URBAN POPULATION AND AMENITIES

65 Pages·2013·0.45 MB·English

URBAN POPULATION AND AMENITIES
David Albouy Bryan Stuart∗
University of Michigan and NBER University of Michigan
[email protected] [email protected]
July 2013
∗For their help and input we thank Jesse Gregory, Andrew Haughwout, Jordan Rappaport, and Will
Strange; conference participants at the 2012 Urban Economics Association annual meeting, 2012 National
TaxAssociationannualmeeting,andthe2013AmericanRealEstateandUrbanEconomicsAssociationan-
nualmeeting; andseminarparticipantsatMichigan, NYUAbuDhabi, theParisSchoolofEconomics, and
the Toulouse School of Economics This paper was previously presented as “Urban Quantities and Ameni-
ties.” Anymistakesareourown. [email protected].
Abstract
We use a frictionless neoclassical general-equilibrium model to explain cross-metro variation in
population density based on three broad amenity types: quality of life, productivity in tradables,
and productivity in non-tradables. Analytically, we demonstrate the dependence of quantities on
amenities through substitution possibilities in consumption and production. Our calibrated model
predictslargeelasticities,consistentwithvariationinU.S.data,andestimatesoflocallaborsupply
and demand. From only differences in wages and housing costs, we explain half of the variation
in density, especially through quality of life amenities. We also show that density information can
provideorrefinemeasuresoflandvalueandlocalproductivity. Ourapproachcanbeusedtostudy
awidevarietyofurbanquantities.
Keywords: Populationdensity,productivity,qualityoflife.
JELNumbers: H2,H4,J3,Q5,R1
1 Introduction
In the United States, population densities vary across space far more than the prices of labor and
housing. Atthemetropolitanlevel,theaverageresidentialdensityofNewYorkisalmost50times
that of Texarkana. Meanwhile, wage levels in the highest-paying metro are not even double that
of the lowest, and housing costs in the most expensive metro average only four times that of the
lowest. In this paper, we examine whether small differences in prices are compatible with large
differences in quantities, like population density, in the neoclassical model of Rosen (1979) and
Roback(1982),withmobilehouseholdsandfirms,andbothtradableandnon-tradablesectors.
Intheneoclassicalmodel,differencesinpricesandquantitiesacrossmetroareasstemfromlo-
calamenities,whichworkthroughthreedifferentchannels: qualityoflife,tradable-sectorproduc-
tivity, and non-tradable-sector productivity. The first two channels determine the extent to which
people follow jobs or jobs follow people, a topic long debated (Blanco 1963, Borts and Stein
1964).1 The third channel determines whether both jobs and people follow availabile housing, a
subjectthathasreceivedmorerecentattention(GlaeserandGyourko2006,Glaeser,Gyourko,and
Saks 2006, Saks 2008). Although researchers regularly use the neoclassical model to examine the
relationship between prices and amenities, they rarely do so to examine the relationship between
quantitiesandamenities. Existingworkoftenimposesstrongrestrictionsonthemodeloraltersits
structure, particularly in the non-tradable sector, and provides only numerical results, which offer
limited intuition (e.g., Haughwout and Inman 2001, Rappaport 2008a, 2008b, Desmet and Rossi
Hansbergforthcoming,Morettiforthcoming).2
Here we consider the relationships between amenities and quantities analytically, using the
canonical neoclassical model with few restrictions. We show how quantity differences depend on
1SeeHoogstra,Florax,andDijk(2005)foraninterestingmeta-analysisofthisliterature.
2HaughwoutandInman(2001)reducethenon-tradablesectortoafixedlandmarket. Rappaport(2008a, 2008b)
constrainsproductivityinthenon-tradablesectortobethesameasinthetradablesector,andassumestheelasticityof
substitutionbetweenfactorsintradableproductionisone. Glaeser,Gyourko,andSaks(2006)andMoretti(forthcom-
ing)useanad-hocpartialequilibriumsupplyfunction,therebyexcludinglaborfromthenon-tradablesector,andforce
householdstoconsumeafixedamountofhousing. DesmetandRossi-Hansberg(forthcoming)constrainelasticitiesof
substitutioninconsumptionandtradableproductiontobeone,andmodelthenon-tradablesectorusingamonocentric
city, where households consume a single unit of housing. Only Rappaport’s work is useful for studying population
density,althoughhisworkisdonenumerically,andisnotlinkedtodatainaclosemanner.
