# 1.4 Hidden Information and Price Discrimination 1

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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, forthcoming. 1.4 Hidden Information and Price Discrimination Second-degree price discrimination is the most widely observed kind of price discrimination. We now take a closer look at it and elaborate on a model that explains why this kind of discrimination emerges, and how it should be done. 1 Consider a profit-maximizing monopolist who deals with a customer who is one of two possible types, with equal probability. 2 Only the customer knows his true type. In other words, the customer s type is his private information. The monopolist only knows the payoff functions of the two possible types, and the probability with which either type occurs. Therefore, price discrimination requires a somewhat sophisticated screening device. While the following model is best suited to analyze a monopolist who faces one customer of unknown type, it can also be interpreted as a pricing problem with a population of customers. However, in an environment with many customers the monopolist may employ more powerful mechanisms that make price offers conditional on their acceptance by all customers, that are ignored here. Also, with many customers there are issues of arbitrage and repeated purchases that are also ignored. The market game is structured as follows: 1. The monopolist sets a uniform nonlinear price function in the form of a menu of price quantity combinations, (T, x), called the sales plan, from which the customer is free to select one: S := {(T 1, x 1 ), (T 2, x 2 ), (0, 0)}. (1.1) The (0, 0) combination is included because market transactions are voluntary; the customer is free to abstain from buying. Of course, x 1, x The customer observes the sales plan and picks that price quantity combination that maximizes his payoff. Payments are made, and the market game ends. Without loss of generality, the component (T 1, x 1 ) is designated for customer type 1, and (T 2, x 2 ) for customer type 2 (incentive compatibility). For convenience these two types will be referred to as customers 1 and 2. Of course, the monopolist could also live with a sales plan where, for example, customer 2 picks (T 1, x 1 ) and 1 picks (T 2, x 2 ), as long as he makes no error in predicting customers rational choice. But then incentive compatibility can be restored simply by relabelling the components of the sales plan. Moreover, one can show that any other format of sales plan can be replicated by an equivalent sales plan of the kind considered here. Therefore, the restriction to incentive-compatible sales plans is without loss of generality. 3 Assumptions Five assumptions are made: A1 (Cost function). Unit costs are constant and normalized to zero. 1 Here we present a two-type version of the continuous-type model by Maskin and Riley (1984a). 2 Equal probability is invoked only in order to minimize notation. 3 This is the essence of the well-known revelation principle in mechanism design theory (see the Chapter on Mechanism Design).

2 2 A2 (Payoff functions). The monopolist maximizes profit π := 1 2 (T 1 + T 2 ), (1.2) and customers maximize consumer surplus x U i (x, T ) := P i (y) dy T for i = 1, 2, (1.3) 0 where P i (x) denotes i s marginal willingness to pay for x. A3 (Declining marginal willingness to pay). P i (x) is continuously differentiable with P i (x) < 0; also P i (0) > 0, and P i (x) = 0, for some x, i {1, 2}. A4 (Single crossing). For all x A5 (Concavity). For all x P 2 (x) > P 1 (x). (1.4) 2P 1 (x) < P 2 (x). (1.5) A4 is called the single-crossing assumption for the following reason: Pick an arbitrary point in (x, T ) space, say x1, T 1, and draw the two types indifference curves that pass through this point. Since the slope of the indifference curves is equal to P i (x), A4 assures that these curves cross only once at this given point, as illustrated in Figure 1.1. And A5 is called concavity because it assures the strict concavity of the objective function, as we show later on. OPTIMAL SALES PLAN The optimal sales plan maximizes π subject to the participation constraints (the outside option of not buying is assumed to yield zero utility), and the nonnegative constraints: and the incentive constraints U 1 (x 1, T 1 ) U 1 (0, 0) = 0, (1.6) U 2 (x 2, T 2 ) U 2 (0, 0) = 0, (1.7) U 1 (x 1, T 1 ) U 1 (x 2, T 2 ), (1.8) U 2 (x 2, T 2 ) U 2 (x 1, T 1 ) (1.9) x 1, x 2 0. (1.10) Conditions (1.6) and (1.8) assure that the component of the sales plan designated for customer 1, (T 1, x 1 ), is dominated neither by the (0, 0) nor by the (T 2, x 2 ) option. Similarly, conditions (1.7) and (1.9) assure that (T 2, x 2 ) is dominated neither by (0, 0) nor by (T 1, x 1 ) An Intuitive Exploration Before dwelling on the formal solution it may be worthwhile to explore the problem with the aid of some graphs assembled in Figure 1.1. Figure 1.1 displays buyers indifference curves in the (T, x) space. The curves labelled 1 resp. 2 are indifference curves of customer type 1 resp. 2. Note, they pass through the origin. Therefore, they display the combinations of (T, x) that keep the customer indifferent between buying and not buying. Curve 1 is below curve 2 because type 1 has a lower valuation at each given x. The slope of

