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(1)

Microeconomic Analysis of Labor Market

F. Langot

PSE & Univ. Le Mans (GAINS) & Cepremap & IZA

2012-2013

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Part I

Motivation

(3)

The recent developments of the microeconomics of the labor market

Rational choices in a market without frictions (hours worked, retirement, education).

The impacts of the search fictions on these choices

How do others markets affect these choice (financial market) Why do microeconomics models useful ?

Beyond the evaluation of an existing policy How to find its optimal design ?

The need of a structural approach

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(4)

”Given that the structure of an econometric model consist of optimal decision rules of economic agents, and that optimal decision rule vary systematically with change in the structure of series relevant of the decision maker, it follows that any change in policy will

systematically alter the structure of econometric models”

(Lucas (1976), p.40)

(5)

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 (1) where y t and x t respectively denote the endogenous and the forcing 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) (2) The solution to this simple model is given by

y t = 1−aρ b x t

x t = ρx t−1 + σε t

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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 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.

(7)

At the opposite, assume that the policy rule change. 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 is

( y t = 1−a b ( ρ 2 ) x t

x t = ρ 2

x t−1 + σε t

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(8)

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

(9)

The link between the Lucas critique and the microeconometric methods of policy evaluation :

The intuition of these methods are :

1 treatment group : for this group, the policy is effective.

2 control group : for these agents, the policy is not effective.

3 Assumption : these two groups have exactly the same

behaviors ⇒ the comparison of the statistics produce by these two groups gives an evaluation of the policy y t+n T − y t+n C . Using our model, this gap is measured by :

Control : y t+n C = b

1 − aρ (ρ) n Treatment : y t+n T = b

1 − a ρ 2 ρ

2 n

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(10)

The main limits with the non-structural approach proposed by econometricians are :

1 The equilibrium with two groups is not the same than an equilibrium with one homogenous group : The policy can be effective, but its equilibrium effect can change the solution of the decision rules. For example :

Control : y t+n C = e b

1 − aρ (ρ) n = b 1 − aρ (ρ) n Treatment : y t+n T = b

1 − a ρ 2 ρ

2 n

such that y t+n C = y t+n T , because ≈ 1 2 n 1−aρ 1−a ( ρ 2 ) .

2 The estimation can not be use to improve the policy : it is an

estimation of an old decision, not a guide of a new policy.

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Ultimately, it [demanding that one’s structural relations be derived from individual optimization] is the only way in which the ”observational equivalence” of a multitude of alternative possible structural interpretations of the co–movements of aggregate series can be resolved”

(Rotemberg and Woodford (1998), p.1) Proposition

To estimate or to calibrate the DEEP parameters {a, b, ρ, σ} using theoretical restrictions.

References : AES special issue (2002)

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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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evaluations

To use the theory to evaluate 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.

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1 Allocation of Employment in a Labor Market without Frictions (2 sessions)

2 Allocation of Employment in a Frictional Labor Market (3 sessions)

3 Job Search and Saving (1/2 session)

4 Optimal Unemployment Insurance (1/2 session)

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Part I

Allocation of Employment in a Labor Market without Frictions

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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.

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.

(17)

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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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,

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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The optimality conditions of this problem lead to :

(1 + τ c,tt = β t U C 0 t (3) (1 − τ w,t )λ t w t = β t U h 0 t (4) λ t = λ t+1 (1 + R) (5) 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 , t) h t = h({w i } i=1 , {τ c,i } i=1 , {τ w,i } i=1 , K 0 , t)

Idea : the actual consumption and the instantaneous labor supply at time t are function of all the dynamics of the real wage and the taxes, for a given initial condition K 0 .

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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 (3) and (4) become :

C t = 1

1 + τ c,t β t

λ t (6)

1 − h t =

1

(1 − τ w,t )w t

β t λ t

σ

(7) Moreover, by iterating on the equation (5) we obtain

λ t = λ 0 /(1 + R) t (8)

(21)

Using (8), (6) and (7) become :

C t = 1

(1 + τ c,t0 [β(1 + R)] t (9) 1 − h t =

1

(1 − τ w,t )w t λ 0

[β(1 + R)] t σ

(10) Consumption and leisure depend on current prices and taxes, but also on the implicit value of the total wealth (forward looking behavior).

