Communities, Competition, Spillovers and Open
Space1
Aaron Strong2 and Randall Walsh 3
October 12, 2004
1The invaluable assistance of Thomas Rutherford is greatfully acknowledged.2Department of Economics, University of Colorado at Boulder3Department of Economics, University of Colorado at Boulder.
Abstract
We explore the impact of both the number and spatial distribution of local jurisdictions on
the decision to commit developable acreage to the provision of open space. Our analysis
differs from the existing literature on the provision of public goods in a number of ways.
First, we demonstrate that the mixed public good nature of open space (in relation to pri-
vate lot consumption) can yield outcomes where a single land rent-maximizing community
over-supplies open space relative to the utility maximizing open space level. Second, by
explicitly incorporating the spatial distribution of open space spillovers, our stylized model
shows how competition can lead not only to inefficient levels of open space provision, but
also to inefficiencies in the spatial distribution of open space. Finally, the efficacy of a
market-based approach to restore open space levels is considered.
Key Words: Open Space, Competition, Land Use.
JEL: H41, R14, Q15
1 Introduction
Open space protection is a focus of local governments across the United States. Communities
are motivated to protect open space both for its amenity and recreation value and as a tool
to manage urban growth. In the 2003 election alone, there were at least 134 ballot initiatives
regarding open space preservation in the United States. Of those, 100, or 75%, were passed
by voters generating over $1.8 billion for land conservation(Land Trust Alliance). With this
ongoing policy focus on land protection, it is important to understand how competition
between jurisdictions affects the provision of open space. To this end, we evaluate the
impacts of competition between jurisdictions on the provision of protected open space in a
spatially differentiated general equilibrium framework.
Open space is an interesting public good for two reasons that we explore in this paper.
First, the input to open space is land. Land is also an input to the private good in question,
namely residential lots. That is, through the provision of open space, we are automatically
increasing the scarcity of land for residential development. We know from hedonic studies
that proximity to undeveloped land can have a marked impact on housing prices.1 Hence,
open space protection impacts housing prices through two channels – the amenity value of
open space increases the value of proximate houses and additional land protection leads to
reductions in the supply of residential land. The second key characteristic of open space
as a public good is that the spatial distribution of open space is equally as important as
the level of provision. This spatial link implies that even when a large quantity of land
is allocated to open space, it is possible that only a few residents will benefit due to sub-
optimal distribution(al) patterns. For instance, from the perspective of households located
near the urban core, the amenity benefits of large quantities of open space at the urban
fringe may be much lower than those provided by small parks in the immediate vicinity.
1Examples of this include: Riddel (2001), Bolitzer and Netusil (2000) and Schultz and King (2001)
1
To explore these issues, a general equilibrium model incorporating homogeneous resi-
dents and land rent maximizing spatially delineated jurisdictions is constructed. Our analy-
sis within the framework of the model starts by considering the implications of different
spatially delineated competition regimes on open space allocation, jurisdiction land values
and welfare. We then consider the efficacy of market based and command and control policy
approaches for addressing inefficiencies in open space provision.
Our basic problem is to model how jurisdictions in close proximity compete in scalable
public goods with spillover benefits to the residents of neighboring jurisdictions. Our ap-
proach draws on several strands of the literature. Key concepts include: the link between
urban spatial structure and amenities; the capitalization effect of public goods when there
are spillovers; and the role of a jurisdiction/social planner’s decision in the allocation of
public goods.
First, consider the relationship between urban spatial structure and open space ameni-
ties. Begin with an area of land that is slated for residential development. The question
that this paper asks is “are more or fewer jurisdictions adventageous to the optimal provi-
sion of open space?” Our approach builds on earlier work by Brueckner (1983) in which he
explicitly models the trade off between land in the residential building footprint and land
in yard space within the context of the monocentric city model. We extend this work by
incorporating the spillovers of communal undeveloped land from one jurisdiction to another
and consider the impact of differing competition structures. Wu and Plantinga (2003) also
consider the impact of environmental amenities on urban spatial structure. Their work
focuses on how municipalities may develop given an exogenous environmental amenity such
as a hill or stream. Our work differs from this by considering an endogenous environmental
amenity.
