Answer to Question #196192 in Microeconomics for Lena

Question

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.

1. 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.
2. Consider again that consumers can only buy after receiving an ad. Suppose there is an avdertising company that offers the firms to coordinate the targeting of their ads. The company suggests to inform all consumers with a location between 0 and 0.4 the product of the firm at location 0 and to all consumers between 0.6 and 1 the product of the firm at location 1. Determine the optimal prices for both firms if they accept this offer. What are the resulting profits?

Answer 1

(1)

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$=½+\frac{(Pi-Pj)}{2t}$

(2)

$Pi= P1+txi$

$=0.4+0.4\times 0$

$=0.4$

$Pj=P2+t(1-xi).$

$=0.5+0.6(1-1)$

$=0.5$

profit$=pj-pi$

$=0.5-0.4=0.1$