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Federal Reserve Bank of Chicago
Firing Costs and Business Cycle
Fluctuations
Marcelo Veracierto
WP 2003-29
Firing Costs and Business Cycle Fluctuations∗
Marcelo Veracierto
Federal Reserve Bank of Chicago
This version: July, 2004
Abstract: This paper considers a real business cycle model with establishment
level dynamics and uses it to analyze the effects of firing taxes. It finds that the
firing taxes have significant consequences on business cycle fluctuations. The
largest effects are on aggregate employment, which becomess less variable and
more persistent. While the business cycle effects of firing taxes are sizable, their
welfare consequences are completely dominated by their steady state effects.
∗
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
There is a large literature evaluating the consequences of firing costs on long-run labor
market outcomes. Its main purpose: To explain the differences in unemployment rates
observed between Europe and the U.S. While this an important objective and there has been
considerable progress in this line of research, this paper changes the focus of the analysis
and evaluates the effects of firing costs on business cycle dynamics. There are two reasons
for doing this. First, policy makers often justify introducing firing costs as an effective tool
for reducing the magnitude of economic downturns. Thus, it seems important to evaluate
the rationale behind this claim. Second, there are features of the data, besides long-run
labor market outcomes, that firing costs might help explain. In fact, employment protection
levels do seem to be related to business cycle differences across countries: Figure 1 plots the
standard deviation of output against the employment protection level for seventeen OECD
countries, showing that they are negatively related. While this relation is highly suggestive,
it does not provide evidence that firing costs affect business cycle dynamics in any important
way. A third factor (e.g. high risk aversion) could be generating lower output fluctuations and
leading countries to adopt higher employment protection levels. Thus, a negative relation in
Figure 1 could be obtained while firing costs have no effects. To determine the effects of firing
costs on business cycle fluctuations, analysis is needed. This paper provides such analysis:
It describes a real business cycle model with establishment level dynamics, introduces firing
costs and evaluates their effects.
The model is a stochastic version of Veracierto [16], which in turn is based on Hopenhayn
and Rogerson [9]. 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 establishments that use capital and labor as inputs into a decreasing
returns to scale technology. Establishments are subject both to idiosyncratic and aggregate
productivity shocks. In the benchmark case, both capital and labor are freely movable across
establishments.
Once the benchmark model is parametrized to U.S. data, firing taxes ranging from one
month to one year of wages are introduced. The paper finds that firing taxes have consid-
erable business cycle effects. In particular, firing taxes equal to one year of wages reduce
the standard deviation of output by 10.7%, the standard deviation of investment by 14.7%
and the standard deviation of employment by 30.6%. Also, the firing taxes make aggregate employment more persistent: its first order autocorrelation increases from 0.66 to 0.71.
These findings suggest that firing costs could play a significant role in explaining differences
in business cycle fluctuations across countries.
This is not the first paper analyzing the effects of firing costs on business cycle dynamics.
In a partial equilibrium setting, Campbell and Fisher [4] studied how firing costs affect the
aggregate behavior of a large number of establishments subject to idiosyncratic productivity
shocks and a shock to the aggregate wage rate. Their focus was on the volatility of job
destruction relative to the volatility of job creation, finding that firing costs increase it.
Cabrales and Hopenhayn [3] analyzed a similar type of model except that the aggregate
shock was in the aggregate productivity level instead of the wage rate. Contrary to Campbell
and Fisher [4], they found that the firing costs decrease the volatility of job destruction
relative to the volatility of job creation. This paper differs from Campbell and Fisher [4] and
Cabrales and Hopenhayn [3] in that it performs a general equilibrium analysis where both
aggregate productivity and the wage rate are changing. In terms of the relative volatility
of job destruction, this paper obtains results that are closer to Cabrales and Hopenhayn [3]
than to Campbell and Fisher [4].
Current work by Samaniego [15] is more closely related. Samaniego also considers a
version of Veracierto [16] and performs a general equilibrium analysis.1 However, he studies how firing taxes affect the deterministic transitionary dynamics after a large persistent
change in aggregate productivity. This paper, on the contrary, computes the full stochastic
equilibrium of a real business cycle model. An advantage of this approach is that it allows to
evaluate how firing taxes affect standard business cycle statistics. Another advantage is that
it allows to assess the welfare benefits of reducing business cycle fluctuations. Despite the
differences, all these papers share a basic result: Firing taxes are found to lower the response
1
The models differ in that Samaniego [15] allows for endogenous entry and exit while this paper treats
them as exogenously determined. Another difference is that this paper gives firms a “quits allowance” before
being subject to firing taxes, while Samaniego doesn’t.
2
of the economy to aggregate productivity changes.
The rest of the paper is organized as follows. Section 2 describes the environment. Section
3 defines a competitive equilibrium and describes the computational strategy. Section 4
calibrates the benchmark economy. Finally, Section 5 introduces firing taxes and evaluates
their effects. An appendix provides the proofs to all the claims made in the paper.
2. The economy
The economy is populated by a unit measure of ex-ante identical agents with preferences
given by
E
∞
X
β t [ln ct + v(lt )] ,
t=0
where ct is consumption, lt is leisure and 0 < β < 1 is the discount factor. Every period
agents are endowed with ω units of time. Given an institutionally determined workweek of
length equal to one, leisure can only take values ω or ω − 1.2
Output, which can be consumed or invested, is produced by a large number of establishments with production function given by
θ
yt = ezt st gt nγ ,
t
where zt ∈ Z is an aggregate productivity shock, st ∈ S = {0, smin , ..., smax } is an idiosyncratic productivity shock, gt is capital, nt is labor, θ > 0, γ > 0, and θ+γ < 1. The aggregate
productivity shock zt ∈ Z, which is common to all establishments, follows a finite Markov
process with transition matrix H. The idiosyncratic productivity shock st ∈ S also follows
a finite Markov process, but with transition matrix Q. Realizations of st are assumed to be
independent across all establishments and st = 0 is assumed to be an absorbing state. Since
there are no fixed costs of operation, establishments will exit only when their idiosyncratic
productivity becomes zero. Every period ν new establishments are exogenously born. The
distribution over initial idiosyncratic productivity levels is given by ψ.
2
In order to analyze the effects of firing taxes it is important to assume that labor is indivisible: It allows
to associate changes in the labor input of establishments with changes in employment.
3
3. Competitive equilibrium
In this section I describe a competitive equilibrium where establishments are subject to
firing taxes and the proceeds are rebated to households as lump sum transfers. Following
Hopenhayn and Rogerson [9], firing taxes are modelled as a tax on reducing employment.
In particular, whenever an establishment makes its current employment level nt lower than
(1 − q)nt−1 it must pay a tax rate τ on the difference. Observe that q is a policy parameter
specifying a contraction rate below which establishments are not subject to firing taxes.
Hereon, I will refer to q as the “quit rate of workers”.3
In order to define a competitive equilibrium I will index the history of an individual
establishment by sa = (s0 , ..., sa ) ∈ S a+1 , where sj is the idiosyncratic productivity that the
establishment had when it was of age j. Also, the history of aggregate productivity levels
since date 0 will be denoted by z t = (z0 , ..., zt ) ∈ Z t+1 , where zj is the aggregate productivity
level that the economy had at date j.
Following Hansen [7] and Rogerson [14], I assume that agents trade employment lotteries. This makes the preferences of the representative household linear with respect to the
probability of working η t .4 The problem of the representative household at date 0 is then
given by the following equation:
(
max ln c0 − αη 0 +
∞
XX
t=1
zt
#)
" t
£
¡ t¢
¡ t ¢¤ Y
β t ln ct z − αη t z
H(zj−1 , zj )
(3.1)
j=1
subject to:
¡ ¢
¡ ¢
¡
¢ X
¡
¢
pt (z t ; zt+1 )bt+1 z t , zt+1
ct z t + kt+1 z t − (1 − δ) kt z t−1 +
zt+1
¡ ¢ ¡ ¢
¡ ¢ ¡
¢
¡ ¢
¡ ¢
¡ ¢
≤ wt z t η t z t + rt z t kt z t−1 + bt z t + Dt z t + Tt z t
3
(3.2)
The parameter q will be actually calibrated to the quit rate of workers since, in practice, establishments
do not have to pay firing taxes on quits. Assuming a positive q is not only a considerable gain in realism,
but will make the problem of computing a competitive equilibrium tractable.
4
In particular, α in equation (3.1) is given by v(ω) − v(ω − 1).
4
b0 = 0, k0 , and z0 given,
where kt is the capital owned by the household, pt (·, zt+1 ) is the price of an Arrow security
which delivers one unit of the consumption good if zt+1 is realized, bt (·, zt+1 ) are the purchases
of this type of security, wt is the wage rate, rt is the rental rate of capital, Dt are profits and
Tt are the lump sum transfers from the government.
