the same fundamental questions addressed when complete and symmetric information was assumed. Early work which clarified the nature of pollution externalities and their relation to the definition and existence of property rights helped to distinguish techniques which could be used to control the externalities. Since the 8 commodities creating the negative externality had no natural market which could be relied upon to allocate effectively, methods of control naturally focused on corrective taxes which would generate a price for the externality generating good, or some form of quantity restriction like quotas or standards. 7 ^These works Include Meade (1952). Gordon (1955). Scott (1956), Buchanan and Stubblebine (1962), and Baumol and Oates (1975). In Canada, the Department of the Environment sets national air-quality standards, although the provinces actually regulate stationary sources of air pollution. The Canadian Clean Air Act (1971) gives the federal government regulatory authority in order to avoid "pollution havens" within a given province. The national emission standards for air contaminants are based on the potential danger these contaminants pose to human health. 7 In 1984, federal and provincial governments agreed to cut sulphur dioxide emissions to one half their 1980 levels by 1994. This implied a target of 20 kg/ha/yr., or a reduction of 2.3 million tonnes/yr. In 1977, the US Clean Air Act created 274 Air Quality Control Regions. The Environmental Protection Agency (EPA) was responsible for maintaining or Improving air quality in those regions. To satisfy its mandate, the EPA defined ambient air quality standards for several criteria pollutants. Including sulphur dioxide. If a state had an air quality control region that was ln contravention of the federal standard, then the state would have to devise a State 22 The standards chosen by regulatory authorities usually have a non-economic role to perform, rather than acting purely as an allocative device. The uniform standards set by regulatory authorities in the US under the auspices of the Clean Air Act were classified as either primary or secondary standards. The primary standard for a criterion pollutant are based on health requirements. They are set so that the most vulnerable members of society are protected. The secondary standards are set so as to protect such things as commercial crops and property. Standards may be classified further as either design standards or performance standards. This is an important distinction because, unlike with the design standard, the performance standard allows the firm to meet the standard by choosing the best available abatement technology, changing its input mix, or by changing, the process it uses to produce its goods. This is an important distinction for us because we make an explicit decision to abstract from this specific choice problem. This means that we will impose design standards on the firm In order to reflect the fact that this has been a popular method of control. We make no claim as to the superiority or desirability of this form of regulation. Besanko (1987), presents an interesting paper which addresses "the economic consequences of performance and design standards in a stylized model of regulation." 8 He finds that under most circumstances, the performance standard is the preferred method of regulation. However, he shows that even under conditions of symmetric information, there may be circumstances under which a design standard is preferred. Implementation Plan (SIP) which would ensure compliance. This would usually involve mandating a pollution abatement technology. The Clean Air Act calls for existing emission sources to use a reasonably available control technology and for new emission sources to use the best available control technology. Besanko (1987). pg. 1 a 23 He suggests that this provides partial evidence of why design standards seem to be the preferred method of regulation in many settings. If a regulator Is concerned with total surplus less damages, instead of simply the sum of emission and pollution damage costs, then it may be that design standards are preferred. This will depend upon enforcement cost differences, the parameters of the model and the structure of the regulated industry. Since the model we study in this thesis has a broader mandate than emission and damage costs, we may be satisfying the requirements for the design standard to be an optimal instrument. Other methods of control studied in the literature besides taxes, subsidies and standards include marketable permits, or pollution certificates. Recent theoretical work by Krupnick, Oates and Van de Verg (1983) and M Gartland and Oates (1985) c developed the ideas of Montgomery (1972) who in turn owes an intellectual debt to Dales (1968) and Arrow (1969). The use of marketable permits relies on certain types of pollutants (ones that are not used up in consumption) being perfect substitutes in individual utility functions. This allows for the possibility that these pollutants might be able to sustain markets in rights in which there are both many sellers and many buyers. The intuition of how these markets might work is as follows. The pollution regulator sets a standard for air quality (either emission standards or ambient standards). Permits are distributed to firms in a manner which will rationalize the standard but not the current level of emissions or air-quality. Firms must hold enough permits to cover all of their emissions (or the effect of their emissions). The firm compares its marginal costs of abatement with the cost of purchasing an 24 additional permit. If price is lower than marginal abatement costs, the firm purchases an additional permit. This method has the potential of minimizing the total cost of attaining a given air-quality standard. This claim has also been tested empirically with mixed results. permits viewpoint, see Atkinson and Tletenberg (1982). An For a pro- alternative view is presented in Russell (1986). Of more Importance to us is the work done on pollution control which incorporates the existence of limited and possibly asymmetric information. This sequence of articles begins with Weitzman (1974). In the previous section, we noted that two natural methods for controlling pollution externalities were to regulate either the price or the quantity of the externality generating good. Weitzman (1974) re-examines the desirability of either of these two modes of control, but within the context of limited information. pondered the following problem in regulation: Weitzman If society wished to regulate the production of a good (e.g., sulphur dioxide), what would be the "best" means of doing so? Should prices be faced and let the firm choose its own profit maximizing level of output, or should the quantity be chosen and then allow firms to pursue cost minimizing behaviour? From a theoretical viewpoint, where specific curvature properties and certainty are assumed, these two control methods are exactly the same. Yet, we typically observe both kinds of regulation. Weitzman is concerned with explaining why this might be so. The model Weitzman uses involves choosing quantity so as to maximize the net social benefit associated with the production of some good, under conditions of uncertainty. The regulator's problem is captured in the following expression: 25 (2.1) max q where: E[B(q,a) - C(q,9)] E 3 expectations operator q = output B = benefit function C s cost function a,Q = independently distributed uncertainty parameters The uncertainty is captured by the parameters a and 9 which are assumed to be independently distributed. The regulator does not know nor does it observe the realized values of a and 6 until after its output choice Is made. These two parameters are meant to capture the difference between public and private information regarding benefits and costs: "an information gap". Even engineers most closely associated with production would be unable to say beforehand precisely what is the cheapest way of generating various hypothetical output levels. How much murkier still must be the centre's ex ante conception of costs, especially in a fast moving world where knowledge of particular circumstances of time and place may be required. True, the degree of fuzziness could be reduced by research and experimentation but it could never be truly eliminated because new sources of uncertainty are arising all the time. 9 An optimal solution under these circumstances would be a contingency message insuring that marginal benefits and marginal costs were equal ex post This means that the value for the optimal price or quantity would be a function of the information revealed, the realized values of a and 9, and would ensure that marginal benefits equal marginal costs. (2.2) B{q*(9,a).a} = C,{q*(e,a),9} = p*(9.a) q Wedtzman (1974). pg. 480. 26 If this were the case, then any uncertainty ex ante has been eliminated ex post Because of the Inherent complexity In offering a schedule of different outputs, Weitzman (1974) argues that such a solution is not feasible. Thus, the problem Weitzman addresses in one in which a single price or quantity is chosen in the face of the uncertainty. In general, there are three influences which interact to determine which method of control is optimal: 1. The curvature of the benefit function. 2. The curvature of the cost function. 3. The variance of pure unbiased shifts in the marginal cost function. When the benefit function is curved or the cost function linear, the quantity method of control is superior. The reason this happens Is because, with marginal costs linear, any difference between ex ante and ex post marginal costs, while using a pricing rule, will bring about large~errors. In addition, a strongly curved benefit function means a high degree of risk aversion towards deviations in quantity. Thus, only a quantity rule would seem optimal under these circumstances. When the benefit function is close to linear, the marginal benefit function will be very flat. This, Weitzman claims, is a good reason "to name it as a price and let producers find the optimal output level themselves, after eliminating the uncertainty from costs." 11 So far we have focused on choosing either price of quantity. However, Weitzman ^his type of contingency solution Is employed by all of the Bayesian approaches used In the information literature. Consequently, It seems subject to the criticism which Weitzman (1974) levels against his optimal contingency message, namely that it is too complicated to be employed In practice. 1 1 Weitzman (1974). pg. 485. 27 suggested that a superior method of regulation under the circumstances might be to regulate both price and quantity. This line of investigation was taken up by Roberts and Spence (1976). They present a simple model in which a controlling authority is trying to minimize the sum of expected clean-up costs and the expected damages associated with emissions from several firms. To accomplish this task, the regulator is able to use two regulatory instruments. The first Is a system of taxes composed of a subsidy and penalty, and the second Is the choice of a global number of licenses. The licences are tradable in a perfectly competitive market. If the firm's emissions are greater than the number of licenses held by the firm, then the firm would receive a penalty on the difference. If licenses are greater than emissions, then the firm receives a subsidy. Thus the firm is trying to choose a level of emissions and the number of licenses to hold which will m i n i m i z e its overall costs. The controlling authority must choose a subsidy, penalty and a global number of licenses in order to minimize the sum of expected damages and costs. The costs are a function of the firms' emissions and a random variable. It is assumed that the firms know, or can discern their true cost functions. The intuition of why the regulation of price and quantity is desirable may be developed with the aid of figure 1. Suppose that a regulatory authority must maximize the difference between benefits and damages before some uncertainty is resolved. do this the regulator evaluates ex ante marginal benefits, MB MD. lt To and marginal damages, The regulator finds that x* is an optimal level of emissions given the expected benefit and damage functions. To Implement this choice, the regulator could impose a tax equal to t dollars/unit of emissions. feels that the firm's emissions will be x*. 28 When faced with this tax, the regulator The firm Is not faced with the same uncertainty. It knows that its true marginal benefit function is M B . The regulator too will discern this ex post, but when the 2 pollution control policy is set, the regulator perceives marginal benefits as M B j . When the firm faces t, it would decide to choose x, - too much pollution, not enough clean up, since Xj > If the regulator were to implement x* using licenses, then the firm would only produce x* emissions. Ex post, however, this would mean too little pollution, x* < x P It should be obvious from this exercise that use of one or the other regulatory instrument causes deviations from the ex post socially optimal level of pollution, x lt when the firm's costs turn out to be different than what the regulator expected, but the deviations away from this optimum are in opposite directions. Roberts and Spence (1976) show that an increase in social welfare is obtainable when both instruments are employed together. The regulator would issue a global number of licenses, which would restrict the total level of emissions to the ex ante social optimum, x*. It would choose a tax/subsidy scheme so that, if costs turned out to be different from those expected when the global quota was set, then there would still exist some residual incentive for the firm to adjust emissions toward the ex post social optimum. Kwerel (1977) analyzes the same pollution problem posed by Roberts and Spence (1976), but attempts to deal directly with the information problem facing the regulator. This information problem arises because the regulator does not know the firm's cost of clean-up. Neither does the regulator know the true level of environmental damages. The difference between these two informational gaps is that the firm has every incentive to exploit its superior knowledge. 29 Figure 1: REGULATING PRICE AND QUANTITY -1 ^ e>M5Sic*£ 30 Faced with this asymmetric Information, the regulator asks the firm to report its entire cost function. The regulator then sets a fixed quota of pollution licenses. The firm has an incentive to reveal information in a manner which enhances its profits. If the regulator could elicit truthful information (get firms to self-select), then it would know firm-specific control costs - the uncertainty would be resolved. It could then use that information to develop a regulatory policy which would result in the socially optimal level of emissions. The problem is to develop an incentive mechanism which results in truth-telling. The focus of this paper is on the incentive scheme rather than on the regulatory policy, as it was in Roberts and Spence (1976). There is one extra round of communication between the regulator and the firms in the Kwerel model that was missing in Roberts and Spence. To develop the truth-telling mechanism, Kwerel introduces two different regulatory instruments used in the control of emissions: a license scheme: and, a pure effluent charge or per unit tax. Under a pure licensing scheme, the information which firms report is used by the regulator to set the total number of licenses. Firms then have an incentive to provide information which will cause the regulator to increase the number of licenses, since this results in a lower equihbrium price for licenses. The lower price for licenses reduces total costs, since total costs are shown to be a decreasing function of price. Therefore, it Is always optimal for the firm to report clean-up costs which are higher than their actual level. Exactly the opposite incentive exists when a pure effluent charge is used. In this case costs are an increasing function of the charge. By reporting lower costs (recall that the firm actually reports an entire function), each firm achieves a lower charge. Obviously there is an incentive to report costs which are below the actual level of clean-up costs. 31 To balance these two tendencies and get firms to reveal the true value of their clean up cost function, Kwerel suggests combining the two regulatory instruments into a single scheme. Under this new option, the authority would choose a number of permits and a subsidy, and simultaneously announce that any firm holding more permits than the level required to cover its current emissions would receive the subsidy. The regulator then announces that it will choose the number of permits and the emission subsidy so that marginal damages equal marginal costs. It then asks the firms to report their control cost functions. Kwerel shows that this incentive scheme results in a socially optimal level of emissions and that with this pattern of emissions, total costs of clean-up for each firm are minimized. Therefore, it is in each firm's best interests to report its true control cost function as long as all other firms report theirs. The problem of truth-telling, as we have discussed already, Is the central focus of the self-selection literature (so-called hidden characteristic problems). Following this basic theme, Dasgupta, Hammond and Maskin (1980), extensively criticize and re-work Kwerel's incentive scheme. They claim that Kwerel's truth-revealing mechanism relies on the existence of a perfectly competitive market for pollution licenses, something which implies that there are many firms Involved in the market. In the case of sulphur dioxide emissions this may be plausible, but for other situations perfect markets are rare. 12 As Montgomery (1972) observed, the perfect market hypothesis of many participants in the market is based on emissions from different sources being perfect substitutes in the social objective function. If this is not true, then we are back to the situation described by Arrow (1969) where each firm faces its own price for a permit instead of the price generated by the operation of the perfectly competitive permits market. See Hahn (1984) for a discussion on some of the consequences of imperfect markets. 32 Dasgupta et al (1980) develop an Incentive scheme which will maximize net benefits. The scheme is an adaptation of a Groves mechanism where the controlling authority develops and issues a common tax scheme which transforms each firm's objective function, net of tax payments, into the social objective function. The tax schedule is constructed to ensure each firm's report about its costs is tmthful, regardless of what any other firm reports. Baron (1985a,b) is the first author to employ the techniques developed in the information and regulation literatures in a setting of pollution control. These models differ most obviously from Dasgupta et al (1980) in that they examine a monopoly regulation problem; Dasgupta et al (1980) do not allow the firm's private information to influence damages. If damages are a function of output, then the firm's private information surely must influence damages. It is also a fact that, like with most Groves mechanisms, there is a budget balance problem for the regulator. Also along these lines. Baron (1985b) notes that Dasgupta et al (1980) do not consider how the scheme will impact on firm profits. There is no guarantee that the firm will receive a fair return. Thus, the scheme may not be Individually rational for many firms. Baron (1985a) uses the solution techniques from Baron and Myerson (1982) to solve a non-localized pollution problem in which a single regulated firm produces a socially desirable good, electricity, and a socially undesirable good, sulphur dioxide. Both the price of the monopoly good and the pollution generated by the firm are regulated. Two distinct regulatory bodies, who may cooperate or act as rivals, each pursuing a separate mandate, perform the regulatory tasks. The cooperative regulatory setting combines the mandates of the regulators into a single objective function. The regulatory game begins with the regulator asking the firm to make a report regarding its privately-held information. This information is modelled as a one-dimensional parameter which is continuous on its support [0,1]. 33 The Information advantage held by the firm concerns only the efficiency of the firm's abatement technology (all participants know the cost function). All other Information is common knowledge. The regulator has some prior regarding the possible choices for the parameter. The differences between Baron's two models are discussed more later on in the text. Using this report, the cooperative regulator sets a two-part pricing scheme, a standard, and an emission tax. Recalling Weitzman (1974), this may be termed a complete contingent message. It identifies a different system of prices, standard, and emission tax for every possible type of firm. Naturally then, this mechanism seems subject to the same criticisms Weitzman levelled against his own work. The regulatory variables are chosen in order to maximize the "social" objective function which is made up of the expected value of a weighted sum of producer and consumer surplus less a weighted sum of damages, abatement costs, and emission tax payments, where the expectation is taken over the firm's type. The revelation principle is invoked so that the regulators* choices are further constrained by the incentive compatibility constraints, which are a manifestation of the revelation principle. Baron (1985a) uses a local approach and verifies that the global constraints are satisfied. To do this, he assumes the objective function Is convex in type. Faced with this set of contracts, the firm chooses the one which will ensure the most profit, given its type. Because of the incentive compatibility constraints, this choice of contract for the firm turns out to be the contract actually designed for its true type. That is, the report it makes is actually the truth. The final constraint on the regulators' choices is the individual rationality constraint, which ensure that the firm receives a fair return, and is willing to participate in the scheme. 34 Given this regulatory game, Baron (1985a) shows that the full information solution is implementable when the regulators cooperate. He shows also that the emissions tax is a dominated regulatory instrument and should be set equal to zero. This seems to be the result of the fact that the standard and two-part pricing scheme are adequate for balancing the regulatory objectives: 1. The marginal reduction in damage from abatement with the marginal cost of abatement. 2. The marginal reduction in damage achievable through a reduction in output with the net benefit from that output. 13 ' The conflict between the regulators is made acute because the externality is nonlocalized. This means that the pollutant is produced in one area and then impacts on a totally different area. Since the interests of the different areas are represented by separate regulators, there is a possibility of conflict arising. In the non-cooperative Stackelberg game, the environmental regulator is modelled as the leader. In this game, the environmental regulator first solves for the reaction function of the public utility regulator. The environmental regulator exploits this strategic advantage when choosing the standard and emissions fee. The environmental regulator then offers a standard and tax for each possible type. The public utility regulator, taking the standard and tax as given, chooses an optimal two-part pricing scheme as a function of the firm's type, so it actually offers a menu of prices. Finally, the firm chooses a contract to maximize its profits. Given this new game, the emission tax has a role to play in generating truthful information. The full information solution is no longer Implementable and the firm Baron (1985a). pg. 559. 35 earns positive rents on its information. Baron (1985b) also shows that, because of these rents, the firm prefers rivalry, while the public utility regulator prefers cooperation. This latter result is based on the fact that consumer surplus receives a higher weight in the public utility regulator's objective function than does producer surplus. And finally, the environmental regulator prefers rivalry because it gains the strategic advantage of a leader and no longer has to temper its regulatory choices by considering how they will impact on the mandate of the public utility regulator. It's as though a constraint was removed from the environmental regulator's maximization problem. Baron (1985b) addresses the same regulatory choice problem as encountered in the cooperative scenario of Baron (1985a). The major modelling change is that unlike Baron (1985a), the firm's icliosyncratic information parameter now enters the abatement cost function, as well as the emission function. In this new scenario, marginal abatement costs are assumed to be increasing in the firm's type. In this article. Baron shows once again that the optimal flat tax (one that does not vary with emissions) is zero, so that It is a dominated regulatory instrument. Further, he shows that if the tax is a function of emissions, then it has a role to play in getting the firm to reveal its privately-held information. The final paper we shall examine in this section is Spulber (1988). Spulber examines environmental regulation in a multi-firm setting in which there is interdependence between the product market equilibrium and the optimal regulatory policy, which is composed of effluent quotas and taxes. The firms are heterogeneous in that their abatement costs differ. These costs are a function of output and an emission control input. The information parameter enters the function by affecting the emission control input, not output. The regulator does not know this firm's specific information and must structure its environmental policy so that this information is revealed. 36 This limited iriformation is important because the regulator does not know output levels when it sets its regulatory policy. Yet. production costs and ultimately the price of output are being influenced by the costs of abatement. Thus, interactions between the product market and the regulatory policy should be important. Spulber develops a necessary and sufficient condition which ensures that the full information effluent levels are obtained. "The condition is that, at the full information optimum, net benefits from the product market (consumer surplus minus total production costs and minus pollution damages) must exceed the costs of inducing truth-telling by firms." 14 2.5 Conclusion We focus exclusively on hidden characteristic problems, suppressing completely any hidden action problems. This literature survey attempted to capture the flavour of adverse selection problems. For example, the reader should be aware that information revelation occurs only at some cost to the principal and that these optimal limited information contracts are typically distorted away from their full information counterparts. The technical aspects of the problem that we focus on or use include the revelation principle and the local approach to solving the global incentive compatibility constraints. 14 Spulber (1988). pg. 1. 37 C H A P T E R IH THE M O D E L AND COOPERATIVE SOLUTION 3.1. Introduction The model Is essentially that of Baron (1985a). It Is dominated by the actions of three distinct economic actors, each pursuing different and possibly conflicting mandates. The first economic actor is a monopoly firm which produces a joint product. The first component of the joint product is a socially desirable good like electricity. The second component of the Joint product Is a socially undesirable nonlocalized externality like sulphur dioxide (the main contributor to acid rain in North America). The firm's output of electricity Is regulated directly by the second economic actor in our model, the public utility authority or commission. This regulator chooses a two-part pricing scheme, made up of a fixed charge and a variable charge, which fixes the firm's level of output for a given demand function. The final decision maker in our model is the environmental regulator who chooses a design standard to be imposed on the firm's abatement technology. In this manner, the firm's output of sulphur dioxide also is regulated directly. Obviously the choices made by the two regulators will influence the firm's performance, but more interestingly, we show how the choice of one regulator will influence the choice of the other regulator as it designs its optimal regulatory policy. The type of common agency being considered here is Intrinsic in the terminology of Bernheim and Whinston (1987). The influence of any single economic actor on another depends on how we model the institutional framework within which they interact. This framework becomes even more important when we compound the complexity of the regulatory process by acknowledging the existence of information asymmetries between the various decision makers. Therefore, the specific characteristics of the institutional relationship 38 profoundly influence the optimal regulatory policies that prevail In equilibrium, and consequently, the well-being of the regulators and the firm. Thus, the changing institutional framework will be the central focus of this thesis. 