Value & Capital, CHAPTER XV, Section 5

This section continues explaining some of the details of how "the problem of maximizing the present value of the production plan is formally identical with the problem of maximizing the surplus of receipts over costs in the static problem of the firm."  For given interest rates, future costs enter into the analysis as discounted costs.  Outputs at different dates are treated as different outputs.  "With these adjustments," the author states, "the whole static theory of the firm still holds.  We have nothing to do but translate."

For conditions of equilibrium, the author identifies three kinds of conditions, "corresponding to the three 'elementary' forms of variation."  These are as follows:
(1)  For any two dates, the marginal rate of substitution between an output at those two dates must equal the ratio of the discounted prices.
(2) For any two dates, the marginal rate of substitution between an input at those two dates must equal the ratio of the discounted prices.
(3)  The marginal rate of transformation between any input and output pair must equal the ratio of their discounted prices.

The author goes on to explain that certain other equilibrium conditions stated by other writers are special cases of these three.  In particular, he notes that "the often stated rule that the current rate of wages equals the discounted value of the marginal product of current labour is a special case of our third condition."  Similarly, a rule noted by Wicksell -- that the interest rate equals the relative marginal productivity of waiting -- is a special case of the first condition.

Also, Keynes's statement that "short-period supply price is the sum of marginal factor cost and marginal user cost" corresponds to a combination of the first and third conditions.  Finally, another rule from Keynes, that the marginal cost of a unit of input equals "the present value of the stream of output increments made possible by the marginal input" appears to be (although not stated explicitly in the text) a combination of the second and third conditions.

The author concludes the section by noting briefly that in some cases, groups of inputs or outputs must be chosen in fixed proportions, but "little is to be gained by paying a great deal of attention to these cases ... at this stage of our inquiry."