Question

3. If the production function is given by y=k:13 where yt where yt is the output per worker and kt is the capital per worker. Suppose that the saving rate (s) equals the depreciation rate (8) plus 0.4. Find the steady state capital, output, investment, and consumption? Hint: all steady state values are functions of either (s) or (8)

Answer 1

Given, production function, yt = kt^(1/3), s = $\delta$ + 0.4

The steady state of capital k* is defined at which capital stock, k is constant, so $\bigtriangleup$ k = 0

$\bigtriangleup$k = sf(k) - $\delta$ k

0 = sf(k) - $\delta$ k

sf(k) = $\delta$ k

s / $\delta$ = k* / f(k*)

($\delta$ + 0.4) /  $\delta$ = k* / k*^(1/3)

Therefore, k* ^(2/3) = ($\delta$ + 0.4) /  $\delta$

k* = (($\delta$ + 0.4) /  $\delta$ )^(3/2)

c* = f(k*) - $\delta$ k* = k*^(1/3) - $\delta$ k* = (($\delta$ + 0.4) /  $\delta$ )^(1/2) - $\delta$ (($\delta$ + 0.4) /  $\delta$ )^(3/2)

output, yt= f(k*) = (($\delta$ + 0.4) /  $\delta$ )^(1/2)

I* = $\delta$ k* / f(k*) = ($\delta$ + 0.4)