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Time-Varying Money Demand and Real

Balance E¤ects

Jonathan Benchimolyand Irfan Qureshiz

March 18, 2020

Abstract

This paper presents an analysis of the stimulants and consequences of

money demand dynamics. By assuming that household’s money holdings and

consumption preferences are not separable, we demonstrate that the interest-

elasticity of demand for money is a function of the household’s preference to

hold real balances, the extent to which these preferences are not separable

in consumption and real balances, and trend in‡ation. An empirical study

of U.S. data revealed that there was a gradual fall in the interest elasticity

of money demand of approximately one-third during the 1970s due to high

trend in‡ation. A further decline in the interest-elasticity of the demand for

money was observed in the 1980s due to the changing household preferences

that emerged in response to …nancial innovation. These developments led to

a reduction in the welfare cost of in‡ation that subsequently explains the rise

in monetary neutrality observed in the data.

Keywords: Time-Varying Money Demand, Real Balance E¤ect, Welfare

Cost of In‡ation, Monetary Neutrality.

JEL Classi…cation: E31, E41, E52.

Please cite this paper as:

Benchimol, J., and Qureshi, I., 2020. Time-varying money demand and

real balance e¤ects. Economic Modelling, 87, 197-211.

This paper does not necessarily re‡ect the views of the Bank of Israel or the Asian Development

Bank. The authors are grateful to Sushanta Mallick (Editor), Zahid Ali, André Fourçans, Edward

O¤enbacher, seminar participants at the Lahore University of Management Sciences (LUMS), and

four anonymous referees for their constructive comments and suggestions. This research did not

receive any speci…c grant from funding agencies in the public, commercial, or not-for-pro…t sectors.

yBank of Israel, Research Department, Jerusalem, Israel.

zAsian Development Bank, Macroeconomics Division, Metro Manila, Philippines. Correspond-

ing author. Email: iqureshi@adb.org.

1

1 Introduction

Since the 1980s, the long-standing empirical theories that connect several alterna-

tive monetary aggregates to movements in prices and interest rates have gradually

evolved (Friedman and Kuttner, 1992). Speci…cally, the application of the frame-

work proposed by Lucas (2000) led Ireland (2009) to the detection of important

changes in the interest semi-elasticity of money demand in the period following the

1980s. For many decades, the monetary policy theory literature was focused on the

implications of the interest-elasticity of money demand and the role this played in

determining the e¤ectiveness of monetary policies (Tobin, 1956; Laumas and Lau-

mas, 1969; Vernon, 1977). King (1999) and Friedman (1999) con…rmed the limited

e¤ectiveness of monetary policy as a consequence of a moneyless economy while the

…ndings of Woodford (2000, 2003, 2008) contradicted this result.

Most of the debate in this domain focused on the interest semi-elasticity of

money demand, which is essentially concerned with monetary neutrality (Lucas,

1996). As this long and lively debate demonstrated, the extent to which money can

in‡uence the interest rate and welfare cost of in‡ation could change over time. In

this paper, we document and assess the causes and macroeconomic consequences of

the time-varying relationship between interest rates and money. We derive a general

micro-founded interpretation of the familiar log-linear money demand relationship

described in Lucas (2000), which is aligned with that employed by Ireland (2009).

The interest semi-elasticity of money demand is described as a function of the

household’s preferences to hold real balances and substitute consumption and real

balances, steady-state gross in‡ation, and interest rates. Therefore, the expression

enables us to capture the structural channels that may have stimulated the changes

in the money demand observed in the empirical literature.

An application of such a micro-founded money demand framework allows the

quanti…cation of the welfare cost of in‡ation by linking it with the structural pa-

rameters that drive the interest semi-elasticity of money demand. The subsequent

framework can pin down the parameters of interest in this equation, both through

examining the …rst moments in the data and direct estimation.

Our empirical estimation of the money demand equation based on the quarterly

U.S. data covering the period 1959 to 2008 reveals that there was a decline in the

interest semi-elasticity of money demand and a subsequent fall in the welfare cost

of in‡ation during this period. The benchmark results con…rm the analysis o¤ered

by Ireland (2009), who found a semi-elasticity below 2 as well as a smaller welfare

cost estimate of modest departures from Friedman’s zero nominal interest rate rule

for the optimum quantity of money during the post-1980s era.

Allowing for time variation in the money demand function using recursive esti-

mates reveals a gradual fall in the interest elasticity of money demand of approx-

imately one-third during the 1970s due to both trend in‡ation and an increase in

interest rates. A further decline in the interest-elasticity of the demand for money

was observed in the 1980s due to the changing household preferences that emerged

in response to …nancial innovation. The latter in‡uenced the household’s prefer-

ences to hold real balances and their willingness to substitute real balances and

consumption. In combination, our results suggest that the entire shift in money

2

demand could be attributed to the evolution of trend in‡ation, interest rates, and

changes in the household’s preferences, thereby explaining the results found in Ire-

land (2009) and Lucas (2000).

These developments led to a reduction in the welfare cost of in‡ation that

subsequently explains the rise in monetary neutrality observed in the data. Our

time-varying estimates of money demand show that the welfare cost of 10 percent

in‡ation decreased from 0.92 percent of income in the 1960s to under 0.20 percent

of income in the 1990s.1Since household’s preferences and trend in‡ation enter

the IS equation through various structural parameters, changes in these parameters

may have broader macroeconomic consequences. A comparison of the reactions of

output to an interest rate shock between pre-1979 and post-1980s periods based

on a vector autoregression (VAR) indicates that the impact elasticity of monetary

policy roughly halved. An interest rate shock had approximately 35% less impact

on output in 1980 than it did during the pre-1979 period. The fall in the house-

hold’s preferences to hold real balances and substitute between consumption and

real balances altered key parameters in the IS curve. Therefore, changes that a¤ect

the traditional money demand relationships may also explain a proportion of the

rise in monetary neutrality observed in the data.

This paper adds to the existing debate in multiple ways. It provides a micro-

founded interpretation of the interest semi-elasticity of money demand and the

welfare cost of in‡ation. This extends the work of many scholars (Cagan, 1956;

Lucas, 1981; Meltzer, 1963; Sidrauski, 1967; Fischer, 1981; Cooley and Hansen,

1989; Dotsey and Ireland, 1996; Lucas, 2000; Ireland, 2009; Miller et al., 2019). The

identi…cation of the changes in the semi-elasticity and the welfare cost can explain

the contrasting welfare estimates presented in the existing literature (Broaddus

and Goodfriend, 1984; Reynard, 2004; Ireland, 2009; Lucas and Nicolini, 2015).

Belongia and Ireland (2019) proposed alternative monetary measures that preserve

these long-standing relationships and add to the theoretical explanations, such as

those based on Baumol-Tobin style inventory-theoretic models of money (Attanasio

et al., 2002; Alvarez and Lippi, 2009), or insurance against idiosyncratic liquidity

shocks (Berentsen et al., 2015), all of which equate changes in household behavior

to the breakdown in money demand relationships.

The changing household’s substitution preferences between consumption and

real balances and the corresponding empirical results extend the existing litera-

ture on estimates of real balances through constant elasticity of substitution (CES)

money-in-the-utility function (MIUF) speci…cation (Holman, 1998; Finn et al., 1990;

Poterba and Rotemberg, 1987; Benchimol and Fourçans, 2012, 2017). While Ireland

(2004) and Woodford (2003) found that the weight of real balances was of a negli-

gible size, our time-varying estimation highlights how this weight was larger during

the 1960s and 1970s before falling to zero from the mid-1980s onwards. Broadly

1For example, Ireland (2009) found a welfare cost for a 10% in‡ation rate of less than 0.25%

of income. Lucas (2000) found a welfare cost for 10% in‡ation of just over 1.8% income. Fischer

(1981) found a welfare cost for 10% in‡ation between 0.2% and 0.3% income. Cooley and Hansen

(1989) found that a welfare cost of 10% in‡ation is about 0.4% of GDP using a cash-in-advance

version of the business cycle model. Miller et al. (2019) found a welfare cost for 10% in‡ation of

just over average 0.27% income.

3

speaking, since real balances enter directly in the dynamic IS, determining in‡ation

and output dynamics through this channel may be relevant during this period, and

this …nding complements that of Castelnuovo (2012) and Benchimol and Fourçans

(2017). This e¤ect is combined by a higher elasticity of substitution between con-

sumption and real balances, implying that household’s preferences are not fully

separable in either time period.

The …ndings also explain the shifts in the welfare cost of in‡ation and connect

both the household behavior and changes in the U.S. macroeconomic dynamics

through the money demand function. The time-varying aspect of the semi-elasticity

contributes to the money demand instability (Khan, 1974; Judd and Scadding, 1982;

Tesfatsion and Veitch, 1990; Hafer and Jansen, 1991; Miller, 1991; Lütkepohl, 1993;

Chen, 2006; Ireland, 2009; Hall et al., 2009; Inagaki, 2009; Jawadi and Sousa, 2013;

Lucas and Nicolini, 2015; Miller et al., 2019). These results indicate that the single-

valued approach to approximating the welfare cost of in‡ation presented in previous

literature captures only the sample average at each point in time.

The introduction of trend in‡ation in the model augments the interest semi-

elasticity of money demand debate by enriching the model along the lines of various

papers (Hornstein and Wolman, 2005; Amano et al., 2007; Ascari and Ropele, 2007;

Kiley, 2007; Ascari and Ropele, 2009; Ascari and Sbordone, 2014). The rise in trend

in‡ation is one of the primary reasons for the fall in the semi-elasticity due to the

rise in the opportunity cost of holding money. By highlighting how high trend

in‡ation a¤ects the semi-elasticity and, therefore, the welfare cost of in‡ation, the

outcomes of our analysis are original and provide several policy recommendations.

Finally, this paper presents an alternative channel by which it is possible to

explain the decline in monetary policy e¤ectiveness that was observed in the post-

1980s period. Mallick and Mohsin (2010, 2016) found that in‡ation has an impor-

tant permanent e¤ect on the real economy in several ways including consumption,

investment, and the current account. Our model, which also incorporates the cash-

in-advance constraint (CIA), mimics these …ndings since it identi…es trend in‡ation

as a key driver of real e¤ects. However, in our framework, the transmission works

through the money demand channel. Boivin and Giannoni (2002) concluded that

changes in the monetary policy rule were responsible for the variations that were

observed in the impulse responses. Pancrazi and Vukotic (2019) found that the

decline in the e¤ectiveness of monetary policy could be attributed to the evolution

of labor market properties. Instead, we show that the changes in the household’s

preferences that were observed may explain a large portion of the decline in the

e¤ectiveness of monetary policy in the short-term. These changes are larger for the

short-run and decline over the medium-to-long run, a result that converges with the

…ndings of Pancrazi (2014).

