What Do Micro Price Data Tell Us on the Validity of the New Keynesian Phillips Curve?

Álvarez, Luis J. Working Paper What Do Micro Price Data Tell Us on the Validity of the New Keynesian Phillips Curve? Economics Discussion Papers, No. 2007-46 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Álvarez, Luis J. (2007) : What Do Micro Price Data Tell Us on the Validity of the New Keynesian Phillips Curve?, Economics Discussion Papers, No. 2007-46, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/17969 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Álvarez Banco de España Abstract: The New Keynesian Phillips curve (NKPC) is now the dominant model of inflation dynamics. In recent years, a large body of empirical research has documented price-setting behaviour at the individual level, allowing the assessment of the micro-foundations of pricing models. This paper analyses the implications of 25 theoretical models in terms of individual behaviour and finds that they considerably differ in their ability to match the key micro stylised facts. However, none is available to account for all of them, suggesting the need to develop more realistic micro-founded price setting models. JEL: E31, D40 Keywords: Pricing models, micro data, Phillips Curve, hazard rate. Correspondence: E-mail: [email protected] This paper has been prepared for the Symposium “The Phillips Curve and the Natural Rate of Unemployment”, organised by the Kiel Institute for the World Economy, 3-4 June 2007. I would like to thank Ignacio Hernando, Pablo Burriel, an anonymous referee of the Banco de España Working Paper Series and seminar participants at the Kiel Symposium and at Banco de España for helpful comments and suggestions and all members of the Eurosystem Inflation Persistence Network, particularly those involved in the analysis of micro data, for extensive discussions during the last years. I would also like to thank Emmanuel Dyne, Johannes Hoffman, Peter Klenow, Fabio Rumler and Roberto Sabbatini for providing data on the hazard rates of their respective countries. www.economics-ejournal.org/economics/discussionpapers © Author(s) 2007. This work is licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany 1. Introduction In recent years, there have been considerable advances in the theoretical modelling of inflation. A new generation of models has emerged, characterised by pricing equations derived from the optimising behaviour of forward looking firms, in a framework of nominal rigidities and imperfect competition. Aggregation over individual decisions leads to relations linking inflation to some measure of real activity, in the spirit of the traditional Phillips Curve, although with firm micro-foundations. This new Keynesian Phillips Curve (NKPC) is now the dominant approach to price modelling and variants of it are routinely used as the supply block of dynamic stochastic general equilibrium (DSGE) models, which are increasingly popular in academic macroeconomics and policy making institutions. There is also growing recognition that the understanding of price stickiness can be improved by examining pricing behaviour at the micro level, where pricing decisions are actually made. Individual information on price setting allows determining to which extent the assumptions used in deriving theoretical models are actually realistic, which helps discriminate among competing models. Micro evidence is also an aid in solving problems of observational equivalence that are sometimes present in the analysis of aggregate time-series data. For instance, as is well known, the popular Calvo (1983) pricing model can be distinguished from the quadratic adjustment model of Rotemberg (1982) on the basis of micro data. Empirical evidence on pricing policies at the microeconomic level had remained quite limited until recent years. Indeed, most quantitative studies with individual price data were quite partial and focussed on very specific products. Fortunately, a large and growing body of empirical research aimed at improving the understanding of the characteristics of the inflation process is now available. Following Bils and Klenow (2004), numerous authors have analysed datasets of the individual prices that are used to compute consumer price indices (CPIs) and producer price indices (PPIs), mostly within the context of the Eurosystem Inflation Persistence Network (IPN). Following Blinder (1991), a significant number of central banks have conducted surveys on price setting behaviour, including those participating in the IPN. The aim of this paper is to survey recent work on micro price data, focussing on those aspects related to the conformity of assumptions used in pricing models put forward in the literature3. After this introduction, the remainder of this paper is organised as follows. Section 2 discusses the main features of micro CPI and PPI datasets, as well as survey data. Section 3 presents the main micro implications of 25 pricing models. Sections 4 and 5 refer to the analysis of frequencies and hazard rates of price adjustment and section 6 to heterogeneity in the frequency 3 For an overview of IPN results on micro data, see Álvarez et al. (2006). More detailed IPN summaries on individual consumer prices are provided in Dhyne et al. (2006) and Sabbatini et al. (2007), which also consider producer price data. Vermeulen et al. (2007) summarises producer price data, whereas Fabiani et al. (2006) and the book by Fabiani et al. (2007) give an overview of results on survey data in the euro area and Lünnemann and Mathä (2007) compare survey results in the euro area with those in other countries. Angeloni et al. (2006) and Gaspar et al. (2007) discuss the implications of micro IPN findings for macroeconomic modelling and the design of monetary policy. 2 of price change. Sections 7 and 8 are devoted to assessing the relevance of time dependent and forward looking behaviour and section 9 presents available evidence on imperfect competition. The paper ends with a section of concluding remarks. 