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 P, : 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 P4 (we assume A is also chosen by the city) h: The height of the building in floors (we use a continuous measure), T'C : The total cost for a building of height A, given by the sum of fixed cost F (the land) and variable costs VC Py =100, P4 =50, VC = 0.5h2 1. 2. 3. Write down an expression for P that involves Pys, P4, and . What is the dollar value of P when A =0, A = 0.2, A = 0.4, and A =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 7 of the developer that involves revenues, variable costs (use expression above), and fixed costs for any given average price P and fixed cost F (b) 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 (c) Substitute in the optimal choice of A* into the profit expression in (a) to find the maximized profits 7*, which will depend on the average price P and fixed cost F (d) Demonstrate mathematically that the maximized profits 7* fall with the affordable share A and explain why this is sensible (hint: write P explicitly in the expression for 7*) (e) Assume that F' = 3,200. What will the developer's actual profits be for A=0, A=0.2, A=04, and A =17 Brooklyn The second potential development site is in Sunset Park, Brooklyn. This is a working class community with large populations from China, Mexico, the Middle East and more. It is a bit removed from the bustle of Manhattan and has limited transit options. As a result, prices are lower, at p = 30. Costs are lower also, with site acquisition and preparation giving rise to fixed costs of I' = 165 and a cost parameter @ = 2. This site also happens to be zoned at h=8. 9. Using the approach from before, calculate for the Sunset Park site the number of units that would be built and the profits associated with the benchmark free market choice, the affordable housing case, and the building as of right case (reflecting the zoning). Which approach does the developer choose? Queens Qur third potential development is in Long Island City, Queens. This is outside of Manhattan but still has good access to Midtown and some nice parks nearby along the East River, so price is p = 40. The site also has high fixed costs of F = 266. The land isn't expensive, but the site is on an old industrial location near Newtown Creek (a hazardous super-fund site) and so requires extensive clean-up before construction can begin. The marginal costs of building are low in this outer-borough, reflected in the parameter o = 2. Zoning here limits as of right building to h=5. 10. Using the approach from before, calculate for the Long Island City case the number of units that would be built and the profits associated with the benchmark free market choices, the affordable housing case, and the building ag of right case (reflecting the zoning). Which approach does the developer choose? 11. What do these cases teach us about the costs and benefits of the housing policies considered here? The city eould remove zoning entirely, It could allow building only according to the existing zoning, Or it could have the voluntary participation in affordable housing as here. How did these choices affect outcomes of total housing produced under the various regimes