Two firms, A and B, engage in Bertrand price competition in a market with invers

Question

Two firms, A and B, engage in Bertrand price competition in a market with inverse demand given by P = 12 - Q. Firm A has a higher marginal cost: cA > cB . Whenever a firm undercuts the rival’s price, it has all the market. If a firm charges the same price as the rival, it has half of the market. If a firm charges more than the rival, it has zero market share.

(a) Suppose that cA = 8 and cB = 3. Find the Nash equilibrium of this game (pA,pB) where one of the firms charges its marginal cost

(b) Suppose that there are not 2, but n firms with different marginal costs. Any number of firms may also have equal marginal costs. Can we have a marginal cost structure where one firm earns a positive profit? Can we have a marginal cost structure where more than one firm earns a positive profit? Just explain the intuition without the math

Explanation / Answer

a. When cA = 8, and cB = 3,

I assume here prices can only take discrete values ,like 1,2, 3...

For given marginal costs, cA>cB, the best strategy for firm A would be to at pA=cA ,since if charges any price pA<cA , it will incur a loss.

So, The best response for A would be pA=cB, but then B's profit is (pBcA)/2 as both will be getting half profit and thus firm B can profitably deviate to pB=cB

So,pB=cA for some small , since we are working with discrete numbers, So for pB = cA -1 = 8 -7 = 7, Firm B wiill get the whole market and will get the profit of (7 - 3)Q, and their is no positive deviation possible for both A abd B .

Thus pA=8 and pB = 7,and firm B getting the full market, is the Nash Equilibrium.

Yes, we can have a marginal cost structure where one firm earns a positive profit as since no firm in the market will produce below the marginal cost , and in case of n firms , if (n-1) has a maginal cost greater than 1 firm , then the best strategy for those (n-1) firms will be to produce at there mc which is greater than firm 1's MC , So, firm 1 by CHARGING ANY PRICE PA = MC(n-1) -   , can earn a positive profit.

No, we cann't have a marginal cost structure where more than one firm earns a positive profit as suppose In a industry there are m+n firms , and the MCm>MCn , then the best strategy for m firms will be to charge Pm = MCm , But n firms can charge Pn = MCm - , can capture the entire market , but then 1 firm from those n firms , will charge P = MCm - 2 , can acquire the entire market from n-1 firm , then any firm from (n-1) firms will decrease the price to P = MCm - 3, to capture the market , this will continue till MCm - = MCn . and thus at P = MCn , no firm will earn a positive profit.

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