A train station is located next to a large tract of land owned by a housing developer. The developer would like to build houses on this land, but noise from the train station reduces the value of land. The more trains come and go$,$ the lower is the amount of profit that the developer makes. Let $T$ be the number of trains that pass by the station per day and let $H$ be the number of houses that the developer builds. The train company's total profit is $60T-T^2,$ and the developer's total profits are $84H-H^2-TH.$ Below you need to figure out outcomes corresponding to different institutional structures.

a$.$ "Mind your own business": Suppose that no bargains can be struck between the train company and the developer and that each can decide on its own level of activity. How many trains pass by every day? Given that number of trains that pass by the station, what is the number of houses that maximizes the developer's profit? What is the profit of each company $(8$ points$)$

b$.$ "Not in my backyard": Suppose that a local ordinance makes it illegal for trains to use that track because they impose an externality on the developer. How many houses would the developer build? What is its profit? $(8$ points$)$

c$.$ "Pay for your damage": Suppose that a law is passed that makes the train company liable for all damages to the developer's property values. How many houses would the developer build? How many trains would pass by every day? What are companies profits? $(8$ points$)$

d$.$ "Merger": Suppose that the housing developer purchases the train company. How many houses the developer would build now? How many train would pass by$?$ What is its profit? $(8$ points$)$

e$.$ "Bargaining": Consider Part a outcome. Suppose the train company and the developer remain independent. Could the developer convince the train company to reduce the train traffic? If yes, how much the developer should pay the train company and how many trains per day should the developer ask to eliminate? $(8$ points$)$