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Working Paper Series
The Optimal Rate of Inflation with
Trending Relative Prices
WP 09-02
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/economic_
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Alexander L. Wolman
Federal Reserve Bank of Richmond
The Optimal Rate of Inflation
with Trending Relative Prices∗
Alexander L. Wolman†
March 3, 2009
Working Paper No. 09-02
Abstract
The relative prices of different categories of consumption goods have
been trending over time. Assuming they are exogenous with respect to
monetary policy, these trends imply that monetary policy cannot stabilize
the prices of all consumption categories. If prices are sticky, monetary policy
then must trade off relative price distortions within different categories
of consumption. Optimally, more weight should be placed on stabilizing
goods and services prices that are less flexible. Calibrating a simple stickyprice model to U.S. data, we find that slight deflation is optimal, even
absent transactions frictions leading to a demand for money. Optimality of
deflation derives from the fact that relative prices have been trending up
for services, whose nominal prices seem to be less flexible.
JEL Classification: E31, E52, E58
Keywords: relative price trends; sticky prices; optimal rate of inflation
∗
The author thanks the late Palle Andersen, Mike Dotsey, Craig Burnside, Fabio Ghironi,
Kevin Huang, Andy Levin and seminar participants at the Bank of Canada, the B.I.S. and the
Federal Reserve Board and conference participants at the Federal Reserve System Committee on
Macroeconomics and the CIREQ Conference on Multi-Sector Models for helpful comments. Fan
Ding and Jon Petersen provided excellent Research Assistance. This paper does not necessarily
represent the views of the Federal Reserve System or the Federal Reserve Bank of Richmond.
†
Federal Reserve Bank of Richmond, alexander.wolman@rich.frb.org.
1. Introduction
Analyses of the optimal rate of inflation typically use one-sector models. The
two nonneutralities of money that are most frequently considered in these analyses involve transactions frictions which can be alleviated by holding money, and
price stickiness. Transactions frictions lead almost inevitably to optimal deflation
— the Friedman rule — whereas price stickiness typically leads to optimality of
approximately stable prices.1 Among policymakers and to some extent among
researchers, a consensus has been reached that the transactions costs associated
with approximately stable prices are low. Together with the benefits of stable
prices with respect to price stickiness, this has led to a corresponding consensus
that over the long run, approximately stable prices are the right objective for
monetary policy to target.
One-sector models are the obvious starting point in macroeconomics, and for
some issues they may be sufficient. Whether they are sufficient for determining the
optimal rate of inflation is the question considered here. The inflation rate is an
aggregate of the rates of price change for many goods and services. In turn, there is
heterogeneity in those individual rates of price change. This heterogeneity means
that stabilization of the price level cannot achieve stabilization of all individual
prices. But stabilization of individual prices is what is optimal in sticky price
models. To the extent that all individual prices cannot be stabilized because of
heterogeneity that is exogenous with respect to monetary policy, it is not obvious
that stabilization of the price level will be optimal. This issue has been studied
in the context of cyclical fluctuations, by Huang and Liu [2005], Benigno [2004],
Erceg and Levin [2006], and Aoki [2001].2 Our concern here is the average inflation
rate. Heterogeneity of price changes across sectors — or categories of consumption
— is a trend phenomenon as well as a cyclical one. As such, the monetary authority
cannot achieve zero average rates of price change for all consumption goods. If
prices are sticky then it is infeasible to eliminate distortions associated with price
dispersion, even in the absence of shocks.3
In a simple model with two consumption goods, I study the determinants of
1
When we refer to price stickiness, we refer to some inflexibility in the level of nominal prices,
in contrast to the assumption of indexation. With indexation the steady state inflation rate is
irrelevant.
2
Carlstrom, Fuerst and Ghironi [2006] study the conditions under which monetary policy
rules generate determinacy in a two-sector model.
3
Shirota [2007] performs a similar analysis in a model with an input-output structure, and
Wolman [forthcoming] provides a survey on monetary policy with relative price variability.
2
the optimal rate of inflation when there is an exogenous trend in the relative price
of the two goods. That trend results from trends in relative productivities for
producing the two goods. I find that the principle determinant of the optimal
inflation rate is interaction between the relative price trend and differential price
stickiness across sectors. It is optimal to require less price adjustment of goods
whose prices adjust less frequently. The expenditure-share weighted inflation rate
is then disproportionately influenced by the relative price changes of those goods
with more flexible prices.
Using personal consumption price and quantity data for the United States, I
calibrate the technology processes and compute the optimal inflation rate. Optimality implies deflation at around four-tenths of a percent per year. Deflation
is optimal because I assume that prices adjust less frequently in the sector with
an increasing relative price. This assumption is consistent with the description of
U.S. data in Bils and Klenow [2004]: prices of services adjust less frequently than
prices of goods, and the relative price of services is rising over time. Imposing less
price adjustment on services means that goods prices must fall at a greater rate
than services prices rise, implying deflation overall.
The theme of this paper is related to Bosworth’s [1980] description of a dilemma
for monetary policy in the 1970s. According to Bosworth, positive relative price
shocks for flexible price goods (commodities) were accommodated, letting the
flexible prices rise in nominal terms while allowing the sticky-price goods to keep
prices fixed. This was appropriate in that it minimized the distortions associated
with price stickiness. Unfortunately it resulted in a higher inflation rate, and the
lack of monetary policy credibility meant that expected inflation rose along with
the actual inflation rate.4 The 1970s involved a level-change rather than a trend
in relative prices, but the policy response resulted in an inappropriate persistent
increase in inflation. That episode suggests an important caveat to the technical analysis in this paper: while there may be sound theoretical arguments for
deviating from zero inflation, in practice those deviations may have unintended
consequences. Thus, policymakers should require a high threshold of evidence
before purposefully deviating from near-zero inflation.
In the next section, I briefly describe U.S. data on price changes by consumption sector, as motivation for the modeling that follows. Section 3 lays out a
flexible price model in order to provide basic intuition for how I generate the
trending relative price and what some of its implications are. Section 4 describes
4
The discussion of credibility and expected inflation is not in Bosworth [1980], but seems like
a natural updating of Bosworth’s interpretation given the work of the last twenty-five years.
3
the sticky price model, which is then used in section 5 to compute the optimal rate
of inflation. Section 6 concludes and speculates on related topics and extensions
to the framework used here.
