Economics has been defined as the science of allocating scarce resources among competing ends. Much of the microeconomic theory encountered in a first semester graduate course is concerned with the static allocation problem faced by firms and consumers. Techniques of constrained optimization, in particular the method of Lagrange multipliers, are employed in developing the theory of the firm and the consumer.

The optimal harvest of renewable resources or extraction of exhaustible resources is inherently a dynamic allocation problem; that is, the firm or resource manager is concerned with the best harvest or extraction rate through time. It turns out that the method of Lagrange multipliers can be extended to intertemporal or dynamic allocation problems in a relatively straightforward fashion. This “discrete-time” extension of the method of Lagrange multipliers serves as a useful springboard to the “continuous-time” solution of dynamic allocation problems via the maximum principle. The method of Lagrange multipliers and its various extensions reduce the original optimization problem to a system of equations to be solved. Solving this system of equations, unfortunately, can often be exceedingly difficult, especially for dynamic problems. There are also technical problems concerning sufficiency conditions for the solutions so obtained (and also pertaining to the existence of a solution to the given problem). In these notes we will normally consider problems which are simple enough that these difficulties are minimized.

Before presenting the dynamic techniques we will briefly review the method of Lagrange multipliers within the context of allocating scarce resources among competing ends at a single point in time.