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Spatial equilibrium with local public goods
Regional Scie~ce and Jrban Economics 7 (1977) 197-216. © North-Holland
SPATIAL E Q U I L I B R I U M W I F H L O C A L P U B L I C G O O D S
Urban Land Rent, Optimal City Size and the Tiebout Hypothesis Oscar FISCH The Ohio State University, ~lumbus, O H 43210, U.S.A.
Received M,ay 1976 This paper extends the standard model of urban land rent to consider the spatial equilibrium conditions in a local pL~blJcgoods market as hypothesized by Charles M. Tiebout. An analysis is made of the spatial dimensions of public goods, their degree of "localness' and their impact on land values, it is shown that the optimal population size of the community (Tiebout's sixth assumption) is simultaneously derived with the optimal supply of local public goods and local taxes. It is also shown that land rent is a poor output indicator of Tiebout's equilibrium conditions and that tht capitalization assumption is not the appropriate test for his hypothesis.
1. latrotlaetioa Almost two decades ago, Tiebout (1956) advanced the hypothesis that public expenditure theory simplifies itself at the local level, as people 'vote with their feet' and move to homogeneous communities. 1 This homogeneity secures through the political process- collective a c t i o n - that the urban dweller gets the tax-expenditure program that maximizes his and everybody's utility function. In examining briefly Tiebout's hypothesis, Samuelson (1958), after recognizing that this hypothesis 'goes some way toward solving the pro~lem', rejected the whole approach on three speculative grounds: first, that existing urban communities are not homogeneous and, even given that a majority cannot get rid of a minority, the existing conflict is still unresolved; second, that population heterogeneity may also be a positive argument in the individual's utihty function; and third, that there exist, the disturbing ethical question 'as to whether groups of like-minded individuals shall be free to run-out on their social responsibilities'. The last argument is clearly a distributive problem that still today disturbs many people. This argument has been used in support o:~ some recent court decisions that have shaken the foundations of local governments. The distributive dimensions were formalized b y - among o t h e r s - Coleman (1968) and 1dssuml~tio;t 3. 'There are a large number of communities in which the consumer-voter may ehc~se to live.' (Tiebout, op. cir., p. 419.)
198
O. Fisch, Spatial equilibrium with local public goods
Rawls (t 971): and an economic analysis was put forward by Arrow (1971) in a set of sharply delineated propositions. But it is Samuelson's first two arguments that question the workings of Tiebout's hypothesis in the real world. To his casual observation in 195t~ that existing urban communities are not homogeneous, there was more than casaal evidence: to the contrary, compiled, coincidentally, at the same time. Wood (1958) Jn an exhaustive study of suburban communities showed that a high degree of homogenization within communities was indeed the result of local politics due to jurisdictional fragmentation of metropolitan areas. In a later study, Williams et al. (1065), the historical trend of Wood's findings was verified and it was contended further that community specialization with respect to class or status, life-style or wealth, is the main force behind the resistance to intercommmaity cooperation (heterogeneization among communities) in metropolitan areas. Granted that the process of homogenization within urban communities is taking place- due ,~o a high degree of geographic mobility of the populationthe existing conflict in the theory of pub!it finance is reduced significantly at the local level, leaving the conflict unsolved at only higher jurisdictional levels of government, or equivalently, at broader spatial dimensions of public goods: metropolitan, state and national. After Samuelson's brief review, very little attention was paid, in theoretical terms, to Tiebout's hypothesis m,.fil two rexzent articles: one by Buchanan and Goetz (1972) and the other by Ellick~on (1973). Both formalize Tiebout's sixth assumption, 2 i.e., that, as the preference maps of the local resident-consumers are now known, the political process which leads to the desired basket of local public goods carries, implicitly,, an optimal (finite) population requirement (optimal city size). Both analyses were carried out with the implicit or explicit thesis of the 'congestion' of the public good: 'crowding' of the local public good either in the consumption side (Theory of Clubs) or in the production side. Further, in the latter case,, Ellickson stated that cr :~wding in the production of local public goods is a required condition for the existence of optimal partitions of the population. Oates (1969) advanced in an empirical study his capitalization assumption of local public expenditures and local taxes as a way of testing Tiebout's hypo2A,~sumplion 6. ' i 7 0 1 • every pattern of community service set by, say, a city man~ager ~vho follows the preferences of the older residents of the coml aunity, there is an optimal commun ity size.' (Tiebout, op. cit., p. 419.) In trying to explain his sixth assumption Tiebout argues that such a case ' . . . implk:s l[~t some factor or resource is f i x e d . . . ' . And, in order to support his case, he advanc.es three dilicrent types of exarnple~: firsL limited amot nt of residentm| land, i.e., fixed supply of a private good (land); second, institutiona~ lh~ ~ts on land density, i.e., zoning ordinawa;s constraining the consumption of land; and t b i d , congestion or crowding of a 'public good', i.e., limited capacity of the local beach (lheory of clubs), We will show in the third section theft the first type o fixity i~, suificient for this s~xth assumption to hold.
O. Fisch, Spatial equilibrium with local ~ublie goods
199
thesis. 3 Since then, many writers have criticized Oates' findings on different ~t,,xnds, but mainly in the specification of the statistical model or in statistical procedures [among others Ellickson (1971), Pollakowski (1973) and Edel and Sclar ~.lg,~t)]. However, these criticisms are raised without taking issue with the theoretizal p,-~blem of considering real e~tate values as an output indicator of the process o," co..~munity homogenization - the core of Tiebout's hypothesis. It is the purpose ~f this paper to assess Tiebout's hypothesis in a more theoretical setting. First, I want to examine the spatial equilibrium cor~ditions and urban land rents under ~he impact of local public services. Second, I want to examine the locational con~.'!'~ns of local public services and the spatial dimensions of exclusion or 'apprcoriability' so important to the mode of economic organization and its relatio, qhip to such concepts as externalities or neighbo~'hood effects (spillovers) in or~Icr to clarify definitional aspects of local public goods. Third, I want to show that ,.,dhering to Tiebout's c,riginal contention in explaining his sixth assumption i.e., that it is sufficient that some factor or resource be fixed - such as the fixed supply of residential land (private good) in the urban community-can lead to a fin/~e population as an optimal size. Fourth, I want to show that the political process ~;,hich leads to an optimal level of local public goods, simultaneously leads to an optimal size of local population, and thereby show global conditions instead of merely marginal ones for such optimal solutions. Finally, I will show that property value is a very poor output indicator of Tiebout's hypothesis. The plan of the paper is the following: in the second section 1 derive the spatial equilibrium conditions of an homogeneous population in a residential ring, reformulating a model by Solow (1973) in order to include a given spatial distribution of local public goods. Embedded in the solution is the soatial pattern of consumption of the private good (residential land) and its rent function. In the third section I will analyze the spatial characteristics of lc,~l public goods and their relation to concepts such as exclusion and spillover (neighborhood) effects. It is in this section also, that I show the simultaneous procedure used to arrive at an optimal level of public goods and of local population. Fv rther, it is shown that the fixed supply of the private good (land) is a sut~icient condition f e r a finite population. Section 4 derive:, a direct measure ¢~f the total land market val,res of the community and the average market value ~i" heusehold's ~'lf this is the c a ~ and if (as the Ticl'~ut model suggests) individuals con ;id,~: the qtlaliW ot local r~ublic services in making locational decisions, we would expect to ti~0d that, ~:~ther th~ng':' ?eing equal (ir~cluding tax rates) across communities, an increased expendilurc per pt~pil would result in higher property values.' [Oates (19¢,9, p. 962~]. Oat¢,; ~scd expenditure p~:',r pupil as a proxy for the output of educational ~rvk.'es. Later research ~im!imps have shown ~ a t both are uncorrelated. Kiesling (1967, p. 366) says ' . , , the relationshir of performance t,:, per pupil expenditure has been found to be. e×cep.~ in large urban sch¢.oi districts, di,,ai~pointingly weak. This would imply, among other things, that the u~iJi,'ati~n of per capita c,:~st figures for an index of p~b[ic performanc~ is a ~ighly dangerous prac~i,,e'. A furl d i ~ u ~ i ( m of the Oates paper would carry us far beyond our subject.
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O. Fisch, Spatial equilibrium with local publie goods
real estate holdings in that community and I show the lack of correspondence between Tiebout's hypothesis and the capitalization assumption. Finally, the last section summarizes the results of the paper.
2. The model: Spatial equilibrium with local public goods The main thrust of Tiebout's hypothesis, mainly related to his third assumption, is the homogenization of the local population as a resuR of 'voting with their feet', eLpolitical process that rests on the assumption of' high geog::aphic mobility (Tiebout's first assumption). The final output of this process is that urban dwellers will choose that urban community-among m a n y - w h i c h offers the tax-expenditure program that maximizes their welfare, thus providing the revelatory demand functions. This is in contrast with the higher jurisdictional public goods case where the individual has every reason not to provide such a function. in theoretical terms, this homogenization process simplifies the analysis of public economics at the local level, because it allows, without making violence to the analysis, the introduction of the simplifying assumption that individuals have the same income and the same utility function. This allowance circumvents the definitional difficulties of interpersonally comparable utilities which permeate and block the theoretical analysis of public goods in fgeneral. In the model we are developing here, we consider that our city is a 'city of equals': equal tastes and equal income. The model is a reformulation of the standard model of urban spatial equilibrium, a reformulat:ion that makes possible the explicit consideration of local public: goods. Physically, the city is a circular one with radius xl. It has aw inner ring with radius Xo which land is used up by the production functions c.f the CBD. The outer ring is composed of homogeneous residential land, and its total population N commutes to the CBD in order to deliver labor inputs. Every individual, besides consuming the two private goods, urban land I and the composite consumption good c, consumes allso a public good g. The utility derived from the consumption of this public good may vary with distance x ~ [ X o , X l ] . The urban dweller's utility function is additive separable, with diminishing marginal utility in every argument, u(c, 1, g(x)) = In [:lag(x)q. The urban dweller's income y after tax is assumed to be totally exhausted by the two private: goods plus commuting transportation costs, y = e + r(x)1+ t(x).
