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.

Value & Capital, Chapter VI, Section 4

In this section Hicks discusses some of "the above difficulties" -- apparently referring to difficulties of satisfying the conditions of equilibrium when there are economies of scale.  One way of proceeding in our analysis is by "sacrificing the assumption of perfect competition."  When a firm is to some extent a monopolist, it can set a price above its marginal cost.  This may be a necessary condition of profitability, because average cost could sometimes be greater than marginal cost; but the problem with extending the assumption of monopoly too far is that, "Under monopoly the stability conditions become indeterminate; and the basis on which economic laws can be constructed is shorn away."

His conclusion, essentially, is that the only way out of the situation ("this wreck" as he calls it) is to assume that most of the markets that we will analyze are not significantly different from perfectly competitive markets.  Thus if prices exceed marginal costs by some percentage, we will suppose that these percentages are "neither very large nor very variable."  We will also suppose that diminishing marginal costs are rare, and therefore that marginal costs generally increase with output at the equilibrium point.

Hicks acknowledges that this assumption is a "dangerous step," that may restrict "to a serious extent" the problems that our analysis will be able to address.  He is doubtful, though, that the problems thus excluded are even "capable of much useful analysis by the methods of economic theory."

Value & Capital, Chapter VI, Section 3

This section consists of a discussion of the validity of the assumption, made in the previous section, that production has decreasing returns to scale, namely increasing marginal and average cost, and diminishing marginal and average product, as the scale of production increases.

Hicks lists two considerations that sometimes lead to criticism of the assumed conditions.  One consideration is "the frequent conviction of entrepreneurs themselves" that they have decreasing average costs.  The other consideration is that of indivisibility of certain types of investment in factors of production.

Hicks explains that for short-run problems, the existence of "fixed equipment or plant of the firm, which has been built up in the past, and is likely to be to some extent unique" can cause a situation of a factor of production being combined with resources that the firm cannot purchase on the open market.  He argues that this kind of situation can tend to cause diminishing returns, or increasing costs.

For long-run problems, the argument for increasing marginal cost follows from "the increasing difficulty of controlling an enterprise, as its scale of production grows."

Hicks devotes the final paragraph of the section to discussing the implications of having conditions on both marginal cost and average cost.  As he notes, "Marginal costs must rise as the firm expands, in order to ensure that its expansion stops somewhere."  But this condition alone is not enough to specify where the expansion stops.  The firm can be expected to sell at a price equal to marginal cost, but this marginal cost must not be too close to its minimum;  otherwise marginal cost would be below average cost, and the firm would be selling at a loss.