Market and Game Dynamics

where xij(p) is the optimum allocation of good i to agent j under the price vector p and wij is the j’s endowment of good i. Last lecture, we showed that f(p) has the following two properties: • Homogeneity (H): f(αp) = f(p), ∀α > 0 • Walras Law (W): p f(p) = 0 In this lecture, we introduce a dynamic process for updating the price vector which converges to the equilibrium price for all excess demand functions that satisfy the gross substitutability condition. Gross Substitutability (GS): If pi ≤ pi, ∀i, and ∃j such that pj < pj , then fi(p) > fi(p) ∀i 6= j in other words, dfi dpj > 0, i 6= j. Tatonnement Process: If the excess demand satisfies (H), (W), and (GS) then the following process, which is called Tatonnement process, converges to the market clearing price vector, p. dpi dt = G(fi(p)),