The model with a representative buyer

E.E. Schlee J. of Economic Behavior Org. 44 2001 347–362 351

2. The model with a representative buyer

We begin with the representative buyer version of the model. The firm produces and sells a perishable good in each of two periods. Its quality is represented by a parameter θ ∈ R + . We let x denote the quantity sold during period 1, and let X denote the period 2 quantity. The time-invariant cost function for producing the good is c., an increasing and differentiable function. We let y and Y denote consumption of a composite good in each period. Each buyer has preferences over the two goods over the two periods representable by an time separable quasilinear utility function, ux, θ + y + δuX, θ + Y , where δ is the discount parameter. 10 Neither side of the market initially knows θ , but both know that it has two point support, {θ L , θ H } with θ L θ H . We let ρ denote the common prior belief that θ = θ H . After period 1, all agents observe a real-valued signal z whose distribution is described by a density function z 7→ f z; x, θ , depending on aggregate consumption, x, and quality, θ . Agents then update beliefs according to Bayes’ rule; the posterior belief that θ = θ H after observing z is ρz, x = ρ f z; x, θ H ρ f z; x, θ H + 1 − ρ f z; x, θ L . We will assume that larger consumption is more informative, in the sense of Blackwell 1951. More formally, an information structure is a pair of densities, {f .; x, θ } θ ∈{θ L ,θ H } that gives the state-conditional distribution of the signal for each x. Definition 1. { f .; x ′′ , θ } θ ∈{θ H ,θ L } is more informative than {f .; x ′ , θ } θ ∈{θ H ,θ L } if R F ρz, x ′′ f z, x ′′ dz ≥ R F ρz, x ′ f z, x ′ dz for every continuous, convex func- tion F .: [0, 1] → R, where f z, x ≡ ρ f z; x, θ H + 1 − ρ f z; x, θ L , the uncon- ditional density of z. In words, one experiment is more informative than the another if the expectation of any convex function of posterior beliefs is higher under the former: intuitively, a more informative experiment leads to a riskier distribution of posterior beliefs. 11 As a simple example, suppose that the agent can observe a signal that takes on two values: he receives good news with probability λ and bad news with probability 1 − λ. Suppose that with good news his posterior is ρ g and with bad news it is ρ b . If his value function from a game is V , then the expected utility with partial information is λV ρ g + 1 − λV ρ b . If, however, the agent receives no additional information, then his expected utility is simply V λρ g + 1 − λρ b . The expected utility with information is higher if and only if V is convex. This version of the definition of “more informative” is especially convenient since we need only show that the firm’s value function from the second period market is convex in beliefs to show that it prefers better quality information. We now restrict the information structure to ensure that larger consumption levels are more informative, one of the sufficient conditions for introductory pricing. 10 In the multiple buyer case, writing utility as a function of aggregate consumption is of course an abuse of notation. 11 See Kihlstrom 1984 for further discussion of equivalent definitions of more informative experiments. 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