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Price Competition with Dynamic Loyalty
Eric Anderson
Graduate School of Business University of Chicago
(E-mail: [email protected])
Nanda Kumar School of Management
University of Texas at Dallas (E-mail: [email protected])
September 2004
The paper has benefited from discussions with Birger Wernerfelt, Ram Rao and Sridhar Moorthy. The usual disclaimer applies.
Price Competition with Dynamic Loyalty
ABSTRACT
Many analytic marketing models have considered the implications of selling to
consumers exogenously assumed to be either loyals or switchers (Narasimhan, 1988; Raju et al
1990, Chen et al 2001, Chen et al 2002, Iyer and Pazgal 2003). In this paper we examine a
market where consumer loyalty is endogenously determined by past purchase experience and
refer to this as dynamic loyalty. We assume that after a purchase experience a fraction of
switching consumers become loyal to the brand last purchased. We conceptualize firm strength
as a firm’s ability to convert switchers to dynamic loyal consumers and consider an asymmetric
dynamic duopoly with a strong (i.e. large fraction become loyal) and weak firm.
Introducing dynamic loyalty leads to several new results compared to extant static
promotion models. Surprisingly, we find that while a strong firm has more dynamic loyal
consumers and larger market share it offers a lower average price than a weak firm. Under
certain conditions, a strong firm may promote more often than a weak firm. One might also
expect that increased strength of a rival leads to lower profits, but we show that profits of the
strong firm may increase if the weak firm creates more dynamic loyalty. We benchmark the
dynamic model against a case where firms are myopic and find a differential impact on each
firm’s profits. The weak firm benefits from myopia and earns lower profits in the dynamic
game. However, profits of the strong firm may increase in the dynamic game when the rival is
very weak. Finally, we obtain predictions on serial and contemporaneous price correlation that
we relate to empirical studies.
KEY WORDS: Price Promotions, Promotion Depth and Frequency, Loyalty
1. Introduction
It is often presumed that a firm with more loyal consumers charges higher average prices
and promotes less frequently than rival firms. This is a prediction in promotion models such as
Narasimhan (1988) and follows from the fact that loyal consumers are more “locked in” or face
switching costs. These models begin with the assumption that some consumers are ex-ante loyal,
however, for many consumers loyalty is created via experience with the product and we refer to
this as dynamic loyalty. When developing a pricing policy in such an environment, one must
balance offering low prices to induce trial and build loyalty versus offering high prices to capture
profits on loyal, repeat buyers. In this paper, we analyze a market with these dynamics and show
that a firm with more loyal consumers may charge a lower average price and in some cases may
promote more frequently.
We consider a dynamic promotion model with two types of loyal consumers. Static loyal
consumers always prefer one firm and never purchase from a rival firm. Dynamic loyal
consumers are initially indifferent but may become loyal after a purchase. Importantly, we allow
firms to differ in their ability to generate dynamic loyalty. Analogous to the extant literature we
refer to a firm with the ability to generate more dynamic loyal consumers as strong and its
competitor as weak.1 A strong firm has greater incentive to offer very low prices to generate trial
but will raise prices once a loyal base of consumers is established. This results in a cycle of
high-low pricing to generate trial and capture profits on repeat purchases. In contrast, a weak
firm creates less positive feedback from a purchase experience and has fewer dynamic loyal
consumers. This decreases the incentive to offer very low prices to induce trial, which raises the
1 As explained later, we will also allow firms to differ in the number of static loyal consumers.
- 1 -
average price of the weak firm. In part, this explains why a strong firm may offer lower average
prices than a weak firm.
Unlike static promotion models, we find that a strong firm may promote more frequently
than a weak firm. In static promotion models, promotion frequency is often dictated by a short-
run trade-off between increased margin versus lower market share. In a dynamic environment,
firms must consider future costs and benefits and these considerations lead to our new finding.
In particular, if a firm does not promote then it will not attract new, price sensitive consumers.
The firm’s loyal base of consumers erodes due to the lack of new consumers and attrition of old
consumers. Thus, failure to promote leads to an expected future loss of loyal consumers. These
dynamic effects are more significant for a strong firm that can generate more dynamic loyal
consumers. When the expected future cost of not promoting is sufficiently large a strong firm
may promote more frequently than a weak firm.
Somewhat surprisingly, we also find that a strong firm may benefit when its weaker rival
becomes more effective at creating dynamic loyalty. Holding the market size fixed, if a weak
rival is able to generate more dynamic loyal consumers then its market share and average price
increase. Because prices are strategic complements the strong firm reacts by increasing its
average price. While the strong firm loses market share to its weaker rival, we show that the
increase in price may lead to an overall increase in profits for both firms. Importantly, this result
only holds when the firms are very asymmetric in their ability to generate dynamic loyal
consumers (i.e., very strong vs. very weak). If the firms become more symmetric then an
increase in dynamic loyalty for a weak firm decreases the profits of the strong firm.
When both firms are perfectly myopic and place zero weight on future profits, our model
simplifies to a static promotion model. We use this model as a benchmark to assess the impact
- 2 -
of dynamics on competitive behavior and firm profits. We find that profits of the weak firm
decrease in the dynamic game. But, when the weak firm has few dynamic loyal consumers
profits of the strong firm may increase in the dynamic model. The intuition for this finding is
that myopia causes the weak firm to engage in more intense price competition, which lowers the
average price of both firms. For the myopic strong firm, lower prices and lost market share lead
to a decrease in profits. Thus, myopia only benefits both firms when they both generate
sufficient dynamic loyal consumers, which enables them to avoid intense price competition.
Our dynamic model also allows us to analyze both contemporaneous and serial price
correlation, which is not possible in static promotion models. To relate our findings to market
outcomes, we recognize that firm strength is correlated with market share. We find that
contemporaneous price correlation is negative when firms have symmetric strength (equal
market shares) and becomes positive when firms are asymmetric (large vs. small market share).
In practice, the positive correlation is analogous to a low market share brand mimicking the
pricing strategy of a leading national brand and the negative correlation is consistent with two
leading national brands promoting in different weeks (e.g. Coke and Pepsi). We also find that
serial price correlation is negative when firms are symmetric (equal market share). As the weak
firm generates less dynamic loyalty, serial correlation decreases for the strong firm and increases
for weak firm and eventually becomes positive.
Our finding that a strong firm may offer lower average prices is consistent with data from
the peanut butter and stick margarine categories. Van Oest and Frances (2003) and Seetharaman
et al (1999) show that the two leading national brands in these categories charge a lower average
price than the third ranked national brand (based on market share). To establish that these brands
also generate more dynamic loyalty, we note that Seetharaman et al (1999) report that many
- 3 -
consumers in these categories exhibit state dependence, which is identical to dynamic loyalty in
our model. Further, Van Oest and Frances (2003) show that brands with more market share have
more state dependence. Together, this implies that leading national brands in these categories
may have more dynamic loyal consumers. In contrast, Seetharaman et al (1999) find that
ketchup is a category with few dynamic loyal consumers, which suggests that high market share
may be due to static rather dynamic loyal consumers. In this case, our model predicts leading
brands should offer a higher price and reassuringly, the average prices of the top three ketchup
brands are perfectly correlated with market share.
To relate our model to empirical studies of price competition, we simulate data from our
model and estimate the price reaction functions proposed by Leeflang and Wittink (1992, 1996).
We find that the price reaction coefficients are negative when two strong firms compete, but are
positive when a strong firm competes against a weak firm. We relate this result to analysis of
dishwashing liquid by Kopalle et al (1999). Their analysis shows that price reaction coefficients
are typically negative when firms have similar market share but tend to be positive when a high
share brand competes with a low share brand. Our results are consistent with this empirical
evidence because firm strength in our model is correlated with market share.
