Hours worked, employment rate, Search and Matching
F. Langot
Univ. Le Mans (GAINS-TEPP & IRA) Institut Universitaire de France
Paris School of Economics Cepremap (ENS-Paris)
IZA
flangot@univ-lemans.fr
2019-2020
Time schedule of the course :
Thursday, November 21, from 1 : 30 pm to 4 : 30 pm Friday, November 22, from 9 : 30 am to 12 : 30 pm Friday, November 29, from 9 : 30 pm to 12 : 30 pm
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Motivation
Motivations
Characteristics of the equilibrium models (micro/macro) for the labor market
Rational choices (individual choices)
1 Labor supply : extensive margin (to work or not, e.g.
retirement) and intensive margin (hours worked per worker) 2 Labor demand : productivity, labor costs
Market coordination
1 Market without frictions : the price (wage) as the good signal 2 Introducing search frictions to account for unemployment and
wage inequalities
Feedback : the impacts of the search frictions on hours worked, retirement...
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Motivations
Why do equilibrium models useful ?
How to understand the impact of labor market policies (taxation, MW, SS, UB,...) ?
How to evaluate existing policies with a description of each channel through which they modify the equilibrium ? Beyond the evaluation : How to find the optimal design of a policy ?
⇒ Not a simple impact assessment in a world without social interactions !
⇒ We need of a structural approach
Motivation : The Lucas Critique (1976)
”Given that the structure of an econometric model consist of optimal decision rules of economic agents, and that op- timal decision rule vary systematically with change in the structure of series relevant of the decision maker, it fol- lows that any change in policy will systematically alter the structure of econometric models” (Lucas (1976), p.40) The debate : reduced forms vs. structural model
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Motivation : The Lucas Critique (1976)
Consider the simple representation of the log–linearized version of a forward-looking model
y t = aE t y t+1 + bx t where |a| < 1
where y t and x t respectively denote the endogenous and the exogenous variables. x t is assumed to follow a rule of the form
x t = ρx t−1 + σε t with ε t ∼ iid (0, 1) The solution to this simple model is given by
y t = 1−aρ b x t The impact of the policy (x) on decision (y)
x t = ρx t−1 + σε t The policy rule... and the shock (ε)
Motivation : The Lucas Critique (1976)
This reduced form may be used to estimate the parameters of the model
y t = αx b t
x t = ρx b t−1 + b σε t
Using this model, one can predict the impact of an innovation ε t = 1/ b σ in period t + n :
y t+n = α b ρ b n
This forecast is true if and only if the policy rule does not change, ie. if and only if ρ is stable.
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Motivation : The Lucas Critique (1976)
At the opposite, assume that the policy rule changes.
ρ becomes ρ/2.
This reduced form of the model becomes ( y t = αx b t
x t =
b ρ 2
x t−1 + σε b t
whereas the true model becomes ( y t = 1−a b
( ρ 2 ) x t
x t = ρ 2
x t−1 + σε t
Motivation : The Lucas Critique (1976)
Proposition
There is a gap between the naive prediction y t+n = α b
ρ b 2
n
= b
1 − aρ
ρ b 2
n
and the “true” prediction
y t+n = b 1 − a ρ 2
ρ b 2
n
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Motivation : The Lucas Critique (1976)
Proposition
We need to estimate or to calibrate the DEEP parameters {a, b, ρ, σ} using theoretical restrictions.
⇒ The reduced form model is not really useful.... except if a = 0,
i.e. if agents have no expectations ! ! !
Motivation : The Lucas Critique (1976)
The implication of the Lucas critique
1 The economist must find micro-foundations to the equilibrium in order to identify both the structural parameters and the policy parameters.
This is necessary to forecast the implications of a policy change.
2 For the policy maker, the Lucas critique implies that the equilibrium of the economy is the equilibrium of a game : the private agents expect the policy changes and then the
discretionary policy becomes inefficient (see Kydland and Prescott [1977] : Rules versus discretion)
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Methodology : from microeconomic behaviors to policy evaluations
How to use the theory in order to evaluate a policy reform To propose an equilibrium model with rational expectations.
To estimate this model before an unexpected policy reform.
To use the model to predict the impact of the “new rules”
(the reform) ⇔ Policy experiments are useful to test theory.
Theory allows to find the optimal design of the policy.
Plan
1 Allocation of employees, hours per workers and participation (1 session)
2 Allocation of employees and wage inequalities (1 session)
3 Policy evaluation using a structural model (1 session)
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Hours worked and employment rate
Part I
Allocation of employees, hours
per workers and participation
Hours worked and employment rate The optimal allocation of Employment
What are the major questions that the labor market theory can deal with ?
The impact of schoks/changes on the total hours worked : How do transitory shocks change the optimal allocation of labor market ?
How do permanent changes in taxes change the effort at work ? How do institutional changes change the ”chance” to be employed ?
How to explain the retirement choices ?
Is it possible to explain the contrasting experiences of different OECD countries ?
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Hours worked and employment rate The optimal allocation of Employment
The Neo-Classical Theory : Assumptions and Notations
The economy is populated by a large number of identical households whose measure is normalized to one.
Each household consists of a continuum of infinitely-lived agents.
At each period there is full employment : N t = 1, ∀t (⇔ an analysis of the intensive margin).
Firm sales goods in a competitive market.
The markets of input factors are competitive : the price of each input factor is equal to its marginal productivity.
Financial market is perfect : households can borrow and save
freely.
Hours worked and employment rate The optimal allocation of Employment
The Neo-Classical Theory : Household Behaviors
The representative household’s preferences are
∞
X
t=0
β t U (C t , 1 − h t )
0 < β < 1 is the discount factor.
C t is the per capita consumption
1 − h t the leisure time, h t the number of hour worked.
The contemporaneous utility function is assumed to be increasing and concave in both arguments
Agent have the opportunity to save : they can trade asset.
The amount of asset at time t is denoted by K t .
