Download On Income Velocity of Money, Precautionary Money Demand and

On Income Velocity of Money, Precautionary Money
Demand and Growth.
Jana Hromcová∗†
June 15, 2006
Abstract
A stochastic growth model with money introduced via a cash-in-advance constraint is used to analyze the behavior of the income velocity of real monetary
balances and money demand. Agents can purchase consumption goods only using
government issued money. The cash-in-advance constraint may become nonbinding
because of the uncertainty about the realization of the state of the economy. We
find that the precautionary money demand may introduce significant changes into
the volatility of the income velocity if it happens almost always. Its presence can
also alter the relationship between the average growth rate of money supply and
the average growth rate of the economy.
Keywords: Cash-in-advance; Income velocity; Precautionary money demand; Money
and Growth
JEL Classification: E40
∗
Universitat de Girona, Departament d’Economia i Empresa, Campus de Montilivi, 17071 Girona,
Spain; tel: +34 - 972418236; fax: +34 - 972418032; email: [email protected]
†
This paper has benefited from the suggestions, comments and discussions with Jordi Caballé. I am
grateful to Cruz Ángel Echevarrı́a, Josefina Monteagudo, Carmelo Rodrı́guez Álvarez, Hugo Rodrı́guez
Mendizábal, Laura Romero and the participants of SED Meeting 2000, Macroeconomic workshop at Universitat Autònoma de Barcelona, Universitat de Alicante and Universitat de Girona for their comments.
This research was undertaken with support of the European Commission’s Phare ACE Program 1995 and
the Spanish Ministry of Education and Science SEJ2004-03276. Financial support from IVIE is gratefuly
acknowledged.
1
1
Introduction
In this paper we study the conditions under which a precautionary money demand occurs
and changes in the behavior of the income velocity of money that it influences. Qualitative
analysis of this effect was performed in Svensson (1985) and Hromcová (1998), in a model
without and with capital, respectively. Here we evaluate the changes quantitatively in a
model with capital and more sources of uncertainty than the previous literature.
There are several effects that may contribute to the variability of the income velocity
of money. One is the reaction of real balances to the changes in the nominal interest
rates.1 When the nominal interest rate increases, it is less attractive to hold money and
agents look for some alternative means of payment. This fact is captured in the literature
by allowing for ’cash’ and ’credit’ goods, as for example in Lucas (1984), Lucas and
Stokey (1983, 1987) and Jones and Manuelli (1995).2 In such cash-credit models agents
have a possibility of substituting between two ways of acquiring consumption goods.
Therefore, when the opportunity cost of holding money increases, agents can switch from
cash consumption towards credit consumption. This affects the real balances, and the
income velocity of money varies accordingly.
Other types of models introduce changes into the money demand exogenously, via a
modified cash-in-advance constraint, see for example Canzoneri and Dellas (1998), Collard, Dellas and Ertz (2000) or Caballé and Hromcová (2001). In these models agents
are allowed to exchange a fraction of the current period income for consumption without
using money. When this fraction increases, individuals economize on their real balances
holdings and this implies a change in the money velocity.
As Lucas (1988) shows, standard cash-in-advance models can explain systematic changes
in velocity due to income and interest rates, but not due to other factors. Trends in velocity are attributed to financial innovations and technological improvements in the financial
sector and to the fact that the monetary policy influences the decisions of agents to devote
resources to the creation of money substitutes (see for example Cole and Ohanian (2002)
or Gordon, Leeper and Zha (1998) for surveys on papers which deal with these topics).
Another approach to model variable income velocity of money can be found in Svensson
(1985). Letting the goods market open before the financial exchange takes place, agents
are forced to make their decisions on the money holdings before the state of the economy
is known. Such information structure may lead to a precautionary money demand in some
states of the world, since agents may end up holding more money than they actually need
in order to purchase consumption goods. In this way their cash-in-advance constraint
may become nonbinding. Therefore, uncertainty might be a source of fluctuations of the
money velocity. Calibrating the model of Svensson (1985) one finds that in states of
the world when cash-in-advance constraint becomes nonbinding velocity varies but the
1
Plotting velocity and interest rates, as done for example in McGrattan (1998a), it can be seen that
these two variables exhibit a similar pattern.
2
In the words of Lucas (1984), ’cash’ goods are consumed using money and ’credit’ goods are exchanged
directly for agents savings or are acquired issuing private securities.
2
quantitative changes are negligible.
Hodrick, Kocherlakota and Lucas (1991) combine the approach of Lucas (1984) and
Svensson (1985). They consider a cash-credit good model that under a certain parameter
choice reduces to a cash-only good model, a version of the model of Svensson (1985) with
two kinds of shocks. They conclude that the cash-only model exhibits small variability.
In the cash-credit model velocity varies because agents substitute between cash and credit
goods, and the cash-in-advance constraint for the cash good almost always binds.
