Formación para toda la vida
Campus Monterrey
Número 6
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COMPARATIVE STATICS UNDER UNCERTAINTY
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Jorge Ibarra Salazar
Laura Razzolini
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Abril 1995
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Serie de Documentos de Trabajo
del Departamento de Economía
División de Administración y Finanzas
Departamento de Economía
Sucursal de Correos J
Monterrey N.L. 64849
Tel. (81) 8358-2000, Ex. 4306 y 4307
Comparative Statics Under Uncertainty*
by
Jorge Ibarra - Salazar
Department of Economics
Instituto Tecnológico y de Estudios Superiores de Monterrey
Sucursal de Correos J
Monterrey N.L. 64849 MEXICO
[email protected]
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and
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Laura Razzolini
Department of Economics and Finance
The University of Mississippi
University, MS 38677, U.S.A.
[email protected]
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Abstract: One of the main objectives of models studying economic decision-making under
uncertainty is to perform comparative statics. The following two problems are often analyzed: (i)
explain the effect that a change in a parameter present in the payoff function exerts on the optimal
choice of the decision variable, while not affecting the distribution of the random variable; (ii)
explain the effect that marginal changes in the distribution of the random variable exert on the
optimal choice of the decision variable, while not affecting the value of the payoff function. In this
paper we develop a model to consider how a change of a parameter in the payoff function may
affect both the optimal choice of the decision variable and the distribution of the random variable.
This entails to consider explicitly the relationship between the parameter and the distribution of the
random variable. The general model is motivated by three examples: portfolio allocation model,
demand for insurance model and a price - taking firm decision model.
(*) We thank J. Hadar and T.K. Seo for helpful comments.
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1.
Introduction
The purpose of this paper is to extend the traditional comparative statics analysis under risk,
by developing a general model that considers the effects that a change in a parameter, endogenous
to the model, may exert on both the value of the payoff function and the distribution of the random
variable.
Two are the approaches mostly followed in the literature to analyze the effects that a
changing economic environment exert on the optimal plan of actions undertaken by a risk averse
agent. On one hand, we have models that investigate how a change in a parameter of the objective
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function affects the agent's optimal choice; on the other hand, we have models that analyze how a
change in the underlying uncertainty affects the agent's choice. In the first class of models dealing
with a parametric change, it is common practice to impose monotonicity restrictions on the
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appropriate measure of risk aversion, in order to obtain unambiguous results about the direction of
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the change in the optimal value of the choice variable. Some widely cited examples of this are,
among others, Sandmo's model [1971] analyzing the behavior of a competitive firm under
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uncertainty; Mossin's insurance model [1968]; Arrow's portfolio model [1971] with one risky and
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one safe asset; Batra and Ullah's model [1974] analyzing the competitive firm's choice of inputs
under price uncertainty; and Leland's model [1972] of the firm facing uncertain demand. In all
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these examples, the parameter causing the change in the decision variable affects only the value of
the payoff or objective function, whose expected utility the agent maximizes. Examples from the
above are the fixed cost or taxes in the models of the firm, wealth or riskless rate of return in the
portfolio model, wealth or per unit price of coverage in the insurance model. In all these cases, the
parameter does not cause changes in the distribution of the random variable.
In the second class of models dealing with a change in uncertainty, two are the approaches
mostly followed: the transformation approach and the stochastic dominance approach. According
to the first, the original random variable is parametrized so that mean preserving spread (MPS)
shifts or changes in the mean are obtained (Sandmo [1971], Arrow [1971], Leland [1972], Batra and
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Ullah [1974], Ishii [1977] and Meyer and Ormiston [1989]). With the transformation of variable
approach it is also possible to analyze the effects of first degree stochastic dominance (FSD) and
second degree stochastic dominance (SSD) shifts in the distribution of the random variable, so that
the changes in uncertainty are not necessarily mean preserving (Hadar and Russell [1974], Cheng,
Magill and Shafer [1987], Meyer [1989] and Ormiston [1992]). According to the stochastic
dominance approach, on the other hand, general marginal changes in the distribution of the random
variable are represented as FSD, SSD or MPS shifts. This technique has been applied both to
general models (Rothschild and Stiglitz [1970], Hadar and Russell [1978], Kraus [1979] and Meyer
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and Ormiston [1985)]), and to specific examples (Rothschild and Stiglitz [1971], Hadar and Seo
[1991, 1992] and Ibarra - Salazar [1996]). Such changes in uncertainty, if explained at all, have
been justified as induced by a change in a parameter, exogenous to the model (Hadar and Russell
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[1978]). Such a parameter, however, is assumed not to alter the value of the payoff function. It is
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assumed to affect only the distribution of the random variable. Similarly, in the transformation
approach, such transformation is defined as a parametric one, but the parameter is not included in
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the payoff function in any meaningful way.
