Competitive Firm and the Theory of Input Demand under Price Uncertainty
Author(s): Raveendra N. Batra and Aman Ullah
Source: The Journal of Political Economy, Vol. 82, No. 3 (May - Jun., 1974), pp. 537-548
Published by: The University of Chicago Press
Stable URL: http://www.jstor.org/stable/1829844
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Competitive Firm and the Theory of Input
Demand under Price Uncertainty
Raveendra N. Batra and Aman UI1ah
SouthernMethodistUniversity
In this paper, we examine the behavior of the competitive firm faced
with making input-hiring decisions under conditions of price uncertainty.
Unlike Sandmo and Baron, among others, we show categorically that a
marginal increase in uncertainty stimulates a decline in the firm's output,
provided the absolute risk aversion is decreasing. Furthermore, the
implications of a change in expected price remain unchanged, but those
of a change in input price need not be the same as in the certainty case.
Specifically, the input demand function is downward sloping only if the
production function is well behaved.
The purpose of this paper is to develop systematically the theory of input
demand for a competitive firm facing price uncertainty. For the sake of
simplicity, we postulate a two-input model and exa mine the implications
of uncertainty for the firm's output and input demand under the
assumption that the competitive firm, facing given input prices, seeks to
maximize the expected utility from profits. Earlier contributors to the
theory of the firm have either approached the question from the side of
product markets,1 or, in examining the firm's behavior from the side of
factor markets, they have assumed that the firm seeks to maximize
expected profits and is thus risk neutral.2 The focus of this paper is on the
We are indebted to J. Hadar and W. Russell for useful discussions on the subject
matter of this paper.
1 See, for example, McCall 1967; Horowitz 1970, chap. 12; Sandmo 1971; and
Leland 1972.
2 See, for example, Smith 1960; Tisdell 1968, chap. 3; and Rothenberg and Smith
1971. One exception is that of Horowitz, who uses the expected utility approach to
analyze the implications for the firm's decision making of uncertainty in the production
function (not in the product price) in terms of a long-run model allowing for variability
in the use of both capital and labor. Apart from the fact that Horowitz is concerned with
different questions, his analysis is incomplete because he does not get involved in the
comparative-statics properties of his model. It may be mentioned in passing that in
another chapter, Horowitz anticipates some of the results derived recently by Sandmo
(1971), especially the results concerning the output level of the risk evader and the
implications of a change in fixed costs.
537
538
JOURNAL OF POLITICAL ECONOMY
expected utility approach, which permits a more general formulation of
the firm's attitude toward risk.
For the main tools of analysis, we draw upon the recent seminal work
by Sandmo (1971), who has examined under price uncertainty the
behavior of the competitive firm from the side of the product market.
However, in contrast to Sandmo's results which are obtained from a
short-run model of the firm where the capital stock is given, our results
are derived from a long-run model where all factors of production are
variable.3 Furthermore, Sandmo (1971) is unable to derive categorical
effects of a marginal increase in uncertainty whereas we show that an
increase in uncertainty leads to a decline in the firm's output, provided
absolute risk aversion is decreasing.
I.
Assumptions
and the Basic Model
Unless otherwise specified, the following assumptions will be maintained
throughout the paper.
1. The firm is a price taker in the probabilistic sense. That is to say,
the firm is not large enough to influence its subjective probability
distribution about the sales price, which is assumed to be a random variable. Furthermore, the prices of the two inputs, capital and labor, are
given to the firm.
2. The decisions concerning the volume of output and hence the
utilization of inputs must be made prior to the knowledge of the market
price.
3. The firm seeks to maximize expected utility from profits, and its
attitude toward risk can be described by a von Neumann-Morgenstern
utility function, which among other things requires that the firm's
decisions are made by one individual.4
Let U stand for utility and ir for profits. Then the firm's utility function
is given by
U=
U(rc)
(1)
where U'(ic) > 0 and U"(ir) ,' 0, depending on whether the firm is
risk averse, risk neutral, or a risk preferrer.5 Let q = output, K = capital,
3 Actually, the short-run results of Sandmo can be derived as special cases of our results.
This assumption is central to much of the recent work in the theory of the firm under
uncertainty (see, for example, McCall 1967, 1971; Sandmo 1971; and Leland 1972).
Acceptance of this utility function implies acceptance of the axioms underlying the
existence of the von Neumann-Morgenstern utility function. For a lucid analysis of
these axioms, see De Groot (1970) and Horowitz (1970, chap. 12).
