The impact of a capital shock on Solow Model using a Cobb Douglas Production Function.

Suppose we have an economy described by the Solow growth model, with Cobb-Douglas production function (Y=F(K,AL) = K^α (AL)^1-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 saving rate s = 0.225 of output Y per year.

Suppose that the economy is initially on its balanced growth path, but that a major disruption to the economy suddenly destroys 75% of its capital stock.

i) How long before output per worker exceeds output per worker on the eve of the disruption?

ii) How long before output per worker returns to within 10% of its steady-state growth path level?

iii) How long before output per worker returns to within 1% of its steady-state growth path level?

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AL' = e^xt N while N' = N (1+n) ^t (3)

So, from (2) (3), we can get

Y'/ N' = (K' AL')^0.5/ N' = {[0.5s(K N) ^0.5 + 0.25 (1-d) K] e^xt N}^0.5/ N (1+n) ^t (4)

i) Y/N = Y'/ N' ...