Introduction

Market mechanism and the private ownership of means of production are the two major instruments by which the working of a capitalistic society may be efficiently organized. On the one hand, markets stimulate information concerning demand and supply conditions that is relevant for an efficient allocation of scarce resources. The private ownership of means of production, on the other hand, creates a social environment in which individual members are motivated to seek their own profit or pleasure in accordance with the rules set by society.

Such an argument implicitly presupposes that all the scarce resources limitational to the economic activities engaged in by the members of the society may be privately appropriated, without involving significant costs either in the administration of such private ownership or in the resulting efficiency as for the productive organization of economic activities. However, in most contemporary capitalistic societies, a significant portion of scarce means of production is not necessarily privately appropriated, as typically illustrated by the existence of a large class of means of production usually termed social overhead capital.

Social overhead capital comprises all those scarce resources which are put in use for the members of the society, either free of charge or at a negligible price. They are either produced collectively by the society, as in the case of social capital such as highways or bridges, or simply endowed within the society, as in the case of natural capital such as air, water, and so forth.

Introduction

The theory of production, as typically described by Samuelson [2, (IV, 57–89)], is primarily concerned with the optimum allocation of factors of production that minimizes the total cost for each output, and with the nature of the cost curves derived from production processes under neoclassical hypotheses. However, it is customary in econometric studies of production structure to specify the form of production functions, up to a certain parametric class (such as Cobb-Douglas or Constant Elasticities of Substitution), and then estimate the parameters, through the cost curves which are usually derived by minimization of total cost. It is of some interest to see if production functions are uniquely determined by curves of minimum total cost and to characterize the class of total cost curves which are derived from production functions with neoclassical properties. This dual determination of production functions from cost curves has been established by Shephard, and in the present note we are interested in extending some of his results as well as formulating explicitly the conditions for cost curves that are derived from neoclassical production processes by a minimization of total cost.

The structure of cost functions

The model of production dealt with in this note consists of one output and a finite number of inputs, say 1, …, n. The structure of production is characterized by specifying the set of all combinations of inputs which result in a given quantity of outputs.

Introduction

In the present paper we are interested in the growth process in a two-sector model of capital accumulation and show that balanced growth equilibria are globally stable under the neoclassical hypotheses.

The neoclassical model of economic growth, as it has been developed by Solow and Swan, is formulated in terms of the aggregate production function. The aggregate production function specifies the relationship between output and factors of production, and output is assumed to be composed of homogeneous quantities identical with capital, or at least price ratios between output and capital are assumed constant. The economy we are concerned with in this paper, on the other hand, consists of two types of goods, investment-goods and consumption-goods, to be produced by two factors of production, capital and labor; prices of investment-goods and consumption-goods are determined so as to satisfy the demand requirements. It will be assumed that capital depreciates at a fixed rate, the rate of growth in labor is constant and exogenously determined, capitalists' income is solely spent on investment-goods, that of laborers on consumption-goods, and production is subject to the neoclassical conditions. Under such hypotheses, then, it will be shown that the state of steady growth exists and the growth process, starting at an arbitrary capital and labor composition, approaches some steady growth. If the consumption-goods sector is always more capital-intensive than the investment-goods sector, then the steady growth is uniquely determined and it is stable in the small as well as in the large.

In 1954 or thereabouts, I received in the mail a letter and a manuscript from Japan. The letter came from a young man, Hirofumi Uzawa, who identified himself as a graduate student in mathematics at the University of Tokyo. The manuscript referred to an unpublished paper by Leonid Hurwicz and myself, distributed as a RAND Corporation paper, which proposed what amounted to an iterative method of solving concave programs based on a partial analogy to the competitive market. We had only been able to show that the method was locally stable; if the starting point was sufficiently close to the optimum, the method would indeed converge to it. Uzawa's manuscript showed elegantly and simply that the Euclidean distance to the optimum was bound to be decreasing and therefore our process must converge globally. The letter stated with characteristic modesty that undoubtedly the author's paper contained errors, and he would be grateful to have them pointed out. Of course, there were no errors. I immediately looked over what small research budget I then had from the Office of Naval Research and found enough to invite him to spend the next year at Stanford. How fortunate to have had the opportunity of making such a good decision even once in a lifetime.

