1981 | OriginalPaper | Chapter
Geometric Interpretation of the Duality Between Cost and Production Function
Author : Ronald W. Shephard
Published in: Cost and Production Functions
Publisher: Springer Berlin Heidelberg
Included in: Professional Book Archive
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The geometric duality between cost and production function may be described in simple terms as follows: Let the plane of Figure 1 be the plane of the factor space, considering only two factors of production. The coordinate axes X1, X2 are used Interchangeably for amounts x1, X2 and prices p1, p2 of these factors. In this coordinate system the production curve ψ(U,X1X2) =1 (U constant) and the unit cost curve Q(U,X1X2) =1 (U constant) are approximately sketched. Let p1’, p2’ be some arbitrarily given prices of the factors of production. These prices define a direction OR, where the coordinates p1; p2 of R are proportional to p1’, p2’, that is p1= τp1 and p2 = τp2’. The proportionality factor τ is taken so that $$Q\left( {U,\,{p_{1}},\,{p_{2}}} \right) = \,T.\,Q\left( {U,{p_{1}}',{p_{2}}'} \right) = 1 $$ which can be done since Q Is homogeneous of degree one In the prices and the point R(p1, P2) lies on the unit cost surface. The amounts x1, x2 of the factors of production which minimize cost, for given values U, P1’, p2’, define a point P(x1, x2) on the production curve ψ(U,X1X2) = 1.