Accepted Manuscript. Procurement strategies with quantity-oriented reference point and loss aversion. Ruopeng Wang, Jinting Wang

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1 Accepted Manuscript Procurement strategies with quantity-oriented reference point and loss aversion Ruopeng Wang, Jinting Wang PII: S (16) DOI: /j.omega Reference: OME 1813 To appear in: Omega Received date: 15 November 2016 Revised date: 10 May 2017 Accepted date: 12 August 2017 Please cite this article as: Ruopeng Wang, Jinting Wang, Procurement strategies with quantity-oriented reference point and loss aversion, Omega (2017), doi: /j.omega This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

2 Highlights To give an explicit form of reference point, which is quantity-oriented in relation to retailer s operational decisions To address how the order quantity varies under the quantity-oriented reference point consideration To show that the optimal order quantity when the retailer is loss averse is strictly less than the risk neutral case under reference point The quantity that the loss aversion under reference point ordered is more than, equal to or less than the classical single-period model To demonstrate some managerial findings which are in contrast to results in classical models 1

3 Procurement strategies with quantity-oriented reference point and loss aversion Ruopeng Wang 1,2, Jinting Wang 1, 1. Department of Mathematics, Beijing Jiaotong University, , Beijing, China; 2. Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, , Beijing, China Abstract This paper considers a single-product, single-period inventory management problem in which the retailer is loss averse with adaptive quantity-oriented reference points. The impact of the loss degree and the quantity-oriented reference point is investigated jointly on the optimal ordering quantity and the profit maximization of the retailer. It shows that there exists a unique optimal order quantity while maximizing the expected utility. The optimal solution can be readily determined via a numerical approach, and it has explicit forms for some special distribution functions. The loss-averse retailer s order quantity is always less than loss-neutral decision maker s when the reference dependency is under consideration. However, the optimal order quantity of the loss-averse system with quantity-oriented reference point is more than, equal to or less than that of the classical system. Furthermore, the structural properties and sensitivity analysis of the optimal solution are addressed. Several important managerial insights are presented by extensive numerical experiments, and finally, some conclusions and future work are given. Keywords: Inventory management, Procurement strategies, Loss-aversion, Reference point. Supported by the National Natural Science Foundation of China (Grant Nos and ). Corresponding author. jtwang@bjtu.edu.cn; Fax:

4 1 Introduction Procurement strategy is crucial to the retailers for revenue management and inventory control. The single period inventory problem, known as the newsboy problem, has been studied extensively due to its significant role in providing a very useful framework for making decision on advanced ordering in the context of practical applications, such as in fashion, daily perishable goods, sporting and various service industries. Recently, there is a growing body of research that considers reference points (i.e. when people frame a problem around an initial wealth) and loss aversion (i.e. when people tend to overweigh losses with respect to comparable gains) which are two essential aspects of prospect theory (PT) on firms pricing strategies, see for example, Heidhues and Koszegi (2005, 2008, 2014), Koszegi and Rabin (2006), Chen et al. (2016) and Baron et al. (2015), among others. In these works, the strategic decision-makers not only aim at the objective of expected profit maximization or expected cost minimization, but also take both the risk implications and reference level into consideration in realistic scenarios. Roughly speaking, the related work on this topic can be categorized into two streams. One stream is to analyze the impact of risk preference on the newsboy problem, see Wang and Webster (2010), Eeckhoudt et al. (1995), Keren and Pliskin (2006). To characterize the decision-maker s behavior such as risk-averse (e.g. Rubio-Herrero et al. (2015), Wu et al (2009), Ray and Jenamani (2015)), the expected utility theory (EUT), the mean-variance analysis and the value-at-risk (VaR), or conditional value-at-risk (CVaR) criteria have been adopted. Another stream is the studies of reference point based on prospect theory, see Long and Nasiry (2014), Nagarajan and Shechter (2014) and references therein. More recently, He and Zhou (2011, 2014) addressed to investigate portfolio choice model featuring a reference point in wealth and probability weighting under prospect theory. And Chen et al. (2016) considered a periodic review stochastic inventory model and analyzed the joint inventory and pricing decisions of a firm with a memory-based reference price. Baron et al. (2015) studied the stochastic reference points that represent consumers beliefs about possible price and product availability and obtained several insights into how consumers loss aversion affects the firm s optimal operational policies. To the best of our knowledge, only a few papers have jointly investigated reference point and loss aversion in the inventory problem, see for example, Arkes et al. (2008, 2010), Arkes (2008, 2010), Shi et al. (2015) and Baron et al. (2015). They adopted a dynamic reference point adaptation approach, which is called stochastic reference points. However, they did not address how to set a 3

5 reference point explicitly although the issue of how to determine the reference point is an important problem in the experimental economics. The existing studies generally assume that the reference point is given over time horizon and is not affected by decision variations. Indeed, most firms are cautious and they set reference points according to order quantity in making procurement. It is crucial to study how to set the quantity-oriented reference point with retailer s loss aversion. In this paper, we aim to study the loss-averse decision-making problem with quantity-oriented reference points from the view of behavioural theory. This is a first attempt to jointly consider these factors and the objective is to maximize its utility under the quantity-oriented reference points by selecting the optimal order quantity. This work is closely related to the pioneer works Arkes et al. (2008, 2010), Shi et al. (2015) and Baron et al. (2015), but with several major differences. Shi et al. (2015) considered a dynamic adaptive reference point which is state-dependent. By choosing suitable updating coefficients, they characterized the firm s optimal inventory and contingent pricing policies. Arkes et al. (2008, 2010) adopted a dynamic reference point which moves in a manner consistent with prior outcomes, shifting upward following a gain and downward following a loss. Baron et al. (2015) captured how consumers loss aversion with endogenized stochastic reference points influences the firms operational decisions. Although those loss-averse models with reference points provide theoretical guidance to managers on their procurement strategies, but the explicit form of the reference point is not addressed in their works, which is a limitation in various applications. Our research is distinct from these works in making a joint consideration of the manager s loss attitude and the updating reference point which is quantity dependent. This is crucial to the decision-maker since the revenue maximization is associated with quantity-oriented reference points and the follow-up optimal order quantity. Besides, we also discuss the impact of the degree of loss attitude and the quantity-oriented reference point on the optimal ordering quantity and the expected utility. Evidently, if the quantity-oriented reference point is considered in the inventory control problem, the structure of ordering quantity may be different from those models without considering. In this paper, we aim to shed light on the following questions: (1) Does the optimal order quantity to the problem under consideration exist? Is it unique? (2) How does the loss-aversion influence the order-quantity of a loss-averse retailer with reference point? (3) How does the reference point affect the optimal decisions, and what is the impact of the retailer s loss-aversion? 4