Uzawa promptly took a leading role in the development of economic theory. His earliest work took off from current topics, such as concave programming and the existence and stability of competitive equilibrium.

The papers collected here deal with various topics in economic theory, ranging from preference and consumption, duality and production, equilibrium, capital, and growth, to the theory of social overhead capital. They all have a common theme: to try to formulate the working of economic forces in a capitalist economy in terms of mathematical models and to explore their static, dynamic, and welfare implications.

Most of the papers were published during the 1960s and a few in the early 1970s. Economic theory since then has made significant progress, both in the refinement and elaboration of analytical methods and in the enlargement of the scope of economic inquiries. In particular, institutional and social aspects of the contemporary economic systems have been explicitly incorporated into the formal framework of the economic theory, and their implications for the mechanism of economic processes have been fully explored. However, the analytical methods developed in the papers and the theoretical implications derived thereby still seem to retain value, particularly in tracing the origin of some of the contemporary works in economic theory.

This volume represents work done while I was affiliated with Stanford University, the University of California at Berkeley, and the University of Chicago. During this period, I was extremely fortunate in being able to work with Professor Kenneth J. Arrow, whose intellect influenced almost all the work I was then doing. I should like to take this opportunity to acknowledge my intellectual and personal indebtedness to him.

Introduction

The purpose of this note is to show the equivalence of two fundamental theorems – Walras's Existence Theorem on the one hand and Brouwer's Fixed-Point Theorem on the other.

Walras's theorem is concerned with the existence of an equilibrium in the Walrasian system of general equilibrium and has been a problem of some importance in formal economic analysis since his work appeared in 1874–7. It was, however, not until Wald's contributions, and, that the existence problem was rigorously treated. Recent contributions, in particular those of Arrow and Debreu, McKenzie, Nikaidô, and Gale, have shown that Walras's theorem is essentially a necessary consequence of Brouwer's Fixed-Point Theorem. The latter theorem, first proved by Brouwer in 1911, also bears a fundamental importance in mathematics. It may be hence of some interest to see that Brouwer's theorem is in fact implied by Walras's theorem. It would indicate the reason that the general treatment of the existence problem in the Walrasian system had to wait for the development of the twentieth century mathematics.

Walras's Existence Theorem

According to Gale and Nikaidô, Walras's theorem may be formulated as follows.

Let there be n commodities, labeled 1, …, n, p = (p1, …, pn) and x = (x1, …, xn) be a price vector and a commodity bundle, respectively. Price vectors are assumed to be nonzero and nonnegative; commodity bundles are arbitrary n-vectors.

Introduction

One of the basic problems in economic planning, in particular in underdeveloped countries, is concerned with the rate at which society should save out of current income to achieve a maximum growth. It is closely related to the problem of how scarce resources at each moment of time should be divided between consumer's goods industries and capital goods industries. In the present paper, we shall analyze the problem in the framework of the two-sector growth model as introduced by Meade, Srinivasan, and Uzawa. We shall abstract from the complications which would arise by taking into account those factors such as changing technology and structure of demand, the role of foreign trade (in particular, of capital imports), and tax policy that are generally regarded as decisive in the determination of the course of economic development. Instead, we shall focus our attention on evaluating the impact of roundabout methods of production upon the welfare of society, as expressed by a discounted sum of per capita consumption. However, since our primary concern is with economic planning in underdeveloped countries, we shall depart with respect to one important point from the two-sector growth model as formulated in which is, in general, concerned with an economy with fairly advanced technology and relatively abundant capital; namely, we shall postulate that a certain quantity of consumers' goods (per capita) is required to sustain a given rate of population growth.

Overview

Utilitarianism is a philosophical thesis two centuries old. It judges collective action on the basis of the utility levels enjoyed by the individual agents and of those levels only. This is literally justice by the ends rather than by the means. Welfarism is the name, coined by Amartya Sen, of the theoretical formulation of utilitarianism, especially useful in economic theory and other social sciences. Its axiomatic presentation, developed in the last three decades, is the subject of Chapters 2 and 3.