(2.2)
O. Fisch, Spatial equilibrium with Iocalpubh," goods
201
In (2.2) prices are in terms of the price of t~ e composite good. Land prices (rent) are given by r(x), an unknown function, and transportation cost~ by t(x), a known function. In maximizing (2.1) subject to (2.2), we have from the usual first-order conditions,
c = r~"
(2.3)
From (2.3) and in making the corresponding substitution into (2.2), (2.4) and (2.5) show that every individual, in each locatior x, spends a fixed proportion of his income net of transportation in urban land, allocating the rest to the consumption of the composite good.
o'O'-t(x)),
(2.4)
c(x) = ( 1 - o ) O , - t ( x ) ) ,
(2.5)
I(x)r(x) =
where o =/~/(0t+fl). Replacing (2.4) and (2.5) into the utility function (2.1), we have that u(x) = In {(1 -a)'aP(.v - t(x)) ~+Pg(xf /r(x)P} .
(2.6)
In our city of equals, and in equilibrium, no individual should have incentive to move. In other words, in equilibrium, as ev, ry tLrban dweller should be indifferent in relation to his final location, his utility must be constant over the geography of the city. Then it follows that du(x) ----. dx
=
0 =
~t+ fl dt(x) y - t ( x ) dx
fl dr(x) r(x) dx
+ .........
y dg(x) . g(x) dx
(2.7)
From (2.7) we get, in separable form, the fundamental differential equation of the urban rent function, dr(x) r(x)
1 dt(x) o y-t(x)
? dg(x) fl g(x)
(2.8)
In solvipg (2.8) we have that In r(x) = In ( y - t ( x ) ) I/~ + In g(x) ~ +C.
(2.9)
The constant of integration C is solved in terms of the values of the variablc~ involved at the boundary Xo. At this initia! point~ t(Xo) = 0 in the simpl@ing
O. Fisch, Spatial equilibrium with localpul,lt: goods
202
assumption that there are no transportation costs within the CBD. It follows then that C
=
In r ( x o ) - l n y l l ° - l n g(xo) ~'l~.
(2.1o)
From (2.9) and (2.10), we have that the unit price (ren0 of land is given by (2.4),
r(x) = r(xo)[(y- t(x))[ylt'+ P~/ffg(x)/g(xo)) rip.
(2.11)
Substituting (2.11) into (2.4) we have that the individual's consumption of land per household at distance x is given by
l(x) = a f l'g(xo)rl#/r( xo)(y-- t(x))'//J g(x)r//J.
(2.12)
The fixed supply of land at distance x, in an infinitesimal ring of width dx, is s(x), 0 < s(x) < 2nx. The demal~d for land at distance x is l(x)n(x)dx, where n(x) is the number of urban dwellers living in the infinitesimal ring of width dx. We assume that in equilibrium the market of urban land is cleared; then it follows that
s(x) dx = l(x)n(x) dx.
(2.13)
The total aggregate demand has the following integral constraint: U =
n(x) dx,
(2.14)
where N is the size of the urban population. From (2.12), (2.13) and (2.14), and making the corresponding substitution, we have that the boundary condition r(xo) is computed by solving
ag(xo)~lPfll'N-r(xo) j ~ s(x)(y- t(x))=lOg(x)~/~ dx = O.
(2.15)
[cm the equilibrium condition (2.7), the individual's utility is invariant in relation to distance x. It follows that, in computing the utility of any one individual at any particular distance, we know everybody's level of utility u at any distance xo Using Xo as that particular distance, from (2.6) we have u =
(Xo) = In
In this approach, the output of the systerr is the utility itself (2.16), ailowhlg a dir:ct measurement of benefits of public programs. ~qs. (2.1t), (2.12), (2.15) and (2.16) are the critical ones in analyzing in equilibrium the impact of any public program, that we will express in terms of g(x), population size N and distance x,
O. Fisch, Spatial equiiibrimn with local public goods
203
r(N, g(x), x) = r(N, g(x), Xo)[(y- t(x))/y] (~+tmp(g(x),'g(Xo))"ha, (2.17)
l(N g(x),x) - try[r(N,g(x),xo)[(y-t(x))]y]'iPlg(x)/,~(Xo)) ~i~, (2.18) r(N,g(x),xo) = N a y / j ~ s(x)[(y-t(x))/y]'/P(g(x)/g(.%)) r/tJdx,
(2.19)
u(N, g(x)) = In { ( 1 - a ) ' [ j'~gs(x)(y-t(xW/ag(x) '/a dx]a/Na}. (2.20) From (2.20) we will define a new welfare function,
U(N, g(x)) = I~ s ( x X y - t(x)~'/t~g(x)';tJ dx/N,
(2.21)
where
U(N, g(x)) = U[u(N, g(x))] = [e"~N'g~'/(l --¢r)~]l/t~ is a monotonic transformation of the original utility function (2.1). Eqs. (2.17), (2.18) and (2.19) give the spatial equilibrium conditions of the residential ring as functions of population size, the spatial distributioa of local public goods and distance. Eq. (2.21) gives the individual's and eve, ybody's level of welfare in the community, as a fimction of population size and the spatial distribution of public goods. We can turn our attention now to the political process- collective action- implicit in Tiebout's sixth assumption.
3. The political process: Local public goods and optimal city size In order to differentiate between the level and the spatial distribution of the public physical output on the one hand, and on the other, the degree of spatial exclusiveness of the public physical output, I will define the following: h(x) is the public physical output located in the infinitesimal ring of width dx at distance x. Then, the total public physical output G of the city is given by G ~ ~ ' h(z) dz.
(3.1)
Given (3.1), a public output density fimction can be defined as p(::) = h(z)/G. Also, a spillover function can be defined as agx, z), where w(-, z) weights the spiUout effect of the public physical output h located at z, and w(x,.) weights the spillin effect to the heusehold located at x. In the specification of this spillover function rests the complex definitional aspect of the local public good. In one direct/on, it introduces the spatial dimensions of the concept of property rights, dimensions that are re~ated to the mode of eeonomie organization. In the other direction, it introduces the degree of'localness' of the public gc,od: in ,;pite of non-rivalriness of the public good,
204
O. Fisch, Spatial equilibrium with localpublic goods
we have a loss in welfare due only to the locational fixities of both the physical output of the public program and of the consumer of the public good, a loss that is a function of the physical distance separating production and consumption. 4 Let us assume that
w ( x , z ) = 1, for all z = x ;
z,x_~xi,
(3.2)
w(x, z) = 0, for x > xl,
(3.3)
w(x,z) = 0 ,
(3.4)
for
z > xl.
Assumption (3.2) means that there are no spiUouts from our community and (3.3) means that there are no spillins into our community; both assumptions are in strict correspondance with Tiebout's fifth assumption. 5 If we assume further that w(., z) - 0,
for all x # z;
x, z _~ x t ,
(3.5)
then we have that there are no spiUovers among residential rings and that the public physical output h is supplied at z exclusive of any other ring. Because of the continuous spatial distribution of population (2.13) and (2.18) only one household is located in an infinitesimal ring of width dx at distance z. Therefore, the degree of appropriability is optimal and perfectly subject to the pricing system. It follows that g(x) = h(x) and g(x) is a publicly supplied private good in this case, a case that does not need our attention here. For a more general case, let us assume now that w(., z) > 0,
for some x # z;
x, z _~ xt.
(3.6)
(3.6) means that there are spillovers (neighborhood effects) at least among some residential rings and that the public physical output h is supplied at z without any exclusivity in relation to at least one other ring. Equivalently, for the household located at z, the degree of appropriability is not optimal and therefore I define h(z) as a public good. It does not follow that every household located at x ~ z is appropriating the total amount of h(z)° Because of the physical distance between the location of ~he public physical output at z and the location of the 4'Non-rivalness in consurrption.., does not mean that some subjective benefit must be derived, or even that precisely tt.e same product quality is available to boLh. Consumer A who lives close to the police station has better protection than B who lives far away. Yet, the two consumption acts are non-rival, and we deal with a social good.' [Musgrave (IO69,p. 126)]. ~Assumption 5. 'The public services supplied exhibit no external ecot~omiesor d~onomies be~.weencommunities? fTiebout, op. cir., p. ,Hg.)
O. Fisch, Spatial equilibrium with local public goods
205
household at x, I can also define w as a loss or distance dissipation function where O-
1, for all
z#x;
z,x~xl.
(3.7)
At every particular location x e [Xo, x~], I define the total amount of public goods g as
g(x) = ~:' w(x, z)h(z) dz;
~ E [Xo, x,].
(3.8)
And from (3.1) we have that
g(x) = 6 ~:' w(x, z)p(z)
(3.9)
dz = o r ( x ) ,
whele
v(x)=E[w(x,z)lp(z)],
for
xe[x0,x~],
0
1,
E being the operator expected value of the function w. In the particular case where there are no dissipatory effects, then ;v(x,.) = 1, for all
z ~ xl.
(3.10)
It follows from (3.9) that, in this case,
E[w(x, z) I p(z)] = 1 - , g(x)
=
G,
(3.11)
and there is no spatial impact of local expenditures on public goods in the residential ring. If we relax (3.2) and (3.3), and define w(x, .) = 1 for all z and w(-, z) = 1 for all x, then the definitional boundaries between local public goods and higher jurisdictional levels of public goods (i.e., national defense) becomes blurred and a new conflicting situation arises. [See Alan Williams (1967).] Given eq. (3.9), I proceed to redefine the results of the preceedirag sectIon as follows"
r(N, G,x) = r(N, G, xo)[O,-t(.~))/y]~*+a~'a(v(x)/v(xo)) ;'/a,
(3.12)
I(N, G, x) = ay/r(N, G, xo)[~)'- t(x))/Y]'"o(v(x)/v(xo)) )'/t~,
(3.13)
r(N, G, xo) = aNy/~'~ s(x)[(y-t(x))/yl=l#(v%x)/v(xo)) ~/# dx.