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In this paper we bring together these two types of comparative statics problems, by
developing a model where the possible relationship between a parameter included in the payoff
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function and the distribution of the random variable is explicitly considered. In this way, we
analyze how a change in a given parameter may affect both the value of the payoff function and the
distribution of the random variable, inducing, for instance, an FSD shift in the distribution.
Therefore, under this more general set - up, the final result of comparative statics caused by a
change in the parameter may differ from the outcome of the traditional analysis for the case of
parametric changes. For the special case in which the parameter does not exert any effect on the
distribution of the random variable, the standard comparative statics results are obtained.
The effect that a parameter can have on the random variable has been previously recognized
by Hymans [1966] and more recently by Ibarra - Salazar [1996], limited to the specific model of a
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price - regulated or price - taking firm facing uncertain demand. In both models, the given price is
the parameter that affects both the value of the profit/payoff function and the distribution of the
random demand. We extend this analysis by deriving a general specification to study the change in
the optimal value of the control variable induced by a change in the parameter. The standard result
of comparative statics analysis of a parametric change is shown to be a special case of such general
specification.
The paper is organized as follows. In Section 2 the general model is developed. Section 3
2.
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illustrates applications of this general framework and Section 4 concludes.
The model: a general approach to comparative statics under uncertainty
A common economic decision model, used in the literature in many specific applications,
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considers a decision maker choosing the optimal value of the choice variable, y, taking the random
variable, x, and a parameter, β,1 as given. The objective of the agent is to maximize the expected
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utility of a payoff function, z(x,y;β). This function might represent wealth, or profit, depending on
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the model considered. Such function is assumed to be twice continuously differentiable, with
zyy(x,y;β) ≤ 0.
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In what follows, we will assume that the parameter β not only affects the value of the
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function z(x,y;β), but it can also induce a first degree stochastic dominance (FSD) shift in the
distribution of the random variable x. Let F(x;β) be the cumulative distribution function of the
random variable x, and f(x; β) the density function. We assume that F is monotone in β, with
continuous first derivatives. Such monotonicity assumption guarantees that changes in the
parameter will cause FSD shifts in the distribution of the random variable.
The problem for the agent can be written as:
+∞
maxy E U(z(x,y;β)) = maxy
∫ U(z(x, y; β)) f (x;β) dx.
−∞
1
β can also be interpreted as a vector representing a family of parameters.
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(1)
The utility function is assumed to be twice continuously differentiable, with U′(z) ≥ 0 and
U″(z) ≤ 0. Further, U(z) is bounded for any finite z. Assuming an interior solution exists, the first
and second order conditions of problem (1) are respectively:
∂EU( z)
= U y (y) =
∂y
+∞
∫ U′ (z) z (x, y;β) f (x;β) dx = 0,
+∞
2
∂ EU( z)
∂y
2
(2)
y
−∞
= U yy ( y ) =
∫ [ U′ ( z ) z
2
yy
( x , y; β ) + U′′ ( z) [ zy ( x , y; β )]
−∞
] f (x;β) dx < 0.
(3)
The assumptions about the utility function, together with zyy ≤ 0 are sufficient to guarantee that the
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second order condition (3) is satisfied. Solving the first order condition, we can derive the optimal
value for the control variable y, as a function of β, the parameter of the model: y = y(β).
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To analyze the effects that a change in the parameter β has on the control variable y, we can
apply the implicit function theorem and derive
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Uyβ ( y )
∂y
,
=−
Uyy ( y )
∂β
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where the sign of the denominator is negative by the second order condition. Therefore, the
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comparative statics problem reduces to computing and determining the sign of the term Uyβ ( y ) .
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We have:
+∞
U yβ ( y) =
∫ { U ′ ( z) z
yβ
}
( x, y; β) + U ′′ ( z) zβ ( x, y; β) z y ( x, y; β) f ( x; β) dx
−∞
+∞
+
∫ {U′ (z) z (x, y;β)}
y
−∞
∂f ( x; β )
dx.
∂β
(4)
Denoting the first term in the right hand side of equation (4) by η and integrating by parts
the second term, we obtain:
U yβ ( y) = η + U ′( z) z y ( x, y; β)
∂F( x; β)
∂β
+∞
−∞
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+∞
-
∫ { U′ ( z) z
yx
} ∂F(∂βx;β) dx .
