Money in the Production Function: A New Keynesian DSGE Perspective

Abstract

This article checks whether money is an omitted variable in the production process by proposing a microfounded New Keynesian Dynamic Stochastic General Equilibrium model. In this framework, real money balances enter the production function, and money demanded by households is differentiated from that demanded by firms. Using a Bayesian analysis, our model weakens the hypothesis that money is a factor of production. However, the demand of money by firms appears to have a significant impact on the economy, even if this demand has a low weight in the production process.

Full text

Money in the Production Function: A New
Keynesian DSGE Perspective
Jonathan Benchimol†
July 2015
Abstract
This article checks whether money is an omitted variable in the production
process by proposing a microfounded New Keynesian Dynamic Stochastic
General Equilibrium (DSGE) model. In this framework, real money balances
enter the production function, and money demanded by households is dif-
ferentiated from that demanded by firms. By using a Bayesian analysis, our
model weakens the hypothesis that money is a factor of production. How-
ever, the demand of money by firms appears to have a significant impact on
the economy, even if this demand has a low weight in the production process.
Keywords: Money demand, Money supply, DSGE mdoels, Bayesian estima-
tion.
JEL Codes: E23, E31, E51.
This paper does not necessarily reflect the views of the Bank of Israel. I thank Jess Benhabib,
Akiva Offenbacher, André Fourçans and two anonymous referees for their helpful advice and
comments.
†Bank of Israel, Jerusalem, Israel. Email: jonathan.benchimol@boi.org.il.
1

Please cite this paper as:
Benchimol, J., 2015. Money in the production function: A New Keynesian
DSGE perspective. Southern Economic Journal 82 (1), 152–184
2

1 Introduction
The theoretical motivation to empirically implement money in the production
function originates from the monetary growth models of Levhari and Patinkin
(1968), Friedman (1969), Johnson (1969), and Stein (1970), which include money
directly in the production function. Firms hold money to facilitate production on
the grounds that money enables them to economize on the use of other inputs and
saves costs incurred by running short of cash (Fischer, 1974).
Real cash balances are at least in part a factor of production. To take a
trivial example, a retailer can economize on his average cash balances by hiring
an errand boy to go to the bank on the corner to get change for large bills
tendered by customers. When it costs ten cents per dollar per year to hold an
extra dollar of cash, there will be a greater incentive to hire the errand boy,
that is, to substitute other productive resources for cash. This will mean both
a reduction in the real flow of services from the given productive resources and
a change in the structure of production, since different productive activities
may differ in cash-intensity, just as they differ in labor- or land-intensity.
Milton Friedman (1969)
Considering real money balances to be a factor of production has numerous
implications. Money would have a marginal physical productivity schedule like
other inputs, firms’ demands for real balances would be derived in the same way
as other factor demand functions, changes in the stock of money would affect real
output–contrary to the classical dichotomy which implies that money is neutral–
and real balances might explain part of the growth rates of total factor productivity
or the residual.
Sinai and Stokes (1972) present a very interesting test of the hypothesis that
money enters the production function, suggesting that real balances could rep-
resent a missing variable that contributes to the attribution of the unexplained
residual to technological changes. Ben-Zion and Ruttan (1975) conclude that as a
factor of production, money seems to play an important role in explaining induced
technological changes.
Short (1979) develops structural models based on Cobb and Douglas (1928) and
generalized translog production functions, both of which provide a more complete
theoretical framework to analyze the role of money in the production process.
The empirical results obtained by estimating these two models indicate that the
relationship between real cash balances and output, even after correcting for any
simultaneity bias, is positive and statistically significant. The results suggest that
it is theoretically appropriate to include a real cash balances variable as a factor
input in a production function in order to capture the productivity gains derived
from using money.
3

You (1981) finds that the unexplained portion of output variation virtually van-
ishes with the inclusion of real balances in the production function. In addition
to labor and capital, real money balances turn out to be an important factor of
production, especially for developing countries. The results of Khan and Ahmad
(1985) are consistent with the hypothesis that real money balances are an impor-
tant factor of production. Sephton (1988) shows that real balances are a valid fac-
tor of production within the confines of a constant elasticity of substitution (CES)
production function. Hasan and Mahmud (1993) also support the hypothesis that
money is an important factor in the production function and that there are poten-
tial supply side-effects of interest rate changes.
Recent developments in econometrics regarding cointegration and error-correction
models provide a rich environment in which to reexamine the role of money in the
production function. Moghaddam (2010) presents empirical evidence indicating
that in a cointegrated space, different definitions of money serve as an input in the
Cobb and Douglas (1928) production function.
At the same time, Clarida et al. (1999), Woodford (2003), and Galí (2008) de-
velop New Keynesian Dynamic Stochastic General Equilibrium (DSGE) models
to explain the dynamics of the economy. However, none of the studies on New
Keynesian DSGE models use money as an input in the production function.
This article departs from the existing theoretical and empirical literature by
specifying a fully microfounded New Keynesian DSGE model in which money
enters the production function. This feature generates a new inflation equation
that includes money. Following Benchimol and Fourçans (2012), we introduce the
new concept of flexible-price real money balances and, in order to close the model,
a quantitative equation. We also analyze the dynamics of the economy by using
Bayesian estimations and simulations to confirm or reject the potential influence of
money in the dynamics of the Eurozone and to determine the weight of real money
balances in the production process. By distinguishing between money used for
productive and nonproductive purposes (Benhabib et al., 2001), this paper intends
to solve the now old and controversial hypothesis about money in the production
function proposed by Levhari and Patinkin (1968) and Sinai and Stokes (1972),
and to more deeply analyze the role of these two components of the demand for
money (demand from households and firms).
After describing the theoretical set-up in Section 2, we calibrate and estimate
five models of the Euro area using Bayesian techniques in Section 3. Impulse re-
sponse functions and variance decomposition are analyzed in Section 3.4, and we
study the consequences of money in the production function hypothesis by com-
paring the monetary policy rules of models in Section 4. Section 5 concludes, and
Section 6 presents additional results.
4

2 The model
The model consists of households that supply labor, purchase goods for consump-
tion, and hold money and bonds, as well as firms that hire labor and produce and
sell differentiated products in monopolistically competitive goods markets. Each
firm sets the price of the good it produces, but not all firms reset their respective
prices each period. Households and firms behave optimally: Households maxi-
mize their expected present value of utility, and firms maximize profits. There is
also a central bank that controls the nominal interest rate. This model is inspired
by Smets and Wouters (2003), Galí (2008), and Walsh (2017).
2.1 Households
We assume a representative, infinitely lived household, that seeks 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
different goods. This requires that the consumption index, Ct, be maximized for
any given level of expenditure. 8t2Nand, conditionally on such optimal behav-
ior, the period budget constraint takes the form
PtCt+Mn,t+Mp,t+QtBtBt1+WtNt+Mn,t1+Mp,t1, (2)
where Ptis an aggregate price index; Mn,tand Mp,tare nominal money held for
nonproductive and productive purposes, respectively; Btis the quantity of one-
period, nominally risk-free discount bonds purchased in period tand maturing
in period t+1 (each bond pays one unit of money at maturity and its price is Qt,
so that the short-term nominal rate itis approximately equal to ln Qt); Wtis the
nominal wage; and Ntis hours worked (or the measure of employed household
members).
The above sequence of period budget constraints is supplemented with a sol-
vency condition, such as 8tlim
n!∞Et[Bn]0.
Preferences are measured using a common time-separable utility function. Un-
der the assumption of a household period utility given by
Ut=eεu
t C1σ
t
1σ+γeεn
t
1νMn,t
Pt1ν
χN1+η
t
1+η!, (3)
5

consumption, labor supply, money demand, and bond holdings are chosen to
maximize Eq. (1), subject to Eq. (2) and the solvency condition. This Money-in-
the-Utility (MIU) function depends positively on the consumption of goods, Ct,
positively on real money balances, Mt
Pt, and negatively on labor Nt.σis the coeffi-
cient of the relative risk aversion of households 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 ηis the inverse of the elasticity of work effort with
respect to the real wage (inverse of the Frisch elasticity of the labor supply).
The utility function also contains two structural shocks: εu
tis a general shock
to preferences that affects the intertemporal substitution of households (preference
shock) and εn
tis a shock to household money demand. γand χare positive scale
parameters.
This setting leads to the following conditions1, which, in addition to the budget
constraint, must hold in equilibrium. The resulting log-linear version of the first-
order condition corresponding to the demand for contingent bonds implies that
ct=Et[ct+1]1
σ(itEt[πt+1]ρc)σ1Et∆εu
t+1, (4)
where the lowercase letters denote the logarithm of the original aggregated vari-
ables, ρc=ln (β), and ∆is the first-difference operator.
The demand for cash that follows from the household optimization problem is
given by
εn
t+σctνmpn,tρm=a2it, (5)
where mpn,t=mn,tptare the log-linearized real money balances for nonpro-
ductive purposes, ρm=ln (γ)+a1, and a1and a2are the resulting terms of
the first-order Taylor approximation of ln (1Qt)=a1+a2it. More precisely, if
we compute our first-order Taylor approximation around the steady-state interest
rate, 1
β, we obtain a1=ln 1exp 1
β
1
β
e
1
β1
and a2=1
e
1
β1
.
Real cash holdings has a positive relation with consumption, with an elasticity
equal to σ/ν, and a negative relation with the nominal interest rate (1
β>1, which
implies that a2>0). Below, we take the nominal interest rate as the central bank’s
policy instrument.
In the literature, due to the assumption that consumption and real money bal-
ances are additively separable in the utility function, the cash holdings of house-
holds do not enter any of the other structural equations: Accordingly, the equation
above becomes a recursive function of the remainder of the system of equations.
However, as in Sinai and Stokes (1972), Subrahmanyam (1980), and Khan and
Ahmad (1985), because real money balances enter the aggregate supply, we will
1See Appendix A.
6