Establishments maximize expected discounted profits net of firing taxes. The problem of
an establishment of age a and idiosyncratic history sa (when the aggregate history is given
by z t ) is described by the following equation:
¡
¢θ
¡
¢γ
¡ ¢
¡
¢
¡ ¢
¡
¢
¡
¢
max {ezt sa ga,t sa , z t na,t sa , z t − wt z t na,t sa , z t − rt z t ga,t sa , z t − τ fa,t sa , z t
+
" j
∞
XXX Y
j=1 sa+j z t+j
#
¡
¢θ
¡
¢γ
pt+h−1 (z t+h−1 ; zt+h ) [ezt+j sa+j ga+j,t+j sa+j , z t+j na+j,t+j sa+j , z t+j
h=1
¡
¢
¡
¢
¡
¢
¡
¢
−wt+j z t+j na+j,t+j sa+j , z t+j − rt+j z t+j ga+j,t+j sa+j , z t+j
#
" j
¡ a+j t+j ¢ Y
−τ fa+j,t+j s , z
Q(sa+h−1 , sa+h ) }
]
h=1
subject to:
¡
¢
¡
¢
¡
¢
na+j,t+j sa+j , z t+j ≥ (1 − q) na+j−1,t+j−1 sa+j−1 , z t+j−1 − fa+j,t+j sa+j , z t+j ,
¡
¢
fa+j,t+j sa+j , z t+j ≥ 0,
(3.3)
(3.4)
¡
¢
na−1,t−1 sa−1 , z t−1 given,
where f is the amount of firing done by the establishment. Observe that the establishment
cannot reduce its employment level below its previous period employment level (net of quits)
without firing workers and paying the associated taxes. Although the above problem was
defined for any initial condition, it must be the case that
¡
¢
na−1,t−1 sa−1 , z t−1 = 0, when a = 0,
5
(3.5)
since establishments are born with zero previous period employment. Also, observe that
at t = 0, establishments of age a and history sa take their previous employment level
na−1,−1 (sa−1 , z −1 ) as given.
In order to aggregate the behavior of all establishments it will be important to describe the
distribution µ of establishments across ages a and idiosyncratic histories sa . This distribution
satisfies the following equations:
µa+1 (sa+1 ) = Q(sa , sa+1 )µa (sa ), for every a ≥ 0 and sa+1 ,
¡ ¢
µ0 s0 = νψ(s0 ).
Observe that the number of establishments of age 0 and productivity s0 is given by the
arrival of new establishments ν times the probability of drawing an initial productivity equal
to s0 .
The consumption good market clearing condition is then given by
¡ ¢
¡ ¢
¡
¢ XX zt
¡
¢θ
¡
¢γ
ct z t + kt+1 z t − (1 − δ) kt z t−1 =
e sa ga,t sa , z t na,t sa , z t µa (sa ).
(3.6)
a≥0 sa
This condition states that aggregate consumption plus aggregate investment must be equal
to the production of all establishments.
The capital market clearing condition is
XX ¡
¢
¡
¢
ga,t sa , z t µa (sa ) = kt z t−1 .
(3.7)
a≥0 sa
That is, the total amount of capital rented by the establishments must be equal to the stock
of capital supplied by the families.
Similarly, the market clearing condition for the labor market is given by
XX
¡
¢
¡ ¢
na,t sa , z t µa (sa ) = η t z t .
a≥0
sa
6
(3.8)
The securities market clearing condition is simply
¡
¢
bt+1 z t+1 = 0,
(3.9)
since households are identical.
As was already mentioned, the government rebates to the households all the firing taxes
collected from the establishments. The budget constraint of the government is then the
following:
XX ¡
¡ ¢
¢
Tt z t = τ
fa,t sa , z t µa (sa ).
a≥0
(3.10)
sa
Finally, the profits received by the representative household must be equal to the profits
made by all the establishments in the economy:
¡ ¢ XX zt
¡
¢θ
¡
¢γ
¡ ¢
¡
¢
Dt z t =
[e sa ga,t sa , z t na,t sa , z t − wt z t na,t sa , z t
a≥0 sa
¡ ¢
¡
¢
¡
¢
−rt z t ga,t sa , z t − τ fa,t sa , z t ]µa (sa ).
(3.11)
3.1. A quasi-planner equilibrium
While the competitive equilibrium with firing taxes described above seems a difficult object
to analyze, it can be simplified quite substantially. It is straightforward to show that if {ct ,
kt+1 , η t , gt , nt , ft , bt+1 , wt , rt , pt , Dt , Tt }∞ is a competitive equilibrium, then {ct , kt+1 , η t ,
t=0
gt , nt , ft }∞ solves the following quasi-planner problem:
t=0
(
max ln c0 − αη 0 +
∞
XX
t=1
zt
#)
" t
£
¡ t¢
¡ t ¢¤ Y
β t ln ct z − αη t z
H(zj−1 , zj )
(3.12)
j=1
subject to
¡ ¢
¡ ¢
¡
¢
ct z t + kt+1 z t − (1 − δ) kt z t−1
XX h
¡ a t ¢θ
¡ a t ¢γ
¡ a t ¢i
¡ ¢
zt
e sa ga,t s , z na,t s , z − τ fa,t s , z µa (sa ) + Tt z t
≤
(3.13)
a≥0 sa
¡
¢
¡
¢
¡
¢
na,t sa , z t ≥ (1 − q) na−1,t−1 sa−1 , z t−1 − fa,t sa , z t ,
7
(3.14)
XX
¡
¢
¡ ¢
na,t sa , z t µa (sa ) = η t z t
(3.15)
a≥0 sa
XX ¡
¢
¡
¢
ga,t sa , z t µa (sa ) = kt z t−1
(3.16)
a≥0 sa
¡
¢
fa,t sa , z t ≥ 0,
¡
¢
na−1,t−1 sa−1 , z t−1 = 0, for a = 0
k0 , z0 , and
(3.17)
(3.18)
©
¡
¢ª
na−1,−1 sa−1 , z −1 a,sa given.
The converse is also true. If {ct , kt+1 , η t , gt , nt , ft }∞ solves the above quasi-planner
t=0
problem for some stochastic process Tt and the following condition is satisfied
XX ¡
¡ ¢
¢
fa,t sa , z t µa (sa ),
Tt z t = τ
(3.19)
a≥0 sa
then {ct , kt+1 , η t , gt , nt , ft , bt+1 , wt , rt , pt , Dt , Tt }∞ is a competitive equilibrium for some
t=0
{bt+1 , wt , rt , pt , Dt }∞ .5
t=0
3.2. A recursive competitive equilibrium
In order to compute a competitive equilibrium it will be useful to work with a recursive
formulation to the quasi-planner equilibrium described above. Since the quasi-planner problem (3.12) is convex, establishments that have different idiosyncratic histories and\or ages
but that have identical previous period employment and current idiosyncratic productivity
levels will be treated as being identical by the quasi-planner, i.e. they will be assigned the
same contingent employment plan.6 As a result, in the recursive formulation that follows,
I will index establishments by their previous period employment level u and their current
idiosyncratic productivity level s.
The individual state of the representative quasi-planner is then given by the stock of
capital k and a measure x describing the distribution of establishments across types (u, s).
5
Appendix A provides a formal proof for this equivalence result.
6
For a proof, see Appendix B.
8
The aggregate state of the economy is given by the economy-wide capital level K, the
economy-wide distribution of establishments X, and the aggregate productivity shock z.
The problem faced by the representative quasi-planner is given by the following dynamic
programming problem:
v(z, K, X, k, x) = max
c,η,g,n,i
(
ln c − αη + β
X
)
v(z 0 , K 0 , X 0 , k0 , x0 )H(z, z 0 )
z0
subject to
Z n
o
c+i≤
ez sg (u, s)θ n (u, s)γ − τ max [0, (1 − q)u − n(u, s)] dx + T (z, K, X)
Z
n(u, s)dx ≤ η
(3.21)
g(u, s)dx ≤ k
(3.22)
k0 = (1 − δ)k + i
(3.23)
Z
x0 (U 0 × {s0 }) =
Z
T (z, K, X) =
0
0
0
X (U × {s }) =
Z
Z
(3.20)
(u,s):n(u,s)∈U 0
Q(s, s0 )dx + χ (0 ∈ U 0 ) νψ (s0 )
τ max [0, (1 − q)u − N(u, s; z, K, X)] dX
K 0 = (1 − δ)K + I (z, K, X)
(u,s):N(u,s;z,K,X)∈U 0
Q(s, s0 )dx + χ (0 ∈ U 0 ) νψ (s0 )
(3.24)
(3.25)
(3.26)
(3.27)
where χ is an indicator function that takes value equal to one if the argument is true and
zero otherwise. Observe that, aside from the aggregate state of the economy (z, K, X),
the representative quasi-planner takes the economy-wide employment decision rule N and
economy-wide investment decision rule I as given.
In a recursive competitive equilibrium, expectations must be rational:
N(u, s; z, K, X) = n(u, s; z, K, X, K, X)
9
and
I (z, K, X) = i (z, K, X, K, X) .
That is, the economy-wide decision rules N and I must be generated by the decision rules
n and i of the representative quasi-planner.
3.3. Computational strategy
Observe that, conditional on n, k, and z, the optimal capital allocation rule g is obtained
R
by maximizing aggregate output ez sg θ nγ dx subject to the feasibility constraint (3.22).
