4 An Affordable Housing Problem (20 Points) We consider a developer that wants to build a building. Notation is as follows : Py : This is the free market price of one of the floors P4 : The \"affordable price\" the developer is allowed to sell a floor at to winners of a city-run lottery (we assume this is chosen by the city) P : The average price of all floors sold (some at market, some affordable) A: Fraction of the floors sold at affordable price P, (we assume ) is also chosen by the city) h: The height of the building in floors (we use a continuous measure), TC : The total cost for a building of height h, given by the sum of fixed cost F' (the land) and variable costs VC Py = 100, Py = 50, VC = 0.5h2 1. 2. 3. Write down an expression for P that involves Pyg, Pa, and A, What is the dollar value of P when A =0, A = 0.2, A = 0.4, and )\\ = 17 We will now guide you through the developer's problem when she can choose height h freely for a given affordable share A (and so also given average price P for floors sold ) (a) Write down an explicit expression for profits = of the developer that (b (d (e ) involves revenues, variable costs (use expression above), and fixed costs for any given average price P and fixed cost F Use the expression from (a) to show how the developer chooses height h to maximize profits and then solve for an explicit expression for the optimal height the developer will choose, h*, which will be a function of the average price P and fixed cost F Substitute in the optimal choice of A* into the profit expression in (a) to find the maximized profits =*, which will depend on the average price P and fixed cost F' Demonstrate mathematically that the maximized profits =* fall with the affordable share A and explain why this is sensible (hint: write P explicitly in the expression for 7*) Assume that I = 3,200, What will the developer's actual profits be for A=0,A=02,A=04, and A =17 (f) If the developer will build so long as profits are non-negative (i.e. not strictly less than zero), what is the largest fraction of affordable units A the city can demand and get the developer to actually build the units? 4. Before the developer would agree to conclude the deal with the city, she commissioned one more market study to confirm the price at which she could sell the market rate units. To her dismay, she learned that the prospective market rate tenants would not be willing to pay the full $100 market price if they have to share an entry with the \"affordable\" tenants. For now we leave to the side what attitudes give rise to this and (for the purposes of this problem) assume the market study is accurate. The developer announces plans to implement a \"poor door\" a separate and less attractive entryway for the \"affordable\" tenants which the market study says will allow the developer to get the initially projected $100 for market rate units. The study says that if the entry is shared, the market rate they can get is only $87.50. The fixed cost remains F' = $3, 200, since while they have to add a door they can now use cheaper finishings. (a) Calculate the new average prices P if the new market rate without a poor door is Py, = $87.50 for A =0, A= 0.2, A =04, and A = 1. (b) The city is not happy. It finds the attitudes of market rate tenants toward the \"affordable\" tenants disturbing. But it has to take se- riously the possibility that the project won't go forward if it insists there be no poor door and keeps the share of affordable units required the same, with the consequence that no new affordable units will be produced (at least for now). It could tacitly ok the poor door. Or it could veto the poor door but reduce the fraction of affordable units required of the developer. If it did this last, how far would it have to reduce the fraction of units that are affordable? [Hint: This is not hard just look at the P' you have already calculated and plug them back into the 7* from before.|