Determinants of efficient growth boundaries with balanced budgets
and stochastic rents

Jyh-Bang Jou*
Department of Economics and Finance (Albany), Massey University, Campus,
New Zealand
Graduate Institute of National Development, National Taiwan University, Taiwan*

Correspondence to: Jyh-Bang Jou, School of Economics and Finance (Albany), Massey University,

New ZealandTel: 64-9-4140800 ext. 9429
Fax: 64-9-4418156
E-mail: J.B.Jou@massey.ac.nz
1Determinants of efficient growth boundaries with balanced budgets
and stochastic rents

Abstract

This article extends Ding et al. (1999) to investigate the design of growthboundary controls in an open city in a framework where a congestible public good is
priced at average cost.

Unlike Ding et al., we consider that urban rents evolve

stochastically over time and that the regulator can slacken the boundary control whenurban rents move upward. We provide three novel testable implications, namely, theregulator will set a more stringent boundary control policy if (i) developers expect
urban rents to grow less rapidly; (ii) developers expect urban rents to be more volatile;and (iii) it costs less for the regulator to implement the control policy.

Keywords: balanced budgets, growth boundaries, real options, stochastic rents
JEL Classification: G13; H23; R52; R58

2

Determinants of efficient growth boundaries with balanced budgets
and stochastic rents

I. Introduction

As a city boundary expands with population growth, supply of public goods may
become a financial burden for the city government because the cost for supplying
them may increase substantially with spatial expansion.

At the same time, the city

government may need to maintain balanced budgets such that it may opt to pricepublic goods at average cost and simultaneously restrict the city boundary.1
While Ding, Knaap, and Hopkin (1999) have investigated the design of growthboundary controls in a monocentric open city framework where a congestible public
good is priced at average cost, we depart from Ding et al. in the following respects.
First, Ding et al. assume that urban rents grow deterministically over time, such thatthe date at which land is developed is also non-stochastic.

In contrast, we assume

that urban rents evolve stochastically and, thus, the development timing is also
1

Urban sprawl can have many adverse effects on local communities as a result of car-dependentcommunication on the urban fringe, congested highway systems, and less green-space (Juergensmeyer
and Roberts, 2003). Consequently, many local communities employ growth management controls torectify these negative impacts. These control programs include: The time or phased growth control,
as developed by Ramapo, New York; the population control strategy listed in the Boca Raton plan in
Florida; and the urban growth boundary control employed in the Oregon and Minnesota states.
Alternatively, some communities employ policies to raise the cost of development in order to curb
growth. These policies include municipal exactions, regulatory delays in the approval process (Mayoand Sheppard, 2001), and impact fees. See the review paper by Brueckner (2007), which investigates
the effects of three commonly employed interventions, including urban growth boundaries, FAR (floorarea ratio) restrictions, and cost-increasing regulations.
3stochastic.

Second, Ding et al. assume that the growth control policy is timeinvariant, while we allow the regulator to have the option to slacken the control whenurban rents move upward.

Our assumption is more consistent with the existingland-use planning system, which requires local governments to periodically change
city boundaries as population grows (see, e.g., Ding et al., 1999; Jaeger and Plantinga,2007). Finally, we assume that the regulator incurs costs to implement the control
policy, while Ding et al. abstract from them.

Similar to Ding et al., we find that the regulator will set a more stringent
boundary control policy if either the development cost, the agricultural rent, thecommuting cost, or the stock of infrastructure increases.

Our paper contributes tothe literature by pointing out that the regulator will tighten the boundary control ifdevelopers expect urban rents to grow less rapidly or become more volatile, or if theimplementation cost decreases.

This article assumes that the regulator may need to use growth boundary controls
to correct inefficiencies associated with the pricing of a public good.

This contrasts

with several articles that employ the amenity-creation model, which assumes thatdevelopment exerts a negative externality on city residents.

These articles include

Cooley and LaCivita (1982), Brueckner (1990), Engle, Navarro and Carson (1992),
Helsley and Strange (1995), Sakashita (1995), Brueckner and Lai (1996), Sasaki
4

(1998), Turnbull (2004a), and Jou and Lee (2008).

This article also contrasts withseveral articles that employ the supply-restriction model, which argues that housing
price increases with the decrease of housing supply created by growth controls.

See,for example., Brueckner (1995) and Brueckner and Lai (1996).

The remaining sections are organized as follows.
basic assumptions of our model.

The next section presents the

In Section III, we derive the choice of development

timing and the city boundary in competitive equilibrium as well as the rule for aregulator to design the efficient boundary control.

In Section IV, we employ

plausible parameters to investigate how various forces that are related to the demandand supply conditions of the real estate market affect both the equilibrium and theefficient city boundary.

The final section concludes and suggests directions for

future research.

II. The Model

Our basic model extends the models of Capozza and Helsley (1990) and Ding et al.
(1999). Consider a linear open city with all employment (the CBD) located at oneend.

The city has perfectly competitive factor and product markets.

