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Independent and joint provision of optional public services
ELSEVIER Regional Science and Urban Economics 25 (1995) 411-425
Independent and joint provision of optional public services Kenichi Tsukahara* Ministry of Construction, 2-I-3 Kasumigaseki Chiyodaku, Tokyo 100, Japan Received December 1993; final version received October 1994
Abstract While 'essential' public services such as police and fire protection are provided in virtually every city, 'optional' public services, such as museums or stadiums, are often found in larger cities. By using a simplified urban land use theory model, this paper examines why optional public services are offered and why cities organize to jointly provide these services. Results suggest that optional public services are provided when their cost is 'small' relative to the cost of essential public services. An expansion of the model suggests that when optional public services are jointly provided, cities vary in size as model parameters vary. Keywords: Public services provision choice; Joint provision; City size JEL classification: R12; H42
1. Introduction L a r g e cities are often o b s e r v e d to provide public services such as stadiums, m u s e u m s and c o n v e n t i o n centers, in m a n y cities independently, while small cities often do not. In o t h e r words, some public services are o p t i o n a l in that they are not essential for a city's existence. Small cities do h o w e v e r occasionally provide optional public services. This m a y occur w h e n * Correspondence to: Kenichi Tsukahara, c/o Embassy of Japan, Indonesia, JL. M.H. Thamrin 24, Jakarta, Indonesia. Tel: +62-21-325140; Fax: +62-21-325460. 0166-0462/95/$09.50 ~ 1995 Elsevier Science B.V. All rights reserved SSDI 0166-0462(95)02093-4
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a coalition of cities jointly provides the services to reduce the financial burden experienced by any single city. City governments therefore face three optional public services alternative provision choices: non-provision, independent provision, and joint provision. This paper uses urban land use theory and club theory to examine the conditions under which cities optimally choose one of these three alternatives. Arnott (1979) introduced the provision of local public goods into an urban land use model examining optimal city size supporting an allocation of households. Sakashita (1987) introduced models of urban land use in public good location. His insightful work suggested that a land market can produce a socially optimal public facility location. Kuroda (1989) extended Sakashita's model to the two-city joint provision case. Abdel-Rahman and Fujita (1993) provide an approach that, while not directly related to the provision of public goods, is applicable to explain variety in public good provision within cities and across city types. In the context of club theory, Brueckner and Lee (1991) analyze multiproduct club optimality in the presence of economies of scope. They formally analyze trade-offs between cost-complementary gains and efficiency losses between single and multiproduct clubs. This paper assumes the existence of essential (EPS) and optional (OPS) public services types. EPS are essential in that a city must provide the services in order for the city to exist. In contrast, a city can exist without OPS and provides them only when their presence increases city residents' utility. We analyze various city system models in which cities are formed by profit-maximizing developers. We initially assume, in Section 2, that cities cannot organize to provide OPS jointly. OPS are provided (not provided) if the cost is 'small' ('large') relative to that of EPS. In Section 3 we allow for the joint provision of OPS in a two-city model. We assume under joint provision that the OPS facilities are physically located in one of the cities and that this city receives full benefit from OPS while the other city receives a partial spill-over benefit. This joint provision occurs when its cost is 'intermediate' relative to that of EPS. In the joint provision case the city in which OPS are located is larger in population than the city receiving OPS spill-over benefits. Section 4 concludes the paper and implications for future research directions are given.
2. The independent provision model We consider a closed economy consisting of a system of linear cities whose total population is given by ,~. We assume that all households in the economy are identical and each household works in and commutes to a central business district (CBD). All households have the same productivity,
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yO > 0, which is fixed exogenously. Each household is free to choose the city in which it resides as well as its location within the city. There are two types of public services, EPS and OPS. EPS must be provided in every city while OPS are optional. In addition we introduce the following set of assumptions:
Assumption 1 (Linear city). Cities are built along a long, narrow strip of land with unit width. Each city has a central business district (CBD) to which all households in the city commute to work. The city expands in both directions from the CBD. The formation of the city is depicted in Fig. 1.
Assumption 2 (Service level and cost of EPS). The service level of EPS, E 1 > 0, and the cost of EPS, F 1 > 0, are given exogenously.
Assumption 3 (Service level and cost of OPS). The service level of OPS is expressed by E. When OPS are provided, E = E 2 > 0, and the cost of OPS is F 2 > 0, where E 2 and F 2 are given exogenously.1 Otherwise E = 0.
Assumption 4 (Common utility function). All households in the economy have identical utility functions U(z,s, E), where z is the amount of consumption of the composite good, and s the lot size of the house. The composite good is chosen as the numeraire, so its price is unity. In order to obtain explicit results we use the following utility function: log U(z, s, E) =-a log z +/3 logs + 3' log(E1 + E ) ,
(1)
where a > O , / 3 > 0 , 3' > O a n d a + / 3 = l .
Assumption 5 (Linear transport cost). The transport cost function is given by T(r)= tr, where t > 0, while r represents household distance from the CBD. r
< I rf
I CBD
I )
r
rf
rf : Boundary of City Fig. 1. Formation of a city.
1The main problem of this paper is whether OPS are provided independently, provided jointly, or not provided. Determining the investment level of OPS would further complicate the analysis without adding much to the understanding of the problem. For an analysis of the variable investment level of OPS for the independent provision case, see Tsukahara (1993).
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We suppose that each city will be developed by a profit-maximizing city developer who can freely enter or exit the market. City developers rent the land for the city at the opportunity cost, which for simplicity is normalized to zero. Each city developer chooses the population size of the city, N, and determines the public services provision so as to maximize its city development profit. If a city provides both EPS and OPS, we call it a type A city; otherwise, we call it a type B city. As noted above, households choose a city that maximizes their utility level. Thus, in equilibrium, the utility level will be equalized across all cities in the economy. We denote this common utility level by U, which is called the national utility level. We also assume that the number of households in any developed city is small compared with the economy's total population and that a sufficiently large number of cities are developed. This implies that each developer considers that its behavior does not affect the market. Thus even though the equilibrium utility level will be determined endogenously, each developer behaves as a utility taker. 2.1. The residential choice behavior
Based on these assumptions, we state the behavior of each household. Each household freely chooses the city where its resides, its location within the city, the amount of land it occupies and the amount of consumption of the composite good. The bid rent function is given by qj(r, U , E ) = m a x
{ Y-
T ( r ) - Z(s, U , E )
= a~/t~[3(r - T(r))t/~(E1 + E ) ~/~ e -U/s ,
(2)
where Z ( s , U , E ) is the solution to U ( z , s , E ) = - a l o g z + ~ l o g s + 3' log(El + E) = U for z, and Y is the income of each household. Then we define the bid-max lot size as follows: s(r, U, E ) = [3(Y - T(r))/qJ(r, U, E ) = a - ~ / t 3 ( r - T(r))-"/t3(E 1 + E ) -~/~ e U/t~ .
