consider the output function isgrvenbys-ArK74\".andcap talacumulationis givenby

Question

consider the output function isgrvenbys-ArK74".andcap talacumulationis givenby grows at 4%, Lt grows at 2% per year. Find out Suppose At grows at 196, = 0.5, Kt the growth rate of Y Does this production function imply the constant return on scale? Solve for the steady state level of capital. How does the steady state level depend on the savings rate (s) and depreciation of capital ( (try to explain by graphs) 1) 2) 3) 4) Compute the steady state level of capital, given -0.5, s-0.05, = 0.1, A-4, L-3.5

Explanation / Answer

1) At grows at 1% means that At+1 = (1 + 1%) At = 1.01 At

Kt grows at 4% means that Kt+1 = (1 + 4%) Kt = 1.04 Kt

Lt grows at 2% means that Lt+1 = (1 + 2%) Lt = 1.02 Lt

Yt+1 = At+1 (Kt+1)^0.5 (Lt+1)^(1-0.5)

Yt+1 = 1.01 At . (1.04 Kt)^0.5 . (1.02 Lt)^0.5

Therefore, Yt+1 / Yt = 1.01 . (1.04)^0.5 . (1.02)^0.5

Yt+1 = 1.04 Yt = (1 + 4%) Yt

So, the growth rate of Yt is 4%.

2) Yes, it is a constant return on scale, provided alpha is constant.

The return is not dependent on the time t.

3) The steady state level of capital is when the additional investment fetches no extra increase in output.

Yt = At . Kt^(0.5) . Lt^(0.5)

Yt+1 = 1.04 Yt

Kt+1 = 1.04 Kt

Some part s of the increased capital goes into savings.

So, the effective capital available is Kt+1e = (1.04 - s) Kt

Also, depreciation of capital means that the value of the capital keeps reducing after every year.

Ktval = Kt / (1 + delta)

K1val = K1 / (1 + delta) = (1.04 - s) K0 / (1 + delta)

K2val = K2 / (1 + delta) = (1.04 - s) K1 / (1 + delta)

K2val = (1.04 - s) (1.04 - s) K0 / (1 + delta) . (1 + delta)

K2val = K0 (1.04 - s)^2 / (1 + delta)^2

In general, Kt = K0 . (1.04 - s)^t / (1 + delta)^t

delta Yt = delta Ktval

Yt+1 - Yt = Ktval+1 - Ktval

1.04 Yt - Yt = K0 . (1.04 - s)^(t+1) / (1 + delta)^(t+1) - K0 . (1.04 - s)^t / (1 + delta)^t

1.04 Yt - Yt = K0 . (1.04 - s)^t / (1 + delta)^t . ((1.04 - s) / (1 + delta))

0.04 Yt = Kt ((1.04 - s) / (1 + delta))

4) Substitute the value of Yt

0.04 At . Kt^0.5 . Lt^0.5 = Kt ((1.04 - s) / (1 + delta))

0.04 . (3.5)^0.5 = Kt^0.5 ((1.04 - 0.05) / (1 + 0.1))

0.04 . 1.871 = Kt^0.5 (0.99/1.1)

Kt^0.5 = 12.03

Kt = 144.65 is the steady state level of capital

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