Omega 37 (2009) 93–105 www.elsevier.com/locate/omega
The loss-averse newsvendor problemଁ Charles X. Wanga,∗, Scott Websterb
aSchool of Management, State University of New York at Buffalo, Buffalo, NY 14260, USA bWhitman School of Management, Syracuse University, Syracuse, NY 13244, USA
Received 23 December 2005; accepted 8 August 2006 Available online 4 October 2006
Abstract Newsvendor models are widely used in the literature, and usually based upon the assumption of risk neutrality. This paper uses loss aversion to model manager’s decision-making behavior in the single-period newsvendor problem. We find that if shortage cost is not negligible, then a loss-averse newsvendor may order more than a risk-neutral newsvendor. We also find that the loss-averse newsvendor’s optimal order quantity may increase in wholesale price and decrease in retail price, which can never occur in the risk-neutral newsvendor model. ᭧ 2006 Elsevier Ltd. All rights reserved.
Keywords: Inventory; Loss aversion; Newsvendor model
1. Introduction there are many counter examples indicating that man- agers’ order quantity decisions are not always consistent The newsvendor problem is one of the fundamental with maximizing expected profit. For example, Kahn models in stochastic inventory theory [1]. The newsven- [2] finds that Chrysler held larger inventory stocks than dor’s objective is to choose an optimal order quantity competitors such as GM and Ford before early 1980, to balance his cost (or disutility) of ordering too many which seemed to imply the stockout-avoidance behav- against his cost (or disutility) of ordering too few. Be- ior (see also [3,4]). We refer to the deviation of the cause of its simple but elegant structure, the newsvendor newsvendor’s optimal order quantity from the profit problem also applies in various settings such as capac- maximization order quantity as decision bias in the ity planning, yield management, insurance, and supply newsvendor problem. chain contracts. One major reason why decision bias may exist in The standard newsvendor model is based upon risk the newsvendor problem is that a manager may have neutrality so that managers will select an order quan- preferences other than risk neutrality [5]. In this paper, tity to maximize expected profit. However, in practice, we attempt to use an alternative choice model—loss aversion—to describe the newsvendor decision bias. Loss aversion states that people are more averse to ଁ This manuscript was processed by Associate Editor Teunter. ∗ losses than they are attracted to same-sized gains. There Corresponding author. Tel.: +1 716 645 3412; fax: +1 716 645 5078. are two main reasons why we use loss aversion to de- E-mail addresses: [email protected] (C.X. Wang), scribe the decision bias in the newsvendor problem. [email protected] (S. Webster). First, loss aversion is both intuitively appealing and
0305-0483/$ - see front matter ᭧ 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.omega.2006.08.003 94 C.X. Wang, S. Webster/ Omega 37 (2009) 93–105 well supported in finance, economics, marketing, and total supply chain performance is suboptimal. Hence, organizational behavior (see [6,7] for more detailed our research findings also lend insights into how loss reviews). This paper is especially motivated by two aversion contributes to supply chain inefficiency and empirical studies of managerial decision-making un- may lead to new policies for mitigating these effects. der uncertainty. The first study by MacCrimmon and This paper is organized as follows. In Section 2, Wehrung [8] is based on questionnaire responses from we briefly review the literature related to our research. 509 high-level executives in American and Canadian In Section 3, we analyze our single-period loss-averse firms, and interviews with 128 of those executives in newsvendor model. In Section 4, we analyze the effects 1973–1974. The second study by Shapira [9] is based of changing parameter values. Finally, in Section 5, we on interviews with 50 American and Israeli executives draw our conclusions and identify opportunities for fu- in 1984–1985. Managers’ decision-making behavior in ture research. both studies is consistent with loss version. Second, in contrast with the widely applications and empirical 2. Related newsvendor literature supports of loss aversion in other fields, unfortunately, the development of loss aversion to describe man- The traditional newsvendor model is based upon ager’s newsvendor decision bias is still in its early risk neutrality; managers place orders to maximize ex- stages. As far as we know, Schweitzer and Cachon [5] pected profits (e.g., [1,10,11]). Recently, a number of is the only paper studying a loss-averse newsvendor attempts have been made to extend our understanding problem. They show that a loss-averse newsvendor of the newsvendor model. Lippman and McCardle [12] (with no shortage cost) will order strictly less than a study the competitive newsvendor problem. Petruzzi