Welfare effects of floor area ratio regulation on landowners and residents with different levels of income

Journal of Housing Economics 46 (2019) 101656

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Welfare effects of floor area ratio regulation on landowners and residents with different levels of income

T

Yoshihiro Takeda, Tatsuhito Kono , Yang Zhang ⁎

Tohoku University, Aoba 09, Aoba-ku, Sendai 980-8579, Japan

ARTICLE INFO

ABSTRACT

Keywords: Congestion toll Floor area ratio regulation Heterogeneous households Land use regulation Property tax residential segregation

This paper explores how Floor Area Ratio (FAR) regulations affect residents with income disparity and absentee landowners in a congested closed city. Theoretical results show that (1) an increase in FAR in a central zone may harm the utility of suburban residents due to the residential segregation pattern of heterogeneous people whereas it always increases the utility in a city with homogeneous households, and (2) how an increase in FAR changes land rents depends on the current FAR and the relative location of the zone where FAR increases. Numerical results clarify the effects of optimal FAR regulations on residents and absentee landowners. In addition, theoretical result (1) denoted above is numerically verified. Furthermore, it is found that optimal FAR gives higher benefits to high income households than to low income households regardless of the location pattern.

JEL classification: R1 R2

1. Introduction Today, many cities in the world suffer from severe traffic congestion, which wastes a massive amount of time and fuel. Many cities have imposed land use regulations such as regulations on building size or height, or lot size, which can adjust urban externalities including traffic congestion. However, cities have heterogeneous agents. So, even if the regulations increase the social welfare, all the agents will not necessarily increase their respective welfare. When implementing the regulations, we need to consider how each agent is affected. We theoretically clarify the welfare effect of floor area ratio (FAR) regulation on heterogeneous households with income disparity and landowners in a congested closed city. In addition, we numerically simulate FAR regulation to explore the distribution of wealth among agents. Traffic congestion is an externality because an additional motorist pays only her private cost and not the marginal social cost generated by her role in increasing other motorists’ travel time. The first-best policy against such externality is to impose a congestion toll that varies at each point such that every motorist would face the marginal social cost. However, it is difficult to implement it because of enormous implementation and operation costs, as indicated by Rouwendal and Verhoef (2006). Furthermore, toll payers, who are generally welfarelosers, reject the policy of congestion tolling. Because of the difficulty of implementing the first-best congestion

⁎

pricing, the policy should be replaced with some second-best policies. In real society, city governments in most cities impose land use regulations which can serve useful practical alternatives to congestion toll policies as the second-best policies. The efficiency of land use regulations has been analyzed by recent papers. The efficacy of regulating on building sizes and city sizes in a congested city was theoretically and numerically evaluated by comparing welfare gain from laissez-faire (Bertaud and Brueckner, 2005; Kono et al., 2012). Rhee et al. (2014) and Kono and Joshi (2017) show the efficiency of land use control in the model combined with production. Urban growth boundary (UGB) regulation is analyzed as an effective second-best remedy for unpriced traffic congestion (Brueckner, 2007). Policies including land use regulations should consider the distribution of wealth among agents as well as the efficiency. Indeed, policies are politically acceptable if they can be designed in such a way that a large majority of the population will benefit. Kono et al., (2010), assuming a closed two discrete city model, show a necessity for minimum as well as maximum FAR regulations to maximize social welfare. Kono and Joshi (2012) explore the difference in optimal regulation between an open city and a closed city. Joshi and Kono (2009) target a growing city with congestion. Kono and Joshi (2017) address a city with agglomeration economies and transport congestion. Kono and Kawaguchi (2017) introduce cordon pricing in a city with land use regulation. These papers show that different regulations are required in different second-best situations due to the

Corresponding author. E-mail address: [email protected] (T. Kono).

https://doi.org/10.1016/j.jhe.2019.101656 Received 17 September 2017; Received in revised form 15 September 2019; Accepted 3 October 2019 Available online 04 October 2019 1051-1377/ © 2019 Elsevier Inc. All rights reserved.

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effects of the specific distortions, and that it is important for policy makers to check what distortions are relevant before determining land use regulations. However, these papers as well as a recent book by Kono and Joshi (in press) mainly focus on efficiency, although the welfare of land owners and residents are compared partly. The current paper focuses on the welfare of income heterogenous residents. To design land use regulations, policy makers should pay attention to low-income residents in particular, because even in developed countries lowest-income residents suffer from poverty. The distribution of wealth of urban policies has been analyzed by many papers (e.g., Dubin et al., 1992; Arnott et al., 1994; Buyukeren and Hiramatsu, 2015).1 Similarly, the distribution of wealth of land use regulations has been analyzed. For instance, Brueckner (1995), investigating urban growth controls about the welfare of landowners and that of households, show that the growth controls harm consumers while enriching landowners. This result clearly shows that we should differentiate residents from landowners for welfare analysis. The effects of a FAR regulation differ across locations. Kono et al. (2012) numerically demonstrate that optimal FAR regulation decreases land rent in areas close to the central business district (CBD) while it increases that in other areas. The geographical difference in changes in land rent indicates that the benefits of landowners depend on which area's land they own, and that the location of the households also matters for changes in the welfare of the households if heterogeneous residents reside separately. Heterogeneous households choose different locations to live. In particular, income heterogeneity yields a residential segregation pattern in which population groups are sorted into different zones (Fujita, 1989). However, past theoretical studies on land use regulation have not considered heterogeneous people. The purpose of the present paper is to investigate the effects of land use regulations on heterogeneous households and landowners, using a simple setting of a monocentric model with three residential zones and a CBD. To focus on the distribution effects of land use regulation on residents and landowners, absentee landownership is assumed. The three zones are connected by bridges. The commuting costs are incurred only when crossing congested bridges. This simple setup allows the derivation of a number of analytical results although it is less realistic than the continuous-space framework used in several previous works of land use regulations. We analyze a case of homogeneous households and that of heterogeneous households to compare them. After obtaining such theoretical analyses, we demonstrate numerical simulations to clarify the welfare effects of FAR regulations on residents and landowners. Theoretical results show the effects on the economy of changes in FAR in only a single zone. FAR should be optimized in all the zones simultaneously to maximize the welfare. This optimization is conducted numerically in the current paper, and the optimal situation is compared to the market equilibrium situation. The setting of geography plays an important role in determining the properties of optimal land use regulations. For example, by setting mixed-use zones in a non-monocentric city having residents with idiosyncratic tastes, and in a system of cities with homogeneous residents, respectively, Anas and Rhee (2007) shows that an expansive UGB may be necessary, which contradicts the traditional conclusions based on a monocentric city model. However, even if we use a monocentric city model, the results should be reasonably valid for any modifications on the model as long as the fundamental relationship between residential

density and commuting costs is preserved (Brueckner, 2007; Sridhar, 2007). For example, even if a city has multiple centers, in the areas where all (or most) residents commute to one center or through one transportation node such as a station to the multiple centers, the fundamental relationship between the residential areas and the commuting center (or the transportation node) is the same as that in a monocentric city. Our target is to explore the effects of land use regulation on residents with different levels of income. Since these effects have not been clarified even in a monocentric city yet, we set a monocentric city with residents with different income levels. The current paper focuses on only one city, and in particular residential areas of the city. As Anas and Pines (2008, 2012) show, if we consider more than one city, optimal land use regulations can differ from those when only one city is supposed. Furthermore, in a system of cities, because the population distribution across cities changes, we can take account of the effect of agglomeration economies arising from various reasons. Agglomeration patterns due to a framework of new economic geography models (e.g., Krugman, 1991; Fujita and Thisse, 2009) have been studied intensively.2 However, the number and the extent of studies on land use regulations in a framework of a system of cities are very limited, so they remain for future studies. However, optimal conditions on land use regulations derived from setting one city should be valid even for a system of cities. Even when we focus on only one city, agglomeration economies also affect land use regulations in particular in business areas. RossHansberg (2004) takes account of the existence of agglomeration spillovers of firms to explore zoning. Rhee et al. (2014), Zhang and Kockelman (2016), and Kono and Joshi (2017) consider the existence of agglomeration economies and traffic congestion to explore land use regulations. These papers do not set a point CBD, and instead consider a CBD with some area to explore land use regulations in the CBD.3 In contrast, the current paper ignores business areas, and focuses on the residential areas. Agglomeration economies in business areas basically do not affect optimal patterns of land use regulations within residential areas, although they affect the optimal size of the total residential areas (i.e., how land should be allocated to each land use purpose.) The remainder of this paper is organized as follows. Section 2 develops a city model in two cases: homogeneous households and heterogeneous households. Section 3 shows the analysis of the effects on residents and landowners among zones. Numerical simulations are presented in Section 4. Section 5 concludes the paper. 2. The model 2.1. The city and three regimes A city is composed of the CBD (zone 0) and three residential zones labelled 1, 2 and 3. The CBD is located on the left and the three zones line up in numerical order. Each zone is connected to the next one by a bridge. A city model with two or three discrete zones is often used for urban policy analyses, e.g., Anas and Pines (2008), Brueckner and Helsley (2011) and Brueckner (2014). The fixed areas of the zones are given by a1, a2 and a3. The land within each zone is homogeneous. We assume a monocentric closed city. A closed city means that no resident migrates to or from another city. One of typical reasons is that the households could not change the city in which they work. The commuting cost ti, which households in zone i (=1, 2, 3) pay, is the sum of the travel expenses and the time cost, both of which are incurred only when crossing bridges. Cars from more distant locations increasingly

