OPTIMAL PRICING AND INVENTORY MANAGEMENT FOR A LOSS AVERSE FIRM WHEN FACING STRATEGIC CUSTOMERS

. This paper considers the joint inventory and pricing decision problem that a loss averse ﬁrm with reference point selling seasonal products to strategic consumers with risk preference and decreasing value. Consumers can decide whether to buy at the full price in stage 1, or to wait till stage 2 for the salvage price. They may not get the product if the product is sold out in stage 2. The ﬁrm aims to choose a base stock policy and ﬁnd an optimal price to maximize its expected utility, while consumers aim to decide whether to buy or wait strategically for optimizing their payoﬀs. We formulate the problem as a Stackelberg game between the ﬁrm and the strategic consumers in which the ﬁrm is the leader. By deriving the rational expectation equilibrium, we ﬁnd both the optimal stocking level and the full price in our model are lower than those in the classical model without strategic consumers, by which leads to a lower proﬁt. Furthermore, it is shown that the reimbursement contract cannot alleviate the impact of strategic behavior of customers while the ﬁrm’s proﬁt can be improved by the price commitment strategy in most cases. Numerical studies are carried out to investigate the impact of strategic customer behavior and system parameters on the ﬁrm’s optimal decisions.

1. Introduction. Nowadays, competitions between firms become more fierce and therefore, various promotion strategies have been developed to obtain competitive advantages and to increase the market share. For instance, November 11th, the so-called Singles Day in China, has been becoming a yearly shopping carnival that almost all the online retailers provide various promotion strategies to customers. On November 11th, 2016, billions of customers rushed into Tmall (a B2C Website), and the sales exceeded 178 billion US dollars in twenty-four hours. In physical stores, since product life cycle (PLC) is limited and the holding cost is expensive, the firm will also stimulate customers to buy products by providing discounts in the end of every year. Consequently, strategic customers will prepare the shopping list in advance and wait for the big sales. They are strategic in the sense that they choose to buy it immediately or to wait for the future discount, based on the prediction of the firm's pricing and promotional strategies. This is quite different from the traditional sales and as Aviv and Pazgal (2008) mentioned that, if retailers overlook the presence of strategic customers, up to 20% of the overall profits will be lost. It is crucial for firms to incorporate these strategic customers into their decisions.
In the literature considerable attentions have been focused on the modeling of strategic customers, while the firm is usually assumed to be risk neutral. Whereas, in practice, there is much evidence that the firm's pricing and inventory decisions are not always consistent with profit-maximizing goals. Recently, there has a growing body of literature that attempts to adopt some alternative risk preference approaches to character this setting. Among them, Prospect Theory (PT) in the field of behavior economics is one of the most well-known methodologies that incorporates cognitive and psychological factors to describe decision-maker's behavior under uncertainty. We refer interested readers to [17], [19] for comprehensive surveys on PT. There are two essential points to model firms' decisions by prospect theory, i.e., reference point and loss aversion, see for example, [14,15,16], [20] and [3], among others. The main concern in this setting is to take both risk implications and reference levels into consideration in realistic scenarios which is different from the case of risk-neutral decisions.
However, there has been little research combining the loss-averse firm with reference effect and strategic customers behavior in an inventory model. In view of this gap in the literature, this paper aims to investigate an inventory model by simultaneously taking these two aspects into account. We address three research questions in this study: (1) How should the loss averse firm determine optimal inventory and pricing decisions in the presence of strategic customers? (2) How do various parameters, e.g. firm's loss-aversion levels, market decisions, customer characteristics, affect the firm's inventory and pricing decisions? (3) How should the loss averse firm mitigate the strategic customer behavior and under what condition will these efforts increase the firm's profit?
The above questions motivate us to investigate the loss averse firm how to make inventory and pricing decisions in the presence of strategic customers. To the best of our knowledge, this seems to be the first paper in the literature to model strategic customers' behavior with decreasing value and loss averse firm simultaneously. The aim of the firm is to maximize the expected utility by deciding how many products to stock and how much price to set before the selling season begins, and customers wish to maximize individual utility. The feature of this paper includes: i) Strategic customers' behavior with decreasing value: Customers are strategic and they decide whether to buy at full price in period 1 or to buy the product at salvage price in period 2. The customers' valuation of the product is decreasing in selling time. ii) Customers with risk preference: Risk neutral, risk aversion and risk seeking customers are all taken into consideration. iii) Loss-averse firm: the firm is loss averse with reference points.
