Embed Size (px)
Citation preview
Urban Density and the Substitution of Market Purchases
for Home Production∗
Daniel Murphy
University of Michigan
November 18th, 2013
Abstract: This paper proposes a new microfoundation for the existence of dense urban areas. Proximity between consumers and service providers in dense areas facilitates demand for market services due to a low time cost of purchasing services. Urban residents share the factor inputs used to produce services by purchasing on the market the services that their suburban counterparts produce at home. The model predicts that economic areas of above-average density have above-average labor supply and above-average prices. The paper presents evidence in support of this new explanation. Specifically, residents who live near service establishments supply more time to the labor market, own fewer durables for home production, and pay higher land rents, even conditional on income, transportation time to work, and other covariates.
Keywords: urban density, home production, consumer demand, regional labor supply.
JEL: D11, R13, R23
∗ Thanks to Chris Barrington-Leigh, Eric Chyn, Alan Deardorff, Jonathan Dingel, Erik Hurst, Lutz Kilian, Andrei Levchenko, Nathan Seegert, and seminar participants at the Urban Economics Association Conferences in Ottawa and Atlanta for helpful comments and discussions. Contact: Darden School of Business, Charlottesville, VA 22903. Email: [email protected].
1
I. Introduction
Urbanization tends to accompany economic growth. This well-documented regularity raises the
following questions: Is there a fundamental underlying cause of growth and urbanization? If so,
what are the precise mechanisms through which urbanization facilitates economic activity? The
early literature on this topic, exemplified by Krugman (1990), views urbanization as a natural
byproduct of industrialization due to increasing returns and low transportation costs associated
with spatial proximity between manufacturers. Such a view is fully consistent with the
experience of the American economy during the majority of the Twentieth Century.
As the economy has continued to grow, however, manufacturing has plummeted as a
share of GDP and services risen (see, e.g. Buera and Kaboski 2012). Cities have flourished
despite the decline of manufacturing, which raises the question of which additional mechanisms
can explain the sustained existence of spatially proximate economic activity. Indeed, as Duranton
and Puga (2004) note,
“While increasing returns are essential to understand why there are cities, it is hard to think of any single activity or facility subject to large enough indivisibilities to justify the existence of cities. Thus, one of the main challenges for urban economists is to uncover mechanisms by which small-scale indivisibilities (or any other small-scale nonconvexities) aggregate up to localised aggregate increasing returns capable of sustaining cities. We can then regard cities as the outcome of a tradeoff between agglomeration economies or localised aggregate increasing returns and the costs of urban congestion”
A recent literature has focused on the ability of cities to provide amenities in the form of
consumer goods and services. Glaeser, Kolko, and Saiz (2001) suggest that urban environments
offer amenities such as a rich variety of consumer services, including high-end restaurants and
theaters. Similarly, George and Waldfogel (2003) document that certain specialty services, such
as theaters and newspapers, are found exclusively in urban areas.
This paper extends the recent work on cities as centers of consumption and offers a new
microfoundation for agglomeration based on the sharing of factors of service production through
market purchases. Specifically, proximity between consumers and service establishments in
dense areas is associated with a low time cost of purchasing services. Due to this low time cost
of obtaining market services, urban consumers purchase on the market services that their
suburban counterparts produce at home. Total service consumption may be the same between
urban and suburban areas, but in urban areas it is more efficient to obtain services on the market
2
rather than through home production due to the scarcity of land and the relatively low time cost
of obtaining market services.
When urban consumers obtain services on the market they effectively share the factors of
production. For example, meal preparation at home requires each consumer to own his or her
own cooking supplies. A restaurant uses a certain amount of kitchen space, oven capacity, and
other factor inputs to serve many customers in a day. By purchasing restaurant services, these
customers share factor inputs. The restaurant acts as an intermediary by collecting payments for
services to pay the factors of production.
Existing explanations for the existence of urban density have been based on either
production-based benefits or consumption-based benefits. For example, Glaeser, Kolko, and
Saiz (2001) note that traditional explanations have focused on the production benefits of cities,
and they propose that an important benefit of cities is proximity between consumers and a rich
variety of services. The market purchases-based microfoundation that I propose combines
features of both production-based and consumption-based explanations. Production is efficient
in dense areas because residents share the factor inputs, leaving them idle for less time. This
efficient sharing is facilitated by high consumer demand for services arising from a low time cost
of shopping.
Thus the consumption benefit of urban density permits the efficient sharing of factor
inputs such as land and durables. As in Glaeser et al, consumption benefits are based on
proximity between consumers and services. The distinction between the Glaeser et al
explanation and the market-purchases-based microfoundation is that the latter refers to the
substitution of market purchases for home production, while the former refers to consumption of
services by urban residents that are generally not consumed by suburban and rural residents.
Furthermore, the cost of urban density in the market purchases-based microfoundation that I
propose is the scarcity of productive land. Traditionally urban economists have viewed the
primary cost of urban density to be disamenities associated with congestion. While the market-
purchases-based microfoundation is fully consistent with costs of congestion, the existence of
such costs is not necessary to rationalize the coexistence of urban and suburban areas.
As with the sharing-based microfoundations outlined in Duranton and Puga (2004),
sharing of production factors through market purchases makes dense areas more efficient at
providing market services. However, the new microfoundation proposed below differs from
3
other sharing-based mechanisms in its predictions for the behavior of market prices across
economic regions. As Rosenthal and Strange (2004, p 2146) note, microfoundations based on
sharing, matching, and learning have nearly indistinguishable predictions for how prices and
output vary across cities because agglomeration-based productivity gains are assumed to be
neutral toward the factors of production.
The market purchases-based mircofoundation, in contrast, features a fundamental
tradeoff in economic geography. Land, which is necessary to produce services, also increases
the time cost of obtaining market services. The model developed below predicts that as
population density increases and available land falls, the time cost of shopping falls and
consumers purchase rather than home produce a range of services. The price of land is high
relative to the wage because density by definition is associated with scarcity of land relative to
labor. Thus, any agglomeration-related benefits are also associated with land scarcity and high
rent-to-wage ratios.
Dense areas are associated with above-average output prices due to above-average land
prices. This stands in contrast to the canonical model in Roback (1982), which predicts that the
deviation of prices in a city from the national average is higher than the deviation of wages only
when quality of life is high. In a study based on the Roback model, Albouy (2012) finds that
predicted measures of quality of life are positively correlated with population density and with
the number of service establishments in a city, and uncorrelated with some direct measures of
quality of life such as crime rates. One interpretation of the Albouy result is that people enjoy the
amenities associated with density more than they dislike congestion and crime. The model below
offers a complementary interpretation that urban density-induced market purchases are to some
extent directly responsible for the price/wage differential that the Roback model labels “quality
of life”.
Section 2 discusses concrete examples illustrating the intuition behind how urban density
facilitates demand for market produced goods and services. The informal discussion offers two
key predictions. First, people who live in close proximity to service establishments supply more
time to the labor market than those who do not. Second, urban residents own fewer durables for
home production than do their suburban counterparts.
Section 3 demonstrates that these predictions are consistent with evidence from Census
data. Single, childless residents in dense Public Use Microdata Areas (PUMAs) in Manhattan
4
work more hours per week than do single, childless residents of suburban parts of the New York
City metro area. Also, residents of dense PUMAs own fewer automobiles and have fewer rooms
per resident in their homes/apartments. The evidence that labor supply depends on proximity to
service establishments complements a recent literature on cross-region variations in labor supply.
Rosenthal and Strange (2008) document that hours depend on the density of workers of the same
occupation in an individual’s area of work; and Black, Kolesnikova, and Taylor (2012)
document that labor force participation of female workers depends on metropolitan area
commuting time.
The evidence in Section 3 motivates the theoretical model in Section 4 an economy that
features a time cost of shopping which is falling in urban density. The baseline model of a
closed, single-region economy (Section 4.1) formally develops the intuition in a simple setting.
Labor supply and market purchases are decreasing in the amount of available land (and
increasing in density), which is assumed to be exogenous. As land size increases, the economy
benefits from the additional factor of production, but it incurs a larger time cost of shopping and
thus loses some of the benefits of sharing factor inputs through market purchases. Section 4.2
presents a more realistic setup in which agents can sort into regions of different land size. This
more complicated model with endogenous density and spatial equilibrium ensures that the
agglomeration benefits of density are exactly offset by the cost of scarce land. As in the closed
economy model, density is associated with more time working on the market and less home
production. Both models also offer an additional prediction that land rents and market prices are
decreasing in the amount of land per capita.
Section 5 tests this prediction of high land prices in dense areas using a similar
methodology to that in Albouy and Lee (2011). Consistent with the model’s prediction, and
consistent with the findings in Albouy and Lee, individuals who live in areas with high service
establishment density pay higher housing rents, conditional on housing characteristics and
conditional on their proximity to work.
