GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.69

Long-Run Impacts of Inflation Tax with
Endogenous Capital Depreciation

Seiya Fujisaki and Kazuo Mino

July 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Long-Run Impacts of Inflation Tax with
Endogenous Capital Depreciation
Seiya Fujisaki∗and Kazuo Mino†
July, 2009

Abstract
This paper examines the long-run impact of inflation tax in the context
of a generalized Ak growth model in which the rate of capital depreciationis endogenously determined. We assume that the rate of capital depreciation positively depends on capital utilization rate and negatively depends
on maintenance expenditures. Money is introduced via a cash-in-advanceconstraint that may apply to the maintenance expenditures as well as toconsumption and investment spendings. We find that the long-run effects
of inflation tax are more complex than those obtained in the monetary Ak
growth model with a fixed capital depreciation rate. In particular, the relation between inflation and growth is highly sensitive to the specification of
the capital depreciation technology as well as to the forms of cash-in-advanceconstraints.
Keywords: maintenance expenditures, endogenous depreciation, inflation tax,
cash-in-advance constraints∗

Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka,
Osaka, 563-0043, JAPAN (email: ege010fs@mail2.econ.osaka-u.ac.jp)†
Institute of Economic Research, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, 606-8501 JAPAN (email:mino@kier.kyoto-u.ac.jp)

1

Introduction

It has been claimed that the activity of maintaining and repairing equipment and structures is large relative to investment and it would be a substantial substitute with new investment.1 Considering this fact, several authors introduce maintenance costs and endogenous capital depreciation into
the standard models of growth and business cycles: see Aznar-Màrquez andRuiz-Tamarit (2004), Guo and Lansing (2007), Licandro and Puch (2000),
Licandro, Puch and Ruiz-Tamarit (2001) and McGrattan and Schmitz Jr.
(1999). These studies show that introducing maintenance expenditures mayalter both dynamic behavior and the stationary-state characterization of themodel economy in a substantial manner.
The purpose of this paper is to explore the long-run impacts of inflation tax in the context of a generalized Ak growth model in which the rateof capital depreciation is endogenously determined. Following the existing
literature mentioned above, we assume that the capital depreciation rate
positively depends on the rate of capital utilization and negatively depends
on maintenance expenditures. Money is introduced via a cash-in-advanceconstraint that may apply to the maintenance spendings as well as to consumption and investment expenditures. We find that the long-run effects
of inflation tax are more complex than those obtained in the monetary Ak
growth model with a fixed capital depreciation rate.2 In particular, the relation between inflation and growth is highly sensitive to the specification of
the capital depreciation technology as well as to the forms of cash-in-advanceconstraints.

2

Model

We assume that the rate of capital depreciation, δ, depends positively on therate of capital utilization, u, and negatively on maintenance expenditures per
capital stock, z/k :³ z´δ = δ u,, δ1 > 0, δ2 < 0,(1)kwhere z denotes maintenance expenditures and k is capital stock. To ensure
the second-order conditions for the optimization problem shown below, we
1

See McGrattan and Schmitz Jr. (1999) and Mullen and Williams (2004).
The standard Ak growth model with cash-in-advance constraints are studied by Chenand Guo (2008a, b), Jha, Yip and Wang (2002), Li and Yip (2004) and Suen and Yip(2005).
22

assume that function δ (u, z/k) is strictly convex in u and z/k. The production technology is given by an Ak production function such thaty = Auk, A > 0,(2)where y denotes aggregate output. Namely, the ratio of output and theutilized capital is fixed.
We consider a competitive, representative-agent economy. The optimization problem for the representative household is given by the following:
Z ∞
1−σ
−1
−ρt c
max
e
dt, ρ > 0, σ > 01−σ0subject toṁ = y − c − v − z − πm + τ,(3)k̇ = v − δk,(4)

c + φ1 v + φ2 z ≤ m,

0 ≤ φ1 , φ2 ≤ 1,

(5)

together with the initial holdings of m and k. Here, c denotes consumption,m real money balances, v investment spending, π rate of inflation, and τ isa lump-sum transfer (lump-sum tax if it is negative) from the government.
In addition, the household’s income y and the capital depreciation rate δ aregiven by (1) and (2), respectively. In this problem, (3) is the flow budget
constraint for the household, (4) describes capital formation and (5) specifies
the cash-in-advance constraint. We assume that the cash constraint is applied
to the entire consumption expenditure as well as to parts of maintenance andinvestment spendings.
The current-value Hamiltonian function for the household’s optimization
problem is given byH =h³ z´ ic1−σ − 1+ q (Auk − c − v − z − πm + τ ) + λ v − δ u,k1−σk+θ (m − c − φ1 v − φ2 z) ,where q and λ respectively denote the implicit prices of m and k, and θ isa Lagrangian multiplier. The control variables in this problem are c, u, v
and z, while the state variables are m and k. The necessary conditions foran optimum are the following:c−σ − q − θ = 0,

