Macroeconomic Dynamics and Policy

Macroeconomic Dynamics and Policy

F. Langot

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

November, 2009

Part I

Motivation

Motivation : The Lucas Critique (1976)

”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)

Motivation : The Lucas Critique (1976)

Consider the simple representation of the log–linearized version of a forward-looking model

y

= aE

y

+ bx

where

< 1 (1) where y

and x

respectively denote the endogenous and the forcing variables. x

is assumed to follow a rule of the form

x

= ρx

+ σε

with ε

iid (0, 1) (2) The solution to this simple model is given by

½

y

=

x

x = ρx + σε

Motivation : The Lucas Critique (1976)

This reduced form may be used to estimate the parameters of the

model

y

= αx

x

= ρx

+

σε

Using this model, one can predict the impact of an innovation ε

= 1 in period t + n :

y

= α

ρ

This forecast is true if and only if the policy rule does not change,

ie. if and only if ρ is stable.

Motivation : The Lucas Critique (1976)

At the opposite, assume that the policy rule change. ρ becomes ρ/2. This reduced form of the model becomes

(

y

= αx

x

=

³bρ 2

´

x

+ σε

whereas the true model is

(

y

=

1−a

(

) x

x

=

2

¢

x

+ σε

Motivation : The Lucas Critique (1976)

Proposition

There is a gap between the naive prediction y

= α

µ

ρ

2

¶_n

= b

1

aρ

ρ

2

¶_n

and the “true” prediction

y

= b 1

a

2

¢ µ

ρ

2

¶_n

Motivation : The Lucas Critique (1976)

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)

To estimate or to calibrate the DEEP parameters

b, ρ, σ} using theoretical restrictions.

References : AES special issue (2002)

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)

Part II

A basic framework : the RBC

model

The RBC model : the production function

All firms have access to the same technology :

Y

= A

F (K

, X

N

) = A

K

(X

N

)

where

K

is the capital stock determined at date t

1. The law of motion of this stock is given by :

K

= (1

δ)K

+ I

N

is the total hours determined at date t

A

is the technological shock observed at date t,

X

is the exogenous technical progress which is

The RBC model : the technological shock

If all the markets are walrasian, the technological shock can be measured by the Solow reisual (Solow (1957)) :

Let y

= Y

/X

and k

= K

/X

then y

= A

F (k

, N

) By differentiating this equation, we obtain :

∆A

A

= ∆y

y

∂F

∂k

k

y

∆k

k

∂F

∂N

N

y

∆N

N

If all the markets are walrasian, then

t

kt

yt

=

t

= α

and

∂Ft

∂Nt

Nt

yt

=

= 1

α.

The RBC model : the households

Preferences

u(C

, L

) =

(

θ log(C

) + (1

θ) log(L

) (

)

1−σ

Ct denotes the consumption Lt denotes the leisure

The time allocation is constrained by : T

= L

+ N

The budgetary constraint is given in a complete market economy :

C

+

p

q

dA

q

+ w

N

where

p

q

dA

represents the value of contingent

claims at time t yielding q units of the consumption good

The RBC model : general equilibrium

The general equilibrium is definied by the set of functions

, K

, N

, Y

u⁰_Ct = βEt

·µ

1−δ+αYt+1

Kt+1

¶ u_C⁰_t+1

¸

u⁰_Lt

u_C⁰_t = (1−α)Yt

Nt

Kt+1 = (1−δ)Kt+Yt−Ct

Yt = AtK_t^α(XN,tNt)^1−α

log(At+1) = ρlog(At) + (1−ρ) log(¯A) +²t

This system gives the equilibrium dynamics of C , K , N , Y ,

The RBC model : resolution method

In order to solve this non-linear dynamic system, we take its first-order Taylor series approximation around the steady-state.

We assume that the utility function is u = θ log(C

) + (1

θ) log(L

)

We obtain the following linear equations :

−bct = Et[−bct+1+a1bkt+1+a2bnt+1+a3bat+1+o(x²)]

b

ct+b1bnt = b2bkt+b3bnt+b4bato(x²)

bkt+1 = c1bkt+c2bnt+c3bat+c4bct+o(x²) bat+1 = ρbat+b²t+1+o(x²)

The RBC model : resolution method

Then, this leads to the first order vector system M

s

= M

s

+ M

η

where s

= [b c

,

k

,

a

]

where η

= [b ²

, w

, w

, w

]

with w

= E

[x

]

x

, for x = c, k , a.

