# Pricing Strategy and Quick Response Adoption System with Strategic Customers

Authors: Junfeng Dong; Li Jiang; Yan Jiao
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This study determined the competitive advantage of a quick response (QR) system when a firm faces forward-looking customers with heterogeneous and uncertain valuations for a product, uncertain demand, and two selling periods. We identify two classes of pricing strategies, namely, no-price commitment strategy and price commitment strategy. Interestingly, the unique equilibrium is proven to exist if and only if most customers have high tastes on a product’s value. We also prove that when customers possess beliefs about the markdown in the second period being smaller enough, a firm obtains a high profit with price commitment; otherwise he obtains a high profit without price commitment. Moreover, we distinguish the competitive advantage of a QR system from two strategies. When a firm uses no-price commitment strategy, the value of QR system in the first period decreases and in the second period increases with customer’s strategic behavior. When a firm provides price commitment, the value of QR system in the first period may increase, decrease, or decrease first and then increase with customer’s strategic behavior. And the value of QR in the second period under price commitment strategy decreases or rises first and then decreases with customer’s strategic behavior.


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## Body

## 1. Introduction

The value of various products, such as electronic products and fast fashion, decreases in one selling season. Customers who focus on the value of products choose to make a purchase when the product value is high. Others who aim for reduced prices choose to wait for markdowns. Therefore, firms prefer to vary prices over time to increase profit. Pricing policies are generally divided into price commitment and no-price commitment. In the first policy, firms commit to a markdown price before the selling season. In the second policy, customers only become aware of the price in the selling period when they arrive. They are not aware of the markdown price. The current study focuses on the selection of these strategies when a firm sells a new product. The influence of various factors on the strategy choices of a firm is also investigated.These products utilize a quick response (QR) system, which recently received considerable attention and is widely used in practice. Firms are unaware of exact demands and cannot produce appropriate quantities because of demand uncertainty. Thus, firms attempt to reduce lead times. If they stock out during the selling season, they have a rapid response alternative that can fill an order quickly. However, the issues of whether firms should always use a QR system and the selling period when a firm should use a QR system have not been resolved. We distinguish competitive advantage in the QR system between the two classes of pricing strategies.In this study, a firm faces forward-looking customers with heterogeneous and uncertain valuations for a product, uncertain product demand, and two selling periods (high-value period and low-value period). A firm can maximize expected profits from sales by continuously changing prices over the course of the season (price commitment or no-price commitment) with or without a QR system in different selling periods. Therefore, a firm has six decision-making options: no-price commitment with a two-period QR system (StrategyN2), price commitment with a two-period QR system (Strategy C2), no-price commitment with a one-period QR system (Strategy N1), price commitment with a one-period QR system (Strategy C1), no-price commitment without a QR system (Strategy N0), and price commitment without a QR system (Strategy C0). We formulate two-period models under six different strategies and provide sufficient conditions that guarantee the existence of a unique equilibrium. A substantial number of studies on consumer behavior and multipored pricing assume uniformly distributed valuations; these studies do not establish the existence of equilibrium for general distribution [1]. In the current study, we prove the existence of equilibrium under common conditions. We focus on the effect of strategic behavior of customers on competitive advantage from a QR system. Result indicates that when a firm does not provide price commitment, the competitive advantage obtained from a QR system increases in a low-value period but decreases in a high-value period with the strategic behavior of customers. We found that when a firm provides price commitment, the strategic behavior of customers’ strategic behavior variably influences competitive advantage from a QR system under different postponed production costs in a QR system. We also argue the choice of optimal strategy through a numerical study.The rest of this study is organized as follows. Section2 provides a literature review. Section 3 describes the model with strategic consumers. Section 4 establishes the model with a two-period QR system under price commitment. Section 5 provides an analysis on a firm’s strategy selection under different marketing environments. Section 6 concludes the paper.


## 2. Literature Review

Our work involves several key components, which include pricing policy, quick response system, and strategic customer behavior. We review the relevant literature on these components.The first stream of literature focuses on pricing policy. Two general pricing policies, namely, price commitment and no-price commitment, are related to this study. The first pricing policy is also known as fixed preannounced pricing. In this policy, a firm announces or commits to a specific price path before the selling season. Early works on this type of policy include those of Coase [2] and Stokey [3]. Yin and Tang [4] develop a model based on a situation when a firm preannounces a price path at the beginning of the selling season and customers arrive in accordance with a Poisson process. Liu and Van Ryzin [5] assume that all customers are risk-averse and present at the beginning of the season. The researchers find that capacity decisions can be more important than price in terms of influencing strategic customer behavior. Whang [6] studies the role of demand uncertainty when a retail firm announces a pair of declining prices for two selling periods. Correa et al. [7] explore preannounced pricing schemes of a firm with strategic customers who are uncertain about the initial inventory level. The second pricing policy is also called dynamic pricing policy. In this policy, customers are not aware of future prices or the price path at the beginning of the selling season. Early works on this topic include Besanko and Winston [8]. Aviv and Pazgal [9] consider two pricing strategies, namely, inventory-contingent discounting strategies and preannounced price paths. Zhang et al. [10] derived optimal dynamic pricing strategy by Pontryagin’s maximum principle and then designed an effective algorithm to obtain the optimal dynamic pricing strategy and replenishment cycle simultaneously. Papanastasiou and Savva [11] investigate how social learning affects strategies under preannounced pricing and responsive pricing strategies. Shum et al. [12] examine the impact of cost reduction under dynamic pricing, price commitment, and price matching. Various studies on dynamic pricing policy (without price commitment) are mainly based on inventory-contingent discounting. By contrast, in the current study, a firm’s dynamic pricing path is mainly affected by customers’ decision in the first selling period. Customers are classified according to a firm’s pricing based on the assumption of the present study. Therefore, a firm’s profit in the second period is affected by the preferences of customers who choose to wait.Another related stream of literature focuses on a QR system. A number of studies [13–15] consider firm decisions under a QR system when customers are nonstrategic. Recent works, such as that of Serel [16], examine a single-period inventory model for multiple products in a QR system with a budget constraint; these studies show that this factor increases order size. Caro and Martínez-de- Albéniz [17] address the effect of QR under competition between two retailers. Choi [18] analyzes the carbon footprint taxation scheme in a QR system; they found that a properly designed carbon footprint taxation scheme can enhance environmental sustainability. Lin and Parlaktűrk [19] find that manufacturers who offer a quick response to only one retailer are optimal when demand variability is moderate. Wang et al. [20] find that responsiveness is favorable under demand uncertainty or under conditions of weak product competition. Gong et al. [21] compare a quick response supplier and a regular supplier; they show that supply source diversification or high supplier reliability increases optimal profit and lowers selling price. However, these works did not consider the influence of strategic consumer behavior on a QR system. In practice, some products are suitable for a QR system, whereas others are not. Choi [22] explores the value of QR with the consideration of a bounded rational human manager in the fashion retailing company. A few studies analyze the factors that affect the selection of this strategy, as well as the influence of these factors. In the present study, we investigate the main influencing factor that affects the selection of this strategy.The extensive literature on strategic purchasing behavior of customers begins with the study of Coase [2], who found that customers will wait for low prices of durable goods in a monopolist market. In recent years, strategic consumer behavior was incorporated into traditional operational models. Existing studies found that a firm’s decision-making can be affected by strategic consumer behavior. For example, Liu and Van Ryzin [5, 23], Lai et al. [24], and Jerath et al. [25] focus on how strategic consumers anticipate future surplus driven by firm inventory decisions. Yin et al. [26] and Su [27] study firm pricing decisions when consumers anticipate future markdowns. Wu et al. [28] consider a retailer’s markdown pricing and inventory decisions in multiple seasons wherein consumers can learn from reference prices, which may help them decide when to purchase. Liu and Zhang [29] examine a heterogeneous market with two firms; they differentiate products and emphasized the role of product quality and the value of price commitment. Huang and van Mieghem [30] use a newsvendor framework in modeling click tracking practice of strategic website customers. Li et al. [31] employ a structural estimation model to analyze strategic customers; they determine that the presence of strategic customers does not necessarily hurt firm profit. A few studies focus on the effect of strategic consumer behavior on choosing a QR system. The current study explores this aspect. Yang et al. [32] analyze the impact of quick response on supply chain performance for various supply chain structures with strategic customer behavior and found that the value of the QR system would be greater in centralized supply chains systems than in decentralized systems when the extra cost of quick response was relatively low.To the best of our knowledge, the problem of when to choose the strategy of price commitment or a QR system by considering the influence of strategic consumer behavior was not addressed in previous studies.