This paper extends the analysis done in Hromcová (1998) to a model with two kinds
of shocks and evaluates quantitatively the effect of precautionary money demand on the
equilibrium of the economy. Similarly to Hodrick, Kocherlakota and Lucas (1991), we
calculate some sample statistics (in order to find the best calibration for the parameters of
the model), but otherwise the focus of the present work is quite different. We concentrate
on determining the circumstances in which the cash-in-advance constraint switches from
binding to nonbinding due to the state of the world and on evaluating quantitative changes
precautionary money demand introduces into the properties of the income velocity. There
is only one kind of consumption good in the model, cash good. Nevertheless, the model
is capable of reproducing the variability of the money velocity found in the data, because
capital acts like credit good. Average levels of velocity similar to the ones found in the
data can be generated. We also find that the money-growth relationship is affected by
the precautionary money demand.
The model does not admit a closed form solution and has to be solved numerically.
We apply the method of parameterized expectations described in Den Haan and Marcet
(1990). The reason why we choose this technique is that the algorithm allows to work
with nonbinding constraints without any additional difficulty.
Our numerical exercise suggests the following properties of the model in equilibrium.
The existence of nonbinding constraints in equilibrium may increase the volatility of
the income velocity up to 30%. Average velocity may increase up to 3%. With the
liquidity constraint always binding, inflation has a slight positive effect on average growth
rate of the economy. If precautionary money demand appear and idle cash is used to
enhance the capital accumulation, average growth rate of the economy may increase with
the occurrence of more nonbinding constraints, i.e. with decreasing average growth rate
of money supply. That means that the money-growth relationship may reverse by the
presence of precautionary money demand in equilibrium.
The remainder of the paper is organized as follows. The model is described in section
2. In section 3 we describe the numerical algorithm. In section 4 we calibrate the model
to the US economy and compare some empirical and simulated statistics. In section 5 we
analyze the equilibrium behavior of the economy. Final conclusions are summarized in
section 6.
3
2
Model Description
2.1
The Household Problem
We consider a representative agent economy. There is one consumption good which must
be paid using currency. The economy has an infinitely lived representative household with
preferences represented by the utility function
!)
(∞
1−θ
X
c
−
1
j
(1)
Et
β j−t
1
−
θ
j=t
where cj is the consumption in period j and θ > 0 is the index of relative risk aversion.
The production and trade is characterized like in Lucas and Stokey (1983). Each
household is composed of a worker-shopper pair. The timing of the events is the following.
Agents enter the period t with a certain amount of monetary balances Mt and bonds Bt ,
carried over from the previous period, and the capital stock kt , which is the result of
the previous periods investment. A representative worker produces via the production
function
y t = At k t
(2)
where At is the shock to technology.3 We assume that the logarithm of technology shocks
follows an autoregressive process,
ln At+1 = (1 − ρA ) ln Ā + ρA ln At + εA,t+1 ,
(3)
where Ā > 0 is the steady state value of the technology, 0 < ρA < 1 and εA,t is white
noise, εA,t+1 ∼ N (0, σA2 ). Capital depreciates at a rate δ. The investment evolves as
it = kt+1 − (1 − δ)kt .
Prior to any trading agents learn the state of the economy At . The production takes
place and yt is realized. The worker stays at the place of production during the whole
period. Goods market opens and consumption takes place. The shopper visits various
stores to acquire consumption goods carrying the monetary balances of the household.
Purchases are subject to the liquidity constraint
ct ≤
Mt
.
pt
(4)
The part of the production which is not consumed is invested into capital. Next, the
monetary transfer Xt = (µt − 1) Mt from the government takes place. We denote µt as
the gross growth rate of money supply. We assume that the logarithm of monetary shocks
follows an autoregressive process,
ln µt+1 = (1 − ρµ ) ln µ̄ + ρµ ln µt + εµ,t+1 ,
3
(5)
To justify the constant return assumption we typically interpret the capital stock as a broad measure
that may include also human capital. The class of models that reduce to an AK model are discussed in
chapter 4 of Barro and Sala-i-Martin (1995). See also McGrattan (1998b) for the defence of AK models
in growth theory.
4
where µ̄ > 0 is the steady state value of the money supply growth rate, 0 < ρµ < 1 and
εµ,t is white noise, εµ,t+1 ∼ N (0, σµ2 ). At the end of the period asset market opens. Agents
can purchase one-period pure discount bonds paying Bt+1 units of money at period t that
pay Rt+1 Bt+1 units of money when they mature in t + 1. Therefore, Rt+1 is the gross
nominal interest rate between period t and t + 1. Bonds are in zero net supply.4 Monetary
balances Mt+1 to be carried over to the next period are chosen. The budget constraint
that agents are facing is thus
ct + kt+1 − (1 − δ)kt +
Mt Rt Bt Xt
Mt+1 Bt+1
+
≤ At k t +
+
+
.
pt
pt
pt
pt
pt
(6)
The real wealth of a household in period t is in the right hand side of (6) and consists
of the current period output, real monetary balances, real return on bonds and the real
lump sum transfer from the government. Current period wealth is used to consume, to
invest into capital, and to save in the form of money and bonds, as can be seen in the left
hand side of (6).
