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new price. Similarly, S may represent the former number of sales for a given period; AS may stand for the contemplated increase of sales during a period of similar length with lower price; and S + AS the total sales ? during this second period. The gross income (price multiplied by sales) will then change from PS to (P AP) (S + AS). The expenses per sale may decrease, remain stationary, or increase with increasing sales. The gross or total expenses are certain to increase. Let us therefore represent the total expenses by E, and the increase of total expenses accompanying AS increase of sales by AE. With the reduction of price from P to P-AP

. expenses will increase from E to E + AE. The net profits will change from PS-E to (P-AP) (S + AS)

P (+ ΔΕ).

(E + AE). The reduction obviously will not be made unless the new net profits are greater than or as great as the old, unless

(P AP) (S + AS) — (E+AE) > PS E.

And these reductions will continue as long as the left side of the expression is greater than the right, ceasing when the expression becomes an equation.

There is another factor of considerable importance. If the decreased price and the increased number and amount of sales involve greater expense, then more capital must be used in running the business. This increment increase of capital will not be applied unless the net profits accruing on the increment business will yield per year the market rate of interest on the additional capital. It is not enough, therefore, that there should be an increment increase of profit. This increment increase of net profit must be equal to or greater than the market rate of interest on the increment increase of capital. Otherwise the additional capital will be invested elsewhere. If C is the capital formerly used, AC the increment extension of investment, and i the market rate of interest for the period assumed, our expression becomes for an average year of that period :

1 Cf. Cournot, Recherches sur les Principes Mathématiques de la Théorie des Richesses, pp. 57 and 58; translation, pp. 53 and 54.

(P AP) (S + AS) — (E+AE) - ZAC = PS E.

There is, in addition, an element of risk, which has not been taken into account. When the entrepreneur, acting on his best judgment, determines to reduce a price, he is, nevertheless, not certain that sales will respond according to his hopes. He therefore will not make the reduction unless he believes the chances to be in favor of enough additional profit to compensate for this risk. If AR is his estimate of the importance or value of the risk, our expression becomes :

(P AP) (S + AS) — (E + AE) - AC — AR = PS E. If the entrepreneur is of the gambling type,-i.e., if his coefficient of caution is negative, and the hope of gain appeals more strongly to him than the fear of loss, risk may serve as an added inducement to the change. In that case AR becomes positive, and we have:

(P AP) (S + AS) – (E+AE) — iAC + AR PS E.

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If, now, we assume this equation to apply only during a limited period of time (as a period of one year), we have an expression for the results that will accrue during that period from a projected increment decrease (or decrement) in price. The actual numbers that must enter into the solution will be different for each kind of business, but the same equation fits all. The form of the expression is such that it fits equally well industries of increasing, decreasing, or proportionate returns (industries of which the relative expenses decrease, increase, or 'remain the same with increasing sales); that it fits equally well businesses whose sales are sharply responsive to variations in price and those whose trade varies little, even with considerable price changes.

There is but one other factor in price determination which it is desired to introduce. It may sometimes happen that to lower a price during a period of years will not of itself pay, that during this period the profits will be less than if the price had been maintained. Nevertheless, it is felt by the managers that ultimate profits will be greater because of the reduction. The reason may be, as in the case of a trade-marked article, that it becomes known to a larger number of people. It may happen that the corporation purposing to reduce a price is one of a group of competing firms, hoping, by means of the reduction, to drive its rivals out of the field, and eventually, as a monopoly, secure higher prices than at first. Or the company planning reduction may be a railroad looking to the ultimate upbuilding and prosperity of the territory it serves, to enable it to secure larger traffic and perhaps higher rates in the future. It may be that such a railroad company will hasten to reduce its rates, in order to discourage the building of competing roads, and thus will be able to prevent so great a permanent fall of rates as would otherwise be inevitable. The principle is, of course, the same, even if the reduction, instead of making possible, eventually, higher rates than before, merely forestalls still greater reductions. In any of these cases the proposed reduction is an investment in succeeding years at the expense of the present. Into the decision as to whether a proposed change shall be adopted enters, therefore, the rate of interest. The ultimate profits must not only be greater than they would else be, they must not only exceed what they would be by as much as the price reduction lessens present profits, but they must compensate for the preceding loss with an equivalent future gain plus interest. In other words, the discounted value of the expected additional or increment income two, three, four, etc., years hence (assuming the previous period of price reduction to be a year), must equal or exceed the loss during the preceding period.' It is evident that the lower is the rate of interest, the more likely is the reduction to be made. A low rate of interest means that future gains and losses are discounted but little, are appreciated keenly. During a period of low interest rates, comparatively large present sacrifices will be made for comparatively small future gains. With the period of reduction extending, as

1 See Marshall, Principles of Economics, third edition, pp. 540 and 541.

2 Since this paper was written, an article has appeared in the Quarterly Journal of Economics, May, 1908, by Professor Alvin S. Johnson, dealing especially with the relations of interest to monopoly price, and elaborating for monopolies the point above set forth. It is intended here, however, to give the principle a broader application, inclusive of competitive, as well as of monopolistic, price-making.

we have assumed, through a year, it would be necessary to determine the mean point in the year of loss concentration, and similarly the points of gain concentration in succeeding years. Without going into these refinements, we may give an approximate statement of the equation. Letting AG, be the excess gain for the second year, AG, the excess for the third, etc., we have :

(P AP) (S + AS)–(E + AE) - AC — AR +

+. + = PS E.

AG2

AG: +

+ (1 + i) (1 + 2)2

Expressed in words, this means that the reduction will be made if the resulting profit, plus the discounted value of future excess gains, sufficiently exceeds the profit at the existing price. More accurately stated, it means that price reduction will take place and will continue as long as the resulting gross income, minus the resulting expense,

1 See Fisher, Nature of Capital and Income, chap. xiii., on Value of Capital.

2 The best development of this principle is to be found in The Rate of Interest by Professor Fisher, Macmillan, 1907.

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minus the market rate of interest on the necessary increment increase of capital, minus the increment of risk, plus the discounted value of the future excess gains, is greater than the gross income at the existing price, minus the existing expense.' It may be pointed out that AG, is itself a function of future prices, future sales, and future expenses, and similarly with AGg. But it would seem needlessly complicated to restate the equation with the suggested substitutions. Should the problem contemplate an increase in price, it is only necessary to change the sign of AP (giving P+AP instead of P - AP in the original equation), and to consider AC, AR, AE, AG2, AG3, etc., as also changed in sign. A reduction may be considered as a negative increase, an increase as a negative reduction.

In concluding, it may be well to emphasize the fact that price-making, whether monopolistic or competitive, is based on the same general principles, and that its conditions are expressible in the same formula.

HARRY G. BROWN.

YALE UNIVERSITY.

1 Transposing:(P AP) (S + AS) — (E +AE) > PS EtiAC + AR

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AS

This gives:PS SAP + PAS E— AE > P3 E + AC + AR .....i

AG, AGE PAS > SAP + SE +iAC + AR

(1 + 1) (1 + 1)2

AG, AGE
SAP + AE + AC + AR

(1 + i) (1 + )2

Р Similarly:

AG2 SAP > PAS — AE - iAC — AR +

(1 + 2)

+ (1 + 1)2

i)

AG, AGE
PAS AE — iAC — AR +7 +

(1 + i) AP

+ (1 + i)?

AG

+

:

+

s

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