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Hours worked and employment rate The optimal allocation of Employment
The Neo-Classical Theory : Household Behaviors
Each household chooses {C t , h t , K t+1 |t ≥ 0} to maximize utility subject to the budget constraint
(1 + R)K t + (1 − τ w,t )w t h t + T t − K t+1 − (1 + τ c,t )C t ≥ 0 (λ t )
w t is the real wage.
R is real interest rate (constant over time for simplicity), τ c,t is the consumption tax rate,
τ w,t the labor income tax rate,
T t is a lump-sum transfer from the government,
Hours worked and employment rate The optimal allocation of Employment
The Neo-Classical Theory : Household Behaviors
If we use λ t to denote the Lagrange multiplier associated to budget constraint at time t, the Lagrangian of the consumer problem is
L =
∞
X
t=0
β t U(C t , 1 − h t ) +
∞
X
t=0
λ t
(1 + R)K t + (1 − τ w,t )w t h t + T t
−K t+1 − (1 + τ c,t )C t
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Hours worked and employment rate The optimal allocation of Employment
The Neo-Classical Theory : Household Behaviors
The optimality conditions of this problem lead to :
(1 + τ c,t )λ t = β t U C 0 t (1) (1 − τ w,t )λ t w t = β t U h 0 t (2)
λ t = λ t+1 (1 + R) (3)
These 3 equations and the budgetary constraint allow to define the solution as follow
C t = C ({w i } ∞ i=1 , {τ c,i } ∞ i=1 , {τ w,i } ∞ i=1 , K 0 ) h t = h({w i } ∞ i =1 , {τ c,i } ∞ i =1 , {τ w,i } ∞ i=1 , K 0 )
Proposition : the consumption and the labor supply at time t are
function of the complete dynamics of the real wage and the taxes,
for a given initial condition K 0 .
Hours worked and employment rate The optimal allocation of Employment
The Neo-Classical Theory : Household Behaviors
Assume for simplicity that the preferences are separable : U (C t , 1 − h t ) = ln C t + σ
σ − 1 (1 − h t ) σ−1 σ σ > 1 Then (1) and (2) become :
C t = 1
1 + τ c,t β t
λ t (4)
1 − h t =
1
(1 − τ w,t )w t
β t λ t
σ
(5) Moreover, by iterating on λ t = λ t+1 (1 + R), we obtain
λ t = λ 0 /(1 + R) t (6)
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Hours worked and employment rate The optimal allocation of Employment
The Neo-Classical Theory : Household Behaviors
Using (6), Equations (4) and (5) become :
C t = 1
(1 + τ c,t )λ 0
[β(1 + R)] t (7)
1 − h t =
1
(1 − τ w,t )w t λ 0
[β(1 + R)] t σ
(8) Consumption and leisure depend on current prices and taxes, but also on the implicit value of the total wealth (λ 0 )
⇔ forward looking behavior.
⇔ Permanent Income Theory
Hours worked and employment rate The optimal allocation of Employment
The Neo-Classical Theory : Household Behaviors
Assuming T t = K 0 = 0, the budgetary constraint can be rewritten as follow :
∞
X
t=0
(1 + R) −t [(1 + τ c,t )C t + (1 − τ w,t )w t (1 − h t )]
=
∞
X
t=0
(1 + R) −t (1 − τ w,t )w t
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Hours worked and employment rate The optimal allocation of Employment
The Neo-Classical Theory : Household Behaviors
Then, using (7) and (8), we have :
∞
X
t=0
(1 + R) −t
"
(1 + τ c,t ) (1+τ 1
c,t )λ 0 [β(1 + R)] t +(1 − τ w,t )w t
1
(1−τ w,t )w t λ 0 [β(1 + R)] t σ
#
=
∞
X
t=0
(1 + R) −t (1 − τ w,t )w t
Hours worked and employment rate The optimal allocation of Employment
The Neo-Classical Theory : Household Behaviors
Then, using (7) and (8), we have :
⇔
∞
X
t=0
" 1 λ 0 β t
+(1 − τ w,t )w t
1
(1−τ w,t )w t λ 0 [β(1 + R)] t σ
(1 + R) −t
#
=
∞
X
t=0
(1 + R) −t (1 − τ w,t )w t
⇔
∞
X
t=0
β t h
1 + (1 − τ w,t )w t λ 0 [β(1 + R )] −t −σ
(1 − τ w,t )w t λ 0 [β(1 + R)] −t i
=
∞
X
t=0
β t
[β(1 + R )] −t (1 − τ w,t )w t λ 0
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Hours worked and employment rate The optimal allocation of Employment
The Neo-Classical Theory : Household Behaviors
Then, using (7) and (8), we can deduce the value of λ 0 :
0 =
∞
X
t=0
β t n
1 + (1 − τ w,t )w t λ 0 [β(1 + R)] −t 1−σ
− [β(1 + R)] −t (1 − τ w,t )w t λ 0
o
The solution takes the implicit form
λ 0 = Λ({w i } ∞ i=1 , {τ w,i } ∞ i=1 )
Lucas and Rapping (1969) propose to use this model in order
to explain the short run dynamics of the employment : the
impact of transitory shocks
Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Permanent Shocks : Theoretical Results
Assume that all wage w t are multiplied by a single quantity w e t = αw t . Then, with τ w,t = 0, e λ 0 is the solution of
0 =
∞
X
t=0
β t
1 +
w e t λ e 0 [β(1 + R)] −t 1−σ
− [β(1 + R)] −t w e t e λ 0
⇔ 0 =
∞
X
t=0
β t
1 +
αw t λ e 0 [β(1 + R)] −t 1−σ
− [β(1 + R)] −t αw t e λ 0
⇒ there is a solution iff λ e 0 = b λ α 0 , because in this case we have
0 =
∞
X
t=0
β t (
1 + αw t
λ b 0
α [β(1 + R)] −t
! 1−σ
− [β(1 + R)] −t αw t
b λ 0
α )
⇔ 0 =
∞
X
t=0
β t
1 +
w t b λ 0 [β(1 + R)] −t 1−σ
− [β(1 + R)] −t w t b λ 0
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Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Permanent Shocks : Theoretical Results
Because w e t e λ 0 = w t b λ 0 , where b λ 0 is the lagrange multiplier in a economy with the initial wage, we deduce from
1 − h t = 1
w t λ b 0
[β(1 + R)] t σ
= 1
w e t e λ 0
[β (1 + R)] t σ
that the labor supply does not change after a permanent shock.
