Menu Costs, Calvo Fairy, Inflation and Micro Facts Etienne Gagnon∗ Federal Reserve Board First version: April 2007 This version: September 2008

Abstract This paper investigates whether extensions of the Calvo and menu-cost models that include idiosyncratic technology shocks are consistent with key features of individual consumer price adjustment. The comparison of the models focuses on three facts pertaining to the impact of inflation on the setting of consumer prices. First, the average frequency of consumer price changes initially rises slowly with the level of inflation, becoming more responsive when annual inflation gets beyond 10-15 percent. Second, the distribution of nonzero price changes contains a large number of both small and large price increases and decreases at low, medium, and high levels of inflation. Third, the average magnitude of price increases and decreases varies little with the duration of price spells. The menu-cost model is consistent with the first and third facts, but the Calvo model, while inconsistent with these two facts, provides a much better fit of the distribution of price changes. JEL classification: E31, D40. Keywords: Calvo model, menu costs, price setting, consumer prices.

∗

E-mail: [email protected]. I thank Ben Eden, Oleksiy Kryvtsov, and workshop participants at the Federal Reserve Board and the Macroeconomics of Price Setting conference at the Rimini Center for Economic Analysis for their insightful comments. Matthew Denes provided excellent research assistance. The views expressed in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System, or any other person associated with the Federal Reserve System. All errors and omissions are mine.

1

The New Keynesian framework has become the main workhorse for the analysis and design of monetary policy, and plays an increasingly central role in the study of business cycles. At the heart of this class of models is the assumption that nominal rigidities — most notably price stickiness — are preventing resources from being allocated efficiently. In recent years, large panels of individual prices, in particular those assembled for the purpose of computing price indices, have been made available to researchers worldwide. The ensuing stream of empirical research has significantly broadened knowledge about the prevalence of price stickiness, and the characteristics of individual price changes.1 While there is a growing consensus about the basic empirical facts, the debate is far from settled on how to best embed price stickiness inside macroeconomic models, and in particular on whether one can construct a price-setting model consistent with the microevidence that has plausible macroeconomic implications. There are currently several competing models emphasizing different mechanisms by which prices fail to adjust, such as menu costs, incomplete or costly-to-process information, search frictions, consumer anger at price changes, market share concerns, uncertain demand, contracts, or random opportunities to optimize prices. The choice of a particular model is important, however, as they often differ in terms of their predictions about key elements, such as the output-inflation trade-off, the degree of exchange rate pass-through, or the effectiveness of monetary policy. In this paper, I assess the ability of two popular models of price rigidity — the Calvo and menu-cost models — to match key features of individual consumer prices behaviors. The features considered pertain to the impact of inflation on individual price-setting decisions, an aspect that has received relatively little attention in the literature due to the lack of micro data with extensive inflation coverage. In the model developed by Calvo (1983), firms face a constant probability of optimizing their price. This model is a popular example of a time-dependent model in which the timing of price changes is exogenous to the firm and constant over time.2 The Calvo model enjoys much popularity in the macroeconomic literature thanks to its tractability, and its ability to generate impulse responses that are consistent with VAR literature on monetary and technology shocks.3 In this model, firms must incur a fixed cost whenever they adjust their 1

For the United States, see Bils and Klenow (2004), Klenow and Kryvtsov (2008), and Nakamura and Steinsson (2007). For the euro area, see Dhyne et al. (2005), and the references provided therein. Other studies using CPI micro data include Eden (2001) for Israel, and Gagnon (2006) for Mexico. 2 The exogeneous arrival of adjustment opportunities is sometimes describe in terms of random visits of the “Calvo fairy.” 3 Recent applications of the Calvo price-setting mechanism include, among many others, Christiano,

2

nominal price. The menu-cost model is part of the state-dependent class of models because the timing of price changes is chosen by firms themselves. My paper develops a common economic environment to more directly compare both price-setting mechanisms. An central feature of this environment is the presence of idiosyncratic productivity shocks, giving rise to a distribution of lumpy price adjustments that includes be both positive and negative changes. The models are judged on their ability to match three key features of the data. These features are documented using item-level Mexican CPI data covering the January 1994 to June 2002 period. The sample begins one year before the Tequila crisis, a period of low and stable inflation, and ends a few years after the burst of inflation brought by the crisis had subsided. Details of the assemblage of the data set are provided in Gagnon (2006). I first compare the ability of the models at matching the level of the average frequency and magnitude of price changes at various levels of inflation. When inflation is below roughly 10-15 percent, Gagnon (2006) found that the monthly frequency of price changes is weakly correlated with inflation in the Mexican CPI data. By contrast, the average magnitude of price changes correlates strongly with inflation. This finding is consistent with evidence from Klenow and Kryvtsov (2008), who show that most of the variation in U.S. inflation over the past two decades, a period of low and stable inflation, can be traced back to variation in the average size of price changes. When inflation is greater than 10-15 percent, movements in both the average frequency and magnitude of price changes become important determinants of inflation. The menu-cost model considered in this paper does a surprisingly good job in this respect, producing an initially slow rise in the frequency of price changes that accelerates as higher levels of inflation are considered. The standard Calvo model is by construction incapable of generating a rise in the frequency of price changes. Movements in the frequency of price increases perfectly offset that of price decreases, leaving the overall frequency of price changes unaffected. To give the Calvo model better chances of matching the facts, I also explore an alternative specification in which the exogenous probability of changing prices is a function of the level of inflation. This variant of the model provides a fit similar to the menu cost-model, although its success hinges on strong assumptions. I next assess the models’ ability to match the distribution of nonzero price changes at low, medium and high levels of inflation. I first show that even when annual inflation is as Eichenbaum, and Evans (2005), Smets and Wouters (2005), and Woodford (2003).

3

high as 50 percent, small price changes represent a large share of price changes. As noted by several authors in low inflation environments (e.g. Kashyap (1995)), the standard menucost model has difficulties generating small price changes; firms wait until the benefit of changing their price is at least as large as the menu cost, which rules out small price changes unless menu costs are small. This shortcoming of the menu-cost model thus extends to high-inflation environments as well. The Calvo model, on the other hand, generates a large number of small price changes because the mass of firms that have not updated their price is an exponentially-decreasing function of duration. A large fraction of firms selected to change their price have therefore short price spells, which tends to be associated with small price changes. Moreover, the presence of a small number of firms that not been visited by the Calvo fairy in a large number of periods helps this model to generate fat tails, which is also consistent with the data. The last feature of the data that I document is the relationship between the duration of price spells and the magnitude of price changes. In the steady state of the Calvo model, firms increase their price on average by the amount of cumulated inflation since their last nominal adjustment. Positive price changes are increased on average by even more. In the Mexican data, the average magnitude of price changes, increases and decreases are rather insensitive to price spell duration. This finding complements the investigation conducted by Eden (2001) using Israeli data, and Klenow and Kryvtsov (2008) using US data. The Calvo model is inconsistent with this finding as it predicts a rise in the size of price changes and increases over time, and a decline in the size of price decreases. In the menu-cost model, on the other hand, the average magnitude of price increases and decreases is fairly insensitive to duration. This happens because the magnitude of price increases and decreases are tight to the width of the Ss band, which is not related to duration per se. The paper is structured as follows. In Section 1, I present the models, and in particular the price-setting mechanisms used by firms. A formal discussion of how the models are solved is relegated to the appendix. In Section 2, I provide a brief description of the Mexican data used in this project. Section 3 contains the main results of the paper. It details the ability of each model to match the three empirical facts. The last section discusses the modelling features necessary to match all three facts, and offers some concluding remarks.

