$(10$ pts$).$ Suppose the production function in a small economy is: $Y=A{K}^{0\mathrm{.}8}{L}^{0\mathrm{.}2}.$

$($a$)$ Does this function exhibit constant returns to scale? Why or why not? Prove your answer.

$($b$)$ Derive the formulas for the marginal product of capital, MPK$,$ and the marginal product of labor, MPL$.$ Are they diminishing when $K$ and $L$ are increasing? Why or why not and how do you know it$?$

$($c$)$ If the economy's total output is $$1,000,$ the total factor productivity, $A=40,$ and labor supply is fixed at $\frac{?}{\text{b}\text{a}\text{r}}(L)=25,$ what are the long$-$run equilibrium real wage, $w,$ and the rental price of capital, $r_c?$

$w=$

$r_c=$

$2$