Regulation versus Taxation: Efficiency of Zoning and Tax Instruments as Anti-Congestion Policies Regulation versus Taxation: Efficiency of Zoning and Tax Instruments as Anti-Congestion

Regulation versus Taxation:

Efficiency of Zoning and Tax Instruments as Anti-Congestion Policies

Hyok-Joo Rheea and Georg Hirteb

aCorresponding author

Department of Public Administration, Seoultech

232 Gongneung-ro

Nowon-Gu, Seoul 139-743, Republic of Korea

Phone: +822-970-6494, Fax: +822-971-4647

E-mail: [email protected]

bFaculty of Traffic Sciences, TU Dresden,

01062 Dresden, Germany, E-mail: [email protected]

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Regulation versus Taxation: Efficiency of Zoning and Tax Instruments as Anti-Congestion Policies

Abstract Using a general equilibrium spatial model with household heterogeneity, we examine working mechanisms and efficiencies of zoning (regulation of floor area ratios and land-use types) and tax instruments (tolls and property taxes), and extend instrument choice to the externalities caused by the congestion of infrastructure. When the instrument design is optimal, zoning can be more efficient than, as efficient as, or less efficient than congestion charges, depending on the relative strength of road and nonroad congestion. However, when there are deviations from the optimum for any reasons, zoning not only becomes inferior to congestion charges but is also likely to reduce welfare. This latter possibility is serious, because it is almost unimaginable to consider city planning without land use regulations. In fact, we provide a global platform that extends the instrument choice theory of pollution control to various types of externalities and to a wide range of policy deviations beyond cost–benefit uncertainties.

Keywords: infrastructure congestion, zoning, road tolls, property tax, instrument choice, household heterogeneity

1. Introduction There has been a long-standing debate among economists on the relative efficiency of

price and quantity regulation instruments to control market failures such as externalities (e.g., Weitzman, 1974, and Kaplow and Shavell, 2002, on pollution control in environmental economics). We focus on this topic in a spatial economy and compare the efficiency of tax instruments (property taxes and congestion charges) and zoning (floor area ratio regulations and land-use type regulations) as policy instruments to control the simultaneous congestion of road and nonroad infrastructure.

According to the benefit view of property tax, these instruments are closely related. Property tax is a tax price paid by residents for local public services, and zoning is a physical mechanism forcing residents to consume “at least some minimum amount of housing” that is taxed for the cost of local public service provision (Hamilton, 1975: 206). Because each additional resident entails additional costs of public service provision, the property tax is an efficient congestion charge that is indispensable when nondistortionary head tax is unavailable (Hoyt, 1991; Krelove, 1993). Similarly, lot-size zoning is equivalent to a congestion charge via its effect on density and on account of duality (Pines and Sadka, 1985; Wheaton, 1998).

Despite this duality in the first-best world, there exist divergent views on the efficiency of land use regulations (LURs) and the policy prescription administered to cope with congestion externalities. In urban economics, the LURs or equivalently zoning hereafter refers

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to regulation of floor area ratios (FARs), lot-size regulation, and regulation of land use types either in built-up areas or at urban boundaries. To survey the divergent views on the efficiency of LURs first, authors driven mainly by empirical studies state that the cost of LURs is huge (Lee, 1999; Cheshire and Sheppard, 2002; Bertaud and Brueckner, 2005). Specifically, urban growth boundaries (UGB), although considered second best in some studies (Kanemoto, 1977; Arnott, 1979, Pines and Sadka, 1985), are thought to improve welfare negligibly at most (Brueckner, 2007) or even be absolutely harmful (Anas and Rhee, 2006). In some cases, an expansionary growth boundary is recommended instead (Anas and Rhee, 2007; Anas and Pines, 2008).

However, there is a sizable body of literature that favors the LURs. In the planning literature for compact development, the efficiency of development controls is not something to be disputed, and the controls are required to “grow smart” (Ewing et al., 2007). According to theoretical studies by urban economists, the LURs are first best (Pines and Sadka, 1985; Wheaton, 1998), almost as efficient as the first-best policy mix (Rhee et al., 2014), second best (Pines and Sadka, 1985; Kono et al., 2012) and even third best (Pines and Kono, 2012).

A similar complication arises with respect to regulatory prescriptions to fight against congestions. To control metropolitan-wide transportation externalities, Kono et al. (2012) rec-ommends increasing the FAR at the city center (a minimum FAR regulation) and lowering the FAR at the suburbs (a maximum FAR regulation) in the closed city. When the city is open, only the maximum FAR regulation is required (Kono and Joshi, 2012). However, in case of (negative) population externalities, such as noise, privacy and congestions, the prescription is reversed for the closed city (Kono et al., 2009). There are at least four reasons for these diverging views. The first is the type of externalities analyzed. Different externalities could require opposing prescriptions. Transport and population externalities are two examples in the literature (e.g., Kono et al., 2012, versus Joshi and Kono, 2009). However, there are usually multiple externalities, which is known to affect optimal tax rates (Parry and Bento, 2002), but this has not yet been studied in urban economics. The second and third reasons are whether the model is of partial or general equilibrium and whether the city is monocentric or nonmonocentric, respectively. In the monocentric city with fixed demand for travel, lot-size control is a perfect substitute for the first-best Pigouvian pricing (Oron et al., 1973; Wheaton, 1998). However, this does not hold if travel demand is endogenous. The fourth reason is household heterogeneity. When marginal utilities of income (MUIs) differ, the standard Pigouvian pricing is no longer first best, unless a special analytical remedy is administered, such as interhousehold income transfers (De Palma and Lindsey, 2004). Interestingly, it has yet not been studied whether externality pricing coupled with income transfers would continue to be first best in the spatial model with heterogeneous households. The survey, therefore, suggests that many of the divergent views arise from differences of study design adopted by the authors, and some of the confusion is an artefact due to the limited model setup used for analysis. We resolve this complication and confusion simply by reformulating the problem using generalized models. We analyze various LUR issues using a significantly extended model that is closely related to the real world but is not yet specified in

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that context. These extensions include a general equilibrium nonmonocentric model, two types of congestion, a basket of different regulatory instruments, mixed land use for production and residences, and household heterogeneity. We derive the first-order formulas for the welfare change associated with various pol-icies (taxes and LURs). The formulas reveal the essential differences between the model with and without household heterogeneity and between taxes and LURs. First, we find that some confusion and complications in urban and transportation economics are attributable to covari-ance terms of the Feldstein (1972) type. One finding is that the standard policy prescriptions (congestion charges and income transfer) are not guaranteed to be first best in the spatial model with household heterogeneity. Because we have explicit formulas for the welfare change, we can precisely measure how significant the covariance terms are in welfare change. Indeed, they account for a significant proportion of the total welfare change. Second, the precise formulas for the derived welfare change suggest how taxes and LURs work differently in the urban spatial economy. The formulas enable us to resolve the conflicting FAR prescriptions, and show the similarities and dissimilarities of various policies’ working mechanisms. For example, the formulas suggest how taxes and LURs should be set to improve welfare and how inefficiencies of the second-best policies might arise. Specifically, LURs have additional terms showing real estate market distortions, and the huge cost of im-precise LURs is attributable to these terms. Because LURs are a type of quantity regulation, we can readily recast the efficiency issues within the instrument choice framework of environmental economics. However, unlike the existing instrument choice literature, we treat the externalities present in public facilities of infrastructure rather than those in individual, private facilities, and we introduce imprecision into the policies themselves rather than into the costs and benefits of pollution abatement. This analytical twist is so useful that we can evaluate any type of policy deviation from optimal policies; the uncertainties of the costs and benefits of pollution abatement are a special case of our approach. The explicit formulas by which to measure the welfare cost of deviations suggest that the superiority of prices to quantities in both planning and pollution control literature stems from the same cost–benefit structure of welfare change that is uniquely associated with each type of policy. In fact, Weitzman’s (1974) famous formula “ B C′′ ′′+ ” holds only near the optimum; when the analytical setting becomes complicated and the policy change is discrete, the formula easily loses its predictive power.

Indeed, numerical simulations not only qualify much of the theoretical study on zoning but also confirm the analytical conjecture concerning price superiority. For example, when optimal policies (first-best and second-best policies) are known, LURs are less efficient than, as efficient as, or more efficient than tax instruments. This finding means two things. First, the second-best efficiency of LURs holds true only for a small range of city population. This result is in contrast to the exiting literature, which claims or implicitly accepts the proposition that zoning is almost as efficient as Pigouvian tolls. However, our finding also means that pricing could be less efficient than zoning, which has not been reported in the literature. At the same time, once we allow for the failures, such as imprecise zoning and taxation, the welfare cost of deviation is shown to be so large for zoning failures that it is not only efficient but also safe to

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use tax instruments rather than LURs. The rest of the paper is organized as follows. Section 2 presents the analytical model. Two types of infrastructure externalities are modeled: road congestion and nonroad congestion. We characterize the potential first-best city in the presence of these externalities, where different MUIs of heterogeneous households are equalized, infrastructure congestion is priced, and infrastructure capacities are sized to obtain balanced budgets. In Section 3, we characterize and compare the second-best policies such as traffic congestion charges, property taxes, and LURs. In Section 4, we conduct numerical simulations and explore the consequences of deviations from optimal policies. Section 5 concludes. A glossary appears at the end of the paper. 2. The Model

We model two types of infrastructure congestion: road and nonroad congestion. Nonroad congestion arises in infrastructure for services that do not require land per se, such as those related to water, sewage, policing, and fire management. We do not consider local public services that require land for service provision, such as schools and parks. When we use the term infrastructure hereafter, the term includes both road and nonroad congestible infrastructure. For the analysis, we need a separate treatment of infrastructure in order to resolve the conflicting prescription of the maximum and minimum FARs in the literature.

