Dynamic Pricing in the Presence of Competition with Reference Price Effect

Transcription

1 Applied Mathematical Sciences, Vol. 8, 204, no. 74, HIKARI Ltd, Dynamic Pricing in the Presence of Competition with Reference Price Effect S. Kachani, Y. Oumanar,2 and N. Raissi 2 Department of Industrial Engineering and Operations Research Columbia University, New York, United States 2 Department of Applied Mathematics Mohammed V University, Rabat, Morocco Copyright 204 S. Kachani, Y. Oumanar and N. Raissi. 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 presents a reference price model for competition in a duopoly market for a single product. We derive Markov-Perfect equilibrium pricing policies. We provide a closed-form heuristic and demonstrate that it is very close to the equilibrium. We also derive closed-form solutions for pricing policies of a retailer who is an optimizer when its competitor follows one of three different suboptimal policies, and prove monotone convergence of these solutions. Finally, we use our results to analyze the impact of competition by comparing the revenue from the equilibrium policy with the one from a non-competition policy. Mathematics Subject Classification: 90C39, 90C46, 9B24, 90B99 Keywords: Dynamic Pricing, Reference Price, Management Decision making, Dynamic Optimization Introduction Traditional literature in pricing assumes that there is no interaction between the firms and their customers. While this assumption is often crucial to the

2 3694 S. Kachani, Y. Oumanar and N. Raissi analysis of the models, it can be very unrealistic. As a result, more sophisticated pricing models that incorporate customer behavior have emerged in recent years. A branch of the dynamic pricing literature has taken a particular interest in the concept of a reference price. Elmaghraby and Keskinocak 2003 noted that this is an important element that, in the past, was largely missing from the academic literature and price optimization softwares. For businesses that have repeated interaction with customers, price expectation is formed through past experiences. Subsequently, the utility of consumption is affected when there is a mismatch between the current and the expected price, and when the reference price is below the observed price, customers perceive gains and purchasing becomes more attractive Hardie et al The most common framework for the formation of the reference price is an exponentially weighted average of the past observed prices, where consumers are given memory parameters associated with past prices see Winer 985, 986, Kalwani et al. 990, Briesch et al. 997, Putler 992 and Greenleaf 995 for more details. For a review of the theoretical foundations and modeling of the reference price concept we refer the reader to Winer 988. Recent papers focus on developing dynamic pricing models with reference price effect to find optimal pricing strategies that would maximize retailers profit. This paper presents a dynamic pricing model in the presence of competition incorporating the concept of customer s reference price. The concept of reference price provides a good framework for modeling competition. The economics literature models competition in the market by directly assuming that the demand of a product is a function of it s competitor s price, in addition to its own price. We propose an alternative model where the retailers compete through their influences on the reference price. 2 Dynamic Pricing Model We consider the case of two retailers each selling a single product For a Multi- Product case, refer to Kachani et al. 204a. The demand for the product from the store i is a function of the price p and the reference price r and is denoted by D i p, r. It is given by where a i, b i, and c i are positive. D i p i, r = a i b i p i + c i r p i, for i =, 2 Competition is modeled through the dynamics of the reference price. the reference price has a memory coefficient α as in the monopoly case Kachani et al. 204b. We let µ i be the influence factor of store i on the reference price. We can view µ i as a measure of the store i s presence in the market.

3 Dynamic pricing with competition and reference price effect 3695 The dynamics of the reference price in this case are: where 0 α <. gp, p 2, r = αr + αµ p + µ 2 p 2 The retailer chooses a pricing policy that maximizes its total revenue. Hence, the Bellman equation we need to solve for the optimal pricing policy is given by: V r = sup D p, rp + βv gp, p 2, r p 3 Optimal pricing policy with a suboptimal competitor In this section, we derive the optimal pricing policy for a retailer when: its competitor s price is constant its competitor s price is a linear function of the retailer s own price its competitor follows a myopic pricing policy We also prove the monotone convergence in the three cases. 3. Competitor s price is constant Theorem 3. The optimal pricing policy for a retailer whose competitor s price is constant at p 2 is given by p r = c + 2 ααβγ µ r + a + αβδ µ + 2p 2 α 2 βγ µ µ 2 2b + c α 2 βγ µ 2 2 where γ = 2 α 2 βµ 2 b α 2 β + c α 2 β αβµ α b α 2 β + c α 2 β αβµ α 2 c 2 α2 βµ δ = a c + 2a ααβγ µ + 2p 2 αβγ 2b + c α + αµ µ 2 2b + c αβ c αβµ 2 α 2 βγ µ 2 ɛ = a + αβδ µ 2 + 4p 2 αβµ 2 b + c δ + p 2 αγ µ 2 + a αγ µ 4 βb + c α 2 βγ µ 2