h(w t ) = h( w e t ) ∀t
Substitution effect ⇒ to work more (the price of the leisure increases)
Wealth effect ⇒ to work less (each hour worked is more valuable)
⇒ Substitution effect is compensated by the Wealth effect (log
utility for consumption)
Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Transitory Shocks : Theoretical Results
In order to analyze the impact of a transitory shock, we assume that ∂λ ∂w 0
t ≈ 0 : the impact of the change in one wage is negligible when we consider all the life cycle.
⇒ We can neglect the variation in λ 0 , leading to
∂h t
∂w t
=
∂ h 1 −
1
(1−τ w,t )w t λ 0 [β(1 + R)] t σ i
∂w t
= σ (w t ) −1 (1 − h t )
⇒ ∂ h t
∂w t
w t
h t
= σ 1 − h t
h t
The larger σ, the larger the change in labor supply following a transitory shock. Hours are sensitive to transitory wage shocks (the business cycle)
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Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
The RBC model : the behaviors after a technological shock
1 After a technological shock, the labor productivity ↑ ⇒ labor demand ↑.
2 The production ↑ because the technological frontier shift-up and because the total hours increase.
3 At equilibrium, this adjustment is possible if the raise in labor productivity (the real wage) gives the incentive to supply more labor.
4 This is possible if the shock is transitory : intertemporal substitution ⇔ large opportunities today in the labor market, leading the agents to work more today than tomorrow when these opportunities disappear.
5 The saving opportunity allows the households to smooth these
additional gains ⇒ ↑ persistence of the shock.
Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
The RBC model : the IRF
0 20 40 60
0 0.5 1 1.5 2
Produit
Trimestres
% Dev.
0 20 40 60
0.2 0.4 0.6 0.8 1
Consommation
Trimestres
% Dev.
0 20 40 60
−2 0 2 4 6
Investissement
Trimestres
% Dev.
0 20 40 60
0 0.5 1 1.5
Capital
Trimestres
% Dev.
0 20 40 60
−0.5 0 0.5 1 1.5
Heures
Trimestres
% Dev.
0 20 40 60
0.2 0.4 0.6 0.8 1
Productivit‚
Trimestres
% Dev.
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Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Long Run Changes : the Growing Trend in Taxation
With constant taxes, the initial lagrange multiplier of the budgetary constraint is given by :
0 =
∞
X
t=0
β t
1 +
(1 − τ w )w t λ 1 0 [β(1 + R)] −t 1−σ
− [β(1 + R)] −t (1 − τ w )w t λ 1 0
⇒ λ 1 0 = (1−τ b λ 0
w ) .
A constant tax has the same impact than a permanent
shock : no impact.
Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Long Run Changes : the Growing Trend in Taxation
Assume now that the tax on wages grows.
If the tax grows, such that τ w,t+1 > τ w,t , then
0 =
∞
X
t=0
β t n
1 + (1 − τ w,t )w t λ 0 [β(1 + R)] −t 1−σ
− [β(1 + R)] −t (1 − τ w,t )w t λ 0
o
⇒ λ 0 6= (1−τ b λ 0
w,t ) because λ 0 must be a constant.
Thus (1 − τ w,t )w t λ 0 6= w t b λ 0 implying that h t (τ w,t > 0) 6= h t (τ w,t = 0)
Time varying taxes have an impact on the long run trend in hours. In which direction ?
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Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Long Run Changes : the Growing Trend in Taxation
Assume that w t = w , t = 1, 2, τ w,1 = 0 and τ w,2 = τ w :
0 =
1 +
w λ b 0
1−σ
− w b λ 0
+ β
(
1 + w λ b 0
β(1 + R)
! 1−σ
− w b λ 0
β(1 + R) )
0 = n
1 + (w λ 0 ) 1−σ − w λ 0
o + β
( 1 +
(1 − τ w )w λ 0
β(1 + R) 1−σ
− (1 − τ w )w λ 0
β(1 + R) )
0 =
1 +
(1 − τ w )w e λ 0
1−σ
− (1 − τ w )w e λ 0
+β (
1 + (1 − τ w )w e λ 0
β(1 + R)
! 1−σ
− (1 − τ w )w e λ 0
β(1 + R) )
Thus, we deduce :
⇒ b λ 0 < λ 0 < λ b 0
1 − τ w = λ e 0
⇒ (1 − τ w )λ 0 < b λ 0
Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Long Run Changes : the Growing Trend in Taxation
Then, by reporting the previous results in the FOC of the demand for leisure, we deduce :
Period 1 : b λ 0 < λ 0 ⇒ 1 − h 1 =
1 w λ 0
[β(1 + R)] t σ
< 1 − b h 1 = 1
w λ b 0
[β(1 + R)] t σ
⇔ h 1 > b h 1
Period 2 : (1 − τ w )λ 0 < b λ 0 ⇒ 1 − h 2 =
1
(1 − τ w )w λ 0
[β(1 + R)] t σ
> 1 − b h 2 = 1
w λ b 0
[β(1 + R)] t σ
⇔ h 2 < b h 2
Labor supply declines with the trend in the tax-wage.
⇔ The best response of the agent is to work when taxes are low (intertemporal substitution).
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Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Why do Americans work so much more than European ?
0.15 0.2 0.25 0.3
0.35 Total hours FR GE UK US
0.25 0.3 0.35 0.4
0.45 Hours per worker
0.55 0.6 0.65 0.7
0.75 Employment rate
Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Prescott (2004) : the tax wedge 1 + TW t = (1+τ 1−τ f ,t )(1+τ c,t )
w,t
1960 1970 1980 1990 2000 2010 2020
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
FR UK US GE
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Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Prescott (2004) : the increase in taxes.
