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Optimal fiscal zoning when the local government is a discriminating monopolist
Regional
Science
and Urban
Economics
22 (1992) 579-596.
North-Holland
Optimal fiscal zoning when the local government is a discriminating monopolist Thomas
J. Miceli”
The University of Connecticut, Storrs C7: USA Received January
1990, tinal version
received
December
1990
Monopoly zoning by local governments has been studied extensively since the original article by White [1975, in: E. Mills and W. Oates, eds., Fiscal zoning and land use controls (Lexington Books, Lexington, MA)]. However, little or no attention has been paid to the possibility of monopolistic discrimination under such a regime. The question is: When will zoning multiple lot sizes be feasible given that buyers with different valuations of land cannot be coerced into purchasing particular lots? The literature on imperfect price discrimination by monopolists is brought to bear on this question. It is shown that the resulting zoning strategy may or may not involve multiple lot sizes. The factors determining the optimal strategy are studied in detail, with particular emphasis on the nature of the market for undeveloped land in the community.
1. Introduction Monopoly zoning by local governments has been studied extensively by economists since the original article by White (1975). In the standard view, communities zone large lots in an attempt to extract tax revenues from entrants in excess of the cost of providing them with services. Existing residents of the community are thereby enriched at the expense of newcomers in a strategy White labels ‘fiscal squeeze zoning’.’ The exercise of monopoly power by local communities naturally suggests the possibility of ‘price’ discrimination. In the case of zoning this would take the form of setting different lot sizes for different consumers in an effort to extract a greater surplus from buyers with a higher valuation of land. While White considered this possibility, she argued, first of all, that it is not supported by the evidence. In fact, the evidence to which she refers suggests Correspondence to: Thomas J. Miceli, College of Liberal Arts and Sciences, University of Connecticut, 341 Mansfield Road, CT 06269-1063, USA. *I wish to thank several anonymous referees for comments that greatly improved this paper. Any remaining errors, of course, are my own. ‘Efforts to find empirical evidence for this sort of activity by local governments have been weakly successful. See, e.g., Hamilton (1978) Windsor (1979, chapters 4 and 5), Fischel (1980) and Rose (1989). 01660462/92/$05.00
0
1992-Elsevier
Science Publishers
B.V. All rights reserved
580
T.J. Mice/i,
Optimal fiscal
Table Minimum Minimum
zoning
1
lot sizes in New Jersey communities,
lot size (acres)
o-o.24 0.25-0.49 0.50-0.99 l-2.99 3 or more Source: Windsor
Percent
1970.
of total acreage
5.2 9.8 19.3 53.8 12.0 (1979, table 3-6, p. 58).
that communities often zone multiple lot sizes.’ In particular, out of 300 New Jersey communities in the New York metropolitan area, ‘( . .) about three quarters of the communities zoned 75% or more of their land for development in a single lot size. Most of the remainder divided their vacant land into regions set aside for two different, but similar lot sizes’ [White (1975, p. 46)]. Windsor (1979) also provides evidence on minimum lot sizes set by New Jersey communities (table l), and although the data are averages across all communities in the sample, they show considerable variation in lot size. A further impediment to discriminatory zoning noted by White is the question of how segmentation of the market can be achieved. In particular, when will a buyer ever purchase a large lot when smaller ones are available? Recently, a large literature has arisen to examine monopolistic discrimination given imperfect sorting of buyers [e.g., Sappington (1983) Cooper (1984), Maskin and Riley (1984), and Salant (1989)]. The sorting device employed in these models is self selection, whereby buyers are free to select their optimal product variety from among those offered. For example, products of varying quality may be offered according to a schedule of prices that exploits consumers’ differing marginal rates of substitution of quality for money.3 However, because the self-selection constraint permits only imperfect discrimination, it turns our that in some cases it is not optimal for the monopolist to offer multiple varieties (or, in this case, lot sizes). And, when multiple varieties are offered, they are typically characterized by allocative distortions. This paper represents the first attempt to apply the techniques of the literature on imperfect discrimination to monopoly zoning. The analysis proceeds as follows. Section 2 describes the model and assumptions. Section 3 then considers the case where land is owned by competitive developers within the community so that lots sell at their opportunity cost. The optimal ‘Note, however, that the existence of multiple lot sizes does not necessarily imply discrimination, but instead may be due to historical or other causes. %ee Phlips (1983) for further examples of imperfect sorting strategies, including sorting along both spatial and temporal dimensions. Phlips also provides descriptions of actual business practices in support of the theory.
T.J. Midi,
Optimal
fiscal zoning
581
discrimination solution is derived in this case, first when perfect discrimination (sorting) is possible (as a benchmark), and then when self-selection must be employed. Section 4 then considers the case where the undeveloped land is owned by a single developer. Again, both the perfect discrimination and the self-selection solutions are examined. Section 5 asks which scenario, competitive developers or a single developer, is more profitable for the community given its optimal zoning strategy. Finally section 6 concludes.
2. Setup of the model Consider a community with a fixed amount of undeveloped land that is owned by a single, or multiple landowner/developers. The community, however, controls development; in particular, it sets lot sizes to extract the maximum fiscal squeeze from newcomers. To be sure, communities employ other means of extracting revenue from developers, for example, subdivision exactions, including contributions of land for schools or parks [White (1979), Fischel (1985)], or the sale of transferable development rights [Mills (1980) Carpenter and Heffley (1982)]. However, the focus in this paper is exclusively on the use of zoning for this purpose. The source of the community’s monopoly power is, first, the fact that newcomers face restricted alternatives, owing to the difficulty of adjusting existing jurisdictional boundaries [Epple and Romer (1989) Henderson of new communities (19831, and costs associated with the formation [Henderson (1980), Epple et al. (1988)]. Second, the community is assumed to possess some unique feature or amenity relative to neighboring communities for which residents will pay a premium. This might take the form of superior access to the central business district, or high quality public services. Undeveloped land within the community, however, is assumed to be homogeneous [in contrast to the model in Epple et al. (1988)]. Given this setting, consider potential entrants to the community. To keep the analysis simple, their utility will take the separable form U(x, 1,e)=x + u(l, e), where x is consumption of all other goods (and the numeraire), 1 is the lot size purchased by the household, and 0 is a parameter capturing the household’s valuation of land relative to other goods. The function u(l,fI) is assumed to have the following properties:
(24 (2b)
(24
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T.J. Miceli, Optimalfiscal zoning
That is, land yields positive but diminishing marginal utiiity, and a higher 0 increases the buyer’s valuation of land, both absolutely and at the margin.4 Buyers with higher B’s are therefore referred to as ‘higher demanders’. Note finally that the public services provided by the community also enter the buyer’s utility function, but as they are fixed, they are suppressed in (1) and throughout without loss of generality. Let the budget constraint of type 0 buyers be given by y(B) = x + R(1) + T(l), where y(8) is income, R(l) is the purchase price of a lot of size 1, and T(I) is the total tax payment associated with a lot of size 1. It will be convenient to substitute the budget constraint into the buyer’s utility function in (1) to obtain U(l,O)=y(B)-R(l)-T(l)+u(l,t?).
(1’)
Suppose that a buyer of type 0 has a next-best utility level of U(O) obtainable elsewhere in the metropolitan area. Equating U(I,@ and U(O) thus implicitly defines the buyer’s maximum willingness-to-pay for land plus taxes in the community: y(B)-U(O)-R(l)-7’(l)+u(l,8)=0.
(3)
The specific forms of R(l) and T(1) in this equation depend upon the nature of developers in the community, and the tax practices of the local government. These will be discussed in turn. Considering first the community, suppose it has a fixed amount of undeveloped land, L, of homogeneous quality that it wishes to have developed. Moreover, suppose that the land is identical to undeveloped land in other communities in the metropolitan area. Given its monopoly power, the community seeks to extract rents from new residents through the property tax, which is used to finance public services. Let the cost of services per person in the community be a constant c. Thus, public services are private rather than public goods at the margin.5 For simplicity, I assume 4All of the results in the paper would hold for the general utility function U(x,I,O) if, in addition to positive marginal utilities of x and I, the following hold: (i) U,/U, is decreasing in I; (ii) UB >O, and (iii) U,/U, is increasing in 8. These correspond to conditions (2b))(2d) in the text, and are standard in the self-selection literature [Cooper (1984)]. ‘This is consistent with the model in Henderson (1980). Note that it assumes, in contrast to White (1975) that the cost of public services per household is not a function of a lot size, thereby insuring that the household’s ‘fiscal surplus’ is an increasing function of lot size (i.e., tax base). This may not be the case, if, for example, households with more school-age children tend to purchase larger lots. Windsor (1979) in fact finds this to be the case in his study of New Jersey communities in 1970, though he still finds that the fiscal surplus is generally increasing in lot size.
