DISCRETENESS AND NOMINAL RIGIDITY DO SUPERMARKET PRICES MOVE TOO by sanmelody

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									DISCRETENESS AND NOMINAL RIGIDITY: DO SUPERMARKET PRICES MOVE TOO

MUCH?


Benjamin Eden and Matthew S. Jaremski1


Vanderbilt University


Revised, December 2010


           We assess the ability of the cross sectional price distribution to react to shocks from the
point of view of a Prescott “hotels” type model, using a sample of 435 products in 75 stores over
121 weeks. We argue that the cross sectional distribution is flexible in spite of the price
repetition observed in the data. From the point of view of our model the question is why prices
move so much. We outline the possibility that prices are used to manage inventories.




Key Words: Price Discreteness, Price Dispersion, Price Rigidity, Nominal Rigidity.



JEL CODES: E00, L11




1
    We thank Maya R. Eden, Philip J. Glandon and Virgiliu Midrigan for comments on an earlier draft of this paper.
                                                                                                                      2


1. INTRODUCTION


         Are nominal rigidities important for understanding the real effects of money? This
question cannot be asked in terms of menu cost models because in these models money has real
effects only if prices are rigid. It cannot be asked in terms of a Lucas’ confusion model because
in this model money has real effects only if prices are flexible. The question requires a model in
which money may have real effects for a variety of reasons, including some form of nominal
rigidity.
         We do not have such a model off the shelf. Eden (1994) use a flexible price version of
Prescott (1975) “hotels” model in which money has real effects because of informational
constraints that arise as a result of sequential trade. We argue that this model can be augmented
to allow for nominal rigidities as the question requires and use it as the baseline model. The
reason for the real effects of money in the augmented model is important. There is more room for
policy intervention if nominal rigidity is the main cause of money non-neutrality.
         From the point of view of the uncertain and sequential trade (UST) model in Eden (1994)
the behavior of the cross sectional price distribution is important but the behavior of the price in
an individual store is not. This is also true in other models that address the money neutrality
issue. See for example, Caplin and Spulber (1987), Caplin and Leahy (1991) and Head et.al.
(2010). We therefore focus on the behavior of the cross sectional price distribution.
         We use a balanced panel of stores that belong to a Chicago based chain. The panel
contains retail price observations for 435 products within 75 stores. In the data, price dispersion
is pervasive but the fraction of prices equal to the mode is large (75% on average) and stable
over weeks. We therefore characterize the cross sectional price distribution by its mode.2
         The cross sectional mode seems to be highly flexible. When looking at the mode for the
435 products we find that in an average week 62% of the weekly revenues are obtained from
selling items whose cross sectional mode had changed in the current week. There is also
considerable uncertainty about the level of the cross sectional mode. In an average quarter the




2
 Our focus on the cross sectional mode has an important side benefit. By the very nature of being the most common
price posted by the 75 stores, it is relatively immune to the data errors described in the literature. See for example,
Campbell and Eden (2007) and Eichenbaum, Jaimovich and Rebelo (forthcoming).
                                                                                                                  3


average standard deviation of the cross sectional mode is an order of magnitude higher than the
standard deviation of the weekly money supply (M1) over a quarter.
        To shed light on why there are so many mode changes and why the mode’s
interetemporal variations are so high, we assemble some facts about it. First, we find that most of
the changes in the cross sectional mode are temporary, suggesting that coordinated sales play a
significant role in the chain’s price level across time. Second, we find relatively few distinct
nominal prices each week. These two observations are consistent with the hypothesis that the
store manager chooses prices out of a menu that offers few alternatives. We find however that
the seemingly restrictive menu does not inhibit the ability of the chain to adjust the cross
sectional price distribution in any measurable way and is thus flexible.
        We then outline a sequential trade model that allows for nominal rigidity and can account
for the observation in the first sections of the paper. Nominal rigidity arises in our model because
of deliberation costs. The model advances the hypothesis that the chain manager reduces
deliberation costs by restricting the choice of store managers to few prices and this leads to the
price repetition observed in the data.
        Our approach is similar to Kehoe and Midrigan (KM, 2007) and Eichenbaum, Jaimovich
and Rebelo (EJR, forthcoming) who allow the cost of changing the price to depend on the type of
price change.3 These “modified menu cost approaches” have been interpreted as an argument for
deemphasizing temporary price changes when studying the reaction to macro shocks. This
interpretation may be the result of the focus on the individual store. By studying the cross
sectional distribution of prices, we reach almost the opposite conclusion, as temporary price
changes are often coordinated across stores and result in many temporary mode changes.4 Our
argument is in line with Chavalier and Kashyap (2010) and Glandon (2010) who argue that
temporary prices are important from a macroeconomic point of view.




3
  KM assume that a temporary change in the price (i.e. sale) is cheaper than changing a regular price, whereas, EJR
assume that changing a price plan that determines the most common price in a given quarter (the reference price) is
costly but otherwise, changes in prices are costless.
4
  We focus on the most common price across stores in a given week (the cross sectional mode defined for a product-
week cell) while EJR focus on the most common price in the store over the period (defined for a product-store-
period cell).
                                                                                                    4


2. DATA

        The Dominick’s Finer Foods (DFF) database arose through a joint venture between the
Chicago-area supermarket chain and the University of Chicago Graduate School of Business.
The partnership started in September of 1989 to document a price randomization experiment, but
continued even after the experiment was completed in 1995. The data make use of scanner
technology that electronically records items as they were purchased in each store. The
observations were aggregated weekly to give the transaction price, quantity sold, and profit
margin for about 3,500 products in 100 store locations. Each product is labeled by a unique
Universal Product Code (UPC) which documents the product’s manufacture, size, and type. For
instance, a 6pk of Coke has a different UPC than a 6pk of Pepsi, a 6pk of Diet Coke, or a 2 liter
Coke. The data does not contain a complete summary of UPCs sold by the chain, but covers
nearly 30% of all dollar sales.
        To date, several papers have used the DFF sample to study pricing. Most recently, Kehoe
and Midrigan (2007) study menu costs effects for regular and sale prices. Chevalier, Kashyap
and Rossi (2003) address demand’s impact on retail prices, quantity sold, retail margins, and
wholesale prices, during holidays. Earlier papers by Dutta, Bergen, and Levi (2002) and
Peltzman (2000) examine the response of prices to cost shocks. These studies test various aspects
of price change, but typically ignore the chain aspect of the data by choosing price data from a
single store.
        When dealing with the DFF source, the major decision one must make is what to do with
missing information which occurs when no units were sold or when a UPC was temporarily
replaced by another UPC denoting a new size, flavor, color or even holiday packaging. For
instance, Mint Flavored M&Ms replace regular M&Ms during the Christmas season. There are
several ways to handle these gaps. Chevalier, Kashyap and Rossi (2003) fill the UPCs with their
replacements in order to maximize the sample period, while Midrigan (2006) and Peltzman
(2000) take observations from other stores in order to obtain the maximum number of products.
Since we want weekly data within many stores, we fill missing observations using the data
around the gap. First, we drop any UPC with a gap longer than a month. Second, if the price is
the same on both sides of the gap, we assume it did not change during the gap, and fill the
“missing” price from the previous week’s price. Next, we look at gaps in which prices are not the
                                                                                                                      5


same on both sides. We replace all single week gaps with previous week’s price and drop gaps
that are longer than a week. Finally, any UPC-Store cell with unfilled gaps are dropped from the
sample.
         Keeping this filling process in mind, there is a tradeoff between the length of the sample
and the number of unfilled gaps. We chose a period of 121 weeks (from 01/12/95 until 05/14/97)
that maximizes the number of balanced price observations. This yields a sample of 435 UPCs
within 75 stores. The sample period spans the final two years of the sample thus avoiding the
randomized pricing experiment that occurred in the first few years of the study.


