Irreversible Investment, Capacity Choice,
And the Value of the Firm
Robert S. Pindyck
Sloan School of Management
Massachusetts Institute of Technology
WPl802-86
June 1986
IRREVERSIBLE INVESTMENT,
CAPACITY CHOICE. AND THE VALUE OF THE FIRM*
by
Robert S. Pindyck
Massachusetts Institute of Technology
Revised:
June 1986
*This paper was written while the author was Visiting Professor of Economics
and Fellow, Institute of Advanced Studies, at Tel-Aviv
University. I am
grateful to the National Science foundation for research support under Grant
No. SES-8318990, and to Saman Maid, Julio Rotemberg, and Lawrence Summers
for helpful discussions and comments.
IRREVERSIBLE INVESTMENT.
CAPACITY CHOICE,
AND THE VALUE OF THE FIRM
by
Robert S. Pindyck
ABSTRACT
A model of capacity choice and utilization is developed
consistent with value maximization when investment is irreversible
Investment requires the full
and future demand is uncertain.
value of a marginal unit of capacity to be at least as large as
its full cost. The former includes the value of the firm's option
not to utilize the unit, and the latter includes the opportunity
We show that for
cost of exercising the investment option.
moderate amounts of uncertainty, the firm's optimal capacity is
much smaller than it would be if investment were reversible, and a
large fraction of the firm's value is due to the possibility of
We also characterize the behavior of capacity and
future growth.
capacity utilization, and discuss'implications for the measurement
of marginal cost and Tobin's q.
1.
Introduction.
When
investment
uncertain, a
firm's
"killing," of
is
irreversible
investment
an option
and
using
expenditure
that
information
rule "Invest when the marginal value
the
Instead the
purchase
marginal
installation cost
and
value
by an
the
conditions are
exercising, or
the
and cost
to re-evaluate the desirability
value must
be included
As a result, the standard investment
of a
unit of
installation
of
One gives up
evolving demand
This lost option
as part of the cost of the investment.
as
demand
involves
information (about
and/or timing of the expenditure.
large
future
(the option to productively invest).
the option of waiting for new
conditions),
and
unit
amount equal
capital is
at least as
cost of the unit" is not valid.
must
to the
exceed
value of
the
purchase and
keeping the firm's
option to invest alive -- an opportunity cost of investing.
This aspect of investment has been explored in
and most
notably in
show that with even
opportunity cost
the recent
moderate
can be
an emerging literature,
paper by McDonald and Siegel
levels
of
uncertainty,
the
(1986).
value
They
of this
large, and an investment rule that ignores it will
be grossly in error.
Their calculations, and those in the related papers by
Brennan and
(1985)
Schwartz
and Maid and Pindyck
cases projects should be undertaken only
when
(1985), show that in many
their
present
value
is at
least double their direct cost.'
The existing
literature has
been concerned
involving discrete projects, e.g. whether
examines the
implications of
large a factory to build.
decision.
This
to build
with investment decisions
a factory.
This paper
irreversibility for capacity choice, e.g. how
In particular I focus on
the marginal investment
provides a simple and intuitively appealing solution to the
- 2 optimal capacity problem, as well as insight into the sources
of
the
firm's
value.
In addition,
and evolution
it clarifies issues related to the
measurement of long-run marginal cost, and
the interpretation
and measure-
ment of Tobin's q.
A
firm's
capacity
choice
is optimal when the value of the marginal
unit of capacity is just equal to the total cost
cost includes
involves
two
this we
the unit.
steps.
capacity must be determined,
To do
First,
An analysis
the
given that
This total
value
the firm
of capacity choice
of a marginal unit of
already has
capacity K.
must account for the fact that if demand unexpectedly falls,
the unit of capacity may be unutilized.
invest in the marginal
unit must
Second, the value of the
option.
option to
be determined (it will depend in part on
the value of the marginal unit itself), together with the
exercising the
unit.
the purchase and installation cost, plus the opportunity cost
of exercising the option to buy
therefore
of that
In essence, this
decision rule for
decision rule is the solution to
the optimal capacity problem.
Because a marginal unit of capacity need not
more
when
demand
fluctuates
stochastically.
be utilized,
This might suggest that the
firm should hold more capacity when future demand is uncertain,
the opposite
is true.
The reason
is that
opportunity cost of exercising the option
Although the
value of
the marginal
increases even more, so the net
capacity. 2
Indeed,
for many
such that firms should
models would suggest.
hold
to
far
invest
is to
product markets
less
but in fact
uncertainty also increases the
unit increases,
effect
it is worth
reduce
in a
this opportunity cost
the
the volatility
capacity
than
marginal unit.
firm's optimal
of demand is
standard investment
This model
choice also has implications for the valuation
of capacity
The value of a firm has two components:
of firms.
capacity, and
model
Solutions of the
the future.
suggest that
for typical
of installed
firm's options to install more capacity in
of the
the value
the value
plausible
parameter values
options" should
account for more
based
firms, "growth
on
that half (and in some cases much more than half) of market value.
have to
stresses the
cited above,
This paper, like others
productively invest.
These options are important assets of firms.
Firms hold them even if they are price-takers in product
and they
ownership of land or natural resource reserves.
resources and
possibly scale,
all of
But more general-
expertise, reputation,
market posi-
have been built up over time,
which may
undertake investments
enable it to productively
What gives
particular production techno-
It may be a patent on a
ly, a firm's managerial
tion, and
and input markets,
can account for a good fraction of their market value.
firms these options?
logy, or
options that firms
that individuals
or other
firms cannot undertake.3
The
next
section
lays
out
a
model of capacity choice with
simple
irreversible investment, based on the assumption
market value.
