Journal of Insurance Issues, 2002, 25, 2, pp. 127–141.
Copyright © 2002 by the Western Risk and Insurance Association. All rights reserved.
and Demand for Loss Reduction
Larry Y. Tzeng and Jen-Hung Wang*
Abstract: This paper extends the research about the impact of an increase in back-ground risk from cases with one decision variable to those with two decision variables. We apply the results of Eeckhoudt and Kimball (1992) to examine the comparative statics of an increase in background risk on demand for loss reduction that depends on market insurance and self-insurance together. We find that individuals with decreasing absolute risk aversion and decreasing absolute prudence demand more loss reduction in the face of an increase in background risk, although they may not demand more market insurance or self-insurance. [Key words: background risk, market insur-ance, self-insurinsur-ance, loss reduction, absolute prudence]
rrow (1965) and Pratt (1964) introduced measures of risk aversion for von Neumann-Morgenstern utility functions. Specifically, for a von Neumann-Morgenstern utility function u, they proposed to be an absolute measure of risk aversion. Since then, the celebrated Arrow-Pratt measures of risk aversion have become indispensable tools for analysis of decisions under uncertainty in the expected utility framework. However, the Arrow-Pratt theory of risk aversion “tends to” compare risky situations with situations of certainty, while actual economic agents are more likely to be comparing two situations of uncertainty. Indeed, it is known that the
* Larry Y. Tzeng (E-mail: email@example.com) is associate professor of the Department of Finance at National Taiwan University. Jen-Hung Wang (E-mail: jenhung@ mba.ntu.edu.tw) is a doctoral student in the Department of Finance at National Taiwan University. The authors appreciate the helpful comments of the editor and anonymous referees. The financial support by the National Science Council of Taiwan is gratefully acknowleged.
Ar u'' u' ---=
Arrow-Pratt measures of risk aversion are too weak for making compari-sons between risky situations (Cass and Sriglitz, 1972; Hart, 1975). Ross (1981) addresses this problem and proposes a stronger measure of risk aversion to handle comparisons between risky situations. Ross’s strong risk aversion condition, though powerful, is very stringent. Kihlstrom, Romer, and Williams (1981) discuss the same question from a somewhat different perspective. They try to show that the Arrow-Pratt theory of risk aversion can be applied to make comparison between risky situations if some more conditions are imposed. Whence opens a stream of research on background risks.
In ordinary life, risks are, in fact, multiple. Since markets (e.g., capital, insurance) are simply incomplete, the risks can be divided to endogenous risks, from which an agent can deliberately choose his exposure level, and exogenous risks or background risks, which are not under the control of the agent. As a consequence, choices about endogenous risks usually are made as the agent faces one or more exogenous background risks simulta-neously. For example, consider the problem of investment in risky assets. The individual has random wealth, , where is wealth from his portfolio of risky assets and has an endogenous risk—the individual decides his investment level—and is income subject to human-capital risks or endowment subject to social risks, which are background risks since the individual has no way to protect himself from such risks.
Kihlstrom, Romer, and Williams (1981) and Nachman (1982) show that the Arrow-Pratt theory of risk aversion applies not only to risks imposed on a world of certainty, but also to risks added to preexisting uncertainty (background risk), as long as the following two conditions hold. The first is a statistics condition of risks: the added risk must be independent of the preexisting uncertainty. The second is a behavior condition of utility func-tions: at least one of the utility functions involved should exhibit decreasing absolute risk aversion.
Pratt and Zeckhauser (1987) introduce the concept of proper risk aversion or properness, which guarantees that an undesirable risk—that is, an such that —always remains undesirable in the presence of an independent undesirable background risk, , whether the wealth is fixed or independently random. However, properness has some disadvantages: The global necessary and sufficient conditions for properness are complicated; the local necessary condition, , is simple but not globally sufficient. (Anyway, it is worth remembering that all mixtures of risk-averse exponential, power, and logarithmic functions exhibit properness.) Besides the inconvenience in tractability, properness has another disadvantage: it does not imply that being forced to face an undesirable risk always reduces the optimal
invest-x˜+y˜ x˜ y˜ x˜ Eu(ω x˜+ ) u ω≤ ( ) y˜ ω r''( ) r' ωω ≥ ( )r ω( )
ment in a security with an independent return. Kimball’s standardness will remedy this shortage.
