Question

Consider the following numerical example using the Solow growth model. Suppose that

F (K, N) = zK1/2N1/2

Furthermore, assume that 5% of the capital is lost each period due to depreciation, the population grows by 1% each period, the consumer in this economy saves 20% of his income and the total factor productivity is z = 2. The unit period is one year.

1. Find the steady state per-capita quantity of capital (k*), production (y*) and consumption (c*). [5 pts]

2. Find the steady state quantity of capital per worker that maximize consumption per worker in this model. [4 pts]

3. Derive the golden rule steady state per-capita consumption (c**), production (y**) and saving (s**). [6 pts]

Answer 1

(1)

$F (K, N) = zK^{1/2}N^{1/2}$

Given

$\delta$ =5%

$n$ =1%

$s$ =20%

$z$ =2, $\alpha$=$\frac{1}{2}$

We know,

From steady state condition:

$k^*=$ steady state capital.

$=(\frac{sz}{n+\delta})^\frac{1}{1-\alpha}$

$=(\frac{0.2\times 2}{0.01+0.05})^\frac{1}{1-\frac{1}{2}}$

$=44.44$

$\therefore F(.)=z(k^*)^\alpha$

$=2\times(44.44)^\frac{1}{2}$

$y^*=13.33$

$c^*=y-s y=(1-s)y=0.8\times 13.33$

$=10.66$

(2)

From steady state condition:

$k^*=$ steady state capital.

$=(\frac{sz}{n+\delta})^\frac{1}{1-\alpha}$

$=(\frac{0.2\times 2}{0.01+0.05})^\frac{1}{1-\frac{1}{2}}$

$=44.44$

(3)

Golden rule

$k_{gr}=(\frac{\alpha}{\delta+n})^\frac{1}{1-\alpha}$

$=(\frac{\frac{1}{2}}{0.05+0.01})^\frac{1}{1-\frac{1}{2}}$

$=69.44$

$y^{**}=2(69.44)^\frac{1}{2}=16.66$

$c^{**}=16.66-(0.01+0.05)69.44$

$=12.49$

$s^{**}=y^{**}-c^{**}$

$=16.66-12.49$

$=4.17$

Savings rate

$=\frac{4.17}{16.66}\times100$%$=25.03$%