Introduction

The literature on index numbers is so vast that we can cover only a small fraction of it in this chapter. Frisch (1936) distinguishes three approaches to index number theory: (i) ‘statistical’ approaches, (ii) the test approach, and (iii) the functional approach, which Wold (1953, p. 135) calls the preference field approach and Samuelson and Swamy (1974, p. 573) call the economic theory of index numbers. We shall mainly cover the essentials of the third approach. In the following two sections, we define the different index number concepts that have been suggested in the literature and develop various numerical bounds. Then in section 4, we briefly survey some of the other approaches to index number theory. In section 5, we relate various functional forms for utility or production functions to various index number formulae. In section 6, we develop the link between ‘flexible’ functional forms and ‘superlative’ index number formulae. The final section offers a few historical notes and some comments on some related topics such as the measurement of consumer surplus and the Divisia index.

Price indexes and the Konüs cost of living index

We assume that a consumer is maximizing a utility function F(x) subject to the expenditure constraint where x ≡ (x1, …, xN)T ≥ 0N is a non-negative vector of commodity rentals, p ≡ (p1, …, PN)T ≫ 0N is a positive vector of commodity prices and y > 0 is expenditure on the N commodities.

Introduction

The study of labour supply and commodity demands has, for the most part, proceeded on separate lines. There has been an extensive literature, much of it inspired by the work of Sir Richard Stone (see, for example, Stone 1954), on the estimation of commodity demand systems; and there has been a recent growth of interest in labour supply equations. However, there have been relatively few attempts to estimate jointly labour supply and commodity demand relationships. At a theoretical level, the main contribution to linking these two aspects of household decision-making has been in the work on household production. The ‘activities’ approach, developed particularly in Becker's theory of the allocation of time (1965), provides considerable insight, but has not been widely adopted in empirical research. In this paper we build on Becker's theoretical work and develop the activities approach as the basis for an econometric investigation of the joint determination of labour supply and commodity demand in the United Kingdom.

We begin in section 2 by discussing the household allocation of income and time, and relating it to the theory of rationing. To illustrate the extension of the standard consumer demand model, we take in section 3 the case of the linear expenditure system. This provides the basis for the empirical work, which uses data on expenditure by commodity category, and hours of work, contained in the Family Expenditure Survey for the United Kingdom.

One of the most significant developments in economics over the past twenty years has been the increasing extent to which economists have been prepared to apply the basic tools of consumer theory to areas other than just the demand for goods. A particularly notable example is the analysis of labour supply, where utility theory has been successfully used in the empirical analysis of a wide range of phenomena, including the supply decisions of primary and secondary workers, the decision whether or not to participate, and the type of behaviour which results from the complex rules of modern tax and social security systems. For a discussion of this material see, e.g. Killingsworth (1981) or Deaton and Muellbauer (1980, chapter 11). More generally, the ‘characteristics’ or household production model has been applied to a wide variety of economic problems. Amongst the earliest examples is Gorman's famous 1956 paper on eggs, although it was Lancaster (1966a, b) whose work firmly established the methodology in the literature. In part one of this volume, the chapter by Theil and Laitinen can be interpreted as a characteristics model with the transformed goods as the characteristics, but much wider applications are possible. In particular, the model has been applied to the analysis of human capital formation, of fertility, of the use of time, of sexual and racial discrimination, of quality, and of health, to name only a few topics.

Introduction

In the general equilibrium model of Chapters 1–3, some of the critical features of a capitalist economy that are responsible for crises are absent. The purpose of those chapters was to study Marxian value theory; the questions of crisis can be studied somewhat independently of that theory. In this chapter, a model is proposed that permits an exposition of various Marxian and neo-Marxian crises: in particular, the profit-squeeze crisis, the realization or underconsumptionist crisis, and the fiscal crisis. To do this, we need to introduce a distinction between ex ante and ex post investment and savings, a government sector, and a reserve army of the employed.

The chapter begins with an exposition of a model of Marxian simple reproduction, and then proceeds to study extended reproduction. We do not propose that the models studied in this chapter are definitive, or that the ideas lying behind them are original. In fact, more than the other chapters, this chapter represents only a “foundation” to a study of an aspect of Marxian economics, rather than an extension or elaboration of a body of Marxian theory. Some of the most important attributes of capitalism, which contribute to crises, are not modeled here, such as the role of money. Thus, the chapter should be taken simply as an exposition of some of the classical Marxian ideas.

Methodology

Although a book in mathematical Marxian economics is no longer a unique phenomenon, its author must still confront the opinion held in many circles, both Marxian and non-Marxian, that such an endeavor is a contradiction in terms. Two lines of defense are available: (1) that Marx himself was not against the use of mathematical methods; (2) that regardless of Marx's position, these methods are appropriate to aid in understanding the social phenomena with which Marx was concerned. Although what Marx believed on this question should not settle the issue, if we consider Marxism to be a science and not a religion, it nevertheless appears that Marx was a supporter of the use of mathematical methods in economics. This is shown by the work of Leon Smolinski (1973), who studied Marx's unpublished as well as published manuscripts for his views on the matter. Smolinski reports there was “not a single injunction against mathematical economics [in] Marx's published or unpublished writings.” Moreover, Lafargue attributes to Marx the statement: “A science becomes developed only when it has reached the point where it can make use of mathematics” (Smolinski, p. 1201). Still, the opposing circumstantial evidence remains that Marx made very little use of formal mathematics (beyond arithmetic) in his work. As Marx studied algebra and calculus quite extensively in his later years, why did he not use these tools?

