# An Economy with a Cobb-Douglas Production Function

Suppose we have an economy described by the Solow growth model, with a Cobb-Douglas production function (Y=F(K,AL) = K^Î±(AL)^-Î±), 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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#### Solution Preview

i) After a slowdown, k tends to a constant k*, y tends to a constant y*

k* = [s/(d+n+x)]^[1/(1-α)] = [0.225/(0.045+0.01+0)]^[1/(1-0.5)] = 16.74

y* = [s/(d+n+x)]^[ ...

#### Solution Summary

The expert examines an economy with a Cobb-Douglas production function. The paths over time after the slowdown are given. The solution answers the question(s) below.