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Spatial externalities and land use regulation: an integrated set of multiple density regulations

Kono, Tatsuhito;Joshi, Kirti Kusum

2017 Journal of Economic Geography

doi: 10.1093/jeg/lbx021

Abstract In a continuous city with three distinct land use zones consisting of business, condominiums and detached houses, we derive the formulae which simultaneously optimize regulations on building size, lot size and the three zonal boundaries under the existence of agglomeration economies and traffic congestion. The formulae show that the optimal regulations require a combination of downward and upward adjustments to the market sizes of buildings within both the business zone and the condominium zone, followed by minimum lot size regulation in the housing zone. The outer boundaries of the condominium and housing zones should be regulated inward whereas the optimal business zone boundary regulation depends on the trade-off between agglomeration economies and traffic congestion costs. 1. Introduction Land use regulations are common urban policies for managing spatial externalities1 caused by high concentration of population in cities. Spatial externalities can be positive or negative. For instance, concentration of workers enhances communication and thus facilitates exchange of innovative ideas (see Rauch, 1993; Ciccone and Hall, 1996; Duranton and Puga, 2001; Moretti, 2004). In contrast, concentration of population also has negative effects such as traffic congestion. In order to internalize such spatial externalities, local governments generally intervene in the urban space market through simultaneous enforcement of multiple regulations on building size and lot size, and by zoning the city into different land uses. For example, the City of Portland in the USA has imposed zonal land use restrictions and floor area ratio regulation (FARR) as well as an urban growth boundary (UGB) regulation.2 This article explores how multiple land use regulations should be simultaneously imposed across a city under the existence of agglomeration economies and traffic congestion. Considering a monocentric and continuous city with three distinct land use zones consisting of business, condominiums and detached houses, we derive the formulae—composed entirely of observable variables—which allow simultaneous optimization of regulations on building size, lot size and the three zonal boundaries across the city. In addition, from the formulae, we derive some theoretical properties of optimal land use regulations with new interpretation. Land use regulations are used as practical alternatives to superior policies—such as congestion pricing against traffic congestion—which are often politically infeasible, as discussed in Lindsey and Verhoef (2001) or King et al. (2007), or incomplete for unavoidable reasons such as enormous implementation costs. In fact, current practical applications of congestion pricing are far from the first-best congestion pricing, and moreover, the applications are limited to a few advanced cities only (e.g., London, Milan, Oslo, Singapore, and Stockholm). In contrast, land use regulations are very common worldwide. In the USA, 92% of the jurisdictions in the 50 largest metropolitan areas have zoning ordinances of one kind or another in place, and only 5% of the metropolitan population live in jurisdictions without zoning (Pendall et al., 2006). However, it is not an easy task for local governments to rationally design optimal land use regulations because they have to take into account the change in the distortions caused by the regulations as well as spatial externalities. Land use interventions and their effects on the welfare level of urban residents have been discussed in many previous studies (see Brueckner (2009) for a survey of theoretical analyses, or Evans (1999) for a survey of empirical investigations). However, although most real-world cities are simultaneously regulated under multiple rules, previous studies examine one land use regulation at a time (or at most, one regulation in the presence of UGB regulation). In related literature, many papers including Pasha (1996) and Wheaton (1998) explore lot size regulation but ignore business area zoning or building size regulation, while Helpman and Pines (1977) and Stull (1974) determine optimal zoning between manufacturing and residential areas without considering regulations on building size or lot size, and Rossi-Hansberg (2004) simulates the same in the existence of agglomeration spillovers of firms. Arnott and MacKinnon (1977), Bertaud and Brueckner (2005) and Brueckner and Sridhar (2012) study the welfare cost of building size regulation in isolation. When optimality of regulation is analyzed by considering only one regulation despite the presence of other regulations, we might actually obtain a nonoptimal result. For example, Kanemoto (1977) shows that in a congested monocentric city, a properly chosen UGB is welfare improving whereas Pines and Sadka (1985) and Wheaton (1998), using the same model, show that controlling lot size3 without UGB can lead to a first-best policy, implying that in the presence of lot size regulation, UGB regulation is unnecessary. This demonstrates that simultaneous consideration of multiple land use regulations gives a different optimal solution than when addressing a single regulation. Some recent papers have demonstrated the need of multiple regulations or at least variations within a single regulation. For instance, Joshi and Kono (2009), Kono et al. (2010, 2012), and Pines and Kono (2012) show that optimal FARRs consist of maximum and minimum FARRs in combination. By definition, floor area ratio (FAR) is the ratio of the total floor area of a building to the size of the plot on which the building is built. FARR is the most common form of building size regulation. Optimal regulations such as those achieved in the aforementioned studies are second-best policies substitutable for the first-best policies. Indeed, according to numerical simulations in Kono et al. (2012), optimal FAR and UGB regulations increase social welfare by approximately 70–85% of the welfare gain of the first-best policy. However, these studies do not consider what happens to the ‘optimal’ regulation if, for example, in addition to the FARR, lot sizes of detached dwellings are also regulated or land use zoning is also in place. Against this backdrop, this article addresses an important research gap on simultaneous imposition of multiple land use regulations across a city. We consider a monocentric city with three distinct land use zones—consisting of business, condominiums and housing lots—which closely resemble land use observed in real-world cities. The city has agglomeration economies in the business zone and traffic congestion across the city. Three points are noted for this setup. First, the setting of geography and residents’ preference plays an important role on the properties of optimal land use regulations. For example, by setting mixed-use zones in a monocentric and non-monocentric city having residents with idiosyncratic tastes, and in a system of cities with homogeneous residents, respectively, Anas and Rhee (2006, 2007) and Anas and Pines (2008) show that an expansive UGB may be necessary, which contradicts the traditional conclusions based on a monocentric city model. A monocentric city with homogeneous residents and firms, which we adopt, is a basic model. The results from a monocentric city model should be reasonably valid for any modifications on the model as long as the fundamental relationship between population (or employment) density and commuting costs is preserved (Brueckner, 2007; Sridhar, 2007). Secondly, distinction between the central city residential land use and that in the suburb is important for the study of certain urban phenomena. For example, using a two-zone model, Brueckner and Helsley (2011) relate urban blight with urban sprawl, and determine price-based corrective policies as well as a UGB regulation to shift population from the suburb back to the central city. However, Anas and Pines (2013), assuming local public goods with scale economies, demonstrate that this conclusion is not true in a system of cities, as population should be shifted by UGBs being imposed to create more and smaller cities. In any geographical setting, distinction between