This paper examines the monetary growth implications of combining Stockman's cash-in-advance constraint on consumption and capital goods and an endogenous rate of time preference that is an increasing function of real wealth. The cash-in-advance constraint imposes an investment tax that reduces steady state consumption and capital. However, endogenous time preference wealth effects link the real and monetary sectors to yield a Mundell-Tobin effect. Cash-in-advance constraint effects dominate endogenous time preference wealth effects so that monetary growth reduces steady state capital and consumption.cash-in-advance constraints

Eric Kam

Paul Missios

English

Research Papers in Economics

Wealth effects in a cash−in−advance economy 
Eric Kam Paul Missios
Ryerson University Ryerson University
Abstract
This paper examines the monetary growth implications of combining Stockman's
cash−in−advance constraint on consumption and capital goods and an endogenous rate of
time preference that is an increasing function of real wealth. The cash−in−advance constraint
imposes an investment tax that reduces steady state consumption and capital. However,
endogenous time preference wealth effects link the real and monetary sectors to yield a
Mundell−Tobin effect. Cash−in−advance constraint effects dominate endogenous time
preference wealth effects so that monetary growth reduces steady state capital and
consumption.
We wish to thank Avi Cohen, David Laidler, Arman Mansoorian, Bill Scarth, John Smithin and an anonymous referee for their
comments. The standard caveat applies.
Citation: Kam, Eric and Paul Missios, (2003) "Wealth effects in a cash−in−advance economy." Economics Bulletin, Vol. 5, No.
2 pp. 1−7
Submitted: January 14, 2003.  Accepted: March 30, 2003.
URL: http://www.economicsbulletin.com/2003/volume5/EB−03E00001A.pdf1 Introduction
Stockman (1981) imposes a cash-in-advance constraint on purchases of consumption and capital
goods, and consequently monetary growth acts as an investment tax, which lowers the production
and demand for steady state consumption and capital. More recently, Kam (2002) models the rate
of time preference as an endogenous function that depends positively on real wealth. The resulting
wealth eﬀects link the monetary and real sectors, converting savings into steady state consump-
tion and capital. This paper demonstrates that the cash-in-advance constraint eﬀect dominates
endogenous time preference wealth eﬀects. Monetary growth decreases steady state consumption
and capital, which reverses the real sector implications of the Mundell-Tobin eﬀect.
The optimizing underpinnings of the inﬁnitely-lived, representative agent model have been
integrated into several macroeconomic applications over the past four decades.1 One standard
assumption is that the rate of time preference and the discount factor applied to the lifetime ﬂow of
utility are exogenous. This assumption is responsible for many of the limiting conclusions obtained
in this framework. Speciﬁcally, it implies monetary superneutrality by ﬁxing the real interest rate
and the marginal product of capital, which removes the link between the real and monetary sectors
and insulates the capital stock and consumption from the eﬀects of monetary growth (Sidrauski
1967a,b).
Uzawa (1968) is an attempt to endogenize time preference and derive a more general represen-
tation of behavior. However, this time preference speciﬁcation has been criticized and generally
dismissed because of the assumptions necessary to ensure steady state stability.2 Uzawa models
time preference as an increasing function of instantaneous utility, which itself depends positively
on current consumption. Saving is modeled as an increasing function of real wealth, which implies
that the representative agent becomes increasingly impatient as consumption increases. This rep-
resentation of behavior is counterintuitive and also contradicts the theory of savings as a decreasing
function of wealth.3
A rate of time preference that is an increasing function of real wealth resolves the counterin-
tuitive preference criticism, permitting microeconomic foundations to generate a monetary growth
model that is characterized by steady state stability and the Mundell-Tobin eﬀect. Monetary growth
yields substitution eﬀects that lower the initial value of real wealth and raise the opportunity cost
of holding real balances. This lowers steady state real balances and the rate of time preference but
augments savings. Endogenous time preference wealth eﬀects reinforce the substitution eﬀect and
convert additional savings into steady state consumption and capital. Optimizing behavior induces
interdependence across time preference, wealth and savings so that the Mundell-Tobin eﬀect is the
result of reinforcing substitution and wealth eﬀects, which has recently been empirically veriﬁed by
Woodward (1992), Shrestha et al. (2002), and Rapach (2003).