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cost and expenditure shares, tax rates, and separate margins of substitution in consumption and in
bothtypesofproduction. Thesubstitutionmarginsreflectthreeseparatebehavioralresponsesthat
lead to higher densities, including the willingness of households to crowd into existing housing,
shifts in production away from land-intensive goods, and the construction of housing at greater
heights. Urban quantities depend on these substitution possibilities in a first-order manner, while
prices do not. Using a pre-set calibration of the United States economy from Albouy (2009),
our results suggest that substitution possibilities in the non-tradable sector, including housing, are
particularlyimportant.
Theanalyticalexercisemapsreduced-formelasticitiesoftenestimatedintheliterature,e.g.,of
local labor or housing supply, to more elementary structural parameters. This mapping reframes
partial-equilibriumshiftsinsupplyanddemandasgeneral-equilibriumresponsestoamenitychanges,
e.g.,anincreaseinlabordemandismappedtoanincreaseintradable-sectorproductivity. Thecal-
ibrated model implies that quantities respond much more than prices to differences in amenities
overthelongrun. Themodelproduceslarge(positive)labor-supplyelasticitiesthatareremarkably
consistent with estimates found in Bartik (1991) and Notowidigo (2012), and even larger (nega-
tive) labor-demand elasticities consistent with estimates in Card (2001). Moreover, our numbers
are consistent with the stylized fact that population density varies by an order of magnitude more
thanwagesandhousingcostsacrossmetroareas.
Our research complements work on agglomeration economies, which examines the reverse
relationship of how population affects amenities, especially productivity. For example, we model
howareaswithhigherqualityoflifebecomedenser,therebymakingthemmoreproductivethrough
agglomeration. Agglomeration then creates a multiplier effect, whereby higher density increases
productivity, bringing forth even higher density and productivity. We also consider the possibility
of greater density reducing quality of life through congestion. Under our calibration, we find that
these multiplier effects are potentially important, magnifying or dampening long-run behavioral
responsesupto15-percent.
We apply the model empirically by using it to relate observable prices to population density
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in 276 American metropolitan areas using Census data. The pre-set calibration does remarkably
well, explaining half of the variation in population density through quality of life and tradable-
sectorproductivitypredictedbytwosimplemeasuresofwagesandhousingcosts. Ourcalibration
fits the data better than those that ignore substitution possibilities, e.g., in consumption or non-
tradableproduction,orassumethattheyareallunitelastic,asinaCobb-Douglaseconomy.
If the calibration produces accurate elasticity values, variation in population density not ex-
plained by quality of life may substitute for missing data on land prices and help to identify pro-
ductivityinthetradableandnon-tradablesectors. Thisapproachsuggeststhatmetroareassuchas
NewYork,Chicago,andHoustonhaveratherproductivenon-tradablesectors,atleasthistorically.
On the other hand, metros such as San Francisco and Seattle have far less productive non-tradable
sectors,despitehavingveryproductivetradablesectors.
Our last exercise explores the relative importance of different amenities in explaining where
people live. A variance decomposition suggests that quality of life explains a greater fraction of
population density than does tradable-sector productivity, even though the latter varies more in
value and affects wage and housing costs more. This conclusion is reinforced if population den-
sity increases tradable-sector productivity or reduces quality of life. Productivity in non-tradables
explains density more than the other types of amenities, although this may have much to do with
how it is measured. We also simulate how population density might change if federal taxes were
made geographically neutral. This tends to increase the influence of tradable-sector productivity
andmultipliersfromagglomerationfeedback.
Thegeneral-equilibriummodeloflocation,withhomogenousagents,providesadifferentpoint
of view than dynamic partial-equilibrium models of location with heterogenous agents (e.g., Ken-
nan and Walker 2011). Partial-equilibrium approaches typically do not consider how wages and
housing costs depend on population sizes. Moreover, the focus of such work is to explain migra-
tiondecisionsovershortandmediumrunhorizons,whileweexaminepopulationdifferencesover
theverylongrun.
The rest of the paper is organized as follows: Section 2 introduces the model and discusses
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analytical results. Section 3 provides results from the calibrated model. Section 4 estimates long
run elasticities of labor and housing demand and supply. Section 5 provides new productivity
estimatesanddecomposesthedeterminantsofpopulationdensity. Section6concludes.