T.J. Miceli, Optimal fiscal
zoning
583
that the tax is assessed in proportion to the value of the lot purchased by new residents, rather than the value of the lot plus improvements as in White (1975) (if improvements are proportional to the lot size, then the analysis would be unaffected by their inclusion). The tax rate, t, is taken as fixed. For example, it may be the rate needed to balance the budget for the services provided to existing residents prior to development. In contrast, the goal of monopoly zoning is to set lot sizes to extract a fiscal transfer from new residents in excess of the cost of the services they consume. Next consider the characterization of developers in the community. Two scenarios will be considered. In the first, land is owned by competitive developers [Henderson (1985, p. 223)]. Note that this is not inconsistent with monopoly power by the community, given that land within the community is perfectly substitutable, and that the local government holds a monopoly over zoning rights. In this case, the price of land will fall to its opportunity cost, p, and the purchase price of a lot in (1’) will be pl, while taxes collected per lot will be tpl. In the second scenario, the undeveloped land in the community is owned by a single developer. The developer can therefore attempt to exercise monopoly power in selling lots as in Edelson (1975), but subject to control of lot sizes by the local government. In this case, R(I) is set by the developer to maximize profits (given I), and taxes collected per lot are tR.
3. Monopoly zoning with competitive developers In this section I consider monopoly zoning by the community with competitive landowners/developers. For simplicity, let there be a discrete number of buyer types indexed by i= 1,2,. . . , I, such that higher indices designate higher demand buyers, i.e., Bj> 0, for j> i. Also, let n, be the number of buyers of type Bi who purchase a lot (i.e., the number of lots of this type sold), where Ni is the number of potential type Bi buyers (hence, rri 5 NJ, and li is the lot size purchased. As noted, taxes collected from each of these buyers is tpl,. Thus, the community’s problem is to maximize the total fiscal surplus from new entrants, equal to Cini(tpli-cc), subject to the constraints xn,l, 5 L, and U(li,Oi) 2 U(Oi). The choice variables of the problem are the lot size for each type, li, and the number of lots of each size to zone, n,. I examine the solution to this problem under two assumptions: (a) buyers’ types (ei) are observable, and each type can be required to purchase the lot size designed for him or her (perfect discrimination);6 and
61n addition, recontracting among buyers is prohibited for the case of perfect discrimination. The fact that recontracting cannot be prevented in practice is one reason self-selection is required for discrimination in the land market.
584
T.J. Mice/i, Optimal fiscal zoning
(b) buyer’s types purchase particular
3.1.
are unobservable, lot sizes (imperfect
The perfect discrimination
and/or buyers discrimination).
cannot
be forced
to
solution
The perfect discrimination solution serves as a benchmark that will aid in motivating the imperfect discrimination solution. In examining the solution to community’s problem, first consider the choice of Zi. Since each buyer type can be required to purchase a particular lot size, then all types will be driven to their reservation utility, U(0,). That is, each type will be on his or her willingness-to-pay curve defined in (3). For the case of competitive landowners, as noted, the cost of land is pl, so we may rewrite (3) as
T = _Y(Qi) - D(di)-p/i + U(li, di),
(4)
where T represents the maximum tax payment that a type oi buyer will pay for a lot of size Ii. The slope of 71 is given by aT/aZi= v,(li, ei) --p, which is initially positive and then becomes negative for large 1. Note that one expects the intercept of T, ~(0~) - u(6,), to be negative, implying that the buyer is not willing to pay a positive amount for very small lots. Finally, given (2), aTJalj > aTJa1, for t3j > tJi, implying, as one would expect, that the optimal lot size increases in 8. Given the characteristics of T and the fact that tpl, is linearly increasing in li, to maximize tax revenues per lot the community should set li to solve T = tpl,. The resulting lot sizes for two buyer types, denoted Ii*, i = 1,2, are shown graphically in fig. 1. Note first that, given y(Qi)- D(ei) ~0, there will generally be two points where tpl, intersects T. It is clear from the graph, however, that the lot size which maximizes tpli is the larger of the two. As a result, the lot size for the higher type buyer is larger than for the lower type, and, as drawn, both are larger than the buyers’ optimal lot sizes. The latter result need not hold, however, for example, if tpl is very steep. Indeed, if tp is large, there need not be an intersection at all, in which case that buyer type could not be induced to enter the community. Turning to the optimal choice of ni, note that a fiscal surplus is realized for all Bi for whom tpl,-c > 0. Hence, if the quantity of undeveloped land allows, ni= Ni for all di that produce a fiscal surplus. That is, if CiNiljr~ L, then ni= Ni for all Bi. Suppose, however, that this condition does not hold. Consider, for example, the case of two buyer types, and suppose that N,lg < L< N,lT + N,l:. In this case, since the fiscal surplus from the higherbuyer type exceeds that from the lower-buyer type (i.e., tpl,* -c > tpl: -c), and then set n, =(L- N,14)/1: (ignoring the integer confirst set n,=N,, straint on ni). That is, first zone enough large sized lots for all
585
T.J. Miceli, Optimal fiscal zoning
Y(q)
-
Y@,)
-
Fig. 1. Perfect discrimination
solution
with competitive
developers.
of the high type buyers, and then zone the remaining land for buyers. Suppose alternatively that N,lf 2 L. In this case, all land zoned for high type buyers. Thus, even if there exist other buyer whom tplj+-c>O, they will be excluded by the community’s zoning
3.2. The imperfect discrimination
low type should be types for policy.
solution
The reason that buyers in the perfect discrimination solution must be required to purchase the lot size set for them (and why recontracting must also be prohibited) can be seen by examining fig. 1. Notice that, because higher levels of utility are associated with downward shifts of the willingnessto-pay curves, type 0, buyers would realize a higher level of utility by purchasing the lot size intended for type Oi buyers. This problem will exist whenever the intersection point between Tl and T2 falls to the left of the optimal point for O1 buyers.’ In this section, I assume that this condition holds, for if it does not, the solution in the preceding section can be implemented by the community without coercion. In order to discriminate among buyers given the above problem, the community must induce self-selection. That is, the lot size intended for any buyer type must be preferred by him or her to all other lot sizes available. A
‘This condition is satislied if y(O,)-ii is either less than y(O,)-_(O,), larger. There seems no obvious reason either to accept or reject this condition
or not too much a priori.
586
T.J. Miceli, Optimal fiscal zoning
standard result in the self-selection literature is that the optimal self-selection solution involves extracting all of the surplus from the lowest demand buyer, while higher demand buyers are left indifferent between their consumption bundle and that intended for the next lowest type [Cooper (1984), Salant (1989)]. In the case of two buyer types, this implies that 1, will still solve T1 =tpl,, while I, must be at a point on the tpl locus that yields no less utility than that obtained by 8, buyers if they purchase 1:. If it is profitable to accommodate both types of buyers (see below), two solutions are possible: (a) the optimal I,, denoted &, is greater than 1: but less than 1;; or (b) ?, = 1:. The first is a separating solution in which different sized lots are zoned for the two buyer types, while the second is a pooling solution in which both types purchase the same sized lot. The separating solution is depicted in fig. 2(a). The dashed curve labelled Fi; represents the maximum willingness-to-pay for 8, buyers such that they receive the same utility as if they purchased 1:. The point where this curve i, is the intersects tpl, i;, thus yields the same utility as 1:. Consequently, community’s optimal choice, as it yields the highest tax payment from 8, buyers while leaving them no worse off than at 1:. The pooling solution is shown in fig. 2(b), where 1; =I:. The logic of this solution is identical to that in the previous one. The only difference is that the willingness-to-pay curve for 6, buyers that goes through 1: intersects tpl at a smaller lot size than 1:. Thus, f2=l: yields the highest tax revenue from the set of lot sizes satisfying self-selection. In this case, only one lot size is zoned for both buyer types. What is the intuition for the difference between these two cases? In both, O2 buyers can be induced to choose a lot size other than 1: only by offering at least as much utility as at 1:. In case (a), this occurs at a larger lot than 1: since T2 - tpl (defined to be the buyer’s ‘tax surplus’) is maximized to the right of 17. In contrast, T2- tpl is maximized to the left of 1: in case (b), so raising 1 above 1: reduces this surplus (and hence, utility). Thus, a necessary condition for a separating solution in this model is that 1: be less than the value of 1 that maximizes T2 - tpl. As suggested above, it is not always profitable for the community to accommodate both (all) buyer types given imperfect discrimination. In particular, it might be better off excluding O1 buyers so that self-selection is unnecessary and zoning only for 8, buyers. This will be true if the additional tax revenue obtainable from zoning only large lots (IT) offsets the lesser number of entrants. Specifically, exclusion of 6,‘s will be desirable if n,tpl:>n,tpl:+n,tpi;, or simply r~~(l~--i~)>n,l:.~
‘This assumes n2 is equal under the two solutions. It need not be, e.g., if L>N,I, in one or both cases. I have not discussed the choice of n, and n2 here as the argument is identical to that in the perfect discrimination case.
