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Working Paper Series
Plant Level Irreversible Investment and
Equilibrium Business Cycles
MarceloVeracierto
Working Papers Series
Research Department
Federal Reserve Bank of Chicago
bruary 1998 (WP-98-1)
FEDERAL RESERVE BANK
OF CHICAGO
P la n t L e v e l Irre v e rs ib le In v e s tm e n t a n d
E q u ilib riu m
B u s in e s s C y c le s
Ma r celo Veracierto*
C o r n e ll U n iv e r s ity a n d
F e d e r a l R e s e r v e B a n k o f C h ic a g o
Original Draft: M a y , 1996
This version: February, 1998
A b stra c t: This paper evaluates the importance of microeconomic irreversibil
ities for aggregate dynamics using a general equilibrium approach. To this end
a real business cycle model of establishment level dynamics is formulated and
analyzed. Investment decisions are subject to irreversibility constraints and con
sequently, axe of the (S,s) variety. This complicates the analysis since the state
of the economy is described by an endogenous distribution of agents. The paper
develops a computational strategy th at makes this class of (S,s) economies fully
tractable. Contrary to what the previous literature has suggested, investment
irreversibilities are found to have no effects on aggregate business cycle dynamics.
* I would like to thank Fernando Alvarez, V. Chari, Huberto Ennis, Larry Jones, Tryphon
Kollintzas, Nobu Kiyotaki, John Leahy, and an anonymous referee for useful comments, as well
as seminar participants at Cornell University, Federal Reserve Bank of Chicago, Federal Reserve
Bank of Minneapolis, University of Iowa, Universidad Torcuato Di Telia, University of Western
Ontario, University of Virginia, the 1997 Canadian M S G meetings, the 1997 SED meetings, and
the 1996 UCLA-Federal Reserve Bank of Minneapolis-Comell University Conference. I would
also like to thank the Institute for Empirical Macroeconomics and the Center for Analytic
Economics for their support. The views expressed herein are those of the author and not
necessarily those of the Federal Reserve Bank of Chicago or the Federal Reserve System.
1. I n t r o d u c t i o n
The empirical literature has provided substantial evidence of investment irreversibilities at
the establishment level. Analyzing the behavior of a large number of manufacturing estab
lishments over time, Caballero, Engel and Haltiwanger [7] determined th at establishments
axe much more inclined to expanding their stock of capital than to reducing it. Ramey and
Shapiro [21] presented more direct evidence. Using data from the equipment auction of an
aerospace firm, they estim ated the wedge between the purchase and resale price for different
types of capital. They found th at machine tools sell at about 31% of their purchase value,
while structural equipment sell at only 5%. These estimates indicate surprisingly large levels
of irreversibilities in investment.
Since the early and influential paper by Arrow [2] there has been substantial theoretical
microeconomic work on irreversible investment (see Dixit and Pindyck [11] for a survey). A
common result is th at irreversibilities are extremely im portant for establishment level dy
namics. For example, Abel and Eberly [1] analyzed the problem of a firm facing a resale
price of capital which is lower than its purchase price. They showed th at the optimal in
vestment decision of the firm is a two-triggers (S,s) policy, and characterized the associated
range of inaction as a function of the wedge between the purchase and resale price of capital.
They found that even small irreversibilities can have a large impact on the range of inaction,
substantially affecting the investment dynamics of the firm.
While the microeconomic consequences of investment irreversibilities are well understood,
their macroeconomic implications are not. To evaluate the importance of investment irre
versibilities for aggregate dynamics, this paper formulates and analyzes a real business cycle
model of establishment level dynamics. The basic framework is analogous to the neoclassical
stochastic growth model with indivisible labor analyzed by Hansen [17] and for a particular
parametrization the model reduces to his. Output, which can be consumed or invested, is
produced by a large number of establishments th at use capital and labor as inputs into a
decreasing returns to scale production technology. Establishments receive idiosyncratic pro
ductivity shocks that determine their expansion, contraction or death. Establishments are
also subject to an aggregate productivity shock which generates aggregate fluctuations in
the economy. For simplicity, both entry and exit are treated as exogenous.
While labor is assumed to be perfectly mobile across establishments, capital is not. Once
capital is in place at an establishment there are costs associated with detaching and moving
it. These costs imply th at a fraction of the productive services of capital are lost in the
process of uninstalling it. This is analogous to the case analyzed by Abel and Eberly [1],
where the sale price of capital is lower than its purchase price.
Computing the stochastic general equilibrium dynamics for this (S,s) economy is a chal
lenging task. The difficulty stems from carrying an endogenous distribution of heterogeneous
agents as a state variable (which is a highly dimensional object). This paper develops a com
putational strategy that makes this class of problems fully tractable. The method involves
keeping track of long histories of (S,s) thresholds as state variables, instead of the current
distribution of agents in the economy. The convenience of this alternative state space is th at
standard linear-quadratic approximation techniques can be directly applied.1
To evaluate the importance of investment irreversibilities for macroeconomic dynamics,
1Other papers that analyze stochastic general equilibrium dynamics in (S,s) economies include: Caplin
and Leahy [9], Dotsey, King and Wolman [13], Fisher and Hornstein [15], and Thomas [24].
economies with different degrees of irreversibilities axe calibrated to U.S. data and their
aggregate fluctuations compared. The results are striking. Investment irreversibilities play no
im portant role in aggregate dynamics: economies ranging from fully reversible to completely
irreversible investment generate almost identical aggregate business cycle fluctuations. The
only way in which investment irreversibilities m atter is for establishment level dynamics.
In principle, investment irreversibilities could affect aggregate business cycle fluctuations if
aggregate productivity shocks were variable enough, but this would require an implausibly
large variability of measured solow residuals.
Previous work on the importance of microeconomic irreversibilities for macroeconomic
dynamics includes Bertola and Caballero [3, 4], Caballero and Engel [5, 6], and Caballero,
Engel and Haltiwanger [7]. None of these studies performs an equilibrium analysis. In some
cases, the optimal decision rule of an individual establishment facing an ad-hoc stochastic
process for prices is derived; the behavior of a large number of such establishments then
aggregated to study aggregate investment dynamics. In other cases, no economic struc
ture is used: even the investment decision rules of establishments are directly assumed. On
the contrary, this paper analyzes the general equilibrium dynamics of an explicit economic
environment. This is an important methodological innovation. To fully capture the implica
tions of microeconomic irreversibilities for macroeconomic dynamics, a general equilibrium
analysis is required.
In fact, the conclusions in this paper axe dramatically different from those in the pre
vious literature. In a celebrated paper, Caballero, Engel and Haltiwanger [7] found th at
non-linear adjustments at the establishment level substantially improve the ability of their
aggregate investment equation to keep track of U.S. aggregate investment behavior, specially
when aggregate investment is far from its mean. Caballero, Engel and Haltiwanger inter
preted this result as evidence th at investment irreversibilities have im portant implications for
macroeconomic dynamics. In particular, that irreversibilities are crucial in generating brisk
expansions and contractions in aggregate investment. Since this has become the dominant
view thereafter, this paper will devote considerable attention to contrasting both methods
of analysis. It will be argued th at the Caballero-Engel-Haltiwanger approach presents a se
rious limitation: it assumes th at im portant features of investment decisions are invariant to
the adjustment costs th at establishments face. As a consequence, it can lead to the wrong
answers. In fact, this paper shows that when the CEH method is applied to our model econ
omy, their same conclusions are obtained. But when the analysis is restricted to investment
behavior which is fully consistent with the economic environment th at establishments face,
microeconomic irreversibilities have no effects on aggregate dynamics.
The paper is organized as follows: Section 2 describes the economy, Section 3 discusses the
computational method, Section 4 parametrizes the model, Section 5 describes the business
cycles of a benchmark economy, Section 6 discusses the importance of investment irreversibil
ities for business cycle fluctuations, and Section 7 concludes the paper.
