Question

Consider the following growth model and answer the questions below.

(1) S=I

(2) S=sY

(3) Y=F(K,AL), λY=F(λK,λAL)

(4) K^'=I-δK 0<δ<1

(5) L^'=nL n>0

(6) A^'=gA g>0

Endogenous (6): , , , , ,  

Predetermined (3): K, L, A

Initial Conditions (3): K_0, L_0, A_0

Exogenous (5): n, g, s, δ

1. Show that equations (1)-(6) reduce to the two intensive form equations

y=f(k)

k^'=sy-[n+g+δ]k

where k=K/AL , y=Y/AL  

2. Show in a diagram how an increase in the rate of depreciation affects the steady-state levels for k and y.

3. Thinking about how an increase in the rate of population growth n would also impact the steady-state levels for k and y, comment only in words (no diagrams or equations) about how a society could offset the damaging impacts of an increase in the rate of population growth by being proactive and seeking to change the rate at which its tools depreciate.  

Answer 1

1. From equation (3), we get Y = F(K, AL)

Now, Y/ AL = F(K/AL, AL/AL)

=> y = f(k, 1)

=> y =f (k)

From equation (4 ), we get that, K' = I - K

Now, substituting I from (1) and (2) we get

K' = sY - K

=>k' = sy - k [Dividing bot side by AL]

Differentiating equation (5) and (6), we get n as the growth rate of labor, and g as the growth rate of technological progress.

this also affects the steady state of capital. Hence the above equation looks like

k' = sy -(n + g + )k

For steady state, k' = sf(k) - (n + g +)k

=> k* = s/(n + g + )

2.

The initial steady-state level of capital is k*, where the savings function intersects the depreciation of capital. Now if we consider an increase in the depreciation of capital the (n + g + )k curve shifts upwards and forces the steady state of capital to reduce at k*'. So, with an increase in the depreciation rate the steady state of capita decreases in an economy. The steady-state level of y also decreases.

3.

An increase in the rate of population growth has a similar effect on the steady-state of capital. The steady state of capital reduces, whereas the steady state level of y decreases. The use of more modern technological tools would help to reduce the depreciation in the steady state of capital, with an increase in the population growth rate.