By premultiplying the preceding equation by the matrix M

, we obtain :

s

= Js

+ Rη

with

k

,

a

predetermined

The RBC model : resolution method

Let Q, the matrix of the eigenvectors, such that Q

JQ = Λ with dim(Q) = 3

3 and where Λ is the matrix of the eigenvalues, we denote :

z

= Q

s

and φ

= Q

Rη

This leads to the following system :





z

z

z





=





λ

0 0 0 λ

0 0 0 λ









z

z

z





+





φ

φ

φ





where, in the general case (dim(Λ) > 3

3) λ

is a diagonal

matrix formed of all the eigenvalues less than unity.

The RBC model : resolution method

Then the saddle path is determined by the set of restrictions : z

= λ

z

+ φ

The rational expectation of this equation leads to : z

= λ

E

h

z

i

=

z

= λ

E

z

∀i

As z

<

and lim

λ

= 0, we get the saddle path : z

= 0

q

k

+ q

c

+ q

a

= 0 Then, using these restrictions, we get after substitution :

· b

k

= A

· b

k

+ Bv

The RBC model : calibration

The unknown parameters are

Ψ

=

A, α, δ, ρ, σ

dim(Ψ

) = 7 the fraction of time spent working to N = 1/3, Aggregate data : K /Y = 10 and I/C = 0.25,

the capital’s share of output to α = 0.36

= 1

α, The following set of steady state restrictions

C =Y −I I =δK 1 =β

·

1−δ+αY K

¸

Y =AK^αN^1−α θ= (1−α)Y

NC

If we normalize Y = 1, these restrictions give

= 0.8,

I

= 0.2, A = 1.2, δ = 0.021 and β = 0.985.

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

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.

0.5 1 1.5

Heures

% Dev.

0.6 0.8 1

Productivit‚

% Dev.

Tab.:US economy versus K/P/82

Correlation between GDP and Variablex 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

Part III

An extension of the RBC

model : Labor market frictions

RBC and labor market analysis

Puzzles

The magnitude of the employment fluctuations The dynamic of the real wage

The Persistence of the employment dynamics

Models

Employment lotteries : Hansen (1985), Rogerson (1988).

Efficiency wages : Danthine and Donalson (1990), Collard and De La Croix (2000).

Matching and wage bargaining : Merz (1995), Langot (1995 and 1996), Andolfatto (1996)

Endogenous job creation and job destruction : Den Han, Ramey and Watson (2000)

RBC and non-walrasian labor market

N

is the number of jobs that are matched with a worker, U

= 1

N

is the measure of unemployment rate.

Workers search passively : e is constant,

V

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

at which new job matches form is governed by an aggregate-matching technology.

The law of motion for aggregate employment is defined by :

N

= (1

s)N

+ M

with M

= ΥV

(eU

)

RBC and non-walrasian labor market : the firms

All firms have access to the same technology : Y

= A

(h

N

)

K

where Y

, N

h

and K

respectively denote the output, the total hours, the capital stock. The stochastic process of the

technological shock, A

, is :

log A

= ρ

log A

+ (1

ρ

) log A + ²

where

< 1 and ²

(0, σ

).

As firms are owned by households, they discount expected future

values at rate β

.

RBC and non-walrasian labor market : the firms

Let

) be the maximum expected value of the firm in state Ω

. This value function must satisfy the following recursive relationship,

V

(Ω

) = max

{Vt,It,Nt+1,Kt+1}

©

A

K

(N

h

)

w

h

N

I

−ωV_t

+ βE

·

λ

λ

)

¸¾

s.t. N

= (1

s)N

+ Φ

V

(3)

K

= (1

δ)K

+ I

(4)

where Ω

=

, N , A , Φ , w , h

summarizes each firm’s state.

RBC and non-walrasian labor market : the firms

The optimal decisions for the firm give conditions that govern the intertemporal allocation of investment in both physical capital and vacancies. They are given respectively by

λ

= βE

·

λ

µ

α Y

K

+ 1

δ

¶¸

ω

Φ

= βE

·

λ

λ

µ

F

h

w

h

+ (1

s )ω Φ

¶¸

Φ

=

is the exogenous probability for each firm that a vacant

position becomes a filled position.

RBC and non-walrasian labor market : the firms

Given the production function, we have F

(1

α)[Y

/N

h

]

Since the ratio of capital stock to labor input K

/N

h

k

is determined by the rental cost, r

= αA

k

, it is straightforward to deduce that

F

= (1

α)α

A

1 1−α

t

r

α α−1

t

It is assumed that the weight of a worker is ‘small’, F

thus being

taken as given by the firm during the bargaining process.