2.1 Summary Statistics
         For reporting purposes, the UPCs have been aggregated into 15 basic categories. The first
column of Table 1 provides the number of UPCs in each category, whereas the second column
lists the fraction of “sale price” observations, i.e. temporary price reductions.5 The original data
contain an indicator denoting sale prices. In addition to these indicators, we define a sale price as
any price that decreased and returned to its original level within two weeks. Clearly evident from
the last column of the table is the importance of sales prices in the DFF data. A product is on sale
25.6% of the time and there is a great deal of variation across categories suggesting that sales are
more common in certain groups of products such as soft drinks.6
         Table 1 also describes the fraction of prices that are changed in an average week. Overall,
prices seem very flexible and 38% change in an average week. As expected, sale prices greatly
increase the frequency of price change. When we replace sales with their regular price, the
fraction of price changes declines to 7.7%.
         New prices are even more flexible than other prices. 57.1% of young prices with age less
than or equal to 3 weeks change in an average week. This frequency of price changes is about
50% higher than the frequency of a price change when all prices are included, suggesting that the
probability of a price change decreases with age. Moreover, new prices seem to be important to
consumers, as the fraction of weekly revenues that the chain receives from newly set prices is



5
  Throughout the rest of the paper, the term “sales” will only refer to temporary price reductions. We will be explicit
when we address the quantity of goods sold or the revenue of a product.
6
  The average frequency of sales for products other than soft drinks is about 21% that is still higher than the
frequency reported by Nakamura and Steinsson (2008) who find that roughly 14% of food observations are sales.
                                                                                                                   6


also much higher than the unweighted frequencies. In an average week 66% of revenues were
made from selling goods whose prices has been changed in the current week.


Table 1: Sample Statistics
                                                                                                        Fraction of
                                                                                                         Weekly
                                            %         Revenue                                            Revenue
                                  # of     Sales      Share of                                          from New
         Category                UPCs      Prices      Chain         Price Change Frequency               Prices
                                                                     All        Young         NS
Analgesics (ana)                    3      16.7%        0.2%        25.8%       42.1%        6.4%          35.8%
Cheeses (che)                      60      23.0%       12.8%        37.0%       54.2%        8.8%          56.0%
Cookies (coo)                      66      23.6%        9.7%        31.8%       48.9%        7.2%          50.2%
Crackers (cra)                     18      27.8%        2.3%        36.6%       47.1%        7.5%          46.1%
Dish Detergent (did)               8       18.2%        0.7%        26.6%       44.9%        2.9%          37.5%
Front End Candies (fec)           36       11.8%        2.0%        15.4%       42.7%        5.2%          17.6%
Frozen Dinners (frd)               5       20.3%        0.4%        28.7%       44.4%        4.0%          31.4%
Frozen Entrees (fre)               24      21.6%        3.5%        29.9%       47.2%        4.2%          44.3%
Frozen Juices (frj)                17      27.2%        3.5%        41.7%       52.4%       10.4%          61.0%
Fabric Softeners (fsf)              4      20.8%        0.8%        26.6%       44.2%        3.1%          36.5%
Laundry Detergents (lnd)            6      15.3%        3.4%        28.9%       51.8%       10.9%          54.3%
Oatmeal (oat)                      20      10.0%        2.6%        15.4%       35.3%        3.6%          20.3%
Refrigerated Juices (rfj)         45       23.1%       17.7%        39.0%       59.6%       12.0%          66.4%
Soft Drinks (sdr)                 114      38.0%       39.7%        56.7%       67.0%       10.0%          80.5%
Soaps (soa)                         9      15.6%        0.7%        20.4%       44.9%        4.3%          25.9%
All Products                      435      25.3%       100.0%       37.7%       56.7%        8.1%          65.9%
Notes: Price change frequency is the fraction of prices that changed in an average week. “All” denotes the frequency
of price changes for all prices, “Young” denotes prices less than or equal to 3 weeks, and “NS” denotes the
calculations after replacing sale prices with their regular prices.



2.2 The Cross Sectional Mode
        Does the cross sectional mode represents the cross sectional price distribution? Figure 1
plots the fraction of price and mode changes over weeks. The correlation between the two is
almost perfect (0.989). Since most of the price changes are temporary, the high correlation can
occur only if there are many temporary changes in the cross sectional mode.
                                                                                                                7




                             Figure 1: The Frequency of Price and Mode Changes Over Weeks



        Table 2 provides measures of flexibility for prices and modes. The frequency of price

changes is higher than the frequency of mode changes but the two are surprisingly similar. In a

given week, 34.7% of modes change compared to 37.7% of prices. The frequency of mode

change is also significantly higher for young modes and lower once sale prices are replaced by

regular prices. Note that the fact that the frequency of mode changes drops from 34.7 to 5.5 once

sale prices are replaced suggests that many cross sectional mode changes are temporary.7




7
  We also computed the frequency of change for the median price, the lowest price and the highest price. The
statistics were 39%, 42% and 31% respectively, indicating that the distribution is flexible by most measures.
                                                                                                      8


Table 2: Price and Mode Changes
                  Frequency of Change                                               Fraction of
                                                                                   Weekly Revenue
                            All                Young                  NS             from New
Prices                     37.7%               56.7%                 8.1%                66%
Modes                      34.7%               51.7%                 5.5%                62%

         Figure 2 provides additional support to the representative price view of the mode. For
starters, Figure 2a examines the fraction of stores that post the five most common prices in an
average week. The most common price (the mode) is ranked 1, the second most common price is
ranked 2 and so on. The fraction of stores that post the mode fluctuates around 75% never falling
below 65%. The average fraction of prices at the second most common price is 12% with a range
between 8-15 percent. The fact that at least 84% of prices are at one of the five most common
prices also suggests that prices are discrete, i.e. there are few nominal prices each week.
         Figure 2b describes the percentage of prices above, at, and below the mode. The figure
suggests a division into three price groups: High, medium and low. The first high-ranking group
(stores 1-22) has on average 60% of the prices at the mode, 33% above the mode and 7% below
the mode. The medium group (stores 23-61) has on average 86% of the prices at the mode, 6%
above the mode and 8% below the mode. The low group (stores 62-75) has on average 67% of
the prices at the mode, 6% above the mode and 27% below the mode. Despite the variation of
prices above and below the mode, the mode itself seems to be important for all stores, not just
the majority.
         We conclude that the mode is important for all stores and the fraction of stores that post
the mode is stable over weeks. The frequency of cross sectional mode change is just under the
frequency of price change and many mode changes are temporary.
                                                                                 9




                A. Percentage of Prices at Most Common Prices




B. Fraction of Prices at the Mode, Below the Mode and Above the Mode by Store.


         Figure 2: The Cross Sectional Mode as a Representative Price
                                                                                                                       10


3. EXCESS VOLATILITY?


          We distinguish between cross sectional price dispersion and intertemporal price
dispersion. The first is a measure of the dispersion across stores in the same week, and the
second is the dispersion across weeks in a quarter (13 weeks period). In Table 3, we calculate the
dispersion for individual prices and modes as well as the dispersion in the prices of two baskets
(store and chain). Each basket is defined by the average quantity sold from each UPC during the
sample period across all stores. The store basket is then evaluated at the store’s weekly prices
and the chain basket is evaluated at the weekly mode.
          Examining the cross sectional dispersion measures in the first row, it is evident that the
dispersion of individual store’s prices is significantly higher than the dispersion of the basket.
The average standard deviation is 4.2% for individual prices but 1.9 % when aggregated into the
basket. This suggests that the average correlation among prices is positive.8 The intertemporal
measures provided in the second row show that prices vary much more over a quarter than within
a week. For instance, the deviation of individual prices triples from 4.2% to 13.9%. As before,
the baskets are significantly less volatile than individual prices, but the different between prices
and modes is insignificant.
          The table goes on to illustrate the importance of sales by examining soft drinks
separately. On the whole there is almost no difference between the cross sectional price
dispersion of soft drinks and that of the entire sample; however, the intertemporal measures are
considerably higher. The table thus suggests that sales create larger price differences across time
than across stores in a single week, a fact that motivates our model’s use of sales.




8
    When prices are    iid the variance of an individual price should be 435 times the variance of the unweighted mean:
               435
                            Var(X)
Var ( 1 435)         Xi =          . In our sample, the average variance is only about 5 times the variance of the basket,
               i=1
                             435
suggesting a positive correlation among prices.
                                                                                                                  11


Table 3: Price Dispersion Measures
            Individual Prices      CS Modes                           Store Basket             Chain Basket
              All     Drinks     All   Drinks                         All    Drinks            All    Drinks
    CS        4.2       4.3        -      -                           1.9      1.8              -        -
 Intertem.    13.9     17.2     13.5    17.9                          3.8      6.2              4      6.6
Notes: The Table reports the standard deviation of log prices for a variety of groups. The first row is the cross
sectional price dispersion. The second row is the intertemporal price dispersion over a quarter. The average standard
deviation of “Individual Prices” is weighted using each UPC’s average fraction of revenue. The first cell in this row
is the average standard deviation in a UPC-Week cell. The pair of cells under “CS Modes” are the standard
deviations of the cross sectional mode over a quarter averaged over UPCs and quarters. The cells under “Store
Basket” are the average standard deviation of the basket of goods sold by the store. The cell under “Chain Basket”
are the average standard deviation for the basket sold by the chain where each UPC is valued by the cross sectional
mode.


