Multiproduct firms and price-setting: Theory and evidence from U.S. producer prices

Author's Accepted Manuscript

Multiproduct Firms and Price-Setting: Theory and Evidence from U.S. Producer Prices Saroj Bhattarai, Raphael Schoenle

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S0304-3932(14)00077-4 http://dx.doi.org/10.1016/j.jmoneco.2014.05.002 MONEC2699

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Journal of Monetary Economics

Received date: 2 February 2012 Revised date: 2 May 2014 Accepted date: 5 May 2014 Cite this article as: Saroj Bhattarai, Raphael Schoenle, Multiproduct Firms and Price-Setting: Theory and Evidence from U.S. Producer Prices, Journal of Monetary Economics, http://dx.doi.org/10.1016/j.jmoneco.2014.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Multiproduct Firms and Price-Setting: Theory and Evidence from U.S. Producer Prices

$,$$

Saroj Bhattaraia , Raphael Schoenleb,∗ a Pennsylvania

b Brandeis

State University University

Abstract Using micro-data on U.S. producer prices, we establish three new facts about price setting by multi-product firms. First, firms selling more goods adjust prices more frequently but on average by smaller amounts. Moreover, their fraction of positive price changes is lower and the dispersion of price changes is higher. Second, price changes within firms are substantially synchronized, which plays a dominant role in explaining pricing dynamics. Third, firms selling more goods have greater within-firm synchronization of price changes. A model with trend inflation and firm-specific menu costs where firms are subject to idiosyncratic and aggregate shocks matches the empirical findings. Keywords: Multi-product firms; Number of Goods; State-dependent pricing; U.S. Producer prices. JEL Classification: E30; E31; L11.

1. Introduction

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Using micro-data underlying the U.S. Producer Price Index (PPI), we provide new evidence on several key statistics of price changes systematically varying with the number of goods produced by multi-product firms. Our main empirical findings are that as firms produce more goods: the frequency of price change is higher while the size of price changes is lower; the fraction of price changes that are increases is lower; the fraction of small price changes and the dispersion of price changes is higher; and the degree of synchronization of price changes within firms increases. As a byproduct, we document a substantial degree of synchronization of price changes across products within a firm, relative to synchronization with other firms in the same sector. We then present a multi-product menu cost model that can match our key empirical facts. Our findings help to distinguish between competing models of price adjustment, and in particular, gauge the extent of the so-called “selection effect” in menu cost models. Making a distinction between models of price adjustment is of first-order importance in macroeconomics: even small changes in assumptions about price setting can imply vastly different real effects from nominal shocks. Our analysis of how the number of $ We thank, without implicating, Fernando Alvarez, Jose Azar, Alan Blinder, Thomas Chaney, Gauti Eggertsson, Penny Goldberg, Oleg Itskhoki, Nobu Kiyotaki, Kalina Manova, Marc Melitz, Virgiliu Midrigan, Emi Nakamura, Woong Yong Park, Sam Schulhofer-Wohl, Kevin Sheedy, Chris Sims, Jon Steinsson, Mu-Jeung Yang and workshop and conference participants at the Board of Governors of the Federal Reserve System, the CESifo Conference on Macroeconomics and Survey Data, the Midwest Macro Meetings, Recent Developments in Macroeconomics at Zentrum f¨ ur Europ¨ aische Wirtschaftsforschung (ZEW), the New York Fed, Princeton University, the Swiss National Bank, the Royal Economic Society Conference, and the XXXV Simposio de la Asociaci´ on Espa˜ nola de Econom´ıa for helpful suggestions and comments. $$ This research was conducted with restricted access to the Bureau of Labor Statistics (BLS) data. The views expressed here are those of the authors and do not necessarily reflect the views of the BLS. We thank project coordinators, Ryan Ogden, and especially Kristen Reed, for substantial help and effort, as well as Greg Kelly and Rosi Ulicz for their help. We gratefully acknowledge financial support from the Center for Economic Policy Studies at Princeton University. ∗ Corresponding author at Brandeis University, 415 South Street, Waltham, MA 02454, United States, Tel.: +1 617 680 0114. Email address: [email protected] (Raphael Schoenle)

Preprint submitted to Elsevier

May 12, 2014

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goods in a firm relates to pricing decisions is moreover of independent interest: 98.55% of all prices in the PPI are set by firms with more than one good, which contrasts with the standard macroeconomic assumption of price-setting by single-product firms. Our analysis exploits a particular feature of the micro data from the Bureau of Labor Statistics (BLS) that underlie the calculation of the U.S. PPI. In the data, there is substantial variation in the number of goods per firm, with a median number (std. deviation) of 4 (2.55) goods. This enables us to compute price change statistics according to the number of goods. While doing so, we find that pricing behavior is systematically related to the number of goods per firm. First, we study the frequency and size distribution of price changes, two key statistics that can discipline the real effect of monetary policy shocks. As is well-understood, higher frequency of price changes leads to lower real effects of monetary shocks in models with nominal rigidities. We find that the frequency of price adjustment increases monotonically and substantially as the number of goods increases. For example, firms that have between 1 to 3 goods on average during their time in the data have a median monthly frequency of price change of 15%. Firms that have more than 7 goods have a median monthly frequency of 23%. At the same time, the average magnitude of price changes, conditional on adjustment, monotonically decreases with the number of goods. This holds for both upwards and downwards price changes. The magnitude of price changes ranges from 8.5% for firms with 1 to 3 goods to 6.5% for firms with more than 7 goods. Two other statistics – the fraction of small price changes and their kurtosis – are key moments related to modeling assumptions in menu cost models. We find that small price changes are highly prevalent in the data, and become even more prevalent when the number of goods increases. The fraction of small price changes is 38% for firms with 1 to 3 goods and increases to 55% for firms with more than 7 goods. We also find that there is substantial dispersion in the size of price changes that increases with the number of goods. For example, the coefficient of variation of absolute price changes and the kurtosis of price changes increase as the number of goods per firm increases. The mean kurtosis for firms with 1 to 3 goods is 5 while for firms with more than 7 goods it is 17. In terms of macroeconomic implications, such evidence directly relates to the strength of the “selection effect” in menu cost models: Firms with prices far from their optimal prices are more likely to adjust prices given adjustment frictions, especially when a shock pushes prices even further from their optimal levels. The more prices absorb shocks, the smaller the real effects of monetary policy shocks. In this context, Golosov and Lucas Jr. (2007) have argued that a strong selection effect generates smaller amounts of monetary nonneutrality in menu cost models than in time-dependent pricing models. Midrigan (2011) challenged this conclusion by showing that Golosov and Lucas’s model did not match the fraction of price changes that are small as well as the excess kurtosis in the size of price changes. He then showed that a menu cost model that matches these features could dramatically decrease the strength of the selection effect, and thereby increase the degree of monetary non-neutrality in menu cost models. In recent work, Alvarez and Lippi (2014) present a model that features menu costs and multi-product firms with an arbitrary number of goods - a generalization from the two goods in Midrigan (2011). In their model, the initial impact of a monetary shock becomes smaller and impulse responses become more stretched out, leading to larger real effects, as the number of goods increases. The broad evidence in this paper confirms the predictions for several moments of the price distribution implied by Midrigan (2011) and Alvarez and Lippi (2014). Second, our data provide strong evidence for synchronization of price adjustment decisions within firms. The extent to which price changes in the economy are synchronized is again a highly relevant statistic for monetary models because it informs us about the importance of strategic complementarities for pricing decisions. As emphasized recently by Carvalho (2006) and Nakamura and Steinsson (2008), strategic complementarities can amplify real effects of nominal shocks. To study synchronization, we estimate a multinomial logit model to relate individual adjustment decisions to the fraction of price changes of the same sign within a firm, within the same industry, and other economic fundamentals. We find that when the price of one good in a firm changes, there is a large increase in the probability that the price of another good in the firm changes in the same direction. A one-standard deviation increase in the firm fraction of other goods changing price in the same direction is associated with a 14.75 percentage point higher probability of upwards and a 8.82 percentage point higher probability of downwards price adjustment.