3.2 The Objective Functions It isn't difficult to understand that the essential characteristic of this model, the characteristic which fosters the possibility for cooperative and non-cooperative behaviour between the regulators, is the non-localized nature of the pollutant. fact that acid rain can travel and The "impact" on areas far-distant from its source enhances the basic conflict in the model. Essentially there are two distinct geographic regions. Is a firm and a group of consumers who In the first region, there receive the service offered by the firm. In the second region there is another distinct group of consumers, but the only things they receive from the firm are the emissions associated with the firm's production, not the electricity. The public utility regulator is concerned with the well-being of the firm and the group of consumers who occupy only region one. Its mandate is completely exclusive of any Influences that occur in region two; i.e., any environmental damages which might occur in region two. The environmental regulator, on the other hand, is concerned with the consumer group occupying region two because they receive all of the damages/pollution from region one, yet this regulator also is concerned about region one in that the firm incurs abatement costs and the environmental regulator regards this as a cost of improving air quality. A conflict may arise between the regulators because the firm produces a pollutant which impacts exclusively on region two, and 39 since the environmental regulator may choose a policy to benefit the group in region two at the expense of the firm and the group of consumers in region one, the public utility regulator may behave non-cooperatively, or act as a rival when setting its regulatory policy. The public utility regulator chooses a two-part pricing scheme, composed of a variable charge, p, and a fixed charge, T, in order to maximize a weighted sum of producer and consumer surplus. 15 Choices regarding prices are based qn what the regulator knows about market demand conditions and the firm's costs. These costs are composed of operating costs, abatement costs (both fixed and variable), and fixed capital costs. We assume that all the functional forms in the model are common knowledge (in the sense of Harsanyi (1968)). 16 It is assumed that the firm holds better Information on its abatement costs and the efficiency of its abatement technology than either of the two regulators. Even though the regulators know the functional forms of the functions relevant to their regulatory decisions, the firm still possesses superior information about its local environment. The regulators, being even more detached from the day-to-day workings of the firm, are unable to observe the firm's ability to apply or install the abatement technology. For example, some engineers may be more adept at installing and maintaining complex scrubbers than are others. The regulators realize that varying degrees of expertise exist, but even if they were to expend considerable effort to ascertain the specific characteristics of the firm, say through some auditing procedure, they could not do so. Baron (1985a) rationalizes the use of this type of social welfare function by suggesting that this Is what utilities actually do in practice. He quotes Richard S. Bower, a former New York State utilities commissioner as saying. The agency objective is to maximize the weighted sum of consumer and producer surplus through time, and the weight accorded producer surplus, not all of which remains within the Jurisdiction, will be small If it is a state regulatory agency." (Baron (1985a), pg. 557). common knowledge assumption Is very Important. It means that all the players know the functional forms, but it also means that all the players know that all the other players know the functional forms. An infinite number of statements of this type defines what Is meant by common knowledge. Common knowledge Is an assumption about what players know the other players know. See Myerson (1985) for an excellent explanation and Rubinstein (1989). 40 An obvious conflict arises between the firm and the public utility regulator because the regulator needs-this firm-specific information on abatement costs in order to choose a set of prices which will maximize social welfare. The firm, on the other hand, wants to promote producer interests over consumer interests, not social welfare. Therefore, the firm has the ability (a hidden characteristic) and now a motive ("anti-social" interests) to act strategically when revealing its cost information to the regulator. A similar divergence of interests exists between the environmental regulator and the firm, and we will discuss this regulatory environment shortly. It is well established In the information literature that games of this type, between a principal and an agent, may be simplified greatly by invoking a result known as the revelation principle. 17 In heuristic terms, the revelation principle ensures that there is no loss of generality associated with focusing on or considering only contracts (here composed of a two-part pricing scheme and a design standard) which Involve the firm reporting its true type. In our model, consistent with most of the literature, the firm's type Is modelled as a one-dimensional parameter which summarizes all the firmspecific information. Invoking the revelation principle (as we discussed in the literature survey) allows for a tremendous simplification of our model since the firm simply reports its type, rather than some other more complex strategy. In this chapter, a report may be either of two values, w and w . A report, w e {w^wj, comes from the same set as the firm's t 2 true parameter, w e f^.wj. If the firm reports its true parameter, then we have w = w. Since the regulator Is unable to observe w directly, it may be the case that w * w so that the firm reports a parameter value, either w or w , which Is different from t 2 The revelation principle was developed by Dasgupta, Hammond and Maskin (1979), as well as Harris and Ravlv(1981) and Myerson (1979). For an excellent Intuitive presentation see Harris and Tbwnsend (1985). 17 41 its true parameter value, again either w or w . The regulator must base its regulatory t 2 policy on w e (w„wj, the reported parameter. Our problem is simplified even further by assuming that abatement effort, output and the level of emissions are all directly and costlessly observable. This assumption is made in order to preclude any hidden action (moral hazard) problems. This allows us to focus exclusively on the resolution of the hidden characteristic problem. In practical situations of environmental control, however, these problems cannot be subsumed. Thus, this is an area toward which further research needs to be directed. The Market We now are in a position to formalize much of the intuition contained in the preceding paragraphs. Assume that the firm faces a market for its product which is characterized by a simple demand curve, whose functional form is common knowledge. (3.1) Q(p) = r„ - r l P or, equivalently p(q) = (ryrj - (1/rjq where: r > 0 and T i 0 > 0 The public utility regulator will choose a variable charge, p(w), and a fixed charge, Ttw), both of which are a function of the report made to the regulator. Consumers are assumed to be willing to pay the fixed charge, regardless of its value. There are N consumers each paying T°(w). Therefore, the "fixed" revenue of the firm will be NT°(w) = T(w). 42 The Firm The firm's cost function is influenced by the level of output, q(w), a design standard, s(w), which is chosen by the environmental regulator, and the firm's true type, w e (wj.wj. For reasons of tractability, we assume that the firm's cost function takes the following quadratic form. The first term in (3.2) is just variable operation cost. The second term might be labelled variable abatement costs, while the third is fixed abatement costs. The final term is fixed capital costs. (3.2) C(q(w),s(w),w) = cq(w) + aq(w)s(w) + [fbws(w) }/2] + K 2 where: c>0;a>0;b>0 w.w e {w Wj} lt This functional form for total costs identifies exactly how the firm's idiosyncratic information enters the problem. Even though the functional form is common knowledge, the specific value for w e {w^w-J is unknown and unobservable. The firm must voluntarily reveal this Information through its report. The actual, realized level of costs sustained by the firm depend on which value it reports and on the true value of the parameter. We shall assume that Wi < w 2 so that a firm of type w will sustain x lower fixed abatement costs, ceteris paribus, than a firm of type w . By introducing 2 idiosyncratic information In this manner, we are assuming that a specific firm knows more about the fixed costs of abatement than does either regulator. This approach to modelling the firm's idiosyncratic information is a bit different than what is encountered in the literature, and so, it needs some Justification. 43 Many models In the literature model asymmetric information as part of the 18 firm's operating costs so that marginal costs are firm specific. completely discernable once the functional form is known. Fixed costs are The idea behind this method of modelling is that the firm has been monitoring and adjusting its production process and thus its variable costs constantly. Because of the constantly changing atmosphere, it is not hard to Imagine that the firm would know more about Its marginal costs or its marginal abatement costs than would a regulator. Clearly, this has been and remains a relevant way to proceed. However, at least within the context of the problem we discuss In this thesis, ignoring fixed costs and more specifically fixed abatement costs is incorrect. Thus, to understand more about the regulator's problem, we examine the case of unknown fixed costs. The abatement cost function is defined as the difference between the total cost of production when there is no design standard, versus the total cost of production after the imposition of a standard. The specific form of the abatement cost function in (3.3) follows directly from equation (3.2). (3.3) A(q(w),s(w),w) = C(q(w),s(w),w) - C(q(w),0,w) = aq(w)s(w) + [bws(w) /2] 2 where: a > 0; b > 0 w,w e {wi.wj Fixed abatement costs, [bws(w) /2], are costs of abatement which do not vary with a output. Since examples of the regulatory framework we are examining include the case in which an abatement technology is added to an existing power plant, we typically are dealing with a retro-fit situation. Under these circumstances, costs which do not vary with output include the following. 1. Facilities and buildings. ^Notable exceptions to this observation obviously include the seminal contribution of Baron and Myerson (1982). 44 2. Engineering and supervision. 3. Purchased equipment (Including taxes and freight charges). 4. Costs of land and working capital. 5. Site preparation. 6. Piping Installation and painting. 7. Handling and erection. 8. Auxiliaries and duct work. Our assumption that the firm will be able to evaluate these costs more effectively than a regulator is based in part on the rule-of-thumb nature in which cost estimates are arrived at by regulators. If rule-of-thumb estimates are refined, then the regulator typically needs site-specific observations, and the provision of these is subject to the discretion of the firm. There is an even more subtle point here. By making the claim that the firm knows more about its fixed abatement costs and about the efficiency of its abatement technology, we are assuming that, for a given design specification, the firm will know more about how well its production facilities will mesh with the design specifications. Given this potential for inefficiency between the design specification and the existing facilities, there is yet another potential source for inefficiency. Typically the firm also will know more about how closely the installation of the abatement technology meets the design specifications (design standards). From this exposition, it should be relatively clear that the firm will know more about the efficiency of its abatement technology, but also about the costs of the technology. The two are related because, for example, the degree to which the design specifications augment the production process, or the degree to which a firm is able to meet the design specifications (within some allowable error), would depend upon, among other things, how much money is spent on such things as engineering and 45 supervision, site preparation, etc. It will depend also upon the costs expended on such things as duct work and auxiliaries, etc. As a result of this interdependency, the firmspecific information represented by the one-dimensional parameter w e {w Wj} will 1( enter both total costs and the emission function. We shall discuss the emission function when we focus on the environmental regulator's objective function. One of the implications of equation (3.3) is that Afq'.s"^) < A(q°,s°,w ). This 2 means that abatement costs are strictly increasing in type. It is this property which causes total costs to be characterized in a similar fashion. Since this result is important in some of our proofs, the reader should be aware of its key role; it seems to be a natural way in which to represent the idea of a 'lower cost" firm. The firm in this model has output (price) regulated as well as some of its costs. However, because the regulators' choices for the regulatory variables are a function of the firm's report, w e (w^Wj), the firm effectively chooses a price when it decides which parameter value to report. The regulator specifies a set of prices, one for each possible type, and the firm picks from this menu. Since the firm knows that the information it reports will be used to set prices and the design standard, its choice is not a trivial one. The firm wants to choose a parameter report which will maximize its profits. Assuming the firm is risk neutral, expected profits are as follows. 7c(w.w) = p(w).q(w) + T(w) - C(q(w).s(w),w) (3.4) = p(w)-q(w) + Ttw) - cq(w) - aq(w)s(w) - [bws(w) /2] - K 2 where: w.w e {Wj.wJ p(w).q(w) = P(q(w)).q(w) = (iyrj.qfw) - (l/rj-qtw) 46 2 Even though the regulators do not know the actual value of the firm's idiosyncratic parameter, they realize that the firm will report its type as one of two possible values. The firm knows its true parameter value before the game begins. The regulators also have prior information about the parameter values, but only the probabilities of each type occurring. This information is common knowledge with, (3.5) 9, = Pr (w = w) and + Q = 1. 2 The Public Utility Regulator The public utility regulator maximizes the expected value of a weighted sum of consumer and producer surplus, where the expectation is taken over the firm's type. The term, Y(q(w)), is a measure of total willingness-to-pay (TWP). In our model, this will be measured as the area under the demand curve. [dY(q(w))/3q(w)] = p(w). The difference between TWP This fact implies that and the total revenue of the firm, p(w)-q(w) + T(w), gives an expression for consumer surplus. Producer surplus is weighted by the parameter fj e [0,1] and reflects the relative importance of producer surplus vis-a-vis consumer surplus in the regulator's objective function (see footnote #2). (3.6) From this specification, we see clearly the dichotomy which the non-localized externality creates. The objectives of the public utility regulator reflect the fact that the producer's, and more importantly the consumers', benefits are independent of any environmental damages. Even though the firm produces the externality, none of it 47 impacts on the consumer group represented by the public utility regulator. The public utility regulator proceeds as if there are no environmental damages to worry about. Having presented the objective function of the public utility regulator, it is now convenient to highlight the main differences between our model and those defined in Baron (1985a,b). The most obvious difference is that our model does not contain an emissions tax. This is a direct accommodation of Baron's result that a flat tax on emissions is a dominated regulatory instrument, and would not be used in the cooperative solution. We have simply ignored the tax, even in the non-cooperative scenario (chapter IV). The second major design change involves the firm's cost function. In Baron (1985a), the only source of asymmetric information was the efficiency of the firm's abatement activity. This we include as equation (3.7) below, but we introduce also firm-specific Information into the cost function (equation (3.2)). Baron (1985b) also does this (as do Baron arid Myerson (1982)), but the asymmetry in his model involves variable abatement costs. The difference, expressed in general functional forms, may be summarized as follows. Baron (1985a) A(q(w).s(w)) = a(q(w).s(w)) + b(s(w)) Baron (1985b) A(q(w),s(w),w) = a(q(w),s(w),w) + b(s(w)) Thesis Model A(q(w),s(w),w) = a(q(w),s(w)) + wb(s(w)) Even with this slight change in the model, the results are distinctly different from either of Baron's models. The Environmental Regulator 48 The other principal in this model is the environmental regulator, who is concerned with minimizing a weighted sum of expected environmental damages and firm abatement costs. Again, the expectation Is taken over the firm's type. Like the public utility regulator, the environmental regulator chooses its regulatory variable based upon which type the firm reports. Since this parameter summarizes important information on idiosyncratic abatement efficiencies and abatement costs, it is important for the regulator to receive truthful information. The environmental regulator views emissions as a random variable. As a result, observing the level of emissions, and knowing the level of output and the design standard, does not allow the regulator to infer the firm's true type. The random nature of emissions (in part caused by, say, weather influences) causes the regulator to be uncertain about whether it is observing abatement inefficiencies caused by the firm's type or simply stochastic influences. The expected value of emissions is given in (3.7) and is a function of the firm's output level, q(w), a design standard, s(w), and the true value of the idiosyncratic parameter. (3.7) X(q(w).s(w).w) = tw(q(w)-s(w)) where: q(w) > 0, x > 0 V w e {w^wj q(w) £ s(w) w.w e {w^wj The observed (or actual) level of emissions is a function of output, the design standard, and the realized value of a random parameter, e. This parameter, having its own distribution, induces a distribution on actual emissions. It could be a random 49 variable representing design efficiencies/inefficiencies. That is, it could be the degree to which a particular production process conforms to a particular scrubber design. The distribution of e is influenced by the firm's type, which could reflect the degree to which the firm is able to meet the design specifications. However, we assume that all of these stochastic influences impact on the distribution of actual emissions, but that it has a non-moving support. This has the implication that observing emissions does not allow the regulator to infer the firm's type. 19 One way to think about's' is to determine how many units of output are foregone when funds are redirected to an abatement technology of a certain design. The design standard itself is a complex blueprint of the technology. Given Input costs, and the current production technology, 's' would represent the number of units foregone. The more stringent the technology, the larger the number of units foregone. As a result, q(w) - s(w) represents units of unabated output. The parameter x then translates output into units appropriate for emissions. The parameter, w e {w^w-J, is simply an efficiency parameter. Expected environmental damages are a function of the same three variables as expected emissions. The only difference is that damages contain an extra parameter, h, which would translate emissions into environmental damages. (3.8) D(q(w),s(w),w) = hcw(q(w) - s(w)) where: h > 0, t > 0, w.w 19 One distribution for e {WJ.WJJ fmig»<nnn which would yield the specification ln s(w)). l) 50 (3.7) is the gamma. X - r(tw(q(w) The environmental regulator attempts to minimize a weighted sum of expected damages and abatement costs, where the weight is given by <|> e 10,1]. There is nothing special about this interval, it Is simply convenient to specify < | > in this manner. (3.9) Given all of this Information, we now can model the objective function of the cooperative scenario as the difference between (3.6) and (3.9) (The actual specification of the objective function is given later on in the text, after we have discussed the relevent constraints on the maximization procedure.). Notice that the regulator wants to maximize this function by choosing a set of contracts, {p(w),T(w),s(w)}, one for each reported type, w e {w^wj. 3.3 Incentive Compatibility Even though we have determined the objective functions of all the economic actors in the model, we still must deal with the essence of the problem. We have identified already one of the driving forces behind the model as the non-localized externality. Another important characteristic which motivates the economic players Is the asymmetry of information between the firm and the regulators. This is something with which the regulators must deal before they can make the proper regulatory choices. The Incentive compatibility constraints are designed to ensure that the firm reports its true type. The intuition of section 3.2 is formalized in the following set of constraints. 51 Type w. n(p(w ),T(w ),s(w ),w ) £ TtfofwJ.TfwJ.sfwJ.wJ Type w JtfotwJ.TtwJ.sfwJ.wJ ^ 7t(p(w).T(w),s(w),w) 2 1 1 1 1 1 1 1 2 Appending this set of constraints implies that the optimal contracts will satisfy global incentive compatibility, since we constraints may have only two possible types. This set of be written in a more compact form in which the arguments of the function have been suppressed. (3.10) 3.4 7t(w„w,) £ ittwj.wj V i,j = 1,2 Individual Rationality The regulator must ensure that the contracts it offers to the firm do not preclude the firm from earning some reservation level of profits, say normal profits. If profits fall below this critical level, then the firm could shut down; it has the option of nonparticipation. This added restriction on the set of possible equilibrium outcomes is captured by the individual rationality constraints (IR). Thus, if the firm reports its type tmthfulh/, then it will receive at least its reservation profit level, which for simplicity will be taken to be zero. (3.11) re(w„wj > 0 V 1 = 1,2 These constraints incorporate the assumption that the firm knows its type before the regulatory policy is formulated. If this were not so, then we would deal with expected profit, where the expectation is taken over 8,, the probability of occurrence of any particular type. 3.5 Two-type Full information: Cooperative Solutions 52 3.5.1 Introduction As a bench mark case, we will present a characterization of the optimal solution when the regulator is endowed with full information. To augment our understanding, it is helpful to highlight an implication associated with using the particular functional form we have chosen for the abatement cost function in equation (3.3). For what follows, we assume that s(wj > O.for all w,. If A(q°,s°,w) < A(q°,s°,wJ, then it is easy to show that 1 7e(w°,wJ > K[W°,WJ. A profit function which satisfies this property is said to be decreasing ln type (DE). When this property is satisfied, a type-Wi firm will always receive higher profits than a type-w, firm if the contracts for both types are the same. This occurs even if the contract is designed for a type-w firm; i.e., w° = w . 2 3.5.2 2 The Full Information Solution: Cooperative Assumptions Full information means that the regulator knows the firm's actual type. We want to find the optimal contract when the regulator does not have to deal with an incentive problem. Another way to think about this is to suggest some naivete on the part of the regulator. What would an optimal contract look like if the regulator proceeded as if it believed that firms would always report their type truthfully. In either case, the regulator still must ensure that the individual rationality constraints are met. This set of circumstances leads to the following maximization problem for the cooperative solution. 20 20 Because of the relationship between price and quantity via the market demand function, it doesn't matter which variable we rnaxlmlre: with respect to. Therefore, for convenience, we have chosen to maximize with respect to q. Using the inverse demand function, we have p(wj = P(q(w)). Note also that, ln thefirst-orderconditions, [3Y(q(wJ)/aq(wJl = p(wj. 53 (3.12) Max q(w).T(w) s(w) Wlw) = Y(q(w)) - p(w)-q(w) - Tfw) + pjt(w) D(q(w),s(w),w) - 4A(q(w).s(w),w) subject to: ir(w,w) > 0 V w e {Wj.w-J Examination of the set of first-order conditions (See appendix I, section I) allows us to establish our first result. If p < 1. then all of the Individual rationality constraints will be binding. This implies that full information optimal profits are zero for both types, jr^fwj.w,) = 0 for all 1 = 1,2. This follows from the fact that u, = (1-0) > 0, V i = 1,2, where u, is the Lagrange multiplier on the individual rationality constraint for the firm if it is of type w, e {w^wj. This result may seem a bit puzzling at first as an examination of the objective function (3.12) shows that the firm's profits enter positively. So it Isn't immediately clear that profits should be driven to zero. Recall that, when given a market demand curve, any increase in producer surplus must come at the expense of consumer surplus. If p < 1, then consumer surplus receives a higher weight In the objective function than does producer surplus so that a redistribution of one dollar from consumers to the producer represents a net welfare loss to the regulator. Therefore, at the optimum, the regulator prefers to drive profits down to the point where the individual rationality constraint is binding, a minimum requirement for the firm's participation. Employing the first-order conditions, we can derive simple expressions for the optimal regulatory variables. (3.13) p(w,) = c + as (wj + (|)as(wJ + hxw, n n FI 54 V i=1.2 The optimal full information per unit charge stated in (3.13) is chosen so that it covers marginal operation costs, c + as"(w,); marginal damages, hxw,; and, the portion of marginal abatement costs considered by the regulatory authority, <t>as(w,). From a n practical standpoint, the cooperative solution probably would have <|> = 0, since there doesn't seem to be any reason to "double-count" the effects of marginal abatement costs in the objective function. Even with this adjustment, it becomes clear that the price per unit contains an implicit tax on consumption in the amount of the marginal 21 damages. This portion of the price is a payment for environmental damages caused by production. Note however, that all of these environmental damages are assumed to occur outside the Jurisdiction In which consumers are paying this "extra" charge. Evidently then, these consumers are receiving none of the benefits arising from abatement, even though they are paying for the abatement, a peculiarity in the pricing arrangement obviously generated by cooperation between the regulators, rather than full information. The optimal design standard is defined implicitly to satisfy the~ following. (3.14) hxw, = (1-H|>){aq(w,) + bw^wj} n V 1 = 1.2 The standard equalizes marginal damages, the term on the lefthand side, and weighted marginal social costs, the terms on the righthand side. Using the demand function, and the expression for variable charge, we can derive easily the optimal expression for output. (3.15) q [w) = r - rjc + (l+<|))as(w) + hxwj n n o 1 V i = 1.2 This interpretation was first made by Baron (1985a). Since we replicate some of his results, our use of his terminology is duly noted. 21 55 Little insight into the problem can be gained from this expression, yet we need it in order to solve for an explicit expression for the optimal standard. This expression figures prominently in our later results. (3.16) s (wj = n hxw, - a(l+<|>){r - r\(c 0 + hxwj} V 1 = 1,2 (l-HWfbw, - r^a+ct.)