T.J. Micek,
Fig. 2(a). Separating
Fig. 2(b). Pooling
4. Monopolistic
Optimal fiscal zoning
solution
solution
with competitive
with competitive
581
developers.
developers.
zoning with a single landowner
In this section I consider the case where the undeveloped land in the community is owned by a single developer. This situation corresponds to the case examined in Henderson (1980), who focuses on risk-sharing and strategic aspects of the problem, and Edelson (1975). It should be emphasized that communities in which the local government and the developer have monopoly power are probably rare, though a notable example seems to be Honolulu, Hawaii [Rose and LaCroix (1989)]. The essentials of the model remain the same as in the previous section except that now, the developer
588
T.J. Miceli, Optimal fiscal zoning
attempts to extract rents from newcomers to the community through of land. As in section 3, I consider the community’s optimal zoning under both perfect and imperfect discrimination.
his sale strategy
4.1. Perfect discrimination As above, perfect discrimination requires that buyer types be observable, and that buyers can be induced to purchase the lot size designed for them. This again allows all buyers to be driven to their reservation utility. Consequently, the purchase price of a lot is determined by the buyer’s willingness-to-pay as defined by (3). As noted, property taxes per lot in this case are tR. Thus, setting T = tR in (3) and solving for Ri = R(/,, 0,) yields Ri=(l/(l
+t))[y(Qi)-O(Oi)+~(li,Oi)]
for
all i.
(5)
Total profits for the developer, given discrimination among buyer types, are therefore x,n,( Ri - pl,). The problem for the community in this case is the same as that in the previous section: to zone the undeveloped land, L, such that the fiscal surplus from newcomers is maximized. Given (5), the objective function is c ), w h’ic h again is maximized subject to &nili 5 L (note that, in this xini(tRiversion of the problem, the constraint U(li,Si)= U(ei) is implicit in the definition of Ri). In addition, we must add the constraint that the developer at least cover his opportunity costs for each type of buyer that he sells to; conditions for ni =Ni and that is ni(Ri -pl,) 20 for all 1.9 The first-order Ii > 0 are given by tRi-C-~li+~i(Ri-pli)~O, t(aRj/ali)-1+~i(aRi/ali_p)=O,
(6) i=l,&...,I,
(7)
where 8Ri/81i =(1/l + t)o,(&, Oi) from (5); 2 is the multiplier on the land constraint, and pi is the multiplier on the developer’s profit for type i buyers. Suppose for now that the developer earns a profit on all lot sizes for which ni > 0, so that pi = 0 for all i. In that case, (6) becomes tRi - c - ivli 2 0, which says that lots should be zoned for all buyers of type Bi for whom the fiscal surplus at least covers the opportunity cost of the lot, where the latter is 9An alternative stategy for the community, not examined here, is to allow the developer to set lot sizes to maximize his profits, but then require him to pay a ‘development fee’, which in the limit can equal his total profits. This would then be distributed among the existing residents [Henderson (1985, p. 223)]. If feasible. this approach presumably would be less distortionary than extracting revenue through fiscal zoning.
T.J. Miceli,
589
Optimal fiscal zoning
$
Y I
Fig. 3. Perfect discrimination
solution
with a monopolistic
developer
given by li times the shadow price ;1. Thus, all entrants must contribute a strictly positive fiscal surplus. Alternatively, if tR, -c - Ali < 0, ni = 0.” AS for li, when nLi=O, (7) reduces to (t/l + t)U,(li,ei)=n. That is, the marginal revenue from an additional unit of land is equated to its shadow price. This further implies that the marginal valuation of land, u,(l,,8,), is equated for all buyer types. Thus, at the optimum, lot sizes are set so that all land yields the same marginal revenue for the community. Fig. 3 shows the solutions for 1: and 1; graphically for the case of two buyer types, where the curves labelled tRi represent the maximum tax payment for each type of buyer as a function of lot size. IT> 1: follows from the fact that 0, buyers value land more highly at the margin than 0r buyers do. Briefly consider the impact of the developer’s profit constraint being binding. Note first that the condition for n, >O is unaffected since if n,(R,-&)=O, the tinal term in (6) drops out as it did above when pi=O. As for Ii, if pi >O, the final term in (7) remains. This term may be positive or negative, depending on whether Ii is set below or above the lot size that the developer would set to maximize his profit. For example, if c?RJ& -p > 0, Ii is too small from the developer’s perspective. The impact in (7) is that the community adjusts Ii upward (thereby increasing the developer’s profit to zero) relative to the case where the profit constraint is not binding. The reverse is true if 8Ri/~li -p < 0.
4.2. Imperfect Once “‘Note
again
discrimination it can be seen from fig. 3 that,
that even if n I(R.-plJ=O, I
(6) would still reduce
in the absence to tR,-c-2,20
of coercion for n,>O
on
590
7-J. Miceli, Optimal fiscal zoning
the part of the developer or the community, 8, buyers will not voluntarily purchase the larger lot designed for them.” Rather, they will choose 1: which yields higher utility (as indicated by the dashed curve). Consequently, self-selection might be desirable in this case as well. As noted above, implementing self-selection for the case of two buyer types involves extracting all of the surplus from low types, but leaving high types indifferent between their lot size and that intended for low types. Thus, RI continues to be defined by (5) while R, is found by setting U(R,, 1,,8,) equal to U(R,, I,, 0,) (the self-selection constraint). Solving the resulting equation for R, and substituting for R, from (5) yields
where the final term in parentheses is positive given ug >O. The community’s maximization problem in this case is identical to that in the perfect discrimination case except that R, is defined by (8) rather than (5). The first-order conditions for the IZ~are therefore identical to those in (6), and hence are omitted (assume that ni =Ni for i= 1,2). Also, assume that developers earn a positive profit so that the developer’s profit constraint can be ignored (if the constraint is binding, an adjustment similar to that described above would be made). In that case, the first-order conditions for I, and 1, are given by
011+ th(l,, e,) - 3.5 0.
(10)
Following Salant (1989) there are five possible solutions to these conditions: (i) I, =Z,=O; (ii) 1, =I,>O; (iii) 1i >O, IZ=O; (iv) I, =O, I,>O, and (v) I, >O, I, >O, 1, #I,. In order to make the problem non-trivial, solution (i) is ruled out by assumption. Consider next solution (ii), which is a pooling solution. For the case of two buyer types, a pooling solution is never optimal in this version of the problem (in contrast to the version in section 3). To see this, equate l1 and l2 in (9) and (10) and substitute out A. This yields
(11) But this violates the assumption that uiB>O. Intuitively, pooling come from grouping together two or more buyer “As in section footnote 7 above.
3 this relies on the fact that
CR, intersects
tR,
the benefits of types of lower
to the left of (tR7.I:).
See
T.J. Miceli, Optimalfiscal
zoning
591
demand in order to extract a larger surplus from the highest demand buyer [Cooper (1984), Sappington (1983)]. Since, in such a solution, the highest demand buyer is never a member of the pool, pooling is never optimal with only two buyer types. As noted, this was not the case in the version of the model in section 3 where a pooling solution was possible. This was due to the fact that the community was restricted to solutions on a ‘tax line’ in that formulation, given that taxes were assessed on the opportunity cost of land. Solution (iii), which is an exclusion solution in which only lots for low demand buyers are zoned, can also be ruled out. This follows from the fact that high demand buyers offer the highest tax payment per lot; hence it would never pay to exclude them. Formally, such a solution implies the following condition [given I, >O and IZ =0 in (9) and (lo)]:
Note that the left-hand side is positive by v,@> 0, while the right-hand negative by vII < 0 and vlB> 0. Hence, (12) implies a contradiction. The remaining possible solutions are (iv), exclusion of low demand and (v) self-selection.12 The condition for exclusion to be optimal is (~2/%)C~#x ~2)-vI(o,
WI > v,a 01)-&Cl,,
02).
side is buyers,
(13)
As in the exclusion solution in section 3, the community benefits by its ability to extract all of the surplus from high demand buyers, while the cost is the lost revenue from low demand buyers (given that tR, -c>O). Thus, as above, exclusion is more desirable as n2/nl increases, all else equal. When (13) does not hold, self-selection is optimal. In this solution, the community will always zone 1, larger than I,. To prove this, assume the reverse is true (recall that 1, = 1, has already been ruled out). Then, from (9) and (10) we have v,(1,,8,)-v,(l2,e2)=(n2ln,)Cv,(l,,e2)-v,(1,,8,)1.
(14)
But since the left-hand side is negative by uL1O, and the right-hand side is positive by v10> 0, we have a contradiction, which proves that 1, > 1,. The lot sizes and tax payments under self-selection (denoted by a ‘ ^‘), along with the perfect discrimination solution (for sake of comparison), are shown in fig. 4 (The comparison is based on the case where pi =0 for i= 1,2.) Note first that T1~1: [which follows from (7) and (9)]. This is a standard “Note that two other solutions, both noted in section 3, are possible. One is where all e2 buyers are accommodated, but only some 0,‘s. This partial exclusion solution might occur if CR, -c-l/, =O. The other possibility is where all 0,‘s are excluded, and only some f3,‘s are allowed to enter (which might occur if tR, --c -11, =O).