(3.14)
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O. Fisch, Spatial equlh'briumwith local pubtic goods
And central to the collective action process, the welfare function is redefined as
U(N, G) = G~I~ ~ s(x)(y-t(x))'lPv(x) ~/tJdx/N.
(3.15)
In (3.15) as in (2.2) the household's income is alter-tax income, y = p-/'/,
where .P is pre-tax income a n d / / i s local tax. This local tax is defined as a percapita load, n
=
c(631N,
where C = C(G) 6 is defined as a well-behaved cost function of the local public sector, with C'(G) > 0 and C"(G) ~_ O, in the relevant segment of the U-shaped marginal curve. It follows then that
y
=
i,-[c(63/2v].
0.I0
Eqs. (3.15) and (3.16) give the required information to Tiebout's city manager to compute (simultaneously) the optimal level of local public goods and the optimal size of the bzcal population. From the first-order condition of (3.15) and making the corresponding substitution of disposable income given by (3.16), we have that 7
OU(N, C.~ 8G
=
0 -.. ~N ~'o s(x)(y-t(x)ymv(x) m dx,
(3.~7)
-o~ac'(6") I~: s(x)(y-t(x))('-P)lPv(x) ~Ip dx = O,
~U(N, 63 ON
0 ~ #2V~ s(x)O,-t(x))'%(x) '~pdx,
(3.1s)
-aC(G) ~,~o t s(x)(y-t(x))('-tJ)tPv(x) '/~ dx = O.
~In the specification of this cost function rests the critical theoretical differe~o with Ellickson 0973). In my case C(G) = C(G, N), due to my assumption that OC(G,N)/~N = O, in order to show that without 'crowding' in the production of public goods, Tiobout's sixth assumption still holds. ~For a discussion of xt free and endogeneouslydefined, and of why (y-t(xDXdxlldN) ,= 0 or equivalently ( y - t ( x ~ ) X d x t / d G - ' ) - - 0 holds° S¢¢ Fiseh (19/5).
O. Fisch, Spatial eqMlibrium with Iocai public goods
207
For the second-order (necessary and sufficient) ~:onditions for a maximum, • e following m a t ~ :
[
~2U(N, G)/~N z
~zU(N, G)/i~N~G]
(3.19)
~v(.,v, ~ ) / ~ 2 J'
must be negative semi-definite, i.e., that the roots 2 from the determinantal equation,
i OzU(N, G)/DN2-2 o 2U(N, G)ION ~G
D2U(N,G)/DNDG
= 0,
(3.20)
o" U(N, G)IOG2 -
are zert or negative. It is shown in the apr~ndix that sufficient conditions hold. From ¢~.17), (3.18) and the appendix, and defining the following density function: s(x)[O'* - t(x))/y* ]U'- P~/Pv(x)~/P
fo(x) = jIx, xo s(x)[(y._t(x))/y.]~,:~/pv(x)~/p dx'
(3.21)
and defining as the optimal disposable income: y* = .o- [c(s*)l.'~*l,
(3.22)
we can now state:
Proposition 1. The optimal level of local public goods G* and the o,, .',real size of the local population N* are given b:, the ,imuitaneous solutioa ~] the following system: oLC(G*)- fiN*y*E[O'* -- t(x))/y* I/~(x)l = O, • 6 . c ' ( ~ * ) - rjv*r*~(y*-t(x))/.~.* Ifo(x)] = o.
Also, it follows from (3.17)and (3.18) that IC(a*)/G*] C'(G*)
fl ~,
(3.23)
From (3.23) we can now state
Proposition 2. At the optanal level of provL~'ion of local public good~, the ratio between average cost and marginal cost of the p~blic good is equal to the
208
o. Fisch, Spatial equilibrium wtth Iocal public goods
ratio between the utility weight of residential land and the utility weigbit of the public good. From the simultaneous solution of G* and N*, y* is also defined. Then, from (3.12), (3.13) and (3.14), the spatial equilibrium in the city at the optimal level of local public goods and population size is given by
r(N*, G*, x) =
r(N*,
a*, Xo)[(y*-
t(x))[y*l(=*o)lP(v(x)[v(Xo)) ffp, (3.12')
I(N*, G*, x)
=
ry*lr(N*, a*, Xo)[(y*-
=tt X| r(N*, G*, Xo) = aN y/S~o s(x)[(y*- t(x))[y*]=/P(v(x)/v(xo))'itJ dx.
(3.14') As local zoning ordinances have, as a main output, the physical regulation (Io: size, type of structure and land use) of urban land, then (3.13') gives the critical input for exclusion purposes in residential areas. We see then, as exclusion activities by zoning ordinances are possible at some finite cost (costs of enfi~rcement and policing, costs ~ hich are ignored here), property rights are reinforced through collective action. In a way, because the inability to exclude stops, the creation of a 'market' ~ la Tiebout for local public goods is feasible, and possibly it is the expected output of the political process that is taking place in the fragmented jurisdictions of the metropolit,'m areas. 4. The capitalization assumption Tiebout did not clarify in his original formulation whether the local tax is an income tax or a property tax. Nor did he clarify the legal arrangements of urban land ownership- land tenure- he envisioned in the optimal supply of local public goods. And, as we will see immediately, land tenure should be introduced implicitly into our analysis in order to measure the impact o[ local ~.axes. If the local tax is an income tax, we can see that the results of the preceeding sectk,ns apply to both cases of legal arrangements: tenants with c ksentee landlords, and homeowners. if :he local tax is a property tax totally shiftable to land, then land tenure is an important input into our analysis and the results of the preceeding section ~hould be re-examined. In the extreme case of a city of tenants with absentee landlords, the tenants having the political power to tax and to distribute local public goods, then, if the local tax is totally shifmbl~ to land, the lower boundary of local public
O. Fisch, Spatial equilibrium wIJhlocal public goods
209
expenditures is .at least the total value of urban land. This gives some support to the claim that property tax is a progressive tax under this legal a~'rangement and illuminates the historical and political controversy of property qualifications on voting that had in James M a d i s o n - a m o n g o t h e r s - a staunch defender, s But the degree of shiftability of property taxes is still an unsettled theoretical issue. Despite some grounds gained by the classical economists, t~e incidence of property taxes on urban land values remains an obscure area in the analysis of our economic c,rganization. It is in relation to this issue that Oates advances his capitalization assumption of local public expenditures and taxes, 9 as a corroborative of Tiebout's hypothesis. It is the purpose of this section ~o show that property value is a poor output indicator- of the optimal level ,of local public goods and taxes, and of the optimal size of the population. In other words, I ~vant to show that the capitalization assumption is alien to Tiebout's hypothesis and to its explicit homogenization process. In order to neutralize in our analysis the incidence of local taxe,, we will transform our city of tenants of the second section into a city of homeowners, envisioning in our city of equals that the whole residential rir~g is owned by a corporation, with the residents as stockholders, receiving as div;zlends the economic rent of the urban residential land. As the dividend of residential rent is reintroduced as personal income, I allow for continuous recontracting in the residential land market. I will show that there exists an equilibrium in that market and that the recontracting l:,rocess is a stable system. We define the total urban land rent of the initial round of the l~.'contracting process as
Ro(G, ¥) = ~ XO x,o s(x)r(G,N, x) dx,
(4.1)
and from (3.12), (3.14) and (4.1), and making the corresponding substitutions, we have that
Ro(G, N) = oXj'oEl(Yo- t(x))/yo l:o(x)],
(4.1')
where the density function ~s )'~(x) --. s(x)[O'--t(x))/y,l'/av(x)~/a/j,o f''' s(x)[(y~- t(x);/yi]'/av(x)r'adx. *James Madison wrote: '1 t England, at this day, if elections were open to all classes of ~ p l e , the propel'ty of Mnd¢d proprie(ors would be unsure . . . . Landholders; ought to have a share in the govermnent, to support these invaluable intercs¢ . . . . They ought to be so constituted as to protect the minority of the opulent against the majorily.' [Cited in Gaffney
(1972, p. 408)3 ~See tbotnote 3 ab'we.
O. Fisch, Spatial equilibrium with ,bcalpublic goods
210
The dividend D accruing at the end of this initial round to every residentstockholder is, from (4.1'), given by Do
=
~yoEo,
(4.2)
where
~, = E[(y,- t(e))/y~ l Y,(x)]. This dividend given by (4.2) produces a first adjustment of personal income and a m.~westimate of land rent and dividend, Yt = Yo+ Do = yo(1 + o'/~o),
(4.3)
D1 = a.r'lE1,
(4.4)
Y 2 = Yo + D~. = yo(l + E 1 + 0 2 E t Eo).
(4.5)
In general, at the nth round of adjustment, we have that (4.6)
O~_ 1 = ¢ry,_tE,_l,
y,/yo
= l+crF-~-t+~raF',-t/~-:'.+...+e'/~,-~/~,-2."/~fl~o
•
(4.7)
In order to test if the terms of the series in (4.7) are monotonically decreasing, we compute ~he ratio between the nth and the ( n - l)th term {ratio test), °'n~n-lEn-2""EIE°--0"~
0 <: I.
