Pure exchange seems quintessentially academic, a storybook portrayal of economic interaction far removed from what we observe in all but the most primitive of societies. Production, and the organization of production by firms, must surely be regarded as the very essence of a modern, capitalistic economy. You might imagine, therefore, that allowing for production would lead to a drastic modification of the theory of general equilibrium. But you would be wrong because, at least as the theory is currently formulated, adding production changes very little.

Of course, the fact that not much changes is a signal that the general equilibrium theory of production does not amount to much. The passive, price-taking firm in Walrasian equilibrium provides little scope for entrepreneurial skill or managerial know-how. But, having acknowledged these shortcomings, it is equally important to give the general equilibrium treatment of production its due. Viewed on its own terms, it is an intellectual tour de force — the first (some would say the only!) successful attempt to build a model in which consumers and firms interact in a logically consistent fashion.

The goal of this chapter is neither to disparage the general equilibrium theory of production nor to claim more than the theory can deliver. Rather I emphasize what is perhaps the most attractive feature of this theoretical point of view, the clear parallel which it draws between economies of pure exchange and those with production, a parallel emphasized both through choice of notation and through the use of an (appropriately modified) net trade diagram.

This book is the outgrowth of a course on general equilibrium theory I have taught to second year graduate students for the past dozen years or so. A few intend mathematical economics as their primary specialty, but most do not. This book is designed to meet the needs of both types of student. In my experience, it has been quite effective.

What specialist and nonspecialist alike need from a book like this is motivation: Where are we heading and why? The first three chapters provide that motivation, building the case that general equilibrium theory is indeed worth learning. What makes the approach work is a reliance on examples rather than theorems and proofs. Examples provide an excellent, and rather painless, opportunity to acquire facility with the abstract notation and concepts of general equilibrium theory. Applications are nontrivial and often unusual: overlapping generations, contingent commodities, indivisibility, local public goods, and hedonic theory. Exercises provide the opportunity to test and strengthen competence.

Although motivation is paramount, these first three chapters accomplish more than that. The reader finishing this introductory material is familiar with important facts about vector spaces and their duals, and sensitive to their geometry. She has learned how to translate the central concepts of general equilibrium theory into the formal language of mathematics, and the formality seems less abstract. And she has a stockpile of examples to draw upon to illustrate the theory building which follows.

The remaining five chapters build upon this foundation, making the transition from examples to rigorous theory.

In most models with a finite number of consumers the core is larger than the set of Walrasian equilibria. What is this telling us? Perhaps that the core is a flabby concept, Walrasian equilibrium precise. But, more likely, this discrepancy signals a problem with the Walrasian hypothesis of price-taking behavior. In most situations where the Walrasian model has been applied, consumers can influence price. The core recognizes the associated opportunities to haggle and bargain. Walrasian equilibrium assumes them away.

Of course, economists have an answer to this criticism. We are not that naive! Although we illustrate the competitive model with an Edgeworth box, the typical applications we have in mind involve not two consumers but many, so many that each consumer has a negligible influence on price. In 1964 the game theorist Robert Aumann made a bold suggestion: if economists intend their models of competition to apply in situations where consumers have negligible influence on price, why not reformulate the model to be consistent with this tacit assumption? The opening two paragraphs of his paper state the case with great force and clarity:

The first part of this book stresses broad concepts and examples, the second rigorous theory. Topology allows us to make the transition. Why does topology play this critical role? Essentially because topology allows us to make headway in characterizing qualitative properties of a model when quantitative information is missing. Why not seek better quantitative information instead? This goes to the heart of the matter, to the basic question of what competitive theory is really about. In modeling a competitive economy we want to assume as little as possible about what consumers and firms are like because it is the absence of such information which renders planners impotent and markets worth having. At least that is what we, as economists, have claimed from Adam Smith on. Modern general equilibrium theory can be faulted for not going far enough in this direction, but it certainly does try.

This chapter looks at individual consumers and firms in isolation, ignoring for now how they interact with one another. The main issues concern their response to prices and other aspects of their environment. The questions we ask are deliberately qualitative, not quantitative. Not what formula describes the demand of a consumer with Cobb-Douglas utility, but rather: Does the consumer have a best response to her environment? If the best response is not unique, what more can we say about the set of best responses? Does the response vary continuously when the environment changes? Without topology we cannot get far in addressing these questions. With topology we can say quite a lot.

Introduction

The purpose of this chapter is to extend the radial input-based efficiency measures in Chapter 3, where efficiency of the observed input vector is judged relative to technology, given fixed outputs, to the case in which input efficiency is judged relative to the revenue indirect production technology. In this case, outputs are restricted but not fixed, and they may vary as long as their revenue meets or exceeds a prespecified target. In contrast, in Chapter 3 outputs are assumed exogenously fixed.