T.J. Miceli, Optimal fiscal zoning
592
$
Fig. 4. Comparison
of perfect
discrimination and self-selection developer.
solutions
with
a monopolistic
result in self-selection models. The reason is that, by lowering fr below 17, a higher selling price (and hence, higher tax payment) can be extracted from high demand buyers while still satisfying self selection [Cooper (1984)]. However, the result is unique for monopoly zoning models which typically argue that lots will be set too large. Large lots are set for high demand buyers compared to their full information levels (i.e., fZ > I;). While this is the standard monopoly zoning result, it is unique for self-selection models where the highest demand buyer is always offered his or her full information bundle. The reason for the difference here is the presence of the land constraint. Suppose for instance that I:=i,. The land constraint then implies that N,i; + N,Q< N,I:+ N,lT= L (where n,= Ni, i= 1,2, is assumed in both solutions). But this cannot be the case if it is optimal for all land to be developed, as we have assumed. Therefore, in order to satisfy the land constraint, the ‘excess’ land from low demand buyers compared to the perfect discrimination solution [i.e., N,(IT-i;)] is reallocated to high demand buyers, thus requiring that & >lq. This further implies that the shadow price of land (A) is lower under the selfselection solution than under the perfect discrimination solution by (10). This, of course, makes sense, since the value of an incremental amount of land is lower when self-selection is required (as opposed to coercion) in order to induce high demand buyers to purchase larger lots than those intended for low demand buyers.
5. A comparison of the two models In this
section
I compare
the
models
presented
in the
preceding
two
593
T.J. Miceli, Optimal fiscal zoning
$
Fig. 5. Comparison
of competitive
and monopolistic
developer
models.
sections in regard to the revenue each produces for the community. For the comparison to be sensible, I consider only solutions to both models where all land is sold.13 Consider first a comparison of the perfect discrimination solutions. Comparing (3) and (5) reveals that 7;=tR,+(R,-pl,).
(15)
That is, the maximum tax payment a buyer is willing to make when land is sold competitively equals the maximum he or she is willing to pay in taxes plus land rents when the developer is a monopolist. Eq. (15) implies that when Ri - pl, = 0, T = tRi. In addition, Ri - pl, = 0 implies tRi - tpl= 0. Thus, the curves L&, tRi, and tpl all intersect at the same point, which coincides with the optimal lot size for buyer i under perfect discrimination in model 1 (the competitive developer model), denoted here 1:(l). This is shown graphically in fig. 5. The optimal lot size for buyer i in model 2 (the monopolistic developer model), denoted l:(2), is also shown. Given the constraint that R,-ppli20 in this model, /F(2) must lie between 1: and l:(l) (i.e., the region where tRi 2 tpl). Note that, because IT(2)517( l), the tax payment per lot sold to type tli buyers is at least as large in model 1 as in model 2. More important, though, is the tax per unit of land. This is given in the graph by the slope of a ray from the origin to the relevant point on the tax revenue curves. For model 1, ‘%pecitically, consider the case of two buyer types where model I (competitive developers) nZ=N, and n, =(L-N,/,)/I, (single developer), ni = Ni, i = 1,2.
lots are zoned for both types. In (i.e., n, SN,), and in model 2
594
T.J. Miceli, Optimal fiscal zoning
this is simply the slope of tpl, or to, for any sized lot. For model 2, it is the slope of a ray to tRi (the dashed line in fig. 5). Note that this slope is greater than tp for any /T(2) strictly between I; and IF(l). This implies that taxes per unit of land for buyer i are higher in model 2 than in model 1 (note that average taxes are maximized for any lot where the ray through the origin is tangent to tRi). The reason is that, when the monopolistic developer is able to earn rents on land sales in model 2, the tax base per lot, Ri, rises above the tax base when land is sold competitively, pl,. As a result, overall taxes collected are at least as large in model 2 as in model 1 (given that all land is sold in both). This is true regardless of how the total undeveloped land is allocated between the two buyer types in the two models. In model 1, taxes per unit of land are tp for all types of buyers, while in model 2, taxes per unit of land are tR,/l, for buyers of type Oi. But for all buyers permitted entry in model 2, Ri -pli 20, or tR,/l, 2 tp. Thus, average taxes on land sold to any type of buyer in model 2 are at least as large as those on land sold to any type of buyer in model 1. It therefore follows that at least as much tax revenue is raised in model 2 as in model 1. This is despite the fact that lot sizes are generally larger in model 1 for any buyer type. Intuitively, in the model with a competitive land market, a portion of buyers’ overall willingness-to-pay to enter the community (on land plus taxes) is competed away by land sellers, rather than accruing partially to the developer as profits, and partially to the community in enhanced revenue, as in model 2. Finally, note that the preceding argument holds for a comparison of any solutions to the two models, including the imperfect discrimination solutions, provided all of the land is developed in both models. This follows from the fact that all solutions to model 1 lie on the tax line where average taxes are tp, and all solutions to model 2 are subject to the constraint that Ri - pli 2 0 for all i. The results in this section suggest that it may actually be in the interest of the local government to promote monopolization of the land market, perhaps through regulations that facilitate cooperation among landowners. In fact, Rose and LaCroix (1989) suggest that something like this has happened in Honolulu, where they argue that the city government facilitates the maintenance of a land cartel. 6. Conclusion This paper has provided a rigorous analysis of monopolistic discrimination by local communities in their zoning of lot sizes for new development. The model examined situations in which the community could coerce new entrants to buy the lot sizes intended for them (perfect discrimination), as well as the more realistic case where buyers must be allowed to self select
7-J. Miceli, Optimal”/iscal
zoning
595
their lot sizes (imperfect discrimination). In the second case, the optimal zoning strategy did not always entail multiple lot sizes. In some cases, only a single lot size was zoned for purchase by all buyer types, while in others, the single lot size was set large enough to exclude low-demand buyers even when they would contribute a strictly positive fiscal surplus. The role of landowners/developers proved to be important in the model. In particular, it was shown that when a single developer owns the undeveloped land, tax revenues extracted from new entrants will generally be higher than when the land is owned by multiple, competitive developers. The reason is that the single developer is able to increase the tax base of entrants by raising the sale price of lots above their opportunity cost. This result suggested the interesting possibility of monopolized local governments and developers cooperating in the determination of land-use policies in local communities.
References Carpenter, B. and D. Hetlley, 1982, Spatial-equilibrium analysis of transferable development rights, Journal of Urban Economics 12, 238-261. Cooper, R., 1984, On allocative distortions in problems of self selection, Rand Journal of Economics 15, 568-577. Edelson, N., 1975, The developer’s problem, or how to divide a piece of land most profitably, Journal of Urban Economics 2, 349-365. Epple, D. and T. Romer, 1989, On the flexibility of municipal boundaries, Journal of Urban Economics 26, 307-3 19. Epple, D., R. Filimon and T. Romer, 1988, Community development with endogenous land use controls, Journal of Public Economy 35, 133-162. Fischel, W., 1980, Zoning and the exercise of monopoly power: A reevaluation, Journal of Urban Economics 8, 283-293. Fischel, W., 1985, The economics of zoning laws (Johns Hopkins University Press, Baltimore, MD). Hamilton, B., 1975, Zoning and property taxes in a system of local governments, Urban Studies 12, 205-211. Hamilton , B., 1978, Zoning and the exercise of monopoly power, Journal of Urban Economics 5,116130. Henderson, J.V., 1980, Community development: The effects of growth and uncertainty, American Economic Review 70, 894910. Henderson, J.V., 1985, The Tiebout model: Bring back the entrepreneurs, Journal of Political Economy 93, 2488264. Maskin, E. and J. Riley, 1984, Monopoly with incomplete information, Rand Journal of Economics 15, 171-196. Mills, D., 1980, Transferable development rights markets, Journal of Urban Economics 7, 63-74. Phlips, L., 1983, The economics of price discrimination (Cambridge University Press, Cambridge, MA). Rose, L., 1989, Urban land supply: Natural and contrived restrictions, Journal of Urban Economics 25, 3255345. Rose. L. and S. LaCroix, 1989, Urban land price: The extraordinary case of Honolulu, Hawaii, Urban Studies 26, 301-314. Salant, S., 1989, When is inducing self-selection suboptimal for a monopolist?, Quarterly Journal of Economics 109. 391-397.
