The Granular Origins of Agglomeration *Shinnosuke Kikuchi

Daniel G. O’Connor

MIT

MIT
October 10, 2024

Abstract
A few large firms dominate many local labor markets. How does that granularity affect the geography of economic activity? And what does it mean for the efficiency offirm entry? To answer these questions, we propose a new economic geography modelfeaturing granular firms subject to idiosyncratic shocks. We show that average wages
increase in the size of the local labor market due to that granularity, and provide a sufficient statistic for the contribution of our mechanism. We further prove that too few
firms enter in equilibrium. Using Japanese administrative data on manufacturing, we
provide evidence consistent with our mechanism and quantify it. Our mechanism
implies that markets with around 2 firms per sector have an elasticity of wages topopulation of 0.05 and firms capture only 85% of their contribution to production in
profits. In large markets like Tokyo, the elasticity is around 0.001, and firm entry is approximately efficient. Enacting optimal place-based industrial policy would increase
the number of firms in modest-sized cities by more than 30% and actually decrease
the number of firms and people in Tokyo.

* We thank Daron Acemoglu, Treb Allen, Kosuke Aoki, David Atkin, David Autor, Arnaud Costinot,

Dave Donaldson, Enrico Moretti, Tomoya Mori, Bob Staiger, Iván Werning, and Nathan Zorzi for their helpful comments. We also thank seminar participants at Dartmouth, Hitotsubashi, MIT, RIETI, and UTokyo.
We thank Satoshi Ichikawa and Tomoko Yamaguchi for excellent research assistance. This study is a partof the project “Macroeconomy and Automation” undertaken at the Research Institute of Economy, Tradeand Industry (RIETI). Kikuchi acknowledges permission to access the microdata of the Census of Manufacture (CoM) and the Economic Census for Business Activity (ECBA), granted by the Ministry of Economy,
Trade and Industry and the Ministry of Internal Affairs and Communications in Japan. We thank Kazunobu
Hayakawa for providing us with a crosswalk file between the HS product code and the product code in the
CoM survey used in Baek et al. (2021).

1

1

Introduction

A few large firms dominate many local labor markets. Kodak accounted for almost aquarter of Rochester, New York’s city payroll at its peak. Toyota hires a large proportionof the workforce in its headquarter-city Toyota. And even in a large city like Seattle,
software engineers are at the mercy of Microsoft. With these giants, a shock to a single
firm can hurt the entire market. If Microsoft has a bad year and lays off a large proportionof its software engineers, those workers might end up unemployed or at a low-wage job
in another industry. How does this exposure to firm-specific shocks shape where workers
and firms are located? And is there room for place-based policies to insulate labor markets
from a single firm’s influence?
In this paper, we study these questions both theoretically and empirically. Our analysis builds on the basic idea that in a small market, if an individual firm has a bad shock,the workers have nowhere else to go. They are stuck at that unproductive firm. By contrast, in a large market, when a single firm becomes less productive, workers can moveto another firm that is doing better and use their skills more effectively. Thus, large markets provide a “constant market for skill” as Marshall (1920) said and Krugman (1992)
formalized. This labor market pooling mechanism implies that larger markets use labor
more effectively than small markets, so that there are benefits to agglomeration when
individual firms matter.
Central to our contribution is a new economic geography model with a continuumof sectors, each of which have a finite number of (heterogeneous) firms, and workers
that are imperfectly mobile across both firms and sectors. With this general model, we
can derive a sufficient statistic for the contribution of granularity to the wage premiumof large cities. We can then transparently tie the theoretical mechanism to the empiricalevidence and policy implications.
We start by showing that, in a granular world, there are increasing returns to scalebecause firms use labor more effectively in large markets. In particular, firms in large
markets expand their employment more in response to productivity shocks, and so theyuse more workers while they are productive. Therefore, average labor productivity ishigher. To see why that is, consider a sector with only one firm. To expand after a good
productivity shock, that firm will need to attract workers from other sectors. Since workers have sector-specific skills, and there are costs to attracting people from other backgrounds, this will be very difficult. Microsoft in Seattle would need to offer jobs to people
who do not have a strong background in software engineering, for example. By contrast,
if a firm hires a small share of the market because the market is very large, it can ex-2

pand by poaching workers from other firms in the same sector. We also show that theexternality disappears as the region becomes very large. That is because, once a market
is sufficiently large, firms no longer have any difficulty finding the workers they need.
Therefore, adding more firms will not improve efficiency.
We go beyond this intuition and show that a sufficient statistic for how effectively asingle firm uses labor is the covariance of that firm’s log productivity and its log employment. Then the productivity of the entire market is just the average (employmentweighted) value of that covariance across firms. Thus, we can confirm our intuition of
how the mechanism operates and quantify the contribution of granularity to the wage
premium of large cities in a transparent, theory-consistent way.
Our last theoretical result considers the implications for policy. We show that when
there are region-wide increasing returns to scale, too few firms enter in equilibrium. Thatis because, when firms are granular, they know their entry affects wages. Not only will
their entry increase average wages in a region, but it will increase wages precisely when
the firm would like to hire more workers because their own attempt to increase theiremployment drives up the wage. That induces a correlation between wages and idiosyncratic firm shocks, which further depresses firm profits conditional on entry since profit
functions are convex in wages and productivity. Thus, there is room for place-based subsidies on firm entry, especially in locations where granularity matters. Our result also
derives the implied size of the subsidy as a function of the sufficient statistic for increasing returns to scale.
Our empirical analysis focuses on Japan, where we have a panel of all manufacturing
establishments with at least 4 employees every year. This data includes employment,
payroll, and shipment by 6-digit product category. We define a sector as a 2-digit JSIC
industry and a region as a commuting zone. Throughout the paper, we will use the termlabor market to refer to a commuting zone and sectoral labor market to refer to the sector
in a single commuting zone. Half of all sectoral labor markets have 13 or fewer firms,
and the median HHI is 1860.91, which suggests that larger firms play an important role.
There would need to be 5.5 firms of the same size in a single market to imply an HHI thatlarge.
We start by providing evidence that granularity matters. We show that the variance oflog payroll in a sectoral labor market is decreasing in the number of firms in that sector,
suggesting, consistent with Gabaix (2011), that individual firms are subject to idiosyncratic shocks. And those shocks average out in larger markets. We also show that average
wages have a large variance, suggesting that the extra payments from firms are not arbitraged away by workers moving in and out of the sector. Thus, moving sectors must be3

costly.
We then provide evidence consistent with our mechanism as suggested by the sufficient statistic. We begin by showing that the variance of log employment at a single firm ina large sectoral labor market is larger on average than a similarly situated firm in a small
sectoral labor market. That suggests, if those firms are subject to similar shocks, that thefirm in the larger market has an easier time expanding in response to good shocks andshrinking in response to bad ones. To provide more direct evidence, we construct revenue
productivity shocks to each firm using that firm’s exposure to real exchange rate movements. A great benefit of our Japanese data is that we have access to product-specificshipments at the establishment level so this is possible. Consistent with our mechanism,
firms that already hire a large portion of the sectoral labor market expand their employment less in response to these shocks.
Finally, we calibrate our geography model to match the manufacturing activity in the
256 commuting zones in Japan. Ideally we would directly estimate the average covariance
between log productivity and log employment by market size, but we do not observe theproductivity of every firm. Instead, we use our empirical results to infer the distribution.
We start by setting labor mobility costs of moving across firms and sectors to match how
firm employment responds to the revenue productivity shocks. Intuitively, we find thecost of moving across firms by how much employment of a firm small relative to its sector
responds to the shock. We determine the cost of moving across sectors by observing how
much employment at firms large relative to their sector responds. We then calibrate thesize of the idiosyncratic and sector-wide shocks to match the variance in log payroll bysectoral labor market across different regions. The size of the idiosyncratic shocks are
determined by how quickly the variance of log payroll falls with the size of the region.
We solve the model using the exact hat algebra of Dekle et al. (2008) to avoid explicitly
calibrating location-specific parameters on entry costs, productivity, and amenities.
Granular labor market pooling implies that markets with an average of 1.9 firms per
sector have the highest elasticity of wages to population at 0.051. Furthermore, firm profits are only 84.0% of the firm’s marginal contribution to production. For context, Combes
et al. (2011) argue that agglomeration externalities together imply an elasticity somewhere
between 0.02 and 0.05, while more recent research like Kline and Moretti (2014) suggest
the elasticity in manufacturing sectors could be as large as 0.4. Granularity matters lessfor large regions like Tokyo. There, the elasticity of wages to population is around 0.001,
and firm entry is approximately efficient. We simulate our model with the optimal subsidy on firm entry and find that the number of firms in modest-sized cities increases bymore than 30%. The number of firms in Tokyo actually decreases, even though there is a4small positive subsidy. That is because workers leave Tokyo for smaller cities in responseto the subsidies.
The rest of the paper is organized as follows. We give a short review of the literature
below. In section 2, we present the model of a single location and show our theoreticalresults. We provide empirical evidence for the mechanism in Section 3. We embed themodel of a single location in a quantitative model of economic geography in Section 4
which we then quantify in Section 5. We present the quantitative results along with the
implications of implementing the optimal policy in Section 6 before concluding in Section
7.

Related Literature
The literature on the spatial agglomeration of economic activity is rich and varied.
In an early contribution, Marshall (1920) proposed three reasons why firms might locate
around other firms: labor market pooling, access to intermediates, and the sharing ofideas. Subsequent theory papers have formalized these ideas and offered other potential mechanisms (Miyauchi, 2018; Davis and Dingel, 2019). Duranton and Puga (2004)
provide a new way to think about all of the mechanisms in their review. Many empirical studies have shown that there are benefits from agglomeration (Andersson et al., 2014;
Kline and Moretti, 2014; Greenstone et al., 2010) and have also analyzed the coagglomeration patterns of sectors to infer the relative importance of different theoretical mechanisms(Ellison and Glaeser, 1997; Ellison et al., 2010). Rosenthal and Strange (2004) review theevidence.
Our paper focuses on a particular mechanism that would fall under the broad umbrella of labor market pooling. There are many microfoundations with different mechanisms of how labor market pooling can lead to agglomeration benefits (Andersson et al.,2007; Papageorgiou, 2022). Our model builds on the basic theoretical framework of Krugman (1992). Krugman considers a setting with a finite number of ex-ante identical firms
where labor is perfectly mobile within a labor market but not across.1 Overman and Puga
(2010) extend Krugman’s model to include multiple sectors and then test the predictions
about where those sectors should be located. Other papers test these predictions in different settings (de Almeida and de Moraes Rocha, 2018; Nakajima and Okazaki, 2012). We
provide a new theoretical model with ex-ante heterogeneous firms and imperfect labormobility across firms and sectors. This allows us to derive a general sufficient statistic
for the strength of this mechanism. Then we can provide evidence consistent with the
1 Duranton and Puga (2004) also reviews and discusses the model.

5mechanism and quantify its importance.
More recent work looks for direct evidence of the labor market pooling mechanism.
Moretti and Yi (2024) show that workers who are laid off in large labor markets have aneasier time finding work as compared to workers in small markets. This is consistent withthe basic theory we lay out, though our evidence focuses on the firm response rather than
the worker side. Conte et al. (2024) shows that when firms in large markets can moreeasily expand in response to productivity shocks, more volatile firms will sort into larger
markets. We abstract from firm sorting but demonstrate how granularity could explain
why firms in larger markets can expand more in response to productivity shocks.
We build on a large literature recently inspired by Gabaix (2011) that looks to quantify what the granular nature of firms means for economic activity and optimal policy.
Gabaix (2011) shows that shocks to individual firms could explain nationwide fluctuations. Bernard et al. (2018) gives a framework for thinking about how a few importantfirms could shape the nature of international trade. Gaubert and Itskhoki (2021) discuss what granularity means for the observed comparative advantage of countries, andGaubert et al. (2021) studies what that implies for optimal policy. We contribute to this literature by considering granularity’s implications for the geography of economic activity
and optimal place-based policy.
Finally, our quantitative model builds on the computational geography literature reviewed in Redding (2022). Two notable early contributions are Allen and Arkolakis (2014)
and Redding (2016). We demonstrate how these models can be extended to include granular firms subject to idiosyncratic shocks in a tractable way. We also consider how the
optimal policy changes in that setting (Fajgelbaum and Gaubert, 2020).

2

How Does Granularity Lead to Agglomeration?

The goal of this section is to demonstrate how granularity leads to higher wages
in larger markets through the labor market pooling mechanism discussed by Marshall(1920). To focus on that, we model the sectoral labor markets of a single location with agiven population. We then endogenize where people live in Section 4 to demonstrate theimplications for agglomeration.
Our model in this section will allow us to identify the key mechanism through which
granularity implies agglomeration benefits. This will then guide our empirics as we lookfor evidence consistent with that mechanism. In order to isolate the effects of granularity
most transparently, we focus on an environment that is neoclassical conditional on firm
entry and delay a discussion of how the results change with imperfect competition to6

t=0

Mass miof firms pay
fixed costto enter

t=1

Sector s andex-ante prod.
shocks zisn
revealed
for each firm nWorkers freely
choose initiallaborallocation Lisn

t=2

Ex-postprod. shocks
aisn (ω ) revealed

Workers adjustlabor Lisn (ω )subject tofrictionsFigure 1: Timing of model
section 2.5.

2.1

Environment

We focus on a region i with a mass ℓi of workers and a continuum of sectors s ∈ S . The
sectors produce perfectly substitutable goods but hire in distinct sectoral labor markets.
Timing. There are three time periods t ∈ {0, 1, 2}. In period 0, a mass mi of establishments pay a fixed cost of the traded final good in order to enter. Each firm is randomly
assigned a sector s and then gets an ex-ante productivity draw z from some known distribution. Thus each sector ends up with a finite number of firms Nis that differ in theirproductivity. This captures the long run differences in firm size and will determine how
exposed different markets are to short-run idiosyncratic firm shocks in period 2.
After observing those initial productivity draws, a representative worker freely allocates her labor Lisn across sectors s and firms n in period 1. This captures the long run
decision-making of a worker free to direct her search or make training decisions to workat certain firms in certain sectors of her region.
Then, in period 2, the state of the world ω ∈ Ω is revealed. This determines the shortrun productivity shocks to each firm. The representative worker adjusts how much labor
she supplies to each firm, Lisn (ω ), subject to frictions of moving labor across firms andsectors on the shorter time scale. Firms then produce and sell their goods. The modeltiming is summarized in Figure 1.

7

Workers. The representative worker is risk neutral.2 She gets utility Ui from consuming
a freely traded final good, ci , and enjoying a location-specific amenity ui ,Ui = ui ci .

(1)

The representative worker is endowed with one unit of labor that she supplies to themarket inelastically and allocates among all the active firms. In period 1, the worker
can freely set her units of labor across sectors s ∈ S and firms n ∈ Nis taking as given
each firm’s ex-ante productivity zisn . In particular, she chooses her vector of labor supply
Li ≡ { Lisn }s,n in the set of feasible labor allocations L, i.e.,Li ∈ L ≡

′
Li |

Z

′
Lisn ds ≤ 1

∑

S n∈N


.

(2)

is

The period 1 labor decision captures the long-run labor supply decision of a worker
who lives in this location. The perfect substitutability of labor across firms and sectors
captures that, in the long run, the representative worker can freely make education decisions and direct her search to work at particular firms.
Then, at the beginning of period 2, the state of the world ω is revealed. This determines ex-post productivity shocks for firms. In response, the representative worker canreallocate her units of labor across firms and sectors subject to moving frictions. In particular, labor must be in some feasible set, L( Li ),Li (ω ) ∈ L( Li ),(3)

where Li (ω ) ≡ { Lisn (ω )}s,n and L( Li ) is defined by,L( Li ) =L′i (ω )|



Z
S

Lis =

Lis · gS

′ (ω )Lis

Lis

ds ≤ 1,∑ Lisn ,

n∈Nis

′
Lis
(ω ) =∑ Lisn · gN

n∈Nis



′ (ω )
Lisn

Lisn


.

Lis is how much labor the worker assigned to sector s in period 1, Lis (ω ) is how much effective labor the worker assigns to sector s in state of the world ω, and the functions g N (·)
2 We explain how this model of a representative worker can map onto a model of heterogeneous workers

with comparative advantage for working at certain firms and sectors below.

8Figure 2: Feasible Labor Allocations

Firm 2 labor

Firm 2 laborThis depicts the feasible labor set for a single sector when labor cannot move between sectors andthere are 2 firms in that sector. The set of feasible labor choices in period 2 is a function of the labor
choice in period 1.

