Onderzoeksrapport nr. 7G06
THE MARKET VALUE RULE AND THE PERFECTLY COMPETITIVE
MARKET STRUCTURE UNDER UNCERTAINTY
by
G. VAN HERCK!!e
x Assistant, Department of Applied Economics, Katholieke Universiteit Leuven.
We gratefully acknowledge the suggestions and criticismof Prof. R. De Bondt,
Prof. E. Famat Prof. M. Van Acoleyen and Prof. L. Vanthienen. Our gratitude
goes likewise to T. Vermaelen for his remarks. All errors remain our
responsibility.
Wettelijk depot
D/1976/2376/15
THE MARKET VALUE RULE AND THE PERFECTLY COMPETITIVE MARI<.BT
STRUCTURE UNDER UNCERTAINTi
by
G. VAN HERCK
0. INTRODUCTION.
In order to explain and predict the behavior of the firm, a number
of theories have been developed, but only fairly limited considerations have
been given to the effects of risk.
In the literature on the theory of the firm under uncertainty, O"Q.e
generally maximizes expected utility of profits, assuming one variable to
be stochastic; see e.g.
LRAND [ 10} considering uncertain demand and
SANDMO [ 16 ] , TURNOVSKY [ 21 J and BATRA-ULLAH [ l ] analyzing the case of u,ncertain output price.
Considering a normal distribution for the uncertain variables, FAMA [ 4]
developed a two parameter (mean-variance) portfoliomodel.
He derived
a~
equilibrium market share value equation in risk-return terms, assuming that
all investors are risk-averse expected utility maximizers.
In a perfect
capital market, he defined the market value rule as follows : management
attemps to maximize the current market value of the outstanding securities
•
. '
It implies that without debt financing, what will be assumed, the market value of the existing shares will be maximized.
The main purpose of this study is to gain insight in the perfectly
competitive market structure and to improve decision making under uncertainty with the market value rule as the objective function.
A~ ready
a felll' authof:s
have been exploring this topic.
STEVENS [ 19] considered a perfectly competitive firm facing an uncertain selling price.
He concluded that despite the risk in the system,
- 2 -
the condition for cost minimization is identical to the one derived under
certainty.
He further incorrectly claimed that the cost function is non-
linear in a risky world when there are constant returns to scale.
SCHRAMM and SHERMAN [ I 7] demonstrated that a value maximizing firm
will prefer greater sales and grow at a faster rate than a profit maximizing
one, when the discount rate is a negative function of the sales and the
growth rate.
KORKIE [ 8] wrote a critical comment using the market value expression
to illustrate the incorrectness of the principle of increasing uncertainty
for a linear stochastic demand function.
total expected revenue increases, ·•·
This principle states that "as
seems natural to expect that the
riskiness of total revenue will increase ••• ".
None of them described the special characteristics of a perfectly
competitive market structure under uncertainty.
As will be proven, STEVENS'
incorrect statement about the cost function is a consequence of this.Their
results are sometimes critically dependent on rather restrictive assumptions;
see e.g. SCHRAMMand SHERMAN's assumption about the
nega~ivity
of the rela-
tion between the discount rata and the sales or growth rate and STEVENS' consideration of only uncertainty in the output price. They derive only a few
results whereas it is our belief that much more can be done.
This study consists of three parts.
First, we discuss the relevance of the market value rule.
assuming ,nerfectly compe.titive
~markets
Second,
and uncertain prices, the equilibrium
share values are derived in terms of firm variables such as sales, costs,
in-and outputs.
This derivation will indicate how risk can be taken into
account in the decision making process.
Third, in analyzing the derived share value equation and considering
it, relative to profit maximization as a profit-risk function, the following
results are derived characterizing the me.!m-variance uncertainty perfect
market structure.
- 3 -
It will be proven that in an uncertainty framework, the curvatures
of .the revenue-risk functions and factor cost-risk equations correspond
in general with the certainty perfectly competitive market structure.
The
differences between the certainty and uncertainty analysis can be explained
by the inequality between the prices and the average and marginal revenue(or factor cost-) risk and by the strict dependence of the slope of the
total revenue - (and factor cost-) risk functions on the riskiness.
Next, we demonstrate the existence of externalities in a multi-industry economy.
An expanding firm becomes more risky whereas the risk qf
the off-setting firm decreases with the same amount, such that the equilibrium value remains the same.
Finally, it will be shown that the certainty relation between
co~stant
returns to scale and a linear cost-risk function remains unchanged under
uncertainty, thus disproving STEVEN's claim referred to earlier.
I • IRE NARKET VALUE RULE AND THE FIRM'S OBJECTIVE FUNCTION
The market val4e rule states that the firm attempts to maximize the
current market value of the payments to its current owners.
The words
11
current ov.-ners" should be stressed, i.e. not for future
owners. In perfect capital markets new securities to finance some operating
decisions are always sold at prices that "fully reflect" these operating decisions.
Because of the "second separation principle" [ 4] which implies that
the optimal operating decisions are independent, or separable from the shareholders' tastes, management will without looking at the individual utility
functions, make the same investment and operating decisions that each stockholder wcruld make if he were running the firm himself. So, ro.a.nagement just
has to maximize the current market V'alue.
-
1;. -
Comparing the market value rule with objectives used in traditional
price theory, one should make a distinction between the certainty and uncertai~ty
case.
Under certainty, profit is the commonly used goal.
Because the cer-
tainty market value rule reduces to ma:x:imizing the present value of future
prof
, there is a strong correspondence with the ordinary neoclassical
profit goal.
While the search for an alternative to profit maximization as an explanation of firms' behavior has been motivated by a des
lism,
for greater rea-
is surprising that in developing a theory of the firm such a limited
consideration has been given to the effects of risk.
Not only therefore should research be situated in a risky world but
also becaus'e some authors [ 12 1 [ ! 7 J pointed out that uncertainty and. risk
aversion in a profit-maximizing context
rr~y
account for many of the pheno-
mena so far in the literature attributed to other than profit maximizing
goals, like sales maximization.
