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NBER WORKING PAPER SERIES
OPTIMAL PRICE INDICES FOR TARGETING INFLATION UNDER INCOMPLETE
MARKETS
Rahul Anand
Eswar S. Prasad
Working Paper 16290
http://www.nber.org/papers/w16290
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
August 2010
We are grateful to Kaushik Basu, Gita Gopinath, Karel Mertens, Parul Sharma, Viktor Tsyrennikov
and Magnus Saxegaard for helpful comments and discussions. We received helpful comments from
seminar participants at Cornell University, the IMF and the Reserve Bank of India. This research was
supported by a grant from the International Growth Centre's Macroeconomics Program. The views
expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau
of Economic Research.
NBER working papers are circulated for discussion and comment purposes. They have not been peer-
reviewed or been subject to the review by the NBER Board of Directors that accompanies official
NBER publications.
© 2010 by Rahul Anand and Eswar S. Prasad. All rights reserved. Short sections of text, not to exceed
two paragraphs, may be quoted without explicit permission provided that full credit, including © notice,
is given to the source.
Optimal Price Indices for Targeting Inflation Under Incomplete Markets
Rahul Anand and Eswar S. Prasad
NBER Working Paper No. 16290
August 2010
JEL No. E31,E52,E61
ABSTRACT
In models with complete markets, targeting core inflation enables monetary policy to maximize welfare
by replicating the flexible price equilibrium. In this paper, we develop a two-sector two-good closed
economy new Keynesian model to study the optimal choice of price index in markets with financial
frictions. Financial frictions that limit credit-constrained consumers’ access to financial markets make
demand insensitive to interest rate fluctuations. The demand of credit-constrained consumers is determined
by their real wage, which depends on prices in the flexible price sector. Thus, prices in the flexible
price sector influence aggregate demand and, for monetary policy to have its desired effect, the central
bank has to stabilize price movements in the flexible price sector. Also, in the presence of financial
frictions, stabilizing core inflation is no longer equivalent to stabilizing output fluctuations. Our analysis
suggests that in the presence of financial frictions a welfare-maximizing central bank should adopt
flexible headline inflation targeting—a target based on headline rather than core inflation, and with
some weight on the output gap. We discuss why these results are particularly relevant for emerging
markets, where the share of food expenditures in total consumption expenditures is high and a large
proportion of consumers are credit-constrained.
Rahul Anand
IMF, Asia and Pacific Department
700 19th Street, N.W.
Washington, DC 20431 U.S.A.
ranand@imf.org
Eswar S. Prasad
Department of Applied Economics and
Management
Cornell University
440 Warren Hall
Ithaca, NY 14853
and NBER
eswar.prasad@cornell.edu
1
1. Introduction
The global financial crisis has led to a vigorous debate about the appropriate objectives for
monetary policy. For instance, it has been posited that a narrow version of inflation targeting
(IT) could pose risks if it implies that potential asset bubbles are ignored by central banks.
The emerging consensus appears to be that the IT framework has delivered price stability
and should be retained but that central banks should use prudential regulation and other
policy tools to counteract asset price bubbles. Whether or not IT is the chosen framework,
central banks around the world view low and stable inflation as a primary, if not dominant,
objective of monetary policy.
What is the right price index that should be the focus of the inflation objective? This is a
central operational issue in implementing not just IT but any version of monetary policy.
Two key issues about the choice of price index are--determining the level of inflation that is
consistent with the notion of price stability and determining the appropriate price index. In
this paper, we focus on the task of analytically determining the appropriate price index for
markets with financial frictions in general and emerging markets in particular.
In the literature, the choice of price index has been guided by the idea that inflation is a
monetary phenomenon. It has been suggested that core inflation (excluding food, energy
and other volatile components from headline CPI) is the most appropriate measure of
inflation (Wynne, 1999). The logic is that fluctuations in food and energy prices represent
supply shocks and are non-monetary in nature. Since these shocks are transitory and
volatile and do not reflect changes in the underlying rate of inflation, they should not be a
part of the inflation targeting price index (Mishkin, 2007, 2008).
Previous authors have used models with price and/or wage stickiness to show that the
choice of this price index is consistent with a welfare maximization objective. Existing
models have looked at complete market settings where price stickiness is the only source of
distortion (besides monopoly power). Infrequent price adjustments cause mark-ups to
fluctuate and also distort relative prices. In order to restore the flexible price equilibrium,
central banks should try to minimize these fluctuations by targeting sticky prices
(Goodfriend and King, 1997, 2001). Using a variant of a New Keynesian model, Aoki
(2001) has shown that under complete markets targeting inflation in the sticky price sector
leads to welfare maximization and macroeconomic stability. Targeting core inflation is
2
equivalent to stabilizing the aggregate output gap as output and inflation move in the same
direction under complete markets.