RBC and non-walrasian labor market : insurance

There is a distinction between jobs and workers flows in the model :

H1)

jobs flows are governed by the matching/separation process,

workers flows are determined exogenously by a game of

“musical chairs”.

⇒

At the beginning of each period, the entire workforce is

“shuffled” randomly across the given set of jobs (before any

trading occurs). There are N

productive jobs at the beginning of

period t, which represents the probability of employment for each

household in any period.

RBC and non-walrasian labor market : insurance

Risk-adverse workers choose to purchase B

units of

unemployment insurance at a price τ

, that delivered B

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

= τ

B

(1

N

) B

Assuming zero profit condition for the unemployment

insurance company implies τ

= 1

N

.

RBC and non-walrasian labor market : insurance

In each period the household chooses a level of insurance, B

, and makes its consumption C

and saving choices q

, contingent upon its employment status (z = n, u ).

Let denoted the vector of control variables

=

B

, C

, q

and Ω

the vector of state variables, the problem faced by each household can be stated recursively by :

W(Ω^Ht) = max

C_t

n Nt

³

Utⁿ+βEt

h

W(Ω^H,n_t+1) i´

+ (1−Nt)

³

Ut^u+βEt

h

W(Ω^H,u_t+1) i´o

RBC and non-walrasian labor market : insurance

This maximization is subject to the constraints referring to each state on the labor market :

Ctⁿ+τtBt+ Z

pt+1qⁿt+1dAt+1 ≤ qt+πt+wtht (5) Ct^u+τtBt+

Z

pt+1q^ut+1dAt+1 ≤ qt+πt+Bt (6)

where

p

q

dA

represents the value of contingent claims

at time t yielding q

units of the consumption good at time

t + 1, in a given state A

. π

represents pure profits due to the

non-walrasian structure of the labor market.

RBC and non-walrasian labor market : insurance

The first order conditions with respect to B

, C

, and q

are respectively :

(1−Nt)λ^ut(1−τt) = Ntλⁿtτt (7)

∂Ut^z

∂C_t^z = λ^zt z = n,u (8) β∂W(Ω^H,z_t+1)

∂q_t+1^z f(At+1|At) = λ^ztpt+1 z = n,u (9)

The envelop theorem implies that :

∂W(Ω

)

∂q

= λ

z = n, u

Because τ

= 1

N

, we have λ

λ

= λ

(Rational

RBC and non-walrasian labor market : insurance

Then, concavity and differentiability of the value function imply :

∂U

∂C

= ∂U

∂C

= λ

and q

= q

= q

This result shows that, given the complete set of insurance markets, the worker’s saving choice does not depend on its state on the labor market. We have

Ω

= Ω

= Ω

RBC and non-walrasian labor market : insurance

Then, the difference between the two budget constraints (equations (5) and (6)) gives the optimal choice of insurance :

B

= w

h

(C

C

)

Consequently, we get the following budget constraint for a representative household :

N

C

+ (1

N

)C

+

p

q

dA

= q

+ π

+ N

w

h Remark that the equilibrium price of asset is given by :

β λ

f (A

) = p

RBC and non-walrasian labor market : Workers

Then, the program solved by the representative household can be summarized by the following dynamic problem :

W(Ω^Ht) = max {^Ctⁿ,C_t^u}

n

NtUtⁿ+ (1−Nt)Ut^u+βEt

h

W(Ω^Ht+1) io

NtCtⁿ + (1−Nt)Ct^u=Ntwth+ Πt

Nt+1 = (1−s)Nt+ Ψt(1−Nt)

where Ψ

= M

/(1

N

), 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

denotes the financial income :

Πt = V(Ω^Ft)−βEt

·λt+1

λt V(Ω^Ft+1)

¸

Z

RBC and non-walrasian labor market : Consumption dynamics

Consequently, the worker’s optimal behavior is completely characterized by

λ

= 1

C

z = n, u (10)

where λ

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 physical capital. This decision gives the intertemporal allocation of consumption :

λ

= βE

[λ

(1

δ + r

)] (11)

The Wage Setting Rule : Constant Hours per worker

The marginal value of employment for the representative household is given by (Ψ

= M

/(1

N

)) :

∂W (Ω

)

∂N

= λ

B

+ U

U

+ (1

s

Ψ

) βE

"

∂W(Ω

)

∂N

#

B

= w

h

(C

C

) refers to the optimal choice of unemployment insurance.

