$4.(40$ points$)$ Consider the fish industry on an island. There are $160$ consumers on the island, each with the utility function $\backslash ($ U$($q$,$ y$)=$y$+10$ q$-5$ q$^\{2\}\backslash ),$ where $\backslash ($ q $\backslash )$ is the consumption of fish and $\backslash ($ y $\backslash )$ is a composite good measuring the total expenditure on consumption of all other goods. Thus, the price of good $\backslash ($ y $\backslash )$ is $1.$

$($d$)(5$ points$)$ With free entry and exit of fishing firms, what is the industry's long run supply curve? Putting together the market demand curve you found in part $($a$)$ and the industry supply curve, what is the equilibrium quantity in the long run? How many firms operate in this equilibrium?

$($e$)(5$ points$)$ What is the total of all consumers' surpluses? What is the total producer surplus?

$($f$)(5$ points$)$ Now suppose the government levies a tax of $2\mathrm{.}5$ dollars per unit of fish. Find the new short$-$run equilibrium with the same number of firms as in the original long run equilibrium.

What is the price paid by the consumers? What is the price received by the firms? What is the equilibrium quantality?

$($g$)(5$ points$)$ In the new short$-$run equilibrium with the same number of firms as in the original long run equilibrium. How much tax revenue does the government collect? What is the profit of each firm? Will any firms want to exit in the short run?

$($h$)(5$ points$)$ With free entry and exit of fishing firms, the market will reach a new long run equilibrium, what is the price paid by the consumers? What is the price received by the firms? How many firms are active? What is the profit of each firm? What is the aggregate loss of consumer surplus?