Discussion Papers In Economics
And Business

Fiscal Rules in a Monetary Economy: Implications for
Growth and Welfare
Tetsuo Ono
Discussion Paper 18-27

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANFiscal Rules in a Monetary Economy: Implications for
Growth and Welfare
Tetsuo Ono
Discussion Paper 18-27

August 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANFiscal Rules in a Monetary Economy: Implications for
Growth and Welfare∗
Tetsuo Ono†
Osaka University
August 27, 2018

Abstract
This study considers two fiscal rules, a debt rule that controls the debt-toGDP ratio, and an expenditure rule that controls the expenditure-to-GDP ratio,
in a monetary growth model with financial intermediation. Tightening fiscal rules
promotes economic growth and thus benefits future generations. However, there
could be two equilibria of the nominal interest rates, and the welfare eﬀects of
the rules on the current generation are diﬀerent between the two equilibria. In
particular, the eﬀects of a decreased debt-to-GDP ratio depend on its initial ratio;
a low (high) ratio country has an incentive (no incentive) to reduce the ratio further
from the viewpoint of the current generation’s welfare. This result oﬀers a reason
for diﬃculties with fiscal reform in countries with already high debt-to-GDP ratios.
Keywords: Fiscal Rule; Government Debt; Economic Growth.
JEL Classifications: E62, E63, H63, O42.

∗

This work was supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid
for Scientific Research (C) Grant Number 18K01650.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. Tel.: +81-6-6850-6111; Fax: +81-6-6850-5256. E-mail: tono@econ.osaka-u.ac.jp

1

Introduction

The emergence and persistence of large fiscal deficits and public debt in many industrial and developing countries in the last few decades have raised concerns about fiscal
sustainability and led to calls for appropriate adjustment with the use of fiscal rules (International Monetary Fund, 2009; Schaechter et al., 2012). The fiscal rules control public
spending and/or public debt issuance, which in turn aﬀects allocation of resources acrossgenerations (Heijdra and Ligthart, 2000; Yakita, 2008; Fernandez-Huertas Moraga andVidal, 2010). In particular, recent studies suggest that fiscal austerity programs create
trade-oﬀs between a negative short-turn eﬀect and a positive long-run eﬀect in output(Bom and Ligthart, 2014), and the corresponding trade-oﬀs between current generations’loss and future generations’ benefit in terms of welfare (Glomm, Jung, and Tran, 2018).
The studies mentioned above are based on non-monetary growth models and thus, pay
little attention to monetary factors. However, fiscal rules may influence monetary variables
through the financial markets. For example, increased government debt is associated withincreased money supply through open market operations. This influences the inflationand nominal interest rates, and thus, may aﬀect the real activity and welfare acrossgenerations through allocative functions of financial markets (Bhattacharya et al., 1997;
Schreft and Smith, 1997, 1998). Therefore, consideration of monetary factors is necessary
for analyzing the fiscal rule eﬀects on current and future generations.
Schreft and Smith (1997, 1998) introduce a role for private banks that provide liquidity following Diamond and Dybvig (1983) into Diamond’s (1965) growth model withoverlapping generations. In this framework, agents are subject to stochastic relocations,and only currency can be transported between locations. Agents seek to liquidate theirholdings of bonds and capital to obtain currency once they have relocated. Under thisenvironment, Schreft and Smith (1997, 1998) consider the eﬀects of monetary policy on
growth and welfare across generations via financial markets. However, they assume no
direct government expenditures and taxes, and say nothing about the eﬀect of fiscal rules
on growth and welfare. Bhattacharya et al. (1997), Espinosa-Vega and Yip (1999, 2002),
and Hung (2005) partly overcome this limitation by investigating the fiscal spending ruleeﬀects on economic growth, but these studies assume no tax or debt rule (Bhattacharya
et al., 1997), or no fiscal deficit (Espinosa-Vega and Yip, 1999, 2002; Hung, 2005).1
1