( Lis1 , Lis2 )Firm 1 labor

Firm 1 labor(a) Possible allocations in period 1, L(b) Possible allocations in period 2, L( Li )and gS (·) summarize the costs of moving labor across firms and sectors, respectively.
Lis · gS ( Lis (ω )/Lis ) is the amount of total labor that the representative agent needs
to supply to sector s to get Lis (ω ) amount of sector s labor when Lis labor is originallysupplied. Similarly, Lisn · g N ( Lisn (ω )/Lisn ) is the amount of sector s labor that the representative agent needs to supply to firm n to get Lisn (ω ) amount of effective labor when
Lisn labor was originally supplied. We assume that g N (1) = gS (1) = 1 so that the initial
allocation Lisn (ω ) = Lisn is feasible. g N and gS are increasing and convex to capture thefact that the representative agent needs to give up time in order to supply labor to eachfirm and that moving a lot of labor between firms becomes increasingly difficult. We illustrate an example two firm sector with no sectoral mobility in Figure 2. In period 1, theworker faces a linear trade-off between supplying labor to firms 1 and 2. Then, in period
2, the initial allocation remains feasible, but moving labor between the two firms can leadto a loss because the worker might have firm-specific skills or the worker might lose time
while searching for a job.
This model nests a number of standard models from the literature. While the originalmodel in Krugman (1992) features a single sector, we can nest a multi-sector version asa limit case. In that setting, there is perfect mobility across firms within a sector, Lisn ·g N ( Lisn (ω )/Lisn ) = Lisn (ω ), and no mobility across sectors, Lis · gS ( Lis (ω )/Lis ) = Lis if
Lis (ω ) ≤ Lis and infinity otherwise. Our assumption also allows for the nested constant9

elasticity labor supply model in Berger et al. (2022) where workers have comparativeadvantage working for certain firms and sectors. That comparative advantage takes aparticular form to keep the model tractable. More generally, we allow for any model ofthe labor market as long as the deadweight loss from moving labor into a sector in thesecond period is proportional to the percentage change in the size of that sector from theinitial allocation. The same must hold for moving across firms.
Firms. There is a continuum of potential firm entrants. To enter, a firm must pay a fixed
cost ψi > 0 in terms of the freely traded final good in period 0. Those firms are then randomly assigned a sector. Thus, while a mass of firms enter, each sector only has a finite
number of firms. We denote by Nis ≡ |Nis | the (finite) number of firms in sector s. We
denote the probability mass function of Nis by p N (·, m) when a mass of m firms enter.
The average number of firms across all of the sectors needs to equal the number of firmR
entrants, i.e. mi = S Nis ds = ∑∞N =0 N p N ( N, mi ). We also assume that p N ( N, m ) is differNN′
p N ( N, m).
entiable in m and for m > m, m′ p N ( N, m′ ) first order stochastic dominates mThis assumption rules out firm entry that is overly biased towards small sectors, and wewill have more to say about it below. Firm n in sector s then gets an ex-ante productivity draw zisn from a distribution Fiz (·) which we assume is continuous and regularly
varying.3
In period 2, each firm n gets an idiosyncratic productivity shock ãisn (ω ), a sector-wide
productivity shock Ãis (ω ), and produces the final good yisn (ω ) according to,yisn (ω ) = aisn (ω ) f (ℓisn (ω )),where aisn (ω ) ≡ zisn ãisn (ω ) Ãis (ω ) is the total productivity shock to firm n and ℓisn (ω )is the total amount of labor firm n hires. We assume that f (·) is CES with elasticity of
production to labor 1 − η ∈ (0, 1), i.e. f (ℓ) = ℓ1−η . Throughout we assume that the expected number of employees at a firm is finite. The idiosyncratic productivity shocks are
independent and identically distributed with finite variance and E[log ãisn (ω )] = 0. Similarly, sector-wide shocks are independent and identically distributed with finite varianceE[log Ãis (ω )] = 0.
3 Formally, L : (0, ∞ ) → (0, ∞ ) is regularly varying if lim

10

L( ax )+
x →∞ L( x ) ∈ R for all a > 0.

Market Clearing. Total expected production in the location is
Yi = E#"Z∑ aisn (ω ) f (ℓisn (ω ))ds ,

S n∈N

(4)

is

where expectations are taken with respect to ω, the number of firms, and the initial productivity draws. We assume that the law of large numbers applies across this continuumof sectors so that realized production is always Yi . Therefore, goods market clearing requires that total consumption plus the amount of the final good used as an investment
must equal expected production,ci ℓi + ψi = Yi .
(5)
In the labor market, labor demanded needs to equal the individual labor supplied byeach worker multiplied by the number of workersℓisn (ω ) = Lisn (ω )ℓi .

2.2

(6)

Decentralized Equilibrium

Labor Supply. Workers choose their initial labor allocation Lisn in period 1 and theirsubsequent labor allocation Lisn (ω ) in period 2 to maximize expected utility, taking wages
as given. We normalize the price of the final good to 1, so workers solve the problem,
Li , {Li (ω )} ∈argmax′′ui ci

Li ∈L, L′i (ω )∈L( Li )s.t. ci ≤Z′(ω )ds,∑ wisn (ω ) Lisn

S n∈N

(7)

is

where wisn (ω ) is the equilibrium wage for firm n in sector s state of the world ω.
Labor Demand. We normalize productivity so that the price of every good is 1. Thenfirms maximize profits taking wages and prices as given,′′ℓisn (ω ) ∈ argmax aisn (ω ) f (ℓisn(ω )) − wisn (ω )ℓisn( ω ).

(8)′ (ω )ℓisn

Entry. We assume that firms enter up to the point that expected profits are equal to
the fixed cost of entering. Defining πisn (ω ) ≡ aisn (ω ) f (ℓisn (ω )) − wisn (ω )ℓisn (ω ) as the11profits firm n earns in the state of the world ω, we can write this,
ψi =

E

R


π
(
ω
)
ds
∑
isn
S n∈Nis.
mi(9)

The numerator is the total amount of profits earned by firms in location i. The denominator is the total measure of firms that enter, so the ratio is a firm’s expected profit before
any firm realizes its sector or productivity shocks.
Definition 1. A local equilibrium consists of wages wisn (ω ), labor supply decisions Lisn ,Lisn (ω ), entry decisions mi , and labor demand decisions ℓisn (ω ) such that,• Workers maximize utility taking wages as given, (7);• Conditional on entry, firms maximize profits taking prices and wages as given, (8);• Firms enter up to the point that expected profits are equal to the fixed cost of entering, (9); and• Goods and labor markets clear, (5), (6).

2.3

Granular Origins of Agglomeration

We now demonstrate how granularity implies average wages increase in the size of themarket. We will proceed in two steps. First, we show that wages increase in population
if, in markets with more firms, the average firm can adjust its employment more easily
in response to shocks. This is the key mechanism, and our empirics in section 3 provide
evidence consistent with it. Throughout this paper, we will refer to this mechanism aslabor market pooling.
Second, we return to granularity. We provide sufficient conditions under which ourmodel leads to labor market pooling, and thereby agglomeration benefits. And finally, weshow that the mechanism disappears in the limit as the number of firms goes to infinity,
highlighting the fact that this is a fundamentally granular phenomenon.
We derive all of our results in this paper by doing a second-order approximation to
expected production around the point with no ex-post productivity shocks. This keeps
the model tractable and makes the mapping between theory and empirics easier and moretransparent.
Labor Market Pooling and IRS. As an intermediate step, we express region-wide production as a function of the number of workers ℓ and the number of firms m,

12

Yi (ℓ, m) ≡ E"Z#∑ aisn (ω ) f (ℓisn (ω ))ds .

S n∈N

is

This object differs from equilibrium production since firm entry is not endogenized. However, in the next lemma, we confirm that understanding this object is sufficient for understanding agglomeration both qualitatively and quantitatively.
Lemma 1. For small ex-post shocks, if Yi (ℓ, m) has increasing returns to scale, i.e. Y1id log wdYi (αℓ,αm)
>
dα
α =1

1, then average wages are increasing in population, i.e. d log ℓ i > 0. In particular,i

1 dYi (αℓ,αm)
−1
Yi
dα
α =1

d log wi=dY (αℓ,αm)
d log ℓi1−η− 1 i
Yi

dα

α =1



−1

.

Thus, we can focus on the region-wide production function. Lemma 1 clarifies how
our estimates of increasing returns to scale map onto the benefits of agglomeration.
In our next proposition, we lean on the second-order approximation to find a sufficient statistic for the strength of those increasing returns to scale. We will use this statistic
in three ways. First, we use it as a stepping stone in our proof that granularity leads to increasing returns to scale. Second, it elucidates the mechanism through which granularity
causes increasing returns to scale. Thus, we will be able to look for evidence consistent
with that mechanism. Third, we use the statistic to quantify our mechanism. We candetermine how much of the urban wage premium is accounted for by granularity rather
than some other externality emphasized by the literature by focusing on this statistic.
Proposition 1. For small ex-post shocks, Yi (ℓ, m) features increasing returns to scale if the(employment-weighted) average covariance between log employment and log productivity shocks
is increasing in the number of firms, i.e.
d
dm

"Z

∑

S n∈N

is

#
ℓisn
Cov (log aisn (ω ), log ℓisn (ω )) ds > 0.
ℓi

In particular,
Yi (ℓ, m) ≈ zi ℓ1−η mη Ω(m),1/ηwhere zi = E[zisn ]η and1−ηΩ(m) ≡ E[ aisn (ω )] +
2

Z

∑
S

n∈Nis

ℓisn
Cov (log aisn (ω ), log ℓisn (ω )) ds.
ℓi

13

(10)

The proof of proposition 1 is in the appendix, but we give a brief sketch here. The
proof proceeds in three steps. We introduce a variable β so that the productivity of firm n
is zisn ( ãisn (ω ) Ãis (ω )) β . First, we solve the model when β = 0 so that there are no ex-post
shocks. Second, we do a second order approximation to regional production in β around
0. We find that Yi ≈ zi ℓ1−η mη Ω(m) where Ω(m) is definedΩ(m) = E[ aisn (ω )] + (1 − η )

Z

∑
S

n∈Nis

ℓisn
Cov (log aisn (ω ), log ℓisn (ω )) ds
ℓi

1−η
−
Ψ ({Var(log ℓisn (ω ))}, {Var(log ℓis (ω ))}) ,2(11)

where Ψ(·, ·) is the measure of deadweight loss that results from moving labor between
firms and sectors in the second period. In the third step, we note that when wages are set
competitively, they optimally trade off the deadweight loss with the productivity, leading
to the simplification
Ψ ({Var(log ℓisn (ω ))}, {Var(log ℓis (ω ))}) =

Z

∑
S

n∈Nis

ℓisn
Cov (log aisn (ω ), log ℓisn (ω )) ds,ℓicompleting the proof.
Proposition 1 provides a clear interpretation of how labor market pooling leads toincreasing returns to scale. Consider a firm n that never adjusts its worforce in response
to idiosyncratic productivity shocks so that
Cov(log ℓisn (ω ), log aisn (ω )) = 0.
That firm would hold onto a large number of workers when it has bad productivity
shocks and not expand to take advantage of good productivity shocks. Therefore, its
average labor productivity would depend solely on its average productivity shock. By
contrast, if the firm were to expand after a good productivity shock and shrink after a badshock, its average labor productivity would increase because it hires more workers when
its more productive. Thus, that firm could produce more goods on average while hiring
the same average number of workers simply by increasing its covariance.
Proposition 1 then says that there are increasing returns to scale if increasing the size ofthe market improves how efficiently labor reallocates across firms in response to productivity shocks, properly weighted by the importance of each firm. Thus, if firms are better
able to expand their employment in response to good productivity shocks in a larger market, then larger markets will be more productive. We will look for direct evidence of this14

in Section 3.
Implicit in Proposition 1 is the fact that the average covariance is only a function of
the number of firms m. Thus, our empirics will use the number of firms as the relevantmeasure of the size of the market.
Granular Labor Market Pooling. Here, we complete the proof that there are increasing
returns when firms are granular. The final assumptions are stated below.
Assumption 1. Labor is imperfectly mobile across sectors, i.e. gS′′ (·) > 0, and firms are
subject to idiosyncratic shocks, i.e. Var(log ãisn (ω )) > 0.
Then we have the following result.
Proposition 2. Suppose ex-post shocks are small, and Assumption 1 holds. Then average wages
d log w
are increasing in population, i.e. d log ℓ i > 0. And as the population goes to infinity, theseid log w

agglomeration effects disappear, i.e. limℓi →∞ d log ℓ i = 0.
i

We start by giving a basic intuition for why there are increasing returns to scale using Proposition 1. Consider how firm n in sector s responds to an idiosyncratic, ex-post
productivity shock ∆ log aisn (ω ). To first order, employment responds according to
"
#
ηS
ℓisn
1
1−
∆ log ℓisn (ω ) ≈∆ log aisn (ω ),ηN + η
η + η N + ηS ℓisg′′ (1)1g′′ (1)1(12)

where η N := gN′ (1) and ηS := gS′ (1) are the inverse of the supply elasticities across firms
NS
within a sector and across sectors, respectively.
In a small region, where the mass of firms mi is small, chances are that there are very
few other firms in sector s. Therefore, firm n hires a large share of the labor force in sectors, i.e. ℓisn /ℓis is large. In that case, the firm’s employment does not respond much after a
productivity shock because it already hires a large proportion of the sectoral labor force,and to expand it would have to attract workers from other sectors. Therefore, the firm
does not effectively scale up in response to a productivity shock and ends up using labor
inefficiently. In other words, in a small market, workers are stuck working at the samefew firms whether or not they are productive in that state of the world. In a market witha large mass of firms m, firm n’s share of the sector s labor force is smaller. Therefore, firmn’s labor responds more in response to productivity shocks because it can take workers
from other firms in the sector, and labor is used more efficiently.

15

Turning to the proof sketch, we now allow for productivity shocks to every firm. Then
labor at firm n in sector s is given by
"
#
ηS
1
ℓisn′
∆ log aisn (ω ) −∆ log aisn′ (ω ) .
∆ log ℓisn (ω ) =η + ηNη N + ηS + η n ′∑∈Nis ℓis
Then some tedious algebra, left to the appendix, shows that the employment weighted
average covariance is given by
Z

ℓis
ℓisn
1
σS2
Cov(log aisn (ω ), log ℓisn (ω ))ds =
∑
η
+
η
+
η
S ℓi n∈N ℓis
N
S
is
"
!#
Z
1
ηS
ℓis
2
1−
,
+
HH Iis ds
σNη + ηNη + η N + ηS
S ℓi

2 is the variance of the log
where σS2 is the variance of log sector-wide shocks and σNidiosyncratic shocks. This variance depends on the number of firms in the sector only2through the HH Iis ≡ ∑n∈Nis ℓisn /ℓis . Thus, the degree of increasing returns to scale isηS
d log Ω(m)1=−d log mη + η N η + η N + ηS

d
m dm

hR

ℓis
S ℓi HH Iis ds

Ω(m)

i
2
σN
.

If the average HHI across sectors decreases as the number of firms m increase, Proposition
1 is satisfied, and there are increasing returns to scale. The expected HH I of a given sectors is decreasing in the number of firms Nis , so the last thing we need to check is that firm
entry across sectors is not too weird. For example, suppose that there were only two
sectors with firms in them: one with two firms and another with one firm. Then if all
future firm entry goes into unoccupied sectors, the average HHI would actually increaseas the average number of firms in occupied sectors decreases. Our assumption that theNdistribution induced by mp( N, m) is increasing in first order stochastic dominance exactly
rules out these strange cases.
The speed with which average HH I decreases depends on the distribution of entrants
across sectors, p N ( N, m), and the ex-ante productivity distribution Fis (z). The shape ofp( N, m) especially matters for low m. As m becomes larger, the ex-ante productivity distribution matters more. As discussed in Gabaix (2011), if Fis (z) has thin tails, HH Iis decreases approximately at the rate of Nis−1 . Then the agglomeration externality is strong
when there are a small number of firms and the HHI falls quickly with new entrants.
However, the HHI quickly approaches zero, at which point the average HHI cannot fall16

further. For example, if a sector has one firm and adds another ex-ante identical firm,
the HHI drops from 1 to 0.5. If a sector already has 100 identical firms, the HHI is 0.01,
and doubling the number of firms only decreases it to 0.005, not increasing productivity
much. Intuitively, that is because if there are already 100 firms in a sector, it is easy for any
firm to expand by attracting workers from the other 99 firms. Adding more firms does
not have much effect.
−ξ
If Fis (z) has thick tails,4 HH Iis decreases at a rate of Nis where ξ ∈ (0, 1). In that case,the weighted HHI does not fall as quickly, and large firms continue to be constrained intheir ability to respond to productivity shocks. Thus, the externality is not as strong forsmall markets, but continues to matter for medium-sized, and even large, cities. Nonetheless, the externality dies out as the number of firms becomes large, and the average HHI
approaches zero.
As suggested by Assumption 1, the preceding argument for increasing returns to scale
can fail for two reasons. The first possibility is that firms are not subject to idiosyncratic
2 = 0, and the average variance of log marginal product does notshocks. In that case, σNchange as the market gets larger. That is because, if all shocks are sector-wide, for a firm
to expand in response to a productivity shock, it will have to attract workers from othersectors since every other firm in the sector is trying to expand as well. We can test thisexplicitly in the data. If firms are subject to idiosyncratic shocks, then the variance of log
total payments to labor will decrease in the size of the sectoral market since the shocks toeach of the firms will average out. The other possibility is that workers are freely mobile
across sectors, i.e. ηS = 0. In that case, the firms are not really granular as the relevant
labor market is the entire commuting zone.

2.4

Under Entry of Firms

In this section, we consider what this force for agglomeration means for policy. Conditional on firm entry, the model is competitive, so Yi (ℓ, m) is not only the equilbrium levelof average production for the market with ℓ workers and m firms. It is also the optimallevel of production given m. Therefore, the only possible source of inefficiency is firm
entry. Rewriting the goods market clearing condition (5), the first best level of entry miFB
4 Formally, if F ( z ) is a pareto distribution, i.e. F ( z ) = 1 − az−ζ , then 1 < ζ (1 − η ) < 2. Then
is
is



1
ξ = 2 1−ζ (1 − η )

17


.

solves the following problem,
miFB ∈ argmax Yi (ℓi , m′ ) − ψi m′ .