The economy is populated by a large number of identical households whose measure is normalized to one.
Each household consists of a continuum of infinitely-lived agents.
At each period there is full employment : N t = 1, ∀t.
The utility function is assumed to be increasing and concave in both arguments and it shows conventional separability between consumption C and leisure 1 − h :
∞
X
t=0
β t U(C t , 1 − h t ) = ln C t + σ ln(1 − h t ) σ > 0 β ∈]0; 1[
The capital stock K t is rented to firms at net price (r t + δ),
where 0 < δ < 1 is the depreciation rate of capital.
Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Prescott (2004) : the tax increase in Europe.
Each household chooses {C t , h t , K t+1 |t ≥ 0} to maximize her intertemporal utility subject to the budget constraint
(1 + R t )K t + (1 −τ w,t )w t h t +T t − K t+1 −(1 + τ c,t )C t ≥ 0 (λ t ) R t = (1 − τ k )r : effective interest rate
τ k , τ c , τ w : capital, consumption, labor income tax rates T : is a lump-sum transfer from the government.
w : real wage.
Each firm has access to a Cobb-Douglas production technology to produce output and seeks to maximize the following profit flow :
π t = A t K t α (h t ) 1−α − (1 + τ f ,t )w t h t − (r t + δ)K t 0 < α < 1 τ f : payroll taxes.
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Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Prescott (2004) : the tax increase in Europe.
Therefore, the labor market equilibrium is then determined by : (1 − α) Y t
h t =
(1 + τ f ,t )(1 + τ c,t ) 1 − τ w,t
σ(1 − h t ) −1 C t
⇔ MPH t = (1 + TW t ) × MRS(H/C ) t where MPH t and MRS(H/C ) denote respectively
the marginal product of an hour worked
the marginal rate of substitution between hours worked and consumption
The tax wedge is defined by :
1 + TW t = (1 + τ f ,t )(1 + τ c,t ) 1 − τ w,t
In a economy without taxes, we have TW t = 0
Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Prescott (2004) : the tax increase in Europe.
How to quantify the disincentive impact of growing taxes on labor supply ?
Under the assumption that the model is a good approximation of the household behaviors, the observed data can be
generated by this model.
⇒ the data integrate the optimal reactions of the agent following the increase in taxes.
Let ∆ t defined by
MRS(H/C ) t = (1 − ∆ t )MPH t
From the point of view of the econometrician, ∆ t is the residual of the FOC estimated with an omitted variable, TW t .
∆ t < 0 : agents react as if there are employment subsidies ↑ h
∆ t > 0 : agents react as if there are employment taxes : ↓ h
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Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Prescott (2004) : the tax increase in Europe.
Using observed data, we can compute ∆ t . If the correlation between ∆ t and observed taxes is large, then the increase of taxes explain the decline of the labor supply.
The time series of ∆ t are computed for the six countries of our sample. Theory gives the restriction
∆ t = 1 − σ(1 − h t ) −1 C t (1 − α) Y h t
t
Y t is measured by the aggregate production per capita, C t by the aggregate consumption per capita
h t by the average number of hours worked per employee,
α = .4 and σ = 2 are calibrated.
Hours worked and employment rate
Permanent and Transitory Shocks : the Optimal Behaviors
Prescott (2004) : the tax increase in Europe.
Mean wedges Hours and Taxes
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05 0 0.05 0.1
r∆FOC
τ=0 τ>0
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
0.2 0.25 0.3 0.35 0.4 0.45 0.5
h norm taxl taxc
Left : We take 1980 as normalization year. − : ∆ t,TW=0 . −− : ∆ t,TW>0 . Right : − : τ w ; −− : τ c and o : h.
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Hours worked and employment rate
Matching frictions, employment rate and LMI
Beyond Prescott (2004) : additional distortive wedges in Europe.
The role of changes in taxes is remarkable over 1960-1985 ⇒ their incorporation reduces the size of the wedge over time and across the countries.
Various additional factors can explain the labor wedges (∆) : On the other side, the negative impact of the LMI, like the worker’s bargaining power (Barg ), and the average
unemployment benefits (ARR), on the performance of European labor markets after 1980 is well documented (Blanchard and Wolfer (1999)).
By modifying the ”chance” to be at work, the LMI also change the number of hours worked.
⇒ We need for a model explaining both extensive and intensive
margins
Hours worked and employment rate
Matching frictions, employment rate and LMI
The trend in LMI : Barg & ARR
1960 1970 1980 1990 2000 2010 0.1
0.2 0.3 0.4 0.5 0.6 0.7
FR UK US GE
1960 1970 1980 1990 2000 2010 0.05
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
FR UK US GE
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Hours worked and employment rate
Matching frictions, employment rate and LMI
A non-walrasian explanation of the employment gaps
Employment dynamics
N t+1 = (1 − s )N t + M t with M t = ΥV t ψ (1 − N t ) 1−ψ V t is the number of vacancies, Υ a scale parameter and the separation rate s is fixed over time
Labor market tightness is given by θ t = 1−N V t
t , f t = Υθ ψ t is the job finding rate
q t = Υθ t ψ−1 is the job filling probabilities.