Let λt and ηt be the non-negative Lagrange multipliers associated with the budget
constraint (6) and the cash-in-advance constraints (4), respectively. Let
mt =
Mt
pt
be the real monetary balances. The equations that characterize the equilibrium are the
first order conditions
c−θ
= λt + ηt ,
t
λt mt = βEt [(λt+1 + ηt+1 ) mt+1 ] ,
λt = βEt [λt+1 (At+1 + 1 − δ)] ,
βRt+1
λt m t =
Et (λt+1 mt+1 )
µt
(7)
(8)
(9)
(10)
and the following transversality conditions:
lim Et β t+j λt+j kt+j+1 = 0,
j→∞
Bt+j+1
t+j
lim Et β λt+j
= 0.
j→∞
pt+j
(11)
(12)
The Lagrange multipliers associated with the budget and cash-in-advance constraints,
λt and ηt can be interpreted as the marginal utility of wealth and the marginal utility
of real balances, respectively. The first order condition on consumption (7) indicates
that the existence of binding liquidity constraint drives a wedge between the marginal
utility of wealth and the marginal utility of consumption since the wealth cannot be used
instantaneously to buy consumption. The first order condition on nominal balances (8)
is the equation for pricing money. Money is priced like other assets, once its return has
4
The role of bonds in this economy is to define the nominal interest rate.
5
been appropriately defined in terms of the liquidity services it provides. The first order
conditions on capital (9) and bonds (10) embody the costs and profits associated with
investing one marginal unit of wealth in capital and bonds, respectively.
The income velocity of money is defined as the ratio between real output and real
balances
yt
.
vt =
mt
Definition: Given the set of initial conditions M1 , k1 , B1 , the equilibrium consists of
stochastic processes {ct , Mt+1 , kt+1 , Bt+1 , Rt+1 , pt , At , µt }∞
t=1 such that
(a)
a
representative
household
chooses
the
stochastic
sequences
in
order
to
maximize
the
expected
discounted
utility
(1)
subject
{ct , Mt+1 , kt+1 , Bt+1 }∞
t=1
to the budget constraint (6) and the cash-in-advance constraint (4),
(b) markets for goods, money and bonds clear in every period
ct + kt+1 − (1 − δ)kt = At kt ,
Mt+1 = Mt + Xt = µt Mt ,
Bt+1 = 0.
3
3.1
(13)
(14)
(15)
Model Transformation and Numerical Solution
Model Transformation
We define ĉt as the ratio of consumption to capital, γt as the gross rate of growth of
capital, and m̂t as the ratio of real balances to capital
ĉt =
ct
kt+1
mt
, γt =
and m̂t =
.
kt
kt
kt
Then we define corresponding Lagrange multipliers as
λ̂t = λt ktθ and η̂t = ηt ktθ .
These new variables are stationary and in the absence of uncertainty have a steady state.
The system of equations that characterize the equilibrium can be rewritten in terms of
these variables, see Appendix.
6
3.2
Solution Method
We are interested in equilibria in which a precautionary money demand occurs. That
means that we will work with equilibria where the cash-in-advance constraint becomes
nonbinding in some states of the world. The method of parameterized expectations described in detail in Den Haan and Marcet (1990) will be applied. This technique is useful
in our case as there is no additional difficulty when working with nonbinding constraints.
The application of this numerical technique to our model is described in more details in
the Appendix.
4
Calibration and Model Performance
4.1
A Look at the Data
We will calibrate the model to match the quarterly US data.5 Suitable data is available
for the period 1964:03 to 2005:04. As discussed in Gavin and Kydland (1999) and Johnson
(1997) there is a break in the data around 1980. They find that the correlations of real
variables do not change much, but the statistics for nominal variables may change in
an important way. This break coincides with the end of an increasing trend in the M1
velocity. The change in the FED monetary policy is reflected in the period of stable
velocity which we pick for our analysis. After 1999 the income velocity of money behaves
differently again. One could plot the M1 velocity to inspect visually the mentioned facts.
We divide the data into three subperiods, 1964:03 to 1979:04, 1980:01 to 1998:04,
1999:01 to 2005:04, and observe how the statistics change within particular intervals. The
variables of interest are the output per worker, consumption per worker, prices, monetary
aggregates M1, growth rates of these variables, nominal interest rate and the income
velocity of M1. Several correlation coefficients which change around 1980 also change
around 2000. The ones that almost do not change are the correlations between the
money growth rate and all other variables, growth rate of output and all other variables,
nominal interest rate and inflation rate, output and consumption, and output and prices.
The correlation coefficients reported in Table 1 are calculated using Hodrick-Prescott
filtered data. The ones that change in a dramatic way are highlighted in bold. The
correlations between output and velocity, nominal interest rate and velocity, prices and
velocity seem to have a break around 1980 and also around 2000. Correlations between
consumption and velocity, money and prices, money and interest rate, and money and
velocity have a significant change around 1980.6 The volatility of the velocity also changes
between subperiods, with a break around 1980, σv1964:03 to 1979:04 = 0.05, σv1980:01 to 1998:04 =
0.25, σv1999:03 to 2005:04 = 0.19, being 0.19 in the whole sample. Taking this into account,
5
The data are taken from http://research.stlouisfed.org/fred2.
