Introduction Model Analysis of the equilibrium
Trade and Labor Market:Felbermayr, Prat, Schmerer (2011)
Davide Suverato1
1LMU University of Munich
Topics in International Trade, 16 June 2015
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 1 / 26
Introduction Model Analysis of the equilibrium
Trade with labor market imperfections
because of labor market imperfections,
• lower number of jobs than under perfect labor market:labor demand < labor supply =⇒ unemployment
• every match yields a strictly positive surplus:the wage is a splitting device for the surplus of a match.
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 2 / 26
Introduction Model Analysis of the equilibrium
Trade with labor market imperfections
let search and matching frictions be the cause of labor marketimperfections
search frictions:
• firms do not hold an infinite number of vacancies because it iscostly
• workers do not have perfect knowledge of all the vacancies inthe market
matching frictions:
• probability that a worker finds a job < 1
• probability that a vacancy is visited by a worker < 1
• every period a match is destroyed with a probability > 0
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 3 / 26
Introduction Model Analysis of the equilibrium
Trade with labor market imperfections
Davidson Martin Matusz (1999),differences in labor market frictions between sectors
• determine the relative price
PX
PY=
2 (ρ+ bX ) + 1
2 (ρ+ bY ) + 1
• determine the relationship between job finding probability andunemployment / employment ratio at the sector level
e lXLsX = bXLeX , ekXKsX = bXKeX
e lY LsY = bY LeY , ekY KsY = bY KeY
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 4 / 26
Introduction Model Analysis of the equilibrium
Trade with labor market imperfections
Therefore,differences in labor market frictions between sectors
• are a source of comparative advantage
Assume that bX < b?X and bY = b?Y thenP?XP?Y
> PXPY
,
the domestic economy specializes in the production andexport of good X .
• determine differences in sectoral unemployment
Assume that bX > bY then,sector X is characterized by higher unemployment µX > µY .
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 5 / 26
Introduction Model Analysis of the equilibrium
Trade with labor market imperfections
An increase of trade openness determines a loss of jobs,when the domestic economy specializes in the sector that ischaracterized by relatively higher unemployment.
• the pattern of specialization depends on across–countrycomparisons of country–specific and sector–specific labormarket frictions
• which is the sector with relatively higher unemploymentdepends on comparisons of sector–specific labor marketfrictions within the domestic economy only!
Notice: there exists an equilibrium in which the increase in tradeopenness does not change the job finding probability.
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 6 / 26
Introduction Model Analysis of the equilibrium
Trade with labor market imperfections
There is more to say about the effect of trade on the probability offinding a job; in Davidson et al. (1999) it is a function of:
– domestic relative factor endowment L/K– terms of trade P? (1− T )– frictions in the domestic labor market bX , bY
Felbermayr, Prat, Schmerer (2011) introduce search generatedunemployment into a 1–sector Melitz’s trade model.
Helpman, Itskhoki (2010) introduce search generatedunemployment into a 2–sector trade model, with Melitz’smonopolistic competition in the tradable sector.
They show the relationship between average productivity andprobability of finding a job.
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 7 / 26
Introduction Model Analysis of the equilibrium
Felbermayr, Prat, Schmerer (2011)
• 1 sector=⇒ changes in unemployment cannot arise from reallocation
of the workforce across sectors, but across firms!
• C.E.S. love for variety + monopolistic competition +endogenous entry + fixed cost of export=⇒ trade induces selection of less productive firms out of
the market, which increases sector average productivity ↑
• labor market frictions + Nash bargaining=⇒ wage and job finding probability will be a function of
the sector average productivity.
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 8 / 26
Introduction Model Analysis of the equilibrium
Framework
• 1 sector consists of producers with horizontally differentiatedvarieties
• 1 type of agent: workers with one indivisible unit of labor each– workers can be employed or unemployed– if unemployed, they search for a job.
• search & matching frictions, wage set through Nashbargaining
• monopolistic competition with fixed costs that induceincreasing returns to scale
• ex–ante investment leads to endogenous entry
• fixed costs of export induce selection into the export market
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 9 / 26
Introduction Model Analysis of the equilibrium
Preferences, demand and production
• Consumers value consumption of the aggregate good
Y =
[M−
1σ
∫ω∈Ω
q (ω)σ−1σ dω
] σσ−1
, σ > 1
let the price of the aggregate good being the numeraire, then
• Consumers’ demand
q (ω) =Y
Mp (ω)−σ
• Production, under market clearing
q (ω) = ϕ (ω) l (ω)
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 10 / 26
Introduction Model Analysis of the equilibrium
Firm behavior
• each firm optimally decides to be a monopolist ϕ ≡ ω
• marginal revenue in the domestic market
mrD (ϕ) =(
1−∣∣∣∂p(ϕ)∂q(ϕ)
q(ϕ)p(ϕ)
∣∣∣) pD (ϕ)
• marginal revenue in the (symmetric) export market
mrX (ϕ) =(
1−∣∣∣∂p(ϕ)∂q(ϕ)
q(ϕ)p(ϕ)
∣∣∣) pD(ϕ)τ for τ > 1.
