$1.$ A monopolist's total costs are given by $\backslash ($ c$($q$)=20+10$ q$+$q$^\{2\}\backslash )$ and she faces the demand curve $\backslash ($ q$=200-2$ p $\backslash ).$

a$.$ What output will the monopolist sell, and at what price?

b$.$ Calculate the monopolist's profits.

c$.$ What output level, if produced, would maximize social surplus?

d$.$ Calculate the deadweight loss due to this monopoly.

e$.$ At the output chosen by the monopolist

i$.$ What is the price elasticity of demand?

ii$.$ What is the marginal revenue?

f$.$ Suppose instead that the demand curve was $\backslash ($ q$=5-\backslash$frac$\{$p$\}\{8\}\backslash )$

i$.$ Verify that the industry is a natural monopoly $($demand intersects $\backslash ($ A C$($q$)\backslash )$ while $\backslash ($ A C$($q$)\backslash )$ is falling$).$

ii$.$ What output will the monopolist sell?

iii. What is the efficient output level $($i$.$e$.$ social surplus maximizing$)?$

iv$.$ If we forced the monopolist to produce the efficient output level, how much subsidy would we need to provide the monopolist so they break even?