2. Inflation and Sectoral Price Changes
The empirical inflation rate I focus on is that associated with the price index for
personal consumption expenditure in the United States. Henceforth I will refer to
this index as the PCE price index, and to its rate of change as PCE inflation. PCE
inflation data are constructed by the Commerce Department’s Bureau of Economic
Analysis from underlying price and quantity data for a large number of categories
of goods and services. In turn, the price data for those underlying categories
are constructed from more direct observation of prices on an even larger number
of specific items (i.e., goods and services). The latter construction is performed
mainly by the Bureau of Labor Statistics in the Department of Labor. For the
most part, the same item prices that form the basis for PCE inflation also form the
basis for the more widely known CPI inflation, which is produced by the Bureau
of Labor Statistics. I focus here on PCE inflation because the methodology used
to produce the PCE inflation numbers corresponds more closely to notions of price
indices suggested by economic theory.
Figure 1 plots the price indices for the three first-level components of personal consumption expenditure (durable goods, nondurable goods and services),
together with the overall PCE price index. Each component differs somewhat
from overall inflation. Services prices have risen much more than the overall index, averaging 4.2 percent compared to 3.42 percent for overall inflation. Durables
price changes have generally been below PCE inflation, averaging 1.4 percent. The
main distinguishing feature of nondurables price changes — which have averaged
3.2 percent — is that they have been more volatile than PCE inflation. Figure 1
shows that the differences in rates of price change across sectors have cumulated
significantly over time: the price index for services has risen by a factor of more
than 12 since 1947, whereas the price index for durables has risen by less than a
factor of three.
Figure 2 plots expenditure shares for durable goods, nondurable goods, and
services from 1947 to the present. Whereas the expenditure share for durable
goods has fluctuated narrowly, between 10 and 18 percent, the shares of nondurables and services have respectively risen and fallen dramatically. In the first
quarter of 1947 services accounted for only 31%, and nondurable goods accounted
4
for 56% of personal consumption expenditure. In the first quarter of 2008 services
accounted for 60% and nondurable goods for only 29% of personal consumption
expenditure. The expenditure shift is striking, but it is not something I address
in this paper.5 Instead, I focus on the fact that in the last ten to fifteen years the
expenditure shares appear to have stabilized, at around 11 percent for durable
goods, 29 percent for nondurable goods, and 60 percent for services. It is this
“stationary” period that generates the facts used to calibrate the model below.
Note that although expenditure shares have stabilized, the trend in relative prices
has not disappeared.
A maintained assumption in this paper is that monetary policy has perfect
ability to control the inflation rate and no ability to affect relative prices across
sectors. These assumptions are extreme and probably incorrect, but I believe they
are useful for thinking about the optimal average inflation rate in an environment
where there are trends in relative prices. Trends in relative prices are apparent
in Figure 1, and Figure 3 illustrates them from a different perspective. Figure 3
displays “zero-inflation” price indexes for each category of consumption relative to
the overall price index. That is, the lines in Figure 3 represent the component price
indexes that would have yielded a zero overall inflation rate each quarter, assuming
that relative prices and expenditure shares followed their historical paths. This
figure shows that a zero inflation policy since 1947 would have implied trend
decreases in both durable and nondurable goods prices, and a trend increase
in services prices. In one-sector sticky price models, zero inflation eliminates
the relative price distortion on average. With multiple sectors and relative price
trends, zero inflation can be consistent with large relative price distortions.
3. A Competitive, Flexible Price Model
In this section I present a model with two sectors, each producing a distinct
final consumption good (or view them as a good and a service). Production of
the two goods is potentially subject to different rates of technological progress.
One response to the data in Figure 2 might be that one should not be looking
for an equilibrium with constant expenditure shares. Over the last ten years,
however, it appears that expenditure shares have stabilized even as relative prices
have trended more sharply. Just as importantly, analyzing steady states is a
5
Ding and Wolman [2005] investigate whether shifting expenditure shares can help explain
the time series properties of inflation. Greenwood and Uysal [2004] propose a model in which
expenditure shares shift with the introduction of new goods.
5
much more straightforward exercise than analyzing equilibria in which a sector
disappears asymptotically.
For optimal policy purposes I want to study a sticky-price version of the model.
However, the steady state relationships are messy enough that it will be useful
to work through the flexible price model first. In the flexible price version the
monetary policy problem that motivates us is absent. The model consists of an
infinitely lived representative household and two representative competitive firms.
Households are endowed with labor and have preferences over the goods produced
by the two firms. Each firm produces goods using a technology that is linear in
labor. The firms hire labor from households in a competitive economywide labor
market.
3.1. Households
The utility function of the representative household over a consumption index ct
and leisure lt is
∞
X
{
β j u(ct+j , lt+j )},
(3.1)
j=0
with
u(ct , lt ) = log(ct ) + χlt .
(3.2)
Labor supplied by households is nt = 1 − lt . The consumption-leisure trade-off
yields
wt = χct ,
(3.3)
where wt is the real wage. The consumption index is a Cobb-Douglas aggregate
of cg,t and cs,t , which are consumption of the two goods (“good” and “service”):
(1−ν)
ct = cνg,t cs,t
, ν ∈ (0, 1) .
(3.4)
Optimal choices of the two goods, which have prices Pg,t and Ps,t , solve the following problem:
i
h
(1−ν)
max cνg,t cs,t
s.t. Pg,t cg,t + Ps,t cs,t = E,
where E is nominal expenditure, taken as given for the moment. The implied
demands are
6
cg,t = ν (Pg,t /Pt )−1 ct
(3.5)
cs,t = (1 − ν) (Ps,t /Pt )−1 ct .
(3.6)
and
The price level for the consumption composite is
¶ν µ
¶1−ν
µ
Ps,t
Pg,t
Pt =
.
ν
1−ν
(3.7)
Money demand is given exogenously by
Mt = Pg,t cg,t + Ps,t cs,t ,
which implies
Mt = Pt ct .
Interaction of money demand with sectoral heterogeneity is an interesting topic,
but I do not pursue it here.
3.2. Firms
Firms produce output according to a linear technology, and there are sectorspecific productivities. So, for the two types of firms, the production functions
are
cg,t = zg,t ng,t and
cs,t = zs,t ns,t ,
(3.8)
(3.9)
where zg,t and zn,t are the productivity levels, ng,t and ns,t are labor supplied to
the two sectors, and nt = ng,t + ns,t . With competitive labor and product markets,
the common wage equals the marginal product of labor in each sector,
wt = zg,t Pg,t /Pt
wt = zs,t Ps,t /Pt ,
(3.10)
(3.11)
and thus relative technologies pin down the relative price of one good in terms of
the other:
Pg,t
zs,t
=
.
