J Optim Theory Appl (2008) 138: 27–44 DOI 10.1007/s10957-008-9368-4

Dual-Role Based Pricing in a Dynamic and Competitive Environment G.E. Fruchter

Published online: 25 April 2008 © Springer Science+Business Media, LLC 2008

Abstract The dual role of price, as a product attribute signaling quality and as a measure of sacrifice, serving as a benchmark for comparing the utility gains from superior product quality, is now well established in the marketing and economic literature. However, knowledge about their long-run impact and the influence of competition on these effects still remains very sparse. In the current paper, with reference to a dynamic and competitive framework, an analytical model is proposed to help determining optimal decision rules for price incorporating both roles. The main results are as follows: (i) The optimal pricing policy is determined as a Nash equilibrium strategy. (ii) The resulting equilibrium price is higher than an equilibrium that disregards the carryover price effects. (iii) For a symmetric competition, we provide normative rules on how firms should set prices as a function of the perceived quality; particularly, how the price should be set initially, when there is little product familiarity and the perceived quality is low, and how this price should vary as the perceived quality increases. (iv) At steady state, we find that the level of equilibrium margin, in percentage terms, decreases with the elasticity of demand with respect to the brand price, but this decrease is moderated by the elasticity of demand with respect to the brand perceived quality, the cross elasticity of demand with respect to the competitor’s perceived quality and the effect of the competitor’s current price on the firm’s perceived quality deterioration. Keywords Marketing · Price · Price signaling quality · Perceived quality · Differential games · Nash equilibrium

Communicated by G. Leitmann. The author thanks Konstantin Kogan for helpful discussions and comments. G.E. Fruchter () Graduate School of Business Administration, Bar-Ilan University, Ramat-Gan 52900, Israel e-mail: [email protected]; [email protected]

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1 Introduction Over the past several decades, researchers have adopted different perspectives to explain the relationship between price and quality. Some inquiries have focused on the relationship between objective product quality and price (Refs. [1–3]). However, with the proliferation of the perspective that purchase decisions are not based on objective facts, but rather on subjective beliefs, the emphasis has shifted from objective product quality to quality as perceived by consumers, what is called perceived quality (e.g. Refs. [4–11]). For a good review on this subject, see Ref. [12]. Consumers use of price to impute lasting product quality perceptions has been demonstrated in numerous empirical studies (e.g. Refs. [10, 13–16]). Reference [17] studied this phenomenon in an international context and found that it is not culture bound; the relative and absolute importance of these cues in imputing quality perceptions were found to be consistent across cultures. There is ample evidence in the marketing and economics literature that consumers view high prices as indicators of high perceived quality and brand reputation (Refs. [9, 18–24]). However, apart from buying a familiar brand for its perceived quality, value is also sought by the consumer in that brand being available at a cheaper price (Refs. [25, 26]). Thus, price has both long-run and short-run effects on sales. In other words, price serves two distinct roles in consumers’ purchasing decisions. First, as a product attribute, price affects the perceived quality. Second, as a measure of sacrifice, price serves as a benchmark for comparing utility gains with superior product quality. The question is: What should the optimal price be in order to maximize the firm profits, taking the dual role of price into account? Reference [27] assumes that, while the fact that consumers may purchase less of a brand when its price rises is immediately observed, the opposite effect, due to the perceived quality increase when the price rises, is neither easily extractable nor observable. The reference provides an empirical approach to account, in an unbiased way, for this long-term price effect on the consumers choices. Despite the aforementioned and far-reaching evidence of diverse price effects on consumers and efficient empirical approaches to measure them as yet the phenomenon has not been modeled to develop efficient pricing strategies. The current paper is among the first analytical attempts to study simultaneous price effects, within the framework of a dynamic and competitive environment, and its intent is to contribute to the marketing literature. First, we formally integrated long-run price effects on sales through perceived quality together with the usual short-run price effects on sales. Second, assuming the perceived quality as a form of goodwill, we modified the well-known dynamic model of Ref. [28] to include competition and price effects. By formulating a differential game, we were able to develop Nash equilibrium strategies that form pricing decision rules for firms over time, incorporating the dual role of pricing. This equilibrium has major theoretical and strategic implications. It highlights the difference between (a) the influence of competition, when only the role of price as “being a measure of sacrifice” is incorporated and (b) the influence of competition, when the dual role is considered. In addition, it addresses the relative short-run and long-run price effects on the equilibrium strategy. Moreover, for a symmetric competition, we provide normative rules on how firms should

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set prices as a function of perceived quality, particularly as regards the initial price, when little product familiarity exists and the perceived quality is low, and how price change as the perceived quality increases. From the modeling point of view, the differential game proposed in this paper recalls analogous models proposed in the quantitative marketing literature (see for example the survey by Ref. [29]). The remainder of the paper is organized as follows: In Sect. 2, we present the model and the notation. In Sect. 3, we derive the equilibrium pricing policies. In Sect. 4, we characterize the steady-state equilibrium. In Sect. 5, we illustrate the solutions by numerical examples. In Sect. 6 we provide marketing implications. Finally, in Sect. 7, we present our concluding remarks and future directions.

