数理解析研究所講究録 1443 巻 2005 年 81-94

81

A short-run and long-run analysis of a quasi-stochastic monetary economy
by*Tamotsu Nakamura
Graduate School of Economics, Kobe University
Kobe University
2-1 Rokkodai
Kobe 657-8501
Japan
Phone & Fax Number: +81-78-803-6843
Email: nakamura@econ.kobe-u.ac.jp

February 2005
Ab stract
To analyze the liquidity trap in a dynamic optimization framework, most studies assume thatmoney has inherent utility. Instead of assuming money-in-utility, this paper considers

uninsurable idiosyncratic risks as the source of money demand to investigate the short-run
economic fluctuations as well as the long-run economic growth. Individuals face uncertainty

over the return to capital, and hence invest both physical capital and money as a risk
diversification. To distinguish the effect of intertemporal substitution from that of risk-aversion,
we utilize a non-expected utility maximization approach. in the sho t-run, due to uncertainty,
the economy may falls into the liquidity trap in which an increase in money supply does notpush down the interest rate because the money demand based on precautionary motives absorb
all the money. In the long-run, there exists the optimal growth rate of money supply, which
depends not only on the degree of risk-aversion but also crucially on the elasticity ofintertemporal substitution.

I would like to thank Professors Hideyuki Adachi, Chiaki Hara, Tetsugen Haruyama, andseminar participants at ffie Kyoto University. The first author gratefully acknowledges the Kobe
University 21st Century COE Program, the research grant from the Japanese Ministry of
Education and Science. The authors are solely responsible for remaining errors
$*$

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1. Introduction
As is well-known, the Japanese economy has been in the serious slump for more than a decadewith very-low nominal interest rates and low inflation or even deflation, as Figure-l shows.

1

This kind of situation is treated as a special case as a “liquidity trap” in the IS-LM models.
—––$\ovalbox{\tt\small REJECT} 11]024168$$|\underline{---2460462\ovalbox{\tt\small REJECT}---\mathrm{N}\mathrm{o}\min \mathrm{a}\mathrm{l}\mathrm{G}\mathrm{D}\mathrm{P}\mathrm{G}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{e}-\mathrm{l}\mathrm{n}\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{e}}|\underline{-- \mathrm{N}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{I}r’ \mathrm{t}\mathrm{e}\mathrm{r}\overline{\mathrm{e}}\mathrm{s}\mathrm{t}\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{e}-------}----$-

Figure-l Recent Movements of GDP, Inflation Rates, and Nominal Interest Rates in Japan

There is the growing literature that focuses on this problem based upon rigorous andexplicit microfoundations. For example, Benhabib, Schm$\mathrm{i}\mathrm{t}\mathrm{t}$-Grohe and Uribe $(2001, 2002)$ ,Buiter and Panigirtzoglou (2003), and Ono $(1994, 2001)$ , capturing the interactions between

forward-looking prices and the agents’ intertemporal maximizing behavior, discuss thepossibility of liquidity trap and the economic policies to avoid or to escape from the trap. In

essence, they are variants of the seminal model of Brock (1975), where money has inherent

utility. Hence, similarly to Brock, most studies on the liquidity trap assume an endowment
1

The inflation rates here are the GDE (Gross Domestic Expenditure) deflators ($=\mathrm{G}\mathrm{D}\mathrm{P}$deflators)

while the nominal interest rates are the yields of the short-term (13 week) government. The dataon the GDP and deflator are taken from the Cabinet Office homepage$(\mathrm{h}\mathrm{t}\mathrm{p};//\mathrm{w}\mathrm{w}\mathrm{w}.\mathrm{c}\mathrm{a}\mathrm{o}.\mathrm{g}\mathrm{o}.\mathrm{j}\mathrm{p}/)$83

economy, i.e., no capital accumulation.
Instead of assuming money-in-utility, this paper focuses on the money demand as a risk

diversification measures. Of course, people are happy with money. But, most of their happinesscomes from its purchasing power, and hence indirectly from consumption. Cash-in-advancemodels capture this feature of money. As Keynes (1936) emphasizes, however, people havemoney as the asset especially in an uncertain world since the value of money is considered morestable than that of other assets. Introducing productivity shocks to physical capital, the model

presented in this paper derives the precautional demand for money and discussed the possibilityof liquidity trap in the short-run.

