CHAPTER XII

FIRST APPROXIMATION IN TERMS OF FORMULAS

§1. Case of Two Years and Three Individuals

In this chapter, the four principles constituting thefirst approximation, previously expressed verbally 1 andgeometrically, 2 will be expressed algebraically. Inasmuchas the equations, the solutions of which express the solu-tion of the interest problem, are necessarily numerousand complicated, we shall first consider a simplified spe-cial case where there are to be considered only two yearsin which there is income and three individuals whoborrow or lend. We shall then pass to the general casewhere there are any given number of years and any num-ber of individuals.

In the simplified case, we assume, therefore, that eachindividual’s degree of impatience for this year’s over nextyear’s income can be expressed as dependent solely onthe amount of this year’s and next year’s income, theincomes of all other future years being disregarded.We also assume, for simplicity, that the income of eachof the two years is concentrated at the middle of theyear, making the two points just a year apart, and thatborrowing and lending are so restricted as to affect onlythis year’s and next year’s income.

‘In Chapter V, §9, and in Chapter IX, §9, first of the three lines,in each case.

“In Chapter X, §15.

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