SFB 649 Discussion Paper 2009-042
The Cost of Tractability and the Calvo Pricing
Assumption
Fang Yao*
* Humboldt-Universität zu Berlin, Germany
This research was supported by the Deutsche Forschungsgemeinschaft through the SFB 649 "Economic Risk".
http://sfb649.wiwi.hu-berlin.de
ISSN 1860-5664
SFB 649, Humboldt-Universität zu Berlin Spandauer Straße 1, D-10178 Berlin
SFB
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The Cost of Tractability and the Calvo Pricing Assumption
Fang Yao�
Humboldt Universität zu Berlin
September 3, 2009
Abstract
This paper demonstrates that tractability gained from the Calvo pricing assumption iscostly in terms of aggregate dynamics. I derive a generalized New Keynesian Phillips curvefeaturing a generalized hazard function, non-zero steady state in�ation and real rigidity. An-alytically, I �nd that important dynamics in the NKPC are canceled out due to the restrictiveCalvo assumption. I also present a general result, showing that, under certain conditions,this generalized Calvo pricing model generates the same aggregate dynamics as the gen-eralized Taylor model with heterogeneous price durations. The richer dynamic structureintroduced by the non-constant hazards is also quantitatively important to the in�ation dy-namics. Incorporation of real rigidity and trend in�ation strengthen this e¤ect even further.With reasonable parameter values, the model accounts for hump-shaped impulse responsesof in�ation to the monetary shock, and the real e¤ects of monetary shocks are 2-3 timeshigher than those in the Calvo model.
JEL classi�cation: E12; E31Key words: Hazard function, Nominal rigidity, Real rigidity, New Keynesian Phillips
curve
�I am grateful to Michael Burda, Carlos Carvalho, Heinz Herrmann, Michael Krause, Thomas Laubach andAlexander Wolman, other seminar participants at the Deutsche Bundesbank and in Berlin for helpful comments. Iacknowledge the support of the Deutsche Bundesbank and the Deutsche Forschungsgemeinschaft through the CRC649 "Economic Risk". All errors are my sole responsibility. Address: Institute for Economic Theory, HumboldtUniversity of Berlin, Spandauer Str. 1, Berlin, Germany +493020935667, email: [email protected].
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Contents
1 Introduction 2
2 The Model 42.1 Representative Household . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Firms in the Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Pricing Decisions under Real Rigidity . . . . . . . . . . . . . . . . . . . . 52.2.2 Pricing Decisions under Nominal Rigidity . . . . . . . . . . . . . . . . . . 72.2.3 Non-zero-in�ation Steady State . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Derivation of the New Keynesian Phillips Curve . . . . . . . . . . . . . . . . . . . 92.3.1 New Keynesian Phillips Curve . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 The NKPC with Trend In�ation (g) . . . . . . . . . . . . . . . . . . . . . 11
3 Analytical Results 123.1 Relationship between the Calvo and the Generalized NKPC . . . . . . . . . . . . 123.2 The Generalized Calvo Model and the Generalized Taylor Model . . . . . . . . . 13
4 Numerical Results 134.1 The General Equilibrium Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.1 E¤ects of Increasing Hazard Functions . . . . . . . . . . . . . . . . . . . . 154.3.2 E¤ects of Real Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3.3 E¤ects of Trend In�ation . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3.4 Real E¤ects of the Monetary Shock . . . . . . . . . . . . . . . . . . . . . 18
5 Conclusion 20
A Deviation of the Marginal Cost 23
B Deviation of the New Keynesian Phillips Curve 24
C Proof for Proposition 1 27
D Proof for Proposition 2 29D.1 The Generalized Calvo model (GCM) . . . . . . . . . . . . . . . . . . . . . . . . 29D.2 The Generalized Taylor model (GTM) . . . . . . . . . . . . . . . . . . . . . . . . 29
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1 Introduction
The Calvo pricing assumption (Calvo, 1983) has become predominant in the world of appliedmonetary analysis under nominal rigidity. The main argument for using this approach, however,is solely based on its tractability. In recent years, detailed micro-level data sets have becomeavailable for researchers. Empirical work using these data sets1 generally reach the consensusthat, instead of having economy-wide uniform price stickiness, the frequency of price adjust-ments di¤ers substantially within the economy. In addition, the Calvo assumption also impliesa constant hazard function of price setting, meaning that the probability of adjusting prices isindependent of the length of the time since last adjustment. Unfortunately, constant hazardfunctions are also largely rejected by empirical evidence from the micro level data. Cecchetti(1986) used newsstand prices of magazines in the U.S. and Goette et al. (2005) apply Swissrestaurant prices. Both studies �nd strong support for increasing hazard functions. By con-trast, recent studies using more comprehensive micro data �nd that hazard functions are �rstdownward sloping and then mostly �at, interrupted periodically by spikes (See, e.g.: Campbelland Eden, 2005, Alvarez, 2007 and Nakamura and Steinsson, 2008).
Given this con�ict between theory and empirical evidence, it is important to understand towhich extent the constant hazard function is innocuous for the in�ation dynamics and implica-tions of monetary policy.
To tackle this question, I construct a generalized time-dependent pricing model and derivethe New Keynesian Phillips curve (NKPC) featuring an arbitrary hazard function, non-zerosteady state in�ation and real rigidity. The resulting NKPC includes components, such as laggedin�ation, future and lagged expectations of in�ation and real marginal costs. This version of thePhillips curve nests the Calvo case in the sense that, under a constant hazard function, e¤ectsof lagged in�ation exactly cancel those of lagged expectations, so that, as in the Calvo NKPC,only current real marginal cost and expected future in�ation remain in the expression. In thegeneral case, however, both lagged in�ation and in�ation expectations should be presented in thedynamic structure of the Phillips curve. In light of this result, we learn that lagged in�ation andlagged expectations are not extrinsic to the forward-looking pricing model. They are missing inthe Calvo setup, only because the restrictive pricing assumption has them canceled out.
Furthermore, I present a general result that, up to log linearization approximation, the gen-eralized Calvo pricing model based on a �exible hazard function implies the same aggregatedynamics as the general Taylor framework with heterogeneous price durations. In the litera-ture, both frameworks have been commonly applied to study the e¤ects of heterogeneous pricestickiness on aggregate dynamics. Dixon and Kara (2005), for example, generalize the simpleTaylor-wage-contract model to explicitly account for the presence of varying contract lengths,and Carvalho (2005) models heterogeneity in price stickiness by introducing continuous Calvosticky price sectors. Both of these works �nd that the presence of a small portion of highlyrigid sectors leads to more persistent in�ation and larger real e¤ects of monetary shocks. In thispaper, I show that under a certain condition regarding the relationship between the distributionof price durations and the hazard function, these two frameworks imply the same aggregatedynamics. This result has an important empirical implication that the aggregate data can be
1See: e.g. Bils and Klenow (2004), Alvarez et al. (2006), Midrigan (2007), Nakamura and Steinsson (2008)among others.
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used to uniquely identify both hazard functions and the distribution of sticky prices from eitherof these two frameworks.
When simulating the complete general equilibrium model, I combine the generalized NKPCwith a standard IS curve and an exogenous nominal money growth rule. The simulation resultsshow that, even without real rigidity and trend in�ation, the increasing hazard function helpsto increase both persistence of in�ation and output gap. When introducing some degree of realrigidity, the generalized NKPC gives rise to substantially di¤erent in�ation dynamics, namely,the impulse response of in�ation to a nominal money growth shock becomes hump-shaped.Moreover, non-zero trend in�ation ampli�es this e¤ect even further. The economic intuitionbehind these results is that, on the one hand, increasing hazard function postpones the timingof the price adjustment. On the other hand, strategic complementary makes earlier adjusting�rms choose a small size for the adjustment, while the later adjusting �rms make a larger priceadjustment. In another words, the increasing-hazard pricing together with some degree of realrigidity not only a¤ect the timing of the price adjustment, but also the average magnitude of�rms� adjustments, leading to a hump-shaped response. Trend in�ation ampli�es this e¤ecteven further, because high trend in�ation causes relative prices to disperse quickly. Last but notleast, when the real e¤ects of monetary policy shocks are measured by the accumulative impulseresponses of the real output gap, models with an increasing hazard function generate real e¤ectsof monetary policy which are 2-3 times larger than those in the corresponding Calvo model.
