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Monopoly Price Discrimination Firm must have ability to sort consumers and prevent resale. Definition: (due to Pigou (1920)) First degree (Perfect) price discrimination Seller charges each consumer equal his/her willingness to pay. Second degree price discrimination Price differs depending on the number of units bought (but not across consumers) Third degree price discrimination Different purchasers are charged different but constant price for each unit of the good bought. First-Degree Price Discrimination - Let ri(x) be consumer i’s maximum willingness to pay for x units of good1 - Consumers have quasilinear utility ui(x), hence demand is given by p = ui( x ) 1 Solution to ui(0) + y = ui(x) - ri(x) +y. Assuming ui(0), ri(x) = ui(x) The monopolist want to offer a specific price and output combination (ri* xi* ) for each i that Max ri – c(xi) r i , xi s.t. ui(x) ≥ ri Constraint is binding at optimal * ( xi* ) F.O.C.: u′ i ( xi ) = c′ ri* = ui ( xi* ) The monopolist who can perfectly price discriminate is producing at Pareto efficient allocation and as * ( xi* ) , it is producing at the same level as p = u′ i ( xi ) = c′ a competitive firm. Diagram Second-Degree Price Discrimination - Assume that there are 2 consumers Low-demand consumer 1 with u1(x1) + y1 High-demand consumer 2 with u2(x2) + y2 ( x) < u′ Assume that u1(x) < u2(x) and u1′ 2 ( x) The monopolist knows that there are low-demand and high-demand consumer but cannot distinguish them. - Suppose that consumer i demands xi and spends ri amount of money. For the monopolist, the expenditure and consumption for the consumers 1 and 2 are (r1, x1) and (r2, x2) respectively. If different types end up choosing different quantity and hence different expenditure levels, the following 4 inequalities must be true u1(x1) − r1 ≥ 0 u2(x2) − r2 ≥ 0 } Participation constraint u1(x1) − r1 ≥ u1(x2) − r2 u2(x2) − r2 ≥ u2(x1) − r1 } Self selection constraint Rearranging (1) r1 ≤ u1(x1) (2) r1 ≤ u1(x1) − u1(x2) + r2 (3) r2 ≤ u2(x2) (4) r2 ≤ u2(x2) − u2(x1) + r1 Profit maximization of the monopolist implies at least one inequality from (1) and (2) is binding and at least one inequality from (3) and (4) is binding One can show that from our assumptions about the utility functions, (1) and (4) are binding. Monopolist’s profit will then be π = [r1– c(x1)] + [r2– c(x2)] = [u1(x1) – c(x1)] + [u2(x2) − u2(x1) + u1(x1) – c(x2)] At the maximum, F.O.C. u1′ ( x1 ) - c′ ( x1 ) + u1′ ( x1 ) - u′ 2 ( x1 ) = 0 u′ ( x2 ) = 0 2 ( x 2 ) - c′ Low-demand consume at a point where MU>MC. Recall that he is already charged the maximum amount he is willing to pay High-demand consumes at level where MU = MC. If this were not true then the monopolist can increase x2 by one unit and charges consumer 2 his marginal utility from that extra consumption. Since MU > MC, the monopolist will make extra profit. Note that (4) is still satisfied with equality. Diagrammatic Example Third Degree Price Discrimination Example: N separate markets, i = 1,2,….,N N N i =0 i =0 Monopolist max ∑pi Di ( pi ) - C ( ∑Di ( pi )) FOC. for each i : Di ( pi ) + pi Di′ ( pi ) = C′ ( X ) Di′ ( pi ) N where X = ∑Di ( pi ) is the total quantity consumed i =0 (X = x1 + x2 +…..+ xN) The monopolist equates MR(x1) = MR(x2) =………= MR(xN) = MC In term of elasticity, pi - C′ (X ) - Di -1 = = pi pi Di′ ( pi ) εi εi is the elasticity of demand in market i, charges more for market with less elastic demand