We show that a one-sector AK model of endogenous growth with the most generalized cash-in-advance constraint is able to account for (i) the observed long-run negative relationship between the nominal growth rate of money and the income velocity of money, (ii) the empirically ambiguous effect of changing inflation on the economy's output growth, and (iii) the divergent growth experience of countries that start with similar macroeconomic conditions.

Jang-Ting Guo

Shu-Hua Chen

English

Research Papers in Economics

On the growth and velocity effects of money 
Shu-Hua Chen Jang-Ting Guo
Shih Chien University University of California, Riverside
Abstract
We show that a one-sector AK model of endogenous growth with the most generalized
cash-in-advance constraint is able to account for (i) the observed long-run negative
relationship between the nominal growth rate of money and the income velocity of money,
(ii) the empirically ambiguous effect of changing inflation on the economy's output growth,
and (iii) the divergent growth experience of countries that start with similar macroeconomic
conditions.
Discussions with Richard Suen which led to this paper are gratefully acknowledged. We also thank Been-Lon Chen, Sharon
Harrison, Chen-Ming Hsu, Junxi Zhang and seminar participants at the 2007 Annual Conference of the Asia-Pacific Economic
Association for helpful comments and suggestions. Part of this research was conducted while Guo was a visiting research fellow
at Academia Sinica, Taipei, Taiwan, whose hospitality is greatly appreciated. Of course, all remaining errors are our own.
Citation: Chen, Shu-Hua and Jang-Ting Guo, (2008) "On the growth and velocity effects of money." Economics Bulletin, Vol.
5, No. 13 pp. 1-7
Submitted: April 1, 2008.  Accepted: April 2, 2008.
URL: http://economicsbulletin.vanderbilt.edu/2008/volume5/EB-08E50014A.pdf1 Introduction
This paper is concerned with the following three macroeconomic stylized facts: (i) there exists
a discernible long-run negative relationship, both over time and across countries, between the
growth rate of nominal money supply and the income velocity of money (Palivos et. al., 1993);
(ii) the empirical evidence on the output-growth e⁄ect of money/in￿ ation is mixed as this
correlation can be negative (Roubini and Sala-i-Martin, 1992; De Gregorio, 1993; Barro, 1995;
and Gylfason and Herbertsson, 2001), positive (Gomme, 1993; and Bullard and Keating, 1995),
or zero (Levine and Renelt, 1992; and Clark, 1997); and (iii) countries like the Philippines and
South Korea, which had similar macroeconomic conditions in 1960, exhibited divergent growth
experience in the next 30 years (Lucas, 1993). Here, we show that a prototypical one-sector
AK model of endogenous growth with the most generalized cash-in-advance (CIA) constraint
is able to provide a theoretical explanation for these stylized facts simultaneously.
In an earlier work, Chen and Guo (2007) examine a similar monetary endogenous growth
model in which the entire consumption purchases and a non-negative proportion of gross in-
vestment must be ￿nanced by the household￿ s real balances (a la Wang and Yip, 1992). It
turns out that the growth and velocity e⁄ects of money are closely linked with the number
and location of the economy￿ s balanced growth paths (BGP). In particular, when there are
two BGP equilibria, the low-growth equilibrium path displays negative growth and veloc-
ity e⁄ects of money, whereas the high-growth BGP equilibrium exhibits positive e⁄ects of
money/in￿ ation on output growth and velocity. However, the positive velocity e⁄ect of money
is not qualitatively consistent with the empirical ￿ndings documented in Palivos et. al. (1993).
Motivated by this inconsistency with international data, we extend Chen and Guo￿ s study
by considering the most generalized liquidity constraint. Speci￿cally, in addition to investment,
the fraction of the household￿ s consumption expenditures that are ￿nanced by its money
holdings is allowed to take a positive value of smaller than one. Moreover, in order to explain
stylized fact (iii) regarding growth divergence from the same initial condition, our analysis is
restricted to parametric speci￿cations that possesses dual balanced growth paths. In this case,
the velocity e⁄ect of money/in￿ ation depends not only on the relative strength of two opposing
forces dubbed the portfolio substitution e⁄ect and the intertemporal substitution e⁄ect, but
also on whether the CIA-constrained proportion of consumption purchases is higher or lower
than that of gross investment. We ￿nd that in contrast to Chen and Guo (2007), the high-
growth equilibrium path, along which the intertemporal substitution e⁄ect dominates, exhibits
1a negative comovement between the money velocity and the nominal growth rate of money
provided the fraction of consumption spending subject to the liquidity constraint is smaller.