} To solve for the optimal expression for the fixed charge, we use the fact that the individual rationality constraints are binding for all types when p < 1. This implies that ^(w„wj = 0, V 1 = 1,2. It is then a matter of some algebra to establish the expression for the fixed charge. (3.17) T"(wJ = [bw,s (wJV2] + K - VP^wJ n V i = 1,2 where: VP^wJ = q {<))as (wj + hxwj > 0 n n The first two components to the right of the equality sign in (3.17) are simply fixed abatement costs and fixed capital costs, respectively. The final term, variable profit, is simply variable revenue less variable costs. The fact that variable profits are always positive suggests that, in the cooperative solution whether we have limited or full information, it will be the case that the fixed charge is less than the firm's fixed costs. This result is contrary to what we usually see in the regulation literature where the fixed charge Is chosen to cover fixed costs exactly. The reason this occurs, however, is quite simple and it highlights one of the influences of cooperation. Since the expression for price contains an implicit tax, it always will be greater than the firm's marginal operating costs. So in this case, price is greater than it would be at the point where output is chosen efficiently; i.e., when marginal cost is just equal to price. Since price is greater than the firm's marginal costs, variable 56 revenues exceed variable costs. Thus, cooperation between the regulators creates positive variable profits for the firm. In setting the variable charge, the regulators equate marginal social cost to price, rather than just to the firm's marginal cost. If variable profits are positive, then covering all of the firm's fixed costs would generate positive economic rents for the firm. As long as p < 1, this would result in a net loss to the regulator. To keep the Individual rationality constraint binding, the regulator reduces the fixed charge to cover only the difference between fixed costs and variable profits. In the full Information solution, the fixed charge acts as a lump sum transfer between consumers and producers which ensures zero profit for the firm. There is one other interesting aspect to this expression. Suppose that marginal damages were extremely high at the optimal solution, so that the price was comparatively large. This could generate large variable profits, which may also result in the fixed charge being negative; i.e., an explicit tax on the firm. Thus, it may be the case that significant threshold effects could cause very high variable charges but relatively low fixed charges. Thus, the damage function (its functional form) could influence not only the absolute values of the fixed and variable charges, but also the relative values of the fixed and variable charges. There are three main things to remember from this section. First we must remember the form of the optimal expressions for the control variables. Second, we must remember that all of the individual rationality constraints are strictly binding. Third, and the main reason for looking at limited information solutions, is that the full information solution assumes that the regulator will receive truthful information. However, it is very easy to show that the optimal solution does not satisfy incentive compatibility. Specifically, we can substitute our expressions for the regulatory control variables into the profit function and show the following. 57 u™(w ,wj = [bs (w ) /2](w -w ) > 0 (3.18) 2 n a 2 2 1 If the firm's type is private information, but the regulators naively believe that the firm will report its true type, then a low type firm has an incentive to lie about its type, thus falsifying the regulator's belief. 3.5.3 Limited Information: The Cooperative Assumptions main difference between the full information problem and information problem is embodied ln the constraints. problem, we the limited In the limited information simply use the incentive constraints defined in ( 3 . 1 0 ) . The objective function changes by virtue of the fact that the regulator no longer knows the firm's true type. Therefore, the regulator must maximize expected social welfare. Notice also that the objective function is a function of only the true parameter values. This follows directly from the revelation principle. 2 ( 3 . 1 9 ) Max W - Z qfwJ.TtwJ s(wj [eJYlqlwJ) - P(q(wJ)-q(wJ - T(wJ + PJC(WJ - D(q(wJ,s(wJ,wJ - <|iA(q(wJ,s(wJ,wj}] subject to: Tcfw^wJ £ 7t(wj,wJ K{w ,wl > 0 t V i,j = 1 , 2 V i a 1,2 This problem is a constrained optimization problem which we can handle using standard Lagrangian techniques. The results depend upon which of the constraints are binding and which are not. Therefore, we need to point out that the u„ i = 1 . . . . 4 , 58 are the Lagrange multipliers on the four coristraints set out in (3.19). We assume that (3.19) has an interior solution. Proposition 3.1: At the solution to (3.19), the design standard must be nonincreasing in type. Proof: See appendix I, section HI. Proposition 3.2: i) If it is optimal to have the firm receive distinct standards, s*(wi) > s*(wj, then only the upward incentive compatibility constraint is binding. ii) If it is optimal to have the firm receive identical standards, s*(wj = s*(wj, then both the upward and downward incentive compatibility constraints will be binding. iii) At the solution to (3.19), the individual rationality constraint is binding for the firm only if its type is w . 3 Proof: See appendix I, section DJ. Proposition 3.1 establishes some feasibility conditions for our solution. At the very least, we know that the standard must be non-increasing. Proposition 3.2 then breaks up our solution into two possibilities, one where the standards are distinct and one where they are not. This distinction becomes important when we derive the actual expressions for the optimal design standard. Interestingly, derivation of the optimal variable charge (quantity) does not depend upon the pattern of binding constraints. This observation Is then instrumental In explaining the bunching results we obtain In the next section. For clarity, however, assume that for the remainder of this section, the conditions of the model are such that it is optimal to offer the firm distinct standards, depending upon the firm's type. In due course, we will specify the conditions under which distinct standards are ensured. 59 To understand why the Individual rationality constraint is binding for the firm only if it is a high type, recall that in the full information solution, a firm would always want to report its type as w , regardless of its true type. As a result, the regulator 2 knows that it needs to provide a truth-telling incentive to a firm only if its type is not w . When P < 1, this means that the regulator would ensure that profits of a type w 2 2 firm are zero, just as in the full information scenario. When the incentive constraint is strictly binding in this manner, it means that the firm is just mdifierent between lying and telling the truth. As is common in models of adverse selection, we assume that if all the incentive for the firm to lie is removed, then the firm will report truthfully. Once we know the pattern of binding constraints, we can solve the rest of the model quite easily. From the first-order conditions, we get the optimal variable charge. p*(w) = c + as*(wj + <t>as*(w,) + hxw, (3.20) V i = 1.2 The optimal limited information variable charge is chosen so that it covers marginal operation costs, c + as*(wj: marginal damages, hxw,; and, the portion of abatement costs considered by the regulatory authority, <t>as*(wj. For clarity, notice that we have assumed that s*(wj * s'fwj. Notice also that the form of these limited information prices is exactly the same as the form of the full information variable charges, equation (3.13). In fact, the two sets of prices would be exactly the same If we could claim that s*K) = s (w ) and that s*(wj = s (wj. F1 n 1 s*(w,) hxw! . (l+<t>){r - r\(c + a o hxwj} {l-HfrK-r^a+il)) + bw,} (3.21) e{(i+<|))bw - n d - H u M + (l-PKl-GJbK-wJ 2 2 60 A quick examination of s (wj (equation 3.16) shows that s*(wj and s (w ) are in n FI l fact equivalent. Given this, it is easy to show that the variable charges are equivalent as well: p*(wj = p^wj. The same cannot be said about the high type standard. Some simple algebra will establish that only if p = 1 or 0 = 1 will s*(wj) = s (wj, and n 2 correspondingly p*(wj = p^wj. As we assume P < 1 and 9 < 1, It is optimal to 2 distort the standard and variable charge of a high type firm. Proposition 3.3: At the solution to (3.19), only one standard and variable charge are distorted away from their full information counterparts, namely: s*(wj < s"(w ) and p*(wj < p (w ). n l 2 Proof: See appendix I, section HI. This is a standard result found throughout the principal-agent literature. If a firm turns out to be a low type, then it will receive a contract-which has the variable charge and standard equivalent to their full information counterparts (efficient). If a firm turns out to be a high type, then it faces a lower standard and price. To understand our problem fully, we must be able to explain this pattern of distortions. To do this, we want to solve for the optimal fixed charges. In the full information solution, we were able to solve for the fixed charges in a relatively easy manner because both individual rationality constraints were strictly binding. In this limited information problem, we can do this only for the highest type firm since proposition 3.1 stated that the individual rationality constraint is binding only for the firm if its type is w . Then, using the fact that the incentive compatibility 2 constraint is binding upwards, and the natural recursMty of our problem, we can solve for the optimal fixed charge as a function of the standard for the firm If it is a low type. 61 (3.22a) T^wJ = [bw s*(w ) /2] + K - VP*(wJ 2 a 3 where: VP*(wJ = q*(wj{(t>as*(w) + hxwj > 0 2 T*^) (3.22b) = [bw s*(w ) /2] + K - VP*(wJ + [bs*(w ) /2](w -w ) 2 1 2 1 2 a 1 where: VP*(wJ = q*(w){<|>as*(w) + hxwj > 0 1 1 Focus first on (3.22a). the expression for the fixed charge chosen by the regulator and levied against consumers on behalf of the firm when its type is w, a high cost 2 firm. This fixed charge has three separate components, and is very similar in form to T^Wj) which had the same three components. The first term is fixed abatement costs, which are a function of the optimal design standard. The second term is simply fixed capital costs. The third term is variable profit. The regulator again chooses the fixed charge to cover only the difference between total fixed costs and variable profit. This will ensure participation by the firm in the regulatory, scheme while keeping rents to a minimum. Significant differences between the limited information solution and the full information solution arise when we consider (3.22b). It contains all the same terms as (3.22a) plus one extra. We shall call this term, [bs*(w ) /2](w -w ), a pure 2 a 2 1 information rent. To understand why this is an information rent, break this term up into two separate components. The first component, [bw s*(w ) /2], measures the 2 2 fixed abatement costs of a firm when it is of type-w . 2 2 The second component, [bw s*(w ) /2], measures the fixed abatement costs a type-Wj firm would sustain if it a 1 2 reported its type as w 2 and faced standard s^wj. In essence, this measures the benefit to the firm from lying. By calculating this difference and adding it to the fixed charge, the regulator exactly removes the firm's incentive to lie. But of course, this 62 costs the regulator something; no longer are economic profits zero if the firm turns out to be of type w lf as they were In the full Information scenario: Tt'lw^wJ = [bs*( )V2](w -w,l > U. W2 a The variable charge and design standard offered to a type-v^ firm are exactly the same with limited Information as they were with full Information. solution is efficient or non-distortionary. So in this sense, the This reflects the regulator's desire to take advantage of the most cost-efficient firm. By offering a full information price and design standard, the size of the available surplus is maximized. The regulator maintains this efficiency by offering the firm a fixed charge which is simply a lump sum transfer from consumers to the producer. The unwanted, but necessary side effect is that this generates information rents for the firm. Another way to understand this result is to explore the structure of the model more carefully. Note first that our model has the characteristic that the fixed charge enters in a linear fashion. This means that the objective function is quasi-linear. When this characteristic of the objective function is not present, the efficiency result would appear as a marginal condition. The gradient of the objective function (In 3 dimensions) would equal the gradient of the profit function (the constraint function). So. In this sense, the result is non-distortionary. Our result Is even stronger. Not only do we have this marginal condition satisfied, we also have this condition satisfied at the same values as in the full Information solution. This follows from the quasilinearity. To use the regulatory variables in this manner makes good intuitive sense, yet we have stated already that the contract offered to a type-w firm is distorted in each 2 of its components (Proposition 3.2). penultimate paragraph? Why Why Is this optimal, given our explanation in the doesn't the regulator always choose the variable charge 63 and design standard so that they are equivalent to their full Information counterparts, and adjust the fixed charge to ensure Information revelation? To understand why there Is a distortion is to understand exactly how the information externality influences the problem. The answer lies in the equilibrium expression for economic rent given above. The regulator wants to ensure that expected rents are as small as possible, while satisfying all the constraints of the problem. Since the regulator does not have an incentive problem for a high type firm, the regulator ensures that the individual rationality constraint is binding for this type. which would yield this zero-profit result. There are many different contracts One of them obviously would be the full Information result. However, a type-Wj firm will reveal its privately-held information only if it receives a rent, but the regulator wants to reduce this payment by as much as possible. To reduce the rents accruing to the low type firm, the regulator distorts s*(wj away from s"(wj, while maintaining (IC) and (IR). This also explains why s (w ) F1 2 > s*(wj, since rents are lowered by reducing s*(wj. By distorting the design stan- dard, the equilibrium variable charge also is distorted away from the full information solution. The degree to which the contracts are distorted away from full information values depends on the regulator's prior information, Q and 0 , and the difference x 2 between the types (w - wj. Examination of the equilibrium expressions will bear this a out. 3.6 Monotonicity and Bunching Properties A separating solution has the characteristic that for each possible type which the firm can report, there is associated with this type a unique contract. An implication of uniqueness is that when the firm chooses an optimal contract, its type will be revealed completely to the regulator. That is, every type can be separated out using the optimal set of contracts. Such a solution may be contrasted with a bunching (pooling) solution in which the firm's type cannot be determined simply by 64 observing which contract the firm chooses at the optimum. This mdetermlnacy occurs because not every type receives a unique contract. Some types (two or more) may receive equivalent contracts. Typically, in models involving more than two types, we might observe separation of some types and bunching of others. However, in models involving only two types, such as the model we are discussing here, we either have a bunching optimum or a separating optimum. In this section, we are interested in characterizing what determines whether any two distinct types will be separated out or bunched together. to show how In addition, we want our monotonicity and bunching properties differ from those found in models which employ only two choice variables. The debate about whether to bunch types or separate them is important to the regulator because bunching and/or separation ultimately influences the rents to be paid to the firm when it Is of any particular type. We have shown already that the regulator would like to keep these rent payments as small as possible. Related to the debate about bunching versus separation are the monotonicity properites of the optimal solution. Proposition 3.1 establishes that the design standards must be non-increasing In type. If It is optimal to offer distinct standards, then bunching would be ruled out. The regulators always would find it optimal to separate out both types. (Recall that complete separation was a characteristic of the mU-information solution). Likewise, if any other component of the optimal contract is found to be strictly monotone in type, then bunching would be ruled out. Our Intuition regarding the relationship between bunching and monotonicity properties can be sharpened by introducting the concept of assignment monotonicity. It states that if any component of the optimum contract is equivalent for two distinct types, then all other components of the contract will be equivalent for the same two types. Therefore, if assignment monotonicity Is satisfied, two distinct types are bunched if and only if they are offered the same value for one of the components of the contract. As a result 65 of this "if and only if' statement, we now enquire as to whether or not bunching is ever optimal, for this will determine the monotonicity properties of the model. Proposition 3.4: At the solution to (3.19), the regulator always finds it optimal to separate out types: i.e., bunching is ruled out and two separate contracts always are offered. Proof: Suppose that contrary to the proposition we have {p*(w),T*(w),s*(w)) = 1 1 1 (p*(w),T*(w),s*(w)). Since p*(Wj) = p*(wj), it follows from (3.20) that c + (l+^as'fwj 2 2 J + hrwj = c + (l+^Jas'fw,,) + htw . a Using the assumption that s*(wj) = s*(wj to simplify the above equation, we obtain w, = w , a contradiction. Q.E.D. 2 This proposition establishes that bunching will never occur as a solution to the cooperative model. Interestingly, this proposition also establishes that, at the optimum, it never can be that (p*(w),s*(w)) = (p*(w),s*(w)). As yet however, we don't actually l 1 2 2 know what kinds of monotonicity characteristics our model does display at the optimal solution. Our next proposition clarifies these characteristics. Proposition 3.5: If the optimal design standards offered to the firm are equivalent, then the variable charge will be decreasing in type; the quantity will be Increasing In type; and, the fixed charge can be increasing, constant, or decreasing in type with T*(w,) | T*( ) as ( J - ( J w - Wi W 2 W W2 a 2 ^-r^as+irp-r^c+ad+^s} r\hT Proof: See appendix I, section DJ. Before we discuss the implications of propositions 3.4 and 3.5, we should at least be clear about the conditions under which the design standards can be equal. 66 At this point, we know already that Incentive compatibility requires the standard to be non-increasing in type. Regardless of whether the standards are strictly increasing or constant in type, the optimal solution is characterized by separation. However, we have seen that the optimal rents to be paid to the firm in equilibrium are determined solely by the design standard. Therefore, it is the design standard alone which generates incentive compatibility. As a result, when the standards are equivalent, there is both upward and downward binding incentive compatibility. In contrast, most screening models involving just two instruments, binding upward and downward incentive compatibility would indicate bunching and the contract would exhibit assignment monotonicity. In our model, proposition 3.4 establishes that even if we have equivalent standards, the solution is still characterized by separation. It is for this reason that we want to establish the conditions under which the design standards are equivalent. Proposition 3.6: i) At the solution to (3.19), a necessary and sufficient condition to have s*(w) = s*(wj = s is to have (fy/OJ 5 C* where: x c* = {hTd+ad+^rJ-Id-H^wJ - bq+flUi+wr.-r^c)} (1-P)b{htw - aU+Mr^tc+hxwJ)} 1 ii) When (fy/GJ < C*. then ejhw, - a(i+<t>)rr.-r(c+h'rw))} + e,{htw, - aU+^fr.-rjc+hwj)} l S - 1 O.Kl-HtiJbw.-r.d-^J^d-BJbCw.-w,)} + 9 {(l-H))bw -r (l+(t))2a2} a a i Proof: See appendix I, section m. This result does not lend itself well to economic interpretation, but it is the kind of result we would expect to find in this type of model. If the ratio of probabilities 67 passes some critical level, then the expected benefit from separation outweighs the expected costs associated with separation. The benefits include the gain accruing to the regulator as a result of its ability to exploit the low cost, highly efficient firm. The costs of separation include the expected rents to be paid to the low type firm in order to get it to reveal its type voluntarily. Now that we have established the condition under which the standards will be equivalent, we need to explore the importance of the result. That is, why is this set of three propositions a contribution to the body of knowledge. To see why this might be so, imagine that we have a regulatory problem very similar to the one we have described except that now the regulator only has two instruments to control. Assume further that these instruments are T and s. It has been mentioned already that there is a strong relationship between the incentive compatibility constraints, bunching, and the monotonicity properties of our model. With the aid of figure 2, we can show this relationship clearly. First, note that the set of axes of the figure are the two choice variables used by the regulator. by jt,. Next define the iso-profit curves for a type W i firm as those indexed This upward sloping curve identifies those combinations of s and T for which the firm receives the same level of profit. Similarly, we index the iso-profit curves associated with a type w firm as 1%. With this particular combination of T and s, a 2 type w, firm would receive profits J^. Notice that the slope of the iso-profit curve for a type w firm is steeper than the slope of the iso-profit curve for a type W! firm at A . 2 This is done by assumption i n order to demonstrate what Is meant by the marginal rate of substitution ordering property. When this property is assumed, then at any particular point like A , the marginal rate of substitution of T for s i n the firm's profit function is higher for a type w firm than for a type ^ 2 that profit Increases in a northwesterly direction. 68 firm. For simplicity, assume G i v e n t h e c o n t r a c t A , w h i c h i s d e s i g n e d for a type w l f i r m , we are interested i n deterrnining w h i c h set of c o n t r a c t s i s incentive c o m p a t i b l e for a type w 2 firm. This set o f c o n t r a c t s w i l l c o n t a i n a n y c o n t r a c t w h i c h g e n e r a t e s a profit level for a type w f i r m greater t h a n o r e q u a l to v^. 2 T o identify t h i s set, c o n s i d e r p o i n t B i n figure C l e a r l y t h i s c o n t r a c t w o u l d m e e t o u r r e q u i r e m e n t s for a f i r m of type w . However, 2 s u c h a c o n t r a c t w o u l d n o t satisfy u p w a r d incentive c o m p a t i b i l i t y b e c a u s e t h e profit level for a type w, f i r m at p o i n t B i s s t r i c t l y greater t h a t J V c o n t r a c t s a s s o c i a t e d w i t h A , B d o e s n o t satisfy Incentive T h e r e f o r e , the set compatibility. of It b e c o m e s c l e a r t h a t , once we fix c o n t r a c t A , t h e o n l y c o n t r a c t s w h i c h are In the set of Incentive c o m p a t i b l e c o n t r a c t s are t h o s e c o n t a i n e d i n the a r e a b e t w e e n a n d jtj above p o i n t A. Incentive c o m p a t i b i l i t y a l o n g w i t h t h e a s s u m p t i o n r e g a r d i n g t h e o r d e r i n g o f the m a r g i n a l r a t e s of s u b s t i t u t i o n g e n e r a t e s a p a r t i c u l a r s t r u c t u r e o n t h e o p t i m a l c o n t r a c t s which has nothing to do with m a x i m i z a t i o n per se. That Is, the monotonicity p r o p e r t i e s o f t h e o p t i m a l s o l u t i o n are d i c t a t e d b y the s t r u c t u r e of t h e m o d e l r a t h e r t h a n through maximization. Notice t h a t if the set of c o n t r a c t s i s g o i n g to satisfy the incentive c o m p a t i b i l i t y c o n s t r a i n t s , t h e n the c o m p o n e n t s o f the c o n t r a c t are o r d e r e d automatically. T h u s , if T c > T , w e k n o w i m m e d i a t e l y f r o m Incentive c o m p a t i b i l i t y a n d A t h e m a r g i n a l rate o f s u b s t i t u t i o n o r d e r i n g p r o p e r t y t h a t Sc > s . A W h a t p r o p o s i t i o n s 3.4 a n d 3.5 i n d i c a t e i s t h a t t h i s type of m o n o t o n i c i t y is not g e n e r a t e d b y t h e incentive c o m p a t i b i l i t y c o n s t r a i n t a n d t h e m a r g i n a l rate o f s u b s t i t u t i o n o r d e r i n g p r o p e r t y w h e n there are three c h o i c e v a r i a b l e s i n s t e a d of two. We c a n state t h e monotonicity. result i n a slightly different m a n n e r b y u s i n g assignment I n t h e m o d e l w h i c h h a s o n l y two i n s t r u m e n t s , the m a r g i n a l rate o f s u b s t i t u t i o n o r d e r i n g p r o p e r t y a n d the incentive e n s u r e t h a t a s s i g n m e n t m o n o t o n i c i t y i s satisfied. c o m p a t i b i l i t y c o n s t r a i n t s together However, i n o u r m o d e l w i t h b o t h incentive c o m p a t i b i l i t y a n d the m a r g i n a l rate of s u b s t i t u t i o n o r d e r i n g p r o p e r t y , 69 2. Figure 2: INCENTIVE COMPATIBILITY AND MONOTONICITY 70 assignment monotonicity may be violated.* The solution can have equivalent standards 2 yet distinct fixed and variable charges. As a consequence, the only equilibrium which occurs in our model results in complete separation. 3.7 Summary and Conclusions This chapter has presented the basic structure of the model we are going to analyze. Some interesting results flow from the analysis even at this stage. It seems that changing the nature in which the information asymmetry enters the model greatly influences the role played by each regulatory variable. This is hardly surprising. For example, in Baron (1985a), where the information parameter enters only the emission function, the variable charge and standard are equivalent to their full information counterparts and the firm earns zero rents. This is because the firm's idiosyncratic information only becomes valuable when there is a positive emissions tax. In Baron's model, the rents can be eliminated by setting the tax equal to zero. cooperative model generates full Information solutions. Thus, the In our model, the firm's idiosyncratic information is more influential so that even with cooperative assumptions, the firm is able to generate information rents. Introducing such a small change as entering the information parameter into the cost function completely changes the results. ^it Is easy to show that our model satisfies the marginal rate of substitution ordering property by examining the profit function. «(w„wj = p(wjq(wj + T(wJ - cq(wj - aq(wjs(wj - (bw^(w,)V2| -K. Fix w, = w so that p(wj a p; q(wj eq: s(wj a s: and. T(wJ a T. Now write the dproflt function as jt*(p,q,s.Tw) = p-q +T - cq - aqs - [bws /2| K. Now substitute for p = P(q). to get jr* = P(q)q + T - cq - aqs - (bwsV21 - K dir* = q|dP/dql-dq + P(q)dq + d T c-dq ^q-dc - qc-da - as-dq - aq-ds - bws-ds - Iws /2| db - [bs /2|-dw - dK. We know that da=db=dw=dk=dc=0, so we get dn* = q[dP/dql-dq + Hq)dq + dT - c-dq - as-dq - aq-ds - bws-ds = 0 for any lso-proflt curve. Now evaluate 2 2 a [dT/ds| I = bws + aq, and dq=0 bws + aq Idq/dslj =0 q[9P/3qJ + P - c - as Clearly, both of these are ordered by type for a given q,s and T. As w changes, so too will the slopes of these lsoproflt curves. This means that this model satisfies the marginal rate of substitution ordering property. 71 In Baron (1985b), the information parameter is Introduced into the variable cost component. In this case, full information contracts are not obtainable, but the job of information revelation is conducted by both the design standard and the variable charge. In our model, the fixed charge Is the only component of the contract which contains an information rent term. The other components, if they are distorted at all, would be distorted because of the information externality; i.e., as a result of the regulator trying to reduce the cost of information revelation. The other novel aspect of this introductory chapter is the manner in which our no-bunching result was derived. Whereas it is normal to have only separation when there are only two types, our Intuition regarding how our result was found suggests that we will have only separation even when there are n distinct cost types. This is one of the topics In chapter V. 72 CHAPTER IV T H E NON-COOPERATIVE M O D E L 4.1 Introduction In this chapter, we extend the analysis to situations in which the regulators behave non-cooperatively and determine how this change in behaviour influences the nature of the optimal contracts. In the process of characterizing the optimal contracts, we want to try to explain some of the stylized facts, in particular, why so little cooperation between agencies involved in environmental control is observed, and why firms might actually prefer more stringent environmental controls. The possibility of non-cooperative behaviour is promoted by the fact that the externality is non-localised. Since all damages from pollution impact on consumers who live outside the jurisdiction of the public utility regulator, but within the Jurisdiction of the environmental regulator, it stands to reason that the policies of the public utility regulator may be at odds with those of the environmental regulator. It Is this basic conflict which we want to capture in our non-cooperative game. There are many different ways in which the non-cooperative relationship between the regulators could be modelled. In Baron (1985a), it Is argued that the environmental regulator typically exhibits a larger degree of strategic influence than the public utility regulator. Specifically, Baron claims that the environmental regulator has "the autonomy and authority to adapt independently a pollution control policy". 