Without imposing restrictions on the utility function and the probability distributions, we show the impact of multiple uncertainty (and each single uncertainty) and change in risk aversion on each input demand. In so doing, we emphasize the importance of the relationship between the inputs in this impact. Moreover, the paper provides technical contributions.uncertainty, firm

Moawia Alghalith

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 3 
  
The theory of the firm under multiple uncertainties 
  
 
 
Moawia Alghalith  
St. Andrews University 
Abstract 
Without imposing restrictions on the utility function and the probability distributions, we show the impact of multiple 
uncertainty (and each single uncertainty) and change in risk aversion on each input demand. In so doing, we 
emphasize the importance of the relationship between the inputs in this impact. Moreover, the paper provides technical 
contributions. 
 
Citation: Moawia Alghalith, (2010) ''The theory of the firm under multiple uncertainties'', Economics Bulletin, Vol. 30 no.3 pp. 2075-2082. 
Submitted: May 15 2010.   Published: August 12, 2010. 
 
     1. Introduction
In the absence of hedging, the vast majority of the studies in uncertainty considered a single
source of uncertainty such as price uncertainty or cost uncertainty. In contrast, theoretical studies
in multiple uncertainty are scarce. Viaene and Zilcha (1998) considered multiple uncertainty;
however they employed a single-input production function and since, as we will show in this
paper, the decision analysis is sensitive to the relationship between the inputs, their results
are limited. Chambers and Quiggin (2003, 2001) and Dalal and Alghalith (2009) investigated
price and output uncertainty; however, they did not analyze the input demand. Therefore
the relationship between the inputs was irrelevant to their models. Moreover, they employed
restrictive assumptions. Even with single uncertainty, none of the previous studies, including
Batra and Ullah (1974) and Pope (1980), showed the impact of uncertainty and change in risk
aversion on input demand; and thus they did not show the role of the relationship between the
inputs in this impact.
Other studies relied on the assumption of supermodularity in deriving comparative statics re-
sults. Examples include Athey (2002) and Milgrom and Shannon (1994), among others. However,
these results require the assumption of decreasing absolute risk aversion (DARA). In addition,
they mainly dealt with a single source of uncertainty. Similarly, Gollier and Pratt (1996) relied
on the assumption of DARA.
In this paper, using a general framework (no restrictions on the utility functions and proba-
bility distributions), we provide comparative statics under multiple uncertainty. In so doing, we
show the impact of multiple uncertainty (and each single uncertainty) and change in risk aversion
on each input demand. We also show the impact of uncertainty on the optimal inputs ratio and
the input productivity. In doing so, we highlight the importance of the relationship between the
inputs (whether substitutes, complements, or independent) in this impact. We show that the
relationship between the inputs are important both in production and in utility.
2. The model
The proﬁt function is given by
˜ π = ˜ pf (x1,x2) − ˜ w1x1 − w2x2,
where ˜ p is the output price with mean ¯ p, f is a neoclassical production function, x1 is the risky
input, ˜ w1 is its price with mean ¯ w1, x2 is the non-risky input with price w2.1 The risk averse
ﬁrm maximizes the expected utility of the proﬁt
Max
x1,x2
Eu(˜ π),
1The results hold if both inputs are risky and/or the output is random (the proofs are similar); for brevity, we
focus on output price uncertainty and one risky input.
