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Kieler Arbeitspapiere • Kiel Working Papers
1312
Real Wage Rigidities and the Cost of
Disinflations:
A Comment on Blanchard and Galí
by Guido Ascari and Christian Merkl
February 2007
Institut für Weltwirtschaft an der Universität Kiel
Kiel Institute for the World Economy
Kiel Institute for the World Economy
Duesternbrooker Weg 120
24105 Kiel (Germany)
Kiel Working Paper No. 1312
Real Wage Rigidities and the Cost of
Disinflations: A Comment on Blanchard
and Galí
by
Guido Ascari and Christian Merkl
February 2007
The responsibility for the contents of the working papers rests with the
authors, not the Institute. Since working papers are of a preliminary
nature, it may be useful to contact the authors of a particular working
paper about results or caveats before referring to, or quoting, a paper. Any
comments on working papers should be sent directly to the authors.
Real Wage Rigidities and the Cost of Disin‡ations:
A Comment on Blanchard and Galí
Guido Ascari
University of Pavia
Christian Merkly
IfW and CAU Kiel
February 1, 2007
Abstract
This paper analyzes the cost of disin‡ation under real wage rigidities in a micro-
founded New Keynesian model. Unlike Blanchard and Galí (2007) who carried out a
similar analysis in a linearized framework, we take non-linearities into account. We
show that the results change dramatically, both qualitatively and quantitatively, for
the steady states and for the dynamic adjustment paths. In particular, a disin‡ation
implies a prolonged slump without any need for real wage rigidities.
JEL classi…cation: E31, E52.
Keywords: Disin‡ation, Sticky Prices, Real Rigidities
Address: Department of Economics and Quantitative Methods, University of Pavia, Via San Felice
5, 27100 PAVIA, Italy. Tel: +39 0382 986211; e-mail: gascari@eco.unipv.it
yAddress: Kiel Institute for the World Economy (IfW), Düsternbrooker Weg 120, 24105 KIEL, Ger-
many. Tel: +49 431 880 260; e-mail: christian.merkl@ifw-kiel.de
1 Introduction
In a very insightful paper Blanchard and Galí (2007) (BG henceforth) advocate the
introduction of real wage rigidities in the standard new Keynesian (NK) model. They
show that real wage rigidities would generate both more realistic policy trade-o¤s, by
breaking what BG called the divine coincidence, and a more realistic empirical behavior
of in‡ation, by generating in‡ation inertia.
In order to show an example of these two previous features brought about by the
introduction of real wage rigidities, in Section 4, BG look at the cost of a classical
monetary policy experiment: a disin‡ation (from 4% to zero).
In this note, we show that, like others in the literature, the analysis of the real e¤ects
of a disin‡ation in BG is ‡awed because it abstracts from non-linearities, being based
on the log-linear formulation of the standard NK model. Such a procedure is clearly not
suited for analyzing the response of the model after a disin‡ation, because the standard
NK model is non-linear, giving rise to non-superneutrality of money. A disin‡ation
experiment is therefore a movement from one steady state to a di¤erent one and cannot
be analyzed by log-linearizing the model around one of the two steady states.
It may be argued that a log-linear analysis is valid in an approximated sense if the
model is "almost" linear. This paper demonstrates that this is not the case. Indeed,
we show that the results in Section 4 in BG are inaccurate both qualitatively and
quantitatively.
2 The Model
The model is as in BG, that is, a standard NK model where:
(i) Firms produce a di¤erentiated product using the following production function1
Yt=FN1
t(1)
where Yis output, and Fand Nare non-produced2and labor inputs, respectively;
1Throughout the paper, capital letters refer to levels, whereas small letters denote the logarithm of
a variable.
2We deviate slightly from the notation by BG who use the letter Mfor the non-produced good, which
we reserved for money.