Further extensions are undertaken by assuming variations in structural parameters (Gomis-Porqueras,
2000), various types of private banking systems (Matsuoka, 2011; Paal, Smith, and Wang, 2013), thepresence of banking crisis (Antinolfi and Keister, 2006), multiple production sectors (Ghossoub and1A notable exception is Schreft and Smith (2002), who consider the consequence ofa declining stock of public debt. In their model, there are two separate entities: thetreasury and central bank. The treasury finances public expenditures by labor income
taxation and public debt issues. The central bank creates money, acquires public bondsin the capital market, and rebates all interest earned on its holding of public debt to thetreasury, as in the cases of Japan and the United States. In particular, the central bank’sbalance sheet constraint requires that the value of the bank’s outstanding liabilities (i.e.,the monetary base) does not exceed the value of its holding of government bonds. Within
this framework, Schreft and Smith (2002) show that there could be two equilibria of thenominal interest rates, because the earned interest prevails over the Laﬀer curve property
with respect to the nominal interest rate, and that the equilibria are Pareto ranked.
Their analysis provides the welfare implications of a debt rule (i.e., a declining stock ofpublic debt), but the analysis is static in nature, because physical capital accumulation
is abstracted away from their model. Thus, their model provides no growth implicationsof the debt rule and its impact on welfare.
To address these issues, this study extends the model of Schreft and Smith (2002) byassuming Barro (1990)-type public production services as an engine of economic growth.
This assumption leads to AK technology as in Espinosa-Vega and Yip (1999, 2002),
Hung (2005), and Bhattacharya, Haslag, and Martin (2009). In addition, we assume
two fiscal rules that are widely used in industrial and developing countries: a debt rule,
which keeps the debt-to-GDP ratio at a constant rate, and an expenditure rule, whichkeeps the expenditure-to-GDP ratio constant (Budina et al., 2012). Within this extended
framework, the present study demonstrates how the two fiscal rules aﬀect growth andnominal interest and inflation rates, and how they in turn aﬀect welfare across generations.
The main results are as follows. First, there are two nominal interest rates that satisfy
the treasury’s budget constraint, but the lower one may violate the central bank’s balance
sheet constraint. In particular, the lower nominal interest rate is less likely to satisfy thebalance sheet as the two fiscal rules are strengthened because the tightened fiscal discipline
lowers the government bond issues, and thereby the central bank’s holding of government
bonds. Thus, the choice of fiscal rule determines the number of equilibria of the nominalinterest rates. However, the growth path of physical capital is always unique regardlessof the number of equilibria of the nominal interest rates, since the debt rule yields a oneReed, 2014), changing demand for cash (Schreft and Smith, 2000; Ghossoub and Reed, 2010), the US
dollar as a choice of storage value (Antinolfi, Landeo, and Nikitin, 2007), and allocation of capital andworkers (Ghossoub and Reed, 2017).

2to-one relationship between government debt and physical capital. This result is in sharp
contrast to those of previous studies that show multiple equilibrium paths of physicalcapital in the absence of a debt rule (Schreft and Smith, 1997, 1998; Espinosa-Vega andYip, 1999, 2002).
Second, when there are two equilibria, the welfare eﬀects of the two fiscal rules are
diﬀerent between the two equilibria. A decrease in the expenditure-to-GDP ratio lowers(raises) the nominal interest rate and thus, raises (lowers) the expected utility of eachgeneration in a low (high) nominal interest rate equilibrium. A decrease in the debt-toGDP ratio also has opposing eﬀects on the two equilibria through the nominal interest
rate, but its eﬀect depends on the initial debt-to-GDP ratio. Furthermore, the decrease
has an additional eﬀect in that it reduces the crowding-out eﬀect of government debt,
promotes capital accumulation, and thereby benefits future generations.
The analysis shows that when the debt-to-GDP ratio is initially high, a decrease in theratio leads to (a) a Pareto improvement in a low equilibrium, and (b) a loss to the current
generation and a gain to future generations in a high equilibrium. The result is reversed
when the ratio is initially low. A corollary of this result is that when the equilibrium is
unique and is distinguished by the higher nominal interest rate, the current generation
has no incentive (an incentive) to cut the debt-to-GDP ratio when the ratio is initiallyhigh (low). This result provides a possible explanation for the observed polarization ofdebt-to-GDP ratios among developed countries (OECD, 2018). The result also oﬀers areason for diﬃculties with fiscal reform in countries with already high debt-to-GDP ratios,such as Greece, Italy, and Japan.
The results described above are obtained under the assumption that the degree ofrelative risk aversion is below one, as in Bhattacharya et al. (1997) and Schreft andSmith (2002). The later part of this study makes an alternative assumption that thedegree of relative risk aversion is above one, and shows that the equilibrium is unique.
This analysis also shows that a decrease in the expenditure-to-GDP ratio raises the utilityof the current generation; and that a decrease in the debt-to-GDP ratio raises (lowers) the
utility of the current generation when the ratio is initially high (low). Thus, the welfare
eﬀects are entirely diﬀerent in the two cases of low and high relative risk aversion. Thissuggests that agents’ attitude toward risk is key to evaluating the welfare eﬀects of the
fiscal rules across generations.2
2