Kimball (1993) develops the concept of standard risk aversion or standardness, which guarantees that a loss-aggravating risk always aggra-vates an independent undesirable risk. The results also hold whether wealth is fixed or independently random. Kimball shows that, given monotonicity and concavity of the utility function, decreasing absolute risk aversion and decreasing absolute prudence are jointly necessary and suf-ficient for the property of standardness.1 Thus, the inconvenience of prop-erness argues for standardness: not only are the conditions for standardness easy to check, but standardness also entails unambiguous comparative statics results when facing introduction of an independent undesirable risk.
Gollier and Pratt (1996) define risk vulnerability as a property of utility functions that guarantees that adding an unfair background risk—an with —in wealth makes risk-averse individuals behave in a more risk-averse way with respect to any other independent risk, whether the background wealth is fixed or independently random. Risk vulnerability not only entails, like standardness, unambiguous comparative statics results when facing introduction of an independent unfair risk, it is in fact the necessary and sufficient condition to ensure such result.
Gollier and Pratt obtain a necessary and sufficient condition for global risk vulnerability. The best sufficient condition the authors found was the local properness condition, . It is very convenient and worth remembering that all risk-averse HARA utility functions2 are risk vulnerable (indeed proper).
So far, the exogenous background risks are considered to be immuta-ble. What is considered is the effect of introducing a background risk. A more general problem would be to consider the effect of a change in the background risk. It seems natural that an exogenous deterioration in background risk, say background wealth, will cause an individual to take more care elsewhere. When the increase in background risk is not an addition of an independent risk, what conditions entail that risk-averse individuals will take more care elsewhere? Eeckhoudt, Gollier, and Schlesinger (1996) address this problem. Specifically, they examine the deterioration of background risks in the form of general first-degree sto-chastic dominance (FSD) and second-degree stosto-chastic dominance (SSD). They derive the necessary and sufficient conditions on utility function for each of these two types of background risk changes to imply more risk-averse behavior on the part of the individual. For the case of FSD changes, the condition is decreasing absolute risk aversion in Ross’s stronger sense. For the case of SSD changes, the condition is locally risk-vulnerable pref-x˜ E x˜( ) 0≤
erence in Ross’s stronger sense. The conditions are fairly restrictive upon preferences. However, as Eeckhoudt et al. (1996) noted, “they [the condi-tions] place canonical limits upon appropriate utility representations if it is believed that individual acts in a more risk-averse manner whenever the distribution of background wealth deteriorates.” Guiso, Jappelli, and Ter-lizzese (1996) lend some empirical supports for such belief from Italian survey data.
Besides the papers cited above, which are more theoretical in nature, several articles analyze applications of the comparative statics of an increase in background risk. Doherty and Schlesinger (1983a, 1983b) dis-cuss whether full insurance is optimal for risk-averse individuals in the presence of background risk. Duffie and Zariphopoulou (1993) find that agents facing uncertainty in labor income reduce their investments in risky assets. Eeckhoudt and Kimball (1992) show that an individual with decreasing absolute risk aversion and decreasing absolute prudence increases his demand for insurance when faced with an increase in back-ground risk.
Although previous research has provided many ingenious findings on the impact of an increase in background risk, most of the literature discuss-ing this issue focuses on cases with just one decision variable. This paper intends to extend the research on the impact of an increase in background risk from cases with one decision variable to those with two decision variables. Specifically, we examine the comparative statics of an increase in background risk on demand for loss reduction, which depends on an individual’s decisions regarding market insurance and self-insurance. We find that an individual will demand greater loss reduction when facing an increase in background risk if and only if his preference exhibits decreasing absolute risk aversion and decreasing absolute prudence. This result is analogous to the results of Eeckhoudt and Kimball (1992), which discusses cases with one random variable. We further find that whether the individ-ual demands more market insurance or self-insurance depends on whether market insurance and self-insurance are normal factors3 for producing loss reduction.
The rest of the paper is organized as follows: Section 2 explains the model. Section 3 discusses the comparative statics results, followed by the conclusion.
