This note shows that full price indexation is not optimal in the long-run, in the New Keynesian model under trend inflation and price staggering à la Calvo. Moreover, we show that more price stickiness may increase steady state welfare, if price indexation is partial.Indexation, Trend Inflation, New Keynesian model

Guido Ascari

Nicola Branzoli

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 1 
  
The long-run optimal degree of indexation in new Keynesian models with 
price staggering à la Calvo 
  
 
 
Guido Ascari  
University of Pavia, Department of Economics 
Nicola Branzoli  
University of Wisconsin Madison
Abstract 
This note shows that full price indexation is not optimal in the long-run, in the New Keynesian model under trend 
inflation and price staggering à la Calvo. Moreover, we show that more price stickiness may increase steady state 
welfare, if price indexation is partial.
 
Citation: Guido Ascari and Nicola Branzoli, (2010) ''The long-run optimal degree of indexation in new Keynesian models with price staggering 
à la Calvo'', Economics Bulletin, Vol. 30 no.1 pp. 482-493. 
Submitted: Sep 10 2009.   Published: February 08, 2010. 
 
     1 Introduction
Indexation of wages and prices was the subject of substantial literature in macro-
economics in the era of high in￿ ation (see the seminal paper by Gray, 1976) and
it has subsequently been neglected. Quite recently, however, many authors
started to introduce various forms of indexation in the New Keynesian model,
within the Calvo price staggering framework.
However, they do it in a completely ad hoc manner. First, the nowadays most
popular form of indexation embedded in these models is the so-called backward-
looking indexation. The main reason is empirical in order to have a lagged term
in the New Keynesian Phillips Curve to match the in￿ ation persistence in the
data. Second, indexation is most of the times assumed to be full, which avoids
the problems arising with positive in￿ ation in steady state (see Ascari, 2004).
In the empirical estimates of these models, however, indexation is usually found
to be only partial.1
This note follows the spirit of Gray (1976) by treating indexation as a policy
parameter. That is, we deal with the following question: given that we assume
backward looking indexation (as most of the recent literature), is it optimal to
set full indexation? We think that this approach may be helpful in understand-
ing how indexation shape trade-o⁄s in the model, in￿ uencing the normative
conclusions one can obtain using the New Keynesian approach.
We decide to focus on the well-known Christiano et al., 2005 (CEE hence-
forth) medium-scale model, which is becoming a benchmark model for the lit-
erature, because it is theoretically quite rich and empirically successful. We
will then ask which are the values of price and wages indexation that maximize
welfare in the deterministic Ramsey steady state of the CEE model.
2 The Model
The main features of the model are: (i) Households: habit persistence in con-
sumption, money in the utility function, each household comprises all the type
of labors and owns capital stock, sticky wages a la Calvo; (ii) Firms: Cash-in-
advance constraint on wage payments, monopolistic competition, price sticki-
ness a la Calvo, standard Cobb-Douglas production function plus a ￿xed cost to
guarantee zero pro￿t in equilibrium, variable capacity utilization, adjustment
costs in investment; (iii) Government expenditure is ￿nanced through lump-
sum taxes and seigniorage. We use the same functional forms, notation and
calibration of SGU (2004).
In this note we focus on indexation, which in the CEE model takes the
popular form of backward-looking indexation. In other words, those prices (and
wages) that can not change are automatically updated accordingly to the level
of price (wage) in￿ ation in the previous period.
1See, e.g., Christiano et al. (2005) for the U.S., Smets and Wouters (2003) and for the
Euro area.
1For a detailed description of the model see for example Schmitt-GrohØ Uribe
(2004) ( SGU henceforth ).
3 The Ramsey Steady State
Both CEE and SGU assume that wages and prices are fully indexed, that is
indexation is 100%. In this note we ask the model what are the values of
￿ 2 [0;1] (i.e., degree of price indexation) and of ~ ￿ 2 [0;1] (i.e., degree of
wage indexation) that maximize the steady state welfare of the representative
