Question

Two local manufacturing firms have a combined demand and total cost functions given by: Q = 105-P TC1=5Q+0.5Q12 TC2=5Q2+0.5Q22 If they cannot successfully collude and instead produce where market price equals marginal cost, what would be their total output? What would each firms profit be?

Answer 1

Solution :- Q₁ = 105 - P₁

P₁ = 105 - Q₁ (inverse demand function)

Total cost for firm 1= 5Q + 0.50Q₁²

Marginal cost (MC) for firm 1 = 5 + 2Q₁ (First derivative of total cost function)

Equating Price (P) and Marginal cost (MC) for firm 1 in the given question,

105 - Q₁ = 5 + 2Q₁

105 - 5 = 2Q₁ + Q₁

100 = 3Q₁

Q₁ = 100 / 3

Q₁ = 33.33 (approx)

Similarly Q₂ (Quantity of output for firm 2) would be also 33.33 because total cost function, marginal cost function and demand function for the firm 2 would be same as that of the firm 1.

Accordingly, Total output = 33.33 + 33.33

= 66.66 (Rounded off to 66.67)

Conclusion :- Total output in the given question = 66.67 units (approx).