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Elements of an economic justification for municipal zoning
JOURNAL
OF URBAN
ECONOMCS
Elements
14, l68- 183 (I 983)
of an Economic Justification for Municipal Zoning THEODOREM. CRONE
Federal Reserve Bank of Philadelphia, Philadelphia, Pennsylvania 19105
Received November 18, 198I ; revised April I, 1982
The possibility that externalities will produce a nonconvexity in the social production set limits the application of both the Cease and Pigou solutions for achieving an optimal allocation of land resources.Conditions on relative land prices are derived which indicate that external effects are strong enough to introduce a nonconvexity into the production set. These conditions were not fulfilled in a sample of single-family and multifamily dwellings in Foster City, California. This does not preclude the possibility that they are fulfilled in casesof more severe external effects.
1. INTRODUCTION Since the 1967 article by Crecine et al. [6], several studies have appeared estimating the effects of municipal zoning on residential property values [8, 9, 12, 15, 19, 211. None of these studies, however, situates the discussion of zoning within the context of a general theory of externalities. The present study provides a link between the theory of optimal resource allocation with externalities and the empirical literature on zoning. In the main body of economic literature, two remedies are generally proposed for achieving a Pareto optimal allocation of resources when externalities are present in the economy. The first is to define property rights more precisely so that all activities which affect others are incorporated into the market process (the Coase solution). The second method is to tax or subsidize activities which cause external harm or benefit (the Pigou solution). However, the possibility that externalities will produce a nonconvexity in some firms’ production sets or in the social production set presents a limitation for both solutions. This limitation on the Coase and Pigou solutions provides the basis of an economic justification for municipal zoning. If the external effect of one activity upon another is localized and strong enough to create a nonconvexity in the individual or social production set, separation by zoning may be as efficient as any other method in achieving an optimal allocation of land resources. 168 0094-I l90/83 $3.00 Copyright All rights
0 1983 by Academic Press, Inc. of reproduction in any form reserved.
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2. LIMITATIONS OF THE COASE AND PIGOU SOLUTIONS TO LOCALIZED EXTERNALITIES It has been suggested that a complete definition of property rights as envisioned by Coase could be achieved by issuing permits to engage in a specified level of a polluting or nuisance-causing activity [4, 71. Since these permits could be purchased and retired from use, the pollution level would become a choice variable for firms adversely affected by the pollutant. This presents a potential difficulty for a competitive market in pollution permits. Under plausible assumptions about the effect of pollution upon output, the production sets of affected firms are not convex [20].’ Proof of the existence of a competitive equilibrium, however, depends upon convexity of production sets defined over firms’ choice variables. Of course, feasible production sets can always be confined to the convex region by limiting the number of permits issued, so that an equilibrium can be assured [S]; but there is no guarantee that this equilibrium will represent a Pareto-optimal resource allocation.* Marketable pollution rights may be useful as a least-cost method of maintaining a given level of pollution [14], but they offer no short-cut method of arriving at the optimal level. Under a Pigouvian tax scheme, affected firms take the pollution level as given, so that the nonconvexity associated with marketable pollution rights does not arise. Moreover, if the individual production sets are convex, proper Pigouvian taxes will sustain a Pareto-optimal allocation of resources [3, 161.The proper Pigouvian tax, however, must be calculated according to the marginal damage caused by the offending activity when all activities are undertaken at their optimal levels and in their optimal locations. With the possibility of so many externalities, no one would claim that the economy has been at such an optimum. This problem of calculating the proper Pigouvian taxes would be of little practical significance if Pigouvian-type taxes based upon actual marginal damage and iteratively adjusted would converge to the true Pigouvian taxes and lead to an optimal allocation. Such iteratively adjusted taxes would be analogous to competitive prices and would converge to their optimal values when the conditions for a unique and stable equilibrium exist. But one of the conditions for the existence of a unique competitive equilibrium is the convexity of the social production
‘A firm’s production set would be nonconvex, for example, if output diminished at an increasing rate with the pollution level up to a point of relative saturation and at a decreasing rate thereafter. *This is true not only because the restricted number of permits issued in any locality will determine the equilibrium price and pollution level, but also because the retirement of permits from use is a public good benefiting all affected firms, a classic free-rider situation.
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set.3 This condition may not be fulfilled when externalities are present, for sufficiently strong externalities can themselves generate a nonconvexity in the social production set [ 1, 2, 10, 181.Given a nonconvex social production set and any normal social welfare indifference map, an iterative adjustment of prices and, if necessary, Pigouvian-type taxes could lead to any one of several local welfare maxima. If the external effects causing the nonconvexity are localized or if they dissipate with distance, the actual equilibrium outcome will depend upon the relative location of activities and need not be the global optimum. Since the optimal location of activities will depend upon the social welfare function, the allocation of land resourcesby political decisions such as zoning may result in a higher level of community welfare than a strict dependenceupon free markets supplemented by Pigouvian-type taxes. 3. LOCALIZED EXTERNALITIES, NONCONVEXITIES, AND THE PRICE OF LAND In an economy characterized by a nonconvex production set, assumptions about markets for inputs and outputs yield general conclusions about the way in which land prices vary with the mix of activities. If there is a nonconvexity in the social production set, the production frontier at some point must be convex to the origin. And if the price ratios remain fixed, the total value of output achieves a local minimum at some point other than the extremes of the production frontier.4 Using this fact, we can derive conditions on the parameters of an hedonic price model which indicate not only whether an externality is present but also whether it is severe enough to create a nonconvexity in the production set. Since land is immobile, the damage from localized externalities should be capitalized in its price. If developers have knowledge of future environmental quality or land-use mix of a neighborhood, they include this information in their offer price for land. In zoned communities, that price will be the value of the marginal product of land in its zoned use. Furthermore, at any location, resources other than land are invested up to the point where the price of the resource equals the value of its marginal product, given the presence of any external effects from neighboring uses. The model of land prices developed below is based on these general propositions and the following specific assumptions. (Pl) Two goods, x, and x1, are produced in a single location (S) which, in the absenceof externalities, is more productive for each of the two goods than other available locations in the community. ‘This condition is not required to show that proper Pigouvian taxes will sustain an optimal allocation of resources. All that is required for that proof is the convexity of each firm’s production set. 40f course, total value could reach local minima at the extremes of the frontier as well.
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(P2) The production of at least one of the goods creates a negative externality in the production of the other. (P3) Production inputs at S include land, of fixed supply AT and another resource, N. (P4) Prices for the outputs, x, and x2, p, and p2, respectively, are exogenous and fixed. This assumption is reasonable if we think of x, and x2 as being sold on a foreign market or in a larger market which includes many other producers, and where prices are determined competitively. (P5) The price of the input N, p,, is exogenous and remains fixed for the same basic reasons that p, and jiz are fixed. (P6) Not only is the total supply of land in S fixed, but the relative supply of land in S for the production of x, and x2 is determined by zoning regulations, so that the price of land used to produce x, (p,,) may differ from the price of land used to produce x2 (pa2). Without these zoning restrictions, of course, competition would eliminate the price differential. Assumptions Pl, P2, and P3 imply production functions for x, and x2 as follows:
XI =f(4
N,P,)
and x2
=g(A,,4, x,>.
A, represents the amount of land used for x,, and A, the amount used for x2. Likewise, N, and N2 represent the amounts of the resource N used for x, and x2. Assumption P2 on the existence of a negative externality implies that af/ax, -C0 and/or ag/ax, c 0. If all land is employed in the production of one of the two goods, A= A, + A,, and since the total amount of land is fixed, units can be resealed so that A= 1. If the proportion of land used in production of x, is designated by p, total land inputs for each good will be A, = P
and
A, = 1 - p.
If the resource-land ratios for x, and x2 are n, and n2, respectively, then total inputs of N for each good are
N, =
pn,
and
N2 = (1 - p)n,.
