§1. Case of Two Years and Three Individuals

IN this chapter, the four principles constituting the first approximation, previously expressed verbally17 and geometrically,18 will be expressed algebraically. Inasmuch as the equations, the solutions of which express the solution of the interest problem, are necessarily numerous and complicated, we shall first consider a simplified special case where there are to be considered only two years in which there is income and three individuals who borrow or lend. We shall then pass to the general case where there are any given number of years and any number of individuals.

In the simplified case, we assume, therefore, that each individual's degree of impatience for this year's over next year's income can be expressed as dependent solely on the amount of this year's and next year's income, the incomes of all other future years being disregarded. We also assume, for simplicity, that the income of each of the two years is concentrated at the middle of the year, making the two points just a year apart, and that borrowing and lending are so restricted as to affect only this year's and next year's income.

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Let f1 represent the marginal rate of time preference19 for this year's over next year's income for Individual 1 (this is the slope at Q of a Willingness line relatively to the 45° line). Let his original endowment of income for the two years20 be respectively

c1' and c1''.

(These are the longitude and latitude of P in Chapter X.) This original income stream, consisting merely of the two jets, so to speak, c1', c1'', is modified by borrowing this year and repaying next year. The sum borrowed this year is called x1' (this is the horizontal shift from P to Q). To represent the final income of this year this sum x1' is therefore to be added to the present income c1'. Next year the debt is to be paid, and consequently the income finally arrived at for that year is c1'' reduced by the sum thus paid. For the sake of uniformity, however, we shall regard both additions to and subtractions from pre-existing incomes algebraically as additions. Thus, the addition x1' say $100, to the first year's income is a positive quantity, and the addition, which we shall designate by x1'', to the second year's income, is a negative quantity—$105. The first year's income is, therefore, changed from

c1' to c1' + x1',

and the second year's from

c1'' to c1'' + x1''.

(Just as c1' and c1'' are the longitude and latitude of P in Chapter X, so c1' + x1' and c1'' + x1'' are those of Q.) Edition: current; Page: [290] By the use of this notation we avoid negative signs and so the necessity of distinguishing between the expressions for loans and repayments or for lenders and borrowers.