Question

Question 1 10 pts The (inverse) market demand function in a homogeneous product Cournot duopoly is as follows: P = 200 - 10(Q1+Q2). The total cost functions are TC = 100 + 40Q1 for firm one and TC = 80 + 60Q2 for firm two. 1.(4 points) Determine the reaction function for each firm. 2. (2 points) Calculate each firm's equilibrium level of output. 3. (2 points) Calculate the market equilibrium price. 4.(2 points) Calculate the profit each firm earns in equilibrium.

Answer 1

Question 1.

P= 200-10(Q1+Q2)

Total revenue for firm 1= TR1= P x Q1= 200Q1-10Q1(Q1+Q2)

MR1= Differentiation of TR1 wrt Q1= 200-20Q1-10Q2

Total revenue for firm 2= TR2= P x Q2= 200Q2-10Q2(Q1+Q2)

MR2= Differentiation of TR2 wrt Q2= 200-10Q1-20Q2

TC of firm 1= 100+40Q1

MC for firm 1(MC1)= Differentiate TC wrt Q1= 40

TC of firm 2= 80+60Q2

MC for firm 2(MC2)= Differentiate TC wrt Q2= 60

1.

For reaction of firm 1:

MR1= MC1

200-20Q1-10Q2= 40

20Q1+10Q2= 160

2Q1+Q2= 16 Equation 1

Q1= (16-Q2)/ 2 Reaction curve of firm 1

For reaction curve of firm 2:

MC2=MR2

60= 200-10Q1-20Q2

Q1+2Q2= 14 Equation 2

Q2= (14-Q1)/2 Reaction curve of firm 2

2.

For equilibrium level of output: Solve equation 1 and 2

Multiply 2 in equation 1 and then subtract it from equation 2:

Q1+2Q2-4Q1-2Q2= 14-32

-3Q1 = -18

Q1 = 6 Equilibrium quantity of firm 1

Put Q1= 6 in reaction curve of firm 2:

Q2= (14-6)/2= 8/2= 4 Equilibrium quantity of firm 2

3.

For equilibrium price:

P= 200-10(Q1+Q2)= 200-10(6+4)= 200-100= 100 Equilibrium price

4.

Profit for firm 1= P x Q1 - (100+40Q1)

Profit of firm 1 = 100 x 6 - 100-40(6)= 600-100-240= 260

Profit of firm 2= P x Q2 - 80-60Q2= 100 x 4 - 80-60(4)= 400-80-240= 80