PS1

Question 1

How do you calculate the maximum price you should pay for a house with constant rent?

Answer 1

Use the perpetuity formula:
𝑃 = 𝐴 / 𝑟
where:
𝐴 = annual rent payment
𝑟 = interest rate

Question 2

How does a maintenance cost of $500 per year affect the price you should pay?

Answer 2

Subtract the maintenance cost from annual rent before applying the perpetuity formula:
𝑃 = (𝐴 − 𝑀) / 𝑟
where 𝑀 is the maintenance cost.

Question 3

What happens if rent grows at a rate of 𝑔 each year?

Answer 3

Use the growing perpetuity formula:
𝑃 = 𝐴 / (𝑟 − 𝑔)
where 𝑔 is the annual growth rate of rent.

Question 4

How do you adjust for maintenance when rent is growing?

Answer 4

𝑃 = 𝐴 / (𝑟 − 𝑔) − 𝑀 / 𝑟

Question 5

If you sell the house after 4 years, does the maximum price you should pay change?

Answer 5

No, because the present value of future rent payments remains the same:

Question 6

What is the consumer’s budget constraint in the monocentric model?

Answer 6

Question 7

What is the equilibrium condition for household location choice in the monocentric model?

Answer 7

This means that the rate of change of housing price with respect to distance is determined by commuting costs and housing consumption.

Question 8

How can the equilibrium condition be rewritten for better intuition?

Answer 8

Question 9

How do rising commuting costs affect the bid-rent function?

Answer 9

Higher commuting costs increase the premium for land near the CBD.
The bid-rent curve shifts upward near the CBD.
More people prefer to live closer to reduce commuting expenses.

Question 10

What happens to housing demand as transportation costs rise?

Answer 10

1. Higher land prices: The cost of land increases, making housing more expensive.
2. Lower net income at each location: Commuting costs reduce disposable income, limiting how much can be spent on housing.

This steepens the bid-rent curve, leading to increased population density near the CBD.

Question 11

How does rising transportation cost affect urban density?

Answer 11

More people will be concentrated in less land, increasing density in central areas.
Density decreases farther away but increases near the CBD.

Question 12

How does rising transportation cost affect the urban fringe?

Answer 12

The city contracts in size, with the boundary shifting inward from 𝑥 to 𝑥′. Suburban areas become less attractive as commuting costs rise.

Question 13

How do rents at the CBD change when transportation costs rise?

Answer 13

Rents at the CBD increase because people avoid commuting costs by living downtown. The equilibrium utility must decrease, which happens through higher prices near the center.

Question 14

How does an ‘open city’ change the impact of rising transportation costs?

Answer 14

In an open city, people can migrate in and out. Higher transportation costs cause out-migration. The bid-rent curve shifts inward, contracting the city’s total size. The slope of the bid-rent curve increases, but the vertical intercept decreases.

Question 15

How does the bid-rent function shift in an open city?

Answer 15

People leave the city due to lower utility. The bid-rent function shifts inward. The slope remains the same, but the intercept decreases due to lower overall demand for housing.

Question 16

How do you calculate the maximum price you’re willing to pay for a house with constant rent?

Answer 16

Use the perpetuity formula:

Question 17

How does selling the house in 4 years impact your willingness to pay today?

Answer 17

You discount each year’s rent and the final selling price using the discount factor

Question 18

How do annual maintenance costs affect the house valuation?

Answer 18

Subtract maintenance costs from the annual rent before applying the perpetuity formula:

Question 19

How do you calculate house value if rent grows at a rate g?

Answer 19

Use the growing perpetuity formula:

Question 20

What is the budget constraint for housing and commuting in a monocentric city?

Answer 20

Question 21

What condition determines where households locate?

Answer 21

The spatial equilibrium condition:

suggesting that housing prices fall as distance increases.

Question 22

What happens to housing prices when transportation costs increase?

Answer 22

Bid-rent curve shifts up, meaning higher prices near the city center.

Question 23

How does rising commuting cost affect city density?

Answer 23

Closer to the city: Density increases because people want to live closer.
Further away: Density decreases, shrinking the urban area.

Question 24

What are the two types of rent control?

Answer 24

First-generation rent control: Hard rent caps.
Second-generation rent control: Limits rent increases.

Question 25

What are the benefits of rent control?

Answer 25

Lower rents for controlled units. Doesn't require subsidies. Helps more people than zoning/subsidy changes.

Question 26

What are some drawbacks of rent control?

Answer 26

Lower incentives for maintenance and construction. Misallocation of households (e.g., small families in large units). Reduces mobility as people avoid moving.

Question 27

How does rent control impact monopolistic vs. competitive housing markets?

Answer 27

In a competitive market, rent control causes shortages and deadweight loss (DWL). In a monopolistic market, rent control reduces market power and eliminates DWL.

Question 28

Why does rent control lead to shortages?

Answer 28

Quantity demanded > Quantity supplied at the rent-controlled price, reducing available rental units.

Question 29

How do you determine the edge of a city in a monocentric model?

Answer 29

Set land rent equal to the agricultural rent: Solve for 𝑥 to find the urban boundary.

Question 30

What happens if landowners restrict urban expansion?

Answer 30

Bid-rent curve shifts up. Higher land values inside the boundary. Lower housing availability.

