Theoretical Economics Letters, 2013, 3, 292-296
http://dx.doi.org/10.4236/tel.2013.35049 Published Online October 2013 (http://www.scirp.org/journal/tel)
Cash-in-Advance Constraint with Status
in a Neoclassical Growth Model
Ken-ichi Kaminoyama1*, Taketo Kawagishi2
1Department of Economics, Doshisha University, Kyoto, Japan
2Faculty of Economics, Tezukayama University, Nara, Japa n
Email: *firstname.lastname@example.org, email@example.com
Received August 22, 2013; revised September 25, 2013; accepted October 7, 2013
Copyright © 2013 Ken-ichi Kaminoyama, Taketo Kawagishi. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
In this paper, we assume that a cash-in-advance (CIA) constraint itself depends on relative income, which implies the
status. This constraint means that agents with higher income are more creditworthy and can make purchases with fewer
money holdings. Under this assumption, we construct a one-sector neoclassical growth model and show that there exists
a unique steady state that has saddle-path stability without specifying each function. Furthermore, we examine the ef-
fects of money growth on capital accumulation. If the status elasticity of CIA constraint is large, the Tobin effect can
arise. In contrast, if it is small, the anti-Tobin effect can arise.
Keywords: Cash-in-Advance Constraint; Status; Money Growth; Neoclassical Growth Model; Tobin/Anti-Tobin Effect
There is a growing macroeconomic literature that exam-
ines the effects of inflation (money growth) on capital
accumulation. Tobin  regards money as a substitute
for capital and concludes that money growth accelerates
capital accumulation, known as the Tobin effect. There-
after, many studies have discussed the effects of money
growth in the context of cash-in-advance (CIA hence-
forth) constraints. For example, Clower  and Lucas 
show that if the CIA constraint applies only to consump-
tion, then money growth has no effect on capital accu-
mulation in the long run, known as the superneutrality of
money. On the other hand, Stockman  considers a
standard neoclassical growth model with the CIA con-
straint applying to both consumption and investment, and
shows that money growth decreases capital accumulation.
This is because after the period of higher inflation the net
rate of return on capital falls. In this case, the level of
capital and the money growth rate are negatively corre-
lated. This is referred to as the anti-Tobin effect.
Recent studies on neoclassical growth models with
CIA constraints capture the role of status in terms of a
social device providing priority in the nonmarket goods
sector, seen in Cole et al. .1 In this context, Chang et
al. , Gong and Zou , and Chang and Tsai  in-
troduce status, defined as capital holdings, into prefer-
ences.2 Under the Clower-Lucas-type CIA constraint,
Chang et al.  confirm that money growth and the
steady-state level of capital are positiv ely correlated. It is
because higher inflation increases the cost of money
holdings, that the agent shifts his/her assets from money
to capital, which provides utility. On the other hand,
Gong and Zou  employ the Stockman-type CIA con-
straint and show that whether money growth promotes
capital accumulation or not depends on the measure of
the agent’s desire for status.3 In a similar vein, Chang
and Tsai  find that when the status seeking effect
dominates the inflation tax effect, money growth and the
steady-state capital stock are positively correlated under
the general-type CIA constraint, which means that whole
consumption and a positive fraction of investment are
1Zou  interprets utility from capital in terms of the spirit of capital-
ism, based on Weber , and shows that endogenous growth can arise
even if the interest rate is smaller than the time preference rate.
2This type of preferences had already been constructed mathematically
by Kurz .
3Gong and Zou  analyze the effects of money growth in the case o
the Clower-Lucas-type CIA constraint and that of the Stockman-type
CIA constraint. Under the Clower-Lucas-type CIA constraint, Gong
and Zou  obtain the same results as in Chang et al. .
*Corresponding a uthor.
opyright © 2013 SciRes. TEL
K. KAMINOYAMA, T. KAWAGISHI 293
purchased using real money balances.
In this paper, on the other hand, we capture the status
in terms of social credibility when making purchases.
Specifically, we embody this concept by assuming that
individuals with higher income (higher status) are more
creditworthy and can make purchases with fewer money
holdings. Avery et al.  show that high income indi-
viduals use cash and cash plus checks for a smaller frac-
tion of their total transactions than low income individu-
als; and Wolff , Kessler and Wolff , and Ken-
nickell and Starr-McCluer  find that the fraction of
household wealth held in liquid assets decreases with
income and wealth. Hence, in the present study, we as-
sume that the CIA constraint itself depends on relative
income, which implies status, and that this CIA constraint
applies to consumption and investment. Under such a
CIA constraint, we consider a neoclassical growth model
and clarify how status has an impact on the relationship
between the rate of money growth and the steady-state
level of capital, as well as a uniqueness of the steady
state and its stability.
