Lectures on Urban Economics

31 Oct Lectures on Urban Economics

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Jan K. Brueckner

The MIT Press Cambridge, Massachusetts London, England

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Brueckner, Jan K. Lectures on urban economics / Jan K. Brueckner.

p. cm. Includes bibliographical references and index. ISBN 978-0-262-01636-0 (pbk. : alk. paper) ISBN 978-0-262-30031-5 (e-book) 1. Urban economics. I. Title. HT321.B78 2011 330.09173’2—dc22

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Contents

Preface

1 Why Cities Exist 1.1 Introduction 1.2 Scale Economies 1.3 Agglomeration Economies 1.4 Transport Costs and Firm Location 1.5 The Interaction of Scale Economies and Transportation Costs in the

Formation of Cities 1.6 Retail Agglomeration and the Economics of Shopping Centers 1.7 Summary

2 Analyzing Urban Spatial Structure 2.1 Introduction 2.2 Basic Assumptions 2.3 Commuting Cost 2.4 Consumer Analysis 2.5 Analysis of Housing Production 2.6 Population Density 2.7 Intercity Predictions 2.8 Summary

3 Modifications of the Urban Model 3.1 Introduction 3.2 A City with Two Income Groups 3.3 Commuting by Freeway 3.4 Adding Employment Outside the CBD 3.5 Durable Housing Capital 3.6 Cities in Developing Countries 3.7 Summary

4 Urban Sprawl and Land-Use Controls 4.1 Introduction

4.2 Empirical Evidence on the Spatial Sizes of Cities 4.3 Market Failures and Urban Sprawl 4.4 Behavioral Impacts of Urban Sprawl 4.5 Using Land-Use Controls to Attack Urban Sprawl 4.6 Other Types of Land-Use Controls 4.7 Summary

5 Freeway Congestion 5.1 Introduction 5.2 Congestion Costs 5.3 The Demand for Freeway Use 5.4 Traffic Allocations: Equilibrium and Social Optimum 5.5 Congestion Tolls 5.6 Choice of Freeway Capacity 5.7 Application to Airport Congestion 5.8 Summary

6 Housing Demand and Tenure Choice 6.1 Introduction 6.2 Housing Demand: The Traditional and Hedonic Approaches 6.3 The User Costs of Housing 6.4 Tenure Choice 6.5 Down-Payment Requirements, Tenure Choice, and Mortgage Default 6.6 Property Abuse and Tenure Choice 6.7 Summary

7 Housing Policies 7.1 Introduction 7.2 Rent Control 7.3 Housing-Subsidy Programs 7.4 Homelessness and Policies to Correct It 7.5 Summary

8 Local Public Goods and Services 8.1 Introduction 8.2 The Socially Optimal Level of a Public Good 8.3 Majority Voting and Voting with One’s Feet 8.4 Public-Good Congestion and Jurisdiction Sizes 8.5 Capitalization and Property-Value Maximization 8.6 Tax and Welfare Competition 8.7 Summary

9 Pollution 9.1 Introduction 9.2 Pollution from a Single Factory and Governmental Remedies 9.3 Bargaining as a Path to the Social Optimum: The Coase Theorem 9.4 Cap-and-Trade Systems 9.5 Evidence on Air Pollution and Property Values 9.6 Summary

10 Crime 10.1 Introduction 10.2 The Economic Theory of Crime 10.3 Additional Aspects of the Theory 10.4 How to Divide a Police Force Between Rich and Poor Neighborhoods 10.5 Summary

11 Urban Quality-of-Life Measurement 11.1 Introduction 11.2 Theory: The Roback Model 11.3 Measuring Urban Quality of Life 11.4 Additional Issues 11.5 Summary

Exercises References Index

Preface

This book offers a rigorous but nontechnical treatment of major topics in urban economics. The book is directed toward several potential audiences. It could be used as a main textbook, or possibly a supplementary book, in an undergraduate or master’s-level urban economics course. It could be used as background reading in a PhD-level course in which students would also read technical journal articles. It could also be read by economists or researchers in other fields seeking to learn what urban economics is about.

To make the book accessible to a broad group of readers, the analysis is mostly diagrammatic. A few chapters make use of some simple formulas and a bit of algebra, but calculus is almost absent. Even though the treatment is nontechnical, the analysis of urban topics attempts to rely on rigorous economic reasoning. The orientation is conceptual, with each chapter presenting and analyzing economic models that are relevant to the issue at hand. In contrast to the cursory theoretical development often found in undergraduate textbooks, the various chapters offer thorough and exhaustive treatments of the relevant models, with the goal of exposing the logic of economic reasoning and teaching urban economics at the same time. Because of its conceptual orientation, the book contains very little purely descriptive or factual material of the kind usually found in textbooks. Instructors wishing to expose students to such material could supplement the book with other readings. Some topics not associated with sharply defined models, such as urban poverty, receive no coverage.

Exercises are presented at the back of the book, for possible use when it is employed as an undergraduate text. They develop numerical examples based on the models presented in the chapters. Footnotes throughout the chapters point to exercises that are relevant to the current discussion.

In view of the nature of the book, the list of references is not particularly extensive. No attempt is made to provide exhaustive citations of the literature on each topic. Instead, one or two representative citations might be given as part of the discussion of standard material that is well accepted among urban economists. However, when a specific idea advanced by a particular author is discussed, the appropriate citation is included. Although the citations are not exhaustive, readers seeking more

exposure to a topic can always find references to the literature in the works that are cited.

Since the book has grown out of my own research on a variety of topics in urban economics, the references include an unavoidably large number of my own papers. Other researchers should recognize that this pattern does not reflect an opinion about relative contributions to the field. Again reflecting my own interests within urban economics, the book contains less material on the New Economic Geography, an area of active research in the field since the early 1990s, than would a book written by an NEG researcher. The material relevant to NEG is confined to chapter 1.

The book’s suitability as a text for an undergraduate course in urban economics, or for a series of such courses, would depend on the length of the course(s). All the chapters can be covered in the undergraduate urban sequence at UC Irvine, which runs for two quarters of 10 weeks each. In a semester-length course of 15 weeks, some of the chapters would have to be dropped, and only about half of the book could be covered in single- quarter course.

