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Federal Reserve Bank of Chicago
Establishments Dynamics, Vacancies
and Unemployment: A Neoclassical
Synthesis
Marcelo Veracierto
WP 2009-14
Establishments Dynamics, Vacancies and
Unemployment: A Neoclassical Synthesis∗
Marcelo Veracierto
Federal Reserve Bank of Chicago
November 2009
Abstract: This paper develops a Walrasian equilibrium theory of establishment dynamics and matching frictions and uses it to analyze business cycle fluctuations. Two scenarios are considered: one in which the matching process is subject
to congestion externalities and another in which it is not. The paper finds that
the scenario with congestion externalities replicates U.S. business cycle dynamics
much better than the scenario with efficient matching. Reallocation shocks improve the empirical behavior of the model in terms of microeconomic adjustments
but have little consequences for aggregate dynamics.
∗
This paper originated in a conversation with Randall Wright and was heavily influenced
by it. I have also benefited from the comments of Bjoern Bruegemann, Mario Cozzi, Nicolas
Petrosky-Nadeau, Michael Elsby and numerous seminar participants. The views express here
do not necessarily reflect the position of the Federal Reserve Bank of Chicago or the Federal
Reserve System. Address: Federal Reserve Bank of Chicago, Research Department, 230 South
LaSalle Street, Chicago, IL 60604. E-mail: mveracierto@frbchi.org. Phone: (312) 322-5695.
1
Introduction
The purpose of this paper is to evaluate if standard neoclassical theory can be used to explain the observed behavior of
establishment dynamics, vacancies and unemployment both at growth and business cycle frequencies. To this end, the
paper constructs a real business cycle model that blends three important strands in the literature: 1) the Hopenhayn and
Rogerson (15) model of establishment dynamics, 2) the Mortensen and Pissarides (18) matching model, and 3) the Lucas
and Prescott (16) islands model. A key feature of the model is that it fully relies on classical price theory: All prices,
including that of labor, are determined in Walrasian markets.
The economy is populated by a representative household that values consumption and leisure. Output, which can be
consumed or invested, is produced by a large number of spatially separated establishments that are subject to aggregate and
idiosyncratic productivity shocks. The amount of hiring that an establishment can undertake is constrained by the number
of recruitment opportunities that it has available. Unemployed workers can become employed only if they gain employment
opportunities. Recruitment opportunities for establishments and employment opportunities for workers are jointly produced
by a neoclassical recruitment technology that uses unemployed workers and the consumption good as inputs of production.
Following the matching literature, the recruitment technology is allowed to be subject to production externalities: The total
number of unemployed workers in the economy and the aggregate amount of recruitment expenditures affect its productivity.
Any of the workers with employment opportunities can be hired by any of the establishments with recruitment opportunities.
The paper defines and fully characterizes a recursive competitive equilibrium for this economy. It shows that an
equilibrium can be constructed by solving a social planning problem with side conditions. The social planner solves a
standard utility maximization problem subject to feasibility constraints, except that it takes as given the total number
of unemployed workers and the aggregate amount of recruitment expenditures that enter the recruitment technology as
external effects. At equilibrium, these variables must be generated by the social planner’s optimal decision rules. The
recruitment opportunities and employment decision rules of the establishments are also characterized. In particular, they
are shown to be of the (S,s) variety. This, together with the assumptions that the idiosyncratic productivity shocks take
a finite number of values and that the aggregate productivity shocks are sufficiently small, implies that the distribution
over establishment types has a finite support. As a consequence, the social planner’s problem can be formulated in terms
of a finite number of state and decision variables. This is an important result: Despite the model’s complexity, simple
linear-quadratic methods can be used for computing a recursive competitive equilibrium.
The paper then evaluates how well the model is able to explain the data. Two versions are considered: A version
without external effects in the recruitment technology and a version with external effects. Both versions are calibrated to
identical U.S. long-run observations. Some parameter values are closely related to the neoclassical growth model and are
calibrated to reproduce similar observations (e.g. the capital/output ratio, the investment/output ratio, etc.). The rest of
the parameters are chosen to reproduce observations on establishment dynamics (e.g. the size distribution of establishments,
job creation and destruction rates, etc.), worker flows (e.g. the separation rates, the hazard rate from unemployment, etc.),
and vacancies (e.g. the vacancy rate, recruitment costs, etc.). When an aggregate productivity shock of empirically relevant
magnitude is introduced, the paper finds that the version without external effects in the recruitment technology fails to
reproduce the data: The aggregate fluctuations that it generates are too small. On the contrary, the version with external
1
effects generates aggregate fluctuations of reasonable magnitude. Thus the paper indicates that, when looked through the
eyes of neoclassical theory, there is empirical support for the hypothesis of congestion externalities in the matching process.
While the paper has a strong empirical focus it also makes a theoretical contribution to the literature on equilibrium
unemployment. This literature has been dominated by two main strands: the Mortensen-Pissarides (18) matching model
and the Lucas-Prescott (16) islands model. The Mortensen-Pissarides model is extremely useful for analyzing vacancies and
unemployment and has been extended to incorporate business cycle fluctuations (e.g. Andolfatto (4), Merz (17), Shimer
(20), Hall (13), Hagedorn and Manovskii (9), etc.) and, more recently, establishment dynamics (e.g. Acemoglu and Hawkins
(1), Cooper et. al (6), etc.). However, the model has a significant drawback: It introduces free parameters in the wage
determination process. Even in the simplest version of the model it is unclear what value to use for the Nash bargaining
parameter. In versions with aggregate fluctuations and establishment dynamics, the degrees of freedom multiply since it
is possible for the Nash bargaining parameter to vary systematically with the state of the economy or of an individual
establishment. The Lucas-Prescott model does not suffer from these difficulties since wages are determined in Walrasian
markets.1 However, there is no notion of vacancies in that model: Firms behave as if they could hire any number of workers
at the island specific competitive wage rate. That is, firms do not need to undertake any type of active recruitment effort
in order to fill their job openings. This paper avoids these limitations: By blending together the Mortensen-Pissarides
model and the Lucas-Prescott model, it delivers a framework for analyzing vacancies and unemployment in which all prices
are fully determined by preferences and technology. Incorporating the Hopenhayn-Rogerson model is also important since
establishments dynamics are the counterpart to worker flows and vacancies. The result is a comprehensive theory of labor
market dynamics.
The paper is organized as follows. Section 2 describes the economy. Section 3 describes a recursive competitive
equilibrium. Section 4 characterizes a recursive competitive equilibrium and describes how to compute it. Section 5
calibrates the two versions of the model. Finally, Section 6 presents the results. An appendix provides proofs to the most
important claims made in the paper.
2
The economy
The economy is endowed with a measure one of workers. A worker is a capital good that does not depreciate and can
not be produced. During any period of time a worker can be in either of two states: employed or unemployed. Employed
workers produce the consumption good while unemployed workers produce home goods. Employed workers can be freely
transformed into unemployed workers. However, unemployed workers can only be transformed into employed workers using
a costly technology. All workers are subject to an idiosyncratic productivity shock called a quit shock, that makes them
temporarily unproductive as employed workers. A worker that quits needs to spend a full period of time unemployed before
regaining his productive capacity. The probability that a worker quits at the beginning of the following period depends on
his current employment status: It is equal to π n if the worker is currently employed and it is equal to πu if the worker is
1 The
Lucas-Prescott model has been used, among other things, to study the effects of labor market policies (e.g. Alvarez and Veracierto
(3)), business cycle dynamics (e.g. Veracierto (23)), occupational mobility (e.g. Kambourov and Manovskii (11)), and rest unemployment (e.g.
Alvarez and Shimer (2)).
2
currently unemployed.
The economy is populated by a representative household with preferences given by
(∞
¸)
X ∙ C 1−σ − 1
t
t
E0
β
,
+ ϕUt
1−σ
t=0
(1)
where Ct is consumption, Ut is the total number of unemployed workers, ϕ > 0, σ > 0 and 0 < β < 1.
The consumption good is produced by a large number of establishments. Each establishment has a production function
given by
yt = ezt st F (nt , kt ) ,
where zt is an aggregate productivity shock, st is an idiosyncratic productivity shock, nt is the number of employed workers,
kt is physical capital, and F is a continuously differentiable, strictly increasing, strictly concave and decreasing returns to
scale production function that satisfies the Inada conditions. The idiosyncratic productivity shock st takes values in a
finite set S and follows a Markov process with monotone transition matrix Q. Realizations of st are independent across
establishments and st = 0 is an absorbing state. Since there are no fixed costs of operation, exit takes place only when the
idiosyncratic productivity level becomes zero. In every period of time a measure
of new establishments is exogenously
born. Their distribution over initial productivity shocks is given by ψ. The aggregate productivity shock follows an AR(1)
process given by
zt+1 = ρzt + εt+1 ,
(2)
where 0 ≤ ρ < 1, and εt+1 is i.i.d., normally distributed, with variance σ 2ε and zero mean.
The number of employed workers nt at an establishment is given by
nt = nt−1 + ht − ft ,
where ht are the gross employment increases (i.e. hirings) and ft are the gross employment reductions (i.e. firings). All the
workers that are fired become unemployed. Because of the exogenous quit of employed workers, ft is effectively constrained
as follows
π n nt−1 ≤ ft ≤ nt−1.
The number of new hires ht is limited by the number of recruiting opportunities jt that the establishment has at the
beginning of the period, i.e.
ht ≤ jt .
(3)
Unemployed workers can become employed only if they are transformed into workers with employment opportunities.
Workers with employment opportunities et+1 and recruiting opportunities jt+1 are jointly produced using the following
recruitment technology:
et+1
= G(at , ut , At , Ut ),
(4)
jt+1
= H(at , ut , At , Ut ),
(5)
3
where at are recruitment expenditures (in the consumption good), ut are unemployed workers, At is the aggregate amount
of recruitment expenditures in the economy, and Ut is the total number of unemployed workers in the economy.2
The recruitment technology satisfies the following assumptions: 1) G and H are continuously differentiable, 2) G and
H are increasing in (at , ut ), 3) G and H are homogenous of degree one with respect to (at , ut ) and homogeneous of degree
zero with respect to (At , Ut ), 4) G and H are concave in (at , ut ), 5) and G satisfies that
G (at , ut , At , Ut ) ≤ ut , for every (at , ut , At , Ut ) .
(6)
Observe from equation (6) that not all unemployed workers that enter the recruitment technology are transformed into
workers with employment opportunities:
xt+1 = ut − G (at , ut , At , Ut ) ,
is the number of unsuccessful candidates that the recruitment technology generates.
3
Recursive competitive equilibrium
The state of the economy is given by the quintuple (z, K, E, X, μ), where z is the aggregate productivity level, K is the
aggregate stock of capital, E is the aggregate number of workers with employment opportunities, X is the aggregate number
of unsuccessful candidates, μ (s, l × j) is a measure of establishments over individual states (s, l, j), and (E, X, μ) satisfies
that3
Z
l μ (s, dl × dj) + E + X = 1.
(7)
There are three competitive sectors in the economy: a households sector, an establishments sector, and a recruitment
industry.
Households earn income from renting capital to the establishments and from the aggregate profits made by the establishments sector.4 They spend their income on consumption, on investment and on renting unemployed workers. The
individual state of a household is the amount of capital that it owns κ. The household’s problem is described by the
following Bellman equation:
B(κ, z, K, E, X, μ) = max
{c,i,m}
½
¾
c1−σ − 1
+ ϕm + βE [B (κ0 , z 0 , K 0 , E 0 , X 0 , μ0 ) | z]
1−σ
(8)
subject to:
c + i + ru (z, K, E, X, μ) m ≤ rk (z, K, E, X, μ) κ + Π (z, K, E, X, μ) ,
κ0
= (1 − δ) κ + i
(K 0 , E 0 , X 0 , μ0 ) = L (z, K, E, X, μ) .
2 Observe
(10)
(11)
that when GA , GU , HA or HU are strictly positive, the recruitment technology is subject to production externalities.
3 Equation
(7) implies that either E or X could be removed from the aggregate state vector. However, this would complicate the definition of
a recursive competitive equilibrium.
4 Each
(9)
household is assumed to own one share of each establishment in the economy.
4
where ru is the rental rate of an unemployed worker, rk is the rental rate of capital, Π are the aggregate profits made
by the establishments sector, i is investment, m is the number of unemployed workers that the household rents, and L is
the law of motion for the endogenous state of the economy. Equation (9) is the budget constraint of the household, and
equation (10) is the law of motion for its stock of capital. The household’s optimal decisions are c = c (κ, z, K, E, X, μ),
i = i (κ, z, K, E, X, μ), and m = m (κ, z, K, E, X, μ) for consumption, investment and unemployed workers, respectively.
The establishments rent capital, purchase workers with employment opportunities (up to the number of recruitment
opportunities that they have at the beginning of the period), sell unemployed workers (up to their previous-period employment level), and purchase next-period recruitment opportunities. The individual state of an establishment is given by
a triple (s, l, j), where s is its current idiosyncratic productivity level, l is its previous-period employment level and j is
its recruitment opportunities at the beginning of the period. The establishment’s problem is described by the following
Bellman equation:
W (s, l, j, z, K, E, X, μ) =
max
{f,h,k,n,v}
{ez sF (n, k) + pu (z, K, E, X, μ) f − pe (z, K, E, X, μ) h
(12)
−rk (z, K, E, X, μ) k − pv (z, K, E, X, μ) v
"
#
X
0
0 0 0 0
0
0
0
0
0
+E
q (z, K, E, X, μ, z ) W (s , l , j , z , K , E , X , μ ) Q (s, s ) | z }
s0
subject to
n = l+h−f
(13)
πn l
≤ f
(14)
f
≤ l
(15)
h ≤ j
(16)
l0
= n
(17)
j0
= v
(18)
(K 0 , E 0 , X 0 , μ0 ) = L (z, K, E, X, μ) .
(19)
where pu is the price of an unemployed worker, pe is the price of a worker with employment opportunities, pv is the price of
a next-period recruitment opportunity, q (·, z 0 ) is the price of an Arrow security that delivers one unit of the consumption
good if the next-period aggregate productivity level is equal to z 0 , n is the number of employed workers, k is the capital
level, f are the firings, h are the hirings, and v are the purchases of next-period recruitment opportunities. The constraints
(13)-(16) have been described in the previous section. The establishment’s optimal decisions are n = n(s, l, j, z, K, E, X, μ),
k = k(s, l, j, z, K, E, X, μ), f = f (s, l, j, z, K, E, X, μ), h = h(s, l, j, z, K, E, X, μ), and v = v (s, l, j, z, K, E, X, μ), for
employed workers, capital, firings, hirings and next-period recruitment opportunities, respectively.
The recruitment companies sell workers with employment opportunities and next-period recruitment opportunities.
They also buy and sell unemployed workers and rent them to the households sector. The individual state of a recruitment
company is a pair (e, x), where e is its number of workers with employment opportunities at the beginning of the period,
5
and x is its number of unsuccessful candidates. The problem of a recruitment company is given as follows:
R (e, x, z, K, E, X, μ) = max
{a,b,d,u}
{pe (z, K, E, X, μ) d + pv (z, K, E, X, μ) b
(20)
+pu (z, K, E, X, μ) [x + e − d − u] + ru (z, K, E, X, μ) u − a
+E [q (z, K, E, X, μ, z 0 ) R (e0 , x0 , z 0 , K 0 , E 0 , X 0 , μ0 ) | z]}
subject to
d ≤ (1 − π u ) e
b = H(a, u, A, U )
e0
= G(a, u, A, U )
x0
= u − G(a, u, A, U )
A = A (z, K, E, X, μ)
U
= U (z, K, E, X, μ)
(K 0 , E 0 , X 0 , μ0 ) = L (z, K, E, X, μ) .
where d is the number of workers with employment opportunities that the recruitment company sells, b is the number of
next-period recruitment opportunities that the recruitment company sells, u is the number of unemployed workers that the
recruitment company owns, a are the expenditures that the recruitment company makes, A are the aggregate recruitment
expenditures in the economy, and U is the aggregate number of unemployed workers. Observe that, since unemployed
workers quit at the rate π u , d cannot exceed (1 − π u ) e. Also observe that the recruitment company can sell as unemployed
workers all of its unsuccessful candidates x and any of its unsold workers with employment opportunities e − d. The
recruitment company’s optimal decisions are a = a(e, x, z, K, E, X, μ), b = b(e, x, z, K, E, X, μ), d = d(e, x, z, K, E, X, μ),
and u = u(e, x, z, K, E, X, μ), for recruitment expenditures, next-period recruitment opportunities, sales of workers with
employment opportunities, and unemployed workers, respectively.5
A recursive competitive equilibrium can now be defined.
Definition 1 A recursive competitive equilibrium (RCE) is a set of value functions B(κ, z, K, E, X, μ), W (s, l, j, z, K, E,
X, μ), R(e, x, z, K, E, X, μ), a set of individual decision rules c(κ, z, K, E, X, μ), i (κ, z, K, E, X, μ), m (κ, z, K, E, X, μ),
n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ), f (s, l, j, z, K, E, X, μ), h(s, l, j, z, K, E, X, μ), v (s, l, j, z, K, E, X, μ), a(e, x, z,
K, E, X, μ), b(e, x, z, K, E, X, μ), d(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ), a pair of aggregate decision rules A(z, K, E, X,
μ), U (z, K, E, X, μ), an aggregate law of motion L (z, K, E, X, μ), an aggregate profits function Π(z, K, E, X, μ), and a set of
price functions rk (z, K, E, X, μ), ru (z, K, E, X, μ), pu (z, K, E, X, μ), pe (z, K, E, X, μ), pv (z, K, E, X, μ), q(z, K, E, X, μ,
z 0 ), such that:
(i) the value function B (κ, z, K, E, X, μ) solves the households’ Bellman equation and c(κ, z, K, E, X, μ), i(κ, z, K, E, X,
μ), and m (κ, z, K, E, X, μ) are the associated decision rules,
5 Sections
1.1-1.3 in the Technical Appendix provide first-order and envelope conditions for the household, establishment and recruitment
company’s decision problems, respectively.
6
(ii) the value function W (s, l, j, z, K, E, X, μ) solves the establishments’ Bellman equation and n(s, l, j, z, K, E, X, μ),
k(s, l, j, z, K, E, X, μ), f (s, l, j, z, K, E, X, μ), h(s, l, j, z, K, E, X, μ), and v(s, l, j, z, K, E, X, μ) are the associated decision
rules,
(iii) the value function R(e, x, z, K, E, X, μ) solves the Bellman equation of the recruitment companies and a(e, x, z, K, E,
X, μ), b(e, x, z, K, E, X, μ), d(e, x, z, K, E, X, μ), and u(e, x, z, K, E, X, μ) are the associated decision rules,
(iv) the prices of the Arrow securities satisfy that
σ
q(z, K, E, X, μ, z 0 ) = β
c (K, z, K, E, X, μ)
,
c (K 0 , z 0 , K 0 , E 0 , X 0 , μ0 )σ
where (K 0 , E 0 , X 0 , μ0 ) = L (z, K, E, X, μ),
(v) the capital rental market clears, i.e.
K=
XZ
s
k(s, l, j, z, K, E, X, μ)μ (s, dl × dj)
(vi) the rental market for unemployed workers clears, i.e.
u(E, X, z, K, E, X, μ) = m (K, z, K, E, X, μ)
(vii) the ownership market for unemployed workers clears, i.e.
u(E, X, z, K, E, X, μ) = X + E − d(E, X, z, K, E, X, μ)
XZ
+
f (s, l, j, z, K, E, X, μ)μ (s, dl × dj)
s
(viii) the market for workers with employment opportunities clears, i.e.
XZ
h(s, l, j, z, K, E, X, μ)μ (s, dl × dj)
d(E, X, z, K, E, X, μ) =
s
(ix) the market for next-period recruitment opportunities clears, i.e.
XZ
b(E, X, z, K, E, X, μ) =
v(s, l, j, z, K, E, X, μ)μ (s, dl × dj)
s
(x) the market for the consumption good clears, i.e.
c (K, z, K, E, X, μ) + i (K, z, K, E, X, μ) + a(E, X, z, K, E, X, μ)
XZ
=
ez sF [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)] μ (s, dl × dj)
s
(xi) the aggregate decision rules are generated by the optimal individual decisions, i.e.
A (z, K, E, X, μ) = a(E, X, z, K, E, X, μ)
U (z, K, E, X, μ) = u(E, X, z, K, E, X, μ)
(xii) the aggregate law of motion is generated by the optimal individual decisions, i.e.
(K 0 , E 0 , X 0 , μ0 ) = L (z, K, E, X, μ)
7
is given as follows:
K 0 = (1 − δ) K + i (K, z, K, E, X, μ)
E 0 = G [a(E, X, z, K, E, X, μ), u(E, X, z, K, E, X, μ), A (z, K, E, X, μ) , U (z, K, E, X, μ)]
X 0 = u(E, X, z, K, E, X, μ) − E 0
XZ
μ0 (s0 , L × J ) =
Q (s, s0 ) μ (s, dl × dj) + ψ (s0 ) I (L × J )
s
where
B(s,L×J )
B (s, L × J ) = {(l, j) : n(s, l, j, z, K, E, X, μ) ∈ L and v(s, l, j, z, K, E, X, μ) ∈ J }
and where I (L × J ) is an indicator function which takes a value of one if (0, 0) ∈ L × J , and a value of zero otherwise.6
4
Characterization and computation of a RCE
Due to the external effects in the recruitment technology a RCE is generally inefficient and must be solved for directly.
The high dimensionality of the sate space makes this a daunting task. However, it can be simplified considerably. This
section provides a solution method that can be easily implemented in actual computations. The method relies on two
key properties of a RCE. First, that it can be characterized as the solution to a dynamic programming problem with side
conditions.7 Second, that in a neighborhood of the deterministic steady state, the dynamic programming problem can be
represented as having a finite number of state and decision variables. The following subsections explain these properties in
detail.
4.1
The myopic social planner’s problem
Consider the problem of a social planner that seeks to maximize utility subject to the economy’s feasibility constraints.
However, the social planner is myopic in the sense that he does not fully internalize the effects of his decisions on the output
produced by the recruitment technology. In particular, the myopic social planner takes the recruitment technology as being
the following:
³
´
E 0 = G A, U, Â, Û
³
´
J 0 = H A, U, Â, Û
where E 0 are next-period workers with employment opportunities, J 0 are next-period recruitment opportunities, A are
recruitment expenditures, U are unemployed workers, and  and Û are exogenous productivity shocks. The shocks  and
Û evolve according to the following stochastic process:
³
´
 =  z, K̂, Ê, X̂, μ̂
6 It
is straightforward to verify that if (E, X, μ) satisfy equation (7), then (E 0 , X 0 , μ0 ) also satisfy equation (7).
7 The
dynamic optimization problem depends on exogenous parameters, wich in turn depend on the solution to the dynamic optimization
problem. Finding a RCE is then reduced to solving a fixed point problem on those parameters. This basic strategy for solving for a competitive
equilibrium in an economy with externalities is already familiar to the literature, though in much simpler contexts (e.g. Kehoe, Levine and
Romer (12), Jones and Manuelli (10), etc.).
8
³
´
Û = Û z, K̂, Ê, X̂, μ̂
³
´
³
´
K̂ 0 , Ê 0 , X̂ 0 , μ̂0 = L̂ z, K̂, Ê, X̂, μ̂
³
´
where z is the aggregate productivity level, and K̂, Ê, X̂, μ̂ are variables that lie in the same space as (K, E, X, μ).
The state of the myopic social planner is then given by the state of the economy (z, K, E, X, μ) and by the variables
³
´
K̂, Ê, X̂, μ̂ , which are sufficient statistics for predicting the future behavior of  and Û . The problem of the myopic
³
´
social planner facing a stochastic process Â, Û , L̂ is described by the following Bellman equation:
V (z, K, E, X, μ, K̂, Ê, X̂, μ̂) = max
subject to
C +I +A≤
XZ
s
U ≤X +E−
μ0 (s0 , L × J ) =
XZ
s
XZ
s
ez sF [n(s, l, j), k(s, l, j)] μ (s, dl × dj)
(21)
(22)
k(s, l, j)μ (s, dl × dj) ≤ K
(23)
³
´
v(s, l, j)μ (s, dl × dj) ≤ H A, U, Â, Û
(24)
s
XZ
s
´ i¾
h ³
C 1−σ − 1
0
0
0
0
0
0
0
0
0
+ ϕU + βE V z , K , E , X , μ , K̂ , Ê , X̂ , μ̂ | z
1−σ
XZ
XZ
s
½
h(s, l, j)μ (s, dl × dj) +
XZ
s
f (s, l, j)μ (s, dl × dj)
h(s, l, j)μ (s, dl × dj) ≤ (1 − π u ) E
(25)
(26)
n (s, l, j) = l + h (s, l, j) − f (s, l, j)
(27)
h (s, l, j) ≤ j
(28)
π n l ≤ f (s, l, j)
(29)
f (s, l, j) ≤ l
(30)
K 0 = (1 − δ) K + I
³
´
E 0 = G A, U, Â, Û
³
´
X 0 = U − G A, U, Â, Û
(31)
{(l,j): n(s,l,j)∈L and v(s,l,j)∈J }
Q (s, s0 ) μ (s, dl × dj) + ψ (s0 ) I (L × J )
³
´
 =  z, K̂, Ê, X̂, μ̂
³
´
Û = Û z, K̂, Ê, X̂, μ̂
³
´
³
´
K̂ 0 , Ê 0 , X̂ 0 , μ̂0 = L̂ z, K̂, Ê, X̂, μ̂ .
9
(32)
(33)
(34)
(35)
(36)
(37)
where equations (22)-(34) are feasibility constraints and equations (35)-(37) describe the stochastic process that  and Û fol³
´
low over time.8 The myopic social planner’s decision rules are C = C m z, K, E, X, μ, K̂, Ê, X̂, μ̂ , I = I m (z, K, E, X, μ, K̂,
³
´
Ê, X̂, μ̂), n = nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), k = km s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , f = f m (s, l, j, z, K, E, X, μ, K̂,
³
´
³
´
Ê, X̂, μ̂), h = hm s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , v = vm s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , U = U m (z, K, E, X, μ, K̂, Ê, X̂,
³
´
μ̂), A = Am z, K, E, X, μ, K̂, Ê, X̂, μ̂ , for consumption, investment, establishment employment, establishment capital,
establishment firings, establishment hirings, establishment recruitment opportunities, unemployment and recruitment expenditures, respectively.
The following proposition provides a characterization of the decision rules to the myopic social planner’s problem.
Proposition 2 Let {C m , I m , nm , km , f m , hm , v m , U m , Am } be the solution to the MSP’s with exogenous stochastic process
³
´
Â, Û , L̂ . Then, there exist thresholds nm (s, z, K, E, X, μ, K̂, Ê, X̂, μ̂), n̄m (s, z, K, E, X, μ, K̂, Ê, X̂, μ̂) and v̄ m (s, z, K, E, X,
³
´
μ, K̂, Ê, X̂, μ̂) and a shadow capital price function rk z, K, E, X, μ, K̂, Ê, X̂, μ̂ such that, for every s > 0 and l + j > 0:
⎧
o ⎫
n
⎨ min (1 − πn ) l + j, nm (s, z, K, E, X, μ, K̂, Ê, X̂, μ̂) , ⎬
o
n
nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) = max
,
⎭
⎩
min (1 − π n ) l, n̄m (s, z, K, E, X, μ, K̂, Ê, X̂, μ̂)
n
o
v m (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) = max v̄m (s, z, K, E, X, μ, K̂, Ê, X̂, μ̂) − (1 − πn )nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), 0 ,
n
o
hm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) = max nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) − l, 0
n
o
f m (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) = max l − nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), 0
i
h
³
´
ez sFk nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), km (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) = rk z, K, E, X, μ, K̂, Ê, X̂, μ̂ ,
³
´
Proof. In the economy in which Â, Û , L̂ truly represent exogenous productivity shocks to the recruitment technology, the Welfare Theorems apply. In this case the problem described by equation (21) is the social planner’s problem
and its solution can be decentralized as a recursive competitive equilibrium in which prices are functions of the state
(z, K, E, X, μ, K̂, Ê, X̂, μ̂). The claim then follows from characterizing the optimal decision rules to the associated establishments’ problem given by equation (12).9
This proposition is important because it can be used to reduce the dimensionality of the decision variables in the myopic
social planner’s problem: Instead of choosing functions nm , vm , hm , f m and km defined over the infinite number of triples
(s, l, j), the myopic social planner can be restricted to choose thresholds nm , n̄m and v̄m defined over the finite number of
singletons s.
The next proposition states that if the solution to a myopic planner’s problem satisfies certain side conditions, then it
is a RCE.
8 Observe
that if equations (35)-(37) were substituted by  = A and Û = U , the solution to this planning problem would be the Pareto optimal
allocation.
9 For
details see Sections 2 in the Technical Appendix.
10
Proposition 3 Let {C m , I m , nm , km , f m , hm , v m , U m , Am } be the solution to the myopic social planner’s problem with
³
´
exogenous stochastic process Â, Û , L̂ .
Suppose that,
 (z, K, E, X, μ) = Am (z, K, E, X, μ, K, E, X, μ) ,
Û (z, K, E, X, μ) = U m (z, K, E, X, μ, K, E, X, μ) .
In addition, suppose that
³
´
K̂ 0 , Ê 0 , X̂ 0 , μ̂0 = L̂ (z, K, E, X, μ) .
satisfies that
K̂ 0 = (1 − δ) K + I m (z, K, E, X, μ, K, E, X, μ) ,
Ê 0 = G
[Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ) ,
Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ)] ,
X̂ 0 = U m (z, K, E, X, μ, K, E, X, μ) − Ê 0 ,
XZ
μ̂0 (s0 , L × J ) =
Q (s, s0 ) μ (s, dl × dj) + ψ (s0 ) I (L × J ) ,
s
where
B(s,L×J )
B (s, L × J ) = {(l, j) : nm (s, l, j, z, K, E, X, μ, K, E, X, μ) ∈ L and v m (s, l, j, z, K, E, X, μ, K, E, X, μ) ∈ J } .
Then, there exists a RCE {B, W , R, c, i, m, n, k, f , h, v, a, b, d, u, A, U , L, Π, rk , ru , pu , pe , pv , q} such that
c (K, z, K, E, X, μ) = C m (z, K, E, X, μ, K, E, X, μ)
i (K, z, K, E, X, μ) = I m (z, K, E, X, μ, K, E, X, μ)
n(s, l, j, z, K, E, X, μ) = nm (s, l, j, z, K, E, X, μ, K, E, X, μ)
f (s, l, j, z, K, E, X, μ) = f m (s, l, j, z, K, E, X, μ, K, E, X, μ)
h(s, l, j, z, K, E, X, μ) = hm (s, l, j, z, K, E, X, μ, K, E, X, μ)
v(s, l, j, z, K, E, X, μ) = vm (s, l, j, z, K, E, X, μ, K, E, X, μ)
A (z, K, E, X, μ) = Am (z, K, E, X, μ, K, E, X, μ)
U (z, K, E, X, μ) = U m (z, K, E, X, μ, K, E, X, μ) .
Proof. It follows from comparing the necessary and sufficient conditions for a RCE with the necessary and sufficient
conditions to the myopic social planner’s problem.10
1 0 For
the details, see Section 3.4 in the Technical Appendix.
11
4.2
State space characterization
When there are no aggregate productivity shocks to the economy (i.e. when z is identical to zero), a deterministic steady
state can be defined. In particular, a myopic steady state is given by an aggregate state (K ∗ , E ∗ , X ∗ , μ∗ , K̂ ∗ , Ê ∗ , X̂ ∗ , μ̂∗ )
that replicates itself under the myopic planner’s optimal decision rules.11 Characterizing the invariant distribution μ∗ of a
myopic steady state will turn to be crucial for characterizing the state-space when the economy is subject to small aggregate
productivity shocks.
From equation (34) and Proposition 2, observe that the invariant distribution μ∗ must satisfy the following equation:
∗
0
μ (s , L × J ) =
where
XZ
s
{(l,j):n∗ (s,l,j)∈L and v∗ (s,l,j)∈J }
Q (s, s0 ) μ∗ (s, dl × dj) + ψ (s0 ) I (L × J ) ,
⎧
⎫
⎨ min {(1 − π ) l + j, n∗ (s)} , ⎬
n
n∗ (s, l, j) = max
,
⎩
⎭
min {(1 − π ) l, n̄∗ (s)}
(38)
n
v∗ (s, l, j) = max {v̄ ∗ (s) − (1 − π n )n∗ (s, l, j), 0} .
(39)
The following proposition characterizes a support to the invariant distribution μ∗ in terms of the finite number of steady
state thresholds n∗ , n̄∗ and v̄∗ .12
Proposition 4 Let M be a natural number satisfying that
(1 − πn )M max {n̄∗ (smax ) , v̄ ∗ (smax )} < min {n∗ (smin ) , v̄ ∗ (smin )} .
Define the set N ∗ as follows:
½
∗
N = ∪
s∈S
(40)
o¾
n
k ∗
k ∗
k ∗
∪ (1 − π n ) n (s) , (1 − πn ) n̄ (s) , (1 − π n ) v̄ (s)
∪ {0} .
M−1
k=0
Then, the set
½
¾ ½
¾
∗
∗
0
P = (s, l, j) : s ∈ S, l ∈ N , and j ∈ 0∪ {max [v̄ (s ) − (1 − π n ) l, 0]} ∪ ∪ {(s, 0, 0)}
∗
s ∈S
s∈S
is a support of the invariant distribution μ∗ .
Proof. See Appendix A.
Observe that Proposition 4 not only constructs a support P ∗ for the invariant distribution μ∗ , but determines that it is
a finite set.
In order to analyze off-steady state dynamics it will be useful to define nt , n̄t , and v̄t , as the threshold functions chosen
at date t. In addition, it will be useful to define the following minimum distance:
ε = min |a − b|
1 1 From
Proposition 3 we know that if (K ∗ , E ∗ , X ∗ , μ∗ ) = (K̂ ∗ , Ê ∗ , X̂ ∗ , μ̂∗ ), this myopic steady-state constitutes a steady-state equilibrium.
See Sections 4.1 and 4.3 in the Technical Appendix for explicit steady state equillibrium conditions and a computational algorithm.
1 2 In
(41)
the statement of the proposition smax and smin denote the largest and smallest positive values for s, respectively.
12
subject to
a, b ∈ D∗ and a 6= b,
where
D∗ = N ∗ ∪
½
∪
s∈S
o¾
n
.