3 1.4 Hidden Information and Price Discrimination 3 T T 2 T 2 2 ˆT 2 T T x 1 x 1 x 2 1 x Figure 1.1: Customers Indifference Curves these indifference curves is equal to the marginal rate of substitution which is also called the marginal willingness to pay P i (x) dt i dx = P i (x i ), i {1, 2}. i Ui (x i,t i )=0 As usual, the marginal willingness to pay is declining. At x i = xi it is equal to marginal cost (which has been normalized to zero). Therefore, the indifference curves reach a maximum at the points x 1 = x1, x 2 = x2 where the marginal willingness to pay is equal to marginal cost. Students are sometimes puzzled by the fact that we allow the marginal willingness to pay to be negative. However, this is a consequence of normalizing marginal cost to be equal to zero. Here, a negative marginal willingness to pay only means that the marginal valuation is below marginal cost. 4 All indifference curves that are shifted below curve 2 (resp. curve 1), such as indifference curves 2 and 2, exhibit higher utility. If the monopolist could observe customers type he would obviously choose (x 1 = x1, T 1 = T1 ) and (x 2 = x2, T 2 = T2 ). However, if customers type is not observable this scheme is not incentive compatible because customer 2 would choose the combination (x1, T 1 ) which gives him a higher utility than the designated combination (x2, T 2 ). While maintaining the combination (T1, x 1 ) one can achieve incentive compatibility and induce participation only by choosing a combination (T 2, x 2 ) from the shaded area in between indifference curves 1 and 2. Therefore, given (T1, x 1 ), the best feasible combination designated for type 2 is ( ˆT 2, x2 ). However, this menu of price quantity combinations can be further improved. Indeed, if one reduces x 1 slightly below x1, say to x 1, one loses very little in revenue from type 1, T 1, since the indifference curve of type 1 is flat at (T1, x 1 ). However, this change permits a substantial increase in the price that one can charge type 2, T 2, from ˆT 2 to T2. These observations suggest the following properties of the optimal solution: 1. Offer type 1 less than the efficient quantity, x 1 < x1 ( distortion at the bottom property). 2. Offer type 2 the efficient quantity, x 2 = x2 ( no distortion at the top property). 4 The normalization requires a transformation of the price variables. Exercise 1.1 offers some guidance on how to explicitly carry out this transformation.