Our model builds on a vast literature of price promotion models in marketing that analyze
competition in markets composed of loyal and switching consumers (e.g., Chen et al 2001, Chen
et al 2002 and Iyer and Pazgal 2003). Following Varian (1980), Narasimhan (1988) and Raju et
al (1990) we model price promotions as a mixed strategy equilibrium. A key difference in our
model is that we analyze a dynamic, infinite period game and endogenize loyalty of some
consumers. In particular, new consumers are initially indifferent between the competing
products but may become loyal to the lower priced brand after their first purchase. The idea that
- 4 -
consumers exhibit (stochastic) loyalty to the last brand purchased has been documented in
numerous empirical studies (Guadagni and Little 1983, Lattin 1987, Vilcassim and Jain 1991,
Bucklin and Lattin 1991, Krishnamurthi and Raj 1991, Papatla and Krishnamurthi 1992,
Seetharaman et al 1999). We incorporate this feature in a dynamic model and examine the long-
term effect of promotions. While several features of our model are consistent with extant static
promotion models we focus on new findings that are driven by dynamic competition.
In a related paper, Villas-Boas (2004) considers a two period model where consumers are
initially uncertain about how well a product fits their preferences. After a purchase experience
consumers learn about the true fit of the product. Our model has a similar feature in that some
consumers are initially indifferent and a fraction of these consumers become loyal. While we
focus on the effect of such behavior on firms’ promotional strategies Villas-Boas (2004) focuses
on “the competitive effects of the potential informational advantages of a product that has been
tried by the consumer” (p. 142 Villas-Boas 2004).
This paper also builds on theoretical models of price competition in markets with
consumer switching costs (e.g., Klemperer 1987, Farrell and Shapiro 1988, Beggs and
Klemperer 1992). However, similar to Anderson, Kumar and Rajiv (2004) and Padilla (1995)
we endogenize these switching costs by relating them to past market outcomes. While Anderson
et al (2004) and Padilla (1995) examine tacit collusion in a symmetric duopoly we focus on
asymmetric firms who differ in their ability to generate future loyal consumers and analyze the
implications for price competition.
Our work also adds to the empirical literature on stochastic consumer choice behavior in
marketing and economics which documents both dynamics and state dependence in consumer
preferences (e.g., Heckman 1991; Papatla and Krishnamurthi, 1996; Erdem and Keane, 1996).
- 5 -
Our approach complements these findings as we study how such consumer dynamics affects
competitive behavior. Finally, our approach complements empirical work that documents
competitive price response (Kopalle et al, 1999; Leeflang and Wittink, 1992; 1996) by providing
a framework to explain empirically observed variations in the sign of price response coefficients.
The remainder of this paper is organized as follows. In §2 we explain our assumptions of
consumer and firm behavior and solve the dynamic game. In §3, we present our results and in §4
we present a model extension. In §5 we relate our model to empirical evidence and we conclude
with a brief discussion in §6.
2. Model
We consider a market with two competing firms that we label s for strong and w for
weak. Each firm sells one product in multiple periods to three types of consumers: static loyal
consumers, dynamic loyal consumers and switchers. We assume both firms have constant
marginal cost equal to zero. Every period a unit mass of switching consumers and a mass of ls
and lw static loyal consumers enter the market and we assume ls ≥ lw. Each consumer buys at
most one unit each period and lives for two periods, which results in overlapping generations
(OLG). Thus, in each period (1+ ls + lw) consumers enter and exit the market and the market size
is always 2(1+ ls + lw).2 In each period, static loyal consumers buy from firm j ∈ {s,w} if pj ≤ r
and never buy from the competing firm. We allow the preferences of switching consumers to
change over time. In the first period of their life, switching consumers buy from the firm that
offers the lowest price provided it is less than or equal to r. In the next period of their life, if firm
j offers the lowest price in the previous period then θj switching consumers become dynamic
- 6 -
loyal consumers to that firm and (1-θj) consumers remain switchers. We assume that the strong
firm is able to generate more dynamic loyalty (θs ≥θw) and that dynamic loyal consumers buy
from firm j ∈ {s,w} if pj ≤ r and never buy from the competing firm.
An example helps to clarify market structure and dynamics. Let pjt equal the price
offered by firm j in period t. If the firms offer ps1 = ½ and pw1 = ¾ in period 1 then all switching
consumers buy from the strong firm. In period 2 there are θs dynamic loyal consumers for the
strong firm and zero dynamic loyal consumers for the weak firm. The strong firm sells to 2ls +
θs loyal consumers and the weak firm sells to 2lw loyal consumers and both firms compete for the
(2-θs) switching consumers. If the firms offer ps2 = 1 and pw2 = ¼ in period 2 then in period 3
there are θw dynamic loyal consumers for the weak firm and zero dynamic loyal consumers for
the strong firm. The strong firm sells to 2ls loyal consumers and the weak firm sells to 2lw + θw
loyal consumers and both firms compete for the (2-θw) switching consumers.
There are two possible states in our model that we label state 0 and state 1. In state 0, the
strong firm has θs dynamic loyal consumers while in state 1 the weak firm has θw dynamic loyal
consumers. Let Vjk equal the continuation payoff to firm j in state k ∈ {0,1} and Fjk(p) equal the
CDF of each firm in each state. In state k each firm mixes over prices in the range [ , )kp r and at
most one firm offers p = r in each state. In state 0, the continuation payoffs are:
( ) ( )( ) ( )( ) ( )0 0 0 02 2 1 1s s s s w w s wV p l F p F p V F p Vθ θ δ⎡ ⎤ ⎡= + + − − + − +⎣ ⎦ ⎣ 0 1s ⎤⎦ , (1)
( ) ( )( ) ( )( ) ( )0 0 0 12 2 1 1w w s s s w sV p l F p F p V F p Vθ δ⎡ ⎤ ⎡= + − − + − +⎣ ⎦ ⎣ 0 0w ⎤⎦
. (2)
2 An alternative interpretation is there are 2(ls + lw) consumers in all periods and a unit mass of switching consumers, who live 2 periods, enter each period.
- 7 -
In equations 1 and 2, the first term represents the current period payoff and the second
term represents the discounted expected future payoff. If the strong firm offers the lowest price
(probability 1-Fw0(p)), then the strong firm’s current payoff is p(2ls + θs + 2-θs) and in the next
period firms are again in state 0. In contrast, if the weak firm offers the lowest price (probability
1-Fs0(p)), then the weak firm’s current payoff is p(2lw + 2-θs) and in the next period firms
transition to state 1. The continuation payoffs in state 1 are:
( ) ( )( ) ( )( ) ( )1 1 1 02 2 1 1s s w w w s wV p l F p F p V F p Vθ δ⎡ ⎤ ⎡= + − − + − +⎣ ⎦ ⎣ 1 1s ⎤⎦ , (3)
( ) ( )( ) ( )( ) ( )1 1 1 12 2 1 1w w w w s s w sV p l F p F p V F p Vθ θ δ⎡ ⎤ ⎡= + + − − + − +⎣ ⎦ ⎣ 1 0w ⎤⎦ . (4)
The interpretation of these equations is analogous to equations 1 and 2. We solve
equations (1)-(4) for Fjk(p) to obtain the pricing policies of the strong and the weak firm in each
state.
( ) ( )( ) ( )
00
0 1
1 21 ,
2w w
ss w w
V l p0F p
p V Vδ
θ δ− −
= − ≤ <− − −
p p r (5)
( ) ( ) ( )( ) ( )
00
0 1
2 1 1,
2s s
ws s s
p l V0F p
p V Vδ
θ δ+ − −
=− + −
p p r≤ < (6)
( ) ( ) ( )( ) ( )
11
0 1
2 1 1,
2w w
sw w w
p l V1F p
p V Vδ
θ δ+ − −
=− − −
p p r≤ < (7)
( ) ( )( ) ( )
11
0 1
1 21 ,
2s s
ww s s
V l p1F p
p V Vδ
θ δ− −
= − ≤ <− + −
p p r . (8)
To fully characterize the equilibrium we need to identify six unknowns (Vjk , kp ) and this
implies that there are six relevant equations. A necessary condition for an equilibrium is that the
lower bound of the support has zero mass for each firm in each state and this leads to four
equations:
- 8 -
( ) 0, { , }, {0,1}jk kF p j s w k= ∀ = = . (9)
The final two equations are determined by whether firm j has a mass point on r in state k.