(13)

m′

The following proposition demonstrates that the optimal policy calls for a subsidy on
firm entry when there are increasing returns to scale.
Proposition 3. Suppose that shocks are small, and Assumption 1 holds. Then the optimal policy
d log Ω(m)is an ad valorem subsidy on firm entry of η1 d log m m=m > 0 where Ω(m) is given by equationi(10) in Proposition 1. And as the population goes to infinity, the optimal subsidy converges to 0.
This result is the normative counterpart of Proposition 2. We sketch the proof here.
Taking the first-order condition associated with the net production maximization problem gives the requirement that the marginal product of another firm needs to equal the
marginal cost of entry in the first best,∂Yi (ℓi , miFB )= ψi .
∂m
In equilibrium, on the other hand, free entry implies that total profits divided by the
number of firms equals the marginal cost of entry. Total profits are simply the value of
total production less payments to labor, so with a subsidy of s,
Yi (ℓi , miE ) − wi ℓi= (1 − s)ψi ,miE
where wi is average wages and miE is equilibrium entry. To find the optimal subsidy, we
note that because wages are competitively set, workers are paid their marginal productiso that wi = ∂Y∂ℓ . However, there are increasing returns to scale so thatiYi (ℓi , mi ) <∂Yi∂Ymi + i ℓi .
∂mi
∂ ℓi

Y (ℓ ,m E )−w ℓ∂Yi
i i
It follows that i i mi
< ∂m= ψi in equilibrium. Therefore, the optimal policy is aii
subsidy on firm entry, i.e. s > 0. This subsidy then converges to 0 as the market becomes
infinitely large because the increasing retuns to scale of Yi disappear in the limit.
The first welfare theorem breaks in this setting because we do not have Walrasianentry. In a competitive equilibrium, firms need to take prices and wages as given, but inour model, we allow firms to ask what their profits would be conditional on entry. That
requires a firm internalizing the effect that their entry will have on wages. Firms know

18that by entering, wages will be higher. Not only will wages be higher on average, but they
will be higher precisely when the firm would like to hire more workers because their own
idiosyncratic shocks affect wages. This leads to under entry, especially in small markets
where adding one more firm will have a big effect on the distribution of wages.
A skeptical reader might wonder why we depart from competitive entry. The reason
is simple. An equilibrium with competitive entry does not exist. To illustrate this mosttransparently, consider a special case of our model where workers are freely mobile acrossfirms within a sector s and all productivity shocks are idiosyncratic. A competitive firm
would then maximize profits taking the sector-wide wages wis (ω ) as given. Solving forthe optimal labor choice and plugging that back into the profit equation (8), we find thatthe profits a firm n would earn in state of the world ω is1− η1πisn (ω ) = η (1 − η ) η aisn (ω ) η wis (ω )

1− η

− η

.

Taking a second order approximation to profits around the no ex-post shock equilibrium
implies thatE[πisn (ω )] ≈ π isn1−η
Cov(log aisn (ω ), log wis (ω ))
ζs −
η2



where ζ s is some sector wide constant and π isn are firm profits when the ex-post shocks
are 1. That is, profits are decreasing in the covariance between firm productivity andequilibrium wages. This is because profit functions are convex, so firms appreciate thevariance that comes when productivity and wages are not correlated.
For a firm that is currently operating, wages are correlated with their shocks because
their attempt to hire more workers drives up the wage. On the other hand, a potential
entrant’s productivity shocks are not correlated with wages. Therefore, given the current distribution of wages, the potential entrant expects strictly higher profits than theoperating firm. But in any competitive equilibrium, operating firms must expect weakly
positive profits while potential entrants must expect weakly negative profits. This is notpossible when potential entrants expect strictly higher profits than current entrants, andtherefore, no competitive equilibrium exists.

2.5

Extensions to Imperfect Labor Markets

Before turning to the empirical evidence for our mechanism, we briefly discuss how
our results change if labor markets are not perfectly competitive. Since there are a finite

19

number of firms in each sectoral labor market, and they face upward sloping labor supply
curves in the short run, firms have market power that they could exploit. To shed light on
the novel aspects, we have ignored that conduct up to this point. But we characterize themodel when firms internalize that power and engage in Cournot competition in appendix
B, as in Berger et al. (2022). We assume that firms can commit to their wage offers. Thus,
firms internalize that if they were to lower their wages, the representative worker would
decrease her time spent at that firm in the long run. Since labor is perfectly substitutablein the long run, this implies that the average wage markdown is zero, but the markdown
could vary in response to ex-post productivity shocks.
Proposition 1 no longer holds because labor does not move between firms to efficiently
trade off deadweight loss with the productivity gains of moving workers to more productive firms. In particular, a firm does not expand as much in response to productivity
shocks since it increases its markdown to exploit its increased market power. However,equation (11) continues to hold. And we show that
Ψ ({Var(log ℓisn (ω ))}, {Var(log ℓis (ω ))}) <

Z

∑
S

n∈Nis

ℓisn
Cov (log aisn (ω ), log ℓisn (ω )) ds.
ℓiThis implies that the covariance of log employment and log productivity understates how
productively firms are using workers. We further show that in the limit the equilibrium
converges to the competitive one, so the covariance accurately describes how efficiently
labor is used. Thus, the observed decline in the covariance between log productivity andemployment will overstate the productivity benefits.
However, for given parameters, the market will use labor less efficiently if firms compete oligopsonistically. And in the limit, as the number of firms goes to infinity, they use
workers just as efficiently as the competitive case. Thus, for a given set of parameters,
monopsony increases the gains from being in a large market when interacted with ourmechanism. We include the exact assumptions necessary to recover an adapted proposition 2 in the appendix.
Proposition 3 then becomes more complicated. The first best policy would featuresubsidies on wages along with subsidies on firm entry. If the planner could not undo
the distortion on wages, then workers are paid less than their marginal product and theefficiency of firm entry is ambiguous.

20

3

Empirical Evidence

Section 2 showed how granularity can lead to increasing returns to scale. In this section, we provide evidence that it does using administrative data from Japan. We start byshowing that large firms subject to their own idiosyncratic shocks dominate hiring withina sector in many commuting zones. We also show that people rarely move across sectors,
confirming the conditions of Proposition 2. We then provide evidence of our mechanism
as presented in Proposition 1. In particular, we see how firm employment responds to idiosyncratic revenue productivity shocks constructed from exchange rates and trade data.

3.1

Data

In mapping the model to the data, we need to lay out what is a region and what is a
sector. We interpret a region i as a commuting zone and denote by I the set of 256 commuting zones in Japan.5 We map sectors s ∈ S from the theory onto 2-digit manufacturing industries in the data.67 Then we interpret each establishment in the data as its own
independent firm in the theory as we do not allow multi-establishmet firms. Throughout
this section, we will use the term establishment.
Japanese Census of Manufactures Data. Our primary data source is the Census of Manufacture (CoM) in Japan for the manufacturing sector. The Ministry of Economy, Trade,and Industry (METI) conducts the Census of Manufacture annually to gather information on the current status of establishments in the manufacturing sector. Specifically, thiscensus covers all manufacturing establishments in years when the last digit of the survey
year is 0, 3, 5, or 8. For other years, the census covers all establishments with at least
4 employees in Japan. The CoM survey was not conducted in 2012 and 2016, and instead, another survey, the Economic Census for Business Activity (ECBA) was conductedby METI and the Ministry of Internal Affairs and Communications for data in 2011 and
5 To construct time-consistent commuting zones from municipalities in Japan, we first follow Kondo(2023) to convert municipalities in each year into time-consistent municipality groups. Japan has 1,724
municipalities as of June 2023, including 6 municipalities in the Northern Territories. We drop these 6
municipalities as the CoM data does not cover them. We then use the converter in Adachi et al. (2020) toconvert these municipality groups into commuting zones.
6 To construct time-consistent industries, we use a crosswalk provided by RIETI to convert categories
into the 2011 codes. For example, manufacturing of iron and steel industry is one of the 2-digit industries.
7 In the theory, we assume a continuum of sectors, but in practice, we only use 23 industries. We do thisbecause non-regular workers in Japan move freely within these broad sectors so that this is the best notionof a labor market. By assuming a continuum of sectors, we abstract away from the granularity of sectors,
understating the influence of individual firms.

21

2015.8 We use the ECBA survey to substitute the CoM survey in 2011 and 2015. The
detailed sample construction is in Appendix E.
This data has two advantages. First, we observe panels of all the establishments withat least 4 employees. This feature allows us to compute a variety of volatility measuresat the establishment level as well as commuting zone and sector-level variables.9 Second, we observe yearly shipment values by detailed product categories for each establishment. This enables us to construct establishment-level exposure to foreign exchange
rate changes using product shipment share and national, product-destination level export
data.
UN Comtrade Data We supplement the CoM data with bilateral trade flows by productyear level from UN Comtrade data to compute the share of destination countries of theJapanese export at the product-year level. See the Appendix E for a detailed data construction.
Penn World Table Data We construct real exchange rates using data for the nominal
exchange rate (“xr”) and price level of exports (“pl x”) from Penn World Table (Feenstra
et al., 2015).

3.2

Summary Statistics

Table 1 presents summary statistics for the sectoral labor markets. The average number of establishments is 53, but more than half of all sectoral labor markets have 13 or
fewer establishments. The employment HHI distribution suggests individual establishments have an outsized role in many labor markets. The median HHI is 1860.91. There
would need to be 5.5 establishments of the same size in a single market to get an HHI thatlarge. Employment is similarly sparse, with half of sectoral labor markets having 346 employees or fewer. See Figure 3 for a histogram of the entire distribution of establishments
and employment.
We also report the summary statistics for a number of variables we use in our subsequent analysis in Table 1. Log average wage is the average salary of workers in the
sectoral labor market. The variance of log wage growth and that of log payroll growth
are computed over time for each sectoral labor market. These measure the volatility of
8 The ECBA survey covers all establishments, including establishments in non-manufacturing sectors,

but we focus on establishments in the manufacturing sector to be consistent with the CoM survey.
9 One further advantage, compared to the US LBD data, is that we can separately identify single establishments within each of the 47 prefectures.

22

wages and payroll of a single market, respectively. The log variance of log wage growthis smaller than that of log payroll growth, suggesting that some of the variance of log payroll growth is driven by workers moving between sectors to take advantage of economic
opportunities as they arise. However, the variance of log wages is not zero. This suggeststhat economic opportunities are not completely arbitraged away by workers moving between sectors. Instead, workers in a sector that has a good shock see an increase in theirwages.
In fact, workers are slow to move across sectors. Only 6% of full-time workers switch
jobs within a year, and of those who switch jobs, 76.9% stay in the same industries.10 11
Table 1: Summary Statistics: Sectoral Labor Market

Num. of Estab.
Employment HHI
Employment
Log Avg. Wage
Log Variance of Log Wage Growth
Log Variance of Log Payroll Growth
Sample size

MeanMedianStd. Dev.

53.19
3222.53
1651.48
5.78
-4.40
-2.97
4,800

13.00
1860.91346.50
5.82
-4.46
-3.05

176.563270.30
4771.52
0.37
1.29
1.38

Note: The tables show the summary statistics for sectoral labor markets. Each sectoral labor market is a2-digit manufacturing industry, commuting zone pair. The variance of log wage growth, the variance of log
payroll growth, and the weighted sum of log employment growth are in log units. The variance of log wage
growth and log payroll growth are computed for a single sectoral labor market over time. The weighted
average of log employment growth is computed by first taking the variance of log employment growth foreach establishment. The variances are then averaged together weighted by each establishment’s average
payroll share.

3.3

Firms Are Subject to Idiosyncratic Shocks

In presenting the summary statistics, we showed that large establishments dominate
many sectoral labor markets and that worker movement across sectors does not arbitrage
away all wage differences. Here we check the final condition of Proposition 2: establishments are subject to idiosyncratic shocks.
10 Data is from the Japanese Panel Study of Employment Dynamics, which was provided by Recruit
Works Institute. The sample is the full-time workers in 2015, and the sample size is 22,965. When computing the mobility across sectors conditional on changing jobs, we use the data for workers who have everchanged jobs in their careers.
11 The probability that a worker would stay in the same industry if she were randomly choosing a new job
is 48.4%. We compute this number using the HHI of 2-digit sectoral employment within each commuting
zone and take the weighted average using the total employment of each commuting zone.

23

Figure 3: Histogram of Number of Establishments and Employment in each Secotral Labor Market(a) Number of Establishments(b) Total Employment

Notes: The figures show the histogram of the number of establishments and employment in each local labormarket, respectively. The unit of observation is the local labor market, a pair of a JSIC 2-digit industry, anda commuting zone. To be visible, I collapse all the local labor markets with the statistics larger than the top
5% percentile to be in the same bin. The top 5% percentile is 210 for the number of establishments and 7,107
for the total employment,

24

Figure 4: Volatility of Payroll and Average Wage Growth and Number of Establishments(a) Payroll(b) Average Wage

Note: The panels show the binned scatter-plots and histograms of the relationship between volatility oflog growth in total labor payment (top) and in average wage (bottom) and the number of establishments
across local labor markets in Japan estimated by (15). We also show the histograms of both variables. The
unit of observation is the local labor market, a pair of JSIC 2-digit industries, and a commuting zone. The
vertical axis is the log variance of log growth in total labor payment (top) and average wage (bottom) over
1990-2016 in each local labor market. The horizontal axis is the number of establishments in log, averaged
over 1986-2016 in each local labor market.

25

In order to do this, we construct a measure of volatility at the sectoral labor market
level. For each sectoral labor market, we start by computing the total labor payroll ineach period. We then compute one-year log growth rate and take its variance over timeto get our measure. We use the variance of the growth rate rather than the variance inlevels as a conservative estimate of the uncertainty in labor payments from one period tothe next. This differences out persistent differences in growth rates that workers could
expect.
We regress our measure of volatility in labor demand against the number of establishments in the sectoral labor market to see if larger labor markets have less volatile demand
for labor. We use the specification,log Var log∑ wisn (ω )ℓisn (ω )!!!

= β ln Nis + µi + µs + ε is

(14)

n∈Nis

where µi and µs are commuting zone fixed effects and industry fixed effects, respectively.
We show the bin scatter plot in Figure 4(a) . As one can see, there is a strong negative relationship, consistent with our story that establishments are subject to idiosyncratic shocks.
Small sectoral labor markets with few establishments have very volatile demand for labor as a shock to a single establishment greatly affects the overall demand for labor. By
contrast, in a large market, there are many establishments, all subject to their own idiosyncratic shocks. So the idiosyncratic shocks “average out” to some extent and the demand
for labor remains relatively constant.
We do a similar exercise with the variance in log wage growth at the sectoral labor market level to ensure that this variance in labor demand is not arbitraged away by workers
moving between sectors. We compute the sector-specific average wage by dividing thetotal labor payment by the number of workers. We then find the variance of log growthin the same way we did for the total payroll. We run the regression,log Var log∑n∈Nis wisn (ω )ℓisn (ω )∑n∈Nis ℓisn (ω )!!!

= β ln Nis + µi + µs + ε is(15)

where µi and µs are commuting zone fixed effects and industry fixed effects, respectively.
The bin scatter plot is in Figure 4(b). We find a similar negative relationship: the varianceof log wages is higher in smaller markets because there are fewer establishments and ashock to one establishment will greatly affect labor demand and it is not arbitraged away
by worker movement.

26

3.4

Firms in Larger Markets Expand Employment More Easily

Having confirmed all of the sufficient conditions of Proposition 2 for there to be increasing returns to scale, we now look for explicit evidence of our mechanism. We do thisby looking for evidence of Proposition 1: the weighted covariance of log employment
and log productivity of establishments is larger in markets with more establishments.
This would provide evidence that larger markets are more productive because workers
can move to where their labor is most needed.
3.4.1

Observational Evidence

We start by providing observational evidence. If establishments have an easier time
finding the workers they need after a good shock in large markets, then an establishment
in a larger market should have a higher variance of log employment than a similar establishment in a small market. We check this by constructing a measure of establishment
employment volatility.
Figure 5: Volatility of Establishment-level Employment Growth and Number of Establishments

Note: The figure shows the binned scatterplots of the relationship between the volatility of establishmentlevel employment growth and the number of establishments across local labor markets in Japan. We also
show the histograms of both variables. The unit of observation is the local labor market, a pair of a JSIC
2-digit industry, and a commuting zone. The vertical axis is the average of establishment-level log varianceof log growth of employment over 1986-2016 in each local labor market. The horizontal axis is the numberof establishments in log, averaged over 1986-2016 in each local labor market.