Allowing the capital stock to change, the paper also investigates the relationship between
growth and inflation in the long-run, Intertemporal substitution plays a key role in the analysis.
In order to distinguish the intertemporal substitution from the risk-aversion, we use KrepsPorteus non-expected utility preferences instead of time- and state-separable isoelastic

preferences.

The organization of the rest of this paper is as follows. Section 2 presents the simple

stochastic optimization model of an individual with a non-expected utility preference. Section 3
investigates the short-run and long-run properties of a quasi-stochastic macroeconomy, The
final section provides some concluding remarks.

2. The Model
Consider an economy that consists of a continuum of identical individuals: each owns a firmand produces a homogenous good according to a stochastic production function:$dY(t)$ $=AK(t)[dt+odz(t)]$where$K\acute{(}t$

with

$A$

$>0$ ,(1)) is the individual’s capital stock. In each period, the deterministic flow of eachfirm’s production is $AK(t)dt$ . In addition to this deterministic part, there is also a stochastic

84

component of production $AK$ (t)o&(t) due to idiosyncratic technology shocks, where $dz(t)$ isa Wiener process with mean zero and unit variance, and parameter a is the instantaneous
standard deviation of the technology shock.
Since the rate of return to capital of each firm is equal to the marginal product of capital,

it becomes(2)$r(t)=A[dt +\sigma dz(t)]$ ,Each individual knows that each firm faces its idiosyncratic risks, and hence wants to hedge therisks by holding money as a risk diversification. Although these risks are assumed to beuninsurable at the individual level, there is no aggregate uncertainty assuming that theindividuals’ risks are cancelled out each other.
The budget constraint of the representative consum er is given by$P(t)C(t)dt+dM(t)+P(t)I(t)dt=r(t)P(t)K(t)$ ,(3)

where $P(t)$ is output price, $dM(t)$ is the nominal money demand, $C(t)$ is consumption and$I(t)$

is the fixed investment. Assuming no depreciation in physical capital for simplicity, thecapital per capita evolves according to the following:$dK(t)=I(t)dt$ .

(4)

The budget constraint in real term is expressed as$C(t)dt+ \frac{dM(t)}{P(t)}+dK(t)=r(t)$

(l) .

(5)

Defining the total nominal asset as $W(t)$ , or$W(t)=M(t)+P(t)K(t)$ ,(6)

the real asset becomes$w(t)=m(t)+K(t)$ ,where $w(t)\equiv W(t)/P(t)$ andbe rewritten a$\mathrm{s}$(7)$\mathrm{w}(\mathrm{t})\equiv M(t)/P(t)$. Hence, the budget constraint in real term can

85$dw(t)=[A(w(t)-m(t))-\pi(t)m(t)-C(t)]dt-\sigma A(w(t)-m(t))dz$.

(8)This can also be expressed as follows:$dw(t)=[r_{R}w(t)-r_{N}(t)m(t) -C(t)]dt-\sigma A(w(t)-m(t))dz$ ,where$r_{R}=\mathrm{A}$(9)is the mean of a real interest rate, $r_{N}(t)=A$ $+\pi(t)$ is the mean of a nominalinterest rate.
The utility of the individual depends only on consumption $C(t)$ . To distinguish the effectof intertemporal substitution from that of risk-aversion, we employ a non-expected utility

maximization setllp. 2 We assume that at point in time$t$

the individual maximizes theintertemporal objective $V(t)$ by recursion,$f([1- \gamma]V(t))=(\frac{1-\gamma}{1-1/\epsilon})C(t)^{1-1/\epsilon}h+e^{-\rho/l}f([1-\gamma]E_{l}V(t+h))$

(10)

,

where the function $f(x)$ is given by$f(x)=( \frac{1-\gamma}{1-1/\epsilon})x^{(1-1/\epsilon)/(1-\gamma)}$

In (10),

$h$

.

