Exchange clear Pure exchange: model where all of the Teemu Olvio - PDF document

• Partial equilibrium: Mat-2.142 Seminar on optimization Microeconomics prices assumed to remain fixed (except studied one) • General equilibrium: Session 13: all prices are variable and all markets Exchange clear Pure exchange: model where all of the Teemu Olvio economic agents are consumers Kari Vatanen S ystems S ystems Analysis Laboratory Analysis Laboratory Session 1 - Student presentation Session 2 - Student presentation Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 1 Seminar on Microeconomics - Fall 1998 / 2 • The aggregate excess demand function: • Agent i’s consumption bundle: n = ( 1 k xi xi ,..., xi ) = ω − ω ∑ z p ( ) [ xi p p i ( , ) ] i = i 1 • Walrasian equilibrium: • Walras’s law: For any price vector p , ( *, * ω ≤∑ ω we have pz(p) ≡ 0; i.e., the value of the ∑ ) xi p p i i i i excess demand is identically zero. S ystems S ystems Analysis Laboratory Analysis Laboratory Session 3 - Student presentation Session 4 - Student presentation Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 3 Seminar on Microeconomics - Fall 1998 / 4 • Market clearing: If demand equals • Desirability: If , then supply in k-1 markets, and , then pi = 0 ( ) > 0 pk > 0 zi p demand must equal supply in the for i=1,…,k. kth market. • Free goods: If p * is a Walrasian • Equality of demand and supply: If all equilibrium and (p*) ≤ 0 0, then . goods are desirable and p* is a * = 0 pj zj That is, if some good is in excess supply Walrasian equilibrium, then z ( p* )=0. at a Walrasian equilibrium it must be a free good. S ystems S ystems Analysis Laboratory Analysis Laboratory Session 5 - Student presentation Session 6 - Student presentation Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 5 Seminar on Microeconomics - Fall 1998 / 6 1

• Existence of Walrasian equilibria: If • Brouwer fixed-point theorem: If f : − → Sk 1 Rk z : is a continuous function − → − Sk 1 Sk 1 is a continuous function that satisfies Walras’ law, pz ( p ) ≡ 0, from the unit simplex to itself, there is Sk − 1 then there exists some p* in such Sk − 1 some x in such that x = f(x ). that z ( p* ) ≤ 0. S ystems S ystems Analysis Laboratory Analysis Laboratory Session 7 - Student presentation Session 8 - Student presentation Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 7 Seminar on Microeconomics - Fall 1998 / 8 Definitions of Pareto efficiency Equivalence of weak and strong Pareto efficiency • Weak Pareto efficiency – A feasible allocation x is weakly Pareto efficient allocation if there is no feasible • If preferences are continuous and allocation x’ such that all agents strictly prefer monotonic, then an allocation is weakly x’ to x Pareto efficient if and only if it is strongly • Strong Pareto efficiency Pareto efficient – There is no feasible allocation x’ such that all agents weakly prefer x’ to x and some agent strictly prefer x’ to x S ystems S ystems Analysis Laboratory Analysis Laboratory Session 9 - Student presentation Session 10 - Student presentation Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 9 Seminar on Microeconomics - Fall 1998 / 10 Walrasian equilibrium Edgeworth box • An allocation-price pair ( x , p ) is a Consumer 2 good 2 Walrasian equilibrium if max u 1 ( x 1 ) – the allocation is feasible x1, x2 � ∑ ∑ x i = ∑ ∑ ω ω i such that u 2 ( x 2 ) ���� û 2 ≥ – each agent is making an optimal choice from x 1 + x 2 = ω ω 1 + ω ω 2 his budget set • If x i ’ is preferred by agent i to x i , then Consumer 1 px i ’ > p ω ω i good 1 S ystems S ystems Analysis Laboratory Analysis Laboratory Session 11 - Student presentation Session 12 - Student presentation Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 11 Seminar on Microeconomics - Fall 1998 / 12 2

Second Theorem of Welfare Economics First Theorem of Welfare • x * is a Walrasian equilibrium for the initial endowments ω ω i = x i * for i = 1, … , n Economics – if x * is a Pareto efficient allocation in which each agent holds a positive amount of each • If ( x , p ) is a Walrasian equilibrium, then x good and if preferences are convex, continuous is Pareto efficient and monotonic • If a competitive equilibrium ( p , x’ ) exists from the initial endowments ω ω i = x i *, then, in fact, (p, x *) is a competitive equilibrium S ystems S ystems Analysis Laboratory Analysis Laboratory Session 13 - Student presentation Session 14 - Student presentation Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 13 Seminar on Microeconomics - Fall 1998 / 14 Calculus Calculus • Pareto efficiency – A feasible allocation x * is Pareto efficient if • Market equilibrium and only if x * solves the following n – If ( x *, p*) is a market equilibrium with each maximization problems for i = 1, …, n consumer holding a positive amount of every good, then there exist a set of numbers max u i ( x i ) ( λ 1 , …, λ n ) such that: (x i g ) n such that ∑ ∑ x ig ω ≤ ω g g = 1, …, k i=1 D u i ( x *) = λ i p * i = 1 , …, n u j (x j *) u j (x j ) j ≠ i ≤ S ystems S ystems Analysis Laboratory Analysis Laboratory Session 15 - Student presentation Session 16 - Student presentation Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 15 Seminar on Microeconomics - Fall 1998 / 16 Welfare maximization Social welfare function • If x* is Pareto efficient allocation (x i * >> 0) • Social utility W(u 1 ,…, u n ) • If utility functions are concave, continuous and monotonic functions max W(u 1 ( x 1 ),…, u n ( x n )) • Then there is some choice of weights a i * n such that ∑ ∑ x ig ω ω g g = 1, …, k such that x* maximizes ∑ ∑ a i * u i ( x i ) subject i=1 ≤ to resource constraints – now a i * = 1/ λ i where λ i is the i th agent’s • If x * maximizes a sosial welfare function, then x * is Pareto efficient marginal utility of income S ystems S ystems Analysis Laboratory Analysis Laboratory Session 17 - Student presentation Session 18 - Student presentation Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 17 Seminar on Microeconomics - Fall 1998 / 18 3

Relationships • Competitive equilibria are always Pareto efficient • Pareto efficient allocations are competitive equilibria under convexity assumptions and endowment redistribution • Welfare maxima are always pareto efficient • Pareto efficient allocations are welfare maxima under concavity assumptions for some choice of welfare weights S ystems Analysis Laboratory Session 19 - Student presentation Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 19 4