Money and Risk Aversion in a DSGE framework : A Bayesian Application to the Eurozone

Abstract

We present and test a model of the Eurozone, with a special emphasis on the role of risk aversion and money. The model follows the New Keynesian DSGE framework, money being introduced in the utility function with a non-separability assumption. Money is also introduced in the Taylor rule. By using Bayesian estimation techniques, we shed light on the determinants of output, inflation, money, interest rate, flexible-price output and flexible-price real money balance dynamics. The role of money is investigated further. Its impact on output depends on the degree of risk aversion. Money plays a minor role in the estimated model. Yet, a higher level of risk aversion would imply that money had significant quantitative effects on business cycle fluctuations.

Full text

Money and Risk in a DSGE Framework:
A Bayesian Application to the Eurozone
Jonathan Benchimoland André Fourçans†
March 2012
Abstract
We present and test a model of the Eurozone, with a special emphasis
on the role of risk aversion and money. The model follows the New Keyne-
sian DSGE framework, money being introduced in the utility function with a
non-separability assumption. Money is also introduced in the Taylor rule. By
using Bayesian estimation techniques, we shed light on the determinants of
output, inflation, money, interest rate, flexible-price output and flexible-price
real money balance dynamics. The role of money is investigated further. Its
impact on output depends on the degree of risk aversion. Money plays a mi-
nor role in the estimated model. Yet, a higher level of risk aversion would
imply that money had significant quantitative effects on business cycle fluc-
tuations.
Keywords: Euro area, Money demand, Risk aversion, Bayesian estimation,
DSGE models.
JEL Codes: E31, E51, E58.
Economics Department, ESSEC Business School and CES, University Paris 1
Panthéon-Sorbonne, 106-112 Boulevard de l’Hôpital, 75647 Paris Cedex 13. Email:
jonathan.benchimol@essec.edu
†Economics Department, ESSEC Business School and THEMA, Avenue Bernard Hirsch, 95021
Cergy-Pontoise Cedex 2, France. Corresponding author Phone: +33-1-34433017, Fax: +33-1-
34433689. Email: fourcans@essec.edu
1

Please cite this paper as:
Benchimol, J., Fourçans, A., 2012. Money and risk in a DSGE framework: a
Bayesian application to the Eurozone. Journal of Macroeconomics, 34(1), 95-
111.
2

1 Introduction
Standard New Keynesian literature analyses monetary policy practically without
reference to monetary aggregates. In this now traditional framework, monetary
aggregates do not explicitly appear as an explanatory factor neither in the out-
put gap and inflation dynamics nor in interest rate determination. Inflation is
explained by the expected inflation rate and the output gap. In turn, the output
gap depends mainly on its expectations and the real rate of interest (Clarida et al.,
1999; Woodford, 2003; Galí and Gertler, 2007; Galí, 2008). Finally, the interest rate
is established via a traditional Taylor rule in function of the inflation gap and the
output gap.
In this framework, monetary policy impacts aggregate demand, and thus in-
flation and output, through changes in the real interest rate. An increase in the
interest rate reduces output, which decreases the output gap, thus decreases in-
flation until a new equilibrium is reached. The money stock and money demand
do not explicitly appear. The central bank sets the nominal interest rate so as to
satisfy the demand for money (Woodford, 2003; Ireland, 2004).
The money transmission mechanism may also emphasize the connections be-
tween real money balances and risk aversion. First, there may exist a real balance
effect on aggregate demand resulting from a change in prices. Second, as indi-
viduals re-allocate their portfolio of assets, the behavior of real money balances
induces relative price adjustments on financial and real assets. In the process, ag-
gregate demand changes, and thus output. By affecting aggregate demand, real
money balances become part of the transmission mechanism. Hence, interest rate
alone is not sufficient to explain the impact of monetary policy and the role played
by financial markets (Meltzer, 1995, 1999; Brunner and Meltzer, 1968).
This monetarist transmission process may also imply a specific role to real
money balances when dealing with risk aversion. When risk aversion increases,
individuals may desire to hold more money balances to face the implied uncer-
tainty and to optimize their consumption through time. Friedman alluded to this
process as far back as 1956 (Friedman, 1956). If this hypothesis holds, risk aver-
sion may influence the impact of real money balances on relative prices, financial
assets and real assets, affecting aggregate demand and output.
Other considerations as to the role of money are worth mentioning. In a New
Keynesian framework, the expected inflation rate or the output gap may "hide"
the role of monetary aggregates, for example on inflation determination. Nel-
son (2008) shows that standard New Keynesian models are built on the strange
assumption that central banks can control the long-term interest rate, while this
variable is actually determined by a Fisher equation in which expected inflation
depends on monetary developments. Reynard (2007) found that in the U.S. and
the Euro area, monetary developments provide qualitative and quantitative in-
3

formation as to inflation. Assenmacher-Wesche and Gerlach (2007) confirm that
money growth contains information about inflation pressures and may play an
informational role as to the state of different non observed (or difficult to observe)
variables influencing inflation or output.
How is money generally introduced in New Keynesian DSGE models ? The
standard way is to resort to money-in-the-utility (MIU) function, whereby real
money balances are supposed to affect the marginal utility of consumption. Kre-
mer et al. (2003) seem to support this non-separability assumption for Germany,
and imply that real money balances contribute to the determination of output and
inflation dynamics. A recent contribution introduces the role of money with ad-
justment costs for holding real balances, and shows that real money balances con-
tribute to explain expected future variations of the natural interest rate in the U.S.
and the Eurozone (Andrés et al., 2009). Nelson (2002) finds that money is a signifi-
cant determinant of aggregate demand, both in the U.S. and in the U.K. However,
the empirical work undertaken by Ireland (2004), Andrés et al. (2006), and Jones
and Stracca (2008) suggests that there is little evidence as to the role of money in
the cases of the United States, the Euro area, and the UK.
Our paper differs in its empirical conclusion, resulting in a stronger role to
money, at least in the Eurozone, when risk aversion is high enough. It differs also
somewhat in its theoretical set up. As in the standard way, we resort to money-
in-the-utility function (MIU) with a non-separability assumption between con-
sumption and money. Yet, in our framework, we specify all the micro-parameters.
This specification permits extracting characteristics and implications of this type
of model that cannot be extracted if only aggregated parameters are used. We
will see, for example, that the coefficient of relative risk aversion plays a signifi-
cant role in explaining the role of money. We test the model and estimate the risk
aversion parameter over the sample period. As risk aversion can be very high in
short periods of time, but cannot be estimated over such short periods, we test the
model again by calibrating a higher risk aversion parameter (twice the previously
estimated value). This strategy allows us to compare the impact of both levels of
risk aversion on the dynamics of the variables.
Our model differs also in its inflation and output dynamics. Standard New
Keynesian DSGE models give an important role to endogenous inertia on both
output (consumption habits) and inflation (price indexation). In fact, both dynam-
ics may have a stronger forward-looking component than an inertial component.
This appears to be the case, at least in the Euro area if not clearly in the U.S. (Galí
et al., 2001). These inertial components may hide part of the role of money. Hence,
our choice to remain as simple as possible on that respect in order to try to unveil
a possible role for money balances.
Finally, Backus et al. (1992) have shown that capital appears to play a rather
4