Hours worked and employment rate
Matching frictions, employment rate and LMI
Household decisions : consumption/saving
W h (N t , K t ) = max
c t ,K t+1
log(c t + G t ind − c ) + χ log(G t col ) +N t (−σ l h 1+η 1+η ) + (1 − N t )Γ u +βW h (N t+1 , K t+1 )
s.t. I t (1 + τ i,t ) + c t (1 + τ c,t ) = (1 − τ w,t ) h
w t h t N t + (1 − N t ) b e t
i +π t + (1 − τ k,t )r t K t
K t+1 = K t (1 − δ) + I t
N t+1 = (1 − s)N t + f t (1 − N t )
where e b t = ρ t w t h t and c indicates the presence of a “subsistence”
term in consumption
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Hours worked and employment rate
Matching frictions, employment rate and LMI
Firm decisions : costly hiring process
V f (N t ) = max
V t ,K t
K t 1−α (A t N t h t ) α − w t h t N t − r t K t − ω t V t + β λ t+1
λ t
V f (N t+1 ) s.t. N t+1 = N t (1 − s ) + q t V t
The FOC on vacancies V t and the envelop condition lead to
ω = β λ t+1
λ t
∂V f (N t+1 )
∂N t+1
∂N t+1
∂V t
| {z }
=q t
⇒ ω q t
= β λ t+1
λ t
∂V f (N t+1 )
∂N t+1
∂V f (N t )
∂N t
= αK t 1−α (A t h t ) α N t α−1 − w t h t + β λ t+1
λ t
∂V f (N t+1 )
∂N t+1
∂N t+1
∂N t
| {z }
=(1−s)
⇒ ω q t
= β λ t+1
λ t
αK t 1−α (A t h t ) α N t α−1 − w t h t + (1 − s ) ω q t+1
Hours worked and employment rate
Matching frictions, employment rate and LMI
Employment surplus and wage bargaining
Employment surplus : S t =
∂W h
∂N t
/λ t
+ ∂V f
∂N t
Nash bargaining
{w t , h t } = Argmax
∂W h
∂N t /λ t
1− t
∂V f
∂N t t
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Hours worked and employment rate
Matching frictions, employment rate and LMI
Employment surplus and wage bargaining
Using φ t = 1− t
t , solution is :
w t h t = (1 − t )
α Y t
N t
+ ω t
(1−s) q t
1 − φ φ t+1
t
(1−τ w,t+1 ) (1−τ w,t )
+ φ φ t+1
t
(1−τ w,t+1 ) (1−τ w,t ) θ t
| {z }
BS
+ t
(1 + τ c,t )
(1 − τ w,t ) (C t − c)
Γ u + σ l
h 1+η t 1 + η
+ ρ t w t h t
| {z }
RW
BS = Bargaining surplus and RW = Reservation wage
σ l h η t (1 + τ c,t )
(1 − τ w,t ) = α Y t
N t h t
1 C t − c
The same equation than in the walrasian case ` a la Prescott.
Hours worked and employment rate
Matching frictions, employment rate and LMI
France with US tax rates
1960 1970 1980 1990 2000 2010 0.26
0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46
Hours per worker (h)
data counterfactual benchmark
1960 1970 1980 1990 2000 2010 0.6
0.65 0.7 0.75 0.8 0.85
0.9 Employment rate (N)
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Hours worked and employment rate
Matching frictions, employment rate and LMI
France with US LMIs
1960 1970 1980 1990 2000 2010 0.26
0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46
Hours per worker (h)
data counterfactual benchmark
1960 1970 1980 1990 2000 2010 0.6
0.65 0.7 0.75 0.8 0.85
0.9 Employment rate (N)
Hours worked and employment rate
Matching frictions, employment rate and LMI
France with US tax rates and US LMIs
1960 1980 2000
0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46
Hours per worker (h)
data counterfactual benchmark
1960 1980 2000
0.6 0.65 0.7 0.75 0.8 0.85
0.9 Employment rate (N)
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Hours worked and employment rate
Matching frictions, employment rate and LMI
A measure of the output losses in France
1980 1985 1990 1995 2000 2005 2010 2015
Output in efficiency units
1 1.05 1.1 1.15 1.2 1.25
French policy variables
Optimal policy variables
Optimal LMI, French taxes
Optimal taxes, French LMI
Hours worked and employment rate Participation choice : the retirement choice
Participation choice : the retirement choice
In this first part, we have focussed on the labor supply of the active individuals : the number of hour worked per worker and the number of employees.
There is another margin : the number of labor market participants in the population.
Two important decisions drive this extensive margin :
1 The retirement decision which determines the employment rate of the older workers.
2 The education decision which determines the employment rate of the young workers.
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Hours worked and employment rate Participation choice : the retirement choice
Employment rate over the life cycle
15−24 ans 25−54 ans 55−59 ans 60−64 ans
0 10 20 30 40 50 60 70 80 90 100
Rang des pays déterminé par le taux d’emploi des 60−64 ans
%
Taux d’emploi sur le cycle de vie − hommes − 2006
Fra Bel Ita PB
All Fin Esp
Can Dan RU EU
Sue Jap
Hours worked and employment rate Participation choice : the retirement choice
Retirement Choice
Taxes on labor incomes have also an impact on the retirement age decision
0 1 2 3 4 5 6 7 8 9 10
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
Taxe implicite sur la prolongation d’activité (indice)
Taux de non−emploi des 55−65 ans
Fra
Can
E−U
Jap Sué
R−U
Esp
P−B
All
Bel
Ita
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Hours worked and employment rate Participation choice : the retirement choice
Retirement Choice
Assumptions and notations :
Let T and R denote respectively the age of the death and of the retirement age,
Let w denotes the real wage
Let r and δ denote respectively the interest and subjective discount rate.
We assume that there is no uncertainty, and that agents can save and borrow in perfect financial markets.
At this stage, we assume that Social Security does not exist ⇔
There is no distortions.
Hours worked and employment rate Participation choice : the retirement choice
Retirement Choice
Preferences : U =
U (c (t)) if t ∈ [0, R) U (c (t)) + v if t ∈ [R, T ]
where v denotes the leisure when the individual is a retiree.
The problem of the agent is :
c(t),R max Z T
0
e −δt U (c (t))dt + Z T
R
e −δt vdt
s .t.
Z T 0
e −rt c(t)dt = Z R
0
e −rt wdt
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Hours worked and employment rate Participation choice : the retirement choice
Retirement Choice
The FOC are :
U 0 (c(t)) = λe −(r −δ)t ve −δR = λwe −rR
⇒ v
|{z}
Opportunity Cost
= U 0 (c (R))w
| {z }
Marginal return
In order to determine R, it is necessary to determine c (R).