5 Arrow (1965) has shown that if the utility function is to satisfy the von NeumannMorgenstern axioms, it must be bounded from above. This means that as Xrapproaches
infinity, the utility function must become concave, so that at least in the range close to
the upper bound of the utility function, U"(7r) < 0. This also suggests that for the firm
to be risk neutral [U"(7r) = 0] or a risk preferrer [U"(7r) > 0], i should be restricted
to a finite value.
4
539
COMPETITIVE FIRM
and L = labor. Then the production function of the firm is given by
q=f(K,L).
(2)
The only assumption that we make about the production function is
that fK and fL, the marginal products of capital and labor, respectively,
are positive and that the isoquants are convex to the origin, so that
- fULL
2fKfLIKL
> 0, which, it may be noted, does not
ffKK
necessarily imply that fij > 0 (i = L, K).6 The firm's total cost of
production is given by wL + rK, so that its profit is given by
it=pq-wL-rK
(3)
where p is product price, which is assumed to be a random variable with
density function f (p) and mean E (p) = i. We assume that p is nonnegative.
In view of (2) and (3), the expected utility from profit is furnished by
E{U[pf (K, L) - wL
-
rK]},
(4)
where E is the expectations operator. The firm chooses input quantities
so as to maximize expected utility from profits. The first-order conditions
for the maximum are
8E(U) = E[U'(7c)(PIfL- w)] = 0
aL
(5)
BE(U) = E[U'(it)(pIK - r)] = 0.
OK
(6)
and
The second-order conditions for the maximum are given by
=
2(U)
w)
Al
=
E[U"(7r) (pfL
A2
=
E[U"(7r)(pfK -r)
a2E(U)
aK~l)=
-
)
< 0,
(7)
+ PfKKU'(7E)] < 0,
(8)
+ PILLU'(7t)]
and
a2E(U) .2E(U)
aL2
aK2
aE(U)
_
]2
aL (K9
> 0;
(9*
or
A1A2 -B2
D>
(9)
where
B-
a2 E(U)
aL aK
-
E[U"(r)(pfK
-
r)(PfL
-
w)
+ PfKLU'(7t)]-
6 See, for example, Hadar (1970, chap. 3, pp. 26-27) on this
point.
(10)
540
JOURNAL OF POLITICAL ECONOMY
Before we proceed further with our analysis, it is necessary first to
briefly examine the second-order conditions for the attainment of maximum expected utility. In the certainty case, utility maximization requires
-i
that fij < 0 and ILLfKK
L > 0 (i = L, K), which means that the
production function is strictly concave.' We will now show that if the
firm is risk averse, strict concavity of the production function is sufficient
but not necessary to ensure expected utility maximization.
Expanding (9*) by using (7), (8), and (10), we obtain
D = {E(U"a2)E(U"b 2)
-
[E(U"ab)]2}
+ {E(p U') [fKKE(U"a 2)
+
where a
=
+
fLLE(U"b 2)
{[E(pU)] 2(fLLIKK -IKL)}
pIL- w, b = pfK- r, and U'
-
2IKLE(U"ab)]}
(11)
U'(ir).8 From the first-
order conditions given by (5) and (6), sincefL and fK are nonrandom
and since w and r are given,
W
fLE[U&(7)p]
r
fKE[U&(Q)P]
(12)
= fL
fK
Using this in a and b, we get
a = (pfL - w) =
bA
= fL
(PfK
r),
(13)
so that a2 = b2fL /fK2. Using this, the first term in (11) reduces to zero,
and D becomes D = -E[U"b IK] (2fL fKfKL -fL fKK
KI LL) +
- IKL).
For second-order conditions to be satisfied,
[E(pU')]2(ILLIKK
Al and A2 from (7) and (8) should be each negative, whereas D should
be positive. If the firm is risk averse, then U"(i) < 0, in which case,
Al and A2 may be negative even iffKK andILL are positive. This immediately suggests that the production function need not be strictly
concave for (7) and (8) to be satisfied. This also holds true for (9), because
D may be positive even if (ILL IKK - IKL) < 0, provided, of course, that
isoquants are convex to the origin so that (2LIfKfKL
-f ffKK
> 0. Thus, if the firm is risk averse, the concavity of the prof2fLL)
duction function is no longer necessary for the firm to be at the optimum.
However, concavity is sufficient because it implies fij < 0 as well as
convexity of the isoquants toward the origin.
If the firm is risk neutral, then concavity is both necessary and
sufficient for (7), (8), and (9) to be satisfied, because here U"(i) = 0.
