Examples for economists with DERIVE: general equilibrium for a Cobb-Douglas exchange economy

In a Cobb-Douglas exchange economy with m consumers and n commodities, each consumer maximises a utility function subject to a budget constraint, specified in terms of the consumer's endowment vector. An equilibrium price system has the property that the corresponding utility-maximising demands are such that excess demand is zero. The advantage of the Cobb-Douglas model in general equilibrium theory is that, like the Leontief model of an economy with production, everything can be calculated exactly. Indeed, the equilibrium price system is an eigenvector corresponding to a unit eigenvalue of a square matrix of order n associated with the matrix of endowments and of utility parameters. This is explained briefly in Luenberger (1995). Related results and historical precursors appear in Gale (1960). Computational aspects are discussed in Eaves (1985). A complete account, with BASIC programs for implementing solutions, is given by Afriat (1987).