The Economics of Space 433: Lectures 11 and 12 Spatial ...€¦ · The Economics of Space: Lecture...

The Economics of Space 433: Lectures 11 and 12

Spatial Forces: Agglomeration and Amenity Spillovers

Costas Arkolakis1

1Yale University

17 February 2020

The Economics of Space: Lecture 11-12, c© Costas Arkolakis 1

Recap: Spatial Forces

I This class focuses on spatial forces: productivity and amenity spillovers

I Recall, productivity in i written Ai = AiLαi

I We could (as we have in the past) assumed α = 0.I Where does α > 0 come from, and why does it matter?

I Recall, amenity in i written ui = uiL−βi

I We could (as we have in the past) assumed β = 0.I Where does β > 0 come from, and why does it matter?

I In this lecture we will review the role of spatial forces on determining the spatialequilibrium

I These forces play the role of centrifugal and centripetal forcesI They intensify agglomeration and dispersion in a region

I They shape not only the intensity of economic activity but for many urbaneconomists are the key to understanding how economic activities in modern cities(or the cities themselves thereof) are shaped

The Economics of Space: Lecture 11-12, Introduction c© Costas Arkolakis 2

Recap: Spatial Forces

I This class focuses on spatial forces: productivity and amenity spillovers

I Recall, productivity in i written Ai = AiLαi

I We could (as we have in the past) assumed α = 0.I Where does α > 0 come from, and why does it matter?

I Recall, amenity in i written ui = uiL−βi

I We could (as we have in the past) assumed β = 0.I Where does β > 0 come from, and why does it matter?

I In this lecture we will review the role of spatial forces on determining the spatialequilibrium

I These forces play the role of centrifugal and centripetal forcesI They intensify agglomeration and dispersion in a region

I They shape not only the intensity of economic activity but for many urbaneconomists are the key to understanding how economic activities in modern cities(or the cities themselves thereof) are shaped

The Economics of Space: Lecture 11-12, Introduction c© Costas Arkolakis 3

Roadmap

I Evidence for Agglomeration Economies

I Understanding Agglomeration

I Productivity and Spatial Spillovers

I Measuring Agglomeration Economies

I Preliminary Evidence on Amenities

I Understanding and Measuring the Effect of Amenities

I Estimating Productivities and Amenities

The Economics of Space: Lecture 11-12, Evidence for Agglomeration Economies c© Costas Arkolakis 4

Agglomeration Economies

I When we assume α > 0, we assume agglomeration economiesI An economy where the productivity in a location is affected by local or nearby

employmentI Agents (in this case both consumers and firms) benefit by locating near other

agents

I Firms benefit through lower marginal cost, consumers through lower prices

I Goals:I Demonstrate evidence for agglomeration economiesI Understand how agglomeration economies workI Illustrate the importance of agglomeration economies in the spatial model





agentsI Firms benefit through lower marginal cost, consumers through lower prices



Agglomeration in the Data

I As of 2010, largest 100 US cities accounted for 20% of the population butaccounted for 0.4% of the land areaI Alternatively, census defined ‘‘urban’’ areas accounted for 80.7% of the population

but accounted for 3% of the land areaI Globally, urban areas account for 54.3% of population but 2.4% of land area

I In the US, per capita GDP increasing in log population density across metro areas(Glaeser 2010)


Evidence: Wage vs. Population (2015)

Data taken from the census.gov and bls.govThe Economics of Space: Lecture 11-12, Evidence for Agglomeration Economies c© Costas Arkolakis 10

Evidence: County Average Wages (2015)

Data from bls.govThe Economics of Space: Lecture 11-12, Evidence for Agglomeration Economies c© Costas Arkolakis 11

Evidence: County Employment (2015)

Data from census.govThe Economics of Space: Lecture 11-12, Evidence for Agglomeration Economies c© Costas Arkolakis 12

Evidence: Zipcode Income per Capita (2015)

Data from the American Community Survey


Evidence: Zip code Employment (2015)

Data from the American Community Survey


G-Econ data: Employment (top) GDPpc (bottom) (2005)


