Urban Economics
Problem Set #3
due Tuesday, Apr 5, in class
(1)
The third largest city of a country has a population of 2.5 million. Using the rank
size rule, what is the population of the 10
th
largest city?
Answer: The ranksize rule states that N
i
=C/R
i
, where N=population and R=rank
of the
i
th city. C is a constant, i.e., the population of the largest city.
If the third largest city has a population of 2.5m, the largest one has 7.5m. Thus,
the 10
th
largest city has a population of 0.75m.
(2)
Assume an urban cluster’s income (I) is given by the function I=4N, where N
denotes the cluster’s population in million. The cluster’s diseconomies of
aggregation (C) are given by C=0.5N
2
.
(a) Calculate the city’s utility maximizing population.
(b) At this population, how large is the resulting utility?
(c) Calculate the resulting utility if the population were on million higher and one
million lower than the optimum.
Answer:
(a)
if I=4N
MI=4
if C=0.5
N2
MC=N
MI=MC
N=4
(b)
at N=4
I=16 and C=8
U=IC=8
(c)
at N=5
I=20 and C=12.5
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 Spring '11
 Storchmann
 Economics, Utility, World population, Central Place Theory

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