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Effects of the property tax in urban areas
JOURNAL
OF URBAN
ECONOMICS
21,146-165
(1987)
Effects of the Property Tax in Urban Areas’ ROBERT
C. STEEN
Department of Economics, Miami University, Oxford, Ohio 45056 Received November 11,1983; revised June 3,1985
1. INTRODUCTION The purpose of this study is to construct and use a general equilibrium model of a metropolitan area to investigate the effects on efficiency, equity, and urban spatial structure of the property tax system and three alternative systems of urban public finance. The study is divided into two papers. In this paper the model and solution techniques are presented and the methods of determinin g parameter values are discussed. Then the model is used to compare the effects of a property tax system with the effects of a lump-sum tax system. In the second paper (Steen [12]), the model is used to examine the metropolitan form of government and the Tiebout-Hamilton system of urban public finance. Previous work on the property tax has shown that the effects of the tax depend on one’s perspective. Models which have been used to investigate the tax from a local perspective have ignored an important aspect of the current property tax system in urban areas: jurisdictional fragmentation and inter-jurisdictional mobility.2 In this paper it is shown that a general equilibrium model of an urban area with multiple jurisdictions and nonidentical households leads to different conclusions about the effects of the property tax on efficiency, equity, and urban spatial structure from a model with only a single jurisdiction. The reason is that the property tax induces substantial changes across jurisdictions in household location, as upperincome households attempt to avoid subsidizing the public service consumption of lower-income households. The model developed in this paper includes different levels of public services among the jurisdictions and requires a balanced budget for each jurisdiction. Since households consider the public service/tax package of ‘I am grateful to Edwin S. Mills and Kenneth Small for guidance in my dissertation research, from which this paper is drawn. Financial support for this research was provided by a grant from the Sloan Foundation to the Princeton University Ekonomics Department. * Tiebout models, of course, take this into account, but they have other serious shortcomings. See Steen [12]. 146 0094-1190/87 Copyright All rights
$3.00
0 1987 by Academic Press, Inc of reproduction in any form reserved.
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each jurisdiction in making residential location decisions, this paper investigates the differential, rather than the absolute, effects of the property tax. That is, since residential location choices depend on both taxes and public services, it is not reasonable to compare a model in which households pay for their public services with property taxes to a model in which they do not pay for public services. Lump-sum, or head, taxes are used as the point of comparison for two reasons.The first is the usual reason that a lump-sum tax is the only nondistorting tax and thus serves to highlight the distorting effects, on locational choices as well as on housing consumption, of the property tax. The second reason is that a lump-sum tax has been proposed by Mills [7], Oates [9], and others as the best way of financing local public services. Oates [9], for example, has written that a head tax is appealing “because it implies that individual decisions concerning the consumption of local public goods would be based, as in competitive markets, on real opportunity costs.” By a simple inversion of the arguments concerning the differential effects of the property tax, one can consider the results of substituting a lump-sum tax system for the current property tax system. Perhaps the greatest difficulty in analyzing the effects of the property tax from an urban perspective is that a metropolitan area is a complex simultaneous system in which many factors (e.g., household location decisions, equilibrium land rents) may be affected substantially by changes in the method of financing local public services. Thus a general equilibrium model is required. Analytic solution of even the simplest general equilibrium model of an urban area is difficult, however, and is probably impossible for a model which captures the essential features of the problem. The difficulty in this caseis not merely the complexity of such a model but also that functions will be discontinuous at jurisdiction boundaries. Fortunately, recent developments in computer technology allow numerical solution of such a general equilibrium model at modest cost. The ability to use a complex model is not the only benefit of using numerical solution, however. In considering the effects of any method of urban public finance, it is important to determine not only the qualitative effects but also the magnitude of the effects. Numerical solution offers quantitative as well as qualitative results. Most past research on the property tax has focused on its absolute incidence. The traditional view has been that the land portion of the tax is borne by landowners, since supply of land is approximately fixed, while the portion on capital in housing has excise tax effects. That is, since the net-of-tax return on capital must be equal in all uses, a residential property tax increases the gross return on capital in housing, resulting in an increase in the rental cost of housing. Viewing the supply of capital to the housing industry as perfectly elastic, the portion of the tax on capital thus is borne
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by housing consumers. Further, these increases in the rental cost of housing distort consumer choice, resulting in a deadweight loss. Mieszkowski [6] states a new view of the property tax which emphasizes the national perspective. Ignoring taxes on land, assuming that the supply of capital is perfectly inelastic, and viewing the tax as a national tax at a uniform rate on all capital, Mieszkowski shows that the full extent of the incidence falls on owners of capital. He also recognizes the existence of local variations in the tax rate and shows that these variations have both positive and negative excise tax effects. That is, the tax increases (decreases) the cost of capital in areas with a tax rate above (below) the national average rate. Thus the incidence of variations in local tax rates falls on some combination of owners of immobile factors and consumers of nontraded goods. Mieszkowski argues, though, that the results of local variations in tax rates will have no net effect on the incidence of the tax if there is little correlation between local tax rates and the level and distribution of income in the different jurisdictions. McLure [5] has pointed out, however, that from a local point of view, any positive local tax rate will increase the cost of capital in that jurisdiction above what it would be in the absenceof a local property tax (assuming the jurisdiction is small enough that a change in its tax rate will not have a significant effect on the average national rate). From a local perspective, then, the full amount of the local tax rate, not just the variation from the national average, will have excise tax effects, so that the tax has the effects suggested by the traditional view. The feasibility of using a general equilibrium model to study problems of urban public finance has been demonstrated by Amott and MacKinnon [I]. Using an extremely simple model which assumesidentical households, only one jurisdiction, and no public services, they find the incidence of the tax to be approximately that suggestedby the traditional view. 2. THE MODEL The model is a generalization of the monocentric city model of urban economics to include a simple public sector. The urban area is divided into concentric rings about the city center, with each ring representing an independent local governmental unit or jurisdiction. The number of jurisdictions, m, and the area and location of each jurisdiction are exogenous. To approximate more closely the continuous nature of locational choice within jurisdictions, the entire urban area is further divided into small concentric rings around the city center. Thus each jurisdiction is divided into a number of rings, or zones. The quantity of public services provided per household varies across jurisdictions but is constant within a given jurisdiction. The local public
PROPERTY
TAX
IN URBAN
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149
service, G, is a collectively provided private good. It is produced at a constant marginal cost per household using only a composite numeraire good, C, with the same simple production function in each jurisdiction: Gk
=
k = l,...,m
cgk
where Cgk is the quantity of the composite good used as an input in jurisdiction k. With property tax finance, the tax bill for housing occupied by a household of type i in ring j, lJj, is qj = t$Hij where Hij is the endogenous number of units of housing occupied by such a household, Pj is the endogenous rent per unit of housing in j, and t: is the endogenous property tax rate for the jurisdiction k in which ring j is located. Budget balance requires gkNk
” =
c pj~HijNij jEJk
i
where Jk is the set of the indices of all rings which lie in jurisdiction k, Sk is the exogenous provision of public services per household in k, Nk is the endogenous total population of k, and Nlj is the endogenous number of households of type i living in ring j. The use of property taxes requires fiscal zoning for the existence of equilibrium, so eachjurisdiction except the first (the central city) has an exogenous zoning rule which specifies gk, the minimum household consumption of housing in k.3 If a lump-sum or head tax, L,, is used instead of the property tax, budget balance requires that the head tax equal the per household cost of producing the public services. Households are assumed to be identical in all respects except income. Each household chooses a residential location and quantities of housing and the composite good so as to maximize utility subject to budget, time, and zoning constraints. The household does this by determining its optimal location within each jurisdiction and then choosesthe jurisdiction in which it can attain the highest utility. Utility is assumed to be a CES function of the market goods, housing and the composite good, multiplied by the nonmarket goods, leisure and public services. The consumer’s problem, then, is to choose a residential location (which determines the quantity of 3The purpose of fiscal zoning is to establish a minimum property tax bill for residents of the community by establishing a floor on consumption of housing. In practice, communities use a combination of building codes and minimum lot size requirements since it is not legal to require a minimum level of housing consumption itself. See Mills [7].
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ROBERT
C. STEEN
leisure) and the quantities of housing and the composite good, so as to maximize
subject to r, = Pj(l + t;)Hij
+ L, + Cij + q,f”X,
Ej = F - qfXj Hij 2 Sk
where Cij is the consumption of the composite good by a household of type i residing in ring j, Ej is the leisure of a household living in j, q is income of a household of type i, ql? and qf are the annual money and time (in hours) costs, respectively, of commuting 1 mile round trip, Xj is the radial distance from the city center to the center of ring j, and F is total time available per day to a household. Under the property tax regime, if the zoning constraint is binding, the demand functions are simply
Cij = (y, - p,(l + ti)F& - qi”?.). Under the head tax regime or under the property tax regime when the zoning constraint is not binding, the demand functions are Hij =
~~/(l+P)p.*-l/(l+P)~.j* J
D g/('+P)y,t
Cij
=
=
D
lJ
where q
= r, - qyxj - L,
q* = p,(l + tk”) D
=
a;/(l+P)Pj*P/(l+P)
+
#/('+P) c
Landowners produce housing under conditions of perfect competition. At any distance from the city center they use the profit-maximizing input combination and rent housing to the highest bidder. The production
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function is assumed to be CES. It is convenient to work with the unit cost function which is
where u is the elasticity of substitution, P, and Prj are the exogenous rental price of structure and the endogenous land rent, respectively, and 6, and 6, are constants. The demand for structure and land per unit of housing can be found using Shephard’s Lemma. Landowners are exogenous in that they live outside the urban area and are assumed to spend all rents outside the urban area. To make the urban border endogenous, it is assumed that there exists an agricultural sector which will bid R,, the exogenous agricultural rent, for land anywhere in the city. Land which is not worth at least this amount in urban uses will be rented for agriculture. For simplicity, land which is used for agriculture is not taxed. In equilibrium, no household is able to attain a higher utility level by moving, and thus households of type i which reside in different rings (of the same or different jurisdictions) must attain the same utility level: qj 2 q,
if Nij > 0, Nid = 0, all i, j, d
qj = q.,
if Nij > 0, Njd > 0, all i, j, d.
