Preliminary version: Please do not distribute or quote without authors’ permission.
Lumpy Price Adjustments:
Structural Estimation of the Menu Cost Function
Wilko Letterie
and
Øivind A. Nilsen*
October 9, 2014
Abstract
We analyze empirically the micro-foundations of pricing behavior. The intertemporal profit
function we consider accounts for various functional forms of menu costs. Norwegian plant
level data are used concerning monthly producer prices. A two-step estimation routine is
employed to identify the menu cost function parameters. In the first step an ordered probit
model allows to acquire parameters that are related to the decision to adjust prices (i.e. the
extensive margin). In the second step, structural parameters are identified by estimating a
model for the price adjustment size (the intensive margin). We find that fixed menu costs are
negligible. Instead linear adjustment costs, and to some degree convex costs, are important to
understand the dynamic pattern found in micro level producer price data.
Keywords: Price Setting, Micro Data.
JEL classification: E30, E31.

Acknowledgements: The paper was partly written while Nilsen was visiting the Center for
Macroeconomic Research – University of Cologne (CMR), whose hospitalities are gratefully
acknowledged. We also like to thank for valuable comments during presentations at Mannheim
University and the 5th Cologne Workshop on Macroeconomics (CMR 2014). The usual disclaimer
applies.

Maastricht University, School of Business and Economics, Department of Organization and
Strategy,
P.O.
Box
616,
6200
MD
Maastricht,
The
Netherlands,
Email:
[email protected]
*
Department of Economics, Norwegian School of Economics, N-5045 Bergen, Norway. E-mail
address: [email protected]; IZA-Bonn
1. Introduction
Prices affect consumers and firms. They determine how much one gets out of income and
profit. Price setting also affects policy makers, because it has an impact on the effectiveness
of macro-economic policies. In fact, classical models in economic theory predict that if prices
and wages were fully flexible a “monetary change results only in proportional changes in
prices with no impact on real prices or quantities” (see Romer, 2012). However, we observe
in practice that nominal shocks have real effects in the short run, and the reason for this lies in
the fact that prices are sticky. Thus, in macro-economic research it is important to understand
how flexible prices are.
Various studies have started to introduce price rigidity in economic theory. Often
nominal prices have been assumed to be sticky at the micro level. Usually these assumptions
are very stylized. For the most commonly used macroeconomic models accounting for price
rigidity it is often assumed that producers in the economy only change prices at a given time
randomly, so-called Calvo pricing (Calvo, 1983).1 In this model a lag in price adjustment at
the micro level is introduced that is technically attractive, but that does not tell us much about
the structural causes of persistency in prices. Mankiw and Reis (2002) use an alternative
model formulation where prices are free to change, but where new information can only be
obtained randomly at a given time. In a recent work by Mackowiak and Wiederholt (2009), it
is instead assumed that firms are free to choose when information is to be obtained, but that
the capacity to process new information is limited.
1
Prices also play an important role in macroeconomic models with intermediate goods. The
producer level price adjustment responding to shocks to production costs and demand for
intermediate goods is transmitted to the consumer level prices. Cornille and Dossche (2008)
show that the degree of producer price rigidity will be decisive in an inflation-targeting
central bank. In addition, 60 percent of the value of a consumer good is generated on the
producer level in industrialized economies (Burstein et al. 2000).
1
Price rigidity may also be caused by menu costs (cf. Sheshinki and Weiss, 1977,
1983). Menu costs are motivated by the fact that changing prices induces direct costs
(repricing, new promotional materials, new promotions) or indirect costs (annoyance among
consumers, etc.). Such menu costs are related to price changes, such that patterns of price
adjustment can be described as “zeroes and lumps”. Indeed, descriptive evidence from micro
data suggests that there are several consecutive periods where no price changes occur, and
then one observes significant changes for a short period (Álvarez et al., 2006; Dhyne et al.,
2006; Vermeulen et al., 2012). Such patterns may be explained by non-convex or fixed menu
costs. At the same time, rather small price changes occur frequently as well. Such small
adjustments might stem from convex adjustment costs. For instance, in the model by
Rotemberg (1982) deviations from optimal prices induce quadratic costs.
With detailed data on product prices, production costs and quantities it should be
possible to learn more about what the main reasons for the price changes of firms’ products
are. A problem in all the empirical research related to pricing, is access to good
microeconomic data. Some of the earliest work with microeconomic data is Cecchetti (1986)
who analyzed price adjustments related to various news and weekly magazines. Carlton
(1986) analyzed how the prices of goods in the association between firms were adjusted,
while Blinder (1991) based his study on interviews with business leaders. In a rather recent
paper from Sweden by Carlsson and Skans (2012), the authors use price data at the product
unit level of industrial manufacturers along with labor costs to investigate the micro
