class: center, middle, inverse, title-slide # Econ 414 - Urban Economics ## Spatial Equilibrium Across Cities ### Marcelino Guerra ### March 2-4, 2021 --- # Introduction * Before, we studied the AMM model to understand prices and population density levels within cities. The Rosen-Roback model is the core tool for understanding prices and population distribution across metro areas * Remember the equation: - Income + Amenities - Housing costs - Transportation costs * Now, income and amenities (e.g., nice weather, low crime rates) differ across space. To compensate for that added complexity, metropolitan areas are treated as a single homogeneous entity - everyone who lives in the area have the same costs of housing and transportation, as well as amenity levels * Both regional housing prices and incomes are likely to vary with amenities * The goal is to analyze the trade-off between income, amenities and housing costs across cities using the Rosen-Roback model * In this framework, it is also possible to compare quality-of-life across areas --- class: inverse,center, middle # Empirical Regularities Across Cities --- # Happiness in US Metro Areas <iframe src="maps/happiness.html" style="width: 1200px; height: 500px; border: 5px" alt=""> --- # Income and Housing Costs
--- # Amenities and Income I
--- # Amenities and Income II
--- class: inverse,center, middle # The Rosen-Roback Model --- # Consumer Analysis * We extend the consumer's utility function to include amenities `\(a\)`. The same way we simplified housing as "quantity of floor space" `\((q)\)`, we treat `\(a\)` as a single index that represents lower crime rates, nice weather, proximity to the ocean, etc. Consumers get utility from amenities. * All consumers have the same preferences that can be represented by `\(U(.)\)`. Including amenities, the new consumer's utility function is `\(U(c,q,a)\)`. * We assume that there are no costs of changing residences across cities, and we ignore intracity commuting costs * Cities vary according to the quantity of an endowed amenity `\(a\)`. Consumers in a given location (with amenities `\(a\)`) will choose `\(c\)` and `\(q\)` to maximize utility given their budget constraint `\(y=pq+c\)` * Associated with this maximization problem is the indirect utility function `\(V(.)\)`. It is easier to see how incomes and prices vary with amenities **considering the utility level that consumers achieve earning `\(y\)` and paying `\(p\)` for housing: `\(V(y,p,a)\)`** * `\(V(.) \downarrow\)` when `\(p \uparrow\)`, `\(y \downarrow\)` or `\(a \downarrow\)` * `\(V(.) \uparrow\)` when `\(p \downarrow\)`, `\(y \uparrow\)` or `\(a \uparrow\)` --- # Consumer Analysis .pull-left[ * Consumers have the same preferences and are free to move without costs. Hence, also for the analysis across cities, we rely on the idea of spatial equilibrium * Spatial equilibrium implies consumers equally well off across regions: `\(V(y,p,a)=\bar{u}\)`. If some location offers higher utility levels compared to others, workers will move there and push incomes down or housing prices up until utility levels are equalized * The analysis is done with indifference curves that show different combinations of `\(p\)` and `\(y\)` that yield the same utility `\(\bar{u}\)`, conditional on a given amenity level ] .pull-right[![](figs/indiff.jpg)] --- # Firm Analysis .pull-left[ * Firms use labor `\(y\)` and "real estate" (floor space) `\(p\)` to produce the commodity `\(c\)` (bread). The price of floor space is common across residential and business uses, and we assume constant returns to scale - the cost per unit of bread is the same, no matter how much output a firm produces * Costs may also depend on amenities. Thus, `\(C(y,p,a)\)`. `\(C\)` is increasing in `\(y\)` and `\(p\)`, and may increase or decrease with amenities. It is also possible that `\(C\)` is independent of `\(a\)`. * Spatial equilibrium implies firm profits equal across regions. The zero-profit condition holds when `\(C(y,p,a)\)` is equal to `\(1\)` - the cost of a unit of c. This condition generates iso-profit curves: given the amenity level, which combinations of `\(y\)` and `\(p\)` lead to zero profit ] .pull-right[![](figs/firm.jpg)] --- # High-amenity x Low-amenity areas <br/ > * Putting indifference curves and iso-profit curves together, one can answer the main question of interest: How do real-estate prices and incomes differ between high-amenity and low-amenity regions? * There are three possible cases: 1. Amenities **do not affect** firms' costs - Only one iso-profit curve 2. Amenities **decrease** firms' costs - Iso-profit curve moves upward - Example: Lack of snow storms 3. Amenities **increase** firms' costs - Iso-profit curve moves downward - Example: Clean air --- # Case 1 .pull-left[ * In the low-amenity area `\(a_{0}\)`, the intersection point occurs at `\(p_{0}\)` and `\(y_{0}\)`. Those values satisfy two requirements. First, paying `\(p_{0}\)` for housing and earning `\(y_{0}\)`, consumers are