Value & Capital, CHAPTER VII, Section 5

This section further discusses the possible relationships among product and factors described in the previous section, where two factors A and B are used to produce one product X.  As the opening paragraph explains, "It is most likely that A and B will be complements, next most likely that no complementarity will be present and no regression, least likely of all that there will be regression.  The reasons for this all hang together."

The discussion then explores a limiting case in which complementarity must necessarily exist. This case assumes that there is no effect on marginal cost from the "fixed 'productive opportunity' of the enterprise" -- no economies of scale that result in cost savings for expanded production, and also no increase in marginal costs with output.  Hicks makes the following argument for A and B being complementary:

Since marginal cost is constant, the increase in product due to a simultaneous proportionate increase in both factors (the marginal product of the two factors taken together) must be constant.  But this joint marginal product is made up of four parts:

1. the marginal product of A with B constant;
2. the increment (or decrement) of this marginal product due to the simultaneous increase in B.  It will be an increment if A and B are complementary, a decrement if they are substitutes;
3. the marginal product of B with A constant;
4. the similar increment (or decrement) due to the increase in A.  To this the same rule applies.

He then claims that as the quantities of factors employed expand, the first and third of these parts must decline (this is because marginal product must be diminishing for an equilibrium to exist). But by assumption the whole does not decline.  Therefore the decline in 1 and 3 must be offset by increments in 2 and 4. Therefore A and B must be complementary.

So in this special case, in which the entrepreneur's "fixed opportunity" does not have a  limiting effect on the scale of production, the two factors must be complementary.  As soon as the fixed opportunity actually does something to limit expansion, the situation changes and the two factors are not necessarily complements.  (They still could be, if their join marginal product declines slowly.)

When might two factors employed to make a single product be substitutes?  For the case where output is variable, Hicks says this can only happen if
(1) "the fixed resources of the entrepreneur must make an appreciable contribution to production,"
(2) "the factors must be such that they would be close substitutes in the production of a given output."
He doesn't define "close" substitutes in the text, but I infer this to mean that the marginal rate of substitution is relatively high. (It's also worth noting that a footnote here says, "Thus in the case of constant costs and two factors, the two factors are necessarily complements in the production of a variable output, and necessarily substitutes in the production of a constant output.  This is a paradoxical situation, which may easily lead to misunderstandings unless we are careful about it."  He then states that it is more convenient not to regard the case of constant costs as the standard case, but as a limiting case in which the effect of the entreprenurial resources vanishes.  In the variable output case, a pair of factors employed by a single firm will ordinarily tend to be complementary.)

At this point, Hicks is able to provide an interpretation of the regression relationship in the current context:  If factor A and product X are regressive, then factors A and B must be substitutes.  From the preceding discussion, it follows that when A and X are regressive, the fixed resources of the entrepreneur must be playing a significant role in limiting production.  He then argues that this limiting effect, together with the regression relationship, imply that factor A must be especially suited to small-scale production, and factor B must be suited to production on a larger scale.  In this case, a decline in the price of A can lead to more employment of A, thereby leading to smaller-scale production and hence a decline in output.  Hicks then concludes, "Regression turns out to be a phenomenon of increasing returns;  one which is just consistent with perfect competition if the fixed entrepreneurial resources are important enough.  Still, it does not yet appear to be a possibility of which we need take much account."