4 4 3. Set T 1 in such a way that type 1 is made indifferent between buying and not buying ( no surplus at the bottom property). 4. Set T 2 in such a way that type 2 is indifferent between (x 1, T 1 ) and (x 2, T 2 ), which entails that type 2 prefers buying to not buying, unless it is optimal to set x 1 = Formal Solution We now turn to the formal characterization of the optimization problem: 1 ( ) max T1 + T 2, s.t. (1.6) (1.10). {T 1,T 2,x 1,x 2 } 2 SOME PRELIMINARIES Luckily, that problem can be simplified by eliminating two constraints. Indeed, among the participation constraints only the lower type s constraint (1.6) binds, and among the incentive constraints only the upper type s constraint (1.9) binds. (A constraint does not bind if eliminating it from the optimization program does not affect the solution.) Therefore it is claimed that one can eliminate constraints (1.7) and (1.8) without loss of generality. What makes us come to this conclusion? Obviously, the above analysis of Figure 1.1 suggests this. Therefore, we take it as a working hypothesis. Of course, this is only justified if the solution of the restricted optimization program turns out to also satisfies the eliminated constraints which we will confirm. SOLUTION OF THE RESTRICTED PROGRAM The restricted program restricted by eliminating constraints (1.7) and (1.8) can be further simplified due to the following results. Lemma 1.1. The optimal sales plan exhibits T 1 = x1 0 T 2 = T 1 + P 1 (y) dy, (1.11) x2 x 1 P 2 (y) dy. (1.12) Proof. We have noted (but not yet proved) that the upper type s incentive constraint and the lower type s participation constraints are binding. If a constraint binds, then it is satisfied with equality at the optimal sales plan. Therefore, (1.6) entails (1.11). Using this result concerning T 1 if (1.9) and (1.6) bind, one has 0 = U 2 (x 2, T 2 ) U 2 (x 1, T 1 ) = which entails (1.12), as asserted. x1 0 P 1 (y) dy + x2 x 1 P 2 (y) dy T 2, (1.13) These price functions have a nice interpretation: 1. The low type is charged his maximum willingness to pay for x 1.

5 1.4 Hidden Information and Price Discrimination 5 2. The high type pays the same as the low type for the first x 1 units plus his own maximum willingness to pay for the additional x 2 x 1 units. Therefore, the high type makes a net gain if x 1 > 0, simply because he obtains the first x 1 units at a bargain price. In view of Lemma 1.1 we can now eliminate the T -variables in the monopolist s objective function and state the restricted program in the form of the unconstrained optimization problem (except for nonnegativity) max x 1,x 2 0 The Kuhn Tucker conditions of the restricted program are ( 1 x1 x2 ) 2 P 1 (y) dy + P 2 (y) dy. (1.14) 2 0 x 1 φ(x 1 ) := 2P 1 (x 1 ) P 2 (x 1 ) 0 and φ(x 1 )x 1 = 0, (1.15) P 2 (x 2 ) 0 and P 2 (x 2 )x 2 = 0. (1.16) The T s are obtained by inserting the optimal x s into (1.11), (1.12). As you can confirm easily, A5 assures the strict concavity of the objective function. It also entails that φ(x 1 ) is strict monotone decreasing with φ(x) < 0 for some x. This assures that (1.15) has a unique solution; however, since φ(x) may be negative everywhere, one may get the corner solution x 1 = 0. Also, P 2 is declining, with P 2 (0) > 0. Therefore, (1.16) has the unique solution x 2 > 0, implicitly defined by P 2 (x 2 ) = The Optimal Sales Plan Proposition 1.1. The optimal sales plan exhibits P 2 (x 2 ) = 0, x 2 > 0 (no distortion at top), (i) x 2 > x 1, T 2 > T 1 (monotonicity), (ii) P 1 (x 1 ) > 0 (distortion at bottom), (iii) U 1 (x 1, T 1 ) = 0 (no surplus at bottom), (iv) U 2 (x 2, T 2 ) 0 with > x 1 > 0 (surplus at top unless x 1 = 0). (v) The optimal prices are computed in (1.11) and (1.12). Proof. First we characterize the solution of the restricted program (1.14) and then show that it also solves the unrestricted program. We have already shown that the Kuhn Tucker conditions have a unique solution that exhibits x 2 > 0 and x 1 0. (i): Follows immediately from the fact that P 2 (x 2 ) = 0, x 2 > 0, as already noted above, and the fact that marginal cost is constant and has been normalized to zero. (ii): By Lemma 1.1 the T s have the asserted monotonicity property if and only if it holds for the x s. We first prove weak monotonicity: Suppose x 1 > x 2, contrary to what is asserted. Since x 2 > 0, one has also x 1 > 0. Therefore, the first part of condition (1.15) is satisfied with equality, and one has, using the single-crossing assumption A4, 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 ).