The role of relative risk aversion is also investigated by Espinosa-Vega and Yip (1999, 2002), GomisPorqueras (2000), and Bhattacharya, Haslag, and Martin (2009), who assume a balanced government
budget and no deficit. The present study diﬀers from their studies in that it analyzes the role of relative

3

The organization of the rest of this paper is as follows. Section 2 presents the model.
Section 3 characterizes a competitive equilibrium and Section 4 analyzes the welfare eﬀects
of fiscal rules when the degree of relative risk aversion is below one. Section 5 considers
the case in which the degree of relative risk aversion is above one. Section 6 provides
concluding remarks.

2

Model

The model is based on that developed by Schreft and Smith (1997, 1998, 2002). Consider
a discrete-time, infinite-horizon economy that starts from period t = 0. The economy
consists of two identical islands, each inhabited by an infinite sequence of overlapping
generations. Agents in each generation live for two periods, youth and old age. Eachlocation contains a continuum of ex ante identical young agents with a unit mass. In
addition, in period t = 0, there is an initial old generation in each location.
There are three assets in this economy: capital, money, and government bonds. It
is assumed that one unit of current consumption invested at time t becomes one unit ofcapital at time t + 1. The capital obtained in period t + 1, denoted by Kt+1 , is then used
for production in period t + 1. It is also assumed that capital fully depreciates in the
production process.
The nominal supplies of money and government bonds in each location in period t are
denoted by Mt and Bt , respectively. Bonds mature in one period, and each bond issued
in period t is a sure claim to It units of currency in period t + 1. Thus, It is the grossnominal rate of interest. We let pt denote the period-t price level, and mt ≡ Mt /pt andbt ≡ Bt /pt denote the period-t supplies of money and government bonds in real terms,
respectively. The initial old agents at each location are endowed with the initial per capita
capital stock, k0 > 0, and the initial per capita money supply, M−1 > 0.

2.1

Private Agents

Each agent is endowed with one unit of labor in youth and nothing in old age. Youngagents supply their labor force inelastically to firms, earn real wage w, and pay labor
income tax at the rate τ ∈ (0, 1). Agents are assumed to value only second-period
consumption, denoted by c. Thus, all of their after-tax income is saved for future consumption. Let u(c) denote the common utility function of all agents. Specifically, werisk aversion in the presence of government deficit and investigates how the fiscal rule eﬀects depend on
relative risk aversion.

4

assume the following form:(c)1−σ − 1u(c) =,
1−σwhere σ(> 0) denotes the degree of relative risk aversion. A higher σ is associated witha higher degree of relative risk aversion. We assume σ ∈ (0, 1) in the main analysis, andleave the case of σ > 1 to the analysis in Section 5.
At the beginning of each period, agents cannot move between or communicate acrosslocations. Goods can never be transported between locations. Thus, goods, labor, andasset market transactions occur within each location at the beginning of each period.
After this trade is conducted, some randomly selected fraction π ∈ (0, 1) of young agentsis chosen to be moved to the other location.
It is assumed that only currency can be transported between locations and that limited communication prevents the cross-location exchange of privately issued liabilities.
Therefore, relocated agents seek to liquidate their holdings of bonds and capital in orderto obtain currency. The relocation plays the role of liquidity preference shocks, as in
Diamond and Dybig (1983). Banks emerge to insure agents against these shocks. The
fraction π is constant across periods and known by all agents, and the probability of being
relocated is independently and identically distributed across young agents. Thus, there isno aggregate uncertainty.

2.2

Government

Government bonds and money are issued by the government sector, which comprises two
separate entities: the treasury and central bank. The treasury finances its expenditureby levying labor income tax on the young agents and by issuing government bonds with anominal value of Bt in period t. The real value of government bonds at the end of period
t is Bt /pt . Government bonds can be held by either private agents or the central bank.
The real value of government bonds demanded by the private agents and by the central
bank is denoted by bpt and bct , respectively. The stock of government bonds demanded inperiod t isbt ≡ bpt + bct .

