III. Uncertainty and Asymmetric Risk Attitudes
Our benchmark equilibrium fully characterized, we can now assess the effect of uncer- tainty on the firms’ behavior. This section investigates the effects of asymmetries in
perceived risk and in risk aversion between the firms. It is shown that uncertainty induces risk averse firms to compete less aggressively. This results in a lower industry output than
under certainty. We also show that asymmetries in risk and in risk attitude could confer sufficient “information” and “attitude” advantages to allow one of the firms to produce
more under uncertainty than under certainty.
We begin our investigation with the effect of uncertainty on the best-response func- tions.
Proposition 1. Let R
iu
x
j
; z be firm i’s best-response function given a
i
, g
i 2
. a Assuming a
i
5 0: R
iu
x
j
; z 5 R
ic
x
j
; Assuming a
i
. 0: b R
iu
x
j
; z , R
ic
x
j
for 0 x
j
d 2 c; c R9
iu
. R9
ic
. Proof. According to Eq. 10, the firm maximizes:
W
i
~ x
iu
; x
ju
5 ~d 2 c 2 x
iu
2 x
ju
x
iu
2 ~ a
i
2g
i 2
x
iu 2
12 The first order condition associated with this optimization problem is:
~d 2 c 2 x
ju
2 ~2 1 a
i
g
i 2
x
iu
5 0. 13
Rearranging, we obtain the best-response function: R
iu
~ x
ju
; a
i
, g
i
5 d 2 c
2 1 a
i
g
i 2
2 x
ju
2 1 a
i
g
i 2
14 From Eq. 14, it becomes clear that: a R
iu
x
j
; z [ R
ic
x
j
if a
i
5 0; b R
iu
x
j
; z , R
ic
x
j
; and c R9
iu
5 212 1 a
i
g
i 2
. R9
ic
5 21 2 if a
i
. 0. Q.E.D.
The first part of the proposition simply states that the best-response function of a risk neutral firm is not affected by the uncertainty. The last two parts of the proposition imply
that the best-response function of the risk averse firm shifts inward. By Eq. 14, we can deduce that the best-response function of firm i and firm j actually rotate inward around
the coordinates 0, d 2 c and d 2 c, 0 as shown in Fig. 1 where aeb and ced are the best-response functions under certainty. Thus uncertainty induces the risk averse firm to
bring less to market. Another implication of the proposition is that the effect of uncertainty depends on the rival’s expected level of output: the greater the expected output of the rival,
the smaller the effect of uncertainty on the firm’s level of output. This reflects Leland’s 1972 principle of increasing uncertainty that links the level of potential losses to the
level of profits. In this case, if the rival’s expected output is small, the residual demand faced by the firm will allow for higher profits. Because the firm has more to lose, it will
be more affected by uncertainty. The effect of uncertainty on the strategy of the firm is directly related to the coefficient of risk aversion of the firm. As in Baron 1970, risk
aversion renders the firm less aggressive in its output decision.
The first two elements of the above proposition are consistent with results derived in Fishelson 1989. However, Fishelson’s best response functions shift downward in a
440 B. Larue and V. Yapo
parallel fashion. In light of this, we assessed the robustness of our results by using a general Von Neumann-Morgenstern utility function.
Proposition 2. Assuming firm i has a general Von Neumann-Morgenstern utility function and R9
iu
x
j
and R9
ic
x
j
are the slopes of the best-response functions of firm i under uncertainty and certainty respectively. a R9
iu
x
j
Þ R9
ic
x
j
; b If U- 5 0, R9
iu
, 0, R0
iu
5 0.
Proof. The first order condition to the utility maximization problem can be represented by:
Ep9
i
~ x
i
U9
i
~p
i
5 Ep9
i
~ x
i
1 covU9
i
~p
i
, p9
i
EU9
i
~p
i
5 0 15
Using the definition of covariance, the above condition can be rearranged in a more intuitive manner:
5
Ep9
i
~ x
i
2 g
i
z EU9
i
~p
i
«
j
EU9
i
~p
i
5 0. 16
Now, unless utility is linear in profits, the second term on the left-hand side will be a function of x
i
. Fishelson 1989 obtains his result see his equations 3 and 4 that the slope of the best-response function remains the same as under certainty by assuming that the
second term is independent of x
i
.
