Slide 1Slide 2Slide 3How are scarce resources assigned to alternative uses?Slide 5How are scarce resources assigned to alternative uses?the beginnings of a mathematical theoryAugustin Cournot (1801-1877)Slide 9Slide 10Slide 11Leon Walras (1834-1910)Leon Walras (1834-1910)Irving Fisher (1867-1947)Irving Fisher (1867-1947)[ Irving Fisher (1867-1947)[….]Arrow-Debreu-McKenzie TheoremKenneth ArrowGerard DebreuExchange Market Model (without production)Exchange Market Model (without production)Competitive (or Walrasian) Market EquilibriumArrow-Debreu Theorem 19546.896: Topics in Algorithmic Game TheoryLecture 13bConstantinos 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 supplyequilibrium pricesthe beginnings of a mathematical theoryAugustin Cournot (1801-1877)notion of a demand functionD = F(p), where p is the price; F(.) is assumed continuous, and it is taken as an empirical proposition that the demand function is decreasinganalysis of a monopoly:- profit-maximizing producer with cost f(D), for production D; discusses decreasing, constant and increasing cost functions- equations determining equilibrium priceduopoly model:two rival producers of a homogeneous productunlimited 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 equilibriumtâtonnement: price adjustment mechanismIrving 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)1891Stock 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, 1983Exchange Market Model (without production)Consider a marketplace with:traders (or agents)goods (or commodities) assumed to be infinitely divisibleUtility function of trader i:Endowment of trader i:non-negative realsamount of goods trader comes to the marketplace withconsumption set for trader ispecifies trader i’s utility for bundles of goodsUnder this price vector, each trader would like to sell some of her endowment and purchase an optimal bundle using her income from what s/he sold; thus she solves the following program:Suppose the goods in the market are priced according to some price vector . Programi(p)Note: If ui is continuous and is compact, then the above program has a well-defined optimum value.Exchange Market Model (without production)Competitive (or Walrasian) Market Equilibriumtotal demandtotal supplyDef: A price vector is called a competitive market equilibrium iff there exists a collection of optimal solutions to Programi(p), for all i = 1,…, n, such that the total demand meets the total supply, i.e.Arrow-Debreu Theorem 1954Theorem [Arrow-Debreu 1954]: SupposeThen a competitive market equilibrium exists.(i) is closed and convex (iii a) (iii b) (iii c) (ii) (all coordinates
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