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lecture12_slides - Econ 121 Intermediate Microeconomics...

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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 12 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 Primitives of the model Two consumers, labeled A and B ; two goods, labeled 1 and 2 A A Consumer A's bundle: .x1 ; x2 / B B Consumer B 's bundle: .x1 ; x2 / B B A A Utility functions: uA.x1 ; x2 / and uB .x1 ; x2 /, for A and B , respectively A A B B Endowments: ! A D .!1 ; !2 /, ! B D .!1 ; !2 / Net supplies: B A !1 D !1 C !1 ; A B !2 D !2 C !2 : Feasible allocations B B A A An allocation is a vector x D ..x1 ; x2 /; .x1 ; x2 //, that is, a consumption bundle for each consumer. An allocation x is feasible if B A x` C x` D !`; for ` D 1; 2: Feasible allocations can be viewed as points in the Edgeworth box. Edgeworth box xB1 xB2 w2 xA2 xA1 w1 x2A x1B Indifference curves in the Edgeworth box B uB = const uA = const A x2B x1A Goal of analysis We address the following questions: 1. Which allocations can emerge as outcomes of some efficient trading process? 2. Which allocations would a social planner like to implement? 3. Which allocations can emerge as outcomes of some decentralized, competitive market? 4. Which policies can a social planner introduce to induce a desirable market outcome? Pareto efficiency An allocation x is Pareto dominated by an allocation x 0 if for each consumer i D A; B , ui .x 0i ; x 0i / 2 1 i i ui .x1 ; x2 /; with strict inequality for some consumer i . A feasible allocation is Pareto efficient if there does not exist a feasible allocation x 0 that Pareto dominates x . Pareto efficiency Two equivalent interpretations of Pareto inefficiency: Gains from trade: if an allocation x is Pareto inefficient, there are mutually beneficial gains from trade. Thus, we "expect" A and B to negotiate some mutually beneficial terms of trade that would make them both better off at some allocation other than x . Social planner: if an allocation x is Pareto inefficient, a benevolent social planner would not like to implement that allocation, as he can find some other allocation that makes all agents weakly better off, and at least some agent strictly better off. A Pareto improvement xB1 xB2 w2 xA2 xA1 w1 Pareto efficiency In the Edgeworth, can plot the set of Pareto efficient allocations. This set is called the contract curve. If preferences are monotonic, convex and smooth then: A feasible allocation x is Pareto efficient if and only if the indifference curves of A and B are tangent at x in the Edgeworth box. The Contract Curve B uA uB A Utility Possibility Set The utility possibility set, U , is the set of all pairs of utility levels B B A A .uA.x1 ; x2 /; uB .x1 ; x2 //; where x ranges over all feasible allocations. The utility possibility set of consumer i , U i , is the set of all utility levels i i ui .x1; x2/ of consumer i , where x ranges over all feasible allocations. Utility Possibility Set uB U UA uA Characterization of Pareto efficiency For each utility level uB in consumer B 's utility possibility set U B consider the maximization problem P.uB / below: A A B B .x1 ;x2 ;x1 ;x2 / max A A uA.x1 ; x2 / subject to B B uB .x1 ; x2 / A B x1 C x1 A B x2 C x2 uB D D !1 !2 Characterization of Pareto efficiency B B A A Theorem. Let x D ..x1 ; x2 /; .x1 ; x2 // be an allocation. (a) If x is Pareto efficient then x solves the maximization problem P.uB / for some uB 2 U B . (b) Conversely, if for some uB the allocation x solves the maximization problem P.uB / and the utility functions are continuous and strictly increasing, then x is Pareto efficient. Characterization of Pareto efficiency Proof of (a). Let x be a Pareto efficient allocation and let us show that it solves the maximization problem corresponding to uB D uB .x B /. We will prove the contrapositive, i.e., we will show that if x is not a solution of P.uB / with uB D uB .x B /, then x is not Pareto efficient. Indeed, if x does not solve the maximization problem corresponding to uB D uB .x B /, then there exists a feasible allocation x 0 such that uA.x 0A/ > u.x A/ and uB .x 0B / and hence x is not Pareto efficient. uB D uB .x B /; Characterization of Pareto efficiency Proof of (b). Again, we will prove the contrapositive. If x is not Pareto efficient, then there exists a feasible allocation x 0 that Pareto dominates x . This means x 0 is feasible and satisfies uA.x 0A/ and uA.x A/ uB .x B /; uB .x 0B / with at least one strict inequality. Characterization of Pareto efficiency Proof of (b). First case: uA.x 0A/ > uA.x A/ and uB .x 0B / uB .x B /. Fix any uB in B 's utility possibility set with uB uB .x B /. Then, we must also have uB .x 0B / uB . Therefore, allocation x 0 satisfies all the constraints of the maximization problem. Since uA.x 0A/ > uA.x A/ we conclude that x cannot solve the maximization problem corresponding to uB . Characterization of Pareto efficiency Proof of (b). Second case: uA.x 0A/ uA.x A/ and uB .x 0B / > uB .x B /. By the strict monotonicity of uA (and the continuity of uB ) we can find a feasible allocation x 00 such that uA.x 00A/ > uA.x 0A/ and uA.x A/ uB .x 00B / > uB .x B /; and so we are back to the first case. Q.E.D. Characterization of Pareto efficiency Back to the maximization problem that characterizes the contract curve: A A B B .x1 ;x2 ;x1 ;x2 / max A A uA.x1 ; x2 / subject to B B uB .x1 ; x2 / B A x1 C x1 A B x2 C x2 uB D D !1 !2 If we could replace the inequality by an equality, then we could apply the method of Lagrange. Digression: Lagrange's method Consider the maximization problem in n variables and m constraints. .x1 ;:::;xn / max f .x1; : : : ; xn/ subject to g1 .x1; : : : ; xn/ g2 .x1; : : : ; xn/ D D c1 c2 gm .x1; : : : ; xn/ D cm Digression: Lagrange's method The Lagrange method is to introduce m new variables 1; : : : ; m , called Lagrange multipliers, and solve the first order conditions of the Lagrangean function L, defined as L.x1 ; : : : ; xn; C 1; : : : ; m/ D f .x1; : : : ; xn/ C c1/ C : : : C m .gm .x1 ; : : : ; xm / 1 .g1 .x1 ; : : : ; xn / cm/: Digression: Lagrange's method Thus, the Lagrange method is to solve: @L .x1; : : : ; xn; @x1 @L .x1; : : : ; xn; @xn and 1; : : : ; m/ D 0 1; : : : ; m/ D 0 @L .x1; : : : ; xn; @ 1 @L .x1; : : : ; xn; @ m 1; : : : ; m/ D 0 1; : : : ; m/ D 0 when you take the partial derivative with respect to the lagrangean multiplier, it simply yields the original constraint Note that the last m conditions are equivalent to g1.x1 ; : : : ; xn/ D c1; : : : ; gm.x1; : : : ; xn/ D cm. Digression: Lagrange's method If f; g1; : : : ; gm are differentiable and .x1 ; : : : ; xn/ is a solution of the maximization problem such that the m vectors @g1 .x1; : : : ; xn/; : : : ; @x1 ::: @gm . .x1; : : : ; xn/; : : : ; @x1 . @g1 .x1 ; : : : ; xn// @xn @gm .x1; : : : ; xn// @xn are linearly independent, then there exist 1 ; : : : ; m such that .x1; : : : ; xn; 1 ; : : : ; m/ solves the first-order conditions of the Lagrangean. Hence, the Lagrange method provides necessary conditions for constrained optima. Under suitable assumptions of f; g1 ; : : : ; gm, these conditions are also sufficient. Characterization of Pareto efficiency Back to the maximization problem that characterizes the contract curve: A A B B .x1 ;x2 ;x1 ;x2 / max A A uA.x1 ; x2 / subject to B B uB .x1 ; x2 / D B A x1 C x1 A B x2 C x2 uB .w/ Lagrange mult. / 1/ 2/ D D !1 .w/ Lagrange mult. !2 .w/ Lagrange mult. Thus, the Lagrangean is A A B B L.x1 ; x2 ; x1 ; x2 ; ; 1; 2/ D A A uA.x1 ; x2 / C B B uB .x1 ; x2 / uB C C 1 B A x1 C x1 !1 C 2 B A x2 C x2 !2 Characterization of Pareto efficiency First-order conditions: @L D0 A @x1 @L D0 A @x2 @L D0 B @x1 @L D0 B @x2 @L D0 @ @L D0 @ 1 @L D0 @ 2 H) H) H) H) H) H) H) M U1A C M U2A C M U1B C M U2B C 1 D0 D0 1 H) H) H) H) M U1A M U2A M U1B M U2B D D D D 1 2 2 D0 D0 1 2 2 B B uB .x1 ; x2 / D uB A B x1 C x1 D !1 B A x2 C x2 D !2 Characterization of Pareto efficiency Therefore, MRS A D that is, M U1A D A M U2 1 2 D M U1B D MRS B ; M U2B a feasible allocation is Pareto efficient if and only if the marginal rate of substitution is equalized across all consumers. Recalling that the MRS is the slope of the indifference curve, the above condition is just the tangency condition we saw in the Edgeworth box. 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 A B B A competitive equilibrium is an allocation ..x1 ; x2 /; .x1 ; x2 // and prices .p1; p2/ that satisfies the following two conditions: A A I. Optimization: the bundle .x1 ; x2 / solves .x1 ;x2 / max uA.x1; x2/ subject to p1x1 C p2x2 and likewise for consumer B . II. Market clearing: B A x1 C x1 B A x2 C x2 A A p 1 !1 C p 2 !2 ; D !1 D !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/ be a strictly positive vector of prices. If .x; p/ is a competitive equilibrium then x is Pareto efficient. The most important result of neo-classical Economics. Benchmark for understanding role of policy making. The First Welfare Theorem This is so important that we will see two proofs of this result! Today we will present a simple proof for the case in which utility functions are smooth, strictly increasing and quasi-concave (i.e. preferences are convex). We will also assume that x is strictly positive (all coordinates are positive). Proof when preferences are smooth and convex In a competitive 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 MRS and A A A .x1 ; x2 / D p1 p2 p1 : p2 MRS Therefore, B B B .x1 ; x2 / D B B A A MRS A.x1 ; x2 / D MRS B .x1 ; x2 /: Also, by the market clearing condition, x is a feasible allocation. Hence, the allocation x satisfies the necessary first-order conditions of the maximization problem that characterizes Pareto efficiency. Since we are assuming that the preferences are convex the first-order conditions are also sufficient conditions, and hence x is Pareto efficient, as required. Q.E.D. ...
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