Prices are Sticky after All

Recent studies say prices change every four months. Economists have interpreted this high frequency as evidence against the importance of sticky prices for the monetary transmission mechanism. Theory implies that if most price changes are regular, as they are in the standard New Keynesian model, then this interpretation is correct. But, if most price changes are temporary, as they are in the data, then it is incorrect. Temporary changes have two striking features: after a change, the nominal price returns exactly to its pre-existing level, and temporary changes are clustered in time. Our model, which replicates these features, implies that temporary changes cannot offset monetary shocks well, whereas regular changes can. Since regular prices are much stickier than temporary ones, our model, in which prices change as frequently as they do in the micro data, predicts that the aggregate price level is as sticky as in a standard model in which micro level prices change once every 12 months. In this sense, prices are sticky after all.

price changes in this series are temporary and have two distinctive features: after a temporary change, the nominal price returns exactly to its pre-existing level, and temporary changes are clustered in time-for example, a change in one direction is usually followed quickly by another change in the opposite direction. These two distinctive features imply that for individual price series, even though there is a great deal of high-frequency price ‡exibility (in that actual prices change frequently), there is also a great deal of low-frequency price stickiness (in that the trend price changes infrequently).
Standard New Keynesian models have only one type of price change and thus have no hope of generating the pattern of price stickiness observed in the micro data. In particular, these models can generate either highly ‡exible prices at both high and low frequencies or highly sticky prices at both high and low frequencies. What they cannot generate is what we see in the data: highly ‡exible prices at high frequencies and very sticky prices at low frequencies.
We extend the standard model to allow …rms to temporarily deviate from a sticky pre-existing price. We quantify our model and show that it reproduces the empirical micro pattern of regular and temporary price changes. We then study our model's implication for the degree of aggregate price stickiness in response to monetary policy shocks. It is quite di¤erent from that of a New Keynesian model. Unlike that model, our model, which is consistent with the micro data pattern, still implies large real e¤ects from monetary shocks.
The key prediction of our model is that temporary micro price changes cannot o¤set monetary shocks well, whereas regular changes can-but they do so only infrequently because regular prices are much stickier than temporary prices. To measure how sticky our model says aggregate prices are, we translate its results into those of a standard model. Our model, in which micro prices change as frequently as they do in the data, predicts that the aggregate price level is as sticky as it is in a standard model in which micro prices change once every 12 months. Aggregate prices, the model says, are sticky after all.
Our study is based on the salient features of the micro price data. To document these features, we study two sets of data: monthly price data from the U.S. Bureau of Labor Statistics (BLS) and weekly data from a chain of grocery stores, Dominick's Finer Food retail chain. Our primary focus is on the statistics from the larger, more comprehensive BLS data set, but we use the Dominick's data to demonstrate the robustness of that analysis. We …nd that the Dominick's data set is also informative, despite its limited coverage, because these data are collected weekly and thus can better measure high-frequency price movements than the monthly data of the BLS. Moreover, the Dominick's data include quantities sold as well as prices, so that we can examine the extent to which temporary changes account for a disproportionate amount of goods sold.
We document in both data sets the two types of micro price changes and quantify their distinctive features. Recall that one of two key features of temporary price changes is that after such a change, the nominal price typically returns exactly, to the penny, to its level before the change. In the Dominick's data, this event occurs 80% of the time; in the BLS data, 50% of the time. In contrast, in both data sets, regular price changes almost never return the nominal price to its pre-existing level. The other key feature of temporary price changes is that they are clustered over time. In both data sets, for example, the probability that a temporary price spell ends in any particular period is about 50%. This means that a temporary change in one direction is often quickly followed by a change in the opposite direction.
Consider now the intuition for our main result that the aggregate price level is sticky even though micro prices change frequently. The model produces price series for individual goods similar to that in Figure 1. In the model, therefore, prices change frequently, but most of those changes re ‡ect temporary deviations from a much stickier regular price. When a …rm changes its price temporarily in a given period because of an idiosyncratic shock, it is also able to react to changes in monetary policy. These responses are, however, short-lived.
And whenever the price returns to the old price, it no longer re ‡ects the change in monetary policy. Moreover, since temporary price changes are highly clustered in time, they are less able to o¤set persistent changes in monetary policy. For example, a …rm that changes its prices four times in January and not at all the rest of the year is less able to respond to persistent money supply changes than a …rm that also changes its prices four times a year, but spreads those changes out over the year to, say, once a quarter. For these two reasons, even though micro prices change frequently, the aggregate price level is sticky. Our key insight is that what matters for how the aggregate price level responds to low-frequency changes in monetary policy is the degree of low-frequency micro price stickiness. Since in the data there is substantial low-frequency price stickiness, the aggregate price level is sticky as well.
Our model of temporary and regular price changes is motivated by evidence on the pricing practices of actual …rms. In particular, Zbaracki et al. (2004Zbaracki et al. ( , 2007 provide evidence that pricing is done at two levels: upper-level managers (at headquarters) set list prices, while lower-level managers (at the store level) choose the actual transaction (posted) prices. These researchers …nd that the managerial costs of changing list prices are much greater than the physical costs of changing posted prices. Moreover, lower-level managers must e¤ectively pay a time cost to be allowed to depart from the regular price set by the upper-level managers.
This interaction between lower-and upper-level managers is illustrated, for example, in the following quotation from an interview with a sales manager (Zbaracki et al. 2004, p. 524): I was a territory manager so I had no pricing authority. The only authority I had was to go to my boss and I would say, "OK, here is the problem I've got." He would say "Fill out a request and we will lower the price for that account." So that is how the pricing negotiations went. At that time I went up the chain to make any kind of adjustments I had to make . . . . My …ve guys have a certain level [of discount] they can go to without calling me. When they get to the certain point they have to get my approval.
We model this two-level decision-making process in a simple, reduced-form way. We assume that retailers set two prices: a list price and a posted price. The posted price is the price at which goods are sold. In our model, the list price matters because charging a posted price other than the list price entails a …xed cost. We think of this cost as standing in for the cost of the lower-level manager obtaining approval from the upper-level manager for temporarily charging a price that deviates from the list price. Changing list prices themselves entails another …xed cost, the managerial cost of upper-level decision making. We assume that the cost of changing posted prices is zero. (We also solved a version of the model with a …xed cost of changing posted prices and got results nearly identical to those presented here. For simplicity, we focus on the model without such costs.) Our model is purposely chosen to be a parsimonious extension of the standard menu cost model of, say, Golosov and Lucas (2007) and Midrigan (2007). Our extension has a di¤erent technology for changing prices, as we have described, and the addition of temporary shocks, which give a motive for temporary price changes. Speci…cally, in our model, …rms are subject to two types of idiosyncratic disturbances: persistent productivity shocks and transitory shocks to either the cost or the elasticity of demand for the …rm's product. The latter shocks are meant to capture in a simple way an idea popular in the industrial organization literature: that …rms face demand for their products with time-varying elasticity. Our simple extension allows the model to produce patterns of both temporary and regular price changes that are similar to those in the data.
A sizable literature in industrial organization has suggested explanations for the temporary price discounts (or sales) which account for the majority of temporary changes in the data. 2 Unfortunately, all of these explanations are about real prices and, hence, cannot explain a striking feature of the data: that the nominal price, after a temporary price discount, often returns exactly to the nominal pre-existing price. Indeed, this feature is the subtle low-frequency price stickiness which is at the heart of our results. Namely, even though there is a large amount of high-frequency variation in prices associated with temporary price changes, there is much less low-frequency variation, which is ultimately what matters for how aggregate prices respond to low-frequency variation in monetary policy.
On the empirical side, our work here is most closely related to that of Bils and Klenow (2004) and Nakamura and Steinsson (2008). As we have noted, Bils and Klenow show that the frequency of all price changes is fairly high, about once every 4.5 months in the BLS data. Nakamura and Steinsson study the same data and show that once temporary price cuts are removed, prices change infrequently, about every 8-11 months. The implicit rationalization for removing temporary price cuts is that they are somehow special and, to a rough approximation, can be ignored when determining the amount of price stickiness in the data.
Some may interpret our results as providing a theoretical rationale for removing temporary price cuts from the data, as Nakamura and Steinsson (2008) and Golosov and Lucas (2008) have done. Thus, here we brie ‡y use our theory to evaluate a similar procedurediscarding all temporary changes in favor of just regular prices. We …nd that compared to our model, this regular price procedure only slightly overstates the degree of aggregate price stickiness.
Our work is also related to the empirical work of Hosken and Rei¤en (2004) and Eichenbaum, Jaimovich, and Rebelo (forthcoming), who document that there is signi…cant low-frequency price stickiness in the micro data despite the large high-frequency price variation.
On the theory side, Guimarães and Sheedy (forthcoming) o¤er an alternative explanation for temporary price discounts (sales) arising from …rms pursuing mixed-price strategies.
Finally, Rotemberg (forthcoming) o¤ers another explanation for why temporary prices return to their previous level. His work shows how costs to the …rm of changing list prices-costs that act similarly to menu costs-can arise from the preferences of consumers.

The Pattern of Price Changes in the U.S. Data
We begin by documenting how prices change in our two U.S. data sets: the BLS monthly data and the Dominick's weekly data. Here we describe several regularities, or facts, that we see in these data. These facts help clarify the distinction between temporary and regular price changes and illustrate their properties. We use these facts to motivate our model.

A. The Data Sets
The BLS data set is the CPI Research Database used by Nakamura and Steinsson (2008). This data set contains prices for thousands of goods and services collected monthly by the BLS for the purpose of constructing the consumer price index (CPI) and covers about 70% of U.S. consumer expenditures. For our use in this work, Emi Nakamura and Jon Steinsson kindly computed statistics for us by applying an algorithm we devised (which we describe in the appendix) to the CPI Research Database. These statistics are available in their forthcoming work.
The Dominick's data set includes nine years (1989-97) of weekly store-level reports from 86 stores in the Chicago area on the prices of more than 4,500 individual products, organized into 29 product categories. The products available in this data base range from nonperishable foodstu¤s (for example, frozen and canned food, cookies, crackers, juices, sodas, and beer) to various household supplies (for example, detergents, fabric softeners, and bathroom tissue) as well as pharmaceutical and hygienic products. (For a detailed description of the data and Dominick's pricing practices, see the work of Hoch, Drèze, and Purk (1994), Peltzman (2000), and Chevalier, Kashyap, and Rossi (2003).)

B. Categories of Price Changes
To identify a pattern of price changes in the data, we wrote a simple algorithm which categorizes each change as either temporary or regular. We de…ne for each product an arti…cial series called a regular price series. This price is essentially a running mode of the original series. Given this series, every price change that is a deviation from the regular price series is de…ned as temporary, whereas every price change that coincides with a change in the regular price is de…ned as regular.
An intuitive way to think about our analysis is to imagine that at any point in time, every product has an existing regular price that may experience two types of changes: temporary changes, in which the price brie ‡y moves away from and then back to the regular price, and much more persistent regular changes, which are changes in the regular price itself. Our algorithm is based on the idea that a price is regular if the store charges it frequently in a window of time adjacent to that observation. The regular price is thus equal to the modal price in any given window surrounding a particular period, provided the modal price is used su¢ ciently often in that window. The algorithm is somewhat involved, so we relegate a formal description to the appendix.

C. The Facts
In Figure 2, we illustrate the results of applying our algorithm to several particular price series from the Dominick's data. On each of the four graphs, for each of the four products, the dashed lines are the raw data (the original posted prices), and the solid lines are the regular price series constructed with our algorithm. Just a glance at these graphs makes several facts about price changes clear: across the board, price changes are frequent and large, but most of them are temporary, and most temporary prices return to the pre-existing regular price. Table 1 reports statistics summarizing the facts about price changes that result from applying the algorithm. The …rst column of data is statistics from the BLS data set, in which all statistics are computed at a monthly frequency. We report revenue-weighted averages of the corresponding statistics at the level of product categories. (The product level statistics are available from Nakamura and Steinsson (forthcoming).) The second column of data in Table 1 is statistics from the Dominick's data set, in which all statistics are computed at a weekly frequency and each good weighted by its revenue share. The third column is statistics constructed by us from the Dominick's data by sampling the weekly data at a monthly frequency; here, as with the BLS data, we do not weight individual products by their revenue share.
Among the monthly BLS data statistics, we highlight some of the key features that motivate our model. First, most price changes in the data are temporary: 72% of price changes are temporary, with an average duration of about two months (1/.53). Second, about 50% of the time, the nominal price after a temporary price change returns to the exact nominal level it had before the change (the old regular price). Because of this feature of the data, although overall the frequency of price changes is large (22% of all prices change every month, so the average duration is 4.5 months), the frequency of regular price changes is much smaller, 7% per month, with an average duration of about 14.5 months. 3 The feature of the data that will become especially signi…cant to this analysis is that micro prices have a subtle type of low-frequency price stickiness. For example, the table shows that 75% of the time during a year, …rms charge exactly the same price for a good, namely, its annual mode. (Note that this statistic does not depend on our algorithm for identifying regular price changes.) When we combine that fact with the fact that …rms change prices once every 4.5 months, we see that prices tend to come back often to the same nominal level.
We will show, using our model, that it is this feature of the data that allows monetary shocks to have sizable e¤ects despite the high frequency of price changes.
Consider now the second data column in Table 1, where we report similar statistics for the Dominick's weekly data. The basic patterns evident in the BLS data are even more evident here. Nearly all price changes are temporary (94%), and after such changes, 80% of these prices come back to the pre-existing price. As a result, even though prices change relatively frequently, once every 3 weeks, the regular price changes occur much less often, once every 8 months. Unlike the BLS data, the Dominick's data also contain information on quantities. Using those data, we …nd that a disproportionate fraction of goods are sold during periods of temporary prices: even though temporary price changes occur only about a quarter of the time, almost 40% of goods are sold during these periods.
When comparing the Dominick's data to the BLS data, note that some of the di¤erence is coming from the frequency of sampling: Dominick's data are sampled weekly; the BLS data, monthly. To illustrate the role of sampling, we also sampled the Dominick's data at the monthly frequency and report the results in the third data column of Table 1. Doing so dramatically lengthens the implied duration of price spells, from 3 weeks (1/.33 weeks) in the weekly data to nearly 3 months (1/.36 months) in the monthly data. This di¤erence illustrates our contention that, at least with Dominick's data, the monthly sampled prices miss many of the high-frequency movements in prices that are reversed within a month.

A Menu Cost Model with Temporary Price Changes
Now we build a menu cost model with temporary price changes and use it to evaluate the relationship between the frequency of micro price changes and the degree of aggregate price stickiness. Here, we describe the model, quantify it, and demonstrate that it does a much better job of reproducing the pattern of changes in the data than the standard model does.
Our model is a simple extension of the standard menu cost model of Golosov and Lucas (2007). To account for the pattern of temporary and regular price changes in the data, we make two additional assumptions.
First, motivated in part by the work of Zbaracki et al. (2004) on the pricing practices of …rms, we assume that …rms choose two prices for each good: a list price P Lt as well as a posted price P t that the consumer faces. Intuitively, we think of the list price as the price set by the upper-level manager and the posted price as the price actually charged to the consumer. The posted price will equal the list price unless the lower-level manager takes a costly action to make it di¤er. Formally, the list price is relevant because every time the …rm posts a price that di¤ers from the list price, the …rm must incur a …xed cost : As a result, the posted price will deviate from the list price infrequently, only when the bene…t from doing so exceeds the …xed cost. We assume that changing list prices is costly and entails a …xed cost . (For simplicity, we set the cost of changing the posted price to zero and in an appendix available on request show that allowing for such a cost has virtually no e¤ect on our results.) Second, we allow for both transitory and permanent idiosyncratic productivity shocks.
The transitory and permanent shocks will help the model deliver the temporary and regular price changes in the data. As Golosov and Lucas (2007) do, we think of these shocks as a stand-in for all the idiosyncratic forces that make changing prices optimal for …rms.

A. Setup
Formally, we study a monetary economy populated by a large number of in…nitely lived consumers and …rms and a government. In each time period t, this economy experiences one of …nitely many events s t: We denote by s t = (s 0 ; : : : ; s t ) the history (or state) of events up through and including period t. The probability, as of period 0, of any particular history s t is (s t ). The initial realization s 0 is given.
In the model, we have aggregate shocks to the economy's money supply and idiosyncratic shocks to a …rm's productivity. In terms of the money supply shocks, we assume that the (log of) money growth follows an autoregressive process of the form where is money growth, is the persistence of , and " (s t ) is the monetary shock, a normally distributed i.i.d. random variable with mean 0 and standard deviation : We describe the idiosyncratic shocks below.

Consumers and Technology
In each period t, the commodities in this economy are labor, money, a continuum In this economy, consumers consume, trade bonds, work, and hold real money balances.
They also own the capital stock and rent it to intermediate good producers. The consumer problem is to choose consumption c(s t ); nominal labor l(s t ); investment x(s t ); nominal money balances M (s t ); and a vector of bonds fB(s t ; s t+1 )g s t+1 to maximize utility ; l s t subject to the budget constraint is the nominal wage, (s t ) are nominal pro…ts, and R (s t ) is the rental rate of capital. Note that capital is subject to adjustment costs, the size of which is determined by : Here we have complete, contingent, one-period nominal bonds. We let B(s t+1 ) denote the consumers' holdings of such a bond purchased in period t and state s t with payo¤s contingent on some particular state s t+1 in t + 1. We let Q(s t+1 js t ) denote the price of this bond in period t and state s t .
Consider, next, the technology for the intermediate good producers. The producer of intermediate good i uses the technology where y i (s t ) is the output of good i, k i (s t ) is the amount of capital it rents, l i (s t ) is the labor input, and m i (s t ) is the amount of materials used in the production process. The good-speci…c productivity has two components: a transitory one, z i (s t ) ; and a permanent one, a i (s t ): The transitory component follows a Markov process. The permanent component evolves according to where " i (s t ) is an i.i.d. shock (or innovation) to the permanent part of productivity with a distribution that we describe below. To keep the model stationary, we assume that intermediate good …rms exit with probability e and are replaced by …rms with a i (s t ) = 1: Below we describe the problem that the intermediate good …rms solve.
Consider, last, the …nal good producers. These …rms, who are perfectly competitive, purchase a continuum of intermediate goods and sell a …nal good to consumers and intermediate good …rms. The problem of a …nal good …rm is to choose the amount of each intermediate good y i (s t ) to purchase in order to maximize subject to the …nal good production function, We de…ne P (s t ) as the price of the …nal good, P i (s t ) as the price of good i purchased from an intermediate good …rm, and as the elasticity of substitution among intermediate inputs.
The solution to this problem is, then, which will be the demand function faced by the producer of intermediate good i: Here, is the minimum cost of producing one unit of the …nal good and, because of perfect competition, the …nal good's price.

The Intermediate Good Firm Problem
Consider, now, the problem of an intermediate good …rm in this economy. The …rm has a …xed cost, measured in units of labor, of changing its list prices. We refer to this cost as a menu cost. Let P L;i (s t 1 ) denote the …rm's list price from the previous period. This list price is a state variable for the …rm at the subsequent state s t : The …rm has two sets of pricing decisions. It can leave the list price unchanged at no cost, or it can pay a …xed cost and change the list price to P Li (s t ). A …rm can then either pay nothing and charge the list price P i (s t ) = P L;i (s t ) or pay and charge any price other than the list price. (We think of as an upper-level managerial cost of changing the list price and as the cost of a lower-level manager deviating from that list price, say, by o¤ering a temporary discount.) In our model, …rms face a mixture of idiosyncratic shocks-permanent and transitory.
Here …rms typically use a list price change to respond to the more permanent shocks and temporarily deviate from this list price in order to respond to transitory shocks.
To write the …rm's problem formally, …rst note that the …rm's period nominal pro…ts, excluding …xed costs at price P i (s t ); are is the nominal cost of producing one unit of intermediate good i and y i (s t ) is given by (5), where is a constant that depends on the parameters of the production function. The present discounted value of pro…ts of the …rm, expressed in units of period 0 money, is given by where L;i (s t ) is an indicator variable that equals one when the …rm changes its list price (P L;i (s t ) 6 = P L;i (s t 1 )) and zero otherwise, and T;i (s t ) is an indicator variable that equals one when the …rm temporarily deviates from the list price (P i (s t ) 6 = P L;i (s t )) and zero otherwise.
In expression (7), the term W (s t ) L;i (s t ) is the labor cost of changing list prices, which we think of as the menu cost, and W (s t ) T;i (s t ) is the cost of deviating from the list price.

Equilibrium
Consider, now, this economy's market-clearing conditions and the de…nition of equilibrium. The market-clearing condition on labor, requires that the sum of the labor used in production and the costs of making both list and temporary price changes equals total labor. The market-clearing condition for the …nal good is Note, for later use, that real output (value-added) at base period prices can be written as The market-clearing condition on bonds is B(s t ) = 0: An equilibrium for this economy is a collection of allocations for consumers fc i (s t )g i , x (s t ) ; and l(s t ); prices and allocations for …rms fP i (s t ); y i (s t ), l i (s t ), k i (s t ), m i (s t )g i ; and aggregate prices W (s t ); P (s t ); and Q(s t+1 js t ); all of which satisfy the following conditions: (i) the consumer allocations solve the consumers' problem; (ii) the prices and allocations of …rms solve their maximization problem; (iii) the market-clearing conditions hold; and (iv) the money supply processes satisfy the speci…cations above.
For convenience, we write the equilibrium problem recursively. At the beginning of s t ; after the realization of the current monetary and productivity shocks, the state of an individual …rm i is characterized by its list price in the preceding period, P L;i (s t 1 ); its permanent productivity component, a i (s t ); and the transitory productivity component, z i (s t ): We normalize all of the nominal prices and wages by the current money supply and let and use similar notation for other prices. With this normalization, we can write the state of an individual …rm i in s t as (p L; 1;i (s t ); a i (s t ); z i (s t )): Let (s t ) denote the measure over all …rms of these state variables. The only aggregate uncertainty is money growth, and the process for money growth is autoregressive; therefore, the aggregate state variables are [ (s t ); (s t )]: Dropping explicit dependence of s t and i; we write the state variables of a …rm as (p L; 1 ; a; z) and the aggregate state variables as S = ( ; ): Let the static gross pro…t function, normalized by the current money supply M; be denoted by where v (a i ; z i ; S) is the unit cost of producing good i and y(p i ; S) is the quantity demanded of good i. Let 0 = ( ; S) denote the transition law on the measure over the …rms'state variables.
Because of discounting, …rms never change a list price in a given period without selling at that price during that particular period. In any period, therefore, a …rm has three relevant options: leave its old list price unchanged and pay 0 to charge a price equal to its list price (N ), leave its old list price unchanged and pay to charge a price di¤erent than its list price (T ), or pay to change the list price and sell at that new list price (L).
In any period, the value of a …rm that does nothing (N )-does not change its price and sells at its existing list price-is (Here the expectations are taken only with respect to the idiosyncratic shocks a and z: Since these shocks are idiosyncratic, the risk about their realization is priced in an actuarially fair way. Of course, our formalization is equivalent to having an intertemporal price de…ned over idiosyncratic and aggregate shocks and then simply summing over both of those.) The value of a …rm that charges a temporary (T ) price p T 6 = p L; 1 is and that of a …rm that changes its list (L) price is Inspection of the value function V T makes clear that the optimal temporary price is static and is chosen so that the marginal gross pro…t d p (p; a; z; S) = 0; so that the optimal temporary price is simply Note that this temporary price is a simple markup over the nominal unit cost of production and is the frictionless price, that is, the price which a ‡exible price …rm would charge when faced with such a unit cost. In contrast, if the list price is changed, then the optimal pricing decision for the new list price, p L ; is dynamic. In particular, p L will not typically equal p T .
As (9)  The Workings of the Model Our model works di¤erently from existing menu cost models because of the ability of a …rm in our model to use temporary price changes to respond to shocks. To provide intuition for our model's predictions, we now describe the …rm's optimal decision rules-in particular, when the …rm chooses to make a temporary price change and when it chooses to make a list price change. Brie ‡y, we show that …rms use changes in the temporary price primarily to respond to temporary shocks and use changes in the list price to respond to permanent shocks.
Consider the …rm's optimal decision rules in a quantitative version of the menu cost model (the details of which will be described later). These rules are a function of the individual states, namely, the normalized list price p L; 1 = P L; 1 =M , the permanent productivity level a; and the transitory productivity level z; as well as the aggregate state variable-the money supply growth rate-and the distribution of …rms .
We illustrate the …rm's optimal decision rules in Figure 3. This …gure shows two decision rules in the (z; p L; 1 ) space for a given level of permanent productivity: the list price p L (z); conditional on the …rm's choice to change the list price, and the temporary price p T (z); conditional on the …rm's choice to set a temporary price. Note that since the permanent productivity shock has a unit root, the list price and the temporary price are nearly identical. Recall also that the temporary price is identical to the frictionless price, the price that would prevail if prices were not sticky. This frictionless price decreases one-for-one with productivity shock increases. This …gure also shows that a …rm chooses to charge a temporary price when two conditions are met: the temporary shock is either su¢ ciently high or su¢ ciently low, and the old list price is close to what would be optimal when the temporary shock is at its mean.
This pattern of temporary price-setting arises from two features of our quantitative model: the cost of changing the list price is not that much higher than the cost of deviating from the list price, and temporary productivity shocks have high mean reversion. Brie ‡y, if the list price is far from the level that is optimal when the temporary shock is at its mean, then the …rm realizes it will likely have to change its list price soon anyway, so it simply changes its list price in order to respond to the transitory shock. Figure 4 shows a simulation of shocks and a …rm's decision rules in our quantitative model for 7 years (84 months). Panel A shows the three shocks. For convenience, we report the inverse of the log of the permanent and transitory productivity shocks (so that these shocks can be interpreted as percentage changes in costs) and the log of the money supply.
Panel B shows the posted and list prices. Comparing these two panels, we see that, for the most part, …rms seem to change list prices in order to o¤set permanent productivity shocks and money shocks and temporarily deviate from the list prices in order to o¤set transitory productivity shocks.
Panel C shows how posted prices di¤er from frictionless prices in the model. Clearly, even though posted prices change frequently, they do not o¤set all the money shocks. For example, from month 50 to month 80, posted prices change 7 times. Even so, the posted price at the end of this period is considerably above the frictionless price, and this di¤erence is mostly due to a persistent decline in the money supply.
Panel D makes clear that the regular prices constructed by our algorithm often agree with the list prices from the theoretical model, although not always.

B. Quanti…cation and Prediction
We want to use the facts about price changes that we have isolated in the two U.S. data sets as the basis for our model and its evaluation. To do that, we must quantify the model with values from U.S. data. Here we describe how we choose the model's functional forms and parameter values. We then investigate whether our parsimonious model can be made to account for the facts about prices that we have documented. We …nd that it can.

Functional Forms and Parameters
We set the length of the period in our model as one month and, therefore, choose a discount factor of = :96 1=12 : We assume that preferences are given by We choose = :39 and ! = :94: As Chari, Kehoe, and McGrattan (2000) show, such a choice of parameters is consistent with estimates from the money demand literature. We choose the value of ; the disutility of labor parameter, to ensure that without aggregate shocks, consumers supply one-third of their time to the labor market. We set the depreciation rate of capital to be consistent with 10% annual depreciation. We choose the adjustment cost on capital so that the standard deviation of investment relative to that of output is equal to 3, a number similar to that in the U.S. data.
For the …nal good production function, we set ; the elasticity of substitution across These values imply that the capital share in value-added is 1/3 and the material share in gross output is 47%. (Recall that to calculate the material share in gross output, we need to divide 1 v by the markup =( 1):) These are consistent with the work of Basu (1995) and Nakamura and Steinsson (forthcoming). We choose the exit rate of intermediate good …rms e = 1:8%: Gertler and Leahy (2008) argue that this probability of exit is consistent with the rate at which products are replaced in the Nakamura and Steinsson (2008) data.
We would like to isolate the real e¤ects of exogenous monetary policy shocks as a simple way of measuring the degree of nominal rigidity in the model. A popular way to do so is as in the work of Christiano, Eichenbaum, and Evans (2005) and Gertler and Leahy (2008), who study the response of the economy to shocks in the money growth rate. We adopt the interpretation of Christiano, Eichenbaum, and Evans (2005), who extract the process for the exogenous component of money growth that is consistent with the monetary authority following an interest rate rule. 4 In that spirit, we set the coe¢ cients in the money growth rule by …rst projecting the growth rate of (monthly) M1 on current and 24 lagged measures of monetary policy shocks. 5 We then …t an AR(1) process for the …tted values in this regression and obtain an autoregressive coe¢ cient equal to .61 and a standard deviation of residuals of m = .0018. (We also redid the analysis in this paper using a Taylor rule and obtained similar results that are available on request. We focused on the money growth rule because the degree of aggregate price stickiness is easier to interpret under a money growth rule than under an interest rate rule.) The rest of the parameters are chosen so that the model can closely reproduce the salient features of the micro price data we have described: , the (menu) cost the …rm incurs when changing its list price; , the cost of deviating from the list price; and the speci…cations of the productivity shocks.
Consider, …rst, the speci…cation of the permanent productivity shocks. The distribution of the innovations " i (s t ) for these requires special attention. Midrigan (2007) shows that when " i (s t ) is normally distributed, a model like ours generates counterfactually low disper-sion in the size of price changes. Midrigan argues that a fat-tailed distribution is necessary in order for the model to account for the distribution of the size of price changes in the data.
We …nd that a parsimonious and ‡exible approach to increasing the distribution's degree of kurtosis is to assume, as Gertler and Leahy (2008) do, that productivity shocks arrive with Poisson probability a and are, conditional on arrival, uniformly distributed on the interval [ ; ]: We follow this approach and assume that where i (s t ) is distributed uniformly on the interval [ ; ]. The productivity process thus has two parameters: the arrival rate of shocks a and the support of these shocks : Paying special attention to the distribution of these shocks is necessary because this distribution plays an important role in determining the real e¤ects of changes in the money supply. Golosov and Lucas (2007) show, for example, that the e¤ects of monetary shocks are approximately neutral when productivity shocks are normally distributed. But as Midrigan (2007) shows, with a fat-tailed distribution of productivity shocks, shocks to the money supply have much larger real e¤ects because changes in the identity of adjusting …rms are muted as the kurtosis of the distribution of productivity shocks increases.
Consider, next, the process for the transitory productivity shocks. To keep the model simple, we assume that these shocks follow a Markov chain, with z t 2 f z; 0; zg referred to as the low, medium, and high productivity values, with transition probabilities Here, the subscripts l and h indicate the low and high productivity values. Hence, l is the probability of experiencing a decrease in productivity from 0 to z; and h is the probability of experiencing an increase in productivity from 0 to z: Finally, s is the probability of staying in a non-medium state. Our parameterization of these shocks thus has four parame-ters f z; s ; l ; h g: Here, l and h govern the probability of temporary price increases and decreases, while s determines the duration of temporary price changes.
We choose these parameters to minimize the distance between 13 moments in the data and the model listed in panel A of Table 2. These include the facts about temporary and regular price changes, as well as other measures of the degree of low-and high-frequency price variation in the BLS data we have discussed. These moments include the frequency of monthly price changes (posted and regular), the fraction of price changes that are temporary, the fraction of periods with temporary price discounts, the proportion of returns to the old regular price, the probability of a temporary price spell ending, the fraction of prices at and below the annual mode, as well as the size and dispersion of price changes (as measured by the interquartile range, or IQR), both including and excluding temporary price changes.
Panel B of Table 2  The frequency of posted price changes is high: 22% in both the data and the model; the frequency of regular price changes is much lower: 6.9% in both the data and the model.
Most price changes are temporary: 72% in the data and 76% in the model. Temporary prices often return to the regular price that existed before the temporary change: 50% in the data and 70% in the model. We also see that temporary price changes are transitory: the probability that a temporary price spell ends is equal to 53% in the data and 54% in the data. Periods with temporary prices account for 10% in the data and 11% in the model, and most of these periods are ones with temporary price declines (6% in both the data and the model).
So far we have used a speci…c algorithm to distinguish between temporary and regular price changes. We think of these price changes as characterizing high-and low-frequency price variation. We would like to argue that our broad conclusions carry through for other ways of distinguishing high-and low-frequency price variation. Another way to see that there is considerable low-frequency price stickiness is to consider the fraction of times a good's price is equal to the annual mode. (This alternative measure of low-frequency price stickiness has been suggested by Hosken and Rei¤en (2004) and was also used recently by Eichenbaum, Jaimovich, and Rebelo (forthcoming).) We apply this measure to both the data and the model, and in both it is quite high: 75% in the data and 73% in the model. When prices are not at their annual mode, they tend to be below the annual mode more often than above it in both the data and the model (13% of the time in the data and about 17% in the model).
In sum, we see that our conclusions that the data exhibit considerable low-frequency price stickiness are robust to alternative ways of examining the data.
Following Golosov and Lucas (2007) and Midrigan (2007), we also examine the size and dispersion of price changes. The mean size of all price changes and regular price changes is high in both the data and the model (11% for all of them). So is the dispersion of these changes as measured by the IQR: 9% for all price changes in both the model and the data and 8% for regular price changes in both.

C. A Comparison with the Standard Model
We next compare the patterns of low-and high-frequency price stickiness in the standard model which has one type of price change with the same patterns in our model. We show that, unlike our model, the standard model cannot simultaneously reproduce the high-frequency price ‡exibility (that is, the low level of high-frequency price stickiness) of micro price changes and the low-frequency price stickiness of micro price changes observed in the data.
To demonstrate that, we consider a sequence of standard menu cost models. In each of these models, we vary the frequency of micro price changes and keep the size and interquartile range of price changes equal to those in the data. We convert this frequency into months and consider it a measure of the degree of high-frequency price stickiness. Then for each model, we simulate a long price series and apply our algorithm to construct the regular price series. We compute the frequency of these regular price changes, convert it into months, and consider it a measure of the degree of low-frequency price stickiness.
The results are displayed in Figure 5. The curve in panel A shows that if micro prices are highly sticky in the standard model, then regular prices are too; the degrees of highand low-frequency stickiness match. This is not the pattern we have seen in the data. That pattern-and the pattern produced by our model-is represented in panel A by a large dot.
In the BLS data and in our model, prices have a low degree of high-frequency stickiness, about 4.5 months, but they also have a high degree of low-frequency (regular) price stickiness, about 14.5 months.
We also do an analogous experiment with the standard model for our alternative measure of low-frequency price stickiness, the fraction of prices at the annual mode. The results of that experiment, displayed in panel B of Figure 5, are quite consistent with the results of the regular price experiment. This consistency strongly suggests that our conclusions are not dependent on the exact way in which we measure low-frequency price stickiness or the details of our algorithm that de…nes regular prices.

The Degree of Aggregate Price Stickiness
We have shown that our menu cost model with temporary price changes can reproduce the main features of the BLS micro price data-and much better than a standard model can.
We And we evaluate one way in which researchers have tried to continue to use the standard model despite its di¢ culty with temporary price change data.

A. A Measure of Aggregate Price Stickiness
For this analysis, we must somehow measure the degree of aggregate price stickiness in our model. We want a measure that captures how slowly the aggregate price level reacts to a change in the money supply. The slower is this reaction, the greater will be the di¤erence between the impulse response for the price level and the impulse response for the money supply. Based on that logic, we choose as our measure of aggregate price stickiness the di¤erence (actually, the integral of the di¤erence) between the impulse response of money and prices to an innovation to the money supply relative to the average impulse response of money (over the …rst two years following the innovation). Notice that our measure has a one-to-one relationship with the real e¤ects of money, as measured by the integral of the impulse response of output to an innovation in the money supply. To express our measure in convenient units, we express it as the frequency of micro price changes in a standard model (without temporary price changes) that produces this same di¤erence between the impulse response of money and prices. We …nd this measure convenient because the units are in terms of the key parameter of the standard New Keynesian model and is, thus, well understood.

B. Impulse Responses in Our Model
We begin by examining the impulse responses to a monetary policy shock in our model.
In Figure 6, we show the impulse responses to a shock to the money growth rate in period 1 that, in the limit, produces a 5% increase in the level of the money supply. In panel B of Figure 6, we see that on the shock's impact, the model implies that output increases by about 6:8% and then declines, reaching half of its peak value in about 8 months. Consumption follows a similar pattern, although its response is only half that of output. All other variables (investment, labor, use of intermediate inputs) have a similar qualitative pattern, so are not reported here.
The column in panel A of Table 3 labeled our model summarizes some of the key properties of the model. The …rst entry reminds us that, at the micro level, prices are ‡exible, in that they change once every 4.5 months. The second entry quanti…es the degree of aggregate price stickiness to be about 40%, measured, again, as the average di¤erence between the impulse response of the money supply and the price level over the …rst 24 months after the impulse divided by the average impulse response of the money supply during this period.
To get some sense of this number, note that if aggregate prices never moved after a monetary shock, our aggregate price stickiness measure would be 100%; whereas if prices instantly adjusted to the ‡exible price level, our measure would be 0%: The table's last two entries quantify how the aggregate price stickiness in our model manifests itself in output: it leads to a sizable output response of about 3% and a half-life of that response of about 8 months.

C. A Comparison of the Degree of Price Stickiness in our Model with that in the Standard Model
For some perspective on our model's aggregate price implications, we now compare them to those of the standard menu cost model used in the literature (as in, for example, the work of Midrigan (2007) and Gertler and Leahy (2008)). The implications are very di¤erent.
The standard model we use here is a special case of our model in which we eliminate the option of temporary changes by setting = 1: We also eliminate the transitory productivity shocks to make our analysis parallel to the existing approaches, which use only one type of idiosyncratic shock. We choose parameters governing the permanent productivity process and the size of the menu cost so that the standard model matches the frequency, average size (11%), and interquartile range of price changes (9%) in the data.
In Figure 7 we report how the aggregate degree of price stickiness in the standard model varies with the degree of micro price stickiness. The …gure shows that the standard model implies a one-to-one relationship between micro price stickiness and aggregate price stickiness. In particular, when micro prices change frequently, the degree of aggregate price stickiness is low. Conversely, when micro prices change infrequently, the degree of aggregate price stickiness is high.
This is quite a contrast to the predictions of our model. In the data, prices change The impulse responses of the standard model that exactly match the degree of price stickiness in our model are displayed in Figure 8. Clearly, the impulse responses of output and prices in our model are nearly identical to those in a standard model with 12 month micro price stickiness.
In sum, even though our model is consistent with frequent micro price changes, it still predicts a quite sticky aggregate price level.

D. Evaluating the Existing Approach in the Standard Model
As we have emphasized, the micro data exhibit two types of price changes: regular changes, which happen at a moderately low frequency, and temporary deviations from regular prices, which happen at a high frequency. We have dealt with this phenomenon by building a model that accounts for both types of price changes. Other researchers have chosen not to explicitly model both but instead use a standard model with only one type. To deal with temporary changes, these researchers discard all temporary price changes from the data and set the frequency of price changes in the model to match the frequency of regular price changes in the data. This approach, which we refer to as the regular price approach, can, at best, only roughly approximate the data. Since the vast majority of quantitative exercises use this approach, we …nd it informative to evaluate just how rough an approximation it delivers.
Returning to Figure 7, we see that a standard model using the regular approach predicts a degree of price stickiness of about 46%, or about 1.15 times greater than in our model. We conclude that the regular price approach only slightly overstates the degree of price stickiness and the real e¤ects of monetary policy shocks. We thus …nd that the results of Golosov and Lucas (2007), Midrigan (2007), and Nakamura and Steinsson (2008), who have used the regular price approach in studying menu cost models, do not greatly overstate the real e¤ects of monetary shocks.
In work available upon request, we …nd a similar result for the regular price approach in a Calvo model with a constant hazard of price changes: the regular price approach does not greatly overstate the real e¤ects of monetary shocks.

E. Explanation
The key prediction of our model is that temporary price changes, although very frequent, do not allow the aggregate price level to react to monetary policy shocks. We now provide some intuition for this result and argue that this phenomenon is explained primarily by the distinctive features of temporary price changes that we have identi…ed. We also show that strategic interactions emphasized by Guimarães and Sheedy (forthcoming) play no role in our result.

Intuition
We claim that the nature of temporary price changes does not allow the aggregate price level to react to monetary policy shocks. To gain some intuition for that explanation, let us walk through the model's pricing process again.
First, consider the response of the aggregate price level to a one-time change in the log of the money supply equal to m in period 1; assuming that each …rm's desired posted and list prices increase by this amount. For simplicity, assume that the probability that any given …rm changes its list price in any given period is constant and equal to L : Similarly, assume that the probability that any given …rm has a temporary deviation from its old list price is constant and equal to T : The change in the aggregate price level can be written, to a …rst approximation, as where p t is the change in the log of the aggregate price level and p it ; the change in the log of an individual …rm's price. In any period after the shock, there are three types of …rms: those that have already reset their list prices, those that currently have a temporary change, and those that are still charging the list price in e¤ect prior to the change in the money supply.
The change in the aggregate price level is, therefore, equal to where L;t is the measure of …rms that have reset their list price as of period t and T;t is the measure of …rms that have a temporary change in period t: Clearly, T;t = T ; since the measure of …rms that have a temporary price change is constant. To compute L;t ; notice that L …rms have reset their list prices one period after the shock, L + (1 L ) L in the second period (the …rms that have done so in the …rst period plus those that do so in the second period), L + (1 L ) L + (1 L ) 2 L in the third period, and so on. So we have Consider now two extreme scenarios. Suppose …rst that all price changes are list price changes, so that L is high and T = 0: If L is high, then the measure of …rms that have had a chance to react to the monetary shock quickly increases with t: For example, if L = 0:2, the fraction of …rms that have reset their list price and permanently responded to the money shock is equal to 89% after 10 periods. Hence, if all price changes are list price changes, the aggregate price level is quite ‡exible.
Now suppose, however, that all price changes are temporary. Since after temporary changes, the nominal price reverts to the pre-existing list price, these changes only allow …rms to temporarily react to a monetary policy shock. If L = 0 and T = 0:2; the measure of …rms that have responded to the monetary shock is the same, 20%; forever. Thus, if all price changes are temporary price changes, the aggregate price level is quite sticky. Notice that our example makes clear that a key di¤erence between temporary and list price changes is that temporary changes are special: since they return the nominal price back to its pre-existing list price, they allow …rms to only temporarily respond to a change in monetary policy.
Finally, the second distinctive feature of temporary changes is their clustering. To see the role of this feature, note that every episode of temporary deviation from a list price involves two price changes: one that initiates the temporary change and another that reverses it. Hence, if T = 0:2 and L = 0; the frequency of price changes is twice as high and equal to 2 T : The second round of price changes does not allow …rms to react to monetary policy shocks and thus does not lower the stickiness of the aggregate price level (even though it does lower the degree of micro price stickiness).

On the Unimportance of Strategic Interactions
Based on recent work by Guimarães and Sheedy (forthcoming), some may conjecture that the intuition for our results comes from strategic interactions in price-setting, interactions that arise from our use of materials as a factor of production and from our use of capital and interest-elastic money demand. In particular, some may think that the reason our model predicts a large amount of aggregate price stickiness is that a given …rm that undertakes a temporary price change in our model has little incentive to respond to a money shock. The logic is that since this given …rm purchases inputs from other …rms with sticky prices, this …rm's input costs are sticky. We show that this conjecture is false: our results are not driven by strategic complementarity in price-setting.
To see this, consider an economy in which there is no such complementarity. In this no strategic complementarities economy, we exclude materials and capital from production and make money demand interest-inelastic by introducing money using a cash-in-advance constraint. To see that these changes eliminate any strategic complementarities, note that, given our log-linear preferences, the nominal unit cost of production is given by the nominal wage that satis…es (10) W s t = P s t c s t = M s t ; where the second equality follows from the cash-in-advance constraint. Equation (10) makes clear the precise sense in which the model has no strategic interactions in price-setting: the desired price of any given …rm is a function of only an exogenously given process, the money supply, and not of the actions of other …rms.
We report the results from this economy in panel B of Table 3. As we did with our original economies, we here consider two versions of the no strategic complementarities economy: one with temporary price changes and one without them. Again we ask, What degree of micro price stickiness in the economy without temporary changes reproduces the degree of aggregate price stickiness in the model with temporary changes? Table 3 shows that the answer is again about once every 12 months.
We therefore conclude that the special nature of temporary price changes in our original model, rather than general equilibrium considerations, accounts for our results. This stands in sharp contrast to the results of Guimarães and Sheedy (forthcoming), who emphasize the role of strategic substitutability of sales. In the variation of our model considered here, a given …rm's incentive to have a temporary price change is independent of the actions of other …rms, and we nevertheless …nd that temporary price changes do not greatly contribute to the ‡exibility of the aggregate price level.

Robustness Exercises Using Dominick' s Data
We now consider two extensions of our model that use statistics from the Dominick's data set rather than the BLS data set. Both extensions provide a check on the robustness of our results. In one extension, we simply reparameterize our model to be consistent with Dominick's data. In the other, we also replace the transitory productivity shocks in our model with transitory shocks to the elasticity of demand for the …rm's product, leaving all else unchanged. We …nd that our results based on the BLS data are robust to both of these extensions.

A. Weekly Data with Quantities
Our use of the Dominick's data is motivated by two concerns about the BLS data.
One concern is that because the BLS data are monthly, by construction, they miss much of the high-frequency movements in prices that are reversed within a month (such as temporary sales). Because the BLS data set misses these movements, when we use it to set parameters in our model, the resulting model may vastly overstate the degree of price stickiness. Our second concern is that since the BLS data set has no information on quantities, it leaves open the possibility that almost all purchases are made when goods are on sale, so that the prices when goods are not on sale, the regular prices, are essentially irrelevant. This possibility might also imply that our exercise with BLS data, which is silent on the quantity facts, vastly overstates the degree of price stickiness.
The Dominick's data set allows us to address both of these concerns. Since these data are weekly (and since Dominick's resets its prices only once a week), this data set does not miss high-frequency movements in prices. Also, the quantity information in the Dominick's data set allows us to investigate whether the vast majority of purchases are made when a good is on sale.

B. Model with Demand Shocks
We have described the Dominick's data set in detail earlier, so we begin this analysis by describing our other extension: a modi…cation to our model to include a di¤erent kind of shock, a demand shock. This extension is motivated by theory and data. The motivation from theory is that a common explanation in the industrial organization literature for temporary price changes is intertemporal price discrimination in response to time-varying price elasticities of demand.
In particular, the idea is that …rms willingly lower markups in periods during which a large number of buyers of products happen to have high elasticities.
The motivation from data to include a demand shock is based on two observations. First, temporary price cuts are associated with reductions in price-cost margins. (See, for example, the work of Chevalier, Kashyap, and Rossi (2003).) Second, as we have documented, periods with temporary price cuts account for a disproportionately large fraction of goods sold. Taken together, these features suggest that, in the data, the demand elasticity that …rms face is time-varying, and this feature leads …rms to have time-varying markups.
Motivated by both theory and data, then, we introduce time-varying elasticities by having consumers with di¤ering demand elasticities and by including good-speci…c shocks to preferences. We assume the same preferences here as in the earlier model, so that utility is still given by (2). Now our consumers belong to a two-member household. Speci…cally, each household supplies labor as a single entity to the market, but now values consumption according to where c A (s t ) and c B (s t ) are composite goods given by Our interpretation of these preferences is that they represent a household with two members, consumer A and consumer B; who jointly supply labor but have separate consumption bundles. We suppose that consumer B has a higher elasticity of substitution across goods, > .
Moreover, consumer B's weight on the di¤erent goods, z i (s t ) ; is a random variable. Note that z i (s t ) is speci…c to any particular good but common to all households. In this sense, z i (s t ) is a good-speci…c demand shock, i.i.d. across goods. We assume here, as earlier, that z i (s t ) is a Markov process (which we describe in detail below). This structure generates timevarying elasticities of demand in a representative household environment. The general idea of allowing for a type of heterogeneity while avoiding the complication of recording individual state variables is similar to that of Lucas (1992).
The household's problem here is to choose how much consumption to allocate to each consumer, as well as how much to invest, work, and purchase state-contingent securities subject to the following budget constraint: where P F (s t ) is the price of the …nal good used for investment, and all other variables are as de…ned in the earlier model.
This problem yields the following consumer demand functions for the di¤erent varieties of goods: where the aggregate price index P (s) = P A (s t ) P B (s t ) 1 and The producer of intermediate good i in this economy uses basically the same technology as before: One big di¤erence is that we eliminate the transitory productivity process and include only the permanent component of the …rm's productivity a i (s t ) that, as earlier, evolves according to (4).
Consider, next, the …nal good …rms. These …rms are perfectly competitive and pur- The problem of a …nal good …rm is to choose the amount of each intermediate good q i (s t ) to purchase in order to maximize which yields the demand function Notice that, for simplicity, we have assumed that the elasticity of substitution for producing the …nal good, , is the same as the elasticity of substitution for type A consumers.
A useful feature of the resulting demand function is that it has time-varying elasticity.
Clearly, as the demand shock z i (s t ) increases, so too does the relative demand for good i coming from the high-elasticity (type B) consumers. Indeed, the total demand elasticity for good i increases with z i (s t ): A bit of algebra shows that this total demand elasticity increases productivity, speci…cally with a i (s t ) and > 0. Intuitively, then, we know that a positive productivity shock leads the …rm to lower the price for its good and increases the relative demand of the high-elasticity consumers.
The assumptions about the technology of price adjustment are the same here as in the earlier model. The problem of the intermediate good …rm is, too, except that its production function is now given by (13) and the demand function is now given by (16).
The equilibrium de…nition for this economy is similar to that in our earlier model, except that now the market-clearing condition for the …nal good is and the market-clearing condition for each intermediate good is Note, for later use, that real output (value-added) at base period prices can be written as

C. Quanti…cation and Prediction
Now we have two variations of our model with temporary price changes, which we can use to study the Dominick's data. One variation, the productivity shocks model, is the model we studied earlier with the BLS data; the other is the economy with transitory demand shocks instead of transitory productivity shocks just described, the demand shocks model. We must now parameterize both of these models in order to reproduce features of the Dominick's data before we can have them simulate responses to monetary shocks. In both models, we assume that the length of a period is one week, the frequency with which the Dominick's data are sampled.
In both models, we assign the same functional forms and parameters as earlier for the production function, the rate at which capital depreciates, the rate at which …rms exit, and so on. For example, we set = 1 (1 0:01) 1 4 ; so that the monthly rate at which capital depreciates is 1%. We set e = 1 (1 0:018) 1 4 ; so that 1.8% of …rms exit in any given month (4 weeks). We adjust all other parameters, including the persistence and standard deviation of money growth shocks, similarly. We calibrate the size of the capital adjustment costs to again ensure that the standard deviation of investment relative to that of output is equal to 3, a number similar to that in the U.S. data. All of these parameters are reported in Table 4.
The rest of the parameters are chosen so that the two models can closely reproduce the salient features of the Dominick's data. We describe these separately for each model.

Productivity Shocks Model
For the model with productivity shocks only, the key parameters are , the (menu) cost the …rm incurs when changing its list price; , the cost of having a temporary deviation from the list price; as well as the speci…cations of the transitory and permanent productivity shocks. The moments we choose to pin down these parameters are the same moments we used earlier with the BLS data. Here, we also ask the model to account for two quantity facts: the fraction of goods that sold in periods of temporary price changes and the fraction sold in periods with temporary price discounts. We compute these facts from the Dominick's data by applying an algorithm to that data set similar to the algorithm used with the BLS data, in order to identify regular price changes.
These values, as measured in the data and in the productivity shocks model, are nearly identical. (See Table 5.) Recall that price changes are much more frequent in the Dominick's data than in the BLS data. The Dominick's frequency of price changes is 33% per week (in both the model and the data) compared to 22% per month in the BLS data. This higher frequency of price changes re ‡ects the much larger role that temporary price changes play in grocery stores and at the weekly frequency: 94% of price changes are temporary in the Dominick's data (95% in the model). Notice also that regular price changes are much less frequent here (2.9% per week in both the model and the data), an outcome of the fact that most temporary price changes (80% in the data and 88% in the model) return to the old regular price.
Our productivity shocks model also reproduces well the other moments that describe the pattern of low-and high-frequency price variation in the Dominick's data. The fraction of periods with temporary prices is high both in the model (24%) and in the data (25%), and most of these are periods with temporary price discounts (20% in both the data and the model). Although prices change very frequently, a substantial proportion of prices are equal to the annual mode (58% in the data, 55% in the model). Deviations from the annual mode are mostly downward (30% of prices are below the annual mode in both the model and the data). Moreover, the model accounts well for the mean and dispersion of the size of price changes in the data. Notice that price changes are on average somewhat larger (17% in absolute value in the model and in the data) than regular price changes (11% in both the model and the data) and more dispersed.
Notice, …nally, in the last two rows of Table 5, that this version of the productivity shocks model does a good job of reproducing the quantity facts in the Dominick's data. In both the data and the model, periods with temporary price changes account for a disproportionate amount of goods sold. Even though prices are temporary 24% of the weeks in the data and 25% of the time in the model, these periods account for 39% of the goods sold in the data and 36%, in the model. Periods in which prices are temporarily below the regular price account for the bulk of these sales: 35% of goods are sold during such episodes in the model and 33% in the data. The reason the model does so well at reproducing these quantity facts is that its demand elasticity ( = 3) is consistent with the price elasticities of demand in the data.
Again, Table 4 (panel A) reports the parameters that allow the productivity shocks model to achieve this …t. Notice that now the cost of a regular price change (1.89% of the …rm's steady-state pro…ts) is roughly twice as high as the cost of temporarily deviating from the list price (1.01% of the …rm's steady-state pro…ts) and that both types of shocks are now more frequent relative to those in the BLS experiments.

Demand Shocks Model
For the model with demand shocks, we must choose the new parameters, those that describe the new shocks.
In this demand shock economy, the optimal markup of an intermediate good …rm is a function za : To reduce the dimensionality of the state-space, we assume thatz = za is a random variable that takes two values,z t 2 f0; zg. (Essentially, this makes the z shock correlated with a in a way that ensures that givenz t ; the optimal markup is not separately a function of a.) Moreover, we normalize z = 1; since this parameter is not separately identi…ed; it plays the same role as ; the parameter determining the relative weight of consumers. Finally, we assume that the law of motion forz t is given by Here, h is the probability that a …rm withz t = 0 (a …rm that sells only to type A consumers) will havez t+1 = 1 next period (and thus sell to both types of consumers). Moreover, s is a parameter that governs the persistence of the high demand state.
In addition to h ; s ; we must also choose ; the weight of type A (low-elasticity) consumers, as well as and , the demand elasticities of the two types of consumers. We set = 6; an upper bound of estimates of demand elasticity in the industrial organization literature, and jointly choose ; h ; s ; as well as the parameters determining the size of the price adjustment costs and the evolution of permanent productivity shocks, in order to match the size, frequency, and persistence of temporary price changes and the quantity facts in the Dominick's data set. The set of moments we use to parameterize the model with demand shocks is thus the same as for the productivity shocks model.
Returning to Table 5, we see that the model with demand shocks …ts the Dominick's data almost as well as does the productivity shocks model. Once again, this model reproduces well the frequency of regular and all price changes and the size and dispersion of the two types of price changes, as well as the fact that periods with temporary price changes account for a disproportionate fraction of goods sold.
Panel B of Table 4 lists the parameter values that produce this …t. Notice that the weight on high-elasticity (type B) consumers, 1 , is equal to 0.11, the value required to account for the fact that as much as 39% of the output is sold in periods with temporary price changes (which occur in the model mostly in states withz t = 1; since this state is more transitory). Moreover, a demand elasticity of = 2 for low-elasticity (type A) consumers is necessary for the model to account for the average size of all price changes in the data of 17%.

D. The Degree of Aggregate Price Stickiness
Finally, we study the degree of aggregate price stickiness in our Dominick's models to see whether our earlier results using BLS data are supported. We …nd that they are.
We begin with the productivity shocks model, which simply uses our model and the standard model with parameters chosen to mimic the Dominick data. We …rst shock money growth in the models in the same way as in the models based on the BLS data: an innovation that leads to an eventual increase of 5% in the money supply. We then calculate the degree of aggregate price stickiness in our (reparameterized) model and …nd that a standard model with a frequency of price changes of 6.2 months produces the same degree of aggregate price stickiness as in our model. In panels A and B of Figure 9 and in Table 6, we display the responses of prices and output in the two models.
We get a similar result from the demand shocks model. Here, a standard model with a frequency of price changes of 7.3 months produces the same degree of aggregate price stickiness as does the demand shocks model. In panels A and B of Figure 10 and in Table 6, we see that the responses of prices and output to a monetary shock are quite similar in the two models.
We thus conclude that the exact source of price changes (transitory demand or productivity shocks) does not matter much for our result: in both models, the degree of aggregate price stickiness is fairly high-aggregate prices change only every 6 or 7 months-even though micro prices in the Dominick's data change much more frequently (every 3 weeks).
These results are consistent with our earlier results using the BLS data. Since the frequency of price changes in the Dominick's data is much higher than in the BLS data (3 weeks vs. 4.5 months), we view this 6-to 7-month number as a conservative lower bound on the degree of price stickiness in the economy as a whole. In this sense, as with the BLS data, we conclude that prices are sticky after all.

Conclusion
Standard New Keynesian models imply that if prices change frequently at the micro level, then aggregate prices are not sticky. In the micro data, prices do change frequently, but our models, which are consistent with the micro data, still imply that aggregate prices are sticky. How is this possible? The answer is that the one-to-one relationship between the frequency of micro price changes and the degree of aggregate price stickiness breaks down when there are two types of price changes, regular and temporary, as there are in the data. Brie ‡y, temporary price changes have two striking features that distinguish them from regular changes: after a temporary change, prices tend to return to their pre-existing levels, and temporary price changes are clustered in time. These features imply that even though temporary price changes happen with high frequency, micro prices also have a type of low-frequency price stickiness that generates stickiness in aggregate prices.

Appendix: The Algorithm to Construct the Regular Price
Here we describe, …rst intuitively and then precisely, our algorithm for constructing a regular price series for each product in the data. We have applied this algorithm ourselves to the Dominick's data set and have asked Emi Nakamura and Jon Steinsson to apply it to the BLS data set.
Our algorithm is based on the idea that a price is a regular price if the store charges it frequently in a window of time adjacent to that observation. We start by computing for each period the mode of prices p M t that occur in a window which includes prices in the previous …ve periods, the current period, and the next …ve periods. 6 Then, based on the modal price in this window, we construct the regular price recursively as follows. For the initial period, set the regular price equal to the modal price. 7 For each subsequent period, if the store charges the modal price in that period, and at least one-third of prices in the window are equal to the modal price; then set the regular price equal to the modal price. Otherwise, set the regular price equal to the preceding period's regular price.
We want to eliminate regular price changes that occur when the store's posted price does not change, but only if the posted and regular prices coincide in the period before or after the regular price change. To do that, if the initial algorithm generates a path for regular prices in which a change in the regular price occurs without a corresponding change in the actual price, then we replace the last period's regular price with the current period's actual price for each period in which the regular and actual prices coincide. Similarly, we replace the current period's regular price with the last period's actual price if the two have coincided in the previous period. Now we provide the precise algorithm we use to compute the regular price and describe how we apply it.
1. Choose parameters: l = 2 (= lag, or size of the window: the number of months before or after the current period used to compute the modal price. For the Dominick's data, we set l = 5 weeks), c = 1=3 (= cuto¤ used to determine whether a price is temporary), a = :5 (= the number of periods in the window with the available price required in order to compute a modal price).
We apply the algorithm below for each good separately: Notes 1 We de…ne the regular price series precisely using an algorithm described in detail in the appendix. Loosely speaking, the regular price series is a type of running mode of the original prices, and the temporary prices are simply deviations from that running mode.
3 Note that this duration of 14.5 months is higher than the corresponding 8-11 month number of Nakamura and Steinsson (2008), primarily because our algorithm takes out temporary price increases as well as temporary price decreases, or sales, that Nakamura and Steinsson focus on. Even though temporary price cuts account for most of the temporary price changes, the number of price increases is large enough relative to the number of regular price changes to in ‡uence the resulting frequency.
4 Speci…cally, Christiano, Eichenbaum, and Evans (2005) specify an interest rate rule in their empirical work as R t = f ( t ) + " t ; where R t is the short-term nominal rate, t is an information set, and " t is the monetary policy shock. They interpret the monetary authority as adjusting the growth rate of money so as to implement this rule. They then identify the process for money growth in their vector autoregression (VAR) which is consistent with this interest rate rule. That process is well-approximated by an AR(1) similar to the one we use. 5 The results we report here use a new measure of shocks due to Romer and Romer (2004), which is available for 1969-96. We have also used the measure of Christiano, Eichenbaum, and Evans (2005) and get very similar results.        Figure 10: Impulse responses to a monetary shock in economy with demand shocks using Dominick's data