Regional
Science and Urban
Economics
OPTIMAL
16 (1986) 241268.
LOCATION
OF PUBLIC
Area Dominance Masahisa
NorthHolland
FACILITIES
Approach
FUJITA*
University of Pennsylvania, Philadelphia, PA 19104, USA Received
June 1984, final version
received
May 1985
This paper examines the optimal location of public facilities under the influence of the land market. Our approach is based on the concept of areadominance, which is a spatial analogy to the concept of stochasticdominance. This concept enables us to conduct a pairwise comparison of possible locations for a public facility. For a certain class of community shapes, it also determines the optimal location of the public facility independently of the utility function, tax rate or externality level.
1. Introduction In the literature on local public goods in urban economics, communities are usually treated as points in space. Thus, the problem of where to locate public facilities within a community does not arise. In urban land use models, public goods are usually treated as pure public goods [e.g., Barr, (1972), Arnott (1979)], or assumed to be located continuously over the city [e.g., Helpman, Pines and Borukhov (1976), Kanemoto (1980)]. In practice, however, many important public goods come from facilities located at discrete points (e.g., schools, hospitals and parks). Although the problem of locating public facilities has received considerable attention, most of the existing works concern applied facility location models.’ The location of public facilities generally affects the distribution of land rent in the community. In the long run, it will also affect the distribution of households in the community. Hence, in determining the optimal location of public facilities, the interaction between the land market and the location of public facilities must be considered. Sakashita (1986) presents an insightful analysis of the optimal location of (pointpattern) public facilities under the influence of the land market. *This research has been supported by NSF grant SES 8502886 (U.S.A.) which is gratefully acknowledged. The author is also grateful to Hiroyuki Koide, Tony E. Smith and David E. Wilson for their valuable comments. ‘For recent works on public facility location, see Lea (1981), Thisse and Zoller (1983) and Johansson and Leonardi (1986). 016f&O462/86/$3.50
0
1986, Elsevier Science Publishers
B.V. (NorthHolland)
M. Fujita, Optimal location of puhlicfacilities
242
One of Sakashita’s conclusions concerns a linear community with a single public facility. For this case, he shows the equilibrium utility of households to be maximum when the facility that provides a positive service locates at the center. Similarly, the equilibrium utility of households is maximized when the facility that provides a negative service locates at an edge. Since this conclusion is intuitively so clear, Sakashita called it a Columbus’ egg. This paper extends Sakashita’s seminal work in a number of ways. First, we show that Sakashita’s basic conclusion holds under a more general utility function. Second, we consider both households and landlords in order to determine the socially optimal, or Paretooptimal, location of the public facility. And, we show that the same conclusion holds for the Paretooptimal location. Finally, and most importantly, we extend Sakashita’s work to twodimensional communities. Our approach is based on the concept of areadominance, which is a spatial analogy to the concept of stochasticdominance. The concept of areadominance enables us to conduct a pairwise comparison of possible locations for a public facility. For a certain class of community shapes, it also determines the optimal location of the public facility independently of the utility function, tax rate or externality level. Section 2 discusses the set of assumptions used to describe the residential choice behavior of households under the influence of a public facility. Section 3 defines the bid rent function and bidmax lot size of households. Section 4 considers the case of a closed community under absentee land ownership. Here, we examine the competitive equilibrium land use and compensated equilibrium land use of the community under a given location of the public facility. We also define the Paretooptimal location of the facility, as the location which maximizes the net revenue subject to a utility constraint. Section 5 generally defines the center and edges of a community with an arbitrary shape. The definition is a purely geometric one based on the concept of areadominance. In section 6, we show that if the center and edges exist, then both utility and net revenue are maximum when the public facility locates at the center (in the case of a positive service) or at an edge (in the case of a negative service). In section 7, we show that for a class of symmetric community shapes, the center and edges can be identified by using the concept of symmetricdominance. Finally, in section 8, we discuss future research directions. 2. Residential
choice of households
Imagine a residential community with a fixed domain, D. Let us consider the case of the closed community model, in which M identical households reside and land is owned by absentee landlords.’ The community govern‘Possible section 8.
extensions
such
as public
land
ownership
and
open
community
are discussed
in
M. Fujita, Optimal location of public facilities
243
ment is planning to construct a public facility of some size at some location in its domain, which will provide an externalitylike service to the residents. Suppose the government has constructed the facility of size K at some location x E D, and imposes a lumpsum tax G on each household. If G < 0, it is in the form of an income subsidy. To examine the longrun equilibrium land use pattern of the community, we assume that given this action of the government, each household optimally chooses its location and the lot size for its house in the competitive market so as to maximize its utility subject to a budget constraint. Let r be the distance from the public facility. Then, the amount of service, E(r), received from the facility by a household at distance r can be expressed as follows:3 E(r) = f(r; K, M).
(2.1)
Since the community area is homogeneous except for the influence of the public facility, the utility function of each household can be expressed as U(z, s, E(r)), where z represents the amount of composite consumer good, and s the lot size of the house. The composite good is chosen as the numkuire, so its price is 1. The income for each household is represented by Y If a household locates at distance r from the facility, the budget constraint is given by z + R(r)s= YG, where R(r) is the unit land rent at r. Thus, we can express the residential choice of each household as max U(z, s, E(r)), r,z,s
subject
to
z + R(r)s = YG.
(2.2)
Without loss of generality, we can assume that O
and
Assumption I (wellbehaved utility function). The utility function is defined for each positive combination of z, s, and E, and is continuously differentiable. Under each value of E> 0, function U( ., ., E) is strictly quasiconcave and strictly increasing, and indifference curves in the zs space do not cut the axes. Assumption 2 (normality of land). Under each fixed value income effect on the ordinary demand for land is positive.
of E> 0, the
With respect to the service provided by the facility, we say that if aU/aE is positive, E(r) represents a positive service; if aU/aE is negative, a negative ‘Generally, a public facility may provide more than one kind of service or externalities, and hence E(r) could be a vector. For simplicity, however, we assume here that E(r) is a single number representing the composite effects of the facility.
