Math Honors Paper - Tu Nguyen - 2015

Incentivizing Development to Maximize Property Value

Tu Nguyen

Advisor: Dr. Marc Ordower

Presented to the Department of Mathematics

in partial fulfillment of the requirements

for a Bachelor of Science degree with Honors

Randolph College

Lynchburg, Virginia

May 6, 2015

Abstract

Greenfield development is considered the solution to address two problems in the United States:

increasing housing demand because of the growing population and expanding local government

budget deficits as a result of the recent global financial crisis. Greenfield development can tackle

these problems through effective zoning practices. This paper develops time-dependent

dynamical models to test how using incentives might help build high-quality and livable

communities while maximizing property value. The results show that there is an inverse

relationship between the number of parklands and the total property value. As a result, there is a

trade-off between having more parklands and maximizing property value.

1

Table of Contents

Chapter 1: The Two Problems …………………………………………………………………... 3

Chapter 2: Greenfield Development …………………………………………………………….. 4

Chapter 3: Traditional Zoning vs Performance Zoning …………………………………………. 6

Chapter 4: Models ……………………………………………………………………………….. 8

Chapter 5: Results ……………………………………………………………………………… 17

Chapter 6: Discussion ………………………………………………………………………….. 38

References ……………………………………………………………………………………… 41

2

Chapter 1

The Two Problems

It was estimated that the United States population would grow by almost 58 million people

during the period 2003-2025, which is about one sixth of the current population. Because of the

limited availability of natural resources, particularly land, a problem arises: How can we find

housing for this new population? Many people see infills, the redevelopment and revitalization of

obsolete and underutilized buildings, as a method to meet the increasing demand for housing.

However, Heid (2004) was afraid that infill strategies cannot happen fast enough to make any

difference by 2025 [1]. In addition, infill redevelopment is often costly because it involves

several steps ranging from purchasing land, removing existing buildings, and cleaning up

environmental contamination onsite [2] [3]. As a result, infill redevelopment is not the ideal

solution to satisfy the new demand for housing.

In addition, it is found that since 2010, there have been 47 municipal bankruptcy filings. Many

local governments across the U.S. still face steep budget deficits [4]. The question is, how can

these financially struggling municipalities find a way to get out of the deficit?

These two completely different problems, fortunately, have a similar solution. These two issues

can be addressed at the same time by practicing greenfield development.

3

Chapter 2

Greenfield development

Greenfield development refers to the process of creating planned communities on previously

undeveloped land in the vicinity of a city. The land used is usually farmland. Following is a list

of characteristics of a greenfield:

Be rural or really low-density land.

Contain significant natural, cultural, or agricultural resources.

Be located outside recognized urban limits.

In addition, Heid (2004) also noted that greenfield development is capable of building high

quality, diverse living environments and is also the most practical and affordable method to build

without sprawl [1]. Furthermore, it has been proven to cost less than infill development [2].

As shown, greenfield development is a viable solution to the problem of finding housing for the

new population. In addition, it also helps tackle the second problem of local government budget

deficits by increasing property value.

Campbell (1951) found that property taxes are the main mechanism through which local

governments raise revenues. The basic formula to determine property taxes is

Property tax=Determined market value of the property× Current tax rate .

Consequently, to increase the amount of property taxes, a local government can either raise the

current tax rate or try to increase the market value of all the properties in the area. The first

4

method of raising the current tax rate is not feasible since it often faces opposition from local

residents. On the other hand, the second method can be achieved through effective zoning

practices.

5

Chapter 3

Traditional Zoning vs Performance Zoning

Zoning is an urban planning tool used by local governments to better relocate their resources and

to segregate uses that are incompatible to protect the property value. Land uses are often divided

into three main categories: residential, commercial, and industrial. In addition, there are several

zoning approaches: Traditional or Euclidean zoning, performance zoning, incentive zoning, and

form-based zoning. The two common types that will be discussed in this paper are Euclidean

zoning and performance zoning.