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Moreover, our results show that such synchronization within the firm is much stronger than within the industry. We also document that the number of goods and the degree of within-firm synchronization strongly interact in determining individual price adjustment decisions. In particular, we find that the strength of within-firm synchronization increases monotonically with the number of goods. Again, this result holds both for upwards and downwards adjustment decisions. At the same time, we find that the strength of synchronization within the same industry decreases monotonically as the number of goods increases. This finding is complementary to the work of Boivin et al. (2009), as it locates the incidence of pricing decisions at the level of multi-product firms relative to their industries. Next, on the theoretical side, we develop a state-dependent pricing model of multi-product firms. We find that varying only one parameter – the firm-specific menu cost – allows us to qualitatively match all the trends observed in the data. Firms in our model face idiosyncratic productivity shocks and an aggregate inflation shock. The productivity shocks are correlated across goods produced by the same firm. Moreover, there is a menu cost of changing prices which is firm-specific and there are economies of scope in the menu cost technology. The main theoretical results of the model are driven by economies of scope in the menu cost technology: firms will change prices of some products even though their prices are not very far out of line from the desired price because prices of other products need to be changed, and the firm might as well change the prices of all products since it is changing some already. As the number of goods increases, this mechanism directly leads to a higher frequency of price changes and a lower mean absolute, positive, and negative size of price changes. It also implies that the fraction of small price changes is higher for multiproduct firms. Moreover, with trend inflation, firms adjust downwards only when they receive substantial negative productivity shocks. Since the firm adjusts prices when the desired price of one item is very far from its current price, a higher fraction of downward price changes becomes sustainable. The model also delivers synchronization of price changes: when we compare a 2-good to a 3-good firm, the model predicts that both positive and negative adjustment decisions become more synchronized within firms. This effect is again due to economies of scope in the menu cost technology: they increase the probability of simultaneous adjustment decisions. In particular, positive adjustment decisions become more synchronized than negative ones due to shocks from upward trend inflation as the number of goods increases. Finally, correlation of shocks among goods within a firm amplifies the synchronization of adjustment decisions. Our empirical work is related to the recent literature that has analyzed micro-data underlying aggregate price indices.1 Our paper contributes the first account of price-setting dynamics from the perspective of the firm. Two other recent papers have also used the same data to uncover interesting patterns. Nakamura and Steinsson (2008) show that there is substantial heterogeneity across sectors in the PPI in the frequency of price changes. Matching groups between CPI and PPI data, they also find that the correlation in the frequency of price changes between the groups is quite high. Goldberg and Hellerstein (2009) document that price rigidity in finished producer goods is roughly the same as in consumer prices including sales and that large firms change prices more frequently and by smaller amounts compared to small firms. Our results are complementary to theirs. Neither of these papers however, contain a systematic analysis from the perspective of the firm, and in particular, how price-setting dynamics differ according to the number of goods produced by firms. Our empirical results are also related to findings from papers that use analyze pricing behavior by retail or grocery stores, such as Lach and Tsiddon (1996), Fisher and Konieczny (2000) or Lach and Tsiddon (2007). Moreover, Midrigan (2011) provides some empirical support for his model based on scanner data from a chain of grocery stores in Chicago (Dominick’s). One contribution of our paper is to provide much more broad-based empirical evidence supporting exactly the types of modeling features that Midrigan (2011) emphasized. On the theoretical side, our model is related to work by Sheshinski and Weiss (1992), Midrigan (2011), and Alvarez and Lippi (2014). Our main contribution relative to Sheshinski and Weiss (1992) and Midrigan (2011) is to consider and systematically analyze price-setting as we vary the number of goods produced by 1 For a survey of this literature, see Klenow and Malin (2010). Most of this literature has focused on the U.S. or the Euro Area. For an analysis of emerging markets, see Gagnon (2009).

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firms, from 1 to 3 goods, going beyond the case of a 2-good firm. Thus, we are able to study trends in some price setting statistics like synchronization by considering 2-good vs. 3-good firms which is not possible by only comparing 1-good and 2-good firms. Alvarez and Lippi (2014) use stochastic control methods to characterize the price setting solution of a multi-product firm producing an arbitrary number of goods. Our relative contribution is that we solve a model with stochastic inflation and also allow idiosyncratic shocks across goods produced by the same firm to be correlated. Importantly, we can also generate additional predictions for the direction and synchronization of price changes. We then match all the empirical trends in these moments qualitatively using a calibrated model. 2. U.S. Producer Price Data and Multi-Product Firms

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We use monthly producer price micro-data from the dataset that is normally used to compute the PPI by the BLS. While we refer the reader for details to the Online Appendices A and C and the existing descriptions of the PPI data in Nakamura and Steinsson (2008) and Goldberg and Hellerstein (2009), we describe a key feature of the data in the next subsection that is relevant to our analysis of pricing by multi-product firms: there is substantial variation in the number of goods across more than 28, 000 firms. This allows us to study how key pricing statistics vary with the number of goods. We also note that we use PPI instead of CPI micro data mainly since we have in mind a model where actual producing firms, not retailers such as supermarkets or grocery stores, set prices. A similar analysis of producer pricing decisions is not feasible with CPI data since the CPI sampling procedure does not map to the production structure of the economy. Instead, it maps to stores, so-called “outlets,” which may sell goods from any number of firms, including imports. This makes pricing a complicated web of decisions that involves the whole distribution network. Moreover, it is generally also not possible to identify the producing firms for specific CPI items. The CPI data only sometimes record the manufacturer of a good. On a technical level, there is also too little variation in the number of goods in the CPI data to perform an analysis like for the PPI as we discuss below. 2.1. Identifying and Grouping Multi-Product Firms

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We pin down the multi-product dimension of firms by using firm identifiers, and then simply computing the average number of goods from a firm over the time periods when the firm has at least one good in the data. We find that the median (mean) number of goods per firm is 4 (4.13) with a standard deviation of 2.55 goods. The minimum number of goods is 1, the maximum is 77. We use this variation to group firms into four bins according to the average number of goods that firms have while in the data: a) bin 1: firms with 1 to 3 goods, b) bin 2: firms with more than 3 to 5 goods, c) bin 3: firms with more than 5 to 7 goods, and d) bin 4: firms with more than 7 goods. Thus, firms in higher bins sell a greater number of goods than firms in lower bins. A similar strategy of binning has also been used in Gopinath and Itskhoki (2010), who look at the relationship between frequency of price changes and exchange rate pass-through. Note that we have many firms who have non-integer numbers of goods due to the averaging of the monthly number of goods for each firm. This is another reason to have bins. We summarize the variation in the number of goods in our data in more detail in Table 1. The median (mean) number of goods per firm across by bins is 2 (2.2), 4.0 (4.0), 6.0 (6.1), and 8.0 (10.3) respectively. The dispersion in the number of goods is higher in bin 4, with for example, a standard error of 0.11. The table also shows that while the majority of firms, around 80%, fall in bins 1 and 2, there are a substantial number of firms in bins 3 and 4 as well. In fact, since firms in bins 3 and 4 set more prices per firm, they account for a much larger share of all prices in the data. Firms in bins 3 and 4 set around 40% of all prices in our data. We verify that the distribution of the number of goods is not systematically driven by firm size: there is no clear trend in terms of median employment per good across these different bins as Table 1 shows.2 2 We

include detailed controls for firm size later in the paper to check the robustness of our results.

