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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Price Stickiness in Ss Models:
New Interpretations
Ricardo
J.
of Old Results
Caballero
Eduardo M.R.A. Engel
Working Paper 07-07
Febi-uary 19, 2007
RoomE52-251
50 Memorial Drive
Cambridge,
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MA
paper can be downloaded without charge from the
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-
"
Price Stickiness in Ss Models:
New Interpretations of Old Results
Ricardo
MIT
J.
Eduardo M.R.A. Engel
Yale University and NBER
Caballero
and
NBER
February
2007^
19,
Abstract
What
is
the relation between infrequent price adjustment and the
The answer
the aggregate price level to monetary shocks?
dynamic response
of
to this question ranges from
a one-to-one link (Calvo, 1983) to no connection whatsoever (Caplin and Spulber, 1987).
The purpose
behind
of this paper
wide range of
this
which
in this area,
price rigidity.
We
The
in
this result
is
to provide a unified
results.
In doing so,
framework to understand the mechanisms
we propose new
turn suggest the kind of Ss model that
first
show that when
is
result
we
revisit
price stickiness
is
is
microeconomic
level.
measured
We
likely to generate substantial
Caplin and Spulber's monetary neutrality model.
in
not a consequence of aggregation, but
stickiness at the
is
interpretations of key results
also
terms of the impulse response function,
is
due instead to the absence of
show that the
"selection effect," according to
which units that adjust their prices are those that benefit the most,
sufficient to
account for the higher aggregate
models. Instead, the key concept
to the aggregate price response.
is
ized
JEL
is
neither necessary nor
5s-type models compared to Calvo
the contribution of the extensive margin of adjustment
The aggregate
the microeconomic frequency of adjustment
Keywords: Aggregate
flexibility of
price-
if
price level
and only
if
is
more
this
flexible
term
is
than suggested by
positive.
price stickiness, adjustment hazard, adjustment frequency, general-
Ss model, extensive margin, Calvo model,
strategic complementarities.
classification: E32, E62.
'Respectively:
caball@mit.edu; eduardo.engel@yale.edu.
We
thank Ruediger Bachmann, Olivier Blan-
chard, Bill Brainard, Mike Golosov, Oleksy Kryvtsov, John Leahy, Virgiliu Midrigan, and participants at
the Conference on "Microeconomic Adju.stment and Macroeconomic Dynamics" (Study Center Gerzensee,
October 20, 2006); the New York Workshop on Monetary Economics (NY Fed, November 17, 2006), and the
Duke macroeconomics workshop for helpful comments on an earlier draft of this paper (September, 2006)
titled "Price Stickiness in Ss Models; Basic Properties." The authors thank the NSF for iinancial support.
Introduction
1
Understanding the response of the aggregate price
central questions in
rigidity is
monetary economics. Since the
a microeconomic
nomic pricing behavior,
level to
monetary shocks
origin of almost
is
among
the
any aggregate nominal
have been many studies documenting microeco-
rigidity, there
in particular the frequency of
microeconomic price adjustment.^
For these studies to be relevant for macroeconomic policy, we need to understand the
mapping from infrequent
price adjustment to aggregate price stickiness.
We
already
know
that this mapping can be surprising. Caplin and Spulber (1987), henceforth CS, construct an
insightful exam.ple
where there
is
no relation between these two measures. They combine a
one-sided Ss model of microeconomic price adjustment with a specific form of asynchronous
adjustment of individual prices (the assumption of a uniform cross-section), and obtain
an aggregate price
level that
responds one-for-one to monetary shocks.
aggregate price stickiness in their model
thereafter
—the impulse response
is
Thus there
is
no
one upon impact and zero
—despite the fact that the frequency of microeconomic price adjustments can take
any value.
In a related recent result,
Golosov and Lucas (2006) show that the sluggishness of the ag-
gregate price response to monetary shocks
is
overestimated
model with a Calvo model, where adjustment
price imbalances.
That
is,
is
when approximating a menu-cost
infrequent but uncorrelated with the size of
the frequency of microeconomic price adjustments underestimates
the flexibility of the aggregate price level in Ss models. Similarly, Bils and Klenow (2004)
report that the flexibility of aggregated price series in U.S. retail data
than suggested by the frequency of price adjustments observed
in
is
significantly higher
microeconomic data: they
estimate a median monthly frequency of price adjustments of 0.21, while one minus the
first-
— a natural measure of aggregate price
equal the adjustment frequency—
almost
order autocorrelation of the aggregate inflation series
flexibility
which
in their
Calvo setting should
is
four times as large.
Aside from these illustrative examples,
is
there anything
more general that can be
said
about the connection between the frequency of microeconomic adjustment and the degree of
aggregate price level?
flexibility of the
What
are the precise channels through which microe-
conomic inaction and aggregate price stickiness are connected
of this paper
some
is
in
Ss models? The purpose
to answer these questions and, along the way, offer
new
interpretations for
old results.
^See Bils and Klenow (2004)., and Nakamura and Steinsson (2006a) for the U.S., and Dhyne et al. (2006)
and Fabiani et al. (2006) for a summary of the findings of the impressive set of country studies sponsored
by the ECB's Inflation Persistence Network.
There are two new interpretations and mechanisms we discuss
out. First, the
effect
main
result in
CS
is
that in their context a
on aggregate output, despite the
fact that at
The standard
units do not adjust their prices.
smaU monetary expansion has no
any given instant most microeconomic
interpretation of this resuh
erases the impact of microeconomic stickiness.
Our
first
point
depends crucially on the concept of price stickiness being used.
concept
is
paper that stand
in this
is
If,
is
that aggregation
that this interpretation
at the
micro
level, this
defined in terms of the frequency of price adjustments, then the statement
correct, since
money
neutral at the aggregate level despite the fact that most firms do not
is
adjust their prices at the micro
stickiness, in
is
level.
Yet
if
we have
in
mind a more standard
definition of
terms of the impulse response to monetary shocks, then the usual interpretation
of Caplin and Spulber's
money
neutrality result changes dramatically, as
now
it
reflects the
absence of microeconomic stickiness.
Second, Golosov and Lucas (2006) coined the expression "selection effect" to describe the
feature of
Ss type models that adjusting firms are those that need
with Calvo's model, where adjustment
is
it
most, in sharp contrast
completely random. They argue informally, and
others have followed, that this feature explains the higher aggregate flexibility of Ss type
models over Calvo models,
for a given
that while the selection effect
behind their additional
is
frequency of microeconomic adjustment.
present in most Ss type models,
it is
show
not the key concept
flexibility.
Instead, the key concept
is
that of an extensive margin.
