Econ 330: Urban Economics

# Econ 330: Urban Economics
## Lecture 8
### John Morehouse
### 30 January, 2020

---

# Lecture 8: Neighborhood Choice II

---

# Schedule

## Today

1. .hi.purple[Model of Neighborhood Sorting]

2. .hi.purple[Minimum Lot Size]

3. .hi.purple[Discussion]

## Upcoming

- .hi.slate[HWII] __due in class Feb 6th__

- .hi.slate[Reading] (Chapter IV & V _ToTC_)
  
  - .hi.slate[Midterm] week 6 -- on the horizon

---

# Neighborhood Sorting

Last class we asked:

1. Who gets desirable neighbors?

2. Will there be segregated or integrated neighborhoods?

3. Will there be sorting or mixing with respect to income, age, race, or some combination of those factors?

4. What are the implications for the price of land in various neighborhoods?

---

# Intro

We will focus on positive externalities (for now). .hi.slate[Assume]

- These increase with income and education level
  
--
  
.hi[Question]:

- What is the income mix of neighborhoods - segregated or integrated?

## Model

- Two neighborhoods: A and B, each with 80 lots

- Two income groups (high and low), each with 80 households

- Only difference between the neighborhoods is income mix

---

# Model

In this model, individual choices to stay or move are determined by the _rent premium_

- __Rent Premium__ (for neighborhood A): `\(RP = R(A)-R(B)\)`

- Premium for workers might be different by type: `\(RP_{high} \neq RP_{low}\)`

- IE, the benefit of living close to high types might vary by type

--
  
--
.hi.slate[Assume]
--

- __Land will be allocated to the highest bidder__

- Everyone in the same neighborhood pays the same rent/price

---

# Segregation Eq

.col-left[
<img src="lecture_8_files/figure-html/seg_eq1-1.svg" style="display: block; margin: auto;" />
]

- Suppose we start at 40 HH's in neighborhood A. This is a .hi[perfectly intergrated] equilibrium

- The RP is 0 for both groups, so households are indifferent between neighborhoods (no incentive to move)

]

---

# Segregation Eq

.col-left[
<img src="lecture_8_files/figure-html/seg_eq2-1.svg" style="display: block; margin: auto;" />
]

- Suppose we start at 40 HH's in neighborhood A. This is a .hi[perfectly intergrated] equilibrium

- The RP is 0 for both groups, so households are indifferent between neighborhoods (no incentive to move)

- What happens if there is a small "shock" to the equilibrium and a few high income households move to neighborhood A?

]

---

# Segregation Eq

.col-left[
<img src="lecture_8_files/figure-html/seg_eq3-1.svg" style="display: block; margin: auto;" />
]

- If 5 high income HH's move into A, `\(RP(high) > RP(low)\)`

- This means high income HH's are willing to bid more for neighborhood A even if they are in neighborhood B. So what happens?

]

---

# Segregation Eq

.col-left[
<img src="lecture_8_files/figure-html/seg_eq4-1.svg" style="display: block; margin: auto;" />

]

- If 5 high income HH's move into A, `\(RP(high) > RP(low)\)`

- This means high income HH's are willing to bid more for neighborhood A even if they are in neighborhood B. So what happens?

- More blues move in (and since they bid higher, they get to live in neighborhood A). This is called a _self reinforcing effect_ or positive feedback loop.

]

---

# Segregation Eq

.col-left[
<img src="lecture_8_files/figure-html/seg_eq5-1.svg" style="display: block; margin: auto;" />
  - __Axiom 2__: _.hi[Self-reinforcing effects] generate extreme outcomes_ `\(\implies\)` we end up at a fully segregated eq of all 80 high inc HH's in nbhd A

]

- If 5 high income HH's move into A, `\(RP(high) > RP(low)\)`

- This means high income HH's are willing to bid more for neighborhood A even if they are in neighborhood B. So what happens?

- More blues move in (and since they bid higher, they get to live in neighborhood A). This is called a _self reinforcing effect_ or positive feedback loop.

]

---

# Integration Eq
.col-left[
<img src="lecture_8_files/figure-html/int_eq_1-1.svg" style="display: block; margin: auto;" />
]

- Is the story the same here?

]

---

# Integration Eq
.col-left[
<img src="lecture_8_files/figure-html/int_eq2-1.svg" style="display: block; margin: auto;" />
]

- Is the story the same here?

