ECON 414: Urban Economics
Homework 2: Suggested Answers
Marcelino Guerra
Monocentric city model [45 Points]
The city’s edge [10 points]
Suppose that the urban land rent function is given by \(r(x)=80-x\), where \(x\) is the distance in miles to the city’s Central Business District. The agricultural sector is willing to pay \(r_{a}=20\) for productive land. Compute the radius \(\bar{x}\) and the urban land area (the area of a circle with radius \(\bar{x}\)). Finally, what is the population density in this urban area? Assume that the city has 100,000 inhabitants.
The city’s edge
Answer
The city’s land area is the result of a competition between housing developers and farmers. The equilibrium is represented by the intersection of the land-rent curve \(r(x)\) and \(r_{a}\), and the resulting \(x\) determines the edge of the city \((\bar{x})\). Hence, \(r(x)=80-x=20=r_{a} \rightarrow \bar{x}=60\). The total urban area is the area of the circle \(\pi\bar{x}^{2}=60^{2}\pi=3600\pi\). Finally, the population density is \(\cfrac{100,000}{3600\pi}=\cfrac{27.77}{\pi}\).
A change in commuting costs [15 points]
i) Let’s consider the effect of an increase in the commuting-cost parameter \(t\). Suppose that initially, the city was in equilibrium just as in 1.1. However, the local government decided to raise the metro ticket price to cover the city’s network of radial roads’ maintenance costs - remember, we assume that consumers use the same transport mode. What happens when \(t\) increases? Write down the whole process that makes the land rent curve rotate. What happens in downtown? What changes in the suburb?
ii) Assume the new land rent function of developers after the increase in \(t\) is \(r(x)=120-2x\). What is the new city’s edge \(\bar{x}\)? What is the value of \(x\) (distance from the CBD) when the old and new developers’ land rent function intersect (call it \(\hat{x}\))? What is happening with the rent bids from 0 to \(\hat{x}\) and from \(\hat{x}\) to the new \(\bar{x}\), compared to the old rent curve? Finally, what is the meaning of \(\hat{x}\)?
The effect of an increase in commuting costs
i)
With higher commuting costs, the old spatial equilibrium no longer holds since suburban commuters can be better off moving closer to the city center. As they move to downtown, that increases the demand for housing in the city center, bidding up housing prices near the CBD - the opposite happens at suburban locations. As a result, the housing-price curve rotates in a clockwise direction.
Now, the housing prices are higher near the center and lower in the suburbs. Therefore, the competition between builders for land in downtown increases, and for suburban ground decreases. Hence, the land-rent curve rotates just like the housing-price curve. In the end, the city’s land area shrinks, apartments are smaller, and buildings in downtown are taller, increasing the density there. Since the land area in the suburbs shrinks, the change in population density is ambiguous (\(D=\frac{\text{Total Population}\downarrow}{\text{Land Area}\downarrow}\)).
ii)
The new city’s edge is \(120-2x=20\rightarrow \bar{x}=50\). \(\hat{x}\) is the intersection between the old and new bid-rent curve: \(80-x=120-2x\rightarrow \hat{x}=40\). From \(0\) to \(40\), developers are willing to pay more for land than before. From \(40\) to \(50\), there is a weaker competition between developers. Therefore, \(\hat{x}\) can be interpreted as a boundary that separates the areas close to the CBD from the suburbs.
Single-family and multi-family homes in Chicago-IL [20 points]
Last semester, one of the ECON 414 students shared this article on Piazza: “Chicago area has the largest homes among US metros — but apartment sizes are shrinking”. Reading the article, you will find interesting information such as:
’‘Among the country’s 20 largest metropolitan areas, Chicago takes first place for having the biggest homes. […] When it comes to apartment sizes though, the city falls toward the bottom of the list, with its average apartment growing smaller over time.’’
Here you find a table with some data about house and apartment sizes across US metro areas in 2010 and 2019.
Can you find an explanation for that trend basing your answer on what you know about the Monocentric City Model?
Hint: think about i) the developers’ perspective (e.g., why do they build up?) ii) the intercity analysis (for example, the effects of an increase in population) iii) the across cities analysis - e.g., downtown Chicago has taller buildings compared to central Phoenix. Finally, it would be best if you borrow some ideas from the real estate agent Sam Jenkins. After all, he has skin in the game.
Answer
You can base your answer on many ways. First, when there is an increase in the demand for downtown apartments, that bids up housing prices and, therefore, land rents. You can think about the demand side in the following terms: population increase/migration or increase in commuting costs, with people moving from the suburbs to the center. With land rents more expensive, developers will react to that building up and shrinking apartments.
