working paper
department
of economics
ON THE MISUSE OF THE PROFITS-SALES RATIO
TO INFER MONOPOLY POWER
Franklin M. Fisher
Number 364
February 1985
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
/on the misuse of the profits-sales ratio
TO infer monopoly power^
Franklin M. Fisher
Number 364
February 1985
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/onmisuseofprofitOOfish
On the Misuse of the Profits-Sales Ratio
to Infer Monopoly Power
Franklin M. Fisher
Department of Economics, E52-359
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
ABSTRACT
A large literature tests the structure-conduct-performance paradigm by
regressions which use the profits-sales ratio as the dependent variable. That
ratio is used because, under constant returns, it is generally supposed that
profits/sales equals the Lerner measure of monopoly power. Profits/sales
ratios are also used to make inferences about monopoly power for individual
firms.
This paper shows, however, that unless proper account is taken of the
valuation of capital and the treatment of capital costs, profits/sales will be
a very substantially inaccurate surrogate for the Lerner measure.
Such proper
account requires Hotelling valuation of capital goods which real firms do not
use.
The full analytics of the problem are worked out and some theorems proved
relating the error to the cost of capital, the growth rate of the firm, and the
depreciation method used.
Examples are used to show that the error is poten-^'
tially enormous, and the error is shown in general to be related to the variables
used on the right hand side of the regression studies referred to above.
The
conclusion is reached that such studies are totally worthless.
That conclusion
will doubtless spark a certain amount of controversy as did the conclusion of
the parallel paper on the accounting rate of return on assets or equity (Fisher
and McGowan, American Economic Review, March 1983).
i
SEP 3
1985
REC£3Ve3
ON THE MISUSE OF THE PROFITS SALES RATIO
TO INFER MONOPOLY POWER
Franklin M. Fisher
Massachusetts Institute of Technology*
1.
Introduction
It is popularly supposed that,
Lerner measure of monopoly power
—
—
under constant returns,
(price - marginal cost)/price
is equal to the ratio of profits to sales.
once
that,
one
Moreover,
shows
This paper
leaves the simplest one-period model,
generally not the case.
the
this
is
the errors can be quite large
and are likely to be systematic.
The
reason
is not hard to find.
Constant
means
returns
constant returns to all factors including capital, so that unless
proper account is taken of capital costs, the profits-sales ratio
will simply misstate the Lerner measure.
Assuming,
city that the prices of capital goods are constant,
costs
involved
for simpli-
the
capital
will be depreciation and foregone interest
opportunity cost of capital).
(the
costs
The reported value of such
depends on the way in which the firm values its capital stock. It
should
therefore
generally
ling
come
as no surprise that capital
costs
not be correctly measured unless the firm uses
valuation in which capital goods are valued at the
will
Hotelpresent
the remaining net revenue streams they generate
using
the firm's risk-adjusted discount rate (Harold Hotelling).
Real
value
of
firms do not do this.
This capital-theoretic problem with the easy measurement
monopoly
power
difficulty
has a familiar ring to it.
Precisely the
occurs with the use of the accounting rate of
of
same
return
on capital value or on equity to infer monopoly profits.
Indeed,
of the profits-sales ratio to the Lerner
measure
relations
the
resemblance to the relations of
bear
a striking
rate
of return on capital to the economic rate of return studied
in
my earlier paper with John J.
Those
McGowan (Fisher and
sought a refuge from our results in the
who
accounting
the
McGowan)
use
the
of
profits-sales ratio instead of the accounting rate of return have
not found one (William Long and David Ravenscraft; Stephen Martin
The analysis of monopoly power cannot be so cheaply
1983, 1984).
accomplished.
with the issues surrounding the accounting rate
As
the
turn,
existence
profits-sales
witz,
of capital-measurement problems
presents a slashing
attack,
re-
with
the
LeiboMartin
and
28-32) lays out some of what is involved.
pp.
of
ratio has been noticed in the literature.
particular,
in
(1983,
2
But, de-
Martin's assertion that such problems are well known (Mar-
spite
tin 1983,
p.
32), they do not appear to be well understood.
The
full analytics of the problem appear not to have been spelled out
nor has the importance of the problem been realized.
That
importance
literature
able
the
substantial.
very
large
using the profits-sales ratio as the dependent
vari-
is
There is
in regressions purporting to test various
structure-conduct-performance paradigm.
Martin,
a
propositions
(See,
of
for example,
1983, Leonard Weiss, and David J. Ravenscraft.) Yet that
variable
dependent
is
subject to substantial
capital-valuation problems studied here.
from
error
Moreover,
the
contrary to
facile assertion that one can assume such errors are distri-
the
buted independently of the right-hand side variables used in such
for example,
studies (See,
Martin 1983,
it is shown
32),
p.
below that such independence cannot be assumed.
Of course,
about the monopoly power of individual
inferences
such
use is sometimes made.
a
independence
make
justify the use of the profits-sales measure to
not
would
even the unwarranted assumption of
Commission's
Line-of -Business
The staff of the
firms.
Yet
Federal
Trade
program proposed to use the
data
generated by that program to identify targets of antitrust prosebased on profits-sales ratios (Long
cution
ants
to
pointed
the Antitrust Division of the Justice
to
high
monopoly power.
cites
1983
known)
£i. ^il)
for
profits-sales ratios as
Thus,
F.
M. Scherer
i
and consult-
Department
convincing
have
proof
of
(one of those whom Martin
the proposition that these
problems
are
well
states (Scherer p. 620)
IBM's normal strategy was to set prices exceeding manufacturing costs by at least a factor of
four. If this does not reflect monopoly power, what
does?
Apart from the fact that such a statement ignores IBM's
tial
non-manufacturing costs,
so far as
I
glosses over the question of capital costs.
a need for a
can tell,
Evidently,
substanit at best
there is
systematic consideration of what is involved here.
2.
Formal Model: Hotellina Valuation
suppose the firm produces output, x, from capital
To begin,
and labor inputs,
K and L,
respectively, according to
a
produc-
tion function:
X
(1)
= F(K,
L)
assumed once continuously diff erentiable.
L a flow, as is x.
The time variable, t, has been omitted.
firm faces an inverse demand curve,
The
where p is price.
ble
inputs)
is w,
The wage of labor
Begin
given by p = p(x),
(which stands for all varia-
and the price of a unit of capital is
considering the simplest case in which
by
norma-
The interest cost of capital is r.
lized to be unity.
invests
Here, K is a stock and
in a single project starting at time 0.
the
In that
firm
case,
the firm maximizes its present value, given by:
oo
J
(2)
)
[pF(K, L)
differentiation
Denoting
-
by
wL]e ^^ dt
- K
subscripts,
.
the
conditions are:
R'Fj^ = w
(3)
all t
and
oo
(4)
j
R'Fj,e ^^ dt = 1
where subscripts denote partial differentiation and;
first-order
R'
(5)
is
marginal
= xp' (x)
+ p
•
Note that R',
revenue.
like the other
variables,
depends on t.
Now,
gives the first-order condition with
(4)
capital as of time
It
respect
the moment at which the capital is bought.
0,
is not hard to see that a similar condition will hold for any
later time, with the right-hand side of
(4)
replaced by an appro-
priate value for
a
unit of older capital.
Indeed,
firm values
a
unit of capital of age
according to a
the
dule,
^0.
to
where V(0)
V(0),
= 1,
suppose that
sche=
and it is natural to assume V(oo)
Then consistent planning will require that V(.) satisfy:
oo
J
(6)
R-F^e""^^^"®^ dt = V(e)
e
so
that V(e)
of
a
is the present value as of
marginal unit of capital.
This,
of course,
valuation (or "economic valuation"), and
which satisfies
(6)
by V*
( .
)
Now
that
V{.)
I
shall denote the V(.)
Obviously,
V*(0) = 1, and
I
shall
0.
assume that the firm does use Hotelling
=
is Hotelling
reserving the unstarred symbol for
,
general valuation functions.
assume that V*(oo) =
of the future benefits
V*(.) and differentiate
(6)
with
valuation,
respect
to
so
0,
obtaining:
R'Fj,
(7)
The
right-hand
= - V*' (0)
side of
(7)
+ rV*(0)
is readily seen to be the
instanta-
consisting of depreciation plus the
cost of capital,
neous
puted interest on the funds tied up in
Since
(6)
and
(7)
unit of capital.
a
are equivalent,
given
(4)
im-
4
and the
fact
that V*(0) = 1, the firm (or the analyst) can use either one.
If
then optimal behavior is characterized totally
in
(7)
is used,
—
of values defined at time
terms
Martin emphasizes (Martin 1983,
later marginal revenue products.
on the left-hand side of
sented
(0)
,
requires forecasting
In any event,
(7)
what is
buy
schedule V*
If the firm
and sell capital of different ages according
( . )
,
repre-
is the full marginal benefit
of an additional unit of capital of age 0.
at time
could
30), but which is quite mis-
p.
since Hotelling valuation, V*
leading,
property of "myopia" which
a
then the right-hand side of
long-run marginal cost of capital of age
to
the
would be the true
(7)
at time 0.
if Hotelling valuation
It is therefore not surprising that,
the prof its-to-sales ratio
is used and constant returns assumed,
does in fact equal the Lerner measure.
To see this, assume that
is homogeneous of degree one.
Denoting profits/sales at
F(K,
t
L)
by M(t)
:
px - wL - [rV*(t) - V*'(t)]K
=
M(t)
(8)
px
where
again it must be remembered that all the variables on
right-hand
side
of
simplicity, of r).
