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The incidence of classified property taxes in a three-sector model with an imperfectly mobile population
JOURNAL
OF URBAN
The
ECONOMICS
2!i,
11-92
(1989)
Incidence of Classified Property Taxes Three-Sector Model with an Imperfectly Mobile Population
in a
AMARK.PARAIANDJOHNH.BECK' Department
of Economics, SUNY College at Fredonia, Fredonia, New York I4063; School of Business, Gonzaga University, Spokane, Washington 99258 Received
July 7,1986;
revised
September
and
8,1986
This paper analyzes the incidence of a classified property tax, imposing different effective tax rates on residential, commercial, and industrial property, as compared to a uniform property tax, taxing capital and land in all sectors at the same rate. Our model includes three sectors, commercial, industrial and residential, and three factors of production, land, labor, and capital. Capital is assumed to be perfectly mobile, but we consider varying degrees of labor mobility, allowing the possibility that part of a classified property tax may be shifted to workers through lower wages. 0 1989 Academic
Press, Inc.
INTRODUCTION Under a classified property tax, different classes of real property, such as residential, commercial, industrial, and agricultural, are taxed at different effective tax rates. The common pattern has been to tax residential property at a lower effective rate than business property. This has been politically attractive because there are more voters who are homeowners than business owners, and therefore reducing the visible tax burden on residents by raising the burden on business property gains votes for politicians. However, in the long run if capital is mobile among jurisdictions the burden of a tax on business property does not fall on the owners of capital. As has been shown by Mieszkowski [12], capital moves out of a jurisdiction in which it is taxed at a higher rate than elsewhere so that owners of capital suffer no reduction in their after-tax rate of return in the long run; the tax is shifted to owners of less mobile factors of production, land and possibly labor, and, if the output of the taxed business is sold in a local market, to consumers. This analysis leads to the question of the determinants of the shares of the tax burden falling on landowners, wage earners, and consumers. Per‘We are grateful to Edwin Mills and an anonymous Gokarn provided research assistance, and the Center Western Reserve University, provided financial support. John H. Beck.
referee for helpful comments. Subir for Regional Economic Issues, Case Address correspondence to Professor
0094-1190/89
$3.00
Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
78
PARAI AND BECK
haps the most recent systematic discussion of the incidence of a classified property tax is Sonstelie’s [l&16] analysis. However, his analysis has two major limitations. First, it considers only two classes of property, commercial and residential. This might be reasonable for Sonstelie’s 1978 study of the District of Columbia [15], which has little manufacturing industry, but is inappropriate for other jurisdictions which may tax commercial and industrial property at different rates. Second, Sonstelie includes only two factors, land and capital, in the production functions in his model. As noted by Sonstelie [16, p. 841, this rules out the important possibility that part of the tax may be shifted to workers in the form of lower wages. The possibility of backward shifting to labor is limited by labor mobility when the classified tax is imposed by only one small jurisdiction within a regional labor market, but, if classification is adopted on a statewide basis, the potential for backward shifting to labor is enhanced. THE MODEL We assume that the local economy has three sectors-residential housing, commercial business, and manufacturing-denoted by subscripts H, B, and I, respectively. There are three factors of production-capital, land, and labor-denoted by K, L, and M, respectively. All three factors are used in the commercial and industrial sectors, but housing services are produced with just capital and land. Thus the production functions of the three sectors are given by X, = &(Km
4,)
0)
X, = X,(&z,
L,, h)
(2)
X, = X,(K,,
L,, W).
(3)
Capital is assumed to be perfectly mobile in the long run, so the net rate of return on capital, r, is taken to be exogenous. The total supply of land within the taxing jurisdiction is assumed to be fixed at L.
L= L,+L,+
L,.
(4)
It is assumed that land may be freely reallocated among the three sectors without impediments such as zoning regulations, so the net rental rates of land are equalized at a value denoted by R. Both R and the wage rate, w, are endogenous. The output of the industrial sector is traded in a national market. Therefore, its price, PI, is exogenous. For convenience, we take PI = 1 as our numeraire. Capital goods constitute part of the output of the industrial sector; therefore, the price of capital goods is also taken as Pr = 1. Land
CLASSIFIED
79
PROPERTY TAXES
values are determined as the net rental price capitalized at a discount rate equal to the exogenous net rate of return on capital. Therefore, the price of land is R/r. We assume that property, both land and capital, in the residential, commercial, and industrial sectors is taxed at rates t,, t,, and t,, respectively. Therefore, total property tax revenues are given by
1 [
+ t, K, + ;LB
1 [
+ t, K, + FL1
1 .
(5)
The assumption that the property tax applies to land and all forms of capital within each sector at the same rate may need some justification. Two recent studies of property tax incidence, Sullivan [17] and Schoettle [14], have treated the property tax as applying to real property (land and structures including equipment permanently affixed to the land) but not to tangible personal property (inventories and portable equipment). The treatment of tangible personal property in the United States varies from state to state. According to the “1982 Census of Governments” [21, pp. xvi-xvii], eight states (Delaware, Hawaii, Illinois, New Hampshire, New York, North Dakota, Pennsylvania, and South Dakota) exempted all tangible personal property from local general property taxes in 1931. However, commercial and industrial inventories and equipment were fully taxable in 12 states (Arkansas, Georgia, Indiana, Kansas, Louisiana, Missouri, North Carolina, Ohio, Oklahoma, Texas, Virginia, and West Virginia). Our analysis is thus applicable to the latter states. The utility of residents of the taxing jurisdiction is assumed to be a function of their household’s consumption of housing, xH, and other consumer goods, xc, and hours of labor supplied, m. These utility functions are identical except for an additive term, Y, which reflects differences in their tastes for living in that jurisdiction. For example, the value of Y would be larger for persons with strong family ties and lower for persons with a weaker attachment to the locality. The differences in the value of v give rise to the “imperfect mobility” referred to in the title of this paper. The corresponding indirect utility function is
v(v) = U(PH, PC, w, y) + v,
(6)
where PH, PC, and w are the prices of housing, other consumer goods, and labor, respectively. “Full income,” Y, is the sum of nonlabor income, y, which is assumed to be exogenous and identical for all households, and the wage rate multiplied by the maximum hours of labor the household could supply, 11, or Y =y + wp.
(7)
80
PARAI AND BECK
Household demands for housing and for other consumer goods and household labor supply are given by
XH = XH(Y, w, PC, p,) xc = Xc(Y, w, PC, p,) m = m(y, w, PC, P,).
(8)
(9) 00)
The number of households choosing to live in the jurisdiction is determined by the number of households for whom u( PH, PC, w, Y) + Y 2 V*, where v* is the utility level any household could attain living outside the jurisdiction. Letting f(v) be the frequency distribution of Y and F( ) be the integral of f( ), the number of households living in the jurisdiction, N, is a function of prices within the jurisdiction: N = N(P,,
PC, w,Y)
= F(oo) - F[V*
- u(PH, PC, w,Y)].
(11)
The market demands for housing and other consumer goods within the jurisdiction are equal to the individual household demands for these goods multiplied by the number of households. Therefore, the commodity market equilibrium conditions are given by: X, = Nx,
(12)
Xc = Nx,.
(13)
The labor supply is equal to the individual plied by the number of households: M=Nm.
household labor supply multi-
(14
The “other consumer goods” supplied in the jurisdiction consist of a fixed amount of services of the local commercial sector combined with manufactured goods. Thus, the production function for “other consumer goods” can be expressed as a fixed coefficients function with the inputs being the outputs of the commercial and industrial sectors. Because a limitless supply of manufactured goods is available on the national market, the limiting input will always be the services of the local commercial sector. Thus the quantity of other consumer goods supplied in the jurisdiction can be expressed as
where 7r is the quantity
of local commercial
services required to produce
CLASSIFIED
PROPERTY TAXES
81
one unit of “other consumer goods.” We assume that local commercial services are measured in units that initially have a price of one and that (1 - n) is the quantity of manufactured goods required to produce one unit of “other consumer goods.” Thus, the price of these goods can be expressed as PC = (1 - T)P, + TP&
(16)
Thus, the costs of the local commercial sector which distributes consumer goods will be included in the price of these goods. Competitive equilibrium in the three sectors implies that
PH = R&L,
+ r&m
PI = 1 = R,C,, + r&
P, = R&
(17) + WC,,
+ rBCKB+ womb,
(18) (19)
where Cij is the cost minimizing quantity of input i required per unit of output m sector j, given the input prices and the production functions specified above. Rj = R(l + tj/r) and 7 = r + fj are the gross (including tax) land rent and rate of return on capital, respectively, in sector j.’ The Cij = Cij(Rj, rj, w) (for i = L, K, M, j = H, B, Z) are homogeneous of degree zero in all input prices. Equilibrium in the land and labor markets can be expressed as L = JWL” M = x,c,,
+ 4&l + x&f,.