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While the number of goods per firm in the data due to sampling does not map one to one into the actual number of goods priced by firms, it is important to emphasize that there is a monotonic relationship between the number of goods in our data and the actual number of goods per firm. On the one hand, this is due to the BLS sampling procedure. The BLS sampling design in the “disaggregation” stage is such that all the economically important products tend to be sampled with probability proportional to their sales.3 In addition, the BLS pays special attention to cover all distinct product categories if they exist in a firm allowing some discretion in sampling when there are many products in a firm. Thus, if a firm has more products, more products will be sampled on average. On the other hand, our strategy of binning goods into the ranges leaves some room for potential errors of sampling into the “wrong” bin and allows us to average out such errors when we calculate our statistics of interest. When at the cut-off to an adjacent bin, the BLS might randomly sample one good more or less than is representative according to the sampling scheme. As long as this happens randomly, our statistics of interest should still be representative of firms with more goods as we move to higher bins. Our empirical results also validate our approach of binning: our choice of binning leads to results that our theoretical model in all cases predicts would be indeed identified with an increasing number of goods per firm. 3. Pricing Behavior of Multiproduct Firms

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In this section, we show how pricing by multi-product firms does not resemble at all pricing of multiple single-product firms considered as one unit. 3.1. Aggregate Price Change Statistics We first show how key aggregate price change statistics vary with the number of goods.

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3.1.1. Frequency of price changes We compute the frequency of price changes separately for each bin. This allows us to trace out how it is a function of the number of goods per firm. We compute the frequency in each bin as the fraction of price changes for a representative good of a representative firm. That is, after computing the fraction of all monthly price changes over the life-time of a single good, we calculate the median frequency over all goods within a firm. This gives us one number per firm. Then, we report the mean, median, and standard error of frequencies across firms in a given bin. We use this standard error to compute 95% confidence intervals throughout the paper.4 We compute upper and lower bounds of the bands as ±1.96 ∗ std. error. We find strong evidence that the frequency of price changes is a function of the number of goods per firm. Figure 1 shows this graphically: the mean frequency of monthly price changes increases with the number of goods per firm. It increases from 20% in bin 1 to 29% in bin 4. The relationship is monotonic across bins except for the mean frequency of price changes for bins 1 and 2. Also, note that 64% of these changes are positive price changes in bin 1 going down to 61% in bin 4, as one would expect under trend inflation, and as summarized in Figure 2.5 Clearly, these trends show that multi-product firms are not the same as an aggregate of multiple single-product firms. 3 We know from Bernard et al. (2010) and Goldberg et al. (2010)’s Table 4 that large firms are multi-product firms with substantial value of sales concentrated in a few goods. We present an analogous table for our dataset in Online Appendix C. Results suggest that sampling is likely to monotonically capture the actual number of economically important goods. 4 In this exercise, we do not count the first observation as a price change and assume that a price has not changed if a value is missing, following Nakamura and Steinsson (2008). We have also verified that left-censoring of price-spells is not a problem, and that our trends are the same when we take means across goods at the firm level. 5 One can interpret the frequency not only as the monthly probability of a price change, but also in terms of the duration of price spells. Inverting these frequency estimates implies that the mean duration of a price spell decreases from 5 months in bin 1 to 3.4 months in bin 4 while the median duration decreases from 6.7 months to 4.3 months.

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3.1.2. Size and distribution of price changes Further, we observe a clear relationship between various measures of the distribution of price changes and the number of goods in our data: firms with more goods change their prices by smaller amounts, price changes are more dispersed and there are more small price changes. First, we consider the absolute size of log price changes as a measure of the magnitude of price changes, conditional on adjustment. When we aggregate from the good to the firm to the bin level as before, we find that the absolute size of price changes decreases with the number of goods produced by firms as it goes down monotonically from 8.5% in bin 1 to 6.6% in bin 4. Figure 3 shows this graphically. This relationship holds even when we separate out the price changes into positive and negative price changes, conditional on adjustment. Figure 4 shows that the size of positive price changes decreases with the number of goods while the size of negative price changes increases with the number of goods. Thus, in general, firms with a greater number of goods adjust their prices by smaller amounts, both upwards and downwards. Second, we consider two other statistics that describe the distribution of price changes and that are crucial to the relevance of the selection effect: the fraction of “small” price changes and the kurtosis of the distribution. We define a price change as small if |Δpi,j,t | ≤ κ|Δpi,t |, where i is a good in firm j, and κ = 0.5, following Midrigan (2011). That is, a price change is small if it is less in absolute terms than a specified fraction of the mean absolute price change in a firm. After creating an indicator variable, we report the fraction of small price changes for each bin. We find that small price changes are quite prevalent. Figure 5 shows that the fraction of small price changes increases from 38% in bin 1 to 55% in bin 4. Thus, small price changes also become more prevalent when firms produce many goods. Our analysis also provides broad-based empirical evidence consistent with a leptokurtic distribution of shocks that can generate large real effects of monetary shocks as posited in the multi-product menu cost literature. We provide such evidence by computing the kurtosis of price changes, which is the ratio of the fourth moment about the mean and the variance squared: i 1  μ4 where μ = (Δpi,j,t − Δpj )4 4 σ4 T − 1 i t=1

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T

(1)

where in a given firm j with n goods, Δpi,t denotes the log price change of good i, Δp the mean price change and σ 4 is the square of the usual variance estimate. We find strong evidence that price changes are leptokurtically distributed. Again, the extent of this is a function of the multi-product nature of firms. Figure 6 shows that the mean kurtosis of price changes increases with the number of goods produced by firms as it goes up from 5.3 in bin 1 to 16.8 in bin 4. Consistently with a more dispersed price change distribution for multi product firms, we also document in Figure 7 that both the 1st and the 99th percentiles of price changes take more extreme values as the number of goods increases. Finally, we find strong trends in the coefficient of variation suggesting that multi-product firms price differently than an aggregate of single-product firms. To compute the coefficient, we pool our data at the firm level. Then, we compute the ratio of the standard deviation to the mean of absolute price changes for an item, take the mean across items in a firm, and then means across firms in a bin. Table 1 shows that the coefficient of variation monotonically increases with the number of goods from 1.02 in bin 1 to 1.55 in bin 4. 3.1.3. Robustness Our aggregate results above are completely robust to controlling for various factors that might lead to trends across bins. All the results in this section can be found in Online Appendix A. First, we show that the different goods bins are not picking up effects of differential firm size. We demonstrate this for the sake of conciseness only for the frequency and size of price changes. When we control linearly for the size of firms in a regression, using the total number of employees per firm as a measure of size, we find that our trends across bins continue to hold. Figures A.1 and A.2 show this result for the frequency and size of price changes. Next, we show in several steps that various kinds of heterogeneity across bins does not affect our results. As a first pass, we present the distribution of firms across bins and industries at the 2-digit NAICS level. 6

