EECKHOUDT,GOLLIER,AND SCHLESINGER The Risk-averse (and Prudent)Newsboy Table 1 Summary Comparative Statics for the Risk-averse Newsboy Exogenous Change Utility Assumptions Change in Order* MIR risk averse and prudent indeterminate MIR for s or only risk averse and prudent decrease MIR for c≥only risk averse and prudent increase Simple spread across o risk aversion alone decrease increase risk aversion risk aversion only decrease increase initial wealth DARA increase increase salvage price v risk aversion only increase increase re-order costc risk aversion only increase increase initial order cost c risk aversion only indeterminate increase initial order cost c DARA decrease increase newspaper price p risk aversion only indeterminate increase newspaper price p DARA indeterminate increase newspaper price p CARA decrease increase newspaper price p decreasing partial risk aversion increase add background risk risk aversion only indeterminate add background risk prudent indeterminate add background risk CARA no change add background risk DARA and DAP decrease where Z*=Z(a,and ZT=Z(T,aF).Since H(a) (0.25,0.75.All other cost and price parameters are is concave,(14)implies that ac<ar;the optimal the same as previously.Calculations show that the op- newspaper order for a risk-averse newsboy will fall if timal order in this case is 37 newspapers.A change in risk increases via a simple spread across aF. the support of the demand distribution to 0(0,100), As an example,consider the case in which or =u, with the same probabilities of {0.25,0.75,is a mean- where u denotes E(),and define x =u+k[-u],k preserving simple spread across a*=37.From our pre- >0.Let G denote the distribution of 0x.For k>1,G is vious example,we know that the optimal order thus a simple spread of F across af.A recent paper by Ger- falls to a*=7,following this particular increase in risk. chak and Mossman (1992)shows that ac in this situation for the risk-neutral newsboy.14 However,our result above shows that a <af for the risk-averse 5.Summary and Conclusion newsboy.15 We have looked at some comparative statics for the As an example of the extent of the effects of an in- classic newsboy problem.Obviously,static models only crease in risk,consider our example at the end of $2, can capture a part of reality.How risk aversion works with constant absolute risk aversion.Let risk aversion in a more realistic dynamic setting is worthy of future be r=0.001,but suppose that demand is now given examination.Indeed,examples show that increasing the by 6E{30,90),with the corresponding probabilities number of periods has an ambiguous effect on the op- timal newspaper order for the risk-averse newsboy.1 14 Gerchak and Mossman(1992)show,in general,that a=(1-k)u +k for a risk-neutral newsboy and for which represents a par- e For example,under constant absolute risk aversion,zero time dis- ticular simple spread across rather than a simple spread count and an identical,independent two-point density of demand for across c. each period,the optimal order is easily shown to be independent of 1Similarly,other qualitatively deterministic results for the Gerchak the number of periods.We should also point out,however,that there and Mossman(1992)model do not follow in the case of a risk-averse are some problems in using expected utility within multi-period mod- newsboy. els.See,for example,Epstein and Zin(1989). MANAGEMENT SCIENCE/Vol.41,No.5,May 1995 793EECKHOUDT, GOLLIER, AND SCHLESINGER The Risk-averse (anid Prudenit) Newsboy Table 1 Summary Comparative Statics for the Risk-averse Newsboy Exogenous Change Utility Assumptions Change in Order c MIR risk averse and prudent indeterminate MIR for a <? a F only risk averse and prudent decrease MIR for a ? a* only risk averse and prudent increase Simple spread across a* risk aversion alone decrease increase risk aversion risk aversion only decrease increase initial wealth DARA increase increase salvage price v risk aversion only increase increase re-order cost C risk aversion only increase increase initial order cost c risk aversion only indeterminate increase initial order cost c DARA decrease increase newspaper price p risk aversion only indeterminate increase newspaper price p DARA indeterminate increase newspaper price p CARA decrease increase newspaper price p decreasing partial risk aversion increase add background risk risk aversion only indeterminate add background risk prudent indeterminate add background risk CARA no change add background risk DARA and DAP decrease whereZ* = Z(aF, 4F) andZ = Z(T, 4*). Since H(a) is concave, (14) implies that a* < a*; the optimal newspaper order for a risk-averse newsboy will fall if risk increases via a simple spread across a*F* As an example, consider the case in which aF = w, where ,u denotes E(0), and define K = /1 + k[j - tt], k > 0. Let G denote the distribution of 6K. For k > 1, G is a simple spread of F across a*. A recent paper by Ger￾chak and Mossman (1992) shows that acG = aF in this situation for the risk-neutral newsboy.'4 However, our result above shows that a* < a* for the risk-averse newsboy." As an example of the extent of the effects of an in￾crease in risk, consider our example at the end of ?2, with constant absolute risk aversion. Let risk aversion be r = 0.001, but suppose that demand is now given by 0 E { 30, 90 }, with the corresponding probabilities '4 Gerchak and Mossman (1992) show, in general, that C* = (1 - k), + kca for a risk-neutral newsboy and for ?k, which represents a par￾ticular simple spread across A rather than a simple spread across ca. 15 Similarly, other qualitatively deterministic results for the Gerchak and Mossman (1992) model do not follow in the case of a risk-averse newsboy. { 0.25, 0.75 }. All other cost and price parameters are the same as previously. Calculations show that the op￾timal order in this case is 37 newspapers. A change in the support of the demand distribution to 0 E { 0, 100 }, with the same probabilities of { 0.25, 0.75 }, is a mean￾preserving simple spread across a* = 37. From our pre￾vious example, we know that the optimal order thus falls to a* = 7, following this particular increase in risk. 5. Summary and Conclusion We have looked at some comparative statics for the classic newsboy problem. Obviously, static models only can capture a part of reality. How risk aversion works in a more realistic dynamic setting is worthy of future examination. Indeed, examples show that increasing the number of periods has an ambiguous effect on the op￾timal newspaper order for the risk-averse newsboy.'6 16 For example, under constant absolute risk aversion, zero time dis￾count and an identical, independent two-point density of demand for each period, the optimal order is easily shown to be independent of the number of periods. We should also point out, however, that there are sorhe problems in using expected utility within multi-period mod￾els. See, for example, Epstein and Zin (1989). MANAGEMENT SCIENCE/VOL 41, No. 5, May 1995 793