The Effect of Transportation and Housing Subsidies on Urban Sprawl

Qiao CAI; Zou WANG; Lingling XIAO

doi:10.21078/JSSI-2018-237-12

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Journal of Systems Science and Information ›› 2018, Vol. 6 ›› Issue (3) : 237-248. DOI: 10.21078/JSSI-2018-237-12

The Effect of Transportation and Housing Subsidies on Urban Sprawl

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Abstract

This paper analyses theoretically the effect of transportation and housing subsidies on urban sprawl, modal choice decisions and urban spatial structure using a spatial general equilibrium model in a monocentric city with two transport modes. Our analysis shows that public transit subsidy leads to urban shrink, whilst subsidizing automobile and housing make the city sprawl. We also find the effects of the other factors on urban sprawl, such as households income and demand, rural land rent, the income tax rate, the total fixed cost of public transit and automobile and the travelling marginal cost of public transit and automobile. Furthermore, this paper also studies how to maximize the urban-area-wide spatial equilibrium utility level.

Key words

monocentric city / subsidies / urban sprawl / spatial equilibrium

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Qiao CAI , Zou WANG , Lingling XIAO. The Effect of Transportation and Housing Subsidies on Urban Sprawl. Journal of Systems Science and Information, 2018, 6(3): 237-248 https://doi.org/10.21078/JSSI-2018-237-12

1 Introduction

Since the 1970s, the rate of urban land sprawl in the world is generally higher than the urban population growth rate, which indicates that urban land is expanding rapidly^[1]. As the largest developing country in the world, China's urban land use is facing a severe situation in the world^[2]. The speed of land urbanization in China is greater than that of population urbanization. In the process of urbanization, especially, the continuous blind sprawl of China's big cities is one of the five most serious problems. Moreover, land urbanization is faster and the situation of land out of control is taking place^[3].

China's urban construction land has expanded rapidly, especially, the scale and speed of urban sprawl are unprecedented in the eastern coastal areas^[4]. In the 1990s, with the acceleration of industrialization, China's urbanization also entered quick development stage^[5], and large-scale population urbanization has also resulted in a large number of agricultural land conversions to urban land^[6]. From 1981 to 2002, China's urban construction land area expanded rapidly from 7, 438 to 25, 973 square kilometers^[7]. After 2000, with China's accession to the WTO and the improvement of urban development, the pace of urban construction and urban sprawl also accelerated further. However, the rapid can cause a large number of agricultural land loss^[8]. Moreover, most of the surrounding cities have the cultivated land of fertile soil and higher productivity, urban sprawl is bound to lead to loss of high-quality cultivated land and aggravates the contradiction between supply and demand of cultivated land^[9]. In addition, urban sprawl encroaches upon a large number of cultivated land and ecological land, which directly leads to a sharp decrease of quantity and quality of cultivated land and eco-environmental problems^[10]. Many domestic scholars have studied the impact of land use change and urban sprawl on ecosystem^[11-15], so the impact of urban sprawl on loss of cultivated land and ecosystem function has been confirmed. How to optimize the allocation of limited land resources and achieve the coordinated development of urban sprawl and ecosystem is a severe challenge for land use in China. On November 3, 2014, the Ministry of Land and Resources and the Ministry of Agriculture jointly released the Notice on Further Making the Work of Generally Permanent Land Use Planning, it is the first-time clear requirement for Beijing, Shanghai, Guangzhou and other 14 major cities to delimit the boundaries. There are many cities with the urban space structure expanding like standing pancake so that large area enclosure occupies a large number of high-quality cultivated land. Meanwhile, it is the measure that avoids the blindly excessive urban sprawl and protects cultivated land and ecology.

If we want to control urban sprawl effectively, we need to study determinants that drive the spatial growth of cities. Potential determinants are selected based on extensive literature review from four categories: Economic, social, demographic, and natural. Multivariable linear regression and variances partitioning are employed to identify and compare the determinants of urban sprawl and their relative importance across the 30 metropolitans. Results show that the identified determinants vary with metropolitans but present some similarities. Economic growth, industrial development, and economic structural transformation are the key economic determinants. Population and labor growth, population urbanization, and population structure act as essential demographic determinants. Primary social determinants include infrastructure construction, energy consumption, and real estate development, while dominant natural determinants are the amount and topology of available land^[16].

In literature, most of the works focus on a closely related issue between the effect of transportation subsidies and housing subsidies on urban sprawl. Subsidies can reduce commuting cost and encourage urban sprawl. Many studies proved that transportation subsidies can create urban sprawl^[1,17,18]. Those present that if commuters do not pay the full cost of their travel, they are willing to travel a greater distance and make more frequent trips. In addition, housing is the primary motivation for residents moving to the periphery^[19]. Housing subsidies can reduce the residents' housing cost so that they can afford larger houses. Therefore, larger houses will inevitably need to occupy more land resources and result in intensifying urban sprawl. Brueckner^[18] studied the effect of transportation subsidies on urban sprawl by constructing a single-mode model. Su and DeSalvo^[17] provided an extension of Brueckner's model with a two-mode model to the impact of transportation subsidies on urban sprawl. Based on these, we introduce housing subsidies into our model to analyze the effects both of transportation and housing subsidies on urban sprawl. Here, we present a theoretical analysis of the effects. Moreover, we also study how to maximize the urban-area-wide spatial equilibrium utility level.

2 Subsidies and Urban Sprawl

2.1 Urban Spatial Model

This subsection presents a monocentric urban spatial model with two transportation modes. Here, we assume that a public transit line is parallel to a highway connecting home to workplaces. The total transportation costs can be formulated as follows:

\begin{aligned} M_{i} = f_{i} + (1 - α_{i}) t_{i} x, i = 1, 2, \end{aligned}

(1)

where

i = 1

means public transit,

i = 2

denotes private car,

f_{i}

is the fixed cost;

t_{i}

is the marginal cost of travel;

x

is the distance from CBD;

α_{i}

(

0 < α_{i} < 1

), is the subsidized proportion to mode

i

We assume that

f_{1} < f_{2}

(1 - α_{1}) t_{1} x > (1 - α_{2}) t_{2} x

. Then there exists a point

\hat{x}

where traveler cost is the same regardless of transportation modes. From Equation (1), we can get

\hat{x} = \frac{f_{2} - f_{1}}{(1 - α_{1}) t_{1} - (1 - α_{2}) t_{2}}

. Obviously, public transit has lower fixed cost than private cars but with a higher marginal cost. It means that travelers living closer to CBD choose public transit and travelers living near fringe choose auto.