4 3696 S. Kachani, Y. Oumanar and N. Raissi To prove this result, we postulate that the value function of retailer is given by V r = γ r 2 + δ r + ɛ The Bellman equation becomes V r = sup p [ b c + α 2 βγ ]p 2 +[a + c r + 2r ααβµ γ + αβµ δ + 2p 2 α 2 βµ µ 2 γ ]p +βαr + αµ 2 p 2 2 γ + αr + αµ 2 p 2 δ + ɛ 3 Since the competitor s price p 2 is constant, the optimal price can be obtained by simply optimizing the Bellman equation 3 over p. Solving V r p = 0 yields the optimal pricing policy 2. Then, by substituting 2 into 3, and collecting the coefficients of the polynomial, we can solve for γ, δ and ɛ and get the desired results. Next, we need to establish that the postulate is valid by showing that the coefficients of the value function are nonnegative real numbers. The expression for γ is the smaller root of the quadratic equation α 2 βµ 2 γ 2 b α 2 β + c α 2 β αβµ αγ + c 2 4 Suppose that 4 has real roots, then it is easy to see that γ is positive because the coefficients of the quadratic and the constant term are positive, whereas the coefficient of the linear term is negative. To show that the roots are real, we rewrite the term inside the square root as b α 2 β + c α 2 β αβµ α 2 c 2 α 2 βµ = α 2 βb 2 α 2 β + 2b c αβα µ + µ + c 2 βα + µ α 2 which is nonnegative. To show that δ is well defined and positive, we need to show that This is true if and only if γ < 2b αβ + c 2 αβ2 µ βµ 2 α 2 βµ 2 b α 2 β + c α 2 β αβµ α b α 2 β + c α 2 β αβµ α 2 c 2 α 2 βµ <2b αβ + c 2 αβ2 µ βµ b 2αβ + α 2 β c αβ µ2 α βµ < b α 2 β + c α 2 β αβµ α 2 c 2 α 2 βµ

5 Dynamic pricing with competition and reference price effect 3697 Which holds because the left hand side is negative. Finally, ɛ is well defined and positive if and only if γ < b + c α 2 βµ 2 Which is again true because b and c are positive and b α 2 β + c α 2 β αβµ α < 2b + c In addition, we can also derive the expression for the transition of the reference price under the optimal pricing policy. It is given by g r = 2b + c α + c αµ r + αµ a + αβδ µ + 2b + c p 2 µ 2 2b + c α 2 βγ µ 2 6 Finally, the steady state reference price is given by: r = a µ + αβδ µ 2 + 2b + c p 2 µ 2 2b + c 2 µ 2 αβγ µ 2 = a αβµ + p 2 2b αβ + c 2 αβ2 µ βµ µ 2 2b αβ + c 2 2αβ µ µ βµ The last equality is from substituting the expressions for δ, and using 4 to complete the square in the denominator Competitor s Price is a Linear Function of the Retailer s Price Proposition 3.2 The optimal pricing policy for a retailer whose competitor s price is a linear function of its own price, namely, p 2 = θp + ρ, with θ > 0 and ρ > 0, is given by p r = c + 2 ααβγ µ + θµ 2 r + a + αβδ µ + θµ α 2 βγ µ 2 µ + θµ 2 ρ 2b + c α 2 βγ µ + µ 2 θ 2 8 where γ = 2 α 2 βµ + µ 2 θ 2 b + c α 2 β c ααβµ + µ 2 θ b + c α 2 β c ααβµ + µ 2 θ 2 c 2 α2 βµ + µ 2 θ 2 δ = a c + 2 ααβγ µ + µ 2 θ + 2 αβγ µ 2 2b α + c µ + µ 2 θ + α2 µ µ 2 θρ 2b αβ + c 2 βµ + µ 2 θ α2 µ µ 2 θ 2 α 2 βγ µ + µ 2 θ 2 ɛ = 4 βb + c α 2 βγ µ + µ 2 θ 2 a + αβδ µ + µ 2 θ 2 +4 αβµ 2 b + c δ + a αγ µ + µ 2 θρ + 4b + c α 2 βγ µ 2 2ρ 2