center and suburb is, therefore, a key factor. Thirdly, assuming the presence of unpriced traffic congestion and agglomeration spillovers of firms, we analytically explore the second-best adjustment to firm and residential density through land use regulations that include not only zoning but also regulations on building size and lot size. In a related paper, Rossi-Hansberg (2004), besides addressing optimal zoning between business and residential areas, also examines the first-best distribution of workers and residents in a city in the presence of spatial production externalities and uncongested commuting costs. This first-best distribution is achieved by location-variable labor subsidies, which are equivalent to transportation subsidies (Helsley and Strange, 2007). However, the optimal location-varying building size regulation, which is targeted by the current paper, cannot achieve the first-best distribution of workers because of the residual deadweight losses in the regulated building size markets (see Figure 3 in the current article). In a more recent paper, Rhee et al. (2014) also treat land use regulations in cities with agglomeration economies and traffic congestion. A key difference compared with our article lies in the geographical setting. The theoretical model of Rhee et al. (2014) assumes only two discrete zones with mixed land uses allowed in each zone,4 whereas our model addresses a continuous city with three different land use zones. Applying optimal control theory to the continuous city, our model achieves optimal density regulation that changes continuously from the center. In addition, we separately treat FARR and lot size regulation because building-size regulation necessarily generates deadweight loss caused by the regulation itself (see Kono et al., 2012) whereas lot size regulation has no deadweight losses (see Wheaton, 1998). Under building size regulation, households can choose their optimal floor size within the regulated buildings. That is, building size regulation controls population density indirectly, whereas lot size regulation does this directly. In addition, we design optimal regulations on multiple zonal boundaries between the business zone, condominium zone and lot housing zone. We also show by how much the building size, the lot size and the zone sizes should differ from those determined at the market equilibrium. The differences are shown in the formulae composed of empirically observable economic variables. All these results are useful for policy makers to design optimal density and zonal regulations in a monocentric city.5 The remaining portion of the article is organized as follows. In Section 2, we present and explain key results even before we introduce our model later in Section 3 so as to facilitate understanding of our inherently complicated theoretical exercise on optimal simultaneous regulations. Section 4 derives necessary conditions for maximizing social welfare using regulations. Section 5 explores optimal regulations on FAR, lot size and multiple zonal boundaries, using the necessary conditions. Finally, Section 6 concludes the article. 2. Key results As mentioned above, our monocentric city model is divided into business zone consisting of firms, condominium zone and lot housing zone. The objective of our study is to find optimal FARR and lot size zoning in the presence of optimal zonal boundaries. In other words, using FARR, lot size zoning and zonal boundary controls as planning tools, this study aims to determine optimal firm density in the business zone and optimal population density in the residential zones. The city population, and thereby the worker population, is fixed. Workers in the business zone (or firms) benefit from agglomeration economies, but residents across the city (including workers in the business zone) also face congestion externality while commuting to and within the business zone. With this setup, our results are summarized as follows. Result 1 (Optimal firm density in business zone).When agglomeration economies are weaker than traffic congestion costs, the optimal firm density in the business zone tends to be higher (resp. lower) in the more central (resp. peripheral) locations relative to the market firm density, economizing on commuting costs. The reverse case is similarly explained. Result 2 (Optimal size of business zone).Enlargement of the business zone decreases agglomeration economies but reduces total commuting distance and hence traffic congestion costs; the optimal size of the business zone depends on the net effect. Result 3 (Optimal population density in residential zones).Relative to the market population density in the residential zones consisting of condominiums followed by detached houses, the optimal population density is higher (resp. lower) in the more central (resp. peripheral) locations. Result 4 (Optimal size of residential zones).In the optimal case, the residential zones are more compact relative to the market equilibrium case, economizing on traffic congestion costs. Note that minimum (resp. maximum) FARR leads to upward (resp. downward) adjustment to market building size and hence density. Likewise, minimum lot size zoning leads to lower population density. The following sections derive these results rigorously and interpret them intuitively. 3. The model 3.1. The city The model city is closed, monocentric and linear with a width of unity and size defined by m∈[−MH,MH], where m denotes distance from the city center. We assume that each zone is sufficiently large relative to the size of a lot such that the location of a lot or building can be expressed in terms of m. Following a real-world urban land use pattern, and as depicted in Figure 1, which shows only the right half of the symmetrical city, the city is divided into the following three zones in successive order: (i) the central business district (CBD) or business zone, Zone B (m∈[0,MB]), consisting of office buildings, (ii) the condominium zone, Zone C (m∈[MB,MC]) and (iii) extending to the urban boundary, the lot housing zone, Zone H (m∈[MC,MH]), consisting of single-family houses. Figure 1 View largeDownload slide The model city. Figure 1 View largeDownload slide The model city. Zone B and Zone C are regulated by FARR. We assume that all buildings are built on lots of equal size which is normalized to unity; therefore, the FAR of a building is equivalent to its total floor supply. Let Fk′( k′∈{B,C}) denote the FAR of a building in zone k′. Likewise, the lots within Zone H are regulated; let fH denote the lot consumption per household in the zone. The city is inhabited by identical households. One member of each household commutes to the CBD, where all firms, and thereby jobs, are located. The city population is identified with the number of households, denoted 2N¯, which is divided equally between the two halves of the symmetric city. The city is closed, which implies that 2N¯ is exogenously fixed. Buildings in Zone B and Zone C are constructed by developers whereas, following most of the previous studies (e.g., Pines and Sadka, 1985), we ignore housing capital in Zone H, assuming that land is directly consumed by the residents. One possible interpretation is that households construct their houses using composite goods. Finally, we assume the so-called public land ownership under which residents share the city land equally. Hereafter, we basically model the right half of the city unless it is unavoidable to model both sides. 3.2. Externalities and regulations We consider two types of externalities: (i) agglomeration economies that arise from communication between firms in Zone B and (ii) traffic congestion across the city. To address these two externalities, FARR, lot size zoning and zonal regulation on zonal boundaries are imposed. The policy variables are as follows. Definition 1 (Policy Variables). The city government regulates (1) FAR at each location in Zone B and Zone C, i.e., Fk′(m), (k′∈{B,C}), m∈[0,MC], (2) lot size at each location in Zone H, i.e., fH(m), (m∈[MC,MH])under lot size regulation, and (3) three zonal boundaries, i.e., Mk, (k∈{B,C,H}). Firm density in Zone B and population density in Zone C are adjusted only by FARR, whereas in Zone H, only lot size regulation adjusts population density. As shown in Kono et al. (2012), FARR generates price distortions whereas lot size regulation generates no price distortion. 