1 Dubin et al. (1992) show that the commuters exposed to high levels of traffic congestion are more likely to favor growth controls, using real voting data. Regarding congestion pricing, Arnott et al. (1994) examining the welfare effects of tolls on heterogeneous commuters with different costs of travel time, their preferred arrival time at work, and the costs they incur from early and late arrival, show that the toll policy has different effects on heterogeneous drivers. Buyukeren and Hiramatsu (2015) show that since marginal utility of income is higher in suburbs, redistribution of tax revenue among residents generates decentralizing effects by increasing housing demand.

2

Some recent new economic geography papers extend one-dimensional economies to two-dimensional economies (e.g., Ikeda et al., 2012; Ikeda et al., 2014). 3 The history of theoretical studies on land use regulation is reviewed in Kono and Joshi (2019). 2

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Fig. 1. City shape.

join traffic while moving towards the CBD. Fig. 1 shows the shape of the city. Developers build dwelling units on land rented from the absentee landowners. The total supply per unit area of land in zone i is denoted Fi, which is controlled by FAR regulation in each zone. We explore the effect of land use regulations on households and landowners. First, we assume that all households are homogeneous as in most previous research of land use regulations. Next, we assume two income groups with which a residential segregation pattern arises. In summary, this paper analyzes the following two cases. (Homogeneous Case) All households are homogenous. People move freely between zones, so utility level is the same among all zones. (Heterogeneous Case) Households with two exogenously set levels of income live in the city. People move freely between zones, so the utility level for households with each level of income is common in equilibrium. In the analysis, we set three regimes: Regime 1 is market equilibrium, Regime 2 is social optimal and Regime 3 is parochially-optimal. The social optimal in Regime 2 implies the maximization of the simple sum of monetary utilities of the people and the profits of absentee landlords. Regime 3 is the maximization of the weighted sum of the monetary utilities of residents (100%) and the profits of absentee landlords (80%). Regime 3 can represent the case in which the regulations are determined based on placing less weight on landlords’ welfare. In this sense, this maximization is parochial. We carry out analytical analyses and numerical analyses. Regime 3 is only considered in numerical simulations. Regarding Regimes 1 and 2, in the analytical analyses, we do not directly compare the outcomes because the level of the outcomes depends on the parameters and the functions. These are not specified in our analytical section, so it is impossible to compare them in terms of scale. Instead, by using comparative statics, we explore how residents’ utilities and land rents change when exogenous FAR changes in both the homogenous case and the heterogenous case. With this result, we can discuss how the social welfare, which is composed of utilities and land rents, changes when the government changes the FAR from a certain level of FAR including the market equilibrium FAR and why it changes so. This information is useful for the government to design Regime 2.

households with high income. For simplicity, we regard the population of the city as the number of households. Each type of residents in zone i (=1, 2, 3) has identical utility functions as

ugi = (cgi, qgi ), g

{l, h},

(1)

where ν(cgi, qgi) is a utility function of numeraire composite goods cgi and housing square footage qgi. cgi includes all consumer goods except for floor space. ugi is the level of utility of group g of residents in zone i. We assume ∂ν/∂cgi > 0 and ∂ν/∂qgi > 0, i.e., the marginal utility of each good is positive.4 From these conditions, we can derive

cgi (ugi , qgi )/ ugi > 0, cgi (ugi , qgi)/ qgi < 0,

(2)

where cgi(ugi, qgi) is a function of a solution of Eq. (1) for cgi. The budget constraint of each resident of group g ∈ {l, h} living in zone i is expressed as i

wg Lgi = cgi + pi qgi +

k=1

k 1k ,

(3)

where wg is wage rate, Lgi is working time,pi is the price per square foot i of housing in zone i and k = 1 k 1k is the total travel expenses from zone i to the CBD. The composite good cgi is moved among zones freely, and the price of the good is one as numeraire. The total time T¯ is expressed as i

T¯ = l¯g + Lgi +

k=1

Mk

1k ,

(4)

where l¯g is leisure time, which is the total available time less the labor time and the total travel time, and Mk 1k is the travel time from zone k1 to zone k . For simplicity, we assume l¯g is fixed. Then, solving Eq. (4) for Lgi and substituting the result into Eq. (3) and arranging the obtained equation yields

yg

tgi = cgi + pgi qgi , g

{l , h }

(5)

where yg is the generalized income, wg (T lg ) , and tgi is commuting i i cost from zone i to the CBD, where tgi = k = 1 k 1k + k = 1 g · Tk 1k , Tk 1k wl· Mk 1k and g = wg / wl ( 1) . ωg is the ratio of value of time wg of group g ∈ {l, h} income households to that of group l income households. The bid rent of group g ∈ {l, h}, pgi is mathematically expressed as

2.2. Household behavior with absentee landownership

max pgi = (yg

To investigate the distribution effects of FAR regulation on residents and landowners, we model household behavior with absentee landownership. Land rent revenue is not returned to households. The welfare of residents is measured by the household utility and the welfare of landowners is measured by the revenue of land rent. We analyze a homogeneous case and a heterogeneous case. However, except for income difference, the model is the same. Hence, we first show the heterogeneous households’ behaviors. The number of households with income level g ∈ {l, h} are exogenously given as Ng. Group l represents households with low income, and h represents

tgi

cgi)/ qgi s. t . eq. (1) g

{l , h }

qgi, cgi

(6)

The highest bidder can use the land, so the highest floor bid rent, pgh or pgl, is the equilibrium price pi per square foot of housing in zone i. From Eq. (6) and using utility levels ugi, pgi and qgi are expressed as

pgi = p (yg

tgi, ugi), qgi = q (yg

tgi, ugi), g

{l, h}.

(7)

4 In addition, we assume a concave utility function, i.e., a well-behaved function.

3

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Based on the first order conditions of solving the maximization problem of (6) and using the implicit function theorem, we derive the properties as

qgi (yg

tgi, ugi)/ tgi > 0, qgi (yg

tgi, ugi )/ ugi > 0.

Considering that households can choose a residential location, the household utility across zones should be equal to the equilibrium utility level u, which is endogenously determined in a closed city, expressed as

(8)

That is, consumption of housing square footage is increasing in tgi and ugi. Removing subscript g from all the variables and setting g = 1, the heterogeneous model will be the homogeneous model.

vi = u

Developers are price-takers and they combine housing capital and land to produce residential buildings. Si(Fi) represents the capital required to produce floor space aiFi. The sum of developers’ net profits from the total floor space supply in the city is given by 3 i=1

ai ·(Fi pi

Si (Fi )

ri ),

(9)

where ri is the land rent, and the price of capital is normalized at unity. Under FAR regulation, the level of floor space supply Fi is exogenously given to developers. As a result of competition among developers, the land rent is therefore expressed as

ri = Fi pi

Si (Fi ),

because

= 0.

(10)

tg1 =

2.4. The profits of absentee landowners

t2 = t3 =

2 k=1

k=1

(12)

k 1k

+ T01 (N ) + T12 (N2 + N3) + T23 (N3).

