A Short-Run Model for the Coal Industry - Henderson, James M. (The Review of Economics and Statistics, 1955)
T HE recent development of linear programming has made possible a new type of economic analysis.1 M1ethods have been developed to allow the computation of numerical solutions for theoretical problems involving economic choice. Linear programming provides a general mathematical model which is applicable to problems that can be expressed as the maximization (or minimization) of a linear form subject to a system of linear inequalities. Many economic problems can be placed in this general format, and linear programming can be used for the analysis of a plant, a firm, an industry, a national economy, or the world economy. The general methods of linear programming are used here in the formulation and solution of a short-run model for the coal industry. The present application is formally similar to the transportation problems formulated by Hitchcock and Koopmans.2 The model for the coal industry contains two separate, but interrelated, programming problems. The data of the model are spatially distributed demands for coal, the capacities of spatially distributed deposits of coal, and the unit costs of deliveries from the deposits to the demand locations. The levels of the deliveries are the variables of the first of the programming problems; they are selected to minimize the cost of meeting the demands subject to the capacity restrictions for the coal deposits. The variables of the second programming problem are the delivered prices of coal at the demand locations and the unit royalties earned by the various deposits. The values of these variables are selected to maximize total revenue net of royalty payments subject to the condition that every possible delivery must yield a non-positive profit. The optimum solutions of these two problems provide a complete description of perfectly competitive short-run equilibrium for the coal industry. Once the model has been presented, it is implemented with factual information for the demands, capacities, and costs; and numerical solutions are computed for both problems for