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

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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 (and Prudent)Newsboy In this paper,we also consider the effects of two types at salvage price v<c.The newsboy is allowed to obtain of increases in risk on the optimal newspaper order:(i)additional newspapers if demand exceeds his original the addition of an independent risk to the newsboy's order,but at a higher cost c.A natural assumption is background wealth,and(ii)an increase in the riskiness that 0 sv<c <cs p.3 The newsboy,facing an un- of newspaper demand.The additional risk in back-certain demand at his newsstand,has to determine a, ground wealth,even if it entails a zero mean,will usually the number of newspapers to order ex ante.The random cause a change in the optimal newspaper order for the variable is fully characterized by its cumulative dis- risk-averse newsboy.This is due to wealth effects which tribution function F(),which is assumed to have its are not present in the case of a risk-neutral newsboy. support contained in [0,T].The newsboy then is en- An increase in the riskiness of newspaper demand is dowed with the following payoff function: examined in a recent paper by Gerchak and Mossman Z(0,a)=zo po-ca +v max(0,a-0) (1992),but only for a particular transformation of the random variable for newspaper demand and only for -c max(0,0-a), (1a) a risk-neutral newsboy.In this paper,we show how the Gerchak and Mossman results do not necessarily or equivalently, apply for risk-averse newsboys.Although our compar- Z-(6,a)=20+(p-)0-(c-v)a ative-static effects with respect to increases in demand risk generally are ambiguous,some fairly simple re- f0≤a, Z(0,a) (1b) strictions on preferences and/or risk increases are Z,(0,a)=z0+(p-c)0+(e-c)a shown to lead to qualitatively deterministic results otherwise. A key component in our model is the piecewise-linear, kinked payoff function.The endogenous kink occurs The newsboy's preference functional over final wealth where newspaper demand exactly equals the newboy's distributions is assumed to be of the expected-utility initial newspaper order.Thus,we have a partitioning type,with u(.)denoting the utility of wealth.The of the states of nature into those where an extra initial newsboy is assumed to be (weakly)risk averse and we order yields a net benefit and those yielding a net cost. thus define u to be increasing and concave.+For ana- This partition proves helpful in interpreting our com- lytical ease,u also is assumed to be thrice differentiable. parative-static results.Moreover,this type of payoff The objective for the newsboy is as follows: function can be applied in a wide array of economic- and financial-market settings such as production- incentive models (e.g.,Kanbur 1982),writing covered 3 Note that if &=p,the newsboy only can buy additional papers at call options on an underlying asset(e.g.,Hull 1993)or their selling price,which is equivalent from his perspective to not being able to buy additional papers.Function Z(a,)is then single- models of deductible insurance (e.g.,Eeckhoudt et al. peaked at =o.Note also that if d=c,we trivially obtain a=0 as 1991).2 an optimal initial order for the newsboy.It is also possible to have a nonlinear newspaper cost c(o)with c'>0 and c"s0,so long as v 2.The Basic-Model <c'(a)<f,for all o.Also,we can introduce nonlinear revenue p() with p'()>&.However,since these complications do not change Consider a newsboy with initial wealth zo who buys the basic qualitative results that follow,we maintain the simpler as- newspapers at a unit price c and resells them at price p sumptions of a constant cost c and price p. >c.All unsold newspapers are returned to the publisher 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"s0 everywhere and u"<0 2 Although the payoff function is similar,the cost of changing the somewhere in every open wealth interval.Whether F is the actual control variable (i.e.exercise price or deductible level)for the last two objective distribution function or represents subjective newsboy problems is dependent on the underlying random variable,which probabilities is not a focus here.Utility can be of the von Neumann- makes it more difficult to analyze.See Eeckhoudt et al.(1991)for the Morgenstern,Savage or Anscombe-Aumann type.However,we do analysis of a particular case of this problem. 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 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 < c. The newsboy is allowed to obtain additional newspapers if demand exceeds his original order, but at a higher cost c. A natural assumption is that 0 ? v < c < C ? p.3 The newsboy, facing an uncertain demand 6 at his newsstand, has to determine a, the number of newspapers to order ex ante. The random variable 6 is fully characterized by its cumulative distribution function F(6), which is assumed to have its support contained in [0, T]. The newsboy then is endowed with the following payoff function: Z(6, a) = zo + pO - ca + v max(O, a - 0) -c max(O, 0- a), (la) or equivalently, Z_(6, a) = zo + (p - v)6 - (c - v)a if 0 a a Z(6, ac) = (ilb) Z+(6, a) = zo + (p - )60 + (c - c)a otherwise. The newsboy's preference functional over final wealth distributions is assumed to be of the expected-utility type, with u( ( ) denoting the utility of wealth. The newsboy is assumed to be (weakly) risk averse and we thus define u to be increasing and concave.4 For analytical ease, u also is assumed to be thrice differentiable. The objective for the newsboy is as follows: 3 Note that if c = p, the newsboy only can buy additional papers at their selling price, which is equivalent from his perspective to not being able to buy additional papers. Function Z(a, .) is then singlepeaked at 0 = a. Note also that if c = c, we trivially obtain a* = 0 as an optimal initial order for the newsboy. It is also possible to have a nonlinear newspaper cost c (a) with c' > 0 and c" < 0, so long as v < c'(a) < c, for all a. Also, we can introduce nonlinear revenue p(0) with p'(0) > 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 (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 1995

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 < C. Observe that a* separates the random demand variable 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 newspaper 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 capacity 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 newspapers. 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 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 0<a*,exceeds the 3.Comparative Statics of Cost and average increase in marginal utility for Z+(0,a*),with Price Changes? >8 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. 6<a*,since marginal utility is decreasing,and is not If the newsboy exhibits constant absolute risk aver- affected for 0>a*.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*, the individual will have a higher payout and thus be richer, which affects marginal utility through a wealth effect. Marginal utility is lowered for 0 < a*, since marginal utility is decreasing, and is not affected for 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* is reduced, leading to a higher u'(Z+). Thus, it follows from (3) that a higher C^ will lead to a higher initial newspaper order. The effect of an increase in the initial per unit cost c on the optimal order is not so clearly determined. Al- 7 For the most part, our results in this section are presented heuristically. A more formal mathematical derivation of these results appears in Eeckhoudt et al. (1992), which is available from the authors. though an increase in c increases the dollar cost and reduces the dollar benefit of increasing a, it also reduces wealth by the amount acc in all states of the world; so that both u'(Z_(6, a*)) and u'(Z+(6, a*)) will rise for all 0. Unlike the risk-neutral newsboy, the risk-averse newsboy exhibits a wealth effect. If u'(Z+) increases by a large extent, this wealth effect could be dominant; i.e., it is possible for marginal utility of benefits (i.e., the second term in the first-order condition (3)) to actually rise. In such a case, it is feasible for the optimal newspaper order to increase. However, if we assume that preferences satisfy DARA, then the average increase in marginal utility for Z_ (0, a*), with 0 < a*, exceeds the average increase in marginal utility for Z, (0, a*), with 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

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