Cournot Duopoly with inverse demand function P=500-2Q, how do I find TR, TC, MR, and MC? cost functions are?

Question

C1=25+2(Q1)^2

C2= 20+(Q2)^2

Answer 1

You have asymmetric costs of firms in this case:

Market demand:

P=500-2(Q1+Q2)

Q=Q1+Q2

First firm:

TC=25+2Q^2

MC=(TC)'=4Q

TR1=Q*P=Q1*(500-2(Q1+Q2))= 500Q1-2Q1^2-2Q1Q2

MR1=(TR1)'=500-4Q1-2Q2

Equilibrium:

MR2=MC2

500-4Q1-2Q2=4Q1

8Q1=500-2Q2

4Q1=250-Q2

Q1=(250-Q2)/4  (firm1 reaction function)

Second firm:

TC=20+Q^2

MC=(TC)'=2Q

TR2=Q*P=Q2*(500-2(Q1+Q2))= 500Q2-2Q2^2-2Q1Q2

MR2=(TR2)'=500-4Q2-2Q1

Equilibrium:

MR2=MC2

500-4Q2-2Q1=2Q2

250-3Q2=Q1

(250-Q1)/3=Q2 (firm2 reaction function)

Solve reactions:

(250-Q1)/3=Q2

Q1=(250-Q2)/4

(250-Q2)/4=250-3Q2

250-Q2=1000-12Q2

11Q2=750

Q2=750/11≈68

Q1=500/11≈45

Q=68+45=113

P=500-113*2=500-226=274

VC1=2Q1^2=4050

VC2=Q2^2=4624

TR1=18632

TR2=12330

TC1=25563

TC2=12789

Profit1= - 6931

Profit2= -459

Revenue for whole market acting as monopolist:

TR=P*Q=500Q-2Q^2

MR=(TR)'=500-4Q