J. LINTNER [ 12] proved for a price-setting
firm that the expected quantity to be produced and sold
~ill
be greater the
greater uncertainty regarding sales and the more risk averse the management.
In a total different context SCHRAMM and SHERMAN [ 17] concentrate on the influence of advertising and research on the riskiness of the profit stream.
By assuming the discount rate to be a negative function of sales and the
growth rate, they found that there is no need for the sales and growth
maximizing objectives to explain the behavior of the firm because a firm in
a risky situation will be motivated to get a higher sales and growth rate
level.
In the model presented only situations are considered in which the
firm is able to summarize its future results by looking at only two pararr..eters of the distribution and thus ignoring other as pee ts of the distribu-,
tion. One special case here assumed in which such an approach
legitimate, is when all distributions are normal.
However the fundamental results
- 5 -
obtained under this assumption can be generalized to all other two-parameter symmetric stable
distributions~
from which the normal one is a special
type [ 4] •
Because of this restriction, ROTHSHILD and STIGLITZ I J6] recently
criticised the mean-variance treatment of the uncertainty problem :
11
The
answers of mean-variance anlysis are spurious ; they hold only if the utility function or the class of distributions is
arbi~rarily
restricted".
First, it should be mentioned that our approach, although it is a
mean-variance one,in no way is restricted to the case of quadratic utility
functions, which has been shown to be theoretical unattractive because it
implies negative marginal utility of consumption (or of profit) beyond some
point.
Second, it's typical that the general treatment in the literature
retnains almost completely restricted to the case of only one variable stqchastic at the same time, which in our eyes is a much stronger arbitrary
restriction.
Last but most important, the uncertainty literature maximizes almqst
1
always the expected utility of profit. Management just examines the ex~
pected utility associated with every possible decision and then chooses the
one that maximizes expected utility.
From a decision- making point of view
the firm faces the impossible task of examining in complete detail the probability distribution of profit.
STEVENS [ 19j, SCHRAMM, SHERMAN [ 17], and KORKIE [8]also were maximizing the market value of the firm. Although the basis for their
objective function was the SHARPE-LINTNER [ 11 J { 18] market valuation
equation, the way they tried to introduce "the theory of the fir.m" and to
set up the objective function is incomplete but this does not influence
their results. The market risk and risk premium should also be expressed
in firm variables. Since they assume that an individual firm cannot
influence them, the:tr results remain valid.
- 6 -
2. THE SHARE VALUE EQUATION IN TERMS OF SALES AND COSTS - AN INDICATION TO
PR4CTICAL RISK QUANTIFICATION.
Under the two basic assumptions that all investors are risk averse
and maximizing expected utility, the equilibrium share-value (or firm value)
equation will be shown to be a function of the sales and cost amounts of
the firm and their riskiness.
Although there is an infinite time-horizon, the investor has to make
a decision only at time 1.
At that time each investor i should choose the
optimal values of his immediate consumption (c 1i) and of his investments
(X .. , the fraction investor i holds of the market value P. of firm j at
lJ
J
time 1) th~t maximize the expected utility of consumption at time l and 2,
subject to the constraint that the total of his current consumption and
his investment equals the market value of his wealth at time I (w i).
1
Assuming that for any ·choice of Xij' the distribution of c i is
2
normal, the expected utility associated with any combination of eli and a
!Jrobability distribution of c 2 i can be summarized in terms of a function [ LJ, 1
,.._,
2"'
U. (c 1 . , E(c .), a (c .)). Expected utility is assumed to be an increasing
1
ll
.
21
2l
functim1 of consumption and a decreasing one of the variance of consumption
(ceteris paribus).
Because of the infinite horizon, profit,sales and costs are assumed
to be constant perpetuities which requires that zero-growth is assumed.
-
Let E(V.) be the expected income for the current shareholder of firm
l
j at time 2 ••• oo,
The expected consumption for investor i at time 2 is thus
- 7 -
[ I
x..•
J
l.J
E(V,)
J
and the expected return for the shareholder in firm j is
E(R.)
J
E(V.)
J
=
P.
J
So far, the model corresponds with FAMA's [4] portfoliomodel.
The mathematical formulation of the expected utility maximization is
[ 3]
max. U.l. [c l,.,
X ••
1
N
N
~
l:
X. ;_E.(V .)
j =1 l.J .:. . J
'
j=l
N
l: X ••
s=l J.J
x.
,...,
~s
""' Vs) 1
cov (V.'
J
l.J
eli
s.t.
wli
-
N
1,;
j=l
X ••
l.J
P. -eli
J
"" o.
In order to relate this model to the theory of the firm, what's our
purpose, some assumptions about the market structure and firm characteristics are required.
- 8 -
All markets are perfectly competitive.
of individual firms have no effect on prices.
This means that decisions
However therefore the other
firms in the market must react on one firm's changes.
This fundamental
reaction principle has been violated frequently in the past in studies of
JENSEN and LONG [ 7 ] ,
STIGLITZ [ 20] and
STEVENS [ 19 ] •
Because a finite number of firms does not prohibit perfect competiit is assumed that there are N firms in the economy. This ecotion [ 6}
'
nomy is split up into industries. The stH.dy will be done from the point of
of one firm j belonging to an industry of n firms. Each firm is
view
producing one good. Firm j has a neoclassical production function
q
= q(K,L)
which is concave.
Uncertainty is restricted to all market prices, i.e. the output price
(p) and the labor (1) and capital input price (k), the distributions of
which are normal (and the investor's expectations about it are assumed to
be the same).
These market output and input prices are statistically inde-
pendent both on the level of the firm and on the aggregated level.
Because of the gene.ral equilibrium context, we also make some assumptions about the input markets,and tflieiin relation to the output markets.
exists many kinds of inputs e.g. high - or low qualified personnel.
