Question

Consider an economy such that Output per worker: yt = 2k£.5 Capital per worker: kt Saving rate: s = 0.2 Depreciation rate: 8 = 0.15 Workforce growth rate: gN = 0.25 sy-(8+g)k¢ Evolution of capital per worker: kt+1- kt = 1+gN In the steady state, capital per worker does not change over time. Let k* denote the steady state level of capital per worker In the steady state, capital per worker is а) 8 b) 4 c) 2 d) 1 Which one of the following statements is correct? In the steady state, capital per worker is higher when workforce grows at a faster rate a) b) In the steady state, capital grows at a faster rate when workforce grows at a faster rate. Both a and b c) None of the above d) 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 + g)*kt)/(1 + g) = 0, where d is delta for depreciation rate, ang g is workforce growth rate (= g_N)

So, s*yt - (d + g)*kt = 0

Or s*yt = (d + g)*kt

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

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

Substituting for s and d, 2*0.2*k*^0.5 = (0.15 + 0.25)*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 + g)

So, k*^0.5 = 2s/(d+ g)

Or k* = (2s/(d + g))² = 4s²/(d + g)²

Notice that there exists an inverse relationship between g and k*, that is, a negative relation between the workforce growth rate and capital per worker in steady state. This clearly means that if workforce grows at faster rare, that is, if the workforce growth rate is higher (value of g is higher), k* will be lower.

So, capital per worker will indeed be lower (not higher)

Further, since capital = capital per worker*number of workers

Growth rate of capital = growth rate of capital per worker + growth rate of workers

Growth rate of capital = 0 + g = g

Growth rate of capital per worker is 0 in steady state as it does not change. So capital is positively related with workforce.

Thus, correct option is (b).

Q3. We already know that in steady state, capital per worker does not change.

From above part, we also saw that capital per worker changes by rate g, so capital does change overtime.

If capital and labor, both change overtime, so will the output. Thus, correct option here is (a) capital per worker does not change overtime.