If market inverse demand is pQ a bQ and the firm produces

If market inverse demand is p(Q) = a - bQ and the firm produces according to C(Q) = cQ + dQ^2, determine the firm\'s optimal quantity, price and profit level. (Assume that a, b, c, d > 0.) What conditions on b and d must hold in equilibrium?

Solution

p = a - bQ

Revenue, R = p x Q = aQ - bQ²

Marginal revenue, MR = dR / dQ = a - 2bQ

Total cost, C = cQ + dQ²

Marginal cost, MC = dC / dQ = c + 2dQ

The optimality condition requires that MR = MC. Or,

a - 2bQ = c + 2dQ

a - c = (2d + 2b) x Q

Q = (a - c) / (2b + 2d)

p = a - bQ = a - [b x (a - c) / (2b + 2d)]

Profit, Z = R - C

= aQ - bQ² - cQ - dQ²

= (a - c)Q - (b + d)Q²

= (a - c) x [(a - c) / (2b + 2d)] - (b + d) x [(a - c) / (2b + 2d)²

Required condition: (b + d) must not be equal to zero