First, we residualize each establishment’s annual employment by establishment and27year-fixed effects.
∆ ln ℓn,t,t+1 = α ln ℓn,t + ηt + εℓn,twhere ∆ ln ℓn,t,t+1 is the changes in log employment of establishment n from year t to t + 1,
ln ℓn,t is the log employment of establishment n from year t, and ηt is a year fixed effect.
We control log employment to capture the fact that large and small establishments are
systematically different. Small establishments might be growing in an expected way assuggested by some papers studying establishment dynamics, such as Hopenhayn (1992).
Second, we compute yearly changes of the estimates of εℓn,t,t+1 , ∆bεℓn,t,t+1 as follows:∆bεℓn,t,t+1 ≡ b
εℓn,t+1 − bεℓn,t .
This gives a measure of unexpected growth in employment for establishment n.
Finally, we take variance of ∆bεℓn,t,t+1 , (Var ∆bεℓn,t,t+1 ), over time within each establishment n to get a measure of employment growth variance relative to expected growth patterns. We then pool the establishments in a sectoral labor market by taking the weighted
average of Var(∆bεℓn,t,t+1 ), weighted by each establishment’s median employment overthe sample period. This gives a measure of employment volatility of establishments inthe sectoral labor market.
We estimate the following log-linear modelℓisn· Var(∆bεℓn,t,t+1 ) = δ log Nis + µi + µs + ε isn∈Nis ℓis∑(16)

where log Nis is the number of establishments in commuting zone i and industry s. µi andµs are commuting zone fixed effects and industry fixed effects, respectively.
Figure 5 shows the result in binned scatter plots. One can see the clear positive relationship implying that comparing two similarly situated establishments, the establishment in the larger labor market will have a higher variance of log employment.
3.4.2

Quasi-Experimental Evidence

Finally, we look for more direct evidence that establishments in large markets can expand more easily in response to productivity shocks. In particular, we construct establishmentlevel demand shock from the establishment’s product mix, country-level destinations,and destinations’ real exchange rates. We then examine how that “adjusted real effective exchange rate shock (AREER)” affects an establishment’s employment share within
sectoral labor markets.

28Specification Our econometric specification is an empirical version of equation (12). It
is as follows:∆ ln ℓn,t,t+1 = β 1 ∆µn,t,t+1 + β 2 (∆µn,t,t+1 · sn,t−1 ) + X′n,t Γ + ζ n + ζ t + ε nt ,(17)

where ∆ ln ℓn,t,t+1 is the log change in the employment of establishment n from year t
to t + 1, ∆µn,t,t+1 is the negative demand shock for establishment n, sn,t−1 is the payroll
share of establishment n in year t − 1 within the sectoral labor market.12 Xn,t is a vector
of covariates at the establishment level, including an establishment age and its square,
lagged payroll share, and lagged export ratio relative to total shipment.13 ζ n and ζ t are
fixed effects for establishments and year, respectively. ε n,t is the error term.
Establishment-Specific Negative Labor Demand Shock We proxy the establishmentspecific revenue productivity shock ∆µn,t,t+1 by an establishment-level exposure to a realeffective exchange rate shock, which we define as follows,14∆µn,t,t+1 = EXPn ×∑ ωn,c · ∆REXc,t,t+1
JPN

!
(18)

c

where EXPn is the median export share relative to the total shipment of establishment n
over the period.15 ωn,c is the median exposure of establishment n to country c over time,
where the time-specific exposure ωn,c,t ≡ ∑ p ωn,p,t · ω p,c,t is the share of product shipmentof establishment n to country c in year t. Since CoM does not report establishment-specificexport destinations, we use ωn,p,t , the share of shipment of establishment n in 6-digit
product category p in year t and ω p,c,t , the share of Japanese export to country c out of
JPN
total shipment in 6-digit product category p from the UN Comtrade data. ∆REXc,t,t+1 is
the change in the real exchange rate of Japanese Yen to the currency in the country c fromJPN
t to t + 1. Therefore, positive ∆REXc,t,t+1 means JPY appreciation against the currency
12 We look at the interaction between the shock and the payroll share of the establishment in the period

before because that is the specification suggested by equation (12). Noting that in larger markets, establishments tend to have lower payroll shares then confirms statistic (ii ) of Proposition 1.
13 Establishment ages are not surveyed in the CoM data. [fill]
14 In the context of Japan, there are several papers, which study the effect of exchange fluctuations on
employment responses(Hosono et al., 2015; Yokoyama et al., 2021). Some recent studies on the effect of theexchange rate on employment using firm-level exposure to trade (Nucci and Pozzolo, 2010; Ekholm et al.,2012; Yokoyama et al., 2021) using firm-level export share. Similar to ours, Dai and Xu (2017) use firm-level
heterogeneity of trade partners and the heterogeneous fluctuations of exchange rates across currencies in
the context of Chinese manufacturing sectors. Our specification leverages the establishment-level productmix, product-country-level export, and country-level exchange rate fluctuations.
15 The CoM survey asked the ratio of exports in each establishment only after 2001, so we use the median,rather than the lagged value.

29

in country c, so that ∆µ p,t,t+1 and ∆µn,t,t+1 are negative demand shock for product andestablishment, respectively.
Definition of Employment Types We examine the effect of a JPY appreciation on employment across establishments. We define executives with compensation and permanent employees (“sei-shain”) as regular workers who typically work full-time with anindefinite contract. We define non-regular workers as the sum of part-time workers andworkers dispatched from temporary help agencies.
Sample Restriction We restrict the sample to 2001 to 2013 because the export share andemployment by employment types are available in the CoM data since 2001, and thetiming of the survey has changed from December in the previous year to June in 2014. We
drop the case where the shock is JPY depreciation, when the AREER shock is negative,
as we expect heterogeneity in responses of employment to positive and negative shocks.
Our final sample in this analysis is an unbalanced panel of 163,121 unique establishmentsin the manufacturing sector from 2001 to 2013.
Table 2: Summary Statistics: Establishment
Mean
Employment
51.11
Payroll (in millions JPY)
223.56
Payroll (in log, JPY)
8.75
Share of Non-Regular Workers
0.34
Log Changes in Employment-0.01
Log Changes in Regular Emp.
-0.01
Log Changes in Non-Regular Emp.
-0.00
Emp. Share within LLM
0.02
Payroll Share within LLM
0.02
Sample size
1,164,363

Std. Dev.

p10

p50

p90

155.861029.06
1.42
0.23
0.19
0.27
0.51
0.08
0.08

6.00
11.71
7.07
0.07
-0.18
-0.22
-0.51
0.00
0.00

18.00
56.22
8.63
0.29
0.00
0.00
0.00
0.00
0.00

104.00408.77
10.62
0.70
0.16
0.20
0.51
0.05
0.05

Note: The tables show the summary statistics for the data used in the analysis across establishments.

Summary Statistics: Establishment Panel Table 2 shows the summary statistics forpanels of the establishments in our analysis. The average and median establishments
have 51 and 18 workers. The average payroll is 224 million in JPY (in 2015 value). On
average, the share of non-regular workers is 34%. The changes in employment share aresymmetric with the median of zero, but the volatility comes from non-regular employment with a standard deviation of 0.51, rather than regular employment with a standard
30

deviation of 0.27. This is consistent with the findings in the previous literature in Japan.16
The average share of employment as well as payroll within a local labor market is 2%.
Results: No Interaction Before showing our main results on the role of the market sizein responses to shocks, we present evidence that our establishment-specific shocks havean impact on establishment outcomes. Table 3 shows the result without the interaction
term. Column (1) uses log changes in sales, Column (2) uses log changes in total employment, Column (3) uses log changes in regular employment, and Column (4) uses
log changes in non-regular employment. Column (1) shows that the 1% of the negative
exchange rate shock decreases sales by 3.5%. Column (2) shows that the employment
declines by 0.3%. This decline is heterogeneous across employment types: Column (3)shows that regular employment declines only by 0.3%, while Column (4) shows that establishments adjust non-regular employment by 2.6%. This is consistent with the findingsin Yokoyama et al. (2021) that a JPY appreciation reduces the sales and non-regular employment of exporters.
Table 3: Effects of JPY Appreciation on Establishment Sales and Employment

AREER ShockObservations
Covariates
Year FEs
Establishment FEs

Dep. Var.: Log Changes
Employment by Types
Sales
Employment Regular Non-Regular
-3.46
-0.25
-0.29
-2.62
(0.17)(0.09)(0.12)(0.23)
1,164,363
1,164,363
1,164,363
1,164,363
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓

Notes: This table shows the relationship between JPY appreciation and various outcomes at the establishment level. Column (1) uses log changes in sales, Column (2) uses log changes in total employment, Column(3) uses log changes in regular employment, and Column (4) uses log changes in non-regular employment.
The running variable is the adjusted real exchange rate shock at an establishment level. All columns include
the following covariates: lagged share of non-regular workers, lagged payroll share within each local labormarket, the establishment’s age square, and the sum of the shock to other establishments within each local
labor market. All columns include establishment fixed effects and year-fixed effects. Standard errors arerobust against heteroscedasticity.

Results: Roles of Size in Response After confirming that our measure of shock does
actually affect the establishment’s sales and non-regular employment, we examine the
16 See Morikawa (2010) and Kambayashi (2017) for the evidence that firms adjust labor more flexibly for

non-regular workers.

31

Table 4: Effects of JPY Appreciation on Changes in Non-Regular Employment
Dep. Var.: Log Changes in Non-Regular Emp.
(1)
(2)
(3)
(4)
AREER Shock
-2.62
-2.98
-0.31
-0.55
(0.23)(0.27)(0.44)(0.44)AREER Shock × Log Payroll
-1.08
-1.12
(0.14)(0.15)AREER Shock × Payroll Share
3.35
8.26
(1.26)(1.41)AREER Shock × (Payroll Share > 3%)2.43(0.50)
Observations
1,164,363 1,164,363 1,164,363
1,164,363
Covariates✓✓✓✓
Year FEs✓✓✓✓
Establishment FEs✓✓✓✓
Notes: This table shows the relationship between JPY appreciation and non-regular employment growth at
the establishment level. The dependent variable is the log growth of employment share within local labormarkets. The running variable is the adjusted real exchange rate shock at an establishment level. The log
payroll share is normalized by subtracting the average log payroll in the entire sample when it interacts
with the AREER shock. All columns include the following covariates: lagged share of non-regular workers,
lagged payroll share within each local labor market, the establishment’s age square, and the sum of theshock to other establishments within each local labor market. All columns include establishment fixed
effects and year-fixed effects. Standard errors are robust against heteroscedasticity.

32

heterogeneous responses by the establishment’s sizes. Table 4 shows the results. The
dependent variable is the log changes in non-regular employment. The running variable
is the adjusted real exchange rate shock at an establishment level. All columns include thefollowing covariates: lagged share of non-regular workers, lagged payroll share within
local labor markets, lagged log payroll, and the establishment’s age square. All columns
include establishment fixed effects and year-fixed effects. Regressions are weighted by
the establishment’s payroll. Standard errors are robust against heteroscedasticity.
Column (1) runs the regression without the interaction, which replicates Column (3)in Table 3 that a JPY appreciation decreases non-regular employment. Column (2) adds
the interaction with payroll share within a local labor market. The positive estimate implies that the establishments with higher payroll share respond less to the shock, which isconsistent with condition (ii ) of Proposition 1. Quantitatively, if the plants differ in their
payroll share by 10% pt, the elasticity decreases by 0.34, which is about 11% (=0.34/2.98)
of the initial elasticity of 2.98.
However, we may find that the interaction between the shock and a establishments’s
payroll share is positive simply because larger establishments are different from smaller
establishments–independent of their size relative to the market. For example, larger establishments may adjust labor less because they can adjust other factors more easily.
Therefore, the estimate may pick up heterogeneity in responses by the absolute size,
rather than relative size.
To address this concern, Column (3) adds the interaction term with the lagged log
payroll of the establishment.17 This means that we compare the elasticities of employment
by payroll share within a local labor market, across establishments with the same absolute
sizes. The estimate is now 8.26, which is larger.
Column (4) uses the interaction of the shock with the dummy variable, which takes
one if the payroll share within a local labor market is larger than 3%. which is roughly
the 80% percentile value in the sample. Again, the estimate is positive and implies that
establishments with larger payroll share within a local labor market respond less to theshock. Therefore, establishments in a larger market can expand their employment moreeasily.

4

A Quantitative Model of Granular Economic Geography

In this section, we describe an empirical version of the model in Section 2 that allows us to estimate the strength of the mechanism using the regressions in Section 3. We
17 We normalize the log payroll by subtracting the average log payroll share.

33then embed this labor market model in a standard model of economic geography with
I locations i ∈ I = {1, . . . , I } with imperfect labor mobility across regions, production
externalities independent of granularity, and amenity externalities. We do this by introducing a migration decision at time t = −1 allowing workers to choose where to live.
We then assume that workers are stuck in their region as moving in response to 1 year
productivity shocks is negligible in the Japanese context.

4.1

Parametric Restrictions

To fully specify the model, we need to specify the probability mass function of firm
entrants p N (·, m), the distribution of the ex-ante productivity shocks Fiz (·), and the costs
to moving across firms and sectors, g N (·), gS (·). We continue to use the second orderapproximation, so we do not need to fully specify the distribution of the ex-post idiosyncratic and sectoral shocks Fa (·), FA (·).
Firm Entry Distribution. We assume that if a mass of mi establishments attempt to enterin location i, the number of realized entrants in any sector s is distributed Poisson withparameter mi . The probability mass function isp N ( N, mi ) =( m i ) N e − mi.
N!

Ex-ante Productivity Shock Distribution. Following Gabaix (2011), we assume that theex-ante productivity shocks, zisn are distributed according to a Pareto distribution withshape parameter λ and scale parameter zi .
zi is the natural productivity of location i. It is subject to increasing returns to scale
independent of our granular mechanism which take the form,zi = zi (ℓi )γz ,(19)

where γz is the agglomeration elasticity independent of our mechanism and zi is the fundamental productivity of location i.

34

Labor Mobility Frictions. The labor mobility frictions take the CES form frequently
used in the labor literature.18 We have thatgN

′ (ω )
Lisn

!

=

′

′ (ω )
Lisn

! 1+κ
κ

,

′

Lisn

Lisn

for individual firms. For moving across sectors, we havegS

′ (ω )Lis!

′′ (ω )
Lis=! 1+ ν κ
ν

.

′

Lis

1+κ

Lis

These imply that the elasticity of substitution across firms is κ and the elasticity of substitution across sectors is ν. It is more difficult to move across firms in different sectors thanin the same sector if ν < κ.

4.2

Migration Decision

We allow the amenities in location i to be a function of fundamentals and spillovers.
In particular,(20)
ui = ui (ℓi )γu ,where ui is the fundamental amenities in location i and γu is a measure of the externality.
un captures fundamental differences across locations like weather and the beauty of the
nearby area. γu captures interactions between people like traffic.
Each worker has an idiosyncratic preference for each location i, ε i . The utility that theworker gets from living in location i is Ui ε i . Each worker can move freely and so lives
in her utility maximizing location. The preference shocks are distributed according to aFréchet distribution with shape parameter θ. Therefore, the number of workers who livein location i is
 θ
Ui
ℓi =
ℓ,
(21)
U
where ℓ is total population, and,"U=

∑ (Ui )

#1
θ

θ

.

(22)

i ∈I

18 Berger et al. (2022) use the same parametric restriction and show it is equivalent to a roy model withFréchet productivity draws across sectors and firms.

35

Definition 2. A Decentralized Equilibrium consists of population ℓi , average wages wi , fundamental utility Ui , and firm entry mi in each labor market, along with wages wisn (ω ),labor supply decisions Lisn , Lisn (ω ), and labor demand decisions ℓisn (ω ) such that,• Workers choose their utility maximizing location taking average wages as given,(21), (22);• Amenities and productivity are consistent with spillovers, (19), (20); and• Average wages, firm entry, wages, labor supply decisions, and labor demand decisions are part of a local equilibrium.

5

A Quantification of Granular Scale Effects

In this section, we calibrate the quantitative model economic geography model introduced in Section 4 using accepted parameters from the literature and our empirical resultsfrom Section 3. Proposition 1 makes clear that we would ideally match the covariance oflog productivity and log employment of individual firms. Unfortunately, we do not observe the entire distribution of productivity shocks to firms, so we infer it. In particular,
we calibrate the elasticity of employment to idiosyncratic productivity shocks using ourestimates of how firm employment responds to productivity shocks. We then calibrate
the distribution of sector-wide shocks and idiosyncratic shocks to match the distributionof log payroll across different sectoral labor markets as described below.

5.1

Calibration

We summarize how we calibrate all of the parameters in Table 5. We discuss eachparameter in more detail below.
Labor Supply and Demand Elasticities. We calibrate the labor supply and demand
elasticities to match our estimates of how firm employment and sales respond to exporting productivity shocks. The details are in the appendix, but we go through the brief
intuition here. To calibrate the degree of decreasing returns to scale η, we compare the coefficient on how labor responds interacted with the size of the market with the coefficienton how sales responds interacted with the size of the market using the specification ofcolumn (2) in Table 4. Intuitively, this identifies the degree of decreasing returns to scaleby finding how much more a firm in a large market can produce because it can expand
its employment more. We calibrate the supply elasticity across firms η N to rationalize the36

A. Labor Supply and Demand Elasticities
Parameter Value Description
κ
7.54
Supply elasticity across firms
ν
0.456 Supply elasticity across sectors
η0.203 Decreasing returns to scale
B. Productivity Shock Parameters
Parameter Value Description
λ8.517 Ex-ante productivity Pareto tail2
σN
0.191 Variance of log idiosyncratic shocks
2
σS
0.019 Variance of log sectoral shocks
C. Migration and Spillover Parameters
Parameter Value Descriptionθ3
Migration elasticity
γu-0.25 Amenity spillovers
γz
0.03
Other productivity spillovers

SourceTable 4

SourceVariance of log sector payroll

SourceRedding (2016)
Japanese housing share
Combes et al. (2011)

Table 5: Calibration summary.
increase in labor in response to the shock for a firm small relative to its market. Finally, wecalibrate the elasticity of subsitution across sectors by comparing how much employment
in a firm small relative to its market responds compared to in a firm that is larger. Thismeasures how much harder it is to attract workers from other sectors. For this calibration,
we use the specification in column (3) of Table 4 to control for the overall size of the firm.
The results are in Panel A of Table 5. The firm substitution parameter is κ = 7.54,
suggesting the workers move relatively easily across firms within a sector. However, it
is smaller than the value of 10.82 that Berger et al. (2022) estimate in the US context andmuch larger than the 1.02 Felix (2021) finds in the Brazilian context. We find an elasticity
of substitution across sectors of ν = 0.456 similar to the estimate of 0.42 that Berger et al.
(2022) find and smaller than the 0.80 Felix (2021) finds. And finally, we find η = 0.203.
2 , and σ2 to match the regionProductivity Shock Parameters. We jointly calibrate λ, σNSwide average variance of log total payments to labor in sector s. This is matching Figure4(b) on average. In the data, that isZℓis
Var log
S ℓi∑ wisn (ω )ℓisn (ω )

n∈Nis

37

!!
ds.

