Heterogeneous buyers: different preferences and private quality signals

E.E. Schlee J. of Economic Behavior Org. 44 2001 347–362 355 second period sales in response to information. 20 The following Theorem summarizes the results from this section thus far. It follows from Propositions 1–3. Theorem 1. If S1 holds then the seller charges an introductory price in one of the senses of Proposition 2. In the single buyer interpretation of the representative buyer model, linear pricing may not be an appropriate assumption. We now consider nonlinear pricing with a single buyer main- taining S1. Let RX, ρ be the buyer’s reservation price for a quantity X. Given quasi- linear utility, RX, ρ = E[uX, θ ] − E[u0, θ ]. The seller’s second period problem is ˜ V s ρ ≡ max X≥ {RX, ρ − cX}, We assume that E[u0, θ ] does not depend on ρ. Here the buyer’s second period expected utility from the equilibrium always equals E[u0, θ ]+M, so that information is valueless to the buyer and hence the buyer never experiments. Let ˜ W s x ≡ R ˜ V s ρz, xf z, x dz. In period 1, the seller chooses x to maximize Rx, ρ −cx+δ ˜ W s x . We compare the maxi- mizers of this expression, denote it by Q, with the maximizer of the seller’s first period profit, x , assumed to be unique. It follows immediately from Milgrom and Shannon 1994, Theo- rem 4 that each element of Q will at least equal x if ˜ W s . is increasing. Since S1 ensures that higher consumption is more informative, we have by Definition 1 that ˜ W s . is increasing if ˜ V s . is convex. Convexity of ˜ V s . follows from linearity of R in ρ see note 20. Hence the firm sells more than x . Moreover, it is immediate from the concavity of u that the average price, Rx, is lower than the average price that maximizes first period profit. In that sense the firm engages in introductory pricing. We summarize these facts in the following proposition. Proposition 4. Suppose S1 holds, and that the monopolist seller perfectly discriminates. Then x ∗ ≥ x for any x ∗ ∈ Q ; moreover average price falls in the sense that Rx ∗ x ∗ ≤ Rx x .

4. Heterogeneous buyers: different preferences and private quality signals

We have assumed thus far that all agents observe a public signal z before period 2. One implication of this public information assumption is that the seller faces a deterministic demand at the time it sets the second period price. In that case, linearity of the inverse demand in beliefs ensures that the seller’s value function is convex note 20. If, however, each buyer observes a private quality signal, then the firm’s second period demand is no longer deterministic: the quantity sold at each price depends on the unobserved realization of the buyers’ signals. This complication means that linearity of the inverse demand function no longer suffices for the seller to value information; accordingly, we will need further 20 More formally, we can write the seller’s value function as V s ρ = max X≥ {gX, ρX − cX} where g., ρ is the inverse demand function. Under quasilinear utility, gX, ρ = ρ∂uX, θ H ∂X + 1 − ρ∂uX, θ L ∂X , which is affine in ρ. Since the maximum of any collection of convex functions is itself convex Rockafeller, 1970, p. 35, the proposition follows. 356 E.E. Schlee J. of Economic Behavior Org. 44 2001 347–362 restrictions to get introductory pricing with quasilinear utility. In this section we show that if each buyer’s demand function is convex in beliefs, then the firm will value information, and we give conditions that ensure such convexity for quasilinear utility. For the remainder of this section, suppose for simplicity that returns to scale are constant and let k denote the seller’s unit cost. Let N denote the number of buyers. In period 1 the firm sets a price and each buyer n picks a consumption, x n . Buyer n observes a private signal, z n , of the good’s quality given by z n = ψx n , θ + ε n , where ε n is the realization of a random variable with density g. satisfying the restrictions in S1, so that higher consumption is more informative. 