6
A Taylor expansion of U9
i
p
i
around E[p
i
] will help us show that, generally speaking, the slope of the best-response function is affected by
uncertainty.
U9
i
~p
i
5 U9
i
~Ep
i
1 U 0
i
~Ep
i
p
i
2 Ep
i
1 U-
i
~Ep
i
2 p
i
2 Ep
i 2
1 R. 17
Limiting the approximation to the terms appearing in Eq. 17 and multiplying by «
j
, we obtain:
U9
i
~p
i
«
j
5 U9
i
~Ep
i
«
j
1 U 0
i
~Ep
i
p
i
2 Ep
i
«
j
1 U-
i
~Ep
i
2 p
i
2 Ep
i 2
«
j
18 Noting that p
i
2 E[p
i
] 5 2g
i
x
i
«
j
, and that E{[p
i
2 E[p
i
]]
2
, «
j
} 5 cov[p
i
2 E[p
i
]]
2
, «
j
, which is in turn equals to g
i 2
x
i 2
cov«
j 2
, «
j
and taking the expectation of Eqs. 17 and 18, we get:
EU9
i
~p
i
5 U9
i
~Ep
i
1 U-
i
~Ep
i
2 g
i 2
x
i 2
19
5
By definition, covU9
i
, p9
i
5 E[U9
i
2 E[U9
i
]p9
i
2 E[p9
i
]]. Since p9
i
5 d 2 c 2 2x
i
2 x
j
2 g
i
«
j
and p9
i
2 E[p9
i
] 5 2g
i
«
j
, covU9
i
, p9
i
5 2g
i
E[U9
i
2 E[U9
i
]«
j
] 5 2g
i
E[U9
i
«
j
] 1 g
i
E[E[U9
i
]«
j
] 5 2g
i
E[U9
i
«
j
].
6
A parallel shift of R
i
[ is possible if the objective function of the firm had been W
i
5 E[p
i
] 2 a
i
2g
i
=varp
i
. Uncertainty would have had the same effect as a rise in the unit cost of production since E[p
i
]x
i
2 a
i
g
i 2
5 0.
Asymmetries in Risk Attitude: Duopoly 441
EU9
i
~p
i
«
j
5 2U 0
i
~Ep
i
g
i
x
i
1 U-
i
~Ep
i
2 g
i 2
x
i 2
cov~«
j 2
, «
j
20 Inserting Eqs. 19 and 20 into Eq. 16 confirms that its second left-hand side term is
indeed a function of x
i
. It can be shown that U-
i
[ 5 0 is a sufficient condition to have linear best-response functions and hence avoid potential multiple equilibria.
7
Thus, under rather general conditions, the introduction of uncertainty will change the slopes of the
best-response functions but it need not transform best-response functions that are linear under certainty into nonlinear ones.
Q.E.D.
