Regional Equilibrium Unemployment with
Entrepreneurs and Agglomeration
Externalities∗
Guillaume Wilemme
Sciences Po, Paris
May 2014
Work In Progress
Abstract
I study the role of agglomeration economies and urban costs in explaining the
regional differences in job opportunities and unemployment rates. A theoretical
model characterized by search frictions, entrepreneurial choice and managerial cost
is sufficiently rich to describe various economic mechanisms linking the unemployment rate and the region’s size. First, the urban cost channel makes the job-finding
probability increase in the region’s size through the cost of housing, so that freely
mobile individuals are indifferent between living in any region. Second, a higher
region’s size implies a higher number of firms created. Agglomeration economies
make them more productive, leading to a higher level of job creation.
JEL Code: J21, J61, L26, R13, R23
Keywords: agglomeration, entrepreneurs, regional job creation, search frictions,
urban costs
1
Introduction
In this paper, I discuss the interaction between regional labor markets characterized
by search frictions and the size of regions, accounting for agglomeration economies and
urban costs. Each of these eects translates into an increasing relation between the
regional job-nding rate and the number of residents. Whereas the urban cost eect is
already documented in the literature, the contribution in understanding an agglomeration
economies eect is original.
Disparities in job opportunities and unemployment rates across regions are important,
both in European countries and in the United States. They are also persistent over time
and stronger than the dierences of the national unemployment rate along the business
cycle.1 Understanding the reasons of this phenomenon is crucial to think of public policies.
The regional unemployment rates relative to the national rate are in most cases nonstationary. This suggests that an empirical analysis of regional unemployment rates
requires to account for the dynamics of other economic variables. Here, I sustain the idea
that the region's size interacts with the regional unemployment rate.
I build a model of the labor market with search frictions where individuals choose
between a worker's career or an entrepreneur's career. The demand for housing is inelastic
and the cost of producing houses is convex, so that the cost of living is increasing in
the region's size. This is the source of the urban costs. The source of agglomeration
economies is the presence of external economies of scale as rst mentioned by Marshall
(1890). The productivity of each worker is increasing in the number of rms in the region.
Entrepreneurs can create several jobs facing a convex managerial cost. Job vacancies and
unemployed workers meet according to a matching technology with constant returns
to scale. Agents can quit the region and other can enter. Individuals are indierent
accept to live in a region with high unemployment probability if the cost of housing is
low enough to compensate, this is the urban costs eect. A higher size of the regional
population induces a higher number of rms and so a gain in productivity. Under the
main assumption that entrepreneurs face a convex managerial cost, the labor market will
become more protable to workers and the employment rate nally increases. This is the
agglomeration economies eect.
The theoretical model is a convenient framework to study several place-based policies
related to the labor market. It could also provide plausible dynamics of the economy out
of the steady state. I claim that the model can be easily extended in dierent directions,
which I plan to develop.
1 Empirical
others.
facts can be found in Lutgen, Van der Linden (2013) and Kline, Moretti (2013) among
2
Literature Review.
Regional disparities in the labor market was rst emphasized by
Blanchard and Katz (1992). They handle the dynamics of the regional unemployment
rate, the regional wage and the region's size simultaneously in a VAR model. Very recent
papers analyze regional unemployment due to search frictions on the labor market. Kline
and Moretti (2013) show that place-based policies can improve welfare in a normative
approach. My objective is to have rather a positive approach by considering interregional
migration for instance and to obtain some quantitative results based on empirical data.
Lutgen and Van der Linden (2013) focus on a particular search technology in their models.
They consider dierently searching a job in his region of residence and in other regions.
Beaudry, Green and Sand (2013) manage to build a model tractable for estimation on
U.S. data, but they require many simplifying assumptions.
My model relies on the entrepreneurial choice of individuals. It originally comes
from Fonseca, Lopez-Garcia, Pissarides (2001) and Pissarides (2002). These papers build
a model with individuals heterogeneous in entrepreneurship whereas they are homogeneous in my model. Lucas (1978) rst considered a model with decreasing returns in
management.
A model with agglomeration economies and urban costs is built in Helpman (1995),
who exploits the Krugman (1991) idea of increasing returns in production. Some research
has been conducted to estimate the size of agglomeration economies and urban costs
(Combes, Duranton, Gobillon (2011), Combes, Duranton, Gobillon (2013) among others).Urban economists have also worked on the spatial sorting of individuals according to
entrepreneurship, as Glaeser, Ponzetto, Tobio (2011), Behrens, Duranton, Robert-Nicoud
(2013).