## 3. Model Description

### 3.1. Firm and Consumers

We consider a seller who sells a product in a single selling season. The selling season for a product is divided into two consecutive periods. The first period is denoted as periodH and the second period is period L. The product is valued by customers at vi in different periods. We assume vH>vL throughout. Thus, the product has higher value in period H than that in period L. A firm has two potential production opportunities, namely, early production and postponed production. Early production is far advanced in a selling season and market size is unknown. We assume N to be a random variable with positive support, distribution function F(·), and density f(·). Postponed production occurs during the selling season and market size is known [33]. Early production incurs unit cost c, whereas postponed production incurs a higher unit cost cqr≥c. We assume that production has a sufficiently short lead time during the latter opportunity. Hence, we assume that cqr≥c.Customers arrive at the beginning of the selling season. Customers have heterogeneous tastes on a product’s valuevi, which is denoted by θ; this factor is private information of each customer [29, 30]. We assume that θ follows distribution function G(·) and density g(·), which is common knowledge. The prices offered in period H are denoted by pH (a regular-season price) and pL (a markdown price) in period L. A firm simultaneously determines price pH and quantity q at the beginning of each period. Customers have opinions about the markdown in the second period; that is, they believe that the price in period H is rpH (0<r<1).We introduce a discount factor (or a waiting cost) to simultaneously model strategic and nonstrategic customers [31], which is denoted by δ (0≤δ≤1). We can interpret δ as the degree of patience to wait for the markdown or the level of strategic behavior. A high δ indicates patient or strategic consumers. If δ=0, customers do not anticipate the opportunity to purchase in period L, whereas if δ=1, they anticipate lack of discounts of future purchases.


### 3.2. Firm Decision to Use or Not to Use QR System and Price Commitment

Thus, consumers and the firm take part in a game: consumers decide when to buy a product (in periodH or in period L); the firm decides whether to use a QR system and whether to provide price commitment. In our model, the firm’s decisions include price regular-season price pH, markdown price pL, and quantity q. Price pH can be observed by a customer before the selling season. Price pL can be observed by the customer before the selling season if the firm provides price commitment. Otherwise, price will be observed in period L. Quantity q is private information of a firm and cannot be observed by a customer. Firms may select from six decision-making strategies (as shown in Table 1). The sequence of events in different strategies is depicted in Figures 1(a)–1(f).Table 1
Six decision-making options.
Strategy
Price commitment
QR system
PeriodH
PeriodL
No-price commitment with two-period QR system (N2)
X
√
√
Price commitment with two-period QR system (C2)
√
√
√
No-price commitment with one-period QR system (N1)
X
√
X
Price commitment with one-period QR system (C1)
√
√
X
No-price commitment without QR system (N0)
X
X
X
Price commitment without QR system (C0)
√
X
X
√: choosing the strategy; X: not choosing the strategy.Figure 1
(a) StrategyN2. (b) Strategy C2. (c) Strategy N1. (d) Strategy C1. (e) Strategy N0. (f) Strategy C0.
(a)
(b)
(c)
(d)
(e)
(f)


### 3.3. Consumer’s Decision to Wait, Buy, or Not Buy

Customers arrive at the beginning of the selling season. They obtain information, such as regular-season price (pH) or markdown price pL, when the firm provides price commitment. When a firm does not provide price commitment, consumers believe that the price in period L is rpH (0<r<1). We can interpret 1-r as markdown amplitude. High r implies that customers assume a low markdown in the second selling period. The value of r can be influenced by advertising, the popularity of a product, and the type of product (r is low if the product is fast fashion).Customers decide to purchase products at that time or wait for the next selling period that provides the highest expected surplus. If a firm does not provide price commitment, the surplus to a consumer purchasing in periodH is θvH-pH+ and the surplus waiting to be purchased in the period L is δ(θvL-rpH)+. Hence, if θvH-pH+>δ(θvL-rpH)+, the customer will purchase in period H; otherwise, he or she will wait for the next period. From θvH-pH+>δ(θvL-rpH)+, we obtain θ>max⁡(1-δr)pH/vH-δvL,pH/vH. Therefore, when r<vL/vH and θ>(1-δr)pH/vH-δvL or when r≥vL/vH and θ>pH/vH, the customer will purchase in period H. The customer purchases in period L if and only if θvL-pL>0; that is, θ>pL/vL.If a firm provides price commitment, the surplus to a consumer purchase in the first period isθvH-pH+, and the surplus waiting to be purchased in the second period is δ(θvL-pL)+. Hence, if θvH-pH+>δ(θvL-pL)+, a customer will purchase in period H; otherwise, he or she will wait. From θvH-pH+>δ(θvL-pL)+, we obtain θ>pH-δpL/vH-δvL. In this situation, pH-δpL/vH-δvL≥pH/vH, that is, pL/vL≤pH/vH, is always satisfied. Otherwise, the demand in period L will be below zero. In periodL, the customer purchases if and only if θvL-pL>0; that is, θ>pL/vL.Thus, given market sizeN, we obtain the expected demand in period H as (1)DH=NΘH,where ΘH=1-G1.The expected demand in periodL is given by(2)DL=NΘL,where ΘL=G1-G2. Denote Θ=ΘH+ΘL=1-G2.Therefore, if a firm does not provide price commitment andr<vL/vH (we use the symbol r_ to denote this condition), G1=G1r_=G(1-δr/vH-δvLpH) and G2=G2r_=G(pL/vL). If a firm does not provide price commitment and r≥vL/vH (we use symbol r- to denote this condition), G1=G1r-=G(pH/vH) and G2=G2r-=G(pL/vL). If a firm provides price commitment (we use the symbol C to denote this condition), G1=G1C=G(pH-δpL/vH-δvL) and G2=G2C=G(pL/vL). Hence, G2=G2r_=G2r-=G2C=G(pL/vL).


## 3.1. Firm and Consumers

We consider a seller who sells a product in a single selling season. The selling season for a product is divided into two consecutive periods. The first period is denoted as periodH and the second period is period L. The product is valued by customers at vi in different periods. We assume vH>vL throughout. Thus, the product has higher value in period H than that in period L. A firm has two potential production opportunities, namely, early production and postponed production. Early production is far advanced in a selling season and market size is unknown. We assume N to be a random variable with positive support, distribution function F(·), and density f(·). Postponed production occurs during the selling season and market size is known [33]. Early production incurs unit cost c, whereas postponed production incurs a higher unit cost cqr≥c. We assume that production has a sufficiently short lead time during the latter opportunity. Hence, we assume that cqr≥c.Customers arrive at the beginning of the selling season. Customers have heterogeneous tastes on a product’s valuevi, which is denoted by θ; this factor is private information of each customer [29, 30]. We assume that θ follows distribution function G(·) and density g(·), which is common knowledge. The prices offered in period H are denoted by pH (a regular-season price) and pL (a markdown price) in period L. A firm simultaneously determines price pH and quantity q at the beginning of each period. Customers have opinions about the markdown in the second period; that is, they believe that the price in period H is rpH (0<r<1).We introduce a discount factor (or a waiting cost) to simultaneously model strategic and nonstrategic customers [31], which is denoted by δ (0≤δ≤1). We can interpret δ as the degree of patience to wait for the markdown or the level of strategic behavior. A high δ indicates patient or strategic consumers. If δ=0, customers do not anticipate the opportunity to purchase in period L, whereas if δ=1, they anticipate lack of discounts of future purchases.


