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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 Dy-

namic Stochastic General Equilibrium (DSGE) model. In this frame-

work, real money balances enter the production function, and money

demanded by households is di¤erentiated from that demanded by

…rms. By using a Bayesian analysis, our model weakens the hypoth-

esis that money is a factor of production. However, the demand of

money by …rms appears to have a signi…cant impact on the economy,

even if this demand has a low weight in the production process.

Keywords: Money in the production function, DSGE, Bayesian

estimation.

JEL Codes: E23, E31, E51.

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.

Bank of Israel and EABCN, POB 780, 91007 Jerusalem, Israel. Phone: +972-2-

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

Benhabib, Akiva O¤enbacher, André Fourçans and two anonymous referees for their help-

ful advice and comments. This paper does not necessarily re‡ect the views of the Bank of

Israel.

1

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 ‡ow of services from the given

productive resources and a change in the structure of production,

since di¤erent productive activities may di¤er in cash-intensity,

just as they di¤er in labor- or land-intensity.

Milton Friedman (1969)

Considering real money balances to be a factor of production has nu-

merous implications. Money would have a marginal physical productivity

schedule like other inputs, …rms’demands for real balances would be derived

in the same way as other factor demand functions, changes in the stock of

money would a¤ect 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 represent 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,

2

even after correcting for any simultaneity bias, is positive and statistically

signi…cant. 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.

You (1981) …nds that the unexplained portion of output variation virtu-

ally vanishes 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 re-

sults of Khan and Ahmad (1985) are consistent with the hypothesis that

real money balances are an important factor of production. Sephton (1988)

shows that real balances are a valid factor of production within the con…nes

of a constant elasticity of substitution (CES) production function. Hasan

and Mahmud (1993) also support the hypothesis that money is an impor-

tant factor in the production function and that there are potential supply

side-e¤ects 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, di¤erent de…nitions 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)

develop 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 in‡ation

equation that includes money. Following Benchimol and Fourçans (2012), we

introduce the new concept of ‡exible-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 con…rm or

reject the potential in‡uence 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 …rms).

3

After describing the theoretical set-up in Section 2, we calibrate and es-

timate …ve models of the Euro area using Bayesian techniques in Section 3.

Impulse response functions and variance decomposition are analyzed in Sec-

tion 3.4, and we study the consequences of money in the production function

hypothesis by comparing the monetary policy rules of models in Section 4.

Section 5concludes, and Section 6presents additional results.

2 The model

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

sumption, and hold money and bonds, as well as …rms that hire labor and pro-

duce and sell di¤erentiated products in monopolistically competitive goods

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

reset their respective prices each period. Households and …rms behave op-

timally: Households maximize their expected present value of utility, and

…rms maximize pro…ts. There is also a central bank that controls the nomi-

nal interest rate. This model is inspired by Smets and Wouters (2003), Galí

(2008), and Walsh (2017).

2.1 Households

We assume a representative, in…nitely lived household, that seeks to maxi-

mize

Et"1

X

k=0

kUt+k#;(1)

where Utis the period utility function and < 1is 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 expen-

ditures among di¤erent goods. This requires that the consumption index, Ct,

be maximized for any given level of expenditure. 8t2Nand, conditionally

on such optimal behavior, the period budget constraint takes the form

PtCt+Mn;t +Mp;t +QtBtBt1+WtNt+Mn;t1+Mp;t1;(2)

where Ptis an aggregate price index; Mn;t and Mp;t are 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

4

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

solvency condition, such as 8tlim

n!1 Et[Bn]0.

Preferences are measured using a common time-separable utility function.

Under the assumption of a household period utility given by

Ut=e"u

t C1

t

1+e"n

t

1Mn;t

Pt1

N1+

t

1 + !;(3)

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.is the coe¢ cient of the relative risk aversion of households or the

inverse of the intertemporal elasticity of substitution, 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 (inverse

of the Frisch elasticity of the labor supply).

The utility function also contains two structural shocks: "u

tis a general

shock to preferences that a¤ects the intertemporal substitution of households

(preference shock) and "n

tis a shock to household money demand. and

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

implies that

ct=Et[ct+1]1

(itEt[t+1]c)1Et"u

t+1;(4)

where the lowercase letters denote the logarithm of the original aggregated

variables, c=ln (), and is the …rst-di¤erence operator.