The rest of the paper is structured as follows. In Section 2, we derive the money

demand curve from micro-foundations that include positive trend in‡ation. Section

3 presents the welfare loss derivations, Section 4 discusses the empirical …ndings, and

Section 5 studies the consequences of the money demand curve on the welfare cost

of in‡ation and the resulting reduction in the impact of monetary policy. Section 6

concludes the paper and o¤ers suggestions for future research. Finally, additional

supporting results and data are provided in the appendix.

4

2 The Theoretical Framework

2.1 The Model

The economy consists of a continuum of households, in which the representative

household seeks to maximize the following objective function:

E0"1

X

t=0

tUCt;Mt

Pt

; Nt#(1)

where Ctis the quantity consumed of the single good, Mt=Ptdenotes holdings of

real money balances and serves as a unit of account, and Ntdenotes worked hours.

We consider the speci…c case in which the period utility is given by the following

functional form:

UCt;Mt

Pt

; Nt=X1

t

1N1+'

t

1 + '(2)

where represents the relative risk aversion of households (or the inverse of the

intertemporal elasticity of substitution), 'is the inverse of the elasticity of work

e¤ort with respect to the real wage (Frisch elasticity), and Xtis a composite index

of consumption and real balances de…ned as:

8

>

>

<

>

>

:

Xt=(1 )C1v

t+Mt

Pt1v1

1v

for v6= 1

Xt=C1

tMt

Ptfor v= 1

(3)

with vrepresenting the (inverse) elasticity of substitution between consumption and

real balances, and the relative weight of real balances in utility, as presented in

Greenwood et al. (1988).

The composite index Xtre‡ects the non separability property of the utility

function2given the values of the parameters and v. The parameter captures the

“direct e¤ect”of money or the marginal utility of money valued at the steady-state.

The parameter v, which represents the (inverse) elasticity of substitution between

consumption and real balances, captures the “indirect e¤ect” or the cross-partial

derivative of money with consumption.

Changes in these parameters have very general interpretations. A variation

in may represent shocks to transactions technology – shocks that change the

utility value of money relative to that of consumption expenditures (Koenig, 1990).

Thus, …nancial innovation that reduces transaction costs may be captured by this

2In non separable utility functions, the marginal utility of consumption directly depends on

variations of real money balances and allows us to investigate the e¤ects of variations in real money

on the economy (Benchimol, 2016). In contrast, a separable utility function leaves consumption,

and the economy, indi¤erent to variations in real money balances (Benchimol, 2014). Under a

separable utility, the equilibrium values of real variables are determined independently of real

money balances and of any implemented monetary policy (Galí, 2015).

5

parameter . On the other hand, a variation in vcaptures the preference changes

for a household to substitute money and consumption.3

Maximization of the objective function (Eq. 1) is subject to a sequence of ‡ow

budget constraints given by:

PtCt+QtBt+MtBt1+Mt1+WtNtTt(4)

where Ptis the price of the consumption good, Wtis the nominal wage, and Btis

the quantity of one-period nominally risk-less discount bonds purchased in period t

and maturing in period t+ 1. Each bond pays one unit of money at maturity and

its price is Qt.Ttrepresents lump-sum additions or subtractions to period income.

Let the total …nancial wealth at the end of period tbe de…ned as t=Bt+Mt.

The budget constraint (Eq. 4) can then be written compactly as:

PtCt+Qtt+ (1 Qt)Mtt1+Mt1+WtNtTt(5)

Written like Eq. 5, one readily sees the opportunity cost of investing resources

in money rather than bonds. The bond price, Qt, determines the interest rate such

as it=ln (Qt), where itis the short-term nominal interest rate and is equal to

=ln ()in the steady-state.

We assume a representative …rm whose technology is described by a production

function given by:

Yt=AtN1

t(6)

where Atrepresents the level of technology and at= ln (At)is assumed to evolve

exogenously according to some stochastic process.

Maximizing the objective function (Eq. 1) subject to the ‡ow budget constraint

(Eq. 5), the necessary …rst-order conditions for any tcan be written as:

Mt

Pt

=Ct(1 exp (it))1

v

11

v

(7)

Qt=Et"Ct+1

CtvXt+1

XtvPt

Pt+1 #(8)

N'

tXv

tCv

t(1 )1=Wt

Pt

(9)

2.2 Motivation for Positive Trend In‡ation

The relevant equations of the model are log-linearized around a zero-growth and

non-zero in‡ation steady-state, which has been shown to be an important feature

3Adding a transaction/liquidity cost, using a shopping time model (with money) or a CIA

constraint, as in Mallick and Mohsin (2010, 2016), would be functionally equivalent to the MIU

speci…cation (Feenstra, 1986). In a way, these mechanisms are succinctly captured by the para-

meters and v.

6

of the U.S. economy.4In this sense, solving for the non-zero trend in‡ation may

yield a more realistic representation of the structural model.

Under the steady-state assumptions, the Euler equation (Eq. 8) can be written

as:

Q = (10)

where Qrepresents the steady-state bond prices and the steady-state gross in-

‡ation. This contains straightforward economic intuition. Under zero in‡ation,

=1, the price of the bond today is exactly equal to the utility weight the house-

hold attaches to its return (which is one). The household has no incentive to save

or dissave to let their marginal utility di¤er across periods. This return is a¤ected

by the gross in‡ation return.

By using Eq. 8, the steady-state money demand relationship can also be sim-

pli…ed such as:

M

P C =

( ) (1 )1

v

=m(11)

The expression suggests that not only does the ratio of steady-state level of money

holdings with respect to consumption decrease in the weight of real balances, trend

in‡ation reduces this ratio as well. This occurs since it raises the opportunity cost of

holding money. More broadly, M

P C may also be de…ned as the inverse consumption

velocity. Assuming that consumption may equal output in steady-state in this

model, the parameter mmay be interpreted to be the key determinant of the

quantitative importance of monetary-non-neutrality in the model (Galí, 2015).

The choice of the CES MIUF and the relaxation of positive trend in‡ation a¤ect

the ratio of real balances with respect to consumption in the steady-state.

2.3 Deriving the linearized system

The …rst order condition (Eq. 7) is log-linearized around the steady-state, and

conditions (Eq. 9 and Eq. 10) are imposed to yield the following money demand

relationship.

mtpt=+ctit(12)

where mt= ln (Mt),pt= ln (Pt),ct= ln (Ct)and it=ln (Qt).

Focusing on the parameters, =

v()may be interpreted as the semi-elasticity

of money with respect to interest rates, and the constant, , is found to be equal

to 1

vh

+ ln

1ln

i, as demonstrated in Appendix A.1.

The key parameter is a function of ,and v, while the constant term is a

function of ,,v,, and . An increase in trend in‡ation, , or the elasticity of

4Ascari and Sbordone (2014) construct a generalized new Keynesian model that accounts for

positive trend in‡ation. In this model an increase in trend in‡ation is associated with a more

volatile and unstable economy and trends to destabilize in‡ation expectations. Hornstein and

Wolman (2005), Kiley (2007), and Ascari and Ropele (2009) show that when appropriately con-

sidered, positive trend in‡ation substantially alters the models’ structural equations and the de-

terminacy region. Amano et al. (2007) study how the business cycle characteristics of the model

(i.e., persistence, correlation, and volatility) vary with trend in‡ation. Ascari and Ropele (2007)

analyze how optimal short-run monetary policy changes with trend in‡ation.

7

substitution, v, would work to reduce as well as the constant term . The steady-

state interest rate positively a¤ects the constant, but does not directly a¤ect the

semi-elasticity. Finally, the ratio of real balances to consumption, , reduces the

constant term.

2.4 How does it connect with the literature?

The unit elasticity of consumption is consistent with the long-run estimate in Lucas

(1988). Considering the special case of zero trend in‡ation by setting = 1 and

ignoring the constant term delivers the money demand curve obtained in Andrés

et al. (2002).

The relationship derived in Eq. 12 can also be written in the following familiar

log-linear form (Lucas, 2000):

ln (mt) = it(13)

Eq. 13 may then be interpreted as linking the log of mt, which represents the

ratio of nominal money balances to nominal income, to the level of it. It is also

related to the money demand function postulated by Cagan (1956):

ln (mt) = ln (B)it(14)

Setting ln (B) = in Eq. 14 returns the form described in Eq. 13. Connecting

this with the …ndings shared by Ireland (2009), who suggests this functional form

to …t better the post-1980s data, suggests that it may be relevant to estimate and

pin down the parameters describing Eq. 13 to better approximate the welfare cost

of in‡ation, as well as identify the sources behind these changes.

The remaining equations are linearized to obtain the following expressions:

wtpt=yt+'nt+!it(15)

yt=Et[yt+1]1

(itEt[t+1]!Et[it+1]) (16)

where Eq. 15 and Eq. 16 are the labor supply curve and the Dynamic Investment-

Saving (IS) relationship, respectively. It is further shown in Appendix A.2 that

!=(v), where =()m

+()mand mrefers to the steady-state of the ratio

of real money balances with respect to consumption. As shown in Appendix A.3,

!also enters the IS curve, making Eq. 16 sensitive to trend in‡ation and money.

The sign of (v)in Eq. 15 and Eq. 16 determines the sign of the e¤ect of

the nominal interest rate on labor supply. Trend in‡ation a¤ects these short-run

relationships through entering the term !. Under standard calibration of the model

considered in Galí (2015), high trend in‡ation seeks to dampen !, thus in‡uencing

the e¤ect of changes in the interest rate on both labor supply and consumption.

Moreover, when v > (implying ! > 0) the reduction in real balances induced

by an increase in the nominal rate lowers the marginal utility of consumption (for

any given ct), lowering the quantity of labor supplied at any given real wage. In

Eq. 16 the anticipation of a nominal rate increase (and, hence, of a decline in real

balances), lowers the expected one-period-ahead marginal utility of consumption

8

(for any expected ct+1), which induces an increase in current consumption (in order

to smooth marginal utility over time).

Since real balances enter this equation directly, they may be relevant in deter-

mining in‡ation and output dynamics. As demonstrated in Appendix A.4, the e¤ect

on output can be extracted from the model using the production function (Eq. 6),

the money demand curve, the labor supply curve, and the Dynamic IS:

yt=1 + '

(1 ) + '+at!(1 )

(1 ) + '+it(17)

where under standard calibrations of ,and ', the e¤ect of interest rates to output

depends on !. Since this parameter itself is a convolution of trend in‡ation and the

function of the weight of real balances, as well as the degree of substitutability in

the utility function, changes in these parameters a¤ect the degree of interest rate

shocks on output. Hence, changes common to those that a¤ect money demand may

also in‡uence the e¤ect of changes in interest rates on output.