(9)
(8)
By
depend on t
(3)
and
wL + [rV*(t)
-
(7)
(with
the
for
and Euler's Theorem:
V*'(t)]K
= R' (LF^
Li
so that
exception,
the
+ KF.,
JS.
)
= R'x
,
p - R'
M(t)
(10)
= -
=
1/h
This is the Lerner measure
where h is the elasticity of demand.
full exploitation of monopoly power on the part of
assuming
the
firm.
Note, however, that this assumes that Hotelling valuation is
used.
If
(as
is nearly always the case for real
valuation schedule is used,
other
,
any
hold
and
firms)
then (10) will not
it will not follow that profits/sales equals the Lerner measure.
To go more deeply into this,
somewhat.
tion
Let G(t,
0)
= R'Fj^
G(t, t)
recalling
be the marginal revenue product at
Then, in the previous notation:
time t of capital of age 9.
(11)
nota-
it will aid to simplify
both marginal revenue
the
marginal
physical product of capital will generally depend on
t.
G(t, t)
the stream of net benefits which flow from
a
marginal
is
again
thus
that
unit of capital acquired at time 0.
two
arguments,
I
and
write G(t,
0)
as
having
because such a stream will generally depend both
on the calendar date,
t,
at which the benefits are received and
on the age of the capital involved,
0.
We must now consider the
nature of such dependence.
In full generality,
t is not
it
is
one-hoss
the marginal product of capital at time
necessarily independent of the age of the
—
a special case
shay
—
capital.
then capital is essentially
type with capital of different
substitutes until one vintage evaporates.
vintages
of
If
the
perfect
More generally, there
technical change embodied in the capital stock or
be
may
may be physical depreciation.
a
there
Hence the net benefit stream from
particular unit of capital may depend on the capital's vintage,
t - 6,
and thus on its age, 9, given the date, t.
however,
is
only
depend
as important for our purposes
least
At
fact that benefit streams will
the
on capital age but also on
calendar
demand may change over time and so may wages.
disembodied technical change.
outputs
different
as
In response,
such
not
effects,
generally
time.
Thus,
There may also be
the firm may choose
and different employment levels at
different
This will make marginal revenue and the marginal product
times.
of capital vary over time.
Only in the quite unrealistic case of
a totally stationary environment will this not be so.
consider
Now,
in
and
(6)
match
(7)
marginal
certainly
.
the generalization of Hotelling valuation as
If depreciation plus imputed interest
benefits on all types of
(a
G(t, 0)
,
then
so that Hotelling valuation, V*(t, 0)
mea-
must generalize to:
(7)
= -V*(t,
to
condition
sufficient for profits/sales to equal the Lerner
sure under constant returns)
(12)
capital
are
9)
+ rV*(t, 9)
is:
oo
V*(t, 9) =
(13)
/
G{t, u)e"^^"~®^ du
9
Note
that the depreciation term in (12) takes no account of
the
fact that when capital is one year older it will be a year later.
In effect,
the firm consistently planning at t behaves as though
it faced a schedule of capital prices,
V*(t, 0), at which it can
buy or sell capital of differing ages.
feature of (12) and (13) is
important
The
Hotelling
that
valuation in the model which leads to profits/sales equalling the
Lerner measure, depends not merely on capital age, 9, but aJLso pp
calendar time
This has
i..
the following implication.
Even when such dependence does not occur,
expect
we do not
firms to use Hotelling valuation and the depreciation sche-
real
dules which correspond to it.
has
stream
a
For example,
benefit
where the
positive bulge some time after the
investment
is
Hotelling valuation would require the firm to write up the
made,
value of existing capital by taking negative depreciation as that
bulge gets closer.
thing.
Obviously, real firms do not do this sort of
the present case,
In
however,
the fact that Hotelling
depends directly on calendar time would require
valuation
firms
to adopt different depreciation schedules for otherwise identical
capital of different vintages solely because wage rates or demand
changes
over time.
practice,
In principle,
they obviously do not.
studies
will
profits/sales
use
Hotelling
firms ought to do
this;
in
The hope that firms in actual
depreciation
make
thus
and
equal the Lerner measure under constant returns is
plainly forlorn.
more word must be added before proceeding.
One
seen,
the case of Hotelling valuation requires
even
profits
to
capital
if the profits-sales ratio is to equal the
sure.
Because
In
r
As we have
be adjusted by subtracting the imputed
practice,
—
this is seldom,
interest
Lerner
done
if ever,
the risk-adjusted cost of capital
accounting
—
on
mea-
directly.
is not
known.
studies which make any adjustment for the opportunity
most
of
cost
capital do so by leaving it out of the computation of profits
term in the value of capital divided by
and
putting
the
right-hand side of any regression.
a
sales
If Hotelling
on
valuation
used and if the capital-output ratio had no other effect on
were
the Lerner measure,
then the coefficient of that variable
In practice, as Leibowitz
be r.
(p.
8)
would
points out, that coeffi-
cient is often found to be significantly negative.
I
shall comment later on a possible reason for such a
For the present,
ing.
imputed
find-
simply consider the procedure of leaving
interest out of costs and placing the value
of
capital
divided by sales on with profits/sales as the dependent variable.
Suppose that the resulting regression coefficient for that variable is b.
ble
This is equivalent to subtracting
r
times that varia-
from the dependent variable and obtaining a regression coef-
ficient
on the right-hand side of b - r.
Such
a
subtraction,
however, amounts to evaluating profits/sales allowing for imputed
interest at the correct interest rate,
valuation
of capital.
r,
but with the firm's own
In what follows,
I
assume that to
have
been done and consider the consequences for estimating the Lerner
measure
as though it were done explicitly.
Studies or opinions
which make no such adjustment are obviously hopeless.
10
3.
A Stationary Environment: The Most Favorable Case
Suppose then that firms do not use Hotelling valuation.
Is
it nevertheless possible that profits/sales will equal the Lerner
measure under constant returns.
useful
To examine this question, it is
to go to the most favorable case,
to ignore
the
issues
just raised, and to assume that the firm exists in an essentially
stationary environment
dar
firm,
and
time.
one
in which benefits do not depend on calen-
(To make this
consistent with net investment by
must suppose that demand shifts in
a
very special way
the firm invests to secure the same marginal revenue in
periods.) In this case, we can replace G(t,
0)
the
by f(9).
all
Allowing
for this change in notation:
f(e)
(14)
= - v*'(e)
+ rv*(e)
Now denote the amount of investment which the firm makes
time u by I(u).
one
(For simplicity,
type of capital;
I
at
assume that the firm uses only
generalization is easy.)
Then,
for
any
valuation schedule, V(.), the sum of total depreciation and total
imputed interest at time t is given by:
t
^-
(15)
/ I(u)
[-V
(t-u)
+ rV(t-u)]
du
-oo
On the other hand, total net returns to capital, the depreciation
and
imputed interest corresponding to Hotelling
given by:
t
(16)
^ "
I(iJ)f(t-u)
J
-oo
11
du
valuation,
are
capital costs are subtracted in
Since
understatement
overstates
costs
(B
measure.
(B
Lerner measure,
the
A)
>
means
profits,
Only
if
B
while an overstatement of
such
Lerner
the
A will profits/sales give
=
an
profits/sales
that
means that profits/sales understates
A)
<
such costs
of
calculating
correct
the
result.
4.
The E rror Formula
expression
the
for
measure
Lerner
instructive
very
is
It
in
now
consider
to
difference between
the
precise
the
profits/sales
stationary-environment
and
the
under
case
consideration.
Let
the true value of
be
L*
L* = (p - R')Pf where p is price.
Lerner
the
measure.
Then
Equivalently,
px = R'x/(1 - L*)
(17)
Let
measure.
L
the profits/sales ratio
be
Then, using
(15),
(16), and
B - A
shows
estimate
lated
that
supposed
Lerner
(17):
- A) (1 - L*)
R'x
px
This
the
=
L - L* =
(18)
(B
—
the error in the use of
profits/sales
of the Lerner measure is very unlikely to
as
an
uncorre-
be
This
with variables which affect the Lerner measure.
is
because L* itself appears on the right-hand side of (18)
Indeed,
be
side
the
only possibility of avoiding this result would
if the appearance of x in the denominator of
of
(18)
the numerator.
the
somehow cancelled out the appearance of
This does not seem a very realistic
12
right-hand
(1
- L*)
in
possibility,
we shall now see that it is definitely not true in the
and
most
interesting leading cases.
we must explicitly account for
see this,
To
physical
the
depreciation of capital and the time it takes for new machines to
be installed and running.
I
do this by altering the description
the technology slightly and assuming that one unit of capital
of
of age 9 is equivalent to a fixed number of units,
it is natural to measure capital
Then
capital.
a
one-hoss shay
—
corresponds
capital
to
no deterioration while the capital is in
a
constant
for
a (9)
Consideration of
good.)
(The case of
(7)
0^9^
and
the life
above,
(14)
use
the
of
however,
that the marginal revenue product of capital of age 9 must
shows
be
—
efficiency
in
each of which has the same marginal product.
units,
of new
a (6),
f(9),
so a
(9)
is proportional to f (9)
and we might
renormalize by weighting capital of age 9 by f(9).
as
well
From (16), B
is then the capital stock in the new efficiency units.