To summarize, we have the following contributions in our paper. 1. We are the first to consider the loss-averse firm with reference point and strategic customers with decreasing valuation jointly in a two-stage inventory model. On one hand, the strategic customers may make timing decision to purchase. Whereas, his valuation is V in stage 1 while the valuation diminishes to δV in stage 2. On the other hand, the overall utility of the loss-averse firm depends on the prospects of two components: one is intrinsic utility, the other one is gain-loss utility. The former reflects the actual expected benefit, and the latter represents the impact of reference point on the firm. This new feature results in more complicated formulation for equilibrium price and optimal stocking level in the framework of the rational expectations equilibrium.
2. We address how strategic customer behavior influences the system optimal decisions. We show that strategic customer behavior leads to a smaller optimal stocking quantity, a lower optimal full price and a lower expected profit in contrast to the known results in classical models. Moreover, it is found that the more risk the customer prefers or the lower the value decreases, the worse strategic customer behavior impacts. Meanwhile, the more loss averse the firm is, the higher full price and the lower stocking level, which leads to a higher expected profit.
3. We quantify that the reimbursement contract cannot alleviate the negative influence of strategic customer behavior and it is unable to enhance the firm's profit essentially. However, it may induce customers' willing to pay at a higher price by maintaining the same stocking level of products. Interestingly, it is shown that the firm's profits may increase by providing price commitment strategies, which induce early purchases of customers.
The paper is organized as follows. Section 2 reviews the related literature. Section 3 outlines our assumptions and notations, and establishes the model setting. In Section 4, we first analyze the form of the optimal order quantity policy under different cases. The sensitivity analysis is addressed by analyzing the influence of salvage valuation, the selling price and the degree of loss aversion to the stocking quantity. Section 5 shows the alleviation effect of customer behavior under reimbursement contract and price commitment strategies. Numerical results are reported in Section 6, which examine how customer behavior and firm's characteristics affect the optimal decisions. Finally, concluding remarks and future directions are given in Section 7.
2. Literature review. We are interested in studying how strategic customers behavior influence loss-aversion firm's decisions when the customer has decreasing value with risk preference. Related issues have been addressed in the area of revenue management, but joint consideration of loss-averse firm and strategic customer has not been studied. We now provide a brief review of the most relevant literature.
The first stream of work is the literature on joint pricing and inventory management considering strategic customer behavior. Earliest work dated back to [8], who studied joint the pricing and inventory control of durable goods monopolist when facing with strategic customers. Later, [5] and [36] contributed some important insights on pricing of durable good monopolist. From then on, the interest in modeling strategic customers behavior and studying the effects on firms' decisions are gaining increasing attentions. Many papers focused on the conditions under which different strategies optimized the firm's profit and their effects on how inventory decisions complemented. For instance, [2] and [34] focused on the impact of strategic customer on dynamic pricing. They proposed optimal pricing at the market with forward-looking customers and examine a market with heterogeneous myopic and strategic customers, respectively. [6,7] considered that the firms intentionally understock products to produce rationing risk which induces strategic customers to buy earlier than they would otherwise intend. [35] investigated the sellers who use commitments to a particular quantity and compensations to customers during stock-outs. [1] developed a deterministic pricing and replenishment model in which the retailer advertise a fixed price and the selling schedule, and the strategic customers incurred holding or shortage costs. [4] investigated a dynamic programming approach to determine firm's optimal pricing under commitment in presence of strategic heterogeneous customers. For a comprehensive review of the literature on pricing and inventory management in case of strategic customers, readers are referred to [32] and [39]. More recent work on inventory management including [27], [26] and [28]. All the above studies assumed that the decision-makers are riskneutral in a single profit-maximizing firm setting, and they do not consider reference dependence effects. As we will show, joint consideration of the reference dependence for the loss averse firm and customers risk preference has a non-negligible impact on the analytical complexity and performance of the problem.