2. Data sources The evidence considered in this paper refers to quantitative datasets made up of the individual transaction prices that are compiled by national statistical offices to compute CPIs and PPIs and qualitative one-off surveys on pricing behaviour, mostly carried out by central banks. These quantitative datasets have the clear advantage that they are representative4 of consumer expenditure and industrial production, in contrast with earlier evidence5 that had a narrow focus, in terms of products, types of outlets and cities considered. Moreover, datasets contain a huge number of monthly price quotes, which may add up to several millions, and extend for several years. Figure 1 Examples of individual price trajectories Consumer prices Producer prices 45000 50000 55000 60000 65000 11-91 11-92 11-93 11-94 11-95 11-96 11-97 11-98 6000 10000 14000 18000 Piece of furniture Refined petroleum 90 140 190 240 290 01-93 01-94 01-95 01-96 01-97 01-98 01-99 01-00 01-01 70000 73000 76000 79000 82000 Fruit Electrical appliance Source: Álvarez and Hernando (2006) for consumer prices and Álvarez et al. (2008) for producer prices. Prices in pesetas. Figure 1 displays some paths of actual price series corresponding to consumer and producer prices in Spain, similar to those found in other countries. [See e.g. Baumgartner et al. (2005]. There are three features worth highlighting. First, most individual prices remain unchanged for several months. Second, prices are not typically reset with a fixed periodicity and, third, there is marked heterogeneity across products in the frequency of price change. A complementary approach to analyse price setting practices is to survey firms directly. Surveys offer unique information on some aspects of pricing policies, such as the information set used or the reasons that justify delays in price adjustments. The approach may be considered controversial, since firms could lack incentives to respond truthfully. However, questionnaires used [See e.g. Fabiani et al. (2007) for the precise questionnaires of euro area countries] have 4 Data confidentiality reasons have prevented full coverage in some countries. See references in table 2 for the precise goods and services covered in each study. 5 Prominent examples include Cecchetti (1986) on newsstand prices of magazines, Carlton (1986) on producer prices of intermediate products used in manufacturing and Lach and Tsiddon (1992) on retail food product prices. 3 avoided problematic questions that could lead firms to conceal the truth6. Moreover, response rates were not low, so selectivity biases probably played a minor role. Finally, answers could be sensitive to the precise wording of questions, the order in which they appear, and the setting in which the questions were answered. Nonetheless, the fact that questionnaires in different countries differ in these aspects7 but produce similar results suggests that the quantitative importance of these concerns is likely to be small. 3. Predictions of pricing models he implications of 25 pricing models, so as to check T aim of this section is to briefly review the them against micro data in the rest of the paper. We focus on the following 4 dimensions: the frequency of price adjustment, the hazard rate (i.e. the probability that a price )( t pwill change after k periods, conditional on having remained constant during the previous k-1 periods) }...|Pr{)( 211 tktktktkt pppppkh = = =≠= −+−+−++ ), the consideration of heterogeneous possibility of allowing for non-rational behaviour. To the extent possible, we also present the Phillips curves implied by these models. behaviour in the frequency of price change, and the .1 Sticky contract or information models ticky contract or information models [Lucas (1972), Fischer (1977), Mankiw and Reis (2002), Lucas (1973) islands model firms have imperfect knowledge of the price level and rationally 3 S Carvalho (2005), Reis (2006) and Maćkowiack and Wiederholt (2007)] imply continuous price adjustment, so that the hazard rate is zero for prices aged more than one period and there is no heterogeneity across firms in the frequency of price change. This is also the case for the heterogeneous Mankiw and Reis (2002) model proposed by Carvalho (2005), since heterogeneity refers to the frequency of updating the information set. Further, in these models firms set prices optimally, subject to informational constraints, so there is no room for irrational behaviour. In estimate it on the basis of the price of its good, solving a signal extraction problem. The Phillips curve is given by )( 1yyE tttt − + = − γ π π is the inflation rate, tt E π 1−t π ,where is the expectation of t π ,conditional on the information set up to t-1, t y is output and yis tre el nd output. In this mod , inflation is driven by its past expectation and the output gap. 6 In some cases, firms may be secretive about prices of their business to business transactions, to avoid tipping off the competition. Moreover, illegal collusive behaviour cannot be expected to be reported in a survey. 7 Cross country methodological differences also exist in quantitative micro data analyses, although their importance seems to be minor. Indeed, Dhyne et al. (2006) use a common 50 product basket for euro area countries and the US and find that quantitative differences with the respective full samples are quite small. 4 Table 1 Frequency (f) o f p rice change Hazard rate Heterogeneity in f Non optimality of price setters Sticky information Carvalho (2005) No Not allowed Fischer (1977) No Not allowed Lucas (1972) No Not allowed Maćkowiak and Wiederholt (2007) No Not allowed Mankiw and Reis (2002) No Not allowed Reis (2006) No Not allowed Menu costs Danziger (1999) No Not allowed Dotsey et al.(1999) [1] No Not allowed Nakamura and Steinsson (2007) [1] increasing or increasing and then non-increasing No Not allowed Sheshinski and Weiss (1977) No Not allowed Time de p endent Álvarez et al. (2005) Yes Not allowed Aoki (2001) Yes Not allowed Bonomo and Carvalho (2004) No Not allowed Calvo (1983) No Not allowed Carvalho (2006) Yes Not allowed Gali and Gertler (1999) Yes Allowed Sheedy (2005) [2] Unrestricted No Not allowed Taylor (1980) No Not allowed Taylor (1993) [3] Yes Not allowed Wolman (1999) No Not allowed Generalised indexation Christiano et al. (2005) [4] No Allowed Smets and Wouters (2003) [4] No Allowed Convex costs of ad j ustment Kozicki and Tinsley (2002) No Not allowed Rotemberg (1982) No Not allowed Consumer an g er Rotemberg (2005) [1], [5] No Not allowed Predictions of pricin g models Notes: [1] No closed form of the hazard rate available [2] The hazard rate cannot be zero at any period and f<1 [3] Assuming that the distribution of contracts is uniform over [1,N] [4] All agents behave non optimally with a given frequency [5] Hazard rate for non-time varying distributions ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =10 11 )( kif kif kh ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =10 11 )( kif kif kh ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =10 11 )( kif kif kh ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =10 11 )( kif kif kh ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =10 11 )( kif kif kh ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =10 11 )( kif kif kh kkh ∀= θ )( ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ > = <> =Δ ∗ ∗ ∗ vkif vkif vkif kh 0 1 0 )( ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =∗ ∗ dkif dkif kh 0 1 )( kkh ∀= θ )( ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ =−+ =1 1)1( )( kif kif