(3)
Since all land within the city boundary is used by households, the land rent at each location coincides with that location's equilibrium bid rent. 2.2. Behavior o f city developers
Given the national utility level U, each developer chooses the optimal city population and the optimal OPS level (E 2 or zero). The developer receives a revenue of N Y °, where N is the population of the city and y0 is the productivity of each household. The developer faces costs in providing
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public services and the m i n i m u m residential cost of the city, which is defined as the minimum of the total transport cost in the city and the city's total composite good cost for achieving the national utility level U for all N residents. That is, the developer in effect pays a wage to residents that provides residents with the national utility level U. A city developer chooses the population density, lot size at each distance r and city fringe rf, optimally. The minimum residential cost of the city, C(U, N , E ) , can be obtained from the following minimization problem: C(U, N , E ) =- rain
rf,n(r),s(r)
2
f?
[T(r) + Z(s(r), U, E)]n(r) d r ,
s.t. s(r)n(r) ~< 1 for r <~rf, and 2
fo"
n(r) dr = N ,
(4)
where n(r) represents the population density at r. Solving the minimization problem (4), we obtain the minimum residential cost of the city: 2 C ( U , N , E ) = D N I+"(E 1 "~ E) -y e t: ,
(5)
where D = 2-~(1 + f l ) - l a - a f t -t~t~" Given the above, the profit of the type A city developer is HA(U, N ) =- N Y ° - CA(U, N ) - F 1 - F 2 ,
(6)
where CA(U, N ) ~ C(U, N , E2). Since the developer choose the optimal city population, the developer's problem is expressed by max HA(U, N ) =- N Y ° - CA(U, N ) - F 1 - F 2 . N>~O
(7)
Applying the first-order condition to (7) yields the optimal population of the type A city: 3 NA(U ) = D-l/a(1 + f l ) - ' / ~ Y ° ' / ~ ( E 1 + E2) ~/~ e - v / s ,
(8)
while substituting NA(U ) into (6) we obtain the maximum profit of a type A city developer for each value of U: H A ( U , NA(U)) = GY°('+~)/~(E1 + E2) ~'/~ e -U/~ - F 1 - F 2 ,
(9)
2 For m o r e detailed calculations and an explanation of the m i n i m u m residential cost, see Fujita (1989, section 5.4). 3 SiPce dZHA/dN2< 0 for all N, uniqueness of the solution is assured.
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where G = D-1/ofl(1 + ~)-(1+~)//3. Similarly, the minimum residential cost of a type B city and the profit of the type B city developer is HB(U, N ) =- N Y ° - CB(U, N ) - F, ,
(10)
where CB(U, N ) =- C(U, N , 0). Thus the type B city developer's problem is expressed by max II~(U, N ) =- N Y ° - C , ( U , N ) - F 1 , N~o
(11)
Applying the first-order condition to (11) yields the optimal population of the type B city: NB(U ) = D -~/~(1 +/3) -l/~ y0~'~El~/t3 e-V/~ ,
(12)
while substituting NB(U ) into (10), we obtain the maximum profit of a type B city developer under each value of U: IIB(U, Na(U)) = GY°~'+~"~ E~/~ e -v/~ - F l .
(13)
Note that the developers' maximum profit for both city types is decreasing in U monotonically.4 We also note that the optimal population o f a type A city is always larger than that o f a type B city under each value of U.5 This is intuitively clear since each household in a type A city enjoys more public services while occupying less land than type B city residents. 2.3. Equilibrium f o r the independent provision m o d e l
In this subsection we examine equilibrium existence conditions for each city type. Since city developers can freely enter and exit the market, no city developer can in equilibrium obtain positive profit. First, we consider the necessary existence conditions for type B cities: IIB(U B, NB(Ua) ) = 0
(14)
IIA(U B, NA(Ua) ) ~ 0.
(15)
and
Conditions (14) and (15) mean that under the zero profit utility level of a type B city (UB) , type A cities can, at best, achieve zero profit. Solving (14) for U, we obtain the zero profit utility level of type B cities: 4This is readily shown by dHA/dU < 0 and dflB/dU < O. 5By subtracting NB(U ) in (12) from NA(U) in (8), it is obvious that NA(U) -Nn(U)>0.
K. Tsukahara / Regional Science and Urban Economics 25 (1995) 411-425
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Us =/3 log/3 - (1 +/3) log(1 +/3) + 3' log E, - log D + (1 +/3) log yO _/3 log F 1
(16)
.
Substituting (16) into Us in (15), we obtain the existence condition of type B cities: F2/F, >IE,~/a[(E2 + E,) v/~ - E[/a].
(17)
The existence condition for type B cities, expression (17), requires that the relative cost of OPS (F2/F1) is large. Intuitively, given a fixed value of F2, a small FI results in a small equilibrium city population. Thus, the per capita cost (F2/NB) of providing OPS is too large for small population cities to bear. Satisfying the following conditions is necessary for the existence of type A cities: //A(G,
(a8)
= o
and (19)
IIs(U A, NB(UA) ) <~0 .
Solving (18)for U we obtain the zero profit utility level of a type A city: UA = fl log 13 - (1 +/3) log(1 +/3) + 3' log(E2 + El)
-
-
+ (1 +/3) log y0 _/3 log(F1 + F2).
log D (20)
Substituting (20) into UA in (19), we obtain the existence condition for type A cities: F2/F1<~ E-(v/~[(E2 + E,) :'/~ - E;/~].