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

lem is given by

"n

t+ctmpn;t m=a2it;(5)

where mpn;t =mn;tptare the log-linearized real money balances for nonpro-

ductive purposes, m=ln ()+ a1, and a1and a2are the resulting terms of

the …rst-order Taylor approximation of ln (1 Qt) = a1+a2it. More precisely,

1See Appendix A.

5

if we compute our …rst-order Taylor approximation around the steady-state

interest rate, 1

, we obtain a1= ln 1exp 1

1

e

1

1

and a2=1

e

1

1

.

Real cash holdings has a positive relation with consumption, with an

elasticity equal to =, and a negative relation with the nominal interest

rate ( 1

>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

balances are additively separable in the utility function, the cash holdings of

households do not enter any of the other structural equations: Accordingly,

the equation above becomes a recursive function of the remainder of the sys-

tem 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 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 bal-

ances enter the aggregate demand equation without entering the production

function.

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

to the optimal consumption-leisure arbitrage implies that

wtpt=ct+ntn;(6)

where 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 for nonproductive

use (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

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

di¤erentiated good, but they all use an identical technology, represented by

the following Money-in-the-Production function2

Yt(i) = e"a

te"p

tMp;t

Ptm

Nt(i)1n(7)

2This approach is similar to Benchimol (2011) and Benchimol (2011).

6

where exp ("a

t)represents the level of technology, assumed to be common to

all …rms and to evolve exogenously over time, and "p

tis a shock to …rm money

demand.

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

gregate price level, Pt, and aggregate consumption index, Ct, as given. As

in the standard Calvo (1983) model, our generalization features monopolis-

tic competition and staggered price setting. At any time t, only a fraction

1of …rms, where 0< < 1, can reset their prices optimally, whereas the

remaining …rms index their prices to lagged in‡ation.

2.3 Price dynamics

Let us assume a set of …rms that do not reoptimize their posted price in

period t. As in Galí (2008), using the de…nition of the aggregate price level

and the fact that all …rms that reset prices choose an identical price, P

t,

leads to Pt=P 1"

t1+ (1 ) (P

t)1"1

1". Dividing both sides by Pt1and

log-linearizing around P

t=Pt1yields

t= (1 ) (p

tpt1):(8)

In this set-up, we do not assume that prices have inertial dynamics. In-

‡ation results from the fact that …rms reoptimize their price plans in any

given period, choosing a price that di¤ers from the economy’s average price

in the previous period.

2.4 Price setting

A …rm that reoptimizes in period tchooses the price P

tthat maximizes

the current market value of the pro…ts generated while that price remains

e¤ective. We solve this problem to obtain a …rst-order Taylor expansion

around the zero-in‡ation steady state of the …rm’s …rst-order condition, which

leads to

p

tpt1= (1 )

1

X

k=0

()kEtcmct+kjt+ (pt+kpt1);(9)

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

its steady-state value, mc =, and = ln ("= ("1)) is the log of the

desired gross markup.

7

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 de…ned as Yt=R1

0Yt(i)11

"di"

"1; it follows

that Yt=Ctmust 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]1(itEt[t+1]c)1Et"u

t+1:(10)

Market clearing in the labor market requires Nt=R1

0Nt(i)di. Using Eq.

7leads to

Nt=Z1

00

@Yt(i)

e"a

te"p

tMp;t

Ptm1

A

1

1n

di (11)

=0

@Yt

e"a

te"p

tMp;t

Ptm1

A

1

1nZ1

0Pt(i)

Pt"

1ndi; (12)

where the second equality (Eq. 12) follows from the demand schedule and

the goods market clearing condition. Taking logs leads to

(1 n)nt=yt"a

tm"p

tmmpp;t +dt;(13)

where mpp;t =mp;t ptare the log-linearized, real money balances for pro-

ductive purposes and dt= (1 n) ln R1

0Pt(i)

Pt"

1ndi, where di is a

measure of price (and therefore output) dispersion across …rms3.