3.1 Do Prices Move Too Much Relative to the Money Supply?
         Table 3 showed that the standard deviation of the cross sectional mode over a quarter is
13.5%. However, by itself, the number does not provide much information and must be
compared to something. In a direct analogy to Shiller (1981), we ask: do supermarket prices
move too much to be justified by changes in the money supply?
         We couch excess volatility in terms of the new Keynesian model pioneered by Blanchard
and Kiyotaki (BK, 1987). In this model, prices are set at the beginning of the period at a level
that is proportional to the beginning of period money supply:                        , where        is the
beginning of period money supply,             is a constant and       is the dollar price. Because
                            and     is a constant, we get:                                    . This equality
allows us to compare intertemporal price variation to intertemporal money supply variation.
         The BK model assumes that each agent sells a single good, but because this assumption is
unrealistic in grocery stores, we consider two cases: The single good is an individual UPC and
the single good is a basket of the goods sold by the chain. The length of the period is determined
by the degree of price rigidity. We adopt the perfect flexibility assumption as a benchmark and
assume that the length of the period is a week.
         Using a rolling window of 13 weeks, Figure 3 compares the standard deviation of prices
and baskets to the standrad deviation of the log of the money supply (weekly data not seasonally
                                                                                                                        12


adjusted).9 Figure 3A graphs the weighted average of the standard deviations of the cross
sectional modes, whereas Figure 3B graphs the standard deviation of the log price of the chain
basket.10 Focusing on the “All Prices” series, prices (including sales) move too much relative to
the BK model or any other model in which money supply shocks dominate. On the other hand,
the “No Sales” series is more volatile than M1 for the average mode but similar to M1 for the
chain’s basket.
           We note that the excess volatility of the mode is similar in magnitude to the excess
volatility of stock prices measured by Shiller (1981). Shiller found that the ratio of the standard
deviation of stock prices to the ex-post rational price is 6 in his earlier sample and 13 in the later
sample. Here we find that the ratio between the standard deviation of the cross sectional mode
and the standard deviation of M1 is 8 when using a simple average of the standard deviation and
10 if we use a weighted average.




9
 To define the rolling standard deviations, let = the share in revenue of UPC index ;         = the mode of UPC
 in week ;                   . We also use       to denote the standard deviation of the variables in the vector .
The rolling standard deviation of UPC in the 13 weeks following week is:                                  . The

rolling standard deviation for the representative or average UPC is:                     . The price of the basket in


week       is:                 , where     is the average quantity sold from UPC    (average over all weeks). The

rolling standard deviation of the basket in week     is:                                    . Figure 8A is a plot of
       and figure 8B is a plot of       , where              .
10
     Note that the chain basket has a lower deviation because it is a weighted average of the cross sectional modes.
                                                                                                          13




        A. Cross Sectional Mode (weighted average of the standard deviations in the following 13 weeks)




                             B. Basket of the Average Units Sold From Each UPC

                         Figure 3: Standard Deviations in Rolling 13 Weeks Windows


       Providing an additional illustration for the role of sales, Figure 4 plots the log of the price
of the chain basket with and without sales. The difference between the actual price of the basket
and the price without sales has an average of 8.1% and a standard deviation of 4.6%.
                                                                                                                14




                  Figure 4: The Log of the (cross sectional mode) Price of the Basket With and Without Sales.



4. “SALES” IN MODELS WITH FLAT EXPECTED PROFITS FUNCTION


              Sales play a significant role in varying the cross sectional mode value of the chain basket,
but this is only part of the story. The chain can react to shocks by changing the fraction of stores
that post the sale price.11 We illustrate this mechanism by an example in which the cross
sectional mode changes only once a year but the average price perfectly adjusts to weekly
changes in M1.
              We assume that the chain manager uses a menu of two prices: A high price P H and a low
price P L . The chain manager changes the price menu (P L ,P H ) only once a year. Knowing the
ex-post money supply, he chooses these prices so that both are in the equilibrium range:
         j
L    P
             Mt    H , where L is the lower bound of the normalized price (dollar price divided by the
weekly money supply) distribution and H is the upper bound of the distribution. The inequalities
hold for all the 52 weeks ( t ) in the year in which the menu is held constant. We assume that the
equilibrium price range H L is sufficiently large. When the expected profit function is
relatively flat the seller may not care much about the average price in the chain. But because we




11
   Glandon and Jaremski (2010) show that Dominick’s stores responded to Wal-Mart’s entry by increasing their
frequency of sales.
                                                                                                                   15


want to illustrate how a rather rigid menu can lead to a flexible price distribution, we assume
that the chain wants to keep its average normalized price at the level of 1 in all weeks.
        It is useful to express the dollar prices in terms of the average money supply (M1) during
                                                                                           52
the year: The high price is P H = M , the low price is P L = M , M = ( 1 52)                    M t is the average
                                                                                          t=1

of M1 over the year and        < are parameters. In week t , the manager chooses the fraction of
stores 0     t    1 that will post the low price. Thus,   t   P L + (1   t   )P H =   t   M + (1      t   ) M = Mt .
This leads to:

(1)                                      t   =(   ) /(        ),
where      = Mt       is given. Figure 5 illustrates two possible solutions to (1): ( , ) = (1.2,0.95)
                  M
and ( , ) = (1.05,0.8) . In the first, the cross sectional mode is the low price whereas in the
second, it is the high price. In both cases, on average about 80% of the prices are equal to the
cross sectional mode.
        This example shows that if the difference between the highest and the lowest price on the
menu is about 25% then the chain can perfectly adjust to weekly changes in the money supply
even if the menu changes once a year. More frequent changes in the menu are required for
perfect adjustment if the difference between the highest and the lowest price is small. In the data
the average (weighted by revenue share) difference between the highest and the lowest price in a
UPC-week cell is 21.6%, suggesting that the chain can achieve almost perfect adjustment to the
money supply simply by varying the fraction of stores that post the low price.
                                                                                                       16




             Figure 5: The Fraction of Stores Posting the Low Price (   t)   Under Two Alternatives.


       The example illustrates that from the point of view of models with price dispersion the
amount of excess volatility is understated when valuing the basket by the cross sectional modes.
It also demonstrates that there does not have to be tension between the assumption that
temporary price reductions are relatively cheap and the claim that sales are important from a
macro point of view. We now turn to discuss the question of why temporary changes in prices
may be relatively cheap.

5. PRICE DISCRETENESS


       Price discreteness is a form of nominal rigidity. A rather benign example is the
requirement that the price of a unit of an indivisible good (like toothbrush) should be expressed
in whole cents. We may expect that a price of a unit that costs 1 dollar may not change when the
money supply increase by a fraction of a percent even if there is no cost of changing prices,
simply because it is not possible to increase the price by a fraction of a cent. We also think that
this kind of rigidity does not lead to serious real effects because otherwise the Treasury
department will issue smaller denominations.
       Discreteness in the Dominick’s data is more serious than the whole cent restriction. As
suggested by the data’s documentation, the price discreteness seen in Figure 2A is likely the
result of stores choosing their prices out of a price menu provided by the chain management. In
                                                                                                    17


fact, Dominick’s seems to have two levels of chain management above the store managers. There
are several possible reasons for choosing prices in this hierarchical way.


External effect story: Price dispersion is “bad” from the point of view of maximizing the chain’s
profits. Individual stores have better information about the price that will maximize their own
profits but fail to take into account the effect of their choice on price dispersion in the chain. The
chain management is worried about both cross sectional dispersion and variations over time and
tries to limit the choice of the stores by requiring that the price be chosen out of a menu with
relatively few alternatives.


Returns to scale story: There are returns to scale in collecting information. The chain
management collects information about the price distribution in other chains and demand
conditions. It summarizes this information through a menu of prices that is handed down to the
stores. The store uses the information in the menu together with its own information about the
realization of idiosyncratic shocks to make a price choice. For example, the store may want to
post a “low price” when the level of inventories is high and a “high price” when the level of
inventories is low. But the store does not have the information about the price that will lead to
the change in inventories in the desired direction. This information is in the menu: The lowest
price will on average lead to an increase in the quantity sold and a reduction in inventories.