That
is,
price increase resulting from the rise in the fraction of agents adjusting
fall in
We
of the additional
upwards and the
the fraction of agents adjusting downwards, both as a consequence of an additional
monetary impulse.
It
contrasts with the intensive margin, which describes the additional
price increase (or reduced price decrease) of those firms that were going to adjust anyway.
Only the
intensive margin
are strictly positive.
is
active in the Calvo model, while in
This extensive margin
of agents reaching the trigger thresholds in
is
Ss type models both margins
a generalization of the effect of the
number
Ss models discussed by Bar-Ilan and Blinder
(1992).3
Section 2 revisits the Caplin and Spulber model. Section 3 begins our study of the relation
of aggregate price flexibility
and the frequency
of
microeconomic adjustment, considering a
simple extension of the Caplin and Spulber model that includes the Calvo model as a hmiting
case. Sections 4
and
5 are the core of the paper: they describe the key results in the context
^Incidentally, while Golosov and Lucas (2006) used the expression "selection effect" to describe the
mechanism behind their finding, they do point out in their formal analysis that the key difference between
the Calvo model and their particular Ss model is the change in the number of agents that adjust in response
to a monetary impulse. The point, however, is that selection and extensive margin are not synonyms.
of a generalized
6 concludes
and
Ss model, which includes standard Ss models
is
as a particular case. Section
followed by an appendix.
Caplin and Spulber Revisited
2
This section recreates and
clarifies
the source of the Caplin and Spulber monetary neutrality
provides a natural starting point for the topics we cover in later sections.
result. It also
The Model
2.1
Let us focus on the aspects of the model which are relevant to our concerns, skipping the
derivation of the underlying microeconomic rules or a discussion of general equilibrium as-
which are largely orthogonal to the issues we address
pects,
Dotsey, King and
There
exists a
Wolman
(1999) for useful references on the steps
continuum of firms indexed by
z
e
[0, 1].
Stokey (2002) and
(see, e.g.,
we
Let pu and
skip).
p*f.
denote the (log of
the - henceforth omitted) actual and target price, respectively, both for firm
CS
i
at time
t.^
In
there are no idiosyncratic shocks and, leaving aside inessential constants:
Pit
where
rrii
denotes the
Aggregate output,
money
yt, is
stock.
of
m^ are continuous and
level, pt,
= mt-Pt,
defined as
=
pudi.
/
there are no frictions in microeconomic price adjustment, p^
and money
is
neutral.
Suppose instead that there
and hence firms adopt Ss
increasing.
proportional to (the log of) real balances:
Pf
If
(1)
The sample paths
yt
with the aggregate price
= ^t
is
=
p*t
=
mj, so that pt
= mt
a fixed cost of adjusting individual prices
rules in setting their prices.
As
usual,
it is
convenient to define a
state variable:
Xit
*ln a
model with stochastic adjustment
=
costs, p*^
conditional on the current state of the economy,
P^t
is
if its
-
P*if
defined formally as the price the agent would choose,
current adjustment cost draw
is
equal to zero.
The adjustment
S — s.
rule
such that when xu reaches
is
s
—
S, the firm increases the price
by
This large adjustment catches up with the accumulated monetary expansion since the
previous adjustment and anticipates some of the expansion that will take place before the
next adjustment (recall that
p*j is
mj plus a constant; furthermore,
equal to
convention used in generalized Ss models, this constant
X
is
is
following the
chosen so that the price imbalance
zero immediately after firms adjust their prices).
In CS, firms' adjustments are not perfectly synchronized because the initial cross-section
distribution of actual prices
distribution of x
is
is
non-degenerate.
uniform over the entire
(s
In particular,
—
S,
0]
CS assume
interval.
It
that the initial
turns out that under
the monotonicity and continuity assumptions for the sample paths of money, this uniform
distribution
is
invariant:
While the position
time, the cross-section distribution remains
of individual firms in state space changes over
unchanged and uniform over
taken from CS, illustrates the variation over time oi r
in the
=
x
+
(s
— 5, 0].
S, iov an agent
absence of idiosyncratic shocks, the distance between agents on the
i\
Figure
1,
note that
circle
remains
unchanged.
Figure
1:
Tlie Caplin
and Spulber Model
S..S
r,(U
r,(U)
^(t,)
2.2
Main Result
The main
result in
CS
is
that in this context a small monetary expansion has no effect on
aggregate output, despite the fact that at any given instant most microeconomic units do
not adjust their prices. To see this result, note that a monetarj' expansion of A?n triggers
the adjustment of Am./{S
The change
—
and each
s) firms,
aggregate price level
in the
of these firms increases its price
by {S —
s).
simply the product of these two terms:
is
Ap=^I^x{S-s) = Am
(2)
and hence
Ay =
The standard
nomic
interpretation of this result
Our
stickiness.
first
point
is
0.
that aggregation erases the impact of microeco-
is
that this interpretation depends crucially on the concept
of price stickiness one has in mind.
concept
If this
defined in terms of the frequency of
is
microeconomic price adjustments, then the statement
is
money
correct, since
is
neutral at
the aggregate level despite the fact that most firms do not adjust their prices at the micro
Yet
level.
if
one has in mind a more standard definition of stickiness,
terms of the impulse
in
response to monetary shocks, then the usual interpretation of Caplin and Spulber's
neutrality result changes, as
now
reflects the
it
A¥e support our claim in two steps.
representative firm
i,
barrier.
Only
if
x
is
absence of microeconomic stickiness.
Consider
first
the price-response Apj(Am,,3:) of a
with state variable x to a small monetary shock of size
X by Am^ leads to no adjustment
close
if
the firm
enough to
—S
s
is
money
at a distance larger
than
Am. A
shift of
Am from the trigger
does the firm adjust, from (approximately)
s
-S
toO.^
Therefore:
iix> s- S + Am,
{0,
S—
and
it
s,
otherwise,
follows that:
if
0,
x
>
s
- S + Am,
Api(Am.,x)
Am
{S
To obtain a measure
tion,
we average
of
— s)/Am,
microeconomic
otherwise.
flexibility in
terms of the impulse response func-
this expression over all possible values of x.
the different values of x
is
The obvious candidate
the average time-distribution (ergodic distribution) of the state
^Strictly speaking, the adju'stment
is
from x —
Am
to 0, but s
— S -
x
+
Am
adjusting firms that s — 5 is a good approximation and simplifies the expressions.
-^
does not depend on this approximation.
