Sasaki (1991)

Ann Reg Sci (1991) 25:271-285 - - - T h e Annals o f - -

Regi0nal Science © Springer-Verlag 1991

Interjurisdictional commuting and local public goods

Komei Sasaki

Research Center for Applied Information Sciences, Tohoku University, Katahira 2, Sendal 980, Japan

Received October 1990 / Accepted in revised form July 199t

Abstract. This paper deals with the provision and finance of local public goods, where free riding across communities emanates from interjurisdictional commuting. In a two-city model with inter-city commuting incorporated, the relation- ship between a market equilibrium and an optimizing solution is investigated. In- struments for achieving Pareto-efficient locations, production and commuting are proposed.

1. Introduction

The seminal paper by Tiebout (1956) demonstrated that if each individual freely chooses a community for residence and each jurisdiction provides public goods locally, an efficient resource allocation may be obtained, because "voting with one's feet" substitutes for the market force. There have been both theoretical and empirical studies around the Tiebout hypothesis. Notwithstanding this hypothesis, it is widely recognized that free migration generally fails to Iead to a Pareto- efficient economy because of congestion in the consumption/provision of local public goods and/or the fiscal externality stemming from rent-sharing. The phenomenon of free riding across communities makes the provision and financing of local public goods more complicated. In reality, cross-town commuting is common and a vital part of community life. Many people work in different communities from where they live, which means they consume some types of local public goods (or services) provided by the jurisdictions of the communities in which they work. The public highway, transportation system, parks and library services are mostly used by such persons without their making any contribution to the provision of such services.

Over the last 20 years, every developed country has experienced the formation of "bedroom communities" or "dormitory suburbs" around large or even medium-sized cities. More than half the workers residing in bedroom communities commute to a locally central city. Local public goods are thus subject

272 K. Sasaki

to congestion according to the number of their users. Residents in a central city may be forced to pay more or higher taxes to finance more public goods to mitigate or eliminate the congestion. On the other hand, when a communi ty attracts commuters from other communities that communi ty has a relative advantage in production, so that wage rates in that communi ty become higher. For such agglomeration economies, many firms locate and operate in large cities, which means that more land space is occupied by firms and, as a result, the land rent is increased. Consequently, a locally central city becomes less desirable as a place to live, and the number of people residing in suburban communities and commuting to the central city increases further. Commut ing has rarely been analyzed in the literature on local public goods. ~ In the present paper, we use a simple two-city model with inter-city commuting incorporated to investigate the relation- ship between a market equilibrium and an optimizing solution, and propose instruments for achieving Pareto-efficient locations, production, and commuting.

The structure of the paper is as follows. In Sect. 2. the model is presented and a market equilibrium is defined. Section 3 examines the condition for Pareto-efficient resource allocation. Sect ion4 is intended to compare a competitive equilibrium with a Pareto-efficient solution.

2. The model and market equilibrium

Imagine a region containing two cities, which are contiguous but jurisdictionally independent of one another. Let us assume the efficiency in production is much higher in city I for the reasons historically given - e.g., there is a communicat ion node facilitating transportation to other regions, and/or more social capital stock related to industrial activities has accumulated, and/or the natural environment is more favorable - so that city 1 attracts commuters f rom city 2. There are N households (workers), who have homogeneous tastes and are employed in either of the two cities: thus,

N = N ~ +N2~ +N22 , (1)

where N/j is the populat ion residing in city i and working in city j . N2~ is thus the number of workers commuting f rom city 2 to city 1.2

In this region k assorted firms operate to produce various commodities for export. The production function of a f irm in either city is assumed to be represented by

X~ = J~ (n 1,1~) (2)

X2 = f2 (n2,/2)

i Schweizer (1985) is one of the exceptions who recognize a free-ride of commuters from a central city in consuming some types of local public goods. 2 The possibility of "cross hauling" of commuters is precluded here. It is technically possible to ad- mit cross hauling between two cities such that both N12 and Nz~ are positive. However, in the context of Pareto-efficient allocation developed in Sect. 3, the optimizing conditions necessarily exclude a "cross hauling" solution in a general circumstance. See Sakashita (1988).