Section 6 relates the empirical and theoretical results to the existing literature on the
dependence of labor supply and land prices on urban density.1
1 This paper relates more broadly to a burgeoning literature on the consumption benefits of urban density. For example, Couture (2013) estimates the value of living near a range of service varieties. Couture’s estimates are
Section 7 contains concluding
remarks.
5
2. Density and Market Production of Services
The intent of the informal discussion in this section is to convey intuition that will map into the
empirical results of Section 3 and the formal model of Section 4. Consider first a woman living
in a small apartment in a dense area of Manhattan whose occupation is unspecified. She does not
own a car or washing machine, and her kitchen is small enough that cooking is limited to only
basic meals. Does the lack of such durables imply that she does not commute, wash her clothes,
or eat large meals? Certainly not. Assuming her income is high enough, she likely commutes by
taxi or subway, gives her laundry to the cleaners at the end of the block for cleaning, and eats out
at the myriad restaurants within a minute’s walk from her apartment. In other words, she
purchases on the market many of the goods and services that she consumes.
Now consider a man living and working in an isolated suburban community elsewhere in
the country whose occupation is likewise unspecified. He owns a car, a washing machine, and a
large kitchen. Due to the fact that there are very few service establishments within a short
driving distance, he cooks at home often. Also, he drives himself to work and washes his clothes
himself, so that many of the services he consumes are produced using his own labor within his
own home.
So far our story is a familiar example of home versus market production that has
appeared before in the literature focusing on the dynamic evolution of labor supplied to the
market versus labor used at home (Rogerson 2008, for example). However, the key nuance to
observe is that the home-producing man and the market-producing woman both exist, across
space, in equilibrium. The woman chooses to live in Manhattan and purchase goods on the
market, while the man chooses to live in the suburbs and produce services for himslef. One
reason for the woman’s choice to purchase services on the market is the fact that she does not
own durables that can be used for home production. But there is another reason, which is that
services are readily available for the woman to purchase. Taxi cabs drive by her house every
minute, the cleaner is at the end of the block, and corner markets and restaurants are all within a
minute’s walk. Market production for her requires almost none of her own time or energy;
rather, it requires only the expenditure of her income.
conditional on individuals’ choices of durable ownership and place of residence. Therefore, the relative consumption benefit of cities may be even larger to the extent that residents of very dense areas can forego the cost of owning certain durables (such as cars) and land for home production.
6
For the suburban man, on the other hand, purchasing market services is difficult. There is
little variety in terms of restaurants in his neighborhood, and even the nearest laundromat is
miles away and requires customers to wash their own clothes using coin-operated machines. Of
course, home production is also fairly time-consuming, but it is less expensive than would be
travelling to purchase laundering services (or waiting a half hour for a cab to arrive at his house).
Therefore the man produces more of his goods at home, leaving less time for his labor to be
devoted to market production and therefore leading to a lower market income.
If income in the suburbs is low due to the lower market labor supply, why does the man
not simply move to Manhattan, earn a higher income, and purchase services on the market? One
possibility is that the man may not need a higher income since he purchases less on the market.
In other words, his total consumption is the same as if he lived in Manhattan because he is
compensated for lower income with more time to produce at home. A second possibility is that
he has utility over land and space, which is limited in the city. The tradeoff for a higher income
in Manhattan is higher land and service prices, and indeed since the man lives in the suburbs in
equilibrium it must be the case that, given his employment opportunities in each region, his
utility is at least as high with a lower market income, more land, and more time devoted to home
production.
Of course this very simple story has abstracted from trade and other real-world features
of urban economies. A more extensive story could be told to incorporate these features, and
indeed the model presented below can likewise be extended along these lines. The simple story
has two strong implications, however, which are easily tested in the data: Consumers in dense
areas allocate more time to market work and own fewer durables.
3. Within-City Differences in Labor Supply and Durable Ownership
The story of Section 2 suggests that labor supply should rise and durable ownership should fall
as the time cost of shopping falls. The challenge in the empirical work is that the time cost of
shopping is likely to be a nonmonotonic function of urban density. The time cost of shopping is
indeed likely to be lower in Manhattan, where most market services are within a stone’s throw
from residences, than in a suburb in which obtaining market services requires a longer commute.
Among less dense areas, however, higher density is not necessarily associated with lower time
costs of shopping. The time cost of traveling to a Mexican restaurant in Denver, CO, for
7
example, is likely to be similar to the time cost of doing so in less dense Casper, WY; both
because traveling a few extra miles by car takes so little time, and because there is less traffic
and congestion in Casper. Density is more likely to be relevant for the time cost of shopping
when comparing two very dense urban areas (walking to a restaurant or a Laundromat is easier in
Manhattan than in Brooklyn, and driving is inconvenient in both locations), or when comparing a
very dense area with other regions.
An additional complication is that we cannot approximate an economic region as the
simple closed economy discussed in Section 2 because trade is an especially prominent feature of
local economies. The presence of trade implies that features other than service proximity could
explain choices of labor supply or durables ownership. In Section 6, I discuss alternative
explanations for the empirical results presented below.
Data. The data on hours and other individual-level characteristics, including PUMA of
residence and durables ownership, are from the 2000 Census 5 percent Integrated Public Use
Microdata Sample (IPUMS). Density at the PUMA level is defined as the number of service
establishments per square mile. The Zip Code Business Patterns dataset at the U.S. Census
provides the number of establishments within a zip code by NAICS industry definition. I match
zip codes to PUMAs using the geocorr2k software maintained by the Missouri Census Data
Center, and I total the number of service establishments in a PUMA to construct a measure of
establishment density for each PUMA.
My definition of service establishments includes those services for which consumers can
produce at home or on the market. Specifically, I classify all restaurants, laundromats,
convenience stores, and fruit, fish, meat, and vegetable markets as service establishments.2
I limit the sample to residents of the New York City metropolitan area. There are few
PUMAs in the United States for which services are within walking distance and the time cost of
shopping is significantly lower than that of other PUMAS. Most of these dense PUMAs are in
the New York City metropolitan area, and focusing on a single metropolitan area removes any
The
results presented below are robust to alternative classifications which include a broader range of
service establishments since high-density areas tend to have high concentrations of all types of
service establishments.
2 I include food markets under the assumption that they provide food storage services for consumers. Urban consumers visit food markets more frequently and store less food at home, while suburban consumers visit grocery stores infrequently but purchase more for storage in each visit.
8
concerns that cross-PUMA results could be driven by differences across metropolitan areas. The
sample is also limited to single, childless, employed individuals between the ages of 21 and 55.
Individuals who are married or have children are likely to select into residential areas based on
reasons related to school quality and are more likely to share durables for home production than
to share them with others through their market purchases. Finally, I include only non-
institutionalized individuals with at least a high school degree.
Density Classifications. I classify PUMAs as highest, high, medium, and low density.
As Table 1 demonstrates, there are distinct differences in density across the groups. The five
PUMAs in Lower Manhattan (3805, 3807, 3808, 3809, and 3810) are classified as highest
density. Each is ranked among the top five in service establishment density by any measure,
each has over twice the service density of the sixth-ranked PUMA. The two PUMAS to the West
and Northwest of Central Park (3806, 3802) are classified as high density. The medium-density
PUMAS are in Upper Manhattan (3801, 3803), Brooklyn (4004, 4017), Jersey City (702), and
Queens (4101, 4102).
Results. Column (1) of Table 2 shows that living in a highest-density PUMA is
associated with a statistically significant 3.80 additional hours worked per week (relative to
living in a low-density PUMA), and living in a high-density PUMA is associated with 2.97
additional hours of work per week. This substantial dependence of work time on service
establishment density may reflect the fact that workers in high-density PUMAs live closer to
work, and therefore have more time available for working. However, the estimates are only
slightly lower when conditioning on an individual’s travel time to work (not shown), suggesting
that work hours depend on factors other than proximity to work.
The dependence of hours on density is lower but still economically and statistically
significant when also conditioning on income, occupation, education, and age (column 2).3
3 The inclusion of additional demographic controls has little effect on the estimates in columns (2), (4), and (6).
The
lower conditional estimates suggest that high-income individuals and those who work in
occupations which require more work hours select into living in dense areas. Glaeser et al
(2001) propose that such selection may be due to the rich variety of services in city centers. An
additional possibility is that selection into dense areas is for the purpose of efficiently obtaining
basic services on the market rather than producing them at home. The subtle distinction between
these explanations is that the latter refers to the substitution of market purchases for home
9
production, while the former refers to consumption of services by urban residents that are
generally not consumed by suburban and rural residents.