6 3698 S. Kachani, Y. Oumanar and N. Raissi The proof to this proposition is similar to that of theorem 3.. Using the same postulate for the optimal value function, and substituting p 2 = θp + ρ into, we get V r = sup p [ b c + α 2 βγ µ + µ 2 θ 2 ]p 2 +[a + c r + αβµ + µ 2 θ2rαγ + δ + 2 αγ µ 2 ρ]p +βαr + αµ 2 ραrγ + δ + αγ µ 2 ρ + ɛ 9 Similarly, solving V r p = 0 yields the optimal pricing policy 8. Then, by substituting 8 into 9, and collecting the coefficients of the polynomial, we solve for γ, δ, and ɛ and get the desired result. Following the same approach in the proof of theorem 3., we can easily verify that γ, δ, and ɛ are well defined, positive and real. The transition of the reference price under the optimal pricing policy is given by g r = 2b + c α 2 βγ µ + µ 2 θ 2 αµ + µ 2 θa + αβδ µ + µ 2 θ +c rµ + µ 2 θ + α2 µ µ 2 θ + 2 αµ 2 ρ + 2b rα + αµ 2 ρ 0 And the steady state reference price is given by: r = a µ + µ 2 θ + αβδ µ + µ 2 θ 2 + 2b + c µ 2 ρ 2b + c α 2 βγ µ + µ 2 θ 2 = a αβµ + µ 2 θ + µ 2 2b αβ + c 2 βµ + µ 2 θ + α2 µ µ 2 θρ 2b αβ + c 2 + βµ + µ 2 θ 2αβ µ µ 2 θ 3.3 Competitor is a Myopic Optimizer A retailer who is a myopic optimizer prices the product to maximize the current-period revenue, i.e., p 2r = arg max D 2 p, rp = a 2 + c 2 r 2b 2 + c 2 2 Proposition 3.3 The optimal pricing policy for a retailer whose competitor is a myopic optimizer is given by p r = a + c r + αβµ 2rαγ + δ + 2 αγ µ 2 φ + rψ 2b + c α 2 βγ µ 2 3