Interjurisdictional commuting and local public goods 273

where Xi, ni and li are output, labor and land inputs, respectively of a firm operating in city i. f i is assumed to show increasing returns to scale until a cer- tain input level of labor and land, say t~ i and / / i s reached and beyond that level, decreasing returns to scale. Notwithstanding the assumption of assorted firms, the production efficiency is assumed to be higher in city 1 than in city 2 for the differences in location-specific factors described above when the level of input is high, i.e.,

f l (n, l) >f2(n,/') for n_> n (1) given 1"

and

f1(ft, l ) > f 2 ( h , l ) for l>_l(h) given

(3)

Letting k i denote the number of firms operating in city i(i = 1,2), it follows that

gin1 = N i l + N 2 1

k2n2 = N22

If the market price of an export good is p, regardless of which city produces it, the profit of a firm operating in city i is represented by

zri= p X i - w i n i - r i l i i = 1,2 , (4)

where w i is the wage rate and r i the land rent in city i. Each firm decides on its location and the amount of labor and land inputs given [w i, r i, ki; i = 1,2} so as to maximize the profit. It thus holds in the competitive market situation that

Wi = fin (17i, li)

, , = f~(ni, 13

i = 1,2 .

(5)

Each household receives its income from three different sources. Income from work is the first source. Secondly, it is assumed that residents in the two cities have equal shares in the stocks of k firms, wherever they reside, so that profits are equally distributed to each household. Thirdly, the total land rent in a city is redistributed equally to the city's residents provided that the total land is shared equally by the residents. The distributed income therefore differs among three types of households classified in (1). Letting Yij denote the distributed income of a person residing in i and working in j , we see that

zr rl L 1 Yll = wl + - - +

N N~t

zr rzL 2 Yzl = wl +- - '~ (6)

N N21 +N22

274 K. Sasaki

n r2L 2 Y22 = W2 + - - + - -

N N21 +N22

where n = kt nl +k2n2, and L i the total land size in city i. The utility level of a household depends on the consumption of the composite

good x, residential land size h, and two types of local public goods Z and G. Z denotes the level of service of the local public good that affects the environment for commuting and working, e.g. highway, railway, park, and library, while G is the provision level of local public good affecting the residential environment, e.g. school, police service, fire department, sewage, and public health. Arguments entering an individual's utility function are thus Z, provided by a city where he or she works, and G, provided by a city where he or she resides, i.e.,

u ( i , j ) = u (xij, hij, Zj, Gi) (7) ( i , j ) = (1, 1), (2,1), (2, 2) .

The land space in each city is used largely for either firms' input or private living space, so that

k 1l 1 +N~I hl~ = L1

k212 + N 2 I h21 +N22h22 = L 2 (8)

Each household is assumed to pay two kinds of tax to the municipal authority in whose catchment area it is based: one is poll tax and the other is property tax levied in respect of the land owned by each household. The budged constraint on each household is hence represented by:

Yll = xll +rl hll + ~ + t~ r~L~ N.

Y21 = x21 +rzhal + r z4 t2r2L2

N21 +N22

Y22 = x22 + ?'2 h22 + ~2 -~ t2r2L2

N21 +N22

+ 0 (9)

where the price of the composite good is fixed at unity, zi and t i are poll tax and ad valorem land tax rates in city i, respectively, and 0 the cost of commuting from city 2 to city 1. Throughout the paper it is assumed that the transportion service for commuting is provided at a constant average cost equal to 0.