2. T h e m o d e l e c o n o m y
The economy is populated by a continuum of ex-ante identical agents with names in the unit
interval. Their preferences are described by the following utility function:
E £ P [log (ct) + v (U)]
t= 0
(1)
where ct and lt axe consumption and leisure respectively, and 0 < /? < 1 is the subjective
time discount factor. Every period agents receive a time endowment equal to u>. Following
Rogerson [22] and Hansen [17], it is assumed th at there is an institutionally determined
workweek of fixed length which is normalized to one, so leisure can only take values u> or
u -1 .
Output, which can be consumed or invested, is produced by a large number of estab
lishments. Each establishment uses capital (k) and labor (n) as inputs into a production
technology given by:
yt = eZtstketn l
(2)
where 6 + 7 < 1, st is an idiosyncratic productivity shock and zt is an aggregate productivity
shock common to all establishments. Realizations of the idiosyncratic productivity shock
st take values in the set {0,1, A} and are independent across establishments. Over time, st
follows a first order Markov process with transition m atrix II, where ir (s, s') is the probability
that st+i = s' conditional on st = s. This process is assumed to be such that: 1) starting
from any initial value, with probability one st reaches zero in finite time, and 2) once st
reaches zero, there is zero probability th at st will receive a positive value in the future.
Given these assumptions, it is natural to identify a zero value for the productivity shock
with the death of an establishment.2 The aggregate productivity shock zt follows a law of
motion given by:
zt+ 1 — pzt
+ £t+1
(3)
2Given that there are no fixed costs to operate an establishment already created, exit will take place only
when the idiosyncratic productivity shock takes a value of zero.
where 0 < p < 1, and
et
is i.i.d. with variance a \ and zero mean.
Labor is perfectly mobile in this economy, but capital is not. On one hand, the amount of
capital k t+i in place at an establishment at date t + 1 must be decided at period t before the
realization of s t+1 becomes known. On the other hand, investment is partially irreversible:
whenever capital is detached from the floor of an establishment it loses a fraction (1 —
its remaining productive services. To be precise, let 0 <
6
q)
of
< 1 be the depreciation rate of
capital. In order to increase an establishment’s next period stock of capital A:t+i above its
current level net of depreciation (1 —
6 )k t,
an investment of fct+i — (1 —
8 )kt
units is needed.
On the contrary, when an establishment decreases its next period stock of capital k t+ i below
its current level net of depreciation (1 —
only a fraction q of (1 —
6 )k t — k t + i.
6 )k t,
the amount of investment goods obtained is
The parameter
q
is a measure of the degree of the
investment irreversibilities in the economy and will play a crucial role in the analysis.
Every period, agents receive an endowment of new establishments which arrive with zero
initial capital in place. Initialvalues for s across new establishments are distributed according
to V>. This exogenous birth of new establishments compensates the ongoing death of existing
establishments (as they get absorbed into zero productivity) and results in a constant long
run number of establishments.3
The presence of idiosyncratic productivity shocks and irreversible investment at the es
tablishment level suggests indexing establishments according to their current productivity
shocks
s
and current stock of capital
k.
In what follows, a measure
Xt
over current pro
3 Even though the entry and exit decisions of establishments are not endogenously determined in this
economy, it seems important to incorporate them at least exogenously. A significant probability of death will
probably affect how establishments respond to aggregate productivity shocks in the presence of investment
irreversibilities.
ductivity shocks and capital levels will describe the number of establishments of each type
at period t .4* Also, a measurable function n t will describe the number of workers across es
tablishment types, a measurable function <&+! will describe the next period stock of capital
across establishment types, and r)t will denote the fraction of the population that works.
Feasibility constraints consumption as follows:
ct < J { e zts
k en t (k ,s)7
■ [&+1 (k ,s ) - (1 - (5)fc]
+
where
q
Q
J(1 -
8) g t (k ,s )
q
Q [gt+x
7T(s,0)
(k ,s )
-
(1 - <5)A:]
}dxt
(4)
d x t- i
() is an indicator function that takes value 1 if its argument is positive, and value
(the irreversibility parameter) otherwise. The first term, is the sum of output minus
investment across all types of establishments, taking into account the capital losses due to
the investment irreversibilities. The second term on the right hand side corresponds to all
those establishments that were in operation the previous period and die during the current
period (transit to an idiosyncratic shock equal to 0), getting to sell a fraction q of their stock
of capital g t ( k , s ) net of depreciation.
Similarly, the total number of workers at establishments is constrained not to exceed the
fraction of the population that works r)t:
J Tit ( k , s )
(5)
d x t < 7]t
Finally, the law of motion for the measure x t must be consistent with the capital decisions
4The measure x t will be defined only over positive productivity levels, i.e.
establishments that die.
xt
does not keep track of
at the plant level. T hat is, for every Borel set B:
x t+i ( B , s ' ) =
J
t t (s
, s ')
dxt +
v ip ( s ')
x(0
€ B)
(6)
(k,s): gt+i(k,s)€B
where
x
0 is an indicator function that takes value 1 if its argument is true, and zero
otherwise. In words, the number of establishments that next period have a stock of capital
in the set
and a productivity shock
B
s ',
is given by the sum of two terms: 1) all those
establishments that transit from their current shocks to the shock
period stock of capital in the set B , and 2) in the case that 0 €
B,
s'
and choose a next
all new establishments
that arrive with an initial productivity shock s ' (note that new establishments are born with
a zero initial stock of capital).
Following Hansen [17] and Rogerson [22], agents are assumed to trade employment lotter
ies. These are contracts that specify probabilities of working, and allow agents to perfectly
diversify the idiosyncratic risk they face. Since agents are ex-ante identical, they all chose
the same lottery. As a consequence, the economy has a representative agent with utility
function:
i=0
where
a
=
v
(uj )
—
v{ui
%]
(7)
— 1) (see Hansen [17] and Rogerson [22] for details ). Since this
is a convex economy with no externalities nor other distortions the competitive equilibrium
allocation can be solved by analyzing the Social Planner’s problem with equal weights, which
is given by maximizing (7) subject to (3), (4), (5), and (6).
3. C o m p u t a t i o n
This section describes the computational approach. The method is novel and constitutes
an important contribution of this paper. However, readers less interested in computational
methods and more interested in substantive results can proceed directly to Section 4 with
no loss of continuity.
The state of the economy is given by the current aggregate productivity shock z, the
current measure x across establishment types, the previous period measure
y
across estab
lishment types, and the previous period investment decisions d (z t ,x t , xt_i and g t respectively
in terms of the previous section notation).5* The Social Planner’s Problem can then be de
scribed by the following Bellman equation:
V (d , x, y
,z) =
M A X { ]n c-a r) + 0 E
V { d \ x \ y ',
z')}
(8)
subject to
c< J
{ e z s k 9n (k ,s)7
•
+
\g ( k , s ) —
(1 - 6)fc]
j ( l - 6 ) d ( k ,s )
J
n ( k ,s )
dx < rj
q
Q [g (k ,s) -
7r(s,0)
dy
(1 — 6)fc] } d x
(9)
(10)
5The notation is changed to avoid time subscripts which complicate the recursive formulation of the
economy.
x '( B ,s ') =
7T(5,s')
I
dx
+
V Ip (s') X
(0 €
B)
(11)
(k,s): y(fc,s)€B
z'
d' = g
(12)
y' = x
(13)
=
pz
+
(14)
e
where the maximization is over n () and g (). Note the high dimensionality of the state space
which seems to preclude any possibilities of computing a solution.6 Below, I will show that
this difficulty is only apparent: the problem becomes fully tractable once it is redefined in
terms of a convenient set of variables.