1where u is a Neumann-Morgenstern utility function. The ﬁrst-order conditions are
(¯ pf1 − ¯ w1)Eu
￿ (˜ π
∗) + f1Cov(u
￿ (˜ π
∗), ˜ p) − Cov(u
￿ (˜ π
∗), ˜ w1) = 0, (1)
(¯ pf2 − w2)Eu
￿ (˜ π
∗) + f2Cov(u
￿ (˜ π
∗), ˜ p) = 0. (2)
The second-order conditions of the maximization problem are
H11 = Eu
￿￿ (˜ π
∗)(˜ pf1 − ˜ w1)
2 + f11Eu
￿ (˜ π
∗) ˜ p < 0,
H22 = Eu
￿￿ (˜ π
∗)(˜ pf2 − w2)
2 + f22Eu
￿ (˜ π
∗) ˜ p < 0,
and
|H| =
￿ ￿ ￿ ￿
H11 H12
H12 H22
￿ ￿ ￿ ￿ > 0,
where
H12 = Eu
￿￿ (˜ π
∗)(˜ pf1 − ˜ w1)(˜ pf2 − w2) + f12Eu
￿(˜ π
∗) ˜ p.
3. The impact of the uncertainty
In this section we will establish the impact of the cost/output price risk or both risks on the
optimal level of each input. That is, we will compare these values in the presence of both risks
to their corresponding values under certainty, output price uncertainty, and cost uncertainty.
Proposition 1. The introduction of output price and cost uncertainty reduces the optimal
level of each input if f12 ≥ 0 and ˜ p and ˜ w1 are statistically independent.
Proof. Let ¯ x denote the optimal level of the input under certainty (in the absence of both
risks). From (1)
¯ pf1
￿
x
∗
1,x
∗
2
￿
− ¯ w1 > 0,
since Cov(u￿ (˜ π∗), ˜ p) < 0 and Cov(u￿ (˜ π∗), ˜ w1) > 0. But from the necessary condition for proﬁt
maximization under certainty
¯ pf1 (¯ x1,¯ x2) − ¯ w1 = 0.
Hence, f1
￿
x∗
1,x∗
2
￿
> f1 (¯ x1,¯ x2). Similarly, using (2), we can establish f2
￿
x∗
1,x∗
2
￿
> f2 (¯ x1,¯ x2).
Totally diﬀerentiating f1 and f2, we obtain
￿
f11 f12
f12 f22
￿￿
dx1
dx2
￿
=
￿
df1
df2
￿
,
and thus
dx1 =
f22df1 − f12df2
f11f22 − f2
12
,dx2 =
f11df2 − f12df1
f11f22 − f2
12
.
Since df1 > 0 and df2 > 0 in response to the risks, dx1 < 0 and dx2 < 0 if f12 ≥ 0.￿
The result is intuitively appealing since the risky input should fall (due to risk aversion)
in response to multiple uncertainty, while the non-risky input falls in response to output price
uncertainty (this can be easily clariﬁed by a single-input production function). However, the
inputs being non-complements in production (f12 ≥ 0) will guarantee that each input will not
increase when the other input falls in response to the uncertainty. Thus the change in each
input is the net result of two eﬀects: the uncertainty and the technological relationship between
2the two inputs (substitutes, complements, or independent). The two eﬀects will be in the same
direction if the two inputs are substitutes and hence the net eﬀect is a decrease in each input
demand.
It is worth noting that the importance of the relationship between the inputs was not captured
by Viaene and Zilcha’s model since they employed a single-input production function. Moreover,
contrary to the case of output price uncertainty or multiplicative output uncertainty where the
optimal output falls regardless of the technological relationship between the inputs, we implied
that the impact of multiple uncertainty on the optimal output is indeterminate if f12 < 0.
Proposition 2. The uncertainty reduces the optimal input ratio, x1/x2, if the production
function is homothetic and ˜ p and ˜ w1 are statistically independent.
Proof. With certainty f1/f2 = w1/w2, but with uncertainty
f1
f2
=
¯ w1
w2
+
Cov (u￿ (˜ π∗), ˜ w1)
w2Eu￿ (˜ π∗)
,
thus d( f1/f2) > 0 when uncertainty is added. For a homothetic production function, f1/f2
= g(x2/x1) where g is a monotonic function. Thus d(x1/x2) < 0 in response to the uncertainty.￿
Proposition 3. The average productivity of the risky input increases in response to the
uncertainty if the production function is homogeneous and ˜ p and ˜ w1 are independent.
Proof. If the production function is homogeneous of degree r, then by Euler’s Theorem the
average productivity of x1 can be written as
f
x1
=
1
r
￿
f1 + f2
x2
x1
￿