This book has grown out of 30 years of teaching urban economics to undergraduates and PhD students, and I’m grateful to all my students for the opportunity to refine my views on the subject. As for assistance with the book itself, I’m indebted to Nilopa Shah, one of my PhD students at UC Irvine, for her expert work in preparing the figures. I also thank various reviewers for helpful suggestions that improved the book in many places.

1 Why Cities Exist

1.1 Introduction

In most countries, the population is highly concentrated in a spatial sense. For example, cities occupy only about 2 percent of the land area of the United States, with the rest vacant or inhabited at very low population densities. Even in countries that lack America’s wide-open spaces, spatial concentration of the population can be substantial, with much of the land vacant. This chapter identifies some forces that lead to the spatial concentration of population. Thus, it identifies forces that help to explain the existence of cities.

Depending on their orientation, different social scientists would point to different explanations for the existence of cities. A military historian, for example, might say that, unless populations are concentrated in cities (perhaps contained within high walls), defense against attack would be difficult. A sociologist might point out that people like to interact socially, and that they must be spatially concentrated in cities in order to do so. In contrast, economic explanations for the existence of cities focus on jobs and the location of employment. Economists argue that certain economic forces cause employment to be concentrated in space. Concentrations of jobs lead to concentrations of residences as people locate near their worksites. The result is a city.

The two main forces identified by economists that lead to spatial concentration of jobs are scale economies and agglomeration economies. With scale economies, also known as “economies of scale” or “increasing returns to scale,” business enterprises become more efficient at large scales of operation, producing more output per unit of input than at smaller scales. Scale economies thus favor the formation of large enterprises. Since scale economies apply to a single business establishment (say, a factory), they favor the creation of large factories, and thus they favor spatial concentrations of employment.1

Whereas scale economies operate within a firm, without regard to the

external environment, agglomeration economies are external to a firm. Agglomeration economies capture the benefits enjoyed by a firm when it locates amid other business enterprises. These benefits include potential savings in input costs, which may be lower when many firms are present, as well as productivity gains. A productivity effect arises because inputs (particularly labor) may be more productive when a firm locates amid other business enterprises rather than in an isolated spot. The mechanisms underlying these effects will be explained later in the chapter.

Transportation costs also influence where a firm locates, and they can lead to, or reinforce, spatial concentration of jobs. This chapter explains several different ways in which transportation costs affect the formation of cities. The last section explores a special kind of agglomeration force: the kind that causes the clustering of retail establishments and the creation of shopping malls.

1.2 Scale Economies

The role of scale economies in the formation of cities can be illustrated with a simple example. Consider an island economy that produces only one good: woven baskets. The baskets are exported, and sold to buyers outside the island. The inputs to the basket-weaving process are labor and reeds. With reeds growing everywhere on the island, the basket-weaving factories and their workers can locate anywhere without losing access to the raw-material input.

The basket-weaving production process exhibits scale economies. Output per worker is higher when a basket-weaving factory has many workers than when it has only a few. The reason is the common one underlying scale economies: division of labor. When a factory has many workers, each can efficiently focus on a single task in the production process, rather than carrying out all the steps himself. One worker can gather the reeds, another can prepare the reeds for weaving, yet another can do the actual weaving, and still another can prepare the finished baskets for shipment.

Figure 1.1 Scale economies.

The production function for the basket-weaving process is plotted in

figure 1.1. The factory’s output, Q, is represented on the vertical axis, and the number of workers, L, on the horizontal axis. Since the reed input is available as needed, it does not have to be shown separately in the diagram. Because the curve shows basket output increasing at an increasing rate as L rises, scale economies are present in basket weaving. This fact can be verified by considering output per worker, which is measured by the slope of a line connecting the origin to a point on the production function. For example, the lower line segment in the figure (call it A) has a slope of QA/LA, equal to the rise (QA) along the line divided by the run (LA). The higher line segment (call it B), which corresponds to a factory with more workers (LB as opposed to LA), has slope QB/LB, equal to output per worker in the larger factory. Since line B is steeper than line A, output per worker is higher in the larger factory, a consequence of a finer division of labor.

Using this information, consider the organization of basket weaving on the island economy. If 100 workers are available, the question is how these workers should be grouped into factories. Consider two possibilities: the formation of one large 100-worker factory and the formation of 100 single-worker factories (involving “backyard” production of baskets). The natural decision criterion is the island’s total output of baskets, with the preferred arrangement yielding the highest output. The answer should be

clear: given the greater efficiency of workers in large factories, the 100- worker factory will produce more output than the collection of 100 backyard factories.

Table 1.1 Basket output of the island economy.

But it is useful to verify this conclusion in a more systematic fashion.

Let α be output per worker in a factory of size 1, equal to the slope of a line like A in figure 1.1 with LA = 1. Let β be output per worker in a factory of size 100, equal to the slope of a line like B with LB = 100. From the figure, it is clear that β > α. Now consider table 1.1.

The island economy’s total output of baskets equals (number of factories) × (workers per factory) × (output per worker). Table 1.1 shows that this output expression is largest with one large factory. It equals 100β, which is larger than the total output of 100 α from the collection of backyard factories, given β > α.

Since the island economy gains by having one large basket factory, the economy would presumably be pushed toward this arrangement, either by market forces or by central planning (if it is a command economy). But once one large factory has been formed, the basket workers will live near it, which will lead to the formation of a city.

This story is highly stylized, but it captures the essential link between scale economies and city formation, which will also be present in more complicated and realistic settings. But something is missing from the story. It can explain the formation of “company towns,” but it cannot explain how truly large urban agglomerations arise.

To see this point, consider a more realistic example in which the production process is automobile assembly. This process clearly exhibits scale economies, since assembly plants tend to be large, typically employing 2,000 workers or more. Thus, an assembly plant will lead to a spatial concentration of employment, and these auto workers (and their

families) will in turn attract other establishments designed to serve their personal needs—grocery stores, gas stations, doctor’s offices, and so on. The result will be a “company town” with the auto plant at its center. But how large will this town be? In the absence of any other large employer, its population may be limited in size, say to 25,000. The upshot is that, while scale economies by themselves can generate a city, it will not be as large as, say, Chicago or Houston. In order to generate such a metropolis, many firms must locate together in close proximity. For this outcome to occur, agglomeration economies must be present.