To understand the rationale for the transformed problem, itwill be convenient to analyze
the structure of the problem that establishments face at the competitive equilibrium. The
individual state of an establishment is given by its current productivity shock
current stock of capital
k.
s
and its
The problem of an establishment with individual state (k ,s)
when the aggregate state is (d , x ,
y, z)
is given by:
6The state of the economy could be simplified considerably. Instead of carrying d and y as state variables,
7r(s, 0) d y could be used instead. The formulation in (8) is selected since it is more closely
related to the computational method employed. In any case, carrying x as a state variable is unavoidable
and this is a highly dimensional object.
f ( l —6 ) d (k , s) q
10
J(k, s;d, x,y,z) =
M A X { ezs kerC —w (d , x , y , z ) n — [fc' —(1 —6)k] Q [&' —(1 —6) fc]
+
E [i
(d, x , y , z \ d', x ' , y \ z')
J (k\
s';d', x', y ' , z'))
(15)
subject to:
s' ~ n (s)
z'
=
pz
(16)
+ e'
(17)
(<f, x', y ’) = H {d, x, y , z )
(18)
where w () isthe equilibrium wage rate, i () axe the equilibrium prices ofArrow securities, H ()
isthe equilibrium law ofmotion for the aggregate state of the economy, and the maximization
is over the scalars n and
k '.
Note that the decision rule for capital that corresponds to the
solution of this Bellman equation isof the (S,s) variety.7 It is characterized by a pair of lower
and upper capital thresholds a(s), A ( s ) such that:
7The optimal k' decision for an establishment that expands (A/ > (1 —6)k) is given by:
M A X { E [i (...) J ( k \ s ',...)]
-
k'}
Note that the a(s) that solves this problem is independent of k.
Similarly, the optimal k r for an establishment that contracts (k r < (1 —8)k) is given by:
M A X { E [i
(...) J
(k\
s',...)] -
qk' }
The A ( s ) that solves this problem does not depend on k either. The decision rule (19) is then obtained.
fc' =
o(s),
if (1 —
6 )k
=
A { s),
if (1 -
S )k >
=
(1 —
otherwise
where the dependence of a ( s ) and
A
6 )k,
< o(s)
(19)
A(s)
(s) on the aggregate state of the economy has been
suppressed to simplify notation (Figure 1 shows a picture of this decision rule). Note that
there is a pair of lower and upper threshold
productivity shock
s.
(a
(s),A (s)) for every possible idiosyncratic
Hereon we will denote (a,.A) as being the vector
(a ( s ) , A
(s))a=1A
across idiosyncratic shocks.
Our strategy will be to keep track of long histories of (a,A ) as state variables instead of
the actual distributions x and
x
and
y
y,
and use them to construct approximate distributions for
using the law of motion (6).8 In principle, as we make the length of the history of
(a, A ) arbitrarily large we would obtain an arbitrarily good approximation for x and
y.
An
important question will be how large to make this length in practice (I will return to this
question below). Our solution method willrequire solving independently for the deterministic
steady state of the economy. Appendix A describes how this is performed.
Let (a, A) denote the history of thresholds {at,A t}(_l jT., for some large horizon
where
( a u A t)
T,
correspond to the thresholds chosen t periods ago. Also, let (ac,A c) be the
thresholds corresponding to the current period. Since we know that the optimal decision
rules of establishments are of the (S,s) variety, there is no loss of generality in defining the
Social Planner’s problem directly in terms of choosing the current thresholds (ac,A c) and
8Note that the previous period thresholds (a, A ) define the previous period decision rule
(19).
12
d
according to
the fraction, of people th at work 77 as follows:9
V { a ,A , z)
=
{In [c (a, A , z , a c, A c,ri)
MAX
- a i 1 + /3 E V ( a ' , A z ' ) } }
(20)
subject to:
a t +1 (s)
for t = 1,2, ...,T —1 and s = 1, A
—
at
=
«c (s).
for s = 1, A
^ t+ i(5) =
A (s),
for t = 1,2, ...,T —1 and
=
Ac (s),
for
2! =
p z + e'
a ' l (s )
A \(s)
(5) )
s
s =
(21)
1, A
= 1, A
(22)
where equations (2 1 ) update tomorrow’s histories given the current threshold choices.
The function
c ( a , A , z , a c, A c,Tj)
given the history of thresholds
gives the maximum consumption that can be obtained
(a, A ) ,
the current aggregate productivity shock z , the current
choices of thresholds (ac, A c), and the decision of how many agents to currently put to work
77. Formally, c (a, A , z , a c, A c, 77) is given by the following static labor allocation problem:
c (a, A ,
-
[g(k,s) -
z , a c, A c, 77)
(1 - 8)k] Q [ g ( k ys ) - (1
=
-
MAX j
{ e*s
k en (k,
s )7
6)k] } d x + J ( 1 - 6) d ( k , s ) q x ( s , 0 ) d y
9Note that problem (20) reduces to the original problem (8) as T goes to infinity.
13
(23)
subject to:
n (k , s ) d x < r f
(24)
n ( k , s ).
where the maximization is with respect to the function
The functions
g, x , d,
and
y
in (23) and (24) are determined by
( a , A , z , a c , A c)
in the
following way:
(i) The current investment decision rule
g(k,s)
g
is implied by the current thresholds (ac, A c):
=
ac(s),
if (1 —S ) k
< a c( s )
=
>lc(s),
if (1 —6 ) k
>
-Ac(s)
x
is obtained by initializing this
(25)
= (1 —6 ) k , otherwise
(ii) The current measure across establishment types
measure T periods ago (ary) to be the deterministic steady state measure
x*,
and updating
it recursively by iterating on the law of motion:
xt_i
(B ,s')=
J
(fc.s):
n (s,s') dxt +
x ( 0 € l? )
(26)
gt(k,s)€B
for t = T , T —1 , 1 . The (approximate) measure
The investment decision rules
vip (s')
(gt ) t
x
is then given by
periods ago (for
t
=
x 0.
T ,T — 1,
1 ), which axe used
in this law of motion are the ones determined by the corresponding thresholds
(at, A t )
in the
history (a, A ) -
9t ( k , s )
=
at(s),
if (1 -
=
A t (s),
if (1
6)k < at(s)
- 6 ) k > A t ( s)
= (1 —6 ) k , otherwise
14
(27)
(iii)
The previous period measure across establishment types
decisions over current capital levels across establishment types
d
y,
and the previous period
are those returned as
X\
and <71 in (ii).
Note th at the Social Planner’s problem in equation (20) has linear constraints, and th at
the deterministic steady state values for the (endogenous) state variables are all strictly
positive. We can then perform a quadratic approximation to the return function about the
deterministic steady state, leaving us with a standard linear quadratic (L-Q) problem which
can be solved by ordinary value function iteration .10
Let us now return to the question of how long the history of thresholds (a, A ) should
be to get a good approximate solution to the original problem (8). It is not difficult to
show th at there exists a length
J
for thresholds histories such th at solving by L-Q methods
the planner’s problem (20) corresponding to length J , gives exactly the same solution as
solving by L-Q methods the planner’s problem (20) corresponding to any other length T
> J
(Appendix B explains the intuition for this result).11 It follows th at the only approximation
error introduced by the solution method stems from the quadratic approximation and not
from keeping track of a finite history of thresholds.
4. P a r a m e t r i z a t i o n o f t h e m o d e l
This section describes the steady state observations used to calibrate the parameters of the
model economy. In this section, the irreversibility parameter
q
will be assumed fixed at
10The quadratic approximation is obtained by imposing zero errors of approximation of the return function
at the grid points that lie just above and below the steady state grid points computed in Appendix A. T his
procedure to obtain numerical derivatives follows closely the one described in Kydland and Prescott [16].
11A formal proof to this claim is available upon request as a technical appendix. It can also be found in
Veracierto [25].
15
some particular value. Given a fixed
P ,6 ,j,S ,a ,v,iJ j(l),X ,
q,
the rest of the parameters we need to calibrate are
the transition matrix II, and the parameters determining the driving
process for the aggregate productivity shock:
p
and
a\.
The first issue we must address is what actual measure of capital will our model capital
correspond to. Since we are interested in investment irreversibilities at the establishment
level it seems natural to abstract from capital components such as land, residential structures
and consumer durables. The empirical counterpart for capital was consequently identified
with plant and equipment. As a result, investment was associated in the National Income
and Product Accounts with non-residential investment. On the other hand, the empirical
counterpart for consumption was identified with personal consumption expenditures in non
durable goods and services. Output was then defined to be the sum of these investment and
consumption measures. The annual capital-output ratio and the investment-output ratio
corresponding to these measures are 1.7 and 0.15 respectively. The depreciation rate
6
was
selected to be consistent with these two magnitudes.