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We find that no particular industry substantially dominates a bin, as summarized in Table A.1. Most firms belong to the manufacturing sector, and typically bins 1 and 2 contain the vast majority of firms. Notable exceptions are NAICS 22 (utilities) which contains a very high proportion of firms that fall in bin 4, and NAICS 62 (health care and social assistance) where almost half the firms are in bins 3 and 4. This broadly flat composition across industries also holds at more disaggregated levels, as shown in Tables A.2 and A.3. As a more rigorous test of whether heterogeneity is the driving force of our results, we show that the same price-setting trends across bins hold even after controlling for industry fixed effects at 2-, 3-, and 4digit NAICS sectoral levels. Figures A.3 and A.4 are in manufacturing, it is also natural to wonder if our results are valid only for these sectors. This is not the case as we show in Figures A.5 and A.6. The trends in frequency and size remain whether we take out manufacturing sectors or whether we compute these statistics separately for each two-digit manufacturing sector. As a final step to account for the effect of heterogeneity through regression, we explicitly use a set of detailed fixed effects as a filter. That is, we filter out month-, product-, and firm-level fixed effects from price changes, with product-level fixed effects defined at the 4-, 6- and 8-digit PPI product codes. Then, we ask how much in the variation these fixed effects can explain. We estimate the following specification regarding variation in |Δp|, the absolute size of price changes: |Δp|i,f,p,t = α0 {DM onth

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m m=12 }m=1

P roduct F irm + α1 Di,f,p,t + α2 Di,f,p + i,f,p,t

(2)

where i denotes an item, f a firm, p a product at the 4-, 6-, and 8-digit level, and t time. Dummies are for months, products, and firms. We show in Table A.4 that the variation in price changes explained by these fixed effects is at most 29%. Importantly, trends across bins are also not due to a particular kind of heterogeneity: varying elasticities of demand or degrees of substitution. We find that the positive relationship between frequency of price changes and number of goods, and the negative relationship between size of price changes and number of goods continue to hold even when we tightly control for varying elasticities of demand or substitution. We verify this by first picking firms that sell goods in a narrow product code provided by the BLS at the 6-digit level, excluding firms which sell in multiple product codes. Then, we compute the median fraction of price changes, the absolute size of price changes, and the number of goods sold by the firms at a point in time. Third, we run two regressions: first, we regress the frequency of price changes on the number of goods and second, the absolute size of price changes on the number of goods. We take the median of the estimated coefficients across all product codes, and then report summary statistics of these medians over time. Tables A.5 and A.6 present our results. Finally, we verify in detail that our results on small price changes are robust to various alternative definitions, as well as the concerns brought forward in recent work by Eichenbaum et al. (2013). Recall that in our baseline results, we defined small price change as |Δpi,j,t | ≤ κ|Δpi,t | where i is a good of firm j and κ = 0.5. In Table A.7 we show that our results continue to hold for κ = 0.10, 0.25, and 0.33. Our results also continue to hold if we define a price change as small relative to the industry mean, or relative to the good level mean. Table A.7 also shows that our results are strongly robust if we define small price changes in terms of absolute values. That is, price changes that are less than 0.25%, or 0.5%, or 1%. Our findings are robust to the recent concerns brought forward by Eichenbaum et al. (2013): measurement error, for example induced by bundling or uncontrolled forms of quality changes, could falsely generate many small price changes. However, we find that this is not a problem for our PPI data. When we exclude the six-digit NAICS sectors that likely encompass the problematic ELIs from the CPI identified by Eichenbaum et al. (2013), our overall, estimated fraction of small price changes remains essentially unchanged across definitions. The reason is that many of the ELIs that may be problematic in the CPI are not in the set of manufactured producer goods which are the bulk of our data. Additionally, we try focusing only on price changes that are bigger than one penny, or larger than 0.01%, or 0.1%. This leaves our results mostly unchanged even though the overall fraction somewhat falls by up to 3.5%. 3.2. Adjustment Decisions and Synchronization To further improve our understanding of individual pricing decisions, we go beyond providing aggregate statistics, and estimate a discrete choice model of pricing decisions. We find some important results: 7

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First, there is substantial synchronization of price changes within a firm which suggests a vital role played by strategic complementarities in price-setting. Second, there is substantial synchronization of individual adjustment decisions at the firm level relative to the industry, with strong trends as the number of goods increases. Third, we find evidence for elements of state-dependent pricing in response to fundamental economic variables.

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3.2.1. Synchronization Within Firms and Across Industries First, results show substantial synchronization of individual adjustment decisions at the firm level and especially relative to the industry, with strong trends as the number of goods increases. To arrive at this result, we estimate a multinomial logit model of the following form: P r(Yi,j,t = 1, 0, −1|Xi,j,t = x) = Φ(βXi,j,t )

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(3)

where Yi,j,t is an indicator variable for upwards, no, or downwards price changes of good i produced by firm j at time t, with 0 as the base category. We use estimates of β to report marginal effects on the change in the probability of adjustment, given one-standard-deviation changes around the mean of Xi,j,t . We focus on estimation separately by bins which distills out trends of how multi-product firms are different from singleproduct firms. However, since the results for the PPI as a whole are of independent interest, we estimate the model on pooled data across all good bins as well. Our explanatory variables Xi,j,t include three sets of variables: First, we try to capture the extent of synchronization in price setting at the firm and the sectoral level. To this purpose, we use the fractions of same-signed price changes within the firm, and the same six-digit NAICS sector, excluding the price change of the good we are trying to explain. Second, we include a dummy for product replacement where we can identify it by changes in the base price at resampling. Finally, we control for energy and food-inclusive CPIs, the number of employees in the firm, industry fixed effects, month fixed effects, time trends and as a measure of marginal costs, we also include the average price change of goods in the same firm and six-digit NAICS sector. We supplement our data with monthly inflation rates from the OECD “Main Economic Indicators (MEI).” We use both CPI inflation including food and energy prices, as well as excluding food and energy prices. Since we find no qualitative difference in our subsequent results, we only report results from the inclusive CPI measure. We find that there is pervasive synchronization both at the firm level and the industry level, exhibiting clear trends. The strongest synchronization is at the level of the firm: When the fraction of price changes of the same sign in a firm changes around its mean by one standard deviation, the pooled data tell us that the probability of a downward price change of a good increases by 8.82 percentage points, while it increases for an upward price change by 14.73 percentage points.6 Table 2 shows this result, as well as the strong trends across bins: as the number of goods increases, the statistical and economic importance for adjustment decisions of what happens within a firm monotonically increases. When the fraction of price changes of the same sign changes by one standard deviation around the mean, the effect goes up from 5.38 percentage points to 15.78 percentage points for negative price changes, and from 9.61 percentage points to 24.24 percentage points for positive price changes. These large complementarities in adjustment decisions within firms plus the strong trends underline our message that multi-product firms price differently than a collection of single-product firms. The trends also suggest that the importance of firm-specific shocks becomes more important as the number of goods increases. Comparing the explanatory power from a regression with and without firm-specific variables also emphasizes their importance: when we omit the firm-specific fraction and the average price change of goods from an otherwise unchanged regression, the R2 goes down from 48% to 30%. The R2 still continues to show a positive trend with the number of goods. The importance of the multi-product dimension for pricing is further reinforced when we consider synchronization with decisions in the industry: There, we find that synchronization is much weaker, especially 6 To

avoid cluttering, we do not report p-values, but all the results we report are statistically significant.