minor role in the business cycle. To simplify the analysis and focus on the role of
money, we therefore do not include a capital accumulation process in the model,
as in Galí (2008).
We differ from existing theoretical (and empirical) analyses by specifying the
flexible price counterparts of output and real money balances. This imposes a
more elaborate theoretical structure, which provides an improvement on the liter-
ature and enriches the model.
We also differ from the empirical analyses of the Eurozone by using Bayesian
techniques in a New Keynesian DSGE framework, like in Smets and Wouters
(2007), while introducing money in the model. Current literature attempts to in-
troduce money only by aggregating model parameters, therefore leaving aside
relevant information. Here we estimate all micro-parameters of the model under
average (estimated) and high risk aversion. This is an important innovation and
leads to interesting implications.
In order to assess further the role of money we also incorporate and estimate
different Taylor rules (without and with different money variables) and analyse
their impact on the dynamics of the model with the two levels of risk aversion.
In the process we unveil transmission mechanisms generally neglected in tra-
ditional New Keynesian analyses. Given a high enough risk aversion, the frame-
work highlights in particular the non-negligible role of money in explaining out-
put variations.
The dynamic analysis of the model sheds light on the change in the role of
money through time in explaining fluctuations in output. It shows that the impact
of money is stronger in the short than in the long run.
Section 2 of the paper describes the theoretical set up. In Section 3, the model
is calibrated and estimated with Bayesian techniques and by using Euro area data.
Variance decompositions are analysed in this section, with an emphasis on the
impact of the coefficient of relative risk aversion. Section 4 presents alternative
introductions of money in the Taylor rule. Section 5 concludes and the Appendix
presents additional theoretical and empirical results.
2 The model
The model consists of households that supply labor, purchase goods for consump-
tion, hold money and bonds, and firms that hire labor and produce and sell differ-
entiated products in monopolistically competitive goods markets. Each firm sets
the price of the good it produces, but not all firms reset their price during each
period. Households and firms behave optimally: households maximize the ex-
pected present value of utility, and firms maximize profits. There is also a central
bank that controls the nominal rate of interest. This model is inspired by Galí
5

(2008), Walsh (2017) and Smets and Wouters (2003).
2.1 Households
We assume a representative infinitely-lived household, seeking to maximize
Et"∞
∑
k=0
βkUt+k#, (1)
where Utis the period utility function and β<1 is the discount factor.
We assume the existence of a continuum of goods represented by the interval
[0, 1]. The household decides how to allocate its consumption expenditures among
the different goods. This requires that the consumption index Ctbe maximized for
any given level of expenditures. Furthermore, and conditional on such optimal
behavior, the period budget constraint takes the form
PtCt+Mt+QtBtBt1+WtNt+Mt1, (2)
for t=0, 1, 2..., where Wtis the nominal wage, Ptis an aggregate price index
(see Appendix A), Ntis hours of work (or the measure of household members
employed), Btis the quantity of one-period nominally riskless discount bonds
purchased in period tand maturing in period t+1 (each bond pays one unit of
money at maturity and its price is Qtwhere it=ln Qtis the short term nominal
rate) and Mtis the quantity of money holdings at time t. The above sequence of
period budget constraints is supplemented with a solvency condition, such as 8t
lim
n!∞Et[Bn]0.
In the literature, utility functions are usually time-separable. To introduce an
explicit role for money balances, we drop the assumption that household pref-
erences are time-separable across consumption and real money balances. Prefer-
ences are measured with a CES utility function including real money balances.
Under the assumption of a period utility given by
Ut=1
1σ(1b)C1ν
t+beεm
t(Mt/Pt)1ν1σ
1νχ
1+ηN1+η
t, (3)
consumption, labor, money and bond holdings are chosen to maximize Eq. 1 sub-
ject to Eq. 2 and the solvency condition. This CES utility function depends posi-
tively on the consumption of goods, Ct, positively on real money balances, Mt/Pt,
and negatively on labour Nt.σis the coefficient of relative risk aversion of house-
holds (or the inverse of the intertemporal elasticity of substitution), νis the inverse
of the elasticity of money holdings with respect to the interest rate, and can be seen
as a non separability parameter, and ηis the inverse of the elasticity of work effort
6

with respect to the real wage.
It must be noticed that νmust be lower than σ. If ν=σ, Eq. 3 becomes a stan-
dard separable utility function whereby the influence of real money balances on
output, inflation and flexible-price output disappears. This case has been studied
in the literature. In our model, the difference between the risk aversion parameter
and the separability parameter, σν, plays a significant role.
The utility function also contains a structural money demand shock, εm
t.band
χare positive scale parameters.
As described in Appendix A, this setting leads to the following conditions,
which, in addition to the budget constraint, must hold in equilibrium. The result-
ing log-linear version of the first order condition corresponding to the demand for
contingent bonds implies that
ˆ
ct=Et[ˆ
ct+1](ˆ
ıtEt[ˆ
πt+1]) /(νa1(νσ)) (4)
(1a1) (νσ)
νa1(νσ)Et[∆ˆ
mt+1ˆ
πt+1]+ξt,c,
where ξt,c=(1a1)(νσ)
(1ν)(νa1(νσ)) Et∆εm
t+1and by using the steady state of the first
order conditions a1
1=1+b
1b1
ν(1β)ν1
ν. The lowercase (ˆ) denotes the log-
linearized (around the steady state) form of the original aggregated variables.
The demand for cash that follows from the household’s optimization problem
is given by
ν(ˆ
mtˆ
pt)+νˆ
ct+εm
t=a2ˆ
ıt, (5)
with a2=1
exp(1/β)1and where real cash holdings depend positively on consump-
tion with an elasticity equal to unity and negatively on the nominal interest rate.
In what follows we will take the nominal interest rate as the central bank’s policy
instrument. In the literature, due to the assumption that consumption and real
money balances are additively separable in the utility function, cash holdings do
not enter any of the other structural equations: accordingly, the above equation
becomes recursive to the rest of the system of equations.
The first order condition corresponding to the optimal consumption-leisure
arbitrage implies that
ηˆ
nt+(νa1(νσ)) ˆ
ct(νσ) (1a1) ( ˆ
mtˆ
pt)+ξt,m=ˆ
wtˆ
pt, (6)
where ξt,m=(νσ)(1a1)
1νεm
t.
Finally, these equations represent the Euler condition for the optimal intratem-
poral allocation of consumption (Eq. 4), the intertemporal optimality condition
setting the marginal rate of substitution between money and consumption equal
to the opportunity cost of holding money (Eq. 5), and the intratemporal optimality
7