Gavin and Kydland (1999) perform a Wald test for stability and find statistically significant changes
in correlation coefficients between output and prices, output and money, and output and velocity around
1980 (they use a sample from 1959:01 to 1994:04). Johnson (1997) points out that correlations between
output and velocity, money and interest rate, and money and inflation are different before and after 1980
(the sample he uses ranges between 1972 and 1993).
6
7
together with the fact that the money velocity in our theoretical model is stationary we
choose to calibrate the model for the period 1980:01-1998:04.
4.2
Calibration
To assign parameters to our model we set some values as implied by the data: the steady
state gross growth rate of the economy to 1.004, steady state gross growth rate of money
supply (M1) to µ̄ = 1.014, discount factor β = 0.99, and the depreciation of capital to
δ = 0.08. These values imply capital-output ratio of 2.6. Given the risk aversion, all other
steady state levels can be determined endogenously from the equilibrium equations. We
will consider that the risk aversion parameter takes values in the interval θ ∈ [0.3, 3]. 7
Solow residuals are calculated using data for real GDP, taking quarterly variations in
the capital stock to be approximately zero, as in Cooley (1997). The correlation coefficient
ρA = 0.99 and the standard error of technology shocks σA = 0.008. For the monetary
shocks, the correlation coefficient is ρµ = 0.57 and the standard error is σµ = 0.013.
4.3
Model Performance
In this section we calculate and compare several sample statistics obtained from empirical
and simulated series. Means, correlation coefficients and standard errors of the variables
mentioned in section 4.1 are computed. Simulated series are obtained employing empirical
technology and monetary shocks.8 The relative risk aversion parameter takes values
θ = {0.3, 0.4, 0.5,...,2.8, 2.9, 3}. All other parameters take values as described in the
calibration section. In order to compare the simulated and empirical statistics, all series
are Hodrick-Prescott filtered, using the parameter λ = 1600 for the filter.9 Our model
is very simple and the main purpose of this exercise is to choose the most appropriate
value for the relative risk aversion. Similarly to Cooley and Hansen (1989) we want
to concentrate on the study of the properties of the model, rather than match all the
statistics.
When considering maximum and minimum values for the statistics obtained for the
given range of parameters, the sample statistics fall into the interval for 16 out of 27
statistics calculated for the model, see Table 2 (intervals comprising the empirical values are marked in bold). In general the model exhibits quite low average level for the
nominal interest rate. It fails to replicate enough variability in the nominal interest rate,
and predicts too much variability in the inflation rate. The correlations between velocity, nominal interest rate, inflation rate, growth rate of consumption (consumption) and
growth rate of output (output) are quite well explained, except for their correlations with
7
For example, for the relative risk aversion θ = 2.5, steady state level of the technology is Ā = 0.1,
steady state income velocity v̄ = 6.2, steady state nominal interest factor R̄ = 1.03. For θ = 0.85,
Ā = 0.094, v̄ = 6.6 and R̄ = 1.023.
8
Technology shocks are constructed using equation (3) employing estimated correlation coefficient
and errors, and the mean Ā that corresponds to the chosen value of the relative risk aversion. Monetary
shocks are actual money supply growth rates found in the data.
9
When applying the Hodrick-Prescott filter, simulated series are transformed into original variables
first.
8
the money growth rate (monetary aggregates). Even if correlation between the growth
rate of output and growth rate of money supply is negative in the model and positive
in the data, in both cases the two variables are practically uncorrelated. The model can
generate the average money velocity found in the data and displays reasonable variability
of the income velocity of money.
In this paper we are concerned with properties of money velocity. We want to choose
the calibration that yields to velocity volatility close to the one found in the data. Therefore we may set the relative risk aversion to θ = 0.85. However, it would be welcome to
explore the properties of the model for the relative risk aversion over unity.10 It comes
out that for θ > 2, velocity volatility keeps around 0.17. We thus choose θ = 2.5, because
for this value the average velocity coincides with the empirical one and the volatility is
still quite close to the empirical value. Nevertheless, throughout the analysis of the equilibrium behavior of the model we will use several values for relative risk aversion, taking
as benchmarks θ = 0.85 for θ < 1 and θ = 2.5 for θ > 1. Table 2 reports the statistics
reproduced by the model for these particular values of the relative risk aversion, again
marking in bold simulated statistics that fit the data. The model performs better for
θ = 2.5 than for θ = 0.85.
5
5.1
Discussion of the Equilibrium Results
Precautionary Money Demand
Money demand is given by equilibrium equations and it is a function of the current
monetary and technology shock, and the parameters of the economy. Take into account
equation (10). Imagine that the shocks always take their steady state value. Gross growth
rate of real balances and of Lagrange multiplier associated to the budget constraint will
be constant. We can see that when the steady state level of money growth rate increases,
the steady state value of the nominal interest rate must also increase. This means that
when µ̄ increases, R̄ increases and it is less attractive to hold money. Once agents have
the money, it is more attractive to spend it on consumption. Increasing the steady state
growth rate of money supply will result in a binding cash-in-advance constraint in more
states of the world. On the other hand, decreasing µ̄ will allow us to find nonbinding
liquidity constraints in more periods.
