Introduction

The stability of multiple markets was first discussed by Professor J. R. Hicks; and he derived from his stability conditions the following three rules about changes in the price system: (1) If the demand for a commodity is increased, then its price rises necessarily. (2) If all commodities are substitutes for one another, then all prices will rise whenever the demand for one of them increases. (3) Further, on the same assumption that all commodities are substitutes for one another, it can be proved that an increased demand for a commodity raises the prices of all the other commodities proportionately less than the price of the commodity. But, since the Hicksian method of stability analysis is but an implicit form of dynamic analysis and therefore imperfect or incorrect, we must reexamine what conditions are necessary and sufficient in order that a system of multiple exchange should be stable dynamically; and this has been done by P. A. Samuelson and O. Lange.

Samuelson, 1948, has tried to show how the problem of stability of equilibrium is intimately tied up with the problem of deriving the rules about the way in which the price system will react to changes in various data or parameters.

Introduction

We have already two non-linear models of trade cycles which seek in non-linearities an explanation of the maintenance of trade cycles: that is, the Kaldor–Yasui and the Hicks–Goodwin models. The most controversial of the assumptions underlying these two models are those concerning the determinants of investment decisions. The rate of investment is assumed in the Kaldor–Yasui theory to depend on the level of income (the profit principle) and in the Hicks–Goodwin theory on the rate of change in income (the acceleration principle). But Dr Kalecki's analysis of the correlation between the rate of investment, the level of income, and the rate of change in income shows that a better approximation is obtained if investment is considered as a function, both of the level and of the rate of change in income, than of either of them only. On this basis neither of the Kaldor–Yasui and the Hicks–Goodwin models can claim to be the best representation of actual cycles. In this article we shall be concerned with a more realistic model; namely, a synthesis of our predecessors.

Assumption

To begin with, we state the Kaldor and the Hicks–Goodwin investment functions in precise terms.

1 When I was writing the original Japanese version of DKR, there was no work which used computers to solve differential equations, though just after its publication I saw Morehouse, Strotz, and Horwitz, 1950 in Econometrica, a report of an application of an analogue-computer to an inventory oscillation model. Economists' works were mainly confined to linear models, especially when they were concerned with systems containing many economic variables. At that time already, we knew basic mathematical properties concerning the stability of equilibrium, the existence of limit cycles, their stability and instability, and so on, which are valid for a fairly large class of non-linear systems. Once they are applied to general equilibrium models, however, they are not very fruitful. We have to be satisfied with rather limited analyses, as chapter 4 above evidences.

In my 1977 work, however, I have developed a simultaneous difference equation version of the model of tatonnement, which may be regarded as the prototype of the work discussed below in this article. The general equilibrium point of the economy is found to be a fixed point of the tatonnement process in terms of these difference equations.

Introduction

It seems to be a generally supported view that the theory of stability has smoothly developed from Walras, through Hicks, to Samuelson who is the originator of the mathematical framework of contemporary stability theory. In terms of the terminology used in this volume, Samuelson concentrates his interest upon the stability of the equilibrium point, while Hicks' object is to deal with economic motion through weeks. Walras is nearer to Hicks than Samuelson in the sense that he is also concerned with economic motion or path rather than with a stationary or static point but differs from Hicks because the path examined is a sequence of temporary equilibria in the case of Hicks, while it is, as will be shown in section III below, a disequilibrium path in the case of Walras.

Arrow and Hahn, in their now classic work (1971, pp. 4–5), who regard Walras' stability theory or theory of tatonnement as ‘rather clumsy’, criticize it in the following way. Walras considers that the markets carry out tatonnement in some definite order. In the first market the price is adjusted so that supply and demand are equal, regarding other prices as given. This change in the price will affect supply and demand in all other markets.

Dynamic economic theory

What subjects should economics be concerned with and what should be their scope? These are very difficult problems which invite many controversies; but, still, it is true that economics ultimately aims to explain all economic phenomena. Such an aim determines how economics is to be constructed. Those phenomena which economics assumes without analysis must not be economic phenomena but are ones which should be elucidated by other disciplines. It is obvious that if economics assumes an economic fact without examination, its aim, which is to deal with all economic phenomena, will never be satisfied. Thinking in this way, those economists who supported general equilibrium theory finally dug their way into a non-economic world on which economic phenomena are finally based. When successful in the business of reducing economic relationships to non-economic ones they think that they have obtained the ultimate principles of explanation and they call those non-economic phenomena by the descriptions of the data that economists take as given. Individuals' tastes, techniques of production, endowments of factors of production, prevailing expectations about the future, and various institutions will be counted as data.