It is well known that the perfectly competitive firm reduces its output under uncertainty if it is risk averse. We now want to verify whether this result extends to the oligopolistic
firm with constant absolute risk aversion. Proposition 3. 1 Let x
iu
a
i
, a
j
; g
i
, g
j
be the sales of the individual firms. a x
iu
, x
ic
if a
i
1g
i 2
; b x
iu
, x
ic
if a
i
. 0 and a
j
5 0. 2 Assuming symmetry in risk perceptions g
i
5 g
j
5 g and that 0 , a
i
, 1g
2
, then x
iu
. x
ic
if a
j
. 2a
i
1 2 a
i
g
2
. Proof. The Nash equilibrium quantities under uncertainty are given by:
x
iu
5 a
iu
~a
i
, a
j
, g
i
, g
j
z ~d 2 c 21
where a
iu
5 1 1 a
j
g
j 2
[2 1 a
i
g
i 2
2 1 a
j
g
j 2
2 1] and i Þ j. Note that the coefficients of risk aversion and the level of risk for both firms enter explicitly in the equilibrium
quantities. This contrasts with the expression derived for the best-response functions. From Eqs. 5 and 21, we can assert that a
ic
5 13. After some rearranging, it can be shown that:
a
iu
5 1
3 1
a
j
g
j 2
z ~1 2 a
i
g
i 2
2 2a
i
g
i 2
9 1 6~a
i
g
i 2
1 a
j
g
j 2
1 3a
i
g
i 2
a
j
g
j 2
22 From Eq. 22, it is clear that a
iu
, 13 if a
i
1g
i 2
for all a
j
0. It is also evident that a
iu
a
i
, 0, g
i
, g
j
5 13 1 2a
i
g
j 2
, 13 for all a
i
. 0. This proves part 1 of the proposition. If we impose symmetry on the level of risk, Eq. 22 simplifies to:
a
iu
5 1
3 1
a
j
g~1 2 a
i
g
2
2 2a
i
g
2
9 1 6~a
i
1 a
j
g
2
1 3a
i
a
j
g
4
23 A careful inspection of Eq. 23 reveals that:
7
The expression cov~U9
i
, «
j
EU
i
~p
i
5 2 g
i
z
3
U 0
i
~Ep
i
x
i
2 U-
i
~Ep
i
2 g
i
x
i 2
cov~«
j 2
, «
j
U9
i
~Ep
i
1 U-
i
~Ep
i
2 g
i 2
x
i 2
4
will have two roots unless the third derivative of the utility function vanishes.
442 B. Larue and V. Yapo
a
iu
1 3
if a
j
~1 2 a
i
g
2
2 2a
i
0. 24
This condition is respected if and only if: a
i
g
2
1 and a
j
ca
i
5 2a
i
1 2 a
i
g
2
. Q.E.D.
The above proposition states that Sandmo-like behavior on the part of the firm is observed if the rival firm is risk neutral. Of particular interest is the fact that the level of
output of the risk averse oligopolistic firm is not necessarily smaller under uncertainty. The condition under which the firm’s output is higher under uncertainty amounts to a large
risk aversion differential between the “slightly” risk averse firm and its rival. The size of that risk aversion differential is in turn function of the level of risk. More specifically, the
output of the firm rises under uncertainty if it is “not too risk averse,” a
i
1g
2
, and if the rival firm is “sufficiently risk averse,” that is: a
j
ca
i
5 2a
i
1 2 a
i
g
2
. Note that c0 5 0, c
u
a
i
31g
2
5 `, c9 5 21 2 a
i
g
2 2
. 0. This means that the minimum level of risk aversion on the part of firm j that will allow firm i to increase its output under
uncertainty is a positive and increasing function of firm i’s level of risk aversion. Fig. 2 illustrates the combinations of risk aversion coefficients required for firm 1 to
maintain its certainty level of output under uncertainty. The closest schedules to the origin are associated with higher levels of risk. The space to the left of each schedule maps the
region over which the level of output of firm 1 is greater under uncertainty than under certainty. Thus, the higher the level of risk g
2
, the smaller the region in the a
1
, a
2
space that will support an equilibrium in which the output of firm 1 is higher under uncertainty
than under certainty. The following proposition investigates the effect of uncertainty on the industry’s level
of output. Proposition 4. Let x
u
a
i
, a
j
, g
2
and p
u
a
i
, a
j
, g
2
be the industry sales and market price when the firms face a symmetric level of risk. If a
i
. 0, then x
u
, x
c
and p
u
. p
c
, a
j
0. Corollary 4.1. a
i
. 0, a
j
5 0 f p
u
5 p
iu
1 p
ju
. p
c
.
Figure 2. Iso-certainty output schedules for different levels of risk g
2
5 2, 1, 0.5. Asymmetries in Risk Attitude: Duopoly
443
Proof. We must show that a
u
5 a
iu
1 a
ju
, 23 for a
i
. 0 and a
j
0. For the case least favorable to the proposition i.e., when the rival firm is risk neutral a
j
5 0, we have:
a
u
5 2
3 2
a
i
g
2
9 1 6a
i
g
2
. 25
From Eq. 25, it is apparent that for a
i
. 0, a
u
a
i
, a
j
, g
2
, 23. The corollary follows from the fact that the industry output will lie between the Cournot and monopoly level.