However, if the firm is a risk preferrer, so that U"(n) > 0, then the
concavity of the production function becomes a necessary but not a
sufficient condition for the firm to be at the optimum.
7 See, for instance, Henderson and Quandt 1971, pp. 61-69.
8
Whenever the equation is too long, we will write U' for U'(7r).
COMPETITIVE FIRM
541
Suppose the firm is risk averse. Then the convexity of the isoquants
toward the origin along with (12) imply that, even with uncertainty about
the product price, the firm minimizes the unit cost of production for
whatever output it chooses to produce. As with the certainty case, the
marginal rate of substitution between the factors equals the factor price
ratio at the optimum, even when the price is seen by the firm as a random
variable. This is very interesting, because in the following section we show
that the risk-averse firm produces an output at which the expected value
of the marginal product of each factor exceeds the given input price. In
other words, the firm produces an output at which expected price exceeds
marginal cost, but the level of output is one where the average cost is
minimum.
II.
Optimal Input Demand and Output under Uncertainty
In this section, we will show that under price uncertainty, the riskaverse competitive firm hires smaller quantities of inputs and thus produces
lower output than in the case where p is known with certainty. The problem may be formulated in terms of this question: What are the optimal
input demand and output under uncertainty as compared with the case
where the product price is known with certainty to be equal to the mean
of the probability distribution?
The first-order conditions (5) and (6) can be rewritten as
E [U'(n)pfL] = E [U'(7t)w]
(5*)
and
E[U'(Qt)pfK]
E[U '(i)r].
=
(6*)
Subtracting E[U'(lr)ufL] from both sides of (5*), we obtain
E[U '(n)fL(p - ia)] = E[U'Qr) (w
-
fL)]-
(14)
Applying the expectations operator to both sides of the profit equation (3)
yields E(Zr) = E(p)q - wL - rK or E(7r) - ,q = -(wL + rK). Substituting the expressionfor - (wL + rK) in (3) then furnishes ir = E(Zc) +
(p - It)q. If the utility function is strictly concave, so that U"(7r) < 0,
then U'(ic) < U'[E(7t)] for p ? jy, or by multiplying through by
(p - ,)fL, U'(7c)fL(p - it) < U'[E(7r)]fL(p - yi) for all p. Taking
expectations on both sides and noting that U'[E(m)] is a given number
and thatfL is nonrandom, we obtain
E[U'(t)fL(p
-
y)] < 0.
Using this result in (14) implies that E[U'(t) (w
-
(15)
IfL)]
<
0, and since
marginal utility is always positive, w < yfL. Similarly, we can show that
r < HIK.
542
JOURNAL OF POLITICAL ECONOMY
In other words, under uncertainty equilibrium, the expected marginal
value productivity of each factor exceeds its price, or, what is the same
thing, E(p) exceeds marginal cost. By contrast, the certainty equilibrium
is defined by the equality between marginal value productivity of each
factor with its price. Furthermore, under certainty conditions, fij < 0
is necessary to satisfy the second-order conditions for utility maximization.
Thus, iffij < 0, it is clear that under uncertainty, the optimal quantity
demanded of each input is lower than the certainty case. It then follows
that the optimal output produced under uncertainty will also be lower
than the certainty output.
Until now, our analysis has been concerned with the implications for
optimal input demand and output of what Sandmo calls the overall
impact of uncertainty. We will now consider the effects of a marginal
increase in uncertainty by defining the increased variability of the
density function of the price in terms of a "mean preserving spread."9
Let us define p* as p* = yp + 0 where 0 and y are the shift parameters
which initially equal zero and unity, respectively. Then the mean
preserving spread type of shift in the density function ofp* leaves the mean
E(p*) unchanged,
that is, dE(p*) -dE(yp
+ 0) = yudy + dO = 0, or
-u
d-
(16)
dy
Using p* instead of p,
7r = (py + 0)f(K,
i
and the first-order conditions are replaced by
L) - wL
rK;
-
E{U'(it)[(py
+ O)fL
w]} = 0,
-
and E{U'(7t)[(py +
r]} = 0. Differentiating these conditions
totally with respect to y, and using (16), we obtain
O)fK -
A
OL
ay
ay
and
Bi
OK
B
aL
OK
+ A2-=
ay
ay
(17)
H,
(18)
H2
where A1, A2, and B1 have been defined before, and where
-H1
= E[qU"(7r)(pfL
-
w)(p
-H2
= E[qU"(7r)(pfK
-
r)(p
-
p)] + E[U'(t)fL(P
-j)]
(19)
and
-
I)]
+ E[U'(7r)fK(p-
e)],
(20)
9 Defining a change in uncertainty in terms of a change in the probability distribution
while keeping its mean constant is quite common with economic theorists nowadays (see,
for example, Sandmo 1970, 1971, 1972; Rothenberg and Smith 1970; and Rothschild
and Stiglitz 1970, 1971).