(6)

−q + λ − θφ1 = 0,

(7)

3

³ z´
qA − λδ1 u,
= 0,
k
³ z´
−q − λδ2 u,− θφ2 = 0,kq̇ = q (ρ + π) − θ,h³ z´
³ z´ ziλ̇ = λ ρ + δ u,
− δ2 u,− qAu,kk k
θ (m − c − φ1 v − φ2 z) = 0, θ ≥ 0, (m − c − φ1 v − φ2 z) ≥ 0,

(8)
(9)
(10)
(11)
(12)

along with (3) , (4) , the initial conditions and the transversality conditions:lim qme−ρt = 0;

lim λke−ρt = 0.

t→∞

t→∞

Note that (12) displays the Kuhn-Tucker conditions for the cash-in-advanceconstraint.
The market equilibrium condition for the final goods is
y =c+v+z(13)

and the real money balances change according toṁ = m (µ − π) ,(14)

where µ denotes a given growth rate of nominal money stock. We assume
that there is neither public debt nor the government’s spending, so that a
newly created money is distributed to the household as a lump-sum transfer.
Hence, the government’s flow budget constraint is given by µm = τ.

3

Balanced-Growth Characterization

Given our specification of the model economy, it is easy to see that thebalanced-growth equilibrium are characterized by the following conditions:
k̇
ẏ
v̇
ż
ṁ
ċ
= = = = =
= g,
c
k
y
v
z
m
q̇
λ̇
θ̇
= = = γ,qλθwhere g and γ are common growth rates that are constant over time on thebalanced-growth path. As a result, the balanced-growth equilibrium requiresthat the capital utilization rate, u, and the maintenance expenditures per
capital, z/k, are also constant.
4

First, note that the balanced-growth conditions mean the following:
γ = −σg,(15)π = µ − g.

(16)

Equation (15) comes from (6) , and (16) is given by (14). Condition (7) yields
θ
1 ³
q´
=
1−
.
(17)
λ
φ1
λ
Using (17) and (9) , we obtainq
φ1 δ2 (u, x) + φ2=,λφ2 − φ1(18)

where x = z/k. Consequently, (8) and (18) giveδ1 (u, x) = A

φ1 δ2 (u, x) + φ2
φ2 − φ1(19)This equation represents the relationship between the optimal levels of capitalutilization rate, u, and the maintenance spending rate, x (= z/k) . Notice thatsince we have not used the balanced-growth conditions to derive (19) , thisrelation also holds out of the balanced growth path.
From (10), (17) and (18) , we obtain
µ
¶
q̇
1 λ
= ρ+π−
−1
q
φ1 q
·
¸
φ2 − φ11= ρ+π−−1 .
φ1 φ1 δ2 (u, x) + φ2
Using (15), (16) and (19) , we see that the above equation is rewritten as
½
·
¸¾
1
1
A
g=
ρ+µ−
−1 .
(20)
1−σ
φ1 δ1 (u, x)Similarly, (11) and (18) yield:·¸
φ1 δ2 (u, x) + φ2λ̇
= ρ + δ (u, x) − δ2 (u, x) x −Au.
λφ2 − φ1
In view of (15) and (19) , the above is expressed asg=1{δ1 (u, x) u − ρ − δ (u, x) + δ2 (u, x) x} .
σ(21)

To sum up, (19) , (20) and (21) may determine the steady-state levels ofx, u and g.
5

4

Long-Run Impacts of Inflation Tax

To clarify our analysis, we now specify the depreciation function asδ (u, x) =δ0 uε,1 + βxε > 1, β > 0,

δ0 > 0.

In the above, the capital depreciation rate is fixed at δ0 when ε = β = 0.
Our specification is a slightly modified version of the depreciation functionused by Licandro and Puch (2000) and Guo and Lansing (2008). Given theabove functional form, the steady-state conditions (19) , (20) and (21) arerespectively expressed by the following:·¸βδ0 uεεδ0 uε−1A=
φ2 − φ1,(22)
1 + βxφ2 − φ1(1 + βx)2½ ·¸¾
1 A (1 + βx)
1−1 −ρ−µ ,(23)g=
σ − 1 φ1εδ0 uε−1
·µ
¶
¸
δ0 uε
1
β
g=
ε−1−
−ρ .
(24)
σ
1 + βx 1 + βx
We examine the effects of a change in the money growth rate, µ, on the balanced growth path under alternative forms of the cash-in-advance constraint.
Before analyzing the above conditions, it is worth remembering the main
findings in the standard Ak growth model with a constant capital deprecation rate. In the models with fixed depreciation, it is shown that thebalanced-growth path is uniquely determined and a rise in the growth rateof money supply depresses the balanced-growth rate, as long as the elasticity of intertemporal substitutability in consumption, 1/σ, is less than one.3
In contrast, if 1/σ > 1, then there may exist dual balanced-growth paths
and a higher money growth rate raises the growth rate of income on thebalanced-growth path with a higher growth rate.
We find that when 1/σ > 1, multiple balanced-growth paths may emerge
in our model as well. To emphasize the effects of endogenizing capital depreciation, in what follows, we focus on the case where 1/σ < 1.
Case (i): φ1 = φ2 = 0
First, we assume that the cash-in-advance constraint binds consumption
expenditures alone. In this case q = λ for all t ≥ 0, and thus (8) and (9)respectively become εδ0 uε−1 = A (1 + βx) and βδ0 uε = (1 + βx)2 . Theseequations giveεu=
(1 + βx) ,
(25)
βA
3

