summerschool‘spatial economicsandimperfect markets ... · wage diﬀerential -0.0025*** -0.0179...

Summer School ‘Spatial Economics and Imperfect Markets: Empirical Researches’, July 2013 1

Summer School ‘Spatial Economics and Imperfect Markets:

Empirical Researches’

Lect 3: Agglomeration and urban systems

Kristian BehrensCanada Research Chair in Regional Impacts of Globalization

UQAM; CIRPEE; and CEPR


Making location choices endogenous.

Until now, people do not move. While this is an assumption that can be justified

in an international context – where mobility remains relatively low – it is much

harder to defend in a regional context where people are much more mobile.

There is, e.g., a lot of mobility in the US. Gross migration flows across counties

are substantial: 47,236,430 million people moved between counties from 1990 to

2000. However, the net migration is ‘only’ 6,027,648. Furthermore, the bulk of

relocations is within states (average of 502,986 intra-state vs 8,665 inter-state)

Note that, as for trade flows, there is a gravity relationship for migration flows

(first noticed by Ravenstein, 1858). When estimating it, amenity differences, wage

differences, and rent differences have the expected sign. Wage differences are very

strongly correlated with rent differences (standard equilibrium outcome).


Coefficient Beta coeff. T-stat p-val

Origin population 0.0001*** 0.1035 12.33 0.000

Destination population 0.0001*** 0.0854 16.70 0.000

Distance -0.0545*** -0.0833 -26.24 0.000

Amenity differential -1.4003*** -0.0078 -7.42 0.000

Wage differential -0.0025*** -0.0179 -7.30 0.000

Median value differential 0.0001*** 0.0128 5.26 0.000

R2 0.0182

N 721,231

Table 1: Determinants of county-to-county migration flows in the US, 1990–2000


What about firms?

Population movements are important. Note, however, that firms do not move

much (e.g., Duranton and Puga, 2001, for France). Most firm relocations are

short distance (within metro areas) or internationally (going to China). Not much

in between and the few that move inter-state or inter-provincial do so due to

life-cycle considerations.

The bulk of firm ‘movement’ is due to entry and exit, and there is quite a lot going

on along that margin.

Following developments in ‘new economic geography’ in the wake of Krugman

(1991) and lots of other authors, we now make location choices endogenous.

The spatial equilibrium is the outcome where agglomeration forces – pushing to-

wards the spatial concentration of firms and workers – are balanced by dispersion

forces – pushing against the spatial concentration.


0.0

5.1

.15

.2kd

ensi

ty

−10 −5 0 5 10ln(relocation distance)

Figure 1: Kdensity of firm relocation distances, Canada 2001–2005.

(Behrens, 2012)


The theory of agglomeration is now very well understood. People and firms con-

centrate spatially to exploit various agglomeration economies:

– ‘natural’ locational advantages

– sharing, matching, and learning externalities (Duranton and Puga, 2004)

– large local markets (Krugman, 1991)

Not all economic activity concentrates in a single location because of

– ‘natural’ locational advantages

– rising urban costs (i.e., land prices) since land in any location is in fixed supply

– dispersed demand to be served and tougher competition in larger markets

Little work has until now tried to quantity the trade-off between agglomeration

and dispersion forces and how they depend on spatial frictions: the costs of moving

goods, people, and ideas across space.


The key problems.

[1] Equilibrium multiplicity: NEG models usually have multiple equilibria. It

is a priori unclear which equilibrium is the relevant one. This problem can

be solved with a calibrated model that takes the observed distribution of

population as the initial equilibrium.

[2] Catastrophic change: Small changes in parameter values (e.g., trade costs)

can have large impacts on population distributions (Krugman, 1991). This

makes these models unstable and hard to work with numerically. The fix is

to add heterogeneity to make the models ‘smooth’.

[3] Replicability of key stylized facts: Models need to be able to cope with key

stylized facts linked to cities, trade, and migration. For example, most ship-

ments are local, the size distribution of cities exhibits strong regularities, net

migration flows are way smaller than gross migration flows, etc.

Only few serious attempts at developing quantifyable NEG models (Mion, 2004;

Hanson, 2005; Redding and Sturm, 2008; Combes and Lafourcade, 2011).


Theoretical considerations.

I now extend the model we have seen in the previous lectures to: (i) include cities;

and (ii) allow for free mobility of utility-maximizing agents. The key questions I

then look at are:

[1] How important are spatial frictions in shaping spatial economic structure (city

size distribution; individual city sizes)? See also Desmet and Rossi-Hansberg

(2012), and Redding (2012).

