Introductory pricing with identical buyers

352 E.E. Schlee J. of Economic Behavior Org. 44 2001 347–362 S1 z = ψx, θ +ε, where ε is a random variable with density function g. continuously differentiable on R that satisfies the monotone likelihood ratio property MLRP, namely g ′ εgε is decreasing on {ε|gε 0}, the latter assumed to be bounded; ψ., θ is continuously differentiable on R + for each θ ; ψx, θ H − ψx, θ L is nonnegative and strictly increasing in x with ψ0, θ H = ψ 0, θ L . The MLRP and the monotonicity of ψx, θ H −ψx, θ L ensure that ρz, x, the posterior belief that quality is high, is an increasing function of the signal realization z. We now show that under S1, higher consumption levels are indeed more informative. For continuity of exposition, we relegate the proofs to an appendix. Proposition 1. Suppose that S1 holds. If x ′′ x ′ , then {f .; x ′′ , θ } θ ∈{θ H ,θ L } is more informative than {f .; x ′ , θ } θ ∈{θ H ,θ L } .

3. Introductory pricing with identical buyers

We begin with the second period problem. We assume that buyers are identical in both period 1 and period 2: they have the same preferences, income, and beliefs. As noted in Section 1, we have two different interpretations in mind for this assumption. The first is that there is only one buyer. In that case the buyer will take into account the effect of its period 1 consumption on the distribution of the signal. Alternatively, there could be a large number of identical small buyers. In this case of course a individual buyer does not have any influence on aggregate consumption, and hence on the public signal. Thus an individual buyer does not experiment and simply chooses first period consumption to maximize first period expected utility. Since buyers are identical, we can describe the second period demand as the demand of a representative agent. The second period demand function, D, is given by DP , ρ ≡ arg max X∈ [0,MP ] {ρuX, θ H + 1 − ρuX, θ L + M − PX}, where P is the good’s price and M the aggregate income. This demand is single-valued and decreasing in P . Let V s denote the seller’s value function giving the second period expected profit from an equilibrium with common posterior ρ: V s ρ ≡ max P ≥ {PDP , ρ − cDP , ρ}. Turning to the first period, the posterior ρ is of course unknown since it depends on the realization of the signal z. Under S1 the density function for the signal is f z, x ≡ ρ gz − ψx, θ H + 1 − ρ gz − ψx, θ L . The expected payoff to the seller in the second period, given a choice of x in period 1, is W s x ≡ R V s ρz, xf z, x dz. The seller’s first period objective is thus π s x, p, δ, ρ = px − cx + δW s x . Recall that c. is the cost function. We summarize the first period behavior of buyers with a demand correspondence, d., ρ , which is upper semicontinuous and decreasing in the sense that “r p, z ∈ dr, ρ , and x ∈ dp, ρ ”, imply z ≤ x. In the case of a large number of identical buyers d., . = D., . , since buyers in that case behave myopically in period 1. In the single E.E. Schlee J. of Economic Behavior Org. 44 2001 347–362 353 buyer case, however, the buyer will not generally behave myopically, but will experiment; and an experimenting buyer may well have a multivalued demand. To see this, let V b denote the buyer’s second period value function giving the equilibrium expected utility as a function of the posterior, and let W b x ≡ R V b ρz, xf z, x dz denote the buyer’s expectation of that value from the viewpoint of period 1. The buyer’s first period demand is a solution to max x∈ [0,Mp] {E [ux, θ ] + M − px + δW b x} . 12 The problem arises because the buyer’s objective function may not be concave. The potential for nonconcavity was pointed out by Radner and Stiglitz 1984, who showed for a single agent problem that the expected value of information is not a concave function of the “amount” of information here parameterized by the first period consumption level. Since we have a game, and not a single agent problem, their result is not directly applicable here. Nevertheless, an earlier draft of this paper Schlee, 1996b, Proposition 7 proved that, if the buyer values information 13 and if our information structure assumption S1 holds, then W ′′ b 0 0 — that is, the expected second period utility is a strictly convex function of first period consumption on an interval containing zero. Hence, if the discount factor δ is high enough, then the buyer’s marginal willingness to pay, E[∂ux, θ ∂x] + δW ′ b x , must be increasing in x over some interval; this fact in turn implies that the buyer’s demand must be multivalued for at least one price. 14 Despite potentially being a correspondence, the demand is otherwise well-behaved; in particular, it will be decreasing in price, a crucial property for establishing introductory pricing. Turning to the seller’s first period choice, a further complication arises in the single buyer case: even if the buyer’s demand is single valued, it is hard to give intuitive conditions ensuring that the seller’s optimal first period price is unique. In a static problem, of course, uniqueness follows from imposing curvature conditions on demand. 