(13)

3 i=1

Ni ,

k=1

k=1

+

g T01 (Ng

+ Ng ) +

g T12 (Ng 2

+ N g ),

k 1k

+

g

T01 (Ng + Ng ) +

g

T12 (Ng 2 + Ng ),

k 1k

+

g

T01 (Ng + Ng ) +

g

T12 (Ng 2 + Ng ) +

3

Ng1· qg1 (yg

tg1, ug ) = a1 F1,

Ng 2· qg 2 (yg

tg 2, u g ) + N g 2 · q g 2 y g

(

(19a) (19b) g

T23 (Ng 3) ,

(21)

(

)

t g 3, ug = a3 F3,

)

t g 2, ug = a2 F2,

(22) (23)

Ng1 + Ng 2 = Ng ,

(24)

Ng 2 + Ng 3 = N g ,

(25)

where g ∈ {l, h}, g′ ∈ {l, h} and g ≠ g′. g represents the group of households which live in zone 1 and 2, while g′ represents the group of households which live in zone 2 and 3. Due to free migration, utility with each level of income should be equal across zones, as

vgi = ug v g i = ug

i

{1, 2}, i

{2, 3},

(26a) (26b)

where g ∈ {l, h}, g′ ∈ {l, h} and g ≠ g′. Since both groups live in zone 2 simultaneously, (27) must hold.

pg 2 = p g 2 (14)

Similar properties hold in the heterogeneous model too. Eq. (15) represents the total population constraint while Eq. (16) represents the population equilibrium condition in each zone.

N=

2

N g 3·q g 3 y g

The commuting cost t1 is constant because all the city residents N cross the bridge between zone 0 and zone 1, whereas t2 and t3 are endogenously determined by the distribution of the population among the residential zones. Travel time cost Ti-1i is an increasing function because of congestion, i.e.,

T23 (N3) > 0. N3

k 1k

(18)

+ N g ),

where g ∈ {l, h}, g′ ∈ {l, h} and g ≠ g′ and the low-level wage is set to be 1 by changing unit of time without loss of generality. Under a residential segregation pattern, the population equilibrium conditions are expressed in Eqs. (21)-(25).

(11)

+ T01 (N ) + T12 (N2 + N3),

T12 (N2 + N3) > 0, (N2 + N3)

g T01 (Ng

(20)

k 1k

3

k=1

tg 3 =

Regarding the travel cost model, we explain the homogeneous model, first. We assume that the automobiles are the only mode of commuting. For the homogeneous model, Ni is the number of household residing in zone i(=1, 2, 3) and ti represents the commuting cost which the residents living in zone i pay. Travel monetary expense i 1i , which includes fuel expense, can depend on congestion level. However, it mainly depends on the distance. So, for simplicity, we assume i 1i depends only on the distance x i 1i . Time cost Ti 1i of crossing a bridge depends on the level of congestion which is determined by population residing in zones beyond the bridge, and hence Ti 1i is a function of Ni. ti is

+ T01 (N ),

(17)

{1, 2, 3}.

2

tg 2 =

2.5. Travel cost and market equilibrium conditions

01

+

01

tg 2 =

The revenue of land rent in zone i is expressed as Ri = ai ri . Then, to investigate the welfare of absentee landowners in each zone, the changes in land rent associated with FAR are explored. We discuss this in Proposition 2.

t1 =

i

In the heterogeneous model, there are two groups of people with high income and with low income in the three zones. Residents with higher bid rent can reside in each zone. Hence, income differences tend to yield a residential segregation pattern in a city. We focus on only simple segregation patterns. That is, in our setting, residents with a certain level of income g ∈ {l, h} live close to the CBD, whereas residents with the other level of income g′ ∈ {l, h: g′ ≠ g} live close to the urban growth boundary. To investigate the interaction among the two heterogeneous groups of households, we explore the situation where heterogeneous households live together in zone 2. In other words, zone 1 is occupied by one group, and zone 2 is shared by the two groups. Zone 3 is occupied by another group.5 In our model, the area size of each zone is freely defined exogenously. So, to represent the situation where the group which lives closer to the center resides in a very small area, we can reduce the area of zone 1. The following analyses hold regardless of the sizes of the three zones. Considering that the value of time differs between two groups, the commuting cost, tgi and t g i , is expressed as

2.3. Developers’ behavior under FAR regulation

=

(16)

Ni = ai Fi / qi (i = 1, 2, 3).

5

(27)

Other types of segregation patterns are possible in reality. For example, the border of the two income groups may be in zones 1 or 3. In addition, integration zones can appear to a large extent in reality. The current paper focuses on simple segregation patterns, so we leave other realistic patterns for future studies.

(15) 4

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Table 1 Notational glossary. Policy variables Floor space at i Urban Growth Boundary

Endogenous variables Fi x¯

Exogenous variables

Utility of households Transportation cost Population in zone i

ugi tgi Ngi

Total population Generalized income Exogenous leisure

N yg l¯g

Housing square footage Numeraire composite goods Price per square foot of housing Bid rent per square foot of housing

qgi cgi pi pgi

Working time Total time Travel monetary expense Area size of zone i

Lgi T¯

Eq. (27) means that both groups present the same bid rent to absentee landowners although they obtain different utility levels in the equilibrium. In the homogenous case, we can solve population in each zone and utility level from the system composed of (15) and (16). In the heterogenous case, we can solve one group population in zone 1, another group population in zone 3, and the two groups’ populations in zone 2 and two utility level from the system composed of (21)-(25) and (27). We assume that the numbers of agents such as consumers and landowners are large so that they cannot form a collusion, and the effects of every agent on markets are negligible. As a result, absentee landowners and developers are in perfect competition. The variables in the model are summarized in the following table. Table 1

too large FAR can even violate the resource constraint because too much material for buildings is demanded. So, the feasibility of the regulated FAR is a necessary condition when we discuss the welfare effects of FAR regulations. The signs of changes in household utility, population and commuting cost associated with FAR can be mathematically obtained with the total differential of four Eqs. (28)-(31). In the analysis, exogenous variables are N, y, a1, a2, anda3, endogenous variables are u, N1, N2, andN3, and policy variables are F1, F2, andF3. Applying the implicit function theorem to Eqs. (28)-(31) yields Proposition 1. These results are also summarized in Table 2. Proposition 1. (Changes in residents’ utility associated with FAR in the homogenous case). As long as the regulated floor area is feasible, in the case of homogeneous households, a larger FAR in zone i (=1, 2, 3) leads to an increase in residential utility in equilibrium. See Appendix A for the proof. Two examples of infeasible regulated floor areas are: (1) The current FAR is large enough to surpass the resources constraint, and (2) The land rent is negative because of excessively large buildings, leading to the situation where landowners do not supply the land. These have already been discussed below Eq. (31). Intuitively, Proposition 1 shows that as long as the regulated FAR is feasible, a larger FAR in zone i (=1, 2, 3) produces a larger total floor space, thereby every resident can increase his floor space through migration, leading to an increase in his utility. Interestingly, a larger FAR in zone 3 increases the equilibrium residential utility, even though it increases congestion on the bridge connecting zone 2 and zone 3. It can be interpreted that a positive effect of a larger floor space in zone 3 surpasses a negative effect of congestion on the bridge. This property can be straightforwardly proved only by checking the utility of residents in zone 1, in which the commuting cost does not change because all residents cross the bridge which the residents in zone 1 use. Since N1 < 0 with the areas of all the zones fixed, when the regulated FAR in F3 zone 3 is increased, population in zone 1 will decrease. Therefore, residents’ lot sizes in zone 1 must increase. This is why the utility level in zone 1 will increase. All residents hold the same utility level in equilibrium. This implies that residents’ utility in zone 3 should increase even though the increased FAR also increases the travel time of the residents through increased congestion. In other words, the positive effect due to the expanded lot size overcomes the negative effect due to the increased commuting cost. We next investigate land rent revenues of absentee landowners.

3. Welfare effects of FAR regulation This section analyzes the effect of a change in FAR in each zone on residents and landowners in the city theoretically. As we show in Section 2.1, in the analytical analyses, we do not directly compare the outcomes between Regime 1 (market equilibrium) and Regime 2 (optimal FAR regulation). Instead, by using comparative statics, we explore how residents’ utilities and land rents change when exogenous FAR changes in both the homogenous case and the heterogenous case. Section 3.1 investigates the homogeneous case. Section 3.2 examines the heterogeneous case. 3.1. Homogeneous case This section investigates changes in residential utility and changes in land rents associated with changes in FAR. These changes involve endogenous changes in zone population and commuting costs. The model is expressed by Eqs. (7), (10), (11)-(13), (15), and (16). First, substituting Eqs. (11)-(13) and Eq. (7) into Eqs. (15) and (16) yields the following system of equations.

