Content uploaded by Jonathan Benchimol

Author content

All content in this area was uploaded by Jonathan Benchimol on Mar 24, 2019

Content may be subject to copyright.

Risk aversion in the Eurozone

Jonathan Benchimol

January, 2014

Abstract

We propose a New Keynesian Dynamic Stochastic General Equi-

librium (DSGE) model where a risk aversion shock enters a separable

utility function. We analyze …ve periods from 1971 through 2011, each

lasting for twenty years, to follow over time the dynamics of several

parameters such as the risk aversion parameter; the Taylor rule coef-

…cients; and the role of the risk aversion shock in output, in‡ation,

interest rate, and real money balances in the Eurozone. Our analysis

suggests that risk aversion was a more important component of output

and real money balance dynamics between 2006 and 2011 than it was

between 1971 and 2006, at least in the short run.

Keywords: Risk aversion, Output, Money, Eurozone, New Keyne-

sian DSGE models, Bayesian estimation.

JEL Classi…cation Number: E23, E31, E51.

Please cite this paper as:

Benchimol, J., 2014. Risk aversion in the Eurozone. Research in

Economics 68 (1), 39–56.

Bank of Israel, Research Department, POB 780, 91007 Jerusalem, Israel. Phone:

+972-2-6552641. Fax: +972-2-6669407. Email: jonathan.benchimol@boi.org.il. This

paper does not necessarily re‡ect the views of the Bank of Israel. I would like to thank

André Fourçans, Christian Bordes, Laurent Clerc, Marc-Alexandre Sénégas, and the two

anonymous referees for their useful comments.

1

1 Introduction

The New Keynesian model, as developed by Galí (2008) and Walsh (2017),

brings together three equations to characterize the dynamic behavior of three

key macroeconomic variables: output, in‡ation, and the nominal interest

rate. The output equation corresponds to the log-linearization of an opti-

mizing household’s Euler equation, linking consumption and output growth

to in‡ation-adjusted return on nominal bonds— that is, to the real interest

rate. The in‡ation equation describes the optimizing behavior of monopolis-

tically competitive …rms that either set prices in a randomly staggered fashion

(Calvo,1983), or face explicit costs of nominal price adjustment (Rotemberg,

1982). The nominal interest rate equation, a monetary policy rule of the kind

proposed by Taylor (1993), dictates that the central bank should adjust the

short-term nominal interest rate in response to a trade-o¤ between changes

in in‡ation and output, and changes in the past interest rate.

Following optimization of household preferences, relative risk aversion

explicitly enters the …rst two equations. It has often been calibrated, esti-

mated, and analyzed as a simple and constant parameter in the literature

(Christiano et al.,2014). Although they deal with issues largely related to

risk aversion by considering log-utilities, some studies are not able to analyze

constant relative risk aversion (Iacoviello,2005). Yet, its role in the economic

dynamics of the Eurozone has not been analyzed further, at least not includ-

ing a relative risk aversion shock in a microfounded new Keynesian Dynamic

Stochastic General Equilibrium (DSGE) model.

Alpanda (2013) highlights the important role played by risk aversion

shocks in the US between 2006 and 2011. However, few studies quantify this

link, or even consider risk aversion as a shock in a New Keynesian DSGE

framework applied to the Eurozone. No studies use Bayesian techniques— as

Fernández-Villaverde (2010) does— to analyze the role of the risk aversion

shock in output, in‡ation, interest rate, and money dynamics in the Euro-

zone.

Constant relative risk aversion may change across di¤erent time periods.

Evidence of time-varying risk aversion can be found in simple applied analy-

sis carried out by Donadelli and Prosperi (2012). For instance, risk aversion

varies in response to news about in‡ation (Brandt and Wang,2003). How-

ever, what is the contribution of a risk aversion shock to the dynamics of

output, in‡ation, and money, over time? Building a simple New Keynesian

DSGE model including a time-varying relative risk aversion variable should

be able to answer this question.

As relative risk aversion measures the willingness to substitute consump-

tion over di¤erent periods, lower the level of risk aversion, more willing the

2

households are to substitute consumption over time. Results on the relation-

ship between relative risk aversion and equity risk premium can be found in

Bansal and Yaron (2004). Additionally, Wachter (2006) and Bekaert et al.

(2010) show that an increase in risk aversion involves an increase in equity

and bond premiums, and may either increase or decrease the real interest

rate through a consumption smoothing e¤ect or a precautionary savings

e¤ect, respectively. Bommier et al. (2012) further show that risk aversion

enhances precautionary savings. These studies con…rm the potential link

between money holdings, output, and risk aversion.

However, most new Keynesian DSGE models do not include money in

agents’utility (MIU). Ireland (2004), Andrés et al. (2006), and Barthélemy

et al. (2011) do not analyze the link between relative risk aversion of house-

holds and the dynamics of key macroeconomic variables.

However, Benchimol and Fourçans (2012) establish a signi…cant link be-

tween money, output, and risk aversion. They show that real money has a

signi…cant role with regard to output if the relative risk aversion level is suf-

…ciently high. Even though they study the role of the level of risk aversion in

a non-standard MIU function (CES), they do not include a study of the mi-

crofounded risk aversion shock for the Eurozone, without non-separabilities.

As in other studies, we consider a new Keynesian DSGE model including

standard shocks: a price-markup shock, a monetary policy shock, and a

technology shock. To analyze the role of risk aversion in the dynamics of

other variables, we consider a time-varying relative risk aversion including a

risk aversion shock.

Additionally, we consider a money equation to take account of the be-

haviors of national central banks (before 1999) and the European Central

Bank (after 1999), and to close the model with as many historical variables

as exogenous shocks.

This article contributes to the literature in several ways. First, we analyze

the role of a microfounded risk aversion shock in the dynamics of a New

Keynesian DSGE model. Second, a completely microfounded model with a

risk aversion shock is an original development, both in terms of …ndings as

well as an estimation technique.