[2] How important are spatial frictions for macro outcomes (productivity advantage

of cities; toughness of competition in cities)? See also Del Gatto et al. (2012), and

Holmes et al. (2012).

We know that lower trade frictions have a dispersive effect in two-region models of

NEG with urban costs (Helpman, 1998). We also know that lower urban frictions

have an agglomerative effect. Yet, we know to date very little about magnitudes.

This is where quantitative NEG models become important.


Introducing urban structure.

I look at the simplest case of monocentric cities. As I show later, this is not a bad

approximation provided we have a degree of freedom in the model.

Assume that there are K cities. Total population L ≡ ∑r Lr is given and fixed,

endogenously determined mass Lr of identical workers in city r.

Cities are monocentric, disk-shaped, land used for housing only.

Each agent consumes one unit of land, is endowed with hr units of time (used

for work and commuting to the CBD). There are cross-city differences in labor

supplies and incomes, but no within-city heterogeneity (see Lecture 1).

Commuting is costly and reduces labor supply (Murata and Thisse, 2005). Effec-

tive labor supply of a worker at distance |x| from the CBD is:

sr(x) = hre−θr|x|, θr > 0


Total effective labor supply at the CBD:

Sr(Lr) =2πhr

θ2r

[1−

(1 + θr

√Lr/π

)e−θr

√Lr/π

]

with S′r > 0 and S′′

r < 0.

Efficiency loss due to commuting is increasing in θr (urban frictions specific to

each city, which is important in the empirical application).

The within-city spatial equilibrium requires agents to be indifferent across loca-

tions. Being closer to the CBD reduces commuting costs, which must thus increase

land rents that agents pay. In equilibrium, wages − commuting costs − land rents

are equalized within each city.

Workers own equal shares of land, equal claims to aggregate profits:

Er ≡ (wr − commuting costs− land rent) +ALRr

Lr+

Π

L


Timing of the model.

0. Mobile workers/consumers choose utility-maximizing locations

1. Firms decide whether or not to enter (entry occurs until expected profits

are zero). Entrants in r pay a sunk cost F , ‘discover’ a variety, draw their

productivity 1/m from the distribution Gr

2. Entrants decide whether or not to produce

3. Sufficiently productive firms ‘survive’ and maximize (domestic) operating

profit

4. (Open economy only) Sufficiently productive firms export and maximize op-

erating profits from export markets

Steps 1–4 are the same as in the trade model (modulus the difference in Lr and

Sr). Taking the distribution of population as given, we can work easily through

the model to get results that are very similar to what we had before.


One important aspect is the indirect utility in different cities, since location deci-

sions will be based on that utility. It can still be expressed as follows:

Ur =κ1

Λr

, where Λr =κ2τrrα

Lr

Srmd

r

is the expenditure-weighted average markup faced by consumers in city r. Cities

with high urban costs – low average labor supply – will have ceteris paribus less

firms and thus higher markups.

[1] Higher productivity cutoff (1/mdr) → agglomeration force

[2] Lower effective labor supply per capita (Sr/Lr) → dispersion force

In standard models where agents are identical, in equilibrium these two forces are

balanced to ensure people get the same utility regardless of where they are.


Migration decisions are based on differences in utility that can be achieved in

different cities. Utility Ur above is what I refer to as market utility since it is

based on prices, rents and wages only.

We know from empirical studies that two other important factors influence migra-

tion decisions: (i) local amenities (both observed and unobserved ones); and (ii)

idiosyncratic taste differences across heterogeneous individuals.

I extend the model to take these differences into account. More precisely, assume

that mobile workers have heterogeneous tastes for locations (Tabuchi and Thisse,

2002; Murata, 2003).

The linear random utility of worker ν is given by

V νr = Ur +Ar + ξνr

where Ar are location-specific ‘amenities’ (both observed and unobserved) that

are valued identically; and ξνr is an idiosyncratic term.


Under a double exponential distribution of ξνr , individual ν’s choice probability for

city r is given by

Pr = Pr

(V νr ≥ max

s 6=rV νs

)=

exp((Ur +Ar)/β)∑s exp((Us +As)/β)

,

where β is linked to taste heterogeneity.

A spatial equilibrium is a fixed point where

Pr =Lr∑s Ls

=Lr

L.

In words, given the current distribution of the population, the share of people who

optimally choose city r is equal to the share of people living in city r.

Note that this is a steady state where net migration is nil. Of course, there can

be gross migration that is compatible with that steady state.


Taking the model to the data.

