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About the article
Published Online: 2013-05-08
Published in Print: 2013-01-01
This paper is a revised version of chapter 3 of my Ph.D. dissertation.
fn 3 provides an example of such a model.
In principle, there may exist multiple valid versions of the above conditional expectation, all of which coincide over a full measure subset of Θ. I focus on the standard version in which whenever the monopolist sells signal s to type θ′ only.
The following setting, borrowed from Rayo and Segal (2010), delivers the above reduced-form preferences. After observing the consumer’s signal s, the social contact must either “accept” or “reject” the consumer. Acceptance is worth υ(θ) to the consumer and θ – r to the social contact, where r is an outside option drawn from a uniform distribution over Θ. If Θ is normalized to [0, 1], the probability of acceptance is .
Note that the second term in h vanishes as θ → θH and F (θ) → 1.
Note that υ′(0) does not exist when α < 1. Theorem 1 remains valid, however, because υ′(θ) is well defined for all other types.
One can only hope for uniqueness almost-everywhere because profits are not affected by whether the boundary points of the intervals in are included or not in the pools.
For a formal definition of D1 equilibria for a continuum of types, see Ramey (1996).
D1 is a stronger requirement than the “Intuitive Criterion” (Cho and Kreps 1987). Unfortunately, the Intuitive Criterion does not guarantee a unique equilibrium when the sender has more than two types (see, for example, Fudenberg and Tirole 2000).
If Lizzeri’s monopolist could charge a report-contingent price, but could not verify the seller’s type directly or use stochastic disclosure rules, his setting would be the same as the present one (with his sellerin the place of my consumer) except for a crucial difference: his seller’s preferences are not single-crossing (namely, . In this case, one can see from problem (I) (Section 2.1) that the monopolist would obtain the same expected profits, , under any truthful filter.
In other words, our papers intersect only when the two populations in Damiano and Li (2007) have identical distributions, utility is linear, and virtual values are monotone, in which case both models are identical and deliver perfect sorting. Another difference between the two papers is methodological. Damiano and Li use properties of supermodular functions to obtain a suffcient condition for perfect sorting, and local variations in the matching schedule to obtain a necessary condition. In contrast, I focus on the overall covariance structure between virtual marginal values and consumer types to find optimal pooling regions.
Diaz-Diaz and Rayo (2012) study the intermediate case of imperfect competition. They find excessive quality consumption and “Veblen effects:” goods with identical intrinsic quality sold at different prices. Daughety and Reinganum (2010) study private provision of public goods when consumers have a status motive and Vikander (2010) studies advertising when a status motive is also present. Glazer and Konrad (1996) argue that signaling wealth is a primary motivation behind charitable contributions.
The two papers also differ in their methodology. I use optimal nonlinear pricing methods, whereas Moldovanu et al. use methods of optimal contest design.
The Stanford Fund, 1998–99, gsbwww.uchicago.edu/campaign, 01/2003. See Harbaugh (1998) for further discussion of charitable donations.
Notice also that we might have P (D) = P (D′) for two different intervals D, D′ ∈ .