4

1

Economic Environment

The economy consists of three types of agents. An infinitely-lived representative household supplies labor and consumes a basket of differentiated consumption items. These items are produced by a continuum of monopolistically competitive firms renting a labor input in a perfectly competitive market, and subject to idiosyncratic menu cost and technology shocks. Finally, there is a monetary authority whose only role is to expand the money supply at a constant rate g. Calibration issues aside, the economic environment in the Calvo and menu-cost specifications considered in this paper differ only in terms of the distribution of menu costs. The analysis focuses on the predictions of these models in stationary environments, leaving the comparison of these models’ transitional dynamics to future work.

1.1

Households

The problem of the representative household is to choose a sequence for consumption, {Ct }, and hours worked, {Nt }, in order to maximize present discounted utility, max

{Ct ,Nt }

∞ X

β t U (Ct , Nt ) ,

t=0

subject to a budget constraint, Pt Ct = Wt Nt + Pt Πt , and a simple money demand, Pt Ct = Mt . The budget constraint states that consumption spending equals the sum of a household’s labor income and profits received from firms. The variable Pt denotes the price index at time t, while Wt is the nominal wage rate. Real profits, Πt , are remitted every period by firms. Following Golosov and Lucas (2007), I assume that utility is separable, logarithmic in consumption and linear in labor: U (Ct , Nt ) = log Ct − ψNt .

5

Under the above assumptions, the intratemporal Euler equation implies that the wage rate is proportional to the stock of money. Wt Wt = . Pt C Mt

ψ=

Consumption is a composite of differentiated items aggregated using a CES function C=

µZ

(cj,t )

μ−1 μ

dj

¶

μ μ−1

.

Consequently, the demand for individual items must satisfy cj,t =

µ

pj,t Pt

¶−μ

C,

and the model-implied CPI is given by Pt =

µZ

1−μ

(pj,t )

dj

¶

1 1−μ

.

Note that the household’s problem is fully static, so that all dynamics in the model will originate from the firms problem. In the stationary equilibrium considered, all aggregate real variables are constant, while aggregate nominal variables grow at the same rate as the money stock.

1.2

Intermediate Firms

There is a continuum of measure one of monopolistically competitive firms. At the beginning of the period, each firm draws an idiosyncratic productivity shock, εj,t , and a menu cost, ξ j,t , expressed in units of labor. Firms then decide whether to retain their previous period nominal price, or to incur the menu cost and select a new nominal price. Once a price is set for the period, firms must fully satisfy demand at that price. The objective of the firms is to maximize their present discounted sum of profits. For simplicity, I assume that the production function of the j − th firm is linear in labor, yj,t = cj,t = φj,t nj,t .

6

where φj,t is the level of technology. Because there is no capital accumulation in the model, the only dynamic linkage is the presence of nominal price rigidities. I assume that a firm’s labor productivity evolves according to ¯ + ρ log φ log φj,t = (1 − ρ) log φ j,t−1 + εj,t , ¡ ¢ where idiosyncratic productivity shocks are drawn from the normal distribution N 0, σ 2ε . ¡ ¢ ¯ be the log-deviation from the nonstochastic mean of the distribuˆ = log φ /φ Let φ j,t j,t ´ ³ ˆ j,t−1 is N ρφ ˆ j,t−1 , σ 2 . The stationary ¯ The distribution of φ ˆ j,t conditional on φ tion φ. ε

distribution of labor productivities, expressed in log-deviations, is ˆ j,t ∼ N φ

µ 0,

σ 2ε 1 − ρ2

¶

.

Absent menu costs, firms would simply set their price equal to a markup over their marginal cost: μ ψ . pi,t = μ − 1 φi,t In this particular case, pi,t is lognormally distributed across firms, resulting in a price index equal to ³ ¢´ μ ψ 2 2 ¡ 2 . σ / 1 − ρ exp (1 − μ) Pf = e ¯ μ−1φ

It will prove convenient to define pˆj,t = log (pj,t /Pf ) as the log deviation of prices from this fully-flexible price index in solving the menu-cost version of the model. 1.2.1

(Truncated) Calvo model

The Calvo and menu-cost specifications of the model differ in terms of the distribution of menu costs, but are otherwise identical. In the baseline Calvo model, firms face a constant probability of optimizing their price every period, 1−θ. The Calvo model is equivalent to a menu-cost model in which menu costs are iid over time, taking the value 0 with probability θ, and infinity with probability 1 − θ. = ξ Calvo j,t

(

0 with probability (1 − θ) +∞ with probability θ

Depending on parameter values, a solution to the baseline Calvo model might not 7

exist. Under positive inflation, an ever-decreasing subset of firms whose price has not been optimized in a long time may monopolize an increasingly large share of demand, making arbitrary large negative profits along the way. This problem is most likely to arise if consumer demand is very elastic, and firms face a low probability of reoptimizing their price. As a solution around this problem, I consider a truncated version of the model.4 For the first T − 1 periods following its last nominal price change, a firm faces a constant probability 1 − θ of freely optimizing its price. If the firm has not changed its nominal price after T − 1, then it automatically gets to reoptimize is price in the T − th period. t , and Let M Cj,t be the real marginal cost of firm j at period t, that is M Cj,t = φ1 W j,t Pt vt+l = Uc (ct+l , nt+l ) /Uc (ct , nt ) be the ratio of marginal utilities of consumption at period t + l relative to period t. The problem of a price-optimizing firm can be expressed as max ∗ pj,t

Et

"T −1 X l=0

l

(βθ) vt+l

µ

# ¶ p∗j,t ∗ − M Cj,t+l Yj,t+l . Pt+l

In a stationary equilibrium with constant aggregate quantities, the solution to this problem simplifies to hP i T −1 μ )l M C ∗ E (βθπ t j,t+l pj,t l=0 μ = . (1) PT −1 l μ−1 ) Pt μ−1 l=0 (βθπ When setting their price relative to the distribution, firms take into account the expected path of their marginal cost over the next T − 1 periods. Note that their effective discount factor, βθπ μ , can exceed one in this environment if inflation and the elasticity of substitution across items are sufficiently high. This does not pose a problem in my environment because the truncation ensures that the expectation is always finite. In the Calvo model described so far, the frequency of price changes, θ, is a treated as an exogenous parameter. As discussed in Gagnon (2006), the frequency is influenced by factors such as the level of inflation, or changes to value added taxes in the Mexican data. To offer the Calvo model a better chance of fitting the empirical evidence, I consider a simple extension in which θ is a function of the steady-state level of inflation. This variant of the model continues to be characterized by the complete randomness of price adjustment 4 One mechanical solution is to linearize the equilibrium conditions around a zero-inflation steady state, but this approach suffers from several drawbacks for inference away from the steady state (see Levin and Yun (2007) for a discussion). An alternative approach pionneered by Yun (1996) and popular in the business cycle literature is to assume that nominal prices are automatically indexed to past or steady-state inflation. This method counterfactually implies that all nominal prices change every period.