More precisely, suppose that, at time t,
to
capital of vintage u is L(t,
u)
the labor assigned
and that the output produced
thereby is:
(19)
x(t, u)
=
F(f(t - u)I(u), L(t, u))
Then (19) states that the effects of capital age can be
as a capital-augmenting technical change.
constant
returns
marginal
revenue is not changing)
and
theorem on capital aggregation (see,
559-60)
Since we are assuming
all labor has the same
,
captured
wage
(and
since
it follows from a well-known
for
example,
Fisher,
that total output is given by the production function:
13
pp.
= F(B,
X
(20)
where
is defined in
B
employed by the firm.
measured
capital
L)
(16)
and L is the total amount of
Note that
(7)
and
labor
imply that we have
(14)
so that the marginal revenue product of
B
is
unity.
Now consider changes over time with w and R' held
Since
F(.,
.)
constant.
is homogeneous of degree one and L will be chosen
to have marginal physical product equal w/R', the ratio of L to B
It follows that x/B will also be constant and
will be constant.
that R'x = mB for some constant m.
Substituting in (18) yields
B - A
L - L* =
(21)
(1 - L*)
mB
The constant,
R'
,
m,
depends on w and on R',
it is determined by the technology.
firms with the same true Lerner measure,
values
profits/sales
taken
In particular, different
L*, will have different
and different firms with the same value of m
of m,
different
have
but, given w and
values
of L*
This means that
.
error
in
cannot
be
the
as a substitute for the Lerner measure
be independent of the Lerner measure itself or of
to
will
the
variables assumed to influence that measure.
It
is
case
(and
below)
.
the
special
particular
helpful in interpreting the numerical examples
given
Suppose the technology is Cobb-Douglas, so that
F(B, L)
(22)
Then
illuminating to evaluate m in a
= B^L-"-"^
fact that the marginal revenue product of B
implies that R'x = B/a, so that m = 1/a and (21) becomes
14
is
unity
B ~ A
L - L*
(23)
= a
(1 - L*)
B
appearance of the Cobb-Douglas parameter in this way is
The
no accident.
Returning to the more general case of any constant-
returns production function,
and
tion,
Euler's
Theorem,
profit-maximiza-
fact that the marginal revenue product of B
the
is
unity imply:
B = R'x - wL
(24)
,
so that
1/m = 1
(25)
- wL/R'x
,
or one minus labor's share of the value of output when that value
is in terms of marginal revenue rather than of price.
Cobb-Douglas
production
this
case,
function,
As in the
will depend on the parameters
but it should come as no surprise that the
errors which come from the mismeasurement of capital bulk
when
(true)
the
of
larger
capital costs are a large share of total costs
than
they do when that share is small.
In
the
case in which the aging of capital does not have
capital-augmenting
effect on its productivity,
are less readily come by.
It remains generally
a
simple
formulae
true,
however,
that the error depends on the Lerner measure and on the nature of
the technology.
case,
the
It is obvious that, as in the capital-augmenting
error must be absolutely larger,
more important capital costs are as
I
and
a
ceteris paribus the
fraction of total costs.
shall return to a discussion of the error formula
also to the use of the Cobb-Douglas example.
15
For the
below,
pre-
however,
sent,
want to put aside the leading but special case
I
capital-augmenting
of
general
the
changes due to capital age and return
to examine the question
case
of
whether
to
without
it is possible that B = A and profits/sales
Hotelling valuation,
equals the Lerner measure.
5.
The
S im plest
Case:
Exponential Growth
To examine the circumstances under which B = A, define
t
K* =
(26)
/
I(u)
du
,
-oo
the total value before depreciation of the firm's capital
stock.
Then A/K* is average depreciation and imputed interest per dollar
of
capital
put in place in the past,
while B/K* is
similar
a
average of net returns to capital. For profits/sales to equal the
Lerner
these two averages have to be the
measure,
obviously
magnitudes
happens
in
if each weight,
applies
equal
to
that would be the case if
the two averages;
used Hotelling valuation.
firm
I(u)/K*,
This
same.
I
now investigate
the
circum-
the
stances in which the two averages will be equal even though their
components are not.
I
shall consider only the simplest and most favorable
case,
that of exponential growth in which:
(27)
I(u)
= e^"
this is at least a case in which one can imagine the stationarity
assumption
to hold with demand and all variables growing at
same rate g
(which does not make it a realistic
16
case,
the
however).
As we shall now see, even in this unreasonably favorable case, it
is
generally
not
that profits/sales
true
equals
the
Lerner
measure.
To see this, define:
so
= e"5^(B - A)
C(g)
(28)
that
is equivalent to profits/sales
=
C(g)
the
we see that, in the exponen-
Letting 9 = t-u,
Lerner measure.
equalling
tial growth case:
oo
C(g)
(29)
=
J
- V*
[rV*(e)
'
(0)
-rV(e)
+
V (0)
]
e~^® de
oo
- f (9)]e 5® d9
[f (9)
where
(30)
f(.)
f (0)
=
-V
+ rV(9)
(9)
should be interpreted as that stream of benefits which would
make
V(.)
(the valuation actually used by the firm)
the
appro-
priate Hotelling valuation.
Theorem 1.
In
the exponential case,
Lerner measure if g = r.
Proof
:
Consider
(29)
at g =
profits/sales equals
the
'
r
On the one hand, since the price
.
of a unit of new capital has been normalized to be unity,
17
oo
(31)
j
f(e)e~^® de = V*(0) =
1
On the other hand.
oo
(32)
/
[-V(e) + rV(G)]e ^® de
= -
V(e)e"^®| =
1
V
for the same reason.
This
accounting
result
parallels that for the
relation
here the cost of capital,
economic rate of return.
return on the margin.)
r,
said.
ing
this
95)
(Of course,
r
except
,
is involved instead
of
the
is the economic rate of
When the growth rate is equal to the cost
of capital, everything comes out all right.
however,
the
rate of return and the economic rate of return in the
exponential growth case (see Fisher and McGowan, p.
that
between
In the present case,
is just about the only positive thing that can be
It is not even true that the weak result that the account-
and
economic rates of return are on the same
side
of
the
growth rate has a parallel here.
This
C(g)
can most easily be seen by observing that
(29)
shows
to be the present value of the difference between two bene-
fit streams, f(.) and f(.), discounted at interest rate g.
Since
there
signs
is nothing to prevent that difference from changing
an arbitrary number of times,
C(g)
deed an infinite number of) roots,
18
can easily have several
(in-
must be one.
Fur-
of which
r
ther, C(g) will generally not be monotonia in g.
and
are unrestricted except by
f(.)
between
and f(.)
firms
reasons,
by considering one way in
this
f(.)
accelerate
are
which
relation
the
is likely to be restricted.
obvious
For
likely to use depreciation schedules
depreciation.
depreciation
all
it may be possible to get a bit fur-
however,
In practice,
than
this is
(32),
These propositions are exemplified below.
that can be said.
ther
and
(31)
So long as f(.)
If
they are able
so
to
which
accelerate
to make the value of their capital stock
as
never
greater and sometimes less than would be the case under Hotelling
depreciation, then
Theorem
definite result does emerge.
a
Suppose
2.
that V(9) £ V*
(6)
It is:
for all 9 2
0,
with
the
strong inequality holding for some set of values of 9 of non-zero
measure.
for growth rates sufficiently close
Then,
zero,
to
profits/sales overstates the Lerner measure.
Evaluate (29) at
Proof.
g
= 0.
Then:
CO
GO
But
V(oo)
=
C(0)
(33)
the
first
=
=
integral
must
[V
J
(9)
-V*'(9)]
integral is zero,
V*(oo).
be
d9+
/
[V* (9)
since V(l)
Under the given
positive.
r
Hence,
=1
-V(9)] d9
V*(l)
=
assumption,
at
profits/sales overstates the Lerner measure.
g =
0,
the
B
>
.
and
second
A
and
By continuity, the
same result is true for growth rates sufficiently close to zero.
19
In
the proof depends on observing that,
essence,
growth rate, the firm is taking depreciation equal to
at
thfe
zero
initial
value of a single capital good, and this is the same using either
valuation.
It follows that all that matters is imputed interest,
and accelerating depreciation must reduce that.
Note that essentially the same proof leads to:
Theorem 3. Suppose that one of two otherwise identical firms uses
a
depreciation
always
less
schedule
which results in
a
valuation
capital
than or equal than that of the other and
non-trivial time periods.
ficiently close to zero,
less
for
Then, for identical growth rates suf-
the firm using accelerated depreciation
will have profits/sales greater than the other firm.
Note that, at low rates of growth, accelerating depreciation
does
not increase the profits/sales ratio.
Of course
this
A simi-
because of the steady-state nature of what is going on.
lar
result
is
holds for the accounting rate of return (Fisher
and
McGowan, p. 86)
As one might expect,
rates of growth,
lar,
the opposite result holds at very high
for high enough rates of growth, a firm accelerating depre-
ciation
relative to Hotelling valuation will have
understating the Lerner measure.
g goes to infinity,
the
values
of
To
profits/sales
see this, observe that, as
the only things that matter in C(g) will
depreciation
installed capital stock.
the
In particu-
although this is less interesting.
and foregone
interest
on
newly-
But the value of new capital stock
same in all valuation systems,
so the sign of C(g)
only on the difference in the depreciation being taken,
20
be
is
depends
and C{g)
will be negative under the stated assumption.
identical
otherwise
two
of
relative
Similarly, if one
depreciation
accelerates
firms
then at high enough rates of growth that
to the other,
firm will have a lower profits/sales ratio.
is monotonic-
It is tempting to conclude from this that C(g)
ally decreasing in the growth rate in such
but this need not be true.
tion cases,
that depreciation is higher,
means
Accelerated depreciation
but it also means that fore-
therefore,
Even in such cases,
gone interest is less.
