DEA Universit´e de Paris 1
Equilibre g´en´eral, chˆomage et assurance compl`ete
Le Mod`ele d’Andolfatto [AER,1996]
Franc ¸ois Langot
GAINS (Le Mans) & Cepremap
19 f´evrier 2004
Trade on the Labor Market
– N t is the number of jobs that are matched with a worker,
– U t = 1 − N t is the measure of unemployment rate.
– Workers search passively : e is constant,
– V t is the total number of new jobs made available by firms,
– each of the vacancies incurs a flow cost equal to ω.
– s is the exogeneous jobs destructions.
– The rate M t at which new job matches form is go- verned by an aggregate-matching technology.
The law of motion for aggregate employment is defined by :
N t+1 = (1 − s)N t + M t with M t = ΥV t ψ (eU t ) 1−ψ
The Firms’ Behavior (1)
All firms have access to the same technology : Y t = A t (h t N t ) α K t 1−α
where Y t , N t h t and K t respectively denote the output, the total hours, the capital stock. The stochastic process of the technological shock, A t , is :
log A t = ρ A log A t−1 + (1 − ρ A ) log A + ² A,t where |ρ A | < 1 and ² A → N (0, σ ²
A).
As firms are owned by households, they discount ex- pected future values at rate β λ λ
t+1t
.
Let V (Ω F t ) be the maximum expected value of the firm in state Ω F t . This value function must satisfy the following recursive relationship,
V (Ω F t ) = max
{V
t,I
t,N
t+1,K
t+1}
½
A t K t α (N t h t ) 1−α − w t h t N t − I t
−ωV t + βE t
λ t+1
λ t V (Ω F t+1 )
s.t. N t+1 = (1 − s)N t + Φ t V t (1)
K t+1 = (1 − δ)K t + I t (2)
where Ω F t = {K t , N t , A t , Φ t , w t , h t } summarizes each
firm’s state.
The Firms’ Behavior (2)
The optimal decisions for the firm give conditions that govern the intertemporal allocation of investment in both physical capital and vacancies. They are given respecti- vely by
λ t = βE t
λ t+1
α Y t+1
K t+1 + 1 − δ
ω
Φ t = βE t
λ t+1 λ t
F 2,t+1 h t+1 − w t+1 h t+1 + (1 − s)ω Φ t+1
Φ t = M V
tt
is the exogenous probability for each firm that a vacant position becomes a filled position.
Given the production function, we have F 2,t ≡ (1 − α)[Y t /N t h t ]
Since the ratio of capital stock to labor input K t /N t h t ≡ k t is determined by the rental cost, r t = αA t k t α , it is straightforward to deduce that
F 2,t = (1 − α)α
1−ααA t
1−α1r t
α−1αIt is assumed that the weight of a worker is ‘small’, F 2,t
thus being taken as given by the firm during the bargai-
ning process.
Full-Insurance and Aggregation Rule (1)
– There is a distinction between jobs and workers flows in the model :
H1) jobs flows are governed by the matching/separation process,
H2) workers flows are determined exogenously by a game of “musical chairs”.
=> At the beginning of each period, the entire work- force is “shuffled” randomly across the given set of jobs (before any trading occurs). There are N t pro- ductive jobs at the beginning of period t, which re- presents the probability of employment for each hou- sehold in any period.
– Risk-adverse workers choose to purchase B t units of unemployment insurance at a price τ t , that delivered B t units of the consumption good whenever she is unemployed within the period.
In a perfectly competitive unemployment insurance markets, the profit flow for an insurance company is given by :
Π A t = τ t B t − (1 − N t ) B t
Assuming zero profit condition for the unemploy-
ment insurance company implies τ t = 1 − N t .
Full-Insurance and Aggregation Rule (2)
In each period the household chooses a level of in- surance, B t , and makes its consumption C t z and saving choices q t+1 z , contingent upon its employment status (z = n, u).
