Why is central Paris rich and downtown Detroit Poor? An amenity- based theory

Why is central Paris rich and downtown Detroit Poor? An amenity- based theory y: rueckner, Thisse, and Zenou Presented by: arber IV, ei, Chen & Tkachenko 1. Introduction Although some American central cities have rich enclaves, high- income residents in U.S. urban areas tend to live in the suburbs. This pattern is often reversed, however, outside the U.S. A prime example is the Paris metropolitan area, where the central city has higher average income than the surrounding suburbs. One of the contributions of the monocentric- city model developed by Alonso (1964), Mills (1967) and Muth (1969) is to give insight into the effect of income on household location. The model identifies two opposing forces. First, the high housing consumption of the rich means that they are more strongly attracted than the poor by low housing prices in the suburbs. However, rich households also have a high opportunity cost of time and thus a high commuting cost per mile. Therefore, the rich value accessibility to the central business district (CD) more than the poor. The theory shows that the net effect of these forces hinges on the behavior of the ratio of commuting cost per mile, t, and housing consumption, q. If t/q rises with income, then the accessibility effect dominates, and the rich tend to live at central locations. If t/q falls with income, then the effect of higher housing consumption dominates, and the rich tend to live in the suburbs. From now on we define this ratio t/q as the one reflecting conventional forces. Given this locational indeterminacy, the model is consistent with the variety of location patterns observed in real- world cities. For example, t/q might fall with income in the U.S. while rising with income in France, which would explain the location patterns in the two countries. However, the required difference in the behavior of the t/q ratio across countries seems implausible. Therefore, it is important to consider other explanations for the observed location patterns. The explanation proposed in the present paper links the location of different income groups to the spatial pattern of amenities in the city. To draw this link, the analysis adopts a key assumption, namely that the marginal valuation of amenities raises sharply with income. In addition, it is assumed that the conventional forces discussed above favor suburban location of the rich. Two cases are then considered: 1) amenities fall rapidly with distance to the CD; 2) amenities fall slowly or even increase with distance to the CD. 1

In the first case, the amenity advantage of the center along with the high amenity demand of the rich generates a force that pulls the rich toward the center more strongly than the poor. If this force is powerful enough, it can dominate the conventional forces that pull the rich outward, leading them to locate in the center. However, if the center s amenity advantage is weak or negative, then the amenity force will be insufficient to overcome the conventional forces, and the suburban location of the rich will be maintained. The urban amenities that underlie this theory can be classified into three categories. Natural amenities are generated by an areas topographical features, including rivers, hills, coastline. Historical amenities are generated by monuments, buildings, parks, and other urban infrastructure from past eras that are aesthetically pleasing to current residents of the city. While natural and historical amenities are largely exogenous, modern amenities are endogenous, with their levels depending on the current economic conditions in a neighborhood, especially the local income level. Such amenities might include restaurants, theaters, and modern public facilities such as swimming pools and tennis courts. The theory sketched above focuses on natural and historical amenities. Since these amenities are exogenous (they are discussed in Section 2), they can be viewed as a causal factor in determining the pattern of location by income. The first case discussed above, where exogenous amenities decline rapidly with distance to the center, might correspond to Paris. Its historical monuments, parks, boulevards, fine architecture, and river scenery give central Paris a large amenity advantage over the suburbs. If the amenity demand of the rich is strong enough, such an advantage might be sufficient to draw rich households to central locations, reversing the U.S. pattern. y contrast, since an American urban area like Detroit lacks the rich history of Paris, the central- city s infrastructure does not offer appreciable aesthetic benefits. This means that no amenity force is working to reverse the conventional forces that draw the rich to the suburbs. As a result, central Detroit is poor. Section 3 shows that the major lessons of the exogenous- amenities model of Section 2 are mostly unaffected when endogenous, modern amenities are added to the analysis. This modification is carried out by assuming that neighborhood income also counts as an amenity, capturing the variety of ways in which income influences the quality of life in an area. It is shown that when the rich value the income amenity more highly than the poor, multiple equilibria may exist, with the rich living either in the center or the suburbs. In effect, the high income (and thus strong endogenous amenities) of their own neighborhood makes the rich reluctant to leave it wherever it might be located. However, the analysis shows that if the center has a strong exogenous- amenity advantage, then the only equilibrium has the rich living in the center. The last step in the discussion is to consider why the pattern of exogenous amenities differs across cities. A central city s natural amenities are partly governed by transportation needs, which often dictate proximity to a body of water. On the other hand, historical amenities are determined mainly by past government decisions regarding investment in urban infrastructure. Section 4 presents a discussion of these factors, attempting to explain why exogenous amenities are more favorable in European than American central cities. 2