Deliberation costs: We may assume that the cost of deliberation (or of making choices) is an
increasing function of the number of elements in the choice set. Because of increasing returns to
scale, the chain management reduces the deliberation cost for the store management by reducing
the number of elements in the choice set. In what follows we adopt this explanation for
discreteness.


       Discreteness may explain the attraction to the intertemporal mode stressed by EJR. To
illustrate, we consider the following parable. There are 6 cities. A large city, say New York, with
75% of the total population and 5 cities of equal size each with 5% of the total population.
Individuals learn their preferred location at the beginning of each period. The preferred location
does not depend on their current location or previous history and the distribution of preferences
                                                                                                                 18


is consistent with the steady state distribution: 75% of the population prefers New York and 25%
of the population prefers one of the 5 smaller cities (5% per city). Agents can costlessly change
their location. The probability of a location change is 1 minus the probability that the agent will
choose the city that he is already in. This is, 1 - 0.75 = 0.25 if he currently resides in New York
and 1 - 0.05 = 0.95 if he resides in one of the small cities. Thus those who live in New York are
much less likely to change their location even when past history does not matter for the location
choice.
           Prices in our data spend most of the time at the mode and therefore the mode is large in
the same sense that New York is large. Eden and Jaremski (2009) estimated the probability of a
price change and use this type of reasoning to include a dummy that indicates whether the last
week price is equal to the cross sectional mode. This dummy variable is highly significant and
indicates that prices that are already in the mode are much less likely to change. This result is in
line with the findings in EJR and speaks for the importance of discreteness in the data.



6. IS PRICE DISCRETENESS AN IMPORTANT SOURCE OF NOMINAL RIGIDITY?


           In Section 4, we argued that price discreteness does not impair the ability of the chain to
adjust to money supply shocks. We now assess the chain’s ability to adjust to unobserved real
shocks.


6.1 The Shock Accumulation Hypothesis
           A restrictive menu is likely to lead to the accumulation of shocks and an eventual change
in the menu. It is therefore likely to produce many small price changes that are followed by a
large change. Figure 6 plots the percentage change in the price of the basket over weeks. The rate
of change varies substantially, but there are relatively few large changes.12 The largest change is
only 11%, and only 6 weeks have a change larger than ±6% . This pattern is not consistent with a
restrictive menu.




12
     When looking at the absolute value of the rate of change we get an average of 4% and a median of 3% per week.
                                                                                                                  19




            Figure 6: Percentage Change in the (Cross Sectional Mode) Price of the Basket over Weeks


         Additionally, if price discreteness causes nominal rigidity, there should be a positive
relationship between the length of “inaction”, the probability of an “action”, and the magnitude
of the “action”. The reason for this relationship is that a long period of “inaction” is likely to lead
to an accumulation of shocks and a large deviation from the desired price. Recent studies have
largely contradicted this hypothesis by finding that the probability of a price change and its size
are not positively related to the age of the price.13
         We examine the probability and size of an “action” as functions of a price’s age in Figure
7. Panel A plots the correlation between the age of the price and a price change dummy in each
Store-UPC cell (i.e. a 6pk of Coke at Store 101). A clear majority of cells have a negative
relationship, and therefore old prices are not more likely to change even when we account for
heterogeneity. Panel B plots the mean and median of the absolute size of the price change by
age. The correlation between size and age in our sample is negative and small suggesting that
stores do not make larger changes to old prices.




13
  Campbell and Eden (2007) find a negative relationship between the age of a price and the probability that it will
change whereas Eden (2001) and Ellis (2009) find that the correlation between age and size of the price change is
close to zero. Klenow and Kryvstov (2008) find no correlation between the probability and size of a price change
and its age.
                                                                                                        20




A. Within UPC-Store Cell Correlations Between the Age and (a) the Price Change Dummy and (b) the Mode
                                           Change Dummy.




          B. The Absolute Size of the Mean and the Median of Price Changes by the Prices’ Age

                                     Figure 7: Age and Reactions
                                                                                                            21


6.2 The Distribution of New Prices
        Looking at the distribution of new prices may provide another way of judging the
importance of nominal rigidity. In Prescott “hotels” type models, prices are set at the beginning
of the period on the basis of all available information about the distribution of demand. If some
prices cannot be changed we should find a difference between the distribution of newly set prices
and the distribution of all prices. For example, if because of nominal rigidities some stores post
prices that are not in the equilibrium range, then we should find that new prices are less dispersed
than all prices. This prediction is even stronger in menu cost models. In these models, ex-ante
identical sellers who make a price change choose the same price and cross sectional price
dispersion arises only because old prices were set at different dates.
        Figure 8 describes the distribution of relative prices defined as the log of individual prices
divided by the average within the UPC-Week Cell. Figure 8A describes the distribution when
looking at the entire sample while Figure 8B describes the distribution when looking at the
subsample of new prices. The distributions are almost identical, suggesting that nominal rigidity
is not an important reason for price dispersion.14




14
  This conclusion is consistent with Campbell and Eden (2007) who found more price dispersion among young
prices than among old prices.
                                                                                                                 22




A. The Distribution of Relative Prices (the log of individual prices minus the log of the average price in the UPC-
                                                     Week Cell)




                                       B. The Distribution of New Prices.

                                  Figure 8: The Distribution of Relative Prices
                                                                                                                     23


7. POSSIBLE EXPLANATIONS


           We start with models that assume ex-ante identical buyers and their equilibria can be
characterized by the way the expected profit of the individual seller varies with the posted price.
Figure 9 illustrates three alternatives. The flat expected profit function labeled (a) is the function
that a competitive seller in a Prescott (1975) and Burdett and Judd (1983) type models face in
equilibrium under the assumption that the probability distribution of the demand shock is
continuous. The discontinuous function labeled (b) is the expected profit function that a seller in
a Prescott type model will face when the distribution of demand is discrete. The bell shaped
function labeled (c) is the expected profit function that a monopolistically competitive seller
faces in the standard new Keynesian model.


Expected profits                              (a)




                 (b)          (b)                   (b)                   (b)



                                                                            (c)



                                 P1t                  P 2t                P 3t                 P 4t         pt   1

            Figure 9: Expected Profit Functions: (a) Perfectly Flat, (b) Approximately Flat and (c) Bell Shape


           A model characterized by a bell shaped function will have difficulty in explaining the
failure of the shocks accumulation hypothesis. A perfectly flat function model can account for
many of the observed behavior of individual prices.15 But once we allow for some menu type
costs, these models will have difficulty in explaining the similarity between the behavior of the
cross sectional mode and individual prices. According to these models, a change in the cross
sectional price distribution requires a change in a fraction of the prices (the prices that are no


15
     See, Eden (2001), Baharad and Eden (2004) and Head et.al. (2010).
                                                                                                               24


longer in the support of the equilibrium price distribution) and therefore individual prices should
move less than the mode while in the data they move slightly more than the mode.
        Figure 10, reproduced from Eden (1994), illustrates this argument. Suppose that the
equilibrium price distribution shifted to the right and the chain wants all its prices to be in the
new equilibrium range [P1t ,PZt ]. Since only the stores with inherited prices pt 1 < P1t must change
prices, only a fraction A of the stores must change prices. (Note that, A = 1 D = C + B ). This
implies that when the cost of changing prices is small, the probability that an individual price
will change must be considerably lower than the probability that the cross sectional mode will
change.



      Fraction
      of stores




                                                C

                                      A



                                                      D
                                                                                      B



                                  P1t 1 P1t                                 PZt   1         PZt
  Figure 10: A Shift in The Equilibrium Distribution Requires a Change in a Fraction      A = 1 D = C + B of all
                                                      Prices.