## 3.2. Firm Decision to Use or Not to Use QR System and Price Commitment

Thus, consumers and the firm take part in a game: consumers decide when to buy a product (in periodH or in period L); the firm decides whether to use a QR system and whether to provide price commitment. In our model, the firm’s decisions include price regular-season price pH, markdown price pL, and quantity q. Price pH can be observed by a customer before the selling season. Price pL can be observed by the customer before the selling season if the firm provides price commitment. Otherwise, price will be observed in period L. Quantity q is private information of a firm and cannot be observed by a customer. Firms may select from six decision-making strategies (as shown in Table 1). The sequence of events in different strategies is depicted in Figures 1(a)–1(f).Table 1
Six decision-making options.
Strategy
Price commitment
QR system
PeriodH
PeriodL
No-price commitment with two-period QR system (N2)
X
√
√
Price commitment with two-period QR system (C2)
√
√
√
No-price commitment with one-period QR system (N1)
X
√
X
Price commitment with one-period QR system (C1)
√
√
X
No-price commitment without QR system (N0)
X
X
X
Price commitment without QR system (C0)
√
X
X
√: choosing the strategy; X: not choosing the strategy.Figure 1
(a) StrategyN2. (b) Strategy C2. (c) Strategy N1. (d) Strategy C1. (e) Strategy N0. (f) Strategy C0.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.3. Consumer’s Decision to Wait, Buy, or Not Buy

Customers arrive at the beginning of the selling season. They obtain information, such as regular-season price (pH) or markdown price pL, when the firm provides price commitment. When a firm does not provide price commitment, consumers believe that the price in period L is rpH (0<r<1). We can interpret 1-r as markdown amplitude. High r implies that customers assume a low markdown in the second selling period. The value of r can be influenced by advertising, the popularity of a product, and the type of product (r is low if the product is fast fashion).Customers decide to purchase products at that time or wait for the next selling period that provides the highest expected surplus. If a firm does not provide price commitment, the surplus to a consumer purchasing in periodH is θvH-pH+ and the surplus waiting to be purchased in the period L is δ(θvL-rpH)+. Hence, if θvH-pH+>δ(θvL-rpH)+, the customer will purchase in period H; otherwise, he or she will wait for the next period. From θvH-pH+>δ(θvL-rpH)+, we obtain θ>max⁡(1-δr)pH/vH-δvL,pH/vH. Therefore, when r<vL/vH and θ>(1-δr)pH/vH-δvL or when r≥vL/vH and θ>pH/vH, the customer will purchase in period H. The customer purchases in period L if and only if θvL-pL>0; that is, θ>pL/vL.If a firm provides price commitment, the surplus to a consumer purchase in the first period isθvH-pH+, and the surplus waiting to be purchased in the second period is δ(θvL-pL)+. Hence, if θvH-pH+>δ(θvL-pL)+, a customer will purchase in period H; otherwise, he or she will wait. From θvH-pH+>δ(θvL-pL)+, we obtain θ>pH-δpL/vH-δvL. In this situation, pH-δpL/vH-δvL≥pH/vH, that is, pL/vL≤pH/vH, is always satisfied. Otherwise, the demand in period L will be below zero. In periodL, the customer purchases if and only if θvL-pL>0; that is, θ>pL/vL.Thus, given market sizeN, we obtain the expected demand in period H as (1)DH=NΘH,where ΘH=1-G1.The expected demand in periodL is given by(2)DL=NΘL,where ΘL=G1-G2. Denote Θ=ΘH+ΘL=1-G2.Therefore, if a firm does not provide price commitment andr<vL/vH (we use the symbol r_ to denote this condition), G1=G1r_=G(1-δr/vH-δvLpH) and G2=G2r_=G(pL/vL). If a firm does not provide price commitment and r≥vL/vH (we use symbol r- to denote this condition), G1=G1r-=G(pH/vH) and G2=G2r-=G(pL/vL). If a firm provides price commitment (we use the symbol C to denote this condition), G1=G1C=G(pH-δpL/vH-δvL) and G2=G2C=G(pL/vL). Hence, G2=G2r_=G2r-=G2C=G(pL/vL).


## 4. Model with Two-Period QR System under Price Commitment

In the quick response regime, a firm can procure inventory before and after receiving a forecast update prior to the start of the selling season [1, 34]. We define the equilibrium to pricing-inventory-purchasing game without price commitment as follows.Definition 1.
Equilibrium with rational expectations and nonzero production to the game between strategic consumers enables the firm to satisfy the following conditions.(1)
The firm sets the price and inventory to maximize expected profit, given that some consumers purchase in periodH and other consumers purchase in period L.(2)
Consumers believe that a markdown (1-r) could occur in period L before the selling season and purchases may happen in periods H given selling price pH and purchase in periods L given selling price pL.
A firm’s expected profit without price commitment under two-period QR strategy and as a function of the initial inventory procurement (q) and price (pH, pL) is(3)ΠN2pH,pL,q=EpHDH+pLDL-cq-cqrDH+DL-q+=pHμΘH+pLμΘL-cq-cqrLq,where L(q) is the expected lost sales function (excess demand above q):(4)Lq=EDH+DL-q+=μΘ+Θ∫0q/ΘFxdx-q.
Thus, the equilibrium conditions are(1)
(pHN2∗,pLN2∗,qN2∗)=argmaxq,pH,pLΠN2(pH,pL,q);(2)
ΘH∗=1-G1∗;(3)
ΘL∗=G1∗-G2∗.
Most studies on consumer behavior and multipored pricing assume uniformly distributed valuations [1]. These works do not establish the existence of equilibrium for general distribution. We attempt to prove the existence of equilibrium with increasing generalized failure rate (IGFR) distributions and obtain the optimal solution under different r (Theorems 2 and 3).Theorem 2.
Whenr<vL/vH and G(·) are IGFR distributions, the firm’s optimal solution is given in the following situations. (i)
WhenvL1-δr/vH-δvL<ψ^, the firm’s profit function has a local maximum value point. Equilibrium solution (pHN2∗,pLN2∗,qN2∗) at maximum value point exists and is characterized by (5), (6), and (7).(ii)
WhenvL1-δr/vH-δvL>ψ^, vL1-δr/vH-δvL<ψpH=pHN2∗,pL=pLN2∗,q=qN2∗ and ΠN2(pHN2∗,pLN2∗,qN2∗)>Π~L(p~L,q~) are satisfied. The firm’s optimal solution is also (pHN2∗,pLN2∗,qN2∗).(iii)
Otherwise, the firm’s optimal solution is(p~L,q~).(5)ΘHr_-g1r_1-δrpH-pLvH-δvL=0,(6)ΘLr_-g2vLpL-M=0,(7)cqrF-qΘ-c=0,where(8)ψ=1g1r_22g2+ΘLr_g2g2′2g1r_+g1r_′ΘHr_g1r_,ψ^=min⁡ψ,Π~Lp~L,q~=max⁡Π~LpL,q=pLμG-2-cq-cqrLq,Mc,cqr=cqrμμ+∫0q/ΘFxdx-qΘFqΘ.Proof.
The proof of this theorem is presented in the Appendix.Situation (iii) in Theorem2 is an extreme situation if and only if most customers have a low taste on a product’s value. If this situation occurs, a firm’s optimal decision is to sell only in a low-value period, which is inconsistent with actual situations. This condition demonstrates that if the firm wants to use a two-period pricing strategy, consumers should pay sufficient attention to product value, that is, demand distribution must satisfy Situation (i) or (ii). The rest of the paper focuses on Situations (i) and (ii) in Theorem 2; that is, the firm’s optimal solution is (pHN2∗,pLN2∗,qN2∗).Theorem 3.
Whenr≥vL/vH and G(·) is an IGFR distribution, the firm’s profit function has a maximum value point. The equilibrium solution (pHN2∗,pLN2∗,qN2∗) at the maximum value point exists and is characterized by (9), (10), and (11):(9)ΘHr--pHvHg1r-=0,(10)ΘLr--g2vLpL-M=0,(11)cqrF-qΘ-c=0,where M(c,cqr)=cqr/μμ+∫0q/ΘF(x)dx-q/ΘF(q/Θ).Proof.
The proof of this theorem is presented in the Appendix.We obtainF(q/Θ)=1-c/cqr from (7) or (11). Therefore, M(c,cqr) only depends on the cost of quick response (cqr), the cost of early opportunity (c), and the distribution function of market size. ∫0F-1(1-c/cqr)F(x)dx-(cqr-c/cqr)F-1(cqr-c/cqr) strictly increases in the rate of early production cost and postponed production cost (c/cqr). M(c,cqr) may remain unchanged when cqr is increasing and c is decreasing. Based on (5) and (6), if M(c,cqr) is fixed, the optimal price in the second period is fixed. Therefore, a suitable value of c^qr exists given other parameters. When cqr>c^qr, we obtain pLN2∗<c^qr. The QR strategy in period L is not a good choice under such a condition.Lemma 4.
Whenr<vL/vH, the optimal price in period L(pLN2∗) strictly decreases in r, and the optimal price in period H(pHN2∗) and optimal profit (ΠN2∗) strictly increase in r.Proof.
The proof of this theorem is presented in the Appendix.Based on Lemma4, when r≥vL/vH, the firm can obtain the maximum profit under Strategy N2. Therefore, firms try to raise the value of r through advertising or other methods to improve profit.Special Case. We obtain ψ^=4 when G(·) is a uniform distribution. vL1-δr/vH-δvL<ψ^ is satisfied. Hence, the equilibrium solution (pHN2∗,pLN2∗,qN2∗) always exists, and Situations (ii) and (iii) in Theorem 2 will never occur.When
G
(
·
)
Is a Beta Distribution
(
θ
∈
B
e
(
α
,
β
)
). Beta distribution is a family of continuous probability distributions defined on the interval [0,1] and parameterized by two positive shape parameters denoted by α and β.(1)
Whenα=1, β=1; the beta distribution turns into a uniform distribution on [0,1]. Thus, equilibrium solution (pHN2∗,pLN2∗,qN2∗) always exists.(2)
Whenα>1, β=1; that is, the number of customers who have high tastes on a product’s value is always greater than those with low tastes [when x<y, g(x)<g(y)]. In this condition, ψ^>4. Therefore, equilibrium solution (pHN2∗,pLN2∗,qN2∗) always exists.(3)
Whenβ≠1, that is, the number of customers who have high tastes on a product’s value may be smaller than that with low tastes [when x<y, g(x)>g(y) may happen], in this condition, ψ^>0. Therefore, all situations in Theorem 2 may occur.