household.
Following the literature initiated by Gray (1976) we treat indexation as a
policy parameter. Consider the problem of a benevolent planner that seeks to
maximize the present value of the utility choosing the optimal level of in￿ ation,
price and wage indexation.
Schmitt-GrohØ and Uribe (2007) has partially investigated this issue. In this
same model they let vary the price-indexation parameter between 0 and 1 and
look for the optimal Ramsey steady state in￿ ation. Their main result is that
the trade-o⁄ between the Friedman Rule and the in￿ ation stability is resolved
in favor of in￿ ation stability, with a steady state level of in￿ ation close to zero.
When indexation increases toward 1, reducing the long-run e⁄ect associated with
in￿ ation, the optimal level of in￿ ation moves toward the Friedman rule2. In light
of their analysis we expect to ￿nd that a benevolent planner maximizing over
in￿ ation and the indexation parameters, would solve both ine¢ ciencies setting
indexation equal to 1 and the in￿ ation rate equal to the discount rate, i.e. the
Friedman Rule.
Surprisingly the solution of the Ramsey problem is:
￿￿
Ra = ￿ ~ ￿
￿
Ra = 1 ￿￿
Ra = 0:95
i.e. the Friedman rule, full wage indexation but partial price indexation. To
investigate the trade-o⁄ behind the Ramsey solution we study the steady state
in the next section.
4 Steady State Analysis
We expected to ￿nd full indexation (i.e., ￿,~ ￿ = 1) to be optimal in steady state3.
The e⁄ect of indexation is visualized in Figure 1, that shows welfare as a function
of ￿ and ~ ￿ (prices and wages indexation respectively): This graph reveals that no
indexation gives the worst outcome. Figure 2 instead uncovers our main result.
While steady state welfare is ever increasing in the wage indexation parameter,
~ ￿; for a given ~ ￿; it is ￿rst increasing and then decreasing in the price indexation
parameter: It follows that for any ~ ￿; full price indexation is never optimal.
2See SGU (2007) for a further discussion on this topic.
3For the sake of simplicity , we analyze the case of positive trend in￿ation, calibrating the
model to the US average in￿ation (4.2%) as SGU. Note however that the arguments provided
are indipendent from the sign of in￿ation, given that it is di⁄erent from 0.
2Result 1 The maximum steady state welfare level is attained at full wage in-
dexation, but partial price indexation, that is ~ ￿ = 1 and ￿ = 0:88.
Corollary The value of ￿ = 0:88 maximizes steady state welfare, for any level
of ~ ￿
The corollary of Result 1 suggests that the source of optimality of partial
indexation is to be found in other ine¢ ciencies rather than the nominal frictions
in the model. In order to explain our intuition we follow the line of argument
in King and Wolman (1996). King and Wolman (1996) focus their attention
on the average mark-up in the economy. The average mark-up is a measure of
the monopolistic distortion in the whole economy, i.e., a lower average mark-
up should be associated with a higher welfare level. Moreover in all the New
Keynesian models, monopolistic competition implies the existence of steady
state ine¢ ciencies.
The average mark-up can be expressed as
Pt
MCt
=
￿
Pt
~ Pt
￿ 
~ Pt
MCt
!
(1)
where MCt represents nominal marginal costs, Pt is the aggregate price level
and ~ Pt is the optimal reset price.
The average mark-up is hence given by two factors: 1) the ￿price adjust-
ment gap￿ , de￿ned as the ratio of the general price level to the price charged
by resetting ￿rms; 2) the ￿marginal mark-up￿ , de￿ned as the mark-up of the
resetting ￿rms.
In a steady state with full indexation, or with no long-run in￿ ation, all prices
and wages would be the same. So there would be no "price adjustment gap" and
the average mark-up would equal the marginal mark-up, in turn simply given
by the Lerner coe¢ cient. Whenever there is partial price indexation this is no
longer true, and the steady state exhibits price dispersion. In this environment
Pt is lower than ~ Pt and a price adjustment gap emerges. Indeed with partial
indexation, positive in￿ ation mechanically erodes the relative price set by ￿rms
in past periods. This can be seen in the steady state version of the equation
that de￿nes the general price level4
Pt =
￿Z 1
0
Pi;t
1￿￿di
￿ 1
1￿￿
=
h
￿￿
(1￿￿)￿
t￿1 P
1￿￿
t￿1 + (1 ￿ ￿) ~ Pt
1￿￿
i 1
1￿￿
=) ss
=) 1 =
2
4￿￿(￿￿1)(1￿￿) + (1 ￿ ￿)
 
