BASEMENT
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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
TAKEOVER DEFENSES AND SHAREHOLDER VOTING*
David Austen -Smith
University of Rochester
and
Patricia C. O'Brien
Massachusetts Institute o£ Technology
WP# 1823-86
September 195
MASSACHUSETTS
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
INSTITUTE OF
TAKEOVER DEFENSES AND SHAREHOLDER VOTING*
David Austen -Smith
University of Rochester
and
Patricia C. O'Brien
Massachusetts Institute of Technology
WP# 1823-86
*
September 19i
The authors gratefully acknowledge helpful conversations
with Jeff Banks and Rick Ruback. Remaining errors are
solely the authors' responsibility.
Takeover Defenses and Shareholder Voting*
David Austen-Smith
Department of Political Science
University of Rochester
Rochester NY 14627
and
Patricia C.O'Brien
Sloan School of Management
M.I.T.
Cambridge
MA 02139
September 1986
*The authors gratefully acknowledge helpful conversations with
Remaining errors are solely the authors' responsibility.
Jeff
Banks and Rick Ruback.
Abstract
Why do shareholders vote for anti-takeover devices which apparendy lower the value of dieir
firm?
We address this question by constructing an agenda-setting model in which rational,
informed, and value-maximizing shareholders vote on requests for such devices
self-interested
management with employment opportunities
outside the firm.
made by
a
We find sufficient
conditions for the value of the firm to decline as a result of a request, although
it is
approved by
shareholders. In our model, the apparendy paradoxical voting behavior occurs because the
expected takeover premium
is
reduced more by rejection of the request than by approval.
1.
Introduction
Why do shareholders vote for anti-takeover devices which apparendy lower the value of their
firm?
We address this question by constructing a model in which rational, informed, and
value-maximizing shareholders vote on requests for such devices made by a self-interested
management with employment opportunities
outside the firm.
We describe conditions under which
the value of the firm declines as a result of the request, although
it is
approved by shareholders. In
our model, the apparendy paradoxical voting behavior occurs because the expected takeover
premium is reduced more by rejection of the request than by
approval.
A large and increasing number of amendments to corporate charters are specifically designed to
increase the cost of transferring control.
amendments
in
DeAngelo and Rice (1983)
report over
250 proposed
1974 through 1979. Linn and McConnell (1983) find generally increasing
incidence of such proposals
among NYSE
firms over the period from 1960 to 1980.
Responsibility Research Center Usts over 200 in 1985 alone.
the motives for instituting anti-takeover charter
amendments
The
Investor
The two major hypotheses describing
are
commonly described as
"management entrenchment" and "shareholder interests."^ According
to the
management
entrenchment view, incumbents are interested in job security and seek protection from the takeover
market, to the detriment of shareholders.
Two suggested explanations for shareholder approval of
entrenching antitakeover devices are: (1) for a majority of shareholders, the costs of becoming
informed about the effects of defensive charter amendments exceed any potential benefits, and
uninformed shareholders consistenUy give
their proxies to
wish to maintain friendly relations with management
management; and
(2) large shareholders
to ensure the benefits of future business,
and
large shareholders control sufficient shares to be pivotal in the vote. Shareholder irrationality is
sometimes offered as a
third alternative.
The stockholder interests hypodiesis recognizes a free-rider problem in
collective action
by
shareholders (Grossman and
Han
(1980), Jarrell and Bradley (1980)). Shareholders have
difficulty colluding to extract larger
premia from takeover bidders, so antitakeover devices benefit
shareholders by enforcing a level of collusion in takeover negotiations. Since defensive charter
amendments
benefit shareholders, there
them. However, there
no inconsistency
in rational shareholders voting for
evidence that shareholders benefit from such amendments.
is Uttle
DeAngelo and Rice (1983)
is
find statistically insignificant negative abnormal returns around the
public announcement of proposed antitakeover amendments. Linn and McConnell (1983) find
positive returns around the board meeting date at
insignificant negative returns around the
find negative returns
classified
which amendments are proposed, and
proxy mailing
date. Jairell,
accompanying the announcement of "shark
Poulsen and Davidson (1985)
repellents", viz. supermajority,
board and authorized preferred amendments.
In the context of our model, the
management entrenchment and shareholder interests
hypotheses do not necessarily lead to different predictions about shareholder value. Anticipatory
takeover defenses raise the costs of acquiring control of the firm, thereby entrenching management,
but lead to a higher
potential
premium for shareholders
managers contribute
if
a bid succeeds.
We presume that different
different value-added to a given firm,
Only managers capable of producing higher value-added than
the
by virtue of skill or experience.
incumbent can mount successful
takeover bids. Whether shareholders gain or lose as a result of the incumbent's defenses depends
upon the quality of the incumbent management relative
opportunities for
employment outside
The paper is organized
3.
and the incumbent's
the firm.
as follows. In section 2
discuss the results in section
to potential bidders,
we develop
the model, and
we summarize and
Section 4 concludes. All proofs are confined to Appendix A.
Appendix B consists of an example
that illustrates
our results.
2.
Model
We are interested in managerial control of a given firm with fixed assets and financial structure.
There
is
M of different managerial types capable of running the firm, different types
a continuum
A manager's type is the common expectation of
contributing differendy to the value of the firm.
owners and the outside market of the value added
the incumbent manager's type, and
types on
M
is
common knowledge
question. Because
we
Assume
function.
M
is
the
whole real
that
line.^
The
m e M denote
distribution F(-) of
and induces a distribution over possible values of the firm
let F(-) also
that the support of
F
The firm is owned by
in
we can,
describe the distribution over managers of possible firm
is
connected, and that
F
is
We additionally assume that lim.^_^^[l-F(t)]/f(t) < <»:
diversified,
manager. Let
take the value of the assets and the financial structure to be fixed,
without loss of generality,
values.
assume
by
to the firm
smooth. Let
this is
f(-)
be the density
not a strong restriction.-^
a set of shareholders holding fully diversified portfolios. Being fully
each shareholder
is
interested only in
maximizing the expected value of the
firm. This
expected value depends both on the incumbent manager's type and on the expected premium paid
the owners, conditional
Heuristically, the
firm
is
two
periods.
on the firm being taken over by some
model of takeovers we have
in
mind
is
alternative
draws a potential alternative manager fi-om the
set
management
the following.
A manager of type m currendy controls the firm.
to
The
lifetime of the
In the second period, nature
M, who can choose whether or not to make
a
takeover bid for the firm. Since there are no takeover bids possible beyond the second period, the
maximum value of the firm in this last period is
control,
i.e.
the second period manager's type.
given by the value added by the manager then in
We suppose that there is no utility value to making
a bid per se, so that, given the incumbent manager's type m, only managers of types n
worthwhile to make any takeover
find
it
will
make an
offer.
Hereafter,
bid. If there is
we assume
>
m could
any cost to bidding, then no type n <
m
there is such a cost, but, to avoid cluttering the notation
unnecessarily, suppose the cost constant and take
it
to
be implicit in the value of m. Hence, given
the distribution F, the probability that a
is
[l-F(m)]
nothing).