6 Motivated by these practical questions, this paper investigates the case that the retailer lossaversion with quantity-dependent reference point and the aim is to optimize the expected utility with stochastic demand. Several important managerial insights are derived and presented in this paper. The effect of updating reference point on the loss-averse retailer s decisions is studied, and it is proved that there is a unique optimal strategy. We use a simple reference payoff for procurement, which is a convex combination of the maximal possible payoff and the maximal potential loss. The maximal payoff is calculated as the profit of selling all products and the maximal loss means that none of the inventory can be sold. The contributions of our paper can be summarized as follows. We are the first to give an explicit form of the quantity-oriented reference point. The quantityoriented reference point reflects the decision-maker s individual sensitive behavior on order quantity. This is meaningful for the retailer to decision-making. By exploiting quantityoriented reference point, we identify the impact of loss-aversion on the decision strategy. We address how the optimal order quantity varies with the change of the quantity-oriented reference point. We show that the optimal order quantity when the retailer is loss averse is strictly less than the case when the retailer is loss neutral under reference point. However, the optimal order quantity of the loss-averse system with quantity-oriented reference point is more than, equal to or less than that of the classical system. We demonstrate some findings that are in contrast to the known results in classical models. The optimal procure quantity is decreasing in wholesale price w, which is the same with the classical risk-neutral model. But the incremental rate varies according to the degree of the loss-aversion of the retailer. We quantify that the decision bias exists in the loss-averse system with quantity-oriented reference point and show how the loss-averse decision bias interacts with the system conditions. Since the loss-averse retailer s order quantity is different from the profit-maximizing quantity, the total system performance is suboptimal. Hence, our research findings also shed insights on how loss-aversion contributes to supply chain and what policies to mitigate these effect. Intuitively, the manager s optimal procurement quantity should increase in salvage cost when there is no stockout cost. However, this is not true in our model. The interpretation is that, a higher salvage price v induces a larger reference level, which in turn leads to lower the order quantity. 5

7 Our findings suggest that it is critical to take into account the reference point when the retailer procures. These findings make a meaningful complement to current literature on procurement strategies with adaptive reference points. The remainder of the paper is organized as follows. In Section 2, we provide a brief review of the related research. Section 3 outlines our assumptions and notations, then establishes the model setting. In Section 4, we analyze the solution of the optimal order quantity policy under different cases. Section 5 analyzes the influence of salvage valuation, selling price whole cost and the degree of loss aversion on the optimal order quantity. Numerical results are represented in Section 6. Finally, concluding remarks and future directions are given in Section 7. 2 Literature review There are three streams of literature that are closely related to our work: inventory management, risk management and reference-dependence. In this section, we briefly review the previous literature in these areas and compare our research with these existing works. The first stream is on inventory management, which started at the beginning 20th centenary when manufacturing industries grew rapidly. As far as we know, the first paper on mathematical models of inventory management was Harris (1913) (reprinted in 1991). Since then, the management of the inventories by the firms or retailer of single product with stochastic demand over a single period setting has been extensively studied in the literature. This problem is also known as the newsboy or newsvendor problem (see Silver et al. (1998) and Zipkin (2000)). The standard newsboy model is based on risk-neutrality and the objective is to maximize the expected profit by finding an appropriate balance between expected income and expected costs (see Porteus (2002) and Whitin (1955)). Many variants of the basic model have been proposed throughout the years, addressing additional complex and realistic situations to the problem. For example, Petruzzi and Dada (1999) presented a single-period approach for additive and multiplicative demand that optimized expected profit. They indicated how to select stocking policy based on the effect of price in the stochastic demand and derived a closed, analytical solution to determine the optimal value. Federgruen and Heching (1999) considered a multi-period model with backlogging and price-dependent stochastic demand and analyzed the optimal decision. The interested readers are referred to the thorough and excellent literature reviews on the single-period inventory problem which are presented by Khouja (1999), Qin et al. (2011), and Petruzzi and Dada (1999). 6