Hence, the following approximate relation among aggregate output, em-

ployment, real money balances, and technology can be written as

yt="a

t+m"p

t+ (1 n)nt+mmpp;t:(14)

An expression is derived for an individual …rm’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 e"a

te"p

tMp;t

Ptm

(1 n)Ntn(16)

="a

t+m"p

t+mmpp;t + ln (1 n)nnt;(17)

3In a neighborhood of the zero-in‡ation steady state, dtis equal to zero up to a …rst-

order approximation (Galí,2008).

8

and the marginal product of real money balances,

mpmpt= ln @Yt

@Mt

Pt!(18)

= ln e"a

te"p

tme"p

tMp;t

Ptm1

Nt1n!(19)

="a

t+m"p

t+ ln (m)+(m1) mpp;t + (1 n)nt;(20)

we obtain an expression of the marginal cost,

mct= (wtpt)mpntmpmpt(21)

=wtpt2 ("a

t+m"p

t)(2m1) mpp;t

(1 2n)ntln (m(1 n)) :(22)

Using Eq. 14, we obtain an expression of ntsuch that

nt=1

1n

(yt"a

tm"p

tmmpp;t):(23)

Plugging Eq. 23 into Eq. 22 leads to an expression of the marginal cost

mct= (wtpt) + 2n1

1n

yt+1nm

1n

mpp;t

ln (m(1 n)) 1

1n

"a

tm

1n

"p

t;(24)

where Eq. 24 de…nes 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+kpt+k)mpnt+kjtmpmpt+kjt, we

obtain

mct+kjt= (wt+kpt+k) + 2n1

1n

yt+kjt+1mn

1n

mpp;t+k

1

1n

"a

t+km

1n

"p

t+kln (m(1 n)) (25)

=mct+k+2n1

1nyt+kjtyt+k(26)

=mct+k"2n1

1n

(p

tpt+k);(27)

9

where Eq. 27 follows from the demand schedule, Ct(i) = Pt(i)

Pt"Ct, com-

bined with the market-clearing condition (yt=ct).

Substituting Eq. 27 into Eq. 9and rearranging terms yields

p

tpt1= (1 )

1

X

k=0

()kEt[cmct+k] +

1

X

k=0

()kEt[t+k];(28)

where = 1n

1n+"(2n1) 1.

Finally, combining Eq. 8with Eq. 28 yields the in‡ation equation

t=Et[t+1 ] + mc cmct;(29)

where cmct=mctmc is the real marginal cost gap and mc = (1)(1)

is

strictly decreasing in the index of price stickiness, , the measure of decreasing

returns, n, and the demand elasticity, ".

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 price setting, average real marginal cost can be expressed as

mct= (wtpt)mpntmpmpt(30)

= (yt+ntn) + 2n1

1n

yt+1mn

1n

mpp;t

1

1n

"a

tm

1n

"p

tln (m(1 n)) (31)

=+n

1n

(1 )yt1 +

1n

"a

tm

1 +

1n

"p

t

+1m

1 +

1nmpp;t nln (m(1 n)) ;(32)

where the derivation of Eqs. 31 and 32 makes use of the household’s optimal-

ity condition (Eq. 6) and the (approximate) aggregate production relation

(Eqs. 14 and 23).

Knowing that > 0,0n1, and 1, it is obvious that

(1 n)++2n1>0. However, the sign of (1 (1 + )mn)coming

from Eq. 32 appears unde…ned. In fact, it con…rms some studies from Sinai

and Stokes (1975,1977,1981,1989) concluding that 1n>(1 + )m>

m. If this is the case, then 1(1 + )mn>0.

Furthermore, under ‡exible prices, the real marginal cost is constant and

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

t, as

10

the equilibrium level of output under ‡exible prices,

mc =+n

1n

(1 )yf

t1 +

1n

"a

tm

1 +

1n

"p

t

+1m

1 +

1nmpf

p;t nln (m(1 n)) ;(33)

implies

yf

t=y

a"a

t+y

p"p

t+y

mmpf

p;t +y

c;(34)

where

y

a=1 +

+n(1 ) (1 n)

y

p=m(1 + )

+n(1 ) (1 n)

y

m=n+m(1 + )1

+n(1 ) (1 n)

y

c=(1 n) (ln (m(1 n)) + n)

+n(1 ) (1 n):

Subtracting Eq. 33 from Eq. 32 yields

cmct=+n

1n

(1 )ytyf

t+1m

1 +

1nmpp;t mpf

p;t;

(35)

where ytyf

tis the output gap, and mpp;t mpf

p;t is the real money gap,

where money is used here only for production purposes. By combining Eqs.