Residents

located D miles from the CBD must allocate their income, y (t ) , so as to pay for5the numeraire good, land rents, R , and transportation costs D , where  ( 0 )At date t 0 an absentee landowner, who

represents the commuting cost per mile.

is also a risk-neutral developer, faces a discount rate denoted by .2

The landownerhas a parcel of vacant land (normalized at one unit) at location D, which yields a rent
equal to  ( 0 ), such that it is never optimal for the landowner to abandon thevacant land. At any time t 0 , the landowner is able to develop the vacant land at
the cost of F , which is fully irreversible.3

As in Ding et al. (1999), we assume that there exists a congestible public good, z ,which must be provided to all residents of the urban area, but is rival in consumption.
Examples of these kinds of goods include education, water and waste-water services,and police and fire protection (Ding et al., 1999).

The public good is produced witha fixed stock of urban infrastructure, k , and the variable input, m , such that the
public good production function can be specified asz G (k , m, n) ,(1)

where n represents the urban population and G/m 0 and G/n 0 ,
implying that the level of the public good rises with variable inputs and decreases

2

We can generalize our model into a risk aversion environment in the manner of Cox and Ross (1976).
McFarlane (1999) argues that investment on land development will be fully irreversible if demolition
costs are extremely high. Similarly, Riddiough (1997) suggests that irreversibility is a reasonableassumption for real estate, in which the physical asset is long-lived and switching costs to alternative
uses are quite high. Turnbull (2005) argues that the irreversibility assumption may not be realistic,
but provides an analytically tractable solution.
3

6

with population.

Assuming that input prices are unit, and that local governments adjust thevariable input to maintain z fixed at a constant level, z , a public good variable
cost function may be specified asm n2 k .

(2)

Equation (2), which indicates that the cost of maintaining a constant level of the
public good rises with population, resembles that used by Brueckner (1997) and in theliterature on clubs.

Following Ding et al., we assume that local governments set the price of acongestible public good at average cost rather than marginal cost so as to yield abalanced budget.

Consequently, city residents may become better off when the city

population is reduced through the application of a growth boundary control.

The

control has no effect on the utility level of consumers, however, because we assume
that the city is too small. All consumers have identical utility functions, which haveas arguments a numeraire good, c , land, , and the public good, z .

If eachresident consumes one unit of land, the price of the numeraire good is one, and thecosts of the public good are paid by landowners, then using the budget constraint, we
can construct a representative utility function as given by7

v ( c, , z ) v( y (t )) R D, , z ) u (t ) ,(3)

where under open city assumptions, resident utility must equal exogenous utility
levels. From Equation (3), we can construct a bid rent function as given by
R ( D, x(t )) x(t ) D,

(4)

where y (t ) x(t ) c , and the stochastic factor x(t ) follows the arithmetic
Brownian motiondx(t ) dt d (t ), 0, 0,

(5)

where  is the constant drift rate,  is the instantaneous volatility of x(t ) , andd (t ) is an increment to a standard Wiener process.4

The specification of Equation(4) indicates that the income of residents is the sum of two parts: the fixed part ofc and the stochastic part of x(t ) .

Our assumptions differ from those of Ding et al. (1999) in the following respects.
First, we assume that urban rents evolve stochastically as in Equation (5), while Ding
et al. assume that these rents are deterministic.

Second, we allow a regulator to set atime-contingent boundary control policy, while Ding et al. assume that the control is4

Our specification is the same as that assumed in Capozza and Helsley (1990), Capozza and Li (1994),
and Capozza and Sick (1994). The arithmetic Brownian motion of urban rents is supported by theempirical data, but suffers the shortcoming that urban rents may be negative. See a thoroughdiscussion in Capozza and Li (1994).
8

time invariant.5

Finally, we assume that the regulator incurs costs to implement thecontrol policy, while Ding et al. abstract from them.

With these differences, we canderive several testable implications that are novel to the literature, as shown later.

Given that the public good is inefficient priced, local governments can thus
implement two broad types of policies to correct this (Turnbull, 2004a): (i) Fees, or
special exactions, imposed on newly developed property; and (ii) direct quantity
restrictions in the form of urban growth boundaries, greenbelt preserves, land banks,
or development moratoria. We will focus only on the design of growth boundary
controls.

An individual landowner will choose a date at which to develop vacant land.
The social planner, who anticipates the development timing chosen by the developer,
can set an efficient level for the city boundary.

We will sequentially investigate thedeveloper’
sa
ndt
hes
oc
i
a
lpl
a
nne
r
’
sde
c
i
s
i
onin the following section.

III.

Choices of Development Timing

We assume that a regulator implements a growth boundary control policy at thecurrent time t 0 .

Therefore, this policy is anticipated by developers when they5

Ding et al. (1999) allow the regulator to periodically change the control in their extended model thatallows the regulator to undertake lumpy infrastructure investment.
9

look to develop at any future date.

Suppose that we denote x(0) as x , and T as

the time of development here and subsequently.

