$11$:$26$

Q$1(45$ points$)$ An airport is located next to a large tract of land owned by a housing developer. The developer would like to build house on this land, but noise from the airport reduces the value of the land. The more planes that fly, the lower is the amount of profits that the developer makes. Let x be the number of planes that fly per day and let $y$ be the number of houses that the developer builds. The airport's total profits are, and the developer's total profits are Let us consider the outcome under various assumptions about institutional rules and about bargaining between the airport and developer.

$($a$)$ "Free to choose with no bargaining". Suppose that no bargains can be struck between the airport and the developer and that each can decide on its own level of activity. No matter how many houses the developer builds, what is the number of planes that maximizes profits for the airport? What is the profit of the developer? What is the total profit?

$($b$)$ "Strict prohibition". Suppose that a local ordinance makes it illegal to land planes at the airport because they impose an externality on the developer. Then no planes will fly. How many houses the developer will build and what is the total profit?

$($c$)$ "Lawyer's paradise". Suppose that a law is passed that makes the airport liable for all damages to the developer's property values. The airport's profits after it has paid damages will be $.$ The developer's profit after it receives the payment of damages will be Please calculate each agent's optimal choice and total profits under this situation.

$($d$)$ "The conglomerate" Suppose that a single firm bought the developer's land and the airport and managed both to maximize joint profits. Please calculate the total profits and the optimal planes and buildings under this situation.

$($e$)$ "Dealing". Suppose that the airport and the developer remain independent. If the original situation was one of "free to choose", could the developer increase his net profits by bribing the airport to cut back one flight per day if the developer has to pay for all of the airport's lost profits? To maximize his own net profits, how many flights per day should he pay the airport to eliminate?