(3.12)
Ps,t
zg,t
7
3.3. A Steady State (possibly with inflation)
In a steady state there is constant inflation, constant real growth of the aggregate
c, and a constant trend in the price ratio of the two goods. In order for there to
be a steady state equilibrium, growth rates of the exogenous money supply and
of the exogenous technology parameters must be constant.
I use the following notation for growth rates, normalizing all exogenous variables to equal 1 when t = 0:
Mt = (1 + μ)t
¡
¢t
zg,t = 1 + ζ g
(3.13)
t
zs,t = (1 + ζ s ) .
Expenditure shares of the two goods are constant:
Pg cg
= ν
Pc
Ps cs
= 1 − ν.
Pc
One can easily determine the values of all other variables in steady state.
Combining the labor supply equation (3.3) with (3.10), then using (3.12) and
(3.7), we have
¡
¢¡
¢νt
ct = χ−1 ν ν (1 − ν)1−ν 1 + ζ g (1 + ζ s )(1−ν)t .
The real wage is then
¡
¢¡
¢νt
wt = ν ν (1 − ν)1−ν 1 + ζ g (1 + ζ s )(1−ν)t .
(3.14)
From (3.10) and (3.11), prices of the two goods relative to the overall price index
are
¶(1−ν)t
µ
¡ ν
1 + ζs
1−ν ¢
.
Pg,t /Pt = ν (1 − ν)
1 + ζg
and
¶νt
µ
¡ ν
1−ν ¢ 1 + ζ g
.
Ps,t /Pt = ν (1 − ν)
1 + ζs
Finally, using (3.5) and (3.6) output of the two goods is given by
8
and
¡
¢t
cg,t = (ν/χ) 1 + ζ g ,
cs,t =
µ
1−ν
χ
¶
(1 + ζ s )t .
(3.15)
(3.16)
As for nominal variables, the price level Pt grows at the rate of money growth
divided by the rate of consumption growth:
1+μ
¢ν
1+π = ¡
,
(3.17)
1 + ζ g (1 + ζ s )1−ν
¡
¢
the sector g price Pg,t grows at rate (1 + μ) / 1 + ζ g , and the sector s price grows
at rate (1 + μ) / (1 + ζ s ) . The growth rate of Pt is the model’s true inflation rate—
that is, the rate of change of the price of one unit of c. United States PCE
inflation is well approximated by an expenditure-share weighted average of the
price changes for different categories of goods and services. We can easily compute
the model’s equivalent of PCE inflation in the same manner:
¸
∙
ν
1−ν
1 + π pce = (1 + μ)
.
(3.18)
+
1 + ζg 1 + ζs
Because it is the standard empirical measure, I will emphasize the PCE inflation
rate in the numerical results below, though I will also report the true inflation rate.
As long as the productivity differential across sectors is not huge, PCE inflation
will be a good approximation to the true inflation rate.
4. The Model with Sticky Prices
In the flexible price model presented above, the strongest form of monetary neutrality holds. It does not “matter” that prices of different goods change at different
rates in a steady state. Now suppose that each sector is made up of a large number of monopolistically competitive firms, and that there is some sort of staggered
price setting in each sector. In this case, different nominal rates of price change
across sectors mean that it is infeasible for the monetary authority to stabilize all
nominal prices. This type of problem is familiar from the work of Erceg, Henderson and Levin [2000] (for sticky nominal wages and prices), Erceg and Levin
[2006] Huang and Liu [2005], and Aoki [2001]. However, each of those papers
9
is concerned with short run fluctuations in relative prices.6 If prices are indeed
sticky in nominal terms, so that there is a relative price distortion associated with
non-zero steady state inflation, then the optimal steady state rate of inflation represents a nontrivial problem for the monetary authority. In this section I describe
a sticky-price version of the model presented above. I reinterpret cg,t and cs,t as
aggregates of a continuum of monopolistically produced goods whose prices are
less than perfectly flexible.
4.1. Households
Up until the definition of the good and service, the description of household behavior is unchanged. Now however, the good and service are composites made up of
Dixit-Stiglitz aggregates of a continuum of differentiated products, with elasticity
of substitution ε:
∙Z 1
¸ε/(ε−1)
(ε−1)/ε
ck,t =
ck,t (z)
dz
, for k = g, s,
0
which leads to the demands
ck,t (z) =
µ
Pk,t (z)
Pk,t
¶−ε
ck,t , for k = g, s.
Combining the two levels of demands (4.1) and (3.5) yields
¶−ε µ
¶−1
µ
Pg,t
Pg,t (z)
cg,t (z) = ν
ct
Pg,t
Pt
and
µ
Ps,t (z)
cs,t (z) = (1 − ν)
Ps,t
¶−ε µ
Ps,t
Pt
The price level for the consumption aggregate is
µ
¶ν µ
¶1−ν
Ps,t
Pg,t
Pt =
,
ν
1−ν
6
¶−1
ct .
(4.1)
(4.2)
(4.3)
(4.4)
Loyo [2002] points out that if there were multiple units of account, nominal item prices could
remain constant while the relative price of the units of account did the adjusting, along the lines
of Friedman’s case for flexible exchange rates. In the same way, multiple units of account could
eliminate the problem with trending relative prices addressed here.
10
and the price indexes for the composite good and service are
∙Z 1
¸1/(1−ε)
1−ε
Pk,t =
[Pk,t (z)] dz
, for k = g, s.
0
Substituting into the price index for the consumption aggregate, we have
⎛ hR
i1/(1−ε) ⎞ν ⎛ hR
i1/(1−ε) ⎞1−ν
1
1
1−ε
1−ε
⎜ 0 [Pg,t (z)] dz
⎟ ⎜ 0 [Ps,t (z)] dz
⎟
Pt = ⎝
⎠ ⎝
⎠ .