Let denoted the vector of control variables C t = {B t , C t z , q t+1 z } z=n,u
and Ω H,z t the vector of state variables, the problem faced by each household can be stated recursively by :
W (Ω H t ) = max
C
t½
N t
µU t n + βE t
·W(Ω H,n t+1 )
¸¶+(1 − N t )
µU t u + βE t
·W(Ω H,u t+1 )
¸¶¾This maximization is subject to the constraints referring to each state on the labor market :
C t n + τ t B t +
Zp t+1 q t+1 n dA t+1 ≤ q t + π t + w t h t (3)
C t u + τ t B t +
Zp t+1 q t+1 u dA t+1 ≤ q t + π t + B t (4)
where
Rp t+1 q z t+1 dA t+1 represents the value of contingent
claims at time t yielding q z t+1 units of the consumption
good at time t + 1, in a given state A t+1 . π t represents
pure profits due to the non-walrasian structure of the
labor market.
Full-Insurance and Aggregation Rule (3)
The first order conditions with respect to B t , C t z , and q z t+1 are respectively :
λ u t = λ n t = λ t (5)
∂U t z
∂C t z = λ z t z = n, u (6) β ∂ W(Ω H,z t+1 )
∂q t+1 z f (A t+1 |A t ) = λ z t p t+1 z = n, u (7) The envelop theorem implies that :
∂ W(Ω H t )
∂q t = λ t ∀ z = n, u
Then, concavity and differentiability of the value function imply :
∂U t n
∂C t n = ∂U t u
∂C t u
q n t+1 = q t+1 u = q t+1
This result shows that, given the complete set of insu- rance markets, the worker’s saving choice does not de- pend on its state on the labor market. We have :
Ω H,n t+1 = Ω H,u t+1 = Ω H t+1
Full-Insurance and Aggregation Rule (4)
Then, the difference between the two budget constraints (equations (3) and (4)) gives the optimal choice of insu- rance :
B t = w t h − (C t n − C t u )
Consequently, we get the following budget constraint for a representative household :
N t C t n + (1 − N t )C t u +
Zp t+1 q t+1 dA t+1 = q t + π t + N t w t h Remark that the equilibrium price of asset is given by :
β λ t+1
λ t f (A t+1 |A t ) = p t+1
The Worker’s Behavior (1)
Then, the program solved by the representative hou- sehold can be summarized by the following dynamic pro- blem :
W (Ω H t ) = max
{C
tn,C
tu}
½
N t U t n + (1 − N t )U t u + βE t
·W(Ω H t+1 )
¸¾N t C t n + (1 − N t )C t u = N t w t h + Π t
N t+1 = (1 − s)N t + Ψ t (1 − N t )
where Ψ t = M t /(1 −N t ), denotes the transition rate from E → U .
In addition, we can abstract from the securities market as it plays no role in our analysis (given the representative framework, these claims are not traded in equilibrium).
Then, let Π t denotes the financial income : Π t = V (Ω F t ) − βE t
λ t+1
λ t V (Ω F t+1 )
= q t + π t −
Zp t+1 q t+1 dA t+1
The Worker’s Behavior (2)
Consequently, the worker’s optimal behavior is com- pletely characterized by
λ t = 1
C t z z = n, u (8) where λ t is multiplier associated with the representative household’s budget constraint.
The optimal decision for the firm gives conditions that govern the intertemporal allocation of investment in phy- sical capital. This decision gives the intertemoral alloca- tion of consumption :
λ t = βE t [λ t+1 (1 − δ + r t+1 )] (9)
where r t ≡ αY t /K t .
The Wage Setting Rule (1)
Hours per worker are constant
The marginal value of employment for the representa- tive household is given by (Ψ t = M t /(1 − N t )) :
∂ W (Ω H t )
∂N t = λ t B t +U t n −U t u +(1 − s − Ψ t ) βE t
∂ W (Ω H t+1 )
∂N t+1
– B t = w t h − (C t n − C t u ) refers to the optimal choice of unemployment insurance.
– λ t is the budget constraint Lagrange multiplier.