The production functions become Xl =f{p,
Pn,, g[(l
- Ph (1 -
PM)
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THEODORE M. CRONE
and x2
=
do
-
PI,
(1
-
dn29
f[PY
Pa
Profits for each industry in location S are:
and n2
-Pa2(l
=j-Qx2
- P> -Pn(l
- Pb2.
If output markets are competitive and landowners are able to retrieve the value of any special productive capacity of their land in higher land prices, profits are zero. Setting V, and r2 equal to zero, the total value of output (V) at location S is expressed by
V = p,x, + F2x2
= P~]P + April
+ ~~~(1 - P) +PJ
- ph.
0)
Production frontiers, of course, are drawn on the assumption that the total amount of each resource is fixed. Thus far, no such restriction has been placed on N and no constraints have been placed on n, and n 2. This is realistic since the resource-land ratio is likely to vary between x, and x2. Also, n r and n 2 may vary with p if the marginal product of N depends upon the extent of the externality. Instead of confining the amount of N employed in S to a constant, consider fl to be the total amount of the resource available in S, where
N= max[pn, + (1 - pb,l. P
By definition, then, Nis a constant. Profit-maximizing firms located in S employ N up to the point where p,( af/aN,) = jF2(ag/aN,) = jr,,. Any portion of F not used could be sold on the open market at the fixed price pm.Alternatively, the value of that portion of fl which is not used can be considered a cost saving to the firms located in S. This cost saving is equal to
pn, - (1 - dn21.
K[fl-
When this saving is incorporated into (l), the total value of output becomes I/= pa,p + &pn,
+j-fJN-
+ pa2(l
- P) +&‘,(I
pn, - (1 - dn21
- PJn2
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JUSTIFICATION
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or
I/= [Pa,P +Pa2(l- dl +fv*
(2)
Since p,,m is a constant, the total value of output will reach a local minimum when the bracketed expression in (2) reaches a minimum. That expression, labeled V* below, is a direct function of p and of land prices, p,i and pa2. If there are localized externalities, these land prices also will be functions of p. Let pa, = h(p) andp,, = k(p), then v* = p * h(p)
+ (1 - p) . k(p).
(3)
Applying the first and second order conditions for a local minimum for I’* and imposing the requirement that 0 -Cp < 1, we obtain the following conditions for a nonconvex production set:
[
[h(p)-k(p)]+p
1
?$-$
(34
+$=o,
(3b) o<
b(P)
-
h(P)]-
dh/ap
ak/aP
- dk/ii’p
< 1
’
(34
The fulfillment of these conditions depends upon the relative changes in land prices as the mix of activities varies. No empirical study of zoning has attempted to estimate these relative changes. These studies have concentrated on the effect of neighboring land uses on the value of single-family homes and often have embodied the unstated assumption that external effects are unilateral from other land uses to single-family residences. The use of a binary variable to indicate the presence of an externality in some studies precludes any estimate of how prices change with the mix of activities [6, 12, 15, 191.And those studies which measure external effects by the proportion of land devoted to non-single-family uses (p in our model) have not attempted to estimate the net effect on land prices from the close proximity of different uses [21, 91. It is the net effect, however, which is crucial in determining whether separation by zoning is necessary or appropriate in an economy characterized by externalities. In order to determine whether conditions (3a) and (3~) are satisfied for some value of p, the functions h(p) and k(p) must be specified. Land prices
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THEODORE
M. CRONE
will be determined: (a) by the equilibrium prices for land in the economy as a whole, which are assumed to be fixed; (b) by the special productive capacity of land in location S; (c) and if there is an externality, by p, the proportion of land used in the production of x,. The simplest assumption which embodies these three determinants of land prices is that they are linear functions of p. w
= 8, + &P,
(4)
and
w = 02+6s.
(5)
The values of 8, and 0, depend on (a) and (b), and the signs and magnitudes of the parameters 6, and 6, depend on (c). These last two parameters indicate the direction and magnitude of the external effect, which is assumed to be proportional to the amount of land used for the activity generating the externality. The linear form of h(p) and k(p) permits only one local minimum for V* in the range 0 < p < 1. A quadratic form implies a similar restriction. Only higher-degree polynomials would allow for multiple local minima.5 Given these linear specifications of h(p) and k(p) conditions (3a)-(3c) can be restated in terms of the parameters of (4) and (5). Since the second derivatives of the price functions are zero, condition (3b) becomes 6, - 6, > 0.
@a)
Solving (3a) for p and substituting that value into (3~) yields the following conditions:
e, - 8, - 6, > 0
(6b)
and 26, - s, > ‘A linear form for h(p) and k(p) minimum or maximum in the range that V* is a thud-degree polynomial maximum in the relevant range. In paper the coefficient on the square different from zero.
e2- 8,.
implies that V* is quadratic in p with 0 < p < 1. A quadratic form for h(p) in p with only one possible minimum the preliminary stages of the empirical of the zoning variable was not found
(64 only one possible and k(p) implies and one possible research for this to be significantly
ECONOMIC JUSTIFICATION
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175
These conditions summarize the three essential questions in assessingthe efficiency of zoning. (1) Is the economy characterized by negative externalities which would lead to a misallocation of land resources in the absence of any government intervention in the market? In terms of the parameters of the model, a positive response to this question implies that 6, > 0 or 6, < 0. All previous empirical studies on zoning have posed the question of whether one or the other of these inequalities exists. (2) If negative externalities exist, are there also positive externalities, and is the net effect such that a separation of interacting land uses is efficient? According to the model, the net effect will be negative and a separation of activities will be efficient if 6, - 6, > 0. No empirical study thus far has sought to measure this net effect. (3) Is the net effect of the externality so severe that it produces a nonconvexity in the social production set? This implies that all three conditions, (6a), (6b), and (6c), are fulfilled. These conditions are sufficient to establish the existence of a nonconvex production set. If they are fulfilled, iteratively adjusted Pigouvian-type taxes cannot ensure the proper separation of activities, and zoning may well represent a more efficient solution to the externality problem. 4. ESTIMATION
OF AN EMPIRICAL
MODEL
Theoretically, the present analysis of localized externalities and land prices applies to any two interacting activities. Price data on parcels of underdeveloped land would offer the most direct test of whether conditions (6a) through (6~) are fulfilled. Since most sales of real property involve both land and structures, however, this direct approach was not feasible. The need to account for the value of structures in the estimations dictated that the empirical analysis of externalities be confined to differing types of residential land use. The data on commercial and industrial structures are simply too meager to estimate their value with any kind of hedonic price equation. In effect, the two goods in the foregoing analysis, x, and x2, will be represented by housing services provided by multifamily structures and by single-family structures, respectively. Although the external effects associated with other conflicting land uses (e.g., industrial and residential uses) are presumably more severe,the restriction to these two types of residential use is not a serious problem. Zoning regulations assume that multifamily units create negative externalities for nearby single-family residents. And in their empirical work, George Peterson and William Stull have estimated that the presence of multifamily structures has a negative effect on the prices of single-family homes [15, 211.
176
THEODORE
M. CRONE
Hedonic price equations have been used in the past to estimate the effects of different land uses on the prices of single-family homes. The model presented here will expand upon those former studies by including an hedonic price index for multifamily dwellings. This will allow a comparison of the effects of the land-use mix on both types of properties and a measure of the net external effect of mixing land uses. The foregoing theoretical discussion was confined to a two-good world. Therefore, the empirical model presented here is meant to apply to strictly residential locations which can include both single-family and multifamily dwellings. The model specifies that PRICE OF DWELLING = a0 + a,(LOT SIZE MF) + a,(LOT SIZE SF) + a3(ZONE MF
x
LOT SIZE MF)
+ a4 (ZONE MF
x
LOT SIZE SF)
+ CBj(MF,) J
+ C~k(sFk) + ~7 k
where LOT SIZE MF = The size of the lot if the dwelling is multifamily, LOT SIZE SF = The size of the lot if the dwelling is single-family, ZONE MF x LOT SIZE MF = Proportion of the neighborhood zoned multifamily times the lot size if the dwelling is multifamily, ZONE MF x LOT SIZE SF = Proportion of the neighborhood zoned multifamily times the lot size if the dwelling is single-family, Mq = The structural characteristics of multifamily dwellings, and SF, = The structural characteristics of single-family dwellings. The model assumesthat neighborhood externalities can affect the prices of both single-family and multifamily dwellings. It has long been assumed that the congestion, noise, lack of open space, and other factors associated with the presenceof multifamily dwellings decreasethe value of single-family homes. This model leaves open the possibility that the same factors also decreasethe value of multifamily dwellings. Some of the coefficients in this regression model can be identified with the parameters discussed in the previous section. If good 1 (x,) represents multifamily housing services and good 2 (x2) represents single-family housing services, the following coefficients in the model will correspond to
ECONOMIC
JUSTIFICATION
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177
the parameters in Eqs. (4) and (5); a, = 8,;
a2 = e2;
a3 = 6,;
a4 = 6,.
