QUESTION 2 (URBAN RESIDENTIAL LAND USE AND ZONING) Consider a linear city with housing: . Employment and consumption of non-housing goods take place at a single location r =0, the Central Business District (CBD). . Every resident commutes to the CBD everyday and gets an exogenous wage of y = 100. . In addition to non-housing goods, individuals living in the city consume housing. . Preferences are represented by a utility function U(c, q), where e is the consumption of non-housing goods and q is the consumption of housing. This function is assumed to be Cobb-Douglas: U(c, q) = ciqi (1) . Assume that the cost of commuting is strictly monetary and increases linearly with distance to the CBD, so that a resident living at distance x (in km) from the CBD incurs a commuting cost of TT. Assume that 7 = 10. . Land covered by the city is endogenously determined in the model and is represented by the segment on the positive real line between [0, I]- . Residents are assumed to be identical, with an exogenous daily utility level O deter- mined outside the model. This is an open city. Assume that 0 = 6 Let P(r) be the rental price of housing at a distance r from the CBD, the representative consumer's budget constraint is: y = tr+ P(x)q + e (2) 100 = 10r + P(x)q + c The consumer's problem is: Max cigs (3) subject to: 100 = 10r + P(x)q + e Note: Relative to the standard consumer problem studied in other economics courses, there are two main differences here. First, residents must choose their location of residence as well as allocate their disposable income optimally between housing and non-housing goods. Second, the price of housing, and thus the budget constraint they face, varies with their location choice.How do we solve this model? Step 1. Solve the optimal budget allocation between housing and non-housing goods at each location (for a given I). Step 2. Obtain housing prices by ensuring that, with each consumer allocating optimally his disposable income, utility is equalized across locations in the city and is equal to their outside option U = 6 (this is the free mobility condition). 2.1: No consumer substitution . For now, assume that housing consumption is fixed and equal to 1 for every resident. . P(r) can be interpreted as the rental price of one unit of land in the city. The only difference between houses is the distance to the CBD. . Assume that there is competing land use (agriculture) with rent equal to P = 14 (exogenous). 2.1.1) Solve for the residential bid-rent function. Tip: use the fact that utility is equalized across all locations in the city and equal to the outside option (i.e. U(c, 1) = 6 V x). 2 2.1.2) Show that rent is decreasing with distance from the CBD. 2.1.3) Solve for the equilibrium extent of the city, I. Tip: use the fact that, at the city limit, one unit of land is equal to agricutural land rent (ie. P(T) = P = 14). 2.1.4) Show that the equilibrium extent of the city increases when transportation cost de- creases. 2.1.5) Calculate aggregate land rent in the city. 2.1.6) Draw a graph to illustrate all quantities. 2.2: Consumer substitution . Now, we allow for consumer substitution. Residents must choose their location of residence as well as allocate their disposable income optimally between housing and non-housing goods. . In 2.2.1 and 2.2.2, you are asked to solve the optimal budget allocation between housing and non-housing goods at each location (for a given I). . In 2.2.3 and 2.2.4, you are asked to obtain house prices by ensuring that, with each consumer allocating optimally his disposable income, utility is equalized across locations in the city and is equal to their outside option U = 6 (free mobility condition).2.2: Consumer substitution . Now, we allow for consumer substitution. Residents must choose their location of residence as well as allocate their disposable income optimally between housing and non-housing goods. . In 2.2.1 and 2.2.2, you are asked to solve the optimal budget allocation between housing and non-housing goods at each location (for a given I). . In 2.2.3 and 2.2.4, you are asked to obtain house prices by ensuring that, with each consumer allocating optimally his disposable income, utility is equalized across locations in the city and is equal to their outside option ( = 6 (free mobility condition). 2.2.1) Write down the consumer's problem. 2.2.2) Using the first-order conditions, derive an expression for utility maximization stating that the marginal utility of more housing per amount spent should be equal to the marginal utility of non-housing consumption. 2.2.3) Write down the free mobility condition. 2.2.4) In equilibrium, the condition derived in 2.2.2 (utility maximization) and the free mo- bility condition must hold. Use these two equations together with the budget constraint to solve for housing prices as a function of r and exogenous parameters of the model. 2.2.5) True or False. Housing prices decrease with the distance from the CBD. 2.2.6) True or False. Houses are smaller the closer you get to the CBD. If you live downtown Montreal, this one should be obvious