Archive for the 'Nash Equilibrium' Category

The Equilibrium of Each Is the Result of the Equilibrium of All, or, the Rational Expectation of Each is the Result of the Rational Expectation of All

A few weeks ago, I wrote a post whose title (“The Idleness of Each Is the Result of the Idleness of All”) was taken from the marvelous remark of the great, but sadly forgotten, Cambridge economist Frederick Lavington’s book The Trade Cycle. Lavington was born two years after Ralph Hawtrey and two years before John Maynard Keynes. The brilliant insight expressed so eloquently by Lavington is that the inability of some those unemployed to find employment may not be the result of a voluntary decision made by an individual worker any more than the inability of a driver stuck in a traffic jam to drive at the speed he wants to drive at is a voluntary decision. The circumstances in which an unemployed worker finds himself may be such that he or she has no practical alternative other than to remain unemployed.

In this post I merely want to express the same idea from two different vantage points. In any economic model, the equilibrium decision of any agent in the model is conditional on a corresponding set of equilibrium decisions taken by all other agents in the model. Unless all other agents are making optimal choices, the equilibrium (optimal) choice of any individual agent is neither feasible nor optimal, because the optimality of any decision is conditional on the decisions taken by all other agents. Only if the optimal decisions of each are mutually consistent are they individually optimal. (Individual optimality does not necessarily result in overall optimality owing to interdependencies (aka externalities) among the individuals). My ability to buy as much as I want to, and to sell as much as I want to, at market-clearing prices is contingent on everyone else being able to buy and sell as much as I and they want to at those same prices.

Now let’s take the argument a step further. Suppose the equilibrium decisions involve making purchases and sales in both the present and the future, according to current expectations of what future conditions will be like. If you are running a business, how much inputs you buy today to turn into output to be sold tomorrow will depend on the price at which you expect to be able to sell the output produced tomorrow. If decisions to purchase and sell today depend not only on current prices but also on expected future prices, then your optimal decisions now about how much to buy and sell now will depend on your expectations of buying and selling prices in the future. For an equilibrium in which everyone can execute his or her plans (as originally formulated) to exist, each person must have rational expectations about what future prices will be, and such rational expectations are possible only when those expectations are mutually consistent. In game-theoretical terms, a Nash equilibrium obtains only when all the individual expectations on which decisions are conditional converge.

Here is how Tom Schelling explained the idea of rational – i.e., convergent – expectations in a classic discussion of cooperative games.

One may or may not agree with any particular hypothesis as to how a bargainer’s expectations are formed either in the bargaining process or before it and either by the bargaining itself or by other forces. But it does seem clear that the outcome of a bargaining process is to be described most immediately, most straightforwardly, and most empirically, in terms of some phenomenon of stable and convergent expectations. Whether one agrees explicitly to a bargain, or agrees tacitly, or accepts by default, he must if he has his wits about him, expect that he could do no better and recognize that the other party must reciprocate the feeling. Thus, the fact of an outcome, which is simply a coordinated choice, should be analytically characterized by the notion of convergent expectations.

The intuitive formulation, or even a careful formulation in psychological terms, of what it is that a rational player expects in relation to another rational player in the “pure” bargaining game, poses a problem in sheer scientific description. Both players, being rational, must recognize that the only kind of “rational” expectation they can have is a fully shared expectation of an outcome. It is not quite accurate – as a description of a psychological phenomenon – to say that one expects the second to concede something; the second’s readiness to concede or to accept is only an expression of what he expects the first to accept or to concede, which in turn is what he expects the first to expect the second to expect the first to expect, and so on. To avoid an “ad infinitum” in the description process, we have to say that both sense a shared expectation of an outcome; one’s expectation is a belief that both identify the outcome as being indicated by the situation, hence as virtually inevitable. Both players, in effect, accept a common authority – the power of the game to dictate its own solution through their intellectual capacity to perceive it – and what they “expect” is that they both perceive the same solution.

If expectations of everyone do not converge — individuals having conflicting expectations about what will happen — then the expectations of none of the individuals can be rational. Even if one individual correctly anticipates the outcome, from the point of view of the disequilibrium system as a whole, the correct expectations are not rational because those expectations are inconsistent with equilibrium of the entire system. A change in the expectations of any other individual would imply that future prices would change from what had been expected. Only equilibrium expectations can be considered rational, and equilibrium expectations are a set of individual expectations that are convergent.

