MONOPOLY (2) Second Degree Price Discrimination

Transcription

1 1/22 MONOPOLY (2) Second Degree Price Discrimination May 4, 2014

2 2/22 Problem The monopolist has one customer who is either type 1 or type 2, with equal probability. How to price discriminate between the two types Payoff functions are common knowledge but the customer s type is his private information. Problem: sorting? bunching? exclusion?

3 3/22 Monopolist sets price function (S: sales plan ) S := {(T 1, x 1 ), (T 2, x 2 ), (0, 0)}. Customers have freedom of choice. Customers choose (T i, x i ) S. Payment and delivery. Market Game

4 Assumptions Constant unit cost, normalized to zero. Payoff functions Monopolist maximizes π := 1 2 (T 1 + T 2 ). Customers maximize U i (x, T ) := x P i (y)dy T, for i = 1, 2, 0 where P i (x) =MRS, and (due to normalization) price means price net after deduction of MC. 4/22

5 5/22 Assumptions (Continued) Diminishing MRS: P i (x) is monotone decreasing and P i (0) > 0, P i (x) = 0, for some x, i = 1, 2. Single Crossing: P 2 (x) > P 1 (x), for all x Concavity 2P 1 (x) P 2 (x) (will be discussed later...)

6 6/22 Optimal Price Function max T s,x s T 1 + T 2 (s.t.) U 1 (x 1, T 1 ) U 1 (0, 0) = 0 (1) U 2 (x 2, T 2 ) U 2 (0, 0) = 0 (2) U 1 (x 1, T 1 ) U 1 (x 2, T 2 ) (3) U 2 (x 2, T 2 ) U 2 (x 1, T 1 ) (4)

7 7/22 Illustration T T 2 T 2 2 ˆT 2 T T x 1 x 1 x 2 1 x

8 8/22 Auxiliary Result Working hypotheses: constraints (1), (4) are binding; (2) and (3) are not binding. Lemma For given x 1, x 2 the optimal price function is: T 1 = x1 T 2 = T x2 P 1 (y)dy x 1 P 2 (y)dy.

9 9/22 Proof Follows from the two constraints (1), (4), written as equality constraints (binding!). 0 = U 2 (x 2, T 2 ) U 2 (x 1, T 1 ) = x1 0 P 1 (y)dy + x2 x 1 P 2 (y)dy T 2, Therefore, one can eliminate T 1, T 2 from the above optimization problem, and obtains:

10 10/22 Kuhn Tucker Conditions Restricted Program ( x1 x2 ) max 2 P 1 (y)dy + P 2 (y)dy. x 1,x x 1 (2P 1 (x 1 ) P 2 (x 1 )) 0 and (...) x 1 = 0 P 2 (x 2 ) 0 and P 2 (x 2 )x 2 = 0.

11 11/22 Proposition The optimal price function has the following properties: 1 P 2 (x 2 ) = 0, x 2 > 0 (no distortion at top) 2 P 1 (x 1 ) > 0 (distortion at bottom) 3 U 1 (x 1, T 1 ) = 0 (no surplus at bottom) Results 4 U 2 (x 2, T 2 ) 0 with > x 1 > 0 (surplus at top unless x 1 = 0)

12 12/22 1) x 2 > 0, P 2 (x 2 ) = 0 2) Assume, per absurdum that x 1 x 2 = x 1 > 0 and This contradicts 1). 0 = 2P 1 (x 1 ) P 2 (x 1 ) < 2P 2 (x 1 ) P 2 (x 1 ) = P 2 (x 1 ) P 2 (x 2 ). Proofs

13 13/22 3) Let x 1 > 0, x 1 = x 2 (the case x 1 = 0 is trivial) = P 1 (x 1 ) = 1 2 P 2(x 1 ) > 1 2 P 2(x 2 ) = 0. 4) U 1 (x 1, T 1 ) = 0 and U 2 (x 2, T 2 ) 0 obvious. 5) The ignored incentive constraint is satisfied, since U 1 (x 2, T 2 ) U 1 (x 1, T 1 ) = = 0. x2 x 1 P 1 (y)dy (T 2 T 1 ) x2 x 1 (P 1 (y) P 2 (y)) dy