Total housing cost is given as follows:

\begin{aligned} H = (1 - β) r q, \end{aligned}

(2)

where

q

is land consumption;

r

is land rent;

β

is the subsidized proportion of house and

0 < β < 1

, then

(1 - β) r q

is the portion of housing cost paid by residents.

A household's quasi-concave utility function is

\begin{aligned} v = v (c, q), \end{aligned}

(3)

where

c

is nonland, non-transportation expenditures.

We assume a household subject to the following constraint:

\begin{aligned} y = θ y + c + (1 - β) r q + f_{i} + (1 - α_{i}) t_{i} x, \end{aligned}

(4)

where

y

means the household's exogenous income;

θ

is the income tax rate. The problem of the household is to maximize Equation (3) subject to Equation (4). We can get

\begin{aligned} \frac{v_{q}}{v_{c}} = (1 - β) r . \end{aligned}

(5)

We assume all urban households' income and utility function are the same. Consequently, there is no household that wants to move so that the urban is in spatial equilibrium.

\begin{aligned} v [(1 - θ) y - (1 - β) r q - M_{i, q}] = u, \end{aligned}

(6)

where

u

is the urban-area-wide spatial equilibrium utility level.

Since

M_{1} = M_{2}

\hat{x}

, and the urban boundary conditions can be formulated as follows:

\begin{aligned} r_{1} (\hat{x}) = r_{2} (\hat{x}), \end{aligned}

(7)

\begin{aligned} r_{2} (\bar{x}) = r_{A}, \end{aligned}

(8)

where

\bar{x}

is city fringe at which the land rent equals to agriculture

r_{A}

The urban population is

\begin{aligned} \int_{0}^{\hat{x}} \frac{δ x}{q_{1}} d x + \int_{\hat{x}}^{\bar{x}} \frac{δ x}{q_{2}} d x = L, \end{aligned}

(9)

where

σ

is the number of radians in a circle available for urban residential use, and

L

is the urban population. Especially, we assume that the urban population is the same as the number of the urban households in the article. The condition ensures the urban population fits the urban boundary.

Since government subsidy is limited, it should be constrained to balance countries' budgets

\begin{aligned} ϕ θ y L + G = α_{1} \int_{0}^{\hat{x}} \frac{t_{1} x^{2}}{q_{1}} d x + α_{2} \int_{\hat{x}}^{\bar{x}} \frac{t_{2} x^{2}}{q_{2}} d x + β (\int_{0}^{\hat{x}} r_{1} x d x + \int_{\hat{x}}^{\bar{x}} r_{2} x d x), \end{aligned}

(10)

where the left-hand side is urban-area government revenues devoted to urban-area transportation and residential housing, and the right-hand side is urban-area government expenditures on transportation and residential housing.

ϕ θ y L

is the share of urban-area tax revenues for transportation and residential housing, where

ϕ

is the tax share

(0 < ϕ < 1)

, and

G

is intergovernmental grants devoted to transportation and residential housing. The first term of the right-hand side is the public transit costs paid by the government. The second term of the right-hand side is the auto costs paid by the government. The third term of the right-hand side is the housing costs paid by the government.

2.2 The Effects of Transportation and Housing Subsidies

In Brueckner's original study^[18], the effect of transport subsidies is discussed in a single-model; Su and DeSalvo^[17] extends it into a two-mode model. Here housing subsidies are introduced into our two-mode model to analyze the effects of both transport and housing subsidies on urban sprawl.

To obtain the comparative static effects of these variables on land rent and land consumption, we get the totally differentiate of Equation (6):

\begin{array}{lll} v_{c} [(1 - θ) d y - y d θ + r q d β - (1 - β) r d q - (1 - β) q d r - d f_{i} \\ - (1 - α_{i}) x d t_{i} - (1 - α_{i}) t_{i} d x + t_{i} x d α_{i}] + v_{q} d q = d u . \end{array}

(11)

Substituting (5) into (11) and rearranging, we get

\begin{aligned} \begin{array}{lll} - v_{c} (1 - β) q d r & = & d u - v_{c} (1 - θ) d y + v_{c} y d θ - v_{c} r q d β \\ + v_{c} (1 - α_{i}) t_{i} d x + v_{c} (1 - α_{i}) x d t_{i} - v_{c} t_{i} x d α_{i} + v_{c} d f_{i} . \end{array} \end{aligned}

(12)

Then the following effects of change in exogenous variables on

r

can be derived from (12):

\begin{aligned} \begin{aligned} \frac{\partial r_{i}}{\partial u} = - \frac{1}{v_{c} (1 - β) q_{i}} < 0, \frac{\partial r_{i}}{\partial x} = - \frac{(1 - α_{i}) t_{i}}{(1 - β) q_{i}} < 0, \frac{\partial r_{i}}{\partial θ} = - \frac{y}{(1 - β) q_{i}} < 0, \\ \frac{\partial r_{i}}{\partial f_{i}} = - \frac{1}{(1 - β) q_{i}} < 0, \frac{\partial r_{i}}{\partial t_{i}} = - \frac{(1 - α_{i}) x}{(1 - β) q_{i}} < 0, \\ \frac{\partial r_{i}}{\partial α_{i}} = \frac{t_{i} x}{(1 - β) q_{i}} > 0, \frac{\partial r_{i}}{\partial β} = \frac{r_{i}}{1 - β} > 0. \end{aligned} \end{aligned}

(13)

According to Hicksian demand slope,

\partial q_{i} / \partial r_{i} |_{u} < 0

, we can get

\begin{aligned} \begin{aligned} \frac{\partial q_{i}}{\partial x} = {(\frac{\partial q_{i}}{\partial r_{i}})}_{u} \frac{\partial r_{i}}{\partial x} > 0, \frac{\partial q_{i}}{\partial y} = {(\frac{\partial q_{i}}{\partial r_{i}})}_{u} (\frac{\partial r_{i}}{\partial y}) < 0, \\ \frac{\partial q_{i}}{\partial θ} = {(\frac{\partial q_{i}}{\partial r_{i}})}_{u} \frac{\partial r_{i}}{\partial θ} > 0, \frac{\partial q_{i}}{\partial f_{i}} = {(\frac{\partial q_{i}}{\partial r_{i}})}_{u} \frac{\partial r_{i}}{\partial f_{i}} > 0, \\ \frac{\partial q_{i}}{\partial t_{i}} = {(\frac{\partial q_{i}}{\partial r_{i}})}_{u} \frac{\partial r_{i}}{\partial t_{i}} > 0, \frac{\partial q_{i}}{\partial α_{i}} = {(\frac{\partial q_{i}}{\partial r_{i}})}_{u} \frac{\partial r_{i}}{\partial α_{i}} < 0, \frac{\partial q_{i}}{\partial β} = {(\frac{\partial q_{i}}{\partial r_{i}})}_{u} \frac{\partial r_{i}}{\partial β_{i}} < 0. \end{aligned} \end{aligned}