23 This autonomy suggests that the environmental regulator can choose either to set a design standard after the public utility regulator has moved to set its menu of fixed and variable charges; i.e., after the firm's true type has been revealed, or to set a " B a r o n (1985a). pg. 561. 73 design standard before the public utility regulator has moved to set its menu of fixed and variable charges. In the first sequence of moves, the public utility regulator acts as the leader, while in the second sequence, the environmental regulator acts as the leader. From the discussion so far, it isn't obvious immediately which sequence of moves will provide the environmental regulator with the highest payoff. When acting as a follower, the environmental regulator is able to free-ride on the information revealed through the regulatory process, setting the standard to gain the biggest advantage from the Information on the firm's type. Yet, as a leader in the sequence of moves, the environmental regulator could move to influence the public utility regulator's choice of menu, and thereby bias this choice in its favour. Obviously, the relevant comparison for the environmental regulator, when choosing to act as a follower or a leader in the regulatory game, Is between the value of the strategic advantage it gains when acting as a leader and the value of the savings in Information costs it would earn as a follower. ~" This debate is based upon the presumption that the environmental regulator would commit to a single design standard when setting its regulatory policy. In Baron (1985a) however, it is argued that, when the environmental regulator acts as a Stackelberg leader, it will want to commit to a menu of standards rather than just a single standard. Under these circumstances, it is easy to show that it is always optimal for the environmental regulator to act as a leader. The reasoning behind this claim is simple. For any set of design standards which are optimal for the environmental regulator when it acts as a follower, there exists the same set of standards available to the environmental regulator when acting as a leader. This suggests that there is nothing to be gained by the environmental regulator when acting as a follower, since any of the information which could be revealed in the regulatory process; Le., the firm's true type, could be anticipated by the environmental regulator 74 when designing its initial menu choice. With this Information, and to the extent that we believe that the environmental regulator can commit to a menu of standards, the optimal game form from the environmental regulator's viewpoint can be described as follows. 51. Nature determines a type for the firm according to the exogenously given prior probabilities. 52. The environmental regulator chooses a menu of design standards in order to minimize the expected value of a weighted sum of damages and firm abatement costs. 53. Taking the menu of standards as fixed and unalterable, the public utility regulator chooses a pricing scheme, a menu of fixed charges and a menu of variable charges, in order to maximize a weighted sum of expected producer and consumer surplus subject to incentive compatibility and individual rationality constraints. 54. The firm makes a profit maximizing report of its type to the regulator, thereby choosing a specific contract composed of a design standard, a variable charge and a fixed charge. S5. The game ends. One controversial component of this game structure enters at stage three above. The way the game is set up right now, the public utility regulator is responsible for ensuring revelation. Since economice rents are a function of only the design standard, the cost of revelation ln the form of economic rent is set by the environmental 75 regulator. The public utility regulator chooses a fixed charge which contains this information rent, so in this sense it pays the costs of revelation set by the environmental regulator. Thus, it is entirely possible that the public utility regulator could make itself better off by allowing the firm to pursue a strategy which doesn't result in the revelation of its true type. Another way to view the problem is to recognize that public utility realizes that the information regarding type which Is revealed during the regulatory process will be used against it. Thus, there is an Incentive for the public utility regulator to "make a deal" with the firm explicitly, or implicitly through its regulatory process. Both of these paragraphs suggest that there is an incentive for the public utility regulator to implement a contract which violates the incentive compatibility constraints. If this Is true, then the game structure presented above may be inappropriate since it assumes that the public utility regulator will choose a two-part pricing scheme which satisfies the incentive compatibility constraints. This assumption can be seen in (4.1). One way to address this problem would be to re-model the regulatory relationship as a hierarchy. Thus, the environmental regulator would engage the public utility regulator in a revelation game to ensure that it was in the public utility regulator's best interests to play a revelation game with the firm. In this manner the environmental regulator would guarantee that the firm's true type would be revealed during the regulatory process. In this thesis, we are going to assume that the public utility regulator willingly pays the costs of revelation. Thus, (4.1) should contain the incentive compatibility constraints as part of the maximization problem. As is usual when solving games of this type, we proceed by backward induction. 76 Thus, the solution is sub-game perfect. This means that the strategy profile induces a Nash equilibrium at every stage of the game. We shall introduce two notions of rivalry between the regulators. They correspond to different specifications for the objective function of the environmental regulator. The objective function of the public utility regulator, however, remains the same in either specification; we examine its problem first. When choosing its menu of fixed and variable charges, the public utility regulator takes as given the menu of standards set by the environmental regulator. Therefore, the public utility regulator's problem is to 2 (4.1) Max W = q(wJ,T(wJ £ [8iY(q(wj) - p(w>q(wj - T(wJ + pft(§(wj,wj}] subject to: ft(s(wj,wj > ft(§(w,),wj 7i(s(wJ.wJ > 0 1= 1,2 ij= 1.2 where: sfWi) and §(wj forms the menu of fixed standards. Before this maximization problem is actually solved, we want to focus on the Incentive compatibility and Individual rationality constraints. To this end, define a menu of contracts, x, in the following manner. x a {x, I x, = (p(wJ,T(wJ,s(wj); 1 = 1,2}. Each x, Is referred to as a contract. Given this notation, profits may as ^(x^wj when a firm of type reports its type as w . t 77 be represented Equivalently, Ttfxj.Wj) represents the profits a type w, firm would receive upon reporting Its type as w . 2 In chapter m, propositions 3.1 and 3.2 defined a particular relationship between the design standard and the Incentive compatibility constraints. Because the firm's idiosyncratic information influenced only fixed abatement costs, the design standard plays a major role in determining the value of the firm's private information. Therefore, it isn't too surprising to claim that incentive compatibility requires the design standards to be non-increasing in type. This will be established below. We showed also in chapter HI that the fixed charge was used to generate a transfer of surplus from the consumers to the producer. This lump sum transfer ensured that the firm is compensated for revealing Its privately-held information. However, the amount of the transfer that needed to be made was ultimately determined by the level of the standards. The importance of this observation for the non-cooperative game should not be underestimated. What this means is that the ability to generate contracts which satisfy the incentive compatibility constraints Is determined by the environmental regulator's choice of standards, and this responsibility cannot be transferred to the public utility regulator. Since the public utility regulator cannot choose the standard, the required monotonicity of the standard, which we prove In the next lemma, becomes a constraint on the environmental regulator's minimization problem. Lemma 4.1: If x satisfies incentive compatibility, then the standards will be non -increasing in type. Proof: Without loss of generality, we have supposed that w, < w . By (IC), we have 2 (1) Jt(x Wi) lf > refxjj.wj 78 and (2) Jtf.Xa.wJ £ nte^wj Subtracting the RHS of (2) from the LHS of (1) and the LHS of (2) from the RHS of (1), we obtain jHxi.wJ - Ttlx^wJ £ JtfXj.wJ - TCfXj.wJ. (3) Substituting for n and cancelling common terms gives [bWasKP/s) - [bw s(w ) /2] 3> rbw s(w ) /2] - [bw s(w ) /2] 2 1 Thus (3) holds a 1 2 2 2 1 2 s ^ ) - s(wj > 0. Q.E.D. With this monotonicity result, constraining the environmental regulator's choice, the importance of the public utility regulator In generating contracts which satisfy the incentive compatibility and individual rationality constraints should not be overlooked. It Is the public utility regulator which ultimately bears the costs associated with meeting the constraints. Since the costs ultimately show up as rents, and the public utility regulator wishes to reduce rents, the public utility regulator's choices are in part driven by the pattern of binding individual rationality and incentive compatibility constraints. Our next result establishes part of this pattern. Lemma 4.2: If x solves the public utility regulator's problem in (4.1), then the individual rationality constraint is binding for a type w firm, and 2 there is binding upward adjacent incentive compatibility. Proof: Assume that x satisfies the incentive compatibility constraints and that 7t(x,w) 2 > 0. 2 In addition, we assume that there does not exist a feasible x such that W(x) > 79 Wtx), where W( • ) is the public utility regulator's objective function in (4.1) and x is a menu of contracts. Specifically, define x as follows. x^x, I M (1) (pfwJ.TfwJ-e.sfw,)); i = 1.2} This information implies that = Tcfx^Wj) - e n(x ,w ) = Ttfxj.wj - e n(x ,w ) = jilxj.Wj) - e A&.wJ = J C I X J . W J - e. jt(Xi,Wj) (2) a 1 and a a In addition, e is chosen sufficiently small such that TtCxj.wJ > jtfxj.Wj) £ 0. Therefore, the menu of contracts x satisfies the individual rationality constraints. The menu of contracts x will be feasible if the incentive compatibility constraints are met. We have assumed that x is feasible since it solves (4.1). This implies (3) nfxi.wj 2 J I C X J . W ! ) and Jtfxj.wJ > iifx^wj Assume that x does not satisfy the incentive compatibility constraint for w,. (4) nik^) < jcl&a.w!) Given (2) above, we may write (4) as nfXj.Wi) - e < nfxj.wj - e Thus, AtXj.wJ < TCfXj.W!). But this contradicts (3). A similar sequence of steps will establish a contradiction of 80 (3) In the case of w = w . a Therefore, x must be feasible since this contradiction will hold for any x, e x that causes a violation of the incentive compatibility constraints. Since we have established feasibility, we have to show the optimality of x. Consider the expression for consumer surplus: CSlX.wJ = Y(q(wj) - pfwj-qfwj - ItwJ, for i = 1,2, where Ytqfwj) is total willingness-to-pay. This suggests that CS&.wJ = Y(q(wj) - p(wj-q(wj - (TfwJ-e), which in turn implies that CS&.wJ = CS(x,,wJ + e for all x, E x, x, e x, and w, e Using this simplification and using (2), it is easy {WJ.WJ}. to show that W(x) - W(x) = i 0, {-(TtwJ-e) + 3(7t(x,,w)-e) + TfwJ - Pre^.w,)} 1 2 = X 9, {e(l-P)} > 0. Thus, x could not have been an optimal contract. As long as nfxj.Wj) > 0, we can always find a feasible x such that W(x) - W(x) > 0. Part 2: Using the same style of proof, assume that contrary to the lemma, ^(x^wj > Tcfxj.W!). Our claim implies that there does not exist a feasible x such that W(x) > W(x), where feasibility implies the joint satisfaction of the incentive compatibility and individual rationality constraints. Define x as where x t = (pfwJ.KwJ-e.sfw!)) and x, = (plwJ.KwJ.sfwj). Obviously then, x, = x,. From our results above, we know that x is feasible since the contract offered to the type w in type. We 2 firm is not altered, and we know that the profit function is decreasing also choose e sufficiently small such that 81 Using our expressions for consumer surplus, we can evaluate W(x) - W(x). W(x) - W(x) = (^(-(TfwJ-e) + ptffc.wj-e)) + e^-tTfwJ) + pij^.w,)))) - (e^-fTfw,)) + ptffei.wj)) + e (-(TtwJ) + pfjc^wj))) a Thus, W(x) - W(x) = e,(e(l-P)) > 0. Therefore x could not have been an optimal contract. As long as jcbq.wj > refe/Wi), we can always find an e sufficiently small so that x is feasible and W(x) > W(x). Thus, the upward Incentive compatibility constraint will be binding at the solution to (4.1) . Q.E.D. This lemma allows us to suggest a simplified maximization problem for the public utility regulator. Specifically, we will replace the Incentive compatibility constraints with binding upward incentive compatibility and the individual rationality constraints with binding individual rationality for a type w firm. It is then important a to verify that the solution to this relaxed problem satisfies the neglected Incentive compatibility and individual rationality constraints. We do this in lemma 4.3. With this caveat, and the simplifications we have derived, the public utility regulator's maximization problem in (4,1) becomes 2 (4.2) Max W q(wJ.T(wJ = E [e,{Y(q(wj) - p(wj-q(wj - T(wJ + pft(§(wj.wj}] subject to: 82 ft(§(w,),W,) = itfslwJ.W!) rcfsfwJ.Wj) S 0 where: §(w ) and s(Wj) forms the menu of fixed standards. x Here we have suppressed the other components of the contract, q(w,) and T(wJ, in the profit function. This maximization yields a set of first-order conditions (appendix n, section I), and from these we can derive optimal expressions, which are in fact reaction functions, for the fixed charge, the level of output and the variable charge. (4.3) q&wj) = r - r\(c + as(wj) V i = 1,2 o (4.4) p(§(wj) = c + as(wj V i = 1,2 Since the upward incentive compatibility constraint and the individual rationality constraint for a type w firm are binding, we can solve directly for an expression of a the optimal fixed charges. T&w,)) = [bw s(w ) /2] + K + [bs(w ) /2](w -w ) 2 1 2 1 2 2 1 (4.5) Tfsfwj) = [bwjsfwj ^] + K 2 These expressions are used by the public utility regulator in solving its problem. The derivations for (4.3) - (4.5) are contained in appendix n, section I. 4.2 Extreme Rivalry The environmental regulator has been endowed with a significant degree of strategic independence. Given this assumption, we have argued that when the environmental regulator commits to a menu of standards, then acting as a leader in 83 the regulatory game Is preferred to acting as a follower. One notion of rivalry that is consistent with this game form is extreme rivalry (ER). Under extreme rivalry, the objectives of the two regulators are completely independent. This seems to reflect the current regulatory environment in which the policies of the public utility regulator and those of the environmental regulator are completely uncoordinated. When there is extreme rivalry between the regulators, the environmental regulator would solve the following problem : 24 (4.6) Min Z m = {D{q(s(wj),s(wJ,wJ + <j>A{q(s(w)),s(w),w}} 2e, 1 1 l s(wj subject to: S(WJ 5 S(WJJ) where: q(s(wj) is the reaction function of the public utility regulator given In (4.3). From the first-order conditions, it becomes obvious that we should break-up our analysis into two different cases, much as we did in chapter III. The two parts correspond to a strictly monotonlc standard and a weakly monotonic standard. For the moment, we assume that s^HwJ > s^rwj. The alternate case will be discussed shortly when we focus on the monotonicity and bunching properties of the model. With a strictly decreasing standard, the Lagrange multiplier on the constraint will be zero so that we can solve directly for the optimal design standards using the firstorder conditions. parameter restriction bw, > 2a ensures that Z(wJ attains a global minimum at s(wj V 1=1.2. This Implies that the abatement cost function Is strictly convex In the design standard. 9 84 (4.7) hxw.d+aTJ - <]>a(r - I^c) 0 s E R { W l ) <t>(bw, - 2r a ) = 2 1 It is then a matter of substitution to establish the optimal prices. Explicit expressions for the variable and fixed charges could be obtained by substituting the relevant standard defined by (4.7) into the expressions given In (4.8) below. p (w,) = c + as (wJ (4.8) V 1 = 1.2 ER ER T^wO = [bw s(w,) /2] + K + lbs (w ) /2](w -w ) ER 2 2 EH 2 1 :i 1 T^wJ = [bw s (w ) /2] + K 2 ER 2 4.3 2 Mild Rivalry Another notion of "rivalry" which is consistent with, the game form set out in section 4.1 is mild rivalry (MR). When the regulators act as mild rivals, we assume that the environmental regulator's objective is expanded to include the welfare of consumers in both jurisdictions as well as the firm. The environmental regulator doesn't really care about the public utility regulator's welfare, per se, as the game form Is still non-cooperative, yet its mandate is wide enough to include both jurisdictions. The only difference between mild and extreme rivalry is that the objective function of the environmental regulator changes. With mild rivalry, the value function of the public utility regulator is included In the ojective function of the environmental regulator, reflecting concern for both producer and consumer surplus. One way in which to view this change is to suggest that the assumption of extreme rivalry might reflect the objectives of a federal agency concerned with environmental Issues, and the assumption of mild rivalry might represent the objectives 85 of a state or provincial agency worried about environmental Issues. The rationale for the difference between these two types of rivalry Is based on the fact that state officials may have a greater stake than federal officials in the welfare of both regulatory jurisdictions and would therefore include producer and consumer surplus in the decision making process. This would be particularly true if state officials were elected. Under mild rivalry, the firm's true type is unknown when the environmental regulator chooses the menu of standards. However, the environmental regulator correctly anticipates the best response of the public utility regulator and of the firm to its decision regarding the standards. So the regulator in choosing first, solves the following problem. 2 (4.9) Max = Z [e {U (q(s(wJ,s(wJ,T(s(wi)),wJ MB i - D(q(s(wj),s(wJ.wJ - <|A(q(s(w)),s(w),w)}] S ( W , ) 1 1 1 subject to: s(w,) £ sfwj where: U"* a Value function of the public utility regulator. Assuming again that the regulator finds it optimal to offer distinct standards, we can solve for the optimal design standards directly. See appendix n, section in. hew, - a(l+<t>)(r - r\c) o (4.10a) s^wj = (l-HObw! - (l+2<)))ar 2 l (4.10b) s^HwJ = G2 {hrw - a(l+<(.)(ro - rlC)} ((l+<t))bw - (l+2<|))ar} + (l-pnXwa-wje! 2 2 2 86 l The values for the variable charges and the fixed charges are obtained by substituting (4.10a) and (4.10b) Into the relevant expressions in 4.11 below. p V j = c + as^wj V M (4.11) i = 1.2 = [bw . (wl) /2] + K + Ibs (w )V2](w -w ) T^wJ MR MH 2 2 1 3 1 T H w J = [bw s (w ) /21 + K MB a 2 2 At this point, it is convenient to show that the solutions to the extreme and mild rivalry problems are in fact feasible. This means that the equilibrium solutions satisfy the neglected constraints. Lemma 4.3: At the solution to the relaxed problem in (4.2), the neglected individual rationality constraint and incentive compatibility constraint are satisfied in both mild rivalry and extreme rivalry. Proof: To show this, assume that contrary to the lemma the neglected downward incentive compatibility constraint is not satisfied so that Substituting for gives 0 < [bw s (w ) /21 - [bw s (w ) /2]. ER I But this implies that w 2 < w lt 1 ER 2 2 l 2 a contradiction, so it must be that ^ X j . w J A (x .w ). ER 1 2 Now assume that contrary to the lemma TC^HXJ.W,) < 0. 87 Substituting for > gives Ibs (w ) /2](w -w ) < 0, a contradiction. Thus, the optimal solution to the extreme ER 2 1 2 rivalry problem constraints. 1 satisfies the Incentive compatibility and individual rationality A similar sequence of steps will verify the result in the case of mild rivalry. Q.E.D. One of the main differences between the solution to the mild and extreme rivalry games and the solution to the cooperative game is the fact that the variable charge no longer contains an implicit tax on consumption, since neither marginal damages nor the weighted marginal abatement cost term appear in the public utility regulator's problem. This is because the public utility regulator solves its problem independent of the objectives of the environmental regulator. The variable charge is required to cover only the marginal costs of the firm, not marginal social cost. The fixed charges are changed In a similar fashion. No longer are variable profits driving a wedge between fixed costs and the fixed charge. In this scenario, the fixed charge exactly covers the fixed costs of the firm (costs which do not vary with output) if it turns out to be of type w . 2 If the firm is a low cost type, then in addition to fixed costs, it will receive an information rent, which was discussed in chapter three. Everything else equal, the amount of the rent generated under mild rivalry versus extreme rivalry will depend upon the relative sizes of the design standards. Proposition 4.1: When the environmental regulator offers distinct standards, then the equfiibrium standards, variable charges and fixed charges are higher under extreme rivalry than under mild rivalry. Specifically, this means: i) s fw,) > s^HwJ ii) p'Vj ER > P^wJ V V 88 1 = 1.2. 1 = 1,2. Ill) T^wJ > T^w) V I= 1.2. Proof: See appendix n , section IV. The reason the design standard is less stringent when the regulators act as mild rivals is because the public utility regulator's objective function is strictly decreasing in the standard. To the public utility regulator, a zero standard would be optimal. The environmental regulator on the other hand must consider the impact of environmental damages. Since damages are a strictly decreasing function of the standard, a zero standard is not feasible. If the environmental regulator pursues a policy independent of the public utility regulator, then it will choose a standard which weighs damages and abatement costs. Clearly this standard would be positive. Thus, the public utility regulator's welfare necessarily falls once the standard Is chosen. If the environmental regulator considers this drop in welfare as important, as it would In the case of mild rivalry, then the value of the standard which it chooses must be lower than the value of the standard it would choose when acting independent of the public utility regulator, as it would in the case of extreme rivalry. The lower standard under mild rivalry reduces the firm's marginal costs and as a result, the variable charge is correspondingly lower, p (w,) > p^wj. ER demand curve then generates the level of output, q^wj The market < q (wJ. Since the design MR standard is lower, fixed abatement costs also will be lower under mild rivalry. But in addition to this, the lower standard reduces the value of the firm's private information. This implies that the information rent required to get a firm of type w x to reveal its privately held information Is correspondingly lower. We can be even more specific now regarding the ranking of the standards. There are four different institutional frameworks to consider: i) extreme rivalry; 11) mild rivalry; iii) cooperation; and, iv) full information. 89 Proposition 4.2: The design standards Imposed on the firm vary as the institutional structure changes. In particular, we can order the standards in the following simple manner. 1) s^fwj > s (w,) = s*(w,) > s^w,) n ii) s ^ w j > s (wj > s*(wj > s (wJ n MR Proof: See appendix II, section IV. In the Stackelberg game, the environmental regulator uses the reaction function of the public utility regulator to predict exactly how the public utility regulator will react to the choice of standard. In the case of extreme rivalry, the environmental regulator uses its strategic advantage to promote its well-being at the expense of the public utility regulator. With mild rivalry, however, the environmental regulator uses its strategic advantage to promote the well-being of itself and that of the public utility regulator. This strategic Independence In choosing the standard is something which is missing in the cooperation game. In addition, it should be obvious that in the mild rivalry game, the welfare of the public utility regulator is double counted. This may suggest one reason why this type of coordination is not feasible; yet, at least from the public utility regulator's perspective, highly desirable. The full information, cooperative solution is included in the comparison presented In proposition 4.2 for completeness. In some sense, it is inappropriate to compare this full information solution to the Stackelberg solutions because the full information solution was derived using the cooperative assumptions. Yet it is interesting to note that the differences between the Stackelberg games are such that the full information solution still falls between them. To a certain extent we can compare the full information cooperative solution to that of extreme rivalry. This is because both design standards turn out to be independent of the regulators' prior information. Thus it is the case that the extreme rivalry standard is undistorted by any information considerations, like those we discussed In chapter DJ. Any differences 90 between the full Information standard and the standard attributed to extreme rivalry arise purely from strategic effects. That Is. the difference arises because of the strategic effects Involved in moving to a Stackelberg game. Given proposition 4.1. and the ordering of the levels of output, we now are in a position to state the ordering of institutional structure according to the preferences of the various economic actors. We begin by showing the relationship between mild and extreme rivalry from the perspective of the firm and assuming that distinct standards have been offered. In the next section, we will state the conditions under which this will happen. Proposition 4.3: If the firm turns out to be a high type, then it is mdifferent to the institutional structure. If the firm turns out to be a low cost type, it will prefer that the regulators act as extreme rivals. Proof: See appendix n, section V. At first this result may seem an anomaly. The reason this might be so is because proposition 4.2 establishes that the firm faces its most stringent environmental control when the regulators act as extreme rivals. Yet, this is exactly the scenario which the firm prefers the most. This Is because the value of the firm's private information, ceteris paribus, is higher when the standard is higher. The regulators are willing to pay a higher rent because the value of a low-cost firm rises as the standards rise. This result is predicated on the fact that the fixed charge is always paid voluntarily by consumers for the service offered by the firm. Since the firm knows that Its fixed costs of abatement are going to be covered, it doesn't mind facing a higher standard. The only way for the firm to earn rents under this pricing scheme is by selling its private information. When environmental control is important, which is reflected in the higher standards, the firm is able to extract larger rents from the regulator. 91 Obviously this rationale does not apply to the firm If it is a high type. This Is because the firm's idiosyncratic Information is worth nothing to the regulator in equilibrium. Indeed, there is no Incentive for a high type firm to mlsreport its type. As a result, a type-w firm will be indifferent to the prevailing institutional structure. 2 Proposition 4.3 provides a partial understanding of why we might observe proenvironment firms. But the result is based on voluntary payment by consumers and the fact that all damages impact on areas outside the jurisdiction of the public utility regulator. If we were to allow damages within this jurisdiction, it seems that there would be a weakening of the result, but the essence of the argument would hold. In fact, we may even observe pro-environment firms and anti-environment consumers within the same jurisdiction. Just as in the case of the firm, the public utility regulator also show a clear preference for institutional structure. Its preference is summarized in the next proposition. Proposition 4.4: The public utility prefers mild rivalry. Proof: See appendix n, section V. The public utility regulator's preferences are related directly to the cost of information revelation. Because it is maximizing a weighted sum of expected consumer and producer surplus, where the weight on producer surplus is less than one, the public utility regulator will prefer a situation where consumer surplus Is as large as possible, and producer surplus guarantees the firm's participation. It stands to reason that the higher rents generated under extreme rivalry lead to reduced welfare for the public utility regulator. Indeed, it follows directly from this proposition that the size of the surplus is highest when there is mild rivalry. 