Assuming, for simplicity, that r = δ, implying c(t) = c, ∀t, the budgetary constrain of the agent leads to :
c (R) = c = w R R
0 e −rt dt R T
0 e −rt dt = w 1 − e −rR 1 − e −rT The optimal age of retirement is then given by :
v = U 0
w 1 − e −rR 1 − e −rT
w
Hours worked and employment rate Participation choice : the retirement choice
Retirement Choice
Assume now that there is a Social Security system. The agent budgetary constraint becomes
Z T 0
e −rt c (t)dt = Z R
0
e −rt (1 − τ )wdt + Z T
R
e −rt p (R)dt The FOC are :
U 0 (c (t)) = λe −(r−δ)t ve −δR = λ
(1 − τ )we −rR − p(R)e −rR + Z T
R
e −rt p 0 (R)dt
⇒ v
|{z}
Opportunity Cost
= U 0 (c (R))
(1 − τ )w − p(R) + Z T−R
0
e −rt p 0 (R)dt
| {z }
For a given U 0 (c(R)), the marginal return can be taxed
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Hours worked and employment rate Participation choice : the retirement choice
Retirement Choice
We can then define this tax on the continued activity tax : TI = τ w
|{z}
Effective Tax
+ p(R)
| {z } Implicit
Tax
− Z T −R
0
e −rt p 0 (R)dt
| {z }
Bonus
The SS system is actuarially fair if and only if τ w + p(R) =
Z T−R 0
e −rt p 0 (R)dt implying no distortion on the labor supply
v = U 0 (c (R SS? ))w ⇒ R SS? = R
Hours worked and employment rate Participation choice : the retirement choice
Retirement Choice
Because for all SS system, we have c = w 1 − e −rR
1 − e −rT because τ w 1 − e −rR
1 − e −rT = p(R) e −rR − e −rT 1 − e −rT If the SS system is actuarially fair, the optimal age of retirement is the same than in a economy without SS At the opposite, if the SS system is not actuarially fair, but under the constrain that the SS system has no deficit, the optimal age of retirement is given by :
v = U 0 (c)[(1−τ )w −p] ⇔ v = U 0 w 1 − e −rR SS 1 − e −rT
!
[(1−τ )w −p]
If there is no bonus when the agent delays her retirement age
⇒ R SS < R
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Frictional Labor Market : employment rate and inequalities
Part II
Allocation of employees and
wage inequalities
Frictional Labor Market : employment rate and inequalities
Plan of the part
Section 1. Equilibrium search theory : a partial equilibrium approach to generate explain wage inequalities
Burdett & Mortensen (1998)
Section 2. Equilibrium search meets human capital theory : an explanation of wage inequalities
Mortensen (2003)
Section 3. Equilibrium search meets matching model : a general equilibrium approach allowing to evaluate policy reforms
Mortensen (2003)
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Frictional Labor Market : employment rate and inequalities Equilibrium search
Intertemporal Choices and Job Search Strategy
Finding a good job is an uncertain process which requires both time and financial resources.
The intertemporal arbitration :
1 In any given week the worker may receive a job offer at some wage w .
2 The decision she faces is whether to accept that offer and forego the possibility of finding a better job, or to continue searching and hope that she is fortunate enough to get a better offer in the near future.
The worker chance in the draw of a wage offer can generate
wage inequalities.
Frictional Labor Market : employment rate and inequalities Equilibrium search
Preferences and opportunities
We assume that individuals derive utility from consumption. There is no saving. Intertemporal preferences are given by :
E 0
Z ∞ 0
e −rt y (t)dt
where E 0 is the expectation operator conditional at time 0, r ∈ [0, 1] the subjective discount rate and y (t) the after-tax income from employment or unemployment compensation :
y(t ) =
w(t) employed worker b(t) unemployed worker
Job offer is drawn from a wage offer distribution F (w ), which denotes the probability of receiving a wage offer between the lower wage of the distribution w and w (t)
F (w ) = Prob(w (t) ≤ w )
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Frictional Labor Market : employment rate and inequalities Equilibrium search
Job search with on-the-job search
Assume that unemployed and worker search for a new job (on-the-job search). The values are
rV (w ) = w + λ 1 Z w
w
[V (x) − V (w )]dF (x) − s [V (w ) − U]
r U = b + λ 0
Z w w r
[V (x) − U ]dF (x)
where λ 0 and λ 1 denote the two exogenous arrival rates of wage offers, for the unemployed and the employed, respectively.
s is the exogenous separation rate.
Frictional Labor Market : employment rate and inequalities Equilibrium search
On-the-job search : the reservation wage
The lowest acceptable offer for unemployed workers is then defined by U = V (w r ).
The unemployed worker’s reservation wage is given by : w r = b + (λ 0 − λ 1 )
Z w w r
[V (x) − V (w r )]dF (x)
= b + (λ 0 − λ 1 ) Z w
w r
1 − F (x)
r + s + λ 1 [1 − F (x)] dx if λ 0 > λ 1 the unemployed worker have a competitive advantage in the search activity : the reservation wage is greater than unemployment benefits.
if λ 0 = λ 1 , then w r = b : the option to search a better job has the same value for an employed worker than for a unemployed.
It is optimal to take the first offer and to jump on-the-job.
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Frictional Labor Market : employment rate and inequalities Equilibrium search
The Reservation Wage : computational details
1/ d
dx {[V (x) − V (w r )][1 − F (x)]}
= dV (x)[1 − F (x)] − [V (x) − V (w r )]dF (x) 2/ [[V (x) − V (w r )][1 − F (x)]] w w r = 0
⇒
Z w w r
[V (x ) − V (w r )]dF (x) = Z w
w r
dV (x)[1 − F (x)]dx Using
rdV (w ) = 1−λ 1 dV (w ) Z w
w
dF (x )
| {z }
=1−F (w)
−sdV (w ) ⇒ dV (w ) = 1
r + s + λ 1 [1 − F(w )]
we deduce :
w r = b + (λ 0 − λ 1 ) Z w
w r
[1 − F (x)]
r + s + λ 1 [1 − F (x)] dx
Frictional Labor Market : employment rate and inequalities Equilibrium search
The Diamond Paradox
Assume that the distribution of wage offers is not exogenous : how firms can set optimally the wages ?