1
(ii) Firms’pricing is described by the usual Calvo mechanism, where is the fraction
of …rms not adjusting their price in any given period;
(iii) Households have the following instantaneous and separable utility function
UCt;Mt
Pt
; Nt=C1
t
1+dmMt
Pt1
1dn
N1+'
t
1 + '(2)
where Cis composite consumption (with elasticity of substitution between di¤erent
types of goods equal to ").
(iv) BG assume the following partial adjustment model for the real wage: wt=pt=
(wt1=pt1) + (1 )mrst, where mrstis the marginal rate of substitution between
consumption and labor supply in logarithms and wt=ptis the real wage in logarithms.
Accordingly, we add the same real wage rigidities to the model, but in a non-linear
fashion, that is
Wt
Pt
=Wt1
Pt1
(MRSt)1. (3)
The Technical Appendix describes all model equations in detail. Note that we add
real money balances in the utility function because a disin‡ation describes a path for
the money supply and therefore we do need money demand. Finally, the benchmark
calibration is as in BG, and the money demand parameters are calibrated accordingly
to Chari et al. (2000) (CKM, henceforth).3
3 Disin‡ation
3.1 Steady State E¤ects
The obvious starting point to analyze a disin‡ation experiment is to look at the steady
state, since the standard NK model is non-linear and non-superneutrality arises. In this
respect BG write:
3The benchmark calibration is explained thoroughly in the Technical Appendix. That is: = 0:99;
= 0:5; = 0:025; ' = 1; = 0:9; the elasticity of substitution in consumption between di¤erent types
of goods is set to 10; in order to match the empirical estimates of money demand in CKM == 2:564;
dm= 0:063832;while dnis calibrated so that Nis equal to 0.33 in a zero in‡ation steady state. We
also experimented with log-utility in consumption as in BG with no substantial di¤erence in the results.
The qualitative results of this paper do not depend in any way on the chosen calibration, unless stated.
2
BG Statement 1 "As is well known, the standard NKPC implies the presence of a
long-run trade-o¤, however small, between in‡ation and the output gap. (...) in
the standard NK model, disin‡ation implies a permanently lower-level output." (p.
47)
BG use the standard linearized Phillips curve to make their point:
t=t+1 +yt(4)
Dropping the time indices implies a positive long-run trade-o¤ between in‡ation and
output: y=1
. This conclusion is an artifact of the model being linearized around
zero in‡ation, as shown in Figure 1.4Indeed, while it is true that the tangent of the
curve in the graph at zero in‡ation exhibits a positive slope equal to 1
;the relationship
between steady state output and in‡ation is non-linear:The e¤ects of non-linearities are
quite powerful and turn up very quickly, inverting the relationship from positive to
negative (see Ascari and Rankin, 2002, Ascari, 2004 and Yun, 2005).5
Quite obviously the strength of the steady state e¤ects due to the non-linearities
depends on the parameters governing them, and in particular ; and '. In this respect,
we show the graphs for the two values of (the degree of decreasing returns to labor)
used by BG, and for = 0:5(probability of not re-setting the prices), as in BG and Bils
and Klenow (2004), as well as = 0:75, by far the value most commonly used in the
literature, see e.g. Galí (2003).
Our simulations show that non-linearities make the steady state relationship between
in‡ation and output more complex than described by BG. Indeed, it may be positive
only for very small level of in‡ation, if = 0:025; or it can instead reach a maximum
for sizeable positive in‡ation levels, if = 0:33 (7.1% if = 0:5, 3.2% if = 0:75).
It follows that the long-run e¤ects of the BG disin‡ation experiment, i.e., from 4% to
zero, are ambiguous and can be sizeable, depending on the calibration chosen. Finally,
it is worth noting that the long-run e¤ects depend very much on the particular starting
4In Figure 1, steady state output at zero in‡ation was normalized to one, and quarterly in‡ation
rates are annualized.
5In the language of Graham and Snower (2004), BG only take the "time discounting e¤ect" into
account, whereas they ignore the "employment cycling" (product cycling for sticky prices) and "labor
supply smoothing" (production smoothing for sticky prices) e¤ects.
