Urban Population and Amenities

77 Pages·2015·1.68 MB·English

Urban Population and Amenities:
The Neoclassical Model of Location
David Albouy Bryan Stuart∗
University of Illinois and NBER University of Michigan
November 16, 2015
Abstract
We use a neoclassical general-equilibrium model to explain cross-metro variation in
population,density,andlandsupplybasedonthreeamenitytypes: quality-of-life,pro-
ductivity in tradables, and productivity in non-tradables. This elucidates commonly-
estimated elasticities of local labor and housing supply and demand. From wage and
housing-cost indices, we explain half of observed density and total population varia-
tion,andfindjobsfollowpeoplemorethanpeoplefollowjobs. Land-areaanddensity
data are used to estimate elasticities of housing and land supply, and improve land-
rents and local-productivity estimates. We show how relaxing land-use regulations
andneutralizingfederaltaxeswouldaffectdifferentcities.
Keywords: Population, population density, land supply, amenities, agglomeration,
housingsupply,locallabormarkets.
JELNumbers: R23,R12,R31,H2
∗Albouy: [email protected]; Stuart; [email protected]. For their help and input we thank David Agrawal,
RebeccaDiamond,JesseGregory,AndrewHaughwout,JordanRappaport,andWillStrange;conferenceparticipants
at the 2012 Urban Economics Association annual meeting, 2012 National Tax Association annual meeting, 2013
American Real Estate and Urban Economics Association annual meeting, 2013 Canadian Economics Association
annualmeeting,2013NationalBureauofEconomicResearchSummerInstitutemeetinginUrbanEconomics,2013
HousingUrbanLaborMacromeetinginAtlanta;andseminarparticipantsatCalgary,Cornell,theClevelandFederal
Reserve, GeorgiaStateUniversity, IEBBarcelona, theKansasCityFederalReserve, Michigan, Minnesota(Applied
Economics),NYUAbuDhabi,theParisSchoolofEconomics,Purdue,SciencesPolitiques,andtheToulouseSchool
of Economics. During work on this project, Albouy was supported by the NSF grant SES-0922340 and Stuart was
supportedbytheNICHD(T32HD0007339)asaUMPopulationStudiesCenterTrainee. Thispaperwaspreviously
presentedas“UrbanQuantitiesandAmenities.” Anymistakesareourown.
1 Introduction
Academicsandpolicymakershavelongsoughttounderstandthehouseholdlocationdecisionsthat
shape our world. Here we examine whether a standard neoclassical model – pioneered by Rosen
(1979) and Roback (1982) – can predict massive differences in population levels and densities
across metropolitan areas. The model assumes that areas differ in their local characteristics, or
“amenities,”andthathouseholdsarefullymobile,makingitparticularlyappropriateforexplaining
long-run outcomes. This is a surprisingly new application of the model, which has been used
primarilytoinferthevalueofamenitiestohouseholdsandfirmsusingobservedwagesandhousing
rents. Because the model relies on standard neoclassical elements and readily available data, it is
anaturalbenchmarkforunderstandingmorecomplexexplanationsoflocationchoice,andmaybe
appliedinmanysettings,historicallyandinternationally.
The model involves a system of cities with three factors – labor, capital, and immobile land –
and two outputs – a good tradableacross cities, and ahome good that isnot. Local amenities vary
in three dimensions: quality-of-life for households, and trade-productivity and home-productivity
for firms. The first two dimensions address the classic problem of whether jobs follow people or
peoplefollowjobs,whilethethirdaddresseswhetherbothjobsandpeoplefollowhousingorother
non-traded goods (Glaeser and Gyourko 2005, Glaeser, Gyourko, and Saks 2006, Saks 2008).
The cross-sectional method we employ assesses the relative importance of these dimensions in
determiningwherepeopleliveanddoesnotdependontimingassumptionscriticalinstudiesbased
ontime-seriesevidence(e.g.,Blanco1963;Hoogstra,Florax,andDijk2005).
In section 2, we derive structural relationships between prices and quantities, such as popula-
tion, and the three amenity types. These relationships depend on cost and expenditure shares, tax
rates, land supply, and substitution responses in consumption and production. Our model com-
plements the core urban economics literature on agglomeration and congestion by looking at how
the the latter affect population, completing the feedback loop. Furthermore, we show how data on
populationdensitymaybeusedinconjunctionwithwageandhousing-costdatatoidentifyhome-
productivity,andimproveestimatesoftrade-productivityandtypicallyunavailablelandvalues.
