Question

1.     Assume that the market demand and the costs of the duopolists are:

P=120-0.4(Q_A + Q_B)

TCA=5Q_A

TCB= 0.2Q²_B

Also assume that firm B is the sophisticated leader, Determine:

1.     The reaction curve of A

2.     The reaction curve of B

3.     The profit function of A

4.     Stackelberg equilibrium output level for firm A

5.     Stackelberg equilibrium output level for firm B

6.     The market price.

Answer 1

Given Information

$P=120-0.4(Q_A + Q_B)\\ TCA=5Q_A\\ TCB= 0.2Q^2_B$

Calculate Marginal Revenue

For Firm A

$P=120-0.4(Q_A + Q_B)$

$TR =Price\times Quantity$

$TR = (120-0.4(Q_A + Q_B)\times Q_A\\ TR = 120\times Q_A - 0.4\times Q_A^2 - 0.4Q_AQ_B$

Now derivate TR with respect to Quantity we get,

$MR = 120 - 0.8Q_A- 0.4Q_B$

 

For Firm B

$P=120-0.4(Q_A + Q_B)$

$TR =Price\times Quantity$

$TR = (120-0.4(Q_A + Q_B)\times Q_B\\ TR = 120\times Q_B - 0.4\times Q_AQ_B- 0.4Q_B^2$

Now derivate TR with respect to Quantity we get,

$MR = 120 - 0.4Q_A- 0.8Q_B$

Calculate Marginal cost for both the firms.

For Firm A

$TCA=5Q_A$

derivate TC with respect to quantity to calculate MC, We get

$MC = 5\\ TCB= 0.2Q^2_B$

derivate TC with respect to quantity to calculate MC, We get

$MC = 0.4Q_B$

1)

For best response function or for reaction function put MR = MC

For Firm A

$MR_A − MC_A\\ 120 - 0.8Q_A- 0.4Q_B = 5\\ 0.8Q_A+ 0.4Q_B =115\\ 0.8Q_A =115 - 0.4Q_B\\ Q_A=143.75 - 0.05Q_B$

Reaction Function for Firm A is $Q_A=143.75 - 0.05Q_B$

2)

For Firm B

$MR_B − MC_B\\ 120 - 0.4Q_A- 0.8Q_B = 0.4Q_B\\ 1.2Q_B = 120 - 0.4Q_A\\ Q_B = 100 - 0.333Q_A$

Reaction Function for Firm B is $Q_B = 100 - 0.333Q_A$

3)

Profit Function of A

$Profit = TR_A - TC_A\\ Profit for A = 120\times Q_A - 0.4\times Q_A^2 - 0.4Q_AQ_B - 5Q_A\\ Profit for A = 115Q_A - 0.4\times Q_A^2 - 0.4Q_AQ_B$

4)

The Stackelberg leader will choose its output Q_A to Max its profits, s.t. reaction function of the firm B

$π_A =120\times Q_A - 0.4\times Q_A^2 - 0.4Q_A\times (100 - 0.333Q_A) - 5Q_A\\ π_A = 115Q_A - 0.4\times Q_A^2 - 0.4Q_A\times (100 - 0.333Q_A)\\ π_A = 115Q_A - 0.4\times Q_A^2 - 40Q_A + 0.133Q_A\\ π_A = 75.132Q_A - 0.4\times Q_A^2$

derivate function with respect to Q_A and put equals to 0, we get

$75.132 - 0.8\times Q_A = 0\\ 75.132 = 0.8\times Q_A\\ Q_A = 93.9165$

5)

Put Value of Q_A  = 93.9165 in Equation $Q_B = 100 - 0.333Q_A$  we get

$Q_B = 100 - 0.333Q_A\\ Q_B = 100 - 0.333\times (93.9165)\\ Q_B = 100 - 31.3023\\ Q_B = 68.697$

6)

market price

$P=120-0.4(Q_A + Q_B)$

Put value Q_A  = 93.9165 and Q_B = 68.697, we get

$P=120-0.4(93.9165 + 68.697)\\ P=120-0.4(162.6141)\\ P=120 - 65.0456\\ P=54.9543$