We focus on how much open space is provided under different competition regimes.
Each competition regime implies different patterns of spatial capitalization. We model
2
these spillovers as a continuous analog to the work of Cremer, Marchand, and Pestieau
(1997). In their work, they consider how two neighboring municipalities decide to allocate
a single non-scalable public good such as a recreation center or stadium. They consider a
Nash equilibrium in public good provision and find that although the efficient level of the
public good is not typically provided in the non-cooperative equilibrium, there does exist
a cooperative system such that both municipalities share the cost and construct a single
public good – the efficient outcome of their model. Our model constructs reaction functions
for each jurisdiction in open space and suggests a result similar to that of Cremer et al.
(1997). Specifically, that when the spillovers of open space provision cannot be captured,
in general, competition will lead to under provision.
In order to highlight capitalization effects in our model, we fix the boundaries of the
development region. As suggested by Brasington (2002), if the jurisdictional boundary may
fluctuate, capitalization may not occur since more land can be allocated if the boundary is
not fixed, driving down the price to marginal cost. Thus, communities located at the center
of a metropolitan area may have greater capitalization effects than edge cities. Further,
if the boundaries are allowed to fluctuate, any jurisdiction may capture all of the amenity
rents that a potential buyer may have for the amenity.
The role of the jurisdiction in our model is that of a land rent maximizer. There is a large
body of work on the role of property value maximization in the provision of efficient public
good levels. Early examples of this literature include: Sonstelie and Portney (1978), Son-
stelie and Portney (1984), Bruekner (1983), and Epple and Zelenitz (1984). These papers
outline assumptions that equate property value maximization with the efficient provision
of public goods. In contrast, we find that under a variety of competition assumptions, in-
cluding a single jurisdiction, property value maximization does not equate to the efficient
provision of public goods. This is the result of the dual nature of open space and the fact
that an individual jurisdiction may not be able to capture all of the amenity benefits of pro-
vision. Thus, by explicitly incorporating the input to public goods production, the results
3
of the previous work is overturned.
2 The Model
The model is comprised of a set of N identical households, J spatially delineated developable
regions of area Aj and a set of K jurisdictions each controlling a development district com-
prised of one or more development regions. Households choose consumption of a numeraire
good (which includes housing but not the residential lot) whose price is normalized to 1, a
location in one of the development regions and conditional on location choice, a quantity
of land (residential lot size). We abstract from the notion of housing in order to reduce the
jurisdictions decision on vertical development and concentrate only on the horizontal aspect
of development. The relationship between lots, regions, and districts is shown in Figure 1.
Households choose development region j to maximize their indirect utility function:
Vj = V (Pj , Qj , Y ), (2.1)
where Y is the shared income level, Pj is the price of land in region j and Qj measures the
open space amenity in region j. This choice of price-environmental quality pair is similar to
the choice faced in jurisdictional competition models where households face a price-public
good pair. The open space amenity level is the spatially weighted sum of the amount of
open space Oj in the given region and its neighboring regions. The level of the open space
amenity in region j is given by:
Qj =∑j′∈J
φj′,j ∗ (Oj′); (2.2)
where φj′,j is a weighting matrix that defines how the contribution of open space to envi-
ronmental quality decays as a function of distance from region j. Because all individuals
are identical, in equilibrium, prices adjust such that Vj is identical across all regions. Con-
4
ditional on region choice, demand for lot size is given by Roy’s identity.
Dj = D(Pj , Qj , Y ) (2.3)
Jurisdictions control a set of development regions, which are aggregated into a district,
and choose the quantity of open space to provide in each region subject to the constraint
that developable land in a given region Lj is given by Lj = Aj − Oj . A jurisdiction’s rent
from a given region,Πj is then given by Pj ∗ Lj . Where Pj is determined by the market
clearing conditions:
V (Pj , Oj , Y ) = V (Pk, Ok, Y ); ∀j, k, (2.4)
D(Pj , Oj , Y ) ∗ nj = Aj −Oj ; ∀j, (2.5)
and,
∑J
nj = N. (2.6)
We consider a closed model for simplicity of analysis.2 The equilibrium outcome that arises
from competition between rival jurisdictions in quantity of open space is characterized as a
Nash Equilibrium in which each jurisdiction plays the rent maximizing strategy contingent
on the actions of their competitors.
