Embed Size (px)
Citation preview
The Cyclicality of Sales and Aggregate Price Flexibility
– Appendix∗–
For online publication
Oleksiy Kryvtsov†and Nicolas Vincent‡
October 2019
∗The views expressed here are ours, and they do not necessarily reflect the views of the Bank of Canada.We thank the editor, Veronica Guerrieri, and three referees, for their valuable comments and suggestions thathelped to improve the paper. We would like to thank Susanto Basu, Nicoletta Berardi, Marty Eichenbaum,Gee Hee Hong, Kim Huynh, David Jacho-Chavez, Nir Jaimovich, Alejandro Justiniano, Jim MacGee, SergioRebelo, Leonard Sabetti, Gregor Smith, Joe Vavra and participants at the 2015 Canadian MacroeconomicStudy Group, the 2015 Banque de France Price Setting and Inflation Conference, the NBER Price Dynamics2014 Workshop, the 2014 Shanghai Macroeconomic Policies and Business Cycles Conference, the Bank ofCanada 2014 Fellowship Learning Exchange, the 2014 Meetings of the European Economics Association, the2014 Dynare Conference, and the Universite de Montreal, University of Toronto, Queen’s University, DukeUniversity, Mannheim University, Riksbank, Uppsala University, and Banque de France seminar series forhelpful comments and suggestions. We also thank Ainslie Restieaux and colleagues in the Prices Divisionat the Office for National Statistics (ONS) for valuable feedback regarding the U.K. CPI data, and BrendanWilliams at the U.S. Bureau of Labor Statistics for providing us the series based on the U.S. CPI data. Allestimates and analysis in this paper, based on data provided by ONS, are by the authors and not by the U.K.Office for National Statistics. Claudiu Motoc, Tony Chernis, Anderson Nzabandorra and Minnie Cui providedsuperlative research assistance. We are grateful to Glen Keenleyside for extremely helpful editorial assistance.Vincent acknowledges financial support from the Fondation HEC Montreal.†Bank of Canada, 234 Wellington Av., Ottawa, Ontario K1A 0G9, Canada. [email protected].‡HEC Montreal, Department of Applied Economics, 3000 Cote-Sainte-Catherine, Montreal, Quebec H3T
2A7, Canada. Email: [email protected]
Contents
A Data Appendix 3
A.1 U.K. ONS Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
A.2 Sales and food products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
A.3 Sales and business cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
A.4 Boostrapped standard errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
A.5 Sales frequency for selected categories . . . . . . . . . . . . . . . . . . . . . . . 10
A.6 Cyclicality of sales at the category level . . . . . . . . . . . . . . . . . . . . . . 11
A.7 Regional regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
A.8 Clearance sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
A.9 Probit regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
A.10 Retailer type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
B Model Appendix 19
B.1 Household’s dynamic optimization problem . . . . . . . . . . . . . . . . . . . . 19
B.2 Intermediate input producers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
B.3 Retailers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
B.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
B.5 Full system of equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . . 23
B.6 Price dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
B.7 Steady-state properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
B.8 Elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
B.9 Model Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
B.9.1 Convex adjustment cost for discounts and shopping time . . . . . . . . . 31
B.9.2 Price of shopping time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
B.9.3 Model with materials as intermediate input in retail production . . . . . 32
C Measuring the impact of discounts on the aggregate price level 35
2
A Data Appendix
A.1 U.K. ONS Data
Here we provide a more detailed description of the consumer price index (CPI) micro dataset
from the United Kingdom’s Office for National Statistics (ONS), which is publicly available.
We also describe some adjustments we made to the raw data to ensure their suitability for
our exercises.
To construct the consumer price index (CPI), the Office for National Statistics (ONS)
surveys the prices for goods and services that are included in the household final monetary
consumption expenditure component of the U.K. National Accounts.1 The survey includes
prices for more than 1,100 individual goods or services a month, collected locally from more
than 14,000 retail stores across the United Kingdom. The survey excludes the housing portion
of consumer prices, such as mortgage interest payments, house depreciation, insurance and
other house purchase fees.2 Also, expenditures for purposes other than final consumption are
excluded, e.g., those for capital and financial transactions, direct taxes, and cash gifts.3
Goods and services in the CPI are classified into 71 classes, according to the international
(European) classification of household expenditure, Classification of Individual Consumption
by Purpose (COICOP). A CPI class represents a basic group category, such as “Meat,” “Liq-
uid Fuels” or “New Cars”; and each CPI class is divided into finer categories, “items.” For
each item and stratum (given by region and shop type pairing in the U.K. data), ONS dataset
provides sampling weights that reflect products’ relative importance in households’ consump-
tion expenditures. These weights exhibit small variation over time due to annual revisions
to capture permanent changes in the expenditure composition of households’ consumption
baskets.4
Prices are collected across 13 geographical regions (e.g., London, Wales, East Midlands).
There are four levels of sampling for local price collection: locations, outlets within location,
items within section and product varieties. For each geographical region, locations and outlets
are based on a probability-proportional-to-size systematic sampling with a size measure based
on the number of employees in the retail sector (locations) and the net retail floor space
(outlets). The data set contains around 150 locations with an average of more than 90 outlets
per location.
Representative items are selected based on a number of factors, including expenditure size
1Detailed descriptions of the data underlying the CPI, the statistical methodology used, collection andvalidation of prices, and calculation of weights, can be found in ONS (2014). The price quote data areavailable via the ONS website: http://www.ons.gov.uk/ons/datasets-and-tables/index.html.
2These prices are used to construct the retail prices index (RPI).3CPI inflation was used by the government for its inflation target starting in December 2003. Before then,
the index was published in the United Kingdom as the Harmonised Index of Consumer Prices (HICP).4Weights are calculated based on the Household Final Monetary Consumption Expenditure (HFMCE) and
ONS Living Cost and Food Survey (LCF).
3
and product diversity, variability of price movements, and availability for purchase throughout
the year (except for certain goods that are seasonal). There are currently over 510 items in
the basket. Examples of representative items include: onions, men’s suit, single bed. Finally,
for each item-outlet-location, individual products and varieties are chosen by price collectors
based on their shelf size and regular stock replenishment.
Most prices are collected monthly, except for some services in household and leisure groups,
and seasonal items. For the purpose of this paper, the sample period includes 212 months,
from February 1996 till September 2013. The total raw number of observations is over 24.4
million, or about 115,000 per month, though we make some adjustments that are described
later in order to make the database amenable to analysis.
In addition to posted prices, the data set also contains information about some charac-
teristics of goods during price collection. Prices for goods that are on special offer (available
to all consumers) or on temporary sale comprise 6.4% of observations in the data set (4.4%
weighted). It is important to note that these “sales” have to be available to everyone (i.e.,
coupons and discounts that require a loyalty card are not taken into account) and on a sin-
gle purchase (e.g., discounts implied by “buy-two-get-one-free” promotions are not recorded).
Forced substitutions happen 8.0% (5.5% weighted) of the time, with about a quarter (three-
quarters) of them corresponding to substitutions for items that are non-comparable (compa-
rable) to previously priced items. An item can be temporarily out of stock (2.2% or 1.5%
weighted) or permanently missing (0.5% or 0.3% weighted). Finally, a small subset of goods
has distinct seasonal patterns and is treated separately: they include some items of clothing,
gardening products, holiday products and air fares. For those seasonal goods for which prices
are not available, such as clothing, gardening and food, prices are imputed based on prices
observed at the end of the previous season or based on prices observed for in-season goods in
the same item category; in addition, weights are adjusted in accordance with the availability
of such goods throughout the year.
We make some adjustments to the ONS database to make it suitable for our analysis.
First, we delete observations that are not coded as valid by the ONS. Second, we deal with
product substitutions by splitting the price time series of a given item every time we encounter
a substitution flag. The resulting benchmark data set contains a total of 20.7 million observa-
tions across about 2.3 million unique items. It should be noted that for our empirical analysis,
we will mostly focus on items that have at least ten price quotes (17.1 million observations).
Value-added taxes (VAT) are included in the price quotes. The implication is that any
change in the VAT rate will automatically lead to a price change. VAT changes, however,
were very few over our sample period. The only exceptions are three major changes to the
standard VAT rate in the later part of the sample. First, in response to the Great Recession,
the Conservative government announced a widespread reduction in the VAT rate from 17.5%
to 15% effective 1 December 2008. Then, the rate was brought back to 17.5% starting 1
4
January 2010. Finally, the standard VAT rate was raised from 17.5% to 20% on 4 January
2011. We control for these events in our analysis.5
A.2 Sales and food products
We show that controlling for a linear time trend in the regression of sales frequency on
unemployment, as it is done in Coibion, Gorodnichenko, and Hong (2015), yields only a
weak relationship for food products, but does not alter the strong positive correlation at the
aggregate level (Tables A.1, A.2).
Table A.1: Regressions at the aggregate level, United Kingdom
Aggregate Food
W W UW UW W W UW UW
ut 0.179*** 0.233*** 0.286*** 0.328*** 0.341*** 0.110** 0.574*** 0.207***
Linear time trend N Y N Y N Y N Y
R2 0.42 0.45 0.26 0.26 0.40 0.55 0.55 0.72
Notes: OLS regressions. CPI data are from the U.K. Office for National Statistics, sample period is fromFebruary 1996 to September 2013. The dependent variable is the fraction of sales flag in a given month. ’W’indicates that the series is weighted using CPI weights, ’UW’ is unweighted. *** and ** indicate statisticalsignificance at the 1% and 5% levels, respectively.
5Moreover, VAT changes are more akin to regular price changes, while our focus is on temporary sales. Forthis reason, they have little incidence on our results.
5
Tab
leA
.2:
Con
trib
uti
onof
sale
sto
the
pri
ceof
food
inth
eU
nit
edK
ingd
om
Contribution
ofsa
les,
all
food
Contribution
ofsa
les,
by
basicclass
Contribution
ofsa
les,
by
region
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
Fra
ctio
nof
sale
sγt
γt
γt
γit
(cla
ss)
γit
(cla
ss)
γit
(cla
ss)
γit
(reg
ion)
γit
(reg
ion)
γit
(reg
ion)
–1.15
–1.20
–1.04
–1.03
–0.97
–1.09
–1.05
–0.95
–1.54
Fix
edeff
ects
Y(c
lass
)Y
(cla
ss)
NY
(reg
ion)
Y(r
egio
n)
N
Month
dum
mie
sN
YY
Y(c
lass
)Y
(cla
ss)
NY
(reg
ion)
Y(r
egio
n)
N
Lin
ear
tim
etr
end
NN
YN
Y(c
lass
)N
NY
(reg
ion)
N
Month
fixed
effec
tsN
NN
NN
Y(c
lass
)N
NY
(reg
ion)
R2
0.7
10.7
40.8
20.5
80.6
20.9
20.6
20.6
60.5
7
#obs
216
216
216
2373
2373
11
2592
2592
12
Note
s:Sale
sare
defi
ned
as
V-s
hap
eddis
counts
of
10%
or
more
.R
egre
ssio
nty
pes
are
indic
ate
din
colu
mns.