1.3 Agglomeration Economies

Agglomeration economies can be either pecuniary or technological. Pecuniary agglomeration economies lead to a reduction in the cost of a firm’s inputs without affecting the productivity of the inputs. Technological agglomeration economies raise the productivity of the inputs without lowering their cost. Simply stated, pecuniary economies make some inputs cheaper in large cities than in small ones, while technological economies make inputs more productive in large cities than in small ones. 1.3.1 Pecuniary agglomeration economies

The labor market offers examples of pecuniary agglomeration economies. Consider a big city, with a large concentration of jobs and thus a large labor market, where many workers offer their labor services to employers. Suppose a firm is trying to hire a specialized type of worker, with skills that are rare among the working population. With its large labor market, a few such workers might reside in a big city, and one presumably could be hired with a modest advertising effort and modest interviewing costs. However, the labor market of a small city probably would contain no workers of the desired type. This absence would force the employer to conduct a more costly search in other cities, and to bring job candidates from afar for interviewing. The firm might also have to pay relocation costs for a hired worker, adding to its already high hiring costs. Thus, by locating in a big city, a firm may lower its cost of hiring specialized labor. The existing employment concentration in the big city thus attracts even more jobs as firms locate there to reduce their hiring costs. The big city then becomes even bigger as a result of this agglomeration effect.

Locating in a big city could also reduce the cost of inputs supplied by other firms (as opposed to labor inputs supplied by individual workers).

For example, the large market for commercial security services in a big city would support many suppliers of security guards. These suppliers would compete among one another, driving down prices and thus lowering the cost of security services used in protecting office buildings or factories. Big-city competition could also reduce the prices of other business services—legal and advertising services, commercial cleaning and groundskeeping, and so on. The same effect could also arise in the context of locally supplied physical inputs, such as ball bearings for a production process or food inputs for the company cafeteria, where competition among big-city suppliers would reduce input costs. As in the case of hiring costs, the job concentration in a big city is self-reinforcing: once jobs become concentrated, even more firms will want to locate in the big city to take advantage of the lower input costs it offers.

In some cases, a small-town location might mean that a particular business service is entirely unavailable locally, just like the specialized worker discussed above. For example, a firm might require specialized legal services (help with an antitrust issue, for example), and there might be no local law firm with such expertise. The firm would then have to pay high-priced lawyers to travel to its headquarters from their big-city offices, or else would have to develop the required expertise “in house” at high cost. In this case, higher input costs arise from the sheer local unavailability of the service in the small city rather than from the low degree of competition among local suppliers.

Firms also purchase transportation services, which are used in shipping output to the market and in shipping inputs to the production site. A firm can reduce its output shipping costs by locating near its market, and it can reduce its input shipping costs by locating near its suppliers. A big city, with its many households, is a likely market for output, and it may also host many of a firm’s input suppliers. In this situation, the firm can minimize its shipping costs for both output and inputs by also locating in the big city. Note that the resulting pecuniary agglomeration benefit is slightly different from those discussed above. Instead of facing a lower unit price of transportation services in the big city (a lower cost per ton- mile), the firm benefits from being able to purchase less of these services because of its proximity to the market and to suppliers.2

Another observation is that this scenario “stacks the deck” in favor of the transportation-cost argument for a big-city location. Suppose instead that input suppliers are in a different location, while the market is still in the big city. Then, locating there will save on output shipping costs, but the inputs will have to come from afar. In this case, the location with the lowest total transport cost may not be in the big city, so that transport-

related agglomeration economies may not be operative. This case is discussed in detail later in the chapter. 1.3.2 Technological agglomeration economies

Technological agglomeration economies arise when a firm’s inputs are more productive if it locates in a big city, amid a large concentration of employment, than if it locates in a small city. To understand how such an effect can arise, suppose that a high-technology firm spends substantial sums on research and development. This spending leads to new technologies and products, which the firm can then patent, allowing it to earn revenue from licensing its discoveries. Suppose for simplicity that the firm’s output is measured by the number of patents it generates per year, and that the only input is labor, measured by the number of engineers the firm employs.

The production function is plotted in figure 1.2, with the output (patents) again on the vertical axis and the input (engineers) on the horizontal axis. The figure shows that research and development exhibits scale economies, although this feature isn’t crucial to the story. The figure also shows two different production functions, which apply in different situations. The lower curve is relevant when the firm locates in a small city where few other high-tech firms are present. The upper curve is relevant when the firm locates in a big city amid a large number of other high-tech firms. The greater height of the upper curve indicates that the firm produces more patents, for any given number of engineers, when it locates in a big city. Thus, engineers are more productive, generating more patentable ideas, in the big city.

Figure 1.2 Technological agglomeration economies.

This beneficial effect could be a result of “knowledge spillovers”

across high-tech firms, which are a type of externality. While engineers within a given firm collaborate intensively in producing patentable ideas, spillovers arise when contact between engineers in different firms also stimulates this productive process. For example, engineers from different high-tech firms might socialize together, sharing a pitcher of beer at a Friday “happy hour.” Although the engineers wouldn’t want to divulge their company’s particular secrets, the happy-hour discussion might cover more general ideas, and it might get the engineers thinking in new directions. At work the following week, this stimulation may start a process that eventually leads to patents that wouldn’t have been generated otherwise. Thus, the engineers end up being more productive because the large high-tech employment concentration allows them to interact with peers doing similar work in other companies.

Although a big city is likely to have a concentration of high-tech employment, allowing knowledge spillovers to occur, some big cities may not have many high-tech firms. Thus, the big city/small city distinction may be less relevant for knowledge spillovers than it was for pecuniary agglomeration economies. Instead, what may matter is the extent of the city’s employment concentration in the industry in which such spillovers occur. If a small or medium-size city happens to have a big employment concentration in the relevant industry, it will offer strong technological agglomeration economies for industry firms despite its limited size.

Might some kinds of knowledge spillovers occur across different industries, so that a city where many different industries are represented is also is capable of generating technological agglomeration economies? For example, might knowledge spillovers arise between manufacturers of medical equipment and producers of computer software? This type of linkage seems possible, and to the extent that it exists, the overall employment level in a city (rather than employment in a firm’s own industry) might be the source of technological agglomeration economies. City size may then capture the extent of such economies, as in the case of pecuniary agglomeration economies.

As will be discussed further below, empirical research on agglomeration effects sometimes finds evidence of such a link between and worker productivity and total employment (and thus city size). “Urbanization economies” is a name sometimes given to this effect. But the empirical evidence for a link between productivity and own-industry employment in a city is much stronger (this effect is referred to as “localization economies”). Thus, technological agglomeration economies appear to operate more strongly within industries than across industries.

