Handbook of Corporate Finance Empirical Corporate Finance Volume 1

Ch. 3: Auctions in Corporate Finance 107

which should be zero atz=t. Thus, we get

(25)

(

v(t, t)−bF(t)

)g(t|t)

G(t|t)

=bF

′

(t).

Sincev( 0 , 0 )=0, we have the boundary conditionbF( 0 )=0. The differential
equation can then be solved^20

bF(t)= (26)

∫t

0

v(y, y) dL(y|t),

whereL(y|t)=exp(−

∫t

y

g(x|x)
G(x|x)dx).
It is easy to check thatL(·|t)is in fact a probability distribution function on[ 0 ,t],
so that the expression for the equilibrium bid is an expected value with respect to some
probability measure.

3.3.3. Revenue ranking and the linkage principle

With affiliated signals, revenue equivalence no longer holds. The ascending auction
generates at least as much expected revenue to the seller as the second-price auction,
which in turn generates at least as much expected revenue as the first-price auction.
While a direct comparison is possible, the so-called “Linkage Principle” provides a
fundamental insight. Consider an auctionAin which a symmetric equilibrium exists,
and suppose that all bidders are bidding in accordance with this symmetric equilibrium
except possibly bidder 1, who has a signaltbut bids as though her signal werez(zcould
equalt). SupposeWA(z, t)denotes the expected price that is paid by that bidder if she
is the winning bidder. Then the Linkage Principle says that of any two auctionsAand
BwithWA( 0 , 0 )=WB( 0 , 0 ), the auction for whichW 2 i(t, t)(i.e., the partial derivate
with respect to the second argument evaluated with both arguments att) is higher will
generate the higher expected revenue for the seller.
With the benefit of the Linkage Principle, it is easy to see why the first-price auc-
tion generates higher revenue than the second-price auction. In the first-price auction,
a bidder with signaltbidding as if the signal werezwould paybF(z)conditional on
winning, i.e.,W 2 F(z, t)=0 for alltandz. On the other hand, in the second-price auc-
tion, the corresponding expected payment isE[bS(Y 1 )|t 1 =t,Y 1 <z], whereY 1 is the
highest signal among the otherN−1 bidders. It can be shown that given thatbS(·)is
an increasing function, affiliation implies thatE[bS(Y 1 )|t 1 =t,Y 1 <z]is increasing in
t. Hence, the second-price auction generates higher expected revenue.
An important implication of the Linkage Principle—especially for corporate finance
purposes—is that the seller can raise her expected price (revenue) by committing to
release to all bidders any information relevant to valuations. More formally, if the

(^20) The first-order condition is only a necessary condition. It can be shown thatPi(z, t)is indeed maximized
atz=tif the signals are affiliated.