Aggre
gate
regre
ssio
n(c
olu
mns
1-3
)is
pt
=c
+βγt+X′ tΦ
+e t
,w
her
ept
isth
em
ean
sale
-pri
cete
rmfo
rall
food,
andγt
isth
efr
act
ion
of
sale
sin
montht,
andXt
are
month
dum
mie
sand
linea
rtr
end.
Panel
regre
ssio
ns
arepit
=αi+βγit
+X′ itΦi+e it,
wher
epit
are
the
mea
npri
cegap
for
food
basi
ccl
assi
(colu
mns
4-6
)or
for
regio
ni
(colu
mns
7-9
)in
per
iodt
andγit
isth
efr
act
ion
of
sale
sfo
rth
at
class
or
regio
nin
montht,
andXit
are
month
dum
mie
s,ca
tegory
-sp
ecifi
clinea
rtr
end,
and
fixed
effec
ts.
Panel
regre
ssio
ns
are
wei
ghte
dby
CP
Isa
mple
wei
ghts
.A
llco
effici
ents
are
stati
stic
ally
signifi
cant
at
1%
level
.
6
A.3 Sales and business cycles
Table A.3 provides standard deviations and correlations for the fraction of sales, real con-
sumption and hours worked in the U.S. We use the following U.S. business cycle variables (all
monthly):
• lt : (Log) Hours worked per capita, total adult population (16+). Total hours
worked, all private [series CES0500000056], BLS. Total civilian uninstitutionalized pop-
ulation, 16 and over [series LNS10000000]. BLS.
• ct : (Log) Real consumption (non-durables and services) per capita, adult
equivalent. U.S.: Real consumption, non-durables [from FRED, series PCESC96].
BEA. Real consumption, services [from FRED, series PCENDC96]. BEA. Adult-equivalent
population = (Population ≥ 16) + 0.5 * (Population ≤ 15). Total population, all ages
[from FRED, series POPTHM]. BEA. Total civilian uninstitutionalized population, 16
and over [series LNS10000000]. BLS.
Table A.3: Standard deviations and correlations
Business cycle United States
variable BLS sample
Standard deviations
γt, ppt 0.44
ut, ppt 2.84
lt, % 3.05
ct, % 1.64
Correlations
corr(γt, ct) –0.635
corr(γt, lt) –0.444
Notes: Table provides standard deviations and correlations. The fraction of sales γt and unemployment rateut are expressed in percentage points, the remaining variables are in log %. The fraction of sales is based onv-shapes. Business cycle variables include ut : unemployment rate; lt : (Log) Hours worked per capita, totaladult population (16+); ct : (Log) Real consumption (non-durables and services) per capita, adult equivalent.The sample period is January 2000 to December 2011.
A.4 Boostrapped standard errors
In the main text we provide a number of alternative standard errors to account for possible
serial correlation and/or heteroskedasticity in the residuals. However, because the small
7
sample properties of these methods are known to be problematic under some cases, we also
compute bootstrapped standard errors according to the following methodology:
1. Run the base regression described in Table 3 on the actual data to obtain parameter
estimates as well as the residuals εγt .
γt = α+ βut +X′t γ + εγt
where γt is the fraction of products on sale in period t, ut is the unemployment rate,
and Xt is a vector of controls such as calendar month dummies and a time trend.
2. Run an AR(1) regression for the unemployment rate using actual data, and save the
residuals εut :
ut = δ + ρut−1 + uut
3. From the empirical distributions of εγt and εut , draw N = 5, 000 samples of residuals
with replacement. For every sample n, the two time series of randomly drawn residuals,εγn,t
andεun,t
, are of length T ′ = B+T , where T is the number of observations used
in the base regression of Step 1 and B is a burn-in parameter set to 500.
4. Take a given sample n. For each t from 1 to T ′:
(a) Using the randomly drawn residual εun,t, generate:
un,t = δ + ρun,t−1 + εun,t
(b) Using the value of the unemployment rate un,t generated in the previous step as
well as the randomly drawn residual εγn,t, generate
˜γn,t = α+ βun,t + εγn,t
5. For each bootstrap sample n, run the base regression using the last T observations:
γn,t = α∗n + β∗nun,t + εγt
and save the standard error for coefficient of interest, β∗n, which we denote as σβn .
6. Report the average standard error across the 5,000 bootstrap samples,∑5,000
n=1 σβn .6
Table A.4 reports the alternative standard errors alongside the point estimates and OLS
standard errors that were provided in the main text.
6Another approach is to find the x2% and 1− x
2% quantiles of the distribution of 5,000 β∗n in order to compute
the x% confidence interval. The conclusions are similar using either method. Also, we find the estimated biasof the estimated coefficientβ is negligible.
8
Table A.4: Alternative standard errors
OLSestimate
Standard Errors
OLS Robust C-O N-W Bootstrap
UK
mean, V-shaped, no trend 0.544 0.024∗∗∗ 0.029∗∗∗ 0.057∗∗∗ 0.047∗∗∗ 0.068∗∗∗
mean, V-shaped, linear trend 0.464 0.023∗∗∗ 0.023∗∗∗ 0.040∗∗∗ 0.033∗∗∗ 0.059∗∗∗
mean, flag, no trend 0.393 0.024∗∗∗ 0.024∗∗∗ 0.053∗∗∗ 0.040∗∗∗ 0.063∗∗∗
mean, flag, linear trend 0.358 0.026∗∗∗ 0.024∗∗∗ 0.052∗∗∗ 0.041∗∗∗ 0.061∗∗∗
US - BLS
mean, no trend 0.177 0.017∗∗∗ 0.016∗∗∗ 0.018∗∗∗ 0.021∗∗∗ 0.045∗∗∗
mean, linear trend 0.233 0.025∗∗∗ 0.023∗∗∗ 0.024∗∗∗ 0.030∗∗∗ 0.043∗∗∗
US - Vavra (2014)
mean, no trend 0.338 0.032∗∗∗ 0.030∗∗∗ 0.044∗∗∗ 0.043∗∗∗ 0.048∗∗∗
mean, linear trend 0.320 0.033∗∗∗ 0.033∗∗∗ 0.048∗∗∗ 0.046∗∗∗ 0.048∗∗∗
US - Anderson et al. (2017)
fixed weights, no trend 0.151 0.021∗∗∗ 0.013∗∗∗ 0.053∗∗∗ 0.026∗∗∗ 0.038∗∗∗
fixed weights, linear trend 0.035 0.013∗∗∗ 0.009∗∗∗ 0.032∗ 0.017∗∗ 0.023∗
BLS weights, no trend 0.067 0.013∗∗∗ 0.010∗∗∗ 0.030∗∗∗ 0.018∗∗∗ 0.024∗∗∗
BLS weights, linear trend 0.073 0.014∗∗∗ 0.011∗∗∗ 0.032∗∗∗ 0.018∗∗∗ 0.024∗∗∗
BLS weights, sales flag, no trend 0.219 0.016∗∗∗ 0.011∗∗∗ 0.030∗∗∗ 0.019∗∗∗ 0.029∗∗∗
BLS weights, sales flag, linear trend 0.152 0.014∗∗∗ 0.014∗∗∗ 0.022∗∗∗ 0.021∗∗∗ 0.023∗∗∗
Notes: Linear regressions of the fraction of products on sale on the unemployment rate (ut). “OLS estimate”:regression estimates from Table 3 above, with cases specified in rows. Alternative standard errors are reportedin separate columns: “OLS”: OLS standard errors from Table 3; “Robust”: Huber/White/sandwich estimator;“C-O” (Cochrane-Orcutt) standard errors assume that the errors follow an AR(1) process; “N-W”: Newey-West standard errors assume that the errors are heteroskedastic and follow an AR(4) process; “Bootstrap”:OLS-HAC standard errors across bootstrap samples. *** p<0.01, ** p<0.05, * p<0.1 denote significance levelscorresponding to alternative standard errors.
9
A.5 Sales frequency for selected categories
The volatility and cyclicality of sales are not driven by just a few large categories. The top-
left plot of Figure A.1 shows the evolution of the fraction of price quotes with a sales flag
aggregated using equal weights across categories, as opposed to the weighted statistic reported
so far. Another concern could be that some changes in the collection of data or the definition
of sales coincided with the onset of the recession and therefore are leading to spuriously
identify a high degree of cyclicality. While the fact that the results hold across multiple sales
filters should allow us to dismiss this concern, we further note that the cyclicality of sales is
not a feature that is common to all categories, and therefore it is unlikely to be generated
mechanically. Figure A.1 shows a significant rise in the incidence of sales (here defined as the
fraction of items with a sales flag) for categories such as “Clothing and footwear,” “Furnishings
and other household equipment,” and “Personal care.” Yet, for other categories such as
“Books” or “Housing rents and maintenance,” there is no significant rise in sales around
2007–09.
It should also be noted that the rise in the sale frequency is broad-based: the number
of products that experienced a rise in their sale frequency between the periods 2005–06 and
2009–10 is almost three times as high as those that saw a drop.
Figure A.1: Frequency of sales (sales flag), selected categories
.05
.06
.07
.08
.09
Une
mp.
rat
e (d
ashe
d)
.02
.03
.04
.05
Sal
es fr
eque
ncy
(sol
id)
19971999
20012003
20052007
20092011
All products − unweighted
.05
.06
.07
.08
.09
Une
mp.
rat
e (d
ashe
d)
.02
.03
.04
.05
Sal
es fr
eque
ncy
(sol
id)
19971999
20012003
20052007
20092011
Clothing and footwear
.05
.06
.07
.08
.09
Une
mp.
rat
e (d
ashe
d)
.05
.06
.07
.08
.09
Sal
es fr
eque
ncy
(sol
id)
19971999
20012003
20052007
20092011
Furnishings and hhld eqpt
.05
.06
.07
.08
.09
Une
mp.
rat
e (d
ashe
d)
.01
.02
.03
.04
Sal
es fr
eque
ncy
(sol
id)
19971999
20012003
20052007
20092011
Personal care
.05
.06
.07
.08
.09
Une
mp.
rat
e (d
ashe
d)
0.0
05.0
1.0
15S
ales
freq
uenc
y (s
olid
)
19971999
20012003
20052007
20092011
Housing rents & maintenance
.05
.06
.07
.08
.09
Une
mp.
rat
e (d
ashe
d)
0.0
5.1
.15
Sal
es fr
eque
ncy
(sol
id)
19971999
20012003
20052007
20092011
Alcoholic bevs.