foundations of different assumptions about sources of price rigidities. Using a reduced form
model, these authors find that the Swedish data indicate limited support for the conclusions
found by Mankiw and Reis (2002), and Mackowiak and Wiederholt (2009), while the results
seem to be reasonable in light of the time-dependent Calvo model.
2
However, Lein (2010) recently found that models of price adjustment gain significant
explanatory power when state-dependent variables are added. This result hints at the
relevance of menu cost models. Menu costs are assumed to affect firm decisions in an
analysis of Dhyne et al. (2011). In that study a reduced form threshold pricing model of the
(S,s) type is used (cf. Sheshinksi and Weiss, 1977, 1983). Other studies have made an effort to
estimate structural parameters underlying firm pricing decisions. Levy et al. (1997) find that
the labor cost of workers spending time on changing prices, referred to as direct physical
pricing costs, are about 0.7% of annual revenues. Including indirect costs as well Slade (1998)
finds that changing prices costs approximately 1.7% of revenues for saltine crackers. Using
Spanish supermarket data Aguirregabiria (1999) finds similar estimates for costs of changing
prices. Midrigan (2011), using supermarket data as well, finds that his model calibrations
suggest price adjustment costs of about 2%. Willis (2000) finds similar costs of about 2-4%
for changing magazine prices using the data employed by Cecchetti (1986).
Zbaracki et al. (2004) find evidence that costs of changing prices may vary with the
size of the price adjustment. The larger the change the more managerial time is spent on the
pricing decision, and, in addition, internal firm communication increases. Furthermore, the
firm is also likely to incur higher cost of negotiation and communication with customers to
explain the decision. Though several studies exist, to the best of our knowledge only a few
have made an effort to obtain structural estimates for fixed, linear and quadratic cost
components in the menu cost function. A priori, we see no reason to exclude linear costs,
which have not received much attention previously. Furthermore, Zbaracki et al. argue that
fixed costs are small. In fact, they observe various scholars have found that fixed menu costs
are not high enough to cause price rigity. Another functional shape that potentially allows for
zeroes in price change data is a linear menu cost function, as such adjustment costs may
3
induce a kink in the first-order derivative at zero, and thus might cause a lumpy adjustment
pattern.
We focus on firms’ pricing behavior using a unique and relatively unexplored
Norwegian micro dataset. The data are based on micro level data from Statistics Norway
(SSB). The primary source is surveys sent to firms, where monthly prices (and prices
changes) are observed for several products. Firms are repeatedly surveyed. Thus, the new data
is a panel with monthly observations for the period 2002-2009. These firm/product level data
are matched with annual firm-level production income- and costs, investments and labour
stock data.
The method we use can be described as structural estimation as the estimated
parameters enable us to trace back parameters in the optimization models of firms’ price
decisions. An advantage of our approach compared to calibration based methods (see for
example Midrigan 2011) is that our assumptions can be tested statiscally. Our goal is to first
set up an optimization model of a firms’ dynamic profit function. This model includes a
function for the menu costs explicitly. In fact, we consider simultaneously three specifications
for the shape of menu costs: fixed, linear and quadratic (convex) costs. We estimate the model
with the existing data such that the menu cost function can be identified. An ordered probit
model allows us to acquire parameters that are related to the decision to adjust prices (i.e. the
extensive margin). Next, we obtain a deeper insight into the structural parameters by also
estimating a model for the size of price adjustment (the intensive margin). We correct for
selection bias at the second estimation stage. Our estimates reveal that fixed menu costs are
negligible. Instead, convex costs and in particular linear costs are found to be important to
understand the dynamic pattern found in micro level price data. By having tested the
assumptions regarding firms’ price-setting statistically, our results provide empirical evidence
4
on the micro foundations of dynamic stochastic general equilibrium models (see also Carlsson
and Skans, 2012).
This manuscript continues as follows. In section 2 we present the model. The
estimation method is depicted in section 3. The data are described in section 4. We present the
estimation results in section 5, and finally we conclude in section 6.
2. The Model
We extend a static price setting model by incorporating menu costs for prices. The idea is to
employ a menu cost function that is capable of replicating empirical features of the data. The
model is similar to labour and capital demand models as developed by for instance Abel and
Eberly (1994, 2002). In these models the size and timing of adjustment is determined by q, i.e.
the shadow value of a unitary change in the decision variable. Our approach is to estimate
various versions of the model based on an approximation of this q.2
We start our analysis assuming each plant produces N i goods and that the price of
each product is decided independently. The plant maximises the present value of discounted
cash flow, given by
(1)
   1 s 
P
Vit  Et   
    Zit  s , Pijt  s   C  Pijt  s 

 s 0  1  r  j1, Ni 




  