1
After parametrizing the model to reflect the U.S. economy, section 3 shows that quantities
respond much more to amenities than prices do. This is consistent with density varying much
more across metros than wages and rents. The model also highlights the particular importance of
housingandothernon-tradedsectorsinaccommodatingpopulationbycreatingspaceforliving.
In section 4, we map commonly estimated reduced-form elasticities – e.g., of local labor or
housing supply – to underlying structural parameters, and recast partial equilibrium shifts in sup-
ply or demand as general equilibrium responses to amenity changes. The parametrized model
quantifies long-run relationships which are difficult to estimate credibly. We obtain large elas-
ticities that are broadly consistent with several estimates from the literature, suggesting those are
consistentwithobservedcross-sectionaldifferences.
Section5assesseshowwelltheneoclassicalmodelexplainspopulationdifferencesacross276
U.S. cities in two steps. In the first step, we use the parametrized model plus quality-of-life and
trade-productivity estimates from Albouy (Forthcoming) to predict population density, and find
that it explains half of the observed variation. Under the assumption that the lack of fit is due to
home-productivity differences, we demonstrate visually how to infer those differences from den-
sityandpricedata. Alternatively,weuseanon-linearregressionmodelwithcross-metrovariation
in land-use regulation and geography to estimate city-specific differences in efficiency and factor
substitution in the non-traded sector. These estimates conform to predictions that regulations and
rugged terrain impede efficiency and factor substitution — with plausible magnitudes — and im-
provethemodel’sfit. TheseresultsreinforcepanelestimatesfromSaks(2008)andSaiz(2010)of
howlocallaborandhousingsupplyelasticitiesvaryacrosscities.
Insection6,weuseinferredlandvaluesandmeasuresofmetropolitanlandareastoestimatethe
own-priceelasticityoflandsupplyandcity-specificdifferencesinlandendowments. Theestimates
find lower land endowments and elasticities in regulated and rugged areas. Using these estimates,
section 7 examines predictions based on wage and rent data (without reference to density), and
finds that the model predicts half of the observed differences in total population. Furthermore,
quality-of-life explains population density more than trade-productivity. We demonstrate with
2
counter-factualexercisestheeffectsofneutralizingregulatoryconstraintsorthegeographiceffects
oftaxation,dramaticallyincreasingthesizeofseverallargecities,whileshrinkingmanyothers.
Tothebestofourknowledge,wearethefirsttoassesshowwelltheneoclassicalmodelexplains
cross-metro population differences. Economists have used two approaches to study household lo-
cation decisions. One combines elements of the Rosen-Roback general-equilibrium model with
various alterations, particularly in consumption and the housing sector.1 Ours goes beyond much
of thiswork theoreticallyfrom having moregeneral productiontechnologies. We alsospecifically
account for every metro’s population in terms of three amenity dimensions, rather than just ratio-
nalize an overall distribution, such as Zipf’s Law. This makes our accounting of urban population
much more detailed relative to Desmet and Rossi-Hansberg (2013) and Lee and Li (2013), and
counter-factualexercisesmoreexactandilluminating.
Thesecondapproachusespartialequilibriumsearchmodels,whereworkersmoveinresponse
to price and amenity differentials, plus idiosyncratic preferences for certain places (e.g., Kennan
and Walker 2011). These models describe household decision-making in greater detail, while ab-
stracting from issues central to general equilibrium models, like how wages and housing costs de-
pend on population. These forward-looking models are more designed for understanding changes
over time — and the frictions that may lead to path dependence — than for explaining the ob-
served cross section. Given the myriad dimensions along which general and partial equilibrium
modelsmightdiverge,akeycontributionofthispaperisprovidingabenchmarkagainstwhichwe
can compare different location choice models. Understanding the performance of the benchmark
model is important in assessing the role of other elements, like preference heterogeneity and path
dependence. Shortcomingsinthecoremodelhighlightusefultopicsforfutureresearch.
1HaughwoutandInman(2001)simplifythenon-tradedsectortoafixedlandmarket. Rappaport(2008a, 2008b)
constrainsproductivityinthetradedandnon-tradedsectorstobethesame,andassumestheelasticityofsubstitution
betweenfactorsintradedproductionisone. Glaeseretal. (2006),Diamond(2013),andMoretti(2013)useapartial
equilibriumhousingsupplyfunction.DesmetandRossi-Hansberg(2013)constrainelasticitiesofsubstitutionintraded
production to be one, and model the non-traded sector using a mono-centric city at a fixed density. The latter four
papersassumeeachhouseholdconsumesasinglehousingunit,andprecludeanalyzingdensity. Ahlfeldtetal. (2012),
whofocusonwithin-citylocationchoices,constrainelasticitiesofsubstitutionindemandandtradedproductiontobe
one.LeeandLi(2013)andSua´rezSerratoandZidar(2014)assumeallelasticitiesofsubstitutionareone,andexclude
laborfromnon-tradedproduction.