+ WL‘,
(20) (21)
In analyzing the incidence of a classified property tax, we follow a differential tax incidence approach. We assume that initially all tax revenues are raised by a tax at a uniform rate t applied to all classes of property.3 Then the tax rate on class of property i is increased by dti and the rates on the other two classes of property, j and k, are changed with dtj = dt, determined so as to keep total tax revenues unchanged. For example, if the tax rate on residential property is increased with an offsetting uniform change in taxes on industrial and commercial business *We assume that there are no taxes (e.g., state or federal corporate taxes) other than the local property tax. 3Sullivan [17,18] also examined the incidence of taxes on industrial and residential property, respectively, in a differential tax incidence framework, but he assumed that the industrial or residential property taxes replaced a nondistortionary land tax. The policy alternatives more frequently considered are a uniform property tax and a classified property tax. Thus, we think our analysis of the incidence of a classified tax replacing a uniform tax is more relevant.
82
PA&U
AND
BECK
property, O=dt,
[
KH+
+t,[dK,
$L H] + t,[dK, + d( +LB)]
In terms of proportionate
+ d( +&)I
+ dt,[K,
+ FL,]
+ df,[K,
+ ;LB]
+ t,[dX,
+ d( fL[)].
changes
0 = [p,x,(x;
+ P,*) + P,X,(X,* + P,*/a) + x,x,* - wM( M* + w*)] +(r/t)[(P,X, + x1 - wM)T,* + P,X,T,*]
or -T,*
= -T$ +XrXI*
= {(t/r)[P,X,(X,* - wM(M*
+ w*)]
+ P,*) + P,X,(X,* + P,X,T,*}/(P,X,
+ P+) + XI - wM),
(22)
where an asterisk denotes proportionate changes; e.g., X$ = dX,/X,, except Tj* = dtj/q = rj* is defined to be the proportionate change in the gross rate of return to capital in sector j in response to a change in the tax rate on that class of property. The properties of production functions [l, pp. 503-508; 4, p. 241 give us Cc = S,,u/,( Czj = B,ju~,(R~ CGj = e,,t&(
rj* - Rf) - rj*)
+ e,,ui,(
W*
+ @,,u&(w*
Rj* - w*) + BKjlJJg
-
Ri*)
- Q*) ‘,* - w*)
(23) j = B, I, H,
where uli represents Allen’s partial elasticity of substitution between factors k and i in sector j and 8,, represents the share of factor k in the value of output of sector j. Note that r&, = 0 = CT,& = ufK and Rr = R* + I;*. Differentiating (8) through (15) and substituting, we obtarn:
+ (e, x, = hfff + E~)PH*
+ +)PC*
+ (eHw + e,)w*
xc? = (%H + eH)PH* + (eC- + q-)PC* + (ecw + e,)W* M* = (emH + eH)PH* + (e,, + Q)PC* + (emw + ew)w*,
(24) (25) (26)
where eH, eC, and E, are the elasticities of N with respect to PH, PC, and w, respectively, and the eij (i = H, C, m; j = H, C, w) are the elasticities
CLASSIFIED
PROPERTY
TAXES
83
of individual demand for housing and other consumer goods and individual supply of labor with respect to PH, PC, and W. Differentiating (16) through (19) and using the conditions for cost minimization, we obtain: P; - T; = eLHR* - (e,, + e,,>T,* PB* - (e,, + e&T,*
= e,,R*
+ eMrw*
= +~2
- (e,, + e&T,*
= e,g* Differentiating
(27)
(20) and (21) and substituting
+ eMBw*. (23)
(28)
(29)
and (26) gives:
where XLj = Lj/L and X, = Mj/M are the shares of land and labor, respectively, employed in sector j. Equations (24), (25), and (27) through (31) are a set of seven differential equations involving seven variables, X,, X,, X,, PC, PH, w, and R, and three parameters, t,, t,, and t,. The incidence of a tax increase on residential property offset by a change in the tax on industrial and commercial business property can be determined by solving this system of equations in terms of T$ using the tax revenue neutrality constraint given by (22). Similarly, the incidence of a tax increase on industrial (commercial business) property offset by a change in the tax rate on housing and commercial (industrial) property can be determined using an equation analogous to (22) solved in terms of T,*(T,*). PARAMETER VALUES The expressions resulting from the solution of these equations are cumbersome and dilkult to interpret. Therefore, rather than presenting a solution in general symbols for the parameters of the model, we have substituted plausible numerical values for the parameters and derived a
84
PAR41 AND BECK
numerical solution using the matrix multiplication and inversion procedures of TSP. We assume that the exogenous after-tax rate of return on capital is r = .06 and the initial uniform tax rate is t = .02. The production functions in the housing, industrial, and commercial business sectors are assumed to be Cobb-Douglas functions, and therefore all of the elasticities of substitution, ui, are equal to one. 4 The other parameter values used and initial values of endogenous variables are shown in Table 1. Sonstelie [15, p. 2591 used values of 675 and .325 for the factor shares of capital and land, respectively, in the production function of the residential sector function and values of .6159 and .3841 for the factor shares of capital and land in the commercial sector. In contrast, Sullivan [18, p. 2421 attributed only 14.36% of housing costs to land. We compromise by using values of I+, = .8 and e,, = .2. Kendrick and Grossman [ll, p. 271 cite an estimate of labor’s share in trade of .85. Dividing the commercial sector nonlabor share of .15 among capital and land in the same proportions as housing gives the capital and land shares of .12 and .03, respectively, shown in Table 1. The value of .756 for labor’s share in manufacturing is based on an estimate cited in [ll, p. 271. Kendrick [lo, p. 371 reports that land accounts for only 1.7% of property in the manufacturing sector; accordingly, the capital and land shares in manufacturing are apportioned as shown in Table 1. We approximate the share of the value of consumer goods added in the commercial sector by the ratio of GNP originating in the wholesale and retail trade industries to personal consumption expenditures on durable and nondurable goods. The “National Income and Product Accounts” [20, pp. 83, 2271 for 1976 give a value of .4869 for this ratio. Therefore, we use a value of n = .487. The “Census of Population” [19, p. 4221 gives the mean family income in metropolitan areas for 1969 as $11,991, and mean wage and salary income ?he Cobb-Douglas production function is widely accepted as a good approximation for manufacturing. The assumption of a Cobb-Douglas production function for the commercial sector is based on convenience and our inability to find any estimates of production functions for this sector in the literature. There is substantial literature on production functions for housing. Assuming a Cobb-Douglas production function (and therefore a unitary elasticity of substitution) for housing conflicts with much of this literature, but the literature gives conflicting results. Sonstelie [15, p. 2591 cites an estimate of the elasticity of substitution between land and capital which is greater than one (- 1.2, estimated by Smith [25]). On the other hand, Polinsky and Ellwood [13, p. 2011 report an estimate of - .45 for the elasticity of substitution, and Schoettle [14, pp. 364-3651, based in part on the Polinsky and Bllwood study used a value of -.5 for the elasticity of substitution between sites and “other inputs” (including capital and labor) in the production function for real property. In light of conflicting estimates, some greater and some less than one, for the elasticity of substitution for the housing production function, we do not think our assumption of a Cobb-Douglas function is unreasonable.