In the literature, some cases of this general hazard pricing model have been studied indi¤erent contexts. Wolman (1999) presents preliminary results, showing that in�ation dynamicsare sensitive to hazard functions under di¤erent pricing rules. Mash (2003) constructed a generalpricing model that nests both the Calvo and Taylor models, and showed that implicationsfor optimal monetary policy based on those limiting cases are not robust to the change inthe hazard function. Whelan (2007) and Sheedy (2007) focus on the relationship between theshape of hazard functions and in�ation persistence. The most closely related work is fromCarvalho and Schwartzman (2008), who study the relationship between the heterogeneity ofsticky prices (sticky information) and monetary non-neutrality. They �nd that heterogeneity inprice stickiness leads to roughly 3 times larger monetary non-neutrality.
The remainder of the paper is organized as follows: in section 2, I present the model withthe generalized time-dependent pricing and derive the New Keynesian Phillips curve; section3 shows analytical results regarding new insights gained from relaxing the constant hazardfunction underlying the Calvo assumption; in section 4, I simulate the complete DSGE modelwith some commonly used parameter values in the literature and then present the simulationresults; section 5 contains some concluding remarks.
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2 The Model
In this section, I present a DSGE model of sticky prices based on both nominal and real rigidities.The scheme of nominal rigidity in the model allows for a general shape of the hazard function.A hazard function of price setting is de�ned as the probabilities of price adjustment conditionalon the spell of time elapsed since the price was last set. Real rigidity is introduced similarlyas in Sbordone (2002), who incorporates upward-sloping marginal cost as a source of strategiccomplementarity.
2.1 Representative Household
The representative in�nitely-lived household deduces utilities from the composite consumptiongood Ct, its labor supply and the real money holding Md
t =Pt, and it maximizes a discountedsum of utilities of the form:
maxfCt;Md
t ;Lt;Bt+1gE0
" 1Xt=0
�t�C1��t
1� � � �HLt1+�
1 + �+ �
Mlog
�Mdt
Pt
��#
Here Ct denotes an index of the households�s consumption of each of the individual goods Ct(i)following a constant-elastisity-of-substitution aggregator (Dixit and Stiglitz, 1977).
Ct ��Z 1
0Ct(i)
��1� di
� ���1
; (1)
where � > 1, and it follows that the corresponding cost-minimizing demand for Ct(i) and thewelfare based price index Pt are given by
Ct(i) =
�Pt(i)
Pt
���Ct (2)
Pt =
�Z 1
0Pt(i)
1��di
� 11��
(3)
For simplicity, I assume that household supplies homogeneous labor units (Lt) in an enocomy-wide competitive labor market.
The �ow budget constraint of the household at the beginning of period t is
PtCt +Mdt +
BtRt�Md
t�1 +WtLt +Bt�1 +
Z 1
0�t(i)di: (4)
Where Bt is an Arrow-Debreu security of one-period bond and Rt denotes the gross nominalreturn on the bond. �t(i) represents the nominal pro�ts of a �rm that sells the good i. I assumethat each household owns an equal share of all �rms. Finally this sequence of period budgetconstraints is supplemented with a transversality condition of the form lim
T!1Et
hBT
�Ts=1Rs
i> 0.
The solution to the household�s optimization problem can be expressed in three �rst ordernecessary conditions:
�HLt�C�t =
Wt
Pt; (5)
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This equation gives the optimal labor supply as a function of real wage.
1 = �Et
"�CtCt+1
��
RtPtPt+1
#; (6)
The Euler equation tells us the relationship between the optimal consumption path and assetprices.
�M
Mt
Pt=
C�t1�R�1t
; (7)
Finally, the demand of real money balance is determined by weighting between the bene�ts andcosts of holding money.
2.2 Firms in the Economy
In the economy, there is a continuum of monopolistic competitive �rms, who use labor as thesingle input to produce good i.
Yt(i) = ZtLt(i)1�a (8)
where Zt denotes an aggregate productivity shock. Log deviations of the shock zt follow anexogenous AR(1) process zt = �z zt�1 + "z;t, where "z;t is white noises and �z 2 [0; 1). Lt(i) isthe demand of labor by �rm i. Following equation (2), demand for intermediate goods is givenby:
Yt(i) =
�Pt(i)
Pt
���Yt (9)
2.2.1 Pricing Decisions under Real Rigidity
In Appendix (A), I derive the economy-wide optimal relative price, which is the ratio betweenthe average optimal price chosen by the adjusting �rms and aggregate price index. Note thateven through the individual optimal prices are not the same due to the fact that marginal costsgenerally depend on the amount produced, we can still derive the aggregate optimal relative-priceratio at period t from the average marginal cost in the economy.
P �tPt=
��
� � 11
1� a
� 1�a1�a+�a
Y�+�(1�a)+a1�a+�a
t Z� 1+�1�a+�a
t (10)
To show how real rigidity a¤ects price setting in this model, I log-linearize the relative priceequation (10). De�ne xt = logXt � log �X as the log deviation from the steady state, up to a loglinearization approximation, one can show that the log deviation of the relative price is equalto the log deviation of the economy-wide marginal cost, which in turn is a linear function of logdeviations of output gap and the technology shock.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
gammak1Real Rigidity
Strategic Neutrality
Figure 1: Real Rigidity, when � = 1; � = 0:5 and � = 10
brpt = cmct = (�1yt � �2zt)where :
=1
1� a+ a��1 = a+ �+ �(1� a)�2 = 1 + �
Parameters and �1 have the economic interpretation as the measure of real rigidity. is theelasticity of relative prices to the change in real marginal cost, while �1 measures the sensitivityof real marginal cost to the change in the output gap. Following Woodford (2003), price-settingdecisions are called strategic complementarity when �1 < 1. When we assume that the monetaryauthority controls the growth rate of the nominal aggregate demand dt, then at equilibrium wehave yt = dt� pt. In this case, price adjustments are �sticky�even under a �exible price setting,because relative price reacts less than one-to-one to a monetary shock. On the other hand, pricesetting decisions can be dubbed strategic substitutes when �1 > 1, so that relative price reactsstrongly to monetary policy shocks.
Now we can discuss how changes in the labor share a a¤ect the magnitude of real rigidity ofprice setting in the model. When setting a equal to zero, creating a linear production technology,then = 1 and �1 = �+�. Under the standard calibration values in the RBC literature ( � = 1and � = 0:5 ), the real rigidity parameter �1 is equal to 1:5 and price decisions are strategicsubstitutes. When the value of a rises, the real rigidity parameter becomes smaller, and pricedecisions turn into strategic complementarity.
In Figure (1), I plot values of and �1 against values of a, while setting � = 1; � = 0:5 and� = 10. In this special case, the sensitivity of real marginal cost to the change in the outputgap �1 is not a¤ected by the labor share, while decreases fairly quickly as a becomes larger.This means that, given the parameter values, real rigidity is mainly driven by the sensitivity of
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the relative price to changes in real marginal cost, and the degree of real rigidity is decreasingin a. Only with a modest value of the labor share (around 0:1), real rigidity drops below thestrategic neutrality threshold.
2.2.2 Pricing Decisions under Nominal Rigidity
In this section, I introduce a general form of nominal rigidity, which is characterized by anarbitrary hazard function. Many well known price setting models in the literature can be shownto have the incorporation of a hazard function of one form or another. The hazard function inthis price setting is de�ned as the probability of price adjustment conditional on the spell oftime elapsed since the price was last set. I assume that monopolistic competitive �rms cannotadjust their price whenever they want. Instead, opportunities for re-optimizing prices dependon the hazard function hj , where j denotes the time-since-last-adjustment and j 2 f0; Jg. Jis the maximum number of periods in which a �rm�s price can be �xed. To keep the modelgeneral, I do not parameterize the hazard function, so that the relative magnitudes of hazardrates are totally free. As a result, this model is able to nest a wide range of staggered pricingNew Keynesian models.
Dynamics of the vintage distribution In the economy, �rms�prices are heterogeneous withrespect to the time since their last price adjustment. I call them price vintages, while the vintagelabel j indicates the age of each price group. Table (1) summarizes key notations concerningthe dynamics of vintages.
Vintage Hazard Rate Non-adj. Rate Survival Rate Distributionj hj �j Sj �(j)
0 0 1 1 �(0)
1 h1 �1 = 1� h1 S1 = �1 �(1)...
......
......
j hj �j = 1� hj Sj =j
�i=0�i �(j)
......
......
...J hJ = 1 �J = 0 SJ = 0 �(J)
Table 1: Notations of the dynamics of price-vintage-distribution.
