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The effects of zoning on structure and land markets
JOURNAL
OF URBAN
ECONOMICS
The Effects
to,
27 I- 285 (198 1)
of Zoning on Structure Land Markets
RONALD E. GRIESON AND JMS
and
R. WHITE’
Department of Economics, Umversrfy of Calrfornra. Santa Crux, Calrforma 95064 Received July 23, 1979; revised May 5. 1980 Zoning takes three general forms: constraints on density. lot size, and allowable use. This paper demonstrates that the three restrictrons doffer m theu effects on structure and land markets. Density zoning is equivalent to a monopoly restriction that can be used to raise all land values. Allowable use restticttons affect land values differentially. Large-lot restrictions affect the price for housing paid by individual households drfferentially. Conndenng external&es does not alter these baste differences. The rmplications for empirical studies of zoning are discussed.
INTRODUCTION The recent literature on zoning typically chooses one, or a combination, of three general techniques to represent “zoning” without recognizing that they may have different market effects. For example, Courant [I] models large-lot zoning as a combination of density and allowable use restrictions, while Ohls er al. [6] model a pure allowable use restriction. They attribute their disparate results to differences in their assumptions about housing and land markets [ 1, 71, neglecting the possibility that their dissimilar zoning restrictions may have dissimilar market effects. This paper demonstrates that there are significant variations in the market effects of the three zoning methods. Allowable use restrictions, density restrictions, and large-lot requirements are each introduced into a model of housing and land markets, allowing a comparison of zoning techniques that is independent of modelling assumptions. The comparison is made in a model of urban land allocation developed by Grieson [2]. The three general zoning methods can be distinguished from among the diverse tools by zoning boards to control land use. The first method is an allowable use restriction: A restriction on the quantity of land available for a certain use, for example, apartments are often restricted to specified zones in a community. The second is a density restriction whereby restrictions are imposed on the quantity (height, floorspace, or volume) of structure per unit land. The third is a minimum input requirement of which the most common ‘The authors are Professor, Umversrty Columbia University, respectively
of California
at Santa Gnu, and Ph.D candidate,
‘71 0094-l
1uO/8l/06027l-l5$02.00/0
Copvnght d 191 I by Acadermc Press, Inc All rrgbts of rrproducrwn m uy form resewed
272
GRIESON
AND
WHITE
form is large-lot zoning. Most zoning regulations are combinations of these three general restrictions. The analysis initially abstracts from externalities. When externalities are included, the conclusion about variations between zoning methods is unchanged. Adding externalities, which are omitted in much of the literature on the metropolitan-wide effects of zoning, alters some of the individual market effects, particularly of allowable use restrictions, but the basic differences between techniques remain. The conclusion discussesthe implications of these results for understanding the incidence of zoning and for empirical studies of zoning. The recent literature on the market effects of zoning models a variety of constraints. Ohls et al. [6] analyze an allowable use restriction within a two-sector model of the urban land market. Structure markets, i.e., prices, quantities, and densities, are not considered. Courant [l] and Moss [5] analyze density restrictions, which they assert are equivalent to minimum lot size requirements, without considering externalities. Moss models a pure density constraint, meaning a constraint applied exclusively on the capitalland ratio used in the production of housing. Courant models a density constraint in this sense,but combines it with a restriction on land use, i.e., no reallocation of land is allowed between the restricted and unrestricted uses. M. White [9] imposes minimum land (large lot), or minimum land and capital consumption requirements. These models differ considerably in complexity and their assumptions about urban structure and land markets. These modelling disparities may mask the fundamental differences between the zoning restrictions. A HOUSING AND LAND MARKET MODEL WITH ZONING In this paper the three zoning techniques will be separately introduced into a simultaneous model of housing and land markets. Although housing markets are focused on here the results apply to all types of structure. The approach is general equilibrium in the sense that effects in one community or housing market will be traced through all metropolitan housing and land markets. The usual implicit assumption is made that all other prices are fixed, e.g., labor, capital. The model of land use employed is Grieson’s [2], in which a closed urban area with respect to land is initially assumed and land is homogeneous.* Housing (structure) is produced from capital which is subject to diminishing returns when added to the fixed factor, land. Competition sets long-run profits equal to zero. In equilibrium there is a uniform price of land 2Homogeneous means either identical or equal in value, in production.
273
MARKET EFFECTS OF ZONING
throughout the urban area and only structure types that can cover all costs, including land, locate in the urban area. Thus, different structure types may simultaneously locate in a homogenous land area. For now, all demand price cross elasticities are assumed to be zero; and furthermore, the urban area is not completely open with respect to all types of structure (elasticities of demand are less than infinite for at least two types of structure in the relevant price range). Demand for different structure types is given by
P, =r;(Q,)v
Ii'
i = l,...,
< 0,
n,
where P, and Q, are the price and aggregate quantity of structure type i, respectively. Density (amount of structure per unit land) is
where T, is the amount of land devoted to structure type i. The capital (construction) cost of structure on a unit of land is
c, = WA)9
WI,) ’ 03
and
CAq, > ’ 0.
The above cost functions include only the cost of density, i.e., the cost of structure on a given site, not the cost of land. In competitive equilibrium four conditions must be satisfied. Profit maximization requires that density be set such that P, = MC, = c:(qJ.
(3)
Cost minimization and a zero long-run profit condition imply that marginal cost equals the average total cost of structure including land cost, or
cT4,) =
c,(%) + Y q I
(4)
Equilibrium in the land market implies one value, V, for land across all sites and that the quantity of land demanded, 2:=,11:(y), equals the aggregate suppl~,~ T, or
y = v,
for all i,
(5)
31n this paper it is assumed that iW/aV = 0; however, as long as 03 > U/al/ 2 0, all of the results hold. As will be seen, allowing T to change with V implies that zoning would affect V less than with a fixed T, but the direction of change is unaffected.