Mainly inspired by Smets and Wouters (2007) and Galí (2008), our model

explores the role of risk aversion in in‡ation, output, interest rates, real

money balances, as well as in ‡exible-price output.

Speci…c emphasis will be placed on how the risk aversion shock impacts

the dynamics of these key variables over time. We use Bayesian techniques,

as in An and Schorfheide (2007), to estimate …ve subsamples of the Eurozone

between 1971 and 2011, each for a period of twenty years. This framework

allows us to successively analyze the informational content of the last two

3

crises (subprime and sovereign debt crises) in comparison with other crises

that occurred between 1971 and 2006 in the Eurozone.

Bayesian estimations and dynamic analyses of the model, with impulse

response functions and short- and long-run variance decomposition following

structural shocks, yield di¤erent relationships between risk aversion and other

variables. This approach sheds light on the importance of risk aversion, and

its impact on output and real money balances during the last …ve years (2006

to 2011).

This original focus on the last forty years highlights that the e¤ect of risk

aversion shocks on output and real money balances are stronger in recent

years than in the distant past. It also shows that the role of monetary policy

with regard to output in the short run has decreased in recent years.

Finally, using modern theoretical and empirical tools, this study explores

a fundamental question about the role of the perception of economic risks—

the ability of households to consume now or later— in the dynamics of the

main economic variables for the Eurozone.

The remainder of the paper is organized as follows. Section 2describes

the theoretical setup. In Section 3, the model is calibrated and estimated us-

ing Eurozone data. Impulse response functions and variance decompositions

are analyzed in Section 4. Section 5concludes, and the Appendix presents

additional theoretical and empirical results.

2 The model

The model consists of households that supply labor, purchase goods for con-

sumption, and hold money and bonds; and …rms that hire labor, and produce

and sell di¤erentiated products in monopolistically competitive goods mar-

kets. Each …rm sets the price of the good it produces, but not all …rms

reset their price during each period. Households and …rms behave optimally:

households maximize the expected present value of utility, and …rms maxi-

mize pro…ts. Additionally, there is a central bank that controls the nominal

rate of interest.

2.1 Households

We assume a representative in…nitely lived household, seeking to maximize

Et"1

X

k=0

kUt+k#(1)

4

where Utis the period utility function and < 1is the discount factor. The

household decides allocation of its consumption expenditure among di¤erent

goods. This requires that the consumption index Ctbe maximized for any

given level of expenditure (Galí,2008). Furthermore, conditional on such

optimal behavior, the period budget constraint takes the form

PtCt+Mt+QtBtBt1+WtNt+Mt1(2)

where t= 0;1;2:::,Ptis an aggregate price index; Mtis the quantity of money

holdings at time t;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 is priced at Qt, where it=ln Qtis the

short-term nominal rate); Wtis the nominal wage; and Ntdenotes hours of

work (or the measure of household members employed). The above sequence

of period budget constraints is supplemented with a solvency condition.1

Preferences are measured with a common time-separable utility function

(MIU). Under the assumption of a period utility given by

Ut=C1t

t

1t

+

1Mt

Pt1

N1+

t

1 + (3)

consumption, money demand, labor supply, and bond holdings are chosen

to maximize Eq. 1, subject to Eq. 2and the solvency condition. This MIU

utility function depends positively on the consumption of goods, Ct, and real

money balances, Mt

Pt; and negatively on labor, Nt.t>0is the time-varying

coe¢ cient of the relative risk aversion of households (or the inverse of the

intertemporal elasticity of substitution), de…ned as t=+"r

t, where "r

tis a

risk aversion shock (detailed in Section 3.1). is the inverse of the elasticity

of money holdings with respect to the interest rate, and is the inverse of the

elasticity of work e¤ort with respect to the real wage. and are positive

scale parameters.

This setting leads to the following conditions2, which must hold in equi-

librium, in addition to the budget constraint. The resulting log-linear version

of the …rst-order condition corresponding to the demand for contingent bonds

implies that

ct=Et[ct+1]1

t

(itEt[t+1]c)(4)

where ct= ln (Ct)is the logarithm of aggregate consumption, itis the nomi-

nal interest rate, Et[t+1]is the expected in‡ation rate in period t+ 1 with

knowledge of the information in period t, and c=ln ().

1Such as 8tlim

n!1

Et[Bn]0, in order to avoid Ponzi-type schemes.

2See Appendix 6.A

5

The demand for cash that follows from the household’s optimization prob-

lem is given by

tctmptm=a2it(5)

where mpt=mtptare the log linearized real money balances, m=

ln () + a1, and a1and a2are resulting terms of the …rst-order Taylor

approximation of ln (1 Qt) = a1+a2it.

Real cash holdings depend positively on consumption with an elasticity

equal to t

and negatively on the nominal interest rate.3We consider nominal

interest rate as the policy instrument of the central bank.

The resulting log-linear version of the …rst-order condition corresponding

to the optimal consumption-leisure arbitrage implies that

wtpt=tct+ntn(6)

where wtptcorresponds to the log of the real wage, ntdenotes the log of

hours of work, and n=ln ().

Finally, these equations represent the Euler condition for the optimal in-

tratemporal allocation of consumption (Eq. 4), the intertemporal optimality

condition setting the marginal rate of substitution between money and con-

sumption equal to the opportunity cost of holding money (Eq. 5), and the

intratemporal optimality condition setting the marginal rate of substitution

between leisure and consumption equal to the real wage (Eq. 6).

2.2 Firms

Backus et al. (1992) have shown that capital appears to play a rather minor

role in the business cycle. To simplify the analysis and focus on the role of

risk, we do not include the capital accumulation process in this model, as in

Galí (2008).