• segmented markets =⇒ it is optimal mrD (ϕ) = mrX (ϕ)
pX (ϕ) = τpD (ϕ) ⇐⇒ qX (ϕ) = τ−σqD (ϕ)
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 11 / 26
Introduction Model Analysis of the equilibrium
Revenue
• revenue from the domestic market rD (ϕ) = YM p (ϕ)1−σ
• revenue from the export market rX (ϕ) = τ1−σrD (ϕ)
• total revenue r (ϕ) =[1 + e (ϕ) τ1−σ] rD (ϕ)
where e (ϕ) = 1 if and only if the firm exports.
inverse demand: p (ϕ)1−σ =(YM
) 1−σσ qD (ϕ)
σ−1σ
clearing: ϕl (ϕ) = qD (ϕ) + τqX (ϕ) =(1 + e (ϕ) τ1−σ) qD (ϕ)
revenue is an increasing, concave, log–linear function ofemployment
r (ϕ, l (ϕ)) =
[Y
M
(1 + e (ϕ) τ1−σ)] 1
σ
(ϕl (ϕ))σ−1σ (1)
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 12 / 26
Introduction Model Analysis of the equilibrium
Search and matching frictions
• there are u unemployed workers and v vacancies,define θ = v
u the labor market tightness
• the probability that a firm matches with a worker is decreasingin l.m.t. m (θ) , m′ < 0
• the probability that a worker matches with a firm is θm (θ)increasing in l.m.t.
• holding a vacancy has a cost c > 0
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 13 / 26
Introduction Model Analysis of the equilibrium
Law of motion for employment
• firms and workers separate with probability s = δ + χ− δχ,that is because of:
– a firm destruction shock that occurs with probability δ
– a job destruction shock that occurs with probability χ
the next period employment l ′ for a for a firm that employs lworkers and holds ϑ vacancies is∗:
l ′ = (1− χ) l + m (θ)ϑ (2)
∗assuming a continuous measure of employees and vacancies.
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 14 / 26
Introduction Model Analysis of the equilibrium
Firm inter–temporal problem
define the firm profit:
π (ϕ, l) = r (ϕ, l)− w (ϕ, l) l − cϑ (ϕ, l)− fD − e (ϕ) fx
Firms are risk neutral, so they choose the number of vacancies thatmaximizes the expected discounted lifetime flow of profit:
Π (ϕ, l) = maxϑ>0
1
1 + r
π (ϕ, l) + (1− δ) Π
(ϕ, l ′
)(3)
subject to:the revenue (1)the law of motion for employment (2).
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 15 / 26
Introduction Model Analysis of the equilibrium
Inter–temporal optimality, firm
define the value of the marginal job: J (ϕ, l) = ∂Π(ϕ,l)∂l
compute the f.o.c. for the optimality of vacancy posting:
c = (1− δ) m (θ) J(ϕ, l ′
)(4)
A firm will hold vacancies up to the point in which the expectedvalue of hiring the marginal worker is equal to the cost of postingthe marginal vacancy.
