APPENDIX TO CHAPTER XIII
§1 (to Ch. XIII, § 9)
Rate of return over cost expressed in the notation of the calculus
IN the notation of the calculus, the rate of return over cost, called in the text r_{1}', is defined in terms of the partial differential quotient with the opposite sign of next year's income with respect to this year's income of Individual 1. That is, by definition
and
.
Analogous formulas express the remaining r's for Individuals 1, 2... n.
§ 2 (to Ch. XIII, § 9)
Rate of return over cost derived by differential equations
The magnitudes of 1 + r_{1}', 1 + r_{1}'',..., 1 + r_{1}^{(m)}, or
,
may be expressed in terms of y_{1}', y_{1}'',..., y_{1}^{(m)} by differentiating the equation for the effective range of choice, f_{1} (y_{1}', y_{1}'',..., y_{1}^{(m)}) = 0.
§ 3 (to Ch. XIII. § 7, also § 9)
Mathematical proof that the principle of maximum present value is identical with the principle that the marginal rate of return over cost is equal to the rate of interest.
THE mathematical proof that the principle of maximum present value of optional income streams is identical with the Investment Opportunity Principle B or that the rate of marginal return over cost is equal to the rate of interest is as follows:
The present value V_{1} of any income stream y_{1}', y_{1}'',...,y_{1}^{(m)}, of Individual 1 or their combined discounted value is
The condition that this expression shall be a maximum is that the first differential quotient shall be zero. That is,
This last equation expresses the relations which must exist between dy_{1}', dy_{1}'',..., dy_{1}^{(m)}, in order that the income stream, y_{1}', y_{1}'',... y_{1}^{(m)}, may have the maximum present value.
This condition contains within itself a number of subsidiary conditions. To derive these, let us consider a slight variation in the income stream, affecting only the income items pertaining to the first two years, y_{1}', and y_{1}'' (the remaining items, y_{1}''',..., y_{1}^{(m)}, being regarded for the time being as constant) and let us denote the magnitudes of dy_{1}' and dy_{1}'', under this assumption of restricted variations, by dy_{1}' and dy_{1}''. Then, under the condition assumed of constancy of y_{1}''', y_{1}^{iv},..., y_{1}^{(m)}, dy_{1}''', dy_{1}^{iv},..., dy_{1}^{(m)}, are equal to zero, and the equation becomes
From this, it follows directly that
But the left-hand member of this equation is by definition one plus the marginal rate of return over cost. Since we have designated the rate of return over cost by r_{1}' we may substitute 1 + r_{1}' for the expression , and write the above equation thus:
1 + r_{1}' = 1 + i',
or thus:
r_{1}' = i'.
In other words, the condition that the marginal rate of return over cost is equal to the rate of interest follows as a consequence of the general condition that the present value of the income stream must be a maximum. This proposition and its proof are analogous to those in regard to desirability or wantability, which have already been discussed in the Appendix to Chapter XII, that the condition of maximum wantability is equivalent in the condition that the marginal rate of preference is equal to the rate of interest.
The same reasoning may be applied to any pair of successive years. Thus, if we assume variations in y'' and y''', without any variations in the other elements of the income stream, y', y^{iv},..., y_{1}^{m}, the original differential equation becomes
or = 1 + i'',
or 1 + r_{1}'' = 1 + i'',
or r_{1}'' = i''.
All this reasoning implies, in using the differentiation process, that there is continuous variation, and that, at the margin, it is possible to make slight variations in any two successive years' incomes without disturbing the incomes of the other years.
§ 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.
§ 5 (to Ch. XIII, § 9)
Maximum total desirability is found when rate of time preference is equal to the rate of interest
IN the last section was outlined a geometric proof that the income stream possessing the maximum present value is such that the rate of interest (connecting each pair of successive years) is equal to the rate of return over cost (for the same pair of successive years).
The geometric method also supplies a simple proof that the maximum total desirability, or wantability, is to be found in the case of that income stream which satisfies the above mentioned condition and, at the same time, has a rate of time preference equal to the rate of interest. In geometric terms for two dimensions this means that this most desirable income position or point is where the Market line, which is tangent to the Opportunity line, is tangent to a Willingness line.
Consider two parallel Market lines, one tangent to the Opportunity line and the other somewhat nearer the origin; and consider the two points Q and S where these two are respectively tangent to a Willingness line. We are to prove that the total desirability or wantability of Q is greater than that of S. Draw a straight line from the origin through S and produce it until it cuts the first Market line at, say, T.
It is evident, of course, that of all the points on any given Market line the point of tangency with a Willingness line is the most desirable income position. Therefore, Q is more desirable than T. We assume that the Willingness lines are such that the farther we recede along a straight line from the origin the more desirable the income situation. Therefore, T is more desirable than S.
Therefore, Q being more desirable than T, and T than S, Q is more desirable than S, which was to have been proved.
§ 6 (to Ch. XIII, § 9)
Walras and Pareto
WALRAS and Pareto probably deserve more attention in interest theory, as in general economic theory, than they have received.
Walras' interest theory forms an integral part of his theory of general economic equilibrium. His solution consists of a demonstration that the problem comprises a number of independent equations exactly equal to the number of unknowns, and that the mathematical solution of these simultaneous equations is a counterpart of the economic process by which the unknowns are determined in the market. There is thus no reasoning in a circle in the Walras system. The number of equations is exactly equal to the number of unknowns.
Walras' treatment of the problem of the determination of the rate of interest is very detailed and highly mathematical. For readers who are not familiar with his treatment I venture to attempt a brief summary. Walras assumes a market for capital goods as well as for services. He assumes that the prices of capital goods depend on the prices of their services. Since some capital goods last longer than others and all are subject to risk, he makes allowance for depreciation, amortization, and insurance.
His treatment combines the subjective and objective elements in a simple and direct manner. His cost of production equations correspond, in a general way, to my opportunity principles. His equation for the demand for savings corresponds, likewise, to my impatience principles.
Pareto's analysis of the problem of the rate of interest is along the lines laid down by Walras, although he was evidently not fully satisfied with Walras' treatment. Neither he nor Walras has developed a systematic theory of income but he shows, in effect, that the substitution of one income stream for another or, as he says, the "transformation in time" is only a particular case of a more general transformation which is dealt with in the theory of production. His indifference equations for consumers correspond in a general way to my impatience principles and the analogous equations for the obstacles in his treatment of production correspond likewise to my opportunity principles.
The fundamental differences between the approach of Walras and Pareto on the one hand and mine on the other seem to be four:
(1) Walras and Pareto determine the rate of interest simultaneously with all the other unknowns of the problem—the quantities of the commodities exchanged and the services used in their production and the prices of the commodities and the services, while I try to isolate the interest problem by assuming that most of such unknowns have already been determined and confine my discussion to the special factors directly affecting the rate of interest.
(2) They both treat what I call interactions or intermediate services along with the ultimate factors—our desires or tastes (les gôuts) and the obstacles which must be overcome to satisfy them—while I try at the outset to get the interactions canceled out, leaving only the income stream and (labor) sacrifice.
(3) Neither Walras nor Pareto has elaborated the concept and principles of an income stream.
(4) Neither has elaborated the concept or principles of opportunity as a choice from among a series of income streams, although it is, in part, implied in Pareto's treatment.