Discussion Paper No. 622

DECREASING MARGINAL IMPATIENCEIN A MONETARY GROWTH MODEL

Ken-Ichi Hirose
andShinsuke Ikeda

November 2004

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Decreasing Marginal Impatience in a
Monetary Growth Model
Ken-Ichi Hirose1 and Shinsuke Ikeda2
Otaru University of Commerce and Osaka University
October 2004

1 The Faculty of Commerce,

Otaru University of Commerce, 3-5-21 Midori,Otaru, Hokkaido 047-8501, Japan. Telephone: +81(134)27-5365. E-mail:hirose@res.otaru-uc.ac.jp
2 Corresponding author. The Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. Telephone:+81(6)6879-8568. Fax: +81(6)6878-2766. E-mail: ikeda@iser.osaka-u.ac.jp

Abstract
Unlike the standard assumption that the degree of impatience, measured by
the rate of time preference, is increasing in wealth, empirical studies support
that impatience is marginally decreasing. By introducing decreasing marginal
impatience into the neoclassical monetary growth model á la Sidrauski, weshow that (i) consistently with empirical results, an increase in the core rateof inflation reduces capital stocks in a steady state; and that (ii) its long-run
welfare cost is larger than predicted with increasing or constant marginal
impatience, implying that estimates of the inflation cost which have so far
been obtained by assuming constant time preference may be underestimates.
Keywords: Decreasing marginal impatience, time preference, inflation, theTobin eﬀect
JEL classification: D90, E00.

1

Introduction

In the theory of endogenous time preference, one of the most controversial
assumptions is that the degree of impatience, measured by the rate of time
preference, is marginally increasing in wealth. Although existing empiricalstudies, e.g., Lawrance (1991), Becker and Mulligan (1997), Samwick (1997),
and Barsky et. al. (1997), commonly report that the degree of impatienceis marginally decreasing in wealth or income, few research has so far been
conducted on implications of decreasing marginal impatience. This contrasts
to the huge accumulation of research on increasing marginal impatience (see,e.g., Epstein (1987)). The main reason for this is that marginal decreasing impatience ceteris paribus destabilizes optimal consumption dynamics.
However, Das (2003) and Hirose and Ikeda (2004) recently show that, evenwith decreasing marginal impatience, equilibrium economic dynamics can bestable in neoclassical growth models where capital input displays decreasing return. By using the resulting, dynamically well-behaved model, it is ofparticular importance to work out policy implications of decreasing marginal
impatience.
By introducing decreasing marginal impatience into the neoclassical monetary growth model á la Sidrauski (1967), the purpose of this exercise isto re-examine the real eﬀects of an increase in the core rate of inflation.
Many empirical studies have reported that inflation harms capital accumulation (e.g., Fischer (1993), De Gregorio (1992), Barro (1995), and Jones andManuelli (1995)). Theoretical studies, however, have shown diﬀerent results
since Tobin (1965) claimed that inflation enlarges capital and production in
the long run by causing portfolio shifting away from real money balances
to real capital. Especially Epstein and Hynes (1983) reproduced the Tobineﬀect by using a general equilibrium model with endogenous time preference.1 However, the result might depend crucially on their assumption that
impatience is marginally increasing.
With decreasing marginal impatience, we indeed show that, consistently
with the empirical stylized fact, a higher inflation rate results in smaller longrun capital stocks. Intuitively, higher inflation reduces real money balances,
1

The Tobin eﬀect has been also obtained by using overlapping generations models(e.g., Weil (1991)). The Tobin eﬀect in these models is associated with the nonneutrality of debts, which diﬀers from Tobin’s argument. Introducing explicitly a transactionfacilitating function of money (e.g., Stockman (1981)) weakens the validity of the Tobineﬀect. For a survey, see Orphanides and Solow (1990).

1

which raises the steady-state time preference and hence the real interest rate
under decreasing marginal impatience. The long-run capital stock should
thus be reduced.
As an important welfare implication, it is also shown that the resulting
long-run welfare cost of inflation is larger than predicted with increasing or
constant marginal impatience. If actual time preference displays decreasing
marginal impatience as empirically reported, the result implies that misspecifying impatience as marginally increasing or constant leads to underestimating the long-run welfare costs of inflation. For example, by using constanttime preference models, Jones and Manuelli (1995) and Lucas (2000) reportthat the welfare cost of inflation is quantitatively very modest. They may,
however, underestimate the actual values.
The rest of the paper is organized as follows. Section 2 presents a monetary growth model with decreasing marginal impatience. Section 3 examines
the eﬀects of inflation on capital accumulation. Section 4 concludes the paper.

