On Aggregating Linear Programming Models of Production

Abstract

T HIS PAPER presents some aggregation results for linear programming models of the firm. It is assumed that the production decision problem for each firm in some regional, industrial, or typical firm aggregate can be represented by a linear programming model. By means of the duality theorem for dual linear programs, it is shown that under suitable conditions a single linear programming model for the aggregate is equivalent to a direct aggregation of the solutions of a set of individual firm models. Conditions sufficient for this equivalence are proportional variations of resources1 and behavioral bounds; proportional variation of net return expectations among all firms in the aggregate; and, finally, common technical coefficients which appear in the constraints on the firms' decisions. It is shown that imperfections in an atomistic market introduced by imperfect knowledge, uncertainty, or government policies may increase the number of aggregate models needed to describe an entire industry such as agriculture. An example is explored and implications of the model for equilibrium of an open economy sketched out.

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