The other equilibrium conditions are that land and housing are rented to the highest bidder, that each jurisdiction has a balanced budget, and that the quantity of land used in housing in each ring must be no greater than the total land available for residential purposes in that ring. 3. SOLUTION TECHNIQUE The solution technique is a generalization of a method used by Muth [8] and others to solve models with identical individuals and no public services. For such a model the method is as follows. Start with a trial price for housing at the center of the city. Then use the unit cost function for housing and Shephard’s Lemma to determine the land per unit of housing, and use the demand function for housing to find the quantity of housing per household. Multiply these quantities to get land per household. Then the number of households which can reside in the first ring is the available residential land divided by land per household. Since all households must attain the same utility level in equilibrium, the price of housing in all other rings can be determined from the housing bid-rent function. Then the number of households in each ring can be calculated as above. If marginal transportation costs are positive, the housing bid-rent will decrease with distance; thus land bid-rent, which is an increasing function
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ROBERT C. STEEN
of housing bid-rent, also will decline with distance. If the exogenous number of households has been housed exactly at the distance at which the implied land rent has fallen to the exogenous agricultural rent, the trial price is an equilibrium one. If too many (or too few) households have been housed, lower (or raise) the trial price and start again. The generalization of this method for a multiple jurisdiction-multiple household model without taxes is as follows. Equilibrium requires that no household type, or group, lives closer to the center of the city than another which has a steeper bid-rent curve (see Henderson [4]). So arrange the groups in decreasing order of steepnessof their bid-rent curves.4Assign the group with the steepest bid-rent curve to the first ring and house all of this group in the manner described above for the single-group case. When this group has been completely housed, assign the group with the second steepest bid-rent curve to the remaining land in the last ring occupied by the first group. Since members of both groups must pay the same rent per unit, this establishes the utility level for the second group, and the rest of the group can be housed in the usual manner. Continue this procedure with successive groups until the land rent equals the agricultural rent. If exactly the exogenous number of households has been housed, the trial price is an equilibrium one. If not, adjust the trial price appropriately and start over. The same method can be used for a multigroup, multijurisdiction model with lump-sum taxes, since the equilibrium tax amount is known in advance. For any model with property taxes, an equilibrium must be found in the above manner for given trial values of property tax rates for each jurisdiction. Then the deficit or surplus is calculated for each jurisdiction and new trial rates are determined. In most cases the best method of adjusting a tax rate is simply to use the tax rate which would have balanced the budget in the old equilibrium. The new trial rates are used to calculate a new equilibrium, surpluses and deficits are calculated, and new trial rates are determine. This continues until an equilibrium is found in which each jurisdiction has a balanced budget. In practice the budget is considered balanced if the surplus or deficit is less than two-tenths of 1% of expenditures in that jurisdiction. In most casesfour or five iterations are sufficient to attain this level of accuracy. 4. PARAMETER VALUES The model is calibrated to approximate an urban area in 1970 of approximately 600,000 population. Census data were assembled for all urban areas which met the following criteria in 1970: (a) population of ?his solution technique is appropriate only when households can be ordered a priori in terms of the steepnessof their bid-rent curves. The assumptions of the model presented here ensure that this can be done.
PROPERTY TAX IN URBAN AREAS
153
500,000 to 700,000, (b) only one central city, and (c) at least 10% of the urbanized area population residing outside the central city. The seven urban areas meeting these criteria are Akron, Birmingham, Dayton, Forth Worth, Oklahoma City, Rochester, and Sacramento. Whenever feasible, median values for these cities are used for parameter values. The choices for the number of household types and the number of jurisdictions for a baseline solution are interrelated. The baseline model is intended to represent the current method of urban public finance. In particular, it is assumed that not all households are able to find a jurisdiction which provides exactly the household’s desired public service/tax package. This means that the number of household types should be larger than the number of jurisdictions. A choice of three jurisdictions and five household types is sufficient to capture the essential features of the problem. Rather than representing merely three communities, the three jurisdictions should be thought of as representing a central city and two types of suburbs. One suburban jurisdiction represents middle to upper-middle income suburbs while the other represents upper-middle to upper income suburbs. With three jurisdictions and five household types, the degree of heterogeneity is no greater than in most actual suburban towns. Housing consumption, for example, varies by less than 25% in the first suburb and by less than 30% in the second. The property tax case was used to determine jurisdiction borders for the baseline solution. The central city boundary was chosen to include slightly more than 50% of the population of the urban area, approximately the median percentage for the seven urbanized areas. The border between the two suburbs was chosen so that the suburban population was divided approximately equally between them. The boundary between the central city and the first suburb is between zones 70 and 71, while the boundary between the two suburbs is between zones 96 and 97. These boundaries were maintained for all other solutions.5 The border between the urban area and the agricultural area is endogenous, as discussed in Section 2. A residential property tax rate of about 25% of the rental cost of housing is commonly accepted stylized fact. Accordingly, the levels of per household provision of public services, gk, were chosen so as to result in property tax rates of about 25%. The resulting expenditures per household were $390 in the single-jurisdiction case (discussed below) and $250, $460, and $600 for the central city and the first and second suburbs, respectively, in the three-jurisdiction case. For comparisons of property tax equilibria with head tax equilibria, the same &‘s were used in both solutions. ‘To incorporate endogenous boundaries would make the problem intractable. Further, the work of Ellickson [2,3] suggests that the political and economic forces working to maintain existing political boundaries are quite powerful.
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ROBERT C. STEEN
The zoning constraints were chosen so that in equilibrium they are binding on some but not all households in each suburban community, reflecting the observation that this is the casein practice. There is no zoning constraint for the central city. The methods of determining the remaining parameter values are discussed in the Appendix. 5. RESULTS There are several factors which will affect the welfare of a given household type under the property tax system. They include (1) the excessburden due to tax-induced distortions of consumer choice, (2) subsidization of the public service consumption of lower-income households by higher-income households, and (3) changes in relative land rents across jurisdictions. To help isolate the influence of these various factors, several models of increasing complexity are considered. The first models to be discussed include only a single jurisdiction and single household type. Then models with a single jurisdiction and multiple household types are considered, and finally models with multiple jurisdictions and multiple household types are discussed. The measure of welfare change is the general equilibrium analog of the compensating variation, the amount of additional income a household must have to be as well off in the new situation as in the original situation6 Since urban land rents are paid to landowners who do not reside in the urban area, the change in differential land rents must be added to the compensating variation to get a measure of the total change in welfare. Before the results are presented, the basic features of the baseline equilibria are mentioned here. Figure 1 shows the equilibrium allocation of households to residential locations in the baseline equilibrium for the three-jurisdiction model. Group 1, which consists of the lowest-income households, has the steepest bid-rent curve (labeled BRl in Fig. 1) and thus resides closest to the center, in zones l-22. All of groups 1 and 2 and about 60% of group 3 live in the first jurisdiction, the central city, while 40% of group 3 and 71% of group 4 live in the first suburb in this equilibrium. The remainder of group 4 and all of group 5, the highest-income households, live in the second suburb. The boundary between the central city and the first suburb is between zones 70 and 71, while the boundary between the
6The unusual compensating variation neglects the effects that the additional household income will have on prices. The general equilibrium version takes this into account by determining the lump-sum transfers which, in a general equilibrium model, will restore all households to their original utility levels.
PROPERTY TAX IN URBAN AREAS
I 10
I
I 30
I
I 50
I
155
! 70
90
110
130
ZONE
FIG. 1. Allocation of household types to zones. BRi = bid-rent curve for household group i.
two suburbs is between zones 96 and 97. The equilibrium tax rates are 0.25277, 0.25067, 0.24626 for jurisdictions 1, 2, and 3, respectively. As shown in Table l(c), population density at the city center in zone 1 is 73.2 households per acre. The gross-of-tax rent is $3305 per unit of housing, net-of-tax housing rent is $2638, and land rent is $5029 per acre of raw land. The households living in zone 1, members of group 1, consume 0.21 units of housing. Housing in this zone is produced using 21.02 units of structure and 0.065 acres of land per unit of housing. Land rent falls monotonically to $410 at the edge of the central city, resulting in declines in net-of-tax and gross-of-tax housing rents to $2081 and $2607, respectively. As land rent falls, land is substituted for capital in housing production, resulting in an increase in land per unit of housing to 0.356 acres, and a decrease in capital per unit of housing to 17.59 units at the edge of the central city. Location in the first zone of the first suburb is more attractive than at the outer edge of the central city for households of type 3, due to an 84%jump in per household public service provision with a negligible increase in commuting costs. For households of type 3 to attain the sameutility level in both jurisdictions, there must be a large jump in land rents from zone 70 to zone 71, with corresponding increases in housing rent and in the use of structure per unit of housing. A similar situation occurs at the boundary between the two suburbs. The zoning constraint of 0.7 units of housing is
156
ROBERT C. STEEN TABLE 1 Results for Sample Zones from Property Tax Solutions
Zone: Group 1 35 ‘d 71 96 97 118 135
3 3 3 3 3 3
1 35 70 71 96 97 118 135
1 3 4 4 5 5 5
1 35 70 71 96 97 118 135
1 2 3 3 4 4 5 5
Population density
H
Housing rent
Land
Gross Net
rent
(a) Single jurisdiction-single 34.7 0.58 3486 2783 19.7 0.62 3171 2531 9.4 0.67 2863 2286 9.2 0.67 2855 2279 4.6 0.70 2647 2113 4.5 0.71 2639 2107
3
unit of housii Structure
Land
household type 7605 21.88 3553 20.38 18.88 1317 1275 18.84 507 17.80 486 17.75
0.050 0.082 0.159 0.163 0.307 0.316
(b) Single jurisdiction-multiple 111.6 0.20 3547 2842 19.0 0.63 3144 2519 7.1 0.87 2848 2282 6.9 0.87 2840 2275 2.8 1.21 2653 2125 2.8 1.21 2646 2120 1.4 1.26 2495 1999 .-
household types 8855 22.23 3398 20.30 1290 18.85 1251 18.81 548 17.87 528 17.84 221 17.07
0.045 0.084 0.161 0.165 0.291 0.298 0.549
(c) Multiple jurisdiction-multiple 73.2 0.21 3305 2638 14.4 0.48 2904 2318 3.9 0.72 2607 2081 9.6 0.70 2885 2307 4.2 0.91 2687 2149 6.2 0.90 2806 2252 2.8 1.21 2647 2124 1.7 1.25 2523 2025
household types 21.02 5029 19.08 1530 410 17.59 19.00 1450 632 18.02 1112 18.66 17.87 544 272 17.23
0.065 0.144 0.356 0.149 0.264 0.179 0.292 0.474
binding on households of type 3 in the first suburb, as is the 0.9 constraint for type 4 households in the second suburb. Data for the baseline equilibrium for the single jurisdiction-five household-type model are also presented in Table 1. As discussed below, in equilibrium the population is packed closer to the center of the city than in the three-jurisdiction equilibrium because there are no jurisdiction boundaries to provide fiscal incentives for suburbanization. Also becauseof the lack of jurisdiction boundaries, land rents, housing rents, and capital input per unit decreasemonotonically as distance increases, all the way to the edge of the urban area. Otherwise the basic features of this solution are similar to those for the three-jurisdiction equilibrium.
