Introduction

It cannot be pretended that the theory of stability developed historically with an eye to the considerations discussed in the previous chapter. Nevertheless, that discussion provides a useful vantage point from which to view the development of the subject. In turn, such a review allows us naturally to build an understanding of the steps which lead to ever more satisfactory stability models.

In my view, there have been four major developments in the history of modern stability analysis. These are (1) the realization that the subject was one which had to be studied in a context with a formal dynamic structure; (2) the realization that global, rather than simply local, results could be obtained; (3) the introduction of non-tâtonnement processes; and (4) closely related to this, the insight that attention paid to specifying the disequilibrium processes involved could lead to far more satisfactory results than could be obtained by restricting the excess demand functions. In some ways, the analyses resulting from each of these steps made increasing use of the economic underpinnings of the stability problem (largely, but not exclusively, Walras' Law) and led to correspondingly more and more satisfactory results.

The purpose of the following appendixes is to prove the theorems that were stated in the text. Appendix A reviews the mathematical notions and results that are required. Appendixes B, C, D, and E are devoted to the proofs of the propositions of Chapters 1, 2, 3, and 4, respectively.

Although the exposition is intended to be self-contained, a fair knowledge of real analysis, as exemplified by Bartle (1964, Sections 1-8, 11, 15, and 16), is required from the reader. Some familiarity with conventional general equilibrium theory, as provided by a graduate course in mathematical economics, is not necessary but is nevertheless recommended.

Introduction

Economists, particularly economic theorists, are most often concerned with the analysis of positions of equilibrium. This is most obviously true in microeconomics, where general equilibrium theory stands as the most complex achievement of rigorous analysis; but it is becoming true of macroeconomics as well, where it has become increasingly popular in rational expectations models to assume that markets always clear.

Less attention is given to disequilibrium. In microeconomics, the subject of the stability of general equilibrium is in poor repute. Too many economists (including economic theorists, who should know better) apparently believe that stability theory means tâtonnement - a branch of the subject that died in 1960 and was long ago superseded.1 They regard it as overformal and empty of results, save under the most extreme ad hoc restrictions, and without much relation to the rich and complex world of real economies.

For macrotheorists the concentration on equilibrium manifests itself in other ways. Aside from the rational-expectations-market-clearing position already mentioned, one currently fashionable branch of the subject investigates fixed-price, quantity-constrained equilibria. Such investigations can be very fruitful, but they are not truly disequilibrium investigations, although they are sometimes misnamed as such. They are analyses of equilibria that are non-Walrasian.

A significant outcome of this analysis concerns the existence of a monetary Walrasian equilibrium in the short run. We found indeed that whenever the traders' learning processes involve short-run rigidities of their expected real interest rates - which is the case in traditional neoclassical macroeconomic models - a short-run Walrasian equilibrium in which money has positive value may not exist. Real balance effects are then the essential regulating mechanisms of the economy, but they may be too weak to equilibrate the markets. As a corollary, in order to guarantee the existence of a short-run monetary equilibrium, one has to appeal to a stabilizing mechanism that was neglected by conventional neoclassical monetary theorists, namely, the intertemporal substitution effects that are induced by a variation of the traders' expected real interest rates. In the class of models that we have considered, this required basically that some or all traders' expectations about future prices and interest rates are to a large extent insensitive to the variations of current prices and rates of interest. The fact that such restrictions on anticipations are rather implausible shows that the existence of a short-run Walrasian monetary equilibrium raises more problems that neoclassical theorists like to believe.

Walras’ Law

I now move closer to modeling the interaction of agents, building on the theory of the individual agent set forth in the previous two chapters. I begin with a consideration of Walras' Law. Certainly, one expects to find some version of Walras' Law holding for this economy, and, indeed, some version does hold; however, there are some points of special interest as to just what that version is.

Walras' Law in its usual form states that the total value of all excess demands is zero. Here, the excess demands involved will be those for commodities (including bonds), shares, and money. But it is not so clear precisely how the result will turn out. To begin with, demands in this model are distinguished by the dates at which agents expect to exercise them; they are not static. Will Walras' Law hold as a statement about the demands planned for any future moment or only as a statement about the value of all future plans? In fact, the result applies to either case; this is because agents expect at every instant to exchange commodities or shares for money of equal value.