Introduction

Before beginning the technical discussion of the theory of the falling rate of profit, it is worthwhile recalling Marx's intellectual project in proposing his theory. Falling-rate-of-profit theories were a standard part of the armor of classical economics. The theories of Ricardo and Malthus were driven by diminishing returns in the natural productivity of the earth: As society was forced to adopt inferior land for agriculture, an increasing part of the economic surplus would be absorbed as rent, with a correspondingly smaller part available for profits. Hence, the rate of profit would fall as a natural, immutable consequence of a growing population, independent of what social and economic system prevailed.

Marx's aim was to show, on the contrary, that the rate of profit would fall as a consequence of the specific laws of motion of capitalist economy. As with so many other questions, he spurned general laws (that is, laws that purported to apply to all modes of production) and sought to locate developments such as a falling rate of profit in a historically specific context. Thus Marx proposed a falling-rate-of-profit theory that was driven by the specific form of technical change he conceived of as taking place under capitalism. There is no diminishing returns aspect to the argument. Although it will be shown in this chapter and the following one that Marx's theoretical conjecture was incorrect, his general methodological insight – that any crisis theory should be specific to the mode of production it seeks to describe – still stands.

The need for microfoundations: methodology

For the most part, discussion of the Marxian falling rate of profit (FRP) theory is marked by lack of attention to microeconomic detail. Precisely, how do the anarchic actions of atomistic capitals give rise to a falling rate of profit? Marx's discussion of this issue in Capital, Volume III, was formulated in a microeconomic way, as we pointed out in the last chapter. Briefly, the profit-maximizing urge of capitalists directs them to replace workers with machinery, which raises the organic composition of capital, which lowers (or produces a tendency to lower) the profit rate. Whether or not this argument is correct, it must be admitted that it is microeconomic in this sense: It claims to deduce a macroeconomic phenomenon, itself quite beyond the ability of any individual capitalist to realize, from the anarchic (competitive) behavior of atomized economic units. This type of economic reasoning, of deducing aggregate economic effects from the behavior of individual economic units, was employed by economists of all ideological bents in the nineteenth century. It is, indeed, one of the hallmarks of why Marxism is scientific socialism. The outcome of socialism (and of capitalist crisis) was argued, by Marx and Engels, not to be a utopian solution (and crisis fortuitous), but the predictable outcome of social forces that eventually were reducible to the actions of individuals and classes of individuals.

Introduction

We have shown that the traditional Marxian notion that equilibrium prices must be the vector of equal-profit-rate prices can be imbedded in a general equilibrium framework using the idea of reproducibility, at least in a linear, Leontief model. The task now is to investigate how robust these ideas are if we consider more general technologies. In particular, we shall focus on two important ideas: the existence of reproducible solutions for economies with more general specifications of production, and the validity of the fundamental Marxian theorem – that exploitation is synonymous with positive profits. In Chapter 3 we investigate a third important idea – whether equilibrium prices equalize profit rates among sectors in these more general models.

The answer is that the Marxian concepts, as discussed in Chapter 1, remain tractable, and the important theorems do generalize to a production environment in which capitalists face general, convex production sets. Thus, the ideas developed by Morishima (1973), Wolfstetter (1973), Okishio (1961), von Weizsäcker (1973), Maarek (1975), Brody (1970), and others for linear, Leontief models, and generalized by Morishima (1974) to von Neumann production models, are shown here to depend not at all on the activity analysis specification of production. Convexity suffices to define the concept of exploitation, and to verify the equivalence of profit making and exploitation.

Carrying out the analysis for convex production sets is instructive for another reason: It makes clear precisely which aspects of classical Marxian value theory are not robust, but incidental to the main story.

Marx's project: where do profits come from?

A main task of a theory of value for capitalist economy, for Marx, was to answer the question: Where do profits come from? Perhaps it is easier to comprehend why this question was of a somewhat paradoxical nature by comparing capitalism to the two major precapitalist, classstratified modes of production: feudalism and slavery. Under slavery, the class of slave owners forcibly expropriated unpaid labor from slaves. Thus, slave owners lived off slaves in a most obvious way. Under the feudal organization of production, the expropriation of the surplus was almost as obvious. Serfs performed the corvée for several days a week and worked several days a week on their own plots. The product of their plots was theirs for consumption; the labor time spent on the corvée was transformed into goods expropriated directly by the lord. Again, there can be no confusion concerning the locus of surplus production, and the locus of its expropriation. In both slavery and feudalism, the key to the division of society into a rich, expropriating class and a poor, expropriated one was the existence of a coercive institution for the exchange of labor.

What was puzzling to Marx concerning capitalism was this: The institution of labor exchange was not coercive. Nevertheless, one class became incredibly rich, and the other remained impoverished. Marx insisted on modeling capitalism as a regime in which markets are fair, an idea he tried to capture in his value theory by requiring commodities of equal value to be exchanged for each other.