In order to understand the conditions under which agents make a precautionary money
demand we display the results graphically. In Figure 1 we plot the numerically calculated
border lines, the lines that separate the binding and nonbinding regions, for various values
of the relative risk aversion. Given the combination of the technology and monetary
shocks at t, (At , µt ), we can determine from Figure 1 if the agents keep some money in
their pockets after the consumption session was completed, or they spend it all. When
the money growth rate µt is sufficiently high, individuals never make the precautionary
10
Different studies arrive to different conclusions about the value for the relative risk aversion. For
example Canzoneri and Dellas (1998) propose a reasonable range between 1 and 4. Lucas (2003) argues
that it should be below 2.5 and the evidence leads many authors to use the value at or near 1 at
applications.
9
money demand because the opportunity cost of holding money is high, as argued in the
first paragraph of this section.
As known from previous research, Hromcová (2005), the slope of the border line alters
for θ < 1 and for θ > 1. For a given monetary policy and a low (high) technology shock we
are more likely to find nonbinding cash-in-advance constraint when θ > 1 (θ < 1). Agents
with high θ want to smooth their consumption path. When there is a positive technology
shock, a direct way of isolating the effect of the shock on consumption is to make the
cash-in-advance constraint binding, so that consumption is prevented from reacting too
much to output fluctuations. For θ = 1 the border line is flat, and if µt > β agents spend
all monetary balances previously accumulated, and if µt < β they make a precautionary
money demand.
Concerning the monetary shocks, the lower is the growth rate of money supply, the
more likely are the agents to leave their cash-in-advance constraint nonbinding. When
the money supply grows at a higher rate than the trend, prices increase at higher rate
than the trend, and the nominal interest rate increases too. Agents only carry the least
necessary monetary balances because inflation acts like tax on consumption. When the
growth rate of money supply is low enough, inflation tax diminishes and agents may leave
their liquidity constraint nonbinding. This behavior is not affected by the value of the
relative risk aversion.
Taking closer look at the scatter points in Figure 1 we can see that our model predicts the existence of nonbinding constraints in equilibrium for contractionary monetary
policies. Only for θ = 0.3 slightly positive growth rate of money supply may lead to
precautionary money demand.
5.2
Money and Growth Relationship
In our exercise we can also see some interesting properties in the relationship between
money and growth. Let us mention first the relationship between the average growth rate
of money supply and the average growth rate of the economy, and then we will have a
look at correlations between monetary aggregates and output.
In the absence of uncertainty money is neutral in our model. However, in the presence
of shocks we observe that when the cash-in-advance constraint is always binding, average
growth rate of the economy is positively related with the average growth rate of money
supply. The opposite happens when agents make a precautionary money demand in many
periods, average growth rate of output and µ̄ may become negatively related. Thus we
have a model with Tobin effect, that may reverse when nonbinding liquidity constraints
occur in equilibrium. To understand this result we recall what happens with idle cash
balances. When µ̄ is very low, there may appear nonbinding cash-in-advance constraints
in equilibrium and unspent balances are used for additional capital accumulation and
increase growth rate of output at the particular time period. When nonbinding constraints
appear in more states of the world, we may expect an increase in the average growth rate of
the economy. As the steady state growth rate of money supply increases, less nonbinding
constraints occur and the additional capital accumulation is disappearing. When µ̄ is
sufficiently high and no precautionary money demand is made in equilibrium, we end up
10
with a model in which money (inflation) has positive effect on the growth rate of output.
We must note that the average level of technology and money supply growth rate are
not related. We depict the described relationship between the average growth rate of
the economy and the average level of the money supply growth rate for various values of
relative risk aversion in the left row of Figure 2.
We also observe that the relationship between output per capita and monetary aggregates is changing under the presence of the precautionary money demand. More nonbinding constraints in equilibrium make the effect of monetary policy on output stronger.
The same happens when the cash-in-advance constraint is always binding and the average money supply growth is higher. Thus we observe U shape curves which relate the
correlation between output and money with average growth rate of money supply, see the
right row of Figure 2.11
5.3
Velocity Analysis
Income velocity of money vt is the ratio between real output and real monetary balances,
as defined in section 2.1. Notice that in terms of the transformed model we can write
vt =
At
.
m̂t
(16)
We want to evaluate the changes in the income velocity of money and its volatility due to
the fact that agents make a precautionary money demand. Our benchmark case will be an
economy with the lowest money growth rate under which the cash-in-advance constraint
is binding in all periods. Lowering µ̄ will lead to precautionary money demand in more
periods.
In most of the cases the volatility of the money velocity increases with increasing
number of nonbinding periods. For example for θ = 0.85 the precautionary money demand
may induce up to 18% increase in σv (when µ̄ = 0.99) with respect to the benchmark value
(µ̄θ=0.85
bench = 1.026). For θ = 2.5 the precautionary money demand may induce up to 30%
increase in σv (when µ̄ = 0.9885) with respect to the benchmark value (µ̄θ=2.5
bench = 1.04).
However, this behavior is not valid in general, as can be seen from Table 2. For low
relative risk aversions (under 0.8) we may find no such regularity between precautionary
money demand and the variability of the income velocity.
