This book presents a coherent and systematic exposition of the mathematical theory of the problems of optimization and stability. Both of these are topics central to economic analysis since the latter is so much concerned with the optimizing behaviour of economic agents and the stability of the interaction processes to which this gives rise. A basic knowledge of optimization and stability theory is therefore essential for understanding and conducting modern economic analysis. The book is designed for use in advanced undergraduate and graduate courses in economic analysis and should, in addition, prove a useful reference work for practising economists.

Although the text deals with fairly advanced material, the mathematical prerequisites are minimised by the inclusion of an integrated mathematical review designed to make the text self-contained and accessible to the reader with only an elementary knowledge of calculus and linear algebra. We strongly urge the reader to peruse the material contained in the mathematical review before proceeding to the main text. Furthermore, Chapter 1 on convexity is to some extent a reference chapter, and can be regarded as an extension to the mathematical review in that it presents certain fundamental properties of convex sets and functions which are used throughout the text. The reader with a basic knowledge of convexity may begin with Chapter 2, where the theory of static optimization is developed. Chapter 4 on comparative statics and duality can be read immediately after Chapter 2 if so desired.

SETS AND MAPPINGS

A set can be regarded as a collection of objects viewed as a single entity. The objects in the collection are called elements or points of the set. If a is an element of set A we write a∈A; a∉A denotes that a is not an element of set A. A set can be defined by listing its elements, A = {1,2,3}, or by stating a common property of its elements; A = {x:x has property P} denotes the set consisting of all objects x that have property P. When two sets A and B have identical elements they are equal, A = B: A ≠ B then denotes that the elements of A and B are not completely identical with each other. A set B is a subset of the set A, denoted B ⊂ A, if for all b∈B, b∈A. Thus, a set is also defined to be a subset of itself. When B ⊂ A and B ≠ A, B is a proper subset of A. Clearly, A = B if, and only if, A ⊂ B and B ⊂ A. The set which contains no elements is called the null or empty set, denoted ø. The null set is a subset of every set. A set can have other sets as its elements. For example, we can consider a set X = {A, B, C} whose elements A, B, C are themselves sets. Here we call X a family of subsets.

INTRODUCTION

In any static optimization problem, the objective function and the constraint functions will contain certain parameters and the optimal solution will depend on the values taken by these parameters. Thus, if any particular parameter value is altered, then we should expect the optimal choice of control variables and the maximum value of the objective function to change. The determination of the effects of parameter variations on the optimal choice of control variables and the maximum value of the objective function is referred to in the economics literature as comparative statics analysis. Section 4.2 is devoted to comparative statics analysis.

Closely related to comparative statics analysis is the theory of duality. At the heart of duality theory in economics is the notion of ‘equivalent representations’. Following Epstein (1981) we may say that:

Thus, in economics, duality refers to the existence of ‘dual functions’ which, under appropriate regularity conditions, embody the same information on preferences or technology as the more familiar ‘primal functions’ such as the utility or production function. Dual functions describe the results of optimizing responses to input and output prices and constraints rather than global responses to input and output quantities as in the corresponding primal functions.

Introduction

Rate-of-return regulation of a profit-seeking firm not only invites an inefficient choice of inputs to produce any given level of output, it may also cause output levels to be chosen inefficiently. Much attention has been accorded the technical input distortion discussed in Chapter 8. We focus in this chapter on the second distortion, the inefficient output levels resulting from nonoptimal relative prices for multiple outputs. Our approach is to see whether firms will adopt second-best welfare-maximizing prices when seeking profit or other goals while subject to profit regulation. We shall find that pricing efficiency is not reliably motivated in firms regulated under Hope guidelines.

An early analysis of rate-of-return regulation by Wellisz (1963) stressed the resulting economic inefficiency in pricing, specifically between peak and off-peak prices. Averch and Johnson (1962) also described a pricing distortion that rate-of-return regulation might induce in a multiproduct firm. Bailey (1973) extended Wellisz's peak-load pricing analysis and asked whether input distortions beyond those due to inefficient pricing would arise and whether inefficient pricing between peak and off-peak periods would result if firms sought to maximize output instead of profit. Waverman (1975) has since confirmed that, along with peak and off-peak pricing distortions, the Averch and Johnson (1962) type of bias between capital and other inputs would also result. Needy (1975) provided a systematic description of output distortions in the profit-maximizing, rate-of-return regulated firm. Pricing distortions due to both monopolistic reliance on demand elasticities and bias toward capital were set out in Bailey and White (1974), Eckel (1983), Sherman and Visscher (1979, 1982a), and Srinagesh (1986).