.05
.06
.07
.08
.09
Une
mp.
rat
e (d
ashe
d)
0.0
5.1
.15
Sal
es fr
eque
ncy
(sol
id)
19971999
20012003
20052007
20092011
Books
.05
.06
.07
.08
.09
Une
mp.
rat
e (d
ashe
d)
0.0
02.0
04.0
06S
ales
freq
uenc
y (s
olid
)
19971999
20012003
20052007
20092011
Restaurants and cafes
.05
.06
.07
.08
.09
Une
mp.
rat
e (d
ashe
d)
0.0
5.1
.15
.2S
ales
freq
uenc
y (s
olid
)
19971999
20012003
20052007
20092011
Audio & Video
Notes: Sample period is 1996:02–2013:09. U.K. Office for National Statistics data.
10
A.6 Cyclicality of sales at the category level
We explore more on composition and differences by sector. Table A.5 reports coefficients
from two regressions at basic class level. Column (1): Sales frequency on monthly aggregate
unemployment rate and a time trend (t-statistics in Column 2); Column (3): Sales frequency
on the log of quarterly category-level real consumption and a time trend (t-statistics in Column
4). Sales are identified using the ONS sales flag and must be at least 10% in size.
We merge the results from these regressions with three types of sector-specific information.
First, we add the data on life expectancy for durable goods, obtained from Bils and Klenow
(1998) for the United States (Column 6 in the Table). Second, we add 5-firm concentration
ratios for the United Kingdom in 2004 from Mahajan (2006), see Column (7). Finally, we
compute class-specific moments for price-adjustment variables: inflation, frequency of price
changes, and absolute size of price changes, their means and standard deviations over time,
using posted and regular prices (Columns 8 through 17).
We find that more durable goods and goods with more concentrated businesses tend
to have more countercyclical sales. These correlations are relatively weak with p-values in
regression on unemployment rate of 0.027 and 0.016, respectively; and in regression on real
consumption 0.162 and 0.224 respectively, owing to the small number of observations for which
we matched the data for durability (12) and concentration ratios (29).
In regressions on price-adjustment moments, only two of them result in significant corre-
lations across basic classes. Goods with smaller mean absolute size of price changes or those
with more volatile frequency of price changes (for either poster or regular prices) tend to
be more countercyclical, with coefficients significant at 1% level or better for regressions on
unemployment rate and 5% or better for regressions on real consumption. The number of
matched goods in these regressions is 35.
11
Tab
leA
.5:
Cycl
ical
ity
ofsa
les
by
bas
iccl
ass,
Un
ited
Kin
gdom
coicop
_id
Basic
class
Class
weight
Freq
uency
of sa
les
Durabil
ity
5‐firm
concen
trati
on ra
tiomean( t)
mean(frt)
mean
(|DP
t|)
mean( t)
mean(frt)
mean
(|DP
t|)
std( t)
std( t)
std(frt)
std(frt)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
coeff
t‐stat
coeff
tstat
posted
regular
posted
regular
1010
1BR
EAD & CER
EALS
4.98
1.05
11.20
‐1.10
‐6.92
0.03
131
0.23
0.20
17.1
0.27
0.16
12.3
0.74
0.57
0.08
0.07
1010
4MILK, CHE
ESE & EGGS
4.12
0.75
10.07
‐0.45
‐2.82
0.02
232
0.23
0.17
15.7
0.26
0.14
11.3
0.92
0.81
0.09
0.08
1010
9FO
OD PR
ODU
CTS
1.00
0.98
7.87
‐0.06
‐0.40
0.03
239
0.19
0.18
18.5
0.23
0.14
13.1
0.97
0.71
0.09
0.07
1010
7VE
GETAB
LES INCLUDING
POTA
TOES AND OTH
ER TUBE
RS
4.50
0.84
7.51
‐0.47
‐3.03
0.03
00.06
0.34
19.6
0.08
0.30
17.9
2.57
2.37
0.06
0.06
1020
2MINER
AL W
ATER
S, SOFT
DRINKS
AND JUICES
2.74
1.41
7.03
‐1.47
‐7.58
0.03
475
0.18
0.20
17.1
0.21
0.16
11.7
0.93
0.66
0.11
0.10
5060
1NON‐DURA
BLE HO
USEHO
LD
GOODS
1.69
1.57
6.94
‐1.93
‐6.86
0.04
30.13
0.21
17.7
0.19
0.16
12.7
1.23
0.75
0.12
0.10
2010
2WINE (IN
C PERR
Y)2.90
1.16
6.31
‐1.41
‐4.78
0.06
750
0.19
0.26
17.6
0.30
0.17
12.8
1.27
0.75
0.13
0.10
1203
01JEWELLERY
CLO
CKS AN
D WAT
CHES
1.78
0.89
6.18
‐0.71
‐2.76
0.03
17.8
160.32
0.12
24.6
0.41
0.09
22.1
0.88
0.60
0.07
0.06
1010
5OILS & FAT
S0.62
1.60
5.74
‐0.76
‐1.63
0.03
888
0.28
0.21
16.0
0.33
0.16
10.8
1.89
1.39
0.12
0.11
1201
02AP
PLIANCE
S, ART
ICLES &
PRODU
CTS FO
R PERSONAL
CA
RE
5.79
0.68
5.47
‐0.59
‐3.71
0.03
740
0.13
0.17
21.3
0.23
0.13
16.6
0.74
0.58
0.08
0.06
5040
0GLASSWAR
E, TAB
LEWAR
E AN
D HO
USEHO
LD UTENSILS
1.94
0.85
4.42
‐0.69
‐3.42
0.05
017
.726
0.01
0.16
28.7
0.16
0.11
27.8
1.54
0.89
0.08
0.07
1010
8SU
GAR
, JAM
, HONEY, SYR
UPS,
CHOCO
LATE AND
CONFECT
IONER
Y
3.42
0.46
4.16
‐0.20
‐2.13
0.01
881
0.30
0.14
14.4
0.34
0.12
10.6
0.76
0.47
0.08
0.07
1203
02OTH
ER PER
SONAL
EFFEC
TS0.97
1.19
3.71
‐0.07
‐0.17
0.05
717
.526
0.14
0.19
24.4
0.31
0.13
23.2
1.66
1.06
0.08
0.06
2010
1SPIRITS
1.68
1.39
3.64
‐0.97
‐1.74
0.09
250
0.15
0.34
11.3
0.23
0.20
8.1
2.35
1.03
0.17
0.13
4030
1PR
ODU
CTS FO
R TH
E RE
GULAR
MAINTENAN
CE AND RE
PAIR OF
DWELLING
2.80
0.44
3.41
‐0.39
‐2.79
0.02
00.24
0.11
18.8
0.27
0.10
17.0
0.89
0.79
0.08
0.08
1010
2MEA
T6.73
0.67
3.39
‐0.83
‐4.39
0.05
017
0.28
0.25
18.3
0.34
0.19
14.2
0.93
0.77
0.06
0.06
9050
3MISC. PRINTED MAT
TER AN
D STAT
IONER
Y AN
D DR
AWING
MAT
ERIALS
1.77
0.35
3.35
‐0.14
‐1.30
0.01
112
0.16
0.09
23.5
0.20
0.09
22.3
0.70
0.65
0.06
0.06
9030
4PETS AND RE
LATED PR
ODU
CTS
INCLUDING VETER
INAR
Y AN
D OTH
ER SER
VICE
S FO
R PETS
2.41
0.26
3.31
‐0.29
‐3.74
0.01
136
0.23
0.13
13.1
0.25
0.12
10.5
0.65
0.48
0.07
0.07
1020
1CO
FFEE, TEA
, COCO
A0.97
0.56
3.08
‐0.46
‐2.17
0.03
80.20
0.24
15.1
0.22
0.19
10.3
1.46
0.94
0.13
0.12
9010
1EQ
UIPMEN
T FO
R TH
E RE
CEPTION & REPRO
DUCT
ION
OF SO
UND & PICTU
RE
1.76
1.55
2.89
‐1.23
‐1.81
0.10
311
.927
‐0.70
0.31
15.9
‐0.46
0.22
16.9
1.96
1.92
0.11
0.08
7020
1SPAR
E PA
RTS & ACC
ESSO
RIES
1.83
0.26
2.81
‐0.14
‐1.43
0.01
23.0
340.17
0.11
13.3
0.19
0.10
12.4
0.63
0.57
0.08
0.08
2010
3BE
ER1.60
0.87
2.69
‐1.48
‐3.10
0.06
550
0.11
0.30
14.5
0.18
0.22
11.6
1.55
1.15
0.13
0.12
6010
2OTH
ER M
EDICAL
PRO
DUCT
S TH
ERAP
EUTIC AP
PLIANCE
S AN
D EQ
UIPMEN
T
1.36
0.17
2.53
0.11
1.39
0.00
92.9
14
0.08
0.06
21.6
0.08
0.05
19.5
0.67
0.56
0.04
0.04
1010
3FISH
1.30
0.94
2.35
‐0.09
‐0.15
0.05
436
0.42
0.26
18.7
0.53
0.19
13.3
1.33
0.78
0.07
0.06
Une
mploymen
t rate
Real
consum
ption
posted
prices
regular p
rices
12
5050
0TO
OLS AND EQ
UIPMEN
T FO
R HO
USE AND GAR
DEN
1.79
0.26
2.03
‐0.54
‐2.06
0.02
07.5
110.18
0.11
21.7
0.23
0.09
20.7
1.29
1.22
0.07
0.07
3010
2GAR
MEN
TS14
.16
0.37
2.03
‐0.46
‐2.40
0.06
93.1
14‐0.25
0.24
32.7
0.19
0.16
28.0
2.95
0.95
0.07
0.06
9030
1GAM
ES TOYS AND HO
BBIES
4.77
0.35
1.89
‐0.43
‐1.91
0.04
05.0
23‐0.05
0.16
27.1
0.11
0.12
25.5
1.30
1.01
0.09
0.08
3010
3OTH
ER ART
ICLES OF CLOTH
ING
& CLO
THING ACC
ESSO
RIES
0.88
0.26
1.83
‐0.40
‐3.06
0.02
90.04
0.14
34.5
0.17
0.10
30.4
2.04
1.03
0.07
0.06
5010
1FU
RNITURE
, FURN
ISHINGS
6.49
0.31
1.29
‐0.53
‐1.40
0.15
29.6
50.17
0.27
23.5
0.38
0.13
22.0
3.51
1.06
0.08
0.05
6010
1PH
ARMAC
EUTICA
L PR
ODU
CTS
1.41
0.05
0.56
0.04
0.46
0.01
957
0.11
0.13
19.6
0.14
0.10
17.2
0.99
0.68
0.08
0.07
9030
2EQ
UIPMEN
T FO
R SPORT
CA
MPING AND OPEN‐AIR
RECR
EATION
1.06
0.06
0.27
‐0.15
‐0.57
0.05
423
‐0.06
0.14
27.4
0.14
0.09
25.4
1.38
0.76
0.08
0.05
5020
0HO
USEHO
LD TEXTILES
2.13
0.06
0.22
‐0.12
‐0.37
0.10
36.2
15‐0.01
0.21
25.3
0.20
0.10
23.6
2.22
0.58
0.07
0.05
1010
6FRUIT
2.51
0.04
0.17
0.08
0.33
0.03
136
0.02
0.37
23.8
0.08
0.33
22.8
3.14
2.61
0.09
0.08
3020
0FO
OTW
EAR INCLUDING
REPA
IRS
2.62
‐0.03
‐0.17
‐0.16
‐0.70
0.06
02.6
25‐0.14
0.18
28.2
0.20
0.12
22.9
1.84
0.67
0.07
0.05
9030
3GAR
DEN PLANTS AND
FLOWER
S1.53
‐0.09
‐0.35
0.16
0.61
0.02
30.07
0.15
29.0
0.12
0.13
28.1
1.75
1.60
0.07
0.06
weighted mean
0.60
‐0.56
0.05
2.6
260.08
0.21
22.0
0.22
0.15
18.8
1.67
0.95
0.08
0.07
med
ian
0.56
‐0.45
0.04
6.8
310.15
0.18
19.6
0.21
0.13
17.0
1.30
0.78
0.08
0.07
(A) regression coeff o
n (1)
‐0.546
0.05
10.00
90.35
41.48
8‐0.034
‐0.188
1.96
6‐0.036
‐0.123
‐0.006
10.497
12.489
p
‐value
0.79
20.02
70.01
60.31
00.15
00.00
20.73
60.12
30.00
00.09
90.96
50.00
10.00
1
(B) regression coeff o
n (3)
‐1.866
‐0.025
‐0.005
‐0.175
‐1.672
0.02
30.00
3‐1.412
0.02
80.07
60.11
0‐11.14
4‐11.96
6 p‐value