(if
The
we make
manager minimally capable of making a takeover bid
the bidding-cost explicit,
we would have
to write [l-F(m+e)]: this adds
qualification "minimally" here refers to the possibility that the incimibent will contest
m
any bid by erecting takeover defenses: being a manager of type n >
for
making a successful bid
for the firm. If a bid is
controls the firm for the remainder of
incumbent remains
Given
this
its life.
model of takeovers,
If there is
first t^je
surely in control of the firm in the first period
enterprise is taken as given,
view such defenses
successful, then the
no bid or if there
m, and the expected premium
derive this expected premium, consider
is
is
necessary but not sufficient
new manager
an unsucessful bid, the
is
the first period expected value of the firm
loss of generality, let discount factors be equal to
The manager
made and
is
in control.**
the inciunbent manager's value added,
is
arrives
it is
in the
one for the manager and
is
simply the
sum of
second period. Without
all
To
shareholders.
incumbent manager's objective function. Since he
and the investment and
financial structure of the
necessary only to examine his second period payoff.
both takeover defenses.
We
as effectively increasing the cost of making a successful takeover bid.
It is
endowed with two control
variables, x
and
y,
natural, then, to describe both types of defense as nonnegative real variables: (x, y)
absence of any defense, managerial types n >
€ R^^.
In the
m will successfully bid for the firm: if there is any
defense, a successful bidder must be capable of overcoming the defenses in addition to improving
on m. Therefore, given defenses x and
to win.
As yet
takeover bid must be at least equal to [m+x+y] in order
there is nothing to distinguish the
potential bidder's perspective the
different
y, a
two
two types of defense and, indeed, from any
sorts are indistinguishable.
They
are,
however, quite
from the incumbent manager's perspective.
Defense x
is
anticipatorv
:
it
can be put in place onlv in the
first
period, prior to any potential
bidder appearing. There are no direct costs to implementing x. However, the manager
to ask the shareholders explicitiy for a level of x. Majority voting
among
is
obliged
the shareholders
determines whether or not the manager's request
granted Amendments are not permitted, and so
is
the shareholders can only accept or reject the manager's proposal.^ In addition, there
cost
is
home by
the
manager
granted This cost
is in
indication that the
the shareholders;
terms of his outside value
on being ousted from
this
for instance,
is
however
X?
> 0]
x = x?; and
that the
the utility payoff he can expect to receive in
the firm.
if
more
Our presumption
in the
is that efforts
absence of an explicit offer
is
on
an
interested in personal security than the welfare of
lowers his outside market value. Let x? denote a request for a level of
anticipatory defense. If x? is approved
defense
-
managers to erect takeover defenses
manager is,
and
an indirect
for requesting anticipatory defense, irrespective of whether this request
the second period, conditional
the part of incumbent
is
it
is
by shareholder vote, then the implemented level of this
not approved then x
above discussion implies
=
0. Evidently,
that the states of the
x?
=
implies x =
world [x =
I
Note
0.
x? = 0] and [x =
I
are distinct: this is crucial to our model.
The second type of takeover defense
implemented oiJy to
available to the manager, y,
responsive
it is
:
second period conditional on such a bid
fight an explicit takeover bid in the
materializing. Unlike anticipatory defenses, the
is
manager is
free to
implement any
level of
responsive defense without the necessity of obtaining shareholder approval. There are, though,
direct costs to
engaging
in a takeover fight. Let c(y)
be the direct costs, measured in units of utility,
bome by the incumbent management in implementing
c'(y)
0:
>
and c"(y) >
a responsive defense of y:
for all y. For technical reasons,
none of our principal
results
depend on
this."'
it
is
convenient to assume lim.y_^Q
any successful bidder
in the
second period must pay
at least
any more be paid? Because shareholders are interested only
pay e >
more than [m+x+y]
to
in
[m+x+y]
c'(y)
=
x and
y,
we
argued
to acquire the firm. Will
maximizing wealth,
persuade them to remove the incumbent
pay any more than necessary to take over the
0,
'
In the heuristic discussion of the takeover process above, given defenses
that
assume c(0) =
firm. Therefore, in the limit,
it is
sufficient to
No acquirer wishes to
epsUon
will
be zero.
Hence, the premium over and above his managerial type, m, that the incumbent secures for the
~ conditional
shareholders
~
is
on losing a takeover
[m+x+y]
make offers, and any such
will
the second period
is
control of the firm,
therefore given by [1
co(x,
(1)
where W(-)
J J
<
-
incumbent obtains, conditional on losing
utility the
I
I
-
c(y),
0-
from the
able to extract
is
acquirer, decreasing
~
utility
in the
V(m). To make the problem
We argued above that,
is
presumed common knowledge,
nontrivial,
we
I
m) >
n.
If this
is
—
latter
in
any
on the
0.
suppose V(m) > W(0,
y, if there is
takeover attempt.
a bid then
is
it
I
is
m).
will be successful.
At the
indeed given. However, since
The incumbent's
as is his type. Likewise,
of the potential bidder in the second period
of type n
premium he
second period, the manager stays in control and receives a
given x and
who makes the
W2 < 0,
the incumbent's outside
responsive, the manager's selection of y will depend, inter alia,
managerial type n
0,
and the cross-effects of this
beginning of the second period, the anticipatory defense x
defense y
>
ultimately, at an ever-increasing rate
former are non-increasing. Assume also that W(0,
no takeover bid
Thus
Wj
increasing concave in the takeover
efforts to secure anticipatory defenses in the first period;
If there is
and x?. Assume
^^ ^^so assume that lim.jj9_^<„ W2 = -«».
value conditional on losing his current position,
was
probability of takeover in
given by:
y X?) = W(x+y, x? m)
W ^2 -
The
>
F(m+x+y)].
the manager's gross outside value given x, y
is
and
is
fixed, only managerial types of n
offer will be successful.
Given x and y, the net second period
W
and y as
precisely [x+y]. Moreover, again taking x
second period despite defenses x and y
battie in the
on
the particular
utiUty and cost schedules are
once nature has made her draw, the type
common knowledge.
Therefore, any potential bidder
capable of calculating the incumbent manager's optimal credible response y to a bid by
response
is sufficient to
presumption of credibility
~
beat n's best offer, then
n will
make no offer at
all.
~
If
n
because of the bidding cost and the
is
capable of topping the incumbent's
make
best response, then n will
the smallest offer necessary to
win control of the
finn.
And even
though the incumbent loses surely, by definition of credibility he will bear the costs of fighting,
and the winner
c(y),
will
remains to determine the
pay the premium [x+y]. Hence the
From
Wj
we
y
I
x?).
obtain:
=
Wi
aco/ay
(4)
a2o)/ay2
dw/dy =
>
It
and define:
(d(x,
r^
I
(3)
Setting
{0, x?},
y*(x X?) = argmax.y g
(1),
argument goes through.
of credible responsive defenses.
set
Let X? and x be given, x €
(2)
earlier
=
- c';
Wi 1
-
<
c"
Vy.
0,
implicidy defines y*(x
and lim.y_^Q
c'(y)
=
I
By
x?).
by assumption,
(4), y*(-IO is
y*(-l-)
>
unique for
all x?,
x>
0.
Since
0.
We now define, Vn > x+m:
v(n,
(5)
The schedule
X m) = V(m)
I
-
v(-) describes the
c(n-x-m).
maximum
second period
utility
the incumbent
given he fights and defeats any bidder of type n (again, no managerial type
offer). Define, for
n(x
(6)
I
x?)
X? >
=
and x €
[inf.n
[inf.n
GSTotice that,
Then
bid.
I
< n(x
I
v(n, x
!•)
=
co(x,
it is
I
(7)
how premiums paid to
n*(x
I
X?)
There are two cases:
= x + y*(x
(a)
n*(x
I
I
I
x?)], if
>
V(m) > W(x+y*(x
will
make an
I
x?), x?
m);
I
m)], otherwise.
