Question

Consider an economy such that Output per worker: yt = 2k0.5 Capital per worker: kt Depreciation rate: 8 = 0.4 Saving rate: s = 0.2 Evolution of capital per worker: kt+1- kt = syt - Skt In the steady state, capital per worker does not change over time Let k* denote the steady state level of capital per worker Question 1.In the steady state, capital per worker is a)8 b)4 c)2 d)1 Question 2.Which one of the following statements is correct? a) In the steady state, capital per worker doubles when the saving rate doubles. b) In the steady state, capital per worker doubles when the depreciation rate decreases by %50. c) Both a and b d) None of the above. Question 3.In the steady state, a) Capital per worker does not change over time b) Capital does not change over time c) Output does not change over time d) All of the above

Answer 1

Solution:

Q1. In the steady state, as already mentioned in the question that the capital per worker does not change. Thus, kt+1 = kt = k*

So, kt+1 - kt = 0

s*yt - d*kt = 0, where d is delta for depreciation rate

Using yt = 2*kt^0.5, we have s*(2*kt^0.5) = d*kt

We can write this as 2sk*^0.5 = dk* (as kt (= kt+1) = k* in steady state)

Substituting for s and d, 2*0.2*k*^0.5 = 0.4*k*

k*/k*^0.5 = 0.4/0.4

k*^0.5 = 1 or k* = 1

Thus, correct option is (d) 1.

Q2. In general terms, we can find the steady state level of capital per worker as:

k*/k*^0.5 = 2s/d

So, k*^0.5 = 2s/d

Or k* = (2s/d)² = 4s²/d²

Now, if saving rate doubles, k* = 4*(2s)²/d² = 4*(4s²/d²) so, capital per worker increases by more than double, it becomes 4 times (it quadruples).

Similarly, if depreciation rate decreases by 50%, so new depreciation rate is 0.5d, then

k* = (4s²/(0.5d)²) = 4*(4s²/d²) which is again 4 times (not 2 times)

So, correct option is (d) none of the above.

Q3. We already know that in steady state, capital per worker does not change. Since, yt = 2*kt^0.5, that is output per worker depends entirely on capital worker, if capital per worker is unchanged, even output per worker is unchanged.

Further, capital per worker = capital/number of workers, so if number of workers does not change (and capital per worker anyway does not change), capital will also not change in steady state.

With unchanged capital and unchanged number of workers, output will clearly be unchanged, as it depends on these two factors only.

Thus, correct option is (d) All of the above.