It differs
from previous
that firms
maximize their
models of this type, e.g. Brennan
and Schwartz (1985) and McDonald and Siegel (1986), in that the focus
the marginal
investment decision,
discrete project.
show how
rather than
is on
the decision to invest in a
As the model is developed, a numerical example is used to
the marginal
value of capital, the opportunity cost of investing,
and the firm's optimal capacity depend on current demand and uncetainty over
future demand.
Remaining
sections of the paper use the model to study the
value of the firm, the behavior
of capacity
and capacity
utilization over
- 4 time, and implications for the measurement of marginal cost and Tobin's q.
2.
A Model of Capacity Choice.
Consider the
investment decisions
of a
firm that might have monopoly
power, and faces the following demand function: 4
P = 8(t) -
Q
(1)
(The firm might be a price-taker, in which case
= 0.)
Here 8(t) evolves
over time according to the following stochastic process:
where dz
s(t) a
Eqn.
8dt +
=
d
is the
serially
(2)
says
dz
(2)
increment of a Weiner process, i.e. dz =
(t)(dt)12 ' , with
uncorrelated
random variable.
that
the
and
current
normally
value
of
distributed
(and thus the current demand
function) is known to the firm, but future values of 8 are
lognormally distributed
Thus
with a
even
though
information
changing),
future
demand
is
unknown, and are
variance that
grows with the time horizon.
arrives
time
always
over
uncertain,
(the
as
firm
is
observes 8
the case in most
real-world markets.5
Denote by Q*e
suppose some
the correlation of
asset or
portfolio of
perfectly correlated with 8, so that
8
with -the
market
portfolio.
Now
assets exists with a stochastic return
if x
is the
price of
this asset, it
evolves according to:
dx =
By the
xdt +
xdz
CAPM, the expected return on this asset must be
% is the market price of risk.
tage rate
of change
of 8,
We will assume that
is less
than p.
that if this were not the case, no firm in
,
= r + fe.ma, where
the expected percen-
(It will
become clear later
the industry
would ever install
- 5 any capacity.
No matter
what the current level of
, firms would always be
better off waiting and simply holding the option to install
capacity in the
future.)
=
The
Denote the difference between
firm's
to
.-
initially
and over time, its
optimal investment spending on new production
capacity.
I
with no
is
-
determine,
firm starts
problem
and c by 8, i.e.
assume that the
capacity, so that at t = 0 it decides how much initial
capacity to put in place.
Later
it
may
or
may
not
add
more capacity,
depending on how demand evolves.
For simplicity
I assume
and capital in place
important
one
--
does not
is
capital in place can
small because
it.
that new capacity can be installed instantly,
that
depreciate.'
by one
no alternative
Thus a factory built to
assumption --
investment is irreversible.
be sold
it has
Another
firm to
and an
That is, although
another, its
scrap value is
use than that originally intended for
produce widgets
can only
be used
to produce
widgets, so if the demand for widgets falls, its market value will fall.
The
fact
that
investment
opportunity cost associated with
forecloses the
irreversible
adding capacity
implies that there is an
-- adding
capacity today
possibility of adding it instead at some point in the future
(or never at all).
install
is
capacity
Put another
at
various
way,
the
firm
points
in
the
currently
has
options to
future (options that can be
exercised at the cost of purchasing the capital), and by installing capacity
now, it closes those options.
The optimal rule is to invest until the value
of a marginal unit of capital is equal to its total cost -- the purchase and
installation cost, plus the value of the option on the unit.7
A few more details are needed to finish specifying the model:
unit of capital can be bought at a fixed price k per
unit;
(ii)
(i) each
each unit
- 6 of capital
in place provides the capacity to produce one unit of output per
time period;
(1/2)czQ2 .
and (iii)
In
general c
0), and
if a
K*(e) = (8 - c
8(K)
>
(2Y+c
has
and/or
of comparison,
For purposes
=
firm
2
an
operating
cost
C(Q)
=
cQ +
can be zero, but if Y = 0 (so the firm
we require C 2 > 0 to bound the firm's size.
is a price-taker),
(
the
0,
note that
if future
initial capital stock would be
the firm's optimal
-
rk)/(2Y+c
2 )K
+ c, + rk.
2
), i.e. the
We
uncertainty over future demand,
firm should
will
demand were certain
see
only if
with moderate amounts of
that
the optimal
add capacity
is considerably
capital stock
smaller than this.
The Value of a Marginal Unit of Capacity.
To solve
the firm's investment problem we first determine the value of
an incremental unit of capacity.
capacity K,
we want
to find
That is, given
V(K).
K.)
the present
value of
profits from the marginal unit.
Because
is just
firm already has
the value to the firm of an incremental unit,
which we denote by
This
that the
(Note that
V(K) is a
function of
the expected
the
unit
as
well as
flow of incremental
does
not
have
to be
utilized, future incremental profits are a nonlinear function of 8, which is
stochastic.
In particular, given
a
current
capacity
K,
the incremental
profit at any future time t resulting from a marginal unit of capacity is:
7t (K) = max [0, (t - (2Y+c
2 )K
- c)]
(3)
Thus AV(K) can be written as:
AV(K) = Sfr (K;8)f(8,t)dee- tdt
oo
where f(8,t)
is the density function for
discount rate, and
(4)
t
tt(K;O)
at time t, p is the risk-adjusted
is given by eqn.
(3).
It is difficult, however,
to evaluate (4) directly.
Instead we
obtain
In addition, the rate p might not be known.