I
n(x
I
x?) will
x?)
m+x
>
manager m
make a
for all x, x?
will fight
> 0).
and defeat the
successful offer for the firm,
differ.
successful bidders vary, define, for x?
x?)
x?)
x?)
credible that the incumbent
the other hand, any bidder of type n
see
I
I
m) by assumption, n(x
although the premiums they have to pay will
To
y*(x
V(m) = W(n-m, x?
x?),
< x+m
obtain,
{0, x?}:
because V(m) > W(0,
for all types n
On
I
n'
manager can
>
and x €
{0, x?):
+ m.
> n(x
I
x?) and (b) n*(x
I
x?)
< n(x
I
x?): case (a) occurs if
and
only
V(m) < W(x+y*(x
if
I
x?
x?),
Figure
I-).
under the assumption
illustrates these
1
that
x = x? >
0.
[HGURE 1 ABOUT HERE]
Consider Figure 1(a) and
incumbent manager first
let
n e (n(x x?), n*(x
I
n to extract
to fight
I
x?)].
the bidder-type exceed n*(x
premium,
in this case (virtually) equal to [n*(x
his outside value, co(x, n*(x
Notice in
this event,
assumed
although the
to
consume
may
I
I
n-m-x
co(x,
I
and
Now
x?): this is clearly credible.
incumbent will optimize by fighting to extract a
x?)
-
m], and then leaving the firm and obtaining
I
do not pay
all that
they are willing to pay for the firm. In
maximum value of the firm is n in
the surplus, n
-
[n*(-l-)
m]. Since the
-
new manager is
the second period, the
life
of the firm
is
only two periods,
one thinks of the new manager as representing some other firm, then
this
be viewed as a windfall gain to the owners of the acquiring firm.
In Figure 1(b), the situation
bidders pay a
will
a best response for the
x?)-m-x x?).
this last case that bidders
this is reasonable. If
surplus
x?): then again the
I
it is
(virtually) all n's willingness-to-pay for the firm,
then to leave the firm to collect his net outside value,
let
Then
is
premium of [n*(x
I
analogous to the second case described for
x?)
-
1(a).
All successful
m], irrespective of their type, and only types n
> n(x
I
x?)
make takeover bids.
For future reference, for
actually paid
over
m
(8)
all
n > n(x x?) and for any x
I
by the successful bidder of type
n.
Then
> 0,
let 7c(n,
x x?) denote the premium
I
for any successfiil bid, the
premium paid
is:
7t(n,
X
I
X?)
= min.[n
-
m, n*(x x?)
I
-
m].
We are now in a position to specify the incumbent manager's second period expected payoff
schedule as a function of anticipatory defenses,
(9)
U(x
I
X?)
x:
= F(n(x x?))-V(m) + 5-[F(n*(x
I
+
{5-[l-F(n*(x
I
X?))]
+
I
x?))
-
F(n(x
[l-5]-[l-F(n(x
I
I
x?))]-J co(x,
x?))]}-co(x,
n-m-x x?)-f(n)dn
n*(x
I
I
x?)-m-x x?)
I
Figure 1(a)
V, 0)
V(m)
W(x,x?lm)
n(x
I
x?)-m-x
n*(x x?)-m-x
I
Figure 1(b)
V, CO
V(ra)
W(x,x?lm)
n*(x
I
x?)-m-x
n(x
1
x?)-m-x
°
where 5 =
n(x
1 iff
I
< n*(x
x?)
of the firm under management
I
x?) and 5
=
otherwise. Similarly, the
period expected value
first
m is given by:
or
S(x X?) =
(10)
I
m + 5-[F(n*(x
+
and 5
x?))
I
{5-[l-F(n*(x
I
-
F(n(x
X?))]
+
x?))]-J [n-m]-f(n)dn
I
[l-6]-[l-F(n(x
I
x?))]}-[n*(x
I
x?)
-
m]
defined as before.
is
The responsive defense y is chosen optimally by
the
management, independent of shareholder
approval and given the bidder type that appears in the second period. Anticipatory defenses require
shareholder approval and have to be set in the
period optimization problem
(11)
is
first
period. Let x
therefore to choose a request, x?
=
>
Then
x?.
the manager's first
0, to:
max.U(xlx?)
subject to: S(x
Clearly, if the
I
x?)
> S(0
manager asks x? =
I
x?).
constraint says that the expected value of the firm conditional
be no less than
its
value
if
However,
0, then the constraint trivially binds.
the request
is
rejected, given x?
>
if
this is
granted
~ we have
in general that
S(0
Because the incumbent manager's
0.
I
then the
on the request being approved must
decision problem in the second period depends, inter alia, on his request, x?, in the
whether or not
x? >
0)
* S(0
I
first
x?) for any x?
>
period
0.
~
In other
words, the relevant alternative that rational shareholders have to compare against the manager's
request
is
not the situation prior to any request, S(0
reject the request,
S(0 x?). This
If the constraint
I
binds at
rejecting the request Since
is
I
0),
but the situation that would result
Appendix
some x? > 0, then shareholders
managers are not so
are indifferent
indifferent, they
between accepting and
can insure acceptance in such
strict
preference on the part of
A shows, such a perturbation is always available.
the constraint binds, then the
they
the central idea of the model.
circumstances by slighdy perturbing the request, x?, to induce
shareholders: as
if
management request
is
approved. Alternatively,
So we assume
we can
that if
adopt the
convention that in cases of shareholder indifference, shareholders always "vote with the
10
management
.
We turn now to some results.
3.
Results
There arc thrce possible conditions
beginning of the second period: (1) the manager
at the
requested no anticipatory defenses, (2) the manager
shareholders, and (3) the manager's request
second period response
on the
first-period
Proposition
Corollary
If
1:
1:
Let x
was approved. Proposition
= x? >
0.
Then y*(0
1
0)
if
is
I
> y*(0
x?)
is
I
> y*(x
I
x?)
>
is
x?).
maximizing response
never larger after a request, even
premium
we show
if
1, total
is
defenses
rejected. Since total defenses determine
successful bidder emerges.
if the
1 is
that the second-period
request
is
denied, than
Thus, the request carries no implicit threat of "scorched earth" responses
Rather, as
0.
a successful bid, this implies that shareholders can expect a smaller
An interesting feature evident from Proposition
is
puts an ordering on the
1
approved. However, from Corollary
premium if they reject the manager's request and a
response
x?)
rejected by shareholders, the second-period
the request
premium received in
I
> y*(0
(anticipatory plus responsive) are smaller if the request
the
was voted down by
which maximizes the manager's outside value, conditional
Letx = x?>0. Then x + y*(x
than
a request which
outcome.
a first-period request
strictly larger
to a takeover bid
made
if
if
maximizing
no request is made.
shareholders deny it
below, the request alters the manager's incentives to fight for a higher
a successful bidder appears.
Anticipatory and responsive defenses are substitutes in their effects on the
minimum bid
11
Generally, the higher the level of anticipatory defense approved by
required to acquire the firm.
shareholders in the
Comparative
first
statics
period, the lower will be the manager's choice of responsive defenses.
on the maximizing second-period response (Appendix A,
that this response is non-increasing in the size of the first-period request
the first-period request, except
its
two arguments and
separability,
it
when
the manager's gross outside utility
the first-period request
follows fixjm Corollary
is
denied.