For the second set of analyses, we incorporate a market-based policy mechanism. It is
assumed that an income tax τ is assessed on each household. The income tax is used to
finance a per acre subsidy on open space provision. A further discussion of the income tax is
provided below. Finally, to close the model and simplify the welfare analysis, jurisdictional
2Qualitatively the results of an open city model as well as an intermediate migration scenario providesimilar results.
5
land rents are recycled equally to all households. This allows us to focus exclusively on the
welfare of the households. These assumptions yield the following two budget constraints:
PjLj + Xj = (Y + (∑j
πj)/N)(1− τ) (2.7)
τ(Y +∑j
πj/N)N =∑j
subsidyOj (2.8)
The complexity of the model’s spatial Nash equilibrium precludes analytical solutions.3
We therefore adopt a numerical strategy for analyzing the implications of the model. Toward
this end, it is assumed that household utility takes a nested constant elasticity of substitution
(NCES) form. Environmental quality and lot size are placed in one nest and numeraire
consumption in the other. Thus, residents maximize utility with choice of development
region:
U(Xj , Lj , Qj) = (αXρj + (1− α)(βLγ
j + (1− β)Qγj )ρ/γ)1/ρ (2.9)
subject to the budget constraint:
Xj + PjLj = (Y +∑j
πj/N)(1− τ). (2.10)
The environmental quality function is assumed to take the form:
Qj =∑j′∈J
φ0/√
distj′,j (2.11)
We calibrate the numerical model to a benchmark. In this benchmark, individuals spend
70% of their income on consumption of the numeraire good, leaving 30% for housing lot
consumption. Further in the benchmark, individuals live on approximately 14 acre lots and
approximately 16 of the land in any development region is allocated to open space. We
3Cremer et al. (1997) are able to derive analytical results in a two location model with a non-scalablepublic good and no explicit consideration of space.
6
assume an elasticity of substitution on the upper level of the nesting structure of 1.2 and
an elasticity of substitution between land and open space of 0.8. Under this specification,
the lot-open space bundle has a stronger substitutes relationship with the numeraire than
do open space and lot size in the lower nest.
The development area is defined as follows. We assume that there are 100 one acre
regions of developable land arranged in a square 10 X 10 grid and parameterize to a popu-
lation of 342.4 Note also that household income is normalized to one. This corresponds to
a baseline utility level of 1 using the calibrated share form of the NCES utility. That is, if
the resident purchases the baseline lot-numeraire bundle with the environmental quality as
specified under the baseline assumptions the individual will achieve a utility of 1.
3 Numerical Method
We solve for the Nash Equilibrium in the model using a diagonalization method. In our
model, once the choice of open space for each development region is made by the juris-
dictions, all other variables are uniquely determined by the market clearing equilibrium
conditions. Here, individual jurisdictions compete in both quantity of residential land and
environmental quality. From a given choice of open space, we derive the open space amenity
associated with all development regions using the weighting matrix defined in equation 2.2
with the weights declining with the sqareroot of distance.
The solution algorithm is as follows. For each jurisdiction, we initialize the amount of
open space that they provide to an initial level. Next, consider first one of the jurisdictions.
Taking all other jurisdictions open space allocations as given, the jurisdiction maximizes
rents by choosing a level of open space in each of the regions that it controls. Next, we
iterate over each of the jurisdictions in the economy identifying their optimal reponse and
4This population corresponds with the benchmark condition on land consumption.
7
updating the open space levels in their region.5 We continue to iterate over the set of
jurisdictions until the system converges. For the social planner’s problem (shared utility
maximization), we simply iterate over choices of open space in each parcel until the system
converges to a maximal shared utility level.
4 Baseline Results and Welfare Calculations
We first consider the case of a social planner maximizing the shared utility level of the
residents. Results are presented in Figure 4a and Table 4. Given our calibration, under the
socially optimal outcome, in the aggregate there is a 14.5% allocation of open space in the
development area.6 Because fewer regions benefit from spillovers when land is protected
near the boundary, this open space is not evenly distributed over the development area.
