Does the Henry George Theorem provide a practical guide to optimal city size?

American Journal of Economics and Sociology, The,  Nov, 2004  by Richard Arnott

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The economy is centrally planned by a benevolent despot who decides how to allocate the economy's resources so as to maximize the common utility level of its residents. In particular, she must decide on the population of each city, as well as the allocation of output produced by the city's residents between consumption, transport service, and the local public good. In deciding on each city's population, she faces a tradeoff; in a more populous city, the cost of the local public good is divided among a larger population but at the same time average commuting distance is increased.

The following notation is employed:

N     population of a city
Y     per capita output
C     per capita consumption
P     units of the pure local public good
t     units of transport service required to move an individual a
      unit distance
x     distance from the CBD
b     distance of the urban boundary from the CBD
U     utility function
s(x)  shadow land rent at x
r(x)  market land rent at x

The planner chooses the geometry of cities so as to minimize aggregate commuting costs at every level of population. It is assumed that the optimal allocation entails no aggregate land scarcity, so that cities are circular. Furthermore, land in nonurban use is unutilized and therefore has no scarcity rent. For a city of population N, the resource constraint is

(1) NY = NC + [[integral].sup.b.sub.0](tx)2[pi]xdx + P.

This indicates that aggregate output of the generic good goes toward aggregate consumption, aggregate transport services, and units of the pure local public good. N residents require N units of land, which entails an urban radius of b(N) = [(N/[pi]).sup.1/2)]. Evaluating Equation (1) yields

(2) NY = NC + 2/3 [t[pi].sup.-1/2] [N.SUP.3/2] + P.

To simplify the algebra, let m = 2/3 [t[pi].sup.-1/2]. The planner chooses N, C, and P to maximize utility U = U(C, P), subject to the resource constraint (written for convenience in per capita terms). The corresponding Lagrangean is

(3) L = U(C, P) + [lambda](Y - C - [mN.sup.1/2] - P/N),

for which the first-order conditions are

(4) N:[lambda](- m/2 [N.sup.-1/2] + P/[N.sup.2]) = 0

(5) C:[U.sub.c] - [lambda] = 0

(6) P:[U.sub.p] - [lambda]/N = 0.

From Equation (4),

(7) P = m/2 [N.sup.3/2] = ATC/2,

where ATC is aggregate transport costs in the city. Thus, whatever the level of public good, optimal population is that for which expenditure on the public good equals one-half aggregate commuting costs.

The shadow rent on land at a distance x from the CBD is the resource saving from having an extra unit of land there. Moving an individual from the boundary of the city to the extra unit of land at x would result in a resource saving t(b - x). The shadow rent on land at the boundary of the city equals the shadow rent on land in nonurban use, which equals zero. Hence,

(8) s(x) = t(b - x).

Integrating s(x) over the area of the city gives aggregate shadow land rents (ASLR):

(9) ASLR = [[integral].sup.b.sub.0] s(x)2[pi]xdx

= m/2 [N.sup.3/2].