The problem for an individual

landowner is to determine the point at which it is optimal to develop, so as tomaximize the expected net present value of one unit of vacant land, which is given by
T 

E0 
e 
d e (x () D nk )d e T F .

0

T



(6)

Equation (6) indicates that the expected net present value of returns to the unit plot of
land at a particular location D is the sum of the returns to vacant land received until
time T , plus the expected present value of land rent beginning at the time ofdevelopment, less the expected present value of the sum of the cost paid for the public
good and the up front developing costs.

We can rewrite Equation (6) as:
 

E0 
e (x() D nk )d e T F 
.



T




(7)

In Equation (7), the first term is the expected present value of agricultural rents,
assuming that the vacant land is never converted to urban use, while the remaining
term is t
hede
ve
l
ope
r
’
sopt
i
onva
l
uet
ode
l
a
yde
ve
l
opme
nt
. The problem for thedeveloper is then equivalent to choosing an optimal date to maximize this option
value and, therefore, can be written as:

10 
V ( x) max E0 
e [x( ) D nk ]d e T F 
.



T
T


(8)

The solution for V ( x) must satisfy the fundamental differential equation of optimal
stopping given by:

1 2 2V ( x)
V ( x)
V ( x) 0.
2
2

x

x

(9)

Substituting the term ex into Equation (9) yields the quadratic equation given by:
1 2 2( )  0.
2(10)

The solution to Equation (9) is then given by:
V ( x) A1e1x A2e2 x ,where

A1 and(11)

A2 are constants to be determined, and 
1 and 2 are,

respectively, the larger and smaller roots of Equation (10): 122 0,
1  2  2  2 
 1
2  2  2 2 22 0.
 

(12)

A landowner chooses the timing of development, which is characterized by x* ,the critical value of x that triggers property development.

This critical value, andA1 and A2 in Equation (11), are obtained from the boundary conditions given by:
lim V ( x) 0,(13) x*  D  nk
V ( x* )    2    F ,
     

(14)

x

11

and

V ( x* ) 1
 .

x


(15)

Equation (13) is the limit condition, which states that the value of the option todevelop vacant land is worthless, as the state of nature is extremely poor, i.e., when
the stochastic factorxapproaches minus infinity.

Equation (14) is thevalue-matching condition, which states that, at the optimal timing of development, alandowner should be indifferent as to whether vacant land is developed or not.
Equation (15) is the smooth-pasting condition, which requires that the landowner not
obtain any arbitrage profits from deviating from the optimal timing of development.

Solving Equations (13) to (15) simultaneously yields:*
1A1  e 1x ,

1

(16)

A2 0,(17)

 1
x* ( D, n) F kn D   .
 
1

(18)

and

Differentiating x* ( D, n) with respect to its underlying parameters yields thefollowing results.

12

Proposition 1. Developers will delay development ( x* ( D, n) increases) if: (i) they
incur larger development costs ( F increases); (ii) they receive more rents fromvacant land (  increases); (iii) they expect urban rents to become more volatile ( increases); (iv) the regulator increases the stock of infrastructure ( k increases); (v)urban residents incur larger commuting costs (  increases); (vi) they expect urban
rents to increase less rapidly over time (  decreases); (vii) their land is located far
away from the CBD ( D increases); and (viii) city population grows ( n increases).

The reason for Proposition 1 is as follows.

When deciding whether to develop,

developers must compare the opportunity cost from delaying development (including
the option value from waiting and the value of vacant land), with the net value fromdevelopment.

They will delay development either because the opportunity cost

increases (scenarios (ii) and (iii)), the net value from development decreases(scenarios (i), (iv), (v), (vii), and (viii)), or the opportunity cost is decreased by lessthan the decrease in the net value from development (scenario (vi)).
Substituting A1 given by Equation (16) and A2 given by Equation (17) into
Equation (10) yields the option value from waiting as given by:*
1
V ( x)  e1 ( x x ( D,n )) .

1

(19)

The value of vacant land and that of urban land are thus respectively given by

13

*
 1
Va ( x, D, n)   e1( xx ( D,n )) , if x x* ( D, n) ,
 
1

(20)

and

 x  D nk
Vu ( x, D, n)    2   F , if x x* ( D, n) .
    
Having determined the optimal development strategy ( x* ) at a given location D ,we now follow Fujita (1982) and Turnbull (1988) and ask: Given a state of nature
denoted by x(t ) , what will be the location D* ( x(t )) , which will be associated withthe state-contingent optimal timing of development?

A developer, however,understands the urban growth process and acts with rational expectations.

In

particular, he knows that the city grows outward over time and thus realizes that his
land will lie at the city’s boundary at the time of conversion.

Since the city is linear,

the city’s population n is thus equal to the distance to the boundary, given that eachresident consumes one unit of land.

In other words, in equilibrium when land at

location D is converted, the city’s population n will be equal to D .