N1· q1 (y

t1, u ) = a1·F1,

(28)

N2· q2 (y

t2 (N2, N3), u) = a2 ·F2,

(29)

N3· q3 (y

t3 (N2, N3), u) = a3·F3,

(30)

N1 + N2 + N3 = N.

i 1i

ai

(31)

The above reduced system of equations can analyze the utility of residents, who do not own land. The equation we have not used within the model is Eq. (10), which determines the landowners’ profit. Eq. (10) does not affect the residents’ utility u because that determines the land rents, which are not returned to the residents in our model. In other words, residents’ welfare is determined if F1, F2 and F3 are given exogenously. F1, F2 and F3 are infeasible if they are too huge. For example, if the regulated FAR is too large, the land price can even be negative (see Proposition 2). If the land price is negative, landowners will not supply the land for buildings. The necessity of positive land price for FAR regulations is discussed also in Pines and Kono (2012) and Kono et al. (2010). Or, a

Table 2 Comparative static analysis for homogenous case.a Endogenous variables policy variables

u

N1

N2

N3

F1 F2 F3

+ + +

+ – –

? + –

? – +

Note that a question mark means that the effect of the policy variable on the endogenous variable is ambiguous. a All results are obtained in Appendix A. 5

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First, we analyze a change in land rent in a zone associated with an increasing FAR in that zone. As Appendix B shows, differentiating Eq. (10) and some arrangements yield a change in land rent in a zone associated with an increasing FAR in that zone:

ri = Fi

Fi · qi

ti c u + i· Fi u Fi

+ Pi

Si Fi

heterogeneous residents in our discrete zone model because of the impossibility of using the envelope theorem in the discrete zones. To clarify a residential segregation in a continuous space model, Fujita (1989) analyzes the relative steepness of bid rent function with respect to the distance from the CBD. This paper uses this theory as the first approximation in the following way. We investigate the relative steepness of bid rent function in zone 2 with respect to travel distance from the CBD. Furthermore, we assume that the obtained relative steepness holds even in zone 1 and zone 3. The households with a higher bid rent live in each zone. We explore the case that heterogeneous households live together in zone 2. The relative steepness of bid rent between the heterogeneous people in zone 2 is obtained as

(i = 1, 2, 3). (32)

Fi· Pi / Fi < 0

The first term implies Fi · ∂Pi/∂Fi, in which ∂Pi/∂Fi in Eq. (32) is a change in bid rent associated with a change in FAR, which is proved to be negative i.e., ∂Pi/∂Fi < 0 (See Appendix B for a proof). The last term Pi Si / Fi , in Eq. (32) is the marginal change in the deadweight loss caused by the FAR regulation. When the current FAR Fi is smaller than the market equilibrium FAR, denoted as FiM hereafter, the sign of the Si / Fi > 0 , noting that Pi = Si / Pi at market equilibrium. term is Pi When the current FAR Fi is larger than FiM, the sign of the term is Pi Si / Fi < 0 . At market equilibrium, the term is zero as Pi Si / Fi = 0 . Using the above properties, in the case of Fi FiM , considering ∂Pi/ Si / Fi 0 , the sign of ∂ri/∂Fi in Eq. (32) is negative. In ∂Fi < 0 and Pi Si / Pi > 0 . So, the sign of ∂ri/∂Fi depends on the case of Fi < FiM , Pi Si / Pi . Then, we can sumthe magnitude of Fi · ∂Pi/∂Fi relative to Pi marize these properties as Proposition 2 (1). A change in land rent in a zone associated with an increasing FAR in a different zone is expressed as (see Appendix B for a derivation)

ri = Fj

Fi · qi

ti c u + i· Fj u Fj

(i, j = 1, 2, 3, i

(33)

i

M12 (x12) + x12

12 (x12 )

1 ql2

x12

1 . qh2 (34)

The signs of changes in land rent ∂ri/∂Fj depend on the first term ∂ti/ ∂Fj in the parenthesis and the second term ∂ci/∂u · ∂u/∂Fj in Eq. (33). Using inequality condition (2) with Proposition 1, we can conclude that the second term is always positive, i.e., ∂ci/∂u · ∂u/∂Fj > 0. Accordingly, if the first term ∂ti/∂Fj is positive, the sign of Eq. (33) is negative, as shown in the following lemma. Lemma 1. If ∂ti/∂Fj ≥ 0, then ∂ri/∂Fj < 0 (i, j = 1, 2, 3,

wh wl

See Appendix C for the detailed derivation. Considering the fact that yh > yl and supposing that housing is a normal good, the consumption of housing square footage of high income households in zone 2 is larger than that of low income households. That property is mathematically expressed as qh2 ql2 > 0 . It implies that the last term in Eq. (34) is always positive. Because the first term can take both signs, the following two patterns are set as residential segregation patterns. Location pattern (I) Low income households live close to the CBD and high income households live in the suburb. Location pattern (II) High income households live close to the CBD and low income households live in the suburb. In location pattern (I), the slope of bid rent of high income households is steeper than that of low income ones (i.e., Expression (34)>0). Location pattern (II) holds the reverse. For the heterogeneous model, we investigate changes in utility, population and commuting cost associated with FAR as in the homogeneous model. First, substituting Eqs. (15)-(17) and Eq. (7) into Eqs. (21)-(23) yields the following system of equations.

j).

Fi· Pi/ Fj

Pl2 w qh2 = l x12 qh2 ql2

Ph2 x12

j).

Lemma 1. is interpreted as follow. If the commuting cost in a zone (say, zone i) increases with a change in FAR regulation in another zone (say, zone j), the land rent in that zone decreases. From Lemma 1, we can use the signs of changes in commuting cost ∂ti/∂Fj to clarify the signs of changes in land rent ∂ri/∂Fj. The commuting cost in any zone closer to the CBD than the zone where FAR increases always goes up because the number of commuters passing through the zone increases, or does not change in zone 1 because the number of commuters crossing the bridge does not change. Thereby, using Lemma 1, the land rent in any zone closer to the CBD than the zone where FAR increases goes down. On the other hand, the effect of the land rent in any zone outer than the zone where FAR increases is ambiguous because of the two terms with opposite signs, ∂ti/∂Fj < 0 and ∂ci/∂u · ∂u/∂Fj > 0. Then, we can summarize these properties, i.e., dri / dFj < 0 (i, j = 1, 2, 3, i < j) , as Proposition 2 (2).

Ng1· qg (yg

tg1, ug ) = a1·F1,

Ng 2 qg 2 (yg

tg 2 (Ng 2), ug ) + Ng 2 qg 2 y g

(

(35)

(

N g 3·q g 3 y g

(

)

t g 3 Ng 2, Ng 3 , u g

t g 2 (Ng 2), u g

) = a2·F2,

) = a3·F3,

(36) (37)

where g ∈ {l, h}, g′ ∈ {l, h} and g ≠ g′. Location pattern (I) implies g = l and g = h . Location pattern (II) means g = h and g = l . Common bid rent for two kinds of residents in zone 2 is expressed as

yl

tl2 (x 01, x12 , Ng 2) ql2*(yl , tl2, ul ) pl2

c (ql*2, ul )

=

yh

th2 (x 01, x12 , Ng 2) qh2*(yh , th2, uh)

c (qh*2 , uh )

,

ph2

(38) where superscript * implies the amount households choose. We also consider the population constraints (24) and (25). In the analysis, exogenous variables are yg , y g , Ng , N g , a1, a2, anda3, whereas endogenous variables are ug , u g , Ng1, Ng 2, Ng 2 , andNg 3, where g ∈ {l, h}, g′ ∈ {l, h} and g ≠ g′. Applying implicit function theorem to the system yields Proposition 3. These results are summarized in Table 3, too. Proposition 3. (Changes in utility associated with FAR in the heterogeneous case). As long as the regulated floor area is feasible,6 in the case of heterogeneous households, regardless of the location pattern, a larger FAR in zone i(=1, 2, 3) leads to an increase in the utility of group of households g, which lives in zones 1 and 2. A larger FAR in zone j(=2, 3) leads to an increase in the utility of group of households g′, which lives in

Proposition 2. (Changes in land rent associated with FAR). (1) When the current FAR Fi is larger than or equal to market equilibrium FAR FiM, a larger FAR in zone i (=1, 2, 3) leads to a decrease in land rent in zone i (=1, 2, 3) . (2) A larger FAR in zone j (=1, 2, 3) leads to a decrease in land rent in zone i (=1, 2, 3, i < j) , regardless of the size of the current FAR Fj. 3.2. Heterogeneous case The heterogeneous case has residential segregation patterns. Households which bid a higher rent can reside in each zone. In order to characterize residential segregation patterns, we need to compare the bid rent among households. However, it is difficult to compare the difference in the bid rents in each discrete zone between two kinds of

