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  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 ๐‘ฆ = ๐ด๐‘˜
    ๐›ผ
    .
  3. A country is described by the Solow model, with a production function of ๐‘ฆ = ๐‘˜
    1/2
    . Suppose
    that ๐‘˜ 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, ๐ด, and the same rate of depreciation, ๐›ฟ. Assuming that the value of
    ๐›ผ 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 ๐›ผ were 2/3?
  5. In a country, output is produced with labor and physical capital. The production function in perworker terms is ๐‘ฆ = ๐‘˜
    1/2
    . The depreciation rate is 1%. The investment rate (๐›พ) is determined as
    follows:
    ๐›พ = 0.10 ๐‘–๐‘“ ๐‘ฆ โ‰ค 10
    ๐›พ = 0.20 ๐‘–๐‘“ ๐‘ฆ > 10
    Draw a diagram showing the steady state(s) of this model. Calculate the values of any steady
    state levels of ๐‘˜ and ๐‘ฆ. Also, indicate on the diagram and describe briefly in words how the
    levels of ๐‘ฆ and ๐‘˜ behave outside of the steady state. Comment briefly on the stability of the steady state(s).

Sample Solution

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