8 The second stream of literature related to our work is on risk management, which includes three research approaches: the expected theory (EUT), mean-variance (MV) and value-at risk (VaR) or conditional value-at-risk (CVaR). For the frame work of the expected utility, Schweitzer and Cachon (2010) studied a loss-averse newsvendor problem and showed that a loss-averse newsvendor will order strictly less than a risk-neutral newsvendor, but they did not consider the shortage cost. Wang and Webster (2010) used loss aversion to model retailer s decision making behavior in the single-period newsvendor problem and found that if shortage cost is not negligible, then a lossaverse newsvendor may order more than a risk-neutral one. The results showed that the risk-verse newsvendor behaves differentially from the risk-neutral counterpart. Keren and Pliskin (2006) presented how risk aversion impacts a newsvendor who faces uniform demand. Eeckhoudt et al. (1995) demonstrated that the optimal order quantity of a risk-averse newsvendor will be less than a risk-neutral newsvendor, and the optimal order quantity is decreasing in risk aversion coefficient. And Agrawal et al. (2000) addressed the optimal order quantity decreases with increasing risk aversion. Kazaz and Webster (2015) analyzed the impact of supply uncertainty and risk aversion on the price-setting newsvendor problems. They showed a risk-averse newsvendor will order less than a risk-neutral newsvendor when demand is uncertain, whereas the opposite is generally the case when supply is uncertain. Wang et al. (2009) offered three difference classes of utility function that studied the newsvendor problem within the expected utility theory. Although a concave and increasing utility function could reflect decision-maker s risk-averse behaviour, and also could represent the degree of the diminishing utility in wealth, the risk attitude and diminishing wealth were neglected. For the mean-variance measure, Lau (1980) was the first to exploit a mean-variance payoff criterion within the newsvendor model. After that, Chen and Federgruen (2000) presented the mean-variance analysis for a single-period inventory problem that optimized a utility function for risk-neutral, risk-averse and risk-seeking decision maker. Wu et al (2009) focused on the impact of stock costs of a mean-variance criterion and showed that the risk averse newsvendor may order less than the risk-neutral newsvendor if the stockout cost is positive, which will never happen if the stockout cost is zero. Rubio-Herrero et al. (2015) investigated the mean-variance model with a risk parameter. They identified conditions for the concavity of this risk-sensitive performance measure and the uniqueness of the optimal solution under the additive demand model. Meanvariance measure is to address the trade-off between the expected return (mean) and the variation of return (variance). However, it suffers an inherent theoretical flaw in which both upside and 7

9 downside variations from the mean are seen as risk. Furthermore, variance is only suitable for the case where the outcome distribution is close to an asymmetric distribution. Chiu and Choi (2013) reviewed the literature on the mean-variance analytical models. For the work of VaR (or CVaR) criteria, Ozler, Tan and Karaesmen (2009) utilized VaR as the risk measure and investigated one-stage, multiproduct system with VaR measure. Gotoh and Takano (2007) studied the risk-averse newsvendor problem under the CVaR criterion. They claimed that downside risk measures including CVaR are tractable due to the convexity property. Chen et al. (2009) focused on a CVaR criterion and investigated how the optimal price and stock quantity changed when varying the η quantile of the objective function. Later, Choi et al. (2011) showed that CVaR actually represented a trade-off between the expected profit and a certain risk measure. Recently, some studies use the CVaR to model the risk-averse newsvendor, see for example, Wu et al. (2014), Dai and Meng (2015) and Xu et al. (2015). The third stream of literature which is closely related to our paper is the reference dependent. People often develop their own gains and losses of a fair assets rather than their final net assets, which is referred to as reference dependent, and therefore make choice in terms of the reference point. The reference-dependence was developed by Kahneman and Tversky s prospect theory (PT) (1979) and have had huge impacts on research in many fields. Long and Nasiry (2014) and Nagarajan and Shechter (2014) introduced prospect theory into the study of the decision bias in the newsboy problem. Their aim is to explain the ordering behavior observed in experiments on the newsvendor problem and got some interesting results. Shi et al. (2015) adopted a dynamic updating reference point. They derived the semi-analytical solution of the model. There have been studies that integrated reference price effects into inventory models, see, for example, Chen et al. (2016), Baron et al. (2015) and reference therein. 3 Model description and preliminaries We consider a single-product, single-period inventory problem with random demand and lossaverse retailer. We adopt the commonly used reference-dependence to model the retailer s lossaverse behaviour. At the beginning of the selling season, the loss-averse retailer (newsvendor) inspects the initial inventory level and then makes an order quantity decision. The random demand D is a positive variable with a known distribution function F ( ) and a density function f( ). It is assumed that there is no capacity restrictions on the purchase quantity. There is no lead time, 8

10 which means any order purchased by the retailer at the beginning of the selling season will be fulfilled immediately. All unsold products will be salvaged at price v and shortage cost is not considered. Since our purpose is to get insights into the impact of the retailer s loss-aversion on the optimal ordering policies, we only consider a simple case where the initial inventory level is zeros. Note that the case of positive initial inventory level can be analyzed in a similar way. Moreover, we suppose that there is a fixed selling price p, a fixed purchase cost w. Table 1 summarizes parameters and variables used throughout the paper; other notations will be defined as needed. Notation w p v D F (x) G(x) Q Π R ( ) U( ) E( ) α β Q Q 0 Q α Description Unit purchase cost price Table 1: Parameters and variables Unit selling price (retail price) of the end product Unit salvage cost, v may be positive or negative, in which represents disposal fee or holding cost Nonnegative and independent random variable, with a cumulative distribution function (CDF) F(D) Cumulative distribution function, characterizing the demand, and tail distribution is F (x)] = 1 F (x) Partial expectation of the random variable D, which is defined as G(x) = x 0 Df(D)dD Decision variable denoting quantity ordered by the retailer Operating profit of the retailer Utility function of the retailer Expectation operator Parameter (α 1) that reflects the degree of loss aversion for the retailer Parameter (0 β 1) that decides reference level Optimal decision for the classical single-period inventory problem Optimal order quantity for the loss-neutral retailer under the quantity-oriented reference point Optimal decision for the loss-averse retailer under the quantity-oriented reference point To make sense of our model, we specify the following assumptions: Assumptions (A1) The cumulative distribution function (CDF) F (x) is continuous, differentiable, invertible, and strictly increasing over I. (A2) p > w > v. Assumption A1 is a general assumption in the classical single-period inventory problem for simplicity of calculation, Assumption A2 ensures the marginal profit of the retailer to be nonnegative. For completeness, we now review the classical single-period inventory problem. Assume that the retailer s order quantity is Q, then quantity sold is min{q, D}, and thus the unsold quantity is (Q min{q, D}) +. The total profit of the classical single-period inventory problem is expressed 9