Am
to weigh
is
sufficiently small for
Of
course, the limit as
variable x for a given firm. Denoting this density by hsix)
—=
-
Am
have:
hE{x)dx,
^^
Am
^
(3)^
^
'
intuitively obvious (for a proof see Section 3) that a firm's ergodic density in
It is
uniform on
—
(s
5,0], so that h^ix)
The impulse response
and adds up
IRF
/
J,_s
we
to
constant and equal to 1/(5*
5s (and Calvo) type models
is
main measure
summarized by the
of
initial
microeconomic price
—
is
s).
typically decreases monotonically
Thus, much of the persistence of the
to one because of long run neutrality.
monetary shocks
define our
of
is
CS
response to such a shock.
We
therefore
flexibility as:
Am
It
follows that there
is
no price stickiness at the microeconomic
/s-S+Am
__s
In other words,
if
we
_
^^
Am
q
„
level since:
1
X
-i-
follow a firm over time and
S
-s
dx
=
1.
draw the histogram
of its marginal
responses to a monetary shock, Ap,t/Am., most of the observations pile up at zero.
a small fraction of observations pile up at (5
response
is
much
larger than one-for-one.
1
Property
1
T
tends to
(The IRF
in
its first
over time of
Apu/Am,
^ Apu
Am
infinity.
Caplin-Spulber has no microeconomic price-stickiness)
In Caplin and Spulber: J^cs^cro
It is
Yet
corresponding to times where the
The average value
T ^^
t=i
converges to one as
— s)/Am,
^
i
easy to extend the above result to the complete impulse response function, beyond
element. Denote by IRF""^''° the average price response of a firm at time k to a small
monetary shock
Am in period
zero,
normalized by the size of the shock.
jp-pCS, micro
We
then have:
^""^'c™^
Thus, not only
but any reasonable definition of price- flexibility based on the entire
IRF, assigns no microeconomic stickiness
The second
step in our study of the
and aggregate price
see equation (2)
in the
IRF
stickiness. It follows
in
CS
context.
Caplin and Spulber connects microeconomic
from our derivation of Caplin and Spulber's
—that the aggregate response of the price
one-for-one
upon impact and zero
however,
that the actual cross-section distribution has
is
level to
thereafter. This is the well
little
monetary shocks
known CS
to
result
do with
also
is
Our
point,
this result.
Once
result.
the model has no microeconomic stickiness, the macroeconomic result follows regardless of
what the
result that
the Ergodic
Theorem
(see, e.g.,
monetary shock
Am
Walters (1982).
can be defined
To obtain an aggregate measure
all
We
analogous to JT™"^™
is
quite general and follows from
sketch the proof of this result next.
that the aggregate price response
as:
Am
of price flexibility,
possible cross-sections f{x)\ fi{x), f2{x),
flexibility
is
we have
for the state variable,
J
over
turn to this issue next.
micro and macro stickiness are the same
Given a cross-section /(x)
to a
We
like.
Relation between Micro and Macro Stickiness
2.3
The
cross-section distributions look
we need
, fn{x).^
to average the above expression
The measure
of aggregate price
therefore defined as:
n
^--™ = ^u-,.F(A),
(5)
fc=i
where w^ denotes the weight
of the k-th cross section, with iVk
Denoting the weighted average of
defined in (4)
is
linear,
we
all
cross-sections
by
/.4(:r),
>
and
'}22=i'^^k
course,
no averaging
have:
is
needed
for the
the same in this case (anduniform on
''The actual
number
of cross-sections
formal statement and proof.
1-
and noting that the operator
^— = HIa) = J ^^^^^^Ux)dx.
Of
=
We
is
assume a
(s
—
5",
0]).
since
More
all
cross sections fk{x) in (5) are
generally, however, such
and not countable, thus measure theory
number for illustrative purposes.
infinite
finite
CS model,
(6)
is
an average
required for a
under rather weak conditions and, by the Ergodic Theorem,
exists
firm's ergodic density, hsix), considered in (3). Since /yi(a;)
=
is
equal to the individual
hsix), comparing (3) and (6)
yields the following property:
Property 2 (Macro and micro price
Macro- and microeconomic
same
in
flexibility are
always the same)
price flexibility, as measured by
-rmicro
Furthermore, a straightforward extension of the derivation of
and micro
over
Our
all
im,pulse response to a
is
all cross sections,
shows that the macro
this result
monetary shock are the same
obtained by averaging over
(Where
at all lags.
the
CS model now
and the micro response by averaging
follows from Properties
Property 3 (The source of aggregate price
When
price stickiness
is
nomic
level.
2.4
The Role
flexibility in
1
and
2:
Caplin and Spulber)
defined in terms of the impulse response function, the source of
aggregate price flexibility in Caplin and Spulber
is
the absence of stickiness at the microeco-
of the Cross-Section
apparent from the previous property that the uniform cross-section distribution
does not vary over time,
is
The answer
the role played by this assumption?
the
same
at all
CS ensured
moments
in time.
is
in
CS
So what
has nothing to do with the absence of aggregate price stickiness in their model.
is
macro
individual time-series.)
interpretation claim for the
It is
are the
^
any (stationary) macroeconomic model:
'T-macTO
response
and J^^^^°
JP"^"^''°
that by choosing a distribution that
that the response of the
economy
For most Ss-type models this
to aggregate shocks
not the case, since
is
the cross-section varies endogenously over time, and so does any reasonable measure of price
flexibility (for
example, the one defined
generahzation of
CS
Consider the
CS
now
(see Caballero
in (4)).
To
illustrate this point,
and Engel (1991)
for
more
we develop a simple
details).
setting, except for the initial distribution of firms' state variable,
covers only half the inaction range: the
initial distribution is
Ss bands are normalized
uniform on [—3/2, —1/2].
8
to
5—
s
=
2
which
and the
Figure
p and
2:
- '
'
p with
m
strat.
in
an extension of Caplin-Spulber
compl.
4-
20
40
30
50
60
time
The
solid line in Figure 2 depicts the evolution of the price level
assumption that
no firm adjusts
m
its
grows linearly over time at the rate
price
and the aggregate price
reach the trigger level s
— S and
the last firm adjusts, a
new period without
We
first
Initially,
there
is
does not change.
a period where
Eventually, firms
the aggregate price level rises twice as fast as money. After
price adjustment begins,
and so
on.
note that the ergodic density for a single firm continues to be uniform on the
entire inaction range, (—2,0].
one as
in the
^pnacro
^
2
level
/i.
under the additional
Thus our measure
of micro price-flexibihty,
J^™^'^'^°^ is
equal to
standard version of Caplin and Spulber, which from Property 2 implies that
as WCll.