29 and 35, we obtain our …rst equation relating in‡ation to its next-period

forecast, output gap, and real money balances gap,

t=Et[t+1 ] + xytyf

t+ mmpp;t mpf

p;t(36)

where

x=+n(1 n) (1 )

1n+"(2n1) (1 )1

and

m=1nm(1 + )

1n+"(2n1) (1 )1

:

The second key equation describing the equilibrium of the New Keynesian

model is obtained from Eq. 10:

yt=Et[yt+1]1(itEt[t+1]c)1Et"u

t+1:(37)

11

Henceforth, Eq. 37 is referred to as the dynamic IS equation.

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

Rearranging Eq. 5yields

mpn;t =

yta2

itm

+1

"n

t:(38)

From Eq. 10, we obtain an expression for the natural interest rate,

if

t=c+Ethyf

t+1i:(39)

Therefore, from Eqs. 39 and 38, we obtain an expression of the money

demand of …rms in the ‡exible-price economy such that

mpf

n;t =

yf

ta2

Ethyf

t+1im+ca2

+1

"n

t(40)

The last equation determines the interest rate through a smoothed Taylor-

type rule,

it= (1 i)(t) + xytyf

t+Mk;t+iit1+"i

t;(41)

where and xare policy coe¢ cients re‡ecting the weight on the in‡ation

and output gaps and the parameter 0< i<1captures the degree of interest

rate smoothing. "i

tis an exogenous ad hoc shock accounting for ‡uctuations

in the nominal interest rate. is an in‡ation target and Mk;t is a money

variable that is de…ned 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 1describes Mk;t’s functional forms.

In the literature, money is generally introduced through a money growth

variable (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 signi…cant 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=etMt;(42)

12

k Mk;t

1 0

22mpp;t mpf

p;t

33mpn;t mpf

n;t

44mptmpf

t

55mpp;t mpf

p;t+6mpn;t mpf

n;t

Table 1: The money variable in the Taylor rule

where Mtrepresents the total nominal money stock and etis an exogenous

time-varying velocity process de…ned in the next section. Taking logs, Eq.

42 leads to

yt=mpt+t=mpn;t +mpp;t +t(43)

The corresponding ‡exible-price economy equation is similar (Eq. 46) to

the previous relation.

2.6 DSGE model

Our DSGE model consists of eight equations and eight dependent variables:

in‡ation, nominal interest rate, output, ‡exible-price output, real money bal-

ances held for production purpose, its ‡exible-price counterpart, real money

balances held for nonproduction purpose, and its ‡exible-price counterpart.

Flexible-price economy

yf

t=y

a"a

t+y

p"p

t+y

mmpf

p;t +y

c(44)

mpf

n;t =

yf

ta2

Ethyf

t+1im+ca2

+1

"n

t(45)

mpf

p;t =yf

tmpf

n;t t(46)

Sticky-price economy

t=Et[t+1 ] + xytyf

t+ mmpp;t mpf

p;t(47)

yt=Et[yt+1]1(itEt[t+1]c)1Et"u

t+1(48)

mpn;t =

yta2

itm

+1

"n

t(49)

mpp;t =ytmpn;t t

13

it= (1 i)(t) + xytyf

t+Mk;t+iit1+"i

t(50)

As we have …ve historical variables, we have …ve microfounded shocks:

technology shock ("a

t), shock to household money demand ("n

t), shock to …rm

money demand ("p

t), short-term interest rate or monetary policy shock ("i

t),

and preference shock ("u

t).

De…nition 1 8j2 fa; n; p; i; ug,"j;t =j"j;t1+j;t, where jis an au-

toregressive coe¢ cient of the AR(1) processes and j;t follows a normal i.i.d.

process with a mean of zero and standard deviation of j.

Following the literature (Benk et al.,2008;Lothian,2009), the veloc-

ity process, t, depends essentially on money shocks. Then, we choose the

following speci…cation for the time-varying velocity process.