where the index i refers to a firm, the index j refers to a product, and the index t refers to a
month. The expression   Z it  s , Pijt  s  denotes the firm’s profit function for a product j. The
We do not specify a full DSGE model. This is done in order to focus on firms’ pricing
decisions and not let the analysis be affected by possible miss-specifications or problems in
other parts of the macro economy.
2
5
monthly discount rate is given by
1
and Zit denotes a vector comprising demand and
1 r
technology shocks, for instance. The menu cost function for prices is given by
C p  Pijt  
(2)
P
ijt
D
2
2




c p  Pijt 
c p  Pijt 

P




  aP  bp Pijt  
P

D

a

b

P

P
 ijt 1

 ijt 1 
ijt
P
p
ijt




2  Pijt 1 
2  Pijt 1 




where DijtP  ( DijtP  ) is a dummy variable equal to 1 for price increases (decreases) and equal to
zero otherwise. Hence, this expression allows for potential asymmetric costs.
Menu costs may, or may not depend on the size of the price change. Fixed cost of
adjustment are given by a P and a P . The costs of price changes consist of producing new
price lists and monthly supplemental price sheets, and informing interested parties. These are
the classical menu costs as considered theoretically by Sheshinki and Weiss (1977, 1983).
Typically such physical costs are independent of the size of the price changes (Levy et al.,
1997; Zbaracki et al., 2004).
Some costs of changing prices may depend on the size of the price adjustment. The
larger the change the more managerial time is spent on the price change decision. Decision
cost and internal firm communication increase for larger price changes. In addition, the firm is
also likely to incur higher cost of negotiation and communication with customers (Zbaracki et
al., 2004). To account for such costs we consider two functional forms allowing costs to vary
with the size of the price change. We consider linear and convex costs. A priori, there should
be no reason to exclude linear costs. Furthermore, Zbaracki et al. argue that fixed costs are
small. In fact, they observe various scholars have observed that fixed menu costs are not high
6
enough to cause price rigidity.3 Another functional shape that potentially allows for zeroes in
price change data is a linear menu cost function. Linear costs are represented by the parameter
bP .4 Convex cost components are given by the expressions multiplied by the parameters c P
and c P . To see the consequences of the various types of menu costs we solve the model in
line with Abel and Eberly (1994, 2002).5 The first order conditions inform us that prices
behave according to the following rules
(3)
Pijt
Pijt 1

2cP aP
1 P
qijt  bP if qijtP 
 bP .
cP
Pijt 1


This expression tells that a price increase occurs if the marginal profits obtained by the change
of price are positive. Similarly, for a price reduction, we have
(4)
Pijt
Pijt 1
2cP aP
1 P
P

qijt  bP if qijt  
 bP
cP
Pijt 1


If the marginal value of a price change does not satisfy the conditions in equations (3)
and (4) a price will not be adjusted, i.e.
Pijt
Pijt 1
 0 . In all of the above cases, the shadow value
of a price is given by
(5)
3
4
s
 
  Z it  s , Pijt  s 
C P  Pijt  s   
1
1



q  Et   

 

 s 0  1  r  
Pijt  s
1 r
Pijt  s



p
ijt
See the references in their footnote 2.
We do not allow for asymmetry in the linear menu cost component, bP , as we cannot
identify such asymmetric costs from the  0P parameter which we present in equation (6)
below. Only the difference between bP and bP can be estimated (a linear cost for positive and
one for negative price adjustments, correspondingly).
5
Various types of adjustment costs and their consequences have been reviewed by
Hamermesh and Pfann (1996). See also Abel and Eberly (1994) for a theoretical exposition.
Though these studies deal with factor demand, the implications are identical.
7
This expression denotes how a unitary change in the price of product j affects the value of the
firm. The two main elements are in the inner brackets of equation (5) and relate to the
marginal profit and the marginal menu cost function, respectively. The first element reveals
that a price change influences marginal profits in future periods. In addition, a change in price
saves menu costs in the future as depicted by the second term.
Equations (3) and (4) show that if fixed menu costs are absent, i.e. a P = a P = 0, then
the model is still capable of explaining the presence of zeroes in the price change data. The
linear cost term bP generates price rigidity. If -bP  qijtP  bP the firm will not adjust its price.
Strikingly, if a P = a P = 0 we will see not so many large price changes in the data. Minor
deviations from the thresholds qijtP  bP and qijtP  bP will induce very small price changes.
Hence, linear costs also make a firm abstain from changing prices. Typically such costs
induce many zeroes in price change data, but actual changes can still be small. However, if
fixed costs are present, i.e. aP  0 and aP  0 , small price changes are infrequent, and the
tails of the price change distribution will become thicker. Fixed costs cause lumpy price
changes because the thresholds in equations (3) and (4) increase in absolute value. Then firms
will not adjust prices for quite some time, and once adjustment takes place the price change
will be large. Now consider the convex costs parameters c P and c P . Such costs provide an
incentive to smooth price changes. In fact, convex costs make larger adjustments costly.
Instead of making large price changes immediately firms will only make relatively small price
modifications, and make a full response to a shock in several smaller steps. This can be seen
from equations (3) and (4), as a higher c P and c P will decrease the response of the price
change to the fundamental variables.
8
3. Estimation
To be able to estimate the model depicted in the previous section, we have to approximate the
shadow value of a price. We follow a strategy that is common in factor demand models and it
was first proposed by Abel and Blanchard (1986). It appears from equation (5) that the
shadow value q is a function of the first order derivative of the profit function with respect to
prices. It is straightforward to show that, assuming a Cobb-Douglas production technology
with flexible labour input components and an iso elastic demand equation, amongst others the
wage rate determines q.6
In addition, q is a function of the first order derivative of the menu cost function with
respect to prices. In empirical factor demand models with quadratic costs components (Abel
and Blanchard, 1986; Gilchrist and Himmelberg, 1999) it has been a standard assumption to
abstract from this second part of the q expression. This simplification has been motivated by
the fact that if the adjustment is small, the derivative of the quadratic adjustment cost
expression with respect to prices can be disregarded, because it is proportional to the square
2
 Pijt 
of the price change rate, i.e. 
. Due to this, if the price change rate is small, this
 Pijt 1 