Using the notation de�ned in Table (1), and also denoting the distribution of price durationsat the beginning of each period by �t = f�t(1); �t(2) � � � �t(J)g, we can derive the ex postdistribution of �rms after price adjustments (~�t)
~�t(j) =
8<:JPi=1hj�t(i) , when j = 0
�j�t(j) , when j = 1 � � �J(11)
Intuitively, those �rms that reoptimize their prices in period t are labeled as �vintage 0�, andthe proportion of those �rms is given by hazard rates from all vintages multiplied by theircorresponding densities. The �rm left in each vintage are the �rms that do not adjust their
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prices. When period t is over, this ex post distribution ~�t becomes the ex ante distribution forthe new period �t+1: All price vintages move to the next one, because all prices age by oneperiod.
As long as the hazard rates lie between zero and one, dynamics of the vintage distributioncan be viewed as a Markov process with an invariant distribution �, obtained by solving �t(j) =�t+1(j) It yields the stationary vintage distribution �(j) as follows:
�(j) =
j
�i=0�i
J�j=0
j
�i=0�i
=SjJ�j=0Sj
, for j = 0; 1 � � �J (12)
Let�s assume the economy converges to this invariant distribution fairly quickly, so thatregardless of the initial vintage distribution, I only consider the economy with the invariantdistribution of price durations.
The Optimal Pricing under Nominal Rigidity In a given period when a �rm is allowedto reoptimize its price, the optimal price chosen should re�ect the possibility that it will notbe re-adjusted in the near future. Consequently, adjusting �rms choose optimal prices thatmaximize the discounted sum of real pro�ts over the time horizon during which the new priceis expected to be �xed. The probability that the new price is �xed is given by the survivalfunction, Sj , de�ned in Table (1).
Here I setup the maximization problem of an average adjustor as follows:
maxP �t
EtJ�1Pj=0
SjQt;t+j
�Y dt+jjt
P �tPt+j
� TCt+jPt+j
�Where Et denotes the conditional expectation based on the information set in period t, andQt;t+j is the stochastic discount factor appropriate for discounting real pro�ts from t to t + j.Note that here P �t is de�ned as the average optimal price chosen by the average adjusting �rm.Therefore TCt denotes the average total costs of producing output Y dt . The representativeadjusting Firm maximizes pro�ts subject to demand for intermediate goods in period t+j giventhat the �rm resets the price in period t, (Y dt+jjt).
Y dt+jjt =
�P �tPt+j
���Yt+j ;
It yields the following �rst order necessary condition for the optimal price:
P �t =�
� � 1
J�1Pj=0
SjEt[Qt;t+jYt+jP��1t+j MCt+j ]
J�1Pj=0
SjEt[Qt;t+jYt+jP��1t+j ]
(13)
MCt denotes the average nominal marginal costs of adjusting �rms. The optimal price is equalto the markup multiplied by a weighted sum of future marginal costs, where weights depend onthe survival rates. In the Calvo case, where Sj = �j , this equation reduces to the Calvo optimalpricing condition.
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Finally, given the stationary distribution �(j), aggregate price can be written as a distributedsum of all vintage prices. I de�ne the vintage price which was set j periods ago as P �t�j . Followingthe aggregate price index equation (3), the aggregate price is then obtained by:
Pt =
J�1Pj=0
�(j)P �1��t�j
! 11��
(14)
2.2.3 Non-zero-in�ation Steady State
If I assume that the gross growth rate of nominal money stock is g, then the steady state ischaracterized by constant real variables and a growing path of all nominal variables at the rateg. Because the aggregate price level increases with trend in�ation in the steady state, �rms needto adjust their prices so that the relative prices are close to the optimal ratio speci�ed below. Ifwe de�ne X as the steady state value of variable X, then the optimality condition (13) can berewritten as:
�p�t =�
� � 1
JPj=0
�jS(j) �Y �P �t+j
JPj=0
�jS(j) �Y �P ��1t+j
mc =�
� � 1
JPj=0
�jS(j) �Y �P �t g�j
JPj=0
�jS(j) �Y �P ��1t g(��1)jmc
�p�t�Pt
=�
� � 1mc
26664JPj=0
�jS(j)g�j
JPj=0
�jS(j)g(��1)j
37775 (15)
As seen in Equation (15), the optimal relative price ratio is equal to the constant markupmultiplied by the real marginal cost along with an extra term, which re�ects how fast trendin�ation erodes the relative prices in the economy. When the gross in�ation rate is equal toone, this term is also equal one. In this case, we have the standard static price setting equation.However, when trend in�ation is greater than one, it follows that the extra term is also greaterthan one, meaning that the adjusting �rms want to �front-load�their price adjustments in orderto hedge the risk that they may not adjust again in the near future. As a result, they adjusttheir prices more than those in the case of zero in�ation. The higher relative price, in turn, leadsto lower steady state output and hence, induces an additional welfare loss caused by the steadystate in�ation.
2.3 Derivation of the New Keynesian Phillips Curve
In this section, I derive the New Keynesian Phillips curve for this generalized model. To do that,I �rst log-linearize equation (13) around the steady state with the trend in�ation (��). This ismotivated by King and Wolman (1996) and Ascari (2004), who show that trend in�ation playsan important role in both the long-run and the short-run dynamics.
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The log-linearized optimal price equations are obtained by
p�t = Et
"J�1Pj=0
(�g�)j S(j)
(cmct+j + pt+j)# ; (16)
where :
=
J�1Xj=0
(�g�)j S(j) and cmct = a+ �+ �(1� a)1� a+ a� yt �
1 + �
1� a+ a� zt
In a similar fashion, I derive the log deviation of the aggregate price by log linearizing equation(14).
pt =J�1Pk=0
�(k) p�t�k; where �(k) =�(k)g(��1)k
J�1Pk=0
�(k)g(��1)k(17)
2.3.1 New Keynesian Phillips Curve
To reveal implications of the general hazard function on the in�ation dynamics, I derive thegeneralized NKPC from equations (16) and (17). To keep the equation as simple as possible, I�rst derive it without trend in�ation, i.e. g = 1. After some tedious algebra, I obtain the NewKeynesian Phillips curve as follows2:
�t =J�1Pk=0
�(k)
1� �(0)Et�k
J�1Pj=0
�jS(j)
cmct+j�k + J�1P
i=1
J�1Pj=i
�jS(j)
�t+i�k
!
�J�1Pk=2
�(k)�t�k+1; where �(k) =
J�1Pj=k
S(j)
J�1Pj=1S(j)
; =J�1Pk=0
�jS(j) (18)
At the �rst glance, this Phillips curve is quite di¤erent from the one in the Calvo model.It involves not only lagged in�ation but also lagged expectations that were built into pricingdecisions in the past. All coe¢ cients in the NKPC are derived from structural parameterswhich are either the hazard function parameters or the preference parameters. When J = 3, forexample, then the NKPC is of the following form:
2The detailed derivation of the NKPC can be found in the technical Appendix (B).
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�t =1
(�1 + �1�2)cmct + �1
(�1 + �1�2)cmct�1 + �1�2
(�1 + �1�2)cmct�2
+1
�1 + �1�2Et
���1cmct+1 + �2�1�2
cmct+2 + ��1 + �2�1�2
�t+1 +
�2�1�2
�t+2
�+
�1�1 + �1�2
Et�1
���1cmct + �2�1�2
cmct+1 + ��1 + �2�1�2
�t +
�2�1�2
�t+1
�+
�1�2�1 + �1�2
Et�2
���1cmct�1 + �2�1�2
cmct + ��1 + �2�1�2
�t�1 +
�2�1�2
�t
�� �1�2�1 + �1�2
�t�1
where : = 1 + ��1 + �2�1�2
In this example, we see more clearly how current in�ation depends on marginal costs, laggedin�ation and a complex weighted sum of lagged expectations. All coe¢ cients are expressed interms of hazard rates (�j = 1� hj) and a preference parameter �:
2.3.2 The NKPC with Trend In�ation (g)
When I derive the NKPC by log-linearizing pricing equations around a steady state with non-zerotrend in�ation, it can be shown that the resulting Phillips curve has the exact same structure asthe one without trend in�ation. However, trend in�ation a¤ects the magnitude of all coe¢ cientsin the NKPC. Again, using the example where J = 3, we obtain
�t =1
mct +
1
mct�1 +
1
mct�2
+ 1Et
���1g
�
mct+1 +
�2�1�2g2�
mct+2 +
� 1
�t+1 +�2�1�2g
2�
�t+2
�+ 2Et�1
���1g
�
mct +
�2�1�2g2�
mct+1 +
� 1
�t +�2�1�2g
2�
�t+1
�+ 3Et�2
���1g
�
mct�1 +
�2�1�2g2�
mct +
� 1
�t�1 +�2�1�2g
2�
�t
�(19)
� 3�t�1 (20)
(21)
1 =1
�1g��1 + �2�1g2��2; 2 =
�1g��1
�1g��1 + �2�1g2��2
3 =�1�2g
2��2
�1g��1 + �2�1g2��2; = 1 + ��1g
� + �2�1�2g2�
In this case, trend in�ation (g) enters every coe¢ cient in the Phillips curve, and hence ithas not only a signi�cant impact on the steady state, but a¤ects the in�ation dynamics in acomplex way as well. In general, 1 and 2 are decreasing in g, while 3 is increasing in g. Sothe changes in trend in�ation alter the relative importance between the forward-looking andbackward-looking terms in the Phillips curve.