In addition to knowledge spillovers, several other channels for such an agglomeration effect can be envisioned. When a city’s employment in a particular industry is large, the existence of a large labor pool makes replacement of workers easy. As a result, unproductive workers can be fired with little disruption to the firm, since they can be immediately replaced. Recognizing this possibility, employees will work hard, achieving higher productivity than in an environment in which shirking on the job is harder to punish with dismissal.

The large labor pool also gives employers a broad range of choice in hiring decisions, which may make it harder for any individual to secure a first job in the industry. Workers may then have an incentive to improve their credentials via additional education and training, and these efforts would lead to higher productivity. Thus, the existence of a large labor pool may raise productivity for workers trying to get a job as well as for those worried about losing one.

A third channel could arise through the phenomenon of “keeping up with the Joneses.” In a city with high employment in a particular industry, workers may be likely to socialize with employees working for other firms in their industry, as was noted above. In addition to making comparisons within their own firm, workers may then judge their achievements against those of friends in other firms. This comparison may spur harder work as employees try to “look good” in the eyes of a broader social set. Thus, in addition to being driven by knowledge spillovers, the higher worker

productivity associated with technological agglomeration economies could arise from these three other sources.

A vast empirical literature tests for the existence of agglomeration economies. For a survey, see Rosenthal and Strange 2004. Early work investigated the connection between worker productivity and both own- industry and total employment in a city (see Henderson 1986, 2003). As was noted earlier, such studies often find evidence of an own-industry effect for knowledge-intensive industries, in which spillovers are likely to occur, without finding much evidence of a total-employment effect. Another empirical approach relates worker productivity to employment density in a city (Ciccone and Hall 1996). Yet another approach links the birth of new firms to own-industry employment, on the belief that business startups are more likely to occur in areas offering agglomeration economies (Rosenthal and Strange 2003). A similar approach relates employment growth to own-industry employment and other agglomeration factors (Glaeser et al. 1992). Other research focuses more explicitly on knowledge spillovers by considering patent activity. A patent application must cite earlier related patents, and some work shows that cited patents tend to be from the same city as the patent application, indicating local knowledge spillovers (Jaffe, Trajtenberg, and Henderson 1993). Other research relates the level of patenting activity to a city’s employment density (Carlino, Chatterjee, and Hunt 2007). This literature offers an overwhelming body of evidence showing the existence of agglomeration economies, mostly of the technological type.

1.4 Transport Costs and Firm Location

As was seen above, transportation-cost savings can be viewed as a type of pecuniary agglomeration effect, which may draw firms to a large city when both its market and suppliers are located there. However, when the market and suppliers are far apart, the firm’s location decision is less clear. To analyze this case, consider a situation in which a firm sells its output to a single market, presumably located in a city, and acquires its input from a single location distant from the market. Viewing the input as a raw material, let the input source be referred to as the “mine.” Moreover, suppose that the mine and the market are connected by a road that can be used for shipments (see figure 1.3), and that the firm’s factory can be located anywhere along this road, including at the market or at the mine. The mine and the market are D miles apart.

Figure 1.3 Mine versus market.

Figure 1.4 Transport costs.

Shipping costs exhibit “economies of distance” in the sense that the

cost per mile of shipping a ton of material declines with the distance shipped. Figure 1.4 illustrates such a relationship, with the vertical axis representing cost per ton and the horizontal axis representing shipping distance (denoted by k). The figure shows shipping cost as having two components. The first is “terminal cost,” which must be incurred regardless of shipping distance. This is the cost of loading the shipment onto a truck or a train, and it is represented by the vertical intercept of the line. The second component is variable cost, which equals the product of the fixed incremental cost per mile (equal to the slope of the line) and the distance shipped. Variable cost is thus the height of the line above its intercept. The presence of economies of distance can be seen by drawing a line between the origin and a point on the curve. The slope of this line is equal to cost per mile (analogous to output per worker in figure 1.1). As shipping distance increases, this line becomes flatter, indicating a decline in cost per mile. The reason for this decline is that the fixed terminal cost is spread over more miles as distance increases.3

For the current analysis, it is convenient to use a somewhat less

realistic transport-cost curve that has zero terminal costs but still exhibits economies of distance. Figure 1.5 shows two such curves, with economies of distance following from their concavity (which reflects declining variable cost). The lower curve represents the cost per ton of shipping the output different distances. To understand the upper curve, suppose that the production process entails refining raw materials, with production of the refined output requiring that a portion of the input be removed and discarded. Then, to produce a ton of output, the refining factory requires more than a ton of input. The upper curve represents the cost of shipping this amount of input various distances. In other words, the curve represents the cost of shipping enough input to produce a ton of output.

Figure 1.5 Input and output shipping.

The fact that the unrefined input and the refined output are

qualitatively similar (both being, in effect, dirt) is convenient. This similarity means that the cost of shipping a ton of input or output a given distance will be the same, which ensures that the input shipping cost curve in figure 1.5 (which pertains to more than a ton) will be higher.

Things are more complicated if inputs and outputs are qualitatively different. In the baking of bread, for example, the flour input is compact but the output of finished loaves is very bulky, so that a ton of output is more costly to ship than a ton of input. Figure 1.5 would have to be

modified to fit this case. The information in figure 1.5 can be used to compute the best

location for the factory. If the factory has a contract to deliver a fixed amount of output to the market, its goal is to minimize the total shipping cost per ton of delivered output. This total includes both the cost of shipping the output and the cost of shipping the input, with the latter cost pertaining to enough input to make a ton of output.

Suppose the factory is located k0 miles from the market. Then the output must be shipped k0 miles and the input must be shipped D – k0 miles (recall that D is the distance between the mine and the market). The output shipping cost (per ton) is represented in figure 1.5 by h, and the input shipping cost (per ton of output produced) by g. The total shipping cost per ton of output at location k0 is then h + g. To find the best location, this calculation procedure must be repeated for all k0 values between 0 and D, with the location yielding the lowest total cost chosen. The required steps are cumbersome, however, and the answer is, so far, unclear.

