§9. Reconciling the Numbers of Equations and Unknowns

The reconciliation of these two discordant results is effected by two considerations. One reduces the number of equations. Just as under the first approximation, we have one less independent equation in the two sets expressing the Market Principles than the apparent number, thus making the final net number of equations

3mn + m - n - 1.

The other consideration is quite different. It subtracts from the number of unknowns. This can be done because each r, of which there are n(m - 1), is a derivative from the y's. By definition r is the excess above unity of the ratio between a small increment in the y of next year to the corresponding decrement in the y of this year. The same applies to any pair of successive years. This derivative is, more explicitly expressed, a differential quotient.25

The reader not familiar with the notation of the differential calculus will get a clearer picture of the inherent derivability of the r's from the y's by recurring to the geometric method in Chapter XI. There y' and y'' are shown as the coördinates ("latitude" and "longitude") of the Opportunity line, while r is shown as the tangential slope of that line. It is evident that, given the Opportunity line, its tangential slope at any point is derived from it. It is not a new variable but is included in the variation of y' and y'' as the position on the curve changes.

If, now, we subtract n(m - 1), the number of the r's, Edition: current; Page: [310] from 4mn + m - 2n - 1, we have, as the final net number of unknowns,

3mn + m - n - 1

which is the same as the total net number of independent equations.26 Thus the problem is fully determinate under the assumptions made.

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