CLASSIFIED
PROPERTY TAXES
85
TABLE 1 Parameter Values and Initial Values of Variables Factor Shares e,, = .12 e,, = .03 eMB = .85 e,, = .240 e,, = ,004 e,, = ,156 eKH = .80 e,, = .20 Commercial services share in consumer goodr R = .481 Individual demand and supply elasticities e “H = -0.1 emw = 0.15
ccc = - 0.98235294 e*C = -0.100 eCH = -0.05294118 eHw = 0.84560647 ecw = 1.0943142 emH = 0.1325971 emc = 0.7513838
Household budget and choice variables
y = 975 (exogenous) m = 3,410 ? = 12,057.50 PHxH = 1,808.625 xc = 10,248.875 Sector shares of factor supplies
A,, = .6605 A,, = .0661 A LB = .2134 A, = .6172 P K” = .3431 .3828 A,, = .1420 A,, = .5149
Prices Pr = 1.0 (exogenous) r = .06 (exogenous) Pe = 1.0 PC = 1.0 R = 41,073.86
w = 3.25
as $11,016. This would suggest nonlabor income of $975. Therefore, we use a value of y = 975. The “Handbook of Labor Statistics” [22, p. 1841 reports average hourly earnings of nonsupervisory workers of $3.23 in 1970. This wage rate and the census wage income of $11,016 imply a value of m = 3410 for each household. Given that one person working 40 hours per week for 50 weeks would supply 2000 hours of labor per year; 3410 seems a reasonable estimate of average household labor supply, with one spouse employed full time and the other part time. With y = 975, w = 3.25, and m = 3410, each household in our model has a money income F = 12,057.50. Polinsky and Ellwood [13] give values of .8 and - .7 for the income and price elasticities of demand for housing, respectively. We use these values although Sonstelie [15, p. 2591 and Grieson [7, p. 3761 have used a larger
86
PARAI
AND
BECK
value of - 1.0 for the price elasticity. According to [20, p. 831, rent of housing (excluding such costs of household operation as utilities) constituted 15.36% of personal consumption expenditures. Therefore we assume that the housing and “other consumer goods” budget shares are CY”= .15 and (~c = .85, respectively. Regarding labor supply, Browning and Browning [6, p. 3431 cite a wage elasticity of emw = .15 according to Fullerton. We assume that the effect of changes in nonlabor income on household labor supply is b’m/ay = - .25, which is approximately the sum of the effects on labor supply of husbands and wives estimated by Hausman [8]. The literature is not so helpful in providing values for the own- and cross-price elasticities of “other consumer goods” or the cross elasticities between labor supply and housing and other consumer goods demand. For lack of any better information, we assume that leisure is neither a complement nor a substitute for housing and other consumer goods, so that changes in w have only an income effect on the demand for these goods and changes in PH and PC have only an income effect on the supply of labor. Based on these assumptions and the housing demand and labor supply elasticities and budget shares given above, we calculate the remaining own- and cross-price elasticities shown in Table 1 using the relationships between price and income elasticities and budget shares derived from consumer theory [9, p. 391. Our hypothetical jurisdiction is assumed to have a land area L = 10,000 square miles populated by N = l,OOO,OOO households. These assumptions imply the initial values for the remaining variables shown in Table 1. From (ll), the change in population will be dN =f(V*
- u)[(&@P,)
dP, + (ch/JP,)
dP, f (&~/a,) + (do/aY)
dw dY] . (32)
By the use of Roy’s identity [23, p. 931, the partial derivatives of the indirect utility function with respect to PH, PC, and w may be replaced by au/a Y (which, without loss of generality, we assume equals one) multiplied by the negatives of the quantities of housing, other consumer goods, and leisure. dN=f(V*-u)[-x,dP,--x,dP,-(p-m)dw+dY] =f(V*
- u)[-x,dP,
- xcdPc
- (p - m) dw + pdw]
=f(V*
- u)[-x,dP,
- xcdPc
+ mdw].
In addition
(33)
to being affected by the values of the partial derivatives of u, f(v). In the extreme case of no population mobility, no one is almost indifferent between moving out and remaining in the community. In this case f = 0. dN also depends on the value of the frequency distribution
CLASSIFIED
PROPERTY TABLE
EtTects of Increasing P;/T; 0 100 500 103 104 105 106
P,+/T,*
1.029 1.056 1.346 0.113 0.942 - 4.052 - 0.077
,056 ,062 ,120 -.130 ,029 - .576 - ,097
87
TAXES
2
the Tax Rate on Residential
w’/T;
R./T,*
T,+/T,*
0.293 0.312 0.520 - 0.312 0.179 - 1.213 - 0.103
0.143 0.282 1.731 - 4.433 - 0.291 - 25.260 - 5.384
Property = TB*/T$
-0.909 -0.912 -1.638 1.226 - 0.550 4.163 0.388
Au/T2 809.71 919.59 2095.05 -2996.35 -18.13 -210.82 -4.46
On the other hand, if the population is perfectly mobile, everyone is indifferent between staying or leaving and f = co. In the following section, we present results for intermediate cases with 0 -Cf < cc. RESULTS Table 2 shows the effects of increasing the tax rate on the residential class of property accompanied by changes in the tax rate on industrial and commercial property to hold total tax revenue constant. Tables 3 and 4 show the effects of increasing the tax rates on the industrial and commercial classes, respectively. In each table, each row shows the results for a different assumption about the degree of population mobility, i.e., about the value of f. The change in the burden on landowners is measured by the change in R shown in column (5). The change in the burden on resident worker-consumers can be measured by the change in u shown in the last column, which is calculated from the changes in PH, PC, and w shown in columns (2), (3), and (4). First, we consider the cases with no population mobility, i.e., f = 0, shown in the first rows of each table. Table 2 shows that an increase in the TABLE Effects
f
PJ*/Tf*
0 100 500 103 104 105 106
- 0.543 - 0.541 -0.357 - 0.517 0.221 - 1.007 - 1.441
of Increasing P,*/T,* -
.173 .173 ,158 ,171 ,109 .217 .322
d/T,* -
3
the Tax Rate on Industrial
.323 .323 .318 ,322 .303 .347 ,525
R*/T,* -0.018 - 0.288 -0.872 - 0.132 3.775 4.581 38.198
T,+/T,*
Property = T,*/T,*
-
0.539 0.535 0.183 0.491 0.982 - 1.923 - 9.081
Av/T,* - 817.10 - 823.02 - 1262.30 - 880.18 - 2650.24 200.19 89.84
88
PARAI AND BECK TABLE 4 Effects of Increasing the Tax Rate on Commercial Property
f
P,*/T,*
0
10s
- .224 - ,227 - ,249 ,674 - .219 - ,369
106
- ,111
loo 500
103 104
P,*/T,* .100
.lOl ,111 - ,332 .081 ,013 - ,040
w*/TB*
R*/T,*
,068 - ,070 ,088 - ,694 .037 - ,050 - ,056
- 0.077 - 0.055 0.130 - 8.043 - 0.470 - 2.691 - 6.171
T,*/T$
= TJ’/T$ - 0.207 - 0.216 - 0.274 2.282 - 0.107 0.200 0.267
Au/T; 131.33 147.86 - 283.74 - 5508.88 - 29.85 - 21.74 - 5.05
tax rate on residential property increases the price of housing, but the accompanying reduction in the tax rate on industrial property induces investment which raises the wage rate. Local retailers must pay higher wages to their workers to compete with wage offers from the manufacturing sector. Thus, although the tax rate on commercial property falls, the increase in the wage rate causes the price of consumer goods to rise. The increased wage rate raises local incomes and thereby increases the demand for housing. Thus, housing prices rise by more than the increase in taxes on residential property, as shown by the value of P$/T,* = 1.029 > 1.0. Consequently land rents rise. The increase in the wage rate more than compensates for the higher prices of housing and other consumer goods, as shown by the increase in the value of the indirect utility function. Although there is no change in the total tax revenues collected, the increases in land rents and in the indirect utility function reflect efficiency gains from reductions in the excess burden of the property tax. Because the supply of capital to the jurisdiction is infinitely elastic, partial equilibrium analysis indicates that from the local viewpoint the excess burden of the tax on capital would be minimized if tax rates were inversely proportional to the demand for capital in each sector [3, pp. 367-3701. A uniform tax on capital does not minimize this excess burden. Consequently, Beck [5] argued that a classified property tax, with a lower tax rate on sectors in which the demand for capital is more elastic, reduces the excess burden. Wilson [24] discusses conditions for an optimal classified property tax in a more general model which qualifies Beck’s conclusions but still concludes that from the local perspective a uniform tax is not generally optimal. In the present case, because manufactured goods are sold in a national market, the derived demand for capital in the industrial sector is more elastic and a lower tax rate in that sector is optimal.’ ‘Amott
and Grieson
sets of assumptions.
