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A theorem regarding elasticities of the transactions demand for money
Economics Letters 27 (1988) 151-154 North-Holland
A THEOREM REGARDING FOR MONEY
151
ELASTICITIES
OF THE TRANSACTIONS
DEMAND
Ross MILBOURNE Queen’s University, Kingsion,
Ont., Canada K7L 3N6
Received 16 December 1987 Accepted 17 February 1988
This paper considers the class of inventory-theoretic transactions demand for money models and proves the following theorem: the sum of the income elasticity of money demand plus the absolute value of the interest elasticity is equal to unity. This result is often ignored in empirical simulation studies and Bayesian estimation.
This paper states and proves a key result inherent in all inventory-theoretic, transactions demand for money models. This result is not widely recognized. Moreover, the restriction is often violated a priori in empirical studies. Despite being dominated in rate of return, money is held for its transactions role. General transactions demand theory assumes that wealth is generally held as bonds, and transferred into money when required. Since such transfers are not done continuously there is usually assumed to be some fixed cost to such transfers, denoted below as $a per transfer. Of the standard transactions demand models, Baumol (1982) Tobin (1956) and Barro (1976) are examples of deterministic income and expenditure patterns. Miller and Orr (1966) is an example of transactions demand models with stochastic receipts and expenditures. Fig. 1 displays a sample path for money holdings with uncertain receipts and expenditures. The decision rule is how much to transfer into money when the account reaches zero or some lower threshold (and in the case of Miller
money
0
time
T
Fig. 1. 0165-1765/88/$3.50
0 1988, Elsevier Science Publishers B.V. (North-Holland)
152
R. Milbourne
and Orr, how much decision
to transfer
/ Elasticities of the transactions
out of the cash balance
on how much to transfer
(and when) implies
demand for money
if some upper threshold a given number
is attained).
of transfers
The
over any given
period T, and given all possible receipt and expenditure patterns, this decision implies an expected number of transfers. Thus the decision rule is essentially a decision on the expected number of transfers desired over some period. Expressed
slightly
more
formally,
any sample
path
is given
by the history
of payments
and
receipts over the period. Combined with the choice of transfer rule (or equivalently, expected number of transfers), this uniquely determines each possible sample path. We can denote My,,(t) as representative of paths for which total transfers into money equal Y, which involve n such transfers. It follows that average money holding along such a path is
M Y,n = -
1
r
T
/ o 4,,(t)
dt.
Averaged over all such sample paths, expected money holdings for a given Y depends upon Y, and the choice of the expected number of transfers Z, so that expected money holdings can be written in general
form
My,,
F( Y, ti).
=
(2)
All inventory-theoretic models can be written in this form. For example in the Baumol and Tobin models, F( Y, ti) = Y/25, where Z = n since the models are deterministic. (This follows because with n transfers, Y/n is added at each transfer, and cash is then run to zero before the next transfer). The key to what follows is to note the special property possessed by the function F. Suppose that we double expenditures, keeping the number of transfers constant. In the case of the deterministic models, the result is trivially obvious. Each transfer will have to be twice as large, since their number is fixed, in order to accommodate expenditures. Consequently average money holdings will be twice as large. The same occurs in stochastic environments. For each sample path M,,(t), consider the sample path M2Y,n(t) with expenditures doubled but with identical transfer times: it follows that optimal
transfers
must be twice as large, otherwise the path is not feasible. Thus M,,,(t) = 2M,,(t) It follows that we can write M ZY,n = 2M,,,.
for all l, so that over all such sample paths,
F(Y,
M=
2) = YF(1,
The standard costs
2)-Yf(ti).
optimizing
are minimized.
These
framework costs
money, rA4, where r is the interest Za. Thus agents seek to minimize
C=rYf(ti) +na.
(3) of transactions
consist
demand
of the opportunity
rate differential
between
models is one in which total portfolio cost money
of interest bonds
foregone
by holding
(B), plus transfer
costs,
(4)
The choice of Z depends upon r, Y and u. If we denote vy as the income elasticity of money demand, and n, as the interest elasticity of money demand, the following result must hold.
Theorem. In all transactions demand for money models of the type described above the income elasticity plus the absolute value of the interest elasticity must sum to unity. That is,
153
R. Milbourne / Elasticities of the transactions demand for money ProoJ
First-order
rYf’(
ii) + a = 0,
conditions
from (4) are
(5)
so that first- and second-order solution for ti can be written
conditions
for a minimum
imply
that
f’(Z)
< O,f”(n)
> 0. The
($ 1
n=(f-’
(6)
)
so that
M=Yg$,
( 1
(7)
where g is the composite
function
g=f((fY)-
(8)
It follows that
and consequently dM Y %,=dYz=l+rYg
dMr 77,=yIgM=ryg From
(9)
a
(10)
the implicit
function
theorem,
(( f’)-‘(
.) = l/f”(-) n
so that g’(.)