Accordingly, the conditions for a nonconvex production set correspond to the following conditions on the regression coefficients: -
ff4
’ 0,
(74
lx2 - a, -
a4
’ 0,
(7b)
a3
and 2a, - a4 > a2 - a,.
(74
Data on single-family and multifamily dwellings in Foster City, California were used to estimate the parameters of the model. Foster City is a residential community located approximately 15 miles south of San Francisco. The city is laced with canals and lagoons, and the area was developed to take advantage of its proximity to San Francisco Bay, making it a preferred location for residential development. Data were made available by the Society of Real Estate Appraisers (SREA) on the sale price, lot size, and structural characteristics of 177 single-family dwellings and 24 multifamily dwellings sold in Foster City between 1975 and 1979. To account for inflation, the sale prices of singlefamily dwellings were adjusted to their 1977 levels using data supplied by Federal Home Loan Bank on the averageprice of single-family homes in the metropolitan area. To adjust the prices of multifamily dwellings to their 1977 levels an inflation index was calculated using a random sample of 987 multifamily structures in the San Francisco Bay Area (cf. Appendix A). Since the dwellings in the Foster City sample were identified by street address, it was possible to calculate for each of them the proportion of land within 500 feet zoned for multifamily use. The goal of this study differs from that of previous ones which have attempted to estimate the effect of multifamily structures on the prices of nearby single-family homes [6, 8, 12, 15, 19, 211. In the theoretical discussion above, the net effect on land prices due to the mixing of land uses proved to be the decisive factor in judging the efficiency of zoning. To measure this effect it was necessary to combine the two sets of data on single-family and multifamily dwellings. When two such data sets are combined, the assumption of homoscedasticity of the error term implied in the ordinary least squares method is likely to be violated [22]. Violation of
178
THEODORE M. CRONE
this assumption does not create a bias in the estimates of the regression coefficients, but it does reduce the efficiency of the estimations. Bartlett’s test was applied to the two housing samples [ 171,and it was determined that there was a statistically significant difference in the error variances between the two data sets. Therefore, the variables in each data set were adjusted using estimated error variances. The dependent variables used in the preliminary equations to estimate these error variances and in the final regression equation are explained in Appendix B. The estimated coefficients of the hedonic equation combining both types of housing data are presented in Table 1. The primary concern is with the coefficients on the zoning variables (proportion of neighborhood zoned TABLE 1 Hedonic Equation for Single-family and Multifamily Dwellings, Foster City Dependent Variable: (Price of Dwelling Adjusted to 1977Level) Absolute value of Independent variable Coefficient t-ratio Square feet of living area (single-family average quality) Square feet of living area (single-family good quality) Square feet of living area (single-family excellent quality) Age of dwelling x square feet of living area (single-family) Lot size (single-family) Number of full baths (single-family) Number of half baths (single-family) Number of fireplaces (single-family) Number of garage spaces(single-family) Multifamily dummy (dummy equal to one if dwelling is multifamily) Square feet of rentable area (multifamily average quality) Square feet of rentable area (multifamily I%& w&tY) Age of dwelling x square feet of rentable area (multifamily) Lot size (multifamily) Proportion of neighborhood zoned multifamily x lot size (all dwellings) Proportion of neighborhood zoned multifamily x lot size (multifamily dwellings) (Constant) Note: if2 = 0.88; N = 201.
29.38
(7.88)
28.78
(8.06)
38.03
(7.61)
-0.19
(2.18)
1.30 - 1748.06 4982.87 2906.37 2954.23 - 18111.33
(0.58) (1.79) (1.08) (0.78) (1.95)
55.00
(1.94)
56.49
(1.81)
- 2.48
(1.02)
0.27 - 3.72
(0.03) (2.11)
0.36
(0.03)
25039.29
(2.83)
(2.40)
ECONOMIC JUSTIFICATION
FOR ZONING
179
multifamily x lot size). The first of the zoning variables applies to all dwellings, and its coefficient measuresthe basic effect of multifamily zoning on the per unit price of neighboring land. The estimate indicates that this effect is substantial (- 3.72) and significantly different from zero at the 0.05 level. This result agrees with that of Peterson [ 151 and Stull [21] that multifamily structures are a source of negative externalities for other nearby dwellings. In the tradition of the previous empirical work on zoning, the implication of this result is likely to be misunderstood. This negative coefficient does not necessarily imply that efficiency in the use of land resources requires the separation of these two residential uses either by zoning or some other method. Some measure of the net effect on total land prices is needed to draw that conclusion. The coefficient on the second zoning variable in Table 1 applies only to multifamily dwellings and measures the difference between the negative effect on land prices for nearby multifamily and single-family properties. In terms of the theoretical model, this coefficient corresponds to 6, - 6,. According to condition (6a), a positive value for this coefficient implies that the net effect on land prices of mixing these two residential land uses is negative and that the uses should be separated on grounds of economic efficiency. The estimated coefficient is positive (0.36). The extremely low value of the t statistic, however, does not allow us to reject the hypothesis that the true value is zero and that there is no difference between the effect of multifamily zoning on nearby single-family and other nearby multifamily dwellings.6 5. CONCLUSION Neither the Coase nor the Pigou solution to the problem of externalities envisions the type of direct land-use regulation represented by municipal zoning. But neither of these remedies can guarantee an optimal allocation of resources when externalities are strong enough to cause a nonconvexity in the production set. In the presence of such a nonconvexity, municipal zoning may be as efficient as either of the two generally proposed solutions to externalities. Under some common assumptions about land rents, this study has shown that the existence of a nonconvexity can be inferred from the effect on the total value of land due to mixing two interacting uses as opposed to keeping them separate. An hedonic equation was estimated using data on single6The estimates of per unit land prices from this regression were much lower than the prevailing market prices. For example, the price per square foot of a single-family lot in Foster City was estimated to be $1.30. At this price the average lot of 6000 square feet would cost $7800. Developers report that the actual price of unimproved lots in Foster City is more than four times that figure. The problem of low estimates on lot size variables has plagued earlier empirical studies on single-family homes. Some of Reuter’s estimates on lot size variables were actually negative [19]. Those of Peterson [ 151and Stull [21] were positive. but very low.
180
THEODORE
M. CRONE
family and multifamily dwellings in Foster City, California. Zoning for multifamily structures proved to have a negative effect on the value of nearby property, but the effect on total land prices due to mixing the two uses was not significantly different from zero. The results indicate that economic efficiency does not require the separation of these two conflicting land uses by zoning. If there are no arguments for separating different types of residential land use from the point of view of efficiency, there may be an argument from the point of view of equity. In the United States, zoning has been used principally to protect the single-family home from encroachment by other land uses.This study did find evidence of a negative effect from multifamily homes. Therefore, in a dynamic economy the individual homeowner can suffer a loss due to the introduction of multifamily units into the neighborhood. The protection of homeowners from such a loss may provide sufficient justification for separating different types of residential land use by zoning. Presumably, the most serious external effects in the urban land market are not between different residential uses but between residential and commercial or industrial uses. The proper supplement to this study would be an attempt to determine whether the separation of these other uses through zoning can be justified from the point of view of efficiency in the allocation of land resources. APPENDIX A Unlike the situation with single-family residential property no independent source was available from which to calculate an inflation index for multifamily structures. Therefore, year to year price increases were estimated from the SREA data themselves. From a file of 2186 multifamily dwellings in Marin, San Francisco, San Mateo, and western Alameda and Contra Costa Counties, a random sample of 987 dwellings was generated. On the assumption that the rentable area of the structure has the greatest influence on the selling price, that variable was used as a proxy for the basic characteristics of the dwellings in the sample. The estimation of the inflation index was based on the following model by which prices can be adjusted to any time period.
