98

Irving Fisher—Mathematical investigations

attained and specified in Part I yields the maximum total utility,for, since (Ch. IV, § 2):

dXJ _ dJJ _ dJJ

dA, ~ dm, • P °’ dB, “ dm, ’ Pb ’ ‘ ' '

therefore:

jtfA, _ _ _ dM.

Pa Pb Pm' W

The numerators are the marginal utilities per unit of commodity.To divide by the price is to make the unit of commodity the dollar’sworth. Each fraction is thu s the marginal utility per dollar’s worth.The equation expresses the fact that the rate of increase of utilityfrom spending more money on any one commodity equals the rateof increase for any other. Hence by a familiar theorem of the cal-culus the total utility must he the maximum attainable by any dis-tribution of a fixed income. In like manner the individual dis-tributes his production so that the marginal disutilities in all modesof producing dollar’s worth of commodity are equal so that his totaldisutility is a minimum. Hence the difference between his totalutility and total disutility or his economic gain is a maximum.

§2-

In the distribution of a single commodity over many individualssince :

dU

dA,

therefore,

dXJ dXJ __ cHJ d\J _ dU

dm, - P ° ’ dA, ~ dm, ‘ Pa ’ ’ ‘ ' ; dA n ~ dm/"

dXJ

dU

dA,

dA,

dA n

dTJ

— c?TJ-

- dV

dm,

dm,

dm.

that is, the marginal utilities (when the unit of utility is the marginalutility of money for each individual) are equal and the total utilityis maximum. In like manner the total disutility is a minimum andgain therefore a maximum.

§ 3 .

The first continuous equation may be divided by

dXJ

dm.

and the