Euclidean zoning segregates different categories of land uses into separated zones to avoid

incompatible uses. For example, if an area is zoned residential, no commercial buildings are

allowed to be built in the area. Although it is a common approach in the United States, the

traditional zoning method has faced criticism for several decades for its inflexibility. Reps (1964)

pointed out that the separation of land uses can prevent mixtures of uses even though it is more

desirable [5]. For example, it might be more beneficial to have a grocery store in a residential

area rather than locating all the grocery stores in just one place. As a result, alternatives to the

traditional zoning practice have been developed to address its problems.

An alternative to the traditional zoning method that is commonly used in many countries is

performance zoning. Performance zoning establishes a set of goal-oriented and performance

standards to regulate land development. Some examples of the performance standards are setting

6

a limit on the noise level in a neighborhood and protecting the natural environment. The

performance zoning method provides the flexibility and transparency not inherent in the

traditional method. In addition, it also tends to create more diverse and livable communities.

In this paper, the performance zoning approach was used to create a model of greenfield

development in a small city with the criteria of maximizing property value and enhancing

livability. The next section will cover the models used.

7

Chapter 4

Models

Time-dependent dynamical models are developed in Excel on a grid of size 6x8, with each cell

representing one farm. Different from typical zoning practices, in this paper, four categories of

land uses are included: single-family residential, multi-family residential, commercial, and

parkland. Industrial use is left out because the city considered in this study is relatively small and

thus is not heavily industrialized. In addition, parkland is included to create a livable

neighborhood. It should be noted that each farm may be purchased and developed by a separate

developer. Additionally, each developer is going to make development decisions to maximize his

profit. The city government cannot require how the development should take place.

654321

_ _ + _ _ _ _ _1 2 3 4 5 6 7 8

Figure 1. A 6x8 grid of greenfield land.

In the model, these four uses were abbreviated as below:

P – Parkland.

S – Single-family residential.

M – Multi-family residential.

8

C – Commercial.

In addition, the underscore signs “_” represent the highway and the plus sign “+” signify an

enter/exit to and from the highway. Number 0 is used to indicated an undeveloped cell. As a

result, Figure 2 shows how the grid looks like at time 0.

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0_ _ + _ _ _ _ _

Figure 2. Initial configuration

It is important to note each of these cells may be developed into a 2x2 grid of city blocks and

each block in the 2x2 grid maybe developed to a purpose, independently of how the other blocks

are developed. For example, a cell named “SPCM” represented four street blocks with their

respective order as follows:

S PC M

Figure 3. Street blocks contained in a cell.

From now on, I will refer to each of those small blocks as “blocks” and the cells containing 4

blocks as “cells”. From the definition of a cell as described above, there are 44=256 possible

combinations of blocks in a cell. In other words, every cell can be in one of the 256 possible

states.

9

A developer would only have the incentive to purchase an undeveloped lot if it generates the

highest potential positive profits. The value of a cell to the developer is determined from the

following formula:

Value ¿developer=Development value−Undeveloped value−Development cost .

Where,

The undeveloped value is the cost of purchasing a piece of farmland.

The development cost takes into account all the four costs of developing each street block in a

cell. Generally speaking, building a multi-family residential property from a greenfield land costs

the most while developing parkland is the cheapest.

The development value is the potential value of a cell. It consists of the value determined from

how far away a cell is from the highway and of internal and external values, both of which are

again potential values. Generally speaking, the closer to the highway, the higher value a cell has.

The internal value of a cell is the total value from each of the four blocks in a cell, which is based

on the existence of other blocks in the cell. Each of the 256 possible cells has its own internal

value. For example, let’s consider the cell “SMCC”. The internal value of this cell is calculated

by adding the value of each individual block: S, M, C, and C. The value of C, for example, is

based on the number of S and the number of M in the cell itself.

The external value of a cell, on the other hand, depends on the states of all the cells that are of

one distance away from it. For example, the external value of cell (2,3) is derived from the states

of the following four cells: (2,2), (2,4), (1,3), and (3,3). Let’s consider the following example.