There
To
simplify our risk-expression (cfr. expression [8]) we assume that each,
firm is using one kind of input and that there is a one-to-one correspondence between the in- and output market, i.e. each kind of factor is just
used by one industry and each industry uses only one kind of factor.
Some new concepts are used.
"Cost-r:i.skn refers to the sum of the ex-
pected factor costs (outlays) and the "riskiness" of the factor costs.
In
- 9 -
value terms, this sum is the "true" total factor cost.
~eant
is
unit.
With "riskiness"
the amount of the risk units multiplied with the risk premium per
On the other hand, "revenue-risk" is the difference of the expected
sales and the "riskiness" of the sales.
Finally$ it is assumed that the total factor cost-risk is positive
because ''costsn usually indicate "something forgone" and not "something to
receiven.
When L. and K. stand for the labor respectively capital input of
J
J
firm j, expected income can easily. be related to the firm variables because
it equals expected profit if the firm is completely financed with
[ 4]
E(V.)
J
= E(p.).q.
J
J
- E(l.).L. - E(k.).K.
J
. J
J
J
u
-
E (p )q
j
j
equity~
- C (q)
j
When people are acting according to the assumptions underlying this model,
1
the equili~rium share·value expression is :
[5J
P. ""
J
I
. Rf
"'
{ [ E(p.)q. -
J
J
......,
C(q.)]
J
E(Vt) - Rf pt
az
"'
""'
cov(VrVt)}
cvt>
where
N
[6
J
p
::0::
t
k
j=l
P.
.J
The derivation of the equilibrium prices can be checked in Appendix.
- 10-
N
[ 7l
[ 8]
=
E(V t)
-
~
-
N
""
"""
E(V .) ""
J
j=t
E(p.)q.
J
j=J
J
-
-
N
~
N
E(l. )L. J
j=J
J
n
2
cov(Vj, Vt) "" q. Var(p.) + q.
J
J
J
~
~
j=l
-
E(k.)K.
J
J
N
~
q Var(p.) + q.
cov(p. !i>P )
J
J s
J s=n+l qs
s=l s
~
s~j
2
-
n
~
N
~
-
~
!: L Var(l.) + L.
L cov(L ,1 )
s
s
J s
J
J s=n+l
s=l
+ L. Var(l.) +L.
J
J
J
s~j
n
+ K.
+ K.2 Var(k.)
J
J
J
N
~
K
s=l
s
Var(k.) + K.
J
~
J s=n+l
- -
K cov(k. ,k.)
s
J J
s~j
= FR
[9I
N
N
!:
~
j=l
s=l
cov(V. , V )
J
The symbols FR and MR stand for the
ket risk".
s
11
= MR
firm risk" and respectively the "mar-
The difference E(Vt) - Rf.Pt will be indicated by RP i.e. the
"risk premium". Using [7], it can be seen as the difference of the market
profit and a certainty return.
It follows that [5] can be rewritten as
{ 101
The equilibrium share value is the discounted risk-adjusted expected
shareholder's net income, where the discount rate is the riskfree rate.
'typical for the model is the rislt··il.djustment term.
that FR/MR corresponds ·with
thi."'.
One should note
8-coefficient that measures the security's
-
sensitivity to market-wide events.
ll -
This
S-coefficient has been expressed
in terms of sales- and cost-risks, and is thus more appealing for management •·
The FR-expression provides management a systematic way to quantify
explicitly risk, in contrast with the traditional accounting risk measures
(leverage, asset size, liquidity etc ••• ).
None of these are explicitly
defined in terms of covariances, and they do attempt to highlight several
aspects of the uncertainty associated with the earnings stream of the firm.
They are surrogates for the total variability of return of the firm's
equity.
Transforming the equilibrium share value relationship into an equilibrium risk-return expression,
(see Appendix), we obtain :
~
[ 11 ]
E(R.)
. J
= Rf
+
-
~
cov(S~,s )+cov(L~,L )+cov(~,K)
[ E(S } - E(V ) -
m
J
m
J
m
m
J m
where
[ 12]
R
m
-Lm - -Km
= sm -
is the market return index
N
~
[ 13]
~·
sm =
p.q.
J J
j=l
p
is the market sales return index
t
N
:E
~
[ 14 ]
Vm "" Lm + Km =
l.L.
j=1 J J +
the market cost index being
pt
the sum of the market labor and market capital requirement cost index.
s~·=
J
p.q.
..lJ...
P,
J
L': =
J
LL.
...J....l.
P.
J
ind~xes
K~
J
=
are the corresponding firm
-
12 -
Because with f:ull equity financing, the expected return equals the
cost of capital, the d~r)v90 expreqsion [ ll] indicates a solution for a
highly controversial problem, i.e. the quantification of the discount rate.
This quantification is based on firm-specific values, and some market
indexes.
Particularly when risk is positive, management should add some risk
factor to the riskfree rate.
To define the risk factor one needs some
market indexes to stand for Sm, Lm and Km•
are readily available e.g. the BNP-
index~
Some approximations for those
the wage-rate index, interest
rate index etc •••
3. THE
CURVA~URE
CHARACTERISTICS OF PERFECT COMPETITION UNDER UNCERTAINTY.
When the market prices are uncertain in a mean-variance framework
some new concepts must be introduced to allow for an intuitively appealing
interpretation of the share value equation [ 10} , equivalent with or at
least strongly related. to profit maximization.
The firm value equation can
also be called, a "profit-risk" function because it implies risk-adjusted
profit maximization.
11
In a similar way, we will define "cost-risk 11 and
revenue-risk" as the sum of the expected factor costs (outlays) and the
risk of the factor costs, respectively as the difference of the expected
sales (receipts) and the risk of the sales.
So "profit-risk" equals the
difference between "revenue-risk" and "cost-risk".
Those concepts are intuitively clear.
