§ 5 (to Ch. XIII, § 9)
Maximum total desirability is found when rate of time 'preferenceis equal to the rate of interest
In the last section was outlined a geometric proof that theincome stream possessing the maximum present value is suchthat the rate of interest (connecting each pair of successiveyears) is equal to the rate of return over cost (for the same pairof successive years).
The geometric method also supplies a simple proof that themaximum total desirability, or wantability, is to be found inthe case of that income stream which satisfies the above men-tioned condition and, at the same time, has a rate of timepreference equal to the rate of interest. In geometric terms fortwo dimensions this means that this most desirable incomeposition or point is where the Market line, which is tangent tothe Opportunity line, is tangent to a Willingness line.
Consider two parallel Market lines, one tangent to theOpportunity line and the other somewhat nearer the origin; andconsider the two points Q and S where these two are respectivelytangent to a Willingness line. We are to prove that the totaldesirability or wantability of Q is greater than that of S. Drawa straight line from the origin through S and produce it untilit cuts the first Market line at, say, T.
It is evident, of course, that of all the points on any givenMarket line the point of tangency with a Willingness line is themost desirable income position. Therefore, Q is more desirablethan T. We assume that the Willingness lines are such that thefarther we recede along a straight line from the origin the moredesirable the income situation. Therefore, T is more desirablethan <S.
Therefore, Q being more desirable than T, and T than S, Q ismore desirable than S, which was to have been proved.
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