The production function is Cobb-Douglas

1. Show mathematically that the Cobb-Douglas production function exhibits constant returns to
   scale.
2. Show that when the production function is Cobb-Douglas, output per worker ? = ??
   ?
   .
3. A country is described by the Solow model, with a production function of ? = ?
   1/2
   . Suppose
   that ? is equal to 900. The fraction of output invested is 30%. The depreciation rate is 2%. Is the
   country at its steady-state level of output per worker, above the steady state, or below the
   steady state? Show how you reached your conclusion.
   (More on next page)
4. In Country 1 the rate of investment is 6%, and in Country 2 it is 18%. The two countries have the
   same levels of productivity, ?, and the same rate of depreciation, ?. Assuming that the value of
   ? is 1/3, what is the ratio of steady-state output per worker in Country 1 to steady-state output
   per worker in Country 2? What would the ratio be if the value of ? were 2/3?
5. In a country, output is produced with labor and physical capital. The production function in perworker terms is ? = ?
   1/2
   . The depreciation rate is 1%. The investment rate (?) is determined as
   follows:
   ? = 0.10 ?? ? ≤ 10
   ? = 0.20 ?? ? > 10
   Draw a diagram showing the steady state(s) of this model. Calculate the values of any steady
   state levels of ? and ?. Also, indicate on the diagram and describe briefly in words how the
   levels of ? and ? behave outside of the steady state. Comment briefly on the stability of the steady state(s).

Sample Solution