Reasoning in marginal terms,
the value of the firm not only increases with the output price but also with
net-revenue (in value terms) being the difference of the price per unit
and the
~rginal
change in the sales risk.
factor cost-risk is part of the
tr~e
The
marginal change of the
cost (in value terms).
- 13 -
I. The revenue-risk functions (cfr. Figure I)
--------------------------
Taking into account there are only n firms in the industry of firm
j, the total
[ 15]
TR_
-~
revenue-risl~-
RP
= E (p J. ) qJ.
(T~)
equation
becomes
J
J
N
n
i;
2
- MR [ q • Var (p . ) + q •
J
q
s=l
8
Var(p.) + q.
q
i;
J s=n+i
J
8
s#j
Total revenue-risk is the difference of the expected sales and the riskiness of the sales.
The latter ter~ consists of three parts : q~ Var(p.)
J
J
stands for the variance of the sales of firm j, the other two terms are
the covariances of the sales of firm j with sales of finns of its own
industry- respectively with those belonging to other industries.
Although commen sense suggests that in a risk-averse world, a firm,
given the amount of expected sales. should try to decrease the risk, one
observes thatthe firm-cannot manage risk except with the production amount.
The reason is that riskiness is linear in the quantity produced» and that
all other variables are determined by the market.
The former part of this
statement can be seen from
{ 16]
RP
MR
2
[ q.
J
N
n
Var(j:L) + q.
J
J
i;
s=l
Var(p.) + q.
q
s
i;
q
J s=n+l 8
J
cov(p.,p )}
J
8
=
s,&j
n
RP
q. [
M1J
rAA
J
(q. +
J
N
:E q) Var(p.) + E q 8 cov(p.,'p ) l
s= l 8
J
s=n+ 1
l s
s#j
in which .the term between brackets is a function of the industry output
n
(q. + i; q ).
J
s
It follows that the higher the industry output, the higher
- 14 -
the riskiness.
Proving the linearity of ( 16] results in showing that the
slope, i.e.
N
n
[ 17]
[ (q. +
J
q) Var(p.) +
s
J
s=n+l
};
};
s=l
q
s
cov (i) • ~ p
J
s
)]
s;'j
is constant when q. changes.
J
This follows from taking into account the
"reaction principle" [6 l. According to this principle other firms respond by
precisely off-setting changes on a change of an individual firm, so that
the decisions of that firm have no effect on prices or on industry output.
It is the essence of perfect competition.
in the industry can always use
pr~cisely
This means that the other firm
the incremental quantities of
factors of production demanded or released by a firm to offset precisely
output changes by that firm.
The other implication of the "reaction principle" is that offsetting
output changes by the industry in response to an output change by an indi.vidual firm must also precisely offset any externalities associated with
the firm's new decision [6].
This will be discussed later on in this
study.
Taking the reaction principle into account, the linearity follows
by differentiating [ 17} with respect to q. : (1-1) Var(p.)
J
J
= 0.
The conclusion is that the firm cannot influence the riskiness
sales without changing its output.
So the introduction of risk in
thi~
perfectly competitive firm does not give any additional tools to use.
The
only decision variable is its quantity to produce, and as will be shown
now from the marginal revenue-risk curve, the introduction of the objective
of maximizing
T~
to explain the behavior of the firm does not lead to a
finite optimal production quantity.
The marginal revenue-risk func-
tion will be shown to be constant and given by the market. Then one knows
immediately
that the ARR also is constant and equal to the ~· The
- 15 -
marginal revenue-risk and average revenue-risk can be easily derived from
= E(p.)
J
[ 18]
N
RP
~
:E
MR
[15],
n
1':.
q. cov(p.;p)- MR Var,p.) [q. + :E
s
n+l
J
s
J
J
s=l
s=#:j
Yielding the following inferences!
First, the
~
and
~
are equal.
Their expression is the difference
between the expected output price and the risk of producing one good.
latter is equal for all firms of the same industry.
The
In fact one could ima-
gine two markets : one physical. ou.tput market and a market for the risk of
the output, both perfectly competitive.
Selling one good at the physical
market generates the expected price but also produces some risk, i.e. the
price of risk times its amount.
Also the amount of risk of selling one
good is defined by the market.
Therefore, under uncertainty, the demanq-
risk function
re~~ins
perfectly elastic and the firm has no monopolistic
~rketpower.
Second~
the demand-risk function and the
of the industry production.
~
are a decreasing function
Third~
price.
the ~ and ~ a~e generally
different from the expected
They are higher when :E q cov(p.,p ) is negative and higher: in
s=n+l
absolute value thqn var(p.) [q. +
. J
J
n
~
.l
s
s=l
q ] •
8
s
This would be the ideal indus-
try to enter because ceteris paribus, the higher negative the covariances,
the lower the risk.
The cash sales income would be increased by
ness.
neg~tive
riski-
This occurs only in industries where the sales are negatively cor-
related with the market, i.e. for industries that are not subject to the
collective market mov:croent up-.o:r
Fourth, the slope of the
d~wnwards.
T~
is function of the sign and relative
In the extreme caset ~ and ~ can
become zero and even negative. This is because of the influence of the
riskiness. It follows that even with positive production, the T~ can be
zero or negative. This would certainly require a negative or zero marginal factor cost-risk.
amount of the different risk terms.
The total factor cost-risk equation (TFCR) is the sum of the total
factor cost-risk equation of labor and
, respectively TFCLR and
TFC~.
According to the asstnuptions about the one-to-one correspondence between
[19
and output markets, the total factor cost-ris.k of labor becomes
= E(l.).L.
1
J
+
J
RP
MR
2
n
N
[L. Var(l.) + L. !: L Var(l.) + L. !:
L
J
J
J s=l s
J
Js=n+l 6
s#j
cov<l., l
J
s
)]
Total factor cost-risk is the sum of the expected labor cash outlays and
the riskiness of the labor costs.
2
The latter term consists of three parts
-
L. Var(l.) stands for the variance of the labor costs of firm j; the other
J
J
terms refer to the covariances of the labor costs of firm j with labor
costs of firms of its own industry respectively with those. from other industries.