The annual interest rate was selected to be 4 per cent. This 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 [18]. The discount factor
(3
was chosen to
generate this interest rate at steady state. Given the interest rate i and the depreciation rate
8,
the param eter
9
was selected to match the capital-output ratio in the U.S. economy. The
labor share param eter was in turn selected to replicate a labor share in National Income of
0.64 (this is the standard value used in the business cycle literature). On the other hand, the
preference param eter a was picked such th at 80% of the population works at steady state
(roughly the fraction of the U.S. working age population th at is employed).
16
The transition m atrix II was chosen to be of the following form:
\
1
c
* (1 -C )
(1 -0 (1 -O
(i - 0 ) (i - 0
(* (1 -0
(28)
,
i.e. a process that treats the low and the high productivity shocks symmetrically. The rest
of the parameters to calibrate axe then
<p, £, v , i p ( l ) ,
and A. The parameters C, <P> and A were
selected to reproduce important observations on job creation and job destruction reported in
Davis and Haitiwanger [10]. These are: (i) that the average annual job creation rate due to
births and the average annual job destruction rate due to deaths are both about 2.35%, (ii)
th at the average annual job creation rate due to continuing establishments and the average
annual job destruction rate due to continuing establishments are both about 7.9%, and (iii)
th at about 82.3% of the jobs destroyed during a year are still destroyed the following year.
The parameter
v
determining the number of establishments being created every period was
chosen so th at the average establishment size in the model economy is about 65 employees,
same magnitude as in the data.
Next, we must determine the distribution
ip
over initial productivity shocks. If we would
allow for a large number of possible idiosyncratic productivity shocks, it would be natural to
chose a
ip
to reproduce the same size distribution of establishments as in the data.12 W ith
only two values for the idiosyncratic shocks this approach does not seem restrictive enough
since we can pick any two arbitrary employment ranges in the actual size distribution to
12In general, allowing for a large number of idiosyncratic shocks would permit to calibrate the model
to more realistic establishment level dynamics. However, the associated computational costs would be
unbearable (even for computing deterministic steady states).
calibrate to. For this reason I chose to follow the same principle as in the choice of II and
pick
ip
= (0.5,0.5), i.e. a distribution th at treats the low and the high productivity shock
symmetrically (note th at these choices of II and
ip
imply that at steady state there will be
as many establishments with the low shock as with the high shock).
Finally, we must determine values for
p
and
o\.
The strategy for selecting values for
these param eters was to chose them so th at measured Solow residuals in the model economy
replicate the behavior of measured Solow residuals in the data. 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 7 , minus the sum of the
proportionate change in capital times (1 —7 ). Note th at these changes in measured Solow
residuals do not coincide with changes in the aggregate productivity variable
z
in the model
(the aggregate production function in the model economy is not a constant returns CobbDouglas function in labor and aggregate capital). Using the measure of output described
above and a share of labor of 0.64, measured Solow residuals were found to be as highly
persistent as in Prescott [20] but the standard deviation of technology changes came up
somewhat smaller: 0.0063 instead of the usual 0.0076 value used in the literature. Given a
fixed irreversibility param eter
for p and
q
and the rest of the parameters calibrated as above, values
were selected so th at measured Solow residuals in the model economy displayed
similar persistence and variability as in the data. It happened to be the case th at values of
p
= 0.95 and cr2
=
0.00632 were consistent with these observations in all the experiments
reported below.
Param eters values corresponding to economies with several different possible values for
q
are reported in Table 1.
18
5. B e n c h m a r k a g g r e g a t e fluctuations
There is considerable uncertainty about the choice of an empirically plausible value for the
irreversibility parameter
q.
Even though a number of empirical papers have documented
features of establishment behavior that suggest the presence of investment irreversibilities
at the establishment level (e.g. Caballero, Engel and Haltiwanger [7], Doms and Dunne
[12 ]), almost no attem pt was made to estim ate the magnitude of the irreversibilities that
establishments face. A remarkable exception is Ramey and Shapiro [21 ]. Using data on the
equipment auction of an aerospace firm, they estimated the wedge between the purchase
price and the resale price for different types of capital. Specifically, they estimated that
wedge to be 31% for machine tools and 5% for structural equipment. A difficulty with
their estimates is that they correspond to a single firm and cannot be directly extended
to the whole economy. In any case, they axe indicative of im portant degrees of investment
irreversibilities at the micro level. As a consequence, the economy with
q
= 0.5 will be
selected as a benchmark case but later on results will be reported under a variety of values
for the irreversibility parameter
q.
Table 2 reports summary statistics (standard deviations and correlations with output)
for the aggregate fluctuations of the benchmark economy and compares them to those of the
actual U.S. economy. Before any statistics were computed, all time series were logged and
detrended using the Hodrick-Prescott filter. The statistics reported for the U.S. economy
correspond to the output, investment and consumption measures described in the previous
section, and refer to the period between 1960:3 and 1993:4. For the artificial economy, time
series of length 136 periods (same as in the data) were computed for 100 simulations, the
19
reported statistics being averages across these simulations.
We observe th at output fluctuates as much in the model economy as in actual data.
Investment is about 5 times more variable than output in the model while it is about 4 times
as variable in the U.S. economy. Consumption is less variable than output in both economies
(though consumption is less variable in the model than in U.S. data). The aggregate stock
of capital varies about the same in both economies. On the other hand, hours variability is
only 70% the variability of output in the model economy while they vary as much as output
in U.S. data. Productivity fluctuates less in the model economy than in the actual economy.
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 actual economy) and productivity (which is highly procyclical in the
model while it is acyclical in the data).
We conclude th at the benchmark economy is broadly consistent with salient features of
U.S. business cycles.
6. M i c r o e c o n o m i c irreversibilities a n d a g g r e g a t e d y n a m i c s
In a celebrated paper Caballero, Engel and Haltiwanger [7] analyzed the importance of mi
croeconomic irreversibilities for macroeconomic dynamics using a non-structural empirical
approach. They concluded th at irreversibilities play a crucial role in generating brisk ex
pansions and contractions in aggregate investment dynamics. Since this has become the
dominant view thereafter, Sections 6.1 and 6.2 will describe their approach in detail and
apply it to the benchmark economy. The objective is not only to facilitate comparisons with
20
the previous literature, but to demonstrate the need for a general equilibrium analysis.
Section 6.3 constitutes the core of the paper. It compares the equilibrium business cycles
of economies subject to different degrees of investment irreversibilities. Contrary to what
the Caballero-Engel-Haltiwanger analysis of Section 6.2 suggests, investment irreversibilities
are found to have no effects on aggregate business cycle dynamics.
6.1. The ” Caballero-Engel-H altiwanger” approach
Caballero, Engel and Haltiwanger (hereafter CEH) proposed the following non-structural
method of analysis.13 They defined ’’desired capital”
kf+ l
to be the stock of capital an
establishment would like to carry to the following period if its investment irreversibility con
straint was momentarily removed during the current period. Correspondingly, they defined
’’mandated investment”
et
to be:
e* =
where
kt
kf +1
- (1 -
(29)
6 )k t
is the establishment’s stock of capital at date
t,
and
6
is the depreciation rate of
capital.
CEH assumed th at the investment behavior of establishments could be described by a
’’hazard function”
H (e),
which specifies for each possible e the fraction of its m andated
investment th at an establishment actually undertakes. Letting
ft
be the distribution of
13This same approach has been used by Caballero, Engel and Haltiwanger [8] to analyze the effects of
non-convex adjustment costs in aggregate employment dynamics.
21
establishments across m andated investments at date
t,
aggregate investment
It
is given by:
(30)
I t = f e H (e) f t (e) d e
To empirically implement their model, CEH first had to estimate the desired stock of
capital
kf+1
for each establishment and time-period in their sample.14 This gave CEH an
empirical time series for the distribution of cross-sectional mandated investment
tional form for the hazard function
H
f t.
