monopoly with incomplete information eric maskin and john riley the rand journal of economics, vol....
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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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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)