– The utility function is given by (Andolfatto [AER,96]) : U t z = log (C t z ) + Γ z,t with
Γ n,t = Γ(h t ) Γ u = Γ(e) At equilibrium, if the utility function is additively sepa- rable in consumption and leisure, we have :
C t n = C t u and U t n − U t u = Γ n − Γ u if Γ n < Γ u <=> employed workers are worse off Then we have :
∂ W (Ω H t )
∂N t = λ t w t h+(Γ n −Γ u )+(1 − s − Ψ t ) βE t
∂ W(Ω H t+1 )
∂N t+1
(10)
The Wage Setting Rule (2)
Symmetrically, for the firm this value is given by :
∂ V (Ω F t )
∂N t = F 2,t h − w t h + (1 − s)βE t
λ t+1 λ t
∂V (Ω F t+1 )
∂N t+1
(11) Then, we define the total marginal value of employment measured in units of the consumption good :
S t = ∂V (Ω F t )
∂N t + ∂ W(Ω H t )/∂N t
λ t (12)
Let 0 < ξ < 1 denotes the firm’s share of S t . This sharing rule implies :
ξ ∂W (Ω H t )/∂N t
λ t = (1 − ξ) ∂ V (Ω F t )
∂N t (13) From the firm’s decision problem, we have also :
βE t
λ t+1 λ t
∂ V (Ω F t+1 )
∂N t+1
= ω
Φ t (14)
Thus, equations (13) and (14) yield : ξβE t
λ t+1 λ t
∂ W(Ω H t+1 )/∂N t+1 λ t+1
= (1 − ξ )E t
λ t+1 λ t
∂V (Ω F t+1 )
∂N t+1
= (1 − ξ ) ω
Φ t (15)
The Wage Setting Rule (3)
Finally, combining conditions (10), (11), (13) and (15), the wage equation is given by :
w t h = (1 − ξ )
F 2,t h + ω V t 1 − N t
+ ξ
Γ u − Γ n λ t
The real wage is a weighted average of
(i) a procyclical component (labor productivity and average hiring costs)
(ii) the outside option, which is a decreasing function of λ t , with Γ u > Γ n .
Outside option dynamics
If the wealth increases (<=> ∆λ t < 0)
=> the outside option increases. Thus, it clearly shows
that dynamics of the real wage is a combination of two
procyclical components, (i) labor productivity and hiring
costs, and (ii) the outside option. If the wealth increases,
so does the real wage.
General Equilibrium
The general equilibrium is definied by the set of func- tions {C t , V t , K t+1 , N t+1 , w t , M t , Y t }, solution of the fol- lowing system :
1
C t = βE t
1 − δ + α Y t+1 K t+1
1 C t+1
ωV t
M t = βE t
C t C t+1
(1 − α) Y t+1
N t+1 − w t+1 h + (1 − s) ωV t+1 M t+1
K t+1 = (1 − δ)K t + Y t − C t − ωV t N t+1 = (1 − s)N t + M t
w t h = (1 − ξ)
α Y t
N t + ω V t 1 − N t
+ ξC t (Γ u − Γ n ) M t = ΥV t ψ (eU t ) 1−ψ
Y t = A t K t α (hN t ) 1−α
log(A t+1 ) = ρ A log(A t ) + (1 − ρ A ) log( ¯ A) + ² A,t
This system gives the equilibrium dynamics of C t , V t , K t+1 , N t+1 ,
given the stochastic process A t .
The Wage Setting Rule (1b)
Real wage and hours per worker are bargai- ned
Γ n replaced by Γ(h t ) = Γ n (1 − h t ) 1+η 1 + η
The marginal value of employment for the representa- tive household is given by (Ψ t = M t /(1 − N t )) :
∂ W (Ω H t )
∂N t = λ t w t h t +Γ(h t )−Γ u +(1 − s − Ψ t ) βE t
∂ W (Ω H t+1 )
∂N t+1
(16) – The individual labor supply elasticity is the given by
η −1 (l −1 − 1).
– λ t is the budget constraint Lagrange multiplier.