[2] and Grieson
[7] also derive
an inverse
elasticity
rule under
different
CLASSIFIED
PROPERTY TAXES
89
The first row of Table 3 shows the effects of increasing the tax rate on industrial property when the population is immobile. The increased tax rate reduces capital in the industrial sector, leading to a decline in the wage rate. Reduced demand for housing and other consumer goods combined with the reduction in the tax rate on residential and commercial property causes PH and PC to fall, but the value of the indirect utility function falls due to the decline in w. Landowners also lose from the decline in R. The first row of Table 4 shows the effects of raising the tax rate on commercial property when the population is immobile. The wage rate increases as labor is substituted for capital and land in the commercial sector.6 As one would expect, the increased tax on commercial property raises the price of consumer goods. The increase in the wage rate and reduction in housing costs more than compensate for the rise in the price of other consumer goods, causing a moderate increase in the value of the indirect utility function. Land rents also fall in this case. Turning to the lower rows of Tables 2-4, we first note that the positive values in some rows of column (6) indicate that in some cases raising one tax rate requires increasing the other tax rates as well in order to hold total revenues constant. This occurs when the tax rate on the first class of property is above the revenue-maximizi ng rate for that class of property. In these cases both land rents and the utility of residents fall as raising a tax rate above the revenue-maximizing rate greatly increases the excess burden of the tax. From a partial equilibrium perspective, greater population mobility implies a larger own-price elasticity of demand for housing and other consumer goods. Because the revenue-maximizing point on the “Laffer curve” occurs at lower tax rates the greater the elasticity of demand, we might expect positive values to occur in column (6) of Tables 2 and 4 as population mobility increases. Indeed, this does occur, but not uniformly. Apparently in some cases of greater population mobility general equilibrium effects allow reductions in the tax rate on other classes of property while maintaining constant revenues despite the large own-price elasticities of demand. Another apparent puzzle is that in some cases increased population mobility leads to larger rather than smaller changes in the utility level of residents. For instance, in Table 2 an increase in the tax rate on residential property results in a larger change in utility, Au/T; = 919.59 with f = 100, than in the case of no mobility, in which Au/T; = 809.71 with f = 0. One 6Note that this depends on the partial elasticities of substitution of labor for capital and land being greater than the elasticity of demand for output [l, p. 5081. Because we assumed unitary partial elasticities of substitution in the commercial sector without any empirical support, there may be more reason to be skeptical of this result than of other aspects of this study dependent on parameter values better supported in the empirical literature.
90
PARAI
AND
BECK
would expect that in-migration in response to higher wages would increase the supply of labor and moderate the increases in wages and utility. However, the in-migration also increases the property tax base as the new workers purchase local housing. This allows a larger reduction in the tax rate on industrial property, which leads to more investment and further increases in the real wage. Similarly, in the analysis of an increase in the tax rate on industrial property shown in Table 3, Au/T,* = - 817.10 with f = 0 but Au/T,* = - 1262.30 with f = 500. The increased tax on industry reduces the local wage rate and thereby reduces the utility of residents. In the imperfect mobility case with f = 100, the resulting decline in population reduces the local demand for housing, and the property tax base declines. To maintain constant tax revenues, the reduction in the tax rate on residential and commercial property is less than in the case with f = 0. Consequently, housing and other consumer goods prices do not fall as much. The decline in the wage rate is moderated only very slightly by the slight degree of mobility. The combination of smaller declines in housing and consumer prices with almost the same decline in wages results in a larger decline in utility (for those remaining) with f = 500 compared to f = 0. The cases in which f = lo6 represent almost perfect mobility. With f = 106, a change in local prices and wage rates equivalent to a $1.00 reduction in income would cause the entire population to leave the jurisdiction. The results in the last rows of Tables 2-4 show that, as expected, with almost perfect mobility the utility level of residents is virtually constant. The values of Au/T* vary from - 5.05 to + 89.84 in these cases. In the case of the increased tax rate on industrial property shown in the last row of Table 3, land rents increase as the tax rate on the more land intensive residential and commercial sectors is reduced. CONCLUSIONS By including labor as an imperfectly mobile factor of production and an industrial sector in the local economy, we have demonstrated some possible consequences of classified property taxes not shown in Sonstelie’s [15, 161 model. As expected, increasing the tax rate on industrial property will reduce the wage rate of immobile labor. More specifically, for the plausible parameter values used in our illustration, increasing the industrial property tax rate while reducing the rates on residential and commercial property reduces wages so much that immobile workers are actually worse off despite the accompanying reductions in the prices of housing and other locally purchased consumer goods. Conversely, when the population is immobile, raising residential property tax rates in order to reduce the rates on business property results in efficiency gains which increase the utility of residents.
CLASSIFIED
PROPERTY
TAXES
91
In the model with imperfect population mobility, there is the possibility that the demand for capital in a sector may be so elastic that increasing the tax rate on that class of property reduces tax revenues, requiring accompanying increases in the tax rates on other classes of property as well. In these cases, the increase in tax rates reduces both land rents and the utility of residents. With nearly perfect mobility, the utility level of residents is almost constant. In this case, increasing the tax rate on the less land intensive manufacturing sector and reducing the rate on the more land intensive sectors will increase land rents. REFERENCES 1. R. G. D. Allen, “Mathematical Analysis for Economists,” St. Martin’s Press, New York (1968). 2. R. Arnott and R. E. Grieson, Optimal fiscal policy for a state or local government, J. Urban Econom., 9,23-48 (1981). 3. A. B. Atkinson and J. E. Stightz, “Lectures on Public Economics,” McGraw-Hill, New York (1980). 4. R. N. Batra and F. R. Casas, A synthesis of the Heckscher-Ohlin and the neoclassical models of international trade, J. In?. Econom., 6, 21-38 (1976). 5. J. H. Beck, Tax competition, uniform assessment and the benefit principle, J. Urban Econom., 13,127-146 (1983). 6. E. K. Browning and J. M. Browning, “Public Finance and the Price System,” Macmillan Co., New York (1983). 7. R. E. Grieson, The economics of property taxes and land values: The elasticity of supply of structures, J. Urban Econom., 1, 367-381 (1974). 8. J. A. Hausman, Labor supply, in “How Taxes Affect Economic Behavior” (H. J. Aaron and J. A. Pechman, Eds.), Brookings, Washington (1981). 9. J. M. Henderson and R. E. Quandt, “Microeconomic Theory: A Mathematical Approach,” McGraw-Hill, New York (1971). 10. J. W. Kendrick, “The National Wealth of the United States by Major Sector and Industry,” The Conference Board (1976). 11. J. W. Kendrick and E. S. Grossman, “Productivity in the United States, Trends and Cycles,” Johns Hopkins Univ. Press, Baltimore (1980). 12. P. Mieszkowski, The property tax: An excise tax or a profits tax, J. Public Econom., 1, 73-96 (1972). 13. A. M. Polinsky and D. T. Ellwood, An empirical reconciliation of micro and grouped estimates of the demand for housing, Rev. Econom. Statist., 61, 199-205 (1979). 14. F. P. Schoettle, A three-sector model for real property tax incidence, J. Public Econom., 27, 355-370 (1985). 15. J. Sonstelie, The classified property tax, in “Technical Aspects of the District’s Tax System: Studies and Papers Prepared for the District of Columbia Tax Revision Commission,” U.S. Government Printing Office, Washington (1978). 16. J. Sonstelie, The incidence of a classified property tax, J. Public Econom., 12, 75-85 (1979). 17. A. M. Sullivan, The general equilibrium effects of the industrial property tax: Incidence and excess burden, Reg. Sci. Urban Econom., 14,547-563 (1984). 18. A. M. Sullivan, The general-equilibrium effects of the residential property tax: Incidence and excess burden, J. Urban Econom., l&235-250 (1985).
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19. U.S. Bureau of the Census, “Census of Population: 1970, General Social and Economics Characteristics,” Final Report PC(l)-Cl United States Summary, U.S. Government Printing Office, Washington (1972). 20. U.S. Department of Commerce, “The National Income and Product Accounts of the United States, 1929-76,” Washington (1981). 21. U.S. Department of Commerce, “1982 Census of Governments,” Vol. 2, “Taxable Property Values and Assessment/Safes Price Ratios,” US. Government Printing Office, Washington (1984). 22. U.S. Department of Labor, Bureau of Labor Statistics, “Handbook of Labor Statistics,” Bulletin 2070, Washington (1980). 23. H. R. Varian, “Microeconomic Analysis,” Norton, New York (1978). 24. J. D. Wilson, Optimal property taxation in the presence of interregional capital mobility, J. Urban Econom., 18, 73-89 (1985). 25. B. A. Smith, The supply of urban housing, Q. J. Econom., 40, 389-405 (1976).