use this money demand equation (Eq. 5) to solve the equilibrium of our model.
See, for instance, Ireland (2004), Andrés et al. (2009), and Benchimol and Fourçans
(2012) for models in which money balances enter the aggregate demand equation
without entering the production function.
The resulting log-linear version of the first-order condition corresponding to
the optimal consumption-leisure arbitrage implies that
wtpt=σct+ηntρn, (6)
where ρn=ln (χ).
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 for nonproductive use (Eq. 5), and the
intratemporal optimality condition setting the marginal rate of substitution be-
tween leisure and consumption 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
differentiated good, but they all use an identical technology, represented by the
following Money-in-the-Production function2
Yt(i)=eεa
teεp
tMp,t
Ptαm
Nt(i)1αn(7)
where exp (εa
t)represents the level of technology, assumed to be common to all
firms and to evolve exogenously over time, and εp
tis a shock to firm money de-
mand.
All firms face an identical isoelastic demand schedule and take the aggregate
price level, Pt, and aggregate consumption index, Ct, as 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, where
0<θ<1, can reset their prices optimally, whereas the remaining firms index
their prices to lagged inflation.
2.3 Price dynamics
Let us assume a set of firms that do not reoptimize their posted price in period
t. As in Galí (2008), using the definition of the aggregate price level and the
fact that all firms that reset prices choose an identical price, P
t, leads to Pt=
2This approach is similar to Benchimol (2011) and Benchimol (2011).
7

hθP1ε
t1+(1θ) (P
t)1εi1
1ε. Dividing both sides by Pt1and log-linearizing around
P
t=Pt1yields
πt=(1θ) (p
tpt1). (8)
In this set-up, we do not assume that prices have inertial dynamics. Inflation
results from the fact that firms reoptimize their price plans in any given period,
choosing a price that differs from the economy’s average price in the previous
period.
2.4 Price setting
A firm that reoptimizes in period tchooses the price P
tthat maximizes the current
market value of the profits generated while that price remains effective. We solve
this problem to obtain a first-order Taylor expansion around the zero-inflation
steady state of the firm’s first-order condition, which leads to
p
tpt1=(1βθ)
∞
∑
k=0
(βθ)kEthc
mct+kjt+(pt+kpt1)i, (9)
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.
2.5 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
εdiε
ε1; it follows that Yt=Ct
must hold for all t. The above goods market clearing condition can be combined
with the consumer’s Euler equation to yield the equilibrium condition
yt=Et[yt+1]σ1(itEt[πt+1]ρc)σ1Et∆εu
t+1. (10)
Market clearing in the labor market requires Nt=R1
0Nt(i)di. Using Eq. 7
leads to
Nt=Z1
00
B
@Yt(i)
eεa
teεp
tMp,t
Ptαm1
C
A
1
1αn
di (11)
=0
B
@Yt
eεa
teεp
tMp,t
Ptαm1
C
A
1
1αnZ1
0Pt(i)
Ptε
1αndi, (12)
8

where the second equality (Eq. 12) follows from the demand schedule and the
goods market clearing condition. Taking logs leads to
(1αn)nt=ytεa
tαmεp
tαmmpp,t+dt, (13)
where mpp,t=mp,tptare the log-linearized, real money balances for productive
purposes and dt=(1αn)ln R1
0Pt(i)
Ptε
1αndi, where di is a measure of price
(and therefore output) dispersion across firms3.
Hence, the following approximate relation among aggregate output, employ-
ment, real money balances, and technology can be written as
yt=εa
t+αmεp
t+(1αn)nt+αmmpp,t. (14)
An expression is derived for an individual firm’s marginal cost in terms of the
economy’s average real marginal cost. Using the marginal product of labor,
mpnt=ln ∂Yt
∂Nt(15)
=ln eεa
teεp
tMp,t
Ptαm
(1αn)Ntαn(16)
=εa
t+αmεp
t+αmmpp,t+ln (1αn)αnnt, (17)
and the marginal product of real money balances,
mpmpt=ln ∂Yt
∂Mt
Pt!(18)
=ln eεa
teεp
tαmeεp
tMp,t
Ptαm1
Nt1αn!(19)
=εa
t+αmεp
t+ln (αm)+(αm1)mpp,t+(1αn)nt, (20)
we obtain an expression of the marginal cost,
mct=(wtpt)m pntmpmpt(21)
=wtpt2εa
t+αmεp
t(2αm1)mpp,t
(12αn)ntln (αm(1αn)) . (22)
3In a neighborhood of the zero-inflation steady state, dtis equal to zero up to a first-order
approximation (Galí, 2008).
9

Using Eq. 14, we obtain an expression of ntsuch that
nt=1
1αnytεa
tαmεp
tαmmpp,t. (23)
Plugging Eq. 23 into Eq. 22 leads to an expression of the marginal cost
mct=(wtpt)+2αn1
1αn
yt+1αnαm
1αn
mpp,t
ln (αm(1αn)) 1
1αn
εa
tαm
1αn
εp
t, (24)
where Eq. 24 defines the economy’s average marginal product of labor, mpnt, and
the economy’s average marginal product of real money balances, mpmpt, in a way
that is consistent with Eq. 14.
Using the fact that mct+kjt=(wt+kpt+k)m pnt+kjtmpm pt+kjt, we obtain
mct+kjt=(wt+kpt+k)+2αn1
1αn
yt+kjt+1αmαn
1αn
mpp,t+k
1
1αn
εa
t+kαm
1αn
εp
t+kln (αm(1αn)) (25)
=mct+k+2αn1
1αnyt+kjtyt+k(26)
=mct+kε2αn1
1αn
(p
tpt+k), (27)
where Eq. 27 follows from the demand schedule, Ct(i)=Pt(i)
PtεCt, combined
with the market-clearing condition (yt=ct).
Substituting Eq. 27 into Eq. 9 and rearranging terms yields
p
tpt1=(1βθ)Θ
∞
∑
k=0
(βθ)kEt[c
mct+k]+
∞
∑
k=0
(βθ)kEt[πt+k], (28)
where Θ=1αn
1αn+ε(2αn1)1.
Finally, combining Eq. 8 with Eq. 28 yields the inflation equation
πt=βEt[πt+1]+λmc c
mct, (29)
where c
mct=mctmc is the real marginal cost gap and λmc =Θ(1θ)(1βθ)
θis strictly
decreasing in the index of price stickiness, θ, the measure of decreasing returns, αn,
and the demand elasticity, ε.
Next, a relation is derived between the economy’s real marginal cost and a
measure of aggregate economic activity. Note that independent of the nature of
10

price setting, average real marginal cost can be expressed as
mct=(wtpt)m pntmpmpt(30)
=(σyt+ηntρn)+2αn1
1αn
yt+1αmαn
1αn
mpp,t
1
1αn
εa
tαm
1αn
εp
tln (αm(1αn)) (31)
=η+αn
1αn
(1σ)yt1+η
1αn
εa
tαm1+η
1αn
εp
t
+1αm1+η
1αnmpp,tρnln (αm(1αn)) , (32)
where the derivation of Eqs. 31 and 32 makes use of the household’s optimality
condition (Eq. 6) and the (approximate) aggregate production relation (Eqs. 14
and 23).
Knowing that σ>0, 0 αn1, and η1, it is obvious that σ(1αn)+
η+2αn1>0. However, the sign of (1(1+η)αmαn)coming from Eq. 32
appears undefined. In fact, it confirms some studies from Sinai and Stokes (1975,
1977, 1981, 1989) concluding that 1 αn>(1+η)αm>αm. If this is the case, then
1(1+η)αmαn>0.
Furthermore, under flexible prices, the real marginal cost is constant and given
by mc =µ. Defining the natural level of output, denoted by yf
t, as the equilib-
rium level of output under flexible prices,
mc =η+αn
1αn
(1σ)yf
t1+η
1αn
εa
tαm1+η
1αn
εp
t
+1αm1+η
1αnmpf
p,tρnln (αm(1αn)) , (33)
implies
yf
t=υy
aεa
t+υy
pεp
t+υy
mmpf
p,t+υy
c, (34)
where
υy
a=1+η
η+αn(1σ) (1αn)
υy
p=αm(1+η)
η+αn(1σ) (1αn)
υy
m=αn+αm(1+η)1
η+αn(1σ) (1αn)
υy
c=(1αn) (ln (αm(1αn)) +ρnµ)
η+αn(1σ) (1αn).
11