543
COMPETITIVE FIRM
remembering that initially y
simultaneously, noting that H,
AL
1 and 0 = 0. Solving (17) and (18)
H2fL/fK, and using (9), we obtain
- B1]
H2[(A2IL/fK)
H2E[U'(7r)p](fLIKK
byYD
(21
-KfKL)
DfK
and
AK
ay
H2(A,
-
H2E[U '()p](I
BjfL/fK)
D
ILL
-fLfKL)
(22)
DfK
The signs of aL/ay and aK/ay depend on the signs of H2 andfKL, given
the fact that fij < 0, and, from (9), D is positive. Now the sign of H2
from (20) depends on the signs of (i) E [U'(ir) (p - yu)],which has already
been shown to be negative under risk aversion, and (ii) E[U"(7r) (pfK r) (p - i)]. In the next section, we show that under the hypotheses of
decreasing absolute risk aversion, E[U"(7r)(pfK- r)] is positive. Thus
E[U"(7t) (pIK- r) (p
= E{U"(7) (pfK
=
E [U"(i) (1K
-
)]
r)[P
-
+ (
-
] + E[U"(72)(PIK- r)](
i-
)
(23)
where E{U"(7r)[(pfKr)2fK]} < 0 if firms are risk averse and, as
shown before, (rifK) < ji. Therefore, with E[U"(pfKr)] positive
under decreasing absolute risk aversion, (23) is clearly negative. All this
discussion suggests that the right-hand side of (20) is negative, so that
"2
>
0.
Once H2 is demonstrated to be positive, then for OL/Dyto be negative,
it is clear that (IL KK - IKIKL) is negative. Similarly, for aK/ay < 0,
< 0. Given that fij < 0, then a sufficient condition
(IK ILL - ILKL)
for the marginal impact of uncertainty to be in line with the overall
impact is that fKL > 0, which, of course, is satisfied for several wellknown production functions.'0 However, if (ILIKK - IKIKL)
and
are positive becausefKL < 0, then the marginal impact
(IKILL - fLAL)
of uncertainty will be opposite to the overall impact.
However, although the marginal and overall effects of uncertainty on
the firm's input utilization may not be the same, it still turns out that the
marginal and overall impacts of uncertainty for the firm's output run
in the same direction. Differentiating the production function given by (2)
with respect to y and using (21) and (22), we obtain (aq/ay) = fL(ULIay)+
fK(aKIDy) = {(2LIfKfKL
It
fKK -fKIfLL)E[U(7r)p]H2}/DIK.
-fL
1o A production function is said to be "well behaved"
iffij < 0 andfEL
our results necessarily hold for well-behaved production functions.
>
0. Thus,
544
JOURNAL OF POLITICAL ECONOMY
can be easilyseen that with convexisoquantsand H2 > 0, (Oqlay)
<
0.
Thus, an increase in uncertainty leads to a decline in the firm's output,
if absolute risk aversion is decreasing. Note that Sandmo (1971), among
many others, has been unable to obtain the predictable effects of a change
in variance alone on the risk-averse firm's output.
III.
The Supply Function
This section is devoted to the analysis of the firm's supply function.
Specifically, we wish to determine the firm's output response to a change
in the expected price. The problem may be formulated as this: Let price
be written as p + 0 where 0 again is the shift parameter. A change in 0
shifts the probability distribution to the left or to the right and changes
the expected value of price without altering the shape of the probability
distribution.
Differentiating (5) and (6) with respect to 0 and evaluating the derivatives at 0 = 0, we obtain
A1 g} + B1
O
= -E[U"(&r)(pfL-
w)q + fLU'(7r)]
=
F1 (24)
B1 At + A2 A = -E[U"(7c)(pfK-
r)q + fKU'(7t)]
=
F2. (25)
and
The solution of this system yields
AL _ A2F1 - B1F2
Ao
D
(26)
K = A1F2 - B1Fj
Ao
D
(27)
and
Since A1 and A2 are negative and D > 0, our problem now is to pin
down the signs of B1, F1, and F2. Using (13) in (10) gives us
B1 = E[U"(n)(pfK
= E [U" () (pfL
-
r)
-
w)(IKIIL)
(fLafK)
(28)
+ fKLU'(l)P]
?KLU (T)P].