2. A Model with Exogenous Amenities ",,.. + " = " FOC u = p u Rewrite u e, q, a as u y tx pq, q, a Letting u denote the uniform utility level, this means that Max u{} y tx pq, q, a = u Differenciate the uniform utility condition w. r. t. x This yields the slope of the bid- price function for housing: t + p x q x + p x q x u + q x u + a x u = Plug in u = p u, rearrangement gives: p x = t u t v y tx, p x, a x + a x = + a (x) q(x) q x u q(x) q x Note that the marginal rate of substitution u /u is rewritten as the amenity derivative of the indirect utility function v[y- tx, p(x), a(x)]. v gives the marginal valuation of amenities after optimal adjustment of housing consumption. The standard urban model ignores amenities by assuming v. This gives us p x = <, which indicates that housing price falls with distance to CD. () If a (x)<, the conclusion of the standard model is reinforced. If a (x)>, the price pattern is ambiguous. Assume now that there are two groups of people, the rich and the poor, with y and y denoting their income level respectively, and y > y Let t and t denote the cost of commuting, and assume that t > t. In other words, the opportunity cost of the rich is larger than that of the poor. The distance between the bid- price slopes at x for the two groups can be written as p x p x = t t v y tx, p x, a x + + a x [ q (x) (x) q x v y tx, p x, a x ] q x To evaluate the amenity effect, suppose we have two cities, a reference city and a comparison city. 3

The reference city has the location pattern where the rich live in the suburbs. The city boundary of the reference city is denoted as x For the comparison city, suppose that for each group, income, commuting cost, and utility are the same as in the reference city, and that a x = a x. This in turn implies that the group boundaries are the same in the two cities (i.e. x = x ) Let the common group boundary in the two cities be denoted x, and let the common amenity level at this location be denoted a. In addition, let the common housing price level at x be p, and let the common levels of poor and rich housing consumption be denoted and t + t + a x [ v y tx, p, a t + t + a x [ v y tx, p, a v y tx, p, a ] v y tx, p, a ] Proposition. Suppose that the conventional locational forces dominate in the reference city, so that > holds and the rich live in the suburbs. Suppose further that the marginal amenity valuation rises faster than housing consumption as income increases. Then, the location pattern will be reversed in the comparison city, with < holding and the rich living in the center, whenever is negative and su ciently large in absolute value. 4

3. A model with endogenous amenities Endogenous amenities: The modern amenities available at a given location depend in part on the local income level, being a by- product rather than a determinant of the location patterns of different income groups. Modification of the old model: Utility function: ueqaz (,,, ) a: exogenous amenities z: income of the consumer s neighborhood Modern amenities are assumed to be an increasing function of neighborhood income. Consider a discrete measure of distance of only two locations: center and suburbs. The z value in a given location depends on which income group lives there. Commuting costs at the center =, total commuting costs from the suburbs equal t and t 1 for the poor and rich (suburban distance is normalized to one). Also, assume that dwellings at any location are available in two fixed sizes, q and q1 > q, with the rich choosing the larger size. The poor are assumed to be indifferent to the level of exogenous amenities and neighborhood a z a z income, while the rich value both. u = u = holds when e and q are small; u, u > holds when e and q are large. p denotes the poor s bid- price for housing in the center p denotes the poor s bid- price in the suburbs a exogenous amenities in the center a exogenous amenities in the suburban z exogenous neighborhood income in the center z exogenous neighborhood income in the suburban The poor: The bid- price in the center and suburbs must equalize the utility of the poor in the two locations: u( y p q,q,a, z ) = u( y t p q,q,a, z) 5