Similar to the analysis of the riskiness of sales, one can argue
that give.I1: the amount of labor costs the firm cannot influence the. risk of
the factor costs.
Also,minimization of TFCLR results in (just a minimiza-
- l7 -
tion of TFCL under certainty) the trivial zero-input solutio11.
The MFCLR and AFCLR will be shown to be equal and constant.
N
n
~ L
[ 201
s=l
8
Var(lJ.) +
~
L
s=n+l
s
)
s~j
covd., l
J
s
)]
A few conclusions can be stated again.
First, the AFCLR and MFCLR are equal and defined by the market.
The firm has no monopsonis·tic marketpower at all.
Second, the AFCLR and MFCLR are an increasing function of the industry input.
Third, both are in general different from the expected input price.
Normally they are higher.
N
~
Only in the case that
s=n+l
L
s
- -1 )
CQV(l , ,
J
s
<0
and
N
~~
s=n+l
AFCLR
AFCLR
L
s
covd.,i
J
= MFCLR <
= Mffi'CLR = E
s
)I > L.J
L
Var(i.) +
J
s
.In the extreme cases,
) and AFCLR
= MFCLR ~
Var(i.)
J
then
also can occur that
O.
The conclusiort from this study with respect to the curvature of the
revenue-risk and factor cost-risk functions is that there exists a strong
relationship with the certainty functions.
fact that whereas
uncertai~ty
The differences arise from the
certainty the prices are defined by the market, under
the average or marginal
~evenue-risk
respectively factor cost-
risk are defined in the market, and arc e1ual for all firms of the same
-
industry.
18 -
It follows that the total functions are linear with their slope
depending on the
risk~
/
/
/
/
A~=:MR
- -
AR=:MR=p
R
E(p
/
q
r-I
q
"~
---- ---.. --.....__
I
l
I
Figure I
The revenue- (risk) functions under certainty (uncertainty)
--~---------------------------------------------
- 19 -
TFCL
/
/
E(l.)
J
/
_
... .,. ......... y - -
:-"'-
/
L
Figure 2
L
The factor cost- (risk) equations under certainty (uncertainty).
4. THE EXISTENCE OF EXTERNALITIES.
Firm's total costs may frequently depend upon the output level of
the industry as a whole.
External economies are realized if an expansion
of the industry output lowers the total cost curve of each firm in the industry (external diseconomies is the reverse).
Under certainty, these effects can formally be included by redefining
the cost function C.l.
= ¢.(q.,!
q.).
1.1..].
A simplified example is the following
l.
in which the industry is represented by two competitive firms with total
cost functions given by
! q. and c 2
•
l.
l.
where
- 20 -
(coefficients a , and a are positive). If b
1
2
nomies ; if b > 0 external diseconomies.
< 0, there are external eco-
Of course, a similar effect can occur at the revenue-side : revenue
does not always depend exclusively upon the firm.'s own quantity 'Ciecision,
but also upon the quantity decision of other firms.
Under uncertainty, in a two-parameter model the concept of externalities has been mentioned by FAMA [ 5} : "Specifically, the output decisions
of individual firms in the risky industry can lead to a novel type of externality : when a firm expands or contracts its output of risk, this in
general changes the risk outputs of other firms, even when these other
firms do not explicitly change their output decisions. 11
Fama explained
the nature of those externalities by restricting its analysis to· the production
framewor:k and one risky industry formed by two non colluding firms.
Because we are primarily interested in the input decision of the
,firm, we will concentrate on externalities caused by this decision. And,
indicated by the model where more industries exist, we will assume there
are two industries (I 1 and I ). Each industry consis'ts of two non collu2
ding firms. Firm j and firm k belong to I 1 • Firm 0 an~ firm p belong to
For simplicity, it is assumed here that only the laborprices are un-
1 •
2
certain.
In equilibrium, the market values of these firms are
[ 21 ]
P. ""
J
1
RI {plqj - k!Kj - E(l 1).Lj
-: [
-
(L~
+ L/'k) Var(l 1) + L.J (L0 +Lp )
(Lk2
+ Lklj) Var(i 1) +
cov(l ,1 )] }
1 2
[ 22
J
pk
1
= Rf
RP
{plqk - ki~ - E(ll) Lk - MR [
~(L0 +Lp)
cov(l ,1 )] }
1 2
------1 the indexes 1 and 2 (e.g.
p ) indicate the prices of industry l (I ) and 2 (I ).
1
1
2
- 21 -
The total market value of all risky firms in 1 1
[ 23 f
PII ...
l
Rf. {pl(qj+qk)- ki(Kj+l1t)- E(ll)(Lj+~)-
2
2
RP
MR [ Var(ll)
- "'
+ cov(l ,1 2 ) (L0 +Lp) (Lj+~)]}
1
[Lj+Lk+2Lj~]
From this equation it is clear, as expected, that the value of an industry
depends on the industry-output, and the industry-input.
However, because
of the risk, there is a cross-industry influence : the value of r
depending on the input of r •
2
When also the output is risky, PI
1
is also
would ·;
I
also depend on the output of 1 •
2
Suppose, firm k decides to expand input.
If firm j does not respond,
it's firm-risk changes from
to
It follows that the risk increases, such that this phenomenon is an
external diseconomy.
I
Applying the reaction principle, i.e.
firm· j
contracts its in-
put by one unit, the firm-risk becomes
which is different from the initial firm-risk position.
After off-setting
actions the external diseconomy becomes an external economy provided positive covd' , 1 ) which as mentioned by FAMA [ 5 l can empirically be expected.
1 2
- 22 -
However the resulting quantity of total firm-risk of the industry
remains constant because (L~ + ~ + 2Lj.Lk) = (Lj + ~) 2 such that offJ
setting actions generate the same industry-risk.
So, the conclusion must
be that the firm that initially changes absorbs some risk of the other firm
(j).