(4.8)
o'~-'E~-IE,-2...Et And it follows that Jim
[a"E,_l...Eo]
= 0,
and that the series (4.7) is convergent. We can now state: Pro~osition 3, The recontracting process of the urban land market under the owneJ'~hip arrangement is a stable system. Froth (4.7) and (4.8) we can compute the equilibrium values of every red,dent's
to~a! h~come, lirn y~ = y = yo{J
tl -¢-
+~E[(y-t(~))/ylfo(X)]},
0.9)
211
O. Fisch, Spatial equilibrium with localpublic goods
and the equilibrium values of (corporation assets) the urban land (in rent
flow), lira R,(G, N) = R(G, N) = aNy E[(y-t(x))/k lfo(x)],
(4.10)
R - t , O0
aos well as the equilibrium values of every household's land holdings, lim D,(G, A) = D(G, N) = oy E[O'-t(x))/Ylfo(x)],
(4.11)
where y is after-tax income. In pre-,ax income, (4.11) can be expressed as D(G, IV)
= o(~-1"1)
£[(~-- F l - t(x))/O'- H)[fo(x)].
(4.11')
It follows from (4.9) and (4.11), that the local tax can be treated indifferently, either as a property tax-independent of the degree of shifting- or as an income tax, and the results of this preceeding section can be applied without any change. The result in (4.11') has the following behavior (see appendix): dD C'(G) I_~ Eo_a~/~ } d'~= N t/lEt -'-" <0, dO
C(G) [~ + = -fir
~i
(4.12)
a - ~ } > O.
(4.13)
/l
From (4.12) and (4.13) and from the assumption C(I) = 0, it follows that
D* = Max {D(G, N)) = lira D(G, N) = lim D(G, N), I ~;G~i oo I&N~®
G~I
(4.14)
N--,.co
and we can state: Proposition 4. The value of every househoid's land hohlings attabls a maximum when. the local tax ,ri = O, i.e., when iocal public expenditure is nil ~,r equivalently, when local population size is infinite.
From the results in the third gection, in the optimal solution, as G* > 1 and N* < oo it tollows that D(G*, N*) < D*,
and we can ~ta~e:
(4.15)
O. Fisch, Spatial equilibrium with local public go~ds
212
Proposition 5. At the optimal level of local public goods G*, optimal local tax H* and optimal population size N*, the value of every household' s land holdings is less than its maximum attainable value. It follows from Propositions 4 and 5, that there is a total lack of correspondence between Tiebout's welfare conditions when ~ho local political process reaches equilibrium (optimality) and local property values (land rent).
R(N)
U(N)
G" - G'(~,~) N'- N'(~,G')
~
(b)
(a)
m m
RCN')
S
! ! l l
N"
N
DCt~)
l
N"
N
(c) D ~
! I I ! l
,,
rl'~
, ,L
_
N
Fig. 1. Welfare (a), total land rent (b), and value of household land holdings as functions of population size.
The results of Propositions 4 and 5 are graphically shown in fig. 1. We can see that land value, as an output indicator of the workings of Tiebout's hypothesis, is totally unrelated to a maximum welfare granted by the homogenization process-implicit in that hypothesis-when it reaches its equilibrium size. Additionally, productivity differentials of the local tax as an incentive to move among communities have also a theoretical limitation. Let us assume that we have two cities A and B with identical geography, two homogeneous populations respectively, with identical income, and where flg=#a
and
~A=~n.
O. Fisch, Spatial equil~briurn with local public goods
213
Then, it ibllows from Proposition 2, in the assumption that there is high mobility of technological information in relation to the public sector, t° titat GI = G~
and
C^(G~)=
C~(6"~).
Let us assume further that ~A ~" ~B,
this inequality being sufficient to lead to
Tnerefore, in spite of different tax rates with the same public output, there is no incentive to shop elsewhere or to move out of any of these cities, because both are at the optimal level of welfare and any capitalization differential does not measure that particular optimal level. Urban land rent is a poor indicator of the b.~nefits of nublic programs in general, and in particular, of the workings of the Tiebout hypothesis" homogenization and optimal supply of !, ~cal public goods. Differentials in p~ oductivity of local taxes are not always an incentive to move and we conclude that the capitalization assumption as a test is totally alien to Tiebout's hypothesis. 5. Summary and conclusions
It was shown that, granted the homogenization of the local population, the Tiebo~at hypothesis simplifies the analysis of local public finance. The required demand functions are revealed, a price (~x) is derived and a 'market' of local public goods is the warranted output. Impiicit in the equilibrium conditions, is the simultaneous derivation of an optimal (finite) population size, the fixed supply of land (private good) being a sufficient condition for that equilibrium. It was shown that the spatial dimensions of public goods are an important factor in the degree of 'localness' of the public good. Additionally, the spatial analysis strengthens the linkage between public goods and such concept~ ~s externalities (neighborhood effects) and 'appropriability'. The spatial I-,~s function clarifies the transformation of the public physical output into a local public good and a spatial measure of local pablic goods was defined for appropriate impact analysis on the land rent function. t°Any assumption to the contrary in the public sector makes violence direcdy to Tiebo~t's assumption. It is very difficult to assume a high geographic mobility of the population v. ith its implicit requirement of no cost and complete information, without assuming, z.t the same time, a complete flow of information (technological) in the public sector and access j~Jrisd/cfional boundaries.
O. Fisch, Spatial equilibrium with local public goods
214
It was show.r,, that land rent is a poor output indicator of Tiebout's hypothesis. Granted that the homogenization process is taking place, the Tiebout hypothesis carries implicitly art efficient allocation. But there seems to exist a perceived conflict between efficiency and distributive aspects in the provision of local public goods. Recent (1972) court decisions in California (Serrano vs. Priest), in New Jersey (Robinson vs. Cahill) and court injunctions in Florida against a City Coupcil's instructions to the 'excess' population to pack and move o u t an implicit local police power required to implement Tiebout's sixth assumption - a r e all directly related to Samuelson's disturbing question as to 'whether groups of like-minded individuals shall be free to run out on their social responsibilities and go off by themselves'. It seems that the thzoretical analysis of this perceived conflict is the most needed task in local public finance but it is totally independent of Tiebout's original formulation and concerns.
Appendix 1
From eq. (3.15) it follows that r:-U ~"
G(~- 2,e)/p
•G 2
N
Eo ~2 ~_p
(,___:, ,c,,o,j
},
(A.l', "-'-~ bN =
aGaN-
Y \ - ~ I ] \ N ] [~--]eo-2E, ,
N
N
Y "E, ?
~t" 2E,-
(A.2)
}
Eo ,
(A.3)
where Eo = E[(y-t(x))/ylfo(X)]
was defined in (3.2 I), and
E, = E[O'-t(x))/>,lA(x)], an~'} where the den.fity function f1(x) is defi,-_ed as
s(x)[(y- t(x))/y] (~ - P)IP,~(x) ~IP A(x) = ,i;,'o s-"-(x)[(>,- t(x))/>,],~-~ )/pv(xy/~ ,
(A...O
O. Fisch, Spatial equilibrium ~ith local public goods
215
From (A.I), (A.2) and (A.3), the necessary and sufficient conditions for a relative maximum are that the roots 2 of the determinamal e,iuction
_(~.~+GC"(G)+,~)~ l = 0 i
(A.5)
-(l+~.)
be negative, is that //>~
if
C"(G)=0,
or
fl(C'(G*)+G*C"(G*)) > yC'(G*),
if C"(G) > O.
Refenmees Arrow, Kenneqh J., 1971, A utilitarian approach to the concept of equality in public expenditures, Quarterly Journal of Economics 85, no. 3, Aug., 409-415. Buchanan, J.M. and Ch.J. Goetz, 1972, Efficiency limits of fiscal mobility: An assessment of the Tiebout model, Journal of Public Economics 1, no. 1, April, 25-43. Coleman, James S., 1968, The concept of equality of educational opportunity, Harvard Educational Review 38, 14-22. Edel, M. and E. Sclar, 1974, Taxes, spending and property values: Supply adjustment in a Tiebout-Oates model, Journal of Political Economy 82, Sept./Oct., 941-954. Ellickson, Bryan, 1971, Jurisdi¢-ional fragmentation and residential choice. American Economic Review (Papers and Proceedings) May, 334--339. EUickson, Bryan, 1973, A gcrmralization of the pure theory of public goods, American Economic Review 63, no. 3, June, 417-432. Fisch, Oscar, 1975, Optimal city st;m, land tenure and the economic theory of clubs, Jou:nai cf ~egional Science and Urban Economics, 13~c. Fis, ,, Oscar, 1975, Externalities, the urban rent and populatior' density functions: The case o¢ air pollution, Journal of Environmental Economics 2, no. 2, June. Gaffney, Mason M., 1972, The property tax is a progressive tax, )roceedings of 64th Annual Conference (National Tax Association, Columbus, OH) 408-426. Kieslir~g, H e r ~ n J., 1967, Measuring a local government service: A study of school districts in New York State, The Revie'~ of Economics and Statistics 49, Aug., 356-367. Musgrave, R.A., 1969. Provision for social goods, in: J. Margolis aad H. Guitton, eds., Public economics (St. Martin's Press, New Yorg). Oat~, Wallace E., 1969, The effects of property taxes and local pv.blic spending on )3ropcr:y values: An empir"tcal study of tax capitalization and the "riebout hypothesis. Journal of Po|itica! F.xonomy 77. Pollakrocs~J, Henry O., 1973, The effects; of property tax~ and local public spending on - , , ~ r t y values A comment and furtl',er results, Journal of Political Economy 1~1, July, 994-10OL ~.awis, John, 1971, A t~'w.ory of justice (Hz~,,ard University Press, C~mbridge, MA) ,~,amu¢|son, Paul, 19Y8, Local finance and ~he mathematics of marriage, Apl~n,)~x to: Aspects of public expenditure theorie~, The Re, ~ew of Economics and Statistics 40, Nov., 332 ~338.