V(K)
by solving
What is the value of a factory that
with operating
cost?@
produces I unit of
output per period,
cost (2Y+c 2)K + c, where the output is sold in a perfectly
competitive market at a price
(temporarily
the following equivalent problem:
and
t, and where
costlessly)
if the
the factory
price
can be
shut down
8t falls below the operating
It is shown in the Appendix that the solution to this problem is:
b.Ht;
e < (2Y+c2)K +
AV(K) = {
(5)
b u2 A
where:
P2c
/8 -
+
+
(r-- 2/
2)
(
2CT2
1
- 1(r02
1 r(p, - P2)
b
=
2
r&(P
[-r)
- P2)
r& - =
well as
Y = 0 and
C2
0 to .4.10
so
bVo(K)
=
1.9
2 /2) 2
-
(2Y+c 2 )K + c]
other aspects
.05, k = 10, c
2
] 1
+ 2r2
/2
> 1
1
+ 2ra 2 1-
< 0
(2Y+c 2)K + c1 1I-Pi
A numerical example is useful
AV(K), as
> (2Y+c)K + c1
+ c/r
[(r-6-o 2 /2)
(r-&
:r -r
- -_21r8)_
b
(2Y+c2)K
>0
> 0
P2
to
illustrate
of the
model.
the
characteristics of
For this purpose I choose
= 0, and either Y = .5 and c2
=
0, or equivalently
I vary 8 or K, and consider values of
in the range of
For purposes of comparison, let AVo(K) denote AV(K)
= 8/8 - £(2Y+c 2)K + c]/r for
For our numerical example,
Figure
.4. Observe
shows
that
= 0,
> (2Y+c 2 )K + cl , and 0 otherwise.
Vo(K) = 20(8-K) for 8
K, and 0 otherwise.
V(K) as a function of 8 for K =
V(K)
for
1 and
a =
0, .2, and
looks like the value of a call option -- indeed it
is the sum of an infinite number of European call options (see
Footnote 8).
- 8 As
with
a
call
option,
AV(K)
AV(K) > Vo(K) because the firm
AV(K)
Vo (K);
very
for
is increasing
need not
large
with
utilize its
relative
o, and
for
capacity.
> 0,
As e8 m,
K, this unit of capacity
to
will almost surely be continuously utilized over a long period of time.
Figure 2 shows
V(K) as a function of K for 8 = 2, and
=
0, .1, .2,
Because demand evolves stochastically, a marginal unit of capacity
and .4.
has some positive value no matter how large
is the
existing capital stock;
there is always some chance that it will be utilized over any finite period
The greater is , the more
of time.
the smaller
V(K)
likely it is that
is K, the more
V(K) -
utilized, and so the smaller is
with c
slowly
declines with
the marginal
Vo(K).
When K =
,
K. Also,
unit will be
V(O) =
Vo(K);
= 0, the first marginal unit will always be utilized.
The fact
that
is larger
V(K)
when
> 0 might suggest that the firm
should hold a larger amount of capacity, but just the opposite is true.
shown below,
As
the firm's opportunity cost of exercising its option to invest
in the marginal unit also becomes larger, and by an even greater amount.
The Decision to Invest in the Marginal Unit.
Having valued the marginal unit of
option to
capacity, we
the firm's
invest in this unit, and the optimal decision rule for exercising
the option.
This is analogous to
price k, on a
current price
stock that
V(K).
a
perpetual
call
option
with exercise
pays a proportional dividend at rate
In the Appendix it is shown that its value,
aAe5
and has a
F(K), is:
8 < 8*(K)
tF(K) =
(6)
AV(K)
where:
now value
- k
a = 2b2(*)(P2-P)
;
+
8 > 0'(K)
1(e*)(1-P1
>0
- 9 p1, p2 , and b
value of
are given
under eqn.
(5)
above, and
e9*(K) is
the critical
at or above which it is optimal to purchase the marginal unit of
capacity, i.e. the firm
should
purchase
the
unit
if
>
8*(K),
This
critical value 8*(K) is in turn the solution to :
212
b2(a1-E2)(o*)2+
P
Pi
Eqn. (7) can
calculate
(2+C
k = 0
)K + c1
r
be solved numerically for 8*, and eqn.
(7)
(6) can then be used to
F(K).
Recall our assumption that 6 > 0.
8*(K)
2
6p1
.
> 0, the
Unless
capacity always
capacity.'
exceeds
Thus
the
if firms
The reader can verify that as &6
opportunity cost
benefit,
and
of investing
the
firm
will
in a unit of
never install
in an industry are investing optimally and some
positive amount of investment is taking place, we should observe
As with a call option on
critical value
i.e. the firm
and .4, 8* is
ing
the
as
1 and
=
should increase
increases.
0, .2,
option
to
invest
, so a higher
cf
stock, both
Figure
and .4.
capacity only
2.45 and 3.44 respectively.
firm's
increases with
a dividend-paying
ef(K) increase
for K =
function of
0,
if
When
F(K) and the
3 shows
a
higher
8
For
= 1.5,
= .2
cost of exercis-
in additional capacity is
implies
F(K) as a
= 0,
exceeds 1.5.
The opportunity
> 0.
critical
F(K),
which
value 8*(K).
Also, ' it is easily shown that 8'(K) is monotonically increasing in K.
The Firm's Optimal Capacity.
The function
are such that
rises to
.
>
8*(K) is
the firm's
optimal investment rule; if
and K
(K), the firm should add capacity, increasing K until
Equivalently
we can substitute for b2 (K) and rewrite eqn. (7)
in terms of K*(8), the firm's optimal capacity:
-
E:L,r-8)p9[(2Y+c 2 )K* + c
rsp 1
l-p2 _ [2X+c 2 )K* + cl
= 0, .2 and .4.
value for
--
see Footnote
smaller when future demand is uncertain.
three times as large as when
way
Another
to
see
(For many
10.)