0.38
70.16
20.22
40.63
20.12
10.04
90.99
60.29
40.01
50.33
80.47
40.00
10.00
2
(1)
coeff
coefficient in
regressio
n of sa
les frequ
ency on un
employmen
t rate
(2)
t‐stat
correspo
nding t‐statistic
(3)
coeff
coefficient in
regressio
n of freq
S on
real con
sumption (quarterly)
(4)
tstat
correspo
nding t‐statistic
(5)
Freq
uency of sa
les
freq
uency of sa
les (V‐shaped
disc
ounts o
ver 1
0%)
(6)
Durability
expe
cted
life (in years), U
nited States, Tables 1
‐3 from
"Usin
g Co
nsum
er The
ory to Test C
ompe
ting Bu
siness C
ycle M
odels" (B
ils and
Kleno
w, JPE
199
8)(7)
5‐firm con
centratio
n ratio
top 5 bu
sinesses a
s a percentage of to
tal outpu
t, United Kingdo
m, 200
4(8)
mean( t)
average mon
thly weighted mean price change, posted prices
(9)
mean(frt)
average mon
thly weighted mean fractio
n of price changes, posted prices
(10)
mean(|D
P t|)
average mon
thly weighted mean absolute size of p
rice change (con
d. on change), po
sted
prices
(11)
mean( t)
average mon
thly weighted mean price change, regular prices
(12)
mean(frt)
average mon
thly weighted mean fractio
n of price changes, re
gular p
rices
(13)
mean(|D
P t|)
average mon
thly weighted mean absolute size of p
rice change (con
d. on change), regular p
rices
(14)
std( t)
posted
standard deviatio
n of th
e weighted mean price change, posted prices
(15)
std( t)
regular
standard deviatio
n of th
e weighted mean fractio
n of price changes, posted prices
(16)
std(frt)
posted
standard deviatio
n of th
e weighted mean price change, regular prices
(17)
std(frt)
regular
standard deviatio
n of th
e weighted mean fractio
n of price changes, re
gular p
rices
posted
prices
regular
prices
Une
mploymen
t rate
Real con
sumption
Notes: C
olum
ns (1
) and
(3) rep
ort coe
fficien
ts from
two regressio
ns at the
category level. (1): Sales frequ
ency on mon
thly aggregate une
mploymen
t rate and a tim
e tren
d. (3
): Sales frequ
ency on the log of quarterly category‐level real
consum
ption and a tim
e tren
d. Sales are iden
tified using the ONS sales flag and must b
e at least 1
0% in
size. U
.K. O
ffice fo
r National Statistics d
ata. Row
s (A) and
(B) rep
ort coe
fficien
ts of regressing cyclicality
coe
fficien
ts in
colum
ns (1
) and (3) o
n column‐varia
ble values across b
asic classes.
13
A.7 Regional regressions
Table A.6 shows the results of panel regressions with region fixed effects, in which the depen-
dent variable is the fraction of sales flag for a given region and month. The first regression
yields results very much in line with the benchmark regressions from the main text. The
conclusions are little changed once we include a time trend. However, when we include
time fixed effects, the unemployment rate becomes non-significant, in line with the Coibion,
Gorodnichenko, and Hong (2015) who find little evidence of a relationship between sales and
unemployment in the cross-section.
Table A.6: Panel regressions with regional unemployment rate, United Kingdom
(1) (2) (3)
urt - Regional 0.300*** 0.258*** 0.017
Calendar month dummies Y N N
Linear time trend N Y N
Time fixed effects N N Y
R2 0.34 0.47 0.77
Notes: Panel regressions with region fixed effects. CPI data are from the U.K. Office for National Statistics,sample period is from February 1996 to September 2013. Each observation is a region/month. Thedependent variable is the fraction of sales flag in a given region/month. urt is the unemployment rate inregion r at time t. *** indicates statistical significance at the 1% level based on robust standard errors.
Finally, Table A.7 reports the coefficients from region-specific regressions of the frequency
of sales on the monthly regional unemployment rate and a time trend. The largest semi-
elasticities are recorded for Scotland and East Anglia, and the lowest for London, North and
Yorks & Humber. One can also notice that there is very little variation in the unconditional
frequency of sales across regions, at around 2.5%. The only exception is London, where the
frequency is somewhat lower at 1.9%.
14
Table A.7: Region-specific regressions, United Kingdom
ScotlandEast East South
WalesWest
Anglia Midlands West Midlands
urt 0.415(0.042)
0.415(0.074)
0.397(0.059)
0.369(0.050)
0.366(0.047)
0.316(0.055)
R2 0.56 0.35 0.56 0.44 0.56 0.45
Sales2.4% 2.4% 2.3% 2.5% 2.6% 2.4%
freq.
South NorthLondon North
Yorks &
East West Humber
urt 0.307(0.060)
0.306(0.045)
0.219(0.024)
0.211(0.032)
0.193(0.031)
R2 0.46 0.55 0.45 0.46 0.62
Sales2.4% 2.4% 1.9% 2.5% 2.4%
freq.
Notes: Linear regressions of the frequency of sales on the regional unemployment rate (urt) and a time trend,ran separately for each region. CPI data are from the U.K. Office for National Statistics, sample period is fromFebruary 1996 to September 2013. Newey-West standard errors allowing for a maximum autocorrelation of 4lags are indicated in parentheses.
A.8 Clearance sales
As evidenced from Figure A.2, temporary sales tend to be more prevalent toward the end of the
price quote line, an indication that clearance sales are an important part of retailers’ pricing
strategies in the U.K. To construct these plots, we computed the probability of observing a sale
N months before the end of a quote line.7 Aggregating across all products, this probability
is almost 8% in the last month of a typical quote line, compared to 3.5% three years earlier.
There are wide differences in this pattern across product categories, however. For Clothing
and Footwear, 22% of the observations in the last month are flagged as sales by the ONS,
compared to a frequency of around 8% one year earlier, while the same numbers are 26% and
14% respectively for Audio and Video. Food products, on the other hand, display a very flat
hazard function with no evidence of clearance sales. We also find that the average size of
discounts tends to increase as the end of a quote line approaches.
7A price line can end because the product is replaced by another comparable item at a specific store, becausethe ONS stops collecting its price, or if the store permanently shuts down.
15
Figure A.2: Lifecycle frequency of sales
.03
.04
.05
.06
.07
Pro
babi
lity
0102030Months until end of quote line
All products
0.0
2.0
4.0
6.0
8P
roba
bilit
y
0102030Months until end of quote line
Food products
.08
.1.1
2.1
4.1
6P
roba
bilit
y
0102030Months until end of quote line
Clothing & Footwear
.1.1
5.2
.25
Pro
babi
lity
0102030Months until end of quote line
Audio & Video
Notes: Notes: Sample period is 1996:02–2013:09. U.K. Office for National Statistics data. Probability ofobserving a sale flag N months before the end of a quote line. CPI quote weights are used.
16
A.9 Probit regressions
In addition to the results from the linear probability model of the main text, here we provide
robustness checks using probit regressions instead. Estimation results are summarized in the
top panel of Table A.8, while the bottom panel shows the predicted sales frequencies at various
unemployment rate levels.
Table A.8: Probit regressions at the product level, United Kingdom
(1) (2) (3)
ut 5.89*** 6.839*** 5.922***
Calendar month dummies Y Y N
Categories dummies N Y Y
Region dummies N Y Y
Linear time trend N N Y
Predicted sales frequencies
ut = 0.04 0.025 0.025 0.026
ut = 0.06 0.033 0.033 0.033
ut = 0.08 0.043 0.043 0.042
ut = 0.10 0.054 0.055 0.052
Notes: Probit regressions. CPI data are from the U.K. Office for National Statistics, sample period is fromFebruary 1996 to September 2013. *** indicate statistical significance at the 1% level.
17
A.10 Retailer type
First, we plot in Figure A.3 the incidence of sales for independent stores and chains, against
the unemployment rate. The UK ONS calls “Independent retailers” those have less than 10
outlets, while “Chains” (or “Multiples”) must have at least 10 outlets. It is clear that sales
are much more prevalent at chains. There is also evidence that they may be more cyclical,
though this depends on the filter used.
