$15$ points$)$ Suppose the production function is given by

Y $=$ zK$\backslash$alpha N $(1\backslash$alpha $)$

where N is labour inputs, K is capital, $\backslash$alpha is the factor share of capital,

and z is total factor productivity.

$($a$)$ What is the marginal product of labour, M PN $,$ of the firm?

$($b$)$ Solve the for firms problem for the equation describing the units of

labour the firm will demands for any given wage, w$.$ What is this

function called?

$($c$)$ Does the amount of labour the firm demands go up or down when

the wage goes up$?$ Prove it $($one way is to derive $$N d

$$w $).$

$($d$)$ Let w $=1,$ k $=1,\backslash$alpha $=0\mathrm{.}3,$ and z $=5.$ What is the amount of labour

demanded by the firm? Round to the nearest unit.

$($e$)$ No let w $=1,$ k $=2,\backslash$alpha $=0\mathrm{.}3,$ and z $=5.$ What is the amount of

labour demanded by the firm? Round to the nearest unit