# A Discrete Monetary Economic Growth Model with the MIU Approach

Authors: Wei-Bin Zhang
Journal: Discrete Dynamics in Nature and Society
 (2008)

---

## Abstract

This paper proposes an alternative approach to economic growth with money. The production side is the same as the Solow model, the Ramsey model, and the Tobin model. But we deal with behavior of consumers differently from the traditional approaches. The model is influenced by the money-in-the-utility (MIU) approach in monetary economics. It provides a mechanism of endogenous saving which the Solow model lacks and avoids the assumption of adding up utility over a period of time upon which the Ramsey approach is based.


---

## Body

## 1. Introduction

Modern analysis of the long-term interaction of inflation and capital
formation begins with Tobin’s seminal contribution [1]. Tobin deals with an isolated
economy in which “outside money” (the part of money stock which is issued by the government) competes with
real capital in the portfolios of agents within the framework of the Solow growth
model. Since then, many models of growth model of monetary economies are built
within
the OLG framework (see, [2–5]). This paper introduces money into the growth theory proposed
by Zhang in the early 1990s (see, [6]).In nonmonetary growth theory, monetary values, such as wage, rate of interests, prices
of goods and services, and land rent, are “fast variables” and are determined by balance conditions
of demand and supply of real variables. In frictionless economic systems,
issuing money has no effect on economic growth, at least in the long term. Nevertheless,
financial assets and paper claims often offer alternatives to hold wealth. In
process of exchange and division of labor, money plays an essential role in
modern economy [7–11]. In his well-known paper on long-run
effects of inflationary policies, Tobin [1] showed that an increase in the
level of the inflation rate will increase the capital stock of an economy. Sidrauski [12] constructed an economic model in which no real variable will
be affected by the economy’s inflation rate. We will address the issues by
Tobin and Sidrauski in the alternative framework. Our approach is strongly
influenced by the money-in-the-utility (MIU) approach which was initially proposed
by Patinkin [13] and Sidrauski [12]. In
this approach, money is held because it yields some services and the way to
model it is to enter real balances directly into the utility function. Sidrauski [12] made a
benchmark contribution to monetary economics, challenging Tobin’s nonneutrality
result. He proposed a framework that explicitly allows for an endogenous
treatment of saving behavior. His analytical framework is developed with
Patinkin’s idea of ensuring a well-defined demand function for money by
assuming that the agent’s utility is directly affected by money. This approach
has been widely applied in monetary growth theory (e.g., [14–17]). Rather than following the Ramsey approach, this paper introduces money-in-the-utility
function proposed by Zhang to show interactions between money and economic
growth. The paper is organized as follows. Section 2 defines the model. Section
3 proves that the dynamic system has a unique unstable equilibrium point and simulates
the model. Section 4 examines effects of changes in some parameters on the
equilibrium. Section 5 concludes the study. The appendix generalizes the model
by treating time distribution between leisure and work as endogenous variables.