Introduction

The welfare goal that competitive markets serve can be set out also as an objective for regulation to achieve when competition cannot function well. We have focused attention on expressing such ideal outcomes, taking economic efficiency as our dominant goal. Chapter 2 presented ways to represent economic welfare and Part II was devoted to the study of prices that would make such welfare as large as possible. But Chapter 3 brought out difficulties inducing the pursuit of welfare through regulation, and Part III showed that real-world regulatory institutions are not really designed to pursue that aim. Indeed, the contrast between Parts II and III reveals how far institutions are from idealized conceptions.

We now want to consider what might be done to improve regulatory performance, especially where monopoly power exists because entry is prevented. In Section 11.2 we briefly review ways to persuade present organizations to pursue welfare aims. Since these organizations and their regulation were influenced by contending parties understandably promoting their own interests, it is not surprising that they are not ideal. Without the forces of competition, it can be difficult to induce welfare aims in a firm or industry. But that is not the only problem. Regulatory institutions have grown out of conflict among different consumers and between consumers and enterprise owners or managers. Political and judicial institutions have played an important role in their formation, and outcomes have represented compromises between opposing positions.

Introduction

Market economies face the same problem throughout the world: how to deal with technologies that complicate the smooth functioning of competition. Television, telephone, water, natural gas, electricity, and railroad transportation illustrate large and/or complex technologies, the use of which, for one reason or another, is guided in many countries by administrative institutions rather than competitive markets. Our aim is to examine the circumstances that cause alternative means of regulation to be substituted for competition, and to consider the approaches taken. We shall find that although normative guidelines can be developed for the alternative institutions, incentives to use them are weak or nonexistent. Designing institutions so they will pursue social goals is not a simple matter, and creating real institutions with that aim is even more difficult.

In the United States an unusual solution to the regulatory problem was chosen for many services. The services are still provided by privately owned firms, but those firms are regulated by public agencies. The firms are called public utilities, a title traceable to their nineteenth-century origin, and they are seen as providing goods or services in which the general public has a great interest. The public regulatory agencies that oversee them are commonly operated at the state level, as a Public Service Commission, State Corporation Commission, or similarly titled agency. The vast majority of electricity, natural gas, television, and local telephone service, plus large amounts of water, public transportation, and other services are provided by privately owned and governmentally regulated public utilities.

Introduction

Thus far we have emphasized an efficiency goal while accepting the current distribution of income as satisfactory. When entry is freely allowed into a market, the cross-subsidization that is necessary if income is to be redistributed through prices cannot be sustained. Yet it is sometimes reasonable to affect the distribution of income through public pricing decisions, particularly if income redistribution goals are widely agreed upon and other means of affecting income distribution are not available. Even if income distribution is not being sought as a goal, it may be useful to understand how it is affected by prices. Considering how individual welfare weights can affect prices shows more clearly why the assumptions underlying consumer surplus do not call for income redistribution.

We first examine the idea of anonymous equity. It is consistent with free entry, and if its conditions are met, cross-subsidization among consumers is impossible. We next consider the pricing implications of welfare weights that differ from those in consumer surplus based on the current distribution of income. We show how alternative weights affect optimal prices and can even rationalize cross-subsidization, which means they require entry barriers to be effective. The effect of welfare weights is extended to the case in which consumption of certain “beneficial” goods creates positive external effects and so is to be encouraged. Discussion is initially confined to uniform prices, prices that are proportional to quantity consumed for all consumers.

The regulation or control of monopoly is being reconsidered today in economies as different as those of China and Singapore, or the United States and the Soviet Union. Beneath catchwords like privatization, liberalization, deregulation, or perestroika lies a fuller awareness of the pitfalls in both private and public ownership of monopoly enterprises than ever existed before. Governments seem willing to consider introducing new institutions on the basis of their expected effects rather than their ideological bloodlines alone, and to undertake radical reform. It is a good time to consider the purposes of regulatory institutions and to attend to their design.

There is a vast economics literature on optimal pricing and technological choices. Anyone at all familiar with it must be struck by how it seems to differ from daily experience. One of our aims here is to present a large portion of that admirable normative material, in up-to-date form, and to explain the welfare representations underlying it. But we also want to stress that little has been done to adopt means of pursuing welfar goals in the absence of competition. Our institutions of monopoly regulation were not carefully designed for pursuing economic welfare or efficiency, at least not by today's standards. Experiments with new institutions of monopoly regulation will be valuable in the challenging design task that remains to be accomplished.

This book was begun more than five years ago as a set of notes for a graduate course in regulation at the University of Virginia. Only elementary mathematics, mainly calculus, is employed, frequently in straightforward constrained optimization problems.