Optimal land use regulations in a city with business areas

CHAPTER FIVE

Optimal land use regulations in a city with business areas

5.1 Introduction Previous chapters have targeted land use regulations in residential areas only. However, business zones also can be regulated to enhance positive externalities. In business zones, 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). To make use of such positive externalities and to mitigate negative externalities such as traffic congestion, governments can intervene in the urban space market through simultaneous imposition of multiple regulations on building size and lot size and by zoning the city into different land uses. In real cities, however, such land use regulations have rarely been implemented or seriously considered. This chapter explores how multiple land use regulations should be simultaneously imposed across a city in the presence of agglomeration economies and traffic congestion following Kono and Joshi (2018). In particular, we consider a monocentric city with three distinct land use zones—consisting of business, condominiums, and detached houses in that order—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. Applying optimal control theory to the continuous city with the three distinct zones, we obtain optimal density regulation that changes continuously in each distinct zone. The rigorous derivation is shown in Technical Appendix, but the sketch of the derivation process is demonstrated in the main text. We separately treat floor area ratio (FAR) regulation and lot size (LS) regulation because building-size regulation such as FAR regulation necessarily generates deadweight loss caused by the regulation itself (see Chapter 2), whereas LS regulation has no deadweight losses (see Wheaton, 1998). Under FAR regulation, households can choose their Traffic Congestion and Land Use Regulations https://doi.org/10.1016/B978-0-12-817020-5.00005-4

© 2019 Elsevier Inc. All rights reserved.

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optimal floor size within the regulated buildings. That is, FAR regulation controls population density indirectly, whereas LS regulation does this directly. In addition, we design optimal regulations on multiple zonal boundaries between the business zone, condominium zone, and detached housing zone. Section 5.2 develops a model, and Section 5.3 examines optimal regulations. Section 5.4 presents conclusion. The final subsection provides a technical appendix that shows mathematical treatment of such three-zone model to derive the optimal conditions on land use regulation. We show by how much the building size, the LS, and the zone size should differ from those found at the market equilibrium. The differences are composed of empirically observable economic variables. This result can readily be used to evaluate whether the current density and zonal regulations are optimal or not.

5.2 The model 5.2.1 The city The model city is closed, monocentric, and linear with a width of unity and size defined by m 2 [MH, MH], where m denotes distance from the city center. As depicted in Fig. 5.1, which shows only the right-hand side of the symmetrical city, the city is divided into the following three zones in the given order: (i) the central business district (CBD) or business zone, Zone B (m 2 [0, MB]), consisting of office buildings; (ii) the condominium zone, Zone C (m 2 [MB, MC]); and (iii) extending to the city boundary, the lot housing zone, Zone H (m 2 [MC, MH]), consisting of single-family houses.

Fig. 5.1 The model city. (Source: Kono, T., Joshi, K.K., 2018. Spatial externalities and land use regulation: an integrated set of multiple density regulations. J. Econ. Geogr. 18 (3), 571–598.)

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Zone B and Zone C are regulated by FAR regulation. 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 area. Let Fj ( j 2 {B, C}) denote the FAR of a building in Zone j. Likewise the lots within Zone H are regulated; let fH denote the lot consumption per household in the zone.
identical households divided equally The city is inhabited by 2N between the two halves of the symmetrical city. One member of each household commutes to the CBD where all firms, and therefore jobs, are located. In other words the city population is identified with the number
is exogenously fixed. of households. The city is closed, implying that 2N Buildings in Zones B and C are constructed by developers, whereas we ignore housing capital in Zone H, assuming that land is directly consumed by the residents. We assume 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. 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, FAR regulation, LS regulation, and zonal regulation on zonal boundaries are imposed. The policy variables are (1) FAR at each location in Zone B and Zone C, that is, Fj(m), ( j 2 {B, C}, m 2 [0, MC]) (2) lot size at each location in Zone H, that is, fH(m), (m 2 [MC, MH]); and (3) three zonal boundaries, that is, Mk (k 2 {B, C, H}). Firm density in Zone B and population density in Zone C are adjusted only by FAR regulation, whereas in Zone H, only LS regulation adjusts population density.

5.2.2 Firms’ behavior All firms are located within Zone B, and they have identical production functions. We model single-worker production that uses floor area as an input and labor. The production function is expressed as AX( fB), where A is the communication-based factor productivity function, fB is the perfirm floor area, and X( fB) is the partial production function.a Following Borukhov and Hochman (1977), O’Hara (1977), and Ogawa and Fujita (1980), we assume that each worker communicates inelastically a

As Borukhov and Hochman (1977) note, single-worker production is not so specific. 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) ¼ AΓ(Q/l, 1), where fB  Q/l.

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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. 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 firm during a certain period is normalized to one without loss of generality. With the number
of total workers and thus the number of single-worker firms being 2N,
the total communication trips for each firm is 2N  1. In this case,
which implies that A
is constant because 2N
 1Þ  A,
 1 is A ¼ Að2N constant. The profit for a firm at m, denoted π(m), is then given by
ðfB Þ  gðmÞ  w ðmÞ  rB ðmÞfB , m 2 ½0,MB  π ðmÞ ¼ AX

(5.1)

where g(m), w(m), and rB(m) denote the communication trip cost, wage, and floor rent for the firm at location m, respectively. 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 j x  mj, where jx  mj is the distance between the communicating 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 ð MB GðmÞ ¼ n0B ðxÞ½τjx  mjdx, m 2 ½0,MB  (5.2) MB

where ÐnB0 (x)  ∂ nB/∂ x, which denotes worker density at x,b and 0 nB(m)  m 0 nB (x)dx is the number of total workers working at the firms located between the CBD center and location m. The communication 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 could be paying more (e.g., by driving inefficiently slowly or by consuming more fuel for unnecessary acceleration). Therefore the relation between G(m) and the actual payment g(m) is expressed as an inequality conditionc: gðmÞ  GðmÞ, m 2 ½0,MB  b c

(5.3)

An apostrophe on a variable, hereafter, denotes the derivative of the variable with respect to distance. A similar method is used in Kono and Joshi (2012) and Kono and Kawaguchi (2017).

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One may think that this inequality is not necessary because a rational worker does not pay more than the supply-side cost. Indeed, this expression is going to hold as an equality due to the rational worker’s optimal behavior. Nevertheless, this inequality expression works later for determining the sign of the shadow price (or Lagrange multiplier) of this constraint.d Differentiating the right-hand side of Eq. (5.2) with respect to m yieldse dGðmÞ  G0 ðmÞ ¼ 2nB ðmÞτ, m 2 ½0,MB  dm and an initial condition is obtained as ð MB n0B ðmÞτmdm Gð0Þ ¼ 2

(5.4)

(5.5)

0

Rent bidding among firms yields π(m) ¼ 0. The bid rent is expressed as  
ðfB Þ  gðmÞ  wðmÞ AX , m 2 ½0,MB  (5.6) rB ðgðmÞ, w Þ ¼ max fB fB The first-order condition yields
ðfB Þ + gðmÞ + w ðmÞ ¼ 0
∂X  AX fB A ∂fB

(5.7)

Thus fB(m) is a function of g(m) and w(m). As Eq. (5.7) shows a firm considers only its private communication costs, whereas its proximity to other firms may allow the other firms to economize on their communication costs. This is what Kanemoto (1990) calls “locational externality.” Locational externalities are technological externalities generated by firms through their location selection. Kanemoto (1990) shows locational externalities existing in the case of bilateral trading between firms, say firm A and firm B, as shown in Fig. 5.2. In this situation, when firm A determines its location, it compares advantages (e.g., a reduction in transportation costs) and disadvantages (e.g., an increase in rents) of the location. For example, a move of firm A toward firm B will benefit not only firm A but also firm B through a reduction in transportation costs for both firms. This is a technological externality. d

Because the multiplier implies the value of the supply-side transport cost, the sign should be negative, implying a natural result that an increase in the transport cost has a negative welfare effect. Thus, when we use an inequality condition, this sign is straightforwardly determined using the Kuhn-Tucker condition. Ð e d MB 0 dm MB nB ðxÞ½τ jx  mjdx ¼ ½nB ðMB Þ + nB ðmÞ  ½nB ðMB Þ  nB ðmÞτ ¼ 2nB ðmÞτ, m 2 [0, MB].

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Fig. 5.2 The mechanism generating locational externalities.