## 2. The Model

We present the model in discrete time, numbered from zero, and indexed
byt
=
0
,
1
,
2
,
…
. Time 0
, being referred to the beginning of period 0
, represents the initial situation from which
economy starts to grow. The end of period t
−
1 coincides with the beginning of period t
; it can also be called time t
. We assume that transactions are made in each
period. The model assumes that each individual lives forever. The production
sector in our model is the same as that in the Solow one-sector growth model
[18, 19]. The discrete version of the
Solow-model is referred to by Diamond [20] and Azariadis [2]. It is assumed
that there is only one (durable) good in the economy under consideration.
Households own assets of the economy and distribute their incomes to consume
and save. Exchanges take place in perfectly competitive markets. Production
sectors sell their product to households or to other sectors; and households
sell their labor and assets to production sectors. Factor markets work well;
the available factors are fully utilized at every moment. Saving is undertaken
only by households, which implies that all earnings of firms are distributed in
the form of payments to factors of production, labor, managerial skill, and
capital ownership.LetK
(
t
) denote the capital existing in period t and N the flow of labor services used at time t for production. In this study, we assume N to be fixed. As our model exhibits constant
returns to scale, the dynamics will not be affected if we allow the population
to change at a constant growth rate over time. We use the conventional
production function to describe a relationship between inputs and output. The
function F
(
t
) defines the flow of production at time t
. The production process is described by some
sufficiently smooth function, F
(
t
)
=
F
(
K
(
t
)
,
N
)
. We assume that F is
neoclassical. Introduce f
(
k
(
t
)
)
≡
F
(
k
(
t
)
,
1
)
, where k
(
t
)
≡
K
(
t
)
/
N
. The function f has the following properties: (i) f
(
0
)
=
0;
(ii) f is increasing, strictly concave on R
+,
and C
2 is on R
+
+; f
'
(
k
)
>
0 and f
"
(
k
)
<
0;
and (iii) lim
⁡
k
→
0
f
'
(
k
)
=
∞ and lim
⁡
k
→
+
∞
f
'
(
k
)
=
0.
Let δ
k denote the fixed rate of capital depreciation.
Markets are competitive, thus labor and capital earn their marginal products,
and firms earn zero profits. The real rate of interest, r
(
t
)
, and real wage rate, w
(
t
)
, are determined by markets. Hence, for any
individual firm, r
(
t
) and w
(
t
) are given at each point of time. The
production sector chooses the two variables K
(
t
) and N
(
t
) to maximize its profit. The marginal
conditions are given by(2.1)
r
(
t
)
+
δ
k
=
f
'
(
k
(
t
)
)
,


w
(
t
)
≡
f
(
k
(
t
)
)
−
k
(
t
)
f
'
(
k
(
t
)
)
.We assume that agents have perfect foresight with respect to all future
events and capital markets operate frictionless. The government levies no
taxes. Money is introduced by assuming that a central bank distributes at no
cost to the population a per capita amount of fiat money M
(
t
)
>
0
. The scheme according to which the money stock
evolves over time is deterministic and known to all agents. With μ being the constant net growth rate of the
money stock, M
(
t
) evolves over time according to the following:(2.2)
M
(
t
)
=
(
1
+
μ
)
M
(
t
−
1
)
,

μ
>
0
.At the beginning of period t
, the government brings M
(
t
)
−
M
(
t
−
1
) additional units of money per capita into
circulation in order to finance all government expenditures via seigniorage.
For the seigniorage mechanism to work, injections of the additional units of
money take place before the other markets open. Let m
(
t
) stand for the real value of money per capita
measured in units of the output good, that is, m
(
t
)
=
M
(
t
)
/
P
(
t
)
. Then, we may rewrite the above equation as(2.3)
τ
(
t
)
=
M
(
t
)
−
M
(
t
−
1
)
P
(
t
)
=
μ
1
+
μ
m
(
t
)
.The
representative household receives μ
m
/
(
1
+
μ
) units of paper money from the government through
a “helicopter drop,” also considered to be independent of his money holdings.
The inflation rate, π
(
t
)
, is given by(2.4)
π
(
t
)
=
P
(
t
+
1
)
−
P
(
t
)
P
(
t
)
. From m
(
t
)
=
M
(
t
)
/
P
(
t
) and M
(
t
)
=
(
1
+
μ
)
M
(
t
−
1
)
, we have(2.5)
π
(
t
)
=
(
1
+
μ
)
m
(
t
)
m
(
t
+
1
)
−
1
.According to the definition of k
(
t
)
, per capita physical wealth is equal to k
(
t
)
. Per capita real current income from the
interest payment, r
(
t
)
k
(
t
)
, and the wage payment, w
(
t
)
, is given by(2.6)
y
(
t
)
=
r
(
t
)
k
(
t
)
+
w
(
t
)
.We call y
(
t
) the current
income in the sense that it comes from consumers’ daily toils (payment
for human capital) and consumers’ current earnings from ownership of wealth. As
in [6], the disposable income in real terms, y
^
, is(2.7)
y
^
(
t
)
=
r
(
t
)
k
(
t
)
+
w
(
t
)
+
a
(
t
)
,where a
(
t
)
≡
k
(
t
)
+
m
(
t
)
. We assume
that in each point of time the
consumer’s utility function for holding money, consuming goods, and saving is
represented by the following utility function:(2.8)
U
(
t
)
=
m
ε
0
(
t
)
c
ξ
0
(
t
)
s
λ
0
(
t
)
,

