Nominal frictions and optimal monetary policy.
Haider, Adnan ; Ramzi, Drissi
It is well known in standard economic literature that nominal
frictions have significant impact on the transmission mechanism of
monetary policy. This paper considers a closed economy version of DSGE
model with various nominal frictions vis-a-vis monetary- cum-fiscal
blocks to seek the basic query that how monetary policy impacts while in
the presence of nominal frictions, like price stickiness, staggered
wages, etc. Using Bayesian Simulation techniques, we estimate the model
for the closed economy. Our simulation results show that despite the
apparent similarities of various frictions, their responses to shocks
and fit to data are quite different and there is no agreement on their
relative performance. As a result, Monetary Authorities cannot afford to
rely on a single reference model which contains few nominal frictions of
the economy but need to model a large number of alternative ways
available when they take their decision of optimal monetary policy.
JEL classification: E32, E37
Keywords: DSGE Models, Nominal Trictions, Monetary Policy
1. INTRODUCTION
The modern modeling research on macroeconomics combines
micro-foundations of both households and firms optimisation problems and
with a large collection of both nominal and real (price/wage) rigidities
that provide plausible short-run dynamic macroeconomic fluctuations with
a fully articulated description of the monetary- cum-fiscal policy
transmission mechanism; see, for instance, Christiano, et al. (2005) and
Smets and Wouters (2003). The key advantage of this area of research
over traditional reduce form macroeconomic models, is that the
structural interpretation of their parameters allows to overcome the
famous Lucas critique (1976). Traditional models contained equations
linking variables of interest of explanatory factors such as economic
policy variables. One of the uses of these models was therefore to
examine how a change in economic policy affected these variables of
interest, other things being equal. Based on these advantages there has
been a growing interest in academics, international policy institutions
and central banks in developing small-to-medium, even large-scale, both
closed and open economy DSGE models based on new-Keynesian framework. In
using DSGE models for practical purposes and to recommend how central
banks and policy institutions should react to the short-run
fluctuations, it is necessary to first examine the possible sources, as
well as to evaluate the degree of nominal and real rigidities present in
the economy. As price stability is the primary objective of every
central bank so it is an important task to model inflation dynamics with
its associated nominal rigidities using DSGE models carefully.
Therefore the core objective of this paper is to consider various
nominal frictions, especially price stickiness with its alternative
representations of the inflation dynamics, each one having formal
microeconomic foundations. To learn dynamics of this friction associated
with each representation we considered four competing closed economy
DSGE models: a standard Calvo type pricing model; Hernandez's
(2004) state- dependent pricing model; Mankiw and Reis (2002) standard
sticky information model; and a mixed version of sticky
price-information model. Each model incorporates various other standard
New-Keynesian features such as habit formation, costs of adjustment in
capital accumulation and variable capacity utilisation. While in the
standard Calvo (1983) model, some prices are exogenously fixed for
certain periods and the Phillips curve associated it performs badly to
reproduce the gradual and delayed effects of monetary shocks on
inflation. Mankiw and Reis (2002) propose to replace it with a Sticky
information model. In that last specification, firms face some frictions
while updating their information sets to determine the optimal flexible
price.
However, in the two cases, the frequency of price revisions is
constant and without cost. Price-setters cannot respond to shocks in the
economy between price revisions. In such a context, literature on
state-dependent pricing [e.g. Dotsey, et al. (1999)] allows firms either
to evaluate in every period if it is convenient to change their price
contracts or to keep them unchanged given a random cost. So we also
simulate the performance of the Hernandez's (2004) model which
combines state-dependent and time-dependent features in the firms
pricing scheme is investigated as a natural extension of the Calvo
model. Finally for each model, the Ramsey allocation has been computed,
giving a natural benchmark for welfare comparisons.
The remainder of the paper is structured as follows. Section 2
gives an outline of Common theoretical framework. In Section 3 he main
different responses observed across each specification essentially by
the nature of nominal rigidities. The methodologies and empirical setup
are discussed in Section 4. Section 5 describes the estimation results
and finally we bring to a close in Section 6 with concluding remarks and
possible model extensions.
2. COMMON THEORETICAL FRAMEWORK
The following relationships are common to all models in the
specification of the economy. These specifications are similar to
Christiano, et al. (2005) and Smets and Wouters (2003). The main
features of the closed economy DSGE model are habit formation in
consumption, capital adjustment costs and a large number of shocks
essential for the fit with data. Such a common framework is a mean to
obtain comparable New Keynesian Phillips curves and to explain the main
different responses observed across each specification essentially by
the nature of nominal rigidities.
2.1. Households Preferences
The economy is inhabited by a representative household (h) who
derives its utility from consumption [C.sub.t], and leisure 1-[L.sub.t].
