Lectures_on_Urban_Economics_----_(2_Analyzing_Urban_Spatial_Structure)

2 Analyzing Urban Spatial Structure 2.1 Introduction Looking out the airplane window, an airline passenger landing in New York or Chicago would see the features of urban spatial structure rep- resented in a particularly dramatic fashion. In both of those cities, the urban center has a striking concentration of tall buildings, with build- ing heights gradually falling as distance from the center increases. The tallest buildings in both cities are office buildings and other commercial structures, but the central areas also contain many tall residential build- ings. Like the heights of the office buildings, the heights of these resi- dential structures decrease moving away from the center, dropping to three and two stories as distance increases. Single-story houses become common in the distant suburbs. Although it is less obvious from the airplane, an equally important spatial feature of cities involves the sizes of individual dwellings (apart- ments and houses). The dwellings within the tall residential buildings near the city center tend to be relatively small in terms of square footage, while suburban houses are much more spacious. Thus, although building heights fall moving away from the center, dwelling sizes increase. In walking around downtown residential neighborhoods in Chicago or New York, the traveler would notice another difference not clearly visible from the airplane. Relative to her suburban neighborhood at home, there would be many more people on the streets in these down- town neighborhoods, walking to restaurants, running errands, or heading to their workplaces. This difference is due to the high popula- tion density that prevails in central-city residential areas, which is also reflected in activities on the street. Population density falls moving away from the city center, reaching a much lower level in the suburbs. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

24 Chapter 2 Other important regularities of urban spatial structure aren’t visible at all from an airplane or from the street. These features involve real- estate prices, and they require experience in real-estate transactions, or familiarity with urban data, to grasp. First, whereas vacant lots usually can be purchased for reasonable prices in the suburbs, vacant land near the city center (when it is available) is dramatically more expensive per acre. The same regularity applies to the price of housing floor space: the rental or purchase price per square foot of housing is much higher near the city center than in the suburbs. Consumers aren’t used to thinking about prices on a per square foot basis (focusing instead on the monthly rent or selling price for a dwelling), but any real-estate agent knows that residential prices per square foot fall moving away from the city center. Other regularities involve differences across cities rather than center- suburban differences within a single city. To appreciate these differ- ences, suppose that our traveler is from Omaha, Nebraska. When her plane lands there on her return trip, she will notice that buildings in central Omaha, though taller than those in Omaha’s suburbs, are much shorter than those in the big city she just visited. In addition, if the traveler had access to price data, she would see that a vacant lot in the center of Omaha would be cheaper than one in the center of New York. Economists have formulated a mathematical model of cities that attempts to capture all these regularities of urban spatial structure. This chapter develops and explains the model. But it does so without relying on mathematics, instead using an accessible diagrammatic approach. As will be seen, the urban model successfully predicts the regularities described above. Since the model thus gives an accurate picture of cities, it can be used reliably for predictive purposes in a policy context. For example, the model can predict how a city’s spatial structure would change if the gasoline tax were raised substantially, thereby raising the cost of driving. It can also be used to analyze how a variety of other policies would affect a city’s spatial structure. The model presented in this chapter originated in the works of William Alonso (1964), Richard Muth (1969), and Edwin Mills (1967). Systematic derivation of the model’s predictions was first done by William Wheaton (1974) and later elaborated by Jan Brueckner (1987). 1 1. For a comprehensive treatment of the economics of urban land use, see Fujita 1989. For a more recent book-length treatment, see Papageorgiou and Pines 1998. Glaeser 2008 also contains a chapter on this topic. For a useful overview paper, see Anas, Arnott, and Small 1998. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

26 Chapter 2 entirely of single-person households. The identical-household assump- tion is relaxed below by allowing the city to have two different income groups: rich and poor. The fourth major assumption is that the city’s residents consume only two goods: housing and a composite good that consists of every- thing other than housing. Since the model is about cities, it naturally focuses on housing. Simplicity requires that all other consumption be lumped together into a single composite commodity, which will be called “bread.” 2.3 Commuting Cost Let x denote radial distance from a consumer’s residence to the CBD. The cost of commuting to work at the CBD is higher the larger is x , and this cost generally has two components. The first is a “money” (or “out-of-pocket”) cost. For an automobile user, the money cost consists of the cost of gasoline and insurance as well as depreciation on the automobile. For a public-transit user, the money cost is simply the transit fare. The second component of commuting cost is time cost, which captures the “opportunity cost” of the time spent commuting— time that is mostly unavailable for other productive or enjoyable activities. Because a proper consideration of time cost makes the analysis more complicated, this component of commuting cost is ignored in developing the basic model. However, time cost is needed in analyzing a city that contains different income groups, so it will be re-introduced below. Residence Residence CBD Figure 2.1 Radial commuting. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

