Question

Two firms, Firm 1 and Firm 2, compete by simultaneously choosing prices. Both firms sell an identical product for which each of 100 consumers has a maximum willingness to pay of $40. Each consumer will buy at most 1 unit, and will buy it from whichever firm charges the lowest price. If both firms set the same price, they share the market equally. Costs are given by ??(??)=16??ci(qi)=16qi. Because of government regulation, firms can only choose prices which are integer numbers, and they cannot price above $40.

Answer the following:

a) (0.25 point) If Firm 1 chooses ?1=33p1=33, Firm 2's best response is to set what price?

b) (0.25 point) If Firm 2 chooses the price determined in the previous question, Firm 1's best response is to choose what price?

c) (1 point) If Firm 1 chooses ?1=7p1=7, Firm 2's best response is a range of prices. What is the lowest price in this range?

d) (1 point) Now suppose both firms are capacity-constrained: Firm 1 can produce at most 26 units, and Firm 2 can produce at most 45 units. If firms set different prices, consumers will first buy from the firm charging the lower price. Once that firm's supply is exhausted, consumers will buy from the firm charging the higher price until that firm's supply is exhausted. What is Firm 1's equilibrium profit?

Answer 1

Answer :

Solution (a) :-

If firm 1 choose P1 = 33, firm 2 can choose next lower integer i.e P2 = 32. Since the customers will move to the firm charging lowest price. The best response of firm 2 will be to set price just the next lower integer. Therefore, P2= 32.

Solution (b) :-

If firm 2 chooses to set the price at P2= 32. It will be in the best interest of firm 1 to set price at the next lower integer i.e P1= 31. Since the customers will move to the firm charging lowest price. Its beneficial to charge just a little bit less (next integer) than the other firm.

Solution (c) :-

If firm 1 chooses price p1 = 7. Firm 2 can choose any price between 7 to 16. The lowest price in this will be 7. But we that firms choose price equal to Marginal cost.

Thus, the lowest price in this range is $7.

In this case c(1)= 16q1

MC(1) =16

MC= P= 16 at opimal level.

So a rational producer will choose p =16(maximum charge price)

Solution (d) :-

Firm 1 will produce 26 units and firm 2 will produce 45 units. so a total of 26+45= 71 customers will be served out of 100.

Firm 1 can charge $40 that is the maximum willingness to pay of the customers. Since after 26 units customers will move to firm 2.

Profit of firm 1 = TR-TC

= 40×26 - 16×26

= 1040 - 416

Profit of Firm 1 = 624