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GRIESON
AND
WHITE
and i T,(y) = T. 1=I
(6)
Equations (l), (2), (3). (4), and (6) can be solved for the endogenous variable P,, Q,, q,, q, and V, after substitution of (5) into (4). DENSITY ZONING Zoning, implemented by the technique of a constraint on density, can be introduced into the model by placing a ceiling value on density for one type of structure, say type 1, denoted by 4,. Such a restriction implies that P, > MC,. Since there is no restriction on the type of structure built on a site, land is still allocated to the highest bidder and there remains one uniform land market clearing price, I/. With a density restriction the following system results (eliminating P,):
f(Q I1
) = C*(%)
+ v 41
f,(Q,) = C,‘(d.
i = 2,...,n,
(8)
T, = Q,/q,,
i = 2,...,n,
(11)
and T, + i
q = T.
(12)
1=2
Equation (7), the competitive zero profit condition, sets price equal to average total cost (including land) for the zoned structure. Assuming only two types of structure, where I is density zoned and 2 is a proxy for all other noncompletely open structure markets, and totally
MARKET
EFFECTS
215
OF ZONING
differentiating this system yields:
42, _ dLh'tQ,>C;'<~,) - 4,[Gt&) -.f,tQ,>][f;tQ&-4, IAl
G'td] > 0;
--dQ, _ 4,
[-Q,f;tQ,)
+ C&7,)
-fitQ,)][-&'(d]
z.
I4
,
as MR, ZMC,;
> 42 _ -KtQ, h%[C;tt&) - f,tQ,) - QJi'tQ, )] 20. -4& JAI as MR, z MC,;
6 _ [Qlf;t~,) - W,) +h(Q,)][fAQ,)T,- C;'(e)] z. 7 zjyPI as MR, z MC,;
dT, = [T,fi(Q,) - G’td][-Q,fi’(Q,) 4, I4
+ C;(%) -fitQ,)]
z.
5
as MR, z MC,; dV izjy=
q22C;'tq*)f;(Qz>[Q,~~tQ,)I4
C;(6)
+h(Q,)]
>
;O. as MR, z MC,;
s=f;(Q,)$$O;
and
2 =j;(Q,)%
$ .O
as MR, z MC, ;
where
JAI = df;(Q&'(qz) - 4:f;(Q,)[f;(Q#zMR, = Q,fi'(Q,) + f,tQ, )v
C;'(e)] ~0,
276
GRIESON AND WHITE
and
MC, = c;( 4, ).
Note that V cannot change if the type two structure market is completely open ( f;(Qz) = 0). If the type one structure market is completely open then dV/&, < 0. Here, it is assumed that both markets are partially open. Intuitively, a density constraint is equivalent to a cartel’s restriction of output in a market with many producers. Each lot produces less of the zoned structure, thereby reducing quantity in the market. Market price is bid up and profit rises until MR, = MC, (MR is marginal revenue from the market’s point of view, not that of a single producer).4 The increased profit is capitalized into land value and implies that land is bid away from other uses by use 1 until a market clearing V is reestablished. If the supply of land to all urban uses is not perfectly elastic this new Y is higher, implying an increased density for alI uses. Restriction of density past the point where MR, = MC, reduces land value. The ambiguity in the effect on land values may be removed once it is realized that a density restriction on one type of structure affects alI land values equally. It is not in any landowner’s interest to support density restrictions on any structure (either his own type of structure or others) beyond the point where MR = MC for that structure. It follows that if landowners control the zoning process, then density restrictions raise all land value.
ALLOWABLE
USE ZONING
In the above, zoning was assumed to take the form of a constraint on density. Instead, zoning could be implemented by restricting land use to certain types of structure. Such an allowable use restriction can be introduced into the basic model by imposing a constraint, 2, on the proportion of total land devoted to use one where 0 5 Z I 1. To simplify the analysis it is assumed that this is the only restriction in force: no density constraint exists. The basic model (again assuming two uses) then reduces to the 41n a completely open community (perfectly elastic supplies and demands of all goods and factors etc. except for land) maximking land values also maximizes social welfare. However, in a closed community the two are not the same; those who own land obviously favor land value maximization even if they are also occupiers of housing in the community except perhaps if they anticipate a significant increase in their consumption of housing though this effect is probably not empirically important.
277
MARKET EFFECTS OF ZONING
following:
f;(Q,) = C,‘(q,17 ad
cxq, 1 =
q I
i = 1,2,
(13)
7
(14)
+ v,
a -< FT , 91
(15)
and
i22L
(1 - F)T.
06)
q2
The competitive conditions (13) and (14), indicate price equal to marginal and average total cost for all types of structure: there are no longer any restrictions on how individual producers combine land and capital. Equation (15) sets the derived demand for type one land equal to the restricted supply (assuming the constraint is binding), while (16) guarantees that the supply of remaining land equals the demand for it. The endogenous variables are now Q,, q,, and <. Totally differentiating this system gives:
dQ, _
C;‘(q,)q,T(G’(q,)
-f;(Qz)(l
x-
-h'(Q,)~T)
dQ2 _ -G’(dq,T(G’(q,) -z--
> o
9
I4
dq, _ f,‘(Q,)q,T(C;‘(qd TE42 -= dT
- F)T)
IAl
-.h’(Q,)(l
- r)T)
IAl
-f;(QhJ(C;'(q,)
-
f;(Q, IIT)
> o
I4
7
--dV, _ q:C;'(q1)f;(Q,>T(C;'(q,) -.h'(Q2)(1 - +")
IAl
--w _ -422C;‘(q2)f;(Q2)T(C;‘(q,) -f;(Q,W) d?
JAI
>o
278
GRIESON AND WHITE
and
where
I4 =[C;'(d
-h'(Qdl
-W-][C;'k,)
-f;(QW']
'0.
