Basic properties of growth rates. Use the fact that the growth rate of a variable equals the time derivative of its log to show:

(a) The growth rate of the product of two variables equals the sum of their growth rates. That is, if Z(t) = X(t)Y(t), then Z(t)/Z(t) = [X(t)/X(t)] + [Y(t)/Y(t)].

(b) The growth rate of the ratio of two variables equals the difference of their growth rates. That is, if Z(t) = X(t)/Y(t), then Z(t)/Z(t) = [X(t)/X(t)]−[Y(t)/Y(t)].

(c) If Z(t) = aX(t)

α, then Z(t)/Z(t) = αX(t)/X(t).