ecause the poor are unaffected by a and z, the difference in these variables between center and suburbs has no effect on the bid prices in above equation. Here only require that nonhousing consumption be equal in the two locations: p q = t + p q Rearrange: get the bid- price differential for the poor between center and suburbs p p = t q The Rich: Since the rich care about neighborhood income, their bid prices at the different locations depend on the residential pattern. A: denotes the pattern which the rich in the center and the poor in the suburbs The rich bid- price must satisfy: u( y 1 A p 1 q1,q 1,a A, y 1 ) = u( y 1 t 1 p 1 q1,q 1,a, y ) Under this pattern, z = y 1 and z= y. A p gives the price that a rich person would pay to live amidst the poor in the suburbs, a 1 movement that would upset the assumed residential pattern. Assuming a > a, the center has better exogenous amenities than the suburbs and offers better modern amenities, a consequence of its higher income. Therefore, utility equalization for the rich requires lower nonhousing consumption in the center p 1 A q1 > t 1 + p 1 A q1 Rearrange: the rich s bid- price differential between suburbs and center under pattern A is A A t p p1 > 1 1 q 1 From section 2, he group with the larger bid- price differential lives in the center. So we need to compare the bid- price differentials for the rich and poor to decide whether pattern A is an equilibrium. Assuming t / q > t 1 / q 1 as before, then it is possible that A A t p 1 p1 > = p q p, indicating the rich have the larger bid- price differential, as long as A A p 1 p1 is much larger than t1. For this to occur, the center s exogenous amenity advantage must be sufficiently large. q 1 Intuitive explanation: A rich person moving to the suburbs would forsake exogenous amenities and would also enter a lower- income area, with a consequent loss of modern amenities. If center has a large amenity advantage, these losses will overwhelm the conventional forces that draw the rich to the suburbs, assuring that pattern A is an equilibrium. 6

If the endogenous, modern amenities are dropped from the model, the above argument is unaffected and mirrors section 2. If exogenous amenities are absent, pattern A can still be an equilibrium with the endogenous component operating by itself. In this case, central location of the rich generates an endogenous amenity in the center that is sufficient to maintain the location pattern despite the lure of cheaper housing in the suburbs. : denotes the pattern which the rich in the suburbs and the poor in the center The bid- prices for the rich under this pattern must satisfy: u( y 1 p 1 q1,q 1,a, y ) = u( y 1 t 1 p 1 q1,q 1,a, y 1 ) In this pattern, a > a while modern amenities are worse in the center, the center s net advantage over the suburbs is ambiguous. Nonhousing consumption can be either smaller or p1 larger in the center, implying that the bid- price differential p 1 could be larger or smaller t1 than q. However, as long as p t 1 p1 is less than the poor differential, pattern is an 1 q equilibrium. This outcome holds if exogenous amenities are absent p 1 p1 < t1 / q 1 < t / q The above discussion shows that endogenous amenities introduce the possibility of multiple equilibria. Pattern A and are both equilibria when p 1 A p1 A > t / q > p 1 p1 For the first inequality to be satisfied, a must be large enough relative to a so that the combined exogenous and endogenous amenity advantage of the center under pattern A is large enough to offset the pull of the conventional forces, keeping the rich in the center. For the second inequality to hold, a must not be so large as to offset the combined pull of the conventional forces and endogenous suburban amenities under pattern, keeping the rich in the suburbs. The intuition for the existence of multiple equilibria is that endogenous amenities make the existing location of the rich attractive to them wherever it might be.moving into the poor area requires a sacrifice of endogenous amenities which depresses the bid- price of the rich and tends to maintain the existing location pattern, whether it is A or. The possibility of multiple equilibria may help explain the variety of location patterns in reality. However, this explanation is not as compelling as in the exogenous- amenities case because it requires one to argue that different equilibria are somehow selected in different cities. The notion that strong exogenous amenities in the center ensure central location of the rich, holds both in this section and in previous section. This is because both bid- price differentials are increasing in a, the level of central amenities. As a result, once a becomes sufficiently large, the second inequality is violated, ruling out pattern as an equilibrium. Thus, as in section 2, the rich must live in the center whenever its exogenous amenity advantage is sufficiently large. 7