De…nition 2 t=+s(ms"n;t + (1 ms)"p;t ), where ,sand 0<

ms <1are parameters.

3 Results

As in Smets and Wouters (2007) and An and Schorfheide (2007), we apply

Bayesian techniques to estimate our DSGE models. We test …ve speci…cations

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, tis the in‡ation rate, measured as the yearly

log-di¤erence of the gross domestic product (GDP) de‡ator 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. (2001). mpn;t and mpp;t are the real money demands

of households and …rms, respectively, and are measured as the logarithm

of the Euro-area accounts series4divided by the GDP de‡ator. We detrend

historical variables using a Hodrick-Prescott …lter (with a standard coe¢ cient

for quarterly data of 1600).

4The money demand series of households and …rms 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 su¢ x). The sum of these two aggregates leads to M2.

14

yf

t, the ‡exible-price output, mpf

n;t, the ‡exible-price household real money

balances, and mpf

p;t, the ‡exible-price …rm real money balances, are com-

pletely determined 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 (), the inverse of the

Frisch elasticity of the labor supply (), the Calvo (1983) parameter (), and

the elasticity of household demand for consumption goods ("). is set at

0:9926 so that the annual steady-state real interest rate is three percent and

,, and "are 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 pa-

rameters 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 …ve versions of the model, we adopt the same

priors in each version with the same calibration, depending of the Taylor

rule speci…cation. The calibration of is inspired by Rabanal and Rubio-

Ramírez (2005) and Casares (2007). They choose risk aversion parameters

of 2:5and 1:5, respectively. In line with these values, we regard = 2 as

corresponding to a standard risk aversion (Benchimol and Fourçans,2012;

Benchimol,2014).

We calibrate our central parameter m, the share of money in the produc-

tion 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 n= 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 dis-

tributions for shock persistence parameters; the backward component of the

Taylor rules; output elasticities of labor, n; and real money balances, m,

of the production function that should be less than one.

The scale parameters and are calibrated to 0:44 and one, respectively,

as in Christiano et al. (2005), and the money velocity mean prior () is

calibrated to 0:31 following Carrillo et al. (2007).

The smoothed Taylor rules (i,, and x) 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 pos-

sible behaviors of the central bank into consideration, we assign a higher

15

standard error to the Taylor rule coe¢ cients. The non-standard parameters’

mean priors of the augmented Taylor rules for k= 2 5are 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 accord-

ing to inverted Gamma distributions, with prior means of 0:02. The latter

distribution ensures that these parameters have a positive support. The au-

toregressive parameters are all assumed to follow beta distributions. All these

distributions are centered around 0:75 and we take a common standard error

of 0:1for the shock persistence parameters, as in Smets and Wouters (2003).

The calibration of the parameters entering the time-varying component of

velocity is quite new. The prior mean of sis 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 ms is calibrated to 0:50 and

theoretically constrained between zero and one. Thus, we assume a Beta

distribution for ms–which can be seen as a trade-o¤ parameter between

the two money demand shocks ("n;t and "p;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 under the …ve con…gurations of the Taylor rule is conducted us-

ing the Metropolis-Hastings algorithm5(ten distinct chains of 300,000 draws

each).

The real money balances parameter (m) of the augmented production

function is estimated to be between 0.014 (k= 4) and 0.042 (k= 1). This

result di¤ers from that found by Sinai and Stokes (1972) for the same parame-

ter (0.087).6The prior and posterior distributions are presented in Appendix

Band 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 di¤er-

ent Taylor rule considerations. Here, we test our model under …ve Taylor rules

5See, for example, Smets and Wouters (2003), Smets and Wouters (2007), Adolfson

et al. (2007), and Adolfson et al. (2008).

6Benchimol (2011) estimates mto be 0.064.

7This work has already been done in Benchimol (2011) and Benchimol (2011).

16

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%

nBeta 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

mBeta 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

Normal 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

Normal 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

Normal 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

Normal 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

iBeta 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

Normal 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

xNormal 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

kNormal 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

6Normal 0.50 0.20 0.5621 0.2212 0.2065 0.9204

Normal 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

sNormal 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

ms 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

aBeta 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

uBeta 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

iBeta 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

pBeta 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

nBeta 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

aInvgamma 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

uInvgamma 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

iInvgamma 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

pInvgamma 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

nInvgamma 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

17

(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 pro-

posed distribution to the target distribution is satis…ed. A diagnosis of the

overall convergence for the Metropolis-Hastings sampling algorithm is pro-

vided in Appendix Dand, following Ratto (2008), all estimations are stable.