-
(a7))
stricdy decreasing in
additively separable in
and therefore the premium from a
successful bid, will be at least as large if requested anticipatory defenses are approved as
anticipatory defenses
show
In the special case of additive
that total defenses,
1
is
It is
(a3)
if
no
were requested.
Value-maximizing shareholders consider both the
size of the
premium they can expect from
successful bid, and the probability that such a bid will emerge, in evaluating
how
a
to vote.
Proposition 2 describes the probability of a successful takeover bid, as a function of requested
anticipatory defenses and shareholder approval of the defenses.
Proposition 2:
(a)
n(x
(b)
Vx =
I
X?)
X?
is strictly
increasing in X?, X e {0, x?};
>
X?)
0,
n(x
Proposition 2 states that the
increases with the
bid will emerge
I
> n(0
I
x?)
> n(0
0).
minimum bid which can succeed against incumbent management
amount of anticipatory defense
falls as the
I
requested. Since the probability that a successful
minimum successful bid rises, part
(a)
of Proposition 2 implies that the
probability of takeover declines with increases in the level of requested anticipatory defenses,
whether or not these are approved. Part
(b) implies that the probability
anticipatory defenses are requested, declines
request
is
approved. Notice that part (b)
is
if
there is a request,
of takeover
is
largest if
and may decline more
not an immediate consequence of part
(a):
no
if the
the
change
in
12
state
from
[x
= x? x? >
I
0] to [x
=
I
x? > 0]
is
not incremental because of the "take
it
or leave
it"
nature of the shareholders' decision.
The value
successful bid
to shareholders
is
from anticipatory takeover defense dep)ends on both the premium if a
made, which generally increases with the
level of anticipatory defense,
probability of a successful bid, which generally decreases. Proposition 3
are
unambiguously worse off if they
reject
shows
and on the
that shareholders
management's request for anticipatory defenses than
they were before the request
Forx?>0,
Proposition 3:
The voting
S(
I
0)
> S(0
I
x?).
(incentive compatibility) constraint in the
model requires
that approval of a request
leave shareholders at least as well off as rejection. Proposition 3 states that shareholders are worse
off if they reject the request than they
latter
circumstance, no request,
To complete our description
leave
them worse
shareholder value
off,
if
is
would have been
if
no request had been made. (Note
not available to shareholders deciding
how
to vote
on a
that this
request.)
of the apparent paradox that shareholders vote for defenses which
we require
a comparison of shareholder value if the request
no request is made. Proposition 4 gives a
is
approved with
sufficient condition for the apparent
paradox to occur.
Proposition 4:
V(m) < W(y*(0
1
0),0
=> 3x? €
According to Proposition
his
maximum
I
m)
(0, oo):
4, if the
S(0
manager's
I
0)
> S(x
utility
I
x?)
> S(0
I
x?),
x = x?.
from employment with the firm
is less
than
gross outside utility before any request for anticipatory defenses, then requests exist
which shareholders
will approve, but
which leave them worse off than they were before. Recall
13
that
we assumed the manager's utility from employment in the fiim was
outside
utility.
The
sufficient condition in Proposition
response to a sufficiently high bid
is to fight
larger than his initial gross
4 describes a manager whose optimal
for a higher
premium, and then
to leave for
higher-valued outside opportunities.
With Proposition
which
4,
we have shown
result in the voting behavior
the existence of requested levels of anticipatory defense
we hoped
to describe.
That
is,
rational, informed,
value-maximizing shareholders will approve the defenses, but are made worse off by them. The
sufficient condition for the result
depends on the manager's
utility in
current and alternative
employment.
We have not yet demonstrated, however, that these levels of anticipatory defense
would
be requested by managers with
in fact
no inconsistency between
utilities satisfying the sufficient
utilities satisfying this
condition and
utilities
anticipatory defense, identified in the Proposition, as a best choice. In
example of a manager with
utility satisfying the
anticipatory defense will be approved
request had been made.
It is
The example
is
in
Appendix B, we provide an
will leave
whose
best choice of
them worse off than
no
if
no sense pathological.
worth noting that under some circumstances, shareholders can be made better off by
implementing some anticipatory defense.
is that
generating the level of
condition of Propostion 4,
by shareholders, and
condition. There is
By
Proposition 3, a necessary condition for this to occur
the manager's request, x?, be strictiy interior to the constraint set;
i.e.
S(x
I
x?)
This amounts to the manager's imconstrained best level of anticipatory defense being
> S(0
I
x?).
strictiy less
than shareholders' most-preferred level."
The primary
is that
implication of our
model
for interpreting the empirical
inferences about whether shareholders vote rationally cannot be
shareholder wealth before and after the vote.
management
as an
unambiguous drop
work which has preceded
made from
a comparison of
We describe the alternative to voting with
in shareholder value, not as a return to the pre-proposal
status quo. Therefore, pre-proposal shareholder wealth
is
it
not the correct benchmark for
14
determining shareholder rationality. Our model suggests that empirical investigation of shareholder
voting and takeover defenses should consider the manager's outside employment opportunities,
and the manager's
4.
skill relative to potential bidders.
Conclusion
We have provided a rational choice model of shareholder voting on anticipatory takeover
defenses. In our model,
it
is feasible for
informed, value-maximizing shareholders to approve
measures which leave them worse off than they were before the measures were requested. What
drives this result
is that
a manager, in requesting such measures, lowers his outside market value.
Consequentiy, the manager's optimal response to an actual takeover bid
rejected than if he
had made no request Shareholders recognize
In the model, the manager's type and utility schedule are
question of why any manager would be hired
which decrease the value of the firm
have
in
mind a
signalling
as a signal, and
we
In other words, our
signalling
different if the request
common knowledge.
game which precedes
is
the
not
To this extent,
two periods we
known with
our model
study.
certainty.
At
is
is
how to vote.
this in evaluating
This raises the
who will wish to implement anticipatory
to shareholders.
perhaps the hiring stage, the manager's type
is
defenses
incomplete.
We
this earlier stage,
The request x?
functions
are implicitly assuming the signal fully reveals the manager's type (and utility).
model
is
predicated on the existence of a separating equilibrium to this earlier
game.
Finally,
we
offer
two remarks.
First,
it is
frequentiy asserted
(cf.
Easterbrook and Fischel
(1983)) that the alienability of ownership claims protects shareholders from detrimental
management entrenchment
tactics.
However, unless shareholders
anticipate the proposal of
takeover defenses by management, they cannot alienate their voting claim in response to the
proposal.
SEC proxy
mailing requirements
demand that
precede the proposal date. Once the proposal
is
the record date for shareholder voting
announced, the constituency
is fixed.
Our model
15
suggests that any drop in share value should occur with the announcement of the proposal, not with
the vote.
Second, our model implies that changes in shareholder wealth associated with voting on
takeover defenses
may
be positive or negative, depending on the manager's type and
schedule. In cases where shareholders' wealth
the manager's request If there
is
is
reduced, the reduction should occur at the date of
any detectable change
at the date
positive. This follows
from Proposition 3 and the voting
shareholders' wealth
unambiguously reduced
is
shareholders' wealth under rejection
interpretation, empirical results
is
utility
if
of the vote,
it
should be
constraint: the first states that
they reject a request; the second ensures that
no larger than
their wealth
under approval.
On
this
which find shareholders' wealth declines following implementation
of anticipatory defenses are not evidence of shareholder irrationality or ignorance.
16
Appendix A: Proofs
Proposition
Proof:
1:
As remarked
and lim.y^^
=
c'(y)
Wi(y*(0
(a.1)
= x? >
Let x
0.