3.3. Firms’ behavior All firms are located within Zone B, and they have identical production function. We model single-worker production that uses floor area as an input besides labor. The production function is expressed as AX(fB), where A is the communication-based factor productivity function; fB is the per-firm floor area and X(fB) is the partial production function.6 Following Borukhov and Hochman (1977), O’Hara (1977), and Ogawa and Fujita (1980), we assume that each worker communicates inelastically with workers in the other firms. Although inelastic communication is less realistic, such inelastic bilateral communication trips can represent agglomeration economies in the sense that firms would concentrate more on saving social communication trip costs. This is why this setting has been used in the aforementioned previous theoretical studies. Although we focus only on the right side of the CBD, firms on the right-hand side communicate with firms all over the CBD including those on the left-hand side. For each firm, the number of trips to each other during a certain period is normalized to 1 without loss of generality. With the number of total workers, and thus the number of single-worker firms being 2N¯, the total communication trip for each firm is 2N¯−1. In this case, A=A(2N¯−1)≡A¯, which implies that A¯ is constant because 2N¯−1 is constant. The profit for a firm at m, denoted π(m), is then given by: π(m)=A¯X(fB)−g(m)−w(m)−rB(m)fB, m∈[0,MB], (1) where g(m), w(m) and rB(m), respectively denote the communication trip cost, wage and floor rent for the firm at location m. The communication trip cost g(m) is defined as follows. A worker at m communicates with a worker at x at the cost of τ|x−m|, where |x−m| is the distance between the corresponding firms, and τ is the constant unit-distance cost. The worker communicates with all other workers; so the total communication trip cost borne by a worker at m, say G(m), is given by G(m)=∫−MBMBn˙B(x)[τ|x−m|]dx, m∈[0,MB], (2) where n˙B(x)≡∂nB/∂x, which denotes worker density at x, and nB(m)=∫0mn˙B(x)dx is the number of total workers working at the firms located between the CBD center and location m. A dot over a variable, hereafter, denotes derivative of the variable with respect to distance. The trip cost G(m) is physically determined by the supply-side condition (or transport capacity). Workers must pay at least the supply-side cost, but they might be paying more (e.g., by driving inefficiently slowly or by consuming more fuel for unnecessary acceleration). Therefore, the relation between the communication trip cost G(m) and the actual payment g(m) is expressed as an inequality condition: g(m)≥G(m), m∈[0,MB]. (3)7We differentiate the right-hand side of Equation (2) with respect to m, which yields dG(m)dm≡G˙(m)=2nB(m)τ, m∈[0,MB],8 (4)8and an initial condition is obtained as G(0)=2∫0MBn˙B(m)τmdm. (5) Rent bidding among firms yields π(m)=0. The bid rent is expressed as rB(g(m),w)=max fB(A¯X(fB)−g(m)−w(m)fB), m∈[0,MB]. (6) The first-order condition yields fBA¯∂X/∂fB−A¯X(fB)+g(m)+w(m)=0. (7) Thus, fB is the function of g(m) and w(m). As Equation (7) shows, a firm considers only its private communication costs, whereas its proximity to other firms may allow the latter to economize on their communication costs. This is what Kanemoto (1990) calls ‘locational externality’. Such locational externality can be adjusted by density regulations. However, under the assumption of an endogenous floor area consumption, the optimal FARR cannot lead to the first-best labor distribution.9 Finally, we define wage rate. Denoting the commuting cost from MB to m as tB(m), the wage w(m) should follow w(m)=w(MB)+tB(m), (8) because the wage at each location should compensate for the commuting cost tB(m). 3.4. Developers’ behavior Developers supply buildings in Zone B and Zone C under the FARR. Let πk′d, k′∈{B,C} denote developers’ profit from the construction of a building in zone k′, which is given by πk′d(mk′)=Fk′rk′(mk′)−Sk′(Fk′)−Rk′(mk′), k′∈{B,C},mB∈[0,MB],mC∈[MB,MC], (9) where Sk′(Fk′) denotes the total construction cost of FARR-regulated floor area Fk′. rk′ and Rk′ denote floor rent and land rent, respectively. Note that buildings are constructed on lots of equal size normalized to 1. Considering perfectly competitive developers, the zero-profit condition is given by πk′d(m)=0, k′∈{B,C}, which then yields Rk′(m)=Fk′(m)rk′(m)−Sk′(Fk′(m)), k′∈{B,C},mB∈[0,MB],mC∈[MB,MC]. (10) 3.5. Commuting cost—an external factor To consider congestion externality, we adopt transport functions à la Wheaton (1998) and Brueckner (2007). For simplicity, the commuting cost is divided into two parts: that within the residential area (i.e., Zone C and Zone H) and that within the business area (i.e., Zone B), denoted t(·) and tB(·), respectively (See Figure 1). The unit-distance commuting cost within the residential area t(m) borne by the resident at location m has the following condition: dt(m)dm≡t˙(m)≥ξ+δ[N¯−n(m)ρ(m)]γ,m∈[MB,MH], (11) where n(m) is the total commuter population residing beyond the CBD edge up to location x, and thus N¯−n(m) is the total commuter population that joins traffic at location m on the way toward the CBD. ξ is the free-flow commuting cost factor; δ and γ are positive parameters; and ρ(m) is the road capacity at location m, given exogenously. Equation (11) uses an inequality condition. The left-hand side is the demand-side cost, which a commuter pays, while the right-hand side is the supply-side cost, which is determined physically. Similar to the communication trip cost in the business area, commuters might be paying more than the supply-side cost.10 The total commuting cost is defined as t(m)≥∫MBm[ξ+δ[N¯−n(x)ρ(x)]γ]dx,m∈[MB, MH]. (12) Similarly, the commuting cost within the business zone borne by a worker employed at a firm located at m, tB(m), is defined as tB(m)≥∫mMB[ξ+δ[nB(x)ρ(x)]γ]dx, (13) and dtB(m)dm≡t˙B(m)≤−ξ−δ[nB(m)ρ(m)]γ,m∈[0, MB].11 (14) where nB(m) denotes number of firms located between the city center and location m. The right-hand side in both Equations (13) and (14) implies that there is congestion caused by commuting trips. Note that as distance m from the center decreases, the traffic volume, implied by nB(m), also decreases because most commuters would have already reached their firms.12 3.6. Household behavior Each household worker earns wage w per period by working in the CBD. The household’s expenditure comprises commuting, housing and non-housing commodity costs. Private cars are the only mode of commuting. For simplicity, we assume a quasi-linear utility function for households living in Zone C and H, denoted VC and VH, respectively, which is expressed as Vk″(mk″)=uk″(fk″(mk″))+zk″(mk″),13 k″∈{C,H}, mC∈[MB,MC] and mH∈[MC,MH]; here, uC and uH, respectively, denote household utility derived from the consumption of floor space fC(m) and from the consumption of lot size fH(m), and zk″ is the numeraire non-housing commodity. The income constraint is expressed as zk″+fk″rk″(m)=w(m)+[1/N¯]Φ−t(m)−tB(m), where t(m) is the round-trip commuting cost to the CBD edge borne by a household residing at location m, and Φ is the total profit from the land, that is, total differential land rent. [1/N¯]Φ on the right-hand side implies the assumption of public ownership of land. Using Equation (8), the income constraint can be simplified as zk″+fk″rk″(m)=w(MB)+[1/N¯]Φ−t(m). The total profit from the land, Φ, is expressed as Φ=∫0MB[RB(m)−RA]dm+∫MBMC[RC(m)−RA]dm+∫MCMH[rH(m)−RA]dm, (15) where RA is the agricultural rent, and rH is the land rent in Zone H. The maximization of household utility yields the demand function fC(rC(m)). Substituting this into the utility function yields zC(m)=−uC(fC(rC(m)))+VC(m), m∈[MB,MC]. Regarding Zone H, we obtain zH(m)=−uH(fH(m))+VH(m), m∈[MC,MH]. Note that under the lot size regulation, fH(m) is regulated, and therefore, cannot be chosen by a household, whereas as stated above, fC(m) is the function of rC(m) following utility maximization. Using the demand functions, the residents’ income constraints are expressed as w(MB)+1N¯Φ−t(m)=−uC(fC(rC(m)))+VC(m)+fC(rC(m))rC(m),m∈[MB,MC], (16) w(MB)+1N¯Φ−t(m)=−uH(fH(m))+VH(m)+fH(m)rH(m),m∈[MC,MH]. (17) 3.7. Market clearing conditions and definitions The equality of utilities among locations and market clearing conditions are shown here. First, Equation (18) implies that the household utility