157
PROPERTY TAX IN URBAN AREAS
The qualitative features of the single jurisdiction-single household-type solution, presented in Table l(a), are similar to those for the single jurisdiction-five household-type equilibrium. In this solution all households are of type 3, with median income for the five household types used in the other solutions. Now I turn to the results of the comparisons between the property tax and head tax solutions. The results are presented in Table 3. In this table, “base case” refers to the initial solution and “solution type” refers to the new situation, while PT and HT refer to property tax and head tax, respectively. The single-jurisdiction cases are labeled l/l and l/5, while the three-jurisdiction-five household-type case is labeled 3/5.
TABLE 2 Results for Sample Zones from Head Tax Solutions
Zone Group 1 35 70 71 96 97 118 135
3 3 3 3 3 3
1 35 70 71 96 97 118 135
1 3 4 4 5 5 5
1 35 70 71 96 97 118 135
1 3 4 4 5 5 5
Population density
H
Housing rent
Land Per unit of housing
Gross Net
rent
(a) Single jurisdiction-single 35.5 0.64 19.9 0.69 9.3 0.74 9.1 0.75 4.5 0.79 4.3 0.79 -
Structure Land
household type 2859 9226 2584 4235 2318 1526 2310 1475 2131 566 2124 542
22.32 20.69 19.07 19.03 17.91 17.86
0.044 0.073 0.144 0.147 0.284 0.293 -
(b) Single jurisdiction-five household types 128.3 0.20 2930 10,950 19.2 0.69 2568 4,015 6.9 0.98 2312 1,487 6.8 0.98 2305 1,440 2.7 1.38 2145 617 2.6 1.38 2139 594 1.3 1.44 2010 241
22.14 20.59 19.04 18.99 17.99 17.96 17.14
0.039 0.075 0.147 0.150 0.268 0.275 0.516
22.57 20.44 18.89 19.13 18.11 18.10 17.27
0.041 0.080 0.158 0.140 0.248 0.250 0.459 -
(c) Five jurisdiction-five household types 115.2 0.21 2900 10,213 17.7 0.71 2542 3,689 6.3 1.00 2288 1,331 7.4 0.97 2327 1,590 3.0 1.36 2163 690 3.0 1.35 2161 680 1.5 1.41 2031 285 -
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ROBERT
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Consider the single jurisdiction-single household-type model first. Welfare changes due to subsidization of lower-income households are not relevant here since there is only one household type and one jurisdiction. As can be seen in Tables 1 and 2, under the property tax regime the rental cost of housing gross of the tax is greater in each zone than the rental cost of housing in the same zone in the lump-sum tax solution. The reason is as follows. The rental price of capital is fixed exogenously, so the gross rent on capital must rise by the full amount of the portion of the tax on capital. The supply of land is inelastic, but not perfectly inelastic, so net-of-tax land rents fall by most of the portion of the tax on land, and gross-of-tax land rents increases by less than the amount of the tax throughout the city. From the unit cost function in Section 2, the rise in both input rents results in an increase in the rental cost of housing. This causes an excise tax type of distortion of consumer choice, with all households consuming less housing in the property tax equilibrium than in the lump-sum tax equilibrium. Compared with the head tax regime, incidence under the property tax regime is more favorable to households because they no longer have to pay the lump-sum amount of $390, while housing rents rise by less than the amount of the property tax because of the decline in the net-of-tax land rents. This shifting onto landowners of part of the burden of financing local public services more than offsets for households the welfare loss due to the tax-induced distortion of consumer choice, so household welfare is higher overall in the property tax solution. As shown in Table 3, the compensating variation with the property tax regime is $5.8 per year. That is, households would be as well off under the property tax regime as under the head tax regime if a lump-sum amount of $5.8 were taken from each household. TABLE
3
Comparison of Current System with Head Tax System Solution type: Base case: HHL welfare gain Group 1 Group 2 Group 3 Group 4 Group 5 Total HHL welfare gain Average HHL welfare gain Gain in land rents Total welfare gain Total welfare gain /#HHLS
PT l/l HT l/l
PT l/5 HT l/5
FT 3/5 HT 3/5
5.8
271.9 119.2 12.1 - 107.5 - 211.2
164.5 68.8 6.7 -53.2 - 211.6
1,131,OOO 5.8 - 5,111,424 - 3,986,424 - 20.4
955,000 4.9 - 4,930,164 - 3,914,664 - 20.4
- 976,200 - 5.0 - 11,012,030 - 11,979,230 -61.4
-
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159
Total household welfare gain-the gain per household multiplied by the number of households-is about $1.1 million. Landlord losses due to decreases in differential land rents total more than $5 million per year. Total welfare change-the gains to households minus the losses of landlords-is a loss of about $4 million. To get a better senseof the scale of the welfare loss, it may be helpful to divide the total welfare loss by the number of households in the city, while noting that this is not a measure of welfare change for a typical household since land rents are paid to exogenous landlords. Expressed in this manner, the net welfare loss is slightly more than $20 per household. The effects on spatial structure in the single jurisdiction-single group case are quite modest. The decline in the rental cost of land relative to the rental cost of capital under the property tax regime causes substitution of land for capital in housing production. This substitution of land for capital tends to increase the size of the urban area. However, the increase in the gross rental cost of housing causesa decline in housing consumption which more than offsets the increase in land used per unit of housing. The overall result, then, is a decreasein the total urban area of about 2.3%. Next, consider the single jurisdiction-five group model. The effects described above for the single jurisdiction-single household-type case are still relevant. There are additional effects of the property tax system in this model, however, due to the possibility of subsidization of the public service consumption of lower-income households by higher-income households. Despite the increase in housing rents, it can still be the case that the total cost of the housing/public service bundle is much lower for lower-income households than in the head tax equilibrium. Since they do not consume much housing in any event, lower-income households pay much less in property taxes than the averagecost of the public services,which is their tax bill in the lump-sum tax equilibrium. The upper-income households, of course, are adversely affected by subsidizing the public service consumption of lower-income households. The result, as shown in Table 3, is that the lowest-income households are better off with the property tax by $272 per year while the highest-income households are worse off by $271. The subsidization effects are much smaller for the next-to-lowest and next-tohighest income groups, and their compensating variations are correspondingly smaller. The middle-income households are relatively unaffected. Since the subsidization effects are transfers between households, they wash out on average and the average household welfare gain under the property tax regime is $4.9 annually, very close to the result for the one jurisdiction-one group case. Total household welfare gain is slightly less than $1 million, while differential land rent losses are nearly $5 million. Total welfare loss under the property tax regime is about $4 million, All of these results are very close to those obtained for the single jurisdiction-single
160
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group case. Thus, the main benefit of adding multiple household types to a single jurisdiction model is that it reveals the substantial variation in effects on different household types due to transfer effects across household groups. The overall effects on spatial structure are similar to those in the single jurisdiction-single group case. Again, the change in relative input prices and the decline in housing consumption due to the increase in gross housing rents have opposite effects on overall land consumption. The result is a decline in total urban area of about 2.9% Next consider the three-jurisdiction case. Now the upper-income groups are able to avoid subsidizing the lowest-income groups by moving to the suburbs. In the head tax equilibrium 74% of the population of the urban area lives in the central city, while in the property tax case only 52% of the population lives in the central city. However, the subsidization effects are reduced but not completely eliminated by the migration to the suburbs. In the central city, groups 2 and 3 subsidize the public service consumption of group 1; in the first suburb, group 4 subsidizes group 3; and in the second suburb, group 5 subsidizes group 4. Overall, relative to the single-jurisdiction model, the ability to move across jurisdiction boundaries in this model decreasesthe welfare losses of the upper-income groups and decreasesthe welfare gains of the lower-income groups when the property tax system is compared with the head tax system. However, substantial changes in relative land rents offset part of the effect. An increase in demand for land in the suburbs drives up suburban land rents as upper-income households migrate, while in the central city, since the supply of land is perfectly inelastic, land rents must fall until all land is rented. These changes in land rents are substantial. The net-of-tax land rent in the middle of the central city in the property tax equilibrium is only 41% of the rent at the same location in the head tax equilibrium. The corresponding figure for the first suburb is 92% and it is 193% for the second suburb. That is, the net-of-tax land rent falls only modestly in the first suburb and, in striking contrast to the sharp decline in land rents in the single-jurisdiction models, the net-of-tax rent nearly doubles in the second suburb. The effect on housing rents is significant but not nearly as large, since land cost is a small percentage of housing costs and since input proportions vary with changes in relative prices. Comparing the compensating variation results for the three-jurisdiction case with the single-jurisdiction case,the benefits of the property tax system for the lowest-income group are much smaller in the multiple-jurisdiction case-$165 rather than $272, as shown in Table 3. This result is a combination of a sharp loss in subsidization benefits and a partially offsetting gain due to the decline in relative land rents. The compensating variation loss of the highest-income group under the property tax regime is
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$212, rather than the $271 loss in the single-jurisdiction case. The sharp increase in land rents has offset much of the benefits of reduced subsidization. The effects on groups 2 and 4 are smaller than on groups 1 and 5 and smaller than on the same groups in the single-jurisdiction case. Again, the effects on members of group 3 are very small. The total effect on household welfare is a loss of slightly less than $1 million, or $5 per household on average, in contrast to a gain in household welfare in the single-jurisdiction case of about $5 on average. The reason is that upper-income households live further from the city in the multiplejurisdiction property tax solution, which allows them to avoid the transfer effect but results in an inefficient spatial allocation of land with inefficiently high commuting costs. The decline in differential land rents totals $11 million, but the effect is quite different acrossjurisdictions, as mentioned above. In the central city land rents falI by more than 50%--nearly $14 million-while they exhibit only a modest decline in the first suburb and a sharp increase in the second suburb. Total welfare losses are about $12 million, triple the total loss for the single-jurisdiction case. The dominant factor is the drop in central city land rents due to interjurisdictional mobility. Average total loss in welfare is $61 per household. The effects on spatial structure in the multiple-jurisdiction cases are strikingly different from those in the single-jurisdiction cases.Instead of a decrease in total urban area of between 2 and 3%, in this case the urban area increases by about 30%. The dominant factor is the movement of upper-income households away from the central city.’ The benefit of using a multiple-jurisdiction model, then, is that it reveals changes in household location across jurisdiction boundaries which result (a) in a substantial reduction of the subsidization effects revealed in the single jurisdiction-five group case, (b) in dramatic changes in relative land rents, and (c) in a substantial increase in size of the urban area. As mentioned in Section 1, some economists have suggesteda lump-sum tax as the best method of financing local public services. The above results can be reinterpreted to show the long-run effects of switching to a head tax system. Thus, for example, all the models indicate that an increase in overall welfare will result if this is done and the multigroup models suggest that upper-income households and landowners as a group would benefit from this change, at the expenseof lower-income households. In addition to the overall increase in welfare, a further attraction to many of a head tax system is that the resulting migration of upper-income households back to the central city would cause increases in property values there and would ‘The effects are considerably more complex than in the single-jurisdiction cases.Details are available from the author.