condition setting the marginal rate of substitution between leisure and consump-
tion equal to the real wage (Eq. 6).
2.2 Firms
We assume a continuum of firms indexed by i2[0, 1]. Each firm produces a dif-
ferentiated good but uses an identical technology with the following production
function,
Yt(i)=AtNt(i)1α, (7)
where At=exp (εa
t)is the level of technology, assumed to be common to all firms
and to evolve exogenously over time, and αis the measure of decreasing returns.
All firms face an identical isoelastic demand schedule, and take the aggre-
gate price level Ptand aggregate consumption index Ctas given. As in the stan-
dard Calvo (1983) model, our generalization features monopolistic competition
and staggered price setting. At any time t, only a fraction 1 θof firms, with
0<θ<1, can reset their prices optimally, while the remaining firms index their
prices to lagged inflation.
2.3 Central bank
The central bank is assumed to set its nominal interest rate according to a general-
ized smoothed Taylor rule such as:
ˆ
ıt=(1λi)λπ(ˆ
πtπc)+λxˆ
ytˆ
yf
t+λme
Mt,k+λiˆ
ıt1+εi
t, (8)
where λπ,λxand λmare policy coefficients reflecting the weight on inflation, on
the output gap and on a money variable; the parameter 0 <λi<1 captures the
degree of interest rate smoothing; εi
tis an exogenous ad hoc shock accounting for
fluctuations of the nominal interest rate. πcis an inflation target and e
Mt,kis a
money variable: when k=0, money does not enter the Taylor rule; k=1 to 3
corresponds respectively to the real money gap (difference between real money
balances and its flexible-price counterpart), the nominal money growth and the
real money growth.
2.4 DSGE model
Solving the model (Appendix A) leads to six micro-founded equations and six
dependent variables: inflation, nominal interest rate, output, flexible-price output,
real money balances and its flexible-price counterpart.
Flexible-price output and flexible-price real money balances are completely de-
termined by shocks. Flexible-price output is mainly driven by technology shocks
8

(whereas fluctuations in the output gap can be attributed to supply and demand
shocks). The flexible-price real money balances are mainly driven by money shocks
and flexible-price output.
ˆ
yf
t=υy
aεa
t+υy
mc
mpf
tυy
c+υy
smεm
t, (9)
c
mpf
t=υm
y+1Ethˆ
yf
t+1i+υm
yˆ
yf
t+1
νεm
t, (10)
ˆ
πt=βEt[ˆ
πt+1]+κx,tˆ
ytˆ
yf
t+κm,tc
mptc
mpf
t, (11)
ˆ
yt=Et[ˆ
yt+1]κr(ˆ
ıtEt[ˆ
πt+1]) (12)
+κmp Et∆c
mpt+1+κsm Et∆εm
t+1,
c
mpt=ˆ
ytκiˆ
ıt+1
νεm
t, (13)
ˆ
ıt=(1λi)λπ(ˆ
πtπc)+λxˆ
ytˆ
yf
t+λme
Mt,k+λiˆ
ıt1+εi
t, (14)
where
υy
a=1+η
(ν(νσ)a1)(1α)+η+α,
υy
m=(1α)(νσ)(1a1)
(ν(νσ)a1)(1α)+η+α,
υy
c=(1α)
(ν(νσ)a1)(1α)+η+αln ε
ε1,
υy
sm =(νσ)(1a1)(1α)
((ν(νσ)a1)(1α)+η+α)(1ν),
υm
y+1=a2
ν(ν(νσ)a1),
υm
y=1+a2
ν(ν(νσ)a1),
κx,t=ν(νσ)a1+η+α
1α(1α)(1
θβ)(1θ)(1+(ε1)εp
t)
1+(α+εp
t)(ε1),
κm,t=(σν) (1a1)(1α)(1
θβ)(1θ)(1+(ε1)εp
t)
1+(α+εp
t)(ε1),
κr=1
νa1(νσ),
κmp =(σν)(1a1)
νa1(νσ),
κsm =(1a1)(νσ)
(νa1(νσ))(1ν),
κi=a2/ν,
with a1=1
1+(b/(1b))1/ν(1β)(ν1)/νand a2=1
exp(1/β)1.
The structural money shock and the markup shock1,εm
tand εp
t, the exogenous
component of the interest rate, εi
t, and of the technology, εa
t, are assumed to follow
a first-order autoregressive process with an i.i.d.-normal error term such as εk
t=
ρkεk
t1+ωk,twhere εk,tN(0; σk)for k=fm,p,i,ag.
1The markup shock is introduced and explained in Appendix A.
9

As can be seen, σand νinfluence all macro-parameters. This influence high-
lights the fact that separability and risk aversion are prominent factors involved
in output, inflation, real money balances and nominal interest rate dynamics, as
well as in flexible-price output and flexible-price real money balances. Moreover,
as far as money is concerned, it is the three macro-parameters, υy
m,κmand κmp ,
that are essential to highlight its possible role in the dynamics of the model: these
coefficients determine the weight of money in Eq. 9, Eq. 11 and Eq. 12.
3 Empirical results
As in Smets and Wouters (2003) and An and Schorfheide (2007), we apply Bayesian
techniques to estimate our DSGE model. Contrary to Ireland (2004) or Andrés
et al. (2006), we did not opt to estimate our model by using the maximum of like-
lihood because such computation hardly converges toward a global maximum.
First, we estimate the risk aversion level and all parameters over the sample pe-
riod. Second, we re-estimate all parameters but with a constant risk aversion level
calibrated to approximately twice its estimated value. We also test four specifica-
tions of the Taylor rule under these two alternatives risk aversion levels.
3.1 Euro area data
In our model of the Eurozone, ˆ
πtis the log-linearized detrended inflation rate
measured as the yearly log difference of detrended GDP Deflator from one quarter
to the same quarter of the previous year; ˆ
ytis the log-linearized detrended output
per capita measured as the difference between the log of the real GDP per capita
and its trend; and ˆ
ıtis the short-term (3-month) detrended nominal interest rate.
These Data are extracted from the Euro Area Wide Model database (AWM) of
Fagan et al. (2005). c
mptis the log-linearized detrended real money balances per
capita measured as the difference between the real money per capita and its trend,
where real money per capita is measured as the log difference between the money
stock per capita and the GDP Deflator. We use the M3 monetary aggregate from
the Eurostat database. ˆ
yf
t, the flexible-price output, and c
mpf
t, the flexible-price real
money balances, are completely determined by structural shocks. To make output
and real money balances stationary, we use first log differences, as in Adolfson
et al. (2008).
3.2 Calibration
Following standard conventions, we calibrate beta distributions for parameters
that fall between zero and one, inverted gamma distributions for parameters that
10