Substituting this solution and equation (3.25) into equation (3.20) and then substituting the
resulting expression together with equation (3.21) into the one-period return function
R = ln ct − αη t
allows to write the return function as a function of (z, K, X, k, x, N, n, i).
The problem of the representative quasi-planner can then be written as
v(z, K, X, k, x) = max {R(z, K, X, k, x, N, n, i) + βE [v(z 0 , K 0 , X 0 , k0 , x0 ) | z]}
(3.28)
n,i
subject to equations (3.23), (3.24), (3.26) and (3.27).
The high dimensionality of the state space seems to preclude any possibility of computing
a recursive competitive equilibrium. However, two features of the problem will render it
tractable. The first is the nature of the employment decision rule n. Appendix C shows that
the employment decision rule is fully characterized by a pair of threshold functions n and n
¯
as follows
n(u, s; z, K, X, k, x) =
n(s; z, K, X, k, x), if (1 − q)u > n(s; z, K, X, k, x)
¯
¯
n(s; z, K, X, k, x), if (1 − q)u < n(s; z, K, X, k, x)
(1 − q)u, otherwise
.
(3.29)
Observe that the upper and lower thresholds n and n do not depend on the previous em¯
ployment level u. The (S,s) nature of the employment decision rule is critical for making the
10
decision variables in (3.28) finite dimensional: instead of letting the quasi-planner choose
a generic function n, there will be no loss of generality in constraining it to choose finite
dimensional thresholds n and n and defining the employment decision rule n as in equation
¯
(3.29).
The second property that makes the problem tractable is that, if the aggregate productivity z fluctuations are sufficiently small, along a stationary equilibrium the distribution x
will always have a finite support. To see this more clearly it will be convenient to consider
the deterministic steady state of an economy where the aggregate productivity level z is
constant and equal to zero. Hereon, any variable superscripted with a star (∗) will refer to
its corresponding deterministic steady state value. Before proceeding I state the following
result.
Proposition 3.1. In a deterministic steady state equilibrium, the invariant distribution x∗
has a finite support given by the union of {0} and the following set:
©
ª
©
ª
¯
m∗ = (1 − q)h n∗ (s) s=smin ,...,smax ∪ (1 − q)h n∗ (s) s=smin ,...,smax
h=1,...,Ω(s)
¯
h=1,...,Ω(s)
where Ω(s) is the lowest natural number satisfying that
(1 − q)Ω(s) n∗ (s) < n∗ (smin )
¯
and Ω(s) is the lowest natural number satisfying that
¯
¯
(1 − q)Ω(s) n∗ (s) < n∗ (smin )
Proof: See Appendix D.
Hereon, I will assume that m∗ is a vector conveniently ordered. I will refer to m∗ (j) as
the jth element of m∗ and the total number of elements in m∗ will be denoted by J. Also,
it will be useful to classify the elements of m∗ into three sets: 1) those that correspond
to establishments that expand (set G ∗ ), 2) those that correspond to establishments that
contract (set C ∗ ), and 3) those that correspond to establishments that remain inactive (set
11
I ∗ ). That is, for j = 1, ..., J:
j ∈ G ∗ , if m∗ (j) = (1 − q)n∗ (s) for some s ≥ smin
j ∈ C ∗ , if m∗ (j) = (1 − q)¯ ∗ (s) for some s ≥ smin
n
j ∈ I ∗ , if m∗ (j) = (1 − q)m(j − 1)
(3.30)
Observe that equation (3.30) defines an implicit ordering for m∗ .
Suppose that, at some date t, the state variable xt has a finite support given by mt (and
the singleton {0}), that mt has dimension J (same dimension as m∗ ), that mt is close to m∗
and that
xt ({0} , s) = x∗ ({0} , s) , for every s ∈ S
(3.31)
xt ({mt (j)} , s) = x∗ ({m∗ (j)} , s) , for every s ∈ S and every j = 1, ..., J
(3.32)
xt = 0, everywhere else.
¯
¯
In addition, assume that nt and nt are close to their steady state values n∗ and n∗ .7 Then,
the next period finite support mt+1 will be given by
mt+1 (j) =
(1 − q)nt (s) , if j ∈ G
∗
(1 − q)¯ t (s) , if j ∈ C ∗
n
(1 − q)m (j − 1), if j ∈ I ∗
t
, for j = 1, ..., J,
(3.33)
where s in the first line satisfies that m∗ (j) = (1 − q)n∗ (s) and s in the second line satisfies
that m∗ (j) = (1 − q)¯ ∗ (s). By continuity, mt+1 will be close to m∗ and xt+1 will satisfy that
n
xt+1 ({0} , s) = x∗ ({0} , s), for every s ∈ S
xt+1 ({mt+1 (j)} , s) = x∗ ({m∗ (j)} , s) , for every s ∈ S and every j = 1, ..., J
xt+1 = 0, everywhere else.
7
For s = 0 I assume without loss of generality that n(s) = n(s) = 0, i.e. that the employment thresholds
¯
are identical to their deterministic steady state values.
12
¯
Assuming that nt , nt , mt , kt , it , N t , Nt , Mt , Kt , It fluctuate in a small neighborhood
¯
of their deterministic steady state values, the original representative quasi-planner problem
(3.28) can then be replaced by the following transformed problem:
n
o
e
¯
v(z, K, M, k, m) = max R(z, K, M, k, m, N, N, I, n, n, i) + βE [v(z 0 , K 0 , M 0 , k0 , m0 ) | z]
¯
n,¯ ,i
n
(3.34)
subject to
m0 (j) =
(1 − q)n (s) , if j ∈ G
∗
(1 − q)¯ (s) , if j ∈ C ∗
n
(1 − q)m(j − 1), if j ∈ I ∗
k0 = (1 − δ)k + i
(1 − q)N (s; z, K, M) , if j ∈ G ∗
0
¯
M (j) =
(1 − q)N (s; z, K, M) , if j ∈ C ∗
(1 − q)M(j − 1), if j ∈ I ∗
, for j = 1, ..., J,
, for j = 1, ..., J,
K 0 = (1 − δ)K + I (z, K, M) .
¯
where the decision variables n and n are defined over s ≥ smin .8 The conditions for a recursive
competitive equilibrium now become:
N(s; z, K, M) = n(s; z, K, M, K, M)
¯
N(s; z, K, M) = n(s; z, K, M, K, M)
¯
I (z, K, M) = i (z, K, M, K, M) .
e
The return function R in (3.34) is given by the value of the return function R in (3.28)
that corresponds to the following variables: 1) the discrete distribution x is defined by the
finite support m as in equation (3.31), 2) the employment rule n is defined by the employment
¯
thresholds n and n as in (3.29), 3) the discrete distribution X is defined by the finite support
8
Without loss of generality, n(0) and n(0) are set identical to zero. If τ is sufficiently small relative to
¯
the present discounted value of wages, this will always be the optimal choice.
13
¯
M, and 3) the employment rule N is defined by the employment thresholds N and N. The
advantage of working with the transformed problem (3.34) instead of the original problem
e
(3.28) is that it has linear laws of motion. Since all the endogenous arguments of R take
strictly positive values in the deterministic steady state, a second order Taylor expansion
around the deterministic steady state can be performed to obtain a quadratic return function.
This delivers a linear-quadratic recursive competitive equilibrium structure that can be solved
using standard techniques (e.g. Hansen and Prescott [8]). The assumption that nt , nt , mt ,
¯
¯
kt , it , N t , Nt , Mt , Kt , It fluctuate in a sufficiently small neighborhood of their deterministic
steady state values is satisfied in all the experiments reported in this paper.
4. Parametrization
This section describes the steady state observations used to calibrate the model parameters.9
Since the model will be calibrated to U.S. data and this economy is characterized by low firing
costs, the parameter τ is set to zero.10 Given τ , the rest of the parameters to be calibrated
are β, θ, γ, ν, q, α, δ, the distribution ψ, the transition matrix Q for the idiosyncratic
productivity shocks, and the transition matrix H for the aggregate productivity shock. The
model time period is selected to be one quarter.
The first 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
inventories. As a result, investment is associated in the 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 capitaloutput 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 = 0.0221.
9
10
The calibration procedure follows Veracierto [16] quite closely.
In the next section, the firing cost parameter τ will be increased and its effects analyzed.
14
The annual interest rate is selected to be 4 per cent, which is a compromise between the
average real return on equity and the average real return on short-term debt for the period
1889 to 1978 as reported by Mehra and Prescott [11]. The discount factor β is then chosen
to be 0.99 in order to generate this annual interest rate.
Given the above values for β and δ, and given that the capital share satisfies
θ=
(1/β + δ) K
,
Y
matching the U.S. capital-output ratio requires choosing a value of θ equal to 0.2186. Similarly, γ = 0.64 is selected to generate the share of labor in the National Income and Product
Accounts.
The disutility of work parameter α is an important determinant of aggregate employment
η. Thus, α = 0.94 is picked so that 80 percent of the population works at steady state,
roughly the fraction of the U.S. working age population that is employed.