2 Locational Equilibrium of Quantities, Prices, and Amenities
2.1 Set-up
To explain how prices and quantities vary with amenity levels across cities, we use the model of
Albouy(2009a),whichaddsfederaltaxestothegeneral-equilibriumthree-equationRoback(1982)
model.Thenationaleconomycontainsmanycities,indexedbyj,whichtradewitheachotherand
share a homogenous population of mobile households. Households supply a single unit of labor
intheircityof residence;theyconsumeanumeraire tradedgoodxandanon-traded “home”good
y with local price pj.3 All input and output markets are perfectly competitive, and all prices and
quantitiesarehomogenouswithincities,thoughtheyvaryacrosscities.
Cities differ exogenously in three general attributes, each of which is an index meant to sum-
marize the value of amenities to households and firms: (i) quality of life Qj raises household
utility, (ii) trade-productivity Aj lowers costs in the tradable sector, and (iii) home-productivity
X
Aj lowerscostsinthenon-tradablesector.4
Y
Firms produce traded and home goods out of land, capital, and labor. Within a city, factors
receive the same payment in either sector. Land L is heterogenous across cities, immobile, and
receivesacity-specificpricerj. Eachcity’slandsupplyLj(rj)maydependpositivelyonrj.
Capital K is fully mobile across cities and receives the price ¯ı everywhere. The supply of
capital in each city Kj is perfectly elastic at this price. The national level of capital may be fixed
3Inapplication, thepriceofthehomegoodisequatedwiththecostofhousingservices. Non-housinggoodsare
consideredtobeacompositecommodityoftradedgoodsandnon-housinghomegoods.
4All of these attributes depend on a vector of natural and artificial city amenities, Zj = (Zj,...,Zj ), through
1 K
functionalrelationshipsQj =Q(cid:101)(Zj),AjX =A(cid:103)X(Zj),andAjY =A(cid:102)Y(Zj). Foraconsumptionamenity,e.g.,clement
weather,∂Q(cid:101)/∂Zk >0;foratrade-productionamenity,e.g.,navigablewater,∂A(cid:103)X/∂Zk >0;forahome-production
amenity, e.g., flat geography, ∂A(cid:102)Y/∂Zk > 0. It is possible that a single amenity affects more than one attribute or
affectsanattributenegatively.
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or depend on¯ı. Households N are fully mobile, have identical tastes and endowments, and each
supplies a single unit of labor. Household size is fixed. Wages wj vary across cities because
households care about local prices and quality of life. The total number of households is NTOT =
(cid:80) Nj,whichmaybefixedordeterminedbyinternationalmigration.
j
Households ownidentical diversifiedportfolios of landand capital, whichpay an incomeR =
(N )−1(cid:80) rjLj from land and I = (N )−1(cid:80) ¯ıKj from capital. Total income mj = wj +
TOT j TOT j
R+I varies across cities only as wages vary. Out of this income households pay a linear federal
incometaxτmj,whichisredistributedinuniformlump-sumpaymentsT. Householdpreferences
are modeled by a utility function U(x,y;Qj) which is quasi-concave over x, y, and Qj. The
expenditurefunctionforahouseholdincityj is
e(pj,u;Qj) ≡ min{x+pjy : U(x,y;Qj) ≥ u}.
x,y
Assume Q enters neutrally into the utility function and is normalized so that e(pj,u;Qj) =
e(pj,u)/Qj,wheree(pj,u) ≡ e(pj,u;1).5
Operating under perfect competition, firms produce traded and home goods according to the
functions Xj = Aj F (Lj ,Nj ,Kj ) and Yj = Aj F (Lj ,Nj,Kj ), where F and F are
X X X X X Y Y Y Y Y X Y
concave and exhibit constant returns to scale. We assume Hicks-Neutral productivity. Unit cost in
thetraded-goodsectorofcityj is
c (rj,wj,¯ı;Aj ) ≡ min{rjL+wjN +¯ıK : Aj F (L,N,K) = 1}.
X X X
L,N,K
Similartotherelationshipbetweenqualityoflifeandtheexpenditurefunction,letc (rj,wj,¯ı;Aj ) =
X X
c (rj,wj,¯ı)/Aj ,wherec (rj,wj,¯ı) ≡ c (rj,wj,¯ı;1). Asymmetricdefinitionholdsfortheunit
X X X X
costinthehome-goodsectorc .
Y
5Themodelgeneralizestoacasewithheterogenousworkersthatsupplydifferentfixedamountsoflaborifthese
workersareperfectsubstitutesinproduction,haveidenticalhomotheticpreferences,andearnequalsharesofincome
fromlabor.