The objective is now to endogenize F (w ).
If there is no on-the-job search (λ 1 = 0), every unemployed workers accept to work for w r .
Because all workers have the same characteristics, all firms propose w = w r ,
We then have
dF (w ) = 0, ∀w > w r and F (w r ) = 1 In this case, the option to wait has zero value and w r = b.
There is no equilibrium with wage dispersion.
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Frictional Labor Market : employment rate and inequalities Equilibrium search
A solution to the Diamond Paradox
If there is on-the-job search, the story is different.
An unemployed worker still accepts to work for b.
But, now, employers have two reasons for offering a wage greater than b.
1 First, the firm’s acceptance rate increases with the wage offer, since a higher wage is more attractive.
2 Second, the firm’s retention rate increases with the wage paid by limiting voluntary quits. This leads to an increase the firm value.
⇒ in this case, an job search equilibrium exists with
non-degenerate dispersion of wage offers.
Frictional Labor Market : employment rate and inequalities Equilibrium search
The job-search equilibrium : steady state job flows
the equilibrium rate of unemployment is deduce from s (1 − u)
| {z } In = Inflow into unemployment
= λ 0 [1 − F (w r )]u
| {z } Out = Outflow from unemployment At the equilibrium, we have F (w r ) = 0 and thus
s (1 − u) = λ 0 u ⇒
( u = s+λ s
0
1 − u = s+λ λ 0
0
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Frictional Labor Market : employment rate and inequalities Equilibrium search
The job-search equilibrium : steady state job flows
We denote the steady-state number of employed workers being paid a wage no greater than w by G (w )(1 − u), where G (w ) is the distribution of wage earnings across employed workers and u the overall unemployment rate.
At steady-state the flow of workers leaving employers offering a wage no greater than w equals the flow of workers hired with a wage no greater than w :
λ 0 max{F (w ) − F (w r ); 0}u
| {z }
hirings
= s(1 − u)G (w )
| {z } firings
+ λ 1 [1 − F(w )](1 − u)G (w)
| {z }
volontary quits
Frictional Labor Market : employment rate and inequalities Equilibrium search
The job-search equilibrium : steady state job flows
For the wage distribution, we have G (w ) = sF (w )
s + λ 1 [1 − F (w )]
At the wage w or higher, the hirings can come from From unemployment λ 0 u = λ 0 s
s + λ 0
From jobs with w e < w λ 1 (1 − u)G(w ) = λ 1 λ 0 s + λ 0
sF (w ) s + λ 1 [1 − F (w )]
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Frictional Labor Market : employment rate and inequalities Equilibrium search
Specific Human Capital : the basic model
The probability that a job with wage offer w is accepted by the worker is defined by :
h(w ) = (1 − u)λ 1 G (w ) + λ 0 u
=
s λ 0
s+λ 1
s+λ 0
s + λ 1 [1 − F (w )]
This function increase with the posted wage w . This leads to a arbitration for the firm :
1 High wages make firm more attractive (↑ h(w )) and then leads to reduce its search costs.
2 But high wage reduces the profits (↓ J(w )).
Frictional Labor Market : employment rate and inequalities Equilibrium search
The Wage-Posting Game (0)
Each employer commits to a wage offer.
rJ(w ) = y − w − (s + λ 1 [1 − F (w )])J(w )
The expected discounted present value of the employer’s future flows of quasi-rents once a worker has been hired at wage w is
J(w ) = y − w
r + s + λ 1 [1 − F (w )]
where y represents the productivity of the match.
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Frictional Labor Market : employment rate and inequalities Equilibrium search
The Wage-Posting Game (1)
The ex ante asset value associated with a job is given by :
w≥w max
h(w )
y − w
r + s + λ 1 [1 − F (w )]
where h(w ) stands for the probability that a job with wage offer w
is accepted by the worker.
Frictional Labor Market : employment rate and inequalities Equilibrium search
The Wage-Posting Game (3)
All jobs are ex ante identically productive.
firm can retain the worker if the wage is high, making a lower instantaneous profit,
At the opposite, a another firm can decide to have a large instantaneous profit but large turnover
⇒ There exists a mixed-strategy equilibrium where different wage policies lead to the same profit for each firm.
In this case, the equilibrium is stable.
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Frictional Labor Market : employment rate and inequalities Equilibrium search
The Wage-Posting Game (4)
The solution is such that :
w = w r because nobody accept to work for a lower wage.
1 If λ 0 > λ 1 , then w r > b and the lower wage in the economy is higher than b
2 If λ 0 < λ 1 , then w r < b and the lower wage in the economy is lower than b
3 If λ 0 = λ 1 , then w r = b and the lower wage in the economy is equal to b
Given the monopsony power of the firm, there is a room for
the introduction of a MW.... but be careful : without labor
demand, the impact of the MW on inequalities can be biased.
Frictional Labor Market : employment rate and inequalities Equilibrium search
The Wage-Posting Game (5)
The solution is such that :
Every wage in the support of the equilibrium wage distribution must yield the same profit. Then, we have, ∀w :
h(w )
y − w r + s + λ 1
= h(w )
y − w
r + s + λ 1 [1 − F (w )]
⇔ r→0 y − w
(s + λ 1 ) 2 = y − w (s + λ 1 [1 − F (w )]) 2
⇒ F (w ) =
s + λ 1
λ 1
1 −
r y − w y − w r
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Frictional Labor Market : employment rate and inequalities Equilibrium search
The Wage-Posting Game (6)
The solution is such that :
Because F (w ) = 0 and F (w ) = 1, we have when r → 0 y − w
(s + λ 1 ) 2 = y − w s 2
⇒ w = w r + (y − w r )
"
1 − s
s + λ 1 2 #
Proposition
The parameters {b, λ 0 , λ 0 , s , y} ⇒ {w r , w , F (·)}.