Checking_ this : the initial firm-risk of firm k
[ 27]
Lk {[ Lk + Lj]
-
- -
Var(l 1) + (L0 + LP) cov(l 1,1 2)
}
changes to, already taking into account the off-setting actions
[ 281
and its increase just corresponds with the decrease in the firm-risk of
firm j.
Because the risk of firms in other industries is not influenced,
the market risk as a whole remains constant.
Finally, because the value equations are function of total input of
,the other industry (L + Lp), and because of off-setting actions there,one
0
can conclude that changes in decisions of 1 do not cause changes in ria,k
2
of r , such that those industries are risk-independent. Clearly, the level
1
of the industry input of 1 2 does have an influence, as demonstrated.
The general conclusion is that under our
assumptions~
although e2';-
te.rnalities exist, the industry values remain the same, such that the
conditions for perfect competition are fully satisfied.
5. THE LINK BETWEEN COST- AND PRODUCTION FUNCTIONS UNDER UNCERTAINTY.
It will be shown thatthe neoclassical correspondence between increasing, decreasing and constan ~long-run marginal costs with diseconomies,
respectively economies of scale, and their·absence
uncertainty.
remains still valid under
- 23 -
Following neoclassical literature, the cost-risk function can be
derived by minimizing the cost-risk equation but maintaining a prescribed
output level.
The mathematical formulation is
{ 29]
= E(l.)L.
min TFCR
J
K,L
RP
+ MR
[ Var(l.)
[ L.2 + L.
.}
J
J
J
n
N
!:
!:
+ L.
L ]
s=l s
J s=n+l
L
s
s;&j
covd. ,i
J
n
RP [ Var(k.)
+ E(k.)K. + MR
[ K.2 + K.. z K ]
J
J
J J
J s=l S'
s;&j
)]
s
N
+ K..
J
~
!:
covd~.
K
s
s=n+J
J
,ks ) ]
q = q(K,L)
The first-order conditions are
[ 30]
[ 31
l
E(l.) +!!: [ Var d.)
MR
J
J
[ L.J
N
n
+ !: L ] + z L covd. ,i ) 1
s
J s
s=l s
s=n+l
s;&j
n
E(k.) + RP [ Var(i~.) [ K. + l: K ] +
MR
J
:J
J
8
s=l
- -
N
K cov(k. ,k ) ]
J s
s=n+l s
l:
-
Aq
-
Aq
L
K
s;&j
[32]
q-
q(K,L) = O.
It follows that
{ 33]
qK.
qlt
RP
E(k.) + MR
[ Var(kj) (K. +
J
=
...
E(l.) +
J
J
RP
...
MR [ Var(lj) [ L. +
J
n
z
~~1
.n
l:
s=l
s;&j
N
K ) +
z
K cov(k. ,k )
l
s
s=n+l s
1+
s
L covd., I ) 1
J s
s==n+l s
L
N
!:
J
s
=o
=
0
- 24 -
This expression gives the expansion path in the implicit form.
qK
k
It is a generalization of the certainty expression -- = It also says
qL
1"
that when only the output price would be uncertain, the condition for cost
minimization of certainty will hold under uncertainty as mentioned by
STEVENS [ I 9].
However when the input prices are stochastic also the marginal cost
must be substituted by the marginal cost-risk.
In order to characterize the curvature of the cost function at the
optimum, we will prove that the Lagrangian multiplier equals the marginal
cost-risk :
N
n
+
~
L } +
s=l
s
~
L
s=n+l
s
cov(i.,i )- itq 1}dL~
1
J s
s.rj
Using the first-order conditions,
RP
_
n
E (i . ) + MR [ Var ( 1. ) [ L. + ~ L
J
J
J
s= 1 s
N
l + ~
L cov (i., l
8
J
n+ 1
8
)]
[ 34]
'
One can conclude that A.x > 0 because q1 > 0 and TFC~ was assumed to .be positive. Raving found that the MCR is positive at the optimum, the next
2
d TFCR
2
step is the search for the sign of
dq
From the first-order conditions one can derive
- 25 ...
q dL
L
X
-
0 ... - dq
dA =
the sign of which is defined
dq
D
only by the production function because riskiness was proven to be linear
from which
in production.
This result is identical with the certainty result. It
follows that a linear cost function (~
turns to scale.
= 0)
corresponds with constant re~
The other relationship between concave or convex produc-
tion function with increasing resp. decreasing marginal costs is
maintain~d
also under uncertainty.
This result critically depends on a
reaction principle, in perfect competition.
correct application of the
It therefore contradicts
completely STEVENS's [ 19]finding that constant returnsto scale in production no ·longer is a bar to a determinate profit maximization equilibrium,
because the cost-function under uncertainty is nonlinear.
b~en proven to be incorrect.
The latter has
By intuition, because risk is a linear func-
tion of each input, its influence does not change the original certainty
picture.
6. CONCLUSION
Under the basic assumptions of recent theoretic financial and economic analysis under uncertainty i.e. that investors are risk averse and
expected utility maximizers, an equilibrium market share value equation
was derived, as a function of the firm decision variables : sales, costs,
out- and input.
This equilibrium share value equals the discounted value
of all future net profit adjusted for risk.
Following the market value
The 11 Dn denotes the determinant of the coefficients, used in applying
Cramer's rule ; the second order condition for a regular constrained
minimum requires it to be negative.
- 26 -
rule, management should just maximize this share value.
As compared
to objectives used in traditional price theory, the market value maximization'takes into account in a systematic way riskiness in order
to
clarify
better the behavior of the firm.
The equilibrium share value condition has been transformed into a
risk-return relatiqnship.
This expression is a guideline for management
for adjusting the discount rate for riskiness.
rate~
Starting from a riskfree
the risk adjustment is fully based on firm specific values and a
few market indexes as the B.N.P ••
The remainder of this study was concerned with describing some characteristics of perfect competition under uncertainty.