Each household behaves in a manner which will maximize (7) subject to (6) and (9). In other words, an individual chooses his or her residential location, workplace, amount of composite good, and land area for residence to maximize his or her utility level given the state of [Yi, ri, zi, ti, Zi, Gi, Nij, O: i,j}. The following are derived as the necessary conditions for the optimum:


Ux(X11,h11,Z1,Gl) i

uh(xll ,hll ,Z1,G1) rl

Ux(X21 , h21 , Z1, G2) _ 1

Uh (X21, h21, Z1, G2) r2

Ux(X22' h22' Z2' G2) = Z

b/h (X22, h22, Z2, G2) r2

(1o)

Each municipal authority provides two kinds of public goods within its jurisdiction. Congestion is generated in the consumption of either public good according to the number of consumers. Reflecting the congestion phenomenon, the supply cost function of each public good, is assumed to be represented by (ll) as in Wildasin (1987), providing that in each jurisdiction two kinds of public goods are produced and provided independently of each other. 3

Cll = ClI (Z1,Nll +N21

C12 = C12(G1,Nll )

C21 = {2'21 (Z2, N22)

c22 = c=(c> N21 +N=)

0 C > 0 , 0 2 C > 0 , 0 C > 0 , OZ 0 Z 2 ON

C(O,N) = 0 .

02C

0 N 2 - - > 0

(11)

Each local government finances its provision of public goods from tax revenue: i.e.,

Ct~ +CI2 = q N l l +hr lL1

C12+C22 = r2(N21 + N22) + t2 r2 L2 . 02)

Taking (4), (6), (9) and (12) into consideration, we obtain the resource constraint on the two-city region in the form:

P(klXl + k2X2) = NIlx11 +N21x21 +N22x22-1- 0N21 + Cl l +C12 +C2I +C22 .

03)

Equations (1) through (13) describe a system of two-city region economy. An equilibrium of the system attained in a decentralized economy; where every

household and every firm can choose its location and every market is competitive,

3 An alternative way of representing the congestion effect is to introduce the number of users of public goods in each individual's utility function as a negative argument (e.g., Flatters et al. 1974; Sandler and Tschirhart 1980; Oakland 1972; Berglas and Pines 1981). Also note it is assumed that no land is used for public utilities.

276 K. Sasaki

is characterized by the relations in (14) and (15) in addition to (1) through (12) [except for (7) and (11)].

= u(1, 1) = u(2, 1) = u(2,2) (14)

= nl = zc2 . (15)

That is, under the assumption of a homogeneous population, the same utility level is attained wherever an individual resides and works; under the assumption of identical technology, the same profit is obtained wherever a firm operates. 4 The system representing an equilibrium, the relations in (1) through (15) [except for (7), (11) and (t3)] consists of 30 equations. It is assumed, as in the literature so far (Flatters et al. 1974; Schweizer 1985; Wildasin 1987), that the supply levels of local public goods are exogenously specified. The exogenous parameters to the system are thus {k,N, Lt,L2,Z1,Z 2, 0t, 02, Ol and the variables endogenously determined are IN 11, N21, N22, Is, 12, nl, r/z, kl, k2, xl 1, x21, h 12, h21, h22, Yl 1, Yzl,Y22, wl, W2, rl, r2, Xt, X2,/~, ~, 21,22, tl, t2} 5.

3. Pareto-efficient resource allocation

In this section we will solve an optimizing problem for a Pareto-efficient resource allocation of the two-city system. The optimizing problem is to determine [x/j } {hijt {/il {ni} {k/} IN/j} [Z/} and {Gi}, maximizing u subject to (1), (2), (3), (8), (13) and (14). The Lagrangian equation is defined as

V = U+21 [u(xlt,hlt,ZI,G1)-u]+22[u(x21,h21,Z1,Gz)-U]

+ 23[U(Xzz, h2z, Z2, Gz)-U] + 24[Nlt + N21-kl nd

+ 2 5 [N22 - k 2 rt2] q- 2 6 [ k - k I - ](2]

+ 27 [N-Nil -N21 -N22] + A8 [LI - k l It - N I l hll] (16)

+ 29]L2 -k212 -N21 h21 - N22 h22] + )[i0 Ell (/'11,11) -X l ]

+ )[11 Lf2 (n2,/2)-)(2] + )[12 Lo ]cl X1 +P k2X2 - N i l xll

- NzlX21 -N22x22- ON21 -Cll (Z1, N11 +N21)- CI2(G 1 ,Nil)

- C21 (Z2, N22) - C22 (02, N21 -~ N22)] .