Thus, we see in Figure 4a, there is a greater concentration of open space in the center of
the development area. Under the social planner, the allocation maximizes the value of any
parcel of open space accounting for the spillovers. As we perturb the spillover coefficient, φ0,
not only is the amount of open space perturbed but also the distribution. As we decrease
spillovers, the total amount of open space increases and the slope, as we move from the
center, decreases. That is, the distribution becomes flatter and more uniformly distributed.
Analysis of welfare changes in the model is complicated by the general equilibrium
nature of the simulations. To assess these issues, we adopt a specification of compensating
variation that is consistent with the model. As a baseline utility for the welfare calculations
we use the socially optimal shared utility level. Because households at different locations
consume different bundles of numeraire, land and environmental quality implying different
5Each iteration requires identifying the equilibrium outcome under each possible set of open space levelsand identifying those levels which maximize jurisdiction’s profit.
6Note this outcome is not identical to the calibration used for the utility paramaters becuase in ourcalibration, we have not taken into account the spatial nature of open space provision. Thus, the calibrationis not supportable as an equilibrium even given the pricing structure derived to calibrate the parameters.
8
welfare implications for the uses of land resources, we do not allow households to relocate in
the welfare calculation. Further, because location is fixed and open space level within the
region is fixed, we also do not allow lot size adjustments in the welfare calculation. Thus,
the only portion of households allocations that will be allowed to adjust in the welfare
calculation is that of the numeraire good. Thus, to identify the compensating variation
for a given location under a given regime, we compute the amount of income that would
have to be given to an individual in order to restore their utility level to that of the socially
optimal utility level conditional on that money being spent on the numeraire good. Further,
since incomes may vary across regimes due to the recycling of jurisdictional land rents, we
normalize by the income level, 1 +∑
j∈J Πj , of the regime outcome in question. Thus,
compensating variation (CV) is given by:
U(xj + CV, Lj , Qj) = U(xSOj , LSO
j , QSOj ) = U
SO (4.12)
where the superscript represents the socially optimal allocation and barred variables are
fixed according to the regime and location under consideration. Our welfare measure there-
fore monetizes the welfare loss to the household under a given regime relative to the socially
optimal utility level.
5 Competition without Spillovers
In order to analyze the role of competition in a systematic manner, we begin by isolating the
market power effect of open space provision and abstract from the role of benefit spillovers
across jurisdictions and within jurisdictions. In our model, this amounts to setting φj =
0,∀j ∈ J . First, we consider a duopoly case and evaluate different configurations of land
within the development area. Second, we consider symmetric configurations with increasing
numbers of jurisdictions.
To consider the role of market power in the duopoly case, we assume that a small
9
developer is located along one edge of the development region. Results for this analysis are
contained in Table 1 and Figure 2. The first observation to make is that moving from a
monopoly case to a case with a small competing jursidiction (90%-10% split), has a large
effect on the provision of open space. This small decrease in market power leads to a large
decrease in welfare loss relative to the socially optimal, on the order of 66%, as competition
alleviates the under provision of residential land that occurs in the single jurisdiction case.
As we further decrease this market power in steps of 10%, we see roughly a halving of the
welfare loss at each step. Thus, market power plays a key role in the provision of open
space. With even a small amount of competition, we see a reduction in excess open space
provision by the dominant jurisdiction. This relaxes the monopoly derived housing supply
restriction by 41%.7
As a second approach to the analysis of the impact of market power in the absence
of spillovers, we consider the impact of moving from a single jurisdiction to a model with
multiple symmetric jurisdictions. We progress from a monopoly jurisdiction through the
case of two and 4 jurisdicitons to a case of 100 separate jurisdicitons that we call the
“competitive model.” Results from this analysis appear in Table 2. As with the case of
an increasingly larger competing jurisdiction without spillovers, decreased market power
moves each jurisdicition toward the socially optimal allocation. The largest increase in
welfare from competition comes at the first step, that of moving from the monopoly to the
duopoly. Under this change, we see a weakening of the over provision of open space on the
order of 86%.
