Chapter 3, Problem 10E

(a)

To determine

Determine the growth rate

Explanation of Solution

If k, l and m grows at constant rate, then the growth rate can be calculated as follows:

$\begin{array}{c}\text{y}={\text{k}}^{\left(\frac{1}{3}\right)}\\ \text{=}\left(\frac{1}{3}\right)\times \overline{{\text{g}}_{\text{k}}}\end{array}$

Thus, the growth rate is $\left(\frac{1}{3}\right)\times \overline{{\text{g}}_{\text{k}}}$

(b)

To determine

Determine the value of growth rate

Explanation of Solution

If k, l and m grows at constant rate, then the growth rate can be calculated as follows:

$\begin{array}{c}\text{y}={\text{k}}^{\left(\frac{1}{3}\right)}{\text{l}}^{\left(\frac{1}{3}\right)}\\ \text{=}\left(\left(\frac{1}{3}\right)\times {\text{g}}_{\text{k}}\right)+\left(\left(\frac{2}{3}\right)\times {\text{g}}_{\text{t}}\right)\end{array}$

Thus, the growth rate is $\left(\left(\frac{1}{3}\right)\times {\text{g}}_{\text{k}}\right)+\left(\left(\frac{2}{3}\right)\times {\text{g}}_{\text{t}}\right)$

(c)

To determine

Determine the value of growth rate

Explanation of Solution

If k, l and m grows at constant rate, then the growth rate can be calculated as follows:

$\begin{array}{c}\text{y}={\text{mk}}^{\left(\frac{1}{3}\right)}{\text{l}}^{\left(\frac{1}{3}\right)}\\ ={\text{g}}_{\text{m}}+\left(\left(\frac{1}{3}\right)\times {\text{g}}_{\text{k}}\right)+\left(\left(\frac{2}{3}\right)\times {\text{g}}_{\text{t}}\right)\end{array}$

Thus, the growth rate is ${\text{g}}_{\text{m}}+\left(\left(\frac{1}{3}\right)\times {\text{g}}_{\text{k}}\right)+\left(\left(\frac{2}{3}\right)\times {\text{g}}_{\text{t}}\right)$

(d)

To determine

Determine the value of growth rate

Explanation of Solution

If k, l and m grows at constant rate, then the growth rate can be calculated as follows:

$\begin{array}{c}\text{y}={\text{mk}}^{\left(\frac{1}{4}\right)}{\text{l}}^{\left(\frac{3}{4}\right)}\\ ={\text{g}}_{\text{m}}+\left(\left(\frac{1}{4}\right)\times {\text{g}}_{\text{k}}\right)+\left(\left(\frac{3}{4}\right)\times {\text{g}}_{\text{t}}\right)\end{array}$

Thus, the growth rate is ${\text{g}}_{\text{m}}+\left(\left(\frac{1}{4}\right)\times {\text{g}}_{\text{k}}\right)+\left(\left(\frac{3}{4}\right)\times {\text{g}}_{\text{t}}\right)$

(e)

To determine

Determine the value of growth rate

Explanation of Solution

If k, l and m grows at constant rate, then the growth rate can be calculated as follows:

$\begin{array}{c}\text{y}={\text{mk}}^{\left(\frac{3}{4}\right)}{\text{l}}^{\left(\frac{1}{4}\right)}\\ ={\text{g}}_{\text{m}}+\left(\left(\frac{3}{4}\right)\times {\text{g}}_{\text{k}}\right)+\left(\frac{1}{4}\right)\times {\text{g}}_{\text{t}}\end{array}$

Thus, the growth rate is ${\text{g}}_{\text{m}}+\left(\left(\frac{3}{4}\right)\times {\text{g}}_{\text{k}}\right)+\left(\frac{1}{4}\right)\times {\text{g}}_{\text{t}}$

(f)

To determine

Determine the value of growth rate

Explanation of Solution

If k, l and m grows at constant rate, then the growth rate can be calculated as follows:

$\begin{array}{c}\text{y}={\left(\text{klm}\right)}^{\left(\frac{1}{2}\right)}\\ =\left(\left(\frac{1}{2}\right)\right)\times \left({\text{g}}_{\text{m}}+{\text{g}}_{\text{k}}+{\text{g}}_{\text{t}}\right)\end{array}$

Thus, the growth rate is $\left(\left(\frac{1}{2}\right)\right)\times \left({\text{g}}_{\text{m}}+{\text{g}}_{\text{k}}+{\text{g}}_{\text{t}}\right)$

(g)

To determine

Determine the value of growth rate

Explanation of Solution

If k, l and m grows at constant rate, then the growth rate can be calculated as follows:

$\begin{array}{c}\text{y}={\left(\text{kl}\right)}^{\left(\frac{1}{4}\right)}{\left(\frac{1}{\text{m}}\right)}^{\left(\frac{3}{4}\right)}\\ =\left(\left(\frac{1}{4}\right)\right)\times {\text{g}}_{\text{k}}+\left(\frac{1}{4}\right)\times {\text{g}}_{\text{t}}-\left(\frac{3}{4}\right)\times {\text{g}}_{\text{m}}\end{array}$

Thus, the growth rate is $\left(\left(\frac{1}{4}\right)\right)\times {\text{g}}_{\text{k}}+\left(\frac{1}{4}\right)\times {\text{g}}_{\text{t}}-\left(\frac{3}{4}\right)\times {\text{g}}_{\text{m}}$