(1)

The central bank issues fiat currency. In period t, the per capita value of the monetarybase outstanding is Mt in nominal terms and mt ≡ Mt /pt in real terms. The central bank’sbalance sheet constraint requires
mt ≤ bct ,5(2)

implying that the value of the central bank’s outstanding liabilities does not exceed thevalue of its holdings of government bonds. In addition, the central bank’s holdings ofgovernment bonds are limited by the stock of government bonds outstanding:0 ≤ bct ≤Bt.
pt

As in the United States, the central bank rebates all interest earned on its holdings ofgovernment bonds to the treasury, but retains the bond’s principal (Schreft and Smith,2002). A similar policy is in place in Japan.3 Let Tt and Rt ≡ It pt /pt+1 denote the nominal
value of the rebate in period t and the gross real rate of interest paid on government bonds
between periods t and t + 1, respectively. Then, Tt is
Tt = bct−1 Rt−1 pt − bct−1 pt−1
(
)
pt−1
c
= bt−1 pt Rt−1 −.
ptThus, the real rebate is

(
)
Tt
pt−1
c
= bt−1 Rt−1 −.
pt
ptGiven the central bank’s behavior, the treasury’s budget constraint becomes
Gt = τ Lt wt +Bt
Bt−1 Tt− Rt−1
+ ,
pt
pt−1
pt

where Gt is government expenditure devoted to public services to private producers, τ Lt wt
is labor tax revenue, Bt /pt is the revenue from issuing new bonds, Rt−1 Bt−1 /pt−1 is thedebt repayment costs, and Tt /pt is the rebate from the central bank to the treasury. In
particular, the government expenditure is assumed to follow the rule that the expenditureis a constant fraction of GDP, Gt = ξYt , where ξ ∈ (0, 1). The parameter ξ presents thefirst fiscal rule, on which we focus in the following analysis. Given Lt = 1 and bt ≡ Bt /pt ,the treasury’s budget constraint is rewritten as follows:
)
(
pt−1
c
ξyt = τ wt + bt − Rt−1 bt−1 + bt−1 Rt−1 −
.
pt

2.3

(3)

Firms

There is a continuum of identical firms with a unit mass in each location. They are
perfectly competitive profit maximizers that produce output, Yt , by using private capital,
Kt , labor (i.e., the size of period-t young agents), Lt , and government services to private
3

Source: Bank of Japan Act, Article 53. http://www.japaneselawtranslation.go.jp/law/detail/?id=92&vm=04&re=01(Accessed on August 12, 2008).

6

producers, Gt , as in Barro (1990). Specifically, the aggregate production function in eachlocation is given by
Yt = Ã (Gt )1−α (Kt )α (Lt )1−α ,where Ã > 0 and 0 < α < 1. For simplicity, capital is assumed to depreciate fully after
production.
Each firm chooses capital and labor in order to maximize its profit, Ã (Gt )1−α (Kt )α (Lt )1−α −rt Kt − wt Lt , where rt is the rental price of capital and wt is the real wage rate. Because of
the assumption of competitive markets, each firm takes rt and wt as given. The first-orderconditions with respect to Kt and Lt are
rt = αÃ (Gt )1−α (kt )α−1 = αyt /ky ,wt = (1 − α) Ã (Gt )1−α (kt )α = (1 − α) yt ,where kt ≡ Kt /Lt and yt ≡ Yt /Lt denote per capita capital and output, respectively.
Because the size of each generation is unity, that is, Lt = 1, we have kt = Kt and yt = Yt
for all t.
Given the assumption of the expenditure rule, Gt = ξYt = ξyt , the output yt becomes
yt = Ã (ξyt )1−α (kt )α , or,( )1/αyt = Ã(ξ)(1−α)/α kt .

( )1/αBy setting A = Ã(ξ)(1−α)/α , we obtain the AK technology of the production function, yt = Akt . The first-order conditions with respect to capital and labor are reduced
tort = αA and wt = (1 − α) Akt ,(4)

respectively. Because of the assumption of full depreciation of capital, the following noarbitrage condition holds in each period:
Rt = rt+1 = r ≡ αA.

2.4

(5)

Private Banks

In each period, young agents deposit the entire value of their after-tax income, (1 − τ )wt ,in a bank. Banks use deposits to acquire money, capital, and government bonds. In
addition, banks promise to pay agents who have (have not) relocated, that is, “movers”
n(“non-movers”), a gross real return of dmt (dt ) per unit on their deposits. We assume that

there is free entry into banking and that banks are competitive in the sense that they take7

the real return on assets as given.4 On the deposit side, we assume that intermediariesnare Nash competitors; that is, banks announce deposit return schedules (dmt , dt ), taking

the announced return schedules of other banks as given.
The return schedules must satisfy the following balance-sheet constraint:
mt + bpt + it ≤ (1 − τ ) wt , t ≥ 0.

