Discussion Papers In Economics
And Business

The Comparative Statics
of Equilibrium Derivative Prices
Masamitsu Ohnishi
Yusuke Osaki
Discussion Paper 04-19

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe Comparative Statics
of Equilibrium Derivative Prices
Masamitsu Ohnishi
Yusuke Osaki
Discussion Paper 04-19

November 2004

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe Comparative Statics
of Equilibrium Derivative Prices∗
Masamitsu Ohnishi †Graduate School of Economics, Osaka University;
Daiwa Securities Chair, Graduate School of Economics, Kyoto University
Yusuke Osaki
Graduate School of Economics, Osaka University

Abstract
We examine the conditions for preferences and risks that guarantee monotonicity of equilibrium derivative prices. In a Lucas economy with a derivative,
we derive the equilibrium derivative price under expectation with respect torisk–neutral probability, and analyze comparative statics on the equilibrium
derivative price based on the risk–neutral probability.
JEL Classification: C65, D51, D81, G12.
Keywords: Equilibrium Derivative Price, First–order Stochastic Dominance,
Noise Risk, Risk–Neutral Probability.

∗

The authors thank Marc Bremer and Katsushige Sawaki for their comments and suggestions.
Corresponding Author.
Address:
Graduate School of Economics, Osaka University, 1–7 Machikaneyama–Machi, Toyonaka, Osaka 560–0043, Japan.
E–mail:ohnishi@econ.osaka-u.ac.jp. Phone Number: +81–6–6850–5234. Fax Number: +81–6–
6850–5277†

1

1Introduction

One of the most important questions for optimal portfolio problems is what conditions on preferences and risks guarantee monotonicity of optimal portfolios. The
analysis has been extended to equilibrium asset prices in pure exchange economies bysome studies such as Gollier and Schlesinger (2002) and Ohnishi and Osaki (2004)
because they are consequences of investor portfolio optimization. For details onthese topics, Gollier (2001) provided an excellent survey. It is needles to say thatexamination of these effects on equilibrium derivative prices is necessary because of
the importance of derivatives from both academic and practical viewpoints. However, to our best knowledge, there has been no formal analysis examining them. The
goal of this paper is to examine them.
Our analysis much owes to the previous literature on comparative statics of optimal portfolios. In particular, it has a close relation to Gollier and Schlesinger (1996)
and Kijima and Ohnishi (1996). Gollier and Schlesinger (1996) showed that theaddition of noise risk to portfolio risk lead to the unambiguous comparative static
result on the optimal portfolio with some restrictions on preference. Kijima andOhnishi (1996) determined that two special classes of First–order Stochastic Dominance (FSD) guarantee the desirable comparative static result for decision problem
by a different way from previous studies such as Landsberger and Meilijson (1990)
and Eeckhoudt and Gollier (1995). Cleary, our analysis differs from these previous
literatures, since its concern is not with optimal portfolios but equilibrium derivative
prices. Further, our analysis follows from the previous study of comparative statics
of equilibrium asset prices. In particular, our analysis has a close relation to comparative statics based on risk–neutral probability such as Milgrom (1981) and Ohnishi
and Osaki (2004). Our results are also related to Gollier and Schlesinger’s recent
analysis (2002) in which they made comparative statics based on excess demand
functions.1) Our results are a generalization of results obtained in these previous literatures because of analyzing assets with non–linear payoffs and/or obtaining themunder weaker conditions.
This paper is organized as follows. In Sec. 2, we derive an equilibrium derivative
price in a pure–exchange economy with homogeneous investors, and rewrite it usingrisk–neutral probability. In Sec. 3, we show that shifts in the sense of two specialclasses of the FSD have monotone effects on the equilibrium derivative price. We
examine the effects of additional noise risks on the equilibrium derivative price inSec. 4. In the conclusion, we summarize the results, and give some comments on
1)

Gollier and Schlesinger (2002) discuss some stochastic dominances that guarantee the monotonicity of equilibrium asset prices based on the central dominance introduced by Gollier (1995).
However, these stochastic dominances can be justified, only when its parameter satisfies a certain
condition. We can obtain the results in Sec. 3 without that restriction.

2

future research.