        Some form of discreteness may solve the problem. We start from the case in which the
distribution of demand is discrete. The nominal demand is M t + X t , where M t is the beginning
of period money supply and X t is a transfer payment that may take Z possible realizations:
X1t < X 2t < ... < X Zt . In equilibrium, there are Z cutoff prices: P1t < P 2t < ... < P Zt . A seller who
posts the price P s 1t < p    P st will sell if X t   X st . The seller will therefore never choose a price
that is strictly between the cutoff prices ( P s 1t < p < P st ) because he can increase his price
without affecting the probability of selling. All the cutoff prices yield the same expected profit
                                                                                                     25


( ) and the seller is indifferent among them. The equilibrium expected profit function is
approximately flat as in case (b) of Figure 9. A small increase in the money supply may shift the
equilibrium cut-off prices and sellers who posted the price pt 1 = P st 1 will find themselves
between two cutoff points ( P s 1t < pt 1 < P st ). Because of the discontinuity in the expected profit
function, they may want to make a price change even if the change in the money supply is small.
Thus when the distribution of the demand shock is discrete, small shocks can lead to many price
changes and the frequency of modes changes may equal the frequency of individual price
changes. The discrete version of the model can thus account for the many price changes but it
cannot account for the price repetition observed in the data.


7.1 Using Prices to Manage Inventories
           A UST model in which sellers are indifferent among prices in the equilibrium range
cannot account for many price changes if the cost of changing prices is non-trivial and if the
distribution of demand is close to a continuous distribution. A tie-breaking element may be
necessary. Here we explore the possibility that the desired price depends on the amount of
inventories.
           The intuition may be captured by the following planner’s problem.16 We assume a single
storable consumption good in the 75 stores controlled by the planner. Buyers arrive sequentially,
and each gets one unit of the consumption good. If there are more buyers than units stored, then
all the available supply will be distributed. Otherwise, some units will not be distributed and will
be carried as inventories to the next period. The planner considers alternative distribution
strategies. He can start from store 1 and distribute the units in this store until the store is stocked
out. He can then move to store 2 and so on. An alternative strategy will be to try to equalize the
amount of inventories carried by each store to the next period. To do that the planner may start
distributing units from stores with large stocks and rotate across stores, when all stores reach the
same level of stocks.
           This second strategy of minimizing cross sectional inventory dispersion is optimal when
the marginal carrying cost is increasing (because of the same argument that is used in
intermediate micro text to show that efficiency requires the equalization of the marginal cost


16
     Under certain conditions, the outcome in UST models is efficient. See Eden (1990, 2009).
                                                                                                                     26


across sellers). It suggests that a store with a relatively large amount of stocks will post a
relatively low price. In this respect our model is similar to Aguirregabiria (1999), Fisher and
Hornstein (2000) and Kryvstov and Midrigan (2010) who use different assumptions to motivate
the holding of inventories. This speaks to the robustness of our main result.17


7.2 Discrimination
         Price discrimination can lead to excess volatility. The mixed strategy equilibrium in
Varian (1980) is not consistent with the observation that sales tend to occur in most stores
simultaneously. This is less of a problem in models that focus on the discrimination between
buyers who shop in a single store. In Chevalier and Kashyap (2010) there are some buyers who
are not willing to switch between brands of very similar products (frozen orange juice) and some
buyers who are willing to switch. Sales discriminate against the non-switchers. This type of
discrimination does not explain the (excess) volatility of the cross sectional mode value of the
basket of goods sold by the chain.
         In Hendel and Nevo (2010) some buyers have access to storage technology and some do
not. This may lead to intertemporal price discrimination. But again pure discrimination does not
require the coordination of sales across stores. The findings in Hendel and Nevo are consistent
with the hypothesis that the marginal storage cost is roughly the same across agents, including
buyers with storage technology.

8. A PARTIAL EQUILIBRIUM MODEL


         As was said in the introduction, we would like to have a general equilibrium model in
which nominal rigidities are not the only possible reason for money non-neutrality. This task is
beyond the scope of this paper. Here we adopt a partial equilibrium approach that illustrates how
nominal rigidities can be introduced to existing UST models in a way that nests the flexible price
version: For some parameter values nominal rigidities are not important and the model reverts to
a flexible price UST model that can still generate real effects for surprise changes in money. We




17
  To decentralize the planner’s solution we must allow each store to put different price tags on different units so that
there will be many prices per UPC in each store. Below we assume that a store cannot post more than one price per
UPC in the same week.
                                                                                                      27


also add a distinction between informed (price sensitive) and uninformed buyers in a way that
nests the standard UST model with only informed buyers.
        We build on the UST model with storage in Bental and Eden (1993, 1996). Like other
UST models, this model pins down the equilibrium price distribution but does not yield strong
predictions about individual store’s behavior. We address this problem by assuming an
increasing marginal storage cost.
        We introduce uninformed buyers who spend the money they have in randomly selected
stores. Their role is to allow for the selling of a positive quantity even when the posted price
turns out to be too high for attracting the informed buyers. Informed buyers play a more
important role and they will be called buyers for short whenever confusion is not likely to arise.
The number of active (informed) buyers is random. They arrive in the market sequentially and
buy at the cheapest available offer.
        During trade some stores are stocked out and in equilibrium sellers face a tradeoff
between the probability of selling and the price. There is a single good that may be interpreted as
the basket of goods sold by the chain or an individual UPC. When the supermarket does not sell
the good in the retail phase of trade (to be described shortly), it sells it at a lower wholesale price
at the end of the period. The more realistic case in which unsold goods are carried as inventories
to the next period is described in the Appendix.
        Unlike other UST models here sellers are not indifferent among prices in the equilibrium
range: Some stores may strictly prefer a low price because they have a relatively large amount of
stocks for sale.


The price changing process: We allow for price rigidity and price discreteness using elements
from Sheshinki and Weiss (1977), Calvo (1983) and Reis (2006). The price setting process is
done in three phases. First the upper level chain manager chooses the “super menu” set         . He
can use the inherited set from last week or change it for a fixed cost . The lower level chain
manager chooses the menu for the current week,             . He may choose the inherited menu or
change it at the cost   . Finally, each store can choose a price p      . It may choose the inherited
price or it may change it for the fixed cost   .
        The cost of making a change is primarily a deliberation cost and depends on the number
of alternatives that the choice maker faces. To simplify, we assume that the number of elements
                                                                                                    28


in the menu and in the super menu does not change over time and use the following functions.
The cost of making a price change for the store is     = (# ) , where # A denotes the number of
elements in the set A . The cost of making a menu change is       = (# ,# ) , and thus depends
both on the number of prices that are on the menu (the complexity of the object of choice) and
the number of menus in the super menu set (the number of elements in the choice set). There is
no one that limits the choice set of the upper management. The cost of changing the super menu
is a constant denoted by: .
        For the sake of concreteness, we assume the following specification for the cost
functions:
                                                                      n
(2)                   (# ) = (# ) m ; (# ,# ) = ((# )(# )) ,
where ( ,m,n) > 0 are parameters.
        To motivate this assumption, we start with m = 1 and assume that the cost of making a
change is proportional to the number of alternatives considered. This may arise if the store
manager computes the expected profits for each alternative (at the cost of      per alternative) and
then chooses the best alternative. But in general the manager will employ better search procedure
for the maximum and therefore, 0 < m < 1. Similarly, a lower level chain manager may choose
the menu     by computing the expected profits of (# )(# ) for each of store that is in the chain.
But in general he will do a more efficient search by grouping similar stores and by guessing the
optimal solution for each group of stores. We therefore expect that n will be less than the
number of stores in the chain.
        Nominal rigidity is increasing in the parameters ( ,m,n) . Discreteness defined by (m,n)
contributes to nominal rigidity. To illustrate, we discuss the following special cases.
(a)   is large and m = n = 0 . In this case, the cost of changing the price at both levels is large and
does not depend on the number of alternatives. We may expect infinitely many alternatives on
the menu and infrequent price changes. The probability that two stores will set the same price in
a given week and the probability that a store will change its price and then go back to the same
price are both small.
(b)   is small, m is large and n = 0 . In this case we expect that many stores will post the same
price, frequent changes in the cross sectional mode but little repetition of prices over time.
(c)   is small, m and n are large. In this case, we expect that many stores will post the same
price and price repetition over time will occur.
                                                                                                                                 29


           We agree with EJR that the data can be best described by alternative (c). But as shown
above, the distribution of prices may adjust perfectly to short run variations in the money supply
even in this case, provided that                        is small.