Appropriateness of the core price index in these models relies heavily on the assumption
that markets are complete (allowing households to fully insure against idiosyncratic risks)
so that the central bank only needs to tackle the distortions created by price stickiness.
However, there is compelling evidence that not all agents in the economy may be able to
smooth their consumption (Campbell and Mankiw, 1989, 1990, 1991).1 This observation is
also consistent with the findings of a number of papers rejecting the permanent income
hypothesis. It has been shown that, in the presence of credit-constrained consumers,
policymakers’ welfare objectives are altered and the Taylor rule becomes too weak a
criterion for stability (Amato and Laubach, 2003; and Gali, Lopez-Salido and Valles, 2004) .
Our main objective in this paper is to develop a model to study the welfare implications of
targeting different price indices in an incomplete markets setting and to analytically
determine the appropriate price index to target. A major contribution of this paper is to
study the implication of financial frictions (modeled by the presence of credit constrained
consumers) on the choice of the optimal price index.
Financial frictions that result in credit-constrained consumers have not received much
attention in models of inflation targeting. To examine the significance of financial frictions,
we develop a model with heterogeneous agents, where a fraction of consumers cannot
smooth their consumption—that is, they simply consume their current labor income.2 When
markets are not complete and agents differ in their ability to smooth consumption, their
welfare depends on the nature of idiosyncratic shocks. Thus, this modeling choice also
allows us to look at the welfare distribution under alternative choices of the price index.
Under complete markets, the income distribution following a sector-specific shock does not
matter for the choice of consumption and, hence, welfare. However, under incomplete
markets, household income, which is influenced by the nature of shocks and the price
1 Campbell and Mankiw estimate that in the U.S. nearly 50 percent of income accrues to
consumers who do not smooth their consumption. Muscatelli, Tirelli and Trecroci (2004) find
that about 37 percent of consumers are rule-of-thumb consumers and they account for 59 percent
of total employment. For further evidence on the proportion of credit-constrained consumers in
the U.S., see Jappelli (1990), Shea (1995), Parker (1999), Souleles (1999), Fuhrer (2000), and
Fuhrer and Rudebusch (2004).
2 We introduce this friction in a manner similar to that of Gali, Lopez-Salido and Valles (2004).
3
elasticity of demand for goods, matters for the consumption choice.3 Price elasticity of the
demand for food, which has not attracted much attention in complete market settings,
becomes important under incomplete markets. We show that, through its impact on a
household’s income and expenditure, the low price elasticity of the demand for food is an
important determinant of the optimal choice of price index under incomplete markets.4
We also incorporate other important features relevant to emerging markets into the model.
The share of food in total household expenditures is much higher in emerging markets,
constituting nearly 40-50 percent of household expenditures compared to 10-15 percent in
advanced economies. Low price and income elasticities of food, and low income levels make
the welfare of agents in emerging markets more sensitive to fluctuations in food prices.
Since expenditure on food in total household expenditure is high and demand for food is
relatively inelastic, agents may factor in food price inflation while bargaining over wages.
Through this channel, food price inflation feeds into inflation expectations. Thus, in
emerging markets even inflation expectation targeting central banks have to be concerned
about food price inflation. 5
The key finding of the paper is that in the presence of financial frictions targeting core
inflation (i.e., inflation in the sticky price sector) may not be optimal. Lack of access to
financial markets makes the demand of credit-constrained consumers insensitive to
fluctuations in interest rates. Since their demand depends only on real wages, a link is
established between aggregate demand and real wages. Thus, in the presence of financial
frictions, the relative price of the good produced in the flexible price sector not only affects
aggregate supply but, through its effects on real wages, also influences aggregate demand.
This result is at variance with the prior literature based on complete markets settings. For
instance, in Aoki’s (2001) model, relative prices of the flexible price sector only appear as a
shift parameter of inflation in the sticky price sector. Under incomplete markets, by contrast,
the central bank cannot ignore fluctuations in the price of the good produced in the flexible
3 A negative productivity shock related to a good with a low price elasticity of demand could
increase the income of net sellers of that good and raise the expenditure of net buyers on that
good.
4 A survey by the U.S. Department of Agriculture suggests that the average price elasticity of
food is -0.34 in a sample of 114 countries; this estimate is smaller in absolute terms than the
elasticity normally used in other models, most of which assume a unitary price elasticity.
5 Walsh (2010) documents the high pass-through from food price inflation to non-food inflation in
middle- and low-income countries.
4
price sector if it wants to affect aggregate demand. Financial frictions break the comovement
of inflation and output (as inflation and output may now move in opposite directions).
Stabilizing core inflation is no longer sufficient to stabilize output. Thus, in the presence of
financial frictions targeting flexible headline inflation is a better policy choice.