2

The model with decreasing marginal impatience

We consider an infinitely-lived representative agent in the “money-in-theutility-function” framework. He or she holds wealth a in the form of realcapital k and real money balances m; supplies one unit of labor inelastically;
and maximizes lifetime utility by choosing the time profiles of consumptionc, real money balances, and the real capital stock. His or her problem is
specified as follows:

(1)

max

Z ∞

u (ct , mt ) exp(−∆t )dt,

0

subject to:(2)

˙ t = δ (u (ct , mt )) , ∆0 = 0,∆(3)ȧt = rt at + wt − ct − (rt + π t ) mt + xt , at ≥ 0,
at = kt + mt , kt > 0, k0 = given,2

where a dot represents the time derivative; u (•, •) represents the felicity function; ∆ denotes a cumulative discountR trate with the instantaneous discountrate δ (u) á la Uzawa (1968): ∆t = 0 δ (uτ )dτ ; r the real rate of interest;
w the wage rate; π the rate of inflation; and x lump-sum transfer paymentsfrom the government.
For intertemporal preferences to be well-defined, we follow the literature(e.g., Epstein (1987, p.75)) in assuming that the following standard regularity
conditions are valid: (C1) u < 0; (C2) u is increasing and concave; (C3) u is
log-convex; and (C4) δ (u (c, m)) is concave in (c, m).
In contrast to the existing literature (e.g., Uzawa (1968) and Epstein(1987)), we assume that δ 0 (u) < 0, for which case, as shown later, the degreeof impatience, measured by the pure rate of time preference is marginallydecreasing in wealth.2
Let λ and φ represent the current-value shadow prices for savings andthe discount factor Ω = exp (−∆), respectively. Let g (c, m, φ) represent
generating function u (c, m) − φδ (u (c, m)). Then, the marginal utilities are
given by first derivatives gc and gm , where gc ≡ ∂g/∂c, etc. It is assumed
that gc and gm are all positive.3 4 For analytical simplicity, we also assume
that c and m are separable: gcm = 0.5
The optimal consumption should satisfy:(4)(5)

gc (ct , φt ) = λt ,χ(ct , mt ) ≡um (ct , mt )= rt + πt ,uc (ct , mt )2

Das (2003) proposes an alternative set of regularity conditions in which ∂δ/∂c < 0
itself is imposed as a part of the regurality condition. Our main result below does notchange at all even when we adopt his regularity conditions, instead of conditions (C1)through (C4).
3This assumption is necessary because, from regularity condition (C1), φ, which equals
the utility index as shown later, is negative. Since φ equals u/δ in steady state, thepositivity of gc and gm is satisfied near the steady-state point only when the elasticity ofthe discount rate δ with respect to felicity u is smaller than unity: uδ 0 /δ < 1. Similarassumptions are needed in the case of increasing marginal impatience when u is assumed
to be positive (see, e.g., Uzawa (1968) and Gootzeit et al (2002)).
4
If we adopted Das’s regurality conditions, instead of our conditions (C1) through (C4),the positivity of gc and gm would be ensured. But, unlike what we do later, we could notconsider the case of increasing marginal impatience under the Das regurality conditions inwhich δ 0 < 0 is imposed as a regurality condition.
5
The separability gcm = 0 is satisfied when u and δ are both separable, ucm = 0 andδ cm = 0.

