Suppose the production function for widgets is given by

$q=kl-0\mathrm{.}8k^2-0\mathrm{.}2l^2,$

where $q$ represents the annual quantity of widgets produced, $k$ represents annual capital input, and $l$ represents annual labor input.

a$.$ Suppose $k=10$; graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point?

b$.$ Again assuming that $k=10,$ graph the $M{P}_l$ curve. At what level of labor input does $M{P}_l=0?$

c$.$ Suppose capital inputs were increased to $k=20.$ How would your answers to parts $($a$)$ and $($b$)$ change?

d$.$ Does the widget production function exhibit constant, increasing, or decreasing returns to scale?