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 𝑦 = 𝐴𝑘
   𝛼
   .
3. A country is described by the Solow model, with a production function of 𝑦 = 𝑘
   1/2
   . Suppose
   that 𝑘 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.
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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, 𝐴, and the same rate of depreciation, 𝛿. Assuming that the value of
   𝛼 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 𝛼 were 2/3?
5. In a country, output is produced with labor and physical capital. The production function in perworker terms is 𝑦 = 𝑘
   1/2
   . The depreciation rate is 1%. The investment rate (𝛾) is determined as
   follows:
   𝛾 = 0.10 𝑖𝑓 𝑦 ≤ 10
   𝛾 = 0.20 𝑖𝑓 𝑦 > 10
   Draw a diagram showing the steady state(s) of this model. Calculate the values of any steady
   state levels of 𝑘 and 𝑦. Also, indicate on the diagram and describe briefly in words how the
   levels of 𝑦 and 𝑘 behave outside of the steady state. Comment briefly on the stability of the steady state(s).

Sample Solution

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