Agglomeration: Historical Diversion

I Adam Smith 1776 pin factory specialization exampleI Each worker alone could produce max 20 pins per day, a factory with ten people

produces 48,000 a dayI =⇒ productivity gain of a factor of 240

I Marshall 1890: knowledge spillovers, input-output linkages, labor marketinteractionsI Knowledge spillovers: faster accumulation and dispersion of skills, generation of

new knowledgeI Input-output linkages: shared intermediate inputs with increasing returnsI Labor market interactions: lower risk for a localized industry with wider labor

market for skills


Roadmap








The Economics of Space: Lecture 11-12, Understanding Agglomeration c© Costas Arkolakis 18

Understanding Solutions of Economics withAgglomeration

I In our model, the effects of agglomeration are subtleI Depending on magnitude of α and β, congestion may overwhelm agglomeration

I We write productivity Ai = AiLαi

I Cost to produce yi in units of labor:

yi = AiLαi · Li =⇒ Li =

(yi

Ai

) 11+α

I For α > 0, we see scaling yi by a factor of 2 scales L by a factor of 21

1+α < 2I So agglomeration causes returns to scale in each location


Agglomeration and Many Solutions

I Agglomeration can lead to fascinating (or distressing...) economic outcomes

I Imagine our spatial model, this time without welfare equalization. Fix twosymmetric locations with symmetric trade costs.

I Now assume large agglomeration economies. What are the solutions?

I By symmetry, there are 3.I (1): Everyone in location 1I (2): Everyone in location 2I (3): Half of the people in location 1, half in location 2

I What does this mean for practical economic outcomes?I There could be tipping points of economic activity





I Now assume large agglomeration economies. What are the solutions?I By symmetry, there are 3.

I (1): Everyone in location 1I (2): Everyone in location 2I (3): Half of the people in location 1, half in location 2







I (1): Everyone in location 1

I (2): Everyone in location 2I (3): Half of the people in location 1, half in location 2







I (1): Everyone in location 1I (2): Everyone in location 2

I (3): Half of the people in location 1, half in location 2







I (1): Everyone in location 1I (2): Everyone in location 2I (3): Half of the people in location 1, half in location 2



Detroit Population


Agglomeration and Many Solutions: 2 Location Example

I Consider our spatial model (with welfare equalization). Lets consider the case of:I 2 symmetric locations in terms of geography: A1 = A2, u1 = u2

I Trade costs symmetric, τ12 = τ21 ≡ τ > 1I α > 0 and β = 0I Normalize aggregate labor supply to 1, L = 1

I Claim: We can write system as Lσ−1

2σ−1 (1−α(σ−1))

i = W 1−σ∑

j τ1−σij L

σ−12σ−1 (1+ασ)

jI To be proven on your homework



I Consider our spatial model (with welfare equalization). Lets consider the case of:I 2 symmetric locations in terms of geography: A1 = A2, u1 = u2

I Trade costs symmetric, τ12 = τ21 ≡ τ > 1I α > 0 and β = 0I Normalize aggregate labor supply to 1, L = 1

I Claim: We can write system as Lσ−1

2σ−1 (1−α(σ−1))

i = W 1−σ∑

j τ1−σij L

σ−12σ−1 (1+ασ)

jI To be proven on your homework



I Equilibrium is solutions of: Lσ−1

2σ−1 (1−α(σ−1))

i = W 1−σ∑

j τ1−σij L

σ−12σ−1 (1+ασ)

j

I Some algebra, plus the fact L1 + L2 = L =1, yields

L1︸︷︷︸supply

=[W 1−σ

(Lσγ2

1 + τ 1−σ (1− L1)σγ2

)] 1σγ1 ≡ f (L1)︸︷︷︸

demand

with

L2 =[W 1−σ

(Lσγ2

2 + τ 1−σLσγ2

1

)] 1σγ1 ⇐⇒ W 1−σ =

(1− L1)σγ1((1− L1)σγ2 + τ 1−σLσγ2

1

)and σ = (σ − 1) / (2σ − 1) , γ1 = 1− (σ − 1)α, γ2 = 1 + ασ

I f (L1) is the demand for labor in location 1. L1 is the supplyI We define f (L1)− L1 as the excess demand function (i.e. demand-supply)