PR = crR@n (1 + r,+JD”‘eU, where
PR = the selling price of the dwelling, R = the rentable area of the dwelling, t = the base year to which all prices are to be adjusted (in the SREA sample this was 1977),
(Al)
ECONOMIC JUSTIFICATION
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FOR ZONING
T,+; = the rate of increase (or decrease)in price between period t + i - 1 and period t + i if i > 0 or between period t + i + 1 and period t + i if i -c 0 (since all prices were adjusted to 1977, i = -2, - 1, Aa,
D,+i = a dummy variable equal to one if the period t + i occurs between the sale of the property and the base period t, and equal to zero otherwise. Thus, for the present sample all properties sold in 1975 had dummies equal to one for D,,,, and D,,,,. Properties sold in 1976 had a value of one for D ,976.Analogous dummy values were given to the properties sold after 1977. Taking natural logarithms of both sides of Eq. (Al) yields ln(PR) = lna + Pin(R) + zD,+,ln(l
+ rt+i) + U.
W)
Since three different construction quality categories were represented in the sample of 987 dwellings, three values of p were estimated depending on the quality of the structure. The results of this regression are reported in Table Al. Estimates of (1 + T,+~)are obtained by taking antilogs of the coefficients on the dummy variables as shown in Table Al. This value (1 +- I-~+;) represents the ratio of prices in year t + i to prices in the following or preceding year, depending on whether the year t + i is prior or subsequent to the base year 1977. Compounding for the years 1975 and 1979 produces the ratio of the price for each of the years in the sample with the price in 1977. TABLE Al Estimation of Increase in Price of Multifamily Dwellings 197% 1979 Dependent Variable (Natural Log of the Sale Price) Absolute value of Independent variable Coefficient t ratio Log of rentable area (average quality) Log of rentable area (good quality) Log of rentable area (luxury quality) Dummy for 1975 Dummy for 1976 Dummy for 1978 Dummy for 1979 (Constant) Note: E= = 0.36; N = 987.
0.3498 0.4126 0.4632 - 0.0665 - 0.0785 0.0382 0.0930 4.0075
(16.58) (20.10) (12.22) (0.72) (1.W (1.73) (1.71) (23.75)
182
THEODORE
M. CRONE
APPENDIX B This appendix lists the dependent variables used in the hedonic equations for single-family and multifamily dwellings. Structural Characteristics of Single-Family Dwellings Living Area. Except for the early study by Crecine et al. [6], all the hedonic price models which do not use a proxy for the structural characteristics of single-family dwellings, such as assessedvalue, have included some measure of the size of the house. Most studies have used the square feet of living area as a measure of size, and the same variable is used in this study. The SREA data also included an index of construction quality ranging from poor to luxury. This allowed different coefficients to be estimated for the living area variable according to the quality categories. Age. The age in 1979 was calculated for each house in the sample from the construction data supplied by the SREA. On the assumption that depreciation is directly related to the size of the structure, the age was weighted by the square feet of living area. Baths, Garage, and Fireplaces. Included among the independent variables in each of the regression equations were the number of full baths, half baths, garage spaces,and fireplaces. Basement. While the SREA survey reported the presence of a basement, it did not report its size. Therefore, the regression equations contained only a dummy variable indicating whether or not the structure had a full basement. Structural Characteristics of Multifamily Dwellings Rentable Area. The net rentable area for multifamily structures corresponds to the living area variable for single-family dwellings. A construction quality index was also available for the multifamily dwellings. Age. The age of multifamily dwellings was determined in the same manner as that for single-family dwellings. In the case of multifamily structures, depreciation was assumed to be proportional to the rentable area. Lot Size and Zoning Variables Lot Size. The SREA data contained information on the width and length of the lot so that a lot size variable could be calculated. In a few caseswhere the dimensions of the lot were not available in the SREA data for multifamily dwellings, the information was supplied by the San Mateo County Assessor’soffice. Zone. The proportion of the neighborhood zoned multifamily represents the proportion of land within 500 feet of the dwelling which is zoned for
ECONOMIC
JUSTIFICATION
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multifamily use. This zoning variable was weighted by the size of the lot on the assumption that neighborhood effects are capitalized in land prices. REFERENCES I. W. J. Baumol, On taxation and the control of externalities, Amer. Econ. Reu.. 62, 307-322 (1972). 2. W. J. Baumol and D. F. Bradford, Detrimental externalities and nonconvexity of the production set, Economica, 39, 160- 176 ( 1972). 3. W. J. Baumol and W. E. Oates, “The Theory of Environmental Policy,” Prentice-Hall, Englewood Cliffs, NJ. (1975). 4. R. H. Cease, The problem of social cost, J. Law and Econ., 3, l-44 (1960). 5. R. D. Cooter, How the law circumvents Starrett’s nonconvexity, J. Econ. Theory, 22, 499-504 (1980). 6. J. P. Crecine, 0. A. Davis, and J. E. Jackson, Urban property markets: Some empirical results and their implications for municipal zoning, J. tiw and Econ., 10, 79-99 (1967). 7. .I. H. Dales, “Pollution, Property and Prices: An Essay in Policy-Making and Economics,” Univ. of Toronto, Toronto (1968). 8. D. M. Grether and P. Mieszkowski, The effects of nonresidential land uses on the prices of adjacent housing: Some estimates of proximity effects, J. Urban Econ., 8. l-15 (1980). 9. G. D. Jud, The effects of zoning on single-family residential property values: Charlotte. North Carolina, Lund Econ., 56, 142- 154 (1980). IO. S.-C. Kolm, “Les Non-Convexites d’Extemalite,” Rapport No. 11, CEPREMAP, Paris (1971). I I. M. M. Li and II. J. Brown, Micro-neighborhood externalities and hedonic housing prices, knd Econ., 56, 125-141 (1980). 12. S. M. Maser, W. H. Riker, and R. N. Rosett, The effects of zoning and externalities on the price of land: An empirical analysis of Monroe County, New York, J. Law and Econ., 20, III-132 (1977). 13. 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 (1978). 14. W. D. Montgomery, Markets in licenses and efficient pollution control programs, J. Con. Theov, 5, 395-418 (1972). 15. G. E. Peterson, “The Effect of Zoning Regulations on Suburban Property Values,” Working Paper 1207-24, The Urban Institute, Washington, D.C. (1973). 16. A. C. Pigou, “The Economics of Welfare,” 4th ed., Macmillan Co., London (1932). Models and Economic Forecasts,” 17. R. S. Pindyck and D. L. Rubinfeld, “Econometric McGraw-Hill, New York (1976). 18. R. Portes, The search for efficiency in the presence of externalities, in “Unfashionable Economics: Essays in Honor of Lord Balogh” (P. Streeten, Ed.) Weidenfeld and Nicolson, London (1970). 19. F. H. Reuter, Externalities in urban property markets: An empirical test of the zoning ordinance of Pittsburgh, J. Luw and Econ., 16, 3 13-349 (1973). 20. D. A. Starrett, Fundamental nonconvexities in the theory of externalities, J. Econ. Theoy, 4. 180-199 (1972). 21. W. J. Stull, Community environment, zoning, and the market value of single-family homes, J. Law and Econ., 18. 535-557 (1975). 22. H. Theil, “Principles of Econometrics,” Wiley. New York (1971).