Figure A.3: Lifecycle frequency of sales
.03
.04
.05
.06
.07
.08
Une
mp.
rat
e (d
otte
d)
.005
.015
.025
.035
.045
.055
Sal
es fr
eque
ncy
(sol
id a
nd d
ashe
d)
19971999
20012003
20052007
20092011
2013
V−shaped
.03
.04
.05
.06
.07
.08
Une
mp.
rat
e (d
otte
d)
.005
.015
.025
.035
.045
.055
Sal
es fr
eque
ncy
(sol
id a
nd d
ashe
d)
19971999
20012003
20052007
20092011
2013
Sales flag
Multiples Independents Unemp. rate
Notes: Sample period is 1996:02–2013:09. U.K. Office for National Statistics data. Probability of observing asale flag N months before the end of a quote line. CPI quote weights are used.
18
B Model Appendix
To understand the dynamics of temporary sales and their importance for aggregate fluctua-
tions, we build a general equilibrium business cycle model in which discounts arise endoge-
nously. In order to simplify the exposition, in the main text we lay out a static shopping-time
problem and other key non-standard features of the model. This appendix provides the other
elements of our framework: the complete description of the household’s dynamics optimization
problem, organization of production, the market clearing conditions, a sufficient condition for
the existence of two-price locally unique equilibrium, the full system of equilibrium conditions,
steady state properties, derivations of the key elasticities, and the results of model robustness
exercises.
B.1 Household’s dynamic optimization problem
In each period, the household consumes, works, shops, and trades money, state-contingent
securities, and nominal bonds. Household utility is separable in consumption and leisure.
The aggregate consumption bundle in period t, ct, consists of differentiated varieties of
perishable consumption goods, indexed by j, combined according to a CES aggregator,
ct =
∑j c
1− 1θ
jt
θθ−1
, where θ is the elasticity of substitution across good varieties. House-
holds have access to a shopping technology that converts time into additional consumption.
The total cost of using this technology in period t is Ξt units of time. Let At+1 denote a vec-
tor of Arrow-Debreu securities paying one dollar in t+1, Qt+1|t is the vector of corresponding
prices, Bt denote holdings of nominal bonds paying the household a gross return Rt in period
t+ 1, and let Mt denote money holdings at the beginning of period t.
Households supply working hours, lut, in a monopolistically competitive labor market,
comprised of unions supplying differentiated labor services indexed by u. These varieties
aggregate into a final labor service lt, according to lt =
∑u l
1− 1ϑ
ut
ϑϑ−1
, where ϑ is the
elasticity of substitution across labor varieties. Each union sets wages Wut for its services,
and accordingly faces the following demand: lut =(WutWt
)−ϑlt, where Wt is aggregate wage,
Wt =∑
uW1−ϑut
11−ϑ
.
We assume unions set wages according to Taylor contracts, with wages fixed for Nw
periods. Contracts are perfectly staggered across unions, i.e., in every period, a fraction 1/Nw
of unions reset their wages. Initial bond holdings of unions are such that each union has the
same present discounted value of income. Although unions set different wages and supply
different amounts of labor services, the availability of a complete set of securities and the
separability between consumption and leisure imply that unions make identical consumption,
savings and shopping decisions in equilibrium. We therefore drop the union-specific index
19
hereafter. A similar setup is implemented in Kryvtsov and Midrigan (2013).
At the end of the period, households combine unspent money holdings from the previous
period with labor income, payments on securities, return on bonds, dividends, and transfers,
and split them into cash and bond holdings to be carried over to the next period.
For their household members, unions choose the sequences of total consumption ct, hours
worked lt, cash holdings, Mt, security holdings At+1, and bond holdings Bt to maximize
lifetime expected utility:
E0
∑t
βt [u (ct)− v (lt)] ,
subject to the budget constraint
Mt +Bt +∑
t+1|tQt+1|tAt+1 ≤ Mt−1 − Ptct +Wt [lt − Ξt] +Rt−1Bt−1 +At + Πt + Tt ,
and a cash-in-advance constraint
Ptct ≤Mt , (1)
where Pt is the price of consumption bundle ct, Wt is a wage, Πt are dividends paid, Tt are
lump-sum government transfers.
The optimality conditions are standard: the first-order condition for hours worked implies
v′ (lt)
u′ (ct)=Wt
Pt, (2)
while the first-order conditions for Arrow-Debreu securities and nominal bonds yield
1 = βRtEt
(u′ (ct+1)
u′ (ct)
PtPt+1
),
where Et(·) denotes the expected value with respect to information available through period
t.
We assume that the supply of money M st follows a random-walk process of the form
lnM st = lnM s
t−1 + µt ,
where µt is log money growth, a normally distributed i.i.d. random variable with mean 0 and
standard deviation σµ.
20
B.2 Intermediate input producers
A continuum of competitive intermediate good firms own and invest in capital kt, hire labor
nt and produce a homogeneous good Yt using a Cobb-Douglas technology
Yt = atn1−χt kχt , (3)
where at denotes total factor productivity in period t. The homogeneous good is sold at the
competitive price Ωt to other intermediate good firms to augment their capital stock, and to
retail firms to produce consumption varieties. The intermediate good producer solves
maxYt,xt,nt
E0
∞∑t=0
βtu′ (ct)
Pt· [ΩtYt − Ωt (xt + φt)−Wtnt]
,
subject to the law of motion of the capital stock
kt+1 = (1− δ) kt + xt ,
and the technology constraint (3). Parameter δ denotes the depreciation rate of the capital
stock, and φt is the convex cost of adjusting capital:
φt =φ
2
(kt+1
kt− 1
)2
kt .
The first-order conditions yield the expression for the price of the intermediate input:
Ωt = (1− χ)−(1−χ)χ−χa−1t W 1−χ
t
(Rkt
)χ,
where Rkt denotes the return on capital, Rkt = χΩtYtkt
.
B.3 Retailers
Each retail firm selling brands of variety j is endowed with a linear production technology
that converts the homogeneous intermediate input Yjt into output of brands of variety j, yjt:
yjt = Yjt . (4)
This technology implies that the retailer’s marginal cost of production is equal to the price of
the intermediate input, Ωt.
Denote by Πkjt =
(P kjt − Ωt
)ckjt the period-t retailer’s profit function from selling con-
sumption goods at price P kjt, k = L,H, and Nkjt be the measure of transactions at price P kjt:
NLjt = γjt
(αRjt + αjtf
), NH
jt = (1− γjt)αRjt + αjt (1− fγjt). The problem of the retail store
21
selling variety j is to choose the sequences of its two consumption price points PHjt and PLjt;
the fraction of price discounts γjt; the fraction of bargain hunters visiting its location, αBjt;
intermediate inputs Yjt; and output levels yjt to maximize the present discounted value of its
profits until it can reset prices again:
maxyjt,Pkjτ ,γjτ ,α
Bjτ
Et
t+NP−1∑τ=t
βt−τ
u′ (ct)
Pτ·[NLjτΠL
jτ +NHjτΠH
jτ − κWτγjτ]
, (5)
subject to the supply constraint
yjt ≤ NLjtc
Ljt +NH
jt cHjt ,
and the demand constraints for ckjt (6, main text), the technology constraint (19), the equation
for the probability of bargain hunting (7, main text), the constraint for the number of bargain
hunters in the retailer’s location (8, main text), and Taylor price adjustment constraints, with
NP denoting the duration of regular price contracts.
B.4 Equilibrium
In each period the timing of events is as follows (see model diagram in Figure 10, main text).
First, the aggregate money and TFP shocks are realized, the household head deposits all of
its cash in the checking account, retail firms post their prices and send their price menus to
households. Next, the household head observes for each variety the overall distribution of
prices across locations and the specific prices for only one random retailer; she then chooses
the shopping types, consumption plans, and pays search costs. She also provides debit cards
to shoppers and sends them off to shop. All shoppers are matched with locations, retail firms,
and consumption brands; they bring the purchased consumption goods back to the household.
At some point during the period the household also earns labour income, which cannot be used
for consumption spending in the same period.8 At the end of the period, households combine
unspent money holdings with labor income, return on bonds, dividends, and transfers, and
split them into cash and bond holdings to be carried over to the next period.
The market clearing condition for labor implies that total hours supplied by unions in the
labor market are equal to the total hours demanded by firms:∑u
lut = nt + κ∑j
γjt .
8It is not material how exactly household earns labour income. For example, we could assume that shopperswork during the day, and after the workday they shop using debit cards to pay for consumption. Standardcash-in-advance models typically assume a dedicated type of household members distinct from shoppers whoare responsible for working and earning income.
22
The equilibrium search flow condition states that the total number of shoppers for each
variety j is equal to 1 in each location:
αBjt + αRjt = 1 .
In the symmetric equilibrium, each location receives the same measure of bargain hunters
for each variety, i.e.,
αBjt = αjt ,
where αjt is the probability of bargain hunting that each household assigns to a shopper j.
The market clearing condition for each good variety j states that the total consumption
is equal to the total amount produced:
NLjtc
Ljt +NH
jt cHjt = yjt .
The market clearing for intermediate goods:
xt +∑j
Yjt = Yt .
Finally, capital, money, bond and security markets also clear. A symmetric equilibrium
for this economy is a collection of allocations and prices that satisfy household and firm
maximization, as well as market clearing conditions. Below we provide the full system of
equilibrium conditions in the model with sticky prices, and derive a sufficient condition for
the existence of two-price locally unique equilibrium.
B.5 Full system of equilibrium equations
This section contains the full system of equilibrium conditions in the model with sticky prices.
The model is solved by applying the Blanchard and Kahn (1980) local solution method to
the log-linearized system of stochastic equilibrium equations around the deterministic steady
state. The code and other replication materials are available upon request. Period utility is
u(c)− v(l) = c1−σ
1−σ − ψl1+1/η
1+1/η .
Denote the following auxiliary variables:
ΠLjt =
(PLjt − Ωt
)(PLjtPt
)−θct, ΠH
jt =(PHjt − Ωt
)(PHjtPt
)−θct ,
ΠRjt = γjtΠ
Ljt + (1− γjt)ΠH
jt , ΠBjt = fγjtΠ
Ljt + (1− fγjt) ΠH
jt ,
NHjt = (1− γjt)αRjt + αjt (1− fγjt) , NL
jt = γjt(αRjt + αjtf
).