2
Theoretical Model of a small open regional economy
2.1
Framework
I consider a region populated by a measure n of identical individuals. Time is continuous and death occurs randomly at a Poisson rate δ . Dead individuals are instantaneously
replaced by new-born individuals. At the beginning of their life, agents must choose once
for all between leaving the region or staying. Foreigners can also immigrate in the region.
The measure of residents n is then determined endogenously.
In the region, individuals must choose between a worker's career or an entrepreneur's
career. A worker starts as an unemployed and enjoys a ow b (indierently unemployment benets, home production, leisure) until she gets hired and obtains the wage w. If
the employer dies, the worker becomes unemployed and can search for a new job. An
entrepreneur is able to create a measure α of jobs, facing a convex management cost
3
αη /η with η > 1. Once lled, a job produces y units of output. E denotes the present
discounted value of being an entrepreneur and U of being an unemployed worker. Participating in the regional labor market consists in choosing the best asset :
A = max(E, U )
(1)
In addition, residents rent inelastically one unit of housing that costs h. The capital
market is perfect and r is the interest rate. The expected asset value from living in the
region is then A −
h
,
r+δ
where the second term is the present discounted value of total
housing costs. Leaving the region provides individuals with an exogenous ow return ξ
that they enjoy until their death. The no-arbitrage condition for interregional mobility
is :
(r + δ)A − h = ξ
(2)
One original contribution of this paper is the accounting of the agglomeration and
dispersion forces. Firstly, absentee landlords produce housing with decreasing returns in
the region's size. Perfect competition insures that the price of housing is equal to the
marginal cost denoted h(n), and h(.) is an increasing convex function.
Secondly, we assume external increasing returns to scale at the rm level. The output
of a job depends increasingly on the number of employers in the local economy. If ne
is the number of entrepreneurs, the output of a job is y(ne ), where y(.) is an increasing
function. This hypothesis is ad-hoc, but the model could be easily extended to incorporate
microeconomic foundations as long as the labor market is not involved. For instance, rms
may sell their homogeneous good to retailers that could dierentiate the good, and sold
it on a market with monopolistic competition.2
When deriving the equilibrium, we need some regularity on the productivity function
and the marginal cost of housing to obtain a unique equilibrium. I choose these functional
forms :
h(n) = h(0) + Ah nh
y(ne ) = y(0) + Ay ney
with Ah , Ay , h and y nonnegative constants.
For a pedagogical purpose, I will characterize the equilibrium in absence of search
frictions rst. The model with frictions is studied next.
2 See
Duranton, Puga (2004) for foundations of agglomeration economies.
4
2.2
Frictionless labor market
The absence of frictions implies that workers nd instantaneously a job and vice versa.
Agents are price-takers in wage. If the wage w is higher than unemployment income b,
every worker prefer to nd a job and there is no unemployment. The present discounted
values of being an entrepreneur or a worker satisfy the following Bellman equations :
rE = max {α[y(ne ) − w] −
α
αη
} − δE
η
(3)
(4)
rU = w − δU
In both careers, the agent may die at a hazard rate δ . We assume working to be more
protable than staying at home, w ≥ b. The condition on the parameters is derived
below. The prot-maximizing condition writes :
1
α = [y(ne ) − w] η−1
(5)
At a steady state, the measure of jobs is equal to the measure of workers,
1
[y(ne ) − w] η−1 ne = n − ne
At equilibrium, individuals are indierent between the two careers, E = U :
η
η−1
[y(ne ) − w] η−1
w=
η
(6)
(7)
Necessarily, the wage is equal to the prot of a rm. Combining equations (6) and (7)
provides a relationship between the wage and the region's size, W AE (n). It corresponds
to the
agglomeration economies
measure of employers, so is W
AE
eect on the wage. Indeed, if y is constant in the
(n) in the region's size. The no-arbitrage condition for
migration writes :
η
η−1
max w,
[y(ne ) − w] η−1 − h(n) = ξ
η
(8)
With equation (7), we obtain a second relationship w = ξ+h(n) ≡ W U C (n) associated
to the
urban costs eect on the wage.
If h is constant in the region's size, so is W U C .
Proposition 1 We have the following properties :
• Agglomeration economies lead to an increasing relation between the wage and the
region's size : W AE is an increasing function.