~ P
P
!1￿￿3
5
1
1￿￿
=)
P
~ P
=
 
1 ￿ ￿￿(￿￿1)(1￿￿)
1 ￿ ￿
! 1
￿￿1
(2)
4￿ is the probability of not being able to reset the price, ￿ is the steady state gross in￿ation
rate and ￿ is the elasticity of demand for di⁄erentiated goods.
3Note that, other things equal, lower price indexation decreases the price adjust-
ment gap, since ￿rms will try to shield themselves from the erosion of relative
prices, thus decreasing the average mark-up in (1), and increasing welfare. Here
it is therefore the positive e⁄ect of partial indexation: the lower ￿; the lower
the price adjustment gap and the lower the average mark-up in the economy.
The second e⁄ect instead concerns the marginal mark-up, which is also af-
fected by partial indexation. Firms know that positive in￿ ation erodes both
their mark-up (since nominal marginal costs will increase with in￿ ation) and
their relative prices. This latter erosion would both increases their demand
(which they have to satisfy by assumption) and decreases their per-unit pro￿ts.
Thus, they react by resetting a higher price when they can, so that the lower
the indexation parameter, the higher is the ratio
~ Pt
MCt. This is the negative
e⁄ect of a lower indexation on the average mark-up in the economy: the lower
￿; the higher the marginal mark-up and the higher the average mark-up in the
economy:
These are hence the two con￿ icting forces acting in steady state. As displayed
in Figure 3, for low levels of ￿; the second e⁄ect dominates such that the average
mark-up decreases with ￿; and therefore welfare is increasing with indexation.
However, at a certain point (￿ = 0:88); the two e⁄ects compensate and then
the ￿rst e⁄ect prevails. For ￿ > 0:88 the average mark-up increases, reducing
welfare:
Following the same argument, the second e⁄ect instead always dominates
with regard to wage indexation, given the CEE calibration.
To conclude, with positive trend in￿ ation, partial price indexation can miti-
gate the monopolistic distortion in the steady state, thereby increasing welfare5.
5 Price indexation and price stickiness
It would be interesting to look at the properties of the steady state de￿ned by the
optimal combination of the two indexation parameters, particularly exploring
the comparative statics with respect to some parameters. The most obvious
one to look at is the Calvo parameter of the price setting mechanism. The
parameter ￿ is the probability of not being able to reset the price, and it is set
equal to to 0.6 in SGU 2004.
We expected welfare to be increasing with price ￿ exibility, that is, for a
given ￿ (and ~ ￿), welfare would be decreasing with ￿: Indeed ￿ = 0 means
complete price ￿ exibility. This seems to be the case by looking at Figure 4
that plots welfare as ￿ and ￿ vary. The lower welfare level is given the point
(￿;￿) = (1;0), as expected.
However, welfare is increasing in ￿; for certain values of ￿: That is, in certain
part of the surface in Figure 4, the higher price stickiness, the higher is welfare.
5Note that our arguments are not based on the particular form of indexation but rather
on the presence of positive long-run in￿ation and monopolitic competition, so they are not
a⁄ected by the assumption of indexation to past in￿ation.
4Generally, this holds for quite high values of ￿; as shown in Figure 5, and in
particularly for ￿ = 0:88.
Result 2 Given ~ ￿ = 1 and ￿ = 0:88; steady state welfare is maximized for the
maximum admissible level of price rigidity (i.e., ￿ = 0:993).6
Indeed, for any value of ~ ￿ and for ￿ > 0:76 welfare is increasing in ￿: The
key variable is still the average mark-up in the economy, which is monotonically
decreasing in price stickiness as shown in Figure 6. Marginal mark-up is increas-
ing in ￿ because price-setting ￿rms increase their mark-up when the hazard rate
of prices is lower. However the average mark-up decreases with ￿ because the ef-
fect of the price adjustment dominates for high values of indexation. Therefore,