(21)
This existence condition (21) requires that the relative cost of OPS (F2/FI) is small. Intuitively, given a fixed value of F2, a large F1 results in a large equilibrium city population that reduces the per capita cost (F2/NA) of providing OPS. Combining existence conditions of each city type, (17) and (21), we can conclude as follows: Proposition 1. Only type A (B) cities exist in the economy when the fixed cost o f optional public services is relatively small (large). That is, type A (B ) cities exist when the following relation holds: <~, E-V/t3r,E F2/F,(>~)' , It 2 + E, )w~
-
ET'~]
•
(22)
We note that type A and type B cities coexist only when F 2/F, = E[v/~[(E2 + E,)V/~ _ E~/tJl.
(23)
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Type A City Existence
I O
Type B City Existence
I (F2/FI)*
)
F2/F1
Fig. 2. Parameter space (F2/F~) of each city type existence.
From the above we can depict the parameter space F 2/F~ of each city type existence as in Fig. 2, where (F2/F~)* = E~/~[(E2 + El) ~/~ - E~/~]. This result can be explained intuitively as follows. The optimal population of the city increases with F~.6 Thus, given a fixed value of F2, the per capita cost of providing OPS decreases with F 1 so that providing OPS becomes advantageous.
3. The joint provision model In this section we examine the circumstances under which joint provision of OPS is optimal. To do so we must modify some assumptions of the preceding model. We assume that a city developer can develop two cities, called joint cities, which have their CBD and EPS but which share OPS. 7 We assume that the OPS facilities are located at a CBD of one of the cities. We further assume that residents of the city in which OPS facilities are located (which we call a type C1 city) enjoy the full effect of the OPS while residents of the city in which the OPS facilities are not located (which we call a type C2 city) experience a lower level of benefit. We note that developers have three alternatives: to be a type A city developer (providing OPS independently); to be a type B city developer (not providing OPS); and to be a joint cities developer (providing OPS and forming joint cities). Thus, we change Assumption 3 as follows:
Assumption 3A (Full and partial service level of OPS). OPS have a spill-over effect only when they are provided by joint cities. The level of OPS varies in cities, but is the same within a city. We define the service level of OPS in a type C1 city by E2, while that of a type C2 city by cE 2, where 0 < c < 1. The coefficient c is called the 'spill-over coefficient'. 6 This can be readily shown by dNB(UB)/dF ~> O. 7 In reality m a n y cases of joint provision are observed in urban areas consisting of one central city where a C B D locates and m a n y surrounding cities which do not have a CBD. We treat these urban areas as cities, and do not consider these joint provision cases in this paper.
K. Tsukahara I Regional Science and Urban Economics 25 (1995) 411-425
419
Service Level of OPS I E0
j
E2
I I I I ! I rfl
I
cE 2
I I I I rf I
I -) ~" O1
cE 2
I -) 4" 02
i I t I I rf2
r f2
01 : CBD of Type C1 City 0 2 : CBD of Type C2 City rfl : Boundary of Type C1 City
rf2 : Boundary of Type C2 City
Fig. 3. Formation of cities and service level of OPS.
The physical structure of two cities providing OPS jointly is given in Fig. 3. 3.1. Behaviour o f joint city developers
A joint cities developer chooses the population density, lot size at each distance r and city fringe rf of each type city to maximize profits. Therefore, residential choice behavior in each city type can be obtained by similar procedures as in Subsection 2.1. Then the minimum residential cost of a type C1 city and a type C2 city are G(U, N ) =- C(U, N, E2)
(241)
C2(U, N ) -- C(U, N, cE2).
(251)
and
Given the above, the profit of a type C city developer is / / c ( U , NI, N2) --= (N, + N2)Y ° -
CI(U , N,) - C2(U, N2) - 2 F 1
-
-
F2 ,
(261) where N1(N2) represents the population of a type C1 (C2) city.
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Since the developer chooses city population to maximize profits, the developer's problem is expressed by max
N 1~>0,N2~>0
H c ( U , N 1 , N 2 ) - ~ ( N 1 +N2)Y ° - CI(U, NI) - C2(U, N2) -
2F 1 -
F2 .
(27)
Applying the first-order condition to (27) for N 1 and N 2 yields the optimal population set of the joint cities as follows: NI(U ) = D-'/~(1 + fl)-'/~Y°X/O(E~ + E2) r/¢ e -v/~
(28)
N2(U ) = D-'/~(1
(29)
and +
fl)-l/13Y°l/~(E 1 + cE2) ~/~ e - w ~ .
Substituting N1(U ) and N2(U ) into N~ and N 2, respectively, in obtain the maximum profit for each value of U :
(26),
we
n c ( U , NI(U), N2(U)) = G Y °`x+~,'~ e-W~{(E1 + E2) w~ + (E 1 + c E 2 ) v/13 } - 2F 1 - V 2 .
(30)
3.2. Joint provision m o d e l equilibrium conditions
In this subsection we examine the existence conditions for each type A, B and joint cities (type C1 and C2 cities). Free entry and exit of city developers implies that, in equilibrium, no city developer can obtain a positive profit. The necessary conditions for the existence of type B cities are
riB(Us, N . ( U . ) )
= O,
(31)
uA(us, NA(Us)) <-o
(32)
H c ( U B , N , ( U B ) , Nz(UB) ) <- O .
(33)
and
That is, given the zero-profit utility level for type B cities (Us), type A city developers and joint cities developers cannot earn a positive profit. The three conditions together yield that F 2 / F 1 >! E?~'/~[(E2 + El) ~/~ - E;/~]
(34)
F 2 / F 1 >! E?v/I3[(E, + E2) v/t~ + (E, + cE2) ~/t~ - 2E~'/~].
(35)
and
K. Tsukahara / Regional Science and Urban Economics 25 (1995) 411-425
421
Since the RHS of (34) is always smaller than the RHS of (35), type B cities can exist only when condition (35) holds. Similarly, the following conditions must be satisfied for the existence of type A cities:
rl,, (VA, NAuD) = o, N (UA))
0
(36) (37)
and
rlc(WA, N,(UA), U2(WA)) O,
(38)
which can hold simultaneously if and only if F2/F 1 <~ (E l + cE2)-r/~[(E~ + E2) ~/~ - (E~ + cE2)~'~].
(39)
Therefore, type A cities can exist only when condition (39) holds. Finally, we examine the existence conditions for the joint cities, which are given by HA(Uc, NA(Uc) ) ~<0,
(40)
I I . ( U c , N . ( U c ) ) <~0
(41)
Hc(U c, N, (Uc), N2(Uc) ) = O .
