Value & Capital, Chapter VI, Section 5

This section, the final one in Chapter VI, returns to the case of perfect competition and spells out the conditions for equilibrium in the general case of a firm converting multiple input factors into multiple products.

As in the simple case of a single factor and a single output, we have a relation between the quantities of factors used as inputs and the quantities of products resulting from production.  In this case we can think of the relation as a surface in multiple dimensions.  It will be useful to think of the elevation of such a surface as reflecting a single quantity, so Hicks explains how, for example, "given all the quantities of factors, and all quantities of products but one, the maximum producible amount of the remaining product can be deduced.  Similarly, given all the quantities of products, and all quantities of factors save one, the minimum amount needed of the remaining factor can be deduced."  In a footnote Hicks points out that such a relation will not be defined everywhere, as there will be cases for which "no amount of a remaining factor will be sufficient to produce the given collection of products."

Starting from a set of factor quantities, and the quantities of products that result from using the factors in production, Hicks notes that variations in production can happen in many ways, but they can all be reduced to some combination of three types of variations:

(1) "One product may be increased at the expense of another, i.e. substituted for another at the margin."
(2) "One factor may be substituted for another."
(3) "One factor and one product may be simultaneously increased (or diminished)."

In a footnote, Hicks states that the first two types can actually be reduced to the third.

We naturally assume that the enterprise will seek to maximize its surplus (the value of products it produces minus the costs of the factor quantities required in producing those quantities of products).  This leads to three conditions of equilibrium corresponding to the condition that price must equal marginal cost:

(1) "The price-ratio between any two products must equal the marginal rate of substitution between the two products."  Hicks calls this a "technical rate of substitution" (as it reflects the technology of production rather than happening according to the preferences of a consumer).
(2) "The price-ratio between any two factors must equal their marginal rate of substitution."
(3) "The price-ratio between any factor and any product must equal the marginal rate of transformation between the factor and product (that is to say, the marginal product of the factor in terms of this particular product)."

Next, the conditions for an equilibrium to be stable are as follows.  For stability in the process of transforming a factor into a product, the condition is that of diminishing marginal rate of transformation (or diminishing marginal product);  this carries over directly from the one-factor one-product case.  For substituting one product for another the stability condition is that of increasing marginal rate of substitution, or as Hicks explains, "increasing marginal cost in terms of the other product (marginal opportunity cost)."  For stability in substituting one factor for another, the condition is diminishing marginal rate of substitution.  Hicks explains in a footnote the intuition behind the opposite direction of the product and factor substitution conditions.

Increasing marginal rate of substitution for products, because the total value of products secured has to be maximized;  diminishing marginal rate of substitution for factors, because the total value of factors used has to be minimized.  These conditions are easily verified graphically, if the amounts of other factors and products are assumed given, and the two products (or factors) in question are measured along two axes.

Hicks explains that the stability conditions must hold for a one-for-one substitution or transformation (one factor or product for one other factor or product) but also for group substitutions or transformations.  Also

The marginal rate of substitution between any pair of groups of products must increase, and between any pair of groups of factors must diminish; the marginal rate of transformation between any group of factors and group of products must diminish.

Finally, Hicks discusses the conditions related to the existence of positive surplus.  Instead of a single condition, there are now multiple conditions.  Namely, it must not pay to abandon production of any subset of the set of all products.

Therefore the average cost of producing each product must be rising, and the average cost of producing each group of products must be rising, including the whole group that includes all the products.

Having laid out the conditions for equilibrium in the general case, Hicks will proceed as in part I of the book.  He will assume that the stability conditions and the conditions for positive surplus hold in the neighborhood of an equilibrium point, and he will then derive laws of market conduct for the firm.