[f(0)
al values of g at which C{g)
=0.
The Size and Behavior of the Error: Examples
It
large
enough to know that such
not
is
problems
can
exist,
It is important to know whether they are likely to
however.
instructive
very
Consideration
begin by working out a
to
set
examples.
of
of the leading case discussed above in
It is
which
the
effects of capital aging are capital-augmenting shows that it
useful
to
do this by evaluating
times denote by
use (23)
Z)
(B
- A)/B
(which
I
For a Cobb-Douglas technology,
.
shall
is
some-
we can then
to obtain an order of magnitude for the errors.
Examples are fairly easy to generate.
Each example requires
choice of f(.) subject to the restriction that
oo
(34)
be
and whether they will tend to be correlated with variables
used in regressions purporting to explain monopoly power.
a
-
can change sign more than once, so that there can be sever-
f(9)]
6.
accelerated-deprecia-
/
f (0)
e"^® de
21
= 1
This is done in various ways as described below.
each of the examples below is worked out
addition,
In
different
three
types of depreciation schedule.
—
straight line
for
:!
which
f (9)
The first
has:
v'(e) =
(35)
for
-1/T
v(e) =
(T -
e)/T
where T is the life of the capital good (the point at
T,
becomes and remains zero.
The second type of depreciation is the continuous-time equi-
valent of sum-of-the-years
digits.
It has
T - e
V (9)
(36)
•
=
^~
-
V(9)
e(2T -
=1
-
TV2
d
switch
Define
straight line when the latter method becomes faster.
= 2/T and 9* = T - 1/d,
.
T^
The third type of depreciation is exponential with a
to
9)
X
so that d = 1/{T
-9*).
Then
this
method has
V(9)
(37a)
^ 9 i
for
= - d e~^^
9*
V(9)
= e"*^®
and
e
-d9*
V'(9) = -
(37b)
V(9) = (T - 9) d
e""""^^
T - 9*
for 9*
^ 9 ^
Each
in
T.
of the tables giving the results for
two parts.
rate,
r
=
.2;
In the first half of each
in the second half,
22
r
=
.15.
Z
table,
=
(B
the
- A)/B
is
interest
These seem reasonable
enough for illustrative purposes.
in the rate
differ
r
In practice,
that should be used,
preted as the cost of capital including
for
must be
r
inter-
premium.
a risk
we should expect from the analysis above,
As
firms
of course,
Z
=
A)/B
-
(B
is most often greater in absolute value for growth rates far from
r
than it is for growth rates close to
common,
when
interest
it
would be
absolutely
I
values
2
they
mistake to conclude that those
a
the results
In any event,
small.
then to the question of how to judge
given in the tables are high or
Z
indicated,
problems
the
for
in
As
L-L*
already
error
case.
(In
B-A
1-L*
=
the
Dividing (23) by L* yields
roughly similar results.)
(38)
whether
other capital-augmenting cases will yield
view of (21) and (25),
a
)
(
L*
)
(
L*
B
Obviously, this means that a given value of
a
high,
rule of thumb can be given using the
rough
a
low.
formula (23) for the Cobb-Douglas-capital-augmenting
to
are
percentage points each.
come
of
rates are low than when
less
case are given for growth rates from zero to 30%
exponential
steps of
not
of the Lerner measure are likely to be relatively
although
become
a
this suggests that the problems with profits/sales as an
estimate
severe
although this is not
Since growth rates as high as 15% are
property.
universal
r
(B -
A)/B corresponds
greater percentage error for low values of L* than for high
ones, a fact
I
shall discuss below.
Ravenscraft
of -Business
Cobb-Douglas
(p.
31)
gives a mean value for L from the Line-
data of .0648,
excluding
imputed
interest,
estimates for the United States suggest an
23
while
average
value of
measure
actually
exclusion
given
If one were to believe that profits/sales
about .25.
a
on average and were to
L*
of imputed interest,
about
forget
this would mean that the
in the tables below should be multiplied by
the
values
about 3.6
to
obtain
an order of magnitude for the errors as a fraction of the
Lerner
measure.
Because of the exclusion of imputed
this number is an underestimate.
profits/sales
Lerner
point
whole
The
that
is
of
the
It is therefore of some interest to judge
the
of the error without assuming that L and L* are
orders of magnitude.
same
the
Solving
as a function of
error
close
This is easy to do and leads to roughly
together on the average.
express
however,
paper,
far from being a trustworthy estimate
is
measure.
magnitude
the
p
this
of
interest,
for
(23)
Dividing
L.
L*,
one
can
L,
one
by
obtains:
L - L*
1
=
(39)
a
L
where,
again,
Z
- L
(
L
=
(B
- A)/B.
Z
)
)
(
1
- aZ
This gives the error as a fraction
of the observable profits/sales ratio, L.
the
tables
sufficiently
given above
below and for
a
For the values of
approximately .25,
however,
small that the (underestimated) multiplier
in
Z
is
aZ
of
3.6
remains roughly correct (as do the other multipliers
given in footnote
8)
;
somewhat greater values are required when
is positive, and somewhat smaller ones when
Z
Z
is negative.
In sum, the Line-of-Business data suggest that the values in
the tables below should be multiplied by at least 3.6 to be inter-
preted as fractions of observed average profits/sales ratios.
24
If
that the average profits/sales ratio is
believes
one
true Lerner measure
average
(a
dangerous assumption)
same multiplier gives the error as
a
,
also
the
then
the
fraction of the average true
Lerner measure
Table
gives the values of
1
case in which
f (9)
They
pattern.
A)/B for a one-hoss
the maximum value of f(6)
is chosen to
(B
-
A)/B exhibit the simplest
are all positive for growth rates near zero
decline monotonically
passing through zero,
,
for low growth rates,
we should also expect,
the-years'
digits
value of
- A)/A,
(B
absolutely
lowest
depreciation
1
and
2
and
must,
at
above.
As
as they
This is to be expected, given Theorems
g = r.
shay
thereafter.
Here the values of
satisfy (34).)
-
is constant for 10 years and zero
in all the examples,
(As
(B
the use of sum-of-
absolutely
yields the
highest
with straight line depreciation yielding the
values and
exponential
depreciation
values
somewhere in between the two.
[TABLE
is evident,
It
HERE]
given a multiplier of 3.6,
that the values
in Table 1 for low rates of growth are quite large
given
to
1
be worth attention,
and the magnitudes will increase
progress to later tables.
generated
comfort
with
sum-of-the-years' digits depreciation,
small errors
soon disappear.
Table
(on the order of "only"
1
as
we
Certainly this is true for the values
in the notion that straight line depreciation
relatively
enough
but
leads
20% or so)
any
to
will
happens to be one of the more favorable
cases for the use of profits/sales.
Table
2
shows that even what appears to
minor change in the benefit profile,
25
f{.),
be
a
relatively
can make an enormous
difference
Table
in
the magnitude of the errors.
table,
like
case.
The
that now the benefits from the capital good
only
presents the results for
1,
difference
is
That
a
one-hoss
shay
begin two years after the expenditure for that good is
the future.
At low growth rates,
the values of
tripling
-
(B
A)/B generated by straight-line
for sum-of-the-years' digits.
results
the
profits/sales
about
Even in Table 1,
doubling
and
those
at four percent
correspond to lower bounds for
errors
in
at the latter 's average value of from about 24% to
Now,
80%.
ranging
this results in roughly
depreciation and somewhat less than
exponential
growth,
In
the benefit profile of Table 1 is shifted two years
other words,
into
made.
they correspond to lower
in Table 2,
Even with a
from about 76% to about 145%.
much
bounds
lower
multiplier than 3.6, these are large errors.
[TABLE 2 HERE]
The results in Table
percentage
but they illustrate
errors,
In that table,
also correspond to (absolutely) large
3
10 years at a unit rate of decay
of f(9)
different
phenomenon.
has been chosen to decline exponentially for
f (6)
£ 10) after which
a
f (9)
(B -
f (9)
= Ce
for
Again, the maximum value
becomes zero.
is chosen to satisfy
is that the values of
(in other words,
The new feature in this table
(34).
A)/B are negative at low growth
rates
and increase, rather than decrease monotonically with g.
[TABLE
Table
3
profits/sales
growth.
As
shows
that
it
3
HERE]
is a
mistake
to
conclude
overestimates the Lerner measure at low
Theorem
2
suggests,
26
that
rates
of
if Hotelling depreciation
is
relative to the depreciation methods
accelerated
actually
used
instead of the other way round, then underestimation will result.
this regard that Ravenscraft
in
(Note
(p.
31)
finds
that
the
value of the profits/sales ratio in the Line-of -Business
minimum
data is substantially negative.)
