56

Irving Fisher—Mathematical investigations

14.

In Fig. 14 the stoppers* or pistons S and S' which regulate thequantities in the cisterns for production and consumption, respect-ively, are so connected as to move together, keeping the quantitiesin the two cisterns equal. Furthermore the water pressure on themfrom the tank keeps the level of all three liquids the same—that inthe tank and those in the two cisterns. The lever keeps the mar-ginal utility equal to the marginal disutility, for its pivot is a fixedone and is placed midway between the axes of ordinates. Theresulting determinate equilibrium is subject to three sets of condi-tions :

(1) The quantity consumed equals that produced—a condition pro-

vided for by the duplicate pistons.

(2) There must be a relation between the quantity produced and

its marginal disutility and between the quantity consumedand its marginal utility—the character of the cisterns.

(3) Marginal utility and disutility are equal—the lever.

§ 4. Analytical.

If A„ and A K be the quantities of A produced and consumed,respectively, the conditions of equilibrium are :

) 1 equation,f 2 unknowns.

P(A«) I

2 equations.

2 new unknowns.

<ZU j 1 equation.d A ) no new unknown.

No. equations : l+2 + l = 4.No. unknowns : 2 + 2 + 0=4.

* In practice a more intricate frictionless bellows would be used.