Value & Capital, Chapter VI, Section 2

This section presents an analysis that derives the equilibrium conditions for a firm operating in a perfectly competitive market.  In the simplest case, the firm is able to take advantage of technical opportunities for converting a single factor A into a single output X.  When the firm faces prices for A and X that are given by the market, it will "embark upon production, so long as the total value of the product secured is greater than the total value of factor employed.  Further, it will be to its advantage to produce that quantity of product which will make the excess as large as possible."

Figure 18 illustrates this graphically.  The horizontal axis measures quantities of the input A, the vertical axis measures quantities of the output X, and the curved line illustrates the set of production possibilities (input-output combinations) available to the firm.

Suppose the distance ON corresponds to the quantity of factor being employed as input.  The curve shows that the distance PN gives the quantity of product that results.  Suppose the line PK has slope equal to the ratio of the unit price of factor to the unit price of product.  Then the distance MK represents the quantity of product having the same market value as ON, the quantity of factor consumed in production. (This interpretation of MK may be easier to grasp by keeping in mind the familiar definition of slope as "rise over run.")  Then the distance OK corresponds to the surplus product that the firm generates.  The market value of OK is the net value of receipts minus costs.  As Hicks notes, "The conditions of equilibrium are that OK should be a maximum, and should be positive."

As the text points out, OK is not maximized in Figure 18.  The line PK could be raised (thus increasing OK) until PK is tangent to the production curve, as in Figure 19.

The text goes on to set out the conditions of equilibrium as follows:

(1)  To maximize the excess value of production, the line PK must be tangent to the production curve.  Thus the slope of the production curve at equilibrium must equal the ratio of the price of the factor to the price of the product.  The slope of the production curve also represents the marginal product.  Therefore, as Hicks notes, this condition for equilibrium "can be put in either of two familiar forms:  the price of the factor equals the value of its marginal product, or the price of the product equals its marginal cost."  If this condition did not hold, the firm would find it more profitable to undertake the production process with some other level of input.

(2)  Because OK must be a maximum rather than a minimum, the production curve must be "convex upwards" at the point of tangency (or as one likely heard such a curve described in introductory calculus, "concave down").  This implies that the slope (and therefore the marginal product) must be decreasing at the equilibrium point.

Hicks then discusses the similarity between these two conditions and those derived earlier for the theory of subjective value.

The production curve, as we have drawn it, is remarkably similar in its properties to an indifference curve.  Where we had equality between a price-ratio and a marginal rate of substitution, we now have equality between a price-ratio and a marginal product -- which may be looked on, if we choose, as a marginal rate of transformation.  As for the stability condition, diminishing marginal rate of substitution is replaced by diminishing marginal product.  These two conditions are therefore substantially identical, and by their means we shall be able to construct a theory of the conduct of the firm closely similar to our theory of the conduct of the private individual.

Hicks then discusses a third condition of equilibrium, for which he notes that there is no correspondence in the theory of subjective value.

(3)  The surplus OK must be positive.  This can only be the case if the slope of OP is greater than that of PK.  Since the production curve lies at or below PK, this implies that the slope of OP must be diminishing as P moves to the right.  Since the slope of OP corresponds to the average product, this means that average product must be diminishing.  In a footnote, Hicks derives the equivalent expression of this condition in terms of average cost:

Alternatively, we may argue in the following way.  If there is a positive surplus, price must be greater than average cost.  But price equals marginal cost.  Therefore marginal cost must be greater than average cost.  Therefore the production of an additional unit must raise average cost.  Therefore average cost must be increasing.

Hicks concludes the section by summarizing the equilibrium conditions in the following lists:

1. Price of factor = value of marginal product.           1. Price of product = marginal cost.

2. Marginal product diminishing.                             2. Marginal cost increasing.

3. Average product diminishing.                              3. Average cost increasing.