3 The Welfare Loss Function

One consequence of the changes in the money demand function identi…ed in the

empirical literature concerns the welfare cost of in‡ation. The classic approach

developed by Bailey (1956) and Friedman (1969) treats real money balances as a

consumption good and in‡ation as a tax on real balances. Lucas (1981) and Fischer

(1981) compute such a welfare cost by calculating the area under the money demand

curve, obtaining surprisingly low estimates of in‡ation. However, Lucas (2000),

using the competing money demand speci…cations of Meltzer (1963), which takes

on a log-log form, and Cagan (1956), which takes on a semi-log form, highlights

the fact that these competing money demand speci…cations may have very di¤erent

implications for the welfare cost of in‡ation. Indeed, Ireland (2009) shows that the

welfare cost of in‡ation depends on the speci…cation of the money-demand curve,

together with …nding that a semi-log form proposed by Cagan (1956), which …ts

better with post-1980s U.S. data, generates modest departures from Friedman’s

zero nominal interest rate rule.

In the …rst step, the functional form of the welfare cost function is captured.

To do this, we apply the method of Bailey (1956), and de…ne the welfare cost of

in‡ation as the area under the inverse money demand function – the consumers’

surplus –which can be gained by reducing the interest rate from some level itto

zero and then subtracting the seigniorage revenue itmtto isolate the dead weight

loss. De…ning m(it)as the estimated function, let (mt)be the inverse function

and de…ne the welfare cost function w(it)by:5

w(i) = Zm(0)

m(i)

(x)dx =Zi

0

m(x)dx im (i)(18)

5As all the variables are expressed in time t, we drop timing to facilitate reading in the subse-

quent equations.

9

The second integral shows an alternative way of calculating consumer surplus. It

can be shown that under the money demand speci…cation (Eq. 13), solving Eq. 18

implies the following welfare function:

w(i) = e

1(1 + i)ei (19)

It is worth highlighting the similarities between this welfare function (Eq. 19) under

money demand (Eq. 13) with the welfare function used by Lucas (2000):

w(i) = B

1(1 + i)ei(20)

Setting B=ein Eq. 19 yields the Lucas (2000) welfare function in Eq. 20.

However, the micro-founded money demand and the welfare function derived in

this paper explicitly link the structural parameters of the model with the welfare

function by altering both the semi-elasticity and the constant term in the money

demand curve.

In the second step, the money demand curve is estimated and combined with the

expression similar to Eq. 19 and Eq. 20 to pin down the welfare cost of in‡ation.

To highlight the importance of di¤erent aspects of the money demand function,

we apply the well-known speci…cation of the money demand curve (Lucas, 2000;

Ireland, 2009).

The …rst row in Table 1 pins down welfare at di¤erent levels of in‡ation and

nominal interest rate. The values of and Bcome from Lucas (2000) based on

annual data from 1900 to 1994. His preferred speci…cations set , allows him to

pin down an average value of B= 0:3548 so that ln (B)equals the average value

of ln (m) + i. This, in turn, allows him to calculate the welfare cost of in‡ation.

However, …xing each combination of fln (m); igyields a di¤erent value of B.

In this spirit, Table 1 also lists down the welfare calculations for a ‘minimum’

and ‘maximum’value of Bfollowing Lucas’s calculations of the constant of money

demand. The second panel in Table 1 repeats the same exercise, this time using the

values presented in Ireland (2009), who estimates to be equal to 1:7944 based on

quarterly data from 1980 to 2006. Again, setting the elasticity at this benchmark

generates both the average as well as the upper and lower bound of B.

Table 1 highlights that the di¤erences in the welfare cost of in‡ation using the

same money demand curve may be due to two factors: the value of the semi-

elasticity of money demand, and the constant of regression. Moving from a regime

where high elasticity is estimated to one that is low works to reduce the welfare

cost of in‡ation. Intuitively, a lower elasticity implies a steeper money demand

curve, therefore, a lower area will represent the welfare cost. However, even with

lower elasticity, if the constant of the money demand has increased then this would

work to mitigate some of the fall in welfare due to money demand steepening; in

this sense, the choice of Bmatters for the total welfare –a higher Bfor a given

generates a higher welfare loss. Table 1 reveals that, even if elasticity is reduced

from 7to 1:7944, it may not necessarily correspond to a fall in welfare if the constant

Bswitches from a low value of 0:1805 to a high value of 0:4589.

10

i= 0:03 i= 0:05 i= 0:13

B0% in‡ation 2% in‡ation 10% in‡ation

Benchmark 7 0:3548 0:0972 0:2466 1:1717

Lower Bound 7 0:1805 0:0495 0:1255 0:5962

Upper Bound 7 0:5068 0:1389 0:3524 1:6738

Benchmark 1:7944 0:2795 0:0217 0:0590 0:3633

Lower Bound 1:7944 0:1591 0:0123 0:0336 0:2068

Upper Bound 1:7944 0:4859 0:0378 0:1026 0:6316

Table 1: Welfare Estimates Note: This table outlines estimates of the welfare costs of zero,

two percent, and ten percent annual in‡ation based on the benchmark and upper and lower bound

regression results in Lucas (2000) and Ireland (2009).

In this sense, the money demand and welfare framework derived in this paper

gives a unique micro-founded interpretation to the money-demand curve, and the

corresponding welfare function utilized in Lucas (2000) and Ireland (2009). Viewed

through the lens of this framework, the potential sources behind the di¤erences in

welfare cost obtained in Lucas (2000) and Ireland (2009) are distilled. To answer

these questions, we delve into the data presented in Appendix B, which allows us to

estimate the money demand curve with the intention of unraveling the di¤erences

in welfare and pinpointing the factors that may have generated these shifts in the

money demand curve.

4 Estimating the Money Demand Curve

4.1 Fixed Coe¢ cients

We estimate the money demand curve using quarterly U.S. data spanning 1959–

2008. The beginning of the sample is chosen to coincide with Ireland (2004), while

the end-of-sample dates are chosen to avoid dealing with the Federal Reserve’s

unconventional monetary policy that began in September 2008.

Following Ireland (2009) and Miller et al. (2019), the money-income ratio is

measured by dividing the Cynamon et al. (2006) sweep-adjusted M1 money stock

(M1RS aggregate) by nominal GDP, the three-month U.S. Treasury bill rate, which

serves as the measure of iand matches the risk-free rate, nominally-denominated

bonds that serve as an alternative store of value in theoretical models of money

demand.6

6The de…nition of money, and the related money aggregate’s ability to estimate money demand,

has been the subject of active literature (Diewert, 2013). While our preference for using the M1RS

indicator to empirically document, and to assess the causes and consequences of these evolving

relationships between interest rates and money, is designed to align our …ndings with Ireland

(2009) and Lucas (2000) who rely on similar money aggregates, it is important to mention the

developments in the money demand literature based on the use of Divisia money (Barnett, 1978;

Barnett et al., 1984; Barnett and Chauvet, 2011). In these series of in‡uential papers, Barnett

recommends the use of a superlative index number construction of the user costs, derived from

François Divisia in money aggregate construction to produce a more sophisticated measure of

money that internalizes the rate of interest within its construction. In this way, the …nancial

11

We utilize both static ordinary least squares (SOLS) and dynamic ordinary least

squares (DOLS) estimates of the parameters of the money demand,7linking ln (m)

and i. Therefore, each of the parameter estimates in the following tables comes

from an ordinary least squares (OLS) regression of ln (m)on a constant, the level of

the nominal interest rate i, and leads and lags of i, the quarter-to-quarter change

in the nominal interest rate computed using the Newey and West (1987) estimator

of the regression error variance for various values of the lag truncation parameter

q.89

ln (m) = i (21)

Focusing …rst on the value of , the SOLS and the DOLS estimates are close to

each other and suggest a value between 3:4542 (SOLS) and 3:8561 (DOLS with four

lags and leads), con…rming that the estimated interest elasticity of money demand

di¤ers signi…cantly from zero. However, this number is estimated to be higher than

that of Ireland (2009), who …nds it to be in the 1:81:9range and, at the same time,

is signi…cantly smaller in absolute value than the Lucas (2000) setting of 7. The

constant of regression is estimated to be higher than that estimated in both Ireland

innovation in the economy in the form of new transactions technology or the introduction of

alternative new monetary assets may be incorporated into the construction of the index number,

ensuring that the money demand function remains stable even during periods of high …nancial

innovation – see, e.g. Belongia (1996) for money demand stability, and further evidence from

11 countries by Belongia and Binner (2000). Furthermore, Belongia and Ireland (2019) argues

that the identi…cation of stable money demand functions –when estimated with Divisia quantity

data and their user cost duals – is consistent with the idea that instability reported since the

early 1990s may be more closely associated with measurement error than shifts in the underlying

economic relationships themselves. Belongia and Ireland (2019) identify a stable money demand

function over a period that includes the …nancial innovations of the 1980s and continues through

the Global Financial Crisis (GFC) and Great Recession, suggesting that a properly-measured

aggregate quantity of money can play a role in the conduct of monetary policy. More broadly,

Qureshi (2016, 2018) argue that using M1 and M3, as compared to M2, may be more useful for

policy purposes. Not only do their results present an alternative framework to explain the historical

actions of the Fed, but the subsequent analysis suggests that the bias against the inclusion of money

in mainstream macroeconomic models may be due to an overreliance on an incorrect aggregate.

7Roughly similar results were obtained using alternative techniques such as VECM. These

results are available upon request.

8DOLS has a number of advantages (Stock and Watson, 1993; Hamilton, 1994). First, and

under the assumption of co-integration in the relationships, the DOLS estimates are asymptotically

e¢ cient and asymptotically equivalent to maximum likelihood estimates obtained, for example,

through the method proposed by Johansen (1988). Second, adding leads and lags of ito the

estimated equations controls for possible correlation between the interest rate and the residual

from the co-integrating relationship, linking ln (m)and i. Finally, the conventional Wald test

statistics formed from these DOLS estimates have conventional normal or chi-squared asymptotic

distributions, making it possible to draw familiar comparisons between the parameter estimates

and their standard errors.

9Since we evaluate welfare costs as a percentage of GDP, we need to formally test for the

assumption of unitary income elasticity and, when evidence in its favor is found, impose it and

estimate long-run money demand equations, where the natural logarithm of the money-income

ratio depends on the nominal interest rate given to our micro-founded speci…cation. Perhaps this

is a restriction, but the unit elasticity of consumption is imposed by the theoretical derivation of

the money demand curve –a result consistent with the long-run estimate in Lucas (1988). In any

case, we directly estimate the unit elasticity and we …nd results consistent with those found in

Ireland (2009), i.e. approximately equal to unity.

12

(2009) and Lucas (2000). Table 2 summarizes these results, including estimates of

welfare for various levels of in‡ation, calculated by plugging these numbers into the

derived expression in Eq. 19.

Zero 2%4%10%

in‡ation in‡ation in‡ation in‡ation

mt=it w (0:03) w(0:05) w(0:07) w(0:13)

SOLS 1:6089 3:4542 0:0292 0:0776 0:1455 0:4389

(0.0318) (0.4089)

DOLS, p = 1 1:5921 3:6188 0:0308 0:0816 0:1526 0:4576

(0.0312) (0.4097)

DOLS, p = 2 1:5868 3:7338 0:0319 0:0843 0:1575 0:4701

(0.0304) (0.4105)

DOLS, p = 3 1:5827 3:8303 0:0328 0:0866 0:1615 0:4803

(0.0299) (0.4112)

DOLS, p = 4 1:5826 3:8561 0:0330 0:0871 0:1624 0:4826

(0.0291) (0.4117)

Table 2: Welfare Cost (Percent of Income). Note: This table outlines estimates of the

welfare costs of zero, two percent, four percent and ten percent annual in‡ation based on the

regression results. *** p<0.01, ** p<0.05, * p<0.1.

