EECKHOUDT,GOLLIER,AND SCHLESINGER The Risk-averse (and Prudent)Newsboy Our analysis should be helpful in examining a dynamic Epstein,L.and S.Zin,"Substitution,Risk Aversion and the Temporal model.Given the prominence of the static newsboy Behavior of Consumption and Asset Returns:A Theoretical problem in the literature,our model should prove useful Framework,"Econometrica,57(1989),937-969. Gerchak,Y.and D.Mossman,"On the Effect of Demand Randomness in its own right as well. on Inventories and Costs,"Oper.Res.,40(1992),804-807. In addition to any dollar benefits and costs,the risk- Hanoch,G.and H.Levy,The Efficiency Analysis of Choices Involving averse newsboy also reacts to wealth effects for changes Risk,"Review of Economic Studies,36(1969),335-346 in prices,costs and risk.It thus becomes more difficult Horowitz,1.,Decision Making and the Theory of the Firm,Holt,Rinehart to determine the qualitative effects of changes in these and Winston,New York,1970. parameters for the risk-averse newsboy than for the Hull,J.C.,Options,Futures,aud Other Derivative Securities,Prentice Hall,Englewood Cliffs,N],1993. risk-neutral newsboy.A summary of comparative-static Hymans,S.H.,"The Price Taker:Uncertainty,Utility,and the Supply results is given in Table 1. Function,"International Economic Review,7(1966),346-356. Although many comparative-static results are am- Kanbur,R.,"Increases in Risk with Kinked Payoff Functions,".Eco- biguous under risk aversion,some fairly simple and 10 mic Theory,27(1982),219-228. canonical restrictions on preferences and/or risk in- Kihlstrom,R.E.,D.Romer,and S.Williams,"Risk Aversion with Random Initial Wealth,"Econowetrica,49 (1981),911-920. creases are seen to lead to qualitatively deterministic Kimball,M.S.,"Precautionary Saving in the Small and in the Large," comparative-static results.Given the wide array of Ecou0 netrica,58(1990),53-73. newsboy-type problems,our results should be useful "Standard Risk Aversion,"Econometrica,61 (1993),589-611. in analyzing other settings for which the payoff function Landsberger,M.and I.Meilijson,"A Tale of Two Tails:An Alternative is piecewise linear.17 Characterization of Comparative Risk,"J.Risk and Uncertainty, 3(1990),65-82. This paper was started while Gollier and Schlesinger enjoyed the Lau,H.-S.,'The Newsboy Problem under Alternative Optimization Objectives,"J.of the Operational Res.Society,31 (1980),525- hospitality of CORE,Louvain-la-Neuve,Belgium,and was partly re- 535. vised while Schlesinger was a guest at the IDEI,University of Toulouse. Li,J.,H.Lau and A.Lau,"Some Analytical Results for a Two-product Financial support of the Insurance Chair of the Federation Francaise des Societe d'Assurance at IDEI is gratefully acknowledged.The au- Newsboy Problem,"Decision Sciences,21(1990),5. thors thank Heraklis Polemarchakis,Ed Schlee,five anonymous ref- Menezes,C.and D.Hanson,"On the Theory of Risk Aversion,"In- ternational Economic Review,11 (1970),481-487. erees,and the editor,Robert Clemen,for helpful comments. Meyer,J.and M.B.Ormiston,"Strong Increases in Risk and Their Comparative Statics,"International Economic Review,26(1985), References 425-437. Arrow,K.J.,Aspects of the Theory of Risk Bearing,Yrjo Jahnssonin, Nachman,D.C.,"Preservation of 'More Risk Averse'Under Expec- Helsinki,Finland,1965. tations,"J.Economic Theory,28(1982),361-368. Baron,D.P.,"Point Estimation and Risk Preferences,".the American Pratt,J.,"Risk Aversion in the Small and in the Large,"Econometrica, Statistical Association,68 (1973),944-950. 32(1964),122-136. Britney,R.R.and R.L.Winkler,"Bayesian Point Estimation and and R.J.Zeckhauser,"Proper Risk Aversion,"Econometrica,55 Prediction,"Annals of the Institute of Statistical Mathematics,26 (1987),143-154. (1974),15-34. Rothschild,M.and J.Stiglitz,"Increasing Risk:I.A Definition,"J. Eeckhoudt,L.and M.S.Kimball,"Background Risk,Prudence and Eco0 nic Theory,2(1970),225-243. the Demand for Insurance,"in Contributions to Insurance Eco- -and一，“"Increasing Risk:Il.Its Economic Consequences,"I. nomics,G.Dionne (Ed.),Kluwer Academic Press,Boston,1992 of Economic Theory,3(1971),66-84. 239-254. Sankarasubramanian,E.and S.Kumaraswamy,"Note on Optimal C.Gollier,and H.Schlesinger,"Increases in Risk and Deductible Order Quantity for Predetermined Level of Profit,"Management Insurance,"J.Economic Theory,55(1991),435-440. 5ci,29(1983),512-514. -,and-,"The Risk-Averse (and Prudent)Newsboy," Zeckhauser,R.and E.Keeler,"Another Type of Risk Aversion,"Econ- University of Toulouse Working Paper,1992. 