Reducing the amount of land allowed in a certain use raises the price and therefore the density of use of that land. The supply is increased to other uses, lowering the price of that land and the density of its use. Output of the restricted use declines while the output of the other use increases. Price of the restricted use rises and price of the other use falls. The change in aggregateland value is ambiguous.5 Therefore, in a world without externalities, landowners would oppose allowable use restrictions on other types of structure since they could only lower their own land values. Since such allowable use restrictions are common in practice, the above model of use restrictions, without extemalities, lacks explanatory power. It will be shown later in this paper that adding externalities to the model makes it possible for all land values to rise. LARGE LOT ZONING The third important type of zoning is a minimum land input requirement, large lot zoning. Since a large lot requirement apparently lowers density, it has been asserted that large-lot zoning and density zoning are equivalent (Moss [5] and Courant [ 11).It is first shown that the two restrictions are not equivalent. Then the market effects of large-lot zoning are discussed. The analysis is again in terms of a pure case, i.e., all single-family houses must be on lots of minimum size but no other restrictions exist. The distinction between density and large-lot zoning is illustrated in Fig. 1, the production function for one house. It is assumed that production is homogeneous of degree one and that land is valued only as an input in housing production. 6 By definition, Q requires some minimum amount of capital, represented by the ray OA. Following Moss [5], a density restriction, Q/T 5 2j, is portrayed as the ray OB. Production is restricted to the right of this ray. By contrast, a minimum lot size requirement, at least in its most prevalent form, suburban ‘See Ohls et al. (61on this pomt. %ee J. White [8] for a proof that this assumption does not affect the results. Also, see M. White [9] for a treatment of land solely as a consumption good.
279
MARKET EFFECTS OF ZONING capita1
Lati 0
T*
FIG. I. A density restriction and a large lot requirement.
large-lot zoning, is a minimum land input per house, not a minimum land input per standardized unit of housing. The minimum lot size is shown as T*, with production constrained to the right of this point. Alternatively, a large-lot requirement is a minimum (maximum) constraint imposed on the ratio of land to households (households to land),’ Large-lot zoning entails a change in the decision facing an individual household because lots are now indivisible below T*. Because of the indivisibility, the payment for a house is equivalent to a two part tariff. In order to consume single-family housing a household is required to buy a minimum amount of land, T*. The payment for this land, T*V, is analogous to the “admission fee” element of a two part tariff. Having purchased T* land, the household is free to choose Q. The marginal cost of Q can be regarded as the “marginal price” of a two-part tariff. In what follows it is assumed that the constraint is binding, i.e., all households consume lots of size r. It is also assumed that the marginal cost of Q is constant, C”(q) = 0. This is the same as assuming that production occurs at a point like “a” in Fig. 1. ‘In general, for a large lot requirement to be bmdmg it is not suffrcrent to set a mimmum land input for each producer or building. Producers could merge to evade the constraint, e.g. apartment buildmgs facing a minimum land input could expand in scale until they are again at the unrestricted market density. Thus, large lot zoning can be bmdmg only when apphed to structure such as smgle farmly housing where scale cannot be increased Indefinitely because only one “household” is allowed to a lot
280
GRIESON AND WHITE
The large lot restriction can be written as T* = T,/N,
where q is the aggregate amount of single-family land (all of which is zoned) and N is the number of households occupying single-family housing. The model used to analyze density and allowable use constraints must, therefore, be reformulated to include N. For expository purposes 7” is temporarily fixed. The basis of the reformulation is the household budget constraint’: Y=MX+
T*V+ QC'
for Q > 0, for Q, > 0,
Y=MX+Q,P
where Y is income, X is a vector of other goods, M is the associated price vector, Q is single family housing, and Q, is the best alternative unzoned housing and P is the price of Q,. Housing is consumed in only one location so that Q, = 0 when Q > 0 and vice versa. This budget constraint implies a demand for single-family housing of the form
Q = Q(T*V, C’),
(17)
eaQ < 0.
aQ aT*vco.
The unit price of single-family housing to the ith household equals the average cost of such housing and can be written as A,= -+ ;”
I
C’.
(18)
The unit price of housing is a decreasing function of the quantity consumed. This is in contradistinction to most goods where unit price is fixed from a household’s point of view. Equilibrium in the land market requires that the number of households demanding zoned housing, N, equal the number of lots available or N = T,/T*.
(19)
Economic intuition suggests that one household’s choice of zoned versus %ee J. White [8] for more detal.
281
MARKET EFFECTS OF ZONING
unzoned housing depends on the price of both. This is a discrete choice-it is not a continuous function of the prices. N can be written as a continuous function by assuming many, nonidentical households. Then N = N(C’, T*V, I’),
E<0, W
-arvaN
< OT
(20)
$0.
Equating (19) and (20) and then differentiating gives dV T* --= dT V
-
----
aT*VN
19 -1
where
depending on whether or not the elasticity of P with respect to N is less than or equal to zero (see White [8]). The sign of the change in land price is ambiguous. It is influenced by the magnitude of the elasticity of N with respect to FV. Intuitively, increasing lot size reduces the number of lots available on a fixed amount of land and price adjusts to clear the market. If a larger lot size is relatively effective at driving households out of the zoned housing market then the price of land could fall. If few households are driven out by the more stringent constraint then the price of land wiIl be bid up until the number of households equals the now smaller number of lots. Allowing T, to vary means that all uses compete for land and pay the same price. If large-lot zoning raises what households will pay for singlefamily land then more land will be devoted to single-family housing, the price of land to all other uses rises, and all other densities rise. From the demand function it is clear that increasing P reduces Q,, the quantity of housing consumed by the ith household. Increasing P increases the “admission fee” element of price and, as shown by the budget constraint, reduces income spendable on Q or other goods. This result depends on the constant marginal cost assumption, but this assumption seemsquite plausible for much suburban large lot zoning. EXTENSIONS AND QUALIFICATIONS The results thus far indicate that a density restriction affects all land prices equally, an allowable use restriction affects land prices differentially,
282
GRIESON
AND
WHITE