7 Dynamic pricing with competition and reference price effect 3699 where φ = a 2 2b 2 +c 2 c 2 2b 2 +c 2 γ = 2 α 2 βµ 2 b βα + αµ 2 ψ 2 + c βα + αµ 2 ψα + αµ + µ 2 ψ b βα + αµ 2 ψ 2 + c βα + αµ 2 ψα + αµ + µ 2 ψ 2 /2 c 2 α 2 βµ 2 δ = 2 αβγ µ 2 φ2b α + αµ 2 ψ + c 2α + αµ + 2µ 2 ψ +a c + 2 αβγ µ α + αµ 2 ψ / 2b αβ αβµ 2 ψ +c 2 αβ2 µ 2µ 2 ψ βµ µ 2 ψ 2 α 2 βγ µ 2 ɛ = 4 βb + c α 2 βγ µ 2 a + αβδ µ 2 + 4b + c α 2 βγ µ 2 2φ 2 +4 αβb + c δ + a αγ µ µ 2 φ a Let φ = 2 and ψ = c 2 2b 2 +c 2 2b 2 +c 2. Then, the myopic pricing policy of the competitor can be written as p 2 r = ψr + φ. Since p 2 r is not a function of p, we get the optimal pricing policy 3 by substituting 2 into 2, which is the optimal pricing policy of theorem 3., when the competitor s price is constant. Similarly, by substituting 3 into the value function 3 in the proof of theorem 3., and collecting the coefficients of the polynomial, we solve for γ, δ, and ɛ and get the desired results. Following the same approach in the proof of theorem 3., we can easily verify that γ, δ, and ɛ are well defined, positive and real. The transition of the reference price under the optimal pricing policy is given by g r = 2b 2b + c α 2 βγ µ 2 + c α + αµ 2 ψ + c µ αr + αµ a + αβδ µ + 2b + c µ 2 φ 4 And the steady state reference price is given by: r = = a µ + αβδ µ 2 + 2b + c µ 2 φ 2b µ 2 ψ + c 2 µ 2µ 2 ψ 2 αβγ µ 2 a µ αβ αβµ 2 ψ + 2b + c αβ αβµ 2 ψ c αµ βµ 2 φ / 2b µ 2 ψ αβ αβµ 2 ψ + c 2 αβ 2αβ + 2βµ + 2µ 2 ψ +2 αβµ 2 ψµ + µ 2 ψ 5

8 3700 S. Kachani, Y. Oumanar and N. Raissi 3.4 Monotonocity and Convergence Proposition 3.4 When the competitor prices according to the assumptions of theorem 3., proposition 3.2, or proposition 3.3, the sequence of optimal prices p and reference prices r converge monotonically to their respective steady states.. Competitor s price is constant: Condition 5 implies that p r in 2 and g r in 6 are positive and strictly increasing for all r 0. Since g r is linear in r, it suffices to show that r is positive and finite. It is easy to see from the expression of r in 7 that this is true. 2. Competitor s price is a linear function of the retailer s price: Similarly, we need to show that p r in 8 and g r in 0 are positive and strictly increasing for all r 0, and the steady state price r in is positive and finite. All of these are true if γ in proposition 3.2 satisfies the following inequality: γ < b + c α 2 βµ + µ 2 θ holds because b and c are positive and b + c α 2 β c ααβµ + µ 2 θ < 2b + c 3. Competitor is a myopic optimizer: First, we need to show that p r in 3 and g r in 4 are positive and strictly increasing for all r 0. Since ψ 0 and φ 0, this is true if γ in proposition 3.3 satisfies the following inequality: γ < b + c α 2 βµ holds because b and c are positive, ψ <, and b βα + αµ 2 ψ 2 + c β α + αµ 2 ψα + αµ + µ 2 ψ < 2b + c Finally, the steady state price r in 5 is well defined and positive because 2 αβ 2αβ + 2βµ + 2µ 2 ψ + 2 αβµ 2 ψµ + µ 2 ψ = 2αβ µ 2 ψ µ µ 2 ψ + µ β + 2βµ 2 ψ + 2 µ 2 ψ βµ 2 ψ > 0

9 Dynamic pricing with competition and reference price effect Markov Perfect Equilibrium when both retailers are optimizers In this section, we consider the case where both retailers are optimizers. In this setting, the retailers are playing an infinitely-repeated game. Since they are both maximizing their total discounted revenue, The Markov Perfect MP equilibrium prices can be determined by solving the following Bellman equations simultaneously: V r = sup V p 2 r = sup p 2 D p, rp + βv rgp, p 2r, r D 2 p 2, rp 2 + βv 2 rgp r, p 2, r To solve this system of equations, we postulate that the value function for each retailer is given by V i r = γ i r 2 + δ i r + ɛ i, for i =, 2 and the optimal pricing policy is given by p i r = ζ i r + ν i, for i =, 2 8 First, let us look at the Bellman equation for retailer V r = sup p D p, rp + βv gp, p 2r, r = supa b p + c r p p p +β γ αr + αµ p + αµ 2 ζ 2 r + ν 2 2 +βδ αr + αµ p + αµ 2 ζ 2 r + ν 2 + ɛ = sup p [ b c + α 2 βγ µ 2 ]p 2 9 +[a + c r + 2r ααβγ µ + αβδ µ + 2 α 2 βγ µ µ 2 ζ 2 r + ν 2 ]p +βr 2 α 2 γ + rαδ + ɛ + αµ 2 ζ 2 r + ν 2 2rαγ + δ + αγ µ 2 ζ 2 r + ν 2 The optimal price can be determined by taking the partial derivative of 9 with respect to p, and setting it equal to zero. Solving = 0, we get V r p p r = c + 2βγ µ α α 2 + α 2 ζ 2 µ 2 r + a + αβµ δ + 2 αγ µ 2 ν 2 2b + c α 2 βγ µ 2 By collecting the coefficients of r, and equating them to the postulate in 8, we get: ζ = c + 2βγ µ α α 2 + α 2 ζ 2 µ 2 2b + c α 2 βµ 2 γ ν = a + αβµ δ + 2 αγ µ 2 ν 2 2b + c α 2 βγ µ