244
kf. [email protected]
Optimd kxation ofpublicfacilities
service. When the service from the public facility is a delivered good such as service from a tire station or noise from an airport, expression (2.1) can be accepted as is. However, in the case of a travelledfor good such as a library, school or exercise field, we must consider that E(r) represents the optimal level of the service chosen by a household at r, and U(z, s,E(r)) represents a derived utility function.4 The following example demonstrates this point. Example 1. Suppose the public facility means a library residential choice behavior of each household is given as
of size K. The
max U(z, s, n, t, 4) = a log z + /3log s + y log g(K, M)n(t  a)h + 6 log t,
r,z,s,n,f,fL
subject to
z + R(r)s = Y  G,
and
(2.3)
t, + n(t(r) + t) = f,
where n is the number of trips to the library (per unit time), t the time spent at the library per trip, tl the rest of leisure time, t(r) the trip time, f the total leisure time, and a is a minimumstaylength. All coefficients a, B, y, 6 and a are positive, and 1 > h > 0. Define the derived utility function by U(z,s,r)=max{U(z,s,n,~,t,)(t,+n(t(r)+t)=f}. &f,f( Solving this, lh
[email protected]+
f
t=a+W)
Ih’
and hence, we obtain U(z, s, r) = a log z + b log s + y log E(r),
(2.4) E(r) =g(K, M)c/(a+ t(r))lph,
where c= fil+a’y)(l h)‘hhh(6/y)d’y(l +6/y) function, (2.3) can be restated as (2.2).
(1ts/y). With this derived utility
3. Bid rent function and bidmax lot size In the context of model (2.2), we define the bid rent function of a household as follows: Y(r, u, G) s Y(r, u, G; E(r)) =max
Y  G  Z(s, u, E(r))
Q
4Lea (1981) is credited
with the distinction
between
S
delivered
(3.1)
2
and travelledfor
goods.
245
M. Fujita, Optimal location of public facilities
YG
0
s(r,u,G)
Fig. 1. Bid rent Y(r, U, G) and bidmax
s
lot size S(r, u, G)
where Z(s, u, E(r)) is the solution of U(z,s,E(r))=u for z. By definition, Y(r,u, G) represents the maximum rent per unit land a household would be able to pay for residing at distance r while enjoying a fixed level of utility u and paying tax G. When we solve the maximization problem of (3.1) we obtain the optimal lot size, S(r, u, G) cS(r,u, G; E(r)), which is called the bidmax lot size.’ Graphically, as depicted in fig. 1, bid rent Y(r, u, G) is given by the slope of the budget line which is just tangent to indifference curve u under given E(r), and bidmax lot size S(r,u, G) is determined from the tangency point B. Example 2.
Consider the following loglinear utility function:
U(z, s, E(r)) = c1log z + /I logs + y log E(r),
cr,B>O,
cr+/3=1.
(3.2)
If y is positive (negative), E represents a positive (negative) service. Since Z(s 3u>E(r)) =sB’aE(r)yiae”/a 3 solving (3.1) we have Y(r, u, G) = [email protected]/?( Y  G) liflE(r)aiBe “‘B,
(3.3)
Under Assumptions 1 and 2, it is not difficult to confirm that
av/au< 0,
alulac < 0,
(3.5)
5Note that because of Assumption 1, whenever it exists, the optimal s for the maximization problem of (3.1) is positive. When there is no solution for this maximization problem, we define Y (r, U, G) = 0 and S(r, u, G) = co.
246
M. Fujita, Optimal location
as/au > 0,
of public facilities
aslaG> 0.
(3.6)
That is, bid rent Y is decreasing in both u and G, and bidmax lot size S is increasing in both u and G. In order to examine the effects of r on functions Y and S, we introduce the next two assumptions: Assumption 3 (declining service level).
E(r)
is
continuous
in
r,
and
E’(r) = dE(r)/dr < 0 for all r. Assumption 4 (substitutability between land and environment). (i) If au/aE > 0 (positive service), as/aE< 0, (ii) if au/BE< 0 (negative service), aS/aE> 0,where S = S(r, u, G: E(r)). Assumption 3 is based on empirical observations. Assumption 4 means that in order to sustain a constant utility level, a smaller (larger) lot size is necessary under a better (worse) environment. This assumption is also empirically reasonable.6 Observe that function S in Example 2 satisfies this assumption. Then, under Assumptions 1, 3 , and 4, applying the envelope theorem to (3.1), we have
aul
= ar
igE’(r)$O
au
as
dE $0,
as
 20. aE
as as
au
z=zE’(r)gO
(3.7)
(358)
That is, in the case of a positive (negative) service, bid rent Y decreases (increases) in r and bidmax lot size S increases (decreases) in r. 4. Equilibrium
land use given a public facility location
We consider the community domain D as a set of points on a twodimensional Euclidean space R ‘. For mathematical convenience, we assume as follows: 6Let .$(YG,R,E) be the ordinary demand function for land, and S(u,R,E) the compensated demand function for land. Then, S(r, u, G; E(r)) 3 s*(Y  G, Yl(r, u, G; E(r)), E(r)) s:S(u, Y(r, u, G; E(r)), E(r)). Hence, as
a9 8~
__=+_=+_. aE aRaE
as*
a3ay
as
dE
aRaE
aE
In the case of a positive service, for example, aY/aE is positive. Hence, a sufficient condition for negative aS/aE is to assume that a3/ldR
M. Fujita,
Optimal
location
Fig. 2. Definition
247
~j’puhlic,fhcilitie.~
of L(r. x).