. (3)
We established that the introduction of uncertainty increases f1, f2 and x2/x1; thus f/x1
increases.￿
This result is also intuitive since the increase in marginal productivity increases the average
productivity. It is worth noting that the previous studies did not investigate the impact of
multiple uncertainty on productivity.
Proposition 4. Starting with cost uncertainty, adding output price uncertainty reduces the
optimal level of each input if f12 ≥ 0 and ˜ p and ˜ w1 are statistically independent.
Proof. Rewrite (1) as
Eu
￿ (˜ π
∗)(¯ pf1 − ˜ w1) + f1Cov(u
￿ (˜ π
∗), ˜ p) = 0.
Since Cov (u￿ (˜ π∗), ˜ p) < 0, the equation above implies that
Eu
￿ (˜ π
∗)(¯ pf1 − ˜ w1) > 0.
Deﬁne the sets A and ∼ A such that
A = {w1|¯ pf1 − w1 ≥ 0},
∼ A = {w1|¯ pf1 − w1 ≤ 0}.
3For any w1 ∈ A and w￿
1 ∈∼ A,
pf − w1x
∗
1 − w2x
∗
2 ≥ pf − w
￿
1x
∗
1 − w2x
∗
2; w1 ∈ A, w
￿
1 ∈∼ A.
Since u￿￿ < 0, the inequality above implies
u
￿ (˜ π
∗ (w1)) ≤ u
￿ (˜ π
∗ (w
￿
1)); w1 ∈ A, w
￿
1 ∈∼ A.
Therefore,
S = max
w1∈A
u
￿ (˜ π
∗) ≤ I = min
w1∈∼A
u
￿ (˜ π
∗).
Thus,
S
u￿ (Ep˜ π∗)
≤
I
u￿ (Ep˜ π∗)
,
where Ep denotes the expectation with respect to ˜ p for a given w1. Since S and I are both
positive, there must exist a positive constant t such that
EpS
u￿ (Ep˜ π∗)
≤ t ≤
EpI
u￿ (Ep˜ π∗)
,
so that
tu
￿ (Ep˜ π
∗) ≥ EpS ≥ Epu
￿ (˜ π
∗), w1 ∈ A, (4)
where the last inequality in (4) holds since S is a maximum. Now (4) implies
(¯ pf1 − w1)tu
￿ (Ep˜ π
∗) ≥ (¯ pf1 − w1)Epu
￿ (˜ π
∗), w1 ∈ A, (5)
since ¯ pf1 − w1 > 0 for w1 ∈ A. Similarly,
(¯ pf1 − w1)tu
￿ (Ep˜ π
∗) ≥ (¯ pf1 − w1)Epu
￿(˜ π
∗), w1 ∈∼ A, (6)
since (¯ pf1 − w1) < 0 for w1 ∈∼ A. Thus the inequalities in (5) and (6) hold for all values of w1,
and taking expectations with respect to ˜ w1, we obtain
tEw1 (¯ pf1 − ˜ w1)u
￿ (Ep˜ π
∗) ≥ E (¯ pf1 − ˜ w1)u
￿ (˜ π
∗) > 0. (7)
Since t > 0, (7) implies
Ew1 (¯ pf1 − ˜ w1)u
￿ (Ep˜ π
∗) > 0. (8)
Now, deﬁne a1 ≡ Ew1 (¯ pf1 − ˜ w1)u￿ (Ep˜ π) and thus da1 > 0 in response to the introduction of the
output price risk. Similarly, deﬁne a2 ≡ Ew1 (¯ pf2 − w2)u￿ (Ep˜ π) and thus da2 > 0 in response
to the output price risk. Totally diﬀerentiating a1 and a2 (holding the parameters constant), we
obtain ￿ ¯ H11 ¯ H12
¯ H12 ¯ H22
￿￿
dx1
dx2
￿
=
￿
da1
da2
￿
,
where ¯ H is the Hessian in the absence of the output price risk and ¯ H12 = f12¯ pEw1u￿ (Ep˜ π); thus
dx1 =
¯ H22da1 − ¯ H12da2
¯ H11 ¯ H22 − ¯ H2
12
,dx2 =
¯ H11da2 − ¯ H12da1
¯ H11 ¯ H22 − ¯ H2
12
.
4Therefore dx1 < 0 and dx2 < 0.￿
It is worth noting that this result is stronger than the standard results on aversion to one risk
in the presence of another (see Golllier (2001) and Pratt (1988)). In addition, the proof provides
technical contributions.
Proposition 5. Given the statistical independence between ˜ p and ˜ w1, starting with output
price uncertainty, adding cost uncertainty (i) reduces the optimal level of the risky input (ii)
reduces the optimal level of the non-risky input if the inputs are substitutes (in preferences).
Proof. Deﬁne b1 ≡ Epu￿(Ew1˜ π)(˜ pf1 − ¯ w1) and b2 ≡ Epu￿ (Ew1˜ π)(˜ pf2 − w2); we can show
db1 > 0 and db2 = 0 in response to the cost uncertainty (the proof is similar to the proof of
Proposition 4). Totally diﬀerentiating b1 and b2 (holding the parameters constant), we obtain
￿ ...
H11
...
H12 ...
H12
...
H22
￿￿
dx1
dx2
￿
=
￿
db1
0
￿
,
where
...
H is the Hessian in the absence of cost risk, thus
dx1 =
...
H22db1 ￿ ￿...
H
￿ ￿ ,dx2 =
−
...
H12db1 ￿ ￿...
H
￿ ￿ .
Hence, dx1 < 0; when the inputs are substitutes Eu12 ≡ H12 > 0; thus, dx2 < 0. Also, dx2 ≥ 0
if the inputs are complements (or independent), since Eu12 ≤ 0.￿
The result is also intuitive since the risky input must fall in response to cost uncertainty,
whereas the non-risky input reacts according to the preferences relationship between the two in-