The equilibrium consists of 8NP +NW + 12 equations for 8NP +NW + 12 variables PLjt,
23
PHjt , cLjt, cHjt , γjt, α
Rjt, αjt, z
∗jt, n,t,k,t,x,t, Yt,Pt, ct, lt, Ωt,Wt, Wjt,W
dt , Rkt , Rt including:
- equation for Ωt
Ωt = (1− χ)−(1−χ)χ−χa−1t W 1−χ
t
(Rkt
)χ,
- 2NP equations for ckt,0, ..., ckt,N−1, k = L,H,
ckjt ,=
(P kjtPt
)−θct ,
- cost-minimization condition for nt
1− χχ
ktnt
=Wt
Rkt,
- equation for aggregate wage Wt
Wt =
∑u
(Wu,t)1−ϑ
11−ϑ
,
- equation for “dual” aggregate wage W dt
W dt =
∑u
(Wu,t)−ϑ
1−ϑ
,
- equation for PHt,0
PHt,0 =θ
θ − 1
Et∑t+NP−1
τ=t βτ c1−στ P θ−1
τ ·NHτ,τ−tΩτ
Et∑t+NP−1
τ=t βτ c1−στ P θ−1
τ ·NHτ,τ−t
(1 + ∆H
τ,τ−t) ,
- NP − 1 equations for PHt,1, ..., PHt,N−1
PHt,0 = PHt+1,1 = ... = PHt+NP−1,NP−1 ,
- NP equations for PLt,0, ..., PLt,N−1
PLjt =θ
θ − 1
1
1 + ∆Ljt
Ωt , (6)
where
∆Hjt = − ∂αjt
∂ lnPHjt
ΠBjt
(θ − 1)NHjt P
Hjt c
Hjt
=1− fγjtWtΞ′′ (αjt)
·ΠBjt
(θ − 1)NHjt
, (7)
24
∆Ljt = − ∂αjt
∂ lnPLjt
ΠBjt
(θ − 1)NLjtP
Ljtc
Ljt
=fγjt
WtΞ′′ (αjt)
ΠBjt
(θ − 1)NLjt
, (8)
- NP equations for the shopping time of the marginal shopper z∗jt
Wtz∗jt =
(f − 1) γjtθ − 1
(PLjtc
Ljt − PHjt cHjt
), (9)
- NP equations for the fraction of bargain hunters αjt
z∗jt = Ξ′ (αjt) ,
- NP equations for the fraction of low prices γjt
(αRjt + fαjt
) [ΠHjt −ΠL
jt
]+ κWt =
∂αjt∂ γ∗jt
·ΠBjt , (10)
where
∂αjt∂ γ∗jt
=f
WtΞ′′ (αjt)
(PLjtc
Ljt − PHjt cHjt
)(11)
- NP equations for the fraction of random shoppers of variety j
αRjt = 1− αjt ,
- equation for the price of aggregate consumption Pt
Pt =
1NP
∑NP−1j=0
[NLjt
(PLjt
)1−θ+NH
jt
(PHjt
)1−θ] 1
1−θ,
- cash-in-advance constraint
Mt = Ptct ,
- equation for Wt,0
W1+ϑ/ηt,0 = ψ
ϑ
ϑ− 1
Et∑t+NW−1
τ=t βτ(W ϑτ lτ)1+1/η
Et∑t+NW−1
τ=t βτ (W ϑτ lτ ) c−στ P−1
τ
,
- NW − 1 equations for Wt,1, ..., Wt,NW−1
Wt,0 = Wt+1,1 = ... = Wt+NW−1,NW−1 ,
25
- aggregate resource constraint
xt +φ
2
(kt+1
kt− 1
)2
kt +∑j
(NLjtc
Ljt +NH
jt cHjt
)= Yt ,
- labor market clearing,
nt + κ∑j
γjt =
(W dt
Wt
)−ϑlt ,
- law of motion for capital stock kt+1
kt+1 = (1− δ) kt + xt ,
- optimality condition for capital stock kt+1
1+φ
(kt+1
kt− 1
)= βEt
(ct+1
ct
)−σ PtPt+1
Ωt+1
Ωt
(Rkt+1
Ωt+1+ 1− δ + φ
(kt+2
kt+1− 1
)+φ
2
(kt+2
kt+1− 1
)2)
- return on capital, Rkt
Rkt = χΩtYtkt,
- equation for the risk-free rate
R−1t = βEt
(c−σt+1Pt
c−σt Pt+1
),
- and an exogenous AR(1) process (in logs) for money growth µt = MtMt−1
:
lnµt+1 = (1− ρµ) lnµ+ ρµ lnµt + εµt ,
where i.i.d. innovations εµt drawn from N(0, σ2
µ
).
TFP process at follows a zero-mean exogenous AR(1) process (in logs) :
ln at+1 = ρµ ln at + εat ,
where i.i.d. innovations εat drawn from N(0, σ2
a
).
Approximation of the cost function.
We approximate the cost function z∗jt = Ξ′ (αjt) around the steady state up to second
order:
ξ ln
(z∗jtz∗j
)= αjt − αj , (12)
26
where ξ =z∗j
Ξ′′(αj).
B.6 Price dispersion
There exists a locally unique equilibrium with price dispersion in this model. In this section,
we derive a sufficient condition for the existence of two-price equilibria, expressed in terms of
the expected fixed cost Ξ (α), the probability parameter f , the cost of posting sales κ, and the
demand elasticity θ. A retailer prefers to post two different prices, instead of a single price for
all its brands when the expected returns to retailer from posting an extra fraction of brands
on sale outweighs the opportunity cost of maintaining high average profit margin. That is
the case when the return on searching is high, the cost of posting sales is low, and demand
is sufficiently inelastic. When the sufficient condition is satisfied, there are always varieties
for which the search cost realizations are low-enough so that households choose the shoppers
responsible for that variety to be bargain hunters. In turn, the presence of bargain hunters
validates the retailers’ incentives to post sales. We also show that there is no single-price
equilibrium when the sufficient condition is satisfied.
Consider the case with fully flexible prices; omit indexes for variety and time. Denote
by Π (p) = (p− Ω)(pPt
)−θct the retailer’s current-period profit function at price p, and by
Rt (p) = p( pP
)−θc the current-period revenue function at p. The retailer’s expected profit
and revenue from selling to regular shoppers in period t are
ΠR = γΠ(PL)
+ (1− γ)Π(PH),
RR = γR(PL)
+ (1− γ)R(PH).
Similarly, the retailer’s expected profit and revenue from selling brands to bargain hunters,
ΠB and RB, are the same as above with γ replaced by fγ.
To gain intuition for why it is optimal for retailers to post different prices, consider the
effect of the search constraint (7, main text) on the retailer’s pricing behavior. If this con-
straint were not binding, the retailer would optimally choose—due to concavity of the profit
function—to charge the monopoly price for all its brands, PL = PH = P ∗ ≡ θθ−1Ω, and
collect monopoly profits Π∗ = Π (P ∗).
In contrast, when the retailer can attract bargain hunters, it chooses the number of bargain
hunters, facing constraints (7) and (8) in the main text. In this case, posting a single price
for all of its brands is no longer optimal. Consider a small price perturbation for a retailer
that, instead of posting a single monopoly price P ∗ for all its brands, posts an arbitrarily
lower price PL < P ∗ for a small fraction of brands dγ, attracting a small number of bargain
hunters, dα, and taking the number of random shoppers as given, αR / 1. The variation of
the retail firm’s profit relative to monopolistic profit δΠ∗ ≡ Π−Π∗ is, up to the first order of
27
magnitude, given by
δΠ∗ ≈ Π∗dα+ dγ[Π(PL)−Π (P ∗)− κW
]. (13)
The first term on the right-hand side is the gain in profits due to sales to the extra fraction
dα of bargain hunters. The second term is the loss in profits due to sales of the fraction dγ
of brands at a lower price PL and the total fixed cost of posting sale prices. From the search
constraint (7, main text), we can express dα, up to the first order of magnitude, as
dα ≈ (f − 1) dγ ·1θ−1
(R(PL)−R (P ∗)
)WΞ′′ (0)
. (14)
For a household, the increase in the probability of becoming a bargain hunter, dα, is pro-
portional to the increase in the probability of finding a low price (in known location versus
random location), (f − 1) dγ, and the ratio of the marginal utility of switching consumption
from high to low price, 1θ−1
(R(PL)−R (P ∗)
), to the marginal cost of becoming a bargain
hunter, WΞ′′ (0).
Plugging (14) into the expression (13) for profit variation, we obtain
δΠ∗ ≈ dγ ·Π∗[−Π (P ∗)−Π
(PL)
+ κW
Π∗+
(f − 1)
θ − 1
R(PL)−R (P ∗)
WΞ′′ (0)
].
If the expression in brackets is positive, i.e., if
(f − 1) ·1θ−1
(R(PL)−R (P ∗)
)WΞ′′ (0)
>Π (P ∗)−Π
(PL)
+ κW
Π∗,
then the benefits stemming from the increase in the market share of the retailer from selling
to bargain hunters outweighs the cost of a smaller profit margin from selling at lower prices,
and so the retailer can improve upon single-price monopolistic profit by posting two prices.
Also note that as γ → 0, PH → P ∗ and PL → 1
1+ fθ−1
Π∗WΞ′′(0)
P ∗ < P ∗; i.e., price dispersion
does not disappear as the fraction of sale prices goes to zero. This means that there is no
equilibrium with a single price. The reason why firms find it optimal to deviate from having
a single price for all brands is that for any small but positive measure of brands on sale, the
return for the household becoming a bargain hunter is strictly positive. So if the marginal
expected fixed cost of becoming a bargain hunter is not growing too quickly with the fraction
of bargain hunters (e.g., if Ξ′′ (0) is low enough), then there will always be households with
low-enough fixed cost realizations who choose to become bargain hunters, justifying price
dispersion by retailers.
28
B.7 Steady-state properties
Fraction of discounts. The desirable number of discount prices for a retailer is determined
by the retailer’s first-order condition with respect to the fraction of brands on sale, γ∗jt :
∂NLjt
∂γ∗jt
(ΠHjt −ΠL
jt
)+ κWt =
∂(NHjt +NL
jt
)∂γ∗jt
ΠHjt . (15)
Condition (15) equates the cost of increasing the fraction of sales (left-hand side) with the
return from extra sales (right-hand side). The cost stems from an increased share of trans-
actions at discounted prices and the additional cost of posting price discounts. The return
comes from the gain in market share,∂(NH
jt+NLjt)
∂γ∗jt, which is proportional to γ∗−1
jt and the rate
of increase in shopping cost, ξ. Panel A in Figure B.1 visualizes the trade-off between the
cost and the return from extra discounts as functions of their fraction in all prices. The cost
is increasing as the fraction of discounts raises the share of transactions at discounted prices.
The return decreases as extra sales bring a diminishing number of additional bargain hunters.
Price decisions. In the flexible-price case, the first order condition for P kjt, k = L, His
∂Πkjt
∂P kjt= −
∂(NHjt +NL
jt
)Nkjt∂P
kjt
(fγ∗jtΠ
Ljt + (1− fγ∗jt)ΠH
jt
). (16)
The left-hand side is the marginal profit as a function of P kjt, keeping market share constant.
If bargain hunters are insensitive to discounts, the firm sets this term to zero by setting both
prices at the optimal monopoly price. If bargain hunters are sensitive to prices, the firm can
boost the demand for its product by setting two different prices, both below the monopoly
price. The term on the right-hand side of (16) captures profits from additional market share.
