Question

1. You are asked to analyze each of the following events using the Solow growth model

(the events all happen at time 0):

a) The investment rate rises in Tanzania.

b) Immigration increases the population of France by 10%.

c) An earthquake destroys 10% of the capital stock of Chile. (Hint: does steady state GDP percapita change in Chile?)

d) Malaysia realizes a 10% rise in TFP due to technology transfer.

For each of these:

Draw a Solow diagram to show what happens when the economy is initially in steady state.

Explain how steady-state GDP per capita changes.Use algebra to help in your explanation. Does

steady-state capital per capita change?

Explain how the growth rate of GDP per capita changes at time 0.

Explain how the economy adjusts from the short run to the long run after the change.

Answer 1

Consider the given problem here the production function is given by, “Y = A*K^a*L^1-a.

=> y = Ak^a, where “y=Y/L=output per capita” and “k=K/L= capital stock per capita”.

Now the change in the “k” is given by ‘dk = sy – (n+d)*k, where “n” be the population growth and “d” be the rate of depreciation. So, at the steady state equilibrium the change in “k” is zero.

=> dk = 0, => sy = (n+d)*k, => s*A*k^a = (n+d)*k, => s*A/(n+d) = k^1-a.

=> k^1-a = s*A/(n+d), => k = [s*A/(n+d)]^1/1-a. Now, the steady state output per worker is given by,

y = Ak^a = A*[s*A/(n+d)]^a/1-a., => y = A^1/1-a*[s/(n+d)]^a/1-a.

So, here we can see that “A” be the TFP parameter and is growing at the rate “g”, => “k” and “y” are growing at the rate “g/1-a” and Y=y*L, the output is growing at the rate “n+g/1-a”. Now, if “g=0”, => “k” and “y” are constant and “Y” is growing at the rate “n”.

a).

Now, as the investment rate in “Tanzania” increases by “10%”, => “s” increases by “10%”. So, as “s” increases, => “k” and “y” both increases as “s” is on the numerator for both of them. Consider the following fig.

So, in the above fig the initial investment rate was “s1” and “E1” be the initial the intersection of “s1*y” and “(n+d)*k”. So, the initial “k” and “y” is given by “k0” and “y0”. Now, as the savings rate increases to “s2” implied the investment per capital increases to “s2*y”, => the new equilibrium is “E2” where “k=k1 > k0” and the “y=y1 > y0”.

So, here we can see that the output and capital stock per capita both increases and both of them are growing at the rate “g/1-a” and the output is growing at the rate “n+g/1-a”. So, in the LR the “k” and “y” are growing at the rate “g/1-a”, as “s” increases implied the new equilibrium shift to “E2”, => this transition is SR movement and once the economy reaches to “E2” further starts growing at the rate “g/1-a”.

b).

Now, as the population of France increases by “10%”, => “n” increases by “10%”, => the break even investment increases implied the steady state “k” and “y” both decreases. Consider the following fig.

So, here “n1” be the initial population growth and “n2” post population growth, => the break even investment increases form “(n1+d)k” to “(n2+d)k”, => the “k” and “y” both decreases to “k1” and “y1” respectively. But the growth rate of “k”, “y” and “Y” remain same as before.

So, here the movement “E1” to “E2” be the SR transitive and once the economy reaches to “E2” it continue to grow at the rate “g/1-a”.

c).

Now, as the capital stock destroy,=> the level of “k” decreases less than the steady state level of “k”. Consider the following fig.

So, here we can see that “e” be the initial equilibrium where “change in k” is zero and “k0” and “y0” be the capital stock per worker and output per worker respectively. Now, because of the earth quake the “k” decreases to “k1 < k0” and the “y” decreases to “y1 < y0”. So, at “k=k1” investment per worker is more than the break even investment per worker, => “k” will increases to “k0” and “y” get back to “y0”, => the transition from “k1” to “k0” and “y1” to “y0” is the SR movement. In the LR the economy come back to “E” and is growing at the rate “g/1-a” and “Y” is growing at the rate “n+g/1-a”.

d).

Now, if the TFP increases by “10%”, => the production function per worker will shift up wards. Consider the following fig.

So, here “y=f(k)” be the production function and “E” be the initial equilibrium, => “k=k0” and “y=y0” be the initial capital stock per worker and output per worker. Now as “A” increases by “10%”, => the new production function is “y*=f(k)*”, => given the rate of investment the investment per worker curve will shift to “sy*”, => the new equilibrium is “E2” and the new “k” and “y” is “k1” and “y1”. SO, here the movement E1 to E2 be the SR transition. So, here “k” and “y” are constant and “Y” is growing at the rate “n”.