lecture13_slides

# lecture13_slides - Econ 121. Intermediate Microeconomics....

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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 13 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 Competitive Equilibrium So far we have talked about allocations in an exchange economy without worrying about how these allocations emerge. Now we will consider trade in this economy. i i Let .!1; !2/ designate consumer i ’s endowment and let A B !1 D !1 C !1 B A !2 D !2 C !2 Write p1 for the price of good 1 and p2 for the price of good 2. We will now deﬁne a competitive equilibrium in this economy. A competitive (or Walrasian) equilibrium is an allocation B B A A ..x1 ; x2 /; .x1 ; x2 // and prices .p1; p2/ that satisfy the following conditions: A A I. Optimization: the bundle .x1 ; x2 / solves max .x1 ;x2 / uA.x1; x2/ subject to p1x1 C p2x2  and likewise for consumer B . II. Market clearing: B A x1 C x1 D !1 B A x2 C x2 D !2 A A p 1 !1 C p 2 !2 ; The First Welfare Theorem Throughout we shall maintain the assumption that utility functions are strictly increasing. Theorem. Let x D .x A; x B / be an allocation and let p D .p1; p2/ ¤ 0. If .x; p/ is a competitive equilibrium then x is Pareto efﬁcient. In words, the allocation of every competitive equilibrium is Pareto efﬁcient. The First Welfare Theorem The FWR is one of the most important results of Economics. Benchmark for understanding role of institution design / policy making. The theorem is so important that we will see two alternative proofs of it. One simple proof is for the case in which utility functions are smooth, strictly increasing and quasi-concave (i.e. preferences are convex). Another, much more general, proof is for the case in which utility functions satisfy only a very weak non-satiation condition. Proof when preferences are smooth and convex In a Walrasian equilibrium .x; p/ each consumer maximizes his utility subject to his budget constraint, taking the prices p D .p1; p2/ as given. Under the assumption that the utility functions are smooth and that x is strictly positive, this implies A A MRS A.x1 ; x2 / D and MRS B B B .x1 ; x2 / D p1 p2 p1 : p2 Therefore, A A B B MRS A.x1 ; x2 / D MRS B .x1 ; x2 /: Proof when preferences are smooth and convex Moreover, by the market clearing condition, x is a feasible allocation. Hence, the allocation x satisﬁes all the necessary ﬁrst-order conditions of the maximization problem P.u/ (with u D uB .x B /) that characterizes Pareto efﬁciency (cf. lecture 12). Since we are assuming that the consumers’ preferences are convex, the ﬁrst-order conditions are also sufﬁcient conditions for an optimum. Therefore, x must be a solution of P.u/, and hence Pareto efﬁcient, as was to be shown. Q.E.D. General proof using revealed preference We will prove the contrapositive, i.e. that if x is not Pareto efﬁcient then .x; p/ cannot be a competitive equilibrium. Suppose x is not Pareto efﬁcient. Then there is a feasible allocation x 0 such that either uA.x 0A/ > uA.x A/ and uB .x 0B /  uB .x B /; or uA.x 0A/  uA.x A/ and uB .x 0B / > uB .x B /: General proof using revealed preference For now, let us “cheat” a little bit and assume strict inequality for both A and B: uA.x 0A/ > uA.x A/ and uB .x 0B / > uB .x B /: Suppose .x; p/ were a competitive equilibrium, for some p D .p1; p2/ ¤ 0. By revealed preference, x 0A cannot be affordable at prices p D .p1; p2/, because x A maximizes A’s utility subject to her budget constraint, taking the equilibrium prices p as given. Hence, A A p 1 x 0 A C p 2 x 0 A > p 1 !1 C p 2 !2 : 1 2 Likewise, x 0B cannot be affordable at prices p : B B p 1 x 0 B C p 2 x 0 B > p 1 !1 C p 2 !2 : 2 1 General proof using revealed preference Adding the above inequalities yields ! p1 x 0 A 1 C x 0B 1  C p2 x 0 A 2 C x 0B 2  > p1 ! 2 1 ‚ …„ ƒ  ‚ …„ ƒ  A A B B !1 C !1 C p 2 !2 C !2 But since x 0 is a feasible allocation, x 0A C x 0B D !1 and x 0A C x 0B D !2 2 2 1 1 and therefore, p 1 !1 C p 2 !2 > p 1 !1 C p 2 !2 ; which is impossible. Thus, .x; p/ cannot be a competitive equilibrium and this concludes the proof (except for the “cheating” above). Q.E.D. The missing step in the proof Let us go back to the beginning of the proof, where we have: uA.x 0A/  uA.x A/ and uB .x 0B / > uB .x B /; without loss of generality. Now we will not assume a strict inequality for both. Instead, let us assume uA.x 0A/ D uA.x A/ and uB .x 0B / > uB .x B /: 0 0 Consider the bundle x 00A D .x1A C "; x2A C "/, where " > 0 is arbitrary. Since uA is strictly increasing, then uA.x 00A / > uA.x 0A/ D uA.x A/: The missing step in the proof Then, assuming that .x; p/ is a competitive equilibrium, A A p1x 00A C p2x 00A > p1!1 C p2!2 : 2 1 and B B p 1 x 0 B C p 2 x 0 B > p 1 !1 C p 2 !2 : 2 1 But, the ﬁrst inequality can be re-written as A A p 1 x 0 A C p 2 x 0 A > p 1 !1 C p 2 !2 2 1 .p1 C p2/": The missing step in the proof Since this must hold for every " > 0, we have A A p 1 x 0 A C p 2 x 0 A  p 1 !1 C p 2 !2 : 2 1 Summing the weak inequality for A and the strict inequality for B yields a strict inequality: ! p1 x 0 A 1 C x 0B 1  C p2 x 0 A 2 C x 0B 2  > p1 Then, the proof proceeds as before... ! 1 2 ‚ …„ ƒ  ‚ …„ ƒ  A A B B !1 C !1 C p 2 !2 C !2 : Implicit assumptions behind the FWT (a) No market power (b) Private goods (c) No externalities (d) No information asymmetries (e) Complete markets When either of these assumptions is violated, the FWT breaks down. Signiﬁcance of the FWT is that it provides a benchmark to study market failures and role of institution design / government policy. ...
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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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