need to be constrained to be greater than zero, and normal distributions in other
cases.
The calibration of σis inspired by Rabanal and Rubio-Ramírez (2005) and by
Casares (2007). They choose, respectively, a risk aversion parameter of 2.5 and
1.5. In line with these values, we consider that σ=2 corresponds to a standard
risk aversion while values above that level imply higher and higher risk aversion,
hence our choice of σ=4 to represent a high level of risk aversion, twice the
standard value. Excepted for risk aversion, we adopt the same priors in the two
models.
As in Smets and Wouters (2003), the standard errors of the innovations are
assumed to follow inverse gamma distributions and we choose a beta distribution
for shock persistence parameters (as well as for the backward component of the
Taylor rule) that should be lesser than one.
The calibration of α,β,θ,η, and εcomes from Smets and Wouters (2003, 2007),
Casares (2007) and Galí (2008). The smoothed Taylor rule (λi,λπ,λxand λm) is
calibrated following Gerlach-Kristen (2003), Andrés et al. (2009) and Barthélemy
et al. (2011), analogue priors as those used by Smets and Wouters (2003) for the
monetary policy parameters. In order to observe the behavior of the central bank,
we assign a higher standard error (0.50) and a Normal prior law for the Taylor
rule’s coefficients except for the smoothing parameter, which is restricted to be
positive and below one (Beta distribution). The inflation target, πc, is calibrated
to 2% and estimated. v(the non-separability parameter) must be greater than
one. κi(Eq. 13) must be greater than one as far as this parameter depends on
the elasticity of substitution of money demand with respect to the cost of holding
money balances, as explained in Söderström (2005); while still informative, this
prior distribution is dispersed enough to allow for a wide range of possible and
realistic values to be considered (i.e. σ>v>1).
Our prior distributions are not dispersed to focus on the role of risk aversion.
The calibration of the shock persistence parameters and the standard errors of the
innovations follows Smets and Wouters (2003) and Fève et al. (2010). All the stan-
dard errors of shocks are assumed to be distributed according to inverted Gamma
distributions, with prior means of 0.02. The latter ensures that these parameters
have a positive support. The autoregressive parameters are all assumed to follow
Beta distributions. All these distributions are centered around 0.75, except for the
autoregressive parameter of the monetary policy shock, which is centered around
0.50, as in Smets and Wouters (2003). We take a common standard error of 0.1 for
the shock persistence parameters, as in Smets and Wouters (2003).
11

3.3 Results
As already said, we calibrate first the level of risk aversion to its standard value,
σ=2, and we estimate it. This model version is considered as a benchmark spec-
ification. In the second version, we set σ=4, about twice this estimated value.
This set-up is motivated by Holden and Subrahmanyam (1996). They show that
acquisition of short-term information is encouraged by high risk aversion level,
and that the latter can cause all potentially informed investors in the economy to
concentrate exclusively on the short-term instead of the long-term. Risk aversion
is generally low in the medium and long run while it could be very high in short
periods. As we can’t estimate risk aversion in the short run, we choose to estimate
our model also by setting σ=4, i.e. a high risk aversion level.
Table 1. Bayesian estimation of structural parameters
Priors Posteriors
σestimated σ=4
Law Mean Std. 5% Mean 95% 5% Mean 95%
αbeta 0.33 0.05 0.282 0.378 0.473 0.384 0.484 0.589
θbeta 0.66 0.05 0.657 0.710 0.764 0.673 0.726 0.777
vnormal 1.25 0.05 1.380 1.447 1.518 1.491 1.528 1.568
σnormal 2.00 0.50 1.771 2.157 2.545
bbeta 0.25 0.10 0.085 0.252 0.410 0.084 0.246 0.399
ηnormal 1.00 0.10 0.895 1.053 1.218 0.957 1.120 1.281
εnormal 6.00 0.10 5.807 5.978 6.143 5.815 5.979 6.141
λibeta 0.50 0.10 0.449 0.573 0.700 0.502 0.614 0.726
λπnormal 3.00 0.50 2.856 3.494 4.104 2.874 3.491 4.145
λxnormal 1.50 0.50 1.133 1.872 2.614 1.175 1.923 2.632
λmnormal 1.50 0.50 0.320 1.011 1.681 0.276 0.964 1.635
πcnormal 2.00 0.10 1.733 1.903 2.071 1.739 1.908 2.071
ρabeta 0.75 0.10 0.987 0.992 0.997 0.991 0.994 0.998
ρpbeta 0.75 0.10 0.960 0.973 0.987 0.958 0.972 0.986
ρibeta 0.50 0.10 0.377 0.460 0.540 0.490 0.560 0.631
ρmbeta 0.75 0.10 0.952 0.971 0.991 0.974 0.984 0.995
σainvgamma 0.02 2.00 0.011 0.013 0.016 0.015 0.019 0.022
σiinvgamma 0.02 2.00 0.013 0.018 0.023 0.009 0.012 0.015
σpinvgamma 0.02 2.00 0.003 0.004 0.006 0.003 0.004 0.006
σminvgamma 0.02 2.00 0.023 0.026 0.029 0.024 0.027 0.030
In this section, we present only results with a Taylor rule incorporating the real
money gap ( e
Mt,1 =c
mptc
mpf
t), the most significant money variable as shown in
12

Section 4. The model is estimated with 117 observations from 1980 (Q4) to 2009
(Q4) in order to avoid high volatility periods before 1980 and to take into con-
sideration the core of the global crisis. The estimation of the implied posterior
distribution of the parameters under the two configurations of risk (Table 1) is
done using the Metropolis-Hastings algorithm (10 distinct chains, each of 50000
draws; see Smets and Wouters (2007) and Adolfson et al. (2007)). Average accep-
tation rates per chain for the benchmark model (σestimated) are included in the
interval [0.256; 0.261]and for the high risk aversion model (σ=4) are included in
the interval [0.248; 0.252]. The literature has settled on a value of this acceptance
rate around 0.25.
Priors and posteriors distributions are presented in Appendix B. To assess the
model validation, we insure convergence of the proposed distribution to the target
distribution (Appendix C).
3.4 Variance decompositions
In this section we analyse the forecast error variance of each variable following
exogenous shocks, in two different ways. The analysis is conducted first via an
unconditional variance decomposition (long term), and second via a conditional
variance decomposition (short term and over time).
3.4.1 Long term analysis
The unconditional variance decomposition (Table 2) shows that, whatever the
risk aversion level, the variance of output essentially results from the technology
shock, the remaining from the other shocks. If money plays some role, this role is
rather minor (an impact of 3.07%) under an estimated standard risk aversion.
Table 2. Unconditional variance decomposition (percent)
estimated σ σ =4
εp
tεi
tεm
tεa
tεp
tεi
tεm
tεa
t
ˆ
yt1.65 1.09 3.07 94.18 0.83 0.28 10.38 88.51
ˆ
πt97.66 2.14 0.09 0.12 97.64 1.79 0.24 0.33
ˆ
ıt78.53 19.64 0.64 1.19 74.41 20.67 1.86 3.07
c
mpt1.85 0.91 52.49 44.75 0.83 0.26 60.87 38.04
ˆ
yf
t0.00 0.00 3.06 96.94 0.00 0.00 10.23 89.77
c
mpf
t0.00 0.00 54.42 45.58 0.00 0.00 62.04 37.96
13