(1 − πn )M n∗ (s) , (1 − πn )M n̄∗ (s) , (1 − π n )M v̄∗ (s)
The following proposition characterizes the distribution μt+1 under the assumptions that μt and the finite history of
©
ªM +1
thresholds nt−k , n̄t−k , v̄t−k k=0 are sufficiently close to their steady-state counterparts.
Proposition 5 Let M be defined by equation (40) and ε by equation (41).
Suppose that
¯
¯
¯nt−k (s) − n∗ (s)¯ < ε/2,
(42)
|n̄t−k (s) − n̄∗ (s)| < ε/2,
(43)
|v̄t−k (s) − v̄ ∗ (s)| < ε/2,
(44)
for every s and every 0 ≤ k ≤ M + 1.
Suppose that the distribution μt has a finite support Pt given by
½
¾ ½
¾
0
Pt = (s, l, j) : s ∈ S, l ∈ Nt , and j ∈ 0∪ {max [v̄t−1 (s ) − (1 − π n ) l, 0]} ∪ ∪ {(s, 0, 0)}
s ∈S
where
Nt =
½
M−1
∪
∪
s∈S
k=0
s∈S
o¾
n
∪ {0} .
(1 − π n )k nt−k−1 (s) , (1 − π n )k n̄t−k−1 (s) , (1 − πn )k v̄t−k−2 (s)
(45)
(46)
In addition, suppose that for every (s, l, j) ∈ Pt :
μt (s, l, j) = μ∗ (s, l∗ , j ∗ ) ,
(47)
where (s, l∗ , j ∗ ) is the unique element of P ∗ satisfying that |l − l∗ | < ε/2 and |j − j ∗ | < ε/2 + (1 − π) ε/2.
Then, the distribution μt+1 has a finite support Pt+1 given by
½
¾ ½
¾
Pt+1 = (s, l, j) : s ∈ S, l ∈ Nt+1 , and j ∈ 0∪ {max [v̄t (s0 ) − (1 − π n ) l, 0]} ∪ ∪ {(s, 0, 0)}
s ∈S
where
Nt+1 =
½
∪
s∈S
s∈S
o¾
n
k
k
k
∪ (1 − π n ) nt−k (s) , (1 − π n ) n̄t−k (s) , (1 − πn ) v̄t−k−1 (s)
∪ {0} .
M−1
k=0
Moreover, for every (s, l, j) ∈ Pt+1 :
μt+1 (s, l, j) = μ∗ (s, l∗ , j ∗ )
where (s, l∗ , j ∗ ) is the unique element of P ∗ satisfying that |l − l∗ | < ε/2 and |j − j ∗ | < ε/2 + (1 − π) ε/2.
Proof. See Appendix A.
Proposition 5 plays a crucial role in the solution method to be described below. It implies that if the economy starts
at the deterministic steady-state at t = 0 and the thresholds nt , n̄t and v̄t thereafter fluctuate within a sufficiently small
neighborhood of their steady state values, then the distribution μt will always have a finite support Pt determined by the
13
©
ªM +1
finite history of thresholds nt−k , n̄t−k , v̄t−k k=1 (equations 45 and 46) and its mass at each point in Pt will be given
by the mass of the invariant distribution μ∗ at the corresponding point in P ∗ (equation 47). As a result the state to the
©
ªM+1
myopic planner problem can be defined in terms of the finite history of thresholds nt−k , n̄t−k , v̄t−k k=1 instead of the
distribution μt .
4.3
Solution method
This section redefines the myopic social planner’s problem so that standard solution methods can be applied. For this
purpose, it will be convenient to return to a recursive formulation and define nk , n̄k and v̄k as the thresholds that were
chosen k periods ago (relative to the current period).
Recall from Section 4.2 that the finite history {nk , n̄k , v̄k }M+1
k=1 can be used to construct the current distribution μ
(as long as fluctuations are sufficiently small). Moreover, Proposition 2 states that the current thresholds (n0 , n̄0 , v̄0 )
fully describe the employment rule n, the vacancies rule v, the hiring rule h and the firing rule f . In turn, the employment decision rule n and the aggregate stock of capital K are sufficient for determining the capital allocation rule k.13
This suggests that the state vector (z, K, E, X, μ, K̂, Ê, X̂, μ̂) in the myopic planner’s problem can be replaced by the
©
ªM+1
M+1
b̄k , b̄
vector (z, K, E, {nk , n̄k , v̄k }k=1 , K̂, Ê, n
bk , n
vk k=1 ) and that the decision variables (k, n, v, h, f ) can be replaced by
(n0 , n̄0 , v̄0 ).14
´
³
Also, observe from equation (32) that A can be written as A = g1 E 0 , U, Â, Û for some differentiable function g1
´
³
b and U = U
b , that  can be written as  = g2 E
b for some differentiable function
b0 , U
and, since at equilibrium A = A
R
R
b = 1− n
g2 . Moreover, at equilibrium we have that U = 1 − n dμ and that U
b db
μ. Substituting these expressions and
equations (22)-(33) into the return function in equation (21), the myopic planner’s problem can then be written as follows:15
³
©
ªM +1 ´
M+1
b̄k , b̄
V z, K, E, {nk , n̄k , v̄k }k=1 , K̂, Ê, n
bk , n
v k k=1
(48)
n ³
´
©
ªM+1
M+1
b0 , n
b̄k , b̄
b̄0 , b̄
= max R z, K, E, {nk , n̄k , v̄k }k=1 , K̂, Ê, n
bk , n
vk k=1 , K 0 , E 0 , n0 , n̄0 , v̄0 , E
b0 , n
v0
∙ µ
¶ ¸¾
n
o
0 M+1
M+1
b̄0k , b̄
+βE V z 0 , K 0 , E 0 , {n0k , n̄0k , v̄k0 }k=1 , K̂ 0 , Ê 0 , n
b0k , n
vk
|z
k=1
subject to
1 3 Since
n0k
= nk−1 , for k = 1, ..., M + 1
(49)
n̄0k
= n̄k−1 , for k = 1, ..., M + 1
(50)
v̄k0
= v̄k−1 , for k = 1, ..., M + 1
(51)
capital is freely movable, the myopic social planner allocates the aggregate stock of capital K to equate the marginal producitivity of
capital across all types of islands (s, l.j), subject to the feasibility constraint (23).
1 4 The
e can be removed from the state vector because they are actually redundant (see equation 7).
variables X and X
1 5 Observe
that equation (24) must be used to remove some vacancy threshold v̄ (e.g. v̄ (smin )) from the formulation of the problem, since
it always hold with equality. Similarly, when the deterministic steady state is such that equation (26) holds with equality, it must be used to
remove some lower employment thresshold n (e.g. n (smin )) from the formulation of the problem. Otherwise, equation (26) must be ignored.
14
µ
n
³
oM+1 ¶
©
ªM+1 ´
0 b̄0 b̄0
0
0
b̄k , b̄
bk , nk , vk
vk k=1 ,
K̂ , Ê , n
= L̂ z, K̂, Ê, n
bk , n
k=1
(52)
³
´
M+1
where the vector of decision variables is K 0 , E 0 , {n0k , n̄0k , v̄k0 }k=1 .
Let the optimal decision rule to the above problem be given by
³
M+1
K 0 , E 0 , {n0k , n̄0k , v̄k0 }k=1
´
³
©
ªM+1 ´
M+1
b̄k , b̄
= D z, K, E, {nk , n̄k , v̄k }k=1 , K̂, Ê, n
bk , n
vk k=1 .
The condition for a RCE in Proposition 3 then becomes:
³
´
³
´
M+1
M+1
M+1
L̂ z, K, E, {nk , n̄k , v̄k }k=1 = D z, K, E, {nk , n̄k , v̄k }k=1 , K, E, {nk , n̄k , v̄k }k=1 .
(53)
Observe that there are a finite number of arguments to the return function in equation (48) and that all their values are
strictly positive at the deterministic steady state (except for the aggregate productivity level z). Since R is differentiable,
a Taylor expansion at the deterministic steady state can then be performed to obtain a quadratic objective function. Since
the constraints in equations (49)-(51) are linear, this delivers a standard linear-quadratic RCE structure that can be solved
using standard methods (e.g. Hansen and Prescott (14)).16 The linear decision rule D thus obtained is a good local
approximation and, as long as fluctuations in the aggregate productivity shock z are small, it can be used to simulate and
analyze equilibrium business cycle fluctuations.
5
Calibration
Throughout the rest of the paper the recruitment technology will be given a matching function interpretation in which
employment and recruitment opportunities are produced in pairs at the aggregate level. In particular, the recruitment
technology will be restricted to satisfy that
G(A, U, A, U ) = H(A, U, A, U ),
(54)
for every (A, U ).
Two version of the model economy are considered: One where the matching technology is subject to congestion externalities and another where it isn’t. The matching technology with congestion externalities is given by
A
G(a, u, A, U ) = u
H (a, u, A, U ) = a
1
,
(55)
1
,
(56)
[U φ + Aφ ] φ
U
[U φ + Aφ ] φ
and the matching technology with no externalities is given by
G(a, u, A, U ) = H(a, u, A, U ) =
u.a
1
.
(57)
[uφ + aφ ] φ
1 6 Strictly
speaking, the linear-quadratic structure is obtained only when the aggregate law of motion L̂ in equation (52) is linear. However,
this will be true in equilibrium. In fact, Hansen and Prescott (14) update the linear law of motion L̂ at each value function iteration by imposing
the RCE condition (53) on the linear decision rule D obtained from the iteration.
15
Both matching technologies satisfy equation (54) and all the assumptions made in Section 2. Moreover, they both
aggregate into a standard den Haan-Ramey-Watson (8) matching function
G(A, U, A, U ) =
U.A
1
.
(58)
[U φ + Aφ ] φ
Also, observe that the matching technology with congestion externalities given by equations (55) and (56) captures a
standard assumption in the matching literature: That the aggregate market-tightness ratio A/U determines the rate at
which individual unemployed workers find employment opportunities and the rate at which individual help-wanted ads find
recruitment opportunities.
The rest of this section calibrates the steady states of both versions of the model economy to identical long-run U.S.
observations. Before proceeding it will be necessary to select a model time-period that is both convenient and consistent
with observations.
The Job Openings and Labor Turnover Survey (JOLTS) conducted by the Bureau of Labor Statistics is an important
source of information for two key features of the model: the creation of recruitment opportunities (i.e. job openings) and
the worker turnover process. JOLTS, which is a monthly survey of continuing nonagricultural establishments, defines job
openings as positions for which there is work available, for which a job could start within 30 days, and for which there is
an active recruitment effort taking place (such as advertisement in newspapers, radio and television, posting “help wanted
signs”, interviewing candidates, etc.). Job openings are measured on the last business day of the month. On the contrary,
hirings, which are defined as all additions to the establishments’ payrolls, are measured over the entire month. The vacancy
yield rate defined as the average monthly ratio of hirings to job openings over the entire period 2000-2005 is equal to 1.3
(see Davis et. al, (7)).
Since hirings cannot exceed recruitment opportunities in the model economy (see equation 3), a vacancy yield rate
greater than one can only be obtained through time aggregation. This suggests calibrating to a short time period. However,
computational convenience requires making the time period as large as possible. The largest time period consistent with the
above observation is 3 weeks. The reason is simple: if total hirings turned out to be approximately equal to total recruitment
opportunities, a monthly vacancy yield rate close to 1.3 would be obtained from the simple fact that a month contains 4/3
three-weeks periods. Observe that, since equation (54) implies that recruitment opportunities are equal to E, equation (26)
indicates that a small π u is a necessary condition for total hirings to be close to total recruitment opportunities. In what
follows the time period will thus be tentatively selected to be 3 weeks and πu will be set to zero.17 Moreover, it will be
assumed that total hirings are approximately equal to total recruitment opportunities. Later on it will be verified that this
assumption is correct and that the monthly vacancy yield rate obtained is indeed consistent with the JOLTS measurement.
The next issue that must be addressed is what actual measure of capital should the model capital correspond to. Since
the focus is on establishment level dynamics, it seems natural to abstract from capital components such as land, residential
structures, and consumer durables. The empirical counterpart for capital is then identified with plant, equipment, and
1 7 Observe
that, since establishments invest in recruitment opportunities one period in advance and some of them end-up transiting to lower
idiosyncratic productivity levels (or even exiting), not all existing recruitment opportunities end-up being exercised. Thus, when π u is equal to
zero equation (26) holds with strict inequality. This in turn implies that pe must be equal to pu , since owners of employable workers must be
made indifferent between selling them or keeping them as unemployed workers.
16
inventories. As a result, investment is associated in the National Income and Product Accounts (NIPA) with nonresidential
investment plus changes in business inventories. The empirical counterpart for consumption is identified with personal
consumption expenditures in nondurable goods and services. Output is then defined as the sum of these investment and
consumption measures. The quarterly capital-output ratio and the investment-output ratio corresponding to these measures
are 6.8 and 0.15, respectively. Since at steady state I/Y = δ(K/Y ), these ratios require that δ = 0.005515.
The production function is assumed to have the following functional form:
y = snγ kθ ,
where 0 < γ + θ < 1. Calibrating to an annual interest rate of 4 percent, which is a standard value in the macro literature,
requires a time discount factor β equal to 0.99755. Given this value for β, the above value for δ, and given that the capital
share satisfies that
θ=
(1/β + δ) K
,
Y
matching the U.S. capital-output ratio requires choosing a value of θ equal to 0.2168. Similarly, γ = 0.64 is selected to
reproduce the share of labor in National Income.18 Observe that, since workers are capital goods, the “wage rate” used in
calculating the labor share is given by the user cost (1 − β) pe . In what follows, the value of pe will be normalized to an
arbitrary value and the utility of leisure parameter ϕ will be selected to generate that value.
The values for the idiosyncratic productivity levels s, the distribution over initial productivity levels ψ and the transition
matrix Q are key determinants of the job-flows generated by the model. As a consequence I choose them to reproduce
observations from the Business Employment Dynamics (BED) data set, which is a virtual census of establishments level
dynamics. Since BED data across establishment sizes can be found for the nine employment ranges shown in the first
column of Table 1, I restrict the idiosyncratic productivity levels s to take nine positive values (s1 , s2 , ..., s9 , with si < sj
for every i < j) and choose them so that all establishments with a same idiosyncratic productivity level choose employment
levels in the same range.
The average size of new entrants can be obtained by dividing the total gross job gains at opening establishments by
the total number of opening establishments. Using data between 1992:Q3 and 2005:Q4, I find that the average size of new
entrants is equal to 5.3 employees. Since this is a small number, I restrict the distribution over initial productivity levels ψ
to put positive mass on only the two lowest values of s and choose ψ (s1 ) to reproduce that average size.
Similarly, the average size at exit can be obtained by dividing the total gross job losses at closing establishments by the
total number of closing establishments. Using data for the same time period as above, I find that the average size at exit
is equal to 5.2 employees. Since this is also a small number, I restrict the probabilities of transiting to a zero productivity
level Q(s, 0) to take positive values only at the three lowest values of s. The values for Q(s1 , 0), Q(s2 , 0) and Q(s3 , 0) are
then chosen to reproduce three observations: 1) the average size at exit, 2) the average quarterly rate of gross job losses due
to closing establishments (JLD), which is equal to 1.6%, and 3) the average quarterly exit rate of establishments, which is
1 8 In
the model, γ is not strictly the same as the share of labor in National Income. However, under γ = 0.64 the labor share turns out to be
0.6367.
17
equal to 5.2%.19
The rest of the transition matrix Q is parameterized with enough flexibility to reproduce other important establishment
level observations. The only restriction that I impose is that Q (si , sj ) > 0 only if j = i − 1, j = i or j = i + 1. Since the
rows of Q add to one this introduces 16 parameters (2 parameters each, for i = 2, ..., 8, and 1 parameter each, for i = 1, 9).
Eight of these parameters are chosen to reproduce the shares in total employment across size classes (which provide eight
independent observations). The other eight parameters are chosen to reproduce the shares in total gross job gains across
size classes (which also provide eight independent observations). I must point out that the BED does not tabulate statistics
across size classes in its regular reports. However, these statistics can be found in Okolie (19) (Tables 1 and 3) for the first
two quarters of 2000. These statistics together with the corresponding model statistics are shown in the first panel of Table
1. The second panel reports the average sizes at entry and exit both in the model and the data. We see that the model
does a good job at reproducing these observations. As a test of the model, Table 1 also includes the shares in total gross
job losses across size classes for the first two quarters of 2000 in Okolie (19), and the average quarterly rates of gross job
gains due to expanding establishments (JGE), gross job gains due to opening establishments (JGB), and gross job losses
due to contracting establishments (JLC) reported by the BED for the period 1992:Q3-2005:Q4. Although the fit is not
perfect, we see that the model also does a good job at reproducing these addional statistics.
The exogenous separation rate πn and the number of establishments created every period
are important determinants
of the worker flows in and out of unemployment, so I calibrate them to reproduce this type of observations. In particular,
I target an average monthly separation rate from employment equal to 3.5% and an average monthly hazard rate from
unemployment equal to 46%, which were estimated by Shimer (21) using CPS data between 1948 and 2004. Since the
separation rate of 3.5% is significantly larger than the rate of job losses experienced by establishments, I select a positive
value of π n to reproduce the excess worker reallocation.20 Also, observe that the separation and hazard rates estimated
by Shimer (21) imply a steady state unemployment rate equal to 7.1%. The average size of establishments implied by the
distribution reported in Table 1 thus determine the entry rate of establishments
that is needed to generate an aggregate
employment level N equal to 0.929.
Based on evidence in Barron et. al (5) and Silva and Toledo (22), Hagedorn and Manovskii (9) determined that the
costs of hiring a worker are equivalent to 4.5% of quarterly wages.21 Since total hirings are assumed to be approximately
equal to total recruitment opportunities, the cost of hiring a worker is approximately equal to the price of a next-period
recruitment opportunity pv . This suggest calibrating parameter values to reproduce the following relation:
pv = 0.045 × 4 × (1 − β) pe ,
(59)
where (1 − β) pe represents 3-weeks wages and a factor of 4 is needed to convert them into quarterly wages. Recall that the
price of a worker pe was normalized to an arbitrary value. Thus, equation (59) solely imposes a restriction on pv .
1 9 Since
2 0 Not
the model time period is three weeks, quarterly statistics are constructed following establishments over four consecutive time periods.
surprisingly, my calibrated value of πn is smaller than the quit rate of workers measured by JOLTS (equal to 2% a month), since many
of those separations entail job-to-job transitions that the model abstracts from.
2 1 Since
in their model capital is iddle while a job is open, Hagedorn and. Manovskii (9) add an imputed opportunity cost of capital to the
total costs of hiring a worker. I do not make such adjustment because there is no iddle capital in my model.
18
The steady-state price of a recruitment opportunity pv depends on A , U and the matching function curvature parameter
φ.22 Also, since (by assumption) total hirings are approximately equal to total recruitment opportunities, we have from
equation (58) that
A
Hirings
(60)
≈
1 ,
U
[U φ + Aφ ] φ
which must be equal to the hazard rate of unemployment that we are calibrating to (i.e. the monthly hazard rate of 46%,
estimated by Shimer (21)). Since we are calibrating to a known value of U (equal to 0.071), equations (59) and (60) can be
used to solve for A and φ. The values thus obtained are quite reasonable. In particular, the implied elasticity of the hazard
rate from unemployment G(A, U, A, U )/U to the unemployment-help-wanted-ads ratio U/A turn out to be 0.52 in the case
of congestion externalities and 0.64 in the case of efficient matching. This elasticities are within the range estimated by
previous studies (e.g. Shimer (20), Hall (13), etc.).
As a test of the model Table 2 reports a set of basic monthly statistics both for JOLTS and the model economy that
were not used as calibration targets (except for the vacancy yield rate).23 . We see that the model does a reasonable job at
reproducing not only the vacancy yield rate, but the hiring and separation rates for continuing establishments, the fraction
of vacancies with zero hirings and the fraction of hires with zero vacancies. The low rate of exogenous separations π n
explains the model’s success in reproducing the fraction of vacancies with zero hirings. The reason is that a significant
number of establishments reach the lower thresholds n and start hiring just enough workers to replenish the exogenous
separation of workers. Since the monthly rate of exogenous separation is less than 1%, following Davis et al. (7), I classify
these establishments (and their corresponding vacancies) as having zero hirings.
Observe that the model’s ability at reproducing the JOLTS vacancy yield rate confirms that the strategy of calibrating
to a three weeks time-period and setting π u to zero was justified. In fact, the assumption that total hirings are approximately equal to total recruitment opportunities is verified: Total hirings turn out to be 96% as large as total recruitment
opportunities.
Finally, the parameters ρz and σ 2ε governing the aggregate productivity shock process are selected to reproduce the
empirical behavior of measured Solow residuals in the U.S. economy.24 Defining output and capital as above and using
civilian employment as the labor input, I find that measured Solow residuals are highly persistent and that their quarterly
proportionate changes have a standard deviations equal to 0.0064 over the period 1951:1-2004:4.25 It turns out that values
of ρz = 0.95 and σ ε = 0.0041 are needed to reproduce these observations using the artificial data generated by both versions
of the model economy.26
All calibrated parameter values are summarized in Table 3.
2 2 For
the details, see Section 4.1 in the Technical Appendix.
2 3 JOLTS
2 4 Let
statistics are from Davis et al. (7).
γ e denote the empirical labor share implicit in the National Income and Product Accounts. Proportionate changes in measured Solow
residuals are then defined as the proportionate change in aggregate output minus the sum of the proportionate change in labor times γ e , minus
the sum of the proportionate change in capital times (1 − γ e ).
2 5 Solow
2 6 In
residuals are constructed using an empirical labor share γ e equal to 0.64.
both model economies quarterly Solow residuals are measured with the same empirical labor share γ e used to measure Solow residuals in
19
6
Business cycles
This section uses both versions of the model economy calibrated in the previous section to address three important questions:
1) Can the neoclassical theory developed so far account for U.S. business cycle observations?, 2) Which scenario for the
matching process is empirically more plausible: The efficient matching or the congestion externalities scenarios?, and 3) Is
the model consistent with microeconomic adjustments at the establishment level?
Before turning to these questions it will be useful to describe salient features of U.S. business cycle fluctuations. Table
4 reports business cycle statistics for a number of U.S. time series corresponding to the period 1951:1 to 2004:4. All time
series were logged and detrended using the Hoddrick-Prescott filter with smoothing parameter 1,600 before computing any
statistics.
The upper panel of Table 4 reports standard deviations and correlations with output for GDP (Y ), consumption (C),
investment (I), capital (K), civilian employment (N ), and labor productivity (Y /N ). These statistics are standard in
the RBC literature. They show that consumption, employment and labor productivity fluctuate roughly 61% as much as
output, that capital fluctuates only 43% as much as output, that investment fluctuates 3.3 times more than output, and
that all variables are strongly procyclical except for capital, which is acyclical.
The lower panel of Table 4 reports standard deviations and the cross-correlation matrix for GDP (Y ), employment
(N ), unemployment (U ), help-wanted ads (A), market tightness (A/U ), job finding probability (H/U ), employment exit
probability (F/N ), job creation rate (JC) and job destruction rate (JD). The job finding and employment exit probabilities,
H/U and F/N , are from Shimer (21). The job creation and job destruction rates, JC and JD, are from Davis, Faberman
and Haltiwanger (7) (based on BED data) and correspond to the subperiod 1990:2-2004:4.27 The statistics in the lower
panel of Table 4 have been emphasized in the labor literature (e.g. Hagedorn and Manovskii (9), Shimer (20), Davis and
Haltiwanger, etc.). They show that unemployment and help wanted ads fluctuate about 7.3 times more than output, that
the employment exit probability fluctuates 2.5 times more than output and that the job finding probability is about 40%
more variable than the employment exit probability, and that the job creation rate fluctuates 1.4 times more than output and
that job destruction rate is about 65% more variable than the job creation rate. Unemployment is strongly countercyclical,
help wanted ads and the job finding rate are strongly procyclical, the employment exit probability is countercyclical, and
the job creation and job destruction rates show weak cyclical patterns. Also observe that a “Beveridge curve” is obtained:
help-wanted ads and unemployment are strongly negatively correlated (their correlation is -0.92).
6.1
Efficient matching vs. congestion externalities
Table 5 reports business cycle statistics for the model economy with efficient matching. Time series of length equal to 864
time periods were computed for 100 simulations and then aggregated into a quarterly frequency to obtain 216 quarters of
the U.S. economy.
2 7 Job
creation (JC) corresponds to the sum of gross job gains due to expanding establishments (JGE) and gross job gains due to opening
establishments (JGB). Job destruction (JD) corresponds to the sum of gross job losses due to contracting establishments (JLC) and job losses
due to closing establishments (JLD).
20
data (the same length as the U.S. series). The reported statistics are averages across these simulations. With regard to
standard RBC statistics, we see from the upper-panels of Tables 4 and 5 that the model with efficient matching reproduces
the comovements with output quite well: Except for capital, which is acyclical, all other variables are procyclical. The
model’s performance is not as good in terms of standard deviations, though. We see that investment fluctuations are as
large as in the data but the rest of the variables are much smoother. The largest difference is with consumption, which
fluctuates only 29% as much as in the data. However, this is a standard problem with RBC models. The most disappointing
performance is with employment, which fluctuates only 37% as much as in the data. This smoothness is in turn inherited
by output, which fluctuates only 69% as much as in the U.S. The failure of the model with efficient matching to account
for labor market dynamics is evident in the lower panel. We see that unemployment, help-wanted ads, the job finding and
employment exit probabilities and the job creation and job destruction rates fluctuate too little compared with the data.
Moreover, the model fails to generate a strong “Beveridge curve”: the correlation of unemployment with help-wanted ads
is only -0.61.
We now turn to the model with congestion externalities in the matching process. Table 6 shows the results. We
see that in terms of standard RBC statistics that this version of the model replicates U.S. business cycle observations
at least as well as the economy with efficient matching. Comovements with output are still very similar with the data:
consumption, investment, employment and labor productivity are all procyclical while capital is acyclical. The dimension in
which the economy with congestion externalities outperforms the economy with efficient matching is in standard deviations:
Employment, capital and output go from being 37%, 62% and 69% as volatile as the data to being 93%, 82%, 91% as volatile,
respectively.28 The model with congestion externalities also outperforms the model with efficient matching in terms of labor
market statistics. In the lower panel of Table 6 we see that unemployment, help-wanted ads, the job-finding probability,
the employment-exit probability, the job creation rate and the job destruction rate become much more variable than in the
model with efficient matching. Also the economy now generates a more noticeable Beveridge curve: The correlation between
unemployment and help-wanted ads is -0.76. Although these are improvements over the model with efficient matching, the
empirical performance of the model is far from perfect: 1) help-wanted ads fluctuate 45% more than in the data, 2) the
job-finding probability is 9 times more volatile than the employment-exit probability, while in the data they are only 40%
more volatile, and 3) job destruction is as variable as job creation, while in the data job destruction is 65% more variable.
Before turning to these limitations the rest of this section explores the reasons for the improved performance of the model
with matching externalities.
Observe that there are two differences between the economy with efficient matching and the economy with congestion
externalities. First, as Table 3 indicates, the economies have different parameter values (in particular, the curvature of the
matching function φ and the utility of leisure ϕ are different). Second, although their aggregate matching functions have
identical functional forms (see equation 58), their individual matching technologies are different (compare equations 5556 with equation 57). In order to determine which of these differences drives the result that the economy with congestion
externalities outperforms the economy with efficient matching, Table 7 reports the business cycle statistics for the Pareto
optimal allocation of the economy with congestion externalities. Since the social planner fully internalizes the effects of the
2 8 The
only drawback is with labor productivity, which goes from being 76% as volatile as the data to being only 58% as volatile.
21
congestion externalities, any differences between these statistics and those of the economy with efficient matching reported
in Table 5 can be solely attributed to differences in parameter values. Since Table 7 is very similar to Table 5, we conclude
that the bulk of the differences in business cycle fluctuations between the economy with efficient matching and the economy
with congestion externalities is not due to different parameter values but to the different individual matching technologies.29
In order to determine what feature of the matching technology with congestion externalities is essential for generating
relatively large business cycle fluctuations, it will be useful to consider the following linear matching technology:
G(a, u, A∗ , U ∗ ) = u h
A∗
(U ∗ )φ + (A∗ )φ
i φ1 ,
U∗
H (a, u, A∗ , U ∗ ) = a h
i1 ,
φ
φ φ
∗
∗
(U ) + (A )
(61)
(62)
where A∗ and U ∗ are positive parameters. Observe that if an economy had identical parameter values as the economy with
congestion externalities but its matching technology was described by equations (61) and (62), with A∗ and U ∗ given by
the steady-state values of help-wanted ads and unemployment in the economy with congestion externalities, respectively,
its steady state would be identical to the steady state of the economy with congestion externalities. However, its business
cycles would be different. The reason is that the linear matching technology in equations (61) and (62) has a constant
productivity while the linear matching technology faced by the myopic social planner in the economy with congestion
externalities is subject to “productivity shocks” given by the realizations of the market tightness ratio A/U . Table 8
reports the business cycle statistics for this economy. We see that its business cycles are in fact much larger than in the
economy with congestion externalities: Except for labor productivity and help-wanted ads, all variables are significantly
more volatile than in Table 6. This indicates that the crucial feature generating the relatively large business cycles in
the economy with congestion externalities is the linearity of the individual matching technology given by equations (55)
and (56): The external effects from endogenous variations in the market tightness ratio A/U only serve to dampen the
aggregate fluctuations generated by the economy. This is not surprising. Since equation (26) holds with strict inequality,
the technology for creating workers with employment opportunities in equation (55) is largely irrelevant for business cycles.
On the contrary, the technology for creating recruitment opportunities in equation (56) binds the amount of hiring that the
economy can undertake. Since aggregate market tightness A/U is strongly procyclical in Table 6, the productivity of the
technology for creating recruitment opportunities turns out to be countercyclical. This reduces the response of aggregate
employment to aggregate productivity shocks, leading to lower employment fluctuations in Table 6 than in Table 8. This
also explains why help-wanted ads are more volatile in Table 6 than in Table 8: Help-wanted ads need to respond more to
aggregate productivity shocks to partially compensate for the countercyclical productivity of the technology for creating
recrutiment opportunities.
2 9 From
Tables 6 and 7 we also conclude that introducing policies that achieve the first-best allocation would significantly reduce aggregate
fluctuations in the economy with congestion externalities. See the working paper version (Veracierto xx) for an analysis of such policies.
22
6.2
Reallocation shocks
Section 6.1 showed that the economy with congestion externalities generates much more realistic business cycle fluctuations
than the economy with efficient matching. However, it had several limitations: 1) help-wanted ads fluctuated much more
than in the data, 2) the job-finding probability was several times more volatile than the employment-exit probability (while
in the data it is only slightly more volatile), and 3) the job destruction rate was as variable as the job creation rate (while
in the data it is considerably more volatile).
The purpose of this section is to explore to what extent the model’s performance could be improved by introducing
reallocation shocks that affect the idiosyncratic productivity process. This is a natural starting point since reallocation
shocks directly influence the behavior of job creation and job destruction. Since the job creation and destruction process
have strong implications for help-wanted ads, the job-finding probability and the employment-exit probability, reallocation
shocks have the potential of affecting the behavior of these other variables as well. In what follows, reallocation shocks will
be introduced to make volatility of job destruction (relative to the volatility of job creation) as large as in the data. A key
question will be if these reallocation shocks help improve the model’s performance in other dimensions.
A wide variety of reallocation shocks may be analyzed using the computational approach developed in this paper. For
instance, the reallocation shock considered could affect the dispersion of the idiosyncratic productivity levels s while leaving
the transition matrix Q unchanged. Another possible reallocation shock could leave the idiosyncratic productivity levels s
unchanged while affecting the persistence Q(s, s) of the different idiosyncratic productivity levels s. It turns out that these
types of reallocations shocks do not improve the model’s performance. The reason is that they synchronize the fluctuations
in job creation and job destruction and thus fail to generate their asymmetric volatilities.
In order to break this synchronization the following reallocation shock will be considered. Let S ∗ be the set of idiosyncratic productivity levels and Q∗ the transition matrix that were calibrated in Section 5. The reallocation shock rt analyzed
leaves the set of values for the idiosyncratic productivity levels unchanged at S ∗ but affects the transitions matrix Qt as
follows. For every s and s0 in S ∗ ,
Qt (s, s0 ) =
⎧
⎪
⎪
⎪
⎨
Q∗ (s, s0 ) , if s0 > s,
ert Q∗ (s, s0 ) , if s0 < s,
⎪
⎪
P
P
⎪
⎩ 1−
rt ∗
00
∗
00
0
s00 <s e Q (s, s ) −
s00 >s Q (s, s ) , if s = s.
(63)
Observe that this reallocation shock rt affects the probabilities of transiting to lower productivity levels but not the
probabilities of transiting to higher productivity levels. Any variations in the probabilities of transiting to lower productivity
levels are absorbed by the probabilities of no-change.30
The reallocation shock rt and the aggregate productivity shock zt follow a joint autoregressive process given by
⎡
⎤ ⎡
⎤⎡
⎤ ⎡
⎤⎡
⎤
zt+1
ρzz ρzr
zt
σ zz σ zr
εzt+1
⎣
⎦=⎣
⎦⎣
⎦+⎣
⎦⎣
⎦,
rt+1
ρrz ρrr
rt
σ rz σ rr
εrt+1
(64)
where εzt+1 and εrt+1 are normally distributed with zero mean and unit standard deviation.
3 0 Restrictions
to ensure that the probabilities of no-change Q(s, s) remain positive for every possible realization of the reallocation shock rt
are ignored in equation (63) since these restrictions turn out to be non-binding in the experiments.
23
Allowing the reallocation shock rt to be negatively correlated with the aggregate productivity shock zt is crucial for
generating asymmetries in the job creation and job destruction process. To see this, suppose that the economy is hit
by a negative aggregate productivity shock that is accompanied by higher transition probabilities to lower idiosyncratic
productivity levels. Since these higher transition probabilities generate job destruction, the response of job destruction
to the negative aggregate productivity shock will thus be amplified. In addition, if the larger transition probabilities are
short-lived (i.e. if ρrr is close to zero), the response of job creation to the negative aggregate productivity shock will be
dampened. The reason, is that after its initial fall, the distribution of idiosyncratic productivity levels will be reverting
towards the invariant distribution generated by the transition matrix Q∗ , creating job creation over time. Both effects work
in the same direction: making job destruction relatively more volatile than job creation.31
Given the above discussion, the reallocation shocks will be restricted to be short-lived and perfectly negatively correlated
with innovations in aggregate productivity. In turn, the aggregate productivity shock will be allowed to have the same
persistence level as in the benchmark case. Under these assumptions the general process in equation (64) reduces to the
following specification:
⎡
⎣
zt+1
rt+1
⎤
⎡
⎦=⎣
0.95 0
0
0
⎤⎡
⎦⎣
zt
rt
⎤
⎡
⎦+⎣
σ zz
0
σ rz
0
⎤⎡
⎦⎣
εzt+1
εrt+1
⎤
⎦.