Subtracting Eq. 33 from Eq. 32 yields
c
mct=η+αn
1αn
(1σ)ytyf
t+1αm1+η
1αnmpp,tmpf
p,t, (35)
where ytyf
tis the output gap, and mpp,tmpf
p,tis the real money gap, where
money is used here only for production purposes. By combining Eqs. 29 and 35,
we obtain our first equation relating inflation to its next-period forecast, output
gap, and real money balances gap,
πt=βEt[πt+1]+ψxytyf
t+ψmmpp,tmpf
p,t(36)
where
ψx=η+αn(1αn) (1σ)
1αn+ε(2αn1)(1θ)1
θβ
and
ψm=1αnαm(1+η)
1αn+ε(2αn1)(1θ)1
θβ.
The second key equation describing the equilibrium of the New Keynesian
model is obtained from Eq. 10:
yt=Et[yt+1]σ1(itEt[πt+1]ρc)σ1Et∆εu
t+1. (37)
Henceforth, Eq. 37 is referred to as the dynamic IS equation.
The third key equation describes the behavior of real money balances. Rear-
ranging Eq. 5 yields
mpn,t=σ
νyta2
νitρm
ν+1
νεn
t. (38)
From Eq. 10, we obtain an expression for the natural interest rate,
if
t=ρc+σEth∆yf
t+1i. (39)
Therefore, from Eqs. 39 and 38, we obtain an expression of the money demand
of firms in the flexible-price economy such that
mpf
n,t=σ
νyf
ta2
νσEth∆yf
t+1iρm+ρca2
ν+1
νεn
t(40)
The last equation determines the interest rate through a smoothed Taylor-type
rule,
it=(1λi)λπ(πtπ)+λxytyf
t+Mk,t+λiit1+εi
t, (41)
where λπand λxare policy coefficients reflecting the weight on the inflation and
12

output gaps and the parameter 0 <λi<1 captures the degree of interest rate
smoothing. εi
tis an exogenous ad hoc shock accounting for fluctuations in the
nominal interest rate. πis an inflation target and Mk,tis a money variable that
is defined as follows: money does not enter the Taylor rule (k=1), leading to a
standard Taylor rule; money enters the Taylor rule by the way of one real money
gap (k=2–4); and money enters the Taylor rule by the way of two real money
gaps (k=5).
Table 1 describes Mk,t’s functional forms.
k Mk,t
1 0
2λ2mpp,tmpf
p,t
3λ3mpn,tmp f
n,t
4λ4mptmp f
t
5λ5mpp,tmpf
p,t+λ6mpn,tmp f
n,t
Table 1: The money variable in the Taylor rule
In the literature, money is generally introduced through a money growth vari-
able (Ireland, 2003; Andrés et al., 2006, 2009; Canova and Menz, 2011; Barthélemy
et al., 2011). However, Benchimol and Fourçans (2012) also introduce a money-gap
variable and show that, at least in the Eurozone, it is empirically more significant
than other money variable measures. Such a rule can also be derived from the
optimization program of the central bank as a social planner (Woodford, 2003).
Finally, closing the model requires an additional equilibrium relation. For that
purpose, we use the following quantitative equation:
PtYt=eζtMt, (42)
where Mtrepresents the total nominal money stock and eζtis an exogenous time-
varying velocity process defined in the next section. Taking logs, Eq. 42 leads
to
yt=mpt+ζt=mpn,t+mpp,t+ζt(43)
The corresponding flexible-price economy equation is similar (Eq. 46) to the
previous relation.
13

2.6 DSGE model
Our DSGE model consists of eight equations and eight dependent variables: infla-
tion, nominal interest rate, output, flexible-price output, real money balances held
for production purpose, its flexible-price counterpart, real money balances held
for nonproduction purpose, and its flexible-price counterpart.
Flexible-price economy
yf
t=υy
aεa
t+υy
pεp
t+υy
mmpf
p,t+υy
c(44)
mpf
n,t=σ
νyf
ta2
νσEth∆yf
t+1iρm+ρca2
ν+1
νεn
t(45)
mpf
p,t=yf
tmpf
n,tζt(46)
Sticky-price economy
πt=βEt[πt+1]+ψxytyf
t+ψmmpp,tmpf
p,t(47)
yt=Et[yt+1]σ1(itEt[πt+1]ρc)σ1Et∆εu
t+1(48)
mpn,t=σ
νyta2
νitρm
ν+1
νεn
t(49)
mpp,t=ytmpn,tζt
it=(1λi)λπ(πtπ)+λxytyf
t+Mk,t+λiit1+εi
t(50)
As we have five historical variables, we have five microfounded shocks: tech-
nology shock (εa
t), shock to household money demand (εn
t), shock to firm money
demand (εp
t), short-term interest rate or monetary policy shock (εi
t), and preference
shock (εu
t).
Definition 1 8j2fa,n,p,i,ug,εj,t=ρjεj,t1+ξj,t, where ρjis an autoregressive
coefficient of the AR(1) processes and ξj,tfollows a normal i.i.d. process with a mean of
zero and standard deviation of σj.
Following the literature (Benk et al., 2008; Lothian, 2009), the velocity process,
ζt, depends essentially on money shocks. Then, we choose the following specifi-
cation for the time-varying velocity process.
Definition 2 ζt=ζ+λsλms εn,t+(1λms )εp,t, where ζ,λsand 0<λms <1are
parameters.
14

3 Results
As in Smets and Wouters (2007) and An and Schorfheide (2007), we apply Bayesian
techniques to estimate our DSGE models. We test five specifications of the Taylor
rule (Table 1) under our assumption that money is part of the production function.
3.1 Eurozone data
In our model of the Eurozone, πtis the inflation rate, measured as the yearly log-
difference of the gross domestic product (GDP) deflator between one quarter and
the same quarter of the previous year; ytis output, measured as the logarithm of
GDP; and itis the short-term (three-month) nominal interest rate. These data are
extracted from the Euro-area Wide Model (AWM) database of Fagan et al. (2005).
mpn,tand mpp,tare the real money demands of households and firms, respectively,
and are measured as the logarithm of the Euro-area accounts series4divided by the
GDP deflator. We detrend historical variables using a Hodrick-Prescott filter (with
a standard coefficient for quarterly data of 1600).
yf
t, the flexible-price output, mp f
n,t, the flexible-price household real money bal-
ances, and mpf
p,t, the flexible-price firm real money balances, are completely de-
termined by structural shocks.
Notice that we deal with as many historical variables as shocks.
3.2 Calibration
We estimate all parameters except the discount factor (β), the inverse of the Frisch
elasticity of the labor supply (η), the Calvo (1983) parameter (θ), and the elasticity
of household demand for consumption goods (ε). βis set at 0.9926 so that the
annual steady-state real interest rate is three percent and θ,η, and εare set to 0.66,
one, and six, respectively, as in Galí (2008) and Ravenna and Walsh (2006).
Following standard conventions, we calibrate beta distributions for parameters
that fall between zero and one, inverted gamma distributions for parameters that
need to be constrained as greater than zero, and normal distributions in other
cases.
As our goal is to compare five versions of the model, we adopt the same priors
in each version with the same calibration, depending of the Taylor rule specifica-
tion. The calibration of σis inspired by Rabanal and Rubio-Ramírez (2005) and
Casares (2007). They choose risk aversion parameters of 2.5 and 1.5, respectively.
4The money demand series of households and firms are referenced in the Euro-area accounts as
S1M.A1.S.1.X.E.Z and S11.A1.S.1.X.E.Z, respectively (with the IEAQ.Q.I6.N.V.LE.F2B suffix). The
sum of these two aggregates leads to M2.
15

In line with these values, we regard σ=2 as corresponding to a standard risk
aversion (Benchimol and Fourçans, 2012; Benchimol, 2014).
We calibrate our central parameter αm, the share of money in the production
process, with a prior mean of 0.25 and a large standard error (relative to its prior
mean) of 0.2. Following Basu (1995), we assume that the share of working hours
in the production process is around αn=0.5.
As in Smets and Wouters (2003), the standard errors of the innovations are
assumed to follow inverse gamma distributions and we choose beta distributions
for shock persistence parameters; the backward component of the Taylor rules;
output elasticities of labor, αn; and real money balances, αm, of the production
function that should be less than one.
The scale parameters γand χare calibrated to 0.44 and one, respectively, as in
Christiano et al. (2005), and the money velocity mean prior (ζ) is calibrated to 0.31
following Carrillo et al. (2007).
The smoothed Taylor rules (λi,λπ, and λx) are calibrated following Gerlach-
Kristen (2003), with priors analogous to those used by Smets and Wouters (2003)
and Benchimol and Fourçans (2012). In order to take possible behaviors of the cen-
tral bank into consideration, we assign a higher standard error to the Taylor rule
coefficients. The non-standard parameters’ mean priors of the augmented Taylor
rules for k=25 are calibrated to 0.5, with a large standard error (relative to its
prior mean) of 0.2.
All the standard errors of shocks are assumed to be distributed according to
inverted Gamma distributions, with prior means of 0.02. The latter distribution
ensures that these parameters have a positive support. The autoregressive para-
meters are all assumed to follow beta distributions. All these distributions are
centered around 0.75 and we take a common standard error of 0.1 for the shock
persistence parameters, as in Smets and Wouters (2003).
The calibration of the parameters entering the time-varying component of ve-
locity is quite new. The prior mean of λsis calibrated to one and, because this
calibration exercise is new, we assume a large standard deviation (0.50) and a
normal distribution. The prior mean of λms is calibrated to 0.50 and theoreti-
cally constrained between zero and one. Thus, we assume a Beta distribution for
λms–which can be seen as a trade-off parameter between the two money demand
shocks (εn,tand εp,t). Its standard deviation is not assumed to be very large (0.1)
with respect to its prior mean.
3.3 Estimations
The model is estimated using 52 observations of the Eurozone from 1999Q1 to
2012Q1 and the estimation of the implied posterior distribution of the parameters
16