We now consider the following situation. The initial wealth of the insured has a fixed component W and a random component y, which
follows a distribution function , , where an increase in denotes adding a negative noise—that is, Prob —in risk of
.4 An interpretation may be that the insured has random wealth, which may be income subject to human-capital risks or endowment subject to social risks—in a word, uninsurable risks. The insured also faces a random loss , which follows a distribution function . The individual can either purchase market insurance or spend his money on self-insurance to reduce his loss, but is uninsurable. Market insur-ance and self-insurinsur-ance are defined according to the usage of Ehrlich and Becker (1972), where market insurance is a proportional insurance cover-age and self-insurance is an expenditure to reduce the loss size. We adopt this usage and structure our model in a broader way. First, Ehrlich and Becker assume that the loss follows a Bernoulli distribution, which has only a fixed amount of loss. In our model, we allow a general form of loss distribution and hence allow partial losses. However, we assume that market insurance and self-insurance reduce the loss size proportionally. Note that, from the definitions we adopt, market insurance and self-insurance reduce the loss size but do not change the shape of the net loss.5 Second, Ehrlich and Becker assume that the price of market insurance is independent of the amount of self-insurance—that is, they assume that an individual’s effort for self-insurance has no impact on the market insurance premium. However, insurance companies sometimes provide premium discounts for the risk-management effort of their insured. So, in our model, we do not limit the functional form of the total expenditure of market insurance and self-insurance, allowing the market insurance premium and the expenditure for self-insurance to interact. Our model would reduce to that of Ehrlich and Becker’s if the loss distribution is specified to follow the Bernoulli distribution and the premium on market insurance is indepen-dent of the expenditure of self-insurance.
Formally, let and denote the amount of market insurance pur-chased and the efforts for self-insurance, respectively. Let be the total expenditure on market insurance and self-insurance, and let
be the total effects of insurance coverage and loss reduction due to market insurance and self-insurance. It is natural to assume that both and are increasing functions of and . Thus, the final wealth of the insured, , is , since we assume that the mar-ket insurance and self-insurance reduce the loss size proportionally. Assume that the insured chooses the optimal insurance amount and self-insurance amount to maximize his expected utility , where
and . The model can be written as:
g y( ,ρ) y∈[a b, ] ρ ρ 0≤ [ ] = 1 g y( ,ρ) x∈[0 L, ] f x( ) g y( ,ρ) Q C E Q C( , ) V Q C( , )x˜ E Q C( , ) V Q C( , ) Q C Z W+y˜–x˜+V Q C( , )x˜ E Q C– ( , ) Q∗ C∗ u Z( ) u'( ) 0⋅ > u''( ) 0⋅ <
Assume that the second-order conditions of Equation (1) hold to guarantee interior solutions.6 The optimal insurance and self-insurance amount can be determined by the following first-order conditions of Equation (1): , (2) where (3) And (4) where (5)
In the above equations, and throughout the paper, all subscripts denote partial derivatives and and denote the optimal amount of and
, respectively. M Q C,axEU Q C u Z f g( , ; , , , ) u W( +y–x+V Q C( , )x E Q C– ( , ))f x( )g y ρ( , ) xd dy. 0 L
∫= EUQ UQ(y Q∗ C∗, , )g y ρ( , ) yd =0 a b
∫= UQ(y Q∗ C∗, , ) VQx–EQ [ ]u' W y x( + – +V Q∗ C∗( , )x E Q∗ C∗– ( , ))f x( ) x.d 0 L
∫= EUC UC(y Q∗ C∗, , )g y ρ( , ) yd =0 a b
∫= UC(y Q∗ C∗, , ) VCx–EC [ ]u' W y x( + – +V Q∗ C∗( , )x E Q∗ C∗– ( , ))f x( ) x.d 0 L
∫= Q∗ C∗ Q C
Comparative statics results
We generate comparative statics by taking a derivative with respect to in Equations (2) and (4):
Thus, from Equation (6),
From the second-order conditions of Equation (1), which are assumed to hold, we have
Thus, the signs of and are determined by the signs of nominators of Equations (7) and (8) respectively. That is,
(10) (11) ρ EUQQ EUQC EUCQ EUCC Q∗ρ C∗ρ EUQρ – EUCρ – = Q∗ρ EUQρ – EUQC EUCρ – EUCC EUQQ EUQC EUCQ EUCC ---. = C∗ρ EUQQ –EUQρ EUCQ –EUCρ EUQQ EUQC EUCQ EUCC ---. = EUQQ EUQC EUCQ EUCC 0. > Q∗ρ C∗ρ sign Q∗( ρ ) = sign EU( CρEUQC–EUQρEUCC). sign C∗( ρ ) = sign EU( QρEUCQ–EUCρEUQQ).
From Equations (2) and (4), at the optimal level of Q and C, we have
, and (12)
By comparing Equations (12) and (13), we can find that at the optimal level of Q and C,
From Equations (12), (13), and (14),
at the optimal level of Q and C.