D is a closure Assumption 5 (welldefined community domain). open connected subset of R2, and it is of definite area.’ Let d(x, y) be the Euclidian each x E R2, define
distance
between
two points
of a bounded
x, y E R2. And,
for
(4.1)
r(x) = sup 4x, Y), Y~D
which gives the maximum distance from x to some point in the community.8 Let L(r,x) be the amount of land available in the community at distance r from point x E D, i.e., L(r,x) is the length of the portion of the circle with radius r at center x which lies on the community (refer to fig. 2). Then under Assumption 5, for each x E D, L(r, x) > 0
for
0 < r < r(x),
for
r>r(x).
(4.2) =0 Detine A(r,x)=jL(t,x)dt,
(4.3)
0
which gives the amount of land available in the community from point x9 Let A(D) be the total area of the community. A(r. .u) is strictly increasing in r< r(x), and A(D)=A(r,x)
for
all r_Lr(x).
within radius r Then, from (4.2)
(4.4)
‘The purpose of requiring D to be a closure of an open subset of RZ is to remove all ‘hairlike’ fringes. D is of definite area if, for example, the boundary curve of D is a connection of a finite number of smooth curves. “Since D is a compact set, in the subsequent analysis, the supremum (intimum) is the same with the maximum (minimum) which always exists as long as the function in question is continuous on D. ‘Throughout this paper, l denotes the sign of a Riemann integral. Under Assumption 5. ,4(r. Y) is welldefined for any r?O, XED.
M. Fujita, Optimal location of public facilities
248
In the subsequent analysis, facility size K and population M are always fixed; hence, the service level function E(r) is also fixed. It is assumed that land not occupied by households is used for agriculture, yielding a constant rent R,>O. For simplicity, it is also assumed that a public facility does not require any land. First, we examine the competitive equilibrium pattern of land use under a given tax rate Cc Y and given public facility location x ED. Let u*,R(r), rf, and n(r) be a utility level, land rent curve, residential fringe distance, and household distribution. In the case of a public facility providing a positive service, each bid rent curve is decreasing in r. Considering this, we define (u*, R(r), rf, n(r)) as a competitive equilibrium if and only if !P(r,, u*, G) = R, =
rf
r(x)
R(r) = Y(r, u*, G) =R,
if
Y(r(x), u*, G) CR,,
if
Ul(r(x), u*, G) 2 R,,
for
r 5 rf,
for
r>rf,
n(r) = L(r, x)/S(r, u*, G) =o
(4.5)
(4.6)
for
r 5 rf,
for
r>rf,
(4.7)
an(r)dr=M,
(4.8)
where functions Y and S have been defined from (3.1). Conditions (4.5) and (4.6) imply that R(r) =max { Y(r, u*, G), RA}. And, conditions (4.5)(4.7) together imply that at each location land is occupied by the highest bidder. Condition (4.8) is the population constraint. Fig. 3 depicts the two possible R
RI
(a)
(b)
L Y(r
=f
l(X)
Fig. 3. Equilibrium
r
0
land use patterns
,u*,G)
rf=r(x) (positive
service).
r
249
M. Fujita, Optimal location of public facilities
equilibrium situations. In the first situation, diagram (a), a portion of community land far from the public facility is used for agriculture. In the second situation, diagram (b), the entire community land is occupied by households. In the case of a negative service, each bid rent curve is increasing in r. Thus, we define (u*, R(r), I~, n(r)) as a competitive equilibrium if and only if Y(r,, u*, G) = R,
if
Y(0, u*, G) < R,,
if
Y(0, u*, G) >=R,,
(4.5’) =o
rf R(r) = R,
for
rzr,,
for
r > rf,
(4.6’) = Y(r, u*, G) n(r) =0
for
rgr,,
for
r > rf,
(4.7’) = L(r, x)/S(r, u*, G) r(x)
(4.8’) d n(r)dr=M.
Fig. 4 presents two different equilibrium situations. In diagram (a), land close to the public facility is used for phenomenon can be often found near a noisy airport or incinerator. In the case of either positive service or negative service, as follows.’ O
0
R
(a)
K
r(x)
rf
Fig. 4. Equilibrium
r
l
the situation of agriculture. This polluting garbage
(b)
xf’0
land use patterns
r(x) (negative
t
r
service).
“Lemma 1 can be proved by using the standard boundary rent curve explained in Fujita (1985). It is also briefly explained in Fujita (1986).
RSUt
we can conclude
procedure,
which
is
250
M. [email protected]
Optimal location ofpublicfacilities
Lemma 1. Suppose Assumptions I5 are satisfied, and we arbitrarily fix a tax rate G< Z Then, under each given location of the public facility in the community, there uniquely exists a competitive equilibrium. Let u(x, G) be the equilibrium utility level under tax rate G< Y and public facility location XE D. If we fix the tax rate at some level, then the equilibrium utility level will become highest when the public facility locates at some point in the community. We call such point a umax location, and note that this location is optimal for the residents of the community. In the next section we ask when a umax location is independent of the tax level. In order to determine the socially optimal (i.e., Paretooptimal) location of the public facility, we must consider both households and landlords. For this, we next consider the compensated equilibrium land use of the community. Suppose we fix the utility level at some target level U, and then we ask that for a public facility located at XE D, what tax rate G (or subsidy rate if GCO) will make the equilibrium utility level just equal to the target utility level u? In the case of a positive service, we define (G*, R(r), rf, n(r)) as a compensated equilibrium if and only if Y(r,, u, G*) = R,
if
Y(r(x), u*, G) < R,,
if
Y(r(x), u, G*) 2 R,,
(4.9) = r(x)
rf
R(r) = Y(r, u, G*)
for
r 5 rf,
for
r>rf,
(4.10) =R,
n(r) = L(r, x)/S(r, u, G*)
for
r 5 rf,
for
r>rF,
(4.11) =o
(4.12)
[n(r)dr=M.
In the case of a negative service, we define (G*, R(r), rf, n(r)) as a compensated equilibrium if and only if Y(r,, u, G*) = R,
if
Y(0, u*, G) CR,,
if
Y(0, u*, G) 2 R,,
(4.9’) =o
Tf
R(r) = R,
for
rsr,,
for
r > rf,
(4.10’) = Y(r, u, G*)
M. Fujita, Optimal location
n(r) = 0
for
rsr,,
for
r > rf,
of publicfacilities
251
(4.11’) = L(r, x)/S(r, u, G*) r(x)
1 n(r) dr= M.
