Linear demand formulations for price competition in horizontally differentiated products are sometimes used to compare situations where additional varieties become available, e. g. due to market entry of new firms. We derive a consistent demand system to analyze such situations.Horizontal product differentiation, preferences for variety, market entry

Höffler, Felix

English

Research Papers in Economics

On the consistent use of linear demand systems if
not all varieties are available
Felix H￿› er￿
October 29, 2007.
Abstract
Linear demand formulations for price competition in horizontally di⁄er-
entiated products are sometimes used to compare situations where additional
varieties become available, e.g. due to market entry of new ￿rms. We derive
a consistent demand system to analyze such situations.
JEL-Classi￿cation: D1, L1
Keywords: Horizontal product di⁄erentiation, preferences for variety, mar-
ket entry
In industrial economics, often the following expression is used to model price
competition in horizontally di⁄erentiated products: There are n product varieties
with demand for variety j = 1;::;n; given by:
Dj (p) =
1
n
￿
1 ￿ pj ￿ ￿
￿
pj ￿
Pn
i=1 pi
n
￿￿
: (1)
This formulation goes back to Shubik and Levitan (1971), is used in many models,
and can be found in text books like e.g. Vives (2001), p. 163. The formulation
is analytically tractable and has a very intuitive interpretation: Demand decreases
directly in the own price but additionally if the own price increases above the price
average, where the parameter ￿ describes how closely the di⁄erent markets are
￿WHU - Otto Beisheim School of Management, Burgplatz 2, 56179 Vallendar, Germany, fe-
lix.hoe› er@whu.edu.
I would like to thank Hendrik Hakenes, Martin Hellwig, Stefan Lauermann and in particular Klaus
Schmidt for helpful discussions.
1linked. An important feature of this demand system is that it can be derived from
a representative consumer with quasi-linear preferences that can be represented by
the following utility function (where w denotes the initial wealth of the consumer):1
U =
n X
j=1
qj￿
1
2
 
n X
j=1
qj
!2
￿
n
2(1 + ￿)
2
6
4
n X
j=1
q
2
j ￿
￿Pn
j=1 qj
￿2
n
3
7
5+
"
w ￿
n X
j=1
qjpj
#
: (2)
It is often interesting to compare situations where di⁄erent numbers of varieties
are available. Consider, for instance, the decision problem of a consumer with
preferences for di⁄erent varieties of cereals who goes to the supermarket and realizes
that the varieties m+1;:::;n are sold out. How much does she buy from the varieties
that are available? Or consider a market where some varieties will be o⁄ered if and
only if new ￿rms enter the market. How do prices, quantities and welfare change
if such market entry occurs? Suppose that qj = 0 for j = m + 1;:::n, i.e., some
varieties are not available because they are sold out or entry did not occur. In this
case, a consumer with preferences according to (2) maximizes:
U =
m X
j=1
qj￿
1
2
 
m X
j=1
qj
!2
￿
n
2(1 + ￿)
2
6
4
m X
j=1
q
2
j ￿
￿Pm
j=1 qj
￿2
n
3
7
5+
"
w ￿
m X
j=1
qjpj
#
: (3)
Note that (2) di⁄ers from the utility of a consumer who is not interested in goods
j = m+1;:::n. Such a consumer would also consume qj = 0 for j = m+1;:::n, but
in his utility function m instead on n appears in the product n
2(1+￿)
Pm
j=1 q2
j. This
subtle change a⁄ects the demand for all the other goods. A consumer with utility
functions (2) who cares about goods j = m + 1;:::;n that are not available and is
constrained not to buy them has the following demand functions for the available
varieties j ￿ m:
Dj (p) =
1 + ￿
n
"
1 ￿ pj ￿
￿
n + m￿
 