7The reduction in the over provision of open space is given by: 29.726−23.3529.726−14.500
.
10
6 Competition with Spillovers
We now consider how competition affects the allocation of open space when spillovers are
present. As discussed in the model section, in this experiment we assume that there exist
spillovers between each development region and not just between jurisdicitons. As in the
no spillovers case, we begin by considering the duopoly case and consider the role of market
power with different initial allocations of land. Results are presented in Table 3 and Figure
3. First, with a very small jurisdiction, we see free riding by the small jurisdiction on
the over provision by the large jurisdiction. In fact, the small jurisdiction would like to
expand its boundaries in order to provide even more area for residential lots. Our second
observation is that as market power decreases and jurisdictions become more symmetric,
welfare increases. Although in the limit, we have an underprovision of open space, the
welfare loss is not that great. Finally, decreasing the market power beyond a 70-30 split has
virtually no effect on welfare. Although jurisdicitons are better able to capture spillovers
and there is less market power, these effects move in opposite directions and almost perfectly
offset each other.
The next step in the analysis is to consider the role of symmetric competition in the
presence of spillovers. As above, the analysis begins with a single jursidiction and progresses
through a system of symmetric jurisdictions ending with each parcel of land owned by one
of 100 different jurisdictions.
The analysis assumes that the jurisdictions have perfect information regarding the pref-
erences of potential residents. Thus, in the monopolist case with a fixed population size, we
have a perfectly price discriminating jurisdiction. Under this framework, initial intuition
might suggest that the monopolist will provide the optimal level of open space because the
value of open space is capitalized in the price of the lot and all spillovers are captured by
the monopolist. As we have already seen in the case without spillovers, this is not the case.
Given the dual nature of open space as an input to both the private as well as public good
11
and the fact that there is a fixed a set of individuals in the development area, the closed
city assumption, the monopolist chooses to over-provide the public good in order to restrict
the amount of the private good available on the market. Thus, we may say that, in general,
property value maximization does not necessarily lead to utility maximization. However,
the general shape of the distribution is similar to that of the socially optimal distibution,
that of larger amounts of open space at the center of the development area with decreasing
open space toward the edge of the development area. Results for this analysis appear in
Table 4 and Figure 4.
Next, consider the case of a duopoly. Assume that the two jurisdictions have equal areas
and are symmetric within the development area. Under this specification, not only is the
total amount of open space provided below the efficient level, but the spatial distribution
is also much more inefficient than under the monopolist. Because the jurisdictions are
acting to maximize their land rents, in the Nash equilibrium they will allocate most of
the open space away from where the other jurisdiction is located in order to maximize the
internalization of the spillovers. This implies that most of the open space occurs not at
the center of the development area, but at the center of the district controlled by each of
the two competing jurisdictions. Under this competition regime, each jurisdiction provides
approximately 12.1% of their area to open space. Thus, we see a shift from an over provision
by the monopolist to a slight under provision with the imposition of competition. As
we move to greater competition, these results are further exacerbated in both levels and
distribution. In the competition without spillovers case, we saw a move toward the socially
optimal with increased competition. By introducing spillovers, competition reduces the
ability of the jurisdiction to capture rents from open space provision. There is an inherant
trade off with increased competition between reducing the supply restriction and the ability
of the jurisdiction to capture rents. Thus, as competition increases, open space necessarily
decreases and the inablity to capture benefits dominates, causing an underprovision.
12
7 Market Based Instrument vs. Command and Control
Finally, we consider the effectiveness of a uniform market-based instrument in the presence
of inter-jurisdictional competition and cross-jurisdiction spillovers – identifying the utility
maximizing income tax-open space subsidy pair under each competition regime. Under
this market-based instrument, a per unit open space subsidy is first identified and then in
equilibrium an income tax rate is set which raises exactly the level of revenue needed to
fund the resulting subsidies. The subsidy level is then chosen to maximize the shared utility
level. We compare the outcome under this optimal uniform market-based instrument to a
command and control regulation that mandates a uniform quantity of open space per acre.