As the static and dynamic OLS estimates look quite similar, so do the implied

welfare costs.10 Assuming, as before, that the steady-state real interest rate equals

three percent, so that r= 0:03 corresponds to zero in‡ation, r= 0:05 corresponds to

two percent annual in‡ation, r= 0:07 corresponds to four percent annual in‡ation

and r= 0:13 corresponds to ten percent annual in‡ation. Therefore, the regression

coe¢ cients put the welfare cost of pursuing a policy of price stability as opposed

to the Friedman (1969) rule at less than 0:0292 percent of income, the cost of two

percent in‡ation at less than 0:0776 percent of income, the cost of four percent

in‡ation at less than 0:1455 percent of income, and the cost of ten percent in‡ation

at less than 0:4389 percent of income. Interestingly, Table 2 also provides estimates

of the cost of ten percent in‡ation compared to price stability, w(0:13) w(0:03),

at approximately 0:4097 percent of income. These numbers are still larger than the

Fischer (1981) estimate of 0:30 percent of income, and the Ireland (2009) estimate

of 0:20 percent of income, but close to the Lucas (1981) estimate of 0.45 percent of

income.11

Before delving into sub-sample estimates, we extract the values of trend in‡ation,

steady-state interest and the subjective rate of time preference parameter from the

data, then use the functional forms derived earlier to extract values for the two

parameters in the utility function, vand . Table 3 summarizes the parameters

obtained under the money demand estimates described in Table 2. In‡ation during

10 Both statistically signi…cant at the 1% level.

11 Notice that the time period under question in the current paper is di¤erent from that consid-

ered by Ireland (2009) and Lucas (2000). In that, whereas the present study focuses on quarterly

data spanning …ve decades from 1959, Ireland (2009) focuses only on the post-1980 period, while

the bulk of the Lucas (2000) sample lies before this date.

13

the sample is …xed at 3:5674 percent, which corresponds to 1:0089 in gross terms.

The sample average for the interest rate is found to be 5:430 percent. These numbers

permit the extraction of the elasticity (v) and the weight of real balances versus

consumption in the utility function, which are 9:8968 and 0, respectively. While

we …nd a moderate degree of inter-temporal elasticity, rejecting the restrictive CES

version to represent utility, the evidence presents little evidence of real balances

a¤ecting the utility function for the entire time period of the benchmark estimates

In‡ation () Interest () Elasticity (v) Weight ()

0:99 1:0089 0:0543 16:2189 0:0000

Table 3: Extraction of Deep Parameters Note: This table outlines the values for the

parameters of the utility function based on OLS estimates from table 2 and utilizing equation

(12).

Looking in detail at the elasticity of substitution, these numbers connect with

Holman (1998) who …nd that the estimated exponent of the CES characterizations

is statistically di¤erent from zero in the nested-CES case, as well as with Galí (2015)

who propose this number to be “reasonably large”. Second, given that v6= 1, the

results imply that utility is not separable in either consumption or money. Third,

the share of real balances is in stark contrast to the …ndings of Holman (1998), Finn

et al. (1990) and Poterba and Rotemberg (1987) who …nd evidence of real balances

in utility. Notice that this could be due to a number of reasons, such as due to the

time period under question, or the type of money aggregate used. Indeed, variation

in may also represent shocks in transactions technology –shocks that change the

utility value of money relative to that of consumption expenditures (Koenig, 1990),

which are potentially time-varying. To accommodate these changes, we focus on

estimating the money demand curve around key break-dates.

4.2 Split-Sample Estimates

To deal with potential instabilities, we rely on a split-sample approach to esti-

mate the money demand function. We rely on static and dynamic OLS techniques

to estimate this money demand function for the two periods: 1959:I–1979:IV and

1980:I–2008:II.12 The break in 1980 is chosen to coincide with both the arrival of

Paul Volcker at the Federal Reserve Board and the implementation of the Deposi-

tory Institutions Deregulation and Monetary Control Act of 1980, which are often

identi…ed as key events marking the start of a new chapter in U.S. monetary his-

tory. As before, the end date of 2008:II is chosen to coincide with the collapse of

the Lehman Brothers and the beginning of unconventional policy by the Fed.13 The

detailed results are available in Appendix C.

12 Note that this is because the OLS/DOLS methodology is not equipped to deal with break-

dates and, thus, we simply apply the technique to two separate sub-samples.

13 Estimates based on the crises period (2008:II–2017:IV) suggest estimates of interest-elasticity

to be inconclusive as the estimates are not statistically signi…cant. These results are available

upon request.

14

Table 4 outlines the key parameters of semi-elasticity and the constant –ob-

tained under OLS estimates. It is immediately clear from these numbers that a

split-sample approach around the break-date highlights a large shift in the value of

the semi-elasticity, re‡ecting a ‡attening of the money-demand curve. In this sense,

our results …nd little disagreement with the estimate suggested by Ireland (2009).

However, pre-1979 estimates paint a completely di¤erent picture because elasticity

is found to be close to the estimates suggested by Lucas (2000). Furthermore, a

clear and statistically signi…cant shift in the constant term is also found as the

upper and lower bounds of the estimates are tightly estimated.

Parameters Pre-1979 Post-1980

7:5351 1:5639

(0.5332) (0.0996)

1:2255 1:8289

(0.0295) (0.0061)

In‡ation ()1:0105 1:0076

Interest ()0:0527 0:0582

Substitution (v)6:3539 35:8216

Weight ()0:61050:21034

Table 4: Money Demand Estimation (Extraction of Deep Parameters) Note: This

table outlines the values for the parameters of the utility function based on OLS estimates. ***

p<0.01, ** p<0.05, * p<0.1.

Table 4 also summarizes the parameters obtained under the money demand es-

timates. In‡ation and interest rates are pinned down from the data and vary across

the sample, which is uncontroversial in the literature. These numbers permit the

extraction of the elasticity (v) and the weight of real balances versus consumption

in the utility function. A large variation in these numbers between the two time

periods is also observed. There are signi…cant changes in the elasticity of money de-

mand, a result which is consistent with the …ndings of Ireland (2009), but di¤erent

from those found in Miller et al. (2019).14

First, the elasticity of substitution between consumption and real balances be-

tween the two periods is said to have fallen. Second, the share of real balances in

utility implies a comparable role for money balances in the …rst half and a negligi-

ble role in the second. Indeed, estimates of in the …rst half present values close

to those found in Holman (1998), Finn et al. (1990) and Poterba and Rotemberg

(1987). Holman (1998) …nd liquidity services to have the largest role in the nested-

CES case (ranging from 0:0242 to 0:0319); Finn et al. (1990) …nd that real balances

comprise less than 10 percent of total expenditures; while Poterba and Rotemberg

(1987) estimate that the share of expenditures on consumption is between 0:961 and

0:969. Thus, while our estimates reveal a slightly smaller role for liquidity services

in the …rst half of the sample, the non-zero values do con…rm previous …ndings.

14 Given that we use similar data, this di¤erence may be due to the time-varying approach

undertaken by Miller et al. (2019), which may not be consistent with the DOLS approach usually

employed in the literature (Stock and Watson, 1993).

15

Moreover, since real balances enter directly the dynamic IS, they may be relevant

in determining in‡ation and output dynamics during the …rst half of the sample,

complementing the …ndings of Castelnuovo (2012).

5 Applications

5.1 Explaining Changes in the Welfare Cost of In‡ation

Combining the estimates of semi-elasticity in Section 4.2 with the welfare cost func-

tion derived in Section 3, Table 5 looks at the welfare cost of in‡ation. Plus, the

counterfactual welfare cost is also illustrated when the constant and semi-elasticity

terms in the money demand curve are varied.

Table 5 suggests that both the semi-elasticity of interest and the constant term

are estimated to be higher during the pre-1979 period when compared to their

post-1980 counterparts. The values for the welfare cost of in‡ation are not too

far o¤ from those implied in Dotsey and Ireland (1996) for the pre-1979 sample.

The welfare cost of pursuing a policy of price stability as opposed to the Friedman

(1969) rule at less than 0:0857 percent of income, the cost of two percent in‡ation

at less than 0:2159 percent of income, the cost of four percent in‡ation at less

than 0.3843 percent of income, and the cost of ten percent in‡ation at less than

1:0002 percent of income. Table 5 also provides estimates of the cost of ten percent

in‡ation compared to price stability, w(0:13) w(0:03), which is approximately

0:9145 percent of income –numbers that are still larger than the Fischer (1981)

estimate of 0:30 percent of income, and the Ireland (2009) estimate of 0:20 percent,

and even the Lucas (1981) estimate of 0:45 percent of income. The di¤erences with

Lucas (1981) and Lucas (2000) arise primarily due to our estimate of the constant

term in the money demand curve.

16

Zero 2%4%10%

in‡ation in‡ation in‡ation in‡ation

mt=it w (0:03) w(0:05) w(0:07) w(0:13)

Pre-1979

Static OLS 1:2255 7:5351 0:0857 0:2160 0:3844 1:0003

Switch Elasticity ()1:2255 1:5640 0:0200 0:0545 0:1046 0:3392

Switch Constant ()1:8290 7:5351 0:0469 0:1181 0:2102 0:5471

Switch In‡ation ()0:0803 8:8172 0:3076 0:7623 1:3355 3:3268

Switch Substitution (v)0:0279 1:3366 0:0570 0:1554 0:2993 0:9790

Switch Interest ()0:1152 7:5351 0:2603 0:6556 1:1667 3:0364

Switch Weight ()4:1583 7:5351 0:0046 0:0115 0:0205 0:0533

Combined Weight

& Substitution 0:9728 1:3365 0:0221 0:0604 0:1163 0:3806

Post-1980

Static OLS 1:8290 1:5640 0:0110 0:0298 0:0572 0:185

Switch Elasticity ()1:8290 7:5351 0:0469 0:1181 0:2102 0:5471

Switch Constant ()1:2255 1:5640 0:0200 0:0545 0:1046 0:3392

Switch In‡ation ()0:7304 1:3366 0:0282 0:0770 0:1482 0:4850

Switch Substitution (v)4:0310 8:8172 0:0059 0:0147 0:0257 0:0640

Switch Interest ()0:7237 1:5640 0:0331 0:0900 0:1728 0:5603

Switch Weight ()0:0053 1:5640 0:0679 0:1846 0:3544 1:1493

Combined Weight

& Substitution 1:6104 8:8172 0:0666 0:1650 0:2891 0:7203

Table 5: Welfare Cost (Percent of Income): Counterfactual Experiments Note: This

table outlines estimates of the welfare costs of zero, two percent, four percent and ten percent

annual in‡ation, disaggregating these changes into two time periods, pre-1979 and post-1980. It

further presents the welfare cost of in‡ation using counterfactual values of the underlying para-

meters driving the semi-elasticity and constant of the money demand curve.

