Q$4.$ If the wage rate is $10$ per hour and the cost of capital is $20$ per unit: firm has

the following production function for its main product:

$Q=8L^{1\mathrm{.}5}{K}^{0\mathrm{.}5}$

where Q$=$ output $($millions units$),L=$ labour input $($hours$)$ and K $=$ capital input

$($units$).$

$($a$)$ Find the combination of labour and capital that minimises the cost of

producing $500$ units of output. What is the total cost of producing this level

of output?

$(40$ marks$)$

$($b$)$ If the both inputs are doubled, find the cost$-$minimising level of output and

the associated costs. What can be said about returns to scale? Comment

on the implications for the firm.

$($c$)$ Is each input subject to diminishing returns? Explain.