Table 5 and Figure 5 present the results from this analysis. First note that in the single-
jurisdiction model, because the unregulated equilibrium leads to an overprovision of open
space, the optimal tax and subsidy are both negative. As the first column in Table 5 shows,
in the case of a single jurisdiction, a subsidy of -0.3442 units of income per acrereturns the
system to the socially optimal levels of open space, land rents, income and shared utility
level. This result is fairly intuitive. Because all households receive the same income, the
income tax is identical to a non-distortionary head tax. In equilibrium, each subsidy level
is associated with a specific aggregate open space level. And, because the single jurisdiction
internalizes all spillovers, the jurisdiction chooses the optimal distribution of open space
across locations for each unique aggregate open space level. For comparison, the final column
of table 5 reports the optimal level of open space under a uniform command and control
regulation. Because the command and control regulation does not allow for adjustments to
reflect the variation in spillovers across location, the command and control policy slightly
under-performs the market instrument under a single jurisdiction.8
Once we move from the single jurisdiction case, because of the spatial inefficiencies
8Since we are uniformly fixing the open space percentage for each region in this command and controlregulation, the command and control level is independent of competition regime.
13
that are introduced by spatially discrete competing jurisdictions, it is no longer possible
to return the system to the social optimal via the tax-subsidy instrument. As is shown in
Table 5, once competition is introduced the open space “tax” is replaced by a subsidy. The
required subsidy increases with competition from 0.1775 under the duopoly case to 0.9860
in the competitive model. Further, in contrast to the single jurisdiction case, once spatial
inefficiencies in open space associated with the competing jurisdictions are introduced the
uniform command and control regulation clearly dominates the optimal uniform tax-subsidy
pair.
8 Conclusion
This paper highlights two important types of market failure that link the spatial structure
of jurisdictional competition to the provision of open space. First, because open space is an
essential input to the development of residential lots, when rent-maximizing jurisdictions
are able to exert market power they may provide open space purely as a by-product of their
attempts to drive up prices through supply restrictions. Second, when open space amenities
spillover across jurisdictional boundaries, changes in the spatial distribution of of competing
jurisdictions lead to changes in the ability of these jurisdictions to internalize the benefits
of open space provision.
Our analysis considers the tradeoffs inherent between these two types of market failure.
Specifically, we demonstrate that the incentives for a monopoly jurisdiction to restrict supply
(potentially leading to the provision of open space beyond the efficient levels) provides
an incentive to encourage greater levels of jurisdictional competition while the inability
of multiple spatially fragmented jurisdictions to internalize the spillover benefits of open
space encourages reductions in jurisdictional competition. The results from our numerical
simulations suggest that externalities related to market power are largely ameliorated at
low levels of competition. Given the problem of internalizing amenity spillovers as the
14
number of competing jurisdictions grows, the analysis therefore finds that social welfare is
maximized when jurisdictional competition exists, but at low levels. Although these results
are for a single parameterization of the model, the qualitative results will hold for most
parameterizations of the model.
The final thrust of our analysis considers the efficacy of uniform market-based instru-
ments relative to a uniform command and control regulatory approach in the face of these
competing market failures. In the case of a single jurisdiction, we find that the market
based-instrument is capable of restoring the socially optimal open space allocation and
therefore clearly dominates the command and control regime. However, once jurisdictional
competition is introduced, inefficiencies in the spatial distribution of open space that are
introduced through this competition cause the market-based approach to under-perform
relative to the command and control strategy.
15
Figure 1: Definition of the Development Area
Development Area
Region
District
Lot
16
Figure 2: Open Space Percentage Under Duopoly Competition with Different Levels ofMarket Power and No Spillovers
0.16
0.17
0.18
0.19
0.2
0.21
0.22
(a) 90-10 Split
0.155
0.16
0.165
0.17
0.175
0.18
0.185
0.19
0.195
0.2
(b) 80-20 Split
0.16
0.165
0.17
0.175
0.18
(c) 70-30 Split
0.162
0.164
0.166
0.168
0.17
0.172
0.174
(d) 60-40 Split
17
Figure 3: Open Space Percentage Under Duopoly Competition with Different Levels ofMarket Power with Spillovers
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
(a) 90-10 Split
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
(b) 80-20 Split
0.06
0.08
0.1
0.12
0.14
0.16
0.18
(c) 70-30 Split
0.05
0.1
0.15
(d) 60-40 Split
18
Figure 4: Open Space Percentage Under Competition with Increased Number of Jurisdic-tions with Spillovers
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
(a) Socially Optimal
0.285
0.29
0.295
0.3
0.305
(b) Monopoly
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
(c) Duopoly
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
(d) Quadopoly
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
(e) Perfectly Competitive
19
Figure 5: Open Space Percentage Under Optimal Tax Instrument and Spillovers
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
(a) Socially Optimal
0.11
0.12
0.13
0.14
0.15
0.16
(b) Monopoly
0.06
0.08
0.1
0.12
0.14
0.16
0.18
(c) Duopoly
0.06
0.08
0.1
0.12
0.14
0.16
0.18
(d) Quadopoly
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0.21
(e) Competitive
20
Table 1: Impact of Size of Jurisdiction in Duopoly Case with No SpilloversSocial Optimal Monopoly 10% and
90%20% &80%
30% &70%
40% &60%
Percent Open Space 14.500 29.726 14.67 &23.35
15.06 &20.40
15.49 &18.65
16.00 &17.50
CV as % of income 0.0 2.28% 0.79% 0.34% 0.16% 0.08%Rents 118.086 120.162 12.35 &
107.3524.16 &95.09
35.89 &83.06
47.61 &71.17
Ave Lot Size 0.2500 0.2055 0.239 &0.225
0.241 &0.234
0.245 &0.239
0.2447 &0.2419
Welfare Rank 1 6 5 4 3 2
Table 2: Impact of Competition on Open Space Provision with No SpilloversSocial Optimal Monopoly Duopoly Quadopoly Competitive
Percent Open Space 14.500 29.726 16.636 15.285 14.525CV as % of income 0.0 2.28% 0.06% 0.009% 0.000009%Rents 118.086 120.162 118.731 118.344 118.095Ave Lot Size 0.2500 0.2055 0.2438 0.2477 0.2499Welfare Rank 1 5 4 3 2
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Table 3: Impact of Size of Jurisdiction in Duopoly Case with SpilloversSocial Optimal Monopoly 10% and
90%20% &80%
30% &70%
40% &60%
Percent Open Space 14.469 29.714 0.00 &20.00
5.39 &17.13
8.69 &14.48
10.67 &13.46
CV as % of income 0.0 2.29% 0.36% 0.21% 0.18% 0.18%Rents 118.061 120.134 12.91 &
105.8624.13 &93.62
35.47 &81.83
46.93 &70.13
Ave LotSize 0.2500 0.2055 0.265 &0.237
0.266 &0.246
0.263 &0.251
0.260 &0.254
Welfare Rank 1 6 5 4 2 3
Table 4: Impact of Competition on Open Space Provision with SpilloversSocial Optimal Monopoly Duopoly Quadopoly Competitive
Percent Open Space 14.469 29.714 12.136 9.722 5.781CV as % of income 0.0 2.28% 0.18% 0.55% 2.16%Rents 118.061 120.134 116.984 115.591 111.887Ave Lot Size 0.2500 0.2055 0.2569 0.2640 .02755Welfare Rank 1 5 2 3 4
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Table 5: Comparison of levels of optimal tax with different levels of competition and Com-mand and Control
Monopoly Duopoly Quadopoly Competitive Command & ControlPercent Open Space 14.469 14.597 14.613 14.578 14.980Shared Utility 1.2863 1.2855 1.2854 1.2855 1.2861CV as % of income 0.000% 0.053% 0.061% 0.043% 0.009%Rents 118.061 112.421 117.962 118.022 118.213Ave Lot Size 0.2500 0.2497 0.2496 0.2497 0.2486Optimal Income Tax -1.09% 0.56 1.48% 3.03% NAOpen Space Subsidy/Tax -0.3442 0.1775 0.4728 0.9860 NAWelfare Rank 1 4 5 3 2
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