Excess Demand Function Analysis

I We can analyze equilibrium properties by analyzing the excess demandI If f (L1)− L1 = 0 then this is an equilibriumI If f (L1) > L1 there is an excess demand (and excess supply if <)

I We can also introduce a new concept borrowing from General Equilibrium Theory:StabilityI Start from equilibrium. Study excess demand of country 1 to see if stableI Assume you increase employment. If f (L1)− L1 < 0 equilibrium is stableI Assume you increase employment. If f (L1)− L1 > 0 equilibrium is unstable

I Intuition for unstable solution: if more employment means more demand for laborwages increases and more workers flood inI Clearly unstable!


Roadmap








The Economics of Space: Lecture 11-12, Productivity and Spatial Spillovers c© Costas Arkolakis 43

Spatial Knowledge Spillovers

I Another idea is that agglomeration comes from spatial knowledge diffusion(Fujita Ogawa ’82)I Knowledges diffuses amongst people and distance is an impediment to that

diffusionI How do we model it? Following Lucas Rossi-Hansberg ’02 and Allen Arkolakis Li

’16 we can writeAi = Ai

∑j

e−τAij Lηj

with τAij > 0 for i 6= j , τAii = 0, and η ≥ 0I You may notice that τij = +∞ for i 6= j takes us back to Ai = AiL

ηi

I In other words here not only local population is important but also nearby


Spatial Knowledge Spillovers and Solutions

I Consider our spatial model (with welfare equalization). Lets consider the case of:I 2 symmetric locations A1 = A2, u1 = u2

I Trade costs symmetric, τ12 = τ21 ≡ τ > 1I And also ui = uiL

−βi where β > 0

I Normalize aggregate labor supply to 1, L = 1

I It can be shown that the system can be written as

Lσ(1+βσ)i = W 1−σ

∑Aσ(σ−1)i τ 1−σ

ij A(1−σ)(σ−1)j L

σ(1−(σ−1)β)i

Ai = Ai

∑j

e−τAij Lηj

with τAij > 0 for i 6= jI Same system as before but now Ai is endogenous so you need another equationI Using Theorem 1 in Allen, Arkolakis, Li ’15 you can prove that there is a unique

equilibrium if η ≤ β and 0 < β < (2σ − 1)−1


Spatial Spillovers with No Diffusion


Spatial Spillovers with Some Diffusion


Strength of Spatial Spillovers

I Remarkably, despite the fact that we have spatial spillovers, the possibility ofmultiple equilibria diminishesI The agglomeration forces diffuse across space


Roadmap








The Economics of Space: Lecture 11-12, Measuring Agglomeration Economies c© Costas Arkolakis 50

Challenges in Measuring Agglomeration

I Agglomeration is highly endogenous, so clean estimation is difficultI For example, observe high wages in areas with high population density (e.g., New

York city)I Are wages high because people move there and are more productive because of

productivity spillovers?I Or do people move there because the wages are high?

I All things considered, measuring agglomeration economies is extremely hardI Need some exogenous reliable variation. But there are not many


Measuring Agglomeration Economies

I In a handful of occasion exogenous variations is feasibleI How does the measurement proceed? Recall from the Rosen-Roback model we

have W = Aiui ⇐⇒ Li = (Ai ui )1

β−α W1

α−β . Since

wi = AiLαi =⇒ wi = Ai (Ai ui )

αβ−α W

αα−β

I We therefore can write (similar relationships hold in our general spatial model)

lnwi =β

β − αln Ai +

α

β − αln ui +

α

α− βln W

I In differences

∆i lnwi =

(β

β − α

)∆i ln Ai +

(α

β − α

)∆i ln ui


Empirical Specification for Differential Effects

I Adding measurement error in the regression and finding proper exogenousvariation we can run