11 by Π R (Q) = p min{q, D} + v (Q min{q, D}) + w Q, (3.1) where the first two terms jointly represent the total revenue and the last term indicates the total cost. Throughout this paper, we use the following notation: x + = max{0, x} and x = min{0, x}. The expected profit is therefore given by max Q 0 E[Π R(Q)] = (p v)h(q) + (p w)q. (3.2) The optimal order quantity Q that maximizes the expected profit (3.2) satisfies the following equation: (p w) (p v)f (Q ) = 0. (3.3) Alternatively, we can express Q in a close-form: Q = F 1 (φ), where φ = p w the critical fractile (see for example, Porteus (2002) and Zipkin (2000)). p v [0, 1] is called In this paper, we consider that the retailer is loss-averse with quantity-oriented reference point. From (3.1), we can conclude that for any given Q, the profit function is increasing in D between 0 and Q, and then is a constant in D when D is larger than Q. Thus, the maximum profit is (p w)q. We assume that the retailer s reference level, denoted as V, is a convex combination of his expected maximal profit and maximal potential loss defined as follows V = β(p w)q + (1 β)(v w)q, (3.4) where the parameter 0 β 1 measures the reference level of the retailer. As β increases, the retailer places a larger weight on the profit and a smaller weight on loss, which represents the information about the attitude towards profit and loss. Note that this adaptive formulation has been widely adopted to form references and make decisions in the psychology and marketing literature as well as operations management literature, see, for example, Arrow and Hurwicz (1977) and Popescu and Wu (2007). Remark 3.1 Koszegi and Rabin (2006) suggested that, in an uncertainty environment, the reference point is determined in a personal equilibrium by decision maker s rational expectations, which depend on his actions. In our model, the reference point is the convex combination of maximal possible profit and maximal possible loss. An interpretation of our reference point is that the retailer forms an expectation about unknown payoff that depends on the order quantity. Since the 10

12 reference point presented is a function of the order quantity, we name it as the quantity-oriented reference point. Remark 3.2 When the shortage cost is considered, the retailer s reference point is more complicated. For example, when taking the shortage cost b into consideration (where b is the penalty cost for unit unsatisfied demand), then the maximal possible profit is (p w)q and the maximal possible loss is min{(v w)q, (p w + b)q bd} (where we might assume that demand D [0, D]). The complexity is increased largely for choosing quantity-oriented reference point. Thus, the conclusion may depart from our results. In this paper, the retailer will define gains and losses with respect to the chosen reference point rather than the actual profit. We address that a retailer s overall utility consists of two components: one is intrinsic utility, the other one is gain-loss utility. The former reflects the actual expected benefit, and the later represents the effects of the reference points on the decision-maker. This idea is motivated by Koszegi and Rabin (2006) and Yang, Guo and Wang (2014), where they adopted reference-dependent utility for a customer who decides to join a service facility in queueing system. Thus the overall utility for the retailer is U(Q) = E[Π R (Q)] + E[(Π R (Q) V ) + ] + E[α(Π R (Q) V ) ], (3.5) where the first term E[Π R (Q)] is the mean of profit given by Eq. (3.2), the second term E[(Π R (Q) V ) + ] is the expected gain and the third term E[α(Π R (Q) V ) ] is the expected loss multiplied by the loss-averse parameter under the reference point. The parameter α measures the degree of the retailer loss aversion. We assume that α 1, where α = 1 corresponds to the loss neutral scenario, and α > 1 indicates that the decision maker is loss averse, i.e., he cares more about a loss than an equally sized gain. Remark 3.3 If the intrinsic utility exceeds the setting reference profit, we conclude that the decision maker makes a profit. Otherwise, he has a loss. Since the decision maker is assumed to be loss averse, the loss coefficient α should be under consideration. Substituting (3.4) into the overall utility expression (3.5) yields (see Appendix A) U(Q) = [(2 β)(p v) (w v)] Q + (p v) [2H(Q) + (α 1)H(βQ)]. (3.6) 11

13 In sharp contrast to the classical studies, there are at least two differences and attributions: on the one hand, when the retailer measures the gain and the loss, we consider that the decision maker has his own reference point, and when the profit is over the reference point, he think that he obtains the revenue. Otherwise, he is lost. The overall utility includes not only the expected profit, but also the expected gain and loss. On the other hand, the utility function measures his loss-aversion attitude depending on reference level, which is quantity-oriented. The individual adopts quantity-oriented reference point which can represent his sensitive behavior to order quantity. 4 Performance measure and the optimal decisions In this section, we investigate the optimal ordering quantity under quantity-oriented reference point. The obtained results will be compare with the loss-neutral ones and the classical inventory model. Under the quantity-oriented reference point, loss averse retailer s task is to find out the optimal order quantity to maximize his expected utility, i.e., max U(Q) = 2 [(p v)h(q) + (p w)q] + [(w v) β(p v)] Q + (α 1)(p v)h(βq). Q A comparison of (4.7) and (3.2) shows that three factors of behavior in the retailer s decisionmaking. The first factor, which is reflected in the first term of (4.7), is similar to the classical model. The second factor, is affected by reference level. The third factor, which is expressed in the third term of (4.7), is due to loss aversion. The last two terms reflect double marginalization, e.g., it may be positive or negative. If α = 1, then the retailer is loss neutral and the last term in (4.7) drops out. To maximize the utility function of the retailer, we first provide the structural property of function U(Q) at the following Lemma. Lemma 4.1 Assume a function g(x) is differentiable for x R. For any x 1 < x 2, if g (x 1 ) > g (x 2 ), then g(x) is strictly concave. Proof: Let x 1 < x 2 and 0 < λ < 1. Obviously, x 1 < λx 1 + (1 λ)x 2 and x 2 > λx 1 + (1 λ)x 2. According to Lagrange s mean value theorem, there exists ξ (x 1, λx 1 + (1 λ)x 2 ) such that (4.7) g(λx 1 + (1 λ)x 2 ) g(λx 1 ) = (1 λ)(x 2 x 1 )g (ξ). (4.8) Likewise, there is a ζ (λx 1 + (1 λ)x 2, x 2 ) such that g(λx 2 ) g(λx 1 + (1 λ)x 2 ) = λ(x 2 x 1 )g (ζ). (4.9) 12