6
In this case production QS  L  is determined by QS  L   A  L , demand for a product

D  P  is given by Q D  P   B   P P  . The price of a firm’s product is given by P and Pc
 c
denotes the general price level in the firm’s industry. Profit for a single product is given by
  A, B, P   P  QD  P   w  L , if w denotes the wage for a worker. The wage is exogenous
to the firm. Note that A captures supply shocks and input factors that are predetermined like
capital and high skilled labour. B captures demand shocks. L denotes workers that can be
hired and fired with hardly any adjustment costs. We abstract from inventory. With these
expressions the first order derivative of profit with respect to price in equation (5) can be
obtained. Solving for the optimal price P yields it is a function of the wage rate and the
general price Pc in the industry. We follow Carlsson and Skans (2012) in assuming that it is
sufficient to look at only one margin of adjustment. Their argument is that cost minimization
implies that at the optimum the cost with each possible adjustment margin should be the
same.
9
quadratic term will be negligible. We make this additional assumption as well to keep the
empirical model tractable.
Based on this we conclude that the shadow value is driven by the wage rate for
instance. As q is a discounted value composed of expected values, the assumption is that the
real wage variable that is driving the first order derivative follows an autoregressive model.
Then, as is usual in time series models, future values can be predicted by current values. We
now assume that q can be approximated by
(6)
qijtp   0p   1p ' X ijtp ijtp
The zero mean stochastic terms ijtP are normally distributed with variance  P2 . The error term
in the shadow value equation captures idiosyncratic plant level elements. Across different
months the random terms ijtP are independent. The parameter  0P represents a constant term.
The vector X ijtP contains variables observed by the econometrician and is multiplied by  1P .
X ijtP contains information reflecting the wage rate. To capture the latter variable we assume
the wage rate is given by the firm’s wage bill divided by the number of workers. This
information is only available at a yearly frequency. Hence, the vector contains the wage rate
of the previous year. The vector also includes the monthly sectoral price level. In addition,
monthly, year and sector dummies are included, to capture systematic variation in demand,
supply and real wage components. To control for unobserved heterogeneity we also include
presample averages of the price and wage rate.7
7
We have experimented with latent class models to control for unobserved heterogeneity. In
these models each class is characterized by a different constant term in equation (6). The
estimation routine then estimates the constant term for each class and the probability of a
class. Unfortunately, the technique cannot disentangle the constant terms from the linear
menu cost terms. Hence, we abandoned this approach.
10
Given the approximation of the shadow value it is possible to estimate the parameters
of the model depicted in equations (3) and (4). Our approach is based on a two-step Heckman
type selection estimator (see also Nilsen et al., 2007). The main advantage of this method is
computational tractability. We have also investigated the possibility to obtain the parameters
in a one-step estimation yielding no convergence however. In the next part of this section we
depict how the parameters are obtained in two steps. First, we develop an ordered probit
model for the probability of price increases, maintaining the current price, and price
reductions. The main objective of the first step is to get an estimator for the determinants of
the shadow value of prices. Secondly, we estimate the equations determining the level of the
price adjustment, using selection correction terms based on the estimates obtained from the
ordered probit.
Extensive Margin
Using equations (3) and (4) the log likelihood function is given by the following ordered
probit
 '
 a  


LogL    log   1 X ijt   0  b  2cP  P  
 Pijt 1  

t 1 Pijt  0




T


 '
 a    



+  log 1    1 X ijt   0  b  2cP  P   
 Pijt 1  

t 1 Pijt  0


  


T
(7)

 
   