and a large lot requirement affects housing prices differentially. It has been assumed that there is no substitution between uses and that there are no externalities between uses. Also, location (or transportation costs) have not been included in the models. This section relaxes these assumptions and shows that the basic differences between the restrictions are essentially unchanged. Location, in the form of the standard bid-rent framework, has been avoided for mathematical conciseness. The authors have analyzed zoning constraints in a bid rent framework elsewhere, see [S], with results identical to those in this paper. Briefly, the constraints shift the structure and land bid-rent curves of the restricted use in the same direction that restricted structure and land price are changed herein. Other markets then adjust as they do in this paper. Allowing uses to be partial substitutes does not change the fundamental differences between the three restrictions. Substitution does make increased land prices more likely. Zoning always makes consumers of the restricted use worse off (without externalities), therefore increasing demand for substitutes and their derived demand for land.’ Relaxing the no externalities assumption expands the explanatory power of the models. The results so far suggest monopoly gain as one possible motive for density restrictions and allowable use restrictions on one’s own structure type. Depending on parameters values, large-lot restrictions could also be used to force up land values. Monopoly gain does not explain landowner’s support for allowable use restrictions on other structure types and the zoning behavior of completely open communities, e.g., small suburbs. Externalities give an explanation for such zoning. Suppose, for example, that apartments impose a negative externality, fiscal or physical. on single family housing in a partially open community. An allowable use restriction on apartments, shown in Fig. 2, reduces the supply of apartment land from T to ?T. Without externalities the new values of apartment and single-family land are V’ and V”, respectively, rather than V. If the zoning reduces an externality then the demand for single-family land might increase to D& raising, rather than lowering, the price of single-family land. Thus, in the presence of externalities. single-family landowners might support prohibitions on other uses. Of course. even if single-family land values do not rise, the increase in consumer surplus might ‘For example. in the case of density zonmg, substitution imphes that dq*/dij, -C 0, dT, /di, > 0, and dT,/dij, < 0, not equal to zero as previously, when MR, = MC,. The signs of dqz/dij,, dV/dq,. and dP,,/dij, are ambiguous when MR, = MC,. The reason is that as density, and thus the supply to type 1 structures, is restricted, the substitution effect increases demand for type 2 structures. The increasing type 2 demand means that type 1 may not be able to bid land away from type 2 even though t?ipe I’s willingness to pay IS increasing
MARKET
EFFECTS
283
OF ZONING
Land Price for Apartment Land
0
CT
Ta
T = Total Land
FIG. 2. Land market with externalities
justify an allowable use restriction on efficiency or social maximization grounds. Figure 2 demonstrates that allowable use restrictions which raise the unrestricted use’s land values, due to a negative externality reduction from the restricted use in this case, unambiguously raise all land values.” In contrast, Ohls, er al. [6] conclude that the effect is ambiguous. They assume an initial allocation where FT land is devoted to apartments and a price differential exists between apartment land and single-family land, perhaps because of existing zoning. Reducing apartment land below tT might raise single-family land values, the externality effect, but lower aggregate land values since high-value apartment land is switched to lower-value singlefamily use. In this paper, the level of an existing constraint is not changed; instead, equilibrium without any zoning (in which all land values are equal) is compared to an equilibrium after zoning. In this case there is no ambiguity- all land values increase. Both approaches are useful: the OhIs et al. analysis probably explains much voter behavior, while the approach herein explains the long-run incidence of zoning. The earlier results for density and allowable use restrictions are basically unchanged by consideration of externalities (again, assuming a partially closed community). An increase in land values is more likely, however. If either restriction reduces a negative externality or creates a positive one then demand for all structure types is increased. A completely open community, which faces infinitely elastic demands for its structures, in a metropolitan area is now considered. No zoning policy by “If the unrestricted use creates negative externalities for the restricted use the effect on the restricted use itself will clearly be ambiguous. This is not the case which we or the literature discuss as it is not considered to occur with a very great frequency.
284
GRIESON AND WHITE
such a community can affect the metropolitan-wide market price of structure. The local price of housing can, however, be influenced by the control of externalities. If the control of externalities raises the prices of some structure types (say group A), then their associated land prices also rise. Structure types whose prices are unchanged by externality control (say group B) would be unable to compete for land and, in the long run, would be forced out of the community. If a significant number of small communities behave this way then the market price of all structure types would be increased (group B because of the market wide supply reduction). In the long run, communities could choose the type of structures they want and, if they choose by land price offers, a new equilibrium would be established with communities segregated by use and higher prices for all uses. If type B structures simultaneously receive a positive externality from the presence of type A, mixed use may be maintained but at elevated structure prices and land values. This assumes that land is homogeneous and that there are no spillovers between perfectly competitive communities.” CONCLUSIONS This paper shows that the market effects of zoning depend upon the zoning technique employed. Density zoning is analogous to a monopoly restriction on quantity and affects land values equally across all uses. Allowable use zoning also restricts quantity but affects land values differentially across uses. Large-lot zoning is equivalent to a two-part tariff; it affects land values equally across uses but leads to a unit housing price differential across households. The paper specifies conditions under which the directions of these market effects are unambiguous. These conclusions are derived from an analysis of pure cases.In practice, zoning is often more complex. The three general restrictions are frequently combined, e.g., suburban zoning that prohibits all uses except single-family residences and imposes lot size requirements. Also, all parcels of a given structure type are not zoned equally, e.g., lot size requirements vary across single-family houses. This paper does not examine the many combinations that are possible. It suggests that models of the market effects and incidences of individual zoning policies must take account of the specific mix of restrictions employed. Such models may be constructed from combinations of the pure casespresented herein.
” Spillovers do not change the results. If community A ‘s zoning reduces negative spillovers on community B, then single-family land values would be raised in both communities and apartment structure would be driven out of both communities by the higher price of land.
MARKET EFFECTS OF ZONING
285
REFERENCES I. P. N. Courant, On the effect of fiscal zoning on land and housing values, Note, J. Urban Em., 3, 88-94 (Jan. 76).
2. R. E. Grieson, The economics of property taxes and land values: the elasticity of supply of structures, J. Lirbun Econ., 1, 367-381 (Oct. 74). 3. R. E. Grieson, On the possibility and optimahty of positive rent gradients in the presence of zoning, J. Urban Econ. 1981. 4. E. S. Mills and W E Oates, “Fiscal, Earmng and Land Use Controls,” Lexington Books, Lexington, Mass. (1975). 5. W. G. Moss, Large lot zoning, property taxes, and metropohtan area, J. Urban Econ. 4, 408-427 (Oct.77) 6 J. C Ohls, R. C. Weisberg, and M. J. White, The effect of zoning on land value, J Urban Econ , 1. 428-444 (Oct. 74). 7. J. C. Ohls, R. C. We&berg, and M. J. White, Welfare effects in alternative models of zomng, Note, J Urban Econ., 3, 95-95 (Jan. 76). 8. J. R. White, “The Market Effects of Zoning,” unpubhshed Ph.D. thesis, Columbia Umversity, 1981. 9. M. J. Whne, The effect of zoning on the size of metropohtan areas, J Urban hon., 2, 270-290 (Oct. 75)
OF URBAN
ECONOMICS
The Effects
to,
27 I- 285 (198 1)
of Zoning on Structure Land Markets
RONALD E. GRIESON AND JMS
and
R. WHITE’
Department of Economics, Umversrfy of Calrfornra. Santa Crux, Calrforma 95064 Received July 23, 1979; revised May 5. 1980 Zoning takes three general forms: constraints on density. lot size, and allowable use. This paper demonstrates that the three restrictrons doffer m theu effects on structure and land markets. Density zoning is equivalent to a monopoly restriction that can be used to raise all land values. Allowable use restticttons affect land values differentially. Large-lot restrictions affect the price for housing paid by individual households drfferentially. Conndenng external&es does not alter these baste differences. The rmplications for empirical studies of zoning are discussed.