We can thus

impose n D , replace x* with x(t ) and D with Dd* ( x(t )) , and then transform
Equation (18) into:1
Dd* ( x(t )) 
[ x(t ) a0 ] ,(k )(21)

1 
where a0 F   , and it is required that x(t ) a0 .

1We may interpret Equation (21) as follows.

14

Suppose that the state of nature x(t )and the distance from the CBD, D , is such that x(t ) is greater than the x* given by
Equation (18).

Some landowners will then develop vacant land further away from the

CBD, increasing D until x* rises to x(t ) .

Equivalently, if the distance from the

CBD, D , is less than the Dd* ( x(t )) given by Equation (21), then some landowners
will develop vacant land further away from the CBD until D is increased to the point
Dd* ( x(t )) .6

Since the city grows outward, Dd* ( x(t )) is, in fact, the equilibrium cityboundary at time t .

Differentiating Dd* ( x(t )) with respect to its underlying parameters yields thefollowing results.

Proposition 2. Without any regulation, the city boundary will be contracted( Dd* ( x(t )) decreases) if: (i) developers incur larger development costs ( F increases);(ii) landowners receive higher agricultural rents (  increases); (iii) developers expect
urban rents to be more volatile (  increases); (iv) the regulator increases the stock ofinfrastructure ( k increases); (v) urban residents incur larger commuting costs ( increases); (vi) developers expect urban rents to increase less rapidly over time ( decreases).

The results of Proposition 2 come directly from Proposition 1.

6

Specifically, if

The above argument resembles that of Pindyck (1988), when he explains the timing decision forundertaking a continuous investment project. See also Lee and Jou (2010) for a thorough discussionof this issue.
15

developers delay development, then the equilibrium city boundary will also shrink.
For example, suppose that developers expect urban rents to become more volatile.
As indicated by Proposition 1(iii), developers will then delay development, because
their option value from waiting becomes more valuable.

As a result, the equilibrium

city boundary will shrink, as developers are less inclined to transform vacant land forurban use.

We can compare the above case with the first-best case in which the public goodis priced at the marginal cost, 2nk .

Following similar procedures as above yields

the first-best growth boundary as given by1
 1
D*f ( x(t )) 
[ x(t )   F ] .
(2k )
 
1

(22)

It is obvious that the equilibrium growth boundary shown in Equation (21) is broader
than the first-best level, as shown in Equation (22).

Consequently, it is possible forthe regulator to implement a growth boundary control that enhances the social
welfare.

Since consumer utility is exogenous in our model, the social planner’s goal isthen to choose an efficient control to maximize the total land value, net of theimplementation cost.

To derive total land value, the value expression in Equation (6)

i
si
nt
e
g
r
a
t
e
da
c
r
os
sa
l
ll
oc
a
t
i
onsD i
nt
hepl
a
nne
r
’
sj
ur
i
s
di
c
t
i
on. The city boundary

16

set by the social planner is denoted by Dc ,and thus city population, n will beequal to Dc , given that the open city is linear.

Imposing this constraint yields theobjective function of the social planner as given by:
Dc

Max W ( Dc ) Max [ Vu ( x, D, Dc )dD
Dc

Dc

0

(23)

B

Va ( x, D, Dc )dD ( Dd Dc )2 ] , 0 ,Dc

where the outer boundary of t
hepl
a
nne
r
’
sj
ur
i
s
di
c
t
i
ons
,denoted by B , is assumed tobe sufficiently large such that B Dc .

Inside the brackets of Equation (23), the first

integral is total urban land value, the second integral is total agricultural land value,
and the last term is the implementation cost, which indicates that the cost for the
regulator to monitor the vacant landowners is an increasing convex function of the
distance between the equilibrium city boundary and the controlled boundary.

This isplausible because the farther the distance of a parcel of land located from thecontrolled city boundary, the more the costs (e.g., transportation costs) for the
regulator to monitor the owner of this parcel of land.

The efficient growth control, denoted by Dc* , is found by totally differentiating
W ( Dc ) in Equation (23) with respect to Dc and setting the result equal to zero.
This yields:dWkVu ( x, Dc , Dc ) Va ( x, Dc , Dc )  Dc
dDc

17

(24)

k B
*
 e1 ( x x ( D, Dc )) dD 2
( Dd Dc ) 0 .
D
 c
Differentiating dW / dDc in Equation (24) with respect to Dc yields*
d 2W (2k ) 2 
[1 e1 ( x x ( Dc , Dc )) ]

dDc

(25)

*
*
k2
 [e1( xx ( Dc , Dc )) e1( xx ( B , Dc )) ] 2,

We must restrict  to be positive in order to derive an interior solution for Dc .
The reason is as follows.

Without any implementation cost ( 0 ), Equation (25)indicates that 0 because x x* ( Dd* , Dd* ) x* ( Dc* , Dc* ) (since Dd* is assumed
to be larger than Dc* ), such that the first two terms on the right-hand side of Equation(25) are both positive.

In other words, without any implementation cost, theobjective function for the social planner shown in Equation (23) becomes a strictly
convex function of the level of the boundary control, Dc .

