Communities, Competition, Spillovers and Open Space 1 Aaron Strong 2 and Randall Walsh 3 October 12, 2004 1 The invaluable assistance of Thomas Rutherford is greatfully acknowledged. 2 Department of Economics, University of Colorado at Boulder 3 Department 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 private 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 suboptimal 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. 1 Examples 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 residents and land rent maximizing spatially delineated jurisdictions is constructed. Our analysis 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 approach 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 amenities. 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 provision 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), Sonstelie 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, including 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 provision. 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 A j and a set of K jurisdictions each controlling a development district comprised 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: V j = V (P j, Q j, Y ), (2.1) where Y is the shared income level, P j is the price of land in region j and Q j 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 O j in the given region and its neighboring regions. The level of the open space amenity in region j is given by: Q j = j J φ j,j (O j ); (2.2) where φ j,j is a weighting matrix that defines how the contribution of open space to environmental quality decays as a function of distance from region j. Because all individuals are identical, in equilibrium, prices adjust such that V j is identical across all regions. Con- 4
ditional on region choice, demand for lot size is given by Roy s identity. D j = D(P j, Q j, 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 L j is given by L j = A j O j. A jurisdiction s rent from a given region,π j is then given by P j L j. Where P j is determined by the market clearing conditions: V (P j, O j, Y ) = V (P k, O k, Y ); j, k, (2.4) D(P j, O j, Y ) n j = A j O j ; j, (2.5) and, n j = N. J (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 2 Qualitatively the results of an open city model as well as an intermediate migration scenario provide similar 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: P j L j + X j = (Y + ( j π j )/N)(1 τ) (2.7) τ(y + j π j /N)N = j subsidyo j (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(X j, L j, Q j ) = (αx ρ j + (1 α)(βlγ j + (1 β)qγ j )ρ/γ ) 1/ρ (2.9) subject to the budget constraint: X j + P j L j = (Y + j π j /N)(1 τ). (2.10) The environmental quality function is assumed to take the form: Q j = φ 0 / dist j,j (2.11) j J 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 1 4 acre lots and approximately 1 6 of the land in any development region is allocated to open space. We 3 Cremer et al. (1997) are able to derive analytical results in a two location model with a non-scalable public 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 population 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 jurisdictions, 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 4 This 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, φ 0, 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 5 Each iteration requires identifying the equilibrium outcome under each possible set of open space levels and identifying those levels which maximize jurisdiction s profit. 6 Note this outcome is not identical to the calibration used for the utility paramaters becuase in our calibration, we have not taken into account the spatial nature of open space provision. Thus, the calibration is 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 Π j, of the regime outcome in question. Thus, compensating variation (CV) is given by: U(x j + CV, L j, Q j ) = U(x SO j, L SO j, Q SO j ) = 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 therefore 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 φ j = 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%. 7 The reduction in the over provision of open space is given by: 29.726 23.35 29.726 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 preferences 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 singlejurisdiction 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 8 Since we are uniformly fixing the open space percentage for each region in this command and control regulation, 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 instruments 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 District Region Lot 16