There is another stream of work that focuses on risk analysis. Within this research stream, loss-averse preference has received considerable attention and has become an important research direction. The existing research on this issue can be categorized into two streams. One stream is to analyze the impact of risk preference on the inventory problem, see [37], [11], [18]. The expected utility theory (EUT), mean-variance analysis and value-at-risk (VaR), or conditional value-at-risk (CVaR) criteria, have been adopted to characterize the decision-maker's behavior such as risk-averse (e.g. [31], [38], [30]). Another stream is the studies of reference dependence, see [23], [25] and references therein. [23] and [25] 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. [33] adopt 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. [12,13] addressed to investigate portfolio model by using the reference point method for academics and quantitatively oriented professionals. Recently, [3] studied the stochastic reference points that represent consumers' beliefs about possible price and product availability and obtain a set of insights into how consumers' loss aversion affects the firm's optimal operational policies. However, there is no published papers that consider loss aversion factor in the presence of strategic customers.
There is a wealth of articles in the literature dealing with the adverse impact of strategic customers, for example, capacity rationing ( [22]), price commitment and quantity commitment in supply performance ( [35]), quick response ( [6,7]), pricematch guarantees ( [21]) and reference therein. [22] proposed a rational expectation model to analyze whether it is optimal for a firm to create rationing risk by deliberately understocking products and show that rationing can be a profitable strategy to influence the strategic behavior of customers when the firm had the ability to choose prices. [35] also utilized a rational expectations model of a seller facing strategic consumers. They demonstrated that the seller can improve by promising either that quantities available will be limited (quantity commitment) or that prices will be kept high (price commitment). [6,7] argued that quick response may mitigate the negative consequences of strategic behavior. [21] used counterfactual analysis to estimate the fraction of strategic consumers in the population under different levels of sophistication in consumers perception of future prices. They showed that, contrary to conventional wisdom, the presence of strategic consumers does not necessarily hurt revenues. It is noteworthy that although our paper also focuses on integrating pricing and inventory with strategic customers, we are handling loss-aversion firm taking reference dependence and risk preference customers under decreasing valuation into consideration. In our model, we analyze two strategies for the firm: a reimbursement contract committing to compensate the customers if the price cuts in a certain period of time, and providing price commitment on the product in the whole selling seasons.
The most closest to our model on pricing and inventory control with strategic customer behavior are those studied by [35], [22], [2] and [10] under different assumptions. [35] studied the impact of strategic customers on supply chain performance based on the newsvendor model. They assumed that customers are risk neutral. And they found that the seller's stocking level, selling price and overall profits are lower than those in the classical model without strategic customers. In their paper, a constant customers' valuation to the product is adopted. [22] studied the problem assuming that customers are risk aversion and pointed out that capacity rationing can induce customers to purchase earlier. They assumed that customers' valuation to the product is unchanged. [2] supposed that customers' valuations decline over the selling season. They show that announced fixed-discount strategies perform better than contingent price schemes in the case of strategic customers. They suppose that customers are risk neutral. More recently, [10] addressed joint inventory and pricing decision considering strategic customers with risk preference and decreasing value. They showed that strategic customer behavior leads to a lower optimal ordering quantity, full price and total profit, compared with the classical newsvendor model. All of the above papers study only the effect of strategic customers behavior, either their risk attitude or their valuation changing, whereas our paper considers not only the strategic customers behavior but also the firm's loss-aversion. Table 1 summarizes these literatures to various modeling assumptions.
A monopoly firm sells a single product in two stages (periods) over a selling season: the unit full price p in stage 1, is greater than the unit markdown price in stage 2, denoted by s, where p is decision variable and s is an exogenous parameter, respectively. At the end of the periods, all remaining inventory is cleared at markdown price and any inventory remaining at the end of the second stage has zero value. We refer to stage 1 as the "entire stage" and stage 2 as "salvage stage". The firm seeks to maximize its expected utility by choosing the stocking quantity Q and full price p; the price is publicly known to all anticipates but the stocking quantity is not observed by the individual customer. And customers choose between buying in stage 1 (at full price) or waiting for the sale in stage 2 (at salvage price) to maximize individual expected surplus. They might get nothing in stage 2 if the product is stock out. The sequence of events is depicted in Fig. 1.