A closed metropolitan area is composed of an arbitrary number of zones. The lots inside a zone are treated identically for the purpose of residence, production, and travel. We measure distance by the distance between zone centroids. Thus, we abstract from intrazonal distance issues. Households are free to choose where to live and work; firms are free to choose the location of operations. Cross commuting is widely observed because residences and jobs are intermingled across the metropolitan area. Our focus is on FAR and land-use type regulations; we do not explore urban growth boundaries and greenbelts that, according to Pines and Kono (2012), are redundant when optimal FARs are employed. In line with traditions in this field, we do not consider competition among municipalities. This remains for future research. 2.1. Builders and composite good producers Builders in zone i construct office buildings using land B

iQ and capital BiX

according to a constant returns to scale technology ( , )B Bi i i iB B Q X= . The output iB is a

proxy for the structure services that users of a building enjoy and are measured by the office building’s floor area. A builder has a lot in which the maximum FAR allowed is B

if , which

means Bi

Bi iB f Q≤ . In other words, imposing a maximum FAR in a zone means that the planner

intends to lower the zone’s market FAR below this maximum level. Let the unit land and capital rents be , X

i ir p , respectively. The Lagrangian of the cost minimization problem facing office builders in zone i is

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( ) ( )( , ) Bi i

B X B B B B B BB i i i i i i i ii i iB B Q X B f QrQ p X µ λ− + −= + + , (1)

where 0Biµ > is the marginal cost of iB when the FAR regulation is not binding, and

0Biλ ≥ is the marginal compliance cost of the FAR regulation in zone i . When B

if is set sufficiently high, the FAR regulation is not binding and the problem reverts to the standard cost minimization problem. Similarly, the planner may set the minimum FAR B

if , which requires B

iB

i ifB Q≥ .

There is a second type of builder, who is known as a housing builder. Denote the total floor area of housing in zone i by iH . Similar to the office builders, the housing builders use

land HiQ and capital H

iX to produce housing iH , and the technology obeys constant

returns to scale. We denote the unit rental price of housing traded in zone i by Hip and

measure housing services by the floor area. Firms produce composite good iX using building services iB and labor iM .

( , )X Xi i i i i iX S x B M S x≡= , where ( , )i ix B M is homogenous of degree one in inputs and X

iS is a multiplier, external to each firm, showing the service quality of nonroad infrastructure in zone i . We set the service level as X

iS ≡ ( , , )Xi i iS B H K , where / 0X

i iS B∂ ∂ < , / 0Xi iS H∂ ∂ <

(congestible) and / 0Xi iS K∂ ∂ > , where iK is capacity of nonroad infrastructure. Nonroad

capacity iK is under the planner’s control. For simplicity, we assume that both firms and households use the same nonroad infrastructure. When land use is mixed, this approximation is not a wholly unrealistic assumption. Road congestion is an important class of infrastructure congestion, but we deal with road congestion through commuters’ congested travel. Compared with smaller buildings, larger buildings consume more services provided by local infrastructure, block more sunlight, and produce more network congestion from business trips and trade. The maximum FAR regulations, motivated mostly by these concerns, target the proxy variables ( , )i iB H for their control in practice. The city collects taxes for financing congestible infrastructure. Specifically, the X-good firm’s profit maximization problem is

, ,max (1 )i i i

B Bi i i i i

Xi iX B M

p B w Mp X τ −− + , (2)

where Biτ is the tax rate, and price terms are defined in an obvious way. In this way, we

assume that one unit of the X-good is converted to one unit of capital input BiX in (1) for

building construction. We write a useful differential equation.

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((1 ) )X X

X B B X i ii i i i i i i i i i i

B Bi i

ii

i

S SX dp p B d B dp M dw p x dB dKB K

τ τ ∂ ∂

= + − + ∂ ∂+ +

⋅ (3)

In the absence of tax and infrastructure congestion, Biτ = 0X

idS = , so that we have X B

i i i i i iX dp B dp M dw= + , a familiar general equilibrium equation relating inputs, output, and their prices to one another. 2.2. Households

By household ( , )i j , we mean the representative household living in zone i and working in zone j . We differentiate households by commuting arrangements ( , )i j . For a given residence–work zone pair ( , )i j , the utility maximization problem of household ( , )i j is

, , ,max ( , , , , )

ij ij ij ij

Hij ij ij ij iz h l ijd

u z h l d S + ,

subject to (1 (8) )X Hi ij i ij j

Hi ij ij ijp z p h w t d Dτ+ + = − + , (8 )ij ij ijT g d l= + + ,

where

( ) ( )1 H H B B Xij i i i i i i i i i i i i

ii i ij ij ijij

iir A y P y

ND t F p H p B r R p Kτ τ + + −

≡ + + − +∑∑ ∑ ∑ ,

land rent taxes returned infrastructure costs net income transfer

( , , )H Hi i i iS S H B K≡ .

ijz denotes the composite good X consumed by household ( , )i j , Hip is the unit rental

price of housing in the residence zone i , and ijh is the amount of household ( , )i j ’s

consumption of housing measured by floor area. The subscripts of the other variables are interpreted in the same way. The tax rate H

iτ is charged on housing consumption. ijt is the

traffic congestion charges collected from households commuting between zone i and j .

(.)HiS is the service level of the infrastructure as rated by households and is a function of a

zone’s floor areas (housing and office buildings) as well as the capacity of nonroad infrastructure in that zone. Household ( , )i j commutes ijd days a month and work eight

hours a day, while being paid jw dollars an hour at work zone j . Each household is

endowed with T hours a month, which it allocates for commuting ij ijg d , leisure ijl , and

working 8 ijd . ijg is the daily commuting time between the two zones ( , )i j .

Households own equal shares of the entire land in the metropolitan area, and the land

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rent collected is distributed equally. The metropolitan government collects taxes, uses them for financing infrastructure, and returns what remains to households. it is the traffic congestion

toll for cars on zone i ’s roads; iF is zone i ’s traffic volume. Nonlabor income ijD shows

the fiscal arrangement to be analyzed. The planner uses the lump-sum instrument ijy to

equalize the MUIs of heterogeneous households in the potential first-best regime. If there is a budget deficit, the head tax is collected. ijP , which we explain in the next paragraph, is the

share of household type ( , )i j among the fixed total population N .

The random utility term ij is an identically and independently distributed Gumbel

variate with mean zero and dispersion parameter ζ . The probability that a household most

prefers zone i and j as its home–work zone pair is given by ijP = / expexp ij mnmnV Vζ ζ∑ ,

where ijV is the indirect utility of household ( , )i j . We measure the welfare W of residents

by the expected value of the maximized utilities of the households in this metropolitan area (Small and Rosen, 1981; Anas and Rhee, 2006):

1) ln ex pma ( xij ijij

ij ijW E V Vζ ζ− = = + ∑ . (4)

Households in our model are heterogeneous because they are differentiated by their tastes for the matched pair of home–work zones. The social welfare function is a nonlinear sum of individual utilities. Residential sorting by heterogeneous households is widely observed in metropolitan areas, and the social welfare function (4) is one way of incorporating this heterogeneity. Because households differ inherently in tastes, any policy necessarily has distinct differential impacts on each type of household, and the planner weighs the impacts using the welfare function of the heterogeneous households’ differentiated evaluations. 2.3. Market equilibrium conditions

The left-hand sides of the following equations represent demand and the right-hand sides show corresponding supply.

Building markets, Housing: ij ij ijNP h H=∑ (5)

Office: input demand of X-good firms = iB (6)

X-goods, B Hij ij i i i ij

NP z X X K X+ + + =∑ (7)

Labor market, (8 )i ji jijM NP d=∑ (8)

Land market, H Bi i i iQ Q R A+ + = (9)

Three types of zero profits, Housing and office builders and X-good firms (10)

In (9), we set road capacity iR simply equal to the land area allocated to roads. We fix

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zonal land areas iA . All eight equalities exist in each zone i ; the unknowns are five prices in each zone,

, , , ,X H Bi i i i ir p p p w , and three outputs, , ,i i iH B X , in each zone. We do not list the capital inputs,

,H Bi iX X , of the housing and office builders as unknowns. Once the outputs ,i iB X are known,

the capital inputs are given by the input demand in the relevant markets. Because we have the same number of equations and unknowns, we should be able to solve for all the unknowns. We can express all variables of the system as functions of these eight unknowns and policy variables in each zone. 3. Analysis 3.1. The potential first-best regime The planner maximizes welfare ( )W ⋅ with respect to Pigouvian charges,

infrastructural capacities #1{ , , , , }H B zones

i i i ii iR Kt τ τ = , and income redistribution { } iij jy ∀ :

#1max ({ , , , } ,{ } ),H B zones

i i i i i iji ijR K yW t τ τ = ∀

subject to the market equilibrium conditions and the public budget constraint.

A familiar approach is to form a Lagrangian with all the important constraints, including the market equilibrium conditions, combined by multipliers. However, this does not work because of the sheer complexity of the problem. Instead, we differentiate W with respect to a policy variable and later incorporate the equilibrium conditions. Because the equation system (5)–(10) is composed of eight types of unknowns specified above, we can write the first-order derivative with respect to a fiscal instrument

{ }#1{ , , } ,{ }H B zones

k k k k ij ijt yφ τ τ = ∀∈ as follows:

1,2 1,2

1,2 1,2 1,2

(.)

.

