EECKHOUDT,GOLLIER,AND SCHLESINGER The Risk-averse (and Prudent)Newsboy We first consider a localized increase in risk for the A second comparative-static result on changes in risk prudent newsboy,i.e.the newsboy with u>0.Positive does not require any restriction on the utility function prudence is a weaker condition than the commonly used other than risk aversion.Rather,we restrict the Roth- assumption of nonincreasing absolute risk aversion.In schild and Stiglitz mean-preserving increase in risk in particular,the following results hold: a very simple way that is fairly common in the literature (i)An MIR restricted to the interval [0,@F],i.e.,sat- on risk. isfying G(0)=F()in the interval [a,T],reduces the DEFINITION.A change from distribution F to distri- optimal order;a<f. bution G is a "simple spread across a"if (ii)An MIR restricted to the interval [af,T.],i.e., satisfying G(0)=F(0)in the interval [0,]increases [G(0)-F(0)][a-]≥0. (13) the optimal order;a>a. Note that (13)implies conditional first degree sto- Result(i)follows using Jensen's inequality together chastic dominance on both sides of a:F dominates G with the fact that u'(z)is a convex function of z.The conditional on a and G dominates F if a.If the MIR satisfying condition(i)increases the first integral simple spread also preserves the mean,it is a particular (marginal cost)in the first-order condition(3),without case of an MIR since it follows that condition (13)im- modifying the second one(marginal benefit).Therefore, plies condition(11)in that case.13 Condition(13),which OH/d<0 with the new distribution G.Given the is a single-crossing condition at a,states that some concavity of H,it follows that the order a is smaller weight is taken away from the neighborhood of a and than Property (ii)follows in a similar manner is put in the tails.Landsberger and Meilijson (1990) to (i). propose a different single-crossing condition in the con- In(i)above,the probability of demand's falling short text of linear payoff functions.Because a simple spread of the initial order af is unchanged under G.A risk- across a*is such that the probability of excess capacity neutral newsboy as well as a risk-averse newsboy with is unchanged (F(a*)=G(a*)),the decision of risk- quadratic utility (in both cases the newsboy has u=0) neutral newsboys would not be affected by a simple would not change his optimal order;=a.However, spread across a*,as shown by Equation (4).The the prudent newsboy responds to the increased risk in probability-preserving property of a simple spread al- wealth at low-demand levels by "saving"more of his lows us to focus on the impact of risk aversion alone. money,rather than spending. Integrating both integrals by parts in the first-order condition(3)yields the following: aH (-)u'(Z*)G(a)+(e-c)u'Z)+(c-p-o)J0u"Z-)G()d0 = -(e-)p-(Z.)G()d0<(v-E)w(ZF(at)+(-)w(ZT) +(p-)(Z)d-(-)P-F(0 <-(c-) '亿-)d+(e-c).'(Z)dF=0, (14) (i)G(cF)>F(a)and p-c<p-2+p-0 →ac>af 2p-0- p-c>p-02+(p-2 2p-6-0 G aF Note also that Kanbur uses all three types of restrictions to obtain these results. (ii)G(o)>F(a)and 13 See Hanoch and Levy (1969),Theorem 3. 792 MANAGEMENT SCIENCE/Vol.41,No.5,May 1995EECKHOUDT, GOLLIER, AND SCHLESINGER The Risk-averse (anid Prudenit) Newsboy We first consider a localized increase in risk for the prudent newsboy, i.e. the newsboy with u"' > 0. Positive prudence is a weaker condition than the commonly used assumption of nonincreasing absolute risk aversion. In particular, the following results hold: (i) An MIR restricted to the interval [0, afl, i.e., sat￾isfying G(6) = F(6) in the interval [4, T], reduces the optimal order; a* < F- (ii) An MIR restricted to the interval [a*, T], i.e., satisfying G(6) = F(6) in the interval [0, a*], increases the optimal order; a*> aF Result (i) follows using Jensen's inequality together with the fact that u'(z) is a convex function of z. The MIR satisfying condition (i) increases the first integral (marginal cost) in the first-order condition (3), without modifying the second one (marginal benefit). Therefore, aH / aa I, < 0 with the new distribution G. Given the concavity of H, it follows that the order a* is smaller than a *. Property (ii) follows in a similar manner to (i). In (i) above, the probability of demand's falling short of the initial order a 4 is unchanged under G. A risk￾neutral newsboy as well as a risk-averse newsboy with quadratic utility (in both cases the newsboy has u."' = 0) would not change his optimal order; a G = a F. However, the prudent newsboy responds to the increased risk in wealth at low-demand levels by "saving" more of his money, rather than spending. A second comparative-static result on changes in risk does not require any restriction on the utility function other than risk aversion. Rather, we restrict the Roth￾schild and Stiglitz mean-preserving increase in risk in a very simple way that is fairly common in the literature on risk. DEFINITION. A change from distribution F to distri￾bution G is a "simple spread across a" if [G(6) - F(6)][a - 0] 2 0. (13) Note that (13) implies conditional first degree sto￾chastic dominance on both sides of a: F dominates G conditional on 0 < a and G dominates F if 0 > a. If the simple spread also preserves the mean, it is a particular case of an MIR since it follows that condition (13) im￾plies condition ( 1 1 ) in that case. 13 Condition (13), which is a single-crossing condition at a, states that some weight is taken away from the neighborhood of a and is put in the tails. Landsberger and Meilijson (1990) propose a different single-crossing condition in the con￾text of linear payoff functions. Because a simple spread across a * is such that the probability of excess capacity is unchanged (F(a*) = G(a*)), the decision of risk￾neutral newsboys would not be affected by a simple spread across a*, as shown by Equation (4). The probability-preserving property of a simple spread al￾lows us to focus on the impact of risk aversion alone. Integrating both integrals by parts in the first-order condition (3) yields the following: aH = (V - C)UI(Z*)G(F) + (C- C)U(Z T) + (C - V)(p - V) F U"(Z)G(6)dO ua * ZO aF - (8-c)(p-C 8)1 U'"(Z+)G(6)dO < (v -C )u(Z*)F(fl + (8-c)u'(ZT) sF pF pT + (c - v)(p - v) f U'(Z)F(0)dO - (C^-c)(p - C^) U"(Z+)F(6)dO <-(c - v) f u'(Z)dF + (8-c) f u'(Z?)dF 0, (14)F paF oT < -(c-v ,) ut(Z-, dF + (c^ -c) J ut(Z+ ) dF = O, O (g~~~~~~~~~F (i) G(Ca) > F(Ca) and 2 p - c^- v aG F, (ii) G(Ca) > F(a4) and - C < (p C ) + (p- V) * * 2p - 8 - zv G Note also that Kanbur uses all three types of restrictions to obtain these results. '3 See Hanoch and Levy (1969), Theorem 3. 792 MANAGEMENT SCIENCE/VOl. 41, No. 5, May 1995