Partial Answer Key

2) Physical and Human Capital as Perfect Complements

Consider two countries, Sri Lanka and Japan. Both countries have a production function per person where physical capital per person (k) and human capital per person (h) are the only two inputs in production. Specifically, the production function per person is:

Suppose Sri Lanka has a low rate of physical capital investment and a very low k. Japan on the other hand as a high rate of physical capital investment and a moderate amount of k. Both countries are the same in all other respects, INCLUDING VERY LARGE LEVELS OF HUMAN CAPITAL.

1. 
   Which country do you predict will grow faster in the short term? Why?
   

Japan. Because both countries have large h’s, we would expect physical capital per person to be the relatively scare factor, so only growth in k will allow growth in y. Thus in the short run, Japan should grow faster because its investment rate in physical capital is higher.
b. Which country do you predict will grow faster in the long term? Why?


Sri Lanka. Once Japan has large levels of both k and h, its growth stops -once k = h, further increases in k does nothing to increase y. Since Sri Lanka starts with such an imbalanced situation, where k <<< h, it can grow for a much longer period of time by increasing k though physical capital investment.
3) The Malthusian Model

Consider the Malthusian model, as described by Weil in section 4.1.

1. 
   Suppose that the economy is in steady state when suddenly there is a change in cultural attitudes toward parenthood. For a given income, people now want to have more children. Draw graphs showing the growth rates of population and income per capita over time.
   

At point a, the population size is stable with no growth. With a sudden change in cultural attitudes, the curve relating the population growth rate and income per capita shifts upward. The sudden shift implies that population growth will suddenly be positive (point b). As the population size grows and as income per capita falls, the growth rate of population will fall. At point c, the country will be in a Malthusian steady-state population level with no growth. Income per capita is now at a permanently lower level than before.

2. 
   Let’s look at a Malthusian model with some actual functions. Let’s say that the growth rate of population is given by:
   

and suppose that output is produced using labor and land, according to the equation:

where X is the quantity of land. Assume that X = 1,000,000. Derive the relationship between population, L, and income per capita, y (remember that y ≡ Y/L). Use the equations you have derived to compute the STEADY-STATE values of L and y.

First, we divide both sides of the production function by L and rearrange to get:





Therefore,



In the steady state, the growth rate of population, , is zero. Using this value and rearranging the first equation, we solve for the steady-state value of income per capita:





Substituting in this value into the production function, we back out the value of L^ss as follows:





The steady-state population is 100.

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