(4.12’)
If As in the case of the competitive equilibrium, equilibrium patterns of land use corresponding service. Let us introduce the next assumption.
there are to positive
two and
different negative
Assumption 6 (perfect substitutability between z and s). On each indifference curve u= U(z, s, E) under fixed E, s approaches zero as z approaches infinity. Then, we can conclude
as follows.”
Lemma 2. Suppose Assumptions 16 are satisfied, and we arbitrarily fix a target utility level u E (  CO,CO). Then, for each location of the public facility in the community, there uniquely exists a compensated equilibrium. Let G(x,u) be the tax rate associated with the compensated equilibrium under target utility level u and public facility location x E D. Let TDR (x, u) be the associated total differential rent defined by
TDR(x, u) = f (R(r)  R,)L(r, x) dr,
(4.13)
H
where H = [0, rr] in the case of a positive service, and H = [r,, r(x)] in the case of a negative service. Then, the net revenue NR(x,u) for the rest of the economy can be calculated as follows:‘2 NR(x, u) = TDR(x,
u) + MG(x, u)  C(K),
(4.14)
where C(K) is the cost of the public facility with size K.13 Suppose we fix the target utility at some level. Then, the net revenue will become largest when the public facility locates at some point in the community. We call such point an NRmax location. Since the congestion for the public service is assumed to possibly occur only at the location of the public facility, a lump“Similarly Lemma 2 can be proved by using the procedure with boundary rent curve. See Fujita (1985, i986). “The rest of the economy means all parties of the economy excluding only the residents of the community in question. It includes the absentee landlords. r3When the lot size of the public facility is neglected (or fixed), we can define the size of the facility by its cost. In this case, C(K) = K.
252
M. Fujita, Optimal location of public facilities
sum tax is sufficient in order to achieve the Paretooptimal allocation of households under a given location of a public facility. Therefore, assuming that the benefit (or welfare) of the rest of the economy is increasing in NR, an NRmax location is also a Paretooptimum location. In the next section, we ask when the NRmax location is independent of target utility level.
5. Areadominance
and the definition of the center and edges
This section considers the question, under what shape of the community domain is an absolute definition of the center and edges possible? In other words, what shapes guarantee that both the umax and NRmax locations of a facility producing positive service are at the center of the community domain. Similarly, what shapes guarantee an edge location for a facility producing negative service? To begin, let us first consider the linear community depicted in fig. 5. Since the width (= 1) of the community is negligible in contrast with its length (= 1), we treat it as onedimensional. In this case, it is not difficult to show that both the umax location and NRmax locations are always at the center 4 (for a positive service) or at the edges e, and e2 (for a negative service). This result is derived in the next section, but the problem here is how to generalize the concept of ‘the center’ and ‘edges’ in twodimensional communities. If we observe that the center of a linear community is also its geometric center and gravitational center, we could define the center of a twodimensional community by its geometric center or by its gravitational center. Unfortunately, fig. 6 demonstrates that neither the geometric center nor the gravitational center can always be a umax and NRmax location. A more fundamental characteristic of the center of a linear community is that it commands an amount of land in its each neighborhood which is equal to or Q
k
a
$
9
r
4 e2
1 _
r(a)
Pr(a)
Fig. 5. A linear community.
Fig. 6. Nonoptimal
geometric
center.
7
M. Fujita, Optimal location
of publicjacilities
253
larger than any other point of the community. Similarly, characteristic of an edge of a linear community is that amount of land in its each neighborhood which is less than other point of the community. In order to state the above observation precisely, recall defined by (4.3). Given two points a and b in D, we say that areadominates b, or b is (weakly) areadominated by a if and A(r, a) 2 A(r, b)
for
a fundamental it commands an or equal to any function A(r,x) point a (weakly) only if
all r > 0.
(5.1)
We say that point a strongly areadominates b, or b is strongly areadominated by a if and only if A(r,a)zA(r,
b)
for
all r>O,
A(r, a) > A(r, b)
for
some r.
Next, suppose
that a point
A(r, 4) zA(r,x)
and (5.2)
4 in D areadominates
for
all r>O,
for
all other points
any XE D,
in D: (5.3)
equivalently, A(r, f$) = max A(r, x)
for
all r > 0.
(5.4)
XCD
Then, we call &J the center of the community. Suppose areadominated by all other points in D: A(r, e)sA(r,x)
for
all r>O,
for
any XE D,
that a point
e in D is
(5.5)
equivalently, A(r, e) = min A(r, x) XSD
for
all r > 0.
(5.6)
Then, we call e an edge of the community. In the case of the linear community of fig. 4, for each point x = 4, a, e, and e2, we can draw an areacurve A(r,x) as in fig. 7. From this, we can see that the middle point C$strongly dominates all other locations of the community, and hence it is indeed the center of the community. We can also see that locations e, and e2 are strongly areadominated by all other locations of the community. Hence, e, and e2 are the edges of the community. Given an arbitrary shape of community domain, the center and edges may
M. Fujita, Optimallocationofpublicfacilities
254
A(r ,x)
Fig. 7. Areacurves
in the case of the linear community.
not always exist. There may be more than one edge. If the center indeed exists, however, it is always unique, as we will show in section 7. Thus, from the definition of strong areadominance, we can immediately conclude as follows: Lemma 3. (i)
Under Assumption
the center
of the community
the community, (ii)
each
edge
5, strongly
areadominates
all other locations of
and
of the community
is strongly
areadominated
by all nonedge
locations of the community.
