Complexity

Volume 2017 (2017), Article ID 9712626, 16 pages

https://doi.org/10.1155/2017/9712626

## Implications for Firms with Limited Information to Take Advantage of Reference Price Effect in Competitive Settings

College of Economics and Management, Tianjin University, Tianjin 300072, China

Correspondence should be addressed to Zhanbing Guo

Received 4 November 2016; Revised 15 January 2017; Accepted 7 February 2017; Published 13 June 2017

Academic Editor: Mehrbakhsh Nilashi

Copyright © 2017 Junhai Ma and Zhanbing Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper studies internal reference price effects when competitive firms face reference price effects and make decisions based on partial information, where their decision-making mechanism is modeled by a dynamic adjustment process. It is shown that the evolution of this dynamic adjustment goes to stabilization if both adjustment speeds are small and the complexity of this evolution increases in adjustment speeds. It is proved that the necessary condition for flip bifurcation or Neimark-Sacker bifurcation will occur with the increase of adjustment speed in two special cases. What is more, numerical simulations show that these bifurcations do occur. Then, the impacts of parameters on stability and profits are investigated and some management insights for firms with limited information to take advantage of reference price effects are provided.

#### 1. Introduction

Reference price plays an important role in consumer purchase decisions. In fact, consumers’ past experiences contribute to the building of reference price. Consumers will compare this reference price with the current price and decide the current demand. The theoretical basis for reference price effect is the adaptation-level theory [1]. Adaptation-level theory holds that the expectation-based reference price is the adaptation level, which the current price is judged against [2]. In particular, if reference price is higher than the current price, the current price will be perceived as “low” and then consumers will be prone to buy more and demand increases. On the contrary, if reference price is lower than the current price, the current price will be perceived as “high” and then demand decreases. Nasiry and Popescu [3] provided an example to illustrate this effect. Apple used to sell digital songs at a low price (99 cents) for a long time. When the company raised these prices to $1.29, the drop in sales was higher than expected. These lost sales were explained by the fact that the new price was perceived as very high when compared to the benchmark (99 cents), so demand dropped significantly.

Due to the significant effect of reference price, researchers have provided many construction models for reference price. Kalyanaram and Winer [4] divided reference price into two types: internal reference price and external reference price. Internal reference price of one type of goods is related to its historic prices [5]; customers’ memories of historic prices form the benchmark (internal reference price). External reference price refers to an external standard, which refers to contextual factors, such as prices of competing products and price of the same product in a different location [6]. In addition to purchase frequency, brand loyalty, and involvement levels, Chen [7] demonstrated that self-construal influences the impact of reference cues. Besides the above factors, advertising is often used to influence reference price as an operational management tool [8, 9].

Based on this comprehension, scholars have provided various implications for firms to take advantage of reference price effect [10–17]. These studies provided critical implications for firms to maximize their revenues via making use of reference price effect. However, in these models, firms are assumed to be fully rational; that is, firms not only master all necessary information but also choose the best solution determinately. In fact, there are many decision-makers, who face complex environments, and full information is very costly, even impossible, to master for them. Then, some bounded-rationality based decision-making mechanisms are adopted by these decision-makers [18, 19]. However, implications for these firms to take advantage of reference price effect are still rare.

In this paper, we try to explore some useful implications for firms with limited information to take advantage of reference price effect in competitive settings. We consider a market with two competitive firms; they sell similar products to customers and compete in price. As the full market information is very difficult and costly to master, they make decisions based on partial information. We assume firms make decisions based on marginal profit, which can be estimated by market experiments. When marginal profit is positive, the firm should increase its product’s price. Conversely, negative marginal profit leads to the decrease of price. We assume that firms estimate the marginal profits by a weighted mean value of past few prices rather than instant price to smooth possible deviations. Although we omit estimation deviation in the model, we also use this mechanism for the following reasons. First, weighted moving average is a widely used estimation method [20, 21]. Second, as will be shown in the following text, adopting an appropriate weight parameter, which belongs to , always benefits the stability of the system. Based on the nonlinear dynamic system theory, we study the evolution characteristics of this market and investigate the impacts of relevant factors to explore some managerial implications.