The first-order conditions for an optimum are shown in Appendix A. The MRS condition for a household between composite good and residential land size is:

4 With the relative position of regional production functions assumed above, it is more likely that a firm in city 1 faces the higher wage rate and land rent and operates with more labor and land inputs to satisfy the equal-profit condition (15). 5 The existence of a competitive equilibrium of this system is assumed.

Interjurisdictional c o m m u t i n g and local public goods

11 /gx _ 212

u~ ~ 2s

21 22 212 R x _ U x _

/./21 //22 29

277

(17)

The following interpretation can then be given to the Lagrange coefficients in the context of a competitive equilibrium:

28 = r 1 ; 2 9 = r 2 ; 212 = 1 .

Manipulating the first-order conditions and taking account of (17), we obtain the optimum condition for the provision of local public goods in the form:

11 21 R z U z

N i t _-777]-+N)1 ~ = CI1 z Ux Ux

/./22 j~r22 z _ ~ - - C2iz

Ux 11

UG Nll ~ = C12 G b/x

21 22 /'/G biG

N21 ~ + N 2 2 ~ = C22 G , b/x b/x

(18)

which is the well-known Samuelson condition: the sum of users of the marginal evaluation of public commodity in terms of composite good is equal to the marginal cost of supplying the public good.

The MRS condition for a firm is derived as

fi t _ As 24

f2. 25

(t9)

The associated coefficients are interpreted in the context of the market solution as follows:

24=wi ; 25=w2 ; 210=Pkl ; 2 1 1 = p k 2 •

The relation below is obtained as the optimum condition for the number of residents and commuters.

278 K. Sasak i

"~7 = W l - - X l l - r ~ h ~ - - C I 1 N - - C12N

= WI --X21 -- F2h21 - C l l N -- C22N -- 0

= W2 --X22 -- r 2 h 2 2 - C21N -- C 2 2 N ,

(20)

which states that the marginal net benefit to the existing population (an increase in wage income minus an increase in resource consumption including an increase in the congestion on public good) accruing from an additional entrant to a group is the same at )~7 among the three population groups. If, for instance, the first line is greater than the second and third ones in (20), then the number of residents in city 1 is below optimum, and thus Nil increases while N21 and N22 decrease.

Likewise, the optimum condition for the location of firms is derived as

)~6 = PX1 -w~ n~ -r~ 11

= p X z - w z n 2 - r z l 2 . (21)

Relation (21) implies that the benefit a city receives from the location of an additional firm, an increase in profit, is the same at )~6 between two cities. For example, when the first line in (21) is greater than the second, the number of firms operating in city 1 is relatively small.

In the above optimizing problem, the total population N and total number of firms k are held constant. If they are also variable, then under a Pareto-efficient allocation the relation in (22) is consistent with the Envelope theorem.

d V - 2 7 = 0

d N

d V -)~6 = 0 .

d k

(22)

Taking advantage of (13) and (20) through (22), we obtain

Cll + C12 + C21 +C22 = r iL l +r2L2+(Nl l +N20CllN

+ N~lC12NN22 C21N + (N21 + N22) C22N (23)

The relation in (23) is essentially the same as that in Arnott (1979, p. 82, 3.6), Ar- nott and Stiglitz (1979, p. 487, 2.25), and Berglas and Pines (1981, p. 156, 22) and might be called the extended version of the Henry-George theorem. Actually, where there is not congestion effect in consumption of a public commodity, i.e., C n = 0, (23) is reduced to the Henry-George theorem; namely

Cll + C12+C21 +C22 = r lL 1 +r2L2 , (24)

which states that the total cost of supplying public goods is equal to the total land rents in a region.


4. Comparison of competitive equilibrium and Pareto-efficient solution

This section is intended to investigate whether the conditions for the Pareto- efficiency are met in a decentralized economic system, and, if not, to find some fiscal policy which will enable those conditions to hold in a competitive economy.