In other words the selection of a location by firm A is not socially optimal due to the existence of locational externalities. From the social viewpoint the combined advantages for both firm A and firm B should be compared with the disadvantages for firm A alone when location of firm A is determined. So the market equilibrium locations are sparser than the optimal locations. Such locational externality can be adjusted by land use regulations. In the real world, there are other agglomeration economy factors such as nonmarket-based communications. However, basically, the locations of firms in the existence of agglomeration economy factors are sparser than the optimal locations. Accordingly, our setting can represent other such agglomeration economy factors basically when we target the effect of agglomeration economies on optimal land use regulations. Finally the wage at each location should compensate for the commuting cost. Let TB(m) denote the commuting cost from MB to m; then, wage w(m) should follow (5.8) wðmÞ ¼ w ðMB Þ + TB ðmÞ

5.2.3 Developers’ behavior Developers supply buildings in Zones B and C under FAR regulation. Let π dj ( j 2 {B, C}) denote developers’ profit from the construction of a building in Zone j, which is given by           π dj mj ¼ Fj mj rj mj  Sj Fj ðmÞ  Rj mj , (5.9) j 2 fB,Cg, mB 2 ½0,MB , mC 2 ½MB ,MC  where Sj(Fj(mj)) denotes the total construction cost of FAR-regulated floor area Fj(mj). Likewise, rj and Rj denote floor rent and land rent, respectively. Note that buildings are constructed on lots of equal size normalized to one. Considering perfectly competitive developers, the zero-profit condition is given by π dj (m) ¼ 0 ( j 2 {B, C}), which yields   Rj ðmÞ ¼ Fj ðmÞrj ðmÞ  Sj Fj ðmÞ , j 2 fB,Cg, mB 2 ½0,MB , mC 2 ½MB ,MC  (5.10)

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5.2.4 Commuting cost—An external factor To consider congestion externalities, we adopt Bureau of Public Roads (BPR) functions, which are frequently used in related papers and used throughout this book. 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 by T() and TB(), respectively (see Fig. 5.1). The unit-distance commuting cost within the residential area, T(m), borne by the resident at location m has the following condition:  
 nðmÞ γ N dT ðmÞ 0 , m 2 ½MB ,MH   T ðmÞ  ξ + δ ρðmÞ dm

(5.11)

where n(m) is the total commuter population residing beyond the CBD edge
 nðmÞ is the total number of commuters passup to location x, and thus N ing through location m on the way to the CBD. Likewise, 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. Eq. (5.11) uses an inequality condition. The left-hand side is the demand-side cost, which is paid by a commuter, 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 could be paying more than the supply-side cost.f Next, TB(m) is defined as   dT B ðmÞ nB ðmÞ γ 0 , m 2 ½0,MB   TB ðmÞ  ξ  δ dm ρðmÞ

(5.12)

where nB(m) denotes number of firms located between the city center and location m. The right-hand side in Eq. (5.12) 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.g f

The discussion following Eq. (5.3) also applies here; we replace “rational worker” with “rational commuter.” Note that this unnecessary commuting cost is not equal to congestion pricing. The payment for congestion pricing will be returned to the society, but the unnecessary commuting cost is lost by the society. g This is why Eq. (5.12) differs from Eq. (5.11) in formulation. In Eq. (5.11), N
 nðmÞ denotes commuter population residing beyond location m (m 2 [MB, MH]). In Eq. (5.12), nB(m) denotes commuter population working at firms located over [0, x] (m 2 [0, MB]) that would cross location m while commuting to and from the firms.

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5.2.5 Household behavior Each household worker earns wage w per period by working in the CBD. The household’s expenditure comprises commuting, housing, and nonhousing commodity costs. Private cars are the only mode of commuting. For simplicity, we assume a quasilinear utility function for households living in Zone C and H, denoted VC and VH, respectively, which is expressed as Vi(mi) ¼ ui( fi(mi)) + zi(mi),h where i 2 {C, H}, mC 2 [MB, MC], and mH 2 [MC, MH]. Here, uC and uH denote household utility derived from the consumption of floor space fC(m) and LS fH(m), respectively, and zi is the numeraire nonhousing commodity.  The income constraint is expressed as zi + fi ri ðmÞ ¼ w ðmÞ + 1=N
Φ T ðmÞ  TB ðmÞ ði 2 fC,H gÞ, 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. Note that ½1= N
Φ on the right-hand side of Eq. (5.12) implies the assumption of public ownership of land. Using Eq. (5.8) the income constraint can be simplified as
Φ  T ðmÞ (i 2 {C, H}). Then, Φ is expressed as zi + fi ri ðmÞ ¼ w ðMB Þ + ½1= N Φ¼

ð MB 0

½RB ðmÞ  Ra dm +

ð MC MB

½RC ðmÞ  Ra dm +

ð MH

½rH ðmÞ  Ra dm

MC

(5.13) where Ra is the agricultural rent and rH is the land rent in Zone H.

5.2.6 Market clearing conditions and definitions The equality of utilities among locations and market clearing conditions are shown here. First, Eq. (5.14) implies that the household utility is equal everywhere because households are indifferent regarding locations. VC ðmC Þ ¼ VH ðmH Þ  V , mC 2 ½MB ,MC , mH 2 ½MC ,MH 

(5.14)

Population function n(m) and commuting cost T(m) are both continuous at MC but are not necessarily smooth. To clearly distinguish these functions h

Because 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 FAR regulation. Even in the latter case, the formula describing optimal regulation will essentially be the same. Moreover, if the change in marginal utility with respect to income does not change much according to the regulations, the same properties regarding optimal regulation are obtained.

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before and after MC, we define population functions as nC(m) and nH(m) and commuting costs as TC(m) and TH(m), where the subscripts represent corresponding zones. Next, Eqs. (5.15), (5.16) express market equilibrium in floor space in Zone B and C, respectively, implying that the total floor space consumed is balanced by total floor space supplied. Next, Eq. (5.17) shows that, in Zone H, the households at m consume fH(m) area of lot; therefore the total area consumed is equal to the unit land area supplied. Floor space in Zone B: fB ðgðmÞ, w ðmÞÞn0B ðmÞ ¼ FB ðmÞ, m 2 ½0,MB 

(5.15)

Floor space in Zone C: fC ðmÞn0C ðmÞ ¼ FC ðmÞ where n0C ðmÞ 

∂nC ðmÞ , m 2 ½MB , MC  ∂m

(5.16)

Lot supply in Zone H: n0H ðmÞfH ðmÞ ¼ 1 where n0H ðmÞ 

∂nH ðmÞ , m 2 ½MC , MH  ∂m

(5.17)

Finally, as shown in Eq. (5.18), because one household member works in the CBD, the total number of workers (left side) is equal to the household pop
(right side). ulation N Labor population: ð MB
n0B ðmÞdm ¼ N (5.18) 0

5.3 Optimal land use regulations 5.3.1 Maximizing social welfare The objective of optimal regulations can be denoted as Eq. (5.19) using the social welfare W composed of total utilities:
V W ¼N

(5.19)

where V is defined in Eq. (5.14). Optimal FAR at each location in Zones B and C, that is, Fj(m), ( j 2 {B, C}; m 2 [0, MC]); optimal LS at each location in Zone H, that is, fH(m), (m 2 [MC, MH]); and three optimal zonal boundaries MB, MC, and MH are

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achieved by maximizing W subject to the market equilibrium Eqs. (5.1)– (5.19). Note that regulation controlling MH is same as urban growth boundary (UGB). The optimal control problem is expressed in the following Lagrangiani:
V L¼N   ð MC FC ðmÞ 1 + κC ðmÞ wðMB Þ + Φ + uC ðfC ðmÞÞ  V  fC ðmÞrC ðmÞ  TC ðmÞ dm
fC ðmÞ N MB   ð MH 1 1 w ðMB Þ + Φ + uH ðfH ðmÞÞ  V  fH ðmÞrH ðmÞ  TH ðmÞ dm κ H ðmÞ +
fH ðmÞ N MC 2 ðM ðM B

C

3

½FB ðmÞrB ðmÞ  SB ðFB Þ  Ra dm  ½FC ðmÞrC ðmÞ  SC ðFC Þ  Ra dm 7 6Φ 6 7 0 MB η6 ð MH 7 4 5 ½rH ðmÞ  Ra dm  MC

ð MB + 0

  ðM ð MB B  ϕB ðmÞ G0 ðmÞ  2nB ðmÞτ dm + θ Gð0Þ  2 n0B ðmÞτmdm + ωðmÞ½ gðmÞ 0

0

      ð MB nB ðmÞ γ FB ðmÞ 0 0  nB ðmÞ dm GðmÞdm + λB ðmÞ TB ðmÞ + ξ + δ μB ðmÞ dm + ρðmÞ fB ðmÞ 0 0       ð MC ð MC
 nC ðmÞ γ FC ðmÞ N 0 0 dm + + λC ðmÞ TC ðmÞ  ξ  δ μC ðmÞ  nC ðmÞ dm ρðmÞ fC ðmÞ MB MB  γ     ð MH ð MH
1 0 ðmÞ  ξ  δ N  nH ðmÞ dm + λH ðmÞ TH μH ðmÞ +  n0H ðmÞ dm ρðmÞ fH ðmÞ MC MC ð MB

+ ςn ½nH ðMC Þ  nC ðMC Þ + ςT ½TH ðMC Þ  TC ðMC Þ

(5.20)

where rB(m)  rB(g(m), w(MB) + TB(m)), fB(m)  fB(g(m), w(MB) + TB(m)), fC(m)  fC(rC(m)), and Sj(Fj)  Sj(Fj(m)) ( j 2 {B, C}). Likewise, κ i(m), η, θ, ω(m), ϕB(m), ψ B(m), λk(m), μk(m), ςn, and ςT (i 2 {C, H}, k 2 {B, C, H}) are shadow prices. Because communication between firms takes place bilaterally, the conÐ MB 0 straint, G(0) ¼ 2 0 nB (m)τmdm from Eq. (5.5), is necessary in Eq. (5.20). i

The Lagrangian considers only the right-hand side of the city. However, communication costs in the right side of the city also depend on the left side. Eqs. (5.4), (5.5) include the left-side firms’ communication costs. So, we can first consider dividing them by two for inclusion in the Lagrangian. But the city is symmetrical, so the left side’s symmetrical change should be considered, which can be done by multiplying the equations by two because the communication trips are inelastic. In conclusion, we can consider Eqs. (5.4), (5.5) as they are. This Lagrangian can also be replaced by Hamiltonian, which generates the same first-order conditions.