Second, Walras' Law requires that we value all excess demands. In more primitive models, this is straightforward. Such valuation simply uses the common prices. In the present model, however, individuals can have different prices for the same commodity. Even without the individual price offers considered in the preceding chapter, this is true of the prices which agents expect. Not surprisingly, therefore, Walras' Law requires us to value each agent's excess demands at the prices at which that agent personally expects to (or actually does) act on them.

This investigation has come some distance from its origins in the traditional stability literature. Unfortunately, there is still a long way for further research to go before we have a sound foundation for equilibrium economics. I now review some of the road traveled and then consider the nature of the road ahead.

We now know that, under very general circumstances, in a world of rational agents who perceive and act upon a nonexpanding set of arbitrage opportunities cast up by disequilibrium, the economy will be driven to equilibrium as such opportunities disappear. This will occur if new opportunities in a wide sense do not continue to appear or to be perceived to appear. In fact, stability is achieved even with new opportunities, provided that such new opportunities do not appear as favorable surprises, seized by agents as optima the moment they appear. This result - while not nearly as helpful as would be a parallel result as to the effects of the nonappearance of new exogenous opportunities - is nevertheless a basic waystation. It holds both for competitive economies and for some economies including monopoly or monopolistically competitive elements.

This general (if weak) stability result is free of many of the problems which flawed the earlier literature. Consider first the tâtonnement literature. There, to obtain stability it was necessary to restrict excess demand functions very severely; no such restriction is required in the present context. Further, the entire setting of tâtonnement was unrealistic, requiring the absence of any disequilibrium activity save price adjustment with continual recontracting. That is plainly not the case here.

In the real world, there are many channels through which a banking system may influence economic activity by implementing a specific monetary policy. One way, which was considered in the previous two chapters, is to intervene on the credit market by trying to manipulate the cost of borrowing, or the amount of money that is created when granting loans to the private sector. Another way, which we shall study presently, is the Bank's attempts to influence the economy's liquidity by exchanging illiquid assets such as long-term bonds for liquid assets such as short-term bonds or money.

The model we shall use to take this kind of phenomenon into account is quite simple, and bears some resemblance to popular Keynesian macroeconomic models. The real part of the model is the same as that in the previous chapters. As in Chapter 1, consumers have to decide in each period how much to consume and to save (no borrowing is allowed). But consumers can now save by holding two sorts of assets instead of one: paper money and perpetuities, both of which are issued by a governmental agency, the Bank. In such a context, the Bank can in principle engage in open-market operations by trading perpetuities for money, and vice versa. In particular, the Bank may wish to peg the interest rate, i.e., the reciprocal of the money price of perpetuities, or the money supply.

Introduction

The analysis of the previous chapter proceeded as though agents always expect to be able to complete their planned transactions with no difficulty. This is hardly a reasonable assumption in a disequilibrium model where agents constantly find their plans thwarted. Further, given the absence of any subjective uncertainty, the fact that transactions of all sizes are assumed to be costlessly made leads the otherwise very sensible speculation results of the previous chapter to be of the “bang-bang” sort: agents rush discontinuously from one arbitrage opportunity to another, switching from buying to selling large amounts in pursuit of even very small speculative profits. It is plainly time to deal with such matters.

In fact, there is more than one matter to deal with here; there are three, and they are related in somewhat different ways.

The first issue is the analysis of transaction costs, that is, difficulties in transacting that prevent the “bang-bang” property just described from arising. Such costs may be thought of either as effort which must be expended in the search for trading partners or as a partial substitute for the effects of subjective uncertainty, which is otherwise conspicuously absent from the model. As we shall see, incorporating this kind of transaction difficulty into the analysis is easy to do; it has some interesting consequences, but it raises no very deep problems.

Introduction

The matters discussed in the previous chapter are all quite complex, and it is best to deal with them one at a time where possible. In the present chapter, I explore the behavior of individual agents where disequilibrium consciousness is allowed and production and consumption take place out of equilibrium. To do so, I first ignore the (welcome) complications which arise from the realization by agents that they may not be able to complete their transactions. Further, I postpone consideration of how prices (and price expectations) are set. These and other matters are introduced later on.

I thus begin with the general setup of the model and the analysis of the optimizing behavior of agents who believe that prices will change. The analysis is complicated, but the results are appealing, being both interesting in themselves and required for what follows. The complexity seems inescapable if we are to deal with models in which time is essential and arbitrage over time takes place. No suitable, complete treatment of these matters seems available in the literature.

