Cobb-Douglas

Suppose we have an economy described by the Solow growth model, with a Cobb-Douglas production function (Y=F(K,AL) = K^α(AL)^-α), a capital share of 0.5; with population, labor-augmenting productivity growth, and depreciation rates given by n =0.01 per year, x = 0.02 per year, and depreciation = 0.045 per year; and with a savings rate s = 0.225 of output Y per year.

Suppose that x suddenly and permanently falls from 2% per year to 0% per year.

i) Calculate the paths over time after the slowdown of k, the ratio of capital to effective labor, of y, the ratio of output to effective labor, of K/Y, the capital-output ratio, and of Y/L, output per worker.

ii) How do these paths compare to the paths had the slowdown in productivity growth not occurred?

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...-α)] = [0.225/(0.045+0.01+0)]^[0.5/(1-0.5)] = 4.09
K/Y = (K/AL)/(Y/AL) = k* / y* = 16.74 / 4.09 = 4.09

Since AL = (e^x) L, L = AL / (e^x) = AL / (e^0) = AL

Then Y/L = ...