OF URBAN
The
ECONOMICS
2!i,
11-92
(1989)
Incidence of Classified Property Taxes Three-Sector Model with an Imperfectly Mobile Population
in a
AMARK.PARAIANDJOHNH.BECK' Department
of Economics, SUNY College at Fredonia, Fredonia, New York I4063; School of Business, Gonzaga University, Spokane, Washington 99258 Received
July 7,1986;
revised
September
and
8,1986
This paper analyzes the incidence of a classified property tax, imposing different effective tax rates on residential, commercial, and industrial property, as compared to a uniform property tax, taxing capital and land in all sectors at the same rate. Our model includes three sectors, commercial, industrial and residential, and three factors of production, land, labor, and capital. Capital is assumed to be perfectly mobile, but we consider varying degrees of labor mobility, allowing the possibility that part of a classified property tax may be shifted to workers through lower wages. 0 1989 Academic
Press, Inc.
INTRODUCTION Under a classified property tax, different classes of real property, such as residential, commercial, industrial, and agricultural, are taxed at different effective tax rates. The common pattern has been to tax residential property at a lower effective rate than business property. This has been politically attractive because there are more voters who are homeowners than business owners, and therefore reducing the visible tax burden on residents by raising the burden on business property gains votes for politicians. However, in the long run if capital is mobile among jurisdictions the burden of a tax on business property does not fall on the owners of capital. As has been shown by Mieszkowski [12], capital moves out of a jurisdiction in which it is taxed at a higher rate than elsewhere so that owners of capital suffer no reduction in their after-tax rate of return in the long run; the tax is shifted to owners of less mobile factors of production, land and possibly labor, and, if the output of the taxed business is sold in a local market, to consumers. This analysis leads to the question of the determinants of the shares of the tax burden falling on landowners, wage earners, and consumers. Per‘We are grateful to Edwin Mills and an anonymous Gokarn provided research assistance, and the Center Western Reserve University, provided financial support. John H. Beck.
referee for helpful comments. Subir for Regional Economic Issues, Case Address correspondence to Professor
0094-1190/89
$3.00
Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
78
PARAI AND BECK
haps the most recent systematic discussion of the incidence of a classified property tax is Sonstelie’s [l&16] analysis. However, his analysis has two major limitations. First, it considers only two classes of property, commercial and residential. This might be reasonable for Sonstelie’s 1978 study of the District of Columbia [15], which has little manufacturing industry, but is inappropriate for other jurisdictions which may tax commercial and industrial property at different rates. Second, Sonstelie includes only two factors, land and capital, in the production functions in his model. As noted by Sonstelie [16, p. 841, this rules out the important possibility that part of the tax may be shifted to workers in the form of lower wages. The possibility of backward shifting to labor is limited by labor mobility when the classified tax is imposed by only one small jurisdiction within a regional labor market, but, if classification is adopted on a statewide basis, the potential for backward shifting to labor is enhanced. THE MODEL We assume that the local economy has three sectors-residential housing, commercial business, and manufacturing-denoted by subscripts H, B, and I, respectively. There are three factors of production-capital, land, and labor-denoted by K, L, and M, respectively. All three factors are used in the commercial and industrial sectors, but housing services are produced with just capital and land. Thus the production functions of the three sectors are given by X, = &(Km
4,)
0)
X, = X,(&z,
L,, h)
(2)
X, = X,(K,,
L,, W).
(3)
Capital is assumed to be perfectly mobile in the long run, so the net rate of return on capital, r, is taken to be exogenous. The total supply of land within the taxing jurisdiction is assumed to be fixed at L.
L= L,+L,+
L,.
(4)
It is assumed that land may be freely reallocated among the three sectors without impediments such as zoning regulations, so the net rental rates of land are equalized at a value denoted by R. Both R and the wage rate, w, are endogenous. The output of the industrial sector is traded in a national market. Therefore, its price, PI, is exogenous. For convenience, we take PI = 1 as our numeraire. Capital goods constitute part of the output of the industrial sector; therefore, the price of capital goods is also taken as Pr = 1. Land
CLASSIFIED
79
PROPERTY TAXES
values are determined as the net rental price capitalized at a discount rate equal to the exogenous net rate of return on capital. Therefore, the price of land is R/r. We assume that property, both land and capital, in the residential, commercial, and industrial sectors is taxed at rates t,, t,, and t,, respectively. Therefore, total property tax revenues are given by
1 [
+ t, K, + ;LB
1 [
+ t, K, + FL1
1 .
(5)
The assumption that the property tax applies to land and all forms of capital within each sector at the same rate may need some justification. Two recent studies of property tax incidence, Sullivan [17] and Schoettle [14], have treated the property tax as applying to real property (land and structures including equipment permanently affixed to the land) but not to tangible personal property (inventories and portable equipment). The treatment of tangible personal property in the United States varies from state to state. According to the “1982 Census of Governments” [21, pp. xvi-xvii], eight states (Delaware, Hawaii, Illinois, New Hampshire, New York, North Dakota, Pennsylvania, and South Dakota) exempted all tangible personal property from local general property taxes in 1931. However, commercial and industrial inventories and equipment were fully taxable in 12 states (Arkansas, Georgia, Indiana, Kansas, Louisiana, Missouri, North Carolina, Ohio, Oklahoma, Texas, Virginia, and West Virginia). Our analysis is thus applicable to the latter states. The utility of residents of the taxing jurisdiction is assumed to be a function of their household’s consumption of housing, xH, and other consumer goods, xc, and hours of labor supplied, m. These utility functions are identical except for an additive term, Y, which reflects differences in their tastes for living in that jurisdiction. For example, the value of Y would be larger for persons with strong family ties and lower for persons with a weaker attachment to the locality. The differences in the value of v give rise to the “imperfect mobility” referred to in the title of this paper. The corresponding indirect utility function is
v(v) = U(PH, PC, w, y) + v,
(6)
where PH, PC, and w are the prices of housing, other consumer goods, and labor, respectively. “Full income,” Y, is the sum of nonlabor income, y, which is assumed to be exogenous and identical for all households, and the wage rate multiplied by the maximum hours of labor the household could supply, 11, or Y =y + wp.
(7)
80
PARAI AND BECK
Household demands for housing and for other consumer goods and household labor supply are given by
XH = XH(Y, w, PC, p,) xc = Xc(Y, w, PC, p,) m = m(y, w, PC, P,).
(8)
(9) 00)
The number of households choosing to live in the jurisdiction is determined by the number of households for whom u( PH, PC, w, Y) + Y 2 V*, where v* is the utility level any household could attain living outside the jurisdiction. Letting f(v) be the frequency distribution of Y and F( ) be the integral of f( ), the number of households living in the jurisdiction, N, is a function of prices within the jurisdiction: N = N(P,,
PC, w,Y)
= F(oo) - F[V*
- u(PH, PC, w,Y)].
(11)
The market demands for housing and other consumer goods within the jurisdiction are equal to the individual household demands for these goods multiplied by the number of households. Therefore, the commodity market equilibrium conditions are given by: X, = Nx,
(12)
Xc = Nx,.
(13)
The labor supply is equal to the individual plied by the number of households: M=Nm.
household labor supply multi-
(14
The “other consumer goods” supplied in the jurisdiction consist of a fixed amount of services of the local commercial sector combined with manufactured goods. Thus, the production function for “other consumer goods” can be expressed as a fixed coefficients function with the inputs being the outputs of the commercial and industrial sectors. Because a limitless supply of manufactured goods is available on the national market, the limiting input will always be the services of the local commercial sector. Thus the quantity of other consumer goods supplied in the jurisdiction can be expressed as
where 7r is the quantity
of local commercial
services required to produce
CLASSIFIED
PROPERTY TAXES
81
one unit of “other consumer goods.” We assume that local commercial services are measured in units that initially have a price of one and that (1 - n) is the quantity of manufactured goods required to produce one unit of “other consumer goods.” Thus, the price of these goods can be expressed as PC = (1 - T)P, + TP&
(16)
Thus, the costs of the local commercial sector which distributes consumer goods will be included in the price of these goods. Competitive equilibrium in the three sectors implies that
PH = R&L,
+ r&m
PI = 1 = R,C,, + r&
P, = R&
(17) + WC,,
+ rBCKB+ womb,
(18) (19)
where Cij is the cost minimizing quantity of input i required per unit of output m sector j, given the input prices and the production functions specified above. Rj = R(l + tj/r) and 7 = r + fj are the gross (including tax) land rent and rate of return on capital, respectively, in sector j.’ The Cij = Cij(Rj, rj, w) (for i = L, K, M, j = H, B, Z) are homogeneous of degree zero in all input prices. Equilibrium in the land and labor markets can be expressed as L = JWL” M = x,c,,
+ 4&l + x&f,.