(14)

Brueckner (1987) provides the effect of the spatial utility level on housing consumption, which can be used in our model. So we can obtain

\begin{aligned} \frac{\partial q_{i}}{\partial u} = (\frac{\partial r_{i}}{\partial u} - \frac{\partial M R S_{q c}}{\partial c} . \frac{1}{v_{c}}) {(\frac{\partial q_{i}}{\partial r_{i}})}_{u} > 0, \end{aligned}

(15)

where

\partial {M R S}_{q c} / \partial c > 0

ensures

q

is a normal good. Using Equations (13)

\sim

(14), we can get

v_{c} > 0

To obtain

u

and

\bar{x}

requires the simultaneous solution of Equations (8) and (9), and the solutions depend on

L

y

r_{A}

θ

G

f_{1}

f_{2}

t_{1}

t_{2}

α_{1}

α_{2}

and

β

. To obtain the effects, we totally can make

λ

stand for exogenous variables listed above. Then, we can obtain

\partial u / \partial λ

from Equation (8) and

\partial \bar{x} / \partial λ

from Equation (9).

Getting the first-order differential of (8) with respect to

λ

, we can obtain

\begin{aligned} \frac{\partial u}{\partial λ} = [\frac{\partial r_{A}}{\partial λ} - (\frac{\partial {\bar{r}}_{2}}{\partial λ} + \frac{\partial {\bar{r}}_{2}}{\partial x} . \frac{\partial \bar{x}}{\partial λ})] / \frac{\partial {\bar{r}}_{2}}{\partial u} . \end{aligned}

(16)

Rewrite Equation (9), we get

\begin{aligned} \int_{0}^{\hat{x}} x / q_{1} d x + \int_{\hat{x}}^{\bar{x}} x / q_{2} d x - L / δ = 0. \end{aligned}

(17)

Totally differentiate Equation (17) with respect to

λ

and we can get

\begin{aligned} \frac{\partial \bar{x}}{\partial λ} = \frac{{\bar{q}}_{2}}{x} [\int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} (\frac{\partial q_{1}}{\partial λ} + \frac{\partial q_{1}}{\partial u} . \frac{\partial u}{\partial λ}) d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} (\frac{\partial q_{2}}{\partial λ} + \frac{\partial q_{2}}{\partial u} . \frac{\partial u}{\partial λ}) d x + \frac{1}{δ} . \frac{\partial L}{\partial λ}], \end{aligned}

(18)

where

\partial x / \partial λ = 0

\partial δ / \partial λ = 0

and terms involving

\partial \hat{x} / \partial λ

are canceled out because

\hat{q_{1}} = \hat{q_{2}}

. At the place

\hat{x}

, we have

M_{1} = M_{2}

and

r_{1} = r_{2}

Substitute Equation (16) into (18), we can obtain

\begin{array}{lll} \frac{\partial \bar{x}}{\partial λ} \\ = \frac{\int_{0}^{\bar{x}} \frac{x}{q_{1}^{2}} [\frac{\partial q_{1}}{\partial λ} . \frac{\partial {\bar{r}}_{2}}{\partial u} + \frac{\partial q_{1}}{\partial u} (\frac{\partial r_{A}}{\partial λ} - \frac{\partial {\bar{r}}_{2}}{\partial λ})] d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} [\frac{\partial q_{2}}{\partial λ} . \frac{\partial {\bar{r}}_{2}}{\partial u} + \frac{\partial q_{2}}{\partial u} (\frac{\partial r_{A}}{\partial λ} - \frac{\partial {\bar{r}}_{2}}{\partial λ})] d x + \frac{1}{δ} . \frac{\partial L}{\partial λ} . \frac{\partial {\bar{r}}_{2}}{\partial u}}{\frac{\bar{x}}{{\bar{q}}_{2}} . \frac{\partial {\bar{r}}_{2}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial u} . \frac{\partial {\bar{r}}_{2}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} . \frac{\partial q_{2}}{\partial u} . \frac{\partial {\bar{r}}_{2}}{\partial x} d x} . \end{array}

(19)

Clearly,

\partial \bar{r_{2}} / \partial u < 0

\partial q_{1} / \partial u \cdot \partial \bar{r_{2}} / \partial x < 0

and

\partial q_{2} / \partial u \cdot \partial \bar{r_{2}} / \partial x < 0

, the denominator of (19) is negative.

By making

λ

stand for exogenous variables, we get some propositions.

Proposition 1 Certeris paribus, urban area expands with the increase in numbers of households, i.e.,

\partial \bar{x} / \partial L > 0

Proof Substitute

λ = L

into Equation (19) and using

\partial q_{i} / \partial L = \partial r_{A} / \partial L = \partial {\bar{r}}_{2} / \partial L = 0

, we can get

\begin{aligned} \frac{\partial \bar{x}}{\partial L} = \frac{\frac{1}{δ} . \frac{\partial {\bar{r}}_{2}}{\partial u}}{\frac{\bar{x}}{{\bar{q}}_{2}} . \frac{\partial {\bar{r}}_{2}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial u} . \frac{\partial {\bar{r}}_{2}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} . \frac{\partial q_{2}}{\partial u} . \frac{\partial {\bar{r}}_{2}}{\partial x} d x} > 0. \end{aligned}

(20)

That is proved.