• Urban costs lead to an increasing relation between the wage and the region's size :
W U C is an increasing function.
5
An increase in the region's size induces an increase in the number of employers. Thanks
to agglomeration economies, the productivity increases and the wage follows the same
trend. On the other hand, the more numerous residents, the higher the housing cost.
The wage must increase to maintain the gain from living in the region constant.
Denition 1 A steady-state equilibrium of the frictionless model is characterized by a
wage w, a measure of entrepreneurs ne and a region's size n such that conditions (7), (6)
and (8) are satised.
The equilibrium is non-degenerated if n > ne > 0.
Proposition 2 Under the following conditions, a non-degenerated and stable steady-state
equilibrium of the frictionless model exists.
with b̄ =
b ≤ b̄
ξ ≤ ξ¯
h >
with ξ¯ − h(0) =
η−1
η
η−1
η
y(0) − b̄
η
η−1
η
y(0) − h(0) − ξ¯ η−1
y η
y + η
(i)
(ii)
(iii)
The rigorous proof is given in appendix. Condition (i) implies that the equilibrium
wage is above the unemployment income when the productivity is minimum, meaning
when the measure of entrepreneurs is close to zero. Thus, workers prefer the employment
situation than staying at home. Condition (ii) imposes the equilibrium wage to be higher
than the sum of the minimum cost of housing h(0) and the opportunity cost from leaving
the region ξ . This is necessary for individuals to settle in the region. If the opportunity
cost were too high, people may prefer to leave. Under condition (iii), the urban costs
eect dominates the agglomeration economies eect for a suciently high region's size.
If agglomeration economies were stronger than urban costs for any region's size, the
equilibrium region's size will tend to innity.
2.3
Labor market with search frictions
Vacancies posted by local entrepreneurs and local unemployed workers meet on a market with search frictions. As a consequence, some vacancies are left open and some
workers are left unemployed at any moment. The matching technology is constantreturns-to-scale, so the meeting rate only depends on the market tightness θ, i.e. the
vacancy-unemployed workers ratio. A vacancy meets an unemployed at the Poisson rate
q(θ), where q(.) is a decreasing function whose elasticity is between −1 and 0.3 Thus,
θq(θ) is the rate at which an unemployed meets a job oer. After a meeting, the worker
and the employer decide to match and share the surplus through a Nash bargain.
3 See
Pissarides (2000) for a detailed introduction to the matching function.
6
Expected payos
Each job, either occupied or vacant, requires paying a management cost. It cannot
be shared with the employees.4 Creating a job vacancy yields an expected payo V .
The value of an occupied job is J . We write the Bellman equations relative to the
entrepreneur :
rE = max {αrV −
α
αη
} − δE
η
(9)
rV = q(θ)(J − V )
(10)
rJ = y(ne ) − w − δ(J − V )
(11)
An entrepreneur enjoys the returns from opening a vacancy rV per job. With an
instantaneous probability δ , the entrepreneur looses its asset because of death. A vacancy
is lled at a Poisson rate q(θ). An occupied job produces an output y(ne ) and costs the
labor price w. If the employed worker dies, the job turns vacant. The number α of created
jobs maximizes the value of being entrepreneur, the rst-order condition writes :
1
α = (rV ) η−1
(12)
The higher the value of a vacancy, the more jobs are created. The Bellman equations
relative to the worker are :
rU = b + θq(θ)(W − U ) − δU
(13)
rW = w − δW − δ(W − U )
(14)
When unemployed, the worker gets unemployment benets b. After meeting a vacancy
at a rate θq(θ), she switches to employment of value W . She can also die at a rate δ . An
employed worker earns the wage w. She may loose her asset if she dies and she may join
the pool of unemployed if her employer dies.
The maximized return from being entrepreneur is :
η
η−1
(r + δ)E =
(rV ) η−1
η
(15)
Basic computations lead to these expressions of the value of being unemployed and of
opening a vacancy :
θq(θ)w + (r + 2δ)b
= (r + δ)U(θ, w)
θq(θ) + r + 2δ
q(θ)(y(ne ) − w)
= rV(θ, w, ne )
rV =
q(θ) + r + δ
(r + δ)U =
4 If
(16)
(17)
not, the Nash-bargained wage would depend on the number of jobs per employer. Solving the
problem would be more complex.