(11)

is the economic decision interval,$E_{l}$

is a mathematical expectation conditional ontime-/ information, and $\rho>0$ the subjective discount rate. The parameter $\gamma>0$ measures therelative risk-aversion while the parameterWhen$\gamma=1/\epsilon$$\epsilon$$>0$

is the intertemporal substitution$\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y}.3$, so that $f(x)=x$ , our setup is the standard state- and time-separable expected-

utility setup, which does not allow independent variation in risk aversion and intertemporal
substitutability over2$\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}.4$

For detailed treatment of “recursive utility,” see Duffie and Esptein (1992).

For a detailed discussion on the roles of these parameters and more general preference setups,
see, for example, Kreps and Porteus $(1979, 1979)$ , Epstein and Zin $(1989, 1991)$ , Weil (1989),
3

and Obstfeld
4$(1994\mathrm{a}, 1994\mathrm{b})$.

This paper analyzes the individual’s behavior in the limit as$h$becomes infinitesimally$\mathrm{s}\mathrm{m}\mathrm{a}1$

86

Let $J(w(t))$ denote the maximum feasible level of the expected sum of discounted

utilities. The value function $J(w(t))$ depends on the contemporaneous variable $w(t)$ only.
Applying Ito’s lemma to the maximization of $V(t)$ in (10), we get the following stochastic

Bellman equation:$0= \max_{C,m}\{[(1-\gamma)/(1-1/\epsilon)]C^{1-1/\epsilon}-ff([1-\gamma]J(w))$$+(1-\gamma)f’([1-\gamma]J(w))[J’(w)(\lrcorner 4(w-m)-m\iota-C)+(1/2)J’(w)(\sigma \mathrm{A}(w-m))^{2}]\}$

.

(12)

(For notational convenience, time arguments are suppressed as long as no ambiguity results.)

From (12), the first-order conditions with respect to$C^{-1/\epsilon}-f’([1-\gamma]J(w))J’(w)=0$

$C$

and

$m$

are

.

(13a)

(A $+\pi$ ) $J’(w)+(\sigma \mathrm{A})^{2}(w-m)J’(w)=0$ .

(13b)

Eq. (10)’ form suggests that $J(w)$ is given by$\mathrm{s}$$J(w)=(bw)^{1-\gamma}/(1-\gamma)$

where

$b$

(14)

is a positive constant to be determined. Eqs. (13a) and (13b) become$C=uw$.

(15a)$m^{d}=(1- \frac{\mathrm{A}+\pi}{\gamma(\sigma \mathrm{A})^{2}})w$

where

,

$u\equiv b^{1-\epsilon}$and$m^{d}$.

(15b)is the money demand. Noticing that $A+\pi$ is a nominal interest rate,(15a) shows that the money demand is a decreasing function of the nominal interest rate, risk-

aversion coefficient 7, and the risk
Substituting (14),When$\gamma=1/\epsilon$$(1 5\mathrm{a})$and$\sigma$$(1 5\mathrm{b})$, (10) implies that as hsum of utilities:.

for (12) gives$arrow \mathrm{O}$, $V(t)$ becomes the standard setup of discounted$\mathrm{V}\{\mathrm{t})=E_{l}\{(1-\gamma)^{-1}\int^{\mathrm{o}}C(t)^{1-\gamma}e^{-p(s-l)}ds\}$.

87$u= \epsilon(p+\pi)+\frac{(1-\epsilon)(A+\pi)^{2}}{2\gamma(\sigma \mathrm{A})^{2}}-\pi$

(16a)

,

and therefore(16b)$b=[ \epsilon(p+\pi)+\frac{(1-\epsilon)(\mathrm{A}+\pi)^{2}}{2\gamma(\sigma \mathrm{A})^{2}}-\pi]^{1/(\mathrm{I}-\epsilon)}$Evidently from (16a) together with (15a), whether an increase in the nominal interest rate$\mathrm{A}+\pi$

raises consumption depends crucially on the intertemporal substitution elasticity$\epsilon$.

