《不确定情况下的决策问题》课程教学资料：The Risk-Averse（and Prudent）Newsboy.pdf_大学文库

The Risk-averse (and Prudent)Newsboy Louis Eeckhoudt.Christian Gollier.Harris Schlesinger Catholic Faculties of Mons,7000 Mons,Belgium and Lille,France IDEI,University of Toulouse,31042 Toulouse,France Department of Finance,University of Alabama,Tuscaloosa,Alabama 35487 he effects of risk and risk aversion in the single-period inventory("newsboy")problem are examined.Comparative-static effects of changes in the various price and cost parameters are determined and related to the newsboy's risk aversion.The addition of a random background wealth and of an increase in the riskiness of newspaper demand are also examined.Although many of the comparative effects generally are ambiguous,some fairly simple restrictions on preferences and /or risk increases are shown to lead to qualitatively deterministic results. Newsboy Problem;Inventory;Increase in Risk;Risk Aversion;Prudence) 1.Introduction papers on the optimal order quantity.We extend Bar- Consider a newsboy who must decide how many ron's results to a more general setting,as well as examine newspapers to order in the morning for sale during the the comparative-static effects of changes in the various day.If he orders too many,his costs will be unneces- other cost and price parameters,and of changes in de- sarily high;while if he orders too few,he will have mand risk,for the risk-averse newsboy. missed an opportunity for additional profit.This classic Our comparative statics with respect to changes in single-period inventory problem is often referred to as price are similar in spirit to the early work on the theory the"newsboy problem."In addition to being a problem of the price-taking firm by Hymans (1966).Hymans of consequence to aspiring young newsboys (and examines how changes in price for a risk-averse firm newsgirls),the problem has many analogies with regard facing a random demand affect the firm's output.For a to topics such as plant capacity,overbooking,and target risk-averse firm in general(not necessarily a newsboy), production levels for planned economies. Hymans demonstrates the possibility of downward In this paper,we examine various comparative statics sloping supply curves if"marginal utility declines very (i.e.,a qualitative sensitivity analysis)for the risk-averse rapidly."We find a similar result for the newsboy and newsboy.Although much has been written about the we make this result more precise by analyzing the mea- newsboy problem,relatively little has been written sure of partial relative risk aversion,as defined by Me- about the risk-averse newsboy.It seems to be well nezes and Hanson(1970)and by Zeckhauser and Keeler known that risk aversion will lead to a reduced initial (1970).This allows us to give precise conditions under newspaper order.Unfortunately,only a scattering of which an increase in newspaper prices will lead to an other results can be found,and usually within very spe- increase in the optimal order by the newsboy. cific models.Horowitz (1970),for example,has exer- cises concerning a risk-averse newsboy(okay,so it is a I Actually,Baron does not consider the newsboy problem per se,but hot-dog vendor)for specific utility functions. his short section on piecewise-linear payoff functions can be inter- An early paper looking at more general risk-averse preted to yield the above-mentioned analysis.Britney and Winkler preferences is Baron (1973).Baron examines the com- (1974)and Lau(1980)also examine the optimal order for a risk- averse newsboy,but for particular utility functions and in conjunction parative-static effects of changes in newsboy risk aver- with particular demand distributions.See also Li,Lau and Lau(1990) sion and changes in the salvage value of unsold news- and Sankarasubramanian and Kumaraswamy (1982). 0025-1909/95/4105/0786$01.25 Copyright 1995,Institute for Operations Research 786 MANAGEMENT SCIENCE/Vol.41,No.5,May 1995 and the Management Sciences