ε
0
,
ξ
0
,
λ
0
>
0
,where ε
0 is called propensity to hold money, ξ
0 propensity to consume, and λ
0 propensity to own wealth. Here the specified functional form
already implies the assumption that real balances and consumption are Edgeworth
complements (u
c
m
>
0). If the assumption is replaced with the
assumption of Edgeworth substitutability (u
c
m
<
0), then the dynamic properties may be
affected. Benhabib et al. [21] show how these two assumptions lead to
different dynamic properties of monetary economies in the Ramsey approach.The real price of holding money is1
+
r
(
t
)
+
π
(
t
)
. The budget constraint is given by(2.9)
(
1
+
r
(
t
)
+
π
(
t
)
)
m
(
t
)
+
c
(
t
)
+
s
(
t
)
=
y
^
(
t
)
.Insert (2.7) into the budget constraint(2.10)
(
r
(
t
)
+
π
(
t
)
)
m
(
t
)
+
c
(
t
)
+
s
(
t
)
=
y
a
(
t
)
≡
(
1
+
r
(
t
)
)
k
(
t
)
+
w
(
t
)
,where we use a
(
t
)
=
k
(
t
)
+
m
(
t
)
. Consumers’
problem is to choose money, consumption, and savings in such a way that utility
levels are maximized. Maximizing U
(
t
) subject to the budget constraint (2.10) yields(2.11)
(
r
(
t
)
+
π
(
t
)
)
m
(
t
)
=
ε
y
a
(
t
)
,


c
(
t
)
=
ξ
y
a
(
t
)
,


s
(
t
)
=
λ
y
a
(
t
)
,where(2.12)
ε
≡
ρ
ε
0
,


ξ
≡
ρ
ξ
0
,


λ
≡
ρ
λ
0
,

ρ
≡
1
ε
0
+
ξ
0
+
λ
0
.The expenditure spent on “consuming money,” (
r
+
π
)
m
, is proportional to the potential disposable
income, y
a
(
t
)
, and the (relative) propensity to use money. We
see that m
(
t
) is negatively related to r
+
π
. This relation is assumed in the Tobin model
and the Levhari and Patinkin’s monetary model [22]. In
our approach, this relation results from optimal behavior of households.According to the definitions ofa
(
t
)
,
s
(
t
),
and τ
(
t
+
1
)
, as the consumer determines his/her
savings in period t by maximizing the utility level for that
period, the real wealth changes as follows:(2.13)
a
(
t
+
1
)
=
s
(
t
)
+
τ
(
t
+
1
)
.We have
thus built the model.


