(3) A model of household location choice. We make the following ass...

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(3) A model of household location choice. We make the following assumptions: The city is located on a featureless plain; all land is identical and there are no (dis)amenities present. The city is circular, with all residents commuting from their household to the central business district (CBD) at x = 0. All job opportunities are located in the CBD. We refer to a city of this type as monocentric. There is a dense, radial transportation network such that any resident can commute directly from their household to the CBD at cost tx, where x represents the household's distance to the CBD (in miles) and t represents the roundtrip per-mile cost of transit. All residents earn the same income y and have identical preferences. Residents purchase a quantity q of housing (measured in square feet) at price p, where p = =p(x) is a function of x (distance from the CBD). Since prices vary with location, q will likely do so as well, i.e., q = q(x). Residents' remaining income after housing costs is used to purchase a composite (nu- meraire) good C. Residents' utility is given by u(c,q), with du(e.) 8c > 0, Du(c.a) 89 > 0, 8c- 0, 024(c) 892 0. (a) What is the economic interpretation of the assumption that 0 and Du(c)9) 0? 8c Be (b) What is the economic interpretation of the assumption that 024(99) 8c2 0, 024(c.) < 0? 8q² (c) What is the budget constraint for a resident of this city? (d) Rearrange this budget constraint so that it takes the form c = and substitute the right-hand side in for c in the utility function u(c,q) - this will give a utility function of the form u (c(q),q) where c itself does not appear. (e) Derive the first-order condition of this now-unconstrained one-variable optimization problem.¹ (f) Using the first-order condition, show that (1) What is the economic interpretation of this result? (Remember: The price of the numeraire good C is 1.) (g) Since all residents are identical and earn the same income, it must be the case in equilibrium that all obtain the same (constant) level of utility (which we denote ü), regardless of location x. Formally, this means that max u(c.q) = ü, so that u(c,q) = u c.q for equilibrium values of c and q. Using the same substitution approach from earlier, we can write (2) u (c(q), = ü Noting that both p and q will generally be functions of location x, totally differentiate both sides of this equation with respect to x and show that2 op ax q What is the sign of this derivative? What is the economic interpretation of this result? (h) Now, totally differentiate both sides of equation (2) with respect to income y (viewing P and q as functions of income) and show that What does this result mean?

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