This study provides the following implications for firms with limited information to take advantage of reference price effect in competitive settings. First, there is a stability condition which must be satisfied to keep the dynamic system asymptotically stable. And firms get their maximal profits when the system is stable; the period of pricing cycle when the system loses its stability depends on the weight parameter and customer perception coefficient. Second, when customers use the most recent price as reference price, there is a threshold, which depends on customer perception coefficient and belongs to . The stable region of adjustment speeds increases in weight parameter when the weight parameter is less than this threshold and decreases in weight parameter when the weight parameter is greater than this threshold. What is more, if firms use instant prices to estimate marginal profits, the stable region of adjustment speeds increases in memory parameter. Third, a higher initial reference price relative to the equilibrium price generally benefits both firms when the adjustment process is asymptotically stable. However, this does not hold when the system is unstable, and firms should not improve initial reference prices blindly in this situation.

This paper is organized as follows. Section 2 introduces the related literature. Section 3 presents the dynamic game model, where firms face reference price effects and make decisions based on partial information. Management insights as well as main results from theoretical analysis and numerical observation are provided in Section 4. Finally, conclusions are given in Section 5.

#### 2. Related Literature

This paper is related to the literature on dynamic pricing with reference price effect and implications on taking advantage of reference price effect. The earliest study analyzing the impact of reference prices on pricing strategies was [10]. The author showed that a firm can increase its profit by considering reference price effects. After that, the impacts of reference price effects in various settings have been investigated. Fibich et al. [11] studied a continuous time and asymmetric reference effects model; they showed that the optimal price stabilizes at a steady-state price, which is below the optimal price without reference price effects. Popescu and Wu [12] studied the dynamic pricing problem of a monopolist firm under a discrete time model; they showed that if consumers are loss averse, optimal prices will converge to a constant steady-state price, otherwise, optimal policy cycles. Martín-Herrán and Taboubi [13] investigated whether the results on the efficiency of price coordination in bilateral monopolies still hold when the reference price effect is taken into account. They showed that, for some values of the initial reference price, there is a time interval where channel decentralization performs better than coordination. Zhang et al. [14] studied the dynamic pricing policy for supply chain firms under integrated channels and decentralized channels; they showed that the initial reference price of consumers plays an important role in determining whether price skimming policy or price penetration policy is more profitable. What is more, key parameters effects are also provided. Lin [16] studied the impact of price promotion in a supply chain with reference price effect. It is shown that the reference price effect could mitigate double marginalization effect and improve channel efficiency. This helps us to understand the fact that firms encourage consumers to recall reference price. Hu et al. [22] considered a gain-seeking reference price effect model and identified conditions on parameters such that a high-low pricing strategy is optimal. Besides these studies, some valuable insights are also reached by integrating pricing with other decision variables, such as advertising strategy [8, 23], replenishment policies [24, 25], and preservation technology investment [26]. Our study is significantly different from aforementioned papers in that we consider firms that only master partial information. We model the decision-making mechanism of these firms by a dynamic adjustment based on partial information, and then we explore managerial implications for these firms to take advantage of the reference price effect.

Our work is also closely related to the research on the decision-making mechanism of bounded rational players [27–29]. Among this stream of literature, these studies that investigate the dynamic adjustment of players with limited information in competitive settings are most close to our study. Wu et al. [30], Ma and Xie [31], and Elsadany [32] studied the evolution processes when the bounded rational players use naive expectation. Considering that players are heterogeneous, some scholars studied the influence of players’ different adjustment mechanisms on the dynamics of game model (see [33–37]). Some papers extended the decision mechanisms by considering that firms may refer to more than one period when they make decisions, and then delay decision was utilized (see [38–41]). The impacts of players’ exact estimation on the evolutions of the dynamic system were considered in [42, 43]. Besides the above researches, Ma and Guo [44] studied the dynamic competition when one player adopts “one-period look-ahead” behavior. Ahmed et al. [45] and Agliari et al. [46] investigated the dynamic game process when bounded rational firms apply the gradient adjustment mechanism. Although the reference price effect does exist and play an important role in consumer purchase decisions, as far as we know, implication on how to take advantage of reference price effect in competitive settings has never been investigated for decision-makers with partial information, so we build this model to explore some useful implications.