We assume a continuum of …rms indexed by i2[0;1]. Although each …rm

produces a di¤erentiated good, they all use identical technology, represented

by the following production function

Yt(i) = AtNt(i)1(7)

where At= exp ("a

t)represents the level of technology, assumed to be com-

mon to all …rms, and which evolves exogenously over time, and "a

tis a tech-

nology shock.

All …rms face an identical isoelastic demand schedule and take the aggre-

gate price level Ptand aggregate consumption index Ctas given. As in the

3Since 1

>1,a2>0.

6

standard Calvo (1983) model, our generalization features monopolistic com-

petition and staggered price setting. At any time, t, only a fraction of …rms,

1, with 0< < 1, can reset their prices optimally, while the remaining

…rms index their prices to lagged in‡ation.

2.3 Central bank

The central bank is assumed to set its nominal interest rate according to an

augmented smoothed Taylor (1993) rule such as:

it= (1 i)(tc) + xytyf

t+m(mptmpc)+iit1+"i

t

(8)

where ,x, and mare policy coe¢ cients re‡ecting the weight assigned to

the in‡ation gap, the output gap, and the real money gap, respectively; the

parameter 0< i<1captures the degree of interest rate smoothing; and "i

t

is an exogenous ad hoc shock accounting for ‡uctuations of the nominal in-

terest rate. cis an in‡ation target and mpcis a money target, essentially in-

cluded to account for changes in policies targeting in‡ation (Svensson,1999)

and monetary aggregates (Fourçans,2007). Other studies introduce a rele-

vant money variable in the Eurozone Taylor rule (Andrés et al.,2006,2009;

Barthélemy et al.,2011;Benchimol and Fourçans,2012).

Additionally, mtakes into account the potential money targeting of the

national central bank before the creation of the European Central Bank

(ECB, 1999). After 1999, the ECB followed an explicit money targeting

policy until 2004, called the Two Pillars policy (Barthélemy et al.,2011),

and might have even followed an implicit money targeting policy after that

(Kahn and Benolkin,2007).

3 Empirical results

3.1 DSGE model

Our macro model consists of …ve equations and …ve dependent variables:

in‡ation, nominal interest rate, output, real money balances, and ‡exible-

price output. Flexible-price output is completely determined by shocks.

yf

t=1 +

t(1 ) + +"a

t+(1 )ln (1 ) + nln "

"1

t(1 ) + +(9)

t=Et[t+1 ] + (1 ) (1 ) (t(1 ) + +)

(1 +t)ytyf

t(10)

7

yt=Et[yt+1]1

t(itEt[t+1]c)(11)

mpt=t

yta2

itm

(12)

it= (1 i)(tc) + xytyf

t+m(mptmpc)+iit1+"i

t

(13)

where a1= ln 1e1

1

e

1

1

and a2=1

e

1

1

.

All structural shocks are assumed to follow a …rst-order autoregressive

process with an i.i.d. (independent and identically distributed) normal error

term, such as "k

t=k"k

t1+!k;t, where "k;t N(0; k)for k=fp; i; a; rg.

3.2 Calibration

Following standard conventions, we calibrate beta distributions for parame-

ters that fall between zero and one, inverted gamma distributions for para-

meters that need to be constrained at greater than zero, and normal distri-

butions in other cases.

The parameters of the utility function are assumed to be distributed as

follows. Only the discount factor is …xed at 0:98 in the estimation procedure.

The intertemporal elasticity of substitution (i.e., the level of relative risk

aversion) is set at 2, a mean between the calibrations of Rabanal and Rubio-

Ramírez (2005) and Casares (2007), and consistent with the calibrated value

used by Kollmann (2001) and the value estimated by Lindé et al. (2009).

The inverse of the Frisch elasticity of labor supply is assumed to be approx-

imately 1, as in Galí (2008), and the scale parameters on money and labor

are assumed to be approximately 0:2, as in Benchimol and Fourçans (2012).

The calibration of ,, and "comes from Smets and Wouters (2007),

Casares (2007), and Galí (2008). The smoothed Taylor rule (i,,x,

and m) priors are calibrated following Smets and Wouters (2003), Andrés

et al. (2009), and Barthélemy et al. (2011). To observe both the behavior

of the central bank and risk aversion, we assign a higher standard error

(0:2) and a Normal prior law for the relative risk aversion level and for the

Taylor rule coe¢ cients (including in‡ation and money targets), except for

the smoothing parameter, which is restricted to be positive and less than

one (Beta distribution). The in‡ation target, c, is calibrated to 2percent,

and the money target, mpc, is assumed to be approximately 4percent.

The calibration of the shock persistence parameters and the standard er-

rors of the innovations follow Smets and Wouters (2007). All the standard

errors of shocks are assumed to be distributed according to inverted Gamma

8

distributions, with prior means of 0:01. The latter ensures that these parame-

ters have positive support. The autoregressive parameters are all assumed to

follow Beta distributions. All of these distributions are centered around 0:75,

except for the autoregressive parameter of the monetary policy shock and the

risk aversion shock, which are centered around 0:50, as in Smets and Wouters

(2007). We take a common standard error of 0:15 for the shock persistence

parameters, which is a mean between that of Benchimol and Fourçans (2012)

and Smets and Wouters (2007).

Law Mean Std. Law Mean Std.

beta 0.33 0.10 mnormal 1.00 0.20

beta 0.66 0.10 cnormal 2.00 0.20

normal 2.00 0.20 mpcnormal 4.00 0.20

vnormal 1.50 0.10 abeta 0.75 0.15

"normal 6.00 0.10 pbeta 0.75 0.15

normal 1.00 0.10 ibeta 0.50 0.15

beta 0.20 0.05 rbeta 0.50 0.15

beta 0.20 0.05 ainvgamma 0.01 2.00

ibeta 0.50 0.10 pinvgamma 0.01 2.00

normal 3.00 0.20 iinvgamma 0.01 2.00

xnormal 1.50 0.20 rinvgamma 0.01 2.00

Table 1: Calibration summary

3.3 Eurozone data

In our Eurozone model, tis the detrended in‡ation rate measured as the

yearly log di¤erence of the detrended GDP de‡ator from one quarter to the

same quarter of the previous year; ytis the detrended output per capita

measured as the di¤erence between the log of real GDP per capita and its

trend; and itis the short-term (3-month) detrended nominal interest rate.