10

MPMSMCCC SPMC CCCC

PCCS

Figure 4. The external value of a cell.

The middle cell, “SPPC”, has an external value that is derived from the other four surrounding

cells. Its external value is calculated by adding the value to each block in the cell, namely S, P,

M, and C, from the surrounding cells. Note that the potential external value of a cell is influenced

by surrounding blocks that were already developed.

From these definitions, it can be seen that at time 0, each of the 48 cells may have a positive

potential internal value from some combinations of blocks but only has a zero potential external

value since no cells have been developed yet.

Following is the list of benefit functions:

For internal value:

Benefit ¿P=0.

Benefit ¿S= 12(¿ P)2 −1× (¿S )2+6 × (¿S )+4.5× (¿C ) .

Benefit ¿ M= 12(¿ P)2

+8 × (¿C ) .

Benefit ¿C=4×(¿ S)2+8 × (¿ M ) .

For external value:

Benefit ¿P=0.

Benefit ¿S= 8(¿ P)2 −0.5× (¿S )2+5 × (¿S )+3 × (¿C ) .

11

Benefit ¿ M= 8(¿ P)2

+2.5× (¿C )2 .

Benefit ¿C=2×(¿ S)2+8× (¿ M ) .

Note that in the above cases, when ¿ P=0, the benefit from parkland was set to be equal to 0.

In both cases, Benefit ¿P was assumed to be 0 because parkland would bring no profit to the

developer.

For the internal values, Benefit ¿S was a function of the number of surrounding parklands,

single-family residential properties, and commercial properties. It was believed that Benefit ¿S

would have the highest value if there was one nearby parkland. The higher the number of

parklands, the lower the benefit. In addition, having a dense neighborhood would be beneficial,

but only up to a point. After that, there would be a problem of overcrowding. As a result, a

downward-opening parabola was chosen to measure the influence of surrounding single-family

residential properties. Finally, not surprisingly, the higher the number of nearby commercial

properties, the higher the value to S. A household would love to be able to go shopping and get

groceries within walking distances.

Using the same reasoning, Benefit ¿ M was a function of the number of parklands closeby and

the number of surrounding commercial properties.

Additionally, Benefit ¿C depended on the number of single-family and multi-family residential

properties nearby. Every commercial company would appreciate having a large customer base in

12

the surrounding area as it can make more profits. Hence, the higher the number of surrounding

single-family and multi-family residential properties, the higher the benefit to C.

Finally, the same logic was applied to compute the external values for P, S, M, and C. However,

the benefits derived from surrounding neighborhoods would have a lower multiplier, taking into

account the distance away.

Regarding the costs to a developer, the undeveloped value was fixed at 70 and the development

costs for P, S, M, and C were 2, 5, 10, and 9 respectively.

Finally, the process of running the development is outlined as below:

1. List all the combinations of blocks for each cell.

2. Calculate the potential internal values and the potential external values for each

combination.

3. Weigh in the development cost and the undeveloped value to compute the value to the

developer for each combination.

4. Compare all the values to the developer and determine the maximum positive value.

Choose a combination that has the highest value for the cell being considered.

5. Do steps 1-4 for all the cells in the model.

6. After all the cells have been considered, move on to the next time period. Note that once

a cell has been developed, it will not change throughout the rest of the model.

13

Several models were conducted in this study. For the first model, a developer would develop his

lot rationally, that is by trying to maximize his profit. For the later models, an incentive was

introduced into those models and again development was also conducted rationally. It is believed

that the decisions of developers working independently based on their own profit do not

necessarily coincide with those of the municipal government nor the current and future residents

of this region. As a result, the local government will try to incentivize development through

grants to maximize livability and property value, with access to parkland being used as a

measure of livability. The incentive considered was a grant from the local government to the

developer for developing a parkland within a cell. With this grant, for models 2-4, the

undeveloped value was now lower for all the cell states containing parklands and remained 70

for all other combinations. In addition, a different type of incentive was introduced to model 5,

which helped pay for the cost of building a parkland. In other words, the development cost for P

was 0 in model 5.