OF URBAN
ECONOMCS
Elements
14, l68- 183 (I 983)
of an Economic Justification for Municipal Zoning THEODOREM. CRONE
Federal Reserve Bank of Philadelphia, Philadelphia, Pennsylvania 19105
Received November 18, 198I ; revised April I, 1982
The possibility that externalities will produce a nonconvexity in the social production set limits the application of both the Cease and Pigou solutions for achieving an optimal allocation of land resources.Conditions on relative land prices are derived which indicate that external effects are strong enough to introduce a nonconvexity into the production set. These conditions were not fulfilled in a sample of single-family and multifamily dwellings in Foster City, California. This does not preclude the possibility that they are fulfilled in casesof more severe external effects.
1. INTRODUCTION Since the 1967 article by Crecine et al. [6], several studies have appeared estimating the effects of municipal zoning on residential property values [8, 9, 12, 15, 19, 211. None of these studies, however, situates the discussion of zoning within the context of a general theory of externalities. The present study provides a link between the theory of optimal resource allocation with externalities and the empirical literature on zoning. In the main body of economic literature, two remedies are generally proposed for achieving a Pareto optimal allocation of resources when externalities are present in the economy. The first is to define property rights more precisely so that all activities which affect others are incorporated into the market process (the Coase solution). The second method is to tax or subsidize activities which cause external harm or benefit (the Pigou solution). However, the possibility that externalities will produce a nonconvexity in some firms’ production sets or in the social production set presents a limitation for both solutions. This limitation on the Coase and Pigou solutions provides the basis of an economic justification for municipal zoning. If the external effect of one activity upon another is localized and strong enough to create a nonconvexity in the individual or social production set, separation by zoning may be as efficient as any other method in achieving an optimal allocation of land resources. 168 0094-I l90/83 $3.00 Copyright All rights
0 1983 by Academic Press, Inc. of reproduction in any form reserved.
ECONOMIC JUSTIFICATION
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169
2. LIMITATIONS OF THE COASE AND PIGOU SOLUTIONS TO LOCALIZED EXTERNALITIES It has been suggested that a complete definition of property rights as envisioned by Coase could be achieved by issuing permits to engage in a specified level of a polluting or nuisance-causing activity [4, 71. Since these permits could be purchased and retired from use, the pollution level would become a choice variable for firms adversely affected by the pollutant. This presents a potential difficulty for a competitive market in pollution permits. Under plausible assumptions about the effect of pollution upon output, the production sets of affected firms are not convex [20].’ Proof of the existence of a competitive equilibrium, however, depends upon convexity of production sets defined over firms’ choice variables. Of course, feasible production sets can always be confined to the convex region by limiting the number of permits issued, so that an equilibrium can be assured [S]; but there is no guarantee that this equilibrium will represent a Pareto-optimal resource allocation.* Marketable pollution rights may be useful as a least-cost method of maintaining a given level of pollution [14], but they offer no short-cut method of arriving at the optimal level. Under a Pigouvian tax scheme, affected firms take the pollution level as given, so that the nonconvexity associated with marketable pollution rights does not arise. Moreover, if the individual production sets are convex, proper Pigouvian taxes will sustain a Pareto-optimal allocation of resources [3, 161.The proper Pigouvian tax, however, must be calculated according to the marginal damage caused by the offending activity when all activities are undertaken at their optimal levels and in their optimal locations. With the possibility of so many externalities, no one would claim that the economy has been at such an optimum. This problem of calculating the proper Pigouvian taxes would be of little practical significance if Pigouvian-type taxes based upon actual marginal damage and iteratively adjusted would converge to the true Pigouvian taxes and lead to an optimal allocation. Such iteratively adjusted taxes would be analogous to competitive prices and would converge to their optimal values when the conditions for a unique and stable equilibrium exist. But one of the conditions for the existence of a unique competitive equilibrium is the convexity of the social production
‘A firm’s production set would be nonconvex, for example, if output diminished at an increasing rate with the pollution level up to a point of relative saturation and at a decreasing rate thereafter. *This is true not only because the restricted number of permits issued in any locality will determine the equilibrium price and pollution level, but also because the retirement of permits from use is a public good benefiting all affected firms, a classic free-rider situation.
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M. CRONE
set.3 This condition may not be fulfilled when externalities are present, for sufficiently strong externalities can themselves generate a nonconvexity in the social production set [ 1, 2, 10, 181.Given a nonconvex social production set and any normal social welfare indifference map, an iterative adjustment of prices and, if necessary, Pigouvian-type taxes could lead to any one of several local welfare maxima. If the external effects causing the nonconvexity are localized or if they dissipate with distance, the actual equilibrium outcome will depend upon the relative location of activities and need not be the global optimum. Since the optimal location of activities will depend upon the social welfare function, the allocation of land resourcesby political decisions such as zoning may result in a higher level of community welfare than a strict dependenceupon free markets supplemented by Pigouvian-type taxes. 3. LOCALIZED EXTERNALITIES, NONCONVEXITIES, AND THE PRICE OF LAND In an economy characterized by a nonconvex production set, assumptions about markets for inputs and outputs yield general conclusions about the way in which land prices vary with the mix of activities. If there is a nonconvexity in the social production set, the production frontier at some point must be convex to the origin. And if the price ratios remain fixed, the total value of output achieves a local minimum at some point other than the extremes of the production frontier.4 Using this fact, we can derive conditions on the parameters of an hedonic price model which indicate not only whether an externality is present but also whether it is severe enough to create a nonconvexity in the production set. Since land is immobile, the damage from localized externalities should be capitalized in its price. If developers have knowledge of future environmental quality or land-use mix of a neighborhood, they include this information in their offer price for land. In zoned communities, that price will be the value of the marginal product of land in its zoned use. Furthermore, at any location, resources other than land are invested up to the point where the price of the resource equals the value of its marginal product, given the presence of any external effects from neighboring uses. The model of land prices developed below is based on these general propositions and the following specific assumptions. (Pl) Two goods, x, and x1, are produced in a single location (S) which, in the absenceof externalities, is more productive for each of the two goods than other available locations in the community. ‘This condition is not required to show that proper Pigouvian taxes will sustain an optimal allocation of resources. All that is required for that proof is the convexity of each firm’s production set. 40f course, total value could reach local minima at the extremes of the frontier as well.
ECONOMIC
JUSTIFICATION
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171
(P2) The production of at least one of the goods creates a negative externality in the production of the other. (P3) Production inputs at S include land, of fixed supply AT and another resource, N. (P4) Prices for the outputs, x, and x2, p, and p2, respectively, are exogenous and fixed. This assumption is reasonable if we think of x, and x2 as being sold on a foreign market or in a larger market which includes many other producers, and where prices are determined competitively. (P5) The price of the input N, p,, is exogenous and remains fixed for the same basic reasons that p, and jiz are fixed. (P6) Not only is the total supply of land in S fixed, but the relative supply of land in S for the production of x, and x2 is determined by zoning regulations, so that the price of land used to produce x, (p,,) may differ from the price of land used to produce x2 (pa2). Without these zoning restrictions, of course, competition would eliminate the price differential. Assumptions Pl, P2, and P3 imply production functions for x, and x2 as follows:
XI =f(4
N,P,)
and x2
=g(A,,4, x,>.
A, represents the amount of land used for x,, and A, the amount used for x2. Likewise, N, and N2 represent the amounts of the resource N used for x, and x2. Assumption P2 on the existence of a negative externality implies that af/ax, -C0 and/or ag/ax, c 0. If all land is employed in the production of one of the two goods, A= A, + A,, and since the total amount of land is fixed, units can be resealed so that A= 1. If the proportion of land used in production of x, is designated by p, total land inputs for each good will be A, = P
and
A, = 1 - p.
If the resource-land ratios for x, and x2 are n, and n2, respectively, then total inputs of N for each good are
N, =
pn,
and
N2 = (1 - p)n,.