Since in this case / varies over time, so does the conditional price-flexibility measure
defined in (4). In this example
it
only takes two values, each one of them half the time:
when no
firm
is
adjusting (/(— 2"*")
=
0),
Hf)
2
Thus, what
is
special about
no aggregate price
when some
CS
is
=
within the class of one-sided Ss models
stickiness. It follows
Engel (1993a) that this
firms are adjusting (/(— 2+)
1).
is
from the Average Neutrality Result
not that there
in
is
Caballero and
the case for a broad family of models of this type. Instead, what
special about CS's cross-section distribution
is
of the aggregate price level to
is
that
it is
invariant and hence the response
monetary shocks does not vary over time.
Strategic Complementarities
2.5
Let us conclude this section with an extension that incorporates strategic complementarities.
Equation
(1)
becomes:
*
=
Kt
where the parameter a G
[0,
(1
-
«)"^i
+
«Pi>
1/2) captures the extent to which firms wish to coordinate their
prices.
Firms have stronger incentives to keep their prices
when
is
a
(7)
in line
with those of other firms
larger.
The aggregate change
in prices
— S)At
ments. Equation
(8)
is
the fraction of firms that adjust and
extends
Substituting (7) in
(8)
is
given by:
= Ap*f{s-S)Atx{S-s),
'Ap
where Ap*f{s
during a small time unit At
(2) to this
and solving
more general
for
Ap
(8)
S—
s
the size of their adjust-
setting.
leads to:
when no
firm
is
=
adjusting (/( — 2"'")
0),
Ap =
(2
—
2a)/jAt/(l
—
2a)
when some
firm are adjusting (/( — 2"*")
=
1).
Hence:
when no
•^(/)
=
firm
is
adjusting (/(— 2"^)
=
0),
(9)
{
(2
The dash-dotted
—
2a)/(l
line in
Compared with the
—
2a)
when some
firms are adjusting (/( —
case without strategic complementarities (a
aggregate price level increases,
it
=
The
0, solid line),
flip
side
is
The
0.4.
the aggregate
that
when the
longer periods of inaction and the shorter
but brisker periods with price adjustments cancel each other out so that the
is
=
does so at a faster rate, since a larger fraction of firms
adjust their price in any given time period.
—which
1).
Figure 2 depicts the evolution of the price level when a
price level remains constant during longer periods of time.
J^
=
2"*")
equal to the average of
J-'{f)
—continues to be equal to one.
10
flexibility
index
From Caplin-Spulber
3
What
we expect between the
relation should
flexibility in
to Calvo
price adjustment frequency
menu-cost models? The one-sided Ss models described
and aggregate price
earlier,
with complete
price flexibility at the aggregate level regardless of the adjustment frequency, suggest that
the answer
is
At the other extreme
none.
coincide and aggregate flexibility
however, the answer
CS
In the
equal to the adjustment frequency.
equal to
is
ps-s+^i
1
S-s
s-S
as before, denotes the
fj..,
range, and
money growth
we have assumed that the
Since the price flexibility index
threshold
now modify
s — S, there
time, regardless of
its
rate (assumed constant), {s
(i.e., fi
J^'^^ is
< S—
the inaction
such that there are always firms
s).
we have
one,
is
— S, 0]
that:
the one-sided Ss model and assume that in addition to the trigger
a strictly positive hazard A that a firm adjusts at any point in
is
price imbalance x. Thus,
one-sided Ss models: As
we
S
choice of time unit
that do not adjust within a given period
Let us
generally,
model, the fraction of firms that adjust in one time period, henceforth the
frequency of adjustment index,
where
More
between these two extremes.
in
is
is
the Calvo (1983) model, where both concepts
is
s
we have a model that
nests both Calvo and
tends to minus infinity we obtain the Calvo model, while
if
A
=
are back to CS.
If
f{x,t) denotes the cross-section density at time
f{x,t
+
At)
= {I-
XAt)f{x
+
fiAt,t),
t,
then
s- S <x<0.
This follows from the fact that a firm with a price imbalance x at time
a price imbalance x
at time
t
+ /j,At
at time
t,
Property 2 we know that the time-average of
of.
an individual price
the concept we need to compute
Setting f{-,t
+
At)
=
f{-,t)
A
=
+ At
must have had
and that a fraction XAt of firms with imbalance x + fiAt
adjusts before reaching x because of a Calvo-type shock.
the ergodic distribution
t
all
From
the derivation of
possible cross-sections, /^(a;),
setter.
is
equal to
Let us calculate this average, as
it is
and T.
hE{-)
=
/a(-) in the expression above, using a first-order
11
Taylor expansion and letting
At —
leads to:
>
=
f'^{x)
with
a=
afA{x),
Imposing that the integral of Ja over the inaction range
X/12.
,/-4(^)
=
,.(5-s)_i
one, then yields:
is
s-S<x<0.
'
(10)
Choosing the unit with which we measure time small enough so that the probability of
two Poisson-shocks
for the
same firm
in a given
fraction of firms that adjust in one time period
Fraction of adjusters
where Fa denotes the
c.d.f. for /a-
The
=
A
first
time-period
is
is:
+
(1
—
X)Fa{s
—S+
(11)
fi)
term on the right hand side
The second term
firms that adjust because of a Poisson shock.
we have that the
negligible,
considers,
the fraction of
is
among
those that
did not receive such a shock, the fraction that adjusted because their state variable reached
the trigger s
—
S. It follows that:
Fraction of adjusters
and from
(10)
we have
=
A
+
(1
—
X)fA{s
—
5)/i,
(12)
that:
Hence, since the fraction of agents that adjust before reaching the CS-trigger s
with
A, the
frequency of adjustment index A"^ increases monotonically with
For technical reasons,
flexibility
index
it
T once there
is
is
useful to turn to discrete time
a Poisson term.
We
when
— S grows
A.
calculating the price-
then have (as the impulse
Am
tends to
zero):
^-A
+ (l-A)/^(s-5)(5'-s).
Am,
The
first
term on the right hand side
is
the marginal price increase due to the monetary
shock, for those firms with a Poisson-induced adjustment.
firms to the
monetary shock
is
(14).
one-for-one.
The marginal response
The second term
is
the additional contribution
to inflation from firms that adjust because they reach the trigger barrier s
with the Poisson-induced adjusters, the response of these firms
12
of these
is
—
S. In contrast
{S — s)/Ain
times the
shock that triggers their adjustment, and therefore
A
<
J?^'^
<
1
for
< A <
More
1.
a{S
+ (l-A) Ms-s) _
A
A~
77?,
case)
T^
and again one
does not vary monotonically with
for
=
A
1
(no micro frictions).
increasing for larger values (see Figure
Figure
3:
T and A for
A
\
I
\
\
1
0.7
\
\
\
0.6
It
is
one
for
=
A
(the
CS
decreasing for small values of A and
It is
a Caplin-Spulber model with Poisson Shocks
\
0.8
A.
:i5)
\
3).