kh θ ωθω ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =∗ ∗ dkif dkif kh 0 1 )( ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ > = < = ∗ ∗ ∗ dkif dkif dkif kh 0 1 )( θ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =10 11 )( kif kif kh ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =10 11 )( kif kif kh ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =10 11 )( kif kif kh ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =10 11 )( kif kif kh kkh ∀= θ )( ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ > ≤− =Nkif NkifkN kh 0 )/(1 )( 10 ≤< f )( kh 1=f ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ =−+ =1 1)1( )( kif kif kh θ ωθω ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = =∗ ∗ dkif dkif kh 0 1 )( ∑ ∑ −− = = 11 )( )()( k ii k iii ii k kkh θωθωβ θβ )()( )()( )()()( )121int( 1 1)121int( 11 )121int( 1 11 121 − + −− ++ − + −− + += += += ∑ ∑ ∑ k n k ii k nn k n k ii k iii nii k k Ikkkh θωθωθωβ θωθωθωβ θβθβ 1=f 1=f 1=f 1=f 1=f 1=f 1=f 1=f 1=f 10 ≤< f 10 ≤< f 10 ≤< f 10 ≤< f 10 ≤< f 10 ≤< f 10 ≤< f 10 ≤< f 10 ≤< f 10 ≤< f 10 ≤< f 10 ≤< f 10 ≤< f 10 ≤< f 5 Fischer (1977) introduces price rigidity by assuming that prices are predetermined, but not fixed. That is, contracts set prices for several periods, specifying a different price for each period. Mankiw and Reis (2002) reintroduce the idea that prices are predetermined. Opportunities to adopt new price paths do not arise deterministically, as in Fischer (1977), but stochastically. Each period, a given fraction λ of price setters obtains new information about the state of the economy and computes a new path of optimal prices. The equation for the inflation rate is given by , where the relevant expectations are past expectations of current economic conditions. Reis (2006) inattentiveness model adds to the standard profit-maximisation problem the constraint that agents must pay a cost to acquire, absorb and process information in forming expectations. The model provides a micro-foundation for Mankiw and Reis (2002) and inflation follows a continuous time version of Mankiw and Reis (2002) expression. In these models, prices react with equal speed to all disturbances. In contrast, in Maćkowiack and Wiederholt (2007) model, firms decide optimally what to observe. When idiosyncratic conditions are more relevant than aggregate conditions firms pay more attention to idiosyncratic conditions. Another extension of Mankiw and Reis (2002) model is given by Carvalho (2005), who introduces heterogeneity in firm behaviour. Specifically, each group of firms updates its information set with a different frequency. This does not change the implication of the model in terms of continuous price adjustment or hazard rates, although, interestingly, leads monetary shocks to have substantially larger and persistent real effects. 3.2 Menu costs models In menu costs models, firms must incur a fixed cost to change nominal prices. As a consequence, firms do not adjust prices continuously, but rather when they find it profitable to do so. In the models we consider [Sheshinski and Weiss (1977), Danziger (1999), Dotsey et al. (1999) and Nakamura and Steinsson (2007)] all firms are assumed identical, so there is no heterogeneity in the frequency of price change. Further, firms set prices optimally. Implications for the hazard function differ from model to model. In Sheshinski and Weiss (1977) firms face a constant rate of inflation and find it optimal to adopt a one sided Ss policy. Nominal prices are fixed over intervals of constant duration (d*), which is (ambiguously) affected by inflation. Consequently, the hazard rate, as in Taylor (1980) and Bonomo and Carvahlo (2004), is one for prices aged d* periods and zero for the rest. Within a general equilibrium framework, Danziger (1999) studies a model with menu costs, where each firm’s productivity is exposed to aggregate and idiosyncratic shocks. Prices are determined by a two-sided Ss markup strategy. In the model, the probability that a firm’s price changes is endogenously determined and is independent of the last price adjustment occurred. Consequently, the hazard rate is constant, as in Calvo (1983). In Danziger (1999), the expected duration of a price is higher the higher are menu costs and the discount rate and lower the more )()1( 10 1tt jjt j tt yEy Δ+−+ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − =∑ ∞ = −− απλλ λ λα π 6 uncertain are idiosyncratic shocks and the higher the trend in the money supply. In Dotsey et al. (1999), each firm faces a different menu cost, which is drawn independently over time from a continuous distribution. Within each period, some firms adjust their price, which is identical for all adjusting firms. Positive average inflation ensures that the benefit to changing prices becomes arbitrarily large over time, which makes the number of vintages (v*) of firms in the economy finite. The hazard rate is increasing up to v*, where it is one and then zero. v* is lower the higher is trend mura and Steinsson (2007), also c ical expression is available for e hazard rate, although it is shown that, as the variance of idiosyncratic shocks rises relative to tract, as in Sheshinski and Weiss (1977) and Bonomo and arvalho (2004). In this case, the hazard rate is zero, except in the period (d*) in which the end when the hazard is one. In the Calvo model, there is a constant robability that a given price setter will change its price at any instant, so the hazard rate is inflation. More recent menu cost models, like Naka onsider idiosyncratic productivity shocks. No analyt th the rate of inflation, the hazard function flattens out at longer durations, although it remains steeply upward sloping in the first few periods. 3.3 Time dependent models Time dependent models allow for infrequent price adjustment. Heterogeneity in the frequency of price change is allowed for in some models [Taylor (1993), Galí and Gertler (1999), Aoki (2001), Álvarez et al. (2005) and Carvalho (2006)]. Non rational behaviour is allowed for only in Galí and Gertler (1999). Models generally differ in their predictions for hazard rates. The most common time dependent pricing specifications in the literature are those by Taylor (1980) and Calvo (1983). In Taylor’s model, prices are set by multiperiod contracts and remain fixed for the duration of the con C of the contract occurs, p constant, as in Danziger (1999). As shown by Roberts (1995), the implied New Keynesian Phillips curve of these two models and that of Rotemberg (1982) is the same )( ˆ11 yyEsE ttttttt − + = + =++ δ π β λ π β π where t s ˆis the deviation of log real marginal cost from its steady-state value. The NKPC relates inflation to anticipated future inflation and real marginal cost. This contrasts, for instance, with ankiw and Reis (2002). costs. More