92 Notice that we have not addressed the environmental regulator's preferences over Institutional structure. This Is because we have changed the objective function of the environmental regulator ln moving from extreme rivalry to mild rivalry. Thus, even though the game form is held constant, meaningful welfare comparisons are nonexistent. The ranking of institutions in the manner presented here has been done by Baron (1985a,b). He examined the relationship between cooperation and extreme rivalry, but did not discuss mild rivalry. He got the same kind of ranking for each of the economic actors but the manner in which the results were generated is different. By this we mean that the regulatory variables perform different tasks, but the end results seem to be consistent with those stated in Baron (1985a). Mild rivalry is an important comparison because it represents, for us at least, a more plausible manner in which to view the coordination of regulatory policy. Unlike cooperation, mild rivalry allows for the separation of the regulatory agencies, but like cooperation, the preference structure is such that one of the regulatory bodies would always prefer to defect from the existing relationship. The obvious result, as Baron (1985a) has observed, is that we would expect these regulatory agencies to act as extreme rivals. It remains to show how the cooperative solution fits into this preference structure. Since the cooperative solution falls In between the two endpoints of extreme rivalry and mild rivalry, our intuition suggests that the public utility regulator will prefer mild rivalry to cooperation, as will a low type firm. A high type firm remains indifferent to the institutional framework. Proposition 4.5: The environmental regulator prefers extreme rivalry over cooperation; the public utility regulator prefers mild rivalry over cooperation; and, the firm is indifferent to the institutional framework if its type Is w 2 but prefers extreme rivalry if its type is w 93 P Proof: See a p p e n d i x n, section V . 4.4 Monotonicity a n d B u n c h i n g Properties W e s a w i n c h a p t e r IH t h a t offering the f i r m a single c o n t r a c t r e g a r d l e s s of type was ruled out - p r o p o s i t i o n 3.4. We noted In p r o p o s i t i o n 3.5 that the normal m o n o t o n i c i t y r e s u l t s a s s o c i a t e d w i t h t h e incentive c o m p a t i b i l i t y d o n o t e x t e n d to t h e multi-dimensional problem. T h a t Is, strict m o n o t o n i c i t y i n two of t h e v a r i a b l e s w i t h r e s p e c t to type i s n o t sufficient to Imply strict m o n o t o n i c i t y In the t h i r d . T h e m o n o t o n i c i t y a n d b u n c h i n g properties o f c h a p t e r IV are d i s t i n c t l y different. B o t h extreme r i v a l r y a n d m i l d r i v a l r y p r e s e n t the p o s s i b i l i t y of b u n c h i n g w h i c h w a s n ' t p r e s e n t i n t h e cooperative s o l u t i o n . In t h e cooperative g a m e , t h e I n c l u s i o n of m a r g i n a l d a m a g e s i n the o p t i m a l e x p r e s s i o n o f t h e v a r i a b l e c h a r g e p r e c l u d e d b u n c h i n g . Thus, e v e n If t h e d e s i g n s t a n d a r d s were e q u a l , the v a r i a b l e c h a r g e s w o u l d n o t be. ~ In the non-cooperative model, m a r g i n a l damages do n o t enter the e q u i l i b r i u m e x p r e s s i o n s for v a r i a b l e profits i n either extreme r i v a l r y o r m i l d rivalry. that, f r o m a p u r e l y algebraic perspective, T h i s suggests t h e p o s s i b i l i t y of b u n c h i n g exists. As a r e s u l t , the c o n d i t i o n s u n d e r w h i c h t h e e n v i r o n m e n t a l r e g u l a t o r w o u l d w a n t to offer a single s t a n d a r d , s fwi) = s ^ w j ER = s™, m u s t b e specified. r e g u l a t o r i s s u e s sF*, t h e e x p r e s s i o n s i n (4.8) s a m e t h i n g w o u l d o c c u r to t h e fixed c h a r g e s . environmental suggest t h a t p^fw,) = p ^ w j = p. ER The S i n c e the c o n t r a c t s w o u l d be i d e n t i c a l i n t h i s c a s e , we s a y t h a t the two t y p e s are b u n c h e d P r o p o s i t i o n 4.6: O n c e the together. i) A n e c e s s a r y a n d sufficient c o n d i t i o n for b u n c h i n g to o c c u r at the s o l u t i o n to the extreme r i v a l r y p r o b l e m i s to h a v e ( T . / r j ^ c + (hxa/<t)b)(l+aT) 94 11) When 1) occurs, the equilibrium design standard Is specified as 9j hxWttl+aTJ Aer S = - (Mr, - r c)} + ejhxw.d+arj t - <fratr - r )} B lC e,{<|)fbw - 2r,a)} + 9{<t>(bw - 2r\a)} 2 2 2 1 2 Proof: See appendix n, section IV, The optimal variable and fixed charges could then be determined by substituting s m into the relevant expressions ln (4.8). This result is Interesting because the condition for bunching is independent of the regulator's prior Information. Usually the costs involved in separating out types would enter the bunching condition. In the cooperative game, these probabilities certainly enter the "costs" of separation. These "costs" appear in the form of rents to the firm. The firm must be given an incentive to reveal its privately-held information. This incentive comes in the form of profit - a pure information rent. This rent is created by a lump sum transfer from consumers to the producer. This is one of the roles played by the fixed charge - see (4.8). The point, however, is that the environmental regulator in effect determines the level of rent which must be generated when it sets the design standard. Recall the discussion regarding the relationship between the value of the firm's privately-held information and the design standard. Once the design standard is set, it is the public utility regulator which is responsible for Instituting the transfer of the surplus when it sets the fixed charge. In the non-cooperative game all the costs of separation are borne by the public utility regulator and the environmental regulator does not care about the welfare of the public utility regulator, so the bunching condition is independent of the prior information on type. Up to a point, the environmental regulator doesn't care about the costs of separation. The obvious question after this discussion is why would there ever be bunching if the environmental regulator never has to bear the costs of separation? Because of the manner in which the idiosyncratic parameter enters the model, the design standard 95 Is the only "variable" which Influences rents; therefore, there is a unique relationship between the design standard and the Incentive compatibility constraints. In order to ensure separation, which is the rationale for offering distinct standards, the environmental regulator must ensure that the design standards are non-increasing. Given the environmental regulator's optimal choice decision, the bunching condition given in proposition 4.6 shows when this constraint would be binding. Obviously, if the parameters are such that optlmality suggests s (wJ 5 s^wj, then separation ER would not occur (this can be shown very easily), and there is no reason for the environmental regulator to offer distinct standards. A similar set of circumstances occurs in the case of mild rivalry. will occur whenever s^w,) Bunching = s ^ w j = s"*. The condition under which there will be a single standard when the regulators act as mild rivals is different than the extreme rivalry example. Given our previous discussion, this isn't too surprising. What we find Is that the probabilities of occurrence for either type play an important role since tiie Environmental regulator is now forced to consider the costs of separation. The implications for policy seem obvious, and would suggest a larger opportunity for bunching equilibria than In the case of extreme rivalry. Proposition 4.7: 1) A necessary and sufficient condition for bunching to occur at the solution to the mild rivalry problem is to have [QJ^] £ C**. b(l-p){hxw - aU+fllT.-riC)} l ii) When i) occurs, the equilibrium design standard is specified as e AMR S t = (hTWt - - r )} + e, a(i-Ht>)(r„ lC {hxw 2 aq+wr,, - r )} lC 9{(l+<t>)bw-(l+2<l))ar}+(l-P)b(w-w)e + 9{(l+<t>)bw-(l+2<|>)ar} 2 1 1 a 1 Proof: See appendix n, section V. 96 2 1 1 2 2 1 A further discussion of the bunching properties of the mild rivalry scenario is not really necessary since the essence of the problem was captured in our discussion of proposition 4 . 6 . However, there is one other aspect of policy decisions related to bunching properties which has been neglected in the literature. Suppose that bunching turns out to be the optimal situation in both mild rivalry and extreme rivalry. Although we do not show this formally, it seems that the set of parameters which generates bunching in the case of mild rivalry probably contains the set of parameters which generates bunching in extreme rivalry. This only suggests that bunching could occur in both models for the same set of parameters. Again, suppose this to be true. Is it still the case that the firm prefers the regulators to act as extreme rivals? Another way to formulate the question is to ask if it is always the case that s 88 > s""? It is easy to show that this is not the case. In fact, depending upon the parameters, we could have s ER less than, equal to, or greater than s"*. Therefore, there certainly could be circumstances under which the firm prefers that the regulators get together and coordinate their policies. 4.5 Conclusions We have presented two notions of a non-cooperative game and find marked differences in the characteristics of their solutions. There are also significant departures from the cooperative solution. The mild rivalry game and the characteristics of Its solution are similar in nature to those found in the industrial organization literature dealing with direct and strategic effects. In our case, the Inclusion of the public utility regulator's value function in the environmental regulator's objective function combined with the environmental regulator's strategic advantage causes the environmental regulator to choose a design standard which is even more lax than in full cooperation. This positive strategic effect dominates any negative effects associated with the environmental regulator's own objective function - perhaps somewhat 97 surprisingly. This chapter has also introduced some new bunching properties which will be discussed further in the next chapter. We found that the possibility for bunching existed because of the separation of regulatory responsibilities. In addition, we showed that when the regulators act as extreme rivals, so that the costs of separation are determined by the environmental regulator, but borne by the public utility regulator, design standards in the separating solution are uniformly higher than In mild rivalry. The variable and fixed charges also were higher. 98 CHAPTER V A GENERALIZATION T O N TYPES 5.1 Introduction From a theoretical standpoint, generalization is always a desirable characteristic. This chapter attempts to generalize our results from the previous two chapters to the case in which the firm can be any one of *n' different cost types. Obviously there are many other ways in which generalization may proceed. For example, we could have used generic functional forms rather than the specific ones contained In the thesis. We chose to Increase the number of types because the solution techniques for multidimensional screening problems are not well developed. Indeed, it is unclear whether the results we derived in the previous two chapters carry over to the n-type case. In most cases, our results are completely general. However, the manner in which we solve the problem is different, and accounting for many types sharpens our intuition regarding the optimal solutions and the bunching properties of the various models. The only difference in the models to be discussed In this chapter and those of chapters three and four is that the dimensionality of the type set increases. Now the firm chooses a report w e fw, wj, where n is finite. The firm's true cost parameter also is a member of this set, w e {w! wj. The rest of the model is as discussed in chapters three and four, so we maintain the following assumptions. Maintained Assumptions Al) There is no ex post renegotiation. A2) All functional forms are common knowledge. A3) The weight on producer surplus, p. Is less than one. 99 A4) The revelation principle has been invoked. A5) The firm will report truthfully when all Incentive to lie has been removed. A6) Abatement effort, output, and the level of emissions are all observable at zero cost. A7) Consumers willingly pay the fixed charge. A8) The firm holds superior information on its fixed abatement costs and abatement efficiency. 5.2 The Full information Solution In the full information solution, when regulatory policies are coordinated through cooperation, the regulators proceed as if they had full information regarding the firm's type. The new cooperative regulator now offers the firm a single contract, depending upon the firm's type. This contract is chosen optimally to maximize a weighted sum of producer and consumer surplus less a weighted sum of environmental damages and firm abatement costs, while maintaining the Individual rationality constraints. (5.1) Max W"(w) = Y(q(w)) - p(w)-q(w) - T(w) + pn(w) q(w),T(w) s(w) D(q(w),s(w).w) - ()jA(q(w).s(w),w) subject to: n(w,w) S O V w 8 {Wi wj This is exactly the same objective function we encountered in equation (3.12). The only difference Is that the individual rationality constraints must hold for more types. Not too surprisingly, all of the results for the full information solution when there are two types continue to hold for n types. 1) All the individual rationality constraints are strictly binding. 100 ii) p (wj = c + as (wj + <j>as(wj + hrw, n (5.2) n n iii) q (wj = r - r,{c + (l+4>)as(w) + hew] n n o 1 iv) T"(wJ = [bw s (wJ /2] + K - VP"(wJ n 2 1 V I= 1 n V i = 1 n V i= 1 n VP"(wJ = q {<t)as (wj + hxwi} > 0 n n hxw, - a(l+<|>){r - Tjfc + hrwj} o v) sJ L(wj = n I= 1 V (1-H»{bw, - r^U+iW} n These expressions all contain realizations of w, namely w. There is only one subtle t difference between the solution we have described here and the one associated with the two-type model. When the firm could only report W! or w, we were unable to distinguish clearly 2 the firm's motive behind lying, or at least what things influence this lie. The motive Itself, of course, is profit. With only two types, there is only one lie available to the firm. Thus, it reports the only other parameter value available to it. Consider the general expression for profit when the firm knows that it will face the full Information contract, but also knows that in fact information is limited and asymmetrically distributed. ^(Wj.wJ = p (w ).q"(w ) + T"( ) - cq (Wj) n n J J Wj aq (w )s (w ) - lbw s (w ) /2] - K F, FI J n J 1 2 J V i,j = 1 Upon substitution for the optimal variables, we get (5.3) ^(w^.wj = [bs (w ) /2](w -wJ > 0 FI 2 J J 101 V i < j. n When written In this form, we see that there are two influences which together determine what kind of a lie the firm will report. The first is the difference in the idiosyncratic parameters, Wj-w,, arising as the report Wj is changed. The second is the manner in which the value of the design standard changes as the report w^ is changed. That is, what is the value of [oVlWj.wydWj)]? * We show in the footnote 28 that this partial derivative is positive. Therefore, under the conditions we have described, the firm would always want to report its type as w . As before, then, the n inequality in (5.3) establishes the firm's incentive to misreport when faced with the full information contract, within the context of limited and asymmetric information. The full information contract is said to be non-implementable. It is for this reason that theorists began to focus on the characterization of the limited information solution. 5.3 The Limited Information Solution: Cooperative Assumptions We have seen that the dimensionality of the constraint set has been increased dramatically. As a result, we must endow the regulators with more prior information so that 6, e {0! 0J, and 8, = Pr[w = wj. One of the major difficulties in solving information problems of the type we have described is created by the global incentive compatibility constraints. We have alluded to this several times, but it is here that we feel the bite of the problem. One aspect of this problem obviously is the large number of constraints which presents severe computational problems. Another aspect of the problem is that the pattern of binding l3ji (w ,wj/awjl = bs V,).(d8 (w)/dwJ.{w -w } + [b/21s"(w )» 26 n n r ) 1 T,(c 1 1 1 Let sPivr) a |A/B|. where A = hxw, - aU+4>){r o + htwp) > 0, and B = (1-Ht>){bwj - r,a»(l44)} > O.It is easy to show that [ds"{w)/dw) = {[bt + a(l+$)r"ihT|/B} . {A-[b(l+<fr)|/B}. a If this expression is going to be negative, then the following must be true, {[bt + a(l+<tfr,bil/(l-H>)b} < (A/B). If we multiply the LHS by tw/wj = 1, we get {[A+C}/[B+D|} < (A/B}_ where C = a(l+4>){r„-r,c) and D = a'U+trT,. This inequality will hold if |C/D| < 1A/B1 B s ^ w j . After substituting and rearranging terms, we get [r./r,| < c + (l+4)as"(wj. This may be written as p _ < p"(w,) - hew,, a contradiction. This Implies that (def{vi)/dw) 2 0. Using this information, we can sign [3»t(w .wJ/8wJ = bs"(w )-(as"(w)/awJ-{w ,, 1 wj + [b/2]s"(wp» > 0. 102 J ) r constraints often is not known a priori, so the large number of constraints must be dealt with. Several solution techniques have been developed to handle the dimensionality problem. We have discussed the local approach, as chronicled by Matthews and Moore (1987), in chapter n. They also present their own method for solving the Incentive compatibility constraints In a multi-dimensional screening problem. We are basing our analysis on their technique. While the parameter w can take on values only In (w t wj, it is useful to compute the profits that would be obtained if type' could take on any value In [w wj. For w lf e [w„w ] there exists a X e [0,1] such that w = Xw, + (1-Xhv . Definere(p,T,s,w)= w 1+1 X3t(p,T,s,wJ + (l-X)jc(p,T,s,w ), V (p.T.s) e Rt.. 1+1 Thus, for all (p.T.s) we have profits defined for all possible types in Jwj.wj. This convexiflcation allows us to illustrate our results using simple graphical techniques. From chapter IV, we know that a contract can be defined in the following manner. Let x, e x, where x is a menu of contracts so that x a {x, I x, = (plw^.TfwJ.sCvv,)); I = 1 n}. Each x, is called a contract. This implies the following equivalence statement: n(x„wj a re(w„wj. The convexiflcation performed above also allows us to define and then draw socalled profit curves (as opposed to the more familiar iso-profit curves). The profit curves used in this thesis are based on, and analogous to, the utility curves Introduced by Matthews and Moore (1987). A profit curve shows how the profit associated with a fixed contract varies as the firm's type Is allowed to vary. The contract which is being held fixed, the reference 103 contract, may be or may not be the contract which was originally designed for the type being examined. Figure 3 may help clarify the concept. Fix the contract and e (w wj to vary. Let.x° 3 {p .T°,s } so that 7r(x°,w) = p°q° + T° - cq° - then allow w 0 0 lt aq°s° - [bw(s ) /2] - K. This expression defines a single profit curve when the contract 0 2 Is x°. Notice that w Is free to vary. We have picked an arbitrary reference contract and then forced each type to face this contract. Because of the functional form we are using, other characteristics of our model follow immediately. 1. Every profit curve has a constant negative slope. To see this simply evaluate rcj^.w) to get -[b(s°)/2]. Clearly this slope depends upon 2 the reference contract which defines s°. 2. Because profit curves slope downward, profits for a given contract are decreasing in type (DE), provided s° > 0. (5.4) Jttx°,wJ > jcb^.w,) V i.j Throughout all the subsequent analysis, we maintain the assumption that s(wj > 0 V w, e {w t We wj. pick up our Investigation of the limited information solution by focusing on the full set of incentive compatibility constraints, both upward and downward incentive compatibility. If our optimal contracts were to satisfy (5.5), then we say that they satisfy global incentive compatibility. (5.5) 7t(w,w) £ 7t(w,w) V w.w e {WL.^.W,,} We now can state the unrelaxed problem which must be analyzed. This is called 104 Figure 3: A REPRESENTATIVE PROFIT CURVE 105 unrelaxed because this problem contains the full set of constraints. It will then be our task to present a series of results which allow us to reduce the complexity of the constraint set. After all, as it stands right now, there are n 2 - n incentive compatibility constraints and another n individual rationality constraints. With the present structure, the regulator wishes to 2 n (5.6) Max W = qlwJ.TtwJ, s(wj {eiYfafw,)) - p(wj-q(w.) - TfwJ + PJC(WJ - 1-1 DfafwJ.sfwJ.w,) -(JiAfqfwJ.sfw,), subject to: it(w„w,) £ 7t(Wj.wJ V n(vr w) S O V lf Wj.w, e {w, w, 8 {Wi wj wj The next series of results are necessary steps to a much simpler problem. They allow us to uncover the implications of the structure of the problem so that we are able to see clearly exactly which of our results follow from maximization, and which follow from the constraints. We are then able to present a much simpler, "relaxed" problem. Some of these results were proved in chapter IV for n = 2. Lemma 5.1: Let x be a menu of contracts. If x satisfies incentive compatibility, then the standards will be non-increasing in type. Proof: Without loss of generality, suppose that w, < Wj. By (IC), we have Jt(w„w,) S nfwj.wj (1) and (2) Jt(w,.Wj) S JIIWLWJ) 106 Subtracting the RHS of (2) from the LHS of (1) and the LHS of (2) from the RHS of (1), we obtain (3) *(w„wj -rt(w w,£ n(w wJ lt Jf 7r(w Wj). J( Substituting for n and cancelling common terms gives [bw s(wJ /2] - [bw^(wJ /2] > [bw s(w ) /2] - [bw,s(w)/2] 2 a 2 J J 2 J J Thus, (3) holds <=» s(wj - s(w,) 3: 0. Q.E.D. To gain some intuition for this result, consider figure 4. To capture the essence of the proof, assume that both of the incentive compatibility constraints hold with strict inequalities. Using our notation for profit curves, we getrcf.jq.wj> jr(xj,wj and jt(Xj,w,) > nfo.Wj). When these inequalities are represented graphically, we get distinct profit curves as in figure 4. It is then transparent that s(w,) > S(WJ), because of the differences in the slopes of the profit curves. The other possibilities can be represented similarly. The monotonicity in the standards will be an important characteristic as we progress In our analysis. At this point, we note that this result stems directly from the structure of the constraints. Maximization plays no role. This means that the monotonicity on the standards will be imposed on the optimal solution. Lemma 5.2: Let x be a menu of contracts. If x satisfies incentive compatibility, if the individual rationality constraint is satisfied for a firm of type w , and if n s(wj > 0, then x is feasible. Proof: Since s(wj > 0, the profit curve n(x„, • ) is downward sloping, so Ttlx^w,^) > rclx^wj > 0. By incentive compatibility, we have Tcfx^WnJ £ 7t(x,w.) > 7t(x ,wJ > 0. B n 1 n This establishes that n(x.,w .) > 0. A series of similar steps establishes that Jt^.w,^ a 1 ll 1 107 Figure 4: P R O O F O F L E M M A 5.1 108 0 V i= 1 n-1. Q.E.D. Another way to show this simple result Is to examine figure 5. Once wefixpoint A, then Incentive compatibility tells us that point B must not be below C. Since s(wj > 0, C is above A. This ensures that ntx^.w,,.,) > 0. We are now able to replace all of the *n' individual rationality constraints with a single constraint, refx^wj > 0. Once again, the result follows immediately from the structure of the constraints. It has nothing to do with maximization. Further simplifications follow from the structure of the constraints once we introduce adjacent incentive compatibility (MC). Adjacent Incentive compatibility for a type w, firm involves its adjacent types w w and w,^, and could be written as K(vr w) lt £ nfwj.w,) for all I and j = i-1,1+1. Lemma 5.3: Let x be a menu of contracts. If x satisfies adjacent incentive compatibility (AIC), then it also will satisfy incentive compatibility (IC). Proof: Since the profit curves in our model satisfy the Matthews and Moore singlecrossing property, we refer to their proof of this lemma on pages 444-445 in Matthews and Moore (1987). Q.E.D. Because the profit curves are linear, a particular profit curve crosses any other profit curve only once. Because of this property, satisfying adjacent incentive compatibility guarantees that incentive compatibility is satisfied. Once again maximization plays no part, but we are still able to reduce the number of constraints on our maximization problem. compatibility constraints. Now we need examine only the adjacent incentive If a set of contracts satisfies this restricted set of constraints, then it will satisfy the global set of incentive constraints. 109 Figure 5: PROOF OF LEMMA 5.2 110 There is yet another simplification which follows from the structure of the constraints and maximization. This involves what we shall call binding individual rationality for the highest type firm, which we denote (BERJ. This result follows from the fact that the profit curves are downward sloping so that they satisfy (DE). Lemma 5.4: Let x be a menu of contracts. If x solves the unrelaxed problem In (5.6), then the individual rationality constraint Is binding for w = w. n Proof: Suppose that x solves (5.6) so that W(x) > W(x) holds for all feasible x and x, where W( • ) is the public utility regulator's objective function. Assume that, contrary to the lemma,rc&Cn.wJ> 0. Feasibility of x Define x to be a distorted menu of contracts such that only the fixed charge component of each x, e x is altered. Specifically, define x as follows. (1) x s {x, I x, s (p(wJ,T(w)-e,s(w)); i = 1 1 l n) It should be understood that the undistorted menu of contracts, x, is defined similarly. (2) x s {x, | x, s (p(wJ,T(w),s(w)); 1 = 1 1 ) n} This Information implies that (3) Ttfjq.wJ = jtecj.wj - e for all i = 1 n and n(xj,wj = n(Xj,wJ - e for all (i,j) = 1 In addition, e is chosen sufficiently small such that nfo.wj > nfj^.wj > 0. 111 n. Therefore, the menu of contracts x continues to satisfy the Individual rationality constraints for all w e {wj wj. The menu of contracts x will be feasible if it maintains the Incentive compatibility constraints. We know that x is feasible since it solves (5.6). This implies 7t(x,,wJ S 7t(Xj,wJ (4) for all x,^c, e x and w, e [w^ wj Assume that x does not satisfy the incentive compatibility constraint for at least one w,. This implies (5) A(x,,W|) < Jt(Xj,wJ for some jq.x, e x and w, e ^.....wl Given (3) above, we may write (5) as 7c(x,,wJ - e < jcfxj.wj - e for some x,^c, e x and w, e (w, wj Thus, rc(x,,wj < 7c(Xj,wJ for some x,,^ e x and w, e {yr^,...,^. But this contradicts (4). Therefore, x must be feasible since this contradiction will hold for any x, E x that causes a violation of the Incentive compatibility constraints. Optimality Now consider the expression for consumer surplus: CS(x,,wJ = Y(q(wj) - p(wj-q(wj TfwJ, where Y(q(wj) is total willingness-to-pay. This suggests that CSfj^.w,) = Y(q(wj) p(wjq(wj - (T(W|)-e), which in turn implies that CS(x,,wJ = CS(x,,wJ + e for all x, e x, x, e x, and w, e {w t wj. Using this simplification and using (3), it is easy to show 112 that W(x) - W(x) = S 0, {-(TfwJ-e) + pft^wj-e) + T(wJ - pjcft.w,)} = £ 0, (e(l-p)} > 0. Thus, x could not have been a solution to (5.6). As long as nfx^wj > 0, we can always find a feasible x such that W(x) - W(x) > 0. Q.E.D. Figure 6 gives a graphical description of the proof. From the expression for the profit curve, it should be obvious that [37c(x°,w)/3T°] = 1. This means that a small increase in the fixed charge will shift the profit curve up in a parallel fashion. Now suppose that, contrary to the lemma, the individual rationality constraint for a type w n firm is not binding, yet x solves the unrelaxed problem in (5.6). The hypothesized set of circumstances is depicted in figure 6. When we alter each fixed charge by e, we can define a new set of distorted contracts, x. defined by x are presented In the figure. Obviously The profit curves the pattern of incentive compatibility constraints is preserved, since all the profit curves shift in a parallel fashion and e is the same for all types. Since e can be arbitrarily small, we can choose it to ensure that Jtfcicn.wj 2 0. The figure shows a situation in which we have assumed a strict inequality. By altering the fixed charges in this manner, we are making a lump sum transfer from producers to consumers. Because this transfer is non-dlstortionary, consumer surplus increases by the same amount. Since the weight, p, on producer surplus in the regulator's objective function is less than one, such a transfer unambiguously increases the welfare of the regulator. 