Chapter 5, Problem 6E

(a)

To determine

The ratio of per capita GDP relative to Country U’s steady state.

Explanation of Solution

The Solow economy is a steady state economy where the level of investment and the level of depreciation in the economy would be the same. This means that the new investment in the economy will only be able to repair and restore the level of depreciation in the economy. This maintains the economy in a steady state and that is known as the Solow’s steady state.

In the case of Country U and Country S, the ratio of per capita GDP can be calculated when there is no change in TFP or in the rate of depreciation across countries. The key equation used in this case is as follows:

$\text{y*}=\frac{\text{Y*}}{\text{L*}}={\overline{\text{A}}}^{3/2}{\left(\frac{\overline{\text{s}}}{\overline{\text{d}}}\right)}^{1/2}$

The value of the depreciation and TFP does not change and that means the only value and changes will be the investment rates for the two economies. The equation can be enlarged as follows:

$\begin{array}{c}\frac{\text{y*s}}{{\text{y*}}_{\text{US}}}={\left(\frac{{\overline{\text{A}}}_{\text{s}}}{{\overline{\text{A}}}_{\text{US}}}\right)}^{3/2}\times {\left(\frac{{\overline{\text{s}}}_{\text{ s}}}{{\overline{\text{s}}}_{\text{ US}}}\right)}^{1/2}\\ \frac{\text{y*s}}{{\text{y*}}_{\text{US}}}={\left(\frac{{\overline{\text{s}}}_{\text{ s}}}{{\overline{\text{s}}}_{\text{ US}}}\right)}^{1/2}\\ ={\left(\frac{32.1}{24.7}\right)}^{1/2}\\ =\sqrt{1.299595}\\ =1.141\end{array}$

It is identified that Country S has more investment than Country U, and it means that Country S would be richer than Country U. The difference in the per capita GDP between Countries U and S is calculated to be 14.1 percent than Country U’s level. Similarly, the difference between Country U and all other countries can be calculated and the table can be completed as follows:

Country	Investment rate $\overline{\text{s}}$ (percent)	Ratio of per capita GDP relative to Country U in steady state (percent)
US	24.7	-
Switzerla	32.1	14.1
Hokong	26.2	3.1
Fran	25.0	0.6
Jap	29.0	8.3
South Kore	35.0	19.0
Argentin	16.0	-19.5
Mexic	21.1	-9.2
Thailan	26.4	3.4
Indi	23.3	-2.9
Keny	11.1	-33.3
Ethiopi	10.4	-36.4

Economics Concept Introduction

Economic growth: Economic growth is the increase in the output of goods and services produced per head of the population over a period of time in an economy.

(b)

To determine

The ratio of per capita GDP relative to Country U’s steady state with TFP differences.

Explanation of Solution

When there is difference in TFP of the economy, the equation would change as follows:

$\begin{array}{c}\frac{\text{y*s}}{{\text{y*}}_{\text{US}}}={\left(\frac{\text{1}\text{.022}}{\text{1}}\right)}^{3/2}\times {\left(\frac{\text{32}\text{.1}}{\text{24}\text{.7}}\right)}^{1/2}\\ =1.03\times 1.14\\ =1.178\end{array}$

It is identified that Country S has more TFP than Country U, and it means that Country S would push the steady-state level higher than Country U’s level by 18 percent. Similarly, the difference between Country U’s and all other countries can be calculated and the table can be completed as follows:

Country	Investment rate $\overline{\text{s}}$ (percent)	TFP	Ratio of per capita GDP relative to Country U in steady state with TFP differences (percent)
US	24.7	1.000	-
Switzerla	32.1	1.022	17.8
Hokong	26.2	0.835	-13.0
Fran	25.0	0.688	-34.9
Jap	29.0	0.680	-36.0
South Kore	35.0	0.711	-31.7
Argentin	16.0	0.534	-55.6
Mexic	21.1	0.438	-67.1
Thailan	26.4	0.381	-73.2
Indi	23.3	0.240	-80.3
Keny	11.1	0.178	-91.4
Ethiopi	10.4	0.102	-96.3

(c)

To determine

The percentage gap between the steady-state income ratio and the ratio in 2014.

Explanation of Solution

The income ratio of Country S is 114.7 percent, whereas the steady-state level with TFP differences is 117.8. Therefore, the gap between the current position and the steady state position can be calculated as follows:

$\begin{array}{c}\text{gap between current and steady state}=\frac{\text{Steady state}-\text{Current}}{\text{Current}}\times 100\\ =\frac{117.8-114.7}{114.7}\times 100\\ =\frac{3.1}{114.7}\times 100\\ =0.027\times 100\\ =2.7\end{array}$

Thus, it shows that Country S is slightly below its steady state position. Similarly, the gap between the current and the steady state position for other countries can be calculated and tabulated as follows:

Country	Investment rate $\overline{\text{s}}$ (percent)	TFP	Ratio of per capita GDP relative to US in steady state (percent)	Ratio of per capita GDP relative to US in steady state  with TFP differences (percent)	Gap between current and steady state level
US	24.7	1.000	-	-	-
Switzerla	32.1	1.022	14.1	17.8	2.70
Hokong	26.2	0.835	3.1	-13.0	-15.62
Fran	25.0	0.688	0.6	-34.9	-35.23
Jap	29.0	0.680	8.3	-36.0	-40.90
South Kore	35.0	0.711	19.0	-31.7	-42.61
Argentin	16.0	0.534	-19.5	-55.6	-44.84
Mexic	21.1	0.438	-9.2	-67.1	-63.77
Thailan	26.4	0.381	3.4	-73.2	-74.08
Indi	23.3	0.240	-2.9	-80.3	-79.71
Keny	11.1	0.178	-33.3	-91.4	-87.11
Ethiopi	10.4	0.102	-36.4	-96.3	-94.18

(d)

To determine

The countries ranked based on the transition dynamics.

Explanation of Solution

The countries that have higher differences between the current state and the steady-state level would grow faster and when the countries reach close to their steady-state level, the economic growth would be slower and slower. This is suggested by the transition dynamics in the economy. The ranking will be in reverse order, which means that the economy with the highest growth rate will be Country E and least growth will be Country U.