6.896: Topics in Algorithmic Game Theory Lecture 13b Constantinos DaskalakisMarkets“Economics is a science which studies human behavior as a relationship between ends and scarce means which have alternative uses.” Lionel Robbins (1898 – 1984)!How are scarce resources assigned to alternative uses?How are scarce resources assigned to alternative uses? Prices! Parity between demand and supply equilibrium prices!the beginnings of a mathematical theoryAugustin Cournot (1801-1877)notion of a demand function D = F(p), where p is the price; F(.) is assumed continuous, and it is taken as an empirical proposition that the demand function is decreasing analysis of a monopoly: - profit-maximizing producer with cost f(D), for production D; discusses decreasing, constant and increasing cost functions - equations determining equilibrium price duopoly model: two rival producers of a homogeneous product unlimited competition, communication of markets on single commodity, … Cournot’s Contributions:!question left unanswered by Cournot…Leon Walras (1834-1910) Elements of Pure Economics, or the theory of social wealth, 1874Leon Walras (1834-1910) - goal was to solve the problem that was left open by Cournot, i.e. characterize the price equilibrium of a market with many commodities; - gave system of simultaneous equations for price equilibrium: informal argument for the existence of an equilibrium based on the assumption that an equilibrium exists whenever the number of equations equals the number of unknowns - recognized the need for dynamics leading to an equilibrium tâtonnement: price adjustment mechanism!Irving Fisher (1867-1947) hydraulic apparatus for solving markets with 3 traders with money and a producer of 3 commoditiesFirst computational approach! Irving Fisher (1867-1947) 1891!Stock Market Crash of 1929: [ Irving Fisher (1867-1947) "Stock prices have reached what looks like a permanently high plateau." Market is “only shaking out of the lunatic fringe” [October 21, 1929 (8 days before black Tuesday)] [Shortly before the crisis] ]![….]Arrow-Debreu-McKenzie Theorem Established the existence of a market equilibrium under very general conditions using Brouwer/Kakutani’s fixed point theorem. it is hence highly non-constructive! One of the most celebrated theorems in Mathematical Economics.Kenneth Arrow • Nobel Prize, 1972Gerard Debreu • Nobel Prize, 1983Market Model (without production) Consider a marketplace with: ntraders (or agents) kgoods (or commodities) assumed to be infinitely divisible Utility function of trader i: ui: Rk+−→ R+Endowment of trader i: non-negative reals specifies trader’s utility for bundles of goods ei: Rk+amount of goods trader comes to the marketplace with (common) assumption: uiis continuousMarket Model (without production) Under this price vector, each trader would like to sell some of his endowment and purchase an optimal bundle using her income from what s/he sold, solving the following program Suppose the goods in the market are priced according to some price vector . p ∈ Rk+max ui(x)s.t. p · x ≤ p · eix ∈ Rk+Programi If ui is continuous, then the above program has an optimum (since the constraint set is compact).Competitive (or Walrasian) Market Equilibrium A price vector is called a competitive market equilibrium iff there exists a collection of optimal solutions to Programi, for all i = 1,…, n, such that the total demand meets the total supply, i.e. p ∈ Rk+Xi(p)n�i=1Xi(p) ≤n�i=1eitotal demand total supplyArrow-Debreu Theorem 1954 Suppose: (i) for all traders i, for all price vectors p, Programi has a unique optimum . ˆXi(p)(ii) for all traders i, is continuous. ˆXi(p)Then a competitive market equilibrium exists. e.g. when ui is strictly