In each state, one firm has zero mass on r while the other firm has positive mass on r and this
results in four possible cases. We present analysis of Case 1 and solutions for Cases 2, 3 and 4
are in the Appendix3. We focus on Case 1 because this is the only solution where the strong firm
has a mass point on p = r in state 0 and the weak firm has a mass point on p = r in state 1. This
equilibrium arises when the firms are relatively symmetric, which leads to more strategic
interaction. The equilibrium solutions for Vjk and kp in Case 1 are:
( )( )
( )( )( )
( )( )( )
( ) ( ) ( ) ( )( )( )
2
*0 2
22
4 4 1 1 2
2 2 2 1 1
2 121 2 1 2 2 4 4
12 2
w s w w
w s w s
w s w ww ww
s w s
w s w
l l l l
r l l
l lr lV
l l
l
δ δ
θ δ θ
δθ δθθδ δ δ δ δθ
δδ θ θ
⎛ ⎞⎛ ⎞⎛ ⎞+ + + +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+ − + + +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+ + − −⎜ ⎟+ ⎝ ⎠⎝ ⎠= +⎜ ⎟
⎛ ⎞− ⎛ ⎞+ + + + +⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟+ + −⎜ ⎟⎝ ⎠⎝ ⎠
⎜ ⎟⎜ ⎟⎝ ⎠
(10)
( )( )( ) ( ) ( )( )( )( )
( ) ( ) ( ) ( )( ) ( )( )( )*1 2
2 2 2 2 2 4 22 1
2 1 1
1 2 1 2 2 1 2 4 2 2
s s s w s sw
s ww
s w s w s
l l ll r
lV
l l l
δ θ θ δ δ θ
δ θ
δ δ δ δθ δ
⎛ ⎞⎛ ⎞− + + + + + +⎜ ⎟+ ⎜ ⎟
⎜ ⎟⎜ ⎟+ + +⎝ ⎠⎝ ⎠=− + + + + + + + − wθ θ
(11)
( )( )( )( )
( )( )( )( )( )( ) ( ) ( )( )( )
2
*0 2 22 2
2 1 1 22 1
1 2 2 2
4 1 1 1 1 2 2 2 2
w s ss
s w w ws
s w w s s w
r l ll
r l lV
l l l l
δ θ
δ δ θ θ
δ δ δ θ θ
⎛ ⎞⎛ ⎞+ − +⎜ ⎟⎜ ⎟+
⎜ ⎟⎜ ⎟+ − + − +⎝⎝ ⎠=+ + − − − + − + −
⎠ (12)
- 9 -
( )( )( )( )
( )( )( )( )( ) ( )
( )( )( )
(*1 2
2
2 1 1 22 1
2 2 21 24 1 1 1 1
1 2 2 2 2
w s ss
s w w ws s s
s w
w s s w
r l ll
r l lV
l l
l l
δ θ
δ θ θ)r l θ
δ δ δ
δ δ θ θ
⎛ ⎞+ + +⎛ ⎞⎜ ⎟+ ⎜ ⎟⎜ ⎟⎜ ⎟+ + − +⎝ ⎠= ⎜ ⎟⎛ ⎞+ + + −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− + + − + −⎝ ⎠⎝ ⎠
− + (13)
( ) ( )( ) ( )( )( )( ) ( ) ( )( ) ( )( )
2*0 2
4 4 1 2 2 1 1 2 1
2 1 2 2 1 2 4 2 2w s w w s w w s w w
s w s w w s
r l l l l l l lp
l l l
δ δ δ θ δ δθ
δ δ δθ δ θ θ
+ + + + + + + + + − −=
+ + + + + + + −
δθ (14)
( ) ( )( ) ( )( )( )( )( ) ( ) ( )( ) ( )( )
*1 2
4 1 2 2 2 2 2 1 1
2 1 2 2 1 2 4 2 2w s s w s s s s w s
s w s w w s
r l l l l l lp
l l l
δ δ δ θ δ θ θ θ δ
δ δ δθ δ θ θ
+ + + + + − + + + +=
+ + + + + + + − (15)
To determine whether this case holds in equilibrium we recognize that if firm j has no
mass point in state k, then there exists a set of strategies for firm j that are dominated by *kp . We
define 0wp and 1sp by following equations:
( )0 02 2 2w w w w sr l V p l V 1wδ θ δ+ = + − + (16)
( )1 12 2 2 0s s s s wr l V p l Vsδ θ δ+ = + − + . (17)
Ignoring the strategy of the strong firm, 0wp represents the lowest price that the weak
firm would charge in state 0. But, if the strong firm’s lowest price in state 0 is *0p then strategies
between 0wp and *0p are dominated for the weak firm. Similarly, in state 1 strategies between
1sp and *1p are dominated for the strong firm. The set of dominated strategies must be non-
empty for Case 1 to be feasible: *0w 0p p≤ and *
1s 1p p≤ . To see why this condition must hold,
suppose *0w 0p p> . Since 0wp is the lowest price the weak firm will offer, the lower bound of the
(cont. next page)
3 Case 1 is a general solution to the model considered by Padilla (1995) and Anderson et al (2004). If dynamic loyal
- 10 -
support cannot equal *0p . In the Appendix, we provide the solution to these inequalities as a
function of model parameters and verify that there is a unique solution to the model.
3. Results
In the previous section, we characterized the equilibrium with analytic closed form
solutions. Because analytic expressions for many properties of the model, such as the expected
price, promotion frequency, and price correlation are complex, we use numerical analysis to
derive our results (see Appendix). We fix parameter values to r = 1, δ = 0.9 and θs = 1 and focus
most of our results on θw, which we assume varies from 0 to 1. To simplify the exposition and
highlight the role of dynamic loyal consumers we assume both firms have the same number of
static loyals (lw = ls = l). To allow variation in the relative importance of dynamic loyalty we
consider l {0.01, 0.25, 0.50} and note that l > 0 to ensure there are always some loyal
consumers (l + θ
∈
w > 0). Given these assumptions, the unique equilibrium for these parameter
values is Case 1.
Before presenting the results it is helpful to note that firms internalize two effects in
making their pricing decisions - a direct effect and a strategic effect. Firms have an incentive to
lower price to acquire consumers and then raise price once they become loyal and we refer to
this as the direct effect. A second consideration is how to optimally respond to competitors'
actions. Because prices are strategic complements, when a competitor raises (lowers) price there
is an incentive to raise (lower) price and we refer to this as the strategic effect. As we show next,
many of our findings can be explained by the relative importance of these two effects.
consumers face large switching costs, lw = ls = 0 and θw = θs = θ then we obtain the symmetric equilibrium analyzed in those papers.
- 11 -
Consistent with previous interpretations of mixed strategy equilibria, we define p < r as a
promoted price and p = r as the regular price. We let CC equal contemporaneous price
correlation between the strong and weak firm and SCj equal the serial price correlation of firm j.
In Figure 1, we plot the expected price of both firms and this leads to our first result.
Result 1
If the strong firm has more dynamic loyal consumers (θs > θw) then
a. the expected price of both firms increases when the weak firm has more
dynamic loyal consumers,
b. the expected price of the strong firm is less than the weak firm.
Figure 1: Expected Price of Strong Firm and Weak Firm
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0 0.2 0.4 0.6 0.8 1
θ w
Exp
ecte
d Pr
ice
l s =l w =0.50
l s =l w =0.01
l s =l w =0.25
Weak Firm Strong Firm
- 12 -
A common result in static promotion models is that prices increase with the number of
loyal consumers. Consistent with this prediction, the weak firm raises its price when it is able to
generate more dynamic loyalty. Because prices are strategic complements the strong firm also
raises its price. Together, this explains Result 1a. But, while the strong firm has more loyal
consumers (θs > θw) it offers a lower expected price (Result 1b). Why doesn’t more loyalty lead
to higher prices for the strong firm?
A key difference in our model is that loyalty is endogenous and some consumers become
loyal after a purchase experience. The strong firm offers low prices to attract indifferent
consumers and build a base of loyal consumers. Once these consumers become loyal, they are
willing to pay higher prices and this creates an incentive for the strong firm to offer high prices.
