Economics 201B Second Half Lecture 12 4 22 10 Justifying or Undermining the Price Taking Assumption Many formulations Core Ostroy s No Surplus Condition Bargaining Set Shapley Shubik Market Games noncooperative other noncooperative games Core is the most commonly used The core is the set of all allocations such that no coalition set of agents can improve on or block the allocation make all of its members better o by seceding from the economy and only trading among its members Core is institution free no mention of prices Core convergence means roughly that For economies with a large number of agents core allocations are approximately Walrasian Approximately Walrasian means di erent things in di erent contexts depending on what we are willing to assume Three motivations for the study of the core Walrasian allocations lie in the core Important strengthening of First Welfare Theorem under same minimal assumptions as First Welfare Theorem Positive Strong stability property of Walrasian equilibrium no group of individuals would choose to upset the equilibrium by recontracting among themselves 1 Normative If distribution of initial endowments is equitable no group is treated unfairly at a core allocation Since Walrasian allocations lie in the core this is a Group Fairness Property of Walrasian Equilibrium Core Convergence strengthens Second Welfare Theorem Second Welfare Theorem says every Pareto Optimum is a Walrasian Equilibria with Transfers Core convergence asserts that core allocations of large economies are nearly Walrasian without transfers One version states that core allocations can be realized as exact Walrasian equilibrium with small income transfers Strong unbiasedness property of Walrasian equilibrium Restricting to Walrasian outcomes does not narrow possible outcomes beyond narrowing occurring in the core Normative No hidden implications for welfare of di erent groups beyond equity issues in the initial endowment distribution Normative Assuming distribution of endowments is equitable any allocation that is far from Walrasian will not be in the core and hence will treat some group unfairly Core Convergence justifies Price Taking Core Nonconvergence suggests Price Taking is Implausible The de nition of Walrasian equilibrium contains hidden in plain sight assumption that economic agents act as price takers 2 In real markets we see prices used to equate supply and demand but this does not guarantee Walrasian outcome Agents possessing market power may choose to supply quantities di erent from the competitive supply for the prevailing price thereby altering that price and leading to a non Pareto Optimal outcome If outcome is not Walrasian Welfare Theorems Existence Determinacy would have limited implications for real economies Positive Core convergence and nonconvergence allows us to identify situations in which price taking is more or less reasonable Edgeworth de ned core in 1881 in Mathematical Psychics an ambitious book developing microeconomic theory in mathematical terms Edgeworth criticized Walras thought the core not the set of Walrasian equilibria was best positive description of outcomes from market mechanism In particular the de nition of the core does not impose the assumption of price taking behavior made by Walras Furthermore if any allocation not in the core arose some group would nd it in its interests to recontract Edgeworth thus argues that the core is the signi cant positive equilibrium concept If core is correct positive concept core convergence justi es price taking Core convergence says all trade takes place at almost a single price Agent who tries to bargain 3 cannot in uence prices much and cannot change outcome much argument more compelling with stronger convergence notions If core is correct positive concept core nonconvergence undermines price taking Edgeworth himself argued that in real life the presence of large rms leads to failure of price taking Definition 1 In an exchange economy a coalition is a set S 1 I A coalition S blocks or improves on an exact allocation x by x if i S x i i S i and i S x i i xi The core is the set of all exact allocations which cannot be improved on by any nonempty coalition Notice we follow MWG and require x i i xi for all i S this is analogous to the de nition of weakly Pareto Optimal Natural status quo should be focal need strict improvement to join a coalition to upset the status quo Notice that the de nition of blocking by a coalition does not specify what happens to the individuals outside the coalition One might imagine individuals not in the blocking coalition making a counterproposal to some of those in the blocking coalition the Bargaining Set takes these counterproposals into account 4 r o A 7 d ore Cote EJ aCo r n rflf I r IQO d s t rl 4 c 1 of j pt It is a common mistake to ask at a core allocation what coalition s are active A core allocation is de ned by the fact that no coalition can defeat it Theorem 2 In an exchange economy every core allocation is weakly Pareto Optimal Proof If x is not weakly Pareto Optimal then there exists x I i 1 x i x i i xi Then S 1 I improves on x by x so x is not in the core Theorem 3 Strong First Welfare Theorem In an exchange economy every Walrasian Equilibrium lies in the core Proof Suppose p x is a Walrasian Equilibrium If x is not in the core there exists S I S and x i i S such that i S x i i S i i S x i i x i Since x i Di p p x i p i so p i S x i i S p x i p i i S p i S but i S x i contradiction Therefore x is in the core 5 i S i i Theorem 4 Suppose we are given an exchange economy with L commodities I agents and preferences 1 I satisfying weak monotonicity if x x y then x i y and the following free disposal condition y y i z x i z If x is in the core then there exists p such that I 1 2L p xi i max 1 I i 1 I I 1 4L inf p y xi y i xi max 1 I i 1 I where x I 1 I 2 max x1 xL Equation 1 says that trade occurs almost at the price p and that each xi is almost in the budget set Equation 2 says that the price p almost supports i at xi If we knew the left sides of Equations 1 and 2 were zero then p xi i 0 xi Bi p y i xi p y p i so x is a Walrasian quasiequilibrium Thus every core allocation satis es a perturbation of the de nition of Walrasian Equilibrium agent i s consumption need not lie in his her budget set but it can t be far outside anything strictly preferred neeed not be outside the budget set but it can t be far below the budget frontier 6 Outline of Proof Follow the