– The utility function is U t z = log (C t z ) + Γ z
Symmetrically, for the firm this value is given by :
∂ V (Ω F t )
∂N t = F 2,t h t − w t h t + (1 − s)βE t
λ t+1 λ t
∂ V(Ω F t+1 )
∂N t+1
(17)
The Wage Setting Rule (2b)
Then, we define the total marginal value of employ- ment measured in units of the consumption good :
S t = ∂ V (Ω F t )
∂N t + ∂W (Ω H t )/∂N t
λ t (18)
S t = F 2,t h t − w t h t + (1 − s)βE t
λ t+1 λ t
∂V (Ω F t+1 )
∂N t+1
+w t h t + (Γ(h t ) − Γ u )
λ t + (1 − s − Ψ t ) βE t
1 λ t
∂ W(Ω H t+1 )
∂N t+1
Let 0 < ξ < 1 denotes the firm’s share of S t . This sharing rule gives the real wage such that :
ξ ∂W (Ω H t )/∂N t
λ t = (1 − ξ) ∂ V (Ω F t )
∂N t (19) and the optimal hours per workers are given by :
∂S t
∂h t = 0 (20)
From the firm’s decision problem, we have also : ξβE t
λ t+1 λ t
∂ W(Ω H t+1 )/∂N t+1 λ t+1
= (1 − ξ )E t
λ t+1 λ t
∂V (Ω F t+1 )
∂N t+1
= (1 − ξ ) ω
Φ t (21)
The Wage Setting Rule (3b)
Finally, combining conditions (16), (17), (19) and (21), the wage equation is given by :
w t h t = (1 − ξ)
α Y t
N t + ω V t 1 − N t
+ ξ
Γ u − Γ n (1−h 1+η
t)
1+ηλ t
and the hours per workers is given by : Γ n
λ t (1 − h t ) η = α Y t N t h t
The wage bill w t h t is a weighted average of
(i) a procyclical component (labor productivity and average hiring costs)
(ii) the outside option.
The hours per workers combined two components :
(i) the instantaneous substitution effect given by α 2 (Y t /N t h t ), (ii) the intertemporal substitution and wealth effects
(λ t ).
Outside option dynamics
– If the wealth increases (<=> ∆λ t < 0) => the outside option increases.
– If the hours per workers increases, the outside option
increases : this leads to an increase of the real wage.
General Equilibrium (bis)
The general equilibrium is defined by the set of func- tions {C t , V t , K t+1 , N t+1 , w t , h t , M t , Y t }, solution of the following system :
1
C t = βE t
1 − δ + α Y t+1 K t+1
1 C t+1
ωV t
M t = βE t
C t C t+1
(1 − α) Y t+1
N t+1 − w t+1 h t+1 + (1 − s) ωV t+1 M t+1
K t+1 = (1 − δ)K t + Y t − C t − ωV t N t+1 = (1 − s)N t + M t
w t h t = (1 − ξ)
α Y t
N t + ω V t 1 − N t
+ ξC t
Γ u − Γ n (1 − h t ) 1+η 1 + η
α Y t
N t h t = Γ n (1 − h t ) η C t
M t = ΥV t ψ (eU t ) 1−ψ
Y t = A t K t α (hN t ) 1−α
A t+1 = A ρ t
AA ¯ 1−ρ
Aexp(² A,t )
The Social Welfare Problem
The central planner solves the following problem, where C t = {C t , h t , V t , K t+1 , N t+1 } :
P (Ω t ) = max
C
t{log(C t ) + N t Γ(h t ) + (1 − N t )Γ u + βE t [P (Ω t+1 )]}
K t+1 = (1 − δ)K t + A t (h t N t ) α K t 1−α − C t − ωV t N t+1 = (1 − s)N t + ΥV t ψ (1 − N t ) 1−ψ
The FOC and the envelope theorem give : 1
C t = βE t
1 − δ + α Y t+1 K t+1
1 C t+1
Γ n (1 − h t ) η = α Y t N t h t
1 C t ωV t
M t = βE t
C t C t+1
ψ(1 − α) Y t+1
N t+1 + ψ (Γ(h t+1 ) − Γ u )C t+1 +ω(1 − ψ) + (1 − s) ωV t+1
M t+1
K t+1 = (1 − δ)K t + A t K t α (h t N t ) 1−α − C t − ωV t
N t+1 = (1 − s)N t + ΥV t ψ (eU t ) 1−ψ
Pareto Optimal Allocation
