class: center, middle, inverse, title-slide # Econ 414 - Urban Economics ## Extending the Monocentric Model ### Marcelino Guerra ### February 16, 2021 --- # Enhancing the monocentric city model The monocentric model brings together household consumption, developers production, location, and commute choices, stating that accessibility determines land and housing prices at different locations. Many simplifying assumptions were imposed, so patterns regarding urban land use we observe in reality could be derived. -- From now, we extend that model in the following ways: * Including more elements of travel costs - The opportunity cost of time - Different travel patterns * Considering heterogeneous residents and properties - Two income groups (rich and poor) - Durable housing capital * Adding multiple business districts - Additional jobs dispersed within the city --- class: inverse,center, middle # Other Elements of Travel costs --- # Adding time costs </br> * Travel is costly beyond out-of-pocket expenses * To have a better picture, besides cash payments, one can consider **time costs** -- * Total commuting costs would be `\(T(x) \equiv \underbrace{\delta wx}_{\text{time}} + \underbrace{tx}_{\text{out-of-pocket}}\)` - If an extra mile reduces work time by some fraction `\(\delta\)` of an hour, that also reduces income by `\(\delta w\)` dollars, besides the out-of-pocket expense `\(t\)` -- * What happens with the spatial equilibrium condition after the introduction of commuting as part of an endogenous allocation of time? --- # Back to the price curve .pull-left[* The consumer's budget constraint before was `\(y-tx=pq+c\)` * The housing-price curve's slope before was `\(\cfrac{dp}{dx}-t/q\)` - That means if a resident moves marginally away from the CBD, his/her housing consumption falls as much as commuting costs increase * When you add the time cost `\(y-T(x)=pq+c\)`, the new housing-price curve's slope is `\(-T'(x)/q\)` * As you can see, everything equal, **that change alone does not affect the main properties and predictions of the model** ] .pull-right[] --- # Adding different travel patterns </br> * The urban travel pattern is more complicated than commuting to work - Shopping, recreational, and other personal/family business trips are equally important * **Even living in a hypothetical monocentric city, the location of entertainment, family, and friends matters** -- * However, you can also add richer notions of accessibility to the model. Assume that the jobs are still concentrated in downtown, but residents go shopping every other day, and those retail centers are equally spaced within the city * In that case, **the model's predictions are unaffected**. The location of jobs still leads to a negative housing-price curve, while the location of retail implies a small peak at every retail location. In the end, job accessibility dominates the pattern because travel to employment is much more frequent than trips to shopping -- * Keep in mind that, in reality, accessibility is not the only determinant of housing prices --- class: inverse,center, middle # Heterogeneous residents --- # Different incomes </br> * In the model described so far, all consumers earn the same labor income `\(y\)` * Now we consider two different groups with heterogeneous income levels - rich and poor -, and how that affects residential location patterns - Poor residents earn `\(y_{P}\)` and have utility level `\(u_{P}\)`. Rich residents earn `\(y_{R}>y_{P}\)` and have utility level `\(u_{R}>u_{P}\)` - **The spatial equilibrium applies within groups, but not across groups**. Poor residents get the same utility `\(u_{P}\)` everywhere in the city. Rich residents get the same `\(u_{R}\)` at any location. -- * Housing is essential, so both groups consume a positive amount of it. This implies that in some locations, low-income residents outbid wealthy residents, and in other areas, the opposite happens -- * **The equilibrium with different groups of residents implies perfect segmentation of these groups within the city** --- # Different incomes .pull-left[ * At any location, residents from different groups will want to consume different amounts of housing. Say, `\(q_{R}\)` and `\(q_{P}\)` * That implies housing-price curves with different steepness * Consider `\(\hat{x}\)` the boundary that separates rich and poor residents * At `\(\hat{x}\)`, the housing-price curves of rich and poor must be the same, i.e., `\(p_{R}=p_{P}\)` * If one group is willing to pay strictly more, there would be only one group of residents living on both sides of this point, so there would be no boundary at all ] .pull-right[] --- # Different incomes .pull-left[ * Again, at `\(\hat{x}\)`, rich and poor are willing to pay the same price. Then, `\(p_{P}(\hat{x}, u_{P})=p_{R}(\hat{x}, u_{R})\)` * That equality in the housing-price curve implies that rich residents must consume more