The full price of unit product in classical model, the model with strategic customers and the model with reimbursement contract, respectively in period 1 The stocking quantity in classical model, the model with strategic customers and the model with reimbursement contract, respectively Q, p Decision variables denoting stocking quantity and full price, respectively D Nonnegative and independent random variable, which indicates customers' demand F (x) Cumulative distribution function, characterizing the demand, and tail distribution is Unit procurement cost of the product to the firm V Customers' valuation for the unit production r Customers' reservation price or maximum price which the customers are willing to pay ξr The firm's belief over customers' reservation price ξprob Customers' belief from obtaining the product on the salvage market δ The decreasing rate (0 < δ ≤ 1) λ Customers' risk preference (λ > 0) α The firm's loss aversion (α ≥ 1)

E(·)
Expectation operator U (·) Utility function of the firm x + and x − The maximum and minimal function between 0 and x, respectively. x + = max{0, x} and x − = min{0, x} The market is characterized by random demand D ≥ 0 with a general cumulative distribution with density F (·) and probability density function f (·). Customers have unit demand for the product and homogenous value V in stage 1 while its value is δV in stage 2 (where 0 < δ ≤ 1). We assume that neither reneging of entering customers nor balking customers at present are allowed. That is, all customers are taken into consideration when the sales begin and remain in the market until their demands are satisfied or the selling periods is over. Each unit of product costs is c. The firm keeps a stocking quantity Q and sets a full price p per product prior to period 1 before observing the stochastic demand D. The price p and quantity Q choices influence the customers' to-buy-or-to-wait decisions. The price is known to customers, but the stocking quantity is not observed by the customers. Table 2 summarizes parameters and variables used throughout the paper.
To make sense of our model, we specify the following assumptions: • A1. The demand distribution has an increasing failure rate (IFR), that is, Assumption A1 is a general assumption in the classical single-period inventory problem, which is satisfied by most commonly used distribution functions in supply chain literature including Uniform distribution, Exponential distribution, Normal distribution, Gamma distribution and Weibull distribution, Assumption A2 ensures the marginal profit of the retailer to be nonnegative.
We assume the firm is loss averse and the strategic customers are risk preference with decreasing value. To study customers' strategic interaction and the firm's decision problem, we apply the notion of a rational expectations (RE) equilibrium, which was first proposed by [24]. It states that economic outcomes do not differ systematically from what people expect them to be. This equilibrium concept has also been used on other operations models (see [36], [35] and [6,7], among others). We describe the equilibrium concept in our context by separately considering the customer's and the firm's decision problems.
3.1. Pricing and stocking decisions. In this subsection, we consider the firm's decision problem. The firm needs to determine the optimal stocking quantity Q and full price p to optimize its expected utility. In the classical model with myopic customers, the customers' value is always V and the firm will set the price p = V . The total profit of the classical problem is expressed by where the first two terms represent the total revenue and the last term indicates the total cost.
is cumulative distribution function, and G(·) is the partial expectation, which is defined as For a risk-neutral firm, its expected profit is as follows It is straightforward to verify the expected profit function E[Π 0 (Q)] is concave in Q. We optimize the expected profit and obtain the optimal stocking quantity Q * 0 satisfies the following first-order condition: In a market without any strategic customer behavior, the optimal price and stocking quantity are Henceforth, the p * 0 and Q * 0 are referred to as classical inventory problem price and quantity.
However, customers are strategic and the firm is loss averse in this paper. More specifically, customers may recognize that if the product remains unsold at full price it will be available in the salvage market at a lower price s. The decision of strategic customer is to choose whether to buy the product immediately or wait to buy the product later in the salvage price. Furthermore, the firm in its decision work compares the current profit with the reference wealth and decides to set price and stocking level.

Strategic customers' behavior.
