EECKHOUDT,GOLLIER,AND SCHLESINGER The Risk-averse (and Prudent)Newsboy seen by differentiating the first-order condition(3)with A.Adding Background Risk respect to p: Thus far,the only uncertainty in our model has been 82H demand uncertainty.We now examine the effect of =-(c-v) u"(Z_)dF(0) adding an independent"background risk"'to newsboy wealth.In particular,suppose that zo in equations(1a) +(-c) and(1b)is replaced by zo +where is a statistically 0u"(Z+)dF(0) independent background risk with zero mean,i.e., "white noise."The new problem is to find the =(c-) 0A(Z_)u'(Z_)dF(0) newspaper order that maximizes E[u(Z(0,a)+)] ≡H(a,). The random wealth zo+e,can be thought of as many -(-)A(Z (ZdF(0) (6) things.For instance,the newsboy might also own an where A(z)=-u"(z)/u'(z)denotes the Arrow-Pratt investment portfolio-or at least have a random allow- measure of absolute risk aversion. ance from his parents.Although this added risk is in- The expression A(z)in Equation (6)is called the dependent of 0,it affects the optimal newspaper order, measure of partial relative risk aversion,as defined by which in the case of added background risk we label Menezes and Hanson(1970)and by Zeckhauser and a**,through the wealth effect.In particular,the first- Keeler(1970).Our result can thus be restated:A higher order condition to maximize H(a,)is simply a more newspaper price leads to an increased order if the general version of(3): newsboy's preferences exhibit decreasing partial relative 0。 risk aversion.Compare this to the case of the risk-neutral =-(c-) E,u'(Z-+)dF newsboy,whose optimal order is unaffected by changes in the price,p,as long as the price is above c. T +(-c) E.u'(Z++)dF=0,(7) 4.Increases in Risk and the Prudent Newsboy where E,denotes the conditional expectation operator. In this section,we examine the effects of two particular If utility is quadratic (i.e.,u"M()=0),then marginal types of changes in the degree of risk facing the news- utility is linear and the addition of will have no effect boy:(1)an introduction of risky background wealth for on u'(Z(0,a));and hence a**will be identical to a* the newsboy and (2)an increase in the riskiness of satisfying(3). newspaper demand.In both instances,we show how If newsboy utility satisfies CARA or DARA,then the qualitative effects on the optimal newspaper order w"(·)>0;and hence generally are indeterminate.However,some simple and Eu'(Z(0,a)+E)>w'(Z(0,a)0 canonical restrictions on newsboy preferences and/or on the types of risk changes can lead to deterministic by Jensen's inequality.This convexity of u'is itself in- results. sufficient to qualitatively compare a**and a",as is seen Hymans(1966)examines price changes for a competitive firm with easily in(7).However,in the case of CARA a quali- random aggregate demand(which cannot produce more after demand tatively deterministic comparison can be made.In this is realized),with costs that are continuous,increasing and strictly regard,define the derived utility function (z),as in convex in output.Although a risk neutral firm in Hyman's model has Kihlstrom et al.(1981),by an upward-sloping supply curve,he demonstrates that downward- sloping supply is possible under risk aversion.Our findings above show that Hymans'results can be extended to other cost functions, (z）= u(z+e)dG(e)=E[u(z+)](8) as well as specify an exact relation on risk aversion to insure an upward sloping supply in the newsboy model. where G denotes the cumulative distribution of e. 790 MANAGEMENT SCIENCE/Vol.41,No.5,May 1995EECKHOUDT, GOLLIER, AND SCHLESINGER The Risk-averse (anid Prudenit) Newsboy seen by differentiating the first-order condition (3) with respect to p: ,a2H ra -(c - v) Ou'(Z_)dF(0) laaapQ*J ? (c -c) f Ou'(Z+)dF(6) (c - v) f OA(Z_)u'(Z_)dF(0) CrT -(cc ) fA(Z+)u'(Z+)dF(e) O (6) where A(z) =-u"(z)/u'(z) denotes the Arrow-Pratt measure of absolute risk aversion. The expression OA(z) in Equation (6) is called the measure of partial relative risk aversion, as defined by Menezes and Hanson (1970) and by Zeckhauser and Keeler (1970). Our result can thus be restated: A higher newspaper price leads to an increased order if the newsboy's preferences exhibit decreasing partial relative risk aversion. Compare this to the case of the risk-neutral newsboy, whose optimal order is unaffected by changes in the price, p, as long as the price is above c. 4. Increases in Risk and the Prudent Newsboy In this section, we examine the effects of two particular types of changes in the degree of risk facing the news￾boy: ( 1 ) an introduction of risky background wealth for the newsboy and (2) an increase in the riskiness of newspaper demand. In both instances, we show how the qualitative effects on the optimal newspaper order generally are indeterminate. However, some simple and canonical restrictions on newsboy preferences and/ or on the types of risk changes can lead to deterministic results. I Hymans (1966) examines price changes for a competitive firm with random aggregate demand (which cannot produce more after demand is realized), with costs that are continuous, increasing and strictly convex in output. Although a risk neutral firm in Hyman's model has an upward-sloping supply curve, he demonstrates that downward￾sloping supply is possible under risk aversion. Our findings above show that Hymans' results can be extended to other cost functions, as well as specify an exact relation on risk aversion to insure an upward sloping supply in the newsboy model. A. Adding Background Risk Thus far, the only uncertainty in our model has been demand uncertainty. We now examine the effect of adding an independent "background risk" to newsboy wealth. In particular, suppose that zo in equations (la) and (lb) is replaced by zo + Z, where e is a statistically independent background risk with zero mean, i.e., "white noise." The new problem is to find the newspaper order that maximizes E[u(Z(6, a) + e)] H(a, e). The random wealth zo + (, can be thought of as many things. For instance, the newsboy might also own an investment portfolio-or at least have a random allow￾ance from his parents. Although this added risk is in￾dependent of 6, it affects the optimal newspaper order, which in the case of added background risk we label a* *, through the wealth effect. In particular, the first￾order condition to maximize H (a, e) is simply a more general version of (3): aa =-(c -v)f Eu '(Z_ ? + )dF CVQ** (c- c) f EEu'(Z+ + Z)dF = 0, (7) where E. denotes the conditional expectation operator. If utility is quadratic (i.e., u"'(.) = 0), then marginal utility is linear and the addition of e will have no effect on u'(Z(6, a)); and hence a** will be identical to a* satisfying (3). If newsboy utility satisfies CARA or DARA, then u"'( .) > 0; and hence Eu'(Z(0, a) + e) > u'(Z(0, a)) VO by Jensen's inequality. This convexity of u' is itself in￾sufficient to qualitatively compare a* * and a *, as is seen easily in (7). However, in the case of CARA a quali￾tatively deterministic comparison can be made. In this regard, define the derived utility function u^(z), as in Kihlstrom et al. (1981), by (z= u(z + E)dG(E) = Ej[u(z + e)] (8) where G denotes the cumulative distribution of e. 790 MANAGEMENT SCIENCE/VOl. 41, No. 5, May 1995