As one should expect from Theorem
it
circumstances,
in these
now the least rapid depreciation method,
is
yields
that
3
growth.
the
(absolutely) largest errors at
low
should also come as no surprise that
It
function,
straight
which
f(.)f
corresponds
to Table
3
by delaying
its
increases
the
shift moves Hotelling depreciation closer to the three
examined.
schedules
tion
positive
Such
a
depreciamethods
It makes the errors for some
and those for others negative,
of
the
of the errors for low rates of growth.
values
absolute
rates
shifting
by two years this time reduces rather than
start
line,
but while some
of
the
errors naturally become absolutely small, not all of them do, and
the
size of the errors (or their sign) cannot be predicted with-
out knowledge of the benefit profile,
given in Table
4
This
rate
4
digits
and change sign at rates of growth
in Table
3
{f(0)
are too large (absolutely) because the
27
the
r.
- f(©)}
(29)).
of course, that the values of
to generate them is extreme.
than
other
is because we have now reached cases in which
It may be objected,
used
The results for
depreciation are not monotonic in
can change sign more than once (cf.
given
are
HERE]
also illustrates a new phenomenon.
sum-of-the-years'
growth
These results
4.
[TABLE
Table
f(.).
(B
-
A)/B
example
That example involves a very
rapid exponential decline in benefits, so that benefits are close
to zero for a good deal of the ten-year life of the capital goods
while the firm is forced to depreciate the goods
involved
at
a
non-negligible rate over the entire ten years.
There
an
are three reasons to be careful before offering
First,
objection.
benefits after taxes,
ciation method.
9
such
the example can be though of as one with
including the effects of the chosen depre-
Second, extending the life of the capital good,
as in Table 4 makes things better,
not worse.
Third,
this
as
suggests, the general principles being examplified do not rest on
the choice of particular examples.
for
Table
5
an exponential decay case identical with that of Table
with a life of five years rather than ten.
of
presents the results,
(B -
A)/B
The resulting values
for low rates of growth are
value than those in Table 3,
but
3
smaller
absolute
in
but they still translate into enor-
mous errors in the use of profits/sales.
[TABLE 5 HERE]
is possible to continue with many more
It
examples.
(In-
deed, only the desire to keep benefit shapes, f(.), relatively
and
probably unrealistically
ing
even
simple keeps one from illustratThe conclusion
more complex behavior.)
should now be clear,
liable
—
Profits/sales is
however.
estimate of the Lerner measure.
using it can vary from very small to
they will be,
as
functions
—
to
a
be
drawn
totally unre-
The errors involved
f righteningly
large.
in
Which
what sign they will have, and how they will behave
of
the growth rate cannot be
knowledge of the benefit profile,
28
f(.).
determined
without
No conclusion as to the
monopoly power of
a
firm or group of firms can be drawn from such
a
Further, all these results come from the unreasonably
measure.
favorable stationary-environment case of exponential growth.
7.
Regression St udies and "Random" Errors
This does not end the matter,
however, for profits/sales is
also used as a measure of monopoly power in regression studies in
which it is the dependent variable.
errors in
In such studies,
the dependent variable become part of the error term.
led some practitioners
create
errors
such
them
assume
p.
32)
uncorrelated
with
the
has
to assume that
no problem because one might just
For brevity,
regression.
Martin 1983,
(e.g.
This
variables
well
as
used
the
in
I
shall refer to this as the assumption
a
dangerous procedure.
of "independence".
This
is,
at best,
The assumption
that error terms are uncorrelated with regressors is often
made.
as here, it has no better foundation than the assertion
Usually,
that the investigator has been unable to think of a reason why it
should
not hold.
Where errors are small,
assumption may do little harm.
large indeed,
degression
Where, as here, they can be very
one is risking a great deal.
studies
that
The very fact
a
the
very large variance
make one suspect (although it certainly does
those
that
appear to recover systematic results in
presence of errors which can obviously have
should
such an independence
errors are systematically related to
the
not
prove)
variables
used, even though one would wish to assume otherwise.
In any event, even such a wishful-thinking assumption cannot
be
made in the present case.
The error involved in the use
29
of
profits/sales ratio is systematically related to
the
vari-
the
ables used in regression studies, and the independence assumption
is therefore false.
To
(18)).
This
consider the error formula (23)
see this,
It contains a term in
- L*)
(1
(or
(21)
which does not cancel out.
means that errors will be absolutely larger for firms
of the Lerner measure than for high ones
values
low
effect
or
with
this
(and
will be even more pronounced in percentage terms)
This
.
says that it will not generally be the case that the error can be
taken
to
independent of the variables used to
be
explain
the
Lerner measure.
More precisely, suppose that L* is supposed to be related to
a set of variables, X, by:
L* =
(40)
in
X
P
+6
Assume that
the usual notation for regression.
^
mean
has
zero and is distributed independently of X.
We know that L - L*
- L*)u.
in fact distributed
is in the form
(1
independently of X;
stochastic.
Suppose that
for convenience,
indeed,
For convenience,
independently distributed.
u is
take X to be non-
suppose further that
Then b,
u and
the least-squares
^
are
estimate
obtained from regressing L on X has the property:
E(b)
(41)
Only
=
E{(X'X)'^X'L} =
t
^
(1
- E(u))
if u has mean zero can one assume that this creates no bias
(and no inconsistency
—
the problem does not disappear for large
sample sizes)
No doubt there will be some willing to assume that E(u) = 0,
30
the lack of basis for such an assumption now
but
possibly
make it are asserting that because a
who
Those
exposed.
lies
large
one might as well assume it to
an unknown sign,
be
number
has
zero.
There is no basis for believing that the benefit profiles
characteristic
happen
of
capital goods in the
American
be distributed relative to the
to
economy
depreciation
just
methods
used so as to make the average value of u = a(B - A)/B zero.
fact
(B -
that
The
A)/B can be either positive or negative is not
a
sufficient basis for such a belief.
the absence of such an unwarranted assumption,
In
(also unwarranted)
assumption that
u is
even the
distributed independently
of the variables used in regression analyses will not rescue such
studies.
Assuming that
- E(u)
1
0,
>
then such studies will have
their coefficients biased towards zero if E{u)
zero if E(u)
<
This sort of problem would only be avoided if
0.
the dependent variable were log
such
Even
made
the
in
profits/sales
because
ables
a
- L)
(1
choice of dependent variable
literature
as
—
will
not
—
rescue
commonly
not
studies
a surrogate for the Lerner measure.
it is clear that u itself is correlated with
used
from
and away
>
in such studies.
I
using
This
is
vari-
the
comment briefly on some of
the
effects involved.
(a)
who
recognize
interest
into
Capital Intensity.
—
account
We saw above,
As already observed,
that the opportunity cost of
investigators
capital
needs to be included frequently attempt to
by using the capital/output ratio as a
however,
—
imputed
take
regressor.
that relatively capital-intensive
31
it
firms
While we cannot say
will have relatively high (absolute) errors.
what sign those errors are likely to have,
it is again foolhardy
to suppose that this means they can be assumed zero.
there is some evidence here as to the sign of such
In fact,
errors.
the
If
book value of capital were also the
value (and Hotelling depreciation subtracted from
coefficient
of
risk-adjusted
has
Retelling
profits)
,
the
the capital-intensity variable should equal
the
cost of capital (assuming that
no direct effect on the Lerner measure)
intensity
capital
Leibowitz
.
(p.
8)
points out, however, that such coefficients are often found to be
significantly
negative.
A
distinct possibility is
results from negative values of
Depreciation Methods.
(b)
(B
- A)/B,
methods
that
12
of
with faster depreciaslower
producing algebraically larger errors than
ones at low enough rates of growth.
however,
as in Table 3.
this
It is obvious that the choice
depreciation method influences the errors,
tion
that
Gerald Salamon points
large firms tend to use accelerated
out,
depreciation
This will lead to an upward bias in the coefficient of
methods.
a firm size variable.
Growth Rate.
(c)
As we have seen,
the size of the error is
influenced by the growth rate of the firm.
Studies which do not
include
variable
will
have
errors
correlated with any independent variable correlated
with
growth
rate.
the
growth rate as an explanatory
Firm size or market share appear candidates here,
since they will be influenced by past growth.
Even
tion
above.
the inclusion of the growth rate requires the
that the environment is stationary in the
sense
assumpdescribed
(Departures from stationarity can also be correlated with
32
included variables, but it is hard to say anything specific about
this.)
Risk.
(d)
The value of
difference in the errors.
used obviously makes
r
differs over firms because it is
of capital including a risk premium.
cost
the
Yet
r
At
sizeable
a
low
it
seems likely that firms engaged in
relatively
activities
will have relatively larger errors
(absolutely)
rates,
will
relatively risk-free firms.
Some variables,
growth
risky
than
such as
re-
search and development expenditures, may be related to participation
in risky activities;
others,
such as the variance of firm
or the Hirschman-Herf indahl index may relate to the
sizes
risky
nature of the industry involved.
8.
Conclusion
The conclusion to be drawn from all this must be
can
(p.
91)
on the companion issue of the relations of
accounting and economic rates of return.
them
sedly
I
do no better than to paraphrase the conclusion of Fisher and
McGowan
who
clear.
Economists (and others)
believe that analysis of the profits-sales ratio
much are deluding themselves.
the
will
tell
The literature which suppo-
relates concentration and the Lerner measure does no
such
thing, and examination of the profits-sales ratio to draw conclu-
sions about monopoly power is a totally misleading enterprise.
33
A ppendi x: A Note on the Use of Tobin's q
—
recent papers have used Tobin's q
Several
the
ratio
of
the market value of a company's stock to the value of its capital
—
assets
the dependent variable in regressions intended
as
test hypotheses about monopoly power.
Michael A.
