Question

Solow Growth Model: Suppose a fictional island called San Bruno in the Mediterranean Sea has a production function of Y=F(K,L)=K^0.5L^0.5.  Note that capital per worker is k=K/L and output per worker is y=Y/L. Suppose there is no population growth and no technological improvements in San Bruno. Assume initially L=100 and K=6400. Also assume that the savings rate in this economy is 24%.  There is not international trade and no government taxes or spending. Therefore, Investment equals Savings. I=sY. On a per worker basis i=sy.

a) Calculate the initial levels of k and y.

b) Suppose a massive earthquake strikes the island, decimating the buildings and equipment.  Suppose that K=2500.  Explain what immediate effect this would have on the yand k.

c) Suppose that before the earthquake, when K=6400, the stock of capital was at its steady-state level. Find the rate of depreciation of capital in San Bruno.

d) Calculate the per capita level of consumption at the pre-earthquake steady state and immediately after the earthquake.

e) Assuming an unchanged savings rate, what will happen to the economy over the long-run?  (Use a spreadsheet to show the progress of the economy as I showed you in class and as shown in your text.)

f) Assuming the depreciation rate you found in (c), now find the Golden Rule level of k. (Hint: First derive the MP_k). At the Golden rule what is the savings rate, total output, and total consumption?

g) Draw a Solow model diagram to illustrate the above problem. Label all axes, curves and key points.  Be sure to show the initial pre-earthquake steady state, the post- earthquake state and the Golden Rule steady state as well as the associated levels of output and investment.  (Hint: You will need investment curves for both the original savings rate and the golden rule savings rate.)

Answer 1

Answer:

Given

details of a Solow Growth Model.

production function of Y = F(K,L)=K^0.5L^0.5

capital per worker is k=K/L

output per worker is y=Y/L

initially L=100 and K=6400

savings rate in this economy is 24%

Investment equals Savings. I=sY. On a per worker basis i=sy.

(a) Calculating the initial levels of k and y is as follows:

The production function of the economy is given as

Y = F(K,L)=K^0.5L^0.5

Assuming K=6400 and L=100, we have

Y = (6400)^0.5 (100)^0.5

= 80*10

= 800

The capital per labor is

k= K/L

= 6400/100

= 64

The output per labor is

y = Y/L

= 800/100

= 8

(b) immediate effect this would have on the k and y is:

Assuming K=2500 and L=100, we have

Y = (2500)^0.5 (100)^0.5

= 50*10

= 500

The capital per labor is

k= K/L

=2500/100

= 25

The output per labor is

y = Y/L

= 500/100

= 5

Therefore , the immediate effect of earthquake will reduce per worker capital and labor.

(c) when K=6400 rate of depreciation rate of capital in San Bruno is:

At the steady state level

Here is the depreciation rate of capital.

Assuming s=0.4, y=5, k=25

Therefore,

(d) per capita level of consumption at the pre-earthquake steady state and immediately after the earthquake is:

The per capita level of consumption in steady state is

Assuming after earthquake y=4,s=0.4

(e) Assuming y₀=4, k₀=16, =0.08, s=0.4, we construct the following table:

k	y
16	4	0.32
16.32	4.039802	0.310321
16.63032	4.078029	0.300786
16.93111	4.114743	0.291409
17.22252	4.150002	0.2822
17.50471	4.183864	0.273168
17.77788	4.216383	0.264322
18.04221	4.247612	0.255668
18.29787	4.277601	0.247211
18.54508	4.3064	0.238953
18.78404	4.334056	0.230899

Here the following formulas are used to calculate k and y

(f)  The per worker production function is given as:

Therefore,

At the golden rule consumption

Therefore,

or,

or,

Therefore the golden rule capital is 39.06. Then golden rule output per labor is

the golden rule consumption is

The golden rule savings rate is

(g) The Solow model is given in the figure below:

The golden rule steady state diagram is given in the figure below: