Econ 330: Urban Economics

# Econ 330: Urban Economics
## Lecture 5
### John Morehouse
### January 21st, 2020

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# Lecture V: Rents

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# Schedule

## Today

1) .hi.purple[Intro to Rents]

2) .hi.purple[Rents] .hi.orange[Across] .hi.purple[Cities]

3) .hi.purple[Rents] .hi.orange[Within] .hi.purple[Cities]

## Upcoming

- ‼️ .hi.slate[HWI due] .hi[next class] (thurs, Jan 21)  ‼️
  
    - ⚠️ __No late homeworks will be accepted__

- .hi.slate[Reading] (Chapter IV _ToTC_)
    
--

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# Taking Stock

.hi.slate[First Two Weeks]: Intoduction and .hi.purple[existence, size & growth] (philosophical-ish questions)

.hi.slate[Now]: fundamentals of .pink[location choice theory]. .qa[Questions]

- Why do people choose to live in one place vs another? (SF vs Detroit)

- .hi.slate[Today]: How do these choices impact rental prices (.pink[across cities])

- Conditional on choosing to live Eugene, will individuals .hi[systematically locate] in one neighborhood vs another?
 
  - .hi.slate[Today]: How do these choices impact rental prices (.purple[within city]) ?

.hi.slate[Later]: Formalize this. Learn __basics__ of [discrete choice modeling](https://en.wikipedia.org/wiki/Discrete_choice)

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# Rents: An Overview

source: [Oppurtunity Atlas](https://www.opportunityatlas.org/)

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# Rents: NY

<img src="images/ny.png" width="90%" style="display: block; margin: auto;" />
source: [Oppurtunity Atlas](https://www.opportunityatlas.org/)

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# Rents: Seattle

source: [Oppurtunity Atlas](https://www.opportunityatlas.org/)
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# Rents: Chicago

source: [Oppurtunity Atlas](https://www.opportunityatlas.org/)

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2) .hi.purple[Rents] .hi.orange[Across] .hi.purple[Cities]

3) .hi.purple[Rents] .hi.orange[Within] .hi.purple[Cities]

]

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# Prices across cities

.hi.slate[Easy version] .purple[Supply] and .pink[demand] curves vary across cities (today)

- Equilibrium will be .pink[different across cities] (and hence prices are different)

.hi.slate[Hard Version] Solving for equilibrium when wages respond to population changes as well (not today)

.qa[A1]: .hi[Supply]: variation in local construction costs, land available for development, and land-use regulations

.qa[A2]: .hi.purple[Demand]: variation in available jobs (income), preference for housing consumption

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# Rents: An Overview

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# Urban Housing Supply Curves

In general, supply curves across cities are impacted by: local construction costs, land available for development, and land-use regulations

- .hi[Local construction costs]: shifts .pink[intercept] (labor is more expensive for all firms in one area vs another)

- .hi.purple[Land available for development] and .hi.purple[land use regulations]: slope (changes __marginal cost__) of developing land. .qa[Why?]

.qa[A]: Less land available to develop `$\rightarrow$` .hi[oppurtunity cost of developing increases] for each next plot of land. Prices get bid up faster. Similar intuition with land use regulations

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# Urban Housing Supply Curves

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# Urban Housing Supply Curves

- .hi[pink]: lower construction cost (lower intercept)

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# Urban Housing Supply Curves

- __black__: higher land use regs or less available land for development

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# Example:

.col-left[
- .hi[Seattle]: 
   
`\begin{align*}
R_{SEA} &= 10 + H_{SEA} \\
R_{SEA} &= 25 - 2*H_{SEA}
\end{align*}`

]

.col-right[
- .hi[SF]: 
  
`\begin{align*}
R_{SF} &= 10 + 2*H_{SF}\\
R_{SF} &= 30 - 3*H_{SF}
\end{align*}`

]

1) Solve for equilibrium in both cities

2) Given your answer to 1, and knowledge of the term _.hi[locational equilibrium]_ what can you say must be the case about .hi.purple[wages and or amenity values] in one city vs the other?