2 Model Formulation and Notation We consider two competitors that need to decide on their brands’ pricing strategy over time. Since consumers tend to associate high prices with high quality, the price assigned to the product affects the perceived quality. The existence of carryover price effects leads to an accumulation of perceived quality for brands over time. This necessitates modeling the relationship between price and perceived quality in a dynamic framework. The resulting quality directly impacts the sales of the particular product, and in turn, on its profits. In addition, following the classical demand function, price is negatively related to sales. Thus, price has both long-run effects on sales, through perceived quality, and short-run effects, through the demand function. Alternatively, the long-run and short-run effects of price on sales are a result of the dual role of price as they relate to consumers judgments. First, as a product attribute, price affects perceived quality. Second, as a measure of sacrifice, price serves as a benchmark for comparing utility gains for superior product quality, directly affecting the demand. To better understand how a firm should strategically design a brand’s price incorporating both roles, we have presented a formal decision-making model below. Let xi = xi (t) be the perceived quality of brand i, i = 1, 2, at time t. The variable xi summarizes the previously perceived quality of similar products sold under the same brand name, as well as all past effects of such products’ prices. Thus, this can be thought of as the accumulated goodwill of brand i at time t. Let pi = pi (t) be the price of brand i, i = 1, 2, at time t. Assuming that price of each brand affects its own perceived quality, and thus its goodwill stock, and to model these relationships in a dynamic framework, we extended the well-known model of Ref. [28] to the following dynamic equations, x˙i (t) = ki pi (t) − (εi + ηi pj (t))xi (t),

xi (0) = x0i ,

i = 1, 2.

(1)

The left-hand side of (1) represents the change in perceived quality. The first term on the right-hand side represents the impact of the current brand’s price on the brand’s perceived quality. The second term in (1) reflects the rate at which the perceived quality of the brand deteriorates, taking into account consumer “fatigue” regarding the brand’s goods and the competitor actions. It is assumed that the rate of depreciation is a function of the level of perceived quality (goodwill). We take ki ≥ 0 as

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representing the effect of the current price on the perceived quality and, εi ≥ 0 and ηi ≥ 0, as the rate of fatigue and the effect of the competitor current price, on perceived quality deterioration, respectively. The constants ki , εi , ηi can be empirically determined. For the given ki , εi , ηi , (1) states that a change in perceived quality will be positive (negative) if the impact of the current brand price on its perceived quality ki pi (t), is higher (lower) than the impact of the weariness and current competitor’s actions on its current perceived quality (εi + ηi pj (t))xi (t). Note that the dynamic setting in (1) is designed to capture the effects of pricing as a signaling device of perceived product quality in a competitive environment. Thus, our analysis in the sequel is restricted to first purchases (durable goods).1 To formulate the problem for the firm i, we assume that the rate of sales Si = Si (t) is governed by the long-run components (perceived qualities) xi and xj and the shortrun components (prices) pi and pj as Si = Si (pi , pj , xi , xj ).

(2)

We assume that the rate of sales decreases with the firm price but increases with the competitor’s brand price. In addition, we assume that the rate of sales increases with the brand perceived quality xi , but decreases with the competitor’s brand perceived quality xj . This leads to the following conditions on the first partial derivatives of Si : ∂Si /∂pi < 0,

∂Si /∂pj > 0

and

∂Si /∂xi > 0,

∂Si /∂xj < 0.

(3)

Another assumption is that cost of production of a unit sale of brand i is constant. Let us denote this constant by ci . Thus, the net profit equation at time t is Ri (pi , pj , xi , xj ) = (pi − ci )Si (pi , pj , xi , xj ).