risk-neutral newsvendor. However, the newsvendor’s and Dada [13] provide a survey of the newsvendor decision-making behavior is unclear from their model problem where both price and quantity are set simulta- if (1) shortage cost is considered and (2) some price or neously. Casimir [14,15] study the value of information cost is changing. To fill in this research gap, we extend in the single and multi-item newsvendor problems. Carr their model by considering the shortage cost and sum- and Lovejoy [16] consider a newsvendor who chooses marizing comparative statics of price and cost changes. which customers to serve given his fixed capacity. Dana The purpose of our research is to build a theoretical and Petruzzi [17] study a newsvendor model in which model that characterizes the outcomes of the decision consumers choose between attempting to purchase the bias of a loss-averse newsvendor, i.e., to show when, newsvendor’s product and an exogenous alternative. how, and why loss aversion causes the decision bias Van Mieghem and Rudi [18] introduce newsvendor in the newsvendor problem. We use a simple “kinked” networks, which generalize the classic newsvendor piecewise-linear loss-aversion utility function to study model and allows for multiple products and multiple the single-period newsvendor model. We find: (1) a processing and storage points. Cachon and Kok [19] loss-averse newsvendor will order less than the risk- study a newsvendor model with clearance pricing for neutral newsvendor if he faces low shortage cost and the leftover inventory at the end of the selling season, the more loss-averse, the less his optimal order quan- which contrasts with the traditional assumption of a tity; (2) a newsvendor will order more than the risk- constant salvage value assigned to each unit of unsold neutral newsvendor if he faces high shortage cost and inventory. Mostard et al. [20] and Mostard and Teunter the more loss-averse, the more his optimal order quan- [21] study the newsvendor problem where undamaged tity. We also discuss some comparative statics of price returned products from customers during the selling and cost changes. We find that the loss-averse newsven- season are still resalable before the season ends. dor’s optimal order quantity may increase in wholesale In addition to risk neutrality, some researchers have price and decrease in retail price, which can never oc- attempted to use risk aversion within Expected Utility cur in the risk-neutral newsvendor model. Our research Theory (EUT) to describe the decision-making behav- contributes to the newsvendor literature in two main as- ior in the newsvendor problem. Eeckhoudt et al. [22] pects. First, we quantify the newsvendor decision bias study a risk-averse newsvendor who is allowed to ob- and show how the newsvendor loss aversion decision tain additional orders if demand is higher than his initial bias interacts with the business conditions, e.g., retail order. They find that a risk-averse newsvendor will or- price, wholesale price, shortage cost, and degree of loss der strictly less than a risk-neutral newsvendor. Agrawal aversion, when shortage cost exists. Second, since a and Seshadri [23] investigate a risk-averse and price- loss-averse newsvendor orders from his supplier a dif- setting newsvendor problem. They find that a risk-averse ferent quantity from the profit-maximizing quantity, the newsvendor will charge a higher price and order less C.X. Wang, S. Webster/ Omega 37 (2009) 93–105 95 than the risk-neutral newsvendor if the demand distri- payoff function can be rewritten as bution has the multiplicative form of relationship with = − price. Also, the risk-averse newsvendor will charge a −(x, Q) x wQ if x Q, = lower price if the demand distribution has the additive (x, Q) +(x, Q) (2) = − − − form of relationship with price, but the effect on the Q s(x Q) wQ if x>Q, quantity ordered depends on the demand sensitivity to where s>w− 1 and w ∈ (0, 1). selling price. Another stream of literature closely related to our Lemma 1. For any Q ∈ I, let q1(Q) = wQ and paper deals with loss aversion. There have since been q2(Q) = (1 + s − w)Q/s. If s (1 − w)Q/(sup I − Q), many economic field tests and applications of loss aver- then the newsvendor has two breakeven quantities of sion in financial markets (e.g., [24]), life savings and realized demand, q1(Q) and q2(Q) where if x
+ − ∗ Utility Theorem 2. For any > 1, if (1 s w)F(q2(Q1)) > ∗ ∗ ∗ ∗ wF(q1(Q1)), then Q >Q1 and dQ/d > 0; if ∗ ∗ ∗ (1 + s − w)F(q2(Q )) = wF(q1(Q )), then Q = ∗ ∗ 1 1 ∗ ∗ Q and dQ/d = 0; otherwise, Q 1, there exists a short- and satisfies the following first-order condition: ∗ ∗ age cost s > 0 such that for all s
0.08 0.06 0.04 0.02 0 -0.02 -0.04 0.5)
s, -0.06 (
g -0.08 -0.1 -0.12 -0.14 -0.16 -0.18 0.01 2.01 4.01 6.01 8.01 10.01 12.01 14.01 16.01 18.01 20.01 Shortage cost s
Fig. 2. A plot of g(s,0.5) with s ranging between 0.01 and 20.01 in increments of 0.5. Demand is exponentially distributed.