∆i lnwi =

(β

β − α

)∆i ln Ai +

(α

β − α

)∆i ln ui + εi

I By obtaining coefficient ββ−α we can measure the (differential) impact of

investments in the presence of agglomeration and dispersion spilloversI There is an apparent issue of the correlation of the shocks across markets

I Notice that in this model you can prove that

Li =

(ui Ai

) 1β−α∑

i ′

(ui ′Ai ′

) 1β−α

L

I Benefits of employment in one region can be losses in others


Can we Measure Aggregate Effects

I Can we measure the aggregate impact of ∆Ai?I A related approach is followed by Kline and Moretti ’14I ∆Ai is an investment in infrastructure by the Tennessee Valley Authority that only

affected some countiesI They estimate effects of ∆Ai using the approach above

I To measure aggregate effects they use the Rosen-Roback model: solve modelentirely as function of parameters

W =

[∑j(Aj uj)

1β−α

]β−α

Lβ−a

I In the presence of spatial frictions (e.g trade costs) these expressions are morecomplicated

I We can still explore effects of ∆Ai by tracing effects through trade networkI We will see later how this is done following the analysis of Adao et al ’18


Roadmap








The Economics of Space: Lecture 11-12, Preliminary Evidence on Amenities c© Costas Arkolakis 56

What Are Consumption Amenities?

I Amenities are the compensating differential that explains the locations choice

I Harder to measure: While productivities can be directly measured using outputdata, amenities are backed out to explain location choices of workers

I Examples of amenities (Glaeser Kolko Saiz ’01)I Consumer ServicesI Public ServicesI Housing StockI AestheticsI Climate and Environmental Factors

I Given the resurgence of life in metropolitan centers they are key to ourunderstanding of the economics of space


Preliminary Evidence on Amenities and the City (Glaeseret. al. 2001)


Preliminary Evidence on Amenities and the City (Glaeseret. al. ’01)


Mean Temperature

Mean annual daily temperature. Darker colors represent higher values. Data fromNorth America Land Data Assimilation System Daily Air Temperatures and Heat Index

(1979-2011).


Distance to an Ocean

Distance, in km, to an ocean. Darker colors represent higher values. Data generatedusing QGIS.


Food and Beverage Establishments

Food and Beverage establishments (restaurants and bars) per capita, per squarekilometer. Darker colors represent higher values. Data from County Business Patterns,

2010.


Art and Entertainment Establishments

Arts, Entertainment, and Recreation establishments per capita, per square km. Darkercolors represent higher values. Data from County Business 2010.


Violent Crime

Violent crime [murder, rape, robbery, (aggravated) assault, arson] per capita. Darkercolors represent higher values. Data from National Archive of Criminal Justice,

2010-2014.


Student Teacher Ratio

Student-teacher ratio in public schools. Darker colors represent higher values. Datafrom National Center for Education Statistics, 2000-2010. Missing data due to ‘‘poor

quality’’.


Roadmap



I Productivity & Spatial Spillovers



I Understanding and Measuring the Effects of Amenities


The Economics of Space: Lecture 11-12, Understanding and Measuring the Effects of Amenities c© Costas Arkolakis 66

How to Analyze The Role of Urban Amenities?

I To analyze the role of Urban amenities researchers typically use the urban model.Recall than in Rosen-Roback

Li =

(ui Ai

) 1β−α∑

i ′

(ui ′Ai ′

) 1β−α

L (1)

I We can form a regression

ln Li =1

β − αln ui +

1

β − αln Ai + c

where c is a regression fixed effect ‘absorbing’ common terms




ln Li =1

β − αln ui +

1

β − αln Ai + c

in other words regress population in a location with location characteristics and afixed effect

I A large number of questions that we can answer. For example:I What are the amenities that determine location choice?I How do different groups make differential location decisions?I What determines the fall or the rise of major city centers

I Our ability to answer questions is determined by the availability of micro-data




ln Li =1

β − αln ui +

1

β − αln Ai + c


I A large number of questions that we can answer. For example:

I What are the amenities that determine location choice?I How do different groups make differential location decisions?I What determines the fall or the rise of major city centers





ln Li =1

β − αln ui +

1

β − αln Ai + c


I A large number of questions that we can answer. For example:I What are the amenities that determine location choice?