216
O. .Fi.~:ch,Spatial equilibrium with local public goods
Solow, Ro~rt M., 1973, Congestion cost and the use of land for streets, Bell Journal of Economics and Management Science 4, no. 2, Autumn, 602-618. Tiebout, Ch~a'les M., 1956, A pure theory of local expenditures, Journal of Political Economy 64, Oct., 416--424. Williams, Alan, 1967, The optimal provision of public goods in a system of local government, Journal of Political Economy 74, no. 1, Feb., 18-33. Williams, O.P., H. Herman, Ch. $. Liebman and Th. R. Dye, 1965, Suburban differences and metropolitan policies (University of Pennsylvania Press, Philadelphia, PA). Wood, Robert. C.:. 1958, Suburbia: Its people and their politics (Houghton Mifflin, Boston, MA).
SPATIAL E Q U I L I B R I U M W I F H L O C A L P U B L I C G O O D S
Urban Land Rent, Optimal City Size and the Tiebout Hypothesis Oscar FISCH The Ohio State University, ~lumbus, O H 43210, U.S.A.
Received M,ay 1976 This paper extends the standard model of urban land rent to consider the spatial equilibrium conditions in a local pL~blJcgoods market as hypothesized by Charles M. Tiebout. An analysis is made of the spatial dimensions of public goods, their degree of "localness' and their impact on land values, it is shown that the optimal population size of the community (Tiebout's sixth assumption) is simultaneously derived with the optimal supply of local public goods and local taxes. It is also shown that land rent is a poor output indicator of Tiebout's equilibrium conditions and that tht capitalization assumption is not the appropriate test for his hypothesis.
1. latrotlaetioa Almost two decades ago, Tiebout (1956) advanced the hypothesis that public expenditure theory simplifies itself at the local level, as people 'vote with their feet' and move to homogeneous communities. 1 This homogeneity secures through the political process- collective a c t i o n - that the urban dweller gets the tax-expenditure program that maximizes his and everybody's utility function. In examining briefly Tiebout's hypothesis, Samuelson (1958), after recognizing that this hypothesis 'goes some way toward solving the pro~lem', rejected the whole approach on three speculative grounds: first, that existing urban communities are not homogeneous and, even given that a majority cannot get rid of a minority, the existing conflict is still unresolved; second, that population heterogeneity may also be a positive argument in the individual's utihty function; and third, that there exist, the disturbing ethical question 'as to whether groups of like-minded individuals shall be free to run-out on their social responsibilities'. The last argument is clearly a distributive problem that still today disturbs many people. This argument has been used in support o:~ some recent court decisions that have shaken the foundations of local governments. The distributive dimensions were formalized b y - among o t h e r s - Coleman (1968) and 1dssuml~tio;t 3. 'There are a large number of communities in which the consumer-voter may ehc~se to live.' (Tiebout, op. cir., p. 419.)
198
O. Fisch, Spatial equilibrium with local public goods
Rawls (t 971): and an economic analysis was put forward by Arrow (1971) in a set of sharply delineated propositions. But it is Samuelson's first two arguments that question the workings of Tiebout's hypothesis in the real world. To his casual observation in 195t~ that existing urban communities are not homogeneous, there was more than casaal evidence: to the contrary, compiled, coincidentally, at the same time. Wood (1958) Jn an exhaustive study of suburban communities showed that a high degree of homogenization within communities was indeed the result of local politics due to jurisdictional fragmentation of metropolitan areas. In a later study, Williams et al. (1065), the historical trend of Wood's findings was verified and it was contended further that community specialization with respect to class or status, life-style or wealth, is the main force behind the resistance to intercommmaity cooperation (heterogeneization among communities) in metropolitan areas. Granted that the process of homogenization within urban communities is taking place- due ,~o a high degree of geographic mobility of the populationthe existing conflict in the theory of pub!it finance is reduced significantly at the local level, leaving the conflict unsolved at only higher jurisdictional levels of government, or equivalently, at broader spatial dimensions of public goods: metropolitan, state and national. After Samuelson's brief review, very little attention was paid, in theoretical terms, to Tiebout's hypothesis m,.fil two rexzent articles: one by Buchanan and Goetz (1972) and the other by Ellick~on (1973). Both formalize Tiebout's sixth assumption, 2 i.e., that, as the preference maps of the local resident-consumers are now known, the political process which leads to the desired basket of local public goods carries, implicitly,, an optimal (finite) population requirement (optimal city size). Both analyses were carried out with the implicit or explicit thesis of the 'congestion' of the public good: 'crowding' of the local public good either in the consumption side (Theory of Clubs) or in the production side. Further, in the latter case,, Ellickson stated that cr :~wding in the production of local public goods is a required condition for the existence of optimal partitions of the population. Oates (1969) advanced in an empirical study his capitalization assumption of local public expenditures and local taxes as a way of testing Tiebout's hypo2A,~sumplion 6. ' i 7 0 1 • every pattern of community service set by, say, a city man~ager ~vho follows the preferences of the older residents of the coml aunity, there is an optimal commun ity size.' (Tiebout, op. cit., p. 419.) In trying to explain his sixth assumption Tiebout argues that such a case ' . . . implk:s l[~t some factor or resource is f i x e d . . . ' . And, in order to support his case, he advanc.es three dilicrent types of exarnple~: firsL limited amot nt of residentm| land, i.e., fixed supply of a private good (land); second, institutiona~ lh~ ~ts on land density, i.e., zoning ordinawa;s constraining the consumption of land; and t b i d , congestion or crowding of a 'public good', i.e., limited capacity of the local beach (lheory of clubs), We will show in the third section theft the first type o fixity i~, suificient for this s~xth assumption to hold.
O. Fisch, Spatial equilibrium with local ~ublie goods
199
thesis. 3 Since then, many writers have criticized Oates' findings on different ~t,,xnds, but mainly in the specification of the statistical model or in statistical procedures [among others Ellickson (1971), Pollakowski (1973) and Edel and Sclar ~.lg,~t)]. However, these criticisms are raised without taking issue with the theoretizal p,-~blem of considering real e~tate values as an output indicator of the process o," co..~munity homogenization - the core of Tiebout's hypothesis. It is the purpose ~f this paper to assess Tiebout's hypothesis in a more theoretical setting. First, I want to examine the spatial equilibrium cor~ditions and urban land rents under ~he impact of local public services. Second, I want to examine the locational con~.'!'~ns of local public services and the spatial dimensions of exclusion or 'apprcoriability' so important to the mode of economic organization and its relatio, qhip to such concepts as externalities or neighbo~'hood effects (spillovers) in or~Icr to clarify definitional aspects of local public goods. Third, I want to show that ,.,dhering to Tiebout's c,riginal contention in explaining his sixth assumption i.e., that it is sufficient that some factor or resource be fixed - such as the fixed supply of residential land (private good) in the urban community-can lead to a fin/~e population as an optimal size. Fourth, I want to show that the political process ~;,hich leads to an optimal level of local public goods, simultaneously leads to an optimal size of local population, and thereby show global conditions instead of merely marginal ones for such optimal solutions. Finally, I will show that property value is a very poor output indicator of Tiebout's hypothesis. The plan of the paper is the following: in the second section 1 derive the spatial equilibrium conditions of an homogeneous population in a residential ring, reformulating a model by Solow (1973) in order to include a given spatial distribution of local public goods. Embedded in the solution is the soatial pattern of consumption of the private good (residential land) and its rent function. In the third section I will analyze the spatial characteristics of lc,~l public goods and their relation to concepts such as exclusion and spillover (neighborhood) effects. It is in this section also, that I show the simultaneous procedure used to arrive at an optimal level of public goods and of local population. Fv rther, it is shown that the fixed supply of the private good (land) is a sut~icient condition f e r a finite population. Section 4 derive:, a direct measure ¢~f the total land market val,res of the community and the average market value ~i" heusehold's ~'lf this is the c a ~ and if (as the Ticl'~ut model suggests) individuals con ;id,~: the qtlaliW ot local r~ublic services in making locational decisions, we would expect to ti~0d that, ~:~ther th~ng':' ?eing equal (ir~cluding tax rates) across communities, an increased expendilurc per pt~pil would result in higher property values.' [Oates (19¢,9, p. 962~]. Oat¢,; ~scd expenditure p~:',r pupil as a proxy for the output of educational ~rvk.'es. Later research ~im!imps have shown ~ a t both are uncorrelated. Kiesling (1967, p. 366) says ' . , , the relationshir of performance t,:, per pupil expenditure has been found to be. e×cep.~ in large urban sch¢.oi districts, di,,ai~pointingly weak. This would imply, among other things, that the u~iJi,'ati~n of per capita c,:~st figures for an index of p~b[ic performanc~ is a ~ighly dangerous prac~i,,e'. A furl d i ~ u ~ i ( m of the Oates paper would carry us far beyond our subject.
200
O. Fisch, Spatial equilibrium with local publie goods
real estate holdings in that community and I show the lack of correspondence between Tiebout's hypothesis and the capitalization assumption. Finally, the last section summarizes the results of the paper.
2. The model: Spatial equilibrium with local public goods The main thrust of Tiebout's hypothesis, mainly related to his third assumption, is the homogenization of the local population as a resuR of 'voting with their feet', eLpolitical process that rests on the assumption of' high geog::aphic mobility (Tiebout's first assumption). The final output of this process is that urban dwellers will choose that urban community-among m a n y - w h i c h offers the tax-expenditure program that maximizes their welfare, thus providing the revelatory demand functions. This is in contrast with the higher jurisdictional public goods case where the individual has every reason not to provide such a function. in theoretical terms, this homogenization process simplifies the analysis of public economics at the local level, because it allows, without making violence to the analysis, the introduction of the simplifying assumption that individuals have the same income and the same utility function. This allowance circumvents the definitional difficulties of interpersonally comparable utilities which permeate and block the theoretical analysis of public goods in fgeneral. In the model we are developing here, we consider that our city is a 'city of equals': equal tastes and equal income. The model is a reformulation of the standard model of urban spatial equilibrium, a reformulat:ion that makes possible the explicit consideration of local public: goods. Physically, the city is a circular one with radius xl. It has aw inner ring with radius Xo which land is used up by the production functions c.f the CBD. The outer ring is composed of homogeneous residential land, and its total population N commutes to the CBD in order to deliver labor inputs. Every individual, besides consuming the two private goods, urban land I and the composite consumption good c, consumes allso a public good g. The utility derived from the consumption of this public good may vary with distance x ~ [ X o , X l ] . The urban dweller's utility function is additive separable, with diminishing marginal utility in every argument, u(c, 1, g(x)) = In [:lag(x)q. The urban dweller's income y after tax is assumed to be totally exhausted by the two private: goods plus commuting transportation costs, y = e + r(x)1+ t(x).