Figure 2: Open Space Percentage Under Duopoly Competition with Different Levels of Market Power and No Spillovers 0.2 0.22 0.21 0.2 0.19 0.17 0.195 0.19 5 0.175 0.17 5 0.155 (a) 90-10 Split (b) 80-20 Split 0.174 0.172 0.175 0.17 5 0.17 8 6 4 2 (c) 70-30 Split (d) 60-40 Split 17
Figure 3: Open Space Percentage Under Duopoly Competition with Different Levels of Market Power with Spillovers 0.22 0.2 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0.14 0.12 0.1 0.08 0.06 0.04 (a) 90-10 Split (b) 80-20 Split 0.15 0.14 0.12 0.1 0.1 0.08 0.06 0.05 (c) 70-30 Split (d) 60-40 Split 18
Figure 4: Open Space Percentage Under Competition with Increased Number of Jurisdictions with Spillovers 0.2 0.305 0.14 0.12 0.3 0.1 0.08 0.295 0.06 0.04 0.29 0.02 (a) Socially Optimal 0 0.2 (b) Monopoly 0.285 0.2 0.14 0.14 0.12 0.12 0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 (c) Duopoly 0 (d) Quadopoly 0 0.2 0.14 0.12 0.1 0.08 0.06 0.04 0.02 (e) Perfectly Competitive 0 19
Figure 5: Open Space Percentage Under Optimal Tax Instrument and Spillovers 0.2 0.14 0.15 0.12 0.1 0.14 0.08 0.13 0.06 0.04 0.12 0.02 0.11 (a) Socially Optimal 0 (b) Monopoly 0.14 0.14 0.12 0.12 0.1 0.1 0.08 0.08 0.06 (c) Duopoly 0.06 (d) Quadopoly 0.21 0.2 0.19 0.17 0.15 0.14 0.13 0.12 0.11 (e) Competitive 20
Table 1: Impact of Size of Jurisdiction in Duopoly Case with No Spillovers Social Optimal Monopoly 10% and 20% & 30% & 40% & 90% 80% 70% 60% Percent Open Space 14.500 29.726 14.67 & 15.06 & 15.49 & 16.00 & 23.35 20.40 18.65 17.50 CV as % of income 0.0 2.28% 0.79% 0.34% % 0.08% Rents 118.086 122 12.35 & 24.16 & 35.89 & 47.61 & 107.35 95.09 83.06 71.17 Ave Lot Size 0.2500 0.2055 0.239 & 0.241 & 0.245 & 0.2447 & 0.225 0.234 0.239 0.2419 Welfare Rank 1 6 5 4 3 2 Table 2: Impact of Competition on Open Space Provision with No Spillovers Social Optimal Monopoly Duopoly Quadopoly Competitive Percent Open Space 14.500 29.726 16.636 15.285 14.525 CV as % of income 0.0 2.28% 0.06% 0.009% 0.000009% Rents 118.086 122 118.731 118.344 118.095 Ave Lot Size 0.2500 0.2055 0.2438 0.2477 0.2499 Welfare Rank 1 5 4 3 2 21
Table 3: Impact of Size of Jurisdiction in Duopoly Case with Spillovers Social Optimal Monopoly 10% and 20% & 30% & 40% & 90% 80% 70% 60% Percent Open Space 14.469 29.714 0.00 & 5.39 & 8.69 & 10.67 & 20.00 17.13 14.48 13.46 CV as % of income 0.0 2.29% 0.36% 0.21% % % Rents 118.061 120.134 12.91 & 24.13 & 35.47 & 46.93 & 105.86 93.62 81.83 70.13 Ave LotSize 0.2500 0.2055 0.265 & 0.266 & 0.263 & 0.260 & 0.237 0.246 0.251 0.254 Welfare Rank 1 6 5 4 2 3 Table 4: Impact of Competition on Open Space Provision with Spillovers Social Optimal Monopoly Duopoly Quadopoly Competitive Percent Open Space 14.469 29.714 12.136 9.722 5.781 CV as % of income 0.0 2.28% % 0.55% 2.16% Rents 118.061 120.134 116.984 115.591 111.887 Ave Lot Size 0.2500 0.2055 0.2569 0.2640.02755 Welfare Rank 1 5 2 3 4 22
Table 5: Comparison of levels of optimal tax with different levels of competition and Command and Control Monopoly Duopoly Quadopoly Competitive Command & Control Percent Open Space 14.469 14.597 14.613 14.578 14.980 Shared Utility 1.2863 1.2855 1.2854 1.2855 1.2861 CV as % of income 0.000% 0.053% 0.061% 0.043% 0.009% Rents 118.061 112.421 117.962 118.022 118.213 Ave Lot Size 0.2500 0.2497 0.2496 0.2497 0.2486 Optimal Income Tax -1.09% 0.56 1.48% 3.03% NA Open Space Subsidy/Tax -0.3442 0.1775 0.4728 0.9860 NA Welfare Rank 1 4 5 3 2 23
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