The production functions become Xl =f{p,
Pn,, g[(l
- Ph (1 -
PM)
172
THEODORE M. CRONE
and x2
=
do
-
PI,
(1
-
dn29
f[PY
Pa
Profits for each industry in location S are:
and n2
-Pa2(l
=j-Qx2
- P> -Pn(l
- Pb2.
If output markets are competitive and landowners are able to retrieve the value of any special productive capacity of their land in higher land prices, profits are zero. Setting V, and r2 equal to zero, the total value of output (V) at location S is expressed by
V = p,x, + F2x2
= P~]P + April
+ ~~~(1 - P) +PJ
- ph.
0)
Production frontiers, of course, are drawn on the assumption that the total amount of each resource is fixed. Thus far, no such restriction has been placed on N and no constraints have been placed on n, and n 2. This is realistic since the resource-land ratio is likely to vary between x, and x2. Also, n r and n 2 may vary with p if the marginal product of N depends upon the extent of the externality. Instead of confining the amount of N employed in S to a constant, consider fl to be the total amount of the resource available in S, where
N= max[pn, + (1 - pb,l. P
By definition, then, Nis a constant. Profit-maximizing firms located in S employ N up to the point where p,( af/aN,) = jF2(ag/aN,) = jr,,. Any portion of F not used could be sold on the open market at the fixed price pm.Alternatively, the value of that portion of fl which is not used can be considered a cost saving to the firms located in S. This cost saving is equal to
pn, - (1 - dn21.
K[fl-
When this saving is incorporated into (l), the total value of output becomes I/= pa,p + &pn,
+j-fJN-
+ pa2(l
- P) +&‘,(I
pn, - (1 - dn21
- PJn2
ECONOMIC
JUSTIFICATION
FOR ZONING
173
or
I/= [Pa,P +Pa2(l- dl +fv*
(2)
Since p,,m is a constant, the total value of output will reach a local minimum when the bracketed expression in (2) reaches a minimum. That expression, labeled V* below, is a direct function of p and of land prices, p,i and pa2. If there are localized externalities, these land prices also will be functions of p. Let pa, = h(p) andp,, = k(p), then v* = p * h(p)
+ (1 - p) . k(p).
(3)
Applying the first and second order conditions for a local minimum for I’* and imposing the requirement that 0 -Cp < 1, we obtain the following conditions for a nonconvex production set:
[
[h(p)-k(p)]+p
1
?$-$
(34
+$=o,
(3b) o<
b(P)
-
h(P)]-
dh/ap
ak/aP
- dk/ii’p
< 1
’
(34
The fulfillment of these conditions depends upon the relative changes in land prices as the mix of activities varies. No empirical study of zoning has attempted to estimate these relative changes. These studies have concentrated on the effect of neighboring land uses on the value of single-family homes and often have embodied the unstated assumption that external effects are unilateral from other land uses to single-family residences. The use of a binary variable to indicate the presence of an externality in some studies precludes any estimate of how prices change with the mix of activities [6, 12, 15, 191.And those studies which measure external effects by the proportion of land devoted to non-single-family uses (p in our model) have not attempted to estimate the net effect on land prices from the close proximity of different uses [21, 91. It is the net effect, however, which is crucial in determining whether separation by zoning is necessary or appropriate in an economy characterized by externalities. In order to determine whether conditions (3a) and (3~) are satisfied for some value of p, the functions h(p) and k(p) must be specified. Land prices
174
THEODORE
M. CRONE
will be determined: (a) by the equilibrium prices for land in the economy as a whole, which are assumed to be fixed; (b) by the special productive capacity of land in location S; (c) and if there is an externality, by p, the proportion of land used in the production of x,. The simplest assumption which embodies these three determinants of land prices is that they are linear functions of p. w
= 8, + &P,
(4)
and
w = 02+6s.
(5)
The values of 8, and 0, depend on (a) and (b), and the signs and magnitudes of the parameters 6, and 6, depend on (c). These last two parameters indicate the direction and magnitude of the external effect, which is assumed to be proportional to the amount of land used for the activity generating the externality. The linear form of h(p) and k(p) permits only one local minimum for V* in the range 0 < p < 1. A quadratic form implies a similar restriction. Only higher-degree polynomials would allow for multiple local minima.5 Given these linear specifications of h(p) and k(p) conditions (3a)-(3c) can be restated in terms of the parameters of (4) and (5). Since the second derivatives of the price functions are zero, condition (3b) becomes 6, - 6, > 0.
@a)
Solving (3a) for p and substituting that value into (3~) yields the following conditions:
e, - 8, - 6, > 0
(6b)
and 26, - s, > ‘A linear form for h(p) and k(p) minimum or maximum in the range that V* is a thud-degree polynomial maximum in the relevant range. In paper the coefficient on the square different from zero.
e2- 8,.
implies that V* is quadratic in p with 0 < p < 1. A quadratic form for h(p) in p with only one possible minimum the preliminary stages of the empirical of the zoning variable was not found
(64 only one possible and k(p) implies and one possible research for this to be significantly
ECONOMIC JUSTIFICATION
FOR ZONING
175
These conditions summarize the three essential questions in assessingthe efficiency of zoning. (1) Is the economy characterized by negative externalities which would lead to a misallocation of land resources in the absence of any government intervention in the market? In terms of the parameters of the model, a positive response to this question implies that 6, > 0 or 6, < 0. All previous empirical studies on zoning have posed the question of whether one or the other of these inequalities exists. (2) If negative externalities exist, are there also positive externalities, and is the net effect such that a separation of interacting land uses is efficient? According to the model, the net effect will be negative and a separation of activities will be efficient if 6, - 6, > 0. No empirical study thus far has sought to measure this net effect. (3) Is the net effect of the externality so severe that it produces a nonconvexity in the social production set? This implies that all three conditions, (6a), (6b), and (6c), are fulfilled. These conditions are sufficient to establish the existence of a nonconvex production set. If they are fulfilled, iteratively adjusted Pigouvian-type taxes cannot ensure the proper separation of activities, and zoning may well represent a more efficient solution to the externality problem. 4. ESTIMATION
OF AN EMPIRICAL
MODEL
Theoretically, the present analysis of localized externalities and land prices applies to any two interacting activities. Price data on parcels of underdeveloped land would offer the most direct test of whether conditions (6a) through (6~) are fulfilled. Since most sales of real property involve both land and structures, however, this direct approach was not feasible. The need to account for the value of structures in the estimations dictated that the empirical analysis of externalities be confined to differing types of residential land use. The data on commercial and industrial structures are simply too meager to estimate their value with any kind of hedonic price equation. In effect, the two goods in the foregoing analysis, x, and x2, will be represented by housing services provided by multifamily structures and by single-family structures, respectively. Although the external effects associated with other conflicting land uses (e.g., industrial and residential uses) are presumably more severe,the restriction to these two types of residential use is not a serious problem. Zoning regulations assume that multifamily units create negative externalities for nearby single-family residents. And in their empirical work, George Peterson and William Stull have estimated that the presence of multifamily structures has a negative effect on the prices of single-family homes [15, 211.