Proposition 2 Certeris paribus, urban area expands with the increase in households' income, i.e.,

\partial \bar{x} / \partial y > 0

Proof Substitute

λ = y

into Equation (19) and using

\partial r_{A} / \partial L = 0

\partial L / \partial y = 0

. Equation (19) can be rewritten as follows:

\begin{aligned} \frac{\partial \bar{x}}{\partial y} = \frac{\int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} [\frac{\partial q_{1}}{\partial y} \cdot \frac{\partial {\bar{r}}_{2}}{\partial u} + \frac{\partial q_{1}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial y}] d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} [\frac{\partial q_{2}}{\partial y} \cdot \frac{\partial {\bar{r}}_{2}}{\partial u} - \frac{\partial q_{2}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial y}] d x}{\frac{\bar{x}}{{\bar{q}}_{2}} \cdot \frac{\partial {\bar{r}}_{2}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} \cdot \frac{\partial q_{1}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} \cdot \frac{\partial q_{2}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial x} d x} . \end{aligned}

(21)

Clearly, the denominator is negative, then substituting Equations (13)

\sim

(15) into the numerator, which can be rewritten as follows:

\begin{aligned} \begin{array}{l} \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} \cdot {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} [\frac{1 - θ}{β_{1} q_{1} β_{2} {\bar{q}}_{2}} (\frac{1}{v_{c}} - \frac{1}{{\bar{v}}_{c}}) + \frac{1 - θ}{v_{c} β_{2} {\bar{q}}_{2}} \cdot \frac{\partial M R S_{q c}}{\partial c}] d x \\ + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} [\frac{1 - θ}{q_{2} β_{2}^{2} {\bar{q}}_{2}} (\frac{1}{v_{c}} - \frac{1}{{\bar{v}}_{c}}) + \frac{1 - θ}{v_{c} β_{2} {\bar{q}}_{2}} \cdot \frac{\partial M R S_{q c}}{\partial c}] d x . \end{array} \end{aligned}

(22)

Since

\partial q_{i} / \partial r_{i} |_{u} < 0

\partial {M R S}_{q c} / \partial c > 0

and

1 / v_{c} - 1 / \bar{v_{c}} > 0

, then the denominator and numerator are negative, we get

\partial \bar{x} / \partial y > 0

Proposition 3 Certeris paribus, the urban area shrinks with an increase in the rural land rent, i.e.,

\partial \bar{x} / \partial r_{A} < 0

Proof Substitute

λ = r_{A}

into Equation (19) and using

\partial q_{i} / \partial r_{A} = \partial \bar{r_{2}} / \partial r_{A} = \partial L / \partial r_{A} = 0

, which becomes

\begin{aligned} \frac{\partial \bar{x}}{\partial r_{A}} = \frac{\int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} \cdot \frac{\partial q_{1}}{\partial u} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} \cdot \frac{\partial q_{2}}{\partial u} d x}{\frac{\bar{x}}{{\bar{q}}_{2}} \cdot \frac{\partial {\bar{r}}_{2}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} \cdot \frac{\partial q_{1}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} \cdot \frac{\partial q_{2}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial x} d x} . \end{aligned}

(23)

Since

\partial q_{i} / \partial u > 0

and the denominator of Equation (23) is negative, thus we get

\partial \bar{x} / \partial r_{A} < 0

Proposition 4 Certeris paribus, the urban area shrinks with an increase in the income tax rate, i.e.,

\partial \bar{x} / \partial θ < 0

Proof Substitute

λ = θ

into Equation (19) and using

\partial r_{A} / \partial θ = \partial L / \partial θ = 0

, the first-order differential can be rewritten as follows:

\begin{aligned} \frac{\partial \bar{x}}{\partial θ} = \frac{\int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} [\frac{\partial q_{1}}{\partial θ} . \frac{\partial \bar{r_{2}}}{\partial u} - \frac{\partial q_{1}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial θ})] d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} [\frac{\partial q_{2}}{\partial θ} . \frac{\partial \bar{r_{2}}}{\partial u} - \frac{\partial q_{2}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial θ})] d x}{\frac{\bar{x}}{\bar{q_{2}}} . \frac{\partial \bar{r_{2}}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} . \frac{\partial q_{2}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial x} d x} . \end{aligned}

(24)

The sign of the numerator is ambiguous, so we can substitute Equations (13)

\sim

(15) into the numerator, the numerator can be rewritten as follows:

\begin{aligned} \begin{array}{l} \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} \cdot {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} [\frac{y}{{(1 - β)}^{2} q_{1} \bar{q_{2}}} (\frac{1}{\bar{v_{c}}} - \frac{1}{v_{c}}) - \frac{y}{(1 - β) \bar{q_{2}} v_{c}} . \frac{\partial M R S_{q c}}{\partial c}] d x \\ + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} [\frac{y}{{(1 - β)}^{2} q_{1} \bar{q_{2}}} (\frac{1}{\bar{v_{c}}} - \frac{1}{v_{c}}) - \frac{y}{(1 - β) \bar{q_{2}} v_{c}} . \frac{\partial M R S_{q c}}{\partial c}] d x > 0. \end{array} \end{aligned}

(25)

Since

\partial q_{i} / \partial r_{i} |_{u} < 0

\partial {M R S}_{q c} / \partial c > 0

and

1 / v_{c} - 1 / \bar{v_{c}} > 0

, then the denominator and numerator are positive, we get

\partial \bar{x} / \partial θ < 0

Proposition 5 Certeris paribus, intergovernmental grants do not affect urban area, i.e.,

\partial \bar{x} / \partial G = 0

Proof Totally differentiating Equation (10) with respect to

G

, we have

\begin{aligned} \frac{\partial ϕ}{\partial G} = - 1 / θ y L < 0, \end{aligned}

(26)

which means that

ϕ

falls sufficiently with an increase in

G

so that tax revenues are completely replaced by intergovernmental grants, and there no changes in other endogenous variables. Therefore, we have

\partial \bar{x} / \partial G = 0

Proposition 6 Certeris paribus, the urban area expands with an increase in the fixed cost of public transit, i.e.,

\partial \bar{x} / \partial f_{1} > 0

Proof Substitute

λ = f_{1}

into Equation (19) and using

\partial r_{A} / \partial f_{1} = \partial \bar{r_{2}} / \partial f_{1} = \partial q_{2} / \partial f_{1} = \partial L / \partial f_{1} = 0

, the first-order differential can be rewritten as follows:

\begin{aligned} \frac{\partial \bar{x}}{\partial f_{1}} = \frac{\int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial f_{1}} . \frac{\partial \bar{r_{2}}}{\partial u} d x}{\frac{\bar{x}}{\bar{q_{2}}} . \frac{\partial \bar{r_{2}}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} . \frac{\partial q_{2}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial x} d x} > 0. \end{aligned}

(27)

Since

\partial q_{1} / \partial f_{1} > 0

\partial \bar{r_{2}} / \partial u < 0

and the denominator of (27) is negative, thus we get

\partial \bar{x} / \partial f_{1} > 0

Proposition 7 Certeris paribus, the urban area expands with an increase in the traveling marginal cost of public transit, i.e.,

