Monopoly with Incomplete Information

Monopoly with Incomplete
Eric Maskin and John Riley
The 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
• Introduction
• Simple application: nonlinear pricing
– Price discrimination
– Quantity discount
• Monopoly pricing of product quality and
optimal bundling
• 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’. –
• In this article, parties involved are constrained
by asymmetric information
• 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
• 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.
Simple application: nonlinear pricing
• A buyer of type I has preferences represented
– 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
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
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
–  q1* , R1* , q2* , R2*  =>  q1* , R1* , q2* , Rˆ 2  =>  q1** , R1**
, q2* , R2**
q ,R
, q2* , R2*
, R1** , q2* , R2**
q ,R
, q2* , Rˆ 2
Nonlinear pricing: price discrimination
• Consider more types of the buyers
– The selling procedure may look like the following
Nonlinear pricing: price discrimination
• With <q(vi), R(vi)> optimal for a buyer with
parameter vi, we can write maximized utility
• Combining
• Get
Nonlinear pricing: price discrimination
• Combining
• Obtain
• Thus the expected seller revenue from a buyer
of type vi would be
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
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
Nonlinear pricing: quantity discount
• The payment per unit purchased
– Decreasing in v, and hence in q, iff
• And for all x < v
– Quantity discounts are always optimal for buyers
at the upper tail of the distribution
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 (?)
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
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
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
Monopoly pricing of product quality
and optimal bundling
– The optimal selling strategy can be reinterpreted
• 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)
Conclusion & Comments
• The seller strategies
– Price discrimination
– Quantity discount
– Quality and bundling
• Theoretical work
• Hard to find a meaningful story immediately
while reading