Analyzing Urban Spatial Structure 27 The parameter t represents the per-mile cost of commuting. For a resident living x miles from the CBD, total commuting cost per period is then tx , or commuting cost per mile times distance. For an automo- bile commuter, t would be computed as follows: Suppose that operat- ing the automobile costs $0.45 per mile, a number close to the value allowed by the Internal Revenue Service in deducting expenses for business use of an auto. Then, a one-way trip to the CBD from a residence at distance x costs 0.45 x , and a round trip costs 0.90 x . A resident working 50 weeks per year will make 250 round trips to the CBD. Multiplying the previous expression by this number yields (250)0.90 x = 225 x as the commuting cost per year from distance x . Thus, under these assumptions t would equal 225. 2 The fact that the same commuting-cost parameter ( t ) applies to all residents reflects another implicit assumption of the model: all resi- dents use the same transport mode to get to work. Urban models with competing transport modes (and thus different possible mode choices) have been developed, but they involve additional complexity. Let the income earned per period at the CBD by each resident be denoted by y . Then disposable income, net of commuting cost, for a resident living at distance x is equal to y – tx. This expression shows that disposable income decreases as x increases, a consequence of a longer and more costly commute. This fact is crucial in generating the model’s predictions about urban spatial structure. 2.4 Consumer Analysis As was mentioned earlier, city residents consume two goods: housing and “bread.” Bread consumption is denoted by c , and since the price per unit is normalized to $1, c gives dollars spent on bread (all goods other than housing). Housing consumption is denoted by q , but the physical units corresponding to q must be chosen. The problem is that housing is a complicated good, with a variety of characteristics that consumers value. The characteristics of housing include square footage of floor space in the dwelling, yard size, construction quality, age, and amenities (views, for example). Although a dwelling is then best 2. Note that the model focuses entirely on commuting cost, ignoring the cost of trips carried out for other purposes (such as shopping). These trips might be viewed as occur- ring close to home at a cost that is negligible relative to the cost of commuting. Alterna- tively, the consumer could be assumed to shop on the way home from work at no extra cost, a behavior that appears to be common. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

Analyzing Urban Spatial Structure 29 been reached, only when consumer utility—that is, the value taken by the utility function u ( c, q )—is the same everywhere. Utilities can be spatially uniform only if the price per unit of housing floor space falls as distance increases. Since higher commuting costs mean that disposable income falls as x increases, some offsetting benefit must be present to keep utility from falling. The offsetting benefit is a lower price per square foot of housing at greater distances. Then, even though consumers living far from the center have less money to spend (after paying high commuting costs) than those closer to the CBD, their money goes farther given a lower p , allowing them to be just as well off as people living closer in. The lower p thus compensates for the disadvantage of higher commuting costs at distant locations. This explanation makes it clear that the lower p at distant suburban locations serves as a compensating differential that reconciles suburban residents to their long and costly commutes. Compensating differen- tials also arise in many other economic contexts. For example, danger- ous or unpleasant jobs must pay higher wages than more appealing jobs with similar skill requirements. Otherwise, no one would do the undesirable work. Like the lower suburban p , the higher wage recon- ciles people to accepting a disadvantageous situation. 4 While the compensating-differential perspective is the best way to think about spatial variation in p , another view that may seem easier to understand focuses on “demand.” One might argue that the “demand” for suburban locations is lower than the demand for central locations given their high commuting cost. Lower demand then depresses the price of housing at locations far from the CBD, causing p to decline as x increases. The inverse relationship between p and x can also be derived using an indifference-curve diagram, as in figure 2.2. The vertical axis repre- sents bread consumption ( c ) and the horizontal axis housing consump- tion ( q ). The steep budget line pertains to a consumer living at a central-city location, close to the CBD, with x = x 0 . The c intercept of the consumer’s budget line equals disposable income, which is y – tx 0 for this individual. The slope of the budget line, on the other hand, equals the negative of the price per square foot of housing. Thus, the 4. Because the price of bread is assumed to be the same in all locations (normalized to $1 per unit), it cannot play a compensating role, as p can. Concretely, this assumption means that the prices of groceries and other non-housing goods are the same at all loca- tions in the city. Although this requirement may not be fully realistic, it is likely to hold approximately. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