## 3. The Motion, Equilibrium, and Stability

This section examines dynamic properties of the system. We now find dynamics of
capital and real money. From the definition ofy
a
(
t
) and (
r
(
t
)
+
π
(
t
)
)
m
(
t
)
=
ε
y
a
(
t
)
, we have(3.1)
(
r
(
t
)
+
π
(
t
)
)
m
(
t
)
=
(
1
+
r
(
t
)
)
ε
k
(
t
)
+
ε
w
(
t
)
.Inserting (2.5) in the above equation, we solve(3.2)
m
(
t
+
1
)
=
[
(
1
+
r
(
t
)
)
ε
k
(
t
)
+
w
(
t
)
ε
m
(
t
)
+
1
−
r
(
t
)
]
−
1
m
(
t
)
(
1
+
μ
)
.Inserting (2.1) in the above equation yields(3.3)
m
(
t
+
1
)
=
Ω
m
(
k
(
t
)
,
m
(
t
)
)
,where(3.4)
Ω
m
(
k
(
t
)
,
m
(
t
)
)
≡
[
δ
ε
k
(
t
)
+
ε
f
(
k
(
t
)
)
m
(
t
)
+
1
+
δ
k
−
f
'
(
k
(
t
)
)
]
−
1
m
(
t
)
(
1
+
μ
)
,where δ
≡
1
−
δ
k
. Hence, we can express m
(
t
+
1
) as a unique function of k
(
t
) and m
(
t
)
.From (2.13) and (2.3), we have(3.5)
k
(
t
+
1
)
=
s
(
t
)
−
m
(
t
+
1
)
1
+
μ
,where we use a
(
t
)
≡
k
(
t
)
+
m
(
t
)
. From the
definition of y
a
(
t
) and (2.1), we obtain(3.6)
y
a
(
t
)
=
δ
k
(
t
)
+
f
(
k
(
t
)
)
.Substituting s
(
t
)
=
λ
y
a
(
t
) and (3.6) into (3.5) yields(3.7)
k
(
t
+
1
)
=
δ
λ
k
(
t
)
+
λ
f
(
k
(
t
)
)
−
Ω
m
(
k
(
t
)
,
m
(
t
)
)
1
+
μ
,where we also use (3.3). In summary, we have the following lemma.Lemma 3.1.
The motion ofk
(
t
) and m
(
t
) can be determined by (3.7) and (3.3). For any
positive solution, k
(
t
)  and m
(
t
)
, of difference equations (3.7) and (3.3), all the
other variables are uniquely determined by the following procedure: a
(
t
)
=
k
(
t
)
+
m
(
t
)
→
r
(
t
)  and w
(
t
)  by (2.1) →
y
a
(
t
)  by (3.6) →
c
(
t
)  and s
(
t
)  by (2.11) →
π
(
t
)  by (2.5) →
f
(
k
(
t
)
)
→
F
(
t
)
=
N
f
(
k
(
t
)
)
.We now examine properties of the dynamic system. From (3.7) and (3.3), an
equilibrium point is determined by(3.8)
1
−
[
δ
ε
k
+
ε
f
(
k
)
m
+
1
+
δ
k
−
f
'
(
k
)
]
−
1
(
1
+
μ
)
=
0
,

k
=
δ
λ
k
+
λ
f
(
k
)
−
m
1
+
μ
, where we neglect another possible solution of m
=
0
. It is known that when m
=
0
, the model is identical to the nonmonetary
one-sector model proposed by Zhang [6, Chapter 2]. In the case of m
=
0
, the system has a unique stable equilibrium.From the second equation, in (3.8) we
solve m as a function of k as follows:(3.9)
m
=
λ
(
1
+
μ
)
[
f
(
k
)
−
λ
1
k
]
,where λ
1
≡
1
/
λ
−
δ
>
0
. For m to be positive, it is necessary to require f
/
k
>
λ
1
. Denote by k
* the value of k such that f
/
k
=
λ
1
. As f
/
k tends to be large for small k and small for large k
, we see that k
* always exists. As f
/
k falls in k
, we see that for m to be positive, we should require 0
<
k
<
k
*
. Taking derivatives of (3.9) with respect to k yields(3.10)
d
m
d
k
=
λ
(
1
+
μ
)
(
f
'
−
λ
1
)
.We see that the
sign of the impact of change in the capital intensity is the same as that of f
'
−
λ
1
. Denote by k
*
* the value of k such that f
'
=
λ
1
. As f
/
k
>
f
' for k
>
0
, we conclude 0
<
k
*
*
<
k
*
.Inserting (3.9) into the first equation in (3.8)
yields(3.11)
H
(
k
)
≡
ε
(
δ
k
+
f
)
λ
(
1
+
μ
)
(
f
'
−
λ
1
)
−
f
'
(
k
)
−
μ
+
δ
k
=
0
.It is straightforward to check the following: H
(
0
)
<
0 and H
(
k
*
*
)
>
0
. Hence, there is at least one positive solution
for 0
<
k
<
k
*
*
. For k
≥
k
*
*
, we have(3.12)
ε
(
δ
k
+
f
)
λ
(
1
+
μ
)
(
f
'
−
λ
1
)
−
f
'
(
k
)
<
0
.Hence, if μ
≥
δ
k or μ
−
δ
k is small in the case of μ
<
δ
k
,
H
(
k
) will always be negative. This implies that it
is reasonable to consider that any meaningful solution is subject to 0
<
k
<
k
*
*
. As(3.13)
d
H
d
k
=
ε
(
δ
+
f
'
)
λ
(
1
+
μ
)
(
f
'
−
λ
1
)
−
(
δ
k
+
f
)
ε
f
"
λ
(
1
+
μ
)
(
f
'
−
λ
1
)
2
−
f
"
>
0
,