## 5. Firm’s Strategy Selection under Different Marketing Environments

Two problems are identified in this model based on an earlier discussion. First, does the firm strategy always maximize profit or not? Second, if not, how does the firm change its strategy? These problems are discussed as follows.


### 5.1. When Should Price Commitment Be Provided?

If a firm commits to a two-period price before the selling season, we obtain the optimal model under StrategyC2. In this model, the firm’s expected profit is(12)ΠC2pH,pL,q=EpHDHC+pLDLC-cq-cqrDHC+DLC-q+=pHμΘHC+pLμΘLC-cq-cqrLq,where L(q)=E(DHC+DLC-q)+=μΘ+Θ∫0q/ΘF(x)dx-q, ΘHC=1-G1C, ΘLC=G1C-G2C, and Θ=ΘHC+ΘLC.Theorem 5.
IfG(·) is an IGFR distribution, a firm’s optimal solution when a price commitment is provided before the selling season under a two-period QR strategy has three situations:(i)
WhenvL1-δ2/vH-δvL<ψ^, the equilibrium solution (pHC2∗,pLC2∗,qC2∗) at the maximum value point exists and is characterized by (13).(ii)
WhenvL1-δ2/vH-δvL>ψ^, vL1-δ2/vH-δvL<ψ|pH=pHC2∗,pL=pLC2∗,q=qC2∗ and ΠC2(pHC2∗,pLC2∗,qC2∗)>ΠC(p~L,q~) are satisfied; the firm’s optimal solution is (pHC2∗,pLC2∗,qC2∗).(iii)
Otherwise, the firm’s optimal solution is(p~L,q~).(13)g1C=vH-δvLpH-pLG-1C,δpH-pLvH-δvLg1C+ΘLC+g2vLM-pL=0,FqΘ=1-ccqr,where (14)ψ=1g1C22g2+ΘLCg2g2′2g1C+g1C′ΘHCg1C,ψ^=min⁡ψ,Π~Cp~L,q~=max⁡Π~CpL,q=pLμG-2-cq-cqrLq,Mc,cqr=cqrμμ+∫0q/ΘFxdx-qΘFqΘ.Proof.
The proof of this theorem is presented in the Appendix.Situation (iii) in Theorem5 is identical to Situation (iii) in Theorem 2. Therefore, we do not consider this situation in the rest of the study. We compare profit under Strategy C2 with that under Strategy N2, which leads to the following result.Lemma 6.
r
⌢
∈
0
,
v
L
/
v
H exist, such that the firm obtains a high profit with price commitment when r<r⌢ and obtains a high profit without price commitment when r>r⌢.Proof.
The proof of this theorem is presented in the Appendix.r
⌢ will increase when δ is increasing because ∂G1/∂δ≥0 and ∂G1/∂r≤0. Hence, the following lemma exists.Lemma 7.
Decision boundary pointr⌢ increases with customer’s discount factor δ.


