lecture14_slides

lecture14_slides - Econ 121 Intermediate Microeconomics...

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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 14 Outline of the course I. Introduction II. Individual choice III. Competitive markets IV. Market failure Outline of the course I. Introduction II. Individual choice III. Competitive markets  Exchange economies (Ch. 31) IV. Market failure Failure of the FWT: monopolist Suppose that consumer A behaves as a monopolist. He goes to consumer B and makes him a take-it-or-leave-it offer: “Either we trade at prices .p1; p2/ or we do not trade at all.” Failure of the FWT: monopolist If consumer B accepts he will choose the bundle that maximizes his utility subject to the budget constraint associated with the prices that were offered by A. If consumer B rejects, he will have to consume his endowment. So, he always accepts. Failure of the FWT: monopolist A knows that whatever price offer he makes, consumer B will accept. Thus, A knows that consumer B will choose a bundle on his offer curve. Hence, the monopolist,A, must maximize his utility over all feasible allocations where B ’s consumption bundle belongs to B ’s offer curve. By the first-order conditions (tangency), the allocation that arises under a monopolist is such that A’s indifference curve is tangent to B ’s offer curve, not to B ’s in difference curve (as required by Pareto efficiency). x2
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 Second Welfare Theorem The First Welfare Theorem states that given any competitive equilibrium .x; p/, the allocation x must be Pareto efficient. Thus, competitive markets always allocate goods efficiently. How about the converse? Is it true that for any Pareto efficient allocation x there is a vector of prices p D .p1; p2/ such that .x; p/ is a competitive equilibrium? Second Welfare Theorem The answer is no. First, in a competitive equilibrium, each consumer must be getting at least as much utility as he would get form consuming his endowment. endowment
 Second Welfare Theorem So, let us change the question a bit: Is it true that for any Pareto efficient allocation x that gives each consumer at least as much utility as he gets from his endowment there is a vector of prices p D .p1; p2/ such that .x; p/ is a competitive equilibrium? Second Welfare Theorem The answer is still no, as shown in the following picture. B
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 Second Welfare Theorem The Second Welfare Theorem states that the answer to the former question becomes yes provided we allow a social planner (the government) to design an appropriate tax system. A lump-sum tax/subsidy  is a tax/subsidy that enters additively in the consumer’s budget constraint: i i p1x1 C p2x2  i i p1!1 C p2!2 C : A system of lump-sum taxes/subsidies  i (i D A; B ), is called purely redistributive if  A C  B D 0. Second Welfare Theorem Theorem. Assume preferences are convex. If x is a Pareto efficient allocation then there exist prices p D .p1; p2/ and purely redistributive lump-sum transfers  D .i /i DA;B such that .x; p/ is a competitive equilibrium under  . A partial converse to the First Welfare Theorem. Pareto efficiency is a socially desirable feature for an allocation, but does not pin down a single allocation. A social planner might like some allocations in the contract curve better than others, due to considerations of fairness or some other issues (outside the scope of Economics). The SWT highlights the role of taxes/redistribution as a policy instrument to implement a target allocation in the contract curve—the favorite allocation of a social planner, say. Proof assuming preferences are smooth If x is a Pareto efficient allocation then MRS A.x A/ D MRS B .x B /: Set p2 D 1 and p1 D MRS A.x A/. Hence, MRS equals minus price ratio for both consumers and x clears the market. Now for each i D A; B define  i such that: i i i i p1x1 C p2x2 D p1!1 C p2!2 C  i Summing across i yields A B A B p1.x1 C x1 / C p2.x2 C x2 / D p1!1 C p2!2 C  A C  B and hence  A C  B D 0: Q.E.D. General proof While we will not go over the general proof without assuming smoothness, it is important to note that the only important step in the proof was to find prices that define a budget line that “separate” the upper contour sets of the two consumers in the Edgeworth box. Such “separation argument” can be made with much greater generality, using more powerful tool from mathematics. The main idea is the Separating Hyperplane Theorem, illustrated in the following picture. Separa&ng
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This note was uploaded on 03/01/2012 for the course ECON 121 taught by Professor Samuelson during the Spring '09 term at Yale.

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