1
0.9
larger than one-to-one, so that
precisely, substituting (10) in (14) yields
T'
Also note that
much
/
\
0.5
^
0.4
0.3
0.2
Adjustment frequency
- - 0,1
t*^
1
1
0.1
1
1
0.4
0.5
1
0.2
0,3
Flexibility
1
1
0.6
0.7
Index
1
0.8
0.9
1
\
This simple example illustrates the complex connection between aggregate
the price adjustment frequency.
Whether an
reduces price stickiness depends on
how
and
increase in the frequency of price adjustments
the increase in frecjuency affects the contribution of
the intensive and extensive margins to the flexibility index
The
flexibility
intensive margin refers to firms for which the
J-^.
monetary shock only
affects firms
behavior by increasing the price adjustment of those that would have adjusted anyway, with
or without the monetary shock.
These are the firms
their adjustment, their contribution to JF
is
the
first
The contribution
to J^ of the intensive
T
is
which the Poisson shock triggers
The
extensive
triggered by the marginal
monetary
term on the
margin, by contrast, refers to firms whose adjustment
shock (the impulse), their contribution to
for
is
of (15).
the second term on the
margin increases with
13
r.h.s.
r.h.s. of (15).
A, this is obvious.
What
is
obvious
less
that the contribution of the extensive margin decreases, since the fraction of
is
firms in the neighborhood of the trigger barrier s
at
S, f{s
—
component decreases
small values of A and small for larger values of
which the fraction of firms
becomes smaller and ^mailer
in the
is
constant (and equal to
large (in absolute value) for
is
The reason
A.
While the rate
S), decreases.
which the intensive margin component of J^ increases with A
one), the rate at which the extensive
at
—
the latter
for
neighborhood of the adjustment trigger
that the rate
is
s
—S
decreases
as A grows.
Taking stock, once we incorporate Poisson shocks into the CS environment, we have
that, as expected, the flexibility
more
direction for large values of A:
more responsive
A
for
a given
The reason
(s
is
and frequency of price adjustment indices move
to shocks. Yet, as
— S, 0]
firms adjusting
shown
in
Figure
interval, for small values of
3,
means that the aggregate
which plots
J^'^
and
A both indices move
in
in the
price level
opposite directions.
that for small values of A the weakening of the extensive margin
why
J^^ decreases as
mechanism
The frequency
models, and this lower bound
and
(15),
a =
X/n).
is
of adjustment index
follows that, for A
The Calvo model
<
is
1
and
a lower
bound
To
fj,
for flexibility in
compare
(13)
s) (also recall
that
see this,
< (S —
Ss
s finite,
A^.
obtained by taking the limit as
s
goes to minus
infinity, in
which case
(for all A):
S
4
a more general context in the
achieved by the Calvo model.
T^ >
we have
is
in
and recah that our choice of time period ensures that
It
flexibility,
A increases.
This example also helps motivate a point we develop
following section:
is
A"^ as a function of
dominates over the standard positive relation between adjustment frequency and
explaining
same
lim J^^
— CX)
=
»
The Extensive Margin
s
—lim
A^ =
A.
>— oo
Effect
Let us generalize the model of the previous sections to consider a broader set of shocks and
adjustment
rules.
These extensions are more
to the growth rate of
money
idiosyncratic (productivity
ance aj.
No assumptions
idiosyncratic) shocks.
are
i.i.d.
with
easily
mean
/x
implemented
made
regarding the
Shocks
and variance a\, and firms experience
and demand) shocks v^ which are
are
in discrete time.
common
i.i.d.
with zero
mean and
vari-
distribution across aggregate (and
These shocks are independent across agents and from the aggregate
14
shock.
With
these assumptions, the target price foUows the process:
Ap*^
With no
croeconomic
We
Arut
+ vu-
further changes, and preserving the fixed cost of adjusting prices at the milevel, this
generalize
costs as well,
=
further
it
environment yields a two-sided Ss pohcy
and assume that there are
drawn from a
these adjustment costs,
distribution G(cj).
i.i.d.
Barro (1972)).
idiosyncratic shocks to adjustment
Integrating over
we obtain an adjustment
(see, e.g.,
all
possible realizations of
hazard, A{x), defined as the probability of
adjusting prior to knowing the current adjustment cost draw, with^
<
It
<
A{x)
1,
Vx.
follows that for non-degenerate distributions G(w), A{x)
increasing for x
>
0:
the cost of deviating from the target price
the distance from this price and therefore adjustment
is
is
is
more
decreasing for x
is
<
and
increasing with respect to
likely
when
jx| is
larger.
This
the increasing hazard property (Caballero and Engel, 1993c).
Denoting by f{x,t) the cross section of price imbalances immediately before adjustments
take place at time
t,
we
have:
Apt
4.1
A
= —
/
xA{x)f{x,t)dx.
Basic Inequality
Let ApQ{Am.'^) denote the average (over
response to a monetary deviation of
Am^
all
possible cross-section distributions) inflation
from
its
average growth rate.
the impulse response function with respect to this shock
jr
=
Ap;(A7n'^
=
is
our
The
first
element of
flexibility index:
0).
'See Caballero and Engel (1999) for a detailed discussion of such a model, Dotsey, King and
for
an estimation of
Wolman
dynamic general equilibrium context, and Caballero and Engel (1993b)
a generalized hazard model for prices.
(1999) for an application to prices in a
15
We
have:
= -
Apo(Am'^)
= - [{x-
first
-
(16)
Am'')fA{x)dx.
Am'^ and evaluating
A{x)fA{x)dx+
at
Am'^
=
yields:
xk'{x)fA{x)dx
I
Note that the
Am'^)dx
I\n/)k{x
Differentiating this expression with respect to
J^=
+
/ xA(x)/.4(a:
(17)
I
term on the right hand side corresponds to the adjustment frequency,
A. The second term, which we denote by £, corresponds the contribution of the extensive
margin (more on
this shortly).
Thus we can
write:
7 = A + £.
It
(18)
follows that:
Property 4 (Flexibility and adjustment frequency are the same
In the Calvo model, where K'{x)
More importantly,
it
=
0,
we have
In increasing hazard models we have
for
X
>
The
0.
J'>A.^
It
S =
and
Q
J^*^^^™
=
Calvo model)
^Caivo_
also follows that:
Property 5 (The adjustment frequency
Proof
that
in the
£ >
is
0,
a lower
bound
follows that .tA'(x)
>
for all x,
T>
and therefore
increasing hazard property states that A'(x)
for flexibility)
>
A.
for
and therefore £
x
<
and
A'(a:)
= J xA'{x)f{x)dx >
<
and
I
The above
results
the lines of what
can be extended to the case with strategic complementarities along
we did
in Section 2.5 (see
complementarity parameter a
in (7)
is
Caballero and Engel, 2006).
positive, (18)
When
the strategic
becomes:
T^hiA + £),
^This result also holds
which A(x)
satisfies this
set with positive
if
we work with the weaker concept
property
if
A'{x)
<
for
x
<
measure (under the model's ergodic density).