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As Tauchen and Witte (1984) and Fujita and Thisse (2013) show, the total communication cost in the CBD is expressed by double integrals, so the Lagrangian should consider this constraint specifically in addition to the 0 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) ¼ 0, and
. For Zone C, nC(MB) ¼ 0. For Zone H, nH ðMH Þ ¼ N.
nB ðMB Þ ¼ N 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 LS market due to the distribution of residences, and (iii) commuting costs. The first-order conditions presented in Technical Appendix 5(1) show the relationships among distortions caused by the regulations, agglomeration economies, and congestion. We interpret these relationships in Technical Appendix 5(2).

5.3.2 Optimal FAR and LS regulations This section obtains important properties of the optimal regulations on FAR and LS. First, we consider shadow prices μk(m)(k ¼ B, C, H), which directly show how and by how much the FAR or lot regulations should be imposed. Following Lemma 5.1, which is presented in Technical Appendix 5(3), the motion of μk(m) is illustrated in Fig. 5.3. As shown in Fig. 5.2, two cases arise in Zone B, depending on whether communication costs are greater or smaller than traffic congestion costs: more specifically, whether 2τ + δγ(m)γ1/ρ(m)γ < 0 (Case I) or 2τ + δγ(m)γ1/ ρ(m)γ > 0 (Case II), where m 2 [0, MB]. Case I implies that either communication cost τ or the transportation capacity ρ(m) is sufficiently large, resulting into low traffic congestion, whereas Case II arises when either τ or ρ(m) is significantly small, resulting into severe congestion.j As explained in Technical Appendix 5(3), the sign of μj(mj)( j 2 {B, C},   ∂Sj ðFj ðmj ÞÞ mB 2 [0, MB], mC 2 [MB, MC]) is the reverse of the sign of rj mj  ∂F m , jð jÞ which denotes distortion in the floor space market caused by FAR   ∂Sj ðFj ðmj ÞÞ regulation as shown in Fig. 5.4A. Note that rj mj  ∂F m ¼ 0 if jð jÞ j

Case I is likely to emerge near the city center because the communication cost for business people is generally high, whereas the number of commuters near the city center would be close to zero. Moreover, in most developed cities with high wages, Case (I) is likely to hold in rather broad areas.

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Fig. 5.3 The motion of shadow prices μk(m). (Source: Kono, T., Joshi, K.K., 2018. Spatial externalities and land use regulation: an integrated set of multiple density regulations. J. Econ. Geogr. 18 (3), 571–598.)

Fig. 5.4 Deadweight loss due to (A) FAR regulation (left) and (B) LS regulation (right). Note: Superscripts ‘†’ and ‘∗’, respectively, refer to the market equilibrium and optimal cases. (Source: Kono, T., Joshi, K.K., 2018. Spatial externalities and land use regulation: an integrated set of multiple density regulations. J. Econ. Geogr. 18 (3), 571–598.)

the FAR is unregulated and determined by the market. FAR greater (resp.   ∂Sj ðFj ðmj ÞÞ smaller) than the market FAR implies rj mj  ∂F m < 0 (resp. jð jÞ   ∂Sj ðFj ðmj ÞÞ rj mj  ∂F m > 0). Likewise the sign of μH(m) (m 2 [MC, MH]) is the jð jÞ reverse of the sign of ∂uH∂fðHfðHmðÞmÞÞ  rH ðmÞ, which denotes distortion in the lot supply market due to LS regulation as shown in Fig. 5.4B. Following Lemma 5.1 presented in Technical Appendix 5(3) and Fig. 5.3, we achieve Proposition 5.1 (see Technical Appendix 5.3 for further explanation).

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Proposition 5.1 (Optimal FAR and LS regulation in the presence of optimal zonal boundaries) (1) Business zone: nB ðmÞγ1 < 0 at any m 2 [0, MB]: The optimal firm • Case I implying 2τ + δγ ρðmÞγ density is higher (resp. lower) in the more central (resp. peripheral) locations relative to the market firm density, requiring minimum (resp. maximum) FAR regulation. nB ðmÞγ1 > 0 at any m 2 [0, MB]: The optimal • Case II implying 2τ + δγ ρðmÞγ firm density is lower (resp. higher) in the more central (resp. peripheral) locations relative to the market firm density, requiring maximum (resp. minimum) FAR regulation. (2) Condominium zone: The optimal population density is higher (resp. lower) in the more central (resp. peripheral) locations relative to the market population density, requiring minimum (resp. maximum) FAR regulation. (3) Lot housing zone: The optimal population density is lower relative to the market population density across the zone, requiring minimum LS regulation. It is important to recall that Proposition 5.1 holds when optimal zonal boundaries are simultaneously imposed, which are presented later in Proposition 5.2. Results of both propositions are presented in Fig. 5.5.

Fig. 5.5 Optimal regulations. Note: In Zone B, FAR regulations in bold letters hold in Case I, and those in parenthesis hold in Case II. Regarding the area regulation a decrease in the zone area hold in Case I, and an increase in the zone area holds in Case II. (Source: Kono, T., Joshi, K.K., 2018. Spatial externalities and land use regulation: an integrated set of multiple density regulations. J. Econ. Geogr. 18 (3), 571–598.)

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The implication of Proposition 5.1 is explained as follows. First the combination of maximum and minimum FAR regulations in the business zone results in an efficient labor distribution 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 FAR regulation” is imposed, the total deadweight loss arising from FAR regulation would be greater than that achieved from the combination of maximum and minimum FAR regulations, as also explained in Chapter 2. Whether the minimum or maximum FAR regulation should be enforced in the central (or peripheral) locations in the business zone depends on the trade-off between agglomeration economies and traffic congestion costs. Next, in the condominium zone, the optimal policy of minimum FAR regulation at the central locations and maximum FAR regulation at the peripheral locations would shift population in favor of more central locations and thereby reduce traffic congestion caused by commuters from distant locations. See Chapter 2 for a detailed discussion. In the lot housing zone, however, the optimal regulation requires enforcement of minimum LS regulation across the zone. The optimal regulation addresses deadweight losses (see Fig. 5.3B). Given that the city also has UGB regulation (explained in the next section) that prevents sprawl, minimum LS regulation reduces supply of housing lots and thereby decreases population in the suburb. This would ultimately reduce traffic congestion across the condominium zone and lot housing zone. It is important to note what minimum LS regulation achieves in our model and how. Although minimum allowable LS in the suburb is prevalent in most countries, especially in the United States, our interpretation of the necessity of minimum LS regulation is different. The objective of such regulation 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 LS regulation in the central area. But our model is able to alleviate congestion externality through minimum LS regulation and without sprawl because it allows options for additional higher density in the central area through minimum FAR regulation. Such result is achieved because of simultaneous consideration of multiple regulations. Finally, Lemma 5.1 (in Technical Appendix 5(3)) shows that the optimal conditions of the regulations are composed of observable variables only.

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119

This is useful for policy evaluators to check whether the current regulations are optimal or not.

5.3.3 Optimal zonal boundaries We now examine how a change in zonal boundaries affects the social welfare under regulated FAR and LS. From Lemma 5.2 in Technical Appendix 5(4), which summarizes the related first-order conditions of the welfare maximization, we achieve Proposition 5.2. Proposition 5.2 (Optimal zonal boundaries in the presence of optimal FAR regulation and LS regulation) (1) Business zone: Recalling the cases stated in Proposition 5.1(1), the optimal zone area is smaller than the market equilibrium zone area in Case I, whereas the result is ambiguous in Case II. (2) Condominium zone: The zone boundary should be shrunk relative to the market boundary. (3) Lot housing zone: The zone boundary, which also acts as the city boundary, should be shrunk relative to the market boundary. The implication of Proposition 5.2(1) is explained as follows. Whether the business zone should be more compact or larger than the market zone area depends on the trade-off between agglomeration economies and traffic congestion costs. An enlargement of the business zone decreases agglomeration economies resulting in a decrease in welfare. Simultaneously, it implies that the business area becomes closer for all residents. Accordingly, congested commuting distances decrease, resulting in an increase in welfare. The net effect is clear in Case I that an enlargement of the business zone would certainly reduce welfare, but the net effect is ambiguous in Case II. Proposition 5.2(2) implies that contraction of Zone C decreases the deadweight loss at the outer edge of Zone C by   ∂SC ðFC ðMC ÞÞ rC ðMC Þ  FC ðMC Þ but increases the same at the inner edge ∂FC ðMC Þ   ∂uH ðfH ðMC ÞÞ of Zone H by  rH ðMC Þ . The first exceeds the latter in abso∂fH ðMC Þ lute value. This can also be explained using the equation in Lemma 5.2(2) (see Technical Appendix 5(4)). Proposition 5.2(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).