At time t, its preferences are described by an intertemporal utility
function:
Where,
[U.sub.t](h) = [E.sub.t] ([[infinity].summation over
(j=0)][[beta].sup.j][[U.sub.1] ([C.sub.t], [H.sub.t]) -
[U.sub.2]([L.sub.t])][[epsilon].sup.B.sub.t+j])
Where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where [[beta].sub.t] [member of] (0,1) is the intertemporal
discount factor which describe rate of time preferences, [[zeta].sub.c]
is the inverse of the elasticity of intertemporal substitution in
consumption and [[zeta].sub.L] is the inverse of wage elasticity of
labour supply. We introduce external habit formation for the
optimisation household as [H.sub.t] = v[C.sub.t+j-1] (h) with degree of
intensity indexed by [upsilon], where [C.sub.t+j-1] is the aggregate
part of consumption index.
Utility also incorporates a consumption preference shock
[[epsilon].sup.B.sub.+j] and a labour supply shock
[[epsilon].sup.L.sub.t+j]. [xi] is the scale parameter. As usual, it is
assumed that, [[zeta].sub.c] > 0 and [[zeta].sub.L] > 1.
Each household h maximises its utility function under the following
budgetary constraint:
[B.sub.t](h)/[P.sub.t](l + [i.sub.t]) + [C.sub.t](h) + [I.sub.t] =
[B.sub.t- 1](h)/[P.sub.t]+ (1-[[tau].sub.w,t])
[W.sub.t](h)[L.sub.t](h)[L.sub.t](h) +
[A.sub.t](h)+[T.sub.t](h)/[P.sub.t] +
[r.sup.k.sub.t][u.sub.t](h)[K.sub.t](h) -
[PHI]([u.sub.t](h))[K.sub.t](h)
Where [B.sub.t](h) is a nominal bond, [W.sub.t](h) is the nominal
wage, [A.sub.t](h) is a stream of income coming from state contingent
securities, [T.sub.t](h) and [[tau].sub.w,t] are government transfers
and time-varying labour tax respectively, and
[r.sup.k.sub.k][u.sub.t](h)[K.sub.t] (h) - [PHI]([u.sub.t]
(h))[K.sub.t](h) represents the return on the real capital stock minus
the cost associated with variations in the degree of capital
utilisation. As in Christiano, et al. (2005), the income from renting
out capital services depends on the level of capital augmented for its
utilisation rate. The cost of capacity utilisation is zero when capacity
are fully used ([PHI](1) = 0 and [PHI]'(l), [PHI]"(l)[greater
than or equal to]0).
Separability of preferences and complete financial markets ensure
that households have identical consumption plans. The first order
condition related to consumption expenditures is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
Where [[lambda].sub.t] is the Lagrange multiplier associated With
the budget constraint. First order conditions corresponding to the
quantity of contingent bonds implies that:
[[lambda].sub.t] = (l +
[i.sub.t])[beta][E.sub.t][[[lambda].sub.t+1] [P.sub.t]/[P.sub.t+1]] ...
... ... ... ... (2)
Where it is the one-period-ahead nominal interest rate.
2.2. Labour Supply and Staggered Wage Settings
Each household is a monopoly supplier of a differentiated labour
service. For the sake of simplicity, we assume that he sells his
services to a perfectly competitive firm which transforms it into an
aggregate labour input using the following technology:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The household faces a labour demand curve with constant elasticity
of substitution:
[L.sub.t](h) = [[[W.sub.t](h)/[W.sub.t]].sup.-[mu]w/[mu]w-1][L.sub.t]
Where:
[W.sub.t] = [[integral].sup.1.sub.0] [[W.sub.t](h).sup.1/1-[mu]w
dh].sup.1- [mu]w]
is the aggregate wage rate.
Households set their wage on a staggered basis. Each period, any
household faces a constant probability 1-[[alpha].sub.w] of changing its
wage. In such a case, the wage is set to [[??].sub.t] which is the same
for all suppliers of labour services, taking into account that it will
not be re-optimised in the near future. Otherwise, wages are adjusted
following an indexation rule on past inflation and central bank
objective:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[[pi].sub.t] = [P.sub.t]/[P.sub.t-1] denotes one plus the GDP
deflator inflation rate while [[bar.[pi]].sub.t] = [P.sub.t] /
[P.sub.t-1] denotes one plus the inflation objective of the central bank
and ([[xi].sub.w] is fraction of wage.
Notice that among the fraction of wage setters, which cannot
re-optimise in period t; each nominal wage appears with the same
frequency as in the t-1 distribution after controlling for the common
indexation on inflation rates. This property crucially hinges on the
fact that each wage has an equal probability of being adjusted in a
given period.