(65)
The parameters σ zz and σ rz in equation (65) are selected to reproduce two important observations. First, that the
standard deviation of measured Solow residuals is equal to 0.0064 (the same observation that was used in the benchmark
case). Second, that the standard deviation of job destruction is 65% larger than the standard deviation of job creation.
The parameter values consistent with these observations turn out to be σ zz = 0.00385 and σ rz = −0.07.
Table 9 reports the business cycle statistics for this economy. We see that in terms of standard RBC statistics, that the
economy with reallocation shocks is virtually identical to the benchmark economy with congestion externalities (see the
upper pannel of Table 6). In fact, despite of the fact that the aggregate productivity shock is 6% less variable and that the
reallocation shock is i.i.d., the economy with reallocation shocks is slightly more volatile than than the benchmark case.
There are significant differences in terms of labor market statistics, though. The most obvious is that (by construction)
job destruction is now much more volatile than job creation, bringing the job creation and destruction process closer to
the data. There are significant improvements on other variables as well. In particular, the volatility of help-wanted ads A
and market tightness A/U are more in line with the data and a clearer Beveridge curve is now obtained (the correlation
between unemployment and help wanted ads is -0.81). We see that the reallocation shocks also help reduce the volatility of
the job-finding probability relative to the volatility of the employment-exit probability: the job-finding probability is now
3 times more variable than the employment-exit probability, while it was 9 times larger in the benchmark case. However,
this asymmetry is still too large compared to the data. The reason why it is so hard for the model economy to generate
large fluctuations in the employment-exit probability is that a large component of it is constant over the cycle: It is given
by the exogenous quit rate of workers π n . Endogenizing this margin may improve the performance of the model in this
3 1 The
assymmetric volatilities may not be obtained if ρrr is close to one. If after a negative aggregate productivity shock hits the economy
establishments expect good idiosyncratic productivity levels to be much more transient than before, they will have fewer incentives to invest in
recruitment opportunities after a high idiosyncratic productivity level is realized. As a consequence, the drop in job creation after the negative
aggregate productivity shock hits the economy may actually be amplified.
24
particular dimension.
We have seen that the reallocation shock introduced improve the empirical performance of the model in terms of its job
creation and destruction rates, job-finding and employment-exit probabilities, and help wanted ads. While there is ample
evidence that idiosyncratic uncertainty increases during recessions (e.g. Bloom, 2009; Gilchrist et al, 2009; Bachmann and
Bayer, 2009), it would be desirable to determine the precise empirical nature of reallocation shocks before jumping to any
strong conclusions. The results presented in this section indicates that this may be a fruitful area of research. Having said
this, the reallocation shocks introduced had very minor effects on macroeconomic variables such as output, employment,
consumption and investment. Thus, reproducing the cyclical microeconomic adjustments observed in the data may turn
up unimportant for aggregate business cycle dynamics.
25
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27
Appendix A
Proof of Proposition 4: From equations (38) and (39) we know that an establishment of type (s, l, j) transits to a
next-period type (s0 , l0 , j 0 ), with s0 randomly determined,
l0 = n∗ (s, l, j),
(66)
j 0 = max {v̄∗ (s) − (1 − π n )l0 , 0} .
(67)
and
Define
P (0) = ∪ {(s, 0, 0)} .
s∈S
Since establishments are created with (l, j) = (0, 0), P (0) describes the set of all possible types (s, l, j) of establishments of
zero age.
Define
N (0) = {0} .
Since n∗ (s, l, j) = 0 whenever (l, j) = (0, 0), N (0) describes the set of all possible employment levels of establishments of
zero age.
Starting from N (0) , define recursively a sequence of sets P (m) and N (m) as follows:
½
¾
P (m) = (s, l, j) : s ∈ S, l ∈ N (m−1) , and j ∈ ∪ {max [v̄∗ (s−1 ) − (1 − πn ) l, 0]}
N (m) =
for m = 1, 2, ..., ∞.
½
s−1 ∈S
¾ n
o
∗
∗
∗
∪ {n (s) , n̄ (s) , v̄ (s)} ∪ n: n = (1 − π n ) nm−1 for some nm−1 ∈ N (m−1) ,
s∈S
From equations (38), (39), (66) and (67) we know that P (m) contains the set of all possible types (s, l, j) of establishments
of age m, and that N (m) contains the set of all possible employment levels of establishments of age m.32
By induction, it can be shown that:
½
o¾
n
m−1
k
k
k
N (m) = ∪ ∪ (1 − π n ) n∗ (s) , (1 − π n ) n̄∗ (s) , (1 − π n ) v̄ ∗ (s)
∪ {0} ,
s∈S
k=0
(68)
for m = 1, 2, ..., ∞.
A direct consequence of equation (68) is that N (m−1) ⊂ N (m) , for every m ≥ 1. Thus, the set N (m) in fact contains
all the possible employment levels of establishments of age m or younger and the set P (m) ∪ P (0) contains all the possible
types of establishments of age m or younger. Moreover,
N (m) /N (m−1) = ∪
s∈S
3 2 Observe
o
n
(1 − πn )m−1 n∗ (s) , (1 − πn )m−1 n̄∗ (s) , (1 − πn )m−1 v̄∗ (s) ,
(69)
that the “max” and “min” operators in equation (38) have been disregarded in the construction of the sets P (m) and N (m) . Thus,
the set of actual types of establishments of age m and the set of actual employment levels of establishments of age m are smaller than P (m) and
N (m) , respectively.
28
for m = 1, 2, ..., ∞, where “/” denotes set difference.
In what follows it will be shown that there exists a M < ∞ such that N (M) contains the set of all possible employment
levels of establishments of all ages m = 0, 1, ..., ∞. To prove this it suffices to show that there exists a M < ∞ such that no
establishment of age M + 1 will choose an employment level in the set N (M+1) /N (M) , i.e. all establishments of age M + 1
will choose an employment level in the set N (M) .33
Let M satisfy equation (40). Since 0 < π n < 1, such a M exists.
Let (s, l, j) ∈ P (M+1) .
Suppose that n∗ (s, l, j) ∈ N (M+1) /N (M) . Since N (M+1) /N (M) satisfies equation (69), and M satisfies equation (40), it
follows that
n∗ (s, l, j) ≤ (1 − π n )M max {n̄∗ (smax ) , v̄∗ (smax )} < min {n∗ (smin ) , v̄∗ (smin )} .
Also, since n∗ (s, l, j) satisfies equation (38) and (s, l, j) ∈ P (M+1) , we have that
½
¾
∗
∗
∗
∗
n (s, l, j) ∈ {n (s) , n̄ (s) , (1 − πn ) l} ∪
∪ v̄ (s−1 ) .
s−1 ∈S
(70)
(71)
From equation (71) and the last inequality in equation (70), we then have that
n∗ (s, l, j) = (1 − πn ) l.
Suppose, first, that j = 0.
Suppose that some establishment of age M transits to (s, l, j). From equation (67), this can be the case only if
0 = max {v̄ ∗ (s−1 ) − (1 − πn )l, 0} ,
for some s−1 ∈ S.
But, from equation (70)
n∗ (s, l, j) = (1 − π n ) l < v̄∗ (s−1 ) ,
for all s−1 ∈ S. A contradiction.
Hence, (s, l, j) ∈ P (M+1) does not correspond to an establishment of age M + 1.
Suppose now that j > 0.
Let s−1 be such that (1 − πn ) l + j = v̄ ∗ (s−1 ) (since (s, l, j) ∈ P (M+1) , such an s−1 exists).
Then, from equation (38) we have that
⎧
⎫
⎨ min {v̄∗ (s ) , n∗ (s)} , ⎬
−1
n∗ (s, l, j) = max
,
⎩ min {(1 − π ) l, n̄∗ (s)} ⎭
n
and, therefore, that
n∗ (s, l, j) = (1 − πn ) l ≤ n̄∗ (s) and n∗ (s, l, j) = (1 − πn ) l ≥ min {v̄ ∗ (s−1 ) , n∗ (s)} .
3 3 This
(72)
condition is sufficient because whenever an establishment reaches age M + 1, its age can be reset to M without consequence. This
procedure can be repeated an infinite number of times.
29
The second inequality in equation (72) contradicts equation (70).
We conclude that no establishment of age M + 1 chooses an employment level in the set N (M+1) /N (M) . It follows that
the set P ∗ = P (M+1) ∪ P (0) is a support of the invariant distribution μ∗ .¥
Proof of Proposition 5: Observe that the optimal decision rules at period t − k are given by
⎫
⎧
⎬
⎨ min {(1 − π } l + j, n
n
t−k (s) ,
nt−k (s, l, j) = max
,
⎩ min {(1 − π ) l, n̄
(s)} ⎭
n
and
(73)
t−k
vt−k (s, l, j) = max {v̄t−k (s) − (1 − πn ) nt−k (s, l, j), 0} ,
(74)
for k = 0, 1, ..., M + 1.
a) We will first show that Pt+1 is a support of the distribution μt+1 .
Define the sets At and Bt+1 as follows:
At = ∪
s∈S
o
n
(1 − πn )M−1 nt−M (s) , (1 − π n )M−1 n̄t−M (s) , (1 − π n )M−1 v̄t−M−1 (s) ,
Bt+1 = {l0 : l0 = (1 − π n ) l, for some l ∈ Nt /At } .
Observe that
Nt+1 = Bt+1 ∪
½
¾
∪ {nt (s) , n̄t (s) , v̄t−1 (s)} .
(75)
(76)
s∈S
To show that Pt+1 is a support of the distribution μt+1 it suffices to show that
(s, l, j) ∈ Pt =⇒ nt (s, l, j) ∈ Nt+1 and vt (s, l, j) ∈ 0∪ {max [v̄t (s0 ) − (1 − π n ) nt (s, l, j) , 0]} .
s ∈S
(77)
Let (s, l, j) ∈ Pt .
Suppose, first, that (l, j) = (0, 0).
From equation (73) we then have that nt (s, l, j) = 0 and, from equation (74), that vt (s, l, j) = v̄t (s). Therefore, equation
(77) is satisfied.
Suppose, now, that (l, j) 6= (0, 0).
Then,
s ∈ S, l ∈ Nt and j ∈ 0∪ {max [v̄t−1 (s0 ) − (1 − πn ) l, 0]} .
s ∈S
From equations (73) and (78), we then have that
⎫
⎧
0
⎬
⎨ min {max [v̄
(s
)
,
(1
−
π
)
l]
,
n
(s)}
,
t−1
n
t
nt (s, l, j) = max
,
⎭
⎩
min {(1 − π ) l, n̄ (s)}
n
(78)
(79)
t
for some s0 ∈ S.
As a consequence,
nt (s, l, j) ∈ {(1 − π n ) l, n̄t (s) , nt (s)} ∪
30
½
¾
∪ {v̄t−1 (s )} .
0
s ∈S
0
(80)
From equations (75), (76) and (80) we have that
l ∈ Nt /At ⇒ nt (s, l, j) ∈ Nt+1 .
Suppose that l ∈ At . Without loss of generality assume that
s)
l = (1 − π n )M−1 nt−M (b
s) and l = (1 − π n )M−1 v̄t−M−1 (b
s) can be handled in exactly the same
for some sb ∈ S (the cases l = (1 − πn )M−1 n̄t−M (b
way).
Then, equation (79) becomes
⎧
i
o ⎫
n
h
⎨ min max v̄t−1 (s0 ) , (1 − π n )M nt−M (b
s) , nt (s) , ⎬
o
n
nt (s, l, j) = max
.
M
⎭
⎩
s) , n̄t (s)
min (1 − π n ) nt−M (b
for some s0 ∈ S.
(81)
But from equation (40) and equations (42)-(44), we have that
(1 − π n )
M
nt−M (b
s) < (1 − π n )
M
n̄t−M (b
s) ≤ (1 − π n )
M
n̄t−M (smax ) < v̄t−1 (smin ) ≤ v̄t−1 (s0 ) ,
and that
(1 − π n )
M
nt−M (b
s) < (1 − π n )
M
n̄t−M (b
s) ≤ (1 − π n )
M
n̄t−M (smax ) < nt (smin ) ≤ nt (s) < n̄t (s) .
Therefore equation (81) becomes
⎫
⎧
0
⎬
⎨ min {v̄
(s
)
,
n
(s)}
,
t−1
t
nt (s, l, j) = max
⎩ (1 − π )M n
(b
s) ⎭
n
t−M
0
= min {v̄t−1 (s ) , nt (s)} .
Thus, from equations (76), nt (s, l, j) ∈ Nt+1 .
From equation (74), observe that
vt (s, l, j) = max {v̄t (s) − (1 − π n ) nt (s, l, j), 0}
Thus,
vt (s, l, j) ∈ 0∪ {max [v̄t (s0 ) − (1 − πn ) nt (s, l, j) , 0]} .
s ∈S
Therefore, Pt+1 is a support of the distribution μt+1 .
b) To prove the second part of the Proposition it will be convenient to define the following (one-to-one and onto)
functions.
For every (s, l, j) ∈ Pt :
lt∗ (l, j) = l∗
jt∗ (l, j) = j ∗
31
where (s, l∗ , j ∗ ) is the unique element of P ∗ satisfying that |l − l∗ | < ε/2 and |[(1 − π n ) l + j] − [(1 − πn ) l∗ + j ∗ ]| < ε/2.
Similarly, for every (s0 , l0 , j 0 ) ∈ Pt+1 :
∗
lt+1
(l0 , j 0 ) = l∗
∗
jt+1
(l0 , j 0 ) = j ∗
where (s0 , l∗ , j ∗ ) is the unique element of P ∗ satisfying that |l0 − l∗ | < ε/2 and |[(1 − π n ) l0 + j 0 ] − [(1 − π n ) l∗ + j ∗ ]| < ε/2.
Observe that, by assumption, we have that for every (s, l, j) ∈ Pt :
μt (s, l, j) = μ∗ (s, lt∗ (l, j), jt∗ (l, j)) .
(82)
We need to show that for every (s0 , l0 , j 0 ) ∈ Pt+1 :
¢
¡
∗
∗
μt+1 (s0 , l0 , j 0 ) = μ∗ s0 , lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 ) .
Let (s0 , l0 , j 0 ) ∈ Pt+1 .
(83)
Using equation (82), we have that
μt+1 (s0 , l0 , j 0 ) =
X
Q (s, s0 ) μ∗ (s, lt∗ (l, j), jt∗ (l, j)) + ψ (s0 ) I (l0 , j 0 ) ,
(s,l,j) ∈ Gt (l0 ,j 0 )
where
Gt (l0 , j 0 ) = {(s, l, j) ∈ Pt : nt (s, l, j) = l0 and vt (s, l, j) = j 0 } .
Also observe that
¢
¡
∗
∗
(l0 , j 0 ), jt+1
(l0 , j 0 ) =
μ∗ s0 , lt+1
where
X
¡∗
¢
∗
Q (s, s0 ) μ∗ (s, l∗ , j ∗ ) + ψ (s0 ) I lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 ) ,
∗
0 0
0 0
(s,l∗ ,j ∗ ) ∈ G ∗ (l∗
t+1 (l ,j ),jt+1 (l ,j ))
©
ª
∗
∗
∗
∗
(l0 , j 0 ), jt+1
(l0 , j 0 )) = (s, l∗ , j ∗ ) ∈ P ∗ : n∗ (s, l∗ , j ∗ ) = lt+1
(l0 , j 0 ) and v ∗ (s, l∗ , j ∗ ) = jt+1
(l0 , j 0 ) .
G ∗ (lt+1
To show that equation (83) holds, it then suffices to show that
¡∗
¢
∗
(l0 , j 0 ) = (0, 0) ⇔ lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 ) = (0, 0) ,
∗
∗
(l0 , j 0 ), jt+1
(l0 , j 0 )),
(s, l, j) ∈ Gt (l0 , j 0 ) =⇒ (s, lt∗ (l, j) , jt∗ (l, j)) ∈ G ∗ (lt+1
³
´
∗
∗
(l0 , j 0 ), jt+1
(l0 , j 0 )) =⇒ s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) ∈ Gt (l0 , j 0 ).
(s, l∗ , j ∗ ) ∈ G ∗ (lt+1
´
³
where [lt∗ ]−1 , [jt∗ ]−1 is the inverse function of (lt∗ , jt∗ ).
b.1) Proof of equation (84).
∗
∗
It is a direct consequence of how lt+1
and jt+1
were defined and equations (42)-(44).
b.2) Proof of equation (85).
Let (s, l, j) ∈ Gt (l0 , j 0 ). Then, (s, l, j) ∈ Pt ,
⎫
⎧
⎨ min {(1 − π ) l + j, n (s)} ⎬
n
t
,
l0 = max
⎩ min {(1 − π ) l, n̄ (s)} ⎭
n
32
t
(84)
(85)
(86)
and
(1 − π n ) l0 + j 0 = max {v̄t (s) , (1 − π n ) l0 } .
Observe that (s, lt∗ (l, j) , jt∗ (l, j)) ∈ P ∗ ,
⎫
⎧
⎨ min {(1 − π ) l∗ (l, j) + j ∗ (l, j), n∗ (s)} ⎬
n t
t
n∗ (s, lt∗ (l, j), jt∗ (l, j)) = max
,
⎭
⎩
∗
min {(1 − π ) l (l, j), n̄∗ (s)}
n
and
t
(1 − π n ) n∗ (s, lt∗ (l, j), jt∗ (l, j)) + v ∗ (s, lt∗ (l, j), jt∗ (l, j))
= max {v̄ ∗ (s) , (1 − πn ) n∗ (s, lt∗ (l, j), jt∗ (l, j))} .
Since
|(1 − π n ) l − (1 − πn ) lt∗ (l, j)| < ε/2,
|[(1 − π n ) l + j] − [(1 − πn ) lt∗ (l, j) + jt∗ (l, j)]| < ε/2,
|nt (s) − n∗ (s)| < ε/2,
and
|n̄t (s) − n̄∗ (s)| < ε/2,
it follows that
|n∗ (s, lt∗ (l, j), jt∗ (l, j)) − l0 | < ε/2,
(87)
|[(1 − π n ) n∗ (s, lt∗ (l, j), jt∗ (l, j)) + v ∗ (s, lt∗ (l, j), jt∗ (l, j))] − [(1 − π n ) l0 + j 0 ]| < ε/2.
(88)
and, therefore, that
Since (s0 , l0 , j 0 ) ∈ Pt+1 and [s0 , n∗ (s, lt∗ (l, j), jt∗ (l, j)), v∗ (s, lt∗ (l, j), jt∗ (l, j))] ∈ P ∗ , equations (87) and (88) imply that
∗
lt+1
(l0 , j 0 ) = n∗ (s, lt∗ (l, j), jt∗ (l, j)),
∗
jt+1
(l0 , j 0 ) = v∗ (s, lt∗ (l, j), jt∗ (l, j)).
Since (s, lt∗ (l, j), jt∗ (l, j)) ∈ P ∗ it follows that
∗
∗
(s, lt∗ (l, j), jt∗ (l, j)) ∈ G ∗ (lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 )).
b.3) Proof of equation (86).
∗
∗
Let (s, l∗ , j ∗ ) ∈ G ∗ (lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 )). Then, (s, l∗ , j ∗ ) ∈ P ∗ ,
∗
lt+1
(l0 , j 0 ) = n∗ (s, l∗ , j ∗ )
⎫
⎧
⎨ min {(1 − π ) l∗ + j ∗ , n∗ (s)} ⎬
n
,
= max
⎭
⎩
min {(1 − π ) l∗ , n̄∗ (s)}
n
33
(89)
(90)
and
∗
∗
(1 − πn ) lt+1
(l0 , j 0 ) + jt+1
(l0 , j 0 ) = (1 − π n ) n∗ (s, l∗ , j ∗ ) + v ∗ (s, l∗ , j ∗ )
©
ª
∗
= max v̄ ∗ (s) , (1 − πn ) lt+1
(l0 , j 0 ) .
³
´
Observe that s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) ∈ Pt ,
nt (s, [lt∗ ]−1
and
(l∗ , j ∗ ), [jt∗ ]−1
⎧
o ⎫
n
⎨ min (1 − πn ) [lt∗ ]−1 (l∗ , j ∗ ) + [jt∗ ]−1 (l∗ , j ∗ ) , nt (s) ⎬
n
o
,
(l∗ , j ∗ )) = max
−1
⎭
⎩
min (1 − π n ) [lt∗ ] (l∗ , j ∗ ), n̄t (s)
(1 − π n ) nt (s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ )) + vt (s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ))
n
o
−1
−1
= max v̄t (s) , (1 − πn ) nt (s, [lt∗ ] (l∗ , j ∗ ), [jt∗ ] (l∗ , j ∗ )) .
Also, from equation (77), we have that
h
i
ii
h
h
s0 , nt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) , vt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) ∈ Pt+1
for every s0 .
Moreover,
i
i
h
h
∗
lt+1
(nt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) , vt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) )
= n∗ (s, l∗ , j ∗ )
and
i
i
h
h
∗
jt+1
(nt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) , vt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) )
= v∗ (s, l∗ , j ∗ )
Hence, from equations (89) and (91), we have that
i
i
h
h
∗
lt+1
(nt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) , vt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) )
∗
= lt+1
(l0 , j 0 )
and
i
i
h
h
∗
jt+1
(nt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) , vt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) )
∗
= jt+1
(l0 , j 0 )
It follows that
l0 = nt (s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ )),
j 0 = vt (s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ )).
³
´
Since s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) ∈ Pt it follows that
³
´
−1
−1
s, [lt∗ ] (l∗ , j ∗ ), [jt∗ ] (l∗ , j ∗ ) ∈ Gt (l0 , j 0 ) ¥
34
(91)
(92)
Table 1
Quarterly observations
Panel A: BED data, March 2000 to June 2000
Size
Data
Model
Classes∗
Shares in
Shares in
Shares in
Shares in
Shares in
Shares in
(employees)
Employment
Job Gains
Job Losses
Employment
Job Gains
Job Losses
[1, 5)
6.4%
16.9%
9.7%
6.2%
16.2%
8.6%
[5, 10)
8.1%
13.1%
11.6%
9.0%
13.1%
10.7%
[10, 20)
10.7%
14.9%
13.7%
12.3%
14.1%
10.8%
[20, 50)
16.6%
18.3%
18.2%
16.4%
18.4%
14.7%
[50, 100)
13.1%
11.6%
12.6%
12.0%
12.2%
16.2%
[100, 250)
16.5%
11.9%
14.6%
16.4%
13.1%
13.9%
[250, 500)
9.8%
5.9%
8.5%
9.3%
4.9%
12.4%
[500, 1000)
7.3%
3.5%
5.4%
7.1%
8.1%
5.3%
[1000, ∞)
11.6%
4.2%
5.9%
11.3%
0.0%
7.5%
Panel B: BED data, 1992:3-2005:4
Data
Model
size at entry
5.3
4.4
size at exit
5.2
5.8
JGB
1.7%
1.2%
JGE
6.2%
6.2%
JLD
1.6%
1.6%
JLC
6.0%
5.8%
Exit Rate
5.2%
6.2%
Table 2
Monthly observations
Panel A: CPS data, 1948-2004
Data
Model
Separation rate
3.5%
3.3%
Hazard rate
46.0%
46.9%
Panel B: JOLTS data, 2000-2005
Data
Model
Vacancy rate
2.2%
2.2%
Hiring rate
3.2%
2.9%
Separation rate
3.1%
2.9%
Vacancies yield rate
1.3
1.3
% Vacancies with zero hiring
18.7%
21.9%
% Hiring with zero vacancies
42.3%
39.3%
% Establishments with zero hiring
81.6%
95.8%
% Establishments with zero vacancies
87.6%
97.9%
2
Table 3
Calibrated parameter values
Parameter
Description
Value
entry of establishments
0.0007038
δ
capital depreciation rate
0.0055147
β
discount factor
0.9975517
θ
capital share
0.216757
γ
labor share
0.64
πu
quit rate, unemployed workers
0
πn
quit rate, employed workers
0.00675
φ
curvature, matching function
1.0161441 (efficient matching)
0.734344 (congestion externalities)
ϕ
utility of leisure
0.78439 (efficient matching)
0.805099 (congestion externalities)
ρz
persistence aggregate shocks
0.95
σε
standard deviation aggregate shocks
0.0041
3
Table 3 (Continued)
Calibrated idiosyncratic process
Idiosyncratic Productivity levels:
s0 = 0.00
s1 = 6.3
s2 = 6.7
s3 = 7.7
s4 = 8.6
s5 = 9.4
s6 = 10.9
s7 = 12.1
s8 = 13.1
s9 = 14.3
ψ 0 = 0.00
ψ1 = 0.82
ψ 2 = 0.18
ψ 3 = 0.00
ψ 4 = 0.00
ψ 5 = 0.00
ψ6 = 0.00
ψ 7 = 0.00
ψ 8 = 0.00
ψ 9 = 0.00
Initial distribution:
Transition matrix:
⎛
1
0
⎜
⎜
⎜ 0.026 0.9467
⎜
⎜
⎜ 0.019 0.0147
⎜
⎜
⎜ 0.007 0
⎜
⎜
⎜ 0
0
⎜
Q=⎜
⎜ 0
0
⎜
⎜
⎜ 0
0
⎜
⎜
⎜
0
⎜ 0
⎜
⎜
0
⎜ 0
⎝
0
0
0
0
0
0
0
0
0
0.0273
0
0
0
0
0
0
0.9555
0.0108
0
0
0
0
0
0.0149
0.9582
0.0199
0
0
0
0
0
0.0330
0.9309
0.0361
0
0
0
0
0
0.0777
0.9119
0.0104
0
0
0
0
0
0.0245
0.9571
0.0184
0
0
0
0
0
0.0612
0.9262
0.0126
0
0
0
0
0
0.0325
0.9335
0
0
0
0
0
0
0.0356
4
0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
0
⎟
⎟
⎟
0
⎟
⎟
⎟
0
⎟
⎟
⎟
0
⎟
⎟
⎟
0
⎟
⎟
⎟
0
⎟
⎟
⎟
0.0340 ⎟
⎠
0.9644
0
Table 4
Business Cycle Statistics: U.S. economy (1951:1-2004:4)
A. Macroeconomic variables
Standard deviations
Y
C
I
K
N
Y /N
1.58
0.90
6.76
0.68
1.00
0.99
Correlations with output
Y
C
I
K
N
Y /N
1.00
0.80
0.91
0.05
0.80
0.79
B. Labor market variables
Standard deviations
Y
N
U
A
A/U
H/U
F/N
JC ∗
JD∗
1.58
1.00
12.32
13.89
25.66
7.72
5.46
3.70
6.15
Correlations matrix
Y
N
U
A
Y
N
U
A
A/U
H/U
F/N
JC ∗
JD∗
1.00
0.80
-0.84
0.90
0.89
0.82
-0.57
0.49
-0.30
1.00
-0.87
0.88
0.89
0.87
-0.38
0.21
-0.02
1.00
-0.92
-0.98
-0.92
0.54
-0.43
0.03
1.00
0.98
0.91
-0.56
0.47
-0.26
1.00
0.93
-0.56
0.46
-0.16
1.00
-0.38
0.45
0.06
1.00
-0.13
0.47
1.00
-0.08
A/U
H/U
F/N
JC ∗
5
Table 5
Business Cycle Statistics: Efficient Matching
A. Macroeconomic variables
Standard deviations
Y
C
I
K
N
Y /N
1.09
0.26
6.23
0.42
0.37
0.75
Correlations with output
Y
C
I
K
N
Y /N
1.00
0.75
0.99
0.18
0.96
0.99
B. Labor market variables
Standard deviations
Y
N
U
A
A/U
H/U
F/N
JC
JD
1.09
0.37
5.20
4.58
8.78
5.52
0.50
1.83
1.97
Correlations matrix
Y
N
U
A
Y
N
U
A
A/U
H/U
F/N
JC
JD
1.00
0.96
-0.96
0.76
0.96
0.98
-0.82
0.12
-0.40
1.00
-1.00
0.60
0.91
0.97
-0.72
-0.07
-0.22
1.00
-0.61
-0.91
-0.97
0.72
0.07
0.23
1.00
0.88
0.76
-0.96
0.62
-0.81
1.00
0.97
-0.93
0.28
-0.56
1.00
-0.84
0.16
-0.44
1.00
-0.54
0.76
1.00
-0.70
A/U
H/U
F/N
JC
6
Table 6
Business Cycle Statistics: Matching externalities
A. Macroeconomic variables
Standard deviations
Y
C
I
K
N
Y /N
1.44
0.31
8.26
0.56
0.93
0.58
Correlations with output
Y
C
I
K
N
Y /N
1.00
0.66
0.99
0.16
0.97
0.93
B. Labor market variables
Standard deviations
Y
N
U
A
A/U
H/U
F/N
JC
JD
1.44
0.93
13.68
20.14
31.84
14.57
1.42
4.55
4.77
Correlations matrix
Y
N
U
A
Y
N
U
A
A/U
H/U
F/N
JC
JD
1.00
0.97
-0.96
0.79
0.92
0.97
-0.62
0.08
-0.31
1.00
-0.99
0.73
0.89
0.97
-0.54
-0.03
-0.20
1.00
-0.76
-0.91
-0.98
0.54
0.03
0.20
1.00
0.96
0.85
-0.85
0.48
-0.67
1.00
0.96
-0.77
0.29
-0.51
1.00
-0.67
0.17
-0.38
1.00
-0.60
0.84
1.00
-0.72
A/U
H/U
F/N
JC
7
Table 7
Business Cycle Statistics: Efficient allocation for economy with matching externalities
A. Macroeconomic variables
Standard deviations
Y
C
I
K
N
Y /N
1.15
0.28
6.60
0.45
0.48
0.71
Correlations with output
Y
C
I
K
N
Y /N
1.00
0.75
0.99
0.18
0.95
0.98
B. Labor market variables
Standard deviations
Y
N
U
A
A/U
H/U
F/N
JC
JD
1.15
0.48
4.92
5.39
9.07
5.58
0.60
2.21
2.40
Correlations matrix
Y
N
U
A
Y
N
U
A
A/U
H/U
F/N
JC
JD
1.00
0.95
-0.95
0.72
0.95
0.97
-0.74
0.16
-0.39
1.00
-1.00
0.54
0.86
0.94
-0.60
-0.04
-0.19
1.00
-0.54
-0.87
-0.94
0.60
0.04
0.19
1.00
0.89
0.77
-0.98
0.69
-0.84
1.00
0.97
-0.91
0.39
-0.60
1.00
-0.81
0.28
-0.50
1.00
-0.66
0.82
1.00
-0.73
A/U
H/U
F/N
JC
8
Table 8
Business Cycle Statistics: Efficient allocation for economy with linear matching technology
A. Macroeconomic variables
Standard deviations
Y
C
I
K
N
Y /N
1.82
0.39
10.92
0.71
1.50
0.43
Correlations with output
Y
C
I
K
N
Y /N
1.00
0.71
0.98
0.16
0.99
0.80
B. Labor market variables
Standard deviations
Y
N
U
A
A/U
H/U
F/N
JC
JD
1.82
1.50
24.80
11.39
30.55
28.34
4.16
9.07
8.37
Correlations matrix
Y
N
U
A
Y
N
U
A
A/U
H/U
F/N
JC
JD
1.00
0.99
-0.95
0.34
0.90
0.94
0.03
0.18
-0.06
1.00
-0.96
0.31
0.90
0.95
0.04
0.16
-0.04
1.00
-0.31
-0.93
-0.98
-0.04
-0.16
0.04
1.00
0.63
0.41
-0.39
0.72
-0.63
1.00
0.95
-0.12
0.40
-0.27
1.00
0.03
0.28
-0.14
1.00
-0.58
0.76
1.00
-0.70
A/U
H/U
F/N
JC
9
Table 9
Business Cycle Statistics: Reallocation shocks
A. Macroeconomic variables
Standard deviations
Y
C
I
K
N
Y /N
1.48
0.34
8.42
0.56
0.96
0.57
Correlations with output
Y
C
I
K
N
Y /N
1.00
0.72
0.99
0.17
0.98
0.94
B. Labor market variables
Standard deviations
Y
N
U
A
A/U
H/U
F/N
JC
JD
1.48
0.96
12.78
14.53
25.96
13.19
3.11
3.54
5.80
Correlations matrix
Y
N
U
A
A/U
Y
N
U
A
A/U
H/U
F/N
JC
JD
1.00
0.98
-0.97
0.80
0.93
0.97
-0.40
-0.06
-0.27
1.00
-0.99
0.78
0.92
0.98
-0.34
-0.12
-0.20
1.00
-0.81
-0.94
-0.99
0.34
0.12
0.20
1.00
0.96
0.86
-0.75
0.38
-0.68
1.00
0.97
-0.59
0.15
-0.48
1.00
-0.40
0.02
-0.30
1.00
-0.63
0.93
1.00
-0.74
H/U
F/N
JC
10
Technical Appendix for:
“Establishments Dynamics, Vacancies and Unemployment:
A Neoclassical Synthesis”∗
Marcelo Veracierto
Federal Reserve Bank of Chicago
November 25, 2009
Abstract: This is the Technical Appendix for my paper “Establishments Dynamics, Vacancies and Unemployment: A
Neoclassical Synthesis”.
∗ The
views express here do not necessarily reflect the position of the Federal Reserve Bank of Chicago or the Federal Reserve System. Address:
Federal Reserve Bank of Chicago, Research Department, 230 South LaSalle Street, Chicago, IL 60604. E-mail: mveracierto@frbchi.org. Phone:
(312) 322-5695.
1
1
Equilibrium conditions
1.1
Households’ problem
The household’s Bellman equation is:
B(κ, z, K, E, X, μ) = max
{c,i,m}
½
¾
c1−σ − 1
+ ϕm + βE [B (κ0 , z 0 , K 0 , E 0 , X 0 , μ0 ) | z]
1−σ
subject to:
c + i + ru (z, K, E, X, μ) m ≤ rk (z, K, E, X, μ) κ + Π (z, K, E, X, μ) ,
κ0
= (1 − δ) κ + i
(K 0 , E 0 , X 0 , μ0 ) = L (z, K, E, X, μ) .