113 Unlike our previous three results, which followed directly from the structure of the problem, this result depends upon rnaxlmlzatlon, since we are using the regulator's preference for consumer surplus over producer surplus. With this result, we can replace the weak inequality ln the individual rationality constraint for a firm of type w n with an equality. Using a similar kind of reasoning, we will show that the upward adjacent incentive compatibility constraints can be reduced to binding upward adjacent incentive compatibility constraints (BUAIC). This means that only the incentive compatibility constraint between adjacent types In an upward direction, w, and w 1+1 {w, for all w, e w^i), is binding. This may be represented formally as follows. Lemma 5.5: Let x be a menu of contracts. If x solves the unrelaxed problem In (5.6), then all of the upward adjacent Incentive compatibility constraints will be binding (BUAIC). Proof: Suppose that x solves (5.6). This would imply that W(x) £ W(x) for all x, and x which are feasible. Recall that W( • ) represents the regulator's objective function. Assume that, contrary to the lemma, there exists a w, e {Wj n(x, ,wj where x , ^ +l e x = (x, | x, 3 (pfwJ.TfwJ.sfwj); i = 1 wj such that 7c(x,,w,) > n-1}. Our claim implies that there does not exist a feasible x such that W(x) > W(x), where feasibility implies the joint satisfaction of the individual rationality constraints and compatibility constraints. Definition of x. and the feasibility of x. i) For all m > I, let x,,, 3 x,,,. ii) For all k <, I, let x 3 fofwJ.TfwJ-e.sfwj). k 114 the incentive Figure 6: PROOF OF LEMMA 5.4 T( ) V •»U ,w^ w I- 115 This Implies that x = {x ,xj = This menu of contracts, x. Is distorted from x k In only one component - the fixed charge - but only for those contracts offered to types below w . 1+1 Recall that w, has the characteristic thatrcfx^w,)> n(x, ,wj; the +1 upward adjacent incentive compatibility constraint holds with a strict Inequality, so we must choose e to ensure that TCCX^W,) > J C & . W J S 7r(x, ,wJ. +l Since x is feasible, it satisfies (IR). This implies that x satisfies (IR), since the contract x„ is never distorted and the profit function satisfies (DE). Thus, we can always choose e sufficiently small to ensure that the Individual rationality constraints are not violated, x will be feasible if the incentive compatibility constraints have not been violated. In this regard, the feasibility of x establishes the feasibility of x. for all s > I, since the contracts are defined to be equivalent. Note that, for all k £ i, (1) nfXk.Wj = retx^wj - E. N O W assume that there exists a k < i such that (2) Jt&.wJ < n(x ,wj. After substituting (1) into (2), we get k+1 nfXk.wJ - E < Tttx^.wJ - £. (3) This implies that jcfx^wj < Tcfx^.wJ, a contradiction of the feasibility of x. Therefore, for any k £ 1, the incentive compatibility constraints will be satisfied, and the distorted contract, x, is feasible. Optimalltv Now examine the expression for consumer surplus in W( • ). CS(x,,w,) = rtqlwj) - pfwj-qfwj - TfwJ, for all i E {1 n} where rtqfwj) is total willingness-to-pay. For all m > i, CSfJL.wJ = CSfx^wJ. For all k £ I, CS&.wJ = Yfawj) - p(wj.q(wj - (TfwJE). This specification implies that (4) CSfx^wJ = CSfx^wJ + E. Using (3) and (4), 116 evaluate the difference between W(x) and W(x). n 1 1 W(x) - W(x) = [Z 9(-(T(w )-e) + P&fx^wJ-e)) + Z 9j-(Tfwj) + p(^(x ,w )))]-[Ze (k 1[ m m k*l m-i-fl k k»l n (Ttwj) + p(A(x .wj)) + Z 9 (-(Tfwj) + pGfcu.wj))]. k Thus, W(x) - W(x) = Z eJeU-P)) > 0. m Therefore, x could not have been a solution to the maximization problem in (5.6). As long as there exists an i such that nfowj > Jt(x ,w ) we can always find an e sufficiently small so that x is feasible and W(x) > M W(x). t > Thus, all the upward adjacent Incentive compatibility constraints must be binding at the solution to (5.6). Q.E.D. Figure 7 below gives a sketch of the proof. For some arbitrary type, w„ we assume that the upward adjacent incentive compatibility constraint is not binding, Such a situation Is captured in the figure by the points D and E. Point D is 7t(x,,wJ while E gives 7c(x, ,wJ. +1 Our distorted contract shifts the profit curves down in a parallel fashion. But notice that only those types below w 1+l receive a distorted contract. All of these types are affected so as to ensure that the Incentive compatibility constraints remain satisfied. That Is, the relationship between A > B > C is maintained after the distortion by J > K > C. However, in the figure, we show that the choice of e has caused the upward adjacent incentive compatibility constraint for w, to bind; i.e., TC(X, , +1 • ) and JI(X,, • ) intersect at E so that JHX^WJ = jt(x, ,wj. Since no constraints have +l been violated, the distorted contract is feasible and the reduction in producer surplus creates an increase In consumer surplus which is preferred by the regulator when p < 1. 117 Figure 7: PROOF OF L E M M A 118 5.5 As ln lemma 5.4, this result uses the fact that the regulator is maximizing. Lemma 5.5 allows us to replace half our adjacent constraints, the upward adjacent incentive compatibility constraints, with BUAIC. Lemma 5.6: Let x be a menu of contracts. If x satisfies BUAIC, and s(wj S ... £ sfwj, then x satisfies (IC). Proof: See appendix HI, section I. We are now i n a rather enviable position in that the five lemmas we have established allow us to reduce the complexity of (5.6), and solve a simpler problem. Basically, there are at least three ways In which we could proceed. The first method of solution would be to maximize the objective function in (5.6) subject to BUAIC, BIR,, and monotonicity on the standards. This method Is feasible because of lemma 5.6. This lemma allows us to replace the full set of incentive constraints with BUAIC and monotonicity on the standard. Lemma 5.4 established that we could replace the full set of individual rationality constraints with a single constraint, namely BIR^ Clearly this is a much more manageable problem than is (5.6). Figure 8 shows why the result will work. BIR,, establishes an "endpoint". The monotonicity of the standards ensure nice, increasing-slope profit curves (This avoids any non-adjacent incentive compatibility problems.), and the BUAIC ensures that the ordering of the types is accomplished in an efficient manner - keeping rents to a minimum. A second way to proceed would be to solve the rnaximizatlon problem subject to BUAIC, BIR,,, and downward adjacent incentive compatibility (DAIC). This problem is equivalent to (5.6) because of lemma 5.3. (AIC) and (IC) is established. It is in this lemma that the equivalence of Since BUAIC and DAIC together imply AIC, the equivalence between the approach we have suggested and the problem described in (5.6) is obvious. 119 Figure 8: INCREASING RENTS AND BUAIC 120 The final way to proceed would be to solve the problem ln a series of steps. First, maximize the objective function in (5.6) for fixed standards, subject to BUAIC and BIR^. Then using the optimal values for the variable and fixed charges, derive the value function. Then, with this new objective function maximize by choosing standards subject to the monotonicity restriction on the standards. The solution to this problem also will be a solution to the problem' in (5.6). We have chosen the second approach. Under these circumstances, the regulator wants to solve the following maximization problem. n (5.7) Max W = X {e,{Y(q(w,)) - pfwj-qfwj - TfwJ + q(w,),T(wJ, s(wj pJt(wJ -.1 DfafwJ.stwJ.w} -((lAfqlwJ.sfwJ.wJjJ subject to: re(w„wj = tfw^.wj V w, e {w, niw^vr) 2 Ji(w w,) Mf w.J_ n V w, e {w a wj 7c(w ,wJ = 0 n This gives a total of 2n-l constraints, and may be solved using standard Lagrangian techniques. The first-order conditions for this problem, and the derivations for the optimal expressions of the choice variables are contained in appendix in. To solve the problem defined in (5.7), we use the fact that there is BUAIC and BIR^ These two pieces of information are sufficient to allow us to solve the model for the variable charge, output and the fixed charges. Some complications are encountered when we solve for the optimal design standards for it is here that we must worry about the downward adjacent incentive compatibility constraints (DAIC). For the moment, however, we proceed with our analysis. The variable charge turns out to be of exactly the same form as we encountered in chapter III. 121 p*(wj = c + as*(wj + <(>as*(wj + hxw, (5.8) V i = l....n As a reminder, we note that the.variable charge is chosen to cover the sum of marginal operation costs, c; marginal abatement costs, (l+<t>)as(w,); and, marginal damages, hxw,. It is this latter term which plays such an important role In the bunching properties of the model. In order to find expressions for the fixed charges, we use BUAIC and BLR,,. These results allow us to work recursively to obtain expressions for T*(wJ. From BLR,,, we know that n*(w ,wj = 0. Solving for T*(wJ gives: n (5.9) T*(wJ = [bw s*(wJ /2] + K - VP*(wJ 2 n where: VP*(wJ = q*(wj{<j>as*(wj + hxwj > 0 To determine T*(wJ, we use BUAIC, to get (5.10) ~~ T*(w^) = [bw . s*(w . ) /2] + K - V P ' K J + [bs*(wJV2](w -w .J 2 n I n l n n where: VP*(w.,) = q^w^jfoasMw^) + hxwnJ > 0 n Using this recursive technique for several iterations establishes the general expression for the fixed charge. n-l (5.11) T*(wJ = [bw,s*(wJ /2] + K - VP*(wJ + Z 2 [bs*(w,J /2](w -w, . ) 2 l+k k-l where: VP*(w J = q*(wj{<l)as*(w J + hxw .J > 0 n n 122 n +k 1 In general, the fixed charge is chosen to cover the difference between the firm's fixed costs, [bws*(wJ /2] + K, and variable profits, VP*(w.,) = q*(w.j{<()as*(w.) + hxw . 2 n n n 1 n J, as well as to provide the firm with a lump sum transfer which ensures the truthful revelation of type. This latter observation may be seen clearly by substituting the optimal expressions for the variable charge and the fixed charge into the profit function. This new value function specifies the rents the firm will earn at the solution to (5.7). n-1 n*{w w) = Z [bs»(w,J /2](w -w . ) (5.12) a u 1+k l+k 1 V i= 1 n This is a pure information rent since it expresses exactly the value of the firm's privately held information to the regulator. When there are 'n' possible cost types for the firm, the nature of the information externality can be seen much more clearly than in the case where there are only two possible cost types. Notice that the rents which accrue to a type w, firm are dependent upon the standards which are offered to all types higher than w,. The reason for this is quite simple. Since the regulators cannot observe type, they must get the firm to self-select, reveal Its type. In most cases, lower cost types receive higher rents because self-selection requires a w ^ firm to receive at least as much rent n as a type w. firm. Once rent is paid to a type w. firm, incentive compatibility then n a n t requires at least this amount of rent to accrue to a type w firm, and so on down the 2 line. Thus, the amount of rent accniing to the top of the distribution influences what types lower down in the distribution will receive. This is why we claim there is an information externality. This information externality is captured graphically in figure 8 above. The idea behind bunching different types is exactly related to this information 123 externality. If conditions are right for bunching, and these must be specified, then the regulator finds it optimal to offer different types the same contract. Bunching higher types together means that the separation of lower cost types can be achieved more "cheaply". The Influence of the information externality is reduced because there are fewer iterative steps required to get to the lower cost types at the bottom end of the distribution. Note again, however, that the optimality of bunching happens under very particular circumstances. Yet, from our discussion in chapters HI and IV, we know that certain parameters play an important role. These include the probability of occurrence of any given type, as well as the distribution of types; i.e., the magnitude of the difference between any two cost types. A characteristic typical of many screening models is that separation of types requires different profit levels to accrue to the firm as its type changes. This type of monotonicity in the agent's objective function Is usually ensured by assignment monotonicity; i.e., the fact that monotonicity in one endogenous variable ensures monotonicity in another. However, monotonicity of this sort is usually imposed exogenously through the single-crossing property. Our results show that we can satisfy the Matthews and Moore (1987) single-crossing property and yet violate assignment monotonicity. Bunching implies that there are identical components for any two contracts. We have shown that in the case where there are only two cost types, bunching never occurs under conditions of complete cooperation. Our next result shows that this separation carries over to a model in which we assume that there are n distinct cost types, w, e wj. Whereas it is customary to have only a separating solution when there are just two types, having this characteristic carry over to the n-type case is somewhat novel. Proposition 5.1: Let x, be an optimal contract for a firm of type w,. At the 124 solution to (5.7), x, * X , for any i,j = 1 n. Proof: From equation (5.8), we know that p*(wj = c + as*(wj + ^as'fwj + hxw, = l,...n. Lemma 5.1 establishes that s*[w) £ s*(w ) V i = 1 1+1 equal x,, then s*(wj = s*(wj) is a necessary condition. n-1. V 1 If x, is going to Therefore, assume that this holds, and compare p*(wj and p*(wj). The variable charges will be equal if c + as*(w,) + <t>as*(w,) + hxw, = c + as*(wj) + <|>as*(Wj) + hxw,. contradiction since w, * w,. Therefore, we may This implies that hxw, = hxw,, a conclude that p*(wj = p*(w,) and s*(wj = s*(w,) are mutually inconsistent. As a result, a bunching (pooling) equilibrium is not feasible. Q.E.D. This proposition establishes that regulators always find it optimal to completely separate out each type just as they did in the full information solution. This result holds regardless of the distribution of types, and regardless of how are distributed. In most examples of screening the probabilities models, agents of different characteristics are offered equivalent contracts in order to reduce the overall costs of attaining incentive compatibility so that lower types may be exploited at lower costs. In our model, the same objective is achieved by offering equivalent standards, rather than the entire contract, since optimal profits are a function of the standard only, not all three choice variables (see equation 5.12). Thus, in this sense our separating solution can contain an implicit form of bunching, since separation does not preclude the regulators from offering equivalent standards. Thus,our results can be divided into two different cases, a solution in which the standards are strictly monotone decreasing and a solution in which the standards are weakly monotone decreasing in type. We shall examine only the case ln which the firm is offered distinct standards for each type. Proposition 5.2: If the standards are strictly monotone decreasing at the solution to (5.7). then the DAIC constraints will be satisfied with a strict 125 Inequality. Proof: Suppose that contrary to the proposition, we have BUAIC and BDAIC for some L This implies that jr*(w„wj = jt*(w ,wJ and n*(w wj = ^(w^.w^. for some i. H1 l( Substituting for optimal variable and fixed charges and optimal output gives n-H [bs*(w ) /2](w -w ) + Z 2 1+1 ltl 1 [bs*(w J /2](w a 1+1 1+1+k -w,J = n-i+1 [bs*(w . ) /2](w . -wj + Z [bs*(w,. J /2](w . -w . J. 2 l 1 l 2 l 1 1 1+k 1 2 kpl Cancelling common terms gives Ibs*(w,)/21[w-w.] = rbs*(w ) /2][w,-w,l. 2 2 1 l l 1 l 1 This In turn implies that s*(w,.i) = s*(w,), a contradiction. Since we must have BUAIC, this contradiction establishes that DAIC must be satisfied with a strict Inequality when the standard is strictly monotone decreasing. Q.E.D. Figure 9 presents the Intuition of the proof graphically. When we have BUAIC then this implies A > B. Since the standards are assumed to be strictly monotone decreasing in type, we have s(w ) > s(w,), which implies that B > C and D > E. From H this, corollary 1 follows immediately without proof, and is depicted graphically in figure 10. Corollary 1: If the DAIC constraint between any two types, i and 1+1, at the solution to (5.7) holds with equality, then s*(wj = s*(w ) = s. Itl Proposition 5.2 provides us with the information necessary to solve for the optimal design standards associated with the strictly monotonic solution. It Is easy to see that 126 the form of these standards is equivalent to the expressions we obtained i n chapter HI. This suggests that the solution to the cooperative problem is completely general. (5.13) s*(wj eihTw.-ad+^tr^rtfe+hTwj}] = V i = 1 n i-i 9i{(l-Ht>)bw, - rj(l-Ht>) a } + ( l - p j b f w . - w j 2 Q 2 2 s The solution to our problem, when both the upward and downward adjacent incentive compatibility constraints bind, is more complicated and will not be discussed. Most of the essential insights of the model can be captured without actually solving for the parameter restrictions which generate constant standards. Corollary 1 and figure 10 show what the solution will look like when two standards are equivalent. Offering the same standard to any two or more types reduces the "cost" of separation because the profit curve will have a constant slope over the interval of bunched types. Compare figures 9 and 10. O f course, the profit curve associated with the constant standard may be steeper or more flat than the profit curves associated with the strictly monotonic solution. Under certain circumstances, the slope of the profit curve i n the weakly monotone solution will be such that the "cost" of getting to lower cost types is reduced. There are two main characteristics associated with the solution which may have a constant standard over a pair of types. separation, proposition 5.1. First, it is always characterized by complete When the firm chooses a contract, the regulator always is able to identify the firm's type. A n d related to this fact is the second characteristic which we present as a proposition. Proposition 5.3: If the standards offered types i and i+1 are equivalent, so that s*(wj = s*(w ) = s, then I+1 i) the variable charge is strictly increasing in type, p(wj < p ( w ) ; 1+1 127 Figure 9: PROOF OF PROPOSITION 5.2 128 Figure 10: PROOF OF COROLLARY 1 129 ii) output is strictly decreasing in type, q(w,) > q(w ); and, 1+l iii) the fixed charge may be increasing, constant or decreasing in type with T*(wj | T*(w ) as ( w w " w ) M w J a 1+1 - w, # ^(c-nad^sl-r^as r^t Proof: See appendix m, section I. This result implies that virtually any pattern of monotonicity in the choice variables is consistent with incentive compatibility. There could be a sequence of constant standards, at any point in the distribution of types, and then a sequence of strictly Increasing standards, and then perhaps another sequence of constant standards. While this is happening, the variable charge is chosen in such a manner as to ensure separation. Note that the choice variables always are monotonic in type. That is, once we establish which direction they are moving as type is changed, the only other choice available is to have the variable constant in type. Therefore, our results differ from those of Matthews and Moore (1987) who show that non-monotonicities can arise" in multi-dimensional screening models. Given the solution to the limited information problem, it is easy to show the distortion away from the full information solution which occurs. As In the case of only two types, the design standards and variable charges In the limited information solution are equivalent to their full information counterparts only if the firm turns out to be of type w . For all other types, the standards offered in the limited information, t cooperative solution are less than their full information counterparts. This means also that the variable charges faced by consumers in the presence of limited information also will be lower than In the full information scenario. The reason for this is obvious. Variable charges are lower because marginal abatement costs are lower, and the marginal abatement costs are lower because the environmental design standard is less stringent. 130 5.4 The Umited-lnformatlon Solution: Non-cooperative Assumptions This section extends the results of chapter IV to the case of n possible cost types (The actual maximization problem Is given in appendix IH. section HI.). The structure of the game is the same as in chapter IV. This particular structure was chosen because we have endowed the environmental regulator with enough autonomy to pick the regulatory game it wants. Acting as a leader in the menu game allowed the environmental regulator to achieve its largest strategic advantage, compelling the public utility regulator to extract the firm's privately-held information. The environmental regulator was able to accomplish this while insulating itself from the costs of information revelation. Two factors were of major importance in arriving at this game structure. First, we must assume a high degree of commitment on the part of the environmental regulator. When the public utility regulator and the firm choose their strategies, they both assume that the environmental regulator will not, or cannot, alter its strategy. Of course, this Is somewhat at odds with our assumption regarding the environmental regulator's autonomy. Therefore, we must assume that once the rules of the game are set, they remain unaltered throughout the regulatory game we have described. The second factor of significance Is the type of environmental policy we are imposing. If the game structure we have described is to be optimal, then the environmental regulator must offer a complete contingency of standards. This is why we took special care in noting that a complete menu of standards, one for each possible type, was offered. Of course, having a complete contingency plan when there are n possible cost types, where n is large, would create severe administrative problems. When the regulatory game is structured with the environmental regulator acting as the Stackelberg leader, the reaction functions for the public utility regulator are 131 determined from the following maximization problem. n (5.14) Max W = q(wJ,T(wJ X [e,{Y(q(wj) p(wj-q(wj - T(wJ + PJI(WJ}] 1*1 subject to: 7c(w„wJ = n(w ,wj 1+1 jr(w„wj £ Jrtw^.wJ V wi e l V wl e {w jt(w .wj = n [w 2 w.) n x wj 0 First, note that we have employed the results of the cooperative game discussed earlier. Therefore, unlike chapter IV, we won't have to verify that the solution to the relaxed problem in (5.14) also solves an unrelaxed problem. Just as In chapter IV, we shall break-up our analysis into two sections. The first section assumes that the solution results in complete separation. impltesjthat s(Wj) > ... > s(wj. From our earlier results, we know that this Whereas the analysis is a little more complex, the results are exactly the same as in chapter IV for the separating solution. Therefore, we will summarize the propositions from chapter IV which apply to the case of n types and then focus on the bunching properties. Proposition 5.5: i) The standards are larger under extreme rivalry than under mild rivalry. ii) Only a firm of the highest type is indifferent to institutional structure. All other types prefer extreme rivalry. ill) The public utility regulator prefers mild rivalry. iv) The environmental regulator also prefers extreme rivalry to cooperation, and the public utility regulator prefers mild rivalry to cooperation. Proof: See appendix HI, section n. 132 The reaction functions to be used by the environmental regulator are presented below. The expressions in (5.14) are functions of the given design standard and hold for all w, e {w ...,wj. lt (5.15a) q(s(w,)) = r - r\(c + as(wj) (5.15b) p(§(wj) = c + as(wj o n-1 TtwJ = [bw.sfwJV^ + K + E [bs(w JV2](w -w . ) (5.15c) l 1+k ltk 1 In order to discuss the optimal design standard, we examine the case in which the regulators act as extreme rivals and then the case in which the regulators act as mild rivals. Extreme Rivalry As a natural contrast to both full cooperation and mild rivalry, extreme rivalry allows the environmental regulator to pursue a completely Independent policy. The environmental regulator uses its leader status to its fullest extent, and in so doing solves the following minimization problem. n (5.16) Min s(w,) subject to: Z ER = Xe, 1 {D{q(s(w )),s(w ),wJ + ())A{q(s(w )).s(w ).w }} 1 1 1 i-i s(w,) £ s(w, ) V w, e {wj +1 where: q(s(w,)) = r - r\(c +as(wj). 0 133 wj 1 1 The solution to this problem depends upon the monotonicity properties of the optimal design standard. These properties are determined by the parameters of the model. We shall specify a condition under which any two types may be bunched together. For the moment, assume that the parameters of the model are such that a fully separating equilibrium is optimal. This implies that the Lagrange multipliers for the constraints listed in (5.16) are all zero. This simplification allows us to solve for the optimal standards directly from the first-order conditions. Then, using (5.17a), we could solve for explicit expressions or the variable charge and the fixed charge by substituting for s^wj in (5.17b) and (5.17c), respectively. It is immediate from (5.17) that once we know the monotonicity of one variable, we know immediately the monotonicity of the other two variables. This is a direct extension of the traditional monotonicity results to the multi-dimensional case. However, because the regulatory procedure Is broken up Into two parts, this isn't too surprising. Just as in chapter IV, the optimal solution is independent of the regulators' prior information on the firm's type, 0,. This reflects the fact that the environmental regulator is able to avoid the costs of information revelation, although it is responsible for setting the level of these cost. It is now important to specify the conditions under which bunching might occur. When there are n possible cost types, we will examine the potential for bunching between any two adjacent types, w, and w . 1+l Proposition 5.6: 1) A necessary and sufficient condition for bunching to occur between two types w, and w 1+1> rivalry problem is to have 134 at the solution to the extreme (iyr,) £ c + (hxa/<|)b)(l+ar) 11) When i) occurs, the equilibrium design standard is specified as e,{ hxw.d+arj - <i>a(r - r^)} + e {htw (i+arj - <t>a(r - r^)} a e r s 0 w w 0 e.fofbw, - 2r\a )} + e {<t>(bw - 21^)} = 2 i+1 1+1 Proof: See appendix HI, section n. There Is an even stronger result here, which we state without proof, that is consistent with our arguments from chapter IV concerning bunching. It Is easy to show that the condition specified in' proposition 5.6 as a necessary and sufficient condition for bunching between any two types Is also the condition for bunching all types. This means that when this condition is satisfied, the regulator will simply offer a single standard, regardless of the number of types. This relates to the separation of regulatory tasks. The environmental regulator will offer distinct standards until the parameters of the model are such that it would be optimal for the regulator to order the standards according to s(wj < ... < s(wj. Since these standards cannot guarantee separation in the public utility regulator's stage of the game, then environmental regulator finds it optimal to offer a single standard. Given a single standard, bunching ensues as specified by the equations in (5.17b) and (5.17c). (5.17a) s E R ( W | ) (5.17b) = htwA+aTJ - acMro-riC} <t>(bw, - 2a rj V i=l n 2 p (wJ = c + as (wJ ER V i = l,..,n ER n-l (5.17c) T^wJ = [bw s (wJ /2] + K + Z [bs (w J /2](w -w . ) ER 2 ER 1 2 1 135 1+k 1+k 1 V i= 1 n Mild Rivalry Mild rivalry presents a similar possibility for bunching. Obviously, the conditions under which bunching between any two adjacent types will occur are quite different when the regulators act as mild rivals. Just as in the cooperative model, the regulator's prior information plays an important role in the bunching decision, the environmental regulator must offer equivalent standards before bunching will ensue simply because it moves first in the regulatory game. To determine the optimal standard, the environmental regulator solves n (5.18) Max W"* = S [eiu^tqtstwJJ.sfwJ.TlsfwJ.wJ - D(q(s(wJ),s(wJ,wJ - <|)A(q(s(w)),s(w),w)}] I 1 1 subject to: s(wj 2 s(w ) w, V 1+1 e {w! w .J n where: U MR = Value function of the public utility regulator. When we assume a separating solution, this maximization problem generates an equilibrium standard which may be used to determine equilibrium values for the variable charges and the fixed charges. (5.19a) s (wJ M B = 9[hxw,-a(l+<|)){r-rc}] l o V 1 ^(l-HtObw,- r,(l+2<|>)a} + (i-p)b(w wj g e . 