### 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.

#### 6 Responses to “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”

1. 1 Bradley G Lewis April 12, 2020 at 7:53 pm

And one or more versions of convergent expectations among a large majority of “players” in this game is also often what we see precisely when optimism or pessimism is at its peak–and probably likely to change. As, for example, when average expectations (as indicated by published surveys) by the public several decades ago that common stock returns would indefinitely stay about 17% for year, or earlier after the two large oil prices in the 1970s would mean that gasoline prices would continue to rise by maybe 15% pre year indefinitely. Which, in my humble opinion, strongly backs your point.

2. 2 Warren D. Miller April 13, 2020 at 8:42 am

Edits needed:

In the first sentence of this fine post, change “who” to “whose”.

The last word of the first sentence in the second graf should be “points”, not “point.”

The penultimate sentence of the second paragraph begins with a parenthesis that has no ending parenthesis.

The third sentence of the third pararaph should have the phrase “at which” inserted after the phrase “on the price.” The fourth sentence in this paragraph should have “also on” inserted between “but” and “expected.”

In the first sentence of the last paragraph, the word ‘or’ should be inserted before “if individuals.”

I hope these suggested edits are constructive and helpful.

Best regards–

Warren Miller, CPA, CFA
Lexington, Virginia

3. April 13, 2020 at 8:57 am

Many thanks for your careful reading of my post. I adopted almost all of your suggestions, except the final one, making a similar but slightly different change from your suggestion.

4. April 18, 2020 at 7:24 pm

Thanks, Bradley, for your comment. Not so sure that it backs my point, but certainly when there is convergence on one expectation, the consequences of a disappointment of that expectation are likely to be more serious than when an expectation not widely shared is disappointed.

1. 1 A Tale of Two Syntheses | Uneasy Money Trackback on May 8, 2020 at 3:37 pm

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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