\partial \bar{x} / \partial t_{1} > 0

Proof Substitute

λ = t_{1}

into Equation (19) and using

\partial r_{A} / \partial t_{1} = \partial \bar{r_{2}} / \partial t_{1} = \partial q_{2} / \partial t_{1} = \partial L / \partial t_{1} = 0

, Equation (19) can be rewritten as follows:

\begin{aligned} \frac{\partial \bar{x}}{\partial t_{1}} = \frac{\int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial t_{1}} . \frac{\partial \bar{r_{2}}}{\partial u} d x}{\frac{\bar{x}}{\bar{q_{2}}} . \frac{\partial \bar{r_{2}}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} . \frac{\partial q_{2}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial x} d x} > 0. \end{aligned}

(28)

Since

\partial q_{1} / \partial t_{1} > 0

\partial \bar{r_{2}} / \partial u < 0

and the denominator of (29) is negative, thus we get

\partial \bar{x} / \partial t_{1} > 0

Proposition 8 Certeris paribus, the urban area shrinks with an increase in the public transit subsidies, i.e.,

\partial \bar{x} / \partial α_{1} < 0

Proof Substitute

λ = α_{1}

into Equation (19) and using

\partial r_{A} / \partial α_{1} = \partial \bar{r_{2}} / \partial α_{1} = \partial q_{2} / \partial α_{1} = \partial L / \partial α_{1} = 0

, the first-order differential can be rewritten as follows:

\begin{aligned} \frac{\partial \bar{x}}{\partial α_{1}} = \frac{\int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial α_{1}} . \frac{\partial \bar{r_{2}}}{\partial u} d x}{\frac{\bar{x}}{\bar{q_{2}}} . \frac{\partial \bar{r_{2}}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} . \frac{\partial q_{2}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial x} d x} < 0. \end{aligned}

(29)

Since

\partial q_{1} / \partial α_{1} < 0

\partial \bar{r_{2}} / \partial u < 0

and the denominator of (29) is negative, thus we get

\partial \bar{x} / \partial α_{1} < 0

Proposition 9 Certeris paribus, the urban area shrinks with an increase in the total fixed cost of automobiles, i.e.,

\partial \bar{x} / \partial f_{2} < 0

Proof Substitute

λ = f_{2}

into Equation (19) and using

\partial q_{1} / \partial f_{2} = \partial r_{A} / \partial f_{2} = \partial L / \partial f_{2} = 0

, the Equation (19) can be rewritten as follows:

\begin{aligned} \frac{\partial \bar{x}}{\partial f_{2}} = \frac{\int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} \cdot \frac{\partial q_{1}}{\partial u} (- \frac{\partial \bar{r_{2}}}{\partial f_{2}}) d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} [\frac{\partial q_{2}}{\partial f_{2}} \cdot \frac{\partial \bar{r_{2}}}{\partial u} - \frac{\partial q_{2}}{\partial u} \cdot \frac{\partial \bar{r_{2}}}{\partial f_{2}}] d x}{\frac{\bar{x}}{\bar{q_{2}}} \cdot \frac{\partial \bar{r_{2}}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} \cdot \frac{\partial q_{1}}{\partial u} \cdot \frac{\partial \bar{r_{2}}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} \cdot \frac{\partial q_{2}}{\partial u} \cdot \frac{\partial \bar{r_{2}}}{\partial x} d x} . \end{aligned}

(30)

Substituting Equations (13)

\sim

(15) into the numerator, the numerator can be rewritten as follows:

\begin{array}{lll} - \int_{0}^{\hat{x}} \frac{x}{β_{1} β_{2} q_{1}^{3} \bar{q_{2}} v_{c}} \cdot {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} d x - \int_{0}^{\hat{x}} \frac{x}{β_{2} q_{1}^{2} \bar{q_{2}} v_{c}} \cdot (\frac{\partial M R S_{q c}}{\partial c}) {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} d x \\ - \int_{\hat{x}}^{\bar{x}} \frac{x}{β_{2}^{2} q_{2}^{3} \bar{q_{2}}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} \cdot (\frac{1}{v_{c}} - \frac{1}{\bar{v_{_{c}}}}) d x - \int_{\hat{x}}^{\bar{x}} \frac{x}{β_{2} q_{2}^{2} \bar{q_{2}} v_{c}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} (\frac{\partial M R S_{q c}}{\partial c}) d x > 0. \end{array}

(31)

Since

\partial q_{i} / \partial r_{i} |_{u} < 0

\partial {M R S}_{q c} / \partial c > 0

and

1 / v_{c} - 1 / \bar{v_{c}} > 0

, then the denominator is negative and the numerator is positive, we get

\partial \bar{x} / \partial f_{2} < 0

Proposition 10 Certeris paribus, the urban area shrinks with an increase in the travelling marginal cost of automobiles, i.e.,

\partial \bar{x} / \partial t_{2} < 0

Proof Substitute

λ = t_{2}

into Equation (19) and using

\partial q_{1} / \partial t_{2} = \partial r_{A} / \partial t_{2} = \partial L / \partial t_{2} = 0

, Equation (19) can be rewritten as follows:

\begin{aligned} \frac{\partial \bar{x}}{\partial t_{2}} = \frac{\int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial u} (- \frac{\partial \bar{r_{2}}}{\partial λ}) d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} (\frac{\partial q_{2}}{\partial t_{2}} . \frac{\partial \bar{r_{2}}}{\partial u} - \frac{\partial q_{2}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial t_{2}}) d x}{\frac{\bar{x}}{\bar{q_{2}}} . \frac{\partial \bar{r_{2}}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} . \frac{\partial q_{2}}{\partial u} . \frac{\partial \bar{r_{2}}}{\partial x} d x} . \end{aligned}

(32)

Substituting Equations (13)

\sim

(15) into the numerator, the numerator can be rewritten as follows:

\begin{array}{lll} - \int_{0}^{\hat{x}} \frac{(1 - α_{2}) x^{2}}{β_{1} β_{2} q_{1}^{3} \bar{q_{2}} v_{c}} \cdot {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} d x - \int_{0}^{\hat{x}} \frac{(1 - α_{2}) x^{2}}{β_{2} q_{1}^{2} \bar{q_{2}} v_{c}} \cdot (\frac{\partial M R S_{q c}}{\partial c}) {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} d x \\ - \int_{\hat{x}}^{\bar{x}} \frac{(1 - α_{2}) x^{2}}{β_{2}^{2} q_{2}^{3} \bar{q_{2}}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} \cdot (\frac{1}{v_{c}} - \frac{1}{\bar{v_{c}}}) d x - \int_{\hat{x}}^{\bar{x}} \frac{(1 - α_{2}) x^{2}}{β_{2} q_{2}^{2} \bar{q_{2}}} \cdot (\frac{\partial M R S_{q c}}{\partial c}) {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} d x > 0. \end{array}