Even when conditioning on income and other covariates, there remains a large and
statistically significant dependence of hours on density: living in a highest-density PUMA is
associated with 1.3 additional hours worked per week, and living in a high density PUMA is
associated with 0.96 additional hours worked per week. These results are even stronger when
limiting the sample to workers who earn between $50,000 and $100,000 a year (columns 3
through 6). Thus urban residents work more hours than do suburban residents who are similar in
terms of income and other characteristics.
The conditional dependence of hours on density may arise because urban residents trade
disutility from work for utility from consumption amenities that are available in dense areas.
Such an explanation would be especially plausible if leisure time and consumption of urban
amenities were substitutes in the utility function. However, many urban amenities, including
those discussed in Glaeser et al (2001), require leisure time to enjoy, which suggests that leisure
time and urban amenities are strong complements.
Alternatively, the high work hours may be a result of less time spent on home production,
which frees up time to earn income with which to purchase services on the market. One way to
test the market substitution explanation is to see whether urban residents own fewer durables for
use in home production. The Census data includes information on vehicle ownership and on
rooms in a respondent’s residence, each of which contributes to home production. Residents
home produce transportation when they drive themselves using their own vehicle (rather than
taking a taxi or public transport), and they home produce entertainment when they relax or watch
television in a room in their home (rather than going out to a bar).
Table 3 shows the results from a probit model in which the dependent variable is the
probability of vehicle ownership. Residents of highest-density, high-density, and medium-
density areas are far less likely to own a vehicle than are residents of low-density suburbs; and
residents of highest-density and high-density areas are less likely to own vehicles than are
residents of medium-density areas. The results are similar when conditioning on income and
other individual-level characteristics, and suggest that residents of dense areas obtain
transportation services from public transportation or taxi cabs.
10
Residence in high-density areas is also associated with smaller homes, as measured by
the number of rooms per person in a home. According to Table 4, residence in a highest-density
or high-density area is associated with approximately 2 fewer rooms per person, and living in a
medium-density area is associated with approximately 1 fewer room per person relative to living
in a low-density suburb. These results are highly significant and are robust to conditioning on a
range of individual-level characteristics.
Another way to test the market substitution explanation is to directly examine how urban
residents allocate time to home production. The American Time Use Study (ATUS) provides
information on time allocated to home production, but does not provide data on respondents’
PUMA (or other sub-city location) of residence. Instead, the most detailed geographic
information that is publicly available is whether the respondent lives in a central metropolitan
city. Since density can vary substantially within a central metropolitan city, the estimated
dependence of home production on metropolitan status should be lower than the dependence on
density at the PUMA level. Nonetheless, central cities have more dense areas than do suburbs,
and therefore we would expect residents of central cities to spend less time on average on home
production. The results from the ATUS confirm such a hypothesis: In a sample of childless,
employed respondents between the ages of 21 and 55 interviewed on weekdays between 2003
and 2007, residents of a central city spend fifteen minutes a day less on home production than do
other Americans.4
The dependence of home production on urban density is hardly surprising, but it offers
further support for the interpretation of urban density as a means through which people share
durable goods by purchasing services on the market rather than producing them at home. The
results with respect to the dependence of labor supply on establishment density in Table 2 are
more novel. Combined, they motivate the development of theoretical models in which density is
associated with less home production and more time spent working.
This estimate is highly significant and is robust to conditioning on age,
income, and other covariates.
4 The home production time use variable is constructed by the Bureau of Labor Statistics to include a range of home production activities, including food preparation, home maintenance, vehicle maintenance, and lawn care.
11
4. Theoretical Model
This section first develops a representative agent, closed-economy model in which the number of
residents and density (residents per land unit) are exogenous. The objective is to present a
simple setting that features a density-dependent time cost of shopping and substitutability
between home and market production. In contrast to the alternative models of city structures,
such as Mills (1967) and Lucas and Rossi-Hansberg (2002), the model does not feature a unique
center from which transportation costs are declining. Instead, time costs of shopping are uniform
within a region. Furthermore, trade with other regions is assumed away for simplicity. In the
baseline closed-economy model, the comparative static of interest is the effect of falling land per
resident (increasing density). In the extended model with sorting into different regions, density
is endogenous.
4.1 Baseline Model
Model Setup. A representative agent has preferences over discrete, satiable wants indexed by
𝑖 ∈ ℝ+. Each want is associated with a service that can be home produced or purchased on the
market. 𝑞𝑀(𝑖): ℝ+ → {0,1} indicates whether want 𝑖 is satisfied on the market, and 𝑞𝐻(𝑖): ℝ+ →
{0,1} indicates whether the want is satisfied through home production. The discrete nature of
consumption simplifies the analysis and maintains the focus of the model on whether a given
service is purchased on the market or produced at home (rather than on relative quantities). It is
straightforward to incorporate an intensive margin of consumption, and the results are
qualitatively identical.
The agent is endowed with 𝑡 units of time, which she allocates to market production 𝑡𝑀,
home production 𝑡𝐻, and shopping 𝑡𝑆. Leisure is not identified separately from 𝑡𝑀 + 𝑡𝐻 + 𝑡𝑆
because ‘leisure’ is considered to be simply a label given to activities that can be otherwise
classified as home production, market production, and shopping. In other words, I follow
Lancaster (1966) in assuming that leisure is an activity (𝑡𝑀 , 𝑡𝐻 , or 𝑡𝑆) that indirectly contributes
to a service over which consumers obtain direct utility. It may be useful to think of leisure
activities as time devoted to home production or shopping. Likewise, ‘labor’ is time devoted to
market production.
Time devoted to the market receives a wage 𝑤, and the purchase of a unit of any service
requires 𝛿(𝑙) units of shopping time. 𝑙 refers to the amount of land in the economy, and
12
shopping time is assumed to be increasing in the amount of land, 𝛿′(𝑙) > 0. I assume that the
time cost of shopping is a uniform function of the total amount of land, and I do not explicitly
model the locations of firms and households within the economy because doing so would add
unnecessary complexity to the model.
Technology. Home and market production use a Leontief technology that takes
time/labor and land as inputs. The home production function is
𝑞𝑖𝐻 = min �1𝑎𝑖𝑡𝑖𝐻 ,
1𝑏𝐻
𝑙𝑖𝐻� , (1)
where 𝑡𝑖𝐻 and 𝑙𝑖𝐻 are the amounts of labor time and land devoted to home producing service 𝑖. 𝑎𝑖
is the time cost of home producing service 𝑖, and 𝑏𝐻 is the amount of land required to home
produce any good. The market production function is
𝑞𝑖𝑀 = min �1𝑎𝑀
𝑡𝑖𝑀 ,1𝑏𝑀
𝑙𝑖𝑀� , (2)
where 𝑡𝑖𝑀 and 𝑙𝑖𝑀 are labor time and land used to produce service 𝑖 on the market. 𝑎𝑀 is the time
cost of market production, and 𝑏𝑀 is the amount of land required to produce a unit of any good
on the market.
Leontief technology serves two purposes. First, it simplifies the analysis, and second, it
embodies complementarity in production between labor and land. To ensure that land markets
clear, land input requirements must differ between home and market production (𝑏𝐻 ≠ 𝑏𝑀). In
particular, I impose 𝑏𝐻 > 𝑏𝑀under the assumption that land more productively contributes to the
provision of services in the market. For example, a household kitchen generally uses space to
feed a single family, while a restaurant kitchen may use comparable space to feed hundreds of
customers in a day.5
Heterogeneity in the model takes the form of differences in home labor productivity. This
simple setup delivers the primary mechanism in this paper but abstracts away from multiple
dimensions of heterogeneity that distinguish different services (e.g. land productivity, market
labor productivity, and shopping time costs). The model can be extended to incorporate
5 The lower land cost of market production can be considered a reduced-form way to introduce sharing of land through market production. An alternative, but more complicated, approach is to assume that market services are provided at zero marginal cost (but positive fixed cost) over some range of output to capture the fact that firms often pay a fixed rate to rent land. Costless sharing occurs over any range of zero marginal costs. See Murphy (2012) for a formal treatment of such an approach.
13
additional dimensions of heterogeneity, but doing so adds complexity without contributing much
to the intuition developed in this section.
Consumer Problem. The agent’s utility function is defined over the wants:
𝑈 = � [𝑞𝑀(𝑖) + 𝑞𝐻(𝑖)]𝑑𝑖
∞
0. (3)
As in Buera and Kaboksi (2011), wants are satiable so that the focus of consumption is on the
extensive margin.6
Utility is maximized subject to
𝑤𝑡𝑀 + 𝑟𝑙 = 𝑟𝑙𝐻 + � 𝑝𝑖𝑞𝑀(𝑖)𝑑𝑖
∞
0, (4)
𝑡 = 𝑡𝑀 + 𝑡𝐻 + 𝑡𝑆 , (5)
𝑡𝐻 = � 𝑎𝑖𝑞𝐻(𝑖)𝑑𝑖
∞
0, (6)
𝑡𝑆 = � 𝛿(𝑙)𝑞𝑀(𝑖)𝑑𝑖
∞
0, (7)
𝑙𝐻 = � 𝑏𝐻𝑞𝐻(𝑖)𝑑𝑖.
