Extraction and royalty time path for a non-renewable resource under a monopolist assumption

Question

I am trying to understand the time path of extraction and royalty (price) for a non-renewable resource under a monopolist framework. The problem is that in the main books on environmental/resource economics and in the resources that I could find on internet the "results", either quantitative or qualitative, are given without proof, and the assumptions are not given ("most cases" :-( ).

Using optimal control, I have no problem to find that the Hotelling rule is here applied to the marginal profit over the extraction quantity, or that its discounted value remains constant and it is below the market price. But what are the specific assumptions required to claim that (a) the initial price is higher than under perfect market competition/benevolent social planner; (b) that the depletion time is longer and finally (c) that the growth rate of price is lower than the (constant) growth rate of marginal profit ?

Answer 1

I would say that (a) follows from a downward sloping demand curve that causes marginal revenue to decline faster than price. As you write a monopolist accounts for that and to maximize profits thus extracts less to drive up the price. That also explains why generally exploitation would take longer.

That this is not always the case is shown by Stiglitz's (1974) Monopoly and the Rate of Extraction of Exhaustible Resources where he shows that the extraction path for the monopolist and perfect competition are the same for a constant elasticity demand curve and zero extraction costs. That in turn, has been shown by Gaudet and Lasserre in "On comparing monopoly and competition in exhaustible resource exploitation" to depend on the fact that the resource stock is fixed and hence it is sometimes hard to restrict output to increase the price.

Lewis et al. provide examples where the monopolist may be more excessive than socially optimal