Thus when one firm is risk averse and the other is risk neutral, industry profits are bounded by the profits associated with the Cournot and monopoly certainty equilibria.
Q.E.D.
Fig. 1 illustrates some of the implications of propositions 3 and 4. In this example, we assumed that d 5 5 and c 5 2. The line cb represents the combinations of firm 1 and firm
2 outputs equal to the collusion industry output while the hef schedule maps the Cournot– Nash industry output. When one firm is risk averse and the other is risk neutral, the
industry output will be on the frontier ceb. Consider the case when firm 1 is risk averse and firm 2 is risk neutral. In this instance, the equilibrium will be on the ce segment along
which: x
1u
, x
1c
, x
2u
. x
2c
. The position of the new equilibrium depends on the extent of the rotation of firm 1’s best response function around point a i.e., the extent of firm
1’s level of risk aversion. When both firms are risk averse, the potential equilibria lie inside the ceb frontier and industry profits under uncertainty may rise above or fall below
the certainty benchmark. The latter requires the firms to be very risk averse andor to face a high level of uncertainty. Fig. 1 also displays the inward rotations of both best-response
functions. This symmetric equilibrium is characterized by a reduction of output for both firms such that the collusive industry level of output is achieved. When risk aversion is
asymmetric, it is possible for one risk averse firm to have a higher output level under uncertainty but not for both firms. This reflects the fact that the outputs of the firms are
strategic substitutes. In Fig. 1, equilibria inside the 1, e, b triangle are consistent with a higher level of output for firm 1 and a lower level of output for firm 2. The reverse holds
for equilibria inside the 1, c, e triangle.
Unless both firms are risk neutral, we showed that the level of industry output falls under uncertainty. The iso-industry output schedule under certainty, line hef in Fig. 1, lies
above its uncertainty counterpart, line cb, when both firms have unitary coefficients of risk aversion. When the output of the industry does not fall too much, the aggregate level of
profits increases under uncertainty. Uncertainty tempers the aggressive nature of the Cournot firms but this does not imply that uncertainty necessarily brings about higher ex
post profits for both firms nor does it imply that firms will seek out uncertainty. Nevertheless, it is possible for both firms to experience higher profits under uncertainty.
Proposition 5. Assuming symmetry in risk g
i 2
5 g
j 2
5 g
2
, d 5 5, c 5 2, and 0 , a
i
, 5 1 3=52g
2
, then p
iu
. p
ic
, p
ju
. p
jc
if 0 , a , a
j
, a where a 5 ga
i
; g
2
and a
5 ha
i
; g
2
. Proof. The certainty benchmark profits and the profits under uncertainty are:
p
ic
5 1 and p
iu
5 b
i
~a
i
, a
j
, g ;
9~1 1 a
i
g
2
~1 1 a
j
g
2 2
3 1 2a
i
g
2
1 2a
j
g
2
1 a
i
a
j
g
4 2
. 26
444 B. Larue and V. Yapo
For the profits of both firms to be higher under uncertainty, it must be shown that the following conditions are satisfied: b
i
a
i
, a
j
, g . 1 and b
j
a
i
, a
j
, g . 1. Solving for b
i
2 1 5 0 and b
j
2 1 5 0, we obtain the boundaries on firm j’s risk aversion in terms of firm i’s risk aversion:
g~a
i
; g 5 3 1 2a
i
g
2
2 2a
i 2
g
4
2 3
Î
~1 1 a
i
g
2 3
25 1 a
i 2
g
4
2 5a
i
g
2
, h~a
i
; g 5 23 1 4a
i
g
2
1 5a
i 2
g
4
1 3
Î
~1 1 a
i
g
2 3
~1 1 5a
i
g
2
2~2 1 a
i
g
2 2
. The g[ schedule is a mapping of the levels of firm j’s risk aversion for which the profits
of firm i under uncertainty are equal to the certainty benchmark. The h[ schedule has a similar interpretation for firm j. From the g[ schedule, a region A
i
can be defined in the a
i
, a
j
space in which p
iu
. p
ic
: A
i
5
H
~a
i
, a
j
: 0 , a
i
, 5 1 3
Î
5 2g
2
and a
j
. g~a
i
J
From h[, we can also define a region A
j
in which p
ju
. p
jc
: A
j
5
H
~a
i
, a
j
: a
i
. 0 and a
j
, h~a
i
, 5 1 3
Î
5 2g
2
J
. For the ex post profit of both firms to be higher, there must exist a region A such that A
i
ù A
j
5 A is non-empty. The existence of a
i
, a
j
combinations defining A is proven by noting that: 1 g[ and h[ are continuous functions over the interval [0, 5 1 3=52g
2
], 2 g90 , h90,
8
3 g[ and h[ are respectively convex and concave in the a
i
, a
j
space see Fig. 3. Thus, there exist many combinations of risk aversion levels bounded by: 0 ,
a
i
, 5 1 3=5 2g
2
and ga
i
, g
2
, a
j
, ha
i
, g
2
, 5 1 3=5 2g
2
that support an equilibrium characterized by higher profits for both firms under uncertainty.
