Part II Affordable Housing Project Choice (30 Points) A developer is considering residential projects in three of the five boroughs of New York City. The projects differ in many of the fundamentals, have some common regulatory elements, and the developer has to decide how (or if) to acquire and develop these sites. Revenues in a location depend on the price p that can be for each apartment at a site and the number of units h that the developer provides at that site. The cost side of producing units has a cost structure that is common with fixed costs I and variable costs %hz. Note that the parameters mentioned above (p, hya, F) all vary with the project, given the distinctive characteristics at each site. The developer is aiming to maximize profit at each site. As a \"free market\" benchmark, we can calculate what the developer would choose if she could develop at each site without any constraints. But this is not allowed. At each site, the developer is offered (but not required) to participate in an affordable housing program. That program requires that one-fourth of the units be offered for sale at an 80 percent discount, while the remaining units can be sold at market rates. If the developer enters the affordable housing program, then the developer can choose the number of units h with no further restrictions. If the developer declines to participate in the affordable housing program, then the developer can sell all units at the market price, but is restricted by zoning to build a maximum number of units A \"as of right\" that may vary by site. If the developer builds as of right, no units have to be sold at the affordable price. Before we turn to examining the projects, we want to do some exercises that will help in examining each of them. FOR MAXIMUM CREDIT, SHOW YOUR WORK! 1. Write down an expression using the information from above that shows the profits of the developer when facing no other constraints. 2. In this free market problem, the developer's choice variable is the height of the building h, which is also the number of units. Find the optimal unconstrained h#, solving for it explicitly in terms of the other parameters of the problem (so it will end up in the form hx = ...). 3. Substitute h#* back into the expression for profits to show the maximized level of profits m* that can be attained when h is chosen optimally. This should be of the form 7+ = ..., where the right hand side only involves exogenous parameters. Keep these expressions for h* and 7w+ handy because they will be helpful in solving other elements of the problem. Manhattan. The first site is in Manhattan, on the Upper East Side near First Avenue and 65th Street. This is not among the most chic neighborhoods on the Upper East Side, but its proximity to Midtown and wide variety of restaurants and bars makes it a desirable neighborhood able to command high prices at p = 60. Costs are also high. Land is expensive and an existing tenement will have to be torn down and removed, giving rise to fixed costs of F' = 284, Workers who come into Manhattan expect to earn a premium wage and there are other difficulties in delivering materials and such, so the parameter in the marginal costs for this site is @ = 3. Zoning in this neighborhood allows a developer there to build as of right to h = 8, but a developer who wants to build higher will need to participate in the affordable housing program. The developer makes whichever choice is more profitable (in case of a tie, assume they do the affordable housing). Recall that at the Manhattan site, p = 60, F' = 284, a = 3, and h = 8. 4. In our benchmark unconstrained case, how many units h does the developer build and what are the associated profits? Hint: You should be able to use the expressions you already derived above for h* and m* to answer this. 5. For the Manhattan site, consider the case of participation in the affordable housing program. Noting in that case that three quarters of the units can be sold at the market price and one-quarter at an 80 percent discount, what would be the average price the developer receives per unit under the affordable housing program? 6. Recall that the developer in the affordable housing program has height restrictions removed. Using the average price from above for this developer, what will be the number of units h the developer will build if she participates in the affordable housing program and what will be the associated profits? 7. The last option open to the developer is to build \"as of right\" not higher than the zoning restriction, which at this site is h = 8. If the developer isn't going to participate in the affordable housing program, but instead build as of right, will she want to build the full 8 stories? What are the profits from making this choice? (Hint: To calculate profits here, you can't plug into the former expression for 7* because that was premised on the idea that height was an unconstrained choice. Instead, return to your expression for profits, enter the Manhattan site parameters along with h = h to calculate these.) 8. You are the financial genius running the developers' projects. How should the developer build on this site (if at all)? Which case would have led to the most housing construction? The least