For
Le
+
r
Figure 4 shows K*(8) for
a conservative
-
10
how
invest in a marginal unit, with
are equal.
(7')
industries .2 is
Observe that K
= .4, 8 must be
uncertainty
is much
more than
over future demand affects the
value of
AF(K), the
the option to
V(K) - k, the net (of purchase cost) value
The optimal capacity K*(8) is the maximum
two quantities
k = 0
= 0 before any capacity is installed.
firm's optimal capacity is by comparing
of the unit.
-
which these
K for
eqn. (6) that for 8 > 8', or equiva-
Note from
lently, K < K*, exercising the option to invest maximizes its value, so that
F(K) =
V(K) -
k, but
F(
for K > K,
V(K) - k, and the option to
>
invest is worth more "alive" than "dead."
F(K) and
This is shown in Figure 5, which plots
of K, for 8 = 2 and
is uncertain.
the option
= .2. Recall that
As the figure shows,
V(K) is larger when future demand
if the
opportunity cost
when
2.3 units,
as opposed to
= O. But at these capacity levels the opportunity cost of
investing in a marginal unit exceeds the
value of
of exercising
to invest were ignored (i.e. the firm added capacity until bV(K)
- k was zero), the firm's capacity would be about
1.5 units
V(K) - k as functions
the firm is not maximized.
the largest K for which
F(K) =
net
value
of
the
unit,
The optimal capacity is only K
so the
= .67,
V(K) - k, and the solution to eqn. (7).
3. The Value of The Firm.
As noted above, K*(8) is the capacity level which maximizes
the firm's
- 11 market value,
net of cash outlays for the purchase of capital.
seen algebraically and from Figure 5 by noting
has two
components, the
value of
firm's options to install more
that the
This can be
value of
the firm
installed capacity, and the value-of the
capacity
in
the
future.
The
firm's net
value as a function of its capacity K is thus given by:
K
Net Value = SbV(v)dv +
o
Differentiating with
such that
V(K*) -
F(v)dv - kK
K shows that this is maximized when K = K*
respect to
F(K*) - k = 0.
The value of the firm's installed
integral in
from K
0
eqn.
(8).
to K*,
options to
capacity, V(K*),
is just
the first
In Figure 5 it is the area under the curve
plus the
expand is
under the curve
(8)
K
purchase cost
the second
F(K) from K = K
kK*.
The
integral, which
to
.
As the
value of
in Figure
V(K) - k
the firm's
5 is the area
figure suggests,
the value
of the firm's growth options is a large portion of its total value.
The sensitivity
of firm
value and
its components to uncertainty over
future demand can be seen from Table 1,
total value
for different
firm is only the
0, the
firm is
value of
worth more
values of
for the
larger is the fraction
When
the the
.
of
capacity.
installed
firm's options to expand.
of firm
F(K), and
When a = 0, the value of the
more volatile
unit
K*, V(K*),
Whatever
the value of
is demand.
capacity,
A larger a
and
a much
Also, the larger is a, the
value attributable
to its
growth options.
= .2 or more, more than half of the firm's value is F(K*), the value
of its growth options.
of total
and
its installed
implies a larger value for each
larger value
which shows
value when
Even when
is 1 or less.
= .1, F(K*) accounts for
more than half
(When demand is currently small, it is
-
12
-
the possibility of greater demand in the future that gives the firm
its value.)
for which K* is
And there is always a range of
much of
zero, so that
all of the firm's value is due to its growth options.
As men tioned earlier,
industries.
A
a
=
.2
or
testable implicat ion
more
should
be
of
capital in
not be much more than one half.
place should
value
I
demand
have not
(valuing capital in place
tions
reported
by
capitalizing a
is itself
Kester
estimated the val u,e of
f1ow of
tried to
(1984)
is only
However, calcula-
wit h both of them.
in 5
15 firms
majority of
He
industries by
from this capita 1,
anticipated earni ngs
about 1/5
is the volatility
task).
consistent
it is half or les S of market value in the
this fraction
value of
test either of these implications
place for
capital in
the
A second implica-
the greater
a difficult
are
to
attributable
tion is that this fraction should be small er
of market
for many
of the model is that for firms in such
industries, the fraction
mar ket
typical
cases.
and found
Furthermore,
to 1/3 in industries where demand is more
volatile (electronics, computers), but more than 1/12 in industries with less
volatile demand (tires and rubber, food processing).
4. The Dynamics of Capacity, Capacity Utilization, and Firm Value.
If
the
firm
begins
installs a starting capacity
with
no
K*(8).
capacity,
If
8 then
it initially observes 8 and
increases, it
capacity accordingly, and the value of the firm will rise.
growth options will also rise, but will become
value (see
Table 1).
However if
value will
The value of its
fraction of total
8 decreases, it will find itself holding
more capacity than it would have chosen had
The firm's
a smaller
will expand
the decrease
been anticipated.
fall, and depending on how much 8 decreases, some of
- 13 -
its capacity may become unutilized.
Because capital does not depreciate in this model,
is non-decreasing,
but
will
rise
only
the firm's capacity
periodically.
The
capacity are characterized in Figure 6, which shows a sample
and the
corresponding behavior of K(t).
movements in K(t) is exaggerated.)
K*( 8 o)
K.
The
path for
(t),
(The duration of continuous upward
firm
begins
at
to
by installing
Then 8 increases until it reaches a (temporary) maximum 8
t,, and K is increased accordingly to K.
when 8 again reaches 81.
when
dynamics of
Here it remains fixed
at
until t,
Afterwards K is increased as 8 increases, until t3
begins to decline from a new maximum, and K remains fixed at K.