### 5.2. Firm Optimal Decision without QR Strategy

This section analyzes four other pricing and QR strategies (no-price commitment with one-period QR system, price commitment with one-period QR system, no-price commitment without QR system, and price commitment without a QR system).Under StrategyN1, a firm sets first period price pH and inventory level q before the selling season. Customers decide whether or not to purchase immediately. In this period, products are replenished with cost cqr if the firm is out of stock. Hence, customers’ demands in first period are satisfied. In the beginning of the second period, a firm observes surplus stock and updates its assumptions over demand in the second period. The firm sets second period price pL and customers make purchasing decisions.A firm’s optimal second period pricing policy is defined by(15)pL=pL1q-DH+≥DL1pL2q-DH+<DL1.When unsold stocks at the end of the first period are excessive, the firm cannot sell them out at a suitable price. Hence, a customer chooses the second period’s lowest pricepL1 to maximizeπ=pLΘL; that is, pL1 satisfies(16)ΘH-pLvLg2pL=pL1=0.The lowest price in the second periodpL1 decreases with price in first period pH.When unsold stocks at the end of the first period are comparatively few, a firm can sell these out at a suitable pricepL2 depending on the quality of unsold stocks. Hence, pL2 should satisfy equation q-DH=DL. We then obtain(17)pL2=vLG-11-qx.Based on (16), we obtain(18)∂pH∂pL1=vH-δvL1-γδ2g2vL+pLg2′g1vL2pL=pL1>0.Therefore, a firm’s profit function under strategyN1 is(19)ΠN1pH,pL,q=EpHDH+pLmin⁡q-DH+,DL-cq-cqrLHq=pHμΘH+pL1ΘL1∫0q/Θxfxdx+∫q/Θq/ΘHpL2ΘL2xfxdx-cq-cqrLHq,where LH(q)=E(DH-q)+=μΘH+ΘH∫0q/ΘHF(x)dx-q. When r<vL/vH, G1=G1r_ and G2=G2r_; when r≥vL/vH, G1=G1r- and G2=G2r-.In (19), from ∂pH/∂pL1>0, when pL1 is sufficiently high, ΠN1(pH,pL,q)→0; when pL1 is sufficiently low, ΠN1(pH,pL,q)→0. Similarly, when q→0, ΠN1(pH,pL,q)→pH-cqrμΘH; when q is sufficiently high, ΠN1(pH,pL,q)→0. Therefore, pH∗,q∗ may exist to maximize ΠN1.When a firm chooses a one-period QR strategy with price commitment (StrategyC1), the profit function is(20)ΠC1pH,pL,q=EpHDH+pLmin⁡q-DH+,DL-cq-cqrLHq=pHμΘH+pLΘH∫q/Θq/ΘHFxdx-ΘL∫0q/ΘFxdx-cq-cqrLHq,where LH(q)=E(DH-q)+=μΘH+ΘH∫0q/ΘHF(x)dx-q.Under this strategy, customers who choose to purchase in periodH can always buy the product [the first term in (20)], and customers who choose to purchase in period L may not obtain the product. When DH>q, products prepared before the selling season are sold out in the first period. Thus, consumers cannot buy any product in a no-QR strategy in the second period. A consumer who purchases after the products in stock are sold out in period H can buy products produced under the QR strategy with product cost cqr [the fourth term in (20)]. When DH<q, that is, (q-DH)+, surplus products in period H are q-DH that can be sold in period L and the demand in period L is DL. Therefore, actual sales are the minimum value between (q-DH)+ and DL [the second term in (20)].The firm’s profit function under StrategiesC1 and C2 can be rewritten as (21)ΠC2=Π~-cqrT,(22)ΠC1=Π~-pLT,where (23)Π~=pHμΘH+pLμΘL-cq-cqrμΘH+ΘH∫0q/ΘHFxdx-q,T=μΘL+Θ∫0q/ΘFxdx-ΘH∫0q/ΘHFxdx.By comparing (21) and (22), we obtain Lemma 8.Lemma 8.
If a firm provides price commitment,c^qr1C∈cqr1lC,cqr1uC exists and the one-period QR strategy is a better choice than the two-period QR strategy when cqr>c^qr1C. Otherwise, the two-period QR strategy is a better choice, where pLC2∗(cqr1uC)=cqr1uC and pLC1∗(cqr2lC)=cqr2lC.
Based onpLC2∗(cqr)=cqr, we obtain the upper bound of c^qr1C, which is denoted as cqr1uC. From pLC1∗(cqr)=cqr, we obtain the lower bound of c^qr1C, which is denoted as cqr1lC. From (21) and (22), we have ΠC2(pHC2∗,pLC2∗,qC2∗)=ΠC1(pHC2∗,pLC2∗,qC2∗) when cqr=cqr1uC. We obtain ΠC2(pHC2∗,pLC2∗,qC2∗)<ΠC1(pHC1∗,pLC1∗,qC1∗) because ∂ΠC1/∂pLpHC2∗,pLC2∗,qC2∗=∂ΠC2/∂pL-ApHC2∗,pLC2∗,qC2∗<0. The same procedure may be easily adapted to obtain ΠC2(pHC2∗,pLC2∗,qC2∗)>ΠC1(pHC1∗,pLC1∗,qC1∗) when cqr=cqr1lC. If cqr>cqr1uC, pLC2∗(cqr)<cqr, profit with the QR strategy in period L is certainly lower than that without a QR strategy in period L. If cqr<cqr1lC, pLC1∗(cqr)>cqr, profit without the QR strategy in period L is certainly lower than that with the QR strategy in period L. Hence, some c^qr1C∈cqr1lC,cqr1uC exist when cqr=c^qr1C, ΠC2∗(cqr)=ΠC1∗(cqr).
Under StrategyN0, products will not be replenished when stocks are out in the two periods. In the beginning of the first period, a firm sets price pH and inventory level q. Customers make decisions whether or not to purchase immediately. In the beginning of the second period, a firm observes the surplus stock and updates its assumptions over demand in the second period. A firm sets the second period price pL and customers make purchasing decisions.
Similar to the analysis under StrategyN1, a firm’s profit function under strategy N0 is(24)ΠN0pH,pL,q=EpHmin⁡q,DH+pLmin⁡q-DH+,DL-cq=pHq-ΘH∫0q/ΘHFxdx+pL1∫0q/ΘxΘLfxdx+∫q/Θq/ΘHpL2xΘLfxdx-cq,where (25)pL=pL1q-DH+≥DL1pL2q-DH+<DL1,ΘH-pLvLg2pL=pL1=0,pL2=vLG-11-qx,LHq=EDH-q+=μΘH+ΘH∫0q/ΘHFxdx-q.When r<vL/vH, G1=G1r_ and G2=G2r_; when r≥vL/vH, G1=G1r- and G2=G2r-.
Similar to the analysis under StrategyC1, a firm’s profit function under strategy C0 is(26)ΠC0pH,pL,q=Π~-μΘH+ΘH∫0q/ΘHFxdx-qpH-cqr-pLT,where (27)Π~=pHμΘH+pLμΘL-cq-cqrμΘH+ΘH∫0q/ΘHFxdx-q,T=μΘL+Θ∫0q/ΘFxdx-ΘH∫0q/ΘHFxdx.
We can use the same approach as the proof of Lemma8. By comparing Strategy C0 with Strategy C1, we can obtain Lemma 9.Lemma 9.
If a firm provides price commitment,c^qr2C∈cqr2lC,cqr2uC exists. When cqr>c^qr2C, the decision not to use a QR system is a better choice than the one-period QR strategy. Otherwise, the one-period QR strategy is a better choice, where pLC1∗(cqr2uC)=cqr2uC and pLC0∗(cqr2lC)=cqr2lC.
Therefore, if the firm does not provide price commitment,c^qr1N and c^qr2N exist. When cqr<c^qr1N, ΠN2∗>ΠN1∗>ΠN0∗; when c^qr1N<cqr<c^qr2N, ΠN2∗<ΠN1∗>ΠN0∗; when cqr>c^qr2N, ΠN2∗<ΠN1∗<ΠN0∗.
Under StrategiesN1 and N0, when δ→1 and cqr is sufficiently high, a firm will prepare sufficient products before the selling season. Hence, loss of stocks in period H almost never occurs; that is, LH(q)→0 and q-ΘH∫0q/ΘHF(x)dx→μΘH. Therefore, ΠN1(pH,pL,q)→ΠN0(pH,pL,q). Under Strategy C1 and Strategy C2, when δ→1, we obtain ΘL→0. Only a few customers purchase products in period L. Hence, ΠC1(pH,pL,q)→ΠC2(pH,pL,q). We then obtain Lemma 10.Lemma 10.
Whenδ=1 and cqr is sufficiently high, ΠN1∗→ΠN0∗. When δ=1 and cqr is sufficiently low, ΠC1∗→ΠC2∗.Example 11.
When a customer’s taste for valueθ follows a uniform distribution on [0,1] and vH=30, vL=20, c=8, and r=0.6. N is uniformly distributed in [0,1000]. Figure 2 shows that a firm’s optimal strategy varies in cqr and δ. For example, when δ=0, a firm chooses Strategy N2 if cqr<10.87. A firm chooses Strategy N1 if 10.87<cqr<21.91. A firm chooses Strategy N0 if cqr>21.91. In Lemma 10, when δ=1 and cqr>17, the optimal profit under Strategies N1 and N0 is almost the same. When δ=1 and cqr<22, the optimal profit under Strategies C2 and C1 is almost the same.
When a firm does not provide price commitment, several customers will choose to purchase in periodL if δ is high. Hence, profit increase under Strategy N2 increases more with δ if ΠN2∗>ΠN1∗than under Strategy N1 (or the value of quick response in the second period). Profit increase under Strategy N1 decreases more with δ if ΠN1∗>ΠN0∗ than under Strategy N0 (or the value of quick response in the first period). Therefore, we obtain Lemma 12.Figure 2
Optimal profit for various values ofδ under different cqr.Lemma 12.
IfΠN2∗>ΠN1∗, percentage profit increase ΠN2∗-ΠN1∗/ΠN1∗ rises with δ. If ΠN1∗>ΠN0∗, percentage profit increase ΠN1∗-ΠN0∗/ΠN0∗ decreases with δ.
Based on Lemma12, we can find that c^qr1N increases with δ and c^qr2N decreases with δ.Example 13.
Figure3 shows the percentage profit increase or the value of quick response when a firm does not provide price commitment. If the equilibrium expected profits under Strategy N2 are higher than those under Strategy N1, percentage profit increases with δ for any cqr. By contrast, the value of the quick response in the first period decreases with δ for any cqr.
When a firm provides price commitment, customers would want to purchase in periodL if δ is high. However, ΘH decreases with δ (when δ→1, ΘH→0) under price commitment. Therefore, when ΠC2∗>ΠC1∗, if δ^1∈(0,1) exists and satisfies ∂ΠC2∗/∂δ=∂ΠC1∗/∂δ, the percentage profit increase ΠC2∗-ΠC1∗/ΠC1∗ rises first and decreases with δ. Otherwise, it always decreases with δ. In a similar manner, when ΠC1∗>ΠC0∗, if δ^2∈(0,1) exists and satisfies ∂ΠC1∗/∂δ=∂ΠC0∗/∂δ, percentage profit increase ΠC1∗-ΠC0∗/ΠC0∗ decreases first and then increases with δ. If δ^2<0, it always increases with δ, and if δ^2>0, it always decreases with δ. Therefore, we obtain Lemmas 14 and 16.Figure 3
Effect ofδ on the percentage profit increase under different cqr.Lemma 14.
WhenΠC2∗>ΠC1∗, percentage profit increase ΠC2∗-ΠC1∗/ΠC1∗ or the value of quick response in the second period decreases with δ if δ^1<0; the percentage profit increase ΠC2∗-ΠC1∗/ΠC1∗ rises first and decreases with δ if 0<δ^1<1.
Based on Lemma14, c^qr1C increases with δ and c^qr2C decreases with δ.Example 15.
Figure4 shows the percentage profit increase ((ΠC2∗-ΠC1∗)/ΠC1∗) or the value of quick response in the second period when a firm provides price commitment. If the equilibrium expected profits under Strategy C2 are higher than that under Strategy C1, percentage profit increase reduces with δ when cqr is sufficiently low (i.e., δ^1<0). When cqr is sufficiently high (i.e., 0<δ^1<1), percentage profit increase rises first and decreases in δ.Figure 4
Effect ofδ on the percentage profit increase ΠC2∗-ΠC1∗/ΠC1∗ under different cqr.Lemma 16.
WhenΠC1∗>ΠC0∗, percentage profit increase ΠC1∗-ΠC0∗/ΠC0∗ or the value of the quick response in the first period increases with δ if δ^2<0; it decreases first and increases with δ if 0<δ^2<1; it decreases with δ if δ^2>0.Example 17.
Figure5 shows the percentage profit increase ((ΠC1∗-ΠC0∗)/ΠC0∗) or the value of the quick response in the first period when the firm provides a price commitment. If the equilibrium expected profits under Strategy C1 are higher than those under Strategy C0, percentage profit increase rises with δ when cqr is sufficiently low (i.e., δ^2<0). When cqr is high (i.e., 0<δ^2<1), the percentage profit increase decreases first and then increases with δ. Otherwise, when cqr is sufficiently high (i.e., δ^2>1), the percentage profit increase rises with δ.Figure 5
Effect ofδ on the percentage profit increase ΠC1∗-ΠC0∗/ΠC0∗ under different cqr.Example 18.
We consider different combinations ofδ and cqr while fixing other parameters. We tried different values of other parameters and obtained qualitatively similar results. Figure 6 demonstrates the optimal profit with and without price commitment for various values of r under different combinations of δ and cqr. We can obtain a firm’s optimal strategy in different conditions. For example, when δ=0.6, cqr=11. A firm’s maximum profit can be obtained under Strategy C1 or C2 if r<0.6383. If r>0.6383, maximum profit can be obtained under Strategy N1. If a firm does not provide price commitment, Strategy N2 should be chosen if r<0.614 and choose Strategy N1 if r>0.614.Figure 6
Optimal profit with and without price commitment for various values ofr.