Average value of the money velocity increases as more nonbinding periods appear in
equilibrium in most of the analyzed cases. As the average level of technology does not
depend on the growth rate of money supply, it means that the ratio of monetary balances
to capital is lower on average when agents leave their liquidity constraint nonbinding
sometimes. This fact is consistent with the relationship between money and growth
commented above. As agents decide to demand relatively less monetary balances, they
consume relatively less (consumption to capital ratio will be lower, too), thus they invest
more into capital and enhance the average growth rate of the economy. For example
for θ = 2.5 the precautionary money demand may induce up to 3% increase in average
11
This exercise is performed under the assumption that the errors εµ,t and correlation coefficient ρµ in
the equation (5) are maintained constant, and the average growth rate of money supply µ̄ varies.
11
velocity with respect to the benchmark value. For θ = 0.85 we observe an increase in the
average velocity only when the agents leave their liquidity constraint nonbinding in all
periods.
These results for several risk aversion parameters are shown graphically in Figure 3
and numerically in Table 3.
6
Conclusion
We have used a simple stochastic growth model with money introduced via a cash-inadvance constraint to analyze the behavior of the money demand and the income velocity
of real balances. We have found combinations of technology and monetary shocks that
lead to precautionary money demand. Precautionary money demand happens mostly
because of a decrease in the inflation tax due to low money supply growth rate. In
the present setup and calibration, the effect of technology shocks discussed in Hromcová
(2005) is overcome by the effect of monetary shocks.
We have also found that in the model in which the inflation increases capital accumulation, when the cash-in-advance constraint is always binding, the effect reverses when the
precautionary money demand occurs in some states of the world. Concerning the volatility of the money velocity, the model can deliver empirically plausible velocity fluctuations
when the cash-in-advance constraint always binds. Given that there are only cash goods
in the economy, it implies that capital acts like credit good in Hodrick, Kocherlakota and
Lucas (1991). Precautionary money demand may account for an increase in the variability of the income velocity of money. The effect can be significant in times of an unusually
contractive monetary policy when the liquidity constraint becomes nonbinding in most
states of the world.
12
Appendix: Solution method
The system of equations that characterize the equilibrium using stationary variables
defined in section 3.1, are the first order conditions
ĉ−θ
= λ̂t + η̂t ,
t
i
βγt1−θ h
λ̂t m̂t =
Et λ̂t+1 + η̂t+1 m̂t+1 ,
µt
h
i
λ̂t = βγt−θ Et λ̂t+1 (At+1 + 1 − δ) ,
i
βγt1−θ Rt+1 h
Et λ̂t+1 m̂t+1 ,
λ̂t m̂t =
µt
(17)
(18)
(19)
(20)
the cash-in-advance constraint
η̂t ≥ 0 and η̂t (m̂t − ĉt ) = 0,
(21)
and the equilibria in goods, money and bonds markets
ĉt + γt − (1 − δ) = At ,
Mt+1 = µt Mt ,
Bt+1 = 0.
(22)
(23)
(24)
In order to solve the system (17)-(24) we parameterize expectations in the equations
(18)-(20). We will approximate the expectations by second degree exponenciated polynomials. In the transformed model we have two exogenous state variables, At and µt , which
have predictive power and are included in the parameterization of the expectations. To
proceed with the numerical algorithm ’good’ initial conditions are needed. We take as
initial conditions for the parameters the ones that correspond to the results obtained from
the log-linear approximation around steady state, a numerical method described in Uhlig
(1995).
13
References
[1] Barro, J. Robert and Sala-i-Martin, Xavier, 1995, Economic Growth, McGraw-Hill
[2] Caballé, Jordi and Hromcová, Jana, 2001, The Role of Central Bank Operating
Procedures in an Economy with Productive Government Spending, Working Paper
504.01, Departament d’Economia i d’Història Econòmica, Universitat Autónoma de
Barcelona
[3] Canzoneri, Matthew B. and Dellas, Harris, 1998, Real Interest and Central Bank
Operating Procedures, Journal of Monetary Economics, 42, 471-494