14 The difference of the equation (4.8) and (4.9) is given by λ [g(λx 1 ) + (1 λ)x 1 ] (1 λ) [g(λx 2 ) g(λx 1 + (1 λ)x 2 )] = λ(1 λ)(x 2 x 1 )(g (ξ) g (ζ)). (4.10) Apparently, if ξ < ζ, then g (ξ) > g (ζ). As a result, g(λx 1 + (1 λ)x 2 ) > λg(x 1 ) + (1 λ)g(x 2 ). (4.11) That is, g(x) is strictly concave. Theorem 4.2 For any α 1, the retailer s expected utility function U(Q) is differentiable and strictly concave in Q. Moreover, U(Q) is asymptotically linear with a slop αβ(p v) (w v) < 0. Proof: Firstly, we prove the differentiability and concavity of U(Q). For any demand D, since H(x) = G(x) xf (x), we have dh(q) = Qf(Q) F (Q) Qf(Q), dq = F (Q), (4.12) and dh(βq) dq = β 2 Qf(βQ) βf (βq) βqf(βq)β = βf (βq). (4.13) Substituting (4.12) and (4.13) into the expression of U(Q) in (3.6) and after some algebraic manipulations, we get the first and second derivatives of (3.6) with respect to Q: du(q) dq = (2 β)(p v) (w v) 2(p v)f (Q) β(α 1)(p v)f (βq) = 2 [(p w) (p v)f (Q)] + β(α 1)(p v)f (βq) +(w v) αβ(p v), (4.14) where F (βq) = 1 F (βq). Hence, U(Q) is differentiable for Q R +. For any Q 1 < Q 2, we have F (Q 1 ) < F (Q 2 ) and F (βq 1 ) < F (βq 2 ) due to cumulative distribution function F ( ) is nondecreasing. Also, we can obtain du(q) dq means that U(Q) is differentiable and strictly concave in Q. 13 Q=Q 1 > du(q) dq Q=Q 2, which

15 Furthermore, we have [ (p v)(2h(q) + (α 1)H(βQ) = lim Q + Q U(Q) lim Q + Q ] + 2(p w) + (w v) β(p v) = [(p v)( 2 (α 1)β) + 2(p w) + (w v) β(p v)] = αβ(p v) (w v). (4.15) This completes the proof. Remark 4.3 The concavity of the expected utility function guarantees the existence of an optimal order quantity. Theorem 4.2 shows that there exists an optimal inventory decision Q α to control loss. The differentiability and concavity of U(Q) makes the differential optimization method available to solve the problem. Remark 4.4 The asymptotically linear property indicates that one unit of over-ordering procurement will result in a unit loss αβ(p v) (w v) when the ordering quantity is large enough. In what follows, we will characterize the uniqueness of the optimal order quantity. Theorem 4.5 For any α 1, there exists a unique optimal order quantity Q α that maximizes the retailer s expected utility, which satisfies the following first-order condition 2 [(p w) (p v)f (Q α)] + β(p v) [ (α 1)F (βq α) α ] + (w v) = 0. (4.16) Proof: According to Theorem 4.2, U(Q) is strictly concave, then the optimal procure quantity Q α is unique. From the first-order optimal condition we obtain that Q α is a finite stationary point of du(q) dq in (4.14). Setting du(q) = 0 and we can obtain that the optimal ordering quantity Q α, which satisfies Eq. dq (4.16). Remark 4.6 From Theorem 4.5, we notice that the general close-form expression of Q α is not available, but it has computational advantage. Eq. (4.16) shows that we only need to compute x 0 f(d)dd, which can be easily determined via numerical method. It should be noted that it has explicit forms for some distributions (e.g. uniformly distributed demand). Remark 4.7 From Eq. (4.16), we observe that the optimal solution with quantity-oriented reference point in the expected utility function is more complicated than that of the system without reference point. Theorem 4.5 further indicates that we can obtain the optimal order quantity by a one-dimensional search algorithm when the close-form solution is unavailable. 14