A startlingly di¤erent picture emerges for the post-1980 sample, where the wel-

fare cost of pursuing a policy of price stability as opposed to the Friedman (1969)

rule reads at less than 0:0109 percent of income, the cost of two percent in‡ation

at less than 0:0298 percent of income, the cost of four percent in‡ation at less than

0:0572 percent of income, and the cost of ten percent in‡ation at less than 0:1855

percent of income. Interestingly, Table 5 also provides estimates of the cost of ten

percent in‡ation compared to price stability, w(0:13) w(0:03), which is approxi-

mately 0:1746 percent of income, numbers that are smaller than the Fischer (1981)

estimate of 0:30 percent of income, and close to the Ireland (2009) estimate of 0:20

percent. Broadly, Table 5 points to large changes in welfare across the two time

periods.

Looking at counterfactual evidence, it is clear from Table 5 that not only switch-

ing the elasticity term but also the switch in the constant term has large implications

on the welfare cost of in‡ation. Focusing …rst on the pre-1979 time-period, switch-

ing the elasticity parameter contributes to an almost 50% fall in welfare, while

switching the constant terms generates a fall of approximately 30%. In contrast,

17

opposing results emerge for the post-1980 period.

The underlying factors of the shifts in the money demand curve reveal the true

sources of the changes in the welfare cost of in‡ation. The …rst block in Table 5

pinpoints the welfare cost of in‡ation in the pre-1979 sample, setting each of the

underlying sources at post-1980 values. First, lower in‡ation works to increase both

the elasticity and the constant parameter, increasing the welfare cost of in‡ation.

Second, a shift in the elasticity of substitution generates a rise in the constant term

but a fall in the semi-elasticity of interest term. A switch in steady-state interest

rates generates a larger constant term and, therefore, a larger loss in welfare. Con-

sidering the share of real balances extracted in the post-1980s sample to calculate

the constant term for the pre-1979 sample, we …nd that this generates a large fall in

welfare despite being roughly the same elasticity of interest rates. Our calculations

suggest that the combined e¤ect of a reduction in the weight of real balances and

the elasticity of substitution between real balances and consumption work to reduce

the welfare cost in the …rst sample.

Moving onto the second half of the sample reveals similar insights. Replacing

a higher value of trend in‡ation or a lower value of the steady-state interest rate

works to reduce the semi-elasticity but increases the constant term, generating a

larger fall in in‡ation. The elasticity of substitution generates a larger but a lower

value of the constant. Finally, considering the share of real balances extracted in the

pre-1979 sample to calculate the constant term for the post-1980 sample generates

a large rise in welfare, this roughly matching the welfare costs observed in the …rst

half of the sample.

What might justify these results? First, the evidence in favor of the time de-

pendence of the deep parameters may be interpreted as time-varying preferences by

American households, or as evidence in favor of breaks due to …nancial innovation,

as argued by Castelnuovo (2012). Indeed, Justiniano and Primiceri (2008) enumer-

ate important elements of this transformation, such as the passing of the Deposi-

tory Institutions Deregulation and Monetary Control Act in 1980 –particularly the

demise of regulation Q, and the Garn-St. Germain Depository Institutions Act of

1982 (Hendershott, 1992; Dynan et al., 2006; Campbell and Hercowitz, 2009). These

changes allowed households unprecedented access to external …nancing (Campbell

and Hercowitz, 2009), which was further facilitated by the emergence of secondary

mortgage markets (Peek and Wilcox, 2006; McCarthy and Peach, 2002). Moreover,

access to external …nancing was enhanced by the development of a market for bonds

with below-investment grade ratings (Gertler and Lown, 1999), as well as a decline

in the cost of new equity issuances (Jermann and Quadrini, 2006).

The irrelevance of more traditional money aggregates and the emergence of

complementary sources of …nance for households may imply a weakening of the

semi-elasticity of interest. This has perhaps worked to reduce the welfare cost of

in‡ation. Looking at this argument another way, money holdings yield direct utility

in the model in a standard framework. Since the importance of real balances seems

to decline in the second half of the sample, so does their contribution to welfare.

18

2

4

6

Semi-Elasticity

-1.8

-1.6

-1.4

Constant

1.006

1.008

1.01

1.012

1.014

Inflation (Gross)

0.04

0.06

0.08 Interest Rates

1959.I-1974:IV

1963.I-1978:IV

1967.I-1982:IV

1971.I-1986:IV

1975.I-1990:IV

1979.I-1994:IV

1983.I-1998:IV

1987.I-2002:IV

1991.I-2006:IV

10

20

30

Elasticity of Substitution

1959.I-1974:IV

1963.I-1978:IV

1967.I-1982:IV

1971.I-1986:IV

1975.I-1990:IV

1979.I-1994:IV

1983.I-1998:IV

1987.I-2002:IV

1991.I-2006:IV

1

2

3

4

510-7 Real Balance Share

Figure 1: Time-varying Money-Demand Parameters. Note: This …gure presents esti-

mates of the semi-elasticity of interest rate and the constant in the money demand curve as well

as the underlying parameters and …rst moments from actual data. Evolution of the parameters

constructed by employing seven rolling windows of 16-year constant length. The dotted lines plot

the standard errors of the 5 and 95 percent con…dence intervals.

5.2 Recursive Estimates

It has been documented by several authors that post-WWII U.S. macroeconomic

relationships may be characterized by instabilities that might not even be captured

using a single split-sample approach. The time-varying aspect of the semi-elasticity

has also been discussed in the literature.15 The evolution of …nancial services, in

particular, may be characterized by a gradual change in the behavior of house-

holds. Accounting for the possibly evolving role played by the underlying factors is,

therefore, of crucial importance for achieving correct identi…cation of the underlying

drivers of the changes in money demand.

Following Castelnuovo (2012), we tackle this issue by recursively estimating

the money demand curve with OLS techniques. We estimate the evolution of the

parameters constructed by employing seven rolling windows of 16-year constant

length. We then extract the underlying structural parameters based on time-varying

estimates of semi-elasticity of interest, which are pictured in Fig. 1.

It is apparent from Fig. 1 that changes in the semi-elasticity and the constant

15 See, for instance, Khan (1974), Judd and Scadding (1982), Tesfatsion and Veitch (1990), Hafer

and Jansen (1991), Miller (1991), Lütkepohl (1993), Ireland (2009), Lucas and Nicolini (2015),

and Miller et al. (2019).

19

term in the money demand function occurred gradually, starting well before the

1980s. These terms are seen declining as the sample moves through observations

conditioned to the 1970s –a period accompanied by rising interest rates and in‡ation

– and a gradually-rising elasticity of substitution between consumption and real

balances.

Fig. 1 suggests two large shifts in the semi-elasticity of interest rates, instead

of occurring around the commonly considered split-sample break. The decline in

semi-elasticity occurs when moving from the window dated 1963:I–1978:IV to the

1967:I–1982:IV. The semi-elasticity of interest is observed to decline substantially

from around 5:8715 to 3:9536 during this period. However, the underlying utility

parameters display remarkable stability during this period. Looking closely, this

change in semi-elasticity is attributed to the rise in trend in‡ation from 4.7994 to

6.194 percent. A smaller change in the constant is observed that, given the stability

of the underlying utility parameters, is attributed to the rise in interest rates.

The second sharp fall in the semi-elasticity of interest rates is observed when

moving from the window 1971:I–1986:IV to 1975:I–1990:IV. The semi-elasticity of

interest declines substantially from around 3:1337 to 1:8716 during this period.

However, in this case, both in‡ation and interest rates, while not constant, display

remarkable stability. From the data, in‡ation is averaged at around 5percent, while

interest rates rise only marginally from 7:9401 to 8:2085. In this case, a sharper

change is observed in the elasticity of substitution between consumption and real

balances, which almost doubles from 12:8699 to 23:2613. The share in real balances

in the utility function declines to zero.

On closer inspection, movements in semi-elasticity of interest toward the latter

half of the sample could be attributed to changes in the elasticity of substitution

between consumption and real balances. While the elasticity of substitution works

to reduce the semi-elasticity of interest rates, the decline in the share of real bal-

ances in utility is the key factor behind the decline in the constant term. One

possible explanation for this factor may lie in …nancial innovation increases during

this period. The availability of alternative sources of payments may cause the share

of real balances in utility to fall, as households have a lower reliance on this par-

ticular aggregate. Because households now hold fewer real balances, the degree of

substitutability for those lower levels of real balances falls. With households now

holding a lesser share, they are less inclined to substitute those real balances. For

the limited amount of real balances held that are more valuable than before, the

opportunity cost rises, which a¤ects the welfare cost of in‡ation.

Table 6 outlines the results from the rolling window estimates, tabulating the

values of in‡ation, interest rates, semi-elasticity, and the share of real balances, as

well as the welfare cost of in‡ation observed.

Assuming, as before, that the steady-state real interest rate equals three percent

so that r= 0:03 corresponds to zero in‡ation, r= 0:05 corresponds to two percent

annual in‡ation, r= 0:07 corresponds to four percent annual in‡ation and r=

0:13 corresponds to ten percent annual in‡ation, this means that Table 6 con…rms

the gradual fall in welfare cost of in‡ation at di¤erent levels of interest rates and

in‡ation. Indeed, the welfare cost is found to be declining gradually. Corresponding

to the decline is the semi-elasticity of interest rates, which occurs moderately due

20

Zero 2% 4% 10%

In‡ation In‡ation In‡ation In‡ation

Sample v w (0:03) w(0:05) w(0:07) w(0:13)

1959.I–1974:IV 6:4441 1:2366 0:0476 1:0087 8:2059 0:51060:0741 0:1892 0:3413 0:9230

1963.I–1978:IV 5:8715 1:3432 0:0551 1:0120 7:6646 0:51060:0614 0:1579 0:2868 0:7916

1967.I–1982:IV 3:9537 1:4967 0:0740 1:0155 9:8254 0:51080:0368 0:0971 0:1807 0:5349

1971.I–1986:IV 3:1338 1:6332 0:0794 1:0145 12:8699 0:21010 0:0259 0:0690 0:1297 0:3960

1975.I–1990:IV 1:8717 1:7888 0:0821 1:0127 23:2613 0:51022 0:0136 0:0368 0:0703 0:2252

1979.I–1994:IV 1:3891 1:8450 0:0773 1:0102 35:2411 0:31024 0:0096 0:0262 0:0504 0:1646

1983.I–1998:IV 2:3885 1:7813 0:0612 1:0064 25:2198 0:11022 0:0173 0:0465 0:0882 0:2771

1987.I–2002:IV 1:7525 1:8174 0:0509 1:0059 35:5866 0:61032 0:0124 0:0336 0:0643 0:2070

1991.I–2006:IV 1:6289 1:8191 0:0399 1:0054 39:4494 0:91035 0:0115 0:0313 0:0600 0:1941

Table 6: Rolling Window Estimates Note: This table outlines estimates of the semi-elasticity

of interest rate, the constant in the money demand curve, the underlying parameters and …rst mo-

ments from actual data and the welfare cost of in‡ation. Evolution of the parameters constructed

by employing seven rolling windows of 16-year constant length. Windows: [1959:I – 1974:IV,

1963:I –1978:IV, ..., 1990:I –2006:IV].

to the constant, while the second decline is due to a combined change in semi-

elasticity of interest and the constant term. According to our results, the …rst

change is primarily attributed to a rise in trend in‡ation and interest rates, while

the second shift is attributed to changes in the utility function –in particular to

the changes in the elasticity of substitution between consumption and real balances,

and to the fall of the share of real balances by households.