3(6)λ̇t = (δ (u (ct , mt )) − rt ) λt ,(7)

φ̇t = −g (ct , mt , φt ) ,and the transversality conditions. Note that the optimal φt equals the lifetime
utility obtained from the optimal consumption stream after time t, as canbe seen by solving diﬀerential equation (7) under transversality condition
limt→∞ exp (−∆t ) φt ∆t = 0.
Define the rate of time preference ρ as ρ (c, m, φ) ≡ −d ln Γt /dt|ċ=0 , whereΓ ≡ gc (c, φ) exp (−∆) represents the present-value marginal utility of c.6
Diﬀerentiate (4) and (5) by t and substitute the results into (6) to obtain:(8)ċtgc (ct , φt )(rt − ρ (ct , mt , φt )) ,=−ct
cgcc (ct , φt )

(9)

ṁt
gm (mt , φt )=−(r − ρ(ct , mt , φt )).
mt
mgmm (mt , φt )where
δ 0 (u (ct , mt ))ρ (ct , mt , φt ) = δ (u (ct , mt )) −g (ct , mt , φt ) .
1 − φδ 0 (u (ct , mt ))Money is supplied by the government in the form of “helicopter money:”(10)µmt = xt ,where µ denotes the growth rate of nominal money supply.
Goods are produced by competitive firms with a constant-to-scale technology. Let f denote a per capita production function satisfying fk > 0 and
fkk < 0. From the profit-maximizing behavior of firms, r and w are given as(11)

wt = f (kt ) − kt fk (kt ) ; rt = fk (kt ) .

By substituting equations (10) and (11) into (3), we obtain(12)k̇t = f (kt ) − ct .

6

The rate of time preference can also be defined by using the marginal utility of realmoney balances. Due to weak separability of the Uzawa-type intertemporal preferences,
however, the time preferece rate with respect to m equals that with respect to c. For thedetailed discussions, see Shi (1994).

4

Equations (3) through (12) jointly determine the equilibrium dynamics
for (at , ct , kt , mt , rt , wt , , xt, λt , φt ) . The steady-state solution (c∗ , m∗ , k∗ , φ∗ ) is
determined byδ (u (c∗ , m∗ )) = fk (k∗ ),χ(c∗ , m∗ ) = fk (k∗ ) + µ,(13)

f (k∗ ) = c∗ ,φ∗ =u (c∗ , m∗ ).
fk (k∗ )To obtain a linearized autonomous dynamic system, combine ṁ/m =µ − π with (5) and (9) to getgm(ρ(ct , mt , φt ) − fk (kt )) = µ + fk (kt ) − χ(ct , mt ),mt gmmwhich can be solved for mt as(14)

mt = υ(ct , φt , kt ; µ).

From (11) and (14), the dynamic equations (7), (8), and (12) then reduce to:ċt = −(15)

gc (ct , φt )(fk (kt ) − ρ (ct , υ(ct , φt , kt ; µ), φt )) ,gcc (ct , φt )φ̇t = −g (ct , υ(ct , φt , kt ; µ), φt ) ,k̇t = f (kt ) − ct .

Letting z represent (c, φ, k)0 , this dynamic system can be linearized aroundthe steady state as żt = A (zt − z ∗ );gcrδ 0 uc
0
f
−
kk
gcc
gcc
0


fkk
gm
um
A =  −gc − gm (− χχm ) r − gm (− mgrδmm
)
−g
(
+
f ) ,
m
χm
χm
mgmm χm kk c−10rwhere the coeﬃcient matrix A is evaluated at the steady-state point.
The steady-state equilibrium point is locally saddle-point stable if andonly if det (A) < 0 and trace (A) > 0, for which case matrix A has two
5

positive and one negative characteristic roots. It can be easily shown thatthe saddle-point stability condition is satisfied if and only if
χ 0χ
δ uc fkk > 0,Λ ≡ (1 − χ c )rδ 0 uc − fkk +
χm
χm
2r +

gm rδ0 um> 0.
mgmm χm

In what follows, these conditions are assumed to be met.7 8

3The eﬀect of inflation

Let us now examine the eﬀect of a permanent increase in the core rate of
inflation µ. From (13), the steady-state equilibrium is determined by(16)

δ (u (f (k ∗ ) , m∗ )) = fk (k∗ ),(17)

χ(f (k∗ ) , m∗ ) = fk (k∗ ) + µ.