Yet, as Table 2 shows, the money shock contribution to the business cycle de-
pends on the value of agents’ risk aversion. The estimation with the higher risk
aversion (σ=4) gives interesting information as to the role of money, and more
generally as to the role of each shock in the long run.
These results indicate that a higher coefficient of relative risk aversion increases
significantly the impact of money on output. Yet it does not really change the im-
pact of money on inflation dynamics, essentially explained by the markup shock
whatever the level of risk aversion. The very small role of the money shock on
inflation dynamics is a consequence of the low value of κm,tin Eq. 11, whatever
the level of risk aversion, even though κm,tincreases with σ. By comparison, the
value of κmp in Eq. 12 is significantly higher, and increases no less significantly
with σ(see Table 5 in Appendix D).
If more than 88% of the variance of output is still explained by the technology
shock with the high risk calibration (σ=4), the role of the interest rate shock and
the role of the markup shock decrease whereas the impact of the money shock
increases from about 3% to 10.4%, i.e. is multiplied by a factor of 3.4. Similarly, the
impact of shocks on flexible-price output also depends on the risk aversion level.
The role of the money shock increases with the risk level from about 3% to 10.2%.
Although money enters the Taylor rule, it does not have a significant role in
the dynamics of the interest rate, whatever the level of risk aversion.
Furthermore, following an increase in the risk aversion level, the dynamics of
real money balances and its flexible-price counterpart are to a lesser extent ex-
plained by the technology shock. Unsurprisingly, the variance of these variables
are mainly explained by the money shock.
3.4.2 Short term and through time analysis
The analysis through time (conditional variance) of the different shocks on output
(Figure 1) shows that the impact of the money shock decreases with the time hori-
zon, as for the interest rate shock2. Under high risk aversion, the role of money in
the first periods remains around 22%, i.e. twice the value in the long term (10.38%).
As far as inflation variance is concerned, the markup shock not only dominates
the process but its impact does not change through time in both risk configura-
tions.
In the short term, as shown in Table 3, the monetary policy shock explains
around 83% of the nominal interest rate variance whereas the markup shock ex-
plain less than 17% for the two configurations of risk. For longer terms, there is an
inversion: whatever the risk aversion level, the interest rate variance is dominated
by the price-markup shock and the monetary policy shock explains around 20%
2The conditional variances decompositions figures for the other variables are not shown here
but are available upon request.
14

Figure 1. Conditional forecast error variance decomposition of output
010 20 30 40 50 60
0
20
40
60
80
100 σ esti mated
%
010 20 30 40 50 60
0
20
40
60
80
100 σ = 4
Quarter
%
Mar kup Shock
Inter est Rate Shoc k
Money Shock
Technolog y Shock
of the interest rate variance. Although money is introduced in the Taylor rule, the
money shock has a minor impact on the nominal interest rate variance at any time
horizon.
Table 3. First period variance decomposition (percent)
estimated σ σ =4
εp
tεi
tεm
tεa
tεp
tεi
tεm
tεa
t
ˆ
yt2.16 31.17 7.50 59.16 2.23 11.19 22.38 64.20
ˆ
πt77.72 22.16 0.08 0.03 83.73 16.08 0.13 0.06
ˆ
ıt16.35 83.44 0.14 0.07 16.66 82.99 0.23 0.13
c
mpt1.28 13.76 69.46 15.49 1.09 5.46 77.25 16.20
ˆ
yf
t0.00 0.00 10.56 89.44 0.00 0.00 24.89 75.11
c
mpf
t0.00 0.00 81.72 18.28 0.00 0.00 82.62 17.38
The role of monetary policy on real money balances is different in the short
term: the monetary policy shock explains almost 14% of the variance of real money
balances in the short term under the standard risk aversion (and around 5% under
15

high risk aversion), whereas, under the two configurations of risk, it has a very
small role at longer horizons. Similarly, the technology shock explains around
16% of the real money balances variance in both configurations of risk, whereas
at longer horizons it explains around 45% of the real money variance under the
estimated risk aversion (and around 38% under the high risk aversion).
It is interesting to notice that the same type of analysis applies to the flexible-
price output variance decomposition. In the short term as well as in the long term,
technology is the main explanatory factor. The role of money increases with the
relative risk aversion coefficient in the short term (from a weight of less than 11%
under standard calibration to close to 25% under high risk aversion calibration),
as in the long term, whereas the monetary policy and the price-markup shocks
play no role in the flexible-price output and the flexible-price real money balances
dynamics.
3.5 Interpretation
The estimates of the macro-parameters (aggregated structural parameters) for es-
timated and high risk aversions are reported in Appendix D (Table 5). These
estimates suggest that a change in risk aversion implies significant variations in
the value of several macro-parameters, notably υy
m,κmand κmp - respectively the
weight of money in the flexible-price output, inflation and output equations. More-
over, the weight of the money shock on output dynamics, κsm, and on flexible-price
output, υy
sm, increases with risk aversion, thus reinforcing the role of money in the
dynamics of the model. It is also worth mentioning that the smoothing parameter
in the Taylor rule equation, λi, increases with risk aversion. This may reflect the
idea that the central bank strives for financial stability in crisis periods.
The comparison between the variance decompositions (Table 2 and 3) of the
two model versions illustrates the fact that the role of the money shock on output
and flexible-price output depends crucially on the degree of agents’ risk aversion,
increases accordingly, and becomes significant when risk aversion is high, what-
ever the time horizon. This result highlights the role of real money to smooth
consumption through time, especially when risk aversion reaches certain levels.
Impulse response functions for the two configurations of risk (Appendix E)
highlights the role of risk aversion on the dynamics of several of the model’s vari-
ables. These results also demonstrate the predominant role of the risk aversion
level on the impact of the money shock on output, inflation, and real and nominal
interest rates. The higher the risk aversion level, the greater the reactions to the
shocks.
16