30 Chapter 2 slope of the budget line for the central-city consumer equals – p 0 , where p 0 is the price per square foot prevailing at x = x 0 . Given this budget line, the consumer maximizes utility, reaching a tangency point between an indifference curve and the budget line. In the figure, this tangency point is ( q 0 , c 0 ). Thus, this central-city resident consumes c 0 worth of bread and q 0 square feet of housing. Now consider a consumer living at a suburban location, with x = x 1 > x 0 . This consumer has disposable income of y – tx 1 , less than that of the central-city consumer. As a result, the consumer’s budget line has a smaller intercept than the central-city budget line, as can be seen in the figure. The main question concerns the price per square foot of housing at this suburban location, denoted by p 1 . What magnitude must this price have in order to ensure that the suburban consumer is just as well off as the central-city consumer? The answer is that p 1 must lead to a budget line that allows the suburban consumer to reach the same indifference curve as her central-city counterpart. For this outcome to be possible, the suburban budget line, with its lower intercept, must be flatter than the central-city line. When the budget line is flatter by just the right degree, the utility-maximizing point will lie on the indif- ference curve reached by the central-city consumer, as seen in the figure. But since the slope of the budget line equals the negative of the housing price, it follows that a flatter budget line (with a negative slope closer to zero) must have a lower price. Therefore, the suburban price p 1 must Slope = – p 0 Slope = – p 1 y – tx 1 y – tx 0 c q ( q 0 , c 0 ) ( q 1 , c 1 ) Figure 2.2 Consumer choice. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

32 Chapter 2 aren’t used to thinking about the price per square foot of housing (focusing instead on total rent), the model assumes that they implicitly recognize the existence of such a price in making decisions. For example, a small apartment with a high rent would be viewed as expensive by a consumer, but the individual would be implicitly reacting to the apartment’s high rental price per square foot. Indeed, commercial space is always rented in this fashion, with a landlord quoting a rent per square foot and the tenant choosing a quantity of space. But one might then argue that residential tenants aren’t offered such a quantity choice (they can’t, after all, adjust the square footage of an apartment), making the model’s portrayal of the choice of dwelling size seem unrealistic. The response is that the consumer’s quantity preferences are ultimately reflected in the existing housing stock. In other words, the size of apartments built in a particular location is exactly the one that consumers prefer, given the prevailing price per square foot. Two additional conclusions can be drawn from consumer side of the model. The first concerns the nature of the curve relating the housing price p to distance. The curve is convex, as in figure 2.3, with the price falling at a decreasing rate as x increases. This conclusion follows from mathematical analysis, which shows that the slope of the housing-price curve is given by the following equation: $ p x Figure 2.3 Housing-price curve. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

Analyzing Urban Spatial Structure 33 ∂ ∂ p x = − t q . Therefore, the slope at any location is equal to the negative of commut- ing cost per mile divided by the dwelling size at that location. The convexity in figure 2.3 follows because q increases with x , so that the – t / q ratio becomes less negative (and the curve flatter) as distance increases. The intuitive explanation is that at a suburban location where dwellings are large a small decline in the price per square foot is suf- ficient to generate enough housing-cost savings to compensate for an extra mile’s commute. But at a central-city location, where dwellings are small, a larger decline in the price per square foot is needed to generate the required savings. A second conclusion concerns the spatial behavior of total rent, pq . The question is how the total rent for a small central-city dwelling compares to the total rent for a larger suburban house. The answer is that the comparison is ambiguous. Since p falls with x while q increases, the product pq could either rise or fall with x , with the pattern depend- ing on the shape of the consumer’s indifference curve in figure 2.2. The implication is that the total rent for the suburban house could be either larger or smaller than the rent for the central-city apartment, a conclu- sion that appears realistic. A large body of empirical work confirms the model’s prediction of a link between price per square foot of housing and job accessibility. The approach uses a “hedonic price” regression (explained further in chapter 6) that relates the value of a dwelling to its size and other characteristics, one of which is distance from the city’s employment center. These regressions usually show a negative distance effect. Thus, with dwelling size held constant, value falls as distance rises, which in turn implies a decline in value (and thus rent) per square foot. 6 2.5 Analysis of Housing Production Now that the consumer’s choice of dwelling size has been analyzed, the next step is to ask what the buildings containing those dwellings look like. To address this question, the focus shifts to the activities of housing developers, who build structures and rent the space to consumers. 7 6. For an example of such a study, see Coulson 1991. 7. In reality, developers sell buildings to landlords, who then rent space to consumers, but this intervening agent is ignored. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