  ' X    b  2c   aP   
0
P
 Pijt 1   
  1 ijt
T

 
 
+  log 

 '
t 1 Pijt  0
  


a
  1 X ijt   0  b  2cP  P   




 Pijt 1   






11
where    denotes a standard normal cumulative distribution function. A large number of
the structural parameters in the model can be estimated. However, as mentioned before we
cannot identify the constant term parameter  0 from the linear adjustment cost components b.
In addition, the variance of the error term remains unknown, as is common in probit type
models. As a consequence, the term  P2 has to be set equal to one. This means that all
structural parameter estimates have to be understood as relative to the corresponding standard
deviation. This is not very harmful in terms of interpretation. For instance, if our estimate for
the convex cost of price changes is cP 
cP
P
, then according to equations (3) and (4) its
inverse measures how much of a one standard deviation shock is transmitted into a price
change. Likewise, the scaled parameters aP 
aP
P
and bP 
bP
P
measure how important the
original parameters are in determining the decision whether or not to change price relative to a
one standard deviation shock. From now on a ~ on top of a parameter indicates that the
original parameter is divided by the standard deviation  P . The ordered probit model in
equation (7) allows us to acquire estimates of the following expressions:  1 ,  0  b ,  0  b ,
cP  aP and cP  aP . To construct a proxy for q the estimate for  1 can be used.
Intensive margin
Once the ordered probit is estimated equations (3) and (4) can be used to determine a model
for the size of the price change. This model needs to account for selection. We estimate the
following two equations
12
(8)
Pijt
Pijt 1


 0 b
c

P


 1 X ijt   ijt
c

P
 +

ijt
for price increases and
(9)
Pijt
Pijt 1


0 b
c

P
   X
1

  ijt
ijt
c

P
 +

ijt
for price reductions. The hats above some parameters denote that estimated values have been
used. Equations (8) and (9) allow us to identify the parameter c representing the quadratic
adjustment cost component. With this estimate and those of the ordered probit it is then also
possible to obtain the parameters of the fixed and linear cost terms, a and b , respectively.
The terms ijt and ijt denote zero mean error terms. The expressions ijt and ijt are inverse
Mills ratios. These equal the expected value of the error term in equation (6), conditional upon
being in either the price increase or price reduction regime. These correction terms are given
by
(10)
 '
 a  


  1 X ijt   0  b  2cP  P  
 Pijt 1  





and
ijt 
 '
 

