PROBLEM SET #3
PRODUCT DIFFERENTIATION
QUESTION 1: Horizontal Differentiation
(Note: the defining characteristic of horizontal or spatial models of product differentiation is that
each firm competes for customers only locally, that is, solely with the firms offering similar
products.)
In this question we will consider the model of the Circular City (see, Tirole, Chapter 7) in which
we can examine product differentiation when entry is possible.
Assume that there are a large number of identical potential firms (free entry). Assume also that
there is a uniform distribution of consumers, i.e., in any part of the circle there is a proportionate
number of consumers. For example, in a portion that is a fifth of the entire circle there are 20%
of the entire population of consumers.
Firm A
Firm B
Firm C
Firm E
Firm D
The above drawing gives an example with 5 firms (the picture should show that the firms are
equidistant from one another). Assume that the perimeter of the circle is equal to one. Firms are
located around the circle and travel occurs along the circle.
Consumers wish to buy one unit of the good and have a unit transportation cost of t. Hence,
when a consumer buys from a store that is located at a distance d away from her, her generalized
cost is price + td. Consumers are willing to buy at the smallest generalized cost as long as it does
not exceed the gross surplus (denote that s) they obtain from consuming the good.
Each firm is allowed to locate in only one location.
First assume that there is a fixed cost of entry.
Once a firm is has entered and is located on a point on the circle, it faces a marginal cost c (which
is assumed to be smaller than s).
1.) Write down firm i’s profit letting Di denote the demand faced by firm one. Note profit
will be 0 if the firm does not enter.
Profit of firm i = πi =
(pi – c)Di – f if the firm enters
0 if the firm does not enter
2.) Now consider the two stage game:
First stage: potential entrants simultaneously choose whether or not to enter. (n is the
number of entering firms. Those firms do not choose their location but are automatically
located equidistant from on another
Second Stage: Firms compete in prices given these locations
Note that the free entry assumption implies that equilibrium profit of entering firms is
zero.
First determine the Nash equilibrium in prices for any number of firms and calculate the
reduced-form profit functions. Proceed as follows.
Assume that n firms have entered the market. Because they are located
symmetrically, it makes sense to look for the equilibrium in which they all charge
the same price p. Assume the fixed costs are small enough that there are enough
firms in the market so that the firms are competing amongst themselves. Firm i is
competing with its two neighboring firms.
i)
Write down the condition for a consumer located at a distance x between
0 and 1/n from firm i to be indifferent between purchasing form firm i and
purchasing from the nearest neighbor. Assume that the other firm is
choosing a set price p.
ii)
Solve this indifference condition to derive firm i’s demand. Note that the firm i
will have a demand equal to 2x since it is taking consumers from the space in
between it and it’s neighbor on the right (x) and between it and its neighbor on
the left (another x).
iii)
The firm seeks to maximize profit. Write down the profit function given that the
firm has entered. (it is the same as in (1) but substitute the Di that you just
calculated)
iv)
Now find the first order condition by differentiating with respect to p and then set
pi=p (this will be the Nash equilibrium). Does the profit margin (p-c) increase
decrease or stay constant with n.
v)
But we know that the number of firms is determined by the zero profit condition
which is imposed by free entry. Write down this zero profit condition.
vi)
Now solve for n (the number of firms)
vii)
Now solve for p from the first order condition in iv). Say whether the firm is
producing at or above marginal cost.
viii)
What happens as the fixed cost increases?
ix)
What happens as the transportation cost increases?
x)
What happens as the entry cost or fixed production cost f converges to zero?
.
QUESTION 2: Vertical Differentiation
(Also based on Tirole, this time Chapter 7.5.)
Let s1, s2 be the qualities of firm 1 and firm 2 respectively.
Assume that s2>s1. Let ∆s ≡ s2 – s1 denote the quality differential.
Let the consumer derive utility U=θs – p from consuming one unit of quality s and paying price p
and let θ lie between some constants θ (max) , θ(min) where high θ’s denote consumers with
preferences for high quality.
For prices p1 and p2, we can find the following the demand functions (under appropriate
consumer heterogeneity and if we ensure that each consumer buys one of the two brands):
D1(p1, p2) = ((p2-p1)/ ∆s) - θ(min)
D2 (p1, p2) = θ (max) –(p2-p1)/2
1) Firm one maximizes its profit, which is a function of its quality (s1) and the other firm’s
quality (s2). Write down the profit function of firm one.
2) Find the best response function for firm one. (this is an expression for firm one’s
optimum price given firm two’s price). That is, substitute the demand conditions into the
profit function and find the derivative with respect to p1. Note this will be a function of
p2, c, ∆s, and θ(min).
3) Find the reaction function for firm two by the above method but now this will be a
function of p1, c, ∆s, and θ(max).
4) Solve for the Nash Equilibrium (substitute one firm’s best response function into the
other’s and solve for each price)
5) Substitute these demands and prices into the profit functions that you found in (1) and
give each firm’s profit.
6) What happens if ∆s = 0 (that is, if firms’ products are undifferentiated). What happens to
each profit function as differentiation increases? What should each firm do to its quality
choice to increase its profit.