7.00 6.50
6.00 5.50 ′ s 5.00
4.50 4.00
Shortage cost 3.50 3.00
2.50 2.00 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 Wholesale price w
Fig. 3. A plot of s as a function of wholesale price w ranging between 0.05 and 0.95 in increments of 0.05. Demand is exponentially distributed.
∗ − x = a ∗ = If q2(Q1) 0.30 0.25 0.20 0.15 0.10 , 0.5) s ( 0.05 g 0.00 CV = 0.5 -0.05 CV = 0.25 -0.10 CV = 0.125 -0.15 0.01 2.01 4.01 6.01 8.01 10.01 12.01 14.01 16.01 18.01 20.01 Shortage cost s Fig. 4. A plot of g(s,w) as a function of s with w = 0.5 and s ranging between 0.01 and 20.01 in increments of 0.5. Demand conforms to a truncated normal distribution with coefficient of variation CV = 0.125, 0.25, and 0.5. 3.50 3.00 ′ s CV = 0.125 2.50 CV = 0.25 CV = 0.5 2.00 1.50 Threshold shortage cost 1.00 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 Wholesale price w Fig. 5. A plot of s as a function of wholesale price w ranging between 0.05 and 0.95 in increments of 0.05. Demand conforms to a truncated normal distribution with coefficient of variation CV = 0.125, 0.25, and 0.5. = ∗ − = + − While the complexity of (8) inhibits analytical conclu- z1 (Q1 )/ and observing that g(s ,w) (1 s ∗ ∗ sions, our numerical tests suggest that g(s, w) > 0 w)F(q2(Q )) − wF(q1(Q )) = 0 can be rewritten as ∗ ∗ 1 1 for s>s (and g(s,w)<0 for s The value of the right-hand side depends solely on the On the flipside, the direction of bias does not change terms in brackets, and is unaffected by proportional in- as the newsvendor’s market increases or shrinks as creases or decreases in and (which leave CV un- long as the relative uncertainty in demand is stable. changed). Furthermore, by substituting CV for in the In addition, the numerical results suggest that the right-hand side and taking the derivative with respect to direction of bias is relatively insensitive to changes CV, it can be shown that s is increasing (decreasing) in wholesale price (e.g., observe the flat curves in in CV when z1 > 0 (z1 < 0). This result in combination Fig. 5). The results indicate that a supplier tactic with Fig. 5 implies that s is increasing in demand un- predicated on the direction of bias may be fairly ro- certainty for w ∈{0.05, 0.10,...,0.95}.1 bust with respect seasonal demand patterns and price Several general properties are evident in the above changes. results. First, bias cannot be positive unless shortage Based upon these numerical results, we could classify cost is positive. Second, the direction of bias is inde- products into two broad categories: the low-shortage- pendent of the sensitivity to losses as measured by cost product and the high-shortage-costproduct. We and, given bias is present, the magnitude of bias is in- could list computers, food, and shirts as examples creasing in . Also, if the loss-averse newsvendor orders of low-shortage-cost products and airline meals, ho- the same as the risk-neutral newsvendor, then the loss- tel rooms, and travel packages as high-shortage-cost averse optimal order quantity is unaffected by changes products, e.g., in the airline industry, the cost per ∗ in . In essence, the presence of bias means that at Q1, meal varies between $2 and $12 but the per meal marginal underage and overage loss are not equal, and shortage cost is around $120 [32]. For high-shortage- the term that dominates determines the direction of bias. cost products, a loss-averse newsvendor may order One might suspect that loss aversion is associated with more than the risk-neutral newsvendor, whereas for a smaller order quantity (relative to risk-neutral), or at low-shortage-cost products, loss aversion will cause least an order quantity that differs from risk-neutral. The the newsvendor to order less than a risk-neutral results tell us that this suspicion does not always hold, newsvendor. and that bias depends both on the probability distribu- tion of demand as well as the relative values of price and cost parameters. 