An optimal control problem in economics (Q1178796)

The authors attempt to examine the issues of existence of solutions to the first control problem in resource economics, namely, the optimal extraction of a finite stock of a natural resource under the assumption of zero extraction costs. The motivation for the exercise in my view, however, is entirely misguided. While economists are not known to always pay due attention to the nitty gritty of mathematical details, the authors of the present paper show a clear misunderstanding of the issue involved. Theorem 2.2 and example 2.3, though mathematically correctly derived, are wrong for they are based on the false premise (Proposition 2.1, part iii), that is --- the optimal extraction rate \(q\) at the terminal date must be zero even if the extraction horizon is finite. This is certainly not part of the Maximal Principle from which the proposition is allegedly obtained. Also the importance of Theorem 3.1 lies in demonstrating the usefulness of the condition \(\text{Lim}_{q\to+\infty} U(q)-\alpha q=+\infty\) for the infinite horizon. Economists have always known this since they often invoke \(\alpha=0\), which is sufficient, as part of the characterization of the utility function.