10 3702 S. Kachani, Y. Oumanar and N. Raissi Substituting 8 into 9, and collecting the coefficients of the polynomial, we can solve for γ, δ and ɛ and obtain the following solutions: γ = ζ c ζ b ζ βα + αζ µ + ζ 2 µ δ = a ζ + c ν 2ζ 2b ζ ν αβγ α + αζ µ + ζ 2 µ 2 µ ν + µ 2 ν 2 βα + αζ µ + ζ 2 µ 2 ɛ = a ν b + c ν 2 + αβµ ν + µ 2 ν 2 δ + αγ µ ν + µ 2 ν 2 β Solving 20 for γ, we get: γ = 2b ζ c 2ζ 2 αβµ α + αζ µ + ζ 2 µ 2 25 Finally, we equate 22 to 25 to get αβµ 2b + c α + αζ 2 µ 2 + αc µ ζ 2 2b + c βα + αζ 2 µ 2 2 ζ + c βα + αζ 2 µ 2 2 = 0 26 In addition, we have the following expression for retailer 2: αβµ 2 2b 2 + c 2 α + αζ µ + αc 2 µ 2 ζ 2 2 2b 2 + c 2 βα + αζ µ 2 ζ 2 + c 2 βα + αζ µ 2 = 0 27 By following a similar procedure, we can solve 2 for δ, and equate it to 23 to get 2 α 2 βγ µ 2 b 2 2βα + αζ 2 µ 2 c 2 βµ + 2ζ 2 µ 2 + α2 µ 2ζ 2 µ 2 ν + 2 α 2 βγ µ µ 2 ν 2 + a αβ αβζ 2 µ 2 = 0 28 Similarly, for retailer 2 we have 2 α 2 βγ 2 µ 2 2 b 2 2 2βα + αζ µ c 2 2 βµ 2 + 2ζ µ + α2 µ 2 2ζ µ ν α 2 βγ 2 µ 2 µ ν + a 2 αβ αβζ µ = 0 29 Now, we can solve for the equilibrium pricing policies p r and p 2r by simultaneously solving 26 and 27 for ζ and ζ 2, and 28 and 29 for ν and ν 2. Even though closed-form expressions for ζ and ζ 2 do not exist, we can determine their values by easily solving equations 26 and 27 numerically. Once