In turn, the quarterly quit rate parameter q is chosen to be 6 per cent, which is consistent
with evidence on quits from the Job Openings and Labor Turnover Survey (JOLTS) published
by the Bureau of Labor Statistics.
The transition matrix for the idiosyncratic productivity levels Q is restricted to be a
finite approximation to a continuous process of the following form:
Π(0, {0}) = 1
Π(s, [smin , s]) =
ˆ
1
ˆ
ˆ
Pr {(a + ρs ln s + ε0s ) ∈ [smin , s]} , for s, s ≥ smin
ζ
where a, ρs and ζ are constants, ε0s is an i.i.d. normally distributed variable with mean 0 and
standard deviation σ s , and Π (s, A) is the probability of transiting from s to a next period
value in the set A.11 We then have to determine the four parameters a, ρs , ζ and σ s , the
idiosyncratic productivity levels {smin, ..., smax } and the initial distribution ψ. Since all these
parameters are important determinants of the establishment dynamics of the model, their
11
Observe that Π is basically an AR(1) process truncated at the value of 0.
15
values will be selected to reproduce several features of U.S. establishment dynamics.
One such feature is the distribution of establishments by employment size as reported
by the Census of Manufacturers. In particular, the distribution over initial idiosyncratic
productivity levels ψ is selected so that the invariant distribution x∗ in the model economy
mimics the average size distribution of manufacturing establishments across the census years
1967, 1972, 1977 and 1982, which is reproduced in Table 1. For this purpose, a total of
nine positive idiosyncratic productivity levels are introduced and their values {smin, ..., smax }
are selected so that the (corresponding nine types of) establishments in the model economy
display employment levels in the middle of each of the employment ranges shown in Table
1.12
Another set of observations on (manufacturing) establishment dynamics pertains to jobcreation and job-destruction data. Davis and Haltiwanger [5] reported that, for the period
between 1972:2 and 1988:4, the job-creation rate due to births (JCB) was 0.62% while the
job-creation rate due to continuing establishments (JCC) was 4.77%. They also reported
that the job-destruction rate due to deaths (JDD) was 0.83% while the job-destruction rate
due to continuing establishments (JDC) was 4.89%.13 Since employment is stationary in the
model economy, the model can not match these exact job-creation and job-destruction rates.
Imposing the approximate symmetry observed in U.S. data, I chose instead to match the
following rates: JCB = 0.73, JCC = 4.80%, JDD = 0.73% and JDC = 4.80%. This gives
rise to three independent observations. In order to calibrate the four parameters a, ρs , ζ and
σ s associated to the transition matrix an additional observation is then needed.
The last observation is obtained from Dunne et al. [6] who analyzed establishment
turnover using data on plants that first began operating in the 1967, 1972, and 1977 Census
of Manufacturing. They found that the five-year exit rate among these establishments was
36.2%. Matching this exit rate, together with the job-creation and destruction rates described
above, requires the following parameter values: a = 0.05155, ρs = 0.996, ζ = 1.005 and
σ s = 0.0372. The values for the idiosyncratic productivity levels {smin, ..., smax }, the initial
12
In practice, I normalized the lowest idiosyncratic productivity level smin to one and chose the endowment
of new establishments ν to make the nine employment levels fall in the middle of the employment ranges.
13
These are all quarterly rates.
16
distribution ψ and the transition matrix Q that correspond to this calibration procedure are
provided in Table 2.
Finally, the aggregate productivity shock is constrained to follow a standard AR(1)
process:
z 0 = ρz z + ε0z
where ε0z is an i.i.d. normally distributed variable with mean 0 and standard deviation
σ z .14 The parameters ρz and σ z are selected so that measured Solow residuals in the model
economy replicate the behavior of measured Solow residuals in the data.15 Using the measure
of output described above and a labor share of 0.64, measured Solow residuals are found to
be as highly persistent as in Prescott [13] but the standard deviation of technology changes
is somewhat smaller: 0.0063 instead of the usual 0.0076 value used in the literature. As a
consequence, ρz = 0.95 and σ z = 0.0063 are chosen here.
5. Results
5.1. Steady state effects
This section reports the steady state effects of firing taxes in the deterministic version of the
model economy.16 Providing a steady state analysis is important because it describes how
the firing taxes affect the mean levels around which the economy fluctuates.
Table 3 shows the effects of increasing the firing tax τ from zero to 0.33, one, two and
four quarters of wages.17 We see that the steady state consequences on the job reallocation
14
Instead of selecting a finite approximation to this process (which would determine the finite set Z and
the transition matrix H described in the previous sections) I choose to work with the continuous AR(1)
process directly since the linear-quadratic computational method renders it tractable.
15
Proportionate changes in measured Solow residual are defined as the proportionate change in aggregate
output minus the sum of the proportionate change in labor times the labor share γ, minus the sum of the
proportionate change in capital times (1 − γ).
16
This type of analysis is not novel. A number of papers have evaluated the steady state effects of firing
taxes in a variety of settings. Alvarez and Veracierto [2], Hopenhayn and Rogerson [9], Millard and Mortensen
[12] and Veracierto [16] are only a few examples.
17
Firing costs equal to one year of wages amount to the severance payments that must be given to blue
collar workers with ten years of service in countries with the toughest legislation (Lazear [10]. That is, they
17
process are quite significant. In order to avoid paying firing taxes equal to one year of wages
(τ = 4w), the job destruction rate of continuing establishments (JDC) decreases from 4.80%
to 2.74%. Since establishments prefer to wait until they exit before firing additional workers,
the job destruction rate due to deaths (JDD) increases from 0.73% to 0.80%. In turn,
establishments that receive positive productivity shocks choose to reduce their employment
growth in order to avoid paying firing taxes in the future. This leads to a reduction in the
job creation rate due to continuing establishments (JCC) from 4.80% to 3.05% and in the job
creation rate due to births (JCB) from 0.73% to 0.47%. The fact that establishments do not
respond to the idiosyncratic productivity shocks as much as they do in the absence of firing
taxes leads to a significant production inefficiency: Establishments with low productivity
levels end up employing too many workers and establishments with high productivity levels
end up employing too few workers. This production inefficiency induces agents to substitute
away from market activities towards leisure, leading to a decrease of 2.46% in aggregate
employment. The lower productivity and employment levels in turn lead to a decrease of
3.52% in output, consumption, capital and investment.
The last row of Table 3 shows the welfare effects. In particular, it reports the proportionate increase in consumption that must be given to the representative agent living in the
steady state with firing taxes to make him indifferent with being in the steady state with
no firing taxes. Since the economy without firing taxes is Pareto optimal, we know that
this compensation must be positive. In fact, Table 3 shows that it can be a large number:
According to this measure, the welfare cost of introducing a firing tax equal to one year of
wages is equal to 1.74% of consumption.18
5.2. Business cycle effects
This section constitutes the core of the paper. It analyzes the effects of firing taxes on business cycle dynamics. Before proceeding to the main results, it will be important to determine
represent an upper bound on what is empirically reasonable.
18
This paper obtains a lower welfare cost than Veracierto [16] because it provides a quits allowance before
taxing firing, while Veracierto [16] doesn’t. As a consequence, the firing taxes have smaller effects.
18
the empirical plausibility of the business cycles generated by the benchmark economy with
zero firing costs that was calibrated in Section 4. The first and second columns of Table 4
report business cycle statistics for the U.S. and the benchmark economy (τ = 0), respectively. Before any statistics were computed, all time series were logged and detrended using
the Hodrick-Prescott filter. The empirical measures of output, investment and consumption
reported in the table correspond to the measures described in Section 4, and cover the period
between 1960:1 and 1993:4. For the model economy, time series of length equal to 136 periods (the same length as the U.S. series) were computed for 100 simulations and the reported
statistics are averages across these simulations. Comparing the business cycles generated by
both economies, Table 4 shows that output fluctuates roughly the same amount in the model
as in the U.S. Investment is about 5 times more variable than output in the model, while it
is about 4 times as variable in the data. Consumption is less variable than output in both
economies, however, it is less variable in the model than in the U.S. The aggregate stock of
capital varies about the same amount in both economies. Hours vary less than output in
the model, while they vary slightly more than output in the data. Similarly, productivity
fluctuates less in the model than in the U.S. In terms of correlations with output, we see that
almost all variables are highly procyclical, both in the model and in U.S. data. The only
exceptions are capital (which is acyclical both in the model and the U.S.) and productivity
(which is highly procyclical in the model, but acyclical in the data). Overall, the benchmark economy is found to be broadly consistent with salient features of U.S. business cycle
dynamics, similarly to previous RBC models in the literature.