3.4 Simulations

3.4.1 Impulse response functions

Appendix Cpresents the responses of key variables to structural shocks for

each k.

In response to a preference shock, the in‡ation rate, output, output gap,

…rm real money balances, nominal interest rate, and real interest rate rise,

whereas household real money holdings display a little undershooting process

in the …rst few periods, then return to their steady-state value.

After a technology shock, the output gap, in‡ation rate, nominal interest

rate, …rm real money balances, and real interest rates decrease, whereas

output and household real money balances rise.

In response to an interest rate shock, the in‡ation rate, output, and

output gap fall. Interest rates and …rm money demand rise. A positive

monetary policy shock induces a fall in interest rates due to a su¢ ciently

low degree of intertemporal substitution (i.e., the risk aversion parameter is

su¢ ciently 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 …rms, interest rates, the output

gap, and the real money holdings of …rms decrease, whereas in‡ation and the

real money holdings of households increase. These impulse response functions

are similar to Smets and Wouters (2003) with regard to output, in‡ation, and

interest rates. However, the responses following a shock in the money demand

of households depends on the model speci…cation (see Appendix C).

3.4.2 Variance decompositions

The analysis is conducted via unconditional and conditional variance decom-

positions (see Table 3) to compare the impact of shocks on variables across

the models and over time.

18

For all models, most of the long-run variance in output comes from the

technology shock (around 75%), about one-quarter of the output variance re-

sults 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 monetary policy shock (around 63%), whereas around 28% is

a result of the technology shock. The money demand of …rms impacts output

variance (and its ‡exible-price counterpart) due to the form of the produc-

tion 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 ‡exible-price counterpart (Benchimol,2011). However, in

the short run, the share of ‡exible-price output variance explained by the

shock in the money demand of …rms 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 frameworks

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 in‡ation variance decompo-

sitions shows the overwhelming role of the interest rate shock, which explains

more than 92% of in‡ation 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 in‡ation 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 signi…cant role. The relative importance of each of

these shocks changes over time. For longer horizons, there is an inversion

over time–the preference shock explains almost 75% of the nominal interest

rate variance, whereas the interest rate shock explains less than 21%.

Table 3shows 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 …rms is essentially determined by its correspond-

ing shock (around 68%) as well as that in the money demand of house-

holds (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 …rms is also driven by its correspond-

ing shock (around 60%) and the households’money demand shock (around

30%) and the …rms’money demand variance decomposition changes. The

latter is mainly driven, in the long run, by the technology shock (around

19

Quarter 1 (k= 1) Quarter 1 (k= 2) Quarter 1 (k= 3) Quarter 1 (k= 4) Quarter 1 (k= 5)

a;t u;t i;t p;t n;t a;t u;t i;t p;t n;t a;t u;t i;t p;t n;t a;t u;t i;t p;t n;t a;t u;t i;t p;t n;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

t0.14 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

mpp;t 0.56 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;t 15.23 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

mpf

p;t 1.44 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

mpf

n;t 35.47 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

t0.00 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)

a;t u;t i;t p;t n;t a;t u;t i;t p;t n;t a;t u;t i;t p;t n;t a;t u;t i;t p;t n;t a;t u;t i;t p;t n;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

t0.23 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

mpp;t 0.89 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;t 27.66 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

mpf

p;t 2.12 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

mpf

n;t 43.13 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

t0.00 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 1(k= 1) Quarter 1(k= 2) Quarter 1(k= 3) Quarter 1(k= 4) Quarter 1(k= 5)

a;t u;t i;t p;t n;t a;t u;t i;t p;t n;t a;t u;t i;t p;t n;t a;t u;t i;t p;t n;t a;t u;t i;t p;t n;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

t0.41 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

mpp;t 2.13 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;t 47.69 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

mpf

p;t 4.88 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

mpf

n;t 62.67 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

t0.00 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

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

‡exible-price output variance decomposition. The technology shock, with a

weight of around 91%, is the main explanatory factor of long-run variance in

‡exible-price output. In shorter time frames, ‡exible-price output variance is

mainly explained by the shocks in technology (around 75%) and …rm money

demand. As previously explained, this result is attributable to the functional

form of the production function.