+ WL‘,
(20) (21)
In analyzing the incidence of a classified property tax, we follow a differential tax incidence approach. We assume that initially all tax revenues are raised by a tax at a uniform rate t applied to all classes of property.3 Then the tax rate on class of property i is increased by dti and the rates on the other two classes of property, j and k, are changed with dtj = dt, determined so as to keep total tax revenues unchanged. For example, if the tax rate on residential property is increased with an offsetting uniform change in taxes on industrial and commercial business *We assume that there are no taxes (e.g., state or federal corporate taxes) other than the local property tax. 3Sullivan [17,18] also examined the incidence of taxes on industrial and residential property, respectively, in a differential tax incidence framework, but he assumed that the industrial or residential property taxes replaced a nondistortionary land tax. The policy alternatives more frequently considered are a uniform property tax and a classified property tax. Thus, we think our analysis of the incidence of a classified tax replacing a uniform tax is more relevant.
82
PA&U
AND
BECK
property, O=dt,
[
KH+
+t,[dK,
$L H] + t,[dK, + d( +LB)]
In terms of proportionate
+ d( +&)I
+ dt,[K,
+ FL,]
+ df,[K,
+ ;LB]
+ t,[dX,
+ d( fL[)].
changes
0 = [p,x,(x;
+ P,*) + P,X,(X,* + P,*/a) + x,x,* - wM( M* + w*)] +(r/t)[(P,X, + x1 - wM)T,* + P,X,T,*]
or -T,*
= -T$ +XrXI*
= {(t/r)[P,X,(X,* - wM(M*
+ w*)]
+ P,*) + P,X,(X,* + P,X,T,*}/(P,X,
+ P+) + XI - wM),
(22)
where an asterisk denotes proportionate changes; e.g., X$ = dX,/X,, except Tj* = dtj/q = rj* is defined to be the proportionate change in the gross rate of return to capital in sector j in response to a change in the tax rate on that class of property. The properties of production functions [l, pp. 503-508; 4, p. 241 give us Cc = S,,u/,( Czj = B,ju~,(R~ CGj = e,,t&(
rj* - Rf) - rj*)
+ e,,ui,(
W*
+ @,,u&(w*
Rj* - w*) + BKjlJJg
-
Ri*)
- Q*) ‘,* - w*)
(23) j = B, I, H,
where uli represents Allen’s partial elasticity of substitution between factors k and i in sector j and 8,, represents the share of factor k in the value of output of sector j. Note that r&, = 0 = CT,& = ufK and Rr = R* + I;*. Differentiating (8) through (15) and substituting, we obtarn:
+ (e, x, = hfff + E~)PH*
+ +)PC*
+ (eHw + e,)w*
xc? = (%H + eH)PH* + (eC- + q-)PC* + (ecw + e,)W* M* = (emH + eH)PH* + (e,, + Q)PC* + (emw + ew)w*,
(24) (25) (26)
where eH, eC, and E, are the elasticities of N with respect to PH, PC, and w, respectively, and the eij (i = H, C, m; j = H, C, w) are the elasticities
CLASSIFIED
PROPERTY
TAXES
83
of individual demand for housing and other consumer goods and individual supply of labor with respect to PH, PC, and W. Differentiating (16) through (19) and using the conditions for cost minimization, we obtain: P; - T; = eLHR* - (e,, + e,,>T,* PB* - (e,, + e&T,*
= e,,R*
+ eMrw*
= +~2
- (e,, + e&T,*
= e,g* Differentiating
(27)
(20) and (21) and substituting
+ eMBw*. (23)
(28)
(29)
and (26) gives:
where XLj = Lj/L and X, = Mj/M are the shares of land and labor, respectively, employed in sector j. Equations (24), (25), and (27) through (31) are a set of seven differential equations involving seven variables, X,, X,, X,, PC, PH, w, and R, and three parameters, t,, t,, and t,. The incidence of a tax increase on residential property offset by a change in the tax on industrial and commercial business property can be determined by solving this system of equations in terms of T$ using the tax revenue neutrality constraint given by (22). Similarly, the incidence of a tax increase on industrial (commercial business) property offset by a change in the tax rate on housing and commercial (industrial) property can be determined using an equation analogous to (22) solved in terms of T,*(T,*). PARAMETER VALUES The expressions resulting from the solution of these equations are cumbersome and dilkult to interpret. Therefore, rather than presenting a solution in general symbols for the parameters of the model, we have substituted plausible numerical values for the parameters and derived a
84
PAR41 AND BECK
numerical solution using the matrix multiplication and inversion procedures of TSP. We assume that the exogenous after-tax rate of return on capital is r = .06 and the initial uniform tax rate is t = .02. The production functions in the housing, industrial, and commercial business sectors are assumed to be Cobb-Douglas functions, and therefore all of the elasticities of substitution, ui, are equal to one. 4 The other parameter values used and initial values of endogenous variables are shown in Table 1. Sonstelie [15, p. 2591 used values of 675 and .325 for the factor shares of capital and land, respectively, in the production function of the residential sector function and values of .6159 and .3841 for the factor shares of capital and land in the commercial sector. In contrast, Sullivan [18, p. 2421 attributed only 14.36% of housing costs to land. We compromise by using values of I+, = .8 and e,, = .2. Kendrick and Grossman [ll, p. 271 cite an estimate of labor’s share in trade of .85. Dividing the commercial sector nonlabor share of .15 among capital and land in the same proportions as housing gives the capital and land shares of .12 and .03, respectively, shown in Table 1. The value of .756 for labor’s share in manufacturing is based on an estimate cited in [ll, p. 271. Kendrick [lo, p. 371 reports that land accounts for only 1.7% of property in the manufacturing sector; accordingly, the capital and land shares in manufacturing are apportioned as shown in Table 1. We approximate the share of the value of consumer goods added in the commercial sector by the ratio of GNP originating in the wholesale and retail trade industries to personal consumption expenditures on durable and nondurable goods. The “National Income and Product Accounts” [20, pp. 83, 2271 for 1976 give a value of .4869 for this ratio. Therefore, we use a value of n = .487. The “Census of Population” [19, p. 4221 gives the mean family income in metropolitan areas for 1969 as $11,991, and mean wage and salary income ?he Cobb-Douglas production function is widely accepted as a good approximation for manufacturing. The assumption of a Cobb-Douglas production function for the commercial sector is based on convenience and our inability to find any estimates of production functions for this sector in the literature. There is substantial literature on production functions for housing. Assuming a Cobb-Douglas production function (and therefore a unitary elasticity of substitution) for housing conflicts with much of this literature, but the literature gives conflicting results. Sonstelie [15, p. 2591 cites an estimate of the elasticity of substitution between land and capital which is greater than one (- 1.2, estimated by Smith [25]). On the other hand, Polinsky and Ellwood [13, p. 2011 report an estimate of - .45 for the elasticity of substitution, and Schoettle [14, pp. 364-3651, based in part on the Polinsky and Bllwood study used a value of -.5 for the elasticity of substitution between sites and “other inputs” (including capital and labor) in the production function for real property. In light of conflicting estimates, some greater and some less than one, for the elasticity of substitution for the housing production function, we do not think our assumption of a Cobb-Douglas function is unreasonable.