INTRODUCTION The recent literature on zoning typically chooses one, or a combination, of three general techniques to represent “zoning” without recognizing that they may have different market effects. For example, Courant [I] models large-lot zoning as a combination of density and allowable use restrictions, while Ohls er al. [6] model a pure allowable use restriction. They attribute their disparate results to differences in their assumptions about housing and land markets [ 1, 71, neglecting the possibility that their dissimilar zoning restrictions may have dissimilar market effects. This paper demonstrates that there are significant variations in the market effects of the three zoning methods. Allowable use restrictions, density restrictions, and large-lot requirements are each introduced into a model of housing and land markets, allowing a comparison of zoning techniques that is independent of modelling assumptions. The comparison is made in a model of urban land allocation developed by Grieson [2]. The three general zoning methods can be distinguished from among the diverse tools by zoning boards to control land use. The first method is an allowable use restriction: A restriction on the quantity of land available for a certain use, for example, apartments are often restricted to specified zones in a community. The second is a density restriction whereby restrictions are imposed on the quantity (height, floorspace, or volume) of structure per unit land. The third is a minimum input requirement of which the most common ‘The authors are Professor, Umversrty Columbia University, respectively
of California
at Santa Gnu, and Ph.D candidate,
‘71 0094-l
1uO/8l/06027l-l5$02.00/0
Copvnght d 191 I by Acadermc Press, Inc All rrgbts of rrproducrwn m uy form resewed
272
GRIESON
AND
WHITE
form is large-lot zoning. Most zoning regulations are combinations of these three general restrictions. The analysis initially abstracts from externalities. When externalities are included, the conclusion about variations between zoning methods is unchanged. Adding externalities, which are omitted in much of the literature on the metropolitan-wide effects of zoning, alters some of the individual market effects, particularly of allowable use restrictions, but the basic differences between techniques remain. The conclusion discussesthe implications of these results for understanding the incidence of zoning and for empirical studies of zoning. The recent literature on the market effects of zoning models a variety of constraints. Ohls et al. [6] analyze an allowable use restriction within a two-sector model of the urban land market. Structure markets, i.e., prices, quantities, and densities, are not considered. Courant [l] and Moss [5] analyze density restrictions, which they assert are equivalent to minimum lot size requirements, without considering externalities. Moss models a pure density constraint, meaning a constraint applied exclusively on the capitalland ratio used in the production of housing. Courant models a density constraint in this sense,but combines it with a restriction on land use, i.e., no reallocation of land is allowed between the restricted and unrestricted uses. M. White [9] imposes minimum land (large lot), or minimum land and capital consumption requirements. These models differ considerably in complexity and their assumptions about urban structure and land markets. These modelling disparities may mask the fundamental differences between the zoning restrictions. A HOUSING AND LAND MARKET MODEL WITH ZONING In this paper the three zoning techniques will be separately introduced into a simultaneous model of housing and land markets. Although housing markets are focused on here the results apply to all types of structure. The approach is general equilibrium in the sense that effects in one community or housing market will be traced through all metropolitan housing and land markets. The usual implicit assumption is made that all other prices are fixed, e.g., labor, capital. The model of land use employed is Grieson’s [2], in which a closed urban area with respect to land is initially assumed and land is homogeneous.* Housing (structure) is produced from capital which is subject to diminishing returns when added to the fixed factor, land. Competition sets long-run profits equal to zero. In equilibrium there is a uniform price of land 2Homogeneous means either identical or equal in value, in production.
273
MARKET EFFECTS OF ZONING
throughout the urban area and only structure types that can cover all costs, including land, locate in the urban area. Thus, different structure types may simultaneously locate in a homogenous land area. For now, all demand price cross elasticities are assumed to be zero; and furthermore, the urban area is not completely open with respect to all types of structure (elasticities of demand are less than infinite for at least two types of structure in the relevant price range). Demand for different structure types is given by
P, =r;(Q,)v
Ii'
i = l,...,
< 0,
n,
where P, and Q, are the price and aggregate quantity of structure type i, respectively. Density (amount of structure per unit land) is
where T, is the amount of land devoted to structure type i. The capital (construction) cost of structure on a unit of land is
c, = WA)9
WI,) ’ 03
and
CAq, > ’ 0.
The above cost functions include only the cost of density, i.e., the cost of structure on a given site, not the cost of land. In competitive equilibrium four conditions must be satisfied. Profit maximization requires that density be set such that P, = MC, = c:(qJ.
(3)
Cost minimization and a zero long-run profit condition imply that marginal cost equals the average total cost of structure including land cost, or
cT4,) =
c,(%) + Y q I
(4)
Equilibrium in the land market implies one value, V, for land across all sites and that the quantity of land demanded, 2:=,11:(y), equals the aggregate suppl~,~ T, or
y = v,
for all i,
(5)
31n this paper it is assumed that iW/aV = 0; however, as long as 03 > U/al/ 2 0, all of the results hold. As will be seen, allowing T to change with V implies that zoning would affect V less than with a fixed T, but the direction of change is unaffected.