4. Comparative statics For more detailed conclusions, we focus on the case of normally distributed demand. Due to the central limit To gain more insights, in this section we investigate theorem, normally distributed demand is arguably a rea- comparative statics of price changes on the loss-averse sonable approximation for a wide variety of settings in newsvendor’s optimal order quantity. practice. We see that the likelihood of positive bias is ∗ increasing in shortage cost and decreasing in relative de- Theorem 3. The optimal order quantity Q has the fol- mand uncertainty. The impact of increasing s is not too lowing relationships with price and cost parameters: surprising and is also evident in exponential demand; ∗ as s increases, the potential for loss due to insufficient (i) Q is increasing in s; ∗ ∗ ∗ stock eventually dominates the concern for overage loss (ii) if F(Q) + ( − 1)(F(q2(Q)) − q2(Q) ∗ ∗ ∗ (e.g., when s s ). The role of uncertainty in the di- f(q2(Q)) + (w/p)q1(Q)f (q1(Q))) < 0, then ∗ ∗ rection of bias is perhaps less obvious a priori. Clearly, dQ/dp<0, otherwise dQ/dp 0; ∗ ∗ ∗ uncertainty in demand contributes to costs, and reduc- (iii) if ( − 1)[F(q2(Q)) − q2(Q)f (q2(Q)) + ∗ ∗ ∗ tions in demand uncertainty help improve newsven- F(q1(Q)) + q1(Q)f (q1(Q))]+1 < 0, then ∗ ∗ dor performance. It is interesting that reductions in de- dQ/dw<0, otherwise dQ/dw 0. mand uncertainty may also translate into a favorable effect for the supplier as bias shifts from negative to It is not surprising to see from Theorem 3(i) that positive. the loss-averse newsvendor’s optimal order quantity is increasing in shortage cost. This result also holds in the risk-neutral newsvendor model. However, The- orem 3(ii) and (iii) identify necessary and sufficient 1 Note that z1 is increasing in s . As illustrated in Fig. 5,at conditions under which the optimal order quantity de- CV=0.125 and w ∈{0.05,.010,...,0.95}, s is increasing in CV, so creases in retail price p and increases in wholesale z1 > 0 for CV0.125 and w ∈{0.05,.010,...,0.95}.Ifz1 < 0 for some CV < 0.125 and w ∈{0.05,.010,...,0.95}, then s decreases price w. These results differ from the risk-neutral as CV increases to 0.125, leaving z1 < 0, which is a contradiction. newsvendor model where the optimal order quantity is 100 C.X. Wang, S. Webster/ Omega 37 (2009) 93–105 always decreasing in wholesale price and increasing in recent work of Buzacott et al. [33]. The authors show retail price. that given the mean–variance (MV) criterion, the opti- We next use a few numerical examples based upon mal order quantity may decrease in selling price. Our uniform, normal, and exponential distributions to illus- result in Fig. 7 shows that the property observed by trate the loss-averse newsvendor’s decision bias as retail Buzacott et al. [33] for an MV model extends to the loss price or wholesale price changes (see Figs. 6–8). For aversion framework. uniform distribution, we assume demand X ∈[0,D] The main lesson from the numerical examples in this with a mean D/2=100; for normal, we assume demand and the previous section is that structural results from is a truncated normal random variable with a mean of the risk-neutral newsvendor model do not necessarily = 100 and a standard deviation of ={12.5, 25, 50}; extend to the loss-aversion, and