(4)

Assuming that each firm maximizes the present value of the net profit streams discounted at a fixed discount rate r over an infinite horizon, the equilibrium pricing policy pi∗ solves the following differential game:    ∞ max Πi = Ri (pi , pj , xi , xj )e−rt dt , (5a) pi

0

s.t. x˙i = ki pi − (εi + ηi pj (t))xi ,

xi (0) = x0i ,

i = 1, 2, j = i.

(5b)

3 Equilibrium Pricing Policies By explicitly recognizing that each competitor’s decision is affected by the other’s actions, we derive their equilibrium pricing policies over time. For this objective, we employ techniques of dynamic optimization (e.g. Ref. [32]). This is accomplished by constructing the Hamiltonians for the two firms and then using them to obtain the 1 Differently from durable goods, in experienced goods, consumers learn about the true quality of the

product by consuming it (see for example, Refs. [30, 31]). Thus, in repeat purchases, the effect of pricing as a signaling device is diminished by the learning process.

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necessary conditions for the Nash equilibrium. We define the current-value Hamiltonians Hi , i = 1, 2, as follows: Hi = (pi − ci )Si (pi , pj , xi , xj ) + λii [ki pi − (εi + ηi pj )xi ] j

+ λi [kj pj − (εj + ηj pi )xj ], i = 1, 2, = i.

(6) j

In (6), we introduced new variables λii and λi that are the current adjoint variables.2 They represent the shadow prices associated with a unit change in the values of the j perceived qualities xi and xj , at time t, respectively. In other words, λii (λi ) is the net benefit or loss to the firm from improving the firm (competitor) perceived quality by one more unit at time t. Definition 3.1 We term βii = (xi /Si )∂Si /∂xi

and βij = −(xj /Si )∂Si /∂xj

(7)

as the elasticity of demand with respect to the firm perceived quality and the crosselasticity of demand with respect to the competitor perceived quality. Definition 3.2 We term μi = −(pi /Si )∂Si /∂pi

(8)

as the elasticity of demand with respect to the firm brand price.3 3.1 Open-Loop Nash Equilibrium Pricing Strategies We are now in a position to state the necessary conditions that the equilibrium dynamic pricing strategy of each firm must satisfy. We do this in Theorem 3.1. More exactly, the following theorem finds necessary conditions for a pricing strategy that is a function of time. This type of strategy is termed an open-loop strategy. Theorem 3.1 At any time t, t > 0, the open-loop Nash equilibrium strategy (p1∗ , p2∗ ) of the differential game (5), satisfies the following necessary conditions: pi∗ = ci −

j

ki λii − ηj λi xj Si − , ∂Si /∂pi ∂Si /∂pi

i = 1, 2, j = i,

where 2 The adjoint variables (costates) are assumed to have continuous first derivatives. 3 As is common in the pricing literature (e.g. Ref. [33]), it is assumed that μ > 1. i

(9)

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x˙i = ki pi∗ − (εi + ηi pj∗ )xi ,

xi (0) = x0i ,

λ˙ ii = [r + (εi + ηi pj∗ )]λii − (pi∗ − ci )

∂Si , ∂xi

j j λ˙ i = [r + (εj + ηj pi∗ )]λi − (pi∗ − ci )

∂Si , ∂xj

(10a) lim e−rt λii (t) = 0,

t→∞

(10b)

lim e−rt λi (t) = 0 (10c) j

t→∞

with i = 1, 2, j = i and Si = Si (pi∗ , pj∗ , xi , xj ). For proof, see Appendix. Thus, to find a pair (p1∗ , p2∗ ) that forms an open-loop Nash equilibrium strategy of the differential game (5), we must solve the TPBVP (10) and substitute the solutions j j xi (t), xj (t), λii (t), λi (t), λij (t), λj (t) into (9). To solve (10), the shooting method can be employed for a given specification of the sales rate function Si . 3.2 Myopic vs. Strategic Price The equilibrium condition in (9) includes the case where the duopolists behave myi opically and set prices as pi = μμi −1 ci , for i = 1, 2, according to the Nash-Cournot equilibrium, by maximizing the integrand in (5) and disregarding the dynamic equations (1) or, in other words, disregarding the effect of the current price on the perceived quality in the future or disregarding the carryover price effects. Alternatively, the equilibrium in (9) resembles the myopic decision rule that price markup over marginal cost depends on demand elasticity, except that here we modify marginal costs j