#### 3. Model

In this section, we set up a dynamic system model to represent the evolution process of prices and reference prices. We assume there are two firms (two players) in the market, labeled by , respectively (we use to denote both 1 and 2 unless otherwise specified). They offer similar products (firm offers product ) and form a price game. Firms make decisions in discrete time period . Their demand functions in the absence of reference price effect are where , and . Traditionally, is viewed as the size of the market base of product , denotes the influence of its own price, and denotes the influence of the substitute’s price. We consider fixed unit-production cost and assume that .

Reference price affects customer demand via the magnitude of perceived “gain” or “loss” relative to the reference point [25]. In this paper, we adopt the linear symmetric reference price effect model: . Then, the demand functions in the presence of reference price effects are modeled aswhere is used to represent the effect of the perceived “gain” or “loss.” We follow the notion given in [47] that the effect of unit perceived “gain” or “loss” on demand is lower than that of unit real “gain” or “loss,” so we define for simplicity, where . Profit functions are

To model the evolution of reference prices, we adopt an exponential smoothing model. The exponential smoothing model, stemming from the adaptive expectation model, is the most commonly used updated model for reference price (see [12, 13, 25, 48]):where is the reference price at time , is the actual price at time , and denotes the memory parameter. Memory parameter captures the strength of past prices which the reference depends on. is high when customers have a long memory. is small when customers pay less attention to past prices. If , the reference price becomes the last period’s price; this means customers only have one-period memory.

Given the initial reference prices, the long-term profit maximization problem of each firm iswhere there is a discount rate of future profit when ; discount rate is not considered when . The updated law of reference prices is (4). and are mutually dependent.

Solving the optimal strategy for each firm not only is very complicated but also needs complete information about the whole market. This condition is very tough in the real market. Usually, firms only master limited information and have to make decisions based on limited information. To model the decision-making processes of firms with limited information, in this paper, we adopt the widely used gradient mechanism, which provides a good approximation to the practical adjustment when only marginal profit is available. The gradient mechanism assumes that each firm gets its marginal profit with respect to its price via market experiments in each period:Based on marginal profit, each makes a myopic decision to maximize its profit in the next period. If this marginal profit is positive (negative), firm will increase (decrease) price in the next period (see [31, 32, 49]):where is the speed of adjustment.

We notice that firms may estimate the marginal profit by a weighted mean value of past few prices rather than instant price to smooth possible deviations (also cited as delay decision; see [38–41]). We include this mechanism here and consider one-period delay decision for simplification:where and is the weight parameter or delay parameter. means firm adopts one-period delay decision; otherwise, firm does not adopt delay decision. Then, the dynamic game process of two myopic firms can be modeled as the following dynamic system:

#### 4. Main Results

The main goal of this section is to study the evolution characteristics of system (9) under different settings and provide some management insights for firms to make use of the reference price effect.

##### 4.1. Equilibrium Point and Stability Condition

By setting , and , we can get the fixed points of dynamic system (9). As the boundary fixed points () are meaningless, we only consider the Nash equilibrium point: , where , , and .

To investigate the local stability of , we calculate its linear approximation. The Jacobian matrix of system (9) iswhere , and .

Then, the stability of can be judged by the eigenvalues of at point .

The characteristic polynomial of (10) takes this form: . According to Jury stability criterion, system (9) is asymptotically stable at if the following condition is satisfied (see [36]):where

##### 4.2. The Dynamic Features with respect to Adjustment Speed

Adjustment speed is a parameter reflecting the character of the decision-maker. A radical firm prefers a big adjustment speed with expecting that its profit can increase quickly. A conservative player is more likely to adopt a small adjustment speed to reduce risk. By analyzing the influence of adjustment speeds, we can get the following proposition.

Proposition 1. *Nash equilibrium point of system (9) is asymptotically stable when the adjustment speeds of firms are very small.*

*Proof. *First, let us look at criterion (11) when and ; then, the subconditions of criterion (11) are As all subconditions are continuous in and , all subconditions, except the first and the last one, are still positive when both and are very small. Then, we consider whether the first one and last one hold for small adjustment speeds. Define and , where when and . We can get their linear approximations:It is easy to see that and for very small adjustment speeds. So, the Nash equilibrium point is asymptotically stable.

Now, we consider two special cases for system (9).

*Assumption 2. *Consumers only remember the most recent price in the reference price model ().