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as the number of goods increases. On average, the probability of a downward price change of a good increases by 1.32 percentage points, while the probability of an upward price change of a good increases by 2.27 percentage points when the industry-level fraction of price changes of the same sign moves around the mean by one standard deviation. As the number of goods increases, the effect goes down from 3.07 percentage points to 0.21 percentage points for positive price changes, and from 2.10 percentage points to −1.51 percentage points for negative price changes. Here, it is important to emphasize that our synchronization results are not due to a purely statistical effect. One could think that the variance of the fraction of price changes of the same sign in a firm decreases as the number of goods increases. This could imply a larger estimated synchronization coefficient. However, there is no such trend in the data for the variance of fractions. In fact, the opposite holds true. 3.2.2. State-dependent response to inflation It is a long-debated and important question in monetary economics whether price setting is timedependent or state-dependent. We contribute some evidence by explicitly considering the role played by inflation, an important aggregate shock, for pricing decisions. We find what one would expect from a model where firms adjust prices in a state-dependent fashion. As Table 2 shows, the likelihood of a price decrease decreases with higher CPI inflation while the likelihood of a price increase increases. When CPI inflation changes by one standard deviation around the mean, the probability of a negative price change of a good decreases by 0.21 percentage points, while the probability of a positive price change increases by 0.17 percentage points. The effects are both statistically and economically significant. 3.2.3. Robustness We conduct a battery of robustness tests for these good-level regressions. First, we consider different levels of aggregation for our definition of the industry variable. In conducting this extension, in addition to checking for robustness, we are motivated by the theoretical results in Bhaskar (2002) that synchronization is likely to be higher within groups with higher elasticity of substitution among goods, that is, at a more disaggregated industry level. Table A.8 in Online Appendix A presents results using an industry classification at the 2-digit NAICS level. It shows that at this higher level of aggregation, prices in the pooled data specification are much less synchronized at the industry level, compared to Table 2. Table A.8 also shows results for the four good bins separately. The general result of higher firm level synchronization and lower industry level synchronization as we move to higher good bins is robust to this alternate definition of industry.7 Compared to Table 2 and as predicted by Bhaskar (2002), however, we see the significant extent to which the industry level synchronization has decreased across all bins. Second, we check that clustering standard errors at the industry level does not affect our findings. Third, we use a polynomial function for the size of firms to control for non-linear size effects in the regressions. Our results do not change due to this modification. Finally, we estimate the multinomial logit model only for the adjustment decisions for the largest sales-value item of each firm. Again, our results do not change due to this modification, in particular, with respect to synchronization.8 4. Theory

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Compared to the existing literature, our main theoretical challenge is to explain the various trends we observe in price setting as we vary the number of goods per firm, an analysis that has not been undertaken before. We do so by developing a model of pricing by state-dependent multi-product firms. We find that varying only one parameter – the firm-specific menu cost of price adjustment – allows us to match the trends 7 This result is similar to that of Cornille and Dossche (2008) and Dhyne and Konieczny (2007) who use the Belgian PPI and the CPI respectively. They use the Fisher-Konieczny measure of synchronization and find that prices tend to be more synchronized at a more disaggregated industry level. Neither of these papers look at the level of the firm, however, which is the focus of our paper, and also the factor that quantitatively matters most in our data. 8 The results mentioned in the second robustness paragraph are available upon request from the authors.

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in the data. Since the mapping from the good bins that we construct in the empirical section to the number of goods in the model is not direct, we view our exercise in this section as qualitative in nature. Nonetheless, the results from simulations of our model allow us to validate modeling features that are needed to explain the empirical trends. 4.1. Model We use a partial equilibrium setting of a firm that decides each period whether to update the prices of its n goods indexed by i ∈ (1, 2, 3), and what prices to charge if it updates. Compared to the work of Sheshinski and Weiss (1992) and Alvarez and Lippi (2014) we allow for a stochastic aggregate shock and correlated idiosyncratic shocks across goods produced by a firm, while compared to Midrigan (2011), we solve for equilibrium as we vary n. In particular, the latter variation allows us to make two important contributions. First, we can solve for trends in price-setting behavior with respect to how many goods firms produce. Second, we can compute trends in synchronization of price-setting by comparing 2-good and 3-good firms. Since no measures of synchronization can be computed in the one-good case, the comparison of 2-good and 3-good firms is necessary to model trends in synchronization of adjustment decisions. This focus on synchronization also additionally distinguishes our model from the aforementioned papers. 4.1.1. Production Technology and Demand The firm produces output ci,t of good i using a technology that is linear in labor li,t : ci,t = Ai,t li,t

(4)

where Ai,t is a good-specific productivity shock that follows an exogenous process: ln Ai,t = ρiA ln Ai,t−1 + iA,t

(5)

2 2 where E[iA,t ] = 0 and cov(Ai,t , Aj,t ) = σA for i = j and σA for i = j. i ,Aj The firm’s product i is subject to the following demand:

 ci,t =

pi,t Pt

−θ Ct

i = 1, 2, ..., n

(6)

where Ct is aggregate consumption, Pt is the aggregate price level, pi,t is the price of good i, and θ is the ¯ We also elasticity of substitution across goods. In this partial equilibrium setting, we normalize Ct = C. assume that the price level Pt exogenously follows a random walk with a drift: ln Pt = μP + ln Pt−1 + P,t 380

(7)

where E[P,t ] = 0 and var(P,t ) = (σP )2 . Given our assumption about technology, the real marginal cost of the firm for good i, M Ci,t , is therefore given by: M Ci,t = Wt /(Ai,t Pt ), where Wt is the nominal wage. We ¯ normalize Wt /Pt = w. Given this setup, the firm maximizes the expected discounted sum of profits from selling all of its goods. Total period gross profits are given by: πt =

n   pi,t i

w − Pt Ai,t



pi,t Pt

−θ ¯ C.

(8)

The problem of the firm is to choose whether to update all prices in a given period, and if so, by how much. Whenever it updates prices, it has to pay a menu cost K, which we discuss next.