Consequently, the dynamics of the aggregate wage index is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Each household chooses [[??].sub.t] in order to maximise:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Given that the demand for differentiated labour service for wage
setters who cannot re-optimise after period t, becomes:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Thus, the first order condition for the re-optimised wage verifies:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Let us denote [[??].sub.t] as the real wage. The previous equation
can therefore be rewritten as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
With
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... ... ... (3)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... ... ... (4)
Accordingly, the aggregate wage dynamics leads to the following
relation.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... (5)
2.3. Investment Dynamics
As in Smets and Wouters (2003), we introduce a delayed response of
investment observed in the data. Capital producers combine existing
capital, Kt, leased from the entrepreneurs to transform an input 1,,
gross investment, into new capital according to:
[K.sub.t+1] = (1 - [delta])[K.sub.t]+[1-S
([I.sub.t][E.sup.I.sub.t]/[I.sub.t- 1])][I.sub.t] ... ... ... ... (6)
Where [I.sub.t] is gross investment, [delta] is the depreciation
rate and the adjustment cost function S(*) is a positive function of
changes in investment. S(*) equals zero in steady state with a constant
investment level (S(1) = 0). In addition, we assume that the first
derivative also equals zero around equilibrium, so that the adjustment
costs will only depend on the second-order derivative (S"(*) as in
Christiano, et al. (2005). We also introduced a shock to the investment
cost function, which is assumed to follow a first-order autoregressive
process with an IID-Normal error term: [E.sup.I.sub.t] = [[rho].sub.I]
[E.sup.I.sub.t-1] + [[eta].sup.I.sub.t].
Households choose the capital stock, investment and the capacity
utilisation rate in order to maximise their intertemporal utility
function subject to the intertemporal budget constraint and the capital
accumulation. The first-order conditions result in the following
equations for the real value of capital, investment and the capacity
utilisation rate:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
The value of installed capital depends on the expected future value
taking into account the depreciation rate and the expected future return
as captured by the rental rate times the expected rate of capital
utilisation.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
Where, [R.sup.K.sub.t] is rental rate of capital and
[E.sup.I.sub.t] can be interpreted as a shock to the relative price of
investment while [E.sup.Q.sub.t] accounts for fluctuations of the
external finance risk premium.
2.4. Firms Behaviour
Intermediate goods are produced with a Cobb-Douglas technology as
follows:
[Y.sub.t] (h) = [E.sup.A.sub.t][([CU.sub.t] (h)[K.sub.t-1]
(h)).sup.a] [L.sub.t] [(h).sup.1-[alpha]] - [OMEGA] [for all]h[member
of] (0,1)
Where [E.sup.A.sub.t] is an exogenous technology parameter and
[OMEGA] is a fixed cost. Firms are monopolistic competitors and produce
differentiated products an aggregate final good that may be used for
consumption and investment. This production is obtained using a
continuum of differentiated intermediate goods with the following Dixit
and Stiglitz (1977) production technology:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
Where [mu] = [[theta].sub.P]/[[theta].sub.P-1] and [theta]P > 1
is the elasticity of substitution between differentiated goods. The
representative final good producer maximises profits [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] subject to the production function
(10), taking as given the final good price [P.sub.t] and the prices of
all intermediate goods. The first order condition for this problem is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
Finally, as the sector is perfectly competitive, the zero profit
condition holds and the expression for [P.sub.t] is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
2.5. Government
Public expenditures [??] G are subject to random shocks
[E.sup.G.sub.t]. The government finances public spending with labour
tax, product tax and lump-sum transfers are expressed as:
[P.sub.t][??][E.sup.G.sub.t] - [[tau].sub.W,t][W.sub.t][L.sub.t] -
[[tau].sub.t][P.sub.t][Y.sub.t] - [P.sub.t]T[T.sub.t] = 0 ... ... ...
... (12)
The government also controls the short term interest rate
[R.sub.t]. Monetary policy is specified in terms of an interest rate
rule: the monetary authority follows generalised Taylor rules which
incorporate deviations of lagged inflation and the lagged output gap
defined as the difference between actual and flexible-price output. Such
reaction functions also incorporate a non-systematic component:
[E.sup.G.sub.t].
3. PRICE SETTING MODELS AND OPTIMAL MONETARY POLICY
This section presents the baseline version of the standard Calvo,
State Dependent Pricing (SDP) and the Sticky Information (SI) models of
price setting as different nominal rigidities modelling strategies.
Furthermore, optimal monetary policy principle is also included in this
section.
3.1. Models Based on Price Stickiness
In this section we describe two competing models based on price
stickiness; (a) a standard Calvo (1983) type price stickiness model, and
(b) a recent extension by Hernandez's (2004): state dependent
pricing model. Both these models capture the fundamental notion of
staggered price mechanism and translate them into New Keynesian Phillips
curve.
3.1.1. Standard Calvo (1983) Model
In each period, firms receive a random signal with constant
probability 1 - [[alpha].sub.p] that allows them to change the price
[p.sup.*.sub.t]. This probability is independent across firms and time.
The average duration of a rigidity period is 1/1-[[alpha].sub.p]. If a
firm cannot re-optimise its price, the
price evolves according to the following simple rule: [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII].
Firms that are allowed to change their price maximise expected
profit:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is the marginal value of one unit of money to the household.