2

Equilibrium Derivative Price

Let us consider a static version of a Lucas (1978) economy except for the introduction of a derivative, that is, a two–date pure exchange economy with homogeneous investors. Every investor has an identical expected utility representation
with a strictly increasing, strictly concave, and sufficiently smooth von Neumann–
Morgenstern utility function (vN–M function) u, which means that all required
higher order derivatives are assumed to be exist. Every investor is endowed with w
units of a risk–free asset, one unit of a risky asset and with one unit of a derivative written on it. Let us define that the risk–free asset is the numeraire in theeconomy, and the gross risk–free rate is normalized to one. The risky asset payoffat the final date is a random variable x̃ with a Cumulative Distribution Function(CDF) F . The CDF F of x̃ has a bounded support [a, b] and is assumed to bedifferentiable, that is, the Probability Density Function (PDF) f = F 0 exists. We
consider an economy in which the one–fund separation theorem holds, and therefore
the risky asset can be viewed as the market portfolio. The payoff of the derivativeis defined as a function of the risky asset payoff x and is denoted by p. The payoff
function of the derivative is assumed so that the final wealth in equilibrium given
by w + x + p(x), is an increasing function of x. An economic interpretation of thisassumption is given as follows. Since the supply of the risky asset which is endowedwith one unit for each investor, is considered as the norm for quantity, the supply ofthe derivative is represented by the slope of its payoff function. If the slope of thepayoff function is sufficiently small relative to the risky asset payoff, the assumption
is satisfied. When the payoff function is differentiable, the condition turns out tobe p0 (x) ≥ −1 for all x ∈ [a, b]. Because supplies of derivatives written on market
portfolios are sufficiently small compared to those for market portfolios in actual
financial asset markets, this assumption is permissible.
The investor buys the portfolio (α, β, γ) to maximize his or her expected utility
from final wealth, where (α, β, γ) is the portfolio for the risk–free asset, the risky
asset and the derivative respectively. Let us represent the price of the risky asset bym and the price of the derivative by q. The investor problem is given as follows:2)
P:

max E[u(α + β x̃ + γp(x̃))](α,β,γ)s.t. α + βm + γq = w + m + q.

(1)

Define the Lagrangian L(α, β, γ; λ) := E[u(α + β x̃ + γp(x̃))] − λ(α + βm + γq −2)

The constraint can be considered to be an equality since the objective function is strictlyincreasing.

3

w − m − q), where λ is the Lagrange multiplier. Because the objective function isa strictly concave function and the constraint is linear, the first–order–conditions
meet the necessary and sufficient conditions for the optimality. By the homogeneityof investors, the demand of the assets are equal to the endowment in equilibrium:
α = w, β = 1, γ = 1, that is, a no–trade equilibrium occurs. The solution of
investor’s problem in equilibrium is given as follows:∂L= E[u0 (z(x̃))] − λ = 0∂α∂L= E[x̃u0 (z(x̃))] − λm = 0∂β∂L= E[p(x̃)u0 (z(x̃))] − λq = 0
∂γ

(2)
(3)
(4)

where z(x) is the final wealth in equilibrium defined by z(x) := w + x + p(x), andis an increasing function of x. By Eqs. (2) and (4), the equilibrium derivative priceis given as follows:q=

E[p(x̃)u0 (z(x̃))].
E[u0 (z(x̃))](5)

Let us define the function0u (z(x))f (x)fˆ(x : u, f ) :=,E[u0 (z(x̃))]x ∈ [a, b].

(6)

Rb
Since fˆ(x : u, f ) ≥ 0 for all x ∈ [a, b] and a fˆ(t : u, f )dt = 1, we can regardfˆ(x : u, f ) as a PDF defined on the bounded support [a, b]. By taking the expectation
with respect to the PDF fˆ, the equilibrium derivative price can be rewritten as
q = Ê[p(x̃)],(7)

where Ê denotes the expectation operator with respect to the PDF fˆ. The probRxability F̂ (x : u, f ) := a fˆ(t : u, f )dt, x ∈ [a, b] induced by the PDF fˆ, is calledrisk–neutral probability, since asset prices become equal to the expected values of
their payoffs under risk–neutral probability.