But c" >
first
I-) -
0),
I
To check y*(0 10)^
Wi(y*(0
!•) -
0),
1
I
I
1
I
Wj
>
everywhere,
y*(0 x?), use (3) to obtain:
I
x?
x?),
=
I-)
c'(y*(0
I
c'(y*(0
-
0))
I
x?)).
Then:
x?).
I
I
0.
from assuming
in section 2, the last inequality follows
Suppose y*(0 0) < y*(0
Wi(y*(0
Then: y*(0 0) > y*(0 x?) > y*(x x?) >
0.
Wi(y*(0
<
implies the RHS(a.l)
I
x?), x?
I-)
>
by
0,
W^ ^
<
0,
W12 <
To check y*(0
0: contradiction.
I
and x? >
x?)
> y*(x
I
0.
x?), again use the
order condition (3) to get:
Wi(x +
(a.2)
y*(x x?), x?
I
Suppose y*(0 x?) < y*(x
I
implies LHS(a.2)
Remark
[x
1:
Then
1
1:
I
I
X?)]
Wi(y*(0
> y*(0
Let x = x? >
I
iff
>
c"
0: contradiction.
y*(0 0) = y*(0 x?)
+ y*(x
Corollary
<
x?).
I
I-) -
I
x?),
x?
1-1
=
c'(y*(x
implies RHS(a.2)
0.
x?))
-
c'(y*(0
But x = x? >
I
x?)).
0, so that
II
W12 = 0.
Hence, x = x? >
and
W12 =
imply:
0).
0.
Then
Proof: Use (3) and Proposition
1
[x
+ y*(x
with
Wj j
I
<
x?)]
0.
> y*(0
1
x?).
II
Before proceeding to establish the remaining Propositions,
following comparative
>
I
it is
convenient to report the
statics.
(a.3)
X = X? => dy*(x x?)/dx? =
(a.4)
dy*(0 x?)/dx? = -Wi2/[Wi
(a.5)
x = X? =>
I
I
dco(x, y
I
x?)/dx?
-[Wn+^U^/TWl r^"! < ^^
^-c"]
<
0,
with the inequality
= W^ + W2, which
strict iff
W12 < 0.
a priori has ambiguous sign.
Wj ^ <
17
Note
that at
y*(x x?)
is
I
given by
any
y,
I
(a.5)
I
I
<
W2 < 0.
=
x?)/cix?
I
(a.6)
(a.4).
If
is
0,
Proof: Let X = X? >
0.
as c"(y*(x
x€
{0, x?}.
I
Then, n(x
dA*/d\l = -W2(x+y*(xlx?), x?
A* can change
and we have, V(m) <
X?)
iff
V(m)
-
X
<
sign at
(>)
I-)
=
x?)
I
+ m. And
0, then
since
an*(0 x?)/9x?
I
is
dy*(xlx?)/ax?. Substituting from (a.3) and
(>) x.
=
x?)
is
(<)
x?)
is
differentiable in x?;
>
x),
I-)
=
=
-Wi2(x+y*(x
[v(n*(x
I
and using the
x?),
x
x?),
I
I-) -
x?
I-).
and an(x x?)/ax? >
I
o)(x,
y*(x
I
x?)
I
0.
x?)].
first-order condition (3) gives,
0.
x?), x?
I
Consider x <
I
I
>
most once as x
W(x+y*(x
W(n(x x?)-m, x?
In this case, n(x
= x + y*(x
x?))
t
By
x.
increases,
= x? such
negative to positive. Suppose there exists an x
I
+
1
Consider the difference. A*
Differentiating this with respect to x? (at x?
n*(x
x?)
we obtain:
Let x? >
Therefore,
I
differentiable in x?. If x
I
I
1:
x?)
I
x = x?, then an*(x x?)/ax? =
an*(x x?)/0x > (<)
(a.7)
By definition, n*(x
I.
differentiable in x?, n*(x
collecting terms,
Lemma
y
dco(0,
(a.6)
iff
I-)
(6),
that
x < (>)
n(x
I
x?)
and
x.
is
this
change can only be from
A* = 0. Then
So by
this
value
is
definition (6), n(x
I
unique
x?)
<
(>)
implicitly defined by,
0.
differentiable because
W
is differentiable.
In particular,
(a.8)
an(xlx?)/9x? = -W2(n(xlx?)-m, x?l-)AVi(n(xlx?)-m,x?IO;
(a.9)
lim.x_^ x- [5n(-l-)/3x?]
Vx <
x,
and,
Now consider x > x. By
V(m)
-
c(n(x
(6),
x?)-x-m)
I
-
Again, differentiability of n(x
(3),
we have
(a.lO)
that
an(x
Vx >
I
n(x
I
(0(x,
I
= -W2(n*(x x?)-m, x?
1
x?)
I
x?)-m, x?
I-)-
implicidy defined by,
is
y*(x
l-)/Wi(n*(x
I
x?)
x?) follows
I
x?)
from
=
0.
differentiability
of c and
W.
In particular, using
x,
x?)/8x? = [c'(n(xlx?)-x-m)
-
Wi*(x)
-
W2*(x)]/c'(n(xlx?)-x-m),
18
where Wi*(x) = Wj(n*(xlx?)-ra, x?
lim.x_^ x+ [3ll(-IO/3x?]
(a.l 1)
=1,
I-), i
=
Hence,
2.
[c'(n*(xlx?)-x-m)
Wi*(x)
-
= -W2(n*(xlx?)-m, x?
from another application of
the second equality following
-
W2*(x)]/c'(n*(xlx?)-x-m)
l-)AVi(n*(xlx?)-m, x?
Together, (a.9) and
(3).
argument for n(x x?) being differentiable everywhere when x = x?
the
n(x
and
I
I
X?)
> x+m, Vx, x? >
Let x
(a.9).
Since
A* >
this
x,
and consider
<
0, n*(xlx?)
>
x?)/ax?
(a.lO).
By
for
x = x? < x
all
Wi*(x) =
(3),
>
n(xlx?). Therefore, c"
Wj+W2 < Wj.
all y,
11) complete
from section
immediate from
is
=
c'(y*(xlx?))
2,
(a. 8)
c'(n*(xlx?)-x-m).
implies,
Therefore, the numerator of (a.lO)
Lemma for x = x?. Now
completes the proof of the
reasoning as before gives n(0
I
Lemma
2:
(a.7)
and
Lemma 2:
Let x
= x? >
Proof: (a)
V(m) < W(x+y*(x
cases.
suppose x =
imply that U(x
1
I
I
positive. Since
and x? >
>
x?)/3x?
0.
c'
>
0,
Similar
follows on implicit
II
Then, n(x
0.
is strictly
x?) differentiable in x?, and 3n(0
two
differentiation of (6) for the
Remark
I
(recall that
(a.
c'(n(xlx?)-x-m)>Wi*(x)>0.
(a.l2)
Moreover,
>
That an(x
0).
I-);
I
x?), x?
x?)
I-)
I
x?) and S(x
> n(0
I
x?) are differentiable in x?.
I
x?);
=> V(m) = W(n(x x?)
I
-
m, x?
I-)
by
(6).
There are two
possibilities:
(a.l)
V(m) < W(n*(0
=> W(n(x X?)
I
=> n(x
(a.2)
I
X?)
-
1
m, X?
= n(0
I
By definition
-
I
I
m, x?