(42)
and
Solving (42) for U, we obtain the zero profit utility level of joint cities (Uc) as
U c =/3 log/3 - (1 +/3) log(1 +/3) - log D + (1 +/3) log y0 +/3 log[(E~ + E2) r/~ + (E 1 + cE2) ~/~] - / 3 log(2F~ + F2).
(43)
Substituting (43) into Uc in (40) and (41), we conclude that joint cities can exist only when the following two conditions hold: F2/F , <~E?~'/~[(E1 + E2) ~'/~ + (E, + cE2) ~/t3 - 2E~'/t~]
(44)
F 2 / f l ~ ( E l --~ c E 2 ) - # ~ [ ( E 1 ~- E2) y/o - ( E l --[-cE2)Y/o] .
(45)
and
Combining the above existence conditions of each city type gives us following proposition: Proposition 2. Given the three possible types o f cities, we have that: (i) type A cities exist only when (39) holds; (ii) type B cities exist only when (35) holds; and (iii) joint cities exist only when both (44) and (45) hold.
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That is, only type B cities, type A cities, or type C1 and C2 cities exist in the economy when the fixed cost of optional public services is relatively 'large', 'small', or 'intermediate', respectively. Thus we can depict the parameter space (F2/F1, c) for the existence of each of these city types as in Fig. 4, where (F2/F1)* = Elr/~[(E2 + El) ~/~ - E~/~]. Intuitively, if F 1 is small, then the optimal population of each city type is small and it is not advantageous to provide OPS in smaller population cities. As F~ increases, the optimal population of the city increases. When the population of the city is intermediate, it may be advantageous to jointly provide OPS by sharing the cost of them. At this point the cost of OPS is still too large for cities to bear independently. Then, as F 1 increases further, the optimal population of each city becomes sufficiently large to cover the cost of OPS independently. 3.3. Numerical examples In this subsection we carry out the following simulation analysis in order to obtain explicit solutions. We can solve for seven unknown endogenous variables, that is, the zero profit utility level of each city type UA, UB and F21F1
I I
Type B City Existence
(F21F1)*
Type A C i t y ~ Existence
I ~.
0
c
t
Fig. 4. Parameter space (F2/F1,c) of each city type existence.
K. Tsukahara / Regional Science and Urban Economics 25 (1995) 411-425
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Table 1 Results of simulation
2.5 5.0 10.0
%
~
~
~(%)
~(~)
~(~)
~(~)
3.20 2.85 2.51
3.00 2.85 2.65
3.10 2.90 2.64
0.75 1.50 3.00
2.25 3.00 4.50
1.82 2.73 4.55
1.18 1.77 2.95
Uc, and the optimal population of each type city NA, NB, N 1 and N 2. Then, we compare UA, Ua and Uc. First, we fix the values of the constants a, [3, ~, c, t, E l, E 2, F 2 and y0: a=[3=3,=0.5, F2 = 5
and
c=0.3,
t-=l.0,
E l=l.0,
E2=1.0,
y0=10.
Then we analyze the zero profit utility level with the cost of EPS, F~. The results of the simulation are shown in Table 1. Only those numbers with an underline can exist in equilibrium. Table 1 confirms the results in the previous subsection. It shows that, when the cost of essential public services, F~, is small, then type B cities are formed with small populations. As F~ increases, forming joint cities become advantageous for medium size cities. Finally, a large F~ leads to the formation of type A cities with large populations.
4. Conclusion
The main objective of this paper has been to explain the relationship between types of public services and city size. This paper has modeled the provision of optional public services from two perspectives. First, to determine under what conditions we may expect to observe cities providing these services, and second, to suggest under what conditions cities will jointly provide optional public services. Cities exist in the model because of scale economies in the production of essential public services, while optional public services are provided only if they increase the utility of residents relative to that of non-provision. To model the conditions under which we would expect to observe cities with optional public services, we assumed that profit-maximizing city deyelopers choose city size, provide essential public services and decide whether or not to provide optional public services. We assumed that the cost of providing essential public services is a fixed cost. Thus, given this assumption, when the fixed cost o f essential public services is large, the optimal size o f the city becomes large because their pet capita costs decline
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with city size. Large cities also mean small per capita costs in the provision of optional public services, thus making their provision more likely. Conversely, when the fixed cost of essential public services is small, the optimal city size is small and the per capita cost of providing optional public services is large, thus making their provision less likely. The model was then extended to analyze the conditions under which cities jointly provide optional public services. We assumed that two cities can share the costs of providing the services and that the city residents in which the facilities of the services are located receive greater benefit than residents of the other city. This extension results in a third equilibrium in which joint provision is possible. This third equilibrium is intermediate to the independent provision and non-provision cases. It is interesting to note that when joint provision takes place, the optimal population of the city in which the facilities of the optional public services are located is larger than that of the city receiving lower spill-over benefits. The model thus provides an explanation for variations in city size even when cities are otherwise identical. A number of restricting assumptions were introduced into the model for analytical convenience. Relaxing these restrictions would provide a number of extensions of this model. For example, we limit the number of cities under which joint provision can take place to two, when it may be more realistic to allow the number of cities under joint provision to be endogenous. We also assume that the spill-over effect of the jointly provided service is a parameter when in fact the service level of many types of public services are distance sensitive. Finally, we assumed that optional public services exists only at the city level. However, in reality there are many levels of public services (e.g. national, regional and city level) and introducing optional public service hierarchies would be an important extension of this model.
Acknowledgements The author thanks Masahisa Fujita, John M. Tofflemire, and anonymous referees for their valuable comments on an earlier draft of this paper.
References Abdel-Rahman, H.M., and M. Fujita, 1993, Specialization and diversificationin a system of cities, Journal of Urban Economics 33, 189-222. Arnott, R.J., 1979, Optimal city size in a spatial economy, Journal of Urban Economics 6, 65-89.
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Brueckner, J.K. and K. Lee, 1991, Economies of scope and multiproduct clubs, Public Finance Quarterly 19, 193-208. Fujita, M., 1989, Urban economic theory: Land use and city size (Cambridge University Press, Cambridge). Kuroda, T., 1989, Location of public facilities with spillover effects: Variable location and parametric scale, Journal of Regional Science 29, 575-594. Sakashita, N., 1987, Optimum location of public facilities under the influence of the land market, Journal of Regional Science 27, 1-12. Tsukahara, K., 1993, Efficient provision of multiple public good types in a system of cities, Ph.D. dissertation, University of Pennsylvania, PA.