We use data for 356 continental US metropolitan statistical areas (MSAs) for the

year 2007. We first estimate urban frictions (θr) and trade frictions (γ).

[1] To obtain the city-specific parameters θr, we use

Lrhr

hr

=2π

θ2r

[1−

(1 + θr

√Lr/π

)e−θr

√Lr/π

], (1)

where Sr = Lrhr.

We compute hr as the average number of hours worked per week in MSA r. The

gross labor supply per capita, hr, which is the endowment of hours available for

work and commuting, is constructed as the sum of hr and hours per week spent

for travel-to-work commuting. Given hr, hr, as well as city size Lr, the above

equation can be uniquely solved for the city-specific commuting parameter θr.


23

45

67

89

log(

mod

el−

base

d A

LR)

16 17 18 19 20 21 22 23log(observed adjusted ALR)

Figure 2: Simulated versus observed aggregate land rents in US MSAs

(Behrens, Mion, Murata, and Suedekum, 2012)


The simple monocentric city model might not be the most appropriate specifica-

tion. Large MSAs (e.g., Los Angeles, Atlanta etc.) are usually polycentric.

Theory predicts that as cities grow, they develop secondary business centers to

reduce the average commuting distance (e.g., Lucas and Rossi-Hansberg, 2002).

This raises efficiency per unit of distance commuted and should work as if θr were

lower in our model.

Correlation between θr and the number of employment centers in each MSA for

the year 2000 (Arribas-Bel and Sanz Gracia, 2010) is −0.4282, while the Spearman

rank correlation is −0.5643.

Hence, our monocentric model with city-specific commuting technology captures

the tendency that polycentric cities are more efficient for commuting (i.e., they

have lower per-unit distance commuting costs conditional on the size of the city).


[2] Trade frictions, γ, are estimated from a standard gravity equation system (see

Lecture 2):

Xrs = NEr Ls

∫mx

rs

0prs(m)qrs(m)dGr(m)

= SrLsτ−krs τk+1

ss (ws/wr)k+1

wr

(md

s

)k+1(µmax

r )−1

,

so that

lnXrs = const.+ ξr + ξs − γkdrs + εrs

Here, for simplicity, we just implement the fixed-effects estimation (since the full

structural version requires solving 356 non-linear equations each time the objective

function is evaluated, which is computationally extremely heavy).

We estimate γk = 1.29 using CFS state-level data for 2007 (the CFS metro-level

data, which has some issues, is used by Duranton et al., 2011).

We again consider all that follows for a given value of k.


[3] To solve for the market outcome (given Lr), plug k, τrs, θr, Lr and Sr into

the GE conditions (zero expected profits, labor market clearing, aggregate budget

constraints):

µmaxr =

∑

s

Lsτrs

(τssτrs

ws

wrmd

s

)k+1

Sr

Lr

1

(mdr)

k+1=∑

s

Ssτrr

(τrrτsr

wr

ws

)k1

µmaxs

Ideally, we would solve for wages, wr, and cutoffs, mdr .

Technological possibilities µmaxr , though exogenous are not observable. So we turn

the problem around. We use data on MSA GDP per capita (proportional to 1/mdr

under the Pareto distribution) to obtain wr and µmaxr consistent with equilibrium.

Having solved for the market outcome, we can construct Ur.


[4] Spatial equilibrium: Back out values that support the observed population

distribution as an initial spatial equilibrium:

Pr =Lr

L=

exp(Dr)∑s exp(Ds)

⇒ Dr = (Ur +Ar)/β.

Decompose Ar into observed amenities (Aor) from USDA data and unobserved

amenities (Aur ) obtained as residuals εr from the simple OLS regression

Dr = α0 + α1Ur + α2Aor + εr.

We obtain α1 = 1.75∗∗∗ and α2 = 0.06∗∗, which allows us to retrieve Aur = εr.

Consistent with theory (choice probabilities increase with market based utility and

amenities).

In all that follows, α0, α1, α2, Aor and Au

r held fixed in all counterfactuals!


Elements of model fit.

We ask again how good is the model in replicating empirical facts (data) that have

not been used during the estimation procedure?

We simulate the model at the firm level and construct a large sample of random

firms drawn representatively from the fitted productivity distributions. We draw

about 6.5 million firms (the number of firms in the County Business Patterns in

2007), and rescale employment so that it matches employment from the CBP.

For each firm, we then compute:

– its total sales and size (total employment)

– its productivity and revenue-based productivity (sales divided by employment)

– its shipping patterns (shipments to each MSA, shipments by distance)


Getting k.