See, for example, Chen and Guo (2008b), Li and Yip (2004) and Suen and Yip (2005).

6

implying that the optimal level of capital utilization rate is proportional tothe optimal rate of maintenance spending. Using (8) and (25) , we see thatthe optimal levels of x and u are given by
#
"µ
1
2
µ
¶ 2−ε
µ
¶ε ¶ 2−ε1ε1εu∗ = (βδ0 ) 2−ε
−1 .
(26)
,
x∗ =
βδ0
βA
β
βA
Hence, the capiotal utilization and maintenance spending rates (so the capitaldepreciation rate) stay constant even out of the balanced-growth equilibrium.
The balanced-growth rate is determined as·µ¶¸βδ0 u∗ε
1
g=
ε−1−
−ρ ,
σ
1 + βx∗ 1 + βx∗
where u∗ and x∗ are given by (26) . As well as in the standard Ak growthmodel with the cash-in-advance constraint, our model shows that money
is superneutral as to the balanced growth rate when the cash-in-advanceconstraint applies to consumption alone.
Case (ii): φ1 > 0 and φ2 = 0
Suppose that the maintenance expenditures are free from the cash-inadvace constraint. As in Case (i), if φ2 = 0, equation (25) always holds.
Note that from (7) q generally diverges from λ. Hence, plugging (25) into
(23) and (24) , we obtain the following:
( "
#
)
¶1−ε
µ
1
1 A
ε
g=
(1 + βx)2−ε − 1 − ρ − µ ,(27)
σ − 1 φ1 εδ0 βA
1
g=
σ

·µ

β
ε−1−
1 + βx

¶

µ
δ0

ε
βA

¶ε

¸
ε−1

(1 + βx)

−ρ .

(28)

Remember that we have assumed that σ > 1 and ε > 1. Under theserestrictions, we see that if ε > 2, then the graph of (27) has a negative
slope and that of (28) has a positive slope. Therefore, there exists a uniquebalanced-growth path. It is also easy to see that a rise in money growth rate,µ, shifts down the graph of (27) , so that a rise in µ lowers g and u. As aresult, from (25) both x and δ decrease as well. In contrast, if 1 < ε < 2,then both graphs have positive slopes. This means that these graphs may
have multiple intersections. Furthermore, if the graph of (27) is steeper thanthat of (28) , then a downward shift of the locus of (27) yields simultaneous
increases in u, x and g. In this case, we obtain a positive long-run relation
between money growth and the growth rate of real income.
7

Case (iii): φ1 = 0 and φ2 > 0
In this case, the cash-in-advance constraint does not apply to investment
but it binds the maintenance expenditures. Condition φ1 = 0 means thatq = λ for all t ≥ 0. Thus (8) becomes εδ0 uε−1 = A(1 + βx), which yields therelation between x and u in such a way that1µ ¶ ε−11Au=(1 + βx) ε−1 .
(29)
εδ0
It is assumed that φ2 > 0 and thus (29) presents·¸θ1βδ0 uε=−1 .
q
φ2 (1 + βx)2(30)