under the five configurations of the Taylor rule is conducted using the Metropolis-
Hastings algorithm5(ten distinct chains of 300,000 draws each).
The real money balances parameter (αm) of the augmented production function
is estimated to be between 0.014 (k=4) and 0.042 (k=1). This result differs
from that found by Sinai and Stokes (1972) for the same parameter (0.087).6The
prior and posterior distributions are presented in Appendix B and estimates of the
macro-parameters (aggregated structural parameters) are provided in Appendix
E.
We use Bayesian techniques to estimate our model including money in the
production function (see Table 2). We do not adopt the Short (1979) restriction
involving constant returns to scale in the production function.7
The presence of a money gap in the Phillips curve (Eq. 47) supports different
Taylor rule considerations. Here, we test our model under five Taylor rules (see
Table 1), and the Bayesian estimation of the model with a productive-money gap
(k=2) yields the higher log marginal density (-435.91).
A robustness test regarding the numerical maximization of the posterior kernel
is also conducted and indicates that the optimization procedure leads to a robust
maximum for the posterior kernel. The convergence of the proposed distribution
to the target distribution is satisfied. A diagnosis of the overall convergence for
the Metropolis-Hastings sampling algorithm is provided in Appendix D and, fol-
lowing Ratto (2008), all estimations are stable.
3.4 Simulations
3.4.1 Impulse response functions
Appendix C presents the responses of key variables to structural shocks for each
k.
In response to a preference shock, the inflation rate, output, output gap, firm
real money balances, nominal interest rate, and real interest rate rise, whereas
household real money holdings display a little undershooting process in the first
few periods, then return to their steady-state value.
After a technology shock, the output gap, inflation rate, nominal interest rate,
firm real money balances, and real interest rates decrease, whereas output and
household real money balances rise.
In response to an interest rate shock, the inflation rate, output, and output gap
fall. Interest rates and firm money demand rise. A positive monetary policy shock
induces a fall in interest rates due to a sufficiently low degree of intertemporal
5See, for example, Smets and Wouters (2003), Smets and Wouters (2007), Adolfson et al. (2007),
and Adolfson et al. (2008).
6Benchimol (2011) estimates αmto be 0.064.
7This work has already been done in Benchimol (2011) and Benchimol (2011).
17

Priors Posteriors (k=1) Posteriors (k=2) Posteriors (k=3) Posteriors (k=4) Posteriors (k=5)
Law Mean Std. Mean Std. 5% 95% Mean Std. 5% 95% Mean Std. 5% 95% Mean Std. 5% 95% Mean Std. 5% 95%
αnBeta 0.50 0.20 0.5019 0.2774 0.1810 0.8347 0.4996 0.2773 0.1708 0.8295 0.5002 0.2774 0.1724 0.8317 0.5001 0.2773 0.1754 0.8310 0.5000 0.2773 0.1781 0.8305
αmBeta 0.25 0.20 0.0427 0.0006 0.0000 0.0720 0.0299 0.0003 0.0000 0.0728 0.0222 0.0006 0.0000 0.0471 0.0147 0.0002 0.0000 0.0330 0.0248 0.0003 0.0000 0.0505
νNormal 1.25 0.10 1.3567 0.0912 1.2044 1.5049 1.4009 0.0907 1.2542 1.5501 1.3270 0.0899 1.1795 1.4771 1.3571 0.0911 1.2049 1.5044 1.3715 0.0953 1.2206 1.5257
σNormal 2.00 0.10 1.8465 0.1004 1.6803 2.0096 1.8755 0.1030 1.7055 2.0427 1.8106 0.1083 1.6481 1.9803 1.8465 0.1004 1.6794 2.0086 1.8540 0.1028 1.6873 2.0241
γNormal 0.44 0.05 0.4399 0.0498 0.3578 0.5209 0.4100 0.0531 0.3237 0.4983 0.4585 0.0500 0.3770 0.5402 0.4388 0.0498 0.3560 0.5203 0.4288 0.0528 0.3435 0.5170
χNormal 1.00 0.10 0.9998 0.1000 0.8374 1.1695 0.9998 0.1000 0.8333 1.1625 1.0009 0.1000 0.8358 1.1661 0.9994 0.1000 0.8346 1.1635 1.0010 0.1000 0.8354 1.1612
λiBeta 0.50 0.05 0.6005 0.0519 0.5181 0.6843 0.6121 0.0507 0.5294 0.6949 0.5875 0.0538 0.5022 0.6735 0.5988 0.0520 0.5160 0.6838 0.6028 0.0523 0.5194 0.6883
λπNormal 3.50 0.20 3.4258 0.1981 3.1020 3.7526 3.3998 0.2009 3.0677 3.7269 3.4246 0.2026 3.0852 3.7416 3.4280 0.1988 3.1048 3.7617 3.4208 0.1994 3.0879 3.7408
λxNormal 1.50 0.20 1.4118 0.2037 1.0724 1.7369 1.3933 0.2023 1.0582 1.7226 1.4305 0.2041 1.1101 1.7693 1.4197 0.2019 1.0952 1.7542 1.4129 0.2025 1.0782 1.7435
λkNormal 0.50 0.20 0.6193 0.2134 0.2731 0.9737 0.4643 0.2725 0.0923 0.8320 0.4201 0.2019 0.0863 0.7450 0.3505 0.2323 -0.0116 0.7269
λ6Normal 0.50 0.20 0.5621 0.2212 0.2065 0.9204
ζNormal 0.31 0.10 0.3082 0.0998 0.1425 0.4711 0.3076 0.0998 0.1421 0.4716 0.3110 0.0999 0.1484 0.4739 0.3084 0.0999 0.1432 0.4703 0.3070 0.0998 0.1442 0.4725
λsNormal 1.00 0.50 2.0798 0.2858 1.6270 2.5183 2.0830 0.2694 1.6330 2.5265 2.0754 0.2974 1.6309 2.5201 2.0806 0.2885 1.6297 2.5169 2.0794 0.2652 1.6286 2.5167
λms Beta 0.50 0.10 0.1617 0.0420 0.0958 0.2299 0.1595 0.0398 0.0928 0.2244 0.1654 0.0428 0.0961 0.2330 0.1622 0.0418 0.0949 0.2286 0.1610 0.0411 0.0922 0.2254
ρaBeta 0.75 0.10 0.9398 0.0217 0.9044 0.9753 0.9333 0.0234 0.8955 0.9720 0.9442 0.0202 0.9112 0.9779 0.9399 0.0209 0.9051 0.9757 0.9378 0.0216 0.9019 0.9750
ρuBeta 0.75 0.10 0.9543 0.0267 0.9307 0.9775 0.9423 0.0151 0.9140 0.9722 0.9638 0.0279 0.9446 0.9842 0.9557 0.0142 0.9341 0.9785 0.9515 0.0165 0.9255 0.9777
ρiBeta 0.15 0.01 0.1567 0.0104 0.1397 0.1736 0.1569 0.0104 0.1398 0.1739 0.1562 0.0104 0.1391 0.1729 0.1567 0.0104 0.1397 0.1736 0.1568 0.0104 0.1397 0.1738
ρpBeta 0.75 0.10 0.7656 0.0832 0.6390 0.8979 0.7492 0.0902 0.6186 0.8851 0.7815 0.0763 0.6592 0.9077 0.7659 0.0791 0.6409 0.8956 0.7605 0.0851 0.6335 0.8949
ρnBeta 0.75 0.10 0.8342 0.0520 0.7526 0.9191 0.8405 0.0510 0.7601 0.9235 0.8289 0.0528 0.7445 0.9156 0.8337 0.0508 0.7516 0.9179 0.8368 0.0509 0.7553 0.9222
σaInvgamma 0.02 2.00 0.0071 0.0007 0.0058 0.0082 0.0068 0.0007 0.0056 0.0080 0.0073 0.0008 0.0061 0.0085 0.0071 0.0007 0.0058 0.0082 0.0070 0.0007 0.0058 0.0082
σuInvgamma 0.02 2.00 0.1063 0.0454 0.0592 0.1520 0.0986 0.0200 0.0563 0.1417 0.1124 0.0585 0.0612 0.1635 0.1081 0.0279 0.0609 0.1566 0.1052 0.0280 0.0597 0.1521
σiInvgamma 0.02 2.00 0.0290 0.0046 0.0211 0.0368 0.0258 0.0042 0.0182 0.0331 0.0341 0.0061 0.0240 0.0438 0.0314 0.0052 0.0226 0.0400 0.0300 0.0055 0.0203 0.0392
σpInvgamma 0.02 2.00 0.0164 0.0027 0.0118 0.0209 0.0164 0.0026 0.0118 0.0209 0.0166 0.0028 0.0119 0.0210 0.0164 0.0026 0.0118 0.0208 0.0164 0.0025 0.0118 0.0209
σnInvgamma 0.02 2.00 0.0169 0.0018 0.0139 0.0199 0.0174 0.0018 0.0143 0.0204 0.0166 0.0018 0.0135 0.0195 0.0169 0.0018 0.0139 0.0199 0.0171 0.0018 0.0140 0.0201
Acceptation rate 2[0.18; 0.19]Acceptation rate 2[0.21; 0.22]Acceptation rate 2[0.15; 0.16]Acceptation rate 2[0.21; 0.22]Acceptation rate 2[0.20; 0.21]
Log data density: -437.29 Log data density: -435.91 Log data density: -438.51 Log data density: -438.74 Log data density: -438.14
Table 2: Bayesian estimation of the model
18