4 Money in the Taylor rule
To evaluate further the role of money we analyse different specifications of the
Taylor rule ( e
Mt,kfor k=f0, 1, 2, 3g, as exposed in Section 2.3), first without money,
then with money introduced in three different ways. We thus test both models
with four types of Taylor rules:
- With no money ( e
Mt,0 =0);
- With a real money gap ( e
Mt,1 =c
mptc
mpf
t);
- With a nominal money growth ( e
Mt,2 =b
mtb
mt1);
- With a real money growth ( e
Mt,3 =c
mptc
mpt1).
As shown in Table 4, all the coefficients of the inflation and the output gap
variables, as well as the interest rate smoothing coefficients are significant (Student
tests superior to 1.96) whatever the risk. Yet, the money coefficient is significant
only with the money gap variable ( e
Mt,1).
Table 4. Alternative ECB’s Taylor rules
estimated σ σ =4
e
Mt,0 e
Mt,1 e
Mt,2 e
Mt,3 e
Mt,0 e
Mt,1 e
Mt,2 e
Mt,3
λi0.527 0.573 0.561 0.547 0.545 0.614 0.546 0.5469
(1λi)λπ1.594 1.491 1.463 1.537 1.579 1.345 1.585 1.575
(1λi)λx1.066 0.799 1.018 1.042 1.034 0.741 1.038 1.039
(1λi)λm0.431 0.1360.0840.371 -0.012-0.018
STy
m(%)7.05 7.50 2.23 3.66 22.61 22.38 23.20 23.28
LTy
m(%)2.75 3.07 2.24 2.36 9.56 10.38 9.29 9.15
LMD -629.8 -618.2 -634.9 -635.3 -639.8 -626.5 -646.1 -646.1
estimations are not significant in terms of student tests (t<1.645)
Furthermore, the log marginal density (LMD) of the data measured through
a Laplace approximation indicates that the Taylor rule including this real money
gap performs better than the others, followed by the no-money ( e
Mt,0) case.
These results suggest that if money has to be introduced in the ECB mone-
tary policy reaction function, it should rather be a real money gap variable than
a money growth variable (contrary to what Andrés et al. (2009) and Barthélemy
et al. (2011) found, whereas Fourçans (2004, 2007) didn’t find such a role for money
growth).
Either way, it is interesting to notice that whatever the formulation of the Taylor
rule, the estimated parameters of the whole model are quite similar. This is true
with both levels of risk aversion.
17

The impact of a money shock on output, as shown through the short term (STy
m,
in the first period) and the long term (LTy
m) variance decomposition of output with
respect to a money shock, are also rather similar whatever the Taylor rule (Table 3).
The impact of money increases with the risk aversion coefficient, and is stronger
in the short run than in the long run, especially when risk aversion is high.
5 Conclusion
We built and empirically tested a model of the Eurozone, with two levels of risk
aversion and with a particular emphasis on the role of money. The model follows
the New Keynesian DSGE framework, with money in the utility function whereby
real money balances affect the marginal utility of consumption.
By using Bayesian estimation techniques, we shed light not only on the deter-
minants of output and inflation dynamics but in addition on the interest rate, real
money balances, flexible-price output and flexible-price real money balances vari-
ances. We further investigated how the results are affected when intertemporal
risk aversion changes, especially as far as money is concerned.
Money plays a minor role in the estimated model with a moderate risk aver-
sion. Most of the variance of output is explained by the technology shock, the rest
by a combination of markup, interest rate and money shocks, a result in line with
current literature (Ireland, 2004; Andrés et al., 2006, 2009). However, another cali-
bration with a higher risk aversion (twice the estimated value) implies that money
plays a non-negligible role in explaining output and flexible-price output fluctua-
tions. We also found that the short term impact is significantly stronger than the
long run one. These results differ from existing literature using New Keynesian
DSGE frameworks with money, insofar as it neglects the impact of a high enough
risk factor.
On the other hand, the explicit money variable does not appear to have a no-
table direct role in explaining inflation variability. The overwhelming explanatory
factor is the price-markup whatever the level of risk aversion.
Another outcome concerns monetary policy. The higher the risk aversion,
the stronger the smoothing of the interest rate. This reflects probably the central
bankers’ objective not to agitate markets.
Our results suggest that a nominal or real money growth variable does not
improve the estimated ECB monetary policy rule. Yet, a real money gap variable
(the difference between the real money balances and its flexible-price counterpart)
significantly improves the estimated Taylor rule. This being said, the introduction,
or not, of a money variable in the ECB monetary policy reaction function does not
really appear to change significantly the impact of money on output and inflation
dynamics.
18

All in all, one may infer from our analysis that by changing economic agents’
perception of risks, the last financial crisis may have increased the role of real
money balances in the transmission mechanism and in output changes.
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21

A Solving the model
Price dynamics
Let’s assume a set of firms not reoptimizing their posted price in period t. Us-
ing the definition of the aggregate price level and the fact that all firms resetting
prices choose an identical price P
t, leads to
Pt=hθP1Λt
t1+(1θ) (P
t)1Λti,
where Λt=1+1
1
ε1+εp
t
is the elasticity of substitution between consumption goods
in period t, and Λt
Λt1is the markup of prices over marginal costs (time varying).
Dividing both sides by Pt1and log-linearizing around P
t=Pt1yields
πt=(1θ) (p
tpt1). (15)
In this setup, we don’t assume inertial dynamics of prices. Inflation results
from the fact that firms reoptimizing in any given period their price plans, choose
a price that differs from the economy’s average price in the previous period.
Price setting
A firm reoptimizing in period tchooses the price P
tthat maximizes the current
market value of the profits generated while that price remains effective. This prob-
lem is solved and leads to a first-order Taylor expansion around the zero inflation
steady state:
p
tpt1=(1βθ)
∞
∑
k=0
(βθ)kEthc
mct+kjt+(pt+kpt1)i, (16)
where c
mct+kjt=mct+kjtmc denotes the log deviation of marginal cost from its
steady state value mc =µ, and µ=ln ε
ε1is the log of the desired gross
markup.
Equilibrium
Market clearing in the goods market requires Yt(i)=Ct(i)for all i2[0, 1]and
all t. Aggregate output is defined as Yt=R1
0Yt(i)11
ΛtdiΛt
Λt1; it follows that
Yt=Ctmust hold for all t. One can combine the above goods market clearing
condition with the consumer’s Euler equation (Eq. 4) to yield the equilibrium
22

