THE THEORY OF INTEREST

returns, for each individual, be equal. This condition isfulfilled in the following equations, each correspondingto one individual:

*/ +

*1

l + »

• = 0,

x*

x"

l + »

= 0 ,

*»' +

x{‘

1 + i

= 0 .

§6. Counting Equations and Unknovms

We now proceed to compare the number of the fore-going equations with the number of unknowns, for oneof the most important advantages of an algebraic state-ment of any economic problem is the facility with which,by such a count, we may check up on whether the prob-lem is solved and determinate. There are evidently 3equations in the first set, 3 in the second, 2 in the third,and 3 in the fourth, making in all 11 equations. The un-known quantities are the marginal rates of time prefer-ence, the amounts borrowed, lent and returned, and therate of interest as follows:

fi, /a, fa, or 3 unknowns,

Xi, x 2 ', x a ', or 3 unknowns,x”, x 2 ", x 3 ", or 3 unknowns,

and finally,

i, or 1 unknown,

making in all 10 unknowns.

We have, then, one more equation than necessary.But examination of the equations will show that theyare not all independent, since any one equation in thethird and fourth sets may be determined from the others

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