3. Dune
Arrakis is the only planet that produces spice in the known galaxy, giving it a monopoly over the intergalactic spice market. It can produce $q \ge 0$ dekagrams of spice each year for a cost $c(q) = 3q$.
Spice is useful to travel through space. The galaxy's marginal willingness to pay for the $q^{th}$ dekagram of spice—that is, its inverse demand curve—is $P(q) = 6q^{-1/\theta}$.
(a) Suppose that Arrakis produces spice to maximize its profits, and that $\theta = 2$. How much spice will it produce per year? What markup does Arrakis charge above its marginal cost?
(b) Calculate the first-best level of spice production that maximizes total surplus, defined as spice consumer surplus, $\int_0^q P(x)dx$, net of Arrakis production costs, 3q. How much spice is Arrakis withholding from the market relative to the first-best?
(c) Suppose that $\theta = 4$. Recalculate Arrakis' profit-maximizing level of spice production, its markup above its marginal cost, and the first-best level of spice production.
(d) Suppose that the sand worms do not like spice harvesters. Assume environmental damages of $D(q) = 3q$ from q dekagrams of spice. For $\theta = 2$, what level of spice production now maximizes this new measure of total surplus, defined as spice consumer surplus net of Arrakis production costs and environmental damage? Using the formula in lecture, what optimal tax could attain this objective?
(e) Suppose that we have now learned that sand worms are a keystone species sustaining the ecosystem, and updated our damages estimate to $D(q) = 6q$. Recalculate the first-best level of spice production and the optimal tax on Arrakis. Explain why the optimal tax on spice harvesting is lower than the marginal damage to the sand worms.