substitution (i.e., the risk aversion parameter is sufficiently high), which generates
a high-impact response of current relative to future consumption. This result has
been noted in inter alia, Jeanne (1994), and Christiano et al. (1997).
Following a shock in the money demand of firms, interest rates, the output
gap, and the real money holdings of firms decrease, whereas inflation and the
real money holdings of households increase. These impulse response functions
are similar to Smets and Wouters (2003) with regard to output, inflation, and in-
terest rates. However, the responses following a shock in the money demand of
households depends on the model specification (see Appendix C).
3.4.2 Variance decompositions
The analysis is conducted via unconditional and conditional variance decomposi-
tions (see Table 3) to compare the impact of shocks on variables across the models
and over time.
For all models, most of the long-run variance in output comes from the tech-
nology shock (around 75%), about one-quarter of the output variance results from
the interest rate shock (around 5-20%) and the remaining quarter occurs due to the
other shocks. In the short run, most of the output variance comes from the mone-
tary policy shock (around 63%), whereas around 28% is a result of the technology
shock. The money demand of firms impacts output variance (and its flexible-price
counterpart) due to the form of the production function (Eq. 7). Although we do
not add a constant return-to-scale restriction to the production function, we know
that such a restriction should also attribute a larger role to real money demand
in explaining the variances of output and its flexible-price counterpart (Benchi-
mol, 2011). However, in the short run, the share of flexible-price output vari-
ance explained by the shock in the money demand of firms is important (around
24%). This role decreases over longer horizons (to around 8%) and is in line with
Moghaddam (2010). However, we must temper this result by the fact that we do
not have a money supply shock in our framework, which is similar to the frame-
works found in the literature (Benhabib et al., 2001; Ireland, 2004; Andrés et al.,
2009; Benchimol and Fourçans, 2012; Benchimol, 2014).
A look at the conditional and unconditional inflation variance decompositions
shows the overwhelming role of the interest rate shock, which explains more than
92% of inflation rate variance in the short run. This role decreases over time,
whereas the role of the preference shock increases from around 6% in the short
run to around 20% in the long run (except for k=2). The other shocks play a
minor role in inflation variance.
The variance of the nominal interest rate is dominated in the short run by the
direct interest rate shock (monetary policy shock), whereas the preference shock
does not play a significant role. The relative importance of each of these shocks
19

Quarter 1 (k=1) Quarter 1 (k=2) Quarter 1 (k=3) Quarter 1 (k=4) Quarter 1 (k=5)
ξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,t
yt27.99 1.20 63.45 7.36 0.00 27.69 1.73 63.66 6.91 0.01 29.64 0.66 62.17 7.53 0.00 28.18 0.93 63.68 7.21 0.00 28.14 1.13 63.68 7.05 0.00
πt0.14 6.42 93.04 0.40 0.00 0.75 5.82 92.37 0.96 0.10 0.00 5.90 93.96 0.13 0.01 0.13 5.50 94.05 0.32 0.00 0.22 5.69 93.62 0.46 0.00
it0.42 19.31 78.83 1.44 0.00 1.01 21.62 75.25 2.10 0.03 0.14 15.36 83.55 0.95 0.00 0.39 17.13 81.26 1.22 0.00 0.51 18.63 79.31 1.55 0.00
mp p,t0.56 0.15 4.26 69.92 25.12 0.51 0.13 3.48 70.31 25.57 0.64 0.15 5.09 68.96 25.17 0.62 0.15 4.74 69.42 25.08 0.62 0.15 4.55 68.98 25.71
mpn,t15.23 0.00 53.10 4.65 27.02 15.13 0.02 50.70 4.43 29.73 15.90 0.02 54.14 4.64 25.30 15.36 0.00 54.10 4.52 26.02 15.37 0.00 53.23 4.50 26.90
yf
t75.16 0.00 0.00 24.84 0.00 74.16 0.00 0.00 25.84 0.00 77.19 0.00 0.00 22.81 0.00 76.30 0.00 0.00 23.70 0.00 75.70 0.00 0.00 24.30 0.00
mp f
p,t1.44 0.00 0.00 72.05 26.51 1.41 0.00 0.00 71.29 27.30 1.58 0.00 0.00 72.14 26.28 1.48 0.00 0.00 71.84 26.67 1.50 0.00 0.00 71.22 27.28
mp f
n,t35.47 0.00 0.00 13.96 50.57 33.12 0.00 0.00 14.14 52.74 38.42 0.00 0.00 13.16 48.41 36.29 0.00 0.00 13.23 50.49 35.43 0.00 0.00 13.71 50.87
ζt0.00 0.00 0.00 96.28 3.72 0.00 0.00 0.00 96.52 3.48 0.00 0.00 0.00 96.26 3.74 0.00 0.00 0.00 96.28 3.72 0.00 0.00 0.00 96.27 3.73
Quarter 4 (k=1) Quarter 4 (k=2) Quarter 4 (k=3) Quarter 4 (k=4) Quarter 4 (k=5)
ξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,t
yt52.21 0.70 37.41 9.68 0.00 52.09 1.01 38.12 8.77 0.01 54.11 0.37 35.40 10.11 0.00 52.49 0.54 37.27 9.70 0.00 52.67 0.66 37.59 9.08 0.00
πt0.23 10.82 88.50 0.45 0.00 1.41 8.28 89.03 1.08 0.19 0.01 11.57 88.25 0.14 0.02 0.21 9.64 89.77 0.37 0.00 0.41 9.43 89.64 0.52 0.01
it1.07 50.30 46.28 2.36 0.00 2.43 52.70 41.79 3.03 0.05 0.39 44.01 53.79 1.80 0.01 1.05 46.75 50.04 2.16 0.00 1.31 49.10 47.10 2.49 0.00
mp p,t0.89 0.53 2.12 68.50 27.95 0.87 0.52 1.84 66.25 30.52 0.96 0.46 2.38 69.69 26.50 0.96 0.48 2.28 68.63 27.64 1.01 0.54 2.29 66.53 29.63
mpn,t27.66 0.96 30.40 6.04 34.95 26.83 0.93 28.40 5.37 38.47 29.09 0.86 30.95 6.35 32.75 27.93 0.90 30.83 6.02 34.33 27.79 0.93 30.24 5.68 35.36
yf
t82.34 0.00 0.00 17.66 0.00 82.46 0.00 0.00 17.54 0.00 83.07 0.00 0.00 16.93 0.00 82.73 0.00 0.00 17.27 0.00 83.14 0.00 0.00 16.86 0.00
mp f
p,t2.12 0.00 0.00 68.79 29.08 2.13 0.00 0.00 65.66 32.21 2.25 0.00 0.00 70.67 27.08 2.12 0.00 0.00 69.00 28.88 2.23 0.00 0.00 66.83 30.95
mp f
n,t43.13 0.00 0.00 11.02 45.86 39.91 0.00 0.00 10.40 49.70 46.50 0.00 0.00 10.99 42.52 43.51 0.00 0.00 10.65 45.84 42.72 0.00 0.00 10.44 46.84
ζt0.00 0.00 0.00 95.75 4.25 0.00 0.00 0.00 95.58 4.42 0.00 0.00 0.00 96.07 3.93 0.00 0.00 0.00 95.82 4.18 0.00 0.00 0.00 95.53 4.47
Quarter ∞(k=1) Quarter ∞(k=2) Quarter ∞(k=3) Quarter ∞(k=4) Quarter ∞(k=5)
ξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,tξa,tξu,tξi,tξp,tξn,t
yt75.42 0.36 18.51 5.70 0.00 73.71 0.55 20.40 5.33 0.01 78.20 0.18 15.87 5.75 0.00 75.88 0.28 18.07 5.77 0.00 75.82 0.34 18.59 5.25 0.00
πt0.41 21.60 77.57 0.41 0.00 2.73 13.08 82.92 1.03 0.23 0.05 28.85 70.95 0.12 0.02 0.40 20.68 78.57 0.35 0.00 0.79 18.83 79.89 0.48 0.01
it1.47 77.71 19.58 1.23 0.00 3.33 75.69 19.31 1.64 0.03 0.55 77.30 21.21 0.94 0.00 1.48 76.08 21.27 1.17 0.00 1.82 76.83 20.06 1.28 0.00
mp p,t2.13 1.70 1.71 64.49 29.98 1.93 1.48 1.51 60.49 34.59 2.41 1.72 1.84 67.12 26.91 2.28 1.59 1.79 64.77 29.57 2.39 1.74 1.84 61.34 32.69
mpn,t47.69 2.78 17.88 4.25 27.41 43.79 2.44 17.41 3.75 32.61 51.89 2.75 17.08 4.46 23.82 48.24 2.63 17.79 4.28 27.06 47.42 2.69 17.65 3.89 28.36
yf
t92.00 0.00 0.00 8.00 0.00 91.63 0.00 0.00 8.37 0.00 92.65 0.00 0.00 7.35 0.00 92.14 0.00 0.00 7.86 0.00 92.51 0.00 0.00 7.49 0.00
mp f
p,t4.88 0.00 0.00 64.15 30.96 4.48 0.00 0.00 59.34 36.18 5.50 0.00 0.00 67.26 27.24 4.86 0.00 0.00 64.49 30.65 5.10 0.00 0.00 61.03 33.88
mp f
n,t62.67 0.00 0.00 6.49 30.84 56.28 0.00 0.00 6.30 37.42 68.11 0.00 0.00 6.27 25.62 62.94 0.00 0.00 6.30 30.76 61.63 0.00 0.00 6.01 32.36
ζt0.00 0.00 0.00 95.18 4.82 0.00 0.00 0.00 94.56 5.44 0.00 0.00 0.00 95.86 4.14 0.00 0.00 0.00 95.28 4.72 0.00 0.00 0.00 94.68 5.32
Table 3: Variance decompositions
20