274
GRIESON
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and i T,(y) = T. 1=I
(6)
Equations (l), (2), (3). (4), and (6) can be solved for the endogenous variable P,, Q,, q,, q, and V, after substitution of (5) into (4). DENSITY ZONING Zoning, implemented by the technique of a constraint on density, can be introduced into the model by placing a ceiling value on density for one type of structure, say type 1, denoted by 4,. Such a restriction implies that P, > MC,. Since there is no restriction on the type of structure built on a site, land is still allocated to the highest bidder and there remains one uniform land market clearing price, I/. With a density restriction the following system results (eliminating P,):
f(Q I1
) = C*(%)
+ v 41
f,(Q,) = C,‘(d.
i = 2,...,n,
(8)
T, = Q,/q,,
i = 2,...,n,
(11)
and T, + i
q = T.
(12)
1=2
Equation (7), the competitive zero profit condition, sets price equal to average total cost (including land) for the zoned structure. Assuming only two types of structure, where I is density zoned and 2 is a proxy for all other noncompletely open structure markets, and totally
MARKET
EFFECTS
215
OF ZONING
differentiating this system yields:
42, _ dLh'tQ,>C;'<~,) - 4,[Gt&) -.f,tQ,>][f;tQ&-4, IAl
G'td] > 0;
--dQ, _ 4,
[-Q,f;tQ,)
+ C&7,)
-fitQ,)][-&'(d]
z.
I4
,
as MR, ZMC,;
> 42 _ -KtQ, h%[C;tt&) - f,tQ,) - QJi'tQ, )] 20. -4& JAI as MR, z MC,;
6 _ [Qlf;t~,) - W,) +h(Q,)][fAQ,)T,- C;'(e)] z. 7 zjyPI as MR, z MC,;
dT, = [T,fi(Q,) - G’td][-Q,fi’(Q,) 4, I4
+ C;(%) -fitQ,)]
z.
5
as MR, z MC,; dV izjy=
q22C;'tq*)f;(Qz>[Q,~~tQ,)I4
C;(6)
+h(Q,)]
>
;O. as MR, z MC,;
s=f;(Q,)$$O;
and
2 =j;(Q,)%
$ .O
as MR, z MC, ;
where
JAI = df;(Q&'(qz) - 4:f;(Q,)[f;(Q#zMR, = Q,fi'(Q,) + f,tQ, )v
C;'(e)] ~0,
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GRIESON AND WHITE
and
MC, = c;( 4, ).
Note that V cannot change if the type two structure market is completely open ( f;(Qz) = 0). If the type one structure market is completely open then dV/&, < 0. Here, it is assumed that both markets are partially open. Intuitively, a density constraint is equivalent to a cartel’s restriction of output in a market with many producers. Each lot produces less of the zoned structure, thereby reducing quantity in the market. Market price is bid up and profit rises until MR, = MC, (MR is marginal revenue from the market’s point of view, not that of a single producer).4 The increased profit is capitalized into land value and implies that land is bid away from other uses by use 1 until a market clearing V is reestablished. If the supply of land to all urban uses is not perfectly elastic this new Y is higher, implying an increased density for alI uses. Restriction of density past the point where MR, = MC, reduces land value. The ambiguity in the effect on land values may be removed once it is realized that a density restriction on one type of structure affects alI land values equally. It is not in any landowner’s interest to support density restrictions on any structure (either his own type of structure or others) beyond the point where MR = MC for that structure. It follows that if landowners control the zoning process, then density restrictions raise all land value.
ALLOWABLE
USE ZONING
In the above, zoning was assumed to take the form of a constraint on density. Instead, zoning could be implemented by restricting land use to certain types of structure. Such an allowable use restriction can be introduced into the basic model by imposing a constraint, 2, on the proportion of total land devoted to use one where 0 5 Z I 1. To simplify the analysis it is assumed that this is the only restriction in force: no density constraint exists. The basic model (again assuming two uses) then reduces to the 41n a completely open community (perfectly elastic supplies and demands of all goods and factors etc. except for land) maximking land values also maximizes social welfare. However, in a closed community the two are not the same; those who own land obviously favor land value maximization even if they are also occupiers of housing in the community except perhaps if they anticipate a significant increase in their consumption of housing though this effect is probably not empirically important.
277
MARKET EFFECTS OF ZONING
following:
f;(Q,) = C,‘(q,17 ad
cxq, 1 =
q I
i = 1,2,
(13)
7
(14)
+ v,
a -< FT , 91
(15)
and
i22L
(1 - F)T.
06)
q2
The competitive conditions (13) and (14), indicate price equal to marginal and average total cost for all types of structure: there are no longer any restrictions on how individual producers combine land and capital. Equation (15) sets the derived demand for type one land equal to the restricted supply (assuming the constraint is binding), while (16) guarantees that the supply of remaining land equals the demand for it. The endogenous variables are now Q,, q,, and <. Totally differentiating this system gives:
dQ, _
C;‘(q,)q,T(G’(q,)
-f;(Qz)(l
x-
-h'(Q,)~T)
dQ2 _ -G’(dq,T(G’(q,) -z--
> o
9
I4
dq, _ f,‘(Q,)q,T(C;‘(qd TE42 -= dT
- F)T)
IAl
-.h’(Q,)(l
- r)T)
IAl
-f;(QhJ(C;'(q,)
-
f;(Q, IIT)
> o
I4
7
--dV, _ q:C;'(q1)f;(Q,>T(C;'(q,) -.h'(Q2)(1 - +")
IAl
--w _ -422C;‘(q2)f;(Q2)T(C;‘(q,) -f;(Q,W) d?
JAI
>o
278
GRIESON AND WHITE
and
where
I4 =[C;'(d
-h'(Qdl
-W-][C;'k,)
-f;(QW']
'0.
