Question

pls solve parts d, e, f

Suppose Country X's production function is given by F(K, AN) = 206,05(A, N.)05 where K, is the capital and A, N, is the effective worker. The evolution of the capital stock is given by K+1 = 0.74K, +1, where the depreciation rate is 26%. Additionally, the saving rate is 36%, the population growth rate is 4% and the technological growth rate is 10%. (a) Derive and show that in the Solow growth model, the steady state capital per effective worker needs to satisfy: 0.40(KIAN) = 7.2(KIAN)05 (b) Calculate the steady-state capital per effective worker (K/AN)', the steady-state output per effective worker (Y/AN)', the steady-state consumption per effective worker (CIAN)". (c) Using graphs and your values from part (b), show the steady-state values of output per effective worker, capital per effective worker and consumption per effective worker. Suppose that Country X permanently changes its saving rate to achieve the Golden rule of capital and output per effective worker. (d) Compute the saving rate that would maximize the steady-state consumption per effective worker. (e) Using graphs and your values from part (d) show the new steady-state values of output per effective worker, capital per effective worker and consumption per effective worker, and compare them to the old steady state values. (1) Consider the path from the old steady state to the new one. Compute the value of the output per effective worker and capital per effective worker in the first period after the change in the saving rate.

Answer 1

(a) Solution:

For a Cobb-Dauglas type production function Q= aKbLb-1

the steady-state according to solows growth model we can found

(1 + g)k = (1 - d) k + sakb

where, g= population growth rate, d= depreciation rate, s= saving rate, a= coefficient in the function, b= is the capital share. k is the capital labour ratio.

Thus in the problem it has given that the production function is

Q = 20Kt0.5 (AtNt)0.5

It has also given that, depreciation rate (d) is 26 %, saving rate (s) is 36%, population growth rate(g) is 4%, and technological growth rate is 10 %.

Thus the steady state position according to solow growth model will be

( 1 + g + t) (K/AN) = (1 - d) (K/AN) + s20 (K/AN)0.5

SInce it has given that , g= 0.04, s= 0.36, d= 0.26, and t= 0.10 (technological growth rate)

The stady state will be

( 1+ 0.04 + 0.10) (K/AN) = (1 - 0.26) (K/AN) + 0.36 x 20 (K/AN)0.5

  1.14 (k/AN) = 0.74 (K/AN) + 7.2 (K/AN)0.5

  1.14 (k/AN) - 0.74 (K/AN) = 7.2 (K/AN)0.5

0.40 (K/AN) = 7.2 (K/AN)0.5

Thus it is proved.

(b) Answer: The steady state capital per worker can be obtained as

0.40 (K/AN) = 7.2 (K/AN)0.5

(K/AN) / (K/AN)0.5 = 7.2 / 0.40

(K/AN)0.5 = 18

{(K/AN)0.5}2 = 182

(K/AN) = 324

Output per effective worker can be obtained as

Y/AN = 20 (K/AN)0.5

= 20 (324)0.5

= 20 x 18

= 360

Consumption per effective worker can be obtained as

C/AN = (1 - s) (Y/AN)

= (1 - 0.36) x 360 {given that s= 0.36}

= (0.64) x 360

= 230.4