Algorithms for Equilibrium Prices in Linear Market Models

Abstract

Near the end of the 19th century, Leon Walrus [Wal74] and Irving Fisher [Fis91] introduced general market models and asked for the existence of equilibrium prices. Chapters 5 and 6 of [NRTV07] are an excellent introduction into the algorithmic theory of market models. In Walrus’ model, each person comes to the market with a set of goods and a utility function for bundles of goods. At a set of prices, a person will only buy goods that give him maximal satisfaction.1 The question is to find a set of prices at which the market clears, i.e., all goods are sold and all money is spent. Observe that the money available to an agent is exactly the money earned by selling his goods. Fisher’s model is somewhat simpler. In Fisher’s model every agent comes with a predetermined amount of money. Market clearing prices are also called equilibrium prices. Walrus and Fisher took it for granted that equilibrium prices exist. Fisher designed a hydromechanical computing machine that would compute the prices in a market with three buyers, three goods, and linear utilities [BS00].