recently, Sheedy (2005) has obtained the NK Phillips Curve for a general M Similarly, Wolman (1999) considers a truncated Calvo model, which allows for a constant hazard up to a given horizon d*, in which all firms must adjust, so that the hazard rate is one. The model rules out the possibility of price durations of arbitrarily long length by assuming a zero hazard rate for horizons greater than d*. This model is able to account for inflation inertia, in the sense that lagged inflation appears in the New Keynesian Phillips Curve. Specifically, ∑∑∑ = ++ = − ∞ = ++= J jjtjjtt J jjit iit sE 011 ˆ κπμπϕπ so that current inflation depends of lagged inflation, future expected inflation and real marginal 7 distribution of price durations. In particular, if the hazard rate is increasing, there is structural persistence, in contrast with the Calvo (1983) model. The expression of the Phillips in the general case is tjtt n jjit n iit sE ˆ 1 1 1 κπμπϕπ ++= + = − − =∑∑ Thus, current inflation depends on n-1 lags of past inflation, n expected future inflation rates, and current real marginal cost. Further, if the hazard function has a positive slope then all lags of inflation have positive coefficients. Taylor (1993) allows for heterogeneity in the frequency of price adjustment, by considering the case where the duration of the price contracts varies across different groups of firms. onsidering that the distribution of firms is uniform over durations in [1,N] the hazard rate is turn, Álvarez et al. (2005) introduce an annual Calvo model, hereby the hazard rate is constant every 12, 24, 36 , … periods8 and incorporate it in a mixture change ontinuously- and a sticky sectorin which prices are set as in Calvo (1983). The hazard rate of C monotonically increasing. In w of Calvo agents. The hazard rate of the finite mixture is monotonically decreasing with spikes every 12, 24, 36… periods. Aoki (2001) introduced a heterogenous economy with a flexible sector –in which prices c this model is constant after the second period, as in Galí and Gertler (1999). Carvalho (2006) has generalised this model allowing for n sectors and not imposing the existence of a fully flexible sector. The hazard rate corresponds to a mixture of Calvo price setters and is monotonically decreasing, converging to the hazard rate of the stickiest sector. [Álvarez et al. (2005)]. The generalized NKPC that accounts explicitly for heterogeneity in price stickiness is: ttttt gyyE ψ ϕ π β π + − + =+)( 1 Heterogeneity introduces a new, endogenous shift term ( t g) in the Phillips Curve that can be written as a weighted average of sectoral output gaps, with weights related to the sectoral cks in this model have considerably larger and more persistent real ffects than in identical-firms economies with a similar degree of rigidities. frequencies of price adjustment. Moreover, the coefficient on the aggregate output gap in the Phillips curve also depends on the sectoral distribution of price stickiness. The standard NKPC obtains as a special case when the frequency of price changes is the same across all sectors. Interestingly, monetary sho e The above models rely on forward looking price setters. Galí and Gertler (1999) propose a model that allows for departures from this assumption. Specifically, they assume that a fraction of firms )( ω set prices according to a backward looking rule of thumb. These firms index on last period’s optimal price, rather than on last period’s aggregate price index. This implies that this fraction of firms changes prices continuously, whereas the rest do in with a constant conditional 8 Specifically, 12 )( Ikh θ = and ⎭ ⎬ ⎫ ⎩ ⎨ ⎧= =elsewhere kkif I0 )12/(int12/1 12 8 Calvo model of Álvarez et al. (2005), but not with the rest of models considered in table 1. Note that the presence of these spikes is a reflection of the seasonality that is present in every uantitative micro data study [See e.g. Sabbatini et al. (2007) for the evidence in euro area Figure 3 q countries]. To account for the three stylised facts on hazard rates, Álvarez et al. (2005) propose a parsimonious model made up of several Calvo agents and an annual Calvo agent. As can be seen in the left panel of figure 3, this provides a very accurate representation of individual data. Another possibility, as in Carvalho (2006) would be to consider that economies are made up of numerous sectors and that each of them follows a different Calvo pricing rule. This could be seen as producing a more accurate representation of the data. However, the results in the right panel of figure 3 point to some problems of this alternative hypothesis. Indeed, the aggregation of Calvo price setters misses some features of the hazard function of price changes. First, even considering a high number of sectors, within sector heterogeneity is likely to be present. In general, there will be some price setters who are more flexible than the average of the most flexible group and others that follow stickier pricing policies than the average of the stickiest group. Second, by construction, the hazard of the aggregate does not show annual spikes that are present in the data. Spanish producer prices United States consumer prices Hazard functions for price changes 0.00 0.05 0.10 0.15 0.20 0122436 Finite mixture Empirical hazard 0 0.1 0.2 0.6 0122436 0.3 0.4 0.5 months since the last price change Aggregation of Calvo Empirical hazard months since the last price change Introducing heterogeneity in pricing models leads to precise implications in terms of the hazard of the aggregate. In some cases, such as Álvarez et al. (2005) or Carvalho (200 ), it generates a decreasing hazard rate. Generalising some other models could also lead to decreasing hazards. For instance, generalising Galí and Gertler (1999) to include several types of Calvo agents plus , Mankiw and Reis (2002), Carvalho 005), Reis (2006), Maćkowiack and Wiederholt (2007 ), models with convex costs of 6 a fraction of rule of thumb price setters leads to a monotonically decreasing hazard rate and the same could be obtained introducing heterogeneity in Aoki (2001), Sheedy (2005) and Rotemberg (2005) models for certain parameter values. However, this will not necessarily happen. The aggregation of models that imply continuous price adjustment [sticky contract or information models, such as Lucas (1972), Fischer (1977) (2 adjustment, such as Rotemberg (1982) or Kozicki and Tinsley (2002), or models with widespread and automatic indexation mechanism, such as Christiano et al. (2005) or Smets and 15 Wouters (2003)] cannot generate a decreasing hazard rate. It is also relevant to note that the mixture of hazard rates which are zero for some horizons is also zero for those horizons12. This is a problem for models such as Wolman (1999) or Dotsey et al. (1999). 