Handbook of Corporate Finance Empirical Corporate Finance Volume 1

Ch. 3: Auctions in Corporate Finance 93

If beliefs on values are governed by the uniform distribution, thenG(v)=vN−^1 ,
g(v)=(N− 1 )vN−^2 , and(8)becomes

b(v)=

1

vN−^1

∫v

0

y(N− 1 )yN−^2 dy

=

N− 1

vN−^1

∫v

0

yN−^1 dy

= (9)

N− 1

N

v.

In the particular case whenN =2,(9)implies that equilibrium bidding calls for
bidding half of one’s value—a significant “shading” of one’s bid beneath true value.
Note that in this case, however, the lowest the competitor’s value could be is zero. If
the distribution of values was instead uniform over[ 8 , 10 ], the equilibrium bid would
be( 8 +v)/2—halfway between the lower bound and one’s own valuation.
To see that in general, there is bid shading, notice that we can write

b(v)=

1

G(v)

∫v

0

yg(y) dy=

1

G(v)

∫v

0

ydG(y)

=

1

G(v)

[

yG(y)

∣

∣v

0 −

∫v

0

G(y) dy

]

=v−

∫v

0

G(y)

G(v)

dy

=v−

∫v

0

[

F(y)/F(v)

]N− 1

dy,

where we have used integration-by-parts in the third line.^5 Notice that whileb(v) < v,
sinceF(y) < F(v)within the integral, asN→∞,b(v)→v. In other words, intense
competition will cause bidders to bid very close to their true values, and be left with
little surplus from winning.
How does the seller fare in this first-price auction? We can construct the seller’s ex-
pected revenue by calculating the expected payment by one bidder and then multiplying
that byN. Sticking to the uniform[ 0 , 1 ]distribution for clarity, we have the expected
payment by bidderias

E(Paymenti)=

∫ 1

0

Pr(win)b(y) dy

= (10)

∫ 1

0

yN−^1

N− 1

N

ydy=

N− 1

N(N+ 1 )

.

(^5) Sinced(uv)=udv+vdu, we can write∫udv=uv−∫vdu. This handy trick is used very commonly
in the auction literature.