a
  1 X ijt   0  b  2cP  P  



 Pijt 1  

(11)
 '
 a  


  1 X ijt   0  b  2cP  P  
 Pijt 1  





ijt 
 '
 a  
1    1 X ijt   0  b  2cP  P  
 Pijt 1  












13
where    denotes a standard normal distribution function. Equations (8) and (9) can be
estimated by OLS after replacing  1 , ijt and ijt by the values calculated from the estimates
acquired from the Ordered Probit model. Note that the size of the price, Pijt 1 , does not enter
the equation determining the size of the price change. It does feature in the threshold equation.
As a result we have a meaningful exclusion restriction that facilitates estimating price change
equations using the selection correction terms.
We conclude this section observing that estimation of the Ordered Probit model depicted
previously yields consistent estimates of the parameters and functions of these parameters if
the explanatory variables are uncorrelated with the error terms. Also the standard errors of the
parameter estimates are consistent. OLS estimation of the equations representing the level of
price changes yields consistent parameter estimates if the explanatory variables are
uncorrelated with the error terms. However, the estimates of standard errors are not
consistent. Obviously, the reason is the generated regressor problem. Since there is just one
generated regressor in each equation, t-statistics can still be used to test the hypotheses their
coefficient is equal to zero (Pagan, 1984). Furthermore, we can also trace back estimates of
the other structural parameters. Using the bootstrap value of the confidence intervals of the
parameter estimates we obtain more accurate inference.
4. The Data8
The dataset used has been constructed by combining two different data sources, both obtained
from Statistics Norway (SSB). The price data are raw data from the commodity price index
for the Norwegian manufacturing sector (VPPI), given as monthly price observations. 9 These
8
Parts of this section are based on Bratlie (2013).
Norwegian abbreviation for “vareprisindeks for industrinæringene”, translating into
“commodity price index for the Norwegian manufacturing industry”.
9
14
price observations have been linked to the structural statistics for manufacturing industries,
mining and quarrying, in order to provide a wide amount of information regarding the
companies reporting their prices.
The price data consist of monthly micro data collected by SSB for calculation of the
VPPI. In theory, such a dataset allows us to analyse price rigidity on the individual producer
level. At the aggregate level, the index is measuring the actual inflation on the producer level
and is a key part of the short-term statistics that monitor the Norwegian economy. The VPPI
is closely connected with the PPI, with the main difference being that the former may be
subject to revisions in retrospect. Developments in the Norwegian market, export and import
market is calculated on the basis of this index, together with the PPI and the price index for
domestic first-hand production (PIF) (SSB 2013a). Only data on domestic production will be
used in this analysis.
The VPPI comprises all commodities and services produced by companies within
manufacturing, mining, mining support service facilities, oil and gas extraction, and energy
supply (SSB 2013a). The price quotes are consequently obtained from firms operating in
these sectors. A selection of producers from these sectors report their prices on a monthly
basis, and large, dominating establishments are targeted in order to secure a high level of
accuracy and relevance (Asphjell 2013). The selection of respondents is furthermore updated
on a regular basis, in order to make sure that the indices continuously are being kept relevant
compared to the development of the Norwegian economy (SSB 2013a). The required
information for the PPI, VPPI and PIF are all collected in the same survey, and responses are
collected both through questionnaires and electronic reporting. Compulsory participation
ensures a high response from the questioned producers. To make sure that the indices hold a
high quality the gathered data is subject to several controls aiming at identifying extreme
values and mistyping.
15
The price data are merged with data from industry statistics. The structural business
statistics for manufacturing, mining and quarrying is reported on a yearly basis, and is a part
of SSB’s industry statistics that provides detailed information about the activity in the
specified industries (SSB 2013b). For each establishment represented in the dataset there is
thus information listed on a number of variables related to their economic activity, including
employment numbers, wages and the like. The structural statistics are only given for the
companies within certain industries, and this lays down constraints on the final dataset. As
these structural statistics are linked to price data from the VPPI, the final sample of price
observations only account for a proportion of the full spectrum of industries presented in the
producer price index. Other industrial sections than manufacturing, mining and quarrying, for
example related to agriculture, energy, transportation and service industries, will not be
included in the empirical analysis.
The dataset used in the upcoming analysis consists of 94,212 individual price
observations. The number of establishments is 388, and the total number of unique products
that is produced is 1803. The observations are distributed across 23 different industries
categorized by the SIC2002 standard, and span a time period from 2002 to 2009. On average
a plant produces 6 products in the actual data.
Comparing the data to the European reference literature (summarized by Vermeulen et
al., 2012) shows that Norwegian producers’ pricing pattern is more or less in line with what is
observed for the rest of Europe (see Table 1). We see approximately 73% of zero price
changes. This means that there must be some non-convex menu costs. This contradicts, or