From the above description of the models used in this study, it is relatively straightforward to

realize that these models are actually a slight variation of a cellular automaton. A cellular

automaton is a discrete, abstract computational system of simple, homogenous objects, called

“cells”. These cells have the following characteristics:

1. The cells live on a grid.

2. Each cell has a state. The number of state is generally finite. For the simplest case, there are

two possible states, 0 or 1. In addition, at each discrete time step, each cell is in only one

state.

3. Each cell has a neighborhood. This is typically a set of directly adjacent cells.

14

These cells evolve through a number of discrete time steps according to a set of propagation

rules. These rules specify how a cell will develop based on the current state of its neighboring

cells.

As seen, to model a system using cellular automata, one will have to define the dimension of the

grid, specify the number of states a cell can have, and construct a function that governs the

evolution of each cell based on its neighboring cells’ current state. The cells will then evolve in

discrete time steps according to the set of propagation rules. The definition of a cellular

automaton is relatively straightforward. However, generally the propagation rules must be

chosen carefully to allow researchers to have meaningful and interesting results.

Cellular automata have been used to model and study natural systems. For example, these

biological and physical systems have been modelled effectively using cellular automata: Urban

development, gas behavior, the flow of electricity in a power grid, and life itself.

In this paper, the time-dependent dynamical models do resemble a cellular automaton. The field

in this study is a two-dimensional grid of size 6 by 8. In addition, each cell can have one of 256

possible states. Finally, the propagation rules in this system are a function that maximizes the

value to a developer from a single cell. However, there are some modifications from a typical

cellular automaton. First, instead of having to set an initial configuration for the system at time 1

in a typical cellular automaton, the propagation rules in this study will govern the initial

configuration at time 1. Second, different from a normal cellular automaton, each cell in this

model, after being developed, will not change its state in the next time steps. This means after a

15

cell is developed as “SPCC” at time t, its state will remain “SPCC” at time t+1, t+2, t+3, etc.

Finally, generally most cellular automata do not have a concrete orientation, in the sense that it is

hard to tell the position of a cell in the grid. However, in this project, The Cartesian coordinate

system is used to specify each point in the grid. This modification is needed to tracks the value

that comes from the distance from a cell to the highway.

Chapter 5

Results

16

The following figures show how the first model develops over time:

Model 1 – No incentive:

t = 0

17

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0

_ _ + _ _ _ _ _

t = 1

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 0 0 0 0 0 01 0 0 SSSC 0 0 0 0 0

_ _ + _ _ _ _ _

t = 2

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 SSSC 0 0 0 0 01 0 SSSC SSSC SSSC 0 0 0 0

_ _ + _ _ _ _ _

t = 3

18

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 SSSC 0 0 0 0 02 0 CCCC SSSC CCCC 0 0 0 01 SSSC SSSC SSSC SSSC SSSC 0 0 0