14 14/22 Examples Sorting Let P i (x) := 1 x/i, i = 1, 2. Then: x 1 = 2 3, T 1 = 4 9, x 2 = 2, T 2 = 8 9 Exclusion Let P i (x) := i x, i = 1, 2. Then, x 1 = T 1 = 0, x 2 = 2, T 2 = 2.

15 15/22 Examples Continued Bunching Let P i (x) := θ i (1 x), i = 1, 2, and 1 = θ 1 < θ 2 < 2. Then: x 1 = x 2 = 1, T 1 = T 2 = 1 2. Note, bunching cannot occur if P 2 (x) > P 1 (x) for all x 0, which we assumed here.

16 16/22 Digression: Concavity Consider the twice continuously differentiable function f (x, y). f is (strictly) concave, if f 11, f 22 0 ( ) f11 f det 12 = f f 21 f 11 f 22 f (at least one inequality strict). Hence: Profit function is concave, if 2P 1 (x) P 2 (x), x.

17 17/22 Supplement: Explicit Normalization Suppose marginal cost is equal to c > 0. The following shows how a change of variables leads to the optimization problem stated above. Prior to the change of variables, the optimization problem is (note: the t instead of T and the p i instead of the P i functions) max t 1 + t 2 c(x 1 + x 2 ), {x,t} s.t. the constraints, U i (x i, t i ) 0, U i (x i, t i ) U i (x j, t j ), i, j = 1, 2, where U i (x j, t j ) := x j 0 p i (y)dy t i.

18 18/22 Now define T i := t i cx i, P i (y) := p i (y) c. Inserting these, one obtains max {x,t } T 1 + T 2, s.t. the above constraints, where U i (x j, T j ) = xj 0 (p i (y) c) dy T j = xj 0 P i (y)dy T j as above. Note, the slope of the indifference curve is then equal to P i (x i ) = p i (x i ) c (marginal willingness to pay (MRS) minus marginal cost). Therefore, a negative slope means that marginal cost exceeds the marginal willingness to pay.

19 19/22 Two Part Tariffs A menu of two-part tariffs {(t 1, f 1 ), (t 2, f 2 )} (f : lump-sum, t: price per unit) is a special case of second-degree price discrimination. There the monopolist can only indirectly control quantity, via t. Every feasible menu of two-part tariffs can be translated into a feasible price-quantity menu; but not vice versa. The reason is that under two-part tariffs customers have more leeway, because they can also choose quantity. Therefore, incentive compatibility is more demanding. Generally, the optimal menu of price-quantity combinations cannot be implemented by a menu of two-part tariffs.

20 20/22 Example/Exercise Example Suppose P i (x) := 1 x/i, i = 1, 2. Then the optimal price discrimination exhibits sorting with x 1 = 2 3, T 1 = 4 9, x 2 = 2, T 2 = 8 9, Show that... 1 the optimal two-part tariffs are S = { (t 1 = 1/2, f 1 = 1/8), (t 2 = 0, f 2 = 7/8) }. 2 two part-tariffs are less profitable. 3 Prove that every menu of incentive compatible two part-tariffs can be implemented by an incentive compatible menu of price-quantity combinations, but not vice versa. 4 Why is this the case?

21 21/22 Other Screening Devices In many cases one can use quality differences to screen. This is particularly appealing if goods are indivisible. A case in point is airline pricing. There, screening is typically achieved by requiring advance booking on tourist class tickets, combined with rationing in the form of offering a limited number of tourist class tickets. Another example is priority pricing where the higher price gives priority of service in the event of rationing.

22 22/22 Illustration: Airline Pricing T 1/γ A b s C T t B t s W t 1 W