!•)
x?)
m, X?
of y*(0
-
=> V(m) = W(n(0 x?)
!•)
I
= W(n(0
I
x?)
-
m, x?
-
m, x?
I-),
by
(6)
!•)
x?).
I
V(m) > W(n*(0
=> W(n(x X?)
x?)
x?).
-
m, x?
!•) -
I-)
c(n(0
I
=> V(m) =
x?)
-
m) =
[q)(0,
co(0,
y*(0 x?)
y*(0
I
I
x?)
I
I
x?)
x?).
+ c(n(0
I
x?)
-
m)], by (6)
.
19
y*(0 X?)
a)(0,
I
I
> W(n(0
X?)
I
x?)
-
m, x?
x?)
-
m) > W(n(0
c(n(0
I-) -
I
x?)
-
m).
Therefore,
W(n(x
I
X?)
m, x?
-
c(n(0
-
I-)
I
I
x?)
-
m, x?
I-) -
c(n(0
I
x?)
-
m)
=>n(xlx?)>n(Olx?).
This proves the proposition for case (a).
(P)
V(m) ^ W(x+y*(x
By CoroUary
and x? >
1
W(n*(Olx?)-ni, X?
(a.13)
V(m) =
(0(0,
=> V(m) =
!•)
W(x+y*(x
0,
So by
I-).
x?
x?),
I
I
x?), x?
y*(x x?)
co(x,
I
> W(y*(0
I-)
I
I
x?),
+ c(n(x x?)-x-m), by
x?)
x?
I
(6).
Hence, V(m) >
!•).
(6),
y*(0
I
x?)
I
x?)
+ c(n(0 x?)
I
m).
-
Therefore,
(a.
By
14)
y*(x
co(x,
Proposition
x?)
I
I
x?)
and Corollary
1
-
1,
15)
n(0
X?)
I
> n(x
I
X?)
-
I
LHS(a.l4)
likewise be strictiy positive. Since
(a.
y*(0 x?)
co(0,
c'
>
x?)
I
=
is strictiy
and x = x? >
c(n(0
I
x?)-
m)
-
c(n(x
positive; hence, the
I
x?)-x-m).
RHS(a.l4) must
0, this implies,
X.
Now implicitiy differentiating (a. 13), we obtain Vx? > 0:
an(0 x?)/ax? = -W2(y*(0lx?), x?
I
Using
(a.
10),
(a.l6)
we have Vx? >
[an(x
+
By
(a. 12),
this
term
-
9n(0
I
x?)/ax?]
=
{
1,
-
8)
[W2(x+y*(xlx?), x? l-)/c'(n(xlx?)-x-m)]}.
W2 < 0, and W12 ^ 0,
> W2(y*(0
<
Vx? >0,
positive for y in the neighborhood of zero
I
x?),
•!•)
> W2(x+y*(x
I
By(a.l5)andc">0,
(a. 1
0.
[Wi(x+y*(xlx?), x? l-)/c'(n(xlx?)-x-m)]}
1 -
{[W2(y*(0lx?), x?l-)/c'(n(0lx?)-m)]
is strictiy
17)
x?)/ax?
>
0:
the first term of (a. 16) in {•) isnonnegative
By CoroUary
(a.
I
l-)/c'(n(Olx?)-m)
c'(n(x
I
x?)-x-m) < c'(n(0
I
x?)-m).
x?),
•!•).
.
and, because
lim.y^Q
=
c'(y)
Consider the second term
in
{
•
}
0,
20
Together,
(a.
and
17)
and, as with the
= 0, both terms
[n(x
I
x?)
-
term of
first
in
{
•
n(0
I
18) imply that the second term of
(a.
16), stricdy positive for
(a.
Vx? ^
vanish. Therefore,
}
=
x?)]
J[8n(r
r)/dr
I
-
dniO
1
(a.
y
16) in
in the
{•) is
Vx? >
also nonnegative
neighborhood of zero.
When
x?
0:
r)/ar]dr
> 0,
o
as required (with equality iff x?
Remark
3; Notice that
from [x=x? X? >
I
Remark
0] to
I
S(0
I
II
Lemma 2 is not implied by Lemma
[x=0 X? >
I
1: this is
because the change
in state
0] is not incremental.
the claim
made
2
in section
x?), then there exists a perturbation in x? -- say, x?
—
that, if
such that S(x~
x
I
= x?
x?~) >
x?~).
Lemma 3;
For any x? >
Proof: Given x
in n.
I
0).
Remark 2 and Lemma 2 justify
4: Together,
and S(x x?) = S(0
=
=
0, v(n, x
By W2 < 0, W(y,
x?
Combining Lemmas
Proposition 2:
(a)
n(x
(b)
Proposition 3:
Proof:
being
n(0
0,
< W(y,
I-)
2 and
x?)
3,
Vx = X? > 0,
3,
made given x? >
n(0
I
x?)
I
0).
1
c(n-m). Since
all
y>
0.
c'
>
0, v(n,
So, by (6), n(0
x m)
I
I
x?)
is strictiy
> n(0
1
0),
decreasing
Vx? >
0.
II
we have proved:
n(x
0)
-
I-),
increasing in x?, x e {0, x?}; and,
is strictiy
Forx?>0, S(0
By Lemma
> n(0
m) = V(m)
I
1,
I
x?)
I
I
> S(0
> n(0
I
is strictiy less
> n(0
X?)
0).
I
x?)
> n(0
1
0).
x?).
I
Hence
than
tiie
probability of a successful takeover bid
when x? =
0:
21
(a.
By
[1-F(n(0
19)
Proposition
1,
n*(0
Therefore, by (8),
(a.20)
And if n*(0
x?)
I
n(0
>
0)
I
< n*(0
I
in section 2, only types n
(a.21)
7t(n,
Vn < n(0
Likewise,
(a.22)
7i(n,
By
Proof:
say X?
>
S(x
By Remark
(a.23)
n(-l-)
:t(n,
I
7i(n,
(a. 19) -
assumption,
X?)
make takeover bids.
x?)
=
I
I
x?)
Wj^ <
is
•)
=
1
I
I
x?)
By
x?).
e (n(0
I
an argument
n(0
0),
I
x?)),
x?), x?
I
x?))]-[n*(x
I
II
= m.
W2 = -«>.
and lim.j^9_^^
I
x?))]-an*(x
I
and the assumption
positive for
an*(-l-)/ax?
By
all finite x?.
<
I
I-).
Therefore, for sufficientiy large x?,
Hence, by
(10),
Vx? >
x?*,
x?)-m].
>
for all x?
I
x?)/ax?
[1-F(n(x
I
-
Vx? >
{f(n(x
<
x?)/3x?
(a.7), the first
x?'.
for
x?',
I
x?))
x?))-an(x
all finite
I
<
By
(7)
and
aU
I
x?)
(a.3),
-
x?',
{
•
is
finite
if:
< an(x x?)/ax?-[n*(x
I
x? > x?" >
for
I
x?)/ax?
x?)/ax?-[n*(x
m), the term in
Then, aS(x x?)/ax? <
I
I
term on the RHS(a.23)
for all x? sufficientiy large.
x?))]/f(n(x
I
V(m) > W(0,
that
Therefore, for sufficientiy large x?, aS(,x
(a.24)
Vn
I
I
suppose an*(x x?)/ax? >
1.
0).
aS(x x?)/ax? =
By Lemma
Suppose
I
0.
