APPENDIX
for the amount z of B being y,, we have, as the condition for
equilibrium,
y=zxy+zy,. . . . . . XxX)
The demand equations for A and B being
Nn=A&amp;), =e)... (xi)
and the supply equation of the two products being
ry=e¢lxr,2) . . . . .
ve have only four equations to determine the five quantities
J» Juv Vos x, &amp;.

If

r=n.z , (xiii)
n being constant, that is, if A and B are produced in fixed
proportions, the fifth equation needed for a determinate
solution is supplied.
If, however, x can be varied somewhat without affecting
2, or if some small changes of both x and 2 can be made
without changing the cost of the production process, ze.
without changing y, the conditions considered on p. 66 are
found. Any mode of variation other than that denoted by
equation (xiii) can be reduced to the first-named. Let, then,
an increase n of y correspond to a small increase £ in z,
z remaining unaltered.
Then, for equilibrium, we must have the value of A such
that # is the value of an amount £ We have, in fact,
determined the marginal cost of production of A per unit
in the ratio of » to &amp; This supplies the fifth condition
aeeded for the determination of the five unknown quantities
of the problem, and may be written
vow = 5 ED
t having, in this equation, the value appropriate to an
equilibrium between supply and demand.
A similar mode of expression might be used for the
problem of joint demand, as stated on p. 68. The nature
of the conceptions involved is, however, sufficiently shown
in the above, and their expression in connection with allied
problems, or with the problem of joint products more
lumerous than two, may be left as an exercise for the
student