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4.1.2. Menu Cost Technology In particular, this menu cost comes in the form of a constant, firm-specific cost K(n) > 0 for adjusting one, or more than one, of its prices. This assumption of firm-specificity directly implies that the firm will either adjust all its prices at the same time or adjust none at all. In our dataset, this is a good first-order assumption: conditional on observing at least one adjustment per firm, the total fraction of goods adjusting in a firm is 0.75. A key feature of our adjustment technology is economies of scope: The cost of changing prices depends on the number of goods produced by the firm and in particular, we assume that: ∂ 2 K(n) ∂K(n) > 0 and < 0. ∂n ∂n2

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(9)

These assumptions mean that the cost of changing prices increases monotonically with the number of goods, and that there are increasing cost savings as more and more goods become subject to price adjustment. Therefore, there are economies of scope in the cost of changing prices which will allow us to match the trends in the data. A priori, however, it is unclear if we can match all observed trends by varying only one parameter K: On the one hand, it is very intuitive that economies of scope imply smaller and more frequent price changes as the number of goods increases per firm. On the other hand, other key statistics of the distribution or synchronization could show any trend. How should we understand the nature of these menu costs? We view them very broadly as a general fixed cost of price adjustment, not as the literal cost of relabeling the price tags of goods. Blinder et al. (1998) provides some evidence for this general interpretation. While we do not model the adjustment process in detail, one can for example think about the adjustment technology as a fixed cost of paying a manager to change prices: it is costly to hire him in the first place, but much less costly to have him adjust the price for each additional good. Stella (2011), using a structural estimation approach, provides evidence that indeed a common component in a multi-product setting accounts for 21% to 90% of adjustment cost independent of the number of goods, with the remainder attributable to variable costs. We conclude from the literature that there is empirical evidence supporting our modeling choice. We leave it to future empirical work to further dissect the underlying micro mechanisms. 4.2. Results We next first discuss the computation and calibration of our model, then our theoretical findings.

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4.2.1. Computation We solve the given problem for a firm that produces n = 1, 2, and 3 goods. First, we employ collocation methods to find the policy functions of the firm. Second, given the policy functions, we simulate time series of shocks and corresponding adjustment decisions for many periods. Finally, we compute statistics of interest for each simulation and good, and across simulations. We also estimate a multi-nomial logit model of adjustment decisions using the simulated data to compare our theoretical results with the empirical findings. Online Appendix B provides further details about computation and analysis. We mostly follow the literature in our choice of parameters. Since our model is monthly, we choose a 1 discount rate β of (0.96) 12 . We choose θ to be 4, which implies a markup of 33%. To parametrize the aggregate exogenous processes, we set the trend in aggregate inflation μP to be a monthly increase of 0.21% and the standard deviation σP to be 0.32%. These values are taken from Nakamura and Steinsson (2008). To parametrize the idiosyncratic productivity shock, we use the NBER productivity database covering 459 sectors from 1984 − 1996 and compute for our productivity process an estimate of ρ = 0.96 for the AR(1) parameter and σA = 2% for the standard deviation.9 For the firms with 2 and 3 goods, we use the same values for the persistence and variance of the all idiosyncratic productivity shocks, but following Midrigan 9 A persistent productivity process is a common specification in the literature. For example, Midrigan (2011) uses a random walk process for idiosyncratic productivity shocks.

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(2011), we allow for a correlation of 0.65 among the good-specific productivity shocks. Finally, we use a value of K such that menu costs are a 0.40% of steady-state revenues for the 1-good firm, 0.70% for a 2-good firm, and 0.80% for a 3-good firm. These choices for K allow us to replicate, as shown below, the trends across the good bins that we document empirically. 4.2.2. Findings We find that the model predicts clear and systematic trends in the key price-setting statistics as we increase the number of goods from 1 to 3. The trends align, qualitatively, with our empirical finding that multi-product firms behave systematically different from a collection of multiple single-product firms. We present the main results from our simulations in Table 3. First, we find that the frequency of price changes goes up from 13.89% to 20.21% while the mean absolute size of price changes goes down from 5.42% to 4.18% as we increase the number of goods. Second, the decrease in the absolute size of price changes also holds for both positive and negative prices changes: they go down from 5.53% to 4.34% and from −5.24% to −3.95% respectively. Third, the fraction of small price changes increases from 2.29%, barely none, to 19.54%. Thus, while firms with more goods change prices more frequently, they do so by smaller amounts on average. Fourth, we also see that the fraction of positive price changes decreases from 62.63% to 60.31%. Thus, as in the data, firms with more goods adjust downwards more frequently. Finally, the model predicts that kurtosis increases from 1.45 to 2.03, again consistent with our empirical findings. What is the mechanism behind our results? For simplicity, compare a 1-good firm with a 2-good firm. For the case of a 2-good firm, when the firm decides to pay the firm-specific menu cost to adjust one of the prices, it also gets to change the price of the other good for free. This leads to a higher average frequency of price changes. At the same time, for the 2-good firm, since a lot of price changes happen even when the desired price is not very different from the current price of the good, the mean absolute size of price changes is lower. This smaller mean also implies that the fraction of “small” price changes is higher for the 2-good firm. In fact, for the 1-good firm, which is the standard menu cost model, the fraction of small price changes is negligible because in that case, the firm adjusts prices only when the desired price is very different from the current price. What causes the decrease in the fraction of positive price changes? With trend inflation, firms adjust downwards only when they receive very big negative productivity shocks. With firm-specific menu costs, since the firm adjusts both prices when the desired price of one good is very far from its current price, it is now more sustainable to have a higher fraction of downward price changes. Finally, kurtosis increases as we go from one good to two goods because of a higher fraction of price changes in the middle of the distribution. Next, we address trends in synchronization of individual good-level price changes. Using simulated data, we run the same multinominal logit regression as in the empirical section given by specification (3). Our explanatory variables now include the fraction of same-signed adjustment decisions at time t within the firm and Πt is the inflation rate at time t. It is important to emphasize here the need to go beyond the 2-good case in Midrigan (2011) or Sheshinski and Weiss (1992), and consider a 3-good case. Otherwise, we cannot check if the model predicts trends in synchronization of price changes that are consistent with the empirical findings. We find from our simulations that the strength of synchronization, that is the coefficient estimate in the logit regressions for the fraction of other goods of the firm changing in the same direction, increases as we go from a 2-good firm to a 3-good firm. Table 3 shows this. This effect is first simply due to the underlying economies of scope: they increase the probability of simultaneous adjustment decisions in a firm as the number of goods increases. Second, correlated shocks in our model also play a role for synchronization since, as is intuitive with correlated shocks across goods, desired prices of goods within the firm become more likely to move in the same direction. Importantly, as is the case empirically, this is also the case with both upwards and downwards price changes. In addition, our simulation results make clear that the simulations predict that positive price adjustment decisions are more synchronized than negative adjustment decisions since the synchronization coefficient for upwards price adjustment decisions is always higher than the coefficient for downwards adjustment decisions.