These data are extracted from the AWM (Area Wide Model) database (Fagan

et al.,2001). mptis the detrended real money balance per capita measured

as the di¤erence between real money per capita and its trend, where real

money per capita is measured as the log di¤erence between money stock per

capita and the GDP de‡ator. We use the M3monetary aggregate from the

Eurostat database.

9

3.4 Results

The model is estimated with 160 observations from the …rst quarter of 1971 to

the …rst quarter of 2011, with Bayesian techniques, as in Smets and Wouters

(2007). However, to capture di¤erent policies and risk perceptions in the

Eurozone between 1971 and 2011, and more speci…cally between 2006 and

2011, we divide this large sample into …ve subsamples, each consisting of 80

observations (20 years).

This procedure allows us to analyze …ve di¤erent periods with a su¢ -

ciently large sample, as speci…ed in Fernández-Villaverde and Rubio-Ramírez

(2004). Accordingly, we estimate our model over …ve di¤erent periods: from

1971 Q1 to 1991 Q1 (P1); from 1976 Q1 to 1996 Q1 (P2); from 1981 Q1 to

2001 Q1 (P3); from 1986 Q1 to 2006 Q1 (P4); and from 1991 Q1 to 2011 Q1

(P5).

0.5

0.6

0.7

1.2

1.4

0.7

0.8

0.9

0

0.2

0.4

1.9

2

2.1

3.98

4

4.02

2.6

2.8

0.92

0.94

0.96

0.4

0.45

0.5

0.99

1

1.01

6

6.005

6.01

1.9

2

2.1

0.198

0.2

0.202

0.2

0.21

0.22

0.98

1

P1 P2 P3 P4 P5

0.6

0.8

1

P1 P2 P3 P4 P5

0.35

0.4

0.45

P1 P2 P3 P4 P5

0.9

0.95

1

Figure 1: Bayesian estimation of parameters over the selected periods

The estimation of the implied posterior distribution of the parameters

across the …ve periods (Fig. 1) is performed using the Metropolis-Hastings al-

gorithm (10 distinct chains, each of 100;000 draws). The average acceptance

rates per chain are included in the interval [0:19; 0:22], and the Student’s

t-tests are all above 1:96 in absolute terms.

10

To assess the model validation, we ensure convergence of the proposed

distribution to the target distribution for each period. Appendix 6.Bshows

that the estimation results are valid and that convergence is obtained for all

estimations and all moments.

Distribution of priors and posteriors are presented in Appendix 6.C. It

shows that the maximum posterior distribution reaches the posterior mean

of each estimated parameter. The estimation is relatively well identi…ed, and

the data is quite informative for most of the estimated microparameters.

4 Interpretation

We analyze the forecast error variance decomposition of each variable follow-

ing exogenous structural shocks.

0

10

20

Output

Risk avers ion

0

2

4

6Price-markup

0

20

40

60 Monetary policy

0

50

100 Technology

0

0.05

0.1

Inflation

0

50

100

0

0.05

0.1

0

5

10

15

0

5

10

In ter e st r a te

0

50

100

0

50

100

0

5

10

15

P1 P2 P3 P4 P5

0

20

40

60

Real money

P1 P2 P3 P4 P5

0

5

10

P1 P2 P3 P4 P5

0

10

20

30

P1 P2 P3 P4 P5

0

50

100

Short run v arianc e Long run variance

Figure 2: Variance decomposition over the selected periods

The analysis is conducted via an unconditional variance decomposition to

analyze long-term variance decomposition (the gray bar in Fig. 2), and con-

ditional variance decomposition, conditionally to the …rst period, to analyze

short-run variance decomposition (the black bar in Fig. 2).

11

Fig. 2shows that output is mainly explained by the technology shock

in the long run (approximately 90%), and by the monetary policy shock

(approximately 35%) and the technology shock (approximately 50%) in the

short run. Rest of the variance in output is explained by the risk aversion

shock (approximately 5% from P1 to P4, and more than 15% for P5) in the

short term, whereas risk aversion shock has a limited role in output variance

in the long run.

Fig. 2also shows that, in accordance with the literature, in‡ation is

mainly explained by the price-markup shock, and interest rate variance is

mainly driven by monetary policy in the short run and monetary policy

and price-markup in the long run. Furthermore, most of the variance in

real money balances is induced by the risk aversion shock (approximately

40%) and the monetary policy shock (approximately 25%) in the short run,

whereas in the long run, real money balance variance is mainly driven by the

technology shock. All these results are in line with the literature.

Sections 4.1 to 4.4 present the role of each shock in the ‡uctuations of the

macroeconomic variables, and the response of these key variables to struc-

tural shocks over the study period.

4.1 Price-markup shock

Fig. 2shows that the price-markup shock explains almost all of the variability

in the in‡ation rate, at least in the short run. Its role in the long run on real

money balance ‡uctuations is halved across the study period.

12

020 40

0

0.2

0.4

0.6

0.8 Inflation

020 40

-0.4

-0.3

-0.2

-0.1

0Output

020 40

-0.4

-0.3

-0.2

-0.1

0Output gap

020 40

0

0.2

0.4

0.6

0.8 Nominal interest rate

020 40

-0.2

-0.15

-0.1

-0.05

0

0.05 R e al inte r es t r ate

020 40

-0.8

-0.6

-0.4

-0.2

0Real money balances

P5 P4 P3 P2 P1

Figure 3: Impulse response function with respect to a price-markup shock

Fig. 3shows that from P1 to P5, the impact of the price-markup shock

on output, in‡ation, nominal interest rate, and real money balances is at

least halved. While transitioning from one period to another, the impact

of the price-markup shock on the overall economy reduces. Regardless of

the period under study, after a positive price-markup shock, in‡ation rate

increases, thus, nominal interest rate increases, decreasing real interest rate,

output, output gap, and real money balances.