m ￿
m X
j=1
pj
!#
for j ￿ m ￿ n: (4)
1See Vives (2001), p. 163. Note that there is a typo, where for the last term in the utility
function it reads
P
j qj; while correctly it should be
￿P
j qj
￿2
; since only the latter results in the
demand functions derived by Vives.
2On the other hand, a consumer who does not care about goods j = m + 1;:::;n
and would not buy them even if he could has di⁄erent demand functions for varieties
j ￿ m:
Dj (p) =
1
m
￿
1 ￿ pj ￿ ￿
￿
pj ￿
Pm
i=1 pj
m
￿￿
=
1
m
(1 ￿ pj ￿ ￿ (pj ￿ p)) for j = 1;:::;m;
(5)
where p denotes the "average price". In some recent papers, e.g. Ordover and Sha⁄er
(2007) or Bourreau, Hombert, Pouyet, and Schutz (2007), the latter demand system
has been used to compare a situation in which some goods j = m + 1;:::;n are
not available to one in which they are available and in which the consumer would
buy them. This is not appropriate. It makes a di⁄erence for the demand system
whether a consumer is constrained not to buy some goods or whether he voluntarily
abstains from buying them. As an illustration assume n = 3; m = 2; p1 = p2 = 1
2;
and ￿ = 1
2: Consider a situation where the price p3 is so high that a consumer
with utility function 3 demands q3
￿
1
2; 1
2;p3
￿
= 0; which is the case for p3 ￿ 7
8: The
resulting other quantities are q1 = q2 = 3
16: In contrast, demand system (5), which
captures the situation where the consumer does not care about the third variety,
yields as quantities for p1 = p2 = 1
2: b q1 = b q2 = 1
4: It is easy to check that for
p1 = p2 = 1
2 the formulation (4) yields quantities q1 = q2 = 3
16:
Furthermore, a consistent welfare analysis2 requires that the two demand func-
tions in the two situations are derived from the same consumer, while the demand
functions (4) and (5) are derived from di⁄erent consumers. This is important, in
particular when evaluating normatively the e⁄ects of market entry or entry deter-
rence.
References
Bourreau, M., J. Hombert, J. Pouyet, and N. Schutz (2007): ￿The Compet-
itiveness of Integrated Goods Markets with Vertically Integrated Firms,￿CEPR
Discussion Paper DP6224.
Ordover, J., and G. Shaffer (2007): ￿Wholesale Access in Multi-Firm Mar-
2The authors of the papers cited do not undertake a welfare analysis.
3kets: When is It Pro￿table to Supply a Competitor?,￿International Journal of
Industrial Organization, forthcoming.
Shubik, M., and R. Levitan (1971): ￿Noncooperative Equilibria and Strategy
Spaces in an Oligopolistic Market,￿in Di⁄erential Games and Related Topics, ed.
by H. Kuhn, and G. Szego, pp. 429￿ 447. North Holland Press, Amsterdam.
Vives, X. (2001): Oligoly Pricing: Old Ideas and New Tools. MIT Press, Cam-
bridge, MA, 2. edn.
Appendix
How to derive (4):
Assume, utility is given by (2), but impose that qj = 0 for j = m+1;:::;n:Thus,
the consumer maximizes:
U =
m X
j=1
qj ￿
1
2
 
m X
j=1
qj
!2
￿
n
2(1 + ￿)
2
6
4
m X
j=1
q
2
j ￿
￿Pm
j=1 qj
￿2
n
3
7
5 +
"
w ￿
m X
j=1
qjpj
#
:
The ￿rst order conditions for j = 1;:::;m are:
pj =
@U
@qj
pj = 1 ￿
m X
j=1
qj ￿
n
2(1 + ￿)
2
42qj ￿
2
￿Pm
j=1 qj
￿
n
3
5
qj =
1 + ￿
n
￿
1 + ￿
n
pj +
￿
n
m X
j=1
qj
| {z }
=:X
(6)
Summing over all m yields:
X =
m(1 + ￿)
n
￿
1 + ￿
n
m X
j=1
pj +
m￿
n
X (7)
X =
m(1 + ￿)
n ￿ ￿m
+
1 + ￿
n ￿ ￿m
m X
j=1
pj: (8)
Plugging (8) into (6) then yields the result in the paper.
4

On the consistent use of linear demand systems if not all varieties are available

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