## 5.1. When Should Price Commitment Be Provided?

If a firm commits to a two-period price before the selling season, we obtain the optimal model under StrategyC2. In this model, the firm’s expected profit is(12)ΠC2pH,pL,q=EpHDHC+pLDLC-cq-cqrDHC+DLC-q+=pHμΘHC+pLμΘLC-cq-cqrLq,where L(q)=E(DHC+DLC-q)+=μΘ+Θ∫0q/ΘF(x)dx-q, ΘHC=1-G1C, ΘLC=G1C-G2C, and Θ=ΘHC+ΘLC.Theorem 5.
IfG(·) is an IGFR distribution, a firm’s optimal solution when a price commitment is provided before the selling season under a two-period QR strategy has three situations:(i)
WhenvL1-δ2/vH-δvL<ψ^, the equilibrium solution (pHC2∗,pLC2∗,qC2∗) at the maximum value point exists and is characterized by (13).(ii)
WhenvL1-δ2/vH-δvL>ψ^, vL1-δ2/vH-δvL<ψ|pH=pHC2∗,pL=pLC2∗,q=qC2∗ and ΠC2(pHC2∗,pLC2∗,qC2∗)>ΠC(p~L,q~) are satisfied; the firm’s optimal solution is (pHC2∗,pLC2∗,qC2∗).(iii)
Otherwise, the firm’s optimal solution is(p~L,q~).(13)g1C=vH-δvLpH-pLG-1C,δpH-pLvH-δvLg1C+ΘLC+g2vLM-pL=0,FqΘ=1-ccqr,where (14)ψ=1g1C22g2+ΘLCg2g2′2g1C+g1C′ΘHCg1C,ψ^=min⁡ψ,Π~Cp~L,q~=max⁡Π~CpL,q=pLμG-2-cq-cqrLq,Mc,cqr=cqrμμ+∫0q/ΘFxdx-qΘFqΘ.Proof.
The proof of this theorem is presented in the Appendix.Situation (iii) in Theorem5 is identical to Situation (iii) in Theorem 2. Therefore, we do not consider this situation in the rest of the study. We compare profit under Strategy C2 with that under Strategy N2, which leads to the following result.Lemma 6.
r
⌢
∈
0
,
v
L
/
v
H exist, such that the firm obtains a high profit with price commitment when r<r⌢ and obtains a high profit without price commitment when r>r⌢.Proof.
The proof of this theorem is presented in the Appendix.r
⌢ will increase when δ is increasing because ∂G1/∂δ≥0 and ∂G1/∂r≤0. Hence, the following lemma exists.Lemma 7.
Decision boundary pointr⌢ increases with customer’s discount factor δ.