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Because the city is closed, such expansion results in relocation of some households from the outer edge of Zone C to the edge of the city, noting that fH(m) (m2 [MC, MH]) is fixed by LS regulation. Such relocation increases the commuting cost in Zone H. The relocation decreases the deadweight loss   ∂SC ðFC ðMC ÞÞ fC ðMC Þ at the outer edge of Zone C by rC ðMC Þ  , where ∂FC ðMC Þ fH ðMH Þ 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 5.2(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 with 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 2 {B, C, H}) denotes optimal change (“+” if expansion and “” if shrinkage) in the corresponding zonal boundary relative to the market equilibrium. Lemma 5.2 in Technical Appendix 5 provides optimal conditions for zone boundaries, which are composed of only observable variables. This is useful for policy evaluators to check whether the current regulations are optimal or not.

5.4 Numerical simulation 5.4.1 The setup This subsection presents some numerical examples 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 paper, 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 (see Section 5.3.1).k Our propositions show what properties the optimal regulations possess, compared with the market equilibrium, while k

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 that our three-zone model city is divided into. We were able to numerically solve the market equilibrium only.

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our numerical simulations only show how much a certain level of difference in the level of regulation from the market equilibrium changes the welfare. Regarding the main parameters such as income and the functional forms such as the utility function and the traffic congestion function, we follow the numerical setup of Kono et al. (2012). The parameters are set for the solution to represent a real situation as closely as possible, using real data or the established parameters. We briefly explain the reasons for setting the parameters in the succeeding text. However, it was impossible to calibrate all the parameters using real data. So, some parameters are set arbitrarily. The three zones in the city are divided into small blocks of equal size ε. The blocks are indexed by i in the residential area composed of Zones C and H, where i ¼ 1 is the residential block adjacent to zone B. In Zone B, i ¼ 1 is the city center block. The inner edge of block i is given by mi ¼ 1 + ε[i  1], where mi now represents the distance variable for the corresponding block. We consider a hypothetical city in Japan with 2 million households, that
¼ 200,000. is, N Zones C and H We first solve Zones C and H. The subutility function uk ( fk (mk )), 00 k 2 {C, H}, is specified as uC( fC(m)) ¼ α ln fC and u( fH) ¼ α ln(1/φ  fH), where α ¼ 0.12y, representing 12% of the household income y, which is
Φ, and φ is a positive constant; we adopt defined as y  w ðMB Þ + ½1=N φ ¼ 0.0015. 00

00

00

The maximization of household utility yields rC ¼ α/fC and rH ¼ α/fH, which leads to the following demand functions: fC ¼ exp(kC) and fH ¼ φ exp(κH), where kC  [VC  y + TC + α]/α and κ H  [VH  y + TH + α]/α. The floor production function is specified as FC ¼ ϖ[S(FC)]β, where ϖ ¼ exp(5.87907) and β ¼ 0.749734, both of which are estimated from 112 samples of Japanese buildings; S(FC) is measured in Japanese yen (JPY)/year, whereas FC represents FAR. When FAR is not regulated, rC  ∂SC(FC)/∂FC ¼ 0, which yields FC ¼ θ[θβrC]β/[1β]. Household density is given by PC ¼ FC/fC and PH ¼ 1/fH in the respective zone. Land rent in Zone C is obtained from the zero-profit condition as RC ¼ FCrC  SC(FC). Assuming 20% allocation of land for nonresidential purpose denoted Ð MC by ρ ¼ 0.2, the differential land Ð MH rent is given by ΦC ¼ [1  ρ] MB [RC(m)  Ra]dm and ΦH ¼ [1  ρ] MC [rH(m)  Ra]dm in the respective zone, where Ra denotes agricultural land rent, equal to 20,000,000 JPY per square km.

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Both income y and household utility V are set exogenously, whereas the total commuting cost from block i, Ti, is derived as follows: T1 ¼ 0; 00 Ti+1 ¼ Ti + ε[Tunit k i + ptoll k i], where Tunit k i, k 2 {C, H}, implies unit cost of commuting and ptoll k i denotes congestion toll. We assume 225 round trips per year, and we use a free-flow commuting cost of 17.6 JPY per km, which is estimated by the Ministry of Land, Infrastructure, Transport, and Tourism (MLIT, 2010). Tunit k i is specified as a BPR function using the Japan Society of Civil Engineers (JSCE, 2003) parameters relevant to urban roads in Japan: " "  2:82 ## 56:78 n^i Tunit k00 i ¼ 2  225  17:6 + 1 + 0:48 Υk00 30 00

00

00

00

00

where ΥC, ΥH are road capacity in respective zones given as ΥC ¼ 152, 240 and ΥH ¼ 76, 120; n^i denotes population beyond block i given by P∗ n^i ¼ ρ ik¼i εϑk , and i∗ denotes the outermost block such that n^i∗ ¼ 0. The congestion toll is calculated as ptoll k00 i ¼ n^i ∂Tk00 i =∂^ ni .
and is carried The iterative process starts at i ¼ 1 with T1 ¼ 0 and n^1 ¼ N out conditional on the value of V that should satisfy the equilibrium conditions. In the laissez-faire and toll-regime (or Pigouvian tax) cases, the iteration stops when i reaches a value i∗ such that n^i  0 and n^i + 1 < 0, indicating that
is just accommodated within mi∗ ¼ MH. The the household population N increment in n^i is simply given by n^i + 1 ¼ n^i  ε½1  ρPi . We then check the equilibrium condition rHi∗ ¼ Ra. The iteration is repeated by adjusting V until the equilibrium condition is achieved within a reasonable degree of accuracy. Note that the boundary of Zone C is defined by the condition: RCi rHi and RC(i+1) < rH(i+1). Also recall that income y remains assumed. y is adjusted while solving Zone B as explained next. Zone B Zone B is also divided into several blocks of size ε, which can be adjusted for more accuracy as explained later. Different from the setting in the residential zones, the total number of blocks, say j, is assumed beforehand (this implies that MB is assumed in advance). Then, i ¼ j, j  1, … , 2, 1 denotes the successive blocks. At the outermost block within Zone B where i ¼ j,
and calculating backward, nB(i1) ¼ nBi  [1  ρ]εPBi where nBði¼jÞ ¼ N, PBi denotes firm density given by PB ¼ FB/fB. Note that nBi denotes commuter population working at firms located between the city center and

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123

block i. As distance from the center to the block decreases, the number of commuting trips, implied by nBi, also decreases because most commuters would have already reached their firms. The way commuting cost is calculated is also different from the setup in the residential zones. Recall that the commuting cost TB is measured from the CBD edge, MB. So, TB(i¼j) ¼ 0, and TB(i1) ¼ TBi + ε[Tunit Bi + ptoll Bi], where Tunit Bi is given by "

"  2:82 ## 56:78 nB Tunit Bi ¼ 2  225  17:6 + 1 + 0:48 ΥB 30 where ΥB ¼ 318,560. The congestion toll is calculated as ptoll Bi ¼ ni ∂ Tunit Bi/∂ nBi. Besides MB and income y, wage w(MB) is also assumed beforehand. Adjustments are made to these variables unless the following three conditions are met simultaneously within a reasonable degree of accuracy: : (i) RBj ¼ RCj (adjust ε for more accuracy); : (ii) WB ¼y  NΦ , where Φ ¼ ΦB + ΦC + ΦH; ΦB is total differential land rent in Zone B, and [ΦC + ΦH] is derived from the calculation related to the residential zones; and :

(iii) y¼ N1 +

hÐ

 Ð MB MB 0 0 nB ðmÞ AX ð fB Þ  gðmÞ  TB ðmÞ  0 SðFB ðmÞÞdm  Ra MB

i

1 ½ΦC + ΦH  N

(adjusting y would start a new iteration for the residential zones as well). The last condition is derived from the resource constraint that is obtained by combining equations of rB (from maximization of utility) and RB (from zeroprofit condition) and integrating the result over Zone B. The components of the condition based on the resource constraint are derived next (index i is suppressed where there is no ambiguity). 0:05 (i) Agglomeration: A ¼ χ ½2N  1 , where χ is a positive scale factor. We use χ ¼ 1,500,000. (ii) Communication: G(MB) ¼ 2τN MB, and G_ ðmÞ ¼ 2τnB ðmÞ > 0ðm > 0Þ: Accordingly, Gi+1 ¼ Gi + 2τnBiε, and by backward calculation, Gi ¼ Gi+1  2τnBiε. In fact, MB is assumed beforehand to facilitate this backward calculation. Allowing round trips for communication, τ ¼ 2  17.6 (JPY/km).