Independent and joint provision of optional public services Kenichi Tsukahara* Ministry of Construction, 2-I-3 Kasumigaseki Chiyodaku, Tokyo 100, Japan Received December 1993; final version received October 1994
Abstract While 'essential' public services such as police and fire protection are provided in virtually every city, 'optional' public services, such as museums or stadiums, are often found in larger cities. By using a simplified urban land use theory model, this paper examines why optional public services are offered and why cities organize to jointly provide these services. Results suggest that optional public services are provided when their cost is 'small' relative to the cost of essential public services. An expansion of the model suggests that when optional public services are jointly provided, cities vary in size as model parameters vary. Keywords: Public services provision choice; Joint provision; City size JEL classification: R12; H42
1. Introduction L a r g e cities are often o b s e r v e d to provide public services such as stadiums, m u s e u m s and c o n v e n t i o n centers, in m a n y cities independently, while small cities often do not. In o t h e r words, some public services are o p t i o n a l in that they are not essential for a city's existence. Small cities do h o w e v e r occasionally provide optional public services. This m a y occur w h e n * Correspondence to: Kenichi Tsukahara, c/o Embassy of Japan, Indonesia, JL. M.H. Thamrin 24, Jakarta, Indonesia. Tel: +62-21-325140; Fax: +62-21-325460. 0166-0462/95/$09.50 ~ 1995 Elsevier Science B.V. All rights reserved SSDI 0166-0462(95)02093-4
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a coalition of cities jointly provides the services to reduce the financial burden experienced by any single city. City governments therefore face three optional public services alternative provision choices: non-provision, independent provision, and joint provision. This paper uses urban land use theory and club theory to examine the conditions under which cities optimally choose one of these three alternatives. Arnott (1979) introduced the provision of local public goods into an urban land use model examining optimal city size supporting an allocation of households. Sakashita (1987) introduced models of urban land use in public good location. His insightful work suggested that a land market can produce a socially optimal public facility location. Kuroda (1989) extended Sakashita's model to the two-city joint provision case. Abdel-Rahman and Fujita (1993) provide an approach that, while not directly related to the provision of public goods, is applicable to explain variety in public good provision within cities and across city types. In the context of club theory, Brueckner and Lee (1991) analyze multiproduct club optimality in the presence of economies of scope. They formally analyze trade-offs between cost-complementary gains and efficiency losses between single and multiproduct clubs. This paper assumes the existence of essential (EPS) and optional (OPS) public services types. EPS are essential in that a city must provide the services in order for the city to exist. In contrast, a city can exist without OPS and provides them only when their presence increases city residents' utility. We analyze various city system models in which cities are formed by profit-maximizing developers. We initially assume, in Section 2, that cities cannot organize to provide OPS jointly. OPS are provided (not provided) if the cost is 'small' ('large') relative to that of EPS. In Section 3 we allow for the joint provision of OPS in a two-city model. We assume under joint provision that the OPS facilities are physically located in one of the cities and that this city receives full benefit from OPS while the other city receives a partial spill-over benefit. This joint provision occurs when its cost is 'intermediate' relative to that of EPS. In the joint provision case the city in which OPS are located is larger in population than the city receiving OPS spill-over benefits. Section 4 concludes the paper and implications for future research directions are given.
2. The independent provision model We consider a closed economy consisting of a system of linear cities whose total population is given by ,~. We assume that all households in the economy are identical and each household works in and commutes to a central business district (CBD). All households have the same productivity,
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yO > 0, which is fixed exogenously. Each household is free to choose the city in which it resides as well as its location within the city. There are two types of public services, EPS and OPS. EPS must be provided in every city while OPS are optional. In addition we introduce the following set of assumptions:
Assumption 1 (Linear city). Cities are built along a long, narrow strip of land with unit width. Each city has a central business district (CBD) to which all households in the city commute to work. The city expands in both directions from the CBD. The formation of the city is depicted in Fig. 1.
Assumption 2 (Service level and cost of EPS). The service level of EPS, E 1 > 0, and the cost of EPS, F 1 > 0, are given exogenously.
Assumption 3 (Service level and cost of OPS). The service level of OPS is expressed by E. When OPS are provided, E = E 2 > 0, and the cost of OPS is F 2 > 0, where E 2 and F 2 are given exogenously.1 Otherwise E = 0.
Assumption 4 (Common utility function). All households in the economy have identical utility functions U(z,s, E), where z is the amount of consumption of the composite good, and s the lot size of the house. The composite good is chosen as the numeraire, so its price is unity. In order to obtain explicit results we use the following utility function: log U(z, s, E) =-a log z +/3 logs + 3' log(E1 + E ) ,
(1)
where a > O , / 3 > 0 , 3' > O a n d a + / 3 = l .
Assumption 5 (Linear transport cost). The transport cost function is given by T(r)= tr, where t > 0, while r represents household distance from the CBD. r
< I rf
I CBD
I )
r
rf
rf : Boundary of City Fig. 1. Formation of a city.
1The main problem of this paper is whether OPS are provided independently, provided jointly, or not provided. Determining the investment level of OPS would further complicate the analysis without adding much to the understanding of the problem. For an analysis of the variable investment level of OPS for the independent provision case, see Tsukahara (1993).
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We suppose that each city will be developed by a profit-maximizing city developer who can freely enter or exit the market. City developers rent the land for the city at the opportunity cost, which for simplicity is normalized to zero. Each city developer chooses the population size of the city, N, and determines the public services provision so as to maximize its city development profit. If a city provides both EPS and OPS, we call it a type A city; otherwise, we call it a type B city. As noted above, households choose a city that maximizes their utility level. Thus, in equilibrium, the utility level will be equalized across all cities in the economy. We denote this common utility level by U, which is called the national utility level. We also assume that the number of households in any developed city is small compared with the economy's total population and that a sufficiently large number of cities are developed. This implies that each developer considers that its behavior does not affect the market. Thus even though the equilibrium utility level will be determined endogenously, each developer behaves as a utility taker. 2.1. The residential choice behavior
Based on these assumptions, we state the behavior of each household. Each household freely chooses the city where its resides, its location within the city, the amount of land it occupies and the amount of consumption of the composite good. The bid rent function is given by qj(r, U , E ) = m a x
{ Y-
T ( r ) - Z(s, U , E )
= a~/t~[3(r - T(r))t/~(E1 + E ) ~/~ e -U/s ,
(2)
where Z ( s , U , E ) is the solution to U ( z , s , E ) = - a l o g z + ~ l o g s + 3' log(El + E) = U for z, and Y is the income of each household. Then we define the bid-max lot size as follows: s(r, U, E ) = [3(Y - T(r))/qJ(r, U, E ) = a - ~ / t 3 ( r - T(r))-"/t3(E 1 + E ) -~/~ e U/t~ .