Let . Thus, Equations (10) and (11) can be rewritten as
If the preference of the individual exhibits decreasing absolute risk aversion and decreasing absolute prudence, then an increase in background risk implies .
Proof of Theorem 1From Equation (15), EUQ VQ --- = 0 EUC VC --- = 0. EQ VQ --- EC VC ---= EUQρ VQ --- EUCρ VC ---= EUQρ VQ --- EUCρ VC --- τ = =
sign Q∗( ρ ) = sign( ) sign Vτ × ( CEUQC–VQEUCC).
sign C∗( ρ ) = sign( ) sign Vτ × ( QEUCQ–VCEUQQ).
where is the risk premium that the individual is willing to pay to avoid the revenue uncertainty.
Thus, from equations (20) and (21),
An increase in represents an increase in background risk. Thus, an increase in implies an increase in if the preference of the individual exhibits decreasing absolute risk aversion and decreasing absolute pru-dence by Eeckhoudt and Kimball (1992).7 Further, since the preference of the individual exhibits decreasing absolute risk aversion, an increase in implies an increase in risk aversion of . Thus, an increase in implies an increase in risk aversion of , if the preference of the individual exhibits decreasing absolute risk aversion and decreasing absolute prudence.
Therefore, Equations (12) and (18) can be rewritten as , and . τ x EQ VQ ---– u' Z Q C x y( ( , , , ))f x( )gρ(y,ρ) xd dy. 0 L
∫= v Z Q C x( ( , , ) ρ, ) u Z Q C x y( ( , , , ))g y ρ( , ) yd =u Z Q C x( ( , , ) π ρ– ( )) a b
∫= π ρ( ) v' Z Q C x( ( , , ) ρ, ) = u' Z Q C x( ( , , ) π ρ– ( )) v'' Z Q C x( ( , , ) ρ, ) = u'' Z Q C x( ( , , ) π ρ– ( )) v'' Z Q C x( ( , , ) ρ, ) v' Z Q C x( ( , , ) ρ, ) --- u'' Z Q C x( ( , , ) π ρ– ( )) u' Z Q C x( ( , , ) π ρ– ( )) ---. – = ρ ρ π ρ( ) π ρ( ) v Z Q C x( ( , , ) ρ, ) ρ v Z Q C x( ( , , ) ρ, ) x EQ VQ ---– v' Z Q C x( ( , , ) ρ, )f x( )dx 0 L
∫= 0 τ x EQ VQ ---– v'ρ(Z Q C x( , , ) ρ, )f x( )dx 0 L
Thus, based on Theorem 4 of Diamond and Stiglitz (1974), if there exists an s u c h t h a t f o r a n d f o r , t h e n
. S i n c e a n d
, increases with respect to x. Thus, for and for when . Therefore, we can conclude Theorem 1.
Q. E. D.
In fact, Equations (12) and (13) are similar to the first-order condition of Eeckhoudt and Kimball (1992, their Equation 8), where they analyze background risk and demand for insurance. Since we show that , Equation (18) plays a role like the comparative statics of demand for insurance with respect to background risk. That is, we find decreasing absolute risk aversion and decreasing absolute prudence are essential conditions for unambiguous comparative statics as found by Eeckhoudt and Kimball in the case of demand for insurance.
If the preference of the individual exhibits decreasing absolute risk aversion and decreasing absolute prudence, then an increase in background risk implies
Proof of Theorem 2
From Equation (16) and Theorem 1, we know that .