3
0 2 4 6 8
0.95
0.96
0.97
0.98
0.99
1
1.01
Steady State Output, α = 0.025
Annuali zed Steady State Infl ation (i n %)
Output
0 2 4 6
0.97
0.98
0.99
1
1.01
1.02
1.03 Steady State Output, α = 0.33
Annuali zed Steady State Infl ation (i n %)
Output
θ = 0.5
θ = 0.75 θ = 0.5
θ = 0.75
Figure 1: Steady state relationship between output and (annualized) in‡ation
point. Steady state output changes are very di¤erent when disin‡ating from 8% to 4%,
rather than from 4% to zero.
Some authors may argue that at least in analyzing the steady state properties of the
standard NK model one should allow for indexation. Partial indexation would ‡atten
and move the output peak somewhat to the right. However, only full indexation to
steady state in‡ation would be reconcilable with BG’s linearized equations (see Ascari,
2004).6But complete indexation to steady state in‡ation would lead to an entirely
vertical (‡at in our Figure 1) long-run Phillips curve, thus wiping out any trade-o¤.
3.2 Disin‡ation Dynamics
3.2.1 Standard NK Model
BG Statement 2 Qualitatively: "Hence, at the time of disin‡ation (period 0) output
declines by dy(0) = ((1 )=),remaining at the lower level thereafter, with
no additional transitional dynamics coming into play." (p. 47) Quantitatively: "In
the standard NK model, the real e¤ects of disin‡ations mentioned above tend to be
small, at least for plausible parameter values." (p. 47)
6Indeed, in what follows, we also consider the case where non-resetting …rms automatically index
their prices to the steady state in‡ation rate. This is motivated by the fact that full indexation is the
only way to obtain the standard New Keynesian Phillips curve (i.e., t=Ett+1 +yt, as used by
BG) by log-linearizing the model around the steady state, independently of the steady state in‡ation
rate. See the Appendix.
4
0 5 10 15
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0Output D ynamics
Quarters
% Devi ation from Old Steady State
χ = 0
χ = 1
Figure 2: Output response after a disin‡ation from 4% to 0
BG’s assessment of the e¤ects of a disin‡ation in a standard NK model is based on
a speci…c log-linearized version of the model. Figure 2 shows the output dynamics (in
percentage deviation) in the fully non-linear standard NK model, following a sudden
decrease in the rate of growth of money from 4% to zero, as in BG.7From a qualitative
point of view, it is evident that transitional dynamics comes into play, and they do not
necessarily seem to be at odds with empirical observations: output drops on impact
and then sluggishly returns to its new steady state value after roughly two and a half
years. From a quantitative point of view, the e¤ects are quite big: the slump on impact
is about 3.5% of the starting output level, and output remains below the steady state
value all along the adjustment path. It is worth stressing that this path is obtained for
the standard microfounded NK model and standard calibration values.8
Two channels induces the slump in output. First, a microfounded money demand
implies a higher level of real balances in the new steady state after the disin‡ation. Thus,
the price level has to slow down more than the money supply during the adjustment
phase, and this requires output to fall (see, e.g., Blanchard and Fischer, 1989, chp. 10,
Ascari and Rankin, 2002 and references therein). Second, a state variable emerges when
considering the full non-linear model: the price dispersion term, as shown by Schmitt-
7In Figure 2 , we thus set = 0 and use the benchmark calibration of footnote 3. Moreover, we
plot two paths for two di¤erent cases: no indexation (= 0) and full indexation (= 1) to the steady
state rate growth rate of money. The paths displayed in Figure 2 and onwards are obtained using the
software DYNARE developed by Juillard (1996) and others at CEPREMAP.
8For a thorough analysis of the e¤ects of disin‡ations in various versions of the NK model see Ascari
and Ropele (2006).
5
0 5 10 15 20
-4
-3
-2
-1
0
1
2
3
4Output Dynamics from 8% to 4%, θ = 0.75
Quarters
% Devi ation from Old Steady State
χ = 0
χ = 1
Figure 3: Output response after a disin‡ation from 8% to 4%
Grohé and Uribe (2007). Price dispersion, indeed, has a backward-looking dynamics,
hence delivering adjustment dynamics following a disin‡ation (see equations (49) and
(47) in the Appendix).