Stephen Ross,
(See Eric Lindenberg
to
and
Salinger, and Michael Smirlock, Thomas
Gilligan, and William Marshall.) Such a use has much to recommend
it
to the use of the accounting rate of return or
relative
profits/sales
ratio;
in particular,
problems relating to
the
risk
disappear as the variable itself includes the market's evaluation
of the required risk premium.
Nevertheless, the capital-theore-
problems which eviscerate the meaning of studies
tic
using
the
other two variables also affect those using Tobin's q.
In
appears
the first place,
in
the value of the capital assets
the denominator of q typically does not include
activities
such
conclusion
tends
This
that
are
to bias q upwards in industries
important and may lead
such activities are
correlated
where
erroneous
the
to
the
development
value of intangible assets such as past research and
activities.
which
with
monopoly
power
More
present
directly
paper,
denominator
torically
even
related
to the problems considered
in
the
assets
in
the
the valuation of physical
of q raises serious problems.
taken accelerated depreciation will show
value for their capital stock,
ceteris paribus
have used slower depreciation methods.
an
Firms that have his-
,
a
lower
book
than firms that
Further, the size of such
effect depends on the rate of growth of the
firms
involved.
Yet depreciation methods may be correlated with variables such as
34
firm size.
is certainly true,
It
the capital-valuation problem affects only the
that
of
as pointed out by Salinger
Tobin's
denominator
q and not both the numerator and denominator
does in the case of the accounting rate of return.
as
Hence a
error in capital valuation relative to Hotelling
percent
160)
(p.
it
one
valua-
corresponds to a one percent error in the measurement of q.
tion
It is not fully clear whether that makes q more or less sensitive
to
such
errors than are the accounting rate of return
profits/sales ratio.
enough
as
I
to
pose
a
the
suspect is true, the sensi-
the errors in valuation seem likely to be
tivity is less,
large
Even if,
and
very serious problem
and
quite
possibly
to
vitiate the use of the measure altogether.
Not
even
Tobin's
q fully escapes the problem
that
invest in capital goods for the future stream of benefits
goods will generate.
ment
cost would help,
firms
those
While valuation of assets at true replacefirms' book values usually do not provide
an appropriate valuation for analytical purposes.
35
References
Fisher, Franklin M., "The Existence of Aggregate Production Functions," Econometrica
October 1969, 22, 553-77.
,
McGowan,
and
John J., "On the Misuse
Accounting Rates of Return to Infer Monopoly Profits."
of
Ame-
rican Eco nQmic Review, March 1983, 21, 82-97.
Hotelling,
"A General Mathematical Theory of
Harold,
Journal
pf
September 1925,
2fi,
tion,"
the American
Statistical
Deprecia-
Association
,
340-53.
Leibowitz, S. J., "The Measurement and Mismeasurement of Monopoly
Power," mimeo, March 1984.
Lindenberg,
Eric and Ross,
Stephen, "Tobin's 3 Ratio and Indus-
trial Organization," Journal of gusiness
,
January 1981, 54
,
1-32.
William F.,
Long,
Lean,
Curtis L.,
Wagner,
David F.,
Ravenscraft,
David J., and
Ill, Benefits and Costs of the
Federal
Trade Commission's Line of Business Program, Volume
Analysis
Federal
,
Trade Commission,
Bureau of
I;
SiafJE
Economics,
September 1982.
William
Long,
F.
Accounting
and
Rates
Ravenscraft,
of Return:
David J., "The
Comment."
Misuse
of
Economic
American
Review. June 1984, JA, 494-500.
Martin,
Stephen, Market, Firm, and Economic Performance
Brothers
Center
for the Study of
Financial
.
Salomon
Institutions,
Graduate School of Business Administration, New York University, 1983.
36
Misuse
"The
.,
of
Accounting
Comment," America n Economic Review
David
Ravenscraft,
Line
^M
Salamon,
J.,
.
Rates
June 1984, JA, 501-06.
"Structure-Profit Relationships
Business and Industry Level," Review
of
St a ti st ic s
Gerald
^
February 1983,
^,
"Accounting Rate
L.,
at
the
Economics
(2i
22-31.
of
Return,
and Tests of Economic Hypotheses:
Error,
Return:
of
Measurement
The Case of
Firm
Size," mimeo., June 1983.
Salinger,
Michael A., "Tobin's a.
Relationship,"
tration-Profits
Summer 1984,
Scherer,
H,
Unionization, and the ConcenFand Journal of
Economics
,
159-70.
F. M., Review of F. M. Fisher, J. J. McGowan,
and J. E.
Greenwood,
Folded, Spindled, and Mutilated: Economic Analy-
sis and
Sj.
iIj.
3L..
Journal ol gconoinic Literaturer June
lEMj.
1984, 21, 620-21.
Smirlock,
Michael,
"Tobin's
a
and
Gilligan,
the
Thomas,
and
Marshall,
Structure-Performance
American Economic Review
,
William,
Relationship,"
December 1984, JAr 1051-60.
37
*
am indebted to Paul L.
I
Joskow for helpful
discussions
but retain responsibility for error.
1.
problem is not limited to constant returns and
The
profits-sales ratio.
Any attempt to calculate the Lerner measure
involve at least an implicit measurement of marginal
must
Such a measure,
Leibowitz.)
J.
For simplicity,
(This point is well made
I
assume constant
that seems a questionable assumption when
though
in
returns
as does most of the empirical literature,
throughout this paper,
even
cost.
if direct, must take capital costs properly into
account to produce a valid result.
S.
the
studying
the effect of firm size.
2.
sure
Some recent papers have used Tobin's q as a simple
While that measure
monopoly power.
of
seems
mea-
definitely
superior to either the profits-sales ratio or the accounting rate
of return, its use does not avoid the capital-theoretic difficul-
ties here discussed.
3.
I
comment briefly on this in the Appendix.
There are other problems with Scherer's
statement.
My
normal strategy for many years has been to weigh 150 pounds.
4.
For simplicity,
I
assume that the price of a new unit of
capital is always constant at unity so that there are no gains or
losses stemming from capital price changes.
5.
the
This
can also be shown more laboriously by
optimal assignment of labor in the Cobb-Douglas
considering
case
given
in
which
(19).
6.
This
valuations
proviso
essentially rules out the case
are discontinuous and the valuation used by the
38
firm
is below Hotelling valuation only at isolated points.
For simplicity, the entire analysis and the examples have
7.
been worked out in before-tax terms.
applies if after-tax
analysis
same
the
interpreting
f (.)
and McGowan,
pp.
It is not hard to show that
magnitudes
as the ^£i£x-tax benefit stream.
95-97.)
Of course,
are
used,
Fisher
(See
in that case the examples
would involve different pre-tax f(.) for different
below
depre-
ciation methods.
Since Ravenscraft (like others) does not find a
8.
sion
coefficient
interpretable
for his capital/sales variable that is
such
that
easily
it is not simple further to adjust the
as r,
figure to take account of imputed interest.
ever,
regres-
It is
3.6
how-
clear,
an adjustment for any reasonable value
of
r
would make the appropriate multiplier very much higher.
Ravenscraft
At the maximum value for profits/sales given by
(.5371), the lower bound for the multiplier is approximately .22;
the
minimum
so
that
the
corresponding to the low end of the range is unbound-
multiplier
ed.
value for profits/sales is negative,
For comparative purposes, using Census of Manufactures data
(Ravenscraft,
p.
30)
,
the lower bound for the multiplier at the
mean of L (.2188) is be roughly .89;
is about
these
.34;
at the minimum
(.0096),
at the maximum (.4232),
it is about
it
Even
25.8.
lower multipliers make the errors in the tables below loom
very large.
9.
value of
10.
See
r
footnote
7.
It should be noted,
however,
that a
of .15 may be a bit high for an after-tax example.
It may still be objected that a decay rate of unity
39
is
very high, although how one is to know that without any empirical
evidence
on f
(
. )
If lower decay rates are
is not clear.
used,
depreciation becomes less accelerated relative to
Hotelling
three depreciation methods examined.
In the limit, a zero decay
rate corresponds to a one-hoss shay example as in Tables
As we have seen,
the
1
and 2.
such an example generates positive, rather than
negative errors at low rates of growth.
Correspondingly, as the
decay rate goes from unity down to zero,
the errors at low rates
of growth move upwards from negative values to positive ones.
course
there
is some decay rate at which such errors are
small
(although that rate will not necessarily be the same for each
the
three
depreciation methods used)
.
It would be
to suppose that such a decay rate must be the
however,
Of
of
foolhardy,
"realis-
tic" one.
11. There is also the probability-zero case that u is corre-
lated with the regressors in just such a way as to cancel out the
effects of
12.
A)/B
and
(1
- L*)
.
There is no need to discuss this.
It could also result from a negative correlation of
capital intensity stemming from a systematic
between capital intensity and the shape of f(.).
