IN TERMS OF FORMULAS

Let /i represent the marginal rate of time preference 3for this year’s over next year’s income for Individual 1(this is the slope at Q of a Willingness line relatively tothe 45° line). Let his original endowment of incomefor the two years 4 be respectively

c/ and Ci".

(These are the longitude and latitude of P in ChapterX.) This original income stream, consisting merely ofthe two jets, so to speak, c/, c/', is modified by borrow-ing this year and repaying next year. The sum borrowedthis year is called X/ (this is the horizontal shift fromP to Q). To represent the final income of this year thissum Xi is therefore to be added to the present incomec/. Next year the debt is to be paid, and consequentlythe income finally arrived at for that year is c" reducedby the sum thus paid. For the sake of uniformity, how-ever, we shall regard both additions to and subtractionsfrom pre-existing incomes algebraically as additions.Thus, the addition x/ say $100, to the first year’s incomeis a positive quantity, and the addition, which we shalldesignate by Xi", to the second year’s income, is a nega-tive quantity — $105. The first year’s income is, there-fore, changed from

Ci' to Ci' +, Xi,

and the second year’s from

Ci" to Ci" + Xi".

(Just as Ci' and c/' are the longitude and latitude of P inChapter X, so Ci'+ Xi ' and c" -f x" are those of Q.)

'The relation of /, representing the rate of time preference of anyindividual tc the marginal desirability, or “wantability,” of this year’s,and of next year’s income, is given in the Appendix to this Chapter, §1.

* The term “year” is used for simplicity, but “month” or “day” wouldbe equally admissible and would do less violence to facts.

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