0
<
k
<
k
*
*
,we conclude that there is a unique
solution of H
(
k
)
=
0 for 0
<
k
<
k
*
*
. The two eigenvalues at the equilibrium point, ϕ
1 and ϕ
2
, are given by(3.14)
ϕ
2
−
[
(
δ
+
f
'
)
λ
+
Ω
2
−
Ω
1
1
+
μ
]
ϕ
+
(
δ
+
f
'
)
λ
Ω
2
=
0
,where(3.15)
Ω
1
=
−
[
(
δ
+
f
'
)
ε
m
−
f
"
]
m
(
1
+
μ
)
<
0
,


Ω
2
=
1
+
(
δ
k
+
f
)
ε
(
1
+
μ
)
m
>
0
.We have(3.16)
ϕ
1
+
ϕ
2
=
(
δ
+
f
'
)
λ
+
Ω
2
−
Ω
1
1
+
μ
>
0
,


ϕ
1
ϕ
2
=
(
δ
+
f
'
)
λ
Ω
2
>
0
.We see that the two eigenvalues are
positive. From the definitions of Ω
j and (3.9), we have(3.17)
ϕ
1
ϕ
2
=
(
δ
+
f
'
)
λ
[
1
+
(
δ
k
+
f
)
ε
(
1
+
μ
)
m
]
>
0
,
ϕ
1
+
ϕ
2
=
(
δ
+
f
'
)
λ
+
1
+
(
δ
k
+
f
)
ε
(
1
+
μ
)
m
+
(
δ
+
f
'
)
ε
(
1
+
μ
)
2
−
m
f
"
(
1
+
μ
)
2
>
0
, where we also use m
=
λ
(
1
+
μ
)
[
f
(
k
)
−
λ
1
k
]
. From f
'
−
λ
1
>
0 at the equilibrium point and from the
definition of λ
0
, we have (
f
'
+
δ
)
λ
>
1
. From this inequality and (3.17), we have(3.18)
ϕ
1
ϕ
2
>
1
,

ϕ
1
+
ϕ
2
>
2
,

ϕ
1
,
ϕ
2
>
0
.This implies that at least one of the two eigenvalues is larger than
unit. Hence, the system is unstable. In summary, we have the following theorem.Theorem 3.2.
Let
μ
≥
δ
k or μ
−
δ
k be small in the case of μ
<
δ
k
. The dynamical system has a unique unstable
equilibrium.It should be noted that even ifμ
−
δ
k is large in the case of μ
<
δ
k
, the conclusion of Theorem 3.2 may still hold. We
now demonstrate Theorem 3.2 with simulation. We specify N
=
1 and F
=
A
K
α
N
β
. We have(3.19)
r
=
α
f
k
−
δ
k
,


w
=
β
f
,where f
=
A
k
α
. We have the dynamics as follows:(3.20)
k
(
t
+
1
)
=
(
δ
k
+
f
)
λ
−
Ω
m
(
t
)
1
+
μ
,


m
(
t
+
1
)
=
Ω
m
(
t
)
,where(3.21)
Ω
m
(
t
)
≡
[
(
δ
k
+
f
)
ε
m
+
1
+
δ
k
−
α
f
k
]
−
1
m
(
1
+
μ
)
.We specify the parameters as follows:(3.22)
A
=
0.9
,