3

The First–order Stochastic DominanceLet us consider two different economies, say economy 1 and 2. The payoff of the risky
asset in economy i (= 1, 2), is represented by the random variable x̃(i), and theserandom variables are ordered with respect to the First–order Stochastic Dominance(FSD). We examine the effect of FSD changes in risk on equilibrium derivative prices
using comparative static analysis.
4In this section, we consider the two special classes of FSD: the Monotone Likelihood Ratio Dominance (MLRD) and Monotone Probability Ratio Dominance (MPRD).3)
Since these stochastic dominances imply the FSD, they can be viewed as the specialclasses of FSD.

3.1

The Monotone Likelihood Ratio DominanceThe definition of MLRD is given as follows:
Definition 3.1. x̃(2) dominates x̃(1) in the sense of MLRD if f (y, 2)/f (y, 1) ≥f (x, 2)/f (x, 1) holds for all y ≥ x. We denote it as x̃(2) ≥MLRD x̃(1).
According to Kijima and Ohnishi (1996), we can obtain the following inequality:fˆ(y : u, f (2))
E[u0 (z(x̃(1))]f (x, 2)fˆ(x : u, f (2))
E[u0 (z(x̃(1))]f (y, 2)=≥=E[u0 (z(x̃(2))]f (y, 1)
E[u0 (z(x̃(2))]f (x, 1)fˆ(y : u, f (1))
fˆ(x : u, f (1))(8)

holds for all y ≥ x by the definition of MLRD. Eq. (8) means that risk–neutral
probability F̂ (2) dominates F̂ (1) in the sense of MLRD. Noting that the MLRD isstronger than the FSD, we can obtainq(1) = Ê[p(x̃(1))] ≤ (≥) Ê[p(x̃(2))] = q(2)(9)

for derivatives whose payoff functions are increasing (decreasing).
We summarize the above discussion as the following proposition:
Proposition 3.1. Let us consider two economies with risky asset payoffs by x̃(1)and x̃(2), and denote the equilibrium prices of derivatives written on them by q(1)and q(2). If x̃(2) ≥MLRD x̃(1), then q(2) ≥ (≤) q(1) holds for all derivatives withincreasing (decreasing) payoff functions.

3.2

The Monotone Probability Ratio DominanceThe definition of MPRD is given as follows:
Definition 3.2. x̃(2) dominates x̃(1) in the sense of MPRD if F (y, 2)/F (y, 1) ≥F (x, 2)/F (x, 1) holds for all y ≥ x. We denote it as x̃(2) ≥MPRD x̃(1).
Note that the MPRD is a stochastic dominance that is weaker than the MLRD
but stronger than the FSD, that is, the MLRD implies the MPRD, and the MPRD
implies the FSD, see Eeckhoudt and Gollier (1995) for the proof. By the definitionof MPRD, we have f (x, 2)/F (x, 2) ≥ f (x, 1)/F (x, 1) for all x ∈ [a, b]. We will show
3)

The MPRD is also known as the reversed hazard rate dominance.

5

that risk–neutral probability F̂ (2) dominates F̂ (1) in the sense of MPRD, that is,
we have to obtain the following inequality:4) for all x ∈ [a, b],fˆ(x, 2)u0 (z(x))f (x, 2)
u0 (z(x))f (x, 1)fˆ(x, 1)= Rx 0≥ Rx 0=.
u (z(t))f (t, 2)dt
u (z(t))f (t, 1)dt
F̂ (x, 2)F̂ (x, 1)
a
a

(10)

Whitt (1980) proved that the following statements are equivalent:• x̃(2) dominates x̃(1) in the sense of MPRD;• [x̃(2) | x̃(2) ≤ x] dominates [x̃(2) | x̃(2) ≤ x] in the sense of FSD for allx ∈ [a, b].
Since u0 (z(x)) is a decreasing function of x,
Z x
10E[u (z(x̃(2))) | x̃(2) ≤ x] =u0 (z(t))f (t, 2)dt
a F (x, 2)
Z x
1≤
u0 (z(t))f (t, 1)dt = E[u0 (z(x̃(1))) | x̃(1) ≤ x]
F
(x,
1)
a