11
3 Analytical Results
In this section, I explore the generalized NKPC (18) analytically to show which new insights wecan learn from relaxing the constant hazard function underlying the Calvo assumption.
3.1 Relationship between the Calvo and the Generalized NKPC
The �rst question I want to address is why these lagged dynamic terms are absent in the CalvoNKPC? Are they new to the NKPC? The answer is No. Next, I use a proposition to illustratethis point.
Proposition 1 : When assuming the hazard function is constant over the in�nite horizon, thegeneralized NKPC (18) reduces to the standard Calvo NKPC and the following equation mustalso hold:
�t = Et
�(1� �)(1� ��)
1Pi=0�i�imct+i + (1� �)
1Pi=0�i�i�t+i
�(22)
Proof : see Appendix (C).
By iterating equation (22) backwards, the following equations hold
�t�1 = Et�1
�(1� �)(1� ��)
1Pi=0�i�imct+i�1 + (1� �)
1Pi=0�i�i�t+i�1
��t�2 = Et�2
�(1� �)(1� ��)
1Pi=0�i�imct+i�1 + (1� �)
1Pi=0�i�i�t+i�1
�...
In light of these results, we learn that the generalized NKPC nests the Calvo Phillips curve inthe sense that, given the constant hazard function, the e¤ects of lagged in�ation terms exactlycancel the e¤ects of lagged expectations, leaving only current variables and forward-lookingexpectations on in�ation in the expression. Moreover, lagged in�ation and lagged expectationsare not extrinsic to the time-dependent nominal rigidity model. They are missing in the Calvosetup only because the constant hazard assumption causes them to be canceled out. Therefore,the fully written NKPC of the Calvo model should be of the form:
�t = � �t+1 +(1� �)(1� ��)
�cmct
� ��t�1 � �2�t�2 � � � �
+�Et�1
�(1� �)(1� ��)
1Pi=0�i�icmct+i�1 + 1P
i=0�i�i�t+i�1
�+�2Et�2
�(1� �)(1� ��)
1Pi=0�i�icmct+i�2 + 1P
i=0�i�i�t+i�2
�...
12
3.2 The Generalized Calvo Model and the Generalized Taylor Model
In the literature, modelers of nominal rigidity frequently use two mechanisms�the hazard func-tion and the distribution of price durations� to characterize sticky prices or wages. One isfollowing the idea of Calvo (1983), assuming that probabilities of nominal price adjustments areexogenously given, while the other modeling strategy has its origin from the staggered-contractmodel of Taylor (1980), who assumes some forms of the distribution over price durations. De-pending on the purpose, the generalized Calvo model (GCM) tends to use a �exible hazardfunction (See, e.g. Wolman, 1999), while the generalized Taylor model (GTM) usually appliesa �exible distribution of price durations (See, e.g. Jadresic, 1999). Here I present a generalresult, showing that, up to log linearization approximation, those two models generate the sameaggregate dynamics under a certain regularity condition regarding the relationship between thedistribution of price durations and the hazard function.
Proposition 2 : Up to log linearization approximation, the generalized Taylor model (GTM)and the generalized Calvo model (GCM) imply the same aggregate dynamics, when �(J) thedistribution of price durations in the GTM and hj the hazard function in the GCM correspondaccording to Equation (40).
Proof : see Appendix (D).
Here I prove that, given the same driving forces of in�ation in both models, the aggregateprice in the GCM is equal to the aggregate price in the GTM, so that these two models shouldgenerate the same aggregate dynamics. In light of this result, models assuming an exogenousdistribution of price durations implies an aggregate hazard function, and vice versa. This resultis not only theoretically interesting, but also has important implications for empirical work thatuses those frameworks to study price stickiness (See: e.g. Coenen et al., 2007). It implies that theaggregate data can be used to uniquely identify both hazard functions and the distribution fromeither of these two frameworks. In another words, both models should extract same informationout of the data about the price stickiness. Therefore one can choose to work on one of thosemodels and safely draw conclusions about both distribution and hazard functions.
4 Numerical Results
4.1 The General Equilibrium Model
In the numerical experiment, I study the behavior of in�ation dynamics in a general equilibriumsetup. For this purpose, I close the model by adding a nominal money stock growth rule. Thelog-linearized equilibrium equations are summarized here:
13
�t =J�1Pk=0
W1(k; g)Et�k
J�1Pj=0
W2(j; g)cmct+j�k + J�1Pi=1
W3(i; g)�t+i�k
!�J�1Pk=2
W4(k; g)�t�k+1
cmct =�+ � + a
1 + ��+ �ayt �
1 + �
1 + ��+ �azt
�Et [yt+1] = �yt + ({t � Et [�t+1])
mt = �yt ��
1� � {t
mt = mt�1 � �t + gtzt = �z � zt�1 + �t where �t v N(0; 0:0072)gt = g + ut where ut v N(0; 0:00252)
Where all variable are expressed in terms of log deviations from the non-stochastic steady state.The weights (W1;W2;W3;W4) in the NKPC are de�ned in the equation (18). mt is the realmoney balance, and gt denotes the growth rate of the nominal money stock, which consists ofa constant g and a white-noise shock ut, representing the regular and irregular parts of thestanding monetary policy.
4.2 Calibration
In the calibration, instead of referring to any microeconometric evidence on the hazard function,I parameterize the hazard function a parsimonious way. The reason is that, until now, thereis not yet consensus on the shape of hazard functions in the empirical literature. As discussedin the introduction, it is evident that the shape of hazard functions is changing over time withthe underlying economic conditions. Since the main purpose of the paper is to demonstrate theimpact of varying hazard rates on the in�ation dynamics, I choose to calibrate it based on thestatistical theory of duration analysis. In particular, the functional form I apply is the hazardfunction of the Weibull distribution, which has two parameters:
h(j) =�
�
�j
�
���1(23)
� is the scale parameter, which controls the average duration of the price adjustment, while �is the shape parameter to determine the monotonic property of the hazard function. It enablesthe incorporation of a wide range of hazard functions by using various values for the shapeparameter. In fact, any value of the shape parameter that is greater than one corresponds toan increasing hazard function, while values ranging between zero and one lead to a decreasinghazard function. By setting the shape parameter to one, we can retrieve the Poisson processfrom the Weibull distribution.
In this numerical experiment, I choose �, such that it implies an average price durationof 3 quarters, which is largely consistent with the median price durations of 7 - 9 monthsdocumented by Nakamura and Steinsson (2008). The shape parameter is set in the intervalbetween one and three, which covers a wide range of shapes of the hazard function3. As for
3This range only covers increasing hazard functions because it makes the maximum number of price durationJ well de�ned.
14
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Hazard Function
0 2 4 6 8 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45Weibull Distribution
Tau=0.8Tau=1Tau=1.5 Tau=2
the rest of the structural parameters, I use some common values in the literature to facilitatecomparison the results. In the calibration of the preference parameters, I assume � = 0:9902,which implies a steady state real return on �nancial assets of about four percent per annum. I alsoassume the intertemporal elasticity of substitution � = 1, implying log utility of consumption.I choose the Frisch elasticity of the labor supply to equal 0:5, an estimate commonly found inmicroeconometric studies (See: e.g. Blundell and Macurdy). As for the technology parameters,I set labor�s share to be either 0 or 0:36 to show the e¤ect of real rigidity. The elasticityof substitution between intermediate goods � = 10, which implies the desired markup overmarginal cost should be about 11%. Finally, I set the standard deviation of the innovation tothe nominal money growth rate to be 25 basic points per quarter. For the aggregate technologyshock, I choose �z = 0:95 and the standard deviation of 0.007, in line with commonly used valuesin the RBC literature, for example King and Rebelo (2000).