11 Dynamic pricing with competition and reference price effect 3703 ζ and ζ 2 are determined, γ and γ 2 are given by 22. Equations 28 and 29 can be solved analytically for the closed form solutions of ζ and ζ 2. However, their expressions are cumbersome and are omitted here. Finally, δ and δ 2 are given by 23, and ɛ and ɛ 2 are given by 24. We propose here a heuristic that approximates the MP equilibrium prices for which we provide a closed form expression. We will illustrate numerically that the solutions of this heuristic are very close to the equilibrium prices obtained by numerically solving the implicit equations given above. The heuristic assumes that each retailer maximizes its revenue assuming that its competitor s pricing policy is a function of its price and the reference price will remain constant. Under this assumption, each retailer solves for the best response pricing function p p 2, r and p 2 p, r. This best response function is the same as the optimal pricing policy given by 2. Since δ is linear and ɛ is quadratic in p 2, we rewrite them as δ = δ, p 2 + δ,0 ɛ = ɛ,2 p ɛ, p 2 + ɛ,0 Then, the best response pricing function in 2 can be rewritten as where p p 2, r = A p 2 + B r + C A = αβδ,µ + 2 α 2 βγ µ µ 2 2b + c α 2 βγ µ 2 B = c + 2 ααβγ µ 2b + c α 2 βγ µ 2 C = a + αβδ,0 µ 2b + c α 2 βγ µ 2 The equilibrium pricing policy for this heuristic is the solution of p = A p 2 + B r + C which is given by Where p 2 = A 2 p + B 2 r + C 2 p r = E r + F p 2r = E 2 r + F 2 E = A B 2 + B A A 2, F = A C 2 + C A A 2

12 3704 S. Kachani, Y. Oumanar and N. Raissi E = A 2B + B 2 A A 2, F = A 2C + C 2 A A 2 The transition of the reference price in this case is given by g r = gp r, p 2r, r = [α + αµ E + µ 2 E 2 ]r + αµ F + µ 2 F 2 And the steady-state reference price is given by: r = µ F + µ 2 F 2 µ E µ 2 E 2 To determine the accuracy of the heuristic, we randomly select 0,000 sets of demand parameters and compare the equilibrium prices p r and the total discounted revenue V r. For each set of parameters, we compute: i The percentage difference in revenue: { } V max i r 0 V i r 0 i=,2 Vi r 0 ii The average percentage difference in price during the path from r 0 to r : { } T p i r t p max i r t i=,2 T p i r t t=0 Where T is the first time the reference price reaches the steady state r within a certain tolerance error ε for table, we use ε = 0 6. Table summarizes the data and results of this analysis. We find that the median error of the total revenues and the prices are 0.003% and.4% respectively. The heuristic gives a very accurate estimate of the total discounted revenue, with a mean error of %, and 95 th percentile of only 4.3%. The error on the average difference in price is higher, with a mean of 4% and a 95 th percentile of 22%. 5 Comparison between Equilibrium Policy and Policy that Ignores Competition In this section, we study the effects of competition in our model. We compare the prices and the revenues under the equilibrium policy to the policy where both retailers maximize their revenue without considering the existence of the other. In other words, they follow the optimal pricing policy of the monopoly case Kachani et al Figure shows the transition of the equilibrium

13 Dynamic pricing with competition and reference price effect 3705 Table : Discrepancy between Prices and Revenues of Heuristic and Equilibrium Policy Parameter Distribution a Unif0,00 b Unif0,2 c Unif0,00 α Unif0, β Unif0, µ Unif0, r 0 Unif0,00 Mean St. Dev. 50th PCT 90th PCT 95th PCT Total Revenue.0% %.7% 4.3% Average Price 4.7% % 3.5% 2.6% prices and the reference price, when the demand for the two products are identical, but their prices have different weights µ i on the reference price. As expected, we find that prices are significantly lower when retailers are considering competition. Figure : Transition of Prices and Reference Prices when Retailers Ignore Competition Next, we study the effects of different demand parameters. We compute the loss of revenue when both retailers ignore the competition. The loss of revenue may be negative, meaning that the retailers may receive higher revenue when they both ignore competition. Figure 2 shows the effect of the magnitude of demand a on the revenue loss. When the retailers have equal influence on the reference price, both their revenues increase when competition is ignored, and