To sum up, BG write that the standard NK model cannot capture the empirical
evidence of the negative e¤ects of a disin‡ation. Again, we instead claim that the
linearization is responsible for this feature, and thus not the NK model per se.
Remark: In Figure 2 we plot two paths for two di¤erent cases: no indexation (= 0)
and full indexation (= 1) to the steady state rate growth rate of money. For the
benchmark calibration, the two paths are almost identical, showing that our result
does not depend on the degree of indexation. The reader should in any case keep in
mind that indexation would matter more, whenever e¤ects arising from non-linearities
are stronger. Indeed, given the benchmark calibration in Figure 2, the old and new
steady states are very close. However, whenever the long-run e¤ects are instead sizeable
(because of di¤erent starting in‡ation values and/or di¤erent calibration), indexation
would obviously also matter for the dynamic adjustment path. This is an important
point, exemplifying how long-run e¤ects and short-run dynamics interrelate with each
other in a full non-linear model. Just as an illustration, Figure 3 shows the output
dynamics following a disin‡ation from 8% to 4% when = 0:75, under the two cases of
no and full indexation (see Ascari and Ropele, 2006).
6
3.2.2 Real Wage Rigidities
BG Statement 3 "Hence, a permanent reduction in in‡ation of 4 percentage points
in (annualized) in‡ation lowers the level of output by roughly 50 basis points in
the quarter the policy is implemented, an e¤ ect about 10 times larger than in the
standard model." (p. 48)
BG claim that real wage rigidities: (i) are necessary to obtain a dynamic response of
output after a disin‡ation, and (ii) they increase the impact e¤ect on output and thus
the overall costs of a disin‡ation manifold. We already saw that a dynamic path for
output is obtained in the standard model without the need for any real rigidities.
Figure 4 shows that BG’s assessment of the role that real rigidities play for the dy-
namic adjustment after a disin‡ation is qualitatively right.9Indeed, real wage rigidities
cause stronger and more persistent e¤ects on output. From a quantitative point of view,
however, the e¤ects are by no means of the order of magnitude suggested by BG. In the
extreme case assumed by BG, i.e., = 0:9;the impact e¤ect is only twice as large as in
the standard model. Moreover, during the adjustment, output oscillates and the sud-
den slump is followed by a prolonged boom that partly compensates the initial output
loss, with respect to the case without real wage rigidities, where convergence is instead
monotonic. Moreover, Figure 4 does not suggest a "relatively fast convergence to the
new steady state."
Finally it is worth noting that only very extreme values of tend to have sizeable
e¤ects on the output response, since for values smaller than 0.5, the quantitative e¤ects
of real rigidities are small (more on that in the next subsection).
BG stress the importance of real rigidities for in‡ation dynamics. Indeed, in Section
6 of their article, BG show that real wage rigidities are able to generate in‡ation inertia
and give raise to a log-linearized Phillips curve equation which is very similar to the
ad hoc speci…cation used in the empirical literature. This point can be visualized by
plotting the dynamic response of in‡ation, as in Figure 5. In‡ation indeed displays more
inertia for higher values of : Moreover, for the calibration chosen by BG, i.e., = 0:9;
in‡ation exhibits a hump-shaped response. However, (i) again only extreme values of
9In Figure 4 and in the following ones, we assume full indexation, benchmark calibration and again
4% to zero disin‡ation experiment. Note that the steady state values do not depend on : The paths
for output would be almost identical if we had assumed no indexation.
