Economia V; Instructor: Marco Del Negro

Problem set 5
Solutions

Page 214 Prob 2
1)

	20 Y ago	today	%change
Y	1000	1300	30
K	2500	3250	30
N	500	575	15

$$\frac{\Delta A}{A}=\frac{\Delta Y}{Y}-\alpha L \frac{\Delta N}{N}$$

$$=30\%-0.3(30\%)-0.7(15\%)=10.5$$

$\alpha K\frac{\Delta K}{K}$: capital growth contributed 9%
$\alpha L\frac{\Delta L}{L}$: labor growth contributed 10.5 $\frac{\Delta A}{A}$ (productivity growth) 10.5 b) $\frac{\Delta A}{A}$=30%-(0.5)(30%)-(0.5)(15%)=7.5%
$\alpha K\frac{\Delta K}{K}$: capital growth contributed 15%
$\alpha N\frac{\Delta K}{K}$: labor growth contributed 7.5%
2a) In the steady state:

K_t+1=K_t

$$\frac{K_{t+1}}{N_{t+1}}-K_{t}{N_{t}}=0$$

$$\frac{K_{t+1}}{N_{t+1}}=\frac{K_{t}}{N_{t}}$$

$$\frac{K_{t+1}}{N_{t+1}}=\frac{K_{t}}{N_{t}} \frac{K_{t+1}}{k_{t}}=\frac{N_{t+1}}{N_{t}}$$

$$\frac{K_{t+1}}{K_{t}}=(1+ \lambda )$$

$$\frac{K_{t+1}}{K_{t}}-1= \lambda$$

if $\lambda$ goes down, the growth rate goes down.
b) The steady state is:

$$k^*=(\frac{sA}{\delta + \lambda})^{\frac{1}{1- \alpha}}$$

if $\lambda$ falls k^* goes up. By the production function y goes up then c goes up. ) See figure.
) Golden Rule

$$c^*=Ak^*{}^\alpha-(\lambda + \delta)k^*$$

taking derivatives to maximize:

$$\alpha Ak^*{}^{\alpha -1}- \delta= \lambda$$

Since the solution is:

$$k_{gold}=(\frac{\alpha A}{\lambda + \delta})^{\frac{1}{1-\alpha}}$$

if $\lambda$ goes down k_gold goes up.
3a) The destruction of some of a country´s capital stock in war would have no effect on the steady state because there has been no change in s, f(k), $\lambda$ or $\delta$. Instead, k is reduced temporarily, but eventually returns to the old pre war steady state.
b) Immigration raise $\lambda$ from $\lambda_{1}$ to $\lambda_{2}$.
The rise in $\lambda$ lowers the steady state k, and lowers the steady state consumption per worker.
c) The rise in energy prices reduces the productivity of capital per worker. This causes sf(k) to shift down. Therefore a decline in the steady state consumption per worker falls for 2 reasons:
1) Each unit of capital has a lower productivity.
2) Steady state k is reduced.
d) A temporary rise in s has no effect on the steady state equilibrium.
e) The increase in the size of the labor force does not affect the growth rate of the labor force. So there is no impact on the steady state capital to labor ratio or on consumption per worker. However, because a larger fraction of the population is working, consumption per person increases.
4) Increase in the depreciation rate $\delta$. Since

$$k_{gold}=(\frac{\alpha A}{\lambda + \delta})^{\frac{1}{1-\alpha}}$$

If $\delta$ goes up, k_gold goes down as well as consumption.