15Propose a fractal whose dimension is log5.
2. Propose a fractal whose dimension is log6.
16 Consider a polycentric city with the population distributed according to this data set. The first row marks the longitude and the first column marks the latitude. 1 indicates a resident and 0 indicates vacancy. Demarcate the area into tracts1 to compute the population density D(m), and map out its level set. Which of the three polycentric formations introduced in class best describes the density in this city?
17 Consider a polycentric city with the population distributed according to this data set . It is tabulated in the same way as section 16. There are two subcenters located at m1 = ( 60, 0) and m2 = (60, 0), where the first entry is the longitude and the second is the latitude. Compute the population density D(m) and regress it on the distances from each subcenter as- suming the second form (complements) in class and estimate the population gradients a1 and a2.
1. Find land consumption sU (r) and sD(r) under each dis- tribution.
Endowments are uniformly distributed across the coun- try (rather than only at r = 0 as in the usual monocentric city model), namely, v(r) = v anywhere in the country. A consumer cannot relocate but she picks up her endowment v following the same commuting cost schedule assection 4.
Consumers have homogeneous preferences represented by
u(s(r), z(r)) = log(s(r)) + z(r), (2)
2. Denote the bid-rent function by pU (r) or pD(r) respec- tively. Find the equilibrium for each one of (1).
3. Between the equilibrium with nU (r) and nD(r), which one Pareto improves upon the other?
18 Find Nash equilibria (if any) for N = 3 in Hotelling’s model.
19 Consider a Hotelling’s model with two vendors introduced in class with a unit circle instead of a unit line segment. Consumers are evenly distributed along the circumference. Distance is measured byitem 2 insection 1. Two vendors can pick a location along the circumference. Find the Nash equilibrium.
20 Consider a closed linear monocentric city stretching from r = 0 to N (> 0) with land supply L(r) = 1 for any r. In line with the theme of this segment (interurban economics), interpret the segment [0, N ] as Canada itself rather than one particular city, with r = 0 being the west coast and r = N the east coast for example. Suppose that consumers are exogenously distributed according to either U (r) = 1,or log 4
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