The Risk-averse (and Prudent) Newsboy Louis Eeckhoudt * Christian Gollier * Harris Schlesinger Catholic Faculties of Mons, 7000 Mons, Belgium and Lille, France IDEI, University of Toulouse, 31042 Toulouse, France Department of Finance, University of Alabama, Tuscaloosa, Alabama 35487 The effects of risk and risk aversion in the single-period inventory ("newsboy") problem are examined. Comparative-static effects of changes in the various price and cost parameters are determined and related to the newsboy's risk aversion. The addition of a random background wealth and of an increase in the riskiness of newspaper demand are also examined. Although many of the comparative effects generally are ambiguous, some fairly simple restrictions on preferences and/or risk increases are shown to lead to qualitatively deterministic results. (Newsboy Problem; Inventory; Increase in Risk; Risk Aversion; Prudence) 1. Introduction Consider a newsboy who must decide how many newspapers to order in the morning for sale during the day. If he orders too many, his costs will be unnecessarily high; while if he orders too few, he will have missed an opportunity for additional profit. This classic single-period inventory problem is often referred to as the "newsboy problem." In addition to being a problem of consequence to aspiring young newsboys (and newsgirls), the problem has many analogies with regard to topics such as plant capacity, overbooking, and target production levels for planned economies. In this paper, we examine various comparative statics (i.e., a qualitative sensitivity analysis) for the risk-averse newsboy. Although much has been written about the newsboy problem, relatively little has been written about the risk-averse newsboy. It seems to be well known that risk aversion will lead to a reduced initial newspaper order. Unfortunately, only a scattering of other results can be found, and usually within very specific models. Horowitz (1970), for example, has exercises concerning a risk-averse newsboy (okay, so it is a hot-dog vendor) for specific utility functions. An early paper looking at more general risk-averse preferences is Baron (1973). Baron examines the comparative-static effects of changes in newsboy risk aversion and changes in the salvage value of unsold newspapers on the optimal order quantity.' We extend Barron's results to a more general setting, as well as examine the comparative-static effects of changes in the various other cost and price parameters, and of changes in demand risk, for the risk-averse newsboy. Our comparative statics with respect to changes in price are similar in spirit to the early work on the theory of the price-taking firm by Hymans (1966). Hymans examines how changes in price for a risk-averse firm facing a random demand affect the firm's output. For a risk-averse firm in general (not necessarily a newsboy), Hymans demonstrates the possibility of downward sloping supply curves if "marginal utility declines very rapidly." We find a similar result for the newsboy and we make this result more precise by analyzing the measure of partial relative risk aversion, as defined by Menezes and Hanson ( 1970) and by Zeckhauser and Keeler (1970). This allows us to give precise conditions under which an increase in newspaper prices will lead to an increase in the optimal order by the newsboy. ' Actually, Baron does not consider the newsboy problem per se, but his short section on piecewise-linear payoff functions can be interpreted to yield the above-mentioned analysis. Britney and Winkler (1974) and Lau (1980) also examine the optimal order for a riskaverse newsboy, but for particular utility functions and in conjunction with particular demand distributions. See also Li, Lau and Lau (1990) and Sankarasubramanian and Kumaraswamy (1982). 0025-1909/95/4105/0786$01 .25 Copyright (? 1995, Institute for Operations Research 786 MANAGEMENT SCIENCE/Vol. 41, No. 5, May 1995 and the Management Sciences

EECKHOUDT, GOLLIER, AND SCHLESINGER The Risk-averse (anid Prudenit) Newsboy In this paper, we also consider the effects of two types of increases in risk on the optimal newspaper order: (i) the addition of an independent risk to the newsboy's background wealth, and (ii) an increase in the riskiness of newspaper demand. The additional risk in background wealth, even if it entails a zero mean, will usually cause a change in the optimal newspaper order for the risk-averse newsboy. This is due to wealth effects which are not present in the case of a risk-neutral newsboy. An increase in the riskiness of newspaper demand is examined in a recent paper by Gerchak and Mossman (1992), but only for a particular transformation of the random variable for newspaper demand and only for a risk-neutral newsboy. In this paper, we show how the Gerchak and Mossman results do not necessarily apply for risk-averse newsboys. Although our comparative-static effects with respect to increases in demand risk generally are ambiguous, some fairly simple restrictions on preferences and / or risk increases are shown to lead to qualitatively deterministic results. A key component in our model is the piecewise-linear, kinked payoff function. The endogenous kink occurs where newspaper demand exactly equals the newboy's initial newspaper order. Thus, we have a partitioning of the states of nature into those where an extra initial order yields a net benefit and those yielding a net cost. This partition proves helpful in interpreting our comparative-static results. Moreover, this type of payoff function can be applied in a wide array of economicand financial-market settings such as productionincentive models (e.g., Kanbur 1982), writing covered call options on an underlying asset (e.g., Hull 1993) or models of deductible insurance (e.g., Eeckhoudt et al. 1991).2 2. The Basic-Model Consider a newsboy with initial wealth zo who buys newspapers at a unit price c and resells them at price p > c. All unsold newspapers are returned to the publisher 2 Although the payoff function is similar, the cost of changing the control variable (i.e. exercise price or deductible level) for the last two problems is dependent on the underlying random variable, which makes it more difficult to analyze. See Eeckhoudt et al. (1991) for the analysis of a particular case of this problem. at salvage price v 0 and c" c. However, since these complications do not change the basic qualitative results that follow, we maintain the simpler assumptions of a constant cost c and price p. 4 By "weakly risk averse" we mean either risk averse or risk neutral; thus, u need not be strictly concave. We use the terminology "risk averse" to denote strict risk aversion: u" < 0 everywhere and ui" < 0 somewhere in every open wealth interval. Whether F is the actual objective distribution function or represents subjective newsboy probabilities is not a focus here. Utility can be of the von NeumannMorgenstern, Savage or Anscombe-Aumann type. However, we do assume that utility is state independent. MANAGEMENT SCIENCE/VOL 41, No. 5, May 1995 787