6. Heterogeneity As seen in the previous section, one possible explanation for the downward slope of hazard functions is that there is heterogeneity in pricing behaviour. However, most pricing models assume that all firms are identical. If differences in pricing behaviour exist but are not taken into account this will lead to misspecificied models. Table 4 Austria 37.5 15.5 72.3 8.4 7.1 Belgium 31.5 19.1 81.6 5.9 3 Denmark 5 Heterogeneity in pricing behaviour Monthly frequency of price changes (%). Non energy industrial goodsEnergy 1. Consumer prices Unprocessed food Processed food Services 7.5 17.6 94.6 8.3 7.3 Euro area 28.3 13.7 78 9.2 5.6 Finland 52.7 12.8 89.3 18.1 11.6 France 24.7 20.3 76.9 18 7.4 Germany 25.2 8.9 91.4 5.4 4.3 Italy 19.3 9.4 61.6 5.8 4.6 Japan 71.8 30.8 50.9 22.7 3.9 Luxembourg 54.6 10.5 73.9 14.5 4.8 Mexico 26.4 12.5 54.9 18.7 6.1 Netherlands 30.8 17.3 72.6 14.2 7.9 Portugal 55.3 24.5 15.9 14.3 13.6 Spain 50.9 17.7 n.a. 6.1 4.6 United States 47.7 27.1 74.1 22.4 15 2. Producer prices Food Durable products Energy Non-durable nonfood Intermediate products Capital goods Belgium 20 14 50 11 28 13 Euro area 27 10 72 11 22 9 France 32 13 66 10 23 12 Germany 26 10 94 14 23 10 Italy 27 7 n.a 10 18 5 Portugal 21 18 66 5 12 n.a Spain 24 10 38 10 28 8 Source: Consumer prices: For euro area countries and United States, Dhyne et al. (2006); for Denmark, Hansen and Hansen (2006); for Japan, Saita et al. (2006) and for Mexico, Gagnon (2006). Figures for Mexico refer to the period 2003-2004. For producer prices, Vermeulen et al. (2007) The recent micro evidence consistently finds that price adjustment is heterogeneous across firms. Indeed, as can be seen in table 4, sectors in which companies change prices frequently coexist with others in which firms frequently keep prices unchanged for relatively long periods. 12 Note that if 0)()( 21 == jhjh for some j, then 0)( = jh 16 Some interesting findings arise. Specifically, CPI price adjustments are particularly frequent for energy and unprocessed food products, whereas services prices tend to remain constant for long periods. In turn, processed food products and non-energy industrial goods tend to occupy an termediate ranking. Survey data also show that prices of food and energy are changed more other goods or services (see Álvarez and Hernando (2007a) for Spain). ithin sector heterogeneity is still highly relevant, as can be seen in figure 4. The left panel the Netherlands or Veronese et al. (2005) for Italy) also points out the impact of the type of outlet on the frequency of price adju Indeed, the frequency of price changes is significantly higher in supermarkets and hypermarkets than in traditional shops, suggesting that the structure of the retail sector plays a role in explaining differences in the degree of price adjustment. Analysis of producer prices also finds that energy and food products are also characterised by more frequent price adjustment, whereas capital goods and durables are the stickiest components. Figure 4 Heterogeneity in price adjustment is certainly a feature that needs to be considered in pricing models. Indeed, of the 25 models considered in Table 1 only 5 allow for heterogeneity of price changes and Galí and Gertler (1999) and Aoki (2001) only allow for a limited degree of heterogeneity, since a fraction of price setters adjust prices continuously, whereas the rest do it with an unrestricted and constant frequency. In turn, 3 models allow for quite general heterogeneity: Taylor (1993), Álvarez et al. (2005) and Carvalho (2006). Interestingly, heterogeneity is found to be related to differences in industry characteristics13 uch as costs and market competition. For instance, the frequency of consumer price change in frequently than for W presents the histogram of price durations in the Nakamura and Steinsson (2007) dataset of United States consumer price data. The right panel presents the histogram of price durations based on survey data of NACE 2 euro area industries used in Álvarez and Hernando (2007b). Available evidence (e.g. Jonker et al. (2004) for stment. s depends on the variability of input prices (Hoffmann and Kurz-Kim (2006)) and differences in the cost structure help explain differences in the degree of producer price flexibility (Álvarez et al. (2008) and Cornille and Dossche (2006)), a result also found with survey data (Álvarez and 13 models, such as Dan Bonomo and Carvalho (2004), predict a positive tween the frequency d the vari ce of idiosyncratic shocks. Some theoretical ziger (1999) and relationship be of price change an an 0.1 .2 .3 Fraction 012 24 36 48 60 Price duration (months) United States. CPI Distribution of price durations 0 5 10 15 Percent 0 5 10 15 20 25 Price duration (months) Survey data. NACE2 industries. Euro area countries Distribution of price durations 17 Hernando (2007a, 2007b)). Specifically, the share of labour costs in variable costs negatively affects the frequency of price change -given that wages do not change often-, whereas the share of costs of intermediate goods in variable costs has a positive impact. Regarding market competition, survey evidence shows that higher competition leads to more frequent price changes (Álvarez and Hernando (2007a, 2007b)), a result also found with consumer prices (Lünnemann and Mathä (2005)). 7. Time dependent behaviour Some estimators have been suggested in the literature to measure the relative importance of time-dependent price setters. The one most commonly used was introduced by Klenow and Kryvtsov (2005). Their measure14 is given by )( )( 2 t t KK Var SVarfr π α =, where fr and t fr refer to the mean frequency of price change and frequency at time t, respectively, and )( t SVar and )( t Var π refer to the variance of the size of price change and inflation, respectively tsov (2005) define the numerator of the above expression as the time d . Klenow and Kryv ependent component of the inflation variance, because that would be the value of )( t Var π if the ressed by Dias et al. (2007), it is important to )() frequency of price adjustment were constant. As st notice that the type of staggering that implies , for which . Models with continuous price adjustment and time dependent models of adjustment, so (2 tt SVarfrVar = π is uniform staggering frfrt= predict a constant frequency 1 = KK α . As an alternati easure of time dependent behaviour, Dias et al. (2006) show that the complement of the Fisher and Konieczny (2000) index15 (FK) can be seen as an estimator of the share of firms with uniformly staggered pricing behaviour. Table 5 presents the results of these measures. I neral, both measures point to the relevance uantitative studies also find some specific elements of state dependence. For instance, ve m n ge of time dependent behaviour for countries