+
Utilizing (12) and the new expression for B1 and the expressions for F
and F2 in (26) and (27), we obtain
8L
80
E(U'p) {E(U') + (q/fK)E[U"(pfK D
AK
E(U'p){E(U')
+ (q/fL)E[U"(pfL
-
r)]}(fKfKL
W)]}(fLfKL
-LfKK)
fKfLL)
29
,(2)
(30)
545
COMPETITIVE FIRM
We will show later in this section that E[U"(pfKr)] and E[U"(pfL w)] are positive under the hypothesis of decreasing absolute risk aversion
or increasing absolute risk preference. If we further assume that fi < 0
andfKL > 0, it is clear that aL1ao> 0 and UK/IO> 0. These results
continue to hold if the firm is risk neutral so that U" = 0. In other words,
a rise in the expected price will induce the firm to hire more inputs.
However, iffKL < 0, then the results are not clear even iffij < 0. The
expression for the change in output can be obtained by using (29) and
(30). Thus, noting that (pfK - r)/fK = (PIL -w)IfL,
E(U'p){E(U')
aq
ao
_
+ (q/fK)E[U"(pfK(2ILfKfKL
-fLfKK
r)]}
fKILL)
(31)
D
Evidently, risk neutrality, decreasing absolute risk aversion, or increasing
absolute risk preference are sufficient to ensure that a rise in the expected
price will induce the firm to increase its output. It is interesting to note
that the effects of an increase in price on input demand and output under
price uncertainty are the same as the corresponding results available in
the certainty case.11
It is high time that we define the concepts of the firm's attitude toward
risk. Following Pratt (1964) and Arrow (1965), let us define the firm's
risk attitude (Ra) as Ra(7t) = - [U"(7r)]/[U'(Q)], where Ra(7t) is commonly called the absolute risk-aversionfunction.12 For risk-averse firms,
Ra(7r)is positive, for risk preferrersnegative, and for risk-indifferentfirms
zero. For risk-averse firms, Arrow suggests that Ra(7t) is decreasing in 7t.
This is the hypothesis of decreasing absolute risk aversion. However, if
Ra(7) < 0, it seems reasonable that -Ra(7t) be an increasing function of
7r.If a firm prefers risk, then its risk preference should normally increase
with the rise in profits. This we call the hypothesis of increasing absolute
risk preference.
Following the line of reasoning suggested by Sandmo, we will now
show that under these hypotheses E [U"(7t)(pfK- r)] > 0. Let 7l be the
profit level when p = rTfK.Since (ptfK- r)IfK = (pfLw)/IL, p also
=
equals WIfK. If Ra(7t) > 0 and Ra'(Z) < 0, then - [U"(7r)]/[U'(ic)]
?
Ra(7) < Ra(n) for p
rTfKwhere Ra(fi) is a given number. Multiplying both sides by - U'(7t)(pfK - r), we obtain U"(7)(ptfK - r) 2
In the certainty case, it can be easily seen that aL/ap = [P(fKfKLILJK)]ID,
=
and (aq/ap) = [p(2fLJKIKL
[P(fLfKL
-fKJLL)]ID,
fJKK
fK fLL)]ID.
The effects of a change in p on the input demand and output under certainty as well as
uncertainty are the same in the sense that, in both situations, the effects are determined
by the signs of the same expressions. Now, if the same expressions have different signs
under certainty versus uncertainty, then that is a different matter. However, if we assume
concave production functions, then the effects of price changes in both situations are
similar.
12
Also, see McCall 1971 and Sandmo 1971.
aK/ap
546
JOURNAL OF POLITICAL ECONOMY
Ra(if) U'(ir) (pfK
-
r) for all p. Taking expectations
on both sides and
noting (6), E[U"(t)(pfK
r)] ? Ra(f)E[U'(T)(PfK
r)] = 0.
The same proof also applies to the case where -RRa > 0 and -R' (i)
>
0. Here [U" (r)]/[U' (i)] = -Ra(7t) ? -Ra(ir) for p > rilK, so that
r)] ? -Ra(Tr)E[U'(7t)(pfK
E[U"(rc)(pfK
r)] = 0 for allp. Therefore, E[U"(7t)(pfK -r)]
is positive.