176
THEODORE
M. CRONE
Hedonic price equations have been used in the past to estimate the effects of different land uses on the prices of single-family homes. The model presented here will expand upon those former studies by including an hedonic price index for multifamily dwellings. This will allow a comparison of the effects of the land-use mix on both types of properties and a measure of the net external effect of mixing land uses. The foregoing theoretical discussion was confined to a two-good world. Therefore, the empirical model presented here is meant to apply to strictly residential locations which can include both single-family and multifamily dwellings. The model specifies that PRICE OF DWELLING = a0 + a,(LOT SIZE MF) + a,(LOT SIZE SF) + a3(ZONE MF
x
LOT SIZE MF)
+ a4 (ZONE MF
x
LOT SIZE SF)
+ CBj(MF,) J
+ C~k(sFk) + ~7 k
where LOT SIZE MF = The size of the lot if the dwelling is multifamily, LOT SIZE SF = The size of the lot if the dwelling is single-family, ZONE MF x LOT SIZE MF = Proportion of the neighborhood zoned multifamily times the lot size if the dwelling is multifamily, ZONE MF x LOT SIZE SF = Proportion of the neighborhood zoned multifamily times the lot size if the dwelling is single-family, Mq = The structural characteristics of multifamily dwellings, and SF, = The structural characteristics of single-family dwellings. The model assumesthat neighborhood externalities can affect the prices of both single-family and multifamily dwellings. It has long been assumed that the congestion, noise, lack of open space, and other factors associated with the presenceof multifamily dwellings decreasethe value of single-family homes. This model leaves open the possibility that the same factors also decreasethe value of multifamily dwellings. Some of the coefficients in this regression model can be identified with the parameters discussed in the previous section. If good 1 (x,) represents multifamily housing services and good 2 (x2) represents single-family housing services, the following coefficients in the model will correspond to
ECONOMIC
JUSTIFICATION
FOR ZONING
177
the parameters in Eqs. (4) and (5); a, = 8,;
a2 = e2;
a3 = 6,;
a4 = 6,.
Accordingly, the conditions for a nonconvex production set correspond to the following conditions on the regression coefficients: -
ff4
’ 0,
(74
lx2 - a, -
a4
’ 0,
(7b)
a3
and 2a, - a4 > a2 - a,.
(74
Data on single-family and multifamily dwellings in Foster City, California were used to estimate the parameters of the model. Foster City is a residential community located approximately 15 miles south of San Francisco. The city is laced with canals and lagoons, and the area was developed to take advantage of its proximity to San Francisco Bay, making it a preferred location for residential development. Data were made available by the Society of Real Estate Appraisers (SREA) on the sale price, lot size, and structural characteristics of 177 single-family dwellings and 24 multifamily dwellings sold in Foster City between 1975 and 1979. To account for inflation, the sale prices of singlefamily dwellings were adjusted to their 1977 levels using data supplied by Federal Home Loan Bank on the averageprice of single-family homes in the metropolitan area. To adjust the prices of multifamily dwellings to their 1977 levels an inflation index was calculated using a random sample of 987 multifamily structures in the San Francisco Bay Area (cf. Appendix A). Since the dwellings in the Foster City sample were identified by street address, it was possible to calculate for each of them the proportion of land within 500 feet zoned for multifamily use. The goal of this study differs from that of previous ones which have attempted to estimate the effect of multifamily structures on the prices of nearby single-family homes [6, 8, 12, 15, 19, 211. In the theoretical discussion above, the net effect on land prices due to the mixing of land uses proved to be the decisive factor in judging the efficiency of zoning. To measure this effect it was necessary to combine the two sets of data on single-family and multifamily dwellings. When two such data sets are combined, the assumption of homoscedasticity of the error term implied in the ordinary least squares method is likely to be violated [22]. Violation of
178
THEODORE M. CRONE
this assumption does not create a bias in the estimates of the regression coefficients, but it does reduce the efficiency of the estimations. Bartlett’s test was applied to the two housing samples [ 171,and it was determined that there was a statistically significant difference in the error variances between the two data sets. Therefore, the variables in each data set were adjusted using estimated error variances. The dependent variables used in the preliminary equations to estimate these error variances and in the final regression equation are explained in Appendix B. The estimated coefficients of the hedonic equation combining both types of housing data are presented in Table 1. The primary concern is with the coefficients on the zoning variables (proportion of neighborhood zoned TABLE 1 Hedonic Equation for Single-family and Multifamily Dwellings, Foster City Dependent Variable: (Price of Dwelling Adjusted to 1977Level) Absolute value of Independent variable Coefficient t-ratio Square feet of living area (single-family average quality) Square feet of living area (single-family good quality) Square feet of living area (single-family excellent quality) Age of dwelling x square feet of living area (single-family) Lot size (single-family) Number of full baths (single-family) Number of half baths (single-family) Number of fireplaces (single-family) Number of garage spaces(single-family) Multifamily dummy (dummy equal to one if dwelling is multifamily) Square feet of rentable area (multifamily average quality) Square feet of rentable area (multifamily I%& w&tY) Age of dwelling x square feet of rentable area (multifamily) Lot size (multifamily) Proportion of neighborhood zoned multifamily x lot size (all dwellings) Proportion of neighborhood zoned multifamily x lot size (multifamily dwellings) (Constant) Note: if2 = 0.88; N = 201.
29.38
(7.88)
28.78
(8.06)
38.03
(7.61)
-0.19
(2.18)
1.30 - 1748.06 4982.87 2906.37 2954.23 - 18111.33
(0.58) (1.79) (1.08) (0.78) (1.95)
55.00
(1.94)
56.49
(1.81)
- 2.48
(1.02)
0.27 - 3.72
(0.03) (2.11)
0.36
(0.03)
25039.29
(2.83)
(2.40)
ECONOMIC JUSTIFICATION
FOR ZONING
179
multifamily x lot size). The first of the zoning variables applies to all dwellings, and its coefficient measuresthe basic effect of multifamily zoning on the per unit price of neighboring land. The estimate indicates that this effect is substantial (- 3.72) and significantly different from zero at the 0.05 level. This result agrees with that of Peterson [ 151 and Stull [21] that multifamily structures are a source of negative externalities for other nearby dwellings. In the tradition of the previous empirical work on zoning, the implication of this result is likely to be misunderstood. This negative coefficient does not necessarily imply that efficiency in the use of land resources requires the separation of these two residential uses either by zoning or some other method. Some measure of the net effect on total land prices is needed to draw that conclusion. The coefficient on the second zoning variable in Table 1 applies only to multifamily dwellings and measures the difference between the negative effect on land prices for nearby multifamily and single-family properties. In terms of the theoretical model, this coefficient corresponds to 6, - 6,. According to condition (6a), a positive value for this coefficient implies that the net effect on land prices of mixing these two residential land uses is negative and that the uses should be separated on grounds of economic efficiency. The estimated coefficient is positive (0.36). The extremely low value of the t statistic, however, does not allow us to reject the hypothesis that the true value is zero and that there is no difference between the effect of multifamily zoning on nearby single-family and other nearby multifamily dwellings.6 5. CONCLUSION Neither the Coase nor the Pigou solution to the problem of externalities envisions the type of direct land-use regulation represented by municipal zoning. But neither of these remedies can guarantee an optimal allocation of resources when externalities are strong enough to cause a nonconvexity in the production set. In the presence of such a nonconvexity, municipal zoning may be as efficient as either of the two generally proposed solutions to externalities. Under some common assumptions about land rents, this study has shown that the existence of a nonconvexity can be inferred from the effect on the total value of land due to mixing two interacting uses as opposed to keeping them separate. An hedonic equation was estimated using data on single6The estimates of per unit land prices from this regression were much lower than the prevailing market prices. For example, the price per square foot of a single-family lot in Foster City was estimated to be $1.30. At this price the average lot of 6000 square feet would cost $7800. Developers report that the actual price of unimproved lots in Foster City is more than four times that figure. The problem of low estimates on lot size variables has plagued earlier empirical studies on single-family homes. Some of Reuter’s estimates on lot size variables were actually negative [19]. Those of Peterson [ 151and Stull [21] were positive. but very low.
180
THEODORE
M. CRONE
family and multifamily dwellings in Foster City, California. Zoning for multifamily structures proved to have a negative effect on the value of nearby property, but the effect on total land prices due to mixing the two uses was not significantly different from zero. The results indicate that economic efficiency does not require the separation of these two conflicting land uses by zoning. If there are no arguments for separating different types of residential land use from the point of view of efficiency, there may be an argument from the point of view of equity. In the United States, zoning has been used principally to protect the single-family home from encroachment by other land uses.This study did find evidence of a negative effect from multifamily homes. Therefore, in a dynamic economy the individual homeowner can suffer a loss due to the introduction of multifamily units into the neighborhood. The protection of homeowners from such a loss may provide sufficient justification for separating different types of residential land use by zoning. Presumably, the most serious external effects in the urban land market are not between different residential uses but between residential and commercial or industrial uses. The proper supplement to this study would be an attempt to determine whether the separation of these other uses through zoning can be justified from the point of view of efficiency in the allocation of land resources. APPENDIX A Unlike the situation with single-family residential property no independent source was available from which to calculate an inflation index for multifamily structures. Therefore, year to year price increases were estimated from the SREA data themselves. From a file of 2186 multifamily dwellings in Marin, San Francisco, San Mateo, and western Alameda and Contra Costa Counties, a random sample of 987 dwellings was generated. On the assumption that the rentable area of the structure has the greatest influence on the selling price, that variable was used as a proxy for the basic characteristics of the dwellings in the sample. The estimation of the inflation index was based on the following model by which prices can be adjusted to any time period.
