Knowledge that Transforms

THE ESSENTIAL ARGUMENT of this paper is that the new capital acquired by all the traders in a closed economy in a given period is given a putative or constructive turnover of once for the period by typical accounting procedure. It is possible to demonstrate the truth of this argument in a very convincing way in the case where unit prices are imagined to be constant over time, for in that case the concept of increase in cost value of all traders' stock of goods is clearly seen to be a construct itself in the sense that it is not really an excess of input at cost over output at cost. In any case the output of a given trader in physical units is the same as the physical input of the trader to whom he sells; so in any case the physical rate of output of all traders by trade is equal to their physical rate of input by trade. Moreover, the cost price of one trader at a given time is the selling price of a trader at an earlier stage of production at the same time if there is no price change. Let X, 2, 3, ... , represent the physical outputs (and inputs) at various stages of the productive process, while p1 2, 3, ..., i are the unit prices respectively. Then the money value of total inputs may be stated as Xipi+Xi_pi_-+ · , while the money value of the outputs are Xipi_l+Xi-pi-2_2+ Thus if Xi units of ore, limestone, labor, etc., in the form of pig iron are sold for pi dollars per ton, and the iron was made of cost elements worth p2 dollars per equivalent composite unit, the input of the buyer would be Xlpl dollars, and the output of the seller would be Xlp2 dollars. The difference between the value of total input at cost and total output at cost for all traders would be EXipi -Xipi_l. But this difference may be restated as follows:

A PROBLEM COMMONLY TREATED in monopoly theory is the effect of a shift in demand on monopoly price and output. In most instances attention centers upon the positive or negative character of the shift, and on occasion the accompanying change in elasticity is considered. The direction of shift in demand is usually ignored, however. This would appear to be a serious omission. The following analysis indicates that the direction of shift in demand may have significant bearing on the results and that it is advisable in all cases to give it explicit consideration. 1 That the addition of a constant increment to the quantities that will be taken at various prices (a horizontal shift in demand) is not in general the mathematical equivalent of the addition of a constant increment to the prices that will be paid for various quantities (a vertical shift in demand) is readily demonstrated. Geometrically one need only shift a demand curve to the right by a constant amount at all levels and then shift the same curve upward by a constant amount for all abscissa points to note that the two new demand curves are not the same. Algebraically it may be shown as follows: Let x = F(p) be the equation of the demand curve in which the quantity is expressed as a function of the price and let p =f(x) be the inverse relationship. Then if a small constant amount is added to the quantity that will be taken at any price, the first equation becomes x' = F(p) +Ax and if Ax is small, the inverse becomes approximately p'=f(x) -f'(x)Ax. Hence the amount that would have to be added to the price at each quantity level to yield the practical equivalent of adding a constant amount to the quantity that will be taken at each price depends upon the slope of the demand curve f'(x) at each point. A linear (constant slope) demand curve is thus the only instance in which the addition of a constant amount horizontally is the equivalent of adding a constant amount vertically. The frequent use of linear demand curves in graphic analysis is possibly one reason why the direction of shift in demand has not been given more consideration.

THE FOLLOWING THEORY of business fluctuations is claimed to be simple in the mathematical sense as stated by Jeffreys2 and also to represent the simplest possible dynamic extension of the Walrasian system. It explains the business cycle as a purely speculative short-run equilibrium phenomenon. Assume an economic system consisting of n commodities. The buyers and sellers take into account not the actual but the anticipated price. Assume further with Evans that they form their anticipations upon the prevailing price and the price tendency, i.e., the rate of change of the price in time.' If all relationships are linear (as first approximations), we get, for the demand for the ith commodity,

IN THE JANUARY, 1937, number of ECONOMETRICA I had an article entitled in Theory and Income Taxation in Practice. One of its contentions was that to tax and later to tax income from those savings, involves a subtle form of double taxation. I had made this same contention in 1906 in The Nature of Capital and Income. Long afterward, through Professor E. R. A. Seligman, I learned that John Stuart Mill had also called attention to this double taxation. Apparently he was first to do so. Strange to say, such double taxation, though ably affirmed by many other writers, notably Marshall and Pigou in England and Einaudi in Italy, has not, to this day, been universally accepted. In a forthcoming book on Tax Spendings not Savings, I am including a general review of whole question-if question it be. In course of renewed study involved, I have gradually become conscious of a companion principle. Apparently it has hitherto been overlooked. This principle is that to tax works extensive destruction upon and spendings. Because of this destructiveness, several paradoxes emerge which have both theoretical and practical interest. In an article on this subject published in Taxes, the Tax Magazine, in August, 1941, I have excluded, as unsuitable for such a journal, underlying mathematics, merely asserting that contentions made can be mathematically demonstrated. The present article gives demonstrations referred to. At close of year zero, say 1900, let Co be value of a certain capital-for instance an automobile plant. Let j be rate at which this initial value Co would increase during first year (1901) without taxes; and, for simplicity, let us suppose that said rate continues uniformly for n years, at end of which period-say at end of 1940-the owner of capital dies. Then Coj would be capital-increase in dollars in first year (1901) (called savings in title to this article); and capitalvalue C1 at end of that year would be C1 = Co(l +j). At end of second year, capital value would be C2= Co (1 +j) 2; of rth year (1) C,r = Co(1 + )r,