A func
also had to be specified. CEH chose to work with the
following polynomial form:
H (e)
= £
V,
e*
(31)
t>=0
Substituting (31) in (30) delivers the following expression for aggregate investment:
It
- £
v=0
(32)
<P, M T * 1
where Mtv is the v-th moment of the distribution of mandated investments
ft-
CEH noticed th at when establishments face quadratic costs of adjustment, their optimal
investment behavior is described by a constant hazard function. In this case the ’’linear”
model results: aggregate investment depends only on the first moment of the distribution
of mandated investment (V = 0 in equations 31 and 32). In all other ”non-linear” cases,
higher moments of the distribution can play an im portant role in accounting for aggregate
investment behavior.
14We refer to the original paper for details on how these measures were obtained.
22
Allowing for a constant, (32) gives rise to the CEH regression equation:
I t = fi + ' £ < P v M ? +1 + e t
(33)
t>=0
Using the first five moments of their estimated cross-sectional distribution of mandated
investment, CEH fitted this equation to aggregate investment data using ordinary least
squares. They found th at the hazard function implied by their estimated coefficients
ip
is
highly non-linear. It indicates th at establishments undertake small adjustments in response
to negative m andated investments and that they respond substantially to positive m andated
investments (a behavior consistent with irreversibilities in investment).15 This finding is ex
tremely im portant, it suggests th at microeconomic irreversibilities exist in the U.S. economy
and that the associated non-linear adjustments at the establishment level are consistent with
aggregate investment dynamics.
To evaluate the importance of non-linear adjustments for aggregate investment dynamics
CEH ran a second regression, this time constraining aggregate investment to depend only on
the first moment of the cross-sectional distribution of mandated investments (i.e. setting
V
to zero in equation 33). CEH considered this linear model to be of particular interest since it
corresponds to an economy with quadratic costs of adjustment. CEH believed th at comparing
the aggregate investment behavior predicted by this linear model with the predictions of
the non-linear model would determine the role played by microeconomic irreversibilities in
aggregate investment dynamics.
15 CEH also estimated the hazard function in an alternative and more direct way. They measured H(e)
to be the average actual-investment/mandated-investment ratio across all establishments with mandated
investment e. They found that this alternative hazard function displays qualitatively similar properties as
the one estimated using equation (33).
23
Analyzing the relative performance of both models, CEH found the following results:
1 ) the non-linear model kept track of aggregate investment behavior much better than the
linear model (the absolute values of the prediction errors were always larger in the linear
model than in the non-linear model), and 2 ) this was specially true at brisk expansions
and contractions (the difference between the absolute prediction error of the linear model
and the absolute prediction error of the non-linear model was larger at periods when ag
gregate investment was far from its mean). CEH interpreted these results as evidence th at
microeconomic irreversibilities are crucial for macroeconomic dynamics. In particular, th at
irreversibilities play an important role in generating brisk expansions and contractions in
aggregate investment.
This way of evaluating the importance of microeconomic irreversibilities for aggregate
dynamics presents a serious weakness. The analysis compares the predictions made by the
linear model with the predictions made by the non-linear model, conditional on a sam e
realization of cross-sectional distributions of m andated investments
f t.
However, mandated
investments are not invariant to the investment technology th at establishments face. If
microeconomic irreversibilities were replaced by quadratic costs of adjustm ent, mandated
investments would no longer be the same. As a result, U.S. mandated investments (which
were determined to correspond to an economy with investment irreversibilities) cannot be
used to evaluate how aggregate investment would behave if establishments faced quadratic
costs of adjustment. Sections 6.2 and 6.3 demonstrate th at this type of analysis can in fact
lead to the wrong answers.
24
6.2. Caballero, Engel and H altiwanger visit th e benchm ark m odel
This section applies the CEH analysis to the benchmark economy. The objective is twofold.
First, to explore wether the benchmark economy is broadly consistent with the CEH empirical
findings. Second, to determine what conclusions can be obtained about the importance of
investment irreversibilities for aggregate dynamics.
Applying the CEH approach will involve running a similar set of regressions as CEH, but
on artificial data generated by the benchmark economy. Before doing so, model-counterparts
for the variables defined by CEH must be determined.
Observe that the desired capital of an establishment of type
(k,s)
is given by kf+1 = a(s),
since this is the stock of capital the establishment would chose if q was set to one during the
current period (see footnote 7). As a consequence, its m andated investment is given by e =
a(s)
—(1 —8 )k . The aggregate measure x t of establishments across idiosyncratic productivity
shocks
s
and capital levels
k
can then be used to obtain the cumulative distribution of
establishments across mandated investments levels in the model economy:16
Ft (e) =
J
dxt
{(& ,s): a ( s ) —( 1 —6 ) k < e and fc> 0}
The moments
to be used in the regressions below axe those obtained from this cumulative
distribution.17
16Note that in defining this cross-sectional distribution of mandated investments, we have excluded all
those establishments that died between periods t — 1 and £, as well as those establishments that were born
between periods t — 1 and t (and therefore have A* = 0). This is done to parallel CEH, who worked with a
balanced panel of establishments. However, similar results are obtained if those establishments are included
in the distribution of mandated investments.
17In applying the CEH approach to the benchmark economy we are abstracting from any measurement
problems. We’ll show that the approach can lead to the wrong conclusions even when mandated investments
are correctly measured.
25
To generate artificial data, the empirical realization of Solow residuals for the period
1960:1 to 1993:4 was fed into the model economy.18 The resulting time series for aggregate
investment and the first five moments M" were subsequently used to estim ate the CEH
investment equation (33) by ordinary least squares. The estimated coefficients gave rise to
/
the hazard function shown in Figure 2.19 This hazard function displays similar properties
as those reported by CEH, i.e. it indicates small adjustments to capital surpluses and large
adjustments to capital shortages. This is not surprising since there are substantial investment
irreversibilities in the benchmark model economy and consequently, establishments do follow
(S,s) decision rules.20
Following CEH, the linear model was also estimated. In particular, a second regression
allowing aggregate investment to depend only on the first moment of the distribution of
mandated investments was fitted to the model-generated data.
Before comparing the predictions of the two statistical models, the tim e periods from
the simulations were sorted in a decreasing order from the largest to the smallest realized
absolute deviation of aggregate investment from its mean. Figure 3 plots the difference
between the absolute prediction error of the linear model and the absolute prediction error
of the nonlinear model, across the sorted tim e periods.21 We observe two im portant patterns.
18This constitutes 136 periods of observations. In the experiment, the distribution across establishment
types was initialized to be the deterministic steady state distribution. Then, 136 periods were generated from
the model economy but only the last 100 periods were considered to be the sample period for the regressions.
This was done to minimis the effects of initializing with the deterministic steady state distribution.
19The figure shows the hazard function over the range of relevant mandated investment values (i.e. the
range of values realized in the experiments).
20In the benchmark model economy, the hazard function over the relevant range of mandated investments
is equal to one for positive values of e (establishments adjust to a ( s ) if (1 —6)k < a(s)), and equal to zero
for negative values of e (establishments let the capital depreciate if (1 —S)k > a(s)).
It should not surprise the reader that this is the hazard function that corresponds to q — 0, since it will be
shown in the next section that establishments behave extremely similarly under q = 0.5 than under q = 0.
21The absolute prediction errors were divided by aggregate investment to obtain relative magnitudes.
26
First, in all periods the difference is positive. Second, the difference is large in the first few
periods and then declines towards zero. In other words, the non-linear model predicts the
benchmark economy’s aggregate investment more accurately and this is specially true when
aggregate investment is far from its mean.22 These are exactly the same findings th at CEH
encountered in their empirical analysis. We see that the benchmark economy conforms with
them quite well.
Based on similar findings, CEH concluded that microeconomic irreversibilities play a cru
cial role in generating brisk expansions and contractions in aggregate investment. However,
the analysis performed provides little economic support for making such type of assessment.