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2 The Neoclassical Model of Location
2.1 System of Cities with Consumption and Production
We use the model of Albouy (2009), which adds federal taxes to the general equilibrium three-
equation Roback (1982) model. The national economy contains many cities, indexed by j, which
trade with each other and share a homogeneous population of mobile households. Cities differ
exogenously in three attributes, each of which is an index summarizing the value of amenities;
quality-of-life Qj raises household utility, trade-productivity Aj lowers costs in the traded sector,
X
and home-productivity Aj lowers costs in the non-traded sector. Households supply a single unit
Y
of labor in their city of residence, earning local wage wj. They consume a numeraire traded good
x and a non-traded “home” good y with local price pj. All input and output markets are perfectly
competitive,andallpricesandper-capitaquantitiesarehomogeneouswithincities.
Firms produce traded and home goods out of land, capital, and labor. Land, Lj, is hetero-
geneous across cities, immobile, and receives a city-specific price rj. Each city’s land supply
LjL˜(rj) depends on an exogenous endowment Lj and a supply function L˜j(rj). The supply of
0 0
capitalineachcityKj isperfectlyelasticattheprice¯ı. Labor,Nj,issuppliedbyhouseholdswho
haveidenticalsize,tastes,andowndiversifiedportfoliosoflandandcapital,whichpayanincome
R = (cid:80) rjLj/N from land and I = (cid:80) ¯ıKj/N from capital, where N = (cid:80) Nj is
j TOT j TOT TOT j
the total population. Total income mj = wj +R+I varies across cities only as wages vary. Out
of this income households pay a linear federal income tax τmj, which is redistributed in uniform
lump-sum payments T.2 Household preferences are modeled by a utility function U(x,y;Qj)
which is quasi-concave over x, y, and Qj. The expenditure function for a household in city j is
e(pj,u;Qj) ≡ min {x + pjy : U(x,y;Qj) ≥ u}. Quality-of-life Q enters neutrally into the
x,y
utilityfunctionandisnormalizedsothate(pj,u;Qj) = e(pj,u)/Qj,wheree(pj,u) ≡ e(pj,u;1).
Firms produce traded and home goods according to the function Xj = Aj F (Lj ,Nj ,Kj )
X X X X X
andYj = Aj F (Lj ,Nj,Kj ),whereF andF areweaklyconcaveandexhibitconstantreturns
Y Y Y Y Y X Y
2The model can be generalized to allow nonlinear income taxes. Our application adjusts for state taxes and tax
benefitstoowner-occupiedhousing.
4
toscale,withHicks-neutralproductivity. Unitcostinthetradedgoodsectorisc (rj,wj,¯ı;Aj ) ≡
X X
min {rjL+wjN+¯ıK : Aj F (L,N,K) = 1}. Similartotherelationshipbetweenquality-of-
L,N,K X
life and the expenditure function, let c (rj,wj,¯ı;Aj ) = c (rj,wj,¯ı)/Aj , where c (rj,wj,¯ı) ≡
X X X X X
c (rj,wj,¯ı;1) is the uniform unit cost function. A symmetric definition holds for unit cost in the
X
homegoodsectorc .
Y
2.2 Equilibrium of Prices, Quantities, and Amenities
Eachcityisdescribedbyablock-recursivesystemofsixteenequationsinsixteenendogenousvari-
ables: three prices pj,wj,rj, two per-capita consumption quantities, xj,yj, and eleven city-level
production quantities Xj,Yj, Nj,Nj ,Nj, Lj,Lj , Lj , Kj,Kj ,Kj . The endogenous variables
X Y X Y X Y
depend on three exogenous attributes Qj,Aj ,Aj and the land endowment Lj. The system first
X Y 0
determines prices — where most researchers stop — then, per-capita consumption quantities and
city-level production quantities. The recursive structure vanishes if amenities depend endoge-
nously on quantities, as described below. We adopt a “small open city” assumption and take
nationallydeterminedvariablesu¯,¯ı,I,R,T asgiven.
We log-linearize the generally nonlinear system, as in Jones (1965) to obtain a model that can
be parametrized and examined empirically. The log-linearized system is described below; the full
nonlinearsystemisinappendixA. AppendixBpresentssimulationresultswhichindicatethatthe
log-linearizedmodelreasonablyapproximatesthenonlinearone.