2.1. CIA-Status Constraint
The representative agent faces the following CIA con-
mlcli ll (1)
where is consumption, i is investment, and m is
real money balances defined as nominal money balances
divided by the price level.4 In addition, c and i
the ratios of consumption and investment goods which
require cash, respectively.
From the observations mentioned in the Introduction
(Avery et al. (1987) etc.), we assume that c and
depend on the agent ’s credit when makin g purcha se s:
are private income and average in-
come in the economy respectively, and yy stands for
the agent’s own relative income.5 Note that we regard
relative income as status. Add itionally,
relative strength of the liquidity constraint applying to
investment expenditure. Along the lines of the standard
CIA literature, we assume that the liquidity constraint on
consumption is stricter than that on investment. More-
over, we posit that is decreasing and convex with
From (1) and (2), we have
In the following analysis, we impose (4), referred to as
the CIA-status constraint.
2.2. Optimal Conditions and Dynamic System
The economy is inhabited by a continuum of identical,
infinite-lived agents endowed with a unit of labor. The
size of the population is constant and is normalized to
unity. Each agent consumes a continuum of non-perish-
able consumption goods produced with a simple neoclas-
sical production technology. We assume that each con-
sumption commodity is perfectly complementary. The
representative agent maximizes the following lifetime
is an instantaneous utility function which
<0u and the Inada conditions
. In addition,
is the constant rate of time preference.
The budget constraint of the repr esentative agent is
where is capital stock and
is the rate of inflation.
Output is produced using a neoclassical production func-
, satisfying , and the
0fk , limkf
The law of motion of capital stock is
,given >0.ki k
For simplicity, the depreciation rate of capital is as-
sumed to be zero. In (6),
is the seigniorage that the
agent receives from the monetary authority as a lump-
is the constant, time-invariant money growth
rate. By using
, the nominal money supply,
Assuming a representative agent and a neoclassical
technology, from (4), we find that the CIA-status con-
4To simplify the notation, we drop the time index from each variable.
5Note that, from (1) and (2),
Copyright © 2013 SciRes. TEL
K. KAMINOYAMA, T. KAWAGISHI
where k is average capital stock in the economy.
The representative agent maximizes (5) subject to (6),
(7) and (10). In this problem, the representative agent is
assumed to take the sequences,
, as given. To
derive the necessary conditions for an optimum, we set
up the following current-value Hamiltonian function:
are the shadow prices of real money
balances and the capital stock, respectively, and
the Lagrange multiplier associated with the CIA-status
constraint (10). The first-order conditions for optimiza-
tion are given as follows:
and the transversality conditions for and m are k
capital stock in the economy:
Equation (11) implies that the marginal utility of con-
sumption equals the marginal cost of consumption,
which is the marginal utility of having an additional unit
of real money balances. Equations (12) and (13) together
indicate the evolution of capital over time, where the last
term on the left-hand side in (13) represents the mar-
ginal benefit from the higher income position (i.e. higher
status). This benefit implies that the agent gets relatively
higher credibility and can make purchases with fewer
money holdings. Equation (14) implies that the marginal
value of real money balances equals the marginal cost.
Since the agents are assumed to be symmetric and the
size of the population is unity, in equilibrium, the level of
the agent’s capital stock is equal to the average level of
Additionally, in equilibri
an um, the goods market clears,
d money demand is equal to money supply:
We assume that the CIA const
raint is always binding
equilibrium, as is common in the CIA literature. Thus,
from (10) and (1 7), we get
c 1m .xi (20)
From (7) and (11) - (20), we obta
na in the following dy-
mic system after some manipulation:
uc fk xuc
expresses the elasticity of the CIA con-
ady state and its sta-
raint with repect to status.
2.3. Steady State and Stability
In this subsection, we consider a ste
bility. In a steady state, the economy is characterized by
. Substituting this condition into
(2ting them, we obtain 1)-(23) and calcula
Since the production function,
, satisfies concav-
ity and the Inada conditions, we haveunique solution,
Copyright © 2013 SciRes. TEL
K. KAMINOYAMA, T. KAWAGISHI 295
*, which represents the steady-state level of capital.
Equation (24) implies that when the CIA constraint itself
ends on status
, the steady-state level of
capital is not determined only by the constant rate of time
preference (i.e. the d golden rule
does not hold), even if the CIA constraint applies only to
0x. In this case, the leve
hinges on money growth, that is, the superneutrality of
money is not n the next section, we conduct the
analysis concerning the effects of money growth on
capital stock and consumption.