21 We let D n ., ρ n denote buyer n’s second period demand function, where ρ n is n’s posterior belief. The seller’s ex-post second period profit is πP , ρ 1 , . . . , ρ N = N X n= 1 D n P , ρ n P − k. Under constant returns to scale, of course, seller profit is just the sum of profit earned from each buyer. If each buyer’s second period demand is convex, then so is the seller’s second period objective function and hence its value function note 20; accordingly, expected profit is higher if each buyer has better quality information. To define an equilibrium for this model, write the seller’s second period expected profit Π as a function of x 1 , . . . , x N , the first period choices of buyers: Π P , x 1 , . . . , x N = N X n= 1 Z D n P , ρz n , x n P − kf z n , x n dz n . Let W s x 1 , . . . , x N = max P ≥ Π P , x 1 , . . . , x N . Buyer n’s second period value function is V n b ρ n , P = max 0≤X≤MP {E n [u n X, θ ] − M − PX}, where E n [.] denotes buyer n’s expectations operator. Finally buyer n’s expected value function is W n b x n , P = R V n b ρz n , x n , P f z n , x n dz n . Definition 4. p ∗ , P ∗ , x ∗ 1 , . . . , x ∗ N is a perfect equilibrium if it maximizes p − k X n x n + δΠ P , x 1 , . . . , x N subject to P ∗ ∈ arg max P ≥ Π P , x ∗ 1 , . . . , x ∗ N , and x n ∈ d n p, P , ρ for all n, where d n p, P , ρ ≡ arg max x∈ [0,Mp] {E [u n x, θ ] − M − px + δW n b x, P }. 21 To reduce notation we assume that each buyer faces the same information structure but that the signal realiza- tions are i.i.d. across buyers. E.E. Schlee J. of Economic Behavior Org. 44 2001 347–362 357 This definition says that the second period price maximizes second period profit given first period consumption; that buyers correctly anticipate this price, but take it is given in their first period choice; the firm’s first period price maximizes the sum of first and second period expected profit. If each buyer’s second period demand is convex in beliefs, then second period firm profit, W s x 1 , . . . , x N , will be increasing in x 1 , . . . , x N . So if each buyer’s first period demand is decreasing in the first period price, then the seller will want to charge a lower period 1 price than would maximize its period 1 profit. Since each buyer has an incentive to experiment, the buyer’s first period demand may be multivalued for the same reason discussed in Section 3. We assume that each buyer chooses the largest element of d n p, P , ρ and ¯ d n p, P , ρ denote the maximum selection from d n . Theorem 2 gives general conditions on demands for introductory pricing, whereas Proposition 5 gives preference conditions that rationalize these demand restrictions under quasilinear utility. Theorem 2. If the technology is constant returns to scale, higher first period consumption is more informative for each buyer, each buyer’s second period demand D n P , . is convex in ρ n and each buyer’s first period demand is decreasing in the first period price p, then the firm engages in introductory pricing in one of the senses of Proposition 2. Theorem 2 assumes that second period demand is convex and that first period demand is decreasing in first period price. This second condition is nontrivial: if buyers infer that higher total consumption today may result in a higher price tomorrow, then this reduces the benefit of learning today; hence lowering today’s price may lower today’s consumption for some buyers 22 To rule this possibility out, we consider three alternative assumptions: buyers discount the future heavily; buyers have the same utility function and the optimal second period price is increasing in first period consumption; or that the optimal second period price is independent of first period consumption. Although each of these assumptions constrains how the first period outcome affects the second period, the seller still has an incentive to manipulate buyer learning if buyer demands are strictly convex in beliefs. Moreover, as we shall show, the last assumption is met in three common demand specifications: linear, constant elasticity, and semi-log demands. Proposition 5. Under quasilinear utility , uX, θ +Y , suppose each buyer’s u function sat- isfies u XX 0 and 1u X = u X X, θ H − u X X, θ L 0 subscripts denote partial deriva- tives . Then demand is convex in beliefs if 21u XX 1u X − E [u XXX ]E[u XX ] ≥ 0 where 1u XX X, θ H − u XX X, θ L . Moreover each buyer’s first period demand will be de- creasing in the first period price under any one of the following conditions : i each buyer behaves myopically , i.e. δ = 0 for all buyers; ii buyers have identical preferences and the equilibrium second period price P ∗ is nondecreasing in x 1 , . . . , x N ; iii the equilibrium second period price P ∗ is independent of x 1 , . . . , x N . 