Handbook of Corporate Finance Empirical Corporate Finance Volume 1

Ch. 3: Auctions in Corporate Finance 121

Bulow, Huang and Klemperer (1999)examine the effect of toeholds in a pure com-
mon value environment. They make a significant contribution to the literature on toe-
holds by deriving bid functions for both the second and first-price auctions when both
bidders have positive toeholds. They examine how (for small positive toeholds) bid-
der asymmetry affects the takeover outcome in each auction, and compare expected
revenues in the two auctions when the toeholds are symmetric as well as asymmetric.
We first discuss their setup in some detail, before discussing the intuition for the main
results.
Bulow, Huang and Klemperer (1999)consider a “pure common value” model with
two bidders where each bidder draws an independent signaltifrom a uniform[ 0 , 1 ]
distribution. The value of the target to each bidder isv(t 1 ,t 2 ). Bidderiowns initial
stakeθiin the target, where 1/ 2 >θi >0, fori= 1 ,2. Each bidder bids for the
remaining 1−θifraction of the shares of the target.
In the second-price auction, bidderi’s problem is to choosebito maximize

MaxbiΠi(ti,bi)= (31)

∫b−j (^1) (bi)
0

[

v(ti,α)−( 1 −θi)bj(α)

]

dα+

∫ 1

b−j^1 (bi)

θibidα.

The first-order condition is

1

b′j

[

v

(

ti,b−j^1 (bi)

)

−( 1 −θi)bj

(

b−j^1 (bi)

)]

+

[

1 −b−j^1 (bi)

]

θi−θibi

1

b′j

= 0.

Let us now defineφj(ti)=b−j^1 (bi(ti)), i.e., this defines the pair of signals for bidders
iandjfor which they have the same bid, sincebj(φj(ti))=bi(ti). Similarly, we can
defineφi(tj)=b−i^1 (bj(tj)). Using these definitions, we can rewrite the first-order
condition as

b′j (32)

(

φj(ti)

)

=

1

θi

1

( 1 −φj(ti))

[

bi(ti)−v

(

ti,φj(ti)

)]

,

where we have replacedtjbyφj(ti).
The corresponding first-order condition for bidderjis

b′i (33)

(

φi(tj)

)

=

1

θj

1

( 1 −φi(tj))

[

bj(tj)−v

(

φi(tj), tj

)]

,

where we have used the fact thatv(φi(tj), tj)=v(tj,φi(tj)). Consider a pair oftiand
tjthat in equilibrium bid the same, then we must haveti =φi(tj)andtj =φj(ti).
Using this, the last equation can be rewritten as

b′i(ti)= (34)

1

θj

1

( 1 −ti)

[

bj

(

φj(ti)

)

−v

(

ti,φj(ti)

)]

.

Sincebj(φj(ti))=bi(ti)andb′i(ti)=b′j(φj(ti))φ′j(ti), dividing(34)by ( 32 ), we get

φ′j(ti)= (35)

θi

θj

1 −φj(ti)

1 −ti

.