Walras, Marginal Productivity Theory and the Interdependence of the Economy

Walras's general equilibrium theory was dependent not only upon demand and, therefore, utility but also upon supply and, therefore, diminishing marginal productivity. Here, too, there is much ambiguity in Walras's exposition. In Lesson 20 of the first three editions, his model used constant coefficients of production, which is to say that there is no marginal product because one factor cannot be varied independently of another. Thus, his early exposition of general equilib­rium theory did not have the second underpinning of a full general equilibrium model. Despite this, he stated that the analysis can be extended to include variable coefficients of production. The reader is left to accept that possibility on faith.

Walras recognized the problem and in the late 1800s asked a colleague how he could extend his analysis to include variable factors of production. Thus, in 1900, in the fourth edition, he incorporated variable factors of production and, thereby, the marginal productivity underpinnings of supply. Yet Walras's incor­poration of marginal productivity came six years after Philip Wicksteed had formally developed the marginal productivity concept and had publicized its importance. Because of this, Walras's contribution to marginal analysis on the supply front is open to question. As was the case with marginal utility, his interest was in the supply function that he needed for his general equilibrium theory, not in the production function that underlay it.

Walras was aware of some of the deficiencies of his model. Other problems were not identified or solved for nearly sixty years after 1874, and some are still not solved. To see some of these problems, consider the following questions.

Is a general equilibrium solution possible?

Some individuals thought that by simply counting equations and unknowns, the existence of a general equilibrium could be deduced. Abraham Wald showed in 1933 that that was not the case and that proving the existence of a solution was far more complicated. It was only in 1954 that Gerard Debreu and Kenneth Arrow were able to prove the existence of a general equilibrium solution.

If a solution does exist, is it a solution that is economically meaningful, or will it yield negative prices and quantities?

Just because one can mathematically prove the existence of general equilibrium does not mean that it has any relevance to the real world. Because the connection between general equilibrium and the real world is so tangential, it is not at all clear that the mathematics is relevant. It has been called the celestial mechanics of a nonexistent world.

How does production fit into the Walrasian system?

Although the Walrasian system seems to include production, careful considera­tion reveals that it is primarily a model of exchange and that production has been inappropriately related to it. As long as there are constant returns to scale, this presents no problem; but if there are increasing returns to scale, the model has serious problems.

Will the equilibrium conditions produced by the market in the various sectors of the economy be consistent with a general equilibrium for the entire economy?

Walras thought he had answered this complicated question, but he hadn't. There are strict conditions under which such consistency will be achieved.

The unknowns determined by the market and given by a general equilibrium solution are (1) the prices of final goods, (2) the prices of factors, (3) the quantities of final goods supplied and quantities demanded, and (4) the quantities of factors supplied and quantities demanded. Is there only one set of prices and quantities that will result in equilibrium for the entire economy, or are there many possible equilibria?

Walras recognized the possibility of multiple general equilibria, and general equilibrium analysis still must contend with it. General equilibrium theorists can show the conditions under which there will be a unique equilibrium, but they cannot show that those are the conditions we can expect in the economy. The matter becomes even more complicated when one tries to include expectations in the model, as one does in what are called sunspot models. Multiple equilibria abound in these models. The possibility of multiple equilibria is one of the greatest limitations of applying the general equilibrium model to the real world. How do multiple equilibria make a difference? With multiple equilibria, even though the market solution may be an equilibrium, it need not be the best equilibrium; a preferable equilibrium might exist. Moreover, if a preferable equilibrium exists, a disequilibrium to that preferable equilibrium might actually be preferable to the equilibrium the market achieves.

Is the equilibrium stable or unstable?

An equilibrium is not necessarily stable; if the model is thrown out of equilibrium, will it return to equilibrium? This issue was answered relatively quickly, and the conditions necessary for stability were shown. What was not shown was whether those conditions fit reality. Several events might actually undermine stability. The very process of the market at work may cause shifting mathematical functions that will not result in final equilibrium. In another scenario, a final equilibrium may be reached, but its position may depend upon the path followed by the variables in the system. Thus, different final equilibrium values may be possible.

How will the equilibrium be achieved? Who sets the price, and what happens if there is disequilibrium trading?

Walras struggled with this question, which is now playing a significant role in modern macroeconomic debates. He proposed numerous schemes involving written and oral pledges and a tatonnement process in which an auctioneer (who has since acquired the name the Walrasian Auctioneer) processes all the bids and offers, determines which prices will clear all markets, and only then allows trading. Donald Walker, who has examined these schemes in depth, has con­cluded that the model is fatally flawed, because Walras did not endow it with enough viable features. Walker's conclusion is extremely damaging to the new classical branch of macroeconomics, which bases its analysis on the reasonable­ness of the assumed auctioneer.

These problems are substantial, but they do not undermine Walras's achieve­ment. He set the framework within which many of the best minds in modern economics have posed questions. Issues concerning the existence and stability of a general equilibrium occupied economists well into the 1950s. Other questions are still occupying them. Even though Walras's formulation was less than perfect mathematically, it has been the framework for advanced research since the 1950s.

The source of Walras's success, his use of mathematics, was also the cause of some of the failures of general equilibrium theory. The highly abstract model offered insight into the interrelatedness of the economy, but Walras made no attempt to measure the concepts in his model empirically. They were not designed to be measured; it was theory without empirical application. The difficulty of measuring the concepts has remained a criticism of general equilib­rium theory through modern times. Thus, although it demonstrates the relation­ships existing within an economy in equilibrium, general equilibrium theory does not explain what happens in that economy when the factors that Walras took as fixed actually change.

The conclusion of most scholars is that although the general equilibrium model has tremendous potential for use in answering questions concerning the consequences of alternative economic policies, this potential has yet to be realized. Frank Hahn, a general equilibrium theorist, writes:

It was Adam Smith who first realized the need to explain why this kind of social arrangement does not lead to chaos. Millions of greedy, self-seeking individuals, in pursuit of their own ends and mainly uncontrolled in these pursuits by the State, seem to "common sense" a sure recipe for anarchy. Smith not only posed an obviously important question, but also started us off on the road to answering it.

General Equilibrium Theory as classically stated by Arrow and Debreu [1954 and 1959] is near the end of that road. Now that we have got there we find it less enlightening than we had expected.