The cost of producing machine learning software depends on an individual$$s skill. In particular, if an individual is skilled, then his or her variable cost is c S $($q $)=12$ q $2$ and marginal cost c $$S $($q $)=$ q $.$ In contrast, if the individual is unskilled, then his or her variable cost function is c U $($q $)=2$q $2$ and marginal cost function c $$U $($q $)=4$q $.$ In addition, suppose that machine learning is highly specialized: skill in this specific industry does not translate well to other industries. Therefore, both skilled and unskilled individuals have the same fixed cost to enter the industry, f $=12.$ Therefore, the total cost function of a skilled individual is c$$ S $($q $)=12+12$ q $2,$ while the total cost function of an unskilled individual is c$$ U $($q $)=12+2$q $2.$

$($a$)$ What is the average cost function of a skilled individual? What is the average

cost function of an unskilled individual? $($Hint: Average cost is total cost divided by the output.$)$

$($b$)$ What is the minimum efficient scale for a skilled individual? What is the minimum

efficient scale for an unskilled individual? $($Hint: At the minimum efficient scale, average cost and marginal cost are equal$).$

$($c$)$ What is the lowest price, p S $,$ at which a skilled individual would enter? What is

the lowest price, p U $,$ at which an unskilled individual would enter? $($Hint: average cost is as small as possible at the minimum efficient scale.$)$

$($d$)$ Assume that there is a large $($infinite$)$ number of individuals of each skill level.

What quantity does the industry supply in the long$-$run if p $>$ min$\{$p S $,$ p U $\}?$ What quantity does the industry supply in the long$-$run if p $<$ min$\{$p S $,$ p U $\}?$ Assuming that demand is positive and finite at all prices, what is the long$-$run equilibrium price?

$($e$)$ Does the unskilled individual choose to enter the machine learning industry in the

long$-$run equilibrium? What about the skilled individual? Explain your answer in a sentence.