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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 Then, using (9) and (10), 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

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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 the business cycle

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(24)

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

b λ 0

α [β(1 + R)] −t

! 1−σ

− [β(1 + R)] −t αw t

b λ 0

α )

Because w e t e λ 0 = w t b λ 0 , where λ b 0 is the lagrange multiplier in a economy without taxes, 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.

(25)

Preferences

u(C t , L t ) =

( θ log(C t ) + (1 − θ) log(L t ) ( C t θ L 1−θ t ) 1−σ −1

1−σ

C t denotes the consumption L t denotes the leisure

The time allocation is constrained by : T 0 = L t + N t

The budgetary constraint is given in a complete market economy :

(1 + R t )K t + w t N t − K t+1 − C t ≥ 0

where R t is the capital returns and w t the unit price of labor (the wage).

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All firms have access to the same technology :

Y t = A t F (K t , X N,t N t ) = A t K t α (X N,t N t ) 1−α where

K t is the capital stock determined rented at time t. Its unit price is R t + δ : the financial return plus the paiement of the forgone capital during the period (δ per unit of capital used).

N t is the total hours determined at date t.

A t is the technological shock observed at date t, X N,t is the exogenous technical progress which is labor-augmenting. Its growth rate is denoted by γ x . Profit maximization leads to

A t F K = α Y t

K t = R t + δ and A t F N = (1 − α) Y t

N t = w t

(27)

The general equilibrium is definied by the set of functions {C t , K t+1 , N t , Y t }, solution of the following system :

u 0 C t = βE t

1 − δ + α Y t+1

K t+1

u C 0 t+1

u 0 L t u C 0

t

= (1 − α) Y t

N t

K t+1 = (1 − δ)K t + Y t − C t

Y t = A t K t α (X N,t N t ) 1−α

log(A t+1 ) = ρ log(A t ) + (1 − ρ) log(¯ A) + t

This system gives the equilibrium dynamics of C t , K t+1 , N t , Y t , given the stochastic process A t .

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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.

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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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Correlation between GDP and Variable x σ x(t − 1) x(t) x(t + 1)

GDP 1,8% 0,82 1,00 0,82

Model 1,79% 0,60 1,00 0,60

Consumption

Services 0,6% 0,66 0,72 0,61

Non-durable 1,2% 0,71 0,76 0,59

Model 0,45% 0,47 0,85 0,71

Investment

Total investment 5,3% 0,78 0,89 0,78

Model 5,49% 0,52 0,88 0,78

Employment

Total hours 1,7% 0,57 0,85 0,89

Hour per capita 1,0% 0,76 0,85 0,61

Model 1,23% 0,52 0,95 0,55

Productivity 1,0% 0,51 0,34 -0,04

Model 0,71% 0,62 0,86 0,56

(31)

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.

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(32)

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)

(33)

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

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(34)

Then, by reporting the previous results in the FOC of the demand for leisure, we deduce :

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

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.

(35)

1960 1970 1980 1990 2000 2010 1400

1600 1800 2000 2200 2400

Belgium

Nh/A h

1960 1970 1980 1990 2000 2010 0.86

0.88 0.9 0.92 0.94 0.96 0.98 1

Belgium

N/A

1960 1970 1980 1990 2000 2010 1200

1400 1600 1800 2000 2200

Spain

Nh/A h

1960 1970 1980 1990 2000 2010 0.75

0.8 0.85 0.9 0.95 1

Spain

N/A

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1960 1970 1980 1990 2000 2010 1200

1400 1600 1800 2000 2200 2400

France

Nh/A h

1960 1970 1980 1990 2000 2010 0.86

0.88 0.9 0.92 0.94 0.96 0.98 1

France

N/A

1960 1970 1980 1990 2000 2010 1400

1500 1600 1700 1800 1900 2000

Italy

Nh/A h

1960 1970 1980 1990 2000 2010 0.88

0.9 0.92 0.94 0.96 0.98

Italy

N/A

(37)