Table 6 also provides estimates of the cost of ten percent in‡ation compared

to price stability, w(0:13) w(0:03) at various junctures in time, starting from

approximately 0:9230 in the …rst window and declining to almost 0:1941. The

numbers obtained for each data sample encompass the con‡icting …ndings in the

previous literature. At the same time, these results indicate that the single-valued

approach to approximate the welfare cost of in‡ation in previous literature captures

only the sample average at each point in time.

When combined, our results suggest that the entire shift in money demand could

be attributed to the evolution of trend in‡ation, interest rates, and changes in the

utility function. This o¤ers an alternative explanation for the changes observed in

the traditional money demand relationship.

5.3 Assessing Changes in the Monetary Transmission Mech-

anism

As documented earlier, several authors have presented evidence of large changes

that took place in the U.S. economy during the 1980s. For example, Boivin and

Giannoni (2002) test whether the monetary transmission mechanism has changed.

They examine whether the macroeconomic e¤ects of monetary policy shocks in

the U.S. were di¤erent in the 1980s and 1990s relative to the 1960s and 1970s.

They conclude that changes in the monetary policy rule are responsible for the

change in the impulse response of in‡ation and output. Pancrazi and Vukotic

21

(2019) test whether conventional monetary policy instruments maintained the same

e¤ectiveness to accommodate any undesirable e¤ects of shocks throughout the post-

war period. They too …nd that the e¤ectiveness of monetary policy (its ability

to counteract undesired shocks) has declined, though they identify the changed

properties of the labor market as proving the key contribution to this decline.

Theoretical results suggest that changes common to those that a¤ect money de-

mand may also in‡uence the e¤ect of changes in interest rates on output (Section 2).

Intuitively, since real balances, the elasticity of substitution between consumption

and real balances in the utility function, and trend in‡ation enter the IS equa-

tion, changes in these parameters may a¤ect the linkages between interest rates on

output.

To test these changes from the data, we begin by documenting evidence re-

garding changes in the monetary transmission mechanism for the U.S., replicating,

in essence, the …ndings of Boivin and Giannoni (2002). The baseline empirical

model of the economy is a VAR in variables describing the economy (Zt) as well as

monetary policy (Rt):

Zt

Rt=+A(L)Zt1

Rt1+t(22)

The structural block is described by the vector Zt= [yt; t]0, of output gap (yt)

and the annualized in‡ation rate (t). The policy instrument Rtis assumed to be

the 3-month treasuring bill used earlier.16

To be consistent with recent VAR analyses, we assume that the economy (Zt)

responds only with a lag to changes in the policy instrument (Rt). The recursive

VAR follows closely the notation used in Boivin (2006) and is expressed as:

Zt= 0+

p

X

i=1

Z

1;iZti+

p

X

i=1

R

1;iRti+Z

t(23)

Rt= 1+

p

X

i=1

Z

2;iZti+

p

X

i=1

R

2;iRti+R

t(24)

In particular, we assess the changes in the e¤ects of monetary policy by compar-

ing impulse response functions of the output gap, in‡ation, and the Fed funds rate to

a monetary policy shock using the VAR estimated over two di¤erent subsamples.17

16 Several clari…cations are in order. First, we do not include a commodity price measure since

it is not formally justi…ed by the theoretical model, but is only included to limit the extent of

the price puzzle in this VAR, as discussed in Boivin (2006). Moreover, Christiano et al. (1996)

show that, while including di¤erent indices of price commodity limits the price puzzle, it is not

justi…ed theoretically. Second, in each series, our results remain robust for including the output

gap instead of output growth.

17 Based on evidence listed earlier regarding the conduct of monetary policy, we base our results

on the following subsamples: sample 1 corresponding to 1959:I–1979:IV and sample 2 correspond-

ing to 1980:I–2008:II. While Boivin (2006) …nd slightly di¤erent results when they use 1984 as the

break-point, Stock and Watson (2003) show that this break date is very imprecisely estimated.

They …nd con…dence intervals for the break date that essentially encompass all of the 1980s, hence

justifying our choice for the break-date.

22

10 20 30 40 50 60

-0.3

-0.2

-0.1

0Output

59:I-79:IV

80:I-08:II

10 20 30 40 50 60

0

0.2

0.4

0.6

Inflation

59:I-79:IV

80:I-08:II

10 20 30 40 50 60

0

0.2

0.4

0.6

0.8

1Interest Rate

59:I-79:IV

80:I-08:II

Figure 2: VAR Evidence: Impact of Unit Shock to Interest Rates on Output and

In‡ation. Note: This …gure presents impulse responses to a monetary shock over the two sub-

samples, 1959:I –1979:IV and 1980:I – 2008:II. The solid line plots the impulse response for the

1959:I –1979:IV sample while the dashed line plots the impulse response for the 1980:I –2008:II

time period.

Fig. 2 displays the impulse response functions for an unexpected unit increase in

the 3-month T-bill rate from the identi…ed VAR, summarizing the speci…c changes in

the transmission mechanism discussed in Boivin and Giannoni (2002) and Pancrazi

and Vukotic (2019). It is clear that a unit change in interest rates seems to have

had a dissimilar initial impact on in‡ation and output gap, and is conditional on

the type of time period analyzed.

Similar to Boivin and Giannoni (2002), we also con…rm these changes by com-

paring the di¤erences in the means of the response to interest rates. Both output

and in‡ation display statistically signi…cant di¤erences; the p-values of output and

in‡ation –of 0:0000 and 0:0166, respectively –con…rm the statistically signi…cant

changes in the transmission mechanism, despite roughly the same impact on interest

rates (p-value of 0:8817) across the two time-periods.

To quantify these changes, we construct a measure of the impact elasticity,

denoted by M P as:

MP;t =Pj

i=1 ~yt+i

Pj

i=1 ~{t+i

(25)

23

where the variables ~yand ~{are the impulse response of a one-unit policy innovation

and jis the horizon of the period analyzed. Thus, the combined e¤ect of a unit

change in interest rates on output is the sum of the e¤ect of output divided by

interest rates at each point in time. Taking the average of this number yields

a measure of the impact elasticity of monetary policy on output. The measure of

elasticity is similar to that constructed by Pancrazi and Vukotic (2019). The change

in MP;t conditioned on the two periods is measured as:

MP =M P;pre1979

MP;post1980

(26)

Table 7 summarizes the impact elasticity for di¤erent time-horizons. For the bench-

mark case, where the horizon –represented here in quarters –is relatively shorter,

the value of M P is equal to 1:22. This implies that the e¤ect on output for the

unit monetary policy shock has declined by almost 18% in the second half of the

sample.18 Values vary for the horizon considered. For the 12-period sample, as an

example, this value rises to approximately 1:70, or a 42% reduction in the e¤ect on

output for the unit monetary policy shock. Although lower over the medium-term,

the impact elasticity remains the same. These changes are larger for the short-run,

and seem to decline over the medium-to-long-run; a result that seems to converge

with the …ndings of Pancrazi (2014) who …nds little evidence of these changes in

the medium-term.

Horizon (j) Impact Elasticity Percentage

Pre-1979 Post-1980 MP Reduction

j= 4 0:2350 0:0449 5:2337 80:8932

j= 8 0:2654 0:0945 2:8083 64:3919

j= 12 0:2720 0:1323 2:0559 51:3606

j= 16 0:2740 0:1603 1:7089 41:4838

j= 20 0:2746 0:1807 1:5194 34:1865

j= 24 0:2747 0:1953 1:4067 28:9104

j= 28 0:2748 0:2056 1:3364 25:1717

j= 32 0:2748 0:2128 1:2916 22:5754

j= 36 0:2748 0:2176 1:2629 20:8190

j= 40 0:2748 0:2208 1:2446 19:6551

j= 44 0:2748 0:2229 1:2332 18:9097

j= 48 0:2748 0:2241 1:2262 18:4467

j= 52 0:2748 0:2249 1:2221 18:1708

j= 56 0:2748 0:2253 1:2197 18:0095

j= 60 0:2748 0:2256 1:2184 17:9232

j= 64 0:2748 0:2257 1:2177 17:8803

Table 7: Impact Elasticity Note: This table outlines the impact elasticity of monetary policy

based on equation (25), and by comparing the period before and after the 1980s.

We present an alternative explanation for the decline in impact-elasticity. We

argue that the fall in the share of real balances and a decrease in elasticity of

18 See j= 48 in Table 7.

24

substitution between consumption and real balances a¤ect the key parameters that

determine the degree of monetary neutrality, as shown in the theoretical model.

Due to …nancial innovation, or the availability of alternative sources of payments,

the share of real balances in utility falls as households have a lower reliance on this

particular aggregate. Because households now hold fewer real balances, the degree

of substitutability for those lower levels of real balances falls. For the lesser share

of real balances households now hold, they become less inclined to substitute them.

Since these variables enter the IS equation, changes in these parameters may a¤ect

linkages between interest rates on output.

We calculate the e¤ect on output to changes in interest rate using the theoretical

model. This can be summarized from the Dynamic IS relationship presented in Eq.

16.

m !

Pre-1979

Benchmark 0:279 7:5351 0:0056 0:229 0:0763

Switch In‡ation ()0:2858 8:8172 0:0049 0:2354 0:0784

Switch Weight ()0:31047:5351 0:61060:31040:8105

Switch Substitution (v)0:7973 1:3365 0:016 0:747 0:249

Combined

Weight & Substitution 0:1555 1:3365 0:0031 0:1476 0:0492

Post-1980

Benchmark 0:1562 1:5639 0:0027 0:1488 0:0496

Switch In‡ation ()0:1555 1:3365 0:0031 0:1476 0:0492

Switch Weight ()0:8008 1:5639 0:0138 0:7542 0:2514

Switch Substitution (v)0:31048:8172 0:51060:21040:8105

Combined

Weight & Substitution 0:2858 8:8172 0:0049 0:2354 0:0784

Table 8: Measure of Monetary Neutrality: Counterfactual Experiments Note: This

table outlines the key parameters of the model which underlie monetary neutralities.