While the predictions of the estimated non-linear model (approximately) describe the be
havior of aggregate investment in our benchmark economy, it is not clear what investment
behavior the estimated linear model is capturing.23 In principle, it represents some economy
with quadratic costs of adjustment. But it would be hard to determine an economy consistent
b o th with the constant hazard function of our estimated linear model a n d the mandated
investments of our benchmark economy.24 Even if such economy exists, comparing its aggre
gate investment dynamics with th at of our benchmark economy (as the CEH approach does)
is probably not a fruitful exercise: the economies would differ in so many dimensions, th at no
definite statem ent could be made about the particular role of microeconomic irreversibilities
22The fact that the non-linear model predicts aggregate investment much better than the linear model
should not be surprising. The (S,s) decision rules that establishments follow in the benchmark model can
be much better approximated by fitting a high order polynomial for the hazard function than a constant
function.
23Recall that the predictions of the linear model are made conditional on the mandated investments of the
benchmark economy.
24 As has already been mentioned, mandated investments are not invariant to the investment technology
that establishments face.
27
in aggregate dynamics.25
Given these difficulties with the CEH approach, Section 6.3 will perform an alternative
analysis. It will compare the general equilibrium dynamics of economies subject to different
investment irreversibility levels. An im portant advantage of this approach is th at explicit
economic environments will be specified and th at the aggregate dynamics analyzed will be
fully consistent with these environments. As a consequence, the analysis will be able to
make precise statem ents about the role of investment irreversibilities in'aggregate dynamics.
Interestingly, the conclusions obtained will be in sharp contrast to CEH.
6.3. The effects o f m icro-irreversibilities in aggregate fluctuations
This section evaluates the effects of irreversibilities on aggregate fluctuations by comparing
the equilibrium business cycles of economies subject to different degrees of investment ir
reversibilities. For this to be a meaningful exercise, the economies must be comparable in
important dimensions. We choose them to be identical to the U.S. economy in terms of the
long run means and ratios of Section 4, as well as their stochastic processes for measured
solow residuals. These are all observations th at the real business cycle literature has empha
sized as being im portant for aggregate fluctuations. Controlling for them will help isolate
the effects of investment irreversibilities in business cycle dynamics.
Table 3 provides summary statistics for the business cycles of economies with investment
irreversibility parameters
q ’s
ranging between 1 and 0.26 The results are striking. Irre-
25It must be pointed out that the CEH analysis provides valuable econometric information. It determines
the prediction biases that would be obtained from measuring mandated investments correctly in an economy
with investment irreversibilities, while misspecifying the hazard function to be a constant function. However,
the CEH approach is less useful as an economic analysis.
26Parameter values were given in Table 1.
28
visibilities tend to decrease the variability of output, investment and hours, and increase
the variability of consumption. But these differences are surprisingly small. For example,
the standard deviation of output decreases monotonically as q goes from 1 to 0 (as one would
expect given the adjustment costs introduced), but it goes from 1.41 when
1.39 when
q
q
= 1 to only
= 0. This is a small difference considering that we are moving from the per
fectly reversible case to the complete irreversibilities scenario. Overall, the properties of the
business cycles generated by all these economies are extremely similar. We conclude that,
at least in terms of the standard statistics that the real business cycle literature focuses on,
investment irreversibilities play no crucial role for aggregate fluctuations.
Nevertheless, investment irreversibilities could still be important for features of the busi
ness cycles not captured by the standard RBC statistics. One possibility is that investment
irreversibilities generate brisker expansions and contractions in aggregate investment. An
other possibility is that investment irreversibilities create asymmetries in aggregate fluctua
tions. Figure 4 explores these possibilities: it reports for the economies with q
=
1 and q = 0,
the histograms of the deviations of aggregate investment from trend across all realizations. If
investment irreversibilities generate brisker expansions or contractions, the histogram under
q
= 0 would have fatter tails than under
q
= 1. On the other hand, if investment irre
versibilities generate asymmetries, the histogram under
than under
q =
q =
0 would be more asymmetric
1. However, Figure 4 shows that these histograms are virtually the same.
Investment irreversibilities do not create noticeable asymmetries nor brisk expansions and
contractions.27
Figure 5 searches for other possible differences in aggregate dynamics created by invest
27The analysis of aggregate output, consumption, and hours worked lead to similar conclusions.
29
ment irreversibilities. It reports the realizations of aggregate investment which arise from
feeding into the economies with
q
= 1 and
q —0
the empirical realization of Solow residuals
for the period 1960:1 to 1993:4. We observe that the time series for aggregate investment gen
erated by the economy with complete irreversibilities is almost identical to the one generated
by the economy with perfectly reversible investment. Figure 6 shows the impulse response
functions for output (Y), consumption (c), investment (I), and labor (7 7 ) to a one-time aggre
gate productivity shock of one standard deviation, which correspond to the economies with
q
= 1 and
q
= 0. We also see th at they are almost the same. We conclude th at investment
irreversibilities have no major effects on aggregate business cycle dynamics.
This is a surprising result. Intuition suggests th at investment irreversibilities could gen
erate im portant asymmetries in aggregate fluctuations, as establishments would be much
more reluctant to adjust their stock of capital to negative productivity shocks than to pos
itive productivity shocks. To understand our lack of asymmetries result, we must analyze
the responses of the model economy to aggregate shocks in further detail.
For the economy with
support of the distribution
q =
xt
0.99, Figure 7 shows the impulse response of the capital
to a one-time aggregate productivity shock of one standard
deviation, starting from the steady state support (the steady state capital distribution
x*
is displayed in Figure 8).28 We see that in response to a positive shock, the thresholds
a (l), A (l) and o(A) increase on impact, continue to increase for a number of periods and
eventually decrease, returning gradually to their steady state levels. Instead, the capital
28 We chose to show the behavior of the economy with q = 0.99 over those with a smaller q, since it has
a relatively small capital support (simplifying the figures considerably). However, similar patterns can be
found in the other economies.
Even though it is not shown, zero always belongs to the support of the capital distribution since new
establishments arrive with zero capital in place.
30
levels pertaining to the range of inaction between a (l) and >1(1) are not affected on impact.
They follow the same dynamics as the upper threshold >1(1) but with a lag, which depends
on the number of periods it takes >1(1) to depreciate to the corresponding capital level.
Observe th at the support of the capital distribution responds symmetrically to positive and
negative shocks. It is then not surprising th at the business cycles generated by these shocks
will inherit similar features.29
On a first impression, the symmetric response of the capital support to aggregate shocks
may appear a necessary consequence of the linear quadratic approximation performed. But
this is not generally true. The solution method implies th at current capital thresholds (the
decision variables) are a Unear function of the aggregate shock and the past history of capital
thresholds (the state variables). If the driving aggregate productivity shock is symmetrically
distributed, of course capital th resh o ld s will behave symmetrically. But this does not imply
that the capital su p p o rt will behave symmetrically. In fact, it seems safe to conjecture th at
the symmetry would be lost if aggregate shocks had an empirically implausible large variance.
To be concrete, let consider how the largest point in the capital support would respond
to a large negative shock, starting from its steady state value a*(A). Suppose th at the shock
is so low th at the threshold a(A) decreases on impact below (1 —S)a*(A). The highest point
in the support would then become (1 —S)a*(A), since it would fall in the range of inaction
defined by the new value of a(A). W hat is important to note is th at negative shocks of larger
magnitude would generate no further effects on impact, since (1 —6 ) a * ( A) would stiU fall in
29 Strictly speaking, describing the response of the capital support is not enough. The number of estab
lishments at each of these capital levels and idiosyncratic productivity shocks should also be considered.
However, at any point in time, the number of establishments at each of these capital levels can be read
directly from the corresponding point in the steady state distribution in Figure 8. The reason is that the
process for the idiosyncratic shocks is exogenous and the paths illustrated in Figure 7 do not cross.
31
a range of inaction. On the contrary, there would be no counterpart to this lack of further
responsiveness when shocks are positive. If a positive shock drives
a ( A)
above (1 —6)a*(A),
the highest point in the support would always jump on impact to the new value of a(A).
This would be true no m atter how large the positive shock is.