∞
0 (8)
In the budget constraint (4), 𝑝𝑖 is the price of service 𝑖 and 𝑟 is the rental price of land. Equation
(5) is the agent’s labor supply constraint and equation (6) is the home production time constraint.
In the shopping time constraint, equation (7), 𝛿(𝑙) is the amount of labor time required to
purchase a unit of any service 𝑖 ∈ ℝ+ on the market. Equation (8) is the home production land
constraint. As noted above, the time cost of shopping and the land cost of home production are
identical across services, and the only source of heterogeneity across services is the time cost of
home production.
Having now set up the consumer problem, it is useful to relate the formal model to the
informal story of Section 2. Define 𝑎� = ∫ 𝑎𝑖𝑞𝐻(𝑖)𝑑𝑖∞0 . The urban-dwelling woman has a
relatively low 𝛿 due to close proximity to services and a high 𝑎� due to the fact that she neither
6 See also Murphy, Shleifer, and Vishny (1989), Matsuyama (2000), Zweimueller (2000), and Matsuyama (2002) for examples of utility functions featuring satiable wants.
14
has the space nor owns the equipment necessary to produce services at home. The suburban man
has a high 𝛿 and a low 𝑎� because his situation is by design the exact opposite of the woman’s.
The urban woman also owns fewer durables because she does not use them for home production.
Ownership of durables is omitted from the model for simplicity and because the intuition with
respect to sharing of durables can be captured in the model by the sharing of productive land
through market purchases.
Model Solution. Let 𝜆1, … , 𝜆5 be the multipliers on constraints (4) through (8). The first
order condition for 𝑡𝑀 states that the return to supplying labor to the market, 𝑤, must equal the
shadow value of relaxing the agent’s time constraint:
𝜆1𝑤 = 𝜆2. (9)
The first order conditions for 𝑡𝐻, and 𝑡𝑆 state that the shadow value of relaxing the agent’s time
constraint must also equal the shadow value of relaxing the home production time constraint and
the shopping time constraint:
𝜆2 = 𝜆3, 𝜆4 = 𝜆2. (10)
Likewise, the first order condition for 𝑙𝐻 states that the cost of land, 𝑟, must equal the shadow
value of relaxing the home production land constraint.
𝜆1𝑟 = 𝜆5. (11)
The first order conditions for 𝑞𝑀(𝑖) and 𝑞𝐻(𝑖) state that the value of market purchases (home
production) must be greater than or equal to the cost of market purchases (home production),
where the cost of market purchases includes the price and the time cost of shopping, and the cost
of home production includes the time cost and the land cost.
1 ≥ 𝜆1𝑝𝑖 + 𝜆4𝛿(𝑙), (12)
1 ≥ 𝜆1𝑤𝑎𝑖 + 𝜆5𝑏𝐻. (13)
Substituting equations (9) through (11) into equations (12) and (13) yields the necessary
conditions for consumption of service 𝑖 through market purchase and home production,
respectively:
1 ≥ 𝜆1[𝑝𝑖 + 𝑤𝛿(𝑙)], (14)
1 ≥ 𝜆1[𝑤𝑎𝑖 + 𝑟𝑏𝐻]. (15)
Service 𝑖 is purchased on the market if and only if equation (14) holds and if the cost of
home production, 𝑤𝑎𝑖 + 𝑟𝑏𝐻, exceeds the total cost of purchasing on the market, 𝑝𝑖 + 𝑤𝛿(𝑙),
which can be equivalently written as
15
𝑝𝑖 ≤ 𝑤(𝑎𝑖 − 𝛿(𝑙)) + 𝑟𝑏𝐻. (16)
Condition (16) states that a service will be purchased on the market when its market price is
lower than the value of the time saved by shopping instead of producing the service at home,
plus the cost of land used as a factor input to home production. In the story of Section 2, the
urban woman has a low time cost of shopping 𝛿, and is therefore more likely than the suburban
man to purchase a given service even if they have the same market wage.
Equation (16) also states that, given prices and time costs, an individual will purchase
more on the market (and produce less at home) as his or her wage increases. Note that this
partial equilibrium result is based on first order conditions, rather than on an individual’s budget
constraint. It captures the familiar notion of the opportunity cost of time emphasized in Becker
(1965). For example, a minimum-wage worker will launder his own clothes because the value of
time required to do so is less than the market price of laundry services. If the worker’s wage
increases so that the value of his time exceeds the market price of laundry services, he will
instead purchase the service on the market.
In the model there is a representative agent with a unique wage, which, along with
service prices, is endogenous. Therefore to understand the general equilibrium implications of
equation (16) we must first close the model. The market price of each service is based on cost
minimization by competitive firms:
𝑝𝑖 = 𝑤𝑎𝑀 + 𝑟𝑏𝑀. (17)
Substituting (17) into (14) and (15) implies the following:
Proposition: Service 𝑖 is purchased on the market if and only if equation (14) holds and if the
cost of market purchase is less than the cost of home production,
𝑤𝑎𝑖 + 𝑟𝑏𝐻 > 𝑤�𝑎𝑀 + 𝛿(𝑙)� + 𝑟𝑏𝑀. (18)
It is home produced if and only if equation (15) holds and
𝑤𝑎𝑖 + 𝑟𝑏𝐻 < 𝑤�𝑎𝑀 + 𝛿(𝑙)� + 𝑟𝑏𝑀. (19)
To explicitly solve for the services that are produced at home, assume that time costs of
home production are uniformly distributed over ℝ+ so that the home labor requirement of a
service is equal to its index: 𝑎𝑖 = 𝑖 ∀ 𝑖 ∈ ℝ+. Substituting into equations (18) and (19) implies
that the necessary condition for market production is
16
𝑖 ≥ 𝑎𝑀 + 𝛿(𝑙) −𝑟𝑤
(𝑏𝐻 − 𝑏𝑀), (20)
and the necessary condition for home production is
𝑖 < 𝑎𝑀 + 𝛿(𝑙) −𝑟𝑤
(𝑏𝐻 − 𝑏𝑀). (21)
Condition (21) is also a sufficient condition for home production because any service that
satisfies equation (21) also satisfies (14) and (15).
Let 𝐻 denote the mass of services that are produced at home and 𝑀 denote the mass of
services that are purchased on the market. Then
𝐻 = � 𝑑𝑖
𝑎𝑀+𝛿(𝑙)−𝑟𝑤(𝑏𝐻−𝑏𝑀)
0= 𝑎𝑀 + 𝛿(𝑙) −
𝑟𝑤
(𝑏𝐻 − 𝑏𝑀), (22)
and
𝑀 = � 𝑑𝑖
𝚤̅
𝑎𝑀+𝛿(𝑙)−𝑟𝑤(𝑏𝐻−𝑏𝑀)= 𝚤̅ − �𝑎𝑀 + 𝛿(𝑙) −
𝑟𝑤
(𝑏𝐻 − 𝑏𝑀)�, (23)
where 𝚤 ̅is the highest index of the services 𝑖 ∈ [0, 𝚤]̅ that are consumed in equilibrium. Note that
𝐻 is increasing in the time cost of shopping 𝛿, which captures the simple intuition that
consumers will produce more services at home rather than purchasing them on the market when
shopping is inconvenient.
Given 𝐻 and 𝑀 and the Leontief nature of the production technology, we can determine
the allocation of land across market and home production:
𝑙𝑀 = 𝑏𝑀𝑀, 𝑙𝐻 = 𝑏𝐻𝐻. (24)
The allocation of time across market production, shopping, and home production is
𝑡𝑀 = 𝑎𝑀𝑀, 𝑡𝑆 = 𝛿(𝑙)𝑀, 𝑡𝐻 =12𝐻2. (25)
In (25), time devoted to home production is derived from 𝑡𝐻 = ∫ 𝑖𝑑𝑖𝐻0 , where 𝐻 is equal to the
upper limit on the index of home produced services.
Four equations characterize the equilibrium. The first two are market clearing in the time
and land markets, which are given by equation (5) and 𝑙 = 𝑙𝑀 + 𝑙𝐻. The last two are the masses
of home and market produced services (equations (22) and (23)). These conditions can be
written explicitly as
𝑡 = [𝑎𝑀 + 𝛿(𝑙)]𝑀 +12𝐻2
17
68101.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
l
r
68101
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Increasing Urban Density →
MH
681019
20
21
22
23
24
25
26
l
tM
𝑙 = 𝑏𝑀𝑀 + 𝑏𝐻𝐻
𝐻 = 𝑎𝑀 + 𝛿(𝑙) −𝑟𝑤
(𝑏𝐻 − 𝑏𝑀)
𝑀 = 𝚤̅ − 𝐻.
