This paper studies internal reference price effects when competitive firms face reference price effects and make decisions based on partial information, where their decisionmaking 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 NeimarkSacker 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.
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 adaptationlevel theory [
Due to the significant effect of reference price, researchers have provided many construction models for reference price. Kalyanaram and Winer [
Based on this comprehension, scholars have provided various implications for firms to take advantage of reference price effect [
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 [
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
This paper is organized as follows. Section
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 [
Our work is also closely related to the research on the decisionmaking mechanism of bounded rational players [
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
Reference price affects customer demand via the magnitude of perceived “gain” or “loss” relative to the reference point [
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 [
Given the initial reference prices, the longterm profit maximization problem of each firm is
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 decisionmaking 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:
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 [
The main goal of this section is to study the evolution characteristics of system (
By setting
To investigate the local stability of
Then, the stability of
The characteristic polynomial of (
Adjustment speed is a parameter reflecting the character of the decisionmaker. 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.
Nash equilibrium point of system (
First, let us look at criterion (
Now, we consider two special cases for system (
Consumers only remember the most recent price in the reference price model (
Firms only consider the most recent price in the decisionmaking mechanism; that is, delay decision is not adopted (
Although these two assumptions seem restrictive, they can provide good approximation to some practical scenarios [
Under Assumption
When one adjustment speed is very small, Jury stability condition (
We assume one adjustment speed is very small (e.g.,
We now investigate
Above all, if
Under Assumption
For system (
When delay decision is not considered, system (
Assume that
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 NeimarkSacker bifurcation, respectively, they constitute strong evidence combined with numerical simulations which show that such bifurcations do occur [
Figure
The stable region of adjustment speeds when
The bifurcation diagram of prices and LLE with respect to
We simulate the attractors of system (
Attractor when
Figures
The bifurcation diagram of
The bifurcation diagram of price and LLE with respect to
Parameter basin in
Parameter basin in
Parameter basin in
The average profits with respect to
Above all, we can get that adjustment speeds have obvious impacts on the dynamics of system (
Memory parameter
As it is easy to prove Proposition
Based on Figures
The stable region of adjustment speeds when
The stable region of adjustment speeds when
The stable region of adjustment speeds when
Above all, the stable region increases in memory parameter when delay decision is not adopted. Firms can expand the stable region by encouraging consumers to recall past prices in this case. When consumers only remember the most recent price, the impacts of delay decision depend on the delay parameter and customer perception coefficient. Firms should be cautious in choosing delay parameter in this situation.
Perception coefficient
The equilibrium price
As the proofs for
We show the stable region boundary of
The stable region of adjustment speeds when
The bifurcation diagram of
The average profits with respect to
Above all, equilibrium price and equilibrium profit decrease in customers’ perception coefficient, but the stable region of adjustment speeds increases in it. Compared with the state without reference price effect (
Reference price is determined by customers but also can be influenced by firms’ activities, such as advertising. Reference price in advertisements (advertised reference price) can draw consumers’ attention and influence consumers’ product evaluations [
Figure
Maximal profits and their locations.
Figure  Figure 
Figure 
Figure 
Figure 
Figure 
Figure 


3.84  2.98  1.14  1.06  0.81  0.81 

1  1  0.73  0.71  0.7  0.7 

2.838  2.708  2.762  2.698  2.35  2.434 
The average profits with respect to initial
The average profits with respect to initial
The average profits with respect to initial
We can know that when the Nash equilibrium point is asymptotically stable, (1) profits get their maximal values when both initial pricing and initial reference price are obviously bigger than the equilibrium price. In this situation, advertising is generally beneficial to both firms’ profits if the advertising cost is very small compared with the selling profit. But when the system is in closed invariant curve state or chaos state, (2) firms’ profits get their peak values when initial pricing and initial reference price are close to equilibrium point. A higher initial reference price relative to equilibrium price, despite improving the current profit, also enlarges the vibration of the evolution process.
This paper investigates the effect of internal reference price in competitive settings for firms that only master partial information, where reference prices are assumed to evolve according to an exponential smoothing process, and firms’ decisionmaking mechanism is modeled as a dynamic adjustment based on estimated marginal profits. By investigating the evolution characteristics of this dynamic adjustment and the impacts of key parameters, this study provides the following implications for firms with limited information to take advantage of reference price effect in competitive settings.
First, firms get their maximal profits when this evolution converges to the Nash equilibrium point, so firms should let the stability condition be satisfied. Fortunately, it is proved that the stability condition is satisfied when firms’ adjustment speeds are small, so firms should adopt small adjustment speeds when adjusting their decisions. Second, if customers’ reference prices are the last paid prices, there is a threshold belonging to
There are several extensions deserving further investigation. First, this paper only considers the internal reference price effect; the results can be more practical if both internal reference price effect and external reference price effect are taken into consideration at the same time. Second, it is assumed that firms are myopic in this paper; reference price effects are not directly reflected in their decisionmaking mechanism; researchers can investigate the case where firms try to maximize the sum profits of two or more periods. Then, researchers may get some more interesting results. Third, random factors are unavoidable and play important roles in the real market. Incorporating random factors into the study of reference price effect makes implications more profound and robust, so this is another interesting and challenging research direction.
The authors declare that they have no conflicts of interest.
This research was supported by the National Natural Science Foundation of China (nos. 61273231 and 71571131), Doctoral Fund of Ministry of Education of China (Grant no. 20130032110073), and Tianjin University Innovation Fund.