CLASSIFIED
PROPERTY TAXES
85
TABLE 1 Parameter Values and Initial Values of Variables Factor Shares e,, = .12 e,, = .03 eMB = .85 e,, = .240 e,, = ,004 e,, = ,156 eKH = .80 e,, = .20 Commercial services share in consumer goodr R = .481 Individual demand and supply elasticities e “H = -0.1 emw = 0.15
ccc = - 0.98235294 e*C = -0.100 eCH = -0.05294118 eHw = 0.84560647 ecw = 1.0943142 emH = 0.1325971 emc = 0.7513838
Household budget and choice variables
y = 975 (exogenous) m = 3,410 ? = 12,057.50 PHxH = 1,808.625 xc = 10,248.875 Sector shares of factor supplies
A,, = .6605 A,, = .0661 A LB = .2134 A, = .6172 P K” = .3431 .3828 A,, = .1420 A,, = .5149
Prices Pr = 1.0 (exogenous) r = .06 (exogenous) Pe = 1.0 PC = 1.0 R = 41,073.86
w = 3.25
as $11,016. This would suggest nonlabor income of $975. Therefore, we use a value of y = 975. The “Handbook of Labor Statistics” [22, p. 1841 reports average hourly earnings of nonsupervisory workers of $3.23 in 1970. This wage rate and the census wage income of $11,016 imply a value of m = 3410 for each household. Given that one person working 40 hours per week for 50 weeks would supply 2000 hours of labor per year; 3410 seems a reasonable estimate of average household labor supply, with one spouse employed full time and the other part time. With y = 975, w = 3.25, and m = 3410, each household in our model has a money income F = 12,057.50. Polinsky and Ellwood [13] give values of .8 and - .7 for the income and price elasticities of demand for housing, respectively. We use these values although Sonstelie [15, p. 2591 and Grieson [7, p. 3761 have used a larger
86
PARAI
AND
BECK
value of - 1.0 for the price elasticity. According to [20, p. 831, rent of housing (excluding such costs of household operation as utilities) constituted 15.36% of personal consumption expenditures. Therefore we assume that the housing and “other consumer goods” budget shares are CY”= .15 and (~c = .85, respectively. Regarding labor supply, Browning and Browning [6, p. 3431 cite a wage elasticity of emw = .15 according to Fullerton. We assume that the effect of changes in nonlabor income on household labor supply is b’m/ay = - .25, which is approximately the sum of the effects on labor supply of husbands and wives estimated by Hausman [8]. The literature is not so helpful in providing values for the own- and cross-price elasticities of “other consumer goods” or the cross elasticities between labor supply and housing and other consumer goods demand. For lack of any better information, we assume that leisure is neither a complement nor a substitute for housing and other consumer goods, so that changes in w have only an income effect on the demand for these goods and changes in PH and PC have only an income effect on the supply of labor. Based on these assumptions and the housing demand and labor supply elasticities and budget shares given above, we calculate the remaining own- and cross-price elasticities shown in Table 1 using the relationships between price and income elasticities and budget shares derived from consumer theory [9, p. 391. Our hypothetical jurisdiction is assumed to have a land area L = 10,000 square miles populated by N = l,OOO,OOO households. These assumptions imply the initial values for the remaining variables shown in Table 1. From (ll), the change in population will be dN =f(V*
- u)[(&@P,)
dP, + (ch/JP,)
dP, f (&~/a,) + (do/aY)
dw dY] . (32)
By the use of Roy’s identity [23, p. 931, the partial derivatives of the indirect utility function with respect to PH, PC, and w may be replaced by au/a Y (which, without loss of generality, we assume equals one) multiplied by the negatives of the quantities of housing, other consumer goods, and leisure. dN=f(V*-u)[-x,dP,--x,dP,-(p-m)dw+dY] =f(V*
- u)[-x,dP,
- xcdPc
- (p - m) dw + pdw]
=f(V*
- u)[-x,dP,
- xcdPc
+ mdw].
In addition
(33)
to being affected by the values of the partial derivatives of u, f(v). In the extreme case of no population mobility, no one is almost indifferent between moving out and remaining in the community. In this case f = 0. dN also depends on the value of the frequency distribution
CLASSIFIED
PROPERTY TABLE
EtTects of Increasing P;/T; 0 100 500 103 104 105 106
P,+/T,*
1.029 1.056 1.346 0.113 0.942 - 4.052 - 0.077
,056 ,062 ,120 -.130 ,029 - .576 - ,097
87
TAXES
2
the Tax Rate on Residential
w’/T;
R./T,*
T,+/T,*
0.293 0.312 0.520 - 0.312 0.179 - 1.213 - 0.103
0.143 0.282 1.731 - 4.433 - 0.291 - 25.260 - 5.384
Property = TB*/T$
-0.909 -0.912 -1.638 1.226 - 0.550 4.163 0.388
Au/T2 809.71 919.59 2095.05 -2996.35 -18.13 -210.82 -4.46
On the other hand, if the population is perfectly mobile, everyone is indifferent between staying or leaving and f = co. In the following section, we present results for intermediate cases with 0 -Cf < cc. RESULTS Table 2 shows the effects of increasing the tax rate on the residential class of property accompanied by changes in the tax rate on industrial and commercial property to hold total tax revenue constant. Tables 3 and 4 show the effects of increasing the tax rates on the industrial and commercial classes, respectively. In each table, each row shows the results for a different assumption about the degree of population mobility, i.e., about the value of f. The change in the burden on landowners is measured by the change in R shown in column (5). The change in the burden on resident worker-consumers can be measured by the change in u shown in the last column, which is calculated from the changes in PH, PC, and w shown in columns (2), (3), and (4). First, we consider the cases with no population mobility, i.e., f = 0, shown in the first rows of each table. Table 2 shows that an increase in the TABLE Effects
f
PJ*/Tf*
0 100 500 103 104 105 106
- 0.543 - 0.541 -0.357 - 0.517 0.221 - 1.007 - 1.441
of Increasing P,*/T,* -
.173 .173 ,158 ,171 ,109 .217 .322
d/T,* -
3
the Tax Rate on Industrial
.323 .323 .318 ,322 .303 .347 ,525
R*/T,* -0.018 - 0.288 -0.872 - 0.132 3.775 4.581 38.198
T,+/T,*
Property = T,*/T,*
-
0.539 0.535 0.183 0.491 0.982 - 1.923 - 9.081
Av/T,* - 817.10 - 823.02 - 1262.30 - 880.18 - 2650.24 200.19 89.84
88
PARAI AND BECK TABLE 4 Effects of Increasing the Tax Rate on Commercial Property
f
P,*/T,*
0
10s
- .224 - ,227 - ,249 ,674 - .219 - ,369
106
- ,111
loo 500
103 104
P,*/T,* .100
.lOl ,111 - ,332 .081 ,013 - ,040
w*/TB*
R*/T,*
,068 - ,070 ,088 - ,694 .037 - ,050 - ,056
- 0.077 - 0.055 0.130 - 8.043 - 0.470 - 2.691 - 6.171
T,*/T$
= TJ’/T$ - 0.207 - 0.216 - 0.274 2.282 - 0.107 0.200 0.267
Au/T; 131.33 147.86 - 283.74 - 5508.88 - 29.85 - 21.74 - 5.05
tax rate on residential property increases the price of housing, but the accompanying reduction in the tax rate on industrial property induces investment which raises the wage rate. Local retailers must pay higher wages to their workers to compete with wage offers from the manufacturing sector. Thus, although the tax rate on commercial property falls, the increase in the wage rate causes the price of consumer goods to rise. The increased wage rate raises local incomes and thereby increases the demand for housing. Thus, housing prices rise by more than the increase in taxes on residential property, as shown by the value of P$/T,* = 1.029 > 1.0. Consequently land rents rise. The increase in the wage rate more than compensates for the higher prices of housing and other consumer goods, as shown by the increase in the value of the indirect utility function. Although there is no change in the total tax revenues collected, the increases in land rents and in the indirect utility function reflect efficiency gains from reductions in the excess burden of the property tax. Because the supply of capital to the jurisdiction is infinitely elastic, partial equilibrium analysis indicates that from the local viewpoint the excess burden of the tax on capital would be minimized if tax rates were inversely proportional to the demand for capital in each sector [3, pp. 367-3701. A uniform tax on capital does not minimize this excess burden. Consequently, Beck [5] argued that a classified property tax, with a lower tax rate on sectors in which the demand for capital is more elastic, reduces the excess burden. Wilson [24] discusses conditions for an optimal classified property tax in a more general model which qualifies Beck’s conclusions but still concludes that from the local perspective a uniform tax is not generally optimal. In the present case, because manufactured goods are sold in a national market, the derived demand for capital in the industrial sector is more elastic and a lower tax rate in that sector is optimal.’ ‘Amott
and Grieson
sets of assumptions.