is equal everywhere, which is because households are indifferent regarding locations. VC(mC)=VH(mH)≡V, mC∈[MB,MC], mH∈[MC,MH], (18) Population function n(m) and transportation cost t(m) are both continuous at MC but are not necessarily smooth. So, to clearly distinguish these functions before and after MC, we define nC(m), nH(m), tC(m) and tH(m): nC(m) ≡{n(m)|m∈[MB,MC]} and tC(m) ≡{t(m)|m∈[MB,MC]}, (19) and nH(m)≡{n(m)|m∈[MC,MH]} and tH(m)≡{t(m)|m∈[MC,MH]}. (20) Next, as Equation (21) states, the total floor space consumed is balanced by total floor space supplied. Likewise, Equation (22) expresses the market equilibrium in floor space in Zone C. Next, Equation (23) expresses that in Zone H, the households at m consume fH(m) area of lot; therefore, total area consumed is equal to the unit land area supplied. Floor space in Zone B, fB(g(m),w(m))n˙B(m)=FB(m),m∈[0,MB], (21) Floor space in Zone C, fC(m)n˙C(m)=FC(m) where n˙C(m)≡∂nC(m)/∂m,m∈[MB,MC], (22) Lot supply in Zone H, n˙H(m)fH(m)=1, where n˙H(m)≡∂nH(m)/∂m,m∈[MC,MH]. (23) Finally, as shown in Equation (24), because one household member works in the CBD, the total number of workers (left side) is equal to the household population N¯ (right side). Labor population, ∫0MBn˙B(m)dm=N¯. (24) 4. Maximizing social welfare with land use regulations 4.1. Objective The objective of optimal regulations can be denoted as Definition 2 (see also Definition 1), using the social welfare composed of total utilities, W=N¯V, (25) where V is defined in Equation (18). Definition 2 (Optimal Regulations). Optimal FAR at each location in Zone B and Zone C, i.e., Fk′(m), k′∈{B,C}, m∈[0,MC], and optimal lot size at each location in Zone H, i.e., fH(m), m∈[MC,MH], and three optimal zonal boundaries MB,MC,MH are given by maximizing social welfare subject to the market equilibrium. Mathematically, [F,f,MB,MC,MH]=arg max F,f,MB,MC,MH W subject to eqs. (1) – (25), where F=(FB(mB),FC(mC))∀mB∈[0,MB], ∀mC∈[MB,MC] and f=fH(m)∀m∈[MC,MH]. The first-best optimum can be achieved by levying a Pigouvian toll on commuting equal to the gap between the marginal social cost and the private cost of travel; and a Pigouvian tax on each communication cost equal to the gap between the social marginal and private communication cost. Agglomeration economies enhancing A¯ are fixed because the city population is fixed. Aside from the first best, the current article explores optimization of the social welfare using land use regulations. 4.2. First-order conditions for optimal regulations Regulations affect social welfare W through changes in (i) agglomeration economies arising from communication in the business zone due to the distribution of firms, (ii) deadweight loss in the floor space and lot size market due to the distribution of residences, and (iii) commuting costs. The optimal control problem maximizing the social welfare subject to the market mechanism in Equations (1)–(25) is expressed in the Lagrangian, which is shown in Appendix A. Our model allows simultaneous optimization of FARR in the business and condominium zones, lot size regulations in lot housing zones and three zonal boundaries. The first-order conditions show the relationships among distortions caused by the regulations, agglomeration economies and congestion. We interpret these relationships in Appendix B. 5. Optimal regulations 5.1. Optimal FARR and lot size regulation This section obtains important properties of the optimal regulations on FAR and lot size. First, we consider shadow prices μk(m)(k=B,C,H) which directly show how and by how much the FAR or lot regulations should be imposed. As we show later in Proposition 1, these properties are important to determine land use regulation. Rigorous description of these properties is shown in Lemma 1 in Appendix C. We summarize the motion of μk(m) in Figure 2. Figure 2 View largeDownload slide The motion of shadow prices. Figure 2 View largeDownload slide The motion of shadow prices. From Equations (A.2) and (A.3) in Appendix A, it is evident that the sign of μk'(m)(k'∈{B,C}) is the reverse of Dk′(mk′)≡rk′(mk′)−∂Sk′(Fk′)/∂Fk′, where Dk′(mk′) denotes distortion in the floor space market caused by FARR as shown in Figure 3(a). Note that Dk′(m)=0, if the FAR is unregulated and determined by the market. An FAR greater (resp. smaller) than the market FAR implies Dk′(m)<0 (resp. Dk′(m)>0). Thus, combining Equation (A.2) with Lemma 1(1) while also using η=1 from Equations (A.7) and (A.11) yields Proposition 1(1). Likewise, combining Equation (A.3) and Lemma 1(2) using η=1 again yields Proposition 1(2). Next, DH≡∂uH/∂fH−rH denotes distortion in the lot supply market due to lot size regulation as shown in Figure 3(b). The combination of Equation (A.4) with Lemma 1(3) leads to DH<0 for any m∈(MC,MH], thereby yielding Proposition 1(3). Proposition 1 (Optimal FARR and lot size regulation in the presence of optimal zonal boundaries). Let superscript ‘ †’ refer to the market equilibrium case. Figure 3 View largeDownload slide Deadweight loss due to regulations: (a) FARR (left); (b) lot size regulation (right). Note: Subscripts ‘ †’ and ‘ ∗’, respectively, refer to the market equilibrium and optimal cases. Figure 3 View largeDownload slide Deadweight loss due to regulations: (a) FARR (left); (b) lot size regulation (right). Note: Subscripts ‘ †’ and ‘ ∗’, respectively, refer to the market equilibrium and optimal cases. Business zone:There is at least one sub-zone where FB(m)>FB†(m)and one other sub-zone where FB(m)<FB†(m), where m∈[0,MB]. In the two cases classified by whether communication costs are stronger than traffic congestion costs or not: Case (i) implying −2τ+δγnB(m)γ−1/ρ(m)γ<0 at any m∈[0,MB]:FB(m)>FB†(m) at any m∈[0,m⌣) and FB(m)<FB†(m) at any m∈(m⌣,MB], where m⌣∈(0,MB); Case (ii) implying −2τ+δγnB(m)γ−1/ρ(m)γ>0 at any m∈[0,MB]: FB(m)<FB†(m) at any m∈[0,m⌣) and FB(m)>FB†(mB) at any m∈(m⌣,MB], where m⌣∈(0,MB); Condominium zone: FC(m)>FC†(m) for any m∈[MB,m⌢) and FC(m)<FC†(m) at any m∈(m⌢,MC], where m⌢∈(MB,MC); Lot housing zone: fH(m)>fH†(m) at any m∈[MC,MH]. Case (i) in Proposition 1(1) implies that communication costs are stronger than traffic congestion costs in the business zone. Case (i) arises if the communication cost, τ, is sufficiently large. Case (ii) is the reverse condition, which arises if the transportation capacity, ρ(m), is sufficiently small, implying severe congestion in the business zone. Note that near the city center, the number of commuters is close to zero; so at m∈[0,ε] where ε=˙0, −2τ+δγnB(0)γ−1/ρ(0)γ<0. If ρ(m) is relatively large in the central area, Case (i) emerges because the communication cost for business people is generally high. In most developed cities with high wages, Case (i) is likely to hold. It is important to recall that Proposition 1 holds when optimal zonal boundaries are simultaneously imposed, which are presented later in Proposition 2. Results of Proposition 1 are depicted in Figure 4, and the implication is explained as follows. Figure 4 View largeDownload slide Optimal regulations. Note: For FARR in Zone B, regulations in bold letters hold in Case (i), and regulations in parenthesis hold in Case (ii). For the area regulation in the business zone, a decrease in the zone area hold in Case (i), and an increase in the zone area holds in Case (ii). Figure 4 View largeDownload slide Optimal regulations. Note: For FARR in Zone B, regulations in bold letters hold in Case (i), and regulations in parenthesis hold in Case (ii). For the area regulation in the business zone, a decrease in the zone area hold in Case (i), and an increase in the zone area holds in Case (ii). First, the combination of maximum and minimum FARRs results in an efficient labor distribution in order to optimize the total welfare composed of deadweight loss in the floor market and agglomeration benefits in the CBD. To achieve a certain labor distribution, if only conventional ‘maximum FARR’ is imposed, the total deadweight loss arising from FARR would be greater than that would arise from the combination of maximum and minimum FARRs. Whether minimum FARR is required in central locations or peripheral locations depends on the relative magnitude of −2τ and δγnB(m)γ−1/ρ(m)γ ( m∈[0,MB]). For example, in Case (i), that is, if −2τ+δγnB(m)γ−1/ρ(m)γ<0 (resp. in Case (ii), that is, −2τ+δγnB(m)γ−1/ρ(m)γ>0) in most parts of the business zone, the optimal policy requires enforcement