changes over time. For longer horizons, there is an inversion over time–the pref-
erence shock explains almost 75% of the nominal interest rate variance, whereas
the interest rate shock explains less than 21%.
Table 3 shows that the demands for real money are mainly explained by the
money, technology, and interest rate shocks. In the short run, variance in the
money demand of firms is essentially determined by its corresponding shock (around
68%) as well as that in the money demand of households (around 25%). However,
the variance in household money demand is mainly driven by the interest rate
shock (around 50%), its corresponding shock (around 25%), and the technology
shock (around 15%). In the long run, variance in the money demand of firms is
also driven by its corresponding shock (around 60%) and the households’ money
demand shock (around 30%) and the firms’ money demand variance decompo-
sition changes. The latter is mainly driven, in the long run, by the technology
shock (around 45%), its corresponding shock (around 23-32%), and the interest
rate shock (around 17%).
It is also interesting to note that the same type of analysis applies to the flexible-
price output variance decomposition. The technology shock, with a weight of
around 91%, is the main explanatory factor of long-run variance in flexible-price
output. In shorter time frames, flexible-price output variance is mainly explained
by the shocks in technology (around 75%) and firm money demand. As previ-
ously explained, this result is attributable to the functional form of the production
function.
Although flexible-price real money demand of firms is mainly impacted by
money shocks (regardless of the time horizon), flexible-price real money demand
of households is mainly driven by the technology shock in the long run (around
56%-68%) and its corresponding shock, whereas in the short run, it is mainly
driven by its corresponding shock (household real money demand shock).
4 Interpretation
Do increases in the real money supply increase the productive capacity of the
economy? Empirical and theoretical papers, ranging from Sinai and Stokes (1972)
to Benchimol (2011), have attempted to answer this question by including real
balances in an estimated aggregate production function. Estimates of the output
elasticity of real money, using various definitions of money and various methods,
range from about 0.02 to about 1.0 (Startz, 1984). Since the growth rate of real
money balances is generally between plus or minus seven percent per annum,
these elasticity estimates suggest that fluctuations in the real money supply can
explain increases in aggregate supply on the order of statistical noise in our case.
The output elasticity of real money balances is equal to 0.016 in our best model
21

specification (k=2). For all k, our results are at least ten times lower than those of
Sinai and Stokes (1972, 1975, 1989) and Short (1979).
In our study, we do not assume constant returns to scale: our production func-
tion specification follows the hypothesis of Khan and Ahmad (1985)–decreasing
return-to-scale hypothesis. This hypothesis gives by-construction no influence on
money in the dynamics of the variables, despite its introduction into the produc-
tion function8. However, because of the Cobb and Douglas (1928) assumption
about the formulation of the production function, and even if the elasticity of the
real money holdings of firms is attributed to statistical noise, the money demand
of firms could impact at least output dynamics.
The simulations (see Table 3 and Appendix C) are close to those obtained in the
literature and provide interesting results regarding the potential effect of money
on output and flexible-price output under two different money demands (house-
hold and firm).
Interestingly, and even if money enters into the inflation equation, money shocks
have almost no effect on the variance decomposition of inflation. This result is
common in the literature on money in a non-separable utility function (Ireland,
2004; Andrés et al., 2009; Benchimol and Fourçans, 2012). Moreover, the estimated
velocity means are in line with Carrillo et al. (2007) and do not change across mod-
els (see Appendix E).
Another interesting result is that the shock on firm money demand has an
important influence on flexible-price output and in the Taylor rule. The esti-
mated contribution of firms’ money holdings in our money-augmented Taylor
rule (k=2) seems to be significant: this means that this shock could poten-
tially impact monetary policy. This result does not mean that we should target firm
money demand. It only tells us that this variable could be taken into account for
policy analysis. Because firm money holding shocks impact macroeconomic vari-
able dynamics in our framework, monetary policy should pay particular attention
to the money demand of firms.
5 Conclusion
One of the most unsettled issues of the postwar economic literature involves the
role of money as a factor of production. The notion of money as a factor of pro-
duction has been debated both theoretically and empirically by a number of re-
searchers in the past five decades. The question is whether money is an omitted
variable in the production process.
8Following Benchimol (2011), the decreasing returns to scale hypothesis is preferred over the
constant returns to scale hypothesis. We do not follow the hypothesis of Short (1979), Startz (1984),
Benzing (1989), and Chang (2002) of constant returns to scale for money in the production function.
22

In parallel, New Keynesian DSGE theory, combined with Bayesian analysis,
has become increasingly popular in the academic literature and in policy analy-
sis. The unique contribution of this paper is to build and test a microfounded
New Keynesian DSGE model that includes money in the production function. We
depart from the existing theoretical and empirical literature by building a New
Keynesian DSGE model à la Galí (2008) that includes money in the production
function, and, as a consequence, in the inflation equation (Phillips curve). Closing
the model leads to the new concept of flexible-price real money balances presented
by Benchimol and Fourçans (2012).
Empirical support for money as an input along with labor has been mixed;
thus, the issue appears to be unsettled. This paper, as in Benhabib et al. (2001), dif-
ferentiates between money demanded by households and firms. This distinction
between money that is used for productive and nonproductive purposes seems
to be warranted. By testing our models with Bayesian techniques under different
monetary policy rules, we show that even if the weight of real money balances in
the production function is very low, the firm money shock has an important influ-
ence on flexible-price output and a significant impact on output. Part of this influ-
ence comes from the functional form of the production function (non-separability
between money holdings and working hours).
With respect to our estimation of the weight of money in the production func-
tion, real money balances could be excluded from the production process. How-
ever, considering this hypothesis of money in the production function highlights
the significant role of the firm money shock. Incorporating the real money bal-
ances variable as a factor input in a production function–in order to capture the
productivity gains derived from using money–could lead to important monetary
policy implications.
6 Appendix
A Optimization problem
Our Lagrangian is given by
Lt=Et"∞
∑
k=0
βk(Ut+kλt+kVt+k)#,
where
Vt=Ct+∆Mn,t
Pt
+∆Mp,t
Pt
+Qt
Bt
Pt
Bt1
Pt
Wt
Pt
Nt
23

and
Ut=eεu
t C1σ
t
1σ+γeεn
t
1νMn,t
Pt1ν
χN1+η
t
1+η!.
The first-order condition related to consumption expenditures is given by
λt=eεu
tCσ
t, (51)
where λtis the Lagrangian multiplier associated with the budget constraint at
time t.
The first-order condition corresponding to the demand for contingent bonds
implies that
λtQt=βEtλt+1
Pt
Pt+1. (52)
The demand for cash held for nonproduction purposes that follows from the
household optimization problem is given by
γeεu
teεn
tMn,t
Ptν
=λtβEtλt+1
Pt
Pt+1, (53)
which can be naturally interpreted as a demand for real balances. The latter is
increasing in consumption and is inversely related to the nominal interest rate, as
in conventional specifications.
Working hours following the household optimization problem are given by
χeεu
tNη
t=λt
Wt
Pt
. (54)
From Eq. 51, we obtain
λt=eεu
tCσ
t,Uc,t=eεu
tCσ
t, (55)
where Uc,t=∂Uk,t
∂Ct+kk=0. Eq. 55 defines the marginal utility of consumption.
Hence, the optimal consumption/savings, real money balance, and labor sup-
ply decisions are described by the following conditions:
Combining Eq. 51 with Eq. 52 yields
Qt=βEt"eεu
t+1Cσ
t+1
eεu
tCσ
t
Pt
Pt+1#,Qt=βEtUc,t+1
Uc,t
Pt
Pt+1, (56)
where Uc,t+1=∂Uk,t
∂Ct+kk=1. Eq. 56 is the usual Euler equation for intertem-
poral consumption flows. It establishes that the ratio of marginal utility of
future and current consumption is equal to the inverse of the real interest
24

rate.
Combining Eq. 51 and Eq. 53 yields
γeεn
t
Cσ
tMn,t
Ptν
=1Qt,Um,t
Uc,t
=1Qt, (57)
where Um,t=∂Uk,t
∂(Mn,t+k/Pt+k)k=0
. Eq. 57 is the intertemporal optimality condi-
tion setting the marginal rate of substitution between money and consump-
tion equal to the opportunity cost of holding money.
Combining Eq. 51 and Eq. 54 yields
χNη
t
Cσ
t
=Wt
Pt
,Un,t
Uc,t
=Wt
Pt
, (58)
where Un,t=∂Uk,t
∂Nt+kk=0. Eq. 58 is the condition for the optimal consumption-
leisure arbitrage, implying that the marginal rate of substitution between
consumption and labor is equal to the real wage.
25

B Priors and posteriors
00.5 1
0
0.5
1
1.5
00.04
0
10
20
01.36
0
2
4
01.85
0
1
2
3
00.44
0
2
4
6
8
0 1
0
1
2
3
00.31 0.62
0
1
2
3
02.08 4.16
0
0.5
1
00.2 0.4 0.6
0
2
4
6
8
00.6
0
2
4
6
03.43
0
0.5
1
1.5
01.41 2.82
0
0.5
1
1.5
00.94
0
5
10
15
00.95
0
10
20
00.16
0
10
20
30
00.77
0
2
4
00.83
0
2
4
6
00.05 0.1
0
200
400
00.5
0
20
40
60
00.05 0.1
0
20
40
60
80
00.05 0.1
0
50
100
00.05 0.1
0
100
200
Figure 1: Priors and posteriors of the estimated parameters (k=1).
26

00.5 1
0
0.5
1
1.5
00.03
0
10
20
01.4
0
2
4
01.88
0
1
2
3
00.41 0.82
0
2
4
6
0 1
0
1
2
3
00.31 0.62
0
1
2
3
02.08 4.16
0
0.5
1
00.2 0.4 0.6
0
2
4
6
8
00.61
0
2
4
6
03.4
0
0.5
1
1.5
01.39 2.78
0
0.5
1
1.5
0 1 2
0
0.5
1
1.5
00.93
0
5
10
15
00.94
0
10
20
00.16
0
10
20
30
00.75
0
2
4
00.84
0
2
4
6
00.05 0.1
0
200
400
00.2
0
20
40
60
00.05 0.1
0
20
40
60
80
00.05 0.1
0
50
100
00.05 0.1
0
100
200
Figure 2: Priors and posteriors of the estimated parameters (k=2).
27

