Product differentiation and price competition

3 November 2008

1 Product differentiation
• Our analysis so far has restricted attention to industries offering a homogenous
product. The truth is that outside agriculture, very few industries sell truly
homogeneous products.

• Our analysis of differentiated product markets is motivated by three related

observations:

1. Product characteristics (brands) often are the major competitive strategies

of firms to win market shares.
2. Only a small subset of all possible varieties of differentiated products are
actually produced. For example, most products are not available in all
colours.
3. Consumers purchase a small subset of the available product varieties.

• The last observation is nothing but just the common sense perception that in
many markets, there can exist substantial difference in tastes among consumers.
In this case, there are often profitable opportunities for firms to differentiate
their products to find their own market niches, explaining the first observation.
Nonetheless, according to the second observation, there are forces that limit
the extent of product differentiation. There are cases where firms deliberately
choose to offer similar products. Relatedly, there are apparent market niches
left not exploited by the industry incumbents and by new entrants.

• Understanding the issues involved in these observations is of massive impor-

tance. For, as most people would agree, increasing product variety is central
to our material well-being in modern times. In this and the next two lectures,
we shall analyze the forces dictating the extent of product differentiation in
equilibrium. Equally importantly, we shall also analyze the efficacy of the free
market to supply the socially optimal product variety.

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2 Differentiated product Bertrand oligopoly
• Consider a differentiated product duopoly. There are two firms, each selling
its own brand of the good To simplify, we assume that the marginal cost to
produce either brand is equal to the same c. The (inverse) demand curves are
of the two brands are respectively:
p1 = G − gq1 − γq2 ,
(1)
p2 = G − gq2 − γq1.

• In (1), for a given good, there is the usual negative relationship between price
and quantity. The novelty is that we also assume a negative relationship be-
tween the demand price of a brand and the quantity of the competing brand.

• The parameter g may be termed the own-price effect, which measures how the
price of a brand will vary in response to the quantity sold of the given brand.
• The parameter γ may be termed the cross-price effect, which measures how the
price of a brand will vary in response to the quantity sold of the competing
brand.

• It makes sense to assume g > γ, which is to say that the effect of increasing
q1 on p1 is stronger than the effect of increasing q2 on p1 . That is, the price of
a brand is more sensitive to a change in quantity of the given brand than to a
change in quantity of the competing brand.

• Note that when γ = 0, the demand of the two brands are completely indepen-
dent. In this case, each firm is a monopoly of its own brand, and the profit
maximizing price is simply equal to the monopoly price.

• In the other extreme case of g = γ, the two brands are perfect substitutes, and
the duopoly becomes a homogeneous product duopoly. To see this, note that
at g = γ, we have from (1)

p1 = G − g (q1 + q2 ) ,
p2 = G − g (q1 + q2 ) .

This means that the demand price of a given brand depends only on the sum
of the quantities of the two brands. Furthermore, with the right hand sides of
the two demand curves identical, we must also have p1 = p2 , and the so the
demand system collapses into

p = G − gQ,

where Q = q1 + q2.

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• In general, the degree of product differentiation may be measured by γ, with
γ = 0 denoting maximum differentiation, and γ = g minimum differentiation.
• We should next solve for the NE of the industry, assuming that firms use prices
as strategies. To begin, the profit of firm 1 is
π 1 = (p1 − c) q1 . (2)
The firm chooses a p1 to maximize profit, while holding a fixed belief on its
competitor’s price p2 .
• In (2) , q1 depends on both p1 and p2 as governed by the demand system (1) .
Solving (1) for q1 :
G (g − γ) − gp1 + γp2
q1 = .
g2 − γ 2
Substitute the above into (2) :
G (g − γ) − gp1 + γp2
π 1 = (p1 − c) . (3)
g2 − γ 2

• What is the p1 that maximizes the firm’s profit? The profit function is maxi-
mized with respect to p1 when
∂π 1
= 0 ⇒ G (g − γ) + cg − 2gp1 + γp2 = 0.
∂p1
Solving for p1 :
g (G + c) − γ (G − p2 )
p1 = . (4)
2g
This is firm 1’s best response function, i.e. its profit-maximizing price as a
function of its belief on firm 2’s price p2 .
• We may similarly derive the best response function of firm 2:
g (G + c) − γ (G − p1 )
p2 = . (5)
2g

• Figure 1 plots the two best response functions.

• As in Bertrand competition with a homogeneous product, the best response
functions are upward sloping, meaning that it is optimal for a given firm to
charge a higher price if the competing firm raises its price. Each firm would like
to charge a high price to earn a bigger profit margin. The drawback of raising
prices is that the firm will lose market share to the rival. Hence the optimal
pricing strategy for a firm is to mark up to a level close to the price charged
by the rival. The NE is at where the two best response functions intersect in
figure 1.

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 Figure 1: NE

• Alternatively the NE can also be recovered by solving the system (4) and (5)
simultaneously for p1 and p2 :
g (G + c) − Gγ
p1 = p2 = . (6)
2g − γ
Substituting the above back into the profit function in (3) :
(g − γ)2 (G − c)2 g
π 1 = π2 = . (7)
(2g − γ)2 (g 2 − γ 2 )

• When the demand of the two brands are completely independent of each other,
i.e. γ = 0, the equilibrium price and profit becomes the monopoly price and
monopoly profit respectively:
G+c
p1 = p2 = ,
2
(G − c)2
π1 = π2 = .
4g

• When the two brands are perfect substitutes, i.e. γ = g, the equilibrium price
falls to the marginal cost of production, and the equilibrium profits are down
to 0.
p1 = p2 = c,
π 1 = π 2 = 0.

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 When the two brands are perfect substitutes, the market degenerates into a
homogenous product Bertrand oligopoly, the equilibrium price of which is equal
to the marginal cost of production.

• In general, it can be verified that as the degree of product differentiation in-

creases, i.e. as γ falls from g towards 0, the equilibrium price rises from the
marginal cost to the monopoly price. In the mean time, the profits of the firms
increase from 0 to the maximum monopoly profit.

• Intuitively, the more similar the brands are, the more intense the price compe-
tition would become. When the products are nearly homogeneous, firms would
have to rely on price competition to attract consumers. In equilibrium, the
price will be lowered to nearly the marginal cost of production.

• Firms benefit to have their brands differentiated as much as possible to avoid

price competition. Is it true then given the choice of the degree of product dif-
ferentiation, firms in oligopoly would like to make their brands as differentiated
as possible from their rivals’ brands. Not necessarily.

• The degree of differentiation is often maximized when each brand is designed to

suit those whose tastes lie on the extremes of the spectrum of consumer tastes.
The brands that emerge from this exercise, however, could be of little value to
the majority of consumers. There are clear—cut benefits to design the brand to
suit the taste of the majority to maximize market share. Such issues cannot be
tackled by the present analysis because we have not modeled how differences
in tastes among consumers may give rise to demand curves that relate how the
quantity demanded for the given brand depends on the price charged for the
competing brands. This will be the subject of the next lecture.