§2 (to Ch. XII,§3) Equality of marginal rate of time preference and rate of interest implies that desirability of income stream is made a maximum

ASSUME at first that only two years are considered. The fact that total desirability or wantability of the individual, as reckoned at the beginning, depends on the amount of income this year and next year may be represented by the equation

W'' = F(c' + x', c'' + x''),

where W'' represents his total wantability, and the equation represents this W'' as a function of his income stream consisting of c' + x' this year and c'' + x'' next year. This W'' is represented in Chapters X and XI by the numbers attached to the several Willingness or Wantability lines, each representing a certain level of wantability of Individual 1. But as we shall here consider only one individual, we omit the subscript numbers, 1, 2,..., n. The individual under consideration will attempt to adjust x' and x'' so as to maximize W. We are to prove algebraically that the condition that W shall be a maximum implies also that the rate of interest i shall be equal to the individual's rate of preference f. The condition2 that W shall be a maximum is that the total differential of W or of its equal F(c' + x',c'' + x''), called below F() shall be zero; thus

where the ∂'s represent the partial differentials with respect to x' and x''.

From this equation it follows that

The left-hand number of this equation is 1 + i, as may be seen by differentiating the equation of the loan as originally stated, viz.:

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This differentiation yields= 1 + i.

The right-hand member, being the ratio of this year's marginal wantability to next year's marginal wantability, is by definition equal to 1 + f. Substituting the new value for the right- and left-hand members, we have

1 + i = 1 + f,

whence it follows that i = f, which was to have been proved.

The same reasoning may now be applied to three or more years. The total wantability for any individual is a function of the total future income stream. In other words,

W = F(c' + x', c'' + x'',..., c(m) + x(m)).

The individual tries to make this magnitude a maximum. In terms of the calculus, this is equivalent to making the first total differential equal to zero namely,

This total differential equation is equivalent, according to well-known principles of the calculus to a number of subsidiary equations obtained by making particular suppositions as to the different variations. Let us, for instance, suppose that only x' and x'' vary in relation to each other and that x''', xiv,..., x(m) do not vary. Then in the above equation all terms after the second disappear and the equation reduces, as before, to

So that, again, 1 + i' = 1 + f', and therefore i' = f'.

This expresses the relation between the first and second years. If we wish, in like manner, to express the corresponding connection between the second and third years, let us assume that x' as well as xiv,..., x(m) are constant but that x'' and x''' vary. Then the first term of the equation and all after the third disappear, and the equation reduces to

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In other words, 1 + i'' = 1 + f'', or i'' = f''. Similarly, i = f for every other pair of successive years.

We have here, in mathematical language, the reason that the point of maximum total wantability is also the point at which the marginal rate of time preference for a unit of each year's income over that of next year's income is equal to the rate of interest connecting these two years.3