λ
0
=
0.8
,


ξ
0
=
0.08
,


ε
0
=
0.03
,


α
=
0.35
,


δ
k
=
0.06
,


μ
=
0.03
.It can be shown that (3.14) has a
unique solution as shown in Figure 1.Figure 1
The existence of a unique equilibrium point.The equilibrium values of the
variables are given as follows:(3.23)
k
=
0.790
,


m
=
0.576
,


a
=
1.362
,


r
=
0.307
,


w
=
0.539
,
f
=
0.829
,


c
=
0.138
,


s
=
1.381
.The two eigenvalues, ϕ
1 and ϕ
2
, are given as(3.24)
ϕ
1
=
1.709
,


ϕ
2
=
0.732
.Hence, the equilibrium point is a
saddle point. We simulate the model with the initial point (
k
0
,
m
0
)
=
(
0.7
,
0.3
)
. We simulate the model with 9 periods. We plot the motion in Figure 2. It
should be remarked that as the system is unstable, the system does not converge
to the equilibrium point with the specified initial condition.Figure 2
The motion of physical capital and real money.


## 4. Comparative Statics Analysis

This section studies effects of changes in some parameters on the
equilibrium. It should be remarked that as the system is unstable, when as the
parameters are changed, the system may not move from one steady state to
another even when the system is initially located at a steady state. Different
from the situation when the system has a unique stable steady state, the
comparative statics analysis in the unstable case provides only some insights
into the properties of the dynamic systems.


### 4.1. The Inflation Policy

One of the important issues in monetary growth economics is effects of
change of inflation rate,μ
. We now examine effects of change in μ on the economic equilibrium. It should be
noted that as the system has a unique equilibrium point, the comparative static
analysis examines the shift of the equilibrium point as parameters are changed.
Taking derivatives of (3.11) with respect to μ yields(4.1)
∂
H
∂
k
d
k
d
μ
=
1
+
ε
(
δ
k
+
f
)
λ
(
1
+
μ
)
2
(
f
'
−
λ
1
)
+
1
>
0
,where ∂
H
/
∂
k
>
0 as demonstrated in (3.13). We conclude that as
the Tobin model predicts, as the inflation rate is increased, the per capita
physical capital is increased. From (2.1), we obtain(4.2)
d
f
d
μ
=
f
'
d
k
d
μ
>
0
,


d
r
d
μ
=
f
"
d
k
d
μ
<
0
,


d
w
d
μ
=
−
k
f
"
d
k
d
μ
>
0
.The output and wage rate are increased and the rate of interest is
reduced.From (3.9), we have(4.3)
1
m
d
m
d
μ
=
1
1
+
μ
+
f
'
−
λ
1
f
−
λ
1
k
d
k
d
μ
>
0
.We see that as the inflation rate is increased, the real money is
increased. From a
=
k
+
m
, the total wealth is increased. From (2.11) and π
=
μ at equilibrium, we have(4.4)
c
=
(
r
+
μ
)
ξ
m
ε
.From (4.4), we
have(4.5)
1
c
d
c
d
μ
=
1
r
+
μ
+
1
1
+
μ
+
(
f
'
−
λ
1
f
−
λ
1
k
+
f
"
r
+
μ
)
d
k
d
μ
.As(4.6)
f
'
−
λ
1
f
−
λ
1
k
>
0
,


f
"
r
+
μ
<
0
,a rise in the inflation rate increases the consumption level if the
absolute value of f
"
/
(
r
+
μ
) is relatively small.


### 4.2. The Propensity to Use Money

Taking derivatives of (3.11) with respect to ε
0 yields(4.7)
∂
H
∂
k
d
k
d
ε
0
=
[
1
+
ε
λ
(
f
'
−
λ
1
)
]
(
δ
k
+
f
)
λ
0
(
1
+
μ
)
(
f
'
−
λ
1
)
>
0
.Hence, as ε
0 is increased, k is increased. From (2.1), we obtain(4.8)
d
f
d
ε
0
=
f
'
d
k
d
ε
0
>
0
,


d
r
d
ε
0
=
f
"
d
k
d
ε
0
<
0
,


d
w
d
ε
0
=
−
k
f
"
d
k
d
ε
0
>
0
.The output and wage rate are increased and the rate of interest is
reduced.From (3.9), we have(4.9)
1
m
d
m
d
ε
0
=
−
ρ
−
k
(
f
−
λ
1
k
)
λ
0
+
f
'
−
λ
1
f
−
λ
1
k
d
k
d
ε
0
.As the first two terms in the right-hand side are
negative and the last term is positive, the impact on the real money is
ambiguous. From a
=
k
+
m
, the impact on the total wealth is ambiguous. From (4.4), we
have(4.10)
1
c
d
c
d
ε
0
=
(
f
"
d
k
d
ε
0
+
1
)
1
r
+
μ
+
1
m
d
m
d
ε
0
−
1
ε
0
.