*Assumption 3. *Firms only consider the most recent price in the decision-making mechanism; that is, delay decision is not adopted ().

Although these two assumptions seem restrictive, they can provide good approximation to some practical scenarios [22]. We use to denote the characteristic polynomial of four-dimensional systems. Then, the Jury condition iswhere , , , , , , .

Under Assumption 2, we can get the following proposition.

Proposition 4. *When one adjustment speed is very small, Jury stability condition (15) will be violated with the increase of the other adjustment speed. If , the violation happens with one eigenvalue being minus one. If , the violation occurs with the modulus of a pair of conjugate complex eigenvalues being one.*

*Proof. *We assume one adjustment speed is very small (e.g., ). Then, the Jury stability condition is If , is positive and increases in , and if , is negative and decreases in . So, always decreases in .

We now investigate . is a quadratic function of and when . Solving , we can get two solutions:*Case **1* (). has only one positive solution and () when (). Then, solving , we can get + . decreases in , and when . What is more, () when . We can get that all subconditions are satisfied when , where is an appropriately small positive constant. When , the characteristic polynomial has one eigenvalue close to but less than , one eigenvalue close to but bigger than , and two eigenvalues with the product of their moduli less than . If the characteristic polynomial has a pair of conjugate complex numbers, the moduli of conjugate complex numbers are less than one. When increased from to , changed from positive to negative; then, a single eigenvalue passes through with other eigenvalues still inside the unit circle.*Case **2* ( and ). has two positive solutions and and . Then, we can get that has two solutions and and . Then, Case 2 is similar to Case 1.*Case **3 * and . has two positive solutions and , and . Then, we can get that has only one positive solution , . decreases in and when . What is more, decreases in , when . Above all, all subconditions are satisfied when . When increased from to , the Jury stability condition was violated with and , so a pair of conjugate complex eigenvalues occur with their moduli equal to one.*Case **4 *. has two positive solutions and and . Then, Case 4 is similar to Case 3.

Above all, if , the violation happens with one eigenvalue being minus one. If , the violation occurs with the modulus of a pair of conjugate complex eigenvalues being one.

Under Assumption 3, the following proposition holds.

Proposition 5. *For system (9) without delay decision , if the adjustment speed of one firm is very small, Jury stability condition (15) can be violated by one eigenvalue passing through minus one.*

*Proof. *When delay decision is not considered, system (9) turns to a four-dimensional system. We investigate the case when the adjustment speed of one firm (e.g., firm 2) is a very small positive constant: We can see that decreases in . Solving gives What is more, when , , Assume that is an appropriately small positive constant. When , the Jury condition is satisfied. Then, the characteristic polynomial has one eigenvalue close to but less than , one eigenvalue close to but bigger than , and two eigenvalues with the product of their moduli less than . If the characteristic polynomial has conjugate complex numbers, the moduli of conjugate complex numbers are less than one. As the moduli of eigenvalues are continuous in , when increases from to , changes from positive to negative, and then a single eigenvalue passes through minus one with other eigenvalues still inside the unit circle.

Although a single eigenvalue becoming minus one and the modulus of a pair of conjugate complex eigenvalues being equal to one are necessary conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation, respectively, they constitute strong evidence combined with numerical simulations which show that such bifurcations do occur [50]. In the following text, we will resort to some numerical simulations to demonstrate these propositions given above and explore more useful results. To do the simulations, we take the basic parameter values as follows: , and , and then we can get and .

Figure 1 shows the stable region of adjustment speeds in system (9). Figure 2 gives the bifurcation diagram of prices with respect to when other parameters are fixed, where the blue orbit represents the evolution of , the red orbit represents the evolution of , and the black orbit shows the largest Lyapunov exponent (LLE) of this system. LLE is widely used to mark chaos; it is positive when chaos occurs [51]. From Figures 1 and 2, we can see that the Nash equilibrium point is locally stable when and are small; this is consistent with Proposition 1. At the same time, if one firm increases its adjustment speed to a certain value, system (9) will become unstable and sink into closed invariant curve via Neimark-Sacker bifurcation. A further increase of its adjustment speed will lead to a series of period-doubling bifurcations. System (9) sinks into chaos eventually via period-doubling bifurcation.