§6. Counting Equations and Unknowns

We now proceed to compare the number of the foregoing equations with the number of unknowns, for one of the most important advantages of an algebraic statement of any economic problem is the facility with which, by such a count, we may check up on whether the problem is solved and determinate. There are evidently 3 equations in the first set, 3 in the second, 2 in the third, and 3 in the fourth, making in all 11 equations. The unknown quantities are the marginal rates of time preference, the amounts borrowed, lent and returned, and the rate of interest as follows:

f1, f2, f3, or 3 unknowns,
x1', x2', x3', or 3 unknowns,
x1'', x2'', x3'', 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 they are not all independent, since any one equation in the third and fourth sets may be determined from the others Edition: current; Page: [293] of those sets. Thus, if we add together all the equations of the fourth set, we get the first equation of the third set. (namely, x1' + x2' + x3' = 0). The addition gives

In this equation we may substitute zero for the numerator of the fraction (as is evident by consulting the second equation of the third set). Making this substitution, the above equation becomes

x1' + x2' + x3' = 0,

which was to have been proved. Since we have here derived one of the five equations of the last two sets from the other four, the equations are not all independent. Any one of these five may be omitted as it could be obtained from the others. We have left then only ten equations. Since no one of these ten equations can be derived from the other nine, the ten are independent and are just sufficient to determine the ten unknown quantities, namely, the f's, x''s, x'''s and i.