[MC.sub.t+j] is the real marginal cost and [[tau].sub.t] is a
time-varying tax on firm's revenue. Due to our assumptions on the
labour market and the rental rate of capital, the real marginal cost is
identical across producers.
[MC.sub.t] = [W.sup.(1-[alpha]).sub.R,t] [R.sup.k[alpha].sub.t]/
[E.sup.A.sub.t][[alpha].sup.[alpha]](1-[alpha]).sup.(1-[alpha])
The first order condition for the optimal nominal reset price
[p.sup.*.sub.t] is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
In this study, the aggregate price level which incorporates rule of
thumb price setters evolves according to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This price setting scheme can be written in the following recursive
form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
Accordingly, the aggregate price dynamics leads to the following
relation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The above specification of Calvo price for which, [[xi].sub.[mu]]
equals to 0 is considered as a standard Calvo.
3.1.2. An Extension: State Dependent Pricing Model
Hernandez's (2004) model gives an explicit role for the
average frequency of price revisions in the inflation-output relation,
by including state dependent fluctuations. To be more precise, it
combines state-dependent and time-dependent features in the firms'
pricing scheme. Firms are allowed to choose a higher probability of
price revisions. In that case, they have to pay a lump sum cost which is
random as in Dotsey, et al. (2004). There are two kinds of monopolistic
firms j [member of] [L,H]. The first one revises prices with the lower
probability (1 - [[alpha].sub.H]) in each period but as soon as they
receive the random signal of price revision, they have the possibility
to benefit from faster price revisions (1 - [[alpha].sub.H]) by paying
the cost [zeta], with the probability [[lambda].sub.t]. If not, they can
set a new price without cost but with the lower probability. The second
one always adjusts prices with the higher probability and without cost.
The profit maximisation program respectively in the two cases is
written as follows, supposing [[tau].sub.L] > 0 and [[tau].sub.H] =
0:
If z [member of] L then:
max [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where;
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and the real profits:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
If z [member of] H then:
max [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where;
The first order condition in both cases gives the same optimal
price:
Where;
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The first order condition in both cases gives the same optimal
price:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where;
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Knowing that, the probability of j choosing(1 - [[alpha].sub.H])
is:
[[lambda].sub.t] = 1 - exp(-b[[D.sub.0H,t] - [D.sub.0L,t]])
and the conditional expected random cost is:
Let [V.sub.t] be the mass of all firms z [member of] L that chose
before and up to t (1 - [[alpha].sub.H]) and have not changes their
price since that time. Consider [[mu].sub.t], the mass of firms that
choose(1 - [[alpha].sub.L]) at t and [bar.[mu]], the mass of firms in L.
The dynamics of [V.sub.t] and [[mu].sub.t], is given by the
following equations:
[V.sub.t] = [V.sub.t-1] + [[lambda].sub.t](1-[[alpha].sub.L]
[[mu].sub.t-1] - (1 - [[lambda].sub.t])(1 - [[alpha].sub.H]) [V.sub.t-1]
[[mu].sub.t] = [bar.[mu]] - [V.sub.t]
With for initial conditions, the steady state values of
[[mu].sub.t] and [V.sub.t], respectively [[mu].sub.0] and [V.sub.0].
In this standard version, the aggregate price level evolves
according to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where;
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
With [V.sub.t] - [V.sub.t-1], the mass of firms z [member of] L
choosing (1 - [[alpha].sub.H]) at t.
3.2. The Sticky Information Model
In each period, a randomly chosen fraction of agents updates their
information set. To be more precise, prices are flexible in the sense
that firms are allowed to change them in any periods, but at a different
level than in a full information environment while they do not have the
same information available about the state of the world. Therefore,
prices fixed based on different information coexist in the economy. This
model has the property that its modeling does not depend on the value of
the steady state of inflation.
At t, firms choose the price [P.sup.*.sub.t] using all current
information. Define [P.sub.t], the overall price index. The optimal
price is determined by the solution of the profit maximisation problem:
max [E.sub.t-j] [(1 - [[tau].sub.t]) [p.sub.t] (h) [Y.sub.t] (h) -
M[C.sub.t][P.sub.t]([Y.sub.t](h) + [OMEGA])] p,(h)
where [Y.sub.t](h) is the demand schedule:
[Y.sub.t](h) = [([p.sup.*.sub.t](h)/[P.sub.t]).sup.[mu]/[mu]-1][Y.sub.t]
The first order condition of this program gives the following
relationship between the optimal price Pt (h) and the real marginal cost
M[C.sub.t]:
[p.sup.*.sub.t](h) = [mu]/ 1 - [[tau].sub.t] M[C.sub.t][P.sub.t]
Lets consider the hybrid specification with backward looking agents
as in Gali and Gertler (1999) by adding rule of thumb price setters.