4.3 Simulation Results
To evaluate the quantitative implication for the aggregate dynamics, I apply the standard algo-rithm to solve for the log-linearized rational expectation model.
4.3.1 E¤ects of Increasing Hazard Functions
In the �rst experiment, I study the e¤ects of varying the shape parameter on the equilibriumdynamics without any real rigidity and the trend in�ation. In Table (2), I report second momentsgenerated by the theoretical models, which are di¤erent with respect to the shape of the hazardfunction. Because I use the Weibull hazard function to calibrate the model, I can change theshape of the hazard function by varying the value of the shape parameter � . In this experiment, Ifocus on the comparison between the baseline Calvo case, with a corresponding shape parameterof � = 1, and the increasing hazard models, where � falls in the range between 1.6 and 3. Inall cases, the moments are for a Hodrick-Prescott �ltered time series. For each of these hazardfunctions, two sets of statistics are reported: �rst, the �rst-order autocorrelation coe¢ cient ofdeviations on in�ation, real marginal cost and output; and second, contemporaneous correlationcoe¢ cients between in�ation and real marginal cost. In all models, I use a persistent technologyshock and a transitory monetary shock, whose stochastic properties are speci�ed above.
15
0 5 10 151
0
1
2
3
4
5
6
7
8x 10 3 Inflation (a=0)
0 5 10 151
0
1
2
3
4
5
6x 10 3 Output (a=0)
CalvoIncreasing Hazard
Figure 2: Comparing impulse responses functions
Calvo Model Increasing Hazard Models� 1 1.6 1.8 2 2.5 3
AR(1) � 0.166 0.524 0.537 0.549 0.567 0.576AR(1) y 0.811 0.876 0.874 0.873 0.870 0.868AR(1) cmc 0.169 0.362 0.338 0.318 0.280 0.264
Corr(�; cmc) 0.998 0.977 0.965 0.950 0.915 0.891
Table 2: Second moments of the simulated data (HP �ltered, lambda=1600)
The �rst noteworthy result from the table is that models with increasing hazard rates gen-erate much higher persistence in in�ation than in the Calvo model, ceteris paribus. Secondly,increases in the shape parameter reduces the persistence of real marginal cost and output. In theCalvo case, because in�ation persistence is solely determined by the dynamics of real marginalcost, in�ation persistence cannot exceed persistence of real marginal cost. In the increasinghazard model, however, the autoregressive terms of real marginal cost are brought into thePhillips curve through lagged expectations, and thus, in comparison to the Calvo model, thisnew transmission mechanism propagates more in�ation persistence. Fuhrer (2006) presentedempirical evidence showing that it is di¢ cult to have a sizable coe¢ cient on the driving processin the Calvo NKPC and that a reduced form shock in the NKPC explains a signi�cant portionof the in�ation persistence. We can understand this evidence through the lens of the general-ized NKPC. The problem of the conventional NKPC is essentially caused by ignoring terms likelagged in�ations and lagged expectations. As I show in the analytical result, this is not the casein the more general time-dependent pricing model. The misspeci�ed Phillips curve fails to ex-plain in�ation persistence with its limited structure. Consequently, we either need to introducethe ad hoc backward-looking behavior or a persistent reduced-form shock to achieve a good �tto the data. Last but not least, as shown in the �nal row of the table, the increasing-hazardpricing model also helps to reduce the correlation between in�ation and current real marginalcost, a rather robust feature of the data (See: e.g. Hornstein, 2007).
Figure 2 shows the impulse responses of the Calvo model compared to the increasing-hazardmodel with the shape parameter of 2. The left panel depicts the impulse responses of in�ation
16
0 5 10 150
1
2
3
4
5
6
7
8x 10 3 Calvo Model
0 5 10 151
0
1
2
3
4
5x 10 3 Increasing Hazard Model
a=0a=0.36
Figure 3: Impulse responses of in�ation with real rigidity
while the right panel shows those of the output gap to a 1% increase in the annual nominalmoney growth rate. Without real rigidity and trend in�ation, we observe that, even thoughthe impulse response function of the increasing-hazard model is somewhat more persistent,the general pattern of the impulse responses are the same in both cases, namely, they dropmonotonically back to the steady state.
4.3.2 E¤ects of Real Rigidity
As in�uentially argued in Woodford (2003), real price rigidity plays an important role in in�ationdynamics in addition to nominal rigidity. In this model I introduce real rigidity in a parsimoniousway, following Sbordone (2002). I now set the labor share parameter equal to 0:36. Combiningthis with other parameter values in the model, it implies that the real rigidity parameter ( �1 =a+�+�(1�a)1�a+a� ) equals 0.35, representing a modest level of strategic complementarity.In Figure (3), I compare the impulse responses of in�ation to a transitory money growth
shock with and without real rigidity. The left panel shows the comparison in the Calvo model.Incorporation of real rigidity makes the impulse responses more long-lasting, but still monotonic.By contrast, in the right panel, impulse responses of in�ation in the increasing hazard modelchange substantially with real rigidity. One can see that not only the persistence of the impulseresponse function gets improved, but, more importantly, the shape of it as well. In this case,the IRF becomes hump-shaped with a peak at around the second quarter.
The economic intuition behind this result is that, on the one hand, increasing hazard functionpostpones the timing of the price adjustment, i.e. only a few �rms adjust their prices immediatelyafter a shock, and more and more adjust later on. On the other hand, real rigidity helps toamplify this postponing e¤ect even further. Because price decisions are strategic complementary,when fewer �rms adjust their prices at the beginning phase of the IRF, even the adjusting�rms choose a small size of the adjustment. Afterwards, however, when more �rms reset theirprices, the size of the price adjustment becomes also larger. In another words, the increasing-hazard pricing together with some degree of real rigidity not only a¤ect the timing of the priceadjustment, but also the average magnitude of �rms�adjustments, leading to a hump-shapedresponse.
17
0 5 10 151
0
1
2
3
4
5x 10
3 Inflation to Monetary shocks
0 5 10 152
0
2
4
6
8
10x 10
3 Output to Monetary shocks
a=0,g=1a=0.36,g=1a=0.36,g=1.02
Figure 4: Impulse response functions with real rigidity and trend in�ation
4.3.3 E¤ects of Trend In�ation
In his seminal paper, Ascari (2004) has shown that trend in�ation has important implicationsfor the model�s dynamics when the Calvo pricing model is log-linearized around non-zero trendin�ation. Here I analyze the dynamic e¤ects of trend in�ation in the increasing hazard pricingmodel. Combining these features is an interesting exercise, because, as I have shown in theprevious section, introducing trend in�ation a¤ects all coe¢ cients in the generalized NKPC (SeeEquation 19), and hence it changes the relative importance between the forward-looking andbackward-looking terms in the Phillips curve. As a result, trend in�ation exerts a larger impacton the dynamics of in�ation in the increasing-hazard pricing model than in the Calvo case.
In Figure (4), I show the impulse responses of in�ation and of the output gap to a transitorymoney growth shock in the increasing hazard model. In the left panel, in�ation without realrigidity and trend in�ation reacts to monetary shock monotonically (solid blue line), while thedashed green line depicts the impulse response of in�ation when real rigidity is present. As shownearlier, this line becomes hump-shaped. Furthermore, when I add a non-zero trend in�ation intothe dynamic structure, the hump becomes even more salient and peaks later (red circled line).On the right panel, impulse responses of the output gap show that the real e¤ect of the monetaryshock is more persistent in the case when real rigidity and trend in�ation are presented.
The reason why high trend in�ation ampli�es the e¤ect of increasing hazard functions is, forone, that �rms in the increasing hazard model are more likely to adjust when their prices areold. When presenting trend in�ation, relative prices disperse quickly over time and, as a result,high trend in�ation causes the size of a �rm�s �rst adjustment is increasing in the time since theshock occurred.
4.3.4 Real E¤ects of the Monetary Shock
In the previous sections, I have informally shown that the real e¤ects of the monetary shock islarger in the increasing hazard model than in the Calvo case. Here I introduce a quantitativemeasure of the real e¤ects of money. In Table (3), I report the accumulative IRF of the realoutput gap to a transitory 1% increase in the annual nominal money growth rate. The accumu-lative IRF is the area below the impulse response function over the whole horizon, and it is in
18
the unit of percentage of the steady state level of real output.