ν
1−ν
Now I incorporate price stickiness, in a form which is a generalization of Calvo
and Taylor style time-dependent pricing.7 A firm in sector k will adjust its price
with (exogenous) probability αk,j , where j indexes the number of periods since
the last price adjustment. The set of price adjustment probabilities αk,j , j =
1, ..., Jk − 1 are collected in the vector α , for k = g, s, where Jk is the maximum
˜k
number of periods that a firm in sector k will remain with the same price. From
the vector α one can derive the fractions of firms in period t charging prices set
˜k
in periods t − j, which will be denoted by ω k,j . To do this, note that
ω k,j = (1 − αk,j ) ωk,j−1 , for j = 1, 2, ..., Jk − 1,
and
JX
k −1
ω k,j .
ω k,0 = 1 −
(4.5)
j=1
This system of linear equations can be solved for ωk,j as a function of αk,j . With
this specification of price stickiness, and the further assumptions that (1) there
is a stationary distribution of firms in terms of the age of their price, and (2) all
firms of the same vintage charge the same price, rewrite the price index as follows:
⎛h
i1/(1−ε) ⎞ν ⎛ hP
i1/(1−ε) ⎞1−ν
PJg −1
Js −1
1−ε
1−ε
j=0 ω g,j Pg,j,t
j=0 ω s,j Ps,j,t
⎜
⎟ ⎜
⎟
Pt = ⎝
⎠ ⎝
⎠
ν
1−ν
where Pk,j,t denotes the price charged in period t by a firm in sector k that last
adjusted its price in period t − j. Note that Pk,j,t = Pk,0,t−j .
7
See Wolman [1999] for more details.
11
Assume as before that households also hold money equal to nominal expenditure,
Z
Z
1
Mt =
1
Pg,t (z)cg,t (z) dz +
0
Ps,t (z)cs,t (z) dz.
(4.6)
0
With constant-elasticity demands for each good, the money-demand specification
in (4.6) implies
(4.7)
Mt = Pt ct .
Let λt be the Lagrange multiplier which represents the shadow value of wealth,
λt =
1
.
ct
(4.8)
Households equate the marginal rate of substitution between leisure and consumption to the competitive real wage rate:
(4.9)
wt = χct .
4.2. Firms
Firms produce output according to a linear technology, and there are sectorspecific productivities. So, for each type of firm, the production function is
ck,j,t = zk,t nk,j,t , for k = g, s.
(4.10)
This implies that real marginal cost is unrelated to the scale of the firm and is
simply
ψk,t = wt /zk,t , for k = g, s,
and that nominal marginal cost is
Ψk,t = Pt wt /zk,t , for k = g, s.
Firms that adjust their prices in period t set prices so as to maximize the
expected present discounted value of their profits, using the household’s marginal
utility as a discount factor. That is, they choose Pk,0,t to maximize their market
value,
JX
k −1
j=0
j
β ωj,k λt+j
µ
Pk,0,t − Ψk,t+j
Pt+j
12
¶
ck,j,t+j .
As monopolistic competitors, firms face demands given by (4.2) and (4.3). Thus
the price-setting problem for a firm in sector j is to maximize
JX
k −1
j
β ω j,k λt+j
j=0
µ
Pk,0,t − Ψk,t+j
Pt+j
¶
ν
k
µ
Pk,0,t
Pk,t+j
¶−ε µ
Pk,t+j
Pt+j
¶−1
ct+j
(note the abuse of notation now: ν k is either ν or (1 − ν)). The first-order condition for this problem can be manipulated to isolate the relative price chosen by
the adjusting firm:
³
´ ³
´−ε ³
´−1
PJk −1 j
Ψk,t+j
Pk,t
Pk,t+j
k
ε
β
ω
λ
ct+j
ν
j,k
t+j
j=0
Pt+j
Pk,t+j
Pt+j
Pk,0,t
=
.
³
´
³
´
−ε
−1
PJk −1 j
Pk,t
Pk,t+j
Pt
Pt
k
(ε − 1) j=0 β ω j,k λt+j Pt+j ν Pk,t+j
ct+j
Pt+j
Now replace nominal marginal cost with real marginal cost multiplied by the price
level and use the fact that λt+j ct+j = 1:
Pk,0,t
=
Pt
ε
PJk −1
j=0
(ε − 1)
j
β ωj,k ψk,t+j ν
PJk −1
j=0
k
³
t
β j ω j,k PPt+j
νk
Pk,t
Pk,t+j
³
´−ε ³
Pk,t
Pk,t+j
Pk,t+j
Pt+j
´−ε ³
´−1
Pk,t+j
Pt+j
´−1 .
In order to make progress toward a steady state, use the facts that ψk,t = wt /zk,t
and wt = χct . Together these imply
ψk,t = χct /zk,t
and thus
ε
Pk,0,t
=
Pt
PJk −1
j
³
Pk,t
Pk,t+j
´−ε ³
Pk,t+j
Pt+j
´−1
β ωj,k χ (ct+j /zk,t+j )
³
´−ε ³
´−1 , for k = g, s. (4.11)
PJk −1 j
Pk,t
Pk,t+j
t
(ε − 1) j=0
β ω j,k PPt+j
Pk,t+j
Pt+j
j=0
4.3. Inflationary Steady State
The next step is to derive a steady state, in order to compute welfare for a range
of money-growth rates (and hence inflation rates). I use the same notation for
the exogenous processes as described in (3.13). The principal equations used to
13
derive a steady state are the two optimal pricing equations (4.11), the two price
indices,
)1/(1−ε)
(J −1
k
X
1−ε
ω k,j Pk,j,t
, for k = g, s;
(4.12)
Pk,t =
j=0
and the overall price index (4.4). In this section I use these equations to derive
the model’s steady state. The first two subsections below manipulate the optimal pricing equations and the price indices, respectively. The third subsection
completes the derivations.
4.3.1. Optimal Pricing
Define the rates of price change in steady state as follows:
(4.13)
1 + π ≡ Pt /Pt−1 ,
1 + π g ≡ Pg,0,t /Pg,0,t−1 , and
1 + π s ≡ Ps,0,t /Ps,0,t−1 .
Note that together with the sectoral price index equations, these definitions imply
Pg,t /Pg,t−1 = 1 + π g , and
Ps,t /Ps,t−1 = 1 + π s .
(4.14)
That is, if a firm adjusting its price in t + 1 sets a price that is a multiple 1 + π g
of the price set by a firm that adjusted in t, then the sectoral price index also
increases at rate 1 + π g .
Then rewrite the optimal pricing equations, using these definitions, using the
assumptions about technology shifters in (3.13), and noting that the growth rate
of consumption is the product of the discount factor and the real interest rate:
ct /ct−1 = β (1 + r) .