(33)

Since

\partial q_{i} / \partial r_{i} |_{u} < 0

\partial {M R S}_{q c} / \partial c > 0

and

1 / v_{c} - 1 / \bar{v_{c}} > 0

, then the denominator is negative and the numerator is positive, we get

\partial \bar{x} / \partial t_{2} < 0

Proposition 11 Certeris paribus, the urban area expan

d

s with an increase in the auto subsidies, i.e.,

\partial \bar{x} / \partial α_{2} > 0

Proof Substitute

λ = α_{2}

into Equation (19) and using

\partial q_{1} / \partial α_{2} = \partial r_{A} / \partial α_{2} = \partial L / \partial α_{2} = 0

, Equation (19) can be rewritten as follows:

\begin{aligned} \frac{\partial \bar{x}}{\partial α_{2}} = \frac{- \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial u} \frac{\partial {\bar{r}}_{2}}{\partial α_{2}} d x + + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} [\frac{\partial q_{2}}{\partial α_{2}} \cdot \frac{\partial {\bar{r}}_{2}}{\partial u} - \frac{\partial q_{2}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial α_{2}}] d x}{\frac{\bar{x}}{{\bar{q}}_{2}} \cdot \frac{\partial {\bar{r}}_{2}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} \cdot \frac{\partial q_{1}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} \cdot \frac{\partial q_{2}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial x} d x} . \end{aligned}

(34)

Substituting Equations (13)

\sim

(15) into the numerator, the numerator can be rewritten as follows:

\begin{array}{lll} \int_{0}^{\hat{x}} \frac{t_{2} x^{2}}{β_{1} β_{2} q_{1}^{3} {\bar{q}}_{2} v_{c}} \cdot {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} d x + \int_{0}^{\hat{x}} \frac{t_{2} x^{2}}{β_{2} q_{1}^{2} {\bar{q}}_{2} v_{c}} \cdot \frac{\partial M R S_{q c}}{\partial c} \cdot {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} d x \\ + \int_{\hat{x}}^{\bar{x}} \frac{t_{2} x^{2}}{β_{2} q_{2}^{3} {\bar{q}}_{2}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} \cdot (\frac{1}{v_{c}} - \frac{1}{{\bar{v}}_{c}}) d x + \int_{\hat{x}}^{\bar{x}} \frac{t_{2} x^{2}}{β_{2} q_{2}^{2} {\bar{q}}_{2} v_{c}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} \cdot \frac{\partial M R S_{q c}}{\partial c} d x < 0. \end{array}

(35)

Since

\partial q_{i} / \partial r_{i} |_{u} < 0

\partial {M R S}_{q c} / \partial c > 0

and

1 / v_{c} - 1 / \bar{v_{c}} > 0

, then the denominator and numerator are negative, we get

\partial \bar{x} / \partial α_{2} > 0

Proposition 12 Certeris paribus, the urban area expan

d

s with an increase in the housing subsidies, i.e.,

\partial \bar{x} / \partial β > 0

Proof Substitute

λ = β

into Equation (19) and using

\partial r_{A} / \partial β = \partial L / \partial β = 0

, which becomes

\begin{aligned} \frac{\partial \bar{x}}{\partial β} = \frac{\int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} [\frac{\partial q_{1}}{\partial y} \cdot \frac{\partial {\bar{r}}_{2}}{\partial β} - \frac{\partial q_{1}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial β}] d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} [\frac{\partial q_{2}}{\partial β} \cdot \frac{\partial {\bar{r}}_{2}}{\partial u} - \frac{\partial q_{2}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial β}] d x}{\frac{\bar{x}}{{\bar{q}}_{2}} \cdot \frac{\partial {\bar{r}}_{2}}{\partial u} + \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} \cdot \frac{\partial q_{1}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial x} d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} \cdot \frac{\partial q_{2}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial x} d x} . \end{aligned}

(36)

Substituting Equations (13)–(15) into the numerator, the numerator can be rewritten as follows:

\begin{array}{lll} \int_{0}^{\hat{x}} \frac{x}{{(1 - β)}^{2} q_{1}^{2}} \cdot {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} \cdot (\frac{{\bar{r}}_{2}}{q_{1} v_{c}} - \frac{r_{1}}{{\bar{q}}_{2} {\bar{v}}_{c}}) d x + \int_{0}^{\hat{x}} \frac{{\bar{r}}_{2} x}{(1 - β) v_{c} q_{1}^{2}} \cdot {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} \cdot \frac{\partial M R S_{q c}}{\partial c} d x \\ + \int_{\hat{x}}^{\bar{x}} \frac{x}{{(1 - β)}^{2} q_{2}^{2}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} \cdot (\frac{{\bar{r}}_{2}}{q_{1} v_{c}} - \frac{r_{1}}{{\bar{q}}_{2} {\bar{v}}_{c}}) d x + \int_{0}^{\hat{x}} \frac{{\bar{r}}_{2} x}{(1 - β) v_{c} q_{2}^{2}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} d x . \end{array}

(37)

According to Equation (4), we can obtain

\begin{aligned} \frac{{\bar{r}}_{2}}{v_{c} q_{1}} - \frac{r_{1}}{{\bar{v}}_{c} {\bar{q}}_{2}} = \frac{{\bar{r}}_{2} r_{1} (1 - β)}{v_{q} q_{1}} - \frac{{\bar{r}}_{2} r_{1} (1 - β)}{{\bar{v}}_{q} {\bar{q}}_{2}} > 0 \end{aligned}

(38)

and

\begin{aligned} \frac{{\bar{r}}_{2}}{v_{c} q_{2}} - \frac{r_{2}}{{\bar{v}}_{c} {\bar{q}}_{2}} = \frac{{\bar{r}}_{2} r_{2} β}{v_{q} q_{2}} - \frac{{\bar{r}}_{2} r_{2} β}{{\bar{v}}_{c} {\bar{q}}_{2}} > 0. \end{aligned}

(39)

