# The production function is Cobb-Douglas

Show mathematically that the Cobb-Douglas production function exhibits constant returns toscale.Show that when the production function is Cobb-Douglas, output per worker 𝑦 = 𝐴𝑘𝛼.A country is described by the Solow model, with a production function of 𝑦 = 𝑘1/2. Supposethat 𝑘 is equal to 900. The fraction of output invested is 30%. The depreciation rate is 2%. Is thecountry at its steady-state level of output per worker, above the steady state, or below thesteady 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 thesame levels of productivity, 𝐴, and the same rate of depreciation, 𝛿. Assuming that the value of𝛼 is 1/3, what is the ratio of steady-state output per worker in Country 1 to steady-state outputper worker in Country 2? What would the ratio be if the value of 𝛼 were 2/3?In a country, output is produced with labor and physical capital. The production function in perworker terms is 𝑦 = 𝑘1/2. The depreciation rate is 1%. The investment rate (𝛾) is determined asfollows:𝛾 = 0.10 𝑖𝑓 𝑦 ≤ 10𝛾 = 0.20 𝑖𝑓 𝑦 > 10Draw a diagram showing the steady state(s) of this model. Calculate the values of any steadystate levels of 𝑘 and 𝑦. Also, indicate on the diagram and describe briefly in words how thelevels of 𝑦 and 𝑘 behave outside of the steady state. Comment briefly on the stability of the steady state(s).

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