λ

is the budget constraint Lagrange multiplier.

The utility function is given by (Andolfatto [AER,96]) : U

= log (C

) + Γ

with

½

Γ

= Γ(h

)

Γ = Γ(e )

The Wage Setting Rule : Constant Hours per worker

At equilibrium, if the utility function is additively separable in consumption and leisure, we have :

C

= C

and U

U

= Γ

Γ

if Γ

< Γ

employed workers are worse off Then we have :

∂W (Ω

)

∂N

= λ

w

h + (Γ

Γ

) + (1

s

Ψ

) βE

"

∂W(Ω

)

∂N

#

(12)

The Wage Setting Rule : Constant Hours per worker

Symmetrically, for the firm this value is given by :

∂V(Ω

)

∂N

= F

h

w

h + (1

s)βE

"

λ

λ

∂V(Ω

)

∂N

#

(13) From the firm’s decision problem, we have also :

βE

"

λ

λ

∂V (Ω

)

∂N

#

= ω

Φ

(14)

The Wage Setting Rule : Constant Hours per worker

We define the total marginal value of employment measured in units of the consumption good :

St= ∂V(Ω^Ft)

∂Nt

+∂W(Ω^Ht)/∂Nt

λt

(15)

Let 0 < ξ < 1 denotes the firm’s share of S

. This sharing rule implies :

ξ∂W(Ω^Ht )/∂Nt

λt = (1−ξ)∂V(Ω^Ft)

∂Nt (16)

Thus, equations (16) and (14) yield :

ξβEt

·λt+1

λt

∂W(Ω^Ht+1)/∂Nt+1

λt+1

¸

= (1−ξ)Et

·λt+1

λt

∂V(Ω^Ft+1)

∂Nt+1

¸

= (1−ξ)ω

(17)

The Wage Setting Rule : Constant Hours per worker

Finally, combining conditions (12), (13), (16) and (17), the wage equation is given by :

wth= (1−ξ)

·

F2,th+ω Vt

1−Nt

¸ +ξ

·Γu−Γn

λt

¸

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

, with Γ

> Γ

.

If the wealth increases (⇔ ∆λ

< 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

General Equilibrium

The general equilibrium is defined by the set of functions

{C_t

, V

, K

, N

, w

, M

, Y

1

Ct = βEt

·µ

1−δ+αYt+1

Kt+1

¶ 1 Ct+1

¸

ωVt

Mt

= βEt

· Ct

Ct+1

µ

(1−α)Yt+1

Nt+1

−wt+1h+ (1−s)ωVt+1

Mt+1

¶¸

Kt+1 = (1−δ)Kt+Yt−Ct−ωVt

Nt+1 = (1−s)Nt+Mt

wth = (1−ξ)

· αYt

Nt +ω Vt

1−Nt

¸

+ξCt(Γu−Γn)

Mt = ΥV_t^ψ(eUt)^1−ψ Yt = AtKt^α(hNt)^1−α

log(At+1) = ρAlog(At) + (1−ρA) log(¯A) +²A,t

The LMS model : the IRF

0 10 20 30

0.5 1 1.5 2 2.5

OUTPUT (Y)

QUARTERS

%DEV.

LMS1 LMS2

0 10 20 30

0 2 4 6 8

INVESTMENT (I)

QUARTERS

%DEV.

−5 0 5 10 15 20

VACANCIES (V)

%DEV.

−0.5 0 0.5 1 1.5 2 2.5

TOTAL HOURS (Nh)

%DEV.

The LMS model : the IRF

0 10 20 30

0 0.2 0.4 0.6 0.8 1

LABOR PRODUCTIVITY (Y/Nh)

QUARTERS

%DEV.

LMS1 LMS2

0 10 20 30

0 0.2 0.4 0.6 0.8 1 1.2 1.4

WEALTH (1/λ)

QUARTERS

%DEV.

0.1 0.2 0.3 0.4 0.5 0.6 0.7

REAL WAGE (w)

%DEV.

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2

LABOR SHARE (whN/Y)

%DEV.

US RBC LMS

σ(Y) 1.91 1.08 1.80

X (1) (2) (3) (1) (2) (3) (1) (2) (3)

C 0.43 0.82 0 0.26 0.90 0 0.60 0.86 +1

I 2.70 0.97 0 4.59 0.99 0 3.44 0.93 0

H 0.95 0.67 +1 0.26 0.99 0 0.93 0.95 +1 LP 0.54 0.53 -1 0.75 0.99 0 0.31 0.38 -1 W 0.48 0.43 -1 0.75 0.99 0 0.16 0.46 -1

WH/Y 0.52 -0.15 +4 0 0 0 0.16 -0.30 +3

corr(H,LP) 0.17 0.99 0.08

corr(H,W) 0.06 0.99 0.17

The Social Welfare Problem

In a labor market search model, are the fluctuation Pareto optimal, as in the basic RBC model ?