by adding

ki λii −ηj λi xj ∂S (− ∂pi ) i

, which accounts for long-run price effects. Thus, for example,

if there is a benefit from improving the perceived quality of the firm (λii > 0), and a j loss from improvement of the competitor’s perceived quality (λi < 0), marginal costs must be modified to account for this, causing prices to be higher for products where price signals (influence) quality (thus ki > 0) and has an impact on the competitor’s quality deterioration (ηj xj > 0). j Therefore, we see that the sign of the shadow prices λii and λi determine whether the strategic equilibrium price will be lower or higher than the price of a myopic duopolist facing the same conditions, such as a price that corresponds to λii = 0 and j j λi = 0. If λii > 0 and λi < 0, then the strategic price will be higher. In the next j proposition, we study the signs of λii and λi . 3.3 Shadow Prices and Strategic Price Proposition 3.1 Assume that ∂ 2 Si /∂pi2 ≤ 0. Then: (i)

j

λii (t) > 0 and λi (t) < 0,

(11) ∀t.

(12)

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(ii) The Nash equilibrium pricing strategy of brand i, i = 1, 2, is above the myopic j price (the price that corresponds to λii = 0 and λi = 0). For proof, see Appendix. Proposition 3.1 states that the equilibrium brand’s price, which accounts for the effect of current pricing strategy on the perceived quality in the future and the long-run effects of price will be higher than the myopic price. That is to say, incorporating the long-run effects along with the short-run effects influences price; thus, considering the dual role of price permits higher prices. This result is consistent with a brand differentiation policy that facilitates higher prices, for example compare with industrial organization literature. 3.4 Feedback Nash Equilibrium Pricing Strategy: Symmetric Competition A strategy that is specified as depending on the state of variables rather than directly on t is referred to as a feedback strategy. This is the strategy we look for. More exactly, we want to find pi∗ as a function of xi . Feedback strategies are more attractive; for example, given such a relationship between the price and firm’s perceived quality (the state xi ), the manager can readily determine the price at equilibrium that corresponds to the current perceived quality, which can be measured by asking customers about their perceptions. To be able to derive an explicit form of such a feedback strategy, we take pi∗ = pi∗ (xi ); next, we assume a symmetric competition. Symmetric competition means that the firms have the same parameters in the game described in (5). The aim is to find a symmetric equilibrium, that is, pi∗ = pj∗ = p ∗ .

(13)

This implies xi = xj = x

j

and λii = λj = λ1

j

and λi = λij = λ2 .

(14)

The following theorem presents a general form solution for the feedback Nash equilibrium, which can then be employed to compute a closed form equilibrium strategy by examining the sufficient conditions for equilibrium. Sufficient Conditions In addition to the necessary conditions, we must also verify second-order conditions to ensure that we are indeed maximizing profits. The sufficient conditions are 2∂Si /∂pi + (pi − ci )∂ 2 Si /∂pi2 < 0,

i = 1, 2.

(15)

Theorem 3.2 Consider the differential game stated in (5), for a symmetric competition, and assume that the sufficient condition holds in some neighborhood of p ∗ . Then, the necessary condition (9) defines a unique local time-invariant feedback Nash equilibrium price of the form p ∗ = p ∗ (x, Φ(x), ϕ(x)),

(16)

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where Φ(x) and ϕ(x) satisfy the following backward differential equations, Φ (x)[kp ∗ − (ε + ηp ∗ )x] = [r + (ε + ηp ∗ )]Φ(x) − (p ∗ − c)

∂Si , ∂xi

Φ(x ss ) = λss 1 , ϕ (x)[kp ∗ − (ε + ηp ∗ )x] = [r + (ε + ηp ∗ )]ϕ(x) − (p ∗ − c) ϕ(x ss ) = λss 2 ,

∂Si , ∂xj

(17)

(18)

ss 4 where x ss , λss 1 and λ2 are the steady-state values of the state and costates, and ∗ where Si = Si (p , x).

For proof, see the Appendix. The solution based on Theorem 3.2 is made clear in Sect. 5. Note that the time trajectories of our specific feedback strategy coincides with the open-loop strategy, and it has the form, p(x(t), λ1 (t), λ2 (t)), where x(t), λ1 (t), λ2 (t) are as in (10) for a symmetric case.