7
The value of unemployment U(θ, w) is increasing both in the market tightness θ and
the wage w. A higher market tightness increases the likelihood of nding a job and a
higher wage increases the gain from working. On the contrary, the value of a vacancy
V(θ, w, ne ) is decreasing both in the market tightness θ and the wage w. A higher market
tightness reduces the rate at which a vacancy can be lled and a higher wage reduces the
prot of the job. A higher number of rms ne increases the productivity because of the
positive external eects, so the value of a vacancy rises.
Wage setting
The wage is determined endogenously when a worker and an employer meet. They share
the surplus from matching according to a Nash bargaining game :
w = argmax (W − U )β (J − V )1−β
The rst-order condition yields :
(1 − β)
w − (r + δ)U
y(ne ) − w − rV
=β
r + 2δ
r+δ
(18)
The fraction in the left-hand side is the gain from matching as a worker, W − U , and
the fraction in the right-hand side is the gain as an employer, J − V . By substituting
U and V by their expressions from the Bellman equations U(θ, w) and V(θ, w, ne ), we
obtain the rst equilibrium condition :
(1 − β)
w−b
y(ne ) − w
=β
θq(θ) + r + 2δ
q(θ) + r + δ
(19)
This equation denes the wage setting curve, meaning an increasing relation between
w and θ. When the market tightness is higher, workers gain less from matching because
the value of being unemployed increases. On the contrary, employers gain more for
the opposite reason. Thus, workers have an incentive to bargain more aggressively and
employers less aggressively. The surplus sharing provides workers a higher wage in this
case. A useful relationship between U and V can be derived :
(r + δ)U − b =
βθ
rV
1−β
(20)
Steady-state stock of jobs
The total number of matches is denoted by m, it follows the law of motion :
ṁ = q(θ)(αne − m) − 2δm
(21)
At a rate q(θ), open vacancies of measure αne − m are lled and 2δm jobs are destructed
(either because the employer or the employee dies). This is also equal to :
ṁ = θq(θ) (n − ne − m) − 2δm
8
(22)
The measure of newly-formed matches is also equal to the meeting rate of workers θq(θ)
times the measure of unemployed n − ne − m. The condition for a steady-state stock
of jobs is obtained once substituting the endogenous number of jobs per employer from
(12) :
1
q(θ)
θq(θ)
(rV(θ, w, ne )) η−1 ne =
(n − ne )
q(θ) + 2δ
θq(θ) + 2δ
(23)
This equation is the equality between the number of lled vacancies and the number
of employed workers at any moment. The fraction in the left-hand side is the job-lling
rate and the one in the right-hand side the employment rate. The number of occupied
jobs (LHS) is decreasing in the labor-market tightness because the meeting rate for entrepreneurs is lower and less jobs are created in reaction. It is also increasing in the
number of entrepreneurs, rst from an accounting point of view and, second, because
productivity gains make entrepreneurs create more jobs. On the contrary, the RHS is
increasing in the market tightness because the meeting rate is higher for workers and
decreasing in the number of entrepreneurs because the pool of unemployed is smaller.
Straightforwardly, this equation corresponds to an upward-sloping relation between the
market tightness and the measure of entrepreneurs, taking the wage as exogenous.
Occupational choice
So far, the model can be solved taking both the number of employers and the region's size
as exogenous. The measure of entrepreneurs is obtained by a non-arbitrage condition,
assuming that each individual can become an entrepreneur. At equilibrium, agents are
indierent between being worker or entrepreneur, E = U :
η
η−1
(rV(θ, w, ne )) η−1 = (r + δ)U(θ, w)
η
(24)
This occupational choice condition is a decreasing equilibrium relation between θ and
w. If the wage increases, becoming worker yields more than becoming entrepreneur. The
market tightness adjust to make both choices equivalent, unambiguously it decreases.
This condition is the equivalent of the free-entry condition in the Diamond-MortensenPissarides framework (see Pissarides (2000)).
Free migration of labor
Individuals are freely mobile between regions. With equations (1), (2) and (15), the
no-arbitrage condition between staying and leaving the region writes :
η
η−1
η−1
max (r + δ)U(θ, w),
(rV(θ, w, ne ))
− h(n) = ξ
η
9
(25)
Denition 2 A steady-state equilibrium of the model with search frictions is characterized by a wage w, a labor-market tightness θ, a measure of entrepreneurs ne and a region's
size n such that conditions (19), (23), (24) and (25) are satised.