We now describe the customers' decision problem: they need to decide whether to buy at full price in stage 1 or to wait for sale in stage 2 to maximize individual utility. However, they might get nothing if the product is stock out in stage 2. Assume that customers' perceived belief of obtaining the product in stage 2 is ξ prob . Evidently, the customers' maximal utility when facing an actual full price p is as follows where the first term U (V − p) is a customer's utility from buying at the full price p in stage 1, and the second term U (δV − s) is the utility by buying in stage 2, where there is probability ξ prob that he gets the product and probability 1 − ξ prob that he might not get the product in stage 2 if the product is out of stock. So the customer chooses to buy at full price in stage 1 only and only if U (V −p) ≥ U (δV −s)ξ prob and waits for sale otherwise. At equilibrium, a customer who will be indifferent between buying the product in the first stage or waiting to buy in the second stage. In other words, given expectations ξ prob , the customer's reserve price r should satisfy the equation To facilitate analysis, we assume that customers have a power utility function where λ denotes customers' risk preference. The power utility function is commonly used in the economics literature. Various values of λ correspond to different risk attitude: λ = 1 means that customers are risk neutral, 0 < λ < 1 indicates that customers are risk averse and λ > 1 implies that customers are risk seeking. Under these conditions, we can explicitly express the customer's reservation price as follows In our case, given rational expectations, the game between the firm and the customers decomposes into two separate decision problems: for the customers, a binary choice problem considering whether to buy in the first period or wait to to buy the produce in the second period (based on stockout), and for the firm, utility maximization under stochastic demand. We will show that both of them have unique solutions.
3.3. The loss-averse firm. We consider the loss-averse firm under reference level that seeks to maximize the expectation of its own utility. Let R denote the firm's reference level. The firm will define gains and losses with respect to the chosen reference point rather than the actual profit. We address that a decision-maker's overall profit is a gain-loss utility, which represents the effects of the reference points on his decision. Thus, a firms 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 [20], where they adopted referencedependent utility for a customer who decides to join a service facility in queueing system. Let denote the firm's profit.
Thus the overall utility for the firm is The first term E[Π α (Q, p)] is the expected profit of the firm; the second term E[(Π α (Q, p) − R) + ] is the expected gain and the third term is the expected loss multiplied by the loss-averse parameter α under the reference point. The parameter α measures the degree of the firm loss aversion. We assume α ≥ 1. A large α stands for higher risk aversion. α = 1 corresponds to the risk neutral scenario, while α > 1 indicates that the retailer is loss averse and more care about a loss than an equally sized gain.
Remark 1. If the intrinsic utility exceeds the setting reference profit, we posit that the firm makes a profit. Otherwise, it has a loss. Since the decision maker is risk averse, the risk coefficient is under consideration.
Without loss of generality, we normalize R = 0. Since the firm is unwilling to lose when he makes decisions, zero servers as the reference point that defines firm's gain and loss. Zero reference point does not preserve the diminishing sensitivity property in PT (see [17], [14,15,16], [20]). Because of its simplicity, it is commonly used in the literature.
We let denote the firm's breakeven price-quantity function. If realized demand relative to stocking quantity Q is to low (i.e. D < q(Q)), then the frim faces losses. If realized demand is more than q(Q), then the firm faces gains. We can express the firm's expected utility U (Q, p) in (8) From (10), we see the firm's expected utility under reference point is the expected profit plus expected gain-loss. If α = 1, then the firm is risk neutral and the first term in (10) vanishes.
In sharp contrast to the traditional studies, there are at most two differences and attributions: on the one hand, when the firm 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. On the other hand, the utility function measures his loss-aversion attitude depending on reference level.
For further analysis, we adopt the rational Expectation Equilibrium (RE Equilibrium). At equilibrium, beliefs must be consistent with outcomes. In other words, the firm's beliefs ξ r must coincide with customers' reservation price r, and customers' beliefs over availability probability ξ prob must agree with the actual in-stock probability corresponding to the firm's chosen quantity Q. Following [35] and [10], we know that the optimal stocking quantity Q * α and equilibrium full price p * α satisfy From (11) and (12), we know that if the stocking quantity is publicly known to customer, then the lower price that the customers' willingness to pay in period 1 accompanies with the higher quantity. This is not the case since the stocking quantity is the private information to the firm. 4. Performance measures and the optimal decisions. In this section, we will first study the properties of the equilibrium price and stocking quantity. Then we will compare our model with the classical inventory model with myopic customers and states that strategic customers behavior have effects on the firm's decisions.
