EECKHOUDT,GOLLIER,AND SCHLESINGER The Risk-averse (and Prudent)Newsboy Calculating-"/'for CARA (i.e.,negative expo-B.Increased Demand Risk nential)utility reveals that and u are equally risk We now turn our attention to the qualitative effects of averse at every total wealth level z.Thus i and u must changes in demand risk on the optimal initial order a*. be equivalent utility representations(i.e.,is an affine Kanbur(1982)analyzes a similar model,whose results transformation of u).Hence, can be applied here as well.His results imply that an H(a,)=E[4(Z(8,a)+)]=El(Z(0,a)】.(9) increase in risk in the sense of Rothschild and Stiglitz (1970,1971)generally has an ambiguous effect on a*. Since and u are equivalent utility representations,it However,we obtain unambiguous effects by making follows that a**equals a. some fairly simple restrictions on the increases in risk For the case where newsboy preferences exhibit and/or making some fairly canonical restrictions on DARA,a deterministic comparison between a**and a* preferences. follows if we make the additional assumption on pref- The change in risk is represented by a change in the erences that absolute prudence,defined by Kimball demand distribution F()to the distribution G(). (1990)as Without loss of generality,we assume that the support B(2)="(2) of G is contained in [0,T].Let a denote the optimal (10) u"(z) order with the original distribution F().We say G rep- resents a mean-preserving increases in risk(MIR)of F, is decreasing in wealth. if G and F satisfy the two Rothschild and Stiglitz con- The notion of prudence,u"(.)>0,was introduced ditions: by Kimball(1990),who shows how prudence is suffi- cient for a precautionary savings demand in standard G(0)d0≥ F(0)d0 for all t∈[0，T],(11) intertemporal consumption models.The notion of de- creasing prudence is discussed extensively in Kimball with a strict inequality on a set of positive probability (1993).In particular,DARA and decreasing absolute measure,and prudence(DAP)in tandem are shown to be equivalent G(0)d0= F(0)d0 (12) to the following canonical condition of preferences: "every risk that has a negative interaction with a small Condition (11)represents second-degree stochastic reduction in wealth also has a negative interaction with dominance of F over G while condition(12)preserves any undesirable,independent risk."Here"negative in- the mean.As shown by Rothschild and Stiglitz(1970), teraction''refers to a decrease in expected utility.The these conditions imply that any risk-averse newsboy class of utility functions satisfying DARA and DAP in- will prefer random demand given by F to that given by cludes,for example,the commonly used constant- G.11 However,(11)and (12)alone are not strong relative-risk-aversion class of utility functions (i.e. enough to guarantee that a lower newspaper order will power utility functions and logarithmic utility).10 be chosen under G. Assuming DARA and DAP,Eeckhoudt and Kimball In order to obtain unambiguous results,one typically (1992)prove that the derived utility function as given can make some restrictions on either(i)the utility func- in (8),is uniformly more risk averse than u.It then tion,(ii)the cost/price parameters,or (iii)the MIR follows easily from(9)and(5)that a**<a.In other itself.In addition,one can also select mixtures of these words,the optimal newspaper order will decrease in types of restrictions.12 response to an added background risk,whenever newsboy preferences display DARA and DAP. 1For mean-preserving changes in the distribution function satisfying (12),Rothschild and Stiglitz(1970)show that(11)holds if and only 10 Kimball(1993)shows how DAP is a natural extension of DARA if every risk averter prefers F to G. and how DARA and DAP together imply Pratts'and Zeckhauser's 12 Kanbur obtains unambiguous results only for the special case of (1987)proper risk aversion:given any two independent undesirable quadratic utility functions.His analysis implies the following results risks,their sum is at least as undesirable. in our model: MANAGEMENT SCIENCE/Vol.41,No.5,May 1995 791EECKHOUDT, GOLLIER, AND SCHLESINGER The Risk-averse (anid Pruidenit) Nezwsboy Calculating -u^"/u^' for CARA (i.e., negative expo￾nential) utility reveals that u^ and u are equally risk averse at every total wealth level z. Thus u^ and u must be equivalent utility representations (i.e., u^ is an affine transformation of u). Hence, H(a, i) = E[u(Z(6, a) + i)] = E4[a^(Z(6, a))]. (9) Since u^ and u are equivalent utility representations, it follows that a** equals a*. For the case where newsboy preferences exhibit DARA, a deterministic comparison between a** and a* follows if we make the additional assumption on pref￾erences that absolute prudence, defined by Kimball (1990) as -u ..' (z) B(z) = u",(z) (10) is decreasing in wealth. The notion of prudence, u"' (.) > 0, was introduced by Kimball (1990), who shows how prudence is suffi￾cient for a precautionary savings demand in standard intertemporal consumption models. The notion of de￾creasing prudence is discussed extensively in Kimball (1993). In particular, DARA and decreasing absolute prudence (DAP) in tandem are shown to be equivalent to the following canonical condition of preferences: "every risk that has a negative interaction with a small reduction in wealth also has a negative interaction with any undesirable, independent risk." Here "negative in￾teraction" refers to a decrease in expected utility. The class of utility functions satisfying DARA and DAP in￾cludes, for example, the commonly used constant￾relative-risk-aversion class of utility functions (i.e. power utility functions and logarithmic utility)." Assuming DARA and DAP, Eeckhoudt and Kimball (1992) prove that the derived utility function u^, as given in (8), is uniformly more risk averse than u. It then follows easily from (9) and (5) that a** < a-'. In other words, the optimal newspaper order will decrease in response to an added background risk, whenever newsboy preferences display DARA and DAP. '? Kimball (1993) shows how DAP is a natural extension of DARA and how DARA and DAP together imply Pratts' and Zeckhauser's (1987) proper risk aversion: given any two independent undesirable risks, their sum is at least as undesirable. B. Increased Demand Risk We now turn our attention to the qualitative effects of changes in demand risk on the optimal initial order a*. Kanbur (1982) analyzes a similar model, whose results can be applied here as well. His results imply that an increase in risk in the sense of Rothschild and Stiglitz (1970, 1971) generally has an ambiguous effect on a*. However, we obtain unambiguous effects by making some fairly simple restrictions on the increases in risk and/or making some fairly canonical restrictions on preferences. The change in risk is represented by a change in the demand distribution F(6) to the distribution G(6). Without loss of generality, we assume that the support of G is contained in [0, T]. Let C* denote the optimal order with the original distribution F(6). We say G rep￾resents a mean-preserving increases in risk (MIR) of F, if G and F satisfy the two Rothschild and Stiglitz con￾ditions: j'G(6)dO jF(6)dO forall t E [0, T], (11) with a strict inequality on a set of positive probability measure, and T oT G()d= J F(6)dO. (12) Condition (11) represents second-degree stochastic dominance of F over G while condition (12) preserves the mean. As shown by Rothschild and Stiglitz (1970), these conditions imply that any risk-averse newsboy will prefer random demand given by F to that given by G."1 However, (11) and (12) alone are not strong enough to guarantee that a lower newspaper order will be chosen under G. In order to obtain unambiguous results, one typically can make some restrictions on either (i) the utility func￾tion, (ii) the cost/price parameters, or (iii) the MIR itself. In addition, one can also select mixtures of these types of restrictions.12 " For mean-preserving changes in the distribution function satisfying (12), Rothschild and Stiglitz (1970) show that ( 11 ) holds if and only if every risk averter prefers F to G. 12 Kanbur obtains unambiguous results only for the special case of quadratic utility functions. His analysis implies the following results in our model: MANAGEMENT SCIENCE/VOL 41, No. 5, May 1995 791