The log-linearized model requires several economic parameters, evaluated at the national av-
erage. For households, denote the share of gross expenditures spent on the traded and home good
as s ≡ x/m and s ≡ py/m; denote the share of income received from land, labor, and capital
x y
income ass ≡ R/m, s ≡ w/m, and s ≡ I/m. For firms, denote the cost share of land, labor,
R w I
and capital in the traded good sector as θ ≡ rL /X, θ ≡ wN /X, and θ ≡¯ıK /X; denote
L X N X K X
equivalentcostsharesinthehomegoodsectorasφ ,φ ,andφ . Finally,denotetheshareofland,
L N K
labor, and capital used to produce traded goods as λ ≡ L /L, λ ≡ N /N, and λ ≡ K /K.
L X N X K X
While not necessary, to fix ideas we assume the home good is more cost-intensive in land relative
5
tolaborthanthetradedgood,bothabsolutely,φ ≥ θ ,andrelatively,φ /φ ≥ θ /θ ,implying
L L L N L N
λ ≤ λ . For any variable z, we denote the log differential by zˆj ≡ lnzj − lnz¯ ∼= (zj −z¯)/z¯,
L N
wherez¯isthenationalaverage.
2.2.1 EquilibriumPriceConditions
Sincehouseholdsarefullymobile,theyreceivethesameutilityu¯acrossallinhabitedcities. Firms
earnzeroprofitsinequilibrium. Theseconditionsimply
−s (1−τ)wˆj +s pˆj = Qˆj (1)
w y
θ rˆj +θ wˆj = Aˆj (2)
L N X
φ rˆj +φ wˆj −pˆj = Aˆj . (3)
L N Y
Equations(1)-(3)simultaneouslydeterminethecity-levelpricespˆj,rˆj,andwˆj asfunctionsofthe
three attributes Qˆj,Aˆj , and Aˆj plus cost and expenditure shares and the marginal tax rate. These
X Y
conditions provide a one-to-one mapping between unobservable city attributes and potentially ob-
servableprices. Householdspaymoreforhousingandgetpaidlessinnicerareas. Firmspaymore
to their factors in more trade-productive areas, and they do the same relative to output prices in
morehome-productiveareas. Albouy(Forthcoming)explorestheseconditionsinmoredetail.
2.2.2 ConsumptionConditions
Inchoosingtheirconsumptionxˆj andyˆj,householdsfaceabudgetconstraintandobeyatangency
condition,implying
(cid:0) (cid:1)
s xˆj +s pˆj +yˆj = (1−τ)s wˆj (4)
x y w
xˆj −yˆj = σ pˆj (5)
D
6
where wˆj and pˆj are determined by the price conditions. Equation (5) depends on the elasticity of
substitutioninconsumption,σ ≡ −e·(∂2e/∂p2)/[∂e/∂p·(e−p·∂e/∂p)] = −∂ln(y/x)/∂lnp.
D
Substituting equation (1) into equations (4) and (5) produces the consumption solutions xˆj =
s σ pˆj −Qˆj and yˆj = −s σ pˆj −Qˆj. Because of homothetic preferences, in areas where Qj is
y D x D
higher, but pj is the same, households consume less of x and y in equal proportions, so the ratio
y/x remains constant — similar to an income effect. Holding Qj constant, areas with higher pj
inducehouseholdstoreducetheratioy/xthroughasubstitutioneffect.
Higher values of σ approximate a more general model with greater taste heterogeneity for
D
home goods. In such a model, households with stronger tastes for y sort to areas with a lower
p. At equilibrium utility levels, an envelope of the mobility conditions for each type forms that
of a representative household, with greater preference heterogeneity reflected as more flexible
substitution.3
2.2.3 ProductionConditions
Givenpricesandper-capitaconsumption,outputXˆj,Yˆj,employmentNˆj,Nˆj ,Nˆj,capitalKˆj,Kˆj ,Kˆj ,
X Y X Y
and land Lˆj,Lˆj ,Lˆj are determined by eleven equations describing production and market clear-
X Y
ing. The first six are conditional factor demands describing how input demandsdepend on output,
productivity,andrelativeinputprices:
Nˆj = Xˆj −Aˆj +θ σLN (cid:0)rˆj −wˆj(cid:1)−θ σNKwˆj (6)
X X L X K X
Lˆj = Xˆj −Aˆj +θ σLN(wˆj −rˆj)−θ σKLrˆj (7)
X X N X K X
Kˆj = Xˆj −Aˆj +θ σKLrˆj +θ σNKwˆj (8)
X X L X N X
Nˆj = Yˆj −Aˆj +φ σLN(rˆj −wˆj)−φ σNKwˆj (9)
Y Y L Y K Y
Lˆj = Yˆj −Aˆj +φ σLN(wˆj −rˆj)−φ σKLrˆj (10)
Y Y N Y K Y
Kˆj = Yˆj −Aˆj +φ σKLrˆj +φ σNKwˆj (11)
Y Y L Y N Y
3Roback(1980)discussesthisgeneralizationaswellasthebelowgeneralizationsinproduction.