7
0 5 10 15 20
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2Output D ynamics
Quarters
% Devi ation from Steady State
γ = 0
γ = 0.25
γ = 0.5
γ = 0.75
γ = 0.9
Figure 4: The e¤ect of real rigidities on the output response to a disin‡ation
0 5 10 15 20
-60
-50
-40
-30
-20
-10
0
10 (Annual ized) Inflation D ynami cs
Quarters
Annualiz ed Inflation
γ = 0
γ = 0.25
γ = 0.5
γ = 0.75
γ = 0.9
Figure 5: The e¤ect of real rigidities on the in‡ation response to a disin‡ation
tend to have signi…cant e¤ects; (ii) the numbers are rather disconcerting. In‡ation
falls immediately with little inertia whatsoever in any case: the …rst quarter after the
disin‡ation, in‡ation is -50% (in annualized terms) if = 0;and -24% if = 0:9:
3.2.3 Returns to Scale
BG Statement 4 "Finally, it is worth noticing that [...] the quantitative results above
change signi…cantly if we assume the presence of decreasing returns. Hence, under
our alternative calibration with decreasing returns, the loss of output at the outset
of the disin‡ation is multiplied by a factor of 4 relative to the case with no real
rigidities (compared with a factor of 10 in the case of constant returns). The
smaller initial impact coexists with a larger persistence." (p. 49)
8
0 5 10 15 20 25 30
-10
-8
-6
-4
-2
0
2Output D ynamics
Quarters
% Devi ation from Steady State
γ = 0
γ = 0.25
γ = 0.5
γ = 0.75
γ = 0.9
Figure 6: Decreasing returns to scale to labor and the e¤ect of real rigidities
0 5 10 15 20 25 30
-10
-8
-6
-4
-2
0
2Output D ynamics
Quarters
% Devi ation from Steady State
α = 0.33
α = 0.025
Figure 7: DRTS and ouput response after a disin‡ation from 4% to 0 (= 0:9)
Figure 6 shows clearly that assuming stronger decreasing returns to scale to labor
(DRTS) cause: (i) a higher persistence in the output response; (ii) a downward rescaling
of the e¤ect of real rigidities. From a quantitative point of view, however, the e¤ects are
not of the size described by BG: actually assuming DRTS makes real rigidities virtually
devoid of importance for the output response to a disin‡ation.
Finally, it is worth visualizing the di¤erent paths of the output response for the BG
preferred calibration (i.e., = 0:9) under almost constant and DRTS. With DRTS not
only persistence, but also the impact e¤ect is larger. Note that simply by di¤erentiating
(25) at p. 48 in BG with respect to ; it is easy to check that BG equations would actually
imply the opposite: an increase in would lower the impact e¤ect of a disin‡ation.
9
4 Conclusions
In a stimulating paper BG study the e¤ects of introducing real wage rigidities in a
standard NK model. In Section 4, BG look at disin‡ations. They claim this feature to
be crucial for this class of models to explain the cost of disin‡ation.
In this paper, we show that, like others in the literature, the analysis in BG is ‡awed
because it abstracts from non-linearities, being based on the log-linear formulation of the
standard NK model. Indeed, we show that the results in their Section 4 are inaccurate
both qualitatively and quantitatively, once the full microfounded and non-linear model
is taken into account.
This paper sounds a cautionary note about the log-linearized model as a tool to
analyze disin‡ation experiments theoretically. More generally, we want to advocate the
explicit consideration of the e¤ects of non-linearities, whenever necessary and possible.
5 References
Ascari, Guido (2004): "Staggered Prices and Trend In‡ation: Some Nuisances."
Review of Economic Dynamics, Vol. 7, No. 3, pp. 642-667.
Ascari, Guido, and Rankin, Neil (2002): "Staggered Wages and Output Dynam-
ics under Disin‡ation." Journal of Economic Dynamics & Control, Vol. 26, No.
4, pp. 653-680.
Ascari, Guido, and Ropele, Tiziano (2006): "Disin‡ation in New Keynesian
Models." mimeo, University of Pavia.
Ball, Laurence (1994): "Credible Disin‡ation with Staggered Price-Setting." The
American Economic Review, Vol. 84, No. 1, pp. 282-289.
Blanchard, Olivier J., and Fischer, Stanley. (1989): "Lectures on Macroeco-
nomics." Cambridge, MA.: MIT Press.
