in the theory of value and prices.

i

53

The condition that the incomes are given is:

A 1 . p a + B l . p b + . +M, . p m = K, 1

A . p a + B, . p b + .+M a . p m = K, !

i m

I

A„. p„ + B„ . p b + . +M„ . p m = K„ J

n equations,new unknowns(prices).

The utility functions (the cistern-forms) are:

c?U , d U > dU t-w,, \

= F(A 1 ); —- =F(B,);-; m =F(M.)

dA t

tfB,

tfU

£ =F(A ” ); S„ =F{B " );

d\}

dA ~ F(Aj); dB “ F(B,); ‘ ' ' ' ; dM “ F(Mj)

• S= F(M ">

mn equations.mn newunknowns(marg. ut.).

The principle of proportion is:

rfU . rfU . rfU . rfU _

<£A, ’ dB x ' dC 1 ' dM. t

f?U dV _ d IT _

dA .: '' r/B„ : . : cZM„ ”

dU dTJdA . ‘ <fB n

dU

dM»

=Pa‘Pi‘Pc-

■ Pm

J

n ( m —1)independentequations.

no newunknowns.

Total number of equations: wi + ra + wm + n (m-“ “ unknowns mn -|- m + mn+V

•1) = 2 mn + m= 2 mn + m

Therefore all magnitudes are determinate and the number of thesemagnitudes as well as the number of the equations is twice the num-ber of commodities times the number of individuals plus the num-ber of commodities.

The valuation of money for each individual can be found fromthe equations:

rfXJ_ dA,dm j — p a

(Ch.lV.ga.)

dm ,

etc.