( x , y; β ) + U′′ ( z) zx ( x , y; β ) zy ( x , y; β )
−∞
Notice that the second term to the right hand side of (5) vanishes since
(5)
∂F(+∞ , β ) ∂F(−∞ ,β )
=
= 0,
∂β
∂β
and by assuming that U′(z) zy(x,y;β) is bounded. Letting
+∞
ψ=−
∫ { U ′ ( z) z
yx
} ∂F(∂βx;β) dx ,
( x , y; β ) + U ′′ ( z) z x ( x , y; β ) z y ( x , y; β )
−∞
we can shortly write:
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Uyβ ( y ) = η + ψ ;
so that:
(6)
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∂y
η
ψ
=
+
.
∂β U ( y )
Uyy ( y )
yy
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Equation (6) shows the total change in the control variable, y, induced by a change in the
parameter β. Such total change is given by the sum of two terms: the first, which we will call the
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payoff effect, identifies the change in y directly induced by a variation in β, through the effect that
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the parameter exerts on the payoff function; the second term, the distribution effect, identifies the
change in y brought about by a change in the distribution of the random variable, induced, in turn,
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by the change in β.
If we let
∂F( x; β )
= 0, in which case the parameter β does not affect the distribution of the
∂β
random variable x, we get the standard comparative statics result:
∂y
η
=
.
∂β U ( y )
yy
We now provide a set of sufficient conditions that help in signing those two effects we have
just described. The following section illustrates, with particular models, the way in which these
conditions have been applied in previous articles. We first consider the payoff effect.
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Proposition 1.
Assume zβx = 0, zyx > 0 and decreasing absolute risk aversion.
(a) If zβ ≥ 0 and zyβ ≥ 0 then η ≥ 0.
(b) If zβ ≤ 0 and zyβ ≤ 0 then η ≤ 0.
Proof. After substituting for the absolute risk aversion function, Ra(x), and since zβx = 0, from (4)
we obtain:
+∞
∫
+∞
∫
U′ ( z) zyβ ( x , y; β ) f ( x; β ) dx − zβ ( x , y; β ) R a ( x ) U′ ( z) zy ( x , y; β ) f ( x; β ) dx .
−∞
−∞
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η=
Using the assumption of decreasing absolute risk aversion and zβ ≥ 0, the first inequality follows:
+∞
η≥
∫
U′ ( z) zyβ ( x , y; β ) f ( x; β ) dx − zβ ( x , y; β ) R a ( ~
x)
∫
U′ ( z) zy ( x , y; β ) f ( x; β ) dx ≥ 0 .
−∞
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−∞
+∞
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~
x is the value of the random variable that makes z(x,y;β) equal zero. That is, z( ~
x ,y;β) = 0. The
second inequality is a consequence of the FOC (2) and of assuming zyβ ≥ 0. When zβ ≤ 0 and zyβ ≤
■
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0, both inequalities are reversed.
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When zβx is different from zero, and zβ varies with x, instead of using the absolute risk
aversion function, the partial risk aversion function could be used. The third application in the
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following section shows a model in which zβx > 0, and the partial risk aversion function is used to
sign the payoff effect.
In order to sign the distribution effect, we apply conditions that impose an upper bound on
the appropriate risk aversion index and uses the way in which the parameter β shifts the distribution
of the random variable; whether it causes a favorable or unfavorable FSD shift.
Proposition 2.
Assume R ≡ −
U′′ ( z)  zx zy 

 ≤ 1 and zyx > 0.
U′ ( z)  zyx 
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(c) If
∂F( x; β )
≥ 0 then ψ ≤ 0.
∂β
(d) If
∂F( x; β )
≤ 0 then ψ ≥ 0.
∂β
Proof.
From (5) we see that R ≤ 1 is equivalent to the term in brackets of ψ to be non - negative.
Therefore, the sign of ψ is opposite to that of
∂F( x; β )
∂β
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Depending on particular models, the condition on R has taken the form of either an upper
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bound on the partial or the relative risk aversion index.
From the results of the two propositions above, the payoff and distributions effects go in the
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same direction if the conditions leading to results (a) and (d), or (b) and (c) are met simultaneously.
Otherwise, the effects would run in opposite directions. For example, letting zyβ = 0, if an increase
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in the parameter β causes a positive effect on the payoff function (zβ > 0) and if it causes a favorable
shift in the distribution of the random variable, then both effects indicate an increase on the decision
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3.
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variable.
Some applications
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In this section, we offer three examples to illustrate the applicability of the general result
shown in equation (6). In all the examples, the standard results established in the traditional models
will be reinforced and generalized.
3.1.