Bils, Mark, and Klenow, Peter (2004): "Some Evidence on the Importance of
Sticky Prices." Journal of Political Economy, Vol. 112, No. 5, pp. 947-985.
Blanchard, Olivier, and Galí, Jordi (2007): "Real Wage Rigidities and the New
10
Keynesian Model." Journal of Money, Credit, and Banking, Vol. 39, No. 1,
Supplement, pp. 35-65.
Chari, V. V., Kehoe, Patrick J., and McGrattan, Ellen R. (2000): "Sticky
Price Models of the Business Cycle: Can the Contract Multiplier Solve the Per-
sistence Problem." Econometrica, Vol. 68, No. 5, pp. 1151-1179.
Galí, Jordi (2003): "New Perspectives on Monetary Policy, In‡ation, and the Busi-
ness Cycle." In: Dewatriport, Mathias, Hansen, Lars P., and Turnovsky, Stephen
J. (Eds), Advances in Economics and Econometrics: Theory and Applications,
eighth World Congress, 2003, pp. 151-197.
Graham, Liam, and Snower, Dennis (2004): "The Real E¤ects of Money Growth
in Dynamic General Equilibrium." ECB Working Paper Series, No. 412, November
2004.
Juillard, Michel (1996): "Dynare: A Program for the Resolution and Simulation
of Dynamic Models with Forward Variables." CEPREMAP Working Paper, No.
9602.
Schmitt-Grohé, Stephanie, and Uribe, Martín (2007): "Optimal Simple and
Implementable Monetary and Fiscal Rules." Journal of Monetary Economics,
forthcoming.
Yun, Tack (2005): "Optimal Monetary Policy with Relative Price Distortions."
American Economic Review, Vol. 95, No. 1, pp. 89-109.
11
Technical Appendix
1. Household
Given the separable utility function
UCt(h);Mt(h)
Pt
; Nt(h)=C1
t
1+dmMt(h)
Pt1
1exp()dn
N1+'
t(h)
1 + '. (5)
subject to the budget constraint
PtCt+ (1 + it)1Bt+Mt=WtNtTt+ t+Bt1+Mt1(6)
where itis the nominal interest rate, Btare one-period bond holdings, Mtis the nominal
money supply, Wtis the nominal wage rate, Ntis the labor input, Ttare lump sum taxes,
and tis the pro…t income. The representative consumer maximizes the intertemporal
utility (using the discount factor )
max
fCt;Wt;Bt;MtgE0
1
X
t=0
tUCt(h);Mt(h)
Pt
; Nt(h), (7)
yielding the following …rst order conditions:
Money demand equation:
Um
UC
=dmC
t
m
t
=it
1 + it
(8)
Labor supply equation:
Wt
Pt
=UN
UC
=exp()dnN'
t
1=C
t
= exp()dnN'
tC
t(9)
We introduce real wage rigidities in the same way as BG, that is
Wt
Pt
=Wt1
Pt1
MRS1
t=Wt1
Pt1UNt
UCt1
(10)
Wt
Pt
=Wt1
Pt1
(exp()dnN'
tC
t)1(11)
Euler equation:
1
C
t
=Et Pt
Pt+1 (1 + it)1
C
t+1 (12)
12
2. Firms’pricing
Final good producers use the following technology
Yt=Z1
0
Y
"1
"
i;t di"
"1
(13)
Their demand for intermediate inputs is therefore equal to
Yi;t+j=Pi;t
Pt+j"
Yt+j(14)
1. No indexation
The problem of a price-resetting …rm can be formulated as
max
Pi;t
Et
1
X
j=0
jt;t+jPi;t
Pt+j
Yi;t+jT C r
t+j(Yi;t+j)(15)
s:t: Yi;t+j=Pi;t
Pt+j"
Yt+j(16)
where Pi;t denotes the new optimal price of producer iand T C r
t+j(Yi;t+j)the real total
cost function and t;t+jis the stochastic discount factor (from period tto period t+j).