The answer can be seen immediately, however, if figure 1.5 is redrawn as figure 1.6. This new figure has two origins, one at 0 and the other at distance D, and the input shipping curve is drawn backward, starting at the D origin. Now, total shipping cost (per ton of output) at location k0 is equal to the height of output curve at that point plus the height of the input curve at the same point, which equals the cost of shipping the input D – k0 miles. Therefore, shipping cost per ton of output is equal to the vertical sum of the two curves at any point, and the resulting curve is the upper hump-shaped one in figure 1.6. By inspection, it is easy to see the cheapest location. It is an endpoint location, with the factory located at the mine. All other locations, including the market and points between the mine and market, result in higher total shipping costs.

Figure 1.6 Transport-cost-minimizing location.

The mine is the best location because the production process is a

“weight-losing” process. In other words, more than a ton of input is transformed into a ton of output. In this case, it doesn’t make sense to ship the input at all, since some of the material will be discarded in the refining process. Only the output should be shipped.

This appealingly simple rationale partially obscures the logic of the solution. The logic has two components. First, because of economies of distance, it isn’t economical to ship both the input and the output. In this case, two intermediate-distance shipments would occur, failing to exploit the lower cost per mile of longer shipments. Thus, either the output should be shipped the entire mine–market distance of D miles or the input should be shipped the entire distance. The best choice is the one that is cheaper, and, given the weight-losing nature of the production process, shipping the output all the way costs less. This argument shows that economies of distance, omitted from the partial explanation above, are crucial to the solution.4

Suppose the production process is, instead, weight gaining. An example would be Coca-Cola bottling, in which syrup (evidently produced under secret conditions near the company’s headquarters in Atlanta) is combined with water to produce the finished product. In this weight- gaining case, the heights of the input and output curves in figure 1.5 are reversed, as are the heights in figure 1.6. The hump-shaped curve in figure 1.6 then reaches its low point at the market rather than at the mine, making

the market the best location for the factory (in this case, the bottling plant). The outcome is natural since it would make little sense to ship finished Coca-Cola long distances when its main component (water) is available everywhere.5

In reality, the bottling of soft drinks is indeed a “market-oriented” production process, with bottling plants located in many cities across the United States. The theory says that weight-gaining production processes should all share this feature, while weight-losing processes should be oriented to the mine (or the input). This simple model, however, omits many elements of reality, including the existence of multiple markets and multiple input sites for a given firm, as well as “bulk” differences between inputs and outputs, which (as explained above) complicate the picture.6 Nevertheless, the model shows that transport costs will affect where firms locate and thus will influence the formation of cities.

In the simple case analyzed, this influence can be delineated. The market (presumably a big city) will attract weight-gaining production processes as firms seek to minimize transport costs. Therefore, the city’s existing concentration of jobs and people will attract additional jobs in weight-gaining industries via this transport-related agglomeration force. Weight-losing industries, however, will shun the market and will instead locate at the source of the raw material. Therefore, the existence of a big- city market may spawn separate employment concentrations at faraway natural-resource sites where factories built to serve the market find it best to locate. Concentrated employment in one spot may thus cause additional remote employment concentrations to arise as a result of transport-related forces.

1.5 The Interaction of Scale Economies and Transportation Costs in the Formation of Cities

Although the desire to minimize transport costs can pull a factory toward a particular location, these costs can also play a role in the overall organization of production. In particular, transport costs can help determine whether production is centralized in one large factory or divided among a number of smaller establishments. The analysis in section 1.2 showed that a single large factory was best, but transport costs played no role in that analysis.

To illustrate the interaction between scale economies and transport costs, and to show how this interaction can affect the formation of cities, consider a simple model adapted from Krugman 1991. Suppose that the

economy has five regions and a total population of N, with N/5 people living in each region. The regions are represented as squares in figure 1.7. Each person consumes one unit of a manufactured good, which is produced with scale economies. In this setting, scale economies are best seen via the cost function C(Q), which gives the total cost of producing Q units of the manufactured good. Scale economies mean that the cost per unit of output declines as Q increases, with the factory becoming more efficient as the level of output expands. Thus, C(Q)/Q falls as Q rises.

Figure 1.7 Scale economies versus transport costs.

The economy must produce N units of manufactured good to serve

the population. Two different arrangements of production are possible: centralized and dispersed. Under dispersed production, a small factory, producing N/5 units of output, would be located in each of the five regions. These factories are represented by the small circles in figure 1.7. The cost per unit of output in each factory is C(N/5)/(N/5) ≡ λ. Under centralized production, one large factory, represented by the large circle, would be located in the central region, producing N units of output at a cost per unit equal to C(N)/N ≡ θ. With scale economies, the cost per unit of output is lower in a large factory than in a small one, with θ < λ. The total cost of production for the N units produced in the economy is θN with centralized production and λN with dispersed production, and, since θ < λ, total cost is lower in the centralized case.

Although this conclusion led to the superiority of one large factory in the basket-weaving economy (where the output was exported), another consideration comes into play in the present setting. Since the manufactured good is consumed within the economy, adoption of

centralized production means that output must be shipped from the large factory to the regions where no manufacturing plant is present. These shipments are represented in figure 1.7 by arrows. The need for shipping is avoided, however, in the dispersed case, since each region has its own factory. Let T denote the cost of shipping a unit of output from the large factory to any of the remote regions. Since four regions lack a factory, a total of 4 N/5 units of the large factory’s output (which equals N units) must be shipped, at a cost of 4TN/5. Note that N/5 units of output are consumed within the large factory’s region and thus need not be shipped.

The overall cost of the centralized and dispersed arrangements includes both production cost and transportation cost. With transportation cost equal to zero in the dispersed case, the overall cost is just λN. In the centralized case, the overall cost is θN + 4TN/5. Therefore, the centralized (dispersed) case has a lower overall cost when λN is greater (less) than θN + 4 TN/5. Rearranging, it follows that centralized (dispersed) production is preferred when λ – θ is greater (less) than 4 T/5. The expression λ – θ is the difference between cost per unit of output at the low output level of Q = N/5 and the higher output of Q = N, a positive difference given the presence of scale economies. If scale economies are strong, so that the small factory is much less efficient than the large factory, then λ – θ will be large, and the first inequality is likely to hold, for a fixed T. Conversely, for a fixed value of λ – θ, the first inequality will be likely to hold when T is small. Therefore, centralized production will be favored when scale economies are strong and transports costs are low. This conclusion makes sense since strong scale economies lead to a substantial production-cost advantage for centralized production, while low transport costs mean that this advantage isn’t offset by the cost of shipping the output.7

Although the conclusion of the analysis of basket weaving above is reaffirmed in this case, dispersed production will be preferred in an economy in which transport costs are high relative to the strength of scale economies. In this case, the second of the above inequalities will hold. This situation might describe an undeveloped economy with poor transport linkages, which would be unable to exploit potential scale economies. When economic development improves the transportation system, lowering T, production would shift to the efficient centralized arrangement.