Portfolio decision model
Consider the standard version of Arrow's [1971] portfolio model. Let β denote the original
wealth of the agent. The agent can choose to invest β either in a risky asset, with random rate of
return x, (x ∈ [-1, ∞)), or in a safe asset, with a positive rate of return r. The agent's problem is to
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choose optimally y, the amount of initial wealth to invest in the risky asset, so that the expected
utility of his terminal wealth z is maximized, where z is defined as follows:
z(x,y;β) = y x + [(β - y) (1 + r) + y].
According to the notation of the previous section, β the original wealth, is the parameter affecting
the distribution of the random variable x, as well as the value of the payoff function z. In particular,
we assume2 that
∂F( x; β )
< 0; that is, the distribution of the rate of return on the risky asset
∂β
improves in an FSD sense as the original wealth increases. The more wealth an agent has, the more
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he/she has access to assets with a better distribution of their rate of return: for instance, the agent
may hire a financial analyst to assist him in his investment planning and exclusive investments may
become available.3
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In computing the effect that an increase in wealth (the parameter β) has on the optimal
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choice of the control variable, y, we must recall that β affects the distribution of the random
variable x as well as the value of the function z. We need, therefore, to determine
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Uyβ ( y )
∂y
=−
,
∂β
Uyy ( y )
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and since Uyy ( y ) < 0 by the second order condition, we only need to compute the sign of Uyβ ( y ) .
Applying the result of equation (6) we get:
Uyβ ( y ) = η + ψ ,
where
+∞
η=
∫ { U′′ (z) (1 + r ) (x − r )} f (x;β) dx ,
−1
and
2
3
We also assume that Ex > r for an interior maximum to exists, as standard practice in the portfolio model.
The corresponding fixed cost of such decisions is, for simplicity, not included in the model.
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+∞
ψ=−
∫ {U′ (z) + U′′ (z) (x − r ) y}
−1
∂F( x; β )
dx .
∂β
The standard case, in which the parameter β does not affect the distribution of the random
variable x (i.e.,
∂F( x; β )
= 0), can be easily derived and the standard comparative statics result
∂β
η ≥ 0 will hold, assuming, as in the Arrow's model, that the Arrow - Pratt absolute measure of risk
aversion is non-increasing in z. In our more general case, the consideration of the effect that the
initial wealth β may exert on the distribution of the random return x, may reinforce the standard
3.2.
U′′ ( z)
( x − r ) y , is less than or equal to 1.4
U′ ( z)
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that the partial measure of risk aversion, R p = −
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comparative statics result. In fact, the distribution effect ψ is greater than or equal to zero, provided
Insurance demand model
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Consider Mossin's [1968] model of insurance demand, extended to incorporate the
expenditure in self - protection, as in the models by Ehrlich and Becker [1972] and Shavell [1979].
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The agent's terminal wealth, or payoff function is:
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z(x, y; β) = W + L - x - p y +
xy
- β,
L
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where W + L is the original wealth and L is the wealth component subject to loss; x is the random
loss, 0 ≤ x ≤ L; y represents the amount of coverage purchased by the agent, 0 ≤ y ≤ L; p is the per
unit price of coverage, 0 < p < 1; the premium paid is p y; and
xy
is the indemnity payment.
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Finally, β is the cost of self - protection (Ehrlich and Becker [1972]) or the cost for “taking care” of
oneself (Shavell [1979]). In this application we assume that β is a given parameter; that is, if the
agent buys some coverage, then he has to spend the amount β on self - protection as part of the
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This result, related with an upper bound on Rp, is shown in Cheng, Magill and Shafer [1987].
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contract with the insurance company.5 Alternatively, β can be interpreted as the minimum
expenditure in contractual insurance relations, that regulatory agencies may impose.
Let F(x; β) be the distribution function of the random loss. We assume that
∂F( x; β )
> 0;
∂β
that is, the distribution of the random loss improves in an FSD sense as the expenditure in self protection increases.6
The effect of an increase in the exogenous expenditure in self - protection on the optimal
choice of coverage is given by:
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U yβ ( y)
∂y
.
=−
U yy ( y)
∂β
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To compute the sign of Uyβ ( y ) , we apply the result of equation (6):
Uyβ ( y ) = η + ψ ,
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where
L

x

η =  U′′ ( z)  − p   f ( x; β ) dx ,
L

0
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and
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∫
L
 U′ ( z)
  ∂F( x; β )
 y
x
ψ=− 
+ U′′ ( z)  − p   − 1 
dx.
