I have been given the following setup:

$$ Y=K^\theta (AL)^{1-\theta }$$

Where Y = Output, K = Capital, L = Labour and A = Productivity.

$$ \frac{\dot{L}}{L} = n $$ $$ \frac{\dot{A}}{A} = g $$

The Capital Accumulation Equation has also been given as:

$$ \dot{K} = sY - \delta K $$

Using this, I have calculated expressions for capital per worker and output per worker in the steady state as:

$$ \frac{K}{L} = A(\frac{s}{n+g+\delta })^{\frac{1}{1-\theta }} $$

$$ \frac{Y}{L} = A(\frac{s}{n+g+\delta })^{\frac{\theta }{1-\theta }} $$

The next questions asks: Calculate the growth rates of capital per worker and output per worker that will hold as the economy moves along a steady state growth path.

This is confusing because I was under the impression that growth of capital and labour is the steady state is 0, and then therefore growth of K/L and Y/L should also be 0. However this seems wrong and I believe the answer should be g. Can anyone help please?

  • 2
    $\begingroup$ Hint: Everything in the RHS of the bottom two equations is constant except A. $\endgroup$ – BKay Jan 12 '16 at 20:29
  • $\begingroup$ @BKay So because A grows at rate g, K/L and Y/L also grow at rate g? $\endgroup$ – James Baker Jan 12 '16 at 22:09

If $Y = C \cdot X$ where $C$ is constant and $\frac{\dot{X}}{X} = g$ then we can solve for $\frac{\dot{Y}}{Y}$ as follows: $$ \frac{d}{dt} Y = \frac{d}{dt} C \cdot X = C \cdot \frac{dX}{dt} = C \cdot \dot{X} \Rightarrow$$ $$ \frac{\dot{Y}}{Y} = \frac{C \cdot \dot{X}}{ C \cdot X} = \frac{\dot{X}}{X} = g$$

Therefore, as you concluded, they both grow at rate $g$.

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