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Sappington, D., 1983, Limited liability contracts between principal and agent, Journal of Economic Theory 29, 1-21. Tiebout, C., 1956, A pure theory of local public expenditure, Journal of Political Economy 64, 41-24. White, M., 1975, Fiscal zoning in fragmented metropolitan areas, in: E. Mills and W. Oates, eds., Fiscal zoning and land use controls (Lexington Books, Lexington, MA). White, M., 1979, Suburban growth controls: Liability rules and Pigovian taxes, Journal of Legal Studies 8, 207-230. Windsor, D., 1979, Fiscal zoning in suburban communities (Lexington Books, Lexington, MA).
Science
and Urban
Economics
22 (1992) 579-596.
North-Holland
Optimal fiscal zoning when the local government is a discriminating monopolist Thomas
J. Miceli”
The University of Connecticut, Storrs C7: USA Received January
1990, tinal version
received
December
1990
Monopoly zoning by local governments has been studied extensively since the original article by White [1975, in: E. Mills and W. Oates, eds., Fiscal zoning and land use controls (Lexington Books, Lexington, MA)]. However, little or no attention has been paid to the possibility of monopolistic discrimination under such a regime. The question is: When will zoning multiple lot sizes be feasible given that buyers with different valuations of land cannot be coerced into purchasing particular lots? The literature on imperfect price discrimination by monopolists is brought to bear on this question. It is shown that the resulting zoning strategy may or may not involve multiple lot sizes. The factors determining the optimal strategy are studied in detail, with particular emphasis on the nature of the market for undeveloped land in the community.
1. Introduction Monopoly zoning by local governments has been studied extensively by economists since the original article by White (1975). In the standard view, communities zone large lots in an attempt to extract tax revenues from entrants in excess of the cost of providing them with services. Existing residents of the community are thereby enriched at the expense of newcomers in a strategy White labels ‘fiscal squeeze zoning’.’ The exercise of monopoly power by local communities naturally suggests the possibility of ‘price’ discrimination. In the case of zoning this would take the form of setting different lot sizes for different consumers in an effort to extract a greater surplus from buyers with a higher valuation of land. While White considered this possibility, she argued, first of all, that it is not supported by the evidence. In fact, the evidence to which she refers suggests Correspondence to: Thomas J. Miceli, College of Liberal Arts and Sciences, University of Connecticut, 341 Mansfield Road, CT 06269-1063, USA. *I wish to thank several anonymous referees for comments that greatly improved this paper. Any remaining errors, of course, are my own. ‘Efforts to find empirical evidence for this sort of activity by local governments have been weakly successful. See, e.g., Hamilton (1978) Windsor (1979, chapters 4 and 5), Fischel (1980) and Rose (1989). 01660462/92/$05.00
0
1992-Elsevier
Science Publishers
B.V. All rights reserved
580
T.J. Mice/i,
Optimal fiscal
Table Minimum Minimum
zoning
1
lot sizes in New Jersey communities,
lot size (acres)
o-o.24 0.25-0.49 0.50-0.99 l-2.99 3 or more Source: Windsor
Percent
1970.
of total acreage
5.2 9.8 19.3 53.8 12.0 (1979, table 3-6, p. 58).
that communities often zone multiple lot sizes.’ In particular, out of 300 New Jersey communities in the New York metropolitan area, ‘( . .) about three quarters of the communities zoned 75% or more of their land for development in a single lot size. Most of the remainder divided their vacant land into regions set aside for two different, but similar lot sizes’ [White (1975, p. 46)]. Windsor (1979) also provides evidence on minimum lot sizes set by New Jersey communities (table l), and although the data are averages across all communities in the sample, they show considerable variation in lot size. A further impediment to discriminatory zoning noted by White is the question of how segmentation of the market can be achieved. In particular, when will a buyer ever purchase a large lot when smaller ones are available? Recently, a large literature has arisen to examine monopolistic discrimination given imperfect sorting of buyers [e.g., Sappington (1983) Cooper (1984), Maskin and Riley (1984), and Salant (1989)]. The sorting device employed in these models is self selection, whereby buyers are free to select their optimal product variety from among those offered. For example, products of varying quality may be offered according to a schedule of prices that exploits consumers’ differing marginal rates of substitution of quality for money.3 However, because the self-selection constraint permits only imperfect discrimination, it turns our that in some cases it is not optimal for the monopolist to offer multiple varieties (or, in this case, lot sizes). And, when multiple varieties are offered, they are typically characterized by allocative distortions. This paper represents the first attempt to apply the techniques of the literature on imperfect discrimination to monopoly zoning. The analysis proceeds as follows. Section 2 describes the model and assumptions. Section 3 then considers the case where land is owned by competitive developers within the community so that lots sell at their opportunity cost. The optimal ‘Note, however, that the existence of multiple lot sizes does not necessarily imply discrimination, but instead may be due to historical or other causes. %ee Phlips (1983) for further examples of imperfect sorting strategies, including sorting along both spatial and temporal dimensions. Phlips also provides descriptions of actual business practices in support of the theory.
T.J. Midi,
Optimal
fiscal zoning
581
discrimination solution is derived in this case, first when perfect discrimination (sorting) is possible (as a benchmark), and then when self-selection must be employed. Section 4 then considers the case where the undeveloped land is owned by a single developer. Again, both the perfect discrimination and the self-selection solutions are examined. Section 5 asks which scenario, competitive developers or a single developer, is more profitable for the community given its optimal zoning strategy. Finally section 6 concludes.
2. Setup of the model Consider a community with a fixed amount of undeveloped land that is owned by a single, or multiple landowner/developers. The community, however, controls development; in particular, it sets lot sizes to extract the maximum fiscal squeeze from newcomers. To be sure, communities employ other means of extracting revenue from developers, for example, subdivision exactions, including contributions of land for schools or parks [White (1979), Fischel (1985)], or the sale of transferable development rights [Mills (1980) Carpenter and Heffley (1982)]. However, the focus in this paper is exclusively on the use of zoning for this purpose. The source of the community’s monopoly power is, first, the fact that newcomers face restricted alternatives, owing to the difficulty of adjusting existing jurisdictional boundaries [Epple and Romer (1989) Henderson of new communities (19831, and costs associated with the formation [Henderson (1980), Epple et al. (1988)]. Second, the community is assumed to possess some unique feature or amenity relative to neighboring communities for which residents will pay a premium. This might take the form of superior access to the central business district, or high quality public services. Undeveloped land within the community, however, is assumed to be homogeneous [in contrast to the model in Epple et al. (1988)]. Given this setting, consider potential entrants to the community. To keep the analysis simple, their utility will take the separable form U(x, 1,e)=x + u(l, e), where x is consumption of all other goods (and the numeraire), 1 is the lot size purchased by the household, and 0 is a parameter capturing the household’s valuation of land relative to other goods. The function u(l,fI) is assumed to have the following properties:
(24 (2b)
(24
582
T.J. Miceli, Optimalfiscal zoning
That is, land yields positive but diminishing marginal utiiity, and a higher 0 increases the buyer’s valuation of land, both absolutely and at the margin.4 Buyers with higher B’s are therefore referred to as ‘higher demanders’. Note finally that the public services provided by the community also enter the buyer’s utility function, but as they are fixed, they are suppressed in (1) and throughout without loss of generality. Let the budget constraint of type 0 buyers be given by y(B) = x + R(1) + T(l), where y(8) is income, R(l) is the purchase price of a lot of size 1, and T(I) is the total tax payment associated with a lot of size 1. It will be convenient to substitute the budget constraint into the buyer’s utility function in (1) to obtain U(l,O)=y(B)-R(l)-T(l)+u(l,t?).
(1’)
Suppose that a buyer of type 0 has a next-best utility level of U(O) obtainable elsewhere in the metropolitan area. Equating U(I,@ and U(O) thus implicitly defines the buyer’s maximum willingness-to-pay for land plus taxes in the community: y(B)-U(O)-R(l)-7’(l)+u(l,8)=0.
(3)
The specific forms of R(l) and T(1) in this equation depend upon the nature of developers in the community, and the tax practices of the local government. These will be discussed in turn. Considering first the community, suppose it has a fixed amount of undeveloped land, L, of homogeneous quality that it wishes to have developed. Moreover, suppose that the land is identical to undeveloped land in other communities in the metropolitan area. Given its monopoly power, the community seeks to extract rents from new residents through the property tax, which is used to finance public services. Let the cost of services per person in the community be a constant c. Thus, public services are private rather than public goods at the margin.5 For simplicity, I assume 4All of the results in the paper would hold for the general utility function U(x,I,O) if, in addition to positive marginal utilities of x and I, the following hold: (i) U,/U, is decreasing in I; (ii) UB >O, and (iii) U,/U, is increasing in 8. These correspond to conditions (2b))(2d) in the text, and are standard in the self-selection literature [Cooper (1984)]. ‘This is consistent with the model in Henderson (1980). Note that it assumes, in contrast to White (1975) that the cost of public services per household is not a function of a lot size, thereby insuring that the household’s ‘fiscal surplus’ is an increasing function of lot size (i.e., tax base). This may not be the case, if, for example, households with more school-age children tend to purchase larger lots. Windsor (1979) in fact finds this to be the case in his study of New Jersey communities in 1970, though he still finds that the fiscal surplus is generally increasing in lot size.