From (10) , (16) and (30) , the balanced-growth relation (23) in the generalcase is replaced with
( "
#
)
ε
µ ¶ ε−1
2−ε
1
A
1
βδ0
(1 + βx) ε−1 − 1 − ρ − µ .
(31)g=
σ − 1 φ2εδ0Additionally, by use of (29) , we write (24) as"µ#ε¶ µ ¶ ε−1
1
β
A
1
ε−1+
δ0
g=
(1 + βx) ε−1 − ρ .
σ
1 + βxεδ0(32)Equations (31) and (32) demonstrate that the comparative statics results
are similar to those in Case (ii): again, if ε > 2, then the balanced-growth
path is uniquely determined and a rise in µ depresses x, u and g. If 1 < ε < 2,
a higher µ may increase x, u and g. on the balanced growth path.
Case (iv): 0 < φ1 ≤ 1 and 0 < φ2 ≤ 1
As the special cases mentioned above suggest, if neither φ1 nor φ2 is zero,
we may have a variety of comparative statics results on the balanced growthpath. Notice that (22) is written asµ¶φ1 εδ0 uε−1 φ1 βδ0 uε
1−
+
= A.
φ2 1 + βx
φ2 (1 + βx)2
Hence, if φ2 > φ1 , then x and u satisfying the above equation change in thesame direction, implying that the above gives x = x (u) , x0 > 0. Substituting
this into (23) and (24) , we obtain the relations between g and u that are
similar to (24) and (27) (or (31) and (32)). Thus the effects of a change inµ will be the same as those in Cases (ii) and (iii). If φ2 < φ1 , it is possiblethat (22) yields a negative relation between u and x. If this is the case, thecomparative statics exercises become more complex than in Cases (ii) and(iii), even if assume that 1/σ < 1.
8

5Conclusion

This paper examines the role of endogenous capital depreciation in the context of an Ak growth model with a cash-in-advance constraint. We assume
that the rate of capital depreciation is determined by the rate of capitalutilization and maintenance expenditures. We also assume that the cashin-advance constraint may apply to the maintenance spendings in addition
to consumption and investment expenditures. Our analysis reveals that thelong-run impacts of inflation tax would be more complex than those obtainedin the standard Ak growth model with a fixed rate of capital depreciation.
In particular, the long-term relation between inflation and growth is highly
sensitive to the specification of the capital depreciation function as well as
to forms of the cash-in-advance constraint.4

References[1] Aznar-Màrquez, J. and Ruiz-Tamarit, R. (2004), ”Endogenous Growth,
Capital Utilization and Depreciation”, FEDRA Discussion Paper No.24.
[2] Chen, S-H., and Guo, J-T. (2008a), ”Velocity of Money, Equilibrium (In)determinacy and Endogenous Growth”, Journal of Macroeconomics 30,
1085-1096.
[3] Chen, S-H., and Guo, J-T. (2008b), ”On the Growth and Velocity Effectsof Money”, Economics Bulletin 5, Issue 13, 1-7.
[4] Fujisaki, S. and Mino, K. (2007), ”Generalized Taylor Rule and Determinacy of Growth Equilibrium”, Economics Bulletin 11, Issue 5, 1-7.
[5] Fujisaki, S. and Mino, K. (2009), ”Long-Run Impacts of Inflation Tax inthe Presence of Multiple Capital Goods”, Economics Bulletin 29, Issue
3, 1649-1657.
[6] Guo, J-T. and Lansing, K. (2007), ”Maintenance Expenditures and Indeterminacy under Increasing Returns to Scale”, International Journalof Economic Theory 3, 147-158.
4

This paper has assumed a simple model of Ak technology. The monetary endogenous
growth models have been studied in more general settings such as models with endogenous
labor supply, multiple capital goods as well as with endogenous money supply rules: see,for example, Fujisaki and Mino (2007 and 2009) and Itaya and Mino (2007). It would be
useful to consider the topic of this paper in these more general environments.

9[7] Itaya, J. and Mino, K. (2007), ”Technology, Preferences and the Growth
Effect of Money Supply”, Macroeconomic Dynamics 11, 589-612.
[8] Jha, S., Yip, C. and Wang, P. (2002), “Dynamics in a TransactionsBased Monetary Growth Model,” Journal of Economic Dynamics andControl 26, 611-635.
[9] Licandro, O. and L. A. Puch (2000), “Capital Utilization, Maintenance
Costs and the Business Cycle,” Annales d’Economie et de Statistique58, 143-164[10] Licandro, O., Puch, L.A. amd Ruiz-Tamarit, R. (2001), ”Optimal
Growth under Endogenous Depreciation, Capital Utilization and Maintenance Costs”, Investigaciones Economicas 25, 543-559.
[11] Li, K.F. and Yip, C-K., (2004), ”Monetary Policy and Equilibrium Indeterminacy in a Cash-in-Advance Economy with Investment”, Economics
Bulletin 5, 2, 1-7.
[12] McGrattan, E. and Schmitz Jr., J.A. (1999), “Maintenance and Repair:Too Big to Ignore,” Quarterly Review of the Federal Reserve Bank of
Minneapolis 23, No. 4, 2-23.
[13] Mullen, K.K. and Williams, M. (2004), ”Maintenance and Repair Expenditures: Determinants and Trade-offs with New Capital Goods”,
Journal of Economics and Business 56, 483-499.
[14] Suen, M-H. and Yip, C-K. (2005), ”Superneutrality, Indeterminacy andEndogenous Growth”, Journal of Macroeconomics 27, 579-595

10