16 If the retailer is loss-neutral with quantity-oriented reference point, i.e. α = 1, then the optimal order quantity Q 0 satisfying the following equation: 2 [(p w) (p v)f (Q 0)] + (w v) β(p v) = 0. (4.17) Theorem 4.8 For any α > 1, Q 0 > Q α. Proof: After substituting Q 0 in (4.17) into (4.14), we have du(q) dq Q=Q = β(α 1)(p v)f (βq 0 0) < 0. (4.18) Since U(Q) is strictly concave in Q and du(q) dq Q=Q = 0, we get α Q 0 > Q α. Remark 4.9 Schweitzer and Cachon (2010) and Eeckhoudt et al. (1995) derived the optimal order quantity for a risk-averse newsvendor, which is smaller than that of a risk-neutral one under the EUT. And Wu et al (2009) focused on a mean-variance criterion and showed that the risk-averse newsvendor may order less than the risk-neutral newsvendor if the stockout cost is positive. Theorem 4.8 shows that a loss-neutral decision maker often order more than the loss-averse ones under the quantity-oriented reference point, which is similar to the existing works (e.g., Schweitzer and Cachon (2010), Eeckhoudt et al. (1995), Wu et al (2009) and reference therein). Remark 4.10 Since the optimal ordering quantity of loss-neutral retailer under quantity-oriented reference point can be obtain explicitly expressed by (4.17), we may use Q 0 as an initial solution for the search algorithm. The following theorem shows the difference between the loss-neutral retailer under reference point consideration and the classical single-period inventory problem. Theorem 4.11 For a loss-neutral retailer (with reference point) and the classical retailer (without reference point), there exists a threshold β = w v w v. If β >, then p v p v Q > Q 0; if β < w v, then p v Q < Q 0. Proof: Rewriting (3.3), we obtain 2 [(p w) (p v)f (Q )] = 0. (4.19) The difference of (4.17) and (4.19) is given by 2(p v) [F (Q 0 F (Q )] = β(p v) (w v). (4.20) 15

17 Note that p v > 0, therefore, if β > w v p v, we have F (Q 0) < F (Q ). otherwise, if β < w v p v, we have F (Q 0) > F (Q ). Since F ( ) is a nondecreasing function, we have completed the proof. Remark 4.12 For a neutral retailer, Theorem 4.11 indicates that the optimal order quantity under reference point (loss-neutral retailer) can be more than or less than the results without reference point (risk-neutral retailer), which is influenced by the reference level of the retailer. The threshold w v p v can be found in (3.4). Our next theorem shows that the decision bias exists in a loss-aversion system, i.e., the difference of the optimal order quantity between the classical system and loss-averse system. Theorem 4.13 For any α > 1, there exists a unique β (0, w v ), which satisfies the following equation: β(p v) [(α 1)F (βq ) + 1] = w v. (4.21) If β (0, β ), then Q α > Q ; if β (β, 1), then Q α < Q ; and for β = w v, we have Q = Q 0 = Q α. Proof: We divide our analysis into two parts, i.e. β > w v w v and β < Case I. β > w v p v. Recalling from Theorem 4.8 and Theorem 4.11, we can obtain that for β ( w v, 1), then Q α < Q. Case II. β < w v p v. After substituting Q in (4.19) into (4.14), we have p v p v p v. du(q) dq Q=Q = β(α 1)(p v)f (βq ) + (w v) β(p v). (4.22) Let h(β) = β(α 1)(p v)f (βq ) + (w v) β(p v), then we have dh(β) dβ = (α 1)(p v)f (βq ) β(α 1)(p v)q f(βq ) (p v) < 0. p v p v (4.23) This indicates that h(β) is strictly decreasing in β. We consider the function h(β) in [0, w v ] and have p v h(0) = w v > 0, (4.24) h( w v v ) = (α 1)(w v)f (w p v p v Q ) < 0. (4.25) 16

18 Hence, there must exist a unique β (0, w v p v ), if β (0, β ), then h(β) > 0; if β (β, w v p v ), then h(β) < 0. Thus, if β (0, β ), we have du(q) dq Q=Q > 0. Since U(Q) is concave in Q, we get Q < Q α. Similarly, if β (β, w v du(q) ), p v dq Q=Q < 0, which in turn implies Q > Q α. Therefore, there exits a unique β Q α < Q. (0, w v p v ), if β (0, β ), Q α > Q ; if β (β, 1), For β = w v, it is easy to obtained that p v Q α = Q based on Theorem Theorem 4.13 provides a sufficient condition under which the loss-averse retailer under quantityoriented reference point will order more than (i.e., positive bias), equal to, or less than (i.e., negative bias) the classical single-period problem. More specifically, it shows that there exists a threshold β : (1) for small β, the loss-aversion system will exert more order quantity than the classical system; (2) for large β, the loss-averse system will exert less order quantity than the classical system; p v and (3) for a fixed β, i.e., β = w v, then the loss-aversion system will exert the same order quantity as the classical system. The above results are different from those of in a classical inventory problem, we refer to Schweitzer and Cachon (2010) and Eeckhoudt et al. (1995). Schweitzer and Cachon (2010) analyzed the case of zero shortage cost without reference point and found that the loss-averse newsvendor will always order less than the risk-neutral one. Eeckhoudt et al. (1995) derived that under natural assumption (0 < v < w < p), the optimal order quantity for a loss-aversion newsvendor is less than that of a risk-neutral one. 5 Sensitivity analysis In this section, we provide a complete sensitivity analysis of parameters, i.e., impact of unit wholesale price w, salvage cost v, unit selling price p and loss-averse degree α on the optimal order quantity. At first, we examine impact of the unit wholesale price on the optimal decision. Theorem 5.1 For any α 1, the retailer s optimal procure quantity Q α is strictly decreasing in w. Proof: By the implicit function theorem, from (4.14), we get that dq α dw = d2 U(Q)/dQdw d 2 U(Q)/dQ 2 = 1 d 2 U(Q)/dQ 2 < 0, (5.26) 17