L
  L   ∂β
L
0
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∫
On one hand, the term η is greater or equal to 0, provided that the absolute measure of risk aversion
is non - increasing (see Ehrlich and Becker [1972] and Shavell [1979] for this standard comparative
statics result). That is, considering only the effect that the parameter β has on the payoff function,
the agent will increase the coverage if he is faced with a higher expenditure in self - protection. On
the other hand, Hadar and Seo[1991] have shown that a beneficial FSD change in the loss
5
A frequent example of this is given by the practice of insurance companies to require the installation of
smoke detectors or alarms before agreeing to sell the insurance coverage.
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For a justification of this assumption see Ehrlich and Becker [1972] and Shavell [1979].
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distribution will decrease the optimal coverage purchased, provided that the relative measure of
risk aversion is less than or equal to 1. In the present case, in which a change in the parameter β
induces an FSD shift in the distribution of x, this condition on the measure of risk aversion is,
therefore, also necessary and sufficient to guarantee ψ ≤ 0. Hence, the final effect on the optimal
value of the choice variable, as induced by a change in the parameter β, depends on the relative
magnitude of the payoff and distribution effects.
3.3.
Price - taker firm with uncertain demand
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Consider, finally, the model of a price - regulated or price - taker firm facing uncertain
demand, as in Hymans [1966], Persky [1991), Dionne and Mounsif [1996] and Ibarra - Salazar
[1996]. The payoff function is the firm's profit
x≤y
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 β x − c(y)
z(x,y;β) = 
β y − c ( y )
x > y,
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where β is the exogenous imposed price, x is the random demand, y ≥ 0 is output, and c(y) is the
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cost function.
Let F(x;β) represent the distribution function of the random demand and let us assume that
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∂F( x; β )
> 0; that is, a decrease in the given price causes a favorable FSD shift in the distribution
∂β
of the demand.
Again, in order to determine the comparative statics effect of an increase in the price we
need to compute:
Uyβ ( y ) = η + ψ ,
where
∞
y
∫
∫
∞
∫
η = − c′ ( y ) U′′ ( z) x f ( x; β ) dx + U′′ ( z) [β − c′ ( y )] y f ( x; β ) dx + U′ ( z) f ( x; β ) dx ,
0
y
and
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y
∞
y
∂f ( x; β )
∂f ( x; β )
ψ = − c′ ( y ) U′ ( z)
dx + U′ ( z) [β − c′ ( y )]
dx.
∂β
∂β
0
y
∫
∫
After integrating by parts, the term ψ becomes:
y
∫
ψ = β c′ ( y ) U′′ ( z)
0
∂F( x; β )
∂F( y; β )
dx − β U′ ( z)
.
∂β
∂β
The payoff effect, η, is strictly positive if the partial measure of risk aversion R p = −
U′′ ( z)
x is
U′ ( z )
non - increasing, as shown in Ibarra - Salazar [1996]. Moreover, the distribution effect, ψ, is strictly
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negative if and only if U″(z) < 0 (see Ibarra - Salazar [1996]). Therefore, the sign of Uyβ ( y )
depends on the relative magnitude of the payoff and the distribution effect.
Concluding remarks
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4.
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In this paper we have developed a general model to perform comparative statics under
uncertainty for those cases where a parameter affects not only the value of the payoff function, but
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also the distribution of the random variable. We have shown that the total effect on the control
variable, induced by a change in the parameter, can be decomposed into two components: the
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payoff and the distribution effect. The first identifies the change in the decision variable directly
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induced by the change in the parameter, via the payoff function. The second identifies the change in
the decision variable caused by the change occurred in the distribution of the random variable, in
turn induced by the parametric change. We provide a set of sufficient conditions to sign each of the
effects. In order to determine the direction of variation in the control variable, we impose
restrictions on a risk aversion function. For the payoff effect, we impose decreasing absolute risk
aversion, and for the distribution effect, an upper bound on the appropriate risk aversion index. The
specific restrictions were shown by developing certain models in a variety of fields: portfolio
theory, insurance and the theory of the firm. In particular, we found that the change in the amount
invested in the risky asset, induced by a change in the original wealth, may be underestimated with
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respect to the traditional Arrow's model. For the demand of insurance model, when the agent
invests in self - protection, we found that the change in the optimal coverage, induced by a change
in the expenditure in self - protection, may be ambiguous, depending on the combined magnitude of
the payoff and distribution effects. Finally, the last application confirms the fact that the slope of the
supply curve of a price - taker firm facing an uncertain demand is indeterminate. It depends, in fact,
on the relative magnitude of the payoff and distribution effect, which run in opposite directions.
Furthermore, whenever the parameter exerts no effect on the distribution of the random
variable, our general result reduces to the traditional standard comparative statics analyzed in
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previous studies. Hence, the generality of our model.
References
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Arrow, K., [1971], Essays in the Theory of Risk Bearing, Chicago: Markham.
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T
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