[4] Collard, Fabrice, Dellas, Harris and Ertz, Guy, 2000, Poole Revisited, CEPR, Discussion Paper 2521
[5] Cole, Harold L. and Ohanian, Lee E., 2002, Shrinking Money: The Demand for
Money and the Nonneutrality of Money, Journal of Monetary Economics, 49, 653-86
[6] Cooley, Thomas F., 1997, Calibrated Models, Oxford Economic Policy, 13, 55-69
[7] Cooley, Thomas F. and Hansen, Garry D., 1989, The Inflation Tax in Real Business
Cycle Model, The American Economic Review, 79, 733-748
[8] Den Haan, Wouter J. and Marcet, Albert, 1990, Solving the Stochastic Growth Model
by Parameterizing Expectations, Journal of Business and Economic Statistics, 8, 3134
[9] Gavin, William T., Kydland, Finn E., 1999, Endogenous Money Supply and the
Business Cycle, Review of Economic Dynamics, 2, 347-369
[10] Gordon, David B., Leeper Eric M. and Zha, Tao, 1998, Trends in Velocity and Policy
Expectations, Carnegie-Rochester Conference Series on Public Policy, 49, 265-304
[11] Hodrick, Robert J., Kocherlakota, Narayana and Lucas, Deborah, 1991, The Variability of Velocity in Cash-in-Advance Models, Journal of Political Economy, 99,
258-384
[12] Hromcová, Jana, 1998, A Note on Income Velocity of Money in a Cash-in-Advance
Economy with Capital, Economics Letters, 60, 91-96
[13] Hromcová, Jana, 2005, Precautionary Money Demand in a Cash-in-Advance Economy with Capital, Computational Economics, 26, 51-63
[14] Jones, Larry E. and Manuelli, Rodolfo E., 1995, Growth and the Effects of Inflation,
Journal of Economic Dynamics and Control, 19, 1405-1428
[15] Johnson, Christian A., 1997, Velocity and Money Demand in an Economy with Cash
and Credit Goods, Working Paper, Central Bank of Chile
14
[16] Lucas, Robert E., Jr., 1984, Money in a Theory of Finance, Carnegie-Rochester
Conference Series on Public Policy, 21, 9-46
[17] Lucas, Robert E., Jr., 1988, Money Demand in the United States: A Quantitative
Review, Carnegie-Rochester Conference Series on Public Policy, 29, 137-167
[18] Lucas, Robert E., Jr., 2003, Macroeconomic Priorities, American Economic Review,
93, 1-14
[19] Lucas, Robert E., Jr. and Stokey, Nancy L., 1983, Optimal Fiscal and Monetary
Policy in an Economy without Capital, Journal of Monetary Economics, 12, 55-93
[20] Lucas, Robert E., Jr. and Stokey, Nancy L., 1987, Money and Interest in a Cash-inAdvance Economy, Econometrica, 55, 491-513
[21] McGrattan, Ellen R., 1998a, Trends in Velocity and Policy Expectations, A Comment, Carnegie-Rochester Conference Series on Public Policy, 49, 305-316
[22] McGrattan, Ellen R., 1998b, A Defence of AK Growth Models, Federal Reserve Bank
of Minneapolis, Quarterly Review, Fall, 13-26
[23] Svensson, Lars E. O., 1985, Money and Asset Prices in a Cash-in-Advance Economy,
Journal of Political Economy, 93, 919-944
[24] Uhlig, Harald, 1995, A Toolkit for Analyzing Nonlinear Dynamic Stochastic Models
Easily, Federal Reserve Bank of Minneapolis, Institute of Empirical Macroeconomics
Discussion Paper 101
15
y
-
R
c
0.78
0.61 0.60
0.68
-
γc
-
-
p
−0.56
−0.30 −0.44
−0.50
−0.28
0.20 −0.04
−0.15
0.40
−0.25 0.05
0.08
0.30
0.72 0.31
0.36
y
R
v
γ
-
π
-
v
M
0.17
−0.54 −0.37
−0.27
0.13
−0.46 −0.13
−0.11
−0.04
0.82 0.51
0.39
−0.06
0.77 0.42
0.38
−0.42
−0.28 −0.17
−0.30
0.66
0.50 0.42
0.57
0.29
−0.28 0.01
0.00
−0.08
0.39 0.10
0.13
γ
-
µ
-
-
-
-
0.66
0.20 0.48
0.55
0.14
0.29 0.36
0.26
0.01
−0.64 −0.47
−0.36
0.47
−0.34 −0.20
−0.16
−0.41
−0.83 −0.92
−0.90
-
-
-
-
-
−0.20
−0.34 −0.13
−0.21
0.25
0.62 0.13
0.13
−0.04
0.52 0.34
0.26
−0.33
−0.50 −0.63
−0.50
−0.43
−0.47 −0.23
−0.28
0.10
0.18 0.07
0.09
−0.18
−0.44 −0.37
−0.30
Table 1: Correlation coefficients for several variables in the US data: y− output per
worker, c− consumption per worker, γc − gross growth rate of consumption per worker,
p− consumer price index, R− nominal interest rate (3-month certificate of deposits), v−
M1 velocity, γ− gross growth rate of output per worker, π− inflation factor (change in
consumer price index), µ− gross growth rate of M1 monetary aggregate. The superscript
and subscript respresent the correlation coefficients for the subperiods before and after
the chosen period, under the bar coefficients for the whole range of data are stated,
1964:03−1979:04
1999:01−2005:04 1980 : 01 − 1998 : 04. All data were detrended using the Hodrick-Prescott
1964:03−2005:05
filter.