that consideration of and for exponential, we assume the mean demand alternative choice models may help explain some deci- 1/ =100. Since the newsvendor’s optimal order quan- sion biases in real-world newsvendor problems. For ex- tity is qualitatively similar for various loss aversion ample, Fisher and Raman [3] observe that managers of levels ={2, 3, 4, 5}, we only report the results for a skiwear manufacturer, Sports Obermeyer, order sys- = 2. tematically less than the risk-neutral manager’s order From Figs. 6–8, we find that the loss-averse newsven- quantity, a bias that may possibly be attributed to loss dor exhibits different ordering behavior from the risk- aversion. On the other hand, Patsuris [4] reports that de- neutral newsvendor as retail price p or wholesale price spite the bad economy in 2001, many retail chains con- w changes. More specifically, our numerical results tinue to add stores and order more unnecessary supply. show that the loss-averse retailer’s optimal order quan- Bear Stearns retail analyst Brian Tunick explains this tity may decrease as p increases. This result contrasts phenomenon as the fact that managers have sharehold- with the risk-neutral newsvendor whose optimal or- ers in mind: “Despite the downturn, they keep adding der quantity always increases in p and suggests that stores to get their shares awarded growth valuations by this newsvendor decision bias towards changing retail the market. When a company stops growing, [it gets] price exists under the commonly used uniform, normal, a multiple of 7 to 10 instead of something like 15 to and exponential distributions in practice. Our numer- 20” [4]. In this situation, managers of retailer stores ical results also show that the loss-averse retailer’s may perceive a high shortage cost due to the negative optimal order quantity may increase as w increases. impact on stock price if sales are stunted by insuffi- This result also contrasts with the risk-neutral newsven- cient supply. High shortage cost is conducive to posi- dor whose optimal order quantity always decreases tive bias and may help explain the excessive inventory in w. investment. To consider why the newsvendor’s order quantity We caution that loss aversion is one of multiple in Fig. 7 increases in w ∈[0.90, 0.95] when s = 0.1, possible choice models that may help explain biases suppose that wholesale price is currently 0.90 and the in human decision-making. More work on appropriate supplier increases the price to 0.95. The risk neutral choice models in newsvendor settings is warranted. newsvendor observes that the unit overage cost in- For example, Schweizer and Cachon [5] conduct a creases, unit underage cost decreases, and thus reduces series of experiments that require subjects to solve his order quantity. Now imagine the perceived impact newsvendor problems. They find that subjects, de- of this change by the loss-averse newsvendor given that pending on the experimental setting, may consistently the order quantity remains tentatively unchanged. The order more or less than the risk-neutral newsvendor. lower margin means that the probability of overage While this over and under ordering characteristic is loss increases and, as noted earlier, unit overage cost also present in our results, the authors find that the bi- increases and unit underage cost decreases. All these ases in ordering decisions cannot be explained by loss factors favor a reduction in the order quantity. However, aversion. the lower margin also causes the probability of under- age loss to increase, and in this example, the increase 5. Conclusion is drastic relative to