Then the aggregate price level is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
In each period, firms face a constant probability
(1-[[alpha].sub.p]) of receiving a signal that allows them to change
their price.
The last equation can be rewritten as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Some manipulations allow us to obtain the stationary version of the
previous equation which symbolises the non linear Sticky information
Phillips curve:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Knowing that:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
For computational reasons, the scheme truncates the infinite
horizon in the age distribution of information sets, such as agents set
their prices based on information outdates by J=12 periods (that is to
say 3 years). Therefore, this parameterisation leads us to divide the
previous Phillips curve by the parameter: [omega]= [j.summation over
(j=0] [alpha][(1 - [alpha]).sup.j] j=0
3.3. Market Equilibrium
Aggregate demand is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Market clearing condition on goods market is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Or
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
With: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It measures the price dispersion due to the staggered price
setting. As in the case of the aggregate price index, we can show that
this price dispersion index under Calvo contracts and sticky information
(SI) contracts has respectively the following dynamics:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The aggregate unconditional welfare is defined by:
[u.sub.t] = [[integral].sup.l.sub.0] [u.sub.t](h)dh
We already mentioned that all household have the same consumption
plans. Consequently:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
Where;
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
As for the price dispersion index:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
3.4. Optimal Monetary Policy (Main Principle)
The optimal monetary policy of the Ramsey policy under commitment
consists in maximising the intertemporal households' welfare
([U.sub.t]) subject to a set of non-linear structural constraints of the
model.
To be more precise, a Ramsey equilibrium is a competitive
equilibrium such that:
(i) Given a sequence of shocks, prices, policy instrument and
quantities [P.sub.t]; [R.sub.t];
[Q.sub.tt=0.sup.[infinity]] it maximises the representative agent
lifetime utility, [U.sub.t].
(ii) [i.sub.t] > 0.
In order to analyse essentially the macroeconomic stabilisation
properties of the monetary policy, we assume subsidies on labour and
goods markets are offsetting first order distortions. In that case, the
flexible price equilibrium is Pareto optimal. The Ramsey policy problem
is written using an infinite horizon Lagrangian:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
Where [[lambda].sub.r] is the weight associated to the cost on
nominal interest rate fluctuations. We introduce ah interest rate
objective in this problem in order to make the Ramsey policy
operational. The first order conditions to this problem are obtained
using the symbolic toolbox of Matlab 2008.
4. THE EMPIRICAL SETUP
This section briefly outlines the empirical setup by illustrating
data, choice of priors and estimation methodology used in this paper. We
adopted the empirical approach outlined in Smets and Wouters (2003) and
estimate our augmented DSGE models with sticky prices-information and
wages employing Bayesian inference methods. This involves obtaining the
posterior distribution of the parameters of the model based on its
log-linear state-space representation and assessing its empirical
performance in terms of its marginal likelihood. In the following we
briefly sketch the adopted approach and describe the data and the prior
distributions used in its implementation. We then present our estimation
results in next coming section.
4.1. Data
We consider 7 key macro-economic quarterly time series from 1973ql
to 2004q4: output, consumption, investment, hours worked, real wages,
GDP deflator inflation rate, and 3 month short-term interest rate. Euro
area data are taken from Smets and Wouters (2003) and Euro-stat official
website. Concerning the euro area, employment numbers replace hours.
Consequently, as in Smets and Wouters, hours are linked to the number of
people employed [e.sub.t] with the following dynamics:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Aggregate real variables are expressed per capita by dividing with
working age population. All the data are detrended before the
estimation. Since the model has implications for the log-deviations from
the steady-state of all these variables, so we preprocess the data
before the estimation stage.
4.2. Choice of Priors
In the overall, the set of priors corresponds to the ones in Smets
and Wouters (2003) (see Tables 2 to 5).
4.2.1. Common Parameters
The discount factor [beta] is calibrated to 0.99, which implies
annual steady state real interest rates of 4 percent. The depreciation
rate 6 is equal to 0.0025 per quarter. Markups are 1.3 in the goods
market and 1.5 in the labour market. The steady state is consistent with
labour income share in total output of 70 percent. Shares of consumption
and investment in total output are respectively 0.65 and 0.18.
4.2.2. Calvo and State Dependent Pricing based Model Parameters
Two additional parameters ([[alpha].sub.p] and [[xi].sub.p]). The
parameter [[alpha].sub.p] which determines the probability that firms
are allowed to change their price, has a prior mean of 0.75 and a
standard deviation of 0.0084. Regarding the hybrid specification, the
parameter of partial indexation to lagged inflation follows a
Beta-distribution.
4.2.3. Sticky Information based Model Parameters
Lets suppose the same prior for the previous parameters in that
case and consider [alpha] which is the probability to receive new
information about the state of the economy, follows a Beta-distribution
with the mean of 0.75 and the standard deviation of 0.0512. This
parameter value is also consistent with Mankiw and Reis (2002).