X H Bij ij j ij ijn n n

ij ij ij ijX H Bij n ij ij n ij nn j n n

ij ij ij ijn n nij ij ij ij

ij n ij n ij n ijn n n

V V dw V Vdp dp dpdW P P P Pd p w p p

V Vd d

V Vdr dH dBP P P Pr H d

d d

d B d

φ φ φ φ φ

φ φ φ φ

= =

= = =

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂+ + + +

∂ ∂ ∂ ∂

∑ ∑ ∑ ∑ ∑ ∑ ∑

∑ ∑ ∑ ∑ ∑ ∑ ∑ (11)

We can derive, say, / Xij iV p∂ ∂ (a derivative of indirect utility in the price of the X-good

produced and sold in zone i ) in the first term by applying the envelope theorem to the utility maximization problem. The standard approach is to apply Roy’s identity, which is rather inconvenient in our general equilibrium setup. The result is / X M

ij i ij ijV p c z∂ ∂ = − . This term

contains household ( , )i j ’s MUI, Mijc .

On the other hand, (11) has price and quantity derivatives such as /Xidp dφ , /jdw dφ ,

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/ndH dφ . The differential equation (3) links these terms. Using the market equilibrium conditions (5)–(10), we can simplify (11) to the following intuitive form (Appendix 1):

MEC from more , ( ) MEC from more , ( )i iB H

X H X HX Xi i i i i ii i i i i iM

i ii

B B H Hi

ii

i ii i i ip p

dS S dB S S dHN dW p x N p x N

c d B B H dHυ τ υ τ

φ φ φ

− − ∂ ∂ ∂ ∂

= + + + + + ∂ ∂ ∂ ∂

∑ ∑

( )1,2

ijiii ijii i j

iw F g

ddPdFt Nydφ φ=

+ − −

′ −∑ ∑ +Cov. (12)

Mc ≡ Mij ijij

P c∑ is the average MUI, iN is the number of zone i ’s residents, ( , )i i ig F Rg ≡

is the travel cost in zone i, ( )/ /H Mij i ijij u S cυ ∂ ∂≡ , and ( )/ii j i ij jNP Nυ υ≡∑ (zone i

households’ average marginal utility of nonroad infrastructure service HiS in monetary terms).

Cov is defined as follows:

Cov ( ) ij ijM Mij ij

ijM

dS dSP c c E

dNc dφ φ

− −

≡ ∑ , (13)

where

(1 )( )( ) 8φ φ φ φ

τ

υ υ δ

φ

φ φ

− + + −

∂ ∂+ + −

∂ ∂

≡ + −Hij i

X Hij j ij

j

ijij i

i iij ij ij

H Hi i i i

j ji

kii

dS dw dgdp dpz d dd d d d d

S dH S dB dd

w

H d B

h (14)

( )/ijE dS dφ is the expected value of /ijdS dφ ,

with 1ijkδ = when the origin–destination zone pair ( , )i j contains zone k , and zero

otherwise. When the policy instrument is ijy , δ ijks is added to (14), where 0δ =ks

ij for

( , ) ( , )=i j k s , and zero otherwise. /ijdS dφ is the change in consumer surplus due to changes

in prices and externalities induced by fiscal instrument φ . The first term in (14) shows the consumer surplus change in zone i ’s X-good market triggered by the price change in the X-

good price, Xip . ijw is household ( , )i j ’s value of time. Because /ijdS dφ distributes with

population probabilities { }ijP , and the two multiplied terms in (13) are weighted by these

probabilities, Cov is the covariance of individual MUIs Mijc and the rate of consumer surplus

change /ijdS dφ that a policy change dφ engenders.

When the policy instruments are infrastructural capacities, a new term is added to (12)

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as follows:

X

k

HX k kk k k k

k k

S Sp x N rR R

υ∂ ∂+ −

∂ ∂ for road capacity kRφ = , (15)

X HX Xk kk k k k k

k k

S Sp x N pK K

υ∂ ∂+ −

∂ ∂ for nonroad capacity kKφ = . (16)

Suppose that individual MUIs, Mijc , all happen to be equal, so that we have MM

ijc c=

for all ( , )i j pairs. In this case, we do not need income transfer (i.e., 0ijy ≡ for all ( , )i j pairs) to arrive at the social optimum. In this first-best regime, the marginal external costs are fully priced, and infrastructure (roads and nonroads) are expanded until the marginal expansion costs are equal to the reduced marginal congestion cost. When the service qualities,

, ,1/H Bi i iS S g , are further assumed to stay constant with regard to proportionate changes in

capacities and patronage, the first-best solutions are not only first best, but also self-sufficient (Yang and Meng, 2002). However, individual MUIs, M

ijc , are bound to differ between different ( , )i j pairs (so, MM

ijc c=/ ). In this case, equalizing income transfers between households, { }ijy , makes only

the Cov term in (12) vanish; the second-last term showing households’ spatial re-sorting continues to exist.1 This means that when MUIs differ between households, the standard prescription composed of congestion charges, infrastructural capacities, and income transfers fails to make the first-order welfare change in (12) vanish.

Summary 1 Because of households’ spatial re-sorting, the standard first-best prescription (i.e., Pigouvian tolls, infrastructural capacities, and redistribution) is not guaranteed to be first best; any policy mix is a candidate for the first-best, and only numerical simulations or empirical testing can tell which policy mix is welfare maximizing.

Approximating the covariance term by zero, we can derive a useful insight with regard to the optimal nonroad tolls. Suppose that housing does not interact with offices with respect to infrastructural externalities in the sense that the cross effects /X

i iS H∂ ∂ = / 0Hi iS B∂ ∂ = .

Then, we can write ( )

XX ii i

B Bi

ii

SpB

p xτ

+

∂− ∂

=

,

( )

H Hi i i

Hi

ii

SNH

pτ υ

+

∂− ∂

=

,

which implies

1 In case of de Palma and Lindsey’s (2004) non-spatial model, the spatial resorting is absent, so the congestion pricing together with income transfers appropriately designed could make (12) vanish.

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Zonal output valueper floor area

Zonal infra benefitper floor area

//

//

B B Xi i iH H H H Hi i i i i i i

Xi i

X X X X Xi i i i i i i i i i

Hi i

Hi

iii i

i

B Sp S x B S B Sp x p B

H SN N Hp S H SS H

τ ητ υ υ η

∂∂∂∂

= × ≡ ×

, (17)

where /X Xi i i ip S x B is the value of zonal output per office floor area and /H

i i iiSN Hυ is the

zonal nonroad infrastructure benefits per housing floor area that iN households of zone i experience as a whole. (17) has the following implications:

Summary 2 Assume zero covariance and zero cross effects. 1. Property taxes are proportional to the elasticities of the service qualities in

floor areas. This proportionality is weighted by the value of affected economic welfare, which are output value in the case of offices and residential service utility in the case of housing.

2. There is no a priori reason to believe that business properties should be taxed more or less than housing.

Note the following remarks. First, the existing models à la Anas and Kim (1996) are

special cases in our model, in which the first two summation terms (12) are not considered and income redistribution { } iij jy ∀ is not available (Anas and Xu, 1999; Anas and Rhee, 2006,

2007). These studies set the first-best tolls equal to the standard Pigouvian tolls making only the third summation term of (12) vanish; the correct first-order condition in their studies are the formulas containing the covariance term.

Second, Rhee et al. (2014) show that land-use type regulations together with production subsidies are almost as efficient (i.e., 99%) as the policy mix of congestion charges and production subsidies in the spatial model with traffic congestion and positive agglomeration economies. We can now provide an important but missing link for such a surprising result. This result was due to the presence of the covariance term in (12). In Anas’s (2012) model with household heterogeneity, neither the standard Pigouvian road tolls maximize welfare, nor does the Henry George Theorem hold. Now, (12) shows that the covariance term has caused all the anomalies. The numerical exercise in Subsection 4.2 shows how significant the covariance term is. Third, Anas and Rhee (2006) show numerically that greenbelt reduces welfare no matter how small it may be. This anomaly puzzled Brueckner (2007). Anas and Rhee (2007) and Rhee et al. (2014) show numerically and analytically, respectively, that zero opportunity cost of land has caused the anomaly. (12) shows that their answer is only partial. The complete answer should include the covariance term.2

2 Numerical simulation shows that infinitesimally reducing the radius of laissez-faire cities either increase or decrease welfare even when land cost is set above zero. According to the standard theory, infinitesimal reduction should increase welfare.

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3.2. Second-best regimes Theory In light of the first-order welfare change (12), we may refer to the congestion tolls on building structures as differentiated property taxes levied on congestible local public services (the first two summation terms). That is, we can regard property taxes as a type of congestion charge. In the real world, property taxes are not necessarily sufficiently differentiated among property types and localities. To take an extreme example, let H B

i iτ τ τ= ≡ (constant) for all zones i and all property types. Even in this case, however, we expect the welfare performance of this tax scheme not to be poor. The reason is that although the tax rate τ is fixed, the tax bill, ,H B

i ip pτ τ , is higher in the central business district (CBD) than in suburbs. So, the congestion charge continues to be higher in the CBD than in the suburbs. Because the efficiency performance of zoning turns out to be most salient in this study, we refer readers interested in spatial analysis of the second-best optimal property tax system to other literature (e.g., Kono and Pines, 2013), and proceed to analysis of the Zoning City.