The solution to this problem yields the familiar formula for the standard optimal resetted
price in a Calvo setup
Pi;t ="
"1EtP1
j=0 jt;t+jhP"
t+jYt+jMCr
i;t+ji
EtP1
j=0 jt;t+jhP"1
t+jYt+ji(17)
where MCr
i;t denotes the real marginal costs function.
This can be rewritten as
Pi;t
Pt
="
"1 t
t(18)
where
t=Et
1
X
j=0
()jUC(t+j)Pt+j
Pt"
Yt+jMCi;t+j(19)
t=Et
1
X
j=0
()jUC(t+j)"Pt+j
Pt"1
Yt+j#(20)
The denominator can also be written as:
t=UC(t)Yt+Et
1
X
j=1
()j"Pt+j
Pt"1
UC(t+j)Yt+j#(21)
13
Next, considering the de…nition for t+1 and collecting the term Pt+1
Pt"1yields the
following expression for t
t=UC(t)Yt+Et"1
t+1 t+1(22)
where t+1 Pt+1
Pt.
Doing exactly the same steps for the numerator gives rise to the following expression
for t
t=UC(t)YtMCi;t +Et"
t+1 t+1 (23)
The aggregate price level evolves according to
Pt=Z1
0
Pi;t1"di1
1"
=)(24)
1 = ""1
t+ (1 )Pi;t
Pt1"#1
1"
(25)
2. Partial indexation to long-run in‡ation (LRI)
Under this assumption, a …rm that cannot re-optimize its price updates the price
according to this simple rule:
Pi;t = Pi;t1(26)
where is the steady state in‡ation level and 2[0;1] is a parameter that measures the
degree of indexation. If = 1, there is full indexation, if = 0 there is no indexation
and the problem is the same one as in the previous case. The problem of a price-resetting
…rm then becomes the following
max
p
t(i)Et
1
X
j=0
jt;t+jPi;t j
Pt+j
Yi;t+jT C r
t+j(Yi;t+j)
s:t: Yi;t+j=Pi;t j
Pt+j"
Yt+j(27)
and the …rst order condition (FOC) is
Pi;t ="
"1
EtP1
j=0 jt;t+jhP"
t+jYt+jMCr
i;t+j"ji
EtP1
j=0 jt;t+jhP"1
t+jYt+j(1")ji(28)
This can be rewritten again as
14
Pi;t
Pt
="
"1 t
t(29)
t=Et
1
X
j=0
()jUC(t+j)Pt+j
Pt"
Yt+jMCr
i;t+j"j(30)
t=Et
1
X
j=0
()jUC(t+j)"Pt+j
Pt"1
Yt+j(1")j#(31)
Employing similar substitution as above these two equations can be written as
t=Et
1
X
j=0
()jUC(t+j)Pt+j
Pt"
Yt+jMCr
i;t+j"j(32)
t=UC(t)YtMCr
i;t+"Et"
t+1 t+1 (33)
and similarly
t=Et
1
X
j=0
()jUC(t+j)"Pt+j
Pt"1
Yt+j(1")j#(34)
t=uc(t)Yt+ (1")Et"1
t+1 t+1(35)
The aggregate price level now evolves according to
Pt=Z1
0
Pi;t1"di1
1"
=h(1")P1"
t1+ (1 )Pi;t1"i1
1"=)(36)
1 = "(1")"1
t+ (1 )Pi;t
Pt1"#1
1"
:(37)
3. Technology
Production function:
Yi;t =F
tN1
i;t :(38)
For simplicity, we omit F
t(since we are not explicitly interested in a cost push
shock).
The labor demand and the real marginal cost of …rm iis therefore
Nd
i;t = [Yi;t]1
1(39)
15
MCr
i;t =1
1
Wt
Pt
Y
1
i;t :(40)
Note that now marginal costs depend upon the quantity produced by the single …rm,
given the decreasing returns to scale. In other words, di¤erent …rms charging di¤erent
prices would produce di¤erent levels of output and hence have di¤erent marginal costs.