6 Note that this feasibility is the same as that discussed regarding Proposition 1.

6

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Table 3 Comparative static analysis for heterogeneous case.a Endogenous variables policy variables

ug

ug

Ng1

Ng2

Ng 2

Ng 3

F1 F2 F3

? + +

? + +

+ – –

– + +

+(−) + –

−(+) – +

Note that a question mark means that the effect of the policy variable on the endogenous variable is ambiguous. Whether the sign of the effect of F1 on Ng 3 is positive or negative depends on the sign of the braces on the right hand side of Eq. (D.2-3). But the effects of policy variable F1 on Ng 2 and Ng 3 are the opposite of each other in terms of sign from Eq. (D.2-4), which are expressed by +(−) and −(+) in Table 3. a All results are obtained in Appendix D, in which g, g′ ∈ {l, h: g′ ≠ g}.

zones 2 and 3, whereas the sign of the effect of a larger FAR in zone 1 on the utility of g′ group of households is ambiguous. See Appendix D for the proof. The ambiguity of the sign of Eq. (43), or du g / dF1, implies that, unlike the homogeneous case, a larger FAR in zone 1 does not necessarily lead to an increase in the utility of residents living in the suburb, though residents living in the center always gain from the changes. In the homogeneous case, the utility of all households increases due to a larger total FAR. This effect remains but there is another effect in the heterogeneous case. A condition yielding du g / dF1 < 0 is that ∂qg2/∂ug > 0 in the term Q of Eq. (43) is large enough. Improvement of the utility of group of households g by a larger FAR in zone 1 induces the group of households to spend larger space to live. Then, if the increase in housing space of group g of households is large enough to expel group of households g′ from zone 2 to zone 3, the utility of group of households g′ decrease in spite of the larger FAR in zone 1. In that case, the number of group of households g′ which live in zone 3 increases and consequently they incur higher commuting costs, suffering from more congestion and smaller housing spaces. The occurrence of this negative Eq. (43) is numerically demonstrated in Section 4.3. Regarding changes in land rent for landowners in the heterogeneous case, the signs of the changes are solved using Eqs. (32) and (33). Then, Proposition 2 holds even in the heterogeneous case.

Fig. 2. Housing taxation rate.

Mi

1i

= xi

1i ·

1+

ni Ci 1i

,

(39)

where θ is the travel time incurred while driving 1 km in the case of no congestion, ni denotes the number of households residing beyond zone i and Ci 1i is the capacity of the bridge between zone i−1 and zone i. Fig. 2 To see the benefit for households, we use equivalent variation (EV). EVi is defined as the difference in the expenditure function before and after implementation of the policy. With Cobb-Douglass utility function, EVi can be expressed as.

EVi =

·

Pi B 1

1

·(u A

uB )

·

Pi B 1

1

· u,

(40)

where superscript A implies before the policy and B implies after that. With the EV, changes in household utility are expressed in terms of money. To see the benefit for landowners, we investigate the changes in land rent in each zone. Land rent in each zone before and after implementation of the policy is compared. It is expressed as ai · ri ai ·(riA riB ) . In summary, social benefit of a policy in the heterogeneous case, SB, is expressed as

Proposition 3. is important, so we explain the results using geometry to develop an intuitive understanding.

SB = Ng1·EVg1 + Ng 2·EVg 2 + Ng 2·EVg 2 + Ng 3·EVg 3 + a1 r1 + a2 r2 + a3 r3,

(41)

4. Numerical simulation

where g ∈ {l, h}, g′ ∈ {l, h} and g ≠ g′. We calibrate parameters, using the data in Sendai city Japan. The population is around 1 million. The total number of households N is set at 500,000, which are divided into the number of high income households Nh as 100,000 and that of low income households Nl as 400,000. The parameter α in Cobb-Douglas utility function u = ci · qi1 is set at 80% of the income of 4200,000 yen per year. The parameter β in the housing production function is set at 458.38, and multiplicative factor γ is set at 3.23. Both are estimated with the data of Japanese buildings (Domon et al., 2015). Supposing that the Sendai urban area is divided into 3 zones, the building areas a1, a2 and a3 are set at 3.2 km2, 16.8 km2 and 17.2 km2 respectively. Each bridge length x i 1i is set as 3 km. The unit consumption of travel expenses φis 17.6 yen/km as in Ministry of Land, Infrastructure, Transport and Tourism (MLIT) (2010). The intercept parameter θ in the commuting cost function is the inverse of average speed Vi 1i as 30 km/h. In the BPR function, parameters ɛ and δ are set at 2.82 and 0.48 as in Japan Society of Civil Engineers (JSCE, 2003), which are used for Japanese roads. Road capacity for each bridge C01, C12 and C23 are set at 318,560, 152,240 and 76,120 per-day vehicle/2lanes. These are set to reflect the real main roads in Sendai and the data is derived from MLIT (2010) and JSCE (2003). We set the number of trips to the CBD as 230 round trips per year.

We demonstrate numerical simulations to clarify the effects of optimal FAR regulations on households and landowners. To focus mainly on the benefit disparity among agents, we show the results in the heterogeneous case. Specification and calibration are shown in Section 4.1. Section 4.2 subsequently shows numerical simulations of optimal FAR regulation. Section 4.3, using another set of functions and parameters, yields a case of negative Eq. (43), which is discussed regarding Proposition 3. 4.1. Specification and calibration The utility function and the floor area production function are specified as Cobb-Douglas utility function v = cgi·qgi1 and S (Fi ) = ·Fi , respectively, where α and β are positive multiplicative factors, and γ is set as γ > 1. Note that income y, the area of zone i, a1, a2 and a3 and total population N are exogenously set. Commuting costs tgi are expressed by Eqs. (18), 19a), (19b), and ((20). Travel expenses i 1i are defined as ·x i 1i , where φ is the unit consumption of travel expenses, and x i 1i is bridge length. Travel time Mi 1i is expressed as 7

Journal of Housing Economics 46 (2019) 101656

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The average income of high/low income households is set to differ for the two location patterns. That is, the difference in income and the difference in value of time are set to hold for each segregation pattern. p pl2 For location pattern (I), which should have xh2 >0, average x 12 12 annual income is set as 10,000,000 yen for high income households and 2000,000 yen for low income. For location pattern (II), which should p pl2 have xh2 <0, average annual income is set as 10,000,000 yen for x 12 12 high income and 5000,000 yen for low income. Besides, value of time is set as 5000 yen per hour for high income, and 1000 yen per hour for low income for both location patterns.

while it increases land rent in zone 3. That is, under optimal FAR regulation, landowners who own lands close to the center are worse off and those who own the land close to the boundary are better off, as graphically shown in Fig. 4. Next, we explore the case in which residents have political power. For an extreme case, the social welfare is composed of only residents’ utilities. In this case, the result is simple in most cases and FARs turn out to be all large enough to have zero land prices. Therefore, we explored a mild case in which the social welfare is composed of the utility level of residents and 80% weighted profits of absentee landlords. We call this Regime 3. Formula (41) will be rewritten as follows:

4.2. Numerical simulations of optimal far regulation

SBregime3 = Ng1·EVg1 + Ng 2· EVg 2 + N g 2· EVg 2 + N g 3·EVg 3

We numerically simulate the effects of optimal FAR regulation. In this section, FAR is optimized in all the zones simultaneously to maximize social benefit. The numerical values of important endogenous variables are presented in Tables 2 and 3 for location patterns (I) and (II), respectively. Under optimal FAR regulation, FAR in zone 1 is larger than the market equilibrium FAR while FAR in zone 3 is smaller than the market equilibrium FAR as shown in the top three rows in Table 4. These results are consistent with the results obtained theoretically in Kono et al. (2012), which only assume homogeneous residents. Benefits for each type of residents are shown in Tables 4 and 5 for location pattern (I) and (II), respectively, and those results are depicted in Fig. 3 in terms of amounts per household Table 6. From these tables and figures, we find Main results 1. Main results 1. Under optimal FAR regulation, the benefits for high income households are higher than those for low income households regardless of the location pattern. A change in the total land rent from market equilibrium in optimal FAR regulation is positive, which means that the total land rent is higher under optimal FAR regulation than at market equilibrium. Optimal FAR regulation increases utility for all the residents, as shown in the left two bars in Fig. 3. This is because a larger total floor space close to the center leads to lower commuting costs for households who migrate to zone 1 or larger space for households which still live in the suburb as shown in Section 3.1. The change in land rent, which is described by the second sentence of Main results 1, can be further explored by zone, using Fig. 4. The height of the bars represents land rents by policy, and the red line represents the market equilibrium land rents. From these figures, we find Main results 2. Main results 2. Optimal FAR regulation decreases land rent in zone 1