Notice that a higher liquidity-constrained fraction of investment expenditures is empirically
plausible in light of the recent ￿ndings that consumer credit has increased signi￿cantly in the
past two decades (Ludvigson, 1999), and that the cash holdings (relative to total assets) of
U.S. ￿rms have more than doubled since 1980 (Bates, Kahle and Stulz, 2006).
The remainder of this paper is organized as follows. Section 2 examines an AK model
of endogenous growth with ￿xed labor supply and the most generalized cash-in-advance con-
straint. Section 3 analyzes the output-growth and velocity e⁄ects of changing the rate of
nominal money growth. Section 4 concludes.
2 The Model
We consider a monetary endogenous growth model in which the representative household￿ s
dynamic optimization problem is to maximize its discounted lifetime utility
U =
Z 1
0
c1￿￿
t ￿ 1
1 ￿ ￿
e￿￿tdt; (1)
subject to the budget constraint
ct + it + _ mt = yt ￿ ￿tmt + ￿t ; (2)
where ct is consumption, ￿ 2 (0;1) is the subjective discount rate, and ￿ > 0 is the inverse of
the intertemporal elasticity of substitution in consumption. In addition, it is gross investment,
￿t is the in￿ ation rate, mt denotes the real money balances, and ￿t represents the real lump-
sum transfers that households receive from the monetary authority. Output yt is produced
by
yt = Akt; A > 0; (3)
where kt is the household￿ s capital stock. Investment adds to the stock of physical capital
according to the law of motion
_ kt = it ￿ ￿kt; k0 > 0 given, (4)
where ￿ 2 [0;1] is the capital depreciation rate.
2The representative household also faces the most generalized cash-in-advance (CIA) or
liquidity constraint as follows:
￿cct + ￿iit ￿ mt; 0 < ￿c ￿ 1 and 0 ￿ ￿i ￿ 1; (5)
where ￿c and ￿i represent the fractions of consumption and investment expenditures that must
be ￿nanced by the household￿ s real balances mt. Notice that when ￿c = 1, we recover the
model of Chen and Guo (2007).
The ￿rst-order conditions for the representative household with respect to the indicated
variables and the associated transversality conditions (TVC) are
ct : c￿￿
t = ￿mt + ￿c t; (6)
it : ￿kt = ￿mt + ￿i t; (7)
kt : _ ￿kt = (￿ + ￿)￿kt ￿ A￿mt; (8)
mt : _ ￿mt = (￿ + ￿t)￿mt ￿  t; (9)
TVC1 : lim
t!1
e￿￿t￿ktkt = 0; (10)
TVC2 : lim
t!1
e￿￿t￿mtmt = 0; (11)
where ￿mt and ￿kt are the utility values (or shadow prices) of real money balances and physical
capital, respectively, and  t represents the Lagrange multiplier for the CIA constraint (5),
which is postulated to be strictly binding in equilibrium.
We assume that the nominal money supply is growing at a constant rate ￿ > 0, hence the
resulting seigniorage returned to households as lump-sum transfers are ￿t = ￿mt. Moreover,
clearing in the goods and money markets imply that
ct + it = yt; (12)
and
_ mt = (￿ ￿ ￿t)mt: (13)
On a balanced growth path (BGP) of the economy, output, consumption, capital and real
money balances exhibit a common positive growth rate denoted by ￿. Using the transformed
variables pt ￿
￿kt
￿mt and zt ￿ ct
kt, our model￿ s equilibrium conditions can be expressed as the
3following autonomous dynamical system:
_ pt
pt
=
￿(pt￿1)
￿i + [￿ ￿ g2(zt)]
￿
￿ ￿ A
pt
￿
+ ￿ [1 ￿ g2(zt)](A ￿ ￿ ￿ zt) ￿ ￿￿ ￿ ￿g2(zt)
￿ ￿ g1(pt)g2(zt)
; (14)
_ zt
zt
=
1
￿
￿
g1(pt)
_ pt
pt
+
A
pt
￿ ￿ ￿ ￿
￿
￿ A + ￿ + zt; (15)
where g1(pt) ￿
￿i￿￿c
￿c(pt￿1)+￿i and g2(zt) ￿
(￿c￿￿i)zt
(￿c￿￿i)zt+￿iA.