Although ‡exible-price real money demand of …rms is mainly impacted

by money shocks (regardless of the time horizon), ‡exible-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 de…nitions 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 ‡uctuations 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 speci…cation (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

function speci…cation follows the hypothesis of Khan and Ahmad (1985)–

decreasing return-to-scale hypothesis. This hypothesis gives by-construction

no in‡uence on money in the dynamics of the variables, despite its intro-

duction into the production function8. However, because of the Cobb and

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.

21

Douglas (1928) assumption about the formulation of the production function,

and even if the elasticity of the real money holdings of …rms is attributed

to statistical noise, the money demand of …rms could impact at least output

dynamics.

The simulations (see Table 3and Appendix C) are close to those obtained

in the literature and provide interesting results regarding the potential e¤ect

of money on output and ‡exible-price output under two di¤erent money

demands (household and …rm).

Interestingly, and even if money enters into the in‡ation equation, money

shocks have almost no e¤ect on the variance decomposition of in‡ation. This

result is common in the literature on money in a non-separable utility func-

tion (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 models (see Appendix E).

Another interesting result is that the shock on …rm money demand has

an important in‡uence on ‡exible-price output and in the Taylor rule. The

estimated contribution of …rms’money holdings in our money-augmented

Taylor rule (k= 2) seems to be signi…cant: this means that this shock could

potentially impact monetary policy. This result does not mean that we should

target …rm money demand. It only tells us that this variable could be taken

into account for policy analysis. Because …rm money holding shocks impact

macroeconomic variable dynamics in our framework, monetary policy should

pay particular attention to the money demand of …rms.

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 production has been debated both theoretically and empirically by

a number of researchers in the past …ve decades. The question is whether

money is an omitted variable in the production process.

In parallel, New Keynesian DSGE theory, combined with Bayesian analy-

sis, has become increasingly popular in the academic literature and in policy

analysis. The unique contribution of this paper is to build and test a micro-

founded 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 in‡ation equation

(Phillips curve). Closing the model leads to the new concept of ‡exible-price

real money balances presented by Benchimol and Fourçans (2012).

22

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), di¤erentiates between money demanded by households and …rms.

This distinction between money that is used for productive and nonproduc-

tive purposes seems to be warranted. By testing our models with Bayesian

techniques under di¤erent monetary policy rules, we show that even if the

weight of real money balances in the production function is very low, the

…rm money shock has an important in‡uence on ‡exible-price output and a

signi…cant impact on output. Part of this in‡uence 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

function, real money balances could be excluded from the production process.

However, considering this hypothesis of money in the production function

highlights the signi…cant role of the …rm money shock. Incorporating the

real money balances 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"1

X

k=0

k(Ut+kt+kVt+k)#;

where

Vt=Ct+Mn;t

Pt

+Mp;t

Pt

+Qt

Bt

Pt

Bt1

Pt

Wt

Pt

Nt

and

Ut=e"u

t C1

t

1+e"n

t

1Mn;t

Pt1

N1+

t

1 + !:

The …rst-order condition related to consumption expenditures is given by

t=e"u

tC

t;(51)

where tis the Lagrangian multiplier associated with the budget constraint

at time t.

23

The …rst-order condition corresponding to the demand for contingent

bonds implies that

tQt=Ett+1

Pt

Pt+1 :(52)

The demand for cash held for nonproduction purposes that follows from

the household optimization problem is given by

e"u

te"n

tMn;t

Pt

=tEtt+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 speci…cations.

Working hours following the household optimization problem are given

by

e"u

tN

t=t

Wt

Pt

:(54)

From Eq. 51, we obtain

t=e"u

tC

t,Uc;t =e"u

tC

t;(55)

where Uc;t =@Uk;t

@Ct+kk=0. Eq. 55 de…nes the marginal utility of consumption.