Imagine a city called Zoning City, where the only instruments available are zoning instruments #

1, , ,{ , }H B H zonesk k

Bk k k ks f f f f = , which are residential land shares ks in nonroad land,

maximum FARs ,H Bk kf f , and minimum FARs ,H B

k kf f in each zone k . Superscripts ,H B

denote housing and office (business) buildings, respectively. In the Zoning City, the LURs are fully differentiated over different zones and property types, and infrastructure is financed by head tax. After modifying the utility maximization problem and market equilibrium conditions in accordance with the Zoning City’s setup, the planner maximizes welfare W with respect to the zoning instruments:

#1max ({ , , }, ),H B H B

k k k kzones

k kf f fW fs = (18)

subject to (a) market equilibrium conditions, (b) fixed infrastructural capacities.

Having analyzed how infrastructural capacities could be set, now, we focus on the LURs, while treating the capacities as fixed. The rates of welfare change with respect to the regulatory instruments are shown as follows:

MEC by an extra MEC by an extra i iH B

X H X HX Xi i i i i ii i i i i iM

i ik i i k i i ki i

S S dH S S dBN dW p x N p x Nc ds H H ds B B ds

υ υ ∂ ∂ ∂ ∂

= + + + ∂ ∂ ∂ ∂ ∑ ∑

( ) Size of distorted MEC in traffic rents

CovH B LSik k k k

i

i ki

ii

dF r r Agw FdF s

+ − + − +

∂∂∑

, (19)

13

X H X HX Xi i i i i ii i i i i iM B B Bi i

i ik i i k i i k

S S dH S S dBN dW p x N p x Nc df H H df B B df

υ υ ∂ ∂ ∂ ∂

= + + + ∂ ∂ ∂ ∂ ∑ ∑

( )Marginal compliance cost

CovB B Max B B M FARBik

inii i k k k k

iB

i k

dFd

g Q Qf

w FF

λ δ λ δ∂ + − + +

∂ −∑

, (20)

(.)i

X H X HX Xi i i i i ii i i i i iM H H H

i ik i ii

i k i k

S S dH S S dBN dW p x N p x Nc df H H df B B df

υ υ ∂ ∂ ∂ ∂

= + + + ∂ ∂ ∂ ∂ ∑ ∑

( )Marginal compliance cost

CovH H Max H H Mini i

FARHi

ikk k

kk kH

i

dFw F g Q Qdf

λ δ λ δ ′ −+ − + +

∑

. (21)

Hkf could be either the maximum FAR H

kf or the minimum FAR Hkf for housing,

whichever is applicable. When the maximum FAR regulation is binding, 1, 0Max Minδ δ= = in the second lines of (20)–(21); when the minimum FAR regulation is binding,

0, 1Max Minδ δ= = . The first three terms in each formula of (19)–(21) are the first three terms

in (12) with the Pigouvian taxes set to zero. ,H Bk kr r are unit rents of residential and business

land in zone k, respectively. The terms before the covariance terms are new terms appearing in the Zoning City, which show the regulatory costs arising from distorted real estate markets. Marginal compliance costs, ,k kλ λ , are the Lagrangian multipliers associated with the maximum and minimum FAR regulations, respectively, that appear in (1).

Summary and implications for applied models

Before proceeding, we note that (12) and (19)–(21) were derived by adding individual market effects over different markets using the differential equation (3). Finally, all the market effects cancel out, and only those representing market imperfections show up. In this sense, the first-order welfare changes as formulated in (12) and (19)–(21) are an envelope result of the general equilibrium land-use-transportation model à la Anas and Kim (1996). Because the result is of the envelope type, we do not need all the market information to calculate the welfare impact of a policy; we need only the impacts on market imperfections, already present or newly introduced.

Summary 3 (Analytical structure) 1. Pricing and quantity instruments work on the same variables to reduce

congestion: volume of structures and traffic. 2. However, the LURs have additional terms associated with the distortionary

cost of real estate markets (structures and land). 3. Envelope result: Calculating the general equilibrium welfare change requires

only information of market imperfections, existing or newly introduced.

14

The envelope result has one important implication on the applied research using the land-use and transportation models. One common platform used in this class of models is the spatial input–output framework, in which a regional economy is composed of a closely knit network with interregional and interindustry linkages. 3 One problem with this ambitious modeling effort is the huge and demanding data set required to run the models, which is mostly unavailable. Now, note that the first-order derivatives (12), (19)–(21) have no terms, at least ostensibly, related with interregional and interindustry technical linkages. Reformulating the original production function ( , )X

i i i iX S x B M= so as to consider these linkages, we continue to have the identical derivatives (12) and (19)–(21) due to the zero profits. Therefore, there is some room to compromise the burdensome data requirement with the theoretical rigor of the applied models.

Therefore, we could measure the net welfare gain or loss differently from Echenique et al. (2012) and Jun (2012). For example, Jun (2012) evaluates the welfare impact of the restricted provision of land for factories in the capitol region of Korea. To calculate the production loss, he required regional input–output data. The formulas in (19)–(21) state instead that it is enough to measure the distortionary cost of the real estate markets affected. In the case of land supply restriction, Jun (2012) already has zonal areas kA (second-last term in (19)). Because the only imperfection in his model is traffic congestion, which can be calculated easily in our approach, the only challenging task is to measure the multiplicative term ( )H B

k kr r− in (19).

Implications for conflicting FAR prescriptions The existing literature on the FAR regulations is predominantly monocentric with no

business land use. To link our discussion to the literature and to gain more insight, imagine a city composed of two zones, where zone 1 (CBD) accommodates both office buildings and housing and zone 2 is completely residential. This setup is more general than most existing models, in which the CBD is described as a point, meaning that the whole metropolitan land is used exclusively for residences. Furthermore, suppose that the number of workdays ijd is

fixed at d to be tractable. Assume that LURs 1 1 1 2( ; , ; )B H Hs f f f are the only policies available and that the

covariance terms are negligible. Now, imagine that the city planner slightly adjusts land shares and FARs prevailing in the free markets with the aim of improving welfare. The associated welfare changes are shown as

3 There is voluminous literature on the benefits and costs of smart growth and compact development in the plan-ning field (Ewing and Cervero, 2010; OECD, 2012). However, Echenique et al. (2014) and Jun (2012) are the only studies that, to the authors’ knowledge, analyze together the economic costs and benefits of LURs within one modeling framework of regional economies.

15

W∆ ≈ nonroad ext. 1s×∆ + ( )21 1 1

1

(

2

)

1 2 1 1

0

H Bw dF r r Ad

g s ss

F

+≤

− + −′

∆ ∆

, (22)

W∆ ≈nonroad ext. 1s×∆ + ( )21 2 2 1 1 1 1 1 1

(0

1

)

H H H Max H H Min HHw F g f QdF

dQ f

fλ δ λ δ

+≤

− + ′ ∆ − ∆

, (23)

W∆ ≈nonroad ext. 1s×∆ + ( )21 2 2 2 2 2 2 2 2

(0

2

)

H H H Max H H Min HHw F g f QdF

dQ f

fλ δ λ δ

−≤

− + ′ ∆ − ∆

, (24)

W∆ ≈nonroad ext. 1s×∆ + ( )1 2 2 1 1 1 1 1 1

0

2

1

B B B Max B B Min BBw F g f QdF f

fQ

dλ δ λ δ

≤

′ ∆ − ∆ − +

. (25)

1s∆ is the discrete change in zone 1’s residential land share. The other notations are interpreted

in the same way. Because we have fixed the number of working days ijd at d , zone 1’s traffic

volume is fixed at 1F N≡ at each working day. This is equivalent to the standard urban

model’s setup, in which traffic volume is fixed at N at the CBD point. Because 1F is fixed,

the abovementioned welfare changes do not contain the term 1 d 0/dF φ = , 1 11 ,{ , ,B Hf fsφ ∈

2 }Hf .

Suppose that more land is allocated to residential use in zone 1 (that is, higher 1s with

1 0s∆ > ). Then, the residential land rent there will fall and the business land rent will rise with

1 1 0H Br r− < . In this case, 11 1( ( ) ( )) 0H B sr r− = − × + <∆ , so this land share adjustment is accompanied by the welfare loss from zone 1’s land market. When the residential share in zone 1 is adjusted downward instead, we continue to have 11 1( ( ) ( )) 0H B sr r− = + × − <∆ . Similarly, the last terms in (23)–(25) are easily shown to be always negative for a small adjustment of the LURs. The intuition is that every binding regulation forces the producers to adjust their operation in a direction they do not want, so that there must arise compliance costs for every binding regulation. Suppress the first terms in (22)–(25) as in the existing LUR literature dealing with traffic congestion only. Because the last terms are all negative in (22)–(25), the only way the LURs can improve welfare W is to make the second terms in (22)–(25) positive. This immediately implies, for welfare enhancing, 1 0Hf∆ > from (23), that is, taller residential

structures at the city center (a minimum FAR regulation). In addition, it implies 2 0Hf∆ < from (24), that is, shorter residential structures at the suburb (a maximum FAR regulation). Here, we assume that taller residential structures in the CBD would induce people to relocate from the suburb to the city center, and that shorter residential structures in the suburb would push people from the suburb to the city center, so that the suburb would be less congested with

16

less traffic 2F (i.e., 2 1/ 0HF f∂ ∂ < , 2 2/ 0HF f∂ ∂ > ). This is the economic rationale of the minimum FAR regulations in the CBD combined with the maximum FAR regulations in the suburb in the monocentric literature on metropolitan-wide traffic congestion (e.g., Pines and Kono, 2012). Our theory adds a qualification to this prescription. The prescription is valid only when such an adjustment does not cause the service quality of nonroads at the city center to deteriorate (see the first terms in (19)–(21) and (22)–(25)). When such adjustment congests the CBD too much, it could be even better to decentralize activities from the CBD to other zones, as in Anas and Rhee’s (2007) case of growth boundaries. In this way, our model reconciles two conflicting prescriptions: on the one hand, Kono et al. (2012) and Pines and Kono (2012), who recommend higher FARs (i.e., minimum FAR regulation) at the CBD to control traffic congestion; and on the other hand, Kono et al. (2010), who recommend lower FARs at the same CBD (i.e., maximum FAR) to curb “population” externalities. Optimal adjustment of land uses