EECKHOUDT, 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 straightforward to derive and follows from our assumption that v u'(Z(a*, a*)) > u'(Z+(02, a*)) VO1 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 aversion.6 Let zo = v = 0, c^ = p, and let 0 E { 0, 100 }, where the probability demand is zero is 0.25 and the probability 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 orders 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 constant in z. 788 MANAGEMENT SCIENCE/VOl. 41, No. 5, May 1995

EECKHOUDT,GOLLIER,AND SCHLESINGER The Risk-averse(and Prudent)Newsboy risk aversion optimal order though an increase in c increases the dollar cost and r=0.00000 100 r=0.00001 100 reduces the dollar benefit of increasing a,it also reduces r=0.0001 wealth by the amount ac in all states of the world;so r=0.001 7 that both u'(Z-(0,a*))and u'(Z(0,a*))will rise for r=0.01 all 0.Unlike the risk-neutral newsboy,the risk-averse r=0.1 0 newsboy exhibits a wealth effect.If u'(Z+)increases by Note also that,in our example,the expected profit per a large extent,this wealth effect could be dominant;i.e., paper ordered is 1.Thus,the expected profit level of it is possible for marginal utility of benefits (i.e.,the the newsboy equals the optimal order quantity.For the second term in the first-order condition(3))to actually case where r =0.1,the newsboy is so risk averse that rise.In such a case,it is feasible for the optimal news- he does not order even a single newspaper,for fear of paper order to increase.However,if we assume that losing the cost of 20. preferences satisfy DARA,then the average increase in marginal utility for Z_(0,a*),with 08 Thus,the optimal newspaper order a*will de- crease with an increase in c. The fact that the marginal dollar cost(c-v)and mar- For an increase in the selling price of a newspaper, ginal dollar benefit (c-c)of increasing the initial p,the comparative-static analysis is even more complex newspaper order are constant allows us to easily com- First,from the first-order condition(3),price only enters pute the comparative statics of changing the cost pa- the decision process through Z+and Z_and p does not rameters.For instance,if the salvage value v increases, affect the dollar benefit or cost.In particular,we see the marginal dollar cost of increasing the initial news- from either (1a)or (1b)that both Z+and Z-include paper order a is reduced.However,for any realized the term po.Increasing p leads to higher income in all value of a*,the individual will have a higher payout states of the world,with the extra income proportional and thus be richer,which affects marginal utility to.Thus,an increase in p also has the effect of making through a wealth effect.Marginal utility is lowered for the distribution of final newsboy wealth riskier. 6a*.In other words,u'(Z_)is everywhere sion (i.e.,negative exponential utility),for example,a lower,while u'(Z,)remains unchanged in the first-order rise in p will cause each u'(Z)to fall more precipitously condition(3).It thus follows easily that the optimal than each u'(Z-).As a consequence,it follows from(3) newspaper order a*will increase if the salvage value that the optimal newspaper order will fall.In other rises.There is no wealth effect for the risk-neutral words,we obtain a downward sloping supply curve for newsboy. newspapers.If newsboy preferences exhibit decreasing Similarly,an increase in the re-order cost d affects absolute risk aversion,it is still possible for an increase only the marginal benefit of increasing o.A higher c in p to lead to a reduced newspaper order.For example, increases the marginal dollar benefit(c-c)for raising this will occur in the case where preferences are very a,while at the same time wealth in states of nature slightly daRa and close to constant absolute risk aver- where 6<a*is reduced,leading to a higher u'(Z). sion.If risk aversion decreases quickly enough in wealth, Thus,it follows from(3)that a higher will lead to a in particular at least as quickly in as,the supply higher initial newspaper order. of newspapers will be upward sloping.This is formally The effect of an increase in the initial per unit cost c on the optimal order is not so clearly determined.Al- Note that the Arrow-Pratt measure of absolute risk aversion,-u"(z)/ 7 For the most part,our results in this section are presented heurist- u'(z),is a decay rate for marginal utility.Thus,for a constant dollar ically.A more formal mathematical derivation of these results appears decrease in wealth of ac,DARA implies u'(Z-)rises by a higher per- in Eeckhoudt et al.(1992),which is available from the authors. centage change than each i'(Z.). MANAGEMENT SCIENCE/Vol.41,No.5,May 1995 789