Real E¤ects Calvo Model Increasing Hazard Model (� = 2)a=0 a=0.36 a=0, g=1 a=0.36, g=1 a=0.36, g=1.2
Acc:IRF (%) 0.09 0.26 0.22 0.48 0.56
Table 3: Real E¤ects of A Transitory Monetary Shock) with varying trend in�ation
In the Calvo model without any real rigidity, the real e¤ect of money is only about 0:09% ofreal output in the steady state, while this �gure rises by a factor of 3 when a modest level of realrigidity is present. On the other hand, the increasing hazard model can generate this level of reale¤ects of the monetary shock even without any helping features. When adding real rigidity intothe increasing hazard model, however, real e¤ects rise to 0:48% of steady state real output, andpresenting trend in�ation reinforces real e¤ects even further. All in all, the increasing hazardmodel implies 2-3 times more real e¤ects of the monetary shock than the constant-hazard Calvomodel.
19
5 Conclusion
The central theme of this study is to show that non-constant hazard functions underlying apricing assumption implies very di¤erent aggregate dynamics. To illustrate this point, I derivea general New Keynesian Phillips curve, re�ecting an arbitrary hazard function, trend in�ationand real rigidity.
My main analytical results show that, �rst, the generalized NKPC involves components in-cluding lagged in�ation, forward-looking and lagged expectations of in�ations and real marginalcost, which nests the standard Calvo Phillips curve as a limiting case. When the hazard functionis constant, the e¤ect of lagged in�ation exactly cancels the e¤ects of the lagged expectationterms, so that only current variables and forward-looking expectations remain in the expression.Furthermore, I present a general result, showing that under a certain condition regarding therelationship between the distribution of price durations and the hazard function, the log devi-ation of the aggregate price in the GCM is equal to that in the GTM. In light of this result,hazard functions and random distribution of price durations are closely related concepts, andwhen setting them up accordingly, both models imply the same aggregate dynamics.
In the numerical exercise, I show that in�ation and output are more persistent in the in-creasing hazard model than in the Calvo case. Introducing real rigidity and trend in�ationstrengthens the dynamic e¤ects of the increasing hazard function on in�ation even further. Themodel can account for hump-shaped impulse responses of in�ation to the monetary shock. Thereal e¤ects of the monetary shock implied by the increasing hazard model are 2-3 times higherthan those in the Calvo model. However, due to the calibration strategy I choose in my paper,the numerical results are limited in the class of the monotonic shape of hazard functions. Forfuture research, microfounded hazard functions of price setting behavior is clearly a favorableextension for further exploration of the topic.
20
References
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Ascari, G. (2004), Staggered prices and trend in�ation: Some nuisances, Review of EconomicDynamics, 7(3), 642�667.
Bils, M. and P. J. Klenow (2004), Some evidence on the importance of sticky prices, Journalof Political Economy , 112(5), 947�985.
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Calvo, G. A. (1983), Staggered prices in a utility-maximizing framework, Journal of MonetaryEconomics, 12(3), 383�98.
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Carvalho, C. (2005), Heterogeneity in price setting and the real e¤ects of monetary shocks,Macroeconomics, EconWPA.
Carvalho, C. and F. Schwartzman (2008), Heterogeneous price setting behavior and aggre-gate dynamics: Some general results, Working paper.
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Coenen, G., A. T. Levin, and K. Christo¤el (2007), Identifying the in�uences of nominaland real rigidities in aggregate price-setting behavior, Journal of Monetary Economics, 54(8),2439�2466.
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Dixon, H. and E. Kara (2005), Persistence and nominal inertia in a generalized taylor econ-omy - how longer contracts dominate shorter contracts, Working paper series, European Cen-tral Bank.
Fuhrer, J. C. (2006), Intrinsic and inherited in�ation persistence, International Journal ofCentral Banking , 2(3).
Goette, L., R. Minsch, and J.-R. Tyran (2005), Micro evidence on the adjustment of sticky-price goods: It�s how often, not how much, Discussion papers, University of Copenhagen.Department of Economics.
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Hornstein, A. (2007), Evolving in�ation dynamics and the new keynesian phillips curve, Eco-nomic Quarterly , (Fall), 317�339.
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King, R. G. and S. T. Rebelo (2000), Resuscitating real business cycles, Nber workingpapers, National Bureau of Economic Research, Inc.
King, R. G. and A. L. Wolman (1996), In�ation targeting in a st. louis model of the 21stcentury, Proceedings, (May), 83�107.
Mash, R. (2003), New keynesian microfoundations revisited: A calvo-taylor-rule-of-thumbmodel and optimal monetary policy delegation, Tech. rep.
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22
A Deviation of the Marginal Cost
I assume that there is an economy-wide competitive labor market, and hence intermediate �rmsare price takers in this market. In each period, �rms choose optimal demands for labor inputsto maximize their real pro�ts given wage and the production technology (8).
maxLt(i)
�t(i) =Pt(i)
PtYt(i)�
Wt
PtLt(i) (24)
Real marginal cost can be derived from this maximization problem in the form:
mct(i) =Wt=Pt
(1� a)ZtLt(i)�a
Using the production function (8), output demand equation (9), the labor supply condition (5)and the fact that at the equilibrium Ct = Yt, we obtain the real marginal cost as follows:
mct(i) =1
1� aY�+�(1�a)+a
1�at Z
� 1+�1�a
t
�Pt(i)
Pt
���a1�a
(25)
Because marginal costs depend on the demand of the individual good, the price set by the �rmalso a¤ects the marginal costs of the �rm. Next, �rms determine their optimal prices givenmarginal costs and the market demand for their goods (9)
maxPt(i)
�t(i) = Yt(i)
�Pt(i)
Pt�mct(i)
�The �rst order condition for Pt(i) yields:
P �t (i)
Pt=
�
� � 1mct(i)
The optimal relative price is equal to the markup multiplied by real marginal cost. By substi-tuting the real marginal cost with equation (25), we get the economy-wide average relative pricein the form:
P �tPt=
��
� � 11
1� a
� 1�a1�a+�a
Y�+�(1�a)+a1�a+�a
t Z� 1+�1�a+�a
t (26)
23
B Deviation of the New Keynesian Phillips Curve
Here I derive the NKPC for g = 1, Starting from 16
p�t = Et
24J�1Xj=0
�jSj
(cmct+j + pt+j)35 (27)
= Et
24J�1Xj=0
�jSj
cmct+j35+ Et
24J�1Xj=0
�jSj
pt+j
35 (28)
The last term can be further expressed in terms of future rates of in�ation
J�1Xj=0
�jSj
pt+j =1
pt +
�S1pt+1 + � � �+
�J�1SJ�1
pt+J�1
=1
pt +
�S1pt +
�S1(pt+1 � pt) + � � �+
�J�1SJ�1
pt+J�1
=
�1
+�S1
�pt +
J�1Xj=0
�jSj
�t+j +�2S2
pt+1 + � � �+�J�1SJ�1
pt+J�2
=
�1
+�S1
+�2S2
�pt +
J�1Xj=1
�jSj
�t+j +J�1Xj=2
�jSj
�t+j�1
+�3S3
pt+1 + � � �+�J�1SJ�1
pt+J�2
...
=
�1
+�S1
+ � � �+ �J�1SJ�1
�pt +
J�1Xj=1
�jSj
�t+j
+
J�1Xj=2
�jSj
�t+j�1 + :::+�J�1SJ�1
�t+1
= pt +J�1Xi=1
J�1Xj=i
�jSj
�t+i
The optimal price can be expressed in terms of in�ation rates, real marginal cost and aggre-gate prices.
p�t = pt + Et
24J�1Xj=0
�jSj
cmct+j35+ Et
24J�1Xi=1
J�1Xj=i
�jSj
�t+i
35 (29)
Next, I derive the aggregate price equation as the sum of past optimal prices. I lag equation
24
29 and substitute it for each p�t�j into equation ??
pt = �(0) p�t + �(1) p�t�1 + � � �+ �(J � 1)p�t�J+1
= �(0)
24pt + Et0@J�1Xj=0
�jSj
cmct+j1A+ Et
0@J�1Xi=1
J�1Xj=i
�jSj
�t+i
1A35+ �(1)
24pt�1 + Et�10@J�1Xj=0
�jSj
cmct+j�11A+ Et�1
0@J�1Xi=1
J�1Xj=i
�jSj
�t+i�1
1A35...
