Monopoly with Incomplete Information Eric Maskin and John Riley The RAND Journal of Economics, Vol. 15, No. 2 (Summer, 1984), pp Presented by: Ming Lung Arun Sundararajan, “Nonlinear Pricing of Information Goods,” Management Science, Vol. 50, No. 12 (Dec., 2004), pp

Outline Introduction Simple application: nonlinear pricing – Price discrimination – Quantity discount Monopoly pricing of product quality and optimal bundling 2

Introduction Much work has considered incentive schemes (or “principal-agent” relationship) – In political science and economics, the problem of motivating a party to act on behalf of another is known as ‘the principal–agent problem’. – Wikipedia In this article, parties involved are constrained by asymmetric information 3

Introduction We show that a variety of issues can be viewed as members of a single family of principal-agent problems – Price discrimination via quantity discounts – Monopoly pricing of products of differing quality For each of these problems, the central issue is how to construct a sorting mechanism (?) to extract the greatest possible private gain 4

Introduction Our main contribution is to show that, under a separability assumption, we can draw strong conclusions about the nature of optimal incentive schemes Also shed new light on closely related topics – Optimal income taxation – Monopoly pricing of insurance – Etc. 5

Simple application: nonlinear pricing A buyer of type I has preferences represented by – q is the number of units purchased – T is total spending on the units – p(q; v) is the demand price – Assume that higher levels of v are associated with a higher demand 6

Simple application: nonlinear pricing Selling procedure The profit or “return” to the seller Rewrite the utility function of a buyer of type I – N(q; vi) is the social surplus generated by the sale Selling procedure is then 7

Nonlinear pricing: price discrimination Consider the figure in the next page – First consider only two different buyers – How would the seller change the selling procedure to increase his return – => => 8

9

Nonlinear pricing: price discrimination Consider more types of the buyers – The selling procedure may look like the following figure 10

Nonlinear pricing: price discrimination With ＜ q(v i ), R(v i ) ＞ optimal for a buyer with parameter vi, we can write maximized utility as Combining Get 11 (?)

Nonlinear pricing: price discrimination Combining Obtain Thus the expected seller revenue from a buyer of type v i would be 12

Nonlinear pricing: price discrimination Taking the limiting case of a continuous distribution of types The expectation of R(v) is The seller tries to choose q*(v) to maximize expected return 13

Nonlinear pricing: quantity discount Quantity discount – “one for a dollar, three for two dollars” Quantity premium – “one for a dollar, two for three dollars” – Difficult to enforce Is quantity premium desirable? – Analyze the payment per unit purchased 14

Nonlinear pricing: quantity discount The payment per unit purchased – Decreasing in v, and hence in q, iff And for all x < – Quantity discounts are always optimal for buyers at the upper tail of the distribution 15

Monopoly pricing of product quality and optimal bundling Consider the Marshallian utility function – y is spending on other goods – q is the quality level of the single unit purchased – v represents the strength of preference for quality – z is a dichotomous variable equal to unity with purchase and zero otherwise – B is a set of affordable packages (?) 16

Monopoly pricing of product quality and optimal bundling If a consumer with income level I pays T for a unit of quality level q, rewrite the indirect utility as With little loss of generality, we can define units of quality in such a way that the marginal cost of a unit of quality level q is cq Then the monopolist's problem is identical to the problem considered before 17

Monopoly pricing of product quality and optimal bundling The natural generalization of this problem is to incorporate the choice of both quality q and the number of units purchased, z Then we have 18

Monopoly pricing of product quality and optimal bundling Optimal bundling – If z*(v), q*(v) solve Ρ(v) ≡ F’(v) / (1-F(v)), the hazard rate of F – The expected profit-maximizing selling strategy is where 19

Monopoly pricing of product quality and optimal bundling – The optimal selling strategy can be reinterpreted as Define inverse function x = φ(q) z**(q) ≡ z*(φ(q)) T**(q) = T*(φ(q)) – The monopolist announces that quality level q will be sold in bundles of z**(q) units for a total cost of T**(q) 20

Conclusion & Comments The seller strategies – Price discrimination – Quantity discount – Quality and bundling Theoretical work Hard to find a meaningful story immediately while reading 21