8

opportunities. 1.2.2

Menu-Cost Model

I consider a simple menu-cost model in which the size of the menu cost is constant over time and across firms, that is ξ j,t = ξ. The overall economic environment resembles that of Danziger (1999), who characterized a general equilibrium menu-cost model with idiosyncratic technology shocks, but with a specification of preferences and menu costs similar to Golosov and Lucas (2007). As in the Calvo model, firms maximize the present discounted stream of real profits. It is convenient to express their ³problem ´ recursively in ˆ pˆ be the Bellman order to solve it using dynamic programming techniques. Let V φ; equation of an optimally behaving firm prior to deciding whether or not to change its nominal price. The state of the firm comprises its relative price (relative to Pf , for ease of ˆ The value function is given by computation), pˆ, and its technological deviation, φ. ³ ´ ˆ pˆ = V φ;

max

{no change, change}

n ³ ´ ³ ´o ˆ pˆ , Vc φ ˆ , Vnc φ;

(2)

³ ´ ˆ pˆ is the value function associated with the firm’s decision to keep its previous where Vnc φ; ³ ´ ˆ is the corresponding nominal price and behave optimally in subsequent periods, and Vc φ value function of a firm choosing to optimize its nominal price in the current period. These functions are expressed as Z ³ ´ ³ ´ ³ 0 ´ ³ 0 ´ ˆ ˆ ˆ ; pˆ − g dG φ ˆ |φ ˆ . (3) Vnc φ; pˆ = π φ; pˆ + β V φ and

Z ´ ³ 0 ³ ´ ³ 0 ´ ∗ ˆ ; pˆ∗ − g dG φ ˆ ˆ |φ ˆ − ξW , (4) π φ; pˆ + β V φ pˆ P ³ ´ ˆ pˆ , is the period gross real profits. respectively. The first right hand-side term, π φ; Taking into µaccount the production¶function and the demand curve, it is expressed as ³ ´ ³ 0 ´ −μ ˆ pˆ = Pf exp (ˆ ˆ |φ ˆ gives the π φ; p) − ¯ 1 ˆ W exp (ˆ p )) C. The measure G φ (P f φ exp(φ) P transition probabilities associated with the AR (1) specification of technology deviations adopted earlier. The integrals in equations (3) and (4) give the expected value function in the next period, taking into account price erosion due to inflation, and the distribution of technology shocks. Additional restrictions must be imposed to ensure the existence of ³ ´ ˆ = max Vc φ ∗

9

a unique solution to the firm’s problem. A discussion of these restrictions, as well as a presentation of the algorithm used to solve both the Calvo and menu-cost models, are relegated to the Appendix.

1.3

Calibration

Some parameters of the models are taken directly from the literature while others are chosen to match particular moments of the distribution of price changes under low inflation. In particular, annual inflation is set to 4.5 percent, its average over the last two years of the sample. The elasticity of substitution across items in the consumption basket is set to 7, the same value considered by Golosov and Lucas (2007). The impact on the results of using alternative elasticity parameters is discussed in Section 3.3. The discount factor selected is (1.05)−1/12 , while the disutility of labor, ψ, is chosen so that households work exactly 25 percent of their time absent nominal frictions (i.e. when menu costs are set to zero at all periods). The objects left to calibrate are the variance of technological innovations, σ 2ε , the persistence of technology deviations, ρ, the distribution of menu costs, and the truncation horizon in the Calvo model. For the menu-cost model, the persistence of technology shocks is set to 0.75, a value similar to that implied by Golosov and Lucas’ quarterly calibration (.551/3 ≈ 0.82 per month) but higher than Midrigan’s (0.5 per month). The remaining parameters, ξ and σ 2ε , are then chosen to match the average frequency and magnitude of price changes in the last two years of data. For the Calvo model with constant hazard, the frequency of price changes, 1 − θ, is chosen to match exactly the average frequency over the last two years of data and the truncation horizon is set to twenty-four months. The remaining parameters, σ 2ε and ρ, are chosen to minimize the distance between the corresponding empirical distribution of price changes and the one generated by the model.5 For the variant of the model in which the Calvo parameter is allowed to depend on inflation, I first compute, for each calendar year in my sample, the average monthly inflation rate and frequency of price change. In then regressed the resulting average frequencies on a constant, the average inflation rate, and its square. I finally set the Calvo parameter equal to its projection at each level of steady state inflation considered. 5 I experimented with a range of persistence parameters for both the menu-cost and Calvo specifications. The results are not very sensistive to the particular value selected. Low values of ρ tend  to be associated with high values of σ2ε , leaving the variance of technological deviations, σ2ε / 1 − ρ2 , little changed. It might be useful to consider moments conditional on the duration of nominal prices, rather than simply fitting the distribution of price changes, in order to obtain greater identifying power. I hope to explore this issue in a later version of the paper.

10

2

Mexican Micro Data

The data comprise 3.8 million monthly price quotes collected by Banco de México for computing the Mexican consumer price index. The period covered is January 1994 to June 2002, for an average of about 37,000 quotes per month. The data set was constructed by merging information released in the Mexican government’s official gazette (the Diario Oficial de la Federación) by the central bank. Price quotes correspond to narrowly defined items sold in specific outlets (e.g., corn flour, brand Maseca, bag of 1 kg, sold in outlet number 1100 in Mexico City). In the particular case of clothing-related items, the reported price is an average of a sample of three items pertaining to the same product category and outlet. The data set excludes all items whose price is regulated, such as gasoline and taxi fares. It also leaves aside product categories that were introduced into or disappeared from the CPI basket over the sample period. The sample contains price quotes from 266 product categories representing 63 percent of Mexican consumer expenditures, and collected in forty-six agglomerations. For more information on the data base construction, product coverage, and potential data issues, see Gagnon (2006). The statistics computed in this paper take into account the relative importance each observation in the consumption basket. Following Klenow and Kryvtsov (2008), it is convenient to decompose inflation as π t = f rt · dpt , where frt is the average frequency of price changes, and dpt is the average magnitude of nonzero price changes. If the frequency of price changes is, for example, 25 percent, then the price of items representing a quarter of expenditures in the consumption basket changed during period t. More formally, let fri,t be the fraction of items in product category i for which a nominal price change is observed at period t, and ∆pit be their corresponding P average size. The aggregate frequency of price change, f rt , is then given by i ω i f ri,t , where ω i is the product category weight in ³the sample. The´aggregate average magnitude X of nonzero price changes, dpt , is given by ω i f ri,t ∆pit /f rt . The product category i weights are based on a survey of consumer expenditures.