housing than poor residents at location `\(\hat{x}\)`. So `\(q_{R}(\hat{x})>q_{P}(\hat{x})\)` * Remember the housing price slopes for rich and poor residents are, respectively, `\(-t/q_{R}\)` and `\(-t/q_{P}\)` * Since `\(q_{R}(\hat{x})>q_{P}(\hat{x})\)`, at `\(\hat{x}\)`, `\(-\cfrac{t}{q_{R}(\hat{x})}>-\cfrac{t}{q_{P}(\hat{x})}\)` * Therefore, the housing-price curve should be flatter for wealthy residents than for poor residents ] .pull-right[] --- # Different incomes and time costs .pull-left[ * Indeed, american cities have higher poverty rates in the center, and the wealthiest neighborhoods are in the suburbs * That fact seems to be a puzzle in the model **when you put together different incomes and time costs**. Travel costs are higher for rich residents since they earn a higher hourly salary than the poor residents: `\(\delta w_{R}>\delta w_{p}\)`. You might expect that high-income residents would be willing to pay more for low commuting costs at downtown * There are two explanations (outside the model) for the phenomenon: 1. If wealthy residents want to own more land, that desire will push them to the edge of the city 2. Rich and poor use different transportation modes - rich residents drive cars, and poor residents value public transportation ] .pull-right[] --- class: inverse,center, middle # Heterogeneous properties --- # Durable housing * When we did comparative statics in the monocentric city model - e.g., population increases, commuting costs increases, etc. - we are letting the city be rebuilt entirely from scratch to fit the new features * In practice, buildings last for decades, and we see old buildings next to brand-new ones. Also, new buildings are usually taller than old buildings, resulting in a pattern that violates the predictions of the model -- * So, how do the model's predictions change when we allow housing capital to be durable? * Now, the spatial pattern depends on both location and construction data * At a given date, buildings in downtown will be taller than the ones farther from the CBD * At a given location, new buildings can be taller or shorter compared to the old ones * **CBD develops first** * Overall, durable housing will lead to irregular local building-height, but the global pattern will look pretty much the same as in the simple model --- # Durable housing .pull-left[ * Deterioration of old buildings also incentives income sorting: as time goes by, high-income residents would move to higher-quality newer dwellings, and low-income residents move in to replace the former wealthy residents - In growing cities, new dwellings are located in the city edge, and high-income households will relocate toward the suburbs - Eventually, deteriorated structures will be replaced by new buildings. That makes high-income households to relocate toward the city center * Another important implication is that **the dynamics in growing and declining cities will be different**. Assume that, at the beginning, the city is in equilibrium and has housing stock `\(H_{1}\)` with associated prices `\(p_{1}\)`. Note that the supply curve is very inelastic up to the point `\((p_{1}, H_{1})\)`, and relatively elastic above the current housing stock ] .pull-right[] --- # Durable housing .pull-left[ * Suppose a positive demand shock (from `\(D_{1}\)` to `\(D_{2}\)`) The demand curve moves upward to `\(D_{2}\)` - Assuming there is vacant land, additional housing will be built, and there is a small increase in price. The new equilibrium is `\(H_{2}\)`, `\(p_{2}\)` * Suppose a negative demand shock from (from `\(D_{1}\)` to `\(D_{3}\)`) - The demand curve moves downward to `\(D_{3}\)` - Because housing is durable, housing stock will reduce very slowly - **There is a sharp drop in prices** - The new equilibrium is `\(H_{3}\)`, `\(p_{3}\)` ] .pull-right[] --- class: inverse,center, middle # Multiple CBDs --- # Employment outside of the CBD <br/> .pull-left[* Although employment may be high in the center, jobs are more dispersed within a city, or concentrated in a secondary business district * If jobs are dispersed, the results will be very similar to the ones when we added the shopping travel to the commuting cost * If employment is concentrated in a secondary center, the "polycentric" city will behave just like two connected monocentric cities ] .pull-right[] --- # Employment outside of the CBD * A puzzling feature of polycentric cities is the "wasteful commuting". Under this pattern, **B** would live close to SBD, but commute to work in the CBD. **A** would do the opposite. That is puzzling because if they switch locations, they will pay the same housing price having a shorter commute * Research shows that wasteful commuting is large in U.S. cities. Among possible explanations: *i)* residents may have preferences for particular neighborhoods *ii)* Employment uncertainty *iii)* Multiple workers households .center[]