We prove the Lemma by showing 38 (x
is strictiy
Therefore,
differentiable in x?. In particular,
[1-F(n(x
large.
any y < y*(0
0.
(a.22) yieldthe desired result
m + [1-F(n(x
2, S(-
at
Vn > n*(0
is strict
will
we have V(m) > W(x+y*(x
=
Wj2 <
x?).
Let x = x?. Then, lim.x9_^^ S(x
X?',
I
I
then the inequality in (a.20)
0),
strict iff
x?),
I
7t(n,
>
with the inequality
0),
I
0))].
1
0),
I
010) =
Together, (10) and
Lemma 4:
>
0)
I
[1-F(n(0
< n*(0
x?)
I
Vn >
;t(n,
<
x?))]
I
m].
}
x?)-m]}.
I
x?" sufficientiy
on the RHS(a.23)
of ambiguous sign.
x? >
x?*.
sup.[an*(x
I
Now
x?)/ax?]
=
22
By
assumption, lim.j_^oo [[l-F(t)]/f(0] <
lim.^9_^^[LHS(a.24)] <
(a. 10),
therefore, 3n(x
Vx? >
X?'.
Hence,
y>
(7),
>
x?)/8x?
1
Wj j
Again by assumption,
oo.
< W(y*(0
0),
1
=> 3x? €
By Remark
Proof:
W(y*(0
=
n.
I
X?*), X?*
Hence
I-)
(0, 00)
[1-F(n(x*
I
x?*))]
X?*) > n*(0
I
I
=
x?*)
= x?*)
I
botii
hypothesis, 3n*(x
^
x?"
-
x?'
such that 3S(x
(10).
II
the
is
I
^ S(0
x?)
I
x?),
x=
1
V(m) < W(x*+y*(x*
Then by case
.
[1-F(n(0
>
--
S(0 0) > S(x
:
such that
obtain (x*
By
W2 = -°°:
I
x?)/9x?
I
(a.26)
Given
by
(n,
Vn >
n*(0
(a. 1)
=
x?*))]
=
{x? >
X?.
S(x
I
I
X?*), 7t(n, x*
(i)
Px?
e
C
(ii)
[Vx? €
C
X?)
X? <
:
:
> S(0
I
I
x*
I
2,
I-),
n(x*
I
and V(m) <
x?*)
x?*) >
X = X?}
I
I
x?*)
the request x?* is accepted:
follows from Corollary
=
:c(n,
Jt(n,
7i(n,
x?*)
9t
= n(0
[1-F(n)].
x?*)
x?*) =
I
that S(x*
X?),
for
1,
and the second
I
I
x?*)
I
x?*)
= 0, Vn <
= n-m
By
n.
only types
2,
(8),
•,
x?*).
> S(0
I
x?*). Therefore,
0.
now two cases:
There are
Lemma 4 and
7t(n,
I
I
x*
bid: hence, 7t(n,
n*(0 x?*)],
0,
x?.
x?*), x?*
I
of Lemma
same whether or not
n: the first inequality
and (a.26) imply
(10), (a.25)
C=
Let X
Vn e
>
<
x?)/9x?
from the premise of the Proposition and the choice of x?*. By an argument of section
n > n will make any takeover
from
m)
1
the likelihood of takeover
(a.25)
Also, n*(x*
3x?* >
2,
and lim.x9_^^
Lemma follows from
allx?>x?". Since lim.^^^[l-F(t)]=0, the
and x = x?,
1
imply that lim.x7_^[RHS(a.24)] = «.
x?, togetiier
Therefore, there exists a sufficientiy large value of x?
Proposition 4; V(m)
<
for sufficienUy large x?.
1
and x =
0,
Lemma
Therefore, by
'^^
X?
<
00
00
&
,
S(x
S(x
S(0 0) > m.
I
I
I
X?)
X?)
= S(0
> S(0
To check
hypothesis. Hence, by (6) and y*(0
1
0)
I
I
x?)]
x?)]
=> [S(0 0) > S(x
1
=> [3x?'e
C
:
x?'
this last inequality, recall
>
(Proposition
1),
x?)],
I
by Proposition
& S(0
< 00
1
0)
V(m) < W(y*(0
m < n(0
I
0)
<
«>
>
1
S(x'
I
3;
x?')],
I-)
0),
by
and n*(0 0) > m.
I
II
23
Appendix B: Example
We claim in section 3 of the text that Proposition 4 allows us to infer the existence of
managerial
utilities
under which the manager would ask
anticipatory defense, x?, such that S(0
1
0)
> S(x
I
for,
and shareholders approve, a
x?). In this appendix,
level of
we justify our claim with
an example.
In the interests of computational simplicity, two assumptions of the
example:
viz.
the support of
F
is
model
not the whole real line, and lim.y^Q c'(y)
are violated in the
^ 0.
We argued that
both of these assumptions are technical conveniences, and the example here supports our case.
Also, there are choices of the distribution function
assumptions of the
and which are
text,
F and
arbitrarily closely
the cost function c(y) that
do
satisfy all the
approximated by the functional forms
exploited in the example.
F be uniform on
Let
the manager's type is
V(m) =
the closed interval [p, q].
m = [p+q]/2 = 0; and
.
I
=
1^9
=
[x+y]'^'^
As we demonstrate
is 1.
Without
Therefore,
[x?2]/2
-
1;
and assume
L>
4.
Assume
let,
+ m,
whence,
y*(xlx?)= l-x,Vxe
=
X?
p,
y.
Then, 3co/3y
(b.l)
-
3/2,
W(x+y, X? m) = 2-[x+y]l/2
c(y)
We write L = q
Vx e
later,
0,
[0,1]
Vx>l.
the manager's unconstrained (and constrained) utility-maximizing value of
loss of generality, then,
[0, 1],
(b.2)
n*(xlx?) =
(b.3)
an*(x x?)/ax? =
I
l,
0.
we assume x e
[0, 1].
2i«
When
< W(x+y*(xlx?
V(-)
n(x
!•),
Hence, given V(-) < W(x+y*(x
=
+
n(x
(b.5)
an(x x?)/ax? =
X?)
[3
x?), x?
I
+
[3
I
Suppose X? = X =
I
co(0,n*(OIO)IO) =
(b.7)
n(0 0) = 9/16 and 9n(0
(b.8)
co(0,
2-l =
I
0)
We also have, d(x)(\, y
I
0)
I
=
-
[x+y]-^/^
n*(0 0)-m 0)/dxl =
(b.lO)
a(o(0,n(OIO)-mlO)/9x?=4/3.
at
x? = x =
=
W(n(x x?)-m, x?
-
I
I-)
=
0.
so, V(-)
= 3/2 < W(y*(0
I
0),
I-)
=
2.
Hence,
I
0,
(9/16)
aco(0,
I
and
1
0)/ax?
(b.9)
Hence,
V(-)
l,
= 2^(9/16)
x?)/ax?
I
(cf. (6)),
!•),
Then, y*(0 0) =
0.
I
given by
(x?)2].x?/4.
(b.6)
n(0
is
(x?)2]2/i6,
(b.4)
I
x?)
I
.
=
x?.
15/16.
So, at x?
=x=
0:
1,
0:
(b. 1 1 )
Jaco/ax?-f (n)dn
(b.l2)
lco-f(n)dn
=
[ 1
-
4/3]/L
=
-
1/3L,
and,
From
(9),
(b.i3)
Here, CO*
=
taking
=
[1
-
15/16]/L = 1/1 6L.