with low and moderate inflation and are in line with the stability over time of the frequency of price change reported in the different micro studies. Interestingly, the Klenow and Kryvstov measure points to a very low share of time dependent price setters for Sierra Leone and Mexico, which is to be expected given the high inflation rates in those countries in the period under analysis. Q inflation is associated with higher frequencies of price increases and lower frequencies of price decreases (see e.g. Veronese et al. (2005) for Italian CPI or Stahl (2006a) for German PPI 14 If 0)(,( ≠ − ttt SfrfrSfrCov the Klenow and Kryvtsov (2005) measure may not be in the [0,1] interval. In practice, this term is typically small. See Dias et al. (2007) for a detailed discussion )1( frfr FK t − = )( frVar 15 18 evidence), although the magnitude of the effects is moderate. Indirect tax changes are also found to have an impact on the frequency of price adjustment (see e.g. Aucremanne and Dhyne (2004) for Belgian CPI or Álvarez et al. (2008) for Spanish PPI), although the share of firms that adjust prices following an indirect tax rate change is relatively small. Table 5 Consumer prices Country Dias et al measure Paper Klenow Kryvstov measure Paper Klenow Kryvstov measure 83 Gautier (2006) 92.2 (97.9) Italy 76 Portugal Austria 79 Belgium 82 Cornille and Dossche (2006) 86 (36) Finland 64 Kurri (2007) 98 France 81 Baudry et al. (2007) Germany 87 Luxembourg 52 Netherlands 73 83 Dias et al. (2006) 74 (69) Dias et al. (2006) 92 Spain 85 Euro area 82 United States Klenov and Kryvtsov ( 2005 ) 97(91) Mexico Gagnon (2006) 34.6 (82.7) Sierra Leone Kovanen (2006) 3.1 Klenow-Kryvs s: For Portuguese CPI, figures refer to 1993-1997 and those in brackets to 1998-2000. For French PPI, figures in brack or seasonality, VAT rate changes and euro cash-changeover. For Belgian PPI, figures exclude the months of January and December, whereas those in brackets do not. For Mexi ures refer to the high inflation 1995-1999 period, whereas those in brackets refer to the low inflation 1999-2002 period. Fo I, figures in brackets refer to regular prices including substitutions Producer prices e dependent behaviour. Quantitative micro data tov measure ets control f can CPI fig r US CP Notes: Dias et al.(2005) measures computed as the complement of the median synchronisation ratio presented in Dhyne et al. (2005). Importance of tim Survey data provide an alternative way of determining the relevance of time dependent ehaviour (table 6). Firms have been asked for the strategy they follow when reviewing their he evidence on country studies summarised in Fabiani et al. (2006) generally shows that the dependent pricing strategies and industry characteristics, table 7 presents the results of a mial logit model with Spanish survey n processes, reflecting a higher stability of marginal costs in those industries. Second: the higher competition the lower is the fraction of firms using purely timeb prices. In the typical survey, they were offered the following options: “At specific time intervals”, which can be interpreted as evidence of time dependence, “In response to specific events”, which is in line with state dependent models, and “Mainly at specific time intervals, but also in response to specific events”, which reflects a mixed strategy. In general, results show the coexistence of time and state dependent elements in pricing behaviour at the individual level. T share of firms following mainly time-dependent rules is generally higher for other services than in trade, which, in turn, is higher than in manufacturing. Larger companies also tend to use time dependent rules slightly more often. To shed more light on the relationship between use of time multino data. The following results are worth highlighting: First: time dependent rules tend to be used more the higher is the labour intensity of productio is the degree of perceived 19 dependent rules. This result is consistent with the idea that prices of firms operating in more competitive markets are more likely to react to changes in their environment. Third, small sized firms tend to rely less on time dependent pricing strategies. Table 6 Country Paper Time-dependent Time and state dependent Austria Kwapil et al. (2005) 41 32 Belgium Aucremanne and Collin (2005) 26 40 Canada Amirault et al. (2006) 66 - Estonia Dabušinskas and Randveer (2006) 27 50 Euro area Fabiani et al. (2006) 34 46 France Loupias and Ricart (2004) 39 55 Germany Stahl (2005) 26 55 Italy Fabiani et al. (2007) 40 46 Luxembourg Lünnemann and Mathä (2006) 18 32 Netherlands Hoeberichts and Stokman (2006) 36 18 Portugal Martins ( 2005 ) 35 19 Spain Álvarez and Hernando (2007a) 33 28 United Kingdom Hall et al. (2000) 79 10 United States Blinder et al. (1998) 60 10 Importance of time dependent behaviour. Survey data Share of firms (%) For US: time and state dependent considers periodic price reviews for some products but not for others. For France, the figure corresponds to the one reported in Fabiani et al. (2006) Overall, there seems to be a need to develop more realistic theoretical state dependent models, though, since implications of the most widespread models are at odds with micro data. For stance, menu cost models, assume that firms evaluate their pricing policy every period and set is odel also predicts that firms must change prices continuously. in a new price if they find it convenient. However, in practice, firms do not continuously evaluate their pricing plans. Fabiani et al. (2006) and Lünnemann and Mathä (2007) show that firms review prices infrequently. Indeed, for the euro area as a whole, Fabiani et al. (2006) find that 57% of firms review prices not more than three times a year and only 12% review more than once a month. The modal firm reviews prices once a year, a result also found for non euro area countries (Lünnemann and Mathä (2007)). These results are in line with the predictions of Reis (2006) inattentiveness model, which rationalises infrequent price reviewing. Unfortunately, th m An additional problem for menu costs models is that they are typically among the least recognised theories by firms16, despite their prevalence in theoretical research. Fabiani et al. (2006) report that menu costs rank eight out of ten theories for the euro area and similar results are reported by Lünnemann and Mathä (2007) for other countries. Theories in which information is costly are even ranked lower [Fabiani et al. (2006)] 16 Only 10.9% and 7.6% of Spanish firms state that menu costs and information costs, respectively, are important or very important reasons for deferring price changes. The corresponding figures for implict contracts, coordination failure and explicit contracts are 57.8%, 43.1% and 39.2%, respectively [Álvarez and Hernando (2007a)]. 