(3)
Since all land within the city boundary is used by households, the land rent at each location coincides with that location's equilibrium bid rent. 2.2. Behavior o f city developers
Given the national utility level U, each developer chooses the optimal city population and the optimal OPS level (E 2 or zero). The developer receives a revenue of N Y °, where N is the population of the city and y0 is the productivity of each household. The developer faces costs in providing
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public services and the m i n i m u m residential cost of the city, which is defined as the minimum of the total transport cost in the city and the city's total composite good cost for achieving the national utility level U for all N residents. That is, the developer in effect pays a wage to residents that provides residents with the national utility level U. A city developer chooses the population density, lot size at each distance r and city fringe rf, optimally. The minimum residential cost of the city, C(U, N , E ) , can be obtained from the following minimization problem: C(U, N , E ) =- rain
rf,n(r),s(r)
2
f?
[T(r) + Z(s(r), U, E)]n(r) d r ,
s.t. s(r)n(r) ~< 1 for r <~rf, and 2
fo"
n(r) dr = N ,
(4)
where n(r) represents the population density at r. Solving the minimization problem (4), we obtain the minimum residential cost of the city: 2 C ( U , N , E ) = D N I+"(E 1 "~ E) -y e t: ,
(5)
where D = 2-~(1 + f l ) - l a - a f t -t~t~" Given the above, the profit of the type A city developer is HA(U, N ) =- N Y ° - CA(U, N ) - F 1 - F 2 ,
(6)
where CA(U, N ) ~ C(U, N , E2). Since the developer choose the optimal city population, the developer's problem is expressed by max HA(U, N ) =- N Y ° - CA(U, N ) - F 1 - F 2 . N>~O
(7)
Applying the first-order condition to (7) yields the optimal population of the type A city: 3 NA(U ) = D-l/a(1 + f l ) - ' / ~ Y ° ' / ~ ( E 1 + E2) ~/~ e - v / s ,
(8)
while substituting NA(U ) into (6) we obtain the maximum profit of a type A city developer for each value of U: H A ( U , NA(U)) = GY°('+~)/~(E1 + E2) ~'/~ e -U/~ - F 1 - F 2 ,
(9)
2 For m o r e detailed calculations and an explanation of the m i n i m u m residential cost, see Fujita (1989, section 5.4). 3 SiPce dZHA/dN2< 0 for all N, uniqueness of the solution is assured.
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where G = D-1/ofl(1 + ~)-(1+~)//3. Similarly, the minimum residential cost of a type B city and the profit of the type B city developer is HB(U, N ) =- N Y ° - CB(U, N ) - F, ,
(10)
where CB(U, N ) =- C(U, N , 0). Thus the type B city developer's problem is expressed by max II~(U, N ) =- N Y ° - C , ( U , N ) - F 1 , N~o
(11)
Applying the first-order condition to (11) yields the optimal population of the type B city: NB(U ) = D -~/~(1 +/3) -l/~ y0~'~El~/t3 e-V/~ ,
(12)
while substituting NB(U ) into (10), we obtain the maximum profit of a type B city developer under each value of U: IIB(U, Na(U)) = GY°~'+~"~ E~/~ e -v/~ - F l .
(13)
Note that the developers' maximum profit for both city types is decreasing in U monotonically.4 We also note that the optimal population o f a type A city is always larger than that o f a type B city under each value of U.5 This is intuitively clear since each household in a type A city enjoys more public services while occupying less land than type B city residents. 2.3. Equilibrium f o r the independent provision m o d e l
In this subsection we examine equilibrium existence conditions for each city type. Since city developers can freely enter and exit the market, no city developer can in equilibrium obtain positive profit. First, we consider the necessary existence conditions for type B cities: IIB(U B, NB(Ua) ) = 0
(14)
IIA(U B, NA(Ua) ) ~ 0.
(15)
and
Conditions (14) and (15) mean that under the zero profit utility level of a type B city (UB) , type A cities can, at best, achieve zero profit. Solving (14) for U, we obtain the zero profit utility level of type B cities: 4This is readily shown by dHA/dU < 0 and dflB/dU < O. 5By subtracting NB(U ) in (12) from NA(U) in (8), it is obvious that NA(U) -Nn(U)>0.
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Us =/3 log/3 - (1 +/3) log(1 +/3) + 3' log E, - log D + (1 +/3) log yO _/3 log F 1
(16)
.
Substituting (16) into Us in (15), we obtain the existence condition of type B cities: F2/F, >IE,~/a[(E2 + E,) v/~ - E[/a].
(17)
The existence condition for type B cities, expression (17), requires that the relative cost of OPS (F2/F1) is large. Intuitively, given a fixed value of F2, a small FI results in a small equilibrium city population. Thus, the per capita cost (F2/NB) of providing OPS is too large for small population cities to bear. Satisfying the following conditions is necessary for the existence of type A cities: //A(G,
(a8)
= o
and (19)
IIs(U A, NB(UA) ) <~0 .
Solving (18)for U we obtain the zero profit utility level of a type A city: UA = fl log 13 - (1 +/3) log(1 +/3) + 3' log(E2 + El)
-
-
+ (1 +/3) log y0 _/3 log(F1 + F2).
log D (20)
Substituting (20) into UA in (19), we obtain the existence condition for type A cities: F2/F1<~ E-(v/~[(E2 + E,) :'/~ - E;/~].
(21)
This existence condition (21) requires that the relative cost of OPS (F2/FI) is small. Intuitively, given a fixed value of F2, a large F1 results in a large equilibrium city population that reduces the per capita cost (F2/NA) of providing OPS. Combining existence conditions of each city type, (17) and (21), we can conclude as follows: Proposition 1. Only type A (B) cities exist in the economy when the fixed cost o f optional public services is relatively small (large). That is, type A (B ) cities exist when the following relation holds: <~, E-V/t3r,E F2/F,(>~)' , It 2 + E, )w~
-
ET'~]
•
(22)
We note that type A and type B cities coexist only when F 2/F, = E[v/~[(E2 + E,)V/~ _ E~/tJl.