The MRS conditions in (17) and (19) are met in a competitive economy. In view of (15), the optimum condition for location of a firm (21) is also met in a market equilibrium. Relation (18) is the condition for an optimal supply of public goods where the number of users is given. Different views might exist about whether this Samuelson condition is satisfied or not in market equilibrium. Technically it is possible for local governments to provide the public goods in such a way- that the supply is adequate (18) if they know the preference structures of users. This might be the case, particularly because residents are not required to pay tax according to their marginal evaluation shown in (18) and hence they have no motive for giving a false picture of their preferences for free riding. However, the Samuelson condition is not relevant to the present context, since in this study competitive equilibrium is defined as a situation in which the supply level of local public goods is exogenously imposed.

The condition for population allocation (20) is more important for a comparison with a market equilibrium. Using (6) and (9), relation (20) is rewritten as

(1-t~)rILl n 2 7 = r 1 C I l N - - C 1 2 N

N~ N

(1 - t 2 ) r z L 2 7r = T 2 C l l N - C22 N (25)

N21 +N22 N

(I - t2) r2 L 2 -- T2 C 2 t N - - C 2 2 N •

N21 + N 2 2 N

It is thus our concern to find whether (25) can be met under some tax scheme to satisfy the local government's budget constraint in (12). Where interjurisdictional commuting is not allowed, Wildasin (1987) proposed two propositions on this subject. We therefore start by examining whether his two propositions also hold true in our context. Proposition 1 (Wildasin, p. 1140) states that if local public goods are q u a s i - p r i v a t e a competitive equilibrium will be locationally efficient under pure poll taxation (without land rent tax). For a public good to be quasi-private implies that the cost of providing it is represented in the form

C ( 9 , Z ) = ~ c ( Z ) .

Under this condition and pure poll taxation, (25) is reduced to

280 K. Sasaki

rlL t N ~7 = 2-I Cl i (Z1) - e l 2 (G1)

N~ z~

r2L 2 7r = 272 1211 (Z1) - c22 (G2)

N21 +N22 N

r2L 2 7~ = 2"2 C21 (Z2) - ¢22 (G2)

N21 + N 2 2 N

(25)'

and the budget equation of local government is rewritten in the form

r~ NI~ +N2~ - c~1 ( Z 0 + clz (GO

Nil N2z

T 2 -- C21 (Z2) -t- £'22 (G2) . N:~ +N22

(12)'

It is observed from (25)' and (12)'that Proposition 1 applies in conditions of no inter-city commuting where the rental revenue in a region is equally redistributed

among the total population, i.e., per capita distribution is rlL1 +r2L2. Where N

commuting from city 2 to city 1 takes place, i.e., N21 '# 0, however, the head tax rates determined by (12)' generally does not satisfy (25)': the only exception is when Z1 = Z2 = 0, so that cll (0) = c21 (0) = 0, and thus, pure poll taxation cannot attain an efficient allocation even with common land sharing

I rl Zl ~r2Z2-

Wildasin's Proposition 2 states that Pareto-efficient allocation is achieved with common land sharing by setting poll taxes equal to the marginal costs of providing public goods and then determining land tax rates so as to satisfy government budget constraints. In our context this suggests setting the poll taxes at

2" l = C l l N -t- CI2 N (26)

2-2 = C 2 1 N + C 2 2 N

and then determining t~ and t2 on the basis of (12) given [Z, G, r,N}. We however, observe that under this regime the first line is equal to the third line in (25) but they are, in general, not equal to the second line as far as N2~ * 0 even with common land sharing: the exceptional instances in which CllN = C2IN are those in which the public goods are non-congestion-prone (CllN = C2~N = 0) or quasi- private with Zl = Z2. To sum up, Wildasin's two propositions do not hold where interjurisdictional commuting takes place. This implies that complete dependence on poll tax and/or land tax is not sufficient to ensure efficiency of a competitive equilibrium and therefore some additional instruments are required.


