Question

Consider a monopolistic market with the following inverse demand curve: P=z(32-Q) where z is the quality level. Suppose that the marginal production cost of output is independent of quality and equal to 0. The cost of quality is C(z)=8z².

a) (6 pts.) Calculate the production level that would maximize profits.

b) (7 pts.) Calculate the quality level that would maximize profits.

Answer 1

(a)

Profit(Pr) = TR - TC where TR = Total revenue = P*Q = z(32-Q)*Q and TC = Total Cost = C + Co where Co = cost of producing output = F + 0 where F = some Fixed cost and is constant and as Marginal cost is 0 => Variable cost = 0

=> Co = F and C = Cost of quality = 8z² {Note Fixed cost has no role to play in determining optimal level of output}

=> TC = Total Cost = Co + C = F + 8z²

Thus, Profit(Pr) = TR - TC = z(32-Q)*Q - (F + 8z²)

Maximize : Pr

First order condition :

dPr/dQ= 0 => z(32 - 2Q) = 0 => Q = 16 -------------(1) {Note here d(Pr)/dQ means partial differentiation of Pr with respect to Q}

d(Pr)/dz = 0 => (32 - Q)*Q - 16z = 0 ----------------------(2)

As Q = 16

From (2) we get :

(32 - 16)*16 = 16z => z = 16

Thus we have Q = 16 and z = 16

Hence, the production level that would maximize profits(Q) = 16 units

(b)

As calculated above z = 16

Hence, the quality level that would maximize profits(z) = 16 units.