Introduction

This paper is primarily devoted to a study of the (static) optimality properties (e.g., Pareto-optimality of the equilibria) of certain resource allocation mechanisms. It is shown that one such mechanism (the “greed process”) is optimal in a class of economic environments much broader than the class for which perfect competition is optimal. More specifically, the greed process has the desired optimality properties for all environments from which so-called external economies or diseconomies are absent; unlike perfect competition, the greed process does not presuppose the absence of indivisible goods, of discontinuities, or of increasing returns. However, the greed process lacks the dynamic (stability) properties known to hold for perfect competition, at least in certain special cases.

That the greed process does have certain optimality properties would be of little interest, were it not for the fact that it belongs to a class of informationally decentralized processes and hence shares with perfect competition a feature that has been extolled as one of the main virtues of the classical market mechanism. Still, just because it is designed to cover a broader class of environments, the greed process calls for more information (is informationally less efficient) than the competitive mechanism. To illustrate this, a variant of the latter (called “quasi-competitive”) is constructed and is shown to have the desired optimality properties when the environment satisfies the usual divisibility and convexity assumptions, while requiring less information than does the greed process.

Formulation of the problem

Introduction

In this paper, we wish to discuss the bearing of some recent developments in mathematical economics on the problem of the optimal allocation of resources. We will confine attention here to an economy whose aims are well defined. That is, we assume that the preferences of the economic system can be embodied in a utility function which depends upon the outputs of commodities. For a given technology, the possibilities of different output combinations are restricted by the availabilities of primary resources. The problem of optimal resource allocation is to choose among all the feasible combinations of production processes that combination which maximizes the utility achieved by the economy.

Since the discussion is at a fairly high level of abstraction, the economy being studied may be a nation or some smaller economic system, including a single firm. The assumption that a single utility function represents the objectives of the economy fits best the case of a firm. For a nation, the assumption is less justified, but it provides an introduction, at least, to the more complex problem raised by the presence of many individuals, each of whom judges the workings of the economic system in light of his own utility function. We also avoid the subtle problems involved in defining optimality in the more general case.

The problem of choosing the allocation of primary resources among different productive processes so as to maximize a prescribed utility function is a mathematical one, and its solution in any concrete case can be regarded as a matter of computation.

Introduction

It is intended here to offer a possible characterization of the concept of complete ignorance. Like other formulations, the problem is taken to be that of choice of an action from a given set when the consequences of any action are functions of an unknown state of nature. However, the properties regarded as defining an optimal choice are designed to reflect completely the idea that there is no a priori information available which gives any state of nature a distinguished position. Most importantly, the optimality criterion differs from those in the now more standard subjective probability framework by not presupposing a fixed list of states of nature. As we note shortly, the arguments and conclusions are much closer to Shackle's than to those of Ramsey, de Finetti, and Savage.

The axiom systems of these last authors imply the existence of subjective probabilities as weights to be assigned to the different possible states of nature. These authors thus provide a foundation for the centuries-old use of probability as a guide to action. The concept of complete ignorance can be expressed in this subjective probability framework only by the assignment of equal probabilities to all the states of nature, which is the principle of indifference or insufficient reason implicit in the earliest combinatorial probability calculations of Pascal and Fermat and explicit in Jacob Bernoulli, Bayes, and Laplace.

This paper deals with the application of certain computational methods to evaluate constrained extrema, maxima, or minima. To introduce the subject, we will first discuss nonlinear games. Under certain conditions, the finding of the minimax of a certain expression is closely related to, in fact identical with, the finding of a constrained minimum or maximum. Let us consider then a game (in a generalized sense) where player 1 has the choice of a certain set of numbers x1, …, xm that are constrained to be nonnegative for present purposes and player 2 selects numbers y1, …, yn also constrained to be nonnegative but otherwise unrestricted. The payoff of the game, the payment made by player 2 to player 1, will be a function of the decisions made by the two players, the x's and the y's. This pay off will be designated by φ(x1, …, xm; y1, …, yx, …, yn). To play the game in an ideal way is to find the minimax solution; we know this solution exists under certain conditions. That is, we arrive at a choice of strategies by the two players where player 1 is maximizing his payoff given the strategy of player 2, and player 2 is minimizing the payoff, given the strategy of player 1.

The role of price adjustment equations in economic theory

In this essay, it is argued that there exists a logical gap in the usual formulations of the theory of the perfectly competitive economy, namely, that there is no place for a rational decision with respect to prices as there is with respect to quantities. A suggestion is made for filling this gap. The proposal implies that perfect competition can really prevail only at equilibrium. It is hoped that the line of development proposed will lead to a better understanding of the behavior of the economy in disequilibrium conditions.