T.J. Miceli, Optimal fiscal
zoning
583
that the tax is assessed in proportion to the value of the lot purchased by new residents, rather than the value of the lot plus improvements as in White (1975) (if improvements are proportional to the lot size, then the analysis would be unaffected by their inclusion). The tax rate, t, is taken as fixed. For example, it may be the rate needed to balance the budget for the services provided to existing residents prior to development. In contrast, the goal of monopoly zoning is to set lot sizes to extract a fiscal transfer from new residents in excess of the cost of the services they consume. Next consider the characterization of developers in the community. Two scenarios will be considered. In the first, land is owned by competitive developers [Henderson (1985, p. 223)]. Note that this is not inconsistent with monopoly power by the community, given that land within the community is perfectly substitutable, and that the local government holds a monopoly over zoning rights. In this case, the price of land will fall to its opportunity cost, p, and the purchase price of a lot in (1’) will be pl, while taxes collected per lot will be tpl. In the second scenario, the undeveloped land in the community is owned by a single developer. The developer can therefore attempt to exercise monopoly power in selling lots as in Edelson (1975), but subject to control of lot sizes by the local government. In this case, R(I) is set by the developer to maximize profits (given I), and taxes collected per lot are tR.
3. Monopoly zoning with competitive developers In this section I consider monopoly zoning by the community with competitive landowners/developers. For simplicity, let there be a discrete number of buyer types indexed by i= 1,2,. . . , I, such that higher indices designate higher demand buyers, i.e., Bj> 0, for j> i. Also, let n, be the number of buyers of type Bi who purchase a lot (i.e., the number of lots of this type sold), where Ni is the number of potential type Bi buyers (hence, rri 5 NJ, and li is the lot size purchased. As noted, taxes collected from each of these buyers is tpl,. Thus, the community’s problem is to maximize the total fiscal surplus from new entrants, equal to Cini(tpli-cc), subject to the constraints xn,l, 5 L, and U(li,Oi) 2 U(Oi). The choice variables of the problem are the lot size for each type, li, and the number of lots of each size to zone, n,. I examine the solution to this problem under two assumptions: (a) buyers’ types (ei) are observable, and each type can be required to purchase the lot size designed for him or her (perfect discrimination);6 and
61n addition, recontracting among buyers is prohibited for the case of perfect discrimination. The fact that recontracting cannot be prevented in practice is one reason self-selection is required for discrimination in the land market.
584
T.J. Mice/i, Optimal fiscal zoning
(b) buyer’s types purchase particular
3.1.
are unobservable, lot sizes (imperfect
The perfect discrimination
and/or buyers discrimination).
cannot
be forced
to
solution
The perfect discrimination solution serves as a benchmark that will aid in motivating the imperfect discrimination solution. In examining the solution to community’s problem, first consider the choice of Zi. Since each buyer type can be required to purchase a particular lot size, then all types will be driven to their reservation utility, U(0,). That is, each type will be on his or her willingness-to-pay curve defined in (3). For the case of competitive landowners, as noted, the cost of land is pl, so we may rewrite (3) as
T = _Y(Qi) - D(di)-p/i + U(li, di),
(4)
where T represents the maximum tax payment that a type oi buyer will pay for a lot of size Ii. The slope of 71 is given by aT/aZi= v,(li, ei) --p, which is initially positive and then becomes negative for large 1. Note that one expects the intercept of T, ~(0~) - u(6,), to be negative, implying that the buyer is not willing to pay a positive amount for very small lots. Finally, given (2), aTJalj > aTJa1, for t3j > tJi, implying, as one would expect, that the optimal lot size increases in 8. Given the characteristics of T and the fact that tpl, is linearly increasing in li, to maximize tax revenues per lot the community should set li to solve T = tpl,. The resulting lot sizes for two buyer types, denoted Ii*, i = 1,2, are shown graphically in fig. 1. Note first that, given y(Qi)- D(ei) ~0, there will generally be two points where tpl, intersects T. It is clear from the graph, however, that the lot size which maximizes tpli is the larger of the two. As a result, the lot size for the higher type buyer is larger than for the lower type, and, as drawn, both are larger than the buyers’ optimal lot sizes. The latter result need not hold, however, for example, if tpl is very steep. Indeed, if tp is large, there need not be an intersection at all, in which case that buyer type could not be induced to enter the community. Turning to the optimal choice of ni, note that a fiscal surplus is realized for all Bi for whom tpl,-c > 0. Hence, if the quantity of undeveloped land allows, ni= Ni for all di that produce a fiscal surplus. That is, if CiNiljr~ L, then ni= Ni for all Bi. Suppose, however, that this condition does not hold. Consider, for example, the case of two buyer types, and suppose that N,lg < L< N,lT + N,l:. In this case, since the fiscal surplus from the higherbuyer type exceeds that from the lower-buyer type (i.e., tpl,* -c > tpl: -c), and then set n, =(L- N,14)/1: (ignoring the integer confirst set n,=N,, straint on ni). That is, first zone enough large sized lots for all
585
T.J. Miceli, Optimal fiscal zoning
Y(q)
-
Y@,)
-
Fig. 1. Perfect discrimination
solution
with competitive
developers.
of the high type buyers, and then zone the remaining land for buyers. Suppose alternatively that N,lf 2 L. In this case, all land zoned for high type buyers. Thus, even if there exist other buyer whom tplj+-c>O, they will be excluded by the community’s zoning
3.2. The imperfect discrimination
low type should be types for policy.
solution
The reason that buyers in the perfect discrimination solution must be required to purchase the lot size set for them (and why recontracting must also be prohibited) can be seen by examining fig. 1. Notice that, because higher levels of utility are associated with downward shifts of the willingnessto-pay curves, type 0, buyers would realize a higher level of utility by purchasing the lot size intended for type Oi buyers. This problem will exist whenever the intersection point between Tl and T2 falls to the left of the optimal point for O1 buyers.’ In this section, I assume that this condition holds, for if it does not, the solution in the preceding section can be implemented by the community without coercion. In order to discriminate among buyers given the above problem, the community must induce self-selection. That is, the lot size intended for any buyer type must be preferred by him or her to all other lot sizes available. A
‘This condition is satislied if y(O,)-ii is either less than y(O,)-_(O,), larger. There seems no obvious reason either to accept or reject this condition
or not too much a priori.
586
T.J. Miceli, Optimal fiscal zoning
standard result in the self-selection literature is that the optimal self-selection solution involves extracting all of the surplus from the lowest demand buyer, while higher demand buyers are left indifferent between their consumption bundle and that intended for the next lowest type [Cooper (1984), Salant (1989)]. In the case of two buyer types, this implies that 1, will still solve T1 =tpl,, while I, must be at a point on the tpl locus that yields no less utility than that obtained by 8, buyers if they purchase 1:. If it is profitable to accommodate both types of buyers (see below), two solutions are possible: (a) the optimal I,, denoted &, is greater than 1: but less than 1;; or (b) ?, = 1:. The first is a separating solution in which different sized lots are zoned for the two buyer types, while the second is a pooling solution in which both types purchase the same sized lot. The separating solution is depicted in fig. 2(a). The dashed curve labelled Fi; represents the maximum willingness-to-pay for 8, buyers such that they receive the same utility as if they purchased 1:. The point where this curve i, is the intersects tpl, i;, thus yields the same utility as 1:. Consequently, community’s optimal choice, as it yields the highest tax payment from 8, buyers while leaving them no worse off than at 1:. The pooling solution is shown in fig. 2(b), where 1; =I:. The logic of this solution is identical to that in the previous one. The only difference is that the willingness-to-pay curve for 6, buyers that goes through 1: intersects tpl at a smaller lot size than 1:. Thus, f2=l: yields the highest tax revenue from the set of lot sizes satisfying self-selection. In this case, only one lot size is zoned for both buyer types. What is the intuition for the difference between these two cases? In both, O2 buyers can be induced to choose a lot size other than 1: only by offering at least as much utility as at 1:. In case (a), this occurs at a larger lot than 1: since T2 - tpl (defined to be the buyer’s ‘tax surplus’) is maximized to the right of 17. In contrast, T2- tpl is maximized to the left of 1: in case (b), so raising 1 above 1: reduces this surplus (and hence, utility). Thus, a necessary condition for a separating solution in this model is that 1: be less than the value of 1 that maximizes T2 - tpl. As suggested above, it is not always profitable for the community to accommodate both (all) buyer types given imperfect discrimination. In particular, it might be better off excluding O1 buyers so that self-selection is unnecessary and zoning only for 8, buyers. This will be true if the additional tax revenue obtainable from zoning only large lots (IT) offsets the lesser number of entrants. Specifically, exclusion of 6,‘s will be desirable if n,tpl:>n,tpl:+n,tpi;, or simply r~~(l~--i~)>n,l:.~
‘This assumes n2 is equal under the two solutions. It need not be, e.g., if L>N,I, in one or both cases. I have not discussed the choice of n, and n2 here as the argument is identical to that in the perfect discrimination case.