=f’(Z)/f”(T1)
< 0. Thus
and
9v+h,/=1. In what property model of including elasticity
(11)
0 follows,
(11)
will be referred
to as the sum of elasticities
(SOE)
property.
The
SOE
is a somewhat general property which follows from (3). Examples include the well known Baumol in which nr = 4, n, = - f . The theorem is true for a broader class of models, the ‘demand for money by firms’ model of Miller and Orr (1966). In that model, the scale is 2/3 and the interest elasticity of -l/3. Moreover, aggregation does not necessarily
destroy this result. income distribution;
Barro (1976) aggregates the Tobin model across the SOE property remains under this aggregation.
an economy
with a Gamma
R. Milbourne
154
This SOE property to as bonds. B=
W-
Defining
Defining
/ Elastictties of the transactions demand for money
is not shared by other assets (taken W = M+
together)
which for simplicity
are referred
B as total wealth,
Yf(ii). 6, as the elasticity
of bond holdings,
with respect
to i, it follows that
(12) (13)
(14) If both W and Y are considered
=1+2$19,/
as scale variables
it follows from (12))(14)
that
>1
from (11). If wealth only is considered
as the appropriate
scale variable
=1+$(1+l%l)>l.
Ew+ I‘$,1
In both cases, the sum of the scale elasticities and the absolute value of the interest elasticity is strictly greater than unity for non-monetary assets. Transactions demand holdings for money satisfies the SOE property whereas other assets do not. The above theorem is useful as an identifying restriction (along the lines of another theoretically derived proposition - price homogeneity) for money demand functions. This is especially useful since a number of factors such as financial innovations, interest rate volatility and regime changes have re-opened the issue of the identifiability of money demand functions. This result makes clear that many studies confine parameter estimates to values which violate the theory from which the models
are derived.
Common
example
are models
which impose
an income
elasticity of unity in estimation. Similar contradictions to the theory occur in many Bayesian of transactions demand where initial estimates of parameters violate the SOE property.
studies
References Barre, Robert J., 1976, Integral March, 77-87.
constraints
and aggregation
in an inventory
model of money demand,
Journal
of Finance
31,
Baumol, William J., 1982, The transactions demand for cash An inventory theoretic approach, Quarterly Journal of Economics 66, Nov., 545-556. Miller, M.H. and D. Orr, 1966, A model of the demand for money by firms, Quarterly Journal of Economics 80, Aug., 413-435. Tobin, J., 1956, The interest elasticity of the transactions demand for cash, Review of Economics and Statistics 38, Aug., 241-247.
A THEOREM REGARDING FOR MONEY
151
ELASTICITIES
OF THE TRANSACTIONS
DEMAND
Ross MILBOURNE Queen’s University, Kingsion,
Ont., Canada K7L 3N6
Received 16 December 1987 Accepted 17 February 1988
This paper considers the class of inventory-theoretic transactions demand for money models and proves the following theorem: the sum of the income elasticity of money demand plus the absolute value of the interest elasticity is equal to unity. This result is often ignored in empirical simulation studies and Bayesian estimation.
This paper states and proves a key result inherent in all inventory-theoretic, transactions demand for money models. This result is not widely recognized. Moreover, the restriction is often violated a priori in empirical studies. Despite being dominated in rate of return, money is held for its transactions role. General transactions demand theory assumes that wealth is generally held as bonds, and transferred into money when required. Since such transfers are not done continuously there is usually assumed to be some fixed cost to such transfers, denoted below as $a per transfer. Of the standard transactions demand models, Baumol (1982) Tobin (1956) and Barro (1976) are examples of deterministic income and expenditure patterns. Miller and Orr (1966) is an example of transactions demand models with stochastic receipts and expenditures. Fig. 1 displays a sample path for money holdings with uncertain receipts and expenditures. The decision rule is how much to transfer into money when the account reaches zero or some lower threshold (and in the case of Miller
money
0
time
T
Fig. 1. 0165-1765/88/$3.50
0 1988, Elsevier Science Publishers B.V. (North-Holland)
152
R. Milbourne
and Orr, how much decision
to transfer
/ Elasticities of the transactions
out of the cash balance
on how much to transfer
(and when) implies
demand for money
if some upper threshold a given number
is attained).
of transfers
The
over any given
period T, and given all possible receipt and expenditure patterns, this decision implies an expected number of transfers. Thus the decision rule is essentially a decision on the expected number of transfers desired over some period. Expressed
slightly
more
formally,
any sample
path
is given
by the history
of payments
and
receipts over the period. Combined with the choice of transfer rule (or equivalently, expected number of transfers), this uniquely determines each possible sample path. We can denote My,,(t) as representative of paths for which total transfers into money equal Y, which involve n such transfers. It follows that average money holding along such a path is
M Y,n = -
1
r
T
/ o 4,,(t)
dt.
Averaged over all such sample paths, expected money holdings for a given Y depends upon Y, and the choice of the expected number of transfers Z, so that expected money holdings can be written in general
form
My,,
F( Y, ti).