In Section 5.1, price-independent measures of indirect input-based technical efficiency are developed. These measures take output prices and target revenue as given and measure efficiency as proportional contraction of all inputs. The measures distinguish between scale, congestion and purely technical efficiency. Complete decomposition of the constant returns, strong disposal revenue indirect measure of technical efficiency is provided. All our technical measures are radial in nature, but it is clear that nonradial measures like the Russell measure introduced in Chapter 3 may also be considered. We leave this for the interested reader.

Price dependent input-based measures are discussed in Section 5.2. We introduce input prices and consider the objective of shrinking expenditure on input. This leads to revenue indirect measures of input cost efficiency and input allocative efficiency which, together with one of our technical measures, yields a decomposition of revenue indirect cost efficiency.

In Section 5.3 we discuss measures of benefit effectiveness by comparing the minimum cost with the target revenue it generates. In Section 5.4 comparisons between direct and indirect measures are discussed.

Introduction

In this chapter we model technology with the input correspondence u → L(u). We assume that x ∈ L(u) and measure production efficiency by calculating where in the input set L(u) the input vector x is located. Thus we take the observed output vector u as given and adopt a resource conservation approach toward efficiency measurement. We refer to this approach as input based, since inputs are the choice variables, and we measure efficiency in terms of maximum feasible shrinkage of an observed input vector, feasibility being determined by the input set L(u).

Shrinkage can be given many interpretations. Throughout most of the chapter shrinkage is accomplished radially, i.e., by an equiproportionate reduction in all inputs. In Section 3.1 input-based efficiency measures that are radial and independent of prices are introduced. We obtain a measure of technical efficiency, and show how it can be decomposed into three components that measure the separate contributions of scale efficiency, congestion of inputs, and “pure” technical efficiency. Each of these measures is calculated relative to the input set L(u) that takes output as given.

In Section 3.2 we introduce input prices and develop a radial, price dependent measure of cost efficiency. This measure is then decomposed into two parts, technical efficiency (which itself has three components) and allocative efficiency. This enables us to attribute potential cost saving to elimination of waste and adjustments to the input mix.

A difficulty with radial measurement of technical efficiency is that a radially shrunken input vector need not necessarily belong to the efficient subset of the input set.

Introduction

The purpose of this chapter is to extend the radial output-based efficiency measures in Chapter 4, where efficiency of the observed output vector is judged relative to technology with fixed inputs, to the case in which outputs are judged relative to the cost indirect production technology. In this case, inputs are restricted but not fixed, they may vary as long as their cost does not exceed a prespecified target. In contrast, in Chapter 4, inputs are assumed exogenously given and fixed.

In Section 6.1, measures of indirect output-based technical efficiency are developed. These measures take input prices and planned cost as given and measure efficiency as proportional expansion of all outputs. The (C, S) measure is then decomposed into component measures of scale efficiency, congestion, and purely technical efficiency, all relative to the cost indirect technology.

In Section 6.2 we introduce output prices and obtain a cost indirect output revenue efficiency measure. This measure gauges the extent to which a producer succeeds in maximizing output revenue when constrained by output prices, input prices, and an input budget. This cost indirect revenue efficiency measure is then decomposed into technical and allocative components, the technical component being the measure introduced in Section 6.1.

In Section 6.3 we present a measure of revenue effectiveness, which shows the ability of a producer to convert an input budget into revenue.

In Section 6.4 our artificial data set is used to illustrate some of the cost indirect output efficiency measures developed in this chapter.

Introduction

In this chapter we depart from earlier chapters in that we model technology and measure efficiency relative to what we call price sets. In contrast, Chapters 3 through 6 judged efficiency and modeled technology relative to what might be called quantity sets. These earlier chapters also had in common the assumption that input and output prices are taken as given, i.e. the individual unit is assumed to be a price taker. In this chapter we model the case in which prices are the choice variables and input and output vectors are taken as given.

In evaluating performance relative to price sets rather than quantity sets we are departing from the usual economic approach, which generally takes quantity data as “primal”. In fact, one possible interpretation of the measurement of efficiency in price space is to assume that this is an accounting model, rather than an economic model. Another interpretation is related to what has come to be known as a nonminimal cost function approach. In that approach it is assumed that firms' observed costs and observed or market prices are “distorted” – due to regulation, for example, price controls, imperfect competition, etc. Or firm managers may be utility maximizers rather than cost minimizers. In these cases, firm behavior may be consistent with shadow cost minimization with respect to shadow prices (which may deviate from observed prices). The goal of the nonminimal cost literature is to identify those shadow prices and use their deviation from observed prices as a measure of allocative efficiency.

