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«Abstract: Population density varies widely among U.S. metro areas. A simple, static general equilibrium model demonstrates that moderate differences ...»

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issn 1936-5330

Consumption Amenities and

City Population Density

Jordan Rappaport

August 2006: Revised January 2008

(Previously titled: Consumption Amenities and City Crowdedness

RWP 06-10

Abstract: Population density varies widely among U.S. metro areas. A simple, static

general equilibrium model demonstrates that moderate differences in metro areas’

consumption amenities can cause extremely large differences in their population

density. Such amenities are more strongly capitalized into housing prices than into wages. Empirical results suggest that amenities do indeed help to support high density levels and that amenities are becoming a more important determinant of where people choose to live. Matching the empirical correlation between wages and density requires that amenities cause approximately one fifth of the cross-sectional variation in metro population density.

Keywords: Population density, consumption amenities, quality of life, productivity, urban agglomeration JEL classification: R00, J00, I31 Federal Reserve Bank of Kansas City; 925 Grand Blvd., Kansas City, MO 64198. 816fax 816-881-2199) Email: jordan.rappaport@kc.frb.org. The views expressed herein are solely those of the authors and do not necessarily reflect the position of the Federal Reserve Bank of Kansas City or the Federal Reserve System.

1 Introduction Population density, or “crowdedness,” varies hugely across U.S. cities. Among metropolitan areas with a population of at least 100,000 in 2000, the most crowded (New York City) had a population density forty-nine times that of the least crowded (Dothan, Alabama). The second-most crowded (Los Angeles) had a population density twenty times that of the least crowded. Moderate differences in metro areas’ total factor productivity can account for this variation (Rappaport, forthcoming). Can plausible differences in metro areas’ consumption amenities do so as well? More generally, how important is quality of life relative to productivity in accounting for the distribution of population density across metro areas? And to what extent are amenities capitalized into housing prices versus into wages?

To answer these questions, the present paper lays out and calibrates a simple, static general equilibrium model of city density. Homogenous individuals choose to live and work in one of two local economies. They derive utility from consumption of a traded good, housing, leisure, and a consumption amenity. Firms in each economy produce the traded good and housing using land, capital, and labor. The level of consumption amenities varies exogenously between the two economies. In equilibrium, each economy must offer individuals the same level of utility and provide capital with the same rate of return. The model is a generalization of Rappaport (forthcoming) and is similar to models in Henderson (1974, 1987, 1988), Haurin (1980), Upton (1981), and Haughwout and Inman (2001). The model’s equilibrium embeds the compensation for quality-of-life differences that forms the basis of empirical work in Rosen (1979), Roback (1982), Blomquist et al. (1988), Gyourko and Tracy (1989, 1991), Gabriel and Rosenthal (2004), and Chen and Rosenthal (2006).

The paper’s methodology combines some of the quantitative elements of the dynamic stochastic general equilibrium (DSGE) literature inaugurated by Kydland and Prescott (1982) with the qualitative elements that characterize more theoretical work. The starting point is the assumption of a structural model, including specific functional forms. As with all theoretical work, the model abstracts from numerous characteristics of potential first-order importance. The functional forms are parameterized using a combination of microeconomic estimation results, national aggregate first moments, and correlations among metro-area aggregate variables. The parameterization includes the choice of baseline values as well as a wide range around these with which to conduct a sensitivity analysis. The sensitivity analysis helps to uncover the workings of the model by characterizing the role of each parameter. In contrast to much of the DSGE literature, which uses data to evaluate the parameterized model, the present paper uses the parameterized model to better understand the data. Could observed outcomes have been generated by processes similar to those in the model? If the model is true, how important is one process relative to another?

The paper finds that plausible differences in consumption amenities can indeed cause the observed large differences in density. Under a baseline calibration, a difference in quality of life valued at 30 percent of average consumption expenditure supports the twenty-fold observed difference in density between the second-most and least-crowded metropolitan areas. This difference in quality of life is within the range of several leading estimates. Much smaller exogenous quality-of-life differences can cause the observed differences in density in the presence of agglomeration economies. Empirical results are consistent with variations in consumption amenities being an important determinant of the distribution of density across metro areas. Density is strongly positively correlated with several subjective rankings of metropolitan-area quality of life. And population growth is strongly positively correlated with several measures of exogenous amenities. Matching an estimate of the correlation between wages and density suggests that variations in quality of life account for approximately one-fifth of the cross-sectional variation in density. The model suggests that high amenity levels are capitalized much more into higher housing prices than into lower wages. Low amenity levels, however, are capitalized approximately evenly into lower housing prices and higher wages.

The paper proceeds as follows. Section 2 describes the paper’s empirical motivation:

the wide variations in population density and perceived quality of life across U.S. metro areas. Sections 3 and 4 lay out the model and discuss its parameterization. Section 5 describes the model’s numerical results, both for a baseline parameterization and for several large perturbations to it. It then discusses the implications of allowing productivity and quality of life to themselves endogenously depend on population density. Section 6 presents empirical results that suggest that variations in quality of life indeed help underpin variations in population density but that variations in productivity are a more important cause. A last section briefly concludes.

2 Empirical Motivation Quantifying variations in density requires taking a stand on two issues. The first concerns the correct geographic unit to use to make comparisons. Metro areas are used herein because they best correspond to the local economies that are modeled. In particular, metro areas encompass a well-defined labor market in which people both live and work. The second issue concerns how to deal with the unequal distribution of population within any geographic unit. Raw population density—total population divided by total land area—is the most straightforward way to measure metro-area crowdedness. It describes average density as experienced by parcels of metro land. However, heterogeneous settlement patterns make using raw density problematic. Metro areas are constructed as the union of one or more whole U.S. counties. Often, large portions of such counties are primarily agricultural or unoccupied.