19 by the virtue of U(Q) is strictly concave in Q and F (βq α) > 0. Theorem 5.1 shows that the optimal quantity is strictly decreasing in w. Intuitively, as the unit procurement cost increases, the marginal revenue decreases. Consequently, the optimal procure quantity should decrease. Theorem 5.2 For any α 1, the retailer s optimal procure quantity Q α is strictly decreasing in α. Proof: By the implicit function theorem, from (4.14), we get dq α dα = d2 U(Q)/dQdα d 2 U(Q)/dQ 2 = β(p v)f (βq α) d 2 U(Q)/dQ 2 < 0, (5.27) the last inequality satisfies based on the fact that U(Q) is strictly concave in Q and F (βq α) > 0. Theorem 5.2 confirms the results with respect to Eeckhoudt et al. (1995) and Agrawal et al. (2000) where they adopted the Expected Utility Theorem (EUT) to model risk-aversion. That is, the more loss-averse the decision-maker is, the less quantity he would procure, even if in the presence of the quantity-oriented reference level. Theorem 5.3 For any α > 1, if F (Q α) increasing in selling price p; if F (βq α) decreasing in selling price p. Proof: By the implicit function theorem, from (4.14), we get 2 β dq α dp = d2 U(Q)/dQdp d 2 U(Q)/dQ 2 2 β 2+β(α 1), then the optimal procure quantity Q α is 2 β 2+β(α 1), then the optimal procure quantity Q α is = 2 β 2F (Q α) β(α 1)F (βq α)) d 2 U(Q)/dQ 2. (5.28) If F (Q α) <, since the CDF F ( ) is nondecreasing, we have F 2+β(α 1) (βq α) < 2 β 2+β(α 1). Thus, 2 β 2F (Q α) β(α 1)F (βq α)) 2 β > 2 β β(α 1) β(α 1) 2 β 2 + β(α 1) 2 + β(α 1) 2 β(α 1) = (2 β) 2 + β(α 1) = 0. (5.29) Since U(Q) is concave in Q, we get dq α dp > 0. 18

20 If F (βq α) > 2 β 2+β(α 1), which also implies F (Q α) > to derivation of (5.29), which in turn obtains dq α dp < 0. Theorem 5.4 For any α > 1, if F (Q α) decreasing in v; if F (βq α) 2 β. The rest of the proof is similar 2+β(α 1) 1 β 2+β(α 1), then the optimal procure quantity Q α is 1 β 2+β(α 1), then the optimal procure quantity Q α is increasing in v. Proof: By the implicit function theorem, from (4.14), we get that 1 β dq α dv = d2 U(Q)/dQdv d 2 U(Q)/dQ 2 = β 1 + 2F (βq α) + β(α 1)F (βq α)) d 2 U(Q)/dQ 2. (5.30) If F (Q α) <, since the CDF F ( ) is nondecreasing, we have F 2+β(α 1) (βq α) < Since U(Q) is concave in Q, we get dq α Theorem 5.3, and therefore is omitted here. β 1 + 2F (βq α) + β(α 1)F (βq α)) 1 β < β β(α 1) + β(α 1) 1 β 2 + β(α 1) 2 β(α 1) β(α 1) = (1 β) 2 + β(α 1) = 0. dp 1 β 2+β(α 1). Thus, (5.31) < 0. The rest of the proof is similar to the proof of Theorem 5.3 and Theorem 5.4 provide the sensitivity the optimal order quantity with the system parameters under loss-averse retailer with quantity-oriented reference point. Theorem 5.3 shows that the order quantity under loss-aversion is not always increasing in market price p, and Theorem 5.4 shows that the order quantity under loss-aversion is not always decreasing in v. These results differ from the classical system in which the optimal order quantity is always increasing in market price p and decreasing in v. On the other hand, Theorem 5.3 and Theorem 5.4 provide a sufficient condition under which the order quantity is increasing or decreasing in p and v. 6 Numerical example In this section, we proceed with serval numerical examples based on uniform distribution and exponential distribution to illustrate the loss-averse retailer s decision, such as the selling price, the procurement cost and the salvage cost changes. Further, we perform sensitivity analysis of parameters and illustrate the impact of parameters on the optimal order quantity. The parameter 19

21 are set as follows: p = 5, w = 2, v = 1, β = 0.5. In order to facilitate our analysis, we assume α = 1, 1.5, 2, respectively. For uniform distribution, we assume demand D [D b, D + b]; for exponential distribution, we assume the mean demand E[D] = 50. We report the results for α = 1, 1.5, 2. We conduct eight numerical experiments to illustrate the impact of parameters on the optimal order-quantity. We vary one of the four variables and keep the other three variables unchanged. For example, we keep the selling price p, the salvage cost v and β fixed, but change the procurement cost w from 1 to 3 in increment of 0.1 for α = 1, 1.5, 2 to investigate how it affects the optimal procurement decision, which is shown in Fig. 1. Figs. 1-4, part (a) is uniformly distributed demand and part (b) is exponentially distributed demand optimal order quantity at p=5,v=1 classical problem α=1 α=1.5 α= Purchase cost w (a) Purchase cost w optimal order quantity at p=5,v=1 (b) classical problem α=1 α=1.5 α=2 Figure 1: Optimal order quantity Q α vs wholesale price w for α = 1, 1.5, 2, where (a) for uniformly distributed demand and (b) for exponentially distributed demand. It can be seen in Figs. 1-4 that how the parameters affect the optimal order quantity Q α. As a whole, the more loss averse, the less optimal order quantity, which is shown that the curve of a larger α is under the curve of a smaller in in Figs But compared with the classical model, our model has not such a monotonic property with respect to p and v, which is caused by the quantity-oriented reference point. Fig. 1 indicates that the optimal order quantity is a strictly decreasing function of w for both loss-neutral and loss-averse scenarios. Fig. 2 and Fig. 3 report that, although the optimal order quantity is usually increasing in selling price p and salvage cost v, it is not always the case. Theorem 5.3 and Theorem 5.4 indicate the opposite case. The impact of β on the retailer s optimal 20