From Equations (2) and (4),
x∗ ∂Q---∂Z≤0 x≤x∗ ∂Q---∂Z≥0 x≥x∗ x EQ VQ ---– v'ρ(Z Q C x( , , ) ρ, )f x( )dx 0 L
∫>0 ∂Z ∂Q --- = VQx–EQ ∂2 Z ∂Q∂x --- = VQ>0 ∂Q---∂Z ∂Z ∂Q ---≤0 x≤x∗ ∂Z ∂Q ---≥0 x≥x∗ ∂Z x∗( ) ∂Q --- = 0 EQ VQ --- EC VC ---= sign Q∗( ρ ) = sign V[( QCEC–EQCVC)–(VCCEQ–ECCVQ)]. sign C∗( ρ ) = sign V[( QCEQ–EQCVQ)–(VQQEC–EQQVC)]. sign Q∗( ρ ) = sign V( CEUQC–VQEUCC) VCEUQC=
From Equation (14), the second terms in Equations (22) and (23) are the same. Thus,
From Equations (2) and (4),
From Equation (25), Equation (24) can be rewritten as: VC (VQCx–EQC)u' Z( )f x( )g y ρ( , ) xd dy + 0 L
∫VQVC2 x EQ VQ ---– x EC VC ---– u'' Z( )f x( )g y ρ( , ) xd dy 0 L
∫VQEUCC= VQ (VCCx–ECC)u' Z( )f x( )g y ρ( , ) xd dy + 0 L
∫VQVC2 x EC VC ---– x EC VC ---– u'' Z( )f x( )g y ρ( , ) xd dy 0 L
∫VCEUQC VQEUCC VCVQC–VQVCC ( )x–(VCEUQC–VQECC) [ ]u' Z( )f x( )g y ρ( , ) xd dy. 0 L
∫= – xu' Z( )f x( )g y ρ( , ) xd dy 0 L
∫u' Z( )f x( )g y ρ( , ) xd dy 0 L
∫--- EQ VQ --- EC VC ---= =
Obviously, is always positive. Thus, from Equation (26),
By the same token,
Q.E.D. It is very important to recognize that the comparative statics in Theo-rem 2 are the conditions to determine whether market insurance and self-insurance are normal factors to produce loss reduction. If we consider V(Q,C) and E(Q,C) to be the production function and expenditure function of loss reduction, respectively, then the minimum expenditure to produce a certain level of loss reduction can be analyzed by the following model.
(27) It is easy to show from Equation (27) that
, and . VCEUQC–VQEUCC u' Z( )f x( )g y ρ( , ) xd dy 0 L
∫VC VQCEC VC ---–EQC VQ VCCEQ VQ ---–ECC – × . = u' Z( )f x( )g y ρ( , ) xd dy 0 L
∫sign Q∗( ρ ) = sign V[( QCEC–EQCVC)–(VCCEQ–ECCVQ)]. sign C∗( ρ ) = sign V[( QCEQ–EQCVQ)–(VQQEC–EQQVC)]. Min Q C, E Q C( , ) s.t. V Q C( , )=θ. ∂Q ∂θ --- = (VQCEC–EQCVC)–(VCCEQ–ECCVQ) ∂C ∂θ --- = (VQCEQ–EQCVQ)–(VQQEC–EQQVC)
The above two equations are identical to the conditions in Theorem 2. Thus, an increase in background risk increases the demand for market insurance and self-insurance depends not only on risk preference of the individual,8 but also on whether market insurance and self-insurance are normal factors to produce loss reduction.
Moreover, if the second-order conditions of Equation (27) hold, we can show that:
If the preference of the individual exhibits decreasing absolute risk aversion and decreasing absolute prudence, then an increase in
back-ground risk implies .
Proof of Theorem 3
From Theorem 2, we know that
(28) Equation (28) is nothing but the second-order condition of Equation (27). Therefore,
We can consider that the individual makes his or her decision in two steps when facing an increase in background risk. In the first step, the individual decides whether to increase the demand for loss reduction. Then in the second step he or she decides how to increase loss reduction, if it is so wanted. The risk preference of the individual plays a key role, in the first step, like that in Eeckhoudt and Kimball (1992). In the second step, the normality of factors is essential to determine the final decision.
Theorem 3 shows that an individual with decreasing absolute risk aversion and decreasing absolute prudence always increases his demand for loss reduction with respect to an increase in background risk. Therefore, by means of Theorems 2 and 3 together, we can conclude that the individual increases market insurance (self-insurance) if market insurance (self-insur-ance) is a normal factor for producing loss reduction.
sign V∗( ρ(Q C, )) 0>
sign V∗( ρ(Q C, )) = VQsign Q∗( ρ) V+ Csign C∗( ρ).
This paper applies the finding of Eeckhoudt and Kimball (1992) to analyze comparative statics of an increase in background risk when indi-viduals need to make two decisions together. We examine the comparative statics of an increase in background risk on the demand for loss reduction, which depends on market insurance and self-insurance together. We find that an individual with decreasing absolute risk aversion and decreasing absolute prudence demands more loss reduction when faced with an increase in background risk. Moreover, the individual’s demand for more market insurance and self-insurance depends on whether market insur-ance and self-insurinsur-ance are normal factors for producing loss reduction. Our model can be extended to analyze other problems, such as the inter-action between production and insurance as well as the interinter-action between saving and insurance.
