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

Answer:

$z(t) = (0.9808)^t$

$z(t) = (1.0196)^t$

$z(t) = (0.9808)^t$

$z(t) = (1.03)^t$

$z(t) = (1.0404)^t$

$z(t) = (1.0001)^t$

Step-by-step 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

$z(t) = x(t)y(t)\\z(t) = (1.04)^t.(1.02)^t\\z(t) = (1.04\times 1.02)^t\\z(t) = (1.0608)^t$

(b) z=x/y

$z(t) =\displaystyle\frac{x(t)}{y(t)}\\\\z(t) = \frac{(1.04)^t}{(1.02)^t} = \bigg(\frac{1.04}{1.02}\bigg)^t\\\\z(t) = (1.0196)^t$

(c) z=y/x

$z(t) =\displaystyle\frac{y(t)}{x(t)}\\\\z(t) = \frac{(1.02)^t}{(1.04)^t} = \bigg(\frac{1.02}{1.04}\bigg)^t\\\\z(t) = (0.9808)^t$

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

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

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

$z(t) =\bigg(\displaystyle\frac{x(t)}{y(t)}\bigg)^2\\\\z(t) =\bigg( \frac{(1.04)^t}{(1.02)^t}\bigg)^2 = \bigg(\frac{1.04}{1.02}\bigg)^{2t}\\\\z(t) = (1.0404)^t$

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

$z(t) = (x(t))^{\frac{-1}{3}}(y(t))^{\frac{2}{3}}\\z(t) = ((1.04)^t)^{\frac{-1}{3}}((1.02)^t)^{\frac{2}{3}}\\z(t) = ((1.04)^{\frac{-1}{3}})^t((1.02)^{\frac{2}{3}})^t\\z(t) = (1.04^{\frac{-1}{3}}\times 1.02^{\frac{2}{3}})^t\\z(t) = (1.0001)^t$