puts. Thus the non-risky input decreases (increases) if the inputs are substitutes (complements).
4. The impact of change in risk aversion
In this section we show the impact of an increase in risk aversion on the optimal level of each
input.
Proposition 6. Given the statistical independence between ˜ p and ˜ w1, when risk aversion
increases the optimal level of each input falls if the inputs are substitutes or independent.
Proof. For each ﬁxed value of ˜ p, deﬁne ˆ w1 by pf1 (x∗)− ˆ w1 = 0 and let ˆ π be the proﬁt when
w1 = ˆ w1. Also let x∗ be the optimal input vector for ﬁrm1, respectively. Assume that ﬁrm 1 is
more risk averse than ﬁrm 2. The ﬁrst-order condition for ﬁrm 1 can be written as
Ew1
u￿
1 (˜ π∗)
u￿
1 (ˆ π)
(pf1 (x
∗) − ˜ w1) = 0, (9)
where u1 is ﬁrm 1’s utility function and u￿
1 (ˆ π) is a constant. Equation (9) can be rewritten as
￿
˜ w1< ˆ w1
u￿
1 (˜ π∗)
u￿
1 (ˆ π)
(pf1 − ˜ w1)Γ( ˜ w1)d˜ w1 +
￿
˜ w1> ˆ w1
u￿
1 (˜ π∗)
u￿
1 (ˆ π)
(pf1 − ˜ w1)Γ( ˜ w1)d ˜ w1 = 0, (10)
where f1 = f1 (x∗). The corresponding expression for ﬁrm 2 is
￿
˜ w1< ˆ w1
u￿
2 (˜ π∗)
u￿
2 (ˆ π)
(pf1 − ˜ w1)Γ( ˜ w1)d˜ w1 +
￿
˜ w1> ˆ w1
u￿
2 (˜ π∗)
u￿
2 (ˆ π)
(pf1 − ˜ w1)Γ( ˜ w1)d ˜ w1. (11)
5Subtracting (10) from (11) yields
￿
˜ w1< ˆ w1
￿
u￿
2 (˜ π∗)
u￿
2 (ˆ π)
−
u￿
1 (˜ π∗)
u￿
1 (ˆ π)
￿
(pf1 − ˜ w1)Γd ˜ w1+
￿
˜ w1> ˆ w1
￿
u￿
2 (˜ π∗)
u￿
2 (ˆ π)
−
u￿
1 (˜ π∗)
u￿
1 (ˆ π)
￿
(pf1 − ˜ w1)Γd˜ w1. (12)
An increase in ˜ w1 decreases ˜ π∗. In the ﬁrst integral (pf1 (x∗) − ˜ w1) is positive, Pratt showed
that the term in square brackets is also positive. These inequalities are both reversed in the
second integral. Consequently (12) is positive and thus (11) is positive. By the independence
assumption,
E
u￿
2 (˜ π∗)
u￿
2 (ˆ π)
(˜ pf1 − ˜ w1) = EpEw1
u￿
2 (˜ π∗)
u￿
2 (ˆ π)
(˜ pf1 − ˜ w1) > 0.
Deﬁne α1 ≡ Eu￿
2 (˜ π)(˜ pf1 (x) − ˜ w1); thus dα1 > 0 when risk aversion increases. Similarly, we can
show that α2 ≡ Eu￿
2 (˜ π)(˜ pf2 (x) − w2) > 0 when risk aversion increases. Totally diﬀerentiating
α1 and α2 (holding the parameters constant), we obtain
dα1 = H11dx1 + H12dx2 > 0, (13)
dα2 = H22dx2 + H12dx1 > 0. (14)
From (13) and (14) ￿
H11 H12
H12 H22
￿￿
dx1
dx2
￿
=
￿
dα1
dα2
￿
.
Thus
dx1 =
H22da1 − H12da2
|H|
,dx2 =
H11da2 − H12da1
|H|
,
and thus dx1 < 0 and dx2 < 0 when risk aversion increases.￿
5. Conclusion
This paper highlights the importance of the relationship between the inputs in determining
the impact of multiple uncertainty on the input demand. In the presence of multiple uncertainty,
each input demand is less than its certainty-equivalent level, given the two inputs are substitutes
in production. This is due to two changes. First, the fact that the risky input falls in response
to multiple uncertainty, while the non-risky input falls in response to output price uncertainty.
Second, the inputs being substitutes causes each input to decrease when the other input falls due
to uncertainty. The two changes will be in the same direction if the two inputs are substitutes
and hence the net eﬀect is a decrease in each input demand. Consequently, in the presence of
multiple uncertainty, each input demand is less than the input demand in the presence of only
cost uncertainty, given the two inputs are substitutes in production/preferences. Also, in the
presence of multiple uncertainty, the demand for each input is less than its level in the presence
of only output price uncertainty, given the inputs are substitutes in preferences. Moreover, when
risk aversion increases, both inputs fall if the inputs are substitutes in preferences.
6References
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The theory of the firm under multiple uncertainties

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