Having established the empirical plausibility of the benchmark economy, I now turn to
evaluate the effects of firing costs on business cycle dynamics. The last three columns show
the results. We see that introducing firing costs equal to one year of wages has considerable
effects: The standard deviation of output decreases by 10.7% (from 1.40 to 1.25). The
expenditures components also become less variable, but by different amounts: The standard
deviation of consumption decreases by 4.1% (from 0.49 to 0.47) while the standard deviation
of investment decreases by 14.7% (from 7.07 to 6.03). These are significant effects. However,
they are relatively small compared to the effects on employment, whose standard deviation
decreases by 30.6% (from 0.98 to 0.68). In terms of comovements with output, the effects
19
of firing taxes are generally insignificant. The only sizable effect is on labor productivity,
whose correlation with output increases from 0.91 to 0.99.
The intuition for why firing taxes reduce the variability of aggregate employment by such
a considerable amount is straightforward. The presence of firing taxes leads establishments
to follow the (S,s) decision rule given by equation (3.29). As a consequence, an establishment
that has a previous employment level net of quits (1 − q)u between its lower employment
threshold n and its upper employment threshold n chooses to remain inactive, that is, it
¯
makes its current employment equal to its previous employment level net of quits (1 − q)u.
Since the employment level of an inactive establishment does not respond to aggregate
conditions, the mere upsurge of this type of establishments leads to a reduction in the
variability of aggregate employment.
The ranges of inaction also lowers the employment variability of active establishments.
The reason is that an establishment that adjusts its employment level at a time of a positive
aggregate shock will be concerned that, in the future, it may enter a long period of inaction
during which aggregate productivity will revert to its mean. If the establishment responds
too much to the current aggregate productivity shock it may find itself with an employment
level that is too high for the aggregate productivity levels that will hold later on. For this
reason, active establishments reduce the response of their employment levels n and n to the
¯
aggregate productivity shocks, dampening the fluctuations in aggregate employment.
Table 5 shows the effects of firing taxes on the persistence of aggregate employment. In
particular, it reports the autocorrelation of aggregate employment at lags that vary between
one and five quarters. We see that the autocorrelation at a one quarter lag increases from
0.66 to 0.71 when firing taxes equal to one year of wages (τ = 4w) are introduced.
The effects are even larger at higher lags. For example, the autocorrelation at a three quarters
lag increases from 0.20 to 0.27. These are significant effects. The reason for why firing taxes
increase the persistence of aggregate employment can be found in the creation of the ranges of
inaction. If an establishment enters a range of inaction after having expanded its employment
level in response to an aggregate productivity shock, the effects of the expansion will persist
until the establishment comes out of the range of inaction and adjusts its employment again.
Thus the presence of inactive establishments makes aggregate employment more persistent.
20
Table 6 reports the effects of firing taxes on the cyclical behavior of job creation and job
destruction. We see that when the firing taxes are equal to zero, the job creation rate and
the job destruction rate vary by the same amount and are negatively correlated.19 When
the firing taxes are introduced, the job creation rate and job destruction rates become less
variable and less negatively correlated. An interesting feature in Table 6 is that the firing
taxes decrease the variability of job destruction more than the variability of job creation.
This finding seems to contradict Campbell and Fisher [4], who report that introducing firing
costs increase the relative variability of job destruction. However, Campbell and Fisher
perform a partial equilibrium analysis where the only source of fluctuations is a wage shock.
Aggregate productivity, which is the source of aggregate fluctuations in this paper, is left
unchanged. This is an important difference. Cabrales and Hopenhayn [3] find, in a partial
equilibrium setting, that firing costs increase the relative variability of job creation when the
source of the fluctuations is an aggregate productivity shock instead of the wage rate.
An interesting feature in Tables 4, 5 and 6 is that even low levels of firing taxes can have
substantial effects on business cycle dynamics. In particular, firing taxes equal to one quarter
of wages (τ = w) are found to reduce the standard deviations of output, investment and
labor 70% as much as firing taxes equal to one year of wages (τ = 4w). The similarities are
even stronger when considering the effects on the autocorrelation of aggregate employment,
and the cyclical behavior job creation and job destruction. The reasons for why small firing
taxes can have such significant effects are familiar to the literature. For example, in the
investment decision problem analyzed by Abel and Eberly [1] the derivative of the range of
inaction with respect to the wedge between the purchase and resale price of capital is shown
to be infinite when the wedge is equal to zero. In this paper, the ranges of inaction are
created by a firing tax but the mechanism is the same: Small firing taxes have large effects
on the length of the ranges of inaction. Through their effects on the ranges of inaction, the
small firing taxes have important consequences for the aggregate fluctuations of the economy.
19
The fact that the job destruction rate is more variable than the job creation rate in U.S. data is not
particularly worrisome. Veracierto (1995) shows that incorporating a reallocation shock that is correlated
with the aggregate productivity shock can reproduce that particular feature of U.S. data. However, the
reallocation shock has no important effects on aggregate fluctuations.
21
Observe that, by reducing aggregate fluctuations, the welfare costs of firing taxes could
be lower than those estimated from comparing steady states. However, this effect turns out
to be negligible: Once the business cycle consequences are taken into account, the welfare
costs of firing taxes are virtually the same as those reported in Table 3. There are two
reasons for this. First, most of the reduction in variability takes place in employment. Since
the preferences of the representative agent are linear with respect to this component, there
are no welfare gains from this effect. Second, the volatility of consumption decreases but by
a very small amount. This small effect, together with the relatively low risk aversion of the
representative agent, produces an extremely small welfare gain.
5.3. No tax Rebates
So far, the firing taxes have been rebated to the representative household as a lump sum
transfer. However, many of the firing costs paid by employers in actual countries do not go to
the workers but involve resources that are wasted: Procedural requirements and legal costs
are examples. To assess the effects of this type of firing costs, this section analyzes firing
taxes that are thrown into the ocean. In particular, it analyzes a social planner problem
given by maximizing equation (3.12) subject to equations (3.13) through (3.18), but where
the lump-sum transfers Tt (z t ) in equation (3.13) are set to zero.
Table 7 reports the steady state results. Not surprisingly, the effects of firing taxes on
the job creation and destruction process are the same as when the firing taxes are rebated
as lump sum transfers. However, there are important differences in the rest of the variables.
In particular, firing taxes equal to one year of wages (τ = 4w) reduce employment by only
0.28%, compared to the 2.46% drop reported in Table 3. The reason is that when the firing
taxes are not rebated to the household sector, they generate a large income effect. This effect
cancels the substitution effect from the lower wages and leads to a small change in labor
supply. Despite the small drop in employment, consumption decreases quite considerably
because part of the output is used up in firing workers. Since both consumption and leisure
are smaller than when the firing taxes are rebated as lump-sum transfers, the welfare costs
are much larger: 3.84% instead of 1.74%.
Table 8 reports the business cycle effects. We see that the business cycle fluctuations are
22
virtually the same when the tax revenues are rebated to the households sector as when they
are not (Table 4). There are two reasons for this. First, total tax revenues are small: Even
when τ is equal to one year of wages and the tax revenues are the largest, they represent
only 1.86% of aggregate output (Tables 3 and 7). Second, the tax revenues fluctuate very
little: When τ is equal to one year of wages, their standard deviation is only 0.50. Since the
tax revenues are small and fluctuate very little, it makes no difference if they are rebated to
the households or not: They do not represent a significant source of aggregate fluctuations.
We conclude that determining to what extent the firing taxes are rebated to the household
sector is crucial for evaluating their long-run outcomes, but has no importance for analyzing
their business cycle effects.
A. Equivalence between quasi-planner and competitive equilibria
Let
#
" t
¡ t¢ Y
β t λt z
H(zj−1 , zj ) ,
j=1
"
j
Y
#"
pt+h−1 (z t+h−1 ; zt+h )
h=1
"
j
Y
#
¡
¢
Q(sa+h−1 , sa+h ) φa+j,t+j sa+j , z t+j ,
h=1
j
Y
#"
pt+h−1 (z t+h−1 ; zt+h )
h=1
j
Y
#
¡
¢
Q(sa+h−1 , sa+h ) ξ a+j,t+j sa+j , z t+j
h=1
be the Lagrange multipliers for equations (3.2)-(3.4), respectively. The first order conditions
for a competitive equilibrium are then given by:
¡ ¢
1
= λt z t
ct (z t )
¡ ¢ ¡ ¢
α = λt z t wt z t
X
£
¡ ¢
¡
¢
¡
¢
¤
λt+1 z t+1 H(zt , zt+1 ) rt+1 z t+1 + 1 − δ
λt z t = β
(A.1)
(A.2)
(A.3)
zt+1
¡ ¢
¡
¢
λt z t pt (z t ; zt+1 ) = βλt+1 z t+1 H(zt , zt+1 )
¡
¢θ
¡
¢γ−1
¡ ¢
¡
¢
− wt z t + φa,t sa , z t −
ezt sa ga,t sa , z t γna,t sa , z t
23
(A.4)
XX
sa+1 zt+1
¡
¡
¢ ¡
¢
¡
¢
¢
(1 − q) φa+1,t+1 sa+1 , z t+1 pt z t+1 Q(sa , sa+1 ) ≤ 0, = 0, if na,t sa , z t > 0 (A.5)
¡
¢θ−1
¡
¢γ
¡ ¢
¡
¡
¢
¢
na,t sa , z t − rt z t ≤ 0, = 0, if ga,t sa , z t > 0
ezt sa θga,t sa , z t
¡
¢
¡
¢
¡
¡
¢
¢
−τ + φa,t sa , z t + ξ a,t sa , z t ≤ 0, = 0, if fa,t sa , z t > 0
¡
¢£
¡
¢
¡
¢
¡
¢¤
φa,t sa , z t na,t sa , z t − (1 − q) na−1,t−1 sa−1 , z t−1 + fa,t sa , z t = 0
and equations (3.2)-(3.11).
