Exchange of two commodities and several commodities for one another

Walras first tries to show that it is possible to solve the problem of exchange between two persons of two commodities, and find the equilibrium prices and quantities. Walras “proves” the existence of the equilibrium by counting the (equal) number of equations and of unknowns.

The generalization of this reasoning to many commodities raises difficulties: in par­ticular a generalized direct or indirect barter could no longer lead to a solution. Walras solved the problem by adopting at the same time a concept of equilibrium which rules out arbitrage (OEC VIII: 161-2), and a numeraire. The introduction of the numeraire, that is, “the commodity in terms of which the prices of all the others are expressed” (OEC VIII: 171 [161]), reduces the number of markets and prices and de facto forces all exchanges to be made against or through the numeraire.

Walras also generalizes the budget constraint of each agent in the two-commodity model, where obtaining a commodity required the sale of the other one, with a simul­taneous equilibrium in both markets. In the case of many commodities, if all markets except one are in equilibrium, then the latter is necessarily in equilibrium, to the extent that consumers have met their budget constraints. This is what Oskar Lange called “Walras’s Law” (1942: 51, n.

What Walras secondly tries to show is “that this selfsame problem of exchange of which we just have furnished the theoretical solution is also the problem that is solved [practically] on the market by the mechanism of free competition” (OEC VIII: 173 [163]). Imagine a market in which all commodities and all economic agents are present and where prices are set at random. If by sheer chance every demand and supply are in equilibrium, the problem is solved. Otherwise, other prices should be set so that, through trial and error, by tatonnement, supplies and demands in each market end up coinciding. If, among the newly set prices, at least one assures the equilibrium for a commodity, only the others unbalancing prices are set again. And so on for all markets. However, by bal­ancing markets one after the other, the previously balanced markets end up unbalanced. Walras then distinguishes between the direct effects and indirect effects and assumes that the former are larger than the latter (OEC VIII: s. 130). This makes him think that he obtains convergence towards equilibrium prices and is thus able to formulate the law of price determination in the case of trading several commodities one for the other. It should be noted that exchange is assumed not to take place until after the equilibrium prices have been found for all commodities (OEC IX: 312 [251]).