6. Optimality
of the center and edges
The optimality of the mathematical properties. Lemmu function (i)
4. Tuke integrable
center
and
edges
derives
from
the
following
any two points u, b ED. Let each g(r) be a realvalued on [0, co]. Then, under Assumption 5, we have
Point a areadominates
b if and only if
iL(r,a)g(r)drzbL(r,b)g(r)dr
for any nonnegative
nonincreasing
Vt>O, g(r).
(6.1)
M. Fujita,
(ii)
Optimal
location
ofpuhlic.fucilities
255
Point a is areadominated by b if and only if a)
st Ur, a)g(r)dr 2
r(a)
s L(r, b)g(r) dr f
V t c r(a),
(6.2)
for any nonnegative nondecreasing g(r). (iii) If point a strongly areadominates b, then i L(r, a)g(r) dr >j L(r, b)g(r) dr 0
Vt 2 r(b),
(6.3)
0
for any nonnegative g(r) which is differentiable and strictly decreasing on (0, r(b)). (iv) If point a is strongly areadominated by b, then r(a) i
r(a)
.Ur,ho) dr > i W, b)g(r)dr
vt
(6.4)
for any nonnegative g(r) which is differentiable and strictly increasing on (0, r(a)). The proof of Lemma 4 is given in Appendix A. Recall that u(x, G) represents the utility level in competitive equilibrium under tax rate G and public facility location x E D. Similarly, NR(x, u) represents the net revenue in compensated equilibrium under target utility level u and public facility location x E D. Setting g(r) = l/S( r,u, G), we apply Lemma 4 for the comparison of u(x, G) and NR(x, u) under different x, and can conclude as follows (see Appendix B for proofs): Theorem 1. Under Assumptions l6, suppose that the public facility provides a positive service. I4 Take two locations a, bE D, and fix a tax rate G< Y or a target utility level u. Then, (i) (ii)
if a areadominates b, u(a, G) 2 u(b, G) and NR(a, u) 2 NR(b, u), and if a strongly areadominates b and if the entire community land is used for housing in competitive equilibrium (compensated equilibrium) under facility location a, then u(a, G) > u(b, G) (NR(a, u) > NR(b, u)).
Theorem 2. Under Assumptions 16, suppose that the public facility provides a ne,~trtirr srrricr. Take two locations a. h E D, and ,fiu a tas rate G < Y or a
‘JAwumption . 6 is needed applies Theorems 24 and 6.
only
in the context
of compensated
equilibrium.
The same
note
256
M. Fujita,
Optimal
location
oj’puhlic,facilities
target utility level u. Then, (i) (ii)
if a is areadominated by b, u(a, G) 1 u(b, G) and NR(a, u) 2 NR(b, u), and tf a is strongly areadominated by b and if the entire community land is used for housing in competitive equilibrium (compensated equilibrium) under facility location a, then u(a, G) > u(b, G)(NR(a, u) > NR(b, u)).
The above two theorems are useful for pairwise comparison of candidatelocations for the public facility. Moreover, from Lemma 3 and Theorems 1 and 2, we can immediately conclude as follows: Theorem 3 (optimality of the center). Under Assumptions 16, suppose the public facility provides a positive service. Assume that the community domain D has the center. Then, under any fixed tax rate G < Y or under any fixed target utility level, (i) (ii)
the center is a umax and NRmax location of the facility, and if the entire community land is used in competitive equilibrium (compensated equilibrium) when the facility locates at the center, then the center is the unique umax (NRmax) location of the facility.
Theorem 4 (optimality of edges). Under Assumptions 16, suppose the public facility provides a negative service. Assume that the community domain D has edges. Then, under any fixed tax rate G < Y or under any fixed target utility level, i;!)
each edge is a umax and NRmax location of the facility, and tf th e entire community land is used for housing in competitive equilibrium (compensated equilibrium) when the facility locates at an edge, then only the edges are the umax (NRmax) locations of the facility.
Our definition of the center and edges is, of course, based on very strong conditions. Nevertheless, Theorems 3 and 4 are useful since they identify optimal locations of the public facility independently of the specific utility function, tax rate G, or servicelevel curve E(r).
7. Identification of the center and edges Recall that the circumscribing circle of community with the minimum radius among all circles which diameter of D by 6(D) = sup r(x) = sup d(x, y), XSD
X,YED
domain contain
D is the circle D. Define the
(7.1)
251
M. Fujita, Optimal location of publicJhcilities
which represents the set
the maximum
distance
between
two locations
in D. Define
E,={xED~~(x)=~(D)}.
(7.2)
Then, we can show as follows: Lemma 5. (i) (ii)
Under Assumption 5,
if the center of the community domain D exists, it must be the center of the circumscribing circle of D, and a location x is possibly an edge of the community only if x E E,, that is, only if the maximum distance r(x) from that location equals diameter 6(D) ofD.
The proof is given in Appendix C. Lemma 5(i) implies that if the center exists, it is unique. Although this lemma is useful in identifying the possible locations of the center and edges, it does not answer the question of the existence of the center and edges. The concept of symmetric dominance, or sdominance, introduced below not only helps to identify the possible locations of the center and edges, but is also useful to show that the center and edges actually exist for a class of symmetric community shapes. Let us arbitrarily take two points, a, b E D, a # b. Draw the line H,, which is vertical to the line & and passes the middle of Z (refer to fig. 8). Hold the bside of D (with respect to line Hat,) over the aside of D. In this way, if the bside of D is completely contained in the aside of D, we say that point a (weakly) sdominates point b, or b is (weakly) sdominated by a. That is, let us define D, = the aside (including points on Hat,) of D with respect to line Hab, D, = the bside (including points on Hat,) of D with respect to line Hat,, 0: = the mirror image of D, in R2 with respect to line H,,. H
ab
Fig. 8. Point
a sdominates
h
M. Fujita, Optimal location of public.facilities
258
If D$ c D,, then point a sdominates b, or b is sdominated by a. If D$ c D, and 0: + D,, we say that point a strongly sdominates b, or b is strongly sdominated by a. Under Assumption 5, sets D,, D, and 0: have the same topological properties with D. Set D, u D$ is symmetric with respect to line Hab, and all points of set D, 0; are closer to a than to b. Hence, Dt c D, implies that A(r,a) IA(r, b) for all r>O, and 0: c D, and D$ # D, implies that A(r, a) zA(r, b) for all r >O and A(r, a) >A(r, b) for some r. Therefore, we can conclude as follows: Lemma 6. (i) (ii)
Under Assumption 5, take two points a, b E D. Then,
if a sdominates b, then a areadominates b, and if a strongly sdominates b, then a strongly areadominates b.