4.2 Technology shock

Fig. 2indicates that technology plays an increasingly important role in the

short term for the in‡ation rate and, thus, for the interest rate in the selected

period.

13

020 40

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02 Inflatio n

020 40

0.4

0.6

0.8

1

1.2 Output

020 40

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1 Output gap

020 40

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02 Nominal interest rate

020 40

-0.04

-0.03

-0.02

-0.01

0

0.01 R e al inte r es t r a te

020 40

0.7

0.8

0.9

1

1.1

1.2

1.3

Real money balances

P5 P4 P3 P2 P1

Figure 4: Impulse response function with respect to a technology shock

Fig. 4highlights that in response to a positive technology shock, output

increases but in‡ation decreases, resulting in an increase in real money hold-

ing, and a slight decrease in nominal interest rate and output gap. The im-

provement in technology is partly accommodated by the central bank, which

lowers the nominal and real interest rate, while increasing the quantity of

money in circulation.

Interestingly, note that maximum sensitivity of output to a technology

shock was observed in P1 and P2.

4.3 Monetary policy shock

Fig. 2shows that compared to the previous periods (P1 to P4), monetary

policy has a smaller role in the short run output variability over the last

period (P5)–from around 37% to 22%, respectively. This highlights a switch

from the role of the monetary policy to the role of risk aversion during recent

years. This con…rms the declining in‡uence of European monetary policy

relative to the in‡uence of risk aversion shocks in recent years.

14

020 40

-0.01

-0.008

-0.006

-0.004

-0.002

0Inflation

020 40

-1

-0 .8

-0 .6

-0 .4

-0 .2

0Output

020 40

-1

-0 .8

-0 .6

-0 .4

-0 .2

0Output gap

020 40

0

0.2

0.4

0.6

0.8

1N o min a l in ter e s t ra te

020 40

0

0.2

0.4

0.6

0.8

1Real interest rate

020 40

-1 .5

-1

-0 .5

0Real money bala nces

P5 P4 P3 P2 P1

Figure 5: Impulse response function with respect to a monetary policy shock

Fig. 5indicates that in response to an interest rate shock, in‡ation rate,

output, output gap, and real money balances fall. The nominal and real

interest rates rise. A positive monetary policy shock could also induce a fall

in interest rates due to a low enough degree of intertemporal substitution (i.e.,

the risk aversion parameter is high enough), which generates a large impact

response of current consumption relative to future consumption (Jeanne,

1994;Christiano et al.,1997). Note that in P5, nominal and real interest

rates are the least sensitive of all other periods, suggesting the largest relative

risk aversion compared to all other periods.

4.4 Risk aversion shock

Fig. 2shows that output and real money balance variances have an important

risk aversion shock component. This …nding shows the leading role of relative

risk aversion in the dynamics of output (Black and Dowd,2011) and real

money balances (Benchimol,2011).

Although in‡ation rate, nominal interest rate, and ‡exible-price output

are strong components of output, risk aversion has a minor role to play in

15

the variance of in‡ation and interest rate, and it does not play a role with

regard to the ‡exible-price output (less than 0:2% in the short- and long-run),

which is completely determined by the technology shock. It also shows that

in‡ation and interest rate variances are quasi-una¤ected by the introduction

of the risk aversion shock, allowing these variables to be mainly explained by

the price-markup shock and the monetary policy shock, respectively.

Additionally, Fig. 2shows that risk aversion plays a more signi…cant

role in output dynamics in the last period (P5) than in other periods (P1 to

P4) in the short run. This …nding re‡ects the increasing role played by risk

aversion in more recent years (between 2006 and 2011) as compared to the

past (between 1971 and 2006).

020 40

0

0.002

0.004

0.006

0.008

0.01

0.012 In fla tion

020 40

-0.5

-0.4

-0.3

-0.2

-0.1

0Output

020 40

-0.4

-0.3

-0.2

-0.1

0Output gap

020 40

0

0.1

0.2

0.3

0.4 Nominal interest rate

020 40

0

0.1

0.2

0.3

0.4 Real interest rate

020 40

0

0.5

1

1.5

2Real money balances

P5 P4 P3 P2 P1

Figure 6: Impulse response function with respect to a risk aversion shock

This case is very interesting because Fig. 6shows that a risk aversion

shock leads to a decrease in output and an increase in in‡ation. This implies a

tightening of monetary policy (because of the strong weight that the central

bank places on in‡ation) and its strength depends on the period (strong

monetary policy tightening in P1 and low monetary policy tightening in P5).

The risk aversion shock also implies an increase in real money balances and

real money growth, and a decrease in the output gap. Our risk aversion

16

shock also suggests that uncertainty is counter cyclical (Baker and Bloom,

2013).

Household consumption reduces (decreasing output), and companies in-

crease their price (to face high risk aversion and possibly, low consumption),

which implies an increase in the in‡ation rate, constrained by a tightening

of monetary policy.

Fig. 6exhibits that the risk aversion shock has a longer impact in P5 than

it does in the other periods. This is due to the increase of the autoregressive

parameter of the risk aversion shock, r, over the periods, as shown in Fig.

1. Although the sensitivity of monetary policy with respect to risk aversion

shock is lower in P5 than during other periods, it is more persistent in P5

than in the other periods. This highlights that the central bank reacts less

strongly after a risk aversion shock, but the persistence of the impact of this

risk aversion shock on nominal interest rate is stronger over time.