Analyzing Urban Spatial Structure 35 tional doses of building materials are increasingly consumed in uses that do not directly yield extra floor space. These uses include a stron- ger foundation, thicker beams, and more space devoted to elevators and stairways. The second property of interest is the degree of returns to scale in housing production. In the discussion of scale economies in chapter 1, only a single input was present, and the presence of scale economies could be inferred by simply looking at the graph of the production function. Although the graph is more complicated with two inputs, economies of scale are present in housing production if doubling both the capital and land inputs leads to more than a doubling of floor space. This doubling of inputs is evident in figure 2.5, where it leads, in effect, to the construction of a second identical building adjoining the original one. The question is whether this building has more than twice the floor space of the original building. It might appear that the answer is No, with floor space instead exactly doubling. But that conclusion ignores what might be a slight gain from the fact that the exterior wall of the original building is now an interior wall, which could be thinner. Since this gain is probably small, it is safe to say that housing production exhibits approximate “constant returns to scale,” with scale economies not present in any important way. Figures 2.4 and 2.5 reflect an underlying assumption that has not been made explicit so far. The assumption is that the building com- pletely covers the land area l , leaving no yard or open space around it. This view is logical since consumers have been portrayed as only valuing floor space, so that any land devoted to open space would be wasted. But the assumption is clearly unrealistic, at least for suburban N N Figure 2.5 Constant returns to scale. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

36 Chapter 2 areas where yard space is plentiful. Strictly speaking, the model can be viewed as pertaining to a place, such as Manhattan or central Paris, where there are few yards. The model can, however, be generalized to allow yard space to be valued by consumers and provided by develop- ers, but the resulting framework is more complex. The housing developer will choose the capital and land inputs for his building to maximize profit, leading to a structure of a particular height. Implicitly, the developer also sets the size of the dwellings within the structure, but this decision simply responds to consumer choices. In other words, the floor space in a building is divided up into dwellings of the size that consumers want at that particular location. This division is illustrated in figure 2.6. The revenue earned by the developer is equal to pH ( N , l ), the price per square foot p times the square footage in the building. Input costs consist of the cost of building materials and the cost of land. To match the rental orientation of the model, both inputs are viewed as being rented rather than purchased. Thus, the developer leases land from its owner rather than purchasing it outright, an arrangement that is occa- sionally seen (in China, for example, all land is owned by the govern- ment and is leased to developers). Land rent per acre is thus the relevant input price, and it is denoted by r . The rental rate per unit of building materials is equal to i , 8 and this price is assumed to be independent of where the structure is built. In other words, building materials are N Figure 2.6 Division of a building into dwellings. 8. Note that i could instead be viewed as the annualized cost of materials that are pur- chased rather than rented. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

38 Chapter 2 related to building heights. With the price of capital fixed and land rent rising moving toward the CBD, the land input becomes more expensive relative to the capital input as distance x declines. Producers generally shift their input mix in response to changes in relative input prices, and housing developers are no exception. In particular, as land becomes more expensive compared to capital, developers econo- mize on the land input and use more capital in the production of floor space. But in making this substitution, the developer is building a taller structure (recall figure 2.4). Thus, as land becomes relatively more expensive moving toward the CBD, developers respond by construct- ing taller buildings. Conversely, as land gets cheaper moving toward the suburbs, developers use it more lavishly, constructing shorter build- ings. Overall, then, building height decreases as distance to the CBD increases. This pattern can be seen from a diagram showing cost minimization on the part of the housing developer. In figure 2.7, the capital input is on the vertical axis and the land input is on horizontal axis. The isoquant shows all the capital-land combinations capable of producing a particular amount floor space, say 150,000 square feet. Consider first the choice problem of a developer at a central-city location where x = x 0 and r = r 0 . The iso-cost lines at this central location have slope – r 0 / i , and they are relatively steep since r 0 is high. To produce 150,000 N ( 0 , N 0 ) ( 1 , N 1 ) Q = 150,000 square feet Slope = – r 1 i Slope = – r 0 i Figure 2.7 Cost minimization by housing developer. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