Frictional Labor Market : employment rate and inequalities Equilibrium search
Calibration
Table – Parameters – Mortensen 2002
b s λ 0 = λ 1 y 0.8 0.287 0.142 2
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Frictional Labor Market : employment rate and inequalities Equilibrium search
Wage densities
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
0 0.5 1 1.5 2 2.5 3 3.5
x 10
−3DISTRIBUTIONS : DENSITES
offres, salaires
w f(w)
g(w)
Frictional Labor Market : employment rate and inequalities Equilibrium search
Cumulative distribution functions
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
DISTRIBUTIONS : REPARTITIONS
offres, salaires
w F(w)
G(w)
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Frictional Labor Market : employment rate and inequalities Equilibrium search
Wage dispersion measured in the data
Figure – Executives, non-managerial workers, white-color workers, blue-color workers - France - 2002
2 4 6 8 10
0 0.1 0.2 0.3 0.4
salaire/smic
fréquence
cadres
1 2 3 4 5 6
0 0.2 0.4 0.6 0.8
salaire/smic
fréquence
prof. inter.
1 2 3 4
0 0.5 1 1.5
salaire/smic
fréquence
employés
1 2 3 4
0 0.5 1 1.5
salaire/smic
fréquence
ouvriers
Frictional Labor Market : employment rate and inequalities Human Capital accumulation
Human Capital in a frictional labor market
Acemoglu and Shimer (1999) suggest that search equilibrium is consistent with ex-post endogenous distribution of
productivities.
The logic is the following
Firms are ex ante homogenous : equilibrium is characterized by a wage dispersion.
Higher-paying firm has an incentive to adopt a more capital-intensive technology.
As a consequence, worker employed by firms with high capital are more productive.
Higher wages can be paid in these firms.
The cost to open a highly productive is large : these positions become rare.
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Frictional Labor Market : employment rate and inequalities Human Capital accumulation
Human Capital in a frictional labor market : empirical motivation
Empirical relevance of the link between productivity and wage distributions.
1 Bontemps, Robin and van den Berg (1999) : it is necessary to allow for productive heterogeneity across employers in order to reproduce the observed wage distribution.
2 Posel-Vinay and Robin (2002) : the productivity differentials across firms explain an half of the French low-skilled wage variance, while the remaining part is due to the search frictions.
⇒ This emphasizes the importance of the specific human
capital on the low skilled workers wage distribution.
Frictional Labor Market : employment rate and inequalities Specific Human Capital
Specific Human Capital : a basic model
I simplifies the model used in the quantitative analysis in order to discuss its theoretical properties.
Assumptions :
Firm invests in specific-match capital. This amount of capital is denoted k .
The price of one unit of this capital is p k .
The production of a job is f (k) with f 0 > 0 and f 00 < 0.
After a separation, a destruction or a job-to-job transition, this capital is dissolved.
⇒ this investment is match-specific and can be viewed as a training paid by firms.
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Frictional Labor Market : employment rate and inequalities Specific Human Capital
Specific Human Capital : the basic model (1)
Wages and training investments are set to maximize the expected return to the posting of a vacancy :
w≥w,k≥0 max {h(w ) [J (w , k) − p k k ]}
h(w ) = s λ 0
s+λ 1
s+λ 0
s + λ 1 [1 − F (w )]
with J(w , k) the expected present value of the employer’s future flow of quasi-rent once a worker is hired at wage w :
J(w , k) = f (k ) − w r + s + λ 1 [1 − F (w )]
Each employer pre-commits to both the wage offered and the
extend of the specific capital investment in the match.
Frictional Labor Market : employment rate and inequalities Specific Human Capital
Specific Human Capital : the basic model (2)
For each wage offer w , the optimal investment is given by : f 0 (k) = p k (r + s + λ 1 [1 − F (w )]) = ⇒ k = k(w ) ∀w
= ⇒ Trade-off between low training and low rotation costs : When the wage is high, the expected duration of the match is longer and the period during which the firm can recoup its
investment increases. Therefore, firm specific productivity increases with wages.
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Frictional Labor Market : employment rate and inequalities Specific Human Capital
Specific Human Capital : the basic model (3)
The wage distribution is then given by (assuming that r → 0) : F (w ) =
s + λ 1
λ 1
"
1 − s
f (k(w )) − w − k (w )(s + λ 1 [1 − F (w )]) f (k(w )) − w − k(w )(s + λ 1 )
#
whereas the support of the wag distribution is bounded by w = w r
w = w r + (y − y) + (y − w r ) 1 − s
s + λ 1 2 ! y = f (k(w )) − p k sk (w )
y = f (k(w )) − p k (s + λ 1 )k (w )
Frictional Labor Market : employment rate and inequalities Specific Human Capital
Quantitative evaluation : Calibration of the basic model
Table – Parameters – Mortensen 2002
b α s λ 0 = λ 1 p k
0.3 0.6 0.287 0.142 1
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Frictional Labor Market : employment rate and inequalities Specific Human Capital
Quantitative evaluation : the equilibrium of the basic model
0.2 0.3 0.4 0.5 0.6 0.7
0 2 4 6 8
w
h(w),d(w)
h(w) d(w)
0.2 0.3 0.4 0.5 0.6 0.7
1.6 1.8 2 2.2 2.4
w
(pf(k)−w)/w
(pf(k)−w)/w
0.2 0.3 0.4 0.5 0.6 0.7
1 1.5 2 2.5 3 3.5 4
w
k(w)
k(w)
0.3 0.4 0.5 0.6 0.7
0.0192 0.0194 0.0196 0.0198 0.02 0.0202 0.0204
w
g(w)
g(w)
