This article is concerned with the theory of the "bottleneck" problem experienced by a developing economy, say that of an underdeveloped country which is trying to achieve specified production goals over a fixed time period, where there is competition for restricted resources and limited external aid, including for example competition between the needs for consumer goods and capital goods. The problem is viewed as a generalized type of nonlinear programming problem. Mathematically, a basic difficulty in such a model is of course its enormous size; and it is desirable to find methods of testing approximate solutions. One such test is provided by the duality theory of nonlinear programming, which is shown to apply to this situation, so that the problem can be regarded as either of two distinct problems, which are shown to have the same optimal solution. Economically, these two problems may be described in terms of determining the optimal procedure at any given time during the period in terms of what has already happened, or alternatively in terms of what is required to happen subsequent to that time. The two problems involve respectively maximization and minimization, so that, in particular, upper and lower bounds for the value of production can be calculated for any particular production plan.