1960 1970 1980 1990 2000 2010 1600

1700 1800 1900 2000 2100 2200 2300

UK

Nh/A h

1960 1970 1980 1990 2000 2010 0.88

0.9 0.92 0.94 0.96 0.98 1

UK

N/A

1960 1970 1980 1990 2000 2010 1600

1700 1800 1900 2000

US

Nh/A h

1960 1970 1980 1990 2000 2010 0.9

0.91 0.92 0.93 0.94 0.95 0.96 0.97

US

N/A

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(38)

1980 1985 1990 1995 2000 2005 0.05

0.1 0.15 0.2 0.25 0.3

τ c

Be Sp Fr It UK US 1980 1985 1990 1995 2000 2005 0.1

0.2 0.3 0.4 0.5

τ w

1980 1985 1990 1995 2000 2005 0.1

0.2 0.3 0.4 0.5

τ f

1980 1985 1990 1995 2000 2005 1.6

1.8 2 2.2 2.4 2.6 2.8 3

TW

(39)

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.

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(40)

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.

(41)

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

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(42)

How to quantity 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 . If ∆ t < 0, agents react as if there is employment subsidies ↑ h.

If ∆ t > 0, agents react as if there is employment taxes : ↓ h.

(43)

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.

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(44)

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.

(45)

Various factors can explain the labor wedges (∆) :

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.

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)).

The specification of the panel regression is :

ln(1 − ∆ h,w i ,t,TW =0 ) = a i + b ln(TW i,t − 1) + γBarg i,t + βARR i,t + i,t

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(46)

The long-run decline in the hours worked per employee is mainly due to the increase in the taxes.

Reg (1) Reg (2) Reg (3) Reg (4) Reg (5) Reg (6) Reg (7)

a be .2055 .0370 .2690 .0047 .0348 .0119 .3177

[-.08 ;.49] [.00 ;.07 ] [-.001 ; .53] [ -.05 ; .05] [-.30 ; .37] [-.05 ; .07] [ -.03 ; .66]

a sp .0773 .0054 .1364 -.0427 .1367 .1518 .3300

[-.10 ;.26] [-.01 ;.02 ] [-.016 ; .28] [ -.10 ; .02] [-.05 ; .33] [ .09 ; .20] [ .12 ; .53 ]

a fr .1383 .0236 .1829 .0002 -.0427 -.0359 .1755

[-.06 ;.34] [-.01 ;.062 ] [-.00 ; .37] [ -.04 ; .05] [-.27 ; .18] [-.09 ; .02] [ -.06 ; .42]

a it .1510 -.0094 .1834 -.0100 -.0115 -.0904 .1568

[-.08 ;.38] [-.04 ;.02 ] [-.04 ; .40] [ -.04 ; .02] [-.29 ; .26] [-.12 ;-.06] [ -.12 ; .43]

a uk .2237 .1353 .2712 .1032 .2872 .2576 .4376

[.04 ;.40] [ .11 ;.15 ] [ .11 ; .43] [ .05 ; .15] [ .08 ; .48] [ .22 ; .29] [ .23 ; .64 ]

a us .1045 .0843 .1387 .0519 .2462 .2484 .3239

[.01 ;.19] [.05 ;.11 ] [ .06 ; .20] [ -.00 ; .10] [ .16 ; .33] [ .21 ; .27] [ .23 ; .41 ]

TW -.3571 -.3209 -.3254 -.3627

[-.44 ;-.26] [-.39 ;-.25] [-.39 ; -.25] [ -.45 ; -.27]

Barg -.2666 -.3196 -.1921 -.4062

[-.64 ;.11] [-.68 ; .05] [-.66 ; .27] [ -.86 ; .05]

ARR .0949 .1222 -.2785 -.3107

[-.06 ;.25] [ -.03 ; .28] [-.43 ;-.12] [ -.46 ; -.15]

N 144 144 144 144 144 144 144

R 2 .9248 .9222 .9224 .9237 .8764 .8867 .8892

OLS regression. “TW” is the tax wedge, “Barg” the bargaining power of the workers and “ARR” the average

replacement rate. The confidence intervals (in brackets) are at the 95% level.