Defining "net" revenue as revenue minus its riskiness and "true 11
(factor)costs as costs plus its
riskiness~
the curvature of the certainty
. functions must be adjusted for riskiness i.e. the total revenue - (or fact(j'r cost-) risk functions remain linear with the slope dependent on riskiness.
So the firm has no monopolistic or
monopso~nistic
market power.
The average revenue- (?r factor cost-) risk functions,·generally different
from the expected prices are a decreasing (respectively increasing) function of the industry output(respectively the industry input).
Because of
the existance of some perfectly competitive risk market, the risk of producing one good was proven to be the same for all firms of the same industry.
However, ceteris paribus, when management would set up a new business,
the riskiness could help to choose
an industry i.e. that one that does not
follow the general market movement.
The typical condition for perfect competition under uncertainty in a
mean variance framework i.e. that off- setting changes by the industry in
response to a change by an individual firm must precisely off-set any
externalities with the firm's new decision is also fulfilled.
Externalities
- 27 -
are proven to exist in a multi-industry economy.
The
exp~nding
firm ab-
sorbs the risk of the contracting one, such that the overall amount of
industry (and/or market) risk remains constant.
is depending on the industry-output and - input.
The value of an industry
\.Jhen the prices are uncertain
there is even an cross-industry influence e.g. when the inputprices are uncertain, the value of one industry depends on the input level of the other.
However the industries are risk-independent in the sense that changes of
individual firms of some industry have no effect outside that industry.
Further in characterLzing the curvature of the cost-risk function
it has been demonstrated that the correspondence between increasing, and·
decreasing longrun marginal COStS, with diseconomies and eCOUOmiPS of SCde
remains unchanged under uncertainty.
As can be expected from this general market structure characterization, the introduction of riskiness will change the optimal decisions of
the firm as compared with certainty.
- 28 -
APPENDIX
From f 4
[ 35]
[ 36]
r , forming
ou.J.
- 1J "" 0
ocli
-
au.1
E(V .)
oE(c 2i)
[ 37]
ou.l.
OJJ
the Lagrangian, the first order-conditions are
""'
+
J
N
l;
l.
[41
oE(c 2i)
. E{V.)J
2
-
x.l.S cov(V. ,V )
J
s=J
s
1JP.
J
=0
j=l ••• N
=0
x .. P. -cH
l.J
J
j=l
analysis:
in [ 36]
Substituting [ 35]
ou.
N
~
2
ocr (c2i)
WH-
Following FAMA's
ou.l.
ou.
l.
+
2
N
2 l;
s=l
ocr (c2i)
x.
l.S
-
"'
cov(V.,
V )
J s
-
au.
l.
rc;-i P. ... 0
j=L •• N
J
ou.l.
dividing by
ocr2(c2i)
ou.1
[ 38]
6E(c 2i)
au.l..
o0 i
N
E(V .)
J
+ 2
~
s=l
x.
l.S
cov(V.} V )
J s
ocr 2 (c2i)
The optimal investment
({ti
su.1
P. "" 0 j=l ••• N
J
2
ocr {c2i)
proportions~ :X ••
l.J
~
depend or.: the expected pro-
- 29 -
fit for the shareholders at time 2, the covariances of its profit with
those of all other firms, the market value of the shares at time 1, and
the investor's marginal rate of substitution between mean and variance of
consumption at time 2 and betweenconsumption at time I and variance of consumption at time 2.
m
Taking the market clearing constraint into account, i.e.
~
x .• =l
i=J l.J
j=l ••• N, and aggregating [38] accross all investors i, i=l ••• m, we obtain
an expression for the value of firm j under market equilibrium.
-
N
m
~
E{V.)
+ 2
J
i=l
~
s=J
m
cov(V. , V )
J s
+
~
i=l
so
2
[ 39]
P. ""
J
m do' (c2i)
~
i=l dE(c 2 i)
m do'2(c2.)
.
~
i=l
N
2
"'
E(V.)
~
- -
cov(V., V )
J s
J
~
deli
Defining
N
[ 40 l
~
j=l
P.
J
= pt
[ 41 ]
N
[ 42
l
Using
N
j=l
~
j=l
I 40·]
, [ 41
~
cov(V."l ,Vs )
..
1 , [ 42]
and aggregating [ 39) accross firms
- 30 -
m d<12(c2i)
l:
['•3 ]
p
t
=
crz
i=l dE(c 2i)
• E(Vt) - 2
2
m dO (c2i)
l:
i=l deli
[ 44
J
P. =
J
m
E
i=l
2
m dO (c2i)
l:
i=l
2
Substituting from { 43]
m
l:
i=J
cv t)
li
back into [ 39 ]
2
N
m do (c 2 i)
E
E cov(V. ,V )
dE(c 2 i)
...,
s=l
J. s
• E(V.) - [ i=J
2
E(Vt) - Pt]
J
m do (c2i)
cr2(V t)
:E
2
dO' (c2i)
dE(c 2 i)
2
da (c2i)
deli
i=I
deli
!Taking into account
N
[ 45]
N
E
s=l
cov(V.? V.)
J s
dcr (c 2 i)
:E
i=l dE(c 2i)
m
!:
2
dcr (c 2i)
i=l
Using [ 45]
[ 47}
cov(V., :E
J s=l
vs )
""
cov(Vj'Vt)
2
m
[ 46]
=
P. =
J
I
"' Rf
where Rf is the risk-free borrowing and lending rate.
li
and [ 46] , { 44
J becomes finally
- 31 -
Rearranging [ 47] and substituting E (R.)
J
r 481
E(R.)
J
= Rf
+ ·P:""
RP •
=
E(V .)
P.
J.
J
FR
MR
J
I. Analyzing the risk-premium [ RP J
E(V )
Substituting
-
E(V.)
J
N
t
:E
j=l
~
E(R.)
J
-
N
I:
j=I
P.E{R.)
J
J
in [ 481
N
=
Rf +
P.