Next, we consider the stability of the steady state.
Linearizing the dynamic syste
l of capital
) around the
steady state, we obtain the following relationships among
the three characteristic roots, 1
c* represents the steady-state level of consumption):6
d (26) t
dicate that the dy-
Equations (25) anogether in
ic system has ve and two
values. Since consumption, c, and the shadow price of
real money balances,
, are jumpable variables and
capital stock, k, is a state vriable, the steady state ex-
hibits the saddle-point stability.
Proposition. In a neoclassical growth model with
the CIA-status constraint (4), the
1re exists a
th on Ca
state that is saddle-path stable.
3. Effects of Money Grow
In this section, we clarify how money growth affects the
steady-state level of capital stock and that of consump-
tion in our framework.
From (24), we have
Expressions (27) indicate that when t
of the CIA constraint,
he status elasticity
of th is great
the relative strength e liquidity constraint applying
er (resp. smaller) than
to investment expenditure,
, an increase in the rate of
, induces an increase (resp. a decrease)
in capital stock. The positive (resp. negative) relationship
between the rate of money growth and capital stock can
be considered as the presence of the Tobin (resp. the
These results can be interpreted through the relation-
ship between the benefit of holding additional capital
stock and the cost of purchasing investment goods. Here,
the benefit is generated from the status enhancement ef-
fect, which arises when the agent makes the CIA con-
straint less restricted by investing more. This benefit can
be captured through the parameter
. On the other hand,
the cost is caused by the inflation tax effect, which oc-
curs when the agent purchases investment goods after the
policy of raising money growth indes higher inflation.
This cost can be measured by the parameter
The intuition is as follows. Suppose that the economy
is in the steady state initially, and that the rate of money
growth rises. When
is greater than
, the status en-
hancement effect dominates the inflation tax effect; that
is, the benefit of holding additional capital stock gener-
ated from the status enhancement effect is greater than
the cost of purchasing investment goods caused by the
inflation tax effect. In this case, if the agent holds addi-
tional capital stock, his/her income increases, so that
he/she obtains higher status and holds less money bal-
ances through the CIA-status constraint. This implies that
the agent can avoid paying the higher inflation tax caused
by the money growth on the future consumption, which
leads to an increase in the future real consumption. Thus,
since the net rate of return on capital in utility terms in-
creases, the agent shifts his/her demand from current
consumption to capital stock.7,8 As a result, a rise in the
money growth rate accelerates capital accumulation,
which leads to an increase in output and that in consump-
tion in the long run.
In contrast, when
is smaller than
, the inflation
tax effect dominates the status enhancement effect. Thus,
an increase in money growth depresses the steady-state
leh evel of capital, whicleads to a decreas in output and
that in consumption in the long run.
Proposition 2. In a neoclassical growth model with
the CIA-status constraint (4), if the status elasticity of the
7Actually, as the third effect, there is the inflation tax effect on pur-
chasing current and future consumption goods. When 0x
can extract only this third effect. However, Stockman (1981) focuses
on this case and shows that the net rate of return on capital in utility
(consumption) terms is unaffected by higher inflation. Thus, we ignore
this third effect.
8Note that the agent obtains the higher net rate of return on capital by
investing more and enhancing his/her status.
6Details for the linearization of the dynamic system are available from
the authors on request.
Copyright © 2013 SciRes. TEL
K. KAMINOYAMA, T. KAWAGISHI
Copyright © 2013 SciRes. TEL
) than the relative
CIA constraint is greater (smaller
rength of the liquidity constraint applying to investment
), then a rise in the money
growth rate increases (decreases) the steady-state level
of capital and that of consumption.
This paper has investigated a neoclassical growth model
int and itself depends on relative i
s status. Under this assumption, we
h function, and 2) the
s referee, who gave us help-
while we were writing this
 J. Tobin, “Money and Economic Growth,” Econometrica
Vol. 33, No. 4
with a CIA constra
come, which implien-
have examined how status, which affects the CIA con-
straint, has an impact on the relationship between mo-
ney growth and capital stock, as well as a uniqueness of
the steady state and its stability.
Und er the CIA-status constraint, we have shown that 1)
there exists a unique steady state that has saddle-path
stability without specifying eac
bin or the anti-Tobin effect arises depending on the
magnitude relationship between the status elasticity of
the CIA constraint and the relative strength of the liquid-
ity constraint applying to investment expenditure.
We would like to express our profound gratitude o Ka-
zuo Mino and the anonymou
ful and valuable comments
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