00.5 1
0
0.5
1
1.5
00.02 0.04
0
20
40
01.33
0
2
4
01.81
0
1
2
3
00.46
0
2
4
6
8
0 1
0
1
2
3
00.31 0.62
0
2
4
02.08 4.16
0
0.5
1
00.2 0.4 0.6
0
2
4
6
8
00.59
0
2
4
6
03.42
0
0.5
1
1.5
01.43 2.86
0
1
2
0 1 2
0
0.5
1
1.5
00.94
0
5
10
15
00.96
0
10
20
30
00.16
0
10
20
30
00.78
0
2
4
00.83
0
2
4
6
00.05 0.1
0
200
400
00.2 0.4
0
20
40
60
00.05 0.1
0
20
40
60
00.05 0.1
0
50
100
00.05 0.1
0
100
200
Figure 3: Priors and posteriors of the estimated parameters (k=3).
28

00.5 1
0
0.5
1
1.5
00.010.020.030.040.05
0
20
40
01.36
0
2
4
01.85
0
1
2
3
00.44
0
2
4
6
8
0 1
0
1
2
3
00.31 0.62
0
1
2
3
02.08 4.16
0
0.5
1
00.2 0.4 0.6
0
2
4
6
8
00.6
0
2
4
6
03.43
0
0.5
1
1.5
01.42 2.84
0
0.5
1
1.5
0 1
0
0.5
1
1.5
00.94
0
5
10
15
00.96
0
10
20
00.16
0
10
20
30
00.77
0
2
4
00.83
0
2
4
6
00.05 0.1
0
200
400
00.2 0.4
0
20
40
60
00.05 0.1
0
20
40
60
00.05 0.1
0
50
100
00.05 0.1
0
100
200
Figure 4: Priors and posteriors of the estimated parameters (k=4).
29

00.5 1
0
0.5
1
1.5
00.02 0.04
0
10
20
30
01.37
0
2
4
01.85
0
1
2
3
00.43
0
2
4
6
0 1
0
1
2
3
00.31 0.62
0
1
2
3
02.08 4.16
0
0.5
1
00.2 0.4 0.6
0
2
4
6
8
00.6
0
2
4
6
03.42
0
1
2
01.41 2.82
0
0.5
1
1.5
0 1
0
0.5
1
1.5
0 1 2
0
0.5
1
1.5
00.94
0
5
10
15
00.95
0
10
20
00.16
0
10
20
30
00.76
0
2
4
00.84
0
2
4
6
00.05 0.1
0
200
400
00.2 0.4
0
20
40
60
00.05 0.1
0
20
40
60
00.05 0.1
0
50
100
00.05 0.1
0
100
200
Figure 5: Priors and posteriors of the estimated parameters (k=5).
30

C Impulse response functions
-0.04
-0.02
0
Technology
shock
Inflation (%)
0
0.5
1
Output (%)
-0.1
-0.05
0
Nominal interest
rate (%)
-0.1
-0.05
0
Real interest
rate (%)
-0.02
-0.01
0
Output
gap (%)
-0.4
-0.2
0
Firms'
real money (%)
0
0.5
1
Households'
real money (%)
020 40
-1
0
1
Total money
velocity
Quarters
0
0.2
0.4
Preference
shock
0
0.1
0.2
0
0.5
1
0
0.2
0.4
0
0.1
0.2
0
0.2
0.4
-0.2
-0.1
0
020 40
-1
0
1
Quarters
-2
-1
0
Interest
rate shock
-1
-0.5
0
0
0.5
1
0
1
2
-1
-0.5
0
0
0.5
1
-2
-1
0
020 40
-1
0
1
Quarters
-0.1
-0.05
0
Fir ms' mo ney
demand shock
0
0.2
0.4
-0.2
-0.1
0
-0.2
-0.1
0
-0.1
-0.05
0
-4
-2
0
0
0.5
1
020 40
0
2
4
Quarters
-1
0
1
Households' money
demand shock
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-2
-1
0
0
1
2
020 40
0
0.5
1
Quarters
Figure 6: Impulse response function (k=1)
31

-0.1
-0.05
0
Technology
shock
Inflation (%)
0
0.5
1
Output (%)
-0.2
-0.1
0
Nominal interest
rate (%)
-0.1
-0.05
0
Real interest
rate (%)
-0.04
-0.02
0
Output
gap (%)
-0.4
-0.2
0
Firms'
real money (%)
0
0.5
1
Households'
real money (%)
020 40
-1
0
1
Total money
velocity
Quarters
0
0.2
0.4
Preference
shock
0
0.1
0.2
0
0.5
1
0
0.5
0
0.1
0.2
0
0.2
0.4
-0.2
0
0.2
020 40
-1
0
1
Quarters
-1
-0.5
0
Interest
rate shock
-1
-0.5
0
0
0.5
1
0
1
2
-1
-0.5
0
0
0.5
1
-2
-1
0
020 40
-1
0
1
Quarters
-0.2
-0.1
0
Firms' mon ey
demand shock
0
0.2
0.4
-0.2
-0.1
0
-0.2
-0.1
0
-0.1
-0.05
0
-4
-2
0
0
0.5
020 40
0
2
4
Quarters
-0.04
-0.02
0
Households' money
demand shock
-0.02
-0.01
0
-0.02
-0.01
0
0
0.005
0.01
-0.02
-0.01
0
-2
-1
0
0
1
2
020 40
0
0.5
1
Quarters
Figure 7: Impulse response function (k=2)
32

0
0.005
0.01
Technology
shock
Inflation (%)
0
0.5
1
Output (%)
-0.04
-0.02
0
Nominal interest
rate (%)
-0.1
-0.05
0
Real interest
rate (%)
-0.01
0
0.01
Output
gap (%)
-0.4
-0.2
0
Firms'
real money (%)
0
0.5
1
Households'
real money (%)
020 40
-1
0
1
Total money
velocity
Quarters
0
0.2
0.4
Preference
shock
0
0.1
0.2
0
0.5
0
0.2
0.4
0
0.1
0.2
0
0.2
0.4
-0.2
-0.1
0
020 40
-1
0
1
Quarters
-2
-1
0
Interest
rate shock
-1
-0.5
0
0
0.5
1
0
1
2
-1
-0.5
0
0
0.5
1
-2
-1
0
020 40
-1
0
1
Quarters
-0.04
-0.02
0
Firms' mon ey
demand shock
0
0.2
0.4
-0.2
-0.1
0
-0.1
-0.05
0
-0.04
-0.02
0
-4
-2
0
0
0.5
1
020 40
0
2
4
Quarters
0
0.01
0.02
Households' money
demand shock
0
5x 10-3
0
0.005
0.01
-4
-2
0x 10-3
0
5x 10-3
-2
-1
0
0
1
2
020 40
0
0.5
1
Quarters
Figure 8: Impulse response function (k=3)
33

-0.04
-0.02
0
Technology
shock
Inflation (%)
0
0.5
1
Output (%)
-0.1
-0.05
0
Nominal interest
rate (%)
-0.1
-0.05
0
Real interest
rate (%)
-0.02
-0.01
0
Output
gap (%)
-0.4
-0.2
0
Firms'
real money (%)
0
0.5
1
Households'
real money (%)
020 40
-1
0
1
Total money
velocity
Quarters
0
0.2
0.4
Preference
shock
0
0.1
0.2
0
0.5
1
0
0.2
0.4
0
0.1
0.2
0
0.2
0.4
-0.2
-0.1
0
020 40
-1
0
1
Quarters
-2
-1
0
Interest
rate shock
-1
-0.5
0
0
0.5
1
0
1
2
-1
-0.5
0
0
0.5
1
-2
-1
0
020 40
-1
0
1
Quarters
-0.1
-0.05
0
Firms' mon ey
demand shock
0
0.2
0.4
-0.2
-0.1
0
-0.1
-0.05
0
-0.04
-0.02
0
-4
-2
0
0
0.5
1
020 40
0
2
4
Quarters
-1
0
1
Households' money
demand shock
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-2
-1
0
0
1
2
020 40
0
0.5
1
Quarters
Figure 9: Impulse response function (k=4)
34

-0.04
-0.02
0
Technology
shock
Inflation (%)
0
0.5
1
Output (%)
-0.1
-0.05
0
Nominal interest
rate (%)
-0.1
-0.05
0
Real interest
rate (%)
-0.04
-0.02
0
Output
gap (%)
-0.4
-0.2
0
Firms'
real money (%)
0
0.5
1
Households'
real money (%)
020 40
-1
0
1
Total money
velocity
Quarters
0
0.2
0.4
Preference
shock
0
0.1
0.2
0
0.5
1
0
0.2
0.4
0
0.1
0.2
0
0.2
0.4
-0.2
-0.1
0
020 40
-1
0
1
Quarters
-2
-1
0
Interest
rate shock
-1
-0.5
0
0
0.5
1
0
1
2
-1
-0.5
0
0
0.5
1
-2
-1
0
020 40
-1
0
1
Quarters
-0.1
-0.05
0
Firms' mon ey
demand shock
0
0.2
0.4
-0.2
-0.1
0
-0.2
-0.1
0
-0.1
-0.05
0
-4
-2
0
0
0.5
020 40
0
2
4
Quarters
-0.01
-0.005
0
Households' money
demand shock
-4
-2
0x 10-3
-4
-2
0x 10-3
0
1
2x 10-3
-4
-2
0x 10-3
-2
-1
0
0
1
2
020 40
0
0.5
1
Quarters
Figure 10: Impulse response function (k=5)
35