4 Steady-State Nash Equilibrium In this section, we attempt to better understand the long-term behavior, that is, the behavior of solutions such as t → ∞, thus at steady-state. Such behavior is especially important for well-established brands. Furthermore, we are interested in characterizing the steady-state Nash equilibrium and comparing it with the duopolist’s price, which is either myopic or accounts for only a single price role. Theorem 4.1 Consider the differential game (5) and Definitions 3.1 and 3.2 at steady state. Suppose that the pair (p1∗ss , p2∗ss ) satisfies the set of two algebraic equations (piss − ci ) = piss (μss i −

1 βiiss [1+r/(εi +ηi pjss )]

−

βijss ) [1+(εj +r)/(ηj piss )]

,

i = 1, 2, i = j, (19)

in piss and pjss . Then, (p1∗ss , p2∗ss ) forms a steady-state Nash equilibrium strategy of the above differential game. For proof, see the Appendix. Theorem 4.1 gives us a decision rule on how the level of margin (price) in long-run depends on the direction of substitution (through μss i ) and how the substitution size can be diminished by (a) incorporating the long-run price effects by the elasticity of demand with respect to the firm’s perceived quality and (b) the cross elasticity of demand with respect to the competitor’s perceived quality (through βiiss and βijss ). 4 x ss , λss and λss are solutions of x˙ = 0, λ ˙ 1 = 0 and λ˙ 2 = 0. 1 2

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The theorem also shows how this moderation depends on other model parameters (εi , ηi , εj , ηj ). Note that the steady-state equilibrium does not depend explicitly on the current price effect on the perceived quality, ki . Considering (19), we conclude with some insights summarized in the following proposition. Proposition 4.1 Under the assumptions of Theorem 3.2, at steady state: (i) Higher elasticity of demand with respect to price μss i permits lower equilibrium margin, in percentage terms. (ii) On the other hand, higher elasticity of demand with respect to the firm’s perceived quality βiiss and/or higher cross elasticity of demand with respect to the competitor’s perceived quality βijss leads to a higher equilibrium margin, in percentage terms. This increase is larger when the effects of the competitor’s price on the firm’s perceived quality ηi and/or the fatigue effects εi increase. For proof, see Appendix. Proposition 4.1 shows the strategic implications of our model. Taking into account both price roles, the substitution direction is diminished by considering the additional attribute created by the long-run effects of price, through the increase of elasticity of demand with respect to the firm’s perceived quality and the cross elasticity of demand with respect to the competitor’s perceived quality (through βiiss and βijss ). Moreover, the substitution effect is diminished and our own price level goes up by an increase of the competitor’s actions’ effectiveness. An implication of Proposition 4.1 is that considering the dual role of price, as a product attribute, signaling quality, and as a measure of sacrifice (the constraint role), helps managers to differentiate the product, thus softening competition and charging higher prices. Said differently, the consideration of the dual role of price creates a mechanism of product differentiation.5 4.1 Price—Single Role vs. Dual Role When the price accounts for only the short-run effects by considering only the role of being a measure of sacrifice, we say that the price accounts for only a single role. This means that the sales response function Si only depends on pi and pj . Formally, this is the same as fixing βii = 0 and βij = 0. Considering (19), we conclude with the immediate result. Proposition 4.2 At steady state, a pricing strategy that accounts for only a single role results is a lower equilibrium margin, in percentage terms. For proof, see Appendix. 5 Not surprisingly, our finding is consistent with the industrial organization literature on product differen-

tiation.

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The above result implies that the price level of a duopolist that employs only a single price role is similar to the price level of a myopic duopolist that accounts for the dual role, but ignores the effect of the current price on the future perceived quality. To illustrate our solutions and acquire more insights, we next consider some illustrative examples.

5 Illustrative Examples For the purpose of illustration, we will consider the following specification for the sales response Si :  γi βi xi Si = Si (pi , pj , xi , xj ) = [(ai − bi pi ) + αi (pj − pi )]xi . (20) xj We consider a symmetric case; thus, ai = aj = a,

bi = bj = b,

αi = αj = α,

βi = βj = β,

γi = γj = γ ,

(parameters for the price and perceived quality response function in (20)), ki = kj = k,

εi = εj = ε,

ηi = ηj = η,

(parameters for the dynamic equation in (1)), ci = cj = c = 0,

r = 1 (cost and discount rate parameters).

Considering the necessary condition (9) and the specification (20), letting λ1 = Φ(x) and λ2 = ϕ(x), (16) becomes p ∗ (x) =

a + (kΦ(x) − ηxϕ(x))x −β . α + 2b

(21)

Substituting (21) into (17) and (18), and solving simultaneously backward from the steady-state values leads to the solutions Φ(x) and ϕ(x). Substituting these solutions in (21), we get the feedback solution and price as a function of the perceived quality x. We use the following parameter values for the base case of the numerical example, η = 0.01,

ε = 0.01,

a = 0.1,

b = 0.01,

α = 0.001,

β = 2,

γ = 1,

k = 0.005,

c = 0,

r = 0.1.