Although the sdominance dominance relation is. Next, let X be a (straight) x E X n D, we represent V, = the vertical If set V, n D is convex uconvex, with respect define
relation
is not
always
transitive,
line in the same R2 that contains
line to X through
point
the
area
D. For each
x.
for every x EX n D, we say D is vertically convex, or to axis X (refer to fig. 9). For each x E X n D, we
Fig. 9. D is vconvex
and symmetric
with respect
to axis X.
M. Fujitu,
Optimal
location
259
ofpublicfacilities
Let
B,={blbEB,, XEXCTD}, which subset 9):
is called the vboundary curve of D with respect to axis X. Bx is a of the boundary curve of D. It is not difftcult to see that (refer to fig.
Lemma 7. Under Assumption 5, suppose that D is vconvex and symmetric with respect to axis X. Given XE Xn D, take two points a, bE V,n D. lf a is closer to x than b, then a strongly sdominates b. From
Lemmas
6 and 7, we can immediately
conclude
as follows:
Lemma 8. Under Assumption 5, suppose that D is vconvex and symmetric with respect to axis X. Then, I!)
each XEX n D strongly areadominates all other points of V, n D, f or each x E X n D, both points of B, are strongly areadominated by all other points of V, n D.
Given two points a, b E D, if a strongly areadominates b, we express a> b. Suppose that D is vconvex and symmetric with respect to two different axes X and I: Let C#Ibe the intersection of X and I: If X and Y are vertical, it is not difficult to see that 4 strongly areadominates all other points of D. For example, in the case of fig. 10(a), we can see from Lemma 8(i) that a
and
lim d(bi, 4) =O. i02
(a)
Fig. 10. The intersection
(b)
of axes is the center.
M. Fujita, Optimal location
260
of publicfacilities
Since relation < is transitive and since function A(r,x) is continuous in x under each r, it follows that A(r, 4) zA(r, a) for all r>O and A(r, 4) > A(r, a) for some r; that is, 4>a. Therefore, we can conclude as follows: Theorem 5. Under Assumption 5, suppose that D is vconvex and symmetric with respect to two different axes, X and I: Then, the intersection C$of X and Y is the center of the community. Fig. 11 presents examples of community shapes which are vconvex and symmetric with respect to more than one axis. In each diagram, intersection $J of the axes is the center of the community. Note that Theorem 5 provides a sufficient condition for the existence of the center, but not the necessary condition. It is not difficult to see, for example, that the regular pentagonal star is not vconvex with respect to any axis, but has the center. We can also show that if D is vconvex and symmetric with respect to two different axes, then D is the starred set. However, it must be noted that D may have the center without being a starred set. Concerning edges, from Lemma 8(ii) we can conclude as follows: Lemma 9. Under Assumption 5, suppose that D is vconvex and symmetric with respect to axis Xi, i= 1,2,. . . , m. Then, all edges of the community must be on iB,Yj, the intersection of all vboundary curves B,,, i = 1,2,, . . , m.
(b)
hexagon
I Cc)
circle
x__+x+ Y Y
(d)
regular
trianglr
Fig.
11.Examples
(e)
of community
shapes
(f)
with more than one axis.
M. Fujita, Optimal location
of public facilities
261
Using either Lemma 5(ii) or Lemma 9, we can see that each dot in the diagrams in fig. 11 represents a possible location of an edge of a community; in the case of circle (c), all points on the circle are possible locations of edges. In the case of diagrams (a)gd), it is not difficult to show by using Lemma S(ii) that each dot indeed represents an edge of the community. For example, in the case of diagram (a), a>b>e and hence a>e. This is true for any noncorner point a ED. Thus, e is an edge of the community. Finally, we can show that under Assumptions 16, functions u(x, G) and NR(x,u) are continuous in x on the compact set D.15 Thus, there exists a umax location under each G < X and an NRmax location under each u. Therefore, from Theorem 1 and Lemma S(i), we can conclude as follows: Theorem 6. Under Assumptions 14, suppose that the public facility provides a positive service, and D is vconvex and symmetric with respect to axis X. Fix a tax rate Cc Y or a target utility level u. Then, (i) (ii)
there exist on X a umax location and an NRmax location of the public facility, and zj’ the entire community land is used for housing in competitive equilibrium (compensated equilibrium) when the facility locates at every umax (NRmax) location on X, then there is no umax (NRmax) location outside X.
This theorem is useful in identifying umax and NRmax locations when the community domain is vconvex and symmetric with respect to only one axis.
8. Conclusion We have examined the optimal location of public facilities under the influence of the land market. By introducing the concept of areadominance, we have defined the center and edges of the community. It has been shown that if the community has a center and edge, the optimal location of the public facility is given at the center or edges independently of the utility function, tax rate or externality level. Although these results are interesting, it is obvious that this paper represents only an initial step in conducting a more comprehensive study on the subject. A number of extensions are left for the future. First, although we have examined the case of the closed community with absentee landlords, other cases such as the open community and public land ownership remain to be studied. It is conjectured that the optimality of the center or edges still hold in these cases. Second, it is also interesting to extend the concepts of areadominance, center and edges to general metric spaces (for example, a space with blockdistance). In this case, the notion of the Haar measure provides the natural ‘area’ concept. Third, “Again,
Assumption
6 is necessary
only for the case of compensated
equilibrium.