Since our model exhibits monetary super-neutrality, we limit our analysis to non-
inflationary steady states (long-run price stability) and do not have anything to say about the
optimal level of inflation. We also do not attempt to define optimal policy rules but focus on
evaluating welfare outcomes of different policy rules using alternative measures of inflation.
The paper is organized as follows. In the next section, we present some empirical facts to
further motivate the analysis. In Section 3, we develop a two-sector, two-good model with
heterogeneous agents that encapsulates the features discussed above. In Section 4 we
discuss the main results and in Section 5 we conduct various sensitivity experiments to
check the robustness of our baseline results and also present some extensions of the basic
model. Section 6 concludes the paper.
2. Basic Stylized Facts
We begin by presenting some stylized facts about the share of household consumption
expenditures on food and also various measures of the elasticity of food expenditures. In a
cross-country comparison, emerging markets and advanced countries differ markedly on
these measures. Next, we present data on credit constraints in emerging markets. We also
look at the features of core and headline CPI inflation measures in some emerging and
advanced economies.
Engel’s law states that as average household income increases the average share of food
expenditure in total household expenditure declines. When this idea is extended to
countries, we expect poor countries to have a high average share of food expenditure in
total household expenditure. Figure 1 plots the expenditure on food (as a percentage of
total expenditure) against log real per capital income for the year 1996.6 It shows that
countries with lower per capita income levels have a higher share of expenditure on food
in total household expenditure. In order to examine how emerging markets differ from
6 We use data for 1996 for illustrative purposes since data for a large number of countries were
available for that year.
5
advanced countries, in Table 1 we present recent data on shares of food expenditure in
total expenditure for selected emerging and advanced economies.7 As expected,
expenditure on food constitutes a much larger share of total household expenditure in
emerging markets relative to advanced economies.
Income and price elasticities of the demand for food are important for our analysis.
Figure 2 plots the income elasticity of food against real per capita GDP for the year 1996.
The income elasticity of food is low, suggesting that food is a necessary good. Since
expenditure on food is not a major share of household expenditure in rich countries, the
income elasticity of food is much lower.8 We present the income elasticity of food for
selected emerging market and advanced economies in Table 2. The income elasticity of
food in emerging markets is on average twice as large as that in advanced economies.
Figure 3 plots, for a large sample of countries, the Slutsky own price elasticity of food
against the log real per capita GDP for the year 1996.9 The price elasticity of food
demand is nonlinear, decreasing at low income levels, and then increasing, with a range
from -0.4 to -0.1. We also present data on the Slutsky own-price elasticity of food for
selected countries in Table 2.10 The price elasticity of food is very low (suggesting that
the demand for food is inelastic). As the share of expenditure on food is high in emerging
markets, the price elasticity of food is higher in these economies. However, the overall
value of the price elasticity of food is much lower than what is used in the literature on
inflation targeting. Low price and income elasticities of the demand for food have
considerable significance for the choice of price index.
In order to examine the extent of credit constraints in emerging markets, in Table 3 we
present data on the percentage of the adult population with access to formal finance
7 We looked at household surveys for each country in this table rather than the weight of food in
each country’s CPI index since those weights are changed only occasionally. However, data from
household surveys are available for only a few emerging markets. These data typically cover
expenditure on food consumed at home and don’t include expenditures on beverages and tobacco.
8 A low income elasticity of demand also means that, as family income increases, consumption of
the commodity will not increase by much.
9 The Slutsky own price elasticity is estimated by keeping real income constant.
10 Frisch elasticity values lie between Slutsky and Cournot values and can be considered as an
average own price elasticity.
6
(measured by the share of the population using financial services) in emerging markets.
On average, more than half of the population in emerging markets does not have access
to the formal financial system.
Next, we examine the characteristics of core and headline inflation. We plot the levels
and volatility of inflation for selected advanced and emerging market economies (Figure
4-5). Values of average inflation, average volatility and the persistence of inflation (for
the period March 1991 – September 2009) are reported in Table 4. The two measures of
inflation have very different characteristics in advanced and emerging market economies.
Average inflation (both headline and core) has been higher in emerging market
economies during the period reported. Headline inflation is more volatile than core
inflation in both advanced and emerging market economies. However, the volatility of
both inflation measures is much higher in emerging markets. Core inflation has on
average been more than twice as volatile in emerging market economies compared to
advanced countries. The two measures of inflation exhibit a high degree of persistence in
both sets of economies.11
We also look at the evolution of two price indices over time. It is expected that they
would deviate from each other in the short run (as the core measure is constructed to
eliminate the fluctuations which do not reflect the underlying inflation developments).
However, since transitory shocks (shocks to food and energy) do not change the
underlying trend, headline inflation should return to its original level in a short period
(Mishkin, 2007). In other words, the headline inflation measure should not remain above
the core inflation measure for an extended period.