As a result, the regulator

will either set the efficient boundary at Dc* 0 or Dc* Dd* , and thus the problem inconsideration becomes a trivial one.

We thus consider only the case where 0 .

For this case, Equation (24) indicates that if a regulator slackens the boundary
control ( Dc increases), then the society as a whole receives the value of urban land at
location Dc , Vu ( x, Dc , Dc ) , but suffers the cost of the sum of the following three
items: Va ( x, Dc , Dc ) ,kDc k B 1 ( x x* ( D, Dc )),edD , and 
2 ( Dd Dc ) .
Dc
 

18

The first

term represents the loss of the value of agricultural land at location Dc that istransformed to urban use, the second term represents the costs borne by all existing
urban residents as new urban residents flow in.

The third term represents the loss ofthe option value borne by all owners of vacant land because as the regulator slackens
the boundary control, these landowners will respond to delay development (since
x* ( D, Dc ) / 
Dc 0 ), thus reducing the option value from waiting.7

Finally, the lastterm represents the cost savings associated with slackening the control policy.

We may compare Dc* with Dd* in Equation (21), the equilibrium city boundary,which is determined by the condition that at the city boundary, urban land value is justequal to vacant land value, i.e., Vu ( x, Dd* , Dd* ) Va ( x, Dd* , Dd* ) .

Equation (24)suggests

the

that

the

efficientcityboundary

is

set

by

condition*

kD k B 1( xx ( D, Dc ))Vu ( x, Dc , Dc ) Va ( x, Dc , Dc )  c  edD 2
( Dd Dc ) ,  Dcwhich

implies that if the regulator sets the city boundary at the equilibrium level, then themarginal cost of the control will be larger than the marginal benefit (provided that is sufficiently small).

Consequently, the regulator can set an efficient boundary at alower level to enhance the social welfare.

Differentiating Equation (24) with respect to its underlying parameters yields the7This effect resembles those of the other forms of land use regulation policies, such as density ceiling
control (Jou and Lee, 2007; Turnbull, 1991), a threat to expropriate development rights (Riddiough,1997; Turnbull, 2002), and the threat of a development moratorium (Turnbull, 2004b). See athorough review by Turnbull (2005).
19

following results.

Proposition 3. The regulator should tighten the boundary control if (i) developers
incur larger costs to develop vacant land; (ii) landowners receive higher agriculturalrents; (iii) developers expect urban rents to evolve more volatile; and (iv) it is lesscostly for the regulator to implement the control policy.

Proof: See Appendix A.

The reason for Proposition 3 is as follows.

When the development cost

increases, a regulator should tighten the boundary control so as to avoid somelandowners to incur more costs to convert agricultural land into urban use.

When

agricultural rents increase, the regulator should also tighten the boundary control
because the opportunity cost for landowners to transform vacant land into urban use
increases.

As urban rents become volatile, those owners of vacant land will delay

development such that the regulator can tighten the boundary control to accommodate
this delaying effect.

Finally, as the implementation cost decreases, the regulator

naturally has more incentives to tighten the control.

We may compare our results with those of Ding et al. (1999), who assume that
there exists no uncertainty, i.e., 0 .

Substituting this condition into Equation (10)yields 
1 / , and thus the critical level of x (t ) that triggers development, as20

shown in Equation (18), and the equilibrium city boundary, as shown in Equation (21),will respectively becomex* ( D, n) F nk D ,(26)1
Dd* ( x(t )) 
[ x(t ) F ] .
(k )(27)

andWe provide a rigorous proof for x* ( D) shown in Equation (27) in Appendix B.

Figure 1 illustrates the main difference between our result and that of Ding et al.
(1999) for the case in the absence of any uncertainty.
axis,

In that Figure, the horizontalx(t ) , has one-to-one correspondence with calendar time tx(t ) x(0) t .

becauseThe path Dd* is that shown in Equation (27), and Dc* is the Dcthat satisfies Equation (24).

Ding et al. design the efficient city boundary as the onethat imposes no control before a critical date, t * , and then sets the city boundary at
Dd* ( x(t * )) afterwards.

Without any uncertainty, both of ours and Ding et al.’s are

two different ways to control the city boundary.

Ours is similar to that proposed by

Brueckner (1990), which is more lenient to those landowners whose land is located
far away from the CBD.

Besides, the control also accommodates demand for urban

land because we assume that demand for urban land increases over time such thaturban rents also increase over time.

21

When there is uncertainty, it is not plausible for a regulator to directly control thedevelopment timing such that it is infeasible for the regulator to implement the control
set in Ding et al.

It is, however, still possible to implement the control as addressedin our framework, as shown in the next section.

IV.

Numerical Analysis

We employ plausible parameter values to investigate how the characteristics ofdemand and supply conditions in the real estate market affect the design of efficientboundary controls. The benchmark values we choose are as follows.