Reducing the amount of land allowed in a certain use raises the price and therefore the density of use of that land. The supply is increased to other uses, lowering the price of that land and the density of its use. Output of the restricted use declines while the output of the other use increases. Price of the restricted use rises and price of the other use falls. The change in aggregateland value is ambiguous.5 Therefore, in a world without externalities, landowners would oppose allowable use restrictions on other types of structure since they could only lower their own land values. Since such allowable use restrictions are common in practice, the above model of use restrictions, without extemalities, lacks explanatory power. It will be shown later in this paper that adding externalities to the model makes it possible for all land values to rise. LARGE LOT ZONING The third important type of zoning is a minimum land input requirement, large lot zoning. Since a large lot requirement apparently lowers density, it has been asserted that large-lot zoning and density zoning are equivalent (Moss [5] and Courant [ 11).It is first shown that the two restrictions are not equivalent. Then the market effects of large-lot zoning are discussed. The analysis is again in terms of a pure case, i.e., all single-family houses must be on lots of minimum size but no other restrictions exist. The distinction between density and large-lot zoning is illustrated in Fig. 1, the production function for one house. It is assumed that production is homogeneous of degree one and that land is valued only as an input in housing production. 6 By definition, Q requires some minimum amount of capital, represented by the ray OA. Following Moss [5], a density restriction, Q/T 5 2j, is portrayed as the ray OB. Production is restricted to the right of this ray. By contrast, a minimum lot size requirement, at least in its most prevalent form, suburban ‘See Ohls et al. (61on this pomt. %ee J. White [8] for a proof that this assumption does not affect the results. Also, see M. White [9] for a treatment of land solely as a consumption good.
279
MARKET EFFECTS OF ZONING capita1
Lati 0
T*
FIG. I. A density restriction and a large lot requirement.
large-lot zoning, is a minimum land input per house, not a minimum land input per standardized unit of housing. The minimum lot size is shown as T*, with production constrained to the right of this point. Alternatively, a large-lot requirement is a minimum (maximum) constraint imposed on the ratio of land to households (households to land),’ Large-lot zoning entails a change in the decision facing an individual household because lots are now indivisible below T*. Because of the indivisibility, the payment for a house is equivalent to a two part tariff. In order to consume single-family housing a household is required to buy a minimum amount of land, T*. The payment for this land, T*V, is analogous to the “admission fee” element of a two part tariff. Having purchased T* land, the household is free to choose Q. The marginal cost of Q can be regarded as the “marginal price” of a two-part tariff. In what follows it is assumed that the constraint is binding, i.e., all households consume lots of size r. It is also assumed that the marginal cost of Q is constant, C”(q) = 0. This is the same as assuming that production occurs at a point like “a” in Fig. 1. ‘In general, for a large lot requirement to be bmdmg it is not suffrcrent to set a mimmum land input for each producer or building. Producers could merge to evade the constraint, e.g. apartment buildmgs facing a minimum land input could expand in scale until they are again at the unrestricted market density. Thus, large lot zoning can be bmdmg only when apphed to structure such as smgle farmly housing where scale cannot be increased Indefinitely because only one “household” is allowed to a lot
280
GRIESON AND WHITE
The large lot restriction can be written as T* = T,/N,
where q is the aggregate amount of single-family land (all of which is zoned) and N is the number of households occupying single-family housing. The model used to analyze density and allowable use constraints must, therefore, be reformulated to include N. For expository purposes 7” is temporarily fixed. The basis of the reformulation is the household budget constraint’: Y=MX+
T*V+ QC'
for Q > 0, for Q, > 0,
Y=MX+Q,P
where Y is income, X is a vector of other goods, M is the associated price vector, Q is single family housing, and Q, is the best alternative unzoned housing and P is the price of Q,. Housing is consumed in only one location so that Q, = 0 when Q > 0 and vice versa. This budget constraint implies a demand for single-family housing of the form
Q = Q(T*V, C’),
(17)
eaQ < 0.
aQ aT*vco.
The unit price of single-family housing to the ith household equals the average cost of such housing and can be written as A,= -+ ;”
I
C’.
(18)
The unit price of housing is a decreasing function of the quantity consumed. This is in contradistinction to most goods where unit price is fixed from a household’s point of view. Equilibrium in the land market requires that the number of households demanding zoned housing, N, equal the number of lots available or N = T,/T*.
(19)
Economic intuition suggests that one household’s choice of zoned versus %ee J. White [8] for more detal.
281
MARKET EFFECTS OF ZONING
unzoned housing depends on the price of both. This is a discrete choice-it is not a continuous function of the prices. N can be written as a continuous function by assuming many, nonidentical households. Then N = N(C’, T*V, I’),
E<0, W
-arvaN
< OT
(20)
$0.
Equating (19) and (20) and then differentiating gives dV T* --= dT V
-
----
aT*VN
19 -1
where
depending on whether or not the elasticity of P with respect to N is less than or equal to zero (see White [8]). The sign of the change in land price is ambiguous. It is influenced by the magnitude of the elasticity of N with respect to FV. Intuitively, increasing lot size reduces the number of lots available on a fixed amount of land and price adjusts to clear the market. If a larger lot size is relatively effective at driving households out of the zoned housing market then the price of land could fall. If few households are driven out by the more stringent constraint then the price of land wiIl be bid up until the number of households equals the now smaller number of lots. Allowing T, to vary means that all uses compete for land and pay the same price. If large-lot zoning raises what households will pay for singlefamily land then more land will be devoted to single-family housing, the price of land to all other uses rises, and all other densities rise. From the demand function it is clear that increasing P reduces Q,, the quantity of housing consumed by the ith household. Increasing P increases the “admission fee” element of price and, as shown by the budget constraint, reduces income spendable on Q or other goods. This result depends on the constant marginal cost assumption, but this assumption seemsquite plausible for much suburban large lot zoning. EXTENSIONS AND QUALIFICATIONS The results thus far indicate that a density restriction affects all land prices equally, an allowable use restriction affects land prices differentially,
282
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AND