In contrast, the weak firm does not have the same incentive to engage in high-low pricing. Low
prices do not lead to as much dynamic loyalty and this discourages offering deep discounts.
Overall, this raises the average price of the weak firm.
To further understand Result 1b, we plot the expected price in each state for l = 0.01 in
Figure 2. The expected price of the strong firm is strictly greater in state 0 (θs dynamic loyal
consumers) than in state 1 (zero dynamic loyal consumers), which we interpret as high-low
pricing. But, for low values of θw the weak firm charges a higher price in state 0 (zero dynamic
loyal consumers) compared to state 1 (θw dynamic loyal consumers). Why would the weak firm
offer a higher price when it has fewer dynamic loyal consumers?
The result can be explained by the relative magnitude of the strategic and direct effects
for the weak firm in state 0. The weak firm has zero dynamic loyal consumers in state 0 and
there is little incentive to offer a low price to build future loyalty because θw is small. This
implies the direct effect is small. At the same time, the strong firm has many dynamic loyal
- 13 -
consumers and offers very high prices to capture margin on its large, loyal base of consumers. In
reaction, the weak firm raises its price due to the strategic effect. For low values of θw the
strategic effect is large enough that the weak firm offers a higher price in state 0 compared to
state 1.
Figure 2: Expected Price in Each State for l =0.01
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0.00 0.20 0.40 0.60 0.80 1.00θ w
Exp
ecte
d Pr
ice
Weak Firm State 0 Strong Firm State 0Weak Firm State 1 Strong Firm State 1
To summarize, the direct effect causes the strong firm to offer very low prices to attract
new consumers, which lowers its price. When the strong firm offers high prices to capture
margin on loyal consumers, the weak firm also offers high prices. This strategic response raises
the price of the weak firm. Thus, the incentive of the strong firm to use high-low pricing and the
competitive reaction of the weak firm explain why a strong firm may offer lower average prices
than a weak firm.
- 14 -
To contrast Result 1 with previous research, we note that static promotion models with
switchers who are indifferent and loyal consumers who are locked in predict that a firm with
more loyal consumers offers a higher average price (Narasimhan 1988). In models that consider
the degree to which a consumer is loyal (Raju et al 1990, Rao 1991) a strong firm may offer deep
discounts to steal loyal consumers from a weak rival. Under certain conditions this may lead to
an outcome analogous to Result 1.4 In our model, loyal consumers are locked in and so the
incentive to steal loyal consumers is not present. Instead, a strong firm offers low prices to
attract new consumers and build future loyalty. Thus, we identify a new effect which may lower
the average price of a strong firm.
Static promotion models have also been used to investigate promotion frequency and
predict that a strong firm promotes less frequently than a weak firm (Narasimhan 1988, Raju et
al 1990). Our model yields new insights on promotion frequency that are summarized in Result
2.
Result 2
a. When there are many static loyal consumers, the strong firm promotes more
frequently than the weak firm.
b. Promotion frequency of both firms increases when the weak firm has fewer
dynamic loyal consumers.
Figure 3 illustrates Result 2 and has two interesting properties. First, the curves shift
upwards as l increases and second the strong firm promotes more frequently than the weak firm,
but only for large values of l. Intuition for the first property is analogous to static promotion
4 For example, in Raju et al (1990) if r=1 and consumers loyal to the weak firm are willing to pay a premium greater than 0.237 then the average price of the strong firm is less than the weak firm.
- 15 -
models, such as Narashimhan (1988). As the number of static loyal consumers increases the
average price, E(p), and the average promotional price, E(p|p < r), increase. The short-run
opportunity cost of a price promotion equals the lost margin on loyal consumers (r- E(p|p < r))
and this is decreasing in l. Thus, both firms promote more frequently as l increases.
Figure 3: Promotion Frequency
75%
80%
85%
90%
95%
100%
0 0.2 0.4 0.6 0.8 1
θ w
Prom
otio
n Fr
eque
ncy
l s =l w =0.50
l s =l w =0.01
l s =l w =0.25
Weak Firm Strong Firm
In a static model, each firm makes a short-run trade-off of volume versus margin. In our
dynamic model, firms must also consider future costs and benefits. In particular, the strong
firm’s pricing strategy incorporates the impact of current promotion frequency on future profits.
When the strong firm does not promote (ps = r) there is a short-run gain in margin and a loss in
volume. But there is also a future cost because the strong firm immediately transitions to state 1,
which is a less profitable state with zero dynamic loyal consumers. As l increases the transition
probability from state 1 to state 0 decreases because the gap between the strong firm’s price and
- 16 -
weak firm’s price narrows (see Figure 1). State 1 is a relatively less profitable state for the
strong firm (zero dynamic loyal consumers) and spending more time in that state is costly. The
strong firm reacts to this expected future cost by increasing its current promotion frequency,
which decreases the likelihood of transitioning to state 1. These future costs (i.e. dynamic
effects) are less significant for a weak firm that generates few dynamic loyal consumers. Thus, a
weak firm’s promotion frequency is affected primarily by short-run trade-offs.
Both firms increase promotion frequency when the weak firm has fewer dynamic loyal
consumers (see Figure 3) and the intuition is similar to Result 2a. For the strong firm a transition
to state 1 is more costly when θw is low because price competition is more intense. The strong
firm reacts by increasing its promotion frequency, which reduces the likelihood of transitioning
to this less profitable state. When the strong firm promotes more frequently, the competitive
reaction of the weak firm is to promote more frequently (strategic effect). Together, this
explains why promotion frequency of both firms increases when the weak firm has fewer
dynamic loyal consumers.
Another surprising finding from our model is that a strong firm may benefit when its rival
is able to generate more loyalty. We expect that if a firm can generate more loyalty and repeat
business that own profits will increase. What is unexpected is that a competitor’s profits may
also increase. This finding is summarized in the next result and shown graphically in Figure 4.
Result 3
Profits of the strong firm are increasing for low values of θw and decreasing for
large values of θw.
We first note that an increase in θw decreases market share of the strong firm and this has
a negative impact on profit. However, the direct effect for the weak firm increases with θw,
- 17 -
which leads to more high-low pricing. In state 1, the weak firm offers higher prices and this
benefits the strong firm through the strategic effect. In state 0, the weak firm competes more
aggressively for new consumers, which hurts the strong firm.
Figure 4: Profit of Strong Firm
0.4
1.4
2.4
3.4
4.4
5.4
6.4
7.4
0 0.2 0.4 0.6 0.8 1θ w
Exp
ecte
d Pr
ofit
Stro
ng F
irm
l s
= lw
=0.0
1
15.0
15.0
15.1
15.1
15.2
15.2
15.3
15.3
15.4
15.4
Exp
ecte
d Pr
ofit
Stro
ng F
irm
l s=l w
=0.5
0l s =l w =0.5
l s =l w =0.01
Initially, the increased margin in state 1 compensates for lost market share and more
intense price competition in state 0. This results in an increase in profits for low values of θw.
But, while the strong firm’s average price in state 1 is increasing in θw the rate of increase
diminishes, which implies that the strategic effect is decreasing in θw. For example, in Figure 2
(l = 0.01) the strong firm’s average price in state 1 increases one hundred percent when θw
increases from 0 to 0.10. However, as θw increases from 0.9 to 1.0 the strong firm’s price
increase is less than five percent. Eventually the strategic effect diminishes and profits of the
strong firm decrease for large values of θw.
- 18 -
To benchmark our dynamic model against static promotion models, we consider the case
where firms ignore future profits (δ = 0). In this case, the equilibrium pricing strategy for both
firms is identical to Narasimhan (1988). To illustrate our results we will focus on the case where
firms are perfectly myopic, δ = 0, and compare against our assumed discount factor, δ = 0.9. In
the dynamic game the continuation payoff is V = π + δV and to compare profits with the static
game we focus on π, which is the expected current period profit. The average prices in the
myopic and dynamic cases are plotted in Figure 5 and expected current period profits are plotted
in Figure 6. Our results are summarized as follows.