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greatly mitigate current financial pressures on the central city. Suburban land rents would fall substantially, however, and one would expect suburban landowners to be quite powerful in opposition to such a change. 6. LIMITATIONS
OF THE MODEL
The general equilibrium interactions in an urban economy are sufficiently complex that, even with numerical solution, it is necessary to use a highly stylized model. The model employed in this paper is subject to most of the usual criticisms of the standard monocentric city model. The assumptions of the monocentric city model which are criticized most frequently are (a) that there is only a single employment center, (b) that capital is perfectly mobile, and (c) that locational choice depends only on the tradeoff between commuting costs and housing consumption. The difficulties with these assumptions are well known (for a full discussion, see Wheaton [19]). The monocentric city model analysis of the effect of income on residential location decisions solely in terms of the income elasticities of demand for leisure and housing also has been criticized, by Wheaton [18]. The model presented here maintains the assumptions of a single employment center and perfect capital mobility. The present modeling of locational choice is an improvement over the standard treatment, however, since locational choice is also affected by the available public service/tax packages. However, there are other factors (e.g., spatial externalities) which are ignored. The characterization of the public sector is also highly stylized. It is assumed that the multidimensional output of public services can be viewed as a single good. Congestible public goods and public services which exhibit interjurisdictional externalities are ignored. Further, the level of output of public services is exogenous. These limitations of the model suggest that the results presented above must be interpreted with considerable caution. On the positive side, the model is very useful in aiding understanding of the interrelationships in an urban area. In particular, the results presented here focus attention on interjurisdictional mobility and the resulting effects on land rents. It would be a mistake, however, to use these results to make inferences about the effects of real world policy changes without taking into consideration the numerous factors which have been omitted from the analysis. 7. CONCLUSION Concerning methodology, this study suggeststhe importance of using a general equilibrium model to investigate questions of urban public finance. Although it is possible to determine all the relevant factors with a partial equilibrium approach, interactions between the various factors are likely to be neglected. In particular, the effects of the different factors on changes in
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land rents which have been shown here to be very important, are difficult to determine without a general equilibrium model. Also, it will be very difficult to determine the overall effects of a variety of factors which work in different directions and interact in a complicated fashion. The study also makes clear the importance, when investigating problems in urban public finance, of using a model which has multiple household types and multiple jurisdictions. For example, as shown by the results presented above, a single-household-type model incorrectly indicates that changing from a property tax to head tax finance would have little effect on household welfare, while a single jurisdiction-multiple household model incorrectly suggestsa decline.in the size of the urban area if such a change is made, while overstating the effects on welfare of high- and low-income households. Using a general equilibrium model with multiple jurisdictions and multiple household types, then, suggeststhe following results. Under the property tax system the metropolitan area is substantially larger, with land rents which are lower in the central city and higher in the suburbs, than in the head tax case. Lower-income households are better off at the expense of higher-income households and landowners, and the property tax system is less efficient, with overall welfare substantially lower than with lump-sum taxation. APPENDIX: PARAMETER VALUES The methods used for determining parameters for the housing cost function, the utility function, household income, marginal transportational costs, and the agricultural land rent are discussed in this Appendix. The other parameter choices are discussed in the text. The housing cost function parameters were determined conditionally on the elasticity of substitution using a method suggestedby Muth [8]. Following Muth, the elasticity of substitution, u, is assumed to be 0.75. The procedure involves solving two equations to determine 6, and 8,. The first equation is the ratio of the marginal productivity conditions while the second is a rearrangement of the production function:
S,/6, = (P*L/P,S)““( PJP,) -@-“)‘u
where P, is the annual rental value of an acre of raw land, L is the quantity of land in acres, P, is the rental value of a unit of structure, S is the quantity of structure, and H is the quantity of housing. A unit of housing was taken to be the amount supplied by the typical FHA-financed new home in the six urbanized areas in the data set which had new FHA homes
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(HUD [16]). The median lot size was 0.226 acres and the median market site price was $3966, or $17,543 per acre. The latter was decreased by one-fourth to get an approximate value for raw, i.e., undeveloped, land of $13,157 per acre. Assuming an interest rate of 6% per year, the annual rental value of an acre of raw land (I’,) is $789. The unit of structural expenditure was taken to be $1000. Subtracting the price of 0.226 acres of raw land from the median house price gives structural expenditure of $18,260, so S = 18.26. Again assuming an interest rate of 68, depreciation, maintenance, and repair costs of 3.5% per year, and mortgage insurance, hazard insurance, and miscellaneous costs of 1.5% the annual rental value of a unit of structure (P,) is $110. Substituting these values into the equations above yields S, = 0.0496906 and 6, = 2.41848. A similar method was developed to determine the utility function parameters conditionally on the elasticity of demand for housing. Available evidence (Polinsky [lo]) suggests -0.75 as a reasonable estimate. The procedure is to use this elasticity to determine p, then to use p to determine S,, and to use the condition that 6, and S, sum to one to determine S,. The resulting values were p = 0.48, 6, = 0.966709, and S, = 0.033291. A value of 0.041 was used for y, the exponent for public services in the utility function, indicating that households would spend about 4% of their income on goods currently financed by property taxes if those goods were available only in private markets. A value of 0.8 was used for (Y, the exponent for leisure. Sensitivity analysis has shown that changes in this parameter value have only a small quantitative effect and no qualitative effect on the results. Household income was determined from 1970 Census data [13] for the seven urbanized areas in the following manner. First, income per capita was found for family members and combined with the figures for unrelated individuals. From the 1970 Census of Housing [14], median household size for the seven areas was 3.1 people, so households of 3.1 individuals with the same per capita income were formed. The households were divided into five groups of equal size, with median household incomes of $2850, $6400, $9100, $12200, and $16850. From the Statistical Abstract 1151,average federal tax rates for these income levels in 1970 were about 0.0, 6.8, 9.99, 12.84, and 16.82%, respectively. After-tax incomes, then, were approximately $2850, $6000, $8200, $10650, and $14000, respectively. From National Transportation Statistics of 1972 [17] the marginal cost per mile of operating a standard-sized automobile is 5.856 cents. This is the sum of expenditures over a lo-year period on repairs and maintenance, tires, accessories,gas, oil, taxes on gas and tires, and one-fifth of the annual depreciation, all divided by 100,000 miles. Multiplying by 2 to get the cost of commuting 1 mile round trip and by 250 to get the annual cost gives a marginal cost of about $29 for each mile the household lives away from the
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city center. Assuming an average commuting speed of 20 mph, the annual cost in tune of commuting 1 mile round trip is 25 hr. A wide range of values seemspossible for the agricultural rent for raw land. Small [ll] suggeststhat the values from $7.50 to $750 might occur in different urban areas in different regions of the country. An intermediate value of $200 was used. REFERENCES 1. R. J. Amott and J. G. MacKinnon, The effects of the property tax: A general equilibrium simulation, J. Urban Econom., 4, 389-407 (1977). 2 B. Ellickson, Jurisdictional fragmentation and residential choice, American Econom. Rev. Pap. Proc. (1971). 3. B. Ellickson, The politics and economics of decentralization, J. Urban Econom., 4, 135-149 (1977). 4. J. V. Henderson, “Economic Theory and the Cities,” Academic Press, New York (1977). 5. C. McLure, The “new view” of the property tax: A caveat, Nat. Tax J. (1977). 6. P. Mieszkowski, The property tax: An excise tax or a profits tax? J. Public Econom. (1972). 7. E. S. Mills, Economic analysis of urban land-use controls, in “Current Issues in Urban Economics” (P. Mieszkowski and M. Straszheim, Eds.), Johns Hopkins Univ. Press, Baltimore (1979). 8. R. F. Muth, Numerical solution of urban residential land-use models, J. Urban Econom., 2, 307-332 (1975). 9. W. E. Oates, “Fiscal Federalism,” Harcourt Brace Jovanovich, New York (1972). 10. A. M. Polinsky, The demand for housing: A study in specification and grouping, Econometricu (1977).
11. K. A. Small, A comment on gasoline prices and urban structure, J. Urban Econom., 10, 311-322 (1981). 12. R. C. Steen, EtTects of governmental structure in urban areas, J. Urbun Econom., 21, 166-179 (1987). 13. U.S. Department of Commerce, Bureau of the Census, “1970 Census,” Vol. 1, Washington, D.C. 14. U.S. Department of Commerce, “1970 Census of Housing,” Vol. 1, Washington, D.C. 15. U.S. Department of Commerce, “Statistical Abstract of the U.S., 1971,” Washington, D.C. 16. U.S. Department of Housing and Urban Development (HUD), “FHA Homes, 1970,” Washington, D.C. 17. U.S. Department of Transportation, Transportation Systems Center, “National Transportation Statistics, 1979.” 18. W. Wheaton, Income and urban residence, Amer. Econom. Rev. (1977). 19. W. Wheaton, Monocentric models of urban land use: Contributions and criticisms, in “Current Issues in Urban Economics” (P. Mieszkowski and M. Straszheim, Eds.), Johns Hopkins Univ. Press, Baltimore (1979).