(1)
(2)
(3)
Let λ (κ, z, K, E, X, μ) be the Lagrange multiplier for equation (1). The first order conditions and envelope conditions
are then the following:
−σ
c (κ, z, K, E, X, μ)
= λ (κ, z, K, E, X, μ)
ϕ = λ (κ, z, K, E, X, μ) ru (z, K, E, X, μ)
λ (κ, z, K, E, X, μ) = βE [Bκ (κ0 , z 0 , K 0 , E 0 , X 0 , μ0 ) | z]
£
¤
Bκ (κ, z, K, E, X, μ) = λ (κ, z, K, E, X, μ) 1 − δ + rk (z, K, E, X, μ)
1.2
Establishment’s problem
The establishments’ Bellman equation is the following
W (s, l, j, z, K, E, X, μ) =
max
{f,h,k,n,v}
{ez sF (n, k) + pu (z, K, E, X, μ) f − pe (z, K, E, X, μ) h
−rk (z, K, E, X, μ) k − pv (z, K, E, X, μ) v
"
#)
X
0
0 0 0 0
0
0
0
0
0
+E
q (z, K, E, X, μ, z ) W (s , l , j , z , K , E , X , μ ) Q (s, s ) | z
s0
subject to
n = l+h−f
(4)
πn l
≤ f
(5)
f
≤ l
(6)
h ≤ j
(7)
l0
= n
(8)
j0
= v
(9)
(K 0 , E 0 , X 0 , μ0 ) = L (z, K, E, X, μ) .
(10)
2
Let ξ(s, l, j, z, K, E, X, μ), α(s, l, j, z, K, E, X, μ), χ(s, l, j, z, K, E, X, μ), and η(s, l, j, z, K, E, X, μ) be the Lagrange multipliers for constraints (4)-(7), respectively. The first order conditions and envelope conditions are then the following:
ez sFk [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)] ≤ rk (z, K, E, X, μ) , (= if k(s, l, j, z, K, E, X, μ) > 0)
pu (z, K, E, X, μ) − ξ(s, l, j, z, K, E, X, μ) + α(s, l, j, z, K, E, X, μ) − χ(s, l, j, z, K, E, X, μ)
≤ 0, (= if f (s, l, j, z, K, E, X, μ) > 0)
−pe (z, K, E, X, μ) + ξ(s, l, j, z, K, E, X, μ) − η(s, l, j, z, K, E, X, μ) ≤ 0, (= if h(s, l, j, z, K, E, X, μ) > 0)
ez sFn [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)] − ξ(s, l, j, z, K, E, X, μ)
"
#
X
0
0 0 0 0
0
0
0
0
0
+E
q (z, K, E, X, μ, z ) Wl (s , l , j , z , K , E , X , μ ) Q (s, s ) | z
s0
≤ 0, (= if n(s, l, j, z, K, E, X, μ) > 0)
−pv (z, K, E, X, μ) + E
"
X
s0
q (z, K, E, X, μ, z 0 ) Wj (s0 , l0 , j 0 , z 0 , K 0 , E 0 , X 0 , μ0 ) Q (s, s0 ) | z
#
≤ 0, (= if v(s, l, j, z, K, E, X, μ) > 0)
α(s, l, j, z, K, E, X, μ) [f (s, l, j, z, K, E, X, μ) − π n l] = 0
χ(s, l, j, z, K, E, X, μ) [l − f (s, l, j, z, K, E, X, μ)] = 0
η(s, l, j, z, K, E, X, μ) [j − h(s, l, j, z, K, E, X, μ)] = 0
Wl (s, l, j, z, K, E, X, μ) = ξ(s, l, j, z, K, E, X, μ) − πn α(s, l, j, z, K, E, X, μ) + χ(s, l, j, z, K, E, X, μ)
Wj (s, l, j, z, K, E, X, μ) = η(s, l, j, z, K, E, X, μ)
1.3
Recruitment company’s problem
The Bellman equation of the recruitment company is:
R (e, x, z, K, E, X, μ) =
max {pe (z, K, E, X, μ) d + pv (z, K, E, X, μ) b
{a,b,d,u}
u
+p (z, K, E, X, μ) [x + e − d − u] + ru (z, K, E, X, μ) u − a
+E [q (z, K, E, X, μ, z 0 ) R (e0 , x0 , z 0 , K 0 , E 0 , X 0 , μ0 ) | z]}
3
subject to
d ≤ (1 − π u ) e
(11)
b = H(a, u, A, U )
e0
= G(a, u, A, U )
x0
= u − G(a, u, A, U )
A = A (z, K, E, X, μ)
U
= U (z, K, E, X, μ)
(K 0 , E 0 , X 0 , μ0 ) = L (z, K, E, X, μ) .
Let ς (e, x, z, K, E, X, μ) be the Lagrange multiplier for constraint (11). The first order conditions and envelope conditions
are then the following:
1 = pv (z, K, E, X, μ) Ha (a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ), A, U )
+E [q (z, K, E, X, μ, z 0 ) Re (e0 , x0 , z 0 , K 0 , E 0 , X 0 , μ0 ) Ga (a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ), A, U ) | z]
−E [q (z, K, E, X, μ, z 0 ) Rx (e0 , x0 , z 0 , K 0 , E 0 , X 0 , μ0 ) Ga (a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ), A, U ) | z]
0 = pv (z, K, E, X, μ) Hu (a(e, x, z, K, E, X, μ) − pu (z, K, E, X, μ) + ru (z, K, E, X, μ)
+E [q (z, K, E, X, μ, z 0 ) Re (e0 , x0 , z 0 , K 0 , E 0 , X 0 , μ0 ) Gu (a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ), A, U ) | z]
+E [q (z, K, E, X, μ, z 0 ) Rx (e0 , x0 , z 0 , K 0 , E 0 , X 0 , μ0 ) | z]
−E [q (z, K, E, X, μ, z 0 ) Rx (e0 , x0 , z 0 , K 0 , E 0 , X 0 , μ0 ) Gu (a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ), A, U ) | z]
pe (z, K, E, X, μ) − pu (z, K, E, X, μ) − ς (e, x, z, K, E, X, μ) = 0
ς (e, x, z, K, E, X, μ) [(1 − π u ) e − d(e, x, z, K, E, X, μ)] = 0
Re (e, x, z, K, E, X, μ) = pu (z, K, E, X, μ) + (1 − πu ) ς (e, x, z, K, E, X, μ)
Rx (e, x, z, K, E, X, μ) = pu (z, K, E, X, μ)
1.4
Conditions for a recursive competitive equilibrium (RCE)
The necessary and sufficient conditions for {B, W , R, c, i, m, n, k, f , h, v, a, b, d, u, A, U , L, Π, rk , ru , pu , pe , pv ,
q} to be a RCE is that there exist Lagrange multipliers λ (κ, z, K, E, X, μ), ξ(s, l, j, z, K, E, X, μ), α(s, l, j, z, K, E, X, μ),
χ(s, l, j, z, K, E, X, μ), and η(s, l, j, z, K, E, X, μ) such that equations (12)-(40) hold (equations 41 through 50 are merely
definitional).
−σ
c (κ, z, K, E, X, μ)
= λ (κ, z, K, E, X, μ)
∙
¸
¤
λ [(1 − δ) κ + i (κ, z, K, E, X, μ) , z 0 , L (z, K, E, X, μ)] £
1=E β
1 − δ + rk (z 0 , L (z, K, E, X, μ)) | z
λ (κ, z, K, E, X, μ)
ez sFk [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)] ≤ rk (z, K, E, X, μ) , (= if k(s, l, j, z, K, E, X, μ) > 0)
4
(12)
(13)
(14)
0 = ru (z, K, E, X, μ) − pu (z, K, E, X, μ)
+pv (z, K, E, X, μ) Hu (a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ), A (z, K, E, X, μ) , U (z, K, E, X, μ))
+ (1 − π u ) Gu (a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ), A (z, K, E, X, μ) , U (z, K, E, X, μ)) ×
E [q (z, K, E, X, μ, z 0 ) [pe (z 0 , L (z, K, E, X, μ)) − pu (z 0 , L (z, K, E, X, μ))] | z]
+E [q (z, K, E, X, μ, z 0 ) pu (z 0 , L (z, K, E, X, μ)) | z]
(15)
1 = pv (z, K, E, X, μ) Ha (a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ), A (z, K, E, X, μ) , U (z, K, E, X, μ))
+ (1 − π u ) Ga (a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ), A (z, K, E, X, μ) , U (z, K, E, X, μ)) ×
E [q (z, K, E, X, μ, z 0 ) [pe (z 0 , L (z, K, E, X, μ)) − pu (z 0 , L (z, K, E, X, μ))] | z]
E
"
X
s0
0
0
0
(16)
0
q (z, K, E, X, μ, z ) η (s , n(s, l, j, z, K, E, X, μ), v(s, l, j, z, K, E, X, μ), z , L (z, K, E, X, μ)) Q (s, s ) | z
#
≤ pv (z, K, E, X, μ) , (= if v(s, l, j, z, K, E, X, μ) > 0)
(17)
ez sFn [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)]
"
#
X
0
0
0
0
+E
q (z, K, E, X, μ, z ) ξ (s , n(s, l, j, z, K, E, X, μ), v(s, l, j, z, K, E, X, μ), z , L (z, K, E, X, μ)) Q (s, s ) | z
s0
−π n E
+E
"
X
"
X
s0
s0
q (z, K, E, X, μ, z 0 ) α (s0 , n(s, l, j, z, K, E, X, μ), v(s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
0
0
0
0
q (z, K, E, X, μ, z ) χ (s , n(s, l, j, z, K, E, X, μ), v(s, l, j, z, K, E, X, μ), z , L (z, K, E, X, μ)) Q (s, s ) | z
≤ ξ(s, l, j, z, K, E, X, μ), (= if n(s, l, j, z, K, E, X, μ) > 0)
−pe (z, K, E, X, μ) + ξ(s, l, j, z, K, E, X, μ) ≤ η(s, l, j, z, K, E, X, μ), (= if h(s, l, j, z, K, E, X, μ) > 0)
#
#
(18)
(19)
pu (z, K, E, X, μ) − ξ(s, l, j, z, K, E, X, μ) + α(s, l, j, z, K, E, X, μ) − χ(s, l, j, z, K, E, X, μ)
≤ 0, (= if f (s, l, j, z, K, E, X, μ) > 0)
(20)
n(s, l, j, z, K, E, X, μ) = l + h(s, l, j, z, K, E, X, μ) − f (s, l, j, z, K, E, X, μ)
(21)
h(s, l, j, z, K, E, X, μ) ≤ j
(22)
π n l ≤ f (s, l, j, z, K, E, X, μ)
(23)
f (s, l, j, z, K, E, X, μ) ≤ l
(24)
η(s, l, j, z, K, E, X, μ) [j − h(s, l, j, z, K, E, X, μ)] = 0
(25)
α(s, l, j, z, K, E, X, μ) [f (s, l, j, z, K, E, X, μ) − π n l] = 0
(26)
χ(s, l, j, z, K, E, X, μ) [l − f (s, l, j, z, K, E, X, μ)] = 0
(27)
b(e, x, z, K, E, X, μ) = H [a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ), A (z, K, E, X, μ) , U (z, K, E, X, μ)]
(28)
5
d(e, x, z, K, E, X, μ) ≤ (1 − πu ) e
(29)
0 = [pe (z, K, E, X, μ) − pu (z, K, E, X, μ)] [(1 − π u ) e − d(e, x, z, K, E, X, μ)]
(30)
ru (z, K, E, X, μ) = ϕλ (κ, z, K, E, X, μ)−1
(31)
σ
q(z, K, E, X, μ, z 0 ) = β
c (K, z, K, E, X, μ)
σ
c ((1 − δ) K + i (K, z, K, E, X, μ) , z 0 , L (z, K, E, X, μ))
c (K, z, K, E, X, μ) + i (K, z, K, E, X, μ) + a(E, X, z, K, E, X, μ)
XZ
ez sF [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)] μ (s, dl × dj)
=
(32)
(33)
s
XZ
s
XZ
k(s, l, j, z, K, E, X, μ)μ (s, dl × dj) = K
(34)
v(s, l, j, z, K, E, X, μ)μ (s, dl × dj) = b(E, X, z, K, E, X, μ)
(35)
s
u(E, X, z, K, E, X, μ) = X + E − d(E, X, z, K, E, X, μ) +
d(E, X, z, K, E, X, μ) =
XZ
s
XZ
s
f (s, l, j, z, K, E, X, μ)μ (s, dl × dj)
h(s, l, j, z, K, E, X, μ)μ (s, dl × dj)
(36)
(37)
m (K, z, K, E, X, μ) = u(E, X, z, K, E, X, μ)
(38)
A (z, K, E, X, μ) = a(E, X, z, K, E, X, μ)
(39)
U (z, K, E, X, μ) = u(E, X, z, K, E, X, μ)
(40)
1−σ
B(κ, z, K, E, X, μ) =
−1
c (κ, z, K, E, X, μ)
+ ϕm (κ, z, K, E, X, μ)
1−σ
+βE {B [(1 − δ) κ + i (κ, z, K, E, X, μ) , z 0 , L (z, K, E, X, μ)] | z}
(41)
(42)
W (s, l, j, z, K, E, X, μ) = ez sF [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)]
+pu (z, K, E, X, μ) f (s, l, j, z, K, E, X, μ) − pe (z, K, E, X, μ) h(s, l, j, z, K, E, X, μ)
+E
"
X
s0
−rk (z, K, E, X, μ) k(s, l, j, z, K, E, X, μ) − pv (z, K, E, X, μ) v(s, l, j, z, K, E, X, μ)
#
q (z, K, E, X, μ, z 0 ) W (s0 , n(s, l, j, z, K, E, X, μ), v(s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
(43)
R (e, x, z, K, E, X, μ) = pe (z, K, E, X, μ) d(e, x, z, K, E, X, μ) + pv (z, K, E, X, μ) b(e, x, z, K, E, X, μ)
+pu (z, K, E, X, μ) [x + e − d(e, x, z, K, E, X, μ) − u(e, x, z, K, E, X, μ)]
⎡
⎛
⎜
⎢
⎜
⎢
⎜
⎢
⎜
⎢
⎜
⎢
0
+E ⎢q (z, K, E, X, μ, z ) R ⎜
⎜
⎢
⎜
⎢
⎜
⎢
⎝
⎣
+ru (z, K, E, X, μ) u(e, x, z, K, E, X, μ) − a(e, x, z, K, E, X, μ)
⎤
⎡
⎞ ⎤
a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ),
⎟ ⎥
⎦,
G⎣
⎟ ⎥
⎟ ⎥
A (z, K, E, X, μ) , U (z, K, E, X, μ)
⎤ ⎟ ⎥
⎡
⎟ ⎥
⎟ | z⎥
a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ),
⎦, ⎟ ⎥
u(e, x, z, K, E, X, μ) − G ⎣
⎟ ⎥
⎟ ⎥
A (z, K, E, X, μ) , U (z, K, E, X, μ)
⎠ ⎦
z 0 , L (z, K, E, X, μ)
6
(44)
(K 0 , E 0 , X 0 , μ0 ) = L (z, K, E, X, μ)
(45)
K 0 = (1 − δ) K + i (K, z, K, E, X, μ)
(46)
E 0 = G [a(E, X, z, K, E, X, μ), u(E, X, z, K, E, X, μ), A (z, K, E, X, μ) , U (z, K, E, X, μ)]
(47)
is given by:
X 0 = u(E, X, z, K, E, X, μ) − G [a(E, X, z, K, E, X, μ), u(E, X, z, K, E, X, μ), A (z, K, E, X, μ) , U (z, K, E, X, μ)]
0
0
μ (s , L × J ) =
where
XZ
s
B(s,L×J )
(48)
Q (s, s0 ) μ (s, dl × dj) + ψ (s0 ) I (L × J )
(49)
B (s, L × J ) = {(l, j) : n(s, l, j, z, K, E, X, μ) ∈ L and v(s, l, j, z, K, E, X, μ) ∈ J }
1.5
(50)
Equilibrium allocations and prices
Evaluate equations (12)-(40) at (κ, e, X) = (K, E, X) and eliminate m, a, b, d and u to get:
ru (z, K, E, X, μ) = ϕc (K, z, K, E, X, μ)
σ
(51)
£
£
¤ ¤
1 = E q (z, K, E, X, μ, z 0 ) 1 − δ + rk (z 0 , L (z, K, E, X, μ)) | z
(52)
pu (z, K, E, X, μ) = ru (z, K, E, X, μ)
+pv (z, K, E, X, μ) Hu (A (z, K, E, X, μ) , U (z, K, E, X, μ) , A (z, K, E, X, μ) , U (z, K, E, X, μ))
+E [q (z, K, E, X, μ, z 0 ) pu (z 0 , L (z, K, E, X, μ)) | z]
+ (1 − π u ) Gu (A (z, K, E, X, μ) , U (z, K, E, X, μ) , A (z, K, E, X, μ) , U (z, K, E, X, μ)) ×
E [q (z, K, E, X, μ, z 0 ) [pe (z 0 , L (z, K, E, X, μ)) − pu (z 0 , L (z, K, E, X, μ))] | z]
(53)
1 = pv (z, K, E, X, μ) Ha (A (z, K, E, X, μ) , U (z, K, E, X, μ) , A (z, K, E, X, μ) , U (z, K, E, X, μ))
+ (1 − π u ) Ga (A (z, K, E, X, μ) , U (z, K, E, X, μ) , A (z, K, E, X, μ) , U (z, K, E, X, μ)) ×
E [q (z, K, E, X, μ, z 0 ) [pe (z 0 , L (z, K, E, X, μ)) − pu (z 0 , L (z, K, E, X, μ))] | z]
(54)
ez sFk [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)] ≤ rk (z, K, E, X, μ) , (= if k(s, l, j, z, K, E, X, μ) > 0)
E
"
X
s0
0
0
0
0
q (z, K, E, X, μ, z ) η (s , n(s, l, j, z, K, E, X, μ), v(s, l, j, z, K, E, X, μ), z , L (z, K, E, X, μ)) Q (s, s ) | z
≤ pv (z, K, E, X, μ) , (= if v(s, l, j, z, K, E, X, μ) > 0)
7
(55)
#
(56)
ez sFn [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)]
"
#
X
0
0
0
0
+E
q (z, K, E, X, μ, z ) ξ (s , n(s, l, j, z, K, E, X, μ), v(s, l, j, z, K, E, X, μ), z , L (z, K, E, X, μ)) Q (s, s ) | z
s0
−π n E
+E
"
X
"
X
s0
s0
0
0
0
0
q (z, K, E, X, μ, z ) α (s , n(s, l, j, z, K, E, X, μ), v(s, l, j, z, K, E, X, μ), z , L (z, K, E, X, μ)) Q (s, s ) | z
q (z, K, E, X, μ, z 0 ) χ (s0 , n(s, l, j, z, K, E, X, μ), v(s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
≤ ξ(s, l, j, z, K, E, X, μ), (= if n(s, l, j, z, K, E, X, μ) > 0)
#
#
(57)
−pe (z, K, E, X, μ) + ξ(s, l, j, z, K, E, X, μ) ≤ η(s, l, j, z, K, E, X, μ), (= if h(s, l, j, z, K, E, X, μ) > 0)
(58)
pu (z, K, E, X, μ) − ξ(s, l, j, z, K, E, X, μ) + α(s, l, j, z, K, E, X, μ) − χ(s, l, j, z, K, E, X, μ)
≤ 0, (= if f (s, l, j, z, K, E, X, μ) > 0)
(59)
n(s, l, j, z, K, E, X, μ) = l + h(s, l, j, z, K, E, X, μ) − f (s, l, j, z, K, E, X, μ)
(60)
h(s, l, j, z, K, E, X, μ) ≤ j
(61)
π n l ≤ f (s, l, j, z, K, E, X, μ)
(62)
f (s, l, j, z, K, E, X, μ) ≤ l
(63)
η(s, l, j, z, K, E, X, μ) [j − h(s, l, j, z, K, E, X, μ)] = 0
(64)
α(s, l, j, z, K, E, X, μ) [f (s, l, j, z, K, E, X, μ) − π n l] = 0
(65)
χ(s, l, j, z, K, E, X, μ) [l − f (s, l, j, z, K, E, X, μ)] = 0
XZ
h(s, l, j, z, K, E, X, μ)μ (s, dl × dj) ≤ (1 − π u ) E
(66)
s
e
"
u
0 = [p (z, K, E, X, μ) − p (z, K, E, X, μ)] (1 − πu ) E −
q(z, K, E, X, μ, z 0 ) = β
XZ
s
h(s, l, j, z, K, E, X, μ)μ (s, dl × dj)
(67)
#
(68)
σ
c (K, z, K, E, X, μ)
σ
c ((1 − δ) K + i (K, z, K, E, X, μ) , z 0 , L (z, K, E, X, μ))
c (K, z, K, E, X, μ) + i (K, z, K, E, X, μ) + A (z, K, E, X, μ)
XZ
=
ez sF [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)] μ (s, dl × dj)
(69)
(70)
(71)
s
XZ
s
XZ
s
k(s, l, j, z, K, E, X, μ)μ (s, dl × dj) = K
(72)
v(s, l, j, z, K, E, X, μ)μ (s, dl × dj) = H [A (z, K, E, X, μ) , U (z, K, E, X, μ) , A (z, K, E, X, μ) , U (z, K, E, X, μ)] (73)
U (z, K, E, X, μ) = X + E −
XZ
s
h(s, l, j, z, K, E, X, μ)μ (s, dl × dj) +
XZ
s
(K 0 , E 0 , X 0 , μ0 ) = L (z, K, E, X, μ)
8
f (s, l, j, z, K, E, X, μ)μ (s, dl × dj)
(74)
(75)
is given by:
0
0
K 0 = (1 − δ) K + i (K, z, K, E, X, μ)
(76)
E 0 = G [A (z, K, E, X, μ) , U (z, K, E, X, μ) , A (z, K, E, X, μ) , U (z, K, E, X, μ)]
(77)
X 0 = U (z, K, E, X, μ) − G [A (z, K, E, X, μ) , U (z, K, E, X, μ) , A (z, K, E, X, μ) , U (z, K, E, X, μ)]
(78)
μ (s , L × J ) =
2
XZ
s
{(l,j):n(s,l,j,z,K,E,X,μ)∈L and v(s,l,j,z,K,E,X,μ)∈J }
Q (s, s0 ) μ (s, dl × dj) + ψ (s0 ) I (L × J )
(79)
Characterization of establishments’ decision rules
It will be convenient to truncate the establishments’ problem to a finite horizon T . The truncated problem is given by
W t,T (s, l, j, z, K, E, X, μ) =
max
{f,h,k,n,v}
{ez sF (n, k) + pu (z, K, E, X, μ) f − pe (z, K, E, X, μ) h
−rk (z, K, E, X, μ) k − pv (z, K, E, X, μ) v
"
#)
X
0
t+1,T
0 0 0 0
0
0
0
0
0
+E
q (z, K, E, X, μ, z ) W
(s , l , j , z , K , E , X , μ ) Q (s, s ) | z
s0
subject to
n ≤ l+h−f
(80)
πnl
≤ f
(81)
f
≤ l
(82)
h ≤ j
(83)
l0
= n
(84)
j0
= v
(85)
(K 0 , E 0 , X 0 , μ0 ) = L (z, K, E, X, μ)
for t = 0, 1, ..., T , where
W T +1,T (s, l, j, z, K, E, X, μ) = 0.
In what follows, it will be assumed that
E [q (z, K, E, X, μ, z 0 ) pu (z 0 , L (z, K, E, X, μ)) | z] ≤ pu (z, K, E, X, μ) ,
and that
pu (z, K, E, X, μ) ≤ pe (z, K, E, X, μ) ,
since these are properties that will be satisfied in equilibrium.
9
(86)
Let ξ t,T (s, l, j, z, K, E, X, μ), αt,T (s, l, j, z, K, E, X, μ), χt,T (s, l, j, z, K, E, X, μ), and η t,T (s, l, j, z, K, E, X, μ) be the
(non-negative) Lagrange multipliers for constraints (80)-(83), respectively. From the first-order and envelope conditions we
get for t = 0, 1, ..., T, the following.
ez sFk [nt,T (s, l, j, z, K, E, X, μ), kt,T (s, l, j, z, K, E, X, μ)] ≤ rk (z, K, E, X, μ) , (= if kt,T (s, l, j, z, K, E, X, μ) > 0)
nt,T (s, l, j, z, K, E, X, μ) = l + ht,T (s, l, j, z, K, E, X, μ) − ft,T (s, l, j, z, K, E, X, μ)
(87)
(88)
pu (z, K, E, X, μ) − ξ t,T (s, l, j, z, K, E, X, μ) + αt,T (s, l, j, z, K, E, X, μ) − χt,T (s, l, j, z, K, E, X, μ)
≤ 0, (= if ft,T (s, l, j, z, K, E, X, μ) > 0)
(89)
−pe (z, K, E, X, μ) + ξ t,T (s, l, j, z, K, E, X, μ) − η t,T (s, l, j, z, K, E, X, μ) ≤ 0, (= if ht,T (s, l, j, z, K, E, X, μ) > 0)
ez sFn [nt,T (s, l, j, z, K, E, X, μ), kt,T (s, l, j, z, K, E, X, μ)] − ξ t,T (s, l, j, z, K, E, X, μ)
(
X
q (z, K, E, X, μ, z 0 ) ×
+E
s0
Wlt+1,T
(s0 , nt,T (s, l, j, z, K, E, X, μ), vt,T (s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
≤ 0, (= if nt,T (s, l, j, z, K, E, X, μ) > 0)
E
(
X
s0
q (z, K, E, X, μ, z 0 ) ×
Wjt+1,T (s0 , nt,T (s, l, j, z, K, E, X, μ), vt,T (s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
≤ pv (z, K, E, X, μ) , (= if vt,T (s, l, j, z, K, E, X, μ) > 0)
(90)
o
(91)
o
(92)
αt,T (s, l, j, z, K, E, X, μ) [ft,T (s, l, j, z, K, E, X, μ) − π n l] = 0
(93)
χt,T (s, l, j, z, K, E, X, μ) [l − ft,T (s, l, j, z, K, E, X, μ)] = 0
(94)
η t,T (s, l, j, z, K, E, X, μ) [j − ht,T (s, l, j, z, K, E, X, μ)] = 0
(95)
Wlt,T (s, l, j, z, K, E, X, μ) = ξ t,T (s, l, j, z, K, E, X, μ) − π n αt,T (s, l, j, z, K, E, X, μ) + χt,T (s, l, j, z, K, E, X, μ)
(96)
Wjt,T (s, l, j, z, K, E, X, μ) = η t,T (s, l, j, z, K, E, X, μ)
(97)
WlT +1,T (s, l, j, z, K, E, X, μ) = 0
(98)
WjT +1,T (s, l, j, z, K, E, X, μ) = 0
(99)
©
ªT
Lemma 1 Let ξ t,T (0, l, j, z, K, E, X, μ) t=0 be any sequence of functions satisfying that
E [q (z, K, E, X, μ, z 0 ) pu (z 0 , L (z, K, E, X, μ)) | z] ≤ ξ t,T (0, l, j, z, K, E, X, μ) ≤ pu (z, K, E, X, μ) , for t < T
10
and that
ξ T,T (0, l, j, z, K, E, X, μ) ≤ pu (z, K, E, X, μ)
©
ªT
Then, ξ t,T (0, l, j, z, K, E, X, μ) t=0 together with
nt,T (0, l, j, z, K, E, X, μ) = 0
kt,T (0, l, j, z, K, E, X, μ) = 0
ft,T (0, l, j, z, K, E, X, μ) = l
ht,T (0, l, j, z, K, E, X, μ) = 0
vt,T (0, l, j, z, K, E, X, μ) = 0
αt,T (0, l, j, z, K, E, X, μ) = 0
η t,T (0, l, j, z, K, E, X, μ) = 0
χt,T (0, l, j, z, K, E, X, μ) = pu (z, K, E, X, μ) − ξ t,T (0, l, j, z, K, E, X, μ)
Wlt,T (0, l, j, z, K, E, X, μ) = pu (z, K, E, X, μ)
Wjt,T (0, l, j, z, K, E, X, μ) = 0
for t = 0, ..., T , satisfy equations (87)-(99).
Proof. It is a straightforward verification.
To simplify the subsequent analysis it will be convenient to define two functions k̂ (n, z, s, r) and F̂n (n, z, s, r) as follows.
For n > 0 and s > 0, they are given by:
h
i
ez sFk n, k̂ (n, z, s, r) = r,
h
i
F̂n (n, z, s, r) = ez sFn n, k̂ (n, z, s, r) .
Observe that F̂n (n, z, s, r) is strictly decreasing, that limn→∞ F̂n (n, z, s, r) = 0, and that limn→0 F̂n (n, z, s, r) = +∞.
Lemma 2 Suppose that s > 0 and that l + j > 0. Then,
£
¤
kt,T (s, l, j, z, K, E, X, μ) = k̂ nt,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
nt,T (s, l, j, z, K, E, X, μ) = l + ht,T (s, l, j, z, K, E, X, μ) − ft,T (s, l, j, z, K, E, X, μ)
(100)
(101)
pu (z, K, E, X, μ) − ξ t,T (s, l, j, z, K, E, X, μ) + αt,T (s, l, j, z, K, E, X, μ) − χt,T (s, l, j, z, K, E, X, μ)
≤ 0, (= if ft,T (s, l, j, z, K, E, X, μ) > 0)
(102)
−pe (z, K, E, X, μ) + ξ t,T (s, l, j, z, K, E, X, μ) − η t,T (s, l, j, z, K, E, X, μ) ≤ 0, (= if ht,T (s, l, j, z, K, E, X, μ) > 0) (103)
11
¢
¡
F̂n nt,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
(
X
+E
q (z, K, E, X, μ, z 0 ) ×
s0 >0
ξ t+1,T (s0 , nt,T (s, l, j, z, K, E, X, μ), vt,T (s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 )
−πn E
(
X
s0 >0
0
q (z, K, E, X, μ, z 0 ) ×
|z
ª
αt+1,T (s , nt,T (s, l, j, z, K, E, X, μ), vt,T (s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z}
(
X
+E
q (z, K, E, X, μ, z 0 ) ×
s0 >0
χt+1,T (s0 , nt,T (s, l, j, z, K, E, X, μ), vt,T (s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 )
|z
+Q (s, 0) E [q (z, K, E, X, μ, z 0 ) pu (z 0 , L (z, K, E, X, μ)) | z]
ª
= ξ t,T (s, l, j, z, K, E, X, μ), for t < T
E
(
(104)
¢
¡
ξ T,T (s, l, j, z, K, E, X, μ) = F̂n nT,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
X
s0 >0
(105)
q (z, K, E, X, μ, z 0 ) ×
η t+1,T (s0 , nt,T (s, l, j, z, K, E, X, μ), vt,T (s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
≤ pv (z, K, E, X, μ) , (= if vt,T (s, l, j, z, K, E, X, μ) > 0) , for t < T
ª
(106)
vT,T (s, l, j, z, K, E, X, μ) = 0
(107)
αt,T (s, l, j, z, K, E, X, μ) [ft,T (s, l, j, z, K, E, X, μ) − π n l] = 0
(108)
χt,T (s, l, j, z, K, E, X, μ) [l − ft,T (s, l, j, z, K, E, X, μ)] = 0
(109)
η t,T (s, l, j, z, K, E, X, μ) [j − ht,T (s, l, j, z, K, E, X, μ)] = 0.
(110)
Moreover, there is no loss of generality in assuming that
©
ª
η t,T (s, l, j, z, K, E, X, μ) = max ξ t,T (s, l, j, z, K, E, X, μ) − pe (z, K, E, X, μ) , 0
(111)
χt,T (s, l, j, z, K, E, X, μ) = 0
(112)
αt,T (s, 0, j, z, K, E, X, μ) = ξ t,T (s, 0, j, z, K, E, X, μ) − pe (z, K, E, X, μ)
(113)
Proof. Equations (99)-(110) follow from equations (87)-(99), Lemma 1 and the fact that F satisfies the Inada conditions
(and therefore that nt,T (s, l, j, z, K, E, X, μ) > 0 and kt,T (s, l, j, z, K, E, X, μ) > 0).
Need to show that η t,T (s, l, j, z, K, E, X, μ) can be restricted as in equation (111).
First, consider the case in which j > 0.
If ht,T (s, l, j, z, K, E, X, μ) > 0, from equation (103) we have that
ξ t,T (s, l, j, z, K, E, X, μ) − pe (z, K, E, X, μ) = η t,T (s, l, j, z, K, E, X, μ) ≥ 0,
where the inequality follows from the fact that η t,T (s, l, j, z, K, E, X, μ) is a Lagrange multiplier.
12
If ht,T (s, l, j, z, K, E, X, μ) = 0, from equations (103) and (110) we have that:
ξ t,T (s, l, j, z, K, E, X, μ) − pe (z, K, E, X, μ) ≤ η t,T (s, l, j, z, K, E, X, μ) = 0.
Hence, when j > 0, η t,T (s, l, j, z, K, E, X, μ) must satisfy equation (111).
When j = 0, equation (110) imposes no restriction on η t,T (s, l, 0, z, K, E, X, μ). The only restrictions (from equations
©
ªT
(103) and (106)) are that η t,T (s, l, 0, z, K, E, X, μ) t=0 must satisfy for t ≤ T that
ξ t,T (s, l, 0, z, K, E, X, μ) − pe (z, K, E, X, μ) ≤ η t,T (s, l, 0, z, K, E, X, μ),
and for t < T that
E
(
X
s0 >0
q (z, K, E, X, μ, z 0 ) ×
η t+1,T (s0 , nt,T (s, l, j, z, K, E, X, μ), 0, z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
ª
≤ pv (z, K, E, X, μ) , for (s, l, j, z, K, E, X, μ) such that vt,T (s, l, j, z, K, E, X, μ) = 0.
Thus there is no loss of generality in restricting η t,T (s, l, 0, z, K, E, X, μ) as in equation (111).
Finally, we need to show that χt,T (s, l, j, z, K, E, X, μ) can be restricted as in equation (112).
If l > 0, equation (112) must hold because of equation (109) and because F satisfies the Inada conditions (and therefore
that nt,T (s, l, j, z, K, E, X, μ) > 0 and kt,T (s, l, j, z, K, E, X, μ) > 0).
