Chapter 1.2, Problem 37P

The Solow model of economic growth (ignoring the effects of capital stock depreciation) is

$k\text{'}=\sigma f\left(k\right)-\left(n+g\right)k$, (i)

Where $k$ is capital stock per unit of effective labor, $f\left(k\right)$ is GDP per unit of effective labor, and $\sigma$, $0<\sigma <1$, is the fraction of gross domestic product (GDP) devoted to investment.

The parameters $n$ and $g$, growth rates of labor $L$ and technology A, respectively, appear in the equations

$\frac{dL}{dt}=nL,\frac{dA}{dt}=gA$.

The product $A\left(t\right)L\left(t\right)$ is referred to as effective labor and the output $Y$ of the economy is given by $Y=ALf\left(k\right)$. Assume that the production function $f\left(k\right)$ satisfies the following conditions:

(i) $f\left(0\right)=0,f\left(k\right)>0\text{for}\text{k>0}$,

(ii) $f\text{'}\left(k\right)>0,f\text{'}\text{'}\left(k\right)<0\text{for}\text{k>0}$,

(iii) $\underset{k\to 0}{\lim }f\text{'}\left(k\right)=\infty ,\underset{k\to \infty }{\lim }f\text{'}\left(k\right)=0$.

For example, the function $f\left(k\right)=c{k}^{\alpha }$ where $0<\alpha <1$ satisfies conditions (i), (ii), and (iii).

Draw a phase line diagram of Eq. (i) by sketching the graph of actual investment $\sigma f\left(k\right)$ and break-even investment $\left(n+g\right)k$ on the same set of coordinate axes. Show that Eq. (i) has an asymptotically stable equilibrium solution $k^{\ast }$.

When $k=k^{\ast }$, we say the Solow economy is on its balanced growth path. Show that when $k=k^{\ast }$, the output of the economy grows at the combined growth rates of labor and technology,

$\frac{dY}{dt}=\left(n+g\right)Y$.