In the model, that is


2 (

1
(1 − η ) ηSσS2 +1 + ηN −ηN + η
η + η N + ηS

Z

! )
ℓis
2
HH Iis ds σN.
S ℓi

We choose the parameters to minimize a quadratic loss function of the log differences
weighted by population.
Migration and Spillover Parameters. We take the Fréchet parameter controlling the migration elasticity θ = 3 from Redding (2016), which is consistent with the evidence from
Bryan and Morten (2019) in Indonesia and Hornbeck and Moretti (2020) in the UnitedStates. We set γu = −0.25 to match the average housing spending in Japan.
We choose the strength of other spillovers, γz , to match the estimated agglomeration
externality from Combes et al. (2011), 0.03.

5.2

Exact Hat Algebra

We avoid explicitly calibrating the location-specific entry costs, ψn , fundamental productivities, zn , and fundamental amenities, un , by solving the model in percent changes
from the observed equilibrium in 2019 using the exact hat algebra method pioneered by
Dekle et al. (2008). This method implicitly calibrates those parameters to exactly matchthe average wages, population, and number of firms in Japanese manufacturing sectors.
Details are in Appendix D.

6Quantitative Results

In Section 5, we calibrated the model from Section 4. In this section, we present theresults. We start by plotting Ω(m), which summarizes the degree of increasing returns toscale due to our mechanism. We then turn to demonstrating what the mechanism means
for the geography of economic activity. In particular, we calculate the implied elasticity
of wages to population and the degree of under-entry for each commuting zone. Finally,
we simulate how economic activity would change if the central government put in place
the optimal entry subsidy, which is a place-based industrial policy.

38

Figure 6: log Ω(m)6.1

Increasing Returns to Scale

We plot Ω(m) in Figure 6. As suggested by Proposition 1, the slope of the line isthe degree of increasing returns to scale. And, in the data, this is determined by how
quickly the average covvariance of log productivity and log employment increases as thenumber of firms increases. In the model, it is determined by how quickly the employment
weighted average HHI across sectors decreases. As emphasized by Proposition 2, thecurve levels off as the number of firms gets very large. At that point, granularity ceases
to be an important driver of agglomeration.
Perhaps more surprisingly, we see that the curve is relatively flat in very small regions
as well, i.e. where the average number of firms is less than 1. At that point, new firm
entrants are likely to enter sectors with no firms. Therefore, the employment weighted
average HHI does not change much. Instead, the new sector with an HHI of 1 is simply
averaged in with the other sectors with a high HHI. As the number of firms incrase, newfirms are likely to enter sectors that already have firms. Therefore, those sectors see a
precipitous decline in HHI, and the average HHI decreases as well.

6.2

The Geography of Economic Activity

Using this estimate of Ω(m), we can estimate the contribution of granularity to agglomeration benefits for each labor market. We provide two ways of interpreting it in
Figure 7. Figure 7(b) captures the degree of under entry in each commuting zone in Japanby plotting the ratio of firm profits to a firm’s marginal product. Large and small locations
with very little externalities see firms mostly internalizing their impact on the productive
39

Figure 7: Strength of the externality(a) Elasticity of wage to population(b) Firm Entry Wedge

capacity of the labor market. For example, firms in Tokyo capture 99.6% of their contribution to production in profits. By contrast, in modest sized cities, adding another firm
will have a large impact on the distribution of wages in a sectoral labor market. Firms
are therefore less likely to enter. In fact, in commuting zones with an average of 1.9 firms
across sectors, profits are only 84% of the firm’s marginal product.
Figure 7(a) show the elasticity of the average wage to an increase in population if
the granular externality were the only reason for agglomeration (i.e. γz = 0). This iscalculated according to lemma 1. The elasticity gets as high as 0.051 in locations with anaverage 1.9 firms per sector. The implied elasticity is much smaller for larger locations.
For example, the elasticity is 0.001 in Tokyo. For context, Combes et al. (2011) find thatmost causal estimates of the urban wage premium find an elasticity between 0.02 and0.05 when pooling across locations of all sizes. Other more recent papers like Kline andMoretti (2014) find that the elasticity could be as high as 0.4 in manufacturing industries.
Thus, this mechanism can explain a significant portion of the productivity advantage oflarge locations, though it is not the whole story, especially for very large locations. 19

6.3

Optimal Place-Based Industrial Policy

As we know from Proposition 3, there is under entry in equilibrium. In this section,
we assess the degree of under entry, and see how the geography of economic activity
in Japan would change were the Japanese government to put in place the optimal entry
19 We include these graphs under the assumption that firms compete oligopsonistically in Figure ??. Assuggested by the shape of Ω(m), the elasticity of wages never gets above 0.06. However, the elasticity
remains above 0.05 for even medium sized cities, and the elasticity in Tokyo is around 0.01 rather than
0.001.

40

Figure 8: Optimal Policy(a) Change in Number of Firms(b) Change in Population

subsidy, paid for with a tax propotional to wages. We graph the results for the change in
the number of firms and the change in population in Figure 8.
For the modest-sized commuting zones where the labor market pooling externality isstrongest, the number of firms increases by 38%. This is driven in part by the direct subsidy. But it also happens because of the reallocation of economic activity. Figure 8b showsthat those locations see an increase in population of 9.5%. Meanwhile, large locations likeTokyo actually see a decrease in population. That decrease in population at the top of the
distribution is so large that even though there is now a subsidy on firm entry, the numberof firms also declines.

7

Conclusion

When individual firms matter, there are benefits to agglomeration. Having many firms
all in the same location can allow workers to move to the firm where their skills will be
best used. This has important implications for understanding where economic activity
takes place and optimal policy.
This paper offers simple model that allows for a transparent mapping between thedata and the positive and normative implications. There is still a lot more to be said about
how granularity interacts with the endogenous skill acquisition, heterogeneous firms andsectors, and sorting.

41

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45

A

Proofs for Section 2

In this appendix, we go through and prove the theoretical results in Section 2. We startby begin explicit about what we mean by second order approximation. In particular, weβintroduce a variable β so that the productivity shock to firm n is zisn Ãis (ω ) ãisn (ω ) .
Then production is
Yi (ℓ, m, β) = E#"Z∑ zisn (aisn (ω )) f (ℓisn (ω, β)) ds ,
β

S n∈N

is

where aisn (ω ) ≡ Ãis (ω ) ãisn (ω ). We then do a second order approximation to this in β.
Lemma 2. To first order,∂ log Lisn (ω, β)1=∂β
η + ηN"ηSη +η Nlog aisn (ω ) −η
1 + η +SηN

Lisn′
log aisn′ (ω )n ∈Nis Lis

∑
′

!#
.

Proof. We start by characterizing the worker’s problem. It is
max E

"Z

′
Li ,L′i (ω )′(ω )ds∑ wisn (ω ) Lisn

S n∈N

#

is

such that
1=
1=
′

Lis =

Z

′

S

Z
S

Lis ds,′Lis · gS

′ (ω )
Lis

!
ds,

′

Lis

′

∑ Lisn ,

n∈Nis

′
Lis
(ω ) =′∑ Lisn · gNn∈Nis

′ (ω )
Lisn
′

Lisn

!
,

where we assume the inequalities bind since workers can always increase utility by

46

working slightly more. Taking the first order conditions,wisn (ω ) = λis (ω ) g′N

λis (ω ) = λi (ω ) gS′′ (ω )
Lisn

Lisn′ (ω )Lis′gN"
λis = E λi (ω )

!

Lis

"
0 = E λis (ω )gS

′ (ω )
Lisn

!

′

Lisn
′ (ω )Lis′

L′ (ω ) ′− isn′
gN
Lisn

!

−

Lis

′ (ω )
Lis
gS′
′
Lis

′ (ω )
Lisn
′

!!#

Lisn
!!#
′ (ω )
Lis
′

Lis

+ λis
+ λi .

The wages are determined competitively so thatwisn (ω ) = zisn ( aisn (ω )) β ( Lisn (ω )ℓi )−η .
We can then start by finding the solution when β = 0. Then notice that if Lisn (ω ) =Lisn and wages are constant across all states of the world, we havew = λis (ω ) g′N (1)
λis (ω ) = λi (ω ) gS′ (1)
0 = E λis (ω ) 1 − g′N (1) + λis
λis = E λi (ω ) 1 − gS′ (1) + λi .
With gn′ (1) = gS′ (1) = 1, we find that w = λis (ω ) = λi (ω ), and λis = λi = 0. Then,−η −ηzisn Lisn ℓi= w.

We take the derivative with respect to β and plug in for the value at β = 0. At that′ ( ω, 0) = L′ (0). Thenpoint, Lisnisn
∂ log Lisn (ω, β)∂ log λis (ω, β)∂ log Lisn (ω, β) ∂ log Lisn ( β)log aisn (ω ) − η
=
+ ηN
−
∂β
∂β
∂β
∂β


∂ log λis (ω, β)∂ log λi (ω )∂ log Lis (ω, β) ∂ log Lis ( β)
=
+ ηS
−
.
∂β
∂β
∂β
∂β

47

And for the next first order condition,0=∂λis ( β)
∂β
"

!!
#
′ (ω )′ (ω )
Lisn
L
∂λ
(
ω,
β
)
is
isn
+E
gN
−
g′N
′
′
′
∂β
Lisn
Lisn
Lisn
"
!
#
′ (ω )′ ( ω, β )′ (ω )
Lisn
Lisn
1 ∂Lisn′′gN+ E −λis (ω ) ′
′
′
∂β
Lisn
Lisn
Lisn
#
"


′ (ω )′ (ω )′ (ω )
Lisn
Lisn
Lisn
∂Lisn (ω, β)′′gN+ E λis (ω, β) ′
2
∂β
Lisn
Lisn
Lisn
!


′
′ ( ω, β )∂ log Lisn
∂ log Lisn ( β)∂λis ( β)−= ηN E.
∂β∂β∂β′ (ω )
Lisn!

Similarly,∂λi ( β) ∂λis ( β)
−
= ηS
∂β
∂β

!
′
′ ( ω, β ) ∂ log Lis
∂ log Lis ( β)
E
−
∂β
∂β


These go along with the constraints
0=

Z
S

Lis

∂ log Lis ( β)ds∂β

∂ log Lis ( β)L ∂ log Lisn ( β)= ∑ isn∂β∂βn∈Nis Lis
0=

Z
S

Lis

∂ log Lis (ω, β)ds∂β∂ log Lis (ω, β)L ∂ log Lisn (ω )= ∑ isn
.
∂β
∂β
L
is
n∈Nis
Then notice that if′′∂λi ( β)∂λ ( β)∂ log Lis ( β)∂ log Lisn ( β)
0=
= is
=
=
,
∂β
∂β
∂β
∂β

48

the other equations can still hold. We would then have
∂ log λis (ω, β)∂ log Lisn (ω, β)+ (η + η N )∂β∂β∂ log λi (ω )∂ log Lis (ω, β)∂ log λis (ω, β)
=
+ ηS
∂β
∂β
∂β


′
∂ log Lisn (ω, β)
0=E∂β′ ( ω, β ) ∂ log Lis0=E
∂βZ
∂ log Lis (ω, β)
Lis
ds
0=
∂β
S
log aisn (ω ) =∂ log Lis (ω, β)L ∂ log Lisn (ω )= ∑ isn.
∂β∂βn∈Nis Lis
So then,∂ log λis (ω, β)log aisn (ω ) −
∂β
!
Lisn
1 ∂ log λis (ω, β)log aisn (ω ) −.
η + ηN∂βLis∂ log Lis (ω, β)L1= ∑ isn∂βη + ηNn∈Nis Lis=1
η + ηN

∑

n∈Nis



Plugging in to the second equation∂ log λis (ω, β)∂ log λi (ω )∂ log Lis (ω, β)
=
+ ηS
∂β
∂β
∂β
ηS
∂ log λi (ω )+=∂βη + ηN=1η
1 + η +SηN

L
∑ Lisn log aisn (ω )
is
n∈Nis
ηS

∂ log λi (ω )η +η N+η∂β
1 + η +Sη−ηS ∂ log λis (ω, β)η + ηN∂β!

Lisnlog aisn (ω )n∈Nis Lis∑N!

Therefore,∂ log Lis (ω, β)1∂ log λi (ω )
1
=−
+
∂β
η + η N + ηS∂β
η + η N + ηS

49

!
Lisn
log aisn (ω ) .
∑n∈Nis Lis

And we also get
11 ∂ log λis (ω, β)∂ log Lisn (ω, β)=log aisn (ω ) −∂β
η + ηNη + ηN
∂β
"
!#
ηS
Lisn′
1
η +η N=log aisn (ω ) −log aisn′ (ω )∑ηη + ηN
1 + η +SηN n′ ∈Nis Lis−∂ log λi (ω ).
η + η N + ηS
∂β1

Finally note that,∂ log Lis (ω, β)
ds
∂β
S
"
Z
1
∂ log λi (ω )
1
=
Lis −
+
η + η N + ηS∂β
η + η N + ηS
S

0=

Z

Lis

L
∑ Lisn log aisn (ω )
is
n∈Nis

!#
ds

∂ log λi (ω )= E[log aisn (ω )] = 0.
∂β

Next, we move onto the second order.
Lemma 3. For small ex-post shocks, Yi (ℓ, m, β) is approximately given by(µz )η (ℓi )1−η (mi )η Ω(mi ),forΩ(mi ) ≡ E[ aisn (ω ) β ]ℓisn (0)∂ log ℓisn (ω, β)
2
E
log
a
(
ω
)
β
ds
isn
∑ ℓi
∂β
S n∈N
is
"
#

Z
1−η
ℓisn (0)∂ log ℓisn (ω, β) 2 2ηβ ds−∑ ℓi E
2
∂β
S n∈N
is
"
2 #
Z
1−η
ℓisn (0)∂ log ℓisn (ω, β)
−
ηN
β2 ds
∑ ℓi E
2
∂β
S n∈N
is
"
#

Z
1−η
ℓis (0)∂ log ℓis (ω ) 2 2
−
ηS
E
β ds.
2
ℓi
∂β
S

+ (1 − η )Z1ηwhere µz = E[zisn ].
50

(23)

Proof. We start by solving the model at β = 0. There is then no uncertainty so we propose
a solution where Lisn (ω ) and wisn (ω ) is constant across all ω. Firms maximize profits
taking wages as given. Therefore
zisn (1 − η ) ( Lisn ℓi )−η = wisn .
The worker then maximizes utility taking wages wisn as given. Since labor is freely mobile
in the first period, the worker will only work for firms that offer the highest wages. Thus,
wages must be equalized across all firms, so that1
Lisn =ℓizisn (1 − η )
w

 η1
.

Labor market clearing requires that
1=

1=

Z

∑ Lisn ds

S n∈N

Z

is

∑

S n∈N

is

1−η
w=
η
ℓi



1
ℓi
Z

1
zisn (1 − η ) η
ds
w
!η
1
η

∑ zisn ds
S

.

n∈Nis

We can then plug this into production. This gives
Yi (ℓ, m, 0) =Zzisn ( Lisn ℓi )1−η ds
∑
S
n∈Nis

=

Z

−1



1

1

∑ zisn w η (zisn ) η (1 − η ) ηS 1− η

ds

n∈Nis
1− η

=w

− η

(1 − η )= (ℓi )1−ηη

= (µz ) (ℓi )

Z

1− η
η

Z

1
η

∑ (zisn ) ds

S n∈N
1
η

1− η

is

∑ (zisn ) ds

S n∈N

!

!η

is

( mi )η ,1/η

where µz = E[zisn ] which we can plug in since we assume that the law of large numbers
holds across the continuum of sectors, and the number of firms is mi .