To estimate k, we try to match the empirical size distribution of firms as reported

in the CBP. This yields k = 6.4.

Employment # firms # firms Mean distance Mean distance Mean distance

Observed Model Observed Model Model (wgt)

All 6,431,884 6,431,886 529.6 71.98 739.8

1–19 5,504,463 5,498,328 327.2 38.5 61.2

20–99 769,705 755,275 423.8 157.9 194.4

100–499 141,510 153,021 520.4 556.0 740.3

500+ 16,206 25,255 588.6 1450.6 1519.1

Table 2: Shipment shares and shipping distances – summary for observed and simulated data

As with all models, it is hard to get the tails right. Here, we underpredict the

small firms and somewhat overpredict the large firms. Also, small firms ship much

smaller average distances in our model, while large firms ship longer distances.


050

100

150

200

250

Num

ber

of s

hipm

ents

0 500 1000 1500 2000 2500Miles

Figure 3a: Simulated number of shipments by distance shipped



Figure 3b: Observed number of shipments by distance shipped, CFS microdata

(Hillberry and Hummels, 2008)


020

4060

Tot

al s

hipm

ent v

alue

s

0 500 1000 1500 2000 2500Miles

Figure 4: Simulated value of total shipments by distance shipped



.2.4

.6.8

1A

vera

ge s

hipm

ent v

alue

s

0 500 1000 1500 2000 2500Miles

Figure 5: Simulated average value of shipments by distance shipped



7.3

7.4

7.5

7.6

7.7

7.8

Pric

e pe

r un

it

0 500 1000 1500 2000 2500Miles

Figure 6: Simulated unit delivered prices by distance shipped



Counterfactual analysis.

We run two counterfactuals to provide a quantitative answer to the questions: (i)

how do spatial frictions affect the spatial allocation of population and economic

activity? (ii) how do spatial frictions affect productivity and welfare?

We could a priori run any counterfactual we like. We will look at two ‘extreme

counterfactuals’ that are designed to isolate the contribution of frictions to the

observed spatial structure. Desmet and Rossi-Hansberg (2012) look at similar

questions in a perfect competition model where there is no trade across cities (so

they cannot gauge the impacts of trade frictions).

Scenario 1: Reduce all urban frictions (in all cities) to zero. Note that full ag-

glomeration does not occur since (i) agents have heterogeneous tastes, and (ii)

amenities differ across locations.

Scenario 2: Reduce all inter-city trade frictions to the intra-city levels, i.e., it is

not more costly to ship across cities than within cities.


We run the counterfactuals as follows:

[1] Start from an initial equilibrium. Set, e.g., θr = 0 for all r (the ‘urban frictions

counterfactual’).

[2] Holding population fixed at initial levels, utility changes (since prices, wages

and rents change). Find the new ‘short run’ equilibrium.

[3] Given the new prices, people chose locations. This in turn leads to price

changes, which make people move again.

[4] Iterate until a new fixed point (spatial equilibrium) is achieved.

This procedure can be applied to any counterfactual scenario that we may want to

look at (e.g., the case of high-speed rail investments in the UK between Manchester

and London).


Change

-6.30 to -4.60

-4.60 to -3.86

-3.86 to -3.19

-3.19 to -2.23

-2.23 to 0.11

0.11 to 27.62

Micro Stat. Area

% Population

Figure 7: MSA population change (CF1)



Change

-0.90 to -0.27

-0.27 to -0.17

-0.17 to -0.10

-0.10 to -0.02

-0.02 to 0.14

0.14 to 1.40

Micro Stat. Area

% Productivity

Figure 8: Average productivity change (CF1)



Change

-7.29 to -5.35

-7.91 to -7.29

-8.50 to -7.91

-9.15 to -8.50

-10.27 to -9.15

-15.97 to -10.27

Micro Stat. Area

% Markup

Figure 9: Average markup change (CF1)



−2

02

46

log(

rank

−1/

2)

−3 −2 −1 0 1 2 3 4

RS observed RS counterfactual

Figure 10: Rank-size rule, observed and counterfactual (CF1)



Change

-18.70 to -7.18

-7.18 to -2.86

-2.86 to 1.22

1.22 to 5.47

5.47 to 13.61

13.61 to 105.04

Micro Stat. Area

% Population

Figure 11: MSA population change (CF2)



Change

41.18 to 66.34

66.34 to 72.32

72.32 to 76.34

76.34 to 82.91

82.91 to 89.27

89.27 to 125.47

Micro Stat. Area

% Productivity

Figure 12: Average productivity change (CF2)



Change

-39.96 to -29.26

-42.01 to -39.96

-43.15 to -42.01

-44.97 to -43.15

-46.92 to -44.97

-55.19 to -46.92

Micro Stat. Area

% Markup

Figure 13: Average markup change (CF2)



−2

02

46

log(

rank

−1/

2)

−3 −2 −1 0 1 2 3 4

RS observed RS counterfactual

Figure 14: Rank-size rule, observed and counterfactual (CF2)



Welfare decomposition.

