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- Show mathematically that the Cobb-Douglas production function exhibits constant returns to
scale. - Show that when the production function is Cobb-Douglas, output per worker ๐ฆ = ๐ด๐
๐ผ
. - A country is described by the Solow model, with a production function of ๐ฆ = 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) - 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, ๐ด, and the same rate of depreciation, ๐ฟ. Assuming that the value of
๐ผ 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 ๐ผ were 2/3? - In a country, output is produced with labor and physical capital. The production function in perworker terms is ๐ฆ = ๐
1/2
. The depreciation rate is 1%. The investment rate (๐พ) is determined as
follows:
๐พ = 0.10 ๐๐ ๐ฆ โค 10
๐พ = 0.20 ๐๐ ๐ฆ > 10
Draw a diagram showing the steady state(s) of this model. Calculate the values of any steady
state levels of ๐ and ๐ฆ. Also, indicate on the diagram and describe briefly in words how the
levels of ๐ฆ and ๐ behave outside of the steady state. Comment briefly on the stability of the steady state(s).
Sample Solution
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