This observation seems to suggest that in real-world cities in which road and nonroad congestion is intermingled over a metropolitan area, there is no predetermined direction of adjusting market FARs for welfare improvement. However, we can do a little better. Convert the first-order derivative, say, (19) into a new formula composed of elasticities and shares of external costs as follows:

Total transp.cost of

Agg. landrent of z zone one

,k ii i i

i ki

H Bk i k ii i

i ik i k i k

k kk

i k kk

s

s dH s dBdW EC ECds H ds B ds

s ARALRA

gw F gg s LR s

∂

= +

∂

− + ∂ ∂

∑ ∑

∑

(26)

where ( )/ /H X X Hi i i i i i i i iiEC p x S H N S H Hυ∂ ∂ + ∂ ∂≡ is the external cost present in zone i’s

nonroad infrastructure due to the marginal adjustment in housing stock iH ; BiEC is the

external cost present in zone i’s nonroad infrastructure due to the marginal office stock change

iB ; and [ (1 )]H Bk k k k k kALR r s r s A+ −≡ is the aggregate land rent of zone k. The coefficient

terms before the elasticities in (26) play the role of relative weights when the elasticities are added over all markets and externalities. The FAR regulations have a similar expression.

Total transp. Total co

Floor ar

mpliance cost ofcost in zone zone 's housing builde

a

s

e

r

(.) H Bi i

i i

Hii i i k

i

H Hk i k i

H H Hk i k i k

H HH Hk kk kH H

i k ki k

EC ECf dH f dB

f g fw F g f Q

dWd

g f f Q

f H df B df

λ

+

=

− +

∂∂

∑ ∑

∑

( )Hk

Hk

Hk

Hk

f Qf

∂∂

(27)

17

We can derive a similar formula for the FAR regulation of office buildings, Bkf .

Summary 4 (Optimal adjustment of LURs) The planner should adjust the LURs, , ,H B

k k kf f s , so as to make (26)–(27) positive, while assigning more weight to the terms with higher cost shares and higher elasticities.

In summary, Pigouvian tolls and the property taxes mimicking Pigouvian tolls are only

second best because they cannot consider differences in MUIs, and because property taxes fail to account fully for traffic congestion. In addition, land-use regulations entail compliance costs and, thus, cannot be first best even if they were to account for household heterogeneity. Considering the sheer complexity of the model, we now turn to numerical exercises. Table 1 Reference parameters --------------------------------------------------------------------------------------------------------- Geography and Population

Zone 1 & 5: 7 km, Zone 2 & 4: 5 km, Zone 3: 4 km N = 1.2 million persons (2 dependents/household) Population density: 14.0 persons/hectare on average (endogenous in each zone)

Production X-good producers: 0.8µ = (labor cost share),

1 0.2µ− = (land cost share) Builders of housing and office buildings: Land cost share = 30% H Bρ ρ= = -0.923 (elasticity of factor substitution=0.52)

Hα = 0.875, Bα =0.915 Household-workers

Household income = $50,000/year Housing expenditure = 30% of the household income Utility function: α = 0.4, β = 0.6, Uρ = -0.786, Uα = 0.475 Time endowment T = 500 hours/month

Number of workdays: 20.8d = days/month ζ = 6 (dispersion parameter)

--------------------------------------------------------------------------------------------------------

4. Numerical Examination 4.1. Calibration

We examine a hypothetical metropolitan area, linear in shape, which accommodates a population of 1.2 million in a fully circular nonmonocentric metropolitan area. The population

18

density is 14 persons/hectare. The population is smaller than mid-sized American metropolitan areas and density is set accordingly. We use a Cobb–Douglas function for the X-good producers,

1Xi i i iX S M Bµ µ−= . Housing builders produce housing according to the technology

( ) ( ) ( )1/

1H

H HHHk kk H

HQ XHρρ ρ

α α = + − .

Office buildings kB are produced similarly to housing. We use the utility function 1/

( l nln 1 ) n lUU U

ijH

ij U ij U ij iu z h l Sρρ ρα α α β = − + + + .

With no harm to the major point of the study, we set the number of workdays ijd at 20.8 days

a month.

Table 2. Technical details of the simulations (a) Section 4.2

City type Road budget Non-road budget Road tolls Prop. tax rates

Tolled City Road tolls exactly

cover road budget in each zone.

Prop. taxes exactly cover non-road

budget in each zone. Endogenous1

PT City Prop. tax rev. = sum of metro-wide

road and non-road budgets No road tolls Endogenous

Zoning City, LS only Financed with head tax Neither road tolls nor prop. taxes

Note: Capacities of road and non-roads in the last two rows = those of the Tolled City (b) Section 4.3

City type Road budget Non-road budget Road tolls Prop. tax rates

Tolled City Not balanced Prop. taxes exactly

cover non-road budget in each zone.

Policy variable

Endogenous1

PT City Financed with head and prop. taxes; prop. taxes ≠ sum of road and non-road budgets.

No road tolls Policy

variable Zoning City,

LS only Financed with head tax Neither road tolls nor prop. taxes

1 Although nonroad capacities are fixed here, land rents and the prices of capital inputs are not fixed in our general equilibrium model. So, property tax rates are “endogenous” to cover variable input costs.

We adjust the cost shares and elasticities of substitution according to empirical studies and consumer expenditure surveys (Koenker, 1972; Shoven and Whalley, 1977; Polinsky and Ellwood, 1979; McDonald, 1981; Thorsnes, 1997). We specify the service

19

qualities of infrastructure as XiS = ( )/ [ ] X

X i i ia K B H δ+ and ( )/[ ] HHi H i i iS a K B H δ= + . In

line with Yeoh and Stansel (2013), we choose Xδ from [0,0.11]. More problematic is to set

the coefficients of the function HiS . We resolve the difficulties in setting the coefficients

,, ,X HX Ha aδ δ in such a way that the uniform property tax rate of 0.95%, as applied to the stock value of properties, covers the cost of infrastructure (roads and nonroads together). We use a discount rate of 5% to convert tax rates into flow rates. We use the Bureau of Public Roads function for the congested travel time, ( , )i ig F R . Table 1 contains the parameters used for the simulations. 4.2. Efficiencies when optimal policies are known

We aim to compare the efficiencies of LURs, congestion charges, and uniform property taxes. To this end, we compare four types of city: Zoning City, property tax (PT) city, Tolled City, and Base City. The Tolled City sets congestion charges for congestion of roads and nonroads by fully internalizing the associated external costs while balancing the road and nonroad budgets in each zone separately. In the model with no user heterogeneity, this scheme is not only self-sufficient but also first best. All other types of cities use these road and nonroad capacities in order to control for any experimental noise that might be introduced by differing infrastructural capacities. In the PT City, traffic congestion is not priced and a single uniform rate is applied to both residential and business properties. In addition, the single rate is applied uniformly, irrespective of the location of a structure. This rate is set so as to precisely cover the combined expenditure for roads and nonroads in the metropolitan area. In the Zoning City, the planner knows the optimal LURs and adjusts land shares is and FARs ,H B

i if f in each zone to maximize welfare (4). The expenditure for roads and nonroads is financed with head tax. The Base City is the laissez-faire city where LURs, congestion pricing, and property taxes are not available. Table 2(a) summarizes the major features of the cities.

Figure 1(a) shows the welfare gains that various types of city attain over and above the Base City’s welfare. Zoning and property taxes are shown to be more efficient than, as efficient as, or less efficient than tax instruments (i.e., the Tolled City), as metropolitan population increases. This means two things. First, zoning is almost as efficient as Pigouvian tolls, which coincides with the existing literature. For example, Kono et al. (2012) show that FAR regulations in the monocentric city achieve 80% of the welfare gain obtained by the first-best city. According to Rhee et al. (2014), land share adjustment together with production subsidies achieve 99% of the first-best instruments’ welfare gain. However, the proposition that zoning is almost as efficient as Pigouvian tolls is valid over only a narrow range of metropolitan population.

Second, it is surprising that Pigouvian congestion charges (i.e., the Tolled City) could be less efficient than zoning, which has not been reported in the literature. This cannot occur in the model with no heterogeneity.

20

Table 3. Land use patterns of the Base and Zoning Cities (reference population) (a) Land use

Zone 3 (CBD) 4 5 Business

land Base City 0.15 0.32 0.49 Zoning City 0.14 0.30 0.46

Residential Land

Base City 0.39 0.54 0.50 Zoning City 0.40 0.56 0.53

Note: In the Base City, roads’ land share at the CBD = 1− (business land 0.15 + residential land 0.39) = 0.46 (=46%). (b) Floor area ratios

Zone 3 (CBD) 4 5 Business buildings

Base City 0.69 0.49 0.17 Zoning City 0.70 0.47 0.15

Residential buildings

Base City 0.66 0.45 0.15 Zoning City 0.68 0.43 0.13

Note: The entries for the Zoning City are optimal LURs, strictly binding to every builder and landowner.