Since

\bar{x} > x

\bar{q_{2}} > q_{2} > q_{1}

\bar{v_{q}} > v_{q}

\partial q_{i} / \partial r_{i} |_{u} < 0

, and

\partial {M R S}_{q c} / \partial c > 0

. Therefore we have

\begin{aligned} \int_{0}^{\hat{x}} \frac{x}{q_{1}^{2}} [\frac{\partial q_{1}}{\partial β} \cdot \frac{\partial {\bar{r}}_{2}}{\partial u} - \frac{\partial q_{1}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial β}] d x + \int_{\hat{x}}^{\bar{x}} \frac{x}{q_{2}^{2}} [\frac{\partial q_{2}}{\partial β} \cdot \frac{\partial {\bar{r}}_{2}}{\partial u} - \frac{\partial q_{2}}{\partial u} \cdot \frac{\partial {\bar{r}}_{2}}{\partial β}] d x < 0. \end{aligned}

(40)

Clearly, the numerator is negative. Since both of the denominator and the numerator are negative, thus we can obtain

\partial \bar{x} / \partial β > 0

In summary, we find that public transit subsi

d

y lea

d

s to urban shrink while automobile subsi

d

y has the opposite effect, which accor

d

s the findings of Brueckner, Su and DeSalvo^{[17, 18]}. Also, housing subsi

d

y as the new intro

d

uced determinant could make the city sprawl.

3 Maximization of Equilibrium Utility Level

In Section 2, we assume that all urban househol

d

s are the same with respect to utility function and income and the city therefore can achieve spatial equilibrium. And the previous models and analysis are provided on the premise that is the urban-area-wide spatial equilibrium utility level when maximizing equilibrium utility.

To balance countries' budgets, we have added the constraint Equation (10). Then we can stu

d

y how to maximize the urban-area-wide spatial equilibrium utility level in the constraint condition. We cannot obtain the direct findings, but we derive two conditions contributed to calculation analysis of samples.

Proposition 13 In order to maximize spatial equalization utility, constraint Equations

(41)

and

(42)

must be satisfied. Equations

(41)

and

(42)

are as follows:

\begin{array}{lll} v_{q} (\frac{\partial q_{1}}{\partial r_{1}})_{u} \cdot \frac{t_{1} x}{(1 - β) q_{1}} \\ + v_{c} \cdot \frac{\int_{0}^{\hat{x}} [\frac{α_{1} r_{1} t_{1}^{2} x^{3}}{(1 - β) q_{1}^{2}} \cdot {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} - r_{1} t_{1} x^{2} - \frac{β r_{1} t_{1} x^{2}}{(1 - β)} - {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} \cdot \frac{r_{1}^{2} t_{1} x^{2}}{q_{1}}] d x - \int_{\hat{x}}^{\bar{x}} {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} \cdot \frac{r_{1} r_{2} t_{1} x^{2}}{q_{1}} d x}{\int_{0}^{\hat{x}} r_{1} x d x + \int_{\hat{x}}^{\bar{x}} r_{2} x d x} \\ = 0 \end{array}

(41)

and

\begin{array}{lll} v_{q} (\frac{\partial q_{2}}{\partial r_{2}})_{u} \cdot \frac{t_{2} x}{(1 - β) q_{2}} \\ + v_{c} \cdot \frac{\int_{\hat{x}}^{\bar{x}} [\frac{α_{2} r_{2} t_{2}^{2} x^{3}}{(1 - β) q_{2}^{2}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} - r_{2} t_{2} x^{2} - \frac{β r_{2} t_{2} x^{2}}{(1 - β)} - {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} \cdot \frac{r_{2}^{2} t_{2} x^{2}}{q_{2}}] d x - \int_{0}^{\hat{x}} {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} \cdot \frac{r_{1} r_{2} t_{2} x^{2}}{q_{2}} d x}{\int_{0}^{\hat{x}} r_{1} x d x + \int_{\hat{x}}^{\bar{x}} r_{2} x d x} \\ = 0. \end{array}

(42)

Proof Totally differentiating Equation (10), getting

\begin{aligned} \frac{\partial β}{\partial α_{1}} = \frac{α_{1} \int_{0}^{\hat{x}} \frac{t_{1} x^{2}}{q_{1}^{2}} . \frac{\partial q_{1}}{\partial α_{1}} d x - \int_{0}^{\hat{x}} \frac{t_{1} x^{2}}{q_{1}} d x - β \int_{0}^{\hat{x}} x . \frac{\partial r_{1}}{\partial α_{1}} d x}{\int_{0}^{\hat{x}} r_{1} x d x + \int_{\hat{x}}^{\bar{x}} r_{2} x d x} \end{aligned}

(43)

and

\begin{aligned} \frac{\partial β}{\partial α_{2}} = \frac{α_{2} \int_{\hat{x}}^{\bar{x}} \frac{t_{2} x^{2}}{q_{2}^{2}} . \frac{\partial q_{2}}{\partial α_{2}} d x - \int_{\hat{x}}^{\bar{x}} \frac{t_{2} x^{2}}{q_{2}} d x - β \int_{\hat{x}}^{\bar{x}} x . \frac{\partial r_{2}}{\partial α_{2}} d x}{\int_{0}^{\hat{x}} r_{1} x d x + \int_{\hat{x}}^{\bar{x}} r_{2} x d x} . \end{aligned}

(44)

If travelers choose to travel by public transit, we can have Equation (45) to maximize spatial equalization utility. Equation (45) is

\begin{aligned} \frac{\partial u}{\partial α_{1}} = v_{q} \frac{\partial q_{1}}{\partial α_{1}} + v_{c} \frac{\partial c}{\partial α_{1}} = 0 . \end{aligned}

(45)

Totally differentiate Equation (4) with respect to and solve for, getting

\begin{aligned} \frac{\partial c}{\partial α_{1}} = \frac{\partial β}{\partial α_{1}} r_{1} q_{1} - (1 - β) q_{1} \frac{\partial r_{1}}{\partial α_{1}} - (1 - β) r_{1} \frac{\partial q_{1}}{\partial α_{1}} + t_{1} x . \end{aligned}

(46)