11 In a cross-country study, Walsh (2010) finds that food price inflation is in fact more persistent
than non-food price inflation. This holds for both advanced and emerging market economies,
although he finds that food price inflation is more persistent in emerging markets.
7
To verify this, we examine the two measures of inflation for two representative core
inflation targeting countries – Canada and Thailand.12 In Canada, in the period from the
spring of 1999 to the fall of 2001, headline inflation remained above core inflation for 30
months in succession (Figure 4a). In Thailand, headline inflation has remained above
core inflation for more than 5 consecutive years (Figure 5a). The core inflation measure
excludes a number of expenditure items and is less representative of the cost of living.
Thus, differences in the behavior of headline inflation (ostensibly a more accurate
measure of the cost of living) and core inflation over an extended period may have
important welfare implications.
3. The Model
Our model builds upon a large literature that has developed and analyzed dynamic sticky
price models (Clarida, Gali and Gertler, 1999; Woodford, 1996; Rotemberg and
Woodford, 1997, 1999; Aoki, 2001). The model is rendered more realistic by
incorporating two features that are relevant to all economies but are particularly
important for emerging markets--a fraction of consumers who are credit constrained and
a subsistence level of food consumption. The model has two sectors and two goods—one
type of flexible price good, food (
F
C
), whose prices adjust instantaneously, and a
continuum of monopolistically produced sticky price goods,
)1,0(in indexed )( !zzc
which we call non-food and whose prices adjust sluggishly.13 In
the subsequent discussion, we interchangeably use the term food sector for the flexible
price sector and the term non-food sector for the sticky price sector.
3.1 Households
The economy is populated by a continuum of 1 + λ infinitely lived households,
where
0>
!
, is the continuum of households in the flexible price sector (food sector).
Each household owns a firm and produces one good. They provide labor to the firms in
their respective sector (we assume that labor is immobile across sectors) and consume
12 Canada is an advanced economy that adopted IT in 1991 while Thailand, an emerging market
economy, adopted IT in 2000. Canada targets core inflation excluding food, energy and indirect
taxes. Thailand targets core inflation, which excludes food and energy prices.
13 We model the sticky price sector by a continuum of monopolistic firms so that these firms have
market power and they can set prices. This is done to introduce price stickiness in this sector.
8
both the flexible price good (food) and all of the differentiated sticky price goods (non
food).14 The representative consumer, i, is indexed by f (flexible price sector) and s
(sticky price sector). Household i maximizes the discounted stream of utility
)],([
0
0
i
t
i
t
t
tNCuE
!
"
=
#
(1)
where
)1,0(!
"
is the discount factor. The utility function takes the form:
!
"
#
!#
+
$
$
=
+$
1
)(
1
)(
),(
11 i
t
n
i
t
i
t
i
t
NC
NCu
(2)
where the argument
i
t
C
is the composite consumption index of household i in period t.
i
t
C
includes the flexible price good and the entire continuum of the differentiated goods.
It is defined as
( )
( )
!
!
!
!
!
""
1
1
1
1
1
,
1
1
1
*
,
1
)(1
#
#
#
$
$
%
&
'
'
(
)#+#=
i
ts
i
tf
i
t
CCCC
(3)
where
1
1
0
1
,
)(
!
!
"
#
$
%
&
'
=
(
)
)
)
)
dzzcC
i
t
i
ts
(4)
The elasticity of substitution between the flexible price and sticky price goods is given
by
],0[ !"
#
and
]1,0[!
"
is the weight on food in the consumption index. The parameter
14 We have assumed the immobility of labor for simplicity and to capture the large inter-sectoral
wage differential in emerging markets. Gali, Lopez-Salido and Valles (2004) have demonstrated
in their model that, even with free labor mobility, financial frictions lead to similar results as ours
(aggregate demand going up even when the central bank raises the policy interest rate).
9
θ >1 is the elasticity of substitution between any two differentiated goods,
i
t
N
is the
aggregate labor supplied by household i in period t and
!
is the risk aversion factor
(inverse of elasticity of inter temporal substitution). The parameter
!
is the inverse of
Frisch elasticity and
n
!
is a scaling factor.
The utility function used here is of a generalized Klein-Rubin form.15 This form is
selected to model the role of food in the economy. Since food is a necessity, households
must consume a minimum amount C* of food for survival.16 We assume that all
households always have enough income to buy the subsistence level of food. Even
though the subsistence level food consumption does not bind, it plays a vital role by
altering the elasticity of substitution between food and non-food and the marginal utility
of food and non-food consumption.
