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Irving Fisher—Mathematical investigations

For the new coordinates the substitutions from the table give :fxdy-f-qy = m, -

which reduces to x = — q,

a straight line parallel to the axis of ordinates (fig. 31).

The Auspitz und Lieben curve does not reveal to the eye the spe-cial supposition (that commodity and marginal utility change pro-portionally). If we suppose that marginal utility decreases at a con-stant rate in relation to constant second differences of commodity,the new diagram reduces to a straight line :

x — qy—m = 0,

while the other curves would be :

(y.+ As.+B)’ = C(D-<)’and Xj i = E — Fy,—Gy/.

§4.

The value of Jevons’ diagram consists in the use of a simple andfamiliar system of coordinates (the Cartesian ) as representing thetwo chief economic quantities, and is probably the best for elemen-tary purposes.

The value of Auspitz und Lieben’s diagram together with a“ derivative ” curve* fiot shown above consists chiefly in the ease withwhich maxima are discovered and the clear association of maximawith equality of marginal utilities. It is believed that the thirdmethod will, by means of its applicability to the mechanisms ofPart I, more clearly reveal the interdependence of the many com-modities of many individuals and of their many utilities.

§5.

The properties which are essential to the curve we have adoptedare:—

First. That the curve shall never admit of being intersected twiceby a horizontal line (i. e. that it shall not cease to run in a generalup and down manner), to express the fact for consumption that mar-ginal utility decreases as quantity of commodity increases and forproduction that marginal disutility increases as the quantity of com-modity increases.

* Whose Cartesian coordinates are x„ and y a .

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