I How do different groups make differential location decisions?I What determines the fall or the rise of major city centers





ln Li =1

β − αln ui +

1

β − αln Ai + c


I A large number of questions that we can answer. For example:I What are the amenities that determine location choice?I How do different groups make differential location decisions?

I What determines the fall or the rise of major city centers





ln Li =1

β − αln ui +

1

β − αln Ai + c





Amenities and Urban Life in the Advent of Big Data

I An in-depth, micro, understanding of amenities can be derived using detailedmicro dataI For example researchers have used quality of restaurants and other entertainment

services at the neighborhood level or other disaggregated location (Davis, Dingel,Monras, Morales ’15, Couture Handbury ’15)

I A more in-depth understanding requires to measure the type of varieties consumedin the city (Handbury Weinstein ’15, Handbury ’17)

I Digitization, data-handling, frontier computer-science techniques (machine anddeep learning) key to this analysis (Glaeser, Kominers, Luca, Naik ’18)


Roadmap



I Productivity & Spatial Spillovers





The Economics of Space: Lecture 11-12, Estimating Productivities and Amenities c© Costas Arkolakis 76

Estimating Productivities and Amenities

I The estimated amenities are the residual in the model so that given data it‘‘predicts’’ welfare equalization

I Using data of wages and labor we can estimate productivities, Ai , and amenities,ui , using our model

I We explain how this can be done

I We start by explaining how this work in the Rosen Roback modelI There recall that wi = Ai = AiL

αi . We will use also the labor supply function 1

I Thus, if we have data on wi , Li and a measure of α then we can ‘‘solve’’ for Ai asa residual:

Ai =wi

LαiI Also from 1 we can solve ui using Ai and Li

ui =(Li )

β−α(Ai

)[∑

i′(ui′ Ai′

) 1β−α

]β−αLβ−α


Intuition and the Procedure

I We start by explaining how this work in the Rosen Roback modelI There recall that wi = Ai = AiL

αi . We will use also the labor supply function 1

I Thus, if we have data on wi , Li and a measure of α then we can ‘‘solve’’ for Ai asa residual:

Ai =wi

LαiI Also from 1 we can solve ui using Ai and Li (assume β > α)

ui =(Li )

β−α(Ai

)[∑

i′(ui′ Ai′

) 1β−α

]β−αLβ−α

I Intuition:I higher wages (adjust by spillovers) mean higher productivitiesI Conditional on productivity, higher population, means higher amenity


Estimating Amenities and Productivities in a Model withSpatial Connections

I How do we estimate productivities and amenities when spatial spillovers exist?I We need some data: wi ,Li and trade costs τijI Census has remarkably good information for income and population.I We construct trade costs using the transportation network, tij

I We can use the fundamental welfare equalization condition

W =wi

Piui

I For amenities, take any two locations: ui/uj =Pi/Pj

wi/wj

I Measurements of wages are necessary.I What about Pi? We can exploit the fact that is a function of wages and trade costs

in the modelI How we back out productivities? Notice

λij =

(wiτijAi

)1−σ

∑i

(wiτijAi

)1−σ

I Given wi ,Li and trade costs τij find Ai to solve wiLi =∑

i λijwjLj





W =wi

Piui


wi/wj


in the model

I How we back out productivities? Notice

λij =

(wiτijAi

)1−σ

∑i

(wiτijAi

)1−σ


i λijwjLj





W =wi

Piui


wi/wj


in the modelI How we back out productivities? Notice

λij =

(wiτijAi

)1−σ

∑i

(wiτijAi

)1−σ


i λijwjLj


Estimating Amenities and Productivities: The Procedure

I We will now formally execute the procedure to estimate amenities, ui

I This procedure is detailed in Allen Arkolakis ’14I It gives us Ai , ui .I We can break them into their exogenous components given data for Li and

Ai = AiLαi , ui = uiL

−βi

I We will now present correlations of the estimated ui with observed spatialvariation


Estimated Amenities (Allen Arkolakis ’14)

Decile plot of estimated amenities ui . Darker colors represent higher values.


