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Basic properties of growth rates. Use the fact that the growth rate of a variable equals...

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Advance Math

Basic properties of growth rates. Use the fact that the growthrate of a variable equals the time derivative of its log toshow:

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

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

(c) If Z(t) = aX(t)α, then Ż(t)/Z(t) = αẊ(t)/X(t).

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