+ �(J � 1)
24pt�J+1 + Et�J+10@J�1Xj=0
�jSj
cmct+j�J+11A+ Et�J+1
0@J�1Xi=1
J�1Xj=i
�jSj
�t+i�J+1
1A35
pt =
J�1Xk=0
�(k)
2666664pt�k + Et�k0@J�1Xj=0
�jSj
cmct+j�k + J�1Xi=1
J�1Xj=i
�jSj
�t+i�k
1A| {z }
Ft�k
3777775 (30)
Where Ft summarizes all current and lagged expectations formed at period t.Finally, we derive the New Keynesian Phillips curve from equation 30.
pt =
J�1Xk=0
�(k) pt�k +J�1Xk=0
�(k)Ft�k| {z }Qt
�t =
J�1Xk=0
�(k) pt�k � pt�1 +Qt
= �(0) (pt � pt�1) + �(0)pt�1 + �(1)pt�1 + � � �+ �(J � 1)pt�J+1 � pt�1 +Qt= �(0) (pt � pt�1) + (�(0) + �(1)) pt�1 + �(2)pt�2 + � � �+ �(J � 1)pt�J+1 � pt�1 +Qt= �(0)|{z}
W (0)
�t + (�(0) + �(1))| {z } �t�1+W (1)
(�(0) + �(1) + �(2)) pt�2 � � �+ �(J � 1)pt�J+1 � pt�1 +Qt
...
= W (0) �t +W (1)�t�1 + � � �+W (J � 2)�t�J+2 +W (J � 1)| {z }=1
pt�J+1 � pt�1 +Qt
= W (0) �t + � � �+W (J � 2)�t�J+2 + pt�J+1 � pt�J+2| {z }��t�J+2
+ pt�J+2 � � � �+ pt�2 � pt�1| {z }��t�1
+Qt
(1�W (0))�t = �(1�W (2))�t�1 � � � � � (1�W (J � 1))�t�J+2 +Qt
�t = �J�1Xk=2
1�W (k)1� �(0) �t�k+1 +
J�1Xk=0
�(k)
1� �(0)Ft�k
The generalized New Keynesian Phillips curve is:
25
�t =
J�1Xk=0
�(k)
1� �(0)Et�k
0@J�1Xj=0
�jSj
cmct+j�k + J�1Xi=1
J�1Xj=i
�jSj
�t+i�k
1A
�J�1Xk=2
�(k)�t�k+1; where �(k) =
J�1Pj=k
Sj
J�1Pj=1Sj
; =
J�1Xj=0
�jSj (31)
26
C Proof for Proposition 1
In the Calvo pricing case, all hazards are equal to a constant between zero and one. Let�s denotethe constant hazard as h = 1� � . We can rearrange the NKPC 18 in the following way:
�t +1Xk=1
�k�t�k = (1� �)1Xk=0
�kEt�k
(1� ��)
1Xi=0
�i�imct+i�k +1Xi=0
�i�i�t+i�k
!
�t + ��t�1 + �2�t�2 + � � � = Et
(1� �)(1� ��)
1Xi=0
�i�imct+i + (1� �)1Xi=0
�i�i�t+i
!
+ �Et�1
(1� �)(1� ��)
1Xi=0
�i�imct+i�1 + (1� �)1Xi=0
�i�i�t+i�1
!
+ �2Et�2
(1� �)(1� ��)
1Xi=0
�i�imct+i�2 + (1� �)1Xi=0
�i�i�t+i�2
!... (32)
Then iterating this equation one period forward,
�t+1 + ��t + �2�t�1 + �
3�t�2 � � � = Et+1
(1� �)(1� ��)
1Xi=0
�i�imct+i+1 + (1� �)1Xi=0
�i�i�t+i+1
!
+ �Et
(1� �)(1� ��)
1Xi=0
�i�imct+i + (1� �)1Xi=0
�i�i�t+i
!
+ �2Et�1
(1� �)(1� ��)
1Xi=0
�i�imct+i�1 + (1� �)1Xi=0
�i�i�t+i�1
!...
�t+1 + �(�t + ��t�1 + �2�t�2 � � � ) = Et+1
(1� �)(1� ��)
1Xi=0
�i�imct+i+1 + (1� �)1Xi=0
�i�i�t+i+1
!
+ �Et
(1� �)(1� ��)
1Xi=0
�i�imct+i + (1� �)1Xi=0
�i�i�t+i
!
+ �2Et�1
(1� �)(1� ��)
1Xi=0
�i�imct+i�1 + (1� �)1Xi=0
�i�i�t+i�1
!...
Substitute equation ?? for the term in the brackets on the left hand side of this equation,
27
�t+1 + �Et
(1� �)(1� ��)
1Xi=0
�i�imct+i + (1� �)1Xi=0
�i�i�t+i
!
+�2Et�1
(1� �)(1� ��)
1Xi=0
�i�imct+i�1 + (1� �)1Xi=0
�i�i�t+i�1
!
+�3Et�2
(1� �)(1� ��)
1Xi=0
�i�imct+i�2 + (1� �)1Xi=0
�i�i�t+i�2
!...
= Et+1
(1� �)(1� ��)
1Xi=0
�i�imct+i+1 + (1� �)1Xi=0
�i�i�t+i+1
!
+�Et
(1� �)(1� ��)
1Xi=0
�i�imct+i + (1� �)1Xi=0
�i�i�t+i
!
+�2Et�1
(1� �)(1� ��)
1Xi=0
�i�imct+i�1 + (1� �)1Xi=0
�i�i�t+i�1
!...
After canceling out equal terms from both sides of the equation, we obtain the followingequation:
�t+1 = Et+1
(1� �)(1� ��)
1Xi=0
�i�imct+i+1 + (1� �)1Xi=0
�i�i�t+i+1
!
Iterate this equation backwards and rearrange it, we get the familiar NKPC of the Calvomodel.
�t = Et
(1� �)(1� ��)
1Xi=0
�i�imct+i + (1� �)1Xi=0
�i�i�t+i
!(33)
�t = (1� �)(1� ��)mct + (1� �)�t + ��Et (�t+1)
�t =(1� �)(1� ��)
�mct + �Et (�t+1)
Proof done
28
D Proof for Proposition 2
D.1 The Generalized Calvo model (GCM)
Firstly, I denote j as the time-since-last-adjustment, which is the vintage label used in thegeneralized Calvo model. Furthermore, I de�ne �J as the maximum duration, in which a pricecan be �xed. As a result, in general prices di¤er across vintages (j 2 f0; �Jg), but, in each vintage,the average price shown in the paper is following the equation (13). After log-linearization, ityields:
pGCMt =
�J�1Xj=0
�jSj�JP
j=0�jSj
Et
�dMCt+j� (34)
Next, the log-linearized aggregate price in the GCM�pCt�is obtained by summing over all
vintage prices, weighted by the stationary distribution �(k),
pCt =J�1Xk=0
�(k) pGCMt�k ; where �(k) =Sk
J�1�k=0Sk
, for j = 0; 1 � � �J
pCt =J�1Xk=0
�(k)
266664�J�1Xj=0
�jSj�JP
j=0�jSj
Et�k�dMCt+j�k�
377775 (35)
D.2 The Generalized Taylor model (GTM)
In the GTM, there are �J di¤erent price sectors in which prices are set exactly for J periods. Asin the GCM, �J is de�ned as the maximum duration in which a price can be �xed. As a result,J should range between 1 and �J . In addition, I denote the distribution of price sectors by �(J);with J 2 f1; �Jg:
In each price sector, the price choice is made to maximize the real pro�t over the next J � 1periods
maxPJt
J�1Xj=0
EtfQt;t+j�Y dt+jjt
P JtPt+j
� TCt+jPt+j
�g
given that Qt;t+j = �Et
��YtYt+1
���and Y dt+jjt =
�PJtPt+j
���Yt+j , which are consistent with those
in the GCM.It yields the following �rst order necessary condition for the optimal price in price sector J :
P Jt =�
� � 1
J�1Pj=0
�jEt[Y1��t+j P
��1t+j MCt+j ]
J�1Pj=0
�jEt[Y1��t+j P
��1t+j ]
(36)
29
After log-linearization, it yields:
pJt =J�1Xj=0
�j
J�1Pj=0
�jEt
�dMCt+j� (37)
Within each price sector J , there is exactly a 1=J fraction of �rms that adjust to the new pricepJt . And there is also exactly a 1=J fraction of �rms that still use the one-period-old price, p
Jt�1,
and so on. Therefore the average price in each sector can be expressed as follows:
�pJt =1
J
J�1Xk=0
pJt�k (38)
Next, the aggregate price in the GTM can be obtained by summing over all average prices acrosssectors, weighted by the distribution of price sectors �(J),
pTt =
�JXJ=1
�(J) �pJt
=
�JXJ=1
�(J)
J
J�1Xk=0
pJt�k
!