4. Why exogenous- amenities patterns differ across Cities In the previous sections the authors have developed a model that explains where the rich will locate based upon amenities. The authors show that when amenities are located within the city center, and the rich highly value them, the rich will locate in the center and not in the suburbs. Conversely, when there is few amenities located in the city center the rich will prefer cheaper and bigger housing available in the suburbs. Since the theory of location is determined by the spatial pattern of amenities, the authors next look at why amenity patterns differ across cities. Recall that there are three types of amenities described by the authors: 1) natural amenities, 2) historical amenities, and 3) modern amenities. When addressing the effect of natural amenities, the authors mostly wave their hands. Almost all major cities have developed near waterways, so it is difficult to understand why certain rivers are better than others. However, the authors make an (largely unconvincing) argument that the Seine River is particularly special when it comes to rivers and thus a natural amenity. However, without a way to differentiate which natural amenities are better than others across cities (why is the Detroit river worse than the Hudson, or the Charles, or the Thames?), the explanation of natural amenities is poorly formulated. The authors put forth a stronger argument about natural amenities. They claim that European cities have a longer historical legacy than U.S. cities, which leads to higher historical amenities within the city center. With increased historical amenities draws the rich into living within the city center. As the authors point out, this could be dependent upon governmental structure and spending. In countries that are highly centralized politically (most of Europe), spending should be focused on the city center, compared to countries where there is a decentralized government (the US). Therefore, the paper predicts that the rich will live in city centers where the government is highly centralized and live in the suburbs in countries where government is highly decentralized. This argument is a divergence from the standard model, and lackadaisically adds in government structure to spending. However, the core of the historical amenity argument still holds: cities that have been around longer should see a larger concentration of rich in the city center due to increased historical amenities. This argument can be tested empirically within any country. Therefore we have put together some data to see if this argument has any bearing within the US. elow is a graph of the proportion of income in the city center compared to the suburbs in fourteen major cities within the United States: 8

(" "#$+/""12+"1+#3-+45+ '#$" '" &#$" "#$%&'()*+ &" %#$" %" #$" " %)*)" %)+" %)+%" %)+&" %)+'" %)+(" %)+$" %)+*" %)++" %)+," %)+)" %)," %),%" %),&" %),'" %),(" %),$" %),*" %),+" %),," %),)" %))" %))%" %))&" %))'" %))(" %))$" %))*" %))+" %))," %)))" &",-.(+ -./123" 45673/6" 89" :;-" <./==>"?1@"A71@"?51=5" 4-" C3D63@" </DEF72." 96=1@61" -3=FGEFD" 4F7.1G" -=5H5=1@I" This data is based upon the REIS data from the EA, where wealth is defined as personal income of the city. Personal income is a measurement that looks at the amount of earnings individuals bring into a geographic space. This differentiates itself from the amount produced in a geographic area, meaning that it captures the relative wealth of the inhabitants of an area. This provides a good test for the author s theory. The y- axis represents the relative wealth per capita of the city divided by the relative wealth per capita of the suburbs. This means the higher the value on the y- axis the richer the city center is compared to its suburban counterparts. As we can see from this graph, only one city (New York City) has richer individuals concentrated within the city center, and the share of rich individuals living in Manhattan compared to the boroughs has increased over time. While this graph shows how much of an outlier NYC is, it restricts the ability to view the variation within the other cities. To see what is happening more generally around the United Sates, we can look at this second graph. The axis is the same in this graph as before, however I ve removed NYC as the main outlier. Contrary to the model, from this graph we can see that most cities experience change in the proportion of rich living within the city center over time. While the overall change has been relatively constant, all of these cities (save DC) has experienced urban flight over the past 3 years. 9