3

Results

The sample’s large inflation and product coverage offers a unique opportunity to compare the models’ predictions to the micro data, especially as they pertain to the impacts of 11

inflation on consumer price-setting decisions. In this section, I focus my attention on three key empirical facts that offer some potential to discriminate between the models. The first is how the average frequency and magnitude of price changes relates to the levels of inflation. The second is how inflation affects the cross-sectional distribution of (nonzero) price changes. The last feature is the relationship between the duration of price spells and the magnitude of price changes. As we are about to see, each of these three facts will highlight sharply contrasting predictions of the Calvo and menu-cost models.

3.1

Average Frequency and Magnitude of Price Changes

Figure 1 displays time series of the average frequency and magnitude of price changes of nonregulated goods and services over the sample period, along with the inflation rate. As shown in the upper-left panel, inflation and the frequency of prices changes climbed sharply following the currency devaluation that occurred at the end of December 1994. By April 1995, inflation had reached 75.2 percent (a.r.), while the average frequency of price changes had risen to 58.8 percent, more than double its 1994 average. Interestingly, the frequency of price changes declined little from mid-1996 onward, despite a clear downward trend in the inflation series. As shown in the upper-right panel, this phenomenon can be partly attributed to offsetting movements in the relative occurrence of price increases and decreases, especially after 1999. The relationship between the frequency and inflation is particularly weak for nonregulated goods (not shown), the correlation coefficient at low inflation is not significantly different from zero. A positive relationship is apparent in the case of services, which have a relatively low share of price decreases at low inflation. Contrary to the average frequency, the average magnitude of price changes moved very closely with inflation over the entire sample period, as can be seen in the lower-left panel. The last panel indicates that the average size of price increases and decreases varied little over the sample period, with the exception of price increases around the inflation peak. As discussed in Gagnon (2006), this finding hints that movements in the average magnitude of price changes are primarily driven by changes in the relative occurrence of price increases and decreases, rather than variations in the absolute size of price changes. In order to assess the ability of the models at replicating the above facts, I calibrated them to match key features of the data over the last two years of the sample. The models’ predictions were then recorded for steady state levels of inflation similar to the annual rate of inflation experienced by the Mexican economy. In order to make the predictions of the

12

models and the data more directly comparable, I averaged the statistics over each calendar year in the sample period. The results are presented in Figure 5. By construction, movements in the frequency of price increases are perfectly offset by opposite movements in the frequency of price decreases in the constant-hazard Calvo model, keeping the overall frequency of price changes constant. It is therefore not surprising that this model fails to generate a rise in the frequency of price changes, as seen in the upper-left panel. While this is a relatively mild problem for levels of inflation under 10-15 percent, the model’s fit deteriorates rapidly as inflation gets beyond that range. The model largely underpredicts the rise in the frequency of price increases, which is bounded above by the Calvo parameters. By contrast, the variable-hazard variant of the Calvo model fits the average frequency of price changes much better. This good fit is no surprise, however, given that the Calvo parameter was directly set to match the best quadratic predictor under least-squares. The calibration somewhat underpredicts the initial rise in price increases, but is overall consistent with very mild changes to the frequency of price decreases between the low and high inflation periods. Contrary to the variable-hazard Calvo model, the menu-cost model was calibrated only on the low inflation part of the sample, so that any change in the average frequency constitutes an endogenous response of firms to the economic environment. This model provides a remarkably good fit of the average frequency of price changes, increases, and decreases over the range of inflation considered. The lower-right panel of Figure 2 shows that the initial rise in the frequency of price changes is slowed by opposite movements in the occurrence of price increases and decreases, similar to those observed in the data. As inflation ramps up, price decreases becomes more scarce, and the average frequency of price changes increasingly mimics that of the frequency of price increases. In addition, the menu-cost model offers a much better fit of the average magnitude of price changes, displayed in the lower-right panel, than the constant-hazard Calvo model, shown in the upper-right panel. Under Calvo pricing, the average magnitude of price changes is a linear function of the inflation rate, dpt = π ss /f r. This relationship turns out to be a rather good approximation for as long as inflation is below 10-15 percent. Beyond that level, the approximation deteriorates, as the frequency of price changes plays an increasingly important role.6 Similarly, the model’s average size of price increases 6

My particular calibration of the Calvo model produces price increases and decreases that are smaller on average than those in the data. This happens because the empirical distribution of price changes has fatter tails over the last two years of the sample period than generated by the model.

13

rises with inflation, asymptotically converging π ss /fr. In the data, however, the average magnitude of price increases and decreases is little impacted by the level of inflation. The menu-cost model provides a substantially better fit of this fact because the size of the Ss band does not vary much with the level of inflation in my calibration, resulting in price changes clustering around a positive and negative values similar to the data.

3.2

Distribution of Nonzero Price Changes

In next the models’ fit of the distribution of price changes under low, medium and high inflation. Given that the models are calibrated at steady state, special care was given to choosing subsamples of consecutive months over which inflation was relatively stable. The models’ predictions are plotted in Figure 3, shown as grey bands, along with the empirical distribution, displayed as stars. In the case of the Calvo model, the results are reported for the variable-hazard variant.7 Price changes are rather spread out in the Mexican data. The distributions of price changes under low, medium and high inflation contain a large number of both small and large price changes.8 As noted by several authors (e.g. Kashyap (1995) and Lach and Tsiddon (2007)), the presence of a large number of small price changes is a challenge for standard menu-cost models. They typically generates few small price changes because firms wait until the benefit of making a nominal adjustment outweighs the cost. Similarly, these models generate few large price changes because firms prefer to incur the menu cost rather than let their price deviate far from its optimum. The simple menu-cost model considered in this paper clearly inherits these problems.9 Price changes are tightly clustered around a positive and a negative values. The main impact of inflation on the model’s distribution of price changes is to transfer mass from the negative to the positive cluster. In sharp contrast, the Calvo model has no difficulty generating a large number of both small and large price changes. Under low inflation, my calibration underpredicts the number of small positive price changes, while simultaneously overpredicting the occurrence of 7

The constant-hazard version produces a slightly fatter positive tail as inflation gets high, but its overall fit is very similar to the variable-hazard variant. 8 The low-inflation distribution of price changes is similar to the ones reported by U.S. and euro-area studies with comparable product coverage (e.g. Klenow and Kryvtsov (2008) and Baudry, Bihan, Sevestre, and Tarrieu (2004)). Eden (2001) also reports similarly-spread distributions of price changes under high inflation in Israel for a sample of food products. 9 The reader can refer Golosov and Lucas (2007) and Midrigan (2006) for detailed discussions of this issue in menu-cost models with idiosyncratic technology shocks.