OF URBAN
ECONOMICS
21,146-165
(1987)
Effects of the Property Tax in Urban Areas’ ROBERT
C. STEEN
Department of Economics, Miami University, Oxford, Ohio 45056 Received November 11,1983; revised June 3,1985
1. INTRODUCTION The purpose of this study is to construct and use a general equilibrium model of a metropolitan area to investigate the effects on efficiency, equity, and urban spatial structure of the property tax system and three alternative systems of urban public finance. The study is divided into two papers. In this paper the model and solution techniques are presented and the methods of determinin g parameter values are discussed. Then the model is used to compare the effects of a property tax system with the effects of a lump-sum tax system. In the second paper (Steen [12]), the model is used to examine the metropolitan form of government and the Tiebout-Hamilton system of urban public finance. Previous work on the property tax has shown that the effects of the tax depend on one’s perspective. Models which have been used to investigate the tax from a local perspective have ignored an important aspect of the current property tax system in urban areas: jurisdictional fragmentation and inter-jurisdictional mobility.2 In this paper it is shown that a general equilibrium model of an urban area with multiple jurisdictions and nonidentical households leads to different conclusions about the effects of the property tax on efficiency, equity, and urban spatial structure from a model with only a single jurisdiction. The reason is that the property tax induces substantial changes across jurisdictions in household location, as upperincome households attempt to avoid subsidizing the public service consumption of lower-income households. The model developed in this paper includes different levels of public services among the jurisdictions and requires a balanced budget for each jurisdiction. Since households consider the public service/tax package of ‘I am grateful to Edwin S. Mills and Kenneth Small for guidance in my dissertation research, from which this paper is drawn. Financial support for this research was provided by a grant from the Sloan Foundation to the Princeton University Ekonomics Department. * Tiebout models, of course, take this into account, but they have other serious shortcomings. See Steen [12]. 146 0094-1190/87 Copyright All rights
$3.00
0 1987 by Academic Press, Inc of reproduction in any form reserved.
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each jurisdiction in making residential location decisions, this paper investigates the differential, rather than the absolute, effects of the property tax. That is, since residential location choices depend on both taxes and public services, it is not reasonable to compare a model in which households pay for their public services with property taxes to a model in which they do not pay for public services. Lump-sum, or head, taxes are used as the point of comparison for two reasons.The first is the usual reason that a lump-sum tax is the only nondistorting tax and thus serves to highlight the distorting effects, on locational choices as well as on housing consumption, of the property tax. The second reason is that a lump-sum tax has been proposed by Mills [7], Oates [9], and others as the best way of financing local public services. Oates [9], for example, has written that a head tax is appealing “because it implies that individual decisions concerning the consumption of local public goods would be based, as in competitive markets, on real opportunity costs.” By a simple inversion of the arguments concerning the differential effects of the property tax, one can consider the results of substituting a lump-sum tax system for the current property tax system. Perhaps the greatest difficulty in analyzing the effects of the property tax from an urban perspective is that a metropolitan area is a complex simultaneous system in which many factors (e.g., household location decisions, equilibrium land rents) may be affected substantially by changes in the method of financing local public services. Thus a general equilibrium model is required. Analytic solution of even the simplest general equilibrium model of an urban area is difficult, however, and is probably impossible for a model which captures the essential features of the problem. The difficulty in this caseis not merely the complexity of such a model but also that functions will be discontinuous at jurisdiction boundaries. Fortunately, recent developments in computer technology allow numerical solution of such a general equilibrium model at modest cost. The ability to use a complex model is not the only benefit of using numerical solution, however. In considering the effects of any method of urban public finance, it is important to determine not only the qualitative effects but also the magnitude of the effects. Numerical solution offers quantitative as well as qualitative results. Most past research on the property tax has focused on its absolute incidence. The traditional view has been that the land portion of the tax is borne by landowners, since supply of land is approximately fixed, while the portion on capital in housing has excise tax effects. That is, since the net-of-tax return on capital must be equal in all uses, a residential property tax increases the gross return on capital in housing, resulting in an increase in the rental cost of housing. Viewing the supply of capital to the housing industry as perfectly elastic, the portion of the tax on capital thus is borne
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by housing consumers. Further, these increases in the rental cost of housing distort consumer choice, resulting in a deadweight loss. Mieszkowski [6] states a new view of the property tax which emphasizes the national perspective. Ignoring taxes on land, assuming that the supply of capital is perfectly inelastic, and viewing the tax as a national tax at a uniform rate on all capital, Mieszkowski shows that the full extent of the incidence falls on owners of capital. He also recognizes the existence of local variations in the tax rate and shows that these variations have both positive and negative excise tax effects. That is, the tax increases (decreases) the cost of capital in areas with a tax rate above (below) the national average rate. Thus the incidence of variations in local tax rates falls on some combination of owners of immobile factors and consumers of nontraded goods. Mieszkowski argues, though, that the results of local variations in tax rates will have no net effect on the incidence of the tax if there is little correlation between local tax rates and the level and distribution of income in the different jurisdictions. McLure [5] has pointed out, however, that from a local point of view, any positive local tax rate will increase the cost of capital in that jurisdiction above what it would be in the absenceof a local property tax (assuming the jurisdiction is small enough that a change in its tax rate will not have a significant effect on the average national rate). From a local perspective, then, the full amount of the local tax rate, not just the variation from the national average, will have excise tax effects, so that the tax has the effects suggested by the traditional view. The feasibility of using a general equilibrium model to study problems of urban public finance has been demonstrated by Amott and MacKinnon [I]. Using an extremely simple model which assumesidentical households, only one jurisdiction, and no public services, they find the incidence of the tax to be approximately that suggestedby the traditional view. 2. THE MODEL The model is a generalization of the monocentric city model of urban economics to include a simple public sector. The urban area is divided into concentric rings about the city center, with each ring representing an independent local governmental unit or jurisdiction. The number of jurisdictions, m, and the area and location of each jurisdiction are exogenous. To approximate more closely the continuous nature of locational choice within jurisdictions, the entire urban area is further divided into small concentric rings around the city center. Thus each jurisdiction is divided into a number of rings, or zones. The quantity of public services provided per household varies across jurisdictions but is constant within a given jurisdiction. The local public
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service, G, is a collectively provided private good. It is produced at a constant marginal cost per household using only a composite numeraire good, C, with the same simple production function in each jurisdiction: Gk
=
k = l,...,m
cgk
where Cgk is the quantity of the composite good used as an input in jurisdiction k. With property tax finance, the tax bill for housing occupied by a household of type i in ring j, lJj, is qj = t$Hij where Hij is the endogenous number of units of housing occupied by such a household, Pj is the endogenous rent per unit of housing in j, and t: is the endogenous property tax rate for the jurisdiction k in which ring j is located. Budget balance requires gkNk
” =
c pj~HijNij jEJk
i
where Jk is the set of the indices of all rings which lie in jurisdiction k, Sk is the exogenous provision of public services per household in k, Nk is the endogenous total population of k, and Nlj is the endogenous number of households of type i living in ring j. The use of property taxes requires fiscal zoning for the existence of equilibrium, so eachjurisdiction except the first (the central city) has an exogenous zoning rule which specifies gk, the minimum household consumption of housing in k.3 If a lump-sum or head tax, L,, is used instead of the property tax, budget balance requires that the head tax equal the per household cost of producing the public services. Households are assumed to be identical in all respects except income. Each household chooses a residential location and quantities of housing and the composite good so as to maximize utility subject to budget, time, and zoning constraints. The household does this by determining its optimal location within each jurisdiction and then choosesthe jurisdiction in which it can attain the highest utility. Utility is assumed to be a CES function of the market goods, housing and the composite good, multiplied by the nonmarket goods, leisure and public services. The consumer’s problem, then, is to choose a residential location (which determines the quantity of 3The purpose of fiscal zoning is to establish a minimum property tax bill for residents of the community by establishing a floor on consumption of housing. In practice, communities use a combination of building codes and minimum lot size requirements since it is not legal to require a minimum level of housing consumption itself. See Mills [7].
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leisure) and the quantities of housing and the composite good, so as to maximize
subject to r, = Pj(l + t;)Hij
+ L, + Cij + q,f”X,
Ej = F - qfXj Hij 2 Sk
where Cij is the consumption of the composite good by a household of type i residing in ring j, Ej is the leisure of a household living in j, q is income of a household of type i, ql? and qf are the annual money and time (in hours) costs, respectively, of commuting 1 mile round trip, Xj is the radial distance from the city center to the center of ring j, and F is total time available per day to a household. Under the property tax regime, if the zoning constraint is binding, the demand functions are simply
Cij = (y, - p,(l + ti)F& - qi”?.). Under the head tax regime or under the property tax regime when the zoning constraint is not binding, the demand functions are Hij =
~~/(l+P)p.*-l/(l+P)~.j* J
D g/('+P)y,t
Cij
=
=
D
lJ
where q
= r, - qyxj - L,
q* = p,(l + tk”) D
=
a;/(l+P)Pj*P/(l+P)
+
#/('+P) c
Landowners produce housing under conditions of perfect competition. At any distance from the city center they use the profit-maximizing input combination and rent housing to the highest bidder. The production
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function is assumed to be CES. It is convenient to work with the unit cost function which is
where u is the elasticity of substitution, P, and Prj are the exogenous rental price of structure and the endogenous land rent, respectively, and 6, and 6, are constants. The demand for structure and land per unit of housing can be found using Shephard’s Lemma. Landowners are exogenous in that they live outside the urban area and are assumed to spend all rents outside the urban area. To make the urban border endogenous, it is assumed that there exists an agricultural sector which will bid R,, the exogenous agricultural rent, for land anywhere in the city. Land which is not worth at least this amount in urban uses will be rented for agriculture. For simplicity, land which is used for agriculture is not taxed. In equilibrium, no household is able to attain a higher utility level by moving, and thus households of type i which reside in different rings (of the same or different jurisdictions) must attain the same utility level: qj 2 q,
if Nij > 0, Nid = 0, all i, j, d
qj = q.,
if Nij > 0, Njd > 0, all i, j, d.