Table 8 presents values of m,,, and the value of =!(1)

(1)+'+, which

measures the degree of monetary neutrality implied by the model. It is immedi-

ately clear, comparing the values of pre1979 and post1980, that the transmission

mechanism has changed. Indeed, MP =MP ;pre1979

MP ;post1980 , is estimated to be around 1:58,

lying within the intervals for the VAR at di¤erent horizons, and roughly matching

the average impact-elasticity of monetary policy found earlier (1:7004).

The framework suggests that the changes in the utility function, perhaps due

to …nancial innovation, may not only explain changes in the money demand rela-

tionships and the welfare cost of in‡ation but also a large part of the decline in

monetary policy e¤ectiveness.19

19 While there may be other changes that may explain changes in monetary policy, such as

…nancial dislocations, the saving glut, …nancial globalization and the “dilemma”, among many

others, the paper adds to this list by presenting another explanation for the decline in monetary

policy potency.

25

6 Conclusion

This paper empirically documents and assesses the causes and consequences of the

evolving relationship between interest rates and money. Using a CES MIUF spec-

i…cation, we show that the interest semi-elasticity of money demand is a function

of the household’s preferences to hold real balances and substitute consumption

and real balances, and trend in‡ation. Our results give rise to a general micro-

founded expression for the welfare cost of in‡ation. Our time-varying estimates

based on quarterly U.S. data revealed that there was a gradual fall in the interest

semi-elasticity of money demand and the welfare cost of in‡ation during the period

spanning 1959 to 2006. The interest elasticity of money demand fell by approxi-

mately one-third during the 1970s due to high trend in‡ation, and further fell during

the 1980s due to the changing household preferences that emerged in response to

…nancial innovation. These developments substantially reduced the welfare cost of

in‡ation. We further showed that the changes in the household’s preferences ex-

plained a large part of the decline in the monetary policy e¤ectiveness that was

observed in the post-1980 era.

This paper adds to the …ndings of previous studies in several ways. Our micro-

founded interpretation of the interest semi-elasticity of money demand and the

welfare cost of in‡ation generates clear insights into the structural factors that

underpinned the changes observed in the periods of interest. Finally, the results

indicate that households do not separate their preferences with regards to con-

sumption and real money, and that trend in‡ation, the preference for the present

(discount factor), and this nonseparability preference play a similar role. The more

trend in‡ation or the nonseparability coe¢ cient increases, or the more the discount

factor decreases, the more monetary neutrality increases. Consequently, as money

supply equals its demand at each point of time, monetary neutrality in‡uences two

distinct central bank tools: interest rate decisions and money supply. Monetary

neutrality requires high durable in‡ation, decreased preference for the present, and

an increased household’s preference to substitute money holdings and consumption.

To manage monetary neutrality, the central bank has to decrease trend in‡ation to

reach its in‡ation target in the long run— and being credible— and to change house-

hold’s preferences to prefer the present and substitute less between consumption

and money holdings.

This policy recommendation is twofold. First, the central bank has to concretely

act against high trend in‡ation through conventional or unconventional monetary

policy decisions. Second, the central bank has to in‡uence household preferences

through communication. Doing so, the central bank will manage monetary neutral-

ity in order to avoid instability, increase its credibility, and reinforce its tools.

References

Alvarez, F., Lippi, F., 2009. Financial innovation and the transactions demand for

cash. Econometrica 77 (2), 363–402.

26

Amano, R., Ambler, S., Rebei, N., 2007. The macroeconomic e¤ects of nonzero

trend in‡ation. Journal of Money, Credit and Banking 39 (7), 1821–1838.

Andrés, J., López-Salido, J. D., Vallés, J., 2002. Intertemporal substitution and

the liquidity e¤ect in a sticky price model. European Economic Review 46 (8),

1399–1421.

Ascari, G., Ropele, T., 2007. Optimal monetary policy under low trend in‡ation.

Journal of Monetary Economics 54 (8), 2568–2583.

Ascari, G., Ropele, T., 2009. Trend in‡ation, Taylor principle, and indeterminacy.

Journal of Money, Credit and Banking 41 (8), 1557–1584.

Ascari, G., Sbordone, A. M., 2014. The macroeconomics of trend in‡ation. Journal

of Economic Literature 52 (3), 679–739.

Attanasio, O. P., Guiso, L., Jappelli, T., 2002. The Demand for Money, Financial

Innovation, and the Welfare Cost of In‡ation: An Analysis with Household Data.

Journal of Political Economy 110 (2), 317–351.

Bailey, M. J., 1956. The welfare cost of in‡ationary …nance. Journal of Political

Economy 64 (2), 93–110.

Barnett, W. A., 1978. The user cost of money. Economics Letters 1 (2), 145–149.

Barnett, W. A., Chauvet, M., 2011. How better monetary statistics could have

signaled the …nancial crisis. Journal of Econometrics 161 (1), 6–23.

Barnett, W. A., O¤enbacher, E. K., Spindt, P. A., 1984. The new Divisia monetary

aggregates. Journal of Political Economy 92 (6), 1049–1085.

Belongia, M., Binner, J., 2000. Divisia monetary aggregates: theory and practice.

New York, NY: Palgrave Macmillan.

Belongia, M. T., 1996. Measurement matters: recent results from monetary eco-

nomics reexamined. Journal of Political Economy 104 (5), 1065–1083.

Belongia, M. T., Ireland, P. N., 2019. The demand for Divisia money: theory and

evidence. Journal of Macroeconomics 61 (A), 103–128.

Benchimol, J., 2014. Risk aversion in the Eurozone. Research in Economics 68 (1),

39–56.

Benchimol, J., 2016. Money and monetary policy in Israel during the last decade.

Journal of Policy Modeling 38 (1), 103–124.

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.

Benchimol, J., Fourçans, A., 2017. Money and monetary policy in the Eurozone:

an empirical analysis during crises. Macroeconomic Dynamics 21 (3), 677–707.

27

Berentsen, A., Huber, S., Marchesiani, A., 2015. Financial innovations, money de-

mand, and the welfare cost of in‡ation. Journal of Money, Credit and Banking

47 (S2), 223–261.

Boivin, J., 2006. Has U.S. monetary policy changed? Evidence from drifting coef-

…cients and real-time data. Journal of Money, Credit and Banking 38 (5), 1149–

1173.

Boivin, J., Giannoni, M. P., 2002. Assessing changes in the monetary transmission

mechanism: a VAR approach. Economic Policy Review, 97–111.

Broaddus, A., Goodfriend, M., 1984. Base drift and the longer run growth of M1 :

experience from a decade of monetary targeting. Economic Review, 3–14.

Cagan, P., 1956. The monetary dynamics of hyperin‡ation. In: Friedman, Mil-

ton (Ed.), Studies in the Quantity Theory of Money. Chicago, IL: University of

Chicago Press, pp. 25–117.

Campbell, J. R., Hercowitz, Z., 2009. Welfare implications of the transition to high

household debt. Journal of Monetary Economics 56 (1), 1–16.

Castelnuovo, E., 2012. Estimating the evolution of money’s role in the U.S. mone-

tary business cycle. Journal of Money, Credit and Banking 44 (1), 23–52.

Chen, Y.-T., 2006. Non-nested tests for competing US narrow money demand func-

tions. Economic Modelling 23 (2), 339–363.

Christiano, L. J., Eichenbaum, M., Evans, C., 1996. The e¤ects of monetary policy

shocks: evidence from the ‡ow of funds. Review of Economics and Statistics

78 (1), 16–34.

Cooley, T. F., Hansen, G. D., 1989. The in‡ation tax in a real business cycle model.

American Economic Review 79 (4), 733–748.

Cynamon, B. Z., Dutkowsky, D. H., Jones, B. E., 2006. Rede…ning the monetary

agggregates: a clean sweep. Eastern Economic Journal 32 (4), 661–672.

Diewert, E., 2013. Irving Fisher and index number theory. Journal of the History

of Economic Thought 35 (02), 199–232.

Dotsey, M., Ireland, P., 1996. The welfare cost of in‡ation in general equilibrium.

Journal of Monetary Economics 37 (1), 29–47.

Dynan, K. E., Elmendorf, D. W., Sichel, D. E., 2006. Can …nancial innovation

help to explain the reduced volatility of economic activity? Journal of Monetary

Economics 53 (1), 123–150.

Feenstra, R. C., 1986. Functional equivalence between liquidity costs and the utility

of money. Journal of Monetary Economics 17 (2), 271–291.

28

Finn, M. G., Ho¤man, D. L., Schlagenhauf, D. E., 1990. Intertemporal asset-pricing

relationships in barter and monetary economies An empirical analysis. Journal

of Monetary Economics 25 (3), 431–451.

Fischer, S., 1981. Towards an understanding of the costs of in‡ation: II. Carnegie-

Rochester Conference Series on Public Policy 15 (1), 5–41.

Friedman, B. M., 1999. The future of monetary policy: the central bank as an army

with only a signal corps? International Finance 2 (3), 321–338.

Friedman, B. M., Kuttner, K. N., 1992. Money, income, prices, and interest rates.

American Economic Review 82 (3), 472–492.

Friedman, M., 1969. Optimum quantity of money. Chicago, IL: Aldine Publishing

Co.

Galí, J., 2015. Monetary policy, in‡ation and the business cycle: an introduction to

the New Keynesian framework, 2nd Edition. Princeton, NJ: Princeton University

Press.

Gertler, M., Lown, C. S., 1999. The information in the high-yield bond spread for

the business cycle: evidence and some implications. Oxford Review of Economic

Policy 15 (3), 132–150.

Greenwood, J., Hercowitz, Z., Hu¤man, G. W., 1988. Investment, Capacity Utiliza-

tion, and the Real Business Cycle. American Economic Review 78 (3), 402–417.

Hafer, R. W., Jansen, D. W., 1991. The demand for money in the United States:

evidence from cointegration tests. Journal of Money, Credit and Banking 23 (2),

155–168.

Hall, S. G., Hondroyiannis, G., Swamy, P., Tavlas, G. S., 2009. Assessing the causal

relationship between euro-area money and prices in a time-varying environment.

Economic Modelling 26 (4), 760–766.

Hamilton, J. D., 1994. Time series analysis. Vol. 2. Princeton, NJ: Princeton Uni-

versity Press.

Hendershott, P. H., 1992. The market for home mortgage credit: recent changes

and future prospects. In: Alton, R. (Ed.), The Changing Market in Financial

Services. Springer, Dordrecht, Ch. 3, pp. 99–123.

Holman, J. A., 1998. GMM estimation of a money-in-the-utility-function model: the

implications of functional forms. Journal of Money, Credit and Banking 30 (4),

679–698.

Hornstein, A., Wolman, A. L., 2005. Trend in‡ation, …rm-speci…c capital, and sticky

prices. Economic Quarterly 91 (4), 57–83.