(23)
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Type A City Existence
I O
Type B City Existence
I (F2/FI)*
)
F2/F1
Fig. 2. Parameter space (F2/F~) of each city type existence.
From the above we can depict the parameter space F 2/F~ of each city type existence as in Fig. 2, where (F2/F~)* = E~/~[(E2 + El) ~/~ - E~/~]. This result can be explained intuitively as follows. The optimal population of the city increases with F~.6 Thus, given a fixed value of F2, the per capita cost of providing OPS decreases with F 1 so that providing OPS becomes advantageous.
3. The joint provision model In this section we examine the circumstances under which joint provision of OPS is optimal. To do so we must modify some assumptions of the preceding model. We assume that a city developer can develop two cities, called joint cities, which have their CBD and EPS but which share OPS. 7 We assume that the OPS facilities are located at a CBD of one of the cities. We further assume that residents of the city in which OPS facilities are located (which we call a type C1 city) enjoy the full effect of the OPS while residents of the city in which the OPS facilities are not located (which we call a type C2 city) experience a lower level of benefit. We note that developers have three alternatives: to be a type A city developer (providing OPS independently); to be a type B city developer (not providing OPS); and to be a joint cities developer (providing OPS and forming joint cities). Thus, we change Assumption 3 as follows:
Assumption 3A (Full and partial service level of OPS). OPS have a spill-over effect only when they are provided by joint cities. The level of OPS varies in cities, but is the same within a city. We define the service level of OPS in a type C1 city by E2, while that of a type C2 city by cE 2, where 0 < c < 1. The coefficient c is called the 'spill-over coefficient'. 6 This can be readily shown by dNB(UB)/dF ~> O. 7 In reality m a n y cases of joint provision are observed in urban areas consisting of one central city where a C B D locates and m a n y surrounding cities which do not have a CBD. We treat these urban areas as cities, and do not consider these joint provision cases in this paper.
K. Tsukahara I Regional Science and Urban Economics 25 (1995) 411-425
419
Service Level of OPS I E0
j
E2
I I I I ! I rfl
I
cE 2
I I I I rf I
I -) ~" O1
cE 2
I -) 4" 02
i I t I I rf2
r f2
01 : CBD of Type C1 City 0 2 : CBD of Type C2 City rfl : Boundary of Type C1 City
rf2 : Boundary of Type C2 City
Fig. 3. Formation of cities and service level of OPS.
The physical structure of two cities providing OPS jointly is given in Fig. 3. 3.1. Behaviour o f joint city developers
A joint cities developer chooses the population density, lot size at each distance r and city fringe rf of each type city to maximize profits. Therefore, residential choice behavior in each city type can be obtained by similar procedures as in Subsection 2.1. Then the minimum residential cost of a type C1 city and a type C2 city are G(U, N ) =- C(U, N, E2)
(241)
C2(U, N ) -- C(U, N, cE2).
(251)
and
Given the above, the profit of a type C city developer is / / c ( U , NI, N2) --= (N, + N2)Y ° -
CI(U , N,) - C2(U, N2) - 2 F 1
-
-
F2 ,
(261) where N1(N2) represents the population of a type C1 (C2) city.
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Since the developer chooses city population to maximize profits, the developer's problem is expressed by max
N 1~>0,N2~>0
H c ( U , N 1 , N 2 ) - ~ ( N 1 +N2)Y ° - CI(U, NI) - C2(U, N2) -
2F 1 -
F2 .
(27)
Applying the first-order condition to (27) for N 1 and N 2 yields the optimal population set of the joint cities as follows: NI(U ) = D-'/~(1 + fl)-'/~Y°X/O(E~ + E2) r/¢ e -v/~
(28)
N2(U ) = D-'/~(1
(29)
and +
fl)-l/13Y°l/~(E 1 + cE2) ~/~ e - w ~ .
Substituting N1(U ) and N2(U ) into N~ and N 2, respectively, in obtain the maximum profit for each value of U :
(26),
we
n c ( U , NI(U), N2(U)) = G Y °`x+~,'~ e-W~{(E1 + E2) w~ + (E 1 + c E 2 ) v/13 } - 2F 1 - V 2 .
(30)
3.2. Joint provision m o d e l equilibrium conditions
In this subsection we examine the existence conditions for each type A, B and joint cities (type C1 and C2 cities). Free entry and exit of city developers implies that, in equilibrium, no city developer can obtain a positive profit. The necessary conditions for the existence of type B cities are
riB(Us, N . ( U . ) )
= O,
(31)
uA(us, NA(Us)) <-o
(32)
H c ( U B , N , ( U B ) , Nz(UB) ) <- O .
(33)
and
That is, given the zero-profit utility level for type B cities (Us), type A city developers and joint cities developers cannot earn a positive profit. The three conditions together yield that F 2 / F 1 >! E?~'/~[(E2 + El) ~/~ - E;/~]
(34)
F 2 / F 1 >! E?v/I3[(E, + E2) v/t~ + (E, + cE2) ~/t~ - 2E~'/~].
(35)
and
K. Tsukahara / Regional Science and Urban Economics 25 (1995) 411-425
421
Since the RHS of (34) is always smaller than the RHS of (35), type B cities can exist only when condition (35) holds. Similarly, the following conditions must be satisfied for the existence of type A cities:
rl,, (VA, NAuD) = o, N (UA))
0
(36) (37)
and
rlc(WA, N,(UA), U2(WA)) O,
(38)
which can hold simultaneously if and only if F2/F 1 <~ (E l + cE2)-r/~[(E~ + E2) ~/~ - (E~ + cE2)~'~].
(39)
Therefore, type A cities can exist only when condition (39) holds. Finally, we examine the existence conditions for the joint cities, which are given by HA(Uc, NA(Uc) ) ~<0,
(40)
I I . ( U c , N . ( U c ) ) <~0
(41)
Hc(U c, N, (Uc), N2(Uc) ) = O .