These equations solve for 𝑟,𝐻,𝑀, and 𝚤.̅ The wage is the numeraire.
Assume that the time cost of shopping is a simple linear function of the amount of land,
𝛿(𝑙) = 𝜓𝑙, for some constant 𝜓. For illustrative purposes, Figure 1 shows how market outcomes
depend on urban density under the following parameter values:
𝑎𝑀 = 5 𝑏𝐻 = 3 𝑏𝑀 = 0.25 𝜓 = 0.25 𝑡 = 30.
Appendix A derives the generality of the results.
Figure 1: Effect of Increasing Urban Density on Market Outcomes
The land rent increases as the amount of available land falls, reflecting its relative scarcity. The
ratio of market produced varieties to home produced varieties also increases as the time cost of
shopping falls and consumers find it cheaper to purchase goods on the market rather than to
18
produce them at home. The finding that more market services are available in dense areas is
consistent with the observations in Glaeser et al (2001) and George and Waldfogel (2003) that
some specialty services, such as theatres and newspapers, are found exclusively in urban areas.
Finally, time spent working on the market increases with density, reflecting the shift from home
production to market production.
4.2 Endogenous Density and Spatial Equilibrium
This section shows that the predictions of the closed economy in Section 4.1 are consistent with
a more complicated setup that features endogenous sorting into areas of different land size and
spatial equilibrium. The economy consists of a city with two regions indexed by 𝑗 ∈ {𝐴,𝐵} that
differ only in land size: 𝑙𝐴 ≠ 𝑙𝐵. The city consists of a tradable sector located outside of these
two regions. There is a mass 𝑁 of city residents who work in this tradable sector for an
exogenous wage 𝑤 and endogenously take residence in one of the two regions. Since the focus
of the model is on proximity between residents and service establishment, travel to work is
assumed to be costless.
Within each region, residents can either produce services at home or purchase them on
the market. I do not explicitly model workers’ choice of employment between the tradable
sector and the service sector but instead assume that services are provided by workers who live
outside of the two regions. The cost of service sector workers’ labor in the two regions is
exogenous to the model. This treatment of the service sector limits the focus of the model to
regional choices and market outcomes for people who work in similar industries, and offers a
contrast to the model of Section 3 in which all residents of the city work in the service sector.
City residents are identified by the region into which they sort. A resident of region 𝑗
maximizes
𝑈𝑗 = � �𝑞𝑗𝑀(𝑖) + 𝑞𝑗𝑀(𝑖)�𝑑𝑖
∞
0. (26)
Subject to
𝑤𝑡𝑗𝑀 = 𝑟𝑗𝑙𝑗𝐻 + � 𝑝𝑖𝑗𝑞𝑗𝑀(𝑖)𝑑𝑖
∞
0, (27)
𝑡 = 𝑡𝑗𝑀 + 𝑡𝑗𝐻 + 𝑡𝑗𝑆, (28)
19
𝑡𝑗𝐻 = � 𝑎𝑖𝑞𝑗𝐻(𝑖)𝑑𝑖
∞
0, (29)
𝑡𝑗𝑆 = � 𝛿𝑗(·)𝑞𝑗𝑀(𝑖)𝑑𝑖
∞
0. (30)
𝑙𝑗𝐻 = � 𝑏𝐻𝑞𝑗𝐻(𝑖)𝑑𝑖.
∞
0 (31)
The variable definitions are analogous to those in Section 4.1, where the 𝑗 subscripts indicate
region of residence. The consumer’s budget constraint does not include income from owning
land. Instead, rents from land are assumed to accrue to agents outside of the city. Note that the
wage 𝑤, the time allocation 𝑡, and the time cost of home producing any service 𝑖 ∈ ℝ+, 𝑎𝑖, are
the same for workers in both regions. Variables with subscripts are endogenous and will differ
across regions based on the regions’ different land sizes. For example, even though labor costs of
services are identical in both regions, land rents will differ. Therefore prices of market services
contain a subscript indicating the region of production and consumption.
Consumer optimization yields the result that a resident will home produce service 𝑖 if and
only if
𝑎𝑖 <𝑝𝑖𝑗𝑤
+ 𝛿𝑗(·) −𝑟𝑗𝑤𝑏𝐻 (32)
As in Section 4.1, market production uses a Leontief technology in labor and land. The price of
service 𝑖 in region 𝑗 can be written
𝑝𝑖𝑗 = 𝑎 + 𝑏𝑀𝑟𝑗 , (33)
where 𝑎 is the labor cost that is identical across regions, 𝑏𝑀 is the amount of land required to
produce a unit of any good on the market, and 𝑟𝑗 represents land rents in region 𝑗.
Assume that the time cost of home producing service 𝑖 is 𝑎𝑖 = 𝑖/𝛾, Then the necessary
and sufficient condition for home production can be rewritten by substituting (33) for 𝑝𝑖𝑗 and 𝑖/𝛾
for 𝑎𝑖 in (32):
𝑖 < 𝛾 �𝑝𝑖𝑗𝑤
+ 𝛿𝑗(·) −𝑟𝑗𝑤𝑏𝐻� (34)
Then the mass of home produced services by a consumer in region 𝑗 is
𝐻𝑗 = 𝛾 �𝑝𝑖𝑗𝑤
+ 𝛿𝑗(·) −𝑟𝑗𝑤𝑏𝐻�, (35)
and the mass of services purchased on the market by an agent in 𝑗 is
20
𝑀𝑗 = 𝚤�̅� − 𝐻𝑗 ,
where 𝚤�̅� is the highest index of services that are consumed in 𝑗. Since spatial equilibrium
requires that utility is equal across regions, and since utility equals the mass of services
consumed, equilibrium requires that 𝚤�̅� = 𝚤�̅�.7
Equilibrium. Equilibrium is characterized by the time constraints of residents of regions
𝐴 and 𝐵, budget constraints of residents of regions 𝐴 and 𝐵, land market clearing, the masses of
home and market produced goods in each region, and the spatial equilibrium conditions. These
conditions can be written explicitly as
𝑡 = 𝑡𝐴𝑀 + 𝐻𝐴2 +
𝜓𝑙𝐴𝑀𝐴
.5 (𝚤�̅� − 𝐻𝐴), 𝑡 = 𝑡𝐵𝑀 + 𝐻𝐵2 +𝜓𝑙𝐵𝑀𝐵
.5 (𝚤�̅� − 𝐻𝐵) (36)
𝑤𝑡𝐴𝑀 = 𝑟𝐴𝑙𝐴𝐻 + 𝑝𝐴(𝚤�̅� − 𝐻𝐴), 𝑤𝑡𝐵𝑀 = 𝑟𝐵𝑙𝐵𝐻 + 𝑝𝐵(𝚤�̅� − 𝐻𝐵) (37)
𝑙𝐴 = 𝑁𝐴(𝑙𝐴𝑀 + 𝑙𝐴𝐻), 𝑙𝐵 = 𝑁𝐵(𝑙𝐵𝑀 + 𝑙𝐵𝐻) (38)
𝑙𝐴𝑀 = 𝑏𝑀𝑀𝐴, 𝑙𝐵𝑀 = 𝑏𝑀𝑀𝐵 , 𝑙𝐴𝐻 = 𝑏𝐻𝐻𝐴, 𝑙𝐵𝐻 = 𝑏𝐻𝐻𝐵 (39)
𝐻𝐴 = 𝛾 �
𝑎𝑤
+𝜓𝑙𝐴𝑀𝐴
.5 −𝑟𝐴𝑤
(𝑏𝐻 − 𝑏𝑀)� , 𝐻𝐵 = 𝛾 �𝑎𝑤
+𝜓𝑙𝐵𝑀𝐵
.5 −𝑟𝐵𝑤
(𝑏𝐻 − 𝑏𝑀)� (40)
𝑀𝐴 = 𝚤�̅� − 𝐻𝐴, 𝑀𝐵 = 𝚤�̅� − 𝐻𝐵 (41)
𝚤�̅� = 𝚤�̅� (42)
𝑁 = 𝑁𝐴 + 𝑁𝐵,
(43)
where I have assumed that the time cost of shopping in region 𝑗 is is explicitly falling in the
density of market services in a region, 𝛿𝑗(·) = 𝜓𝑙𝑗/𝑀𝑗.5. The fourteen equilibrium conditions
yield unique values for the endogenous variables 𝑡𝐴𝑀 , 𝑡𝐵𝑀 ,𝐻𝐴,𝐻𝐵,𝑀𝐴,𝑀𝐵 , 𝚤�̅�, 𝚤�̅�, 𝑙𝐴𝑀, 𝑙𝐴𝐻, 𝑙𝐵𝑀, 𝑙𝐵𝐻,𝑁𝐴,
and 𝑁𝐵.