The optimal pricing equations become
µ
εχc0
β (1 + r)
Pk,0,t
=
·
Pt
(ε − 1)
1 + ζk
¶t PJk −1 β j ω
j=0
j,k
³
PJk −1
j=0
14
β(1+r)
1+ζ k
´j
jε
(1 + π k )
³
β j ω j,k (1 + π k )j(ε−1)
1+π
1+π k
´j
, for k = g, s;
(4.15)
4.3.2. From the price indices
Using the shorthand for rates of price change in (4.14), the industry price indices
can be rewritten as
)1/(1−ε)
(J −1
k
X
Pk,t
j(ε−1)
=
ω k,j (1 + π k )
, for k = g, s.
(4.16)
Pk,0,t
j=0
Substituting these expressions into the overall price index yields a relationship
between relative prices in the two sectors:
⎛
i1/(1−ε) ⎞ν
h
µ
¶ PJg −1 ω (1 + π )−j(1−ε)
g,j
g
j=0
⎜ Pg,0,t
⎟
1 = ⎝
(4.17)
⎠
Pt
ν
⎛
i1/(1−ε) ⎞1−ν
h
µ
¶ PJs −1 ω (1 + π )−j(1−ε)
s,j
s
j=0
⎜ Ps,0,t
⎟
⎝
⎠ .
Pt
1−ν
Time does not appear on the LHS of this expression, thus, it must ³
be that
´ν time
Pg,0,t
does not appear on the RHS either. This implies that the trends in Pt
and
³
´1−ν
Ps,0,t
are offsetting, and from (4.15) those trends satisfy
Pt
µ
Pg,0,t
Pt
¶ν µ
Ps,0,t
Pt
¶1−ν
³
´t
¡
¢−ν
ν−1
∝ β (1 + r) 1 + ζ g
(1 + ζ s )
.
Imposing lack of a time trend on
³
Pg,0,t
Pt
´ν ³
Ps,0,t
Pt
´1−ν
(4.18)
, the real interest rate is
¡
¢ν
(1 + r) = β −1 1 + ζ g (1 + ζ s )1−ν .
(4.19)
4.3.3. Completing the Steady State Derivation
Recalling that the money growth rate is 1 + μ, from the money demand equation
we have
µ
¶
1+μ
−1
(1 + r) = β
.
(4.20)
1+π
15
Thus from (4.19) and (4.20) we have the inflation rate:
(1 + μ)
¢ν
.
1+π = ¡
1 + ζ g (1 + ζ s )1−ν
(4.21)
It remains to compute the sectoral rates of price change. From (4.15) we have
¶
µ
Pk,0,t
β (1 + r)
= (1 + π)
Pk,0,t−1
1 + ζk
which implies
¶
µ
Pg,0,t
1+μ
= 1 + πg =
Pg,0,t−1
1 + ζg
¶
µ
Ps,0,t
1+μ
.
= 1 + πs =
Ps,0,t−1
1 + ζs
(4.22)
(4.23)
Relative price levels also need to be computed. The rates of change of relative
prices are given by the last three numbered equations. The relative price levels in
period zero come from (4.15), as functions of c0 :
³
´jε
PJk −1 j
1+μ
β
ω
j,k
j=0
1+ζ k
Pk,0,0
εχc0
·
=
(4.24)
³
´j(ε−1) , for k = g, s.
P0
(ε − 1) PJk −1 j
1+μ
j=0 β ω j,k 1+ζ
k
There are now three equations ((4.24) for k = g, s and (4.17)) in the three
variables c0 , Pg,0,0
and Ps,0,0
. To solve for c0 , substitute (4.24) into (4.17) (also
P0
P0
eliminate 1 + π using our solution):
⎛
1 ⎞ν
"
# 1−ε
−j
³
´
J
−1
g
³
´
JP
−1
g
P
jε
1+μ 1−ε
1+μ
⎜
⎟
ω g,j 1+ζ
β j ωj,g 1+ζ
⎜ ε
⎟
g
g
j=0
j=0
⎜
⎟
−1
· J −1
c0 = χ ⎜
⎟(4.25)
³
´
g
j(ε−1)
⎜ (ε − 1) P j
⎟
ν
1+μ
⎝
⎠
β ω j,g 1+ζ
g
j=0
⎛
JP
s −1
³
´jε
1+μ
⎜
β j ω j,s 1+ζ
⎜ ε
s
j=0
⎜
· J −1
⎜
³
´
j(ε−1)
s
⎜ (ε − 1) P
1+μ
⎝
β j ω j,s 1+ζ
j=0
s
16
"
JP
s −1
j=0
ω s,j
³
1+μ
1+ζ s
1−ν
−j
´ 1−ε
1 ⎞1−ν
# 1−ε
⎟
⎟
⎟
⎟
⎟
⎠
.
The last step is to compute leisure:
lt = 1 − nt .
Labor input (nt ) is the weighted sum of labor input for every type of firm, where
the weights are the fractions of firms:
nt =
Jg −1
X
j=0
Js −1
cg,j,t X
cs,j,t
ω g,j
+
ω s,j
.
zg,t
zs,t
j=0
Using the demand functions (4.2) and (4.3),
" µ
¶−1 µ
¶−ε JX
g −1
Pg,0
Pg,0,0
ωg,j (1 + π g )jε +
n0 = c0 ν
P0
Pg,0
j=0
#
µ
¶−1 µ
¶−ε JX
s −1
Ps,0
Ps,0,0
(1 − ν)
ω s,j (1 + π s )jε .
P0
Ps,0
j=0
To compute n0 using this last equation, use the price ratios
(4.24) and get the other price ratios
Pk,0
P0
Pk,0,0
Pk,0
(4.26)
from (4.25) and
from
Pk,0,0 Pk,0,0
Pk,0
=
÷
,
P0
P0
Pk,0
(4.27)
where the first factor comes from (4.24) and the second factor comes from (4.16):
Pk,0,0
=
Pk,0
(J −1
k
X
j=0
ωk,j
µ
1+μ
1 + ζk
¶j(ε−1) )1/(ε−1)
, for k = g, s.
(4.28)
The steady state derivation is now complete in that for given parameter values
steady state welfare follows from the solutions for ct and nt .