(22)

For these parameters’ values, we displayed the feedback equilibrium solution (21) as well as the equilibrium’s time-trajectory in Fig. 1a. For comparison, we also display the myopic price denoted by pMyopic in Fig. 1a. Moreover, we conducted a sensitive analysis by setting the model’s parameters as in the base case and only changing one parameter each time. We focused on analyzing the effects of model parameters for both p(x) and p(t) = p(x(t)). From the time trajectory p(t), we can especially learn about the level of the steady-state. These results are all displayed in Figs. 1–3. Below, we summarize our findings.

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Fig. 1 Price and perceived quality: base case

Fig. 2 Brand price effect on perceived quality

Fig. 3 Competitor price effect on perceived quality

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Perceived Quality Trajectory of the Equilibrium (Base Case) On the left-hand side of Fig. 1, we display the feedback solution, p(x). We observe that, when the x is low, and under its steady-state value, x ss , it is optimal to set a high price (above its steadystate value), p ∗ (x) > p ss . Then, over time (the arrows in Fig. 1, left-hand side), the perceived quality increases, gradually lowering the brand’s price until the perceived quality reaches its steady-state value. On the other hand, when the perceived quality is high, and above its steady-state value, x ss , it is optimal to set a relatively low price, under its steady-state value, p ∗ (x) < p ss . Then, as the perceived quality depreciates toward its steady-state value, it is optimal to gradually increase the price. Furthermore, following Fig. 1, if p ∗ (x) > p ss , p ∗ decreases much faster to the value p ss than it increases to p ss when p ∗ (x) < p ss . Thus, the equilibrium policy for prices lower than p ss is to increase slowly to p ss as the perceived quality depreciates to the steady-state value x ss . The optimal policy for prices higher than p ss is to quickly decrease to p ss .6 From the comparison with myopic price, we can see that the myopic price is always under the strategic price. Time-Trajectory On the right-hand side of Fig. 1, we describe the time-trajectory for the case of initially low perceived quality, x(0) < x ss . As we can see, the price quickly converges to the steady state level. Brand’s Price Effect on Perceived Quality (k) The left-hand side of Fig. 2 compares the base case of the perceived quality trajectory of price with the trajectory that is represented by a case with a lower price effect on perceived quality, k = 0.001. As indicated in Fig. 2, the trajectory of this case is under the trajectory of the base case. Put differently, the same price level signals a lower quality for a lower price effect on perceived quality. Moreover, the depreciation of the perceived quality increases as we reach the steady-state value. As expected (Theorem 4.1), the right-hand side of Fig. 2 indicates that the steady-state value of price is not influenced by k. However, before reaching the steady-state, as indicated in Fig. 1b (right-hand side), when the effect of price on perceived quality is weaker, the price level is higher. Competitor’s Price Effect on Perceived Quality Deterioration (η) The left-hand side of Fig. 3 compares the base case of the perceived quality trajectory of price with the trajectory that is represented by a case with a higher competitor’s price effect on perceived quality, η = 0.025. As indicated in Fig. 3, the trajectory of this case is under the trajectory of the base case. Thus, the same price level signals a lower quality when a competitor’s action has stronger effects on the firm’s perceived quality deterioration. Again, the depreciation of the perceived quality increases as we reach the steady-sate value. As expected (Proposition 4.1), the right-hand side of Fig. 3 indicates that the price level as regards the steady-state level increases with the competitor’s actions. Similarly we compare the base case of the perceived quality trajectory of price with the trajectory that is represented by a case with lower elasticity of demand with 6 This type of behavior is akin to goodwill according to the Ref. [28] model.

