Question

Solow Model time paths

Sketch figures that describe the time-paths for {kt, Vt, cr, it^ in the following three cases (a) The economy is initially in steady-state. A shock to the economy destroys half of the capital (b) The economy is initially in steady-state. A shock to the economy increases the population growth stock rate permanently to n'> n. (c) The economy is initially in a steady-state with k<k*. The savings rate in the economy changes to s*

Answer 1

(a). The economy is in steady state. A shock to the economy destroys half the capital stock. The formula for stock is as:

$c = (1-s)A^{\frac{1}{1-\lambda }}(\frac{s}{n &plus; \delta })^{\frac{\lambda }{1 - \lambda }}$
A decrease in capital stock results in a shift in the curve (to the bottom).

c' =c/2 낫


(b). A shock to the economy increases population growth rate to n' > n

As n increases, from the equation, we can see that c decreases. If it decreases by c*, the graph will look like this:
(in effect, the graph is similar).


Je (n+6) Kt SAk 志 SAKI ト-A n十ク ,-ス kt


(c). If the savings rate changes to s*, there can be two possible solutions.
The equation plotted changes to $c = (1-s*)A^{\frac{1}{1-\lambda }}(\frac{s*}{n &plus; \delta })^{\frac{\lambda }{1 - \lambda }}$
Where, maximum c will be obtained when s = 1/2 (and is decreasing otherwise). If c increases, there is an upward shift in the graph, and if c decreases, there is a downward shift in the graph.

The amount of shift can be determined easily. Take a look at the graph above. At the equilibrium value of k, if the value of c increases, simply shift the curve by that amount upwards and make similar changes at other values of k.