100 Irving Fisher —Mathematical investigations

capacities for pleasure were great would consume the most in orderto make the aggregate gain in the whole market a maximum). Orwe may destroy all the levers and re-arrange the rear thicknessesuntil the front and back ordinates are made equal.

In like manner the minimum disutility would be attained if allmarginal disutilities were equal. The maximum gain would thenresult. This is the maximum gain obtainable when the amounts ofeach commodity consumed and produced are fixed and given. Ifwe are permitted to rearrange these amounts also, we shall securethe maximum gain when the marginal utilities equal the marginaldisutilities; i. e.

rfU _ tfU _ tfU _

dA lin ~ rfA* * * § ~ dA St n etC ’

Under such a socialistic regime more “necessaries” and less“luxuries” would be consumed and produced than previously.The “rich ” or powerful would produce more and consume less thanpreviously ; the poor or weak would consume more and produceless. Yet for each the marginal utilities and disutilities would beequal.

It is needless to say that these considerations are no plea <forsocialism, but they serve to clear up a subject sometimes discussedby mathematical economists and reconcile Launhardt’s contention*that utility is not a maximum with Auspitz und Lieben’s that it is.The farmer unconsciouslv has reference to equation (4) which is nottrue, the latter to equation (3) which is.f

IV. ELIMINATION OF VARIABLES.

The four sets of equations, Part I, Ch. IV, § 10, can be reduced.

We may substitute for its value F(A,) and thus eliminate all mar-

ginal utilities. Moreover we can get an expression ior p a , p b , etc.,in terms of commodities. First, if m — n the second set of equa-tions are easily solved by determinants! giving :§

* Volkswirthschaftslehre under “ Widerholte TauBch.”

f Auspitz und Lieben appear to overlook this difference of standpoint.Preface, p. xxv.

$ Burnside and Panton. Theory of Equations, p. 251.

§ This equation does not mean that any arbitrary values can he assigned toAi, B,, etc., and the resulting price of A be so simply expressed ; only when A,,Bi, etc. satisfy'aH the conditions of Ch. IV, §10 will the price be expressible asthe quotient of the two determinants.