(2.2)
O. Fisch, Spatial equilibrium with Iocalpubh," goods
201
In (2.2) prices are in terms of the price of t~ e composite good. Land prices (rent) are given by r(x), an unknown function, and transportation cost~ by t(x), a known function. In maximizing (2.1) subject to (2.2), we have from the usual first-order conditions,
c = r~"
(2.3)
From (2.3) and in making the corresponding substitution into (2.2), (2.4) and (2.5) show that every individual, in each locatior x, spends a fixed proportion of his income net of transportation in urban land, allocating the rest to the consumption of the composite good.
o'O'-t(x)),
(2.4)
c(x) = ( 1 - o ) O , - t ( x ) ) ,
(2.5)
I(x)r(x) =
where o =/~/(0t+fl). Replacing (2.4) and (2.5) into the utility function (2.1), we have that u(x) = In {(1 -a)'aP(.v - t(x)) ~+Pg(xf /r(x)P} .
(2.6)
In our city of equals, and in equilibrium, no individual should have incentive to move. In other words, in equilibrium, as ev, ry tLrban dweller should be indifferent in relation to his final location, his utility must be constant over the geography of the city. Then it follows that du(x) ----. dx
=
0 =
~t+ fl dt(x) y - t ( x ) dx
fl dr(x) r(x) dx
+ .........
y dg(x) . g(x) dx
(2.7)
From (2.7) we get, in separable form, the fundamental differential equation of the urban rent function, dr(x) r(x)
1 dt(x) o y-t(x)
? dg(x) fl g(x)
(2.8)
In solvipg (2.8) we have that In r(x) = In ( y - t ( x ) ) I/~ + In g(x) ~ +C.
(2.9)
The constant of integration C is solved in terms of the values of the variablc~ involved at the boundary Xo. At this initia! point~ t(Xo) = 0 in the simpl@ing
O. Fisch, Spatial equilibrium with localpul,lt: goods
202
assumption that there are no transportation costs within the CBD. It follows then that C
=
In r ( x o ) - l n y l l ° - l n g(xo) ~'l~.
(2.1o)
From (2.9) and (2.10), we have that the unit price (ren0 of land is given by (2.4),
r(x) = r(xo)[(y- t(x))[ylt'+ P~/ffg(x)/g(xo)) rip.
(2.11)
Substituting (2.11) into (2.4) we have that the individual's consumption of land per household at distance x is given by
l(x) = a f l'g(xo)rl#/r( xo)(y-- t(x))'//J g(x)r//J.
(2.12)
The fixed supply of land at distance x, in an infinitesimal ring of width dx, is s(x), 0 < s(x) < 2nx. The demal~d for land at distance x is l(x)n(x)dx, where n(x) is the number of urban dwellers living in the infinitesimal ring of width dx. We assume that in equilibrium the market of urban land is cleared; then it follows that
s(x) dx = l(x)n(x) dx.
(2.13)
The total aggregate demand has the following integral constraint: U =
n(x) dx,
(2.14)
where N is the size of the urban population. From (2.12), (2.13) and (2.14), and making the corresponding substitution, we have that the boundary condition r(xo) is computed by solving
ag(xo)~lPfll'N-r(xo) j ~ s(x)(y- t(x))=lOg(x)~/~ dx = O.
(2.15)
[cm the equilibrium condition (2.7), the individual's utility is invariant in relation to distance x. It follows that, in computing the utility of any one individual at any particular distance, we know everybody's level of utility u at any distance xo Using Xo as that particular distance, from (2.6) we have u =
(Xo) = In
In this approach, the output of the systerr is the utility itself (2.16), ailowhlg a dir:ct measurement of benefits of public programs. ~qs. (2.1t), (2.12), (2.15) and (2.16) are the critical ones in analyzing in equilibrium the impact of any public program, that we will express in terms of g(x), population size N and distance x,
O. Fisch, Spatial equiiibrimn with local public goods
203
r(N, g(x), x) = r(N, g(x), Xo)[(y- t(x))/y] (~+tmp(g(x),'g(Xo))"ha, (2.17)
l(N g(x),x) - try[r(N,g(x),xo)[(y-t(x))]y]'iPlg(x)/,~(Xo)) ~i~, (2.18) r(N,g(x),xo) = N a y / j ~ s(x)[(y-t(x))/y]'/P(g(x)/g(.%)) r/tJdx,
(2.19)
u(N, g(x)) = In { ( 1 - a ) ' [ j'~gs(x)(y-t(xW/ag(x) '/a dx]a/Na}. (2.20) From (2.20) we will define a new welfare function,
U(N, g(x)) = I~ s ( x X y - t(x)~'/t~g(x)';tJ dx/N,
(2.21)
where
U(N, g(x)) = U[u(N, g(x))] = [e"~N'g~'/(l --¢r)~]l/t~ is a monotonic transformation of the original utility function (2.1). Eqs. (2.17), (2.18) and (2.19) give the spatial equilibrium conditions of the residential ring as functions of population size, the spatial distributioa of local public goods and distance. Eq. (2.21) gives the individual's and eve, ybody's level of welfare in the community, as a fimction of population size and the spatial distribution of public goods. We can turn our attention now to the political process- collective action- implicit in Tiebout's sixth assumption.
3. The political process: Local public goods and optimal city size In order to differentiate between the level and the spatial distribution of the public physical output on the one hand, and on the other, the degree of spatial exclusiveness of the public physical output, I will define the following: h(x) is the public physical output located in the infinitesimal ring of width dx at distance x. Then, the total public physical output G of the city is given by G ~ ~ ' h(z) dz.
(3.1)
Given (3.1), a public output density fimction can be defined as p(::) = h(z)/G. Also, a spillover function can be defined as agx, z), where w(-, z) weights the spiUout effect of the public physical output h located at z, and w(x,.) weights the spillin effect to the heusehold located at x. In the specification of this spillover function rests the complex definitional aspect of the local public good. In one direct/on, it introduces the spatial dimensions of the concept of property rights, dimensions that are re~ated to the mode of eeonomie organization. In the other direction, it introduces the degree of'localness' of the public gc,od: in ,;pite of non-rivalriness of the public good,
204
O. Fisch, Spatial equilibrium with localpublic goods
we have a loss in welfare due only to the locational fixities of both the physical output of the public program and of the consumer of the public good, a loss that is a function of the physical distance separating production and consumption. 4 Let us assume that
w ( x , z ) = 1, for all z = x ;
z,x_~xi,
(3.2)
w(x, z) = 0, for x > xl,
(3.3)
w(x,z) = 0 ,
(3.4)
for
z > xl.
Assumption (3.2) means that there are no spiUouts from our community and (3.3) means that there are no spillins into our community; both assumptions are in strict correspondance with Tiebout's fifth assumption. 5 If we assume further that w(., z) - 0,
for all x # z;
x, z _~ x t ,
(3.5)
then we have that there are no spiUovers among residential rings and that the public physical output h is supplied at z exclusive of any other ring. Because of the continuous spatial distribution of population (2.13) and (2.18) only one household is located in an infinitesimal ring of width dx at distance z. Therefore, the degree of appropriability is optimal and perfectly subject to the pricing system. It follows that g(x) = h(x) and g(x) is a publicly supplied private good in this case, a case that does not need our attention here. For a more general case, let us assume now that w(., z) > 0,
for some x # z;
x, z _~ xt.
(3.6)
(3.6) means that there are spillovers (neighborhood effects) at least among some residential rings and that the public physical output h is supplied at z without any exclusivity in relation to at least one other ring. Equivalently, for the household located at z, the degree of appropriability is not optimal and therefore I define h(z) as a public good. It does not follow that every household located at x ~ z is appropriating the total amount of h(z)° Because of the physical distance between the location of ~he public physical output at z and the location of the 4'Non-rivalness in consurrption.., does not mean that some subjective benefit must be derived, or even that precisely tt.e same product quality is available to boLh. Consumer A who lives close to the police station has better protection than B who lives far away. Yet, the two consumption acts are non-rival, and we deal with a social good.' [Musgrave (IO69,p. 126)]. ~Assumption 5. 'The public services supplied exhibit no external ecot~omiesor d~onomies be~.weencommunities? fTiebout, op. cir., p. ,Hg.)
O. Fisch, Spatial equilibrium with local public goods
205
household at x, I can also define w as a loss or distance dissipation function where O-
1, for all
z#x;
z,x~xl.
(3.7)
At every particular location x e [Xo, x~], I define the total amount of public goods g as
g(x) = ~:' w(x, z)h(z) dz;
~ E [Xo, x,].
(3.8)
And from (3.1) we have that
g(x) = 6 ~:' w(x, z)p(z)
(3.9)
dz = o r ( x ) ,
whele
v(x)=E[w(x,z)lp(z)],
for
xe[x0,x~],
0
1,
E being the operator expected value of the function w. In the particular case where there are no dissipatory effects, then ;v(x,.) = 1, for all
z ~ xl.