The other equilibrium conditions are that land and housing are rented to the highest bidder, that each jurisdiction has a balanced budget, and that the quantity of land used in housing in each ring must be no greater than the total land available for residential purposes in that ring. 3. SOLUTION TECHNIQUE The solution technique is a generalization of a method used by Muth [8] and others to solve models with identical individuals and no public services. For such a model the method is as follows. Start with a trial price for housing at the center of the city. Then use the unit cost function for housing and Shephard’s Lemma to determine the land per unit of housing, and use the demand function for housing to find the quantity of housing per household. Multiply these quantities to get land per household. Then the number of households which can reside in the first ring is the available residential land divided by land per household. Since all households must attain the same utility level in equilibrium, the price of housing in all other rings can be determined from the housing bid-rent function. Then the number of households in each ring can be calculated as above. If marginal transportation costs are positive, the housing bid-rent will decrease with distance; thus land bid-rent, which is an increasing function
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of housing bid-rent, also will decline with distance. If the exogenous number of households has been housed exactly at the distance at which the implied land rent has fallen to the exogenous agricultural rent, the trial price is an equilibrium one. If too many (or too few) households have been housed, lower (or raise) the trial price and start again. The generalization of this method for a multiple jurisdiction-multiple household model without taxes is as follows. Equilibrium requires that no household type, or group, lives closer to the center of the city than another which has a steeper bid-rent curve (see Henderson [4]). So arrange the groups in decreasing order of steepnessof their bid-rent curves.4Assign the group with the steepest bid-rent curve to the first ring and house all of this group in the manner described above for the single-group case. When this group has been completely housed, assign the group with the second steepest bid-rent curve to the remaining land in the last ring occupied by the first group. Since members of both groups must pay the same rent per unit, this establishes the utility level for the second group, and the rest of the group can be housed in the usual manner. Continue this procedure with successive groups until the land rent equals the agricultural rent. If exactly the exogenous number of households has been housed, the trial price is an equilibrium one. If not, adjust the trial price appropriately and start over. The same method can be used for a multigroup, multijurisdiction model with lump-sum taxes, since the equilibrium tax amount is known in advance. For any model with property taxes, an equilibrium must be found in the above manner for given trial values of property tax rates for each jurisdiction. Then the deficit or surplus is calculated for each jurisdiction and new trial rates are determined. In most cases the best method of adjusting a tax rate is simply to use the tax rate which would have balanced the budget in the old equilibrium. The new trial rates are used to calculate a new equilibrium, surpluses and deficits are calculated, and new trial rates are determine. This continues until an equilibrium is found in which each jurisdiction has a balanced budget. In practice the budget is considered balanced if the surplus or deficit is less than two-tenths of 1% of expenditures in that jurisdiction. In most casesfour or five iterations are sufficient to attain this level of accuracy. 4. PARAMETER VALUES The model is calibrated to approximate an urban area in 1970 of approximately 600,000 population. Census data were assembled for all urban areas which met the following criteria in 1970: (a) population of ?his solution technique is appropriate only when households can be ordered a priori in terms of the steepnessof their bid-rent curves. The assumptions of the model presented here ensure that this can be done.
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500,000 to 700,000, (b) only one central city, and (c) at least 10% of the urbanized area population residing outside the central city. The seven urban areas meeting these criteria are Akron, Birmingham, Dayton, Forth Worth, Oklahoma City, Rochester, and Sacramento. Whenever feasible, median values for these cities are used for parameter values. The choices for the number of household types and the number of jurisdictions for a baseline solution are interrelated. The baseline model is intended to represent the current method of urban public finance. In particular, it is assumed that not all households are able to find a jurisdiction which provides exactly the household’s desired public service/tax package. This means that the number of household types should be larger than the number of jurisdictions. A choice of three jurisdictions and five household types is sufficient to capture the essential features of the problem. Rather than representing merely three communities, the three jurisdictions should be thought of as representing a central city and two types of suburbs. One suburban jurisdiction represents middle to upper-middle income suburbs while the other represents upper-middle to upper income suburbs. With three jurisdictions and five household types, the degree of heterogeneity is no greater than in most actual suburban towns. Housing consumption, for example, varies by less than 25% in the first suburb and by less than 30% in the second. The property tax case was used to determine jurisdiction borders for the baseline solution. The central city boundary was chosen to include slightly more than 50% of the population of the urban area, approximately the median percentage for the seven urbanized areas. The border between the two suburbs was chosen so that the suburban population was divided approximately equally between them. The boundary between the central city and the first suburb is between zones 70 and 71, while the boundary between the two suburbs is between zones 96 and 97. These boundaries were maintained for all other solutions.5 The border between the urban area and the agricultural area is endogenous, as discussed in Section 2. A residential property tax rate of about 25% of the rental cost of housing is commonly accepted stylized fact. Accordingly, the levels of per household provision of public services, gk, were chosen so as to result in property tax rates of about 25%. The resulting expenditures per household were $390 in the single-jurisdiction case (discussed below) and $250, $460, and $600 for the central city and the first and second suburbs, respectively, in the three-jurisdiction case. For comparisons of property tax equilibria with head tax equilibria, the same &‘s were used in both solutions. ‘To incorporate endogenous boundaries would make the problem intractable. Further, the work of Ellickson [2,3] suggests that the political and economic forces working to maintain existing political boundaries are quite powerful.
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The zoning constraints were chosen so that in equilibrium they are binding on some but not all households in each suburban community, reflecting the observation that this is the casein practice. There is no zoning constraint for the central city. The methods of determining the remaining parameter values are discussed in the Appendix. 5. RESULTS There are several factors which will affect the welfare of a given household type under the property tax system. They include (1) the excessburden due to tax-induced distortions of consumer choice, (2) subsidization of the public service consumption of lower-income households by higher-income households, and (3) changes in relative land rents across jurisdictions. To help isolate the influence of these various factors, several models of increasing complexity are considered. The first models to be discussed include only a single jurisdiction and single household type. Then models with a single jurisdiction and multiple household types are considered, and finally models with multiple jurisdictions and multiple household types are discussed. The measure of welfare change is the general equilibrium analog of the compensating variation, the amount of additional income a household must have to be as well off in the new situation as in the original situation6 Since urban land rents are paid to landowners who do not reside in the urban area, the change in differential land rents must be added to the compensating variation to get a measure of the total change in welfare. Before the results are presented, the basic features of the baseline equilibria are mentioned here. Figure 1 shows the equilibrium allocation of households to residential locations in the baseline equilibrium for the three-jurisdiction model. Group 1, which consists of the lowest-income households, has the steepest bid-rent curve (labeled BRl in Fig. 1) and thus resides closest to the center, in zones l-22. All of groups 1 and 2 and about 60% of group 3 live in the first jurisdiction, the central city, while 40% of group 3 and 71% of group 4 live in the first suburb in this equilibrium. The remainder of group 4 and all of group 5, the highest-income households, live in the second suburb. The boundary between the central city and the first suburb is between zones 70 and 71, while the boundary between the
6The unusual compensating variation neglects the effects that the additional household income will have on prices. The general equilibrium version takes this into account by determining the lump-sum transfers which, in a general equilibrium model, will restore all households to their original utility levels.
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I 10
I
I 30
I
I 50
I
155
! 70
90
110
130
ZONE
FIG. 1. Allocation of household types to zones. BRi = bid-rent curve for household group i.
two suburbs is between zones 96 and 97. The equilibrium tax rates are 0.25277, 0.25067, 0.24626 for jurisdictions 1, 2, and 3, respectively. As shown in Table l(c), population density at the city center in zone 1 is 73.2 households per acre. The gross-of-tax rent is $3305 per unit of housing, net-of-tax housing rent is $2638, and land rent is $5029 per acre of raw land. The households living in zone 1, members of group 1, consume 0.21 units of housing. Housing in this zone is produced using 21.02 units of structure and 0.065 acres of land per unit of housing. Land rent falls monotonically to $410 at the edge of the central city, resulting in declines in net-of-tax and gross-of-tax housing rents to $2081 and $2607, respectively. As land rent falls, land is substituted for capital in housing production, resulting in an increase in land per unit of housing to 0.356 acres, and a decrease in capital per unit of housing to 17.59 units at the edge of the central city. Location in the first zone of the first suburb is more attractive than at the outer edge of the central city for households of type 3, due to an 84%jump in per household public service provision with a negligible increase in commuting costs. For households of type 3 to attain the sameutility level in both jurisdictions, there must be a large jump in land rents from zone 70 to zone 71, with corresponding increases in housing rent and in the use of structure per unit of housing. A similar situation occurs at the boundary between the two suburbs. The zoning constraint of 0.7 units of housing is
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ROBERT C. STEEN TABLE 1 Results for Sample Zones from Property Tax Solutions
Zone: Group 1 35 ‘d 71 96 97 118 135
3 3 3 3 3 3
1 35 70 71 96 97 118 135
1 3 4 4 5 5 5
1 35 70 71 96 97 118 135
1 2 3 3 4 4 5 5
Population density
H
Housing rent
Land
Gross Net
rent
(a) Single jurisdiction-single 34.7 0.58 3486 2783 19.7 0.62 3171 2531 9.4 0.67 2863 2286 9.2 0.67 2855 2279 4.6 0.70 2647 2113 4.5 0.71 2639 2107
3
unit of housii Structure
Land
household type 7605 21.88 3553 20.38 18.88 1317 1275 18.84 507 17.80 486 17.75
0.050 0.082 0.159 0.163 0.307 0.316
(b) Single jurisdiction-multiple 111.6 0.20 3547 2842 19.0 0.63 3144 2519 7.1 0.87 2848 2282 6.9 0.87 2840 2275 2.8 1.21 2653 2125 2.8 1.21 2646 2120 1.4 1.26 2495 1999 .-
household types 8855 22.23 3398 20.30 1290 18.85 1251 18.81 548 17.87 528 17.84 221 17.07
0.045 0.084 0.161 0.165 0.291 0.298 0.549
(c) Multiple jurisdiction-multiple 73.2 0.21 3305 2638 14.4 0.48 2904 2318 3.9 0.72 2607 2081 9.6 0.70 2885 2307 4.2 0.91 2687 2149 6.2 0.90 2806 2252 2.8 1.21 2647 2124 1.7 1.25 2523 2025
household types 21.02 5029 19.08 1530 410 17.59 19.00 1450 632 18.02 1112 18.66 17.87 544 272 17.23
0.065 0.144 0.356 0.149 0.264 0.179 0.292 0.474
binding on households of type 3 in the first suburb, as is the 0.9 constraint for type 4 households in the second suburb. Data for the baseline equilibrium for the single jurisdiction-five household-type model are also presented in Table 1. As discussed below, in equilibrium the population is packed closer to the center of the city than in the three-jurisdiction equilibrium because there are no jurisdiction boundaries to provide fiscal incentives for suburbanization. Also becauseof the lack of jurisdiction boundaries, land rents, housing rents, and capital input per unit decreasemonotonically as distance increases, all the way to the edge of the urban area. Otherwise the basic features of this solution are similar to those for the three-jurisdiction equilibrium.