(#$" "#$+/""12+"1+#3-+45+ (" -./123" #'" "#$%&'()*+ #&" #%" 45673/6" 89" :./;;<" =1>"?71>" =51@;5" 4-" A363>" :/@CD72." 96;1>61" -3;DECD" 4D7.1E" -;5F5;1>G" #$" " ()&)" ()*" ()*(" ()*$" ()*+" ()*%" ()*," ()*&" ()**" ()*'" ()*)" ()'" ()'(" ()'$" ()'+" ()'%" ()'," ()'&" ()'*" ()''" ()')" ())" ())(" ())$" ())+" ())%" ())," ())&" ())*" ())'" ()))" $",-.(+ What evidence can we see, if any, of the argument that older cities will be more likely to capture rich citizens due to the city s increased historical amenities? To look at anecdotal evidence for this, we look at the historical cities within the United States (DC, oston, Philadelphia) compared to new manufacturing cities within the US (Cleveland & Columbus). (#$" /"*#("1.2+"#$+3"4"56+ (" #'" "#$%&'()*+ #&" -./1" 23" 456758" 359:;96" 3<=<>8?" #%" #$" " ()&)" ()*" ()*(" ()*$" ()*+" ()*%" ()*," ()*&" ()**" ()*'" ()*)" ()'" ()'(" ()'$" ()'+" ()'%" ()'," ()'&" ()'*" ()''" ()')" ())" ())(" ())$" ())+" ())%" ())," ())&" ())*" ())'" ()))" $",-.(+ Here we can see that there is little evidence that older cities are more likely to attract rich citizens into their city centers. In fact, the city with the most historical amenities, Washington 1

DC, has the least amount of rich citizens living in the city in the sample. This provides little support for the author s theory about historical amenities. If natural amenities provide a poor explanation and there is little evidence that historical amenities carry much explanatory power within countries, then modern amenities must be telling the story. However, as we can see from the sample, the proportion of people leaving the city center can drastically change over time. (#$" /+123-(+"#$+1"4"23+ (" #'" "#$%&'()*+ #&" -./12/" 34" 567"867" #%" #$" " ()&)" ()*" ()*(" ()*$" ()*+" ()*%" ()*," ()*&" ()**" ()*'" ()*)" ()'" ()'(" ()'$" ()'+" ()'%" ()'," ()'&" ()'*" ()''" ()')" ())" ())(" ())$" ())+" ())%" ())," ())&" ())*" ())'" ()))" $",-.(+ This graph shows three cities that have most drastically changed over time: Los Angeles, San Francisco, and Detroit. LA and San Francisco actually started the 7 s with the richer individuals living within the city center. However, over time both cities have seen the rich start to migrate outside the city center. This pattern is troubling for the modern amenities theory. There were certainly high levels of modern amenities in all of these cities if we go back far enough. If amenities is the causal story for why rich move, the question becomes: why do the rich abandon amenities in the city center to travel out to the suburbs? The authors, due to their non- dynamic model, cannot fully account for shifts in patterns. Of course we could come up with anecdotal explanations for these cases (the true amenity in LA & San Fran is the weather, which can be enjoyed everywhere, and Detroit had several race problems making white flight occur, etc.), however this theory struggles to explain changes within the same cities. 11

4. Conclusion This article attempts to remedy the standard model of urban location by taking account of city level amenities. y explicitly adding amenities into the model of location, the authors have created a scenario where location is dependent upon a city idiosyncratic characteristics. Thus it can predict multiple location patterns across cities. The problem with explaining everything is that you also explain very little. While the authors model can predict multiple equilibria, it also has very little leverage in explaining phenomenon. Therefore when asking a question about where people will locate, all this paper can say definitively is: it depends. While the authors see the conditional nature of the theory as a strength, it is also a weakness. If every scenario is possible, this tells us little about any individual case. The authors started this project because they were unsatisfied with the traditional model s ambiguous predictions: it is unsatisfactory from a scientific view to explain real- world complexity by appeal to an ambiguous theory. Yet, their model fairs no better, relying upon equally squishy terms. Furthermore, since this model is not dynamic, it is deterministic. Wherever amenities are located, so too will the rich. Yet empirically we can see that the composition of the city changes over time. If the rich are locating downtown for the amenities in one year, yet moving out the following year, why are they leaving the amenities? To understand flow of population we need a theory of amenities, which is based upon government expenditures. 12