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Monopoly with Incomplete Information

Eric Maskin and John RileyThe RAND Journal of Economics, Vol. 15,

No. 2 (Summer, 1984), pp. 171-196

Presented by: Ming Lung

Arun Sundararajan, “Nonlinear Pricing of Information Goods,” Management Science, Vol. 50, No. 12 (Dec., 2004), pp. 1660-1673

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Outline

• Introduction• Simple application: nonlinear pricing– Price discrimination– Quantity discount

• Monopoly pricing of product quality and optimal bundling

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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

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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

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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.

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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

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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

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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– => => *

2*2

*1

*1 ,,, RqRq **

2*2

**1

**1 ,,, RqRq 2

*2

*1

*1

ˆ,,, RqRq

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*2

*2

*1

*1 ,,, RqRq

**2

*2

**1

**1 ,,, RqRq

2*2

*1

*1

ˆ,,, RqRq

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Nonlinear pricing: price discrimination

• Consider more types of the buyers– The selling procedure may look like the following

figure

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Nonlinear pricing: price discrimination

• With ＜ q(vi), R(vi) ＞ optimal for a buyer with parameter vi, we can write maximized utility as

• Combining

• Get

(?)

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Nonlinear pricing: price discrimination

• Combining• Obtain

• Thus the expected seller revenue from a buyer of type vi would be

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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

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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

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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

v

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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 (?)

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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

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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

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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

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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)

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Conclusion & Comments

• The seller strategies– Price discrimination– Quantity discount– Quality and bundling

• Theoretical work • Hard to find a meaningful story immediately

while reading