5. The Optimal Rate of Inflation
The steady state money growth rate — and hence the steady state rate of inflation
— matter in this model for two reasons. First, with sticky prices, different average
inflation rates correspond to different levels of the monopoly markup in both
17
sectors. Second, and more importantly from the perspective of our motivation,
sticky prices mean that different average inflation rates correspond to different
degrees of relative price distortion within each sector. Here I describe the two sets
of distortions and show how they summarize the channels through which monetary
policy can affect steady state welfare.
5.1. Sectoral Markups
The sectoral markup (M) of price over marginal cost in sector k is given by
Pk,0
Pk,0
=
P0 w0
χP c
µ 0 0
¶
Pk,0 Pk,0,0
−1
.
= (χc0 )
Pk,0,0 P0
Mk =
(5.1)
In a flexible price version of the model, the sectoral markup would simply be
ε/ (ε − 1) . With sticky prices, however, from (4.25), (4.28) and (4.24), it is clear
that the sectoral markups depend on monetary policy (through the money growth
rate μ):
⎛
³
´jε ⎞
(J −1
)1/(1−ε) PJk −1 j
1+μ
µ
¶
j(ε−1)
k
j=0 β ω j,k 1+ζ k
1+μ
ε
⎜ X
⎟
Mk =
·⎝
ωk,j
⎠,
³
´
j(ε−1)
PJk −1 j
ε−1
1
+
ζ
1+μ
k
j=0
j=0 β ω j,k 1+ζ
k
for k = g, s.
(5.2)
With sticky prices, an individual firm would like to achieve the markup ε/ (ε − 1)
in every period (note that Mk is the aggregate sectoral markup). However, if there
is a trend in the price of the firm’s output, then infrequent price adjustment means
that the optimal markup cannot be achieved in every period. Firms optimally
set a high markup (i.e. greater than ε/ (ε − 1)) in the period they adjust, and
then watch it depreciate as long as their price is fixed. For an individual firm, the
markup when adjusting is increasing in the sectoral rate of price change. However,
a greater positive rate of price change also causes the markup to depreciate faster
for nonadjusting firms.8 For the Calvo model (where ω k,j declines geometrically
with j), it is straightforward to show that together the two effects generate a
markup that is convex in the rate of sectoral price change: Mk is minimized at a
8
For additional discussion, see King and Wolman [1996].
18
1+μ
small but positive value of sectoral price change (i.e. 1+ζ
> 1). For the general
g
time-dependent model, no such results are immediate. It is clear however, that if
the price stickiness and demand elasticity parameters are common across sectors,
but productivity growth is not, then the markup-minimizing money growth rates
will differ across sectors. Thus, it will not be possible to choose steady state
inflation in order to minimize the markup in both sectors.
5.2. Sectoral Relative Price Distortions
Relative price distortion refers to the fact that with sticky prices, if prices within
a sector change over time at a constant rate, then at any point in time there will
be more than one price charged by firms within the sector. Because firms charge
different prices, they will also produce in different quantities. This is inefficient
because of the symmetry in preferences and production within a sector: efficiency
dictates that firms within a sector should produce in the same quantity. Define
the relative price distortion within a sector as
Rk =
zk,t nk,t
.
ck.t
This has the interpretation of the ratio of (a), the amount of the sector k composite
that could be produced with the current labor input in sector k to (b), the amount
of the sector k composite that is produced by the current labor input in sector k.
The relative price distortion is influenced by monetary policy, because the
rate of price change in the sector determines the dispersion of prices within the
sector. Using the production functions and demand functions makes precise the
relationship between monetary policy and the relative price distortion:
PJk −1
j=0 ω k,j ck,j,t
Rk =
ck.t
¶−ε JX
µ
¶jε
µ
k −1
1+μ
Pk,0,0
=
ω k,j
,
(5.3)
Pk,0
1
+
ζ
k
j=0
P
is given by (4.28) for k = g, s. The relative price distortion is a
where Pk,0,0
k,0
convex function of (1 + μ) , minimized at μ = ζ k . When μ = ζ k , firms in sector
k never need to change their prices (π k = 0), so the stickiness of prices becomes
unimportant and the relative price distortion is eliminated. This corresponds to
zero inflation in a one-sector model. But in our two-sector model, if there are
19
different rates of productivity growth in the two sectors (ζ g 6= ζ s ) the relative
price distortion cannot be eliminated in both sectors.
5.3. Optimal Steady State Inflation
If the rate of productivity growth differs across sectors, then both the markup and
the relative price distortion require the monetary authority to trade off benefits
to one sector against costs to the other sector when choosing the inflation rate.
In fact, the level of steady state welfare can be expressed as a function of only the
sectoral relative price distortions and markups.
Steady state welfare in period zero is
vss
0
=
=
∞
X
j=0
∞
X
β j {ln(cj ) + χlj }
j
β (j ln ((1 + ζ s ) (1 + γ s ))) +
j=0
∞
X
j=0
β j {ln c0 + χ (1 − n0 )} .
The first term is unaffected by monetary policy, so focus on the second term:
ṽss
0
=
∞
X
j=0
β j {ln c0 + χ (1 − n0 )} .
From the expressions for c0 (4.25) and the sectoral markups (5.2), c0 can be written
as a function of the sectoral markups:
µ
¶−ν µ
¶ν−1
Mg
Ms
−1
c0 = χ
.
ν
1−ν
Also, from expressions for nt (4.26) and the sectoral relative price distortions
(5.3), as well as the sectoral markups (5.1), n0 can be written as a function of the
markups and relative price distortions:9
Pk,0
= χc0 Mk
P0
µ
¶
νRg (1 − ν) Rs
−1
.
n0 = χ
+
Mg
Ms
9
Recall that n0 = nt .
20
(5.4)
When one combines the last two equations, it becomes clear that monetary policy
affects steady state welfare only through its effects on the sectoral markups and
relative price distortions (the following expression omits the constant term 1/χ):
1
{−ν ln Mg (μ; .) − (1 − ν) ln Ms (μ; .)
ṽss
0 (μ; .) =
1−β
¶¾
µ
νRg (μ; .) (1 − ν) Rs (μ; .)
+
.
(5.5)
−
Mg (μ; .)
Ms (μ; .)
Recall that the productivity growth rates show up in the solutions for consumption, leisure and thus the distortions. Therefore, the optimal money growth rate
is sensitive to the productivity growth rates.
6. Numerical Results
I will present two types of results. First I will look at a benchmark calibration in
order to understand how the optimal inflation rate trades off the four distortions
described above. Then I will vary the productivity processes and the degree of
price stickiness to illustrate the sensitivity of the optimal inflation rate to those
parameters.