Results. Table 5 shows the market outcomes in the two regions under the following
parameter values:
𝑡 = 12, 𝑤 = 1, 𝑎 = 2.6, 𝜓 = 0.03, 𝑏𝐻 = 1, 𝑏𝑀 = .035, 𝛾 = 2,
𝑁 = 1.
7 In reality, there may be non-produced and non-traded amenities that are associated with urban or rural living. For example, fresh air and the serenity of isolation are valuable features of rural areas that are not explicitly included in the agent’s utility function. The model could be extended to incorporate place-specific amenities in the utility function without affecting its qualitative predictions.
21
Region 𝐴 has less land, and in equilibrium is the denser region in terms of population per land
area and market services per land area. It also has higher land rents and its residents spend more
time working.
While the smaller region (𝐴) is also denser under the given parameter values, under
alternative parameter values it is possible for the larger region to also be denser. In general,
however, whichever region is denser will also have higher land rents, lower home production,
and higher market production. To see this, insert (39), (41), and (42) into the land market
clearing conditions (38), 𝑙𝐴𝑁𝐴
= (𝑏𝑀𝚤̅+ (𝑏𝐻 − 𝑏𝑀)𝐻𝐴),𝑙𝐵𝑁𝐵
= (𝑏𝑀𝚤̅+ (𝑏𝐻 − 𝑏𝑀)𝐻𝐵),
where 𝚤 ̅is the mass of services consumed (at home and on the market) in each region: 𝚤̅ = 𝚤�̅� =
𝚤�̅� (by spatial equilibrium). Assume that region 𝐴 is denser ( 𝑙𝐵𝑁𝐵
> 𝑙𝐴𝑁𝐴
), and in particular that
𝑙𝐵𝑁𝐵
−𝑙𝐴𝑁𝐴
= (𝑏𝐻 − 𝑏𝑀)(𝐻𝐵 − 𝐻𝐴) > 𝛿
for some 𝛿 > 0. Since 𝑏𝐻 > 𝑏𝑀 (by assumption), it follows that 𝐻𝐵 > 𝐻𝐴. Specifically,
𝐻𝐵 − 𝐻𝐴 > 𝛿(𝑏𝐻 − 𝑏𝑀). (44)
In other words, the denser region has less land available per person, and since spatial equilibrium
requires that the total mass of services per resident is equal across regions, this is only possible if
the denser area uses land more efficiently by producing more of its services on the market rather
than by its residents at their homes.
Since the mass of home produced services per resident in a region (40) is decreasing in
the land rent, it follows that higher masses of home production are associated with lower land
prices. Specifically, substituting (40) into (44) for 𝐻𝐵 and 𝐻𝐴 yields
𝛾 �𝑎𝑤
+𝜓𝑙𝐵𝑀𝐵
.5 −𝑟𝐵𝑤
(𝑏𝐻 − 𝑏𝑀)� − 𝛾 �𝑎𝑤
+𝜓𝑙𝐴𝑀𝐴
.5 −𝑟𝐴𝑤
(𝑏𝐻 − 𝑏𝑀)� > 𝛿(𝑏𝐻 − 𝑏𝑀),
which can be rearranged to yield
𝑟𝐴 − 𝑟𝐵 >𝛿𝛾
+𝜓
𝑏𝐻 − 𝑏𝑀�𝑙𝐴𝑀𝐴
.5 −𝑙𝐵𝑀𝐵
.5�.
The difference in land prices between region 𝐴 and region 𝐵 is an increasing function of the
density difference 𝛿.
22
Relationship between the Model and the Evidence. The empirical evidence in Section 3
establishes a positive relationship between work time and service establishment density,
consistent with the models’ predictions that dense areas have more market services available.
Section 3 also demonstrates that urban residents own fewer durable factor inputs, consistent with
the models’ predictions that urban residents use less land for home production.
The models above offer an additional prediction that land rents and service prices are
higher in dense areas. Traditionally economists have attributed high land rents in dense areas to
the high value associated with living near where people work at the city center, as predicted in
Mills (1967) and, more recently, Lucas and Rossi-Hansberg (2002). The models above offers an
alternative explanation for high land prices in dense areas based on the relative scarcity of land.
Both proximity to work and scarcity as a factor of production are likely to account for the
high land prices observed in reality. However, as Glaeser et al (2001) note, the rise of reverse
commuting suggests that proximity to work is becoming less important as a determinant of urban
land prices. Section 5 provides further support to the notion that urban land prices are based on
density-related factors in addition to proximity to work.
5. The Dependence of Land Prices on Service Establishment Density
This section tests the prediction in the models above that urban density is associated with high
land prices. Other studies have primarily focused on differences in land prices across
metropolitan areas (e.g., Glaeser et al. 2001 and Albouy 2012), and interpret high land prices
relative to wages as being driven by high metropolitan amenities, consistent with the Roback
(1982) model. In a study based on the Roback model, Albouy (2012) finds that predicted
measures of metropolitan amenities are positively correlated with population density and with
the number of service establishments in a city, and uncorrelated with some direct measures of
quality of life such as crime rates. The model above suggests Albouy’s results may be due to
high land prices that are a direct result of land scarcity in urban areas in addition to amenities that
are associated with density.
In contrast to Glaeser et al (2001) and Albouy (2012), I investigate the determinants of
land prices across PUMAs within the New York/Northeastern New Jersey metropolitan area.
Controlling for the metropolitan area helps mitigate concerns that high land prices are simply a
result of amenities such as sunshine and air temperature which are typically treated as uniform
23
within a city. The data and sample are the same as in Section 3. Following Albouy (2012), I
obtain the housing cost differential for individual 𝑖 using a regression of gross rents, 𝑟𝑖, on
controls (𝑌𝑖) for size, rooms, commercial use, kitchen and plumbing facilities, age of building,
homeownership, and the number of residents per room.
log(𝑟𝑖) = 𝑌𝑖𝛽 + 𝜖𝑖. (45)
Rents for homeowners are imputed using a discount rate of 7.85 percent (Peiser and Smith
1985). The residuals 𝜖𝑖 are the rent differentials that represent the amount individual 𝑖 pays for
her apartment/home relative to the average cost of a similar apartment/home in the New York
metro area.
I estimate the dependence of rent differentials on indicators of urban density using the
following specification:
𝜖𝑖 = 𝛼 + 𝛽1Mediumi + 𝛽2Highi + 𝛽3Highesti + 𝑋𝑖𝛾 + 𝑒𝑖,
Where Mediumi, Highi, and Highesti are indicator variables for the density classification of
individual 𝑖’s PUMA of residence, and 𝑋𝑖 is a vector of individual-specific controls, including
the log of the number of minutes spent traveling to work. Column 1 of Table 6 shows that rent
differentials are substantially higher in highest-density and high-density areas than they are in
low-density areas. The dependence of rent differentials on density is only slightly lower when
conditioning on transportation time to work (column 2) and other demographic controls (column
3). The coefficient on transportation time to work is negative, as expected, but there remains a
large dependence of rent differentials on density even when conditioning on transportation time
to work. Column 4 shows that these results are robust to assuming a log-linear, rather than
categorical, dependence on PUMA establishment density.
The main message from Table 6 is that home land rents, as measured by home rent
differentials, strongly depend on service establishment density. The model above suggests that
high land rents in dense areas are due to the fact that land is the relatively scarce factor of
production there.
6. Discussion
This paper proposes an explanation based on substitution from home production to market
purchases in dense areas to account for the following empirical findings:
24
1) Work hours are higher in high-density areas,
2) Urban residents own fewer durables,
3) Land prices are positively associated with service establishment density, even when
conditioning on travel time to work.
Some of these facts alone may have alternative explanations. For example, (1) may be due to
other agglomeration forces such as the rat race effect. However, Rosenthal and Strange (2008)
find a positive dependence of hours on proximity to employees of similar occupations for
professional workers but not for lower-income workers. The dependence of hours on service
establishment density holds at all levels of the income distribution, consistent with the model’s
partial equilibrium prediction that substitution of work time for home production occurs at any
wage level.
Fact (3) could be explained by consumers paying high land rents to live close to PUMA-
specific amenities other than proximity to service establishments. However, proximity to non-
service amenities is unlikely to explain the low durables ownership in dense areas; and, to the
extent that leisure and urban amenities (parks, libraries, etc) are strong complements, proximity
to these other amenities should cause urban residents to work less at a given level of income.
The fact that urban residents work more and own fewer durables suggests that they are to some
extent saving time by producing less at home.
In addition to providing a unified explanation for the empirical regularities presented in
Sections 3 and 5, substitution of market purchases for home production can also help explain the
finding in Albouy (2012) that predicted measures of quality of life are positively correlated with
congestion (as measured by population density) and with the number of service establishments in
a city, and uncorrelated with some direct measures of quality of life such as crime rates. These
quality of life estimates are high when metropolitan areas have high price-to-wage ratios.
