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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 lowvaluation types and B are the highvaluation 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 takeitorleaveit offer of a quantitytariff 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. Seconddegree 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 viceversa. Seconddegree 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 /: Seconddegree 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: Seconddegree 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 selfselection 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 Seconddegree 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 selfselection 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. Seconddegree Price Discrimination
Next, note that the participation constraint of A must bind at the profitmaximizing 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. Seconddegree 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 Seconddegree Price Discrimination
Plugging the expression for t A into the selfselection 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 profitmaximizing menu, the selfselection 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. Seconddegree 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 selfselection 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 : Seconddegree 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 Seconddegree 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 Rearraging 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 Seconddegree 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: Seconddegree 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 . Seconddegree 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 : Firstorder 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 Seconddegree 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 firstorder 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 Seconddegree 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 ; Seconddegree Price Discrimination
The profitmaximizing incentivecompatible 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/: Seconddegree Price Discrimination
Qualitative features that generalize to richer models: No distortion at the top: the highestvaluation type gets the efficient
output. Inefficient output for lowervaluation types. Information rents: all types but the lowest valuation type receive positive
surplus. Lowestvaluation type receives zero surplus.
Fundamental assumption: p B .x/ p A.x/ for all x 0: This is an instance of the socalled SpenceMirlees 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.
 '09
 SAMUELSON
 Microeconomics

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