## Question:

Consider a numerical example using the Solow growth model. Suppose that $F\left(K,N\right)=z{K}^{0.4}{N}^{0.6}$, with $d=0.07$$s=0.2$$n=0.01$, and $z=20$, and take a period to be a year.

A) Determine capital per worker, income per capita, and consumption per capita in the steady state.

B) Now, suppose that the economy is initially in the steady state that you calculated in part A). Then, $s$ increases to $0.3$. Determine capital per worker in each of the $10$ years following the increase in the saving rate.

C) Determine capital per worker, income per capita, and consumption per capita at the new steady state.

D) Discuss your results; in particular comment on the speed of adjustment to the new steady state after the change in the savings rate, and the paths followed by capital per worker.

Under the steady state equilibrium, the production function is computed as a per worker production function. All the variables like capital per worker, consumption per worker, and income per capita are dependent on the capital stock in the economy. At the steady state equilibrium, the change in the capital stock is nil.

A) The per worker production function is:

$\begin{array}{rl}y& =\frac{Y}{N}\\ & =\frac{z{K}^{0.4}{N}^{0.6}}{N}\\ & =20{k}^{0.4}\end{array}$

$\begin{array}{rl}sy& =\left(d+n\right)k\\ 0.2y& =\left(0.07+0.01\right)k\\ 0.2\left(20{k}^{0.4}\right)& =0.08k\\ {k}^{0.4}& =0.02k\\ {k}^{0.6}& =50\\ k& =661.13\end{array}$

So, the capital per worker is: $k=661.13$

The income per capita is: $y=20{\left(661.13\right)}^{0.4}=268.62$

The consumption per capita is:

$\begin{array}{rl}c& =y-sy\\ & =268.62×0.8\\ & =214.89\end{array}$

B) The savings rate is 0.3. So, at steady state, the capital per worker is:

$\begin{array}{rl}sy& =\left(d+n\right)k\\ 0.3y& =\left(0.07+0.01\right)k\\ 0.3\left(20{k}^{0.4}\right)& =0.08k\\ 3.75{k}^{0.4}& =k\\ {k}^{0.6}& =75\\ k& =1295\end{array}$

The capital per worker for 10 years is:

C) The capital per worker is: $k=1295$

The income per capita is: $y=20{\left(1295\right)}^{0.4}=351.51$

The consumption per capita is:

$\begin{array}{rl}c& =y-sy\\ & =251.51×0.7\\ & =246.06\end{array}$

D) As seen from the table above, it will take a lot of time to reach the steady state equilibrium. It will take approximately 41 years.