EECKHOUDT, GOLLIER, AND SCHLESINGER The Risk-averse (anid Prudenit) Newsboy risk aversion optimal order I = 0.00000 100 r= 0.00001 100 r= 0.0001 65 r= 0.001 7 rO=l0.01 1 r = 0.1 0 Note also that, in our example, the expected profit per paper ordered is 1. Thus, the expected profit level of the newsboy equals the optimal order quantity. For the case where r = 0.1, the newsboy is so risk averse that he does not order even a single newspaper, for fear of losing the cost of 20. 3. Comparative Statics of Cost and Price Changes7 The fact that the marginal dollar cost (c - v) and marginal dollar benefit (c^ - c) of increasing the initial newspaper order are constant allows us to easily compute the comparative statics of changing the cost parameters. For instance, if the salvage value v increases, the marginal dollar cost of increasing the initial newspaper order a is reduced. However, for any realized value of 0 a*. In other words, u'(Z_) is everywhere lower, while u'(Z+) remains unchanged in the first-order condition (3). It thus follows easily that the optimal newspaper order a* will increase if the salvage value rises. There is no wealth effect for the risk-neutral newsboy. Similarly, an increase in the re-order cost c affects only the marginal benefit of increasing a. A higher C^ increases the marginal dollar benefit (C^ - c) for raising a, while at the same time wealth in states of nature where 0 a*.8 Thus, the optimal newspaper order a* will decrease with an increase in c. For an increase in the selling price of a newspaper, p, the comparative-static analysis is even more complex. First, from the first-order condition (3), price only enters the decision process through Z+ and Z_ and p does not affect the dollar benefit or cost. In particular, we see from either (la) or (lb) that both Z+ and Z_ include the term p6. Increasing p leads to higher income in all states of the world, with the extra income proportional to 0. Thus, an increase in p also has the effect of making the distribution of final newsboy wealth riskier. If the newsboy exhibits constant absolute risk aversion (i.e., negative exponential utility), for example, a rise in p will cause each u'(Z+) to fall more precipitously than each u'(Z_). As a consequence, it follows from (3) that the optimal newspaper order will fall. In other words, we obtain a downward sloping supply curve for newspapers. If newsboy preferences exhibit decreasing absolute risk aversion, it is still possible for an increase in p to lead to a reduced newspaper order. For example, this will occur in the case where preferences are very slightly DARA and close to constant absolute risk aversion. If risk aversion decreases quickly enough in wealth, in particular at least as quickly in 0 as 0-', the supply of newspapers will be upward sloping. This is formally 8 Note that the Arrow-Pratt measure of absolute risk aversion, -u"(z)/ u'(z), is a decay rate for marginal utility. Thus, for a constant dollar decrease in wealth of ac, DARA implies u'(Z_) rises by a higher percentage change than each u'(Z+). MANAGEMENT SCIENCE/VOl. 41, No. 5, May 1995 789

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 1995

EECKHOUDT, 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 newsboy: ( 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 downwardsloping 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 allowance from his parents. Although this added risk is independent 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 firstorder 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 insufficient to qualitatively compare a* * and a *, as is seen easily in (7). However, in the case of CARA a qualitatively 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