6 6 But this contradicts (i). Therefore, x 2 x 1. Finally, note that if it were possible to have x = x 1 = x 2 > 0, one would need to have P 2 (x) = P 1 (x) = 0 for some x, which contradicts A4. Therefore, the monotonicity is strict. (iii): If x 1 = 0, one has P 1 (x 1 ) > 0, by A3. And if x 1 > 0, condition (1.15) combined with monotonicity (ii) entails, due to x 2 > x 1, P 1 (x 1 ) = 1 2 P 2(x 1 ) > 1 2 P 2(x 2 ) = 0. In either case the low customer gets less than the efficient quantity, P 1 (x 1 ) > 0 (distortion at bottom). (iv): U 1 (x 1, T 1 ) = 0 is obvious from (1.11). (v): U 2 (x 2, T 2 ) 0, with strict inequality if x 1 > 0, follows immediately from (1.12) and monotonicity. Finally, we need to confirm that the restricted program also satisfies the two omitted constraints (1.7) and (1.8). The omitted participation constraint (1.7) is obviously satisfied by (v). And the omitted incentive constraint (1.8) holds for the following reasoning (the last step uses the monotonicity property x 2 > x 1 and the single-crossing assumption A4): This completes the proof. 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 Why it Pays to Distort Efficiency Why is it optimal to deviate from efficiency in dealing with the low type but not the high type? The intuition is simple. The high type has to be kept indifferent between (x 2, T 2 ) and (x 1, T 1 ). This is achieved by charging the high type the price T 1 for the first x 1 units and a price equal to his maximum willingness to pay for x 2 x 1 units. From this observation it follows immediately that profit is maximized by expanding x 2 to a level where the marginal willingness to pay equals the marginal cost, P 2 (x 2 ) = 0 (see Figure 1.1). In turn, starting from P 1 (x 1 ) = 0 (see point (x 1, T 1 ) in that figure), a small reduction in x 1 is costless in terms of forgone profits from the low type (the marginal profit is zero at this starting point). But, at the same time it extends the domain where the high type is charged a price equal to his maximum willingness to pay. Altogether, it thus pays to introduce a downward distortion at the bottom, illustrated by the two-star variables in Figure 1.1. Figure 1.2 provides another useful illustration of these considerations (for a particular parameter specification). There, the optimal x 1 is at the point where the function φ(x 1 ) reaches zero. Customer type 1 is charged a price T 1 equal to the area under P 1, from 0 to x 1. Type 2 gets the quantity x 2 at which P 1 (x 2 ) = 0, and he is charged a price T 2 equal to the non shaded area under P 2, from 0 to x 2. This illustrates how type 2 is charged T 1 for the first x 1 units plus the area under his inverse demand function for the additional x 2 x 1 units. Therefore, the shaded area is 2 s consumer surplus. That surplus is lowered if one reduces x 1, and it vanishes altogether if one sets x 1 = Sorting, Bunching, and Exclusion Finally, note that it is not always optimal to serve both types of customers and, if we slightly change the assumptions, it may not even be optimal to discriminate. Altogether, the optimal price discrimination falls into one of three categories:

7 1.4 Hidden Information and Price Discrimination 7 P i (x), φ(x) 1 P P 1 (x) 2 (x) φ(x) x 1 = x 2 = 2 x Figure 1.2: Optimal Sorting with Two Customers and P i (x) := 1 x/i. 1. Sorting, with 0 < x 1 < x 2, T 1 < T Bunching, or no discrimination, with x 1 = x 2 > 0, T 1 = T Exclusion, or extreme discrimination, where only the high type is served at a price equal to its maximum willingness to pay, with 0 = x 1 < x 2, T 2 > T 1 = 0. Note carefully that Proposition 1.1 excludes only case 2. Example 1.1. Here we illustrate these three cases. 1. 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, and, incidentally, exhibits a declining unit price (quantity discount) T 2 /x 2 = 4 9 < 6 9 = 2 3 = T 1/x Suppose P i (x) := θ i (1 x), i = 1, 2, and 1 = θ 1 < θ 2 < 2. Then it is optimal to abstain from discrimination (bunching). Specifically, x 1 = x 2 = 1, T 1 = T 2 = Suppose P i (x) := i x, i = 1, 2. Then it is optimal to serve only the high type (exclusivity) and take away the entire surplus x 1 = T 1 = 0, x 2 = 2, T 2 = 2. However, bunching is a pure borderline case and cannot occur generically. Why? A necessary condition for bunching is that there exists a quantity ξ > 0 for which P 2 (ξ) = P 1 (ξ) = 0 (which is, incidentally, ruled out by A4). This property cannot survive parameter perturbations; hence, it is irrelevant. Example 1.2. Modify Example 1.1(2) by setting θ 2 > 2. Then φ(x 1 ) = (2 θ 2 )(1 x 1 ) is evidently not decreasing; hence, the monopolist s payoff is not concave. In that case, the Kuhn Tucker conditions have two solutions: 1) x 1 = x 2 = 1 and 2) x 1 = 0, x 2 = 1 (see Figure 1.3). Comparing payoffs shows that the unique maximizer is the corner solution x 1 = 0, x 2 = 1 (exclusion) Generalization* The above analysis of optimal price-quantity combinations generalizes in a straightforward manner to n 2 types with the single-crossing marginal willingness-to-pay functions P 1 (y) < P 2 (y) < < P n (y), y. (1.45)

8 8 P i (x), φ(x) 3 2 P 2 (x) φ(x) P 1 (x) x Figure 1.3: Nonconcavity: P i (x) := θ i (1 x), θ 1 = 1, θ 2 = 3. In particular, if complete sorting is optimal, one can show that the optimal price discrimination exhibits 1. zero consumer surplus for the lowest type only; 2. no distortion at the top only; 3. only local downward incentive constraints bind (customer i 2 is indifferent between (T i, x i ) and (T i 1, x i 1 ); all other price quantity combinations in the optimal sales plan are inferior). Moreover, the optimal sales plan is then completely characterized by the following rules: (n + 1 i)p i (x i ) = (n i)p i+1 (x i ), i {1,..., n 1} P n (x n ) = 0 T 1 = x1 0 T i = T i 1 + P 1 (y) dy, xi x i 1 P i (y)dy, i {2,..., n}. The proof of these assertions is a fairly straightforward extension of the above analysis of the twotypes case. It also generalizes to a continuum of types, as we show in the follow-up analysis in Chapter 9, Section Two-Part Tariffs A Special Case An alternative price-discrimination scheme is to offer a menu of two-part tariffs. A two-part tariff is an affine price function T i (x) := t i x + f i with the constant unit-price t i and the lump-sum f i. This pricing scheme is frequently observed for example in public utilities pricing, in the taxi business, in mobile phone contracts, to name just a few.

13 1.4 Hidden Information and Price Discrimination 13 Exercise 1.3 (Airline Pricing and Sorting II). Consider the airline price discrimination example. How can one change the utility function of tourist customers so that it exhibits aversion to rationing? Can rationing serve as a screening device in that case?

14 14

15 Bibliography BELOBABA, P. [1987], Airline yield management: An overview of seat inventory control, Transportation Science, 21, DANA, J. D. [1998], Advance-purchase discounts and price discrimination in competitive markets, Journal of Political Economy, 106, MASKIN, E. AND J. G. RILEY [1984], Monopoly with incomplete information, RAND Journal of Economics, 15, PIGOU, A. C. [1920], The Economics of Welfare, Cambridge University Press. VULCANO, G., G. VAN RYZIN, AND C. MAGLARAS [2002], Optimal dynamic auctions for revenue management, Management Science, 48,

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

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