Question

Computing growth rates (I): Suppose Xt=(1.04)^t and Yt=(1.02)^tCalculate the growth rate of Zt in each of the following cases:(a) z=xy(b) z=x/y(c) z=y/x(d) z=x^1/2 y^1/2(e) z=(x/y)^2(f) z=x^-1/3y^2/3

Answer 1

`z(t) = (0.9808)^t`

`z(t) = (1.0196)^t`

`z(t) = (0.9808)^t`

`z(t) = (1.03)^t`

`z(t) = (1.0404)^t`

`z(t) = (1.0001)^t`

Explanation:

We are given the following in the question:

`x(t)=(1.04)^t\\y(t)t=(1.02)^t`

We have to find the growth rate z(t) in each of the following cases:

(a) z = xy

`z(t) = x(t)y(t)\\z(t) = (1.04)^t.(1.02)^t\\z(t) = (1.04* 1.02)^t\\z(t) = (1.0608)^t`

(b) z=x/y

`z(t) =\displaystyle(x(t))/(y(t))\\\\z(t) = ((1.04)^t)/((1.02)^t) = \bigg((1.04)/(1.02)\bigg)^t\\\\z(t) = (1.0196)^t`

(c) z=y/x

`z(t) =\displaystyle(y(t))/(x(t))\\\\z(t) = ((1.02)^t)/((1.04)^t) = \bigg((1.02)/(1.04)\bigg)^t\\\\z(t) = (0.9808)^t`

(d) z=x^(1/2) y^(1/2)

`z(t) = (x(t))^{(1)/(2)}(y(t))^{(1)/(2)}\\z(t) = ((1.04)^t)^(1)/(2) ((1.02)^t)^(1)/(2)\\z(t) = (1.0608)^{(t)/(2)}\\z(t) = (1.03)^t`

(e) z=(x/y)^2

`z(t) =\bigg(\displaystyle(x(t))/(y(t))\bigg)^2\\\\z(t) =\bigg( ((1.04)^t)/((1.02)^t)\bigg)^2 = \bigg((1.04)/(1.02)\bigg)^(2t)\\\\z(t) = (1.0404)^t`

(f) z=x^(-1/3)y^(2/3)

`z(t) = (x(t))^{(-1)/(3)}(y(t))^{(2)/(3)}\\z(t) = ((1.04)^t)^{(-1)/(3)}((1.02)^t)^{(2)/(3)}\\z(t) = ((1.04)^{(-1)/(3)})^t((1.02)^{(2)/(3)})^t\\z(t) = (1.04^{(-1)/(3)}* 1.02^{(2)/(3)})^t\\z(t) = (1.0001)^t`