Result 4
If firms are perfectly myopic (δ = 0), then relative to the dynamic model (δ > 0)
a) the average price of each firm increases if the number of dynamic loyal
consumers is sufficiently large,
b) expected current period profits of the weak firm always increase and expected
current period profits of the strong firm increase if the number of dynamic
loyal consumers is sufficiently large.
When firms are myopic they ignore the effect of their current pricing strategy on future
profits. The effect of myopia differs in each state and varies with θw because the number of loyal
consumers changes. In state 0, a myopic strong firm faces many loyal consumers, focuses on
short-run profits and ignores the possibility of building future loyal consumers. Lack of concern
for future profits raises the average price of a myopic strong firm in state 0, which in-turn raises
the price of a myopic weak firm in state 0. In state 1, two related effects lead to a lower average
price in the myopic game for low values of θw. First, competition for switching consumers
lowers both firms average price and second, the intense price competition increases the
occurrence of state 1. Larger values of θw increase the number of loyal consumers in state 1 and
- 19 -
this softens price competition in the myopic game. In addition, prices increase at a faster rate
compared to the dynamic game because firms ignore the possibility of building future loyal
consumers. Eventually myopia leads to higher prices in state 1 compared to the dynamic game.
When θw is low the effects in state 1 dominate and the average price decreases in the
myopic game. That is, the increase in average price in state 0 is outweighed by both the lower
average price in state 1 and the increased occurrence of state 1. As θw increases there is less
price competition in state 1 and a lack of concern for future profits leads to higher prices versus
the dynamic game.
Figure 5: Impact of Dynamics on Expected Price
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.2 0.4 0.6 0.8 1θ w
Exp
ecte
d Pr
ice
l =0.50
l=0.01
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.2 0.4 0.6 0.8 1
θ w
Exp
ecte
d Pr
ice
l =0.5
l =0.01
Myopic Weak Firm Dynamic Weak Firm Dynamic Strong FirmMyopic Strong Firm
Our analysis of myopia highlights two marginal effects in the dynamic game that arise
due to a concern for future profits. First, when the market is composed of primarily switching
consumers there is less price competition, which increases the average price. Second, even when
- 20 -
firms sell to many loyal consumers there is a concern for building a future base of loyal
consumers. This has a marginal effect of lowering prices in the dynamic game.
The impact of myopia on each firm’s profits is shown in Figure 6 and differs for each
firm. Surprisingly, the weak firm always benefits from myopia. While low values of θw lower
the average price of the weak firm, a gain in market share compensates for lost margin. In
contrast, myopia decreases profits of the strong firm for low values of θw. As θw increases price
competition becomes less intense and the strong firm benefits from the higher prices induced by
myopia.
Figure 6: Impact of Dynamics on Expected Profit
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.2 0.4 0.6 0.8 1
Exp
ecte
d Pr
ofit
Dynamic Weak Firm Dynamic Strong Firm
Myopic Weak Firm Myopic Strong Firm
θw
Our model also leads to predictions on serial price correlation of each firm, SCj, and the
contemporaneous price correlation, CC. Our results are summarized as follows.
- 21 -
Result 5
a) Contemporaneous price correlation is positive for low values of θw and
negative for high values of θw.
b) Serial correlation of the weak firm is positive for low values of θw and
negative for high values of θw. Serial correlation of the strong firm is always
negative.
The contemporaneous correlation in prices is depicted in Figure 7 and illustrates Result
5a. When θw is sufficiently small the weak firm’s incentive to build loyalty is not strong and the
strategic effect is dominant. Thus, when the strong firm offers a high price the weak firm tends
to offer a high price and this induces a positive correlation. As θw increases, the direct effect
becomes important for both firms and each uses high-low pricing. In state 0, the strong firm
offers high prices to capture margin on loyal consumers and the weak firm offers low prices to
build loyalty. In state 1, the weak firm offers high prices to capture margin on loyal consumers
and the strong firm offers low prices to build loyalty. This results in a negative
contemporaneous price correlation for large values of θw.
Figure 7: Contemporaneous Correlation
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0 0.2 0.4 0.6 0.8 1
θ w
Con
tem
pora
neou
s Cor
rela
tion
l s =l w =0.50
l s =l w =0.01
l s =l w =0.25
- 22 -
The use of high-low pricing (direct effect) also explains Result 5b, which is shown
graphically in Figure 8. The strong firm always uses high-low pricing to build loyalty and then
capture profits, and this leads to negative serial correlation. For the weak firm, the strategic
effect is dominant for low values of θw. As shown in Figure 2 the average price of the weak firm
is similar in both states, and this leads to a positive serial correlation. For high values of θw the
weak firm uses high-low pricing and serial correlation is negative.
Figure 8: Serial Correlation
-0.55
-0.50
-0.45
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0 0.2 0.4 0.6 0.8 1
θ w
Seri
al C
orre
latio
n
l s =l w =0.5
l s =l w =0.01
l s =l w =0.25
Weak Firm Strong Firm
We have presented several new results and it may be helpful to summarize them here.
First, we find that the average price of the strong firm is less than the weak firm. Second, the
strong firm may promote more frequently than the weak firm. Third, profits of the strong firm
follow an inverted-U shape with respect to the number of dynamic loyal consumers for the weak
firm. Fourth, if the weak firm has few dynamic loyal consumers and firms are myopic then
- 23 -
average prices may decrease. The price decrease may lead to lower profits for the strong firm,
but profits of the weak firm increase. Fifth, the contemporaneous price correlation is positive
when the weak firm has few dynamic loyal consumers and is negative when firms are symmetric
in the number of dynamic loyal consumers. Serial correlation of the strong firm is negative due
to high-low pricing. For the weak firm, serial correlation is positive when it has few dynamic
loyal consumers but is negative when firms are symmetric in the number of dynamic loyal
consumers.
4. Model Extension
In this section we demonstrate the robustness of our model and analyze a game in which
consumers live for three periods rather than two periods. This results in four possible states that
we label k = 0, 1, 2, and 3. In state 0 there are 2θs dynamic loyal consumers for the strong firm
and zero for the weak firm. In state 3 there are 2θw dynamic loyal consumers for the weak firm
and zero for the strong firm. In states 1 and 2 there are θs + θw dynamic loyal consumers who
live for one or two additional periods. We refer to a consumer as “old” if they have one period
remaining and “young” if they have two periods remaining. In state 1, there are θs old
consumers and θw young consumers. In state 2, there are θs young consumers and θw old
consumers.
The analysis of the model is analogous to §2 and details are available from the authors.
We specify eight continuation payoffs and then solve for eight CDFs (2 firms x 4 states) as a
function of and jkV kp . The twelve equilibrium values for and *jkV *
kp require twelve equations
and eight of these are given by ( ) 0jk kF p = . There are four possible states and in each state
exactly one firm has a mass point, which leads to sixteen (42) possible solutions. In our
- 24 -
simulations, we focus on a solution analogous to Case 1 where the strong firm has a mass point
in states 0 and 2 and the weak firm has a mass point in states 1 and 3. In each of these states, the
firm with the mass point offered the lowest price in the previous period and hence acquired a
new group of dynamic loyal consumers.
Analysis of this model, which is available from the authors, demonstrates that our results
hold in a richer, more complex model. The assumption that consumers live for two periods
yields a parsimonious model and our results continue to hold when consumers live for more than
two periods. To illustrate the similarity of the models, we plot the expected price in Figure 9. In
Figure 9a we consider the two extreme states (state 0 and 3) where one firm has 2θj dynamic
loyal consumers and the other firm has zero. Given the similarity of Figure 2 and Figure 9a, it is
not surprising that our results hold in the three period OLG model.
Figure 9: Expected Price in Each State for 3 Period OLG Game
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0.00 0.20 0
Exp
ecte
d Pr
ice
Weak Firm State Weak Firm State
0.90 ) )
(a.40 0.60 0.80 1.00θ w
0 Strong Firm State 03 Strong Firm State 3
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.00 0.20 0
Exp
ecte
d Pr
ice
Weak Firm State 1Weak Firm State 2
- 25 -
(b
.40 0.60 0.80 1.00
θ w
Strong Firm State 1Strong Firm State 2
In Figure 9b, we analyze the intermediate states where each firm has θj dynamic loyal
consumers. We find that firms offer low prices when they have θj young loyal consumers and
zero old loyal consumers to build their loyal base of consumers. If successful at acquiring
switching consumers, firms then transition to the state where they have 2θj dynamic loyal
consumers. At this point, they “harvest” and offer high prices to their loyal consumers.