## 5.2. Firm Optimal Decision without QR Strategy

This section analyzes four other pricing and QR strategies (no-price commitment with one-period QR system, price commitment with one-period QR system, no-price commitment without QR system, and price commitment without a QR system).Under StrategyN1, a firm sets first period price pH and inventory level q before the selling season. Customers decide whether or not to purchase immediately. In this period, products are replenished with cost cqr if the firm is out of stock. Hence, customers’ demands in first period are satisfied. In the beginning of the second period, a firm observes surplus stock and updates its assumptions over demand in the second period. The firm sets second period price pL and customers make purchasing decisions.A firm’s optimal second period pricing policy is defined by(15)pL=pL1q-DH+≥DL1pL2q-DH+<DL1.When unsold stocks at the end of the first period are excessive, the firm cannot sell them out at a suitable price. Hence, a customer chooses the second period’s lowest pricepL1 to maximizeπ=pLΘL; that is, pL1 satisfies(16)ΘH-pLvLg2pL=pL1=0.The lowest price in the second periodpL1 decreases with price in first period pH.When unsold stocks at the end of the first period are comparatively few, a firm can sell these out at a suitable pricepL2 depending on the quality of unsold stocks. Hence, pL2 should satisfy equation q-DH=DL. We then obtain(17)pL2=vLG-11-qx.Based on (16), we obtain(18)∂pH∂pL1=vH-δvL1-γδ2g2vL+pLg2′g1vL2pL=pL1>0.Therefore, a firm’s profit function under strategyN1 is(19)ΠN1pH,pL,q=EpHDH+pLmin⁡q-DH+,DL-cq-cqrLHq=pHμΘH+pL1ΘL1∫0q/Θxfxdx+∫q/Θq/ΘHpL2ΘL2xfxdx-cq-cqrLHq,where LH(q)=E(DH-q)+=μΘH+ΘH∫0q/ΘHF(x)dx-q. When r<vL/vH, G1=G1r_ and G2=G2r_; when r≥vL/vH, G1=G1r- and G2=G2r-.In (19), from ∂pH/∂pL1>0, when pL1 is sufficiently high, ΠN1(pH,pL,q)→0; when pL1 is sufficiently low, ΠN1(pH,pL,q)→0. Similarly, when q→0, ΠN1(pH,pL,q)→pH-cqrμΘH; when q is sufficiently high, ΠN1(pH,pL,q)→0. Therefore, pH∗,q∗ may exist to maximize ΠN1.When a firm chooses a one-period QR strategy with price commitment (StrategyC1), the profit function is(20)ΠC1pH,pL,q=EpHDH+pLmin⁡q-DH+,DL-cq-cqrLHq=pHμΘH+pLΘH∫q/Θq/ΘHFxdx-ΘL∫0q/ΘFxdx-cq-cqrLHq,where LH(q)=E(DH-q)+=μΘH+ΘH∫0q/ΘHF(x)dx-q.Under this strategy, customers who choose to purchase in periodH can always buy the product [the first term in (20)], and customers who choose to purchase in period L may not obtain the product. When DH>q, products prepared before the selling season are sold out in the first period. Thus, consumers cannot buy any product in a no-QR strategy in the second period. A consumer who purchases after the products in stock are sold out in period H can buy products produced under the QR strategy with product cost cqr [the fourth term in (20)]. When DH<q, that is, (q-DH)+, surplus products in period H are q-DH that can be sold in period L and the demand in period L is DL. Therefore, actual sales are the minimum value between (q-DH)+ and DL [the second term in (20)].The firm’s profit function under StrategiesC1 and C2 can be rewritten as (21)ΠC2=Π~-cqrT,(22)ΠC1=Π~-pLT,where (23)Π~=pHμΘH+pLμΘL-cq-cqrμΘH+ΘH∫0q/ΘHFxdx-q,T=μΘL+Θ∫0q/ΘFxdx-ΘH∫0q/ΘHFxdx.By comparing (21) and (22), we obtain Lemma 8.Lemma 8.
If a firm provides price commitment,c^qr1C∈cqr1lC,cqr1uC exists and the one-period QR strategy is a better choice than the two-period QR strategy when cqr>c^qr1C. Otherwise, the two-period QR strategy is a better choice, where pLC2∗(cqr1uC)=cqr1uC and pLC1∗(cqr2lC)=cqr2lC.
Based onpLC2∗(cqr)=cqr, we obtain the upper bound of c^qr1C, which is denoted as cqr1uC. From pLC1∗(cqr)=cqr, we obtain the lower bound of c^qr1C, which is denoted as cqr1lC. From (21) and (22), we have ΠC2(pHC2∗,pLC2∗,qC2∗)=ΠC1(pHC2∗,pLC2∗,qC2∗) when cqr=cqr1uC. We obtain ΠC2(pHC2∗,pLC2∗,qC2∗)<ΠC1(pHC1∗,pLC1∗,qC1∗) because ∂ΠC1/∂pLpHC2∗,pLC2∗,qC2∗=∂ΠC2/∂pL-ApHC2∗,pLC2∗,qC2∗<0. The same procedure may be easily adapted to obtain ΠC2(pHC2∗,pLC2∗,qC2∗)>ΠC1(pHC1∗,pLC1∗,qC1∗) when cqr=cqr1lC. If cqr>cqr1uC, pLC2∗(cqr)<cqr, profit with the QR strategy in period L is certainly lower than that without a QR strategy in period L. If cqr<cqr1lC, pLC1∗(cqr)>cqr, profit without the QR strategy in period L is certainly lower than that with the QR strategy in period L. Hence, some c^qr1C∈cqr1lC,cqr1uC exist when cqr=c^qr1C, ΠC2∗(cqr)=ΠC1∗(cqr).
Under StrategyN0, products will not be replenished when stocks are out in the two periods. In the beginning of the first period, a firm sets price pH and inventory level q. Customers make decisions whether or not to purchase immediately. In the beginning of the second period, a firm observes the surplus stock and updates its assumptions over demand in the second period. A firm sets the second period price pL and customers make purchasing decisions.
Similar to the analysis under StrategyN1, a firm’s profit function under strategy N0 is(24)ΠN0pH,pL,q=EpHmin⁡q,DH+pLmin⁡q-DH+,DL-cq=pHq-ΘH∫0q/ΘHFxdx+pL1∫0q/ΘxΘLfxdx+∫q/Θq/ΘHpL2xΘLfxdx-cq,where (25)pL=pL1q-DH+≥DL1pL2q-DH+<DL1,ΘH-pLvLg2pL=pL1=0,pL2=vLG-11-qx,LHq=EDH-q+=μΘH+ΘH∫0q/ΘHFxdx-q.When r<vL/vH, G1=G1r_ and G2=G2r_; when r≥vL/vH, G1=G1r- and G2=G2r-.
Similar to the analysis under StrategyC1, a firm’s profit function under strategy C0 is(26)ΠC0pH,pL,q=Π~-μΘH+ΘH∫0q/ΘHFxdx-qpH-cqr-pLT,where (27)Π~=pHμΘH+pLμΘL-cq-cqrμΘH+ΘH∫0q/ΘHFxdx-q,T=μΘL+Θ∫0q/ΘFxdx-ΘH∫0q/ΘHFxdx.
We can use the same approach as the proof of Lemma8. By comparing Strategy C0 with Strategy C1, we can obtain Lemma 9.Lemma 9.
If a firm provides price commitment,c^qr2C∈cqr2lC,cqr2uC exists. When cqr>c^qr2C, the decision not to use a QR system is a better choice than the one-period QR strategy. Otherwise, the one-period QR strategy is a better choice, where pLC1∗(cqr2uC)=cqr2uC and pLC0∗(cqr2lC)=cqr2lC.
Therefore, if the firm does not provide price commitment,c^qr1N and c^qr2N exist. When cqr<c^qr1N, ΠN2∗>ΠN1∗>ΠN0∗; when c^qr1N<cqr<c^qr2N, ΠN2∗<ΠN1∗>ΠN0∗; when cqr>c^qr2N, ΠN2∗<ΠN1∗<ΠN0∗.
Under StrategiesN1 and N0, when δ→1 and cqr is sufficiently high, a firm will prepare sufficient products before the selling season. Hence, loss of stocks in period H almost never occurs; that is, LH(q)→0 and q-ΘH∫0q/ΘHF(x)dx→μΘH. Therefore, ΠN1(pH,pL,q)→ΠN0(pH,pL,q). Under Strategy C1 and Strategy C2, when δ→1, we obtain ΘL→0. Only a few customers purchase products in period L. Hence, ΠC1(pH,pL,q)→ΠC2(pH,pL,q). We then obtain Lemma 10.Lemma 10.
Whenδ=1 and cqr is sufficiently high, ΠN1∗→ΠN0∗. When δ=1 and cqr is sufficiently low, ΠC1∗→ΠC2∗.Example 11.
When a customer’s taste for valueθ follows a uniform distribution on [0,1] and vH=30, vL=20, c=8, and r=0.6. N is uniformly distributed in [0,1000]. Figure 2 shows that a firm’s optimal strategy varies in cqr and δ. For example, when δ=0, a firm chooses Strategy N2 if cqr<10.87. A firm chooses Strategy N1 if 10.87<cqr<21.91. A firm chooses Strategy N0 if cqr>21.91. In Lemma 10, when δ=1 and cqr>17, the optimal profit under Strategies N1 and N0 is almost the same. When δ=1 and cqr<22, the optimal profit under Strategies C2 and C1 is almost the same.
When a firm does not provide price commitment, several customers will choose to purchase in periodL if δ is high. Hence, profit increase under Strategy N2 increases more with δ if ΠN2∗>ΠN1∗than under Strategy N1 (or the value of quick response in the second period). Profit increase under Strategy N1 decreases more with δ if ΠN1∗>ΠN0∗ than under Strategy N0 (or the value of quick response in the first period). Therefore, we obtain Lemma 12.Figure 2
Optimal profit for various values ofδ under different cqr.Lemma 12.
IfΠN2∗>ΠN1∗, percentage profit increase ΠN2∗-ΠN1∗/ΠN1∗ rises with δ. If ΠN1∗>ΠN0∗, percentage profit increase ΠN1∗-ΠN0∗/ΠN0∗ decreases with δ.
Based on Lemma12, we can find that c^qr1N increases with δ and c^qr2N decreases with δ.Example 13.
Figure3 shows the percentage profit increase or the value of quick response when a firm does not provide price commitment. If the equilibrium expected profits under Strategy N2 are higher than those under Strategy N1, percentage profit increases with δ for any cqr. By contrast, the value of the quick response in the first period decreases with δ for any cqr.
When a firm provides price commitment, customers would want to purchase in periodL if δ is high. However, ΘH decreases with δ (when δ→1, ΘH→0) under price commitment. Therefore, when ΠC2∗>ΠC1∗, if δ^1∈(0,1) exists and satisfies ∂ΠC2∗/∂δ=∂ΠC1∗/∂δ, the percentage profit increase ΠC2∗-ΠC1∗/ΠC1∗ rises first and decreases with δ. Otherwise, it always decreases with δ. In a similar manner, when ΠC1∗>ΠC0∗, if δ^2∈(0,1) exists and satisfies ∂ΠC1∗/∂δ=∂ΠC0∗/∂δ, percentage profit increase ΠC1∗-ΠC0∗/ΠC0∗ decreases first and then increases with δ. If δ^2<0, it always increases with δ, and if δ^2>0, it always decreases with δ. Therefore, we obtain Lemmas 14 and 16.Figure 3
Effect ofδ on the percentage profit increase under different cqr.Lemma 14.
WhenΠC2∗>ΠC1∗, percentage profit increase ΠC2∗-ΠC1∗/ΠC1∗ or the value of quick response in the second period decreases with δ if δ^1<0; the percentage profit increase ΠC2∗-ΠC1∗/ΠC1∗ rises first and decreases with δ if 0<δ^1<1.
Based on Lemma14, c^qr1C increases with δ and c^qr2C decreases with δ.Example 15.
Figure4 shows the percentage profit increase ((ΠC2∗-ΠC1∗)/ΠC1∗) or the value of quick response in the second period when a firm provides price commitment. If the equilibrium expected profits under Strategy C2 are higher than that under Strategy C1, percentage profit increase reduces with δ when cqr is sufficiently low (i.e., δ^1<0). When cqr is sufficiently high (i.e., 0<δ^1<1), percentage profit increase rises first and decreases in δ.Figure 4
Effect ofδ on the percentage profit increase ΠC2∗-ΠC1∗/ΠC1∗ under different cqr.Lemma 16.
WhenΠC1∗>ΠC0∗, percentage profit increase ΠC1∗-ΠC0∗/ΠC0∗ or the value of the quick response in the first period increases with δ if δ^2<0; it decreases first and increases with δ if 0<δ^2<1; it decreases with δ if δ^2>0.Example 17.
Figure5 shows the percentage profit increase ((ΠC1∗-ΠC0∗)/ΠC0∗) or the value of the quick response in the first period when the firm provides a price commitment. If the equilibrium expected profits under Strategy C1 are higher than those under Strategy C0, percentage profit increase rises with δ when cqr is sufficiently low (i.e., δ^2<0). When cqr is high (i.e., 0<δ^2<1), the percentage profit increase decreases first and then increases with δ. Otherwise, when cqr is sufficiently high (i.e., δ^2>1), the percentage profit increase rises with δ.Figure 5
Effect ofδ on the percentage profit increase ΠC1∗-ΠC0∗/ΠC0∗ under different cqr.Example 18.
We consider different combinations ofδ and cqr while fixing other parameters. We tried different values of other parameters and obtained qualitatively similar results. Figure 6 demonstrates the optimal profit with and without price commitment for various values of r under different combinations of δ and cqr. We can obtain a firm’s optimal strategy in different conditions. For example, when δ=0.6, cqr=11. A firm’s maximum profit can be obtained under Strategy C1 or C2 if r<0.6383. If r>0.6383, maximum profit can be obtained under Strategy N1. If a firm does not provide price commitment, Strategy N2 should be chosen if r<0.614 and choose Strategy N1 if r>0.614.Figure 6
Optimal profit with and without price commitment for various values ofr.