comes in addition to the convex costs suggested by Rotemberg (1982), which would induce
very few zeroes.
*** Insert Table 1 about here ***
16
Table 2 shows the distribution of the monthly prices changes. We see clearly the large
amount of zeroes. These could be caused by both linear and fixed adjustment costs. Note
however, that we also see a mass point of small price changes around the zero, and at the
same time no fat tails, as we would expect to see if there are significant fixed adjustment
costs. Thus, the likely menu cost structure is a combination of convex and linear costs.
*** Insert Table 2 about here ***
Price adjustment contains a seasonal component. Figure 1 depicts that most price changes
take place in January. In addition, according to Figure 2 sectoral differences in price change
frequencies are large.
*** Insert Figures 1 & 2 about here ***
The latter conclusion is supported by Tables 3 and 4, which reveal that price adjustment
patterns vary across product categories. It appears that heterogeneity across sectors is large.
These observations imply that in estimating the price change model controlling for industry
and seasonal specific components is necessary. To that end all equations contain industry and
month dummies.
*** Insert Tables 3 and 4 about here ***
5. Results
When estimating the model using the full data set, our maximum likelihood routine
encountered convergence problems. For that reason we had to reduce the heterogeneity
observed in the data. We excluded sectors producing capital goods. In addition, we trimmed
17
the data.10 Using the resulting data set the routine maximizing the likelihood of the ordered
probit model outlined in Section 3 converged.
The preliminary findings show that a concave relationship exists between q and the
wage rate (see Table 5). We also measure the existence of statistically significant linear menu
costs, 𝑏̃. At the same time we find the product of the fixed costs parameter, 𝑎̃, and the convex
costs parameter 𝑐̃ , to be statistically insignificant. This means that at least one of them is
negligible. Estimating equations (8) and (9) by OLS reveals that convex costs are significant.
Hence, it is the fixed cost parameter 𝑎̃ that is not significantly different from zero. This
finding is in line with our descriptive statistics. They revealed a large amount of zeroes.
However, inactivity can be explained by both linear and fixed menu costs. Our empirical
finding of insignificant fixed menu costs is consistent with the relatively frequent occurrence
of small price changes, and no fat tails in the Norwegian price change data. In the near future
we will use a bootstrap method to get the statistical significance of the three adjustments costs
parameters, 𝑎̃, 𝑏̃, and 𝑐̃ individually.
*** Insert Table 5 about here ***
If fixed menu costs are negligible, then we conclude from equation (2) that for price increases
convex costs are the largest menu cost component when
10
Pijt
Pijt 1
 2
b
1.190
 2
 0.423 . For
c
5.578
In the initial sample prices range between (0.09, 4 835 000) NOK or (0.01, 500 000) EURO.
After removing tails we lost 6 % of the observations. In the sample used for estimation prices
range between (20, 20 000) NOK or (2.50, 25 000) EURO. The number of observations is
44963. The number of establishments, products and sectors are 284, 1124 and 20,
respectively.
18
decreasing prices convex costs are largest when
Pijt
Pijt 1
 2 
b
1.190
 2 
 -0.167 . Holding
c
14.139
this together with Table 2, linear costs dominate convex costs. In less than 1% of the
observations in our data convex costs are larger than linear menu costs. Note also that the
convex menu cost parameter is larger (though insignificant) for price decreases than increases.
Whether the asymmetry is significant is hard to tell because of the large standard error of the
convex costs of negative price changes.
It remains to be seen whether the estimated cost structure with the combination of
linear and convex menu costs for prices is able to explain the aggregate patterns of both the
mean values and the spread of price changes seen over the sample period.
6. Conclusion
Prices are adjusted only infrequently. In this paper we investigate the shape of the menu cost
function that is capable of explaining this feature of the data. Our approach allows for three
different types of menu costs. As usual fixed costs are considered. In addition, we investigate
the role of convex and linear costs. So far, our estimates suggest it is in particular linear costs
that explain micro level pricing dynamics. Various studies have documented the presence of
physical menu costs that are fixed. Usually estimates vary between 1-2% of annual revenues.
Though we do not estimate the size of the costs, our estimates tell that fixed costs are not
significantly different from zero. This is in line with Zbaracki et al. (2004) who find that fixed
costs are small. In fact, their findings and ours support the view that fixed menu costs are not
high enough to cause price rigidity. It should be noted though that Asphjell (2014) finds small
fixed menu costs, which do affect the dynamics of price setting at the micro level. Instead, we
find that linear costs are relevant. Our estimates suggest these costs are the largest component
19
in the menu cost function. These linear menu costs explain the presence of many zeroes in the
price change distribution as well. These type of costs are also in line with the fact that the tails
of the price change distribution are not fat. Linear costs also allow for many small price
changes.
In this paper we have assumed that prices in a plant are decided upon independently. A
more recent strand in the literature assumes that multiproduct firms incur a single menu cost
when it changes various prices (Midrigan, 2011; Alvarez and Lippi, 2014). This assumption
of economies of scope in price adjustment implies that firms have an incentive to synchronise
price adjustment. This approach is capable of explaining two features in the data: (1) in reality