(3.10)
It follows from (3.9) that, in this case,
E[w(x, z) I p(z)] = 1 - , g(x)
=
G,
(3.11)
and there is no spatial impact of local expenditures on public goods in the residential ring. If we relax (3.2) and (3.3), and define w(x, .) = 1 for all z and w(-, z) = 1 for all x, then the definitional boundaries between local public goods and higher jurisdictional levels of public goods (i.e., national defense) becomes blurred and a new conflicting situation arises. [See Alan Williams (1967).] Given eq. (3.9), I proceed to redefine the results of the preceedirag sectIon as follows"
r(N, G,x) = r(N, G, xo)[O,-t(.~))/y]~*+a~'a(v(x)/v(xo)) ;'/a,
(3.12)
I(N, G, x) = ay/r(N, G, xo)[~)'- t(x))/Y]'"o(v(x)/v(xo)) )'/t~,
(3.13)
r(N, G, xo) = aNy/~'~ s(x)[(y-t(x))/yl=l#(v%x)/v(xo)) ~/# dx.
(3.14)
206
O. Fisch, Spatial equlh'briumwith local pubtic goods
And central to the collective action process, the welfare function is redefined as
U(N, G) = G~I~ ~ s(x)(y-t(x))'lPv(x) ~/tJdx/N.
(3.15)
In (3.15) as in (2.2) the household's income is alter-tax income, y = p-/'/,
where .P is pre-tax income a n d / / i s local tax. This local tax is defined as a percapita load, n
=
c(631N,
where C = C(G) 6 is defined as a well-behaved cost function of the local public sector, with C'(G) > 0 and C"(G) ~_ O, in the relevant segment of the U-shaped marginal curve. It follows then that
y
=
i,-[c(63/2v].
0.I0
Eqs. (3.15) and (3.16) give the required information to Tiebout's city manager to compute (simultaneously) the optimal level of local public goods and the optimal size of the bzcal population. From the first-order condition of (3.15) and making the corresponding substitution of disposable income given by (3.16), we have that 7
OU(N, C.~ 8G
=
0 -.. ~N ~'o s(x)(y-t(x)ymv(x) m dx,
(3.~7)
-o~ac'(6") I~: s(x)(y-t(x))('-P)lPv(x) ~Ip dx = O,
~U(N, 63 ON
0 ~ #2V~ s(x)O,-t(x))'%(x) '~pdx,
(3.1s)
-aC(G) ~,~o t s(x)(y-t(x))('-tJ)tPv(x) '/~ dx = O.
~In the specification of this cost function rests the critical theoretical differe~o with Ellickson 0973). In my case C(G) = C(G, N), due to my assumption that OC(G,N)/~N = O, in order to show that without 'crowding' in the production of public goods, Tiobout's sixth assumption still holds. ~For a discussion of xt free and endogeneouslydefined, and of why (y-t(xDXdxlldN) ,= 0 or equivalently ( y - t ( x ~ ) X d x t / d G - ' ) - - 0 holds° S¢¢ Fiseh (19/5).
O. Fisch, Spatial eqMlibrium with Iocai public goods
207
For the second-order (necessary and sufficient) ~:onditions for a maximum, • e following m a t ~ :
[
~2U(N, G)/~N z
~zU(N, G)/i~N~G]
(3.19)
~v(.,v, ~ ) / ~ 2 J'
must be negative semi-definite, i.e., that the roots 2 from the determinantal equation,
i OzU(N, G)/DN2-2 o 2U(N, G)ION ~G
D2U(N,G)/DNDG
= 0,
(3.20)
o" U(N, G)IOG2 -
are zert or negative. It is shown in the apr~ndix that sufficient conditions hold. From ¢~.17), (3.18) and the appendix, and defining the following density function: s(x)[O'* - t(x))/y* ]U'- P~/Pv(x)~/P
fo(x) = jIx, xo s(x)[(y._t(x))/y.]~,:~/pv(x)~/p dx'
(3.21)
and defining as the optimal disposable income: y* = .o- [c(s*)l.'~*l,
(3.22)
we can now state:
Proposition 1. The optimal level of local public goods G* and the o,, .',real size of the local population N* are given b:, the ,imuitaneous solutioa ~] the following system: oLC(G*)- fiN*y*E[O'* -- t(x))/y* I/~(x)l = O, • 6 . c ' ( ~ * ) - rjv*r*~(y*-t(x))/.~.* Ifo(x)] = o.
Also, it follows from (3.17)and (3.18) that IC(a*)/G*] C'(G*)
fl ~,
(3.23)
From (3.23) we can now state
Proposition 2. At the optanal level of provL~'ion of local public good~, the ratio between average cost and marginal cost of the p~blic good is equal to the
208
o. Fisch, Spatial equilibrium wtth Iocal public goods
ratio between the utility weight of residential land and the utility weigbit of the public good. From the simultaneous solution of G* and N*, y* is also defined. Then, from (3.12), (3.13) and (3.14), the spatial equilibrium in the city at the optimal level of local public goods and population size is given by
r(N*, G*, x) =
r(N*,
a*, Xo)[(y*-
t(x))[y*l(=*o)lP(v(x)[v(Xo)) ffp, (3.12')
I(N*, G*, x)
=
ry*lr(N*, a*, Xo)[(y*-
=tt X| r(N*, G*, Xo) = aN y/S~o s(x)[(y*- t(x))[y*]=/P(v(x)/v(xo))'itJ dx.
(3.14') As local zoning ordinances have, as a main output, the physical regulation (Io: size, type of structure and land use) of urban land, then (3.13') gives the critical input for exclusion purposes in residential areas. We see then, as exclusion activities by zoning ordinances are possible at some finite cost (costs of enfi~rcement and policing, costs ~ hich are ignored here), property rights are reinforced through collective action. In a way, because the inability to exclude stops, the creation of a 'market' ~ la Tiebout for local public goods is feasible, and possibly it is the expected output of the political process that is taking place in the fragmented jurisdictions of the metropolit,'m areas. 4. The capitalization assumption Tiebout did not clarify in his original formulation whether the local tax is an income tax or a property tax. Nor did he clarify the legal arrangements of urban land ownership- land tenure- he envisioned in the optimal supply of local public goods. And, as we will see immediately, land tenure should be introduced implicitly into our analysis in order to measure the impact o[ local ~.axes. If the local tax is an income tax, we can see that the results of the preceeding sectk,ns apply to both cases of legal arrangements: tenants with c ksentee landlords, and homeowners. if :he local tax is a property tax totally shiftable to land, then land tenure is an important input into our analysis and the results of the preceeding section ~hould be re-examined. In the extreme case of a city of tenants with absentee landlords, the tenants having the political power to tax and to distribute local public goods, then, if the local tax is totally shifmbl~ to land, the lower boundary of local public
O. Fisch, Spatial equilibrium wIJhlocal public goods
209
expenditures is .at least the total value of urban land. This gives some support to the claim that property tax is a progressive tax under this legal a~'rangement and illuminates the historical and political controversy of property qualifications on voting that had in James M a d i s o n - a m o n g o t h e r s - a staunch defender, s But the degree of shiftability of property taxes is still an unsettled theoretical issue. Despite some grounds gained by the classical economists, t~e incidence of property taxes on urban land values remains an obscure area in the analysis of our economic c,rganization. It is in relation to this issue that Oates advances his capitalization assumption of local public expenditures and taxes, 9 as a corroborative of Tiebout's hypothesis. It is the purpose of this section ~o show that property value is a poor output indicator- of the optimal level ,of local public goods and taxes, and of the optimal size of the population. In other words, I ~vant to show that the capitalization assumption is alien to Tiebout's hypothesis and to its explicit homogenization process. In order to neutralize in our analysis the incidence of local taxe,, we will transform our city of tenants of the second section into a city of homeowners, envisioning in our city of equals that the whole residential rir~g is owned by a corporation, with the residents as stockholders, receiving as div;zlends the economic rent of the urban residential land. As the dividend of residential rent is reintroduced as personal income, I allow for continuous recontracting in the residential land market. I will show that there exists an equilibrium in that market and that the recontracting l:,rocess is a stable system. We define the total urban land rent of the initial round of the l~.'contracting process as
Ro(G, ¥) = ~ XO x,o s(x)r(G,N, x) dx,
(4.1)
and from (3.12), (3.14) and (4.1), and making the corresponding substitutions, we have that
Ro(G, N) = oXj'oEl(Yo- t(x))/yo l:o(x)],
(4.1')
where the density function ~s )'~(x) --. s(x)[O'--t(x))/y,l'/av(x)~/a/j,o f''' s(x)[(y~- t(x);/yi]'/av(x)r'adx. *James Madison wrote: '1 t England, at this day, if elections were open to all classes of ~ p l e , the propel'ty of Mnd¢d proprie(ors would be unsure . . . . Landholders; ought to have a share in the govermnent, to support these invaluable intercs¢ . . . . They ought to be so constituted as to protect the minority of the opulent against the majorily.' [Cited in Gaffney
(1972, p. 408)3 ~See tbotnote 3 ab'we.
O. Fisch, Spatial equilibrium with ,bcalpublic goods
210
The dividend D accruing at the end of this initial round to every residentstockholder is, from (4.1'), given by Do
=
~yoEo,
(4.2)
where
~, = E[(y,- t(e))/y~ l Y,(x)]. This dividend given by (4.2) produces a first adjustment of personal income and a m.~westimate of land rent and dividend, Yt = Yo+ Do = yo(1 + o'/~o),
(4.3)
D1 = a.r'lE1,
(4.4)
Y 2 = Yo + D~. = yo(l + E 1 + 0 2 E t Eo).
(4.5)
In general, at the nth round of adjustment, we have that (4.6)
O~_ 1 = ¢ry,_tE,_l,
y,/yo
= l+crF-~-t+~raF',-t/~-:'.+...+e'/~,-~/~,-2."/~fl~o
•
(4.7)
In order to test if the terms of the series in (4.7) are monotonically decreasing, we compute ~he ratio between the nth and the ( n - l)th term {ratio test), °'n~n-lEn-2""EIE°--0"~
0 <: I.