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(iii) Production: Assuming a one-degree homogeneous Cobb-Douglas function, AΓ ðFB l Þ ¼ AFB αB l 1αB , where l denotes total number of workers, and αB ¼ 0.4, a constant. Then, applying this to our  αB FB Γ ðFB l Þ one-labor firms, X ðfB Þ ¼ l ¼ ¼ ½fB αB . This also yields l ∂X/∂ fB ¼ αB[ fB]αB1. (iv) Floor rent and consumption: Solving zero-profit condition, we
½∂X ðfB Þ=∂fB  ¼ αB ½fB αB 1 , and fB ¼ ½½gðmÞ + wðmÞ= get rB ¼ A
½1  αB 1=αB . ½A (v) Floor production function: FB ¼ ϖ[S(FB)]β, where ϖ ¼ exp(5.87907) and β ¼ 0.749734; S(FB) is measured in JPY/year, whereas FB represents FAR. When FAR is not regulated, rB  ∂SB(FB)/∂FB ¼ 0, which yields FC ¼ θ[θβrB]β/[1β]. (vi) Land rent: RB ¼ FBrB  SB(FB).

5.4.2 Numerical results For the residential zones, we adopt ε ¼ 4 km, whereas the block size in Zone B is kept flexible for reasons explained earlier. We present a total of 10 examples including the laissez-faire case, Pigouvian tax regime, and examples with exogenously set FARs and/or UGB. The Pigouvian tax regime internalizes two kinds of externalities addressed in our paper, namely, traffic congestion and “locational
are fixed because externality.” Agglomeration economies arising from A the population size is exogenously fixed. Regarding the “locational externality” whereby a firm considers only its private communication costs, the social cost is calculated by doubling the communication costs. The welfare achievable by this regime is the first best. In Example 1, residential zone size, (MH  MB), is set at 44 km, representing a UGB case. The Example 2 (20, 10, 15, 30, 2) refers to the case where, relative to the laissez-faire case, the FARs at the central locations of Zone B are increased by 20% whereas the FARs at the farther locations of Zone B are decreased by 10%; the FARs at the central locations of Zone C are increased by 15% whereas those at the farther locations are decreased by 30%; and the LS across Zone H is increased by 2%. The same setup is repeated in another example (Example 3) with MH  MB set at 44 km, representing FAR with UGB policy. The results under different cases are summarized in Table 5.1. The social
V. welfare is expressed in terms of household utility given W ¼ N

Zonal boundary (km) Policy regime

MB

MC 2 MB

MH 2 MB

MH 2 MC

Social welfare per household (V)

Welfare gain (%)

Laissez-faire Pigouvian Ex. 1 residential zone size (MH  MB) set at 44 km Ex. 2 (20, 10, 15, 30, 2) Ex. 3. (20, 10, 15, 30, 2) with residential zone size (MH  MB) set at 44 km

0.354 0.176 0.355

38 40 38

50 44 44

12 4 6

2,606,090 2,709,630 2,660,970

100.00 53.00

0.396 0.390

34 34

50 44

16 10

2,662,850 2,667,450

54.82 59.26

Land use regulations with business areas

Table 5.1 Zonal boundaries and welfare change.

Note: Ex.: Example.

125

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Traffic congestion and land use regulations

Examples show that compared with a single regulation, a set of appropriately chosen regulations leads to higher welfare gain. Densification at the central locations of Zones B and C relative to the laissez-faire case is welfare inducing. Moreover, compactness of the city (or zones) leads to increase in welfare. Earlier in this chapter, we have shown that optimal regulations compose of minimum FAR regulation at the more central locations and maximum FAR regulation at the farther locations in Zone C, followed by minimum LS regulation in Zone H. Figs. 5.6–5.8 illustrate these results (compare Firm density at Zone B

Number of firms per sq. km

9,000,000

Pigouvian tax regime

8,000,000 Laissez faire 7,000,000 Example 3 (20, -10, 15, -30, 2) with residential zone size set at 44 km

6,000,000 5,000,000 4,000,000 3,000,000 2,000,000 1,000,000 0.00

0.01

0.02

0.03

0.04

0.05

Distance from the city center (km)

Fig. 5.6 First density at the central locations of Zone B under selected policy regimes.

Household density at central locations in Zone C

Number of households per sq. km

29,000

Pigouvian tax regime 27,000

Laissez faire

25,000

Example 3 (20, -10, 15, -30, 2) with residential zone size set at 44 km

23,000 21,000 19,000 17,000 15,000 0

1

2 3 Distance from Zone B edge (km)

4

Fig. 5.7 Household density in Zone C under selected policy regimes.

5

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Household density in Zone H No. of households per sq. km

100

Pigouvian tax regime

95

Laissez faire

90

Example 3 (20, -10, 15, -30, 2) with residential zone size set at 44 km

85 80 75 70 65 60 34

36

38

40

42

44

46

48

50

52

Distance from Zone B edge (km)

Fig. 5.8 Household density in Zone H under selected policy regimes.

laissez-faire case with Pigouvian tax regime). Example 3, with the highest relative welfare gain, is included in Figs. 5.6–5.8 for comparison with Pigouvian tax regime. Note that we have found ambiguity in terms of the effect of FAR regulation and boundary regulations on social welfare in the case of Zone B. So regarding Zone B, our numerical examples should be interpreted accordingly.

5.5 Conclusion In this chapter, we have simultaneously optimized multiple regulations—on building size, LS, and zonal boundaries—in a monocentric city with office buildings, condominiums, and single-family dwellings in that order in distinct but adjoining districts. We demonstrate the necessity of both minimum and maximum FAR regulation in both business zone and condominium zone. Although Chapter 2 shows this result in residential zone, this chapter considers office buildings and residential buildings simultaneously. Regarding the optimal FAR regulation in the business zone, 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 FAR regulation at the more central locations and maximum FAR regulation at more peripheral locations. The same

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applies nonambiguously in the case of the condominium zone, followed by minimum LS regulation in the suburb. In the presence of optimal FAR regulation and LS 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. The objective of minimum LS regulation as practiced is to promote lowdensity development, but such a policy contributes to urban sprawl and thus increases congestion costs. However, our model is able to alleviate congestion externality through minimum LS regulation and without sprawl because it allows options for additional higher density in the central area through minimum FAR regulation. Such result is achieved because of simultaneous consideration of multiple regulations.

Technical Appendix 5 Traditionally, urban models have been solved by applying optimal control theory to the continuous area (typically, residential area). The model in this chapter can also be solved with optimal control theory. But one important distinct feature from the traditional urban models is that our urban model is composed of three distinct zones (i.e., business, condominium, and detached housing zones). Hence, we have to combine the three distinct zones, each of which has continuous area, using the boundary conditions between the adjacent zones. (1) Lagrangian and Pontryagin’s maximum principle To obtain the first-order conditions, we integrate the Lagrangian in Eq. (5.20) by parts. After that, differentiating the Lagrangian with regard to the policy variables and the endogenous variables, we obtain the firstorder conditions (A5.1)–(A5.29). These expressions use the following relations: ∂rB/∂ g ¼  1/fB and ∂ rB/∂ w ¼  1/fB. Also note that an apostrophe denotes derivative with respect to distance from the center:   ∂L ∂SB ðFB ðmÞÞ 1 ¼ 0 : η r B ð mÞ  + μB ðmÞ ¼ 0, m 2 ½0,MB  ∂FB ðmÞ ∂FB ðmÞ fB ðmÞ (A5.1)

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  ∂L ∂SC ðFC ðmÞÞ 1 ¼ 0 : η rC ðmÞ  + μ C ð mÞ ¼ 0, m 2 ½MB ,MC  ∂FC ðmÞ ∂FC ðmÞ f C ð mÞ (A5.2) ∂L ¼0: ∂fH ðmÞ   1 ∂uH ðfH ðmÞÞ 1 ¼ 0, m 2 ½MC ,MH   rH ðmÞ  μH ðmÞ κ H ðmÞ fH ðmÞ ∂fH ðmÞ fH ðmÞ2 (A5.3) where, for deriving Eqs. (A5.2), (A5.3), Eqs. (5.16), (5.17) are used respectively, in addition to Eq. (5.14): ∂L ¼0: ∂rC ðmÞ κC ðmÞFC + ηFC ðmÞ  μC ðmÞ

FC ðmÞ df C ðmÞ ¼ 0, m 2 ½MB ,MC  (A5.4) fC ðmÞ2 dr C ðmÞ

∂L ¼ 0 : κ H ðmÞ + η ¼ 0, m 2 ½MC ,MH  ∂rH ðmÞ ∂L
 ¼0:N ∂V

ð MC

FC ðmÞ dm  fC ðmÞ

ð MH

(A5.5)

1 dm ¼ 0 f H ð mÞ

(A5.6)

ð MC ð MH ∂L FC ðmÞ 1 κC ðmÞ κ H ðmÞ ¼0: dm + dm ð m Þ ð ∂w ðMB Þ f f C H mÞ MB MC ð MB ð MB FB ðmÞ FB ðmÞ ∂fB ðmÞ dm + dm ¼ 0 μB ðmÞ η fB ðmÞ2 ∂w ðMB Þ 0 fB ðmÞ 0