Figure 1 Welfare performance

(a) Welfare gain (b) Component welfaresof the Zoning City

b

Welfare change(unit: utils×10,000)

-15

-10

-5

0

5

10

15

20

0.0 0.5 1.0 1.5

Systematic CovarianceSystem.+Cov. Real est. mkt

A

B

C

Millions

$/yr/household

0

200

400

600

1.2 1.5 1.8 2.1 2.4

Tolled City PTBal CityZoning City LS onlyFirst-best

(ref. pop)PT City

Tolls+income transfers

φ

The covariance term explains the apparent anomalies in Figure 1(a). To see this more

clearly, we create a straight path of the LUR policy changes, parameterized by φ , from the

21

Base City to the Zoning City, and path integrate the welfare function along this straight line using (19)–(21) (Yu and Rhee, 2013). In Figure 1(b), 0φ = is the Base City, and 1φ = is the Zoning City. The curve “Systematic” is welfare explained by the first four terms in concert in (19)–(21); the curve “Covariance” is welfare explained by the covariance terms in concert in (19)–(21). The covariance part explains 38% of the welfare gain of the Zoning City over the Base City (i.e., BC/AC=0.38). This share is astonishingly large and is shown to increase even further as population increases.

How do the anomalies arise? In smaller cities with lower externalities, zoning improves on redistribution effects in the presence of household heterogeneity while market distortions are relatively small. With increasing city size causing higher externalities, zoning becomes less and less efficient compared to Pigouvian tolls because redistribution effects become relatively less important and land market distortions or compliance costs increase fast (the dotted curve in Figure 1(b)). The same applies to property taxes. The findings even hold true for a package of zoning policies that might be much more suited to internalize the spatially differentiated types of externalities than any single zoning instrument.

As a reference, we also report a zoning city in which only land shares is are adjusted. The “LS only” curve represents that city in Figure 1(a). Comparing the Zoning City curve with the LS Only curve suggests that the efficiency gain of the Zoning City arises mostly from FAR regulations. The “Tolls+income transfers” represents the city in which Pigouvian charges and income transfers are used together. Figure 1(a) simply shows that any policy not considering household heterogeneity would fail to improve welfare much.

4.3. Efficiencies when optimal policies are not known Setup

In Subsection 4.2, the planner could pinpoint an optimal policy (first-best or second-best policy). In reality, however, the instrument choice and actual implementation are not free of hindrance and complications. Authors cite various types of constraints: imperfect property rights, multiple externalities, market power, unobservable behavior, imperfect information, administrative capacity, and politics (Fischel, 1985: ch.10; Bennear and Stavins, 2007). Further issues are discussed in the political economy of instrument choice (e.g., influence of lobbying groups by Grossman and Helpman, 1994; overview related to transport by Hepburn, 2006). Consequently, the instrument level chosen is likely to deviate from the theoretic optimum. In this subsection, we consider this issue and explore its consequences for the optimal instrument design.

To evaluate the welfare impact of deviations from optimal policies, we make the experimental setting more realistic. Imagine two cities each with 1.2 million inhabitants. The first city, Tolled City, plans to introduce second-best road tolls with Pigouvian property taxes that are already well instituted. The road capacities in this city are the same as those of the Tolled City in Subsection 4.2. Next, we vary road tolls around the second-best optimal tolls while continuing to adjust the Pigouvian property taxes to balance nonroad budgets in each zone. Because tolls do not necessarily balance local road budgets, the welfare of this subsection’s Tolled City is higher than the welfare of the Tolled City in Figure 1(a) with road

22

and nonroad budgets all balanced in each zone. The second city, PT City, tries to improve on the existing property tax system and does

not charge road tolls. This type of city is observed more commonly in the real world and mirrors what local governments can do in the real world. We inherit road and nonroad capacities of the Tolled City in the Subsection 4.2 and vary the spatially uniform tax rates for housing Hτ and business buildings Bτ to maximize welfare. In this exercise, we do not require property tax revenues to balance the nonroad infrastructural budgets. The intention is to check the sensitivity in the setting with a minimal degree of constraint on the metropolitan government’s discretion in the use of property taxes. With this additional degree of freedom, we can focus better on making the central point of this subsection. Table 2(b) summarizes the major features of the city types examined in this subsection. Theory

To measure welfare loss accompanied by the deviations from optimal policies, we parameterize the congestion charges by 0b ≥ , where 0b = corresponds to the Base City and

1b = is the Tolled City. Similarly, we parameterize the LURs by 0φ > , where 1φ = corresponds to the Zoning City and (0,1)φ ∈ corresponds to the Base City.4 We perturb ,b φ around 1 and denote the welfare change by TolledW∆ for the Tolled City and by ZoningW∆ for the Zoning City. By construction, 0TolledW∆ < and 0ZoningW∆ < . We express these differentials up to the second order of the parameters to obtain

22

21

1 ( )2

TolledTolled

b

d WW bdb

=

∆ ≈ ∆ , 2

22

1

1 ( )2

ZoningZoning d WW

d φ

φφ

=

∆ ≈ ∆ , (28)

where ,Tolled ZoningW W are welfare of the Tolled and Zoning Cities, respectively. We collect the terms from (28) under the rubric of B′′ for benefits and C′′ for costs to obtain the formulas containing second-order terms similar to Weitzman (1974) and Laffont (1977). Implicit in our exercise is that uncertainties have been introduced into the policy instruments and that welfare is given for certainty. This means that welfare is ex post welfare. Unlike Weitzman (1974), this twist greatly facilitates the exposition, as we shall see.

There are two reasons why (28) does not help measure the welfare cost of deviations much. First, the general equilibrium nature of our framework is too complex to provide an unambiguous sign from analytics. Second, the policy intervention is rarely marginal. Rather, it is discrete, and so, the local approximation (28) working only near the optimums could be completely wrong for the discrete changes introduced by the Zoning and Tolled Cities. Therefore, Weitzman and Laffont’s second-order formulas, which hold only around the optimums, are not much help in obtaining the welfare differential with discrete changes. We have to rely on alternative ways to evaluate derivations from optimal policies (see Pizer, 2002,

4 The usage of φ here slightly differs from its usage in Figure 1(b). In Figure 1(b), the Base City corresponds to 0φ = .

23

for mitigating climate change). Once again, the first-order welfare changes (12), (19)–(21) show a way to get around

this difficulty. One critical difference between (12) and (19)–(21) is that (12) has two terms, one positive and the other negative, mutually offset inside parentheses, but (19)–(21) do not. Because the terms inside parentheses of (12) cancel out, the welfare change will be small in association with a given change in b, b∆ , away from the optimum ( 1b = ). Note that (12) holds at any equilibrium near or far from the optimum. Figure 2(a) is a graphical rendering consistent with this observation, in which the marginal benefit (MB) curve is located close to the marginal cost (MC) curve.

Figure 2 Deviation from optimal policies

(a) Congestion tolls (b) Land use regulations

$

MB

MC

Optimal(b=1)

Tolls

$

MB

MC

Optimal(Φ=1)

LURs

The case of LURs is not that simple. However, they have an additional term, which is

missing in pricing instruments, showing the distortionary cost of real estate markets. When we move away from the Zoning City to the Base City, this term appears as positive. In the other cases where LURs are binding, this term will appear as negative. This means that when the deviation is discrete (and so, the LURs are binding) this term appears as negative eventually. On the other hand, the other terms (the first three terms in (19)–(21)) measure the overall welfare gain from congestion relief on the way from the Base City to the Zoning City. Conversely, therefore, these terms appear as negative when we move the Zoning City toward the Base City. Otherwise, the sign appears as indeterminate and will depend on the specific manner of deviation.

Therefore, when the change is sufficiently large, the second-last term is likely to have a bigger role, making the overall welfare change of discrete deviations negative. Indeed, Table 3 shows that the Zoning City only slightly perturbs the Base City, and the dotted curve in Figure 1(b) indicates that when the deviation is large, the welfare cost by the second-last terms in (19)

24

–(21) will soon overwhelm all the other effects. Figure 2(b) is a graph consistent with this interpretation. The two curves are located far apart compared to Figure 2(a). Again, note that (19)–(21) hold at any point near or far from the optimum. Once we have a figure resembling Figure 2, we are likely to obtain a larger welfare loss in the Zoning City than in the Tolled City for the same given deviation b φ∆ = ∆ . Simulation

We aim to validate the insight provided by (12) and (19)–(21), or equivalently Figure 2. Indeed, Figure 3 confirms our conjecture. It shows new profiles of welfare gain over the Base City under the reference population of 1.2 million. On the horizontal axis, 1.2 means that policies were set 20% higher than the optimal values. The curve labeled “LS only” is the Zoning City, in which only land shares are perturbed, while FARs are fixed at the levels of Subsection 4.2. Conversely, if we perturb the FARs only and hold land shares fixed at the levels of Subsection 4.2, we obtain a shape closely following the “LS only” curve. So, we omit the “FAR only” curve in Figure 3. The city types relying on the tax instruments rarely experience large welfare losses compared to the second-best optimal welfare (approximately $220). It does not matter much whether the planner fails to pinpoint the optimal tax policies. In contrast, when the planner is wrong about the zoning policy, the cost is huge. Because a small deviation could bring about a strictly negative net benefit, zoning could be even worse than doing nothing.

-800

-600

-400

-200

0

200

400

0.4 0.6 0.8 1 1.2 1.4 1.6

Tolled City PT City Zoning City LS only

Figure 3 Sensitivity of welfare under imperfect information

$/year/household

(FAR+LS)

b

Note: The graphs were drawn under the reference population of 1.2 million.

One problem with Figure 3, however, is that all the policy instruments are inflated or

deflated by the same ratio to obtain the graphs. To explore whether the abovementioned outcome arises from this error proportionality, we conduct another experiment. This time, we

25

allow actual policies to deviate from the optimal policies by setting differing ratios. As pricing instruments are extremely insensitive to deviation (e.g., policy failure or information errors), it is sufficient to check the sensitivity of the LURs against the deviation. Because the city is nonmonocentric with no obvious city center, we continue to assume that the zoning adopted is binding to every builder in every zone, unless it happens to coincide with the Base City’s land-use pattern.