IV.
The Input Demand Functions
We now turn to the effect of a change in the price of any input on the
firm's demand for the two factors. For the sake of simplicity, suppose there
occurs a change in the wage rate but r remains unaltered. Differentiating
the first-order conditions (5) and (6) totally with respect to w alone and
remembering that dr = 0, we obtain A1(aL/aw) + B1(aK/aw) = C1
and B 1(L/aw) + A 2(aK/aw) = C2 where C1 = LE[U"(r) (pfL - w)] +
and C2 = LE[U"(7t)(pfK - r)]. Following
E[U'(7r)]
exactly the
procedure established in the earlier section, we find that
-
A2E(U')fK
+ LE(U'p)E[U"(PfK
aw
-
r)1(fLIKK
-fKfKL)
DfL
A IE (U')fK + [E (U'p)] {LE [ U"(pfL
aK
+
E(U')}(fKfLL
aw
-
W)]
-fLfKL)
(32)
DfL
and
aq
-
{LE[U"(pfK + fKE(U')
(fLfKK
r)](fL
fKK
+fK fLL
2fLIfKfKL)
(33)
-fKfKL)E(U'p)}.
The results, as usual, depend on our hypotheses of the firm's attitude
toward risk as well as the signs Of KL and fi. In the certainty case,
aL/aw < 0, whereas the signs of aK/aw and aq/8w depend on the sign
offKL.13
These results continue to hold when the firm is risk neutral for
then aL/aw < 0, whereas the sign of aqlaw depends, among other things,
on the sign OffKL. In the case of aK/aw, once it is realized that with risk
neutrality A1 now reduces to fLLE[U'(n)p],
it can be clearly seen that
the sign of aK/aw depends only on the sign OffKL.
The results, however, are not so clear and interpretable when the firm
is not risk indifferent. If we assume thatfKL
> 0 andfij
< 0, then under
the hypothesis of decreasing absolute risk aversion or increasing absolute
13
( fLL
In the certainty case, aL/w =
KEfL), and aq/lw = [P(fLfLL
(PfLL)/(fLLfKK
-fKfKL)]I(fLLfK
-
fK.L)l
aK/aw
-fKL)
=
-
(PfL)l
COMPETITIVE FIRM
547
risk preference which ensures that E[U"(7t) (pfK- r)] > 0, aL/aw < 0
and aq/aw < 0. However, the sign of aK/awis still indeterminate because
the second term in (32) is negative, whereas the first term, with A, < 0
from (7), is positive.
The differences between these results and those available under
certainty must now be ostensible. The main differences are:
i) Under uncertainty, the sign offKL also enters into the determination
of the sign of OL/aw,whereas in the certainty casefKL plays no such role,
and as a result the sign of aL/awis unambiguously negative.
ii) Under certainty, the sign of aK/awis determined solely by the sign
offKL, whereas under uncertainty the sign of aK/awis ambiguous.
The crux of this discussion is that with well-behaved production
functions (so that fc < 0 andfKL > 0) and with our hypotheses concerning the firm's risk attitude, a rise in the price of any input induces the
firm to reduce the demand for that input and lower its output. However,
the firm's response toward the demand for the other input is indeterminate.
V.
Conclusions
Using a two-input, long-run equilibrium model of the competitive firm
facing price uncertainty, we have derived the following results in the
foregoing sections.
1. The concavity of the production function is both necessary and
sufficient to ensure the firm to reach an optimum only if the firm is risk
neutral. If the firm is risk averse, concavity is sufficient but no longer
necessary, whereas if the firm is a risk preferrer, the opposite is true,
provided the firm's objective is to maximize expected utility from profits.
2. The risk-averse firm utilizes smaller quantities of inputs and thus
produces a lower output than a firm operating under certainty, even
though the former minimizes the unit cost of production of whatever
output it chooses to produce. Furthermore, an increase in uncertainty
leads to a decline in the firm's output if its absolute risk aversion is
decreasing in profits.
3. The marginal and overall impacts of uncertainty need not be the
same for input demand, although for the output level, they continue to be
the same.
4. The presence of uncertainty makes no difference to the effects of
an increase in price on input demand and the output.
5. In the certainty case, a rise in the price of an input induces the
firm to lower the utilization of that input. In the uncertainty case, however, we need a well-behaved production function to ensure this result.
Furthermore, under uncertainty the effects on the demand for the other
input are unpredictable.
548
JOURNAL OF POLITICAL ECONOMY
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