3. The Analysis
For simplicity, the population of individuals is normalized to unity. Since there is no aggregate

uncertainty, the aggregate output at each point in time becomes$Y=AK$ .

(17)

Also, the differential equation on the time-path of the aggregate asset is$\dot{w}=\mathrm{A}(w-m)-fim-uw$

.

(18)

3-1. The Short-Run Analysis
In this subsection, assuming no capital accumulation we will analyze the short-run dynamics ofthe economy. In what follows, the capital stock is assumed to be fixed at$\overline{K}$, or$K=\overline{K}$. Henceour economy here the stochastic version of endowment economy analyzed in Benhabib,
Schmitt-Grohe and Uribe $(2001, 2002)$ , Buiter and Panigirtzoglou (2003), Ono $(1994, 2001)$ ,and others.

The money market is assumed to be in equilibrium at each point in time,$m^{d}=(1$where$m^{s}$$- \frac{A+\pi}{\gamma(\sigma A)^{2}}$)$(m^{d}+\overline{K})=m^{\Delta}=m$or$( \frac{A+\pi}{\gamma(\sigma A)^{2}})(m+\overline{K})=\overline{K}$

.

(19)

is the money supply. The above equilibrium condition of course determines the

88nominal interest rate and hence the rate of inflation because the real interest rate is fixed at A:$\pi=(\frac{\overline{K}}{m+\overline{K}})\gamma(\sigma A)^{2}-A$.

Figure-2 Dynamics(20)

Real Money Balance in the Short-Run

Assuming that there is no growth of nominal money supply, the differential equation onreal money balance becomes(21)$\frac{\dot{m}}{m}=-\pi=\mathrm{A}-(\frac{\overline{K}}{m+\overline{K}})\gamma(\sigma A)^{2}$

The dynamics of the real money balance can be depicted in Figure-l.

$m^{*}$

in Figure-2 is a

89neoclassical equilibrium in the economy, where there is no inflation, or $\pi=0$ .

Since w$=m+\overline{K},\dot{w}=\dot{m}$. Substituting these two equations into (18) gives
or

tit $=\mathrm{A}\overline{K}-mn-u(m+K)$

where

(22)

,

is the excess supply in the goods market Since$ES\equiv Y-C=A\overline{K}-u(m+\overline{K})$$\dot{m}/m=-\pi$$\frac{\dot{m}}{m}=\frac{ES}{m}-\pi$

ffom (21), the goods market is always in equilibrium, or $ES\equiv Y$ $-C=0$ . However,

only at the neoclassical equilibrium
If the initial money balance$m^{*}$$m$, there is no inflation nor deflation, or $\pi=0$ .

is larger than$m^{\mathrm{r}}$or $m>m^{*}$ , then$m$continues to grow,in other words, the deflation continues, or $\pi<0$ . This trajectory also satisfies all the optimallyconditions including the following transversality condition:$\lim_{larrow\infty}[e^{-\rho l}E_{0}J(w(t))]=0$

.

(23)

As is shown in Figure-2, on this transitional path the rate of deflation $\dot{m}/m=-\pi$ approaches to$A$, and hence the nominal interest rate$r_{N}\equiv A+\pi$

approaches to zero. In other words, theeconomy falls into a liquidity trap.

Proposition 1
In a stochastic monetary economy there exist deflationary equilibria as well as a uniqueneoclassical equilibrium without inflation or deflation. Hence, the economy may fall into

liquidity trap, which is characterized by the deflationary equilibria.

3-2. The Long-Run Analysis
Since $m/w$ is constant over time, $\dot{m}/m=\dot{w}/w=\dot{K}/K$

Substitution of (15a) and$(15\mathrm{b})$into (18) giveson the balanced growth path.

so$g_{\mathrm{w}} \equiv\frac{\dot{w}}{w}=\frac{(1+\epsilon)(A+\pi)^{2}}{2\gamma(\sigma \mathrm{A})^{2}}-\epsilon(\rho+\pi)$There are two polar cases for$\pi\leq-A$$g_{d},$

,

.

