82

Irving Fisher—Mathematical investigations

§ 10 .

It is seen that analytically the treatment of interdependent com-modities differ from that of independent commodities only in this,that the equations which represent the functions have more letters ;i; e. we have

g = F ( A 1 , Bl

N,) instead of = F(A,).

All other equations are just as in Part I. In fact these functionequations are, so to speak, the residuary formulae ; they contain allthe unanalyzed conditions of the problem.

The marginal utilities are (as in Part I) in a continuous ratiowhich is the ratio of prices. Yet there are some peculiar caseswhich could not occur under the suppositions of Part I, viz : thosecases arising when the marginal utility of one or more articles hasno meaning.

If two articles are perfect completing articles, as gun and trigger,there is no such quantity as the marginal utility of triggers alone.There is, however, a marginal utility of a combined gun and trigger.Now there are separate marginal disutilities for producing the gunand trigger. How are all these quantities to be introduced into ourcontinuous proportion of marginal utilities ?

Suppose for a moment there were no difficulty of this sort. Theproportion for each individual would be just as before (Part I,Ch. 1Y, § 10) and might be expressed as follows [G&<? for gun T &ttor trigger] :

Pi,

Pi

r p s i

f p ‘ 1

d\5 —

tfU —

’dU

•eHT

dG„

l <*G„,

l dT K J

P, + Pi

'd\7

clG

+ rfT

P 9 +Pi

dXJ

Finally: It is clear that

dU dU dU

I = Ao + Bf> + . . . +Mm and vU, =7r“ + 35' ( '+ • • • +3^7' ,

dA, dB, dM,

Substituting these values in I v U, = 0 we have after performing the multipli-cation and remembering that a . a — 1 and a.b — a.c— ...=b. c~ . . . =0,

A

d UdAi

+ B

dU

1 dB] +

. . +M

dU1 dM,

= 0

or since prices are proportional to marginal utilities:

A, p„ + B, pi + . . . M, pm = 0.

Likewise for II, III, etc. making n equations.

Conversely we could derive the vector equations from the scalar.