6. (Requires calculus.) A monopolist produces a product whose demand price and p

Question

6. (Requires calculus.) A monopolist produces a product whose demand price and production costs vary with quality s and quantity q according to
P(s, q) = s(1 q)
C(s, q) = s2q.
(a) Calculate the price and quality levels that a monopolist would choose, and the corresponding quantity sold.
(b) Consumer surplus at any {s, q} combination can be derived as 1/2sq2. The corresponding value for profits is (p(s, q)s2)q = (ssqs2)q. Substitute the monopolist’s profit-maximizing
quantity from (a) and then derive optimal quality for that quantity choice (the level of quality
that maximizes consumer plus producer surplus). Show that the monopolist’s actual quality
choice is lower than optimal quality, given the quantity chosen.

Explanation / Answer

I can try the 1st part (a) A monopolist will produce where marginal cost = marginal revenue. Total revenue is P*q = s(1-q)*q = sq - sq^2 Total cost is s^2q Take the first derivitive (with respect to Q): MR = s - 2sq MC = s^2 from there, Solve for q. hope this helps...

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