multiproduct firms do synchronise price changes; (2) one does observe a large frequency of
small price changes within the data. Our model allows for explaining the second observation.
Linear adjustment costs explain small price changes as well. In future research our aim is to
investigate whether price dynamics is affected by simulatenous adjustments of other product
prices within the firm. The model in our paper can be adjusted to see if the thresholds
affecting the decision to adjust or not are affected by the change of other prices. This type of
interrelation would allow for synchronization as well (see Aphjell et al., 2014). In addition,
interrelation of price dynamics is likely to feature that some prices at a multiproduct firm do
not change while others are adjusted. The models by Midrigan (2011) and Alvarez and Lippi
(2014) assuming economies of scope in adjusting prices do not allow for this, while
inspection of the Norwegian data suggests full synchronization does not happen.
20
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23
Figure 1: Monthly Frequency Of Price Changes
Note: Average frequencies are given as number of price changes within a given month divided by the
total number of price quotations in the month.
24
Figure 2: Average Monthly Price Change Frequencies, by Sector
60%
40%
20%
0%
13 14 15 16 17 18 19 20 21 22 24 25 26 27 28 29 31 32 33 34 35 36 37
Note: The sector codes are found in Appendix Table A1.
25
Table 1: Average Monthly Producer Price Changes in European Countries
Frequency of price adjustments
Belgium
France
Germany
Italy
Portugal
Spain
Euro area
Norway
Changes
Increases
Decreases
23.6
24.8
21.2
15.3
23.1
21.4
20.8
22.8
12.8
13.8
11.8
8.5
13.6
12.2
11.6
13.9
10.9
11.0
9.4
6.8
9.5
9.2
9.2
8.9
Fraction of
price decreases
45.9
41.9
44.4
45.0
41.2
43.2
43.8
39.0
Inflation
0.12
0.09
0.09
0.14
0.17
0.17
0.11
0.17
Note: Estimates are given in percent, average share of prices changed per month. For the
other European countries the estimates are taken from Vermeulen et al. (2012). The
Norwegian inflation figure is average monthly change in CPI from 2004 to 2009.
26
Table 2: Distribution of (monthly) price changes (p/p)
0.50 
0.40  < 0.50
0.30  < 0.40
0.20  < 0.30
0.10  < 0.20
0.00 < < 0.10
0,00
-0.10  < 0.00
-0.20  < -0.10
-0.30  < -0.20
-0.40  < -0.30
< -0.40
0,0
0,0
0,1
0,2
1,2
12,4
77,2
8,1
0,6
0,2
0,0
0,0
0.075 
0.050 
0.025 
0.000 <
-0.025 
-0.050 
-0.075 
-0.100 
< 0.100
< 0.075
< 0.050
< 0.025
< 0.000
< -0.025
< -0.050
< -0.075
27
0,8
1,8
3,4
6,5
4,9
1,7
0,9
0,5
Table 3: Monthly Price Change Frequency, by Product Categories
Frequency of price adjustments
Consumer goods
Non-durables, food
Non-durables,
nonfood
Durables
Capital goods
Intermediate goods
Changes
Increases
Decreases
35.4
9.0
20.1
5.7
14.6
3.3
16.6
13.0
29.3
11.2
8.9
17.6
5.4
4.1
11.6
Note: Estimates are given in percent, average share of prices changed per month. How the
different sectors have been grouped in the product categories can be seen in Table A1 in the
appendix.
28
Table 4: Size Of Price Adjustments, by Product Categories
Size of price adjustments
Increases
All items
Consumer goods
Non-durables, food
Non-durables, non-food
Durables
Capital goods
Intermediate goods
Decreases
4.8
4.1
3.7
5.9
5.8
5.5
5.0
3.5
5.1
5.3
4.4
4.2
Note: The estimates are average absolute value of the price changes, given as percentages.
29
Table 5: Estimation results
Column 1
Column 2
coeff
se
coeff
se
Ordered probit results
wt-1
0,130
(0,045) **
0,130
(0,045) **
wt-12
-0,156
(0,048) *
-0,156
(0,048) *
1,190
(0,005) **
** -31900,3
**
ln(a+ * c+)
-44,214
(7245,360)
ln(a- * c-)
-40,250
(1990,766)
B
log L
1,190
(0,005) **
-31900,3
OLS with selection correction
1/c+
0,173
(0,039) **
0,173
(0,039) **
1/c-
0,071
(0,056)
0,071
(0,056)
Parameter estimates
a+
0,000
a-
0,000
B
1,190
(0,005) **
1,190
(0,005) **
c+
5,578
(1,310) *
5,578
(1,310) *
c-
14,139
(11,156)
14,139
(11,156)
Nbr. of observations
Ordered probit
44963
44963
Notes: month- and year-dummies are included in the ordered probit equation.
All the parameters should be thought of as normalized by the standard deviation σp
A ** (*) indicates the estimate is significant at the 5% (10%) level at least.
In brackets standard errors are included.
30
APPENDIX
TABLE A1: INDUSTRIES REPRESENTED IN THE DATASET, 2-DIGIT SIC2002
2-digit
code
13
14
15
16
17
18
19
20
21
22
24
25
26
27
28
29
31
32
33
34
35
36
37
228
1644
18852
264
3540
2064
Share
of
data
set
0.24
1.75
20.0
0.28
3.76
2.19
360
0.38
9744
10.3
3540
60
6312
5868
9228
1104
8664
3.76
0.06
6.70
6.23
9.79
1.17
9.20
9240
1608
1464
9.81
1.71
1.55
2628
2.79
1944
48
5556
252
2.06
0.05
5.90
0.27
Number of
price quotes
Industrial activity
Mining of metal ores
Other mining and quarrying
Manufacture of food products and beverages
Manufacture of tobacco products
Manufacture of textiles
Manufacture of wearing apparel; dressing and dyeing of
fur
Tanning and dressing of leather; manufacture of luggage,
handbags, saddlery, harness and footwear
Manufacture of wood and of products of wood and cork,
except furniture; manufacture of articles of straw and
plaiting materials
Manufacture of pulp, paper and paper products
Publishing, printing and reproduction of recorded media
Publishing, printing and reproduction of recorded media
Manufacture of rubber and plastic products
Manufacture of other non-metallic mineral products
Manufacture of basic metals
Manufacture of fabricated metal products, except
machinery and equipment
Manufacture of machinery and equipment n.e.c.
Manufacture of electrical machinery and apparatus n.e.c.
Manufacture of radio, television and communication
equipment and apparatus
Manufacture of medical, precision and optical
instruments, watches and clocks
Manufacture of motor vehicles, trailers and semi-trailers
Manufacture of other transport equipment
Manufacture of furniture; manufacturing n.e.c.
Recycling
Note: Shares are given as percentages. Industry codes and classifications have been collected from
SSB (2013c) (Norwegian classification SIC2002) and Eurostat (2005) (NACE Rev. 1.1 classification).
31