4.3. Bayesian Estimation Approach
In empirical literature, there are numerous strategies used to
determine the parameters of new-Keynesian DSGE models. These ranging
from pure calibration, e.g., Kydland and Prescott (1982), Monacelli
(2003), over generalised method of moments (GMM) for estimation of
general equilibrium relationships, e.g., Christiano and Eichenbaum
(1992), to full-information based maximum likelihood estimation as in
Altug (1989), Mcgrattan (1994), Leeper and Sims (1994), Kim (2000) and
Irland (2000). Other studies also proposed mixed strategies like
limited-information based methods to explore a key question whether a
DSGE model matches the data with some certain dimensions. For example,
Canova (2002) and Christiano, et al. (2005) used minimum distance based
criterion to estimate VAR and DSGE model impulse response functions.
Further methodological debate can be referred using the following
studies by Diebold (1998), Ruge-Murcia (2003) and Tovar (2008).
Other than these proposed estimation and calibration strategies,
this study uses another estimation approach called Bayesian estimation
approach. This alternative approach is a combination of calibration and
estimation of selected model parameters. The fundamental advantage of
this approach is a batter adoption of the model to the conditions in the
given economy, [see e.g., Smets and Wouters (2003)].
In any empirical modeling exercise, there are three possible
sources of uncertainty; the model itself; the parameterisation condition
of the model and the data. The debate on the issue of uncertainty is the
most important as it provide a difference between frequentist
(classical) and Bayesian approach. In classical approach the probability
of the occurrence of ah event, i.e., the measurement of uncertainty is
associated with its frequency. However, in Bayesian approach, the
probability of ah event is determined by two components; the subjective
believe (prior) and the frequency of that event. For further detail on
this notion, [see for instance Gelman (2006) and Koopman, et al.
(2007)].
The seminal work on DSGE modeling used this approach started with
the study by Landon-Lane (1998), DeJong, et al. (2000), Schorfheide
(2000) and Otrok (2001). This approach has been generalised by Lubik and
Schorfheide (2005) who estimate a DSGE model without providing
restrictions to the determinacy region of the parameter space. Almost
all recent studies on DSGE model has been used this approach, e.g.,
Smets and Wouters (2003), Laforte (2004), Onatski and Williams (2004),
Ratto, et al. (2008), Adolfson, et al. (2008) and Kolasa (2008).
In practical sense, we try to fit out referenced model, which
consists in placing a prior distribution p([GAMMA]) on structural
parameters [GAMMA], the estimate of which are then updated using the
data [y.sup.T] according to the Bayes rule:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
Where p([Y.sup.T]/ [LAMBDA])=L([LAMBDA]/[Y.sup.T]) is the
likelihood function p([LAMBDA]/[Y.sup.T]) is the posterior distribution
of parameters and p([Y.sub.T]) is the marginal likelihood defined as:
p([Y.sup.T]) = [integral]p([Y.sup.T] /[LAMBDA])p([LAMBDA])d[LAMBDA]
... ... ... ... ... (20)
Any DSGE model forms a linear system with rational expectations,
the solution to which is of the form:
[R.sub.t] = [B.sub.1]([LAMBDA])[R.sub.t-1] +
[B.sub.2]([LAMBDA])/[[mu].sub.t] ... ... ... ... ... ... (21)
[[mu].sub.t] = [B.sub.3]([LAMBDA])[[mu].sub.t-1 +
[B.sub.4]([lambda])[[epsilon].sub.t] ... ... ... ... ... ... (22)
Where [R.sub.t] is a vector of endogenous variables, [[mu].sub.a],
is a vector of stochastic disturbances and [[epsilon].sub.t], is a
vector of innovations to stochastic shocks and coefficient matrices
[A.sub.i] depending on
the parameters of the model. The measurement Equations (21) and
(22) linking observable variables used in the estimation with endogenous
variables can be written as:
[Y.sup.T] = [CR.sub.t] ... ... ... ... ... ... ... ... (23)
Where, C is the deterministic matrix. The Equations (21), (22) and
(23) form the state-space representation of the model. The likelihood of
which can be evaluated using Kalman filter. The analytical solution of
the whole system may not be obtain in general, however the sequence of
posterior draws can be obtain using Markov-Chain-Monte- Carlo (MCMC)
simulation methodology. This methodology is briefly discussed in Lubik
and Schorfheide (2005), Gelman, et al. (2006) and Koopman, et al.
(2007). For our estimation setup the random walk Metropolis-Hastings
algorithm is used to generate Morkov-Chains (MC) for the model
parameters.
5. EMPIRICAL RESULTS
The Bayesian framework as discussed in previous section is used in
order to compare the purely forward looking and hybrid Calvo to the
baseline and the hybrid specifications of a truncated Sticky Information
model, under the assumption that the models have equal prior
probabilities.