of minimum (resp. maximum) FARR at the more central locations and maximum (resp. minimum) FARR at the peripheral locations. Recall that Case (i) is more likely to hold in developed cities. Next, in the condominium zone, the optimal policy requires enforcement of minimum FARR at the central locations and maximum FARR at the periopheral locations. Such regulations shift population in favor of more central locations, and thereby reduce traffic congestion caused by commuters from distant locations. See Kono et al. (2010) for a related discussion in detail. In the lot housing zone, however, the optimal regulation requires enforcement of minimum lot size regulation over the zone. As shown in Figure 3(b), DH denotes marginal deadweight loss in the lot size market at the given location. The optimal regulation addresses these deadweight losses. Given that the city also has UGB regulation that prevents sprawl, minimum lot size regulation reduces supply of housing lots and thereby decreases population in the suburb and ultimately reduces traffic congestion across the condominium zones and lot housing zone. 5.2. Optimal zonal boundaries We now examine how a change in each zonal boundary affects the social welfare under regulated FAR and lot size. From Lemma 2 in Appendix D, which summarizes the related first-order conditions of the welfare maximization, we directly derive Proposition 2, which shows optimal zoning conditions composed of only empirically observable variables. Proposition 2 (Optimal zonal boundaries in the presence of optimal FARR and lot size regulation).Let superscript ‘ †’ refer to the market equilibrium case. The optimal zonal boundaries are as follows. Whether the business zone should be more compact or larger than the market size depends on the trade-off between agglomeration economies and traffic congestion costs. More concretely, recalling the two cases defined in Proposition 1(1), MB<MB† in Case (i) whereas whether MB>MB† or MB<MB† is ambiguous in Case (ii); The boundary of Zone C should be shrunk relative to the market boundary. That is, MC<MC†; The boundary of Zone H, which is equal to the urban boundary, should be shrunk relative to the market boundary. That is, MH<MH†. The implication of Proposition 2(1) is explained as follows. Regarding the business zone, an enlargement of the zone decreases agglomeration economies resulting in a decrease in welfare. Simultaneously, it implies that the business area becomes closer for all residents. Accordingly, the congested commuting distances decrease. That implies a decrease in congestion cost of commuting in the residential floor market at location MB. The net effect is ambiguous. Proposition 2(2) implies that contraction of Zone C decreases the deadweight loss at Zone C edge by DC(MC)FC(MC) but increases the same at the inner edge of Zone H by DH(MC). The first exceeds the latter in absolute value. This can also be explained using the equation in Lemma 2(2). Proposition 2(3) implies that a marginal expansion of Zone H decreases social welfare. Expanding Zone H by a unit area means supplying additional lots, thereby increasing the population at the city edge by 1/fH(MH). Because the city is closed, such expansion results in reallocation of some households from the outer edge of Zone C to the city edge, noting that fH(m)(m∈[MC,MH]) is fixed by lot size regulation. Such relocation increases the commuting cost in Zone H by ∫MCMHδγ[[N−n(m)]/ρ(m)]γdm·[1/fH(MH)] (i.e., the first term in the right-hand side of equation of Lemma 2(3)). The relocation decreases the deadweight loss at the outer edge of Zone C by DC(MC)[fC(MC)/fH(MH)], where fC(MC)/fH(MH) implies decrease in the building size at the outer edge of Zone C because of the relocation of [1/fH(MH)] number of residents. This can also be explained using the equation in Lemma 2(3). Finally, we explore whether the total area of the city decreases or not. Although the areas of Zone C and Zone H should be shrunk relative to the market boundary, the optimal size of Zone B is ambiguous in Case (ii). When the congestion costs are large enough compared to agglomeration benefits, a larger Zone B is welfare improving because it reduces traffic congestion by reducing the distance between Zone B and the residential area. In such case, an optimal city can be larger than a market city if +ΔMB−ΔMC−ΔMH>0, where ΔMk(k∈{B,C,H}) denotes optimal change (‘+’ if expansion and ‘−’ if shrinkage) in the corresponding zonal boundary relative to the market equilibrium. 6. Summary and conclusion This article simultaneously optimizes multiple regulations—on building size, lot size and zonal boundaries—in a monocentric city with office buildings, condominiums and single-family dwellings in distinct but adjoining districts. We demonstrate the necessity of both minimum and maximum FARR in both business zone and condominium zone. Although Kono et al. (2010) achieve similar results in their treatment of condominiums, our results differ because we also consider office buildings. Accordingly, in our model, where minimum or maximum FARR should be imposed within the business zone, which has both agglomeration economies and congestion externalities, differs from that in condominium zone which has only congestion externalities. Although the optimal FARR in the business zone is ambiguous in our model, our results suggest that if agglomeration economies are relatively dominant over traffic congestion costs (such as in developed cities), it is more likely that the optimal policy requires enforcement of minimum FARR at the more central locations and maximum FARR at the more peripheral locations. The same applies non-ambiguously in the case of condominium zone, followed by minimum lot size regulation in the suburb. In the presence of optimal FARR and lot size regulation, we also explore optimal zonal boundaries. The optimal size of the business zone is ambiguous depending on the trade-off between agglomeration economies and traffic congestion costs but the outer boundaries of the condominium and housing lot zones should be shrunk. This has an important implication regarding what minimum lot size regulation achieves in our model, and how. Although minimum allowable lot size in the suburb is prevalent in most countries, especially in the USA, our interpretation of the necessity of minimum lot size regulation (see Proposition 2) is different. The objective of such zoning as being practiced is to promote low-density development but such a policy contributes to urban sprawl (Pasha, 1996), and thus increases congestion costs. That is why Pines and Sadka (1985) and Wheaton (1998) suggest maximum lot size regulation in the central area. But our model is able to alleviate congestion externality through minimum lot size regulation and without sprawl because it allows options for additional higher density in the central area through minimum FARR. Such result is achieved because of simultaneous consideration of multiple regulations. Our theoretical results are supplemented by some numerical examples which show how social welfare is affected by changes in the regulation (see Appendix E). The application of our results to real cities has some caveats. In this study, we have not assumed any other means of transport except automobiles. Buyukeren and Hiramatsu (2016), assuming a congested car mode and an uncongested public transit mode, analytically demonstrate that, under certain conditions, an expansionary UGB would be optimal. Hence, our results are not applicable to cities with multiple transport modes as they are. However, many cities have no railroad system, and even if they have one, it does not cover the whole city. Our results are applicable to areas without railroads within cities. In addition, some relevant transport policies might affect land use regulation drastically. Indeed, cordon pricing affects land use regulation (see Kono and Kawaguchi, 2017). Furthermore, although the current article only assumes homogeneous residents, we should explore how land use regulation affects heterogeneous residents before implementing the regulation. Another major limitation is that our model is a static city model, which treats the building stock as fully malleable as if it could be adjusted instantaneously. Fluctuations in productivity could cause fluctuations in demand for space in the city, which implies that the optimal FAR and zonal boundaries could change over time. Indeed, several recent papers have attempted to take account of dynamic aspects. For example, Joshi and Kono (2009) analyze optimal FARR in a dynamic setting in a growing city with two zones with congestion externality. Jou (2012) explores the optimal UGB in a monocentric city with uncertainty. Optimal dynamic regulations can thus be another subject of interest for further research. Footnotes 1 As Hanushek and Quigley (1990) note, fiscal objectives are other justification for land use regulation. 