“The Idleness of Each Is the Result of the Idleness of All”

Everyone is fretting about how severe the downturn that is now starting and causing the worst plunge in the stock market since the 1929 crash is going to be. Much of the discussion has turned on whether the cause of the downturn is a supply shock or a demand shock. Some, perhaps many, seem to think that if the shock is a supply, rather than a demand, shock, then there is no role for a countercyclical policy response designed to increase demand. In other words, if the downturn is caused by people getting sick from a highly contagious virus, making it dangerous for people to gather together to work, then output will necessarily fall. Because the cutback in the supply of labor necessarily will cause a reduction in output, trying to counteract supply shock by increasing demand, as if an increase in demand could prevent the reduction of output associated with a reduced labor force after the onset of the virus, seems like an exercise in futility.

The problem with that reasoning is that reductions in supply are themselves effectively reductions in demand. The follow-on reductions in demand constitute a secondary contractionary shock on top of the primary supply shock, thereby setting in motion a cumulative process of further reductions in supply and demand. From that aggregate perspective, whether the initial contractionary shock is a shock to supply or to demand is of less importance than ensuring that the cumulative process is short-circuited by placing a floor under aggregate demand (total spending) so that the contraction caused by the initial supply shock does not become self-amplifying.

The interconnectedness of the entire economy, and the inability of any individual to avoid the consequences of a social or economic breakdown by making different (better) choices — e.g., accepting a cut in wages to retain employment — was recognized by the most orthodox of all Cambridge University economists, Frederick Lavington, in his short book The Trade Cycle published in 1922 in the wake of the horrendous 1921-22 depression from which the profound observation that serves as the title of this post is taken.

It’s now 60 years since John Nash defined an equilibrium as a situation in which “each player’s mixed strategy maximizes his payoff if the strategies of the others are held fixed. Thus each player’s strategy is optimal against those of the others.” If the expectations of other agents on which other agents are conditioning their strategies (plans) are sufficiently pessimistic, then an unemployed worker may not be able to find employment at any wage, even if it is only a small fraction of the wage earned when last employed. That situation is not the result of a diminution in the productivity of the worker, but of the worsening expectations underlying the strategies (plans) of other agents.

To call unemployment “voluntary” under such circumstances is like calling the reduced speed of drivers in a traffic jam “voluntary.” To suppose that the intersection of a supply-demand diagram provides a relevant analysis of the problem of unemployment under circumstances in which there are massive layoffs of workers from their jobs is absurd. Nevertheless, modern macroeconomics for the most part proceeds as if the possibility of an inefficient Nash equilibrium is irrelevant to the problems with which it is concerned.

There are only two ways to prevent that cumulative decline from taking hold. The first is to ensure that there is an immediate readjustment of all relative prices to a new equilibrium at which all agents are able to simultaneously formulate and execute optimal plans by buying and selling at market-clearing equilibrium prices. Such an immediate readjustment of relative prices to a new equilibrium price vector is, for a multitude of reasons which I have described in my recent paper “Hayek, Hicks, Radner and Four Equilibrium Concepts,” an extremely implausible outcome.

If an immediate adjustment to an unexpected supply shock that would return a complex economy back to the neighborhood of equilibrium is not even remotely likely, then the only way to ensure against a cumulative decline of aggregate output and employment is to prevent total spending from declining. And if total spending is kept from declining in the face of a decline in total output due to a supply shock, then it follows, as a matter of simple arithmetic, that the prices at which the reduced output will be sold are going to be correspondingly higher than they would have been had output not fallen.

In the face of an adverse supply shock, a spell of inflation lasting as long as the downturn is therefore to be welcomed as benign and salutary, not resisted as evil and destructive. The time for a decline in, or reversal of, inflation ought to be postpone till the recovery is under way.

David Glasner
Washington, DC

I am an economist in the Washington DC area. My research and writing has been mostly on monetary economics and policy and the history of economics. In my book Free Banking and Monetary Reform, I argued for a non-Monetarist non-Keynesian approach to monetary policy, based on a theory of a competitive supply of money. Over the years, I have become increasingly impressed by the similarities between my approach and that of R. G. Hawtrey and hope to bring Hawtrey’s unduly neglected contributions to the attention of a wider audience.

My new book Studies in the History of Monetary Theory: Controversies and Clarifications has been published by Palgrave Macmillan