1 Decreasing absolute risk aversion says that the measure of absolute risk aversion is
decreas-ing in wealth, or formally, . It is implied by such behavior as investing more in risky securities as one becomes wealthier and is almost universally considered a reasonable assumption. Kimball (1990) gives the name “prudence” to the sensibility of the optimal choice of a decision variable to risk. Analogously to Arrow-Pratt’s measure of risk aversion, he gives an absolute measure of prudence .
2 A utility function is HARA if the reciprocal of its absolute measure of risk aversion is linear
in wealth. All CARA, CRRA utility functions are HARA.
3 A normal factor is such a factor of production that the demand for this factor increases when
there is an (infinitesimal) increase in output.
4 Note that the support of has been set large enough to cover all relevant outcomes for all
5 Ehrlich and Becker also discuss another type of risk management, self-protection, which
influences the loss distribution rather than the loss size. A further extension of our model to include self-protection is possible but may cloud the current focus of this paper. Thus, we do not consider self-protection in our model for the simplicity of demonstrating our points.
6 The conditions may not hold. However, when the conditions fail, the maximization solution
may not exist or may be a corner solution. Such insurance setting is unordinary and out of our consideration. So we set the assumption to focus attention on relevant cases.
7 The results of Eeckhoudt and Kimball (1992) are more general than what we need here. They
show that the result holds not only for adding a negative noise in background risk, but also for adding any independent undesirable background risk. More than this, most of their paper dis-cusses a more general situation indicating that there is some “positive relationship” between background risk and the other risk—the distribution of background risk conditional upon a given level of insurable loss deteriorates in the sense of third-order stochastic dominance as the amount of insurable loss increases.
∂r ω( ) ∂ω ---<0 p( )ω u'''( )ω u''( )ω ---– = y˜ ρ
8 In Eeckhoudt and Kimball (1992), whether an increase in background risk will increase the
demand for insurance depends only on the risk preference of the individual since there is only one decision variable in their model.
Arrow, K.J. (1965) “Aspects of a theory of risk bearing,” Yrjo Jahnsson Lectures, Helsinki. Reprinted in Essays in the Theory of Risk Bearing (1971). Chicago: Markham Publishing Co.
Diamond, P.A. and J.E. Stiglitz (1974) “Increase in risk and in risk aversion,” Journal
of Economic Theory, 8, 333–361.
Doherty, N. and H. Schlesinger (1983a) “Optimal insurance in incomplete market,”
Journal of Political Economy, 91, 1045–1054.
Doherty, N. and H. Schlesinger (1983b) “The optimal deductible for an insurance policy when initial wealth is random,” Journal of Business, 56, 555–565.
Duffie, D. and T. Zariphopoulou (1993) “Optimal investment with undiversifiable income risk,” Mathematical Finance, 2, 135–148.
Eeckhoudt, L., C. Gollier, and H. Schlesinger (1996) “Changes in background risk and risk-taking behavior,” Econometrica, 64, 683–689.
Eeckhoudt, L. and M.S. Kimball (1992) “Background risk, prudence and the demand for insurance,” in Contributions to Insurance Economics, edited by G. Dionne. Boston: Kluwer, 1992.
Ehrlich I. and G.S. Becker (1972) “Market insurance, insurance, and self-protection,” Journal of Political Economy, 80, 623–648.
Gollier, C. and J.W. Pratt (1996) “Risk vulnerability and the tempering effect of background risk. Econometrica, 64, 1109–1123.
Guiso, L., T. Jappelli, and D. Terlizzese (1996) “Income risk, borrowing constraints, and portfolio choice,” American Economic Review, 86, 158–172.
Kihlstrom, R.E., D. Romer, and S. Williams (1981) “Risk aversion with random initial wealth,” Econometrica, 49, 911–920.
Kimball, M. (1990) “Precautionary saving in the small and in the large,”
Economet-rica, 58, 53–73.
Kimball, M. (1993) “Standard risk aversion,” Econometrica, 61, 589–611.
Nachman, D.C. (1982) “Preservation of ‘more risk averse’ under expectations,”
Journal of Economic Theory, 28, 361–368.
Pratt, J.W. (1964) “Risk aversion in the small and in the large,” Econometrica 32, 122– 136.
Pratt, J.W. and R.J. Zeckhauser (1987) “Proper risk aversion,” Econometrica, 55, 143– 154.
Ross, S.A. (1981) “Some stronger measure of risk aversion in the small and the large with applications,” Econometrica, 49, 621–638.