(42)
and
Solving (42) for U, we obtain the zero profit utility level of joint cities (Uc) as
U c =/3 log/3 - (1 +/3) log(1 +/3) - log D + (1 +/3) log y0 +/3 log[(E~ + E2) r/~ + (E 1 + cE2) ~/~] - / 3 log(2F~ + F2).
(43)
Substituting (43) into Uc in (40) and (41), we conclude that joint cities can exist only when the following two conditions hold: F2/F , <~E?~'/~[(E1 + E2) ~'/~ + (E, + cE2) ~/t3 - 2E~'/t~]
(44)
F 2 / f l ~ ( E l --~ c E 2 ) - # ~ [ ( E 1 ~- E2) y/o - ( E l --[-cE2)Y/o] .
(45)
and
Combining the above existence conditions of each city type gives us following proposition: Proposition 2. Given the three possible types o f cities, we have that: (i) type A cities exist only when (39) holds; (ii) type B cities exist only when (35) holds; and (iii) joint cities exist only when both (44) and (45) hold.
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That is, only type B cities, type A cities, or type C1 and C2 cities exist in the economy when the fixed cost of optional public services is relatively 'large', 'small', or 'intermediate', respectively. Thus we can depict the parameter space (F2/F1, c) for the existence of each of these city types as in Fig. 4, where (F2/F1)* = Elr/~[(E2 + El) ~/~ - E~/~]. Intuitively, if F 1 is small, then the optimal population of each city type is small and it is not advantageous to provide OPS in smaller population cities. As F~ increases, the optimal population of the city increases. When the population of the city is intermediate, it may be advantageous to jointly provide OPS by sharing the cost of them. At this point the cost of OPS is still too large for cities to bear independently. Then, as F 1 increases further, the optimal population of each city becomes sufficiently large to cover the cost of OPS independently. 3.3. Numerical examples In this subsection we carry out the following simulation analysis in order to obtain explicit solutions. We can solve for seven unknown endogenous variables, that is, the zero profit utility level of each city type UA, UB and F21F1
I I
Type B City Existence
(F21F1)*
Type A C i t y ~ Existence
I ~.
0
c
t
Fig. 4. Parameter space (F2/F1,c) of each city type existence.
K. Tsukahara / Regional Science and Urban Economics 25 (1995) 411-425
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Table 1 Results of simulation
2.5 5.0 10.0
%
~
~
~(%)
~(~)
~(~)
~(~)
3.20 2.85 2.51
3.00 2.85 2.65
3.10 2.90 2.64
0.75 1.50 3.00
2.25 3.00 4.50
1.82 2.73 4.55
1.18 1.77 2.95
Uc, and the optimal population of each type city NA, NB, N 1 and N 2. Then, we compare UA, Ua and Uc. First, we fix the values of the constants a, [3, ~, c, t, E l, E 2, F 2 and y0: a=[3=3,=0.5, F2 = 5
and
c=0.3,
t-=l.0,
E l=l.0,
E2=1.0,
y0=10.
Then we analyze the zero profit utility level with the cost of EPS, F~. The results of the simulation are shown in Table 1. Only those numbers with an underline can exist in equilibrium. Table 1 confirms the results in the previous subsection. It shows that, when the cost of essential public services, F~, is small, then type B cities are formed with small populations. As F~ increases, forming joint cities become advantageous for medium size cities. Finally, a large F~ leads to the formation of type A cities with large populations.
4. Conclusion
The main objective of this paper has been to explain the relationship between types of public services and city size. This paper has modeled the provision of optional public services from two perspectives. First, to determine under what conditions we may expect to observe cities providing these services, and second, to suggest under what conditions cities will jointly provide optional public services. Cities exist in the model because of scale economies in the production of essential public services, while optional public services are provided only if they increase the utility of residents relative to that of non-provision. To model the conditions under which we would expect to observe cities with optional public services, we assumed that profit-maximizing city deyelopers choose city size, provide essential public services and decide whether or not to provide optional public services. We assumed that the cost of providing essential public services is a fixed cost. Thus, given this assumption, when the fixed cost o f essential public services is large, the optimal size o f the city becomes large because their pet capita costs decline
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with city size. Large cities also mean small per capita costs in the provision of optional public services, thus making their provision more likely. Conversely, when the fixed cost of essential public services is small, the optimal city size is small and the per capita cost of providing optional public services is large, thus making their provision less likely. The model was then extended to analyze the conditions under which cities jointly provide optional public services. We assumed that two cities can share the costs of providing the services and that the city residents in which the facilities of the services are located receive greater benefit than residents of the other city. This extension results in a third equilibrium in which joint provision is possible. This third equilibrium is intermediate to the independent provision and non-provision cases. It is interesting to note that when joint provision takes place, the optimal population of the city in which the facilities of the optional public services are located is larger than that of the city receiving lower spill-over benefits. The model thus provides an explanation for variations in city size even when cities are otherwise identical. A number of restricting assumptions were introduced into the model for analytical convenience. Relaxing these restrictions would provide a number of extensions of this model. For example, we limit the number of cities under which joint provision can take place to two, when it may be more realistic to allow the number of cities under joint provision to be endogenous. We also assume that the spill-over effect of the jointly provided service is a parameter when in fact the service level of many types of public services are distance sensitive. Finally, we assumed that optional public services exists only at the city level. However, in reality there are many levels of public services (e.g. national, regional and city level) and introducing optional public service hierarchies would be an important extension of this model.
Acknowledgements The author thanks Masahisa Fujita, John M. Tofflemire, and anonymous referees for their valuable comments on an earlier draft of this paper.
References Abdel-Rahman, H.M., and M. Fujita, 1993, Specialization and diversificationin a system of cities, Journal of Urban Economics 33, 189-222. Arnott, R.J., 1979, Optimal city size in a spatial economy, Journal of Urban Economics 6, 65-89.
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Brueckner, J.K. and K. Lee, 1991, Economies of scope and multiproduct clubs, Public Finance Quarterly 19, 193-208. Fujita, M., 1989, Urban economic theory: Land use and city size (Cambridge University Press, Cambridge). Kuroda, T., 1989, Location of public facilities with spillover effects: Variable location and parametric scale, Journal of Regional Science 29, 575-594. Sakashita, N., 1987, Optimum location of public facilities under the influence of the land market, Journal of Regional Science 27, 1-12. Tsukahara, K., 1993, Efficient provision of multiple public good types in a system of cities, Ph.D. dissertation, University of Pennsylvania, PA.