The fixed

component of urban rents is equal to 2.0 ( 2.0 ), the return to vacant land is fixed at
0.1 unit ( 0.1 ), and the development cost is set at 1 unit ( F 1 ).

The factor thatshifts urban rents is expected to grow at 3% per year ( 3% ), and the volatility ofthis shift factor is equal to 20% per year ( 20% ).

As the distance from the CBD

increases by one mile, urban rents decrease by one-tenth unit ( 0.1 ).

We alsoassume that the stochastic factor that affects urban rents, i.e., x(t ) , is equal to 1.

All

developer are risk-neutral and face a discount rate equal to 10% per year ( 10% ),and the stock of infrastructure is normalized at one unit ( k 1 ).
of the planner’s jurisdictions is set at 15 miles ( B 15 ).

22

The outer boundary

Finally, the parameterassociated with the implementation cost, , is set at 1,000 ( 1, 000 ).

Given these parameter values, Table 1 shows that the equilibrium city boundary
is equal to 2.253 miles, i.e., Dd* 2.253 ; the efficient city boundary is equal to 2.225
miles, i.e., Dc* 2.225 , and the total social welfare W * is equal to 91.37, which is
0.85% greater than its counterpart in the absence of any regulation, i.e., 90.60.
Table 1 also shows that F changes in the region (0.5, 1.5),  changes in the region(0, 0.2),  changes in the region (15%, 25%), k changes in the region (0.5, 1.5),
 changes in the region (0.05, 0.15),  changes in the region (1%, 3%), and changes in the region (500, 1500), holding all other parameters at their benchmark
values.

Several important findings derived from Table 1.

First, for all exogenous forces

except for the implementation cost factor ( ), increases in these factors have the same
qualitative effects on both the equilibrium city boundary and the efficient cityboundary.

Specifically, similar to Ding et al. (1999), we find that an increase ineither the development cost, the agricultural rent, or the stock of infrastructure
decreases the optimal size of the city.8

Furthermore, we also find that a decrease in

the commuting cost expands the optimal city size.

8

The last scenario will hold in the framework of Ding et al. if they assume that the marginal cost ofproviding the public good is increasing (rather than decreasing) in the stock of infrastructure.
23

We contribute to the literature by furthermore providing the following testable
implications: The regulator will set a more stringent boundary control policy ifdevelopers expect urban rents to grow less rapidly or to be volatile, or if the regulator
spends less to implement the control policy.

To explain our policy implications more clearly, we use the same benchmark
parameter values to generate Figure 2 (expect for x(t ) ), which presents a series ofstochastic rents x(t ) , the corresponding equilibrium city boundary, Dd* , and theefficient city boundary, Dc* .

Starting from t 0 and x(0) 1 , the figure showsthat the corresponding Dd* and Dc* are equal to 1.344 and 1.315, respectively.
Both the equilibrium city boundary Dd* and the efficient city boundary Dc* will notincrease unless x(t ) exceeds all its historical values.

This occurs at those years:t  0.1, 0.4, 1.3, 1.5, 1.9, 2.3, 3.1, 8.4, 9.3, 9.4, and 9.5.

In other words, theregulator will not alter the city boundary control when urban rents move downward,
but will slacken the control when urban rents move upward.

This type of control,

which pushes the city boundary outward as time goes by,9 is also consistent with theexisting land-use planning system that requires local governments to periodicallychange city boundaries as population grows (See, e.g., Ding et al. (1999) and Jaeger
and Plantinga (2007)).
9

Cumming (2010) reports how property speculators buy up land on the Auckland city outskirts andwaiting until the time was ripe to develop. City governments are thus usually forced to broaden theboundary to meet the increase in demand for urban land.
24

V. Conclusion

This article extends Ding et al. (1999) to investigate the design of growthboundary controls in an open city in a framework where a congestible public good is
priced at average cost. Unlike Ding et al., we consider that urban rents evolve
stochastically over time and that the regulator can slacken the boundary control whenurban rents move upward.

We provide three novel testable implications, namely, theregulator will set a more stringent boundary control policy if (i) developers expect
urban rents to grow less rapidly; (ii) developers expect urban rents to be more volatile;and (iii) it costs less for the regulator to implement the control policy.

Our article considers an open city model and, thus, suffers the shortcomings ofthis consideration.

As Brueckner (1990) indicates, this implies that populationpressure and the resulting excess demand for housing plays no role in determining themarket impact of growth controls.

We can follow the supply-restriction model, as

outlined in both Brueckner (1995) and Helsley and Strange (1995), to take this into
account.

Furthermore, we may follow Brueckner and Lai (1996) to distinguish

renters from landlords in the model. We may then compare the results of this article
with those of the more generalized model.