the increase in overage probability (i.e., from 7% to 45% versus 6–8%). This effect domi- This paper investigates an alternative choice model nates the loss-averse newsvendor’s perceptions and he (i.e., loss aversion) to describe newsvendor’s decision- increases his order quantity. Similar arguments explain making behavior under demand uncertainty. We use a why the order quantity decreases as retail price increases simple “kinked” piecewise-linear loss aversion utility in Fig. 7. Of particular relevance to this paper is the function to study the single-period newsvendor model C.X. Wang, S. Webster/ Omega 37 (2009) 93–105 101 s = 0.1 s = 0.5 s = 1 s = 0.1 s = 0.5 s = 1 Optimal order quantity at w = 1 Optimal order quantity at p = 1 150 250 200 100 150 100 50 50 0 0 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Retail price p Wholesale price w Fig. 6. A plot of newsvendor’s optimal order quantity as a function of retail price p ranging between 1 and 3 in increments of 0.1, and as a function of wholesale price w ranging between 0.01 and 1 in increments of 0.02. Demand conforms to a uniform distribution with D = 100. Loss aversion level = 3. s = 0.1 s = 0.5 s = 1 s = 0.1 s = 0.5 s = 1 Optimal order quantity at w = 1 Optimal order quantity at p = 1 110 150 100 90 130 80 110 70 90 60 50 70 40 50 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Retail price p Wholesale price w Fig. 7. A plot of newsvendor’s optimal order quantity as a function of retail price p ranging between 1 and 3 in increments of 0.1, and as a function of wholesale price w ranging between 0.1 and 1 in increments of 0.05. Demand conforms to a truncated normal distribution with mean 100 and standard deviation 25. Loss aversion level = 3. s = 0.1 s = 0.5 s = 1 s = 0.1 s = 0.5 s = 1 Optimal order quantity at w = 1 Optimal order quantity at p = 1 80 300 70 250 60 50 200 40 150 30 100 20 10 50 0 0 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Retail price p Wholesale price w Fig. 8. A plot of newsvendor’s optimal order quantity as a function of retail price p ranging between 1 and 3 in increments of 0.1, and as a function of wholesale price w ranging between 0.1 and 1 in increments of 0.05. Demand conforms to an exponential distribution with mean 100. Loss aversion level = 3. and show when, how, and why loss aversion causes the averse newsvendor will order less than a risk-neutral decision bias in the newsvendor problem. More specif- newsvendor and the more loss-averse, the less his opti- ically, we find (1) when shortage cost is low, a loss- mal order quantity, and (2) when shortage cost is high, 102 C.X. Wang, S. Webster/ Omega 37 (2009) 93–105 a loss-averse newsvendor will order more than a risk- the other breakeven quantity does not exist in I and neutral newsvendor and the more loss-averse, the more +(q,Q)>0 for all q ∈ (Q, sup I). his optimal order quantity. We also find that the loss- averse newsvendor’s optimal order quantity may in- Proof of Theorem 1. (i) If q2(Q) sup I, then af- crease in wholesale price and decrease in retail price, ter mapping the retailer’s payoff function (2) into the which can never occur in the risk-neutral newsvendor expected utility function (4), we can rewrite (4) as model. In essence, the presence of the decision bias in follows: the loss-averse newsvendor problem can be explained by the fact that the loss-averse newsvendor is more E[U((X, Q))] sensitive to losses than gains and his marginal util- Q ity of underage and overage loss are not equal, and = (x − wQ)f (x) dx the term that dominates determines the direction of 0 bias. sup I Future research on the appropriateness