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The qualitative features of the single jurisdiction-single household-type solution, presented in Table l(a), are similar to those for the single jurisdiction-five household-type equilibrium. In this solution all households are of type 3, with median income for the five household types used in the other solutions. Now I turn to the results of the comparisons between the property tax and head tax solutions. The results are presented in Table 3. In this table, “base case” refers to the initial solution and “solution type” refers to the new situation, while PT and HT refer to property tax and head tax, respectively. The single-jurisdiction cases are labeled l/l and l/5, while the three-jurisdiction-five household-type case is labeled 3/5.
TABLE 2 Results for Sample Zones from Head Tax Solutions
Zone Group 1 35 70 71 96 97 118 135
3 3 3 3 3 3
1 35 70 71 96 97 118 135
1 3 4 4 5 5 5
1 35 70 71 96 97 118 135
1 3 4 4 5 5 5
Population density
H
Housing rent
Land Per unit of housing
Gross Net
rent
(a) Single jurisdiction-single 35.5 0.64 19.9 0.69 9.3 0.74 9.1 0.75 4.5 0.79 4.3 0.79 -
Structure Land
household type 2859 9226 2584 4235 2318 1526 2310 1475 2131 566 2124 542
22.32 20.69 19.07 19.03 17.91 17.86
0.044 0.073 0.144 0.147 0.284 0.293 -
(b) Single jurisdiction-five household types 128.3 0.20 2930 10,950 19.2 0.69 2568 4,015 6.9 0.98 2312 1,487 6.8 0.98 2305 1,440 2.7 1.38 2145 617 2.6 1.38 2139 594 1.3 1.44 2010 241
22.14 20.59 19.04 18.99 17.99 17.96 17.14
0.039 0.075 0.147 0.150 0.268 0.275 0.516
22.57 20.44 18.89 19.13 18.11 18.10 17.27
0.041 0.080 0.158 0.140 0.248 0.250 0.459 -
(c) Five jurisdiction-five household types 115.2 0.21 2900 10,213 17.7 0.71 2542 3,689 6.3 1.00 2288 1,331 7.4 0.97 2327 1,590 3.0 1.36 2163 690 3.0 1.35 2161 680 1.5 1.41 2031 285 -
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Consider the single jurisdiction-single household-type model first. Welfare changes due to subsidization of lower-income households are not relevant here since there is only one household type and one jurisdiction. As can be seen in Tables 1 and 2, under the property tax regime the rental cost of housing gross of the tax is greater in each zone than the rental cost of housing in the same zone in the lump-sum tax solution. The reason is as follows. The rental price of capital is fixed exogenously, so the gross rent on capital must rise by the full amount of the portion of the tax on capital. The supply of land is inelastic, but not perfectly inelastic, so net-of-tax land rents fall by most of the portion of the tax on land, and gross-of-tax land rents increases by less than the amount of the tax throughout the city. From the unit cost function in Section 2, the rise in both input rents results in an increase in the rental cost of housing. This causes an excise tax type of distortion of consumer choice, with all households consuming less housing in the property tax equilibrium than in the lump-sum tax equilibrium. Compared with the head tax regime, incidence under the property tax regime is more favorable to households because they no longer have to pay the lump-sum amount of $390, while housing rents rise by less than the amount of the property tax because of the decline in the net-of-tax land rents. This shifting onto landowners of part of the burden of financing local public services more than offsets for households the welfare loss due to the tax-induced distortion of consumer choice, so household welfare is higher overall in the property tax solution. As shown in Table 3, the compensating variation with the property tax regime is $5.8 per year. That is, households would be as well off under the property tax regime as under the head tax regime if a lump-sum amount of $5.8 were taken from each household. TABLE
3
Comparison of Current System with Head Tax System Solution type: Base case: HHL welfare gain Group 1 Group 2 Group 3 Group 4 Group 5 Total HHL welfare gain Average HHL welfare gain Gain in land rents Total welfare gain Total welfare gain /#HHLS
PT l/l HT l/l
PT l/5 HT l/5
FT 3/5 HT 3/5
5.8
271.9 119.2 12.1 - 107.5 - 211.2
164.5 68.8 6.7 -53.2 - 211.6
1,131,OOO 5.8 - 5,111,424 - 3,986,424 - 20.4
955,000 4.9 - 4,930,164 - 3,914,664 - 20.4
- 976,200 - 5.0 - 11,012,030 - 11,979,230 -61.4
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Total household welfare gain-the gain per household multiplied by the number of households-is about $1.1 million. Landlord losses due to decreases in differential land rents total more than $5 million per year. Total welfare change-the gains to households minus the losses of landlords-is a loss of about $4 million. To get a better senseof the scale of the welfare loss, it may be helpful to divide the total welfare loss by the number of households in the city, while noting that this is not a measure of welfare change for a typical household since land rents are paid to exogenous landlords. Expressed in this manner, the net welfare loss is slightly more than $20 per household. The effects on spatial structure in the single jurisdiction-single group case are quite modest. The decline in the rental cost of land relative to the rental cost of capital under the property tax regime causes substitution of land for capital in housing production. This substitution of land for capital tends to increase the size of the urban area. However, the increase in the gross rental cost of housing causesa decline in housing consumption which more than offsets the increase in land used per unit of housing. The overall result, then, is a decreasein the total urban area of about 2.3%. Next, consider the single jurisdiction-five group model. The effects described above for the single jurisdiction-single household-type case are still relevant. There are additional effects of the property tax system in this model, however, due to the possibility of subsidization of the public service consumption of lower-income households by higher-income households. Despite the increase in housing rents, it can still be the case that the total cost of the housing/public service bundle is much lower for lower-income households than in the head tax equilibrium. Since they do not consume much housing in any event, lower-income households pay much less in property taxes than the averagecost of the public services,which is their tax bill in the lump-sum tax equilibrium. The upper-income households, of course, are adversely affected by subsidizing the public service consumption of lower-income households. The result, as shown in Table 3, is that the lowest-income households are better off with the property tax by $272 per year while the highest-income households are worse off by $271. The subsidization effects are much smaller for the next-to-lowest and next-tohighest income groups, and their compensating variations are correspondingly smaller. The middle-income households are relatively unaffected. Since the subsidization effects are transfers between households, they wash out on average and the average household welfare gain under the property tax regime is $4.9 annually, very close to the result for the one jurisdiction-one group case. Total household welfare gain is slightly less than $1 million, while differential land rent losses are nearly $5 million. Total welfare loss under the property tax regime is about $4 million, All of these results are very close to those obtained for the single jurisdiction-single
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group case. Thus, the main benefit of adding multiple household types to a single jurisdiction model is that it reveals the substantial variation in effects on different household types due to transfer effects across household groups. The overall effects on spatial structure are similar to those in the single jurisdiction-single group case. Again, the change in relative input prices and the decline in housing consumption due to the increase in gross housing rents have opposite effects on overall land consumption. The result is a decline in total urban area of about 2.9% Next consider the three-jurisdiction case. Now the upper-income groups are able to avoid subsidizing the lowest-income groups by moving to the suburbs. In the head tax equilibrium 74% of the population of the urban area lives in the central city, while in the property tax case only 52% of the population lives in the central city. However, the subsidization effects are reduced but not completely eliminated by the migration to the suburbs. In the central city, groups 2 and 3 subsidize the public service consumption of group 1; in the first suburb, group 4 subsidizes group 3; and in the second suburb, group 5 subsidizes group 4. Overall, relative to the single-jurisdiction model, the ability to move across jurisdiction boundaries in this model decreasesthe welfare losses of the upper-income groups and decreasesthe welfare gains of the lower-income groups when the property tax system is compared with the head tax system. However, substantial changes in relative land rents offset part of the effect. An increase in demand for land in the suburbs drives up suburban land rents as upper-income households migrate, while in the central city, since the supply of land is perfectly inelastic, land rents must fall until all land is rented. These changes in land rents are substantial. The net-of-tax land rent in the middle of the central city in the property tax equilibrium is only 41% of the rent at the same location in the head tax equilibrium. The corresponding figure for the first suburb is 92% and it is 193% for the second suburb. That is, the net-of-tax land rent falls only modestly in the first suburb and, in striking contrast to the sharp decline in land rents in the single-jurisdiction models, the net-of-tax rent nearly doubles in the second suburb. The effect on housing rents is significant but not nearly as large, since land cost is a small percentage of housing costs and since input proportions vary with changes in relative prices. Comparing the compensating variation results for the three-jurisdiction case with the single-jurisdiction case,the benefits of the property tax system for the lowest-income group are much smaller in the multiple-jurisdiction case-$165 rather than $272, as shown in Table 3. This result is a combination of a sharp loss in subsidization benefits and a partially offsetting gain due to the decline in relative land rents. The compensating variation loss of the highest-income group under the property tax regime is
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$212, rather than the $271 loss in the single-jurisdiction case. The sharp increase in land rents has offset much of the benefits of reduced subsidization. The effects on groups 2 and 4 are smaller than on groups 1 and 5 and smaller than on the same groups in the single-jurisdiction case. Again, the effects on members of group 3 are very small. The total effect on household welfare is a loss of slightly less than $1 million, or $5 per household on average, in contrast to a gain in household welfare in the single-jurisdiction case of about $5 on average. The reason is that upper-income households live further from the city in the multiplejurisdiction property tax solution, which allows them to avoid the transfer effect but results in an inefficient spatial allocation of land with inefficiently high commuting costs. The decline in differential land rents totals $11 million, but the effect is quite different acrossjurisdictions, as mentioned above. In the central city land rents falI by more than 50%--nearly $14 million-while they exhibit only a modest decline in the first suburb and a sharp increase in the second suburb. Total welfare losses are about $12 million, triple the total loss for the single-jurisdiction case. The dominant factor is the drop in central city land rents due to interjurisdictional mobility. Average total loss in welfare is $61 per household. The effects on spatial structure in the multiple-jurisdiction cases are strikingly different from those in the single-jurisdiction cases.Instead of a decrease in total urban area of between 2 and 3%, in this case the urban area increases by about 30%. The dominant factor is the movement of upper-income households away from the central city.’ The benefit of using a multiple-jurisdiction model, then, is that it reveals changes in household location across jurisdiction boundaries which result (a) in a substantial reduction of the subsidization effects revealed in the single jurisdiction-five group case, (b) in dramatic changes in relative land rents, and (c) in a substantial increase in size of the urban area. As mentioned in Section 1, some economists have suggesteda lump-sum tax as the best method of financing local public services. The above results can be reinterpreted to show the long-run effects of switching to a head tax system. Thus, for example, all the models indicate that an increase in overall welfare will result if this is done and the multigroup models suggest that upper-income households and landowners as a group would benefit from this change, at the expenseof lower-income households. In addition to the overall increase in welfare, a further attraction to many of a head tax system is that the resulting migration of upper-income households back to the central city would cause increases in property values there and would ‘The effects are considerably more complex than in the single-jurisdiction cases.Details are available from the author.