Estimated ui and Amenity Measures

I For each of the previous features (mean temperature, distance to the ocean, etc)x , estimate

ui = axi + γS + εi

ui = app(xi ) + γS + εi

with γS state fixed effects, p(xi ) the percentile of the feature x , and εierrors.

Measure of Amenity a ap

Mean Temperature (F) 0.020 (0.001) 0.52 (0.044)Distance to an Ocean (km) -0.0002 (2.4e-05) -0.42 (0.043)

Food Establishments per cap per sq km -0.34 (0.12) -0.13 (0.018)Art and Entertainment per cap per sq km -6.47(1.55) -0.17 (0.017)

Violent Crime per cap -0.38 (0.063 ) -0.035 (0.016)Student Teacher Ratio 0.024 (0.002) 0.43 (0.021)



I For each of the previous features (mean temperature, distance to the ocean, etc)x , estimate

ui = axi + γS + εi

ui = app(xi ) + γS + εi

with γS state fixed effects, p(xi ) the percentile of the feature x , and εierrors.

Measure of Amenity a ap

Mean Temperature (F) 0.020 (0.001) 0.52 (0.044)Distance to an Ocean (km) -0.0002 (2.4e-05) -0.42 (0.043)

Food Establishments per cap per sq km -0.34 (0.12) -0.13 (0.018)Art and Entertainment per cap per sq km -6.47(1.55) -0.17 (0.017)

Violent Crime per cap -0.38 (0.063 ) -0.035 (0.016)Student Teacher Ratio 0.024 (0.002) 0.43 (0.021)



I All features f together:

ui =F∑

f =1

af p(xfi ) + γS + εi

with γS state fixed effects, p(xfi ) the percentile of the feature xf , and εi errors.

Measure of Amenity af

Mean Temperature (F) 0.19 (0.027)Distance to an Ocean (km) -0.22 (0.025)

Food Establishments per cap per sq km -0.13 (0.009)Art and Entertainment per cap per sq km 0.02 (0.008 )

Violent Crime per cap 0.01 (0.008)Student Teacher Ratio 0.40 (0.012)


References

I Agglomeration Economics, Glaeser 2010.

I The Wealth of Nations, Smith Edwin. New York, N.Y.: Bantam Classic, 2003.

I Principles of Economics: Marshall 1890

I Multiple equilibria and structural transition of non-monocentric urban configurations, Fujita Ogawa1982.Regional Science and Urban Economics, Elsevier, vol. 12(2), pages 161-196, May.

I On the Internal Structure of Cities, Lucas Rossi-Hansberg 2002. Econometrica, Volume70, Issue 4 July 2002.Pages 1445-1476

I Optimal City Structure, Allen Arkolakis Li 2016. Mimeo.On the Existence and Uniqueness of TradeEquilibria, Allen Arkolakis Li 2015. Mimeo

I Local Economic Development, Agglomeration Economies and the Big Push: 100 Years of Evidence from theTennessee Valley Authority, Kline Moretti 2014. Quarterly Journal of Economics, 129(1), 275-331

I Spatial Linkages, Global Shocks, and Local Labor Markets, Adao Arkolakis Esposito 2018. Mimeo

I Consumer City, Glaeser, Kolko, Saiz 2001. Journal of Economic Geography, Oxford University Press, vol.1(1), pages 27-50 January 2001.

I How Segregated is Urban Consumption?, Davis Dingel Monras Morales 2017. NBER Working Papers23822, National Bureau of Economic Research, Inc.

I Urban Revival in America, 2000 to 2010, Couture Jessie 2017. No 24084, NBER Working Papers, NationalBureau of Economic Research, Inc.

I Goods Prices and Availability in Cities, Handbury Weinstein 2015. Review of Economic Studies, OxfordUniversity Press, vol. 82(1), pages 258-296.

I Big Data and Big Cities: The Promises and Limitations of Improved Measures of Urban Life. GlaeserKominers Luca Naik 2018. Economic Inquiry, Vol. 56, Issue 1, pp. 114-137 2018.


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