Figure 9 illustrates these ideas by showing the impulse responses of the highest point in
the capital support to one-time aggregate shocks, ranging from one to twenty standard devi
ations in magnitude.30 Let consider the responses in period one to each of these shocks. We
see th at when shocks axe negative, the largest capital level in the support moves to smaller
values as the shock becomes larger. However, once the shock reaches fifteen standard devia
tions, it stops responding to further shocks. On the contrary, when shocks are positive, this
capital level always moves to higher values as the shock gets larger. This pattern of response
opens interesting possibilities for the creation of asymmetries in aggregate fluctuations. In
particular, it suggests th at aggregate investment would tend to decrease slowly in response
to large negative shocks, and increase sharply in response to large positive shocks.31
In view of these arguments, we must consider the lack of asymmetries the theory predicts
as arising purely from measurement. Measured solow residuals are not variable enough for
investment irreversibilities to create asymmetries in aggregate business cycles: the associated
fluctuations in capital thresholds are small compared with the drift introduced by depreei30 This figure is drawn only for heuristic purposes. If shocks were as large as those shown, the linear
quadratic approximation performed in the paper would probably be of poor quality.
31It should be clear that a(l) would generate similar asymmetries, since it would mimic the behavior of
a(A). The analysis would be somewhat more complicated though, since capital levels in the lower portion
of the range of inactivity would be affected by large fluctuations in a(l). In particular, these points would
collapse into a(l) under sufficiently large increases in a(l), but will not be affected when a(l) decreases. This
effect would tend to reinforce the asymmetries described above.
For economies with g’s smaller than 0.9 the behavior of A(l) becomes irrelevant, since the ranges of
inaction overlap and no establishment has a capital level close to .4(1).
32
ation (actually in none of the simulations reported, the rate of change of thresholds ever
exceeded the rate of depreciation).32
To complete our analysis, we now consider the importance of irreversibilities for plant
level investment dynamics. Figure 10 shows the distribution of plant level gross invest
ment rates (for continuing establishments) across all realizations, under different values for
the irreversibility parameter
q 33
We observe that when there are no irreversibilities
(q —
1.0), a large number of establishments have near-zero net investment (gross investment ap
proximately equals the depreciation of capital).34 We also observe th at a small number of
firms make sharp increases and sharp decreases in their stock of capital.35 On the contrary,
when the irreversibility parameter
q
becomes zero: 1) the number of establishments display
ing close-to-zero adjustment is larger, 2) there are no establishments with sizable negative
investment rates, and 3) capital increases are not as sharp as under
q
= 1.0. Note th at a
substantial mass of establishments have zero gross investment, since a large number of plants
chose to remain inactive because of the investment irreversibility they face (they lie in ranges
of inaction). It is also interesting to note th at the histogram of investment rates that arises
when
q
= 0.95 is very similar to the one under
q
= 0. This conforms with the finding by
Abel and Eberly [1] th at small degrees of irreversibilities can m atter a lot. Figure 10 shows
32This result is related to Dow and Olson [14]. They found that in the real business cycles model of
Hansen [17], aggregate irreversibilities play no role since productivity shocks are not variable enough to
make the non-negativity constraint in aggregate investment binding. An important difference in this paper
is that plant level irreversibilities do bind. However, they bind due to the amount of idiosyncratic risk that
establishments face, not because of the level of aggregate uncertainty in the economy. Aggregate productivity
shocks play a minor role.
33These histograms correspond to the same simulations as those underlying Table 3.
34These are establishments that in the absence of aggregate shocks would like to keep the same stock of
capital they have. With aggregate fluctuations their investment rates do fluctuate, but the grid in Figure 2
is not fine enough to reflect these movements.
35 This is due to the fact that only two (positive) idiosyncratic shocks are considered, and that they are
very persistent.
33
that even though irreversibilities play no im portant role for aggregate fluctuations, they are
crucial for establishment level dynamics.
7. C o n c l u s i o n s
Caballero, Engel and Haltiwanger [7] found th at non-linear adjustments are crucial for ag
gregating mandated investments in the U.S. economy into observed aggregate investment
dynamics. While their findings represent substantial evidence of investment irreversibilities
in the U.S. economy, their analysis is less useful for assessing the importance of investment
irreversibilities in business cycle dynamics. To evaluate the effects of investment irreversibil
ities on aggregate fluctuations, the business cycles of economies subject to different levels of
irreversibilities must be analyzed.
An interesting empirical project would be to study how business cycles vary with invest
ment irreversibilities across actual countries, after controlling for variables such as average
capital-output ratios, capital shares, investment rates, variability and persistence of solow
residuals, etc (i.e. variables th at business cycle theory emphasize as being im portant). How
ever, lack of data on investment irreversibilities makes this exercise unfeasible.
This paper has used economic theory to proxy for this exercise. It has compared the
business cycles of artificial economies th at look exactly the same as the U.S. economy in
terms of their long run observations and solow residuals, but which differ in term s of their
investment irreversibility levels. The results were striking: economies ranging from fully
reversible to completely irreversible investment generate almost identical business cycle dy
namics. The only dimension in which investment irreversibilities m atter is for establishment
34
level dynamics.
Our results suggest th at macroeconomists can safely abstract from investment irreversibil
ities when modeling aggregate business cycle dynamics.
A. A p p e n d i x
This appendix describes the algorithm used to compute the steady state of the deterministic
version of the economy. We will show that the problem is reduced to solving one equation
in one unknown (after the relevant substitutions have been made). First, it must be noticed
that (similarly to the neoclassical growth model) the steady state interest rate is given by:
l+t = i
Fixing the wage rate at an arbitrary value
w,
(34)
the value of the different types of estab
lishments (as a function of w ) can be obtained by solving the following functional equation:
J(fc, s ; w )
—M AX
{
s k 9n r
+■ T~~:
1+*
—w n —[A/ —(1 —6)k]
J (k \
w)*
Q
[A/ —(1 —5) A:]
(5>s ') }
(35)
The solution to this problem is computed using standard recursive methods. Note th at the
solution to this problem also gives the decision rules n(A;, s; w ) and
g(k,
s; w ) as a function
of w .
Given a
w
and the corresponding
g(k,
s; to), a measure
x(w)
and capital levels can be obtained from the law of motion for x:
35
across productivity shocks
x (B ,s';w )=
7r (s,s')
J
d x ( w ) + v ip (s')
x (0 G J3 )
(36)
In practice, this is solved by iterating on this law of motion starting from an arbitrary
initial guess for
Once a
x(w)
x(w).
is obtained and given the previous
n ( k , s; w )
and
g(k,
s; w ) found, we can
solve for the corresponding consumption c(w ) implied by the feasibility condition:
c(w) = j
s k 0n ( k , s ; w ) ' 1
•f /
A wage rate
w
—(1 —6)k] (^[^(fc, s ; w )
■ [g (k, s ; w )
(1
- 6 ) g (k ,s\w ) q
7r(s,0)
dx(w)
— (1
—5)fc]
dx(w)
(37)
corresponds to the steady state value if the marginal rate of substitution
between consumption and leisure is satisfied, i.e.:
c{w)
= -
(38)
This is one equation in one unknown and is solved using standard root finding methods.
The actual computer implementation of this algorithm requires working w ith a finite grid
of capital levels. In all experiments reported in the paper, the number of grid points were
between 1,000 and 1,800.
36
B. A p p e n d i x
This appendix provides intuition for why it is sufficient to carry a finite history of thresholds
when solving the social planner’s problem (20) by L-Q methods.
Suppose th at it takes exactly
J
periods for the steady state upper threshold >1*(1) to
first depreciate below the steady state lower threshold a*(l). For simplicity, assume th at
A*(l) < a*(A). Fixing all other thresholds at their steady state values, consider for example
the effects of a small change in the upper capital threshold >1(1) about its steady state value
>T(1). In what follows it will be argued that within
J
periods, an establishments th at was
affected by this small change in >1(1) will end up with exactly the same capital level as in
the absence of such small perturbation.