Let
¡
¢
¡
¢
ξ a,t sa , z t fa,t sa , z t = 0,
(A.6)
(A.7)
(A.8)
(A.9)
" t
#
¡ t¢ Y
β t λt z
H(zj−1 , zj ) ,
j=1
#
" t
¡ t¢ Y
¡
¢
H(zj−1 , zj ) φa,t sa , z t µa (sa ) ,
β t λt z
j=1
#
" t
¡ t¢ Y
¡ ¢
β t λt z
H(zj−1 , zj ) wt z t ,
j=1
#
" t
¡ t¢ Y
¡ ¢
H(zj−1 , zj ) rt z t ,
β t λt z
j=1
#
" t
¡ t¢ Y
¡
¢
H(zj−1 , zj ) ξ a,t sa , z t µa (sa )
β t λt z
j=1
be the Lagrange multipliers for equations (3.13)-(3.17), respectively. The first order conditions for a quasi-planner equilibrium are then given by equations (A.1)-(A.9) and equations
(3.13)-(3.19).
Establishing that quasi-planner and competitive equilibria are equivalent then amounts
to showing that equations (3.2)-(3.11) are satisfied if and only if equations (3.13)-(3.19) hold.
This is a straightforward verification.
24
B. Determination of the individual state of an establishment
Proposition B.1. Let (h, sh ) be the age and the idiosyncratic history of a particular type
of establishment at date 0. Let (j, sj ) be the age and the idiosyncratic history of another
type of establishment at date 0. Suppose that
¡
¢
¡
¢
nh−1,−1 sh−1 , z −1 = nj−1,−1 sj−1 , z −1 ,
and that sh = sj .
ˆ ˆ ˆ ˆ t=0
Then, the solution {ˆt , kt+1 , η t , gt , nt , ft }∞ to the quasi-planner problem (3.12) has the
c ˆ
following property:
¡
¢
¡
¢
nh,0 sh , z 0 = nj,0 sj , z 0 ,
ˆ
ˆ
¡
¢
¡
¢
ˆ
ˆ
fh,0 sh , z 0 = fj,0 sj , z 0
and for every t > 1, st and z t .
¡
¢
¡
¢
ˆ
gh,0 sh , z 0 = gj,0 sj , z 0 ,
ˆ
¡
¢
¡
¢
ˆ
nh+t (sh , st ), z t = nj+t (sj , st ), z t ,
ˆ
¡
¢
¡
¢
ˆ
ˆ
fh+t (sh , st ), z t = fj+t (sj , st ), z t .
Proof: Suppose not.
¡
¢
¡
¢
ˆ
gh+t (sh , st ), z t = gj+t (sj , st ), z t ,
ˆ
Let
¡ ¢
¡
¢
¡
¢
nψ z 0 = ψ 0 nh,0 sh , z 0 + [1 − ψ 0 ] nj,0 sj , z 0 ,
ˆ
ˆ
0
¡ ¢
¡
¢
¡
¢
ψ
ˆ
ˆ
f0 z 0 = ψ 0 fh,0 sh , z 0 + [1 − ψ 0 ] fj,0 sj , z 0 ,
where
¡
¢
¡
¢
ψ¡ ¢
ˆ
ˆ
g0 z 0 = ψ 0 gh,0 sh , z 0 + [1 − ψ 0 ] gj,0 sj , z 0 ,
¡ ¢
µh sh
,
ψ0 =
µh (sh ) + µj (sj )
25
and, for every t > 1, st and z t , let
¡
¢
¡ ¢
¡
¢ £
¡ ¢¤
¡
¢
nψ st , z t = ψ t st nh+t (sh , st ), z t + 1 − ψ t st nj+t (sj , st ), z t ,
ˆ
ˆ
t
¡
¢
¡ ¢
¡
¢ £
¡ ¢¤
¡
¢
ˆ
ˆ
ftψ st , z t = ψ t st fh+t (sh , st ), z t + 1 − ψ t st fj+t (sj , st ), z t ,
where
¢
¡ ¢
¡
¢ £
¡ ¢¤
¡
¢
ψ¡
ˆ
ˆ
gt st , z t = ψ t st gh+t (sh , st ), z t + 1 − ψ t st gj+t (sj , st ), z t ,
¡ ¢
ψt st =
¡¡
¢¢
µh+t sh , st
.
µh+t ((sh , st )) + µj+t ((sj , st ))
η g e e t=0
Consider an alternative contingent plan {et+1 , et , et , nt , ft }∞ which is identical to the
k
solution to the quasi-planner problem except that
¡
¢
¡
¢
¡ ¢
nh,0 sh , z 0 = nj,0 sj , z 0 = nψ z 0
e
e
0
¡
¢
¡
¢
¡ ¢
ψ
e
e
fh,0 sh , z 0 = fj,0 sj , z 0 = f0 z 0
¡
¢
¡
¢
ψ¡ ¢
e
g
eh,0 sh , z 0 = gj,0 sj , z 0 = g0 z 0
and for everyt > 1, st and z t :
¡
¢
¡
¢
¡
¢
nh+t (sh , st ), z t = nj+t (sj , st ), z t = nψ st , z t
e
e
t
¡
¢
¡
¢
¡
¢
e
e
fh+t (sh , st ), z t = fj+t (sj , st ), z t = ftψ st , z t .
¡
¢
¡
¢
¡
¢
ψ
g
g
eh+t (sh , st ), z t = ej+t (sj , st ), z t = gt st , z t
This alternative plan is feasible and, by the strict concavity of the establishment level
production function, it leads to a larger right hand side to equation (3.13) for every z t .
Hence, consumption can be made larger than under the optimal plan (strictly larger at some
z t ) while aggregate employment is left unchanged. This increases expected utility, leading
to a contradiction.
26
C. Characterization of the optimal employment rule
From equations (3.14), (A.7)-(A.9) we know that
¡
¢
0 ≤ φa,t sa , z t ≤ τ
(C.1)
¡
¢
¡
¢
¡
¢
na,t sa , z t > (1 − q) na−1,t−1 sa−1 , z t−1 =⇒ φa,t sa , z t = 0
¡
¢
¡
¢
¡
¢
na,t sa , z t < (1 − q) na−1,t−1 sa−1 , z t−1 =⇒ φa,t sa , z t = τ
¡
¢
¡
¢
¡
¢
0 < φa,t sa , z t < τ =⇒ na,t sa , z t = (1 − q) na−1,t−1 sa−1 , z t−1
(C.2)
(C.3)
(C.4)
Using equations (A.5), (A.6), and (C.1)-(C.4) we have (when sa > 0) that:
¡
¢
©
£
¡
¢ ¤ª
φa,t sa , z t = min τ , max Ωa,t sa , z t , 0
(C.5)
where
¡
¢
¡ ¢
1
Ωa,t sa , z t = wt z t − (ezt sa ) 1−θ
+
XX
sa+1 zt+1
µ
θ
rt (z t )
θ
¶ 1−θ
£
¡
¢¤− 1−θ−γ
γ (1 − q) na−1,t−1 sa−1 , z t−1 ( 1−θ )
¡
¢ ¡
¢
(1 − q) φa+1,t+1 sa+1 , z t+1 pt z t+1 Q(sa , sa+1 ).
Under a recursive formulation, φa,t (sa , z t ), Ωa,t (sa , z t ) and na,t (sa , z t ) will depend on
(u, s; z, K, X, k, x), where u is the previous period employment na−1,t−1 (sa−1 , z t−1 ) and s is
the current idiosyncratic productivity level sa , while wt (z t ), rt (z t ), and pt (z t+1 ) will depend
on (z, K, X, k, x). Abusing notation, for s > 0, we can rewrite equation (C.5) as follows:
φ (u, s; z, K, X, k, x) = min {τ , max [Ω (u, s; z, K, X, k, x) , 0]}
(C.6)
where
z
Ω (u, s; z, K, X, k, x) = w(z, K, X, k, x) − (e s)
1
1−θ
27
µ
θ
r(z, K, X, k, x)
θ
¶ 1−θ
γ [(1 − q) u]−(
1−θ−γ
1−θ
)
XX
+
(1 − q) φ ((1 − q) u, s0 ; z 0 , K 0 , X 0 , k0 , x0 ) p(z, K, X, k, x; z 0 )Q(s, s0 )
s0
z0
and where K 0 , X 0 , k0 and x0 are given by equations (3.26), (3.27), (3.23) and (3.24) respectively.