22 75 70 classical problem α=1 α=1.5 α=2 optimal order quantity at w=2,v= classical problem α=1 α=1.5 α=2 optimal order quantity at w=2,v= Selling price p (a) Selling price p Figure 2: Optimal order quantity Q α vs market price p for α = 1, 1.5, 2, where (a) for uniformly distributed demand and (b) for exponentially distributed demand optimal order quantity at p=5 and w=2 classical problem α=1 α=1.5 α= salvage cost v (b) optimal order quantity at p=5 and w=2 classical problem α=1 α=1.5 α= salvage cost v (a) (b) Figure 3: Optimal order quantity Q α vs salvage cost v for α = 1, 1.5, 2, where (a) for uniformly distributed demand and (b) for exponentially distributed demand. 21

23 90 optimal order quantity at p=5,w=2,and v=1 110 optimal order quantity at p=5,w=2 and v= classical problem 30 classical problem 30 α=1 α=1 α= α=1.5 α=2 α= Parater β Parater β (a) Figure 4: Optimal order quantity Q α vs β for α = 1, 1.5, 2, where (a) for uniformly distributed demand and (b) for exponentially distributed demand. order quantity is complex in an analytic technique, but Fig. 4 shows that the larger reference level the retailer depends, the smaller he orders. Since the retailer does not consider the impact of the shortage cost, he may make profit at the expense of lowering the order quantity. In order to further show the effects that loss aversion and reference level on the optimal decision and the expected utility, Tab. 2 displays the impact of system parameters on the optimal order decisions for the different reference level β and loss-averse parameter α. We also adopt the uniformly distribution demand and the exponential distributed demand to analyze the optimum performance measure of the objective function. The first column (β) of Tab. 2 is the retailer s reference level; the second column is optimal order quantity (Q ) and utility value (U ) under quantity-oriented reference point; and the other columns are the corresponding value under different degree of lossaversion and demand distributions. Theorem 4.8 indicates that Q α < Q 0, which presents a threshold of Q α. Moreover, Q 0 is given explicitly by (4.17) when the demand distribution is known. Thus, a interval [0, Q 0] which the optimal order quantity lies is obtained. More precisely, we conclude that the optimal order quantity is in the interval [Q, Q 0] for β (0, β ) (based on Theorem 4.13); otherwise, the optimal order quantity lies the interval [0, Q 0]. Hence, we may apply bisection method to solve (4.16). Consequently, the utility can be calculated directly from (3.5). From the numerical results showed in Tab. 2, we further conclude that (1) The optimal order quantity Q α under the quantity-oriented reference point decreases as the (b) 22

24 Table 2: Numerical Results for the uniformly distributed and exponentially distributed demand uniformly distributed demand Exponential distributed demand β Optimal value α = 1 α = 1.5 α = 2 α = 1 α = 1.5 α = 2 β = 0 Q U β = 0.1 Q U β = 0.2 Q U β = 0.3 Q U β = 0.4 Q U β = 0.5 Q U β = 0.6 Q U β = 0.7 Q U β = 0.8 Q U β = 0.9 Q U β = 1 Q U level of loss-aversion parameter α increases except for β = 0. (2) The optimal order quantity Q α under the quantity-oriented reference point decreases as the reference level β increases. (3) The optimal utility value Uα with respect to the quantity-oriented reference point has the same analytic property with the optimal order quantity Q α. 7 Conclusions and future work In this paper, we studied a single-product, single-period inventory problem in which the decision maker is loss-averse and the reference point is quantity-oriented. Our formalization of quantityoriented reference point is distinct from the existing literature on reference point where it be a fixed value or a random variable. The quantity-oriented reference point in our model is a function of the 23

25 order quantity which reflects the decision maker s reference level on maximal possible profit and maximal possible loss. By considering a loss-averse retailer and the quantity-oriented reference point jointly, we characterized the structural properties of the retailer s optimal procurement strategies. Specifically, there exists a unique optimal order quantity for any given reference level and loss-averse degree. The optimal order quantity for the loss-averse retailer with quantity-oriented reference point is less than that of the loss-neutral problem. When comes to the loss-neutral system with reference point and the classical system, the optimal order quantity also affects by reference point. More precisely, if the reference level of the retailer β is larger (e.g. β > w v ), the optimal order quantity of loss-neutral system is greater than that of the classical system; and vice versa, i.e., for a smaller β, the loss-neutral system orders lower than the classical system. Furthermore, we find that the optimal order quantity always decreases as we face an increasing loss-aversion degree. We also conducted sensitivity analysis of the retailer s optimal decision and profit, with regard to retailer s procurement cost, selling price and salvage price, etc. The most interesting findings are about the impact of the selling price and the salvage price. The optimal order quantity may increase or decrease in the selling price and the salvage price depending on the retailer s reference level. The results show that the retailer s reference level is a very crucial factor to determine the sensitivity for the optimal production quantity. There are several directions for further research. First, we assume that there is no shortage cost. However, in reality, the shortage penalty cannot be neglected. In this case, we need to redefine the reference point, e.g., the reference point maybe dependent on the maximal demand or average demand. Second, the dynamic reference point is an interesting topic to be studied and it is worthy of investigating how the results found in this paper change in price-setting problem. It would be practicable to incorporate price dependent demand models and decide jointly on the price and demand. With the quantity-oriented reference point and loss attitude of the retailer, the underlying interaction between the price and the demand needs to be further studied. Third, extending this work to a multi-period setting with coordination of the supply chain is a hard but a promising work. p v References Arkes, H.R., Hirshleifer, D., Jiang D, Lim S., Reference point adaptation: Tests in the domain of security trading, Organ. Behav. Human Decision Processes 105(1),