What is called commut ing tax is introduced for attaining the optimal locations of residents and optimal numbers of commuters on the assumption that the marginal congestion cost of Z in city i is larger than that of city 2. City 1 imposes this tax on those commuting from city 2 at a per capita. 6 Under the commuting tax scheme, the relation in (25) is reduced to

•7 2"1 ( 1 - t l ) r l L 1 = C11 N -- C12N

Nil N

( l - t2) r2 L 2 = 2"2 C l I N - C 2 2 y + a (25)"

N21 +N22 N

=r2 (1 -- t2) r2L 2

C21N-C22N • N21 + Nz2 N

The rate of commuting tax is set equal to the difference in marginal congestion cost of Z between two cities: i.e.,

a = C~1 y - C21N • (27)

Poll tax and land tax rates are determined so as to satisfy the following relations.

2"1 -- 2"2 = C12N -- C22N q- a "F (1 - t l ) r lL1 ( l - t 2 ) r 2 L 2

NI~ N21+ N22 (28)

Cl l -t- C I 2 = T1Nlt +tlrlLl +aN21 ,

C21 + C22 = Z'z(N21 +N22) + t2 r2 L 2 . (29)

By implementing the tax scheme represented in (27), (28) and (29) a Pareto-efficient resource allocation can be attained in a competitive equilibrium. In par- ticular, with common land sharing, if the commuting tax is set by (27), poll tax by (26), and land tax by (29), then the relation in (25)" holds true: this might be an extended version of Wildasin's Proposition 2.

Before closing this section, we will examine whether the Henry-George theorem is satisfied within each city when the numbers of population and firms in two cities as a whole are optimized. Where N and k are chosen optimally, conditions (22) and (23) are satisfied. The Henry-George theorem (23) indicates that if taxes are imposed to internalize the externalities stemming from congestion and commuting according to (27) and (28), whereby the amount of (N~l +N21)Cll N "k-NIIClZN-t-N22C21N+(N21-I-N22)C22 N is collected, then the additional costs of providing local public goods is met by land tax exhausting the total (differen- tial) land rents in a two-city region. Since )~6 = )~7 = 0 under the optimal numbers

6 In practice, the commuting tax might be collected in the form of increased transport cost for commuting: the fare for public transportation is set equal to 0+ a.

282 K. Sasaki

of population and firms in a region, the poll tax rate in each city is derived from (28) as

(1 - t l ) r~ L ~ Zl -- + C11 N + C12N

g .

z2 - (1-t2)r2L2 +.C21N+C22 N . N21 +N22

(30)

Substitution of (30) into (29) yields:

Cll +C12 = riLl +Nll(CllN+CI2N)+aN21 C21 + C22 = r2L2 + (N21 +N22)(C21N + C22N) •

(31)

It is easily seen that the above tax scheme is equivalent to {zl = CIIN+CI2N, T2 = C21N+C22N, a = CllN--C21N, t 1 = t 2 = 1}. This implies that the Henry-George theorem applies in each city in that a municipal authority's expenditure in excess of the poll tax and commuting tax is just covered by the toal land rent in a city. We note, however, that the Henry-George theorem does not, in general, hold within each jurisdiction under common land sharing, since (31) is replaced by

Ci l + C12 =

C21 +C22 =

[Nil (1 - t l)+Ntl}rj L1 +NII (1 - t2) r2L2 N

+ Nll(CIIN+C12N)+ aN21

(Nzl +NzE)(1-t l ) r lL1 + l(N21 +N22)(1 - t2)+NtzlrzL2

N

+ (N21 +N22)(C21N + C22N) •

(31)

Finally, we ~ve a numerical example of an optimum tax scheme in the interci- ty commuting setting. Let the costs of providing local public goods per month be represented in the following linear additive functions:

Cll = 1000 Z I +3000 (Nll +Nzi)

C12 = 500 GI + 1000Nil

Czl = 1000 Z 2 + 1000 N22

C22 = 500 G 2 + 1000 (N21 +N22) •

Assume the following values:

L1 = 50000 (Tsubo; about 165000 m 2)

L 2 = 150000 (Tsubo: about 495000 m 2)


r 1 = 200 (yen per Tsubo/month)

r 2 = 111 (yen per Tsubo/month)

N t l = N21 = N22 = 1000

Z 1 = 200 (unit), Z2 = 100 (unit)

G1 = 100 (unit), G 2 = 200 (unit) .

