### Exploring a Simple Neo-Classical Growth Model

4. One of the predictions of a very simple neoclassical growth model is that without technological change and with a constant savings rate, long-run economic growth is determined by population growth. There is, however, an advantage to a higher savings rate. It will give a higher standard of living.

To see this, suppose that total income is determined by the following production function:

Total Output = 1 * SQUAREROOT(Labor * Capital)
Assume also that:
Labor is constant at 400.
Each year 10% of capital depreciates (is used up).
Savings = Investment
Equilibrium exists when depreciation equals investment.

If capital is 100, the production function says that output will be:

Output = \$square root (100 * 400) = \$10 * 20 = \$200
Output per worker will be \$200/400 or \$.5
Depreciation will be .1 * \$100 = \$10
To be at equilibrium, savings must equal \$10, the amount of depreciation. What savings rate will do that?
\$10 = x* \$200
x = .05 or 5%

So a five percent savings rate will yield an equilibrium with zero growth and a per capita income of \$.5

Can you finish the table?

 Capital Output Output Per Worker Depreciation Equilibrium Savings Rate 100 \$200 \$.5 \$10 5% 400 \$400 \$1 40 10% 900 \$ \$ 90 15% 1600 \$ \$ \$ % 2500 \$ \$ \$ %

5. We can play a little more with this simple production function. It simply says that output is equal to the square root of labor times the square root of capital. But even with this simple equation, we can see some results.

In the early nineteenth century both Thomas Malthus and David Ricardo were pessimistic about population. Suppose we let capital stay constant at 400. Suppose also that if output per worker falls below one, then enough people will starve to stop population growth. Let the population start at 10. What will output per worker be? If population starts growing, how long will it grow before it must stop because output per worker drops to 1?

 Labor Capital Output Output Per Worker \$

Can you explain why this result has made many people, including some economists, argue that population control is needed to promote economic development?

Suppose, however, that the growth of technology, which is zero in this model, depends on the size of the population. This was a position argued by the late Julian Simon. When there are more people, there are more potential innovators. If this view is correct, does it undermine the Malthusian view?

What happens if both capital and labor increase? Find out by increasing each by the same percentage. What do you observe? (Comment: This production function has what is called constant returns to scale. With just tiny alterations it could have either increasing or decreasing returns to scale.)

6. The text mentioned human capital. It did not mention social capital, which some people both in economics and other social sciences think is important for growth. Can you discover what social capital is with an Internet search? How is it different from the cultural values discussed in the last section of this chapter?