2 See http://www.portlandonline.com/auditor/index.cfm?c=28197 (last accessed on 23 March 2015). 3 Pines and Sadka (1985) use housing tax to control lot size. However, this is equivalent to lot size regulation in terms of social welfare. 4 Rhee et al. (2014) numerically simulate optimal regulations in a model with two or three zones on the right side of the city. Although the number of zones is still limited, they allow mixed use of industrial and residential uses in each zone. This point is different from our model. 5 Note that land use regulations can be replaced by equivalent property tax policies (see Pines and Kono, 2012). 6 Single-worker production is not so specific as Borukhov and Hochman (1977) note. If the production function is expressed as AΓ(Q,l), where Γ(Q,l) is one-degree homogeneous production function, Q is the total floor space for a firm, and l is the labor size, then we obtain a production function with one unit of labor, given by AX(fB)=AΓ(Q/l,1), where fB=Q/l. 7 Needless to say, a rational worker does not pay more than the supply-side cost unless she could improve her welfare which does not happen in the stable equilibrium. Hence, this expression is going to hold as an equality as a result of the rational worker’s optimal behavior. Nevertheless, this inequality expression is useful later for determining the sign of the shadow price (or Lagrange multiplier) of this constraint. Because the multiplier implies the value of the supply-side transport cost, the sign is negative, implying a natural result that an increase in the transport cost has a negative welfare effect. This is straightforwardly proved using the Kuhn–Tucker condition if we use an inequality condition. A similar method is used in Kono and Joshi (2012) and Kono and Kawaguchi (2017). 8 ddm∫−MBMBn˙B(x)[τ|x−m|]dx=[nB(MB)+nB(m)−[nB(MB)−nB(m)]]τ=2nB(m)τ, m∈[0,MB]. 9 If the floor area is fixed as in O’Hara (1977) and Ogawa and Fujita (1980), the optimal FARR brings about no deadweight loss in the floor market because the demand is inelastic. But we focus on the deadweight loss brought by the FARR; so the assumption of endogenous elastic floor area is vital for the purpose of our article. To achieve the first-best distribution, as Helsley and Strange (2007) show, transportation subsidies should be imposed. 10 The discussion in footnote 7 also applies here by replacing ‘rational worker’ with ‘rational commuter’. In addition, note that this unnecessary travel cost is not equal to congestion pricing. The payment for congestion pricing will be returned to the society, but the unnecessary travel cost is lost by the society. 11 Differentiating Equation (13) with respect to m reverses its inequality sign because t˙B(m) is negative. 12 This is why formulation of Equation (14) differs from that of Equation (11). In Equation (11), N¯−n(m) denotes household (or commuter) population residing beyond location m (m∈[MB,MH]) that would be commuting to the CBD. In Equation (14), nB(m) denotes commuter population working at firms located over [0,x] (m∈[0,MB]) that would cross location m while commuting to the firms or while returning home. 13 Because the marginal utility with respect to income is constant over locations under the assumption of a quasi-linear utility, the derivation would be simple. In contrast, Pines and Kono (2012) use a general utility function (i.e., u(f,z)) to obtain the optimal FARR. However, in the current article treating multiple regulations, the derivation of the optimal regulations will be very complex if a general utility function is adopted. However, even in the latter case, the formula describing the optimal regulation will essentially be the same. Furthermore, if a change in the marginal utility with respect to income does not change much according to the regulations, the same properties regarding the optimal regulation are obtained. 14 The Lagrangian considers only the right-hand side of the city. However, communication costs in the right side of the city depend on the left side as well. Equations (4) and (5) include the left-side firms’ communication costs. So, we have to first consider dividing them by 2 for inclusion in the Lagrangian. But, because the city is symmetric, the left side’s symmetrical change should be considered, which can be done by multiplying the equations by 2 because the communication trips are inelastic. In conclusion, we can consider Equations (4) and (5) as they are. Further, note that this Lagrangian can be replaced with Hamiltonian, which generates the same first-order conditions. 15 λH(MC)=N¯−nC(MC) is proved simply by using Equations (A6), (A17), (A18) and (A19). 16 Totally differentiating Equation (16) yields −dtC=−[∂uC/∂fC−rC]·∂fC/∂rC·drC+fCdrC, where ∂uC/∂fC=rC because of the first order condition of the utility maximization. So, we obtain −1/fC=drC/dtC. Multiplying both sides by dfC/drC yields −[1/fC]dfC/drC=dfC/dtC. 17 A similar equation to Equation (A33) appears in Kanemoto (1977), Arnott and MacKinnon (1977), Arnott (1979), Pines and Sadka (1985) and Pines and Kono (2012). As explained in the main text, μC(m) expresses the distortion arising from the FARR in our model. In contrast, in the case of those previous papers, except for Pines and Kono (2012), the distortion arises from the allocation of land between road and residential areas, which is fixed in our model. Simultaneously controlling FARR with road area remains a future task. 18 The Harberger’s welfare formula is expressed as dW/dQ=∑iΞi∂Xi/∂Q where Ξi is the distortion (e.g., price minus marginal cost) in market i, X is the output in market i, and Q is the policy variable. 19 Note that market land rents between two adjacent zones are not equalized in the formulation of Lagrangian in Equation (A.1) or subsequent derivations. This should show that Proposition 1 is not derived with zonal boundaries set at market equilibrium. 20 See Lemma 1(1) for definition of Cases (i) and (ii). 21 Programs for the simulation are available upon request. 22 Programs solving the market equilibrium are composed of simultaneous equations only. In contrast, programs solving the optimal solution maximize the social welfare subject to multiple equations (conditions or constraints). In other words, the second-best optimum is a constrained nonlinear optimization subject to numerous constraints multiplied by the number of blocks our three-zone model city is divided into. We were able to numerically solve the market equilibrium only. Acknowledgements We appreciate the editor Kristian Behrens and anonymous reviewers for useful comments. Earlier versions of this paper were presented at an Urban Economic Association session of the North American Meetings of the Regional Science Association in Ottawa, Canada, the Applied Regional Science Conference in Toyama, Japan, and a seminar at Kyoto University. We are thankful to the participants at the meetings for their helpful comments and suggestions, in particular to T. Mori, S. Mun, J. Siodla, and Y. Yoshida. Furthermore, we are very grateful to the late Y. Hayashiyama, who kindly supported and reviewed our research. Despite assistance from many sources, any errors in the paper remain the sole responsibility of the authors. 