(A5.7)

MB

κ C ð mÞ

MC

κ H ðmÞ

where for the last term, note that ∂ w(m)/∂ w(MB) ¼ 1 by virtue of Eq. (5.8): ∂L FB ðmÞ FB ðmÞ ∂fB ðmÞ (A5.8) ¼ 0 : η  μB ðmÞ  λ0B ðmÞ ¼ 0 2 ð m Þ ∂TB ðmÞ fB ðmÞ ∂T fB ðmÞ B ∂L ¼ 0 : λB ð0Þ ¼ 0 (A5.9) ∂tB ð0Þ ð  ð MH ∂L 1 MC FC ðmÞ 1 dm + dm  η ¼ 0 (A5.10) κ ðmÞ κ H ð mÞ ¼0:
MB C ∂Φ N fC ðmÞ fH ðmÞ MC

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Traffic congestion and land use regulations

∂L FB ðmÞ FB ðmÞ ∂fB ðmÞ + ωðmÞ  μB ðmÞ ¼ 0 : η ¼ 0, m 2 ½0,MB  ∂gðmÞ fB ðmÞ fB ðmÞ2 ∂gðmÞ (A5.11) ∂L ¼ 0 : ϕ0B ðmÞ  ωðmÞ ¼ 0, m 2 ½0,MB  ∂GðmÞ ∂L ¼ 0 : ϕB ð0Þ + θ ¼ 0 ∂Gð0Þ

(A5.12) (A5.13)

where for deriving this condition, Eq. (A5.12) is used noting that the latter holds when m ! 0: ∂L ¼ 0 : ϕB ðMB Þ ¼ 0 ∂GðMB Þ

(A5.14)

∂L FC ðmÞ 0 ¼ 0 : κ C ðmÞ  λC ðmÞ ¼ 0, m 2 ½MB ,MC  ∂TC ðmÞ fC ðmÞ

(A5.15)

∂L 1 ¼ 0 : κ H ðmÞ  λ0 ðmÞ ¼ 0, m 2 ½MC ,MH  ∂TH ðmÞ fH ðmÞ H

(A5.16)

∂L ¼ 0 : λC ðMC Þ  ςT ¼ 0 ∂TC ðMC Þ

(A5.17)

∂L ∂L ¼ 0 : λH ðMC Þ + ςt ¼ 0 and ¼ 0 : λH ðMH Þ ¼ 0 ∂TH ðMC Þ ∂TH ðMH Þ (A5.18) ∂L ½nB ðmÞγ1 ¼ 0 : 2τ½θ  ϕB ðmÞ + μ0B ðmÞ + λB ðmÞδγ ¼ 0, m 2 ½0,MB  ρðmÞγ ∂nB ðmÞ (A5.19) ∂L ½N  nC ðmÞγ1 + μ0C ðmÞ ¼ 0, m 2 ½MB ,MC  ¼ 0 : λC ðmÞδγ ρðmÞγ ∂nC ðmÞ (A5.20) ∂L ½N  nðmÞγ1 + μ0H ðmÞ ¼ 0, m 2 ½MC ,MH  ¼ 0 : λH ðmÞδγ ρðmÞγ ∂nH ðmÞ (A5.21) ∂L ¼ 0 : μH ðMC Þ + ςn ¼ 0 (A5.22) ∂nH ðMC Þ

Land use regulations with business areas

∂L ¼ 0 : μC ðMC Þ  ςn ¼ 0 ∂nC ðMC Þ

131

(A5.23)

  γ 
∂L FB ðMB Þ N  λB ðMB Þ ξ + δ ¼ 0 : ½FB rB ðMB Þ  SB ðFB Þ  Ra  + μB ðMB Þ ∂MB fB ðMB Þ ρðMB Þ   γ 
FC ðMB Þ N λC ðMB Þ ξ+δ  ½FC rC ðMB ÞSC ðFC Þ Ra μC ðMB Þ ¼0 fC ðMB Þ ρðMB Þ (A5.24)

To derive (A5.24), we use ϕB(MB) ¼ 0 from Eq. (A5.14) and η ¼ 1 from Eqs. (A5.6), (A5.10): ∂L ¼ 0 : ½FC ðMC ÞrC  SC ðFC Þ  Ra   ½rH ðMC Þ  Ra  ∂MC FC ðMC Þ 1 + μC ðMC Þ  μH ðMC Þ ¼0 fC ðMC Þ fH ðMC Þ

(A5.25)

To derive (A5.25), we used λC(MC) ¼ λH(MC) by virtue of Eqs. (A5.17), (A5.18): ∂L 1 ¼0 ¼ 0 : ½rH  Ra  + μH ðMH Þ ∂MH fH ðMH Þ ∂L ∂L  0,ωðmÞ  0, ωðmÞ ¼ 0, m 2 ½0,MB  ∂ωðmÞ ∂ωðmÞ λB ðmÞ

∂L ∂L ¼ 0, λB ðmÞ  0, 0 ∂λB ðmÞ ∂λB ðmÞ

(A5.26) (A5.27) (A5.28)

noting the sign of the corresponding constraint in Eq. (5.13): λi ðmÞ

∂L ∂L ¼ 0, λi ðmÞ  0,  0, i 2 fC,Hg ∂λi ðmÞ ∂λi ðmÞ

(A5.29)

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. (2) Interpretation of optimal FAR regulation and LS regulation The following discussion interprets optimal conditions of FAR regulations and LS regulation by zone. These conditions can be explained with Harberger’s welfare formula, which measures the welfare change in a distorted economy (see Harberger, 1971).

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Traffic congestion and land use regulations

a. FAR regulation in the business zone First, solving Eqs. (A5.1), (A5.8), (A5.9) and using η ¼ 1 from Eq. (A5.6) and

 ðm  ∂SB ðFB ðxÞÞ FB ðxÞ ∂fB Eq. (A5.10) yields λB ðmÞ ¼ nB ðmÞ + rB ðxÞ  dx, ∂FB ðxÞ fB ðxÞ ∂TB 0 where λB(m) is the shadow price for the commuting cost within the business zone. Next, substituting Eq. (A5.12) into Eq. (A5.11) to cancel out ω(m) and then integrating the result and using boundary condition ϕB(MB) ¼ 0 from ð MB FB ðxÞ ∂fB Eq. (A5.14) yield ϕB ðmÞ ¼ ½N
 nB ðmÞ  μB ðxÞ dx: From fB ðxÞ2 ∂g m Eq. (A5.13), θ ¼ ϕB(0), whereas ϕB(0) is obtained from the aforementioned ð MB FB ðxÞ ∂fB equation involving  ϕB(m), thereby yielding θ ¼ N
+ μB ðxÞ dx: fB ðxÞ2 ∂g 0 Substituting these three equations regarding λB(m),  ϕB(m), and θ as well as

Eq. (A5.1) into Eq. (A5.19) yields ΩB  μ0B ðmÞdm

2 3  6  ðm  ∂SB ðFB ðxÞÞ FB ðxÞ ∂fB ½nB ðmÞγ1 7 6 7 rB ðxÞ dx 62τ + δγ ¼ nB ðmÞ 7dm, 4|{z} ∂FB ðxÞ fB ðxÞ ∂g ρðmÞγ 5 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} <0 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} ¼θϕB ðmÞ>0

>0

m 2 ð0,MB  (A5.30)

Note that μ0B ðmÞdm ¼ μB ðm + dmÞ  μB ðmÞ, where μB(m) is the shadow price for the floor area containing FB/fB number of workers at location m. Therefore Eq. (A5.30) 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 FAR regulation. Recall that we have supposed a symmetrical 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)]γ1/[ρ(m)]γ in Eq. (A5.30). The first term [2τ] is related to the increase in the communication cost for firms located in [0, MB]. The relocation of one worker from m to m + dm increases the communication cost of the firms located in [0, m] but decreases the communication cost of the firms located in [m, MB]. Likewise the symmetrical left-side relocation of one worker from m to mdm increases the communication cost of the firms located in both [0, m] and [m, MB]. Summing up the total communication cost for all firms located

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in [0, m] increases by 2nB(m)τ and that for all the firms over [m, MB] increases
 nB ðmÞ½τ  τ ¼ 0. As a result the change in the total communication by ½N 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 2 [0, m] affects the respective floor market deadweight loss caused by FAR regulation, which is expressed as  ðm  ∂SB ðFB ðxÞÞ FB ðxÞ ∂fB ðxÞ  rB ðxÞ  dx  ½2τ, noting that expresses ∂FB ðxÞ fB ðxÞ ∂gðxÞ 0 the change in the deadweight loss in the floor market (see Fig. 5.3). Next the term δγ[nB(m)]γ1/ρ(m)γ in Eq. (A5.30) is the saving in traffic congestion cost for the firms located over [0, m]. This change in traffic congestion costs also affects the respective floor market distortions caused by FAR regulation, which is expressed as 