(11)

holds for all x ∈ [a, b]. It follows from Eq. (11) and f (x, 2)/f (x, 1) ≥ F (x, 1)/F (x, 1)
that
Z x
Z
Z
F (x, 2) x 0f (x, 2) x 0
0u (z(t))f (t, 2)dt ≤u (z(t))f (t, 1)dt ≤u (z(t))f (t, 1)dt(12)F (x, 1) af (x, 1) aaholds for all x ∈ [a, b]. Eq. (12) means Eq. (10), that is, risk–neutral probabilityF̂ (2) dominates F̂ (1) in the sense of MPRD. We can obtain the following proposition
by an argument similar to that in the previous subsection:
Proposition 3.2. Let us consider two economies with the risky asset payoffs byx̃(1) and x̃(2), and denote the equilibrium prices of derivatives written on them by
q(1) and q(2). If x̃(2) ≥MPRD x̃(1), then q(2) ≥ (≤) q(1) holds for all derivatives
with increasing (decreasing) payoff functions.
Remark 3.1. It should noted that the concavity of u explicitly used in the proofof Prop. 3.2, whereas it does not appear in the proof of Prop. 3.1. This meansthat Prop. 3.2 implicitly holds under more restrictive conditions than Prop. 3.1,
and this requirement is consistent with the fact that the MPRD is weaker than the
MLRD.

4)

Kijima and Ohnishi (1996) obtained this inequality in a different manner and applied it to thedecision problem. However, we give a simpler proof for the completeness of our paper.

6

4

The Addition of Noise Risk

Let us consider the random variables ²̃ such that the following two conditions are
satisfied• the expectations are equal to zero: E[²̃] = 0;
• they are independent from the risky asset payoffs.
We call these random variables noise risk. We examine the effects of additional
noise risk on equilibrium derivative prices using comparative static analysis. In thissection, we consider two cases of additional noise risk: the addition of noise risk to
the endowment and to the risky asset payoff.

4.1

The Addition of Noise Risk to the Endowment

In this subsection, the investor endows the non–tradable component except for theendowment previously considered. The non–tradable component is the noise risk
which is a random variable ²̃. The objective function of investor’s problem consideredin Sec. 2 can be written with (non–tradable) noise risk to the endowment as:
E[u(α + β x̃ + γp(x̃) + ²̃)].

(13)

Let us define the derived utility function by v(x) := E[u(x + ²̃)] (Kihlstrom et. al.,
1981; Nachman, 1982), and rewrite Eq. (13) as:E[v(α + β x̃ + γp(x̃))].

(14)

This means that we can view the investor’s problem under the addition of noise risk
to the endowment as the problem of investor with preference v. The equilibrium
derivative price can be written by using risk–neutral probability:q(v) = Êv [p(x̃)],(15)

where Êv is the expectation operator with respect to the CDF F̂ (x : v, f ).
Kimball (1993) introduced the notion of Standard Risk–Aversion (SRA) concerning vN–M functions, which is the property that both their risk–aversion andprudence are decreasing functions, and proved that derived utility functions induced zero–mean risks are more risk–averse than the original function under SRA,
that is, A(v) = −v 00 /v 0 ≥ −u00 /u0 = A(u) holds, where prudence is defined byP(u) := −u000 /u00 . An equivalent condition for this inequality is given by the condition that there exists an increasing and concave function g such that v = g ◦ u
7

(Pratt, 1964). Differentiating the above equation yields that v 0 /u0 is an increasing
function. Therefore, by a discussion similar to Sec. 3.1,fˆ(y : u, f )E[v 0 (z(x̃))]u0 (y)E[v 0 (z(x̃))]u0 (x)fˆ(x : u, f )=≥=E[u0 (z(x̃))]v 0 (y)E[u0 (z(x̃))]v 0 (x)fˆ(y : v, f )fˆ(x : v, f )(16)holds for all y ≥ x. This means that risk–neutral probability F̂ (x : u, f ) dominatesF̂ (x : v, f ) in the sense of MLRD.
Following to Sec. 3.1, we can obtain:q(u) = Êu [p(x̃)] ≥ (≤) Êv [p(x̃)] = q(v)(17)

for the derivatives whose payoff functions are increasing (decreasing). We summarize
the result of this subsection as the following proposition:
Proposition 4.1. Assume that investor preferences display the SRA. Additions ofnoise risk to endowments decrease (increase) equilibrium derivative prices, whenevertheir payoff functions are increasing (decreasing).