Figure 4 Welfare loss of imprecise zoning

(b) Varying population

$/yr/HH

0

100

200

300

400

500

1.2 1.5 1.8 2.1 2.4

Avg welfare loss Std. dev.

Millions

$/yr/HH

Imprecision of zoning

-1100

-900

-700

-500

-300

-100

100

300

0 3 6 9

(a) Welfare gain and loss

−$148 (avg. of the left panel)

Avg = -$148

Now, we introduce discrete random variables , 1, 2,3i iθ = that could be either positive

or negative. We denote the optimal LURs by is∗ (residential land share), Hif

∗ (FAR of

housing) and Bif∗ (FAR of office buildings). Lacking the precise information of the second-

best LURs, the planner adopts actual land-use controls as follows:

1)(1i iss θ ∗= + , 2(1 ) HHi iff θ ∗= + , 3(1 ) BB

i iff θ ∗= +

Pr( iθ =0) = Pr( iθ =0.07) = Pr( iθ =−0.07)

= Pr( iθ =0.15) = Pr( iθ =−0.15) = 1/5 for 1, 2,3i = .

The random variables are uniformly distributed with equal probability of 1/5. 5 Note that without knowing the exact values of the LURs, this distribution remains conservative. To set

5 This uniform distribution is not new at all. Refer to Pizer (2002: 415, 417) for a similar practice in estimating the costs and benefits of climate change mitigation.

26

the random variables iθ , we toss three dice, each die for each iθ , calculate the associated welfare change from the Base City, and measure the imprecision of zoning, which is defined

to be ( )2 2 2100 ( ) ( ) ( )H H B Bi i i i i ii

s s f f f f∗ ∗ ∗− + − + −∑ . We repeat this independently 20 times.

In Figure 4(a), we plot the 20 welfare changes resulting from this experiment against the degree of regulatory imprecision. Welfare is higher than the Base City in 7 trials (dots above the x-axis) and lower than the Base City in 13 trials (dots below the x-axis). Although the planner is correct on average, without the precision he loses on average $148 (Figure 4(a)) instead of winning more than $200 (Figure 3).

Next, we vary the metropolitan population N , perform the same experiments leading to Figure 4(a), and tabulate the average welfare losses. Figure 4(b) is the result. We calculate the standard deviation of each experiment composed of 20 independent trials and superimpose the standard deviations on the average welfare losses. As in Figure 4(a), the planner always loses on average; the average loss and standard deviation increase as the metropolitan area becomes more congested. We may narrow the range of the values that iθ takes on or we may

use different (symmetric) distributions for iθ . Nonetheless, unless the range is much smaller than 0.15± , we obtain essentially the same results of welfare loss.

Summary 5 1. Depending on policy accuracy, the efficiency ranking of the LURs and tax

instruments varies greatly. 2. When optimal policies are known, zoning could be more efficient than, as

efficient as, or less efficient than Pigouvian tolls. That is, in contrast to the finding in the theoretic literature that zoning is very efficient (Pines and Sadka, 1985; Wheaton, 1998; Kono et al., 2012; Rhee et al., 2014), zoning could be far less efficient than pricing instruments. Conversely, zoning could be more efficient than congestion charges.

3. Policy decision-making and the imperfectness of regulatory information needed for LURs or taxes might cause nonoptimal choice of zoning and taxes. The welfare costs of suboptimal zoning are so huge that zoning is strongly adverse compared to Pigouvian tolls and property taxes. It is even possible that zoning is worse than doing nothing at all.

The experimental results of the LURs coincide with what has been reported in the

literature on metropolitan-wide development controls. For example, building height restriction in Bangalore, India has resulted in a loss of 1.5% to 4.5% of household income (Bertaud and Brueckner, 2005). Similar welfare costs are reported for the greenbelts around metropolitan areas in Seoul, Korea by Lee (1999) and in the UK by Cheshire and Sheppard (2002). One persistent difficulty is that it is not easy to set land use regulations optimally in the real world. The empirical study simply shows this well-known fact.

27

Instrument choice in pollution control and city planning

The formulas (12), (19)–(21) should not come as a surprise, because they are conceivable from Harberger (1971) and are related to the traditional view of instrument choice in environmental economics. Nevertheless, these formulas considerably extend the discussion of existing instrument choice literature; they provide a global platform that extends the instrument choice theory of pollution control to various types of externalities and to a wide range of policy deviations beyond cost-benefit uncertainties.

Strictly speaking, Weitzman’s (1974) famous formula B C′′ ′′+ holds only near the optimums. Many authors adopt this local formula to differentiate the global superiority of different policies (Hoel and Karp, 2002) or resort to graphs to draw global implications and conclusions (Stavins, 1996; Kaplow and Shavell, 2002). Pizer (2002) warns the danger of this practice. He calculates that the welfare differential between prices and quantities is about five times larger with the linearization of Weitzman in contrast to the application of a full welfare analysis.

This practice simply testifies to the difficulty in deriving formal expressions applicable to a discrete policy change. In the process, uncertainties of benefits and/or costs are introduced, only to have the same local characterization of the welfare differential between prices and permits (Laffont, 1977). In contrast, we examine deviations from optimal policies that might be caused by quite a number of different reasons found in the planning and decision process, such as lobbying or information failures. The trick was to incorporate the uncertainties or imprecision into the policies rather than into the costs and benefits of pollution control.

Let us illustrate how nicely (12), (19)–(21) explain the superiority of prices to quantities in the instrument choice literature whose setting is much simpler than ours is. Again, Pizer (1997: 2) reports essentially the same simulations as ours in controlling greenhouse gases and summarizes that “slightly more stringent targets lead to dramatic welfare losses.” This is not a coincidence. Pizer sets the tax price equal to the almost constant marginal benefit of pollution abatement (Figure 5 in his report, a special case of our nonlinear marginal harm schedules). In addition, permits are bound to entail distortionary costs under uncertainties (the second-terms of (19)–(21) in our setting). Thus, the welfare calculus in Pizer’s study has essentially the same mathematical structure as our study of the cost–benefits associated with the two competing instruments, although the mathematical structure is not explicit in his study.

Now, since the instruments solving the equation (12)=0 are first best and his pricing always equals the true marginal damages due to the flat marginal harm schedule, the flat pricing should work better than permits in Pizer, even under (cost) uncertainties. Because quantities entail not a small cost of market distortions, as the second-last terms in (19)–(21) show, the welfare differential should be large between pricing and permits. In other words, the same cost–benefit calculus applies to Pizer’s and our studies; it is natural that pricing works much better than permits in his study and climate control literature in general with stock pollutants, when either one of the instruments is to be used.

Consequently, although (12), (19)–(21) are intended to be discerning for nonstock externalities, they turn out to apply equally well to stock pollutants. Our theory and simulations

28

show that these findings from environmental policies in principle carry over to congestion policies in a very different spatial framework that considers various nonstock externalities, land use, and heterogeneous households. It is in this sense that (12), (19)–(21) and Figure 2 provide a global platform that extends the instrument choice theory of pollution control to various types of externalities and to a wide range of policy deviations beyond cost-benefit uncertainties.

Summary 6 1. The responsibility to control externalities differs between owners of

individual facilities in the case of pollution control and public authorities managing infrastructure in our case. However, similar economics is at work, so that (12), (19)–(21) are equally applicable to the instrument choice in both environmental economics and city planning.

2. The superiority of prices to quantities in controlling externalities stems from the cost–benefit structure of welfare changes, which is uniquely associated with each type of policy.

5. Conclusion We found that some of the well-known results on congestion policies no longer hold if there are heterogeneous households, multiple land uses, and different externalities. It turns out that redistribution effects can be so large that any policy not considering redistribution is far from being first best. This result demands further research to explore whether household heterogeneity has similarly strong implications on optimal policy design for other regulation issues. Furthermore, our result that even in a first-best framework Pigouvian taxes and redistribution to equalize marginal utilities of income do not provide a first-best solution might be relevant for other cases in which re-sorting across heterogeneous household types is feasible (e.g., heterogeneity according to education status, family status, and income type).

The findings concerning zoning are surprising. The finding from the literature that zoning is almost as efficient as Pigouvian tolls is valid only for a narrow range of city population. Furthermore, zoning could be even better than congestion charges. Such findings are not yet known in the literature. The findings point to the significance of redistribution issues when household heterogeneity is present. Then, some of the generally accepted findings are challenged. Whether this holds true for other regulation policies is an open issue.

As policies might not be set optimally for various reasons, such as imperfect information, we provide an evaluation of the efficiency losses of deviations from optimal policy. We conclude from this exercise that enacting zoning is a very risky undertaking because this regulation policy could worsen welfare below the no-policy case with congestion. This issue is serious, considering that zoning is a standard instrument in city and regional planning.

Of course, land-use regulations usually have other objectives too, such as separation of incompatible uses. Our study is limited in this respect. However, even then, the efficiency problem remains and might offset all or part of the benefits that land-use planning intends to reap. The strong distortion that zoning imposes on real estate markets is the reason for the high

29

welfare costs of small deviations from optimal regulation. The same distortionary cost will continue to work in models with other types of externalities, notably, production and consumption externalities, and in models of government competition among municipalities.