16
Model
Statistics
E(v)
E(R)
E(π)
cor (γ, π)
cor (γ, R)
cor (γ, v)
cor (γ, µ)
cor (R, π)
cor (R, v)
cor (µ, v)
cor (µ, π)
cor (µ, R)
cor (γc , µ)
cor (γc , γ)
cor (y, p)
cor (y, R)
cor (y, v)
cor (y, M )
cor (R, p)
cor (M, v)
cor (M, p)
cor (M, R)
cor (c, M )
cor (c, y)
σv
σπ
σR
Data
θ ∈ [0.3, 3]
(6.0,14.6)
θ = 2.5
7.0
(1.018,1.032)
1.030
9.1
1.025
(1.01,1.02)
(-0.86,0.9)
(-0.49,-0.05)
(-0.32,0.58)
1.012
1.014
-0.85
-0.05
-0.32
-0.08
0.29
(-0.15,-0.07)
(-0.03,0.99)
(-0.70,0.98)
(-0.20,0.99)
(0.10,0.99)
(0.74,1)
(-0.55,-0.05)
-0.11
0.56
0.52
0.99
-0.26
0.93
0.34
-0.10
0.91
0.28
0.31
0.91
1
-0.52
-0.20
0.10
-0.15
0.82
0.06
-0.27
0.78
-0.04
0.02
-0.01
0.95
0.49
0.48
(-0.87,0.95)
(-0.55,0.65) -0.53
(-0.56,-0.17) -0.17
(-0.92,0.86) -0.91
(-0.17,0.02)
0.00
(-0.62,0.12)
(-0.20,0.07)
(0.31,0.96)
θ = 0.85
(-0.20,-0.18) -0.19
-0.20
(-0.09,0.12)
0.09
0.96
0.00
-0.37
0.17
0.02
0.009
0.23
(-0.78,0.97)
(0.08,2.7)
(0.01,0.11)
(0.005,0.01)
0.015
0.007
7.0
1.08
1.010
-0.13
-0.17
0.01
0.07
0.42
0.42
-0.23
-0.37
-0.63
0.36
0.48
-0.44
-0.04
0.05
0.13
0.31
-0.92
-0.47
-0.20
0.34
0.60
0.25
0.005
0.016
Table 2: The comparison of some sample statistics of the model with the data; v−
money velocity, R− nominal interest factor, π− inflation factor, γ− gross growth rate
of output per worker, µ− gross growth rate of money supply, γc − gross growth rate of
consumption per worker, c− consumption per worker, y− output per worker, p− price,
M − nominal monetary balances; E(x) denotes average value of x = v, R, π; σx means
the volatility of a variable x = v, R, π. We mark in bold simulated statistics that fit the
empirical ones.
17
NONB
[%]
∆σv
∆E(v)
[%]
[%]
θ
µ̄
0.3
1.016
1.014
1.01
1.008
1.006
1.004
1.002
1
0.996
0
2.5
10
27
40
54
73
89
100
0
-0.8
-2
-3
-4
-5
-7
-11
-34
0
-0.6
-1.8
-2
-2
-3
-4
-5
6
0.5
1.018
1.014
1.01
1.004
1.002
0.999
0.996
0.994
0.993
0
1.2
5
20
30
50
75
95
100
0
-0.3
-0.5
-0.5
-0.4
0.5
2
-6
-12
0
-0.3
-0.5
-0.4
-0.1
1
4
8
10
0.85
1.026
1.014
1.006
1.004
1
0.993
0.994
0.992
0.99
0
4
22
30
43
50
70
80
100
0
-1.3
-0.8
0.3
1.5
3
8
12
18
0
-0.7
-1
-1
-1.5
-2
-2
-2
-0.3
NONB
[%]
∆σv
∆E(v)
[%]
[%]
θ
µ̄
1.2
1.035
1.028
1.014
1.008
1.005
0.998
0.992
0.99
0.9895
0
1
9
18
32
50
77
97
100
0
0.4
8
18
25
56
106
145
160
0
0.1
3
5
5
6
7
10
10
2.5
1.04
1.02
1.014
1.008
1.004
0.998
0.992
0.99
0.989
0
5
10
26
35
50
65
75
82
0
0.3
1
4
6
12
18
27
30
0
-0.2
-0.2
0.5
0.5
0.6
0.8
1.4
2.5
3
1.045
1.022
1.014
1.01
1.008
1.002
0.996
0.992
0.989
0
4
10
20
27
40
51
63
70
0
-0.1
0.6
1
2
6
11
17
23
0
-0.4
-0.5
-0.6
-0.6
0
0.1
0.3
0.9
Table 3: Variations in the volatility of the income velocity and the average money
velocity as a function of the precautionary money demand; θ− relative risk aversion, µ̄−
steady state growth rate of money supply, NONB − fraction of periods in which the
liquidity constraint becomes nonbindig in equilibrium, ∆σv − increase in volatility of the
income velocity of money, ∆E(v)− increase in average money velocity (negative values
mean decrease in respective variables).
18
Figure 1: Lines that separate binding and nonbinding regions for various values of relative
risk aversion and empirical technology and monetary shocks’ combinations.
19
Figure 2: Average growth rate of the economy E(γ), left row, and correlation between
output per worker and monetary aggregates cor(y, M ), right row, as a function of average
growth rate of money supply µ̄ for several values of the relative risk aversion.
20
Figure 3: Average money velocity E(v), volatility of money velocity σv , and a fraction of
periods in which the cash-in-advance constraint becomes nonbinding NONB, as a function
of steady state growth rate of money supply µ̄ for several risk aversion parameters.
21