51

We then take the first derivative. That is
"Z
#
∂Yi
β
1− η
=E
∑ zisn aisn (ω ) ℓisn (ω, β) log(aisn (ω ))ds
∂β
S n∈N
"Z is
#
∂
ℓ
(
ω,
β
)
+E
∑ zisn aisn (ω )β (1 − η )ℓisn (ω, β)−η isn∂β ds .
S n∈N
is

We can then take the second derivative. That is
#
"Z
∂2 Yi
=E
zisn aisn (ω ) β ℓisn (ω, β)1−η log( aisn (ω ))2 ds
∑
∂β2
S n∈N
#
"Z is
∂ℓ (ω, β)log( aisn (ω ))ds
+ 2E
zisn aisn (ω ) β (1 − η )ℓisn (ω, β)−η isn
∑
∂β
S n∈N
is
"Z
2 #

∂
ℓ
(
ω,
β
)
isn
−E
ds
∑ zisn aisn (ω )β (1 − η )η ℓisn (ω, β)−η−1
∂β
S n∈N
is
"Z
#
2 ℓ ( ω, β )∂+E∑ zisn aisn (ω )β (1 − η )ℓisn (ω, β)−η isn∂β2 ds
S n∈N
is

Then we plug in at β = 0. We use the fact thatzisn aisn (ω ) β ℓisn (ω, β)1−η =52

1w (0)ℓisn (ω, 0).
1−η iThen, to second order production is approximately given by,
Yi (ℓ, m, β) ≈ (µz )η (ℓ)1−η (m)η
Z
1
w (0)
+
∑ ℓisn (0)E[log(aisn (ω ))] βds
1−η i
S n∈N
is


Z
∂ log ℓisn (ω, β))+ wi (0)βds∑ ℓisn (0)E
∂β
S n∈N
is
Z
i
h
1 1
2
β2 ds
+
wi (0)
a
(
ω
))
ℓ
(
0
)
E
log
(
isn
isn
∑
21−η
S n∈N
is


Z
∂ log ℓisn (ω, β)+ wi (0)ℓisn (0)Elog( aisn (ω )) β2 ds
∑
∂β
S n∈N
is
"

 #
Z
∂ log ℓisn (ω, β) 2 2
1β− η wi (0)∑ ℓisn (0)E
2
∂β
S n∈N
is
 2

Z
1
∂ ℓisn (ω, β) 2
+ wi (0)
β ds.
∑ E
2
∂β2
S n∈N
is

Then we can make a few substitutions. First, note that1
E[ aisn (ω ) β ] ≈ 1 + E[log( aisn (ω ))] β + E[log( aisn (ω ))2 ] β2 + o ( β3 ).
2Then we also note thatwi (0)ℓ = (1 − η ) (µz )η (ℓ)1−η (m)η .
We also know that to first orderZ∂ log ℓisn (ω, β))ds = 0.
∑ ℓisn (0)E
∂β
S n∈N
is

53

Therefore, we can write,1− η1 + E[ aisn (ω ) β ] − 1Zℓisn (0)∂ log ℓisn (ω, β)+ (1 − η )log( aisn (ω )) β2 ds∑ ℓi E
∂β
S n∈N
is
#
"

Z
1−η
ℓisn (0)∂ log ℓisn (ω, β) 2 2−ηβ∑ ℓi E
2
∂β
S n∈N
is
 2


Z
∂ ℓisn (ω, β) 2
1−η 1
β ds
+
∑ E
2 ℓi S n∈N
∂β2
is

η
1− η
η
= (µz ) (ℓ) (m) E[ aisn (ω ) β ]


 
Z
ℓisn (0)∂ log ℓisn (ω, β)η ∂ log ℓisn (ω, β)+ (1 − η )Elog( aisn (ω )) −
β2
∑
ℓi
∂β
2
∂β
S n∈N
is
 2


Z
1−η
∂ Lisn (ω, β) 2+E
β ds .
∑
2
∂β2
S n∈N

Yi (ℓ, m, β) ≈ (µz ) (ℓ)
η

(m)

η

is

Next, we turn to transforming the last line. We do this by noting that labor needs tosatisfy,
1=
Lis =

Z
S

Lis ds∑ Lisnn∈NisLis (ω )1=
Lis · gS
ds
Lis
S


Lisn (ω )Lis (ω ) = ∑ Lisn · g N
.
L
isn
n∈Nis
Z

54

Then taking the derivative, we find that,
0=

Z

∂Lis
ds
S ∂β

∂Lis
∂Lisn
= ∑
∂β
∂β
n∈Nis






Z  
Lis (ω ) ∂Lis
Lis (ω ) ∂Lis (ω ) Lis (ω ) ′ Lis (ω ) ∂Lis
′
0=
gS
+ gS
−
gS
ds
∂β
∂β
∂β
Lis
Lis
Lis
Lis
S






∂Lis (ω )Lisn (ω ) ∂Lisn
Lisn (ω ) ∂Lisn (ω ) Lisn (ω ) ′Lisn (ω ) ∂Lisn′gN= ∑ gN
+ gN
−
.
∂β
∂β
∂β
∂β
Lisn
Lisn
Lisn
Lisn
n∈Nis
is
We then take the second derivative, substituting in Lis (ω ) = Lis , ∂L
∂β = 0, gS (1) =g N (1) = gS′ (1) = g′N (1) = 1, gS′′ (1) = ηs and g′′N (1) = η N at β = 0. Then

0=Z

∂2 Lisds
S ∂β2∂2 Lis∂2 Lisn
=
∑ ∂β2
∂β2
n∈Nis


Z
Z
∂2 Lis (ω )∂ log Lis (ω ) 2
ds
ds = −ηS
Lis
∂β
dβ2
S
S


∂2 Lisn (ω )∂2 Lis (ω )∂ log Lisn (ω ) 2=− η N ∑ Lisn
.
∑
∂β
∂β2
∂β2
n∈N
n∈N
is

is

Therefore,
1−η
2

Z

 2



Z
∂ Lisn (ω, β) 2
∂ log Lisn (ω ) 2 2
1−η
ηN
β ds
β =−
∑ E
∑
2
∂β
∂β2
S n∈N
S n∈N
is
is


Z
1−η
∂ log Lis (ω ) 2 2
−
ηS
Lis
β ds.
2
∂β
S

Plugging this back into the second order approximation to the production function gives(23).
Then we can come back and prove lemma 1.
Lemma 1. For small ex-post shocks, if Yi (ℓ, m) has increasing returns to scale, i.e. Y1i55

dYi (αℓ,αm)
>
dα
α =1

d log w

1, then average wages are increasing in population, i.e. d log ℓ i > 0. In particular,idY (αℓ,αm)i−1d log wi
α =1
.
dα
=
dYi (αℓ,αm)
d log ℓi
−1
1−η−

dα

α =1

Proof. Production is approximately given by Y (ℓ, m) = ℓ1−η mη Ω(m). Since wages arecompetitively set, they are equal to the marginal product of labor. Thus, wℓ = (1 −η )Y (ℓ, m). Then free entry implies
ψ=Y (ℓ, m)Y (ℓ, m) − wℓ
=η
.
m
m

Thus, we can take the total differential to get
d log Y = (1 − η )d log ℓ + ηd log m +∂ log Ω(m)d log m∂ log md log w = d log Y − d log ℓ
0 = d log Y − d log m.
Solving, we get∂ log Ω(m)∂ log md log w =d log ℓ.
∂ log Ω(m)1 − η − ∂ log m

We complete the proof by noting that∂ log Ω(m)dY (αℓ, αm)
−1 =
.
dα
∂ log mα =1Then we can return to prove Proposition 1.
Proposition 1. For small ex-post shocks, Yi (ℓ, m) features increasing returns to scale if the(employment-weighted) average covariance between log employment and log productivity shocks
is increasing in the number of firms, i.e.
d
dm

"Z

∑

S n∈N

is

#
ℓisn
Cov (log aisn (ω ), log ℓisn (ω )) ds > 0.
ℓi

In particular,
Yi (ℓ, m) ≈ ℓ1−η mη Ω(m),56

where1−η2Ω(m) ≡ E[ aisn (ω )] +

Z

∑
S

n∈Nis

ℓisn
Cov (log aisn (ω ), log ℓisn (ω )) ds.
ℓi

Proof. We know from Lemma 3 thatΩ(mi ) ≡ E[ aisn (ω ) β ]∂ log ℓisn (ω, β)ℓisn (0)2+ (1 − η )log aisn (ω ) β ds
∑ ℓi E
∂β
S n∈N
is
#
"

Z
1−η
ℓisn (0)∂ log ℓisn (ω, β) 2 2−ηβ ds∑ ℓi E
2
∂β
S n∈N
is
"
2 #
Z
ℓisn (0)1−η
∂ log ℓisn (ω, β)
ηN
E
β2 ds
−
∑
2
ℓi
∂β
S n∈N
is
"
#


Z
ℓis (0)1−η
∂ log ℓis (ω ) 2 2
−
ηS
E
β ds.
2
ℓi
∂β
S
Z

And recall from Lemma 2 that"ηS
∂ log ℓisn (ω, β)1
η +η N=log aisn (ω ) −η∂βη + ηN
1 + η +Sη
We then solve for E∂ log ℓisn (ω,β) 2 2
β
∂β

N


. Defining X N ≡

57

Lisn′
log aisn′ (ω )∑n′ ∈Nis Lis

Rℓisn (0)S ∑n∈Nis ℓi E



!#
.


∂ log ℓisn (ω,β) 2 2
β
∂β


ds,

we get#∂ log ℓisn (ω, β) 2 2
β ds
XN =
∑
∂β
S n∈N
is
"
!#2 
ηS
Z
2
β
Lisn′
ℓisn (0) η +η N ds=
E log aisn (ω ) −log aisn′ (ω )
∑
∑
ηS
2
ℓi
(η + η N ) S n∈N
1
+
L
′
is
n ∈Nisη +η Nis"!#Z′ℓisn (0)
η
L
β2
S
isn
E log aisn (ω )2 − 2log aisn (ω )log aisn′ (ω )
=
∑
∑
2
ℓi
η + η N + ηS(η + η N ) S n∈N
L′isn ∈Nis
is

!2 


Z
2
2
Lisn′
β
ℓisn (0) ηS+Elog aisn′ (ω ) ∑∑2ℓiη + η N + ηS(η + η N ) S n∈N
L′isn ∈Nisisℓisn (0)
E
ℓi

Z

β2
=
( η + η N )2

−2

Z

"

∑

S n∈N

β2

(η + η N

)2 η

is

i
ℓisn (0) h
E log aisn (ω )2 dsℓi

ηS
N + η N + ηS


1−

ηS
1
2 η N + η N + ηS

ZMoving onto the variance in sectoral labor, we have XS ≡

R

!2 
Lisn′
ℓis 
E
log aisn′ (ω )  ds.
∑
L
S ℓi
is
n′ ∈Nisℓis (0)S ℓi E∂ log ℓis (ω ) 2 2
β
∂β

Then

ℓis (0)
XS =
E
ℓi
S
Z

=

"

∂ log ℓis (ω )∂ββ2( η + η N + ηS )2

2


Z

#
β2 ds

!2 
ℓisn′
ℓis (0) Elog aisn′ (ω )  ds
∑
ℓi
ℓ
S
is
′
n

58


ds.

Defining X2 ≡ (η + η N ) X N + ηS XS , we haveβ2
X2 =
η + ηN

Z

∑

S n∈N

is

i
ℓisn (0) h
E log aisn (ω )2 ds
ℓi

!2 
ℓis 
ηS
1
ηS
Lisn′
−2
1−
log aisn′ (ω )  ds
E∑
η + η N η + η N + ηS
2 η + η N + ηS
L
S ℓi
is
n′ ∈Nis


!2
Z
β2 ηS
ℓis (0) ℓisn′+
E∑′ ℓis log aisn′ (ω )  ds
( η + η N + ηS )2 S ℓ i
n
β2

β2
=
η + ηNZ

iℓisn (0) h
2
ds
a
(
ω
)
E
log
isn
∑ ℓi
S n∈N
is

!2 
Z
2
β
ℓisn′
ℓis (0) ηS−Elog aisn′ (ω )  ds.
∑
η + η N η + η N + ηS S ℓ i
ℓis
n′
Z

h
i
R
∂ log ℓisn
2 ds.
a
(
ω
)
β
Next, we solve for the weighted covariance. Define X1 ≡ S ∑n∈Nis ℓℓisn E
log
isn
∂β
i
Then we can write


Z
ℓisn
∂ log ℓisn
2
X1 =E
log aisn (ω ) β ds∑
∂βS n∈N ℓi
is
"
!!
#
ηS
Z
2
′
β
L
ℓisn
η +η Nisn
log aisn′ (ω )=E
log aisn (ω ) −log aisn (ω ) ds∑∑
ηS
η + η N S n∈N ℓi
1
+
L
′
is
n ∈Nisη +η NisZiβ2ℓisn (0) h
2
a
(
ω
)
ds
=
E
log
isn
∑ ℓi
η + η N S n∈N
is

!2 
Z
2
β
ℓisn′
ℓis (0) ηS−Elog aisn′ (ω )  ds.
∑
η + η N η + η N + ηS S ℓ iℓis′n
In particular, X1 = X2 . Therefore,Ω(m) = E[ aisn (ω ) β ] + (1 − η ) X1 −

1−η
X2
2

1−η
X1
2
Z
1−η
ℓisn
β
= E[ aisn (ω ) ] +Cov (log aisn (ω ), log ℓisn (ω )) ds,∑
2
S n∈N ℓi

= E[ aisn (ω ) β ] +iscompleting the proof.

59

Proposition 2. Suppose ex-post shocks are small, and Assumption 1 holds. Then average wages
d log w
are increasing in population, i.e. d log ℓ i > 0. And as the population goes to infinity, theseid log w

agglomeration effects disappear, i.e. limℓi →∞ d log ℓ i = 0.
iR
Proof. In order to prove this result, we need to show that S ∑n∈Nis ℓℓisn Cov (log aisn (ω ), log ℓisn (ω )) dsi
is increasing in the number of firms. Notice, from the proof of the previous proposition,β2X=η + ηNiℓisn (0) h
2
a
(
ω
)
ds
E
log
isn
∑ ℓi
S n∈N
is

!2 
Z
2
β
ηS
ℓis (0) ℓisn′−Elog aisn′ (ω )  ds,∑
η + η N η + η N + ηS S ℓ i
ℓ
is
n′
Z

R
where X ≡ S ∑n∈Nis ℓℓisn Cov (log aisn (ω ), log ℓisn (ω )) ds. Then we compute the varii
ances.
ih
E log aisn (ω )2 β2 = E [log aisn (ω )]2 β2 + Var( β log aisn (ω ))2= E β log ãisn (ω ) + β log Ãis (ω ) + Var( β log ãisn (ω ) + β log Ãis (ω ))2= σS2 + σN,

where the last line follows from the fact that the idiosyncratic shock and the sector shock
are independent and their log means are zero. Similarly,
"
!2 
#2
′
′
L
L
isn
isn
log aisn′ (ω ) β  = E ∑β log aisn′ (ω ) + Var
E ∑
L
L
′
′
is
is
n ∈Nisn ∈NisLisn′
β log aisn′ (ω )∑L′isn ∈Nis!

Lisn′
β log ãisn′ (ω )Lisn ∈Nis
2Lisn′= Var β log Ãis (ω ) + ∑Var( β log ãisn′ (ω ))Lisn′ ∈Nis
"

2 #
′
L
isn
2
= σS2 + ∑σN.
Lisn′ ∈Nis= Var β log Ãis (ω ) +

60

∑
′

!

Therefore, we can write


Z


1
1
ηS
ℓis
2
2
2
2
σ + σN −X=σS +HH Iis dsσNη + ηN S
η + η N η + η N + ηS
S ℓi


Z
ℓis
1
ηS
1
2
2
σ +
1−
HH Iis ds σN=,
η + η N + ηS S η + η N
η + η N + ηS S ℓiwhere HH Iis ≡ ∑n′ ∈Nis



Lisn′
Lis

2 

.

R
Then, to complete the prooof, we need to show that S ℓℓis HH Iis ds is decreasing in m.
iThen we also need to show that its derivative converages to a constant. Recall that,
1 η1Lisn = zisn
ℓi
where
w=



1−η
w

1
η

,

1−η η
η
η m (µz )
ℓi

1
η

1
η

and µz = E[zisn ]. Therefore, Lisn = zisn /(mµz ). And we can writeZS

Lis HH Iis ds =

=

Z
S

Lis

( Lisn )2ds∑2
n∈Nis ( Lis )

Z

!

∑

S

n′ ∈N

1/η

zisn′ /(mµz )
is

∑ 

n∈Nis

1/η

1/η

(zisn /(mµz ))22
1/η
∑n′ ∈Nis zisn′ /(mµz )1/η∑n′ ∈Nis zisn′ ∑n∈Nis (zisn )2
Nis

 ds
Nis
1/η 2
S
∑n′ ∈Nis zisn′




1
=
mµz

Z

∞

=

1/η
 ∑n′ ∈N zisn′

1 
N
E
∑
µ z N =0 m
"

N

ds



1/η
∑n∈N (zisn )2 
2  p( N, m) .
1/η
∑n′ ∈N zisn′

#

1/η

2
zisn′ ∑n∈N (z1/η
∑ ′
isn )
I will denote by ψN ≡ E n ∈N
2 . By assumption, ψ N is decreasing in N.
N1/η∑n′ ∈N zisn′∞n
I will also define F ( N, m) ≡ ∑∞n= N m p ( n, m ). Notice that because m = ∑ N =0 N p ( N, m ),this is 1 minus a CDF. Furthermore, F (1, m) = 1 for all m.
Then consider m′ and m′′ with m′ > m′′ . By our assumption on FOSD, F ( N, m′ ) ≥F ( N, m′′ ) for all N. Notice that this must be strict for some N. If not, then mN′ p( N, m′ ) =
N
m′
′′
′
′′
′
′′
m′′ p ( N, m ), i.e. p ( N, m ) = m′′ p ( N, m ). This is contradiction for m > m because it
61

would be impossible for both p( N, m′ ) and p( N, m′′ ) to sum to 1 across all N.
Notice that we can writeZ

1 ∞ N
p( N, m)ψN
Lis HH Iis ds =
µz N∑
S
=0 m

=

1
(ψ1 F (1, m) + (ψ2 − ψ1 ) F (2, m) + (ψ3 − ψ2 ) F (3, m) · · · ) .
µz

Therefore,
X=

=

1 ∞ N
1 ∞ N
′
p
(
N,
m
)
ψ
−
p( N, m′′ )ψN
N
∑
′
′′
µz N∑
m
µ
m
z N =0=01
ψ1 ( F (1, m′ ) − F (1, m′′ )) + (ψ2 − ψ1 )( F (2, m′ ) − F (2, m′′ )) + · · ·µzThen F (1, m′ ) = F (1, m′′ ), ψn − ψn−1 < 0, and F (n, m′ ) − F (n, m′′ ) ≥ 0 for n ≥ 2, with
one strict. Thus,
1 ∞ N
1 ∞ N
′
p( N, m )ψN −p( N, m′′ )ψN < 0.
∑∑′′′
µ z N =0 m
µ z N =0 m
It follows that average HHI is decreasing in the number of firms.
To complete the proof, we need to show that as the population goes to infinity, these
agglomerations effects disappear. In particular, we need to show that
d
− dm
1− η
E[ aisn (ω )] + 2

h

hR

1

ℓis
S ℓi HH Iis ds

12
η +η N +ηS σS + η +η N



i

2
σN

R
η
1 − η +ηNS+ηS S ℓℓis HH Iis ds
i



2
σN

i → 0.