A BGP equilibrium is characterized by a pair of positive real numbers (p￿;z￿) such that
_ pt = _ zt = 0. It is straightforward to derive from (14) and (15) that p￿ is the solution to the
quadratic equation
p￿ = 1 + ￿i
￿￿
1 ￿
1
￿
￿￿
A
p￿ ￿ ￿
￿
+ ￿ +
￿
￿
￿
￿ f(p￿); (16)
and that the expression for z￿ is
z￿ =
1
￿
￿
￿ + ￿ ￿
A
p￿
￿
+ A ￿ ￿: (17)
With (16) and (17), it follows that the common rate of economic growth ￿ and the correspond-
ing income velocity of money V ￿ are given by
￿ =
1
￿
￿
A
p￿ ￿ ￿ ￿ ￿
￿
or ￿ = A ￿ ￿ ￿ z￿; (18)
V ￿ =
A
￿iA + (￿c ￿ ￿i)z￿: (19)
3 Growth and Velocity E⁄ects of Money
As mentioned in the Introduction, we focus on con￿gurations in which the economy possesses
multiple balanced growth paths so that stylized fact (iii) can be accounted for. As in Suen
and Yip (2005) and Chen and Guo (2007), this requires that (i) the intertemporal-elasticity
parameter ￿ < 1, and (ii) a non-zero fraction of gross investment is subject to the liquidity
constraint ￿i 6= 0.1 These two parametric restrictions are thus maintained throughout the
subsequent analyses.2 To examine the existence and number of BGP equilibria, we ￿rst note
1Notice that ￿ < 1 is not inconsistent with empirical evidence from the U.S. aggregate time series, as found
in Eichenbaum, Hansen and Singleton (1988), among others.
2The results for the cases with ￿ ￿ 1 and/or ￿i = 0, where the model exhibits a unique BGP equilibrium,
are available upon request.
4that equilibrium p￿ can be derived from the intersection(s) of f(p￿) in (16) and the 45-degree
line. Figure 1 shows that when ￿ < 1 and 0 <￿i ￿ 1, f(p￿) is a upward-sloping concave
curve. Therefore, depending on the model￿ s structural parameter values, the number of BGP
equilibria can be zero, one or two. When f(p￿) is tangent to the 45-degree line, the economy
exhibits a unique BGP equilibrium characterized by ^ ￿ and ^ p.3 Moreover, it is straightforward
to show that
@f(p￿)
@￿ =
￿i￿
￿ > 0, hence there exists no (two) balanced growth path(s) provided
￿ < (>) ^ ￿. In the case of two BGP equilibria, the equilibrium path with a lower relative
shadow price of capital, denoted as p￿
1, will grow faster than the other that is associated with
p￿
2, that is ￿(p￿
1) > ￿(p￿
2) (see equation 18 where @￿
@p￿ < 0).
3.1 Growth E⁄ect
Figure 1 shows that a higher nominal money growth shifts the locus of f(p￿) upward. This
leads to a theoretical ambiguity regarding the sign for the growth e⁄ect of money. Intuitively,
how money a⁄ects the economy￿ s output growth depends crucially on the relative strength of
two opposing forces. On the one hand, a rise in the money growth rate ￿ generates a higher
in￿ ation, which in turn increases the cost of money holdings. As a result, the representative
household substitutes out of real balances and into capital (the portfolio substitution e⁄ect),
raising the relative shadow price of capital p￿ because of a higher demand
￿
dp￿
d￿ > 0
￿
. This
will reduce its net rate of return and thus the BGP￿ s growth rate. On the other hand, a
higher monetary growth rate ceteris paribus induces the representative household to consume
less and invest more today in exchange for higher future consumption (the intertemporal
substitution e⁄ect). This expands the supply of capital, thereby lowering its relative shadow
price
￿
dp￿
d￿ < 0
￿
. It follows that the economy￿ s output growth rate will rise.4
As shown in Figure 1, starting from the low-growth BGP equilibrium associated with
￿(p￿
2), a higher nominal money growth raises the relative shadow price of capital because of
a dominating portfolio substitution e⁄ect. Consequently, the economy displays a negative
relationship between output growth and money/in￿ ation
￿
d￿(p￿
2)
d￿ < 0
￿
. On the contrary, due
3Using f
0(^ p) = 1 and (16) evaluated at ^ p, it can be shown that that ^ ￿ and ^ p are jointly determined by
^ ￿ =
￿iA
￿iA+(^ p)2 and ^ p =
1
2
￿
1 + ￿i
￿￿ 1
^ ￿ ￿ 1
￿
￿ + ￿ +
￿
^ ￿
￿￿
.