0 etrica,38(1970),661-665 Accepted by Robert T.Clemen;received May 11,1992.This paper has been with the authors 12 months for 2 revisions 794 MANAGEMENT SCIENCE/Vol.41,No.5,May 1995EECKHOUDT, GOLLIER, AND SCHLESINGER The Risk-averse (anid Prudenat) Newsboy Our analysis should be helpful in examining a dynamic model. Given the prominence of the static newsboy problem in the literature, our model should prove useful in its own right as well. In addition to any dollar benefits and costs, the risk￾averse newsboy also reacts to wealth effects for changes in prices, costs and risk. It thus becomes more difficult to determine the qualitative effects of changes in these parameters for the risk-averse newsboy than for the risk-neutral newsboy. A summary of comparative-static results is given in Table 1. Although many comparative-static results are am￾biguous under risk aversion, some fairly simple and canonical restrictions on preferences and/or risk in￾creases are seen to lead to qualitatively deterministic comparative-static results. Given the wide array of newsboy-type problems, our results should be useful in analyzing other settings for which the payoff function is piecewise linear.17 17 This paper was started while Gollier and Schlesinger enjoyed the hospitality of CORE, Louvain-la-Neuve, Belgium, and was partly re￾vised while Schlesinger was a guest at the IDEI, University of Toulouse. Financial support of the Insurance Chair of the Federation Francaise des Societe d'Assurance at IDEI is gratefully acknowledged. The au￾thors thank Heraklis Polemarchakis, Ed Schlee, five anonymous ref￾erees, and the editor, Robert Clemen, for helpful comments. References Arrow, K. J., Aspects of the Theory of Risk Bearinig, Yrjo Jahnssonin, Helsinki, Finland, 1965. Baron, D. P., "Point Estimation and Risk Preferences," J. the American Statistical Associationi, 68 (1973), 944-950. Britney, R. R. and R. L. Winkler, "Bayesian Point Estimation and Prediction," Anniials of the Inistitute of Statistical Matheniatics, 26 (1974), 15-34. Eeckhoudt, L. and M. S. Kimball, "Background Risk, Prudence and the Demand for Insurance," in Conitributionis to Inisuranace Eco￾noniics, G. Dionne (Ed.), Kluwer Academic Press, Boston, 1992, 239-254. , C. Gollier, and H. Schlesinger, "Increases in Risk and Deductible Insurance," J. Econiomiiic Theory, 55 (1991), 435-440. and , "The Risk-Averse (and Prudent) Newsboy," University of Toulouse Working Paper, 1992. Epstein, L. and S. Zin, "Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework," Econiomiietrica, 57 (1989), 937-969. Gerchak, Y. and D. Mossman, "On the Effect of Demand Randomness on Inventories and Costs," Oper. Res., 40 (1992), 804-807. Hanoch, G. and H. Levy, "The Efficiency Analysis of Choices Involving Risk," Review of Econiomiiic Studies, 36 (1969), 335-346. Horowitz, I., Decisionz Makinig anid thze Theony of the Firm, Holt, Rinehart and Winston, New York, 1970. Hull, J. C., Optionis, Futures, anid Othler Derivative Securities, Prentice Hall, Englewood Cliffs, NJ, 1993. Hymans, S. H., "The Price Taker: Uncertainty, Utility, and the Supply Function," Initerniationial Econotmic Review, 7 (1966), 346-356. Kanbur, R., "Increases in Risk with Kinked Payoff Functions," J. Eco￾n1om01ic Thleory, 27 (1982), 219-228. Kihlstrom, R. E., D. Romer, and S. Williams, "Risk Aversion with Random Initial Wealth," Econiomiietrica, 49 (1981), 911-920. Kimball, M. S., "Precautionary Saving in the Small and in the Large," Econiometrica, 58 (1990), 53-73. , "Standard Risk Aversion," Econiometrica, 61 (1993), 589-611. Landsberger, M. and I. Meilijson, "A Tale of Two Tails: An Alternative Characterization of Comparative Risk," J. Risk anid Unicertainity, 3 (1990), 65-82. Lau, H.-S., "The Newsboy Problem under Alternative Optimization Objectives," J. of the Operationial Res. Society, 31 (1980), 525- 535. Li, J., H. Lau and A. Lau, "Some Analytical Results for a Two-product Newsboy Problem," Decisioni Scienices, 21 (1990), 5. Menezes, C. and D. Hanson, "On the Theory of Risk Aversion," In￾ternationial Econiomiiic Reviezw1, 11 (1970), 481-487. Meyer, J. and M. B. Ormiston, "Strong Increases in Risk and Their Comparative Statics," Initerniationial Econiomiiic Review, 26 (1985), 425-437. Nachman, D. C., "Preservation of 'More Risk Averse' Under Expec￾tations," J. Econiom0iic Theory, 28 (1982), 361-368. Pratt, J., "Risk Aversion in the Small and in the Large," Econioniietrica, 32 (1964), 122-136. and R. J. Zeckhauser, "Proper Risk Aversion," Econiomiietrica, 55 (1987), 143-154. Rothschild, M. and J. Stiglitz, "Increasing Risk: I. A Definition," J. Econiom0iic Theory, 2 (1970), 225-243. and , "Increasing Risk: II. Its Economic Consequences," J. of Econiomiiic Thleory, 3 (1971), 66-84. Sankarasubramanian, E. and S. Kumaraswamy, "Note on Optimal Order Quantity for Predetermined Level of Profit," Maniagemenit Sci., 29 (1983), 512-514. Zeckhauser, R. and E. Keeler, "Another Type of Risk Aversion," Econl￾omiietrica, 38 (1970), 661-665. Accepted by Robert T. Clemien; received May 11, 1992. This paper hlas beeni with the authlors 12 nmoniths for 2 revisionis. 794 MANAGEMENT SCIENCE/VOl. 41, No. 5, May 1995