Bloom (2009) simulates a macro uncertainty shock, which produces a

rapid drop in aggregate output, mainly because higher uncertainty causes

…rms to temporarily pause their investment and hiring. Our results suggest

that the impact of the risk aversion shock— a micro uncertainty shock—

on output and output gap is very important during the last period, P5,

as compared to the other periods. As a matter of fact, an important part

of the variation in output is dependent, through the risk aversion shock,

on major shocks such as crises, news, and disasters (Bloom,2009;Baker

and Bloom,2013). Controlling these media-parameters— for instance, by

communication–could attenuate their impact on growth.

The leading role of the risk aversion shock in the dynamics of real money

balance in the short run is another important …nding. Fig. 2indicates that

real money balances are mainly explained by the technology shock (approxi-

mately 80%) in the long run, whereas in the short run, real money balances

are mainly explained by the risk aversion shock and the monetary policy

shock. Last but not the least, in line with Benchimol and Fourçans (2012),

a risk aversion shock drastically increases real money balances response, de-

spite increasing in‡ation and nominal interest rate responses (Fig. 6).

5 Conclusion

Risk aversion as a concept in economics and …nance is based on the behavior

of consumers and investors who are exposed to uncertainty. It is the reluc-

tance of a person to accept a bargain with an uncertain payo¤, rather than

one that o¤ers a more certain, but possibly lower, expected payo¤.

This paper presents a standard New Keynesian DSGE model that includes

17

a risk aversion shock. It shows the involvement of this risk aversion shock in

the dynamics of the economy: it increases in‡ation, decreases output (Fig.

6), and diminishes the impact of the action by the central bank on output

variance, at least in the short run (Fig. 2).

Additionally, risk aversion plays an important role in output and real

money balance dynamics. It is clearly identi…ed that risk aversion plays a

negative role in determining output, whereas it increases real money balances

and real money growth in the initial period (Fig. 6).

Moreover, while estimations are quite robust (Fig. 7to Fig. 12), they

show that risk aversion shock has a stronger impact on output dynamics

during the last twenty years (P5) as compared to other analyzed periods

(P1 to P4). This result is explained by the inclusion of the subprime and

sovereign debt crises in P5 from 2007 to 2011.

This enhanced baseline model shows the importance of such a parameter

to the economy, especially its impact on output, money, and monetary policy.

It also serves to show the importance of controlling shocks to the agents’risk

aversion, for instance, by communication.

6 Appendix

A Solving the model

Price dynamics

Let us assume a set of …rms not reoptimizing their posted prices in period

t. Using the de…nition of the aggregate price level and the fact that all …rms

resetting prices choose an identical price P

t, leads to

Pt=hP 1t

t1+ (1 ) (P

t)1ti1

1t(14)

where t= 1 + 1

1

"1+"p

t

is the elasticity of substitution between consumption

goods in period t, and t

t1is the markup of prices over marginal costs (time

varying). Dividing both sides by Pt1and log-linearizing around P

t=Pt1

yields

t= (1 ) (p

tpt1):(15)

In this setup, we do not assume inertial dynamics of prices. In‡ation

results from …rms reoptimizing their price plans in any given period, and

choosing a price that di¤ers from the economy’s average price in the previous

period.

18

Price setting

A …rm reoptimizing in period tchooses price P

tthat maximizes the cur-

rent market value of the pro…ts generated, while the price remains e¤ective.

This problem resolves and leads to a …rst-order Taylor expansion around the

zero in‡ation steady state:

p

tpt1= (1 )

1

X

k=0

()kEtcmct+kjt+ (pt+kpt1)(16)

where cmct+kjt=mct+kjtmc denotes the log deviation of marginal cost from

its steady state value mc =, and = ln "

"1is 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 de…ned as Yt=R1

0Yt(i)11

tdit

t1; it

follows that Yt=Ctmust hold for all t. One can combine the above goods

market clearing condition with the consumer’s Euler equation (4) to yield

the equilibrium condition

yt=Et[yt+1]1

t(itEt[t+1]c)(17)

Market clearing in the labor market requires Nt=R1

0Nt(i)di. With

the production function (7) and taking logs, one can express the following

approximate relationship between aggregate output, employment, and tech-

nology as

yt="a

t+ (1 )nt(18)

An expression is derived for the marginal cost of an individual …rm in

terms of the economy’s average real marginal cost:

mct= (wtpt)mpnt(19)

=wtpt1

1("a

tyt)ln (1 )

for all t, where mpntde…nes the economy’s average marginal product of labor.

As mct+kjt= (wt+kpt+k)mpnt+kjt, we have

mct+kjt=mct+kt

1(p

tpt+k)(20)

19

where the second equality follows from the demand schedule combined with

the market clearing condition ct=yt. Substituting (20) into (16) yields

p

tpt1= (1 )

1

X

k=0

t+k()kEt[cmct+k] +

1

X

k=0

()kEt[t+k](21)

where t=1

1+t1is time varying to take into account the markup

shock.

Finally, (15) and (21) yield the in‡ation equation

t=Et[t+1 ] + mctcmct(22)

where ,mct=(1)(1)

t.mctis strictly decreasing in the index of

price stickiness , in the measure of decreasing returns , and in the demand

elasticity t.

Next, a relationship is derived between the economy’s real marginal cost

and a measure of aggregate economic activity. From (6) and (18), the average

real marginal cost can be expressed as

mct=t++

1yt1 +

1"a

tlog (1 )n(23)

Under ‡exible prices, the real marginal cost is constant and equal to mc =

. De…ning the natural level of output, denoted by yf

t, as the equilibrium

level of output under ‡exible prices leads to

mc =t++

1yf

t1 +

1"a

tlog (1 )n(24)

thus, implying

yf

t=a"a

t+c(25)

where a=1+

t(1)++and c=(1)(ln(1)+nln("

"1))

t(1)++. Subtracting (26)

from (25) yields

cmct=t++

1ytyf

t(26)

where cmct=mctmc is the real marginal cost gap and ytyf

tis the output

gap. Combining the above equation with (24), we obtain

t=Et[t+1 ] + tytyf

t(27)

where t=(1)(1)(t(1)++)

(1+t), and ytyf

tis the output gap.