4The portfolio substitution e⁄ect (PSE) and the intertemporal substitution e⁄ect (ISE) can be separately
identi￿ed by decomposing
dp￿
d￿ as
dp
￿
d￿
=
￿i
[1 +
￿iA
(p￿)
2
| {z }
PSE > 0
] + [￿
￿iA
￿ (p￿)
2
| {z }
ISE < 0
]
:
5to a stronger intertemporal substitution e⁄ect, the BGP￿ s growth rate and money are positively
correlated along the high-growth equilibrium path
￿
d￿(p￿
1)
d￿ > 0
￿
. Notice that both theoretical
results are consistent with stylized fact (ii) where the existing empirical evidence on the output-
growth e⁄ect of money/in￿ ation is mixed.
3.2 Velocity E⁄ect
To analyze the velocity e⁄ect of money, we ￿rst note that the BGP￿ s consumption-capital ratio
z￿ is positively correlated with the relative shadow price of capital p￿ (see equation 17). As
discussed earlier, an increase in the money growth rate ￿ yields a higher relative shadow price
of capital p￿ when the portfolio substitution e⁄ect dominates along the low-growth equilibrium
path. This will lead to decreases in investment, output, and consumption because of a lower
net rate of return on capital. Since the marginal propensity to consume is smaller than 1, the
decline in consumption will be smaller than that in output. Hence, the consumption-output
ratio rises. Due to the AK production function, the consumption-capital ratio z￿ will rise
as well, thus dz￿
d￿ > 0. Notice that this result is reversed
￿
dz￿
d￿ < 0
￿
in the high-growth BGP
equilibrium as the intertemporal substitution e⁄ect is stronger .
Next, the BGP￿ s income velocity of money V ￿ is negatively (positively) related to z￿ when
￿c > (<) ￿i (see equation 19). The intuition for this ￿nding can be understood by using (3)
and (12) to rewrite the money velocity as
V ￿ =
A
￿cz￿ + ￿i( i￿
k￿)
: (20)
In light of the AK technology, the goods-market clearing condition (12) implies that changing
z￿ generates the same magnitude of opposite movement in the investment-capital ratio i￿
k￿,
that is ￿z￿ = ￿￿
￿ i￿
k￿
￿
. It follows that in response to a rise of z￿, the denominator of (20) will
increase (decrease) if ￿c > (<) ￿i, which in turn produces a lower (higher) money velocity V ￿.
Based on the previous discussions, we ￿nd that the impact of changing ￿ on V ￿ (through
z￿) depends not only on the location of the BGP equilibrium, but also on whether the CIA-
constrained proportion of consumption expenditures ￿c exceeds or falls below the investment
fraction ￿i. Speci￿cally, on the low-growth equilibrium path associated with ￿(p￿
2) and a
stronger portfolio substitution e⁄ect, the velocity e⁄ect of money is negative (positive) when
￿c > (<) ￿i. On the contrary, since the intertemporal substitution e⁄ect dominates along the
high-growth BGP with ￿(p￿
1), the velocity e⁄ect of money is positive (negative) when ￿c > (<)
6￿i.5 The above results imply that stylized fact (i), where the rate of nominal money growth
and the income velocity of money are negatively correlated, can be obtained in either BGP
equilibrium.
4 Conclusion
We have shown that in a one-sector monetary AK model of endogenous growth with the most
generalized cash-in-advance constraint, the presence of two balanced-growth-path equilibria
o⁄ers a plausible explanation for the variations in growth experience between (for example)
the Philippines and South Korea. Moreover, we ￿nd that when the portfolio substitution e⁄ect
dominates and the liquidity constraint is applicable more to consumption purchases, the e⁄ects
of changing in￿ ation on the economy￿ s output growth and velocity are both negative along
the low-growth equilibrium path. By contrast, when the opposing intertemporal substitution
e⁄ect is stronger and the CIA-constrained fraction of gross investment is higher, the economy￿ s
high-growth BGP equilibrium exhibits a positive growth e⁄ect and a negative velocity e⁄ect
of money. As a result, each of the two BGP equilibrium paths in our model is able to account
for the three macroeconomic stylized facts discussed at the beginning of this paper.
5Notice that V
￿ =
1
￿i when ￿c = ￿i (see equation 19). In this case, the income velocity of money is
independent of changes in the nominal money growth rate
dV ￿
d￿ = 0.
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On the growth and velocity effects of money

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