4.2

The Addition of Noise Risk to the Risky Asset Payoff

We examine the effect of additional noise risk to the payoffs of risky assets on
equilibrium derivative prices in this subsection. The addition of noise risk to the
payoff of the risky asset is represented by x̃ + ²̃. Rothschild and Stiglitz (1970, 1971)
introduced the notion of Second–order Stochastic Dominance (SSD) that is definedvia concave functions. One equivalent condition to SSD is given by the addition ofnoise risk such that the conditional expectation is equal to zero. This means thatthe additional noise risk considered in this subsection is a special case of the SSD,
since the SSD does not require the condition of independence.
By the addition of noise risk to the risky asset payoff, the equilibrium price ofthe derivative written on it is given as follows in a discussion similar to Sec. 2 :E[p(x̃ + ²̃)u0 (w + (x̃ + ²̃) + p(x̃ + ²̃))]q(²) =
E[u0 (w + (x̃ + ²̃) + p(x̃ + ²̃))]Ex̃ (E²̃ [(p(x̃) + ²̃)u0 (w + (x̃ + ²̃) + p(x̃ + ²̃)) | x̃]).
=Ex̃ (E²̃ [u0 (w + (x̃ + ²̃) + p(x̃ + ²̃)) | x̃])

(18)

Assuming that the payoff function is differentiable, the following inequality is ob-

8tained for a sufficiently small noise risk in the case of increasing payoff functions:Ex̃ (E²̃ [(p(x̃) + p0 (x̃)²̃)u0 (w + x̃ + +p(x̃) + (1 + p0 (x̃))²̃) | x̃)Ex̃ (E²̃ [u0 (w + x̃ + p(x̃) + (1 + p0 (x̃))²̃) | x̃])Ex̃ (E²̃ [p(x̃)u0 (w + x̃ + p(x̃) + (1 + p0 (x̃))²̃ | x̃])=Ex̃ (E²̃ [u0 (w + x̃ + p(x̃) + (1 + p0 (x̃)²̃) | x̃)])Ex̃ (p0 (x̃)E²̃ [²̃u0 (w + x̃ + p(x̃) + (1 + p0 (x̃))²̃) | x̃])+Ex̃ (E²̃ [u0 (w + x̃ + p(x̃) + (1 + p0 (x̃))²̃ | x̃])Ex̃ (E²̃ [p(x̃)u0 (w + x̃ + p(x̃) + (1 + p0 (x̃))²̃) | x̃])≤,Ex̃ (E²̃ [u0 (w + x̃ + p(x̃) + (1 + p0 (x̃)²̃) | x̃])q(²) '(19)

where the inequality follows from the covariance inequality:5)
E²̃ [²̃u0 (w + x̃ + p(x̃) + (1 + p0 (x̃))²̃) | x̃] ≤ E[²̃]E²̃ [u0 (w + x̃ + p(x̃) + (1 + p0 (x̃))²̃) | x̃] = 0.(20)Using the derived utility function E[v(x̃)] := Ex̃ (E²̃ [u(x + (1 + p0 (x̃))²̃) | x̃]), we canrewrite Eq. (19) by
q(²) ≤

E[p(x̃)v 0 (w + x̃ + p(x̃))].
E[v 0 (w + x̃ + p(x̃))](21)

Assuming that preferences display SRA, we have the following inequality from amanner similar to the previous subsection:
E[p(x̃)u0 (w + x̃ + p(x̃))]E[p(x̃)v 0 (w + x̃ + p(x̃))]≤= q.
q(²) ≤
E[v 0 (w + x̃ + p(x̃))]E[u0 (w + x̃ + p(x̃))](22)

The following inequality holds for the case of decreasing payoff functions in a similar
discussion except for changing the sign:q(²) ≥

E[p(x̃)v 0 (w + x̃ + p(x̃))]E[p(x̃)u0 (w + x̃ + p(x̃))]≥= q,
E[v 0 (w + x̃ + p(x̃))]E[u0 (w + x̃ + p(x̃))](23)

where the derived utility function is defined by E[v(x̃)] = Ex̃ (E²̃ [u(x + (1 − p0 (x))²̃) |x̃]). Assuming the differentiability of payoff functions in this subsection, we have
1 − p0 (x) ≥ 0, for all x ∈ [a, b] by the assumption in Sec. 2. We summarize theresult as the following proposition:
Proposition 4.2. Assume that payoff functions of derivatives are differentiable.
We also assume that noise risk is sufficiently small and investor preferences display
the SRA. Additions of noise risk to risky asset payoffs decrease (increase) equilibrium
derivative prices, whenever their payoff functions are increasing (decreasing).
5)

The covariance inequality (Theorem 4.1 in McEntire, 1984) claims the following statement: if
both f and g are increasing functions, then E[f (x̃)g(x̃)] ≥ E[f (x̃)]E[g(x̃)] holds for every randomvariable x̃.