(47)

In this first part, we have focussed on the intensive margin : the number of hour worked by worker.

There is another margin : the extensive margin, which give the number of labor market participant in the population of the 15-65 years old.

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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(48)

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

(49)

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

49/211

(50)

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.

(51)

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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(52)

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 = 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

(53)

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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(54)

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

(55)

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 :

p + v

U 0 (c ) = (1−τ )w ⇔ 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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(56)

0 2 4 6 8 0.5

1 1.5 2 2.5

3 x 10 5 GAINS

PERIODES

FRANCS

bas moyen haut moyenne

0 2 4 6 8

38 40 42 44 46 48

HEURES TRAVAILLEES

PERIODES

HEURES/SEMAINE

0 2 4 6 8

0 0.5 1 1.5 2

2.5 x 10 5 ACTIF

PERIODES

FRANCS

(57)

log w = α + [school ; exp ; exp 2 ]β + g (Z ) + ε

Diploma All low medium hight

school 0.0722 0.0366 0.0283 0.0377

[25.887] [4.995] [3.051] [30.012]

exp 0.0307 0.0307 0.0386 0.0430

[11.037] [9.676] [5.510] [13.022]

exp 2 -0.000453 -0.000465 -0.000376 -0.000713

[-6.787] [-6.503] [-2.107] [-8.106]

R 2 0.334 0.16 0.30 0.20

t − stat are reported in the brackets.

The investments in Human capital, schooling or training, have significant impact on the real wage of the workers.

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(58)

Estimation 1 Estimation 2 Estimation 3

school 0.0133 0.0282 0.0247

[1.676] [3.541] [3.094]

school ∗ dipm 0.0616 0.0085 0.0085

[7.275] [0.772] [0.773]

school ∗ diph 0.0706 0.0266 0.0278

[9.553] [3.115] [3.265]

exp 0.0354 0.0228 0.0221

[12.77] [7.667] [7.441]

exp 2 -0.000548 -0.000314 -0.000306

[-8.238] [-4.358] [-4.258]

exp ∗ dipm - 0.0116 0.0147

[2.509] [3.163]

exp 2 ∗ dipm - -0.00004 -0.00003

[-0.288] [-0.207]

exp ∗ diph - 0.0321 0.0334

[6.605] [6.889]

exp 2 ∗ diph - -0.000649 -0.000678

[-4.32] [-4.52]

no − certif - - -0.0875

[-4.262]

R 2 0.36 0.39 0.40

The returns of these investment increase with the

educational level.

(59)

We assume that the direct cost of education is zero : the cost is the opportunity cost measured by the wage, w . Then, the objective of the agent is

max

T

Z R 0

e −rt (1 − s (t))h(t)wdt

sc h(t) = ˙ αs (t)h(t) with s(t) = 1 or s (t) = 0

where α denotes the ability to learn. When the agent quits school at age T her amount of human capital is

h(t) ˙

h(t) = α ⇒ d log(h(t))

dt = α ⇒ log(h(t)) = µ + Z t

0

αd τ We deduce that

h(t) = h 0 e αt ⇒ h(T ) = h 0 e αT

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The agent problem can be rewritten as follow : max T

Z R T

e −rt h 0 e αT wdt The FOC is

−e −rT h 0 e αT w + Z R

T

e −rt αh 0 e αT wdt = 0

−e −rT + α e −rT − e −rR

r = 0

α − r

α = e r (T−R) R + 1

r log

α − r α

= T

(61)

Interpretation :

The arbitration between school and work :

−e −rT h 0 e αT w

| {z }

Opportunity cost of the marginal year of schooling

+

Z R T

e −rt αh 0 e αT wdt

| {z }

Discounted present value of this additional year of schooling

= 0

Its appears that T (α) with T 0 (α) > 0. More efficient individuals study longer.