condition
ˆ
yt=Et[ˆ
yt+1]1
νa1(νσ)(ˆ
ıtEt[ˆ
πt+1]) (17)
+(σν) (1a1)
νa1(νσ)(Et[∆ˆ
mt+1]Et[ˆ
πt+1]) +ξt,c.
Market clearing in the labor market requires Nt=R1
0Nt(i)di. By using the
production function (Eq. 7) and taking logs, one can write the following approxi-
mate relation between aggregate output, employment and technology as
yt=εa
t+(1α)nt. (18)
An expression is derived for an individual firm’s marginal cost in terms of the
economy’s average real marginal cost:
mct=(ˆ
wtˆ
pt)d
mpnt(19)
=(ˆ
wtˆ
pt)1
1α(εa
tαˆ
yt), (20)
for all t, where d
mpntdefines the economy’s average marginal product of labor. As
mct+kjt=(ˆ
wt+kˆ
pt+k)mpnt+kjtwe have
mct+kjt=mct+kαΛt
1α(p
tpt+k), (21)
where the second equality follows from the demand schedule combined with the
market clearing condition ct=yt. Substituting Eq. 21 into Eq. 16 yields
p
tpt1=(1βθ)
∞
∑
k=0
Θt+k(βθ)kEt[c
mct+k]+
∞
∑
k=0
(βθ)kEt[πt+k], (22)
where Θt=1α
1α+αΛt1 is time varying in order to take into account the markup
shock.
Finally, Eq. 15 and Eq. 22 yield the inflation equation
πt=βEt[πt+1]+λmctc
mct, (23)
where β,λmct=(1θ)(1βθ )
θΘt.λmctis strictly decreasing in the index of price
stickiness θ, in the measure of decreasing returns α, and in the demand elasticity
Λt.
Next, a relation is derived between the economy’s real marginal cost and a
measure of aggregate economic activity. From Eq. 6 and Eq. 18, the average real
23

marginal cost can be expressed as
mct=ν(νσ)a1+η+α
1αˆ
ytεa
t1+η
1α(24)
+(σν) (1a1) ( ˆ
mtˆ
pt)+ξt,m.
Under flexible prices the real marginal cost is constant and equal to mc =µ.
Defining the natural level of output, denoted by yf
t, as the equilibrium level of
output under flexible prices leads to
mc =ν(νσ)a1+η+α
1αˆ
yf
tεa
t1+η
1α(25)
+(σν) (1a1)c
mpf
t+ξt,m,
where c
mpf
t=ˆ
mf
tˆ
pf
t, thus implying
ˆ
yf
t=υy
aεa
t+υy
mc
mpf
tυy
c+υy
smεm
t, (26)
where
υy
a=1+η
(ν(νσ)a1) (1α)+η+α
υy
m=(1α) (νσ) (1a1)
(ν(νσ)a1) (1α)+η+α
υy
c=(1α)
(ν(νσ)a1) (1α)+η+αln ε
ε1
υy
sm =(νσ) (1a1) (1α)
((ν(νσ)a1) (1α)+η+α) (1ν).
We deduce from Eq. 17 that ˆ
ıf
t=(ν(νσ)a1)Eth∆ˆ
yf
t+1iand by using Eq.
5 we obtain the following equation of real money balances under flexible prices
c
mpf
t=υm
y+1Ethˆ
yf
t+1i+υm
yˆ
yf
t+1
νεm
t, (27)
where υm
y+1=a2(ν(νσ)a1)
νand υm
y=1+a2(ν(νσ)a1)
ν
Subtracting Eq. 25 from Eq. 24 yields
c
mct=φxˆ
ytˆ
yf
t+φmc
mptc
mpf
t, (28)
where c
mpt=ˆ
mtˆ
ptis the log linearized real money balances around its steady
state, c
mpf
tis its flexible-price counterpart, φx=ν(νσ)a1+η+α
1αand φm=
(σν) (1a1).
24

By combining Eq. 28 with Eq. 23 we obtain
ˆ
πt=βEt[ˆ
πt+1]+κx,tˆ
ytˆ
yf
t+κm,tc
mptc
mpf
t, (29)
where ˆ
ytˆ
yf
tis the output gap,c
mptc
mpf
tis the real money balances gap,
κx,t=ν(νσ)a1+η+α
1α(1α)1
θβ(1θ)1+(ε1)εp
t
1+α+εp
t(ε1),
and
κm,t=(σν) (1a1)(1α)1
θβ(1θ)1+(ε1)εp
t
1+α+εp
t(ε1).
Then Eq. 29 is our first equation relating inflation to its one period ahead fore-
cast, the output gap and real money balances.
The second key equation describing the equilibrium of the model is obtained
by rewriting Eq. 17 so as to determine output
ˆ
yt=Et[ˆ
yt+1]κr(ˆ
ıtEt[ˆ
πt+1]) +κmpEt∆c
mpt+1+ξt,c, (30)
where κr=1
ν(νσ)a1,κmp =(σν)(1a1)
νa1(νσ)and ξt,c=κsm Et∆εm
t+1where κsp =
1
νa1(νσ)and κsm =(1a1)(νσ)
νa1(νσ)1
1ν. Eq. 30 is thus a dynamic IS equation
including the real money balances.
The third key equation describes the real money stock. From Eq. 5 we obtain
c
mpt=ˆ
ytκiˆ
ıt+1
νεm
t, (31)
where κi=a2/ν.
25

B Priors and posteriors
The vertical line of Figures 2 and 3 denotes the posterior mode, the grey line the
prior distribution, and the black line the posterior distribution.
Figure 2. Priors and posteriors (σestimated)
0.02 0.04 0.06 0.08 0.1
0
100
200
300
SE_ua
00.05 0.1
0
50
100
SE_ui
00.05 0.1
0
200
400
SE_up
0.02 0.04 0.06 0.08 0.1
0
50
100
150
200
SE_um
0.2 0 .4 0.6 0.8
0
2
4
6
8
alpha
0.5 0 .6 0.7 0 .8 0.9
0
5
10
teta
1.2 1 .4 1.6
0
5
10 vega
1 2 3
0
0.5
1
1.5
si gma
00.5 1
0
1
2
3
4
b
1.5 22 .5
0
1
2
3
4
pb
0.5 11 .5
0
1
2
3
4
ne ta
5.5 66 .5
0
1
2
3
4
epsilon
00.5 1
0
2
4
li1
2 4 6
0
0.5
1
li2
024
0
0.2
0.4
0.6
0.8
li3
-1 0 1 2 3
0
0.5
1
li4
0.4 0 .6 0.8 1
0
50
100
rhoa
0.4 0 .6 0.8 1
0
20
40
rhop
0.2 0 .4 0.6
0
2
4
6
8
rhoi
0.4 0 .6 0.8 1
0
10
20
30
rhom
26

