Equilibrium and the Adjustment to It

In the end we have arrived at a situation where we can show the existence of an equilib­rium under certain, rather strong, restrictions on the characteristics of individuals, and that equilibrium satisfies a minimal efficiency criterion.

But what about the other concept of equilibrium, that of the resting point of a process of adjustment from an out of equilibrium position? If we consider what Arrow (2007) has to say about this, even though his name is inextricably linked with the proof of the exist­ence of such a state, then it is clear that he, like Walras, has kept in mind a process, which would bring the economy to an equilibrium. It is worth observing that Debreu never ventured into the field of stability analysis, with the exception of his contribution to the literature (Debreu 1974), showing that, in the Arrow-Debreu model, stability could not be guaranteed, whereas Arrow wrote a number of papers on the subject. Despite the attention Walras paid to the equilibrium as a solution of a set of equations, he never completely abandoned his idea of the economy as self-organizing into such a state. For example, he observed, when describing the evolution of an economy, that it is “like a lake stirred by the wind, in which the water continually seeks its equilibrium without ever achieving it” (Walras 1877: 310). Or as Arrow himself said recently:

The very concept of equilibrium, economic or otherwise, presupposes a dynamic system which determines change as a function of state variables. An equilibrium is a vector of state vari­ables for which no change occurs. In economics, this is interpreted as a set of quantities and prices for which there is no incentive on anyone’s part to change. The dynamics runs in terms of profit opportunities or incentives to outbid others for scarce commodities or for market opportunities. (Arrow 2007: 1)

Again, Walras when he developed his market models (see Walker 1996) was explicitly concerned with the stability of an adjustment mechanism.

The alternative approach suggested by Walras was the so-called “tatonnement” process. Here prices adjust whenever markets are not in equilibrium and they do so in proportion to the level of excess demands for the commodities. However, in doing this, we encounter a problem with the adjustment approach. If the process of adjustment to equilibrium is a process which happens in the real economy, it must take time, in par­ticular when the production of goods is involved. Alternatively, you could think of the process as a virtual one which happens instantaneously and does not take any time at all. This could be thought of as akin to certain arguments in physics where one thinks of something moving slowly along a surface but adjusting instantaneously back to the surface whenever it leaves it.

Among the epistemological inconsistencies, the most evident is revealed by the dualistic char­acter of the Walrasian theoretical system: in fact, in the versions of this system developed in the second (and third) edition of the Elements, one can find, side by side, two alternative interpre­tations of the tatonnement, the virtual and the effective; these two kinds of processes evolve in two distinct time sets, the “logical” and the “real”, lead to alternative notions of equilibrium, the “instantaneous” and the “stationary”, and finally pertain to different models, the pure- exchange model, on the one hand, and the models with production, on the other.

As Donzelli rightly points out, it is production which makes the Walrasian concept of adjustment in real time implausible. The conflict over the interpretation of the Walrasian price adjustment process is apparent in the acrimonious debate that Walras had with Edgeworth, often through von Bortkiewicz as intermediary (see Walker’s 1996 very com­plete discussion of the debate between Edgeworth and Walras). The latter asserted in a review of Walras’s Elements that Edgeworth was convinced that Walras had described a system which was not functional and that what Walras was describing was a virtual adjustment, and in his review he came down on Edgeworth’s side when he said:

Well, the way to solve the equilibrium equations analysed by M. Walras is absolutely consistent with the idea that Jevons had about the nature of these equations. As to the exchange problem, M. Walras thinks about this in a purely static way, in the sense that the quantities of goods available are fixed, preferences are unchanging and he simply solves the equations by increasing and decreasing prices. (Bortkiewicz 1890: 86)

Edgeworth was evidently convinced that the process described by Walras was virtual since there was none of the “higgling” that he described in his recontracting process. Furthermore, the quantities to be traded and the preferences were not modified while the process was taking place.

I am engaged in absolutely useless exercises in my efforts to demonstrate that the operations of the raising and lowering of prices, of the increases and decreases of the quantities of products produced, etc. on the markets are nothing other than the solution by tatonnement of the equa­tions of exchange, of production and of capital formation. (Letter no. 927 to von Bortkiewicz, in Jaffe 1965)

His conviction that his view was a general one made him object particularly strongly to the idea that adjustment processes could only be applicable in specific institutions. He seemed to assimilate the notion of “free competition” to the tatonnement process and once again he complained to von Bortkiewicz:

I take the almost universal regime of free competition in regard to exchange, that which was described by John Stuart Mill, and which consists in raising the price in the case of the quantity demanded exceeding the quantity supplied and lowering it in the case of the quantity supplied exceeding the quantity demanded, and I demonstrate that the process leads to equilibrium by establishing the equality of the quantities supplied and demanded. Whereupon there is thrown at my head the market for English public debt, the system of English auctions, the system of Dutch auctions etc., etc.

Of course, unfortunately Walras had done nothing of the sort. Indeed, much later it became clear that what he tried to do was not possible. There is a long history of what might be called “centralized” adjustment processes in which all individuals take prices as given and somehow these prices are adjusted as a function of aggregate excess demand for various goods (see, for example, Arrow and Hurwicz 1958; Arrow et al. 1959).