In contrast, when a firm has zero young loyal consumers and θj old loyal consumers
should it offer high prices (“harvest”) or offer low prices (“invest”)? We find that the strong
firm’s expected price is higher in state 1 (θs old loyal consumers) compared to state 2 (zero old
loyal consumers). Similarly, the weak firm offers higher prices in state 2 (θw old loyal
consumers) compared to state 1 (zero old loyal consumers). This indicates that the incentive to
harvest old loyal consumers dominates the incentive to build a loyal base of consumers.
Overall, the extension demonstrates the robustness of our results. It also suggests that
firms will offer high prices in multiple periods to extract profits on their loyal base of consumers.
When the loyal base of consumers is sufficiently low, the firm will offer a series of deep
promotions to establish a loyal base of consumers. Once the loyal base of consumers is
established the cycle repeats.
5. Empirical Evidence
In the introduction, we presented stylized facts from three categories (peanut butter, stick
margarine, and ketchup) and related these to Result 1. In this section, we relate model
predictions to published empirical studies of price promotions. Our strategy is to simulate data
from our model for 1,000 periods for a given vector of parameters. Similar to §3, we vary l and
θw and hold the other parameters constant. For each set of parameters, we estimate the price
reaction functions proposed by Leeflang and Wittink (1992, 1996). This model has been
- 26 -
frequently used to analyze price promotions and we compare our parameter estimates with a
recent study by Kopalle et al (1999). The price reaction coefficients for each firm are graphed in
Figure 10.
Figure 10: Price Reaction Functions
When the weak firm has more dynamic loyal consumers, the expected market share of
the firms is more similar and the price reaction coefficients are negative. In contrast, when θw is
near zero, the strong firm has a much larger market share relative to the weak firm and the price
reaction coefficients are positive. Thus our model predicts that if competing firms are more
symmetric (i.e. θs ≈ θw) then market shares will be equal and the price reactions will be negative.
When firms are asymmetric (i.e. θs > θw) the price reaction coefficient will be positive, the strong
firm will have high market share and the weak firm low market share. Note that this implies a
positive correlation between the relative market share (shares/sharew) and the price reaction
coefficient.
We compare these price reaction coefficients with the findings of Kopalle et al (1999)
who estimate the same model on six brands of dishwashing detergent. The authors find a mix of
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0 0.2 0.4 0.6 0.8 1
θ w
Pric
e R
eact
ion
Coe
ffic
ient
l=0.50
l=0.01
l=0.25
Weak Firm
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0 0.2 0.4 0.6 0.8 1
θ w
Pric
e R
eact
ion
Coe
ffici
ent
l=0.50
l=0.01
l=0.25
Strong Firm
- 27 -
positive
owever, a stronger empirical test is needed to rule out alternative explanations. As
the foc
This paper examines optimal pricing policies in a duopoly when loyalty of some
consumers is endogenously determined h ral new predictions compared to static
promot
, negative and insignificant coefficients, which may not seem surprising since our model
shows that the price reaction coefficients can be positive or negative. To relate our findings with
Kopalle et al (1999) we classify the six dishwashing liquids into two groups based on market
share.5 Organizing their price reaction coefficients in this manner leads to a pattern of results
similar to our model. When brands have similar market share (high share vs. high share or low
share vs. low share) the statistically significant price reaction coefficients are all negative. Our
model offers a similar prediction: if market share of the two firms is similar (strong vs. strong)
then the price reaction coefficient is negative. If a high share brand competes with a low share
brand Kopalle et al (1999) find that the price reactions are more likely to be positive (eight of the
eleven significant coefficients are positive). Again, our model offers a similar prediction: if the
market share of the two firms is asymmetric (strong vs. weak) then the price reaction coefficient
is positive.
This empirical evidence is consistent with our results and offers preliminary support for
our model. H
us of this paper is theoretical, we leave detailed empirical investigation for future work.
6. Conclusion
. T is leads to seve
ion models. We find that a firm with more dynamic loyal consumers may offer lower
average prices and promote more frequently than its rival. We also find that profits of a firm
5 See Table 3 from Kopalle et al (1999). High market share brands are Dawn, Palmolive and Sunlight. Low market share brands are Ivory, C.W. Octagon, and Dove.
- 28 -
may increase when a rival, weak firm is able to generate more dynamic loyal consumers. This
shows that a weak firm may lower the price of both firms and erode industry profits. An
implication for managers is that a leading brand must be cautious using tactics that weaken a
rival. This may lead to the unexpected outcome of lower industry prices and profits for both
firms.
The results also demonstrate that concern for future profits decreases profits of a weak
firm compared to the myopic case. Thus, a weak firm would benefit if it could ignore future
profits.
eriod. An implication of this assumption is that both the contemporaneous
and ser
loyal
and pay
In contrast, a strong firm benefits from forward looking behavior when the weak rival
has few dynamic loyal consumers. The weak firm’s concern for future profits softens price
competition when there are many switching consumers in the market, which raises the average
price of both firms.
To relate static promotion models to data, researchers have assumed that firms play a
static game in each p
ial price correlations are zero. Our dynamic model does not require this assumption and
yields predictions for both contemporaneous and serial price correlations. We show conditions
where the contemporaneous price correlation and serial price correlation can be either positive or
negative. We related our findings to empirical studies from the price promotions literature.
A limitation of our model is that we do not allow for strategic consumer behavior. That
is, a strategic consumer may forgo a low price today if the consumer anticipates becoming
ing a high price tomorrow. To analyze the extent of this issue in our model, we simulate
data and calculate the fraction of dynamically inconsistent choices. A choice is classified as
inconsistent if the current benefits of buying the low-price product are less than the discounted
expected future benefits of buying the other product. For low values of θw and l (e.g. θw = 0 and
- 29 -
l = 0.01) there are no dynamically inconsistent choices, which demonstrates that our results will
extend to a game with strategic consumer behavior. For larger values of these parameters
consumers occasionally make dynamically inconsistent choices and modeling this behavior is a
subject of future research.
This paper incorporates consumer dynamics that are well-established in empirical
marketing studies. The model allows us to highlight the implications of dynamic consumer
behavior for firm pricing strategies and profits. Our model replicates several findings from static
promotion models but also leads to several new, testable results that may be investigated in
future empirical research.
- 30 -
7. References
Anderson, E. T., N. Kumar, S. Rajiv (2004). “A Comment on: Revisiting Dynamic Duopoly with Consumer Switching Costs.” Journal of Economic Theory, 116 (1), 177-186.
Beggs, A. and P. Klemperer (1992), “Multi-Period Competition with Switching Costs,” Econometrica, 60 (3), 651-666.
Bucklin, R. E. and J. M. Lattin (1991), “A Two-State Model of Purchase Incidence and Brand Choice,” Marketing Science, 10 (1), 24-39
Chen, Y., G. Iyer and V. Padmanaban (2002), “Referral Infomediaries,” Marketing Science, V21, n4, pp. 412-434.
Chen, Y. C. Narasimhan, Z. John Zhang (2001), “Individual Marketing with Imperfect Targetability,” Marketing Science, v20, n1, pp. 23-41.
Erdem, T. and M. P. Keane (1996), “Decision-making under uncertainty: Capturing dynamic brand choice processes in turbulent consumer goods markets,” Marketing Science, 15 (1), 1-20.
Farrell, J. and C. Shapiro (1988), “Dynamic Competition with Switching Costs,” RAND Journal of Economics, 19 (1), 123-137.
Guadagni, P. M. and J. D.C. Little (1983), “A Logit Model of Brand Choice Calibrated on Scanner Data,” Marketing Science, 1 (2), 203-38.