Inagaki, K., 2009. Estimating the interest rate semi-elasticity of the demand for

money in low interest rate environments. Economic Modelling 26 (1), 147–154.

29

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

Credit and Banking 36 (6), 969–983.

Ireland, P. N., 2009. On the welfare cost of in‡ation and the recent behavior of

money demand. American Economic Review 99 (3), 1040–1052.

Jawadi, F., Sousa, R. M., 2013. Money demand in the euro area, the US and the

UK: Assessing the role of nonlinearity. Economic Modelling 32, 507–515.

Jermann, U., Quadrini, V., 2006. Financial innovations and macroeconomic volatil-

ity. NBER Working Papers 12308, National Bureau of Economic Research.

Johansen, S., 1988. Statistical analysis of cointegration vectors. Journal of Economic

Dynamics and Control 12 (2-3), 231–254.

Judd, J. P., Scadding, J. L., 1982. The search for a stable money demand function:

a survey of the post-1973 literature. Journal of Economic Literature 20 (3), 993–

1023.

Justiniano, A., Primiceri, G. E., 2008. The time-varying volatility of macroeconomic

‡uctuations. American Economic Review 98 (3), 604–41.

Khan, M. S., 1974. The stability of the demand-for-money function in the United

States 1901-1965. Journal of Political Economy 82 (6), 1205–1219.

Kiley, M. T., 2007. Is moderate-to-high in‡ation inherently unstable? International

Journal of Central Banking 3 (2), 173–201.

King, M. A., 1999. Challenges for monetary policy : new and old. Proceedings -

Economic Policy Symposium - Jackson Hole, 11–57.

Koenig, E. F., 1990. Real money balances and the timing of consumption: an

empirical investigation. Quarterly Journal of Economics 105 (2), 399–425.

Laumas, P. S., Laumas, G. S., 1969. Interest-elasticity of demand for money. South-

ern Economic Journal 36 (1), 90–93.

Lucas, Robert E., J., 1981. Discussion of : Stanley Fischer, "towards an under-

standing of the costs of in‡ation: II". Carnegie-Rochester Conference Series on

Public Policy 15 (1), 43–52.

Lucas, Robert E., J., 1988. Money demand in the United States: a quantitative

review. Carnegie-Rochester Conference Series on Public Policy 29 (1), 137–167.

Lucas, Robert E., J., 1996. Nobel lecture: monetary neutrality. Journal of Political

Economy 104 (4), 661–682.

Lucas, Robert E., J., 2000. In‡ation and welfare. Econometrica 68 (2), 247–274.

Lucas, Robert E., J., Nicolini, J. P., 2015. On the stability of money demand.

Journal of Monetary Economics 73 (C), 48–65.

30

Lütkepohl, H., 1993. The sources of the US money demand instability. Empirical

Economics 18 (4), 729–743.

Mallick, S. K., Mohsin, M., 2010. On the real e¤ects of in‡ation in open economies:

theory and empirics. Empirical Economics 39 (3), 643–673.

Mallick, S. K., Mohsin, M., 2016. Macroeconomic e¤ects of in‡ationary shocks with

durable and non-durable consumption. Open Economies Review 27 (5), 895–921.

McCarthy, J., Peach, R., 2002. Monetary policy transmission to residential invest-

ment. Economic Policy Review, 139–158.

Meltzer, A. H., 1963. The demand for money: the evidence from the time series.

Journal of Political Economy 71 (3), 219–246.

Miller, S. M., 1991. Monetary dynamics: an application of cointegration and error-

correction modeling. Journal of Money, Credit and Banking 23 (2), 139–154.

Miller, S. M., Martins, L. F., Gupta, R., 2019. A time-varying approach of the US

welfare cost of in‡ation. Macroeconomic Dynamics 23 (02), 775–797.

Newey, W. K., West, K. D., 1987. A simple, positive semi-de…nite, heteroskedastic-

ity and autocorrelation consistent covariance matrix. Econometrica 55 (3), 703–

708.

Pancrazi, R., 2014. How bene…cial was the Great Moderation after all? Journal of

Economic Dynamics and Control 46 (C), 73–90.

Pancrazi, R., Vukotic, M., 2019. In‡ation sensitivity to monetary policy: what

has changed since the early 1980s? Oxford Bulletin of Economics and Statistics

81 (2), 412–436.

Peek, J., Wilcox, J. A., 2006. Housing, credit constraints, and macro stability:

the secondary mortgage market and reduced cyclicality of residential investment.

American Economic Review 96 (2), 135–140.

Poterba, J. M., Rotemberg, J. J., 1987. Money in the utility function: an empirical

implementation. In: Barnett, W. A., Singleton, K. J. (Eds.), New Approaches to

Monetary Economics. Cambridge, UK: Cambridge University Press, pp. 219–240.

Qureshi, I., 2016. The role of money in Federal Reserve Policy. The Warwick Eco-

nomics Research Paper Series (TWERPS) 1133, University of Warwick, Depart-

ment of Economics.

Qureshi, I., 2018. Money aggregates and determinacy : a reinterpretation of mone-

tary policy during the Great In‡ation. The Warwick Economics Research Paper

Series (TWERPS) 1156, University of Warwick, Department of Economics.

Reynard, S., 2004. Financial market participation and the apparent instability of

money demand. Journal of Monetary Economics 51 (6), 1297–1317.

31

Sidrauski, M., 1967. Rational choice and patterns of growth in a monetary economy.

American Economic Review 57 (2), 534–544.

Stock, J. H., Watson, M. W., 1993. A simple estimator of cointegrating vectors in

higher order integrated systems. Econometrica 61 (4), 783–820.

Stock, J. H., Watson, M. W., 2003. Has the business cycle changed and why?

In: NBER Macroeconomics Annual 2002, Volume 17. NBER Chapters. National

Bureau of Economic Research, pp. 159–230.

Tesfatsion, L., Veitch, J. M., 1990. U.S. money demand instability A ‡exible least

squares approach. Journal of Economic Dynamics and Control 14 (1), 151–173.

Tobin, J., 1956. The interest-elasticity of transactions demand for cash. Review of

Economics and Statistics 38 (3), 241–247.

Vernon, J., 1977. Money demand interest elasticity and monetary policy e¤ective-

ness. Journal of Monetary Economics 3 (2), 179–190.

Woodford, M., 2000. Monetary policy in a World without money. International

Finance 3 (2), 229–260.

Woodford, M., 2003. Interest and prices: foundations of a theory of monetary policy.

Princeton, NJ: Princeton University Press.

Woodford, M., 2008. How important is money in the conduct of monetary policy ?

Journal of Money, Credit and Banking 40 (8), 1561–1598.

32

Appendix

A Derivations

A.1 Money demand

Taking Eq. 7 in logs yield

mt=1

vln (1 exp (it)) + 1

vln

1(27)

By expanding the …rst term on the LHS we obtain:

mt=1

vln (1 Q) + Q

ln (1 Q)(it) + 1

vln

1(28)

where exp (i) = Qis the steady-state bond price at maturity.

Imposing the steady-state relationship, Q= , leads to:

mt=

v( )it+1

v

v( )ln

+ ln

1 (29)

which is the expression found in Section 2.3.

A.2 Labor Supply

We proceed with deriving the labor schedule in log-deviations from steady-state:

wtpt=ct+'nt+ (v) (ctxt)(30)

To eliminate xt, we …rst derive it using the composite consumption-real money

balances index:

Xt="(1 )C1v

t+Mt

Pt1v#1

1v

(31)

A …rst-order Taylor approximation of Xtaround the steady-state leads to:

xt=(1 )C1v

(1 )C1v+Mt

Pt1vct+

Mt

Pt1v

(1 )C1v+Mt

Pt1v(mtpt)(32)

Plugging this into the labor supply schedule:

wtpt=ct+'nt+ (v)0

B

B

@

ct(1)C1v

(1)C1v+Mt

Pt1vct

+Mt

Pt1v

(1)C1v+Mt

Pt1v(mtpt)1

C

C

A(33)

which can be simpli…ed to obtain:

wtpt=ct+'nt+(v) (ct(mtpt)) (34)

33

where =Mt

Pt1v

(1)C1v+Mt

Pt1v=1v

m

1+1v

mand hence:

=m

1

v

m+m

(35)

Eq. 11 shows:

M

P C =

( ) (1 )1

v

=m(36)

Combining Eq. 35 and Eq. 36, we obtain the following expression:

=( )m

+ ( )m

(37)

Finally, using the money-demand curve, we obtain:

wtpt=ct+'nt+!it(38)

where !=(v).

A.3 Dynamic IS

The Euler equation is log-linearized to obtain:

ct=Et[ct+1]1

(itEt[t+1](v)Et[ct+1]xt+1(ctxt)) (39)

Again, eliminating xtwe get the following expression:

ct=Et[ct+1]1

(itEt[t+1](v)Etct+1 ct[(mt+1 pt+1)(mtpt)])

(40)

As before, ct+1 ct[(mt+1 pt+1)(mtpt)] is eliminated using the money

demand function and imposing the market clearing condition yt=ct:

yt=Et[yt+1]1

(itEtt+1 !Et[it+1]) (41)

A.4 E¤ects of Policy Shocks

To obtain Eq. 17, the production function is log linearized to obtain:

nt=1

1(ytat)(42)

Labor market equilibrium is needed to obtain Eq. 17. Log-linearizing the labor

demand equation:

atnt=wtpt(43)

which, in combination with the labor supply schedule, gives rise to the following

equilibrium condition:

yt+'nt+!it=atnt(44)

Plugging in the Eq. 42 to substitute out ntyields the Eq. 17 where =!(1)

(1)+'+

captures the elasticity of output with respect to interest rates.

is a function of trend in‡ation, the elasticity of substitution and the share of

real balances since these terms enter the convolution in !.

34

B Data summary

Table 9 presents the data used in our empirical exercises.

Variable Data Time-period Source

Interest Rates U.S. Three month Treasury bill rate 1959:I –2008:II FRED

Money-income

ratio

Divide the Cynamon et al. (2006)

sweep adjusted M1 money stock the

M1RS aggregate, by nominal GDP

1959:I –2008:II FRED

Table 9: Data summary Note: FRED stands for the Federal Reserve Economic Data, Federal

Reserve Bank of St. Louis.

C DOLS Estimates of the Split-Sample Estima-

tion

Table 10 presents DOLS and OLS estimates of the split-sample money demand

equation estimation considered in Section 4.2.

Pre-1979 Post-1980

mt=it

SOLS 1:2255 7:5351 1:8920 1:5640

DOLS, p = 1 1:1971 8:1235 1:8226 1:6317

DOLS, p = 2 1:1721 8:6419 1:8201 1:6562

DOLS, p = 3 1:1232 9:6221 1:8215 1:6018

DOLS, p = 4 1:0854 10:335 1:8234 1:5337

Table 10: Robustness of Split-Sample Estimate. Note: This table outlines estimates of

the money demand curve using both SOLS and DOLS estimates.

35