The following theorem characters the property of U (Q, p).
Theorem 4.1. For a given price p and any α ≥ 1, the firm's expected utility function U (Q, ·) is differentiable and strictly concave in Q. Thus, there exist unique optimal stocking quantity Q * α and equilibrium price p * α that maximize the firm's expected utility, which satisfy the the following equations: Moreover, U (Q, ·) is asymptotically linear with a slop (α + 1)(c − s).

Proof. For any realized demand
and hence Substituting (15) and (16) into the formula of U (Q, p) in (10) and after some algebraic manipulations, we get the first and second derivatives of (10) with respect to Q: and Hence, the firm's expected utility function is concave in Q.
At equilibrium, we get that According to U (Q, p) is strictly concave in Q, then the optimal stocking quantity Q * α = arg max Q {U (Q, p)} is unique. From the first-order optimal condition we obtain that Q * α is finite stationary point of (17). Noting that (19) and setting ∂U (Q,p) ∂Q = 0 in (17) and we can obtain that the optimal stocking quantity Q * α and the full price p * α satisfing Eq. (13) and (14).
This completes the proof.
Remark 2. The asymptotically linear property indicates that one unit of overstocking will result in a unit loss (α + 1)(c − s) as stocking quantity is very large.
Remark 3. Theorem 4.1 indicates that the firm's decisions strongly dependent on customers' risk preference and his loss-aversion. However, the close form of equilibrium price and optimal stocking quantity can not be obtained unless for a special distribution. This is different from the risk-neutral case.
Remark 4. We note that the optimal quantity Q * α and equilibrium price p * α are the same as Q c and p c studying in [10] when the parameter α → 1. This is true since the firm is risk neutral and the reference point is zero.
However, the decision-makers might adopt different reference points under complicated circumstance which may be reference-state or quantity-reference or others. Hence, we extends their results.
Now we compare our model with the classical inventory model with myopic customers. (3) and (13), we have

Theorem 4.2. For any
and The difference between F (Q * α ) and F (Q * 0 ) is Since p * α > c > s and δV − s > 0, we have which means Q * α < Q * 0 .

RUOPENG WANG, JINTING WANG AND CHANG SUN
From (14), we have p * α < p * 0 = V . We get that Since for all Q.
Remark 6. Theorem 4.2 indicates that comparing with the classical newsvendor model, strategic customer with loss aversion leads to a lower optimal price, stocking quantity and expected profit when it comes to the firm with reference point. The same results are gotten when customers are risk-neutral or risk averse where the retailer is risk neutral. We extends their results to the case with the customer preference and a loss-averse firm with reference point.

Remark 7.
The interpretations of Theorem 4.2 are intuitive. The firm tries best to induce more customers to make purchase early at full price in period 1 by setting a lower price and less stocking quantity, which will lower the availability of products in period 2. Lower price and less stocking quantity lead to a lower expected profit.
We observe from Proposition (1) that (i) the optimal stocking quantity Q * α , the full price p * α and expected profit are strictly decreasing in customers' risk preference λ. In other words, when customers' value decreasing rate is fixed, the more the customers prefer risk, the smaller the firm's stocking quantity and the lower the full price. This results in a lower profit. Intuitively, if customers prefer risk, they are less eager to buy the product immediately. Then the reserve price of customers will be low, which forces the firm to set a lower full price and a less stocking quantity. (ii) the optimal stocking quantity Q * α , the full price p * α and expected profit also strictly decrease in δ. This result is reasonable because strategic customer valuation discount factor enhancement will stimulate the firm to force strategic customers to buy later and reduce the full price and less stocking quantity.