### 4.3. The Propensity to Save

Taking derivatives of (3.11) with respect to λ
0 yields(4.11)
∂
H
∂
k
d
k
d
λ
0
=
−
[
1
+
(
ε
0
+
ξ
0
)
λ
0
(
f
'
−
λ
1
)
]
ε
(
δ
k
+
f
)
λ
0
λ
(
1
+
μ
)
(
f
'
−
λ
1
)
<
0
.As the propensity to save is increased, the per capita physical wealth
is increased. From (2.1), we obtain(4.12)
d
f
d
λ
0
=
f
'
d
k
d
λ
0
<
0
,


d
r
d
λ
0
=
f
"
d
k
d
λ
0
>
0
,


d
w
d
λ
0
=
−
k
f
"
d
k
d
λ
0
<
0
.The output and wage rate are reduced and the rate of interest is
increased.From (3.9), we have(4.13)
1
m
d
m
d
λ
0
=
ε
+
ξ
λ
0
+
f
'
−
λ
1
f
−
λ
1
k
d
k
d
λ
0
+
(
ε
+
ξ
)
k
(
f
−
λ
1
k
)
λ
λ
0
.As the propensity to save is changed, the impact on the real money is
ambiguous. From (4.4), we have(4.14)
1
c
d
c
d
λ
0
=
(
f
"
d
k
d
λ
0
+
1
)
1
r
+
μ
+
1
m
d
m
d
λ
0
.


### 5. Conclusions

We proposed a one-sector monetary growth model with the MIU approach.
The model is much influenced by the Solow-model, the Ramsey model, the Tobin
model, and the MIU approach in monetary economics. The main deviation from the
traditional approaches is that we proposed an alternative approach to behavior
of consumers. It provides a mechanism of endogenous capital and money. The
dynamics is two-dimensional. In comparison with the Ramsey approach which would
lead to four-dimensional dynamics for a similar problem, the dimension in our
approach is reduced. It should be mentioned that the utility function used in
this study has been applied to different fields of economics by Zhang (e.g., [6]).


## 4.1. The Inflation Policy

One of the important issues in monetary growth economics is effects of
change of inflation rate,μ
. We now examine effects of change in μ on the economic equilibrium. It should be
noted that as the system has a unique equilibrium point, the comparative static
analysis examines the shift of the equilibrium point as parameters are changed.
Taking derivatives of (3.11) with respect to μ yields(4.1)
∂
H
∂
k
d
k
d
μ
=
1
+
ε
(
δ
k
+
f
)
λ
(
1
+
μ
)
2
(
f
'
−
λ
1
)
+
1
>
0
,where ∂
H
/
∂
k
>
0 as demonstrated in (3.13). We conclude that as
the Tobin model predicts, as the inflation rate is increased, the per capita
physical capital is increased. From (2.1), we obtain(4.2)
d
f
d
μ
=
f
'
d
k
d
μ
>
0
,


d
r
d
μ
=
f
"
d
k
d
μ
<
0
,


d
w
d
μ
=
−
k
f
"
d
k
d
μ
>
0
.The output and wage rate are increased and the rate of interest is
reduced.From (3.9), we have(4.3)
1
m
d
m
d
μ
=
1
1
+
μ
+
f
'
−
λ
1
f
−
λ
1
k
d
k
d
μ
>
0
.We see that as the inflation rate is increased, the real money is
increased. From a
=
k
+
m
, the total wealth is increased. From (2.11) and π
=
μ at equilibrium, we have(4.4)
c
=
(
r
+
μ
)
ξ
m
ε
.From (4.4), we
have(4.5)
1
c
d
c
d
μ
=
1
r
+
μ
+
1
1
+
μ
+
(
f
'
−
λ
1
f
−
λ
1
k
+
f
"
r
+
μ
)
d
k
d
μ
.As(4.6)
f
'
−
λ
1
f
−
λ
1
k
>
0
,


f
"
r
+
μ
<
0
,a rise in the inflation rate increases the consumption level if the
absolute value of f
"
/
(
r
+
μ
) is relatively small.