14

small negative changes. The fit of the distribution of price changes is nevertheless relatively good, especially given that the model abstracts completely from the underlying heterogeneity in price setting across products categories. In particular, few price decreases are recorded for services (see Gagnon (2006) for details), creating a bulge of price increases when price changes of goods and services are pooled together. The ability of the Calvo model to generate small price changes comes from the fact that the survival probability of nominal prices decreases exponentially over time. As a consequence, most price changes occur at short durations, that is, before inflation or technology shocks have time to cumulate. The empirical distributions of price changes displayed in Figure 3 present some evidence of fat tails. The excess kurtosis of the low, medium, and high inflation distributions are 2.7, 2.1 and 0.97, respectively. These statistics are in a range similar to the ones reported by Midrigan (2006) for the AC Nielsen and Dominick’s data sets of scanner data, at 0.5 and 2.4, respectively. Midrigan argues that departures from the assumption of Gaussian technological innovations are necessary to generate empirically plausible levels of excess kurtosis in his multi-product menu-cost model. The distributions of price changes from the menu-cost model reported in Figure 3 suffer from the same problem. Their excess kurtosis are -1.6, -1.6, and -1.5, for the low, medium and high inflation simulations, respectively. Interestingly, the Calvo model has no difficulty generating fat tails, even though technology is driven by Gaussian innovations. The corresponding low, medium and high inflation excess kurtosis are 1.5, 2.7 and 3.7, respectively. In the Calvo model, fat tails arise largely because there is a small number of firms that have to wait a large number of periods before optimizing their price. In the simulations, firms that have not updated in a long time post increasingly large price increases as inflation rises. This results in a distribution of price changes whose kurtosis is an increasing function of the level of inflation. As will be seen shortly, the ability of the Calvo model to generate both small price changes and fat tails through a constant hazard rate has stringent implications for the magnitude of price increases and decreases as functions of duration; implications for which the data provide little support. It must also be noted that fat tail statistics are by nature sensitive to outliers. The above discussion illustrates a challenge in the calibration of models with both idiosyncratic technology and menu-cost shocks, such as Midrigan (2006), Konieczny and Rumler (2006) and Caballero and Engel (2007). Assumptions about the distribution of menu costs directly impact the calibration of the technology shock process, and vice-versa. 15

While small price changes require the joint presence of small menu costs and technology shocks, large price changes can be generated by either large technology shocks or the repeated sampling of large menu costs. More micro evidence on the distribution of menu costs and marginal cost across items and over time would certainly be welcome in addressing this issue.

3.3

Average Magnitude of Price Changes and Spell Duration

In the steady state of the baseline Calvo model with constant inflation and no idiosyncratic technology shocks, optimizing firms change their nominal price by the amount of cumulated inflation since their last price change. More formally, a price-optimizing firm increases its price by the amount ∆pi,t = π ss · duri,t , where duri,t is the duration of item i’s price spell. This relationship does not depend on the Calvo parameter, and thus carries to a similar environment with a distribution of Calvo parameters across items. Similarly, it holds in an environment with fixed-duration price contracts, or a menu-cost model with no idiosyncratic supply shocks, such as Sheshinski and Weiss (1977). The magnitude of price changes is then a function of contract length or the size of the menu cost. In the presence of idiosyncratic technology shocks, individual price changes typically differ from the amount of cumulated inflation. Interestingly, if these shocks are mean-zero across firms, then a relationship similar to the one above holds for the average price change in the Calvo model, that is E [dpi |duri ] = π ss · duri . Firms may be hit by positive of negative technology shocks, leading them to post smaller or larger price changes, but as long as the shocks average out, the average magnitude of price changes will be linear in price spell duration. This result does not carry to a menu-cost model with idiosyncratic technology shocks, however. As was clear from the right panels of Figure 3, price increases and decreases cluster around particular values because the width of the Ss band does not depend directly on price spell duration. This diverging prediction of the Calvo and menu-cost models about the shape of the average price change as a function of price spell duration thus offers a potentially powerful tool for discriminating between the two pricesetting mechanisms. In Figure 4, I report the average magnitude of price changes as a function of spell duration for special groups of products, along with two standard deviations confidence intervals.10 It is important that inflation remains stable before and over the period considered, 10

The statistics take into account product category weights. The results obtained by pooling all obser-

16

as any statistic conditional on duration is by nature likely to be sensitive to past events or unusual occurrences. Consequently, the time period considered is limited to July 2000 to June 2002, over which inflation averaged 4.4 percent (a.r.) and was fairly stable. The main finding of Figure 4 is that, for all groups of products considered, the average magnitude of price changes is fairly invariant to spell duration.11 A similar conclusion holds for price increases and decreases, hinting that the mix of price increases and decreases varies little as a function of price spell duration. The main exception is unprocessed food, for which the average absolute magnitude of price increases and decreases falls over the first few months. The average magnitude of price changes is close to zero at short spell durations for unprocessed and processed food, two groups of products with relatively low average inflation (2.8 percent and 2.9 percent, respectively) than the full sample (4.4 percent). The groups nonenergy industrial goods and services share the feature that price changes are, at least initially, larger than zero. This is particularly striking in the case of services, shown in the lower-right panel, for which the average magnitude of price changes and increases are close to each other. This is a direct consequence of infrequent price decreases at all durations. Services had the largest annual inflation rate of all groups considered, 9.4 percent, over the last two years of the sample. Figure 5 displays the same statistics for all nonregulated goods and services in the sample, along with the models’ predictions. The Calvo model is broadly inconsistent with the data. First, the average magnitude of price changes at short duration is not close to zero, as implied by the model. This divergence is largely driven by services, as seen in Figure 4. Second, the average magnitude of price changes is at best mildly rising with spell duration in the data, while the model implies a steeper positive slope. Third, the average magnitude of price increases and decreases are roughly invariant to price spell duration in the data. By contrast, the average magnitude of price increases in the Calvo model has a strong upward trend in the Calvo model. This happens because idiosyncratic shocks and price erosion due to inflation cumulate as duration lengthens, making it more likely that the nominal price ends up well below its optimum. The two effects work in opposite directions in the case of price decreases. Initially, the average (absolute) magnitude of price vations together are very similar. 11 The unusualy small average increase in the price of services at durations equal to eleven months is due to the particular way the cost of education services is computed. Banco de Mexico reports the sum of registration and tuition fees, as these fees cannot be incurred separately. Changes in registration fees, the smallest of the two components, are sometimes recorded a month earlier than changes to tuition fees, resulting in a small price change followed by a larger one.