3.1.1 Flexible Price Sector (Food Sector) Households
Households in the flexible price sector (food sector) do not have access to financial
markets and they consume their wage income in each period.17 So these households are
akin to the “rule of thumb” consumers. Each household in the sector owns one firm and
produces food by linear technology in labor, given by
f
ttftf
NAy
,,
=
(5)
Af,t is a random productivity shock. Since we are interested in analyzing the effects of
sector-specific shocks rather than household-level idiosyncratic shocks, we assume that
all the households in the food sector face the same shock.
15 Expenditure system corresponding to Klein-Rubin utility function is referred to as the Stone-
Geary linear expenditure system; Stone (1954) and Geary (1949).
16 This is also similar to habit persistence with C* being independent of time.
17 There is no storage technology in the model. So consumers in the flexible price sector cannot
smooth their consumption by saving their output. We have made this restrictive assumption to
keep the model tractable. Moreover, Table 3 shows that more than 50 percent of individuals in
emerging markets lack access to formal finance. Basu et al. (2005) have documented that 80
percent of individuals in India’s agricultural sector have no access to formal finance.
10
3.1.2 Sticky Price Sector (Non Food Sector) Households
Households in this sector can buy one period nominal bonds and smooth their
consumption. Each household owns a firm and provides labor to each firm in the sector.
They hold one share in each firm of the sector. Each firm uses a linear technology in
labor given by
)()(
,
zNAzy
s
ttst
=
(6)
where
)( zy
t
is a sticky price good and
)( zN
s
t
is the labor used in the firm producing
good indexed by z ( where
]1,0[ !z
). As,t is a random productivity shock. We assume that
the shock is identical for all households in the non-food sector.
3.2 Consumption Decision
3.2.1 Food Sector Households (Credit Constrained Consumers)
All households in this sector face an identical budget constraint every period (as their
wage income is the same in every period). A representative household maximizes its
lifetime utility given by equation (1) subject to the budget constraint
f
t
f
t
f
tsts
f
tftf
NWCPCP =+
,,,,
(7)
where
f,t
P
is the market price of food,
ts
P
,
is the price index of non-food (defined below)
and
f
W
t
is the nominal wage in the food sector. The optimal allocation for a given level
of spending between food and all the differentiated non-food goods leads to a Dixit-
Stiglitz demand relation. The total expenditure to attain a consumption index
f
t
C
is given
by
f
tt CP
where
t
P
is defined as
[ ]
!
!!
""
#
##
#+=
1
1
1
,
1
,
))(1()(
tstft
PPP
(8)
11
The budget constraint can be written as:
*
,
CPNWCP
tf
f
t
f
t
f
tt
!=
(9)
Demand for the flexible price good is given by
*
,
,CC
P
P
Cf
t
t
tf
f
tf +
!
!
"
#
$
$
%
&
=
'
(
)
(10)
Demand for the sticky price good is given by
f
t
t
ts
f
ts
C
P
P
C
!
"
#
$
$
%
&
'
'
(
)
#=
,
,
)1(
(11)
where
ts
P
,
is the Dixit-Stiglitz price index defined as
!
!
"
"
#
$
%
&
'
(
=
)
1
1
1
0
1
,
)( dzzXP
tts
(12)
)( zX t
is the price of differentiated good indexed on z at time t. Demand for each
differentiated good is given by
f
ts
ts
t
f
C
P
zX
zc
t
,
,
)(
)(
!
"
#
#
$
%
&
&
'
(
=
(13)
The labor supply decision is given by the usual first order condition with respect to
f
t
N
:
t
f
t
f
t
f
t
n
P
W
C
N=
!
"
#
$
)(
)(
(14)
12
3.2.2 Non Food Sector households (Unconstrained Consumers)
Each household in this sector provides labor to each one of the firms in the sector and
also holds one share in each firm. This setting is the one followed by Woodford (2003).18
In this setup, each household faces the same budget constraint each period and hence
chooses the same consumption stream. A representative household maximizes the
lifetime utility given by equation (1) subject to the following budget constraint
*
,11
1
0
1
0
)()()( CPBRdzzdzzNzWBCP
tftt
s
t
s
t
s
tt
s
tt
!+"+=+
!!
##
(15)
where
t
B
represents the quantity of one-period nominal riskfree discount bonds bought in
period t and maturing in period t+1 and
t
R
is the gross nominal interest rate between
period t and t+1.
)( zW
s
t
and
)( zN
s
t
represent the nominal wage prevalent in firm z and
the amount of labor supplied to firm z by the household, respectively.
)( z
s
t
!
is the profit
of firm z. Maximization with respect to
s
t
C
yields the Euler equation
!
"
#
$
%
&
'
=
+
(
+
(
1
1
)()(
t
t
s
tt
s
t
R
CEC
))
*
(16)
where
1!