According to the model above, high price-to-wage ratios are a direct result of land scarcity in the
production of services. The precise extent to which urban amenities and disamenities interact
with land scarcity to determine price-to-wage ratios is left as a topic for future research.
The theory is most applicable to high density areas, including regions in New York City.
While other American cities are less dense than New York City, the theory is especially relevant
to the many cities in other countries which are far denser. Gollin, Jedwab, and Vollarth (2013)
25
document a recent trend of rapid urbanization without industrialization in developing countries.
The theory above helps rationalize the existence and growth of these cities by demonstrating the
benefits of urbanization that can counteract the costs of land scarcity.
7. Conclusion
In the analysis above, urban areas are simply dense economic regions in which the most efficient
manner for consumers to satisfy their wants is for firms to provide services for consumers to
purchase. Suburban areas are sparsely populated economic regions in which consumers most
efficiently satisfy their wants by producing them at home.
Under this view, metropolitan areas and cities consist of adjoined economic regions of
differing densities. There is no unique city center, but rather areas of differing densities within a
city. Glaeser and Kohlhase (2004) suggest that modern cities no longer feature the traditional
role of city centers, and they call for regional models that reflect this new reality. The model
presented above is a step in this direction. The model predicts that labor supply, land prices, and
services prices should vary within a metropolitan area according to the density of a given
economic region (or neighborhood) in the metro area.
These predictions are, to some extent, hardly surprising. Despite the somewhat intuitive
nature of the empirical and theoretical results presented here, this paper is the first to explicitly
model the density-dependent tradeoff between home and market production and its implications
for market prices and quantities. These implications differ from the predictions of standard
models and shed light on why prices are often high in areas that feature disamenities such as
congestion, crime, and pollution.
References
Albouy, David. 2012. "Are Big Cities Bad Places to Live? Estimating Quality-of-Life across Metropolitan Areas" University of Michigan. Albouy, David and Bert Lue. 2011. “Driving to Opportunity: Local Wages, Commuting, and Sub-Metropolitan Quality of Life.” University of Michigan. Becker, Gary. 1965. “A Theory of the Allocation of Time.” The Economic Journal 75(299): 493-517.
26
Buera, J. and J. Kaboski. 2012. “The Rise of the Service Economy.” American Economic Review 102: 2540-2569. Couture, Victor. 2013. “Valuing the Consumption Benefits of Urban Density.” University of Toronto. Mimeo. Duranton, Gilles and Diego Puga. 2004. “Micro-Foundations of Urban Agglomeration Economies,” in J.V. Henderson and J. F. Thisse (eds), Handbook of Regional and Urban Economics Volume 4. Amsterdam and New York: North Holland, 2063–2117. George, Lisa and Joel Waldfogel. 2003. “Who Affects Whom in Daily Newspaper Markets?” Jounrnal of Political Economy, 111(4): 765-784. Glasaer, Edward L., Jed Kolko, and Albert Saez. 2001. “Consumer City.” Journal of Economic Geography 1: 27-50. Glasaer, Edward L. and Janet E. Kohlhase. 2004. “Cities, Regions, and the Decline of Transport Costs.” Journal of Regional Science. 83: 197-228. Gollin, Douglad, Rémi Jedwab, and Dietrich Vollrath. 2013. “Urbanization with and without Industrialization.” Mimeo. Lancaster, Kelvin J .1966. "A New Approach to Consumer Theory." Journal of Political Economy 74(2): 132-157. Lucas, Robert E., Jr. and Esteban Rossi-Hansberg. 2002. “On the internal structure of cities.” Econometrica 70(4):1445–1476. Matsuyama, K. 2000. “A Ricardian Model with a Continuum of Goods under Nonhomothetic Preferences: Demand Complementarities, Income Distribution, and North-South Trade.” Journal of Political Economy 108(6): 1093-1120. Matsuyama, K. 2002. “The Rise of Mass Consumption Societies,” Journal of Political Economy 110: 1035-1070. Murphy, Daniel P. 2012. “A Shopkeeper Economy.” Mimeo. Murphy, Kevin., Schleifer, Andrei., and Robert. Vishny. 1989. “Income Distribution and Market Size, and Industrialization.” Quarterly Journal of Economics 104: 537-64.
27
Ottavanio, Giancarlo. I. P., Takatoshi Tabuchi. and Jacques-François Thisse. 2002. “Agglomeration and Trade Revisited”, International Economic Review 43: 409–436. Peiser, Richard B. and Lawrence B. Smith. 1985. "Homeownership Returns, Tenure Choice and Inflation." American Real Estate and Urban Economics Journal 13: 343-60. Roback, Jennifer .1982. "Wages, Rents, and the Quality of Life." Journal of Political Economy 90: 1257-1278. Rogerson, Richard. 2008. “Structural Transformation and the Deterioration of European Labor Market Outcomes.” Journal of Political Economy. 116(2): 235-259. Rosenthal, Stuart S. and William C. Strange. 2004. "Evidence on the Nature and Sources of Agglomeration Economies," in J. V. Henderson and J. F. Thisse (eds), Handbook of Urban and Regional Economics, Volume 4. Amsterdam and New York: North Holland. Rosenthal, Stuart, and William Strange. 2008. “Agglomeration and Hours Worked.” Review of Economics and Statistics. 90(1): 105–118. Ruggles, S., J.T. Alexander, K. Genadek, R. Goeken, M. Schroeder, and M. Sobek. 2010. “Integrated Public Use Microdata Series: Version 5.0.” University of Minnesota.
Appendix A
This section derives the necessary and sufficient conditions under which an increase in density in
the baseline model causes an increase in the land rent and a fall in home production. First,
substitute (25) into the time constraint (5) for 𝑡𝑀 , 𝑡𝑆, and 𝑡𝐻, and substitute equations (22) and
(23) for 𝐻 and 𝑀.
𝑡 = [𝑎𝑀 + 𝜓𝑙](𝚤̅ − {𝑎𝑀 + 𝜓𝑙 − 𝑟(𝑏𝐻 − 𝑏𝑀)}) +12
{𝑎𝑀 + 𝜓𝑙 − 𝑟(𝑏𝐻 − 𝑏𝑀)}2.
Combine terms to yield
𝑡 = [𝑎𝑀 + 𝜓𝑙]𝚤̅ −12
[𝑎𝑀 + 𝜓𝑙]2 +12
{𝑟2(𝑏𝐻 − 𝑏𝑀)2}. (46)
Likewise, substitute equations (22) and (23) into the land constraint for 𝐻 and 𝑀:
𝑙 = 𝑏𝑀𝚤̅+ (𝑏𝐻 − 𝑏𝑀){𝑎𝑀 + 𝜓𝑙 − 𝑟(𝑏𝐻 − 𝑏𝑀)}.
Combine terms to yield
28
𝑙[1 − (𝑏𝐻 − 𝑏𝑀)𝜓] = 𝑏𝑀𝚤̅+ (𝑏𝐻 − 𝑏𝑀)𝑎𝑀 − 𝑟(𝑏𝐻 − 𝑏𝑀)2. (47)
Note that the land constraint implies that 𝑙 (the inverse of density) and 𝑟 are inversely
related. This relationship captures the fact that, holding constant 𝚤,̅ a fall in 𝑙 must be
accompanied by a rise in 𝑟 to maintain equilibrium in the land market. We can totally
differentiate equations (46) and (47) to formally derive the response of 𝑟 to a change in density.
The total derivative of (46) is
0 = [𝑎𝑀 + 𝜓𝑙]𝑑𝚤̅+ [𝚤̅ − 𝑎𝑀 − 𝜓𝑙]𝜓𝑑𝑙 + (𝑏𝐻 − 𝑏𝑀)2𝑟𝑑𝑟, (48)
and the total derivative of (47) is
𝑑𝚤̅ =1𝑏𝑀
[1 − (𝑏𝐻 − 𝑏𝑀)𝜓]𝑑𝑙 + (𝑏𝐻 − 𝑏𝑀)2𝑑𝑟. (49)
Substitute (49) for 𝑑𝚤 ̅in (48) and rearrange to yield
𝑑𝑟 = −𝜓
(𝑏𝐻 − 𝑏𝑀)2(𝑟 + 𝑎𝑀 + 𝜓𝑙)�𝚤̅+
1𝑏𝑀
[𝑎𝑀 + 𝜓𝑙][1 − (𝑏𝐻 − 𝑏𝑀)𝜓] − 𝑎𝑀 − 𝜓𝑙� 𝑑𝑙.