enjoying the common utility level across regions `\(\underline{u}\)`. * Second, firms earn zero profit since the point `\(p_{0}\)`, `\(y_{0}\)` also lies on the iso-profit curve. The same reasoning can be applied to high-amenity area `\(a_{1}\)` * As one can see, the high-amenity region has a higher real-estate price and lower-income level compared to the low-amenity region ] .pull-right[![](figs/case1.jpg)] --- # Case 2 .pull-left[ * With firm costs decreasing in consumer amenity (no winter storm), there will be two iso-profit curves - in the high amenity area `\(a_{1}\)`, the iso-profit curve shifts upward * Depending on the shift in the iso-profit curve (i.e., depending on how much amenities decrease the firm's costs), the high-amenity region can have both higher incomes and higher housing prices (as you see in the figure), or lower incomes and higher housing prices ] .pull-right[![](figs/case2.jpg)] --- # Case 3 .pull-left[ * In case the amenity is "unproductive", the iso-profit curve shifts downward. For instance, clean air is a valuable amenity to consumers but demands firms to adopt new nonpolluting technologies, and that might increase firm's cost * Again, depending on how much the amenity increases costs - i.e., the shift in the iso-profit curve -, the high-amenity region can have both lower incomes and housing prices (as you see in the figure), or lower incomes and higher housing prices * To summarize, when amenities have only a small role on firm costs, the pattern we saw on case 1 will prevail: the higher amenity region will offer higher housing prices and lower wages ] .pull-right[![](figs/case3.jpg)] --- # Quality of Life in the US <iframe src="maps/qol.html" style="width: 1200px; height: 500px; border: 5px" alt=""> --- # Housing Supply and Productivity Shocks <style type="text/css"> .pull-left2 { float: left; width: 39%; } .pull-right2 { float: right; width: 59%; } .pull-right2 ~ p { clear: both; } </style> .pull-left2[ * In reality, cities differ in terms of housing supply. For instance, different regulatory environment (or natural boundaries) might impact developers, who would respond differently to increase in demand for housing * For instance, consider a productivity shock in both cities A and B. City B has an inelastic supply curve, and when demand shows up, that is translated to higher housing prices. * When the supply curve is very elastic (Sun belt metros), you might see fast-growing areas without increasing housing prices ] .pull-right2[
] --- # People and Prices Across Space The (more sophisticated) Rosen-Roback model serves as the basis for empirical work on cities. The spatial equilibrium model shows how **productivity, housing supply, and amenities** determine changes in **population, wages, and prices** across cities. When you solve the model, three intuitive (and estimable) equations come out, representing the primary relationships to understand urban change: `\begin{equation} Population=f(\underbrace{\text{Productivity}}_\text{+}, \underbrace{\text{Amenities}}_\text{+}, \underbrace{\text{Land Supply}}_\text{+}) \end{equation}` `\begin{equation} Income=f(\underbrace{\text{Productivity}}_\text{+}, \underbrace{\text{Amenities}}_\text{-}, \underbrace{\text{Land Supply}}_\text{-}) \end{equation}` `\begin{equation} Prices=f(\underbrace{\text{Productivity}}_\text{+}, \underbrace{\text{Amenities}}_\text{+}, \underbrace{\text{Land Supply}}_\text{-}) \end{equation}` The population rises with productivity, amenities, and land supply. Income is rising in productivity and falling in both amenities and land supply. Prices increase in productivity and amenities, and decline in land. Also, changes in population, income, and prices can identify amenities, productivity, and land supply changes. --- # The Rise of Sunbelt .pull-left[ * Over the past years, America's fastest-growing metro areas are Atlanta, Dallas, Phoenix, and Houston * Some areas added almost 1 million people between 2000 and 2010 * The figure relates the population growth between 2013 and 2015 * Different theories explain the rise of Sunbelt. Based on the Rosen-Roback model, one can raise three possibilities: 1. Increase in productivity (e.g., capital accumulation, improvements in political institutions) 2. Consumption of amenities (higher desire for mild winters, improvements in public service) 3. Local policies that support new construction of housing ] .pull-right[![](figs/subelt.jpg)] --- # The Rise of Sunbelt .pull-left[ * The rise of Sunbelt cities seems to be more related to a **very elastic housing supply** * The plentiful housing supply reflects the combination of abundant land, freeways, and pro-growth permitting. There is a high tolerance for new constructions there * One can emphasize the Sunbelt's rising productivity, but the combination of a growing population, stagnant wages, and declining housing prices suggest that amenity values were not increasing ] .pull-right[![](figs/subelt.jpg)] --- # House Values and City Growth