- Now, a small movement of high income HH's into A means `\(RP(High) < RP(low)\)`

]

---

# Integration Eq
.col-left[
<img src="lecture_8_files/figure-html/int_eq3-1.svg" style="display: block; margin: auto;" />
]

- Is the story the same here?

- Now, a small movement of high income HH's into A means `\(RP(High) < RP(low)\)`

- So we get pushed back to the initial equilibrium. In this case, intergration is the .hi[only equilibrium]

- Furthermore, integration is a .hi.purple[stable equilibrium]

]

---

# Integration Eq
.col-left[
<img src="lecture_8_files/figure-html/int_eq4-1.svg" style="display: block; margin: auto;" />
.hi.orange[Note]: 80 high income HH's in A is not an EQ because `\(RP(low)>RP(high)\)`. So low incomes will outbid highs and move in

]

- Is the story the same here?

- Now, a small movement of high income HH's into A means `\(RP(High) < RP(low)\)`

- So we get pushed back to the initial equilibrium. In this case, intergration is the .hi[only equilibrium]

- Furthermore, integration is a .hi.purple[stable equilibrium]

]

---

#Mixed Eq

.col-left[
<img src="lecture_8_files/figure-html/mixed_eq-1.svg" style="display: block; margin: auto;" />
]

- What about a story like this?

]

---

#Mixed Eq

.col-left[
<img src="lecture_8_files/figure-html/mixed_eq2-1.svg" style="display: block; margin: auto;" />
]

- What about a story like this?

- Integration eq (40 of each type in each nbhd) is still an equilibrium. Is it .hi[stable]?

]

---

#Mixed Eq

.col-left[
<img src="lecture_8_files/figure-html/mixed_eq3-1.svg" style="display: block; margin: auto;" />
]

- What about a story like this?

- Integration eq (40 of each type in each nbhd) is still an equilibrium. Is it .hi[stable]?

- No. A small deviation away means `\(RP(high) > RP(low)\)`. So highs outbid lows until `\(RP(high) = RP(low)\)` at 45 highs in A and 35 lows.

- Is 45 highs in A stable?

]

---

#Mixed Eq

.col-left[
<img src="lecture_8_files/figure-html/mixed_eq4-1.svg" style="display: block; margin: auto;" />
]

- What about a story like this?

- Integration eq (40 of each type in each nbhd) is still an equilibrium. Is it .hi[stable]?

- No. A small deviation away means `\(RP(high) > RP(low)\)`.  So highs outbid lows until `\(RP(high) = RP(low)\)` at 45 highs in A and 35 lows.

- Is 45 highs in A stable? Yes (you think about why)

]

---

#Mixed Eq

.col-left[
<img src="lecture_8_files/figure-html/mixed_eq5-1.svg" style="display: block; margin: auto;" />
- .hi.orange[Note]: Full segregation here is _not_ an equilibrium for a similar reason to the last example
]

- What about a story like this?

- Integration eq (40 of each type in each nbhd) is still an equilibrium. Is it .hi[stable]?

- No. A small deviation away means `\(RP(high) > RP(low)\)`.  So highs outbid lows until `\(RP(high) = RP(low)\)` at 45 highs in A and 35 lows.

- Is 45 highs in A stable? Yes (you think about why)

]

---

# Eq Defn

To be clear, an _equilibrium_ in this model is a point at which the rent premium is in balance across both groups

- This will hold when the rent premium curves intersect. Except at full segregation
  
--
  - If the `\(RP\)` for the group listed on the axis is _.pink[higher]_ then this will also be an equilibrium because .hi[there is no tendency for change]
  
  - If the `\(RP\)` for the group listed on the axis is _.purple[lower]_ then population dynamics move away from this point

--
  
--

---

# Stable vs Unstable Eq

1) An eq is .hi[stable] if a small movement away will encounter self - .hi[correcting] forces

- An eq is stable if when you move away from it, the pop. dynamics push you back to where you came from

2) A eq is .hi.purple[unstable] if a small movement away will encounter self - .hi.purple[reinforcing] forces

- That is, an eq is untable if when you move away from it, the population dynamics push you even farther than where you came from
  
--

---

# A Heuristic

1) Draw a verticle dashed line at every intersection point

2) For every region between the verticle dashed lines, it must be the case that one of the rent premium curves is above the other