The land rent is falling in available land (increasing in density) whenever
𝚤̅+ (𝑎𝑀 + 𝜓𝑙) �1𝑏𝑀
(1 − (𝑏𝐻 − 𝑏𝑀)𝜓) − 1� > 0 ⇔ 1 > 𝑏𝐻𝜓 + 𝑏𝑀 �1 − 𝜓 − 𝚤̅1
(𝑎𝑀 + 𝜓𝑙)�.
This necessary and sufficient condition is easily satisfied whenever, for example, 𝑏𝐻, 𝑏𝑀 < 1 and 𝜓 ∈ [0,1].
The dependence of home production on density can be seen by totally differentiating
equation (22):
𝑑𝐻𝑑𝑙
= 𝜓 − (𝑏𝐻 − 𝑏𝑀)𝑑𝑟𝑑𝑙
.
Home production is increasing in 𝑙 (decreasing in density) whenever the land rent is falling in 𝑙.
Equation (23) shows that for a given total mass of services 𝚤,̅ market production 𝑀 and time
spend on the market 𝑡𝑀 are falling in 𝑙 when the land rent if falling in 𝑙.
The total mass of services 𝚤 ̅may increase or decrease with density, depending on the
strength of the benefits associated with time saved from lower shopping costs and land use saved
29
by market production instead of home production. To see this, note that land market clearing can
be written as
𝑙 = 𝑏𝑀𝚤̅+ (𝑏𝐻 − 𝑏𝑀)[𝑎𝑀 + 𝜓𝑙 − 𝑟(𝑏𝐻 − 𝑏𝑀)]. (50)
Totally differentiating (50) and solving for 𝑑𝚤/̅𝑑𝑙 yields
𝑑𝚤̅𝑑𝑙
=1𝑏𝑀
[1 − (𝑏𝐻 − 𝑏𝑀)𝜓] + (𝑏𝐻 − 𝑏𝑀)2𝑑𝑟𝑑𝑙
.
If 1 < (𝑏𝐻 − 𝑏𝑀)𝜓, then 𝚤 ̅is increasing in density when 𝑟 is increasing in density. This
sufficient condition states that the relative land savings from market production (𝑏𝐻 − 𝑏𝑀) or the
fall in shopping costs 𝜓 must be sufficiently large for 𝚤 ̅to increase when land falls. In the
theoretical model with endogenous land density, spatial equilibrium requires that benefits of
density perfectly offset the costs of less land.
30
Tables
PUMAService Establishments
per Square Mile N
3810 685 7283808 648 10073809 601 4413807 555 8433805 451 1400
3806 194 11103802 174 177
702 117 3304004 105 3004101 99 3033803 99 1543801 98 2254102 96 1054017 88 156
Low Others 22 (average) 16498
Highest
High
Medium
Table 1: Density Classifications
31
None None50,000-75,000
50,000-75,000
75,000-100,000
75,000-100,000
Regressors (1) (2) (3) (4) (5) (6)
Highest Density 3.803*** 1.283*** 3.436*** 1.788*** 4.892*** 2.907***(0.193) (0.193) (0.346) (0.360) (0.509) (0.547)
High Density 2.966*** 1.215*** 2.126*** 1.674** 3.559*** 2.231**(0.331) (0.317) (0.614) (0.626) (0.825) (0.847)
Medium Density -0.411 -0.416 1.177* 0.782 1.566 -0.364(0.301) (0.279) (0.577) (0.557) (0.937) (0.919)
log(income) 4.614*** 6.063*** 6.899**(0.094) (1.107) (2.425)
log(transportation time to work) -0.686*** -0.777*** -0.356(.0889) (0.169) (0.286)
Age -0.167* -0.736*** -0.509*(0.0697) (0.148) (0.252)
Age^2 0.000 0.007*** 0.004(0.001) (0.002) (0.003)
Occupation Controls YES YES YESEducation Controls YES YES YESR-squared 0.02 0.17 0.02 0.10 0.04 0.13N 23777 23777 5341 5341 2295 2191
Table 2-Dependence of Hours Worked on Service Establishment Density
Dependent Variable: Average Hours Worked per Week
Sample Restrictions on Income
Note: Data are from the 5 percent sample of the 2000 Census Integrated Public Use Microdata Series. Service establishment density is based on data from the Zip Codes Business Patterns. Robust standard errors clustered at the PUMA level are in parentheses. ***, **, and * indicate significance at the 0.1%,1%, and 5% levels, respectively.
32
None None50,000-75,000
50,000-75,000
75,000-100,000
75,000-100,000
Regressors (1) (2) (3) (4) (5) (6)
Highest Density -1.706*** -1.993*** -2.057*** -2.130*** -2.089*** -2.086***(0.0927) (0.115) (0.107) (0.139) (0.130) (0.143)
High Density -1.687*** -1.936*** -2.160*** -2.207*** -2.099*** -2.110***(0.0780) (0.101) (0.116) (0.130) (0.120) (0.142)
Medium Density -1.089*** -1.062*** -1.246*** -1.212*** -1.109*** -1.012***(0.176) (0.177) (0.154) (0.164) (0.240) (0.255)
log(income) 0.330*** 0.613** -0.0375(0.018) (0.234) (0.390)
log(transportation time to work) -0.262*** -0.221*** -0.0988(0.031) (0.044) (0.061)
Age Controls YES YES YESOccupation Controls YES YES YESEducation Controls YES YES YESN 23777 22620 5341 5144 2295 2191
Table 3-Dependence of Vehicle Ownsership on Service Establishment Density
Dependent Variable: Probability of Vehicle Ownership
Sample Restrictions on Income
Note: Data are from the 5 percent sample of the 2000 Census Integrated Public Use Microdata Series. Service establishment density is based on data from the Zip Codes Business Patterns. Robust standard errors clustered at the PUMA level are in parentheses. ***, **, and * indicate significance at the 0.1%,1%, and 5% levels, respectively.
33
None None50,000-75,000
50,000-75,000
75,000-100,000
75,000-100,000
Regressors (1) (2) (3) (4) (5) (6)
Highest Density -1.689*** -1.731*** -2.101*** -1.966*** -2.306*** -2.125***(0.0844) (0.0929) (0.0808) (0.0892) (0.125) (0.139)
High Density -1.520*** -1.629*** -1.721*** -1.725*** -2.162*** -2.097***(0.104) (0.131) (0.179) (0.169) (0.121) (0.125)
Medium Density -1.022*** -0.845*** -1.118*** -0.969*** -1.319*** -1.080***(0.117) (0.119) (0.149) (0.145) (0.134) (0.120)
log(income) 0.383*** 1.405*** 0.493(0.029) (0.217) (0.403)
log(transportation time to work) -0.105*** -0.0790* -0.122*(0.023) (0.034) (0.053)
Age Controls YES YES YESOccupation Controls YES YES YESEducation Controls YES YES YESR-squared 0.15 0.24 0.21 0.27 0.28 0.33N 23777 22620 5341 5144 2295 2191
Table 4-Dependence of Roomsltran in Residence on Service Establishment Density
Dependent Variable: Roomsltran per Resident Sample Restrictions on Income
Note: Data are from the 5 percent sample of the 2000 Census Integrated Public Use Microdata Series. Service establishment density is based on data from the Zip Codes Business Patterns. Robust standard errors clustered at the PUMA level are in parentheses. ***, **, and * indicate significance at the 0.1%,1%, and 5% levels, respectively.
34
Table 5: Market Outcomes
Region 𝐴 Region 𝐵
𝑙𝑗 (exogenous) 1 2
𝑡𝑗𝑀 10.1 6.3
𝑟𝑗 2.6 1.8
𝑀𝑗 1.48 0.05
𝐻𝑗 0.96 2.67
𝚤�̅� 3.41 3.41
𝑁𝑗 0.41 0.59
𝑁𝑗/𝑙𝑗 0.41 0.30
35
Regressors (1) (2) (3) (4)
Highest Density 0.355*** 0.351*** 0.259***(0.044) (0.045) (0.036)
High Density 0.158*** 0.155*** 0.109***(0.071) (0.070) (0.048)
Medium Density -0.004 0.000 -0.002(0.060) (0.059) (0.035)
log(establishment density) 0.202***(0.009)
log(transportation time to work) -0.038*** -0.044*** -0.072***(0.006) (0.005) (0.006)
log(income) 0.214*** 0.233***(0.009) (0.008)
Demographic Controls YES YESR-squared 0.14 0.14 0.24 0.21N 23497 22369 22369 22369
Dependent Variable: log(rent) differential
Table 6-Dependence of Rent Differentials on Service Establishment Density
Note: Data are from the 5 percent sample of the 2000 Census Integrated Public Use Microdata Series. Service establishment density is based on data from the Zip Codes Business Patterns. Robust standard errors clustered at the PUMA level are in parentheses. ***, **, and * indicate significance at the 0.1%,1%, and 5% levels, respectively.
