A competitive industry is in long$-$run equilibrium. Market demand is linear, p $=$ a $$ bQ$,$ where

a $>0,$ b $>0,$ and Q is market output. Each firm in the industry has the same technology with cost

function, c$($q$)=$ k$2+$ q$2.$

$($a$)$ What is the long$-$run equilibrium price? $($Assume what is necessary of the parameters to ensure

that this is positive and less than a$.)$

$($b$)$ Suppose that the government imposes a per$-$unit tax, t $>0,$ on every producing firm in the industry. Describe what would happen in the long run to the number of firms in the industry. What is

the post$-$tax market equilibrium price? $($Again$,$ assume whatever is necessary to ensure that this

is positive and less than a$.)$

$($c$)$ Calculate the long$-$run effect of this tax on consumer surplus. Show that the loss in consumer

surplus from this tax exceeds the amount of tax revenue collected by the government in the

post$-$tax market equilibrium.

$($d$)$ Would a lump$-$sum tax, levied on producers and designed to raise the same amount of tax

revenue, be preferred by consumers? Justify your answer.

$($e$)$ State the conditions under which a lump$-$sum tax, levied on consumers and designed to raise the

same amount of revenue, would be preferred by consumers to either preceding form of tax.