Temporary Monetary Equilibrium Theory

A Differentiable Approach

Kuan-Pin Lin

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781351719766
• Edition: 1
• Topic: Negocios y empresa
• Subtopic: Negocios en general

This title, first published in 1984, considers a temporary monetary equilibrium theory under certainty in a differentiable framework. Using the techniques of differential topology the author investigates the structure of the set of temporary monetary equilibria. Temporary Monetary Equilibrium Theory: A Differentiable Approach will be of interest to students of monetary economics.

Chapter I

INTRODUCTION

The differentiable approach to equilibrium theory originates with the works of Debreu [9], [10] and is followed up by the works of E. and H. Dierker [13] and Smale [53], [55] among others (see also [3], [14], [18] for instance). This approach allows one to directly study the structure of the set of equilibria as well as the existence problem.

From a classical viewpoint of equilibrium theory, the existence problem is simply based upon enumerating the conditions of equilibrium for each agent in the economy and counting the equations describing these equilibrium states. Unlike the mathematical tools of fixed point theorems in algebraic topology, the differentiable approach enables us to attack the equations and their solutions directly provided that the equations describing equilibrium conditions are sufficiently differentiable. Recent works on approximation theorems (for instance, see [29], [38] and [44]) give further justification to the use of the differentiable approach in economics. Once there is a general solution to an economic system, as pointed out by Debreu in [9] and [10], one should also investigate the structure of the set of equilibria. Otherwise, the explanation of equilibrium is totally indeterminate. First, there may exist infinitely many equilibria. Also, the economic system may be unstable in the sense that a small change of economic data would lead to an entirely different set of equilibria. Therefore, it is highly desirable to have an economy for which the set of equilibria is locally unique (i.e., discrete) and stable (i.e., continuous). These properties are also needed if one wants to study the comparative statics of the system. The way of studying these equilibrium properties is provided by differential topology. In a general Walrasian equilibrium framework, Debreu [9] was the first to prove the existence and finiteness of equilibria for a space of economies with variable resources but fixed preferences for each economic agent by using Sard’s theorem of differential topology. Similar results have been obtained for the case of variation of demands as well as variable resources by Dierker [14] and Fuchs [18] among others. Smale [53], [55] studied the same problem from a viewpoint of variable utility functions in a general way and does not require well defined demand functions. In this research, one considers short run or temporary equilibrium in a Walrasian setting from a differentiable viewpoint and studies the equilibrium properties for such models in which some roles of money are also examined.

The basic idea of the short-run or temporary equilibrium model is that each agent’s decisions are made sequentially and may be revised as time progresses according to the information conveyed about the future environment, which is based upon the agent’s current and past observations. “Loosely speaking, whereas the Arrow-Debreu analysis is essentially static, the short-run equilibrium analysis studies the dynamic features of the economy within a short time interval [56].” In a temporary equilibrium analysis, the expectations taken as data of the system link the subsequent spot markets together and play the most important role in determining the equilibrium properties with uncertain information available in the future. This idea can be traced back to the work of Hicks [32] and it is also a very much part of the Keynesian thinking. Recent contributions to temporary equilibrium theory can be found in the works of Arrow and Hahn [2] and Stigum [57], [58] among others.

For a simple case, the structure of temporary equilibrium theory reveals (indirectly) the Veblen-Scitovsky effect through the agent’s price expectations. Veblen [59] argued that some commodities do have the property of demonstration of personal wealth and Scitovsky [50] concluded a result that economic agents judge the price of commodity as an index of quality, that is, price parameters influence the individual’s preferences in an explicit way (see also [36]). Applying utility analysis, Dusansky and Kalman [16], [17] have shown that the derived demand of a Veblen-Scitovsky economy is usually changed non-homogeneously with respect to the proportional change in price system and initial wealth.

As a matter of fact, the basic simplified temporary equilibrium model can be easily extended to incorporate other aspects of economic activity which do not occur in an Arrow-Debreu economy. First, it is very natural to introduce a financial asset such as “money” which serves as the only store of value in a temporary equilibrium framework, and study the properties of a monetary equilibrium. This idea was adopted by Patinkin [47] in his integration of monetary and value theories. For a money economy, in general there is money illusion in the construction in addition to “judging quality by price” and “snob appeal” originated by Scitovsky and Veblen, respectively. This is a generalization of Patinkin’s real balance effect (see also [16]). More recent contributions to temporary monetary equilibrium theory, especially the existence of a monetary equilibrium, can be found in the works of Grandmont [20], Grandmont and Laroque [22], [23], Grandmont and Younés [25], [26], Hool [35], Sondermann [56] and Younés [61], among others. In these works, two assumptions on expectations were imposed. First, each agent’s subjective probability distribution over future events varies continuously with respect to the current and past observations. Secondly, each agent is certain that all future prices (including future price of money) are positive and that future prices of consumption goods and asset, if any, are relatively inelastic with respect to the current and past information available in the economy. Under these assumptions, the existence of a monetary equilibrium has been established in the sense that the equilibrium price of money is positive although money has no intrinsic value for consumption purpose.