Hence, the optimal consumption/savings, real money balance, and labor

supply decisions are described by the following conditions:

Combining Eq. 51 with Eq. 52 yields

Qt=Ete"u

t+1 C

t+1

e"u

tC

t

Pt

Pt+1 ,Qt=EtUc;t+1

Uc;t

Pt

Pt+1 ;(56)

where Uc;t+1 =@Uk;t

@Ct+kk=1. Eq. 56 is the usual Euler equation for in-

tertemporal consumption ‡ows. It establishes that the ratio of marginal

utility of future and current consumption is equal to the inverse of the

real interest rate.

Combining Eq. 51 and Eq. 53 yields

e"n

t

C

tMn;t

Pt

= 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

condition setting the marginal rate of substitution between money and

consumption equal to the opportunity cost of holding money.

24

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

-10

Interest

rate shock

-1

-0.5

0

0

0.5

1

0

1

2

-1

-0.5

0

0

0.5

1

-2

-10

020 40

-10

1

Quarters

-0.1

-0.05

0

Firms' money

demand shock

0

0.2

0.4

-0.2

-0.1

0

-0.2

-0.1

0

-0.1

-0.05

0

-4

-20

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

-10

020 40

-10

1

Quarters

-0.2

-0.1

0

Firms' money

demand shock

0

0.2

0.4

-0.2

-0.1

0

-0.2

-0.1

0

-0.1

-0.05

0

-4

-20

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

-10

Interest

rate shock

-1

-0.5

0

0

0.5

1

0

1

2

-1

-0.5

0

0

0.5

1

-2

-10

020 40

-10

1

Quarters

-0.04

-0.02

0

Firms' money

demand shock

0

0.2

0.4

-0.2

-0.1

0

-0.1

-0.05

0

-0.04

-0.02

0

-4

-20

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

-10

Interest

rate shock

-1

-0.5

0

0

0.5

1

0

1

2

-1

-0.5

0

0

0.5

1

-2

-10

020 40

-10

1

Quarters

-0.1

-0.05

0

Firms' money

demand shock

0

0.2

0.4

-0.2

-0.1

0

-0.1

-0.05

0

-0.04

-0.02

0

-4

-20

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

-10

Interest

rate shock

-1

-0.5

0

0

0.5

1

0

1

2

-1

-0.5

0

0

0.5

1

-2

-10

020 40

-10

1

Quarters

-0.1

-0.05

0

Firms' money

demand shock

0

0.2

0.4

-0.2

-0.1

0

-0.2

-0.1

0

-0.1

-0.05

0

-4

-20

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 speci…c 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 measures, convergence requires both lines to become rela-

tively horizontal and converge 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= 1 k= 2 k= 3 k= 4 k= 5

y

a1,0397 1,0321 1,0496 1,0398 1,0378

y

p0,0443 0,0308 0,0233 0,0152 0,0256

y

m-0,2145 -0,2273 -0,2390 -0,2446 -0,2337

y

c-1,0445 -1,1319 -1,2286 -1,3251 -1,1871

1,3610 1,3388 1,3644 1,3606 1,3518

a20,7828 0,7700 0,7847 0,7826 0,7775

m+ca2

-0,1534 -0,0984 -0,1882 -0,1516 -0,1332

1

0,7370 0,7138 0,7535 0,7368 0,7291

x0,6561 0,6943 0,6743 0,6819 0,6850

m0,1407 0,1578 0,1611 0,1668 0,1601

1

0,5415 0,5331 0,5523 0,5415 0,5393

c

0,0040 0,0039 0,0041 0,0040 0,0040

a2

0,4239 0,4105 0,4334 0,4238 0,4193

m

-0,1566 -0,1014 -0,1915 -0,1547 -0,1363

i0,6005 0,6121 0,5874 0,5987 0,6027

(1 i)1,3685 1,3187 1,4127 1,3753 1,3588

x(1 i)0,5640 0,5404 0,5901 0,5695 0,5612

k(1 i)0,2402 0,1915 0,1685 0,1392

6(1 i)0,2232

exp ()1,3609 1,3602 1,3647 1,3612 1,3593

sms 0,3363 0,3322 0,3433 0,3374 0,3347

s(1 ms)1,7434 1,7506 1,7321 1,7431 1,7446

Table 4: Macroparameter values

37

References

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