It is important to note that this population increase needs to be disproportionately concentrated in the city center - maybe young people with stronger tastes for amenities are moving to downtown (they want to be in the “right location” according to Jenkins). That explanation is outside of the model since we assume people with the same preferences and hold amenity levels constant within the city. If the demand for housing in the suburbs also increases, everything constant, that would shrink suburban houses, but the opposite is happening - single-family homes are getting bigger. That trend might signal the absence of suburban land supply restrictions. Also, according to Jenkins, “[…] The COVID-19 pandemic has exacerbated and crystallized the trend a bit more […]”. Now, people want to have dedicated spaces to home-office (as a grad student who works from home since 2017, I agree with that), increasing the demand for bigger houses. Of course, if you want more space, you go to where the price per square foot is lower.
Monocentric city model - Extensions [55 Points]
Durable housing in growing and declining cities [15 points]
i) Consider an extreme version of the durable housing we saw in class - the first 60 houses built in a city last forever. The demand curve function is given by \(P=70-\frac{2}{3}H\). The cities start in equilibrium with 60 houses. Compute the equilibrium price in this situation.
ii) Now, suppose a factory closed its doors and all the employers were fired. Without a job, they decided to leave the city. Due to this negative shock the demand \(D_{2}\) decreases to \(P=50-\frac{2}{3}H\). What is the new equilibrium price?
Declining cities
iii) Assume there is a positive shock and the new demand \(D_{3}\) is characterized by \(P=100-\frac{2}{3}H\). As you can see, cities A and B have different supply curves. For new housing, city’s A supply curve is given by \(H=2P\), and city B has a supply curve equal to \(H=\frac{2}{3}P+40\). Compute the respective equilibrium quantities and prices for both cities (round to the nearest integer). Do you have an explanation of why would the supply curves be different in different cities? Which supply curve would characterize Houston, and which one can be associated with Santa Clara (Silicon Valley)?
Growing cities
i)
Equilibrium price is \(P=70-\frac{2}{3}60=30\)
ii)
New equilibrium price is \(P=50-\frac{2}{3}60=10\). With durable housing, the supply stays the same, and there is a sharp decrease in housing prices in declining cities.
iii)
City A:
\(P_{A}=100-\frac{2}{3}2P_{A}\rightarrow\frac{7}{3}P_{A}=100\rightarrow P_{A} \approx 43\)
\(H_{A}=2P_{A} \approx 86\)
City B:
\(P_{B}=100-\frac{2}{3}(\frac{2}{3}P_{A}+40)\rightarrow \frac{13}{9}P_{A}=\frac{220}{3}\approx 51\)
\(H_{B}=\frac{2}{3}P_{B}+40 \approx 74\)
As one can see, in equilibrium, city A has a higher housing quantity and lower price than city B. City B supply curve is more inelastic, and this is most likely related to land supply restriction - it could be natural boundaries such as mountains or land-use regulation (a UGB, for instance). In our context, city A characterizes Houston, and city B is Santa Clara.
Two income groups [15 points]
Suppose there are two income groups in the hypothetical monocentric city. A typical wealthy citizen earns 70,000 dollars per year, and poor residents earn 30,000. Assume that both groups have the same out-of-pocket commuting costs \(tx\), and spend 10,000 with good \(c\) per year.
i) Write down the budget constraints for both groups.
ii) We know that both groups’ housing price curves should cross at some point \(\hat{x}\). Otherwise, one group would outbid the other everywhere in the city, and there would be only high-income (or low-income) residents there. We also know that, at \(\hat{x}\), both groups pay the same price per square foot, i.e., \(p_{R}=p_{P}=p\). Write down the quantities of floor space that high-income and low-income residents would consume at \(\hat{x}\), and show that \(q_{R}\) is indeed higher than \(q_{P}\). Also, what is the implication of that in terms of housing-price curve slopes for both groups?
Hint: you want to write \(q_{R}\) and \(q_{P}\) as a function of \(t\hat{x}\) and \(p\).
iii) Let’s say the city’s edge is \(\bar{x}=60\). Suppose the housing-price curves for low-income and high income are \(p_{P}(x)=120-2x\) and \(p_{R}(x)=100-x\), respectively. Find the distance \(\hat{x}\) that separates the city between rich and poor residents. Finally, what is the total suburban land area?