262
M. Fujita,
Optimal location
of public facilities
the concept of areadominance in this paper is a spatial analogy to the concept of the firstorder stochastic dominance. Naturally, it would be appropriate next to introduce the concept of secondorder areadominance. Then, the optimal location of public facilities would be identified for broader classes of community shapes. Finally, it is important to extend the analysis from our examination of a single public facility to the case of multiple public facilities. When all public facilities are identical and hence each household chooses only one facility, the extension of the concept of areadominance to this multiple facility case is straightforward. When all public facilities are not the same, we may need new approaches.
Appendix A: Proof of Lemma 4 A.I.
Proof of Lemma 4(i)
First, we prove the ‘only if’ part. Given
t > 0, let
K = i L(r, a)g(r) dr  i L(r, b)g(r) dr = j (L(r, a)  L(r, b))g(r) dr. 0
0
0
If g(r) is nonnegative and theorem there exists s (0~~5
nondecreasing, t) such that
from
the
second
mean
value
K=g(r+O)~(L(r,n)L(r,b))dr~g(r+0)(A(s,a)A(s,b)). 0
Since a areadominates b, A(s, a) 2 A(s, b), and hence K 2 0. Hence, the ‘only if’ part has been shown. Next, the ‘if’ part can be shown as follows. Suppose that the relation (6.1) holds for any nonnegative nonincreasing g(r). Then, since the function, g(r) = 1, is nonnegative and nonincreasing,
Vt>O,
~L(r,u)g(r)dr~bL(r,b)g(r)dr,
L(r, a) dr 2 i L(r, b) dr,
*i 0
+A(t,a)zA(t,b), *a
vt>o,
0
Vt>O,
areadominates
Thus, we can conclude
b.
as Lemma
4(i).
263
M. Fujita, Optimal location of public facilities
A.2. Proqf of Lemma 4(ii) With the change of integralvariable proved similarly to the previous section.
from
r to z =r(a) r,
this
can
be
A.3. Proof of Lemma 4(iii) Let c$(r) E L(r, a)  L(r, b). Then, K =j L(r, a)g(r)dr0
If a strongly
j c$(r)g(r)dr.
jL(r,b)g(r)dr=
0
0
b, r. 4(r) dr=O for any t 2 r(b). Hence,
areadominates
for any
t 2 r(b), K = j 4(r)(g(r) g(O)) dr = j 4(r) j g’(t) dt dr 0
0
0
= d g’(r) dr [ 4(r) dr  6 g’(r) i 4(t) dt dr
(from integration
by parts)
bg’(r)(A(r,a)A(r,b))dr.
= ig’(l)i&(t)dfdr=
If a strongly areadominates b, then A(r, a) zA(r, b) for all r>,O, and A(r, a) >A(r, b) for some rsr(a), and r(u) zr(b). Then, since A(r, a) and A(r, b) are continuous in r, if g(r) is differentiable and strictly decreasing on (0. r(h)), K>O for any tzr(b). A.4. Proof of Lemmu 4(iu) With the change of integralvariable proved similarly to the previous section.
Appendix B: Proofs of Theorems
from r to z=r(u) r. Q.E.D.
this
can
be
1 and 2
B.I. Proof of Theorem 1 Theorem I(i). (a) Let us fix a tax rate G< Y Assuming that the public facility provides a positive service, for each utility level U, we define distance F(u) by the next relation: Y’(f(u), u, G) = R,
F(u)
=o
if
Y(0, u, G) > R,,
_if
Y(0, U, G) 5 R,.
264
Then,
M. Fujita, Optimal location of public facilities
in r and u,
since Y(r, U, G) is decreasing dF((u)/du < 0
Next, define population
if
(‘4.1)
f(u) > 0. density
p(r, U) = l/S(r, U, G)
function
for
r 5 f(u),
for
r>f(u).
p(r,u) by
W4
=o
Then, since S(r, u, G) is increasing i3p(r, u)/& < 0
in r and u,
dp(r, u)/du
and
For each x E D, define population
function
at each Y< F(U).
(A.3)
M(u, x) by
M(u, xl = j W, x)p(r, 4 dr 0
= 1 L(r, x)p(r, u) dr
[from (A.2) and (4.2)]
0
ES L(r,x)p(r,u)
From
dr
for
any rzmin{f((u),
r(x)}.
(A.4)
(A.l) and (A.3), ~?iV(u,x)/&
if
M(u,x)>O.
(A.5)
Now, set g(r) = p(r, u), which is nonnegative from Lemma 4(i), if a areadominates b, M(u,a)ZM(u,b)
for
any u.
Let ~(a, G) and u(b, G) be equilibrium under public facility location a and (A.4) and (4.8) M(u(a, G), a) = M(u(b,
and nonincreasing
in r. Hence,
(A.6) utility levels in competitive equilibrium Then, since rf= min{f((u), r(x)}, from
b.
G), b) = M > 0.
(A.7)
From (A.6) and (A.7), M(u(b, G), b) = M(u(a, G), a) 2 M(u(a, G), b). Hence, from (A.5), it must be true that u(b,G)Zu(a,G). Therefore, we can conclude that if a areadominates b, then ~(a, G) 2 u(b, G).
265
M. Fujita, Optimal location of public facilities
(b) Let us fix the target utility level at u. Let G(a,u) and G(b,u) be tax rates in compensated equilibrium under public facility location a and b. Then, since each of functions Y(r, u, G) and S(r, u, G) behaves qualitatively the same way in variables u and G, we can conclude, similarly as (a), that if a areadominates b. then G(a, u) 2 G(b, u).
([email protected]
Next, from (4.14), NR(a, u)  NR(b, u) = TDR(u, u)  TDR(b, u) + M(G(u, u)  G(b, u)). (A.9) By definition
TDR(b,u)=‘S(Y(
I, u, G(b, u))  RJL(r,
(A.lO)
b) dr.
0
Under
fixed b and u, we also define a function TDR*(b,
u, G) = J (Y(r, u, G)  R,)L(r,
of G, b) dr,
0
where i(G) is defined
by the next relation:
Y(i(G), u, G) = R,
if
Y(0, u, G) > R,,
if
Y(0, u, G) 5 R,.
