Question

Suppose that the production function is given by 𝑦=0.5√𝐾√𝐿

a) Derive the steady-state levels of output per worker and capital per worker in terms of the saving rate, s, and the depreciation rate, δ.

b) Derive the equation for steady-state output per worker and steady-state consumption per worker in terms of s and δ.

Answer 1

a)Given  Y=0.5√𝐾√𝐿

$\therefore \frac{y}{L}=\frac{0.5\sqrt K\sqrt L}{L}=\frac{0.5\sqrt K}{\sqrt L}$

$y=\frac{Y}{L}=0.5\sqrt \frac{K}{L}$

where y=output per worker and K=capital per worker.

Steady state occurs when $\Delta K=0=sy-\delta K$

$0.5s\sqrt K=\delta K=K=\frac{0.5s}{\delta}$

$K=\frac{0.25s^2}{\delta}$ ..............steady state level of capital per worker.

$y=0.5\sqrt K=\frac{0.25s}{\delta}$.................. steady state level of output per worker

b)

$y=c+i$ where c=consumption per worker

i=investment per worker=$sy=\frac{0.25s^2}{\delta}$

$c=y-i=\frac{0.25s}{\delta}-\frac{0.25s^2}{\delta}$

$c=\frac{0.25s}{\delta}(1-s)$..................... steady state level of consumption per worker