6.1. Calibration
I interpret the model as describing quarterly data. Most of the model’s parameters
are fixed a priori: β = 0.99, χ = 4.5, and ε = 10. The parameters representing
price stickiness are chosen somewhat arbitrarily, but with some input from the
results in Bils and Klenow, showing that goods prices change more frequently
than services prices: I set α = 0 and α = [0 0]0 , meaning that goods prices are
˜g
˜s
fixed for two quarters and services prices are fixed for three quarters.
The parameter ν is equal to the expenditure share of goods in total consumption. I set this parameter to 0.40, which was the 2008 expenditure share of goods
in consumption.
Calibration of the productivity trends is based on the data in Section 2, along
with corresponding data on real consumption expenditure. First I aggregate
durable goods and nondurable goods price changes using their expenditure shares
to get a rate of price change for goods. Over the period 1993 to 2008,
1. The price of goods relative to that of services ‘grew’ at a gross quarterly
rate of 0.9956 (that is, it fell 0.44 percent per quarter).
21
2. Real personal consumption expenditure per capita grew at a gross quarterly
rate of 1.0059 — about 0.6 percent per quarter.
I match up the relative price growth rate to the inverse of the relative productivity shifters’ growth:
1 + ζs
= 0.9956.
(6.1)
1 + ζg
I also match up real consumption growth in the data with the expenditure share
weighted average of real growth in the two sectors, which gives the equation
¢
£
¡
¢¤
¡
1.0059 = ν 1 + ζ g + (1 − ν) 0.9956 1 + ζ g .
Solving for ζ g given v = 0.40 yields
1 + ζ g = 1.00856.
The first equation (6.1) then determines ζ s .
6.2. Benchmark Results
The calibration (and the U.S. data) involves a trend in the relative price of goods
to services, and that trend is exogenous with respect to monetary policy. It
is therefore impossible for the monetary authority to replicate the flexible price
allocation by appropriate choice of the money growth rate. The relative price of cg
to cs needs to change on average, so nominal prices within sector g or sector s (or
both) must change on average. There will then be real effects of price stickiness.
Figure 4 plots welfare, the sectoral relative price distortions and markups, and
the sectoral rates of price change as a function of the PCE inflation rate. In
the background, the steady-state money growth rate μ is being varied to induce
the changes in PCE inflation, according to (3.18). The optimal PCE inflation
rate is approximately −0.4 percent, implying small deflation. Why deflation?
The relative price of services will rise regardless of the inflation rate (see the
top right panel of the figures). At zero inflation, the nominal price of goods
will be falling at approximately the same rate as the nominal price of services
is rising. With symmetric sectors - other than productivity growth— this would
imply approximately equal relative price distortions in the two sectors.10 Recall
10
The relative price distortions in the two sectors would not be equal if the sectors were
symmetric because price decreases have different effects on the markup than price increases of
the same magnitude. We discuss this further in the next subsection.
22
however that cg prices are more flexible than cs prices. Relative flexibility in the
g sector means that a given rate of price change translates into a smaller relative
price distortion. It is optimal then for the g sector to bear more of the burden of
nominal adjustment, and since the relative prices in the g sector are falling, this
means that nominal prices in the g sector should fall more than nominal prices in
the s sector should rise. Panel C illustrates the fundamental trade-off facing the
monetary authority: that it cannot eliminate the relative price distortion in both
sectors. The relative stickiness of s prices is reflected in panels C and D in the
higher degree of curvature in the two distortions for the s sector compared to the
g sector.
6.3. The Relative Price Trend and Differential Price Stickiness
Figures 5 and 6 illustrate how the optimal steady state inflation rate varies with
the degree of price stickiness in sector s and the rate of productivity growth in
sector s. Specifically, for Figure 5, I fix the degree of price stickiness in sector g
as described above (two-quarter pricing) and vary the expected duration of prices
in sector s from one quarter (flexible prices) to six quarters. Panels A and B of
Figure 5 show that as price stickiness in sector s rises to even moderate levels,
the optimal inflation rate becomes negative and nearly stabilizes the price level in
the s sector, imposing the entire burden of relative price adjustment on nominal
prices in the g sector.
There are several other things to note about this figure. When prices are
flexible in sector s — the far left of each panel, there is overall inflation of one
percent (panel A), sector g prices are nearly constant (panel B), relative price
distortions are nearly eliminated (panel C) and the markup is nearly identical
across sectors at its flexible price level (panel D). In this case, it is optimal to
place the entire burden of price adjustment on the flexible price sector. I use
the qualifier “nearly” because, as explained in King and Wolman [1999], a very
small positive trend in prices allows the monetary authority to increase welfare
by decreasing the markup in sector g.
Another interesting feature of Figure 5 is that when price stickiness does not
vary across sectors (expected duration equals two), the optimal PCE inflation rate
is not zero but -0.4%. This is because of the higher expenditure share on services:
with the same degree of price stickiness in both sectors, it is optimal to impose
more of the burden of adjustment on the smaller sector. In this case, sector g
prices must fall more than sector s prices rise, as illustrated in panel B at the
23
point where expected duration equals two.
Panel C and D of Figure 5 shows non-monotonicity in the relative price distortion and markup for sector s. If money growth were held fixed while the duration
of prices increased, then these relationships would be monotonic. However, as
can be seen from panel A, inflation (and hence money growth) varies with the
expected duration, creating the possibility of nonmonotonicity.
For Figure 6, I fix the rate of productivity growth in the g sector at its calibrated value and vary the rate of productivity growth in the s sector. The focal
point in this figure is a growth rate of 0.00915; at that point productivity growth
is identical in the two sectors. Then there is no trend in relative prices (panel B),
and the optimal steady state involves approximately zero inflation (panel A).11
In Figure 6, the sector s relationships have less curvature than those for sector
g. This is because less price stickiness and a smaller expenditure share for sector
g makes it optimal for nominal price changes in that sector to bear most of the
burden of accommodating the different degrees of relative price change implied by
different relative productivity growth rates. Thus, in panel B the sectoral price
change relationship for sector g is steeper than that for sector s.