If l = 0, equation (109) imposes no restriction on χt,T (s, l, j, z, K, E, X, μ) and equation (108) imposes no restriction
on αt,T (s, l, j, z, K, E, X, μ). Observe that equation (102) is satisfied under equations (112) and (113). Also, observe that
the variables
αt+1,T (s0 , nt,T (s, l, j, z, K, E, X, μ), vt,T (s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ))
and
χt+1,T (s0 , nt,T (s, l, j, z, K, E, X, μ), vt,T (s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ))
that enter equation (104) have nt,T (s, l, j, z, K, E, X, μ) > 0 (because F satisfies the Inada conditions) and therefore equations (112) and (113) do not apply to them.
Thus there is no loss of generality in assuming that equations (112) and (113) hold.
Lemma 3 Suppose that s > 0 and that l + j > 0. Then,
£
¤
kt,T (s, l, j, z, K, E, X, μ) = k̂ nt,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
(114)
nt,T (s, l, j, z, K, E, X, μ) = l + ht,T (s, l, j, z, K, E, X, μ) − ft,T (s, l, j, z, K, E, X, μ)
(115)
pe (z, K, E, X, μ) ≤ ξ t,T (s, l, j, z, K, E, X, μ), if ht,T (s, l, j, z, K, E, X, μ) > 0
(116)
13
¢
¡
F̂n nt,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
(
X
+ (1 − π n ) E
q (z, K, E, X, μ, z 0 ) ×
s0 >0
0
ξ t+1,T (s , nt,T (s, l, j, z, K, E, X, μ), vt,T (s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
"
#
X
0
u
0
0
q (z, K, E, X, μ, z ) p (z , L (z, K, E, X, μ)) Q (s, s ) | z
+πn E
ª
s0 >0
+Q (s, 0) E [q (z, K, E, X, μ, z 0 ) pu (z 0 , L (z, K, E, X, μ)) | z]
= ξ t,T (s, l, j, z, K, E, X, μ), for t < T
(117)
¢
¡
ξ T,T (s, l, j, z, K, E, X, μ) = F̂n nT,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
E
(
X
s0 >0
(118)
q (z, K, E, X, μ, z 0 ) ×
max[ξ t+1,T (s0 , nt,T (s, l, j, z, K, E, X, μ), vt,T (s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ))
−pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z}
≤ pv (z, K, E, X, μ) , (= if vt,T (s, l, j, z, K, E, X, μ) > 0) , for t < T
(119)
vT,T (s, l, j, z, K, E, X, μ) = 0
(120)
ξ t,T (s, l, j, z, K, E, X, μ) ≥ pu (z, K, E, X, μ)
(121)
£
¤
ξ t,T (s, l, j, z, K, E, X, μ) − pu (z, K, E, X, μ) [ft,T (s, l, j, z, K, E, X, μ) − πn l] = 0
£
¤
max ξ t,T (s, l, j, z, K, E, X, μ) − pe (z, K, E, X, μ) , 0 [j − ht,T (s, l, j, z, K, E, X, μ)] = 0
(122)
(123)
Moreover, there is no loss of generality in assuming that
ht,T (s, l, j, z, K, E, X, μ) [ft,T (s, l, j, z, K, E, X, μ) − πn l] = 0.
Proof. Equations (114)-(123) are a straigthward consequence of Lemma 2.
Need to show that ht,T (s, l, j, z, K, E, X, μ) and ft,T (s, l, j, z, K, E, X, μ) can be restricted as in equation (124).
First consider the case in which pe (z, K, E, X, μ) > pu (z, K, E, X, μ) .
Suppose that ht,T (s, l, j, z, K, E, X, μ) > 0 and that ft,T (s, l, j, z, K, E, X, μ) > π n l.
From equations (116) and (122)
pe (z, K, E, X, μ) ≤ ξ t,T (s, l, j, z, K, E, X, μ) = pu (z, K, E, X, μ) ,
which is a contradiction. Hence, when pe (z, K, E, X, μ) > pu (z, K, E, X, μ), equation (124) must hold.
Now consider the case in which pe (z, K, E, X, μ) = pu (z, K, E, X, μ).
Suppose that ht,T (s, l, j, z, K, E, X, μ) > 0 and that ft,T (s, l, j, z, K, E, X, μ) > π n l.
Observe, from equation (115), that
nt,T (s, l, j, z, K, E, X, μ) = l + ht,T (s, l, j, z, K, E, X, μ) − ft,T (s, l, j, z, K, E, X, μ)
14
(124)
and, from equation (122), that:
pu (z, K, E, X, μ) = pe (z, K, E, X, μ) = ξ t,T (s, l, j, z, K, E, X, μ)
1) Suppose that ht,T (s, l, j, z, K, E, X, μ) − ft,T (s, l, j, z, K, E, X, μ) ≥ −π n l.
Let
ĥt,T (s, l, j, z, K, E, X, μ) = nt,T (s, l, j, z, K, E, X, μ) − l + πn l
fˆt,T (s, l, j, z, K, E, X, μ) = π n l
Observe that
l + ĥt,T (s, l, j, z, K, E, X, μ) − fˆt,T (s, l, j, z, K, E, X, μ) = nt,T (s, l, j, z, K, E, X, μ),
that
ĥt,T (s, l, j, z, K, E, X, μ) = nt,T (s, l, j, z, K, E, X, μ) − l + π n l
< nt,T (s, l, j, z, K, E, X, μ) − l + ft,T (s, l, j, z, K, E, X, μ)
= ht,T (s, l, j, z, K, E, X, μ) ≤ j
and that
ĥt,T (s, l, j, z, K, E, X, μ) = nt,T (s, l, j, z, K, E, X, μ) − l + π n l
= l + ht,T (s, l, j, z, K, E, X, μ) − ft,T (s, l, j, z, K, E, X, μ) − l + π n l
≥ 0.
From equation (125) we then know that equations (114)-(124) hold.
2) Suppose that ht,T (s, l, j, z, K, E, X, μ) − ft,T (s, l, j, z, K, E, X, μ) ≤ −π n l.
Let
ĥt,T (s, l, j, z, K, E, X, μ) = 0
fˆt,T (s, l, j, z, K, E, X, μ) = l − nt,T (s, l, j, z, K, E, X, μ).
Observe that
l + ĥt,T (s, l, j, z, K, E, X, μ) − fˆt,T (s, l, j, z, K, E, X, μ) = nt,T (s, l, j, z, K, E, X, μ),
that
fˆt,T (s, l, j, z, K, E, X, μ) = l − nt,T (s, l, j, z, K, E, X, μ)
= ft,T (s, l, j, z, K, E, X, μ) − ht,T (s, l, j, z, K, E, X, μ) ≥ π n l
and that
fˆt,T (s, l, j, z, K, E, X, μ) = l − nt,T (s, l, j, z, K, E, X, μ) < l.
From equation (125) we then know that equations (114)-(124) hold.
Lemma 4 considers the case in which t = T .
15
(125)
Lemma 4 Suppose that s > 0, l > 0 and j = 0. Then,
n £
o
¤
ξ T,T (s, l, j, z, K, E, X, μ) = max F̂n (1 − πn ) l, z, s, rk (z, K, E, X, μ) , pu (z, K, E, X, μ)
Proof. Since j = 0, hT,T (s, l, j, z, K, E, X, μ) = 0 and equations (116) and (123) impose no restrictions on ξ T,T (s, l, j, z, K, E,
X, μ). Moreover,
nT,T (s, l, j, z, K, E, X, μ) = l − fT,T (s, l, j, z, K, E, X, μ)
(126)
¤
£
1) Consider the case that F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) ≥ pu (z, K, E, X, μ).
Suppose that fT,T (s, l, j, z, K, E, X, μ) > π n l. Then, from equations (122), (118), (126), and the fact that F̂n is strictly
decreasing in its first argument, we have that:
pu (z, K, E, X, μ) = ξ T,T (s, l, j, z, K, E, X, μ)
¤
£
= F̂n l − fT,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
¤
£
> F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) ,
which is a contradiction. Then, fT,T (s, l, j, z, K, E, X, μ) = π n l and
¤
£
ξ T,T (s, l, j, z, K, E, X, μ) = F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) ≥ pu (z, K, E, X, μ) .
¤
£
2) Consider the case that F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) < pu (z, K, E, X, μ).
Suppose that fT,T (s, l, j, z, K, E, X, μ) = πn l. Then, from equations (118) and (126),
¤
£
ξ T,T (s, l, j, z, K, E, X, μ) = F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) < pu (z, K, E, X, μ) .
But this contradicts equation (121).
Hence, fT,T (s, l, j, z, K, E, X, μ) > π n l and, from equation (122),
¤
£
ξ T,T (s, l, j, z, K, E, X, μ) = pu (z, K, E, X, μ) > F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) .
Lemma 5 Suppose that s > 0, l = 0 and j > 0. Then,
n £
o
¤
ξ T,T (s, l, j, z, K, E, X, μ) = max F̂n j, z, s, rk (z, K, E, X, μ) , pe (z, K, E, X, μ)
Proof. Since l = 0, fT,T (s, l, j, z, K, E, X, μ) = 0 and equation (122) imposes no restriction on ξ T,T (s, l, j, z, K, E, X, μ).
Moreover,
nT,T (s, l, j, z, K, E, X, μ) = hT,T (s, l, j, z, K, E, X, μ) > 0
because F satisfies the Inada conditions.
¤
£
1) Consider the case that F̂n j, z, s, rk (z, K, E, X, μ) ≥ pe (z, K, E, X, μ) .
16
(127)
Suppose that hT,T (s, l, j, z, K, E, X, μ) < j. Then, from equations (123), (118), (127), (116) and the fact that F̂n is
strictly decreasing in its first argument, we have that
pe (z, K, E, X, μ) = ξ T,T (s, l, j, z, K, E, X, μ)
¤
£
= F̂n hT,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
¤
£
> F̂n j, z, s, rk (z, K, E, X, μ) ,
which is a contradiction. Then, hT,T (s, l, j, z, K, E, X, μ) = j and
¤
£
ξ T,T (s, l, j, z, K, E, X, μ) = F̂n j, z, s, rk (z, K, E, X, μ) ≥ pe (z, K, E, X, μ) .
¤
£
2) Consider the case that F̂n j, z, s, rk (z, K, E, X, μ) < pe (z, K, E, X, μ) .
Suppose that hT,T (s, l, j, z, K, E, X, μ) = j. Then, from equations (127) and (118), we have that
£
¤
ξ T,T (s, l, j, z, K, E, X, μ) = F̂n j, z, s, rk (z, K, E, X, μ) < pe (z, K, E, X, μ) .
But this contradicts equation (116).
Hence, hT,T (s, l, j, z, K, E, X, μ) < j and from equations (116) and (123):
¤
£
ξ T,T (s, l, j, z, K, E, X, μ) = pe (z, K, E, X, μ) > F̂n j, z, s, rk (z, K, E, X, μ) .
Lemma 6 Suppose that s > 0, l > 0 and j > 0. Then,
⎧
o ⎫
n £
¤
⎨ max F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) , pe (z, K, E, X, μ) , ⎬
n
o
ξ T,T (s, l, j, z, K, E, X, μ) = min
⎭
⎩ max F̂ £(1 − π ) l, z, s, rk (z, K, E, X, μ)¤ , pu (z, K, E, X, μ)
n
n
Proof. a) First consider the case in which
n £
o
¤
max F̂n (1 − πn ) l + j, z, s, rk (z, K, E, X, μ) , pe (z, K, E, X, μ)
n £
o
¤
≥ max F̂n (1 − πn ) l, z, s, rk (z, K, E, X, μ) , pu (z, K, E, X, μ)
Suppose that hT,T (s, l, j, z, K, E, X, μ) > 0. Then, from Lemma 3, fT,T (s, l, j, z, K, E, X, μ) = π n l.
From equations (116), (118) and the fact that F̂n is strictly decreasing in its first argument, we have that
and that
pe (z, K, E, X, μ) ≤ ξ T,T (s, l, j, z, K, E, X, μ)
¢
¡
= F̂n (1 − π n ) l + hT,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
¢
¡
< F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ)
¤
¢
£
¡
F̂n (1 − πn ) l + j, z, s, rk (z, K, E, X, μ) < F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) ,
which contradict equation (128).
17
(128)
Thus, hT,T (s, l, j, z, K, E, X, μ) = 0 and equation (126) holds.
Considering the two cases in the proof of Lemma 4 we conclude that.
n £
o
¤
ξ T,T (s, l, j, z, K, E, X, μ) = max F̂n (1 − πn ) l, z, s, rk (z, K, E, X, μ) , pu (z, K, E, X, μ)
n £
o
¤
≤ max F̂n (1 − πn ) l + j, z, s, rk (z, K, E, X, μ) , pe (z, K, E, X, μ) .
b) Consider the case in which
n £
o
¤
max F̂n (1 − πn ) l + j, z, s, rk (z, K, E, X, μ) , pe (z, K, E, X, μ)
n £
o
¤
< max F̂n (1 − πn ) l, z, s, rk (z, K, E, X, μ) , pu (z, K, E, X, μ)
(129)
Suppose that fT,T (s, l, j, z, K, E, X, μ) > πn l. Then, from Lemma 3, hT,T (s, l, j, z, K, E, X, μ) = 0.
From equations (122) (118), the fact that pe (z, K, E, X, μ) ≥ pu (z, K, E, X, μ) and the fact that F̂n is strictly decreasing
in its first argument, we have that
pe (z, K, E, X, μ) ≥ pu (z, K, E, X, μ)
which contradict equation (129).
= ξ T,T (s, l, j, z, K, E, X, μ)
¢
¡
= F̂n l − fT,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
¢
¡
> F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ)
¢
¡
> F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) ,
Thus, fT,T (s, l, j, z, K, E, X, μ) = π n l and
nT,T (s, l, j, z, K, E, X, μ) = (1 − π n ) l + hT,T (s, l, j, z, K, E, X, μ)
Suppose that hT,T (s, l, j, z, K, E, X, μ) = 0.
Then, from equations (123) and (118),
pe (z, K, E, X, μ) ≥ ξ T,T (s, l, j, z, K, E, X, μ)
¡
¢
= F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ)
¡
¢
> F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ)
which contradicts equation (129). Hence, hT,T (s, l, j, z, K, E, X, μ) > 0
£
¤
b.1) Consider the case that F̂n (1 − πn ) l + j, z, s, rk (z, K, E, X, μ) ≥ pe (z, K, E, X, μ) .
Suppose that hT,T (s, l, j, z, K, E, X, μ) < j. Then, from equations (116), (123), (118), and the fact that F̂n is strictly
decreasing in its first argument, we have that
pe (z, K, E, X, μ) = ξ T,T (s, l, j, z, K, E, X, μ)
¤
£
= F̂n (1 − π n ) l + hT,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
¤
£
> F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) ,
18
which contradicts equation (129). Then, hT,T (s, l, j, z, K, E, X, μ) = j and
¤
£
ξ T,T (s, l, j, z, K, E, X, μ) = F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) ≥ pe (z, K, E, X, μ) .
¤
£
b.2) Consider the case that F̂n (1 − πn ) l + j, z, s, rk (z, K, E, X, μ) < pe (z, K, E, X, μ) .
Suppose that hT,T (s, l, j, z, K, E, X, μ) = j. Then, from equation (118) we have that
¤
£
ξ T,T (s, l, j, z, K, E, X, μ) = F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) < pe (z, K, E, X, μ) .
But this contradicts equation (116).
Hence, hT,T (s, l, j, z, K, E, X, μ) < j and from equations (116) and (123):
¤
£
ξ T,T (s, l, j, z, K, E, X, μ) = pe (z, K, E, X, μ) > F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) .
In case b) we thus conclude that
n £
o
¤
ξ T,T (s, l, j, z, K, E, X, μ) = max F̂n (1 − πn ) l + j, z, s, rk (z, K, E, X, μ) , pe (z, K, E, X, μ)
n £
o
¤
< max F̂n (1 − πn ) l, z, s, rk (z, K, E, X, μ) , pu (z, K, E, X, μ)
Lemma 7 Suppose that s > 0, l + j > 0. Then,
⎧
o ⎫
n £
¤
⎨ max F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) , pe (z, K, E, X, μ) , ⎬
n
o
ξ T,T (s, l, j, z, K, E, X, μ) = min
⎭
⎩ max F̂ £(1 − π ) l, z, s, rk (z, K, E, X, μ)¤ , pu (z, K, E, X, μ)
n
n
Proof. If follows from Lemmas 4, 5, and 6.
The following assumption will be helpful in stating subsequent Lemmas.
Assumption 3: For every s > 0, l + j > 0, ξ t,T (s, l, j, z, K, E, X, μ) is given by
⎫
⎧
⎨ max ©Λ £(1 − π ) l + j, z, s, rk (z, K, E, X, μ)¤ , pe (z, K, E, X, μ)ª , ⎬
t,T
n
ξ t,T (s, l, j, z, K, E, X, μ) = min
⎩ max ©Λ £(1 − π ) l, z, s, rk (z, K, E, X, μ)¤ , pu (z, K, E, X, μ)ª ⎭
t,T
(130)
n
for some continuous function Λt,T that is strictly decreasing in its first argument and strictly increasing in s.
Lemma 8 Let s > 0, l + j > 0. Suppose that ξ t,T (s, l, j, z, K, E, X, μ) satisfies Assumption 2 and that
ξ t,T (s, l, j, z, K, E, X, μ) > pe (z, K, E, X, μ) .
Then,
¤
£
ξ t,T (s, l, j, z, K, E, X, μ) = Λt,T (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) .
Proof. a) Consider the case that j > 0.
¤
£
Suppose that ξ t,T (s, l, j, z, K, E, X, μ) 6= Λt,T (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) . Since ξ t,T (s, l, j, z, K, E, X, μ) >
pe (z, K, E, X, μ) ≥ pu (z, K, E, X, μ), from Assumption 2 we have that
¤
£
ξ t,T (s, l, j, z, K, E, X, μ) = Λt,T (1 − π n ) l, z, s, rk (z, K, E, X, μ) > pe (z, K, E, X, μ) .
19
But since Λt,T is strictly decreasing in its first argument,
¤
£
ξ t,T (s, l, j, z, K, E, X, μ) = Λt,T (1 − πn ) l, z, s, rk (z, K, E, X, μ)
©
¤
ª
£
> max Λt,T (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) , pe (z, K, E, X, μ) ,
which contradicts Assumption 2.
b) Consider the case that j = 0.
¤
£
Suppose that ξ t,T (s, l, j, z, K, E, X, μ) 6= Λt,T (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) . Since ξ t,T (s, l, j, z, K, E, X, μ) >
pe (z, K, E, X, μ), from Assumption 2 the only possible value left is
ξ t,T (s, l, j, z, K, E, X, μ) = pu (z, K, E, X, μ) .
But ξ t,T (s, l, j, z, K, E, X, μ) > pe (z, K, E, X, μ) ≥ pu (z, K, E, X, μ). A contradiction.
Lemma 9 Suppose that ξ t,T satisfies Assumption 2. Let s > 0 and l + j > 0. Then
©
ª
©
ª
min ξ t,T (s, l, j, z, K, E, X, μ), pe (z, K, E, X, μ) = min ξ t,T (s, l, 0, z, K, E, X, μ), pe (z, K, E, X, μ)
Proof. Since pe (z, K, E, X, μ) ≥ pu (z, K, E, X, μ), from equation (130) we have that
©
¤
ª
£
ξ t,T (s, l, 0, z, K, E, X, μ) = max Λt,T (1 − πn ) l, z, s, rk (z, K, E, X, μ) , pu (z, K, E, X, μ) .
Hence equation (130) can be written as follows:
⎧
⎫
⎨ max ©Λ £(1 − π ) l + j, z, s, rk (z, K, E, X, μ)¤ , pe (z, K, E, X, μ)ª , ⎬
t,T
n
ξ t,T (s, l, j, z, K, E, X, μ) = min
.
⎩
⎭
ξ (s, l, 0, z, K, E, X, μ)
(131)
t,T
Observe that the first term of the min operator in equation (131) is greater than or equal to pe (z, K, E, X, μ). Therefore,
ξ t,T (s, l, j, z, K, E, X, μ) < pe (z, K, E, X, μ) ⇒
ξ t,T (s, l, j, z, K, E, X, μ) = ξ t,T (s, l, 0, z, K, E, X, μ) < pe (z, K, E, X, μ)
Suppose now that ξ t,T (s, l, j, z, K, E, X, μ) ≥ pe (z, K, E, X, μ). From equation (131) we know that
ξ t,T (s, l, j, z, K, E, X, μ) ≤ ξ t,T (s, l, 0, z, K, E, X, μ).
Hence,
pe (z, K, E, X, μ) ≤ ξ t,T (s, l, j, z, K, E, X, μ) ⇒ pe (z, K, E, X, μ) ≤ ξ t,T (s, l, 0, z, K, E, X, μ).
Observe from equation (119) that for t < T , vt,T (s, l, j, z, K, E, X, μ) depends on (s, l, j) only through s and nt,T (s, l, j, z,
K, E, X, μ). This motivates the following definition:
Definition 10 Let t < T and suppose that ξ t+1,T satisfies Assumption 2. For every n ≥ 0 and s > 0 (implicitely) define
v̂t,T (n, s, z, K, E, X, μ) ≥ 0 as follows:
(
X
E
q (z, K, E, X, μ, z 0 ) ×
s0 >0
max[ξ t+1,T (s0 , n, v̂t,T (n, s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z
≤ pv (z, K, E, X, μ) , (= if v̂t,T (n, s, z, K, E, X, μ) > 0)
20
ª
(132)
The following definition will help characterize v̂t,T .
Definition 11 Let t < T and suppose that ξ t+1,T satisfies Assumption 2. For every s > 0, define v̄t,T (s, z, K, E, X, μ) as
follows:
E
(
X
s0 >0
q (z, K, E, X, μ, z 0 ) ×
max[ξ t+1,T (s0 , 0, v̄t,T (s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z
= pv (z, K, E, X, μ)
ª
(133)
From Assumption 2 and Lemma 8, observe that v̄t,T (s, z, K, E, X, μ) > 0 is uniquely determined.
Lemma 12 Let t < T and suppose that ξ t+1,T satisfies Assumption 2. Then, for every n ≥ 0 and s > 0 :
⎫
⎧
⎨ v̄ (s, z, K, E, X, μ) − (1 − π )n, if (1 − π )n ≤ v̄ (s, z, K, E, X, μ) ⎬
t,T
n
n
t,T
,
v̂t,T (n, s, z, K, E, X, μ) =
⎭
⎩
0, otherwise
Proof. It is a direct consequence of Assumption 2, Lemma 8, Definition 10 and Definition 11.
Lemma 13 Let t < T and suppose that ξ t+1,T satisfies Assumption 2. Then, for every n ≥ 0 and s > 0 :
E
(
X
s0 >0
q (z, K, E, X, μ, z 0 ) ×
(134)
max[ξ t+1,T (s0 , n, v̂t,T (n, s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z
⎫
⎧
©P
0
⎪
⎪
⎪
⎪
E
q
(z,
K,
E,
X,
μ,
z
)
×
0
⎪
⎪
s >0
⎬
⎨
ª
0
0
e
0
0
= min
max[ξ t+1,T (s , n, 0, z , L (z, K, E, X, μ)) − p (z , L (z, K, E, X, μ)) , 0]Q (s, s ) | z ,
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎩
pv (z, K, E, X, μ)
ª
Proof. a) If (1 − π n )n ≤ v̄t,T (s, z, K, E, X, μ):
pv (z, K, E, X, μ)
(
X
= E
q (z, K, E, X, μ, z 0 ) ×
s0 >0
max[ξ t+1,T (s0 , 0, v̄t,T (s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z
(
X
q (z, K, E, X, μ, z 0 ) ×
= E
ª
s0 >0
max[ξ t+1,T (s0 , n, v̂t,T (n, s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z
(
X
q (z, K, E, X, μ, z 0 ) ×
≥ E
s0 >0
max[ξ t+1,T (s0 , 0, (1 − π n ) n, z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z
(
X
q (z, K, E, X, μ, z 0 ) ×
= E
ª
ª
s0 >0
ª
max[ξ t+1,T (s0 , n, 0, z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z .
21
(135)
where the first equality follows from Definition 11, the second equality follows from Lemmas 8 and 12, the inequality follows
from Lemma 8 and Assumption 2 (in particular, the fact that Λt+1,T is strictly decreasing in its first argument), and the
last equality follows from Lemma 8.
b) If (1 − πn )n > v̄t,T (s, z, K, E, X, μ), from Lemma 12 we have that v̂t,T (n, s, z, K, E, X, μ) = 0. From Definition 10
we then have that:
pv (z, K, E, X, μ)
(
X
≥ E
q (z, K, E, X, μ, z 0 ) ×
s0 >0
ª
max[ξ t+1,T (s0 , n, 0, z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z .
(136)
Lemma 14 Let t < T and suppose that ξ t+1,T satisfies Assumption 2. Then, for every n ≥ 0 and s > 0 :
E
(
X
s0 >0
⎧
⎪
⎪
⎪
⎨
q (z, K, E, X, μ, z 0 ) ξ t+1,T (s0 , n, v̂t,T (n, s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
E
©P
s0 >0
q (z, K, E, X, μ, z 0 ) ×
⎫
⎪
⎪
⎪
⎬
)
ª
max[ξ t+1,T (s0 , n, 0, z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z ,
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎩
pv (z, K, E, X, μ)
(
)
X
0
0
0
e
0
0
+E
q (z, K, E, X, μ, z ) min[ξ t+1,T (s , n, 0, z , L (z, K, E, X, μ)) , p (z , L (z, K, E, X, μ))]Q (s, s ) | z .(137)
= min
s0 >0
Proof. Observe that for any positive numbers a and b :
a = max {a − b, 0} + min {a, b} .
Hence,
ξ t+1,T (s0 , n, v̂t,T (n, s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ))
= max[ξ t+1,T (s0 , n, v̂t,T (n, s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]
+ min[ξ t+1,T (s0 , n, v̂t,T (n, s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) , pe (z 0 , L (z, K, E, X, μ))].
22
(138)
Therefore,
E
= E
(
X
s0 >0
(
X
s0 >0
q (z, K, E, X, μ, z 0 ) ξ t+1,T (s0 , n, v̂t,T (n, s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
)
q (z, K, E, X, μ, z 0 ) ×
max[ξ t+1,T (s0 , n, v̂t,T (n, s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z
(
X
q (z, K, E, X, μ, z 0 ) ×
+ E
ª
s0 >0
ª
min[ξ t+1,T (s0 , n, v̂t,T (n, s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) , pe (z 0 , L (z, K, E, X, μ))]Q (s, s0 ) | z
⎫
⎧
©P
0
⎪
⎪
⎪
⎪
E
⎪
⎪
s0 >0 q (z, K, E, X, μ, z ) ×
⎬
⎨
ª
0
0
e
0
0
= min
max[ξ t+1,T (s , n, 0, z , L (z, K, E, X, μ)) − p (z , L (z, K, E, X, μ)) , 0]Q (s, s ) | z ,
⎪
⎪
⎪
⎪
⎪
⎪
v
⎭
⎩
p (z, K, E, X, μ)
(
)
X
0
0
0
e
0
0
+E
q (z, K, E, X, μ, z ) min[ξ t+1,T (s , n, 0, z , L (z, K, E, X, μ)) , p (z , L (z, K, E, X, μ))]Q (s, s ) | z .
s0 >0
where the first equality follows from equation (138), and the second equality follows from Lemmas 9 and 13.
Lemma 15 Let t < T and suppose that ξ t+1,T satisfies Assumption 2. Then, for every s > 0 and l + j > 0:
(139)
ξ t,T (s, l, j, z, K, E, X, μ)
⎫ ⎫
⎧
⎧
¤
£
⎪
⎨ F̂ (1 − π ) l + j, z, s, rk (z, K, E, X, μ) + (1 − π ) Ω ((1 − π ) l + j, s, z, K, E, X, μ) ⎬ ⎪
⎪
⎪
n
n
n
t,T
n
⎪
⎪
⎪
, ⎪
max
⎪
⎪
⎪
⎪
⎬
⎭
e
⎨
⎩
+Ψ (z, K, E, X, μ) , p (z, K, E, X, μ)
⎧
⎫
.
= min
⎪
⎨ F̂ £(1 − π ) l, z, s, rk (z, K, E, X, μ)¤ + (1 − π ) Ω ((1 − π ) l, s, z, K, E, X, μ) ⎬
⎪
⎪
⎪
n
n
n
t,T
n
⎪
⎪
⎪
⎪
max
⎪
⎪
⎪
⎪
⎭
⎩
⎭
⎩
+Ψ (z, K, E, X, μ) , pu (z, K, E, X, μ)
where
Ωt,T (n, s, z, K, E, X, μ)
(140)
⎫
⎧
©
P
0
⎪
⎪
⎪
⎪
E
⎪
⎪
s0 >0 q (z, K, E, X, μ, z ) ×
⎬
⎨
ª
0
0
e
0
0
= min
max[ξ t+1,T (s , n, 0, z , L (z, K, E, X, μ)) − p (z , L (z, K, E, X, μ)) , 0]Q (s, s ) | z ,
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎩
pv (z, K, E, X, μ)
(
)
X
0
0
0
e
0
0
+E
q (z, K, E, X, μ, z ) min[ξ t+1,T (s , n, 0, z , L (z, K, E, X, μ)) , p (z , L (z, K, E, X, μ))]Q (s, s ) | z .
s0 >0
and where
Ψ (z, K, E, X, μ) = π n E
"
X
s0 >0
q (z, K, E, X, μ, z 0 ) pu (z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
+Q (s, 0) E [q (z, K, E, X, μ, z 0 ) pu (z 0 , L (z, K, E, X, μ)) | z]
#
(141)
Proof. Since ξ t+1,T satisfies Assumption 2, it follows that Ωt,T (n, s, z, K, E, X, μ) is weakly decreasing in n and, therefore,
that
¤
£
F̂n n, z, s, rk (z, K, E, X, μ) + (1 − πn ) Ωt,T (n, s, z, K, E, X, μ) + Ψ (z, K, E, X, μ)
23
(142)
is strictly decreasing in n.
Also, from Definition 10, observe that nt,T (s, l, j, z, K, E, X, μ) and vt,T (s, l, j, z, K, E, X, μ) in equation (119) satisfy
that
vt,T (s, l, j, z, K, E, X, μ) = v̂t,T [nt,T (s, l, j, z, K, E, X, μ), s, z, K, E, X, μ] .
Using Lemma 14 and following exactly the same arguments as in the proofs of Lemmas 4, 5 and 6 (with equation
¤
£
(142) taking the place of F̂n n, z, s, rk (z, K, E, X, μ) and equation (117) taking the place of equation (118)) then leads to
equation (139).
Lemma 16 For t = 0, ..., T − 1, ξ t,T (s, l, j, z, K, E, X, μ) satisfies equation (139) for every s > 0 and l + j > 0. Moreover,
ξ t,T (s, l, j, z, K, E, X, μ) is decreasing in l and j.
Proof. From Lemma 7, ξ T,T (s, l, j, z, K, E, X, μ) satisfies Assumption 2 (with F̂n playing the role of ΛT,T ). The claim then
follows by induction.
Lemma 17 Let t ≤ T − 1 and s > 0. Define nt,T (s, z, K, E, X, μ) ≤ n̄t,T (s, z, K, E, X, μ) as follows:
¤
£
pe (z, K, E, X, μ) = F̂n nt,T (s, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
¢
¡
+ (1 − π n ) Ωt,T nt,T (s, z, K, E, X, μ), s, z, K, E, X, μ + Ψ (z, K, E, X, μ) ,
¤
£
pu (z, K, E, X, μ) = F̂n n̄t,T (s, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
+ (1 − π n ) Ωt,T (n̄t,T (s, z, K, E, X, μ), s, z, K, E, X, μ) + Ψ (z, K, E, X, μ) ,
(143)
(144)
where Ωt,T (n, s, z, K, E, X, μ) is given by equation (140) and Ψ (z, K, E, X, μ) is given by equation (141).
Then,
⎫
⎧
⎨ min ©(1 − π ) l + j, n (s, z, K, E, X, μ)ª ⎬
n
t,T
nt,T (s, l, j, z, K, E, X, μ) = max
.
⎩ min {(1 − π ) l, n̄ (s, z, K, E, X, μ)} ⎭
n
(145)
t,T
Proof. From Lemma (16), ξ t,T satisfies equation (139)
From equation (117), Lemma (14) and the fact that nt,T (s, l, j, z, K, E, X, μ) and vt,T (s, l, j, z, K, E, X, μ) satisfy that
vt,T (s, l, j, z, K, E, X, μ) = v̂t,T [nt,T (s, l, j, z, K, E, X, μ), s, z, K, E, X, μ] ,
we have that
¤
£
ξ t,T (s, l, j, z, K, E, X, μ) = F̂n nt,T (s, l, j, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
+. (1 − πn ) Ωt,T (nt,T (s, l, j, z, K, E, X, μ), s, z, K, E, X, μ)
+Ψ (z, K, E, X, μ) .
(146)
a) Suppose that (1 − πn ) l > nt,T (s, z, K, E, X, μ). Then, since F̂n is strictly decreasing in n and Ωt,T is decreasing in
n, we have that
¤
£
F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) + (1 − πn ) Ωt,T ((1 − πn ) l, s, z, K, E, X, μ) + Ψ (z, K, E, X, μ)
< pe (z, K, E, X, μ)
⎧
⎫
⎨ F̂ £(1 − π ) l + j, z, s, rk (z, K, E, X, μ)¤ + (1 − π ) Ω ((1 − π ) l + j, s, z, K, E, X, μ) ⎬
n
n
n
t,T
n
≤ max
.
⎩
⎭
+Ψ (z, K, E, X, μ) , pe (z, K, E, X, μ)
24
Since pu (z, K, E, X, μ) ≤ pe (z, K, E, X, μ), frome equation (139) we have that
ξ t,T (s, l, j, z, K, E, X, μ)
⎫
⎧
⎨ F̂ £(1 − π ) l, z, s, rk (z, K, E, X, μ)¤ + (1 − π ) Ω ((1 − π ) l, s, z, K, E, X, μ) ⎬
n
n
n
t,T
n
.
= max
u
⎭
⎩
+Ψ (z, K, E, X, μ) , p (z, K, E, X, μ)
(147)
£
¤
a.1) Suppose that F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) + (1 − πn ) Ωt,T ((1 − πn ) l, s, z, K, E, X, μ) + Ψ (z, K, E, X, μ) >
pu (z, K, E, X, μ).
Then, from equation (147),
ξ t,T (s, l, j, z, K, E, X, μ)
¤
£
= F̂n (1 − πn ) l, z, s, rk (z, K, E, X, μ) + (1 − π n ) Ωt,T ((1 − π n ) l, s, z, K, E, X, μ) + Ψ (z, K, E, X, μ) .