(4.8)
o'~-'E~-IE,-2...Et And it follows that Jim
[a"E,_l...Eo]
= 0,
and that the series (4.7) is convergent. We can now state: Pro~osition 3, The recontracting process of the urban land market under the owneJ'~hip arrangement is a stable system. Froth (4.7) and (4.8) we can compute the equilibrium values of every red,dent's
to~a! h~come, lirn y~ = y = yo{J
tl -¢-
+~E[(y-t(~))/ylfo(X)]},
0.9)
211
O. Fisch, Spatial equilibrium with localpublic goods
and the equilibrium values of (corporation assets) the urban land (in rent
flow), lira R,(G, N) = R(G, N) = aNy E[(y-t(x))/k lfo(x)],
(4.10)
R - t , O0
aos well as the equilibrium values of every household's land holdings, lim D,(G, A) = D(G, N) = oy E[O'-t(x))/Ylfo(x)],
(4.11)
where y is after-tax income. In pre-,ax income, (4.11) can be expressed as D(G, IV)
= o(~-1"1)
£[(~-- F l - t(x))/O'- H)[fo(x)].
(4.11')
It follows from (4.9) and (4.11), that the local tax can be treated indifferently, either as a property tax-independent of the degree of shifting- or as an income tax, and the results of this preceeding section can be applied without any change. The result in (4.11') has the following behavior (see appendix): dD C'(G) I_~ Eo_a~/~ } d'~= N t/lEt -'-" <0, dO
C(G) [~ + = -fir
~i
(4.12)
a - ~ } > O.
(4.13)
/l
From (4.12) and (4.13) and from the assumption C(I) = 0, it follows that
D* = Max {D(G, N)) = lira D(G, N) = lim D(G, N), I ~;G~i oo I&N~®
G~I
(4.14)
N--,.co
and we can state: Proposition 4. The value of every househoid's land hohlings attabls a maximum when. the local tax ,ri = O, i.e., when iocal public expenditure is nil ~,r equivalently, when local population size is infinite.
From the results in the third gection, in the optimal solution, as G* > 1 and N* < oo it tollows that D(G*, N*) < D*,
and we can ~ta~e:
(4.15)
O. Fisch, Spatial equilibrium with local public go~ds
212
Proposition 5. At the optimal level of local public goods G*, optimal local tax H* and optimal population size N*, the value of every household' s land holdings is less than its maximum attainable value. It follows from Propositions 4 and 5, that there is a total lack of correspondence between Tiebout's welfare conditions when ~ho local political process reaches equilibrium (optimality) and local property values (land rent).
R(N)
U(N)
G" - G'(~,~) N'- N'(~,G')
~
(b)
(a)
m m
RCN')
S
! ! l l
N"
N
DCt~)
l
N"
N
(c) D ~
! I I ! l
,,
rl'~
, ,L
_
N
Fig. 1. Welfare (a), total land rent (b), and value of household land holdings as functions of population size.
The results of Propositions 4 and 5 are graphically shown in fig. 1. We can see that land value, as an output indicator of the workings of Tiebout's hypothesis, is totally unrelated to a maximum welfare granted by the homogenization process-implicit in that hypothesis-when it reaches its equilibrium size. Additionally, productivity differentials of the local tax as an incentive to move among communities have also a theoretical limitation. Let us assume that we have two cities A and B with identical geography, two homogeneous populations respectively, with identical income, and where flg=#a
and
~A=~n.
O. Fisch, Spatial equil~briurn with local public goods
213
Then, it ibllows from Proposition 2, in the assumption that there is high mobility of technological information in relation to the public sector, t° titat GI = G~
and
C^(G~)=
C~(6"~).
Let us assume further that ~A ~" ~B,
this inequality being sufficient to lead to
Tnerefore, in spite of different tax rates with the same public output, there is no incentive to shop elsewhere or to move out of any of these cities, because both are at the optimal level of welfare and any capitalization differential does not measure that particular optimal level. Urban land rent is a poor indicator of the b.~nefits of nublic programs in general, and in particular, of the workings of the Tiebout hypothesis" homogenization and optimal supply of !, ~cal public goods. Differentials in p~ oductivity of local taxes are not always an incentive to move and we conclude that the capitalization assumption as a test is totally alien to Tiebout's hypothesis. 5. Summary and conclusions
It was shown that, granted the homogenization of the local population, the Tiebo~at hypothesis simplifies the analysis of local public finance. The required demand functions are revealed, a price (~x) is derived and a 'market' of local public goods is the warranted output. Impiicit in the equilibrium conditions, is the simultaneous derivation of an optimal (finite) population size, the fixed supply of land (private good) being a sufficient condition for that equilibrium. It was shown that the spatial dimensions of public goods are an important factor in the degree of 'localness' of the public good. Additionally, the spatial analysis strengthens the linkage between public goods and such concept~ ~s externalities (neighborhood effects) and 'appropriability'. The spatial I-,~s function clarifies the transformation of the public physical output into a local public good and a spatial measure of local pablic goods was defined for appropriate impact analysis on the land rent function. t°Any assumption to the contrary in the public sector makes violence direcdy to Tiebo~t's assumption. It is very difficult to assume a high geographic mobility of the population v. ith its implicit requirement of no cost and complete information, without assuming, z.t the same time, a complete flow of information (technological) in the public sector and access j~Jrisd/cfional boundaries.
O. Fisch, Spatial equilibrium with local public goods
214
It was show.r,, that land rent is a poor output indicator of Tiebout's hypothesis. Granted that the homogenization process is taking place, the Tiebout hypothesis carries implicitly art efficient allocation. But there seems to exist a perceived conflict between efficiency and distributive aspects in the provision of local public goods. Recent (1972) court decisions in California (Serrano vs. Priest), in New Jersey (Robinson vs. Cahill) and court injunctions in Florida against a City Coupcil's instructions to the 'excess' population to pack and move o u t an implicit local police power required to implement Tiebout's sixth assumption - a r e all directly related to Samuelson's disturbing question as to 'whether groups of like-minded individuals shall be free to run out on their social responsibilities and go off by themselves'. It seems that the thzoretical analysis of this perceived conflict is the most needed task in local public finance but it is totally independent of Tiebout's original formulation and concerns.
Appendix 1
From eq. (3.15) it follows that r:-U ~"
G(~- 2,e)/p
•G 2
N
Eo ~2 ~_p
(,___:, ,c,,o,j
},
(A.l', "-'-~ bN =
aGaN-
Y \ - ~ I ] \ N ] [~--]eo-2E, ,
N
N
Y "E, ?
~t" 2E,-
(A.2)
}
Eo ,
(A.3)
where Eo = E[(y-t(x))/ylfo(X)]
was defined in (3.2 I), and
E, = E[O'-t(x))/>,lA(x)], an~'} where the den.fity function f1(x) is defi,-_ed as
s(x)[(y- t(x))/y] (~ - P)IP,~(x) ~IP A(x) = ,i;,'o s-"-(x)[(>,- t(x))/>,],~-~ )/pv(xy/~ ,
(A...O
O. Fisch, Spatial equilibrium ~ith local public goods
215
From (A.I), (A.2) and (A.3), the necessary and sufficient conditions for a relative maximum are that the roots 2 of the determinamal e,iuction
_(~.~+GC"(G)+,~)~ l = 0 i
(A.5)
-(l+~.)
be negative, is that //>~
if
C"(G)=0,
or
fl(C'(G*)+G*C"(G*)) > yC'(G*),
if C"(G) > O.
Refenmees Arrow, Kenneqh J., 1971, A utilitarian approach to the concept of equality in public expenditures, Quarterly Journal of Economics 85, no. 3, Aug., 409-415. Buchanan, J.M. and Ch.J. Goetz, 1972, Efficiency limits of fiscal mobility: An assessment of the Tiebout model, Journal of Public Economics 1, no. 1, April, 25-43. Coleman, James S., 1968, The concept of equality of educational opportunity, Harvard Educational Review 38, 14-22. Edel, M. and E. Sclar, 1974, Taxes, spending and property values: Supply adjustment in a Tiebout-Oates model, Journal of Political Economy 82, Sept./Oct., 941-954. Ellickson, Bryan, 1971, Jurisdi¢-ional fragmentation and residential choice. American Economic Review (Papers and Proceedings) May, 334--339. EUickson, Bryan, 1973, A gcrmralization of the pure theory of public goods, American Economic Review 63, no. 3, June, 417-432. Fisch, Oscar, 1975, Optimal city st;m, land tenure and the economic theory of clubs, Jou:nai cf ~egional Science and Urban Economics, 13~c. Fis, ,, Oscar, 1975, Externalities, the urban rent and populatior' density functions: The case o¢ air pollution, Journal of Environmental Economics 2, no. 2, June. Gaffney, Mason M., 1972, The property tax is a progressive tax, )roceedings of 64th Annual Conference (National Tax Association, Columbus, OH) 408-426. Kieslir~g, H e r ~ n J., 1967, Measuring a local government service: A study of school districts in New York State, The Revie'~ of Economics and Statistics 49, Aug., 356-367. Musgrave, R.A., 1969. Provision for social goods, in: J. Margolis aad H. Guitton, eds., Public economics (St. Martin's Press, New Yorg). Oat~, Wallace E., 1969, The effects of property taxes and local pv.blic spending on )3ropcr:y values: An empir"tcal study of tax capitalization and the "riebout hypothesis. Journal of Po|itica! F.xonomy 77. Pollakrocs~J, Henry O., 1973, The effects; of property tax~ and local public spending on - , , ~ r t y values A comment and furtl',er results, Journal of Political Economy 1~1, July, 994-10OL ~.awis, John, 1971, A t~'w.ory of justice (Hz~,,ard University Press, C~mbridge, MA) ,~,amu¢|son, Paul, 19Y8, Local finance and ~he mathematics of marriage, Apl~n,)~x to: Aspects of public expenditure theorie~, The Re, ~ew of Economics and Statistics 40, Nov., 332 ~338.
216
O. .Fi.~:ch,Spatial equilibrium with local public goods
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