="
t
t
tP
P
is gross headline inflation. The labor supply decision of the household
to a firm indexed by z is given by
t
s
t
s
t
s
t
n
P
zW
C
zN )(
)(
))(( =
!
"
#
$
(17)
18 Alternatively, we could use the other set up specified in Woodford (2003) in which each
household produces one of the differentiated products and there exist a complete range of
securities through which they can insure fully against idiosyncratic risks. In that formulation also,
each household will choose the same consumption stream and therefore the analysis will be the
same as in the present setting.
13
Demand for the flexible price good is given by
*
,
,CC
P
P
Cs
t
t
tf
s
tf +
!
!
"
#
$
$
%
&
=
'
(
)
(18)
Demand for the sticky price good is given by
s
t
t
ts
s
ts C
P
P
C
!
"
#
$
$
%
&
'
'
(
)
#=,
,)1(
(19)
and the demand for each differentiated good is given by
s
ts
ts
t
s
C
P
zX
zc
t
,
,
)(
)(
!
"
#
#
$
%
&
&
'
(
=
(20)
3.3 Firms
3.3.1 Firms in the Flexible Price Sector (Food Sector)
Firms are assumed to be price takers. Given a market price
tf
P
,
they set their price such
that
tf
f
t
tf
A
W
P
,
,
=
(21)
The supply function for the flexible price firm is obtained by combining equations (5),
(14) and (21), and is given by:
!
"
#
$
%
&
'
(
)
*
=)(
,
,
,
,
f
ttf
tf
tf
n
t
tf
CA
A
y
P
P
(22)
14
The market-clearing condition for food implies
)1( *
,
,,, CC
P
P
CyY t
t
tf
tftftf
!"!
#
++
$
$
%
&
'
'
(
)
===
*
(23)
where we have defined
tt
s
t
f
t
YCCC ==+
!
(24)
It can be considered as the total composite demand and hence equal to supply in
equilibrium.
3.3.2 Firms in the Sticky Price Sector
We follow Calvo (1983) and Woodford (1996) in modeling price stickiness. A fraction
)1,0(!
"
of firms cannot change their price in each period. Firms are free to change the
price at time t; they choose a price
t
X
to maximize the following objective function:
[ ]
!
"
#
$
%
&'
++++
(
=
)
))(()()()(
,,,,
0
)(
zyTCzyzXQEMax
jttjttjtttjtt
j
j
t
zX
t
*+
(25)
where
jt
t
s
t
s
jt
i
jtt
P
P
C
C
Q
+
!
+
+
"
"
#
$
%
%
&
'
=
(
)
,
is the stochastic discount factor and
)(
,zy jtt +
is the output
of firm in period t+j when it has set its price in period t that is given by
jts
jts
t
jtt
Y
P
zX
zy
+
!
+
+
"
"
#
$
%
%
&
'
=
,
,
,
)(
)(
(
(26)
where we have made use of the market clearing conditions
( )
s
ts
f
ts
ts
t
s
t
f
ttt
CC
P
zX
zczczcy
,,
,
)(
)()()( (z) +
!
!
"
#
$
$
%
&
=+==
'
((
)
(27)
15
tst
t
ts
ts
s
ts
f
ts
YC
P
P
CCC
,
,
,,,
)1( =
!
!
"
#
$
$
%
&
'==+
'
(
)*
(28)
ts
ts
t
ts
ts
t
t
Y
P
zX
C
P
zX
zy
,
,
,
,
)()(
)(
!!
""
#
#
$
%
&
&
'
(
=
#
#
$
%
&
&
'
(
=
(29)
The sticky sector price index is expressed by
[ ]
!
!! ""
#
##
##+= 1
1
11
1,, )()1()( zXPP ttsts
(30)
The price
)( zX
t
solves the following first order condition
0)(
1
)()()(
,,
0
=
!
"
#
$
%
&
'
(
)
*
+
,
-
-
+++
.
=
/
zMCzXzyQE
r
jttjttjtt
j
j
t
0
0
12
(31)
where
!
"
#
$
++
+
+
+
+
+
%
&
'
(
)
*
== )(
)(
)(
)(
,
,
,
s
jtjts
jts
jtt
n
r
jt
jt
jt
CA
A
zy
zMC
P
zMC
(32)
and
1!
"
"
is the constant markup over the marginal cost.19
Equations (8), (9), (16), (22), (23), (28), (30), (31) and (32), coupled with a monetary
policy rule to choose the nominal interest rate, jointly determine the equilibrium path of
consumption, output and price index in both the sectors.
19 Since the technology is linear, MCt,t+j = MCt+j That is, marginal cost is independent of the level
of production.
16
3.4 Inflation and Relative Prices
We define the relative prices as follows:
tf
t
tf
x
P
P
,
,
=
, relative price of food,
ts
t
ts
x
P
P
,
,
=
, relative price of non-food; and
t
ts
t
x
P
X=
,
,
relative price charged by firms which are free to choose the price in time t. We define the
gross headline inflation as
1!