25

Appendix A:
Totally differentiating Equation (24) with respect to F , , , and 
rearranging yields
Dc* 1 0 ,

F


(A1)


Dc* 2 0 ,

 

(A2)


Dc* 3 0 ,

 

(A3)


Dc* 4 0 ,

 

(A4)

and

where as shown in Equation (25), 0 , and*1 (dW / dDc ) / 
F 
kDc )
1 ( x x ( Dc , Dc ) 


2 
0 ,
(k )
2
2 (dW / dDc ) /  1 ( x x* ( Dc , Dc ) kDc )0 ,
(k )*1 
1 [( 1 k )( x* ( D , D ) x )e 
1 ( x ( Dc , Dc ) x )
3 
c
c

  
1 

*
k
1
 ( x* ( B, Dc ) x)e1 ( x ( B , Dc )x ) ] 0 ,2(k )
1

given that 

0 ,
1/

and
4 2( Dd Dc ) 0 .

26

Q.E.D.

Appendix B:
In the absence of any uncertainty x() x . Substituting this into Equation(8) yields the objective function of the developer as given by1V ( x) max U (T , x) [ (x(T  D nk )) F ]e T .
T



(B1)

Intuitively, a landowner needs to decide to either develop immediately in the initialperiod or delay development.
0 .

The former occurs if 0 , while the latter occurs if

Suppose that the landowner develops at a time T T * 0 .

Differentiating

Equation (A1) with respect to T , and then setting the result equal to zero yields*

U (T * , x)[ F nk ( x D ) T * ]e T 0 .

T

(B2)

The second-order condition is given by2U (T * , x)T2*e T 0 ,(B3)which holds if 0 .

Equation (B2) indicates that the interior solution for T is given by1
T *  [ F nk D x] 0 ,
(B4)which requires that( F nk ) D x .

(B5)

27

Imposing T * 0 and replacing x by x* yields x* as shown in Equation (26).

Substituting T T * into Equation (B1) yields the option value of waiting as givenby*
V ( x)  2 eT .


(B6)

Q.E.D.

28

Table 1: Efficient Boundary Controls and Efficient Impact Fees.
Benchmark Case: F 1 , 0.1 , 0.2 , k 1 , 0.1 , 0.03 , 2 , x(t ) 1 , 1, 000 , B 15 ,0.1 .

Variation in F
0.50

0.75

1.00

1.25

1.50

Dc*

2.270

2.247

2.225

2.201

2.179

Dd*

2.298

2.276

2.253

2.230

2.208

W*

92.76

92.06

91.37

90.69

90.01

Dc* / Dd*

0.988

0.988

0.987

0.987

0.987

Variation in 
0.00

0.05

0.10

0.15

0.20

Dc*

2.315

2.270

2.225

2.179

2.134

Dd*

2.344

2.298

2.253

2.208

2.162

W*

79.16

85.26

91.37

97.51

103.68

Dc* / Dd*

0.988

0.988

0.987

0.987

0.987

Variation in 
0.150

0.175

0.200

0.225

0.250

Dc*

2.323

2.274

2.225

2.174

2.124

Dd*

2.348

2.301

2.253

2.205

2.156

W*

82.51

86.80

91.37

96.19

101.22

Dc* / Dd*

0.989

0.988

0.987

0.986

0.985

Variation in k
0.50

0.75

1.00

1.25

1.50

Dc*

4.117

2.895

2.225

1.799

1.502

Dd*

4.131

2.916

2.253

1.836

1.549

W*

123.95102.78

91.37

84.43

79.95

Dc* / Dd*

0.997

0.993

0.987

0.980

0.970

Variation in 
0.050

0.075

0.100

0.125

0.150

Dc*

2.318

2.271

2.225

2.179

2.134

Dd*

2.360

2.305

2.253

2.203

2.155

W*

109.95

99.41

91.37

85.16

80.11

Dc* / Dd*

0.982

0.985

0.987

0.989

0.990

29

Variation in 
0.01

0.02

0.03

0.04

0.05

Dc*

2.157

2.193

2.225

2.251

2.275

Dd*

2.182

2.220

2.253

2.282

2.307

*

75.73

83.16

91.37

100.33109.98

Dc* / Dd*

0.989

0.988

0.987

0.987

0.986

W

Variation in 
500

750

1000

1250

1500

Dc*

2.193

2.214

2.224

2.230

2.234

Dd*

2.253

2.253

2.253

2.253

2.253

W*

92.23

91.65

91.37

91.21

91.11

Dc* / Dd*

0.973

0.983

0.987

0.990

0.992

Note: The terms, F , , , k , , , , , B , , Dc* , Dd* , and W * denote the development costs, thereturn to vacant land, the volatility of the stochastic urban rent, the stock of infrastructure, the commuting costs per unit
distance to the CBD, the expected growth rate of the stochastic rent, the fixed component of urban rents, the parameterassociated with the implementation cost, the outer boundary of the planner’s jurisdiction, the discount rate per year, theefficient city boundary, the equilibrium city boundary, and total social welfare.

30

Dd*
Dc*
Dd* ( x(t * ))x (t )x (t * )Figure 1: The efficient city boundary control in the absence of any uncertainty.