Analyzing Urban Spatial Structure 39 square feet of floor space as cheaply as possible, the developer finds the input bundle on the isoquant lying on the lowest possible iso-cost line. This bundle, denoted by ( l 0 , N 0 ), lies at a point of tangency, as can be seen in the figure. In contrast, a developer building 150,000 square feet at a suburban location, where x = x l > x 0 and r = r 1 < r 0 , faces flatter iso-cost lines. His cost-minimizing input bundle is ( l 1 , N 1 ), which has less capital and more land than the central-city bundle. Instead of building a high-rise structure like the one at x 0 , this developer builds a garden-apartment complex. Building heights in the two developments are reflected in the amount of capital per acre of land, given by the ratios N 0 / l 0 and N 1 / l 1 . These ratios are equal to the slopes of the rays shown in the figure that connect the input bundles to the origin. With the central-city ray steeper, it follows that the building at x 0 is taller than the building at x 1 . Thus, building height falls moving away from the CBD. 10 The two main predictions from the producer analysis are that land rent per acre and building height both fall as distance to the CBD increases. Symbolically, r ↓ as x ↑ , building height ↓ as x ↑ . 2.6 Population Density A final intracity regularity is the decline of population density with distance to the CBD, which the model also generates. Population density, denoted by D , is equal to people per acre. But since dwellings contain a single person, D is just dwellings per acre. Figure 2.8 illus- trates the difference between dwellings per acre in the central city and the suburbs. The central-city location has a tall building (with high capital per acre) that is divided into small dwellings, while the subur- ban location has a short building divided into large dwellings. From the figure, dwellings per acre is clearly higher at the central-city loca- tion than in the suburbs. In other words, since suburban buildings have less floor space per acre of land and contain larger dwellings than central-city buildings, the suburbs have fewer dwellings per acre than the central city. Thus, D falls moving away from the CBD. Symbolically, D ↓ as x ↑ . 10. This prediction is confirmed by McMillen 2006 and by other studies. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

Analyzing Urban Spatial Structure 41 (zero) developer profit. The dashed box in the figure contains the crucial real-world fact that drives the entire model: the increase in com- muting cost as distance increases. The logical arrows show how this real-world fact and the two equilibrium conditions combine to generate the various predictions. The increase in commuting cost with distance and the requirement of uniform utility imply that p falls with x , which in turn implies that q rises with x . The zero-profit requirement and the decline in p with x imply that r falls with x . The decline in r then implies that building height falls with x . Finally, the rise in q and decline in building height as x increases imply that D falls with x . x Population density Regression line Figure 2.9 Population-density regression. Spatially uniform utility Spatially uniform (zero) profit Commuting cost ↑ as x ↑ Building height ↓ as x ↑ p ↓ as x ↑ D ↓ as x ↑ q ↑ as x ↑ r ↓ as x ↑ Figure 2.10 Logical structure of the model. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.

42 Chapter 2 2.7 Intercity Predictions As was explained earlier, the model also generates intercity predictions that match observed regularities. For example, the model predicts that large cities will have taller buildings than small cities. To generate these predictions, it is necessary to analyze what might be called the “supply- demand” equilibrium of the city. Basically, the equilibrium requirement is that the city fits its population, or that the “supply” of housing equals the “demand” for it. The size of the land area occupied by a city determines how much housing the city contains. The city’s land area is, in turn, the result of competition between housing developers and farmers for use of the land. Suppose that farmers are willing to pay a rent of r A per acre of land. This agricultural rent will be high when the land is very produc- tive or when the crops grown on it command a high price. Although r A might vary with location, being higher near the delivery points for agricultural output (where transport cost is low), this rent will instead be viewed as constant over space, thus being independent of x . Figure 2.11 shows the graph of r A , which is a horizontal line, along with the downward-sloping urban land-rent curve. Like the housing-price curve in figure 2.3, the land-rent curve is convex, with r decreasing at a decreasing rate as x increases. r A x r $ Housing Farms x Figure 2.11 Determination of city’s edge. Brueckner, J. K. (2011). Lectures on urban economics. MIT Press. Created from utoronto on 2023-10-19 18:36:09. Copyright © 2011. MIT Press. All rights reserved.