ADVANCED MICROECONOMICS (TUTORIAL)

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1 ELMAR G. WOLFSTETTER, MAY 12, 214 ADVANCED MICROECONOMICS (TUTORIAL) EXERCISE SHEET 4 - ANSWERS AND HINTS We appreciate any comments and suggestions that may help to improve these solution sets. Exercise 1 (Second-Degree Price Discrimination). Consider a monopolist with zero costs who serves two costumers with different demand functions. He knows the (inverse) demand functions but he cannot tell the costumers apart (he does not know who is type 1 and who is type 2). 1. Two Part Tariff. Suppose the monopolist can induce price discrimination by using a menu of two part tariffs {(t 1, f 1 ),(t 2, f 2 )}. Tariffs are made up of a fixed price (lump sum price) f i and an unit price t i. Each costumer can choose a tariff and then buy as many units as he wishes according to the conditions. The demand functions are: Question: what are the optimal tariffs? X 1 (p) = 1 p; and X 2 (p) = 2 2p Profit Maximization Problem (PMP) of the monopolist: subject to the following constraints: max π = f 1 +t 1 X 1 (t 1 ) + f 2 +t 2 X 2 (t 2 ) (1) (t 1, f 1 );(t 2, f 2 ) U 1 (t 1, f 1 ) (Type 1 s PC) (2) U 2 (t 2, f 2 ) (Type 2 s PC) (3) U 1 (t 1, f 1 ) U 1 (t 2, f 2 ) (Type 1 s IC) (4) U 2 (t 2, f 2 ) U 2 (t 1, f 1 ) (Type 2 s IC) (5) where PC stands for participation constraint and IC for incentive compatibility constraint. Consumers utility: Xi (t) U i (t, f ) = P i (y)dy (tx i (t) + f ), i {1,2} where the first term on the right hand-side is the consumer s total willingness to pay for X i (t) units of goods. Working hypothesis: constraints (2) and (5) are binding. When constraint (2) is binding we have: X1 (t 1 ) U 1 (t 1, f 1 ) = P 1 (y)dy (t 1 X 1 (t 1 ) + f 1 ) = 1

He knows the (inverse) demand functions but he cannot tell the costumers apart (he does not know who is type 1 and who is type 2). 1. Two Part Tariff.

2 Substitute the inverse demand function P 1 (y) = 1 y and type 1 s demand at unit price t 1 which is X 1 (t 1 ) = 1 t 1, we get: X1 (t 1 ) f 1 (t 1,t 2 ) = P 1 (y)dy t 1 X 1 (t 1 ) = 1 2 t t2 1 When constraint (5) is binding, we have where U 2 (t 2, f 2 ) = U 2 (t 1, f 1 ) X2 (t 2 ) U 2 (t 2, f 2 ) = P 2 (y)dy (t 2 X 2 (t 2 ) + f 2 ) U 2 (t 1, f 1 ) = X2 (t 1 ) f 2 can also be solved as a function of t 1 and t 2 : P 2 (y)dy (t 1 X 2 (t 1 ) + f 1 ) X2 (t 2 ) f 2 (t 1,t 2 ) = P 2 (y)dy t 2 X 2 (t 2 ) + (t 1 X 2 (t 1 ) + f 1 ) X 2 (t 1 ) = 1 2 +t t2 1 2t 2 +t 2 2 We substitute f 1 (t 1,t 2 ), f 2 (t 1,t 2 ), X 1 (t 1 ) and X 2 (t 2 ) into the monopolist s objective function (1) and reduce the the monopolist s PMP to maxπ {t1,t 2 } = 1 2 t t2 1 +t 1 (1 t 1 ) t t2 1 2t 2 +t 2 2 +t 2 2(1 t 2 ) =1 t 2 1 +t 1 t 2 2 The first order conditions are t 1 = 2t = t 2 = 2t 2 = Solving the two conditions, we get t 1 = 1 2 and t 2 =. Substitute them back into f 1 (t 1,t 2 ) and f 2 (t 1,t 2 ), we obtain the optimal fixed fees: f 1 = 1 8 and f 2 = 7 8. Check constraints (3) and (4). Constraint(3) is fulfilled since U 2 (t 2, f 2 ) = 1 8 >. Constraint(4) is fulfilled since U 1 (t 1, f 1 ) = > U 1 (t 2, f 2 ) = 3 8. Therefore, the working hypothesis is confirmed. 2. Second-degree Price Discrimination. Suppose the monopolist offers a menu of price quantity combinations ((,T 1 ),(x 2,T 2 )). A customer who chooses tariff i gets the quantity x i and pays the price T i. Participation is voluntary. The inverse demand functions are given. Determine the optimal tariffs. 2

function of t 1 and t 2 : P 2 (y)dy (t 1 X 2 (t 1 ) + f 1 ) X2 (t 2 ) f 2 (t 1,t 2 ) = P 2 (y)dy t 2 X 2 (t 2 ) + (t 1 X 2 (t 1 ) + f 1 ) X 2 (t 1 ) = 1 2 +t 1 1 2 t2 1 2t 2 +t 2 2 We substitute f 1