In the traditional development of economic theory, the usual starting point is the construction for each individual (firm or household) of a pattern of reactions to events outside it (examples of elements of a reaction pattern: supply and demand curves, propensity to consume, liquidity preference, interindustrial movements of capital and labor in response to differential profit and wage rates). This point of view is explicit in the neoclassicists (Cournot, Jevons, Menger, and successors) and strongly implicit in the classicists (from Smith through Cairnes) in their discussion of the motivations of capitalists, workers, and landlords which lead to establishment of the equilibrium price levels for commodities, labor, and the use of land. The basic logic of Marx's system brings it, I believe, into the same category, although some writers have referred to his theories as being “class” economics rather than “individual” economics.

This book draws together a long series of papers by the two senior authors, alone and in collaboration with each other and with other friends and colleagues, to whose thinking and stimulation we are grateful. Both of us have had a strong primary concern with the workings of the economic system as a mechanism for achieving the optimal allocation of resources. The theme is of course an old one in economic thought; its importance was especially reinforced to us through our teachers, Harold Hotelling and Oskar Lange, and our colleague, Jacob Marschak. The very concept of optimization with resource constraints links the theory with the classical mathematical theory of constrained optimization and the more modern versions of so-called mathematical programming, where emphasis has been placed on inequality constraints.

The important property of the market as a resource allocation mechanism is its decentralization. It has always been assumed in the mainstream of economic theory, though frequently only implicitly, that the transmission of detailed information about tastes or technology is costly and that there is a virtue to systems in which decisions are made at the point where the information already existed. This desire for decentralization has, however, to be reconciled with the need for balance in the economy as a whole, most especially, in the need to respect limitations on the overall availability of resources. The mathematical characterizations of constrained optima, at least in the Lagrangian formulation and its generalizations, suggests the possibility of decentralization.

Introduction

A great deal of work has been done recently on what one may call the static aspects of competitive equilibrium, its existence, uniqueness, and optimality. This work is characterized, in the main, by being based on models whose assumptions are formulated in terms of certain properties of the individual economic units, although in the last analysis it is the nature of the aggregate excess demand functions that determines the properties of equilibria.

With regard to dynamics, especially the stability of equilibrium, much remains to be done. The concept of stability, used already by the nineteenth century economists in its modern sense, did not receive systematic treatment in the context of economic dynamics until Samuelson's paper of 1941. Samuelson, however, did not fully explore the implications of the assumptions underlying the perfectly competitive model. He (as well as Lange, Metzler, and Morishima) focused attention on the relationship between “true dynamic stability” and the concept of “stability” as defined by Hicks in Value and Capital, rather than on whether under a given set of assumptions stability (in either sense) would prevail or not. Even though the Hicksian concept does not, in general, coincide with that of “true dynamic stability,” it is of considerable interest to us for two reasons: first, as shown by the writers just cited, there are situations where the two concepts are equivalent; second, because the equilibrium whose “stability” Hicks studied is indeed competitive equilibrium.

The conditions under which a Walrasian system of multiple markets will be stable have been investigated by a number of authors under the implicit assumption of static expectations. It often has been assumed that expectations based upon an extrapolation of current rates of change (rather than upon the assumption that the future would be like the present) would prevent the system from converging onto its equilibrium position at all. Since interesting results have been scarce and difficult to achieve even in the case of static expectations, it is not surprising that little has been done with the relationship between extrapolative expectations and dynamic stability. In this paper, we shall introduce, under rather restrictive assumptions, a type of extrapolative expectations and we shall test its effects on the stability of a dynamic system.

Excess demands in a multiple market system are usually taken to be functions of the current prices of all goods. Ideally, it would be desirable also to include expected prices for all future time periods and for all individuals and all assets as arguments of the excess demand functions. A theory of such formal generality, however, would be necessarily devoid of much content. Abstractions and simplifying assumptions are necessary. There are many possible expectations functions by which people might relate current and expected prices, and there is a variety of ways to represent plausibly the type of extrapolative expectations which we wish to describe. Our choice was made largely on the grounds of mathematical simplicity.

The increasing span of government control over economic life in the last fifty years has directed the attention of economic theorists to the relative merits of centralization and decentralization in economic decision-making.

If the aim of the economic system is something like the maximization of national income, it may be asked whether it is better to make the economic decisions in a central agency where information relating to the entire system can be used or in the many independent units which characterize a capitalist economy such as ours.

In the course of this great debate, the role of the price system in coordinating many individual decisions has been given stronger and stronger recognition, although the idea itself already appears in Adam Smith's famous “invisible hand.”

There is also growing concern in the field of industrial management with the administration of large business corporations. Again there arises the issue of centralization vs. decentralization. To what extent is it necessary for the efficiency of a corporation that its decisions be made at a high level where a wide degree of information is, or can be made, available? How much, on the other hand, is gained by leaving a great deal of latitude to individual departments which are closer to the situations with which they deal, even though there may be some loss due to imperfect coordination?

A representative collection of management viewpoints on this issue is found in the proceedings of a conference held in The Netherlands some years ago.