$(4$ points$)$ A representative profit$-$maximizing, price$-$taking firm uses a single input $x$ to produce output $Y.$ The price of each unit of $x$ is $$1.$ The firm can produce $0,8,$ or $16$ units of $Y$ by using $0,4,$ or $12$ units of $x,$ respectively. If there are no other possible input$-$output combinations, the production function of the firm is therefore given by:

$0,x=0$

$8,x=4$

$16,x=12$

a$.(1$ point$)$ Derive the firm's total cost function $C(Y).$

b$.(1$ point$)$ Derive the firm's supply function $y(p).$ Explain how you obtain it$.$

c$.(1$ point$)$ Suppose that there are $5$ such price$-$taking firms in the market, and there is no entry and exit of firms. Market demand is given by $Q^D=100-20p.$ Find the equilibrium price and quantity in the market.

d$.(1$ point$)$ Suppose now that instead of the $5$ firms in $($c$),$ there is actually a single monopolist that is five times as large as one of these firms. The monopolist's production function is therefore:

$0,x_M,=0$

$Y_M=f(x_M)=,40,x_M,=20$

$80,x_M,=60$

Assume there is no entry and exit of firms and market demand is given by $Q^D=100-20p.$ What is the equilibrium price and quantity in this market? Compare this outcome with the equilibrium in $($c$)$ and explain why they are the same or different.

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