7
The dependence on input prices is determined by partial (Allen-Uzawa) elasticities of substitution
in each sector for each pair of factors, e.g., σLN ≡ c ·(∂2c /∂w∂r) /(∂c /∂w·∂c /∂r). Our
X X X X X
baselinemodelassumesthatproductiontechnologydoesnotdifferacrosscities,implyingconstant
elasticities;werelaxthisassumptionforthehousingsectorbelow. Tosimplify,wealsoassumethat
partial elasticities within each sector are the same, i.e., σNK = σKL = σLN ≡ σ , and similarly
X X X X
forσ ,aswithaconstantelasticityofsubstitution(CES)productionfunction.
Y
Higher values of σ correspond to more flexible production of the traded good, as firms can
X
vary the proportion of inputs they employ. In a generalization with multiple traded goods sold at
fixed prices, firms would specialize in producing goods for which their input costs were relatively
low. For example, areas with high land costs and low labor costs would produce goods that use
labor intensively. A representative zero-profit condition is formed by an envelope of the zero-
profit conditions for each good, with a greater variety of goods reflected in greater substitution
possibilities.
Arelatedargumentexistsforhomegoods. Ahighervalueofσ meansthathousingproducers
Y
canbettercombinelaborandcapitaltobuildtallerbuildingsinareaswithexpensiveland. Fornon-
housing home goods, retailers would use taller shelves and restaurants would hire extra servers to
make better use of space in cities with expensive land. If all home goods were perfect substitutes,
thenanenvelopeofzero-profitconditionswouldformarepresentativezero-profitcondition.4
Three conditions express the local resource constraints for labor, land, and capital under the
assumptionthatfactorsarefullyemployed:
Nˆj = λ Nˆj +(1−λ )Nˆj (12)
N X N Y
Lˆj = λ Lˆj +(1−λ )Lˆj (13)
L X L Y
Kˆj = λ Kˆj +(1−λ )Kˆj . (14)
K X K Y
4Theconditionthatallhomegoodsareperfectsubstitutesissufficient,butmightnotbenecessary. Analternative
sufficientcondition,whichholdswhenconsideringtradedgoods,isthatrelativepricesoftypesofhomegoodsdonot
varyacrosscities.
8
Equations (12)-(14) imply that sector-specific factor changes affect overall changes in proportion
tothefactorshare. Locallandisdeterminedbythesupplyfunctioninlogdifferences
Lˆj = Lˆj +εj rˆj (15)
0 L,r
withendowmentdifferentialLˆj andlandsupplyelasticityεj ≡ (∂L˜j/∂r)·(rj/L˜j).
0 L,r
Finally,themarketclearingconditionforhomegoodsis
Nˆj +yˆj = Yˆj. (16)
Walras’ Law makes redundant the market clearing equation for traded output, which includes per-
capitanettransfersfromthefederalgovernment.
2.3 Total Population, Density, and Land
The log-linearized model readily separates intensive population differences holding land supply
constant, i.e. density, from extensive differences driven by land supply. If we define population
densityasNj ≡ Nj/Lj,thenthetotalpopulationdifferentialisalinearfunctionofdifferentialsin
∗
density,thelandendowment,andlanddrivenbyrent:
Nˆj = Nˆj +Lˆj +εj rˆj (17)
∗ 0 L,r
whereNˆj andrˆj dependonamenitiesQˆj,Aˆj ,Aˆj butthelandendowmentLˆj doesnot.5
∗ X Y 0
5Inprinciple,landsupplycanvaryontwodifferentmargins. Attheextensivemargin,anincreaseinlandsupply
correspondstoagrowingcityboundary. ExtensivemargindifferencescanbedrivenbythelandendowmentLˆj orthe
0
supply function εj rˆj. At the intensive margin, an increase in land supply takes the form of employing previously
L,r
unused land within a city’s border. The assumption of full utilization, seen in equations (13) and (15), rules out
intensivemarginchanges.
9