We use an example to illustrate Proposition 1. Let the market demand follows an uniform distribution D ∼ U (0, 100), V = 15, c = 5, s = 4, the pattern of the firm's optimal stocking quantity and full price Q and p with increasing the customer's risk preference λ are shown in Fig. 2 in different firm's loss-aversion α and decreasing value δ. There are more insights which we can observe from Fig. 2. Firstly, it can be seen that the optimal stocking quantity Q * α and full price p * α are strictly decreasing in λ. In other words, when customers' value decreasing rate is fixed, the more the customers prefer risk, the smaller the firm's stocking quantity and the lower the full price. Secondly, we can see that there is direct relationship between firm's decisions and strategic customer valuation discount factor. That is, any increase of δ influences firms stocking quantity and full price negatively when there are the same α and δ. Thirdly, the firm's loss-aversion coefficient α affects full price and  stocking quantity inversely. It can be seen that the curve of the smaller α is above on the larger ones in Figure 2 (a) while Figure 2 (b) is on contrary. Fig. 3 indicates the pattern of the firm's optimal stocking quantity and full price Q and p with increasing the customer's decreasing rate δ. Fig. 4 shows that how the expected profit changes with customer's risk preference λ and the decreasing rate δ.  From Proposition 1 and Figs. 2-4, the firm will get full understand on the strategic customer's behavior. Based on this, the firm can adopt strategies to lessen the negative impact of customers behavior: To do better than RE equilibrium outcomes by adopting alternative strategies that serve as performance targets. 5. The effect of the firm's strategy. In section 4, we investigate that impact of strategic customer behavior is critical to the firm. As analyzed above, the RE equilibrium between the firm and strategic customers in the game undermines the firm's performance comparing to the classical model, which is caused by strategic customers behavior. Can the firm do better than RE Equilibrium outcome? Provided that the firm will take some strategies to attract customers to buy early and alleviate strategic customer behavior. Note that there are many strategies which can alleviate strategic customer behavior and improve firms' performance, such as inventory commitment, price guarantee policy, customer reimbursement contract, fast fashion strategy, etc, in which most of them are applied by the firm to encourage the customers to buy more. However, our aim is to induce customers to buy early and alleviate the negative effect of the strategic customers behavior. In this section, we consider two distinct types of widely used strategies in practice: reimbursement contract and price commitment strategy. And we investigate whether and under what conditions it is effective to mitigate the strategic customer behavior. This section tries to address these problems. 5.1. Reimbursement contract. Customer reimbursement contract means that the firm commits to compensate the customers if the price is cut in a certain period after they buy products in period 1. Under reimbursement contract, the firm commits to compensate L to the customers who bought at the full price in period 1 if the price is cut in period 2. We assume that the firm sells the products at full price p in period 1 and then sells the remaining at the salvage price s in period 2. The salvage price s is also exogenous. We will study whether reimbursement contract can mitigate strategic customers behavior. Although [10] also investigated customer reimbursement contract, they considered only the special case that customers are risk neutral. Under the utility function, the maximal customer surplus is as follows: Utilized power utility function, the reserve price of customer satisfies The firm's expected utility is as follows Then, the optimal stocking quantity Q * r and full price p * r are obtained by solving the following problem Similar to the previous discussion, it is easy to verify that the optimal stocking quantity satisfies the following equation where q r (Q * r ) = c−s p * r −s Q * r . The following theorem states the effect of reimbursement contract on the system performance.
Theorem 5.1 indicates that when the firm really decreases the price in period, the reimbursement contract cannot alleviate the negative impact of strategic customers' behavior. The interpretation is natural: if the firm promises that he will compensate L to the customers who bought at full price in period 1 when he really decreases the price to s in period 2, the optimal stocking quantity keeps the same and the full price is L larger than that of without reimbursement contract. Then the firm returns L to the customer who bought in period 1. The firm's optimal profit does not change. This confirms the fact that one gets the benefit, but the price has been paid.

5.2.
Price commitment strategy. We begin by discussing price commitment strategy. Suppose the firm can inform an ex-post price to customers after demand has been realized. The firm promises that the price keeps in the same level, even if the products held in reserve they would have not reduce the price. For a rational decision-maker, it is reasonable to consider the case in which the firm commits the price at exactly R. And customers would be willing to pay R unit product at the period 1. There is no difference in price, no customer is willing to wait for sale. At this case, the firm sets the price p * p = R, and profit function is Similarly, we utilize the zero reference point V and the firm's breakeven function is By substituting the breakeven function (51) into the firm's utility function, we may obtain − cQ * p denote the optimal stocking quantity, equilibrium full price and optimal profit level, respectively. Here, the subscript p stands for "price commitment".
