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lecture22_slides (1) - Econ 121. Intermediate...

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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 22 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 IV. Market failure Monopoly (Ch. 24, 25) Price Discrimination Suppose a monopolist with constant marginal cost of c faces a population that is heterogeneous in their willingness to pay. A franction A 2 OE0; 1 of the population has a downward sloping inverse demand function: p A.x/; while a fraction B D 1 A has downward sloping inverse demand p B .x/: Assume that p B .x/ p A.x/ for all x: Thus, A are the low-valuation types and B are the high-valuation types. Price Discrimination As a benchmark consider first what happens if the monopolist knows each individual's demand type (A or B ). In that case, the monopolist can make a take-it-or-leave-it offer of a quantity-tariff pair .x A; t A/ to consumer A. Likewise, it can make a similar offer .x B ; t B / to consumer B . Price Discrimination The meaning of an offer .x; t/ is x D quantity to be consumed t D payment from the consumer to the monopolist. Thus, t is not a unit price. Rather, it is the total amount that the consumer must pay to consume x units. Price Discrimination If the monopolist knows the consumer's type, he can extract all the consumer surplus. Monopolist solves for the quantity x A such that p A.x A/ D c: Consumer A's surplus from buying x A units is: (1) S A.x A/ D Z xA p A.x/ dx D area below p A./ 0 So, monopolist offers .x A; t A/ to consumer A, where x A solves (1) and t A D S A.x A/. Likewise for consumer B . Price Discrimination Does consumer A accept? Yes. Since he is indifferent between accepting and rejecting, he has no incentive to reject. (Think of t A really as t A minus 1 penny.) Should the monopolist offer more units than x A? No, because in order to induce A to buy any extra units, those units must be priced below marginal cost. Should he charge anything other than t A? No, because under t A consumer A is exactly indifferent between accepting and rejecting. Second-degree Price Discrimination Suppose now the monopolist does not know the demand of each individual consumer. All he knows is that half of the consumers are of type A and the other half are of type B . Still, monpolist can offer a menu f.x A; t A/; .x B ; t B /g and each consumer is free to choose any of the two options. In particular, consumer A can choose .x B ; t B / and vice-versa. Second-degree Price Discrimination Consumer A chooses .x A; t A/ if and only if A's net surplus from .x A; t A/ and 0 A's net surplus from .x A; t A/ A's net surplus from .x B ; t B /: Second-degree Price Discrimination Thus, consumer A chooses .x A; t A/ if and only if S A.x A/ and tA 0; S A.x A/ tA S A.x B / tB: Likewise, consumer B chooses .x B ; t B / if and only if S B .x B / and tB 0; S B .x B / tB S B .x A/ t A: Second-degree Price Discrimination Monopolist chooses the menu f.x A; t A/; .x B ; t B /g to maximizes his overall profit subject to the participation and self-selection constraints. x A ;t A ;x B ;t B max A.t A cx A/ C .1 A/.t B cx B / subject to S A.x A/ S B .x B / S A.x A/ S B .x B / tA tB tA tB 0 0 S A.x B / S B .x A/ tB tA Second-degree Price Discrimination To solve this problem, first note that the participation constraint of B never binds: S B .x B / tB S B .x A/ tA S A.x A/ tA 0; where the first inequality follows from the self-selection constraint of B , the penultimate inequality follows from p B p A and the last inequality from the participation constraint of A. So, we can delete the participation constraint of B from the maximization problem, as it is redundant. Second-degree Price Discrimination Next, note that the participation constraint of A must bind at the profit-maximizing menu, i.e. S A.x A/ t A D 0: Otherwise, can increase t A a bit, without violating PA, PB and SSB. This would increase profits. Second-degree Price Discrimination Thus, the constraints in the maximization problem have been simplified to S A.x A/ S A.x A/ S B .x B / tA tB tA D 0 S A.x B / S B .x A/ tB tA Second-degree Price Discrimination Plugging the expression for t A into the self-selection constraint of B yields: tA tB S B .x B / tB D S A.x A/ S A.x B / S B .x A/ S A.x A/: Next, note that at the profit-maximizing menu, the self-selection constraint of B must bind, i.e. S B .x B / t B D S B .x A/ S A.x A/: Otherwise, we can raise t B a bit without violating any constraints. Second-degree Price Discrimination We thus have the following constraints: tA tB tB D S A.x A/ S A.x B / S B .x B / S B .x A/ C S A.x A/: D Therefore, the self-selection constraint of A is equivalent to S B .x B / which is equivalent to S A.x B / S B .x A/ S A.x A/; Z and thus equivalent to xB .p B .x/ xA p A.x// dx 0; xA xB : Second-degree Price Discrimination Thus, the monopolist maximizes his profit subject to the constraints tA xB tB D S A.x A/ xA S B .x B / S B .x A/ C S A.x A/: D Second-degree Price Discrimination Recall that the monopolist profit is A.t A cx A/ C .1 A/.t B cx B / So, if we plug in the expressions of t A and t B we get A S .x / A A cx A C .1 / S .x / A B B S .x / C S .x / B A A A cx B Re-arraging yields A S A.x A/ 1 A A A B B B B A A A A S .x / S .x / cx C.1 / S .x / cx Second-degree Price Discrimination Thus, monopolist chooses x A, x B to maximize A S .x / A A 1 A A A B B B A S .x / S .x / cx C.1 / S .x / cx B A A A subject to xB x A: Second-degree Price Discrimination To solve this maximization problem, we will guess that the constraint x A x B does not bind at the optimum. We will then solve the relaxed problem of maximizing A S A.x A/ 1 A A A B B B B A A A A S .x / S .x / cx C.1 / S .x / cx without constraints. Finally, we will check that the solution of this relaxed problem does satisfy the constraint x A xB . Second-degree Price Discrimination Let us maximize A S A.x A/ 1 A A A B B B B A A A A S .x / S .x / cx C.1 / S .x / cx : First-order conditions: .S / .x / A 0 A 1 A A .S / .x / B 0 A .S / .x / D c A 0 A .S B /0.x B / D c Second-degree Price Discrimination Recall that surpluses are calculated from the inverse demands as follows: S A.x/ D Z x p A.y/ dy and S B .x/ D 0 Z x p B .y/ dy for all x 0 0: Thus, .S A/0.x/ D p A.x/ and .S B /0.x/ D p B .x/ for all x 0: Plugging into the first-order conditions yields A A p .x / 1 A A p .x / B A p .x / D c A A p B .x B / D c Second-degree Price Discrimination We are not done yet! We still have to check that x A Indeed, xB . p .x / and therefore, p A.x A / A A 1 A A p .x / B A p .x / D p B .x B / A A 0 , ...,, p B .x A/ p A.x A/ , ...,, p B .x A/ p B .x B / which implies p .x / p .x / D A A B B 1 A A 0; xA since p B is downward sloping. xB ; Second-degree Price Discrimination The profit-maximizing incentive-compatible menu is f.x A; t A/; .x B ; t B /g such that p .x / A A 1 A A p .x / B A p B .x B / D c t A D S A.x A/ t B D t A C S B .x B / p .x / D c A A S B .x A/: Second-degree Price Discrimination Qualitative features that generalize to richer models: No distortion at the top: the highest-valuation type gets the efficient output. Inefficient output for lower-valuation types. Information rents: all types but the lowest valuation type receive positive surplus. Lowest-valuation type receives zero surplus. Fundamental assumption: p B .x/ p A.x/ for all x 0: This is an instance of the so-called Spence-Mirlees condition, which is pervasive in Information Economics. ...
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This note was uploaded on 03/10/2012 for the course ECON 121 at Yale.

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