$4.(40$ 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 $1.$

$($a$)(5$ points$)$ Let p denote the price of fish and $\backslash ($ I $\backslash )$ denote the income of a representative consumer. Find the representative consumer's demand function for $\backslash ($ q $\backslash )$ and $\backslash ($ y $\backslash ),$ and find the market demand function for $\backslash ($ q $\backslash ).$

$($b$)(5$ points$)$ The fishing industry consists of several firms. Each firm, to produce and sell anything at all, must get a boat and hire a market stall. The cost per period of owning a boat $($the interest on the money tied up$)$ is $3$ dollars, and the cost of hiring a market stall is $1$ dollar. In the long run, both of these costs are fixed. The variable cost of producing output $\backslash ($ q $\backslash )$ is $\backslash ($ q$^\{2\}\backslash ).$

Write down expressions for each firm's total cost $($TC$),$ average cost $($AC$),$ and marginal cost $($MC$),$ in each case as functions of $\backslash ($ q $\backslash ).$ Find the values of $\backslash ($ q $\backslash )$ that minimize AC$.$ What is the minimized $\backslash ($ A C $\backslash )?$ What is a representative firm's short run supply curve?

$($c$)(5$ points$)$ In the long run, there is free entry and exit of fishing firms. Given that new fishing firms have the same cost as existing firms, what is the price in a long$-$run equilibrium? What is the supply of a representative firm? What is the profit of each firm?

$($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' sarpluses? What is the total producer surplus?

$($f$)(5$ points$)$ Now suppose the govermment 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?