Thus an implication of the model
spurts, when
levels.1 2
demand is rising, and
Firms usually
is
that
only when
increase capacity
investment
occurs
only in
it is rising above historic
only periodically,
often attributed to the "lumpiness" of investment.
and this is
But lumpiness is clearly
not required for this behavior.
Let us now examine the
periods of
expansion, all
firm's
capacity
capacity will
utilization.
be utilized.
Clearly during
When demand falls,
however, some capacity may go unutilized, but only if it falls far enough.
If the firm had unlimited capacity it would maximize current profits by
setting output
at Q* = (8-cj)/(2Y+c 2 ).
However K*(8)
(8-c)/(2Y+c2), and
as shown in Section 2, can be much less even for moderate values of
for 8
in the
range
(K)
(2X+c2)K + c
fixed but will be fully utilized.
8(K).
Thus
8 < 8'(K), capacity will remain
Capacity will go unutilized only when 8 (
In Figure 6 this occurs during the intervals (t.,
The irreversibility
.
of investment
tb)
and (to, t).
induces firms to hold less capacity
as a buffer against unanticipated drops in demand.
As a
result there will
- 14 be periods
of low
demands when capacity is fully utilized. A large drop in
demand is required for capacity utilization to fall below 100%.
The value of the firm will move in
the same
direction as
.
Most of
the time the firm's capacity K will be above K*(8) -- in Figure 6 exceptions
and (t2 ,
are during the intervals (to, t)
t3 ) --
but given
8 and
K, the
firm's value can always be computed from:
K
lV(v;O)dv +
Value =
o
(9)
of the firm's value due to its growth options will also fluctuate
The share
For example as Table
with 8.
F(v;9)dv
K
growing (so
that K
1
= K*(8)),
falling and K > K(8).
during
shows,
this share
periods
when
capacity is
It also falls when
falls.
is
Thus as firms in this model evolve over time, growth
options tend to account for a smaller share of value.
5. The Measurement of Long-Run Marginal Cost.
The measurement of long-run marginal cost and its relationship to price
can be important for
tions
in
industry analyses
particular.
As
shown
in general,
below,
and antitrust applica-
when investment is irreversible,
traditional measures will understate marginal cost and overstate
by which
it differs from price, even in a competitive market.
the amount
This problem
is particularly severe when product markets are volatile.
Suppose
= 0.
,V(K)
Note that
V(K),
8/
Then
F(K) = 0, and the firm sets K (and 0) so that:
- [(2+c 2 )K + cI/r = k
the value
of
a
marginal
(capitalized) marginal operating cost.
8/8 - 2K/r = (c
unit
(10)
of
capacity,
is
net of
Let us rewrite (10) as follows:
+ c2 K)/r + k
(10')
- 15 The left-hand
side of
ized at a rate
(10') is capitalized marginal revenue (
because
The rght-hand
between
capital-
8 =
r -
).
full marginal cost, the capitalized operating cost,
a
plus the purchase cost of
relation
= 0,
t is growing at rate c; with
side is
is
marginal
unit
of
revenue
capital.
and
Eqn.
marginal
is
(10')
the usual
cost when the former is
increasing at a deterministic rate.
Now suppose
condition
> 0, and K = K*(e).
V(K) = k +
- 2YK/r
8/
Observe
that
two
/8 -
adjustments
the firm's
(2Y+c2)K + c/r = k +
- b29z2 + (C2 K + c)/r + k +
marginal cost, the RHS of
value of
from eqn. (5),
the optimality
F(K) can be written as:
dV(K) = b2 8p2 +
or:
Then
must
(11').
be
F(K)
(11)
F(K)
(11')
made to obtain full
(capitalized)
The first term on the RHS of (11')
is the
option to let the marginal unit of capacity go unutil-
ized, and must be subtracted from capitilized operating cost.
The last term
is the opportunity cost of exercising the option to invest.
As we have seen
in Section 2, the last term dominates the first, so
must be smaller
to satisfy
(1'),
and
that K
marginal cost as conventionally measured will under-
state true marginal cost.
If the firm is a price-taker, Y =
0
and
P
=
8.
Price
marginal cost, if the latter is defined correctly as in (11').
ly the first and last terms on the
particularly with
aggregate data.
RHS of
(11') are
will equal
Unfortunate-
difficult to measure,
But if one wishes to compare price with
marginal cost, ignoring them can be misleading.'s
6. Marginal q and Investment.
The q theory of investment says that firms have an
incentive to invest
- 16 whenever
marginal
--
q
the
increase
in
market
of that
marginal unit of capital divided by the cost
But models
unit --
exceeds one.
based on this theory have not been very successful in explaining
investment."
that
value resulting from a
in
There may be sever al reasons for this, but one possibility is
such
models
the
cost
of
a
marginal unit of capital typically
As we have seen, this can
includes only the purchase and installation cost.
grossly understate the true cost of the unit.
In
empirical
(e.g. Abel
applications
and
Blanchard), the change in
market value from a marginal unit of capital is measured
expected profit
clearly
q
>
1
V(K)/k.
does
capital increases
this ignores
But
impl y
not
measure for q is found by
the opportunity
that K should be increased.
noting that
the value
of the
the addition
firm's capital
decreases the value of the firm's growth options
in market value is thus
q
Setting q
by
of a
cost
F(K);
The correct
marginal unit of
in place by
F(K).
V(K),
but
The net change
F(K), so the correct measure for q is:
F(K)]/k
(11)
= 1 yields the optimal K*.
It is
easy to see why marginal q as it is usually measured may fail to
explain investment.
increase
V(K) -
= [8V(K) -
NPV of the
the unit, i.e. the present value of the unit. In
flow from
our notation, q =
as the
V(K),
but
amount.