## 6. Conclusion

We explore price commitment and no-price commitment schemes and their equilibrium solution for a firm that faces forward-looking customers who have heterogeneous and uncertain valuations for a product with or without a QR system in two selling periods. We develop different models based on six decision-making options: a firm provides price commitment or no-price commitment, and the firm uses or does not use a QR system in two selling periods.First, unlike many previous studies on consumer behavior and multipored pricing, we prove that the existence of equilibrium is guaranteed if most customers have a high enough taste on a product’s value; otherwise the firm’s best strategy is to use the one-period selling strategy. Second, when customers possess beliefs about the markdown in the second period being smaller than a certain value, the firm obtains a high profit with price commitment; otherwise he obtains a high profit without price commitment. The certain value increases with customer’s discount factor. Moreover, a stylized analysis of the effects of customers’ strategic behavior on the competitive advantage from a QR system between two classes of pricing strategies is presented. When a firm does not provide price commitment in two selling periods, the competitive advantage from a QR system in a low-value period increases and decreases in a high-value period with customers’ strategic behavior. When a firm provides price commitment, the competitive advantage obtained from a QR system in a low-value period decreases with customers’ strategic behavior if postponed production cost is sufficiently low. Otherwise, it increases first and then decreases with customers’ strategic behavior. The competitive advantage obtained from a QR system in a high-value period increases with customers’ strategic behavior if postponed production cost is sufficiently low and decreases with customers’ strategic behavior if postponed production cost is sufficiently high. Otherwise, it decreases first and then increases with customers’ strategic behavior.Further, future research can extend our analysis in several directions. First, the firm’s pricing and inventory decisions when the value of product continuously decreases with time can be examined. It would be interesting to investigate an optimal time to mark prices down during a selling season. Second, a firm’s decision, when using a quick response system, should be compared with decisions in a traditional system to gain further insights on the impact of strategic customer behavior.



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*Source: 1287162-2017-06-29.xml*