From equations (146) and (144), and using that F̂n is strictly decreasing in n and that Ωt,T is decreasing in n, we have
that
nt,T (s, l, j, z, K, E, X, μ) = (1 − πn ) l < n̄t,T (s, z, K, E, X, μ).
(148)
¤
£
a.2) Suppose that F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) + (1 − πn ) Ωt,T ((1 − πn ) l, s, z, K, E, X, μ) + Ψ (z, K, E, X, μ) ≤
pu (z, K, E, X, μ).
Then, from equation (147),
ξ t,T (s, l, j, z, K, E, X, μ) = pu (z, K, E, X, μ) .
From equations (146) and (143), and using that F̂n is strictly decreasing in n and that Ωt,T is decreasing in n, we have
that
nt,T (s, l, j, z, K, E, X, μ) = n̄t,T (s, z, K, E, X, μ) ≤ (1 − πn ) l.
(149)
From equations (148) and (149) we then have that
(1 − π n ) l > nt,T (s, z, K, E, X, μ) ⇒ nt,T (s, l, j, z, K, E, X, μ) = min {(1 − πn ) l, n̄t,T (s, z, K, E, X, μ)} .
(150)
b) Suppose that (1 − πn ) l ≤ nt,T (s, z, K, E, X, μ). Then, since F̂n is strictly decreasing in n and Ωt,T is decreasing in
n, we have that
¤
£
F̂n (1 − πn ) l, z, s, rk (z, K, E, X, μ) + (1 − π n ) Ωt,T ((1 − π n ) l, s, z, K, E, X, μ) + Ψ (z, K, E, X, μ)
≥ pe (z, K, E, X, μ) ,
and that
Hence,
¤
£
F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) + (1 − πn ) Ωt,T ((1 − πn ) l, s, z, K, E, X, μ) + Ψ (z, K, E, X, μ)
¤
£
≥ F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) + (1 − π n ) Ωt,T ((1 − π n ) l + j, s, z, K, E, X, μ) + Ψ (z, K, E, X, μ) .
£
¤
F̂n (1 − π n ) l, z, s, rk (z, K, E, X, μ) + (1 − πn ) Ωt,T ((1 − πn ) l, s, z, K, E, X, μ) + Ψ (z, K, E, X, μ)
⎧
⎫
⎨ F̂ £(1 − π ) l + j, z, s, rk (z, K, E, X, μ)¤ + (1 − π ) Ω ((1 − π ) l + j, s, z, K, E, X, μ) ⎬
n
n
n
t,T
n
≥ max
.
⎩
⎭
+Ψ (z, K, E, X, μ) , pe (z, K, E, X, μ)
25
From equation (139) we then have that
ξ t,T (s, l, j, z, K, E, X, μ)
⎫
⎧
⎨ F̂ £(1 − π ) l + j, z, s, rk (z, K, E, X, μ)¤ + (1 − π ) Ω ((1 − π ) l + j, s, z, K, E, X, μ) ⎬
n
n
n
t,T
n
= max
e
⎭
⎩
+Ψ (z, K, E, X, μ) , p (z, K, E, X, μ)
(151)
¤
£
b.1) Suppose that F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) +(1 − πn ) Ωt,T ((1 − πn ) l + j, s, z, K, E, X, μ)+Ψ(z, K, E, X,
μ) > pe (z, K, E, X, μ).
Then, from equation (151),
ξ t,T (s, l, j, z, K, E, X, μ)
¤
£
= F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) + (1 − π n ) Ωt,T ((1 − π n ) l + j, s, z, K, E, X, μ) + Ψ (z, K, E, X, μ) .
From equations (146) and (143), and using that F̂n is strictly decreasing in n and that Ωt,T is decreasing in n, we have
that
nt,T (s, l, j, z, K, E, X, μ) = (1 − π n ) l + j < nt,T (s, z, K, E, X, μ).
(152)
¤
£
b.2) Suppose that F̂n (1 − π n ) l + j, z, s, rk (z, K, E, X, μ) +(1 − πn ) Ωt,T ((1 − πn ) l + j, s, z, K, E, X, μ)+Ψ(z, K, E, X,
μ) ≤ pe (z, K, E, X, μ).
Then, from equation (151),
ξ t,T (s, l, j, z, K, E, X, μ) = pe (z, K, E, X, μ) .
From equations (146) and (143), and using that F̂n is strictly decreasing in n and that Ωt,T is decreasing in n, we have
that
nt,T (s, l, j, z, K, E, X, μ) = nt,T (s, z, K, E, X, μ) ≤ (1 − πn ) l + j.
(153)
From equations (152) and (153) we then have that
©
ª
(1 − πn ) l ≤ nt,T (s, z, K, E, X, μ) ⇒ nt,T (s, l, j, z, K, E, X, μ) = min (1 − π n ) l + j, nt,T (s, z, K, E, X, μ) .
(154)
c) Need to show that equation (145) holds.
c.1) Consider the case that
ª
©
min (1 − π n ) l + j, nt,T (s, z, K, E, X, μ) < min {(1 − π n ) l, n̄t,T (s, z, K, E, X, μ)} .
(155)
Suppose that (1 − π n ) l ≤ nt,T (s, z, K, E, X, μ).
Since nt,T (s, z, K, E, X, μ) ≤ n̄t,T (s, z, K, E, X, μ), it follows that (1 − πn ) l ≤ n̄t,T (s, z, K, E, X, μ). Hence, equation
(155) becomes
©
ª
min (1 − π n ) l + j, nt,T (s, z, K, E, X, μ) < (1 − πn ) l.
But (1 − π n ) l ≤ nt,T (s, z, K, E, X, μ) and j ≥ 0. A contradiction.
Thus, (1 − π n ) l > nt,T (s, z, K, E, X, μ) and from equation (150) we conclude that
©
ª
nt,T (s, l, j, z, K, E, X, μ) = min {(1 − π n ) l, n̄t,T (s, z, K, E, X, μ)} > min (1 − π n ) l + j, nt,T (s, z, K, E, X, μ) .
26
c.2) Consider the case that
ª
©
min (1 − π n ) l + j, nt,T (s, z, K, E, X, μ) > min {(1 − π n ) l, n̄t,T (s, z, K, E, X, μ)} .
(156)
Suppose that (1 − π n ) l > nt,T (s, z, K, E, X, μ).
Since j ≥ 0, it follows that (1 − πn ) l + j > nt,T (s, z, K, E, X, μ). Hence, equation (156) becomes
nt,T (s, z, K, E, X, μ) > min {(1 − π n ) l, n̄t,T (s, z, K, E, X, μ)} .
But nt,T (s, z, K, E, X, μ) ≤ n̄t,T (s, z, K, E, X, μ) and (1 − π n ) l > nt,T (s, z, K, E, X, μ). A contradiction.
Thus, (1 − π n ) l ≤ nt,T (s, z, K, E, X, μ) and from equation (154) we conclude that
©
ª
nt,T (s, l, j, z, K, E, X, μ) = min (1 − π n ) l + j, nt,T (s, z, K, E, X, μ) > min {(1 − π n ) l, n̄t,T (s, z, K, E, X, μ)} .
c.3) Consider the case that
©
ª
min (1 − π n ) l + j, nt,T (s, z, K, E, X, μ) = min {(1 − π n ) l, n̄t,T (s, z, K, E, X, μ)} .
Equations (150) and (154) then imply that
©
ª
nt,T (s, l, j, z, K, E, X, μ) = min (1 − π n ) l + j, nt,T (s, z, K, E, X, μ) = min {(1 − π n ) l, n̄t,T (s, z, K, E, X, μ)} .
From cases c1), c2) and c3), we conclude that equation (145) holds.
Lemma 18 Let t ≤ T − 1 and s > 0. Define v̄t,T (s, z, K, E, X, μ) as follows:
(
X
E
q (z, K, E, X, μ, z 0 ) ×
s0 >0
max[ξ t+1,T (s0 , 0, v̄t,T (s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z
= pv (z, K, E, X, μ)
ª
(157)
Then,
vt,T (s, l, j, z, K, E, X, μ) = max {v̄t,T (s, z, K, E, X, μ) − (1 − π n )nt,T (s, l, j, z, K, E, X, μ), 0}
where nt,T (s, l, j, z, K, E, X, μ) is given by equation (145).
Proof. By Lemma (16), ξ t+1,T satisfies Assumption 2. Therefore, Lemma (12) applies. The claim then follows from
Lemma (17) and the fact that
vt,T (s, l, j, z, K, E, X, μ) = v̂t,T [nt,T (s, l, j, z, K, E, X, μ), s, z, K, E, X, μ] .
Lemma 19 Let ξ(s, l, j, z, K, E, X, μ) be the Lagrange mulplier function for the establishments’ problem with infinite planning horizon (i.e. for T = ∞). Then, for every s > 0 and l + j > 0:
ξ(s, l, j, z, K, E, X, μ)
(158)
⎫ ⎫
⎧
⎧
£
¤
⎪
⎨ F̂ (1 − π ) l + j, z, s, rk (z, K, E, X, μ) + (1 − π ) Ω ((1 − π ) l + j, s, z, K, E, X, μ) ⎬ ⎪
⎪
⎪
n
n
n
n
⎪
⎪
⎪
, ⎪
max
⎪
⎪
⎪
⎬
⎭ ⎪
⎨
⎩
+Ψ (z, K, E, X, μ) , pe (z, K, E, X, μ)
⎧
⎫
.
= min
¤
£
⎪
⎨ F̂ (1 − π ) l, z, s, rk (z, K, E, X, μ) + (1 − π ) Ω ((1 − π ) l, s, z, K, E, X, μ) ⎬
⎪
⎪
⎪
n
n
n
n
⎪
⎪
⎪
⎪
max
⎪
⎪
⎪
⎪
⎭
⎩
⎭
⎩
+Ψ (z, K, E, X, μ) , pu (z, K, E, X, μ)
27
where
Ω (n, s, z, K, E, X, μ)
(159)
⎫
⎧
©
P
0
⎪
⎪
⎪
⎪
E
⎪
⎪
s0 >0 q (z, K, E, X, μ, z ) ×
⎬
⎨
0
0
e
0
0
= min
max[ξ (s , n, 0, z , L (z, K, E, X, μ)) − p (z , L (z, K, E, X, μ)) , 0]Q (s, s ) | z} ,
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎩
pv (z, K, E, X, μ)
(
)
X
+E
q (z, K, E, X, μ, z 0 ) min[ξ t+1,T (s0 , n, 0, z 0 , L (z, K, E, X, μ)) , pe (z 0 , L (z, K, E, X, μ))]Q (s, s0 ) | z .
s0 >0
and where
Ψ (z, K, E, X, μ) = π n E
"
X
s0 >0
0
u
0
0
q (z, K, E, X, μ, z ) p (z , L (z, K, E, X, μ)) Q (s, s ) | z
#
+Q (s, 0) E [q (z, K, E, X, μ, z 0 ) pu (z 0 , L (z, K, E, X, μ)) | z]
(160)
Moreover, ξ(s, l, j, z, K, E, X, μ) is decreasing in l and j.
Proof. The claim follows from Lemma 16 and the fact that limT →∞ ξ 0,T = ξ and that limT →∞ ξ 1,T = ξ (see Easley and
Spulber (1)).
Lemma 20 Let n(s, l, j, z, K, E, X, μ), h(s, l, j, z, K, E, X, μ), f (s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ) and v(s, l, j, z,
K, E, X, μ) be optimal decision rules for the establishments’ problem with infinite planning horizon (i.e. for T = ∞). Let
ξ(s, l, j, z, K, E, X, μ) be given by equation (158), Ω (n, s, z, K, E, X, μ) be given by equation (159) and Ψ (z, K, E, X, μ) be
given by equation (160).
Define n(s, z, K, E, X, μ), n̄(s, z, K, E, X, μ) and v̄(s, z, K, E, X, μ) as follows:
¤
£
pe (z, K, E, X, μ) = F̂n n(s, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
+ (1 − π n ) Ω (n(s, z, K, E, X, μ), s, z, K, E, X, μ) + Ψ (z, K, E, X, μ) ,
¤
£
pu (z, K, E, X, μ) = F̂n n̄(s, z, K, E, X, μ), z, s, rk (z, K, E, X, μ)
+ (1 − πn ) Ω (n̄(s, z, K, E, X, μ), s, z, K, E, X, μ) + Ψ (z, K, E, X, μ) ,
E
(
X
s0 >0
q (z, K, E, X, μ, z 0 ) ×
max[ξ (s0 , 0, v̄(s, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) − pe (z 0 , L (z, K, E, X, μ)) , 0]Q (s, s0 ) | z}
= pv (z, K, E, X, μ)
Then, for every s > 0 and l + j > 0:
⎧
⎫
⎨ min {(1 − π ) l + j, n(s, z, K, E, X, μ)} ⎬
n
n(s, l, j, z, K, E, X, μ) = max
,
⎩ min {(1 − π ) l, n̄(s, z, K, E, X, μ)} ⎭
n
h(s, l, j, z, K, E, X, μ) = max {n(s, l, j, z, K, E, X, μ) − l, 0}
28
f (s, l, j, z, K, E, X, μ) = max {l − n(s, l, j, z, K, E, X, μ), 0}
ez sFk [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)] = rk (z, K, E, X, μ) ,
v(s, l, j, z, K, E, X, μ) = max {v̄(s, z, K, E, X, μ) − (1 − π n )n(s, l, j, z, K, E, X, μ), 0} .
Proof. It is a direct consequence of Lemma 3, Lemma 16, Lemma 17, Lemma 18, Lemma 19 and the fact that limT →∞ ξ 0,T =
ξ, limT →∞ ξ 1,T = ξ, limT →∞ n0,T = n, limT →∞ v0,T = v (see Easley and Spulber (1)).
3
3.1
Finding a RCE
The myopic social planner’s problem
³
´
The problem of the myopic social planner facing a stochastic process Â, Û , L̂ is given by the following Bellman equation:
V (z, K, E, X, μ, K̂, Ê, X̂, μ̂) = max
subject to
C +I +A≤
XZ
s
U ≤X +E−
μ0 (s0 , L × J ) =
XZ
s
XZ
s
ez sF [n(s, l, j), k(s, l, j)] μ (s, dl × dj)
(161)
k(s, l, j)μ (s, dl × dj) ≤ K
(162)
³
´
v(s, l, j)μ (s, dl × dj) ≤ H A, U, Â, Û
(163)
s
XZ
s
´ i¾
h ³
C 1−σ − 1
0
0
0
0
0
0
0
0
0
+ ϕU + βE V z , K , E , X , μ , K̂ , Ê , X̂ , μ̂ | z
1−σ
XZ
XZ
s
½
h(s, l, j)μ (s, dl × dj) +
XZ
s
f (s, l, j)μ (s, dl × dj)
h(s, l, j)μ (s, dl × dj) ≤ (1 − π u ) E
(164)
(165)
n (s, l, j) = l + h (s, l, j) − f (s, l, j)
(166)
h (s, l, j) ≤ j
(167)
π n l ≤ f (s, l, j)
(168)
f (s, l, j) ≤ l
(169)
K 0 = (1 − δ) K + I
³
´
E 0 = G A, U, Â, Û
³
´
X 0 = U − G A, U, Â, Û
{(l,j): n(s,l,j)∈L and v(s,l,j)∈J }
Q (s, s0 ) μ (s, dl × dj) + ψ (s0 ) I (L × J )
³
´
 =  z, K̂, Ê, X̂, μ̂
³
´
Û = Û z, K̂, Ê, X̂, μ̂
29
³
´
³
´
K̂ 0 , Ê 0 , X̂ 0 , μ̂0 = L̂ z, K̂, Ê, X̂, μ̂ .
³
´
³
´
The MSP’s decision rules are C = C m z, K, E, X, μ, K̂, Ê, X̂, μ̂ , I = I m z, K, E, X, μ, K̂, Ê, X̂, μ̂ , n = nm (s, l, j, z,
³
´
³
´
K, E, X, μ, K̂, Ê, X̂, μ̂), k = k m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , f = f m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , h = hm (s, l, j,
³
´
z, K, E, X, μ, K̂, Ê, X̂, μ̂), v = v m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , U = U m (z, K, E, X, μ, K̂, Ê, X̂, μ̂), A = Am (z, K, E,
X, μ, K̂, Ê, X̂, μ̂).
3.2
Solution to MSP’s problem
The necessary and sufficient conditions for a solution {C m , I m , nm , k m , f m , hm , v m , U m , Am } to the MSP’s problem with ex³
´
ogenous stochastic process Â, Û , L̂ is that there exist Lagrange multipliers λm (z, K, E, X, μ, K̂, Ê, X̂, μ̂), rk,m (z, K, E, X,
³
´
μ, K̂, Ê, X̂, μ̂), pv,m (z, K, E, X, μ, K̂, Ê, X̂, μ̂), pu,m z, K, E, X, μ, K̂, Ê, X̂, μ̂ , pe,m (z, K, E, X, μ, K̂, Ê, X̂, μ̂) − pu,m (z,
³
´
K, E, X, μ, K̂, Ê, X̂, μ̂), ξ m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , η m (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), αm (s, l, j, z, K, E, X, μ,
³
´
K̂, Ê, X̂, μ̂) and χm s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ (for constraints (161)-(169) respectively) such that equations (170)-
(199) hold.
³
´−σ
³
´
= λm z, K, E, X, μ, K̂, Ê, X̂, μ̂
C m z, K, E, X, μ, K̂, Ê, X̂, μ̂
³
´
⎤
⎡
³
´i
λm z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0 h
³
´
1 − δ + rk,m z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0 | z ⎦
1 = E ⎣β
λm z, K, E, X, μ, K̂, Ê, X̂, μ̂
h
³
´i
ez sFk nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), km s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂
³
´ ³
³
´
´
≤ rk,m z, K, E, X, μ, K̂, Ê, X̂, μ̂ , = if k m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ > 0
(170)
(171)
(172)
³
´−1
³
´
0 = ϕλm z, K, E, X, μ, K̂, Ê, X̂, μ̂
− pu,m z, K, E, X, μ, K̂, Ê, X̂, μ̂
i
³
´
h
³
´
+pv,m z, K, E, X, μ, K̂, Ê, X̂, μ̂ Hu Am z, K, E, X, μ, K̂, Ê, X̂, μ̂ , U m (z, K, E, X, μ, K̂, Ê, X̂, μ̂), Â, Û
i
h
³
´
+ (1 − πu ) Gu Am z, K, E, X, μ, K̂, Ê, X̂, μ̂ , U m (z, K, E, X, μ, K̂, Ê, X̂, μ̂), Â, Û ×
³
´
⎧
⎨ λm z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0
³
´ ×
E β
⎩
λm z, K, E, X, μ, K̂, Ê, X̂, μ̂
³
´
³
´i o
h
pe,m z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0 − pu,m z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0 | z
³
´
⎤
⎡
³
´
λm z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0
³
´ pu,m z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0 | z ⎦
(173)
+E ⎣β
λm z, K, E, X, μ, K̂, Ê, X̂, μ̂
30
i
³
´
h
³
´
1 = pv,m z, K, E, X, μ, K̂, Ê, X̂, μ̂ Ha Am z, K, E, X, μ, K̂, Ê, X̂, μ̂ , U m (z, K, E, X, μ, K̂, Ê, X̂, μ̂), Â, Û
i
h
³
´
+ (1 − πu ) Ga Am z, K, E, X, μ, K̂, Ê, X̂, μ̂ , U m (z, K, E, X, μ, K̂, Ê, X̂, μ̂), Â, Û ×
³
´
⎧
⎨ λm z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0
³
´ ×
E β
⎩
λm z, K, E, X, μ, K̂, Ê, X̂, μ̂
³
´
³
´i o
h
(174)
pe,m z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0 − pu,m z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0 | z
³
´
⎧
⎨X λm z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0
³
´ Q (s, s0 ) ×
β
E
m
⎩ 0
λ
z,
K,
E,
X,
μ,
K̂,
Ê,
X̂,
μ̂
s
o
³
´
η m (s0 , nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), v m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0 ) | z
³
´³
³
´
´
≤ pv,m z, K, E, X, μ, K̂, Ê, X̂, μ̂ = if v m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ > 0
(175)
h
³
´i
ez sFn nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), k m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂
³
´
⎧
⎨X λm z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0
³
´ Q (s, s0 ) ×
+E
β
m
⎩ 0
λ
z, K, E, X, μ, K̂, Ê, X̂, μ̂
s
³
³
´
´ o
ξ m s0 , nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), v m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0 | z
³
´
⎧
⎨X λm z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0
³
´ Q (s, s0 ) ×
−π n E
β
m
⎩ 0
λ
z, K, E, X, μ, K̂, Ê, X̂, μ̂
s
³
³
´
´ o
αm s0 , nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), v m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0 | z
³
´
⎧
⎨X λm z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0
³
´ Q (s, s0 ) ×
E
β
m
⎩ 0
λ
z, K, E, X, μ, K̂, Ê, X̂, μ̂
s
³
³
´
´ o
χm s0 , nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), v m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , z 0 , K 0 , E 0 , X 0 , μ0 , K̂ 0 , Ê 0 , X̂ 0 , μ̂0 | z
´
³
´ ³
≤ ξ m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ , = if nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) > 0
(176)
³
´
−pe,m z, K, E, X, μ, K̂, Ê, X̂, μ̂ + ξ m (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂)
³
³
´
´
≤ η m (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) = if hm s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ > 0
(177)
³
´
pu,m z, K, E, X, μ, K̂, Ê, X̂, μ̂ − ξ m (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) + αm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂)
³
³
´
´
−χm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) ≤ 0 = if f m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ > 0
(178)
³
´
³
´
³
´
nm s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ = l+hm s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ −f m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ (179)
³
´
hm s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ ≤ j
(180)
³
´
π n l ≤ f m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂
(181)
31
³
´
f m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ ≤ l
h
³
´i
η m (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) j − hm s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ = 0
h
i
³
´
αm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) f m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ − π n l = 0
h
³
´i
χm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) l − f m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ = 0
³
´
XZ
hm s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ μ (s, dl × dj) ≤ (1 − π u ) E
(182)
(183)
(184)
(185)
(186)
s
0 =
³
´
³
´i
h
pe,m z, K, E, X, μ, K̂, Ê, X̂, μ̂ − pu,m z, K, E, X, μ, K̂, Ê, X̂, μ̂
"
#
³
´
XZ
m
(1 − π u ) E −
h
s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ μ (s, dl × dj)
(187)
s
³
´
³
´
³
´
C m z, K, E, X, μ, K̂, Ê, X̂, μ̂ + I m z, K, E, X, μ, K̂, Ê, X̂, μ̂ + Am z, K, E, X, μ, K̂, Ê, X̂, μ̂
h
³
´i
XZ
=
ez sF nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), k m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ μ (s, dl × dj)
(188)
s
XZ
s
XZ
s
³
´
k m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ μ (s, dl × dj) = K
(189)
³
´
v m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ μ (s, dl × dj)
h
i
³
´
= H Am z, K, E, X, μ, K̂, Ê, X̂, μ̂ , U m (z, K, E, X, μ, K̂, Ê, X̂, μ̂), Â, Û
U m (z, K, E, X, μ, K̂, Ê, X̂, μ̂) = X + E −
XZ
s
(190)
³
´
hm s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ μ (s, dl × dj)
³
´
XZ
+
f m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ μ (s, dl × dj)
(191)
s
³
´
K 0 = (1 − δ) K + I m z, K, E, X, μ, K̂, Ê, X̂, μ̂
h
i
³
´
E 0 = G Am z, K, E, X, μ, K̂, Ê, X̂, μ̂ , U m (z, K, E, X, μ, K̂, Ê, X̂, μ̂), Â, Û
³
´
h
i
X 0 = U m (z, K, E, X, μ, K̂, Ê, X̂, μ̂) − G Am z, K, E, X, μ, K̂, Ê, X̂, μ̂ , U m (z, K, E, X, μ, K̂, Ê, X̂, μ̂), Â, Û
XZ
0
0
μ (s , L × J ) =
Q (s, s0 ) μ (s, dl × dj) + ψ (s0 ) I (L × J ) ,
s
(192)
(193)
(194)
(195)
B(s,L×J )
where
n
³
´
³
´
o
B (s, L × J ) = (l, j) : nm s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ ∈ L and v m s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂ ∈ J .
³
´
 =  z, K̂, Ê, X̂, μ̂
³
´
Û = Û z, K̂, Ê, X̂, μ̂
´
³
´
³
K̂ 0 , Ê 0 , X̂ 0 , μ̂0 = L̂ z, K̂, Ê, X̂, μ̂ .
32
(196)
(197)
(198)
(199)
3.3
Characterization of myopic planner’s decision rules
Proposition 21 Let {C m , I m , nm , km , f m , hm , v m , U m , Am } be the solution to the MSP’s with exogenous stochastic process
³
´
Â, Û , L̂ . Then, there exist thresholds nm (s, z, K, E, X, μ, K̂, Ê, X̂, μ̂), n̄m (s, z, K, E, X, μ, K̂, Ê, X̂, μ̂) and v̄ m (s, z, K, E, X,
³
´
μ, K̂, Ê, X̂, μ̂) and a shadow capital price function rk z, K, E, X, μ, K̂, Ê, X̂, μ̂ such that, for every s > 0 and l + j > 0:
⎧
o ⎫
n
⎨ min (1 − π n ) l + j, nm (s, z, K, E, X, μ, K̂, Ê, X̂, μ̂) ⎬
n
o
,
nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) = max
⎩ min (1 − π ) l, n̄m (s, z, K, E, X, μ, K̂, Ê, X̂, μ̂)
⎭
n
v m (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂)
n
o
= max v̄ m (s, z, K, E, X, μ, K̂, Ê, X̂, μ̂) − (1 − π n )nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), 0 .
n
o
hm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) = max nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) − l, 0
n
o
f m (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂) = max l − nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), 0
i
h
ez sFk nm (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂), km (s, l, j, z, K, E, X, μ, K̂, Ê, X̂, μ̂)
³
´
= rk z, K, E, X, μ, K̂, Ê, X̂, μ̂ ,
³
´
Proof. In the economy in which Â, Û , L̂ trully represent exogenous productivity shocks to the recruitment technology the
Welfare Theorems apply. In this case the problem described in Section 3.1 is the social planner’s problem and its solution can
be descentralized as a recursive competitive equilibrium in which prices are functions of the state (z, K, E, X, μ, K̂, Ê, X̂, μ̂).
The claim then follows from analyzing the associated establishments’ problem using identical arguments as those in the
proof of Proposition 20.
3.4
Construction of a RCE
Proposition 22 Let {C m , I m , nm , km , f m , hm , v m , U m , Am } be the solution to the MSP’s with exogenous stochastic process
³
´
Â, Û , L̂ .
Suppose that
and that
satisfy the following conditions:
Ê 0
 (z, K, E, X, μ) = Am (z, K, E, X, μ, K, E, X, μ) ,
(200)
Û (z, K, E, X, μ) = U m (z, K, E, X, μ, K, E, X, μ) ,
(201)
´
³
K̂ 0 , Ê 0 , X̂ 0 , μ̂0 = L̂ (z, K, E, X, μ)
(202)
K̂ 0 = (1 − δ) K + I m (z, K, E, X, μ, K, E, X, μ) ,
(203)
= G [Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ) ,
Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ)] ,
33
(204)
X̂ 0 = U m (z, K, E, X, μ, K, E, X, μ) − Ê 0 ,
XZ
μ̂0 (s0 , L × J ) =
Q (s, s0 ) μ (s, dl × dj) + ψ (s0 ) I (L × J ) ,
s
where
(205)
(206)
B(s,L×J )
B (s, L × J ) = {(l, j) : nm (s, l, j, z, K, E, X, μ, K, E, X, μ) ∈ L and v m (s, l, j, z, K, E, X, μ, K, E, X, μ) ∈ J } .
(207)
Then, there exists a RCE {B, W , R, c, i, m, n, k, f , h, v, a, b, d, u, A, U , L, Π, rk , ru , pu , pe , pv , q} such that
c (K, z, K, E, X, μ) = C m (z, K, E, X, μ, K, E, X, μ)
i (K, z, K, E, X, μ) = I m (z, K, E, X, μ, K, E, X, μ)
n(s, l, j, z, K, E, X, μ) = nm (s, l, j, z, K, E, X, μ, K, E, X, μ)
f (s, l, j, z, K, E, X, μ) = f m (s, l, j, z, K, E, X, μ, K, E, X, μ)
h(s, l, j, z, K, E, X, μ) = hm (s, l, j, z, K, E, X, μ, K, E, X, μ)
v(s, l, j, z, K, E, X, μ) = v m (s, l, j, z, K, E, X, μ, K, E, X, μ)
A (z, K, E, X, μ) = Am (z, K, E, X, μ, K, E, X, μ)
U (z, K, E, X, μ) = U m (z, K, E, X, μ, K, E, X, μ) .
Proof. Since {C m , I m , nm , k m , f m , hm , v m , U m , Am } is a solution to the MSP’s problem with exogenous stochastic process
³
´
©
ª
Â, Û , L̂ we know that there exist Lagrange multipliers λm , rk,m , pv,m , pu,m , pe,m − pu,m , ξ m , η m , αm , χm such that equations (170)-(199) hold. Evaluate these equations at (K̂, Ê, X̂, μ̂) = (K, E, X, μ) and use equations (200)-(207) to get
equations (208)-(230).
