Chapter 6, Problem 5NP

a)

To determine

To describe: The steady state values of capital-labor ratio, output per worker and consumption per worker is to be calculated.

Answer to Problem 5NP

The steady-state values are −

Capital-labor ratio = $k_t=36$

Output per worker = $y_t=18$

Consumption per worker = $c_t=12.6$

Explanation of Solution

Given that −

Output per worker = $y_t$

Consumption per worker = $c_t$

Capital stock per worker = $k_t$ ’

When the all above values of economy is constant then it is known as steady state condition.

The following equation will be used to represent the steady state value of the capital-labor ratio −

  $sf(k)=(n+d)k$ ……………………Equ (1)

Given values −

s = 0.3

  $y_t=f(k_t)=3k_t^{0.5}$

n = 0.05

d = 0.1

Put the above values in Equ (1) −

  $0.3\times 3k_t^{0.5}=(0.05+0.1)k_t$

  $\frac{0.9k_t^{0.5}}{0.15}=\frac{0.15}{0.15}{k}_t$

  $6k_t^{0.5}=k_t$

Divide the above equation by $k_t^{0.5}$

  $\frac{6k_t^{0.5}}{k_t^{0.5}}=\frac{k_t}{k_t^{0.5}}$

  $6=k_t^{0.5}$

Square on both sides −

  $6^2={\left(k_t^{0.5}\right)}^2$

  $k_t=36$

Now, the capital-labor ratio = 36

The output per worker is calculated by using the following equation −

  $y_t=f(k_t)=3k_t^{0.5}$

Now, put the calculated value in above equation −

  $y_t=3{\left(36\right)}^{0.5}$

  $y_t=18$

Now, the steady state value for the output per worker = 18

The following equation will be use to represent the consumption per worker −

  $c_t=y_t-(n+d)k_t$ ……………………Equ (2)

Also given that −

Population growth rate, n = 0.05

Depreciation rate, d = 0.1,

Now, put the given and calculated values in Equ (2) −

  $c_t=18-(0.05+0.1)36$

  $c_t=18-5.4=12.6$

The consumption per worker, $c_t=12.6$

Economics Concept Introduction

Introduction:

The ratio between capital and labor is known as capital-labor ratio and the ratio between the output and labor is known as output per worker.

b)

To determine

To describe: The steady state values of capital-labor ratio, output per worker and consumption per worker is to be calculated with saving rate 0.4 instead of 0.3.

Answer to Problem 5NP

The steady-state values are −

Capital-labor ratio = $k_t=64$

Output per worker = $y_t=24$

Consumption per worker = $c_t=14.4$

Explanation of Solution

Given values −

s = 0.4

  $y_t=f(k_t)=3k_t^{0.5}$

n = 0.05

d = 0.1

The following equation will be used to represent the steady state value of the capital-labor ratio −

  $sf(k)=(n+d)k_t$ ……………………Equ (1)

Now, put the given values in above Equ (1) −

  $\begin{array}{l}0.4\times 3k_t^{0.5}=\left(0.05+0.1\right)k_t\\ \frac{1.2k_t^{0.5}}{0.15}=\frac{0.15}{0.15}{k}_t\end{array}$

  $k_t=8k_t^{0.5}$

Divide the both side of above Equ by $k_t^{0.5}$ -

  $\begin{array}{l}\frac{k_t}{k_t^{0.5}}=\frac{8k_t^{0.5}}{k_t^{0.5}}\\ k_t^{0.5}=8\end{array}$

Square both sides −

  ${\left(k_t^{0.5}\right)}^2=8^2$

  $k_t=64$

The capital − labor ratio = $k_t=64$

The output per worker is calculated by using the following equation −

  $y_t=f(k_t)=3k_t^{0.5}$

Now, put the calculated value in above equation −

  $y_t=3{\left(64\right)}^{0.5}$

  $y_t=24$

Now, the steady state value for the output per worker = 24

The following equation will be use to represent the consumption per worker −

  $c_t=y_t-(n+d)k_t$ ……………………Equ (2)

Also given that −

Population growth rate, n = 0.05

Depreciation rate, d = 0.1,

Now, put the given and calculated values in Equ (2) −

  $c_t=24-(0.05+0.1)64$

  $c_t=24-9.6=14.4$

The consumption per worker, $c_t=14.4$

Economics Concept Introduction

Introduction: The ratio between capital and labor is known as capital-labor ratio and the ratio between the output and labor is known as output per worker.

c)

To determine

To describe: The steady state values of capital-labor ratio, output per worker and consumption per worker is to be calculated with 0.8 population growth rate.