Equilibrium analysis
First, we substitute the wage from equation (19) in the steady-state stock of jobs equation
(23) and the occupational choice equation (24). Both operations implicitly dene the
market tightness as, respectively, a function ΘSJ (ne , n) and a function ΘOC (ne ).5
Lemma 1 Functions
ΘSJ and ΘOC are uniquely dened under assumption (i). In ad-
dition, ΘSJ (ne , n) is increasing in ne and decreasing in n and ΘOC (ne ) is increasing in
ne .
Proof.
The solution of the following system in θ and V enables us to dene ΘSJ (ne , n) :
(
(1−β)q(θ)
rV = β[θq(θ)+r+2δ]+(1−β)[q(θ)+r+δ]
(y(ne ) − b)
q(θ)
q(θ)+2δ
1
(rV ) η−1 ne =
θq(θ)
(n
θq(θ)+2δ
− ne )
The rst equation is the characterization (17) of the value of a job V from the Bellman
equations when the wage is taken endogeneous with (19). Given ne and n, it represents
a downward-sloping curve in the plan (V, θ). The second one is the steady-state condition, it corresponds to an increasing curve. A simple static comparative in ne and n
unambiguously concludes on the behavior of the function ΘSJ (ne , n).
ΘOC can be obtained by analyzing the following system in θ and V :
(
(1−β)q(θ)
rV =
(y(ne ) − b)
β[θq(θ)+r+2δ]+(1−β)[q(θ)+r+δ]
η
η−1
βθ
rV + b
(rV ) η−1 = 1−β
η
The second equation is obtained with the occupational choice equation (24) and the surplus sharing equation (20). It is upward-sloping in the plan (V, θ). The two curves cross
each other under assumption (i). Unambiguously, ΘOC (ne ) is increasing in the measure
of entrepreneurs ne .
Given a region's size n, we can solve the system of two equations
θ = ΘSJ (ne , n)
θ = ΘOC (ne )
in θ and ne . Let ΘAE (n) be the market tightness solution.
Lemma 2 Function ΘAE is dened. In addition, ΘAE (n) is increasing in n.
5 Exact
formula are given in appendix.
10
SJ
OC
(ne,n)
(ne)
y(ne)-b
V
y(ne)-b
V
Figure 1: Denition of ΘSJ in a (V, θ) plan. Figure 2: Denition of ΘOC in a (V, θ) plan.
(SJ)
(OC)
AE
(n)
n
ne
Figure 3: Denition of ΘAE in a (ne , θ) plan.
Proof.
Given n, The curve dened by ΘSJ (ne , n) has an asymptote when ne is close
to n. Necessarily, the curves cross each other (see Figure 3).
Figure 3 represents conditions (OC) and (SJ) in a (ne , θ) plan. ΘAE corresponds to
the eect of
agglomeration economies on the market tightness (see Figure 4).
Indeed,
if there were no such gains in agglomeration, the productivity y would be invariant in
the measure of entrepreneurs ne . The function ΘOC (ne ) would then be constant in ne ,
11
implying ΘAE (n) to be constant in n.
Equations (20), (24) and (25) dene the condition :
βθ
ξ + h(n) − b =
1−β
η−1
η
η
(ξ + h(n))
⇔ θ = ΘU C (n)
η−1
(26)
Lemma 3 Function ΘU C is uniquely dened under assumption (ii). In addition, ΘU C (n)
is increasing in n.
Proof.
ΘU C can be obtained by analyzing the following system in θ and V :
 η
 η−1 (rV ) η−1 = βθ rV + b
η
1−β
η
η−1

(rV ) η−1 + h(n) = ξ
η
The second equation is obtained with equation (25). It is upward-sloping in the plan
(V, θ). The two curves cross each other and ΘU C .
It is worth notice that the agglomeration economies eect is absent here. Function
UC
Θ
(n) corresponds to the eect of
urban costs
on the labor market. Without this
channel, h is constant and ΘU C (n) does not depend anymore on the region's size n.
Proposition 3 We have the following properties :
• Agglomeration economies lead to an increasing relation between the market tightness
and the region's size : ΘAE (.) is an increasing function.
• Urban costs lead to an increasing relation between the market tightness and the
region's size : ΘU C (.) is an increasing function.
When the region's size is higher, more individuals become entrepreneurs and the
market tightness tends to decrease. As the agglomeration economies increases prots,
the market tightness tend to increase. This second eect dominates the rst one.
Because of decreasing returns in managing, a higher value of labor market activities
leads to a higher market tightness. A more populated region requires a higher value to
compensate for the housing cost, which implies a higher market tightness.