20

The second key equation describing the equilibrium of the model is ob-

tained by rewriting (20) to determine output

yt=Et[yt+1]1

t(itEt[t+1]c)(28)

Equation (28) is thus, a dynamic IS equation including real money bal-

ances.

The third key equation describes the behavior of real money balances.

From (5), we obtain

mpt=t

yta2

itm

(29)

B Model validation

The red and blue lines in Fig. 7represent an aggregate measure based on the

eigenvalues of the variance-covariance matrix of each parameter both within

and between chains. Each graph represents speci…c convergence measures

and has two distinct lines that represent the results within and between

chains. These measures are related to the analysis of the parameter’s mean

(…rst moment), variance (second moment), and third moment of the model

for the relevant period. Convergence requires both lines for each of the three

measures to become relatively constant and converge to each other.

Diagnoses of the numerical maximization of the posterior kernel indicate

that the optimization procedure was able to obtain a robust maximum for the

posterior kernel. A diagnosis of the overall convergence for the Metropolis-

Hastings sampling algorithm is provided in Fig. 7.

Diagnoses for each individual parameter were also obtained, following the

same structure used for overall convergence. Most of the parameters do not

seem to exhibit convergence problems, notwithstanding that this evidence is

stronger for some parameters than it is for others.

21

7

8

9

10

P1

Firs t moment

8

10

12

14 Second moment

40

60

80

100 Third moment

6

8

10

12

P2

5

10

15

40

60

80

100

7

8

9

10

P3

5

10

15

0

50

100

7

8

9

10

P4

8

10

12

14

40

60

80

0 5 10

x 104

7

8

9

Iterations

P5

0 5 10

x 104

5

10

15

Iterations 0 5 10

x 104

0

50

100

Iterations

Within variance Betw een and within variance

Figure 7: Multivariate Metropolis-Hastings convergence diagnosis

22

C Priors and posteriors

The following …gures present the priors and posteriors of the estimated struc-

tural parameters over the study period.

00.45 0 .9

0

2

00.94

0

10

20

02.5

0

1

2

0 2

0

2

4

00.99

0

2

06.01

0

2

00.2 0.4 0.6

0

5

00.2 0.4 0.6

0

5

0 2

0

1

0 4

0

1

00.54

0

2

4

02.8

0

1

2

01.76

0

1

2

00.73 1.46

0

1

2

00.99

0

50

00.39 0.78

0

5

00.96

0

10

20

00.52 1.04

0

2

4

00.05

0

50

100

00.05

0

50

100

00.05

0

50

100

00.05

0

100

Posterior distribution Prior distribution Pos terior mode

Figure 8: Priors and posteriors of the estimated parameters (P1)

23

00.46 0.92

0

2

00.93

0

10

20

02.6

0

1

2

0 2

0

2

0 1

0

2

06.01

0

2

4

00.2 0.4

0

5

00.21 0.42

0

5

0 2

0

1

2

0 4

0

1

00.52

0

2

4

02.82

0

1

2

01.78

0

1

2

00.430.861.291.72

0

1

2

00.99

0

50

00.42 0.84

0

5

00.95

0

10

00.77

0

5

00.05

0

50

100

00.05

0

50

100

00.05

0

50

100

00.05

0

100

Posterior distribution Prior dis tribution Posterior mode

Figure 9: Priors and posteriors of the estimated parameters (P2)

24

00.43 0.86

0

2

00.94

0

10

20

02.54

0

1

2

02.04

0

2

4

00.99

0

2

4

0 6

0

2

4

00.2 0.4

0

5

00.2 0.4 0.6

0

5

01.97

0

1

0 4

0

1

00.57

0

5

02.91

0

1

2

01.69

0

1

2

00.64 1.28 1.92

0

1

2

00.99

0

50

00.42 0.84

0

5

00.94

0

10

00.66

0

5

00.05

0

100

00.05

0

100

00.05

0

100

00.05

0

100

200

Posterior distribution Prior dis tribution Posterior mode

Figure 10: Priors and posteriors of the estimated parameters (P3)

25

00.43 0.86

0

2

00.93

0

10

20

02.5

0

1

2

02.06

0

2

4

00.99

0

2

0 6

0

2

4

00.2 0.4

0

5

00.21 0.42

0

5

01.95

0

1

03.99

0

1

00.62

0

5

02.95

0

1

2

01.66

0

1

2

00.45 0 .9 1.35 1.8

0

1

2

00.99

0

50

100

00.46

0

5

00.94

0

10

20

00.81

0

5

00.05

0

100

00.05

0

100

200

00.05

0

100

200

00.05

0

100

Posterior distribution Prior distribution Posterior mode

Figure 11: Priors and posteriors of the estimated parameters (P4)

26

00.43 0.86

0

2

00.95

0

20

02.51

0

1

2

02.05

0

2

4

00.99

0

2

0 6

0

2

00.2 0.4 0.6

0

5

00.2 0.4

0

5

01.98

0

1

0 4

0

1

00.56

0

5

02.95

0

1

2

01.66

0

1

2

00.73 1.46

0

1

2

00.99

0

50

00.42 0.84

0

5

00.95

0

10

20

00.9

0

10

00.05

0

100

00.05

0

100

200

00.05

0

100

00.05

0

100

200

Posterior distribution Prior dis tribution Posterior mode

Figure 12: Priors and posteriors of the estimated parameters (P5)

27

References

Alpanda, S., 2013. Identifying the role of risk shocks in the business cycle

using stock price data. Economic Inquiry 51 (1), 304–335.

An, S., Schorfheide, F., 2007. Bayesian analysis of DSGE models. Economet-

ric Reviews 26 (2-4), 113–172.

