APPENDIX TO CHAPTER XII
§ 1 (to Ch. XII, § 1)
Algebraic expression of rate of time preference
IF W signifies wantability, or utility, or desirability and this year's income is signified by X' and next year's by X'', then DW' may be taken to signify the present wantability of, or want for, a small increment, DX' of money this year and will be the want-for-one-more unit of money this year. Also DW'' is the present want for a small increment, DX'', of money available next year, and will be the present want-for-one-more unit of money available next year.
Exact mathematical theory requires that the marginal wants per unit of money are the limits of those ratios when the increments approach zero as their limits or lim and lim ordinarily written in the differential calculus: and which, for short, may be called w' and w''.
The rate of preference, f, for a unit of present money over a unit of next year's may be defined as
§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 condition 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.:
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''', x^{iv},..., 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 x^{iv},..., 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
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.