Thus, the general equilibrium theory which analyses economic phenomena on the basis of a given data complex naturally pursues the question of how economic phenomena will change if there is a change in the given data complex. That is to say that general equilibrium theory is concerned with correspondences between data and economic phenomena.

After DKR was published in Japanese in 1950, a great theoretical development has been made in the three main areas of the general equilibrium theory, concerning existence, uniqueness and stability of general equilibrium. The existence theorem is an achievement of efforts in the first half of the 1950s, while its second half and the first half of the 1960s were devoted to uniqueness and stability, mainly to the latter. In this addendum I sketch the theoretical development after DKR in my own way and show how I see it from my present arrival point.

The existence problem has been solved by formulating Walras' economy in terms of inequalities, rather than equations. This inequality approach is not entirely foreign to Walras, who supports the scarcity theory of value asserting that the price is zero for a ‘free (or non-scarce) good’, that is a commodity for which supply exceeds demand in equilibrium. Walras is concerned with an economy in which an exchange is actually made, though some goods may be free, and distinguishes this general equilibrium (which may be called an essential equilibrium) from the more general one in which goods may all be free and, therefore, not traded.

This section was originally published in the form of an independent article as Morishima, 1956, which may, however, be regarded as an example of the negative counterpart of the text of this volume. In the text we are concerned with an economy in which equilibration is made in terms of prices, while quantities are assumed to depend on prices, whereas in this article it is made in terms of quantities, all prices being implicitly assumed to be kept constant. The problem of price adjustment versus quantity adjustment later became a topic of general equilibrium analysis in the 1960s and the early 1970s.

As far as we measure the time t in an ordinary way by calendar time, the equilibrium to be established at t = ∞ is no more than a kind of utopia which is never realized at an actual point in time. Therefore, where we formulate a tatonnement process in terms of differential or difference equations, we should either transform the calendar time into a tatonnement time which tends to infinity at a certain point of the calendar time, as we have done so in the text, or we should truncate the tatonnement process at a certain point of the calendar time, as Walras did in his Elements.

1 To publish a translation of a book which was published nearly half a century ago in an academically local language we need convincing justifications. This volume originally entitled Dogakuteki Keizai Riron (DKR) was published in Japanese in 1950, by Kobundo, Tokyo. Soon after the war, I returned from the military service to the university and completed my undergraduate study at Kyoto University in 1946. DKR is a thesis written as a report on my graduate research work supported by the Ministry of Education Special Scholarship. The topic suggested by my supervisor, Professor Hideo Aoyama, was the ‘mathematization of Hicks’ Value and Capital (VC)' so that DKR is more or less parallel to it. However, there are a few substantial differences.

First, in VC it is assumed that output equals supply and input equals demand. This basic assumption enables us to use the production theory of the firm as the tool to explain its behaviour in the market. Moreover, this assumption implies that the stocks of input and output commodities do not change. In DKR I was concerned with firms whose inputs and outputs may deviate from the demands for the factors and the supplies of the products, respectively. I had, therefore, to construct a theory of the firm which could explain its production, trading, and inventory consistently.

Secondly, once this line of approach is adopted, it is clear that we cannot be satisfied with a physical input–output theory of the firm of the VC type. Money and securities are close substitutes for inventories of physical commodities. The theory of the firm, therefore, has to be extended so as to explain its behaviour of liquidity preference.

1 Under the slogan of the ‘neoclassical synthesis’ the neoclassical general equilibrium theory and Keynesian economics have both been taught in parallel in two classrooms. In one it is taught that the price mechanism works; a general equilibrium exists at which labour is fully employed and the economy is in a state of Pareto-optimum, while in the other, full employment is taught to be impossible and the market has to be supported and supplemented by conscious and conscientious economic policy activities of the government. These two views should of course be incompatible. As Keynes has pointed out, there must be a hidden hypothesis of ‘Say's law’ behind the neoclassical world; where it is rejected, the theory has to abdicate and be replaced by a new regime in which unemployment is recognized as a long-standing, unremovable state of affairs, that is inevitable. The mechanism of self-regulation of the market does not work where Say's law of the market is negated. It is deeply disappointing particularly for the author that he has to complete this volume with the final section of the Addendum which establishes a thesis that no general equilibrium of full employment is possible unless the equalization of rates of profits between capital goods is ruled out.