We start by showing that ψN → 0 as N → ∞. Then we will show that that implies
S Lis HH Iis ds → 0. Note that because 1 − Fiz is regularly varying, there exists α andslowly varying function L20 such that 1 − Fiz ( x ) = x −α L( x ). Starting with ψN , note that
we can rewrite it,



R

1/η

a N ∑n∈N (zisn )21 ,ψN = E  2
1/η
aN
N
−
1
N
∑ z
n isn

2/η

−1
where a N is definedso that P(zisn > a N ) = N . Then by the strong law of large
1/η

number, N −1 ∑n zisn

1/η

→ E[zisn ] a.s.. If E[(zisn )2/η ] exists, then we can cancel a N and

20 A function is slowly varying if for every a > 0, L( ax ) → 1 as x → ∞.
L( x )

62



2/η
2/η
use the strong law of large numbers to note that N −1 ∑n zisn → E[zisn ]. Then that is
multiplied by N −1 and the part in expectations converges to 0 almost surely.
If the variance does not exist, we need to use Levy’s Theorem.21 By Lévy’s theorem,
1/η∑n∈N (zisn )2
converges in distribution to a non-degenerate distribution. That simply leavesaN2
a N /N . Note thataN2/η= a N P(zisn > a N )2
2
N
1/2 2
α
= a N a−N L( a N ) .
This converges to 0 as a N → ∞ if α > 1. But note that since the mean exists, α must be
greater than 1. Further note that a N → ∞ as N → ∞ otherwise the variance would exist.
Thus, the part in the expectations must converge to 0 in distribution.
Np( N, m) → 0 as m → 0. Then
Note that p( N, m) ≤ 1 for all N and m. Therefore, mtake ε > 0. Since ψN → 0, there exists a N such that ψN < µz 2ε for all N > N. Next note′εthat for every N ′ < N, there exists a m N ′ such that Nm p( N ′ , m) ≤ µz 2Nψ
for all m > m N ′ .
N′
Take m = max N ′ ≤ N m N ′ . Then for m > m,Z

1 ∞ N
Lis HH Iis ds =p( N, m)ψN
µz N∑
S
=0 m
∞
1
1 N N
p( N, m)ψN + ψN ∑ p( N, m)
≤
µz N∑
µz
=0 m
N +1

≤

1 N N
1
p( N, m)ψN + ψN∑
µ z N =0 m
µz

≤

1 N N
ε
p( N, m)ψN +∑µ z N =0 m
2
N

ε
ε
ψN +
2
N =0 2Nψ N

≤ ∑
= ε,
completing the proof.
21 See Durrett (2019).

63

B

Oligopsony

B.1

Environment

Under oligopsony, we assume that firms no longer take as given the wages in every
state of the world. Instead, firms maxmize profits taking as given the labor demanded by
the other firms in its sector and the worker’s opportunities in other sectors. For simplicity,
we use the constant elasticity version of the model, i.e. gS ( x ) = x1+ηS and g N ( x ) = x1+ηN .
The problem of firm n is thenmax
Lisn (ω ),Lisn ,Lis (ω ),Lis ,wisn (ω )E [ aisn (ω ) f ( Lisn (ω )ℓi ) − wisn (ω ) Lisn (ω )ℓi ]

where


wisn (ω ) = λi (ω )(1 + ηS )(1 + η N )"Lis (ω )Lis
 1+ η N

 ηS 

Lisn (ω )
Lisn

η N
,




 !#
Lis (ω ) ηS
Lisn (ω )Lisn (ω )Lisn (ω ) ηN
−λis = E λi (ω )(1 + ηS )−(1 + η N )
Lis
Lisn
Lisn
Lisn
!#
"




Lis (ω ) 1+ηS Lis (ω )Lis (ω ) ηS
+ λi ,
λis = E λi (ω )−(1 + ηS )
Lis
Lis
Lis


and

∑ Lisn′Lis =n′ ∈Nis1=

ZS

Lis ds
Lisn′ (ω ) 1+ηN
Lis (ω ) = ∑ Lisn′ ·Lisn′n′ ∈NisZ
Lis (ω ) 1+ηS
1 = Lis ·
.
Lis
s


B.2

Linear-Quadratic Approximation

We do a quadratic approximation to the profit function and linear approximation to
the constraints around the point with no ex-post shocks just as we did in the competitiveenvironment.

64

B.2.1

Profit Function

DefineΠisn (ω ) = aisn (ω ) f ( Lisn (ω )ℓi ) − wisn (ω ) Lisn (ω )ℓi .
Then, starting with the first order, we have∂Πisn
= aisn (ω ) f ( Lisn (ω )ℓi )∂ log aisn (ω )∂Πisn
= aisn (ω ) f ′ ( Lisn (ω )ℓi ) Lisn (ω )ℓi − wisn (ω ) Lisn (ω )ℓi∂ log Lisn (ω )∂Πisn
= −wisn (ω ) Lisn (ω )ℓi .
∂ log wisn (ω )Then the second order derivatives are∂2 Πisn (ω )= aisn (ω ) f ( Lisn (ω )ℓi )∂ log aisn (ω )2∂2 Πisn (ω )= aisn (ω ) f ′ ( Lisn (ω )ℓi ) Lisn (ω )ℓi∂ log aisn (ω )∂ log Lisn (ω )∂2 Πisn (ω )= aisn (ω ) f ′ ( Lisn (ω )ℓi ) Lisn (ω )ℓi + aisn (ω ) f ′′ ( Lisn (ω )ℓi ) ( Lisn (ω )ℓi )22∂ log Lisn (ω )− wisn (ω ) Lisn (ω )ℓi∂2 Πisn (ω )= −wisn (ω ) Lisn (ω )ℓi∂ log Lisn (ω )∂ log wisn (ω )∂2 Πisn (ω )= −wisn (ω ) Lisn (ω )ℓi .
∂ log wisn (ω )2Then the second order approximation around β = 0 is then11
Πisn ∝ E −ŵisn (ω ) + âisn (ω ) L̂isn (ω ) − η L̂isn (ω )2 − L̂isn (ω )ŵisn (ω ) − ŵisn (ω )222

where x̂ denotes log deviations from β = 0 value. In order to get a quadratic-linear
approximation, we need to transform this function to be completely second order. Thus,
we need a new form for ŵisn (ω ). Note that the first constraint is already log-linear, so thatŵisn (ω ) = λ̂i (ω ) + ηS L̂is (ω ) − Lˆ is + η N L̂isn (ω ) − Lˆ isn .

65

Meanwhile, the other two equations determining labor supply give"
λis = η N (1 + ηS )E λi (ω )and"
λ i = ηS E λ i ( ω )Lis (ω )Lis

Lis (ω )Lis

 ηS 

Lisn (ω )Lisn 1+ η N # 1+ ηS #+ λis .

Substituting out λis we can write2λi  ˆ
λi 1  ˆ
λi − λ̂i (ω ) +λi − λ̂i (ω )λi ( ω )λi ( ω ) 2 η (1 + η )2 
2 

S
S
ˆ
ˆ
= E ηS (1 + ηS ) L̂is (ω ) − Lis +L̂is (ω ) − Lis
2
h
 


i
+ E η N (1 + ηS ) ηS L̂is (ω ) − Lˆ is + (1 + η N ) L̂isn (ω ) − Lˆ isn

 


2 
1
ˆ
ˆ
+ E η N (1 + ηS ) ηS L̂is (ω ) − Lis + (1 + η N ) L̂isn (ω ) − Lisn
2
h



i
= E ηS (1 + η N )(1 + ηS ) L̂is (ω ) − Lˆ is + η N (1 + ηS )(1 + η N ) L̂isn (ω ) − Lˆ isn2 ηS (1 + ηS )2 ˆ+EL̂is (ω ) − Lis
2

 


2 
1
ˆ
ˆ
+ E η N (1 + ηS ) ηS L̂is (ω ) − Lis + (1 + η N ) L̂isn (ω ) − Lisn
2
Then dropping the λi (ω ) and λi as the firm has no effect on it, profits are given by2 
ηS (1 + ηS )2 
1ˆΠisn =
L̂is (ω ) − LisE(1 + η N )(1 + ηS )
2



2 
1
η N (1 + ηS )  ˆˆ+EηS L̂is (ω ) − Lis + (1 + η N ) L̂isn (ω ) − Lisn
(1 + η N )(1 + ηS )
2


1
1
2
2
+ E âisn (ω ) L̂isn (ω ) − η L̂isn (ω ) − L̂isn (ω )ŵisn (ω ) − ŵisn (ω )22Then we can do a first order approximation to the constraints
B.2.2

Constraints

We haveŵisn (ω ) = λ̂i (ω ) + ηS L̂is (ω ) − Lˆ is + η N L̂isn (ω ) − Lˆ isn
66


h
i
λi  ˆ
λi − λ̂i (ω ) = (1 + η N )(1 + ηS )E ηS ( L̂is (ω ) − Lˆ is ) + η N ( L̂isn (ω ) − Lˆ isn )λi ( ω )along withLˆ is =
0=L̂is (ω ) =
0=
B.2.3

Lisn′ ˆLisn′
n ∈Nis Lis

∑
′

Z
S

Lˆ is ds

Lisn
L̂isn′ (ω )n ∈Nis Lis

∑
′

Z
S

L̂is (ω )ds.

Combining Things

Then to first order, λˆ i = λ̂i (ω ) = Lˆ is = Lˆ isn = 0. We can then make those substitutions
and also substitute in for ŵisn (ω ). We can then get
i 1 η
h
i
1 1 + ηS hN
ηS
E L̂is (ω )2 +E (ŵisn (ω ) + L̂isn (ω ))2
2 1 + ηN
2 1 + ηN
1
1
+ E[ âisn (ω ) L̂isn (ω )] − ηE[ L̂isn (ω )2 ] − E[ L̂isn (ω )ŵisn (ω )] − E[ŵisn (ω )2 ]2
2hi
1 1 + ηS
1
= E[ âisn (ω ) L̂isn (ω )] + ηS
E[ L̂is (ω )2 ] − η E L̂isn (ω )2
2 1 + ηN
2
h
i
1 ηN
1
1
+
E (ŵisn (ω ) + L̂isn (ω ))2 − E[(ŵisn (ω ) + L̂isn (ω ))2 ] + E[ L̂isn (ω )]2 1 + ηN22hi
1 1 + ηS
1
= E[ âisn (ω ) L̂isn (ω )] + ηS
E[ L̂is (ω )2 ] + (1 − η ) E L̂isn (ω )2
2 1 + ηN
2
h
i
1 1
−
E (ŵisn (ω ) + L̂isn (ω ))2
2 1 + ηNΠisn =

subject to the constraintL̂is (ω ) =Lisn′
L̂isn′ (ω ),n ∈Nis Lis

∑
′

and
ŵisn (ω ) = ηS L̂is (ω ) + η N L̂isn .
Taking the first order conditions, gives
0 = âisn (ω ) + (1 − η ) L̂isn (ω ) −1
L(ŵisn (ω ) + L̂isn (ω )) + λ L (ω ) isn + λw (ω )η N
1 + ηNLis67

0 = ηS
0=−

1 + ηSL̂ (ω ) − λ L (ω ) + λw (ω )ηS
1 + η N is1(ŵ (ω ) + L̂isn (ω )) − λw (ω )1 + η N isn

Substituting in
0 = âisn (ω ) + (1 − η ) L̂isn (ω ) −+ ηS

1(ŵ (ω ) + L̂isn (ω ))1 + η N isn

LL
1 + ηS
L̂is (ω ) isn + λw (ω )ηS isn + λw (ω )η N
1 + ηN
Lis
Lis



1
Lisn
= âisn (ω ) + (1 − η ) L̂isn (ω ) −(ŵ (ω ) + L̂isn (ω )) 1 + η N + ηS
1 + η N isn
Lis

+ ηS

L
1 + ηS
L̂is (ω ) isn
1 + ηN
Lis

Then we will substitute in for ŵisn (ω ) = η N L̂isn + ηS L̂is . This gives


1
Lisn
0 = âisn (ω ) + (1 − η ) L̂isn (ω ) −(ηS L̂is (ω ) + (1 + η N ) L̂isn (ω )) 1 + η N + ηS
1 + ηNLis
1 + ηS
L
L̂is (ω ) isn
1 + ηNLis
Lisn= âisn (ω ) − η + η N + ηS
L̂isn (ω )
Lis

+ ηS

!
ηS2 Lisn
1 + ηS Lisn
+ ηS
L̂is (ω )+ − ηS −1 + η N Lis
1 + η N Lis




Lisn
1 Lisn0 = âisn (ω ) − η + η N + ηS
L̂isn (ω ) − ηS 1 −L̂is (ω )1 + η N Lis
Lis
Then we defineΨisn ≡1

η + η N + ηS LLisn
is
1 LisnΩisn ≡ 1 −Ψisn .
1 + η N Lis

68

Then some tedious algebra implies




Ψisn′
 L ′
L̂isn (ω ) = Ψisn âisn (ω ) − ηS Ωisn ∑ isn
L ′′n′ Lis 1 + η ∑ ′′ isn Ω
S

n

Lis

isn′′


âisn′ (ω ) .

Then summing up



Ψisn′
 L ′
L̂is (ω ) = ∑ isn
L ′′n′ Lis 1 + η ∑ ′′ isn Ω
S

B.3

n

Lis

isn′′


âisn′ (ω ) .

Adapted Propositions

Next, we need to compute Ω(m). While the final covariance form no longer holds, theformulaΩ(m) = E[ aisn (ω )] + (1 − η )−1−η(η + η N )
2

−

1−η
ηS
2

Z
S

Z
S

Z
S

Lis

Lis


Lisn
Cov L̂isn (ω ), âisn (ω ) dsn∈Nis Lis∑

Lisn
Var( L̂isn (ω ))dsn∈Nis Lis∑(24)

Lis Var( L̂is (ω ))ds

still holds. This is our adjusted Proposition 1.
Proposition 1’. For small ex-post shocks and oligopsonistic firms, Yi (ℓ, m) features increasing
retuns to scale if Ω(m) is increasing in m, where Ω(m) is defined as in equation (24). In particular,
Yi (ℓ, m) ≈ ℓ1−η mη Ω(m).
Next, we need to go through and calculate each of the statistics in order to recover a69version of proposition 2. We have
X1 ≡

=

Lisn
Cov( L̂isn (ω ), âisn (ω ))n∈Nis Lis∑

Lisn
Ψisn Var( âisn (ω ))n∈Nis Lis


∑



Lisn
Ψisn′
 L ′
Ωisn ∑ isn
L ′′n∈Nis Lisn′ Lis 1 + η ∑ ′′ isn Ω− ηS ∑

S

n

Lis

isn′′


Cov( âisn (ω ), âisn′ (ω ))

2 . For n ̸ = n′Note that Var ( âisn (ω )) = σS2 + σN

Cov( âisn (ω ), âisn′ (ω )) = Cov log Ãis (ω ) + log ãisn (ω ), log Ãis (ω ) + log ãisn′ (ω )= σS2 .
Therefore,
2
X1 = (σN+ σS2 )

L
∑ Lisn Ψisn
is
n∈Nis

!

!