Heckman, J. J. (1991), “Identifying the Hand of Past: Distinguishing State Dependence from Heterogeneity.” American Economic Review, 81 (2), 75-79.
Iyer, G. and A. Pazgal (2003), “Internet Shopping Agents: Virtual Co-Location and Competition,” Marketing Science, v22, n1, pp. 85-106.
Klemperer, P. (1987), “Markets with Consumer Switching Costs,” Quarterly Journal of Economics, 102 (2), 375-394.
Kopalle, P. K.; C. Mela and L. Marsh (1999), “The dynamic effect of discounting on sales: Empirical analysis and normative pricing implications,” Marketing Science; 18 (3), 317-332.
Krishnamurthi, L. and S. P. Raj (1991), “An Empirical Analysis of the Relationship between Brand Loyalty and Consumer Price Elasticity,” Marketing Science, 10 (2), 172-183.
Lattin, J. M. (1987), “A Model of Balanced Choice Behavior,” Marketing Science, 6 (1) 48-65. Leeflang, P. S. H. and D. R. Wittink (1992), “Diagnosing Competitive Reactions Using
(Aggregated) Scanner Data,” International Journal of Research in Marketing, 9 (1), 1992, 39-57.
Leeflang, P. S. H. and D. R. Wittink (1996), “Competitive reaction versus consumer response: Do managers overreact?” International Journal of Research in Marketing, 13 (2), 103-119.
Narasimhan, C. (1988), “Competitive Promotional Strategies," Journal of Business, 61 (4), 427-449.
Padilla, A. J. (1995), “Revisiting Dynamic Duopoly with Consumer Switching Costs,” Journal of Economic Theory, 67 (2), 520-530.
Papatla, P. and L. Krishnamurthi (1996), “Measuring the dynamic effects of promotions on brand choice,” Journal of Marketing Research, 33 (1), 20-35.
- 31 -
Raju, J. S., V. Srinivasan, and R. Lal (1990), “Effectiveness of Brand Loyalty on Competitive Price Promotional Strategies,” Management Science, 36 (3), 276-304.
Rao, R. C. (1991), “Pricing and Promotions in Asymmetric Duopolies,” Marketing Science, 10 (2), 131-144.
Seetharaman, P. B., A. Ainslie and P. K. Chintagunta (1999), “Investigating household state dependence effects across categories,” Journal of Marketing Research, 36 (4), 488-500.
Van Oest, R. and P. H. Frances (2003), “Which Brand Gains Share From Which Brands? Inference from Store-Level Scanner Data,” Tinbergen Institute Discussion Paper.
Varian, H. (1980), “A Model of Sales,” American Economic Review, 70 (4), 651-659. Vilcassim, N. J. and D. Jain (1991), “Modeling Purchase-Timing and Brand-Switching
Behavior,” Journal of Marketing Research, 28 (1), 29-41. Villas-Boas, M. (2004), “Consumer Learning, Brand Loyalty and Competition,” Marketing
Science, 23 (1), 134-145
- 32 -
Appendix
Numerical Simulation To initialize the model in t =1, we assume the state is s = 0. We then simulate a game
with one million periods for each vector of parameters. We used the same seed for the random
number generator for each new vector of parameters; our results are not sensitive to this given
the large number of draws. In the symmetric case (θs = θw and lw=ls) the expected profits, prices,
and promotion frequency are symmetric in each state and we use this to verify the accuracy of
our numerical results. Our numerical results are accurate to at least four decimal places (i.e., >
10^(-4)).
Solutions for the Cases
Define 0wp , 0sp 1wp and 1sp as the solution to:
( )0 02 2 2w w w w sr l V p l V 1wδ θ δ+ = + − + (18)
( )1 0(2 ) 2 2 0s s s s sr l V p l Vsθ δ+ + = + + δ (19)
( )0 1(2 ) 2 2w w w w w wr l V p l V 1θ δ+ + = + + δ (20)
( )1 12 2 2 0s s s s wr l V p l Vsδ θ δ+ = + − + (21)
Case 1
In this case and( )0 1wF r = ( )1 1sF r = . Case 1 is feasible if 1sp < *1p and 0wp < *
0p . We
substitute the equilibrium solutions (10)-(15) in these two inequalities to obtain conditions
and respectively. 1C 2C
- 33 -
( ) ( )( ) ( ) ( )( )( )( )
( ) ( )( )
( ) ( ) ( ) ( )( ) ( )( )( )
22
2
2 1 4 1 4 1 2 2
4 4 4 4 2 12 1
2 1 2 21: 0
2 2 2 1 2 2 4 4 2 2
s s s w s s s
s w s w ww s w
s s w s s
s w s w s w s w
l l l
r l l l l ll
l lC
l l l l
δθ θ δ δθ δθ
δθ θ
δθ δ δ θ θ
θ δ δ δθ δ θ θ
⎛ ⎞⎛ ⎞+ − − − + + +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞+ − + − − +⎜ ⎟⎜ ⎟+ − +⎜ ⎟
⎜ ⎟⎜ ⎟⎜ ⎟+ − − + −⎝ ⎠⎝ ⎠⎝ ⎠ >+ − + + + + + + + −
( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( )( )
( )( )( ) ( ) ( ) ( )( ) ( )( )( )
2
2
2
8 1 4 1 1 1 2 1
2 2 2 2 1 1 1
1 2 22 : 0
2 2 2 1 2 2 4 4 2 2
w s w s w w s w s w s
s w w w s s w
w s w
w s s w s w s w
l l l l l l l l l
r l l l l l
lC
l l l l
δ θ θ
δ δ δ θ θ
δ δ θ θ
θ δ δ δθ δ θ θ
⎛ ⎞⎛ ⎞+ − + − + + − + − +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+ + − + − − + − −⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟− − + −⎝ ⎠⎝ ⎠ >+ − + + + + + + + −
Case 2
In this case and . Case 2 is feasible if ( )0 1wF r = ( )1 1wF r = 1wp < *1p and 0wp < *
0p or
equivalently if conditions and hold. To distinguish these solutions from one in the body of the paper, let and
3C 4C*c
jkV * ckp be the equilibrium solution in Case c.