Centralized production would presumably lead to concentrated employment for manufacturing workers, who have not been explicitly included in the analysis so far. This concentration would lead to formation of a large city in the central zone. In the dispersed case, manufacturing workers would be scattered across the regions, with no notable job

concentration arising. Adding manufacturing workers would require some minor modifications to the model, but the conclusion that scale economies can interact with transportation costs in the location of production (and the hence in the formation of cities) would remain unchanged.8

1.6 Retail Agglomeration and the Economics of Shopping Centers

Another type of agglomeration phenomenon is retail agglomeration: the spatial concentration of retail outlets. Cities have long had shopping districts in which stores are concentrated. While such districts were the result of uncoordinated store-location decisions, they have been increasingly supplanted by shopping malls, whose owners “orchestrate” the process of retail agglomeration.

Two forces contribute to retail agglomeration. The first is a desire by shoppers to limit the costs of shopping trips, which include both time costs and out-of-pocket costs (such as automobile expenses, including the cost of gasoline). When a consumer has to visit multiple stores to make a variety of purchases, the cost of the shopping trip is reduced when the stores are in close proximity. Therefore, a multiple-stop shopper would prefer to carry out his or her trip at a shopping district or a mall rather than visiting a sequence of isolated stores. As a result, stores are likely to attract more customer traffic when they are spatially concentrated than when they are dispersed, and this gain can stimulate retail agglomeration.

The second force leading to concentration of stores is the benefit to consumers of comparison shopping, which can arise even when only one purchase is being made. The ability to compare similar products will generate a better purchase decision, raising the benefits from shopping. Comparison shopping, easily done in a shopping district or a mall, is costly when the stores are spatially separated. This added cost may in fact be prohibitive relative to the benefit, making comparison shopping economical only on a visit to a shopping district or a mall. Spatially concentrated stores can thus offer higher shopping benefits than isolated stores, leading to more customer traffic. This gain for stores will again stimulate retail agglomeration.

Price competition between stores selling similar products is also likely to be more intense when the stores are spatially concentrated. This competition, which leads to lower prices, is beneficial for consumers but puts downward pressure on stores’ profits. The resulting loss tends to reduce the attractiveness of spatial concentration from the viewpoint of stores, offsetting some of the gains described above. The fact that owners

of stores seem to prefer locations in malls and shopping districts, however, suggests that the beneficial forces dominate the loss attributable to greater price competition.

The benefits of agglomeration can be viewed as arising from inter- store externalities. For example, shoppers visiting a shoe store in a mall may also visit a clothing store, and vice versa, so that each of the store types gains from the presence of the other type. Such externalities may be weaker, however, between other types of stores. For example, visitors to clothing or shoe stores may have little reason to visit a toy store or a specialty tobacco and pipe store, and vice versa. Inter-store externalities are illustrated in figure 1.8, where the widths of the arrows represent the strength of the beneficial effects between stores.

Figure 1.8 Inter-store externalities.

When retail agglomeration is orchestrated by the owner of a shopping

mall, the strength and the direction of such externalities are taken into account. Since the mall’s owner can charge higher rent to a store when it earns more revenue, and since revenue depends on inter-store externalities, the owner of the mall will want to choose the mix of stores and their sizes taking these externalities into account. The owner will allocate the mall’s fixed square footage to stores in a way that “optimizes” the externality flows, in the proper sense. The mall’s owner can reap more rental income, and thus earn more profit, by doing so.9

1.7 Summary

This chapter has discussed economic reasons for the existence of cities. Scale economies, which favor the creation of large business establishments, are capable of generating a moderate-size company town

oriented around a single large factory. But agglomeration economies, which cause separate firms to locate near one another, are required for job concentrations substantial enough to generate a big city. Technological agglomeration economies, which raise worker productivity, can arise from knowledge spillovers among similar firms locating in close proximity. Pecuniary agglomeration economies, in contrast, reduce the cost of inputs without affecting their productivity. One pecuniary effect is the saving on transportation costs when a firm locates in a city that contains both its market and its input suppliers. But when the suppliers are remote, transport-cost considerations may pull the firm toward a remote location, generating an employment concentration far from the market. Transport costs may also overturn the employment-concentrating effect of scale economies. When output must be shipped to consumers in dispersed locations, it may be better to forsake the gains from scale by putting small factories in these locations. Finally, retail agglomeration is generated by inter-store externalities, which generate gains for individual stores when they locate in close spatial proximity.

1. When used with “scale” or “agglomeration,” the word “economies” means “savings” or “benefits.” 2. Paying high-priced big-city lawyers to travel to a firm’s small-city location fits into this scenario. The firm is transporting its legal input, an outcome that could be avoided by locating in the big city. 3. A diagram like figure 1.4 can be used to illustrate transport mode choice between truck and train. Relative to trains, trucks have low terminal cost (they can be driven directly to a shipment pick-up point), but they have high variable cost, using more fuel and labor per ton shipped than trains. Therefore, the truck line in a diagram like figure 1.4 would start below the train line and eventually rise above it. As a result, a shipper choosing the cheapest mode would select truck for a short-distance shipment and train for a long-distance shipment. 4. If transport costs instead exhibited diseconomies of distance (with the curves convex), then intermediate-distance shipments would be desirable, and the best location would be an intermediate point between the mine and market. This result, which can be seen by redrawing figure 1.6 for the diseconomies case, shows that weight loss in production isn’t sufficient for a mine location to be best. 5. The best factory location may sometimes lie at a “transshipment” point between the mine and the market, where the shipment must be unloaded and reloaded for some reason. Exercise 1.1 considers such a case, assuming that an unbridged river cuts the road between the mine and market, necessitating unloading, transfer to a barge, and reloading of the shipment. 6. With multiple markets and mines, it can be shown that the optimal location is usually some intermediate point, so that a market or mine location is not best. 7. Exercise 1.2 provides a numerical example based on this model. 8. A large theoretical literature has modeled the trade-off between scale economies and transportation cost in a more elaborate and sophisticated fashion than the simple approach from above. For a survey and a synthesis of these models, see Fujita and Thisse 2002. 9. Exercise 1.3 provides a stylized numerical example of a shopping-mall owner’s optimization problem. For a formal analysis of problems of this kind, see Brueckner 1993.

