Solow model - question

[L14]

Given the Cobb-douglas production function
Y=(K)^0.5 * (N)^0.5, consider the economy is in its steady state with saving 10%. The depreciation rate is 10%. If it increases the saving rate to 20%, how long does it take to reach its new steady state(assuming no technological progress and no increase in labour force)?

[kangkan]

Since labour force doesnt change, we assume N=1 .thus k=K

f(k)=rootover (k)

old steady state 0.1* rootover(k)=0.1*k

hence k=1

New steady state

0.2*rootover k=0.1*k

=> k*(new steady state)= 4

Since N=1(by assumption)

K(t+1)=0.2*rootover K(t)-0.1*K(t)

We have to find t such that K(t)=K(t+1)=4.( K(0)=1)

This is a difference equation....put in values for t till you find k(t)=4...i guess there is some compliated formula to solve non linear difference equation,but i dont knw it

[SoniaKapoor]

@kangkan i didnt get how u got t=4....please can u elaborate

[kangkan]

Hi Sonal,the depreciation rate is d=0.1

Accd to the solow formula at steady state s*f(k)=(n+d)*k.

In our case f(k)=rootver k and n=0

Hence it becomes s* rootover(k)=d*k
=> 0.2*rootover k=0.1*k
=>4k=k^2
=>k=4.

Now for the time t, we know that k(t+1)= I(t)-d*k(t)+k(t)
=>k(t+1)=s*y(t)-d*k(t)+k(t) where s=.2 d=0.1 y(t)= rootover(k)

since N=1 by assumption
we can replace k=K

Run it it excel as manually calculating it will take a lot of time

I ran it the equation excel..the answer comes to year 310.

Regards