Chapter 10, Problem 5NP

a

To determine

To calculate:Solow residual for year 2015 and 2016 and growth rate

Answer to Problem 5NP

The growth rate of Solow residual between 2015 and 2016 is $5.01\%$

Explanation of Solution

Given Information:

Capital stock = 30

Employed Labor = 100

For 2015:

Substituting values in above formula:

  $\text{Solow Residual = }\frac{100}{{(30)}^{0.3}{(100)}^{0.7}}$

  $=\frac{100}{(2.77)(25.12)}$

  $=1.437$

Therefore, the Solow residual in2015 is 1.437.

For 2016:

Y = 105

Capital stock = 30

Employed Labor = 100

  $\text{Solow Residual = }\frac{105}{{(30)}^{0.3}{(100)}^{0.7}}$

  $=\frac{105}{(2.77)(25.12)}$

  $=1.509$

Therefore, the Solow residual in the year 2016 is 1.509.

The growth rate of Solow Residual between the two years is calculated as follows:

  $\frac{{\text{Solow Residual}}_{2016}-{\text{Solow Residual}}_{2015}}{{\text{Solow Residual}}_{2015}}\times 100$

  $=\frac{1.509-1.437}{1.437}\times 100$

  $=5.01\%$

Therefore, the growth rate of Solow residual between 2015 and 2016 is $5.01\%$

Economics Concept Introduction

Introduction:

Solow residual is a numerical expression which shows growth of output due to factor inputs. It is usually calculated in chronological manner on yearly basis.

  $\text{Solow Residual = }\frac{Y}{K^a{N}^{1-a}}\mathrm{.......}(1)$

Y = Total output

K = Capital Stock

N= Labor employed

b)

To determine

Relationship between growth in 2015 and 2016.

Explanation of Solution

After including the utilization rate of the factor inputs for interpreting the Solow Residual, the formula for Solow Residual can be written as follows:

  $\text{Solow Residual}=A{u}_K{}^a{u}_N{}^{1-a}\mathrm{.......}(2)$

Where, $u_K$ is the utilization rate of capital and $u_N$ is the utilization rate of labor.

The Solow residuals, calculated in part (a), are the residuals without taking into consideration the utilization rate of the factors of input. Those residuals are now the value ofA(productivity) for the respective years. If the utilization rate of the factor inputs (capital and labor) remains unchanged during 2015-16, then there is no difference in the growth rate of productivities and the growth rate of Solow Residuals for the two years. This can be shown as follows:

  $\text{Growth rate in the Solow Residuals = }\frac{{(A{u}_K{}^a{u}_N{}^{1-a})}_{2013}-{(A{u}_K{}^a{u}_N{}^{1-a})}_{2012}}{{(A{u}_K{}^a{u}_N{}^{1-a})}_{2012}}\times 100$

Since, $A{u}_K{}^a{u}_N{}^{1-a}$ is same for both the years, equation (3) can be simplified as follows:

  $=\frac{u_K{}^a{u}_N{}^{1-a}\left[{(A)}_{2013}-{(A)}_{2012}\right]}{u_K{}^a{u}_N{}^{1-a}{(A)}_{2012}}\times 100$

  $=\text{Growth rate of the productivity (A) between the two years}$

Economics Concept Introduction

Introduction:

Steady state is a situation at which investment is equal to depreciation. It implies all the investment done is used to replace the depreciating capital.

c)

To determine

Relationship between growth in 2015 and 2016 when utilization of labor increases by 3%.

Explanation of Solution

Under the assumption that the utilization of labor ( $u_N$ ) increases by 3% between 2015 and 2016, the growth rate of Solow residual now differs from the growth rate of productivity between the two years. The growth rate of productivity is 5.01% (calculated in part (a)). The growth rate of Solow Residual can be calculated as follows:

  $\text{Growth rate = }\frac{{(A{u}_K{}^a{(u_N+0.03u_N)}^{1-a})}_{2013}-{(A{u}_K{}^a{u}_N{}^{1-a})}_{2012}}{{(A{u}_K{}^a{u}_N{}^{1-a})}_{2012}}\times 100$

  $=\frac{u_K{}^a{u}_N{}^{1-a}\left[{(1.03)}^{0.7}(1.509)-1.437\right]}{u_K{}^a{u}_N{}^{1-a}(1.437)}\times 100$

  $=\frac{(1.0209)(1.509)-1.437}{1.437}\times 100$

  $=7.2\%$

Therefore, the growth rate of Solow residuals is 7.2% after incorporating the increase in utilization rate of labor. The growth rate of Solow residuals is more than the productivity growth. This is so because there has been a growth in utilization rate of labor by 3%.

Economics Concept Introduction

Introduction:

Steady state is a situation at which investment is equal to depreciation. It implies all the investment done is used to replace the depreciating capital.

d)

To determine

Relationship between growth in 2015 and 2016 when utilizationof labor and capital is increased by 3%.

Explanation of Solution

If the utilization rates of labor and capital both increase by 3% between 2015 and 2016 then the growth rate of Solow Residual increases further. The growth rate can be calculated as follows:

  $\text{Growth rate = }\frac{{\left[A{(u_K+0.03u_K)}^a{(u_N+0.03u_N)}^{1-a}\right]}_{2013}-{(A{u}_K{}^a{u}_N{}^{1-a})}_{2012}}{{(A{u}_K{}^a{u}_N{}^{1-a})}_{2012}}\times 100$

  $=\frac{(1.0209)(1.0089)(1.509)-1.437}{1.437}\times 100$

  $=8.16\%$

Therefore, the growth rate of Solow Residuals is 8.16% after incorporating the increase in utilization rates of both labor and capital. We have a higher growth rate of Solow Residuals as compared to the growth rate of productivity. This is so because we observed an increase of 3% each in the utilization rate of labor and capital between the years 2015 and 2016.

Economics Concept Introduction

Introduction:

Steady state is a situation at which investment is equal to depreciation. It implies all the investment done is used to replace the depreciating capital.