Front Page Titles (by Subject) Distribution of a Commodity in Time. - The Theory of Political Economy
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Distribution of a Commodity in Time. - William Stanley Jevons, The Theory of Political Economy 
The Theory of Political Economy (London: Macmillan, 1888) 3rd ed.
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Distribution of a Commodity in Time.
We have seen that, when a commodity is capable of being used for different purposes, definite principles regulate its application to those purposes. A similar question arises when a stock of commodity is in hand, and must be expended over a certain interval of time more or less definite. The science of Economics must point out the mode of consuming it to the greatest advantage—that is, with a maximum result of utility. If we reckon all future pleasures and pains as if they were present, the solution will be the same as in the case of different uses. If a commodity has to be distributed over n days' use, and v1, v2, etc., be the final degrees of utility on each day's consumption, then we ought clearly to have
v1 = v2 = v3 =...=vn.
It may, however, be uncertain during how many days we may require the stock to last. The commodity might be of a perishable nature, so that if we were to keep some of it for ten days, it might become unserviceable, and its utility be sacrificed. Assuming that we can estimate more or less exactly the probability of its remaining good, let p1, p2, p3... p10, be these probabilities. Then, on the principle (p. 36) that a future pleasure or pain must be reduced in proportion to its want of certainty, we have the equations
v1p1 = r2p2 =... = v10p10.
The general result is, that as the probability is less, the commodity assigned to each day is less, so that v, its final degree of utility, will be greater.
So far we have taken no account of the varying influence of an event according to its propinquity or remoteness. The distribution of commodity described is that which should be made and would be made by a being of perfect good sense and foresight. To secure a maximum of benefit in life, all future events, all future pleasures or pains, should act upon us with the same force as if they were present, allowance being made for their uncertainty. The factor expressing the effect of remoteness should, in short, always be unity, so that time should have no influence. But no human mind is constituted in this perfect way: a future feeling is always less influential than a present one. To take this fact into account, let q1, q2, q3, etc., be the undetermined fractions which express the ratios of the present pleasures or pains to those future ones from whose anticipation they arise. Having a stock of commodity in hand, our tendency will be to distribute it so that the following equations will hold true—
v1p1q1 = v2p2q2 = v3p3q3 =... = vnpnqn.
It will be an obvious consequence of these equations that less commodity will be assigned to future days in some proportion to the intervening time.
An illustrative problem, involving questions of prospective utility and probability, is found in the case of a vessel at sea, which is insufficiently victualled for the probable length of the voyage to the nearest port. The actual length of the voyage depends on the winds, and must be uncertain; but we may suppose that it will almost certainly last ten days or more, but not more than thirty days. It is apparent that if the food were divided into thirty equal parts, partial famine and suffering would be certainly endured for the first ten days, to ward off later evils which may not be encountered. To consume one-tenth part of the food on each of the first ten days would be still worse, as almost certainly entailing starvation on the following days. To determine the most beneficial distribution of the food, we should require to know the probability of each day between the tenth and thirtieth days forming part of the voyage, and also the law of variation of the degree of utility of food. The whole stock ought then to be divided into thirty portions, allotted to each of the thirty days, and of such magnitudes that the final degrees of utility multiplied by the probabilities may be equal. Thus, let v1, v2, v3, etc., be the final degrees of utility of the first, second, third, and other days supplied, and p1, p2, p3, etc., the probabilities that the days in question will form part of the voyage; then we ought to have
p1v1 = p2v2 = p3v3 =... = p29v29 = p30v30.
If these equations did not hold true, it would be beneficial to transfer a small portion from one lot to some other lot. As the voyage is supposed certainly to last the first ten days, we have
p1 = p2 =... = p10 = 1:
hence we must have
v1 = v2 =... = v10:
that is to say, the allotments to the first ten days should be equal. They should afterwards decrease according to some regular law; for, as the probability decreases, the final degree of utility should increase in inverse proportion.
THEORY OF EXCHANGE