Question 31

How do we calculate the net gain/loss from zoning restrictions?

Answer 31

Compare rent gain (blue area) vs. rent loss (orange area) in the diagram: Net Gain = Gain from price increase − Loss from reduced land use.

Question 32

Should landowners restrict land at 50 blocks or 65 blocks?

Answer 32

65-block restriction is better: It raises urban land rent but still allows enough land for development. 50-block restriction is too severe: Reduces housing supply too much, lowering total rent revenue.

Question 33

key formulas

Answer 33

Question 34

What is the consumer's budget constraint in the monocentric city model?

Answer 34

Question 35

What is the equilibrium condition for household location choice in the monocentric model?

Answer 35

(This means that the change in land rent per unit distance must equal the marginal commuting cost per unit housing consumption.)

Question 36

How can we rearrange the equilibrium condition to enhance intuition?

Answer 36

(The change in total rent 𝑑𝑃(𝑥)ℎ must equal the change in commuting cost −𝑡𝑑𝑥 to keep the consumer indifferent across locations.)

Question 38

What happens to bid-rent when commuting costs increase?

Answer 38

Rising commuting costs increase the premium for land near the Central Business District (CBD), making the bid-rent curve steeper.

Question 39

What happens to housing demand at each location when transportation costs increase?

Answer 39

Housing consumption h(x) decreases because: 1. Higher land prices reduce affordability. 2. Higher commuting costs lower net income y−tx at each location.

Question 40

What happens to population density when transportation costs rise?

Answer 40

Density increases closer to the CBD and decreases further away, as people seek to minimize commuting expenses.

Question 41

How does an increase in commuting costs affect the urban fringe (edge of the city)?

Answer 41

The urban fringe contracts, shifting inward from x to x′, reducing the city's total developed area.

Question 42

What happens to rents at the CBD when commuting costs rise?

Answer 42

Rents at the CBD increase, as the only way to maintain equilibrium utility is for downtown prices to rise.

Question 43

How does an open city adjust differently from a closed city when commuting costs increase?

Answer 43

In an open city, equilibrium is achieved through migration. Rising transportation costs lead to out-migration, shifting the bid-rent curve inward but maintaining a similar slope.

Question 44

What happens to the bid-rent curve in an open city with rising transportation costs?

Answer 44

The bid-rent curve shifts inward (left), lowering land rents but maintaining a steep slope due to reduced urban population.

Question 45

What does the Problem 2b diagram illustrate in terms of zoning and bid-rent changes?

Answer 45

The diagram shows the impact of restricting the city boundary from 80 blocks to 65 blocks on bid-rent. Blue-shaded area (rent gain): Represents the additional land rent earned by landlords due to the zoning restriction, which increases land prices closer to the CBD. Orange-shaded area (rent loss): Represents the lost land rent from the restricted area (blocks 65 to 80) that is no longer available for development. The shift in the rent function increases the rent curve from R(x) = 100 - x to R(x) = 105 - x, leading to a higher bid-rent in the inner city. Since the rent gain is larger than the rent loss, the restriction benefits landlords financially. Key takeaway: Land-use regulation that limits urban expansion can increase rents in central areas, benefiting landowners but restricting housing supply.

Question 46

Answer 46

This diagram represents the effect of imposing a land-use restriction that reduces the city's boundary from 80 blocks to 65 blocks in a monocentric city model. The Blue Area (Rent Gain): This represents the increase in land rents due to the restriction. Because the city is now constrained to a smaller area, land closer to the Central Business District (CBD) becomes more valuable. The new land rent function shifts upward, increasing landowners' earnings in those areas. The total rent gain is calculated as (5 × 65) = 325. The Orange Area (Rent Loss): This represents the loss in land rent due to the reduced urban boundary. Since land beyond 65 blocks is no longer developable, the land rent that could have been collected from that region is lost. The total rent loss is calculated as (15 × 15) / 2 = 112.5. Net Effect on Landowners: Since the rent gain (325) is greater than the rent loss (112.5), the restriction benefits landowners overall. The restriction limits the land supply, making the remaining land more valuable.

Question 47

Answer 47

This diagram illustrates the effect of further restricting the city's boundary from 65 blocks to 50 blocks in a monocentric city model. Key Changes in the Diagram Blue Area (Rent Gain) → $250$ This represents the increase in land rents due to the more aggressive land-use restriction. Since land is now even more scarce, land rents rise even higher closer to the Central Business District (CBD). The total gain in rent from this restriction is calculated as (5 × 50) = 250. Orange Area (Rent Loss) → $412.5$ This represents the loss in land rent due to the reduced urban boundary. The restriction prevents land beyond 50 blocks from being developed, eliminating potential rent revenue. The total loss is calculated as (15 × 15) / 2 + (20 × 15) = 412.5. Net Effect on Landowners → $-162.5$ Unlike the previous restriction to 65 blocks, this further restriction hurts landowners overall. Since the rent gain (250) is smaller than the rent loss (412.5), landowners experience a net loss of 162.5. This means that reducing the boundary too much limits the amount of rentable land and lowers net rents relative to a 65-block restriction. Conclusion: Landowners prefer the 65-block restriction over the 50-block restriction because the 50-block limit reduces their total earnings. Over-restricting land use can be counterproductive—while it increases rents closer to the city center, it eliminates too much land from the market, causing an overall loss.