Demand: Demand fluctuates because of changes in the money supply and because of shocks to
taste. Both the informed and the uninformed buyers experience taste shocks that determine
whether they want to consume or not. Buyers who realize a desire to consume spend all the
money they have. (In Eden [1994], the buyers are the old generation who do not save).
           Uninformed buyers have partial information about the distribution of prices in alternative
chains but do not know the prices in individual stores. When they want to consume, they make
their choice of a chain on the basis of habits (loyalty) and the sets ( , ) in alternative chains.
The set             is used as a statistic about the cross sectional distribution of prices in the chain and the
set   is used as a statistic about the variation in the distribution over time.
           An uninformed buyer who chooses to buy in chain j spends his entire money holdings in
a randomly chosen store (i, j) if the price in the store is less than or equal to the upper bound of

the price distribution in the chain ( pij                         max{      j   } ) and if the store is not stocked out. Otherwise,
the uninformed buyer chooses another store randomly (that may belong to a different chain) and
keeps doing it until he spends all his money. From the point of view of the individual store, the
nominal demand of the uninformed buyers is a random variable denoted by
d j = d(    j   ,     j   ;   j   ,   j   ) , where (     j   ,   j   ) denotes the vector of menus in competing chains.
           The number of (price sensitive) buyers who want to consume is random and the average
per buyer amount of money spent is random. As a result the amount spent is a random variable
D. We assume that D can take Z possible realizations (indexed s): 0 < D1 < D2 < ... < DZ . For
notational convenience we use D0 = 0 . After the uninformed buyers complete their transactions,
the informed buyers who want to consume arrive at the market one by one. Upon arrival each
buyer sees all advertised price offers that are backed by actual supply. Some posted prices are
not backed because stores may be stocked out.
           From the sellers’ point of view, the demand of the informed buyers arrives in batches.
The first batch of D1 dollars arrives with certainty. After this amount is spent, there are two
possibilities: if D = D1 then trade for the current period ends. If D > D1 , there are some buyers
                                                                                                                       30


who want to consume but could not find the good at the cheapest price. The minimum amount of
unsatisfied demand is: D2 D1 = argmin s{Ds D1} dollars. These dollars are spent on goods that
are offered at the lowest price. Again there are two possibilities: If D = D2 , trade ends. If D > D2 ,
at least additional D3 D2 dollars arrive and so on until there are no unsatisfied buyers or no
stores with unsold supplies.
           In general, batch s consists of          s   = Ds Ds 1 dollars. Figure 11 describes the sequence of
events.


                                                           p               d                     D            p*
 The choice                       The choice         The choice         The arrival         The arrival    Wholesale
 of the super                     of the weekly      of price           of the              of the         buying of
 (quarterly) menu                 menu                                  uninformed          informed       inventories


                                     Figure 11: The Sequence of Events within the Period


Supply: There are J chains and k stores per chain. We use                      jt   to denote the menu of chain j in

week t . The price of store i in chain j in week t is denoted by: pijt                      jt   . Some indices may be

dropped whenever confusion is not likely to arise.
           We allow for storage: NS units that were not sold yield R(NS) units of inventories at the
end of the period, where like a production function, the gross return function R is monotonic and
concave. An example may be the case of limited storage (shelve) space, L . In this case,
R(NS) = NS when NS                 L and R(NS) = L when NS > L . At the end of the period the store sells
the R(NS) units at the wholesale price of p* .
           At the beginning of week t , store (i, j) faces the following predetermined variables:
(Qijt , pijt ,   jt
                        ˜
                      , pt ) , where Q is the quantity available for sale, pijt is the last week price,            is the
               ˜
price menu and p is the distribution of prices across all stores (a description of the number of
units of the good offered at each price by all stores). Variations in inventories across stores arise
as a result of supply shocks.
           The individual store is small relative to the size of each batch and it assumes that it will
sell its entire supply if the state of demand is sufficiently high. Otherwise, if the realization of N
is low, it will sell only to uninformed buyers. The probability of selling to the informed buyers,
                                                                                                                       31

                      ˜
q( pijt ) = q( pijt ; pt ) , depends on the store’s price, p , and the cross sectional distribution of prices
˜
p . From the store’s point of view the function q( p) is a sufficient statistic for the entire price
             ˜
distribution p .
         We assume that the store satisfies demand, unless it is stocked out. The store expects to
sell its entire supply of Q units with probability q( p) . With probability 1 q( p) the store sells
only to uninformed buyers a random amount                    d
                                                                 p      Q and the accumulated stocks of R(NS) units
at the wholesale price p* , where NS = max(Q ( d p ),0) is the amount that was not sold at the

retail phase.
         We assume that the store maximizes the expected profits in the current week. This
simplification abstracts from the effect of the current price choice on future profits. The
Appendix has a dynamic programming formulation that does not use this assumption. We follow
Calvo (1983) in assuming that the store may revise its price with an exogenously given
probability . Whenever a price revision is allowed the store deliberates and spends the fixed
cost   (# ), but because of discreteness not all deliberations result in actual price changes.18
         When the store does not deliberate and uses the inherited price p = pijt , its expected

nominal profits are:

(3)                nd   ( p,Q) = q( p)Qp + (1 q( p)) E {(max( pQ,d)) + ( R( NS )) p* } ,
where NS = max(Q ( d p ),0) and the sub nd is for “no deliberation”.

         When the store deliberates a nominal price change its expected profits are:

(4)                        d ( ,Q) = max p                        nd   ( p,Q)     (# ) .
The store’s unconditional expected profits are:

(5)                         ( pij ,   j   ,Qij ) = (1   )   nd   ( pij ,Qij ) +     d   (   j   ,Qij )


The lower level chain manager deliberates with probability                              . Whenever he deliberates he
maximizes the expected weekly profits of the chain, taking the beginning of period stocks in the



18
  A more satisfactory treatment may assume that the store decides whether or not to review its price on the basis of
the perceived match between the beginning of period stock, Q , and the inherited price, p . For example, a store
with a relatively high p and a relatively high Q will choose to review its price while a store with a relatively low
 p and a relatively high Q may choose not to review it.
                                                                                                                                                                      32


chain, Q jt = (Q1 jt ,...,Qkjt ) , and the inherited prices in the chain, p jt = ( p1 jt ,..., pkjt ) as given. The

expected profits of the chain at the time this manager may act are:

(6)                   G( p j ,     j   ,        j   ,Q j ) = (1      )Gnd ( p j ,          j   ,Q j ) + Gd ( p j ,      j   ,Q j )
where
                                                                                      J
(7)                                            Gnd ( p j ,    j,     j ,Q j ) =                ( pij ,     j   ,Q j )
                                                                                     i=1

(8)                   Gd ( p j ,           j   ,Q j ) = max           j
                                                                          Gnd ( p j , ,            j   ,Q j )       (# ,#      j   )


The upper level chain manager chooses the “super menu” before the lower level manager
chooses the menu. This level of management worries about variations in prices over time while
the lower level management worries about cross sectional price dispersion. We leave the
discussion of this level of management to the Appendix, and for now, assume that the super set
      is exogenously given.
                                                                                     dij
Partial Equilibrium: Store (i, j) sells zij = max                                          ,Qij units to the uninformed, where (dij ,Qij )
                                                                                     pij

are the realizations of the nominal demand ( d ) and the supply shock ( Q). At the beginning of the
sequential trade phase, the nominal supply of store (i, j) at the price p , is:

(9)                       Sij ( p) = pij (Qij                zij )        if p = pij and zero otherwise.


The nominal supply is thus the nominal value of the goods that were not sold to the uninformed
buyers. Summing over chains and stores we obtain the aggregate nominal supply at the price p :
                                                                            J    k
(10)                                                         S( p) =                  Sij ( p)
                                                                           j=1 i=1

          As in other UST models there are Z hypothetical markets that opens sequentially. The
first batch of    1   = D1 (informed) dollars buy in the first market, the second batch of                                                              2   = D2     D1
dollars buy in the second market and so on. But unlike other UST models, here there is typically
more than one price per market.
                                                                                                                                       Ps

          We define the cut-off points Pt = (0                                  P0t < P1t < P2t <,...,< PZt ) by:                               S( p)dp =   s   . Thus,
                                                                                                                                       Ps   1


the net nominal supply of all stores that post prices between Ps 1t and Pst is equal to the nominal
                                                                                                                                                                     33


demand of batch s. The probability of selling at the price Ps 1t < p                                                                   Pst is the probability that
batch s will arrive: q( p) = Pr ob(N                                        N s ) if Ps 1t < p     Pst .
                We now use the cut-off points to describe the sequential trade that occurs after the
uninformed buyers have completed trade. Market 1 opens with probability 1 and satisfies the
demand of the first                            1   informed dollars. The nominal supply at the price below the first cut off
                    P1

point is                 S( p)dp . The first cutoff point is defined in a way that insures the clearing of the first
                    0

                    P1

market:                  S( p)dp =             1   . After transactions in the first market are complete we may have two
                    0

possibilities. Either no more (informed) dollars arrive or additional dollars arrive and open a
second market. In general, market s opens with probability Pr ob(N                                                                      N s ) . The nominal demand
                                                                                                                 Ps

in this market is                   s and the nominal supply in this market is                                            S( p)dp . The definition of the
                                                                                                                 Ps   1


cutoff points insures market clearing.