95Conclusion

Using comparative static analysis, we have shown that equilibrium derivative prices
have some monotone properties for shifts of risky asset payoffs with respect to two
sub–classes of the FSD (Sec. 3), and for additions of noise risks under some restrictions on investor preferences (Sec. 4). These results are a generalization of theprevious studies, such as Gollier and Schlesinger (2002).
We give two comments on future research. First, the analysis of Sec. 4.2. should
weaken the restrictions on noise risks and payoff functions. Although piecewise
linear functions are not differentiable, they are arbitrarily approximated by smoothfunctions. Therefore, the result of Sec. 4.2 holds for derivatives with piecewise linear
functions, which constitute an important class of derivatives because they include
most types of derivatives traded in actual financial asset markets, e.g. vanilla types ofcall and put options. Second, we have to analyze the economy where the raison d’êtreof derivatives is justified.6) Despite a standard setting, risks cannot be transferredamong investors by the derivatives, since the investors do not trade derivatives inequilibrium. This means that the roles which derivatives play, are not clear in our
economy.

References
Eeckhoudt, L. and C. Gollier, 1995. Demand for risky assets and themonotone probability ratio order. Journal of Risk and Uncertainty
11, 113–122.
Franke, G. and R. C. Stapleton and M. G. Subrahmanyam, 1998. Whobuys and who sells options: the role of options in an economy withbackground risk. Journal of Economic Theory 82, 82–109.
Gollier, C., 1995, The comparative statics of changes in risk revisited.
Journal of Economic Theory 66, 522–536.
Gollier, C., 2001, The economics of risk and time. MIT Press, Cambridge.
Gollier, C. and H. Schlesinger, 1996. Portfolio choice under noisy asset
return. Economics Letters 53, 47–51.
Gollier, C. and H. Schlesinger, 2002, Changes in risk and asset prices.
Journal of Monetary Economics 49, 747–760.
6)

In a recent paper, Franke. et. al (1998) justified the raison d’être of derivatives by inducing
a non–linear risk sharing rule among investors being faced with heterogeneous background risks,even if the investors have the linear risk tolerance with an identical slope. Our analysis is different
from theirs since they did not provide any qualitative analysis for equilibrium derivative prices.

10

Kihlstrom, R., D. Romer, and S. Williams, 1981, Risk aversion withrandom initial wealth. Econometrica 49, 911–920.
Kijima, M. and M. Ohnishi, 1996, Further comparative statics for choice
under risk. In: R. J. Wilson, D. N. P. Murthy and S. Osaki (Eds.),
Proceedings of the Second Australia–Japan Workshop on Stochastic
Models in Engineering, Technology and Management, 321–326.
Kimball, M. S., 1993, Standard risk aversion. Econometrica 61, 589–611.
Landsberger, M. and I. Meilijson, 1990, Demand for risky financial assets:
a portfolio analysis. Journal of Economic Theory 50, 204–213.
Lucas, R. E., 1978. Asset prices in an exchange economy. Econometrica
46, 1429–1446.
McEntire, P. L., 1984, Portfolio theory for independent assets. Management Science 20, 952–963.
Milgrom, P., 1981, Good news and bad news: Representation theorems
and applications. Bell Journal of Economics 12, 1089–1122.
Nachman, D. C., 1982, Preservation of “more risk averse” under expectations. Journal of Economic Theory 28, 361–368.
Ohnishi, M. and Y. Osaki, 2004, The comparative statics on asset prices
based on bull and bear market measure, forthcoming in European
Journal of Operational Research.
Pratt, J., 1964, Risk aversion in the small and in the large. Econometrica
32, 122–136.
Rothschild, M. and J. E. Stiglitz, 1970, Increasing risk: I. A definition.
Journal of Economic Theory 2, 225–243.
Rothschild, M. and J. E. Stiglitz, 1971, Increasing risk: II. Its economic
consequences. Journal of Economic Theory 3, 66–84.
Whitt, W., 1980, Uniform conditional stochastic order, Journal of Applied Probability 17, 112–123.

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