7. Conclusion
Given the relative price trend across sectors, the optimal rate of inflation in our
model is driven by heterogeneity in price stickiness. Relative degrees of price
stickiness are likely robust determinants of the optimal rate of inflation in the
presence of relative price trends. However, the analysis here raises the issue of
other ways in which sectoral heterogeneity might interact with monetary policy. I
use this concluding section to suggest a few possibilities for further research along
these lines.
I have taken as given the degree of price stickiness in the two sectors, and
held it fixed with respect to policy changes. Presumably the frequency of price
adjustment responds to the average rate of sectoral price change. It is difficult
to find such a relationship in the Bils and Klenow data, but it may be obscured
by other sectoral heterogeneity, say in the form of idiosyncratic productivity or
cost shocks. It would be interesting to include such firm-level heterogeneity and
11
Although it is feasible to eliminate the relative price distortions in both sectors when the
productivity growth rates are identical, it is optimal to have a bit of inflation in order to decrease
the markup. This is another version of the effect described by King and Wolman (1999).
24
extend the analysis to a state-dependent pricing framework; there the frequency
of price adjustment would naturally respond to the average rate of price change.
As mentioned above, money demand may interact with sectoral heterogeneity.
Here there are no transactions frictions motivating money demand. Of course
those frictions exist. Furthermore, they may differ systematically across sectors,
and they may be correlated with price rigidities. It is unlikely that such heterogeneity would by itself overturn the Friedman rule, but it might have other
interesting implications.
I analyzed a stylized model containing two types of consumption differing only
in the stickiness of prices within the sectors and in productivity growth. There are
other dimensions of heterogeneity which ought to be modeled, and have to some
extent been modeled by others: durability (e.g. Erceg and Levin [2006]), money
demand (just mentioned), intermediate input shares (Huang and Liu [2005]), the
extent of competition, etc.
Although expenditure shares appear to have stabilized, there was a long run
change in expenditure shares at least until the mid-1990s. Ideally the model could
help us to understand shifting expenditure shares over time. A paper along these
lines is Greenwood and Uysal [2004], although they do not consider monetary policy. A model with shifting expenditure shares might be an interesting laboratory
for studying changes in the composition of means of payment.
25
References
[1] Aoki, Kosuke, “Optimal Monetary Policy Response to Relative Price
Changes,” Journal of Monetary Economics 48 (2001), 55-80.
[2] Benigno, Pierpaolo, “Optimal Monetary Policy in a Currency Area,” Journal
of International Economics 63 (2004), 293-320.
[3] Bils, Mark and Pete Klenow, “Some Evidence on the Importance of Sticky
Prices,” Journal of Political Economy, October 2004.
[4] Bosworth, Barry P., “Economic Policy” in Joseph A. Pechman ed., Setting
National Priorities; Agenda for the 1980s. (1980) Washington DC: Brookings
Institution.
[5] Carlstrom, Charles T., Timothy S. Fuerst and Fabio Ghironi, “Does It Matter
(for Equilibrium Determinacy) What Price Index the Central Bank Targets?”
Journal of Economic Theory 128 (May 2006): 214-231.
[6] Ding, Fan and Alexander L. Wolman, “Inflation and Changing Expenditure
Shares,” Federal Reserve Bank of Richmond Economic Quarterly 91 (2005).
[7] Erceg, Christopher, Dale Henderson and Andrew Levin, “Optimal Monetary
Policy with Staggered Wage and Price Contracts,” Journal of Monetary Economics, vol. 46 (October 2000), pp. 281-313
[8] Erceg, Christopher and Andrew Levin, “Optimal Monetary Policy with
Durable Consumption Goods.” Journal of Monetary Economics 53 (October 2006), pp. 1341-1359
[9] Greenwood, Jeremy and Gokce Uysal, “New Goods and the Transition to a
New Economy,” NBER Working Paper 10793, September 2004.
[10] Huang, Kevin, and Zheng Liu, “Inflation Targeting: What Inflation Rate to
Target?” Journal of Monetary Economics 52(8), November 2005, pp. 14351462.
[11] King, Robert G. and Alexander L. Wolman, “Inflation Targeting in a St.
Louis Model of the 21st Century,” Federal Reserve Bank of St. Louis Review,
May/June, 1996
26
[12] King, Robert G. and Alexander L. Wolman, “What Should the Monetary
Authority Do When Prices are Sticky?” in John B. Taylor ed. Monetary
Policy Rules, 1999.
[13] Loyo, Eduardo, “Imaginary Money Against Sticky Relative Prices,” European
Economic Review 46 (2002), 1073-1092.
[14] Ngai, L. Rachel, and Christopher Pissarides, “Structural Change in a MultiSector Model of Growth,” manuscript, LSE.
[15] Shirota, Toyoichiro, “Optimal Trend Inflation and Monetary Policy under
Trending Relative Prices.” Bank of Japan Working Paper Series No. 07-E-1,
January 2007.
[16] Wolman, Alexander L., “Nominal Frictions, Relative Price Adjustment, and
the Limits to Monetary Policy.” Federal Reserve Bank of Richmond Economic
Quarterly, forthcoming.
[17] Wolman, Alexander L., “Sticky Prices, Marginal Cost and the Behavior of
Inflation,” Federal Reserve Bank of Richmond Economic Quarterly 85 (Fall
2000) , 29-48.
27
Figure 1.
Sectoral Prices Indices
14
12
10
services
8
overall pce
price index
6
nondurables
4
durables
2
0
1947 - 1950 - 1953 - 1956 - 1959 - 1962 - 1965 - 1968 - 1971 - 1974 - 1977 - 1980 - 1983 - 1986 - 1989 - 1992 - 1995 - 1998 - 2001 - 2004 - 2007 Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Figure 2.
Consumption expenditure shares
0.7
0.6
services
0.5
0.4
nondurables
0.3
0.2
durables
0.1
0
1947 - 1950 - 1953 - 1956 - 1959 - 1962 - 1965 - 1968 - 1971 - 1974 - 1977 - 1980 - 1983 - 1986 - 1989 - 1992 - 1995 - 1998 - 2001 - 2004 - 2007 Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Figure 3.
"Zero-inflation" Price Indices
1.6
1.4
services
1.2
1
nondurables
0.8
0.6
0.4
durables
0.2
0
1947 - 1950 - 1953 - 1956 - 1959 - 1962 - 1965 - 1968 - 1971 - 1974 - 1977 - 1980 - 1983 - 1986 - 1989 - 1992 - 1995 - 1998 - 2001 - 2004 - 2007 Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
Q1
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@FedFRASER