For the utilitarianist, social cooperation is good only inasmuch as it improves upon the welfare of individual members of society. The means of cooperation (social and legal institutions, such as private contracts and public firms) do not carry any ethical value; they are merely technical devices - some admittedly more efficient than others - to promote individual welfares. For instance, protecting certain rights - say, freedom of speech - is not a moral imperative; it should be enforced only if the agents derive enough utility from it.

This very dry social model rests entirely upon the concept of individual preferences determined by the agents “libre-arbitre” while deliberately ignoring all its mitigating factors (among them education and the shaping of individual opinion by the social environment, as well as kinship, friendship, or any specific interpersonal relation). Ever since Bentham, utilitarianists have been aware of these limitations. Yet, oversimplified as it is, the utilitarian model is easily applicable, and its "liberal" ideology has the force of simplicity. The philosophical debate on utilitarianism is still quite active (see, e.g., Sen and Williams [1982]).

Overview

The economic behavior of a public firm is among the oldest problems of welfare economics. The two main instances where a public firm is called for are the provision of a public good and that of a private good of which the production technology has increasing returns to scale (IRS). Those are cases of natural monopoly. On the economic theory of natural monopolies see Baumol, Panzar, and Willig [1982]; on the optimal provision of a public good see the historical review by Musgrave and Peacock [1958] and the theoretical survey of Milleron [1972]. In both cases, the only efficient organization of production utilizes a single production unit, yet it would be socially wasteful to let this unit be operated by a profit-maximizing monopolist. Thus, there exist the need for a regulated public firm and the compelling normative question of its pricing policy.

Even under decreasing returns to scale (DRS), the case for joint production frequently arises. The workers of a cooperative jointly own its machines, and the partners of a law firm share its patrons. They must choose an equitable scheme to share the fruits of their cooperation. Thus, the general discussion of a public firm involves an arbitrary technology; if it has decreasing returns, we mean that it is operated jointly by the agents with no possibility of duplication (there is a large implicit fixed cost to open a new production unit).

Overview

Nash [1950] proposed to generalize SWOs to more complex choice rules that we call social choice functions (SCFs). The idea is to take into account the whole feasible set of utility vectors to guide the choice of the most equitable one. Accordingly, any two utility vectors may no longer be compared independently of the context (namely, the actual set of feasible vectors), as was the case with SWOs. To understand how this widens the range of conceivable choice methods, think of relative egalitarianism as opposed to plain egalitarianism. Say that two agents must divide some commodity bundle. The egalitarian program simply chooses the highest feasible equal utility allocation (assuming away the equality-efficiency dilemma) independently of the feasible unequal utility vectors.Relative egalitarianism, on the other hand, computes first the utility w, that each agent would derive from consuming alone the whole bundle; then it chooses thisefficient utility vector where the ratio of the actual utility to the highest conceivable utility w, is the same for each agent. In other words, relative egalitarianism equalizes individual ratios of satisfaction (or of frustration) by defining full satisfaction (i.e., zero frustration) in a context-dependent manner. We give a numerical example (Example 1.1), stressing the difference between egalitarianism and relative egalitarianism.

Overview

In Part III we apply the tools forged in Parts I and II to specific microeconomic surplus-sharing problems. The two problems discussed in this chapter are the simplest models of surplus and cost sharing, in which the distributive issue has no obvious answer.

The cost-sharing problem is best thought of as the provision of an indivisible public good, say, a bridge or any such public facility. The issue is how to share the cost of the facility. The only available data are the total cost of building the bridge and the benefits that each agent derives from the facility (a single dollar figure per agent since we assume quasilinear utilities). Another interpretation of the same formal model is the settlement of a bankruptcy. Here the agents are creditors of the bankrupt firm; each agent holds a monetary claim over its estate, but the total value of the estate falls short of the sum of the claims. The problem is to share the estate among its creditors.

The surplus-sharing problem is that of dividing the proceeds of an indivisible cooperative venture among several partners. Think of an orchestra giving a single performance. The receipts must be shared among the musicians; the sharing rule must be based upon the estimated “market value” of each musician, again, a single number per agent representing his opportunity cost of joining the orchestra. Assuming that the receipts exceed the sum of the opportunity costs, how should it be divided among the agents?