§ 4 (to Ch. XIII, § 9)
Geometrical explanation of the proposition expounded in § S of
But the foregoing proof by algebra may not appeal to manystudents 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 Marketline is the same as that of any other income position on that line.
It follows that the present value of any point on a givenMarket line is measured by the intercept of that line on thehorizontal axis, for that intercept evidently measures thepresent market value of one particular point on the Marketline (namely its lower end) and, as just stated, this must havethe same present value as every other point on the Market line.
It follows that as the Market line is moved further away fromthe origin (keeping its direction unchanged) the intercept be-comes greater, and thus the present value of every one of thepoints on the Market line becomes greater correspondingly.
When, therefore, the line is thus moved as far as possible, sothat it thereby assumes the position of tangent to the Opportunityline, the present value of every point on it and, therefore, ofthat point of tangency must be greater than that of any otherpoint on the Opportunity line, since any other such point willnecessarily he on a Market line nearer the origin.
The same proof applies in three dimensions, substitutingOpportunity surface for Opportunity line and Market plane forMarket line. By analogy the proof may be extended to ndimensions.
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