EECKHOUDT,GOLLIER,AND SCHLESINGER The Risk-averse (and Prudent)Newsboy Calculating-"/'for CARA (i.e.,negative expo-B.Increased Demand Risk nential)utility reveals that and u are equally risk We now turn our attention to the qualitative effects of averse at every total wealth level z.Thus i and u must changes in demand risk on the optimal initial order a*. be equivalent utility representations(i.e.,is an affine Kanbur(1982)analyzes a similar model,whose results transformation of u).Hence, can be applied here as well.His results imply that an H(a,)=E[4(Z(8,a)+)]=El(Z(0,a)】.(9) increase in risk in the sense of Rothschild and Stiglitz (1970,1971)generally has an ambiguous effect on a*. Since and u are equivalent utility representations,it However,we obtain unambiguous effects by making follows that a**equals a. some fairly simple restrictions on the increases in risk For the case where newsboy preferences exhibit and/or making some fairly canonical restrictions on DARA,a deterministic comparison between a**and a* preferences. follows if we make the additional assumption on pref- The change in risk is represented by a change in the erences that absolute prudence,defined by Kimball demand distribution F()to the distribution G(). (1990)as Without loss of generality,we assume that the support B(2)="(2) of G is contained in [0,T].Let a denote the optimal (10) u"(z) order with the original distribution F().We say G rep- resents a mean-preserving increases in risk(MIR)of F, is decreasing in wealth. if G and F satisfy the two Rothschild and Stiglitz con- The notion of prudence,u"(.)>0,was introduced ditions: by Kimball(1990),who shows how prudence is suffi- cient for a precautionary savings demand in standard G(0)d0≥ F(0)d0 for all t∈[0，T],(11) intertemporal consumption models.The notion of de- creasing prudence is discussed extensively in Kimball with a strict inequality on a set of positive probability (1993).In particular,DARA and decreasing absolute measure,and prudence(DAP)in tandem are shown to be equivalent G(0)d0= F(0)d0 (12) to the following canonical condition of preferences: "every risk that has a negative interaction with a small Condition (11)represents second-degree stochastic reduction in wealth also has a negative interaction with dominance of F over G while condition(12)preserves any undesirable,independent risk."Here"negative in- the mean.As shown by Rothschild and Stiglitz(1970), teraction''refers to a decrease in expected utility.The these conditions imply that any risk-averse newsboy class of utility functions satisfying DARA and DAP in- will prefer random demand given by F to that given by cludes,for example,the commonly used constant- G.11 However,(11)and (12)alone are not strong relative-risk-aversion class of utility functions (i.e. enough to guarantee that a lower newspaper order will power utility functions and logarithmic utility).10 be chosen under G. Assuming DARA and DAP,Eeckhoudt and Kimball In order to obtain unambiguous results,one typically (1992)prove that the derived utility function as given can make some restrictions on either(i)the utility func- in (8),is uniformly more risk averse than u.It then tion,(ii)the cost/price parameters,or (iii)the MIR follows easily from(9)and(5)that a**<a.In other itself.In addition,one can also select mixtures of these words,the optimal newspaper order will decrease in types of restrictions.12 response to an added background risk,whenever newsboy preferences display DARA and DAP. 1For mean-preserving changes in the distribution function satisfying (12),Rothschild and Stiglitz(1970)show that(11)holds if and only 10 Kimball(1993)shows how DAP is a natural extension of DARA if every risk averter prefers F to G. and how DARA and DAP together imply Pratts'and Zeckhauser's 12 Kanbur obtains unambiguous results only for the special case of (1987)proper risk aversion:given any two independent undesirable quadratic utility functions.His analysis implies the following results risks,their sum is at least as undesirable. in our model: MANAGEMENT SCIENCE/Vol.41,No.5,May 1995 791