5=1:
[au/ax?]ix=x? = f(n)-an/ax?-V(-)
co(x,
n*-m
x?) and
I
+
[F(n*)-F(n)]-[laco/ax?-f(n)dn
+
[f(n*)-an*/ax?
+
[l-F(n*)]-aco*/ax?
a = co(x, n-m
I
-
Given 5=1, dU/dxl >
Hence,
co*-an*/ax?
-
io-an/ax?]
f(n)-an/ax?]-jco-f(n)dn
-
f(n*)-an*/ax?-(0*.
x?). Substituting
[au/ax7]lx=x?=0 = -[(l-p)/L-(9/16-p)/L]/3L +
=
+
from
(b.l)
-
(b.l2) gives,
[1 - (l-p)/L]
7/48L]/L.
[q
- 1 -
at
x = x? =
iff
[q
- 1 -
7/48L] >
0.
By
construction,
L = 2q >
0.
25
[aU/ax?]lx=x?=0 >
q2-q-7/96>0
*=>
<» q>1.07.
Therefore, because
L>
But when x? = x =
0, V(-)
Similarly,
(b.l4)
from
4 by assumption, the manager's
< W(y*(0
(10), taking
[9S/ax?]lx=x?
=
0),
I
I-):
hence
utility is
increasing in x at zero
+
Consider x? = x =
=
(b.l5)
V(-)
(b.l6)
n*(l
(b.l7)
and
(b.l8)
[V(-)
Given
(b.l 5),
au(i
1
from
-
f(n)-9n/ax?]-J[n-m]-f(n)dn
=
1)
n(l
l)/ax?
I
-
co(l,
[aU(l
1
Then from
=
1)
I
y*(l
= n(0
from
1,
I
I
1)
l)/ax?]l5^i
I
\n-m\-dn/dx?]
f(n*)-an*/9x?-[n*-m].
= x? =
(b.l2) at x
-
-
-
0, gives:
0.
some
= W(l+y*(l
3/2
I
i)/ax?
1.
prefer
[l-F(n*)]-an*/8x?
(b.l)
lx=x?=0 = [7/16L]2 >
So shareholders too would
anticipatory takeover defense, relative to having none.
y*(l
(b.l),
1), 1
1)
1
I
1)
=
0.
Hence,
!•),
=
1,
from
(b.2)
and
(b.4),
(b.5),
= c(y*(l
1)]
= [au(l
=
f(n)-an/ax?-[v(-)
=
[l-F(n)]-aco*/ax?,
-
I
1))
=
0.
l)/ax?]l5^Q. Hence,
1
from
(9),
w*] + [i-F(n)]-aco*/ax?
by
(b.l 8).
Substituting for ao)*/ax? gives,
au(i ii)/ax? =
o.
Furthermore, by (b.l7), (b.l 8), and
d2u(l
I
l)/ax?2
=
F uniform,
[l-F(n)]-a2(o*/ax?2
= -3-[q-l]/2L <
Therefore, x
= x? =
1 is
1.
5=1.
+ [F(n*)-F(n)]-[F(n*)-F(n)+[n*-m]-an*/8x?
[as/ax?]
5=
5=1:
[f(n*)-an*/ax?
Substituting where appropriate
if
-
f(n).aco*/ax?
0.
an unconstrained
maximum for the manager
(indeed,
it is
a global
26
maximum).
Proceeding similarly,
aS(l
I
=
l)/ax?
we
obtain ftxjm (10),
[l-F(n)]-an*/ax?
f(n)-an/ax?-[n*-m].
-
Substituting,
aS(l ll)/ax? = -l/L <
0.
Therefore, the shareholders' most prefeired value for x
To check that
the manager's unconstrained optimum, x
and substitute from above to
S(l
I
1)
is strictly
=
[1
=
[q-l]/2q
=
less than the manager's.
1, is
in the constraint set,
we
use (10)
find:
(l-p)/L]
-
= S(0I1).
Therefore, x?
=
surely will be.
1
will be
By
Proposition 3, S(0
S(0 0) = [(l-p)/L
I
=
approved by the shareholders
-
(9/16
-
II
0)
p)/L]-[l
[7/32q]2 + [q-l]/2q
>S(1I1).
I
-
> S(l
I
if it is
requested by the manager
1): in particular,
9/16]/L +
[1
-
(l-p)A.]
~
as
it
27
Footnotes.
1.
DeAngelo and Rice (1983) provide
2.
The use of the term
"type" differs
a thorough discussion of these hypotheses.
somewhat from the usual game-theoretic concept:
"type" indicates a skill level, so that managers of the
Further, allowing infinite
same "type" can have
here,
different utilities.
m is only a technical convenience; a claim we justify via the worked
example of Appendix B.
3.
For example, the uniform, normal,
gamma
and exponential distributions
all satisfy this
restriction.
similar to that used in
4.
This structure
5.
This type of "take
context by
6.
The
level of
is
it
or leave
Romer and Rosenthal
it"
Grossman and Hart (1980).
agenda control has been extensively studied
(1978, 1979).
role of the assumption that lim.y_^Q c'(y)
=
0, is to insure that
explicitiy: this
adds very
assertion is vindicated
for every y
7.
>
strictly positive
Notice
littie
and does not substantively
by the example
in
Appendix B,
in
we have
alter
to consider the
our main
which
results.
comer case
Again,
this
— for computational ease — c'(y) =
0.
that,
because lim.y_^Q
the neighborhood of y
8.
some
y will always be chosen by the manager, whatever the value of x. Consequentiy, we can
use the calculus in our analysis. Without the assumption,
1
in a political
=
c'(y)
=
by assumption, we must have c"(y) >
at least in
0.
Although the model
is
not game-theoretic, the notion of shareholders evaluating the
proposal against what would occur
if
they reject the manager's request
is
closely related to the
game-theoretic idea of (subgame) perfecmess (Selten, 1975).
9.
In the
example of Appendix B, shareholders
However, the manager wants
still
more.
in fact
want some positive
level of defense, x.
28
References
DeAngelo.H. and Rice,E., Antitakeover charter amendments and stockholder wealth, J.Financial
Economics
11 (1983) 329-360.
Easterbrook.F. and Fischel,D., Voting in corporate law,
J.Law and Economics 26 (1983)
395-427.
Grossman,S. and Han,0., Takeover bids, the free-rider problem, and the theory of the
corporation, Bell J.Economics, Spring (1980) 42-64.
Jarrell,G.
and Bradley,M., The economic
offers,
Jarrell,G.,
effects of federal
and
state regulations
of cash tender
J.Law and Economics 23 (1980) 371-407.
Poulsen,A. and Davidson,L., Shark repellents and stock prices: the effects of
antitakeover
amendments
since 1980,
mimeo. Office of the Chief Economist,
SEC
(1985).
Linn,S. and McConnell,!.,
on
common
An empirical
investigation of the impact of 'antitakeover'
stock prices, J.Financial
Romer,T. and Rosenthal,H.,
Economics
amendments
11 (1983) 361-399.
Political resource allocation, controlled
agendas and the status quo.
Public Choice 33 (1978) 27-43.
Romer,T. and Rosenthal,H., Bureaucrats vs
by
direct
Selten,R.,
voters:
on
the political
economy of resource
allocation
democracy. Quarterly J.Economics 93 (1979) 563-587.
Reexamination of the perfectoess concept for equilibrium points
games, International J.Game Theory 4 (1975) 25-55.
in extensive
form
991
I
039
'V,\|
Date Due
Lib-26-67
3
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