="
t
t
t
P
P
, and gross inflation in the sticky price sector as
1,
,
,
!
="
ts
ts
ts P
P
. The relationship between headline and core inflation (inflation in the sticky
price sector) is given by:
t
tsts
t
t
t
ts
ts
ts
ts
x
P
P
P
P
P
P
x
!
!
== "
"
"
"
"
1,,
1
1
1,
1,
,
,
(33)
The system of equations in terms of stationary variables is presented in Appendix I.
3.5 Steady State
We characterize the steady state with constant prices (zero inflation) and no price
stickiness in the economy.20 This implies that
1 and 1
,
=!=!
tst
for all t. Under
symmetric equilibrium, each firm faces the same demand and sets the same price. Thus,
1 and
,== ttst xPX
. Therefore,
r
tts MCx
1
,!
=
"
"
. In the steady state, all firms set a price
which is a constant markup over the real marginal cost.21 We assume that productivity is
the same in both the sectors and normalize it to one.
20 Our model exhibits monetary super-neutrality. Therefore, the level of steady state inflation does
not affect steady state values of real variables.
21 We also compute the welfare gains when the steady state involves a tax rate which is set such
that the steady state level of output in the sticky price sector is efficient. All our results go
through under this alternative characterization of steady state.
17
3.6 Monetary Policy Rule
We assume that the monetary authority sets the short term nominal interest (
t
R
)
according to a simple Taylor (1993) type rule of the following form
)/log()/log()/log()/log(
___
1
_
YYRRRR tyttit
!!!
"
+##+= $
(34)
where
___
and , RY !
are the steady state values of output, inflation and the nominal interest
rate, respectively. The term
i
!
represents the Central Banker’s preference for interest rate
smoothing.
!
"
and
y
!
are the weights on inflation and output gap assigned by the policy
makers.22 We characterize core inflation as the inflation in the sticky price sector,
ts,
!
,
and headline inflation as the over all inflation,
t
!
, for our policy experiments.
We evaluate our model under the following monetary policy regimes:
Strict Core Inflation Targeting: The central bank cares only about interest rate smoothing
and stabilizing inflation in the sticky price sector.
)/log()/log()/log(
_
,
_
1
_
ststit
RRRR !!+=
"
#
$$
(35)
Strict Headline Inflation Targeting: The central bank cares only about interest rate
smoothing and stabilizing headline inflation.
)/log()/log()/log(
__
1
_
!!+=
"ttit
RRRR
#
$$
(36)
22 We include an interest rate smoothing parameter in our monetary policy rule as the benefits of
such smoothing are well documented in the literature (see, e.g., Lowe and Ellis, 1997; Sack and
Wieland, 1999). Various authors have argued that moving interest rates in small steps increases
its impact on the long-term interest rate; it also reduces the risks of policy mistakes and prevents
large capital losses and systemic financial risks. Mohanty and Klau (2004) find that all emerging
market central banks put substantial weight on interest rate smoothing. Clarida et al.(1998) find
that central banks of advanced economies also put a large weight on interest rate smoothing.
18
Flexible Core Inflation Targeting: The central bank cares about interest rate smoothing
and in addition to stabilizing sticky price inflation also tries to stabilize output by
assigning a weight to the output gap (deviation of output from trend).
)/log()/log()/log()/log(
__
,
_
1
_
YYRRRR ty
s
tstit
!!!
"
+##+= $
(37)
Flexible Headline Inflation Targeting: The central bank cares about interest rate
smoothing and in addition to stabilizing headline inflation also tries to stabilize output.
)/log()/log()/log()/log(
___
1
_
YYRRRR tyttit
!!!
"
+##+= $
(38)
3.7 Exogenous Shock Process
We assume that the productivity in the flexible price sector and sticky price sector follow
AR(1) processes
ttfaftf
AA
!"
+=
+,1,
,
t
!
~ i.i.d. (0,
af
!
) (39)
ttsasts AA
!"
+=
+,1,
,
t
!
~ i.i.d. (0,
as
!
) (40)
In the literature, exclusion of food prices from the price index has been justified on the
ground that shocks to food (and energy) prices represent supply shocks. In order to
compare our model with those in the prior literature and also to highlight the role of
adverse supply shocks on the choice of price index, we focus on productivity shocks.
3.8 Competitive Equilibrium
A stationary competitive equilibrium is a set of processes
tttttstf,ttstf
s
t
f
mcxRyyyxxCC
t
, , , , , , , , , , ,
,ts,,,
!!
for t = 0,1,… that remain bounded in
some neighborhood around the deterministic steady state and satisfy equations (52) – (62)
of Appendix I, given the exogenous stochastic processes
tf
A,
,
ts
A,
and the monetary
policy rule given by equation (34).