3 The PMP of the monopolist: subject to the following constraints: max π = T 1 + T 2 (6) (,T 1 ),(x 2,T 2 ) U 1 (,T 1 ) (Type 1 s PC) (7) U 2 (x 2,T 2 ) (Type 2 s PC) (8) U 1 (,T 1 ) U 1 (x 2,T 2 ) (Type 1 s IC) (9) U 2 (x 2,T 2 ) U 2 (,T 1 ) (Type 2 s IC) (1) For each one of the three pairs of demand functions, we always have P 2 (x) > P 1 (x), therefore, type 2 consumer is always the high type (with high demand). Again we solve the problem with the working hypothesis that the lower type s participation constraint (7) and the the high type s incentive constraint (1) bind and in the end confirm that constraints (8) and (9) are satisfied. Consumers s utilities: x U i (x,t ) = P i (y)dy T The binding constraint (7) can be written as: which delivers U 1 (,T 1 ) = P 1 (x)dx T 1 = (11) T 1 (,x 2 ) = P 1 (x)dx (12) From binding constraint (1), we have U 2 (x 2,T 2 ) = U 2 (,T 1 ) where U 2 (x 2,T 2 ) = U 2 (,T 1 ) = Solve for T 2 as a function of and x 2, we have T 2 (,x 2 ) = T 1 (,x 2 ) + P 2 (x)dx = Now we can reduce the PMP of the monopolist to: such that maxπ(,x 2 ) = T 1 (,x 2 ) + T 2 (,x 2 ) = 2,x 2 The Kuhn-Tucker conditions are:, x 2 P 2 (x)dx T 2 P 2 (x)dx T 1 P 1 (x)dx + P 2 (x)dx (13) P 1 (x)dx + P 2 (x)dx = 2P 1 ( ) P 2 ( ),, and (2P 1 ( ) P 2 ( )) = = P 2 (x 2 ), x 2, and x 2 P 2 (x 2 ) = Now consider the three pairs of demand functions separately: 3

4 (a) P 1 (x) = 1 x and P 2 (x) = x = 2(1 ) (1 1 2 ) = 1 3 2,, and (1 3 2 ) = = x 2, x 2, and x 2 (1 1 2 x 2) = whose solution is = 2 3 and x 2 = 2. Plug and x 2 into equations (12) and (13), we get T 1 = 4 9 and T 2 = 8 9. (b) P 1 (x) = 1 x and P 2 (x) = a(1 x), 1 < a < 2 = (2 a)(1 ),, and (2 a)(1 ) = = a(1 x 2 ), x 2, and x 2 a(1 x 2 ) = whose solution is = 1 and x 2 = 1. Plug and x 2 into equations (12) and (13, we get T 1 = 1 2 and T 2 = 1 2. (c) P 1 (x) = 1 x and P 2 (x) = 2 x =,, and ( ) = = 2 x 2, x 2, and x 2 (2 x 2 ) = whose solution is = and x 2 = 2. Plug and x 2 into equations (12) and (13), we get T 1 = and T 2 = Optimality Comparison. A two part tariff {(t 1, f 1 ),(t 2, f 2 )} can be replicated by a price quantity combination {(,T 1 ),(x 2,T 2 )} such that x i = X i (t i ) and T i = f i +t i X i (t i ). This does not generally work the other way round. If you try to replicate the optimal price quantity combination by a two part tariff, then the upper type s incentive constraint is usually violated. He will not choose (t 2, f 2 ) as intended since (t 1, f 1 ) gives him a higher utility. With a two part tariff the upper type can choose the quantity he likes, whereas the quantity is fixed in the price quantity combination. We show this by the example of the demand functions from 1/2a. In order to translate a price-quantity combination into a 2-part tariff, one has to design a two-part tariff, such that the demanded quantities and the total payments are the same. That is our starting point. We take the optimal price quantity combination from exercise 2a. {( 2 {(x t,t t ),(x 2,T 2 )} = 3, 4 ) (, 2, 8 )} 9 9 The corresponding two-part tariff must lead to the same demand: Therefore, 4

(b) P 1 (x) = 1 x and P 2 (x) = a(1 x), 1 < a < 2 = (2 a)(1 ),, and (2 a)(1 ) = = a(1 x 2 ), x 2, and x 2 a(1 x 2 ) = whose solution is = 1 and x 2 = 1.

5 t 1 = P 1 ( ) = = 1 3 t 2 = P 2 (x 2 ) = = Now, in order to generate the same payments as with the price-quantity combination, we must have that f 1 = T 1 t 1 = = 2 9 f 2 = T 2 t 2 x 2 = = 8 9 However, this violates customer 2 s incentive constraint as U 2 (t 2, f 2 ) = P 2 (y)dy t 2 x 2 f 2 = = 1 9 X2 (t 1 ) < U 2 (t 1, f 1 ) = P 2 (y)dy t 1 X 2 (t 1 ) f 1 = = 2 9 which completes the proof (by counterexample) that not each price-quantity combination can be replicated by an equivalent two-part tariff. 5

incentive constraint as U 2 (t 2, f 2 ) = P 2 (y)dy t 2 x 2 f 2 = 1 8 9 = 1 9 X2 (t 1 ) < U 2 (t 1, f 1 ) = P 2 (y)dy t 1 X 2 (t 1 ) f 1 =

1.4 Hidden Information and Price Discrimination 1

1.4 Hidden Information and Price Discrimination 1 To be included in: Elmar Wolfstetter. Topics in Microeconomics: Industrial Organization, Auctions, and Incentives. Cambridge University Press, new edition,