Answer to Question #194562 in Microeconomics for Faisal

Question

Problem 3

Consider a market with two firms. Each firm is located at one end of a line with lenght one.

There is a mass one of consumers. The location of each consumer is given by 0 < x < 1

which is uniformly distributed (with density 1). Firms have no cost of production and set

price simultaneously.

a) Derive the demand for each firm by identifying the location of the indifferent con-

sumer for each price pair. Assume that all consumers know about both products.

b) Write down the profit functions and calculate the Nash equilibrium prices for both

firms.

c) Assume that consumers only know the product if they have received and ad. Suppose

that ads are not targeted and each firm reaches any consumer with probability 0.5

with her ad. Calculate the size of the different consumer segments. Determine the

resulting demand and the new Nash equilibirum prices of the firms.

d) Suppose that the ads are costless. When do the firms make larger profits? With

fully informed consuemers b) or with imperfect ads c)?

Answer 1

a)

There are two firms Firm 1 and firm 2 and they are located at two extremes at 0 and 1. Now, assuming there is one consumer and the location is within 0<x<1. 

It is said that the distribution of consumers is uniform in nature. It is a Hoteling model that signifies the locational equilibrium in a duopoly model. The model shows that customers who stay away from firms have less utility from the ones staying near. Total demand is maximized when a firm caters to all the demand of the customers.

 

Let V be the value of the consuming product and t the transporting costs, Pi be the P of the products from the firms and xi is the center location. So consumers staying near 0 will go to firm 1 and those living near 1 will go to firm 2.

Firm 1’s demand is x and Firm 2’s demand is 1-x

Now, the price of the product will include the transportation costs as well. Therefore, the price for product 1 by F1 will be $Pi= P1+txi$ and for P2 will be $Pj=P2+t(1-xi).$

The utility is calculated by $Ui=V-(p+t[x-ai])$

Now, it has been assumed that consumers have complete information about the products. The demand is equal to the utility of both the firms therefore,

Demand$=½+\frac{(Pi-Pj)}{2t}$

(b)

the consumers are all willing to pay 1 unit of price for each unit of good, (from the density function).

Now quantity share of each firm is 0.5

They will charge unit price as equilibrium price. Claiming more than that, would not be feasible. Claiming less than that will not be efficient as the consumers are willing to pay more. 

Profit function of both the firms is 

π = 0.5p

profit is maximum for the maximum value of p which is 1. 

Hence 1 is the Nash Equilibrium price for both firms. 

(c)

Both A and B can attract customers only by providing ad, and ads can attract only half of the customer base. But they are already at a Nash equilibrium at the midpoint of the (0,1) segment. Therefore both of them will only be able to acquire 0.25 of the markets each. 

Hence the new demand will be 0.25 of the whole customer base for each producer. 

The Nash equilibrium prices will still remain at p = 1.

The profit function here, is π = 0.25p

which is less than the complete information scenario.

(d)

If ads are costless then the firms make more profit when consumers are fully informed. Because of imperfect ads, they can only read half of the total consumer base thus reduces profit.