and effects + [(1 − w)Q − s(x − Q)]f(x)dx of alternative choice models is warranted. In addi- Q tion, future research should consider a supply chain q1(Q) comprised of manufacturer(s) selling to multiple com- + ( − 1) (x − wQ)f (x) dx peting loss-averse retailers. Such investigations may 0 lead to new policies for improving supply chain effi- sup I ciency. Finally, we wish to note that the loss-averse + [(1 − w)Q − s(x − Q)]f(x)dx . (10) nature of the newsvendor might also be captured by q2(Q) making the unit shortage cost an increasing (convex) function of the shortage quantity and the unit salvage After taking the first derivative of (10) with respect to value a decreasing (concave) function of the unsold Q and applying the chain rule, we get: quantity. It would be interesting to see whether or not the newsvendor decision bias is qualitatively different dE[U((X, Q))]/dQ from the loss aversion utility function employed in this =−wF (Q) + (1 + s − w)F (Q) + ( − 1) paper. × − wF(q1(Q)) + (q1(Q) − wQ)f (q1(Q)) Acknowledgments dq1(Q) The authors would like to thank the editor, Ben Lev, × +(1+s−w)F(q2(Q))−[(1−w)Q dQ the associate editor, Ruud Teunter, and three anonymous referees for their valuable feedback. dq (Q) −s(q (Q) − Q)]f(q (Q)) 2 . (11) 2 2 dQ Appendix Since from Lemma 1, q1(Q) = wQ and q2(Q) = (1 + − Proof of Lemma 1. (i) For any q After taking the second derivative of E[U((X, Q))] follows: with respect to Q, we get: dE[U((X, Q))]/dQ 2 [ ] 2 d E U( (X, Q)) /dQ = (1 + s − w)F (Q) − wF (Q) + ( − 1) 2 ×[(1 + s − w)F(q (Q)) − wF(q (Q))], =−(1 + s)f (Q) − ( − 1) w f(q1(Q)) 2 1 (1 + s − w)2f(q (Q)) which is the same as (12). Hence, if q2(Q) > sup I, then + 2 . (13) ∗ s there exists a unique optimal order quantity Q that satisfies the first-order condition (5). − 2 From (12) and (13), and noting s>w 1, we see that Proof of Theorem 2. After substituting Q∗ into [ ] [ ] 1 dE U( (X, 0)) /dQ>0, dE U( (X, sup I)) /dQ<0, dE[U((X, Q))]/dQ,weget and d2E[U((X, Q))]/dQ2 < 0 for all Q ∈ I. Hence, if q2(Q) sup I, then there exists a unique opti- ∗ ∗ dE[U((X, Q ))]/dQ mal order quantity Q that satisfies the first-order 1 = + − ∗ − ∗ + − condition (5). (1 s w)F(Q1) wF(Q1) ( 1) (ii) If q (Q) > sup I, then sup I [(1 − w)Q − s(x − ×[ + − ∗ − ∗ ] 2 q2(Q) (1 s w)F(q2(Q1)) wF(q1(Q1)) . Q)]f(x)dx = 0. Hence, we can rewrite (4) as follows: [ ∗ ] [ ] From (6), dE U( (X, Q1)) /dQ reduces to E U( (X, Q)) Q = (x − wQ)f (x) dx [ ∗ ] dE U( (X, Q1)) /dQ 0 = − [ + − ∗ − ∗ ] sup I ( 1) (1 s w)F(q2(Q1)) wF(q1(Q1)) . + [(1 − w)Q − s(x − Q)]f(x)dx (17) Q q1(Q) + − − + − ∗ ∗ ( 1) (x wQ)f (x) dx. (14) Therefore, if (1 s w)F(q2(Q1))>wF(q1(Q1)), 0 [ ∗ ] [ ] then dE U( (X, Q1)) /dQ>0. Since E U( (X, Q)) is concave,Q∗ Proof of Theorem 3. (i) By the implicit function the- orem, from (5), we get: dQ∗ d2E[U((X, Q∗))]/dQ ds = ds −d2E[U((X, Q∗))]/dQ2 − ∗ ∗ ∗ (p w)q2(Q) ∗ F(Q) + ( − 1) F(q2(Q)) + f(q2(Q)) = s 2 ∗ 2 −d E[U((X, Q))]/dQ > 0. (ii)–(iii) Similarly, we get: w − v F(Q∗) + ( − 1) F(q (Q∗)) − q (Q∗)f (q (Q∗)) + q (Q∗)f (q (Q∗)) 2 2 2 p − v 1 1 Q∗/ p = d d 2 ∗ 2 −d E[U((X, Q))]/dQ and ∗ ∗ ∗ ∗ ∗ ∗ −1 − ( − 1)[F(q2(Q)) − q2(Q)f (q2(Q)) + F(q1(Q)) + q1(Q)f (q1(Q))] Q∗/ w = d d 2 ∗ 2 . −d E[U((X, Q))]/dQ ∗ ∗ ∗ ∗ So if F(Q)+(−1)[F(q2(Q))−q2(Q)f (q2(Q))+ ∗ ∗ [10] Lee HL, Nahmias S. 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Q, from (2), +(q, Q) = (1 + s − w)Q − sq = q = q (Q) = ( + s − w)Q/s 0iff 2 1 . Therefore, we can rewrite (11) as follows: If s (1 − w)Q/(sup I − Q), then q2(Q) sup I. Hence, there exists another breakeven quantity dE[U((X, Q))]/dQ q2(Q) ∈ I. Since +(q, Q) is strictly decreasing in q when q>Q,ifQ