EECKHOUDT, GOLLIER, AND SCHLESINGER The Risk-averse (anid Pruidenit) Nezwsboy Calculating -u^"/u^' for CARA (i.e., negative exponential) utility reveals that u^ and u are equally risk averse at every total wealth level z. Thus u^ and u must be equivalent utility representations (i.e., u^ is an affine transformation of u). Hence, H(a, i) = E[u(Z(6, a) + i)] = E4[a^(Z(6, a))]. (9) Since u^ and u are equivalent utility representations, it follows that a** equals a*. For the case where newsboy preferences exhibit DARA, a deterministic comparison between a** and a* follows if we make the additional assumption on preferences that absolute prudence, defined by Kimball (1990) as -u ..' (z) B(z) = u",(z) (10) is decreasing in wealth. The notion of prudence, u"' (.) > 0, was introduced by Kimball (1990), who shows how prudence is sufficient for a precautionary savings demand in standard intertemporal consumption models. The notion of decreasing prudence is discussed extensively in Kimball (1993). In particular, DARA and decreasing absolute prudence (DAP) in tandem are shown to be equivalent to the following canonical condition of preferences: "every risk that has a negative interaction with a small reduction in wealth also has a negative interaction with any undesirable, independent risk." Here "negative interaction" refers to a decrease in expected utility. The class of utility functions satisfying DARA and DAP includes, for example, the commonly used constantrelative-risk-aversion class of utility functions (i.e. power utility functions and logarithmic utility)." Assuming DARA and DAP, Eeckhoudt and Kimball (1992) prove that the derived utility function u^, as given in (8), is uniformly more risk averse than u. It then follows easily from (9) and (5) that a** < a-'. In other words, the optimal newspaper order will decrease in response to an added background risk, whenever newsboy preferences display DARA and DAP. '? Kimball (1993) shows how DAP is a natural extension of DARA and how DARA and DAP together imply Pratts' and Zeckhauser's (1987) proper risk aversion: given any two independent undesirable risks, their sum is at least as undesirable. B. Increased Demand Risk We now turn our attention to the qualitative effects of changes in demand risk on the optimal initial order a*. Kanbur (1982) analyzes a similar model, whose results can be applied here as well. His results imply that an increase in risk in the sense of Rothschild and Stiglitz (1970, 1971) generally has an ambiguous effect on a*. However, we obtain unambiguous effects by making some fairly simple restrictions on the increases in risk and/or making some fairly canonical restrictions on preferences. The change in risk is represented by a change in the demand distribution F(6) to the distribution G(6). Without loss of generality, we assume that the support of G is contained in [0, T]. Let C* denote the optimal order with the original distribution F(6). We say G represents a mean-preserving increases in risk (MIR) of F, if G and F satisfy the two Rothschild and Stiglitz conditions: j'G(6)dO jF(6)dO forall t E [0, T], (11) with a strict inequality on a set of positive probability measure, and T oT G()d= J F(6)dO. (12) Condition (11) represents second-degree stochastic dominance of F over G while condition (12) preserves the mean. As shown by Rothschild and Stiglitz (1970), these conditions imply that any risk-averse newsboy will prefer random demand given by F to that given by G."1 However, (11) and (12) alone are not strong enough to guarantee that a lower newspaper order will be chosen under G. In order to obtain unambiguous results, one typically can make some restrictions on either (i) the utility function, (ii) the cost/price parameters, or (iii) the MIR itself. In addition, one can also select mixtures of these types of restrictions.12 " For mean-preserving changes in the distribution function satisfying (12), Rothschild and Stiglitz (1970) show that ( 11 ) holds if and only if every risk averter prefers F to G. 12 Kanbur obtains unambiguous results only for the special case of quadratic utility functions. His analysis implies the following results in our model: MANAGEMENT SCIENCE/VOL 41, No. 5, May 1995 791

EECKHOUDT, 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., satisfying G(6) = F(6) in the interval [4, T], reduces 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, a. If the simple spread also preserves the mean, it is a particular case of an MIR since it follows that condition (13) implies 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 context 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 riskneutral newsboys would not be affected by a simple spread across a*, as shown by Equation (4). The probability-preserving property of a simple spread allows 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 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

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 ac0.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 793

EECKHOUDT, 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 0. Let G denote the distribution of 6K. For k > 1, G is a simple spread of F across a*. A recent paper by Gerchak 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 increase 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 particular 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 optimal 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 meanpreserving simple spread across a* = 37. From our previous 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 optimal newspaper order for the risk-averse newsboy.'6 16 For example, under constant absolute risk aversion, zero time discount 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 models. See, for example, Epstein and Zin (1989). MANAGEMENT SCIENCE/VOL 41, No. 5, May 1995 793

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 1995

EECKHOUDT, 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 riskaverse 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 ambiguous under risk aversion, some fairly simple and canonical restrictions on preferences and/or risk increases 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 revised 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 authors thank Heraklis Polemarchakis, Ed Schlee, five anonymous referees, 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 Econoniics, 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. Econ1om01ic 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," Internationial 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 Expectations," 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," Econlomiietrica, 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