Hence, the average density experienced by parcels of metro land can be considerably biased downward for the portion of the metro area where most people actually live.

Average density as experienced by residents is instead used herein to measure metroarea crowdedness. It is constructed as a population-weighted average of raw subunit densities (Glaeser and Kahn, 2004; Rappaport, forthcoming). More specifically, the Census Bureau partitions all U.S. counties into subdivisions. These subdivisions are then further partitioned into the portions of any municipalities that lie within them (many municipalities span multiple subdivisions) along with any remaining unincorporated area. In other words, a county subdivision may have a portion of municipality 1, a portion of municipality 2, as well as some unincorporated land. A neighboring county subdivision may have a different portion of municipality 1, which is treated as a separate observation from the portion in the first county subdivision.

The resulting population-weighted average density suggests that metro-area crowdedness in 2000 varied by a multiplicative factor of forty nine (Table 1). Unsurprisingly, the New York City metropolitan area had the highest density with 18.9 thousand persons per square mile (7.3 thousand per square kilometer). The next-most-crowded metro area, Los Angeles, had a weighted density less than half as large. Among people living in metropolitan areas with population of at least 100,000, the median density was experienced by those living in Omaha. That is, at least half of individuals experienced density greater than or equal to that of Omaha, and at least half experienced density less than or equal to that of Omaha.1 A second, harder-to-quantify, motivation is the perceived wide variation in quality of life across U.S. localities. Quality of life is meant to connote the direct contribution to utility from local consumption amenities. Equivalently, quality of life can be thought of as a local area’s attractiveness to individuals as a place to live abstracting from expected wage and cost-of-living considerations.

An objective way to measure quality of life is to estimate compensating differentials (Rosen, 1979; Roback 1982). The value of a place’s quality of life is inferred as the sum of the expected wage sacrifice and cost-of-living premium households accept to live there.

Top-twenty metro area rankings from two compensating differential studies, Blomquist et al. (1988) and Gyourko and Tracy (1991), are shown in Table 2 Panel A. Many of the top-ranked metro areas seem misplaced. For example, few would probably agree that Pueblo Colorado, a small city of approximately 100,000 that lies 40 miles south of Colorado Springs, is the nicest place to live in the United States. Similarly, Macon GA, Bighamton NY, Roanoke VA, Lackawanna PA, Tallahasse Fl, Shreveport LA, Lancaster PA, and Amarillo TX are unlikely to be among most people’s choices for the nicest places to live in the United States. Conversely, many metro areas that are typically considered to have high quality of life are ranked poorly. Among 253 urban counties in 1980, Blomquist et al. rank San Francisco County number 105, neighboring Marin County number 142, and New York County (Manhattan) number 216. Among 130 large cities in 1980, Gyourko and Tracy rank Miami number 86, Seattle number 104, and Ann Arbor number 115. Numerous other apparent large misrankings are easily identified.

The most likely explanation for such misrankings is that the estimation of compensating differentials includes a serious omitted-variable bias (Gyourko, Tracy, and Kahn, 1999). In particular, it is difficult to distinguish whether observed differences in wages are attributable to locational compensation or to skill differences. A high-amenity metro area that attracted workers with unobserved high skills and observed high wages would incorrectly be inferred to have low quality of life.

An alternative approach to measuring quality of life is to grade localities using subjective An alternative measure of the variation in crowdedness, the raw population density of municipalities with population of at least 100,000 in 2000, shows a similar forty-five-fold multiplicative spread. Raw density of metro areas varied by a multiplicative factor of 437.

criteria. Top-twenty metro area rankings for two such studies are shown in Table 2 Panel B.

Savageau (2000) ranks 327 continental U.S. metro areas in each of seven quality-of-life categories: transportation, education, climate, crime, the arts, healthcare, and recreation. Each of these categories, in turn, is divided into two or more subcategories that can be objectively measured. For example, the transportation category is constructed as a weighted average of daily commute time, public transit revenue-miles, passenger rail departures, interstate highway proximity, nonstop airline destinations, and proximity to other metro areas. The arts category is constructed as a weighted average of number of art museums, museum attendance, per-capita museum attendance, ballet performances, touring artist bookings, opera performances, professional theater performances, and symphony performances. An overall quality-of-life index is then constructed as a weighted average of scores in each of the seven categories. Sperling and Sander (2004) similarly rank 329 continental U.S. metro areas in eight quality-of-life categories.2 3 Model The model uses a static, open-city framework. The world is made up of a national economy and a city economy. The former can be interpreted as the aggregate of numerous city economies. It establishes the reservation level of utility that the city economy must offer mobile individuals. In the parlance of international trade theory, the national economy is “large” and the city economy is “small”. That is, outcomes in the national economy affect outcomes in the city economy but not vice versa. Results from modelling two interdependent economies would be qualitatively similar.

3.1 Individuals Individuals derive utility from consumption of a traded good (x), housing (h), leisure, and consumption amenities (quality). The level of consumption amenities is assumed to vary exogenously between the two economies. Amenities thereby serve as the model’s primary source of crowding.

Savageau (2000) and Sperling and Sander (2004) also include job-opportunity and cost-of-living categories. The overall quality-of-life rankings used herein are recalculated to exclude these.

–  –  –

This specialization characterizes the baseline parameterization below. With a unitary elasticity of substitution, σ = 1, (1b) further reduces to Cobb Douglas utility.

Optimizing behavior by individuals equates the ratio of marginal utility to price within each economy. Individuals must each satisfy a budget constraint that their expenditure does not exceed their wage income plus any non-wage income. Except when specified otherwise, non-wage income is assumed to be zero. Perfect mobility by individuals equates utility levels

between the city and national economies:3

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