§12. Counting Equations and Unknowns

We therefore have as the total number of equations the following:

• n (m - 1) equations expressing Impatience Principle A,
• n (m - 1) equations expressing Impatience Principle B,
• m equations expressing Market Principle A, and
• n equations expressing Market Principle B.

The sum of these gives 2 mn + m - n equations in all.

We next proceed to count the unknown quantities (rates of time preference, loans, and rates of interest): First as to the f's:

For Individual 1 there are f1', f1'',..., f1(m-1), the number of which is m - 1, and, as there is an equal Edition: current; Page: [298] number for each of the n individuals, there will be in all n (m - 1) unknown f's.

As to the x's, there will be one for each of the m years for each of the n individuals, or mn.

As to the i's, there will be one for each year up to the last year, m - 1. In short there will be

n (m - 1) unknown f's,
mn unknown x's,
m - 1 unknown i's,

or 2 mn + m - n - 1 unknown quantities in all. Comparing this number with the number of equations, we see that there is one more equation than the number of unknown quantities.

This is accounted for, as in the simplified case, by the fact that not all the equations are independent. This may be shown if we add together all the equations of the fourth set, and substitute in the numerators of the fractions thus obtained their value as obtained from the third set, namely, zero. We shall then evidently obtain the first equation of the third set. Consequently we may omit any one of the equations in the last two sets. There will then remain just as many equations as unknown quantities, each independent (that is non-derivable from the rest), and our solution is determinate.

In the preceding analysis, we have throughout assumed a rate of interest between two points of time a year apart. A more minute analysis would involve a greater subdivision of the income stream, and the employment of a rate of interest between each two successive time elements. This will evidently occasion no complication except to increase enormously the number of equations and unknowns.

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