C m (z, K, E, X, μ, K, E, X, μ)−σ = λm (z, K, E, X, μ, K, E, X, μ)
´
³
⎧
⎨ λm z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)
1 = E β
×
⎩
λm (z, K, E, X, μ, K, E, X, μ)
³
´i o
h
1 − δ + rk,m z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ) | z
(208)
(209)
ez sFk [nm (s, l, j, z, K, E, X, μ, K, E, X, μ), k m (s, l, j, z, K, E, X, μ, K, E, X, μ)]
≤ rk,m (z, K, E, X, μ, K, E, X, μ) , (= if k m (s, l, j, z, K, E, X, μ, K, E, X, μ) > 0)
34
(210)
0 = ϕλm (z, K, E, X, μ, K, E, X, μ)−1 − pu,m (z, K, E, X, μ, K, E, X, μ)
+pv,m (z, K, E, X, μ, K, E, X, μ) Hu [Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ),
Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ)]
+ (1 − π u ) Gu [Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ),
Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ)] ×
´
³
⎧
⎨ λm z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)
E β
×
⎩
λm (z, K, E, X, μ, K, E, X, μ)
h
´
´i o
³
³
pe,m z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ) − pu,m z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ) | z
´
³
⎤
⎡
´
³
λm z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)
pu,m z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ) | z ⎦
+E ⎣β
λm (z, K, E, X, μ, K, E, X, μ)
(211)
1 = pv,m (z, K, E, X, μ, K, E, X, μ) Ha [Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ),
Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ)]
+ (1 − π u ) Ga [Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ),
Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ)] ×
´
³
⎧
⎨ λm z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)
E β
×
⎩
λm (z, K, E, X, μ, K, E, X, μ)
´
´i o
³
³
h
pe,m z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ) − pu,m z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ) | z
´
³
⎧
⎨X λm z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)
β
E
Q (s, s0 ) ×
⎩ 0
λm (z, K, E, X, μ, K, E, X, μ)
(212)
s
η (s0 , nm (s, l, j, z, K, E, X, μ, K, E, X, μ), v m (s, l, j, z, K, E, X, μ, K, E, X, μ) ,
m
z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)) | z
o
≤ pv,m (z, K, E, X, μ, K, E, X, μ) , (= if v m (s, l, j, z, K, E, X, μ, K, E, X, μ) > 0)
35
(213)
ez sFn [nm (s, l, j, z, K, E, X, μ, K, E, X, μ), k m (s, l, j, z, K, E, X, μ, K, E, X, μ)]
´
³
⎧
⎨X λm z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)
+E
β
Q (s, s0 ) ×
⎩ 0
λm (z, K, E, X, μ, K, E, X, μ)
s
m
0
ξ (s , nm (s, l, j, z, K, E, X, μ, K, E, X, μ), v m (s, l, j, z, K, E, X, μ, K, E, X, μ) ,
o
z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)) | z
´
³
⎧
⎨X λm z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)
−π n E
β
Q (s, s0 ) ×
⎩ 0
λm (z, K, E, X, μ, K, E, X, μ)
s
αm (s0 , nm (s, l, j, z, K, E, X, μ, K, E, X, μ), v m (s, l, j, z, K, E, X, μ, K, E, X, μ) ,
o
z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)) | z
´
³
⎧
⎨X λm z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)
β
Q (s, s0 ) ×
E
⎩ 0
λm (z, K, E, X, μ, K, E, X, μ)
s
χ (s0 , nm (s, l, j, z, K, E, X, μ, K, E, X, μ), v m (s, l, j, z, K, E, X, μ, K, E, X, μ) ,
o
z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)) | z
m
≤ ξ m (s, l, j, z, K, E, X, μ, K, E, X, μ) , (= if nm (s, l, j, z, K, E, X, μ, K, E, X, μ) > 0)
(214)
−pe,m (z, K, E, X, μ, K, E, X, μ) + ξ m (s, l, j, z, K, E, X, μ, K, E, X, μ)
≤ η m (s, l, j, z, K, E, X, μ, K, E, X, μ) (= if hm (s, l, j, z, K, E, X, μ, K, E, X, μ) > 0)
(215)
pu,m (z, K, E, X, μ, K, E, X, μ) − ξ m (s, l, j, z, K, E, X, μ, K, E, X, μ) + αm (s, l, j, z, K, E, X, μ, K, E, X, μ)
−χm (s, l, j, z, K, E, X, μ, K, E, X, μ) ≤ 0 (= if f m (s, l, j, z, K, E, X, μ, K, E, X, μ) > 0)
(216)
nm (s, l, j, z, K, E, X, μ, K, E, X, μ) = l + hm (s, l, j, z, K, E, X, μ, K, E, X, μ)
−f m (s, l, j, z, K, E, X, μ, K, E, X, μ)
(217)
hm (s, l, j, z, K, E, X, μ, K, E, X, μ) ≤ j
(218)
π n l ≤ f m (s, l, j, z, K, E, X, μ, K, E, X, μ)
(219)
f m (s, l, j, z, K, E, X, μ, K, E, X, μ) ≤ l
(220)
η m (s, l, j, z, K, E, X, μ, K, E, X, μ) [j − hm (s, l, j, z, K, E, X, μ, K, E, X, μ)] = 0
(221)
αm (s, l, j, z, K, E, X, μ, K, E, X, μ) [f m (s, l, j, z, K, E, X, μ, K, E, X, μ) − π n l] = 0
(222)
χm (s, l, j, z, K, E, X, μ, K, E, X, μ) [l − f m (s, l, j, z, K, E, X, μ, K, E, X, μ)] = 0
XZ
hm (s, l, j, z, K, E, X, μ, K, E, X, μ) μ (s, dl × dj) ≤ (1 − π u ) E
(223)
(224)
0 = [pe,m (z, K, E, X, μ, K, E, X, μ) − pu,m (z, K, E, X, μ, K, E, X, μ)]
"
#
XZ
(1 − π u ) E −
hm (s, l, j, z, K, E, X, μ, K, E, X, μ) μ (s, dl × dj)
(225)
s
s
36
C m (z, K, E, X, μ, K, E, X, μ) + I m (z, K, E, X, μ, K, E, X, μ) + Am (z, K, E, X, μ, K, E, X, μ)
XZ
=
ez sF [nm (s, l, j, z, K, E, X, μ, K, E, X, μ), km (s, l, j, z, K, E, X, μ, K, E, X, μ)] μ (s, dl × dj)
(226)
s
XZ
k m (s, l, j, z, K, E, X, μ, K, E, X, μ) μ (s, dl × dj) = K
(227)
XZ
v m (s, l, j, z, K, E, X, μ, K, E, X, μ) μ (s, dl × dj)
(228)
s
s
= H [Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ),
Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ)]
m
U (z, K, E, X, μ, K, E, X, μ) = X + E −
+
XZ
s
XZ
s
hm (s, l, j, z, K, E, X, μ, K, E, X, μ) μ (s, dl × dj)
f m (s, l, j, z, K, E, X, μ, K, E, X, μ) μ (s, dl × dj)
Define
A (z, K, E, X, μ) = Am (z, K, E, X, μ, K, E, X, μ)
U (z, K, E, X, μ) = U m (z, K, E, X, μ, K, E, X, μ)
L (z, K, E, X, μ) = L̂ (z, K, E, X, μ)
rk (z, K, E, X, μ) = rk,m (z, K, E, X, μ, K, E, X, μ)
ru (z, K, E, X, μ) = ϕλm (z, K, E, X, μ, K, E, X, μ)−1
pv (z, K, E, X, μ) = pv,m (z, K, E, X, μ, K, E, X, μ)
pu (z, K, E, X, μ) = pu,m (z, K, E, X, μ, K, E, X, μ)
pe (z, K, E, X, μ) = pe,m (z, K, E, X, μ, K, E, X, μ)
i
h
λm z 0 , L̂ (z, K, E, X, μ) , L̂ (z, K, E, X, μ)
q (z, K, E, X, μ, z 0 ) = β
λm (z, K, E, X, μ, K, E, X, μ)
Π (z, K, E, X, μ) = C m (z, K, E, X, μ, K, E, X, μ) + I m (z, K, E, X, μ, K, E, X, μ)
+ru (z, K, E, X, μ) U m (z, K, E, X, μ, K, E, X, μ) − rk (z, K, E, X, μ) K
λ (κ, z, K, E, X, μ) = λm (z, K, E, X, μ, K, E, X, μ)
c (κ, z, K, E, X, μ) = C m (z, K, E, X, μ, K, E, X, μ)
m (κ, z, K, E, X, μ) = U m (z, K, E, X, μ, K, E, X, μ)
i (κ, z, K, E, X, μ) = I m (z, K, E, X, μ, K, E, X, μ) + rk (z, K, E, X, μ) κ − rk (z, K, E, X, μ) K
ξ(s, l, j, z, K, E, X, μ) = ξ m (s, l, j, z, K, E, X, μ, K, E, X, μ)
α(s, l, j, z, K, E, X, μ) = αm (s, l, j, z, K, E, X, μ, K, E, X, μ)
37
(229)
(230)
χ(s, l, j, z, K, E, X, μ) = χm (s, l, j, z, K, E, X, μ, K, E, X, μ)
η(s, l, j, z, K, E, X, μ) = η m (s, l, j, z, K, E, X, μ, K, E, X, μ)
f (s, l, j, z, K, E, X, μ) = f m (s, l, j, z, K, E, X, μ, K, E, X, μ)
h(s, l, j, z, K, E, X, μ) = hm (s, l, j, z, K, E, X, μ, K, E, X, μ)
k(s, l, j, z, K, E, X, μ) = km (s, l, j, z, K, E, X, μ, K, E, X, μ)
n(s, l, j, z, K, E, X, μ) = nm (s, l, j, z, K, E, X, μ, K, E, X, μ)
v(s, l, j, z, K, E, X, μ) = v m (s, l, j, z, K, E, X, μ, K, E, X, μ)
e m
A (z, K, E, X, μ, K, E, X, μ)
E
e
u(e, x, z, K, E, X, μ) = U m (z, K, E, X, μ, K, E, X, μ)
E
a(e, x, z, K, E, X, μ) =
e
H [Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ),
E
Am (z, K, E, X, μ, K, E, X, μ) , U m (z, K, E, X, μ, K, E, X, μ)]
Z
eX
hm (s, l, j, z, K, E, X, μ, K, E, X, μ) μ (s, dl × dj)
d(e, x, z, K, E, X, μ) =
E s
b(e, x, z, K, E, X, μ) =
In addition, define recursively the following value functions:
B(κ, z, K, E, X, μ) =
c (κ, z, K, E, X, μ)1−σ − 1
+ ϕm (κ, z, K, E, X, μ)
1−σ
+βE {B [(1 − δ) κ + i (κ, z, K, E, X, μ) , z 0 , L (z, K, E, X, μ)] | z}
W (s, l, j, z, K, E, X, μ) = ez sF [n(s, l, j, z, K, E, X, μ), k(s, l, j, z, K, E, X, μ)]
+pu (z, K, E, X, μ) f (s, l, j, z, K, E, X, μ) − pe (z, K, E, X, μ) h(s, l, j, z, K, E, X, μ)
+E
"
X
s0
−rk (z, K, E, X, μ) k(s, l, j, z, K, E, X, μ) − pv (z, K, E, X, μ) v(s, l, j, z, K, E, X, μ)
#
q (z, K, E, X, μ, z 0 ) W (s0 , n(s, l, j, z, K, E, X, μ), v(s, l, j, z, K, E, X, μ), z 0 , L (z, K, E, X, μ)) Q (s, s0 ) | z
R (e, x, z, K, E, X, μ) = pe (z, K, E, X, μ) d(e, x, z, K, E, X, μ) + pv (z, K, E, X, μ) b(e, x, z, K, E, X, μ)
+pu (z, K, E, X, μ) [x + e − d(e, x, z, K, E, X, μ) − u(e, x, z, K, E, X, μ)]
⎡
⎛
⎢
⎜
⎢
⎜
⎢
⎜
⎢
⎜
⎢
⎜
+E ⎢q (z, K, E, X, μ, z 0 ) R ⎜
⎢
⎜
⎢
⎜
⎢
⎜
⎣
⎝
+ru (z, K, E, X, μ) u(e, x, z, K, E, X, μ) − a(e, x, z, K, E, X, μ)
⎡
⎤
⎞ ⎤
a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ),
⎟ ⎥
⎦,
G⎣
⎟ ⎥
⎟ ⎥
A (z, K, E, X, μ) , U (z, K, E, X, μ)
⎡
⎤ ⎟ ⎥
⎟ ⎥
⎟ | z⎥
a(e, x, z, K, E, X, μ), u(e, x, z, K, E, X, μ),
⎥
⎣
⎦
u(e, x, z, K, E, X, μ) − G
, ⎟
⎟ ⎥
⎟
⎥
A (z, K, E, X, μ) , U (z, K, E, X, μ)
⎠ ⎦
z 0 , L (z, K, E, X, μ)
Using these definitions, the homogeneity of degree one of H and G with respect to (a, u), and the homogeneity of degreee
zero of H and G with respect to (A, U ), equations (200)-(207) together with equations (208)-(230) imply equations (12)-(50).
38
4
4.1
Steady state equilibrium
Steady state conditions
In order to compute a recursive competitive equilibrium it will be first necessary to compute a steady state equilibrium for
the deterministic version of the economy, i.e. one in which the aggregate productivity level z is equal to zero. This section
describes the conditions that such steady state equilibrium must satisfy. Using equations (51)-(79), Lemma (19), Lemma
(20), the conditon that z = 0 and the condition that the vector (K, E, X, μ) is constant over time, we get the following
steady state conditions.
r=
1
−1+δ
β
(231)
⎧
o ⎫
n
⎨ max F̂n [(1 − πn ) l + j, s, r] + (1 − πn ) Ω ((1 − πn ) l + j, s) + Ψ, pe , ⎬
o
n
.
ξ(s, l, j) = min
⎭
⎩
max F̂n [(1 − π n ) l, s, r] + (1 − π n ) Ω ((1 − π n ) l, s) + Ψ, pu
(
)
X
X
0
e
0
v
Ω (n, s) = min
β max[ξ (s , n, 0) − p , 0]Q (s, s ) , p +
β min[ξ t+1,T (s0 , n, 0) , pe ]Q (s, s0 )
s0 >0
Ψ = πn
X
(232)
(233)
s0 >0
βpu Q (s, s0 ) + Q (s, 0) βpu
(234)
s0 >0
pe = F̂n [n(s), s, r] + (1 − π n ) Ω (n(s), s) + Ψ,
(235)
pu = F̂n [n̄(s), s, r] + (1 − πn ) Ω (n̄(s), s) + Ψ,
X
pv =
β max[ξ (s0 , 0, v̄(s)) − pe , 0]Q (s, s0 )
(236)
(237)
n(s, l, j) = max {min {(1 − π n ) l + j, n(s)} , min {(1 − π n ) l, n̄(s)}}
(238)
h(s, l, j) = max {n(s, l, j) − l, 0}
(239)
f (s, l, j) = max {l − n(s, l, j), 0}
(240)
sFk [n(s, l, j), k(s, l, j)] = r,
(241)
v(s, l, j) = max {v̄(s) − (1 − πn )n(s, l, j), 0} .
(242)
pu = ϕcσ + pv Hu (A, U, A, U ) + βpu + (1 − πu ) Gu (A, U, A, U )β [pe − pu ]
(243)
1 = pv Ha (A, U, A, U ) + (1 − πu ) Ga (A, U, A, U )β [pe − pu ]
(244)
s0 >0
XZ
s
e
u
h(s, l, j)μ (s, dl × dj) ≤ (1 − πu ) G [A, U, A, U ]
"
0 = [p − p ] (1 − π u ) G [A, U, A, U ] −
XZ
s
XZ
s
h(s, l, j)μ (s, dl × dj)
v(s, l, j)μ (s, dl × dj) = H [A, U, A, U ]
39
(245)
#
(246)
(247)
XZ
s
c + δK + A =
n(s, l, j)μ (s, dl × dj) + U = 1
XZ
s
XZ
s
μ (s0 , L × J ) =
XZ
ez sF [n(s, l, j), k(s, l, j)] μ (s, dl × dj)
(249)
k(s, l, j)μ (s, dl × dj) = K
(250)
E = G [A, U, A, U ]
(251)
X = U − G [A, U, A, U ]
(252)
{(l,j):n(s,l,j)∈L and v(s,l,j)∈J }
s
(248)
Q (s, s0 ) μ (s, dl × dj) + ψ (s0 ) I (L × J )
(253)
Observe from Lemma (22) that if {C m , I m , nm , km , f m , hm , v m , U m , Am } is a solution to the MSP’s with exogenous
³
´
stochastic process Â, Û , L̂ , conditions (200)-(207) are satisfied, z is identical to zero, and the aggregate state (K, E, X, μ)
is constant over time, then equations (231)-(253) must hold.
4.2
Invariant distribution
The following Lemma characterizes a support to the invariant distribution μ that satisfies equation (253).1
Lemma 23 Let M be a natural number satisfying that
(1 − π n )M max {n̄ (smax ) , v̄ (smax )} < min {n (smin ) , v̄ (smin )} .
Define the set N as follows:
N =
Then, the set
P=
½
∪
s∈S
(254)
o¾
n
k
k
k
∪ (1 − πn ) n (s) , (1 − πn ) n̄ (s) , (1 − π n ) v̄ (s)
∪ {0} .
M −1
k=0
½
½
¾
¾
(s, l, j) : s ∈ S, l ∈ N , and j ∈ 0∪ {max [v̄ (s0 ) − (1 − πn ) l, 0]} ∪ {0}
s ∈S
is a support of the invariant distribution μ.
Proof. From equations (238) and (242) we know that an establishment of type (s, l, j) transits to a next-period type
(s0 , l0 , j 0 ), with s0 randomly determined,
l0 = n(s, l, j),
(255)
j 0 = max {v̄ (s) − (1 − πn )l0 , 0} .
(256)
and
Define
P (0) = ∪ {(s, 0, 0)} .
s∈S
1 In
the statement of the lemma smax and smin denote the largest and smallest positive values for s, respectively.
40
Since establishments are created with (l, j) = (0, 0), P (0) describes the set of all possible types (s, l, j) of establisments of
zero age.
Define
N (0) = {0} .
Since n(s, l, j) = 0 whenever (l, j) = (0, 0), N (0) describes the set of all possible employment levels of establishments of zero
age.
Starting from N (0) , define recursively a sequence of sets P (m) and N (m) as follows:
½
½
¾
¾
(m)
(m−1)
= (s, l, j) : s ∈ S, l ∈ N
, and j ∈
∪ {max [v̄ (s−1 ) − (1 − π n ) l, 0]} ∪ {0}
P
N (m) =
for m = 1, 2, ..., ∞.
½
s−1 ∈S
¾ n
o
∪ {n (s) , n̄ (s) , v̄ (s)} ∪ n: n = (1 − π n ) nm−1 for some nm−1 ∈ N (m−1) ,
s∈S
From equations (238), (242), (255) and (256) we know that P (m) contains the set of all possible types (s, l, j) of
establishments of age m, and that N (m) contains the set of all possible employment levels of establishments of age m.2
By induction, it can be shown that:
½
o¾
n
m−1
k
k
k
N (m) = ∪ ∪ (1 − πn ) n (s) , (1 − π n ) n̄ (s) , (1 − π n ) v̄ (s)
∪ {0} ,
s∈S
k=0
(257)
for m = 1, 2, ..., ∞.
A direct consequence of equation (257) is that N (m−1) ⊂ N (m) , for every m ≥ 1. Thus, the set N (m) in fact contains
all the possible employment levels of establishments of age m or younger. Moreover,
N (m) /N (m−1) = ∪
s∈S
o
n
(1 − π n )m−1 n (s) , (1 − πn )m−1 n̄ (s) , (1 − π n )m−1 v̄ (s) ,
(258)
for m = 1, 2, ..., ∞, where “/” denotes set difference.
In what follows it will be shown that there exists a M < ∞ such that N (M ) contains the set of all possible employment
levels of establishments of all ages m = 0, 1, ..., ∞. To prove this it suffices to show that there exists a M < ∞ such that no
establishment of age M + 1 will choose an employment level in the set N (M +1) /N (M ) , i.e. all establishments of age M + 1
will choose an employment level in the set N (M ) .3
Let M satisfy equation (254). Since 0 < π n < 1, such a M exists.
Let (s, l, j) ∈ P (M +1) .
Suppose that n(s, l, j) ∈ N (M +1) /N (M ) . Since N (M +1) /N (M ) satisfies equation (258), and M satisfies equation (254),
it follows that
n(s, l, j) ≤ (1 − π n )
2 Observe
M
max {n̄ (smax ) , v̄ (smax )} < min {n (smin ) , v̄ (smin )} .
(259)
that the “max” and “min” operators in equation (238) have been disregarded in the construction of the sets P (m) and N (m) . Thus,
the set of actual types of establishments of age m and the set of actual employment levels of establishments of age m are smaller than P (m) and
N (m) , respectively.
3 This
condition is sufficient because whenever an establishment reaches age M + 1, its age can be reset to M without consequence. This
procedure can be repeated an infinite number of times.
41
Also, since n(s, l, j) satisfies equation (238) and (s, l, j) ∈ P (M +1) , we have that
½
¾
n(s, l, j) ∈ {n (s) , n̄ (s) , (1 − πn ) l} ∪
∪ v̄ (s−1 ) .
s−1 ∈S
(260)
From equation (260) and the last inequality in equation (259), we then have that
n(s, l, j) = (1 − π n ) l.
Suppose, first, that j = 0.
Suppose that some establishment of age M transits to (s, l, j). From equation (256), this implies that
0 = max {v̄ (s−1 ) − (1 − π n )l, 0} ,
for some s−1 ∈ S.
But, from equation (259)
n(s, l, j) = (1 − π n ) l < v̄ (s−1 ) ,
for all s−1 ∈ S. A contradiction.
Hence, (s, l, j) ∈ P (M +1) does not correspond to an establishment of age M + 1.
Suppose now that j > 0.
Let s−1 be such that (1 − πn ) l + j = v̄ (s−1 ) (since (s, l, j) ∈ P (M +1) , such an s−1 exists).
Then, from equation (238) we have that
n(s, l, j) = max {min {v̄ (s−1 ) , n(s)} , min {(1 − πn ) l, n̄(s)}} ,
and, therefore, that
n(s, l, j) = (1 − π n ) l ≤ n̄(s) and n(s, l, j) = (1 − π n ) l ≥ min {v̄ (s−1 ) , n(s)} .
(261)
The second inequality in equation (261) contradicts equation (259).
We conclude that no establishment of age M + 1 chooses an employment level in the set N (M +1) /N (M ) . It follows that
the set P (M +1) is a support of the invariant distribution μ.
4.3
Steady state computational algorithm
This section describes the algorithm used to compute a steady state equilibrium. It will be convenient to do so under the
functional forms that will be used later on. In particular, the production function F is here assumed to have the following
form:
F (n, k) = nγ kθ ,
(262)
where γ > 0, θ > 0, and γ + θ < 1. Observe that under this functional form Fk becomes
Fk (n, k) = θnγ k θ−1
42
(263)
and F̂n becomes:
F̂n (n, s, r) = s
1
1−θ
n
γ+θ−1
1−θ
θ
∙ ¸ 1−θ
θ
γ
.
r
(264)
Lemmas 24-26 provide certain homogeneity results that will be used in the computational algorithm.
Lemma 24 Suppose that F is given by equation (262). Let ξ(s, l, j; pu , pe , pv ), Ω (n, s; pu , pe , pv ), Ψ (pu ), n(s; pu , pe , pv ),
n̄(s; pu , pe , pv ), v̄(s; pu , pe , pv ), n(s, l, j; pu , pe , pv ), h(s, l, j; pu , pe , pv ), f (s, l, j; pu , pe , pv ), k(s, l, j; pu , pe , pv ), v(s, l, j; pu , pe ,
pv ) be the solutions to equations (232)-(242), given prices (pu , pe , pv ). Then,
1−θ
1−θ
ξ(s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv ) = λξ(s, l, j; pu , pe , pv ),
³ 1−θ
´
Ω λ γ+θ−1 n, s; λpu , λpe , λpv
= λΩ (n, s; pu , pe , pv )
Ψ (λpu ) = λΨ (pu )
(265)
(266)
(267)
n(s; λpu , λpe , λpv ) = λ
1−θ
γ+θ−1
n(s; pu , pe , pv )
(268)
n̄(s; λpu , λpe , λpv ) = λ
1−θ
γ+θ−1
n̄(s; pu , pe , pv )
(269)
v̄(s; λpu , λpe , λpv ) = λ γ+θ−1 v̄(s; pu , pe , pv )
(270)
1−θ
1−θ
1−θ
1−θ
1−θ
γ+θ−1
1−θ
γ+θ−1
j; λpu , λpe , λpv ) = λ
1−θ
γ+θ−1
h(s, l, j; pu , pe , pv )
(272)
j; λpu , λpe , λpv ) = λ
1−θ
γ+θ−1
f (s, l, j; pu , pe , pv )
(273)
j; λpu , λpe , λpv ) = λ
γ
γ+θ−1
k(s, l, j; pu , pe , pv )
(274)
v(s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv ) = λ γ+θ−1 v(s, l, j; pu , pe , pv )
(275)
n(s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv ) = λ γ+θ−1 n(s, l, j; pu , pe , pv )
h(s, λ
f (s, λ
k(s, λ
1−θ
γ+θ−1
1−θ
γ+θ−1
l, λ
l, λ
l, λ
1−θ
1−θ
γ+θ−1
1−θ
γ+θ−1
1−θ
1−θ
(271)
for every λ > 0.
Proof. The claim follows from guessing and verifying equations (265)-(275) in equations (232)-(242).
Lemma 25 Suppose that F is given by equation (262). Let μ (pu , pe , pv ) be the invariant distribution that satisfies equation
(253) and P (pu , pe , pv ) be the finite support in Lemma 23, when prices are given by (pu , pe , pv ). Then, for every λ > 0,
³
´
1−θ
1−θ
(s, l, j) ∈ P (pu , pe , pv ) ⇔ s, λ γ+θ−1 l, λ γ+θ−1 j ∈ P (λpu , λpe , λpv )
and
(276)
³
´
1−θ
1−θ
(s, l, j) ∈ P (pu , pe , pv ) ⇒ μ (s, l, j; pu , pe , pv ) = μ s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv .
Proof. Equation (276) is a direct consequence of Lemmas 23 and 24.
From Lemma 23, observe that equation (253) can be written as follows. For every (s0 , l0 , j 0 ) ∈ P (pu , pe , pv ) ,
μ (s0 , l0 , j 0 ; pu , pe , pv ) =
X
X
s (l,j)∈B(s,l0 ,j 0 ;pu ,pe ,pv ):
Q (s, s0 ) μ (s, l, j; pu , pe , pv ) + ψ (s0 ) I (l0 , j 0 ) ,
(277)
where
B (s, l0 , j 0 ; pu , pe , pv ) = {(l, j) : (s, l, j) ∈ P (pu , pe , pv ) , n(s, l, j; pu , pe , pv ) = l0 and v(s, l, j; pu , pe , pv ) = j 0 } ,
43
(278)
and where I (l0 , j 0 ) = 1 if (l0 , j 0 ) = (0, 0), and I (l0 , j 0 ) = 0, otherwise.
³
´
1−θ
1−θ
For the same reason, we have that for every s0 , λ γ+θ−1 l0 , λ γ+θ−1 j 0 ∈ P (λpu , λpe , λpv ),
³
´
1−θ
1−θ
μ s0 , λ γ+θ−1 l0 , λ γ+θ−1 j 0 ; λpu , λpe , λpv
´
³
X
X
1−θ
1−θ
=
Q (s, s0 ) μ s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv
s
where
1−θ
1−θ
1−θ
1−θ
λ γ+θ−1 l,λ γ+θ−1 j ∈B s,λ γ+θ−1 l0 ,λ γ+θ−1 j 0 ;λpu ,λpe ,λpv :
³ 1−θ
´
1−θ
+ ψ (s0 ) I λ γ+θ−1 l0 , λ γ+θ−1 j 0 ,
(279)
³
´ ³
´
´
n³ 1−θ
1−θ
1−θ
1−θ
1−θ
1−θ
B s, λ γ+θ−1 l0 , λ γ+θ−1 j 0 ; λpu , λpe , λpv
=
λ γ+θ−1 l, λ γ+θ−1 j : s, λ γ+θ−1 l, λ γ+θ−1 j ∈ P (λpu , λpe , λpv ) ,
1−θ
1−θ
1−θ
n(s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv ) = λ γ+θ−1 l0
o
1−θ
1−θ
1−θ
and v(s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv ) = λ γ+θ−1 j 0 .
Observe, from equation (280) and Lemma 24, that
³
´
´
n³ 1−θ
1−θ
1−θ
1−θ
B s, λ γ+θ−1 l0 , λ γ+θ−1 j 0 ; λpu , λpe , λpv
=
λ γ+θ−1 l, λ γ+θ−1 j : (s, l, j) ∈ P (pu , pe , pv ) ,
n(s, l, j; pu , pe , pv ) = l0 and v(s, l, j; pu , pe , pv ) = j 0 } .
(280)
(281)
From equations (278) and (281) we then have that
³ 1−θ
´
³
´
1−θ
1−θ
1−θ
λ γ+θ−1 l, λ γ+θ−1 j ∈ B s, λ γ+θ−1 l0 , λ γ+θ−1 j 0 ; λpu , λpe , λpv ⇔ (l, j) ∈ B (s, l0 , j 0 ; pu , pe , pv )
(282)
³ 1−θ
´
1−θ
Using equation (282) and the fact that I λ γ+θ−1 l0 , λ γ+θ−1 j 0 = I (l0 , j 0 ), equation (279) can be written as follows:
³
´
1−θ
1−θ
μ s0 , λ γ+θ−1 l0 , λ γ+θ−1 j 0 ; λpu , λpe , λpv =
³
´
X
X
1−θ
1−θ
Q (s, s0 ) μ s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv + ψ (s0 ) I (l0 , j 0 ) .
s (l,j)∈B(s,l0 ,j 0 ;pu ,pe ,pv ):
But this is the same as equation (277). It follows that
³
´
1−θ
1−θ
μ s0 , λ γ+θ−1 l0 , λ γ+θ−1 j 0 ; λpu , λpe , λpv = μ (s0 , l0 , j 0 ; pu , pe , pv ) .
Lemma 26 Suppose that F is given by equation (262). Let ξ(s, l, j; pu , pe , pv ), Ω (n, s; pu , pe , pv ), Ψ (pu ), n(s; pu , pe , pv ),
n̄(s; pu , pe , pv ), v̄(s; pu , pe , pv ), n(s, l, j; pu , pe , pv ), h(s, l, j; pu , pe , pv ), f (s, l, j; pu , pe , pv ), k(s, l, j; pu , pe , pv ), v(s, l, j; pu , pe ,
pv ) be the solutions to equations (232)-(242), let μ (pu , pe , pv ) be the invariant distribution that satisfies equation (253) and
let P (pu , pe , pv ) be the finite support in Lemma 23, when prices are given by (pu , pe , pv ). Then, for every λ > 0,
Z
XZ
1−θ X
n(s, l, j; λpu , λpe , λpv )μ (s, dl × dj; λpu , λpe , λpv ) = λ γ+θ−1
n(s, l, j; pu , pe , pv )μ (s, dl × dj; pu , pe , pv )(283)
s
XZ
s
XZ
s
XZ
s
s
h(s, l, j; λpu , λpe , λpv )μ (s, dl × dj; λpu , λpe , λpv ) = λ
u
e
v
u
e
v
v(s, l, j; λp , λp , λp )μ (s, dl × dj; λp , λp , λp ) = λ
k(s, l, j; λpu , λpe , λpv )μ (s, dl × dj; λpu , λpe , λpv ) = λ
44
1−θ
γ+θ−1
1−θ
γ+θ−1
γ
γ+θ−1
XZ
s
h(s, l, j; pu , pe , pv )μ (s, dl × dj; pu , pe , pv )(284)
s
.
v(s, l, j; pu , pe , pv )μ (s, dl × dj; pu , pe , pv )(285)
s
k(s, l, j; pu , pe , pv )μ (s, dl × dj; pu , pe , pv )(286)
XZ
XZ
XZ
s
= λ
sn(s, l, j; λpu , λpe , λpv )γ k(s, l, j; λpu , λpe , λpv )θ μ (s, dl × dj; λpu , λpe , λpv )
γ
γ+θ−1
XZ
(287)
sn(s, l, j; pu , pe , pv )γ k(s, l, j; pu , pe , pv )θ μ (s, dl × dj; pu , pe , pv )
s
Proof. We shall prove equation (283). The proofs for equations (284) and (287) are analogous.
From Lemmas 24 and 25 we have that for every (s, l, j) ∈ P (pu , pe , pv ) :
1−θ
1−θ
1−θ
1−θ
1−θ
n(s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv )μ(s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv ) = λ γ+θ−1 n(s, l, j; pu , pe , pv )μ(s, l, j; pu , pe , pv ).
Therefore,
X
1−θ
1−θ
1−θ
1−θ
n(s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv )μ(s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv )
(s,l,j)∈P(pu ,pe ,pv )
1−θ
= λ γ+θ−1
X
n(s, l, j; pu , pe , pv )μ(s, l, j; pu , pe , pv ),
(s,l,j)∈P(pu ,pe ,pv )
and from equation (276),
X
1−θ
1−θ
1−θ
1−θ
n(s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv )μ(s, λ γ+θ−1 l, λ γ+θ−1 j; λpu , λpe , λpv ) =
1−θ
1−θ
s,λ γ+θ−1 l,λ γ+θ−1 j ∈P(λpu ,λpe ,λpv )
X
1−θ
λ γ+θ−1
n(s, l, j; pu , pe , pv )μ(s, l, j; pu , pe , pv ).
(s,l,j)∈P(pu ,pe ,pv )
Thus,
X
n(s, l, j; λpu , λpe , λpv )μ(s, l, j; λpu , λpe , λpv ) =
(s,l,j)∈P(λpu ,λpe ,λpv )
X
1−θ
λ γ+θ−1
n(s, l, j; pu , pe , pv )μ(s, l, j; pu , pe , pv ),
(s,l,j)∈P(pu ,pe ,pv )
which is equation (283).
Two computational algorithms will be described: One for the economy with no externalities and another for the economy
with externalities.
4.3.1
Algorithm for economy with no externalities
In the economy with no externalities the recruitment technology is assumed to be given by
u.a
G (a, u, A, U ) =
[uφ
H (a, u, A, U ) =
1
,
1
,
+ aφ ] φ
u.a
[uφ + aφ ] φ
In this case, we have that
G (A, U, A, U ) = H (A, U, A, U ) =
U.A
1
[U φ + Aφ ] φ
45
=U
(
1
¡ U ¢φ
A
+1
) φ1
=U
(µ
U
A
¶φ
)− φ1
+1
,
Ga (A, U, A, U ) =
Gu (A, U, A, U ) =
(
(
Therefore, equations (243), (244) and (247) become:
1+
1
¡ A ¢φ
U
1
¡ U ¢φ
A
+1
) φ1 +1
,
) φ1 +1
.
(1 − β) pu = ϕcσ + {pv + (1 − πu ) β [pe − pu ]}
v
e
u
1 = {p + (1 − πu ) β [p − p ]}
XZ
s
v(s, l, j)μ (s, dl × dj) = U
(
(
1+
(
1
¡ U ¢φ
A
1
¡ A ¢φ
U
+1
) φ1 +1
) φ1 +1
φ
{pv + (1 − πu ) β [pe − pu ]} 1+φ − 1
φ
{pv + (1 − πu ) β [pe − pu ]} 1+φ
) φ1
The computational algorithm is given by the following steps.
Step 1: Fix pu0 = 1.
Step 2: Choose some pe0 ≥ pu0 (it is convenient for the first choice to be pe0 = pu0 ).
Step 3: Choose some pv0
Step 4: Set j = 0 and solve for the functions ξ(s, l, 0; pu0 , pe0 , pv0 ), Ω(n, s; pu0 , pe0 , pv0 ) and Ψ (pu0 ) that satisfy equations
(232)-(234) when j = 0 and prices are given by (pu0 , pe0 , pv0 ). (This can be done through value function iterations).
Step 5: Solve for the function ξ(s, l, j; pu0 , pe0 , pv0 ) that satisfy equation (232) when j > 0 and prices are given by
(pu0 , pe0 , pv0 ). (Observe that given the output of Step 4, this takes only one value function iteration).
Step 6: For each s, solve for the thresholds n(s; pu0 , pe0 , pv0 ), n̄(s; pu0 , pe0 , pv0 ), v̄(s; pu0 , pe0 , pv0 ) that satisfy equations (235)(237) (This can be done through standard root finding methods),
Step 7: Construct the functions n(s, l, j; pu0 , pe0 , pv0 ), h(s, l, j; pu0 , pe0 , pv0 ), f (s, l, j; pu0 , pe0 , pv0 ), k(s, l, j; pu0 , pe0 , pv0 ) and v(s, l, j;
pu0 , pe0 , pv0 ) as in equations (238)-(242).
Step 8: Construct the finite support P (pu0 , pe0 , pv0 ) as in Lemma 23.
Step 9: For every (s, l0 , j 0 ), construct the set B (s, l0 , j 0 ; pu0 , pe0 , pv0 ) as in equation (278) and solve for the invariant
distribution μ (s, l, j; pu0 , pe0 , pv0 ) that satisfies equation (277) (This can be done recursively).
R
R
R
R
Step 10: Evaluate v (pu0 , pe0 , pv0 ) dμ (pu0 , pe0 , pv0 ), n (pu0 , pe0 , pv0 ) dμ (pu0 , pe0 , pv0 ), h (pu0 , pe0 , pv0 ) dμ (pu0 , pe0 , pv0 ), k (pu0 , pe0 , pv0 )
R
dμ (pu0 , pe0 , pv0 ) and n(pu0 , pe0 , pv0 )γ k(pu0 , pe0 , pv0 )θ dμ (pu0 , pe0 , pv0 ).
Step 11: Define
λ (pu0 , pe0 , pv0 ) =
1
pv0 + (1 − π u ) β [pe0 − pu0 ]
Find the factor λ > λ (pu0 , pe0 , pv0 ) that satisfies that
Z
1−θ X
γ+θ−1
λ
v(s, l, j; pu0 , pe0 , pv0 )μ (s, dl × dj; pu0 , pe0 , pv0 )
s
) φ1
φ
∙
¸( φ
Z
1+φ {pv + (1 − π ) β [pe − pu ]} 1+φ − 1
1−θ
λ
u
0
0
0
u
e
v
u
e
v
=
1 − λ γ+θ−1 n (p0 , p0 , p0 ) dμ (p0 , p0 , p0 )
φ
φ
λ 1+φ {pv0 + (1 − π u ) β [pe0 − pu0 ]} 1+φ
46
(288)
(This can be done using standard root finding methods)
Observe that the left hand side of equation (288) is stricly decreasing in λ, while the right hand side is stricly increasing
in λ. Morever,
lim
LHS (λ) > 0, lim LHS (λ) = 0
lim
RHS (λ) = 0, lim RHS (λ) = 1
e v
λ→λ(pu
0 ,p0 ,p0 )
e v
λ→λ(pu
0 ,p0 ,p0 )
λ→∞
λ→∞
Hence, there exists a unique λ (pu0 , pe0 , pv0 ) > λ (pu0 , pe0 , pv0 ) that satisfies equation (288).