22 Cyert and Degroot 1980 present a model of trial offers in which the uncertainty is purely about consumer tastes for the good and there is no noise in any consumer’s problem: a consumer buys one unit or nothing at all, and if the consumer tries the good, he learns for sure what his reservation price is. Despite these differences between their model and ours, a similar problem arises in their model: a consumer may not buy the good today in response to a lower price if he expects the price to rise tomorrow see Eq. 8 and the surrounding discussion in their paper. 358 E.E. Schlee J. of Economic Behavior Org. 44 2001 347–362 Fig. 1. Demands that imply introductory pricing under private information: the higher inverse demand is flatter and both demands are weakly convex. The first sentence of the proposition simply requires that second period demand be an increasing function of the belief that quality is high. The inequality in the second sentence contains two terms. The second term, −E[u XXX ]E[u XX ], is the Arrow–Pratt measure of the convexity of the inverse demand function in consumption note that the inverse demand function is E[u X ]. The numerator of the first term is the difference in slopes of the high quality and low quality inverse demands. So this inequality is satisfied if both inverse demands are convex in quantity and the higher inverse demand is flatter than the lower, as in Fig. 1. A comparison with Mirman et al. 1993 is illuminating. They find that the firm sets a lower mean first period price if the higher inverse demand is flatter than the lower. Our model agrees with theirs that the firm does not experiment if the two inverse demands are parallel and linear. They do not require any convexity assumptions, but our proposition permits the slope condition to fail provided that demand is sufficiently convex as in our Examples 2 and 3. Of course the rationale for introductory pricing is quite different in their model. In their model the firm experiments to affect its own learning and information is always valuable to it; the issue is to determine when higher output is more informative. In our model the firm uses the price to affect consumer learning and higher consumption is more informative by assumption; the issue is to determine when information is valuable to the firm. The next three examples verify that some frequently used specifications satisfy the condi- tions in Theorem 2: quadratic utility, implying linear interior demands; constant elasticity demands; and semi-log demands. In each case the second period equilibrium price is inde- pendent of first period consumption, allowing for some heterogeneity in buyer preferences. Example 1 Linear demands. Let buyer n’s utility be given by u n X, θ = αX − B n θ 1 2 X 2 , where α 0 and B n . is a positive, decreasing function of θ . Let E n [B n ] = ρ n B n θ H + 1 − ρ n B n θ L . The second period demand is D n P , ρ n = max{ 0, α − P E n [B n ]}, which is strictly convex in ρ n for interior choices. Note moreover that the second period equilibrium price is P ∗ = 1 2 α + k , which is independent of the beliefs E.E. Schlee J. of Economic Behavior Org. 44 2001 347–362 359 of buyers, and hence of first period consumption. Thus each buyer’s first period demand correspondence will be decreasing in the first period price, and the firm will engage in introductory pricing. Example 1 is consistent with Mirman et al. 1993: for two linear demand curves with the higher inverse demand flatter than the lower, the firm prices lower initially. Example 2 Constant elasticity demands. u n X, θ = β n αθ X α with 0 α 1, and β n 0 Buyer n’s demand is D n P , ρ n = P α− 1 − 1 B n ρ n , where B n ρ n = α n E n [θ α ] 1−α − 1 . This demand is strictly convex in ρ n . Aggregate demand is P α− 1 − 1 B , where B is the sum of the B n terms of buyers. Again the equilibrium second period price is independent of beliefs and hence of period 1 consumption. Thus if buyers all have pref- erences of this form, the firm will charge an introductory price. The demands in Example 3 would lead to lower first period output, and hence higher mean period 1 price, in Mirman, Samuelson and Urbano, since the higher inverse demand is steeper than the lower. The next example would lead to neither lower nor higher first period outputs in their model, though it does lead to introductory pricing in ours. Example 3 Semi-log demands. Let u n X, θ = A n θ X + β R X lnω dω, where A n . is a positive, increasing function of θ , and β 0. The demand for buyer n is D n P , ρ n = exp{E n [A n θ ] − P β}, and the aggregate demand is B exp{−P β}, where B is the sum of the exp{E n [A n θ ]β}, terms across buyers. Again each buyer’s demand is convex in beliefs and the second period price is independent of beliefs.

5. Concluding remarks