17

decreases rises with duration because, as technology shocks cumulate, the optimal price is increasingly likely to have wandered below the current nominal price. Over time, however, real price erosion due to inflation makes large declines of the optimum price less likely, so that the absolute magnitude of price decreases eventually diminishes. The menu-cost model provides a much better fit of the data than the Calvo model, as shown in the upper-right panel of Figure 5. The calibration generates a slight rise in the average (absolute) magnitude of price increases and decreases over the first few months, which then quickly levels off. Upon closer inspection, the model does not provide as good a fit for price changes as it does for increases and decreases. While the model’s average magnitude of price changes is similar to the data at short durations, it then levels off around zero as duration lengthens. If there were no aggregate inflation, then the average magnitude of price changes predicted by the model would eventually becoming negative. This aspect deserves a few comments. The model generates a ‘selection effect’ by which price increases are relatively more likely to occur than price decreases at short durations. In this zero-inflation environment, about 35 percent of nominal prices lasting one period end with a price decrease, a proportion that climbs to about 65 percent after twelve periods. This selection effect is tightly linked to the choice of a high elasticity of substitution across items in the consumption basket. The lower-left panel of Figure 5 presents an alternative simulation in which the elasticity is set to a lower value (μ = 2) than the main calibration (μ = 7). The low-elasticity simulation produces a slow rise in the average magnitude with spell duration, which is closer to the data. As seen from Figure 6, the higher this elasticity, the more cone-shaped is the Ss band with respect to a firm’s technology. When the elasticity is high, a productive firm makes large profit losses if it sets its price away from its optimum. If the price is set too high, demand is very low and profits are small. If the price is set too low, then the firm must supply a large quantity, but it makes small or even negative profits on each item sold. The loss in total profits associated with a deviation from the optimal price thus quickly exceeds the menu cost. If instead the firm has low productivity, it must set a high price in order to cover its high costs, leading to an abrupt fall in demand, and low total profits. For any given deviation in the price from its optimum, the change in total profits of a lowproductivity firm is small compared to a productive one, making the Ss band relatively wider. A coned-shaped Ss band generates a ‘selection effect’. If a firm’s last nominal adjustment happened over the last few periods, it is likely to have occurred in the tight portion 18

of the Ss band. The firm likely enjoyed relatively high technology, making the occurrence of a price increase relatively more likely as technology reverts to its mean. If much time has elapsed since the last price change, the relative price likely wandered in the wide portion of the Ss band, making the occurrence of a price decrease more likely as productivity improves. With an elasticity of substitution equal to 2, this selection effect is very weak because the Ss bands are almost parallel, as seen in Figure 6. Therefore, if inflation were zero, the average magnitude of price changes would also be near zero at all durations. For positive inflation rates, such as in the calibration to the Mexican data shown in the lowerleft panel of Figure 5, the average magnitude of price changes is increasing with duration. In this alternative calibration, the model’s predictions are closer to the data than with an elasticity equal to 7, providing an argument for considering calibrations with relatively low elasticity of substitution.

4

Conclusion

The results broadly support some key predictions of the simple menu-cost model with idiosyncratic technology shocks developed in this paper. In particular, this model generates an average frequency and magnitude of price changes similar to that found in the Mexican data under low, medium and high inflation. The model is also consistent with an average magnitude of price increases and decreases that are roughly invariant to price spell duration. The menu-cost model has a well-known caveat, however, which is that it fails dramatically to match the cross-sectional distribution of price changes. As I showed, this failure hold whether inflation is low, medium, or very high. In this respect, the Calvo model fares much better, generating a distribution of price changes containing both a large number of small price changes, and fat tails. The model’s success in this respect hinges on the randomness of adjustment opportunities. The constant hazard generates a large number of price changes at short durations, as well as creating a small but non-negligible mass of firms that have not optimized their price for several periods. The assumption of a constant hazard has important shortcomings, however, such as leading to a misprediction of the average magnitude of price changes as a function of spell duration. A price-setting model merging elements of both the Calvo and menu-cost models may offer an overall better empirical performance. The Calvo model can be interpreted as a menu-cost model with menu-cost shocks, sampled every period from the distribution {0, ∞}. This suggests that specifications with both idiosyncratic technology and menu19

cost shocks might offer a good fit of all facts. Some important steps in this direction have been made recently, in particular by Midrigan (2006).In his multi-product environment, firms pay a single menu cost to change the price of all items that they produce. The cost of changing the first price is positive and large, but the marginal cost of changing the second price is zero, effectively generating a bimodal distribution of (marginal) menu costs. At the moment, there is a lack of evidence about the joint distribution of technology and menu costs. More empirical research on this aspect would certainly be welcome.

20

References Baudry, L., H. L. Bihan, P. Sevestre, and S. Tarrieu (2004). Price rigidity. Evidence from the French CPI micro-data. European Central Bank, Working Paper Series: 384. Bils, M. and P. J. Klenow (2004). Some evidence on the importance of sticky prices. Journal of Political Economy 112 (5), 947—985. Caballero, R. and E. Engel (2007). Price Stickiness in Ss Models: New Interpretations of Old Results. Economic Growth Center, Yale University, Working Papers. Calvo, G. A. (1983). Staggered prices in a utility-maximizing framework. Journal of Monetary Economics 12 (3), 383—98. Christiano, L. J., M. Eichenbaum, and C. L. Evans (2005). Nominal rigidities and the dynamic effects of a shock to monetary policy. Journal of Political Economy 113 (1), 1—45. Danziger, L. (1999). A dynamic economy with costly price adjustments. American Economic Review 89 (4), 878—901. Dhyne, E., L. J. Alvarez, H. L. Bihan, G. Veronese, D. Dias, J. Hoffmann, N. Jonker, P. Lunnemann, F. Rumler, and J. Vilmunen (2005). Price setting in the euro area: some stylized facts from individual consumer price data. European Central Bank, Working Paper Series: 524. Eden, B. (2001). Inflation and price adjustment: An analysis of microdata. Review of Economic Dynamics 4 (3), 607—36. Gagnon, E. (2006). Price Setting during Low and High Inflation: Evidence from Mexico. International Finance Discussion Papers 896, Federal Reserve Board. Golosov, M. and J. R. E. Lucas (2007). Menu costs and phillips curves. Journal of Political Economy 115 (2), 171—199. Kashyap, A. K. (1995). Sticky prices: New evidence from retail catalogs. Quarterly Journal of Economics 110 (1), 245—74. Klenow, P. J. and O. Kryvtsov (2008). State-dependent or time-dependent pricing: Does ˝ it matter for recent u.s. inflation? Quarterly Journal of Economics 123 (3), 863U904. Konieczny, J. D. and F. Rumler (2006). Regular adjustment - theory and practice. European Central Bank, Working Paper Series: 669. Lach, S. and D. Tsiddon (2007). Small price changes and menu costs. Managerial and Decision Economics 28, 649—656. Levin, A. and T. Yun (2007). Reconsidering the natural rate hypothesis in a new keynesian framework. Journal of Monetary Economics 54 (5), 1344—1365. Midrigan, V. (2006). Menu Costs, Multi-Product Firms, and Aggregate Fluctuations. mimeo, Ohio State University. 21

Nakamura, E. and J. Steinsson (2007). Five Facts About Prices: A Reevaluation of Menu Cost Models. mimeo, Harvard University. Sheshinski, E. and Y. Weiss (1977, 06). Inflation and costs of price adjustment. Review of Economic Studies 44, 287—303. Smets, F. and R. Wouters (2005). Comparing shocks and frictions in us and euro area business cycles: a bayesian dsge approach. Journal of Applied Econometrics 20 (2), 161—183. Stokey, N. L. and R. E. Lucas (1989). Recursive methods in economic dynamics. With Edward C. Prescott; Cambridge, Mass. and London:; Harvard University Press. Woodford, M. (2003). Interest and prices. Princeton and Oxford:; Princeton University Press. Yun, T. (1996). Nominal price rigidity, money supply endogeneity, and business cycles. Journal of Monetary Economics 37, 345—370.