Q.E.D.
9
To complement the above proof, an example is constructed to demonstrate that the profits of both firms can rise under uncertainty. This is the case for the pair g[, h[ 5
0.8a
i
, 1.25a
i
. To see this, rewrite Eq. 26 as: b
i
5 1 1 3g
2
~2a
j
2 a
i
1 D
i
~a
i
, a
j
, g 3 1 2a
i
g
2
1 2a
j
g
2
1 a
i
a
j
g
4 2
where D
i
a
i
, a
j
, g 5 5a
j 2
g
4
2 4a
i 2
g
4
1 5a
i
a
j 2
g
6
2 4a
i 2
a
j
g
6
1 4a
i
a
j
g
4
2 a
i 2
a
j 2
g
8
and the level of profit under certainty is 1. From this representation of the profits of firm i under
uncertainty, it can be shown that for a given level of a
i
: 1 b
i
. 1 if a
i
a
j
, 4g
4
and a
j
. 4a
i
5, and 2 b
j
. 1 if a
i
a
j
, 4g
4
and a
j
, 5a
i
4. According to these conditions, a small level of risk can generate a more profitable equilibrium for both firms if there is not too
8
By evaluating the derivatives of g[ and h[ at a
1
5 0, we get g90 5 0.5 , h90 5 2.
9
We replaced our original assumption on demand and cost parameters i.e., d . c . 0 by d 5 5 and c 5 2 because that the gains in tractability and intuition far outweigh the loss of generality. Clearly, equilibria
characterized by higher profits for both firms can be supported by different values of d and c.
Asymmetries in Risk Attitude: Duopoly 445
much asymmetry in the levels of risk aversion between the firms. In the case of symmetrical and unitary risk attitudes and risk levels, we have b
i
5 b
j
5 1.125 . p
ic
5 p
jc
5 1, b 5 2.25 . p
c
5 2. Fig. 3 illustrates the last proposition. The convex schedules represent the combinations
of risk aversion levels that maintain the level of firm 1’s profit to the certainty benchmark for different levels of risk. The combinations to the left of each schedule provide higher
profits for firm 1. The higher profit region shrinks with the level of risk as shown by the respective position of the three schedules. The same interpretation can be given to the
concave iso-certainty profit schedules of firm 2. The ellipses made by both sets of iso-certainty profit schedules define the combinations of risk aversions that support higher
profits for both firms for different levels of risk. The size of the ellipse clearly shrinks with the level of risk.
Instead of representing the region of increased profits for both firms in risk aversion space, we can define it in terms of quantities. Fig. 1 displays the best-response functions
under certainty and under uncertainty as well as the iso-profit curves associated with the Nash equilibrium under certainty. The ellipse formed by these iso-profit curves defines the
region within which the profits of both firms can rise. The particular Nash equilibrium under uncertainty a
i
5 a
j
5 g 5 1 shown in Fig. 1 falls in that region.
Figure 3. Firm 1 and firm 2 iso-certainty profit schedules for g
2
5 2, 1, 0.5.
446 B. Larue and V. Yapo
IV. Uncertainty and Asymmetry in Risk
Kategori