T.J. Micek,
Fig. 2(a). Separating
Fig. 2(b). Pooling
4. Monopolistic
Optimal fiscal zoning
solution
solution
with competitive
with competitive
581
developers.
developers.
zoning with a single landowner
In this section I consider the case where the undeveloped land in the community is owned by a single developer. This situation corresponds to the case examined in Henderson (1980), who focuses on risk-sharing and strategic aspects of the problem, and Edelson (1975). It should be emphasized that communities in which the local government and the developer have monopoly power are probably rare, though a notable example seems to be Honolulu, Hawaii [Rose and LaCroix (1989)]. The essentials of the model remain the same as in the previous section except that now, the developer
588
T.J. Miceli, Optimal fiscal zoning
attempts to extract rents from newcomers to the community through of land. As in section 3, I consider the community’s optimal zoning under both perfect and imperfect discrimination.
his sale strategy
4.1. Perfect discrimination As above, perfect discrimination requires that buyer types be observable, and that buyers can be induced to purchase the lot size designed for them. This again allows all buyers to be driven to their reservation utility. Consequently, the purchase price of a lot is determined by the buyer’s willingness-to-pay as defined by (3). As noted, property taxes per lot in this case are tR. Thus, setting T = tR in (3) and solving for Ri = R(/,, 0,) yields Ri=(l/(l
+t))[y(Qi)-O(Oi)+~(li,Oi)]
for
all i.
(5)
Total profits for the developer, given discrimination among buyer types, are therefore x,n,( Ri - pl,). The problem for the community in this case is the same as that in the previous section: to zone the undeveloped land, L, such that the fiscal surplus from newcomers is maximized. Given (5), the objective function is c ), w h’ic h again is maximized subject to &nili 5 L (note that, in this xini(tRiversion of the problem, the constraint U(li,Si)= U(ei) is implicit in the definition of Ri). In addition, we must add the constraint that the developer at least cover his opportunity costs for each type of buyer that he sells to; conditions for ni =Ni and that is ni(Ri -pl,) 20 for all 1.9 The first-order Ii > 0 are given by tRi-C-~li+~i(Ri-pli)~O, t(aRj/ali)-1+~i(aRi/ali_p)=O,
(6) i=l,&...,I,
(7)
where 8Ri/81i =(1/l + t)o,(&, Oi) from (5); 2 is the multiplier on the land constraint, and pi is the multiplier on the developer’s profit for type i buyers. Suppose for now that the developer earns a profit on all lot sizes for which ni > 0, so that pi = 0 for all i. In that case, (6) becomes tRi - c - ivli 2 0, which says that lots should be zoned for all buyers of type Bi for whom the fiscal surplus at least covers the opportunity cost of the lot, where the latter is 9An alternative stategy for the community, not examined here, is to allow the developer to set lot sizes to maximize his profits, but then require him to pay a ‘development fee’, which in the limit can equal his total profits. This would then be distributed among the existing residents [Henderson (1985, p. 223)]. If feasible. this approach presumably would be less distortionary than extracting revenue through fiscal zoning.
T.J. Miceli,
589
Optimal fiscal zoning
$
Y I
Fig. 3. Perfect discrimination
solution
with a monopolistic
developer
given by li times the shadow price ;1. Thus, all entrants must contribute a strictly positive fiscal surplus. Alternatively, if tR, -c - Ali < 0, ni = 0.” AS for li, when nLi=O, (7) reduces to (t/l + t)U,(li,ei)=n. That is, the marginal revenue from an additional unit of land is equated to its shadow price. This further implies that the marginal valuation of land, u,(l,,8,), is equated for all buyer types. Thus, at the optimum, lot sizes are set so that all land yields the same marginal revenue for the community. Fig. 3 shows the solutions for 1: and 1; graphically for the case of two buyer types, where the curves labelled tRi represent the maximum tax payment for each type of buyer as a function of lot size. IT> 1: follows from the fact that 0, buyers value land more highly at the margin than 0r buyers do. Briefly consider the impact of the developer’s profit constraint being binding. Note first that the condition for n, >O is unaffected since if n,(R,-&)=O, the tinal term in (6) drops out as it did above when pi=O. As for Ii, if pi >O, the final term in (7) remains. This term may be positive or negative, depending on whether Ii is set below or above the lot size that the developer would set to maximize his profit. For example, if c?RJ& -p > 0, Ii is too small from the developer’s perspective. The impact in (7) is that the community adjusts Ii upward (thereby increasing the developer’s profit to zero) relative to the case where the profit constraint is not binding. The reverse is true if 8Ri/~li -p < 0.
4.2. Imperfect Once “‘Note
again
discrimination it can be seen from fig. 3 that,
that even if n I(R.-plJ=O, I
(6) would still reduce
in the absence to tR,-c-2,20
of coercion for n,>O
on
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7-J. Miceli, Optimal fiscal zoning
the part of the developer or the community, 8, buyers will not voluntarily purchase the larger lot designed for them.” Rather, they will choose 1: which yields higher utility (as indicated by the dashed curve). Consequently, self-selection might be desirable in this case as well. As noted above, implementing self-selection for the case of two buyer types involves extracting all of the surplus from low types, but leaving high types indifferent between their lot size and that intended for low types. Thus, RI continues to be defined by (5) while R, is found by setting U(R,, 1,,8,) equal to U(R,, I,, 0,) (the self-selection constraint). Solving the resulting equation for R, and substituting for R, from (5) yields
where the final term in parentheses is positive given ug >O. The community’s maximization problem in this case is identical to that in the perfect discrimination case except that R, is defined by (8) rather than (5). The first-order conditions for the IZ~are therefore identical to those in (6), and hence are omitted (assume that ni =Ni for i= 1,2). Also, assume that developers earn a positive profit so that the developer’s profit constraint can be ignored (if the constraint is binding, an adjustment similar to that described above would be made). In that case, the first-order conditions for I, and 1, are given by
011+ th(l,, e,) - 3.5 0.
(10)
Following Salant (1989) there are five possible solutions to these conditions: (i) I, =Z,=O; (ii) 1, =I,>O; (iii) 1i >O, IZ=O; (iv) I, =O, I,>O, and (v) I, >O, I, >O, 1, #I,. In order to make the problem non-trivial, solution (i) is ruled out by assumption. Consider next solution (ii), which is a pooling solution. For the case of two buyer types, a pooling solution is never optimal in this version of the problem (in contrast to the version in section 3). To see this, equate l1 and l2 in (9) and (10) and substitute out A. This yields
(11) But this violates the assumption that uiB>O. Intuitively, pooling come from grouping together two or more buyer “As in section footnote 7 above.
3 this relies on the fact that
CR, intersects
tR,
the benefits of types of lower
to the left of (tR7.I:).
See
T.J. Miceli, Optimalfiscal
zoning
591
demand in order to extract a larger surplus from the highest demand buyer [Cooper (1984), Sappington (1983)]. Since, in such a solution, the highest demand buyer is never a member of the pool, pooling is never optimal with only two buyer types. As noted, this was not the case in the version of the model in section 3 where a pooling solution was possible. This was due to the fact that the community was restricted to solutions on a ‘tax line’ in that formulation, given that taxes were assessed on the opportunity cost of land. Solution (iii), which is an exclusion solution in which only lots for low demand buyers are zoned, can also be ruled out. This follows from the fact that high demand buyers offer the highest tax payment per lot; hence it would never pay to exclude them. Formally, such a solution implies the following condition [given I, >O and IZ =0 in (9) and (lo)]:
Note that the left-hand side is positive by v,@> 0, while the right-hand negative by vII < 0 and vlB> 0. Hence, (12) implies a contradiction. The remaining possible solutions are (iv), exclusion of low demand and (v) self-selection.12 The condition for exclusion to be optimal is (~2/%)C~#x ~2)-vI(o,
WI > v,a 01)-&Cl,,
02).
side is buyers,
(13)
As in the exclusion solution in section 3, the community benefits by its ability to extract all of the surplus from high demand buyers, while the cost is the lost revenue from low demand buyers (given that tR, -c>O). Thus, as above, exclusion is more desirable as n2/nl increases, all else equal. When (13) does not hold, self-selection is optimal. In this solution, the community will always zone 1, larger than I,. To prove this, assume the reverse is true (recall that 1, = 1, has already been ruled out). Then, from (9) and (10) we have v,(1,,8,)-v,(l2,e2)=(n2ln,)Cv,(l,,e2)-v,(1,,8,)1.