In order to have an Equilibrium Allocation which cor- respond to the Pareto Optimal one, the wage must in- ternalize the search externalities. The Optimal path of employment is such that
ωV t
M t = βE t
C t C t+1
ψ(1 − α) Y t+1
N t+1 + ψ (Γ n − Γ u )C t+1 +ω(1 − ψ) + (1 − s) ωV t+1
M t+1
whereas the Equilibrium path is such that ωV t
M t = βE t
C t C t+1
(1 − α) Y t+1
N t+1 − w t+1 h + (1 − s) ωV t+1 M t+1
w t h = (1 − ξ )
α Y t
N t + ω V t 1 − N t
+ ξC t (Γ u − Γ n ) or, after substitution,
ωV t
M t = βE t
C t C t+1
ξ(1 − α) Y t+1
N t+1 + ξ (Γ n − Γ u )C t+1 +ω(1 − ξ) + (1 − s) ωV t+1
M t+1
Thus Equilibrium level of employment, and of vacancies,
correspond to the Social-Optimal level iff ξ = ψ, the
Resolution Method (1)
In order to solve this non-linear dynamic system, we take its first-order Taylor series approximation around the steady-state. We obtain the following linear equations :
− c
bt = E t [− c
bt+1 + a 1 k
bt+1 + a 2
cn t+1 +a 3 a
bt+1 + o(x 2 )]
c
bt + b 1 v
bt + b 2
cn t = E t [b 3 c
bt+1 + b 4 k
bt+1 + b 5
cn t+1 +b 6 a
bt+1 + b 7 v
bt+1 + o(x 2 )]
k
bt+1 = c 1 k
bt + c 2
cn t + c 3 a
bt + c 4 c
bt + c 5 v
bt + o(x 2 )
c
n t+1 = d 1
cn t + d 2 v
bt + o(x 2 ) a
bt+1 = ρ A a
bt + ²
bt+1 + o(x 2 )
Then, this leads to the first order vector system, where s t = [ c
bt , v
bt , k
bt ,
cn t , a
bt ] 0 :
M 1 s t = M 0 s t+1 + M 2 η t+1
where η t+1 = [ ²
bt+1 , w
cc t+1 , w
cv t+1 , w
ck t+1 , w
cn t+1 , w
ct+1 a ] 0 , with w
ct+1 x = E t [x t+1 ] − x t+1 , for x = c, v, k, n, a. Premultiply the preceding equation by the matrix M 1 , leads to :
s t = Js t+1 + Rη t+1
k
b0 ,
cn 0 , a
b0 predetermined.
Resolution Method (2)
Let Q such that Q −1 JQ = Λ where dim(Q) = 5 × 5, we denote z t :
z t = Q −1 s t and φ t = Q −1 Rη t This leads to the following system :
z t f z t b z t x
=
λ f 0 0 0 λ b 0 0 0 λ x
z t+1 f z t+1 b z t+1 x
+
φ f t+1 φ b t+1 φ x t+1
where λ f is a diagonal matrix formed of all the eigenva- lues less than unity. Then the saddle path is determined by the set of restrictions :
z t,i f = λ f,i z t+1,i f + φ f t+1,i for i = 1, 2 The rational expectation of this equation leads to :
z t,i f = λ f,i E t
·z t+1,i f
¸= ⇒ z t,i f = λ T f,i E t
·z t+T,i f
¸for i = 1, 2 As z t,i 1 < ∞ and lim T →∞ λ T f,i = 0, we get the saddle path :
z t,i f = 0 ⇐⇒
q 1,k −1 k
bt + q 1,n −1
cn t + q 1,c −1 c
bt + q 1,v −1 v
bt + q 1,a −1 a
bt = 0 q 2,k −1 k
bt + q 2,n −1
cn t + q 2,c −1 c
bt + q 2,v −1 v
bt + q 2,a −1 a
bt = 0 Then, using these restrictions, we get after substitution :
k
bt+1
cn t+1
= A
k
bt
cn t
+ Bv t+1
Calibration (1)
– the rate of transition from E → U to s = 0.15, – the average employment ratio to N = 0.57, – the fraction of time spent working to h = 1/3, – the fraction of time spend searching to e = (1/2)h, – the probability that V → N to Φ = 0.9,
– Blanchard and Diamond [1989], ξ = ψ = 0.6, – Aggregate data => K/Y = 10 and I/C = 0.22, – the capital’s share of output to 1 − α = 0.36, – The following set of steady state restrictions
C = Y − I − ωV I = δK 1 = β
1 − δ + (1 − α) Y K
Y = AK 1−α (Nh) α whN
Y = α − ωV Y
1 − β (1 − s) sβ
wh = (1 − ξ)
α Y
N + ω V 1 − N