That is, the size of the suburban city (city 2) is three times that of the central city (city 1); the land rent of city 1 is about 1.8 times that of city 2; half the residents of city 2 commute to city 1; city 1 provides type 2 public commodity more while city 2 provides more of type 2 public commodity. In these circumstances, the following tax scheme (per month) satisfies the Pareto-efficient conditions (27), (28), and (29): a = 2000 (yen), rl = 3000 (yen), 2 2 = 1000 (yen), tl = 0.225, t2 = 0.072. The poll tax in city 1 is three times that in city 2; the sum of poll tax and commuting tax for a commuter to city 1 is the same as the poll tax for a resi- dent of city 1. A commuter has to pay the commuting (transport) cost in addition to poll tax, but benefits from the higher wage rate in city 1 and lower land rent in city 2: the land tax rate in city 1 is about three times that in city 2.

5. Concluding remarks

In this paper we have dealt with the provision and financing of local public goods when there is free riding across communities as a result of interjurisdictional commuting. We set a region consisting of two cities, a locally central city and its "dormitory town" incorporating inter-city commuting. Both a competitive equilibrium and a Pareto-efficient solution were derived, after which we investigated whether the conditions for Pareto-efficiency are met in a competitive economy and tried to work out some fiscal policy to make these conditions hold there if not. Where interjurisdictional commuting takes place, Wildasin's two propositions (1987) concerning the financing of local public goods do not apply, which implies that complete dependence on poll tax and/or land tax alone cannot guarantee the efficiency of a competitive equilibrium, so that some additional instruments are required. What is called commuting tax is introduced for the pur- pose of attaining the optimal locations of residents and optimal numbers of commuters.

Acknowledgements. A previous version of this paper was presented at the Regional Science and Urban Economics Workshop in Tsukuba, I wish to thank Professors Noboru Sakashita and Takahiro Miyao and other participants of the Workshop for their useful comments. I am also grateful to Professor David Batten and three anonymous referees for their helpful and constructive comments on the preliminary version. This research was supported in part by a grant from The Telecommunications Advancement Foundation.

284

Appendix A.

The f irst-order condi t ions f o r (16)

X11;

X2~:

222:

h11:

h21:

h22:

Z~:

z2:

GI:

G2:

Nil :

N2j:

N22:

h i :

n2;

11:

/2:

kl :

k2:

21:

x2:

2 1 u ~ 1 - 2 1 2 N l i = 0

22u~ l - 212N21 = 0

22 23 u~ - A12N= = 0

2iu~ ~-28Nll = 0

22b/21 -- 29N21 = 0

23b/22- 29N22 = 0

& u~ 1 - & u 21 - & 2 C l l z = 0

22 23/g z - - 2 1 2 C 2 1 z : 0

21bl ~ - 212C12 G : 0

&u~ + & u ~ - &2C22~ = 0

2 4 - 27 - 28 h~1 - )q2(xll + Clan + Clzn) = 0

2 4 - 2 7 - 2 9 h 2 1 - 2 1 2 ( X 2 1 + C l I N + C 2 2 N + 0 ) = 0

25 - 27 -- 29 h22 - 2 i2 (222 + C21N + C22N) = 0

- - 2 4 k I + 210fl n = 0

- &k2+ &lfzn = 0

- 2 s k t + 2ref i t = 0

- - 2 9 k Z + 211f21 = 0

- - 2 4 n 1 - - 2 6 - - 2 8 l 1 + 2 1 2 p X 1 = 0

- - 25 H 2 - - 2 6 - - 2912 + 2 1 2 p X 2 = 0

- 2 l o + 2 1 2 p k ~ = 0

- 2 1 ~ + , ~ a 2 P k 2 = 0 .

K. Sasaki

References

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Documents

Sasaki (1991)