(41)

+ 0.8(a1 r1 + a2 r2 + a3 r3)

Based on Regime 3, we obtain the following results. Main results 3. If social welfare is composed of residential utility and 80%-weighted profits of absentee landlords, optimal FAR increases in all three zones in both location patterns, although the size of the increase from the market size in zone 1 is larger than in zones 2 and 3. This result is very easy to intuitively understand. If land rent decreases, residents in the city will be happy due to an increase in lot size. Therefore, the optimal FAR will be increased in each zone when the residents’ utilities are weighted heavily. Simultaneously, an increase in lot size in zone 1 makes congestion worse off, which decreases residents’ welfare. Since the former is larger than the latter, the social welfare increases. FAR regulation in zone 1 being larger than in zones 2 and 3 can reduce congestion. In order to intuitively understand the mechanisms of our model, we give a geometric interpretation for the case of Location pattern (I), in which low-income households live near the CBD, while high-income households live in suburbs. We have six endogenous variables Nl1, Nl2, Nh2, Nh3, ul, uh, which are determined by six equilibrium conditions (24), (25), (35)-(38). The following discussion and graph are based on the specified functions used for our numerical simulations because the theoretical model can have many possible graphs, so the explanation would be very complex. First, substituting (24), (25), (35) and (36) into (37), we obtain a function of only two variables ul and uh. The obtained function shows a negative relationship between ul and uh, which is described by the red solid bold line in the upper graph of Fig. 5. The x-axis of this Figure denotes the utility level of low-income households, while the y-axis denotes the utility level of high-income households. Similarly,

Table 4 Numerical values in location patterns (I).

F1 F2 F3 Ng1 Ng2 Ng 2

Ng3 ul uh p1(103¥/m2/year) p2(103¥/m2/year) p3(103¥/m2/year) R1(103¥/m2/year) R2(103¥/m2/year) R3(103¥/m2/year) qg1(m2) qg2(m2) q g 2 (m2)

q g 3 (m2)

Location pattern (I) Market equilibrium (Regime 1)

Optimal FAR (Regime 2)

146 126 117 122,276 277,724 29,698

174 122 111 144,898 255,102 32,399

Parochially-optimal FAR (Regime 3) 19.2% −3.2% −5.1% 18.5% −8.1% 9.1%

188 132 120 144,783 255,217 32,419

28.8% 4.8% 2.6% 18.4% −8.8% 9.2%

70,302 28,613.5 146,234.0 9.9 7.1 6.1 989.5 618.8 490.9 38.0 49.3 251.9

67,601 28,646.1 146,387.0 9.8 7.2 6.2 913.4 631.7 501.1 38.2 48.7 249.0

−3.8% 0.1% 0.1% −0.6% 1.6% 1.8% −7.7% 2.1% 2.1% 0.6% −1.1% −1.2%

67,581 29,097.5 148,694 9.1 6.7 5.7 687.4 559.3 447.9 41.3 52.7 269.3

−3.9% 0.2% 0.2% −8.1% −5.6% −6.6% −30.5% −9.9% −8.8% 8.7% 6.9% 6.9%

286.3

282.4

−1.3%

305.4

6.7%

8

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Table 5 Numerical values in location patterns (II).

F1 F2 F3 Ng1 Ng2 Ng 2

Ng3 ul uh p1(103¥/m2/year) p2(103¥/m2/year) p3(103¥/m2/year) R1(103¥/m2/year) R2(103¥/m2/year) R3(103¥/m2/year) qg1(m2) qg2(m2) q g 2 (m2)

q g 3 (m2)

Location pattern (II) Market equilibrium (Regime 1)

Optimal FAR (Regime 2)

177 150 136 45,192 54,808 180,810

192 155 123 48,800 51,200 196,949

Parochially-optimal FAR (Regime 3) 8.5% 3.3% −9.6% 8.0% −6.6% 8.9%

209 167 134 49,046 50,954 195,968

18.1% 11.3% −1.5% 8.5% −7.0% 8.4%

219,190 71,083.1 132,590.0 15.2 10.6 8.5 1849.6 1101.7 793.6 124.7 166.1 89.0

203,051 71,102.8 132,711.0 15.1 10.6 8.7 1812.8 1100.9 814.6 125.3 166.2 89.0

−7.4% 0.0% 0.1% −0.5% 0.0% 3.1% −2.0% −0.1% 2.6% 0.5% 0.1% 0.0%

204,032 72,243.7 134,846.0 13.9 9.8 8.0 1484.7 944.5 737.2 135.7 179.9 96.4

−6.9% 1.6% 1.7% −8.6% −7.5% −5.9% −19.7% −14.3% −7.1% 8.8% 8.3% 8.3%

106.7

104.2

−2.4%

113.0

5.9%

Fig. 3. Benefit among agents in location patterns (I) and (II).

substituting (24), (35) and (36) into (38), we obtain a function of only y t two variables ul and uh as uh = yh th2 ul . This equation shows a positive l2 l relationship between ul and uh, which is described by the blue solid thin line. Solving these two equations, we can determine the equilibrium solutions of ul and uh. In other words, the intersection of the two lines in the upper graph of Fig. 5 represents the solution of ul and uh. We call

this stage Step 1. When F1, F2, or F3 increases, the two solid lines in the upper graph shift to the dashed lines. For our numerical simulations, we assume that the wage rate (=value of time) is proportional to the income level. As a y t result, h h2 is constant, thus the blue solid line and the blue dashed yl

t l2

line are the same. Consequently, the red line shifts right-upward, and

Table 6 Benefit among agents in location patterns (I).

Change in land rent 1 (106¥/km2/year) Change in land rent 2 (106¥/km2/year) Change in land rent 3 (106¥/km2/year) Benefit of landowner (106¥/year) EVg1(103¥/year) EVg2 (103¥/year) EV g 2 (103¥/year)

EV g 3 (103¥/year) Sum of EVs (106¥/year) Social Benefit (106¥/year) Social Benefit per Household (¥/year)

Market equilibrium

Optimal FAR

0 0 0 0 0 0 0

−761 129 102 1506.5 2.1 2.0 9.4

0 0 0

1740.0 3246.5 6492.6

0

9.1

9

Parochially-optimal FAR −7.7% 2.1% 2.1%

−3021.0 −594.6 −430.1 −21,607.3 31.7 29.7 151.0 146.2

26,942.8 5335.6 10,671.2

−30.5% −9.6% −8.8%

Journal of Housing Economics 46 (2019) 101656

Y. Takeda, et al.

Fig. 4. Changes in land rent in location patterns (I) and (II).

the equilibrium point shifts from E to E′, E″, orE‴, where single prime, double prime and triple prime, attached to E, represent the equilibria in the cases of changes in F1, F2, or F3, respectively. After ul and uh are solved in Step 1, Nl1, Nl2, Nh2, Nh3 will be solved with (35), (24), (36) and (25) in the lower graph in Fig. 5, in which the

total length of the horizontal axis and the vertical axis represent the numbers of low-income and high-income households, respectively. The allocation of each-income-category residents between two zones is represented by the division of the two axes in the graph. This division is mathematically represented by (24) and (25). The red solid bold line and the blue solid thin line denote (35) and (36). The intersection of the red bold line and the blue solid line represents the equilibrium allocation. According to changes in utilities in response to changes in F1, F2, or F3, the two lines shift. Step 1 and Step 2 in Fig. 5 can be explained as follows. We use F1 as an example. If F1 increases, the utility of low-income households increases (as shown in the upper graph) because low-income households enjoy larger lot size in zone 1, and simultaneously low-income households in zone 2 move to zone 1 (as shown in the lower graph). Some space made by the move of low-income households in zone 2 will be filled by high-income households moving from zone 3. As a result, the number of high-income households in zone 3 decreases (as shown in the lower graph). Consequently, the equilibrium point of population allocation shifts from E to E′ in the lower graph. Similar logic can be used to interpret the effects of F2 and F3. This explanation using graphs is for showing the mechanisms of the model concisely. But this discussion is based on the specified functions for our numerical simulations. Note that Propositions obtained in Section 3 are based on un-specified functions (just on well-behaved functions).

or

0

4.3. Proof of occurrence of a negative Eq. (43) by numerical simulation This section numerically validates the existence of a negative Eq. (43), which implies a decrease in utility of the group that also lives in the suburb when a FAR increases in zone 1. Since the setting in 4.2 cannot generate the result, we use the following utility function.

ugi = f (cgi) +

· qgi

where

z z ·c

f (c ) =

(0

(c > 1) c 1)

(42)

In this setting, as far as vg / g / vg / c = pressed as

qgi = (ug

z )/

and pi = { ·(yg

tgi

/ z < pi , qgi and pi are ex-

z )}/(ug

z ).