( )( ) ( )( )( )
( )( )( ) ( )( )
2*0
2 2 2 1 2 1 22 1 1 2 2
w s s s w s sw
s w w
r l l r l lV
l lθ θ δ δ δθ
δ θ+ − + − + −
= ++ − + −
(22)
( )(( ) ( )(
))
2*1
2 1 21 2 2
w s sw
w w
r l lV
lδθ
δ θ+ −
=− + −
(23)
2*0
21
ss s
lV r θδ
⎛= +⎜ −⎝ ⎠⎞⎟ (24)
2*1
21
ss
lV rδ
⎛= ⎜ −⎝ ⎠⎞⎟ (25)
( )( )( )
2*0
2 12 1s s
s
r lp
lθ δ+ −
=+
(26)
( )( )
2*1
22 2
s s
s w
r lp
lδθ
θ−
=+ −
(27)
- 34 -
Substituting (22)-(27) in the conditions 1wp < *1p and 0wp < *
0p we obtain the following two
conditions:
( ) ( ) ( ) ( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )
( )( )( )( )
2 2
2 2
2 1 4 2 1 1 2 1 1
4 1 2 1 1 1 13: 0
2 1 2 2 2 2
s s w w s w s s
w s w s w s s w
s w s s w
r l l l l l l
r l l l l lC
l l l
δ δ θ δ θ
δ δ δ δ θ δ θ θ
θ θ
⎛ ⎞+ − + + − − − + − − +⎜ ⎟⎜ ⎟⎜ ⎟− − − + − + − − + −⎝ ⎠ >
+ + − + − (28)
( ) ( ) ( ) ( )( )( )( )
( )( )( )
( )( )( )( )
2
2
2 1 2 2 4 4 1
4 4 4 4 2 1
2 1 2 2
2 14 : 0
4 1 1 2 2
s s s s s s w s
s w s w ww
s s w s s
s w
s w s w
l l l
l l l l lr
l l
lC
l l l
δθ δθ θ δθ δθ
δθ
δθ δ δ θ θ
θ
θ
⎛ ⎞⎛ ⎞+ + − − + − + −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞+ − + − − +⎜ ⎟⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟+ − − + −⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+ +⎝ ⎠⎝ ⎠ >
+ + + − (29)
Case 3
In this case and . Case 3 is feasible if ( )0 1sF r = ( )1 1wF r = 1wp < *1p and 0sp < *
0p or
equivalently if conditions and hold where and 5C 6C *cjkV *c
kp are defined as:
3*0
21
ww
l rVδ
=−
(30)
( ) ( ) ( ) ( )( )( )( )( ) ( )( ) ( ) ( )( )( )( )
3*1 2 22
4 1 1 2 1 1 2 2
1 4 1 1 1 2 2 2 2
w s w s w sw
s w w s s w
l r l l l lV
l l l l
δ δ δ θ
δ δ δ θ
+ − + − − + −=
− + + − − + − + −θ (31)
( ) ( ) ( ) ( )( )( )( ) ( )( )( ) ( )( ) ( ) ( )( )
3*0 3 3
4 1 2 1 2 2 1
2 1 2 1 1 1 2 1 1 2 2
s s w s w s w w ws
s w s
l r l l l l l l lV
l l l
δ δ θ
w s wδ δ δ θ δ θ θ
+ + − + + − −=
+ + − − − − − − + − (32)
3*1
21
ss
l rVδ
=−
(33)
( ) ( )( )( )( )( )( ) ( )( ) ( )( ) ( ) ( )( )
3*0 2 2
2 2 2 4 1 2
2 1 2 1 2 1 1 1 2 2
s w s w
s w s w
r l l lp
l l l
δ δ δ θ
s wδ δ θ δ θ θ
− − + − −=
+ + − + − + − + − (34)
- 35 -
( ) ( )( )( )( )( )( ) ( )( ) ( )( ) ( ) ( )( )
3*1 2 2
2 2 2 4 1 2
2 1 2 1 2 1 1 1 2 2
w s w s
s w s w
r l l lp
l l l
δ δ δ θ
s wδ δ θ δ θ θ
+ − + − −=
+ + − + − + − + − (35)
Substituting (30)-(35) in the conditions 1wp < *1p and 0sp < *
0p we obtain the following
conditions:
( ) ( ) ( ) ( )( ) ( )( )( )( )( ) ( )( )( )( ) ( )( )
( ) ( ) ( )( ) ( )( ) ( ) ( )( )( )
2 2
2 2
2 2
2 1 4 2 1 1 2 1 1
4 1 1 2 1 1 1 15 : 0
2 1 2 1 2 1 2 1 1 1 2 2
s s w w s w s s
w s w s w s s w
s s w s w s w
l r l l l l l
r l l l l lC
l l l l
δ δ θ δ θ
δ δ δ θ δ θ θ
δ δ θ δ θ θ
⎛ ⎞+ − + + − − − + − −⎜ ⎟⎜ ⎟⎜ ⎟+ − − + + − − + − − −⎝ ⎠ >
+ + + − + − + − + −(36)
( )( ) ( )( )( ) ( ) ( )( )( ) ( )( )
( ) ( )
( ) ( ) ( )( ) ( )( ) ( ) ( )( )( )2 2
2 2
8 1 4 1
2 1 1 1 2 12
1 1 1
1 2 26 : 0
2 1 2 1 2 1 2 1 1 1 2 2
w s w s w s w s
w s w sw
w s s
w s w
w s w s w s w
l l l l l l l
l l l lr
l l
lC
l l l l
δ δ θ
δ δθ
δ δ δ θ
δ θ θ
δ δ θ δ θ θ
⎛ ⎞⎛ ⎞− + + − − +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞+ + − − − +⎜ ⎟⎜ ⎟⎜ ⎟+
⎜ ⎟⎜ ⎟⎜ ⎟+ − − − − +⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− − + −⎝ ⎠⎝ ⎠ >
+ + + − + − + − + −(37)
Case 4
In this case and . Case 4 is feasible if ( )0 1sF r = ( )1 1sF r = 1sp < *1p and 0sp < *
0p or
equivalently if conditions and hold, where and 7C 8C 4*jkV 4*
kp are symmetric to Case 2 with
labels of the weak and strong firm reversed and the labels of state 0 and state 1 reversed.
( )( ) ( ) ( )( )( )( ) ( ) ( ) ( )( )( )( )( )
( )( )( )( )
2
2
8 1 4 1 1 1
2 1 2 2 2 2 1 1 1
1 2 27 : 0
4 1 1 2 2
w w s s w s w w s
w s s w w w s s w
w s w
s w w s
l l l l l l l l
r l l l l l l
lC
l l l
δ θ
θ δ δ δ θ θ
δ δ θ θ
θ
⎛ ⎞⎛ ⎞+ − + + − − + +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+ + − + − + − − − − −⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟+ − + −⎝ ⎠⎝ ⎠ >+ + + −
(38)
- 36 -
( )( ) ( )( )( ) ( )( )( ) ( )( )
( ) ( )( )( )( )( )
2 2
8 1 4 1
2 1 1 1 2 22
1 1 1
1 2 28 : 0
2 1 2 2 2 2
w w s s w s w s
w s w sw
w s s
w s w
w w s s w
l l l l l l l
l l l lr
l l
lC
l l l
δ δ θ
δ δ δθ
δ δ δ θ
δ θ θ
θ θ
⎛ ⎞⎛ ⎞+ − + − − +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞+ + − − − −⎜ ⎟⎜ ⎟⎜ ⎟+
⎜ ⎟⎜ ⎟⎜ ⎟+ − − + − −⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− − + −⎝⎝ >
+ + − + −⎠⎠ (39)
Uniqueness
To establish uniqueness of the equilibrium we need to show that {1,2,3,4}n∀ = when conditions and hold, and cannot simultaneously hold for 2 1nC − 2nC 2 1nC ′− 2nC ′ {1,2,3,4}n n′∀ ≠ = .Because of
the complexity of the expressions we are unable to analytically establish uniqueness but we are able to confirm uniqueness with extensive numerical simulations (available from authors). As an example, consider the following parameters values:{ }1, 0.9, 1, 0.01, 0.01s w sr l lδ θ= = = = = .
Evaluating conditions C1-C8 at these parameter values we obtain expressions for these conditions which are a function only of wθ . Evaluated at these parameter values we find the
following:
( )1 0, [0, 2.02) (3.09, ] and 1 0, 2.02,3.09 w wC Cθ θ> ∀ ∈ ∪ ∞ < ∀ ∈
2 0, [0,78.10) and 2 0, (78.10, ] w wC Cθ θ> ∀ ∈ < ∀ ∈ ∞
3 0, [0,2.02) and 3 0, 2.02 w wC Cθ θ< ∀ ∈ > ∀ >
4 0, [0,2.02) (3.09, ] and 4 0, (2.02,3.09)w wC Cθ θ< ∀ ∈ ∪ ∞ > ∈
5 0, 116.78 and 5 0, 116.78 w wC Cθ θ< ∀ < > ∀ >
6 0, [0,0.011) and 6 0, 0.011w wC Cθ θ> ∀ ∈ < ∀ >
7 0, [0,78.10) and 7 0, 78.10w wC Cθ θ< ∀ ∈ > ∀ >
8 0, 2.02 and 8 0, 2.02 w wC Cθ θ< ∀ < > ∀ >
Uniqueness: An Illustration
Parameter Region Equilibrium
[0,2.02)wθ ∈ Case 1 (2.02,3.09)wθ ∈ Case 2 (3.09,78.10)wθ ∈ Case 1 (78.10, )wθ ∈ ∞ Case 4
- 37 -
The table above specifies the unique equilibrium for different values of wθ . For each range both
conditions for a single case are satisfied and at least one condition for the remaining cases is not
satisfied. This proves uniqueness. As an example, for [0,2.02)wθ ∈ conditions C1 > 0 and C2 >
0, which proves that Case 1 is an equilibrium. In this range the other cases are not feasible (C3 <
0, C5 < 0, and C7 < 0), which proves that Case 1 is a unique equilibrium.
- 38 -