The optimal pricing decisions for the flexible-price case are visualized in Panel B, Figure B.1.
Shopping time. Bargain hunting entails additional shopping time between zero and z∗.
In equilibrium, the maximal shopping time is equated to the ratio of returns on shopping
∆x and the cost of time W/P to determine the fraction of bargain hunters αB, see Panel
C in Figure B.1. Higher returns or lower opportunity cost of time will push the cut-off for
shopping, leading to an increase in bargain hunters.
29
Figure B.1: Household and firm decisions, steady state
B.8 Elasticities
In the model, the household’s additional shopping time is equal to∫z<z∗t
zdG(z) units of time,
where z is drawn from distribution with c.d.f. G(z). z∗t denotes the maximal shopping time
30
by a bargain hunter. Small variations of total shopping time St around its steady state S are:
Sd lnSt ≈ z∗ (G(z∗t )−G(z∗))
i.e., the additional shopping time stems from changes in the mass of bargain hunters around
the steady state. Given our definitions, G(z∗t ) = αt ≈ α+ ξd ln (z∗t ), we obtain
Sd lnSt ≈ z∗ξ · d ln z∗t (17)
The log-linearized first-order condition (7, main text) for the household’s shopping time im-
plies:
d ln z∗t ≈ d (ln ∆xt)− d ln (Wt/Pt) (18)
Together, (17)–(18) imply that the elasticity of shopping time to the price of time is −ξ z∗S .
Our baseline parameterization implies that the average shopping time S corresponds to 4
hours per week and the average working time L is 37.1 hours per week, yielding z∗ to be 3.1
hours per week. This gives an elasticity of log shopping time with respect to log wages of
–0.14. Aguiar and Hurst (2007) estimate this elasticity in the ATUS sample to be roughly
–0.12.
Aguiar and Hurst (2007) also find that doubling the frequency of shopping trips lowers
the price of consumption by 7 to 10%. In the model, the household’s variation in total price
savings around the steady state is αd (∆xt). Noting that d (∆xt) = ∆x · d (ln ∆xt), (17)–(18)
imply that the elasticity of price to household’s shopping time is −αP∆x
ξ z∗
S
. For the baseline
model, this yields an elasticity of –11.9%.
B.9 Model Robustness
Table B.1 provides the results for alternative versions of the model.
B.9.1 Convex adjustment cost for discounts and shopping time
In the main text, we use counterfactual exercises in which we shut down adjustments of
discounts or shopping time by imposing prohibitively large convex adjustment costs. These
costs are introduced as follows.
Convex cost of adjusting the number of discounts. In the problem of the retailer,
the cost of posting discounts is replaced with: κWt
(γjt +
εγ2 (γjt − γ)2
). Equilibrium condi-
tion (10) is replaced with
(αRjt + fαjt
) [ΠHjt −ΠL
jt
]+ κWt(1 + εγ (γjt − γj)) =
∂αjt∂ γ∗jt
·ΠBjt .
31
Convex cost of adjusting the size of discounts. Add adjustment cost term in the
decision of adjusting the discount price: εδ2
(PLjt/P
Hjt − PL/PH
)2. Equilibrium condition (6)
is replaced with
PLjt =θ
θ − 1
Ωt
1 + ∆Ljt
· exp(εδ(PLjt/P
Hjt − PL/PH
)).
Convex cost of adjusting shopping time. In specification (12), we add quadratic
term:
ξ ln
(z∗jtz∗j
)= (αjt − αj) +
1
2εα (αjt − αj)2 ,
where ξ =z∗j
Ξ′′(αj)and εα =
Ξ′′′(αj)Ξ′′(αj)
.
B.9.2 Price of shopping time
In the main text, the price of shopping time is given by the market wage Wt in front of the
shopping cost∑
j Ξ(αjt) in the household’s expenditure minimization problem (5, main text).
In Section 6, we conduct simulations in models with alternative measures of the price of time.
We consider a plausible alternative, whereby the household head bases her decision of shopper
type on the cost of supplying time, measured by the marginal rate of substitution between
consumption and hours worked, σct + η−1lt. For this version of the model, we replace Wt
with σct + η−1lt, in equilibrium conditions (7)–(9), (11).
We also considered an alternative version of the model where we replace Wt with the
non-sticky real wages of new hires, Wt,0. The results were quantitatively in-between what we
obtained with the two other specifications.
B.9.3 Model with materials as intermediate input in retail production
To asses the effect of strategic pricing complementarities on the retailer’s pricing decisions, we
introduce materials as another intermediate input of production in the retail firm’s technology:
yjt = Y 1−χmjt mχm
jt , (19)
where mjt denotes the quantity of composite material input demanded by firm j.
The brands of variety j are sold to shoppers for consumption or to other retailers as
materials in production. Retail firm j buys brands from other retailers j′ at price PMj′t , and
combines them into a composite material input via CES technology:
mjt =
∑j
[(mj,j′,t
)1− 1θ
]θθ−1
,
32
where mj,j′,t denote intermediate inputs that the firm j acquires from other retailers, j′.
Cost minimization by retailers implies:
- the demand by firm j of materials from firm j′, mj,j′,t is
mj,j′,t =
(PMj′,t
PMt
)−θmjt ,
- the price of intermediate input composite is PMt =∑
j
(PMjt
)1−θ 1
1−θ,
- the demand for composite input according to
1− χmχm
mjt
Yjt=
Ωt
PMt,
- and the modified expression for retailer’s marginal cost
ΩRt = (1− χm)−(1−χm)χ−χmm Ω1−χm
t
(PMt
)χm.
We assume that prices of material inputs are also subject to Taylor-type adjustment con-
straints, and assume that they are reset at the same time as regular prices for consumption
brands. The roundabout production structure and pricing constraints for the price of mate-
rials create strategic pricing complementarities in the retailer’s pricing decisions (Basu, 1995,
Nakamura and Steinsson, 2010).
Denoting total demand for materials by Mt =∑
jmjt, we obtain the following optimal
pricing conditions
- equation for PMt,0
PMt,0 =θ
θ − 1
Et∑t+NP−1
τ=t βτ(P−1τ c−στ
) (PMτ
)θMτ · Ωτ
Et∑t+NP−1
τ=t βτ(P−1τ c−στ
)(PMτ )θMτ
,
- NP − 1 equations for PMt,1 , ..., PMt,NP−1
PMt,0 = PMt+1,1 = ... = PMt+NP−1,NP−1 .
Finally, the modified resource constraint for variety j is
NLjtc
Ljt +NH
jt cHjt +
∑j′
mj′,j,t = Y 1−χmjt mχm
jt .
33
Tab
leB
.1:
Rob
ust
nes
sof
mod
elp
red
icti
ons
Shop
ping
cos
t: O
ppor
tuni
ty
cost
of t
ime
Shop
ping
co
st: W
ages
of
new
hir
es
Stic
ky s
ales
pr
ices
, N
P=12
Pers
iste
nt
mon
etar
y sh
ock,
=0
.6
Low
er re
venu
e sh
are
from
sa
les,
2.0
Low
er a
vg
retu
rn o
n sh
oppi
ng, 2
%
Hig
her a
vg
retu
rn o
n sh
oppi
ng, 8
%
Mat
eria
ls in
pr
oduc
tion,
m
=0.7
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
std( t
)/st
d(C
t)0.
27
0.24
0.38
0.33
0.32
0.43
0.32
0.34
0.24
0.10
std( t
)/st
d(L t
)0.
140.
200.
350.
280.
260.
240.
260.
290.
200.
09
corr
(t ,
Ct)
–0.6
4–0
.53
–0.5
5–0
.55
–0.5
5–0
.63
-0.5
5–0
.60
-0.5
1–0
.39
corr
(t ,
Lt)
–0.4
4 –0
.35
–0.5
2–0
.42
–0.3
7–0
.73
-0.3
7–0
.31
-0.3
7–0
.30
No
disc
ount
s–0
.48
–0.4
8–0
.48
–0.4
8–0
.89
–0.4
8–0
.48
–0.4
8–0
.60
Con
stan
t di
scou
nts
–0.4
8–0
.41
–0.4
7–0
.48
–0.8
9–0
.48
–0.4
9–0
.49
–0.6
4
Cyc
lical
dis
coun
ts–0
.29
–0.2
3–0
.30
–0.3
1–0
.60
–0.3
1–0
.28
-0.3
0–0
.55
% d
iffer
ence
39.7
44.4
35.6
34.9
32.4
34.0
42.3
39.8
14.1
U.S
. Dat
a
B. A
vera
ge re
spon
se to
a –
1% im
puls
e to
mon
ey s
uppl
y
A. T
ime-
seri
es s
tati
stic
s
Base
line
Mod
el ro
bust
ness
Note
s:T
he
table
pro
vid
esre
sult
sof
sim
ula
tions
of
alt
ernati
ve
ver
sions
of
the
model
inth
em
ain
text.
Alt
ernati
ve
ver
sions
dis
cuss
edin
Sec
tions
6.3
–6.4
of
the
main
text,
and
Sec
tion
B.9
inA
pp
endix
.P
anel
Apro
vid
esti
me-
seri
esst
ati
stic
sfo
rth
eU
.S.
data
and
the
model
.P
anel
Bpro
vid
esav
erage
consu
mpti
on
resp
onse
sto
a–1%
impuls
eto
money
supply
over
the
firs
t12
month
saft
erth
esh
ock
.C
olu
mn
(1):
U.S
.data
,(2
):B
ase
line
model
,(3
),(4
):th
epri
ceof
tim
eis
giv
enby
the
opp
ort
unit
yco
stof
tim
eor
by
wages
of
new
hir
es(S
ecti
on
B.9
.2),
(5):
dis
count
pri
cePL t
issu
bje
ctto
the
sam
epri
ceco
nst
rain
tsas
regula
rpri
cePH t
,(6
):se
rially
corr
elate
dm
oney
gro
wth
,(7
):L
ower
calibra
tion
targ
etfo
rth
ere
ven
ue
share
from
sale
sto
2.0
,kee
pin
goth
erta
rget
sth
esa
me,
(8),
(9):
change
calibra
tion
targ
etfo
rpri
cesa
vin
gs
by
barg
ain
hunte
rsto
2%
or
8%
,kee
pin
goth
erta
rget
sth
esa
me,
(10):
intr
oduce
mate
rial
inputs
inre
tail
pro
duct
ion
(Sec
tion
B.9
.3).
34
C Measuring the impact of discounts on the aggregate price
level
In our model, the flexibility of the aggregate price response is due to the increase in the fraction
of sale prices and a shift of consumption weights toward discounted items. Because the shift
in consumption weights is generally not observed by statistical agencies, fluctuations in sale
prices may have important implications for aggregate price measurement. In this section, we
explore empirical importance of this margin.