Also,
(C.7)
φ (u, 0; z, K, X, k, x) = τ
since establishments that receive an idiosyncratic productivity s equal to zero choose zero
employment levels.20
Note that, since
z
− (e s)
1
1−θ
µ
θ
r(z, K, X, k, x)
θ
¶ 1−θ
γ [(1 − q) u]−(
1−θ−γ
1−θ
)
is strictly increasing in u, the solution φ to the functional equation given by (C.6) and (C.7)
will be increasing in u. As a consequence, there exists a unique n(s; z, K, X, k, x) satisfying
¯
z
τ = w(z, K, X, k, x) − (e s)
1
1−θ
µ
θ
r(z, K, X, k, x)
θ
¶ 1−θ
γ [¯ (s; z, K, X, k, x)]−(
n
1−θ−γ
1−θ
)
XX
+
(1 − q) φ (¯ (s; z, K, X, k, x), s0 ; z 0 , K 0 , X 0 , k0 , x0 ) p(z, K, X, k, x; z 0 )Q(s, s0 ),
n
s0
(C.8)
z0
and there exists a unique n(s; z, K, X, k, x) satisfying
0 = w(z, K, X, k, x) − (ez s)
1
1−θ
µ
θ
r(z, K, X, k, x)
θ
¶ 1−θ
γ [n(s; z, K, X, k, x)]−(
1−θ−γ
1−θ
XX
+
(1 − q) φ (n(s; z, K, X, k, x), s0 ; z 0 , K 0 , X 0 , k0 , x0 ) p(z, K, X, k, x; z 0 )Q(s, s0 ).
s0
z0
¯
Observe that n(s; z, K, X, k, x) < n(s; z, K, X, k, x).
Since
n(u, s; z, K, X, k, x) > (1 − q) u =⇒ φ (u, s; z, K, X, k, x) = 0,
20
This is true if τ is sufficiently small relative to the present discounted value of wages
28
)
(C.9)
n(u, s; z, K, X, k, x) < (1 − q) u =⇒ φ (u, s; z, K, X, k, x) = τ ,
0 < φ (u, s; z, K, X, k, x) < τ =⇒ n(u, s; z, K, X, k, x) = (1 − q) u,
and since
φ (u, s; z, K, X, k, x) = w(z, K, X, k, x)
z
− (e s)
1
1−θ
µ
θ
r(z, K, X, k, x)
θ
¶ 1−θ
γ [n(u, s; z, K, X, k, x)]−(
1−θ−γ
1−θ
)
XX
(1 − q) φ (n(u, s; z, K, X, k, x), s0 ; z 0 , K 0 , X 0 , k0 , x0 ) p(z, K, X, k, x; z 0 )Q(s, s0 ),
+
s0
z0
equations (C.8) and (C.9), together with the fact that φ is increasing in u, imply the employment decision rule (3.29).
D. Support of invariant distribution
Proof of Proposition 3.1: That 0 belongs to the support follows from the fact that
new establishments are born with zero previous period employment and from the fact that
establishments that die (i.e transit to s = 0) choose zero employment level.
That the set m∗ belongs to the support follows from the fact that every time that an
establishment of type (u, s) has a next period number of agents different from n0 = (1 − q)u
n
it must be n0 = (1 − q)n∗ (s), if the establishment expands, or n0 = (1 − q)¯ ∗ (s), if the
establishment contracts. Observe that Ω(s) is an upper-bound on the duration of inaction
for an establishment that has just expanded (and has current idiosyncratic productivity
¯
s ≥ smin ). Similarly, Ω(s) is an upper-bound on the duration of inaction for an establishment
that has just contracted (and has current idiosyncratic productivity s ≥ smin ).
29
FIGURE 1
Output fluctuations vs. employment protection
2.5
•
•
•
2
-
standard deviation
•
•
• •
1.5
•
•
•
•
• •
•
•
1
•
0.5
0
0
5
10
15
OECD employment protection index
20
25
Table 1
Size distribution of U.S. manufacturing establishments
Employment Shares (%)
5-9
23.15
10-19
22.82
20-49
24.83
50-99
12.59
100-249
10.05
250-499
3.86
500-999
1.68
1000-2499
0.73
>2500
0.28
30
Table 2
Calibrated idiosyncratic process
Idiosyncratic Productivity levels:
s0 = 0.00 s1 = 1.00 s2 = 1.11 s3 = 1.26 s4 = 1.40
s5 = 1.58 s6 = 1.76 s7 = 1.94 s8 = 2.18 s9 = 2.53
Initial distribution:
ψ 0 = 0.00 ψ 1 = 0.50 ψ 2 = 0.15 ψ 3 = 0.35 ψ 4 = 0.00
ψ 5 = 0.00 ψ 6 = 0.00 ψ 7 = 0.00 ψ 8 = 0.00 ψ 9 = 0.00
Transition matrix:
1.000 0.000
0.088 0.847
0.005 0.084
0.005 0.000
0.005 0.000
Q=
0.005 0.000
0.005 0.000
0.005 0.000
0.005 0.000
0.005 0.000
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.065 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.879 0.032 0.000 0.000 0.000 0.000 0.000 0.000
0.086 0.847 0.062 0.000 0.000 0.000 0.000 0.000
0.000 0.088 0.876 0.031 0.000 0.000 0.000 0.000
0.000 0.000 0.090 0.846 0.059 0.000 0.000 0.000
0.000 0.000 0.000 0.092 0.808 0.095 0.000 0.000
0.000 0.000 0.000 0.000 0.094 0.873 0.028 0.000
0.000 0.000 0.000 0.000 0.000 0.096 0.895 0.004
0.000 0.000 0.000 0.000 0.000 0.000 0.099 0.896
31
Table 3
Steady state effects
τ =0
τ = 0.33w
τ =w
τ = 2w
τ = 4w
output
100.00
99.13
98.38
97.65
96.48
consumption
100.00
99.13
98.38
97.65
96.48
investment
100.00
99.13
98.38
97.65
96.48
capital
100.00
99.13
98.38
97.65
96.48
labor
100.00
99.31
98.99
98.55
97.54
0.0%
0.49%
0.78%
1.13%
1.86%
JCB
0.73%
0.66%
0.58% 0.53%
0.47%
JCC
4.80%
3.71%
3.17% 3.05%
3.05%
JDD
0.73%
0.77%
0.79% 0.80%
0.80%
JDC
4.80%
3.61%
2.96% 2.79%
2.74%
Welfare cost
0.00%
0.35%
0.88% 1.30%
1.74%
taxes/output
32
Table 4
Business cycle effects
U.S. data τ = 0 τ = 0.33w
τ =w
τ = 2w
τ = 4w
A: Standard deviations
output
1.33
1.40
1.35
1.29
1.26
1.25
consumption
0.87
0.49
0.49
0.48
0.47
0.47
investment
4.99
7.07
6.72
6.32
6.12
6.03
capital
0.63
0.50
0.49
0.47
0.46
0.45
labor
1.42
0.98
0.88
0.77
0.71
0.68
productivity
0.76
0.49
0.52
0.55
0.57
0.58
taxes
n.a.
n.a.
2.27
1.27
0.71
0.50
B: Correlations with output
output
1.00
1.00
1.00
1.00
1.00
1.00
consumption
0.91
0.91
0.92
0.92
0.93
0.93
investment
0.91
0.98
0.98
0.98
0.98
0.98
capital
0.04
0.08
0.08
0.08
0.08
0.08
labor
0.85
0.98
0.98
0.99
0.99
0.99
productivity
-0.16
0.91
0.95
0.97
0.98
0.99
taxes
n.a.
n.a.
0.26
0.20
0.29
0.54
33
Table 5
Employment autocorrelation function
τ = 0 τ = 0.33w
τ =w
τ = 2w
τ = 4w
1 quarter
0.66
0.69
0.71
0.71
0.71
2 quarters
0.40
0.44
0.46
0.47
0.47
3 quarters
0.20
0.24
0.26
0.27
0.27
4 quarters
0.03
0.06
0.08
0.09
0.09
5 quarters
-0.09
-0.07
-0.05
-0.04
-0.04
34
Table 6
Job creation and job destruction fluctuations
U.S.
τ = 0 τ = 0.33w
σ(JC)
0.88
0.44
σ(JD)
1.65
corr(JC, JD) -0.37
τ =w
τ = 2w
τ 4w
0.39
0.34
0.32
0.31
0.44
0.37
0.30
0.28
0.27
-0.60
-0.60
-0.57
-0.57
-0.56
35
Table 7
Steady state effects - No tax rebates
τ =0
τ = 0.33w
τ =w
τ = 2w
τ = 4w
output
100.00
99.60
99.12
98.72
98.25
consumption
100.00
99.03
98.22
97.41
96.10
investment
100.00
99.60
99.12
98.72
98.25
capital
100.00
99.60
99.12
98.72
98.25
labor
100.00
99.88
99.90
99.88
99.72
0.0%
0.49%
0.78%
1.13%
1.86%
JCB
0.73%
0.66%
0.58% 0.53%
0.47%
JCC
4.80%
3.71%
3.17% 3.05%
3.05%
JDD
0.73%
0.77%
0.79% 0.80%
0.80%
JDC
4.80%
3.61%
2.96% 2.79%
2.74%
Welfare cost
0.00%
0.89%
1.74% 2.56%
3.84%
taxes/output
36
Table 8
Business cycle effects - No tax rebates
U.S. data τ = 0 τ = 0.33w
τ =w
τ = 2w
τ = 4w
A: Standard deviations
output
1.33
1.40
1.35
1.29
1.26
1.24
consumption
0.87
0.49
0.49
0.48
0.47
0.47
investment
4.99
7.07
6.74
6.32
6.14
6.02
capital
0.63
0.50
0.49
0.47
0.46
0.45
labor
1.42
0.98
0.88
0.77
0.71
0.68
productivity
0.76
0.49
0.52
0.55
0.57
0.58
taxes
n.a.
n.a.