Introduction

Chapters 3 and 4 of this monograph introduce and decompose cost and revenue efficiency, respectively. In this chapter we are interested in profit efficiency, which requires that we simultaneously adjust input and output quantities, given input and output prices. Since we simultaneously wish to increase output quantities and associated revenues and decrease input quantities and their associated costs, we require a different reference technology and perhaps a different means of attaining the frontier of that technology than in Chapters 3 and 4.

In order to judge adjustments of both input and output quantities simultaneously, we model technology with the graph. Specifically, we introduce a series of efficiency measures which judge performance relative to technology as described by the graph. In contrast to most of the efficiency measures introduced in earlier chapters, those introduced here are not radial contractions or expansions of observed data, but rather follow a hyperbolic path to the frontier of technology. More specifically, both inputs and outputs are allowed to vary by the same proportion, but inputs are proportionately decreased while outputs are simultaneously increased at the same proportion.

The fact that we require increases in output quantities and decreases in input quantities to occur at the same proportion or rate, yields a hyperbolic path to the frontier of the graph. One can imagine alternatives: a Russell type measure which allows variation in the rate of increase of individual outputs and in the rate of decrease of individual inputs, or a measure which increases outputs at a proportionate rate which may differ, however, from the proportionate rate of decrease of inputs. These variations are left to the interested reader.

The Objectives of this Study

We have four goals we wish to accomplish in this study. The first two are interlinked, and involve the development of a model of efficient producer behavior, and the simultaneous development of a taxonomy of possible types of departure from efficiency, in a variety of environments. These two goals are accomplished by constructing a variety of production frontiers, and measuring distance to the frontiers. The third goal is the development of an analytical and computational technique for examining the first two. The technique we use, linear programming, is one of several available techniques, but is the one we prefer. The fourth goal is a demonstration of the wide applicability of the approach we take to modeling producer behavior. We meet this fourth goal in a number of ways, but primarily by adopting an attitude throughout the study that this is a study in applied production analysis. We focus on the empirical relevance of producer frontiers and distance tothem. We apply the linear programming techniques to artificial data to illustrate the type of information they can generate. And we frequently suggest problems to which these ideas can be applied.

Conventional microeconomic theory is based on the assumption of optimizing behavior. Thus it is assumed that producers optimize from a technical or engineering perspective by not wasting resources. Loosely speaking, this means that producers operate somewhere on the boundary, rather than on the interior, of their production possibility sets. Producers are also assumed to optimize from an economic perspective by solving some allocation problem that involves prices.

In recent years there has occurred a rapid increase in the volume of published research devoted to the analysis of efficiency in production. A small amount of this research has explored the theoretical foundations of efficiency measurement, but the vast majority of it has been empirical. The empirical research has investigated the nature and magnitude of, and occasionally the influences on, productive efficiency in a wide variety of industries, across a multitude of countries, and over a span of time stretching from Domesday England c. 1086 to the present. Some of the empirical research has been of primarily academic interest, some of it has had considerable public policy relevance, and some of it has been directed to managerial decision making. The research is interdisciplinary, having spread from economics to operations research and to the fields of management science, public administration, and a host of others. It seems to be having an impact in each of these fields.

Our primary motive for writing this book is to provide, in a single source, the theoretical foundations for the measurement of productive efficiency. Most of these foundations have appeared before, over a period of time and in a variety of sources, and we think it worthwhile to bring them together in one accessible source.

Introduction

In this chapter we expose the reader to the production models relative to which efficiency will be evaluated. We include models whose map sets are in quantity space and models whose map sets are in price space. More specifically, the first group of models is divided into those which involve input quantities and output quantities, such as the graph of the technology and the input and output correspondences. The first group of models also includes those which are value constrained. In the value constrained group we include the cost indirect output correspondence and the revenue indirect input correspondence. Examples of models with map sets in price space include the dual input and dual output correspondences. All these models are introduced in Section 2.1.

To avoid an overwhelming number of technicalities, little or no discussion is devoted to the particular properties different models possess. However, in the piecewise linear formulation of these models developed in Section 2.4, the properties relevant for efficiency measurement are derived.

In Section 2.2 we introduce two sets of notions that model variation in the size of operation of a production unit. First we define returns to scale with respect to each of the seven models introduced above, and then we define the notion of returns to diversification. Thus in Section 2.2 both scaling and addition in production are discussed.

Disposability – both weak and strong – is discussed in Section 2.3. In particular, since some outputs may be “bads” and some inputs may cause congestion, we need both concepts. If, for example, outputs are strongly (also termed freely) disposable, then any output can be discarded without resource use or cost.