(43)

We only demonstrate the case of location pattern (I). Using new parameters z and α′ as 400 and 100,000,000 respectively, the simulation results of the changes in increase of floor area ratio in zone 1 from market equilibrium are obtained as follows Table 8.

Fig. 5. Geometric explanation of changes in endogenous variables. 10

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Table 7 Benefit among agents in location patterns (II).

6

2

Change in land rent 1 (10 ¥/km /year) Change in land rent 2 (106¥/km2/year) Change in land rent 3 (106¥/km2/year) Benefit of landowner (106¥/year) EVg1 (103¥/year) EVg2(103¥/year) EV g 2 (103¥/year)

EV g 3 (103¥/year) Sum of EVs (106¥/year) Social Benefit (106¥/year) Social Benefit per Household (¥/year)

Market equilibrium

Optimal FAR

0 0 0 0 0 0 0

−367.9 −8.6 209.8 2292.2 8.6 8.0 1.3

0 0 0

1338.9 3631.1 7262.3

0

Parochially-optimal FAR −2.0% −0.1% 2.6%

−3648.9 −1571.9 −564.1 −38,184.5 161.0 150.0 77.1

1.2

−19.7% −14.3% −7.1%

73.7

45,680.8 7496.3 14,992.6

Table 8 Proof of occurrence of a negative Eq. (D.5-2). F1

Ul

Uh

ql2 (m2)

qh2 (m2)

Nl1

Nl2

Nh2

Nh3

184 188 192 196 200

593,298. 593,304. 593,309. 593,315. 593,320.

3,028,598.1 3,028,589.6 3,028,581.2 3,028,572.7 3,028,564.2

59.2298 59.2304 59.2309 59.2315 59.232.

302.76. 302.759 302.758 302.757 302.756

197,887 202,187 206,487 210,787 215,087

202,113 197,813 193,513 189,213 184,913

13.1485 12.8689 12.5893 12.3097 12.0301

99,986.9 99,987.1 99,987.4 99,987.7 99,988.

In Table 7, as the FAR in zone 1, F1, increases, as shown in the first column, the floor space for low income residents in zone 2, ql2, which is in column 3, increases, i.e., ∂ql2/∂F1 > 0. This increase leads to ∂ul/ ∂F1 > 0. Concurrently, that increase in housing space for low income households decreases the housing space for high income households in zone 2 and consequently they are forced to move to zone 3 (See population movements between zone 2 and zone 3, which are shown in the two columns on the right). That movement causes more congestion and smaller space in zone 3, which leads to a decrease in the utility of high-income households.

the last forty years, with downtowns now being occupied primarily by high-income households. Basically, FAR does not affect the generalized cost much affects the lot size and the equilibrium utility. The last issue is the new urbanism. Partly due to the new urbanism, partly to gentrification, there is a trend to increasing FARs in central areas. This leads to cities becoming more compact. However, this trend forward compactness cannot exclude externalities completely without economic policies. In situations where the first-best policies cannot be implemented to internalize all the externalities, we need to consider second-best policies. Furthermore, centralization of residents does not always improve the social welfare. Whether making cities compact is good or not should be explored while considering market failures.

5. Extensions of our discussion The current paper is based on a monocentric model. As the Introduction states, optimal land use regulations are strongly dependent on the setting of geography. To discusses optimal FAR regulations in other geographical settings such as non-monocentric, polycentric, or dispersed cities, we basically have to set such situations to explore optimal regulations because multiple price distortions are intertwined in a different way depending on the situations. However, even if other price distortions are added to our setting by extensions of the setting, the fundamental mechanisms leading to optimal FAR regulations remain as they are. Another issue is the treatment of landownership. The current paper assumes public ownership. However, in many developed countries including the US and Japan, most higher-income households are owneroccupiers. In this case, actually, we have to consider two additional situations. First, landownership other than public ownership generates another distortion because landowners residing in certain areas pay and receive higher-than-average rents due to their ownership. This distorts land use. So, we have to consider how this distortion changes according to land use regulations. Second, we have to consider the effects of land use regulations on heterogenous income residents. This is related to the political economics of land use regulation, which is explored by many papers (e.g., Hilber and Robert-Nicoud, 2007; Helsley and Strange, 1995) Gentrification is a recent trend of land use patterns, which are caused by high bid rent in the central areas. In the current paper, the land use patterns depend on who can bid higher rents, which are determined by each resident's unit distance generalized travel time and residential lot size. In the major metropolitan areas in the US at least, the locational pattern by income has changed quite dramatically over

6. Conclusion This paper theoretically and numerically analyzes the effects of Floor Area Ratio (FAR) regulations on households with income disparity and landowners in a congested closed city. The heterogeneous case yields different qualitative results of the effects of FAR regulation on households from those in the homogeneous case. Changes in land rent in a zone associated with FAR are determined by the size of the unregulated FAR in that zone or the relative location of the zone where FAR changes. Numerical results clarify the effects of optimal FAR regulations on households and landowners. In particular, using a certain set of parameters, the results show that an increase in FAR in a central zone worsens the utility of suburban residents due to the residential segregation pattern of heterogeneous people. Numerical results also show that under optimal FAR regulation, the benefits for high income households are higher than those for low income households regardless of the location pattern. We consider a parochially optimal case in which social welfare is composed of residential welfare and only 80%-weighted benefits of absentee landlord. Unsurprisingly, simulation results show that FAR regulations in all three zones are increased. Additional research on the topic can be conducted by investigating the different effects with supplementary policies, e.g., cordon pricing7 7 With homogeneous residents, cordon pricing and land use regulations are analyzed in Kono and Kawaguchi (in press).

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Journal of Housing Economics 46 (2019) 101656

Y. Takeda, et al.

and UGB regulation etc. In connection with political decision making processes such as majority voting etc., further research need to explore the mechanisms of the policy making of FAR regulation.

Education, Culture, Sports, Science and Technology (Grant-in-Aid for Scientific Research C 26380285), which are gratefully acknowledged. Despite assistance from many sources, any remaining errors in the paper are the sole responsibility of the authors.

Acknowledgments This research was supported by grants from the Ministry of Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jhe.2019.101656. Appendices Appendix A. Total differential form in the homogeneous case Differentiating four Eqs. (28)-(31) with respect to floor area ratio in zone 1, and arranging the results, we obtain Eqs. (A.1.1)-(A.1.3) and (A.4.1). Note that this comparative static analysis shows changes in the equilibrium solution when the regulated floor areas are given exogenously.

q T dN1 a = 1 · B q3 + N3· 3 · 23 + Aq2 > 0, dF1 W t3 N3 dN2 = dF1

q T a1 · B q3 + N3· 3 · 23 + BD W t3 N3

dN3 a = 1 ·(BD dF1 W

AC

(A.1.1)

AC ,

(A.1.2)

Aq2 ).

(A.1.3)

Differentiating four Eqs. (28)-(31) with respect to floor area ratio in zone 2, and arranging the results, we obtain eqs. (A.2.1)-(A.2.3) and (A.4.2).

q T a2 · q3 + N3· 3 · 23 < 0, W t3 N3

(A.2.1)

q T dN2 a = 2 · q1 A + D + q3 + N3· 3 · 23 > 0, dF2 W t3 N3

(A.2.2)

dN3 = dF2

(A.2.3)

dN1 = dF2

a2 ·(q A + D ) < 0. W 1

Differentiating four Eqs. (28)-(31) with respect to floor area ratio in zone 3, and arranging the results, we obtain eqs. (A.3.1)-(A.3.3) and (A.4.3).

dN1 = dF3

a3 q < 0, W 2

(A.3.1)

dN2 = dF3

a3 (q B + C ) < 0, W 1

(A.3.2)

dN3 a = 3 (q1 B + q2 + C ) > 0. dF3 W

(A.3.3)

