in the theory of value and prices. 59

dTJ

c*U

dTJ

dTJ 1

<*A„,r

dTJ

dB„ ‘ ’dTJ

' dU„,dTJ

*A«.. ‘

rfl)

’ '

dTJ

• -rfM.,,-

(2 m— 1) nindepen-^ dent

equations,no new

^A ff , 2

dR„, i

‘ dA nt

d Bk, 2

• 'iM„r

dTJ

dTJ

dTJ

dTJ

dTJ

dK.:

d^„.n

dA K ,n

dB^,' ‘

' • <*M„. -

unknowns.

~Pa ■

— p b ■■ • •

■ — Pm

+ Pa

+ Pt •

■■+ Pm

No. equations: m + (n— 1) +2mn+ (2m— 1) n=.4mn + m— lNo. unknowns: 2mn + m + 2mn + 0 =4 mn-\-m.

There are just one too few equations. It may not be evident atfirst why the second set does not contain n independent equationsinstead of ( n —1). The point is that any one of these equations canbe derived from the others together with the equations' of the firstset. Thus multiply the equations of the first set by p„ p b , . . . p mrespectively and add the resulting equations arranging as follows :

A*, \ • Pa + B„., 1 • Pb + ■ • ■ 4- M w> i . p m +

+ A,,, 2 ■ Pa + s • Pb + • • • + 5 . p m -t- ^ _

+ . - . - .

“I - A,,, " • Pa "4" B wjn . p b -j- ■ • ■ 4“ . p m

( + p m +

J + A <,»'I ) «4B„,.^ + . . • 4- ! ■ /•'m 4

+ A„. >n • Pa + B„„ . p b + . . . + M K|n . 2^*1 •

Subtracting from this equation the sum of all but the first (say)of the second set, our result is :

A w , i • Pa 4- B„ t • Pb 4~ • • -4“ , . p m =

A„ 1 • Pa + B„,, . Pi, 4- • • • + , . p m

which is the first equation of the second set. This equation is there-fore dependent on the others, or there is one less independent equa-tion than appears at first glance. Hence we need one more equa-tion. We may let:

Pa-= 1.

This makes A the standard of value (of. §5).

No such limitation applies to the equations in Chapter IV.