we might conclude that price stickiness improves welfare by alleviating monop-
olistic distortions. Result 2 is surprising and should warn about the fact that
a wrong calibration of indexation may mislead to the conclusion that nominal
frictions are bene￿cial to the economy.
6 Conclusions
This note shows that full price indexation is not optimal in the long-run in the
New Keynesian model, because of monopolistic distortion. The argument pro-
vided for the optimality of partial indexation is very di⁄erent from the classical
one in Gray (1976). Indeed, we are analyzing the steady state of a microfounded
model, without considering any stochastic supply or demand shocks, that were
instead crucial for Gray￿ s (1976) argument. Moreover, we show that more price
stickiness may increase steady state welfare, if price indexation is partial.
7 References
Ascari, G. (2004) "Staggered prices and trend in￿ ation: Some nuisances" Review
of Economic Dynamics 7, 642￿ 667.
Calvo, G. (1983) "Staggered prices in a utility-maximising framework" Jour-
nal of Monetary Economics 12, 383 ￿ 398.
Christiano, L., M. Eichenbaum and C. Evans (2005) "Nominal Rigidities
and the Dynamic E⁄ects of a Shock to Monetary Policy" Journal of Political
Economy, 113(1).
Gray, J. A. (1976)."Wage indexation: a macroeconomic approach" Journal
of Monetary Economics 2, 221￿ 235.
King, R. G. and A. L. Wolman (1996) "In￿ ation targeting in a st. louis
model of the 21st century" Federal Reserve Bank of St. Louis Quarterly Review,
83￿ 107.
6Given positive steady state in￿ation and partial indexation, there is a maximum value for
￿; such that the ￿rst order condition for reset price is de￿ned (see Ascari, 2004).
5Schmitt-GrohØ S. and Uribe M., (2004) "Optimal Operational Monetary
Policy in the Christiano-Eichenbaum-Evans Model of the U.S. Business Cycle"
NBER wp 10724.
Schmitt-GrohØ, S. and M. Uribe (2007) "Optimal in￿ ation stabilization in a
medium-scale macroeconomic model" Monetary Policy Under In￿ation Target-
ing, Central Bank of Chile, 125 ￿ 186.
Smets, F. and R. Wouters (2003) "An estimated stochastic dynamic gen-
eral equilibrium model of the Euro Area" Journal of the European Economic
Association, 15(1), 1123 ￿ 1175.
Smets, F. and R. Wouters (2007) "Shocks and frictions in U.S. business cycle:
A Bayesian DSGE approach" American Economic Review, 97(3), 586 ￿ 606.
68 Figures
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
-160
-159.5
-159
-158.5
-158
-157.5
-157
-156.5
c tilde
Steady State Welfare
c
Figure 1
70.9
0.92
0.94
0.96
0.98
1
0.65
0.7
0.75
0.8
0.85
0.9
0.95
-156.74
-156.735
-156.73
-156.725
-156.72
-156.715
-156.71
c tilde
Steady State Welfare
c
Figure 2
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
1.1999
1.2
1.2
1.2001
c (c tilde = 1)
A
v
e
r
a
g
e
 
M
a
r
k
-
u
p
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
-156.74
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-156.73
-156.725
-156.72
-156.715
-156.71
c (c tilde = 1)
s
s
 
w
e
l
f
a
r
e
Figure 3
80
0.2
0.4
0.6
0.8
1 0
0.2
0.4
0.6
0.8
1
-168
-166
-164
-162
-160
-158
-156
c
Steady State Welfare
a
Figure 4
90.8
0.82
0.84
0.86
0.88
0.9
0.65
0.7
0.75
0.8
0.85
0.9
0.95
-156.9
-156.85
-156.8
-156.75
-156.7
-156.65
-156.6
a
Steady State Welfare
c
Figure 5
100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.1994
1.1995
1.1996
1.1997
1.1998
1.1999
1.2
a (c = 0.88 )
A
v
e
r
a
g
e
 
M
a
r
k
-
u
p
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-156.72
-156.71
-156.7
-156.69
-156.68
-156.67
-156.66
-156.65
-156.64
-156.63
-156.62
a (c = 0.88)
s
s
 
w
e
l
f
a
r
e
Figure 6
11

The long-run optimal degree of indexation in new Keynesian models with price staggering à la Calvo

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The long-run optimal degree of indexation in new Keynesian models with price staggering à la Calvo

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