_ _ + _ _ _ _ _

t = 4

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 SSSC 0 0 0 0 0

3 0MMM

M SSSCMMM

M 0 0 0 0

2MMM

M CCCC SSSC CCCCMMM

M 0 0 01 SSSC SSSC SSSC SSSC SSSC SSSC 0 0

_ _ + _ _ _ _ _

t = 5

6 0 0 0 0 0 0 0 05 0 0 SSSC 0 0 0 0 04 0 SSCC SSSC SSCC 0 0 0 0

3 CCCCMMM

M SSSCMMM

M CCCC 0 0 0

2MMM

M CCCC SSSC CCCCMMM

M SSCC 0 01 SSSC SSSC SSSC SSSC SSSC SSSC SSSC 0

_ _ + _ _ _ _ _

19

t = 6

6 0 0 SSSC 0 0 0 0 05 0 MCCC SSSC MCCC 0 0 0 0

4MMM

M SSCC SSSC SSCCMMM

M 0 0 0

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M 0 0

2MMM

M CCCC SSSC CCCCMMM

M SSCC MCCC 01 SSSC SSSC SSSC SSSC SSSC SSSC SSSC SSSC

_ _ + _ _ _ _ _

t = 7

6 0 MMMC SSSC MMMC 0 0 0 05 MCCC MCCC SSSC MCCC MCCC 0 0 0

4MMM

M SSCC SSSC SSCCMMM

M CCCC 0 0

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC 0

2MMM

M CCCC SSSC CCCCMMM

M SSCC MCCC MMMC1 SSSC SSSC SSSC SSSC SSSC SSSC SSSC SSSC

_ _ + _ _ _ _ _

t = 8

6 MMCC MMMC SSSC MMMC MMCC 0 0 0

5 MCCC MCCC SSSC MCCC MCCCMMM

M 0 0

4MMM

M SSCC SSSC SSCCMMM

M CCCCMMM

M 0

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC2 MMM CCCC SSSC CCCC MMM SSCC MCCC MMMC

20

M M1 SSSC SSSC SSSC SSSC SSSC SSSC SSSC SSSC

_ _ + _ _ _ _ _

t = 9

6 MMCC MMMC SSSC MMMC MMCC MCCC 0 0


M CCCC 0

4MMM

M SSCC SSSC SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

t = 10

6 MMCC MMMC SSSC MMMC MMCC MCCCMMM

M 0


M CCCCMMM

M

4MMM

M SSCC SSSC SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

21

t = 11


M CCCC


M CCCCMMM

M

4MMM

M SSCC SSSC SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

Total property value was 27,218.5.

For the second model with incentive:

Model 2: Incentive with a grant amount of 15.

t = 0

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0

_ _ + _ _ _ _ _

t = 1

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 0 0 0 0 0 01 0 0 SSSP 0 0 0 0 0

22

_ _ + _ _ _ _ _

t = 2

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 SSSC 0 0 0 0 01 0 SSSC SSSP SSSC 0 0 0 0

_ _ + _ _ _ _ _

t = 3

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 SSSC 0 0 0 0 02 0 CCCC SSSC CCCC 0 0 0 01 SSSC SSSC SSSP SSSC SSSC 0 0 0

_ _ + _ _ _ _ _

t = 4

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 PSSS 0 0 0 0 0

3 0MMM

M SSSCMMM

M 0 0 0 0

2MMM

M CCCC SSSC CCCCMMM

M 0 0 01 SSSC SSSC PSSS SSSC SSSC PSSS 0 0

_ _ + _ _ _ _ _

23

t = 5

6 0 0 0 0 0 0 0 05 0 0 SSSC 0 0 0 0 04 0 SSCC PSSS SSCC 0 0 0 0

3 CCCCMMM

M SSSCMMM

M CCCC 0 0 0

2MMM

M CCCC SSSC CCCCMMM

M SSCC 0 01 SSSC SSSC PSSS SSSC SSSC PSSS SSSC 0

_ _ + _ _ _ _ _

t = 6


4MMM

M SSCC PSSS SSCCMMM

M 0 0 0

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M 0 0

2MMM

M CCCC SSSC CCCCMMM

M SSCC MCCC 01 SSSC SSSC PSSS SSSC SSSC PSSS SSSC SSSC

_ _ + _ _ _ _ _

t = 7


4MMM

M SSCC PSSS SSCCMMM

M CCCC 0 03 CCCC MMM SSSC MMM CCCC MMM MCCC 0

24

M M M

2MMM

M CCCC SSSC CCCCMMM

M SSCC MCCC MMMC1 SSSC SSSC PSSS SSSC SSSC PSSS SSSC SSSC

_ _ + _ _ _ _ _

t = 8



M 0 0

4MMM

M SSCC PSSS SSCCMMM

M CCCCMMM

M 0

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

t = 9



M CCCC 0

4MMM

M SSCC PSSS SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

25

t = 10


M 0


M CCCCMMM

M

4MMM

M SSCC PSSS SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

t = 11


M CCCC


M CCCCMMM

M

4MMM

M SSCC PSSS SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

Total property value was 26,823.