Proposition 4 Under assumptions (i), (ii) and (iii), a steady-state equilibrium exists.
Proof.
Condition (ii) is required for the (AE) curve to be above the (UC) curve when
the region's size is close to 0. Condition (iii) imposes that the (UC) curve is above (AE)
when n becomes high enough. As a result, the two curves cross at least once.
12
(UC)
(AE)
n
Figure 4: (AE) and (UC) in a (n, θ) plan
The existence of an equilibrium requires that agglomeration economies eect dominates
the urban costs eect when the region's size is small enough and inversely. This is a
standard assumption in urban/regional economics. Without this assumption, nobody
would live in the region.
3
Discussion and Extensions
Public intervention
One goal of the paper is to investigate the consequences of public subsidies : housing
subsidies, rm subsidies, and worker subsidies. This is the main eld I am currently
focusing on. The model is tractable for analyzing place-based policies, in which these
transfers could be nanced by taxes on the local labor market.
A two-region model
In the analytical framework, we consider a small economy that does not impact on the
rest of the world, meaning ξ does not depend on the region's size. An interesting work is
to study two regions in a closed economy. Thus, an increase in the number of residents
in one area implies a decline in the other area. We expect to have dierent equilibrium
congurations depending on the superiority of the agglomeration economies eect versus
the urban costs eect.
Heterogeneity
Two types of heterogeneity should be interesting to investigate. The rst extension con13
sists in going back to the spirit of Fonseca, Lopez-Garcia, Pissarides (2001) and Pissarides
(2002). In their model, individuals have dierent managerial skills when they choose to
be an entrepreneur. In my model with two regions, this could lead to sorting according
to this variable. Another extension is to consider two sectors in the labor market (highskilled labor and low-skilled labor for instance). Again, we expect sorting at equilibrium
implying dierent externalities through the agglomeration economies and urban costs
eects.
Eciency
In a normative approach, the decentralized equilibrium has several reasons to be inefcient. Individuals do not internalize the agglomeration economies eect leading to a
suboptimal number of entrepreneurs. They do not also internalize their personal impact
on the housing cost. Lastly, other labor market ineciencies may arise due to the wagebargaining or the managerial cost. A particular study is required to conclude on the
existence of these ineciencies.
Data
In the seminal work of Blanchard and Katz (1992), the model with unemployment predicts
that the relative unemployment rate is stationary.6 Nevertheless, an augmented DickeyFuller test on the time series cannot reject the hypothesis of a unit root in each state in
the U.S. except two according to these authors. In France, I conducted the same analysis
and the conclusions are similar. This work is an attempt to explain this non-stationarity.
6 They
dene the relative unemployment rate as the regional unemployment rate minus the national
unemployment rate.
14
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[14] C. A. Pissarides. Company Start-Up Costs and Employment. Technical report, 2002.
[15] C.A. Pissarides. Equilibrium Unemployment Theory. the MIT press, 2000.
15
A
Proof of Proposition 1
First, I denote F (w) =
η−1
η
η
[y(ne ) − w] η−1 − w. The equilibrium wage w∗ satisfying
equation (7) is a root of F (.). Since F (.) is continuous, strictly decreasing, F (0) > 0
and F (y(ne )) < 0, the intermediate value theorem implies that F (.) admits a unique
root. Unambiguously, (7) denes a relationship between the wage and the measure of
entrepreneurs.
For participating in the labor market, workers must receive a wage higher than the
unemployment income b. The condition b ≤ w∗ is equivalent to F (b) ≥ F (w∗ ) = 0, which
is condition (i) for ne = 0.
The system of equations (7) and (6) dening the agglomeration economies relationship
is equivalent to :
η
η−1
w=
[y(ne ) − w] η−1
η
η1
ηw
n − ne
=
η−1
ne
(27)
w is increasing in ne in the rst equation, and decreasing in the second one. Applying the
intermediate value theorem to a well-chosen function denes a unique solution, taking n
as a parameter. The wage solution W AE (n) is increasing in n (agglomeration economies
eect).