- If the rent prem curve for the group listed on the axis is .hi[higher], then this group will increase in number. Draw rightward arrows on the axis
  
  - If the rent prem curve for the group listed on the axis is .hi.purple[lower], then this group will decrease in number. Draw leftward arrows

---

# A Heuristic

3) If there are rightward arrows pushing toward 100% in one nbhd, then 100% (complete segregation) is an eq even if the rent prem curves do not intersect there

4) For every eq. value, look at its immediate vicinity

- If there are arrows moving towards it, it is a .hi.purple[stable eq]
  
  - If there are arrows moving away from it on one or both sides, it is a .hi[unstable eq]
  
---

1) .hi[Model of Neighborhood Sorting] ✅

- Segregated, Integrated, & Mixed Equilibria
  
  - Stable vs Unstable Equilibria

2) .hi.purple[Minimum Lot Size]

]

3) .hi.purple[Discussion]

]

---

# Lot Size

## Role of lot-sizes

- Land is a .hi[normal good]. Recall from EC201, what does this mean?

- more 💰 `\(\implies\)` higher demand

- Larger lot `\(\implies\)` smaller .pink[premium per unit] of land
  
  - .hi.purple[Pair of low-income] households .pink[outbids] .hi[single high income] household
  
  
---

# Minimum Lot Size

Minimum Lot Size (MLS) zoning increases the premium per unit of land for low-income households

- Low-income households are thus more likely to be outbit for lots by high-income households

- MLS restrictions promote segregation

---

# Example

Consider a developer that is .hi[facing a choice] to develop a plot of `\(60,000 ft^2\)` into:

-  12 lots and sell to low-income HH's for 200k each 
  
    - they are WTP 200k for `\(5000 ft^2\)` of land

- 4 lots and sell to high-income HH's for 500k each
  
    - they are WTP 500k for `\(15000 ft^2\)` of land

1. What does the developer do?
  
  2. How large does the MLS need to be to keep low income households out?
  
--

---

# Example

## Sell to?

- If they sell to L: `\(200k*12 = 2.4m\)`

- If they sell to H: `\(500k*4 = 2m\)`

- They sell to .hi[low]
  
--

---

# Example

## MLS?

- Low, get `\(\frac{200k}{5000ft^2} = 40/ft^2\)`
  
- High, get  `\(\frac{500k}{15000ft^2} = 33.333/ft^2\)`

Set:

`\begin{align*}
\frac{200,000}{lot_l} &= 33.33\\
lot_l &= \frac{200,000}{33.33}\\
lot_l &= 6000ft^2
\end{align*}`

---

1) .hi[Model of Neighborhood Sorting] ✅

- Segregated, Integrated, & Mixed Equilibria
  
  - Stable vs Unstable Equilibria

2) .hi[Minimum Lot Size] ✅

]

3) .hi.purple[Discussion]

]

---

# Racial Integration

---

# Racial Integration

---

# Racial Integration

---

#Racial Integration

---

# Discussion

Common theories for racial segregation (in no particular order)

1) White households have preference for segregated neighborhoods

2) Income and race are strongly correlated, so income segregation contributes to racial segregation

3) MLS zoning excludes low-income HH's

4) __Racial Steering__: Encouraging by real-estate agents, bureaucrats, or property owners reduce access of minority households to certain neighborhoods

---

# So What?

What are the consequences of nbhd segregation? _Spatial Mismatch_

- Inferior access to jobs
  
  - .hi[Inferior access] explains __25%__ of black-white employment gap
    
  - .hi.purple[Inferior acess] explains __31%__ of Hispanic-white employment gap
    
- Mismatching bigger problem in large cities

---

# So What?

Schools and Poverty Traps

- Low education spending `\(\implies\)` low achievement in poor nbhds

- Education more costly in poor nbhds
    
  + family crises
  + security
  + weak prep

- twice highschool dropout rate
  
  - Education for black HS grad eq to ed of white suburban dropout
  
  - High poverty schools: low proficiency rates for math and reading

---

1) .hi[Model of Neighborhood Sorting] ✅

- Segregated, Integrated, & Mixed Equilibria
  
  - Stable vs Unstable Equilibria

2) .hi.purple[Minimum Lot Size] ✅

]

3) .hi[Discussion] ✅

]

---

```r
p_load(pagedown)
pagedown::chrome_print(here::here("008-nbhd_choiceII","lecture_8.html"))
```