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# Example

1) Solve for equilibrium in both cities

`\begin{align*}
\text{SEA}: (H_{SEA}^\star, R_{SEA}^\star) &= (5,15)\\
\text{SF}: (H_{SF}^\star, R_{SF}^\star) &= (4,18)
\end{align*}`

2) Given your answer to 1, and knowledge of the term _.hi[locational equilibrium]_ what can you say must be the case about .hi.purple[wages and or amenity values] in one city vs the other?

- Rental prices are higher in SF. .hi[In equilibrium], utility levels are equalized across cities. Thus, it must be that .pink[either wages] .hi.orange[and or] .purple[amenities] are higher in SF than SEA

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# Stepping Back

One assumption underling the above example:

<center>
 Perfect competition 
</center>

Is this reasonable? .hi.purple[Discuss]

- SF has rent control (not .hi[perfectly competitive]). I am not as sure about Seattle rental market

- In the case of .hi.orange[monopoly], the outcomes here are pretty different. We will do the labor version of this (monopsony) later in the course

---

2) .hi[Rents] .hi.orange[Across] .hi[Cities] ✅

- Supply and Demand variation 
  
  - Eq computation

3) .hi.purple[Rents] .hi.orange[Within] .hi.purple[Cities]

]

---

# The Bid-Rent Curve

The __Bid - Rent Curve__ is the _.pink[relationship between housing prices and the distance of land from the city center]_ .pink[†]

.footnote[
.pink[†] It actually does not have to be the city center -- can be a point of attraction. In this class we will always use the city center though.
]

These curves vary across sectors

- __Consumer Bid rent curve__: .pink[commuting costs]

- Rural Bid Rent: .pink[fertility] of land

- Manufacturing: Accessibility to .pink[consumers] and .purple[suppliers]

- Tech/info: Accessibility to .pink[Information]

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# Housing Prices Model

We now build a simple model of rental/housing prices .hi.orange[within] a city

1) Commuting cost is .hi[only location factor] in decision making

- .pink[All locations] are otherwise identical

2) Only .hi.orange[one member] of household commutes to employment area

3) Only considers the .hi[monetary (not time) cost of commuting]

4) Noncommuting travel is .purple[insignificant]

5) Public services, .hi.purple[taxes, amenities] are the .purple[same everywhere] (implication from 1)

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# Locational Indifference

.hi[Axiom 1]: _Housing prices adjusts until there is locational indifference_ (and prices in general)

- IE: until an increase in rent for a closer location just offsets the lower commuting costs
  
--

.hi.orange[In math]:

`\begin{align*}
\Delta P \cdot h + \Delta x \cdot t = 0
\end{align*}`

- P: .hi[price] of housing (price per square foot)

- h: .hi.purple[amount] of housing (in `$ft^2$`)

- x: .hi.orange[distance] to employment area

]

- t: .hi.green[commuting cost] per mile

]

---

# Slope of the Housing Bid-Rent Curve

If there is locational indifference we can derive the slope of the bid-rent curve:

`\begin{align*}
\Delta P \cdot h + \Delta x \cdot t &= 0
\end{align*}`

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# Slope of the Housing Bid-Rent Curve

If there is locational indifference we can derive the .hi.purple[slope] of the .hi[bid-rent] curve:

`\begin{align*}
\Delta P \cdot h + \Delta x \cdot t &= 0\\
\Delta P \cdot h &= -\Delta x \cdot t
\end{align*}`

---

# Slope of the Housing Bid-Rent Curve

If there is locational indifference we can derive the .hi.purple[slope] of the .hi[bid-rent] curve:

`\begin{align*}
\Delta P \cdot h + \Delta x \cdot t &= 0\\
\Delta P \cdot h &= -\Delta x \cdot t\\
\frac{\Delta P}{\Delta x} &= -\frac{t}{h}
\end{align*}`

.hi.slate[Notice]: `$\frac{\Delta P}{\Delta x}$` is the .purple[slope] of the .pink[bid-rent] curve

