x' + ——= 0. This differentiation yields — yy = 1 -f- i.
i. "y" % CLjC
The right-hand member, being the ratio of this year’s marginalwantability to next year’s marginal wantability, is by definitionequal to 1 + /. Substituting the new value for the right- andleft-hand members, we have
1 + t = 1 +/,
whence it follows that i = /, which was to have been proved.
The same reasoning may now be applied to three or moreyears. The total wantability for any individual is a functionof the total future income stream. In other words,
W = F(c' + s', c" + s",., c (m) + s ( ">).
The individual tries to make this magnitude a maximum. Interms of the calculus, this is equivalent to making the first totaldifferential equal to zero namely,
dW = dx' + dx" +
dx'
dx'
+
mj
dx (m)
dx (m) = 0.
This total differential equation is equivalent, according to well-known principles of the calculus to a number of subsidiaryequations obtained by making particular suppositions as to thedifferent variations. Let us, for instance, suppose that only
x' and x" vary in relation to each other and that x'", x iv , .,
x (m) do not vary. Then in the above equation all terms afterthe second disappear and the equation reduces, as before, to
_d^ = 3F( ) / dF( )dx' dx' / dx"
So that, again, 1 + t' = 1 + /', and therefore i’ = /'.
This expresses the relation between the first and second years.If we wish, in like manner, to express the corresponding con-nection between the second and third years, let us assume that
x' as well as z ( ",., x (m) are constant but that x" and x’"
vary. Then the first term of the equation and all after the thirddisappear, and the equation reduces to
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