When the agent arrive in the labor market, her wage is wh(T (α)) : the longer the studies, the higher the wage.

T increases with R : the longer is the horizon allowing to recoup the eduction costs, the longer is the eduction duration.

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(62)

Empirical studies show that the number of years of schooling have a positive impact on the wage dynamics.

Nevertheless, after this intensive investment in education, empirical studies show also that the wage continue to grow.

There is a learning process during the career of the individuals.

Assume that after the school, agent can learn if she devotes a

fraction l (t) of her time in the training sector.

(63)

Then, the agent maximize her gains discounted over the whole of her life cycle, subject to the human capital accumulation

technology max

l (t)

Z R 0

e −rt A[1 − l(t)]h(t)dt sc . h ˙ = αg (l(t)h(t)) − δh(t) where

α is the worker ability,

δ the depreciation rate of knowledge,

g (.) the production function of the human capital sector, A[1 − l (t)] represent the wage by unit of human capital : A is the potential wage and Al(t) the training cost.

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The hamiltonian of the problem is

H = e −rt A[1 − l(t)]h(t) + λ(t)[αg (l (t)h(t)) − δh(t)]

The FOC, under the assumption that l(t) ∈]0; 1[ :

∂H

∂l (t) = 0 ⇔ −e −rt Ah(t) + λ(t)αh(t )g 0 = 0

∂H

∂h(t) = − λ(t) ˙ ⇔ e −rt A[1 − l (t)] + λ(t)[αl(t)g 0 − δ] = − λ(t) ˙

The first equation shows that the cost of one hour in the training

sector (e −rt Ah(t)) must be equal to its marginal productivity

(αh(t)g 0 (l (t)h(t))) in term of wage λ(t).

(65)

The two FOC lead to :

− λ(t) + ˙ δλ(t) = e −rt A

A particular solution for this differential equation takes the form λ(t) = µAe rt , implying µ = r 1 .

The solution of the homogenous equation − λ(t) + ˙ δλ(t) = 0 is ce δt .

The solution is then λ(t) = ce δt + r A e rt with the terminal condition λ(R) = 0.

We then have

λ(t) = A r + δ e −rt

h

1 − e −(r+δ)(R−t) i

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(66)

λ(t) represents the marginal value of human capital : its decrease during the life cycle, and it is equal to zero at the retirement age R. At this time investment implies a cost without any returns : l(R) = 0.

At the beginning of the life cycle, λ(t) is large : agent invests a lot in training

At the end of the life-cycle, the depreciation can be higher

than the investment implying a decline in the gains.

(67)

The households : U =

" T X

t=1

β t−1 c t 1−γ − 1

1 − γ + λ L 1−η t − 1 1 − η

#

+ θ a 1−ρ T+1 − 1 1 − ρ

| {z }

Altruism

L t = l t h t Leisure augmented by the human capital T 0 = l t + n t + e t

h t+1 = (1 − δ)h t + α(e t h t ) ζ with ζ ∈ (0, 1) a t+1 = (1 + r )a t + wn t h t + χ t b − c t

a t ≥ 0 Financial constraint

where b is the heritage and a T +1 the leg. The leisure is denoted by l t , the number of hour worked n t and training effort e t .

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(68)

Firms and General equilibrium

r = F K (K, H) − δ k

w = F H (K, H)

H =

T

X

t=1

µ t

X

J

n t (J)h t (J)Θ(J)

K =

T

X

t=1

µ t

X

J

a t (J)Θ(J) Y = C + δ k K

C =

T

X

t=1

µ t

X

J

c t (J)Θ(J) Y = AK m H 1−m

where µ t is the share of age-t workers in the population and Θ(J)

the share of characteristic-J workers in the population.