- price is on the verticle axis, distance is on the horizontal. So this is rise over run

---

# Another Derivation

Suppose you have decided that the optimal amount of money to spend on housing and commuting per month is `$M^*$`

- You can allocate this as

`\begin{align*}
P\cdot h + x \cdot  t = M^*
\end{align*}`

- Since we graph the bid rent curve in the (x,P) space, we solve for p:

`\begin{align*}
P\cdot h + x \cdot  t &= M^*\\
P\cdot h &= M^* - x\cdot t
\end{align*}`

---

# Another Derivation

Suppose you have decided that the optimal amount of money to spend on housing and commuting per month is `$M^*$`

- You can allocate this as

`\begin{align*}
P\cdot h + x \cdot  t = M^*
\end{align*}`

- Since we graph the bid rent curve in the (x,P) space, we solve for p:

`\begin{align*}
P\cdot h + x \cdot  t &= M^*\\
P\cdot h &= M^* - x\cdot t\\
P &= \frac{M^*}{h} - \frac{t}{h} \cdot x
\end{align*}`

- Slope: `$\Delta P = 0 - \frac{t}{h} \cdot \Delta x \implies \frac{\Delta P}{\Delta x} = -\frac{t}{h}$`
  
  - Can also take derivative if p w.r.t to x and get the same thing, if that is easier for you

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# No Substitution

.hi.slate[Example] Suppose the following:

- Each household has $800 a month to spend on housing and commuting

- All rental units are the same size, with each HH occupying a rental unit that is 1000 sq ft
  
--

- Monthly commuting cost is $50 dollars per mile from employment center

.qa[Task]: Draw the housing - price curve. Put miles from city center on .hi.orange[x axis] and price per square foot on .hi[y axis]

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# Example: The housing price curve

a: max WTP for a square foot (at center of city)

b: further away from center HH is willing to live

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# Substitution

.qa[Q1]: If you really wanted to live closer to campus -- or an exciting downtown in a big city -- would you be willing to live in a smaller apartment to do so?

.qa[A2]: Substitute housing consumption for .purple[lower commuting cost] (and whatever else being close to the center of the city gets you)

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# Substitution

Let's formalize the mechanism for substitution a bit:

.pink[higher prices] `$\implies$` .purple[higher oppurtunity cost] per square foot of housing (for the consumer)

- As price of rent increases, consumers are likely to substitute (atleast somewhat) towards other goods, decreasing the square footage of housing demanded

- __Housing units closer to city centers are thus likely to be smaller in size__

---

# Adding substitution to the model

`\begin{align*}
\Delta P \cdot h + \Delta x \cdot t &= 0
\end{align*}`

.qa[A3]: No because `$h$` (the quantity of housing consumed) is .hi[independent of distance] from center ($x$)

`\begin{align*}
\Delta P \cdot h(x)  + \Delta x \cdot t = 0 
\end{align*}`

- Where `$h(x)$` is an _increasing_ function of x

- .hi.slate[Ex]: `$h(10) > h(5)$` (the quantity of housing demanded 10 miles from the center exceeds that of 5 miles)
  
--

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# Quick Q

`\begin{align*}
\frac{\Delta P}{\Delta x} = -\frac{t}{h(x)}
\end{align*}`

.qa[Q5] Using the equation above what happens to the .purple[slope of the housing bid-rent] curve as x increases. .hi[Why]?

- Since higher value of x `$\rightarrow$` higher value of h `$\rightarrow$` smaller value of `$\frac{1}{h(x)}$`. This means `$-\frac{1}{h(x)}$` will be _less negative_
  
--

__Let's graph this, to make sure we get it__

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# Model with Substitution Graph

.hi.purple[purple]: no substitution

red: substitution

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2) .hi[Rents] .hi.orange[Across] .hi[Cities] ✅

- Supply and Demand variation across cities
  
  - Eq computation

3) .hi[Rents] .hi.orange[Within] .hi[Cities] ✅

- The bid rent curve for consumers
  
    + Locational Indifference
    
    + With substitution
    
    + Without Substitution

]

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```