The advantage of using a model-based approach to measure the importance of sales lies
in the explicit specification of the mechanisms that can account for their cyclicality. We
argued in the main text that these mechanisms are realistic and can be corroborated by the
evidence on retailers’ discounting and households’ search behaviour. Nonetheless, some of
the simplifying assumptions in the model may be material for the quantitative importance of
sales. For example, our analysis assumes symmetry across consumption sectors; homogeneity
of price discounts in the data, both in terms of frequency and size, and that nominal demand
shocks are the main drivers of business cycles. While the model can be extended along these
dimensions, we undertake a complementary approach based on available micro price data.
Specifically, we quantify empirically the wedge between the standard (CPI) aggregate price
index and the effective price index, assuming that the shift in consumption expenditures
toward discounted items is based on the constant-elasticity-of-substitution demand system.
Unlike the model-based approach, this empirical exercise takes the observed dynamics of price
discounts as given and measures the impact of shutting down those dynamics, keeping all else
equal. We focus on the main idea and results here, leaving the details available upon request.
First, express any posted (log) price of an item i in month t, pi,t , as the sum of the regular
(log) price, pRi,t, and a (log) discount in the event of a sale, ∆i,t:
pi,t = pRi,t + ∆i,t , (20)
By definition, ∆i,t < 0 if there is a sale in month t, and ∆i,t ≡ 0 if there is no sale. This
decomposition allows us to express the aggregate price index as the sum of the regular price
index, pRt , and an aggregate discount term, ∆t:
pt =∑i
ωi,tpRi,t +
∑i
ωi,th∆i,t∆i,t
= pRt + ∆t , (21)
where ωi,t are the usual CPI item weights while h∆i,t are additional weights for items on sale.
To construct the official CPI, statistical agencies implicitly assume that h∆i,t = 1 for all i and
35
t: the weight on each quote is the same, irrespective of whether the item is discounted or
regularly-priced. In light of the cyclicality of sales in the data, we study the implications of
relaxing this assumption and allowing h∆i,t to be above 1 and vary over time.
Ideally, one would estimate the h∆i,t factors using data on both prices and quantities. Such
data, however, are not available for a broad spectrum of products. Therefore, we opt for a
simple alternative: we use the weights implied by a constant-elasticity-of-substitution demand
system, similar to the one in our model. With an elasticity of substitution θ, the additional
weight h∆i,t is given by:
h∆i,t =
(exp pi,t
exp pRi,t
)−θ= (exp ∆i,t)
−θ . (22)
For instance, a value of θ = 5 implies that the weight on a 25% discount should be about 4.2
times higher than the weight on a regular price. We also explored a large scanner dataset for
food to show that CES demand with θ = 5 is a good approximation of the average elasticity
of quantities to discount prices across products within a category.9 This estimate is also
in line with other studies. For example, Glandon (2018) reports that for groceries in the
United States, sales account for 17% of price quotes but 40% of revenue, i.e., one percentage
point of price discounts accounts for 2.7 percentage points of revenue share. Assuming a 20%
discount, Glandon’s results imply a discount weight of 4.1. Using IRI data, Chevalier and
Kashyap (2019) found the elasticities for peanut butter to be above 4.5 across metropolitan
markets, and between 2 and 10 for coffee. In sum, the amount of extra consumption generated
by sales is substantial.
The final step is to use weights ωi,t and h∆i,t to construct a measure of the aggregate discount
term ∆t for the United Kingdom. An implicit assumption in this exercise is that the price
elasticity of quantities estimated across products with regular and sales prices is useful for
inferring the elasticity of aggregate price in response to time variation in the number of sales.10
For example, if Coke goes on sale without Pepsi going on sale, then, under this assumption,
the increase in demand for Coke rises by approximately the same amount as in the case when
both Coke and Pepsi go on sale simultaneously. This assumption is reasonable since in the
data the fraction of discounts is not correlated with regular prices or other discounts within
the same category, and the size of discounts is relatively stable. Both of these features are
accounted for in our model, which explicitly expresses the elasticity of the aggregate price with
respect to the fraction of discounts as a function of additional expenditure weights δ−θ(the
counterpart of h∆i,t in equation 21).
In Figure C.1, we compare our constructed series for three values of θ: 0 (equivalent to
9We also explored an alternative approach where, instead of assuming CES demand, we measure weightfactors h∆
i,t directly from the scanner dataset. This approach yields quantitatively similar results.10We are grateful to an anonymous referee for pointing this out to us.
36
the CPI case, with ω∆i,t = 1), 3 and 5. By construction, each series represents a deviation
from the regular price index (pRt ). Time series are provided for the aggregate as well as three
broad categories.
Figure C.1: Contributions of sales to the effective price level in the United Kingdom.
-.06
-.04
-.02
0%
20022004
20062008
20102012
q = 5 q = 3q = 0
All products
-.1
-.08
-.06
-.04
-.02
0%
20022004
20062008
20102012
Food and beverages
-.2
-.15
-.1
-.05
0%
20022004
20062008
20102012
Household furniture and equipment-.
15-.
1-.
050
%
20022004
20062008
20102012
Personal care services
Notes: The time series are the 12-month centered moving averages of the aggregated discount term defined in(22) in Section (C), for various values of the elasticity of substitution θ. All series represent percentagedeviations of the effective price index from its regular-price counterpart. Discount terms are based onV-shaped discounts and aggregated across products using official quote and stratum weights. “ θ = 0 ”corresponds to the official CPI mesure in that it uses equal weights for regular-priced and discounted pricequotes. The sample period is 1996:02–2012:12, U.K. Office of National Statistics data.
First, there are significant differences in the average levels of the discount term across
values of θ and categories. In the aggregate, the effective price level is 0.9% below the regular
price index for θ = 0 (the official CPI case), but 7% lower when θ = 5. With an elasticity
of subsitution of θ = 3, the average contribution of sales ranges from -3% for Food and
Beverages to –6.2% for Household Furniture and Equipment, reflecting mostly differences in
the prevalence of sales across categories.
Without any fluctuations in the frequency or the size of discounts, these contributions
37
would be stable over time. Figure C.1 makes it clear that they are not. In the aggregate, for
θ = 5, sales contributed to a 3-percentage-point drop in the effective price level in less than
three years, relative to the regular-price index. For Personal care services, the contribution
falls from –5% in 2005 to –17% by 2010. In other words, ignoring sales altogether leads us to
overestimate the effective price inflation of this category by about 2.4 percentage points per
year over this period.11
Comparing those fluctuations to the ones for θ = 0 (i.e. sale and regular quotes get equal
weights) indicates that the official CPI can in some instances significantly underestimate the
role of sales: in this scenario, the range of the discount term ∆t over the sample period is
only about one percentage point for Food and Beverages and Clothing and footwear and 1.8
percentage point for Personal care services. These findings are consistent with Anderson et al.
(2017) who document a small contribution of sales to U.S. inflation measured using BLS micro
price dataset but imposing constant expenditure weights.
An important observation from the empirical analysis (Figure C.1) is that the importance
of discounts is close to what the model predicts. For example, in the empirical analysis, a
1-percentage-point increase in the fraction of sales is associated with a 1.56-percentage-point
decrease in the aggregate price level for θ = 5 , whereas in the model that decrease would
be 0.92 percentage points. This difference can be reconciled by Jensen’s inequality: since
the weights in (22) are convex and increasing in the absolute size of the log discount, larger
sales have a disproportionately larger contribution to the discount term, ∆t. For instance,
recomputing ∆t using only discounts smaller than 30% reduces the elasticity by a six-fold,
from 1.56 to 0.26. Because in our model all sales have the same size, the effect of the spread
in the size of discounts is not captured.
In summary, our findings in this section demonstrate that properly accounting for sales
can have important ramifications for the assessment of the cyclical behavior of aggregate
price and consumption. First, even small fluctuations in the frequency of sales can have
economically-relevant effects on the effective price level, in line with what we found in our
model. Second, the size of these effects depends heavily on the expenditure-switching behavior
of consumers between discounted and regular-price items; this channel is mostly absent from
official consumer price indices. Third, larger discounts provide a disproportionately large
impact on consumption and price levels.
11Freeman et al. (2008) reports the findings from The Rapport Attali in France that introduction of theban on below-cost price discounting led to a decrease in competition between the grocery retail chains andbetween suppliers, resulting in significant increases in price levels. For example, between 1996 and 2004, theprice of food in France (excluding meat and fresh produce) increased by 16%, 3 percentage points more thanthe general price index in France, and faster than food (excluding meat and fresh produce) in the rest of theEuro zone.
38
References
Anderson, Eric, Benjamin A. Malin, Emi Nakamura, Duncan Simester, and Jon Steinsson.
2017. “Informational rigidities and the stickiness of temporary Sales.” Journal of Monetary
Economics 90 (C):64–83.
Basu, Susanto. 1995. “Intermediate Goods and Business Cycles: Implications for Productivity
and Welfare.” American Economic Review 85 (3):512–531.
Bils, Mark and Peter J. Klenow. 1998. “Using Consumer Theory to Test Competing Business
Cycle Models.” Journal of Political Economy 106 (2):233–261.
Blanchard, Olivier J. and Charles. M. Kahn. 1980. “The solution of the linear difference
models under rational expectations.” Econometrica 48:1305–1313.
Chevalier, Judith A. and Anil K. Kashyap. 2019. “Best Prices: Price Discrimination and
Consumer Substitution.” American Economic Journal: Economic Policy 11 (1):126–159.
Coibion, Olivier, Yuriy Gorodnichenko, and Gee Hee Hong. 2015. “The Cyclicality of Sales,
Regular and Effective Prices: Business Cycle and Policy Implications.” American Economic
Review 105 (3):993–1029.
Freeman, Peter, Jayne Almond, Barbara Donoghue, Alan Gregory, Alan Hamlin, and Bruce
Lyons. 2008. “The Supply of Groceries in the UK.” Market investigation, Competition
Commission. The Stationery Office, London.
Glandon, P.J. 2018. “Sales and the (Mis)measurement of price level fluctuations.” Journal of
Macroeconomics 58 (C):60–77.
Kryvtsov, Oleksiy and Virgiliu Midrigan. 2013. “Inventories, Markups, and Real Rigidities in
Menu Cost Models.” Review of Economic Studies 80 (1):249–276.
Mahajan, S. 2006. “Concentration ratios for businesses by industry in 2004.” Economic
Trends 635, Office for National Statistics.
Nakamura, Emi and Jon Steinsson. 2010. “Monetary Non-neutrality in a Multi-Sector Menu
Cost Model.” Quarterly Journal of Economics 125 (3):961–1013.
39
