Question

Consider an industry comprising two firms producing a homogeneous product. The market demand and total cost equations are: P=200-2(Q_1+Q_2 ); 〖TC〗_i=4Q_i, where i = 1, 2.

a. What is each firm’s reaction function?

b. What are the market-clearing price, and output and profit of each firm?

Answer 1

P= 200-2Q₁ -2Q₂

TC of firm 1= TC₁ = 4Q₁

TC of firm 2= TC₂ = 4Q₂

a.

Profit of firm 1= Pr1= PQ₁ - TC₁

Pr1= 200Q₁ -2Q₁²  -2Q₂ Q₁ -4Q₁

Differentiate Pr1 wrt Q₁ :

dPr1/dQ₁ = 200-4Q₁ -2Q₂ -4= 0

4Q₁ +2Q₂ = 196 Equation 1

4Q₁ = 196-2Q₂

Q₁ = (196-2Q₂ )/4 Reaction curve of firm 1

Profit of firm 2= Pr2= PQ₂ - TC₂

Pr2= 200Q₂ -2Q₂²  -2Q₂ Q₁ -4Q₂

Differentiate Pr2 wrt Q₂ :

dPr2/dQ₂ = 200-4Q₂ -2Q₁ -4= 0

4Q₂ +2Q₁ = 196 Equation 2

4Q₂ = 196-2Q₁

Q₂ = (196-2Q₁ )/4 Reaction curve of firm 2

b.

For market clearing output solve equation 1 and 2:

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

8Q₂ +4Q₁ -4Q₁ -2Q₂ = 392-196

6Q₂ = 196

Q₂ = 196/6 Market clearing quantity by firm 2

Use this Q₂ =196/6 in reaction curve of firm 1:

Q₁ = 196/6 Market clearing quantity by firm 1

P= 200-2Q₁ -2Q₂ = 200-2(196/6)-2(196/6)= 200-130.67= 69.33 Market clearing price

Pr1= 69.33 x 196/6 - 4(196/6)= 2264.78-130.67= 2134.11

Pr2= 69.33 x 196/6 - 4(196/6)= 2264.78-130.67= 2134.11