ðm  0

rB ðxÞ 

 ½nB ðmÞγ1 ∂SB ðFB ðxÞÞ FB ðxÞ ∂fB ðxÞ dx  δγ . ∂FB ðxÞ fB ðxÞ ∂gðxÞ ½ρðmÞγ

This concludes the explication of Eq. (A5.30). Next, from Eq. (A5.1), ΨB  μ0B ðmÞdm

  ∂SB ðFB ðm + dmÞÞ ¼ fB ðm + dmÞ rB ðm + dmÞ  ∂FB ðm + dmÞ   ∂SB ðFB ðmÞÞ + fB ðmÞ rB ðmÞ  ∂FB ðmÞ

(A5.31)

recalling that η ¼ 1 from Eqs. (A5.6), (A5.10). The relocation of one worker implies relocation of fB units of floor space. Accordingly, Eq. (A5.31) implies the total change in deadweight loss in the floor market, which arises from the FAR regulation 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 Eq. (A5.30), should be balanced with the change in the deadweight loss in the floor market expressed as ΨB in Eq. (A5.31). This is the explication of the first-order conditions with respect to FAR regulation in the business zone. b. FAR regulation in the condominium zone First, noting that η ¼ 1, combination of Eq. (A5.4) and Eq. (A5.15), by FC ðmÞ μC ðmÞ FC ðmÞ df C ðmÞ cancelling κ C,  ¼ 0. yields λ0C ðmÞ + fC ðmÞ fC ðmÞ fC ðmÞ2 dr C ðmÞ μ ðmÞ ∂SC ðFC ðmÞÞ Rearrangement of Eq. (A5.2) yields  C ¼ rC ðmÞ  . fC ðmÞ ∂FC ðmÞ Substituting this into the earlier equation and integrating the result with respect to m, we obtain

134

ð MC m

Traffic congestion and land use regulations

λ0C ðxÞdx ¼ 

ð MC m

n0C ðxÞdx +

 ð MC  ∂SC ðFC ðxÞÞ 0 df ðxÞ nC ðxÞ C rC ðxÞ  dx, ð x Þ ∂F dT C C ð xÞ m

which leads to
 nC ðmÞ + λC ðmÞ ¼ ½N

ð MC  m

where

relationships are used: 1 df df C
 nðMC Þl and  λC ðMC Þ ¼ λH ðMC Þ ¼ N ¼ C , which is obtained fC dr C dT C from differentiating Eq. (5.16).m Recalling the form of the Lagrangian in Eq. (5.20), [λC(m)] on the lefthand side of Eq. (A5.32) expresses the shadow price for a unit of commuting time at m (m 2 [MB, MC]). Eq. (A5.32) is easily interpreted as follows.n The first term on the righthand side of Eq. (A5.32) is the direct effect of total increase in commuting
 nC ðmÞ. The second time for all commuters passing through m, that is, N term is the effect of change in the per-capita floor area consumption, dfC/dTC, beyond m, which is induced by the increase in commuting cost TC(x) (x 2 [m, MC]). The change in the per-capita floor area consumption, 0 dfC/dTC, multiplied by nC (x) gives the change in the total floor area ∂SC ðFC ðxÞÞ is the marginal F(x) at x. As explained earlier the term rC ðxÞ  ∂FC ðxÞ change in the deadweight loss (or distortion) caused by the FAR regulation (see Fig. 5.4). If FC is determined in perfect competition, ∂SC ðFC ðxÞÞ , and correspondingly the second term on the rightrC ðxÞ ¼ ∂FC ðxÞ hand side of Eq. (A5.32) is zero. However, when the floor area is regulated, ∂SC ðFC ðxÞÞ ; that is, the second term in Eq. (A5.32) is not zero. rC ðxÞ 6¼ ∂FC ðxÞ l

the

 ∂SC ðFC ðxÞÞ 0 df ðxÞ rC ðxÞ  nC ðxÞ C dx ∂FC ðxÞ dT C ðxÞ (A5.32)

following


 nC ðMC Þ is proved simply by using Eqs. (A5.5), (A5.16)–(A5.18). λH ðMC Þ ¼ N h i ∂fC ∂uC C Totally differentiating Eq. (5.16) yields dTC ¼  ∂u ∂fC  rC ∂rC dr C + fC dr C , where ∂fC ¼ rC from the dfC dr C 1 first-order condition of utility maximization. So, we obtain  fC ¼ dTC , which leads to  f1C drdfCC ¼ dT . C

m

n

A similar equation to Eq. (A5.32) appears in Kanemoto (1977), Arnott and MacKinnon (1978), 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 FAR regulation 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 the present model. Simultaneously controlling FAR with road area is shown in Chapter 6 in which cordon pricing is implemented.

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Land use regulations with business areas

Next, we interpret the optimality condition of FAR regulation by combining the first-order conditions. Substituting λC(m) from Eq. (A5.32) into Eq. (A5.20) yields
 nC ðmÞγ1 ½N
 nC ðmÞ ½N ρðmÞγ (A5.33)  
 nC ðmÞγ1 ð MC ½N ∂SC ðFC ðxÞÞ 0 dfC rC ðxÞ  dx nC ðxÞ + δγ ρðmÞγ dT C ∂FC ðxÞ m

ΩC  μ0C ðmÞdm ¼ δγ


 nC ðmÞγ1 =ρðmÞγ ¼ ∂TC ðmÞ=∂n0C ðmÞ. where δγ ½N Next, differentiating Eq. (A5.2) with respect to m yields.   ∂SC ðFC ðm + dmÞÞ 0 ΨC  μC ðmÞdm ¼ fC ðm + dmÞ rC ðm + dmÞ  ∂FC ðm + dmÞ   ∂SC ðFC ðmÞÞ + fC ðmÞ rC ðmÞ  ∂FC ðmÞ (A5.34) Because both ΩC and ΨC are equal to μC0 (m)dm, the right-hand side of Eq. (A5.33) should be equal to the right-hand side of Eq. (A5.34) for the optimality of the FAR regulation; that is, ΩC  ΨC ¼ 0. A similar case is presented in detail in Kono and Joshi (2012). We present only a brief explanation here as follows. The relation ΩC  ΨC ¼ 0 for the optimality of FAR regulation implies that with the relocation of one person from m to m + dm, the welfare increase associated with the deadweight loss in the FAR regulation, that is, ΨC, should be cancelled out by the welfare increase associated with the increased commuting cost, that is, ΩC. Importantly, “ΩC  ΨC” is compatible with Harberger’s welfare formula, which measures the welfare change in a distorted economy (see Harberger, 1971). c. LS regulation in the lot housing zone In the detached housing zone, population density is directly adjusted, whereas in the case of FAR regulation, even if the building sizes are adjusted, per-capita floor area cannot be controlled by the government. Therefore the second term in Eq. (A5.33) exists in the case of FAR regulation but not in the case of LS regulation. To check the difference, we can derive μ0H (m) using Eqs. (A5.5), (A5.16), (A5.18), (A5.21), as
 nH ðmÞγ1

½N μ0H ðmÞ ¼ δγ

ρðmÞγ


 nH ðmÞ ½N

(A5.35)

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Traffic congestion and land use regulations

Eq. (A5.35) does not have a term corresponding to the second term in Eq. (A5.33). However, the other first-order conditions are essentially the same. (3) Lemma 5.1 and Proof of Proposition 5.1 Lemma 5.1 corresponds to the mathematical expressions of Proposition 5.1. Lemma 5.1 (Optimality condition for FAR regulation and LS regulation in the presence of optimal zonal boundaries.o (1) Business zone:The sign of μB(m)(m 2 [0, MB]) depends on the sign of nB ðmÞγ1 , which can be positive or negative; however, there is at least 2τ + δγ ρðmÞγ one location where μB(m) changes sign. Two cases arise: nB ðmÞγ1 • Case (i) implying 2τ + δγ < 0 at any m 2 [0, MB]: μB(m) > 0 at ρðmÞγ  ^ ^ any m 2 0, m and μB(m) < 0 at any m 2 m,MB ; nB ðmÞγ1 : • Case (ii) implying 2τ + δγ γ > 0 at any m 2 [ε, MB], where ε ¼ 0: ρðmÞ  ^ ^ μB(m) < 0 at any m 2 ε, m and μB(m) > 0 at any m 2 m,MB .  _ (2) Condominium zone: μC(m) > 0 at any m 2 MB , m and μC(m) < 0 at any _ _ m 2 m ,MC and μC m ¼ 0. (3) Lot housing zone: μH(mH) < 0 at any m 2 (MC, MH] and μH(MC) ¼ μC(MC). Lemma 5.1 is proved as follows. (1) Business zone: We prove Lemma 5.1(1) in two steps. First, noting that ð MB FB ðmÞ
combination of Eqs. (A5.6), (A5.7) yields η ¼ 1 and dm ¼ N, 0 fB ðmÞ ð MB 0

o

μ B ð mÞ

FB ðmÞ ∂fB dm ¼ 0 fB ðmÞ2 ∂w

(A5.36)

Note that market land rents between two adjacent zones are not equalized in the formulation of the Lagrangian in Eq. (5.20) or subsequent derivations. This should show that Proposition 5.1 is not derived with zonal boundaries set at market equilibrium.