2.28
1.27
0.72
0.50
B: Correlations with output
output
1.00
1.00
1.00
1.00
1.00
1.00
consumption
0.91
0.91
0.92
0.92
0.93
0.93
investment
0.91
0.98
0.98
0.98
0.98
0.98
capital
0.04
0.08
0.08
0.08
0.09
0.08
labor
0.85
0.98
0.98
0.99
0.99
0.99
productivity
-0.16
0.91
0.95
0.97
0.98
0.99
taxes
n.a.
n.a.
0.26
0.20
0.28
0.54
37
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39
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Lawrence J. Christiano, Christopher Gust and Jorge Roldos
WP-02-05
Regulatory Incentives and Consolidation: The Case of Commercial Bank Mergers
and the Community Reinvestment Act
Raphael Bostic, Hamid Mehran, Anna Paulson and Marc Saidenberg
Technological Progress and the Geographic Expansion of the Banking Industry
Allen N. Berger and Robert DeYoung
WP-02-06
WP-02-07
4
Working Paper Series (continued)
Choosing the Right Parents: Changes in the Intergenerational Transmission
of Inequality Between 1980 and the Early 1990s
David I. Levine and Bhashkar Mazumder
WP-02-08
The Immediacy Implications of Exchange Organization
James T. Moser
WP-02-09
Maternal Employment and Overweight Children
Patricia M. Anderson, Kristin F. Butcher and Phillip B. Levine
WP-02-10
The Costs and Benefits of Moral Suasion: Evidence from the Rescue of
Long-Term Capital Management
Craig Furfine
WP-02-11
On the Cyclical Behavior of Employment, Unemployment and Labor Force Participation
Marcelo Veracierto
WP-02-12
Do Safeguard Tariffs and Antidumping Duties Open or Close Technology Gaps?
Meredith A. Crowley
WP-02-13
Technology Shocks Matter
Jonas D. M. Fisher
WP-02-14
Money as a Mechanism in a Bewley Economy
Edward J. Green and Ruilin Zhou
WP-02-15
Optimal Fiscal and Monetary Policy: Equivalence Results
Isabel Correia, Juan Pablo Nicolini and Pedro Teles
WP-02-16
Real Exchange Rate Fluctuations and the Dynamics of Retail Trade Industries
on the U.S.-Canada Border
Jeffrey R. Campbell and Beverly Lapham
WP-02-17
Bank Procyclicality, Credit Crunches, and Asymmetric Monetary Policy Effects:
A Unifying Model
Robert R. Bliss and George G. Kaufman
WP-02-18
Location of Headquarter Growth During the 90s
Thomas H. Klier
WP-02-19
The Value of Banking Relationships During a Financial Crisis:
Evidence from Failures of Japanese Banks
Elijah Brewer III, Hesna Genay, William Curt Hunter and George G. Kaufman
WP-02-20
On the Distribution and Dynamics of Health Costs
Eric French and John Bailey Jones
WP-02-21
The Effects of Progressive Taxation on Labor Supply when Hours and Wages are
Jointly Determined
Daniel Aaronson and Eric French
WP-02-22
5
Working Paper Series (continued)
Inter-industry Contagion and the Competitive Effects of Financial Distress Announcements:
Evidence from Commercial Banks and Life Insurance Companies
Elijah Brewer III and William E. Jackson III
WP-02-23
State-Contingent Bank Regulation With Unobserved Action and
Unobserved Characteristics
David A. Marshall and Edward Simpson Prescott
WP-02-24
Local Market Consolidation and Bank Productive Efficiency
Douglas D. Evanoff and Evren Örs
WP-02-25
Life-Cycle Dynamics in Industrial Sectors. The Role of Banking Market Structure
Nicola Cetorelli
WP-02-26
Private School Location and Neighborhood Characteristics
Lisa Barrow
WP-02-27
Teachers and Student Achievement in the Chicago Public High Schools
Daniel Aaronson, Lisa Barrow and William Sander
WP-02-28
The Crime of 1873: Back to the Scene
François R. Velde
WP-02-29
Trade Structure, Industrial Structure, and International Business Cycles
Marianne Baxter and Michael A. Kouparitsas
WP-02-30
Estimating the Returns to Community College Schooling for Displaced Workers
Louis Jacobson, Robert LaLonde and Daniel G. Sullivan
WP-02-31
A Proposal for Efficiently Resolving Out-of-the-Money Swap Positions
at Large Insolvent Banks
George G. Kaufman
WP-03-01
Depositor Liquidity and Loss-Sharing in Bank Failure Resolutions
George G. Kaufman
WP-03-02
Subordinated Debt and Prompt Corrective Regulatory Action
Douglas D. Evanoff and Larry D. Wall
WP-03-03
When is Inter-Transaction Time Informative?
Craig Furfine
WP-03-04
Tenure Choice with Location Selection: The Case of Hispanic Neighborhoods
in Chicago
Maude Toussaint-Comeau and Sherrie L.W. Rhine
WP-03-05
Distinguishing Limited Commitment from Moral Hazard in Models of
Growth with Inequality*
Anna L. Paulson and Robert Townsend
WP-03-06
Resolving Large Complex Financial Organizations
Robert R. Bliss
WP-03-07
6
Working Paper Series (continued)
The Case of the Missing Productivity Growth:
Or, Does information technology explain why productivity accelerated in the United States
but not the United Kingdom?
Susanto Basu, John G. Fernald, Nicholas Oulton and Sylaja Srinivasan
WP-03-08
Inside-Outside Money Competition
Ramon Marimon, Juan Pablo Nicolini and Pedro Teles
WP-03-09
The Importance of Check-Cashing Businesses to the Unbanked: Racial/Ethnic Differences
William H. Greene, Sherrie L.W. Rhine and Maude Toussaint-Comeau
WP-03-10
A Structural Empirical Model of Firm Growth, Learning, and Survival
Jaap H. Abbring and Jeffrey R. Campbell
WP-03-11
Market Size Matters
Jeffrey R. Campbell and Hugo A. Hopenhayn
WP-03-12
The Cost of Business Cycles under Endogenous Growth
Gadi Barlevy
WP-03-13
The Past, Present, and Probable Future for Community Banks
Robert DeYoung, William C. Hunter and Gregory F. Udell
WP-03-14
Measuring Productivity Growth in Asia: Do Market Imperfections Matter?
John Fernald and Brent Neiman
WP-03-15
Revised Estimates of Intergenerational Income Mobility in the United States
Bhashkar Mazumder
WP-03-16
Product Market Evidence on the Employment Effects of the Minimum Wage
Daniel Aaronson and Eric French
WP-03-17
Estimating Models of On-the-Job Search using Record Statistics
Gadi Barlevy
WP-03-18
Banking Market Conditions and Deposit Interest Rates
Richard J. Rosen
WP-03-19
Creating a National State Rainy Day Fund: A Modest Proposal to Improve Future
State Fiscal Performance
Richard Mattoon
WP-03-20
Managerial Incentive and Financial Contagion
Sujit Chakravorti, Anna Llyina and Subir Lall
WP-03-21
Women and the Phillips Curve: Do Women’s and Men’s Labor Market Outcomes
Differentially Affect Real Wage Growth and Inflation?
Katharine Anderson, Lisa Barrow and Kristin F. Butcher
WP-03-22
Evaluating the Calvo Model of Sticky Prices
Martin Eichenbaum and Jonas D.M. Fisher
WP-03-23
7
Working Paper Series (continued)
The Growing Importance of Family and Community: An Analysis of Changes in the
Sibling Correlation in Earnings
Bhashkar Mazumder and David I. Levine
WP-03-24
Should We Teach Old Dogs New Tricks? The Impact of Community College Retraining
on Older Displaced Workers
Louis Jacobson, Robert J. LaLonde and Daniel Sullivan
WP-03-25
Trade Deflection and Trade Depression
Chad P. Bown and Meredith A. Crowley
WP-03-26
China and Emerging Asia: Comrades or Competitors?
Alan G. Ahearne, John G. Fernald, Prakash Loungani and John W. Schindler
WP-03-27
International Business Cycles Under Fixed and Flexible Exchange Rate Regimes
Michael A. Kouparitsas
WP-03-28
Firing Costs and Business Cycle Fluctuations
Marcelo Veracierto
WP-03-29
8
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