Geometric Interpretation of the Duality Between Cost and Production Function

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.