Model 3: Incentive with a grant amount of 10.

26

t = 0

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0

_ _ + _ _ _ _ _

t = 1

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 0 0 0 0 0 01 0 0 PSSS 0 0 0 0 0

_ _ + _ _ _ _ _

t = 2

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 SSSC 0 0 0 0 01 0 SSSC PSSS SSSC 0 0 0 0

_ _ + _ _ _ _ _

t = 3

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 0

27

4 0 0 0 0 0 0 0 03 0 0 SSSC 0 0 0 0 02 0 CCCC SSSC CCCC 0 0 0 01 SSSC SSSC PSSS SSSC SSSC 0 0 0

_ _ + _ _ _ _ _

t = 4

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 SSSC 0 0 0 0 03 0 MMMM SSSC MMMM 0 0 0 0

2MMM

M CCCC SSSC CCCCMMM

M 0 0 01 SSSC SSSC PSSS SSSC SSSC SSSC 0 0

_ _ + _ _ _ _ _

t = 5

6 0 0 0 0 0 0 0 05 0 0 SSSC 0 0 0 0 04 0 SSCC SSSC SSCC 0 0 0 03 CCCC MMMM SSSC MMMM CCCC 0 0 0

2MMM

M CCCC SSSC CCCCMMM

M SSCC 0 01 SSSC SSSC PSSS SSSC SSSC SSSC SSSC 0

_ _ + _ _ _ _ _

t = 6


28

4 MMMM SSCC SSSC SSCC MMMM 0 0 0

3 CCCC MMMM SSSC MMMM CCCCMMM

M 0 02 MMMM CCCC SSSC CCCC MMMM SSCC MCCC 01 SSSC SSSC PSSS SSSC SSSC SSSC SSSC SSSC

_ _ + _ _ _ _ _

t = 7

6 0 MMMC SSSC MMMC 0 0 0 05 MCCC MCCC SSSC MCCC MCCC 0 0 04 MMMM SSCC SSSC SSCC MMMM CCCC 0 0

3 CCCC MMMM SSSC MMMM CCCCMMM

M MCCC 02 MMMM CCCC SSSC CCCC MMMM SSCC MCCC MMMC1 SSSC SSSC PSSS SSSC SSSC SSSC SSSC SSSC

_ _ + _ _ _ _ _

t = 8



M 0 04 MMMM SSCC SSSC SSCC MMMM CCCC MMMM 0

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC2 MMMM CCCC SSSC CCCC MMMM SSCC MCCC MMMC1 SSSC SSSC PSSS SSSC SSSC SSSC SSSC SSSC

_ _ + _ _ _ _ _

t = 9

29



M CCCC 04 MMMM SSCC SSSC SSCC MMMM CCCC MMMM MCCC

3 CCCCMMM

M SSSCMMM

M CCCCMMM


_ _ + _ _ _ _ _

t = 10

6 MMCC MMMC SSSC MMMC MMCC MCCC MMMM 0


M CCCCMMM

M4 MMMM SSCC SSSC SSCC MMMM CCCC MMMM MCCC

3 CCCC MMMM SSSCMMM

M CCCCMMM


_ _ + _ _ _ _ _

t = 11

6 MMCC MMMC SSSC MMMC MMCC MCCC MMMM CCCC


M CCCCMMM

M4 MMMM SSCC SSSC SSCC MMMM CCCC MMMM MCCC

3 CCCC MMMM SSSCMMM

M CCCCMMM


_ _ + _ _ _ _ _

Total property value was 27,088.

