# Calculate the monopolist’s profit-maximizing price and output.

Consider a monopolist facing demand function Qd(p) = 120 − 2p and having cost function C(Q) = 1 Q2. 4
The monopolist sells all units at the same price (linear pricing).

(a) (5 points) Calculate the monopolist’s profit-maximizing price and output. Calculate the associated consumer surplus and producer surplus.
(b) (5 points) Calculate the deadweight welfare loss due to monopoly.
A monopolist producer of widgets serves the US and European markets. The inverse demands in the US and Europe are each PUS(qus) = 100 − qus and PE(qe) = 80 − qe, respectively, where where qus and qe denote the sales in the US and Europe. The widget monopolist has two plants, one in Europe and one in the US. The plant in Europe has constant marginal cost equal to 10; the US plant has marginal cost functionMC(Q) = Q, where Q denotes the level of production at the US plant. Suppose the monopolist’s transportation cost of shipping the widgets between the US and Europe is 5 per widget.
(a) (3 points) Calculate the socially efficient level of widget sales in the US and in Europe.
(b) (3 points) Assume resale of widgets is impossible (so arbitrage is impossible). Under third-degree price discrimination, what prices should the monopolist set in the US and in Europe to maximize profit?
(c) (3 points) Now suppose, due to a new trade agreement, resale is now possible and anyone can transport widgets between the US and Europe at a cost of 5 per widget. Under third-degree price discrimination, what prices should the monopolist set in the US and in Europe to maximize profit?
(d) (3 points) A recently graduated, recently hired MBA cannot figure out how to account for the arbitrage possible in (c), so he simply sets the same price in both markets. Given uniform pricing, what price should he set to maximize profit?
A monopolist with constant marginal cost of production serves two distinct, independent constant- elasticity demand markets. In market 1 the elasticity of demand, in absolute value, is |η1| = 3 and in market 2 it is |η2| = 4. There are no other variable costs. The monopolist practices third-degree price discrimination.
In market 1 the monopolist sets its per-unit price at pd1 = 180.

(a) (3 points) What discriminatory price does the monopolist set in market 2?

(b) (3 points) Calculate the monopolist’s marginal cost.

Men and women have the same demands for widgets. For men the aggregate widget demand is Qm(p) = 1−p for p ≤ 1 and equals 0 for p > Similarly, for women aggregate widget demand is Qw(p) = 1−p for p ≤1 and equals 0 for p > 1.
The widget monopolist’s marginal cost of selling widgets to men is 0, but, for some unknown reason, its marginal cost of selling to women it is c ∈ [0, 1]. Because widgets are customized to the buyer, there is no possibility of arbitrage between the two groups.

(a) (3 points) If the monopolist uses third-degree price discrimination, what price does it set for widgets to men and widgets to women?
(b) (3 points) A recent court case has forced the monopolist to sell widgets to men and women at the same price (costs still differ, as described). Calculate the monopolist’s profit-maximizing uniform price. (No one can be prevented from purchasing widgets—so if a widget is offered for sale to anyone, widgets must be available to everyone, both to men and to women.)
(c) (3 points) Is social welfare higher under uniform pricing or under third-degree discrimination? Ex- plain.

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