§2. The Six Sets of Formulas Incomplete

All that I shall attempt here is to point out the shortcomings of the six sets of formulas in the second approximation. Impatience Principle A in the second approximation is expressed by formulas of the type:

f = F (y' + x', y'' + x'',..., y(m) + x(m))

indicating that a person's impatience is a function of his income stream as specifically scheduled, indefinitely in the future.

Of course, in the uncertainties of actual life no such specific scheduling is possible. The equation is true but incomplete, as f is properly not only a function of one expected Edition: current; Page: [318] income program but of many possible such programs each with its own series of probabilities, and those probabilities are too vague even to be specifically expressed or even pictured by the person concerned. That is, the average person is merely aware that he is willing, say, to pay five per cent for a $1000 loan because he thinks his future prospects will justify it. He vaguely expects that his present $10,000 income will probably rise to $20,000 within a few years, possibly to $30,000—and possibly not rise at all. He could think of innumerable possibilities and these would imply many other variables than those above cited. Much might depend on the income of others besides himself and on the future size of his family, the state of their health, and other conditions without end. His own future income is the important matter, but that itself is dependent on all sorts of variables on which he will reckon summarily in a rule of thumb fashion.

We could, formally, rewrite for the third approximation the above equation so as to read:

f = F ()

merely refusing to attempt any enumeration of the innumerable variables inside the parentheses—among them being, perhaps, all the variables included in all the equations in the second approximation, including rates of interest as well as numberless other variables such as probabilities not there included. In so far as the latter, or new, variables enter, each of them requires a new equation in order to make the problem determinate. Such new equations would be merely empirical. Among the equations which would be needed, would be those expressing the y's and x's in terms of real income—that Edition: current; Page: [319] is, as the sum of the enjoyable services, each multiplied by its price. This would lead us into the theory of prices and the general economic equilibrium.

Impatience Principle B was expressed in the second approximation by formulas of the type:

i = f.

But now we must face not only one i but a series of i's, according as the market is the call loan market, the 60 to 90 day commercial paper market, the gilt-edge bond market, the farm mortgage market, and innumerable others, for each of which there is its own separate f and i.

These many magnitudes, including the i's, require still other empirical equations impossible to formulate satisfactorily, albeit we know in a general way that the rate on gilt-edge bonds is lower than on risky bonds, the rate on first mortgages lower than that on second mortgages, and that the long term and short term markets do influence each other. But these relations are too indefinite to be put into any equations of real usefulness, theoretical or practical.

Investment Opportunity Principle A was expressed by formulas of the type

f (y', y'',..., y(m)) = 0

This becomes

f () = 0

where the blank parenthesis stands for a multitude of unknowns (and unknowables) which could be discussed ad infinitum and each of which, in so far as it was not already included among the variables entering into our system of equations, would require a new empirical equation of some sort in order that the problem shall be determinate. Moreover, the f equation representing a Edition: current; Page: [320] man's ensemble of income opportunities is a composite of separate opportunities, the full and detailed expression of which would take us again into the theory of prices and general economic equilibrium.

Investment Opportunity Principle B, expressed in the second approximation by equations of the type

i = r,

would have to be replaced by as profuse a variety as replaced the i = f above.

A full statement of the margins of investment opportunity would include the margins of numberless individual enterprises and adjustments in the use of every item of capital. It would again lead us into the theory of prices and general economic equilibrium. Walras and Pareto have formulated systems of equations which do this and in which the theory of interest is merely a part of a larger whole.

Market Principle A, expressed in the second approximation by equations of the type

x1 + x2 +.... + xm = 0,

will remain true only if or in so far as performance of contracts corresponds to promises and expectations. In so far as, because of defaults, the equations fail of being precisely true, no useful mathematical relation expressing that failure seems possible.

The same is true of Market Principle B, expressed under the second approximation by formulas of the type

to say nothing of the fact that this type of equation will take many different forms in view of the variety of i's and in view of the probability factors.

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The only explicit practical inclusion of such probability factors, mathematically, is to be found in the formulas of life insurance actuaries. But these are of little more than suggestive value in our present effort to express the determination of the rate of interest.