The economy has two sectors, manufacturing firms and research universities. The two sectors are described by the production functions

Y = K power 1/2[(1-u)LE] power 1/2 

????E = u E

where u is the fraction of labour force in universities (assume u is exogenous).

1)Write the equation of motion of capital, ????K = sY - ????K, in intensive form.

2)Write down the steady state condition and find the steady state  level of capital per effective worker.

3)Write down the Golden Rule and find the saving rate required to reach the Golden Rule steady state.

4)Suppose the economy is in a steady state and, due to government policies, the value of u increases. What happens to the steady state consumption per worker (compared to the old steady state consumption per worker)?

The steady state condition is -

sy = (?δ? + n + g)k

s (K ^1/2 [(1-u)LE] ^1/2 / L + E) = (?δ? + n + g)k*

Step-by-step explanation

Y = K ^1/2 [(1-u)LE] ^1/2

????E = u E

E is the efficiency of labor. As technology improves, the efficiency of labor increases (for example- with increase in education and skills).

From Y, ?α? = 0.5 and (1 - ?α? ) = 0.5.

Thus, n + g = α = 0.5

1) ????K = sY - ????K

Now, output per effective worker = y = Y/ L + E

= y = K ^1/2 [(1-u)LE] ^1/2 / L + E

capital per effective worker = k = K/ L + E

????K = sY - ????K

????K = sy - ????k (in intensive form)

2) The steady state condition is -

sy = (?δ? + n + g)k

s (K ^1/2 [(1-u)LE] ^1/2 / L + E) = (?δ? + n + g)k*

The level of capital per effective worker in steady state is:

sy = (?δ? + 0.5)k*

k* will be the level of capital per effective worker.

3) In a Cobb-Douglas production function in Solow model,

MPK = ?α? /(K/Y)

The Golden rule steady state condition is:

MPK = (n + g + ?δ?)

MPK = 0.5 + δ

4) Saving rate required to reach the golden rule steady-state is:

s = (n + g + ?δ?)(K/Y)

fro the initial saving rate of

s = (n + g + ?δ?)(k/y)

5) Increase in u will increase the efficiency of labor, E.

let's assume E becomes E* after increase in u.

sy = (?δ? + n + g)k

s (K ^1/2 [(1-u)LE*] ^1/2 / L + E*) = (?δ? + n + g)k*