Figure 3. Priors and posteriors (σ=4)
0.02 0.04 0.06 0.08 0.1
0
50
100
150
SE_ua
0.02 0.04 0.06 0.08 0 .1
0
50
100
150
200
SE_ui
0.02 0.04 0.06 0.08 0 .1
0
200
400
SE_up
0.02 0.04 0.06 0.08 0.1
0
50
100
150
200
SE_um
0.2 0 .4 0.6 0 .8
0
2
4
6
8
alpha
0.5 0 .6 0.7 0 .8 0.9
0
5
10
teta
1.2 1 .4 1.6
0
5
10
15
vega
-0 .2 00.2 0.4 0 .6 0.8
0
1
2
3
4
b
1.5 22 .5
0
1
2
3
4
pb
0.5 11 .5
0
1
2
3
4
ne ta
5.5 66 .5
0
1
2
3
4
epsilon
0.2 0 .4 0.6 0 .8 1
0
2
4
6
li1
2 4 6
0
0.5
1
li2
0 2 4
0
0.2
0.4
0.6
0.8
li3
0 2 4
0
0.5
1
li4
0.4 0 .6 0.8 1
0
50
100
150
200
rhoa
0.4 0 .6 0.8 1
0
20
40
rhop
0.4 0 .6 0.8
0
5
10 rhoi
0.4 0.6 0.8 1
0
20
40
60
rhom
27

C Model validation
The diagnosis concerning the numerical maximization of the posterior kernel indi-
cates that the optimization procedure leads to a robust maximum for the posterior
kernel. The convergence of the proposed distribution to the target distribution is
thus satisfied.
Figure 4. Multivariate MH convergence diagnosis (σestimated)
0.5 11.5 22.5 33.5 44.5 5
x 10
4
6.5
7
7.5
8
8.5 Interval
0.5 11.5 22.5 33.5 44.5 5
x 10
4
6
8
10
12 m2
0.5 11.5 22.5 33.5 44.5 5
x 10
4
30
40
50
60
70 m3
A diagnosis of the overall convergence for the Metropolis-Hastings sampling
algorithm is provided in Figure 4 and Figure 5.
Each graph represents specific convergence measures with two distinct lines
that show the results within (red line) and between (blue line) chains (Geweke,
1999). Those measures are related to the analysis of the parameters mean (in-
terval), variance (m2) and third moment (m3). For each of the three measures,
convergence requires that both lines become relatively horizontal and converge to
each other in both models.
From Figure 4, it can be inferred that the model with standard risk aversion
needs more chain to stabilize m3 (third moment), in comparison with the case
where risk aversion is high (Figure 5).
Diagnosis for each individual parameter (not included but it can be provided
upon request) indicates that most of the parameters do not exhibit convergence
problems.
28

Figure 5. Multivariate MH convergence diagnosis (σ=4)
0.5 11.5 22.5 33.5 44.5 5
x 10
4
6.5
7
7.5
8
8.5 Interval
0.5 11.5 22.5 33.5 44.5 5
x 10
4
6
8
10
12 m2
0.5 11.5 22.5 33.5 44.5 5
x 10
4
20
40
60
80 m3
Moreover, a BVAR identification analysis (Ratto, 2008) suggests that all para-
meter values are stable.
The estimates of the innovation component of each structural shock, respec-
tively for the estimated σand the calibrated σ=4, respect the i.i.d. properties and
are centered around zero. This reinforces the statistical validity of the estimated
model (the corresponding figures can be provided by the authors).
29

D Macro-parameters
Table 5. Aggregated structural parameters
σestimated σ=4
υy
a0,8166 0,6849
υy
m-0,1027 -0,3508
υy
c0,0452 0,0304
υy
sm 0,2294 0,6644
υm
y+1-0,6889 -1,0841
υm
y1,6889 2,0841
κx,t0,1057 0,0960
κm,t0,0108 0,0337
κr0,5741 0,3457
κmp 0,2386 0,7286
κsm -0,5328 -1,3799
κi0,3956 0,3748
λi0,5732 0,6146
(1λi)λπ1,4914 1,3455
(1λi)λx0,7991 0,7412
(1λi)λm0,4314 0,3717
E Impulse response functions
The thin solid line of Figure 6 represents the impulse response functions of the
model with estimated risk aversion while the dashed line represents the impulse
response functions of the model with high risk aversion (σ=4).
After a markup shock, the inflation rate and the nominal interest rise, then
gradually decrease toward the steady state. The output and the output gap de-
crease then increase to their steady state values.
After an interest rate shock, inflation, output and the output gap fall. The real
rate of interest rises. Real money growth displays an overshooting/undershooting
process in the first periods, then rapidly returns to its steady state value.
After a technology shock, the output gap, the nominal and real interest rate,
and the inflation decrease whereas output as well as real money balances and real
money growth rise.
30

Figure 6. Impulse response functions with both risk configurations
0
0.2
0.4
Markup
Shock
Inflation
(%)
-0.4
-0.2
0
Output
(%)
-1
0
1x 10-15
Flexible
Price
Output (%)
0
0.2
0.4
Nominal
Interest
Rate (%)
-0.2
0
0.2
Real Interest
Rate (%)
-0.4
-0.2
0
Real
Money (%)
-0.5
0
0.5
Real Money
Growth (%)
-2
0
2x 10-15
Flexible
Price Real
Money (%)
-0.4
-0.2
0
Output
Gap (%)
-0.4
-0.2
0
Real Money
Gap (%)
020 40
0
0.5
Shock
(%)
Quarters
-0.01
-0.005
0
Technology
Shock
0.5
1
1.5
0.5
1
1.5
-0.04
-0.02
0
-0.04
-0.02
0
0.5
1
1.5
-2
0
2
0.5
1
1.5
-0.01
-0.005
0
-5
0
5x 10-3
020 40
1
1.5
2
Quarters
0
0.01
0.02
Money
Shock
0
0.5
1
0
0.5
1
0
0.05
0
0.02
0.04
0
2
4
-5
0
5
0
2
4
0
0.01
0.02
-0.02
-0.01
0
020 40
0
2
4
Quarters
-0.2
-0.1
0
Interest
Rate Shock
-1
-0.5
0
-2
0
2x 10-16
0
0.5
1
0
0.5
1
-1
-0.5
0
-1
0
1
0
1
2x 10-16
-1
-0.5
0
-1
-0.5
0
020 40
0
1
2
Quarters
31

After a money shock, the nominal and the real rate of interest, the output and
the output gap rise. Inflation increases a bit then decreases through time to its
steady state value.
The flexible-price output and the flexible-price real money balances increase
after a technology shock and after a money shock.
All these results are in line with the DSGE literature, especially with Smets and
Wouters (2003) and Galí (2008).
32