96 S. Dasgupta and R.G. Hansen
to its most highly-valued use. Reserve prices, considered below, may hamper this effi-
cient transfer. Efficiency of auctions under asymmetric beliefs is also not assured (see
Krishna, 2002, for further discussion).
2.4. Revenue equivalence
The result that the second-price auctions and the first-price auction yield the same ex-
pected revenue to the seller is a consequence of the so-called “Revenue Equivalence
Theorem”. What is fascinating about the revenue equivalence of these two auctions is
that such sophisticated models confirm a result which is really quite intuitive: different
mechanisms all yield what is really a “competitive” price, that being the second-highest
valuation. The seller cannot, under these standard auction rules, extract any more rev-
enue than the valuation of the second-highest bidder.
The revenue equivalence result in this independent private value context can be
generalized—not only to encompass a broader class of auctions, but also a more gen-
eral value environment. Suppose that each bidderiprivately observes an informational
variablexi. To simplify notation, we assumeN=2. Assume thatx 1 andx 2 are indepen-
dently and identically distributed with a distribution functionF(xi)and densityf(xi)
over[ 0 ,x ̄]fori= 1 ,2. Letvi=v(xi,xj)denote the value of the object to bidderi,
i= 1 ,2 andi =j.
Consider a class of auctions in which the equilibrium bid function is symmetric
and increasing in the bidder’s signal, and letAdenote a particular auction form. Let
ΠiA(z, x)denote the expected payoff to bidderiwhen she receives signalxi=xand
bids as if she received signalz. Then
ΠiA(z, x)=∫z0v(x, y)f (y) dy−PiA(z),wherePiA(z)denotes the expected payment conditional on bidding as if the signal
werez, and we have used the assumption that the bidders have symmetric and increas-
ing bid functions, so thatiwins if and only ifxj<z. Differentiating with respect toz,
we get:
∂ΠiA(z, x)
∂z=v(x, z)f (z)−dPiA(z)
dz.
In equilibrium,
∂ΠiA(z,x)
∂z =0atz=x, and hence
dPiA(y)
dy
=v(y, y)f (y).Integrating, we get
PiA(x)=PiA( 0 )+∫x0v(y, y)f (y) dy.