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 represented in a particularly dramatic fashion. In both of those cities, the urban center has a striking concentration of tall buildings, with building 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 buildings. Like the heights of the office buildings, the heights of these residential 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 (apartments 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 downtown neighborhoods, walking to restaurants, running errands, or heading to their workplaces. This difference is due to the high population 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.

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 differences, 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 The presentation in this chapter is basically a nonmathematical version of Brueckner’s approach.

2.2 Basic Assumptions

As is true of all economic models, the urban model is based on strategically chosen simplifications, which facilitate a simple analysis.

These simplifications are chosen to capture the essential features of cities, leaving out details that may be less important. Once the model is analyzed and its predictions are derived, greater realism can be added, often with little effect on the main conclusions.

The first assumption is that all the city’s jobs are in the center, in an area called the “central business district” (CBD). In reality, many job sites are outside city centers, scattered in various locations or else concentrated in remote employment subcenters. Thus, although job decentralization (the movement of jobs out of the CBD) is a hallmark of modern cities, this process is initially ignored in developing the model. It therefore applies best to cities of the early to mid twentieth century, in which jobs were more centralized than they are now. However, once the model has been analyzed, it can be realistically modified to include the formation of employment subcenters. As will be seen, many of its lessons are unaffected.

Since the goal is to analyze residential (as opposed to business) land use, the CBD is collapsed to a single point at the city center, so that it takes up no space. The model could easily be modified to allow the CBD to have a positive land area, in which case the nature of land use within the business area would become a focus in addition to residential land use outside the CBD.

The second major assumption is that the city has a dense network of radial roads. With such a network, a resident living some distance from the CBD can travel to work in a radial direction, straight into the center, as illustrated in figure 2.1. In reality, cities are criss-crossed by freeways, which are often used in combination with surface streets to access the CBD, thus leading to non-radial automobile commute paths for many residents. As will be seen below, freeways can be added to the model without changing its essential lessons.

The third major assumption is that the city contains identical households. Each household has the same preferences over consumption goods, and each earns the same income from work at the CBD. For simplicity, household size is normalized to one, so that the city consists entirely of single-person households. The identical-household assumption is relaxed below by allowing the city to have two different income groups: rich and poor.

Figure 2.1 Radial commuting.

The fourth major assumption is that the city’s residents consume only

two goods: housing and a composite good that consists of everything 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.

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 automobile commuter, t would be computed as follows: Suppose that operating 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 0.45x, 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)0.90x = 225x 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 residents 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 described by a vector of characteristics, the model requires that consumption be measured by a single number. The natural choice is square footage, the feature that consumers probably care about most. Thus, q represents the square feet of floor space in a dwelling.

With this measurement choice, the price per unit of housing is then the price per square foot of floor space, denoted by p. For simplicity, the model assumes that everyone in the city is a renter, so that p is the rental price per square foot.3 Note that “rent,” or the rental payment per period, is different from p. It equals pq, or price per square foot times housing consumption in square feet. In digesting the model, it is important to grasp this distinction between the rental price per square foot and the more common notion of rent, which is a total payment.

The consumer’s budget constraint, which equates expenditures on bread and housing to disposable income net of commuting cost, is

c + pq = y – tx.

The budget constraint says that expenditure on bread (which equals c given bread’s unitary price) plus expenditure on housing (“rent,” or pq) equals disposable income. The consumer’s utility function, which gives the satisfaction from consuming a particular (c, q) bundle, is given by u(c, q). As usual, the consumer chooses c and q to maximize utility subject to the budget constraint. The optimal consumption bundle lies at a point of tangency between an indifference curve and the budget line, as will be shown below.

As was explained in section 2.1, one of the regularities of urban spatial structure is that the price per square foot of housing floor space declines as distance to the CBD increases. In other words, p falls as x increases. The first step in the analysis is to show that the model indeed predicts this regularity. The demonstration makes use of a simple intuitive argument, which is then reinforced by a diagrammatic analysis.

The argument relies on a fundamental condition for consumer locational equilibrium. This equilibrium condition says that consumers must be equally well off at all locations, achieving the same utility regardless of where they live in the city. If this condition did not hold, then consumers in a low-utility area could gain by moving into a high-utility area. This incentive to move means that a locational equilibrium has not been attained. The incentive is absent, implying that equilibrium has 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 differentials

also arise in many other economic contexts. For example, dangerous 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 reconciles 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 represents bread consumption (c) and the horizontal axis housing consumption (q). The steep budget line pertains to a consumer living at a central-city location, close to the CBD, with x = x0. The c intercept of the consumer’s budget line equals disposable income, which is y – tx0 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 slope of the budget line for the central-city consumer equals –p0, where p0 is the price per square foot prevailing at x = x0. 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 (q0, c0). Thus, this central- city resident consumes c0 worth of bread and q0 square feet of housing.

Figure 2.2 Consumer choice.

Now consider a consumer living at a suburban location, with x = x1 >

x0. This consumer has disposable income of y – tx1, 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 p1. 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 p1 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 indifference 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 p1 must be lower than the central-city price p0, so that p1 < p0. Figure 2.2 thus establishes that the price per square foot of housing p must fall as distance x to the CBD increases, confirming the previous intuitive argument.

Figure 2.2 contains additional important information about consumer choices. The suburban consumption bundle (q1, c1), which is the point of tangency between the suburban budget line and the indifference curve, can be compared with the central-city bundle (q0, c0). From the figure, this comparison shows that the suburban resident consumes more square feet of housing and less bread than the central-city resident. Therefore, suburban dwellings are larger than central-city dwellings, so that dwelling size q rises as distance from the CBD increases. This substitution in favor of housing and away from bread is the consumer’s response to the decline in the relative price of housing as x increases.5 Recall from above that this pattern was one of the main regularities of urban spatial structure, and the figure shows that the model predicts it.