Figure 1 depicts the determination, where schedule DD0 represents (16)and MM 0 (17). The MM 0 schedule is positively sloping and, under the assumption that Λ is positive, the slope of the DD0 schedule is also positive but
gentler than that of MM 0 . The steady-state equilibrium is determined at theintersection of the two schedules. An increase in µ shifts the MM 0 schedule
upward, bringing the steady state point from E0 to E1 . As illustrated, thesteady state capital stock decreases in response to the rise in the inflation
rate. In fact, from (16) and (17), we can deriveδ 0 um
dk∗
< 0.
=−
χm Λ
dµ

(18)
7

We havedet (A) =

·
¸
gm χc
gc r
gm
(1 −
rδ 0 uc fkk +)rδ 0 uc − fkk ,gcc χmgcc
gc χmtrace (A) = 2r +gm rδ 0 um.
mgmm χm

It can easily be shown that det (A) < 0 ⇔ Λ > 0.
8
It is a standard procedure to assume saddle-point stability for comparative dynamics.
See, e.g., Heijdra (1998).

6

Intuitively, higher inflation reduces real money holdings, which, underdecreasing marginal impatience, ceteris paribus raises the discount rate andhence the interest rate in the steady state. This reduces the capital stockin the long run. This result contrasts to the case of increasing marginal
impatience, in which case (16) is depicted as negatively sloped as illustrated
as schedule II 0 in Figure 1, so that the increase in µ enlarges k ∗ as illustrated
by the movement from E0 to E2 .
Proposition 1: When impatience is marginally decreasing, an increase in
the core rate of inflation reduces the capital stock.
Remark: As seen from (18), whether an increase in µ increases or decreasesk∗ depends crucially on the sign of ∂δ/∂m (= δ 0 um in present case), but noton ∂δ/∂c, that is on whether impatience with respect to m is marginally
increasing or decreasing, but not on whether impatience with respect to c
is marginally increasing or decreasing. When δ depends directly on (c, m)
independently of u, and hence when preferences are weakly non-separable
in the sense of Shi (1994), the same result as in Proposition 1 can be valid
even if impatience is marginally increasing in c, insofar as it is marginallydecreasing in m.
From (13), jointly with the decrease in k∗ , steady-state consumption is
also reduced. Since consumption and real money balances decrease and hence
the discount rate rises in the long run, the steady-state welfare level φ∗ isdefinitely lowered. Note that the new steady-state point E1 , obtained underdecreasing marginal impatience, is located in the south-west to, and hence
dominated by, point E2 , attained under increasing marginal impatience. Thisimplies that the long-run welfare deterioration caused by an increase in µ is
larger than it would when impatience is marginally increasing. More precisely, we have u (c∗ , m∗ )|point E1 < u (c∗ , m∗ )|point E2 and fk (k∗ )|point E1 >fk (k∗ )|point E2 , so that, from (13), φ∗ |point E1 (= u (c∗ , m∗ ) /fk (k∗ )|point E1 ) issmaller than φ∗ |point E2 (= u (c∗ , m∗ ) /fk (k ∗ )|point E2 ). Similarly it is easy to
see that the long-run welfare deterioration is also larger than it would underconstant marginal impatience.
Proposition 2: The long-run welfare cost of inflation under decreasing marginal impatience is larger than predicted with increasing or constant marginal
impatience.
7

If actual time preference displays decreasing marginal impatience as often
empirically reported, Proposition 2 implies that estimates of the welfare costof inflation which are obtained by using models of increasing or constantmarginal impatience are underestimates of the actual cost. For example, byusing constant time preference models, Jones and Manuelli (1995) and Lucas(2000) obtain quantitatively very modest estimates for the welfare cost of
inflation. They may, however, underestimate the actual costs.

4

Conclusions

By introducing decreasing marginal impatience into a neoclassical monetary
growth model, we have shown that, consistently with many empirical studies,a rise in the core rate of inflation reduces the steady-state capital stock. The
resulting long-run welfare cost of inflation is larger than it would if impatienceis marginally increasing or constant. This implies that if actual impatience ismarginally decreasing as empirically reported, misspecifying it as marginally
increasing or constant as in the literature results in underestimating the longrun welfare costs of inflation.

8

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[13] Lawrance, E.C., 1991, Poverty and the rate of time preference, Evidence
from panel data, Journal of Political Economy 99, 54-77.
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cash-in-advance economy, Journal of Monetary Economy 8, 387-393.
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32, 37-53.

10

k

M

I

D
E2
E0

E1

D′
I′
M′

m

Figure 1. Inflation and capital stock under decreasing marginal impatience