This paper analyzes the monetary growth implications of simultaneously modeling two contra-
dictory eﬀects: ﬁrst, a continuous-time, monetary growth transformation of Stockman’s cash-in-
advance constraint on the purchase of consumption and capital goods; second, an endogenous rate
of time preference that is modeled as an increasing function of real wealth. The following section
derives the monetary growth implications of the representative agent model with this endogenous
time preference. Section 3 considers the implications of combining this time preference speciﬁcation
1For a comprehensive review of monetary growth in representative agent models, see Dornbusch and Frenkel (1973)
and Wang and Yip (1992).
2Blanchard and Fisher (1989, 74-5) argue, “::: the Uzawa function, with its assumption (that the rate of time
preference increases in instantaneous utility) is not particularly attractive as a description of preferences and is not
recommended for general use.”
3For the latter, see Mundell (1963), Tobin (1965), Laidler (1969), Begg (1980) and Epstein and Hynes (1983).
1with a cash-in-advance constraint on the purchase of consumption and capital goods. The critical
result is that the cash-in-advance constraint eﬀect dominates endogenous time preference wealth
eﬀects so that monetary growth reduces steady state consumption and capital. Section 4 oﬀers
concluding comments.
2 The Representative Agent Model with Endogenous Time Pref-
erence
Consider a monetary growth model in which real balance holdings do not yield utility but appear in
the deﬁnition of real wealth and the asset accumulation identity. Assume that the discount factor
that is applied to future streams of utility is
¯(v) =
Z t
0
µv(av)dv (1)
where v is a time index and µ is the rate of time preference that is assumed to be an increasing
function of real wealth a. The representative agent maximizes
Z 1
0
u(ct)e
¡
R t
0 µv(av)dvdt (2)
where c is consumption,4 and faces two ﬂow budget constraints
˙ at = f(kt) + x ¡ ct ¡ ¼tmt (3)
at = kt + mt (4)
and a stock budget constraint
lim
t!1
ate
¡
R t
0 rvdv ¸ 0 (5)
where f is a constant returns to scale production function, k is the capital stock, x is the real value
of government transfers that maintain constant real balances following an inﬂationary period, ¼ is
the inﬂation rate, m is the stock of real balance holdings and r is the real interest rate.
Maximizing (2) subject to (3), (4) and (5) yields the ﬁrst-order optimality conditions
uc(c;m) ¡ ¸ = 0 (6)
¡¸
¡
f0(k) + ¼
¢
= 0 (7)
˙ ¸ = ¡¸
¡
f0(k) ¡ µ(k + m)
¢
(8)
where ¸ is the co-state variable.5 In the steady state, from (6) and (7)
4By assumption uc > 0 and ucc < 0:
5The transversality condition is
lim
t!1
at¸te
¡
R t
0
µvdv
= 0
2f0(k) + ¼ = 0 (9)
The resource constraint is given by6
˙ k = f(k) ¡ c (10)
implying in the steady state
c = f(k) (11)
At the same time, from (8)
µ(k + m) = f0(k) (12)
and from (3) and (11)
¼¤ = ¹ (13)
where ¹ is the monetary growth rate.
Linearizing around the steady state levels of consumption, real balances and capital (c¤;m¤;k¤);
characterized by (9), (11) and (12) using (13), yields7
2
6
4
0 0 f00
0 µ0 ½
¡1 0 µ
3
7
5
2
6
4
dc¤
dm¤
dk¤
3
7
5 =
2
6
4
¡1
0
0
3
7
5d¹ (14)
The determinant of the coeﬃcient matrix in (14)
j∆j = f00µ0 < 0 (15)
Therefore, it is possible to determine the eﬀect of monetary growth on steady state consumption
dc¤
d¹
=
¡µµ0
j∆j
> 0 (16)
on real balance holdings
dm¤
d¹
=
½
j∆j
< 0 (17)
and on the capital stock
dk¤
d¹
=
¡µ0
j∆j
> 0 (18)
Monetary growth substitution eﬀects lower the initial value of real wealth and raise the op-
portunity cost of holding real balances. This reduces steady state real balance holdings and the
rate of time preference, but raises savings. Endogenous time preference wealth eﬀects reinforce the
substitution eﬀect and transform the added savings into steady state consumption and capital. The
Mundell-Tobin eﬀect is the result of reinforcing wealth and substitution eﬀects and is generated
without any counterintuitive preference assumptions.8
6Depreciation of capital has been removed with no loss of generality.
7½ = µ
0 ¡ f
00 > 0.
8Kam (2002a) demonstrates that assuming the rate of time preference is an increasing function of real wealth
implies stability.
33 The Representative Agent Model with Endogenous Time Pref-
erence and a Cash-in-Advance Constraint
Stockman models a cash-in-advance constraint that applies to the purchase of capital and con-
sumption goods. In continuous time, this constraint becomes
mt = ct + ˙ kt (19)
Substituting (4) into (3) and (19) gives the constraints
˙ kt = (at ¡ kt) ¡ ct (20)
and
˙ at = f(kt) + xt ¡ ct ¡ ¼(at ¡ kt) (21)
The representative agent maximizes (2) subject to (5), (20), (21) and the stock budget constraint
lim
t!1
kte
¡
R t
0 rvdv ¸ 0 (22)
yielding the ﬁrst-order optimality conditions
uc(c) ¡ (Ã + °) = 0 (23)
°µ(k + m) ¡ Ã
¡
f0(k) + ¼
¢
+ ° = ˙ ° (24)
Ãµ(k + m) + Ã¼ ¡ ° = ˙ Ã (25)
where Ã and ° are co-state variables.9
