Question

A closed economy has two factors of production: capital and labor. The production function is known to exhibit constant returns to scale. The capital stock is about 4 times one yearís real GDP. Approximately 8% of GDP is used to replace depreciating capital. Labor income is 70% of real GDP. Real GDP grows at an average rate of 4% per year. Assume the economy is at a steady state. (a) Is the capital per e§ective worker lower or larger than it would have been in the golden rule steady state? [To receive points on this question, you need to show me your calculations] (b) [growth accounting] If the population grows at a rate 1% per year, Önd what portion of output growth is due to: (i) an increase in capital; (ii) an increase in labor; (iii) an increase in total factor productivity.

Answer 1

A).

Consider the given problem here there are two factor of production, => “L” and “K” and labor income is 70% of the GDP, => the production function is given by.

=> Y = K^0.3*(AL)^0.7, where “AL” be the numbers of effective workers. Now, approximately 8% of the GDP is used to replace the depreciating capital, => dK = 0.08*Y, where “d” be the depreciation rate, => d = 0.08*(Y/K).

The capital stock is about “4 times” of GDP, => K=4*Y, => Y/K = ¼”.

=> d = 0.08*(Y/K) = 0.08*0.25 = 0.02 = 2%. So, the depreciation rate is “2%”.

Now, the production function can be written as, “y=k^0.3”, where “y=Y/AL” and “k=K/AL”.

Now, from “K=4Y”, => K/AL = 4*(Y/AL), => k=4y, => k = 4*k^0.3, => k^0.7 = 4.

=> k = 4^(1/0.7) = 7.25 = steady state level of capital per effective worker. Now, the GDP is growing at the rate 4%, => the growth rate of “AL”, => n+g = 4%.

Now, the at the golden rule level of “k” the following condition mist hold.

=> MPk = (n+g+d), => 0.3*k^(-0.7) = 0.04+0.02 = 0.06, => k^0.7 = 0.3/0.06 = 5, => k=9.97.

So, we can see that the golden rule level of “k” is more than the steady state level of “k”.

B).

Here let’s assume the production function is given by, “Y = A*K^0.3*L^0.7, => the growth accounting equation is given by.

=> g(Y) = 0.3*g(K) + 0.7*g(L) + g(A).

Now, we have given that the population growth is “1%”, => n=0.01=g(L). Now, at the steady state equilibrium “k=K/L=capital per worker” is constant, => K=k*L, => “K” is growing at the rate “L”, => g(K)=g(L)=1%.

=> g(Y) = 0.3*g(K) + 0.7*g(L) + g(A), => g(A) = g(Y) - 0.3*g(K) - 0.7*g(L).

=> g(A) = 4% - 0.3*1% - 0.7*1% = 3%. So, “0.3%” of the output growth is due to increase in “K”, “0.7%” of the output growth is due to increase in “K” and “3%” of the output growth is due to increase in “A”.