=
(2)
All inventory-theoretic models can be written in this form. For example in the Baumol and Tobin models, F( Y, ti) = Y/25, where Z = n since the models are deterministic. (This follows because with n transfers, Y/n is added at each transfer, and cash is then run to zero before the next transfer). The key to what follows is to note the special property possessed by the function F. Suppose that we double expenditures, keeping the number of transfers constant. In the case of the deterministic models, the result is trivially obvious. Each transfer will have to be twice as large, since their number is fixed, in order to accommodate expenditures. Consequently average money holdings will be twice as large. The same occurs in stochastic environments. For each sample path M,,(t), consider the sample path M2Y,n(t) with expenditures doubled but with identical transfer times: it follows that optimal
transfers
must be twice as large, otherwise the path is not feasible. Thus M,,,(t) = 2M,,(t) It follows that we can write M ZY,n = 2M,,,.
for all l, so that over all such sample paths,
F(Y,
M=
2) = YF(1,
The standard costs
2)-Yf(ti).
optimizing
are minimized.
These
framework costs
money, rA4, where r is the interest Za. Thus agents seek to minimize
C=rYf(ti) +na.
(3) of transactions
consist
demand
of the opportunity
rate differential
between
models is one in which total portfolio cost money
of interest bonds
foregone
by holding
(B), plus transfer
costs,
(4)
The choice of Z depends upon r, Y and u. If we denote vy as the income elasticity of money demand, and n, as the interest elasticity of money demand, the following result must hold.
Theorem. In all transactions demand for money models of the type described above the income elasticity plus the absolute value of the interest elasticity must sum to unity. That is,
153
R. Milbourne / Elasticities of the transactions demand for money ProoJ
First-order
rYf’(
ii) + a = 0,
conditions
from (4) are
(5)
so that first- and second-order solution for ti can be written
conditions
for a minimum
imply
that
f’(Z)
< O,f”(n)
> 0. The
($ 1
n=(f-’
(6)
)
so that
M=Yg$,
( 1
(7)
where g is the composite
function
g=f((fY)-
(8)
It follows that
and consequently dM Y %,=dYz=l+rYg
dMr 77,=yIgM=ryg From
(9)
a
(10)
the implicit
function
theorem,
(( f’)-‘(
.) = l/f”(-) n
so that g’(.)
=f’(Z)/f”(T1)
< 0. Thus
and
9v+h,/=1. In what property model of including elasticity
(11)
0 follows,
(11)
will be referred
to as the sum of elasticities
(SOE)
property.
The
SOE
is a somewhat general property which follows from (3). Examples include the well known Baumol in which nr = 4, n, = - f . The theorem is true for a broader class of models, the ‘demand for money by firms’ model of Miller and Orr (1966). In that model, the scale is 2/3 and the interest elasticity of -l/3. Moreover, aggregation does not necessarily
destroy this result. income distribution;
Barro (1976) aggregates the Tobin model across the SOE property remains under this aggregation.
an economy
with a Gamma
R. Milbourne
154
This SOE property to as bonds. B=
W-
Defining
Defining
/ Elastictties of the transactions demand for money
is not shared by other assets (taken W = M+
together)
which for simplicity
are referred
B as total wealth,
Yf(ii). 6, as the elasticity
of bond holdings,
with respect
to i, it follows that
(12) (13)
(14) If both W and Y are considered
=1+2$19,/
as scale variables
it follows from (12))(14)
that
>1
from (11). If wealth only is considered
as the appropriate
scale variable
=1+$(1+l%l)>l.
Ew+ I‘$,1
In both cases, the sum of the scale elasticities and the absolute value of the interest elasticity is strictly greater than unity for non-monetary assets. Transactions demand holdings for money satisfies the SOE property whereas other assets do not. The above theorem is useful as an identifying restriction (along the lines of another theoretically derived proposition - price homogeneity) for money demand functions. This is especially useful since a number of factors such as financial innovations, interest rate volatility and regime changes have re-opened the issue of the identifiability of money demand functions. This result makes clear that many studies confine parameter estimates to values which violate the theory from which the models
are derived.
Common
example
are models
which impose
an income
elasticity of unity in estimation. Similar contradictions to the theory occur in many Bayesian of transactions demand where initial estimates of parameters violate the SOE property.
studies
References Barre, Robert J., 1976, Integral March, 77-87.
constraints
and aggregation
in an inventory
model of money demand,
Journal
of Finance
31,
Baumol, William J., 1982, The transactions demand for cash An inventory theoretic approach, Quarterly Journal of Economics 66, Nov., 545-556. Miller, M.H. and D. Orr, 1966, A model of the demand for money by firms, Quarterly Journal of Economics 80, Aug., 413-435. Tobin, J., 1956, The interest elasticity of the transactions demand for cash, Review of Economics and Statistics 38, Aug., 241-247.