[2] and Grieson
[7] also derive
an inverse
elasticity
rule under
different
CLASSIFIED
PROPERTY TAXES
89
The first row of Table 3 shows the effects of increasing the tax rate on industrial property when the population is immobile. The increased tax rate reduces capital in the industrial sector, leading to a decline in the wage rate. Reduced demand for housing and other consumer goods combined with the reduction in the tax rate on residential and commercial property causes PH and PC to fall, but the value of the indirect utility function falls due to the decline in w. Landowners also lose from the decline in R. The first row of Table 4 shows the effects of raising the tax rate on commercial property when the population is immobile. The wage rate increases as labor is substituted for capital and land in the commercial sector.6 As one would expect, the increased tax on commercial property raises the price of consumer goods. The increase in the wage rate and reduction in housing costs more than compensate for the rise in the price of other consumer goods, causing a moderate increase in the value of the indirect utility function. Land rents also fall in this case. Turning to the lower rows of Tables 2-4, we first note that the positive values in some rows of column (6) indicate that in some cases raising one tax rate requires increasing the other tax rates as well in order to hold total revenues constant. This occurs when the tax rate on the first class of property is above the revenue-maximizi ng rate for that class of property. In these cases both land rents and the utility of residents fall as raising a tax rate above the revenue-maximizing rate greatly increases the excess burden of the tax. From a partial equilibrium perspective, greater population mobility implies a larger own-price elasticity of demand for housing and other consumer goods. Because the revenue-maximizing point on the “Laffer curve” occurs at lower tax rates the greater the elasticity of demand, we might expect positive values to occur in column (6) of Tables 2 and 4 as population mobility increases. Indeed, this does occur, but not uniformly. Apparently in some cases of greater population mobility general equilibrium effects allow reductions in the tax rate on other classes of property while maintaining constant revenues despite the large own-price elasticities of demand. Another apparent puzzle is that in some cases increased population mobility leads to larger rather than smaller changes in the utility level of residents. For instance, in Table 2 an increase in the tax rate on residential property results in a larger change in utility, Au/T; = 919.59 with f = 100, than in the case of no mobility, in which Au/T; = 809.71 with f = 0. One 6Note that this depends on the partial elasticities of substitution of labor for capital and land being greater than the elasticity of demand for output [l, p. 5081. Because we assumed unitary partial elasticities of substitution in the commercial sector without any empirical support, there may be more reason to be skeptical of this result than of other aspects of this study dependent on parameter values better supported in the empirical literature.
90
PARAI
AND
BECK
would expect that in-migration in response to higher wages would increase the supply of labor and moderate the increases in wages and utility. However, the in-migration also increases the property tax base as the new workers purchase local housing. This allows a larger reduction in the tax rate on industrial property, which leads to more investment and further increases in the real wage. Similarly, in the analysis of an increase in the tax rate on industrial property shown in Table 3, Au/T,* = - 817.10 with f = 0 but Au/T,* = - 1262.30 with f = 500. The increased tax on industry reduces the local wage rate and thereby reduces the utility of residents. In the imperfect mobility case with f = 100, the resulting decline in population reduces the local demand for housing, and the property tax base declines. To maintain constant tax revenues, the reduction in the tax rate on residential and commercial property is less than in the case with f = 0. Consequently, housing and other consumer goods prices do not fall as much. The decline in the wage rate is moderated only very slightly by the slight degree of mobility. The combination of smaller declines in housing and consumer prices with almost the same decline in wages results in a larger decline in utility (for those remaining) with f = 500 compared to f = 0. The cases in which f = lo6 represent almost perfect mobility. With f = 106, a change in local prices and wage rates equivalent to a $1.00 reduction in income would cause the entire population to leave the jurisdiction. The results in the last rows of Tables 2-4 show that, as expected, with almost perfect mobility the utility level of residents is virtually constant. The values of Au/T* vary from - 5.05 to + 89.84 in these cases. In the case of the increased tax rate on industrial property shown in the last row of Table 3, land rents increase as the tax rate on the more land intensive residential and commercial sectors is reduced. CONCLUSIONS By including labor as an imperfectly mobile factor of production and an industrial sector in the local economy, we have demonstrated some possible consequences of classified property taxes not shown in Sonstelie’s [15, 161 model. As expected, increasing the tax rate on industrial property will reduce the wage rate of immobile labor. More specifically, for the plausible parameter values used in our illustration, increasing the industrial property tax rate while reducing the rates on residential and commercial property reduces wages so much that immobile workers are actually worse off despite the accompanying reductions in the prices of housing and other locally purchased consumer goods. Conversely, when the population is immobile, raising residential property tax rates in order to reduce the rates on business property results in efficiency gains which increase the utility of residents.
CLASSIFIED
PROPERTY
TAXES
91
In the model with imperfect population mobility, there is the possibility that the demand for capital in a sector may be so elastic that increasing the tax rate on that class of property reduces tax revenues, requiring accompanying increases in the tax rates on other classes of property as well. In these cases, the increase in tax rates reduces both land rents and the utility of residents. With nearly perfect mobility, the utility level of residents is almost constant. In this case, increasing the tax rate on the less land intensive manufacturing sector and reducing the rate on the more land intensive sectors will increase land rents. REFERENCES 1. R. G. D. Allen, “Mathematical Analysis for Economists,” St. Martin’s Press, New York (1968). 2. R. Arnott and R. E. Grieson, Optimal fiscal policy for a state or local government, J. Urban Econom., 9,23-48 (1981). 3. A. B. Atkinson and J. E. Stightz, “Lectures on Public Economics,” McGraw-Hill, New York (1980). 4. R. N. Batra and F. R. Casas, A synthesis of the Heckscher-Ohlin and the neoclassical models of international trade, J. In?. Econom., 6, 21-38 (1976). 5. J. H. Beck, Tax competition, uniform assessment and the benefit principle, J. Urban Econom., 13,127-146 (1983). 6. E. K. Browning and J. M. Browning, “Public Finance and the Price System,” Macmillan Co., New York (1983). 7. R. E. Grieson, The economics of property taxes and land values: The elasticity of supply of structures, J. Urban Econom., 1, 367-381 (1974). 8. J. A. Hausman, Labor supply, in “How Taxes Affect Economic Behavior” (H. J. Aaron and J. A. Pechman, Eds.), Brookings, Washington (1981). 9. J. M. Henderson and R. E. Quandt, “Microeconomic Theory: A Mathematical Approach,” McGraw-Hill, New York (1971). 10. J. W. Kendrick, “The National Wealth of the United States by Major Sector and Industry,” The Conference Board (1976). 11. J. W. Kendrick and E. S. Grossman, “Productivity in the United States, Trends and Cycles,” Johns Hopkins Univ. Press, Baltimore (1980). 12. P. Mieszkowski, The property tax: An excise tax or a profits tax, J. Public Econom., 1, 73-96 (1972). 13. A. M. Polinsky and D. T. Ellwood, An empirical reconciliation of micro and grouped estimates of the demand for housing, Rev. Econom. Statist., 61, 199-205 (1979). 14. F. P. Schoettle, A three-sector model for real property tax incidence, J. Public Econom., 27, 355-370 (1985). 15. J. Sonstelie, The classified property tax, in “Technical Aspects of the District’s Tax System: Studies and Papers Prepared for the District of Columbia Tax Revision Commission,” U.S. Government Printing Office, Washington (1978). 16. J. Sonstelie, The incidence of a classified property tax, J. Public Econom., 12, 75-85 (1979). 17. A. M. Sullivan, The general equilibrium effects of the industrial property tax: Incidence and excess burden, Reg. Sci. Urban Econom., 14,547-563 (1984). 18. A. M. Sullivan, The general-equilibrium effects of the residential property tax: Incidence and excess burden, J. Urban Econom., l&235-250 (1985).
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19. U.S. Bureau of the Census, “Census of Population: 1970, General Social and Economics Characteristics,” Final Report PC(l)-Cl United States Summary, U.S. Government Printing Office, Washington (1972). 20. U.S. Department of Commerce, “The National Income and Product Accounts of the United States, 1929-76,” Washington (1981). 21. U.S. Department of Commerce, “1982 Census of Governments,” Vol. 2, “Taxable Property Values and Assessment/Safes Price Ratios,” US. Government Printing Office, Washington (1984). 22. U.S. Department of Labor, Bureau of Labor Statistics, “Handbook of Labor Statistics,” Bulletin 2070, Washington (1980). 23. H. R. Varian, “Microeconomic Analysis,” Norton, New York (1978). 24. J. D. Wilson, Optimal property taxation in the presence of interregional capital mobility, J. Urban Econom., 18, 73-89 (1985). 25. B. A. Smith, The supply of urban housing, Q. J. Econom., 40, 389-405 (1976).