20 Table 7 Multinomial logit regression. Pr Vatiable Coefficient Standard error z Coefficient Standard error z Labour 3.15 0.55 5.7 2.16 0.58 3.7 Competition -0.12 0.06 -2.1 0.06 0.06 1.1 Demand conditions 0.04 0.03 1.3 0.07 0.03 2.1 Small sized firm -0.48 0.12 -3.9 -0.68 0.13 -5.4 Food -0.61 0.41 -1.5 0.41 0.47 0.9 Consumer non food -0.29 0.38 -0.8 0.54 0.45 1.2 Intermediate -1.52 0.37 -4.1 -0.37 0.44 -0.8 Capital goods -1.34 0.38 -3.5 -0.11 0.45 -0.3 Food trade -0.18 0.41 -0.4 0.23 0.48 0.5 Energy trade 0.05 0.75 0.1 0.23 0.91 0.3 Other trade 0.02 0.37 0.1 0.63 0.44 1.4 Hotels and travel agents 0.27 0.44 0.6 1.09 0.52 2.1 Bars and restaurants -0.56 0.39 -1.4 0.59 0.46 1.3 Transport -0.07 0.37 -0.2 0.66 0.46 1.5 Communications -0.67 0.47 -1.4 -0.21 0.58 -0.4 Constant -0.24 0.38 -0.6 -1.40 0.45 -3.1 ice review Time dependent Time and state dependent Number of observations 1847 Wald chi2 (30) 213.08 Log likelihood -1881.63 AIC 3768.71 BIC 3945.39 Pseudo R2 0.07 Reference group: State dependent. Reference sector: Energy Robust standard errors 8. Forward looking behaviour Survey evidence allows determining to which extent pricing policies of firms are forward oking, as typically assumed in theoretical models. Table 8 presents evidence on forward anadian firms state that they will raise prices in the face of anticipated costs increases. When asked about the reasons for not lo looking pricing behaviour in the surveys of the United States and Canada. The evidence shows substantial departures from the hypothesis of forward looking price setters. In particular, a significant fraction of firms is not affected by changes in the outlook for the national economy. The impact of future inflation is generally more important, although less so than anticipated firm specific costs17. However, only 45% of US firms and 40% of C 17 In Maćkowiak and Wiederholt (2007) firms pay more attention to idiosyncratic shocks than to aggregate conditions if id iosyncratic shocks are more variable than economy-wide ones. 21 changing prices in this context, firms give especial attention to coordination failure and implici and explicit contract explanations. These are also the theories that tend to receive the b support in surveys carried out in other countries [Fabiani et al. (2006) for euro area countries]. Table 8 t roadest United States 1. Do forecasts about the future outlook for the national economy ever directly affect the prices you set? Never 70.5 Ocassionally 15.0 Often 14.5 2.Do forecasts of future economy-wide inflation rates ever directly affect the prices you set? Never 51.8 Ocassionally 19.9 Often 28.3 3. When you see cost or wage increase coming, do you raise your prices in anticipation? Yes or often 44.4 No or rarely 55.6 4. Why do not firms raise their prices in the face of anticipated cost increases? We worry competing firms won't raise their prices 26.4 It would antagonize or cause difficulty for our customers 25.6 Once costs rise, we can raise our prices promptly 14.9 We lack confidence in our cost forecasts 8.3 Contracts or regulation prohibit anticipatory price hikes 6.6 Other 18.2 Canada Yes 40 Other 60 Forward looking price behaviour. American surveys Source: For the United States, Blinder et al. (1998). Question 3 only asked to firms that do not consider cost totally unimportant. Question 4 only asked to firms that do not rise prices in anticipation of cost increases. For Canada, Amirault et al. (2006) Share of firms (%) If you foresee an increase in your future costs (such as raw materials), do you raise your own prices in anticipation? able 9 presents the evidence of European surveys. Again, the existence of a significant share of on the frequency of price change, so firms which use simple rule sof thumb change prices less T firms deviating from full forward looking behaviour is found. Interestingly, some surveys have asked firms whether they follow some simple rule of thumb when setting prices (for instance, changing prices by a fixed percentage) or whether they consider a wide set of indicators that relate to the current environment (backward looking firms) or include expectations on the future economic environment (forward looking firms). It is found that around one third of firms employ some simple rule of thumb when setting prices, in line with Galí and Gertler (1999), a prediction which is not shared by any other theoretical model. However, Álvarez and Hernando (2007a) find that the fact that a firm applies a rule of thumb has a significative negative impact frequently than the rest. In contrast, in Galí and Gertler model (1999) rule of thumb price setters 22 change prices continuously and the rest of companies adjust prices less often. Probably, a rule of thumb whereby firms change prices once a year, in line with aggregate yearly past inflation, would capture inflation dynamics more realistically and would also capture seasonal behaviour. Table 9 Country Rule of thumb Backward looking Forward looking Belgium 37 29 34 Estonia n.a. 59 41 Luxembourg 32 34 34 Portugal 25 33 42 Spain 33 39 28 Past information Past information and forecasts Forecasts Austria 37 51 12 Past information Contemporary information Expectations Germany 23 55 15 Past information Current and future information Italy 32 68 Note: For Germany, rescaled figures from Stahl (2006b) on firms stating that the corresponding information vintage is very important. Forward looking price behaviour. European surveys Share of firms (%) To analyse the relationship between the information set that a firm uses and industry characteristics, Table 10 presents the results of a multinomial logit model with Spanish survey data. Some interesting results are obtained: First, a higher sectoral labour share is associated with a greater reliance on rule of thumb behaviour, reflecting lower uncertainty in total costs developments. Second, the higher is the degree of market competition, the higher is forward looking behaviour. Third, the more relevant are demand conditions the higher is the use of forward looking strategies. Fourth, small sized firms are more likely to adopt some simple rule of thumb. 