PR = crR@n (1 + r,+JD”‘eU, where
PR = the selling price of the dwelling, R = the rentable area of the dwelling, t = the base year to which all prices are to be adjusted (in the SREA sample this was 1977),
(Al)
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FOR ZONING
T,+; = the rate of increase (or decrease)in price between period t + i - 1 and period t + i if i > 0 or between period t + i + 1 and period t + i if i -c 0 (since all prices were adjusted to 1977, i = -2, - 1, Aa,
D,+i = a dummy variable equal to one if the period t + i occurs between the sale of the property and the base period t, and equal to zero otherwise. Thus, for the present sample all properties sold in 1975 had dummies equal to one for D,,,, and D,,,,. Properties sold in 1976 had a value of one for D ,976.Analogous dummy values were given to the properties sold after 1977. Taking natural logarithms of both sides of Eq. (Al) yields ln(PR) = lna + Pin(R) + zD,+,ln(l
+ rt+i) + U.
W)
Since three different construction quality categories were represented in the sample of 987 dwellings, three values of p were estimated depending on the quality of the structure. The results of this regression are reported in Table Al. Estimates of (1 + T,+~)are obtained by taking antilogs of the coefficients on the dummy variables as shown in Table Al. This value (1 +- I-~+;) represents the ratio of prices in year t + i to prices in the following or preceding year, depending on whether the year t + i is prior or subsequent to the base year 1977. Compounding for the years 1975 and 1979 produces the ratio of the price for each of the years in the sample with the price in 1977. TABLE Al Estimation of Increase in Price of Multifamily Dwellings 197% 1979 Dependent Variable (Natural Log of the Sale Price) Absolute value of Independent variable Coefficient t ratio Log of rentable area (average quality) Log of rentable area (good quality) Log of rentable area (luxury quality) Dummy for 1975 Dummy for 1976 Dummy for 1978 Dummy for 1979 (Constant) Note: E= = 0.36; N = 987.
0.3498 0.4126 0.4632 - 0.0665 - 0.0785 0.0382 0.0930 4.0075
(16.58) (20.10) (12.22) (0.72) (1.W (1.73) (1.71) (23.75)
182
THEODORE
M. CRONE
APPENDIX B This appendix lists the dependent variables used in the hedonic equations for single-family and multifamily dwellings. Structural Characteristics of Single-Family Dwellings Living Area. Except for the early study by Crecine et al. [6], all the hedonic price models which do not use a proxy for the structural characteristics of single-family dwellings, such as assessedvalue, have included some measure of the size of the house. Most studies have used the square feet of living area as a measure of size, and the same variable is used in this study. The SREA data also included an index of construction quality ranging from poor to luxury. This allowed different coefficients to be estimated for the living area variable according to the quality categories. Age. The age in 1979 was calculated for each house in the sample from the construction data supplied by the SREA. On the assumption that depreciation is directly related to the size of the structure, the age was weighted by the square feet of living area. Baths, Garage, and Fireplaces. Included among the independent variables in each of the regression equations were the number of full baths, half baths, garage spaces,and fireplaces. Basement. While the SREA survey reported the presence of a basement, it did not report its size. Therefore, the regression equations contained only a dummy variable indicating whether or not the structure had a full basement. Structural Characteristics of Multifamily Dwellings Rentable Area. The net rentable area for multifamily structures corresponds to the living area variable for single-family dwellings. A construction quality index was also available for the multifamily dwellings. Age. The age of multifamily dwellings was determined in the same manner as that for single-family dwellings. In the case of multifamily structures, depreciation was assumed to be proportional to the rentable area. Lot Size and Zoning Variables Lot Size. The SREA data contained information on the width and length of the lot so that a lot size variable could be calculated. In a few caseswhere the dimensions of the lot were not available in the SREA data for multifamily dwellings, the information was supplied by the San Mateo County Assessor’soffice. Zone. The proportion of the neighborhood zoned multifamily represents the proportion of land within 500 feet of the dwelling which is zoned for
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FOR ZONING
183
multifamily use. This zoning variable was weighted by the size of the lot on the assumption that neighborhood effects are capitalized in land prices. REFERENCES I. W. J. Baumol, On taxation and the control of externalities, Amer. Econ. Reu.. 62, 307-322 (1972). 2. W. J. Baumol and D. F. Bradford, Detrimental externalities and nonconvexity of the production set, Economica, 39, 160- 176 ( 1972). 3. W. J. Baumol and W. E. Oates, “The Theory of Environmental Policy,” Prentice-Hall, Englewood Cliffs, NJ. (1975). 4. R. H. Cease, The problem of social cost, J. Law and Econ., 3, l-44 (1960). 5. R. D. Cooter, How the law circumvents Starrett’s nonconvexity, J. Econ. Theory, 22, 499-504 (1980). 6. J. P. Crecine, 0. A. Davis, and J. E. Jackson, Urban property markets: Some empirical results and their implications for municipal zoning, J. tiw and Econ., 10, 79-99 (1967). 7. .I. H. Dales, “Pollution, Property and Prices: An Essay in Policy-Making and Economics,” Univ. of Toronto, Toronto (1968). 8. D. M. Grether and P. Mieszkowski, The effects of nonresidential land uses on the prices of adjacent housing: Some estimates of proximity effects, J. Urban Econ., 8. l-15 (1980). 9. G. D. Jud, The effects of zoning on single-family residential property values: Charlotte. North Carolina, Lund Econ., 56, 142- 154 (1980). IO. S.-C. Kolm, “Les Non-Convexites d’Extemalite,” Rapport No. 11, CEPREMAP, Paris (1971). I I. M. M. Li and II. J. Brown, Micro-neighborhood externalities and hedonic housing prices, knd Econ., 56, 125-141 (1980). 12. S. M. Maser, W. H. Riker, and R. N. Rosett, The effects of zoning and externalities on the price of land: An empirical analysis of Monroe County, New York, J. Law and Econ., 20, III-132 (1977). 13. 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 (1978). 14. W. D. Montgomery, Markets in licenses and efficient pollution control programs, J. Con. Theov, 5, 395-418 (1972). 15. G. E. Peterson, “The Effect of Zoning Regulations on Suburban Property Values,” Working Paper 1207-24, The Urban Institute, Washington, D.C. (1973). 16. A. C. Pigou, “The Economics of Welfare,” 4th ed., Macmillan Co., London (1932). Models and Economic Forecasts,” 17. R. S. Pindyck and D. L. Rubinfeld, “Econometric McGraw-Hill, New York (1976). 18. R. Portes, The search for efficiency in the presence of externalities, in “Unfashionable Economics: Essays in Honor of Lord Balogh” (P. Streeten, Ed.) Weidenfeld and Nicolson, London (1970). 19. F. H. Reuter, Externalities in urban property markets: An empirical test of the zoning ordinance of Pittsburgh, J. Luw and Econ., 16, 3 13-349 (1973). 20. D. A. Starrett, Fundamental nonconvexities in the theory of externalities, J. Econ. Theoy, 4. 180-199 (1972). 21. W. J. Stull, Community environment, zoning, and the market value of single-family homes, J. Law and Econ., 18. 535-557 (1975). 22. H. Theil, “Principles of Econometrics,” Wiley. New York (1971).