14 3706 S. Kachani, Y. Oumanar and N. Raissi the gain is increasing in a. When µ = 0.75, 0.25, both retailers still gain from ignoring competition, and the more influential retailer achieves more gain. However, when all of the influence lies with one retailer, ignoring the competition has no impact on that retailer s revenue, but significantly reduces the revenue of the other. The loss of revenue of the retailer with no influence is increasing in a. Figure 2: Effects of Magnitude of Demand on Revenue Loss Figure 3 shows the effect of the price sensitivity of demand b on the loss of revenue. Similarly, when the retailers have equal influence on the reference price, they receive more revenue when competition is ignored. However, the gain itself is decreasing in the price sensitivity b. When µ = 0.75, 0.25, both retailers gain from ignoring the competition, with the more influential retailer gaining more. Finally, the retailer that does not have any influence on the reference price experiences a loss of revenue when the competition is ignored. The loss in this case is decreasing in b. Figure 3: Effects of Price Sensitivity of Demand on Revenue Loss Figure 4 shows the impact of the magnitude of the reference price effect factor c on the loss of revenue when both retailers ignore competition. We observe

15 Dynamic pricing with competition and reference price effect 3707 that the loss/gain of the revenue is a non-monotonic function of the magnitude of the reference price effect factor c. For instance, when the two retailers have equal influence on the reference price, their revenue gains are initially increasing with c, but become decreaseng once c exceeds a certain threshold. The loss of revenue of the retailer that has no influence on the reference price is increasing in c. Figure 4: Effects of Magnitude of Reference Price Effects on revenue Loss Overall, we find the retailers are better off when they both ignore competition. However, this is only when both retailers have some influence on the reference price. the right panels of figures, 2, and 3 show that the loss revenue of retailer is never positive when µ 0.5. In fact, it is only positive when µ is close to zero in these examples, when µ Conclusion In this paper, we considered a new competitive model where retailers interact through their influences on the customers reference price. In the case of two retailers selling a single product, we derived closed-form solutions for pricing policies for a retailer who is an optimizer when its competitor is following three different suboptimal policies and proved monotone convergence of these policies. When both retailers are optimizers, we provided a system of cubic equations, which can be solved numerically for the MP equilibrium prices. We provided a closed-form heuristic solution that is very close to the MP equilibrium prices. We found that competition reduces revenue in that revenue from equilibrium prices is lower that revenue from the case where both retailers ignore competition and optimize their prices. We intend to extend our model to consider pricing of multiple products in a duopoly market.

16 3708 S. Kachani, Y. Oumanar and N. Raissi References [] Briesch, R. A., L. Krishnamurthi, T. Mazumdar, S. P. Raj, A Comparative Analysis of Reference Price Models. Journal of Consumer research: An Interdisciplinary Quarterly, 997, [2] Elmaghraby, W., P. Keskinocak, Dynamic Pricing in the Presence of Inventory Considerations: Research Overview, Current Practices, and Future Directions. Management Science, 2003, [3] Greenleaf, E. A, The Impact of Reference Price Effects on the Profitability of Price Promotions. Marketing Science, 995, [4] Hardie, B. G. S., E. J. Johnson, and P. S. Fader, Modeling Loss Aversion and Reference Dependence Effects on Brand Choice. Marketing Science, 993, Fall; 2: [5] Kachani, S., Y. Oumanar, and N. Raissi, Dynamic Pricing for Multiple Products with Consumer Reference Price Effect. International Journal of Applied Mathematics and Statistics, 204, Vol. 52, 2, 4. [6] Kachani, S., Y. Oumanar, and N. Raissi, A Closed-Form Solution for a Dynamic Pricing Model with Reference Price Effect. Applied Mathematical Sciences, 204, Vol. 8, 34, [7] Kalwani, M. U., C. K. Yim, H. J. Rinne, and Y. Sugita, A Price Expectations Model of Customer Brand Choice. Journal of Marketing Research, 990, [8] Putler, D. S, Incorporating Reference Price Effects into a Theory of Consumer Choice. Marketing Science, 992, [9] Winer, R. S., A price Vector Model of Demand for Customer Durables: Preliminary Developments. Marketing Science, 985, [0] Winer, R. S., A Reference Price Model of Brand Choice for Frequently Purchased Products. Journal of Consumer Research: An Interdisciplinary Quarterly, 986, [] Winer, R. S., Behavioral Perspectives on Pricing: Buyers Subjective Perception of Price Revisited. Pp in Issues in Pricing: The Theory and Research, Timothy M.D., 988, Lexington, MA: Lexington. Received: April 5, 204