In previous discussion of temporary equilibrium models, the analysis is restricted to the “spot” markets only. It can be extended to allow the existence of future markets such that the current activities of an agent include not only the decision to make immediate consumption, but also the intention to make contracts in the more or less distant future. For a non-monetary economy with futures contracts, Green [27] considered a temporary equilibrium model in which the markets for trading current and future commodities are open at the initial date, and all future commodities are again tradable in the future. Moreover, an extension has been made in [28] to allow bankruptcy with some probability and the agents can also be endowed with preexisting contracts when the market of the current period opens. In addition to the above conditions on expectations, some degree of compatibility of expectations among agents are necessary to ensure the existence of an equilibrium with spot and futures transactions.

In this research, we formulate temporary monetary equilibrium models in a differentiable setting. Using techniques of differential topology recently introduced into the economics literature by Debreu [9], [10] and Smale [53], [55], we establish the properties of local uniqueness and stability of temporary equilibria for “almost all” money economies as well as existence for every money economy. The crucial assumptions are sufficient differentiability of the direct utility functions and expectation functions.¹ Furthermore, the conditions for the existence of temporary monetary equilibrium, as discussed earlier, contain the inelasticity of expectations in the case of pure spot markets, and together with the one which restricts the possibility of abtratige on future markets in the general case. Although these assumptions on expectations are not necessary for the study of the set of temporary monetary equilibria, we include them only for the reason of consistency of the models. We prove that the set of temporary equilibria is a finite set and depends continously on the money economy. These are new results in the literature.

In Chapter II, some basic definitions and assumptions are made and discussed for a monetary economy with spot markets. Among them, a set of classical assumptions are postulated on the future spot market which guarantee the differentiability requirement in the model. The dynamic programming approach used in this section to derive the indirect utility function originates with the works of Stigum [57], [58] and Grandmont [20]. However, they do not use a differentiable viewpoint as we do in this work. We study utility functions directly and our class of utility functions includes those which define continuously differentiable demand functions. Chapter III formulates an (infinite dimensional) space of money economies in which an element is a list of expectations, direct utility functions, money and commodity endowments. In other words, we allow changes of tastes, beliefs and endowments of all agents in the model. We prove the continuous differentiability of expected utility functions with respect to the commodity-money holding and the price system as well.² Moreover, a concept of “extended” monetary equilibrium is introduced, which contains (classical) monetary equilibrium. In Chapter IV the topological concept of “almost all” economies is introduced in the space of money economies. Because of the infinite dimensionality of the space of money economies, the best we can show at this point is that the set of “regular” money economies is “open and dense” in the space of all money economies with respect to an appropriate topology. Two theorems, local uniqueness and stability, are presented and proved for all regular money economies. The former can be obtained without assuming concavity of the utility functions for any agents and the latter follows from an application of the implicit function theorem. Finally, the existence of temporary monetary equilibrium is proved by demonstrating that every money economy is a continuous deformation of an economy with unique equilibrium. The technique we use is degree theory in differential topology. As an extension, in Chapter V, we introduce futures transactions in the above temporary monetary equilibrium analysis. The activities of borrowing and lending may occur by trading commodities and money deliverable in the future against present ones. The precautionary and speculative motives for money holding are obvious in the formulation. This particular chapter can be considered as an extension of Green [27] in the direction of including money and studying the existence, finiteness and stability of temporary monetary equilibria. Finally, in Chapter VI, some suggestions for further research are stated briefly.

¹  Differentiability of expectations is also considered by Fuchs and Laroque [19] for a different model in which no uncertainty is considered.

²  For a related result in a non-differentiable framework, see [20].

Chapter II

A SPOT MONEY ECONOMY

We consider a framework of temporary monetary equilibrium theory with two successive periods where for each period all commodities are immediately deliverable for consumption and “money” is used as the only store of value but gives no direct utility to the agents. In such a “spot” money economy, taking into consideration future (market and individual) uncertainty, money is a link of transferring wealth between periods. The analysis is an “interior analysis” which considers only positive price systems and each agent owns at least a little of each commodity and money. Moreover, the restriction of the analysis to two periods is by no means essential, and the so-called “period” is considered as a synonym of Hicksian “week” in which markets are held on “Monday”. Although it is...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• ABSTRACT
• ACKNOWLEDGMENTS
• Chapter I: INTRODUCTION
• Chapter II: THE MODEL – A SPOT MONEY ECONOMY
• Chapter III: THE SPACE OF MONEY ECONOMIE
• Chapter IV: MAIN THEOREMS
• Chapter V: AN EXTENTION- TEMPORARY MONETARY EQUILIBRIUM THEORY WITH SPOT AND FUTURES TRANSACTI
• Chapter VI: CONCLUSION: SUGGESTIONS FOR FUTURE RESEARCH
• REFERENCES