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Lagrangian and Pontryagin's maximum principle The Lagrangian according to Definition 2 is as follows.14 L= N¯V+∫MBMCκC(m)FC(m)fC(m)[w(MB)+1N¯Φ+uC(fC(rC(m)))−V−fC(rC(m))rC(m)−tC(m)]dm+∫MCMHκH(m)1fH(m)[w(MB)+1N¯Φ+uH(fH(m))−V−fH(m)rH(m)−tH(m)]dm−η[Φ−∫0MB[FB(m)rB(g(m),w(MB)+tB(m))−SB(FB)−RA]dm−∫MBMC[FC(m)rC(m)−SC(FC)−RA]dm−∫MCMH[rH(m)−RA]dm]+∫0MBϕB(m)[G˙(m)−2nB(m)τ]dm+θ[G(0)−2∫0MBn˙B(m)τmdm]+∫0MBω(m)[g(m)−G(m)]dm+∫0MBλB(m)[t˙B(m)+ξ+δ[nB(m)ρ(m)]γ]dm+∫0MBμB(m)[FB(m)fB(g(m),w(MB)+tB(m))−n˙B(m)]dm+∫MBMCλC(m)[t˙C(m)−ξ−δ[N¯−nC(m)ρ(m)]γ]dm+∫MBMCμC(m)[FC(m)fC(rC(m))−n˙C(m)]dm+∫MCMHλH(m)[t˙H(m)−ξ−δ[N¯−nH(m)ρ(m)]γ]dm+∫MCMHμH(m)[1fH−n˙H(m)]dm+ςn[nH(MC)−nC(MC)]+ςt[tH(MC)−tC(MC)], (A.1) where κk″(m), η, θ, ω(m), ϕB(m), ψB(m), λk(m), μk(m), ςn and ςt are shadow prices; k″∈{C,H} and k∈{B,C,H}. Because communication between firms takes place bilaterally, the constraint, G(0)=2∫0MBn˙B(m)τmdm from Equation (5), is necessary in Equation (A.1). As Tauchen and Witte (1984) and Fujita and Thisse (2002) show, the total communication cost in the CBD is expressed by double integrals; so the Lagrangian should treat this constraint specifically in addition to the constraint of G˙(m). This necessity is also intuitive because the boundary condition tB(0) depends on the endogenous labor distribution. The other boundary conditions are as follows. For Zone B, nB(0)=0, and nB(MB)=N¯. For Zone C, n(MB)=0. For Zone H, n(MH)=N¯. To obtain the first order conditions, we integrate the Lagrangian in Equation (A.1) by parts. After that, differentiating the Lagrangian with regard to the policy variables and the endogenous variables, we obtain the first order conditions (A.2)–(A.30). These expressions use the following relations: ∂rB/∂g=−1/fB and ∂rB/∂w=−1/fB. Also note that a dot denotes derivative with respect to distance from the center. ∂L∂FB(m)=0:η[rB(m)−∂SB∂FB(m)]+μB(m)1fB(g(m),w(m))=0, m∈[0,MB], (A.2) ∂L∂FC(m)=0:η[rC(m)−∂SC∂FC(m)]+μC(m)1fC(rC(m))=0, m∈[MB,MC], (A.3) ∂L∂fH(m)=0:κH(m)1fH[∂uH∂fH−rH]−μH(m)1fH(m)2=0, m∈[MC,MH], (A.4) where, for deriving Equations (A.3) and (A.4), Equations (16) and (17) have been used respectively, in addition to Equation (18); (A.5) ∂L∂rH(m)=0:−κH(m)+η=0, m∈[MC,MH], (A.6) ∂L∂V=0:N¯−∫MBMCκC(m)FCfCdm−∫MCMHκH(m)1fHdm=0, (A.7) ∂L∂w(MB)=0:∫MBMCκC(m)FCfCdm+∫MCMHκH(m)1fHdm −η∫0MBFB(m)fB(m)dm+∫0MBμB(m)FB(m)fB(m)2∂fB∂w(MB)dm=0, (A.8) where for the last term, note that ∂w(m)/∂w(MB)=1 by virtue of Equation (8); ∂L∂tB(m)=0:−ηFB(m)fB(m)−μB(m)FB(m)fB(m)2∂fB∂tB−λ˙B(m)=0, (A.9) ∂L∂tB(0)=0:−λB(0)=0, (A.10) ∂L∂Φ=0:1N¯[∫MBMCκC(m)FCfCdm+∫MCMHκH(m)1fHdm]−η=0, (A.11) ∂L∂g(m)=0:−ηFB(m)fB(m)+ω(m)−μBFB(m)fB(m)2∂fB(m)∂g(m)=0, m∈[0,MB], (A.12) ∂L∂G(m)=0:−ϕ˙B(m)−ω(m)=0,m∈[0,MB], (A.13) ∂L∂G(0)=0:−ϕB(0)+θ=0, (A.14) where for deriving this condition, Equation (A.13) is used noting that the latter holds when m→0; ∂L∂G(MB)=0:ϕB(MB)=0, (A.15) ∂L∂tC(m)=0:−κC(m)FC(m)fC(m)−λ˙C(m)=0,m∈[MB,MC], (A.16) ∂L∂tH(m)=0:−κH(m)1fH(m)−λ˙H(m)=0,m∈[MC,MH], (A.17) ∂L∂tC(MC)=0:λC(MC)−ςt=0, (A.18) ∂L∂tH(MC)=0:−λH(MC)+ςt=0, and ∂L∂tH(MH)=0:λH(MH)=0, (A.19) ∂L∂nB(m)=0:2τ[θ−ϕB(m)]+μ˙B(m)+λB(m)δγ[nB(m)]γ−1ρ(m)γ=0, m∈[0,MB], (A.20) ∂L∂nC(m)=0:λC(m)δγ[N¯−nC(m)]γ−1ρ(m)γ+μ˙C(m)=0,m∈[MB,MC], (A.21) ∂L∂nH(m)=0:λH(m)δγ[N¯−nH(m)]γ−1ρ(m)γ+μ˙H(m)=0,m∈[MC,MH], (A.22) ∂L∂nH(MC)=0:μH(MC)+ςn=0, (A.23) ∂L∂nC(MC)=0:−μC(MC)−ςn=0, (A.24) ∂L∂MB=0:[FBrB(g(MB),w)−SB(FB)−RA]+μB(MB)FB(MB)fB−[FCrC(m)−SC(FC)−RA]−λB(MB)[ξ+δ[N¯ρ(MB)]γ]−λC(MB)[ξ+δ[N¯ρ(MB)]γ]−μC(MB)FC(MB)fC(rC(MB))=0. (A.25) To derive Equation (A.25), we used ϕB(MB)=0 from Equation (A.15) and η=1 from Equations (A.7) and (A.11). ∂L∂MC=0:[FC(MC)rC−SC(FC)−RA]−[rH(MC)−RA]+μC(MC)FC(MC)fC(rC(MC))−μH(MC)1fH(MC)=0. (A.26) To derive Equation (A.26), we used λC(MC)=λH(MC) by virtue of Equations (A.18) and (A.19). ∂L∂MH=0:[rH−RA]+μH(MH)1fH(MH)=0, (A.27) ∂L∂ω(m)≥0,ω(m)≥0,∂L∂ω(m)ω(m)=0, m∈[0,MB], (A.28) λB(m)∂L∂λB(m)=0,λB(m)≤0,∂L∂λB(m)≤0, (A.29) noting the sign of the corresponding constraint in Equation (13). λk″(m)∂L∂λk″(m)=0,λk″(m)≥0,∂L∂λk″(m)≥0,k″∈{C,H}. (A.30) The first-order conditions with respect to shadow prices, except for ω(m), λB(m), λC(m) and λH(m), are suppressed because they are obvious. Appendix B. Interpretation of optimal FARR and lot size regulation B.1 FARR in the business zone First, solving Equations (A.2), (A.9) and (A.10) and using η=1 from Equation (A.7) and Equation (A.11) yields λB(m)=−nB(m)+∫0m[rB(x)−∂SB∂FB]FB(x)fB∂fB∂tBdx, where λB(m) is the shadow price for the commuting cost within the business zone. Next, substituting Equation (A.13) into (A.12) to cancel out ω(m), and then integrating the result and using boundary condition ϕB(MB)=0 from Equation (A.15), yields −ϕB(m)=−[N¯−nB(m)]−∫mMBμB(x)FB(x)fB2∂fB∂gdx. From Equation (A.14), θ=ϕB(0), whereas ϕB(0) is obtained from the aforementioned equation involving −ϕB(m), thereby yielding θ=N¯+∫0MBμB(x)FB(x)fB2∂fB∂gdx. Substituting these three equations regarding λB(m), −ϕB(m), and θ as well as Equation (A.2) into (A.20) yields ΩB≡μ˙B(m)dm =[nB(m)−∫0m[rB(x)−∂SB∂FB]FB(x)fB∂fB∂gdx]︸=θ−ϕB(m)>0[−2τ︸<0+δγ[nB(m)]γ−1ρ(m)γ︸>0]dm, m∈(0,MB]. (A.31) Note that μ˙B(m)dm=μB(m+dm)−μB(m), where μB(m) is the shadow price for the floor area containing FB/fB number of workers at location m. Therefore, Equation (A.31) implies a change in the social welfare from the relocation of one worker from location m to location m + dm due to a change in the FARR. Recall that we have supposed a symmetric distribution of firms with respect to the center, and have focused only on the right side of the city. In fact, the relocation should take place symmetrically on both sides. The combined effect of the relocation of one worker on communication costs and traffic congestion is −2τ+δγ[nB(m)]γ−1/[ρ(m)]γ in Equation (A.31). The first term, −2τ, is related to the increase in the communication cost for firms located over [0,MB]. The relocation of one worker from m to m + dm increases the communication cost of the firms located over [0,m], but decreases the communication cost of the firms located over [m,MB]. Likewise, the symmetrical left-side relocation of one worker from –m to –m–dm increases the communication cost of the firms located over both [0,m] and [m,MB]. Summing up, the total communication cost for all firms located over [0,m] increases by 2nB(m)τ and that for all the firms over [m,MB] increases by [N¯−nB(m)][τ−τ]=0. As a result, the change in the total communication cost for the firms in the right side of the city is 2nB(m)τdm. Furthermore, the change in the communication cost for the firms located at x∈[0,m] affects the respective floor market deadweight loss caused by FARR, which is expressed as −∫0mDB(x)[FB(x)/fB][∂fB/∂g]dx·[−2τ], noting that DB(x)≡rB(x)−∂SB/∂FB expresses the change in the deadweight loss in the floor market (see Figure 3). Next, the term δγ[nB(m)]γ−1/ρ(m)γ in Equation (A.31) is the saving in traffic congestion cost for the firms located over [0,m]. This change in the traffic congestion costs also affects the respective floor market distortions caused by FARR, which is expressed as −∫0mDB(x)[FB(x)/fB][∂fB/∂g]dx·δγ[nB(m)]γ−1/[ρ(m)]γ. This concludes interpretation of Equation (A.31). Next, from Equation (A.2), ΨB≡μ˙B(m)dm=−fB(m+dm)[rB(m+dm)−∂SB∂FB(m+dm)]+fB(m)[rB(m)−∂SB∂FB(m)], (A.32) recalling that η=1 from Equations (A.7) and (A.11). The relocation of one worker implies relocation of fB units of floor space. Accordingly, Equation (A.32) implies the total change in deadweight loss in the floor market, which arises from the FARR at m + dm and m. At the optimal condition, the social welfare change due to the change in the communication cost, expressed as ΩB in Equation (A.31), should be balanced with the change in the deadweight loss in the floor market expressed as ΨB in Equation (A.32). This is the interpretation of the first-order conditions with respect to FARR in the business zone. B.2 FARR in the condominium zone First, noting that η=1, combination of Equation (A.5) and Equation (A.16), by cancelling κC, yields λ˙C(m)+FC(m)/fC−[μC(m)/fC][FC(m)/fC2]dfC/drC=0. Rearrangement of Equation (A.3) yields −μC(m)/fC(m)=rC(m)−∂SC/∂FC. Substituting this into the above equation and integrating the result with respect to m, we obtain ∫mMCλ˙C(x)dx=−∫mMCn˙C(x)dx+∫mMC[rC(x)−∂SC/∂FC]n˙C(x)[dfC/dtC]dx, which leads to −λC(m)=−[N¯−nC(m)]+∫mMC[rC(x)−∂SC∂FC]n˙C(x)dfCdtCdx, (A.33) where the following relationships are used: λC(MC)=λH(MC)=N¯−n(MC),15 and −[1/fC]dfC/drC=dfC/dtC, which is obtained from differentiating Equation (16)16. Recalling the form of the Lagrangian in Equation (A.1), [−λC(m)] on the left-hand side of Equation (A.33) expresses the shadow price for the unit travel time at m (m∈[MB,MC]). Equation (A.33) is easily interpreted as follows.17 The first term on the right-hand side of Equation (A.33) is the direct effect of the total increase in travel time for all the commuters, that is, N¯−nC(m), passing through m. The second term is the effect of the change in the per-capita floor area consumption, dfC/dtC, beyond m, which is induced by the increase in the travel cost