Consider a linear city of length L in which the consumers are uniformly distribu

Question

Consider a linear city of length L in which the consumers are uniformly distributed. There are two firms located at the extremes of the linear city: firm 1 is located at the left-hand extreme, and firm 2 is located at the right-hand extreme. Assume that every consumer buys one unit of the product, that transportation cost are linear (td, where d is distance), and that marginal production costs are zero. Given their location, the firms engage in price competition: they set prices simultaneously as in the Bertrand Model.

-Derive the demands faced by every firm.

-Find the equation of the best response function of every firm

-Find the Bertrand-Nash equilibrium set of prices.

Explanation / Answer

. There is a linear city of length one, the [0,1] interval. A unit mass of consumers are uniformly distributed on this interval. Consumers are identical except for their location. Each consumer has unit demand. If a consumer buys from a firm located at distance d at price p, she incurs a transport cost t(d) and her net surplus is strictly increasing.

There are two firms in the market that produce the good. Firms are located

on the unit interval. The unit cost of production is identical across firms and is

independent of location.

The products of the two firms are horizontally differentiated by their location.

Other things being equal, each consumer prefers the product of the firm

located closest to her.

Interpret [0, 1] as the product space. The location of a firm on this interval

indicates the type of product it supplies. The location of a consumer indicates

her most preferred product type. When a consumer buys from a firm whose

product is not her most preferred type, she incurs a psychological cost that

depends on how far removed the purchased product is from her most preferred

type; the transport cost function captures this psychological cost.

Socially optimal solution: Firms locate at 1

4 , 3

4 so as to minimize the total

transport cost of society (and serve all consumers as long as S is large enough).

1. Product Choice with No Price Competition:

Assume prices are fixed so that there is no price competition in the market:

c p1 = p2 = p<S e + t(1)

In this case, all consumers buy, no matter where firms locate. A consumer

strictly prefers to buy from the firm closest to it.

Firms decide on location simultaneously.

Note that payoff of each firm is directly proportional to its market share.

Suppose that firm 1 locates at a point a and firm 2 located at a point 1 b.

Without loss of generality, assume

a 1 b,

i.e., firm 1 locates to the left of firm 2.

1

The market share of firm 1 is

a + 1 b

2

and the market share firm 2 is

1 a + 1 b

2

= b + 1 a

2 .

Unique NE: Minimal Differentiation i.e., a = b = 0.5.

Proof: First, we show that a = b = 0.5 is a NE. To see this note that if

a = 0.5 and firm 2 sets b < 0.5, then only consumers to the right of the mid

point of the interval [0.5, 1 b] buy from firm 2 so that its market share is

1 0.5 + (1 b)

2

= 0.25 + b

2 < 0.5.

Next, we note that a < 1b cannot be a NE. That it because firm 1 can increase

market share by locating at 1 b >a instead, where > 0 is small. Finally,

note that a = 1 b 6= 0.5 cannot be a NE. In such a situation, all consumers

are indifferent between the firms and so each firms gets market share equal to

0.5. If, for instance, a = 1 b < 0.5, then firm 2 can move slightly to the right

and get market share that is higher than 0.5 (everyone to the right of firm 2

will now strictly prefer firm 2).

This was the original result in Hotelling’s paper. He used this conclude that

competition will lead to minimal product differentiation.

Note: This is also one of the basic models of electoral competition in political

economy where the location of a political party signifies its choice of political

platform or its position on public policy and the consumers arereplaced by voters

who are located at their most preferred outcome on public policy. The above

conclusion translates to convergence of political platforms (to the median voter’s

preferred outcome). Assume: t(d) = td2.

Assume that S is large enough so that in equilibrium, all consumers buy.

If a = 1 b, then both firms sell identical products in the market and the

model reduces to a standard homogenous good symmetric Bertrand duopoly.

All consumers buy from the firm with the lower price and if they charge equal

prices, we can suppose that the consumers randomize evenly between the firms

so that each firm sells 0.5 (as long as the price charged is such that all consumers

earn non-negative net surplus).

Now, consider a < 1 b. We first figure out the quantity demanded from

each firm at a pair of prices (p1, p2) assuming all consumers buy. A consumer

located at point y strictly prefers to buy from firm 1 if, and only if,

p1 + t(y a)

2 < p2 + t(1 b y)

2.

If both firms sell positive quantities, then there must exist some consumer located

at some point x [0, 1] who is indifferent between buying from firms 1

and 2. For such x :

p1 + t(x a)

2 = p2 + t(1 b x)

2

so that

x = a + 1 b

2 + p2 p1

2t(1 a b)

and every consumer located to the right of x strictly prefers to buy from firm 2

while consumers to the left of x strictly prefer to buy from firm 1. If the price

difference is large then the expression above lies outside the [0, 1] interval in

which case one firm takes the entire market and the other sells zero.

Therefore, the demand for firm 1’s product product at prices (p1, p2) is given

by

D1(p1, p2) = a + 1 b

2 + p2 p1

2t(1 a b)

,

if t[(1 b)

2 a2] p1 p2 t[(1 b)

2 a2]

= 0, if p1 p2 t[(1 b)

2 a2]

= 1, if p1 p2 t[(1 b)

2 a2]

The demand for firm 2’s product at prices (p1, p2) is given by

D2(p2, p1)=1 D1(p1, p2).

The payoff of each firm is given by

i(pi, pj )=(pi c)Di(pi, pj ).

Confine attention to prices such that both firms sell and prices c. We can

derive the reaction function of each firm i (taking pj as given). The reaction

function for firm 1 is given by:

D1 + (p1 c)

D1

p1

= 0

which yields:

a + 1 b

2 + p2 p1

2t(1 a b) 1

2t(1 a b)

(p1 c)=0

i.e.,

p1 = 1

2

[c + p2 + t{(1 b)

2 a2}].

Similarly,

p2 = 1

2

[c + p1 + t{(1 a)

2 b2}].

Observe that the reaction functions are upward sloping (game of strategic

complementarity).

Solving the reaction functions, we obtain the Nash equilibrium (unique):

pb1 = c + t(1 a b)(1 + a b

3 )

pb2 = c + t(1 a b)(1 + b a

3 ).

Observe following:

1. The above expressions give the correct NE even when a = 1 b so that

we have a homogenous good symmetric Bertrand price competition model.

2. If a < 1 b, then equilibrium prices exceed marginal cost and firms make

positive profit. Thus, product differentiation softens price competition, enables

firms to make money and increases market power.

3. If t = 0 i.e., consumers do not differentiate between the goods, then

independent of how differently the products are positioned, the market is effectively

a homogenous good market and we get the standard price = MC Bertrand

outcome.

4. As t increases i.e., consumers differentiate the goods more, market power

increases.

5. If firms are located symmetrically i.e., a = b = z (say), pb1 = pb2 =

c + t(1 2z). In that case, (1 2z) is the distance between the two firms and as

this increases (product differentiation by product positioning increases), market

power increase

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