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respect to a firm’s perceived quality (βii = 1.5 + 1 < 2 + 1). We obtain that the trajectory of this case is under the trajectory of the base case. Thus, the same price level signals a lower quality when the elasticity is weaker. As expected (Proposition 4.1), the price level as regards the steady-state level increases with the elasticity. Another exercise we did was to compare the base case of the perceived quality trajectory of price with the trajectory that is represented by a case with higher rate of fatigue ε. The trajectory of this case is under the trajectory of the base case. Thus, the same price level signals a lower quality when the rate of fatigue is higher, which again makes sense. Again, as expected (Proposition 4.1), the price level as regards the steady-state level increases with rate of fatigue. 6 Marketing Implications The equilibrium policy has a very interesting pattern. At the beginning, the equilibrium should be set at a high value, and then over time, with the increase of the perceived quality, the brand’s price should be gradually lowered until the perceived quality reaches its steady-state value. The intuition behind this behavior is that initially, when there is little product familiarity and the perceived quality is low, a high price will signal a strong perceived quality and enable its high quality to remain stable by gradually lowering the price until the perceived quality reaches the steady-state value. In the steady-state, it is useful to contrast our results with myopic behavior, such as that of a duopolist that either disregards the carryover price effects or one that accounts for only a single price role (both considers only the short-run effects of price). It is well known that as long as the substitution is high, as a result of competition (or homogeneity), the elasticity of demand with respect to price will remain high, while the margins will remain low. By differentiation, the substitution direction can be moderated. Taking into account both roles of price, we demonstrate that a strategic duopolist can diminish the substitution direction; the substitution direction is accomplished, for example, by the elasticity of demand with respect to the firm’s perceived quality. Thus incorporation of long-run price effects permits higher prices. Moreover, our analysis shows that the level of price at steady-state increases with the effectiveness of the competitor’s actions. The explanation behind this states that the stronger the competition, the more the perceived quality deteriorates. Therefore, the firm needs to set a higher price in order to maintain strong perceived quality. Hence, consideration of the long-run price effects causes a higher steady-state solution even under competition. In other words, for brands where price has a dual role, in the long-run, the issue of competition works in an opposite way than for brands where price serves a single role (when price decreases as a result of competition, c.f. Refs. [34, 35]), or when the duopolist behaves myopically, ignoring the effect of current price on future perceived quality. 7 Conclusions, Limitations and Future Directions In the current paper, we have proposed an analytical model that presents a formal examination of how a firm should design a competitive pricing strategy over time that

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includes both roles of price as an attribute to affect perceived quality and as a sacrifice to benchmark consumers’ utility gains. The main findings of our analytical model are presented below. Integrating long-run price effects on sales through perceived quality together with the usual short-run price effects on sales and understanding perceived quality as a form of goodwill, we modify the well-known Ref. [28] dynamic model to include competition and price effects. By formulating a differential game, we were able to develop Nash equilibrium strategies that create pricing decision rules for firms over time, incorporating the dual role of pricing. The equilibrium has major theoretical and strategic implications. It highlights the difference between the influence of competition when (a) only the role of price as “being a measure of sacrifice” is incorporated and (b) the influence of competition when the dual role is considered. In addition, it addresses the relative price effects, of short- and long-run, on the equilibrium strategy. For a symmetric competition, we provided normative rules on how firms should set prices as a function of perceived quality; particularly, how the price should be set initially, when little product familiarity exists and the perceived quality is low, and how this price should vary as perceived quality increases. Some of our modeling assumptions can be relaxed to address more complex situations. First, we assume that the effect of price on perceived quality is constant. However, since the relative importance of money and product quality may change over time, a future direction will be to revise this assumption and consider this effect to be changing over time in a dynamic way. Second, the analysis presented here was conducted within the framework of a single quality indicator. Since firms also use advertising to signal quality, finding the optimal combination of advertising and pricing strategies in a competitive environment, as well as identifications of situations in which each of those indicators are important, can be an interesting topic for future research. Another direction for future research is to validate the suggested model empirically. This requires data on the involved variables—perceived quality and price— over a given period of time. It will be interesting to compare real pricing decisions against the normative rules our model provides.

Appendix: Proofs

Proof of Theorem 3.1 Considering (6), the necessary first-order optimality conditions are given by ∂Hi /∂pi = 0 or j

Si + (pi − ci )∂Si /∂pi + ki λii − ηj λi xj = 0,

i = 1, 2, j = i

(23)

and the adjoint equations with the terminal conditions, λ˙ ii = rλii −

∂Hi ∂Si = [r + (εi + ηi pj )]λii − (pi − ci ) , ∂xi ∂xi

lim e−rt λii (t) = 0,

t→∞

(24)

J Optim Theory Appl (2008) 138: 27–44 j j λ˙ i = rλi −

41

∂Hi ∂Si j = [r + (εj + ηj pi )]λi − (pi − ci ) , ∂xj ∂xj

lim e−rt λi (t) = 0, j

(25)

t→∞

i = 1, 2, j = i. Equation (23) can be rewritten in terms of elasticity of demand (Definition 3.2) to j j obtain (9), where xi , xj are as in (5) and λii , λi , λij , λj are as in (24) and (25).  Proof of Proposition 3.1 Considering (1), (7) and (8), at steady state, we obtained the following relations, ki piss − (εi + ηi pjss )xiss = 0, λiss i

j ss λi

=

∂Si (piss ,pjss ,xiss ,xjss ) ∂xi (εi + ηi pjss ) + r

,

(27)

∂Si (piss ,pjss ,xiss ,xjss ) ∂xj (εj + ηj piss ) + r

,

(28)

(piss − ci )

(piss − ci )

=

(26)

i = 1, 2, j = i. Considering (27), (28) and (3), we obtain λiss i >0

j ss

and λi

< 0.