pTt =
�JXJ=1
�(J)
J
8>>><>>>:J�1Xk=0
26664J�1Xj=0
�j
J�1Pj=0
�jEt�k
�dMCt+j�k�377759>>>=>>>; (39)
When the aggregate dynamics in the GCM are the same as in the GTM, then the aggregateprice obtained from both models should be equal,
pTt = pct
�JXJ=1
�(J)
J
8>>><>>>:J�1Xk=0
26664J�1Xj=0
�j
J�1Pj=0
�jEt�k
�dMCt+j�k�377759>>>=>>>; =
J�1Xk=0
�(k)
266664�J�1Xj=0
�jSj�JP
j=0�jSj
Et�k�dMCt+j�k�
377775Given the same driving forces of in�ation (dMCt) in both models, we have the same aggregate
30
prices when the weights to the corresponding marginal costs are equal. This yields
�JXJ=1
�(J)
J
8>>><>>>:J�1Xk=0
26664J�1Xj=0
�j
J�1Pj=0
�jEt�k
�dMCt+j�k�377759>>>=>>>; =
J�1Xk=0
�(k)
266664�J�1Xj=0
�jSj�JP
j=0�jSj
Et�k�dMCt+j�k�
377775(40)
where : �(k) =Sk
J�1�k=0Sk
SJ =J�j=0
(1� hj) , for j = 0; 1 � � � �J
Equation (40) gives the exact correspondence between the distribution of price sector �(J)in the GTM and the hazard function hj in the GCM. In principle, one can solve these �J � 1
number of equations of corresponding weights, plus the regularity condition that�JP
J=1
�(J) = 1,
for �(J), then we get the expression of �(J) in terms of hj : Proof done
31
SFB 649 Discussion Paper Series 2009
For a complete list of Discussion Papers published by the SFB 649, please visit http://sfb649.wiwi.hu-berlin.de.
001 "Implied Market Price of Weather Risk" by Wolfgang Härdle and Brenda López Cabrera, January 2009.
002 "On the Systemic Nature of Weather Risk" by Guenther Filler, Martin Odening, Ostap Okhrin and Wei Xu, January 2009.
003 "Localized Realized Volatility Modelling" by Ying Chen, Wolfgang Karl Härdle and Uta Pigorsch, January 2009. 004 "New recipes for estimating default intensities" by Alexander Baranovski, Carsten von Lieres and André Wilch, January 2009. 005 "Panel Cointegration Testing in the Presence of a Time Trend" by Bernd Droge and Deniz Dilan Karaman Örsal, January 2009. 006 "Regulatory Risk under Optimal Incentive Regulation" by Roland Strausz,
January 2009. 007 "Combination of multivariate volatility forecasts" by Alessandra
Amendola and Giuseppe Storti, January 2009. 008 "Mortality modeling: Lee-Carter and the macroeconomy" by Katja
Hanewald, January 2009. 009 "Stochastic Population Forecast for Germany and its Consequence for the
German Pension System" by Wolfgang Härdle and Alena Mysickova, February 2009.
010 "A Microeconomic Explanation of the EPK Paradox" by Wolfgang Härdle, Volker Krätschmer and Rouslan Moro, February 2009.
011 "Defending Against Speculative Attacks" by Tijmen Daniëls, Henk Jager and Franc Klaassen, February 2009.
012 "On the Existence of the Moments of the Asymptotic Trace Statistic" by Deniz Dilan Karaman Örsal and Bernd Droge, February 2009.
013 "CDO Pricing with Copulae" by Barbara Choros, Wolfgang Härdle and Ostap Okhrin, March 2009.
014 "Properties of Hierarchical Archimedean Copulas" by Ostap Okhrin, Yarema Okhrin and Wolfgang Schmid, March 2009.
015 "Stochastic Mortality, Macroeconomic Risks, and Life Insurer Solvency" by Katja Hanewald, Thomas Post and Helmut Gründl, March 2009.
016 "Men, Women, and the Ballot Woman Suffrage in the United States" by Sebastian Braun and Michael Kvasnicka, March 2009.
017 "The Importance of Two-Sided Heterogeneity for the Cyclicality of Labour Market Dynamics" by Ronald Bachmann and Peggy David, March 2009.
018 "Transparency through Financial Claims with Fingerprints – A Free Market Mechanism for Preventing Mortgage Securitization Induced Financial Crises" by Helmut Gründl and Thomas Post, March 2009.
019 "A Joint Analysis of the KOSPI 200 Option and ODAX Option Markets Dynamics" by Ji Cao, Wolfgang Härdle and Julius Mungo, March 2009.
020 "Putting Up a Good Fight: The Galí-Monacelli Model versus ‘The Six Major Puzzles in International Macroeconomics’", by Stefan Ried, April 2009.
021 "Spectral estimation of the fractional order of a Lévy process" by Denis Belomestny, April 2009.
022 "Individual Welfare Gains from Deferred Life-Annuities under Stochastic Lee-Carter Mortality" by Thomas Post, April 2009.
SFB 649, Spandauer Straße 1, D-10178 Berlin http://sfb649.wiwi.hu-berlin.de
This research was supported by the Deutsche
Forschungsgemeinschaft through the SFB 649 "Economic Risk".
SFB 649 Discussion Paper Series 2009
For a complete list of Discussion Papers published by the SFB 649, please visit http://sfb649.wiwi.hu-berlin.de.
SFB 649, Spandauer Straße 1, D-10178 Berlin http://sfb649.wiwi.hu-berlin.de
This research was supported by the Deutsche
Forschungsgemeinschaft through the SFB 649 "Economic Risk".
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correlation" by Stefanie Eckel, Gunter Löffler, Alina Maurer and Volker Schmidt, April 2009.
026 "Regression methods for stochastic control problems and their convergence analysis" by Denis Belomestny, Anastasia Kolodko and John Schoenmakers, May 2009.
027 "Unionisation Structures, Productivity, and Firm Performance" by Sebastian Braun, May 2009. 028 "Optimal Smoothing for a Computationally and Statistically Efficient
Single Index Estimator" by Yingcun Xia, Wolfgang Härdle and Oliver Linton, May 2009.
029 "Controllability and Persistence of Money Market Rates along the Yield Curve: Evidence from the Euro Area" by Ulrike Busch and Dieter Nautz, May 2009.
030 "Non-constant Hazard Function and Inflation Dynamics" by Fang Yao, May 2009.
031 "De copulis non est disputandum - Copulae: An Overview" by Wolfgang Härdle and Ostap Okhrin, May 2009.
032 "Weather-based estimation of wildfire risk" by Joanne Ho and Martin Odening, June 2009.
033 "TFP Growth in Old and New Europe" by Michael C. Burda and Battista Severgnini, June 2009.
034 "How does entry regulation influence entry into self-employment and occupational mobility?" by Susanne Prantl and Alexandra Spitz-Oener, June 2009.
035 "Trade-Off Between Consumption Growth and Inequality: Theory and Evidence for Germany" by Runli Xie, June 2009.
036 "Inflation and Growth: New Evidence From a Dynamic Panel Threshold Analysis" by Stephanie Kremer, Alexander Bick and Dieter Nautz, July 2009.
037 "The Impact of the European Monetary Union on Inflation Persistence in the Euro Area" by Barbara Meller and Dieter Nautz, July 2009.
038 "CDO and HAC" by Barbara Choroś, Wolfgang Härdle and Ostap Okhrin, July 2009.
039 "Regulation and Investment in Network Industries: Evidence from European Telecoms" by Michał Grajek and Lars-Hendrik Röller, July 2009.
040 "The Political Economy of Regulatory Risk" by Roland Strausz, August 2009.
041 "Shape invariant modelling pricing kernels and risk aversion" by Maria Grith, Wolfgang Härdle and Juhyun Park, August 2009.
042 "The Cost of Tractability and the Calvo Pricing Assumption" by Fang Yao, September 2009.