19
3.9 Complete Markets Specification
We follow the setting of Aoki (2001) to study the choice of price index under complete
markets. In this setting all households can insure one another against idiosyncratic
income risks completely. It implies that given same initial wealth each household will
choose an identical consumption sequence.23 Thus, under this complete markets setting
!!
+
=
+
== 11
tt
s
t
f
t
YC
CC
(41)
and aggregate demand is given by
!
!
"
#
$
$
%
&
'
(
)
*
+
,
-
+
=
(
)
*
+
,
-
+
.
+
.
t
tt
t
tRY
E
Y
//
0
1
0
11
1
(42)
Equations (53), (55)-(61) of Appendix I and (41)-(42) define the system of equations that
combined with the monetary policy rule and exogenous stochastic processes
tf
A,
and
ts
A,
determine the equilibrium path of the economy in the complete markets setting.
3.10 Welfare Evaluations
We are interested in the choice of policy rule that yields the highest level of lifetime utility
within the class of policy rules considered.24 In particular, we evaluate policy rules
according to the value of lifetime utility:
),(
0
i
jt
i
jt
j
j
t
i
tNCUEV ++
!
=
"
#
$
for i = f,s (43)
We compute the total welfare of the economy as a weighted sum of households’ welfare
23 Insurance contracts are assumed to be written before households know which sector they are
assigned to. The insurance contracts make the marginal utility of nominal income identical across
the households at any time t.
24 We study the policy rule which is implementable and optimal as defined by Schmitt-Grohe and
Uribe (2007). Implementability refers to the local uniqueness of rational expectations equilibrium
while optimality means that it yields the highest lifetime utility within the class of policy rules
considered.
20
s
t
f
ttotal VVV += *
!
. Formally, we compute
total
V
associated with each policy rule and look
for a policy rule that yields the highest value of
total
V
.
3.11 Solution Method
Following Kydland and Prescott (1982) and King, Plosser and Rebelo (1988), it has
become commonplace to characterize the solution of nonlinear models using
approximation methods, with first-order approximation techniques being the norm.
However, it is now widely accepted that first-order approximation techniques are ill-
suited for the comparison of different policy environments using aggregate utility as a
welfare criterion.25 To enable accurate welfare comparisons across alternative policy
environments, we need at least a second-order approximation of the equilibrium welfare
function (Schmitt-Grohe and Uribe, 2004; Woodford, 2003).26
In recent years, scholars have come up with various methods to produce second-order
accurate approximation to the solutions of DSGE models. Jin and Judd (2002), Collard
and Juillard (2000) and Schmitt-Grohe and Uribe (2004) have used the perturbation
method for second-order and higher-order approximations. Kim and Kim (2003) and
Sutherland (2002) have developed the bias correction method that produces similar
results as the second order perturbation method.
We compute the second-order accurate consumer welfare measure with different
monetary policy regimes as in Schmitt-Grohe and Uribe (2004). To produce an accurate
second-order approximation of the welfare function, we use a second-order
approximation to the policy function. The policy function is approximated using the
perturbation method by employing a scale parameter for the standard deviations of the
exogenous shocks as an argument of the policy function and taking a second-order Taylor
expansion with respect to the state variables as well as the scale parameter. We use an
25 Up to a first-order approximation, lifetime utility, Vt, is equal to its non-stochastic steady state
value. Hence, given the same non-stochastic steady state, all policy rules yield the same amount
of welfare up to a first-order approximation (Schmitt-Grohe and Uribe, 2007).
26See Kim and Kim (2003). However, if one is sure that nonlinearity is small in certain
dimensions one can justify using a first-order approximation by making specific assumptions,
Woodford (2003).
21
approximation algorithm developed by Schmitt-Grohe and Uribe (2004) with suitable
modifications.
3.12 Measuring Welfare Gains
Strict core inflation targeting is regarded as the welfare maximizing policy rule in the
literature. Therefore, we evaluate the welfare gains associated with a particular policy
regime by comparing it to the strict core inflation targeting rule allocation. Let the strict
core inflation targeting rule allocation be denoted by r, and an alternative policy regime
be denoted by a. We define the welfare associated with the core allocation conditional on
the economy being at its non-stochastic steady state at time zero:
),(
0
00
r
t
r
t
t
tr NCUEV
!
"
=
=
#
(44)
where
r
t
C
and
r
t
N
are the consumption and hours of work under the strict core inflation
targeting policy rule. Similarly, the conditional welfare under the alternative regime a is
defined as
),(
0
00
a
t
a
t
t
ta
NCUEV
!
"
=
=
#
(45)
The use of the conditional rather than unconditional expectation is consistent with the