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Time-varying Money Demand and Real Balance Effects

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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 inflation. 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 inflation. 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 financial innovation. These developments led to a reduction in the welfare cost of inflation that subsequently explains the rise in monetary neutrality observed in the data.
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
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
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 ination, 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 conrm 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 inuenced 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 households
preferences that were observed may explain a large portion of the decline in the
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 ination. 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 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
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 Xtreects 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 dened 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=+ctit(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 ination, , 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 ination 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 ination. 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 dene 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)ei (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 derent 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 Co 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 ination 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 ination,
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
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 conrm 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 ination
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 ination 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 ination 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 Ination 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
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 ination 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.
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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 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:
atnt=wtpt(43)
which, in combination with the labor supply schedule, gives rise to the following
equilibrium condition:
yt+'nt+!it=atnt(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
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