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A monopolist produces a product whose demand price and production costs vary with quality s and quantity q according to:

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) = s^2q
(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 1/2sq^2
.
The corresponding value for profits is (p(s, q)−s^2)q = (s −sq −s^2)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.

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