5.1. Model Comparison Based on Marginal Densities
The following Table 1 reports the marginal densities for the all
pricing models. The model with the highest marginal density is the
standard Calvo model over the other specifications. In the overall, it
dominates the sticky information for all specifications in terms of
marginal densities.
In the literature, the motivation for including inertia is largely
empirical and justified theoretically with an assumption that a fixed
proportion of firms has backward-looking price setting behaviour.
Empirically the adequacy of this model, which nests the pure
forward-looking sticky price model and inherits the good properties of
backward-looking behaviour, to data is controversial. In this study
using Bayesian estimation, the introduction of indexation to lagged
inflation is not a necessary condition to reproduce plausible inflation
dynamics as in the standard Calvo model, [see for instance Laforte
(2004); Paustian and Pytlarczyk (2006)].
Some researchers criticise all models built on the sticky price
hypothesis because they would not be at odds with the facts and the
hybrid models would be even worse than the standard ones [see for
example, Mankiw and Reis (2002)]. They advise to replace this
specification by Sticky Information contracts that prevent inflation to
jump immediately after shocks. However under such pricing, the fit is
poor and regarding the Hybrid curve, all expected inflation is
integrated in the price path such as the scheme of indexation is again
of little interest. Moreover, the extension of the maximum age of
outdated information sets from 12 to 22 quarters does not improve very
much the performance of the Sticky Information models such as we
consider 12 quarters represent a good approximation of the infinite sum
in terms of contracts duration. In the overall, the introduction of
indexation, under mixed SI models, does not by itself add more
persistence in the two specifications and basically, the choice of the
price structure seems to be much more important.
5.2. Model Comparison Based on Posterior Distribution
Tables A1 to A7 of Appendix-A,* present information about the
posterior distributions of the two pricing schemes, under different
assumptions. In the standard case (without indexation), while most of
estimated parameters are quite similar, the estimated degree of wage
indexation is significantly high in the Sticky Information model under
Calvo wage contracts (0.76) and low in the Calvo model (0.21).
In the same way, the variance of wave markup is 0.40 in SI model vs
0.19 in Calvo model. We can also note an important difference across the
pricing regarding the persistence degree of the preference shock and its
variance (respectively in Calvo and SI : 2.54 vs 1.95 and 5.30 vs 9.34).
As a result, the Sticky information assumption has different
implications for some key parameters including the ones in the policy
instrument. The degree of inertia is slightly smaller in that this model
as opposed to the Calvo specification. This shows that model parameters
are highly sensitive to both specifications; therefore, it is difficult
to conclude the degree of robustness of each model specification. As
both models can produce ah important degree of persistence such as the
choice of Sticky Price against Sticky Information is not sufficient to
determine dynamics properties of two key variables inflation and output.
5.3. Model Comparison Based on Impulse Responses
Figures B1 to B8 of Appendix-B compare the models' estimated
impulse responses of main variables after one percent increase in key
structural shocks, showing the 90 percent posterior bands and the median
of the posterior densities. Figures B1 show the responses after a
productivity shock. Across both Calvo models, the propagation of the
shock is consistent, though in the hybrid version, the inflation
displays a 'hump-shaped' curvature after the few initial
impact. As opposed to the SI model, the Calvo models can bring down the
policy instrument slightly longer below its steady state in the short
run. In the overall, the short run responses are much stronger under the
Sticky information pricing due to its volatile short run dynamics for
the nominal variables. Indeed, after an initial boost, the variables
more quickly come back towards the long-run values.
Regarding the responses of output and inflation to a Monetary
Policy shock (Figure B2), both specifications lead to a hump-shaped
response of inflation (except for the pure FL Calvo model). First of
all, the standard Calvo model exposes an immediate response of
inflation. Mankiw and Reis (2002) criticise in the fixed prices models,
the absence of delay in the inflation reaction. While it seems to be
only a feature related to the fixed prices forward looking models, the
hybrid Calvo reproduce a reaction of inflation less delayed than the
response of the Sticky Information model. Moreover, this last
specification respect the condition of a more delayed response of
inflation than Output while in the Calvo models the response of
inflation is faster. Indeed, the peak slightly occurs before the one of
Output.
Besides, in the Smets and Wouters (2003) model under Calvo
contracts, the price markup shock is dominant in the inflation driving.
In the Sticky Information models, such shocks lead to responses less
persistent into the main selected variables, returning more quickly to
their Steady State than in the Calvo models (Figures B3 and B4).
Paustian and Pytlarczik (2006) show the estimation of a Calvo model
without markup shock induces a marginal likelihood lower and advance one
explanation for the poor fit could be the inability of Sticky
Information models to match the volatility of inflation as well as the
persistence of inflation and real wages. Indeed, such non structural
shocks play an important role in the inflation persistence, in
particular for the model comparison.
5.4. Welfare Comparison Based on Optimal Monetary Policy
In this section, the Ramsey allocation is computed by solving the
first order approximation of the equilibrium conditions. Figures B1 and
B2 refer to the responses of aggregates after an efficient supply shock.