[ :E
J=l
J
N
}.';
=
·Rf + -
P.
[ j=l
P •• E(R.)
J
J
-
Rf
] •
FR
J
[ 49]
=
pt
·Rf + P.
J
[ E(Rm) -
Rf
J
FR
NR
where
[50]
E(R )
m
= E(Sm)
- E(L ) -
m
E(Km)
or
N
~
j=I
E(V .)
J
= [
N
.,_
I: E(P.)q. j=l
J J
N
N
}.'; E(l.)L. - I: E(k.)K.J
j=I
J J
j=l
J J
l
-
pt
In words the market return index is the difference between the market sales
return index and the market cost indexes.
- 32 -
2. ~nalyzing the market risk
N
N
}.;
}.;
[MRJ
cov(V. ,V )
s
J
j=l s=l
N
N
}.;
}.;
P .• R.
COV ( _..J.__J_
j=I s""t
Pt
[ 51 ]
3. Analyzing the firm risk [FR}
FR=
,
-
=
cov(V., V )
J
t
Expressing cov(R.,R ) in terms of firm characteristics
J
m
p.q.
= cov (.:..J...:l.p
,
cov(R. ,R )
J
m
l.L.
_.L.l
P.
J
J
[52]
X
X
X
J
J
J
'
Sm - Lm - Km)
cov(S. - L. - K., S - L - K)
=
cov(S~, S ) + cov(L~, L) + cov(K~, K)
J
m
J
m
J
m
E(R.)
J
P.
J
=
Substituting [ 50] , [51 ] , [ 521
[ 53]
k.K.
__,L1_
-
m
m
m
in [ 491
X
=.
Rf + [E(S ) - E(L) - E(K) m
m
m
·Rf ] •
cov(S.~S
J
X
X
)+cov(L.,L
)+cov(K.,K)
m
J
cr2 (R )
m
m
J
m
- 33 -
BIBLIOGRAPHY
----------[ J1
R.N. BATRA, A. ULLAH : Competitive Firm and the Theory of Input Demand under Price Uncertainty, Journal of Political Economy, 197, pp. 537-548.
[ 2]
P. BROWN, R. BALL : Some Preliminary Findings on the Association between the Earnings of a Firm, Its Industry, and the Economy, Empirical Research Accounting : Selected Studies, 1967,
Suplement to Vol. 5~ Journal of Accounting Research, pp.
55-17.
[ 3]
R.M. CYERT : Discussion of some Preliminary Findings on the Association between the Earnings of a Firm, Its Industry~ and the
Economy, Empirical Research in Accounting Selected Studies,
1967, pp. 78-80.'
[ 4]
E.F. FAMA, M.H. MILLER : The Theory of Finance, Holt, Rhinehart,
Winston, N.Y., 1972.
[ 5]
E .F. FAMA : Perfect Competition and Optimal Production Decisions
under Uncertainty, The BellJournal of Economics and Management Science, 1972, pp. 509-530.
: [ 6]
E.F. FAMA, A.B. LAFFER : The Number of firms and Competition, The
American Economic Review, 1972.
{ 7}
M. JENSEN and' J. LONG : Corporate Investment under Uncertainty ap.d
Pareto Optimality in the Capital Markets, The Bell Journal of Economics and Management Science, Vol 3, 1972,
pp • I 51- l 74 •
[ 8]
R.M. KORKIE : Theory of the Firm Facing Uncertain Demand: Comment,
American Economic Review, pp. 245-246.
[ 9]
H. E. LELAND : Production theory and the stock market, The Bell Journal of Economics and Management Science, 1974~ pp. 125.-144.
[ 10)
H.E. LELAND : Theory of the Firm Facing Uncertain Demand~ The American Economic Review, 1972, pp. 278-291.
[ 11 ]
J. LINTNER : The valuation of Risky Assets and the Selection of Risky
Investments in Stock Portfolios and Capital Budgets, Review of Economics and Statistics, 1965, pp. 12-37.
[ 12]
J. LINTNER : The Impact of Uncertainty on the nTradi tiona!" Theory
of the Firm : Price-setting and Tax Shifting,
J.W.
MARKHAM and G.F. PAPANEK : Industrial Organization and
Economic Development~ pp. 238-265, 19'70.
- 34 -
[ 13]
J. LONG : Wealth, Welfare and the Price of Risk, The Journal of
Finance, 1972, pp. 419-433.
[ J4]
M. ROTHSCHILD, J. STIGLITZ : Increasing Risk : I. A Definition,
Journal of Economic Theory, 1970~ pp. 225-243.
[ 15]
M. ROTHSCHILD, J. STIGLITZ: Increasing Risk: II : Its Economic
Consequences, Journal of Economic Theory, 1971, pp.66-84.
[ 16
l
A. SANDMO : On the Theory of the Competitive Firm Under Price
Uncertainty, The American Economic Review, 1972, pp.65-73.
[17]
R. SCHRAM, R. SHERMAN: Profit Risk Management and the theory of
the Firm, Southern Economic Journal, 1974, pp. 353-363.
[ 18]
W.F. SHARPE: Capital Asset Prices, Journal of Finance, 1964,
pp. 425-442.
[ 19 1
G. V. STEVENS : On the Impact of Uncertainty on the Value and Investment of the Neoclassical
Arr~rican Economic
Review, June 1974, pp. 319-335.
[ 20]
,J.
[ 21 1
S. TURNOVSKY : Production Flexibi
Price Unve:rtainty and the
behavior of the Competitive firm, International Economic
Review. 1973, pp. 395-413.
[ 22]
L.
STIGLITZ : On the Optimal
of the Stock Market Allocation of
Investment;Quarterly Journal of Economics, 1972, pp.25-60.
VANTHIENEN : Portfolio Theorie en Bedrijfsfinancie:ring, Onderzoeksrapport nr. 7305, Departement Toegepaste Economie,
Katholieke Universiteit Leuven? 1973. 31 pp.