The difference in bread consumption indicates an additional pattern: while occupying a small dwelling, the central-city resident consumes a lot of bread. Concretely, this resident has a nice car, beautiful furniture, and gourmet food in the refrigerator, and takes expensive vacations. The suburban resident’s consumption, in contrast, is skewed toward housing

consumption, with less emphasis on bread. Given that the city only has one income group, this prediction may be not very realistic, and it doesn’t survive the generalization of the model to include multiple income groups. But the (realistic) prediction regarding dwelling-size variation with distance is robust to this generalization, as will be seen below.

So far, the model’s two main predictions are that the price per square foot of housing falls, and that size of dwellings rises, as distance to the CBD increases. These outcomes can be represented symbolically as follows:

With these important conclusions established, several aspects of the preceding analysis deserve more discussion. The consumer has been portrayed as choosing her dwelling size on the basis of the prevailing price per square foot at a given location. Although most consumers 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:

Figure 2.3 Housing-price curve.

Therefore, the slope at any location is equal to the negative of commuting 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 sufficient 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 depending 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 conclusion 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

In reality, developers produce housing floor space using a variety of inputs, including land, building materials, labor, and machinery. As in the consumer analysis, simplicity requires narrowing down the list of choice variables. Thus, the model assumes that floor space is produced with land and building materials alone, ignoring the role of labor and machinery. In one sense, this view isn’t unreasonable, given that the land and materials inputs are present over the entire life of a building, while construction workers and machines (though crucial) are present only for a relatively short time at the outset.

The production function for housing floor space is written as Q = H(N, l), where Q is the floor space contained in a building, N is the amount of building materials (measured in some fashion), l is the land input, and H is the production function. An engineer or an architect would point out that building materials are certainly not a homogeneous category (they include steel, wood, concrete, glass, and so on), but these distinctions are ignored for simplicity in measuring the material input. For convenience, building materials will sometimes be referred to as the “capital” input into housing production.

Several properties of the production function deserve note. The first is the diminishing marginal product of capital. This property means that, with the land input held fixed, extra doses of building materials lead to smaller and smaller increases in floor space. This property makes sense when it is recognized that increasing N while holding l fixed makes the building taller, as can be seen in figure 2.4. Diminishing returns arise because, as the building gets taller, additional doses of building materials are increasingly consumed in uses that do not directly yield extra floor space. These uses include a stronger foundation, thicker beams, and more space

devoted to elevators and stairways.

Figure 2.4 Making a building taller.

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 completely 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 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 developers, but the resulting framework is more complex.

Figure 2.5 Constant returns to scale.

Figure 2.6 Division of a building into dwellings.

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 occasionally seen (in China, for example, all land is owned by the government 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 delivered to any construction site, regardless of its location in the city, at a common price per unit. Combining all this information, the developer’s production cost is equal to iN + rl.

Although i doesn’t vary with location, spatial variation in land rent r is necessary to make developers willing to produce housing throughout the city. The reason is that locations far from the CBD are disadvantageous for development since the price p received by the developer per square foot of floor space is low. In contrast, locations close to the CBD are favorable since the developer can charge a high price per square foot there for his output.

In order for developers to be willing to build housing in all locations, the profit from doing so must be the same everywhere. But with close-in locations offering higher revenue per square foot than suburban locations, profits will not be uniform unless a compensating differential exists on the cost side. With the capital cost fixed, this compensating differential must come from spatial variation in land rent r. In particular, land rent must be lower in the suburbs than at central locations. With r falling as x increases, the revenue disadvantage of the suburbs is offset, and the profits from housing development remain constant over space. Because land rent must do all the work in equalizing profits, given that i is fixed, r must fall with distance much faster than p itself, declining at a greater percentage rate. Therefore, the gap between central-city values and suburban values is wider for r than for p.

As in the consumer analysis, this compensating differential can be viewed as a demand-based phenomenon. Developers will compete vigorously for land in central locations because floor space built there commands a high price. This competition bids up land rents near the CBD. Conversely, developers ‘ lower demand for suburban land, a consequence of the low housing revenue it offers, leads to a lower land rent. Since

competition for land among developers will bid up rent until profit is exhausted, the uniform profit achieved through compensating land-rent differentials is in fact a zero profit level (corresponding to “normal” economic profit).

The model thus predicts another one of the regularities of urban spatial structure: declining land rent (and thus land value) as distance to the CBD increases.9 This pattern, in turn, generates another regularity 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 economize 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 constructing taller buildings. Conversely, as land gets cheaper moving toward the suburbs, developers use it more lavishly, constructing shorter buildings. 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 = x0 and r = r0. The iso-cost lines at this central location have slope –r0/i, and they are relatively steep since r0 is high. To produce 150,000 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 (l0, N0), 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 = x1 > x0 and r = r1 < r0, faces flatter iso-cost lines. His cost-minimizing input bundle is (l1, N1), which has less capital and more land than the central-city bundle. Instead of building a high-rise structure like the one at x0, this developer builds a garden-apartment complex.

Figure 2.7 Cost minimization by housing developer.

Building heights in the two developments are reflected in the amount

of capital per acre of land, given by the ratios N0/l0 and N1/l1. 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 x0 is taller than the building at x1. 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,

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 illustrates 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 suburban location has a short building divided into large dwellings. From the figure, dwellings per acre

is clearly higher at the central-city location 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,

Central city Suburbs (many dwellings per acre)(fewer dwellings per

acre)

Figure 2.8 Population density.

Most empirical testing of the urban model has focused on testing this

prediction about the spatial behavior of population density. Dozens of empirical studies have investigated the relationship between density and distance to the CBD for individual cities all over the world.11 These studies rely on the fact that cities are divided into small spatial zones for census purposes, with the population of the zones tabulated. Once the land area of each zone has been estimated, the zone’s population density can be computed by dividing the population by its area. In addition, the distance from the zone to the CBD can be measured. The result is a point scatter in density-distance space like that shown in figure 2.9. The empirical researcher then runs a regression, which generates a curve passing through the point scatter, as shown in the figure.12 The estimated density curves for the world’s cities are almost always downward sloping, confirming the

prediction of the model. The entire set of intra-city predictions is summarized in figure 2.10,

which shows the logical linkages involved in the predictions. The solid boxes in the figure contain the two fundamental equilibrium conditions in the model: spatially uniform consumer utility and spatially uniform (zero) developer profit. The dashed box in the figure contains the crucial real- world fact that drives the entire model: the increase in commuting 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.

Figure 2.9 Population-density regression.

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Lectures on Urban Economics

31 Oct Lectures on Urban Economics

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