(69)

0 2 4 6 8 4

6 8 10 12

14 x 10 4 GAINS

PERIODES

FRANCS

théorique observé

0 2 4 6 8

30 35 40 45

HEURES TRAVAILLEES

PERIODES

HEURES/SEMAINE

0 2 4 6 8

0 0.5 1 1.5 2

2.5 x 10 5 ACTIF

PERIODES

FRANCS

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(70)

0 2 4 6 8 0

0.5 1 1.5 2

2.5 x 10 5 GAINS

PERIODES

FRANCS

théorique observé

0 2 4 6 8

10 20 30 40 50 60

HEURES TRAVAILLEES

PERIODES

HEURES/SEMAINE

0 2 4 6 8

0 0.5 1 1.5

2 x 10 5 ACTIF

PERIODES

FRANCS

(71)

0 2 4 6 8 0

0.5 1 1.5 2 2.5 3

3.5 x 10 5 GAINS

PERIODES

théorique observé

0 2 4 6 8

0 10 20 30 40 50 60

HEURES TRAVAILLEES

PERIODES

0 2 4 6 8

0 0.5 1 1.5 2

2.5 x 10 5 ACTIF

PERIODES

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(72)

Part II

Allocation of Employment in a

Frictional Labor Market

Job Search and Matching

(73)

Consider an unemployed worker.

Finding a good job is an uncertain process which requires both time and financial resources.

This assumption stands in contrast to the classical model, in which workers and firms are assumed to have full information at no cost about job opportunities and workers.

References : McCall’s (1970),

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(74)

Consider an unemployed worker who is searching for a job by visiting area firms :

1 Although the worker likely has many job opportunities, she has incomplete information as to the location of her best

opportunities.

2 Hence, she must spend time and resources searching, and she must hope she has luck finding one of her better opportunities quickly.

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.

(75)

Each week the worker receives one wage offer.

In order to capture the uncertainty of job offers, we assume that this offer is drawn at random from an urn containing wage offers between w and w.

Draws from this urn are independent from week to week, so the size of next week’s offer is not influenced by the size of this week’s offer.

While I will interpret draws as weekly wage rates, they can be thought of more generally as capturing the total desirability of a job, which could depend on hours, location, prestige, and so on.

For simplicity, assume that all jobs require the same number of hours and are of the same overall quality, so that jobs differ only in terms of the wage.

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(76)

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

(77)

An unemployed worker, in each period t , receives an unemployment benefit b.

The probability of getting a job offer is denoted λdt

This 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 )

Let w r denote the reservation wage. The worker accepts the wage offer w (t) if w (t) > w r , which implies that she earns that wage in period t and thereafter for each period she has not been laid off.

The probability of being laid off at the beginning of the period is sdt.

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In making her decision, the unemployed worker must compare the expected lifetime incomes of accepting or rejecting a particular offer.

rv accept (w ) = w − s

v accept (w ) − v reject rv reject = b + λ

Z w w r

[v accept (x) − v reject ]dF (x )

v reject (w ) is the expected present value of lifetime income if

she rejects a wage offer w and waits for a better offer,

v accept (w ) is the expected present value of lifetime income if

she accepts w ,

(79)

The value function of an acceptable offer can be rewritten as follow v accept (w ) = w + sv reject

r + s

v accept (w ) increases linearly with w , whereas v reject is constant.

For values of w less than w r , v reject is greater than v accept (w ), so the worker is better off rejecting the offer.

For w greater than w r , v reject is less than v accept (w ), so the worker is better off accepting the offer.

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(80)

The reservation wage w r is the value of w which satisfies v accept (w r ) = v reject

This implies that

rv accept (w r ) = w r and then

w r = b + λ Z w

w r

[v accept (x) − v reject ]dF (x)

= b + λ

r + s Z w

w r

[1 − F (x)]dx

(81)

d

dx {[v accept (x) − v reject ][1 − F (x)]} = dv accept (x)[1 − F (˜ x)]

−[v accept (x) − v reject ]dF (x) [v accept (x) − v reject ][1 − F (x)] w

w r = 0

⇒ Z w

w r

dv accept (x)[1 − F (x)]dx = Z w

w r

[v accept (x) − v reject ]dF (x) Because dv accept (x) = r +s 1 , we deduce :

w r = b + λ r + s

Z w w r

[1 − F (x)]dx

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1 Because a higher interest rate implies discounting future earnings more rapidly, an increase in the real interest rate lowers the benefits of waiting for a higher wage. This suggests that the reservation wage will decrease.