22

A A.1

Solving the models Price Setting under Menu Costs

A few additional steps must be taken to ensure that the firm’s problem, as defined by ˆ and pˆ can take any equations (2) − (4) in Section 1.2.2, has a unique A priori, φ ³ solution. ´ ˆ pˆ unbounded. Consequently, standard value in R, making the period profit function π φ; proofs of the existence and uniqueness of a fixed point to the dynamic problem cannot be applied directly. A solution around this problem is to first assume that technological deviations cannot take values than some arbitrary threshold. A new measure of ´ ³ greater 0 ˆ , can be defined as ˆ |φ ˜ φ technological transition, dG ¯ 0¯ ¯ˆ ¯ if ¯φ ¯<φ ¯ 0¯ ³ 0 ´ ¯ˆ ¯ 0 if ¯φ ¯ > φ . ˆ |φ ˆ = ˜ φ dG Z ∞ ¯ 0¯ ³ 0 ´ ⎪ ⎪ ¯ˆ ¯ ⎪ ˆ ˆ ⎪ dG φ | φ if ¯φ ¯ = φ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

³ 0 ´ ˆ |φ ˆ dG φ

¯ φ

£ ¤ ¯ φ ¯ , can be chosen so that the probability of achieving its The domain of φ, noted Φ = −φ, ∗ bounds is arbitrarily small. Similarly, the domain ³ ´ of pˆ and pˆ can be restricted to the com£ ¤ ˆ pˆ is bounded over Φ× P. A further complipact P = p, p . Under these conditions, π φ; cation the need to ensure that pˆ stays in P despite A simple ³ ´the Rdrift³ in0 inflation. ´ solution ³ 0 ´ © ª ˆ pˆ as V φ ˆ ; max p, pˆ − g dG ˆ |φ ˆ . ˜ φ to this problem is to write the integral in V nc φ, One can then define the functional operator T , ³ ´ ³ ´o ³ ´ n ˆ pˆ|V , Vc φ|V ˆ ˆ pˆ = max Vnc φ; (T V ) φ; where

and

Z ³ ´ ³ ´ ³ 0 ´ ¡ © ª¢ ˆ ˆ ˆ |φ ˆ . ˜ φ Vnc φ; pˆ|V = π φ; pˆ + β V φ0 ; max p, pˆ − g dG ³ ´ ˆ = sup Vc φ|V pˆ∗

³ ´ ˆ pˆ∗ |V − ξ W . Vnc φ; P

This operator maps the ³ 0set´of bounded continuous functions into itself. This result follows ˆ |φ ˆ has the Feller property under the distributional assumptions, ˜ φ from the fact that G ³ ´ ˆ pˆ|V inherits continuity and boundedness from V . By the theorem ensuring that Vnc φ; 23

³ ´ ˆ achieves its supremum over its domain P, and thus retains of the Maximum, Vc φ|V from V the properties of being bounded and continuous. Finally, one can show that the operator satisfies the monotonicity and discounting properties, so that Blackwell’s sufficient condition for the existence of a unique fixed point are satisfied (see the discussion surrounding theorems 4.6 and 9.6 in Stokey and Lucas (1989) for a more general treatment). An equilibrium to the economy is computed in the following way. I start by guessing the aggregate price level, from which I derive all aggregate objects entering the firm’s problem. The fixed point of the firm’s problem conditional on the aggregate variables is then computed using standard dynamic programming techniques based on polynomial approximations of Vn and Vc . Once a fixed point has been found, I generate a long Markov chain of prices using the firm’s policy function. These prices are then aggregated to compute the corresponding price index. If this price index differs from my initial guess, it is updated and the procedure is repeated until satisfying numerical convergence.

A.2

Pricing Setting under (Truncated) Calvo

The expression for the price set by optimizing firms in a stationary equilibrium, Equation 1, can written as "T −1 # X p∗i,t 1 − βθπ μ−1 μ = Et (βθπ μ )s M Ci,t+s Pt μ − 1 1 − (βθπ μ−1 )T s=0 Using the functional assumptions for the marginal cost and the distribution of technology shocks, the optimal pricing strategy becomes p∗j,t = Ω(μ,β,θ,π,ψ,φ¯ ) Pt

(

ÃT −1 X

¡ ¢ )! 2s 1 − ρ σ 2ε ˆ i,t + (βθπ μ )s exp −ρs φ C 1 − ρ2 2 s=0

where μ Ω(μ,β,θ,π,ψ,φ¯ ) = μ−1

Ã

1 − βθπ μ−1

1 − (βθπ μ−1 )T

!

(A1)

ψ ¯ φ

Once aggregate consumption, C, has been solved for, the distribution of nominal price changes can then be easily recovered by generating a long Markov chain. The remainder of this section derives an expression for steady state consumption from which other aggregate objects can be easily derived. Two equations are used to solve for C. The first is obtained

24

by rearranging the price index, Pt , defined as Pt =

µZ

1−μ

(pj,t )

dj

¶

1 1−μ

Using the fact price-optimizing firms are selected randomly across states and periods, one gets 1 ÃT −1 Z ! 1−μ X ¡ ∗ ¢1−μ λs dj pj,t−s Pt = s=0

where p∗i,t−s denotes the nominal price set by firm j when it last optimized its price s periods ago. Let µZ ¶ 1 1−μ ¡ ∗ ¢1−μ ∗ p dj Pt = j,t be the price index associated with firms optimizing their price in period t. Plugging the above expression into the previous equation, we get

Pt =

ÃT −1 X s=0

¡ ∗ ¢1−μ λs Pt−s

1 ! 1−μ

where λs is the mass of firms whose last price optimization occurred s periods earlier, measured at the end of the current period after all pricing decisions are made, so that λ0 is the mass of prices that were changed in the current period. Expressions for the λs can be obtained straightforwardly. In the truncated Calvo, a fraction (1 − θ) of firms are randomly selected to change their price in any given period. All firms whose last price change occurred T periods earlier automatically get the chance to reoptimize. The ergodic distribution of durations must thus satisfy the following relationships 1 = λ0 + λ1 + ... + λT −1 λ0 = θ (λ0 + λ1 + ... + λT −2 ) + λT −1 = θ + (1 − θ) λT −1 λi = (1 − θ) λi−1 , for 0 < i ≤ T − 1

´ ³ Solving this system of equations, one obtains λ0 = θ/ 1 − (1 − θ)T from which all other λs can be easily recovered.