30

Model 4: Incentive with a grant amount of 22

t = 0

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0

_ _ + _ _ _ _ _

t = 1

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 0 0 0 0 0 01 0 0 SSSP 0 0 0 0 0

_ _ + _ _ _ _ _

t = 2

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 SSSC 0 0 0 0 01 0 SSSC SSSP SSSC 0 0 0 0

_ _ + _ _ _ _ _

31

t = 3

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 PSSC 0 0 0 0 02 0 CCCC SSSC CCCC 0 0 0 01 PSSC SSSC SSSP SSSC PSSC 0 0 0

_ _ + _ _ _ _ _

t = 4

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 SSSP 0 0 0 0 0

3 0MMM

M PSSCMMM

M 0 0 0 0

2MMM

M CCCC SSSC CCCCMMM

M 0 0 01 PSSC SSSC SSSP SSSC PSSC SSSP 0 0

_ _ + _ _ _ _ _

t = 5

6 0 0 0 0 0 0 0 05 0 0 SSSC 0 0 0 0 04 0 SSCC SSSP SSCC 0 0 0 0

3 CCCCMMM

M PSSCMMM

M CCCC 0 0 0

2MMM

M CCCC SSSC CCCCMMM

M SSCC 0 0

32

1 PSSC SSSC SSSP SSSC PSSC SSSP SSSC 0_ _ + _ _ _ _ _

t = 6

6 0 0 PSSC 0 0 0 0 05 0 MCCC SSSC MCCC 0 0 0 0

4MMM

M SSCC SSSP SSCCMMM

M 0 0 0

3 CCCCMMM

M PSSCMMM

M CCCCMMM

M 0 0

2MMM

M CCCC SSSC CCCCMMM

M SSCC MCCC 01 PSSC SSSC SSSP SSSC PSSC SSSP SSSC PSSC

_ _ + _ _ _ _ _

t = 7

6 0 PMMM PSSC PMMM 0 0 0 05 MCCC MCCC SSSC MCCC MCCC 0 0 0

4MMM

M SSCC SSSP SSCCMMM

M CCCC 0 0

3 CCCCMMM

M PSSCMMM

M CCCCMMM

M MCCC 0

2MMM

M CCCC SSSC CCCCMMM

M SSCC MCCC PMMM1 PSSC SSSC SSSP SSSC PSSC SSSP SSSC PSSC

_ _ + _ _ _ _ _

t = 8

33

6 MMCC PMMM PSSC PMMM MMCC 0 0 0


M 0 0

4MMM

M SSCC SSSP SSCCMMM

M CCCCMMM

M 0

3 CCCCMMM

M PSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

t = 9

6 MMCC PMMM PSSC PMMM MMCC MCCC 0 0


M CCCC 0

4MMM

M SSCC SSSP SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M PSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

t = 10

6 MMCC PMMM PSSC PMMM MMCC MCCCMMM

M 0


M CCCCMMM

M

4MMM

M SSCC SSSP SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M PSSCMMM

M CCCCMMM

M MCCC MMCC

34

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

t = 11

6 MMCC PMMM PSSC PMMM MMCC MCCCMMM

M CCCC


M CCCCMMM

M

4MMM

M SSCC SSSP SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M PSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _


Model 5: Incentive with a grant to cover parkland development cost.

t = 0

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0

_ _ + _ _ _ _ _

t = 1

35

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 0 0 0 0 0 01 0 0 SSSC 0 0 0 0 0

_ _ + _ _ _ _ _

t = 2

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 02 0 0 SSSC 0 0 0 0 01 0 SSSC SSSC SSSC 0 0 0 0

_ _ + _ _ _ _ _

t = 3

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 03 0 0 SSSC 0 0 0 0 02 0 CCCC SSSC CCCC 0 0 0 01 SSSC SSSC SSSC SSSC SSSC 0 0 0