Answer to Problem 5NP

The steady-state values are −

Capital-labor ratio = $k_t=25$

Output per worker = $y_t=15$

Consumption per worker = $c_t=10.5$

Explanation of Solution

Given values −

s = 0.4

  $y_t=f(k_t)=3k_t^{0.5}$

n = 0.08

d = 0.1

The following equation will be used to represent the steady state value of the capital-labor ratio −

  $sf(k)=(n+d)k_t$ ……………………Equ (1)

Now, put the given values in above Equ (1) −

  $\begin{array}{l}0.3\times 3k_t^{0.5}=\left(0.08+0.1\right)k_t\\ \frac{0.9k_t^{0.5}}{0.18}=\frac{0.18}{0.18}{k}_t\end{array}$

  $5k_t^{0.5}=k_t$

Divide the both side of above Equ by $k_t^{0.5}$ -$\begin{array}{l}\frac{5k_t^{0.5}}{k_t^{0.5}}=\frac{k_t}{k_t^{0.5}}\\ 5=k_t^{0.5}\end{array}$

Square both sides −

  ${\left(k_t^{0.5}\right)}^2=5^2$

  $k_t=25$

The capital − labor ratio = $k_t=25$

The output per worker is calculated by using the following equation −

  $y_t=f(k_t)=3k_t^{0.5}$

Now, put the calculated value in above equation −

  $y_t=3{\left(25\right)}^{0.5}$

  $y_t=15$

Now, the steady state value for the output per worker = 15

The following equation will be use to represent the consumption per worker −

  $c_t=y_t-(n+d)k_t$ ……………………Equ (2)

Also given that −

Population growth rate, n = 0.08

Depreciation rate, d = 0.1,

Now, put the given and calculated values in Equ (2) −

  $c_t=15-(0.08+0.1)\times 25$

  $c_t=15-4.5=10.5$

The consumption per worker, $c_t=10.5$

Economics Concept Introduction

Introduction:

The ratio between capital and labor is known as capital-labor ratio and the ratio between the output and labor is known as output per worker.

d)

To determine

To describe: The steady state values of capital-labor ratio, output per worker and consumption per worker is to be calculated with following production function −

  $y_t=4k{0}_t^{0.5}$

Answer to Problem 5NP

The steady-state values are −

Capital-labor ratio = $k_t=64$

Output per worker = $y_t=32$

Consumption per worker = $c_t=22.4$

Explanation of Solution

Given values −

s = 0.3

  $y_t=f(k_t)=4k_t^{0.5}$

n = 0.05

d = 0.1

The following equation will be used to represent the steady state value of the capital-labor ratio −

  $sf(k)=(n+d)k_t$ ……………………Equ (1)

Now, put the given values in above Equ (1) −

  $\begin{array}{l}0.3\times 4k_t^{0.5}=\left(0.05+0.1\right)k_t\\ \frac{1.2k_t^{0.5}}{0.15}=\frac{0.15}{0.15}{k}_t\end{array}$

  $8k_t^{0.5}=k_t$

Divide the both side of above Equ by $k_t^{0.5}$ -$\begin{array}{l}\frac{8k_t^{0.5}}{k_t^{0.5}}=\frac{k_t}{k_t^{0.5}}\\ 8=k_t^{0.5}\end{array}$

Square both sides −

  ${\left(k_t^{0.5}\right)}^2=8^2$

  $k_t=64$

The capital-labor ratio = $k_t=64$

The output per worker is calculated by using the following equation −

  $y_t=f(k_t)=4k_t^{0.5}$

Now, put the calculated value in above equation −

  $y_t=4{\left(64\right)}^{0.5}$

  $y_t=32$

Now, the steady state value for the output per worker = 32

The following equation will be use to represent the consumption per worker −

  $c_t=y_t-(n+d)k_t$ ……………………Equ (2)

Also given that −

Population growth rate, n = 0.05

Depreciation rate, d = 0.1,

Now, put the given and calculated values in Equ (2) −

  $c_t=32-(0.05+0.1)\times 64$

  $c_t=32-9.6=22.4$

The consumption per worker, $c_t=22.4$

Economics Concept Introduction

Introduction: The ratio between capital and labor is known as capital-labor ratio and the ratio between the output and labor is known as output per worker.