2 An increase in the firing rate reduces the expected length of time at a given job, and thus reduces the benefit of waiting for a relatively high wage offer : the reservation wage decreases.

3 If the maximum of the wage offer distribution increases, then incentive to wait for a better job raises : w r increases.

4 Since unemployment compensation acts as a subsidy to search

activity, the worker is willing to wait longer for a high-paying

job and thus increases her reservation wage.

(83)

The exist rate of unemployment is given by λ[1 − F (w r )]. The average duration of unemployment is then given by

UD = 1

λ[1 − F (w r )]

when w r increases, UD increases.

The unemployment rate generated by the model is : s (1 − u)

| {z }

exists from employment

= uλ[1 − F (w r )]

| {z } exists from unemployment

⇒ u = s

s + λ[1 − F (w r )]

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(84)

The population can be divided into 6 age groups , denoted C i for i = 1, ..., 6.

We consider the following age groups :

20 - 34 year old individuals, in C 1 , start working.

Experienced individuals of age 35 - 49, in C 2 , expect to be employed for a long time.

People of age 50 - 54 in C 3 and especially 55 - 59 in C 4 , expect that the duration of the job is short before retirement.

Individuals in age group 60 - 64, in C 5 , can choose to retire early or not.

Finally, people aged 65 and more, in C 6 , are all retirees as 65 is

the mandatory retirement age.

(85)

Let V i e (w ) be the value of the optimization problem for a worker of age C i and paid w ,

V i u the value of the optimization problem for an unemployed worker of age C i ,

V r the value of a retiree.

Let V i (w ) be the value of the optimization problem for a worker of age C i who was employed in the previous period and has today the option to work at wage w

V i (w ) = max {V i e (w ), V i u } for i = 1, .., 5

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Bellman equations can be written as : for i = 1, 2, 3

V i e (w ) = u((1 − τ p − τ b )w , T − h) + β {π i [(1 − λ i )V i (w ) + λ i V i u ] +(1 − π i )[(1 − λ i+1 )V i+1 (w ) + λ i+1 V i+1 u ]

V i u = max

s i

{u((1 − τ p )b i , T − s i ) +β

π i

φ(s i )

Z

V i (w )dF i (w ) + (1 − φ(s i ))V i u

+(1 − π i )

φ(s i+1 ) Z

V i+1 (w )dF i+1 (w )

+(1 − φ(s i+1 ))V i u +1

(87)

for i = 4

V 4 e (w ) = u((1 − τ p − τ b )w , T − h) + β {π 4 [(1 − λ 4 )V 4 (w ) + λ 4 V 4 u ] +(1 − π 4 )[(1 − λ 5 ) max{V 5 (w ), V 5 r } + λ 5 max{V 5 u , V 5 r }]}

V 4 u = max

s 4

{u((1 − τ p )b 4 , T − s 4 ) +β

π 4

φ(s 4 )

Z

V 4 (w )dF 4 (w ) + (1 − φ(s 4 ))V 4 u

+(1 − π 4 )

φ(s 5 ) R

max{V 5 (w ), V 5 r }dF 5 (w ) +(1 − φ(s 5 )) max{V 5 u , V 5 r }

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(88)

for i = 5

V 5 e (w ) = u((1 − τ p − τ b )w , T − h)

+β {π 5 [(1 − λ 5 ) max{V 5 (w ), V 5 r } + λ 5 max{V 5 u , V 5 r }]

+(1 − π 5 )V 6 r 6 V 5 u = max

s 5

{u((1 − τ p )b 5 , T − s 5 ) +β

π 5

φ(s 5 )

Z

max{V 5 (w ), V 5 r }dF 5 (w ) +(1 − φ(s 5 )) max{V 5 u , V 5 r }] + (1 − π 5 )V 6 r 6 V 5 r = u(p 5 , T ) + β

π 5 V 5 r + (1 − π 5 )V 6 r 5

where p 5 denotes the retiree’s pension at age 5.