And if
so that
the desired
Note
that
an
it
will
also
increases, both
increase
in
increase
F(K) and
capital stock falls.
demand
(i.e. in
8) will
F(K), although by a smaller
V(K) rise,
but
V(K)
by less,
But the market value of the firm
increases, so that even if market value data is used
to measure
q, we will
see an increase in q but a decrease in investment.
As mentioned earlier,
_11^__1_
F(K) is unfortunately difficult to measure.
But
- 17 in some cases it may be possible to construct a proxy for
sample
variance
of
demand
fluctuations.
F(K) based on the
The use of such a proxy might
improve the explanatory power of q theory models.
7. Conclusions.
The model
such as
tures.
presented here
adjustment costs,
It
can
numerical methods
easily
simple one
is a
that ignores complications
delivery lags, and lumpiness of capital expendito
extended
be
account
for
For
Of course once
may then be required to obtain solutions.
numerical methods are used, other aspects of the model can
ized.
can be
example, demand
a nonlinear
factors, but
these
also be general9, or
function of
could
follow some alternative sto chastic process, perhaps including jumps.
By treating capital in place as homogeneous, we have been able to focus
on the
marginal investment decision, and clarify the ways in which irreveruncertainty over
sibility of investment and
future demand
Besides yielding a relative-
value and cost of a marginal unit of capacity.
choice, the
ly simple solution to the problem of capacity
straightforward
components.
for
calculating
provides
insight
method
It also
affect both the
the
firm's
into
the
model provides a
market value and its
measurement
of long-run
marginal cost, and Tobin's q.
In many markets output prices fluctuate with annual standard deviations
in excess of 20 percent.
should hold
much less
Our
show
results
capacity than
would be
reversible or future demand were known
market value
of these
firms is
that
due to
in
such
markets, firms
the case if investment were
with certainty.
Also, much
of the
the possibility (as opposed to the
expectation) of stronger demands in the future.
This value may
result from
- 18 patents
and
technical
knowledge,
but
it also arises from the managerial
expertise and infrastructure, and market position that gives these firms (as
opposed to potential entrants) the option to economically expand capacity.
One might ask whether firms correctly compute and take into account the
opportunity cost of
Ignoring such
investing
when
making
capacity
costs would lead to overinvestment.
(1986) have found
that
for
manufacturing
firms,
expansion decisions.
McConnell and Muscarella
market
value
tends to
increase (decrease) when managers announce an increase (decrease) in planned
investment expenditures, which is inconsistent with a systematic tendency to
overinvest. 1 5
But
there
is
anecdotal
evidence that managers often base
investment decisions on present values computed with discount rates that far
exceed those
that would
be implied
by the
CAPM -- diversifiable and non-
diversifiable risk are sometimes confused, and an arbitrary "risk factor" is
often added
to the
discount rate.
It may be, then, that managers use the
wrong method to get close to the right answer.
-
19
-
APPENDIX
Here we derive eqn. (5) for
V(K;8)
and
eqns. (6) and
optimal investment rule and value of the investment option
The value
of a marginal unit of capacity,
an equivalent "incremental project"
that
(7) for the
F(K;8).
-
V(K;8), is found by valuing
produces
I unit
of
output per
period at cost (2Y+c 2 ) + c, which is sold at price 8(t), and where the firm
can (temporarily and costlessly) shut down
value this,
if price
falls below
create a portfolio that is long the project and short
of the output, or equivalently the
correlated with
.
asset or
portfolio of
88ObV
per
assets perfectly
unit of
rational investor would hold the corresponding long position).
this portfolio is i = AV - AVe,
d(AV) -
Vsd8 - &88V
The last term in (.1)
8
+ j[8 - (2Y+c 2 )K - c]
is the cash flow from
for d
Lemma,
d(lV)
=
= rV - rVoe
292
8 Ve8 + (r-s8)ev
the "incremental
time (or no
The value of
- (2Y+c
project;"
j
(2Y+c 2 )K + c, and O otherwise.
is riskless.
Substitute eq. (2)
Setting that return
yields the following equation for
+ j
-
(A.1)
ede + (1/2)tVee(d9) 2.
and observe that the return (A.1)
equal to r
=
and its instantaneous return is:
is a switching variable: j = 1 if 8(t)
Ito's
To
VO units
Because the expected rate of growth of 8 is only
8, the short position requires a payment of
By
cost.
2 )K
- c
V:
- rV =
(A.2)
The solution must satisfy the following boundary conditions:
AV(8=0) = 0
lim
V =a8/8 -
(2Y +cz) + c/r
Or
lim AVe = 1/6
and
V and ave continuous at the switch point 8 = (2Y+c
2 )K
+ c.
The reader
can verify that (5) is the solution to (A.2) and its boundary conditions.
-
Eqn.
(A.2) can
also be
obtained by dynamic programming.
optimal operating policy (
above portfolio.
rE =
-
20
= 0
or 1)
that maximizes
the value
max
{jfe
- (21+c
i.e. the competitive
2
) - c
return r
BVe
-
has two
+
--E d}
dt t
(A.3)
components, the cash flow given by
the first two terms in the maximand, and the expected rate of
capital gain.
= dV - EVode and substituting into (A.3) gives (A.2).
Finally, note
that
V
must be
the solution to (A.2) and the boundary
conditions even if the unit of capacity (the "incremental project")
exist, and could not be included in a hedge portfolio.
is an asset or
dynamics of
portfolio
8.
