Value & Capital, CHAPTER VII, Section 2

This section begins the "disentangling" (mentioned at the end of the previous section) of the possible substitution and complementarity relationships that might exist among commodities that could be products or be factors used in production of other products.  The first step in the analysis is to construct a simple case in which the firm will produce a fixed amount of output and, to do so, it will employ two factors, A and B.  The goal for the firm is to choose the quantities of the factors so as to minimize the cost of production.  Figure 21 illustrates the possible choices.

We assume the production curve is concave up.  This corresponds to the assumption of diminishing marginal rate of substitution between factors.  The line PK represents possible tradeoffs between quantities of the factors A and B, where each pair of quantities on the line has the same total cost, for the given factor prices.  The point P, where PK is tangent to the production curve, represents a position of equilibrium when the ratio of the prices of A and B is MK to PM.

Suppose the price of A were to fall.  Then, the amount of B having an equal value to the quantity ON of A would also fall, from MK to, say, MK1, and the total cost of production (valued in terms of factor B) falls from OK to OK1.  But since PK1 is not tangent to the production curve, the production costs can be reduced by moving to the point P' which is where the line PK2 (parallel to PK1) is tangent to the production curve.

At this new equilibrium, the production costs have been reduced to OK2; less of factor B is employed, and an additional quantity of factor A has been substituted for it.  These results follow just as necessarily as did the expansion of supply of the product when the factor price fell, in the case of one factor and one product.