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Slide 1Slide 2Exchange Market ModelFisher Market ModelCompetitive (or Walrasian) Market EquilibriumArrow-Debreu Theorem 1954Utility FunctionsSlide 8Slide 9Back to the Exchange ModelHardness of Leontief Exchange MarketsSlide 12Excess Demand at prices pJustification of (WL) under Arrow-Debreu Thm conditionsExcess Demand at prices pGross-Substitutability (GS)Differential Form of Gross-Substitutability (GSD)Not all goods are free (Pos)Properties of EquilibriumWeak Axiom of Revealed Preferences (WARP)Computation of EquilibriaTatonnementCorollariesCorollaries6.896: Topics in Algorithmic Game TheoryLecture 15Constantinos DaskalakisRecapExchange Market Modeltradersdivisible goodstrader i has:- endowment of goodsnon-negative realsamount of goods trader comes to the marketplace withconsumption set for trader ispecifies trader i’s utility for bundles of goods- utility functionFisher Market Modelcan be obtained as a special case of an exchange market, when endowment vectors are parallel:in this case, relative incomes of the traders are independent of the prices.n traders with:k divisible goods owned by seller; money mi , and utility function uiseller has qj units of good jCompetitive (or Walrasian) Market Equilibriumtotal demandtotal supplyDef: A price vector is called a competitive market equilibrium iff there exists a collection of optimal bundles of goods, for all traders i = 1,…, n, such that the total supply meets the total demand, i.e.[ For Fisher Markets: ]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 positive)Utility FunctionsCES (Constant Elasticity of Substitution) utility functions:linear utility formLeontief utility formCobb-Douglas formEisenberg-Gale’s Convex Program for Fisher ModelRemarks: - No budgets constraint!- It is not necessary that the utility functions are CES; the program also works if they are concave, and homogeneous- Optimal Solution is a market equilibrium (alternative proof of existence)Complexity of the Exchange ModelComplexity of market equilibria in CES exchange economies.At least as hard as solving Nash Equilibria [CVSY ’05]Poly-time algorithms known [Devanur, Papadimitriou, Saberi, Vazirani ’02], [Jain ’03], [CMK ’03], [GKV ’04],…OPEN!! Back to the Exchange Model -1 0 1 Hardness of Leontief Exchange MarketsProof Idea:Theorem [Codenotti, Saberi, Varadarajan, Ye ’05]: Finding a market equilibrium in a Leontief exchange economy is at least as hard as finding a Nash equilibrium in a two-player game.Reduce a 2-player game to a Leontief exchange economy, such that given a market equilibrium of the exchange economy one can obtain a Nash equilibrium of the two-player game.Corollary: Leontief exchange economies are PPAD-hard.Gross-Substitutability ConditionExcess Demand at prices pWe already argued that under the Arrow-Debreu Thm conditions:(H) f is homogeneous, i.e. (WL) f satisfies Walras’s Law, i.e. (we argued that the last property is true using nonsatiation + quasi-concavity, see next slie)suppose there is a unique demand at a given price vector p and its is continuous (see last lecture)Justification of (WL) under Arrow-Debreu Thm conditionsNonsatiation + quasi-concavity  local non-satiation at equilibrium every trader spends all her budget, i.e. if xi(p) is an optimal solution to Programi(p) theni.e. every good with positive price is fully consumedExcess Demand at prices pWe already argued that under the Arrow-Debreu Thm conditions:(H) f is homogeneous, i.e. (WL) f satisfies Walras’s Law, i.e. suppose there is a unique demand at a given price vector p and its is continuous (see last lecture)Gross-Substitutability (GS)Def: The excess demand function satisfies Gross Substitutability iff for all pairs of price vectors p and p’:In other words, if the prices of some goods are increased while the prices of some other goods are held fixed, this can only cause an increase in the demand of the goods whose price stayed fixed.Differential Form of Gross-Substitutability (GSD)Def: The excess demand function satisfies the Differential Form of Gross Substitutability iff for all r, s the partial derivatives exist and are continuous, and for all p : Clearly: (GS)  (GSD)Not all goods are free (Pos)Def: The excess demand function satisfies (Pos) if not all goods are free at equilibrium. I.e. there exists at least one good in which at least one trader is interested.Properties of EquilibriumLemma 1 [Arrow-Block-Hurwicz 1959]: Suppose that the excess demand function of an exchange economy satisfies (H), (GSD) and (Pos). Then if is an equilibrium price vectorCall this property (E+)Lemma 2 [Arrow-Block-Hurwicz 1959]: Suppose that the excess demand function of an exchange economy satisfies (H), (GS) and (E+). Then if and are equilibrium price vectors, there exists such that i.e. we have uniqueness of the equilibrium rayWeak Axiom of Revealed Preferences (WARP)Theorem [Arrow-Hurwicz 1960’s]: Proof on the boardSuppose that the excess demand function of an exchange economy satisfies (H), (WL), and (GS). If >0 is any equilibrium price vector and >0 is any non-equilibrium vector we haveComputation of EquilibriaCorollary 1 (of WARP): If the excess demand function satisfies (H), (WL), and (GS) and it can be computed efficiently, then a positive equilibrium price vector (if it exists) can be computed efficiently.proof sketch: W. l. o. g. we can restrict our search space to price vectors in [0,1]k, since any equilibrium can be rescaled to lie in this set (by homogeneity of the excess demand function). We can then run ellipsoid, using the separation oracle provided by the weak axiom of revealed preferences. In particular, for any non-equilibrium price vector p, we know that the price equilibrium lies in the half-spaceTatonnement Corollary 2: If the excess demand function satisfies continuity, (H), (WL), (GSD), and (Pos), then the tatonnement process (price-adjustment mechanism) described by the following differential equation converges to a market equilibriumTo show convergence to a price equilibrium, let us pick an arbitrary price


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