25

Note that Pt∗ /Pt is constant in a stationary equilibrium, so that, after rearranging terms, T −1 X λs π s(μ−1) (Pt∗ /Pt )1−μ , 1= s=0

or

Pt∗ = Pt

ÃT −1 X

λs π s(μ−1)

s=0

1 ! μ−1

= Λ(π,θ,μ,T ) .

(A2)

The second equation is obtained by reconstructing the price index using equation (A1): Pt∗ Pt

=

ÃZ µ

p∗i,t Pt

¶1−μ

!

1 1−μ

di

1 ⎛ Ã ( ¢ )!1−μ ⎞ 1−μ ¡ Z TX −1 2s 2 ˆ + 1 − ρ σε = CΩ(μ,β,θ,π,ψ,φ¯ ) ⎝ (βθπ μ )s exp −ρsj,t φ dj ⎠ (A3) . 2 1 − ρ 2 s=0

Merging (A2) and (A3), one finally obtains an expression for steady state consumption 1 ⎛ Ã ( ¡ ¢ )!1−μ ⎞ μ−1 Z TX −1 2s 2 ˆ + 1 − ρ σε ⎝ (βθπ μ )s exp −ρs φ dj ⎠ . C= j,t 2 Ω(μ,β,θ,π,ψ,φ¯ ) 1 − ρ 2 s=0

Λ(π,θ,μ,T )

This expression can be evaluated numerically using a quadrature of the integral. All other aggregate quantities can then be easily recovered. Aggregate consumption is an increasing ¯ and decreasing function of the disutility of work, ψ. function of aggregate technology, φ, A rise in the variance of technological innovations, σ 2ε , or in the persistence of technology deviation, ρ, increases the dispersion of relative prices. Because items in the consumption basket are substitutable, this greater dispersion leads to a decline in the price of the basket, and thus higher steady state consumption.

26

Figure 1: Average Frequency and Magnitude of Price Changes (nonregulated items) a) Frequency of price changes vs inflation

b) Frequency of price changes, increases and decreases

80

80 frequency inflation

60

40

percent

percent

60

20

40

20

0

0

1994 1995 1996 1997 1998 1999 2000 2001 2002

1994 1995 1996 1997 1998 1999 2000 2001 2002

a) Magnitude of price changes vs inflation

b) Magnitude of price changes, increases and decreases

15

20 changes

average change

increases

monthly inflation 10 percent

percent

10

5

0

0

-10

-5 1994 1995 1996 1997 1998 1999 2000 2001 2002

27

-20 1994 1995 1996 1997 1998 1999 2000 2001 2002

decreases

Figure 2: Models’ fit of the average frequency and magnitude of price changes, increases and decreases

Calvo (cst) - magnitude 20

30

10 percent

percent

Calvo (cst) - frequency 40

20 10 0

-10

0

10

20 30 40 inflation (a.r.) Calvo (var) - frequency

-20

50

50

percent

percent

20

20 30 40 inflation (a.r.) Calvo (var) - magnitude

50

0

10

20 30 40 inflation (a.r.) Menu cost - magnitude

50

0

10

0 -10

10 0

10

20 30 40 inflation (a.r.) Menu cost - frequency

-20

50

50

20

40

10

30

percent

percent

10

10

30

20

0 -10

10 0

0

20

40

0

0

0

10

20 30 inflation (a.r.)

40

50

-20

changes (model) increases (model) decreases (model) changes (data) increases (data) decreases (data)

28

20 30 inflation (a.r.)

40

50

Figure 3: Models’ Fit of the Distribution of Price Changes

0.15

Calvo - July 2000 to June 2002

Menu cost - July 2000 to June 2002 0.25

model data

Frequency: 24.9% Inflation: 4.5% Inflation std.: 4.8%

model 0.20

data

Frequency: 24.9% Inflation: 4.5% Inflation std.: 4.8%

0.10 density

density

0.15

0.10

0.05 0.05

0 -50 -40 -30 -20 -10 0 10 percent

20

30

40

0 -50 -40 -30 -20 -10 0 10 percent

50

Calvo - February 1996 to May 1996

20

30

40

50

Menu cost - February 1996 to May 1996

0.15

0.25 model data

Frequency: 30.8% Inflation: 25.7% Inflation std.: 3.6%

model 0.20

data

Frequency: 30.8% Inflation: 25.7% Inflation std.: 3.6%

0.10 density

density

0.15

0.10

0.05 0.05

0 -50 -40 -30 -20 -10 0 10 percent

20

30

40

0 -50 -40 -30 -20 -10 0 10 percent

50

Calvo - January 1995 to March 1995

20

30

40

50

Menu cost - January 1995 to March 1995

0.15

0.25 model data

Frequency: 41.0% Inflation: 50.0% Inflation std.: 2.6%

model 0.20

data

Frequency: 41.0% Inflation: 50.0% Inflation std.: 2.6%

0.10 density

density

0.15

0.10

0.05 0.05

0 -50 -40 -30 -20 -10 0 10 percent

20

30

40

50

29

0 -50 -40 -30 -20 -10 0 10 percent

20

30

40

50

Figure 4: Average magnitude of price changes and price spell duration for special groups of products (July 2000 to June 2002) Processed Food 20

15

15

10

10

5

5

0

0

percent

percent

Unprocessed Food 20

-5

-5

-10

-10

-15

-15

-20

-20

-25

2

4

6

8

10 months

12

14

16

-25

18

2

4

6

20

15

15

10

10

5

5

0 -5

-10 -15

-20

-20 4

6

8

10 months

14

16

18

12

14

16

18

0

-15

2

12

-5

-10

-25

10 months

Services

20

percent

percent

Other Goods

8

12

14

16

18

changes

-25

increases

30

2

4 decreases

6

8

10 months

Figure 5: Average magnitude of price changes and price spell duration Menu cost (elasticity = 7) 20

10

10 percent

percent

(Truncated) Calvo 20

0

-10

-20

0

-10

5

10 duration (months)

-20

15

5

10 duration (months)

Menu cost (elasticity = 2) 20 changes (model) increases (model) decreases (data) changes (data) increases (data) decreases (data)

percent

10

0

-10

-20

5

10 duration (months)

15

31

15

Figure 6: Elasticity of substitution and the Ss band and in the menu-cost model 80

optimal price (μ=7) optimal price (μ=2) Ss band (μ=7) Ss band (μ=2)

60

pˆ (percent)

40

20

0

-20

-40

-60 -50

-40

-30

-20

-10

0

φˆ (percent)

32

10

20

30

40

50