Chapter 9, Problem 5PA

(a)

To determine

The per worker production function.

Explanation of Solution

The production function is given using Equation (1) as follows:

$\text{F}\left(\text{K,L}\right)={\text{AK}}^{\text{α}}{\text{L}}^{\text{1}-\text{α}}$ (1)

 According to the Solow growth model with technological progress, the output per worker is y and the capital per worker is k.

 According to the Solow growth model, the output per worker can be calculated using Equation (3) as follows:

$\text{Output per worker}=\frac{\text{Total output}}{\text{Number of workers}}$ (2)

The output per worker can be calculated by substituting the respective values in Equation (2) as follows:

$\begin{array}{c}\text{Output per worker}=\frac{\text{F}\left(\text{K,L}\right)}{\text{L}}\\ =\frac{{\text{AK}}^{\text{α}}{\text{L}}^{1-\text{α}}}{\text{L}}\\ =\text{A}{\left(\frac{\text{K}}{\text{L}}\right)}^{\text{α}}\\ \text{y}={\text{Ak}}^{\text{α}}\end{array}$

Thus, the output per worker is given as ${\text{Ak}}^{\text{α}}$.

(b)

To determine

The ratio of steady state income in Country R to Country P.

Explanation of Solution

In the steady state, the condition for the steady state value of income is given as follows:

$\text{sy}=\left(\text{δ}+\text{n}+\text{g}\right)\text{k}$ (3)

Substitute the value of y in Equation (3) to obtain the steady state value of capital per worker as follows:

$\begin{array}{c}{\text{sAk}}^{\text{α}}=\left(\text{δ}+\text{n}+\text{g}\right)\text{k}\\ \frac{\text{k}}{{\text{k}}^{\text{α}}}=\frac{\text{sA}}{\text{δ}+\text{n}+\text{g}}\\ {\text{k}}^{1-\text{α}}=\frac{\text{sA}}{\text{δ}+\text{n}+\text{g}}\\ \text{k}={\left(\frac{\text{sA}}{\text{δ}+\text{n}+\text{g}}\right)}^{\frac{1}{1-\text{α}}}\end{array}$

Thus, the steady state value of capital per worker is $\text{k}={\left(\frac{\text{sA}}{\text{δ}+\text{n}+\text{g}}\right)}^{\frac{1}{1-\text{α}}}$.

The steady state value of output per worker can be calculated as follows:

$\begin{array}{c}\text{y}=\text{A}{\left(\frac{\text{sA}}{\text{δ}+\text{n}+\text{g}}\right)}^{\frac{\text{α}}{1-\text{α}}}\\ ={\text{A}}^{\frac{1}{1-\text{α}}}{\left(\frac{\text{s}}{\text{δ}+\text{n}+\text{g}}\right)}^{\frac{\text{α}}{1-\text{α}}}\end{array}$

Thus, the steady state value of capital per worker is $\text{y}={\text{A}}^{\frac{1}{1-\text{α}}}{\left(\frac{\text{s}}{\text{δ}+\text{n}+\text{g}}\right)}^{\frac{\text{α}}{1-\text{α}}}$.

Given that the saving rate in Country R is S_R=0.32 and in Country P is S_P=0.1. The rate of population growth in Country R is n_R=0.01 and in Country P is n_P=0.03. The rate of technological progress, g=0.02 and the rate of depreciation is 0.05.

The ratio of steady-state income per worker in Country R to the Country P can be calculated as follows:

$\begin{array}{c}\frac{{\text{y}}_{\text{R}}}{{\text{y}}_{\text{P}}}={\left(\frac{\left(\frac{{\text{S}}_{\text{R}}}{\text{δ}+{\text{n}}_{\text{R}}+\text{g}}\right)}{\left(\frac{{\text{S}}_{\text{P}}}{\text{δ}+{\text{n}}_{\text{P}}+\text{g}}\right)}\right)}^{\frac{\text{α}}{1-\text{α}}}\\ ={\left(\frac{\left(\frac{\text{0}\text{.32}}{0.05+0.01+0.02}\right)}{\left(\frac{0.1}{0.05+0.03+\text{0}\text{.02}}\right)}\right)}^{\frac{\text{α}}{1-\text{α}}}\\ =4^{{}^{\frac{\text{α}}{1-\text{α}}}}\end{array}$

Thus, the ratio of steady-state income per worker in Country R to the Country P is as follows:

$4^{{}^{\frac{\text{α}}{1-\text{α}}}}$.

(c)

To determine

Comparison of income in the two countries.

Explanation of Solution

 We know that the ratio of steady state income in both countries is given as follows:

 $\frac{{\text{y}}_{\text{R}}}{{\text{y}}_{\text{P}}}=4^{{}^{\frac{\text{α}}{1-\text{α}}}}$ (4)

 When the value of $\text{α}=\frac{1}{3}$, the ratio can be obtained as follows:

 $\begin{array}{c}\frac{{\text{y}}_{\text{R}}}{{\text{y}}_{\text{P}}}=4^{{}^{\frac{\frac{1}{3}}{\left(1-\frac{1}{3}\right)}}}\\ =4^{\frac{1}{{}^2}}\\ =2\end{array}$

 Thus, the income per worker in Country R is two times higher than the income per worker is Country P.

(d)

To determine

The difference in income per worker in both countries.

Explanation of Solution

 It is given that the steady state income in Country R is 16 times greater than the income of Country P.

 $\begin{array}{c}\frac{{\text{y}}_{\text{R}}}{{\text{y}}_{\text{P}}}=4^{{}^{\frac{\text{α}}{1-\text{α}}}}\\ 16=4^{{}^{\frac{\text{α}}{1-\text{α}}}}\\ {\left(4\right)}^2=4^{{}^{\frac{\text{α}}{1-\text{α}}}}\\ 2=\frac{\text{α}}{1-\text{α}}\\ 2-2\text{α}=\text{α}\\ 2=3\text{α}\\ \text{α}=\frac{2}{3}\end{array}$

 Thus, the value of the capital’s share of income should be 2/3. This could be due to the reason that capital includes both human capital and physical capital. There is also a possibility that the factor productivity of both countries would be different largely due to the difference in the values of other parameters which is assumed to be constant in the given situation. This is the reason for the difference in the income gap in the two countries.