(89)

for i = 6

V 6 r5 = u(p 5 , T ) + β π 6 V 6 r 5 V 6 r6 = u(p 6 , T ) + β

π 6 V 6 r 6

Benchmark : the pension is not increased by additional years of working beyond the full pension rate : p 6 = p 5 .

⇔ V e does not increase if the agent decides to postpone retirement (huge tax on continued activity).

Policy : an actuarially fair increase in pension (p 6 > p 5 ) can make the early retirement option undesirable for employed workers.

⇔ Employment is more valuable than any other options : there is now an employment surplus at the early retirement age, which conversely boosts the search intensity before this age.

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(90)

The optimal decision for search intensity is given by : for i = 1, 2, 3, 4

u 0 2 ((1 − τ p )b i , T − s i ) = φ 0 (s i )βπ i Z

V i (w )dF i (w )

− V i u

The marginal disutility of job search activity equals its expected

return, which is captured by the increase in the probability of

getting a wage offer times the expected surplus of employment.

(91)

for i = 5

u 2 0 ((1−τ p )b 5 , T −s 5 ) = φ 0 (s 5 )βπ 5

R

max[V 5 (w ), V 5 r ]dF 5 (w )

− max[V 5 u , V 5 r ]

If the continued activity opportunity is sufficiently attractive after the early retirement age, the employment and the unemployment values converge later, only when the mandatory retirement (C 6 ) is imminent.

The horizon of older workers just before the early retirement age is then broadened.

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Dividend of Actuarially-Fair Pension Adjustments

Table : Incentive schemes and employment rates

Age groups C 1 C 2 C 3 C 4 C 5

Age in years 20-29 30-49 50-54 55-59 60-64

1. Benchmark 0.828 0.867 0.874 0.549 0

2. Retirement Policy 0.830 0.867 0.874 0.714 0.201

(93)

Now assume that worker can search on-the-job. The value to be employed is now

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.

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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.

(95)

Assumptions and notations. Let

V n (w ) denote the value function for an employed worker who earns w ,

V us the value function for a short-term unemployed person who is paid b unemployment benefits,

V ul the value function of a long-term unemployed individual who is paid a minimum social income msi.

Let δ ∈ [0, 1] denotes the probability that a short-term unemployed worker becomes a long-term unemployed worker.

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(96)

These functions are solved by : rV n (w) = w + λ e

Z

w

[V n ( ˜ w ) − V n (w )] dF ( ˜ w ) − s [V n (w ) − V us ] rV us = b + λ s

Z

x s

[V n ( ˜ w ) − V us ] dF ( ˜ w ) − δ h

V us − V ul i rV ul = msi + λ l

Z

x l

h

V n ( ˜ w ) − V ul i

dF ( ˜ w )

(97)

The reservation wage policies x s and x l are derived from the two conditions V n (x s ) = V us and V n (x l ) = V ul , so that :

x s = b + (λ s − λ e ) Z w

x s

[V n ( ˜ w ) − V us ] dF ( ˜ w ) − δ h

V us − V ul i x l = msi + (λ l − λ e )

Z w x l

h

V n ( ˜ w ) − V ul i

dF ( ˜ w ) − s [V us − V ul ] The positive probability of losing the unemployment benefit (δdt > 0) then accounts for a decrease in the short-term unemployed workers’reservation wage, x s .

The possibility of becoming short-term unemployed through employment destruction, which occurs with a probability sdt, implies a fall in the reservation wage of the long-term

unemployed.

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(98)

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, 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.

(99)

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 that lead to an increase the firm value.

⇒ in this case, an job search equilibrium exists with non-degenerate dispersion of wage offers.

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(100)

the equilibrium rate of unemployment is deduce from s (1 − u ) = λ 0 [1 − F (w r )]u

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

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