## 4.2. The Propensity to Use Money

Taking derivatives of (3.11) with respect to ε
0 yields(4.7)
∂
H
∂
k
d
k
d
ε
0
=
[
1
+
ε
λ
(
f
'
−
λ
1
)
]
(
δ
k
+
f
)
λ
0
(
1
+
μ
)
(
f
'
−
λ
1
)
>
0
.Hence, as ε
0 is increased, k is increased. From (2.1), we obtain(4.8)
d
f
d
ε
0
=
f
'
d
k
d
ε
0
>
0
,


d
r
d
ε
0
=
f
"
d
k
d
ε
0
<
0
,


d
w
d
ε
0
=
−
k
f
"
d
k
d
ε
0
>
0
.The output and wage rate are increased and the rate of interest is
reduced.From (3.9), we have(4.9)
1
m
d
m
d
ε
0
=
−
ρ
−
k
(
f
−
λ
1
k
)
λ
0
+
f
'
−
λ
1
f
−
λ
1
k
d
k
d
ε
0
.As the first two terms in the right-hand side are
negative and the last term is positive, the impact on the real money is
ambiguous. From a
=
k
+
m
, the impact on the total wealth is ambiguous. From (4.4), we
have(4.10)
1
c
d
c
d
ε
0
=
(
f
"
d
k
d
ε
0
+
1
)
1
r
+
μ
+
1
m
d
m
d
ε
0
−
1
ε
0
.


## 4.3. The Propensity to Save

Taking derivatives of (3.11) with respect to λ
0 yields(4.11)
∂
H
∂
k
d
k
d
λ
0
=
−
[
1
+
(
ε
0
+
ξ
0
)
λ
0
(
f
'
−
λ
1
)
]
ε
(
δ
k
+
f
)
λ
0
λ
(
1
+
μ
)
(
f
'
−
λ
1
)
<
0
.As the propensity to save is increased, the per capita physical wealth
is increased. From (2.1), we obtain(4.12)
d
f
d
λ
0
=
f
'
d
k
d
λ
0
<
0
,


d
r
d
λ
0
=
f
"
d
k
d
λ
0
>
0
,


d
w
d
λ
0
=
−
k
f
"
d
k
d
λ
0
<
0
.The output and wage rate are reduced and the rate of interest is
increased.From (3.9), we have(4.13)
1
m
d
m
d
λ
0
=
ε
+
ξ
λ
0
+
f
'
−
λ
1
f
−
λ
1
k
d
k
d
λ
0
+
(
ε
+
ξ
)
k
(
f
−
λ
1
k
)
λ
λ
0
.As the propensity to save is changed, the impact on the real money is
ambiguous. From (4.4), we have(4.14)
1
c
d
c
d
λ
0
=
(
f
"
d
k
d
λ
0
+
1
)
1
r
+
μ
+
1
m
d
m
d
λ
0
.


## 5. Conclusions

We proposed a one-sector monetary growth model with the MIU approach.
The model is much influenced by the Solow-model, the Ramsey model, the Tobin
model, and the MIU approach in monetary economics. The main deviation from the
traditional approaches is that we proposed an alternative approach to behavior
of consumers. It provides a mechanism of endogenous capital and money. The
dynamics is two-dimensional. In comparison with the Ramsey approach which would
lead to four-dimensional dynamics for a similar problem, the dimension in our
approach is reduced. It should be mentioned that the utility function used in
this study has been applied to different fields of economics by Zhang (e.g., [6]).



---

*Source: 435787-2008-04-22.xml*