_ _ + _ _ _ _ _

t = 4

36

6 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 04 0 0 SSSC 0 0 0 0 0

3 0MMM

M SSSCMMM

M 0 0 0 0

2MMM

M CCCC SSSC CCCCMMM

M 0 0 01 SSSC SSSC SSSC SSSC SSSC SSSC 0 0

_ _ + _ _ _ _ _

t = 5

6 0 0 0 0 0 0 0 05 0 0 SSSC 0 0 0 0 04 0 SSCC SSSC SSCC 0 0 0 0

3 CCCCMMM

M SSSCMMM

M CCCC 0 0 0

2MMM

M CCCC SSSC CCCCMMM

M SSCC 0 01 SSSC SSSC SSSC SSSC SSSC SSSC SSSC 0

_ _ + _ _ _ _ _

t = 6


4MMM

M SSCC SSSC SSCCMMM

M 0 0 0

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M 0 0

2MMM

M CCCC SSSC CCCCMMM

M SSCC MCCC 01 SSSC SSSC SSSC SSSC SSSC SSSC SSSC SSSC

_ _ + _ _ _ _ _

37

t = 7


4MMM

M SSCC SSSC SSCCMMM

M CCCC 0 0

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC 0

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

t = 8



M 0 0

4MMM

M SSCC SSSC SSCCMMM

M CCCCMMM

M 0

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

t = 9



M CCCC 0

38

4MMM

M SSCC SSSC SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

t = 10


M 0


M CCCCMMM

M

4MMM

M SSCC SSSC SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

t = 11


M CCCC


M CCCCMMM

M

4MMM

M SSCC SSSC SSCCMMM

M CCCCMMM

M MCCC

3 CCCCMMM

M SSSCMMM

M CCCCMMM

M MCCC MMCC

2MMM

M CCCC SSSC CCCCMMM


_ _ + _ _ _ _ _

39


Chapter 6

Discussion

As can be seen from the previous section, there are no parklands in the first model while there

are several parklands in the second model. This helps achieve the goal of having high quality and

healthy living environments. This finding shows that a developer is still capable of building a

livable community while maximizing his own interest.

In addition, if the grant amount to encourage having a parkland is reduced to 10 from the original

amount of 15, the number of parklands decreases while the total property value rises. On the

other hand, if the grant amount is increased to 22, the number of parklands increases while the

total property values falls. With the parameters used in the model, 22 is the maximum grant

amount that can be given because if the grant amount goes over 22, the whole first row will be

developed at time 1, which is not realistic. Recall from the first chapter that besides developing a

livable community, the other goal of the paper is to conduct zoning effectively so that the

property value, and thus government revenues, is maximized. Unfortunately, these two goals

cannot be achieved at the same time. There is a trade-off between having more parklands and

maximizing property value. Figure 5 and Figure 6 demonstrates the inverse relationship between

these two variables.

40

0 5 10 15 20 2526000

26200

26400

26600

26800

27000

27200

27400

Total Property Value

Grant amount

Tota

l pro

perty

val

ue

Figure 5. Total property value as a function of the grant amount.

0 5 10 15 20 250

2

4

6

8

10

12

Number Of Parklands

Grant amount

Num

ber o

f par

klan

ds

41

Figure 6. Number of parklands as a function of the grant amount.

This study has several limitations. First, the greenfield area considered in this study is relatively

small. It might be better to include a larger area. Second, most of the parameters chosen are

arbitrary. Future researchers should consider using actual data to see the impacts of performance

zoning in real life. Finally, in the study, I only consider the influence on a cell’s external value

from cells that are one-distance away. A more generalized model that takes into account the

influence from farther away cells would be worth investigating.

42

References

[1] Heid, J. (2004). Greenfield development without sprawl: The role of planned

communities. Washington, DC: Urban Land Institute.

[2] Porter, M (1995). "The Competitive Advantage of the Inner City". Harvard Business Review:

55–72.

[3] Farris, J. T. (2001). "The barriers to using urban infill development to achieve smart growth".

Housing Policy Debate 12 (1): 1–30.

[4] Governing. (2014). “Bankrupt Cities, Municipalities List and Map”. Retrieved from

http://www.governing.com/gov-data/municipal-cities-counties-bankruptcies-and-defaults.html.

[5] Reps, J. W. (1964). Requiem for zoning. Department of City and Regional Planning, Cornell

University.

43

http://www.governing.com/gov-data/municipal-cities-counties-bankruptcies-and-defaults.html

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Math Honors Paper - Tu Nguyen - 2015