Lisn
Ψisn′

 L ′
Ωisn ∑ isnL′′n∈Nis Lisn′ Lis 1 + ηS ∑n′′ isn Ωisn′′
L

 is

2
Lisn′
Ωisn′ Ψisn′
2 − ηS σN ∑
.
L
Lis
n′
1 + ηS ∑n′′ Lisn′′ Ωisn′′− ηS σS2

∑

is

70



Next we do the variances.
X2 =

Lisn
Var( L̂isn (ω ))n∈Nis Lis

∑

=

Lisn
Cov(Ψisn âisn (ω ) − ηS Ωisn L̂is (ω ), Ψisn âisn (ω ) − ηS Ωisn L̂is (ω )0n∈Nis Lis=

Lisn 2
Ψisn Cov( âisn (ω ), âisn (ω ))n∈Nis Lis

∑

∑

Lisn
Ψisn Ωisn Cov( âisn (ω ), L̂is (ω ))n∈Nis Lis

− 2ηS ∑

Lisn 2Ωisn Cov( L̂is (ω ), L̂is (ω ))n∈Nis Lis!
Lisn 2∑ L Ψisn (σS2 + σN2 )
is
n∈Nis


+ ηS2 ∑
=



Lisn
L ′
Ψisn′

Ψisn Ωisn Cov  âisn (ω ), ∑ isn
L ′′n∈Nis Lisn′ Lis 1 + η ∑ ′′ isn Ω− 2ηS ∑

S

Lis

Lisn 2Ωisn Var( L̂is (ω ))
L
is
n∈Nis
!
Lisn 2∑ L Ψisn (σS2 + σN2 )
is
n∈Nis


!
Ψisn′
L
 L ′

− 2ηS σS2 ∑ isn Ψisn Ωisn ∑ isnL′′n∈Nis Lisn′ Lis 1 + ηS ∑n′′ isn Ωisn′′
Lis



2
Ωisn′ Ψ2isn′
Lisn′

2 
− 2ηS σN ∑

L
′′
Lis
n′
1 + ηS ∑n′′ Lisn Ωisn′′+ ηS2 ∑=nisL
+ ηS2 ∑ isn Ω2isn Var( L̂is (ω ))n∈Nis Lis

71

isn′′


âisn′ (ω )

We next need to find Var( L̂is (ω )). We have
X3 = Var( L̂is (ω ))




Ψisn′
 L ′
= Cov ∑ isn
L ′′n′ Lis 1 + η ∑ ′′ isn Ω
S

n

Lis

isn′′

Lisn′
Ψisn′
L ′′
n′ Lis 1 + η ∑ ′′ isn Ωâisn′ (ω ), ∑

S

2


Ψisn′
 L ′
= σS2 ∑ isn
L ′′n′ Lis 1 + η ∑ ′′ isn Ω
S

n

Lis

isn′′



2


2
+ σN

L

isn

∑
L
′

n ∈Nis

is

′

Ψisn′
1 + ηS ∑n′′Lisn′′Ωisn′′
Lis

72


 .

n

Lis

isn′′


âisn′ (ω )

Putting everything together!
L2Ω(m) = E[ aisn (ω )] + (1 − η )(σN+ σS2 ) Lis ∑ isn Ψisn ds
Sn∈Nis Lis


!
Z
L
Ψisn′

 L ′
− (1 − η )ηS σS2 Lis ∑ isn Ωisn ∑ isn
 ds
L
′′
S
n∈Nis Lisn′ Lis 1 + ηS ∑n′′ isn Ωisn′′
L
 is

2

Z
Lisn′
Ωisn′ Ψisn′2− (1 − η )ηS σN
Lis ∑
 ds
L
Lis
S
n′
1 + ηS ∑n′′ Lisn′′ Ωisn′′
is
!
Z
1−η
L
2
−
(η + η N )(σS2 + σN
) Lis ∑ isn Ψ2isn ds
2Sn∈Nis Lis


!
Z
L
Ψisn′
 L ′

+ (1 − η )(η + η N )ηS σS2 Lis ∑ isn Ψisn Ωisn ∑ isn
 ds
L ′′
S
n∈Nis Lisn′ Lis 1 + ηS ∑n′′ isn Ωisn′′
Lis



2
Z
Ωisn′ Ψ2isn′
Lisn′


2
Lis ∑+ (1 − η )(η + η N )ηS σN
 ds
L
′′
L
S
isn
′
is
n
1 + ηS ∑n′′ L Ωisn′′
is

2
Z

1−η
ηS ((η + η N )ηS − 1) σS2
−
2

Z

Ψisn′
 L ′
Lis ∑ isn
L ′′Sn′ Lis 1 + η ∑ ′′ isn Ω
S

n

Lis

isn′′


 ds
2



−

1−η
2
ηS ((η + η N )ηS − 1) σN
2

Z
S

Lis

L

isn

∑
L
′

n ∈Nis

is

′

Ψisn′
1 + ηS ∑n′′Lisn′′Ωisn′′
Lis


 ds.
(25)

Then instead of the assumption that average HHI is decreasing in the number of firms,
we need to assume that the market power adjusted HHI is decreasing in the number offirms.
Proposition 2’. Suppose equation (25) is decreasing in m. Then average wages are increasing in
d log w
population, i.e. d log ℓ i > 0. And as the population goes to infinity, these agglomeration effectsid log w

disappear, i.e. limℓi →∞ d log ℓ i = 0.
i

73

C

Calibration

We calibrate the model using observed variance of log earnings. We calculate thosevalues in the model here. Throughout this section we will use x̂ for log deviations.

C.1

Competitive Equilibrium

Recall that ŵisn (ω ) = âisn (ω ) − η ℓ̂isn (ω ). Therefore
Var log∑ wisn (ω )ℓisn (ω )

!!

n∈Nis

!

ℓisn 
≈ Var ∑
ŵisn (ω ) + ℓ̂isn (ω )ℓn∈Nis is!
ℓisn 
= Var ∑âisn (ω ) + (1 − η )ℓ̂isn (ω )n∈Nis ℓis
ℓisn
âisn (ω )= Var∑n∈Nis ℓis
"
!# 
1−η
ηS
ℓisn′
+
â (ω ) −âisn′ (ω )
η + η N isnη + η N + ηS n ′∑∈Nis ℓisℓisn 1 + η N= Varâ (ω )∑
η + η N isnn∈Nis ℓis
! 
1−η
ℓisn′
ηS
−
âisn′ (ω )
η + η N η + η N + ηS n ′∑∈Nis ℓis!!

1−η
1 + ηN
ηS
ℓisn′
= Var
−
âisn′ (ω )∑
η + ηNη + η N η + η N + ηSn′ ∈Nis ℓis

Then we break this up using covariances. Notice that

ℓisn′
ℓ ′
âisn′ (ω ), ∑ isn âisn′ (ω )Cov∑n′ ∈Nis ℓisn′ ∈Nis ℓis!

= σS2 +

∑

n∈Nis

ℓisn
ℓis

!2
2
σN
.

Thus,
Var log∑ wisn (ω )ℓisn (ω )

n∈Nis

!!



=

1 + ηN1−ηηS−
η + ηNη + η N η + η N + ηS

74

2 "

σS2 +

Z

!

#

ℓis
2
HH Iis ds σN
.
S ℓi

C.2

Oligopsony

Note thatlog∑ wisn (ω )ℓisn (ω )

n∈Nis

!


ℓisn 
ŵ
(
ω
)
+
ℓ̂
(
ω
)
isn
isn
∑
n∈Nis ℓiℓ 
= ∑ isn ηS ℓ̂is (ω ) + η N ℓ̂isn (ω ) + ℓ̂isn (ω )n∈Nis ℓi≈ℓisn
ℓ̂isn (ω )n ℓis= ηS ℓ̂is (ω ) + (1 + η N ) ∑= (1 + η N + ηS )ℓ̂is (ω )




Ψisn
 ℓ
= (1 + η N + ηS ) ∑ isnℓn ℓis 1 + ηS ∑ ′ isn′ Ω
n

ℓis

isn′


âisn (ω )

Therefore,Z
S

Lis Var log∑ wisn (ω )ℓisn (ω )

n∈Nis



!!


≈ (1 + η N + ηS )2 


Z

2

ℓisn
Ψisn
2
 ds
 σS
ℓ
′
isn
ℓ
n
is 1 + ηS ∑ ′Ωisn′

Lis ∑Sn+ (1 + η N + ηS )2 D

Solving the Quantitative Model

D.1

Summarizing the Equations

ZS

Lis ∑ nℓis

2Yi = zi (ℓi )1−η (mi )η Ω(mi ),



ℓisn
Ψisn
2
 ds
 σN
ℓ
′
isn
ℓis 1 + ηS ∑ ′
Ωisn′
n

ℓis

As shown above, production is given by(26)

where Ω depends on if it is a competitive equilibrium or oligopsony. As payments tolabor are given by
wi ℓi = (1 − η )Yi ,(27)
and firms get the remaining production. Average profits must be equal to the fixed costof entering, therefore
ηYi
ψi =
.
(28)
mi
75



Utility of living in location i is,
Ui = ui wi .

(29)

People are free to move wherever they would like so that


ℓi =

Ui
U

θ

where
"
U=

∑ (Ui )θ

ℓ,

(30)

#1
θ

.

(31)

i ∈I

Then spillovers in amenities and production are
ui = ui (ℓi )γu ,(32)

zi = zi (ℓi )γz .

(33)

and

D.2

Exact Hat Algebra

Rewriting the model into changes from an initial equilibrium, we get the following.
Ŷi = ẑi (ℓ̂i )1−η +γz (m̂i )η

Ω(m̂i mi )
Ω ( mi )ŵi ℓ̂i = Ŷi

(35)

Ŷi
m̂i

(36)

ψ̂i =

Ûi = ûi (ℓ̂i )γu ŵi
 θ
Ûi
ℓ̂i =
ℓ̂
Û
"
#1
θ
ℓi
θ
Û = ∑ (Ûi )
ℓ
i ∈I

D.3

(34)

(37)
(38)

(39)

Counterfactual Computational Algorithm

We choose the equilibrium that has the most number of locations with a nonzero
amount of labor. We take as given Û, and we find what that implies for labor and en76

try in location n. Notice that, combining the equations, we can writeℓ̂i = Ûiθ Û −θ ℓ̂θ= ûi (ℓ̂i )γu ŵi Û −θ ℓ̂θ= ûi (ℓ̂i )γu −1 Ŷi Û −θ ℓ̂θγu − η + γzη Ω ( m̂i mi )(m̂i )
Û −θ ℓ̂.
= ẑi ûi (ℓ̂i )Ω ( mi )Then solving for ℓ̂i ,ẑi ûi (m̂i )ℓ̂i =η Ω ( m̂i mi )

Ω ( mi )

Û

−1

ℓ̂

1
θ



1
1 −γ +η −γ
u
z
θ

.

We then return to the free entry condition,
ψ̂i =

Ŷi
m̂i
Ω(m̂ m )=ẑi (ℓ̂i )1−η +γz (m̂i )η Ω(mi )iim̂i

 1− η + γz
1 −γ +η −γ
1
Ω
(
m̂
m
)
u
z
η −1 Ω ( m̂i mi )ii
= ẑi ẑi ûi (m̂i )η
m̂iÛ −1 ℓ̂ θ θΩ ( mi )Ω ( mi )11(γu − θ )(1−η )+γz  θ − γu +1
1− η + γz1
11Ω(m̂i mi ) θ −γu +η −γz− γu + η − γz1
= ẑi ẑi ûi Û −1 ℓ̂ θ θ −γu +η −γz m̂i θΩ ( mi )1
1 − γ +1
(γu − )(1−η )+γz
u
− 1 θ

 1 1− η + γz
−1 θ
1
−
γ
+
η
−
γ
u
z
− γu + η − γz
− 1 θ θ − γu + η − γzθ= ẑi ẑi ûi Û ℓ̂miΩ ( mi ) θ(γu − 1θ )(1−η )+γz
· (m̂i mi )1 −γ +η −γuzθΩ(m̂i mi )Then we set up the algorithm.
(i) Guess Û.

77

1 − γ +1uθ
1 −γ +η −γuzθ(ii) For each location i, find m̂i such thatẑi ẑi ûi Û −1 ℓ̂1θ 1 1− η + γzθ− γu + η − γzψ̂i(γu − 1θ )(1−η )+γz−
mi

1 −γ +η −γuzθ

1 − γ +1u

Ω ( mi )−1 θθ− γu + η − γz(γu − 1θ )(1−η )+γz
= (m̂i mi )1 −γ +η −γuzθΩ(m̂i mi )1 − γ +1uθ
1 −γ +η −γ
u
z
θ

.

(iii) Then we can find the implied labor in location i,


ℓ̂i =

ẑi ûi (m̂i )η Ω ( m̂i mi )

Ω ( mi )

Û

−1

ℓ̂

1
θ



1
1 −γ +η −γ
u
z
θ

,

and the implied utility
Ûi = ûi (ℓ̂i )γu −η +γz ẑi (m̂i )η

Ω(m̂i mi ).
Ω ( mi )(iv) Then we can back out how Û ′ must have changed based on those implied changes
in utility
#1
"
θ
ℓ
i
θ
′
Û = ∑ (Ûi )ℓi ∈I(v) Update the guess of Û and iterate until convergenceÛ new = αÛ ′ + (1 − α)Û.

D.4

Optimal Policy - Exact Hat Algebra

In the first best, profits should not equal the fixed cost of entering. Instead, the marginal
product of another firm should equal the fixed cost. That is
i
d h
1− η
η
ψi =
z (ℓ ) (mi ) Ω(mi )dmi i i= ηzi (ℓi )1−η (mi )η −1 Ω(mi ) + zi (ℓi )1−η (mi )η Ω′ (mi )ηYi
1 ∂ log Ω(mi )
=
1+
.
mi
η ∂ log mi78

We use x̃ to denote differences from the original equilibrium. We assume that this is paid
for using a tax proportional to income. Therefore, the wages derived here are correct but
the utility needs to be adjusted to take into account the taxes necessary to subsidize firm
entry.
Ỹi = z̃i (ℓ̃i )1−η +γz (m̃i )η

Ω(m̃i mi )
Ω ( mi )(40)w̃i ℓ̃i = Ỹi
ψ̃i =

Ỹi
m̃i


1+

(41)

1 ∂ log Ω(mi )η ∂ log mi(42)Ũi = ũi (ℓ̃i )γu w̃i
 θ
Ũi
ℓ̃
ℓ̃i =
Ũ
#1
"
θ
ℓi
θ
Ũ = ∑ (Ũi )
ℓ
i ∈I

D.5

(43)
(44)

(45)

Optimal Policy Computational Algorithm

Algebra similar to that for the counterfactual exact hat algebra gives


ℓ̃i =

z̃i ũi (m̃i )η Ω ( m̃i mi )

Ω ( mi )

Û

−1

ℓ̂

1
θ



1
1 −γ +η −γ
u
z
θ

.

Free entry implies
 1 1− η + γz1ψ̃i = z̃i z̃i ũi Ũ −1 ℓ̃ θθ− γu + η − γz−mi(γu − 1θ )(1−η )+γz
1 −γ +η −γuzθ(γu − 1θ )(1−η )+γz
· (m̂i mi )1 −γ +η −γuzθΩ(m̂i mi )1 − γ +1u

Ω ( mi )1 − γ +1uθ
1 −γ +η −γ
u
z
θ



−1 θ
θ

− γu + η − γz

1 ∂ log Ω(m̃i mi )1+
η∂ log mi

The computational algorithm then follows the counterfactual one closely, but with theadjusted entry condition.
(i) Guess Û.

79

(ii) Find m̃i such thatz̃i z̃i ũi Ũ −1 ℓ̃

1
θ

ψ̃i
(γu − 1θ )(1−η )+γz−

 1 1− η + γzθ− γu + η − γzmi

1 −γ +η −γuzθ

1 − γ +1u

Ω ( mi )−1 θθ(γu − 1θ )(1−η )+γz
= (m̃i mi )1 −γ +η −γuzθΩ(m̃i mi )− γu + η − γz
1 − γ +1uθ
1 −γ +η −γ
u
z
θ



1 ∂ log Ω(m̃i mi )1+
η∂ log mi.

(iii) Then we can find the implied labor in location i,


ℓ̃i =

z̃i ũi (m̃i )η Ω ( m̃i mi )

Ω ( mi )

Û

−1

ℓ̃

1
θ



1
1 −γ +η −γ
u
z
θ

,

and the implied utility
Ũi = ũi (ℓ̃i )γu −η +γz z̃i (m̃i )η

Ω(m̃i mi ).
Ω ( mi )(iv) Then we can back out how Û ′ must have changed based on those implied changes
in utility#1"θℓŨ ′ = ∑ i (Ũi )θ
ℓi ∈I(v) Update the guess of Ũ and iterate until convergenceŨ new = αŨ ′ + (1 − α)Ũ.

80

E

Data Appendix

E.1

Comtrade DataFirst, we take the annual value of traded goods from 1980 to 2016 across 4-digit product categories in SITC Rev. 2. Second, we convert them into the HS code, using a cleaner
provided by Feenstra and Romalis (2014). Their cleaner resolves inconsistencies in favorof importer’s reports, corrects values known to be inaccurate, accounts for re-exports ofChinese goods through Hong Kong, and includes Taiwan as a trading partner.22 Third,
we combine countries that unify or report jointly for subsets of years in the database.23
Fourth, we convert these SITC Rev.2, 4-digit industrial categories into HS 2007, 6-digit
using the crosswalk provided by the United Nations.24 Finally, we convert these data inthe 6-digit HS 2007 product code into 6-digit product categories used in the CoM data.
We construct new crosswalk files based on the crosswalk provided in Baek et al. (2021).
Sample Construction of Establishment-level Panel Data Our main sample in the regression is a panel of establishments in Japan. First, We drop establishments with lessthan 4 employees since they are not sampled at an annual frequency.25 Second, we use aconverter provided by RIETI to link establishments across time starting in 1986. Finally,
we keep establishments that appear at least 5 years consecutively. This is because wecompute variance over time within establishments and need sufficient observations foreach establishment.26 Our final sample is an unbalanced panel of 724,417 unique establishments in manufacturing sectors from 1986 to 2016.

22 Their cleaner is available here.
23 The countries combined are East and West Germany;

Belgium and Luxembourg; the islands thatformed the Netherlands Antilles; North and South Yemen; and Sudan and South Sudan.
24 The crosswalk is available in the UNSD web page here.
25 The results are robust when we use all the establishments with at least 30 employees.
26 Changing this threshold to a minimum of 10 years does not change our results.

81