Consider that two identical firms (firm 1 and firm 2) in a homogenous-product ma

Question

Consider that two identical firms (firm 1 and firm 2) in a homogenous-product market compete in prices. The capacity of each firm is k. The firms have constant marginal cost equal to 1 up to the capacity constraint. The demand in the market is given by P=10-Q. If the firms set the same price, they split the demand equally. If a firm sets a higher price than his competitor, the demand of the firm becomes residual demand.

a. If each firm’s capacity is 9 (k=9), then what would be market equilibrium price?

b. If each firm’s capacity is 3 and frim 1 charges his price at the marginal cost, then does firm 2 charge the same price? If not, what would be firm 2’s optimal price?

c. If each firm’s capacity is still 3 and firm 2 charges $4 for his price, then what would be firm 1’ optimal price?

d. From result from part b and c, can you say that the Cournot model can be interpreted as a special case of Bertrand model with limited capacity?

Explanation / Answer

a) Market Demand : P = 10-Q

We know from the question that there are Two firms. Let Q1 represent the Quantity demanded of Firm 1 and Q2 be the quantity demanded of firm 2. Hence We can write the Market Demand as

P = 10-(Q1+Q2)

Total Revenue (TR) = Price* Quantity

Total Revenue of Firm 1: P* Q1

TR1 = {10-(Q1+Q2)}*Q1

TR1 = 10Q1 - Q12 -Q1*Q2

MR1 = 10-2Q1 -Q2

At Profit Maximization MC= MR

So we get

MR1 = 10-2Q1 -Q2 = 1

So we get an Equation for Firm 1

2Q1+Q2 =9 ----------------------(1)

Similarly, we need to find the Total Revenue for Firm 2

Total Revenue of Firm 1: P* Q1

TR1 = {10-(Q1+Q2)}*Q2

TR1 = 10Q2 - Q1 *Q2 - Q22

MR1 = 10-Q1 -2Q2

At Profit Maximization MC= MR

So we get

MR1 = 10-Q1 -2Q2 = 1

So we get an Equation for Firm 2

Q1+2Q2 =9 ----------------------(2)

Let's Write the 2 equations together

2Q1+Q2 =9 ----------------------(1)

Q1+2Q2 =9 ----------------------(2)

Multiplying Equation (1) by 2, We get

4Q1+2Q2 =18 --------------------(3)

Subtracting Equation 2 from 3 we get

4Q1+2Q2 =18

Q1+2Q2   =9

-----------------------

3Q1 =9

Q1 = 3

Substituting the Value of Q1 in equation (1) we can get the value of Q2

2*3+ Q2= 9

6+Q2 =9

Q2 =3

Thus the Market Price P = 10- Q1- Q2 = 10-3-3 = 4

Thus the Market Equilibrium Price would be 4.

(b) Even if the firms capacity falls to 3 and Firm 1 charges his price equal to marginal Cost, The optimal price that firm 2 should charge is the same as firm 1 i.e $4. If firm 2 charges less than 4, it would not maximize its profit. If Firm 2 charges more than $4 there is a possibility that it will lose all its customers to firm 1 as both firms sell identical products.

(c) If each firm's capacity is 3 and firm 2 charges $ 4, then the optimal price for firm 1 is $2.5.

We are now in a Bertrand situation.

Let P1 be the price of Firm 1 and P2 be the price of Firm 2

The Market Demand can be written as Q = 10- P

Q = 10- (P1+ P2) where P = P1+P2

Total Revenue for Firm 1 = P1*Q

TR = 10P1 - P1 2 - P1*P2

MR = 10-2P1-P2= MC = 1

We Know P2 = 4

On solving we get the value as P1= $2.5

(d) No.

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