Any shock to the system causes people to move, which has both a positive effect on

welfare (because larger markets have higher productivity and more consumption

diversity), and a negative effect (because larger markets have higher land rents

and lower effective labor supply per capita).

Using the same technique than in Lecture 2, it is easy to show that

Ur

Ur=

hr

hr

(Ls

Ls

) 1k+1(λss

λss

)− 1k+1

When population is immobile, the first two terms vanish and we are back to the

standard result of Arkolakis et al. (2011).

Redding (2012) derives a similar result, but in his case land is endowed and thus

crowding has only a negative impact on utility. In our case, it is ∩-shaped.


In CF1 (urban frictions), cutoffs and trade shares do not change much. Welfare

does not change that much either. Larger cities become somewhat larger (positive

effect), there is a small fall in cutoffs, but there is also the countervailing effect of

smaller hr (negative effect). The net effect is positive, with population weighted

average of 9.4%.

In CF2 (trade frictions), the cutoffs and trade shares change substantially. Welfare

changes a lot. Larger cities become somewhat smaller (negative effect) but there

is the countervailing effect of smaller hr and a (large) fall in cutoffs. The net effect

is positive, with population weighted average of about 54%.

The sources of the welfare gains are very different in the two cases: (i) more

product diversity and lower markups due to more local firms in CF1; and (ii)

more productive firms and higher efficiency due to intercity trade in CF2.


Zero urban frictions: Key findings

[1] Rank-size distribution across MSAs remains almost unchanged (Zipf coeffi-

cient hardly changes).

[2] Total population reshuffling across MSAs is about 4 million people. Mobility

within a stable city size distribution.

[3] Big cities tend to gain, small cities tend to lose population. In line with ‘NEG

expectations’.

[4] Average productivity increases by only 0.04% in the aggregate. Markups fall

by 9.85%.

[5] Changes are very unevenly spread across space. Many MSAs lose population

and productivity.


Zero gravity: Key findings

[1] Rank-size distribution across MSAs again very stable, but a bit more action

than with zero urban frictions.

[2] Larger population reshuffling: 10.2 million people. Trade frictions have a

larger impact than urban frictions!

[3] Now big cities tend to lose, while small cities tend to gain population. Again

in line with ‘NEG expectations’.

[4] On average: markups fall by 40%, productivity goes up by 68%. Changes

much stronger than for zero urban frictions.

[5] Again very unevenly spread across space. Productivity increases everywhere,

despite population losses in some places.


Extensions.

Model allows for many possible extensions. An obvious one is to include more for-

mally sources of agglomeration economies into the model. We do this by assuming

(reduced form) that µmaxr (Lr) is a decreasing function of local population. Hence,

agglomeration economies are modeled as a right-shift in the ex ante productivity

distribution.

This can be viewed as cities providing resources (e.g., specialized services, diver-

sity) that help creating more productive firms. It is also somewhat in the spirit of

‘nursery cities’ (Duranton and Puga, 2001).

Our estimation yields an elasticity of µmaxr with respect to Lr of about 3% (con-

sensus range; e.g., Melo et al., 2010; Puga, 2010). Running the model with ag-

glomeration economies does not change much our quantitative findings (though

3% is a lot in the cross section, it is not in the ‘time series’)


The way forward.

There are various dimensions along which the models I presented need refinement.

The two most important ones (in my view) are linked to heterogeneity on the

workers’ side.

Workers differ in terms of human capital and productivity. It is known that larger

cities are in part more productive because they have more skilled workers (Combes

et al., 2008; Behrens et al., 2012). Sorting across cities is an important issue, and

the structural approach would allow to better understand to what extent sorting

matters for productivity and the spatial structure of the economy.

If workers differ along human capital, they also differ along income. Income het-

erogeneity is important in this type of model with non-homothetic preferences. In

particular, richer workers value diversity more and are less price sensitive, which

should act to reinforce sorting across cities. Quantifying these aspects is the next

logical step in this research agenda.

summerschool‘spatial economicsandimperfect markets ... · wage diﬀerential -0.0025*** -0.0179...

Documents