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and a large lot requirement affects housing prices differentially. It has been assumed that there is no substitution between uses and that there are no externalities between uses. Also, location (or transportation costs) have not been included in the models. This section relaxes these assumptions and shows that the basic differences between the restrictions are essentially unchanged. Location, in the form of the standard bid-rent framework, has been avoided for mathematical conciseness. The authors have analyzed zoning constraints in a bid rent framework elsewhere, see [S], with results identical to those in this paper. Briefly, the constraints shift the structure and land bid-rent curves of the restricted use in the same direction that restricted structure and land price are changed herein. Other markets then adjust as they do in this paper. Allowing uses to be partial substitutes does not change the fundamental differences between the three restrictions. Substitution does make increased land prices more likely. Zoning always makes consumers of the restricted use worse off (without externalities), therefore increasing demand for substitutes and their derived demand for land.’ Relaxing the no externalities assumption expands the explanatory power of the models. The results so far suggest monopoly gain as one possible motive for density restrictions and allowable use restrictions on one’s own structure type. Depending on parameters values, large-lot restrictions could also be used to force up land values. Monopoly gain does not explain landowner’s support for allowable use restrictions on other structure types and the zoning behavior of completely open communities, e.g., small suburbs. Externalities give an explanation for such zoning. Suppose, for example, that apartments impose a negative externality, fiscal or physical. on single family housing in a partially open community. An allowable use restriction on apartments, shown in Fig. 2, reduces the supply of apartment land from T to ?T. Without externalities the new values of apartment and single-family land are V’ and V”, respectively, rather than V. If the zoning reduces an externality then the demand for single-family land might increase to D& raising, rather than lowering, the price of single-family land. Thus, in the presence of externalities. single-family landowners might support prohibitions on other uses. Of course. even if single-family land values do not rise, the increase in consumer surplus might ‘For example. in the case of density zonmg, substitution imphes that dq*/dij, -C 0, dT, /di, > 0, and dT,/dij, < 0, not equal to zero as previously, when MR, = MC,. The signs of dqz/dij,, dV/dq,. and dP,,/dij, are ambiguous when MR, = MC,. The reason is that as density, and thus the supply to type 1 structures, is restricted, the substitution effect increases demand for type 2 structures. The increasing type 2 demand means that type 1 may not be able to bid land away from type 2 even though t?ipe I’s willingness to pay IS increasing
MARKET
EFFECTS
283
OF ZONING
Land Price for Apartment Land
0
CT
Ta
T = Total Land
FIG. 2. Land market with externalities
justify an allowable use restriction on efficiency or social maximization grounds. Figure 2 demonstrates that allowable use restrictions which raise the unrestricted use’s land values, due to a negative externality reduction from the restricted use in this case, unambiguously raise all land values.” In contrast, Ohls, er al. [6] conclude that the effect is ambiguous. They assume an initial allocation where FT land is devoted to apartments and a price differential exists between apartment land and single-family land, perhaps because of existing zoning. Reducing apartment land below tT might raise single-family land values, the externality effect, but lower aggregate land values since high-value apartment land is switched to lower-value singlefamily use. In this paper, the level of an existing constraint is not changed; instead, equilibrium without any zoning (in which all land values are equal) is compared to an equilibrium after zoning. In this case there is no ambiguity- all land values increase. Both approaches are useful: the OhIs et al. analysis probably explains much voter behavior, while the approach herein explains the long-run incidence of zoning. The earlier results for density and allowable use restrictions are basically unchanged by consideration of externalities (again, assuming a partially closed community). An increase in land values is more likely, however. If either restriction reduces a negative externality or creates a positive one then demand for all structure types is increased. A completely open community, which faces infinitely elastic demands for its structures, in a metropolitan area is now considered. No zoning policy by “If the unrestricted use creates negative externalities for the restricted use the effect on the restricted use itself will clearly be ambiguous. This is not the case which we or the literature discuss as it is not considered to occur with a very great frequency.
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GRIESON AND WHITE
such a community can affect the metropolitan-wide market price of structure. The local price of housing can, however, be influenced by the control of externalities. If the control of externalities raises the prices of some structure types (say group A), then their associated land prices also rise. Structure types whose prices are unchanged by externality control (say group B) would be unable to compete for land and, in the long run, would be forced out of the community. If a significant number of small communities behave this way then the market price of all structure types would be increased (group B because of the market wide supply reduction). In the long run, communities could choose the type of structures they want and, if they choose by land price offers, a new equilibrium would be established with communities segregated by use and higher prices for all uses. If type B structures simultaneously receive a positive externality from the presence of type A, mixed use may be maintained but at elevated structure prices and land values. This assumes that land is homogeneous and that there are no spillovers between perfectly competitive communities.” CONCLUSIONS This paper shows that the market effects of zoning depend upon the zoning technique employed. Density zoning is analogous to a monopoly restriction on quantity and affects land values equally across all uses. Allowable use zoning also restricts quantity but affects land values differentially across uses. Large-lot zoning is equivalent to a two-part tariff; it affects land values equally across uses but leads to a unit housing price differential across households. The paper specifies conditions under which the directions of these market effects are unambiguous. These conclusions are derived from an analysis of pure cases.In practice, zoning is often more complex. The three general restrictions are frequently combined, e.g., suburban zoning that prohibits all uses except single-family residences and imposes lot size requirements. Also, all parcels of a given structure type are not zoned equally, e.g., lot size requirements vary across single-family houses. This paper does not examine the many combinations that are possible. It suggests that models of the market effects and incidences of individual zoning policies must take account of the specific mix of restrictions employed. Such models may be constructed from combinations of the pure casespresented herein.
” Spillovers do not change the results. If community A ‘s zoning reduces negative spillovers on community B, then single-family land values would be raised in both communities and apartment structure would be driven out of both communities by the higher price of land.
MARKET EFFECTS OF ZONING
285
REFERENCES I. P. N. Courant, On the effect of fiscal zoning on land and housing values, Note, J. Urban Em., 3, 88-94 (Jan. 76).
2. R. E. Grieson, The economics of property taxes and land values: the elasticity of supply of structures, J. Lirbun Econ., 1, 367-381 (Oct. 74). 3. R. E. Grieson, On the possibility and optimahty of positive rent gradients in the presence of zoning, J. Urban Econ. 1981. 4. E. S. Mills and W E Oates, “Fiscal, Earmng and Land Use Controls,” Lexington Books, Lexington, Mass. (1975). 5. W. G. Moss, Large lot zoning, property taxes, and metropohtan area, J. Urban Econ. 4, 408-427 (Oct.77) 6 J. C Ohls, R. C. Weisberg, and M. J. White, The effect of zoning on land value, J Urban Econ , 1. 428-444 (Oct. 74). 7. J. C. Ohls, R. C. We&berg, and M. J. White, Welfare effects in alternative models of zomng, Note, J Urban Econ., 3, 95-95 (Jan. 76). 8. J. R. White, “The Market Effects of Zoning,” unpubhshed Ph.D. thesis, Columbia Umversity, 1981. 9. M. J. Whne, The effect of zoning on the size of metropohtan areas, J Urban hon., 2, 270-290 (Oct. 75)