16
of "increasing hazard property", according to
and A'(x) >
for
x >
0,
with
strict inequality
on a
where h{x)
is
As
a strictly increasing function.^
before, a positive extensive
margin £
ac-
counts for a larger average response of the price level to monetary shocks than suggested by
the intensive margin A.
Extensions
4.2
In this subsection
of standard
Ss
we
discuss two extensions of the above results. First,
policies,
we
consider the case
where the smoothness assumptions under which we derived
(17)
do not hold. Second, we show that the above inequality extends to the case of conditional
impulse responses.
Standard Ss Policies
4.2.1
The above
derivation assumes that A(x) and
are continuous, which does not hold for
/.4(a:)
standard Ss models, such as Barro (1972), Caphn and Spulber (1987), and other models
where the distribution
when xA{x)f{x) has
of adjustment costs
is
mass
a
jump-discontinuities at Xi,X2,X2,
point. -"^
....
As shown
in the
Appendix,
expression (17) becomes:
J^= lA{x)Ux)dx + JxA'{x)Ux)dx + Y^Xk[A{xt)f{xt)-Mx;)f{x^)],
where
f{x'l)
and f{x^) denote the
values than Xk) and
left,
limit of f{x)
when x approaches
(19)
x^ from the right (larger
respectively.
In particular, for a standard two-sided
policy,
we have
that the
+
- Fa{U)),
price imbalance belongs to [L,
[/],
hand
Fa{L)
side of (19) are equal to
Ss
(1
where
first,
firms'
remain inactive when their
second and third terms on the right
and
|L|/.4(I)
+
UfA{U),
respectively,
so that:^^
^=
where Fa denotes the
It
Fa{L)
c.d.f.
+
(1
- Fa{U)) +
|I|/.4(L)
+ UfAiU),
(20)
corresponding to /a-
follows that for a standard two-sided
A=
Fa{L)
^5 model we have:
+ {1-Fa{U)),
£ = \L\fA{L) + UfA{U).
^h{x)
=
(1
-a)xl[l -ax).
^°See Sheshinski and Weiss (1993) for a comprehensive volume on Ss models of pricing decisions.
^^Bar-Ilan and Blinder (1992) derive a similar expression in the context of one-sided
17
Ss
rules.
Conditional Impulse Responses
4.2.2
As we showed
time variation in the cross-section distribution leads to price-
in Section 2,
Ss models, the
flexibihty indices that vary over time. In generaHzed
of price imbalances
cross section distribution
influenced by the sequence of monetary shocks hitting the economy.
is
IRF
quantify the extent to which the
fluctuates over time,
To
we note that the expressions we
derived above also apply to time- varying measures of price-adjustments and price-flexibility.
Denote by f{x,t) the cross-section of price imbalances at time
firms adjusting their price in period
response function.
We
t
and by
the
!Ft
by At the fraction of
t,
element of period
first
t's
impulse
then have that:
.F,
=
At
+
Et
with
£t=
That
is,
fluctuations in the
IRF
xA'{x)f{x,t)dx.
of generalized
Ss models are driven by changes
fraction of agents that adjust in any given time interval
and
in the extensive margin.
Extensive Margin or Selection Effect?
5
As mentioned
That
is,
earlier,
the term
£
quantifies the importance of the extensive margin effect.
of the additional price increase resulting
adjusting upwards and the
fall
which describes the additional price increase
larger adjustment (after the
rise in
monetary impulse)
It
the fraction of agents
downwards, both as a
contrasts with the intensive margin A,
reduced price decrease) resulting from the
(or
In the Calvo model, only the intensive margin
both margins are
from the
in the fraction of agents adjusting
consequence of the additional monetary impulse.
of those that
is
were going to adjust anyway.
active, while in increasing
hazard models
strictly positive.
This extensive margin
trigger thresholds in
is
a generalization of the
eflPect
of the
number
Ss models discussed by Bar-Ilan and Blinder
Golosov and Lucas (2006) have referred to the additional
of agents reaching the
(1992).^^
flexibility in
More
shock are those that benefit the most from doing
so.
recently,
Ss models
selection effect, arguing that the agents that adjust their prices in response to a
is
in the
As we show below, the
as the
monetary
selection effect
not always an accurate description of the source of the additional flexibility present
in
Ss-
^"For recent papers highlighting the role of the extensive margin see, for example, Klenow and Kryvtsov
(200.5)
and Burstein and Hellwig (2006).
18
type models. For this reason, we instead advocate using the extensive margin index defined
above to characterize
tlie
mechanisms
In order to understand the
X and decompose
its
monetary shocks
additional responsiveness to
we consider a
at work,
in S's-type models.
firm with price imbalance
marginal response to a monetary shock, T{x), into the
sum
of the
contributions of the intensive and exteiisive margins:
where
r]{x)
=
increasing hazard model
a selection
effect
£{x)
=
xA'{x)
J-'{x)
=
is
characterized by
A{x)
+
>
r7(x)
0, yet,
as
we show
£{x) increases monotonically with
normal
to the inverted
A{x)
>
A{x)7]{x),
\x\
next, this does
(which
is
what
means).
Assume A{x) belongs
with A2
=
Example
First
5.1
A(x),
x.
always imply that
n.ot
=
xA'{x)/A{x) denotes the elasticity of the adjustment hazard with respect to
the price imbalance
An
A{x)
=
family:-^"^
-
I
e-^'~^\
(21)
0.
Figure 4 plots the intensive, extensive and total response as a function of the price
imbalance
As
x.
Some simple
the case with
<
1,
that
is,
It is
increasing hazard models, A{x)
all
Xo
>
£{x)
0,
values of x with A(x)
however, the extensive margin
increases.
all
algebra shows that, for
such that Xox^
\x\,
is
<
effect decreases as
1
is
—
is
increasing in
1/e
=
increasing in
\x\
\x\.
for values of
x
For larger values of
0.632.
the magnitude of the price imbalance
therefore not surprising that in the range of price imbalances with sufficiently
high probabihty of adjusting (A(x)
>
1
—
e""^/^
=
monotonically with
0.78), J-'{x) decreases
|x|.i4
For this family of adjustment hazards, the largest marginal price response to a monetary
shock comes from firms with an ex-ante probability of adjusting slightly below 80%. Firms
with higher adjustment probabilities benefit more from changing their price but contribute
less to the
economy's marginal response to a monetary shock.
