Understanding markets: diagrams or equations?
Towards the end of the nineteenth century, different approaches to the study of supply and demand came into sight, relying more heavily upon mathematics than before. Of the many contributions of this period, only the approaches of Alfred Marshall (1890) and Walras (1874 [1988]) are sketched here as they are illustrative of methodological and substantive issues of great import for formalization.
Marshall took up the supply-and- demand diagrams of Cournot and other predecessors to develop a “partial equilibrium” approach, focusing on a single market. Partial equilibrium admits a simple solution where supply equals demand, but is only an approximation, based on the restrictive assumption that changes in the price of a good have repercussions on the quantity of that good only. In contrast, Walras emphasized interdependencies among markets and developed a notion of “general equilibrium” corresponding to equality between supply and demand on each market. A related, though distinct, question is whether and how actual trade practices will drive prices and quantities towards equilibrium; again, Marshall and Walras provided different answers. In Walras’s “tatonnement” process, transactions take place simultaneously, at equilibrium, so that the same prices apply to all traders. Instead, Marshall had in mind bilateral transactions that occur sequentially, in disequilibrium, at prices that may differ from a pair of traders to the other.Neither Marshall nor Walras identified the most appropriate combination of theoretical constructs and mathematical techniques. Marshall’s supply and demand geometry was a static tool that could not account for a dynamics of sequential transactions, and failed to convey the richness and depth of his thought. Its inconsistencies opened the way to criticism, most prominently with a famous attack by Piero Sraffa (1925). Walras, in turn, was unable to answer the question of whether his system admits a solution that is meaningful both mathematically and economically, that is, an equilibrium in which demand equals supply for scarce goods (that is, with positive prices) and demand does not exceed supply for free goods (that is, with a price of zero). Because his proofs were merely based on counting equations and unknowns, they could not rule out negative equilibrium prices or quantities; therefore, the very notion of equilibrium of a system of interrelated competitive markets remained vacuous.