In the steady state, it must be that
°µ(k + m) ¡ Ã
¡
f0(k¤) + ¼
¢
+ ° = 0 (26)
from (24) using ˙ ° = 0,
Ãµ(k¤ + m¤) + Ã¼ ¡ ° = 0 (27)
from (25) using ˙ Ã = 0; and
c¤ = m¤ (28)
from (4) and (20) using ˙ k = 0: Again linearizing around the steady state (c¤;m¤;k¤), now charac-
terized by (11), (23), (26), (27) and (28) using (13), yields10
9The transversality condtions are
lim
t!1
kt°te
¡
R t
0
µvdv
= 0
lim
t!1
atÃte
¡
R t
0
µvdv
= 0
10Let ® = ¹µ
0 ¡ ¸f
00 > 0.
42
6 6
6
6 6
4
ucc 0 0 ¡1 ¡1
0 ® °µ0 (1 + µ) ¡(f0 + ¹)
0 Ãµ0 Ãµ0 ¡1 (µ + ¹)
¡1 f0 0 0 0
¡1 0 1 0 0
3
7 7
7
7 7
5
2
6 6
6
6 6
4
dc¤
dk¤
dm¤
d°
dÃ
3
7 7
7
7 7
5
=
2
6 6
6
6 6
4
0
Ã
¡Ã
0
0
3
7 7
7
7 7
5
d¹ (29)
The determinant of the coeﬃcient matrix in (29)
jAj =
¡
®(µ + ¹) + Ãµ0(f0 + ¹)
¢
+
¡
® + Ãµ0(1 + µ)
¢
+f0[
¡
°µ0(µ + ¹) + Ãµ0(f0 + ¹)
¢
+ °µ0 + Ãµ0(1 + µ)] > 0
Thus, the eﬀect of monetary growth on steady state consumption is
dc¤
d¹
=
¡Ã(f0)2
jAj
< 0 (30)
on real balance holdings is
dm¤
d¹
=
¡Ã(f0)2
jAj
< 0 (31)
and on the capital stock is
dk¤
d¹
=
¡Ã(f0)
jAj
< 0 (32)
If the rate of time preference is modeled as an increasing function of real wealth and is combined
with a cash-in-advance constraint that applies to purchases of consumption and capital goods,
monetary growth lowers steady state consumption, real balances and capital. Thus, endogenous
time preference has no eﬀect on Stockman’s exogenous time preference result that monetary growth
is inversely related to steady state capital accumulation.
The reversal of the Mundell-Tobin eﬀect can also be demonstrated using steady conditions (11)
and (28)
f(k¤) = m¤ = c¤ (33)
or, steady state output, real balances and consumption are equal. Substituting (47) into (1)
¯(v) =
Z t
0
µv (kv + f(kv))dv (34)
Comparing (1) and (34) demonstrates that the rate of time preference, ﬁrst modeled as an increasing
function of real wealth, is changed into a function that depends only on the capital stock. Although
monetary growth initially raises savings, the cash-in-advance constraint eﬀect directs savings away
from the accumulation of steady state consumption and capital.
5The results of the paper are summarized below:
Approach Eﬀect of an increase in monetary growth on steady state
c m k
NO CIA - - -
CIA + - +
The eﬀects of monetary growth are determined by the relative magnitude of diverging eﬀects. The
cash-in-advance constraint applies to purchases of consumption and capital goods, which levies
an investment tax that directs savings away from the accumulation of steady state capital and
consumption. Endogenous time preference wealth eﬀects convert savings from real balance hold-
ings into consumption and capital. The cash-in-advance constraint eﬀect dominates endogenous
time preference wealth eﬀects so that monetary growth has the net eﬀect of reducing steady state
consumption and capital.
4 Conclusions
This paper examines the monetary growth eﬀects of combining an endogenous rate of time prefer-
ence that is modeled as an increasing function of real wealth and a continuous-time representation
of Stockman’s cash-in-advance constraint on consumption and capital purchases. The result is that
monetary growth lowers steady state consumption and capital, which reverses the real implications
of the Mundell-Tobin eﬀect.
As the cash-in-advance constraint applies to purchases of consumption and capital goods, it
imposes a second constraint on the evolution of the capital stock, which implies equality across
steady state real balances, output and capital. The rate of time preference, initially an increasing
function of real wealth, is transformed into a function that depends solely on capital. This nulliﬁes
endogenous time preference wealth eﬀects so that monetary growth aﬀects real variables through
a substitution eﬀect that raises savings and a cash-in-advance constraint that directs savings away
from steady state consumption and capital.
6References
[1] Begg, D. (1980), “Rational expectations and the non-neutrality of systematic monetary policy,”
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[2] Blanchard, O. J., and S. Fischer, Lectures On Macroeconomics (Cambridge, MA: MIT Press,
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of Money, Credit and Banking 5, 141-56.
[4] Epstein, L., and Hynes, J. A. (1983), “The rate of time preference and dynamic economic
analysis,” Journal of Political Economy 91, 611-35.
[5] Kam, E. (2002), “Endogenous time preference and the microeconomic foundations of the
Mundell-Tobin eﬀect,” Ryerson University, Department of Economics Working Paper 26, 1-
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[6] Laidler, D.E.W. (1969), “Money, wealth and time preference in a stationary economy,” Cana-
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7

Wealth effects in a cash-in-advance economy

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Wealth effects in a cash-in-advance economy

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