(29)

∀t ≥ 0.

(30)

Now, we argue that j

λii > 0 and λi < 0,

Assume by contradiction that (30) is not true. Then, there is some t0 < ∞ such that j

λii (t0 ) < 0 or λi (t0 ) > 0. Moreover, since the costates are assumed to be continuous (see Footnote 3), there is ∞ > t1 > t0 such that λii (t1 ) = 0 (and λii (t1− ) < 0)

or λi (t1 ) = 0 (and λi (t1− ) > 0). j

j

(31)

Considering (7) and (8) and (3), at t1 we have λ˙ ii |t=t1 < 0

j and λ˙ i |t=t1 > 0.

(32)

Since the derivative of the costates are assumed to have continuous derivatives (see footnote 3), we have λ˙ ii |t=t − < 0 1

j and λ˙ i |t=t − > 0. 1

(33)

However, since t1− is in the neighborhood of t1 , such that t1 > t1− , (31) contradicts j j our assumption that 0 = λii (t1 ) > λii (t1− ) or 0=λi (t1 ) < λi (t1− ). Thus, (30) is true.

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J Optim Theory Appl (2008) 138: 27–44

Considering (9) and (3) again, we argue that a myopic price (that corresponds to λii = j 0 and λi = 0) should be lower than a nonmyopic price (that corresponds to λii > 0 j and λi < 0, according (30)). To see this, let us consider the contrary. Condition (9) in case of a myopic price becomes pi = c i −

Si . ∂Si /∂pi

j

If a price corresponding to λii = 0 and λi = 0 is higher, the corresponding Si is lower i (because of (3)). Thus, to satisfy (33), the term − ∂SiS/∂p should be higher. However, i according to the assumption of this proposition, −∂Si /∂pi is higher or the same and i is lower. Therefore, we get the contradiction.  thus − ∂SiS/∂p i Proof of Theorem 3.2 Consider the necessary condition (9) and sufficient condition (17). Next, we can apply the implicit function theorem (see Ref. [36], p. 374) to (9) to arrive at the unique value of the equilibrium price p ∗ in terms of x, λ1 and λ2 , p ∗ = p ∗ (x, λ1 , λ2 ).

(34)

Let λ1 = Φ(x)

and λ2 = ϕ(x).

(35)

and ϕ (x)x˙ = λ2 .

(36)

In particular, (35) results in Φ (x)x˙ = λ˙ 1

Now, the theorem follows immediately from (10b) and (10c), after substituting in (34)–(36).  Proof of Theorem 4.1 Consider the relations (26), (27) and (28). The corresponding elasticity of demand with respect to price and perceive quality at steady-state will be ss ss ss ss ss ss ss ss ss μss i = −(pi /Si (pi , pj , xi , xj ))∂Si (pi , pj , xi , xj )/∂pi ,

(37)

βiiss = (xiss /Si (piss , pjss , xiss , xjss ))∂Si (piss , pjss , xiss , xjss )/∂xi ,

(38)

βijss

=

−(xjss /Si (piss , pjss , xiss , xjss ))∂Si (piss , pjss , xiss , xjss )/∂xj .

(39)

Substituting (26)–(28) and (37)–(39) into the necessary condition (9), we obtain 1+

  βijss (piss − ci ) βiiss ss −μ + = 0. + i piss [1 + r/(εi + ηi pjss )] [1 + (εj + r)/(ηj piss )]



Proof of Proposition 4.1 The proof is immediate by considering (19) and the derivations below. Let us denote (p ss − ci ) . Piss = i ss pi

J Optim Theory Appl (2008) 138: 27–44

43

Next, from (19), we obtain Piss =

1 μss i −

βiiss r 1+ ε +η p ss i i j

−

βijss εj +r 1+ ηj piss

.

Taking the partial derivatives with the respective parameters, we conclude with our proposition.  Proof of Proposition 4.2 This proposition immediately follows from (19) and the assumption that the price accounts only for the single role. 

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