40
(B -
relation
\m
c
C
o
CO
O K
E
M
O
f-H
r»
00
00
o
r*-
V
r~
o
00
o
vo
o
o
r»
^
o
i-H
n
CD
CN
o>
in
CN
vo
n
(N
m
CN
CM
cs
Ol
i-(
•-(
i-<
1^
o
QO
re
m
n
n
o
^
(N
r-
o
o
00
a^
i-H
O
<H
n
V
o
o
in
tc
o
o
in
vo
o
o
o
<N
^•
•a-
•-I
I-H
o
o
O
O
i-H
o>
in
^H
in
r-)
^
in
in
vc
vc
o
o
I
I
I
vo
in
CO
vo
•«•
eri
o\
—i
tH
O
^
^
o
o
o
o
^
o
o
^
m
o
o
•-H
in
o
o
o
o
o
I
t^
o
^
CN
r^
r-
r»
r»
00
I
I
o
o
o
o
O
o
.-(
in
rlO
iH
^
o
O
o
1
1
in
vo
r-
«)•
Ol
a
u
r»
4J
C
-r-
re
—1
n
o>
o
o
r»
o
00
o
o
CO
l~
\o
o
o
o
in
in
o
o
•a-
O)
^
o
o
o
o
o
ro
o
00
r~
>-i
o
o
00
in
o
o
o
00
in
o
o
o
1
fN
a
^^^
^-^
<
tfl
o
o
o
n
vo
o
00
O
o
VO
•H
<H
o
r~
rH
o
o
1
00
<H
o>
r~
vo
00
^•
fn
lO
(-«
o
o
o
CO
in
1
1
1
in
CO
rsi
o
n
o
o
1
O
tN
<N
CM
o
o
vo
o
o
1
IN
VO
rs
n
T
in
•H
tn
^T
.^T
o
o
oo
o
n
OO
w
N
CO
1-1
CQ
^«^
0)
iH
^
1
14-1
(U
o
c
re
H
o
o
to
01
3
l-l
r-l
re
0)
re
p-
c
0)
c
o
> o
1
Q.
lu
01
o m
i-i
E
QC
re
40 >•
t-i
o
m
o\
in^»
•-(
iH
fH
<H
o
O
O
^
00
ov
Ol
o
o
00
^•
CO
o
o
o
o
o
o
r>-
"»
m
in
o
o
o
o
iH
r>-
<<r
CN
o
o
o
o
m
n
iH
O
O
O
O
O
O
O
at
f-H
in
cs
o
o
m
o
o
1
1
1
CNJ
O
o
^•invovoo%mom«*>ooo^
CT>o^<Nf'iinr~«cMinoin
oo
cj
O
o
m
o
VO
o
o
in
c>»
o
o
o
m
o>
o
o
I
I
I
I
I
o
00
C>4
•H
o
o
O
o
r>-mr>4ovvomor--incMOCM
Cs»C>»CNfH.-|.iH»-HOOOOO
O
r»-
r>-
o
r^
iH
•-(
o
a
Ol
in
o
n
cs
•«
c
•-<
>-l
re
•<H
o
o
vo
vo
o
^
o
^r
<N
o>
o
o
n
CO
r~
o
o
in
T
VO
o
o
m
o
o
o
in
vo
p~
r~-
^•
e*i
CN
o
o
o
o
f-i
M
^
O
O
o
o
o
o
o
r^
o
o
«N
o
^
o
vo
o
CO
o
o
fH
(N
iH
<a>
vo
r-(
•-(
CO
<H
o
CN
1
1
n
r-i
fS
o
o
1
oo
o
o
m
n
o
o
.-1
CM
vo
CNl
•
•
1
o
n
ir>
c
o
c
in
o\
CM
o%
o\
oi
CM
00
m
vo
fN
o
m
ON
ro
o>
,-t
1-1
i-l
r^
m
in
f-H
m
o
r~
o
vo
CM
o
c
c
X
u
en
CO
k'
j-i
-H
oo
•H
vo
oo
n
^
m
V
m
,
iH
CO
o\
fs
O
o
m
"»
o>
ON
vo
^^
rsi
i-i
i-t
fH
E n
3 0) —
en >- O
rOl
00
o
o
\o
o
n
o
o
CM
vo
i~
O
n
vo
(^
eo
.-I
i-l
o
in
o
n
«o
in
in
O
o
o
o
o
o
o
o
I
I
I
1
I
I
I
^
d
m
vo
m
^
vo
«N
r»
vo
1-^
CM
o
o
^
Ot
o
n
o
•H
o\
n
in
n
1
1
1-1
o
o
o
o
1
1
r^
vo
00
f-i
m
m
o
m
o
^
o
1
1
O
^•
vo
CS
CN
CO
CN
O)
OV
<n
o
o
m
o
1
1
1
VO
CN
00
CN
o
ro
CN
CN
vo
in
iH
CN
CN
in
o
o
I
I
I
VO
r*
en
vo
CN
•
II
u
o
^
bC
TO
vo
^
m
o
o
vo
(U
cs
x:
w
CM
rH
r-<
^
•»
vo
o
a
Ot
H
CN
O
o
in
CNI
fS
o
o
CO
CN
a\
vo
^•
00
o
o
iH
o\
o\
vo
fH
O
O
1
1
1
o
CN
«N
CN
l-i
o
00
i-t
o
"»
o
M
o
I
(U
c
o
u
to
CN
^
^
00
^
O
oo
O
00
VO
o
o
^
CN
I
<
O"
I
a
>1
CO
s
o
Si
cu
td
H
3
o
c
c
o
a.
X
>
cd
0)
a
fH
r»
00
n
in
n
rCN
n
o
•«•
o\
CN
.
CO
03
m
CN
V
^
CN
CO
•H
CO
CN
1-1
o\
o\
rn
iH
VO
in
o\
o
o
a\
^•
o
o
o
o
O
in
^
in
O
o
vo
in
o
iH
I
^
o
n
00
CN
u
n
^
U-J
CO
O
U
E re
::
W
en
OJ
Q)
-H
00
-H
>-
Q
00
r~
«T
•
o
n
iH
^
^•
•
o
o
CN
o
>»
n
o
•
•
o
vo
n
o
m
vo
^
ro
•
o
o\
CT>
VO
CN
•
o
o
iH
CN
CM
•
O
VO
en
vo
i-t
•
o
vo
in
fH
i-l
•
O
•-I
o\
in
o
o
•
o
o
o
o
o
•
CO
H
VO
o
o
m
1
n
vo
O
o
o
a
vo
n
o
o
1
1
1
1
CN
•H
•
n
ov
r-4
•
•
•
CM
II
U
"91
(30
lU
•H.
C
re
I-
CN
VO
cn
-H
o>
«r
m
m
in
in
o
m
n
^1
r>-
CN
^
CN
VO
TT
CN
CN
CN
rH
00
o
o
CN
o
vo
o
00
o
o
^
OV
VO
^
O
O
n
1-1
00
CO
CO
in
m
o
o
o
o
o
I
I
O
CN
CM
fH
CN
o
o
I
I
VO
CN
00
CM
vo
CM
CO
«c
o
c
c
CO
E
3
ij
cn
^j
—
o
(0
oo
-H
01
•
W > O
o
o
1
1
1
1
n
m
vo
in
—1
vo
a\
CN
CO
o\
r~
•-t
o
o
m
o
o
t
n
•»
<9
o
^
m
ro
o
r-
l-H
in
»»•
lu,
r>
r~
in
CM
vo
o^
CN
in
CN
•
o
o
CN
•
o
o
.-i
•
o
u>
o^
CO
\o
CD
(N
»—
1—
o
CN
fH
.
««•
o
o
O
o
o
1
1
1
vo
•
1
•
•
r-i
n
CN
o
o
CM
fvl
CN
o
o
r^
m
lo
o
in
f-H
i-(
o
n
r»
r—
VD
CO
vo
a>
m
n
04
•-(
t-(
I—
I—
CN
00
ro
a\
CO
vo
t-i
n
CO
o
i-<
fH
r-t
o\
o\
IN
^>
n
en
^•
in
•H
lO
O
O
o
vc
o
»-H
co
^r
fN
•J"
00
m
1
II
u
00
00
01
•H
C
10
-H
vo
CN
in
o\
vo
CN
•T
o
o
n
o
1
1
1
»—
i-H
^•
o
o\
CN
CN
vo
in
vo
vo
r»
V
m
CN
in
CN
vo
CN
o\
»*>
i-t
i-H
o
o
o
1
1
1
VO
00
O
CN
vo
00
O
•-<
<-l
r-t
r-(
m
r»
o
o
^•
CN
o
o
o
o
o
o
n
o
T
CN
o
m
CN
00
m
•H
<-i
CN
r~
•H
o
o
o
O
o
CN
CN
«»•
CN
VO
CN
CO
CN
o
o
o
o
o
n
r~
n
o
o
^
en
CN
CD
rH
in
to
iH
iH
fH
o
O
O
r~
rcn
n
CN
'T
m
m
•a-
r-
"O-
o
O
in
CN
O
4-1
CO
CD
1
<u
u
(U
c
n
o
o
CN
O
o
O
CN
O
**^
y-K
tn
00
•H
>H
0)
< o
n
^-^
.-1
n
•H
0)
rH
^
VM
4.1
o
to
H
c
c
o
a.