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This difference in synchronization probabilities is due to positive trend inflation. If we omit positive trend inflation from the model, the difference disappears. The model thus matches our findings on synchronization established in the empirical section. 4.2.3. Robustness In this section, we discuss alternative modeling mechanisms to economies of scope in menu costs. Our main finding is that each alternative mechanism can match some of the trends, but it usually also fails in matching some other trends. This validates the choice of our main, parsimonious model while suggesting which other mechanisms may be important depending on other modeling interests. All detailed results, if not in Online Appendix B, are available on request. First, we investigate whether correlated shocks are the most important feature of the model as opposed to economies of scope in menu costs. When we shut down correlation of productivity shocks by setting σAi ,Aj = 0, in fact we can still match qualitatively all of our trends in pricing statistics.10 We summarize this experiment in Table B.1 in Online Appendix B. Moreover, we have also simulated a model with correlation of the productivity shocks but without complementarities in menu costs. Table B.2 summarizes this modification, showing that correlation alone cannot generate the trends found in the data. Thus, economies of scope in menu costs are critical to generate the trends in the model. In particular, just firm-specific menu costs, without economies of scope and absent correlation, are not enough to replicate the trends from our empirical section. Table B.3 shows the results from a model specification where menu costs are firm-specific, but the per-good menu cost is the same across firms with 1, 2, or 3 goods. Second, we investigate the possibility that perhaps our results are driven by the fact that firms that produce more goods also produce more substitutable goods. We simulate a specification of 2-good and 3-good firms with no economies of scope in the cost of adjusting prices and no correlation of productivity shocks but with an elasticity of substitution among goods that is higher than that of the 1 good firm. While matching the trends in frequency, size, and fraction of positive price changes, we generally find that this specification fails to match trends in the fraction of small price changes, kurtosis, and synchronization as we increase the number of goods. Third, there might be concern that our synchronization results could be due to purely mechanical, statistical reasons. To investigate this possibility, we perform two tests. In the first, we run a Monte-Carlo exercise based on a simple statistical model of price changes. We model price changes as i.i.d. Bernoulli trials with a fixed probability of success. Using simulated time series for an arbitrary number of goods, we estimate our synchronization equation. We find no mechanical trend in synchronization. In the second test, we use the single-good firm from our simulation exercise and model a multi-product firm as a collection of multiple, independent single-product firms. That is, a 2-good firm now is simply a collection of two single-good firms, with no economies of scope in cost of adjusting prices and no correlated shocks. Similarly, a 3-good firm is a collection of three independent single-good firms. When we run our synchronization test with simulated data from these specifications, we find that the model cannot produce synchronization results that are consistent with the empirical findings. For example, the synchronization coefficients estimated using simulated data become smaller as we move from two single-good to three single-good firms. Incidentally, this specification of multiproduct firms as multiple single-good firms fails to match any of our aggregate trends in price-setting, such as the frequency of price changes. We summarize results in Table B.4 in Online Appendix B. These results highlight the need to model a multi-product firm as distinctly different from an aggregate of multiple single-product firms. Fourth, we consider two alternative demand specifications. In the first specification, we allow for non-zero cross-elasticities of demand among the goods per firm, which is precluded in our baseline model with CES demand. Again, we shut down economies of scope and correlation of productivity shocks in our experiments. While this experiment generates a higher fraction of small price changes and greater kurtosis, we find that it cannot account for the trend in the size of price changes. In the second specification, we allow for idiosyncratic 10 It is still worthwhile to include this feature in the baseline model since it is intuitive, has been used frequently in the literature, and does generate higher synchronization of price changes.

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demand shocks as an alternative to productivity shocks. We find that demand shocks are unable to generate the fraction of negative price changes that we observe in the data. The intuition for this result is that large negative demand shocks increase the menu costs relative to current profits, thus making negative price changes too costly. 5. Conclusion

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Our results show that the pricing behavior of multi-product firms is systematically related to the number of goods per firm. A model with firm-specific menu costs where firms are subject to both idiosyncratic and aggregate shocks as well as trend inflation can match the empirical findings. Two directions for future work follow directly from the results: First, since the number of goods is endogenously chosen by firms, it appears important to understand what drives this choice. We conjecture that supplementary data on changes in the scope of firms over the business cycle may provide valuable insights. Second, while the majority of price adjustments happens jointly among goods within firms, this is not always the case. So, to fully match the facts, it will be necessary to introduce a modification to price adjustment technology, such as good-specific menu costs. References Alvarez, F., Lippi, F., 2014. Price setting with menu cost for multiproduct firms. Econometrica 82, 89–135. Bernard, A., Redding, S., Schott, P., 2010. Multi-product firms and product switching. American Economic Review 100, 70–97.

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Bhaskar, V., 2002. On endogenously staggered prices. Review of Economic Studies 69, 97–116. Blinder, A.S., Canetti, E.R.D., Lebow, D.E., Rudd, J.B., 1998. Asking About Prices: A New Approach to Understanding Price Stickiness. Russell Sage Foundation, New York, N.Y. Boivin, J., Giannoni, M.P., Mihov, I., 2009. Sticky prices and monetary policy: Evidence from disaggregated US data. American Economic Review 99, 350–84.

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Carvalho, C., 2006. Heterogeneity in price stickiness and the real effects of monetary shocks. The BE Journal of Macroeconomics (Frontiers) 2. Cornille, D., Dossche, M., 2008. Some evidence on the adjustment of producer prices. Scandinavian Journal of Economics 110, 489–518.

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Dhyne, E., Konieczny, J., 2007. Temporal distribution of price changes: Staggering in the large and synchronization in the small. Research series 200706-02. National Bank of Belgium. Eichenbaum, M.S., Jaimovich, N., Rebelo, S., Smith, J., 2013. How frequent are small price changes? Forthcoming. American Economic Journal: Macroeconomics. Fisher, T.C.G., Konieczny, J.D., 2000. Synchronization of price changes by multiproduct firms: Evidence from Canadian newspaper prices. Economics Letters 68, 271–277.

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Gagnon, E., 2009. Price setting during low and high inflation: Evidence from Mexico. Quarterly Journal of Economics 124, 1221–1263. Goldberg, P., Hellerstein, R., 2009. How rigid are producer prices? Federal Reserve Bank of New York Staff Reports .

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Goldberg, P., Khandelwal, A., Pavcnik, N., Topalova, P., 2010. Multi-product firms and product turnover in the developing world: Evidence from India. Review of Economics and Statistics 92, 1042–1049.

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Golosov, M., Lucas Jr., R.E., 2007. Menu costs and Phillips curves. Journal of Political Economy 115, 171–199. Gopinath, G., Itskhoki, O., 2010. Frequency of price adjustment and pass-through. Quarterly Journal of Economics 125, 675–727. 565

Klenow, P.J., Malin, B.A., 2010. Microeconomic evidence on price-setting, in: Friedman, B.M., Woodford, M. (Eds.), Handbook of Monetary Economics. Elsevier. volume 3 of Handbook of Monetary Economics. chapter 6, pp. 231–284. Lach, S., Tsiddon, D., 1996. Staggering and synchronization in price-setting: Evidence from multiproduct firms. American Economic Review 86, 1175–96.

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Lach, S., Tsiddon, D., 2007. Small price changes and menu costs. Managerial and Decision Economics 28, 649–656. Midrigan, V., 2011. Menu costs, multi-product firms, and aggregate fluctuations. Econometrica 79, 1139– 1180.

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Nakamura, E., Steinsson, J., 2008. Five facts about prices: A reevaluation of menu cost models. Quarterly Journal of Economics 123, 1415–1464. Sheshinski, E., Weiss, Y., 1992. Staggered and synchronized price policies under inflation: The multiproduct monopoly case. Review of Economic Studies 59, 331–59. Stella, A., 2011. The Magnitude of Menu Costs: A Structural Estimation. Working Paper. Federal Reserve Board.

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6. Tables Table 1: Summary Statistics by Number of Goods

1-3 2996 427 17.15 2.21 0.01 0.77 1 3 1.78 2 3 1.02 (0.01) 9111

Mean Employment Median Employment % of Prices Mean # of Goods Std. Error # Goods Std. Dev. # Goods Minimum # of Goods Maximum # of Goods 25% Percentile # Goods Median # Goods 75% Percentile # Goods Coeff. of Var. of Price Changes Number of Firms

Number of Goods 3-5 5-7 1427 1132 155 195 43.53 18.16 4.05 6.06 0.00 0.01 0.34 0.40 3.01 5.01 5 7 3.94 5.91 4 6 4 6 1.15 1.30 (0.01) (0.02) 13577 3532

>7 1016 296 21.16 10.26 0.11 4.88 7.02 77 7.98 8 11.77 1.55 (0.02) 2160

We calculate mean and median employment by taking means and medians of the number of employees per good across firms in a bin/overall category. % of Prices denotes the fraction of prices in the PPI set by firms in each bin. For the coefficient of variation of price changes, we compute it at the level of the firm pooling all price changes. Then, we take medians across firms.

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Table 2: Marginal Effects by Number of Goods, ± 1/2 Std. Dev.

1-3 Goods

3-5 Goods

5-7 Goods

>7 Goods

All Goods

All Goods†

2.10% 5.38% -0.08%

1.41% 6.52% -0.14%

0.89% 9.92% -0.17%

-1.51% 15.78% -0.54%

1.32% 8.82% -0.21%

6% -0.25%

3.07% 9.61% 0.17% 42.85%

2.12% 11.46% 0.10% 47.93%

1.55% 15.94% 0.14% 48.33%

0.21% 24.24% 0.35% 49.10%

2.27% 14.74% 0.17% 47.56%

10.29% 0.16% 29.33%

Negative Change Fraction Industry Fraction Firm πCP I Positive Change Fraction Industry Fraction Firm πCP I R2

The table shows the bin-specific marginal effects in percentage points of a one-standard deviation change in the explanators around the mean on the probability of adjusting prices upwards or downwards. Marginal effects are calculated for a model estimated for each bin separately as well as on the pooled data. All reported effects are statistically significantly different from zero. †Results from omitting firm-specific variables are shown in the last column.

Table 3: Results of Simulations

Frequency of price changes Size of absolute price changes Size of positive price changes Size of negative price changes Fraction of positive price changes Fraction of small price changes Kurtosis 1st Percentile 99th Percentile

1 Good 13.89% 5.42% 5.53% -5.26% 62.92% 2.41% 1.45 -7.24% 7.45%

2 Goods 17.85% 4.60% 4.66% -4.52% 62.03% 15.92% 1.72 -7.94% 8.36%

3 Goods 20.21% 4.18% 4.34% -3.95% 60.31% 19.54% 2.03 -8.68% 9.04%

Synchronization measures: Fraction, Upwards Adjustments Fraction, Downwards Adjustments Correlation coefficient Menu Cost

0.40%

31.29 30.72 0.65 0.70%

38.65 37.72 0.65 0.80%

We perform stochastic simulations of our model in the 1-good, 2-good and 3-good cases and record price adjustment decisions in each case, allowing for correlation of the productivity shocks Ai,t in the multi-good cases. Then, we calculate statistics for each case as described in the text. In the 2-good and the 3-good cases, we report the mean of the good-specific statistics. We obtain the synchronization measure from a multinomial logit regression analogous to the empirical multinomial logit regression. We control for inflation. Menu costs are given as a percentage of steady state revenues.

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7. Graphs

Mean Frequency of Price Changes with 95% Bands 31%

Frequency of price changes

29% 27% 25% 23% 21% 19% 17% 15% 1-3

3-5

5-7

>7

Number of goods Figure 1: Mean Frequency of Price Changes with 95% Bands

Based on the PPI data we group firms by the number of goods. We compute the mean frequency of price changes in these groups in the following way. First, we compute the frequency of price change at the good level. Then, we compute the median frequency of price changes across goods at the firm level. Finally, we report the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96 * std. error across firms.

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Mean Fraction of Positive Price Changes with 95% Bands 66%

Fraction of positive price changes

65% 65% 64% 64% 63% 63% 62% 62% 61% 61% 1-3

3-5

5-7

>7

Number of Goods Figure 2: Mean Fraction of Positive Price Changes with 95% Bands

Based on the PPI data we group firms by the number of goods. We compute the mean fraction of positive price changes in these groups in the following way. First, we compute the number of strictly positive good level price changes over all non-zero price changes for a given firm. Then, we report the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96 * std. error across firms.

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Mean Absolute Size of Price Changes with 95% Bands 9.00%

Absolute size of price changes

8.50%

8.00%

7.50%

7.00%

6.50%

6.00% 1-3

3-5

5-7

>7

Number of goods Figure 3: Mean Absolute Size of Price Changes with 95% Bands

Based on the PPI data we group firms by the number of goods. We compute the mean absolute size of price changes in these groups in the following way. First, we compute the percentage change to last observed price at the good level. Then, we compute the median size of price changes across goods at the firm level. Then, we report the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96 * std. error across firms.

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Figure 4: Mean Size of Positive and Negative Price Changes with 95% Bands

Based on the PPI data we group firms by the number of goods. We compute the mean size of positive price changes in these groups in the following way. First, we compute the percentage change to last observed price at the good level. Then, we compute the median size of price changes across goods at the firm level. Then, we report the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96 * std. error across firms.

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Mean Fraction of Small Price Changes with 95% Bands 60%

Fraction of small price changes

55%

50%

45%

40%

35%

30% 1-3

3-5

5-7

>7

Number of goods Figure 5: Mean Fraction of Small Price Changes with 95% Bands

Based on the PPI data we group firms by the number of goods. We compute the mean fraction of small price changes in these groups in the following way. First, we compute the fraction of price changes that are smaller than 0.5 times the mean absolute percentage size of price changes across all goods in a firm. Then, we report the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96 * std. error across firms.

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Mean Kurtosis of Price Changes with 95% Bands 20

Kurtosis of price changes

18 16 14 12 10 8 6 4 1-3

3-5

5-7

>7

Number of goods Figure 6: Mean Kurtosis of Price Changes with 95% Bands

Based on the PPI data we group firms by the number of goods. We compute the mean kurtosis of price changes in these groups in the following way. First, we compute the kurtosis of price changes at the firm level, defined as the ratio of the fourth moment about the mean and the variance squared of percentage price changes. Then, we report the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96 * std. error across firms.

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Figure 7: Mean First and 99th Percentile of Price Changes with 95% Bands

Based on the PPI data we group firms by the number of goods. We compute the mean size of positive price changes in these groups in the following way. First, we compute the percentage change to last observed price at the good level. Then, we compute the median size of price changes across goods at the firm level. Then, we report the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96 * std. error across firms.

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