This graphs showsthat the city boundary designed by Ding et al. (1999) is to impose no control before t t * such thatthe efficient city boundary is the same as the equilibrium city boundary, Dd* ( x(t )) .
after t t , the regulator then imposes a fixed level of city boundary at*

Dd* ( x(t * )) .

Immediately

By contrast, theefficient city boundary in our framework, Dc* ( x(t )) , is always lower than the equilibrium level,

Dd* ( x(t )) .

31

3

2.5

x(t)
Dd*

2

Dc*

1.5

1

0.5

t (years)

0
0

1

2

3

4

5

6

7

8

9

10

-0.5

-1

-1.5

Figure 2: The Equilibrium City Boundary, Dd* , the Efficient City Boundary, Dc* , and the Stochastic
Component of Urban Rents, x(t ) , as a Function of Calendar Year, t .
change unless x(t ) exceeds x() , for all t .

32

Both Dd* and Dc* will notReferences
Brueckner, J.K., 1990. Growth controls and land values in an open city, Land
Economics 66(3), 237-248.
Brueckner, J.K., 1995. Strategic control of growth in a system of cities, Journal of
Public Economics 57, 393-416.
Brueckner, J.K., 1997. Infrastructure financing and urban development: theeconomics of impact fees, Journal of Public Economics 66, 383-407.
Brueckner, J.K., 2007. Government land-use interventions: an economic analysis,
paper presented at the 4th Urban Research Symposium, World Bank, Washington,
D.C.
Brueckner, J.K., Lai, F.-C., 1996. Urban growth with resident landowners, Regional
Science and Urban Economics 26, 125-143.
Capozza, D., Helsley, R., 1990. The stochastic city, Journal of Urban Economics 28,
187-203.
Capozza, D., Li, Y., 1994. The intensity and timing of investment: the case of land,
American Economic Review 84(4), 889-904.
Capozza, D., Sick, G..,1994. The risk structure of land markets, Journal of Urban
Economics 35, 297-319.
Cooley, T.F., LaCivita, C.J., 1982. A theory of growth controls, Journal of Urban
Economics 12, 129-45.
Cox, J. C., Ross, S. A., 1976. The valuation of options for alternative stochastic
process, Journal of Financial Economics 3(1), 145-166.
Cumming, G., 2010. Pushing the urban boundaries, New Zealand Herald (Feb. 13).

33

Ding, C., Knaap, G. J., Hopkins, L. D., 1999. Managing urban growth with urban
growth boundaries: A theoretical analysis, Journal of Urban Economics 46, 53-68.
Engle, R., Navarro, P., C., Carson, R., 1992. On the theory of growth controls, Journalof Urban Economics 32, 269-283.
Fujita, M., 1982. Spatial patterns of residential development, Journal of Urban
Economics 12(1), 22-52.
Helsley, R. W., Strange, W. C., 1995. Strategic growth controls, Regional Science and
Urban Economics 25, 435-460.
Jou, J-B., Lee, T., 2007. Do tighter restrictions on density retard development? Journalof Real Estate Finance and Economics 34(2), 225-232.
Jou, J-B., Lee, T., 2008. Taxation on land value and development when there arenegative externalities from development, Journal of Real Estate Finance andEconomics 36(1), 103-120.
Juergensmeyer, J. C., Roberts, T. E., 2003. Land use planning and development
regulation Law, West Group, St. Paul, Minnesota
Lee, T., Jou, J-B., 2010. Urban spatial development: a real options approach, Journalof Real Estate Finance and Economics 40, 161-187.
Mayo, S., Sheppard, S., 2001. Housing supply and the effects of stochastic
development control, Journal of Housing Economics 10, 109-128.
McFarlane, A., 1999. Taxes, fees, and urban development, Journal of Urban
Economics 46(3), 416-436.
Pindyck, R.S., 1988. Irreversible investment, capacity choice, and the value of thefirm, American Economic Review 78(5), 969-985.
34

Riddiough, T.J., 1997. The economic consequences of regulatory taking risk on land
value and development activity, Journal of Urban Economics 41(1), 56-77.
Sakashita, N., 1995. An economic theory of urban growth control, Regional Scienceand Urban Economics 25, 427-434.
Sasaki, K., 1998. Optimal urban growth controls, Regional Science and Urban
Economics 28, 475-496.
Turnbull, G.K., 1988. Residential development in an open city, Regional Science and
Urban Economics 18(2), 307-320.
Turnbull, G.K., 1991. A comparative dynamic analysis of zoning in a growing city,”Journal of Urban Economics 29(2), 235-248.
Turnbull, G.K., 2002. Land development under the threat of taking, Southern
Economic Journal 69(2), 290-308.
Turnbull, G.K., 2004a. Urban growth controls: transitional dynamics of development
fees and growth boundaries, Journal of Urban Economics 55(2), 215-237.
Turnbull, G.K., 2004b. Development moratoria, Journal of Housing Economics 13(3),155-169.
Turnbull, G.K., 2005. The investment incentive effects of land use regulations, Journalof Real Estate Finance and Economics 31(4), 357-395.

35