First, we will eliminate K from (2) by using the definition of k. From
the definition, it follows immediately that
k = e--rtl(- 'Yk.
Substitute from 1.(9) into (2) and then substitute the resulting expression
in the formula for k:
(7)
c(t) is proportional to consumption per effective worker. If the marginal
felicity of consumption has constant elasticity u, then
(8)
Let us introduce a new "price" for consumption per effective worker, say
p. We wish to define p so that
p = (e-r 1jP)"q. (10)
We now need a differential equation for p to replace (5) for q. Take
natural logarithms in (10):
log p = u('Yt-log P) +log q.
Differentiate with respect to time:
pjp = p + UT - j'(k) = W - f'(k) 1 (11)
from 1.(11).
