EECKHOUDT,GOLLIER,AND SCHLESINGER The Risk-averse (and Prudent)Newsboy maximize H(a)=E[u(Z(0,a))], (2) a20 k'(u(Z))u'(Z)aF where E denotes the expectation operator. The first-order condition for(2)is +(e-c).K(u(Z+)'(Z+)dF aH Ba =-(c-) u'(Z-)dF <K(uZa,a-(c-)w)dF +(e-c)w'Z+)dF=0 (3) +(e-e)w(Z.)dF The second-order condition for a maximum is straight- =0 (5) forward to derive and follows from our assumption that v<C. Because H is concave in a,Inequality (5)implies that Observe that a+separates the random demand vari- the optimal order a*will decrease as risk aversion rises. able into ranges where an increased order provides a If the newsboy's preferences exhibit the commonly cost or a benefit.The first term on the right-hand side assumed property of decreasing absolute risk aversion of Equation(3)is the marginal cost in expected utility (DARA),the above analysis further implies that of an increase in the initial newspaper order a,whereas wealthier newsboys (i.e.,higher zo)will order more the second term is its marginal benefit.Indeed,c-v is newspapers,ceteris paribus,since the higher initial the net dollar cost of initially ordering one more news- wealth implies lower risk aversion over the support of paper when the demand happens to be less than the the newsboy's distribution of final wealth.5 order.The quantity c-c is the net dollar benefit of the To see the effect that risk aversion can have on the additional initial order when the realized demand is optimal order,consider the following simple example higher than the order. of a risk-averse newsboy,whose preferences satisfy When the newsboy is risk neutral (u"=0),Equation constant absolute risk aversion.Such preferences can (3)yields the well-known solution be represented by the utility function u(z)=-exp(-rz), Fa)=￡-c where r represents the newsboy's degree of risk aver- C-0 (4) sion.6 Let zo=v=0,6=p,and let E{0,100),where the probability demand is zero is 0.25 and the proba- This condition states that the probability of excess ca- bility demand is 100 is 0.75.We set the newspaper price pacity is a constant for all newsboys.This probability at p=28 and newspaper cost at c=20.Straightforward is a simple function of cost parameters and salvage calculations yield the following optimal newspaper or- value.It also is independent of the output price. ders for different levels of risk aversion(assuming paper The case of a risk-averse newsboy implies that F(a*) orders are rounded to the nearest integer): <(c-c)/(c-v).This follows easily from Equation (3)by noting that,whenever u is strictly concave, s The assumption of DARA was postulated by Arrow.This property u'(Z-(01,a*)>4'(Z(a*,a*)>'(Z+(02,a*) is implied if,for a fixed risk,individuals are willing to pay more to 寸01<a*<92. avoid the risk when they are poorer.See Arrow (1965).The two results presented here,that awill fall due to either increased risk Thus,the risk-averse newsboy orders fewer newspa- aversion or decreased wealth under DARA,were shown by Baron (1973)for the case where d=p (equivalently,the case where no pers.More generally,an increase in risk aversion is second newspaper purchase is possible). equivalent to a concave transformation of the utility .The degree of absolute risk aversion is given by r(z)=-u"(z)/ function (see Pratt 1964).Replacing u(Z)with k[u(Z)] u'(z).For the newsboy preferences given above,this measure is con- in Equation (3),where k'(.)>0 and k"(.)<0,yields stant in z. 788 MANAGEMENT SCIENCE/Vol.41,No.5,May 1995EECKHOUDT, GOLLIER, AND SCHLESINGER The Risk-averse (anid Prudenit) Newsboy maximize H(ac) E[u(Z(6, a))], (2) Cy 2- 0 where E denotes the expectation operator. The first-order condition for (2) is H = -(c - v) u'(Z_)dF ? (C- ) u'(Z+)dF = 0. (3) The second-order condition for a maximum is straight￾forward to derive and follows from our assumption that v < C. Observe that a* separates the random demand vari￾able into ranges where an increased order provides a cost or a benefit. The first term on the right-hand side of Equation (3) is the marginal cost in expected utility of an increase in the initial newspaper order a, whereas the second term is its marginal benefit. Indeed, c - v is the net dollar cost of initially ordering one more news￾paper when the demand happens to be less than the order. The quantity C^ - c is the net dollar benefit of the additional initial order when the realized demand is higher than the order. When the newsboy is risk neutral (u" = 0), Equation (3) yields the well-known solution F(a*) = C C (4) This condition states that the probability of excess ca￾pacity is a constant for all newsboys. This probability is a simple function of cost parameters and salvage value. It also is independent of the output price. The case of a risk-averse newsboy implies that F(a*) < (c^ - c) /(c^ - v). This follows easily from Equation (3) by noting that, whenever u is strictly concave, u'(Z-(01, a*)) > u'(Z(a*, a*)) > u'(Z+(02, a*)) VO1 < a* < 02- Thus, the risk-averse newsboy orders fewer newspa￾pers. More generally, an increase in risk aversion is equivalent to a concave transformation of the utility function (see Pratt 1964). Replacing u(Z ) with k[u(Z )] in Equation (3), where k'(*) > 0 and k"(*) < 0, yields H = -(c v) f k'(u(Z_))u'(Z_)dF Oaa rT ? (6- c) f k'(u(Z+))u'(Z+)dF < kf(u(Z(ax*, ca)))[-(c - v) U'(Z_)dF ? (6- c) f u'(Z+)dF] =0. (5) Because H is concave in a, Inequality (5) implies that the optimal order a* will decrease as risk aversion rises. If the newsboy's preferences exhibit the commonly assumed property of decreasing absolute risk aversion (DARA), the above analysis further implies that wealthier newsboys (i.e., higher zo) will order more newspapers, ceteris paribus, since the higher initial wealth implies lower risk aversion over the support of the newsboy's distribution of final wealth.5 To see the effect that risk aversion can have on the optimal order, consider the following simple example of a risk-averse newsboy, whose preferences satisfy constant absolute risk aversion. Such preferences can be represented by the utility function u(z) = -exp(-rz), where r represents the newsboy's degree of risk aver￾sion.6 Let zo = v = 0, c^ = p, and let 0 E { 0, 100 }, where the probability demand is zero is 0.25 and the proba￾bility demand is 100 is 0. 75. We set the newspaper price at p = 28 and newspaper cost at c = 20. Straightforward calculations yield the following optimal newspaper or￾ders for different levels of risk aversion (assuming paper orders are rounded to the nearest integer): 5 The assumption of DARA was postulated by Arrow. This property is implied if, for a fixed risk, individuals are willing to pay more to avoid the risk when they are poorer. See Arrow (1965). The two results presented here, that a:' will fall due to either increased risk aversion or decreased wealth under DARA, were shown by Baron (1973) for the case where e = p (equivalently, the case where no second newspaper purchase is possible). 6 The degree of absolute risk aversion is given by r(z) = -u"(z)/ u'(z). For the newsboy preferences given above, this measure is con￾stant in z. 788 MANAGEMENT SCIENCE/VOl. 41, No. 5, May 1995