Next, we analyze the firm's stocking quantity decision. The following theorem demonstrates that the firm's utility function is unimodal. Thus, the existence and uniqueness of the optimal decisions are obtained.
Theorem 5.2. For any given λ > 0, α ≥ 1, the firm's utility function U p (Q) is quasi-concave in Q, and the optimal stocking quantity is determined by the unique solution to the first order condition Proof. Substituting p * p = R into (52) and after algebraic calculation, we get that the firm's utility under price commitment strategy is Differentiating of (54) yields Setting dUp(Q) dQ = 0, we have Since the left-hand-side is increasing, and the right-hand-side is constant, so the first order condition at most has a unique solution.
Further, we know that Note: The expected profits are the classical inventory model, the proposed model and the model under price commitment strategy in turn. We mark by red and green color when the expected profit of our model is larger than that of under price commitment strategy model. of λ and δ, because the price is committed by the firm in the two periods and the customer buys the products in period 1. Our first observation is that the firm's profit of classical system, loss-aversion system and price commitment system are monotonically decreasing in c. It is natural that the higher production cost leads to lower profits. Observe that the profit of loss-aversion system is decreasing in λ. This feature is that the more risk prefer customers are, they are less eager to get the product immediately or they are optimistic of getting the product in period 2 with a lower price which result in a lower profit. However, in the price commitment system, the price does not change with λ and only depends on rational expectations, which in turn has no impact on the firm' profit. As for δ, the same situation occurs. It seems that the profit in δ appears to be most sensitive to the parameter in λ (the profit in loss-aversion system has relatively minimal variation as a function of λ).
Another interesting observation is the case in which the profit is not monotonic in α in loss-aversion system but monotonically decreases in price commitment system. The reasons are as follows: in the loss-aversion system, the equilibrium price balances the difference between loss averse level and profit; however, the price is committed in price commitment system; the more loss aversion is, the less quantity is stocked, which leads to a lower profit.
The other cases we also examined, including higher salvage price s, lower customers' valuation V . As expected, the expected profit and stocking quantity decrease in the production cost c. And decreasing rate δ and customer risk prefer λ lead to less stocking quantity which in turn results in lower profit in loss-aversion system. The firm should take different strategies to improve expected profit in the presence of strategic customers.

7.
Conclusions and future work. Strategic customer behavior has deeply affected the firm's decisions and has been received extensive attention both in practice and research literature. In this paper we study the joint pricing and inventory control problem not only considering customers' behavior, but also integrating reference-dependent effect into the firm's decisions in a two-stage inventory problem. We apply Prospect Theory (PT) to describe the irrational decision-making behavior under uncertainty. The strategic customers may make timing decision to purchase and the overall utility of the loss-averse firm depends on the prospects. None of the existing literature includes all of the following four critical features in our paper, e.g., Strategic customers behavior with decreasing valuation, Customers with risk preference and loss-averse firm with reference point. We show that strategic customers' behavior leads to a lower optimal price, a lower optimal stocking level and a lower expected profit of the loss-aversion firm compared with the classical system. Furthermore, the more the customers prefer risk, the smaller the firms stocking quantity and the lower the full price will be. We also examine the effect of the reimbursement contract and price commitment strategy on the alleviation of the strategic customer behavior. We conclude that the reimbursement contract cannot alleviate the negative influence of strategic customer behavior and the firms profits may increase by providing price commitment strategies. The proposed model is analyzed numerically in which a uniform distribution demand function is adopted, and some managerial implications have been derived.
There are several directions to which our work may be extended. First, one can study the coordination between manufacturers and firms in addition to consider the interaction between strategic customers and a single firm. In this situation, supply contracts may mitigate the loss-aversion effect and strategic customer behavior, and improve supply chain performance. Second, we have assumed that all the customers are homogenous and have the same valuation function. A more interesting topic is to study the similar problem with heterogeneous customers. Third, if a product is sold by two or many firms, their inventory and pricing decisions should be heavily influenced by competitions and strategic customers' behavior. How to optimize a firm's joint inventory and pricing decisions over a planing horizon with T periods under the situation that customers can predict reference prices based on the firm's past information is also a subject worthy of study.