2 r (5.19b) p ^ w j = c + as^wj 136 V 1 = l,...n 1= 1 n n-l (5.19c) 5.4.1 THwJ = [bw s (wJ /2] + K + Z [bs (w )l /2](w -w . ) MR ; 2 MR 2 l l+k 1+k 1+k 1 Focus on Mild Rivalry When the regulators act as mild rivals, we maintain the non-cooperative structure of the game, but the environmental regulator's strategic advantage is less acute. As a result, equation (5.19) shows us that the characteristics of the solution are distinctly different. The fact that the regulators' prior information again plays an important role in the optimal solution suggests that bunching may be optimal, depending upon the parameters of the model. Proposition 5.7: 1) A necessary and sufficient condition for bunching to occur between two types w, and w _ at the solution to -the mild 1+1 rivalry problem is to have [9 /9 ] ^ C***, where 1+1 C*** = [AA* - B A 1+2 l+2 ] / [ A B - B A J , and l+1 l 1+1 k = i, 1+1, 1+2 A„ = hTw - ad+Mr.-riC) k k-l B,, = (l+<t>)bw - r\(l+2<l>)a + (l-P)b(w -w . )Z8 2 k k k 1 j-i J k = 1, 1+1, 1+2 U) When 1) occurs, the equilibrium design standard is specified as QA + KiKi + 9 A 1+ 0,Bi + 9i+iB I+l Proof: See appendix in, section n. 137 + 9 1+2 +3 B lta 5.5. Conclusions The most interesting results of this chapter deal with the bunching properties of the model. We have shown several different kinds of models in which bunching has been ruled out even when the number of possible types is quite large. In conjunction with this, we have shown some different monotonicity results on the choice variables involved in the regulatory process. The intuition behind these results enhances our understanding of screening models which involve more than two choice variables. The separating solution and the monotonicity properties are intimately related. Our result also seems to depend upon the quasi-llnearity of the firm's objective function, the point at which incentive compatibility comes into play. When a menu of contracts is feasible, we decreasing in type. know that the standards must be monotonlcally non- In all the models involving only two choice variables, the monotonicity on one choice variable compelled the other choice variable to display monotonicity as well. Now, if we were to perform a monotonic transform on the objective function involved in the incentive compatibility problem, in our case the firm's profit function, we would still preserve the pattern of Incentive constraints. In our problem, such a monotonic transform would be to multiply all the variables by type. Profit would become: Il/wJrt(w„wJ = [l/w,]{p(wj-q(wj + T(wJ - cq(wj - aq(wjs(wj - K} - [bs(w,)/2]. 2 This may be written as (1/wJA - [bs(wJ /2J. Just as In our original problem, we would 2 expect incentive compatibility to imply some monotonicity on the "variable" A, not just a single choice variable. In the two choice variable problem, this was a natural result, especially given the single-crossing property. However, when there are more choice variables involved in the problem, the monotonicity on A, which is required by incentive compatibility, can be achieved in many different ways involving p, T, and s. result, however, is that separation always occurs. 138 The C H A P T E R VI CONCLUSIONS 6.1 Summary of Contributions The most obvious objective of this thesis was to examine a multiple principal, single agent asymmetric information model In which the asymmetry of information was modelled as part of fixed costs. In our model, the one-dimensional parameter summarizing the firm's private information entered fixed abatement costs and the firm's emission function. This relatively small change in the model generated significant changes in the solution to the problem. Whereas this was to be expected, the differences themselves were quite interesting. For example, we found that the regulators would choose the fixed charge, when they were cooperating fully, to cover only the difference between fixed costs and variable profits; Le.. the firm's fixed costs are never completely covered by the fixed charge, contrary to most results when there is full-information. The cooperative scenario was also characterized by only a separating solution, regardless of the number of types. The separating role was played by marginal damages, which are strictly increasing in type. When there Is an implicit tax contained in the variable charge, in this case in the form of marginal damages, then it is never optimal to bunch. We found also that the normal assignment monotonicity rules were violated for our model. Strict monotonicity in one variable did not impose similar structure on the other variables as it normally would with assignment monotonicity. As a contrast to the cooperative scenario, in which the above results hold, we presented two plausible regulatory structures which were Intended to flank cooperation on both sides, namely mild and extreme rivalry. 139 The non-cooperative game, with Its two-stage nature presented some more traditional results, but with some Interesting twists. Both extreme rivalry and mild rivalry generated an equilibrium fixed charge which always completely covered fixed costs. As in the cooperative model, low-cost firms also have an information rent term as part of their fixed charge. The existence of this rent term in the fixed charge is different from any model which has the information asymmetry as part of marginal costs, or some other component which varies with output, rather than as part of fixed costs. We showed also that the extreme rivalry game presents a solution which is completely independent of the regulator's prior information. This was Interesting because the solution arose out of a limited information game. The extreme rivalry solution and the mild rivalry solution were similar in that both exhibited bunching. Necessary and sufficient conditions for bunching were stated for each model. The extreme rivalry and mild rivalry games form endpoints around the cooperative game in terms of preferences by the economic actors. The environmental regulator prefers extreme rivalry to cooperation, and would probably veto any attempts at regulatory coordination. This suggests that some third party would have to intervene in order to promote cooperation in some form, if this were deemed desirable. The public utility regulator prefers mild rivalry because this is the scenario under which consumer interests are most heavily favoured. The environmental regulator is able to use its strategic advantage in the promotion of the public utility regulator's welfare, something it was unable to do in the pure cooperative game. When the regulators act as extreme rivals, the design standards which influence the level of pollution are the most stringent. Yet oddly enough, we show that the firm also prefers these stringent design standards. This Is because fixed abatement costs, 140 the source of the firm's Information rents are strictly Increasing In the design standard. To the extent that we believe that asymmetric Information Is a component of fixed abatement costs, this result suggests that regulators should be wary of promoting more stringent standards as they may simply lead to an increase In rents to the firm, not only because the fixed charge must also provide an Information rent which does not arise In the fuU-lnformation solution. This rent then. Is simply a consumer welfarereducing transfer to the firm - something regulator's may deem unnecessary. As with extreme rivarly, mild rivalry exhibited interesting bunching charateristics. Just as in Weymark (1986), the pattern of bunching can be quite varied. There can be a sequence of bunched types, a sequence of separated types and then another sequence of bunched types. Proposition 5.11 defines the conditions under which it Is optimal to have bunching. This brief summary outlines some of the original contributions which this thesis has made to the asymmetric information literature. Many of the results are predicated on the functional forms which are chosen and the assumptions which underlie the model. This fact has suggested many directions in which improvements to the model could take place. These improvements fall under the category of directions for future research. 6.2 Directions for Future Research One of the underlying assumptions of the model was that the design standards (or how well the design specifications are met or maintained) and the level of emissions are always observable and observable at zero cost. Obviously this assumption is somewhat suspect. For example, just because the design standard Is chosen and the required abatement technology Is set in place does not mean that this technology will be maintained properly or even used properly. Likewise it is unrealistic to assume that 141 emissions would be continuously monitored. Both of these observability problems suggest that moral hazard or hidden action problems must be dealt with. The interaction of moral hazard and adverse selection under conditions of changing institutional design is not well understood, yet most problems in the regulation of externalities involve both problems. Thus incorporating additional incentive constraints into the model could prove useful. The analysis in this thesis is an example of partial equilibrium analysis, and it suffers from the fact that maximizing welfare as measured by producer and consumer surplus may actually move regulators or society to lower levels of welfare, especially when other markets are considered. In order to address these problems, direct consideration of consumers' preferences must be modelled, and another sector introduced. This will allow us to move away from producer and consumer surplus as measures of welfare. 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(1986) 'Bunching Properties of Optimal Non-linear Income Taxes", Social Choice and Welfare, Vol. 3, pp. 213-222. 147 APPENDIX I Contents Section I • Objective Function of the Full Information Solution: Cooperative Assumptions • First-order Conditions Section II • Derivation of equation (3.13) • Derivation of equation (3.15) • Derivation of equation (3.16) • Derivation of equation (3.17) Section m • Objective Function of the Limited Information Solution: Cooperative Assumptions • First-order Conditions • Proof of Proposition 3.1 • Derivation of equation (3.20) • Derivation of equation (3.21) • Proof of Proposition 3.2 • Proof of Proposition 3.4 • Proof of Proposition 3.5 148 Section I Objective Function of the Full Information Solution: Cooperative Assumptions Lagrangean associated with problem (3.12). IT =(r„/r )q(w ) - (i/2r )q(w ) 1 l 1 1 a - [(iyrj - (i/rjq(wjl.q(w ) - Ttwj + 1 (l/rjqfwjl-qlwj + TfwJ - cq(wj - aq(wjs(w,) - Ibws(w)/21 2 i s(wj) - <t>(aq(wjs(w,) 1 - p{[(r/r\) 0 K } - hxw,(q(w,) - + [bw,s(w,) /2]) + ^{[(lyrj - (l/rjqM-qfwJ + T(wJ 2 cq(wj - aqfwjsfwj - [bw s(w ) /2] - K } 2 1 1 First-order Necessary Conditions a^/aqCwJ = (l/rjqCwJ + pftr./rj - (2/rj-qlwJ - c - as(w,)} - htw, - <|)as(wj + p J f J o / r j - (2/r )-q(w ) - c - aslwj} = 0 l 1 3L /3TtwJ = - 1 + p + u, = 0 dLT/ds(xv) = p(- aq(wj - bw,s(w,)) + hxw, - <t>(aq(w,) + bw,s(w,)) + u,{- aq(w,) - n bw,s(w,)} = 0 Section II Derivation of Expression (3.13). (3.15). (3.16). and 13.17) Derivation of the variable charge: p(w,), 1 = 1.2. Since the problem Is symmetric, we will solve for p(wj. conditions, dL /dq(w) n Using the first-order = 0. we get for all 1: ( l / r j q f w j + pftryrj - (2/rj.qfarJ - c - as(w,)} - {hxw, + (frasfw,)} + u,{(T./r ) - (2/r )-q(wJ - c - as(w,)} = 0. 1 Substituting for u, = 1 - p, {(T./rj - (l/rj-qfwj - c - as(wj} - {hxw, + (fraslw,)} = 0. Using the definition of the demand function, we get p (wj = c + as (wj + (t)as (wj + hxw, n n n 149 V i=l,2. 1 Derivation of the optimal design standard: s(w,). n From the first-order conditions, dl/VdsfwJ = 0, we get for all i = 1,2: P(- aq(wj - bw,s(wj) + hxw, - <t>(aq(w,) + bw,s(w,)) + u,{- aq(wj - bw,s(w,)} = 0. Substituting for u, gives (- aqfwj - bw,s(wj) + hxw, - <|)(aq(wj + bw,s(wj) = 0 (l+<t>){- aqfwj - bw,s(wj} + hxw, = 0. Equation (3.14) follows directly. Substitute for output q (wj = r - r,{c + (l+<t»)as(w,) n n o + hxwj to get (l+4>){-a[r - rCl+<|))as(wJ - r,(c+hxwj] - bw,s(wj} + hxw, = 0 o v s(w){a(l-Ht>)T -(l+<(>)bwj = -hxw, + 2 l 2 1 s fw) " n (l+4>)ar - aT^l+Mc+hxw,) o hxw, - a(l+<|)){r - r,(c + hxw)} 0 ; — V i= 1,2. d+<|)){bw, - r (i+<t))} 2 ia Derivation of the fixed charge, T"(wJ. In order to determine the optimal expressions for the fixed charges, we use the fact that the individual rationality constraints are binding, regardless of type. P (wj-q (wj + T"(wJ - cq (wj - aq (wjs (wj - lbw,s (wJ /2] - K = 0 n n n n n n 2 Solving for T^wJ gives T'fwJ = [bw,s (wJ /2] + K - VP^wJ n 2 where 150 V 1=1,2 VP^wJ = q {<t)as (w) + hxwj > 0 f1 F, 1 Section in Limited Information: Cooperative Assumptions. Lagrangian associated with problem (3.19). = e {(r„/rjqforj L u 1 - (i/2rjq(wj 2 IffVrj - (i/rjq(wj].q(wj - Ttwj + p{[(r /rj 0 (l/rjq(wj].q(wj + TfwJ - cq(wj - aq(wjs(wj - [bw s(w ) /21 - K} - hxwiqf.wj 2 1 1 sjffj) - <|>(aq(wjs(wj + rbw (wJ /2])} + 9 {(r /rjq(wJ. - (l/2rjq(wj - [fjyrj 2 2 2 lS o (l/rjq(wj].q(wj - TfwJ + p{[(T0/r,) - (l/rjq(wj].q(wj + TfwJ - cqfwj - aqfwjsfwj - [bw s(w ) /2] - K} - hxw (q(wj - sfwj) - <|>(aq(wjs(wj + 2 2 3 2 [bw s(wJ /2])} + p {[(T /r ) - (l/l\)q(wj].q(wj + TfwJ - cqfwj - aqfwjsfwj 2 2 D 1 1 [bw s(w,) /2] - K - [tT„/rj - (l/rjq(wj]-q(wj - TtwJ + cqfwj + aqfwjsfwj + 2 1 [bw s(w ) /2] + K} + iiafttryrj - (l/rjq(wj].q(wj + TfwJ - cqfwj - aqfwjsfwj 2 1 2 [bw s(w ) /21 - K - [tr./rj - (l/rjq(wj].q(wj " TfwJ + cq(wj + aq(wjs(wj + 2 2 2 [bw s(wJ /2] + K ) + u {[(r /rj - (l/rjq(wj]-q(wj + TtwJ - cqfwj - aq(wjs(wj 2 3 2 o [bw s(wJ /2] - K} + u {[(r /rj - (l/rjq(wj]q(wj + TtwJ - cq(wj - aq(wjs(wj 2 4 1 o [bw s(wj /21 - K } 2 2 First-order Necessary Conditions: 3L /8q(wJ u = ^{(1/rjqfwJ + p{(r„/rj - (2/rj-qfwJ - c - as(wj} - hxw! - ()>as(wj} 151 + ii^tr./rj - (2/rj.qK) - c - asK)} - pjdyrj - (2/r,)-q(wj - c as(w,)} + ai/VaqfwJ Uat^/r,) - (2/r,)-q(w ) - c - asfw,)} = 0 1 Oaftl/rjqK) + Pftiyrj - (2/rj-qK) = u.ftiyrj - (2/rj-qfwJ - c - asfwj} - hxw2 - (MwJ) - - c - asfwj} + u j d y r j - (2/r,).q(wJ - c - asfwj) + u {(T./r ) - (2/r,)-q(w ) - c - asfwj} = 0 1 4 2 31/VaTK) = QJ.. 1 + p) + u, - u, + p = 0 ai/78TfwJ = 0 (- 1 + P) - p, + u +' p = 0 aiy/dsfwj = 3 2 2 4 9i{p(- aq(wi) - bwjstwj) + hTWj - <t>(aq(wj + bw^lwj)} + pJ- aq(wj - bwjsfarj} - u j - aqfwj .- bWjSCwj} + u {- aq(Wj) - bWiSK)} = 0 3 dlP/dsfaJ = 9 {p(3 aq(wj) - bw-jsfwj) + hxw 2 (KaqfwJ + bw s(w )} + Uj,{- aq(wj - bwjsfwj} + 1 2 pJ- bws(Wj))} 2 - pJ- aqtwj - aq(wj - bwjsfwj} = 0 Proof of Proposition 3.1. In order for any solution to (3.19) to be feasible, it must satisfy both incentive compatibility constraints. This implies and JC*(w,W) > 7C*(W,W) 2 a 1 2 Adding these two expressions, after substituting for the optimal variables, yields -[w bs*(w,) /2] - [w bs*(w ) /2] S> -[w,bs*(w)/21 - [w bs*(w ) /2] 2 2 1 2 2 2 2 2 s'KHw, - wj ;> s*( ) {w 2 W2 2 2 1 wj s*(w) > s^wj t Q.E.D. Proof of Proposition 3.2 I) Assume that s*(wj * s*(wj, but that the downward incentive constraint and the upward Incentive constraint bind. This implies 152 ^(Wj.wJ = ir*(w ,wj 2 and n*(w ,Wa) = jr*(w,w) 2 1 2 Adding these two expressions, after substituting for the optimal variables, yields -[wbs*(w)V2] - [w bs*(w)V2] = -[wbs*(w)V2] - [w bs*( )V2] 1 1 2 2 S ^ W J ^ W , - Wj} This equality will hold only if w = w, t 2 1 2 = S'CwJ^Wa - Wl Wj} a contradiction. Therefore, both multipliers 2 cannot be positive. Suppose now that u = p t = 0. This would yield the full information solution, 2 which we already know to be Infeasible (see equation 3.18). Therefore, If s*(Wi) * s'fwj. then either (i) u > 0 and x > 0. Assume that, contrary to the proposition, u Uj = 0 or (11) u = 0 and x = 0 and u, > 0. t ^ If this assertion is true, then the first-order condition dL/dT(wi) = 0 can be simplified. u Q (- 1 + x u 3 = GjU-p) + p 2 P) - pa + p = 0 3 > 0, since P < 1 and u > 0, by assumption. 2 => JiffWj.wJ = 0 JI*(W ,W ) = 0. This implies that jt*(w ,Wa) > By (IR). we must have rc'f.Wa.wJ > 0. 1 2 1 Since by assumption s(wj > 0. by (IC) and (DE), we must have ^(w^w,) £ n ' t w a ^ ) > ^(Wij.wJ, a contradiction. This establishes that the only possible combination of multipliers that is feasible Is to have u > 0 and U j = 0, which establishes part i). x 11) Assume that s*(wj = s'f.wj. First suppose that, contrary to the proposition, both Incentive compatibility constraints are satisfied with a strict Inequality. This implies that rt'tw^wj > 7t*(w ,W!) 2 and TC^WJJ.WJ) > jt*fw wj lt . Adding these two expressions, after substituting for the optimal variables, yields -[w,bs*(w)/2] - [w bs*(w ) /2] > -lw bs*(w ) /2] - [w bs*(w,)/21 2 1 2 2 2 2 s*(w ) {w - w j 2 1 a 2 1 > s*(w2) {w - Wi) 2 2 153 2 2 Clearly, this Is a contradiction since we assumed the standards were equivalent. Thus, It Is not optimal to have both ^ = 0 and p = 0. 2 Suppose now that P i = 0 and p > 0. 2 ^•(wj.wj £ JC*(W ,W ) 2 1 and iz*[w ,wj = 2 it*(w„wj Once again, adding these two expressions establishes a contradiction when s'twj = s*(wj since * w . When we let p > 0 and p = 0, we get a similar inequality, and 3 t 2 a similar contradiction. The only possibility remaining Is to have both multipliers strictly positive. This means that if it is feasible to have identical standards, then both incentive constraints must bind. This establishes part ii). ill) If this portion of the proposition Is incorrect, then one of three possibilities must hold. a) Assume that p = p = 0. 3 4 From the first-order conditions p + p = 1 - P > 0. This rules out case a). 3 4 b) Assume that p > 0 and p = 0. 3 4 This implies that ^(wj.wj = 0 andre*(w ,w )> 0. But by (IC) and (DE), we know that 2 2 n*(wj,W!) > TC*(W ,W ), which rules out this case. 2 2 c) Assume that p > 0 and p > 0. 3 4 This implies that ic*(w,w) = Tt'twj.wj = 0, which violates (DE). 1 1 The only remaining possibility is that p = 0 and p > 0, which completes the proof. Q.E.D. 3 4 Derivation of equations (3.20) and (3.21) Using the first-order condition,3L '/3q(w ) = 0, and the fact that p = 0, we get i 1 2 the following: eja/rjqfwj + p{(r0/r,) - (2/rj-qtwj - c - astwj} - hxw 1 - tasfwj + pjoyrj - (2/r )-q(w ) - c - as(w,)} = 0 1 1 Using the first-order condition 3L /3T(w ) = 0, we get Lf 1 eifcryrj - (2/r)-q(w) - c - as( )} + [e^wj/r,] - e^hxw, + 1 l Wl 154 =o eitffVr,) - (l/rj-qlwj - c - as(w,)} - ejhxw, + (frasfw,)} = 0. The equation for q*(Wi) follows directly. If we substitute p(w,) = (r /rj) - (l/rjqfw,), o we get p*(wj = c + (l+<t>)as*(wj) + hxw,. A similar derivation is used to determine p^wj. To derive s*(w), 1 note that from the first-order conditions and proposition 3.1, we know that p^ = p = 0, and that p, = GJl-p}. Upon substitution into the first-order 3 conditions, we get: hxw = (l+<))){aq*(w) + bw^lw,)}. 1 1 Now substitute for q*(Wj) = T - r,{c + (l-HtOas'tw,) + hxwj. Solving for s*(Wj) yields 0 hxw, .(l+<t>){r - r,(c + hxw,)} (l^H-r^U-HM + bw,} a S * (Wl) = o A similar sequence of steps can be used to derive s^wj, except we note that p = 1 - P and p, = (l-GJd-P). This allows a simplification of the first-order condition. 4 0{P(- aq(wj - bw2s(wj) + hxw - ^(aqCwj) + bw2s(wj)} - p,{- aq(wj - bw^fwj} 2 2 pJ- aqfwj - bw s(w2)} = 0 2 This reduces to an even simpler equation. sfwjI-eaPbWj - 8<(>bw + G^l-pjbw, - (1-P)bw} = -0hxw + aqfajte^ + OjJ 2 2 2 155 2 2 + Upon substitution for qfwj, and solving for sfwj, we get s*(wj = e [hxw -a(l+<fr){r-r (c+hxwj}] 2 0 1 l 9 {(l+(l))bw - r(l+<t»)a} + (l-pjd-OJbfwj-w,) 2 2 2 2 1 Proof of Proposition 3.3 To show this result, multiply s (wj by (8 /ej = 1. This gives the following n 2 expression. s (w ) = 3 [ F, h x 2 - r,(c + hrwj}] e(l-H|)){bw - na-Hl+^j} 2 ' ( W{ o a w 1+( 2 r 2 e[hxw-a(l+<|)){r-r(c+hxw)}] 92{(1-Ht>)bw- r,(l+<|))a} + (l-pJd-ejbCwj-wO s*(wj = 2 2 o 1 2 2 2 2 Since the numerators are the same, the result will hold if the denominator of s*(wj Is larger than the denominator of s (w ). This holds if (l-PHl-ejbfwj-Wi) > 0, which FI 2 Is true since w > Wj, p < 1, and 9 < 1. Q.E.D. 2 2 Proof of Proposition 3.5 Part i) of the proposition states that if s^w,) = s'HwJ, then p*(wi) < p*(wj. This follows directly from the definition of the optimal variable charge. p*(wj) = c + (1-H())as*(w) + hxwj 1 p*(wj = c + (l-HOas^wJ + hxw 2 Thus, p*(wj - p*(wj = hx(w! - w j < 0, since w > w^ 2 Part ii) also follows from the definitions, but involves more algebra. Assume that S^W,) = S*(Wj) = s*. T*(W!) | T*(wJ <=> [bw s* /2] + K - VP*(wJ + [bs* /2](w - w j | [bw s* /2] + K - VP*(wJ 2 2 1 2 2 156 2 <=> -VP*(wJ | -VP*(wJ <=> q*(wj[<t>as* + hxwj | q*(wj[<)>as* + hTw ] 3 <=» [r-r{c+(l+<t))as*}-rhTw]-[(t)as* + hxwj] | [^-r^c+d+cWas^-r^TwJ-^as* + hxwj o l <=> 1 1 r^xKwja-twj^i I {w - wj.fr.-rjc+d+itijas^-r^as*} 3 This implies that T*(wJ # T*(wJ as r\hx w -w Q.E.D. 3 t Proof of Proposition 3.6 If s*(wj = s*(wj = s*. then vb > 0 and u £ 0 (See the proof of proposition 3.2). 2 We know that p = 0, p = (1-p) and p = Q^l-P) + p from the first-order conditions 3 4 x 3 for the fixed charge. Using this information, write the first-order condition for s*(wj as eJpG aq*(wj - bw,s*(wj) + hxw t - <|>(aq*(wj + bw^lwj)} + {GJ1-P) + pj{- aq*(wj - bw!S*(wj} - p {- aq*(wj - bw s*(wj} = 0 3 3 Solving for p , we get: 3 Giffl+Maq'lwJ+bw^wJ) - hxwj bs*(wj{w-wj 2 Thus, Pi Ml-PJbs'f.wJfWa-wJ + Gjd+^faq^wJ+bw^^wJ) - hxwj = bs*(wj{w-wj 2 Since p > 0, 3 hxw £ d+<|>){aq*(wj + bwiS*(wj} t Substituting for q*(wj, we get (1) hxw - d+iWairo-r^c+hxwJ] £ -a^l+^T^lwJ + d+<|>)bwjs*(wj. x Before proceeding further, note that 157 hxw, - ad+<W{r - r,(c + hxwj} d+<|)){bw, - r (i+(t>)} s"(wj = o 2 ia Since this expression is strictly positive, this suggests that the denominator and numerator are of the same sign. For simplicity, and without loss of generality assume that both are positive. This allows us to simplify (1) to get (2). , ^ f9 - a(l-H>){r - r,(c + 0 hxw,)} s*fw) > To solve for s*(w,) = s*(wj = s*. substitute the expressions for p, and Uj into the firstNote also that vu = (G^GJd-P). order condition for s(wj. Arranging and collecting terms gives: (3) {ea+ejl-pj+Oa-ejiPM-aqtwJ-bs*} + eJhxwj-^CaqtwJ+bwaS*)} + (-Gxd-PJM-aqfwj) + bs'lp^-pawj = 0 An aside: p,w, - p w„ = ejl-phv, a - (Gj/bs'Hd+MaqfwJ+bw^*) - hxw,}. Substituting this expression into (3), cancelling some terms and solving for s* gives 9,{htw, - a(l+<t>)fr-r(c+hxw))} + 9,{htw, - ad-HiJfr.-r^c+hTwJ)} O.Ki+^jbw.-r.d-HtiJ^d-PJbCw^w,)} + 9 {(l+<t.)bw -r (l-Ht>) a +(l-P)b(w -w,)} 0 3 * 1 1 2 = J a a i a Substituting this expression into (2) above gives a simple expression. 9A - GjA, A, GjB, + 9,Cj - G B 3 2 B, where: A, = hxw, - a(l+<t))(r-r(c+hxw)) o A,, = hxw 2 l 1 a(l+<t))(r-r(c+hxw)) o 1 2 B, = (l-H|))bw -r(l+(t>) a 2 1 2 l B, = (l-H^JbWj-rjd-Ht))^ 2 C, = d-pJbK-w,) Cross-multiplying and cancelling elements gives the expression in the text. 158 {hTd+ad+<t>)rJ-{d^) a rJ - bd+^Jtad-HWC.-^c)} 2 2 d-pjbthtw, - ad+^MT.-rxtc+hxwJ)} 159 A P P E N D I X II Contents Section I • Objective Function of the Non-cooperative Model • First-order Conditions • Derivation of equations (4.3) and (4.4) • Derivation of equation (4.5) Section II • Statement of Extreme Rivalry Problem, equation (4.6) • First-order Conditions • Derivation of equation (4.7) • Second-order Conditions for minimization in (4.6) Section in • Derivation of the Value Function and equation (4.9) • First-order and second-order Conditions Section IV • Proof of Proposition 4.1 • Proof of Proposition 4.2 Section V • Proof of Proposition 4.3 • Proof of Proposition 4.4 and 4.5 • Proof of Proposition 4.6' • Proof of Proposition 4.7 160 Section I Objective Function (4.2). Limited Information: Non-cooperative Assumptions L NC (l/^rjqfw,) - [(r /r ) - (l/rjqM-qto) - T(W,) + 2 = e {(r„/r )q(w ) 1 1 0 1 pfftryr,) 1 -(l/rjqfwjj-qfw,) + Ttw,) - cq(w,) - aq(w,)§(w,) - [bw s(w ) /2] - K } + 2 1 1 (l/CTjqtoP - IttVr,) - d/roqtwjj.qtwj e,{(r„/rjqtwj - (l/rjqtwjl-qtwj + TtwJ - cqtwj - Pi{[(r /r,) - (l/rjqtwjl-qtw,) o [flVr,) - (l/rjqtwjj-qtwj - - Tfwj + p{[(r /r,) 0 aqtwjstwj - [bw §(w ) /2] 2 2 2 -K} + + T(w,) - cq(w,) - aqtwjStw,) - tbw,s(w)/2] - K 2 1 TfwJ + cq(wj + aqtwjstwj + [bw s(w ) /2] + K } 2 1 2 v J l O V r j - (l/rOqtwjlqtwJ +TtwJ - cqtwj - aqtwjStwJ - bw s(w ) /2] 2 2 2 - + K } First-order Necessary Conditions: aL^/aqtw,) = e^tryr,) - a/rjqtw,) - (ryr,) - c - as(w,)} + aL^/aqtwj = e {(r /r ) 2 0 -c- 1 ujtr./r,) + (2/rj.q(wj + p{tr /r,) - (2/rj.qK) 0 - (2/r,)-q(w,) - c - as(w,)} = 0 - d/rjqtwj - (r„/r,) + (2/rj-qtwj astwj) - ujtr^r,) - (2/rj.qfwJ + p{(r /r,) - (2/rj-qfog 0 - c - aStwj} + u {(r /r,) 2 o (2/rj-qtwJ - c - astwj) = 0 aL^/artwj = e,(- 1 + p) + u, = o aL^/aTtwj = et- 1 + p) - u, + p = o 2 2 Derivation of equations (4.3) and (4.4) Using the first-order conditions for fixed charge which give expressions for the Lagrange multipliers, we substitute for the multipliers in the first-order conditions on output to get the following: 161 eja/r^wj} + {pe, + e,(i-p)}{(r„/r ) - (2/r,)q(wj - c - a§(wj} = o 1 8,{r./rj - (l/r,)q(wj - c - as(wj} = 0 q(wj = r - r,(c + as(wj) V i = 1,2 o Using the definition of the demand function, we can solve directly for the variable charge. V 1 = 1,2 p(wj = c + as(wj Derivation of equation (4.5) The reason this derivation is included is because ln this non-cooperative scenario, the variable profits of the firm turn out to be zero, so the optimal fixed charges are a bit different. We use the fact that the individual rationality constraint is binding for the firm if its type is w . a ptwj-qfwj + TtwJ - cq(wj- - aqfwj-sfwa) - Ibw s(w ) /2] - K = 0 2 2 3 TtwJ = Ibw §(w ) /2] + K - ptwj-qfwj) + cqfwj + aqtwj-ifwj 2 2 2 Substituting for p(wj gives VP(wJ. VP(wJ = -{c + astwjjtqfwj} + cq(wj + aqlwjsfwj - 0 Thus the fixed charge exactly covers the firm's fixed costs. TfwJ = (bw §(w ) /2] + K 2 2 2 To establish the fixed charge for the firm if it is type w „ we use the fact that the upward incentive compatibility constraint is strictly binding. IttWj.W,) = 7t(w .W ) 2 1 Solving for Ttw,) gives the following. Ttw,) = [bw,s(w ) /2] + K + pfwj-qfwj + TtwJ - cqfwj + aqfwj-stwj 2 1 [bWi§(w ) /2] - K - plwj-qfwj) + cq(w,) + aqfwj-slw,) 2 2 Substituting for TtwJ and p(wj gives the optimal expression for the fixed charge. Ttw,) = {bw s(w ) /2} + K + (bs(w ) /2}(w -w ) 2 1 2 1 2 162 2 1 Section II Statement of Extreme Rivalry Problem, equation (4.6) 2 = Min 2e, {hm {r -r (c+as(w ))-s(wj} + ^astwjfo-r^c+asfw.))) + [bw,s(wJ /2]}} 2 1 o l 1 1-1 s(w,) subject to: s(wj £ sfwj First-order Necessary Conditions dZ^/asfwJ = eJ-hTWiaT! - hxwi + <t>aT - r ^ a - 2<j>ariS(w) + ttfowjstwj)} + Uj = 0 a 0 1 az^/asfwj = eJ-hTWaaT! - hxw + <(>ar 2 - 2<|>ars(w) + (jjbwjsfwj)} - p = 0 2 o 1 2 t Derivation of equation (4.7) From the first-order conditions, we simply solve for s(wj, since we have assumed that p! = 0. Notice that the 0's drop out. t That is, we could have solved the minimization problem assuming Wj had occurred, and then again assuming w 2 had occurred. (4.7) htw.d+aTj - (|>a(r - r,c) o = <t>fbw, - 2r\a ) 2 Second-order Conditions for the Minlrnizatlon in (4.6) Using the first-order conditions, we get: 3 Z /3s(wJ = - 2 ^ , 2 ER 2 This Implies that bw, > 2a r!. 2 that s ER + <t>bw, > 0; 1 = 1,2. It Is clear from the expression above that, to ensure is well defined, we must have the regulator's objective function strictly convex in the standard, so bWi > 2a r\ Is enough to ensure this. 2 Section in Derivation of the Value Function and Objective Function 4.9 To conserve space, we have combined two steps. What follows is the long163 hand version of (4.9) which contains the value function of the public utility regulator. = e {(r /r )[r -r (c+as(w ))] - (i/2r )[r -r (c+as(w ))] - [ i r . / r j - u / r j i r . a 1 0 1 0 1 l 1 0 1 1 r (c+as(w ))l].[r„-r (c+as(w ))] - [lbw s(w )V2]+K+[bs(w )V2](w -w )] + 1 1 1 1 1 1 2 2 1 p{[(r /r )-(i/r )[r -r (c+as(w ))]].[r -r (c+as(w ))] - [[bw s(w )V2] + K + 0 l 1 0 l 1 0 1 1 1 1 [bs(w ) /2)(w -w )] - clr.-r^c+asfw!))] - asfwjIrvrVc+asfw,))] 3 a a 1 Ibw s(w,) /2] - K } - htwj^-r^c+asfwjj-slwj} - <t>{as(wj[r -r (c+as(w ))]+ 2 o 1 [bw s(w ) /2l}} + Gal^/rOlro-r^c+asfwJ)] 1 - (l/2r )[r -r (c+as(w ))] - 2 2 1 1 1 1 0 1 2 [(r /r )-d/r )[r -r (c+as(w ))]]-[r -r (c+as(wj)] - [bw^fwj ^] + K + 0 1 l 0 3 1 2 0 1 p{[(r /r )-d/r )[r -r (c+as(w ))]].[r -r (c+as(w ))] 0 1 1 0 a 1 0 l - [bw s(w )V2] + K - c[r - 2 D 2 2 IVc+asfwJ)] - astwjlr.-r^c+asfwj))] - [bw s(w ) /2] - K } - hxw {(r 2 3 2 3 o T^c+astwJ) - s ( w j } - ^{astwjlr^ryc+aslwj)] + [bw s(w ) /2]}} + pjsfw,) 2 3 2 - s(wj} First-order Necessary Conditions aw^/asK) = Q [-r a + r a - arVc+asK)) + r a + (2/r )[r -r (c+as(w ))].[-r a] 1 o o o l o 1 1 1 bw,sK) + p{-r a - (2/r )[r -r (c+as(w ))].[-r a] + bw^fwj + cr\a o aT + 0 aTiC + 1 o 1 1 1 2a T s(w ) - bwjsfwj} + TiahTWj + hxw, - <t>aT + ^aT^c 2 1 1 0 + 2<t»ras(w) - <|)bwis(wi)} + pj = 0 2 1 1 aw^/asfwj = Gj-bstwjfwa-wi) + pbsfwjfwa-wj)} + 9 {-r a + r a - aT^c+astwJ) + 3 o o r„a + (2/r )[r„-r (c+as(w ))].[-r a] - b w ^ w j + p{-r a - (2/rjlrv 1 l I 1 0 r (c+as(w ))]-I-r a] + bwjstwj + cr\a - aT + aTjC + 2a r s(w ) 2 1 3 1 0 l bwjslwj} + Tjahxwj + hxw - <t>ar + ^aTjC + 2<))r a s(w ) 2 3 (jjbwjsfw!)} - pi = 0 164 o 1 3 3 . Second-order Sufficient Conditions 3 W /3s(w ) 2 MR 2 1 8 W /3s(w ) 2 MR 2 2 >0 = 0{(l+2<|))ar - (l+iWbwJ = 6{(l+2<»ar - (l+^jbwj - (l-pJbK-wJGi > 0 2 1 1 a 1 1 Both of these are restrictions on the denominators of s^HwJ. . Section IV Proof of Proposition 4.1 In order to show which standard is larger, we will establish a contradiction. To begin, we repeat the expressions for the standards and then cross-multiply. Suppose that s^wj > s^fwj. Then hxw - a(l-M>)(ro - r\c) t hiw,(l+arj - <t>a(r - r\c) ' <^{bw - 2r\a ) o > (l-Htfbw, - (l+2(|>)ar 2 2 1 l This may be written as a - air.