If no, structural policies can have an impact at business cycle frequencies.

The central planner solves the following problem, where

=

, V

, K

, N

:

P

(Ω

) = max

Ct

{log(C_t

) + N

Γ

+ (1

N

)Γ

+ βE

[P(Ω

)]}

K

= (1

δ)K

+ A

(h

N

)

K

C

ωV

N

= (1

s)N

+ ΥV

(1

N

)

The Social Welfare Problem

The FOC and the envelope theorem give :

1 Ct

= βEt

·µ

1−δ+αYt+1

Kt+1

¶ 1 Ct+1

¸

ωVt

Mt = βEt

· Ct

Ct+1

µ

ψ(1−α)Yt+1

Nt+1 −ψ(Γu−Γn)Ct+1

−(1−ψ)ω Vt+1

1−Nt+1 + (1−s)ωVt+1

Mt+1

¶¸

Kt+1 = (1−δ)Kt+AtKt^α(hNt)^1−α−Ct−ωVt

Nt+1 = (1−s)Nt+ ΥVt^ψ(eUt)^1−ψ log(At+1) = ρAlog(At) + (1−ρA) log(¯A) +²A,t

In order to have an Equilibrium Allocation which correspond to the

Pareto Optimal one, the wage must internalize the search

Pareto Optimal Allocation

The Equilibrium path of the labor market is such that

ωVt

Mt = βEt

· Ct

Ct+1

µ

(1−α)Yt+1

Nt+1 −wt+1h+ (1−s)ωVt+1

Mt+1

¶¸

wth = (1−ξ)

· αYt

Nt

+ω Vt

1−Nt

¸

+ξCt(Γu−Γn)

or, after substitution,

ωVt

Mt = βEt

· Ct

Ct+1

µ

ξ(1−α)Yt+1

Nt+1 −ξ(Γu−Γn)Ct+1

−(1−ξ)ω Vt+1

1−Nt+1 + (1−s)ωVt+1

Mt+1

¶¸

Pareto Optimal Allocation

The Optimal path of employment is such that

ωVt

Mt

= βEt

· Ct

Ct+1

µ

ψ(1−α)Yt+1

Nt+1

−ψ(Γu−Γn)Ct+1

−(1−ψ)ω Vt+1

1−Nt+1 + (1−s)ωVt+1

Mt+1

¶¸

whereas the equilibrium allocation solves :

ωVt

Mt

= βEt

· Ct

Ct+1

µ

ξ(1−α)Yt+1

Nt+1

−ξ(Γu−Γn)Ct+1

−(1−ξ)ω Vt+1

1−Nt+1 + (1−s)ωVt+1

Mt+1

¶¸

Thus Equilibrium level of employment, and of vacancies,

correspond to the Social-Optimal level iff ξ = ψ, the

Part IV

Real and nominal shocks : a

DSGE model

Unemployment and price rigidities : the Phillips and Beveridge curves

At each date t a final consumption good is produced by a perfectly competitive firm that combines a continuum of intermediate goods using a C.E.S. technology.

The demand for each intermediate good is given by :

Yj,t^d = µPj,t

Pt

¶₋^1+γ

γ Yt with Pt= µZ ₁

0

P_j,t^−γ1dj

¶−γ

, Yt= µZ ₁

0

Yj,t^d 1+γ1 dj

¶1+γ

where γ

0, and

is the elasticity of substitution

between different goods.

Unemployment and price rigidities : the Phillips and Beveridge curves

The dynamic problem of each intermediate good firm is written recursively as follows :

W(S_j,t^F ) = max

C_j,t

½Pj,t

Pt

³

At(hj,tNj,t)^1−αK_j,t^α −F´

−wj,thj,tNj,t+ (1−δ)Kj,t

−Kj,t+1−φ

2 µ Pj,t

Pj,t−1

−1

¶₂

−ωVj,t+βEt

· Ct

Ct+1

W(S_j,t+1^F )

¸)

s.t. Yj,t ≤ Yj,t^d

Nj,t+1 = (1−s)Nj,t+ ΘtVj,t

where Θ

is the exogenous probability for each firm