As
of
assets
Merton (1977)
function with a portfolio consisting
bonds, and
of the
The Bellman equation is:
J-(0,1)
Expanding d
Consider the
(x)
that
has shown,
only
of
All that is required
replicates
one can
the
did not
the stochastic
replicate the value
asset
x
and risk-free
since the value of this portfolio will have the same dynamics as
AV, the solution to (A.2),
Eqn. (6) for
V must be the value function to avoid dominance.
F(K;O) can be derived
in the
arguments as above, it is easily shown that
'2AFee
+ (r-S)eAF
same way.
Using the same
F must satisfy the equation:
- rF = 0
(A.4)
with boundary conditions:
AF(8=0)
= 0
6F(e=e*)
AFe(8=e*)
where 8*
=
exercising.
=
V(e=8*)
- k
= Ave(=e*)
*(K) is the exercise point, and
V(8*) - k is the net gain from
The reader can verify that eqns. (6) and
to (A.4) and the associated boundary conditions.
___^_____
(7) are
the solution
-
c
(cL=
___^__11__1_1__1_·_I___----.-.
=
21
-
TABLE 1: VALUE OF FIRM
O, Y = .5, r =
= .05,
0*
e
K*
V(K*)
0
.5
1
2
3
4
0
0.5
1.5
3.5
3.5
.1
.5
1
2
3
4
.2
.4
k = 10)
F(K)
VALUE
0
7.5
37.5
87.5
157.5
0
0
0
0
0
0
7.5
37.5
87.5
157.5
0
0.23
1.00
1.82
2.65
0
4.3
33.1
80.4
147.3
0.4
3.4
9.1
20.3
35.5
0.4
7.7
42.2
100.7
182.8
.5
1
2
3
4
0
0.04
0.67
1.37
2.09
0
0.8
24.0
67.1
134.1
3.1
13.5
49.2
94.6
143.7
3.1
14.3
73.2
161.7
277.8
.5
1
2
3
4
0
0
0.-15
0.64
1.22
0
0
5.9
36.2
91.3
25.8
69.7
182.6
307.2
427.5
25.8
69.7
188.7
343.4
518.8
-
22
-
FOOTNOTES
1.
Other examples of this literature include the papers by Bernanke
(1983), Cukierman (1980), Baldwin (1982), and Paddock, Siegel, and
Smith (1984).
In the papers by Bernanke and Cukierman, uncertainty
over future market conditions is reduced as time passes, so firms have
an incentive to delay investing when markets are volatile (e.g. during
recessions).
In the other papers cited above and in the model I
present here, future market conditions are always uncertain. But
because access to the investment opportunity is analogous to holding a
call option on a dividend-paying stock, for any positive amount of
risk, an expenditure should be made only when the value of the resulting project exceeds its cost by a positive amount, and increased
uncertainty will increase the incentive to delay the investment
expenditure.
Thus the results are similar to those in Bernanke
and Cukierman, but for different reasons.
2.
In Abel (1983) and Hartman (1972) uncertainty over future prices leads
to an increase in the firm's optimal capital stock when the production
function is linear homogeneous and there are convex adjustment costs.
The reason is that the marginal revenue product of capital is a convex
function of price, so that as in our model, a marginal unit of capital
is worth more when price is stochastic. However in Abel and Hartman,
investment is reversible, so that no options are closed when the firm
invests, and the opportunity cost of investing is zero.
3.
A complete model of industry evolution would also describe the competitive processes through which firms obtain these options.
Such a model
is beyond the scope of this paper.
4.
Analytic solutions for this model can be obtained for any demand
function linear in 8, i.e. P = 8(t) + f(Q). I use (1) for simplicity.
5.
It is straightforward to also allow for uncertainty over future
operating cost. The qualitative results would be the same.
6.
Relaxing these assumptions makes no qualitative difference in the
results. In fact, allowing for lead times in the construction and
installation of new capacity will magnify the effects of uncertainty.
For a related model that looks specifically at the effects of lead
times, see Maid and Pindyck (1985).
7.
Still another way to see this is to note that the NPV of investing in a
marginal unit of capital might be positive because of possible future
states of nature that are profitable. But that NPV might be made even
larger by investing at a later date (and discounting the cost).
8.
The valuation of a factory that can be temporarily shut down has been
studied by Brennan and Schwartz (1985) and McDonald and Siegel (1985).
Observe from eqn. (3) that the present value of an incremental profit
at future time t is the value of a European call option, with expiration date t and exercise price (2Y+c 2 )K + c, on a stock whose price is
_ ____ ____I_·_
____________
- 23 8, paying a proportional dividend . This point was made by McDonald
and Siegel (1985).
Thus V(K), the value of our "equivalent factory,"
can be found by summing the values of these call options for every
future t. However this does not readily yield a closed form solution,
and I use an approach similar to that of Brennan and Schwartz (1985).
I
9.
With r and
equal, (c = 0 if stochastic changes in 8 are completely
diversifiable (i.e. Qe = 0 so p = r), but
:>0 otherwise. Also, as
can be seen from eqn. (5),
(Y=0, c2 =1)
and (=.5, c2 =0) give the
same marginal value of capital, and the same optimal behavior of the
firm.
In the first case the firm is a price-taker but earns inframarginal rent, and in the second case it has monopoly power.
10.
The standard deviations of annual changes in the prices of such
commodities as oil, natural gas, copper, and aluminum are in the range
of 20 to 50 percent. For manufactured goods these numbers are somewhat
lower, but often 20 percent or higher. Thus a value of
of .2 could
be considered "typical" for simulation purposes.
11.
If
= 0, AV(K) has an expected rate of growth equal
to the riskadjusted market rate. Since the firm's option to invest is perpetual,
there would be no gain from installing capacity now rather than later.
12.