Substituting from Equations (13), (14), and (43) into Equation (46), after some manipulation, pro

d

uces

\begin{aligned} \frac{\partial c}{\partial α_{1}} = [\frac{α_{1} \int_{0}^{\hat{x}} \frac{t_{1} x^{2}}{q_{1}^{2}} \cdot \frac{\partial q_{1}}{\partial α_{1}} d x - \int_{0}^{\hat{x}} \frac{t_{1} x^{2}}{q_{1}} d x - β \int_{0}^{\hat{x}} x \cdot \frac{\partial r_{1}}{\partial α_{1}} d x}{\int_{0}^{\hat{x}} r_{1} x d x + \int_{\hat{x}}^{\bar{x}} r_{2} x d x}] \cdot r_{1} q_{1} - (\frac{\partial q_{1}}{\partial r_{1}})_{u} \cdot \frac{r_{1} t_{1} x}{q_{1}} . \end{aligned}

(47)

And substituting from Equation (47) into Equation (45), which becomes

\begin{aligned} \begin{array}{lll} \frac{\partial u}{\partial α_{1}} = v_{q} (\frac{\partial q_{1}}{\partial r_{1}})_{u} \cdot \frac{t_{1} x}{(1 - β) q_{1}} \\ + v_{c} \cdot \frac{\int_{0}^{\hat{x}} [\frac{α_{1} r_{1} t_{1}^{2} x^{3}}{(1 - β) q_{1}^{2}} \cdot {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} - r_{1} t_{1} x^{2} - \frac{β r_{1} t_{1} x^{2}}{(1 - β)} - {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} \cdot \frac{r_{1}^{2} t_{1} x^{2}}{q_{1}}] d x - \int_{\hat{x}}^{\bar{x}} {(\frac{\partial q_{1}}{\partial r_{1}})}_{u} \cdot \frac{r_{1} r_{2} t_{1} x^{2}}{q_{1}} d x}{\int_{0}^{\hat{x}} r_{1} x d x + \int_{\hat{x}}^{\bar{x}} r_{2} x d x} \\ = 0. \end{array} \end{aligned}

(48)

If travelers choose to travel by automobile, we have Equation (49) to maximize spatial equalization utility. Equation (49) is

\begin{aligned} \frac{\partial u}{\partial α_{2}} = v_{q} \frac{\partial q_{2}}{\partial α_{2}} + v_{c} \frac{\partial c}{\partial α_{2}} = 0 . \end{aligned}

(49)

Totally differentiate equation (4) with respect to

α_{2}

and solve for

\partial c / \partial α_{2}

, getting

\begin{aligned} \frac{\partial c}{\partial α_{2}} = \frac{\partial β}{\partial α_{2}} r_{2} q_{2} - (1 - β) q_{2} \frac{\partial r_{2}}{\partial α_{2}} - (1 - β) r_{2} \frac{\partial q_{2}}{\partial α_{2}} + t_{2} x . \end{aligned}

(50)

Substituting from Equations (13), (14), and (44) into Equation (46), after some manipulation, pro

d

uces

\begin{aligned} \frac{\partial c}{\partial α_{2}} = \frac{\int_{\hat{x}}^{\bar{x}} [\frac{α_{2} r_{2} t_{2}^{2} x^{3}}{(1 - β) {q_{2}}^{2}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} - r_{2} t_{2} x^{2} - \frac{β r_{2} t_{2} x^{2}}{(1 - β)} - {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} \cdot \frac{r_{2}^{2} t_{2} x^{2}}{q_{2}}] d x - \int_{0}^{\hat{x}} {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} \cdot \frac{r_{1} r_{2} t_{2} x^{2}}{q_{2}} d x}{\int_{0}^{\hat{x}} r_{1} x d x + \int_{\hat{x}}^{\bar{x}} r_{2} x d x} . \end{aligned}

(51)

And substituting from Equations (14) and (51) into Equation (49), which becomes

\begin{array}{lll} \frac{\partial u}{\partial α_{2}} = v_{q} (\frac{\partial q_{2}}{\partial r_{2}})_{u} \cdot \frac{t_{2} x}{(1 - β) q_{2}} \\ + v_{c} \cdot \frac{\int_{\hat{x}}^{\bar{x}} [\frac{α_{2} r_{2} t_{2}^{2} x^{3}}{(1 - β) {q_{2}}^{2}} \cdot {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} - r_{2} t_{2} x^{2} - \frac{β r_{2} t_{2} x^{2}}{(1 - β)} - {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} \cdot \frac{r_{2}^{2} t_{2} x^{2}}{q_{2}}] d x - \int_{0}^{\hat{x}} {(\frac{\partial q_{2}}{\partial r_{2}})}_{u} \cdot \frac{r_{1} r_{2} t_{2} x^{2}}{q_{2}} d x}{\int_{0}^{\hat{x}} r_{1} x d x + \int_{\hat{x}}^{\bar{x}} r_{2} x d x} \\ = 0. \end{array}

(52)

To sum up, our results show that we cannot obtain the direct findings but two complex equations. However, we derive two constraint conditions contributed to calculation analysis of samples. Anyway, this paper is only a fundamental exploration in the field of maximization of equilibrium utility level.

4 Conclusion

This article analyzes theoretically effects of transportation subsidies and housing subsidies on urban sprawl in a monocentric urban spatial model. We assume the theoretical model incorporates two transportation modes, and our analysis indicates that the urban area shrinks with an increase in public transit subsidies but expan

d

s when automobile subsidies and housing subsidies increase. Moreover, our comparative static analysis pro

d

uces that the urban area grows with an increase in housing subsidies. In the meanwhile, we also provide some propositions to indicate that how some relevant exogenous variables influence the spatial size of urban areas, among which these findings could contribute to policies makers to control urban sprawl effectively.

References

Publishing order | Descend order by publishing year | Descend order by cited within

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Funding

Humanities and Social Sciences of Beijing Jiaotong University(B16JB00190)

the National Natural Science Foundation of China(71501012)

the Social Science Foundation of Beijing(16GLC054)

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Abstract
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1 Introduction
2 Subsidies and Urban Sprawl
2.1 Urban Spatial Model
2.2 The Effects of Transportation and Housing Subsidies
3 Maximization of Equilibrium Utility Level
4 Conclusion
References
Funding

Received	Accepted	Published
2017-12-27	2018-01-29	2018-06-25
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2018-06-25

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1 Introduction

2 Subsidies and Urban Sprawl

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2.2 The Effects of Transportation and Housing Subsidies

3 Maximization of Equilibrium Utility Level

4 Conclusion

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1 Introduction

2 Subsidies and Urban Sprawl

2.1 Urban Spatial Model

2.2 The Effects of Transportation and Housing Subsidies

3 Maximization of Equilibrium Utility Level

4 Conclusion

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