We expect that this vulnerability of land-use regulations to nonoptimal choice is relevant to policy of other types of regulation applied to various markets. This is already known with respect to misperception (e.g., Kaplow and Shavell, 2002). However, in our study, any deviation for any reason from the optimal regulation causes this robustness problem of the instrument. Accordingly, further research is needed on the local robustness of regulatory instruments around the optimum. We are confident that this issue is replicated in other markets and regulatory policies. Therefore, it could be a very common issue and should be considered when exploring optimal instrument design. Acknowledgement This work was supported by the National Research Foundation of Korea Grant funded by the

Korean Government (NRF-2014S1A5A2A01011212). Authors thank Professor Tony Smith at

the University of Pennsylvania for his help in elaborating the idea of the envelope methodology

of land use-transportation models.

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33

Appendix 1. Derivation of the first-order welfare change

Form the Lagrangian of the household’s utility maximization problem.

( ) ( )(.) (8 )) (1 (8 )M X H Tij ij j ij ij ij i ij i ij

Hi ij ij ij ijL u c w t d D p x p h c H g d lτ= + − + − − + + − + −

We calculate each term of (11) one by one. An example follows.

( ) ( )(.) (8 ) (1 (8 ))M X H Tij ij j ij ij ij i ij i ij ij ij ij ij

HiX

n

ijXn

ijM M nij ij ij ijX

nin in

u c w t d D p x p h c H g d lp

Vp

D Kc x c xp N

τ

δ δ

∂+ − + − − + + − + −

∂

∂ = ∂

∂ − = − = − ∂

We have applied the envelope theorem to have the formula. After calculating this type of de-rivatives, we substitute them into (11). The first seven terms of (11)

1,2 1,2

X H Bij j ij ij ijn n n

ij ij ij ijX H Bij ij n ij n ij nj k n k n k n k

V dw V V Vdp dp dpP P P Pw dt p dt p dt p dt= =

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂∑ ∑ ∑ ∑ ∑ ∑ ∑

1,2 1,2 1,2

(8 )

ij ij ijn n nij ij ij

ij n ij n ij nn k n k n k

XjM M n n

ij ij ij ij ij ijij ij

in

nk k

V V Vdr dH dBP P Pr dt H dt B dtdw K dpP c d P c xdt N dt

δ

= = =

∂ ∂ ∂+ + +

∂ ∂ ∂

− = + −

∑ ∑ ∑ ∑ ∑ ∑

∑ ∑ ∑

( )1,2 1,2

(1 )M MH Bij H H B

n n i in ij nijn n

ij ijij n ij n

nk k

c cdp dpP NH h BPN dt N dt

τ τ δ τ= =

+ − + +∑ ∑ ∑ ∑

( )1,2

Mij n

iji

nj k

nn

c drP AN t

Rd=

−+∑ ∑

1,2 1,2

M MH Hij ij ij iji n i n

ij ijH Hij n ij n

H H B Bin n n in n n

i i k i i k

u c u cS dH S dBP PS H N dt S B N d

p pt

δ τ δ τ= =

∂ ∂∂ ∂+ + + + ∂ ∂ ∂ ∂ ∑ ∑ ∑ ∑

( )

1,2

(

)

8 )

(1

X XjM M M M Mn n n

ij ij ij ij i inj ijij n ij nk k k

M H H BMij nH H B

n n i in ij n nn

n nij

ij n nk k k

dw K dp dpP c c c d c P c xdt N dt dt

c dp dp dpcP NN d

H hdt

Bt N dt

δ

τ τ δ τ=

= − + − −

+ − + +

∑ ∑ ∑

∑ ∑

∑

∑ ∑

( )1,2

MHMij ijn i n n

ij Hn ij nk i i

H Hn n

k k

MHij iji n

in n nn

B Bin

nij H

ij n i k kn

nin

u cdr S dH dHc A PN dt S H dt N dt

u cS dB dBPS B dt N d

R

pt

pδ τ

δ τ

=

∂ ∂+ + + ∂ ∂

∂ ∂+ + ∂ ∂

−

∑ ∑ ∑

∑

∑

∑ ∑

Using (3) and the market equilibrium conditions, we can simplify the above as follows: The first seven terms of (11)

34

X H X HM MX Xi i i i i ii i i i i i

i ii i k i i

B B Hi i

k

Hi i i ip

S S dB S S dHc cp x N p x N

N B B dt N H H dtpυ τ υ τ

∂ ∂ ∂ ∂+ + + + +

∂ ∂ ∂ ∂

=

∑ ∑

+ ( ) ( )(8 )X

jM M M M iij ij ij ij ij ij

ij ijk k

dw dpP c c d P c c xdt dt

− − −∑ ∑ - ( ) ( )1 HH

M M iij

ki

iji jij h dpP c c

dtτ− +∑

+ ( ) ( )H H

M M M Mi i i iij ij ij ij

ii

j ijii

kj

kj

i

S dH S dBP c c P c cH dt B dt

υ υ∂ ∂− + −

∂ ∂∑ ∑ (29)

In the similar fashion, we can calculate the last term of (11).

The last term of (11) ijij

ij k

VP

t∂

=∂∑

1,2

1,2

Mij ijM Mi

ij ij k i ij ij ijij i k k

M Mij ij ijM Mi

i

ij mlk ml ij

ml k

ij mlk mj ij ij k i ij ij ij

ij ij i ijkij

k kl

ml

c gFP Nd F t c c dN t t

c c gFP Nd P F t c P c dN N t t

P y wt

P y wt

δ

δ

=

=

∂ ∂= − + + − − ∂ ∂

∂ ∂= − + + − ∂ ∂

∂∂

∂−

∂

∑ ∑

∑ ∑

∑

∑∑ ∑

( )

( )1,2

1,2

1 MM M M i

ij ij ij k iij i k

i

ijk

mlml ij

m

jM M M Mij ij ij

ij k

M MijMi

ij ij k i ij ijij i ijk

l k

Mij mlk ml ij

ml kk

FcP c c c Nd F tN N t

gc P c c c d

t

gFc cP Nd F t

P y wt

P cyc NP dN N t t N t

w

δ

δ

=

=

∂−

∂

∂

∂= − − + + + ∂

∂− − +

∂

∂ ∂= − + + − ∂ ∂

− ∂

∑ ∑

∑

∑

∑

∑∑ ∑

( ) ( )1 ijk

ijM M M Mij ij ij ij ij ij

ij ijij

k

gP c c Nd NP wc c d

N tδ

∂− − − −

∂∑ ∑

( )1,2

Mi

ii

ijMi i i ij

i kk j

F Pw F g c yt

N t tc

=

∂= − − ∂

∂

∂′ −∑ ∑ (30)

( ) ( ) ijM M M Mij ij ij ij ij ij

ij ij k

ijk ij

gP c c d NP c c

tw dδ

∂− − − −

∂∑ ∑

Combine (29) and (30) to have the desired formula. We can derive the derivatives of the other policy variables in the similar fashion. Appendix 2. Glossary

Abbreviations LUR land-use regulations; FAR floor area ratio; MUI marginal utility of in-come

Zone index , , ,i j k s

Transport ( , )i i i ig g F R≡ travel time in zone i (hour/km), ijg travel time between

zone i and j ; ,i ijt t are tolls per kilometer similarly defined; ,i ijF F

35

traffic volumes similarly defined; iR road capacity in zone i ; iK capac-ity of nonroad infrastructure such as water, sewage, power, gas, and telecom-munications that do not consume land per se

Producers iX composite goods = X-goods; iM labor input to X-good production; iB

building input to X-good production (unit: floor area); ,H Bi iQ Q land inputs

for the production of residential and commercial buildings, respectively; ,H B

i iX X capital inputs (=capital converted from X-goods by one-to-one) for the production of residential and commercial buildings, respectively; iH housing units measured in floor area; iB : business buildings measured in floor area

Household N metro population; iN zone i ’s residents; household ( , )i j : the repre-

sentative household living in zone i and working in zone j ; ijz compo-

site consumed by household ( , )i j ; ijq lot size consumed by household

( , )i j ; T time endowment; ijd number of commuting days of household

( , )i j ; ijV indirect utility function of the household ( , )i j ; W welfare

function = expected maximum utility; ijε idiosyncratic taste term for home-

work zone pairs; λ dispersion parameter of the idiosyncratic terms; ,ij iju V

ordinary and indirect utility functions, respectively; ,M Tij ijc c Lagrangian

multipliers of income and time, respectively; Mc weighted average of Mijc ’s; /T M

ij ij ijw c c≡ ; iw value of time of the travelers in zone i

Prices ir unit land rent in zone i ; iw hourly wage offered in zone i ; Xip price

of composite good = price of capital inputs to housing and business building producers; B

ip unit rental price of office floor area; Hip unit rental price of

housing floor area; iw value of time of zone i 's travelers; ijw value of time of household ( , )i j

Externalities iυ zone i households' marginal utility of nonroad infrastructure;

( , , )Xi i i iS B H K quality of non-road infrastructure in zone i as perceived by

zone i ’s X-good producers; ( , , )Hi i i iS B H K quality of nonroad infrastruc-

ture in zone i as perceived by households living in zone i

Taxes ,B Hi iτ τ property tax rates imposed on one dollar rental price of office build-

ings and housing units, respectively; it , ijt traffic congestion tolls in zone i and for the trips whose O-D is ( , )i j , respectively; ijy income transfer for household (i,j)

36

Regulations is residential land share of zone i ; ,H Hi if f maximum and minimum

floor area ratios, respectively, imposed on housing units in zone i ; ,B Bi if f

maximum and minimum floor area ratio imposed on business buildings in zone i , respectively; ,H H

i iλ λ marginal compliance cost when maximum and minimum FARs, respectively, are imposed on housing units in zone i ;

,B Bi iλ λ marginal compliance cost when maximum and minimum FARs, re-

spectively, are imposed on office buildings in zone i

37