15 In our intertemporal model, however, these curvature conditions are quite difficult to interpret: they would involve third order derivatives of the function W b . , for example. It seems better therefore to allow for multiple solutions, even at the expense of more complex comparative statics derivations. Definition 2. The perfect equilibrium outcome correspondence is et = arg max x,p≥ π s x, p, t δ, ρ s.t. x ∈ dp, ρ . In addition, we define e − t = {x, −p|x, p ∈ et } . Our objective is to compare e1 and e0 — the seller’s optimal first period choice with its myopic choice allowing that the sets are not singletons. The difference between the two objective functions is δW s . ; Proposition 2 gives various versions of the result that if W s . is increasing, then the firm charges a lower price in period 1 than what maximizes its period 1 profit. If the first period demand is single-valued, then the first part follows from Milgrom and Shannon 1994, Theorem 4 after we verify that the relevant “single crossing property” 12 E [.] denotes the expectations operator for the random variable θ . 13 As Schlee 1996a demonstrates, this condition is by no means trivial. The buyer will value information, however, for the case of constant returns and linear demands. 14 Namely the price at which the buyer is indifferent between not buying at all and buying some positive amount. 15 If the demand function is twice differentiable and marginal cost is constant, then 2d ′ p 2 − dpd ′′ p implies that the firm’s objective is strictly quasiconcave in price, and hence that the profit maximizing price is unique. 354 E.E. Schlee J. of Economic Behavior Org. 44 2001 347–362 holds. 16 See Definition 5 in Appendix A. Parts ii and iii progressively strengthen the assumptions on the payoffs to get sharper results. Since equilibrium prices and outputs need not be unique, we first record two definitions of one subset of R n being “higher” than another set. See Shannon 1995 for a more formal discussion of these notions. Definition 3. Let B, A ⊂ R n . B is strongly higher than A, written B ≥ S A , if x ∈ A, y ∈ B, implies that x ∧ y ∈ A and x ∨ y ∈ B. 17 B is completely higher than A, written B ≥ C A if x ≤ y for every x ∈ A and y ∈ B. 18 Proposition 2. i If W s . is increasing and d., ρ is decreasing 19 then e − 1 ≥ S e − 0. ii If in addition to i, W s . is strictly increasing on an interval [0, a] that includes the set of outputs in e 0 i.e. {x|x, p ∈ e0} then e − 1 ≥ C e − 0. iii If in addition to ii W s . and d., ρ are differentiable functions with W ′ s x 0 and ∂dp, ρ ∂p 0 on an open cell of x, p values containing e 0, then x ∗ , p ∗ ∈ e 1 implies p ∗ min{p|x, p ∈ e 0}. Part i is a weak version permitting identical equilibrium outcomes; it does however imply that the highest resp. lowest equilibrium price from e1 is not higher than the highest resp. lowest equilibrium price from e0. If W s . is strictly increasing then any equilibrium first period price is at most equal to the lowest price that maximizes period 1 profit. iii says that if the first period demand is single-valued and if d and W s are strictly monotone in the differentiable sense, then every equilibrium price is strictly lower than any price in e0. To summarize, a sufficient condition for introductory pricing is that W s . is increasing. Now, W s . is increasing if higher consumption is more informative and information is valu- able. By Definition 1, information is valuable if V s . is convex. The following proposition from Schlee 1996a shows that V s . is convex for any technology. Proposition 3 Schlee, 1996a, Proposition 1. V s . is convex for any cost function c. . Intuitively, the buyer’s marginal willingness to pay — the inverse demand — is linear in beliefs with quasilinear utility. Thus, at a given quantity, the expected second period price that the seller can charge is independent of information. Accordingly, the seller’s expected profit is never lower with better information, and is generally higher since it adjusts its 16 If the first period demand is a correspondence then we cannot use the constraint to “substitute out” the quantity variable and must formulate the problem as choosing a vector x, −p. Milgrom and Shannon 1995, Theorem 4 then does not apply since the objective function is not supermodular in x, −p. 17 “x ∧ y” denotes the vector whose ith coordinate is the minimum of the ith coordinate of x and y; x ∨ y is obtained by taking the coordinate-by-coordinate maximum. 18 The notion of “strongly higher”, permits the two sets to be equal; completely higher only permits the two to be equal if they are singletons. To illustrate the definition, our restriction on the first period demand correspondence is that dr, ρ is completely higher than dp, ρ when p r. 19 In the sense that “r p, z ∈ dr, ρ , and x ∈ dp, ρ ”, imply z ≤ x. 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