Linear cost =⇒ firms can always adjust at the steady stateemployment l ≡ l ′, therefore:
ϑ =χ
m (θ)l (5)
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 16 / 26
Introduction Model Analysis of the equilibrium
Inter–temporal optimality, firm
Solve for the value of a job from the inter–temporal problem (3),
when posting is optimal c = (1− δ) m (θ) J (ϕ, l ′)
(1 + r) J (ϕ, l) =∂r
∂l− ∂w
∂ll − w − c
∂ϑ
∂l+
c
m (θ)
∂l ′
∂l
where vacancies are optimally chosen ϑ = χm(θ) l such that
employment is in steady state l ′ = l
J (ϕ, l) =1
1 + r
[∂r
∂l− ∂w
∂ll − w +
(1− χ) c
m (θ)
](6)
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 17 / 26
Introduction Model Analysis of the equilibrium
Inter–temporal optimality, worker
Worker are risk neutral, so they search for a job to maximize theexpected discounted lifetime flow of income:
• value of being employed (and working)
rE (ϕ, l) = w (ϕ, l) + s[U − E
(ϕ, l ′
)]• value of being unemployed (and searching)
rU = bw + θm (θ)[E(ϕ, l ′
)− U
]• steady state l ≡ l ′
E (ϕ, l)− U =w (ϕ, l)− rU
r + s(7)
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 18 / 26
Introduction Model Analysis of the equilibrium
Wage determination
within the period,
the wage is the outcome of a Nash bargaining,
in which the firm bargains with every worker on how to split thesurplus of the marginal job:
βJ (ϕ, l) = (1− β) [E (ϕ, l)− U]
which together with the worker’s surplus (7) yields
(r + s) J (ϕ, l) =1− ββ
(w (ϕ, l)− rU) (8)
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 19 / 26
Introduction Model Analysis of the equilibrium
Wage equation
The system of value of a job (6) and optimality in vacancy posting(4) yields the job creation condition,
(r + s) J (ϕ, l) =
[∂r (ϕ, l)
∂l− ∂w (ϕ, l)
∂ll − w (ϕ, l)
](9)
which in conjunction with the bargaining equation (8) yields thewage equation:
w (ϕ, l) = β
(∂r (ϕ, l)
∂l− ∂w (ϕ, l)
∂ll
)+ (1− β) rU (10)
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 20 / 26
Introduction Model Analysis of the equilibrium
One wage!The wage equation (10) is an O.D.E. in w (l) with a particular
solution w = (1− β) rU + β(
σσ−β
)∂r∂l , which I refer to as (?).
Solving for the revenue (1), allows to compute the monopsony
component of wages ∂w∂l l = − 1
σ
[β(
σσ−β
)∂r∂l
]< 0.
Inserting this result in the job creation condition (9), where
J (ϕ, l) = c(1−δ)m(θ) yields w =
(σ
σ−β
)∂r∂l −
(r+s1−δ
)c
m(θ) , which I
refer to as (??).
The system with the solution of the wage equation (?) yields:
w (ϕ, l) = rU +
(β
1− β
)(r + s
1− δ
)c
m (θ)∀ ϕ, l (11)
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 21 / 26
Introduction Model Analysis of the equilibrium
The Wage curve
Since there is one wage, w ≡ w , which allows to solve for theoutside option of the unemployed rU = bw + β
1−βcθ
1−δ .Inserting this result in the (11) yields the first equilibrium conditionthe Wage Curve:
w =β
1− βc
(1− b) (1− δ)
[r + s
m (θ)+ θ
](12)
The wage curve describes an increasing and convex relationshipbetween wage and labor market tightness.
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 22 / 26
Introduction Model Analysis of the equilibrium
The Labor Demand curve
Look at (??), and use the definition of marginal revenue tosubstitute for ∂r
∂l =(σ−1σ
)p (ϕ)ϕ.
As in Melitz (2003) define ϕ the productivity of the firm withaverage revenue. Then p (ϕ) coincides with the C.E.S.consumption based price index of the aggregate good, which is thenumeraire, so p (ϕ) = 1.
This allows to write the wage (??) as
w =
(σ
σ − β
)ϕ−
(r + s
1− δ
)c
m (θ)(13)
an increasing function of the measure of average productivity ϕ,and a decreasing function of labor market tightness θ.
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 23 / 26
Introduction Model Analysis of the equilibrium
Equilibrium
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 24 / 26
Introduction Model Analysis of the equilibrium
The effect of trade on the average wage
Notice that from now on the solution of the model follows Melitz(2003).
A trade liberalization, τ ↓ or fX ↓, that determines an increase inthe average productivity will lead to:
• higher wage w ↑, both in nominal and in real terms.
• higher labor market tightness θ ↑
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 25 / 26
Introduction Model Analysis of the equilibrium
The effect of trade on unemployment
The job finding probability is increasing in labor market tightnessθm (θ) ↑
The steady state unemployment rate has to satisfy the Beveridgecurve. Let 1− u be the number of workers unemployed, thens (1− u) = θm (θ) u implies:
u =s
s + θm (θ)(14)
A trade liberalization, τ ↓ or fX ↓, that determines an increase inthe average productivity leads to a lower unemployment.
Davide Suverato, LMU Trade and Labor Market: Felbermayr, Prat, Schmerer (2011) 26 / 26