^'See Caballero and Engel (1999) for an application of
i^Both statements above follow from A{x)
8'{x)
=
4A2x(l -
A2a;2)e-^2^' and
r{x) =
=
1
A2x(6 -
-
tliis
e''^^^',
follows that the "selection
family to investment.
£{x)
A\2X^)e-^^-''\
19
It
=
2\2X^e-^^''''
,
A!{x)
=
2A22;e"
Figure
4:
Generalized Ss Model where the Selection Effect does not hold
1.5
'
Extensive margin
Intensive margin
Flexibility
-0.25 -0.2 -0.15 -OJ -0.05
index F(x)
0,05
0.15
0.1
0.25
0.2
price imbalance x
effect"
does not provide the correct intuition underlying the larger price response to monetary
shocks in this example.
5.2
General Insight
The
insight of the preceding
attains its
minimum
example
value at x
=
valid
is
Furthermore, often this
0.
A(x) admits a quadratic approximation
;^A"(0)a;^ in a
xA'{x)
77(0)
We
=
=
x^o A{x)
approach
zero. It follows that, as
to values close to
directions as
\x\
when
|x| is
|x|
present, the selection effect
x
=
0,
is,
=
0,
|x|
= 2A
so that
(22)
increases, r]{x)
grows, so that A'{x), and therefore
must decrease from a value
two
is
increasing in
\x\.
would always hold. Yet there
we have £
zero and
i-5 iA"(0)a:2
large enough.-'^ Thus,
A{x)
is
A"(0)a:-
^^Furthermore, in Caballero and Engel (2006) we show that
of
neighborhood of x
value
effects pull
T{x)
increases. First, firms tolerate less well larger price imbalance
likely to adjust their price, that
neighborhood
minimum
lim
also have that, in general, A(x) approaches one as
ri{x),
X
more broadly. In generalized Ss models, A(x)
and therefore
JF
20
=
if 77(0;)
3A.
is
If this
a second
in opposite
and are more
were the only
effect:
remains close to 2
in
of 2 at
as
|x|
effect
grows
a sufficiently large
large enough, the elasticity
'i]{x)
eventually
is
decreasing in
|.t|.
This leads to a diminishing
contribution of the extensive margin, A{x)'q{x), and points in the opposite direction.
selection effect provides the correct intuition only
values of x.
However the
latter effect
5s-type models, since £{x)
is
is
likely to
when
The
the former effect dominates for
dominate
for large
enough
in
\x\
all
many
decreasing in this range of values while A{x) approaches a
constant value of one.
Property 6 (The Selection Effect and the Extensive Adargin Effect)
In generalized Ss models, the firms that contribute the most to the aggregate price level
due
to the fact that
size of the price
imbalance when
are not necessarily those that benefit the m.ost from> adjusting.
the coTitribution of the extensive
the latter
is
margin decreases with the
Ss models
also holds for standard
two-sided Ss policy and the notation introduced
when
1
^{Am,x) =
{
Am
+
U
+ x/Am,
L,
Am<x<U,
< X <U + Am,
(23)
x>U + Am.
Firms with imbalances slightly above the trigger L and
ones that contribute most to the aggregate price
former, because the shock leads
in the
then have:
L+
1,
would have remained inactive
We
Consider a
L<x<L + Am,
|j|/Am,
0,
1
in discrete time.
deriving (20).
<
X
1,
The
is
sufficiently large.
The above property
shock.
This
them
slightly
level's
below the trigger
U
are the
marginal response to a monetary
to adjust their price
absence of the monetary shock.
upward when they
The
latter,
because
the additional monetary expansion modifies their optimal choice from adjusting their price
downward
X
» U)
to not adjusting at
benefit
In contrast, firms with larger imbalances {x
all.
more from adjusting but contribute
5.3
Second Example
In our
first
example the selection
example where the
selection effect
effect
is
less to
£ =
L
or
the aggregate price response.
does not hold and yet £
present and
«
>
0.
Next we provide an
0.
Aggregate shocks are drawn from a density with zero mean and bounded support, with a
very small variance. Idiosyncratic shocks are drawn from the trimodal distribution depicted
21
Figure
5:
Density of idiosyncratic shocks
1
0.8
S
0,6
0.4
0.2
A-
A
-0.25-0.2-0.15-0,1-0 05
price
in
05
imbalance
01
15
02
25
x
Figure 5}^ With a high probability (0.8 in the figure), the idiosyncratic shock
with a low probability
bounded support that
(0.2 in the figure)
is
far
away from
it
is
which have the same magnitude but opposite
and
either large
The
positive, large
is
a fixed fraction of firms profits, leading to a two sided
The
shown
as in Barro (1972).
Figure
which also shows the corresponding Ss bands (dashed
this figure, there are
case.
The
means,
zero.
Ss policy
6,
differ in their
signs. It follows that idiosyncratic shocks are
and negative, or
cost of adjusting prices
zero while
the realization of one of two densities with
These two densities only
zero.
is
resulting ergodic density
is
no firms at the trigger barriers, so that £
line).
=
in the solid
As can be seen
and J^
= A
all
—have their price imbalance
in
in this
—
in this
in the region
where
leptokurtic shape of idiosyncratic shocks implies that very few firms
simple and extreme example no firm at
hne in
marginal monetary shocks affect their decision to adjust. Firms either have large (absolute)
price imbalances
and adjust independent
of the additional
monetary shock,
or have small
deviations and will not adjust, with or without the marginal shock. Yet the selection effect
is
present since those firms that adjust are, indeed, those with largest imbalances and therefore
those
who most
level is the
same
benefit from doing so, even
though
their marginal contribution to the price
as that of adjusting firms in a Calvo model.
^^This results in a leptokurtic distribution for idiosyncratic shocks, as
Leahy (2006).
22
in
Midrigan (2006) and Gertler and
Figure
6:
Ergodic density and Ss bands
-0-25-0.2-0.15-0.1-0.05
price
015
0.1
02
025
Summary
5.4
In
05
imbalance x
summary, what matters
Ss) models
is
for the additional flexibiUty of increasing
the increase in the fraction of agents adjusting in the direction of the impulse
(and the decrease of agents adjusting
Whether
in the opposite direction),
is
caused by the impulse.
come from agents that
these changes in the decision to adjust
the target
hazard (and standard
are near the tails or
a secondary factor.
Decomposing the
flexibility
and extensive margins,
A
and
index J^ into the
sum
S, provides a simple
responsiveness present in 5s-type models.
The
of the contribution of the intensive
way
larger
is
of quantifying the additional price
£, the larger
is
the contribution to
the aggregate price level of time- variations in the fraction of firms adjusting.