D Robustness checks
Each graph represents specific convergence measures through 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 model parameter means
(intervals), variances (m2), and third moments (m3). For each of the three mea-
sures, convergence requires both lines to become relatively horizontal and con-
verge toward each other in both models9.
6
8
10 First moment
5
10
15 Second moment
0
50
100 Third moment
7
8
9
5
10
15
0
50
100
7
8
9
8
10
12
0
50
100
6
8
10
5
10
15
0
50
100
0 1 2 3
x 105
6
8
10
Iterations 0 1 2 3
x 105
5
10
15
Iterations 0 1 2 3
x 105
0
50
100
Iterations
Figure 11: Metropolis-Hastings’ convergence diagnostics
9Robustness analysis with respect to calibrated parameters is available upon request.
36

E Macro parameters
k=1k=2k=3k=4k=5
υy
a1,0397 1,0321 1,0496 1,0398 1,0378
υy
p0,0443 0,0308 0,0233 0,0152 0,0256
υy
m-0,2145 -0,2273 -0,2390 -0,2446 -0,2337
υy
c-1,0445 -1,1319 -1,2286 -1,3251 -1,1871
σ
ν1,3610 1,3388 1,3644 1,3606 1,3518
σ
νa20,7828 0,7700 0,7847 0,7826 0,7775
ρm+ρca2
ν-0,1534 -0,0984 -0,1882 -0,1516 -0,1332
1
ν0,7370 0,7138 0,7535 0,7368 0,7291
ψx0,6561 0,6943 0,6743 0,6819 0,6850
ψm0,1407 0,1578 0,1611 0,1668 0,1601
1
σ0,5415 0,5331 0,5523 0,5415 0,5393
ρc
σ0,0040 0,0039 0,0041 0,0040 0,0040
a2
ν0,4239 0,4105 0,4334 0,4238 0,4193
ρm
ν-0,1566 -0,1014 -0,1915 -0,1547 -0,1363
λi0,6005 0,6121 0,5874 0,5987 0,6027
λπ(1λi)1,3685 1,3187 1,4127 1,3753 1,3588
λx(1λi)0,5640 0,5404 0,5901 0,5695 0,5612
λk(1λi)0,2402 0,1915 0,1685 0,1392
λ6(1λi)0,2232
exp (ζ)1,3609 1,3602 1,3647 1,3612 1,3593
λsλms 0,3363 0,3322 0,3433 0,3374 0,3347
λs(1λms)1,7434 1,7506 1,7321 1,7431 1,7446
Table 4: Macroparameter values
37

References
Adolfson, M., Laséen, S., Lindé, J., Villani, M., 2007. Bayesian estimation of an
open economy DSGE model with incomplete pass-through. Journal of Interna-
tional Economics 72 (2), 481–511.
Adolfson, M., Laséen, S., Lindé, J., Villani, M., 2008. Evaluating an estimated
New Keynesian small open economy model. Journal of Economic Dynamics
and Control 32 (8), 2690–2721.
An, S., Schorfheide, F., 2007. Bayesian analysis of DSGE models. Econometric Re-
views 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.
Barthélemy, J., Clerc, L., Marx, M., 2011. A two-pillar DSGE monetary policy
model for the euro area. Economic Modelling 28 (3), 1303–1316.
Basu, S., 1995. Intermediate goods and business cycles: implications for produc-
tivity and welfare. American Economic Review 85 (3), 512–531.
Ben-Zion, U., Ruttan, V., 1975. Money in the production function: an interpretation
of empirical results. Review of Economics and Statistics 57 (2), 246–247.
Benchimol, J., 2011. New Keynesian DSGE models, money and risk aversion. PhD
dissertation, Université Paris 1 Panthéon-Sorbonne.
Benchimol, J., 2014. Risk aversion in the Eurozone. Research in Economics 68 (1),
39–56.
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.
Benhabib, J., Schmitt-Grohé, S., Uribe, M., 2001. Monetary policy and multiple
equilibria. American Economic Review 91 (1), 167–186.
Benk, S., Gillman, M., Kejak, M., 2008. Money velocity in an endogenous growth
business cycle with credit shocks. Journal of Money, Credit and Banking 40 (6),
1281–1293.
38

Benzing, C., 1989. An update on money in the production function. Eastern Eco-
nomic Journal 15 (3), 235–239.
Calvo, G., 1983. Staggered prices in a utility-maximizing framework. Journal of
Monetary Economics 12 (3), 383–398.
Canova, F., Menz, G., 2011. Does money matter in shaping domestic business cy-
cles ? An international investigation. Journal of Money, Credit and Banking
43 (4), 577–607.
Carrillo, J., Fève, P., Matheron, J., 2007. Monetary policy inertia or persistent
shocks: a DSGE analysis. International Journal of Central Banking 3 (2), 1–38.
Casares, M., 2007. Monetary policy rules in a New Keynesian Euro area model.
Journal of Money, Credit and Banking 39 (4), 875–900.
Chang, W., 2002. Examining the long-run effect of money on economic growth: an
alternative view. Journal of Macroeconomics 24 (1), 81–102.
Christiano, L., Eichenbaum, M., Evans, C. L., 1997. Sticky price and limited par-
ticipation models of money: a comparison. European Economic Review 41 (6),
1201–1249.
Christiano, L., Eichenbaum, M., Evans, C. L., 2005. Nominal rigidities and the
dynamic effects of a shock to monetary policy. Journal of Political Economy
113 (1), 1–45.
Clarida, R., Galí, J., Gertler, M., 1999. The science of monetary policy: a New Key-
nesian perspective. Journal of Economic Literature 37 (4), 1661–1707.
Cobb, C., Douglas, P., 1928. A theory of production. American Economic Review
18 (1), 139–165.
Fagan, G., Henry, J., Mestre, R., 2005. An area-wide model for the euro area. Eco-
nomic Modelling 22 (1), 39–59.
Fischer, S., 1974. Money and the production function. Economic Inquiry 12 (4),
517–33.
Friedman, M., 1969. Optimum quantity of money. Chicago, IL: Aldine Publishing
Co.
Galí, J., 2008. Monetary policy, inflation and the business cycle: an introduction to
the New Keynesian framework, 1st Edition. Princeton, NJ: Princeton University
Press.
39

Gerlach-Kristen, P., 2003. Interest rate reaction functions and the Taylor rule in the
Euro area. Working Paper Series 258, European Central Bank.
Geweke, J., 1999. Using simulation methods for Bayesian econometric models: in-
ference, development, and communication. Econometric Reviews 18 (1), 1–73.
Hasan, M., Mahmud, S., 1993. Is money an omitted variable in the production
function ? Some further results. Empirical Economics 18 (3), 431–445.
Ireland, P. N., 2003. Endogenous money or sticky prices? Journal of Monetary
Economics 50 (8), 1623–1648.
Ireland, P. N., 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 effect. Mimeo, ENPC-
CERAS.
Johnson, H., 1969. Inside money, outside money, income, wealth and welfare in
monetary theory. Journal of Money Credit and Banking 1 (1), 30–45.
Khan, A., Ahmad, M., 1985. Real money balances in the production function of a
developing country. Review of Economics and Statistics 67 (2), 336–340.
Levhari, D., Patinkin, D., 1968. The role of money in a simple growth model. Amer-
ican Economic Review 58 (4), 713–753.
Lothian, J., 2009. Milton Friedman’s monetary economics and the quantity-theory
tradition. Journal of International Money and Finance 28 (7), 1086–1096.
Moghaddam, M., 2010. Co-integrated money in the production function-evidence
and implications. Applied Economics 42 (8), 957–963.
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.
Ratto, M., 2008. Analysing DSGE models with global sensitivity analysis. Compu-
tational Economics 31 (2), 115–139.
Ravenna, F., Walsh, C. E., 2006. Optimal monetary policy with the cost channel.
Journal of Monetary Economics 53 (2), 199–216.
Sephton, P., 1988. Money in the production function revisited. Applied Economics
20 (7), 853–860.
40

Short, E., 1979. A new look at real money balances as a variable in the production
function. Journal of Money, Credit and Banking 11 (3), 326–339.
Sinai, A., Stokes, H., 1972. Real money balances: an omitted variable from the
production function ? Review of Economics and Statistics 54 (3), 290–296.
Sinai, A., Stokes, H., 1975. Real money balances: an omitted variable from the
production function ? A reply. Review of Economics and Statistics 57 (2), 247–
252.
Sinai, A., Stokes, H., 1977. Real money balances as a variable in the production
function: reply. Journal of Money, Credit and Banking 9 (2), 372–373.
Sinai, A., Stokes, H., 1981. Real money balances in the production function: a
comment. Eastern Economic Journal 17 (4), 533–535.
Sinai, A., Stokes, H., 1989. Money balances in the production function: a retrospec-
tive look. Eastern Economic Journal 15 (4), 349–363.
Smets, F., Wouters, R., 2003. An estimated dynamic stochastic general equilibrium
model of the Euro area. Journal of the European Economic Association 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.
Startz, R., 1984. Can money matter ? Journal of Monetary Economics 13 (3), 381–
385.
Stein, J., 1970. Monetary growth theory in perspective. American Economic Re-
view 60 (1), 85–106.
Subrahmanyam, G., 1980. Real money balances as a factor of production: some
new evidence. Review of Economics and Statistics 62 (2), 280–283.
Walsh, C., 2017. Monetary theory and policy. Cambridge, MA: MIT Press.
Woodford, M., 2003. Interest and prices: foundations of a theory of monetary pol-
icy. Princeton, NJ: Princeton University Press.
You, J., 1981. Money, technology, and the production function: an empirical study.
Canadian Journal of Economics 14 (3), 515–524.
41