9. Imperfect competition One defining characteristic of New Keynesian price setting models is some element of imperfect competition, which provides a price formation story: prices arise from the profitmaximizing decisions of individual firms. Imperfect competition also makes it feasible for some firms not to adjust their price in a given period, in contrast with a perfect competition environment. Table 10 23 Coefficient Standard error z Coefficient Standard error z Labour -2.16 0.54 -4.0 -1.92 0.61 -3.1 Competition 0.21 0.05 4.0 0.21 0.06 3.4 Demand conditions 0.09 0.03 3.0 0.13 0.04 3.6 Small sized firm -0.22 0.12 -1.9 -1.13 0.14 -8.0 Food 0.44 0.40 1.1 0.32 0.43 0.7 Consumer non food 0.35 0.38 0.9 0.30 0.39 0.8 Intermediate 0.65 0.37 1.8 0.51 0.37 1.4 Capital goods 0.26 0.38 0.7 0.04 0.38 0.1 Food trade 0.07 0.41 0.2 -1.05 0.46 -2.3 Energy trade 1.25 0.93 1.3 0.95 1.08 0.9 Other trade 0.14 0.37 0.4 -0.32 0.39 -0.8 Hotels and travel agents 0.70 0.41 1.7 0.99 0.42 2.4 Bars and restaurants 0.29 0.38 0.8 -0.47 0.43 -1.1 Transport -0.13 0.37 -0.4 -0.35 0.39 -0.9 Communications -0.50 0.54 -0.9 0.39 0.45 0.9 Constant -0.35 0.38 -0.9 -0.49 0.40 -1.2 Number of observations 1847 Wald chi2 (30) 253.33 Log likelihood -1852.35 AIC 3768.71 Multinomial logit regression. Information set Backward looking Forward looking BIC 3945.39 Pseudo R2 0.07 Reference group: Rule of thumb. Reference sector: Energy Robust standard errors The various surveys address the issue of how firms set prices using slightly different formulations. Nevertheless, the results of the national surveys can be compared by grouping the answers into three alternatives: “markup over costs”, “price set according to competitors’ rices” and “other”. For the euro area as a whole, a significant share of firms (54%) set their arginal costs, suggesting that they enjoy a non negligible degree of arket power. The fraction of companies setting prices according to those of their competitors des strong support for the view that imperfect competition characterizes most product markets. Imperfect competition, though, seems to be of a more complex k implied by the monopolistic competition model, since there is evidence of e.g. price discrimination. Table 11 p prices as a markup over m m is 27%. Finally, around 19% of the companies state that they do not have autonomous price setting policies. For these firms, the final decision on the price charged is taken by a different economic agent, and this may be the public sector, the parent company, the main customers or the suppliers. Country results, as reported in table 11, provide a similar picture. Overall, survey evidence provi ind than 24 Hoeberichts, M. and A. Stokman (2006). Pricing behaviour of Dutch companies: results of a survey. European Central Bank Working Paper No. 607. Further information Hoff onsumer price adjustment under the microscope: mann, J. and J. Kurz-Kim (2006). C Germany in a period of low inflation. European Central Bank Working Paper No. 652. Further information in IDEAS/RePEc Jonker, N., Blijenberg H. and C. Folkertsma (2004). Empirical analysis of price setting behaviour in the Netherlands in the period 1998-2003 using micro data. European Central Bank Working Paper No. 413. Further information in IDEAS/RePEc Klen Time-Dependent Pricing: Does it ow, P. and O. Kryvtsov (2005). 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"Aggregate Dynamics and Staggered Contracts," Journal of Political Economy, 88(1):1-23. Further information in Taylor, John B. (1993) Macroeconomic Policy in a World Economy: From Econometric Design to Practical Operation. W.W. Norton. Further information Vermeulen, P., Dias D., Dossche, M., E. Gautier, Hernando I., Sabbatini R., and H. Stahl (2007). Price Setting in the Euro Area: Some Stylised Facts from Individual Producer Price 32 Data and Producer Surveys. European Central Bank Working Paper No 727. Further information Veronese, G., Fabiani, S., Gattulli, A. and Sabbatini, R. (2006). Consumer price setting in Italy, Giornale degli Economisti e Annali di Economia, 65(1). Further information in IDEAS/RePEc Vilm prices change in Finland? Micro-level unen, J. and H. Laakkonen (2005). How often do evidence from the CPI, mimeo, Bank of Finland. Further information Wol es, Marginal Cost, and the Behavior of Inflation. man, Alexander L (1999) Sticky Pric Economic Quarterly. Federal Reserve Bank of Richmond. 85(4):9-48. Further information in IDEAS/RePEc Woodford , Michael (2007) Interpreting Inflation Persistence: Comments on the Conference on “Quantitative Evidence on Price Determination”, Journal of Money, Credit and Banking, 39(1):203-210. Further information in IDEAS/RePEc 33 App justment [1] Stylised facts endix 1a Continous price ad Nakamura and Steinsson (2007) Sheshinski and Weiss (1977), Taylor (1980), Bonomo and Carvalho (2004) Calvo (1983), Danziger (1999) and Rotemberg (2005) [2] Dotsey et al. (1999) 0.0 0.1 0 122436 months since the last price change Pr 0. 0. 0. 0. 0. obability of a price change 6 7 8 0.9 1.0 2 3 0.4 0.5 0.0 0.1 0122 Pr 0.2 0.3 0.4 0.5 obability of a pr 0.6 0.7 436 months since the last price change ice change 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0. 0. 1. 0 122436 months since the last price change Probability of a price change 8 9 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0122436 months since the last price change Probability of a price change 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 122436 months since the last price change Probability of a price change 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0122436 months since the last price change Probability of a price change Notes: [1] Lucas (1972), Fischer (1977), Mankiw and Reis (2002), Carvalho (2005), Reis (2006), Maćkowiack and Wiederholt (2007), Christiano et al. (2005), Smets and Wouters (2003), Rotemberg (1982) and Kozicki and Tinsley (2002) [2] Rotemberg (2005): particular case 34 Appendix 1b Gali and Gertler (1999) and Aoki (2001) Sheedy (2005) Taylor (1993) [3] Wolman (1999) Álvarez et al. (2005) Carvalho (2006) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 122436 months since the last price change Probability of a price change 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 01224 months since the last price change Probability of a price change 36 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 122436 months since the last price change Probability of a price change 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 01224 months since the last price change Probability of a price change 36 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 122436 months since the last price change Probability of a price change 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 01224 36 Probability of a price change months since the last price change [3] Assuming that the distribution of contracts is uniform over [1, 40] 35 Appendix 2 Variable Source Comment Labour Industrial, Trade and Services surveys. Instituto Nacional de Estadíitica Labor costs as a percentage of labour and intermediate inputs costs. NACE 3 digit level Compettition Álvarez and Hernando (2007a) Importance of competitors' prices to explain price decreases. Demand conditions Álvarez and Hernando (2007a) Importance attached by firms to demand conditions in explaining price changes. Small sized firm Álvarez and Hernando (2007a) Employment of firms with less than 50 employees. Data definitions for variables used in multinomial logit models 36