Two income groups
i)
High-income group:
\(y_{R}-tx=p_{R}q_{R}+c\rightarrow 70,000-tx=p_{R}q_{R}+10,000 \rightarrow 60,000-tx=p_{R}q_{R}\)
Low-income group:
\(y_{P}-tx=p_{P}q_{P}+c\rightarrow 30,000-tx=p_{P}q_{P}+10,000 \rightarrow 20,000-tx=p_{P}q_{P}\)
ii)
At \(\hat{x}\), both groups pay the same communting costs \(t\hat{x}\). Also, they pay the same housing price \(p_{R}=p_{P}=p\). Let’s rewrite the budget constrains:
\(60,000-t\hat{x}=pq_{R} \rightarrow q_{R}=\cfrac{60,000-t\hat{x}}{p}\)
\(20,000-t\hat{x}=pq_{P} \rightarrow q_{P}=\cfrac{20,000-t\hat{x}}{p}\)
Therefore, \(q_{R}>q_{P}\). The implication of that is the following: the housing-price curve should be flatter for wealthy residents than for poor residents. Comparing the slopes of both housing-price curves you see that, at \(\hat{x}\), \(\cfrac{-t}{q_{R}}>\cfrac{-t}{q_{P}}\). It is important to remember that we are assuming same commuting costs for both income groups. When you add time costs in the commuting costs, that relationship might not hold anymore.
iii)
\(120-2x=100-x \rightarrow \hat{x}=20\). Hence, we will consider downtown land from 0 to 20, and the suburban ground from 20 to 60 \((\bar{x})\). The total urban area is \(60^{2}\pi=3600\pi\). The downtown land area is \(20^{2}\pi=400\pi\). Hence, the suburban land area is \(3600\pi-400\pi=3200\pi\)
Household income and distance from the center [25 points]
You already know that one of the predictions of the Monocentric City Model with two income groups with the same commuting costs is that the high-income residents will live in the suburbs, and the low-income group will locate at the city center. Below is a map of 2,210 census tracts within the Chicago-Naperville-Elgin (IL-IN-WI) metro area. The information about Median Household Income is coming from the American Community Survey 2015-19. To make things simpler, consider the Navy Pier as the metro area’s Central Business District.
i) (Overall) Does this pattern of Median Household Income within Chicago metro area agree with the prediction from the Monocentric city model?
ii) What do you see in the neighborhoods close to the Navy Pier? Does the model have an explanation for this pattern?
iii) Here you have ACS 2015-19 data about the percentage of people living below the poverty line in census tracts within Chicago metro area. Let’s take a look at the relationship between poverty levels and distance from the city center constructing a scatter plot with those variables (y-axis perc_pov and x-axis dist_km). What do you see? Does that make sense/agree with your answer in i)?
i)
Overall, it does. Although you do not see perfect segregation between high-income and low-income, as you go farther from the center, the median household income, on average, increases.
ii)
You do see high median household income close to the city center. The AMM predicts perfect segregation between high-income and low-income within the city, but that is not happening. It is important to remember that the model i) holds amenities constant in the town ii) assumes same preferences within groups. Chicago downtown offers high amenity levels and might attract high-income individuals with stronger preferences for amenities. They might prefer the entertainment options in the center instead of more housing space in the suburban area.
iii)
There is a negative relationship between share of poverty and distance from the city center, although distance only explains 8.5% of poverty’s variation within the Chicago metro area (check the regression below). As the AMM model with two income groups predicts, the share of people living below the poverty line is higher close to downtown.
One explanation (outside the AMM framework) for that pattern is that low-income people relies on public transportation to commute. The pattern also agrees with what we see in the map - as you go further from downtown, the median household income increases.
library(ggplot2)
library(ggthemes)
library(ggrepel)
income<-readRDS("chi_inc.RDS")
ggplot(income, aes(x=dist_km, y=perc_pov)) +
geom_point(color="#6794a7", size=3,alpha=.7) +
stat_smooth(method = "lm", formula =y~x, se=F, colour="#014d64") +
scale_x_continuous(name = "Distance from city center") +
scale_y_continuous(name = "Share of people below the poverty line") +
theme_economist(base_size = 17)+
theme(axis.text=element_text(size=15),
axis.title=element_text(size=15,face="bold"),
panel.grid.major.x = element_line( size=.05, color="white"))
reg<-lm(perc_pov~dist_km, data=income)
summary(reg)##
## Call:
## lm(formula = perc_pov ~ dist_km, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.563 -8.104 -3.286 5.291 60.366
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.94631 0.43587 43.47 <2e-16 ***
## dist_km -0.16169 0.01127 -14.34 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 11.36 on 2200 degrees of freedom
## (8 observations deleted due to missingness)
## Multiple R-squared: 0.08551, Adjusted R-squared: 0.08509
## F-statistic: 205.7 on 1 and 2200 DF, p-value: < 2.2e-16