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137

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 2 [0, MB], or pattern (2) μB(m) is zero all over m 2 [0, MB]. Next, we will analyze the sign of μB0 (m) in Eq. (A5.30). Noting that ∂ fB/∂ w ¼ ∂ fB/∂ g based on Eq. (5.7), substituting Eq. (A5.36) into the ð MB FB ðxÞ ∂fB
+ dx (from Technical Appendix 5 μB ðxÞ equation θ ¼ N fB ðxÞ2 ∂g 0
.
, and substituting into (A5.13) yields ϕB ð0Þ ¼ N (2)-a) yields θ ¼ N Note that ϕB(MB) ¼ 0 from (A5.14). Next, Eqs. (A5.12), (A5.27) show ϕ0B ðmÞ < 0 at any m 2 (0, MB) when ∂ L/∂ ω(m) ¼ 0 due to the complementary slackness. Therefore θ  ϕB(m) > 0, m 2 (0, MB]. This explains why the first parenthesis in Eq. (A5.30) is positive. It thus turns out that " # γ1 ½ n ð m Þ  B the sign of μB0 (m) is the same as that of 2τ + δγ , which can ρðmÞγ be either positive or negative. This concludes the proof of Lemma 5.1(1). (2) Condominium zone: A similar explanation as in the case of the business zone applies. Noting that η ¼ 1, combination of Eqs. (A5.4), (A5.5), ð MC FC ðmÞ ∂fC (A5.6) yields μC ðmÞ dm ¼ 0. To satisfy this the solution fC ðmÞ2 ∂rC MB 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 2 [MB, MC], or pattern (2) μC(m) is zero all over m 2 [MB, MC]. From Eq. (A5.20), ½N  nC ðmÞγ1 , m 2 (MB, MC). Accordingly, μ0C ðmÞ ¼ λC ðmÞδγ ρðmÞγ μ0C (m) < 0 because λC(m) > 0 from Eq. (A5.29), where m 2 (MB, MC). Therefore, we can exclude pattern (2) of the solution of μC(m). Finally, continuous μC(m) satisfies Lemma 5.1(2). (3) Lot housing zone: From Eqs. (A5.5), (A5.16), noting the condition ÐM
 nH ðmÞ. Plugging this into λH(MH) ¼ 0, λH ðmÞ ¼ m H fH 1ðxÞ dx ¼ N  
 nH ðmÞ γ N 0 Eq. (A5.21) yields μH ðmÞ ¼ δγ < 0,m 2 ðMC ,MH Þ. ρðmÞ Eqs. (A5.22), (A5.23) imply μH(MC) ¼ μC(MC), and as proved earlier,   ðm
 nðxÞ γ N μC(MC) < 0 . Therefore μH(m) ¼ μC(MC) δγ dx < 0, ρðxÞ MC m 2 [MC, MH). The results μ0H ðmÞ < 0 and μH(m) < 0 prove Lemma 5.1(3).

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Traffic congestion and land use regulations

Proof of Proposition 5.1 Proposition 5.1 is derived from Lemma 5.1 as follows. From Eqs. (A5.1), (A5.2), it is evident that the sign of μj(m)    ∂Sj Fj ðmÞ . Thus, com( j 2 {B, C}) is the reverse of the sign of rj ðmÞ  ∂Fj ðmÞ bining Eq. (A5.1) with Lemma 5.1(1) while also using η ¼ 1 from Eqs. (A5.6), (A5.10) yields Proposition 5.1(1). Likewise, combining Eq. (A5.2) and Lemma 5.1(2) using η ¼ 1 again yields Proposition 5.1(2). The combination of Eq. (A5.3) with Lemma 5.1(3), noting that κ H(m) ¼ ∂uH ðfH ðmÞÞ  rH ðmÞ < 0 for any 1 from Eq. (A5.5) given η ¼ 1, leads to ∂fH ðmÞ m 2 (MC, MH], thereby yielding Proposition 5.1(3). (4) Lemma 5.2 and Proof of Proposition 5.2 Lemma 5.2 corresponds to the mathematical expressions of Proposition 5.2. Lemma 5.2 (Optimal condition on zonal boundaries in the presence of optimal FAR regulation and LS regulation) (1) Business zone: ½FB rB ðgB ðMB Þ, wÞ  SB ðFB Þ  ½FC rC ðMB Þ  SC ðFC Þ ¼   ∂SB ðFB ðMB ÞÞ FB ðMB Þ rB ðMB Þ  ∂FB ðMB Þ   ∂SC ðFC ðMB ÞÞ FC ðMB Þ,  rC ðMB Þ  ∂FC ðMB Þ   ∂SB ðFB ðMB ÞÞ fB ¼ μB ðMB Þ. In Case (i), where rB ðMB Þ  ∂FB ðMB Þ μB(MB) > 0, and in Case (ii), μB(MB) < 0.p Likewise,   ∂SC ðFC ðMB ÞÞ rC ðMB Þ  fC ¼ μC ðMB Þ < 0; ∂FC ðMB Þ (2) Condominium zone: ½FC ðMC ÞrC  SC ðFC Þ  Ra   ½rH ðMC Þ  Ra      ∂SC ðFC ðMC ÞÞ ∂uH ð fH ðMC ÞÞ FC ðMC Þ  rH ðMC Þ  , ¼ rC ðMC Þ  ∂FC ðMC Þ ∂fH ðMC Þ

p

See Lemma 5.1(1) for definition of Cases (I) and (II).

Land use regulations with business areas

139

  ∂SC ðFC ðMC ÞÞ fC ðMC Þ ¼ μC ðMC Þ; where rC ðMC Þ  ∂FC ðMC Þ   ∂uH ðfH ðMC ÞÞ rH ðMC Þ  fH ðMC Þ ¼ μH ðMC Þ; and ∂fH ðMC Þ μC(MC) ¼  μH(MC) > 0. (3) Lot housing zone: rH  Ra ¼    ð MH  N  nðmÞ γ 1 ∂SC ðFC ðMC ÞÞ fC ðMC Þ + rC ðMC Þ  , δγ dm  ρðmÞ fH ðMH Þ ∂FC ðMC Þ fH ðMH Þ MC   ∂SC ðFC ðMC ÞÞ fC ðMC Þ ¼ μC ðMC Þ > 0. where rC ðMC Þ  ∂FC ðMC Þ Lemma 5.2 is proved as follows. Lemma 5.2(1) is derived from the combination of Eqs. (A5.1), (A5.4), (A5.24), (A5.32) using η ¼ 1 and ð MC FC ðmÞ ∂fC ðmÞ μC ðmÞ dm ¼ 0 (from the proof of Lemma 5.1(2)). FollowfC ðmÞ2 ∂rC ðmÞ MB
and, from Technical Appendix 5(2)-b, ing relations are also used: λC ðMB Þ ¼ N
λB ðMB Þ ¼ N . The inequality conditions involving μk(k ¼ {B, C, H}) are obtained from Lemma 5.1. Next, Lemma 5.2(2) is derived from the combination of Eqs. (A5.2), (A5.4), (A5.5), (A5.25). μC(MC) ¼ μH(MC) is obtained from Eqs. (A5.22), (A5.23). μC(MC) < 0 is from Lemma 5.1(2). Finally, Lemma 2(3) is derived from Eq. (A5.26) with   ðm
 nðxÞ γ N μH ðmÞ ¼ μC ðMC Þ  δγ dx, which is obtained in the proof ρðxÞ MC of Lemma 5.1. This concludes the proof of Lemma 5.2. We now 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) ¼ FCrC(MB)  SC(FC) where the land rents are equal between Zones B and C. Lemma 5.2(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 5.2(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 5.2(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) ¼ rH(MC). Lemma 5.2(2) shows that, in the optimal case, the sign of [FC(MC)rC  SC(FC)  Ra] 

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Traffic congestion and land use regulations

[rH(MC)  Ra] is the same as that of the right-hand side of the equation in Lemma 5.2(2). The right-hand side can be arranged into

   ∂SC ðFC ðMC ÞÞ ∂uH ðfH ðMC ÞÞ FC ðMC Þ   rH ðMC Þ ¼ fC ðMC Þ½rC ðMC Þ ∂FC ðMC Þ ∂fH ðMC Þ ∂SC ðFC ðMC ÞÞ ½nC ðMC Þ  nH ðMC Þ, using the second equation in Lemma 5.2  ∂FC ðMC Þ



rC ðMC Þ 

(2). This implies that the right-hand side is greater than zero because



 ∂SC ðFC ðMC ÞÞ > 0 as denoted in Lemma 5.2(2) and ∂FC ðMC Þ nH(MC)] > 0 because, by definition, a condominium has more rC ðMC Þ 

[nC(MC) 

households than a detached house. Correspondingly, in the optimal case, [FC(MC)rC  SC(FC)  Ra]  [rH(MC)  Ra] > 0. Finally, if the city boundary is determined in the market, then rH(MH) ¼ Ra. Lemma 5.2(3) shows that in the optimal case, rH(MH) > Ra because the right-hand side of the equation in Lemma 5.2(3) is greater than zero.

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