Second-Degree Price Discrimination

Transcription

1 Second-Degree Price Discrimination Lecture 4 Goal: Separating buyers into di erent categories by o ering suitable deals: Screening. Monopolist knows that buyers come in di erent types. Maybe some buyers are sophisticated while others are not. There is no way of telling which consumers are sophisticated. Can the monopolist induce the di erent types to choose di erentially in the market? With linear prices, little hope of this. f prices are lower for one of the types, both types would buy at those prices. Allow for non-linear prices. Maybe selling the good in di erentiated qualities might also be a good strategy.

2 Examples of Second-Degree Price Discrimination or Menu Pricing Quantity discounts: Price of six-pack of Coke less than 6 times price of single can. Di erential xed fee, variable fee combinations: Charging for downloading nternet content to smart phones. Package vs. single ride on Linnanmäki. Closely related: Quality versioning First-class, Business and Economy airfare. Di erent speeds on broadband. nsurance with di erent deductibles.

3 nformation Economics: Adverse Selection or Screening Basic Set-Up: An uninformed party (principal) o ers a menu of alternatives to an informed party (agent). n nonlinear pricing, we take the seller to be the principal and the buyer to be the agent. Proposition consists of a list f q l, t l g L l=1. q stands for a physical allocation to the agent. t is the transfer that the agent makes to the principal. Hence choosing q l, t l means that the agent gets physical allocation q l in exchange for paying t l. Notice that this is not a per unit price but a total price for q l.

4 Adverse Selection or Screening Agent s utility from q depends on her type θ 2 Θ. For most of this section, we assume that there are two types: θ 2 fθ H, θ L g. Quasilinear utility. Agent: Principal: u A (θ, q, t) = θv (q) t. u P (θ, q, t) = t c (q). Here we interpret v (q) as the utility from allocation q. Assume increasing utility with diminishing marginal utility: v 0 (q) 0, v 00 (q) 0 c (q) is the cost of providing allocation (quantity or quality) q. Assume increasing convex cost: c 0 (q) 0, c 00 (q) 0.

5 nformation Economics: Adverse Selection or Screening Seller makes an o er fq l, t l g L l=1. She does not know the type of the buyer (but has a belief on the likelihoods of the di erent types). With two types, set λ = Prfθ = θ H g, 1 λ = Prfθ = θ L g. Since the payo functions are common knowledge, she can also calculate, which buyer types will choose which pairs. Buyer of type θ picks the pair q l, t l that gives her the maximal utility or picks nothing if that gives higher utility. Since each type picks at most one pair, we can restrict the number of alternatives o ered to be at most the number of di erent types of buyers. With two types of buyers θ 2 fθ H, θ L g, enough to consider two pairs f(q 1, t 1 ), (q 2, t 2 )g. Call the pair chosen by θ i as q i, t i for i 2 fh, Lg. Examples: nsurance company screening privately known risk types, Monopoly bank screening projects with privately known success rate, Regulator screening public utilities with privately known marginal cost, etc.

6 nformation Economics: Adverse Selection or Screening Since θ H chooses θ H v q H, t H over q L, t L, we have q H t H θ H v q L t L. Similarly for θ L θ L v q L t L θ L v q H t H. These constraints are called incentive compatibility constraints. f the agent can secure a payo of u by not trading with the principal at all, then we also must have: θ H v q H t H u, θ L v q L t L u. These constraints are known as individual rationality or participation constraints.

7 nformation Economics: Adverse Selection or Screening Summing together: The principal s problem is: max λ t H f(q H,t H ),(q L,t L )g c q H + (1 λ) t L c q L subject to θ H v q H θ L v q L t H θ H v q L t L, t L θ L v q H t H, θ H v q H t H u, θ L v q L t L u.

8 nformation Economics: Adverse Selection or Screening This is the simplest form of a problem under adverse selection. The agent has private information at the time when the principal proposes the contract. This private information gives (at least some type of) the agent some surplus even if the principal make a take-it-or-leave-it o er. Model generates a genuine sharing of surplus. Will the outcome be socially e cient as in the case where the principal knows θ? The more general theory framework encompassing this model is called Mechanism Design. Treated in Microeconomic Theory V in the spring term in detail. First steps outlined in the handout.

9 Back to Non-Linear Pricing The monopolist (principal) knows that there are buyers (agents) of di erent types θ. The monopolist cannot determine the type of a buyer. The monopolist can use non-linear schemes and choose some product characteristics. The monopolist sells goods in packages of size q l costing t l. Hence the buyers choose from the menu f q l, t l g L l=1. There are two types of buyers θ 2 fθ H, θ L g. Denote by λ the fraction of buyers that are of type H.

10 Non-Linear Pricing The buyers utilities are given by u (θ, q, t) = θv (q) t, and assume that with θ H > θ L. v is assumed to be concave. The sellers payo : t c (q), where c (q) is a convex increasing cost function for producing quantity q.

11 Non-Linear Pricing We saw in rst degree price discrimination (with two part-tari s) how the monopolist would choose bq i,bt i for i 2 fh, Lg if the type were observable. bq i is e cient : c 0 bq i = θ i v 0 bq i, bt i = θ i v bq i. Suppose the monopolist attempts this in the case where the type is not observable. Then types θ H would select bq L,bt L. f the seller wants θ H to pick q H, t H and θ L to pick q L, t L from f q H, t H, q L, t L g, it must be that: θ H v θ L v q H t H θ H v q L t L θ L v Sum these together to get: θ H θ L v q H v q L t L, (1) q H t H. q L 0.

12 Non-Linear Pricing Furthermore if the buyer can get 0 by refusing to trade: θ H v q H t H 0, (2) θ L v q L t L 0. The rst two inequalities are called the incentive compatibility constraints. They ensure that each type of buyers chooses the bundle that is intended for them. The latter inequalities are called individual rationality constraints. They ensure that the payo from buying is at least as large as the payo from not buying. The monopolist s problem is to maximize pro t subject to these constraints. max λ t H f(q H,t H ),(q L,t L )g subject to C and R. c q H + (1 λ) t L c q L

13 Non-Linear Pricing Analysis: i) C for H must bind. (f not, then R for H not binding and then you can increase pro t by increasing t H a little). ii) R for L must bind. (f not, increase both prices by the same amount). Use R of type L to solve t L = θ L v q L. Use C of H to solve t H = t L + θ H v q H θ H v q L = θ H v q H θ H θ L v q L. But then: We call θ H v q H t H = θ H θ L v q L > 0 if q L > 0. θ H θ L v q L the information rent of the high type.

14 Non-Linear Pricing Hence the maximization problem becomes: max q H,q Lfλ θ H v q H θ H θ L v + (1 λ) θ L v q L c q L g. q L c q H FOC with respect to q H : Hence q H is chosen e ciently. FOC with respect to q L : λ θ H v 0 q H = c 0 q H. θ H θ L v 0 q L = (1 λ) c 0 q L θ L v 0 q L. From this we see that q L is smaller than the e cient level. This re ects the desire to squeeze the rent of the high type.

15 Non-Linear Pricing Conclusion: Higher types buy larger quantities. High types earn positive information rent. Low types earn no rents and are indi erent between participating and not. Pro t maximizing solution trades o e ciency and rent extraction. Next lecture: Applications of this material.