§4 (to Ch. XIII, § 9) Geometrical explanation of the proposition expounded in § 3 of this Appendix

BUT the foregoing proof by algebra may not appeal to many students as much as the proof by geometry.

We know (see Chapter X, § 3 and Chapter XI, §§ 4 and 5) that the present value of any income position on the Market line is the same as that of any other income position on that line.

It follows that the present value of any point on a given Market line is measured by the intercept of that line on the horizontal axis, for that intercept evidently measures the present market value of one particular point on the Market line (namely its lower end) and, as just stated, this must have the same present value as every other point on the Market line.

It follows that as the Market line is moved further away from the origin (keeping its direction unchanged) the intercept becomes greater, and thus the present value of every one of the points on the Market line becomes greater correspondingly.

When, therefore, the line is thus moved as far as possible, so that it thereby assumes the position of tangent to the Opportunity line, the present value of every point on it and, therefore, of that point of tangency must be greater than that of any other point on the Opportunity line, since any other such point will necessarily lie on a Market line nearer the origin.

The same proof applies in three dimensions, substituting Opportunity surface for Opportunity line and Market plane for Market line. By analogy the proof may be extended to n dimensions.

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