Andrés, J., López-Salido, J. D., Nelson, E., 2009. Money and the natural

rate of interest: structural estimates for the United States and the Euro

area. Journal of Economic Dynamics and Control 33 (3), 758–776.

Andrés, J., López-Salido, J. D., Vallés, J., 2006. Money in an estimated

business cycle model of the Euro area. Economic Journal 116 (511), 457–

477.

Backus, D., Kehoe, P., Kydland, F., 1992. International real business cycles.

Journal of Political Economy 100 (4), 745–775.

Baker, S. R., Bloom, N., 2013. Does uncertainty reduce growth ? Using

disasters as natural experiments. NBER Working Papers 19475, National

Bureau of Economic Research.

Bansal, R., Yaron, A., 2004. Risks for the long run: a potential resolution of

asset pricing puzzles. Journal of Finance 59 (4), 1481–1509.

Barthélemy, J., Clerc, L., Marx, M., 2011. A two-pillar DSGE monetary

policy model for the euro area. Economic Modelling 28 (3), 1303–1316.

Bekaert, G., Engstrom, E., Grenadier, S., 2010. Stock and bond returns with

moody investors. Journal of Empirical Finance 17 (5), 867–894.

Benchimol, J., 2011. New Keynesian DSGE models, money and risk aversion.

PhD dissertation, UniversitéParis 1 Panthéon-Sorbonne.

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.

Black, D., Dowd, M., 2011. Risk aversion as a technology factor in the pro-

duction function. Applied Financial Economics 21 (18), 1345–1354.

Bloom, N., 2009. The impact of uncertainty shocks. Econometrica 77 (3),

623–685.

28

Bommier, A., Chassagnon, A., Le Grand, F., 2012. Comparative risk aver-

sion: a formal approach with applications to saving behavior. Journal of

Economic Theory 147 (4), 1614–1641.

Brandt, M., Wang, K., 2003. Time-varying risk aversion and unexpected

in‡ation. Journal of Monetary Economics 50 (7), 1457–1498.

Calvo, G., 1983. Staggered prices in a utility-maximizing framework. Journal

of Monetary Economics 12 (3), 383–398.

Casares, M., 2007. Monetary policy rules in a New Keynesian Euro area

model. Journal of Money, Credit and Banking 39 (4), 875–900.

Christiano, L., Eichenbaum, M., Evans, C. L., 1997. Sticky price and limited

participation models of money: a comparison. European Economic Review

41 (6), 1201–1249.

Christiano, L. J., Motto, R., Rostagno, M., 2014. Risk shocks. American

Economic Review 104 (1), 27–65.

Donadelli, M., Prosperi, L., 2012. The equity premium puzzle: pitfalls in esti-

mating the coe¢ cient of relative risk aversion. Journal of Applied Finance

and Banking 2 (2), 177–213.

Fagan, G., Henry, J., Mestre, R., 2001. An Area-Wide Model (AWM) for the

Euro area. Working Paper Series 42, European Central Bank.

Fernández-Villaverde, J., 2010. The econometrics of DSGE models. SERIEs

1 (1), 3–49.

Fernández-Villaverde, J., Rubio-Ramírez, J. F., 2004. Comparing dynamic

equilibrium models to data: a Bayesian approach. Journal of Econometrics

123 (1), 153–187.

Fourçans, Andréand Vranceanu, R., 2007. The ECB monetary policy: choices

and challenges. Journal of Policy Modeling 29 (2), 181–194.

Galí, J., 2008. Monetary policy, in‡ation and the business cycle: an intro-

duction to the New Keynesian framework, 1st Edition. Princeton, NJ:

Princeton University Press.

Iacoviello, M., 2005. House prices, borrowing constraints, and monetary pol-

icy in the business cycle. American Economic Review 95 (3), 739–764.

29

Ireland, P., 2004. Money’s role in the monetary business cycle. Journal of

Money, Credit and Banking 36 (6), 969–983.

Jeanne, O., 1994. Nominal rigidities and the liquidity e¤ect. Mimeo, ENPC-

CERAS.

Kahn, G., Benolkin, S., 2007. The role of money in monetary policy: why

do the Fed and ECB see it so di¤erently ? Economic Review Q3, Federal

Reserve Bank of Kansas City.

Kollmann, R., 2001. The exchange rate in a dynamic-optimizing business

cycle model with nominal rigidities: a quantitative investigation. Journal

of International Economics 55 (2), 243–262.

Lindé, J., Nessén, M., Söderström, U., 2009. Monetary policy in an estimated

open-economy model with imperfect pass-through. International Journal

of Finance & Economics 14 (4), 301–333.

Rabanal, P., Rubio-Ramírez, J. F., 2005. Comparing New Keynesian models

of the business cycle: a Bayesian approach. Journal of Monetary Economics

52 (6), 1151–1166.

Rotemberg, J. J., 1982. Monopolistic price adjustment and aggregate output.

Review of Economic Studies 49 (4), 517–31.

Smets, F., Wouters, R., 2003. An estimated dynamic stochastic general equi-

librium model of the Euro area. Journal of the European Economic Asso-

ciation 1 (5), 1123–1175.

Smets, F., Wouters, R., 2007. Shocks and frictions in US business cycles: a

Bayesian DSGE approach. American Economic Review 97 (3), 586–606.

Svensson, L. E. O., 1999. In‡ation targeting as a monetary policy rule. Jour-

nal of Monetary Economics 43 (3), 607–654.

Taylor, J. B., 1993. Discretion versus policy rules in practice. Carnegie-

Rochester Conference Series on Public Policy 39 (1), 195–214.

Wachter, J. A., 2006. A consumption-based model of the term structure of

interest rates. Journal of Financial Economics 79 (2), 365–399.

Walsh, C., 2017. Monetary theory and policy. Cambridge, MA: MIT Press.

30