There are two possibilities. The first case is if after exactly
J
periods the establishment
has not yet made a transition to s = A. In this case, its stock of capital (after
J
periods)
would be given by (1 —<$)J >1(1) (since the establishment would have stayed in the range of
inaction corresponding to
s
= 1). Given th at >1(1) is close to A*(1), and (1 —6 ) J A*(l) <
a*(l), it follows from (19) th at (after
capital to be
k'
= a*(l) (the same
k'
J
periods) the establishment will chose its next period
the establishment would have chosen without the small
perturbation in >1(1)).
The second case is if the establishment makes a transition to
t < J.
s
= A in
t
periods, where
In this case, its capital level would be (1 —6)* A (l) at the time of such transition. Since
>1(1) is close to A*(l) and (1 —S)1 >1*(1) < a*(A), it follows from (19) th at the establishment
will chose a next period stock of capital
same
k'
k1 =
a* (A) at the time of the transition (again, the
the establishment would have chosen without the small perturbation in >1(1))-
37
Note that if a(l) or a(A) would’ve differed slightly from their steady state values at the
time that these actions take place, the argument would still apply. The establishment would
end up within
J
periods with a stock of capital given by a threshold level determined in
a later period, independently of the small perturbation in A(l) made
J
periods ago. As
a consequence no further information is gained from keeping histories more than
J
periods
long, provided that capital thresholds remain in a neighborhood of their steady state values.36
References
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Economic Studies, 63, 581-93.
[2] Arrow, K. 1968. Optimal Capital Policy with Irreversible Investment’in Value, Capital
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Publishing Company.
[3] Bertola, G. and Caballero, R. 1990. Kinked Adjustment Costs and Aggregate Dynamics.
N B E R Macroeconomics Annual.
[4] Bertola, G. and Caballero, R. 1994. Irreversibility and Aggregate Investment. Review
of Economic Studies, 61, 223-246.
[5] Caballero, R. and Engel, E. 1991. Dynamic (S,s) Economies. Econometrica, 59-6,16591686.
36Capital thresholds actually fluctuate very littlealong the simulations analyzed in this paper.
38
[6] Caballero, R. and Engel, E. 1994. Explaining Investment Dynamics in U.S. Manufac
turing: A Generalized (S,s) Approach. N B E R Working Paper # 4887.
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[9] Caplin, A. and Leahy, J. 1997. Aggregation and Optimization with State-Dependent
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[10] Davis, S. and Haltiwanger, J. 1990. Gross Job Creation and Destruction: Microeconomic
Evidence and Macroeconomic Implications. N B E R Macroeconomics Annual, V, 123168.
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University Press.
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[14] Dow, J. and Olson, L. 1992. Irreversibility and the behavior of aggregate stochastic
growth models. Journal of Economic Dynamics and Control, 16, 207-223.
39
[15] Fisher, J. and Hornstein, A. 1995. (S,s) Inventory Policies in General Equilibrium.
Mimeo, University of Western Ontario.
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[20] Prescott, E. 1986. Theory Ahead of Business Cycle Measurement. Federal Reserve Bank
of Minneapolis. Quarterly Review.
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40
[25] Veracierto, M. 1997. Plant Level Irreversible Investment and Equilibrium Business Cy
cles. Institute for Empirical Macroeconomics, Discussion Paper 115. Federal Reserve
Bank of Minneapolis.
41
TABLE 1
IS
1!
M
o
P aram eter V alues
q = 0.99
p
0.99
0.99
0.99
0.99
0.99
0.99
0.99
e
0.2186
0.2185
0.2179
0.2174
0.2173
0.2172
0.2168
Y
0.64
0.64
0.64
0.64
0.64
0.64
0.64
6
0.02206
0.02184
0.02163
0.02146
0.02066
0.01914
0.01622
a
0.94
0.94
0.94
0.94
0.94
0.94
0.94
V
7.257E-5
7.260E-5
7.257E-5
7.273E-5
7.278E-5
7.280E-5
7.287E-5
vd)
0.5
0.5
0.5
0.5
0.5
0.5
0.5
X
1.112
1.127
1.195
1.208
1.210
1.213
1.220
<p
0.92
0.92
0.92
0.92
0.92
0.92
0.92
0.00593
0.00593
0.00593
0.00593
0.00593
0.00593
0.00593
p
0.95
0.95
0.95
0.95
0.95
0.95
0.95
«E2
0.00632
0.00632
0.00632
0.00632
0.00632
0.00632
0.00632
q = 0.95
q = 0.90
q = 0.75
q = 0.50
q =0.00
42
TA BLE 2
U.S. and benchmark fluctuations
U.S. Economy (60:1-93:4)
Benchmark Economy
Std. Deviation
Correlation
Std. Deviation
Correlation
Output
1.33
1.00
1.39
1.00
Consumption
0.87
0.91
0.51
0.91
Investment
4.99
0.91
6.83
0.98
Capital
0.63
0.04
0.49
0.08
Hours
1.42
0.85
0.94
0.98
Productivity
0.76
-0.16
0.51
0.92
43
B usiness cycles across econom ies
S tan d ard D eviations:
- q=l,0
q = 0.99
q = 0.95
q = 0.90
q = 0.75
q = 0.50
q = 0.00
Output
1.41
1.40
1.39
1.39
1.39
1.39
1.39
Cons.
0.49
0.50
0.51
0.51
0.51
0.51
0.52
Investm.
7.09
7.01
6.89
6.85
6.83
6.83
6.82
Capital
0.51
0.50
0.49
0.49
0.49
0.49
0.48
Hours
0.98
0.97
0.95
0.94
0.94
0.94
0.94
Product.
0.49
0.50
0.51
0.51
0.51
0.51
0.52
q = 0.99
q = 0.95
q = 0.90
q = 0.75
q = 0.50
q = 0.00
o
-Q
II
C orrelations w ith O u tp u t;
Output
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Cons.
0.91
0.91
0.91
0.91
0.91
0.91
0.91
Investm.
0.98
0.98
0.98
0.98
0.98
0.98
0.98
Capital
0.08
0.08
0.08
0.08
0.08
0.08
0.08
Hours
0.98
0.98
0.98
0.98
0.98
0.98
0.98
Product.
0.91
0.91
0.91
0.92
0.92
0.92
0.92
44
F I G U B EJL
Decision Rules
45
FIGU R E
2
Hazard Function
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-300
-200
-100
0
100
M a n d a t e d Investment
46
200
300
FIGURE 3
Difference Absolute Prediction Errors
(Linear vs. Non-linear model)
0.4
0.3
I
0.2
t
0.1
!
/wl||A
A
jl|V
I..
K
K
\ 1A Aj 1
_____
M l J U , A...1A
r
'
m
M|
0
l
f
Wl V
a
V
In
'i
(\
kk
A
1
-0.1
0
20
40
60
80
100
R a n k i n g a b s o l u t e i n v e s t m e n t deviations
47
120
F IG U R E 4
A g g r e g a te I n v e s tm e n t D e v ia tio n s fr o m T r e n d
(q=1.0)
(q=.00)
48
F IG U R E S
R e a liz a tio n s o f A g g r e g a te I n v e s tm e n t
49
FI G U R E
6
I m p u l s e R e s p o n s e Functions
(q=1.0)
Periods
(q=.00)
Periods
50
F I G U R E -2
I m p u l s e Response-Capital S u p p o r t
Positive Shock:
Negative Shock:
O
CM
8
§
®
330
330
A(1);
A(1>^
310 --
310..
29G--
290..
270.-
270. r_
a(l)
a(l>
250
10
-H
20
8
§
O
20
30
40
250
30
40
0
10
Periods
Periods
51
F I G U R E-8
497.0
Steady State Capital Distribution
52
F I G U R E 9
I m p u l s e R e s ponses - Highest Point in S u p p o r t
Negative Shock
Positive Shock
53
FIG U R E
10
Establishment Level Investment Rates
(q=1.0)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
‘n T r c o c \ i T - o - r - C M c O T r i n < o r ^ c o o > - r - o
p o o
*
•
o o
•
•
d d d d d o d d d
A
(q=.95)
(q=.00)
54
^
@FedFRASER