Express MCr
i;t as
MCr
i;t =1
1
Wt
Pt
Y
1
i;t (41)
=1
1
Wt
Pt"Pi;t
Pt"
Yt#
1
:
4. Aggregation and price dispersion
The aggregate resource constraint is now simply given by
Yt=Ct(42)
and the link between aggregate labour demand and aggregate output is provided by
Nd
t=Z1
0
Nd
i;tdi=2
4Z1
0"Pi;t
Pt"
Yt#1
1
di3
5=(43)
= [Yt]1
1Z1
0"Pi;t
Pt"#1
1
di
| {z }
st
=st[Yt]1
1
where
st=Z1
0"Pi;t
Pt"#1
1
di (44)
is a sort of tax due to price distortions (and the non-linearity of the aggregator). Schmitt-
Grohé and Uribe (2007) show that stis bounded below at one, so that strepresents the
resource costs due to relative price dispersion under the Calvo mechanism with long-run
in‡ation. Indeed, the higher st, the more labor is needed to produce a given level of
output. Note that stcan also be rewritten as a ratio between two di¤erent price indexes
Ptand e
Pt
st=Pt
e
Pt"
where e
Pt=Z1
0
Pt(i)"di1="
, (45)
16
as in Ascari (2004). Whenever there is price dispersion these two indexes evolve di¤er-
ently from each other, determining a certain dynamics for st;that a¤ects the level of
production negatively . stwould not a¤ect the real variables up to …rst order whenever
there is no trend in‡ation (i.e., = 1) or whenever the resetted price is fully indexed to
any variable whose steady state level grows at the rate .
To close the model we just need to solve for the dynamic of susing (44). This would
depend on the indexation.
1. No indexation
st=Z1
0"Pi;t
Pt"#1
1
di (46)
st= (1 )Pi;t
Pt"
1
+
"
1
tst1(47)
2. Long-run indexation
st=Z1
0Pi;t
Pt"
1
di (48)
st= (1 )Pi;t
Pt"
1
+t
"
1st1(49)
5. System of Equations
The following systems of equations are simulated non-linearly:
1. No Indexation
Equations (8), (9), (12), (18), (22), (23), (25), (39), (40), and (47).
2. Indexation
Equations (8), (9), (12), (18), (33), (35), (37), (39), (40), and (49).
In both cases, the money supply identity equation closes the system
mt1rgmt=mtt, (50)
where rgmtis the rate of growth of money which is reduced under a disin‡ation.
In the presence of a real wage rigidity, equation (9) is replaced by equation (11).
6. Calibration
We calibrate the money demand in the same way as in CKM (pp. 1160 f.). While
they have a non-separable utility function, we used a separable form as in BG.
17
Given the money demand
dmC
t(1 + it) = itm
t(51)
and taking the logarithm
ln mt=ln dm
+
ln Ct1
ln it
1 + it(52)
we obtain the same analytical form as CKM (p. 1161, see equation (25)):
ln Mst
P(st)=ln !
1!+ ln cstln Rst1
R(st)!(53)
To obtain the same interest rate elasticity of money demand, we set = 2:5641
(CKM: = 0:39). To obtain the same output elasticity, we set = 2:5641 as well
(CKM: != 0:94). Furthermore, dmis set to 0:063832.
As in CKM, dnis calibrated in such a way that people devote one third of their
time to work (under zero steady state in‡ation). The elasticity of substitution between
di¤erent product types (") is set to 10.
Furthermore, we use a standard quarterly discount rate of one percent (= 0:99) and
a quadratic disutility of labor ('= 1), see e.g. Galí (2003). The quarterly probability
of not re-setting the prices () is either set to 50 percent (see Bils and Klenow, 2004) or
to 75 percent, as in most of the calibrations in the literature. The degree of decreasing
returns to labor (1) is either 0:975 (BG write that the share of oil in production is
roughly 2.5 percent) or 0:67 (as in CKM) in our calibration.
18