Concerning the productivity shock, the Ramsey allocation generates a
stronger and faster response of real variables and real wage in the
Calvo Model but weaker and slower in the SI model. The associated
interest rate path is much more accommodative in the short term but
reverts very quickly to its initial level.
In the overall, for both models, over longer horizons, the response
of real variables becomes significantly closer in both monetary regimes.
Regarding the labour supply shock, in the Calvo model, the hump-shaped
downward under the Ramsey policy stimulated output, consumption and
investment and leaves quasi-unchanged inflation and real wages. Under
Sticky Information pricing, the effect is weaker and the hump- shaped
stimulates all the aggregates. By contrast, the estimated rule is not
supportive enough to prevent a decrease in real wage and inflation,
above all in the SI model where the interest rate is close to the steady
state value.
Turning now to efficient demand shocks, the increase in consumption
after a preference shock, is more limited under the Ramsey policy than
the alternative rule, and the contraction in investment is stronger. In
the Calvo model, the output decreases in short term under the Ramsey
allocation while inflation and real wage are almost fully stabilised
while in the SI model, the output is stabilised and the real wage
decreases in short term. Under estimated rule, such a shock is
expansionary on output and upward pressures emerge on real wages and
inflation.
For the others demand shocks, the differences noted above are less
pronounced. The responses of output, consumption, investment and real
wages to an investment shock of a government spending shock are
relatively similar under Ramsey policy and he estimated rule, even if
the inflation response is much more muted in the Ramsey allocation (see
Figures B4 and B5 in Appendix).
Figures B6 to B8 refer to inefficient shocks. The transmission of
price markup shocks to the economy is not strongly different under both
monetary regimes which suggest a similar inflation/output tradeoff for
this type of shock. However, in the case of wage markup and external
finance premium shocks, the Ramsey policy is much more restrictive. It
delivers lower real variables and more stable inflation. In the overall,
compared with the estimated Taylor rule, the Ramsey policy accommodates
more strongly the efficient supply shocks, leans more against efficient
demand shocks. In addition, the optimal policy is much more responsive
to labour market shocks than the estimated rule which incorporates only
goods market variables such as inflation and output.
6. CONCLUSION
This paper considers a closed economy version of DSGE model with
various nominal frictions vis-a-vis monetary-cum-fiscal blocks to seek
the basic query that how monetary policy impacts while in the presence
of nominal frictions, like price stickiness, staggered wages, etc. Using
Bayesian Simulation techniques, we estimate the model for the closed
economy. Our simulation results show that despite the apparent
similarities these frictions, their responses to shocks and fit to data
are quite different and there is no agreement on their relative
performance. Both these hypotheses can produce ah important degree of
persistence such as the choice of Sticky Price against Sticky
Information is not sufficient to determine dynamics properties of two
key variables inflation and output. Hence, as a result, monetary
authorities cannot afford to rely on a single reference model which
contains few nominal frictions of the economy but need to model a large
number of alternative ways available when they take their decision of
optimal monetary policy.
Appendices
APPENDIX B
MODEL IMPULSE RESPONSES
[FIGURE B1 OMITTED]
[FIGURE B2 OMITTED]
[FIGURE B3 OMITTED]
[FIGURE B4 OMITTED]
[FIGURE B5 OMITTED]
[FIGURE B6 OMITTED]
[FIGURE B7 OMITTED]
[FIGURE B8 OMITTED]
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* For detail results of this Appendix (Tables A1 to A7) please
contact the authors via email: rdfissi@esg.
Adnan Haider <04phd2009@pide.edu.pk> is PhD Scholar at
Pakistan Institute of Development Economics, Islamabad, Pakistan. Drissi
Ramzi <rdrissi@esg.fr> is Research Assistant at CERESG, ESG
Business School, France.
Authors' Note: Views expressed here are those of the authors
and not necessarily of the institutions, they attached with.
Table 1
Model Comparison
Marginal Density Laplace Appr. Metropolis
Model
State-Dependent -467.549 -466.673
Standard Calvo -472.703 -471.660
Hybrid Calvo -473.223 -472.247
Mixed Standard Sticky Info (J=12) -514.784 -515.292
Mixed Hybrid Sticky Info (J=12) -518.912 -519.466
Standard Sticky Info (J=12) -591.869 -592.952
Hybrid Sticky Info (J=12) -515.399 -515.796
Marginal Density Acceptation Rates
Model
State-Dependent 0.36954
Standard Calvo 0.28976-0.28845
Hybrid Calvo 0.27243-0.27305
Mixed Standard Sticky Info (J=12) 0.23582
Mixed Hybrid Sticky Info (J=12) 0.19194
Standard Sticky Info (J=12) 0.21310
Hybrid Sticky Info (J=12) 0.28529
Table Key: The Mixed Hybrid SI with 22 lags gives the
Marginal Density-514.2014 (Laplace Approximation).