(A.1 1) =o
i(G)
Then, since TDR(b, u) = TDR*(b,
u, G(b, u)),
TDR(b, u) = “j+’ aTDR;l;b, W, u) From
” ‘) dG + TDR*(b, u, G(u, u)).
(3.1) and (A.1 1) aTDR*(b, BG
u, G)
i(c) 8Y = / c?GL(r,b)dr=
F(G) s  ,fr”;“(;) 0
>
,
We also have that i(G)
s L(r, b)/S(r, u, G) dr = A4 0
for
G = G(b, u),
and
dr.
(11)
M. Fujita, Optimal location
266
of public facilities
Thus, r^(G) 1 L(r, b)/S(r, U, G) dr < A4
for
G > G(b, u).
0 SO,
TDR(b,
From
U) 5 M(G(u, u)  G(b, u)) + TDR*(b,
(A.9) and (A.12) and recalling
u, G(a, u)).
(A.12)
(4.2),
NR(a, U)  NR(b, U) 2 TDR(u, U)  TDR*(b,
U, G(u, u))
i(G(n.14) =
i
( y(r, u, G(a,4)  R,dL(r, a) dr *^(G(a.u))

j
0
(Vr, u, G(u, 4)  R.dW, b)dr.
Since g(r) = Y(r, U, G(u, u))  R, is nonnegative and nonincreasing in r
I(@).
If point
a strongly
areadominates
b,
r(u) _I r(b).
(A.13)
If the entire community land is used for housing under facility location a, then r(u) si;(u(u, G), and i)p(r, ~(a, G))/& ~0
for
in competitive
r < ?(~(a, G)).
equilibrium
(A.14)
Hence, if we let g(r) = l/S(r, ~(a, G), G) for all r>,O, then g(r) is differentiable and strictly decreasing on (0, r(b)) and g(r) =p(r, ~(a, G)) for t5 f((u(u,G)). Hence, from (A.4)
Wu(a, G),4 =
s0 L(r, 4p(r, 40, G))dr
= d L(r,u)g(r)dr
[from (4.2) and (A.13)]
> f f,(v, h)g(r) dr
[from
0
Lemma
4(iii)]
M. Fujita,
Optimal
location
cfpuhlic,
267
fkiliiies
= kf(u(a, G), b).
(A. 15)
From (A.7) and (A. 15), M(u(b, G), h) = M(u(u, G), a) > M(u(a, G), b). Thus, from (A.5) u(h, G) G(b, u). Hence, we can see from the analysis of (b) that N(u, U) > N(b, u). Therefore, we can conclude as Theorem l(ii).
B.2. Proof'
of Theorem
This can be proved omitted.
2
quite
similarly
to Theorem
1, and hence
the proof
is
Appendix C: Proof of Lemma 5  sketch
C.I.
Proof
of’ Lemma 5(i)
For each XE R2, r(x) is defined by (4.1). The radius r(D) of set D is defined by r(D) =infxtD I(X). The radius r,(D) of the circumscribing circle of set D is defined by r,(D) = inf,,, z r(x). And, the center d, of the circumscribing circle of set D is such point in R2 that r(4,) =rc(D), which exists uniquely. The proof of Lemma 5(i) is easy when C/J~ED. Suppose 4C~ D. Then, v,(D) = r(D) = r(&),
and
D=A(r(4,),4,)>A(r(4,),x)
for
any xi&.
(A.16)
of Hence,if 4 # 4,, 444,), 4) < max,,, A(r(4,), x), which is the contradiction (5.4). Therefore, we can conclude that if 4, ED and if the center 4 exists, then 4=$~,. Next, suppose that 4, is not in D. If the center 4 exists, from definition of r(4) and r(D) we can show that r(c#j)=r(D). We can also show that if 4, is not in D and if r(D) =r(x) and XE D, then x must be on the boundary of D. Hence if 4, is not in D and if C/Jexists, 4 must be on the boundary of D. Then, since D has interior points, conditions (5.4) cannot be satisfied for small r. Therefore, when 4, is not in D, the center 4 does not exist. Thus, we can conclude as Lemma 5(i).
268
M. Fujita, Optimal location of public fuciliries
C.2. Proof of Lemma 5(ii) By definition, if x is an edge of D, A(r(x), x) 5 A(r(x), y) for any y E D. If x is not in E,, then r(x) <6(D); thus A(r(x), x) = D > A(r(x), y) for any y E E,. Therefore, we can conclude as Lemma 7(ii).
References Arnott, R., 1979, Optimal city size in a spatial economy, Journal of Urban Economics 6, 6589. Barr, J.L., 1972, City size, land rent and supply of public goods, Regional and Urban Economics 2, 67103. Fujita, M., 1985, Existence and uniqueness of equilibrium and optimal land use: Boundary rent curve approach, Regional Science and Urban Economics 15, no. 2, 295324. Fujita, M., 1986, Urban land use theory, in: Fundamentals of pure and applied economics, and for Encyclopedia of economics (Harwood Academic Publishers). Helpman, E., D. Pines and E. Borukhov, 1976, The interaction between local government and urban residential location: Comment, American Economic Review 67, 9961003. Johansson, B. and G. Leonardi, 1986, Public facility location: A multiregional and multiauthority decision context, in: P. Nijkamp, ed., Handbook in regional economics (NorthHolland, Amsterdam) forthcoming. Kanemoto, Y., 1980, Theories of urban externalities (NorthHolland, Amsterdam). Lea, A.C., 1981, Public facility location models and the theory of impure public goods, Sistemi Urbani 3, 345390. Sakashita, N., 1986, Optimal location of public facilities under influence of the land market, Journal of Regional Science, forthcoming. Thisse, J.F. and H.G. Zoller, 1983, Locational analysis of public facilities (NorthHolland, Amsterdam).