Applying the implicit function theorem to Eqs. (28)-(31) yields

q T du a = 1 (q2 + C ) q3 + N3 3 23 + q2 D , dF1 W t3 N3

(A.4.1)

q T du a q1 = 2 q + N3 3 23 , dF2 W N1 q1 3 t3 N3 u

(A.4.2)

q1 q2 du a = 3 , dF3 W N1 q1/ u

(A.4.3)

where W q2 (q1 A + D) + (q1 B + q2 + C )(q3 + N3·( q3/ t3)·( T23/ N3)) > 0 , C N2·( q2/ t2)·( T12/ (N2 + N3)) > 0, D N3·( q3/ t3)·( T12 / (N2 + N3)) >0 . Here, A ≡ (N3 · ∂q3/∂u)/(N1 · ∂q1/∂u) > 0, B ≡ (N2 · ∂q2/∂u)/(N1 · ∂q1/∂u) > 0. The signs of the terms A, B, C, D and W are proved to be positive, using inequality conditions (8) and (14). Consequently, the signs of the right-hand side of Eqs. (A.4.1)-(A.4.3) are all proved to be positive, i.e. du/ dFi > 0(i = 1, 2, 3) as shown in Proposition 1. Appendix B. Differential form of land rent The Lagrangian function expressing Eq. (6) is L = Lagrangian with ci, we obtain

L / ci =

1/ qi

y

i·

ti qi

ci

i (u

v (ci, qi)) , where λi is the Lagrangian multiplier for utility. Differentiating this

ci / u = 0 , which leads to 12

i

=

( ci / u)/ qi (= pi / u ) . Using the envelop theorem and

Journal of Housing Economics 46 (2019) 101656

Y. Takeda, et al.

differentiating Eq. (6) with respect to qi yields pi / ti =

1/qi . To sum up, we obtain following relational expressions p

pi ti

=

1 qi

pi u

=

i

=

1 ci . qi u

i (i = 1, 2, 3) . Using the relational expressions pi / ti = 1/qi and Differentiating Eq. (10) with respect to Fi (i = 1, 2, 3) yields Fi = Fi Fi + Pi Fi i i pi / u = ( ci / u )/qi , ∂pi/∂Fi is expressed as pi / Fi = ( ti/ Fi + ci/ u· u/ Fi)/ qi . Substituting this into the above equation yields Eq. (32). Using inequality condition (2) and Proposition 1, ∂ci/∂u · ∂u/∂Fiin the parenthesis is positive. Furthermore, ti1/ F1 = 0 because N is constant in (11). ∂t2/ dN dN2 + dN3 ) < 0 . Similarly, using (A.3.1) and (A.3.2) leads to ∂t2/∂F2 > 0. In summary, the term ∂F2 > 0 holds because (A.2.1) implies dF1 (= dF2 2 ti / Fi + ci/ u· u/ Fi is positive. That leads to ∂pi/∂Fi < 0. p r Differentiating Eq. (10) with respect to Fj (j = 1, 2, 3 i j) yields Fi = Fi Fi (i , j = 1, 2, 3 i j ) . Using the relational expressions pi / ti = 1/qi ,

r

pi / u =

S

j

( ci / u )/qi , ∂pi/∂Fj is expressed as Eq. (33).

j

Appendix C. Derivation of Eq. (34) Substituting 19a) and (19b) into ((38) yields i k=1

yl

pl2 =

k 1k (xk 1k )

wl·M01 (x 01)

wl·M12 (x12 )

c (ql*, ul )

ql*2 (yl , tl2, ul )

ph2 =

i k=1

yh

k 1k (xk 1k )

wh· M01 (x 01)

wh·M12 (x12)

,

(C.1)

c (qh*, uh )

qh*2 (yh , th2, uh )

.

(C.2)

Differentiating pl2 and ph2 with respect to x12 yields (34). Appendix D. Total differential form in the heterogeneous case Differentiating six Eqs. (35)-(38), (24) and (25) with respect to floor area ratio in zone 1, and arranging the results, we obtain Eqs. (D.2.1)(D.2.4), (D.5.1) and (D.5.2).

dNg1 dF1

dNg 2 dF1 dN g 3 dF1

dNg 2 dF1

=

=

=

=

cg 2 cg 2 a1 F · qg 2 + qg 2 ·(H + YJ ) Zqg1 ug ug

dNg1 dF1

> 0,

(D.2.1)

< 0,

(D.2.2)

1 qg 3 + K ·

dN g 3 dF1

T23 Ng 3

cg 2 a1 q q J+ Zqg1 g 2 g 2 ug

cg 2 T12 q J {(G + I ) (Ng 2 + Ng ) g 2 ug

(qh2

.

ql2 ) F }

qg 2

cg 2 ug

FK

, (D.2.3) (D.2.4)

Differentiating six Eqs. (35)-(38), (24) and (25) with respect to floor area ratio in zone 2, and arranging the results, we obtain Eqs. (D.3.1)(D.3.4), (D.5.3) and (D.5.4).

dNg1 dF2 dNg 2 dF2 dN g 3 dF2 dNg 2 dF2

=

=

cg 2 a2 Eq < 0, Zqg1 g 2 u g dNg1 dF2

> 0,

(D.3.2)

{

qg 2

a2

=

Z qg 3 + K =

dN g 3 dF2

(D.3.1)

T23 Ng 3

+K

cg 2 ug

+(

h

T12 E l )· (N + N ) q (qh2 g2 g g1

}

ql2 ) · C

cg 2 T12 E q (Ng2 + N g ) qg1 g 2 u g

< 0, (D.3.3)

> 0.

(D.3.4)

Differentiating six Eqs. (35)-(38), (24) and (25) with respect to floor area ratio in zone 2, and arranging the results, we obtain Eqs. (D.4.1)(D.4.4), (D.5.5) and (D.5.6).

dNg1 dF3 dNg 2 dF3

=

=

cg 2 EY a3 qg 2 < 0, Zqg1 ug dNg1 dF3

(D.4.1)

> 0,

(D.4.2)

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Journal of Housing Economics 46 (2019) 101656

Y. Takeda, et al.

dN g 3 dF3

dF3

=

Z qg 3 + K

dNg 2

F + qg 2 +

a3

dN g 3

=

dF3

{

T23 Ng 3

+ qg 2

cg 2 ug

T12 (G (Ng 2 + N g )

+(

h

E

+ I) · q

qg 2

g1

cg 2 ug

T12 E l )· (N + N ) q (qh2 g2 g g1

}

> 0,

ql2) H

(C.4.3)

< 0.

(C.4.4)

Arranging the total differential of six Eqs. (35)-(38), (24) and (25) yields the signs of changes in utility associated with FAR as

du g dF1

du g dF1

du g dF2 du g dF2

du g dF3 du g dF3

=

a1 T12 (H + YJ ) (q Zqg1 (Ng 2 + Ng ) h2

=

cg 2 a1 q X Zqg1 g 2 ug

=

cg 2 a2 q , Z g2 u g

=

cg 2 a2 q +( Z g 2 ug

Z

T12 (q (Ng 2 + Ng ) h2

cg 2 ug

X ,

ql2) Q ,

=Y

cg 2 a3 q +( Z g 2 ug

T12 E (q (Ng 2 + Ng ) qg1 h2

l)

h

cg 2 a3 q , Z g2 u g

(D.5.2)

ql2) ,

h

l)

T12 E (q (Ng 2 + N g ) qg1 h2

cg 2 cg 2 XE q + (H + YJ ) qg 2 +( ug qg1 g 2 u g

E

Ng1 qg1/ ug > 0, Q

I

Ng 2

g

(D.5.4) (D.5.5)

ql2) ,

qg 2 + (G + I + YK )· T12 / (Ng 2 + Ng ) > 0 , Y

F+

(D.5.1)

(D.5.3)

=Y

where X

ql2 ) + qg 2

Ng 2 qg 2/ ug > 0, G

qg 2/ t g 2 > 0, J

Ng 2

N g 3 qg 3 / u g > 0, K

qg 2 / (qg 3 + K · T23/ Ng 3) > 0 , l)

h

g

(D.5.6)

T12 E (q (Ng 2 + Ng ) qg1 h2

qg 2/ tg 2 > 0, H

Ng 3

g

ql2 ) > 0,

N g 2 qg 2/ u g > 0,

qg 3 / t g 3 > 0,

g ∈ {l, h}, g′ ∈ {l, h} and g ≠ g′. The signs of the terms E, Q, G, H, I, J, K, X, Y, and Z are proved to be positive, using inequality conditions (8) and (14). Then, substituting these terms into Eqs. (D.5.1)-D.5.6) with inequality conditions ((8) and (14) and qh2 ql2 > 0 , the signs of Eqs. (D.5.1) and (D.5.3)-(D.5.6) are proved to be positive, i.e., dug / dFi > 0(i = 1, 2, 3), du g / dFj > 0(j = 2, 3) where g ∈ {l, h}, g′ ∈ {l, h} and g ≠ g′. But, interestingly, Eq. (D.5.2), i.e., du g / dF1, can be positive or negative. These results are summarized as Proposition 3.

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