If we allow for depreciation, investment will occur more frequently and
even when demand is somewhat below historic highs, but it will still
occur in spurts.
13.
Hall (1986) reports that price significantly exceeds marginal
cost for
most two-digit industries, and finds no explanation for this disparity
consistent with competition.
Hall's test of marginal cost pricing is
based on the relation between the marginal product of labor and the
product wage.
If firms set marginal operating cost equal
to a
(constant) proportion of price, his technique will apply, whatever the
capital stock. But as shown in Section 4, there can be a wide range of
prices for which the firm is capacity constrained, and the ratio of
marginal operating cost to price will vary with price.
14.
For example, the model developed by Abel and Blanchard (1986) is one of
the most sophisticated attempts to explain investment in a q theory
framework; it uses a carefully constructed measure for marginal rather
than average q, incorporates delivery lags and costs of adjustment, and
explicitly measures expectations of future values of explanatory
variables.
But Abel and Blanchard conclude that "our data are not
sympathetic to the basic restrictions imposed by the q theory, even
extended to allow for simple delivery lags." Also, see Summers (1981).
15.
But they find the opposite true for firms in the oil industry, where
there may be a tendency to overinvest in exploration and development.
_L__
REFERENCES
Abel, Andrew B., "Optimal
Investment under
Uncertainty," American Economic
Review, March 1983, 83, 228-33.
Abel, Andrew B., and Olivier J. Blanchard, "The Present Value of Profits and
Cyclical Movements in Investment," Econometrica, March 1986.
Baldwin, Carliss Y., "Optimal Sequential Investment when Capital
Readily Reversible," Journal of Finance, 1982, 37, 763-82.
is Not
Bernanke, Ben S., "Irreversibility, Uncertainty, and Cyclical Investment,"
Quarterly Journal of Economics, February 1983, 98, 85-106.
Brennan, Michael J., and Eduardo S. Schwartz, "Evaluating Natural Resource.
Investments," Journal of Business, January 1985, 58, 135-157.
Cukierman, Alex, "The Effects of Uncertainty on Investment under Risk
Neutrality with Endogenous Information," Journal of Political Economy,
June 1980, 88, 462-475.
Hall,
Robert E., "The Relation Between Price and Marginal Cost in U.S.
Industry," NBER Working Paper No. 1785, January 1986.
Hartman, Richard, "The Effects of Price and Cost Uncertainty on Investment,"
Journal of Economic Theory, October 1972, 5,
258-66.
Kester, W. Carl, "Today's Options for Tomorrow's Growth," Harvard Business
Review, March/April 1984, 153-160.
Maid, Saman, and Robert S. Pindyck, "Time to Build, Option Value, and
Investment Decisions," NBER Working Paper No. 1654, June 1985.
McConnell, John
J., and Chris J. Muscarella, "Corporate Capital Expenditure
Decisions and the Market
Value
of
the
Firm,"
Journal
of Financial
Economics, forthcoming, 1986.
McDonald, Robert, and Daniel R. Siegel, "Investment and the Valuation of
Firms When There is an Option to Shut Down," International Economic
Review, October 1985.
McDonald, Robert, and Daniel R. Siegel, "The Value of Waiting to Invest,"
Quarterly Journal of Economics, forthcoming, 1986.
Merton, Robert C., "On the Pricing of Contingent Claims and the Modigliani
Miller Theorem," Journal of Financial Economics, 1977, 5, 241-49.
Paddock, James L., Daniel R. Siegel, and James L. Smith, "Option Valuation
of Claims on Physical Assets: The Case of Offshore Petroleum Leases,"
MIT Energy Laboratory Working Paper, November 1984, to appear in the
Quarterly Journal of Economics.
Summers, Lawrence H., "Taxation and Corporate
Investment: A q-Theory
Approach," Brookinqs Papers on Economic Activity, 1981-1, 67-127.
FIG
URE 1: VAUE
F
MRGINAL
UNIT OF CAPACITY
(K = 1)
50
45
40
z
35
-j
30
z
I'
25
2
LI
20
w
--
15
10
5
0
0
0.5
1
1.5
2
3
2.5
THETA
FIGURE
2:
MARGINiAL VVALUE OF INSTALLED CAPACITY
( = 2)
40
35
-
30
z
25
-
20
C.
0
-J
15
10
5
0
0
0.5
1
1.5
2
2.5
K
3
3.5
4
4.5
5
FIGURE a3: VALUE OF OPTION TO INVEST IN MARGINAL UNIT OF CAFPCITY
, F(K) = V(K) - k)
(K = 1; + INDICATES *(K=I); FOR 8 >
._
I-
60
50
IJ
Z
40
z0
30
p2
O
0
I.
O
20
O
>
10
0
0
2
1.5
1
0.5
3.5
3
2.5
4
THETA
4:
FIGURE
--
CAPACITY K*(t)
OPTIMAL
--
4
3.5
3
2.5
I
2
1.5
1
0.5
0
.*
0
0.5
1
1.5
2
2.5
THETA
·--
"--
II"-"-`--------
3
3.5
4
4.5
5
5:
FIGURE
OPTIMAL CAPACITY, NET VALUE OF MARGINAL UNIT OF INSTALLED
-'-CAPACITY, AND VALUE OF OPTION TO INVEST IN MARGINAL UNIT
(8 = 2)
30
z
0
25
.
0
z
20
-C
-J
I',.
U
15
L.
0
,(
10
5
I
-.
-5
0.5 K
0
1.5
1
3
2.5
2
4
3.5
K
FI GURE 6: BEHAVIOR OF CAPACITY AND CAPACITY UTILIZATION
A
I
I
I
I1
't o0
,xll------.----1I.---------------
L
tl
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