We
have provided an example where there
example where the selection
effect holds, yet
is
no selection
£ =
0.
We
effect
and £ >
0,
and another
conclude that selection
is
neither
necessary nor sufficient to obtain an aggregate price level that, for a given frequency of price
adjustments, responds more to monetary shocks than the corresponding Calvo model.
6
The
Final
Remarks
recent work of Golosov and Lucas (2006), Burstein (2006),
Klenow and Krystow
Midrigan (2006), Gertler and Leahy (2006), and others has rekindled the
cost type models.
As with many microeconomic
variables, the prices of
interest in
menu
goods and services are
seldom adjusted continuously. However, the implications of microeconomic inaction
23
(2005),
for the
stickiness of the aggregate price level
to important but specific examples.
is
only gradually being understood, usually restricted
In this article
we have
tried to take a step further in
describing the conceptual issues involved in the connection between the frequency of price
adjustments and aggregate price
The contributions
flexibility in generalized
Ss models.
of this paper are pedagogical in nature.
In particular,
we
clarified
the source of the Caplin and Spulber (1987) neutrality result, which does not rely on the
aggregation process bwt on the particulars of microeconomic adjustment in that model.
also
showed that the reason
from an extensive margin
whifch
is
for the additional flexibility of
This
effect.
effect is distinct
We
Ss over Calvo type models comes
from the so-called "selection
effect,"
neither necessary nor sufficient for the additional flexibility.
The models we analyzed were intended
ment, not to
offer
to isolate a particular feature of discrete adjust-
an account of observed price
For that, the mechanisms we
stickiness.
discussed need to be interacted with a variety of other sources of stickiness (Ball and Romer,
On
1990).
this account, the recent
Nakamura and
work
by, e.g.,
Carvalho (2007), Gorodichenko (2006),
Steinsson (2006b), Dotsey and King (2005) and Kleshchelski and Vincent
(2007) seem particularly promising.
Moreover,
in the
main
text
we focused on increasing hazard models, but the
results
we
derived also provide a hint on what kind of models are likely to generate substantial price
rigidity:
These are models with a negative extensive margin
Kehoe and Midrigan
effect.
(2007) provide one such example, where the option to implement temporary sales effectively
generates a decreasing hazard model for small deviations
x.
This feature
is
quantitatively
important since the cross section distribution accumulates a large number of agents
values of x, thus
it is
perfectly possible to generate an
our perspective, however,
is
that one can
still
f <
for small
The important message from
0.
understand the workings of their model with
our framework.
Finally,
it
should be apparent that there
are properties about contexts where
prices
is
just one application.
is
nothing specific to prices
lumpy microeconomic adjustment
At the microeconomic
level,
paper.
in
nature and hence
results.
These
prevalent, of which
many
other important
the essentials of the model in this
^^
^'For a recent application to investment see,
is
fit
is
our
investment decisions, durable
purchases, hiring and firing decisions, inventory accumulation, and
economic variables are lumpy
in
significantly less
e.g.,
Bachmann
et al 2006.
than one and time varying, with a substantial
24
rise
There we argue that
during the late 1990s.
in
the
US
T
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About
MIT
Optimum
Pricing, Inflation,
and the Cost of Price
Press.
Brownian Models
An
University.
in Economics. Manuscript.
Introduction to Ergodic Theory, Springer- Verlag.
26
Appendix: Formal Proofs
Proposition
adjusters.
at
Al
(The Case with Discontinuities) Denote
T
f{x) the cross-section of price imbalances,
ard,
Assume
that
xA{x)f{x)
discontinuous at x^
is
by A{x) the adjustment haz-
the flexibility index
<
<
X2
and
<
A
the fraction of
Xn, with a
jump D^
Xfc.-
D,^x,{A{xt)f{xt)-A{x^)f{x^)],
where f{x'^) and f{x'^) denote the limit of f{x) when x approaches Xk from the right (larger
values than Xk) and left, respectively. Also assume lim,^^±ooxA{x)f{x) = 0. We then have:
A{x)f{x)dx,
00
"
J^
=A+
/•oo
^Dk+
xA'{x)f{x)dx.
/
fc=i
Proof
Ai
=
The expression
(xi_i,Xj),
change
i
=
for
A
l,2...,n+
in the price level
is
1;
To
straightforward.
= —00
with xq
and
A
{x
=
Xr,+i
00.
bj'
J^,
denote
Apo(ii) the
rXi+v
-^^
- v)A{x -v)f{x)dx =
{x
- v)A{x -v)f{x)dx.
basic result from calculus (Leibniz's formula) states that
differentiable
Denoting
due to a monetary shock v we have:
"+!
/
derive the expression for
if
a{v), b{v)
(24)
and g{x,v) are
and
rb{v)
G{v)
=
g{x,v)dx,
/
Ja(v)
i{v)
then
G\v) =
g{b{v),v)b'{v)
-
g{a{v),v)a'{v)
+ f
"
Ja{v)
Applying
this result to calculate the
differentiating w.r.t. v
and evaluating
n +
at v
=
1
integrals in the
71+1
the
the
r.h.s.
of (24),
„xi
^=-5]{x,A(x-)/(x-)-x,_iA(x.+_:)/(.T+_i)}+E /
we assumed
sum of the jumps
sum on
0:
n+l
Since
^ix,v)dx.
C^
[Mx)
+
xA'{x)]fix)dx.
\u-nx^±oo xA{x) f (x) = 0, the first sum on the r.h.s. of (25)
of xA{x)f{x). The expression for J^ now follows.
27
is
(25)
ec^ual to
Corollary Al If the assumptions of Proposition
and positive when x^ > 0, as
negative when Xk <
1
hold and K{x^)f{x^)
the case for
is
—
A{Xf. )f{Xf.
)
is
most Ss-type models, then
T>A.
Corollary A 2 (Caplin and Spulber) To obtain a discrete-time approximation to the Caplin
and Spulber model, we consider f{x) uniform on {s — S — ji, —jj], since the relevant crosssection
place.
is
the one after the aggregate shock (of size n), immediately before adjustments take
We also
assume A(x)
=
if
x e
then implies that:
{s
-
S,0) and A{x)
=
^cs^l^^
o
1
otherwise. Proposition
A
1
(26)
s
uniform distribution on the inaction range is only an approximation to the
ergodic density in discrete time. This explains why we obtain even more flexibility than in
Of
course,
a
the continuous time case.
If
we adapt
(15) to discrete time,
we have:
^
and we obtain (26)
letting
A -^
'
0.
28
3521
7
ga(S-s)
_
'
I
Date Due
Lib-26-67