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Coordination of Price Promotions Within a Product Line
Maxim Sinitsyn
Department of Economic, University of California, San Diego
October 2012
Abstract
This motivating example considers two …rms each producing two products. The products
are horizontally di¤erentiated with respect to one characteristics, for example, fat content.
Each …rm o¤ers one low-fat version of the product and one regular version. As this characteristic becomes more important in the overall consumer demand (as measured by the number of
consumers who will buy only the product with the characteristic they prefer), the …rms put
more emphasis on promoting both versions of their products together. Further treatment of
this problem will involve analysis of …rms’ strategies for more ‡exible demand speci…cation,
namely a nested logit.
1
Model
Lal and Villas-Boas (1998) study the model, in which two manufacturers produce two products,
which they sell to the two retailers. The two retailers then sell the products to the consumers. I am
interested in this second stage of this model. This stage basically models the competition between
two …rms each carrying two substitute products. Following the notation in Lal and Villas-Boas
(1998), the …rms 1 and 2 will set the prices PA1 and PA2 (…rm 1), and PB1 and PB2 (…rm 2). In
Lal and Villas-Boas (1998) notation, A and B referred to the manufacturers. Here, these letters
refer to some common characteristics of the products, for example, A1 and A2 can be regular
Coke and Pepsi while B1 and B2 can be diet Coke and diet Pepsi. The consumers are separated
into 7 segments. The price sensitive consumers will buy the product with the lowest price. The
size of this segment is S. There are 4 segments of loyal consumers who buy only their preferred
1
product. Each of these segments has size I. Finally, there are two segments of consumers who
are loyal to the product characteristic and will choose the …rm that o¤ers the product with their
preferred characteristic at a lower price. The size of these segments is M . Lal and Villas-Boas
(1998) also have 2 segments of size R that consist of consumers loyal to the speci…c …rm. To keep
the model as simple as possible, I will not consider these segments and set R equal to zero. The
reservation price of consumers is r, which I normalize to 1.
First, I will consider the case when the costs of all four products is identical (wA = wB in
Lal and Villas-Boas (1998)). Since, M=I is less than (M + S)=I, the conditions of Proposition
3 in Lal and Villas-Boas (1998) are satis…ed, and the equilibrium strategies are outlined in the
following proposition.
Proposition 1 (Proposition 3 from Lal and Villas-Boas (1998) with r = 1 and R = 0). The
equilibrium strategies of the …rms are
F (p; 1) = F (1; p) = 1
I(1 p)
Mp
S+M
2S+M
2I(1 p)
Mp
F (p; 1) = F (1; p) =
F (p; p) = 1
where p =
I
I+S+M
and p =
I(1 p)
(2S+M )p
2I
2I+M .
for p 2 [p ; 1];
for p 2 [p; p ];
for p 2 [p ; 1],
Proposition 3 provides the formulas for F (p; p) and for the marginal distributions F (p; 1) and
F (1; p). The CDF of the price distribution at any point (pi1 ; pi2 ) can not be identi…ed. However,
it is possible to make some conclusions about the coordination of price promotions without the
2I
I (1 2I+M
)
= 21 . Given that
knowledge of F (pi1 ; pi2 ): First, note that F (p ; 1) = F (1; p ) = 1
M 2I
2I+M
F (p ; p ) = 0, this means that F (p ; 1)
F (p ; p ) = F (1; p )
F (p ; p ) =
1
2.
Therefore, the
pdf of charging a price pair (pi1 ; pi2 ) is positive only in the upper-left and bottom-right corners
of the price support square, that is when one of the prices is higher than p and another price is
lower than p . There is zero probability assigned to the price pairs, in which both of the prices
are smaller than p or both of the price are larger than p . This characterization reveals that the
…rms always promote one of the products deeper than another.
Even though the promotion depths are never identical for the two products, it is possible
that most of the weight can be placed on promotions of similar depths. In order to examine
how the percent of the consumers who are loyal to the product characteristic, M , changes the
2
…rms’ strategies of coordinating promotions, I will examine the variable , which is equal to
F 1; 1+p
2
1
2
F
p+p
2
; 1 . In …gure 1,
represents the di¤erence in the probability of price
pairs falling into region D and the probability of price pairs falling into region B. Region D
represents price pairs with relatively similar depths of price promotions. On the contrary, in
region B, there is a deep discount on one of the products and a shallow discount on another.
Figure 1: Support of F
1
0
0
1
It is possible to write out an explicit formula for
and substituting 1
4I
using the formulas from Proposition 1,
2M for S in order to normalize the total number of consumers to 1.
However, it is not possibly to analytically determine whether @ =@M is greater or smaller than
zero. I performed this analysis numerically for the values of I between 0:01 and 0:24 with a step
of 0:01 and for the values of M between 0:001 and (1
4I)=2 with a step of 0:001. It turns out
that everywhere on this grid @ =@M is greater than zero. This means that as the number of
consumers who are loyal to the product characteristic increases, the …rms are more likely to o¤er
similar promotions for their substitute products.
3
Appendix.
Proof of Proposition 1. Assume that both …rms are charging the prices from a binomial
distribution F (pA ; pB ). If the …rm charges the reservation prices for both of its products, (1; 1),
it will serve only its loyal consumers, and its pro…t is 2I. When the …rm charges identical prices
for both products, (p; p), it will win all the switchers when p is the lowest price. This happens
with probability 1
F (p; 1)
F (1; p) + F (p; p). It will also win the consumers loyal to one of the
characteristics with probability 1
(p; p) = p (2I + M (2
F (p; 1). Then, the expected pro…t from charging (p; p) is
F (p; 1)
F (1; p)) + S (1
F (p; 1)
F (1; p) + F (p; p))) .
When the …rm charges prices (p; 1), it wins all the switchers with probability 1
(1)
F (p; 1)
F (1; p) + F (p; p) and the consumers loyal to the …rst type of product with probability 1
F (p; 1).
Given that it charges 1 for the second type of product, it never wins the consumers who are loyal
to the second type of product. Then, the expected pro…t from charging (p; 1) is
(p; 1) = p (I + M (1
F (p; 1)) + S (1
F (p; 1)
F (1; p) + F (p; p))) + I.
After equating (2) and (1), and setting F (p; 1) = F (1; p), we get p (2I + M (2
p (I + M (1
2F (p; 1))) =
I(1 p)
M p . Substituting this formula in equation
2I(1 p)
1 + F (p; p) , from where F (p; p) = 1
Mp
F (p; 1)))+I, from where F (p; 1) = 1
2I = (p; p), we get 2I = p 2I +
2I(1 p)
Mp .
(2)
F (p; p) = 0 at p = p =
2I(1 p)
p
2I
2I+M .
+S
F (p ; 1) is greater than zero, therefore, another equation,
(p; 1) = 2I with F (p; p) = 0, describes F (p; 1) for p < p . p (I + M (1
I = 2I, from where F (p; 1) = F (1; p) =
S+M
2S+M
I(1 p)
(2S+M )p .
4
F (p; 1)) + S (1
F (p; 1) = 0 at p = p =
2F (p; 1)))+
I
I+S+M .