Because it seems desirable to continue to make the discussion accessible to relatively nontechnical readers without at the same time sacrificing continuity for those interested in a more rigorous treatment, this and succeeding chapters are organized a bit differently from the earlier ones. Each subject is discussed in one or more nontechnical sections. Those sections are followed by one or more technical ones, which are indicated by asterisks.

The aim of this chapter is to study the existence and the properties of steady states in the credit money economy that was analyzed in the preceding chapter, when population is stationary. Such steady states are defined as sequences of short-run equilibria (in the sense of the previous chapter), where the nominal interest rate, the relative prices of goods, and the rate of inflation - and more generally all “real” equilibrium magnitudes - are constant over time. Moreover, traders are assumed to forecast future prices and interest rates correctly at every moment. Although we have shown that the existence of a short-run equilibrium was doubtful in economies of this type, steady states are of independent interest because they may obtain as long-run equilibria of other dynamic (e.g., disequilibrium) processes.

It will be established that quantity theory and the classical dichotomy are valid propositions when applied to steady states. Real magnitudes (the relative prices of goods, the real rate of interest, the traders' consumptions) are determined by the equilibrium conditions of the goods markets. Nominal values (the money prices of goods, the rate of inflation, the nominal interest rate) are determined in turn by looking at the money sector, including the Bank's monetary policy. In particular, when population is stationary, the money prices of goods are proportional at any time to the level of monetary aggregates such as outside money or the Bank's money supply. In other words, prices and monetary aggregates grow at the same rate.

The previous chapter examined an economy in which consumers were constrained to hold nonnegative money balances. The only ways for the Government to alter the money stock were either to engage in fiscal policy by making positive or negative money transfers to the private sector - that type of policy was briefly considered at the end of the chapter - or to trade on the consumption goods markets. That sort of money is often called outside money in the literature.

We will now introduce to the model the possibility for consumers to borrow against future income by selling short-term bonds to a governmental agency, called the “Bank.” Since there is money creation whenever the Bank grants loans by buying bonds from the consumers, the money stock can then vary over time according to the needs of the economy, through the extension of credit.

A useful distinction between “inside” and “outside” money is often developed in credit money economies of this type. The part of the money stock that is “backed” by the Bank's claims on the private sector - that is, that has been issued by the Bank when purchasing consumers' bonds - is called inside money. Outside money is not backed by private debts.

Toward a more sensible model

It is now time to consider developing the models so far considered in the direction of our final goal. We must allow production and consumption to take place out of equilibrium. More important, we must allow agents to realize that they are not in equilibrium and to act on arbitrage opportunities as they occur. This fundamentally requires that agents be permitted to do two things. First, they must recognize that prices may change. Second, they must recognize that they may not be able to complete their desired transactions. In forming their consumption and production plans, agents must take these things into account.

The fact that agents may not be able to complete their transactions is not unrelated to the fact that prices can change. Agents who believe that they face transaction constraints are also likely to believe that prices in markets with such constraints will change. This is particularly likely in a Hahn Process world of orderly markets in which agents can take their own inability to purchase as reflecting a general short supply. Moreover, agents facing such constraints may themselves make price offers to get around them.

Handling all of this is a tall order and is best done in pieces. In the present chapter, I consider in general how the analysis so far developed can be adapted to deal with these matters. The details - which are often very interesting indeed - are explored later on.

In a wide class of social systems each agent has a range of actions among which he selects one. His choice is not, however, entirely free and the actions of all the other agents determine the subset to which his selection is restricted. Once the action of every agent is given, the outcome of the social activity is known. The preferences of each agent yield his complete ordering of the outcomes and each one of them tries by choosing his action in his restricting subset to bring about the best outcome according to his own preferences. The existence theorem presented here gives general conditions under which there is for such a social system an equilibrium, i.e., a situation where the action of every agent belongs to his restricting subset and no agent has incentive to choose another action.

This theorem has been used by Arrow and Debreu [2] to prove the existence of an equilibrium for a classical competitive economic system, it contains the existence of an equilibrium point for an N-person game (see Nash [8] and Section 4) and, naturally, as a still more particular case the existence of a solution for a zero-sum two-person game (see von Neumann and Morgenstern, Ref. [11], Section 17.6).