Unformatted text preview: CHAPTER 4 CITY SIZE 1. Rank Size Rule Economists have observed a constant relationship between city size and its rank (within a country/region) Fir instance if the largest city has a population of 10 million, the second largest has 5m, the third largest has 3.33m and the fourth largest has 2.5m etc. In general, rank
population relationships more or less follow the rule: Ranki = C/Ni, where C is a constant (i.e., the population of the largest city) and N the population of the ith city. The “Rank
Size
Rule” is not normative/causal but is purely descriptive. 2. Utility and City Size € The growth of a city is associated with benefits and costs.
Agglomeration economies (I) increase labor productivity but at a decreasing rate
Agglomeration diseconomies (C) are caused by increasing commuting cost, crime rates, environmental problems. We assume the marginal cost to increase with city size. Over utility is defined as the sum of productivity induced income increases (I) minus agglomeration costs: U = I – C At the margin, that is for the last unit, we state ∂U ∂I ∂C
, e.g., marginal utility equals marginal income minus marginal cost (all =
−
∂N ∂N ∂N
with respect to city size N) ∂U ∂I ∂C
To maximize Utility we set marginal utility equal to zero =
−
= 0 , which ∂N ∂N ∂N
requires that marginal income and marginal costs are equal ∂I ∂C
MI = MC or =
€
∂N ∂N
The Figure below shows a graphical representation of utility maximization: € C,I I C Max Utility N The difference between I and C is maximized when the slopes of both curves are identical. The same relationship can also be expressed by a utility curve where U=I
C: U Max Utility S Sopt €
€ Mathematical Example: Assume income in a city is given by I = 20 S − 0.5⋅ S 2 and agglomeration cost is given by C = 2⋅ S 2 , where S stands for city size (i.e., its population). Overall utility is maximized where marginal income (MI) equals marginal cost (MC). 20 − S = 4 S ⇒ Sopt = 4 €
At a city size of 4, income will be 72, cost equals 32 and utility will reach its maximum, i.e., 40. €
Income
Cost
Marginal
Marginal
Utility
Population S I C Income Cost U=(IC) 19.5
38
55.5 2
8
18 19
18
17 4
8
12 17.5
30
37.5 4 72 32 16 16 40 5
6 87.5
102 50
72 15
14 20
24 37.5
30 1
2
3 Rents and Locational Equilibrium Within a city, rents adjust to guarantee a locational equilibrium where workers are indifferent where to live. Each location will provide the same utility Utility = Income
Cost – Rent Paid Commuting Income Cost Rent Paid Utility Distance (from labor) (e.g., commuting) (U=I
C
R) 100 0 80 20 0 miles 100 20 60 20 5 miles 100 40 40 20 10 miles 3. Systems of Cities Cities may be too large but not too small Assume a workforce of 6m people and three possibilities of how they can live. Option 1: 6 cities with a population of 1m each with a utility of $59/worker (S) Option 2: 3 cities with a population of 2m each with a utility of $79/worker (M) Option 3: 2 cities with a population of 3m each with a utility of $68/worker (L) We will observe the following dynamics: Option 1 is instable and won’t be an equilibrium solution. Suppose there are 6 cities (A,B,C,D,E) of 1m each. There is a strong incentive for a person to move, for instance, A to F, since utility per worker will increase in F. Since, on the other hand, it decreases in A there are more incentives to leave A. At the end of this process A will disappear. Similarly, B and C may disappear leaving only three cities with 2m people each. Since Option 1 is on the increasing branch of the utility curve (i.e., increasing agglomeration results in higher utility) the number of cities and their size will be driven toward the optimum (Option 2). Option 2 is a stable equilibrium. Suppose there are three cities (D,E,F). Any movement from one to the other will make one city larger and the other smaller. Either case will lead to decreasing utility and is not worthwhile. Option 3 is a stable equilibrium. Suppose there are two cities (E,F). Any movement from one to the other will make one city larger and the other smaller. Either case will lead to decreasing utility and is not worthwhile. Although Option 3 is suboptimal, it is a stable equilibrium. 4. Differences in Size One way to explain differing city sizes draws on differences in localization economies (see the Figure below). First, companies with localization economies form clusters. Depending on the extent of these external economies the clusters differ in size. This creates towns of different sizes (in the Figure below, ranging from 120 to 30 jobs) Second, companies that benefit from urbanization economies move toward the larger ones of these clusters making larger towns even larger. In the Figure this relates to an addition of 80 and 20 jobs in large and medium sized cities, respectively. Third, these export
oriented workers will be cause “local employment,” in sectors such as food, education, entertainment, banking or insurance. The ratio of local job per export job varies with the function of the city. Since larger clusters assume functions for smaller clusters (e.g., there cannot be a symphony orchestra or a university in each small town), their ratio is higher than the one for small towns. In the Figure below, there is one local job for every export job in small cities. This ratio is 1.5 for mid
sized cities and 2 for large cities. Different local job ratios further amplifiy the differences in city size. Merging these results with the idea of city size and utility from above yields the following Figure: Locational equilibrium requires that workers in are indifferent between living in one of the three cities. Thus, utility per worker (u*) is identical at each of the three cities. Note that all equilibria (s,m,b) are on the downward sloping branch of the respective utility curve and are thus stable outcomes. 5. Central Place Theory (CPT) Central Place Theory (CPT) was developed by the German geographer Walter
Christaller (1933) and the German economist August Lösch (1941)
Thrust of Central Place Theory:
 Are cities distributed randomly by size through the space economy or is there an
explainable pattern that underlies this distribution?
 A normative economic model that explains the pattern of the distribution of cities in the
spatial economy. Geometric Basics of CPT:
If consumers pay for transportation cost, the optimal market size is a circle that
minimizes the maximum distance to be traveled is a circle (as we know from above). However, circular markets leave some potential customers unserved: The next best shape that comes closest to being a circle and is space filling is the
hexagonal market area. The CPT combines geometrical ideas with different functionality levels of cities.
Each city size is assumed to serve different functions. Or in other words, the threshold
population for certain functions varies. While it takes only about a population of 250
people to sustain a mom & pop food store, a jewelry store requires a population of at least
800 people. Berry and Garrison (1958)1 computed threshold populations for various
businesses in Snohomish County, WA.
Function
Service Station
Church
Restaurant
Post Office
High School
Beauty Shop
Dry Cleaners
Liquor Store
Optometrist
Florist Threshold Population
25
300
307
358
710
788
875
1287
1800
2188 1 Berry, B. and Garrison, W. (1958). Functional bases of the central place hierarchy. Economic Geography, 34,145
154 Or in more general terms: Christaller’s system distinguished three principles.
(1) K=3 marketing principle The Central Place (cluster of highest order) serves it’s own market and one third of each
of the 6 adjacent markets. The term “K=3” denotes the number of markets served:
Markets served K = 1 + 6*(1/3) =3
(2) K=4 transportation principle
This principle serves a larger market area with the existing highway grid. Geometrically,
the hexagon of the central place is turned by 90 degree. Each central place now serves its
own market plus half of each adjacent market.
K = 1+ ½ + ½ + ½ +½ + ½ + ½ = 4 (3) K=7 administrative principle
The central place serves itself plus all adjacent cities: K = 1+ 1 + 1+ 1 + 1+ 1 + 1 = 7 However, the CPT assumes more than only two layers. For instance, the Figures below
show K=3 marketing principle for 3 and 4 layers:
K=3 principle with 3 layers K=3 principle with 4 layers ...
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 Spring '11
 Storchmann
 Central Place Theory, the00, a00

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