(14)
But since the left-hand side is negative by uL1
T.J. Miceli, Optimal fiscal zoning
592
$
Fig. 4. Comparison
of perfect
discrimination and self-selection developer.
solutions
with
a monopolistic
result in self-selection models. The reason is that, by lowering fr below 17, a higher selling price (and hence, higher tax payment) can be extracted from high demand buyers while still satisfying self selection [Cooper (1984)]. However, the result is unique for monopoly zoning models which typically argue that lots will be set too large. Large lots are set for high demand buyers compared to their full information levels (i.e., fZ > I;). While this is the standard monopoly zoning result, it is unique for self-selection models where the highest demand buyer is always offered his or her full information bundle. The reason for the difference here is the presence of the land constraint. Suppose for instance that I:=i,. The land constraint then implies that N,i; + N,Q< N,I:+ N,lT= L (where n,= Ni, i= 1,2, is assumed in both solutions). But this cannot be the case if it is optimal for all land to be developed, as we have assumed. Therefore, in order to satisfy the land constraint, the ‘excess’ land from low demand buyers compared to the perfect discrimination solution [i.e., N,(IT-i;)] is reallocated to high demand buyers, thus requiring that & >lq. This further implies that the shadow price of land (A) is lower under the selfselection solution than under the perfect discrimination solution by (10). This, of course, makes sense, since the value of an incremental amount of land is lower when self-selection is required (as opposed to coercion) in order to induce high demand buyers to purchase larger lots than those intended for low demand buyers.
5. A comparison of the two models In this
section
I compare
the
models
presented
in the
preceding
two
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T.J. Miceli, Optimal fiscal zoning
$
Fig. 5. Comparison
of competitive
and monopolistic
developer
models.
sections in regard to the revenue each produces for the community. For the comparison to be sensible, I consider only solutions to both models where all land is sold.13 Consider first a comparison of the perfect discrimination solutions. Comparing (3) and (5) reveals that 7;=tR,+(R,-pl,).
(15)
That is, the maximum tax payment a buyer is willing to make when land is sold competitively equals the maximum he or she is willing to pay in taxes plus land rents when the developer is a monopolist. Eq. (15) implies that when Ri - pl, = 0, T = tRi. In addition, Ri - pl, = 0 implies tRi - tpl= 0. Thus, the curves L&, tRi, and tpl all intersect at the same point, which coincides with the optimal lot size for buyer i under perfect discrimination in model 1 (the competitive developer model), denoted here 1:(l). This is shown graphically in fig. 5. The optimal lot size for buyer i in model 2 (the monopolistic developer model), denoted l:(2), is also shown. Given the constraint that R,-ppli20 in this model, /F(2) must lie between 1: and l:(l) (i.e., the region where tRi 2 tpl). Note that, because IT(2)517( l), the tax payment per lot sold to type tli buyers is at least as large in model 1 as in model 2. More important, though, is the tax per unit of land. This is given in the graph by the slope of a ray from the origin to the relevant point on the tax revenue curves. For model 1, ‘%pecitically, consider the case of two buyer types where model I (competitive developers) nZ=N, and n, =(L-N,/,)/I, (single developer), ni = Ni, i = 1,2.
lots are zoned for both types. In (i.e., n, SN,), and in model 2
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T.J. Miceli, Optimal fiscal zoning
this is simply the slope of tpl, or to, for any sized lot. For model 2, it is the slope of a ray to tRi (the dashed line in fig. 5). Note that this slope is greater than tp for any /T(2) strictly between I; and IF(l). This implies that taxes per unit of land for buyer i are higher in model 2 than in model 1 (note that average taxes are maximized for any lot where the ray through the origin is tangent to tRi). The reason is that, when the monopolistic developer is able to earn rents on land sales in model 2, the tax base per lot, Ri, rises above the tax base when land is sold competitively, pl,. As a result, overall taxes collected are at least as large in model 2 as in model 1 (given that all land is sold in both). This is true regardless of how the total undeveloped land is allocated between the two buyer types in the two models. In model 1, taxes per unit of land are tp for all types of buyers, while in model 2, taxes per unit of land are tR,/l, for buyers of type Oi. But for all buyers permitted entry in model 2, Ri -pli 20, or tR,/l, 2 tp. Thus, average taxes on land sold to any type of buyer in model 2 are at least as large as those on land sold to any type of buyer in model 1. It therefore follows that at least as much tax revenue is raised in model 2 as in model 1. This is despite the fact that lot sizes are generally larger in model 1 for any buyer type. Intuitively, in the model with a competitive land market, a portion of buyers’ overall willingness-to-pay to enter the community (on land plus taxes) is competed away by land sellers, rather than accruing partially to the developer as profits, and partially to the community in enhanced revenue, as in model 2. Finally, note that the preceding argument holds for a comparison of any solutions to the two models, including the imperfect discrimination solutions, provided all of the land is developed in both models. This follows from the fact that all solutions to model 1 lie on the tax line where average taxes are tp, and all solutions to model 2 are subject to the constraint that Ri - pli 2 0 for all i. The results in this section suggest that it may actually be in the interest of the local government to promote monopolization of the land market, perhaps through regulations that facilitate cooperation among landowners. In fact, Rose and LaCroix (1989) suggest that something like this has happened in Honolulu, where they argue that the city government facilitates the maintenance of a land cartel. 6. Conclusion This paper has provided a rigorous analysis of monopolistic discrimination by local communities in their zoning of lot sizes for new development. The model examined situations in which the community could coerce new entrants to buy the lot sizes intended for them (perfect discrimination), as well as the more realistic case where buyers must be allowed to self select
7-J. Miceli, Optimal”/iscal
zoning
595
their lot sizes (imperfect discrimination). In the second case, the optimal zoning strategy did not always entail multiple lot sizes. In some cases, only a single lot size was zoned for purchase by all buyer types, while in others, the single lot size was set large enough to exclude low-demand buyers even when they would contribute a strictly positive fiscal surplus. The role of landowners/developers proved to be important in the model. In particular, it was shown that when a single developer owns the undeveloped land, tax revenues extracted from new entrants will generally be higher than when the land is owned by multiple, competitive developers. The reason is that the single developer is able to increase the tax base of entrants by raising the sale price of lots above their opportunity cost. This result suggested the interesting possibility of monopolized local governments and developers cooperating in the determination of land-use policies in local communities.
References Carpenter, B. and D. Hetlley, 1982, Spatial-equilibrium analysis of transferable development rights, Journal of Urban Economics 12, 238-261. Cooper, R., 1984, On allocative distortions in problems of self selection, Rand Journal of Economics 15, 568-577. Edelson, N., 1975, The developer’s problem, or how to divide a piece of land most profitably, Journal of Urban Economics 2, 349-365. Epple, D. and T. Romer, 1989, On the flexibility of municipal boundaries, Journal of Urban Economics 26, 307-3 19. Epple, D., R. Filimon and T. Romer, 1988, Community development with endogenous land use controls, Journal of Public Economy 35, 133-162. Fischel, W., 1980, Zoning and the exercise of monopoly power: A reevaluation, Journal of Urban Economics 8, 283-293. Fischel, W., 1985, The economics of zoning laws (Johns Hopkins University Press, Baltimore, MD). Hamilton, B., 1975, Zoning and property taxes in a system of local governments, Urban Studies 12, 205-211. Hamilton , B., 1978, Zoning and the exercise of monopoly power, Journal of Urban Economics 5,116130. Henderson, J.V., 1980, Community development: The effects of growth and uncertainty, American Economic Review 70, 894910. Henderson, J.V., 1985, The Tiebout model: Bring back the entrepreneurs, Journal of Political Economy 93, 2488264. Maskin, E. and J. Riley, 1984, Monopoly with incomplete information, Rand Journal of Economics 15, 171-196. Mills, D., 1980, Transferable development rights markets, Journal of Urban Economics 7, 63-74. Phlips, L., 1983, The economics of price discrimination (Cambridge University Press, Cambridge, MA). Rose, L., 1989, Urban land supply: Natural and contrived restrictions, Journal of Urban Economics 25, 3255345. Rose. L. and S. LaCroix, 1989, Urban land price: The extraordinary case of Honolulu, Hawaii, Urban Studies 26, 301-314. Salant, S., 1989, When is inducing self-selection suboptimal for a monopolist?, Quarterly Journal of Economics 109. 391-397.
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Sappington, D., 1983, Limited liability contracts between principal and agent, Journal of Economic Theory 29, 1-21. Tiebout, C., 1956, A pure theory of local public expenditure, Journal of Political Economy 64, 41-24. White, M., 1975, Fiscal zoning in fragmented metropolitan areas, in: E. Mills and W. Oates, eds., Fiscal zoning and land use controls (Lexington Books, Lexington, MA). White, M., 1979, Suburban growth controls: Liability rules and Pigovian taxes, Journal of Legal Studies 8, 207-230. Windsor, D., 1979, Fiscal zoning in suburban communities (Lexington Books, Lexington, MA).