Step 12: Define U (pu0 , pe0 , pv0 ), A (pu0 , pe0 , pv0 ), K (pu0 , pe0 , pv0 ), Y (pu0 , pe0 , pv0 ), and c (pu0 , pe0 , pv0 ) as follows:
Z
1−θ
U (pu0 , pe0 , pv0 ) = 1 − λ (pu0 , pe0 , pv0 ) γ+θ−1 n (pu0 , pe0 , pv0 ) dμ (pu0 , pe0 , pv0 )
⎫ φ1 +1
⎪
⎪
⎪
⎬
⎧
⎪
⎪
⎪
⎨
1
1
=
¶φ ⎪
µ
e
u
+ (1 − πu ) β (p0 − p0 )] ⎪
e v
A(pu
⎪
⎪
0 ,p0 ,p0 )
⎪
⎪
⎭
⎩1 +
e ,pv
U (pu
,p
)
0
0 0
Z
γ
K (pu0 , pe0 , pv0 ) = λ (pu0 , pe0 , pv0 ) γ+θ−1 k (pu0 , pe0 , pv0 ) dμ (pu0 , pe0 , pv0 )
λ (pu0 , pe0 , pv0 ) [pv0
Y
(pu0 , pe0 , pv0 )
γ
=
λ (pu0 , pe0 , pv0 ) γ+θ−1
Z
sn(pu0 , pe0 , pv0 )γ k(pu0 , pe0 , pv0 )θ dμ (pu0 , pe0 , pv0 )
c (pu0 , pe0 , pv0 ) = Y (pu0 , pe0 , pv0 ) − δK (pu0 , pe0 , pv0 ) − A (pu0 , pe0 , pv0 )
Since λ (pu0 , pe0 , pv0 ) > λ (pu0 , pe0 , pv0 ), observe that U (pu0 , pe0 , pv0 ) > 0 and that A (pu0 , pe0 , pv0 ) > 0.
Step 13: Evaluate the function
f (pu0 , pe0 , pv0 ) = ϕc (pu0 , pe0 , pv0 )σ
+λ (pu0 , pe0 , pv0 ) {pv0 (1 − π u ) β [pe0 − pu0 ]}
− (1 − β) λ (pu0 , pe0 , pv0 ) pu
(µ
U (pu0 , pe0 , pv0 )
A (pu0 , pe0 , pv0 )
¶φ
)− φ1 −1
+1
Step 14: In order to satisfy equation (243), go back to Step 3 with a new value for pv0 until
f (pu0 , pe0 , pv0 ) = 0
(This can be done using standard root-finding methods)
Step 15: If
1−θ
λ (pu0 , pe0 , pv0 ) γ+θ−1
Z
h (pu0 , pe0 , pv0 ) dμ (pu0 , pe0 , pv0 ) ≤ (1 − πu ) G [A (pu0 , pe0 , pv0 ) , U (pu0 , pe0 , pv0 ) , A (pu0 , pe0 , pv0 ) , U (pu0 , pe0 , pv0 )] ,
(289)
and
0 = [pe0 − pu0 ] {(1 − π u ) G [A (pu0 , pe0 , pv0 ) , U (pu0 , pe0 , pv0 ) , A (pu0 , pe0 , pv0 ) , U (pu0 , pe0 , pv0 )]
¾
Z
1−θ
−λ (pu0 , pe0 , pv0 ) γ+θ−1 h (pu0 , pe0 , pv0 ) dμ (pu0 , pe0 , pv0 ) ,
47
(290)
(i.e. if equations (245) and (246) are satisfied), then
pu
= λ (pu0 , pe0 , pv0 ) pu0 ,
pe
= λ (pu0 , pe0 , pv0 ) pe0 ,
pv
= λ (pu0 , pe0 , pv0 ) pv0
are equilibrium prices. (At this point, exit the algorithm).
Step 16: If conditions (289)-(290) are not satisfied, go back to Step 2 with a new guess for pe0 (the search for pe0 can be
implemented within a standard root finding method).
4.3.2
Algorithm for economy with externalities
In the economy with externalities the recruitment technology is assumed to be given by
A
G (a, u, A, U ) = u
H (a, u, A, U ) = a
1
,
1
,
[U φ + Aφ ] φ
U
[U φ + Aφ ] φ
In this case we have that
G (A, U, A, U ) = H (A, U, A, U ) =
U.A
1
=U
[U φ + Aφ ] φ
(
1
¡ U ¢φ
+1
A
) φ1
=U
Ga (A, U, A, U ) = 0
A
Gu (A, U, A, U ) =
Ha (a, u, A, U ) =
1
[U φ + Aφ ] φ
U
1
[U φ + Aφ ] φ
Hu (a, u, A, U ) = 0.
Therefore, equations (243), (244) and (247) become:
(1 − β) pu = ϕcσ + (1 − πu ) β [pe − pu ]
v
1=p
XZ
s
(
1+
1
¡ A ¢φ
U
) φ1
v(s, l, j)μ (s, dl × dj) = U
The computational algorithm is given by the following steps.
Steps 1-10 are the same as in Section 4.3.1.
48
(
(
1
¡ U ¢φ
A
φ
(pv ) − 1
(pv )
φ
+1
) φ1
) φ1
(µ
U
A
¶φ
)− φ1
+1
,
Step 11: Define
λ (pu0 , pe0 , pv0 ) =
1
pv0
Find the factor λ > λ (pu0 , pe0 , pv0 ) that satisfies that
Z
1−θ X
γ+θ−1
λ
v(s, l, j; pu0 , pe0 , pv0 )μ (s, dl × dj; pu0 , pe0 , pv0 )
(291)
s
) φ1
¸( φ v φ
∙
Z
1−θ
λ
(p
)
−
1
0
u
e
v
u
e
v
=
1 − λ γ+θ−1 n (p0 , p0 , p0 ) dμ (p0 , p0 , p0 )
λφ (pv0 )φ
(This can be done using standard root finding methods)
Observe that the left hand side of equation (291) is stricly decreasing in λ, while the right hand side is stricly increasing
in λ. Morever,
lim
LHS (λ) > 0, lim LHS (λ) = 0
lim
RHS (λ) = 0, lim RHS (λ) = 1
e v
λ→λ(pu
0 ,p0 ,p0 )
e v
λ→λ(pu
0 ,p0 ,p0 )
λ→∞
λ→∞
Hence, there exists a unique λ (pu0 , pe0 , pv0 ) > λ (pu0 , pe0 , pv0 ) that satisfies equation (291).
Step 12: Define U (pu0 , pe0 , pv0 ), A (pu0 , pe0 , pv0 ), K (pu0 , pe0 , pv0 ), Y (pu0 , pe0 , pv0 ), and c (pu0 , pe0 , pv0 ) as follows:
Z
1−θ
U (pu0 , pe0 , pv0 ) = 1 − λ (pu0 , pe0 , pv0 ) γ+θ−1 n (pu0 , pe0 , pv0 ) dμ (pu0 , pe0 , pv0 )
1
λ (pu0 , pe0 , pv0 ) pv0
=
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩1 +
γ
K (pu0 , pe0 , pv0 ) = λ (pu0 , pe0 , pv0 ) γ+θ−1
Y
(pu0 , pe0 , pv0 )
γ
=
λ (pu0 , pe0 , pv0 ) γ+θ−1
Z
µ
Z
1
e v
A(pu
0 ,p0 ,p0 )
e v
U (pu
0 ,p0 ,p0 )
⎫ φ1
⎪
⎪
⎪
⎬
¶φ ⎪
⎪
⎪
⎭
k (pu0 , pe0 , pv0 ) dμ (pu0 , pe0 , pv0 )
sn(pu0 , pe0 , pv0 )γ k(pu0 , pe0 , pv0 )θ dμ (pu0 , pe0 , pv0 )
c (pu0 , pe0 , pv0 ) = Y (pu0 , pe0 , pv0 ) − δK (pu0 , pe0 , pv0 ) − A (pu0 , pe0 , pv0 )
Since λ (pu0 , pe0 , pv0 ) > λ (pu0 , pe0 , pv0 ), observe that U (pu0 , pe0 , pv0 ) > 0 and that A (pu0 , pe0 , pv0 ) > 0.
Step 13: Evaluate the function
f (pu0 , pe0 , pv0 ) = ϕc (pu0 , pe0 , pv0 )σ
+ (1 − π u ) βλ (pu0 , pe0 , pv0 ) [pe0 − pu0 ] Gu (A (pu0 , pe0 , pv0 ) , U (pu0 , pe0 , pv0 ) , A (pu0 , pe0 , pv0 ) , U (pu0 , pe0 , pv0 ))
− (1 − β) λ (pu0 , pe0 , pv0 ) pu
Steps 14-16 are the same as in Section 4.3.1.
49
5
Off-steady state dynamics
In this section it will be important to have separate notation for steady state variables. In particular, n∗ , n̄∗ , and v̄ ∗ will
denote steady state threshold functions and μ∗ will denote the invariant distribution. From Lemma 23, we know that μ∗
has a finite support given by
½
½
¾
¾
∗
∗
∗
0
P = (s, l, j) : s ∈ S, l ∈ N , and j ∈ 0∪ {max [v̄ (s ) − (1 − πn ) l, 0]} ∪ {0}
s ∈S
where
∗
N =
½
∪
s∈S
o¾ ½
n
k ∗
∪ (1 − π n ) n (s)
∪ ∪
M −1
s∈S
k=0
o¾ ½
n
k ∗
∪ (1 − πn ) n̄ (s)
∪ ∪
M −1
s∈S
k=0
o¾
n
k ∗
∪ (1 − π n ) v̄ (s)
∪ {0} ,
M −1
k=0
and where M is a natural number satisfying that
M
(1 − πn )
max {n̄∗ (smax ) , v̄ ∗ (smax )} < min {n∗ (smin ) , v̄ ∗ (smin )} .
(292)
In order to analyze off-steady state dynamics it will be useful to define nt , n̄t , and v̄t , as the threshold functions chosen
at date t. In addition, it will be useful to define the following minimum distance:
ε = min |a − b| ,
(293)
subject to
a, b ∈ D∗ and a 6= b,
where
D∗ = N ∗ ∪
½
∪
s∈S
o¾
n
M
M
M
.
(1 − πn ) n∗ (s) , (1 − πn ) n̄∗ (s) , (1 − π n ) v̄ ∗ (s)
The following Lemma characterizes the distribution μt+1 under the assumptions that μt and the finite history of
©
ª
thressholds nt−k , n̄t−k , v̄t−k k=0,1,...,M are sufficiently close to their steady-state values.
Lemma 27 Let M be defined by equation (292) and ε by equation (293).
Suppose that
¯
¯
¯nt−k (s) − n∗ (s)¯ < ε/2,
(294)
|n̄t−k (s) − n̄∗ (s)| < ε/2,
(295)
|v̄t−k (s) − v̄ ∗ (s)| < ε/2
(296)
for every s and every 0 ≤ k ≤ M + 1.
Suppose that the distribution μt has a finite support Pt given by
½
½
¾
¾
0
Pt = (s, l, j) : s ∈ S, l ∈ Nt , and j ∈ 0∪ {max [v̄t−1 (s ) − (1 − π n ) l, 0]} ∪ {0}
s ∈S
where
Nt
=
½
o¾ ½
n
k
∪ ∪
(1 − π n ) nt−k−1 (s)
s∈S k=0
s∈S
½
¾
o
n
M −1
k
∪
∪ (1 − π n ) v̄t−k−2 (s)
∪ {0} ,
∪
s∈S
M −1
∪
k=0
50
M −1
∪
k=0
o¾
n
k
∪
(1 − π n ) n̄t−k−1 (s)
In addition, suppose that for every (s, l, j) ∈ Pt :
μt (s, l, j) = μ∗ (s, l∗ , j ∗ ) ,
where (s, l∗ , j ∗ ) is the unique element of P ∗ satisfying that |l − l∗ | < ε/2 and |j − j ∗ | < ε/2 + (1 − π) ε/2.
Then, the distribution μt+1 has a finite support Pt+1 given by
½
½
¾
¾
0
Pt+1 = (s, l, j) : s ∈ S, l ∈ Nt+1 , and j ∈ 0∪ {max [v̄t (s ) − (1 − π n ) l, 0]} ∪ {0}
s ∈S
where
Nt+1
½
o¾ ½
o¾
n
M −1 n
k
k
=
∪
∪ (1 − π n ) nt−k (s)
∪ ∪
∪ (1 − π n ) n̄t−k (s)
∪
s∈S k=0
s∈S k=0
½
¾
o
M −1 n
k
∪
∪ (1 − π n ) v̄t−k−1 (s)
∪ {0} ,
M −1
s∈S
k=0
Moreover, for every (s, l, j) ∈ Pt+1 :
μt+1 (s, l, j) = μ∗ (s, l∗ , j ∗ )
where (s, l∗ , j ∗ ) is the unique element of P ∗ satisfying that |l − l∗ | < ε/2 and |j − j ∗ | < ε/2 + (1 − π) ε/2.
Proof. Observe that the optimal decision rules at period t − k are given by
⎫
⎧
⎬
⎨ min {(1 − π } l + j, n
n
t−k (s)
nt−k (s, l, j) = max
⎩ min {(1 − π ) l, n̄
(s)} ⎭
n
(297)
t−k
and
vt−k (s, l, j) = max {v̄t−k (s) − (1 − πn ) nt−k (s, l, j), 0} ,
(298)
for k = 0, 1, ..., M + 1.
a) We will first show that Pt+1 is a support of the distribution μt+1 .
Define the sets At and Bt+1 as follows:
At = ∪
s∈S
o
n
M −1
M −1
M −1
nt−M (s) , (1 − π n )
n̄t−M (s) , (1 − π n )
v̄t−M −1 (s) ,
(1 − πn )
Bt+1 = {l0 : l0 = (1 − π n ) l, for some l ∈ Nt /At } .
Observe that
Nt+1 = Bt+1 ∪
½
¾
∪ {nt (s) , n̄t (s) , v̄t−1 (s)} .
s∈S
To show that Pt+1 is a support of the distribution μt+1 it suffices to show that
½
¾
0
(s, l, j) ∈ Pt =⇒ nt (s, l, j) ∈ Nt+1 and vt (s, l, j) ∈ 0∪ {max [v̄t (s ) − (1 − π n ) nt (s, l, j) , 0]} ∪ {0} .
s ∈S
(299)
(300)
(301)
Let (s, l, j) ∈ Pt . Then,
s ∈ S, l ∈ Nt and j = max [v̄t−1 (s0 ) − (1 − π n ) l, 0] , for some s0 ∈ S.
51
(302)
From equation (297) we have that
⎧
⎫
⎨ min {(1 − π } l + j, n (s) ⎬
n
t
.
nt (s, l, j) = max
⎩ min {(1 − π ) l, n̄ (s)} ⎭
n
t
Using equation (302), we then have that
⎫
⎧
0
⎬
⎨ min {max [v̄
(s
)
,
(1
−
π
)
l]
,
n
(s)}
t−1
n
t
,
nt (s, l, j) = max
⎭
⎩
min {(1 − π ) l, n̄ (s)}
n
for some s0 ∈ S.
(303)
t
As a consequence,
nt (s, l, j) ∈ {(1 − π n ) l, n̄t (s) , nt (s) , v̄t−1 (s0 )}
for some s0 ∈ S.
From equations (299) and (300) we have that
l ∈ Nt /At ⇒nt (s, l, j) ∈ Nt+1 .
Suppose that l ∈ At . Without loss of generality assume that
l = (1 − πn )M −1 nt−M (s00 )
for some s00 ∈ S (the cases l = (1 − π n )M −1 n̄t−M (s00 ) and l = (1 − π n )M −1 v̄t−M −1 (s00 ) can be handled in exactly the
same way).
Then, equation (303) becomes
⎧
i
o ⎫
n
h
⎨ min max v̄t−1 (s0 ) , (1 − πn )M nt−M (s00 ) , nt (s) ⎬
n
o
nt (s, l, j) = max
.
⎭
⎩
min (1 − πn )M nt−M (s00 ) , n̄t (s)
But from equation (292) and equations (294)-(296), we have that
(1 − π n )M nt−M (s00 ) < (1 − π n )M n̄t−M (s00 )
≤ (1 − π n )
M
n̄t−M (smax )
≤ v̄t−1 (smin )
≤ v̄t−1 (s0 )
and that
(1 − π n )
M
nt−M (s00 ) < (1 − π n )
M
n̄t−M (s00 )
≤ (1 − π n )M n̄t−M (smax )
≤ nt (smin )
≤ nt (s)
< n̄t (s)
52
(304)
Therefore equation (304) becomes
⎧
⎫
0
⎨ min {v̄
⎬
t−1 (s ) , nt (s)} ,
nt (s, l, j) = max
⎩ (1 − π )M n
(s00 ) ⎭
n
t−M
0
= min {v̄t−1 (s ) , nt (s)} .
Thus, from equation (300), nt (s, l, j) ∈ Nt+1 .
From equation (298), observe that
vt (s, l, j) = max {v̄t (s) − (1 − π n ) nt (s, l, j), 0}
Thus,
vt (s, l, j) ∈
½
¾
0
∪
{max
[v̄
(s
)
−
(1
−
π
)
n
(s,
l,
j)
,
0]}
.
t
n
t
0
s ∈S
Therefore, Pt+1 is a support of the distribution μt+1 .
b) To prove the second part of the Proposition it will be convenient to define the following (one-to-one and onto)
functions.
For every (s, l, j) ∈ Pt :
lt∗ (l, j) = l∗
jt∗ (l, j) = j ∗
where (s, l∗ , j ∗ ) is the unique element of P ∗ satisfying that |l − l∗ | < ε/2 and |[(1 − π n ) l + j] − [(1 − πn ) l∗ + j ∗ ]| < ε/2.
Similarly, for every (s0 , l0 , j 0 ) ∈ Pt+1 :
∗
(l0 , j 0 ) = l∗
lt+1
∗
jt+1
(l0 , j 0 ) = j ∗
where (s0 , l∗ , j ∗ ) is the unique element of P ∗ satisfying that |l0 − l∗ | < ε/2 and |[(1 − π n ) l0 + j 0 ] − [(1 − π n ) l∗ + j ∗ ]| < ε/2.
Observe that, by assumption, we have that for every (s, l, j) ∈ Pt :
μt (s, l, j) = μ∗ (s, lt∗ (l, j), jt∗ (l, j)) .
(305)
We need to show that for every (s0 , l0 , j 0 ) ∈ Pt+1 :
¢
¡
∗
∗
(l0 , j 0 ), jt+1
(l0 , j 0 ) .
μt+1 (s0 , l0 , j 0 ) = μ∗ s0 , lt+1
Let (s0 , l0 , j 0 ) ∈ Pt+1 .
Using equation (305), we have that
μt+1 (s0 , l0 , j 0 ) =
X
(s,l,j) ∈ Gt
Q (s, s0 ) μ∗ (s, lt∗ (l, j), jt∗ (l, j)) + ψ (s0 ) I (l0 , j 0 ) ,
(l0 ,j 0 )
where
Gt (l0 , j 0 ) = {(s, l, j) ∈ Pt : nt (s, l, j) = l0 and vt (s, l, j) = j 0 } .
53
(306)
Also observe that
¢
¡
∗
∗
μ∗ s0 , lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 ) =
where
X
¡∗
¢
∗
Q (s, s0 ) μ∗ (s, l∗ , j ∗ ) + ψ (s0 ) I lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 ) ,
∗
0 0
0 0
(s,l∗ ,j ∗ ) ∈ G ∗ (l∗
t+1 (l ,j ),jt+1 (l ,j ))
©
ª
∗
∗
∗
∗
G ∗ (lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 )) = (s, l∗ , j ∗ ) ∈ P ∗ : n∗ (s, l∗ , j ∗ ) = lt+1
(l0 , j 0 ) and v ∗ (s, l∗ , j ∗ ) = jt+1
(l0 , j 0 ) .
To show that equation (306) holds, it then suffices to show that
¡∗
¢
∗
(l0 , j 0 ) = (0, 0) ⇔ lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 ) = (0, 0) ,
∗
∗
(s, l, j) ∈ Gt (l0 , j 0 ) =⇒ (s, lt∗ (l, j) , jt∗ (l, j)) ∈ G ∗ (lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 )),
³
´
∗
∗
(s, l∗ , j ∗ ) ∈ G ∗ (lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 )) =⇒ s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) ∈ Gt (l0 , j 0 ).
³
´
−1
−1
where [lt∗ ] , [jt∗ ]
is the inverse function of (lt∗ , jt∗ ).
b.1) Proof of equation (307).
∗
∗
It is a direct consequence of how lt+1
and jt+1
were defined and equations (294)-(296).
b.2) Proof of equation (308).
Let (s, l, j) ∈ Gt (l0 , j 0 ). Then, (s, l, j) ∈ Pt ,
⎫
⎧
⎨ min {(1 − π ) l + j, n (s)} ⎬
n
t
l0 = max
,
⎩ min {(1 − π ) l, n̄ (s)} ⎭
n
and
t
(1 − π n ) l0 + j 0 = max {v̄t (s) , (1 − π n ) l0 } .
Observe that (s, lt∗ (l, j) , jt∗ (l, j)) ∈ P ∗ ,
⎫
⎧
⎨ min {(1 − π ) l∗ (l, j) + j ∗ (l, j), n∗ (s)} ⎬
n t
t
n∗ (s, lt∗ (l, j), jt∗ (l, j)) = max
,
∗
⎭
⎩
min {(1 − π ) l (l, j), n̄∗ (s)}
n
and
t
(1 − π n ) n∗ (s, lt∗ (l, j), jt∗ (l, j)) + v ∗ (s, lt∗ (l, j), jt∗ (l, j))
= max {v̄ ∗ (s) , (1 − πn ) n∗ (s, lt∗ (l, j), jt∗ (l, j))} .
Since
|(1 − π n ) l − (1 − πn ) lt∗ (l, j)| < ε/2,
|[(1 − π n ) l + j] − [(1 − πn ) lt∗ (l, j) + jt∗ (l, j)]| < ε/2,
|nt (s) − n∗ (s)| < ε/2,
and
|n̄t (s) − n̄∗ (s)| < ε/2,
54
(307)
(308)
(309)
it follows that
|n∗ (s, lt∗ (l, j), jt∗ (l, j)) − l0 | < ε/2,
(310)
|[(1 − π n ) n∗ (s, lt∗ (l, j), jt∗ (l, j)) + v ∗ (s, lt∗ (l, j), jt∗ (l, j))] − [(1 − π n ) l0 + j 0 ]| < ε/2.
(311)
and, therefore, that
Since (s0 , l0 , j 0 ) ∈ Pt+1 and [s0 , n∗ (s, lt∗ (l, j), jt∗ (l, j)), v ∗ (s, lt∗ (l, j), jt∗ (l, j))] ∈ P ∗ , equations (310) and (311) imply that
∗
lt+1
(l0 , j 0 ) = n∗ (s, lt∗ (l, j), jt∗ (l, j)),
∗
jt+1
(l0 , j 0 ) = v ∗ (s, lt∗ (l, j), jt∗ (l, j)).
Since (s, lt∗ (l, j), jt∗ (l, j)) ∈ P ∗ it follows that
∗
∗
(s, lt∗ (l, j), jt∗ (l, j)) ∈ G ∗ (lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 )).
b.3) Proof of equation (309).
∗
∗
Let (s, l∗ , j ∗ ) ∈ G ∗ (lt+1
(l0 , j 0 ), jt+1
(l0 , j 0 )). Then, (s, l∗ , j ∗ ) ∈ P ∗ ,
∗
lt+1
(l0 , j 0 ) = n∗ (s, l∗ , j ∗ )
⎫
⎧
⎨ min {(1 − π ) l∗ + j ∗ , n∗ (s)} ⎬
n
,
= max
⎭
⎩
min {(1 − π ) l∗ , n̄∗ (s)}
(312)
(313)
n
and
∗
∗
(l0 , j 0 ) + jt+1
(l0 , j 0 ) = (1 − π n ) n∗ (s, l∗ , j ∗ ) + v ∗ (s, l∗ , j ∗ )
(1 − πn ) lt+1
©
ª
∗
= max v̄ ∗ (s) , (1 − πn ) lt+1
(l0 , j 0 ) .
³
´
Observe that s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) ∈ Pt ,
and
⎧
o ⎫
n
⎨ min (1 − πn ) [lt∗ ]−1 (l∗ , j ∗ ) + [jt∗ ]−1 (l∗ , j ∗ ) , nt (s) ⎬
o
n
nt (s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ )) = max
,
⎭
⎩
min (1 − π n ) [lt∗ ]−1 (l∗ , j ∗ ), n̄t (s)
(1 − π n ) nt (s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ )) + vt (s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ))
n
o
= max v̄t (s) , (1 − πn ) nt (s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ )) .
Also, from equation (301), we have that
h
h
h
i
ii
s0 , nt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) , vt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) ∈ Pt+1
for every s0 .
Moreover,
i
i
h
h
−1
−1
−1
−1
∗
lt+1
(nt s, [lt∗ ] (l∗ , j ∗ ), [jt∗ ] (l∗ , j ∗ ) , vt s, [lt∗ ] (l∗ , j ∗ ), [jt∗ ] (l∗ , j ∗ ) )
= n∗ (s, l∗ , j ∗ )
55
(314)
(315)
and
i
i
h
h
∗
jt+1
(nt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) , vt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) )
= v ∗ (s, l∗ , j ∗ )
Hence, from equations (312) and (314), we have that
i
i
h
h
−1
−1
−1
−1
∗
lt+1
(nt s, [lt∗ ] (l∗ , j ∗ ), [jt∗ ] (l∗ , j ∗ ) , vt s, [lt∗ ] (l∗ , j ∗ ), [jt∗ ] (l∗ , j ∗ ) )
∗
= lt+1
(l0 , j 0 )
and
h
h
i
i
∗
jt+1
(nt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) , vt s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) )
∗
= jt+1
(l0 , j 0 )
It follows that
l0 = nt (s, [lt∗ ]
−1
−1
(l∗ , j ∗ ), [jt∗ ]
(l∗ , j ∗ )),
j 0 = vt (s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ )).
³
´
−1
−1
Since s, [lt∗ ] (l∗ , j ∗ ), [jt∗ ] (l∗ , j ∗ ) ∈ Pt it follows that
³
´
s, [lt∗ ]−1 (l∗ , j ∗ ), [jt∗ ]−1 (l∗ , j ∗ ) ∈ Gt (l0 , j 0 ).
56
References
[1] Easley, D. and D. Spulber. 1981. Stochastic equilibrium and optimality with rolling plans. International Economic
Review 22:79-103.
57
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WP-06-24
What Can We Learn about Financial Access from U.S. Immigrants?
Una Okonkwo Osili and Anna Paulson
WP-06-25
Bank Imputed Interest Rates: Unbiased Estimates of Offered Rates?
Evren Ors and Tara Rice
WP-06-26
Welfare Implications of the Transition to High Household Debt
Jeffrey R. Campbell and Zvi Hercowitz
WP-06-27
Last-In First-Out Oligopoly Dynamics
Jaap H. Abbring and Jeffrey R. Campbell
WP-06-28
Oligopoly Dynamics with Barriers to Entry
Jaap H. Abbring and Jeffrey R. Campbell
WP-06-29
Risk Taking and the Quality of Informal Insurance: Gambling and Remittances in Thailand
Douglas L. Miller and Anna L. Paulson
WP-07-01
2
Working Paper Series (continued)
Fast Micro and Slow Macro: Can Aggregation Explain the Persistence of Inflation?
Filippo Altissimo, Benoît Mojon, and Paolo Zaffaroni
WP-07-02
Assessing a Decade of Interstate Bank Branching
Christian Johnson and Tara Rice
WP-07-03
Debit Card and Cash Usage: A Cross-Country Analysis
Gene Amromin and Sujit Chakravorti
WP-07-04
The Age of Reason: Financial Decisions Over the Lifecycle
Sumit Agarwal, John C. Driscoll, Xavier Gabaix, and David Laibson
WP-07-05
Information Acquisition in Financial Markets: a Correction
Gadi Barlevy and Pietro Veronesi
WP-07-06
Monetary Policy, Output Composition and the Great Moderation
Benoît Mojon
WP-07-07
Estate Taxation, Entrepreneurship, and Wealth
Marco Cagetti and Mariacristina De Nardi
WP-07-08
Conflict of Interest and Certification in the U.S. IPO Market
Luca Benzoni and Carola Schenone
WP-07-09
The Reaction of Consumer Spending and Debt to Tax Rebates –
Evidence from Consumer Credit Data
Sumit Agarwal, Chunlin Liu, and Nicholas S. Souleles
WP-07-10
Portfolio Choice over the Life-Cycle when the Stock and Labor Markets are Cointegrated
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein
WP-07-11
Nonparametric Analysis of Intergenerational Income Mobility
with Application to the United States
Debopam Bhattacharya and Bhashkar Mazumder
WP-07-12
How the Credit Channel Works: Differentiating the Bank Lending Channel
and the Balance Sheet Channel
Lamont K. Black and Richard J. Rosen
WP-07-13
Labor Market Transitions and Self-Employment
Ellen R. Rissman
WP-07-14
First-Time Home Buyers and Residential Investment Volatility
Jonas D.M. Fisher and Martin Gervais
WP-07-15
Establishments Dynamics and Matching Frictions in Classical Competitive Equilibrium
Marcelo Veracierto
WP-07-16
Technology’s Edge: The Educational Benefits of Computer-Aided Instruction
Lisa Barrow, Lisa Markman, and Cecilia Elena Rouse
WP-07-17
3
Working Paper Series (continued)
The Widow’s Offering: Inheritance, Family Structure, and the Charitable Gifts of Women
Leslie McGranahan
Demand Volatility and the Lag between the Growth of Temporary
and Permanent Employment
Sainan Jin, Yukako Ono, and Qinghua Zhang
WP-07-18
WP-07-19
A Conversation with 590 Nascent Entrepreneurs
Jeffrey R. Campbell and Mariacristina De Nardi
WP-07-20
Cyclical Dumping and US Antidumping Protection: 1980-2001
Meredith A. Crowley
WP-07-21
Health Capital and the Prenatal Environment:
The Effect of Maternal Fasting During Pregnancy
Douglas Almond and Bhashkar Mazumder
WP-07-22
The Spending and Debt Response to Minimum Wage Hikes
Daniel Aaronson, Sumit Agarwal, and Eric French
WP-07-23
The Impact of Mexican Immigrants on U.S. Wage Structure
Maude Toussaint-Comeau
WP-07-24
A Leverage-based Model of Speculative Bubbles
Gadi Barlevy
WP-08-01
Displacement, Asymmetric Information and Heterogeneous Human Capital
Luojia Hu and Christopher Taber
WP-08-02
BankCaR (Bank Capital-at-Risk): A credit risk model for US commercial bank charge-offs
Jon Frye and Eduard Pelz
WP-08-03
Bank Lending, Financing Constraints and SME Investment
Santiago Carbó-Valverde, Francisco Rodríguez-Fernández, and Gregory F. Udell
WP-08-04
Global Inflation
Matteo Ciccarelli and Benoît Mojon
WP-08-05
Scale and the Origins of Structural Change
Francisco J. Buera and Joseph P. Kaboski
WP-08-06
Inventories, Lumpy Trade, and Large Devaluations
George Alessandria, Joseph P. Kaboski, and Virgiliu Midrigan
WP-08-07
School Vouchers and Student Achievement: Recent Evidence, Remaining Questions
Cecilia Elena Rouse and Lisa Barrow
WP-08-08
4
Working Paper Series (continued)
Does It Pay to Read Your Junk Mail? Evidence of the Effect of Advertising on
Home Equity Credit Choices
Sumit Agarwal and Brent W. Ambrose
WP-08-09
The Choice between Arm’s-Length and Relationship Debt: Evidence from eLoans
Sumit Agarwal and Robert Hauswald
WP-08-10
Consumer Choice and Merchant Acceptance of Payment Media
Wilko Bolt and Sujit Chakravorti
WP-08-11
Investment Shocks and Business Cycles
Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti
WP-08-12
New Vehicle Characteristics and the Cost of the
Corporate Average Fuel Economy Standard
Thomas Klier and Joshua Linn
WP-08-13
Realized Volatility
Torben G. Andersen and Luca Benzoni
WP-08-14
Revenue Bubbles and Structural Deficits: What’s a state to do?
Richard Mattoon and Leslie McGranahan
WP-08-15
The role of lenders in the home price boom
Richard J. Rosen
WP-08-16
Bank Crises and Investor Confidence
Una Okonkwo Osili and Anna Paulson
WP-08-17
Life Expectancy and Old Age Savings
Mariacristina De Nardi, Eric French, and John Bailey Jones
WP-08-18
Remittance Behavior among New U.S. Immigrants
Katherine Meckel
WP-08-19
Birth Cohort and the Black-White Achievement Gap:
The Roles of Access and Health Soon After Birth
Kenneth Y. Chay, Jonathan Guryan, and Bhashkar Mazumder
WP-08-20
Public Investment and Budget Rules for State vs. Local Governments
Marco Bassetto
WP-08-21
Why Has Home Ownership Fallen Among the Young?
Jonas D.M. Fisher and Martin Gervais
WP-09-01
Why do the Elderly Save? The Role of Medical Expenses
Mariacristina De Nardi, Eric French, and John Bailey Jones
WP-09-02
Using Stock Returns to Identify Government Spending Shocks
Jonas D.M. Fisher and Ryan Peters
WP-09-03
5
Working Paper Series (continued)
Stochastic Volatility
Torben G. Andersen and Luca Benzoni
WP-09-04
The Effect of Disability Insurance Receipt on Labor Supply
Eric French and Jae Song
WP-09-05
CEO Overconfidence and Dividend Policy
Sanjay Deshmukh, Anand M. Goel, and Keith M. Howe
WP-09-06
Do Financial Counseling Mandates Improve Mortgage Choice and Performance?
Evidence from a Legislative Experiment
Sumit Agarwal,Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-09-07
Perverse Incentives at the Banks? Evidence from a Natural Experiment
Sumit Agarwal and Faye H. Wang
WP-09-08
Pay for Percentile
Gadi Barlevy and Derek Neal
WP-09-09
The Life and Times of Nicolas Dutot
François R. Velde
WP-09-10
Regulating Two-Sided Markets: An Empirical Investigation
Santiago Carbó Valverde, Sujit Chakravorti, and Francisco Rodriguez Fernandez
WP-09-11
The Case of the Undying Debt
François R. Velde
WP-09-12
Paying for Performance: The Education Impacts of a Community College Scholarship
Program for Low-income Adults
Lisa Barrow, Lashawn Richburg-Hayes, Cecilia Elena Rouse, and Thomas Brock
Establishments Dynamics, Vacancies and Unemployment: A Neoclassical Synthesis
Marcelo Veracierto
WP-09-13
WP-09-14
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