Handbook of Corporate Finance Empirical Corporate Finance Volume 1

Ch. 3: Auctions in Corporate Finance 97

Notice thatPiA( 0 )is the expected payment made by bidderiwith the lowest draw of

the signal. Since the seller’s expected revenue is simply 2 times

∫x ̄

0 P

A(x)f (x) dx,it

follows that all auctions in which the bid functions are symmetric and increasing, and in
which the bidder drawing the lowest possible value of the signal pays zero in expected
value, are “revenue equivalent”.^12
The model considered here is one in which the values of the bidders are “interde-
pendent” in the sense that one bidder’s signal affects the value (estimate) of the other
bidders. The signals themselves, however, are statistically independent. An example of
the value function we considered here would be, for example,v 1 =αx 1 +( 1 −α)x 2
andv 2 =αx 2 +( 1 −α)x 1 , where 1α0. Clearly, the independent private values
model is a special case, in whichα=1. The case ofα= 1 /2 corresponds to a case
of the “pure common value” model, for whichv(x, y)=v(y, x), i.e., the bidders have
identical valuations of the object as a function of both bidders’ signals.

2.5. Reserve prices

As reserve prices have figured in some of the corporate finance literature, it is worth-
while to consider analysis of reserve prices in auctions. Sticking with independent
private values, consider an open auction with two bidders. Suppose that bidder 1 has
valuationv 1 >0 and bidder 2 has valuationv 2 =0. Then the open auction will yield
a price of zero. Better in this case would be for the seller to have a reserve price set
in-between 0 andv 1 so that bidder 1 would still win but pay the reserve. Of course, the
problem with a reserve price is that if it is set abovev 1 no sale will result.
To understand how the reserve price is chosen,^13 let us return to the independent pri-
vate values model withNbidders. Consider any auction formAin the class of auctions
with symmetric increasing bid functions. As above, denote byPA(z)the expected pay-
ment by a given bidder in auctionAwhen she bidsbA(z). If the bidder’s private value
of the object isv, her expected profit is

ΠA(z, v)=G(z)v−PA(z),

whereG(z)=FN−^1 (z). As above, in equilibrium, it must be optimal for the bidder
with valuationvto bidb(v), which requires thatΠA(z, v)is maximized atz=v.This
implies that

g(y)y= (14)

dPA(y)

dy

.

Let us suppose now that a bidder with private valuev∗is indifferent between bidding
and not bidding. For such a bidder (known as the “marginal bidder”), by definition

(^12) Absent reserve prices, the bidder drawing the lowest possible signal will typically be indifferent between
bidding and not bidding.
(^13) Our treatment of the problem here follows that inRiley and Samuelson (1981).