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greatly mitigate current financial pressures on the central city. Suburban land rents would fall substantially, however, and one would expect suburban landowners to be quite powerful in opposition to such a change. 6. LIMITATIONS
OF THE MODEL
The general equilibrium interactions in an urban economy are sufficiently complex that, even with numerical solution, it is necessary to use a highly stylized model. The model employed in this paper is subject to most of the usual criticisms of the standard monocentric city model. The assumptions of the monocentric city model which are criticized most frequently are (a) that there is only a single employment center, (b) that capital is perfectly mobile, and (c) that locational choice depends only on the tradeoff between commuting costs and housing consumption. The difficulties with these assumptions are well known (for a full discussion, see Wheaton [19]). The monocentric city model analysis of the effect of income on residential location decisions solely in terms of the income elasticities of demand for leisure and housing also has been criticized, by Wheaton [18]. The model presented here maintains the assumptions of a single employment center and perfect capital mobility. The present modeling of locational choice is an improvement over the standard treatment, however, since locational choice is also affected by the available public service/tax packages. However, there are other factors (e.g., spatial externalities) which are ignored. The characterization of the public sector is also highly stylized. It is assumed that the multidimensional output of public services can be viewed as a single good. Congestible public goods and public services which exhibit interjurisdictional externalities are ignored. Further, the level of output of public services is exogenous. These limitations of the model suggest that the results presented above must be interpreted with considerable caution. On the positive side, the model is very useful in aiding understanding of the interrelationships in an urban area. In particular, the results presented here focus attention on interjurisdictional mobility and the resulting effects on land rents. It would be a mistake, however, to use these results to make inferences about the effects of real world policy changes without taking into consideration the numerous factors which have been omitted from the analysis. 7. CONCLUSION Concerning methodology, this study suggeststhe importance of using a general equilibrium model to investigate questions of urban public finance. Although it is possible to determine all the relevant factors with a partial equilibrium approach, interactions between the various factors are likely to be neglected. In particular, the effects of the different factors on changes in
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land rents which have been shown here to be very important, are difficult to determine without a general equilibrium model. Also, it will be very difficult to determine the overall effects of a variety of factors which work in different directions and interact in a complicated fashion. The study also makes clear the importance, when investigating problems in urban public finance, of using a model which has multiple household types and multiple jurisdictions. For example, as shown by the results presented above, a single-household-type model incorrectly indicates that changing from a property tax to head tax finance would have little effect on household welfare, while a single jurisdiction-multiple household model incorrectly suggestsa decline.in the size of the urban area if such a change is made, while overstating the effects on welfare of high- and low-income households. Using a general equilibrium model with multiple jurisdictions and multiple household types, then, suggeststhe following results. Under the property tax system the metropolitan area is substantially larger, with land rents which are lower in the central city and higher in the suburbs, than in the head tax case. Lower-income households are better off at the expense of higher-income households and landowners, and the property tax system is less efficient, with overall welfare substantially lower than with lump-sum taxation. APPENDIX: PARAMETER VALUES The methods used for determining parameters for the housing cost function, the utility function, household income, marginal transportational costs, and the agricultural land rent are discussed in this Appendix. The other parameter choices are discussed in the text. The housing cost function parameters were determined conditionally on the elasticity of substitution using a method suggestedby Muth [8]. Following Muth, the elasticity of substitution, u, is assumed to be 0.75. The procedure involves solving two equations to determine 6, and 8,. The first equation is the ratio of the marginal productivity conditions while the second is a rearrangement of the production function:
S,/6, = (P*L/P,S)““( PJP,) -@-“)‘u
where P, is the annual rental value of an acre of raw land, L is the quantity of land in acres, P, is the rental value of a unit of structure, S is the quantity of structure, and H is the quantity of housing. A unit of housing was taken to be the amount supplied by the typical FHA-financed new home in the six urbanized areas in the data set which had new FHA homes
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(HUD [16]). The median lot size was 0.226 acres and the median market site price was $3966, or $17,543 per acre. The latter was decreased by one-fourth to get an approximate value for raw, i.e., undeveloped, land of $13,157 per acre. Assuming an interest rate of 6% per year, the annual rental value of an acre of raw land (I’,) is $789. The unit of structural expenditure was taken to be $1000. Subtracting the price of 0.226 acres of raw land from the median house price gives structural expenditure of $18,260, so S = 18.26. Again assuming an interest rate of 68, depreciation, maintenance, and repair costs of 3.5% per year, and mortgage insurance, hazard insurance, and miscellaneous costs of 1.5% the annual rental value of a unit of structure (P,) is $110. Substituting these values into the equations above yields S, = 0.0496906 and 6, = 2.41848. A similar method was developed to determine the utility function parameters conditionally on the elasticity of demand for housing. Available evidence (Polinsky [lo]) suggests -0.75 as a reasonable estimate. The procedure is to use this elasticity to determine p, then to use p to determine S,, and to use the condition that 6, and S, sum to one to determine S,. The resulting values were p = 0.48, 6, = 0.966709, and S, = 0.033291. A value of 0.041 was used for y, the exponent for public services in the utility function, indicating that households would spend about 4% of their income on goods currently financed by property taxes if those goods were available only in private markets. A value of 0.8 was used for (Y, the exponent for leisure. Sensitivity analysis has shown that changes in this parameter value have only a small quantitative effect and no qualitative effect on the results. Household income was determined from 1970 Census data [13] for the seven urbanized areas in the following manner. First, income per capita was found for family members and combined with the figures for unrelated individuals. From the 1970 Census of Housing [14], median household size for the seven areas was 3.1 people, so households of 3.1 individuals with the same per capita income were formed. The households were divided into five groups of equal size, with median household incomes of $2850, $6400, $9100, $12200, and $16850. From the Statistical Abstract 1151,average federal tax rates for these income levels in 1970 were about 0.0, 6.8, 9.99, 12.84, and 16.82%, respectively. After-tax incomes, then, were approximately $2850, $6000, $8200, $10650, and $14000, respectively. From National Transportation Statistics of 1972 [17] the marginal cost per mile of operating a standard-sized automobile is 5.856 cents. This is the sum of expenditures over a lo-year period on repairs and maintenance, tires, accessories,gas, oil, taxes on gas and tires, and one-fifth of the annual depreciation, all divided by 100,000 miles. Multiplying by 2 to get the cost of commuting 1 mile round trip and by 250 to get the annual cost gives a marginal cost of about $29 for each mile the household lives away from the
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city center. Assuming an average commuting speed of 20 mph, the annual cost in tune of commuting 1 mile round trip is 25 hr. A wide range of values seemspossible for the agricultural rent for raw land. Small [ll] suggeststhat the values from $7.50 to $750 might occur in different urban areas in different regions of the country. An intermediate value of $200 was used. REFERENCES 1. R. J. Amott and J. G. MacKinnon, The effects of the property tax: A general equilibrium simulation, J. Urban Econom., 4, 389-407 (1977). 2 B. Ellickson, Jurisdictional fragmentation and residential choice, American Econom. Rev. Pap. Proc. (1971). 3. B. Ellickson, The politics and economics of decentralization, J. Urban Econom., 4, 135-149 (1977). 4. J. V. Henderson, “Economic Theory and the Cities,” Academic Press, New York (1977). 5. C. McLure, The “new view” of the property tax: A caveat, Nat. Tax J. (1977). 6. P. Mieszkowski, The property tax: An excise tax or a profits tax? J. Public Econom. (1972). 7. E. S. Mills, Economic analysis of urban land-use controls, in “Current Issues in Urban Economics” (P. Mieszkowski and M. Straszheim, Eds.), Johns Hopkins Univ. Press, Baltimore (1979). 8. R. F. Muth, Numerical solution of urban residential land-use models, J. Urban Econom., 2, 307-332 (1975). 9. W. E. Oates, “Fiscal Federalism,” Harcourt Brace Jovanovich, New York (1972). 10. A. M. Polinsky, The demand for housing: A study in specification and grouping, Econometricu (1977).
11. K. A. Small, A comment on gasoline prices and urban structure, J. Urban Econom., 10, 311-322 (1981). 12. R. C. Steen, EtTects of governmental structure in urban areas, J. Urbun Econom., 21, 166-179 (1987). 13. U.S. Department of Commerce, Bureau of the Census, “1970 Census,” Vol. 1, Washington, D.C. 14. U.S. Department of Commerce, “1970 Census of Housing,” Vol. 1, Washington, D.C. 15. U.S. Department of Commerce, “Statistical Abstract of the U.S., 1971,” Washington, D.C. 16. U.S. Department of Housing and Urban Development (HUD), “FHA Homes, 1970,” Washington, D.C. 17. U.S. Department of Transportation, Transportation Systems Center, “National Transportation Statistics, 1979.” 18. W. Wheaton, Income and urban residence, Amer. Econom. Rev. (1977). 19. W. Wheaton, Monocentric models of urban land use: Contributions and criticisms, in “Current Issues in Urban Economics” (P. Mieszkowski and M. Straszheim, Eds.), Johns Hopkins Univ. Press, Baltimore (1979).