tC(x)(x∈[m,MC]). The change in the per-capita floor area consumption, dfC/dtC, multiplied by n˙C(x) gives the change in the total floor area F(x) at x. As explained earlier, the term rC(x)−∂SC/∂FC is the marginal change in the deadweight loss (or distortion) caused by the FARR (see Figure 3(a)). If FC is determined in the perfect competition, rC(x)=∂SC/∂FC, and correspondingly the second term on the right hand side of Equation (A.33) is zero. However, when the floor area is regulated, rC(x)≠∂SC/∂FC; that is, the second term is not zero. Next, we interpret the optimality condition of FARR by combining the first-order conditions. Substituting −λC(m) from Equation (A.33) into Equation (A.21) yields ΩC≡μ˙C(m)dm=−δγ[N¯−nC(m)]γ−1ρ(m)γ[N¯−nC(m)]+δγ[N¯−nC(m)]γ−1ρ(m)γ∫mMC[rC(x)−∂SC∂FC]n˙C(x)dfCdtCdx, (A.34) where δγ[N¯−nC(m)]γ−1/ρ(m)γ=∂tC(m)/∂n˙C(m). Next, differentiating Equation (A.3) with respect to m yields ΨC≡μ˙C(m)dm=−fC(m+dm)[rC(m+dm)−∂SC∂FCFC(m+dm)]+fC(m)[rC(m)−∂SC∂FCFC(m)]. (A.35) Because both ΩC and ΨC are equal to μ˙C(m)dm, the right-hand side of Equation (A.34) should be equal to the right-hand side of Equation (A.35) for the optimality of the FARR; that is, ΩC−ΨC=0. A similar case is interpreted in Kono et al. (2012) in detail. We present only a brief explanation here as follows. The relation ΩC−ΨC=0 for the optimality of FARR implies that under the relocation of one person from m to m + dm, the welfare increase associated with the deadweight loss in the FARR, that is, −ΨC should be cancelled out by the welfare increase associated with the increased travel cost, that is, ΩC. Importantly, ‘ ΩC−ΨC’is compatible with the Harberger’s welfare formula, which measures the welfare change in a distorted economy (see Harberger, 1971).18 B.3 Lot size regulation in the lot housing zone In detached housing zone, population density is directly adjusted whereas in the case of FARR, even if the building sizes are adjusted, per-capita floor area cannot be controlled by the government. Therefore, the second term in Equation (A.34) exists in the case of FARR but not in the case of lot size regulation. To check the difference, we can derive μ˙H(m) using Equations (A.6), (A.17), (A.19) and (A.22), as μ˙H(m)=−δγ[N¯−nH(m)]γ−1ρ(m)γ[N¯−nH(m)]. (A.36)Equation (A.36) does not have the term corresponding to the second term in Equation (A.34). However, the other first-order conditions are essentially the same. Appendix C. Lemma 1 Proof is shown after Lemma 1. Lemma 1 (Optimality condition for FARR and lot size regulation in the presence of optimal zonal boundaries).19 Business zone: The sign of μB(m)(m∈[0,MB]) depends on the sign of −2τ+δγnB(m)γ−1/ρ(m)γ which can be positive or negative; however, there is at least one location where μB(m) changes sign. Two cases arise: Case (i) implying −2τ+δγnB(m)γ−1/ρ(m)γ<0at any m∈[0,MB]: μB(m)>0at any m∈[0,m⌣)and μB(m)<0at any m∈(m⌣,MB]; Case (ii) implying −2τ+δγnB(m)γ−1/ρ(m)γ>0at any m∈[ε,MB], where ε=˙0: μB(m)<0at any m∈[ε,m⌣)and μB(m)>0at any m∈(m⌣,MB]; Condominium zone: μC(m)>0at any m∈[MB,m⌢)and μC(m)<0at any m∈(m⌢,MC]and μ(m⌢)=0where m⌢∈(MB,MC); Lot housing zone: μH(mH)<0at any m∈(MC,MH]and μH(MC)=μC(MC). Lemma 1 is proved as follows. Business zone: We prove Lemma 1(1) in two steps. First, noting that η=1 and ∫0MB[FB(m)/fB(m)]dm=N, combination of Equations (A.7) and (A.8) yields ∫0MBμB(m)[FB(m)/fB2][∂fB/∂w]dm=0. (A.37) This implies that the solution of μB(m) has one of the following two patterns: pattern 1) μB(m) is positive at some m and negative at other m, where m∈[0,MB], or pattern 2) μB(m) is zero all over m∈[0,MB]. In the second step, we will analyze the sign of μ˙B(m) in Equation (A.31). Noting that ∂fB/∂w=∂fB/∂g based on Equation (7), substituting Equation (A.37) into the equation θ=N¯+∫0MBμB(x)FB(x)fB2∂fB∂gdx (from Appendix B) yields θ=N¯, and substituting into (A.14) yields ϕB(0)=N¯. Note that ϕB(MB)=0 from (A.15). Next, Equations (A.13) and (A.28) show ϕ˙B(m)<0 at any m∈(0,MB) when ∂L/∂ω(m)=0 due to the complementary slackness. Therefore, θ−ϕB(m)>0, m∈(0,MB]. This explains why the first parenthesis in Equation (A.31) is positive. It thus turns out that the sign of μ˙B(m) is the same as that of [−2τ+δγ[nB(m)]γ−1/ρ(m)γ] which can be either positive or negative. This concludes the proof of Lemma 1(1). Condominium zone: A similar explanation as in the case of business zone applies. Noting that η=1, combination of Equations (A.5), (A.6) and (A.7) yields ∫MBMCμC(m)[FC(m)/fC2][∂fC/∂rC]dm=0. To hold this, the solution of μC(m) has one of the following two patterns: pattern 1) μC(m) is positive at some m and negative at other m, m∈[MB,MC], or pattern 2) μC(m) is zero all over m∈[MB,MC]. From Equation (A.21), μ˙C(m)=−λC(m)δγ[N−nC(m)]γ−1/ρ(m)γ, m∈(MB,MC). Accordingly, μ˙C(m)<0 because λC(m)>0 from Equation (A.30), where m∈(MB,MC). Therefore, we can exclude pattern 2) of the solution of μC(m). Finally, continuous μC(m) holds Lemma 1(2). Lot housing zone: From Equations (A.6) and (A.17), noting the condition λH(MH)=0, λH(m)=∫mMH1/fH(x)dx=N¯−nH(m). Substituting this into Equation (A.22) yields μ˙H(m)=−δγ[[N¯−nH(m)]/ρ(m)]γ<0,m∈(MC,MH). Equations (A.23) and (A.24) imply μH(MC)=μC(MC), where as proved earlier, μC(MC)<0. Therefore, μH(m)=μC(MC)− ∫MCmδγ[[N−n(x)]/ρ(x)]γdx<0, m∈[MC,MH). The results μ˙H(m)<0 and μH(m)<0 prove Lemma 1(3). Appendix D. Lemma 2 Proof is shown after Lemma 2. Lemma 2(1) is derived from the combination of Equations (A.2), (A.5), (A.25), and (A.33) using η=1 and ∫MBMCμC(m)[FC(m)/fC2][∂fC/∂rC]dm=0 (from the proof of Lemma 1(2)). Following relations are also used: λC(MB)=N¯ and, from Appendix B, λB(MB)=−N¯. The inequality conditions involving μk(k={B,C,H}) are obtained from Lemma 1. Lemma 2(2) is derived from the combination of Equations (A.3), (A.4), (A.6) and (A.26). μC(MC)=μH(MC) is obtained from Equations (A.23) and (A.24). μC(MC)<0 is from Lemma 1(2). Lemma 2(3) is derived from Equation (A.27) with μH(m)=μC(MC)−∫MCmδγ[[N−n(x)]/ρ(x)]γdx, which is obtained in the proof of Lemma 1. This concludes proof of Lemma 2. The equations in Lemma 2 are interpreted following Proposition 2. But first, we compare the optimal zonal boundaries with the market boundaries. If the boundary of Zone B is determined by the market, then FBrB(g(MB),w)−SB(FB)= FCrC(MB)−SC(FC) where the land rents are equal between Zones B and C. Lemma 2(1) shows that, in Case (i), FBrB(g(MB),w)−SB(FB)>FCrC(MB)−SC(FC) because the right-hand side of the first equation in Lemma 2(1) is greater than zero. However, in Case (ii), whether FBrB(g(MB),w)−SB(FB) is greater or less than FCrC(MB)−SC(FC) is ambiguous because the right hand side of the first equation in Lemma 2(1) can be either negative or positive. Likewise, if the boundary between Zone C and Zone H is determined by the market, then FC(MC)rC(MC)−SC(FC)=rH(MC). Lemma 2(2) shows that, in the optimal case, the sign of [FC(MC)rC−SC(FC)−RA]−[rH(MC)−RA] is same as that of the right side of the equation in Lemma 2(2). The right side can be arranged into DC(MC)FC(MC)−DH(MC)=fC(MC)DC(MC)[nC(MC)−nH(MC)], using the second equation in Lemma 2(2). This implies that the right side is greater than zero because DC(MC)>0 as denoted in Lemma 2(2) and [nC(MC)−nH(MC)]>0 because, by definition, a condominium has more households than a detached house. Correspondingly, in the optimal case, [FC(MC)rC−SC(FC)−RA]−[rH(MC)−RA]>0. Finally, if the urban boundary is determined in the market, then rH(MH)=RA. Lemma 2(3) shows that in the optimal case, rH(MH)>RA because the right-hand side of the equation in Lemma 2(3) is greater than zero. Appendix E. Numerical simulations In the Supplementary data, we present some numerical examples21 to demonstrate how social welfare in our model changes with FAR and/or zonal boundaries. This helps understand the property of optimal land use regulation, theoretically achieved in this article, in a quantitative manner. However, the numerical simulation does not completely trace our Propositions. The base simulation model is a market equilibrium model, not our maximization problem (that is, Definition 2).22 Our Propositions show what properties the optimal regulations possess, compared to the market equilibrium, while our numerical simulations only show how much a certain level of difference in the level of regulation from the market equilibrium changes the welfare. © The Author (2017). Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)