-rj a, t where: °* * + «• > otj = htWj - a(l+<t»)(ro - r\c) > 0, since (ro/r,) > c. This follows from P^TWJ = c + as^wj and (T /rj a p ^ . 0 a, = bwi - r\a > 0, since bw! - 2r\a > 0 is required by the second-order 2 2 conditions. a, = ^fbw, - 2r\a ) > 0, by the second-order conditions. 2 Cross-multiplying gives: -[Oaafr; - ^c)] > [a.,0,]. Since both terms in the square brackets on the LHS are positive, this establishes a contradiction. Therefore, it must be that s^HwJ > s ^ w j . Exactly the same style of proof establishes that s^fwj > s^wj. Q.E.D. Proof of Proposition 4.2 Proposition 4.1 already proves that s (wJ > s^wj, and chapter HI shows that ER s"(wj) = s*(w) and 1 s (wj > s*(wj. Therefore, we need only show that s (wJ > s*(wj, n ER s*(wj > s (wJ. and s^lwj > s'Vj. We begin with s (wJ > s*(wj. MR ER 165 1. To show this result, we simply compare the relative magnitudes of the numerators and denominators of equilibrium standards. Assume that the numerator of s*(wj > s^fw,). This Implies - <|>a(ro hiWid+aTj) - r\c) < Iro^ - ad-Ht>){r-r\c} + o a(l-H|))r hxw . 1 1 0 < -a(T -r c) + a ^ r ^ i W i 0 1 r < r^c+^hxw,) 0 (ryr,) < c + <t>hTW! Since (r /r,) = p ^ , and c + ^ f r c W i < c + (l+^las'lwj + hxwj = p'twj, this Inequality o Implies p ^ < p'twj, a contradiction. Therefore, we may conclude that the numerator of s^w,) £ s*(w,). Now assume that the denominator of s*(wj) £ denominator of s^fwj. This implies (l+<|>){bw-(l+<|>)ar} <, <))(bw-2ar) 2 2 1 1 1 1 bwj £ (l+<|)) ar - 2<|>ar 2 2 2 l l bw! £ (l+f)aT„ a contradiction. This is a contradiction because the second-order conditions for the extreme rivalry problem require bwj > 2a r\ (recall that <|> £ 1). Therefore, we may conclude that the 2 denominator of s*(Wj) > the denominator of s^wj. Combining these two results establishes that s^rwj > s*(wj. 2. To show that s (wj) > s^wj, we use exactly the same style of proof. The only ER difference is that we multiply s^HwJ by (Gj/GJ = 1. Assume that the numerator of s*(wj) > numerator of s^wj to get Gjhxwjd+aTJ - <|)a(r0 - I^c)} < G {hxw - aU+«fr){ro-rxc} + aU+$)r,hTwJ. 2 (ryrj 2 < c + (tihxwj, a contradiction. 166 We may conclude that the numerator of s ^ w j > the numerator of s*(wj. If the denominator of s^fwj 2 denominator of s^wj, then we get ejl-HfllbWa-U-HWaTj + U-P^bfo-w,) £ 8 (()(bw -2a r ) 2 2 1 1 (l+<t>)ar £ bw + (l-pjtO^ejbtWa-W!), a contradiction. 2 2 1 a This is a contradiction because d-PHSi/eaJblWa-wJ > 0. and bw 2 specified by the second-order conditions. > d-H> )a r as 2 2 1 From this we may conclude that the denominator of s^twj is strictly less than the denominator of s^wj. Combining these two results establishes that s ^ w j > s*(wj). 3. To show that s'fwj > s ^ w j , compare the numerators. By inspection, we can see that the numerator of s*(Wj) 2 the numerator of s (wJ. MR Assume that the denominator of s*(wj > denominator of s^HwJ. (l-Ht)){bw-(l+<|>)ar} £ (l+itObw! - d+2<t>)ar 2 2 1 1 l -(^aT, 2 0, a contradiction. Combining our two results gives s*(w) > s^HwJ. 1 4. Assume that the numerator of s ^ w j > s*(wj. 9 {hxw - a(l-wt»)(r„ - r\c)} > 8 {hxw - ad-HtffT.-IV:} + ad+^r^xwj. 2 2 2 (IVrj <c+ 2 ^hTWx, a contradiction. Therefore, the numerator of s*(w) 2 s ^ w j . 2 Assume that the denominator of s^wj 2 s ^ w j . 8d+<|>){bw-d+<t>)ar} + d-pJ^bK-wJ £ 8{(l+<l))bw-d+2(|.)ar} + d-P^bfo-wJ 2 2 2 2 l 2 2 -(j) > 0, a contradiction. 2 167 1 Combining these two results establishes that s*(wj > s ^ w j . 5. It remains to show that s (wJ > s (wj. Assume that the numerator of s (wj > ER n n the numerator of s (w.J. ER htw (l+ari) - <|)a(r - r,c) < htw - a(l+<t»){r„-rc} + ad+^hrw.,. o 2 (rjTJ 2 1 < c + ^hcw^ a contradiction. Thus, the numerator of s^HwJ £ the numerator of s (w ). F1 2 Assume that the denominator of s^wj 5 s ^ w j . (l»bw - 2<)>ar £ d+<t>)bw - ( l + ^ W j 2 1 3 2 d + ^ l a ^ i S bw , a contradiction V <>| <, 1. 2 This means that our assertion was Incorrect, allowing us to conclude that the denominator of s^wj > denominator of s^TwJi. Together, these two pieces of evidence allow us to conclude that s ^ w j > s(wa). Q.E.D. n Section V Proof of Proposition 4.3 Since the firm is trying to maximize profits. Its preferences for institutional structure will be determined by its realization of profits under the different regimes. Assuming a separating solution, the individual rationality constraint is binding with a strict equality for a firm of type w in both extreme rivalry and mild rivalry. Since 2 profits are always zero for a high type firm, it will be indifferent to the institutional structure; Le., it is indifferent between mild rivalry and extreme rivalry. Such is not the case for the firm If its type is w,. To show the firm's preference structure, we must derive expressions for profit. JT^WJ.WJ = p^fwj.q^wj + T™(wJ - cq^w,) - a q ^ J s ^ w , ) - [bw s (w ) /2] - K ER 1 168 2 1 Upon substitution for the optimal variables, we get a simple expression for profit. TC^K.WJ = [bs (w )V2](w -w ) ER 2 2 1 A similar procedure establishes optimal profits for the firm when there Is mild rivalry. flw.-w,) = [bs (w )V2](w -w ) MR 2 2 1 Clearly JC^HWLWJ > JC^WLW,) by virtue of the fact that s^HwJ > s^w,). As a result, the firm prefers extreme rivalry. Q.E.D. Proof of Propositions 4.4 and 4.5 To show that the public utility regulator prefers mild rivalry when there is a separating equilibrium, we derive the equilibrium value for the public utility regulator's objective function under extreme rivalry and mild rivalry, and compare these values. U"* = 9 {(l/2r ).(q (w ) ) - I b w ^ w , ) / ^ - K + (3-D{rbs (w )V21(w -w )}} + MR 1 2 1 2 MR 2 1 2 l 0 {(l/2r ).(q (wJ ) - [bw s (w ) /2] - K} MR 2 2 MR 1 2 2 2 Due to the symmetry of the problem, we can write out U in an analogous manner. ER 9 {(l/2r ).(q (w )2) - [bw s (w )V2] - K + (p-l){[bs (w )V21(w -w )}} + U™ = ER 1 ER 1 1 ER 1 1 2 2 1 0 {(l/2r ).(q (w )2) . [bw s V ) /21 - K} ER 2 Subtracting U UM R . R UE ER E 2 1 from U M H 2 gives a simple result. 0 {(i/2r )(q (w P-q (w P) - [bw /2]{s (w )2-s (w ) } + (p-l){[b(w MR = 2 2 l i ER 1 MR 1 1 169 ER 1 a l 2 w, s(wJ}} > 0 ER 2 This difference is strictly positive since (P-l) < 0, q^wj > q (w,), and s (wJ> s^wj. ER ER Since this difference is strictly positive, the public utility regulator prefers mild rivalry to extreme rivalry. Q.E.D. Proof of Proposition 4.6 Assume that s^fwj = s^HwJ = s . m From the first-order conditions, we know that U i £ 0 if [-hxw^i - taw, + <t>aT - <t>2ai\s - <t>aT,c + <t>bw!S ] £ 0. Of course, 2 ER ER 0 this assumes that 0, > 0 and 6 > 0, which is the only interesting case. a This expression reduces to hcw.a+arg - <t>a(r„ - r\c) (1) Now (^(bWi - we need an expression for s . 2ria ) 2 This can be attained from the first-order m conditions, while substituting p from one first-order condition into the other. x gives (2 ) A E R S _ 9j hxwJl+aTj - $a(T - I^c)} + eJhcwaU+aTj - <|)a(r - I^c)} " Gifofbw! - 2r\a )} + 9{(|>(bw - 2ria )} 0 o 2 2 2 2 Substituting (2) into (1) yields OxtAj + 6,^ ( 3 ) ^ GilAj e (B ) + e j i B j 1 1 where: A, = hiWid+aTJ B - (tiair^rj = <))(bw-2ar,) 2 t 1 170 This A, = hxwad+aT,) - <»(r -r ) o B (jjfbWa^aTi) = 3 1 From the second-order conditions, we know that A lt A^ B lt and B are positive. This 2 allows us to simplify the requirement for equivalent standards down to uVr,) £ c + (hta/<|>b)(l+ar) (4) Thus, if (4) Is met, then the environmental regulator will find it optimal to offer a single standard defined by s in (2). m Q.E.D. Proof of Proposition 4.7: Assume that s^wj = s ^ w j = s . From the first-order conditions, we know EH that uj £ 0 if 0!{-r a + r a - aT (c+as ) + r a + (2/r )[r -r (c+as )]-[-r al - bw^** + ER ER o o 1 p{-r a - (2/r )[r -r (c+as )]-[-r al + bw 1!R 1 o 0 1 1 1 o o + cr,a - aT + ER lS 1 1 0 aTiC + 2a T s 2 - ER 1 bw^Wi)} + r,ahxw, + hiw, - <t>aT + <|>ar,c + 2<|>r,as - ^bw^™} £ 0. Assuming Q > 0. this 2 ER 0 x may be simplified to hTw - ad-HWir. - r ) t U J s lC (l+<Mbw, s - (l+2<t>)ar, 2 To derive an expression for s " , use one fist-order condition, to solve for p 1 This gives 1 P (2) p, = 8 t {-bs^.-wJH+p)} + e {hxw (l+aT )-a(l+(t))(r -r c)-(l+(t))bw s - (l+2 t>)a T s }. MR 2 2 o l 1 ! 2 < MR 1 Substituting (2) into the first-order condition [W^/astw,)] = 0, gives ( 3 ) 8, {hxwx - a(l-H|>)(r - T.c)} + 8 o A M R S ~ {htw 2 2 - a(l+<|))(r - r )} o lC 0 {(l+(t>)bw -(l+2(|))ar }+(l-p)b(w -w )8 + e{(l+<|>)bw-(l+2<|>)ar} 2 1 1 2 1 171 2 1 1 2 2 1 Substituting (3) Into (1) above generates the relevant parameter restriction. [ei /ej £ (l^HaU+wr.-r^)} - {hT(i+2<i>)a r} 2 b(l-p){hTw Q.E.D. 172 l - a(l+<|))(ro-r1c)} 1 A P P E N D I X IH Contents Section I • Proof of Lemma 5.6 • Objective Function of the Limited-information Solution: Cooperative Assumptions • First-order Conditions • Derivation of equation (5.8) • Derivation of equation (5.13) • Proof of Proposition 5.3 Section II • Objective Function of the Limited-information Solution: Non-cooperative Assumptions • Proof of Proposition 5.5. Derivation of equation (5.15) • Mild Rivalry • First-order Conditions • Second-order Conditions • Derivation of the optimal standard, equation (5.17a) • Extreme Rivalry • First-order Conditions • Second-order Conditions • Derivation of the optimal standard, equation (5.19a) • Proof of Proposition 5.6 • Proof of Proposition 5.7 173 Section I Proof of Lemma 5.6. The proof is by induction. Let x be defined as follows. x 3 {x, I x, a (p(wJ.T(wJ.s(wj); i = 1 n} Assume x satisfies BUAIC, and s(w,) 2 ... 2 s(wj, but that x violates (IC). This means that there exists a type w, e {w„...,w} such that (1) Jt(x,,wJ < n TC(XJ,WJ for at least one J. Both x, and x, are contained in x. Specifically, we have assumed (2) 7c(x,,wJ = (3) s(wj 2 s ( w j Part A: V i e (1 rt(x, ,w,), +1 V i e {1 n-1}. and n-1}. Suppose that j > i+1 (We will discuss j < i in part B). From (2). we have (4) JC(X,,WJ (5) n(x ,w . ) = n(x^ ,w ). 1+1 = icfx^.w,), and h 1 a 1+1 Since the profit curves are not upward sloping, we know also that (6) Jt(x, ,wJ 2: n(x j,w ). +1 I+ 1+1 First, suppose that J = 1+2 so that the violation of (IC) may be written as follows. (Al) 7c(x,.wJ < 7t(x, ,wJ +2 Using (Al). (4) and (6), we get (7) Ttlx^.wJ > 7t(x,.wJ = &(x,,w,) 2 n(x, ,w ) = 7c(x, ,w,J, which implies (8) A(x, ,wJ > + 1 +2 +1 1+1 +2 jttx^.WjJ. Using (7) we get (9) 7C(X, ,WJ +2 - 7t(X, ,W + 2 l + l ) Ttlx^.W,) - > Jt(X I + 1 ,W 1 + 1 ). Substituting for n. and cancelling common terms yields (10) [(-bw (wJV2) + (bw s(w )V2)] > [(-bw (w )V2) + (bw s(w )V2)] => stw^Hwj-wJ > s(w ) (w -w,) =» s(w ) > s(w ), a contradiction of (3). lS ltl 1+I lS 1+l 1+1 l+1 2 1+1 1+2 2 l+1 From this we may conclude that (Al) was a false assumption. Thus, we can state (I) JC(X,,WJ > n(x, ,wj. This is the first part of the proof by induction. We establish that +2 the result holds for j = i+2, and are then able to show that It holds for j = 1+3. 174 Suppose there exists a smallest j such that (A2) nlXj.wJ > 7t(x,,wJ. From this, we know that 7t(x,,w,) £ Thus from (A2), which Is a violation of n(x . ,wj. j 1 (IC), and from BUAIC, we get (11) 7t(x,,wJ > Jt(x,,wJ £ JtfXj.j.wJ £ 7C(X,. ,W . ) = JCfXj.W,.!). 1 ) 1 This Implies, Jt(x,,w,.j) £ 7t(x.,w) - nfx.,w.). (12) 7t(x,,wJ - =* s(w,) £ sfw,.!), a contradiction of (3). J 1 I J 1 J 1 We know the result Is true for j = 1+1 and j = i+2, and we know that it holds form some arbitrary J. Therefore, we know that it holds for all j > i+1. Part B. Suppose that j < L Again, we establish the result using a proof by Induction. To that end, we first establish the result for i and i-1. We then assume the result holds for i and j+l, and establish that the result holds for 1 and j . (BI) (13) rt(x,,wj < Assume n(x,. wj. lt JcfeM.wJ = jfo.wj , by BUAIC. Subtracting the RHS of (BI) from the LHS of (13) and the LHS of (BI) from the RHS of (13) gives (14) - rt(x,,wj > rcfown) TtfXn.Wn) - rcbcu.wj. After substituting for K, we can show that (14) Implies (15) s{wj > sfw,.,), a contradiction of (3). Thus (BI) Is incorrect and nfxj.wj £ jt(x,.i,wj. Now suppose (B2) Jt(x,,wJ £ Jt(x ,w ) where j + 1 < 1. In addition, assume (B3) 7c(x,,wJ < Tcfxj.w,). j+l 1 This is the violation of (IC). (B2) and (B3) allow us to claim that (B4) JC(X,,WJ > 7C(Xj ,W,). +l By BUAIC, we know that (16) Jt(Xj,w,) = jt(^ ,Wj). +1 (B4) and (16) imply (17) rtlXj.W,) - 7C(^,WJ < Jcfx^j.w,) - 7t(x, „wJ. + 175 Upon substituting for n and cancelling common elements, we find that (17) implies (18) s(w ) > J+1 s(Wj), a contradiction of (3). Therefore, the assumption (B3) could not have been correct. This establishes that TC(X,,WJ > TC(XJ,W,) for all j < i. Combining parts A and B we can claim that there does not exist a type i such that 7t(x,,wJ <retxj.w,)for at least one j, when we have BUAIC and monotonicity on the standards. Therefore, the menu of contracts which satisfies BUAIC and monotonicity on the standards also satisfies (IC). Q.E.D. Lagrangian for the maximization problem in (5.7): L = u Z oic/rjqw - (l/flrjqw* - [tfyrj - u/rjqM-qW - T( J + W 1*1 p{[(r /r,) - (l/rjqfwjl-qfwj + TtwJ - cq(wj - aq(wjs(wj - [bw,s(w,)/21 - K ) 2 o hTw,(q(wJ - s(wj) - <|)(aq(wjs(wj + [bw,s(w,)V2])} + Z p,{[(r /r,) - (l/r,)q(w,)] o •q(wj + Ttw,) - cq(wj - aq(wjs(wj - [bw,s(wJV2] - K - [(r„/r,) - (l/r,)q(w, ,)] + •q(w ) - Ttw ) + cq(w, ) + aqtw.Jstw.J + [bw,s(w ) /21 + K} + p {[(r /r,) a 1+1 I+i +1 1+l n o (l/r,)q(wj]-q(wj + T(wJ - cq(wj - aq(wjs(wj - [bw s(wJ /2] - K } + 2 n Z p .i{[(r /r,) - (l/rjqM-qfwJ n+1 o + Ttw,) - cqtwj - aq(wjs(wj - [bw s(wJ /2] 2 l K - [(r /r,) - (l/TJqtwjl.qtw,.,) + TtwJ - cq(w,.,) - aq(w,.,)s(w,.,) o [bw,s(w,.,)V2] - K } First-order Necessary Conditions: 3L"/3qtwJ = eifcl/rjqtwj + p{(T /r) Q 1 (2/r ).q(w ) - c - as( )} - hxw, - <t>as(w,)} l + p^ffyr,) - (2/r )-q(w ) - c - as(w,)} 1 l 1 Wl p {(r /r,) - (2/r,).q(w,) - c n+1 o as(w,)} = 0 3L /aq(wJ u = e.ftl/rjqtwj + p{(r /i\) - (2/r,).q(w,) - c - as(wj} - hxw, - tasfwj} o 176 l U O V r , ) - (2/rj.qtwJ - c - as(wj} + p j c r . / r j - (2/rj-q(wJ - c as(wj} - vJSTjrj - (2/rj-qtw.) - c - as(wj} + pn+1.l{(ro/r1) as(wj} = 0 Vi = 2 - (2/r )-q(wJ - c 1 n-1. 3L /aq(wJ = Gjd/rjqtwJ + pftryrj - (2/I\).q(wJ - c - asfwj} - htw - <t>as(wj} u n - P»i{tr /rj - (2/rj.q(wJ - c - as(w„)} + u . {(T /r ) - (2/rj-q(wJ - c o 2n 1 0 1 as(wj) = 0 aLF/dTtw,) = (- 1 + p) + u, - u ^ = 0 9l aL /dT(wJ = 9,(- 1 + P) - + p, - u ^ + u ^ , = 0 u 3L /aT(wJ = J- 1 + P) - u ^ + p, + Vi = 2 n-1. =0 u ai/VasfWj) = 8i(p(- aq(w,) - bWiSCwO) + htw - ((ifaqfwj + bw s(w ))} + pJ- aq(Wj) t 1 1 bwjstwj} - p^it- aq(wj - bwaslwj} = 0 dL /ds[vr) = 9,{p(- aq(w,) - bw,s(wj) + hxw, - <j>(aq(w,) + bw,s(w,))} - p,.^- aq(wj u bw,.!s(wj} + p,{- aq(wj - bw,s(wj} - p {- aq(wj - bw s(wj} + vwJn+l bw,s(wj) = 0 V i=2 1+1 aq(wj„- n-1. ai/VasfwJ = 9n{p(- aqfwj - bw s(wj) + hTw - <|>(aq(wj + bw^wj)} - p^f- aq(wj n n bw^sfwj) + p {- aq(wj - bw s(wj} - p ^ t - aq(w„) - bw s(wj} = 0 n n n Derivation of. equation (5.8) From the first-order conditions on Ttw,). we know that p, - u„ = 9^1-p}. We can +1 use this Information to eliminate the multipliers from the first-order condition for q(wj. 9{(l/r)q(w )}+{9 (l-P)+9p}{(r /r)-(2/r )-q(w )-c-as(w )}-e {hTw -<|>as(w )} l l 1 1 l o l 1 1 1 1 1 1 =0 9i{(T /r ) - (l/rj-qK) - c - as( )} -e^hTw, - (tasK)} = 0 0 Wl 1 Substituting for p(wj = ffyrj - (l/rjqK) gives p*(wj = c + (l+((>)as*(w) + h t W i . 1 Using this same technique, and the fact that p, - p,., - p„ + +1 177 = 9,{l-p}, we get a similar set of expressions for i = 2 n-l. eJa/rjqM - + (9,(1-3) + ejiKayrj e.tflVrj Substituting for p(wj - (2/rj.qtwj - c - as(wj} -e,{hTw, - <t>as(wj} = o (l/rj-qfwj - c - as(w,)} -8,{hxw, - <|>as(wj} = 0 = (iyr,) - (l/r,)q(w,) gives p*(wj = c + (l+<)>)as*(wj + hxw,. It is then easy to show that this is also the general form for p*(wj. V write p*(wj = c + (l+<t>)as*(wj + hxw, 1=1 Thus, we may n. Derivation of equation (5.13) We know that when the standards are strictly monotone decreasing, all of the DAIC constraints are satisfied with a strict inequality, proposition 5.2. This means that all of the Lagrange multipliers attached to the DAIC constraints are zero. This simplifies the first-order conditions on s(w,) as follows. 9L /ds(w,) = ejp(- aq(w,) - bw,s(w,)) + hxw - <t>(aq(w,) + bw,s(w,))} + u,{- aq(w,) u t bw,s(w,)} = 0 Using the FOC on Tfw,), we get u, =9,(1-P). Substituting for u, gives hxw, Now = (l+<|>){aq*(wi) + bw s*(w )}. 1 1 substitute for q(Wj) = r„ - r,{c + (1-H(>)as(w,) + hxwj. Solving for s(w,) establishes the result. a(l-H|>){r - r\(c + hxw,)} (l+<|>){-r,a(l+<t>) + bw,} hxw, 11 " o 2 A similar sequence of steps can be used to derive s*(wj, except we note that u, = ijOjU-p). This allows a simplification of the first-order condition. This reduces to an even simpler equation. s(wj{9, + (t>9Jbw, + [(1-P)b(w, - WnJ^le.] = 9,hxw, - aq(wj{9rf) + 8j Upon substitution for output, and solving for s(wj, we get 178 fliftl+flbw, - r\d+<|>)a} + (l-p)b(w,-wj £ 9j 2 2 It is easy to show that the expression for s^wj is just a specific form of s*(wj above. Thus, it remains to show s*(wj. However, it is easy to show that the same sequence of steps can be used to derive s*(wj, which has exactly the same form as s*(wj. wj. Q.E.D. Therefore, we may write s*(wj as above V w, e {w, Proof of Proposition 5.3 Suppose that the standards for two arbitrary adjacent types, i and i+1, are the same. If s*(wj = s*(w ) = s, then p*(wj < p*(w ). This follows Immediatelyfromthe w l+1 definition of the variable charge. Recall that this expression for the variable charge can be derived independent of the multipliers on the downward incentive constraints, so the expression for the variable charge does not change when we have strictly monotone decreasing standards or when we have constant standards between adjacent types, or even when the standards are constant over more than two types. p*(wj = c + d+<|>)as + hxw,. p*(w ) = c + d+<Was + hxw . l+1 Since w 1+1 > w„ we get p*(wj < p*(w ). Then, using the demand function, we get q*(wj l+1 w > q*(w ). 1+1 Now suppose that we examine the fixed charges. Again, the expressions for these variables will not change n-l r(wj = [bw,s /2] + K - q*(wj{<t»as +htwj + [bs /2](w 2 2 - wj + Z [bs*(w ) /2](w -w . ) 2 l+l l+k 1+k l+k and n-l T(w ) = [bw s /2] + K - q*(w){(t)as +hxw } + S [bs*(w,J /2](w -w . ) 2 I+l 2 I+l w I+l From these two expressions, it must be that T*(wJ = T*(w,J « -VP*(wJ = -VP«(w ) H1 <=> q*(wj[<j>as + hxwj = q*(w )[<t>as + hxw ] I+1 1+1 179 l+k 1+k 1 1 [r-r{c+(l+<|))as}-rh.Tw]-[<|)as + htwj = [r^rjc+d+^lasJ-rihTw^J-^as + hxw, ] o 1 1 1 +1 «=> (w - wj^r^as-r^rjc+d+^as} = iMitifwjMwj ] 2 1+1 This implies that T*(wJ| T*(w J ( l+ a S Q.E.D. ^£±f_ W , + 1 " W § {r.-r^c+ad-f^sl-r^as l F l h X Section II Limited- information Solution: Non-cooperative Assumptions L NC = £ e{(r„/r)q(wj 1 1 pflGVr,) K} + S - K - 2 [ffVrj - d/rjq(wj].q(wj - TtwJ + - (l/TJq(wj].q(wJ + TfwJ - cqfwj - aqfwjsfwj - [bw (wJ /21 2 lS u,{[(r/rj - (l/rjq(wj]-q(wj + TfwJ o - cqfwj - aqfwjsfwj - bw,s(wJ /2] 2 [(T /rj) - (l/r,)q(w)]-q(w) - Tfw J + cq(w ) + aq(w J§(w J + 0 2 1 1 + 1 1 + 1 [bw §(w ) /2] _ - (i/2rjq(wj - 1+l + K} - - [bw s(wJ /2] 2 - n lt 1+1 1+ 1+ + un{[(ro/rj - (l/rjq(wj].q(wj + TfwJ - cqfwj - aq(wj§(wj + S u^fftiyrj - (l/rjqfwj].q(wj + TfwJ K} - cqfwj - _ 1-2 aqfwjsfwj - bw s(wJ /2] 2 1 - K} - [(T /rj - (l/rjq(w.j].q(w.J - TfwJ + cq(w,.J 0 1 l + aqtw^Jifw^J + [bw,s(w.J/2] + K } 2 l First-order Necessary Conditions: aL*7aq(wj = ejd/rj-qfwj + pftryrj - (2/rj.q(wj - c - asfwj} + u ^ f i y r j (2/rj-qfwJ - c - asfwj} - un+1{(ro/rj dLT/dqW = e.td/rj-qfwj + p{(r/rj 0 - (2/rj-qfwJ - c - asfwj} = 0 (2/rj-qfwj - c - asfwj} - u,{(r./rj - (2/rj-qfwJ - c - a§(wj} + u,{(ro/rj - (2/rj-q(wJ - c - asfwj} - un+1{(ro/rj (2/rj-q(wJ - c - asfwj} + Pm^fiyrj V i=2 n 180 - (2/TJ-qfwJ - c - asfwj} = 0 - aL"7a (wj = eja/rj-qtwj + q 0 2n 1 1 + p) + U l 1 pjffyrj - - c - asfwj} = 0 - u„ = o +1 1 + p) - P H + p, - p»* + vw, aL^/aTtwj = e,(- - c - a§(wj} - as(wj} + u . {(r„/r ) - (2/rj.q(wJ (2/rj.q(wJ - c aL^/artwj = p{(r/i\) - G/rj-qiwj = o v i= 2 n. aL^/dTtwJ = 9 (- 1 + P) - Un.! + p, + p ., = 0 2n n Derivation of equation (5.14) Using the first-order condition on TtWj), we get u = 9J1-P) + u . x this expression into the first-order condition on qfwj, n+1 Substituting and collecting terms gives eitOVrj - (l/rjqK) - c - asM = o. Using the demand function, we can solve for the reaction function which ultimately defines the variable charge. ptSfwJ.w,) = c + as(w,) Using a similar technique, it is easy to show that the general expression for the variable charge reaction function is given as — p(s(wj,wj = c + as(w,) V 1= 1 n. Derivation of the fixed charges is analogous to what it was In the previous section where we used the pattern of binding constraints to solve recursively for each standard. n-l TtsfwJ.wJ = [bw s(wJ /2] + K + X [bs(w ) /2](w -w . ) 2 2 1 1+k 1+k I+k 1 k-l Extreme Rivalry n Min Z** = 2 9, {hTw,{r-r(c+as(wJ)-s(wj} + ^{astwXro-r^c+aslwJ)) + s(wj o *" 1 ,\ [bw (wJ /2lj/ 2 lS subject to: 181 V s(wj £ s(w ) 1+l E W, {w x W.j} n First-order Necessary Conditions dLf^/dsivr) = 9i-hTwar-hxw,+<|)ar-<l)ar,c-<|)2ras(wJ+(|)bws(wj} + u, - Pn = 0 2 1 1 o l 1 Equation (5.17a): hxw.d+arj - «)>a(r - r^) 0 s E R t w J = (frfbw, - 2r\a ) 2 Second-order Conditions the Minimization Using the first-order conditions, we get: Plf*/ds{wp This implies that bw, > 2a r x . 2 = -2<|>ar + <t>bw £ 0; 1 = 1,2. 2 1 t It is clear from the expression above that, to ensure that s™ is well defined, we must have the regulator's objective function strictly convex in the standard. Comparison of Optimal Solutions Given that the expressions for the standards are exactly equivalent to the case of two variables, the proof of the proposition requires us to show that s (wJ > s ^ w j ER 1=1 V n. If we can do this, then all of the preference orderings are maintained as the proofs of propositions 4.1 and 4.2 demonstrate. Part iv of the proposition requires us to show the relationship between s^HwJ and s*(wj, as well as s (wJ and s*(wj. MH begin with parts i - iii, by showing that s (wJ > s (wJ V i = 1 ER 1) Suppose that s ^ w j > s (w,) V 1 = 1 We n. MR n. Cross multiplying and cancelling terms ER gives: i-i i-i 0 > hxw^bw, - aT, + (l-pjbfw.-wj ZaJ - •a(r -r c){(l-pjb(w -wj Za, a l l + a rj 2 If this is negative, and we know that hxw^l+aTJ > ^a(ro-rj, because of the secondorder conditions - all numbers being positive - then it must be that 182 0 > bw, - 2a r\, 2 a contradiction since the second-order conditions require bw, > 2a r,. Therefore, we 2 may conclude that s^wj > s ^ w j V 1 = 1 n. iv) In this part of the proof, we want to show that s*(w) > s (w,) and that s^HwJ > MR s*(wj V i = l,„.,n. Suppose that the numerator of s ^ w j £ numerator of s*(wj. This implies the following inequality. 9,{hxw, - a(l-H|>)[r - r,(c+hxwj]} < GihtwA+aT,) - (1+aT.) - (l+^aGV^c)} o Cancelling terms gives (l+<t>) 5 0, a contradiction. Compar2e the denominators by assuming that the denominator of s ^ w j < s*(wj. After cancelling terms, we get -a (l+ct>) r > -(l+2<|>)ari 2 2 2 l - a ^ r , > 0, a contradiction. Combining these two results suggests that the numerator of s^fwj > numerator of s*(wj and the denominator of s ^ w j S denominator of s*(wj, suggesting that s ^ w j > s*(wj V I = 1 n. The comparison of s (w,) and s*(wj is done in appendix II, section HI. Q.E.D. ER Proof of Proposition 5.6 Assume that p, = ... = p,_, = p, = ... = ^ = 0. This gives +2 (1) 9,{-hTw,ar,-hxw,+(|)ar-<t)ar,c-<t)2rias(wJ+(()bw,s(wj} + p, = 0 (2) 9, {-hTw, ,ar,-htw,i+<t>ar-<|)ar,c-<t)2r,as(w i)+<l)bw s(w, )}. + p, - p, = 0 (3) 9 {-hTw, aT,-hw H^^ 2 o 2 + +1 ua o + 1+ 1+l +1 - p, = 0 1+2 +2 +1 +1 These first-order conditions Imply (4) - p,, = e, {-hTw, ar,-hxw 44ar -<t»ar,c-<|)2r a s(w,J+<|>bw, s(w,j} (5) p, = 9,{-hxw,ar,-hxw,+(|)ar-<|)ariC-<t)2ras(wJ+<l)bw,s(wj}. 2 + +2 1+2 +3 o 1 +2 2 o 1 If p, > 0. and 9, > 0, then -hxw,ar,-hxw,+<t>ar-<t)arc-<l)2ras(wJ+<t>bw,s(wJ > 0. 2 o 1 1 183 This implies that for u to be non-negative, we must have t hxw.d+aT!) - <|>a(r - r\c) o £ER s S <t>fbw, - s 2r,a) 2 In order to derive an expression for s , we must substitute (4) and (5) into (2) and w solve for s* . This assumes that s(wj = s(w ) = s(w ) = s . From these substitutions, 31 ER l+1 lt2 we get the following simple result. 8A, + 9 A, + 8, A, 1+l +1 +2 ^ 8 A, +2 8 A + e^B^.! + 8, B +2 8A 1+2 where: A, = h x W i d + a T j - <|>a(r - r,c) o A, +1 = hTw^d+aTJ - <t>a(r„ - r\c) = htw^d+aTj) - <t»a(r - I\c) 0 B, = <j>(bw, - 2r\a ) 2 B 1+l = <t»(bw - 2r ) B 1+2 = <|>fbw - 2r ) 2 ia l+1 2 ia lt3 Solving this generates the parameter restriction given in the text. (iyr,) £ c + (hxa/Wd+aT,) Q.E.D. Mild Rivalry When the regulators act as mild rivals, the environmental regulator solves an unconstrained maximization problem which contains the public utility regulator's value function. n w wl = Z{e{(r /r )[r -r (c+as(w ))] - (i/2r )[r -r (c+as(wj)] - [(r /r )-(i/r )[r 2 l 0 1 0 1 1 1 <> 0 1 1 1 0 1*1 n-1 r^c+aslwJJll-lr.-r^c+astw,))] - [[bw s(w ) /2]+K+ Z [bs(w J /21(w -w . )] + 2 1 2 1 i 1+k l+k 1 k»l p{[(r /r )-(i/r)[r -r (c+as(w ))]].[r -r (c+as(w ))] - [[bw,s(w)/2] + K + 2 0 l l 0 1 1 0 l 1 1 n-1 Z [bs(w J /21(w -w . )] - clivrac+asfw,))] - asMliVIVc+asfw,))] - [bw s(w) /2] 2 1 2 1+k 1+k 1 1 184 t - K) - hxw {r-r(c+as(w))-s(w)} - <|>{as(wj[r-r(c+as(w))]+ Ibws(wJ/2]}}} + ^pisfwj - s(w )} 1-1 1 2 o l 1 o 1 1 1 1 H1 First-order Necessary Conditions dw^/asfwj = e,{-ra + r a - arac+asfwj) + r a + (2/r,)[r -r (c+as(wj)].[-r ai 0 0 0 0 1 1 + r,<|>a + r,c<|)a + 2ras(w)<j> + p{-r a - (2/r,)[r - bw,s(w,) - 2 1 o 1 o r^c+asfwjtf-l-r^a] + bw,s(w,) + cl^a - aT + ar\c + 2a T s(wJ-bw s(w,)}+ 2 0 1 1 1-1 r ahxw + hxw, - §aT l 1 + (tiaTjC + 2<t>ras(w) - <t>bw,s(w,)} 2 1 0 l X Gjbslwjfw,- 1-1 w ) + pX 8.bs(w)(w-w.) + u, - u,.! = 0 j-i M dW^/dsiwd 2 l l l 1 = 9{(l+2(|))ar - (l+c|>)bwj > 0 a 1 1 These are restrictions on the denominators of s (wJ. which can be found using the MR first-order conditions. If we assume a separating solution, then we get Equation (5.19a): s M R ( W i ) 9,[hTw,-a(l+<t>){r-rc}] _ o V i =1 1 n i-i ^{(l-HjObw,- r(l+2<t>)a} + (l-PMw.-wJ Z 9j 2 l Proof of Proposition 5.7 From the first-order condition, we get (1) 9,{-r a + r a - aT.fc+asfwJ) + r a + o o o (2/r,)[r -r (c+as(w ))]-[-r al o 1 1 1 bw.sfwj - r<t»a + T ^ a + r^fa. + 2rias(w,)<t) + p{-r0a 2 0 (2/rjlr. - ri(c+as(wJ)]-[-r!a] + bw,s(w,) + cr\a - aT + aT,c + 2ars(w)-bw,s(w,)}+ 2 B 1 1 i-i I^ahxw, + hxw, - tyaT + QaTtC + 2<])ras(wJ - ^bw^Cwj} i-i 2 1 0 w ) + pX gjbsfwJlw.-Wn) + p, - p n = 0 w A similar condition may be written for w, and w,. Write (1) as +l 185 + a X 9,bs(wJ(w,- (2) 0.K, + u, = 0 The other first-order conditions can be written in a similar fashion. (3) 9 K, + p, - u, = 0 (4) + u, - u W +1 +1 +2 Now assume that Pi = ... = P n = p 1+1 = ... = I+a = 0 = 0. We are only considering the possibility of bunching between two adjacent types. From (2), we know that (5) -eft. u, = From (4). we know (6) p, = 0 aK, . Therefore, upon substituting (5) and (6) Into (3). +1 1+ +a we get e^ir™ + 8 A (7) + eft = o. If we assume that s(vr ) = s(w,) = I"*, we can solve this equation to get an expression M for s"*. This turns out to be 0A, + 0, A+l + 01+2^1+2 0B, + 0 + 0 +1 a " = (8) 1 1 + 1 B W l+2 Bn where: Ak = hxw - ad-f^lfr.-rxC) k = i. l+l, i+2 k k-l B,, = (l+4>)bw - ri(l+2<|))a + 2 k (l-fybfWfc-WkjZO, k = i. l+l, i+2 Using (1). we know that if 0, £ 0. then (9) s mj j Wi O [hTw a(l++)(r.-r c}] t r V i= 1 1 e,{(l+<t>)bw,- r(l+2<|))a} + 2 l Substituting (8) for s ^ w j In (9) gives 186 (l-pjb(w,-wj 2 0j J-l n (10) O A + fy+A+i + Qi+A+a + + 9 B 1+2 1+2 <, 9A 9,B, Cross-multipfying and solving for the ratio of probabilities establishes the result le^i/e.*] = C***, where C*" = [Aft* - B A J / K i B , -B1+A1 Q.E.D. 187
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The control of non-localized externalities with asymmetric information Van Egteren, Henry John Bernard 1989
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Title | The control of non-localized externalities with asymmetric information |
Creator |
Van Egteren, Henry John Bernard |
Publisher | University of British Columbia |
Date Issued | 1989 |
Description | This thesis presents a model in which there is a single agent and two principals. The agent is a monopoly firm, while the two principals are regulators who may cooperate when selecting their optimal policies or act as rivals. The specific regulatory problem in which all are involved is the control of acid rain. An environmental regulator chooses a design standard and a public utility regulator chooses a two-part pricing scheme. These choices are made within an environment of limited and asymmetric information. Specifically, we assume the firm knows more about its fixed abatement costs than does either regulator. The firm is able to act strategically when revealing this information. Within the context of this regulation problem, we characterize the equilibrium solutions when the regulators cooperate and when they act as rivals. The non-cooperative game endows the environmental regulator with the status of leader. We also give characterizations for two different kinds of rivalry, extreme rivalry and mild rivalry. In addition, this thesis presents some different results on the bunching properties of these models. |
Subject |
Environmental protection Acid rain |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-02-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0100648 |
URI | http://hdl.handle.net/2429/31372 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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