It is this that led De Vroey, as mentioned earlier, to assert that the only idea that was compatible with this sort of mechanism was to have some sort of auctioneer. Here, an important gulf between what was needed to ensure some sort of stability and what were typically made as assumptions on the individual participants in the economy becomes clear. The conditions that guarantee stability, at least of the tatonnement process, are conditions on the aggregate excess demand of the economy. For example, a condition that guarantees both stability and uniqueness of equilibrium is that all goods be “gross substitutes”, if the price of one good increases the demand for none of the other goods decreases. This is of course not a reasonable empirical condition, everyone can immedi­ately think of goods which are complementary, cars and tyres, for example. Worse, this is a condition on aggregate excess demand while theoretical macroeconomists continued, and continue, to insist that one should limit oneself to assumptions on the rationality of individuals. However, the basic problem is that the structure that we impose on indi­vidual firm and consumers’ choices does not carry over to the aggregate.

To take a simple example, Samuelson (1938) introduced the notion of “revealed pref­erence”, that is, rather than make rather abstract assumptions about the underlying pref­erences of consumers one could impose a simple axiom of consistency on their choices. This, it was thought, would rid us of the unverifiable structure that had been imposed by Walras and Pareto and would mean that we could verify empirically the consistency of individual choices.

The fruitless search for showing that reasonable conditions on individual preferences for consumers or for technologies for firms that would guarantee stability of some plausible price adjustment process was brought to a halt by the results of Sonnenschein (1972), Mantel (1974) and Debreu (1974) himself. (For simplicity, the reference will be to the SMD results from here on.)

What did they show? This needs a little background. It was well established long before these results that aggregate excess demand, that is, aggregate demand minus aggregate supply which we can denote by Z(p), satisfies four conditions:

1. Z(p) is a continuous function of the prices of the l goods in the economy.

2. Z(p) satisfies Walras’ Law that is p ∙ Z(p) = 0, that is, the value of what is sold at any prices must be equal to the value of what is bought.

3. Z(p) is homogeneous of degree 0. Changing the price level, that is, multiplying all prices by a constant does not change excess demand.

4. If the price of any good goes to zero, average aggregate demand ||Z(p)|| becomes infinite.

Each of these conditions is easily derived from the standard assumptions on individuals. In particular, the first reflects the condition that individuals have continuous preferences, they do not suddenly switch their preferences when faced with small changes in the choices with which they are faced. The second comes from the fact that individuals are assumed to satisfy their budget constraints. The third reflects the fact that only relative prices matter for individual choices. The last, for which Walras was mildly criticized by Poincare, comes from the assumption that individuals always prefer more of any good, as Poincare (1901) said, they are assumed to be “infinitely greedy”.

Now, there was a latent idea that the conditions for uniqueness and stability of equilib­rium were somehow related. This was, no doubt, due to the fact that the known sufficient conditions on aggregate excess demand for uniqueness also implied stability. However Scarf (1960) gave an example of an economy with a unique but unstable equilibrium. Furthermore, it was not difficult to construct examples of economies with multiple equi­libria. Therefore, the hope was to use the standard assumptions and to show that they ruled this sort of phenomenon out.

What the SMD results showed was that the only conditions imposed on aggregate excess demand by the standard assumptions on individuals were the four mentioned above. Since we can find economies satisfying these conditions which have multiple and/ or unstable equilibria there was no hope of guaranteeing uniqueness or stability. This meant that simply constructing models on the basis of assumptions on individual ration­ality does not lead to any empirically meaningful restrictions on aggregate behaviour. Indeed many distinguished economists have claimed that uniqueness and stability are minimal requirements for a satisfactory model of an economy. For example, Schumpeter (1939) argued that a satisfactory model must, of necessity, generate a unique equilibrium to be considered as scientific:

The first and foremost task of economic analysis is to explore the properties of that system... What we want to learn before anything else is whether or not the relations known to subsist between the elements of the system are, together with the data, sufficient to determine these elements, prices and quantities, uniquely. For our system is logically self contained only if this is the case: we can be sure that we understand the nature of economic phenomena only if it is possible to deduce prices and quantities from the data by means of those relations and to prove that no other set of prices and physical quantities is compatible with both the data and the relations. The proof that this is so is the magna charta of economic theory as an autonomous science, assuring that its subject matter is a cosmos and not a chaos. (Schumpeter 1939: 41)

Even if we can handle the problem of multiple equilibria, and this poses serious prob­lems for those who are interested in analysing the comparative statics of their model, the stability problem seems much more important. For as Morishima (1984: 68-9) remarked:

If economists successfully devise a correct general equilibrium model, even if it can be proved to possess an equilibrium solution, should it lack the institutional backing to realise an equi­librium solution, then the equilibrium solution will amount to no more than a utopian state of affairs which bear no relation whatsoever to the real economy.

The SMD results put an end to any hope of showing that there is a mechanism which would provide such an institutional backing, at least if one insists on building a general equilibrium model based solely on assumptions about individuals. Hahn (2002) pointed out that, worse, these results that undermined Walrasian general equilibrium theory came from within the group of distinguished theorists who had so successfully promoted the formalization of the theory and who were responsible for perfecting it. Perhaps strangest of all, the SMD results did not arise from a discussion as to whether an economy would converge from an out-of-equilibrium position to such a state. As Hahn observed: “The enterprise was doomed not to be capable of reaching general conclusions in the Walrasian setting. A theorem not directly related to or connected with dynamics did the damage” (Hahn 2002: 24).