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2.2 Equilibrium Conditions
Eachcitycanbedescribedbyasystemofsixteenequationsinsixteenendogenousvariables: three
pricespj,wj,rj,andthirteenquantitiesxj,yj,Xj,Yj,Nj,Nj ,Nj,Lj,Lj ,Lj ,Kj,Kj ,Kj . We
X Y X Y X Y
begin by having these depend on three exogenous attributes Qj,Aj ,Aj and a land supply func-
X Y
tionLj(rj). Inthisscenario,thesystemofequationshasablock-recursivestructure,allowingusto
first determine prices, where most researchers stop, then determine per-capita consumption quan-
tities, and finally, production quantities, including total population. This block-recursive structure
is broken if amenities are made endogenous to quantities, e.g., if Aj = Aj (Nj)α where Aj is
X X0 X0
trade-productivity due to fixed natural advantages, and (Nj)α is due to agglomeration economies.
Endogenizing amenities is more important for comparative statics, e.g., increasing Aj changes
X0
Nj, than measurement, where Nj may be treated as fixed so long as we are satisfied in measur-
ing the composition Aj (Nj)α. Throughout, we adopt a “small open city” assumption and take
X0
nationallydeterminedvariablesu¯,¯ı,I,R,T asgivenforanyindividualcity.6
2.2.1 PriceConditions
Since households are fully mobile, they must receive the same utility across all inhabited cities.
Higherpricesorlowerqualityoflifearecompensatedwithgreaterafter-taxincome,
e(pj,u¯)/Qj = (1−τ)(wj +R+I)+T, (1)
whereu¯ isthelevelofutilityattainednationallybyallhouseholds.
Firms earn zero profits in equilibrium. For given output prices, firms in more productive cities
6Inaclosedcity,wecouldinsteadtakeNj orKj asgiven,andendogenizefactorincomesRj orIj. Intheopen
cityweassumethatthefederalgovernment’sbudgetisgivenbyτ(cid:80) Njmj +T (cid:80) Nj =0,soacitywithaverage
j j
incomereceivesatransferwhichexactlyoffsetsitstaxes.
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mustpayhigherrentsandwages,
c (rj,wj,¯ı)/Aj = 1 (2)
X X
c (rj,wj,¯ı)/Aj = pj. (3)
Y Y
Equations (1), (2), and (3) simultaneously determine the city-level prices pj,rj, and wj for each
city as implicit functions of the three attributes Qj,Aj , and Aj . In equilibrium, these condi-
X Y
tionsprovideaone-to-onemappingbetweenunobservedcityattributesandpotentiallyobservable
prices,obviatingtheneedtoexaminequantities.
2.2.2 ConsumptionConditions
Indecidingtheirconsumptionquantitiesxj andyj,householdsfacethebudgetconstraint
xj +pjyj = (1−τ)(wj +R+I)j +T, (4)
where pj and wj are determined by the price conditions. Optimal consumption is determined in
conjunctionwiththetangencycondition
(∂U/∂y)/(∂U/∂x) = pj. (5)
As we assume preferences are homothetic, Qj does not affect the marginal rate of substitution.
Thus, in areas where Qj is higher, but pj is the same, households consume less of x and y in
equal proportions, holding the ratio y/x constant, similar to an income effect. Holding Qj con-
stant, increases in pj are compensated by increases in wj so that households reduce their relative
consumptiony/xduetoapuresubstitutioneffect.
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2.2.3 ProductionConditions
With prices and per-capita consumption levels accounted for, Levels of output Xj,Yj, employ-
ment Nj,Nj ,Nj, capital Kj,Kj ,Kj , and land Lj,Lj ,Lj are determined by eleven equations
X Y X Y X Y
describing production and market clearing. The first six express conditional factor demands using
Shephard’s Lemma. Because of constant returns to scale and Hicks neutrality, the derivative of
the uniform unit-cost function equals the ratio of the relevant input, augmented by city-specific
productivity,tooutput:
∂c /∂w = Aj Nj /Xj (6)
X X X
∂c /∂r = Aj Lj /Xj (7)
X X X
∂c /∂i = Aj Kj /Xj (8)
X X X
∂c /∂w = Aj Nj/Yj (9)
Y Y Y
∂c /∂r = Aj Lj /Yj (10)
Y Y Y
∂c /∂i = Aj Kj /Yj. (11)
Y Y Y
The next three conditions express the local resource constraints for labor, land, and capital under
theassumptionthatfactorsarefullyemployed.
Nj = Nj +Nj (12)
X Y
Lj = Lj +Lj (13)
X Y
Kj = Kj +Kj (14)
X Y
Equation(13)differsfromtheothersaslocallandisdeterminedbythesupplyfunction,
Lj = Lj(rj), (15)
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