«
(U
CO
0)
^3
W u
> u
,
c
01
c
n
o
m
i~
^
n
fH
o
^
O
n
CO
n
o
1
1
1
O
vo
in
in
o
1
a.
o
rH
CN
VO
T
iH
fH
m
o
m
o
m
vo
^'
CN
VO
r00
fH
O
o
O
o
o
1
1
1
1
1
in
CO
cn
vo
in
in
1
•a-
fH
«N
in
CO
r-
o
o
o
^•
.H
r-
o
O
o
'
n
m
1
Ed
<T>
B
3
W
in
in
cn
00
CO
n
1-
-H
•
•
(0
cio
T-i
LU
0)
PH
Q
CN
VO
CO
CN
CN
CO
o\
fH
<Tl
^
^f
CN
CO
in
iH
CN
CN
fH
•
•
•
in
o
o
o
o
O
o
o
O
m
o
o
1
1
1
1
1
1
1
1
•
r»-
CN
o
o
o
o
1
1
•
o
o
o
o
o
o
o
o
o
o
o
V
CO
m
o
o
o
CN
«N
•
in
CN
•
O
o
•
o
o
*
n
o
o
o>
•
rH
fH
•
O
tvi
II
JJ
^00
(U
c
•H -H
(0 rH
W
(30
r»
vo
vo
vo
in
vo
vo
in
•
•
iH
VO
P~
o
o
^
o
1
1
1
o
o
CN
O
•
vo
^
cn
n
o
1
•
VO
o
00
o
ov
n
CN
o
n
n
CO
CN
in
CN
a\
fH
rH
•
•
•
•
n
o
o
1
1
CO
o
O
O
CO
r~
CO
^
•
•
o
o
1
tn
•H
O
O
•
vo
n
ro
o
•
o
vo
o
ov
in
n
n
vo
fH
fH
rH
•
•
«•
•
o
O
O
VO
CN
CO
CN
n
1
CN
^
CN
o
r^
CM
'T
CO
r-t
1—
•-i
m
a.
U-
CD
o
j-i
CO
E
=
oo
ra
—
0)
W
Q
><
m
o
CO
<N
00
•
o
O
1
1
r-
«N
o\
rn
«-<
•
•
o
o
o
o
o
•
O
fM
0\
r~
o
o
•
r-
ID
*r
ro
V
m
o
o
•
o
o
m
o
o
o
o
o
I
1
1
•
1
CM
o
o
o
f-1
•
vo
m
o
o
o
o
n
o
o
CO
o
o
o
CN
m
o
o
•
1
CM
in
i-H
»—
in
"•
(N
5
o
'^
i-i
o
•
in
o
1
fN
r»
o
r^
vo
-<
O
II
r>-
l-t
ts
«N
o
o
o
o
o
V
o
o
1
1
1
i-H
o
o
fM
in
-I
•
(T,
in
-<
O
CO
<N
o\
tN
-H
O
o»
r»
O
O
CO
o
o
o
O
o
m
o
o
in
T
T
O
o
o
o
o
o
1
1
1
1
1
•
•
•
«o
r~
in
•
CN
CD
I
II
i-i
c
o
•H
d
to
1-t
<u
a
o
^
i-H
CM
«N
r^
iH
o
o
I
I
r~.
in
vo
ri
i-(
m
o
V
o
o
o
m
CM
o
o
o
o
o
I
1
1
1
,-(
o
i-t
vo
CM
\o
^
r»
o
•
•
in
r~
•
r>
\o
o
o
o
•
f«-
tiH
o
o
•
o\
in
CM
o
o
•
^
«H
n
o
o
•
in
^«
ro
f
o
O
o
•
O
o
cn
0)
c
o
o
o
a.
ea
^«»
^
<
uX
to
C>l
o
o
00
o
CO
o
CM
CM
CM
^
VO
CM
CD
CM
u
to
1
•a-
>*•
o
1
(U
r-i
^
IW
O
ts
CO
H
ca
^
3
•o
fH
to
>
<u
:%
JJ
o
c
c
o
(U
.-H
O
Cb
u
>>
o
^
0V
fH
CO
^•
^
1-1
1-1
iH
o
o
O
^
CO
o
o
1
1
1
1
in
in
C3
"fll
o^
iH
iH
tN
r*
-H
m
CO
rH
CO
r-i
VO
CM
VO
"B"
n
O
O
o
o
r*
00
CM
o
o
<-i
r~
iH
O
o
VO
CO
o
o
o
o
m
o
o
o
o
o
o
o
o
lO
o
o
o
o
iH
iH
O
O
O
O
o
o
o
in
ov
o
o
o
iH
r»
CO
iH
O
O
1
'
.H
OJ
Q
to
(U
>-"
IW
CM
o
E
3
CO
tn
vj
to
OJ
to ><
4J
•H
00
•H
a
o
o
^•
o
o
o
O
O
o
o
ro
O
CM
O
o
"«•
.
iH
tN
O
O
VO
o
CM
o
o
o
00
1-1
o
o
r*'
VO
m r*
^• o
o o
o o
o
o
o
o
o
CM
OV
o
o
o
1
1
o\
o\
l-t
o
CM
in
o
o
o
VO
o
o
1
1
1
VO
in
CM
n
o
o
n
o
o
1
1-1
0V
It
u
o
000)
to
-H
ov
CM
CM
O
1
00
o
o
CM
"J"
r*-
^1
^T
CO
VO
^^
iH
l-t
^
o
o
o
o
o
1
1
o
VO
PO
1-1
r»
^•
CM
o
ov
^1
m
CO
VO
o
00
o
o
o
1
o
o
1
CO
CO
CN
o
o
n
VO
1-1
o
o
1
CO
VO
o
o
o
1
o
o
o
o
o
CM
1*
o
o
o
o
VO
o
o
o
o
o
o
m
o
o
o
1
1
o
CM
•
o
tN
CM
T
CM
VO
CM
CO
CM
o
ri
m
n
•-H
IT)
t-i
1-^
•—
O
o
vo
c
G
C
o
c
\D
o
o
fN
m
o
o
o
o
o
o
o
o
o
o
1
1
1
1
1
1
^
1
00
u-i
en
O
4J
tn
tj
-^
y:
1
x:
oc
0)
ov
On
ON
•H
C
CO
-H
w
n
on
1
•
1
•
1
•
1
•
o
o
—t
CN
in
o
o
><
-w
ON
o\
o
o
o
^
o
o
r~
"T
vo
o
o
M
Q
E «
3 01
vo
t~
r^
o
ON
o
o
o>
Ol
vo
•
ON
•-)
vo
vo
•H
•
•
o
o
<r
0\
in
r-l
,-i
i-t
r-t
o
o
o
o
O
o
I
I
I
I
I
<*1
f-H
Oy
CT>
in
in
O
o
o
o
o
1
1
1
in
00
m
«o
00
CN
00
CM
f—
o
VD
.H
00
00
»r
00
•
O
o
o
Iin
o
o
o
•
4>
o
o
V
d
o
o
o
o
I
I
I
00
VD
CSl
.
O
o
Ol
in
i-i
n
o
o
1-1
i-H
vo
<N
•
•
O
o
vo
tN
r-i
vc
^'
o
o
CM
m
o
n
m
o
00
o
CO
en
LD
o
o
.H
VD
n
o
o
•
00
CN
r~
(N
tn
r-
ON
Ol
00
o
o
o
o
r~
in
^T
o
o
•
r>-
vo
o
^H
•
o
o
f-H
rN
in
in
VD
o
^
o
o
p»
On
O
o
o
o
m
n
O
iH
r^
CN
.H
CM
vo
CN
00
CN
O
CM
m
o
o
o
p~
^
o
o
n
.-(
i-H
in
VO
o
o
r>-
CO
CO
CN
00
ren
vo
vo
^"
n
CO
O
o
o
n
o
o
vo
rS
ON
CN
CN
•«
CN
O
iH
ON
o
«»•
iH
CD
ID
U
>-^
0)
c
CO
^.^
/-v
<
1
m n
«
H
^O
00
vo
o
CO
o
O
o
1-1
cs
iH
CN
o.
n
(U
a
Nw^
.-H
U-l
H
xj
ra
0)
—1
XI
o
o
•H
o
3
c
01
c
o
i-l
o.
0]
0)
ffl
>
X
O
^
•H
CN
I-l
m
o
vo
^
O
o
o
^
m
^
o
1
1
1
iH
r^
C
n
c
Q
a.
X
>>
1
)
ON
vw
CO
O
tn
XJ
u
B
D
TO
01
-^
OO
•H
W
>•
Q
m
•-I
.-(
r«-
CO
oo
CN
O
o
o
o
CJN
p«-
vo
r~
p>.
-
CO
"T
rt
•H
O
n
ON
o
o
1
1
-H
n
o
o
r*'
1
vo
n
m
o
o
1
m
in
vo
o
o
o
o
o
I
I
I
1-1
in
ON
^
"O-
r~
d
o
o
1
vo
o
o
o
o
o
O
o
I
1
I
I
in
CN
O
O
o
o
o
o
o
1
1
r-l
CN
o
^
O
o
I
O
o
O
O
^•
vo
»H
O
O
ON
ON
o
o
O
o
o
o
o
n
u
JJ
^i-
in
in
^
o
_f^
CN
CN
CN
in
ON
»H
•
•
oo
•H
re
U
0)
C
-H
-H
jj
&0
m
r»
vo
vo
^
o
O
O
o
1
1
1
1
o
o
CN
o
vo
o
r~
ON
CO
o
m
ON
00
in
VD
vo
CN
^^
CN
•
•
•
o
n
-H
^
O
o
o
1
1
1
n
i-t
•
00
o
•
•
o
o
o
o
oo
o
o
o
o
o
o
o
o
•
o
CN
ON
ON
•H
•
CT>
•
o
o
V
ro
o
•
O
o
VO
CN
CO
CN
m
rin
•
•
O
S'SOk
c
Of+5
MIT LIBRflRlES
3
TDflQ
QD3
Dt,2
Bbfl
j^CLr
C^^
^-5
Oy\
\?0XLk~
c^^^r~