The urban costs channel corresponds to the function W U C (n) = ξ + h(n). An equilibrium satises the system :
w = W AE (n)
w = W U C (n)
Condition (ii) is equivalent to W AE (0) ≥ W U C (0). We need to compare the behavior
of these two functions when n tends to innity. Straightforwardly,
W U C (n)
tends to a nonzero constant.
nh
The wage w dened by (7) is such that w/ney tends to a nonzero constant when ne tends
to innity. We then substitute the wage in (27) by Kney + K 0 (ne ) where K is a nonzero
constant and K 0 (ne )/ney tends to 0. In this equation, taking n as a function of ne , it can
be shown that
n
y +η
η
tends to a nonzero constant when ne tends to innity.
ne
Equivalently, taking ne as a function of n,
ne
tends to a nonzero constant when n tends to innity.
η
n y +η
16
Back in (7), we can show with the same trick that
W AE (n)
y η
tends to a nonzero constant.
n y +η
Under condition (iii), W U C (n) ≥ W AE (n) for n high enough. The intermediate value
theorem helps us to conclude the existence of an equilibrium.
B
Computations in the model with search frictions
By substituting the wage from (19) in (16) and (17), we obtain :
βθq(θ)y(ne ) + [(1 − β)(q(θ) + r + δ) + β(r + 2δ)]b
β[θq(θ) + r + 2δ] + (1 − β)[q(θ) + r + δ]
(1 − β)q(θ)
(y(ne ) − b)
rV =
β[θq(θ) + r + 2δ] + (1 − β)[q(θ) + r + δ]
(r + δ)U =
Functions ΘSJ and ΘOC are implicitly dened by :
θ = ΘSJ (ne , n)
1
η−1
q(θ)
(1 − β)q(θ) (y(ne ) − b)
θq(θ)
(n − ne )
⇔
ne =
q(θ) + 2δ β[θq(θ) + r + 2δ] + (1 − β)[q(θ) + r + δ]
θq(θ) + 2δ
βθq(θ)y(ne ) + [(1 − β)(q(θ) + r + δ) + β(r + 2δ)]b
β[θq(θ) + r + 2δ] + (1 − β)[q(θ) + r + δ]
η
η−1
η−1
(1 − β)q(θ) (y(ne ) − b)
=
η
β[θq(θ) + r + 2δ] + (1 − β)[q(θ) + r + δ]
θ = ΘOC (ne ) ⇔
Proof.
In the plan (w, θ), the (WS) curve is increasing from the support [b, y(ne )[
into [0, +∞[ and the (OC) curve strictly decreasing from the support ]0, ω] into ]0, +∞[.
Thus, if an equilibrium exists, it is unique. A sucient condition is to have b ≤ ω where
ω satises :
which writes
η−1
η
η−1
η
η
(y(ne ) − ω) η−1 = b
η
(y(ne ) − b) η−1 ≥ b. To be fullled for any number of entrepreneurs
ne , we assume this condition to be true at the limit when ne tends to 0. This is condition
(i).
When ne increases, the (WS) and the (OC) curves move to the right. Now, I will
show that ΘJC (ne ) is increasing. Equation (19) provides an explicit form of the wage
function of θ and ne :
w=
β[θq(θ) + r + 2δ]y(ne ) + (1 − β)[q(θ) + r + δ]b
β[θq(θ) + r + 2δ] + (1 − β)[q(θ) + r + δ]
17
We substitute this expression in equation (17) :
rV =
(1 − β)q(θ)
(y(ne ) − b)
β[θq(θ) + r + 2δ] + (1 − β)[q(θ) + r + δ]
(28)
On one hand, equation (28) denes an application F (., .) such that rV = F (θ, ne ).
On the other hand, equations (20) and (??) provide :
η
η−1
βθ
(rV ) η−1 =
rV + b
η
1−β
(29)
It satises the implicit function theorem : there is a function G(.) such that rV = G(θ)
and one can show that G(.) is increasing by dierentiating. Thus, ΘJC (ne ) is solution of
F (θ, ne ) = G(θ). F (θ, ne ) is decreasing in θ and increasing in ne . Necessarily, ΘJC (ne ) is
increasing in ne .
In equation (23), we substitute the value of opening a vacancy rV(θ, w, ne ) by its
expression in equation (28) :
q(θ)
q(θ) + 2δ
(1 − β)q(θ) (y(ne ) − b)
β[θq(θ) + r + 2δ] + (1 − β)[q(θ) + r + δ]
1
η−1
ne =
θq(θ)
(n − ne ) (SJ)
θq(θ) + 2δ
This equation results from the condition for steady-state stock of jobs (SJ). ΘAE (n) is a
market tightness, solution of the system of equations (JC) and (SJ) in θ and ne .
18