A Solow Growth Model with Physical and Human Capitals. (45 points)

Let us consider a Solow growth model augmented with human capital accumulation. The

aggregate output/income Y is produced according to the following production function:

????" = ????????"

????"

( ????"

*++(

where A>0 stands for the productivity level, K stands for the aggregate physical capital, L

denotes the number of workers, H represents the aggregate stock of human capital and

a, ????(0,1) are parameters. Let us assume that the number of workers grows at a constant

rate n>0:

????".* = (1 + ????) ????"

The law of motions for the aggregate physical capital and aggregate human capital from time

t to time t+1 can be written as:

????".* = ????"

6 + (1 ????)????"

????".* = ????"

9 + (1 ????) ????"

where ????"

6 denotes the aggregate investment in physical capital, ????"

9 denotes the aggregate

investment in human capital and ????(0,1) represents the depreciation rate. Let us assume

that the aggregate consumption can be defined as follows:

????" = (1 ????)????"

where ???? denotes the marginal propensity to save. Let us assume that a faction ????(0,1) of the

aggregate saving is invested into physical capital and a faction 1 ???? is invested into human

capital.

a. Write-down the production function in per worker units. (5 points)

b. Write-down the goods' market equilibrium condition and show that it is equivalent to

the equality between aggregate saving and aggregate investment. (5 points)

2

c. Show that the physical capital per worker at time t+1 depends in equilibrium on the

output per worker at time t and the physical capital per worker at time t. (5 points)

d. Show that the human capital per worker at time t+1 depends in equilibrium on the

output per worker at time t and the human capital per worker at time t. (5 points)

e. Derive the steady-state physical capital per worker, the steady-state human capital per

worker and the steady-state output/income per worker. (15 points)

f. Compute the fraction of the saving invested in human capital: 1 ???? that maximizes

the steady-state income per worker. (10 points)