A partial equilibrium is defined for the inherited prices, pt = ( p11t ,..., p1kt ;...; pJ1t ,..., pJkt ) ,
inherited menus,                        t 1    =(     1t 1   ,...,   Jt 1   ) , the super menus             t   =(        1t   ,...,    Jt   ) and the wholesale price
p* . Given these predetermined magnitudes, the vector ( t , pt ,Pt ) and the functions
( ,        nd   ,        d   ,G,Gnd ,Gd ,St ,qt ) are partial equilibrium, if in addition to (3)-(10), the following
conditions are satisfied:
(a) qt ( p) = Pr ob(N                          N s ) when Ps 1t < p                  Pst
(b) pijt = argmax p                      jt
                                               {     nd   ( p,Qijt )} if pijt        pijt ;

(c)        jt   = argmax                  jt
                                               {Gnd ( p jt ,         ,   jt   ,Q jt )} if     jt   jt   ;
       Pst

(d)             St ( p)dp =         s      for all s .
      Ps   1t


                The requirement (a) specifies the probability of selling, (b) and (c) say that a price (menu)
change must maximize expected profit, and (d) are market-clearing conditions.
                                                                                                                       34


The expected profits function: The expected profit function (3) when not deliberating a change is
discontinuous and has a global maximum because it depends on the amount of inventories, Q.
Figure 12 illustrates. The discontinuous line is the expected profits as a function of the inherited
price pt 1 , holding Q constant. The flat line is the expected profit that can be achieved if the
store deliberates.


                   nd   ( pt 1 ,Q 0 )




     d   (Q 0 )




                                        P1t            P2 t             P3t              P4 t
                                                                                                                  pt   1

                         Figure 12: Expected Revenues as a Function of the Inherited Price for a Given   Q = Q0


Money non-neutrality: In this model money surprises may have real effects even when nominal
rigidities are relatively unimportant ( = 0 ). In this case, each store will set the optimal price at
the beginning of the period but it may still not sell its entire supply.19 A low realization of
nominal demand d + D leads to a large amount of unsold goods that will be carried to the next
period as inventories. In Bental and Eden (1993) the accumulation of “undesired” inventories
leads to a decrease in next period production. When > 0 prices at the beginning of the period
will not fully reflect available information and there is room for policy intervention designed to
insure that a large fraction of available supply is sold.
                  We now turn to a discussion of the implication of the model and its ability to account for
some of the observations made here and in the literature.




19
  There is still the question of whether sellers will have incentives to change their prices during trade. This depends
on the nature of the information revealed during trade. In Eden (1994) the information revealed during trade does
not create incentives to change prices and in this sense, prices are perfectly flexible.
                                                                                                             35


The inventory target hypothesis: In Bental and Eden (1993) an increase in inventories leads on
average to a reduction in prices and an increase in the amount sold. By assuming an increasing
marginal storage cost we derive now a similar result for the individual store: A store that
accumulates relatively more inventories will post a relatively low price.20
        We simplify by assuming that (a) d = 0 and there are no uninformed buyers, (b) Z = 2
and there are only two hypothetical sequential markets: market 1 opens with certainty and market
2 opens with probability q2 = Pr ob(N = N 2 ) , (c) The menu has only the two cutoff points:
(P1t ,P2t ) . Under these assumptions we now show the following claim.


                               ˆ
Claim: There is a cutoff point Q , such that a store that deliberates will choose the first market if
       ˆ
Qijt > Q and the second market if Qijt         ˆ
                                               Q.


        To show this Claim, note that when the store chooses the first market its nominal
revenues will be: Qijt P1t . The expected nominal revenues when supplying to the second market

are: q2Qijt P2t + (1 q2 ) p* R(Qijt ). The store will choose market 1 if:

(11)                             q2Qijt P2t + (1 q2 ) p* R(Qijt ) Qijt P1t
We write (11) as:
                                           R(Qijt )
(12)                          F(Qijt ) = (1 q2 ) p*    P1t q2 P2t
                                             Qijt
The function F(Q) is the unit value when not selling the good. The right hand side of (12) is the
difference in the expected price between the two markets. The store prefers market 1 when the
unit value of inventories does not compensate the difference in expected price.
                                                                                               ˆ
        The concavity of R implies that F is a decreasing function and there is a cutoff point Q
such that stores with Qijt     ˆ                                             ˆ
                               Q will choose market 1 and stores with Qijt < Q will choose market 2.

Figure 13 illustrates.




20
  Similar to West (1986) and Bils and Kahn (2000), the approach here yields a positive correlation between
expected sales and inventories.
                                                                                                                         36




P1t      q2 P2t




                                  Stores in                 Stores in
                                  market 2                  market 1

                                                                                                  F(Q)

                                                       ˆ
                                                       Q                                                 Q

      Figure 13: Stores with stocks greater than   ˆ                                                                 ˆ
                                                   Q supply to market 1, while stores with stocks that are less than Q
                                                      supply to market 2

                                                                                 ˆ          ˆ
           In this example, stores behave as if they have a “target stock” level Q . When Q Q they
                                    ˆ
post the low (sale) price. When Q < Q they post a high (regular) price. Note that the target level
of stock is a relative one. Stores post the low price if their stock is high relative to other stores
that choose to change their price.
           We now turn to discuss the failure of the shock accumulation hypothesis.


The lack of correlation between size and age: We define relative Q as the quantity in the store
divided by the average quantity across all other stores (including stores that do not belong to the
DFF chain). When relative Q is serially independent, the desired price in the current period does
not depend on history and there will be no correlation between size and age. We now turn to
discuss the serial correlation in (relative) Q.
           In multi-periods models that allow for storage the amount of current period inventories is
the sum of inventories carried from last week (goods that were allocated to markets that did not
open) and new shipments. There may be a random shock to new shipment as a result of weather
and stock-outs at the wholesale level. The identity of the stores that will get their order filled may
be a random variable that is serially independent. The second source of variation in Q is the
amount of stocks that were carried as inventories from the previous period.
           This second source of variation has two opposite effects on the serial correlation. In
detail, the amount of inventories at the beginning of period t is R(Qt 1 ) with probability q( pt 1 )
                                                                                                      37


and zero otherwise. A store that had a relatively high Qt 1 will post a relatively low price and
therefore the probability of carrying no inventories, q( pt 1 ) , is relatively high. On the other hand
the amount of inventories carried from last period will be relatively high if the store did not sell
in the last period (with probability 1 q( pt 1 ) ). These opposite effects may lead to a low serial
correlation in relative Q, leading to a low correlation between age and size.


The lack of positive correlation between age and the likelihood of a change: In our model the
store may not change its price every week even when         is small because it has only a relatively
few prices on the menu. We now assume that the weekly menu includes 2 prices only: A sale
price and a regular price. The store uses the lower sale price when its relative Q is high and it
uses the regular price otherwise.
        Sales are likely to last for a short time. To see this note that whenever the store posts the
low price it will sell its entire supply and is likely to have a relatively small amount of stocks in
the following period. It will therefore switch to a higher price.
        But even after controlling for sales there is no positive correlation in our data between
age and the probability of making a change. This may be explained under the assumption that
some low prices are not defined as “sale” prices.


A “markup target”: In our model, each of the Z markets has a different markup. A low index
market has a low markup and high probability of selling while a high index market has a high
markup with a low probability of selling. A chain that tries to stay in the equilibrium range
behaves as if it has a “target” level of inventories and a “markup target”. The latter is consistent
with the description of EJR .