(24)

. If the nominal interest rate becomes non-positive, or, then the people has no incentive to hold the physical capital. The individual hold all

the assets in the form of money, or $m/w=1$ . In this case,$g_{w}=\epsilon(\mathrm{A}-\rho)$, which is the same as

the growth rate in the typical AK model.

The other polar case arises when$\pi$becomes equal to or larger than

. In this$\gamma(\sigma \mathrm{A})^{2}-A$

case, evidently from (15b), the individual has no incentive to hold money, and hence $m/w=0$ .
In other words, the economy behaves just like the stochastic non-monetary economy analyzed

by Smith (1996). In this case,$g_{\mathrm{W}} \equiv\epsilon(A-p)+\frac{(1-\epsilon)\gamma(\sigma A)^{2}}{2}$,which is of course exactly the same as the growth rate in Smith’s model.
Suppose that the grow rate of nominal money supply is constant at$\mu$, or $\dot{M}/M=\mu$ .

Then,$\dot{m}$$g_{m}\equiv-=\mu-\pi m$

.

(25)

Hence, the long-run growth rate $g*$ and inflation rate

The determination of $g*$ andshown in Figure-3, where

supplyrate$\pi^{\tau}$$\mu$shifts$g_{m}$$\pi*$$\pi*\mathrm{a}\mathrm{r}\mathrm{e}$determined by (24) and (25).

when the intetemporal substitution is small, or$\overline{\pi}=\gamma(\sigma \mathrm{A})^{2}-\mathrm{A}$$g_{m}$

$<1$

, is

. Since an increase in the growth rate of money-line to the left, it increases the grow rate

. An increase in$\epsilon$$g$.

but decreases the inflationmust be accompanied by the same amount of increase inbalanced growth path. When the intetemporal substitution$\epsilon$,$g_{\mathrm{w}}$on theis small, this is possible only

when the nominal interest rate falls. If the nominal interest rate increases when$\epsilon$

is small, bothcurrent and future consumptions increase, and hence the capital accumulation fall. With a$\mathrm{s}\mathrm{m}\mathrm{a}1$91$\mathcal{E}$, therefore, the inflation rate

fasters the capital accumulationmust decrease because the decreased nominal interest rate$\pi\ell$$g_{u},$.

$g_{w}\ g_{m}$The determination of $g*$ and$\pi^{*}$

when the intetemporai substitution is large, or

shown in Figure-4. In this case, an increase in the growth rate of money supply
both the grow rate

. and the inflation rate . In order for$\pi^{t}$$g$$g_{w}$

to equate to$\mu$$g_{w}$$\epsilon$

$>1$

, is

increases
when

$\mu$

82

increases,$g_{\mathrm{w}^{J}}$should increase. Since the intetemporal substitution$\epsilon$

is large, this is possible

only when the nominal interest rate increases. If the nominal interest rate increases when$\epsilon$

islarge, then current consumption falls while future consumption increases, and hence theincreased interest rate fasters the capital accumulation$g_{u^{r}}$.

$g_{l}"\ g_{m}$Figure-4 Balanced Growth When$\epsilon$

$>1$

93

Proposition 2

An increase in the growth rate of money supPly increases the long-run growth rate.
However, it increases the inflation rate when the intertemporal elasticity of substitution is large,

but increases the inflation rate when the intertemporal elasticity ofsubstitution is small

4. Concluding Remarks
As Keynes (1936) emphasizes, the precautional demand for money does matter in an uncertain

world. In this paper, instead of assuming money-in-utility, we consider uninsurable

idiosyncratic risks as the source of money demand. As a result, it is shown that the individual’s
risk-averse behavior play a key role in deriving the precautional demand for money, which is intum the driving force for putting the economy into a liquidity trap in the short-run.
Intetemporal substitution is another important factor characterizing the individual’s
dynamic preferences. On the balanced growth path, higher growth is accompanied by higher

inflation only when the intetemporal substitution is large. When it is small, on the contrary, seehigher growth with lower inflation. Regardless of the substitution elasticity, however, theincreased money supply growth enhances the growth rate of the economy.

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