Lower prices in high demand periods: In their well known paper, Chevalier, Kashyap and Rossi
(2003) observed that prices do not rise during periods of high demand. This may occur in a
Prescott type model, if demand uncertainty is low during periods of high demand. To see this
point, consider a perishable good that is demanded with probability 0.5 in a "regular" week and
is demanded with probability 1 in a "peak demand" week. Assume that whenever the good is in
demand buyers are willing to pay a high reservation price for it. The cost of the good to the
supermarket chain is 5 dollars. In a competitive environment in which stores make zero profits
                                                                                                     38


on average, the store will charge a price of 10 dollars in a "regular" week in which it sells with
probability 0.5. The store will charge a price of 5 dollars in a "peak demand" week in which it
sells with probability 1. Thus we may observe that the price in a "peak demand" week is lower
than the price that we observe in a "regular" week when the good is actually sold. The intuition is
that in high demand periods the probability of selling is higher on average and therefore the
average price is lower. This explanation can be tested by comparing measures of price dispersion
during holidays and other "peak demand" weeks to the measures during "regular weeks".

9. CONCLUDING REMARKS


       We looked at the behavior of the cross sectional price distribution, from the point of view
of a UST model augmented to allow for nominal rigidities. We augmented the model so that we
could reject the joint hypothesis that nominal rigidities are not important and (nevertheless)
money has real effects. But we did not find evidence against this joint hypothesis. The cross
sectional mode varies much more than the money supply. The probability of a cross sectional
mode change and the absolute size of the change do not increase with its age. The dispersion of
newly set prices is not less than the dispersion of all prices.
       From the point of view of models with equilibrium price dispersion, the question is why
prices move so much? We assume increasing marginal storage cost that breaks the tie between
prices. Under this assumption the store is no longer indifferent between the prices in the
equilibrium range and prefers a low price when it has a large amount of inventories relative to
other stores. Thus, prices change so much because they are used to manage inventories.
       Can we make out of sample predictions about the effect of large money supply shocks?
We doubt that this is possible because in our sample the money supply did not change much and
therefore it is possible that many deliberations did not result in actual changes. For example, it is
possible that the upper level chain management did not change the “super menu” very often
because in many deliberations it concluded that there is enough variety of prices on the menus to
deal with the current environment. This does not mean that it will not make a change in response
to a large shock.
       We think that it is more productive to look for the ability of the price distribution to react
to large shocks in an environment in which large shocks do occur. In our model nominal rigidity
occurs if there are not enough deliberations to insure that all posted prices are in the equilibrium
                                                                                                                       39


range. We may therefore expect that if nominal rigidity is important than cross sectional price
dispersion will increase with the absolute level of the inflation rate. The literature on inflation
and cross sectional price dispersion does not support this hypothesis.21



APPENDIX: A DYNAMIC FORMULATION OF THE CHAIN’S PROBLEM

           In this Appendix we relax the assumption that the chain maximizes current profits and
assume that unsold goods becomes, after depreciation, part of the next period’s stocks.
           To formulate the chain’s problem, we distinguish between a dollar price and a normalized
price, the latter defined as the dollar price divided by an appropriate deflator. The deflator at
week t is denoted by M t and we may think of it as the money supply at the beginning of week t .
The deflator in week t + 1 is M t +1 = (1+ μ)M t , where the rate of change μ is iid . A store that
does not change its dollar price may experience a change in the normalized price. For example, if
the normalized price was 1 at week t , it will become, in the absence of a nominal price change,
 1
  1+ μ   at week t + 1.
           The inherited normalized price is the normalized price that a store will have if it does not
make a nominal price change. Thus if last week price was 1 normalized dollar, this week
inherited normalized price is 11+ μ . Note that the inherited normalized price is different from the
last week normalized price.


The store’s problem: At the beginning of period t , the store (i, j) faces the following
                                                              pijt   1
predetermined variables: mijt ,Qijt = Iijt + x ijt , pijt =              ,   jt
                                                                                    ˜
                                                                                  , pt , where m is the amount (in
                                                              1+ μ
terms of current normalized dollars) that it will distribute as dividends, I is the beginning of
period inventories, x is current exogenously given production, Q = I + x is the quantity available
for sale, pijt 1 is the last week normalized price, pijt is the inherited normalized price (the

normalized price that the store will have if it does not make a nominal price change),                        is the


21
  Reinsdorf (1994) found a negative relationship between inflation and cross sectional price dispersion, during the
Volker disinflation era. Eden (2001) used data from high and moderate inflation periods in Israel and finds no
significant relationship between inflation and cross sectional price dispersion. Ahlin and Shintani (2007) compare
the cross sectional price dispersion before and after the “tequila crisis” in Mexico and found that cross sectional
price dispersion decreased after the crisis while inflation went up dramatically.
                                                                                                              40

               ˜
price menu and p is the distribution of normalized prices across all stores (a description of the
number of units of the good offered at each price by all stores).
         We assume that the store satisfies demand, unless it is stocked out. The store expects to
sell its entire supply of I + x units with probability q( p) . With probability 1 q( p) the store sells
only to uninformed buyers a random amount           d
                                                        p        I + x and its beginning of next period’s
inventories are: R(NS) where NS = max( I + x ( d p ),0) is the amount that was not sold in the

current period.
                                                                                          (I + x) p
         When the store sells its entire supply, the store will distribute ms =                     normalized
                                                                                           1+ μ'
dollars at the beginning of next period. When the store sells only to uninformed buyers, the store
                        max( d, p(I + x))
will distribute mns =                     normalized dollars at the beginning of next period.
                             1+ μ'
         The store forms expectations about next period’s menus and next period’s probability of
selling function. In what follows we treat these expectations as exogenously given random
variables and denote them by:       ', ',q' .
                                                   pijt     1
         The value of the store, V m,I, pijt =                  , , ,q , satisfies the following functional
                                                   1+ μ
equation:

(A1)          V (m,I, pijt , , ,q) = (1      )Vnd (m,I, pijt , , ,q) + Vd (m,I, , ,q)

where,
(A2)                                      Vnd (m,I, p, , ,q) =
                             p                                             p
 m + q( p) EV ms,I'= 0,          , ', ',q' + (1 q( p)) EV mns,I'= R(NS),       , ', ',q'
                           1+ μ'                                         1+ μ'
(A3)              Vd (m,I, , ,q, p* ) = max p {Vnd (m,I, p, , ,q)                 (# )}


                    max( d( , ), p(I + x))
And NS = I + x                             is the amount of stocks carried to the next period when
                              p
                                                (I + x) p
there is no sale to the informed buyers, ms =             is the next period’s dividend distribution if
                                                 1+ μ'
                                                                                                                                                                     41


                                                                            max( d, p(I + x))
there was a sale to the informed buyers and mns =                                             is the next period’s dividend
                                                                                 1+ μ'

distribution if there was no sale to the informed buyers.

The lower level chain manager chooses the menu for the current week taking the initial
conditions of the stores that belong to the chain as given. We use, m jt = (m1 jt ,...,mkjt ) to denote

the current dividends distributed by each of the k stores belonging to the chain; I jt = (I1 jt ,...,Ikjt )

                                                                                       p1 jt         1              pkjt   1
to denote the beginning of period inventories and p jt =                                                 ,...,                 to denote the inherited
                                                                                       1+ μ                         1+ μ
normalized prices. The value of the chain at the point that the lower level chain manager makes
his choice is:

(A4) G(m jt ,I jt , p jt ,   jt   ,   jt   ,qt ) = (1    )Gnd (m jt ,I jt , p jt ,     jt    ,           jt   ,qt ) + Gd (m jt ,I jt , p jt ,         jt   ,qt ) ,

where:
                                                                  k
(A5)                         Gnd (m,I, p jt , , ,q) =                   V (mijt ,Iijt , pijt , , ,q)
                                                                  i=1

(A6)              Gd (m,I, p jt , ,q) = max                  {Gnd (m,I, p jt ,         , ,q)                        (# ,# )}



The upper level chain manager deliberates with probability . At the time he may act, the value

of the chain is given by the following equations.


(A7) H(m jt ,I jt , p jt ,   jt   ,   jt    ,qt ) = (1    )H nd (m jt ,I jt , p jt ,        jt   ,            jt   ,qt ) + H d (m jt ,I jt , p jt ,        jt   ,qt )

(A8)                              H nd (,m,I, p ,        ,    ,q) = G(m,I, p ,                   ,             ,q)
(A9)                H d (m,I, p ,            ,q) = max {H nd (m,I, p ,               , ,q)                         (# ,# )}


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