(6)

In period t, a young agent deposits his or her entire savings, (1 − τ ) wt , with a bank. The
bank acquires mt units of real balances and bpt units of real bond holdings, and makes aninvestment in capital of it .
The announced return schedules must also satisfy the following two constraints. First,
banks promise to deliver the gross real return dm
t to all of their depositors who haverelocated in period t. By the law of large numbers, a fraction π of the bank’s depositors
have relocated; thus, the banks promise a payment per depositor of πdmt (1 − τ ) wt toagents who have relocated in period t. Relocated agents must be given currency. Thus,mbanks face the constraint pt+1 πdmt (1 − τ ) wt ≤ Mt , where pt+1 πdt (1 − τ ) wt represents

the nominal deposit returns to relocated agents, and Mt the bank’s currency holdings at
the beginning of period t + 1. Dividing both sides by pt , the constraint is rewritten asπdmt (1 − τ ) wt ≤ mt

pt
.
pt+1

Second, for the fraction 1 − π of depositors who have not relocated in period t, banks
promise a gross return of dnt per unit deposited. Thus banks owe (1 − π)dnt (1 − τ ) wt to
non-movers, which they pay in period t + 1. We assume It > 1 throughout the analysis;money is dominated in rate of return. Thus, banks do not carry cash reserves between
periods. Non-movers are repaid from the return on the bank’s bond holdings and capital
investments. Thus, dnt must satisfy(1 − π)dnt (1 − τ ) wt ≤ rt+1 it + Rt bpt .
Given that (5) holds, banks are indiﬀerent between investing in bonds and physical capital.
Let γt ≡ mt / (1 − τ ) wt denote the bank’s ratio of reserves to deposits in period t.
This implies that 1 − γt is the ratio of capital investments and bond holdings. Then the4

Alternatively, we can assume imperfect banking systems, as in Matsuoka (2011), Paal, Smith, andWang (2013), and Ghossoub and Reed (2017). However, the present study keeps the standard assumptionof competitive banking systems, since our focus here is on fiscal rules rather than banking systems.

8

above two constraints can be rewritten asγt pt,
π pt+1r (1 − γt )
dnt ≤
,
1−π

dm
t ≤

(7)
(8)

respectively. The constraint in (8) is derived by using (5).
Competition among banks for depositors forces banks to choose return schedules andportfolio allocations to maximize the expected utility of a representative depositor, subject
to the constraints, (6), (7), (8), and γt ∈ [0, 1]. In other words, banks choose γt to
maximize(πγtpt(1 − τ ) wt pt+1
π

)1−σ

1−σ

−1

(
+ (1 − π)r(1−γt )(1 − τ ) wt
1−π

)1−σ

−1

1−σ

subject to γt ∈ [0, 1]. The solution to the problem is
[
]−1
1−π
(1−σ)/σγt = γ (It ) ≡
(It )
+1
,
π

(9)

where the gross nominal interest rate is defined by
It = Rt pt+1 /pt = rpt+1 /pt .

(10)

Thus, a higher nominal interest rate implies a higher inflation rate.
The function γ (·), summarizing the behavior of the bank, gives the reserve-deposit
ratio as a function of the nominal interest rate, It . The function γ (·) has the following
properties: γ ′ (·) < 0, γ(0) = 1, γ(1) = π, limI→∞ γ (·) = 0 if σ ∈ (0, 1). In particular,γ ′ (·) < 0 implies that higher nominal interest rates create higher opportunity costs ofholding reserves and thus, a lower reserve-deposit ratio for the bank. Given the bank’s
optimal reserve-deposit ratio, equations (7) and (8) imply thatγ (It ) rγ (It ) pt
r=,
π rpt+1π It
1 − γ (It )
dnt =
r,
1−π

dm
t =

(11)
(12)

n
where dmt ≤ dt holds whenever It > 1 is satisfied.

We can now write the maximized expected utility of a representative depositor inperiod t as a function of the nominal interest rate, It , and the capital stock, kt , as
]1−σ
]1−σ
[
[
π
1 − π 1 − γ (It )γ (It )V (It , kt ) =+,r (1 − τ ) (1 − α) Aktr (1 − τ ) (1 − α) Akt
1 − σ πIt
1−σ
1−π
9

or,
[r (1 − τ ) (1 − α) Akt ]1−σ
V (It , kt ) =
1−σ

{

[
σ

(π)

γ (It )It}]1−σ
+ (1 − π) [1 − γ (It )]
σ

1−σ

.
(13)

Movers are made worse oﬀ whereas non-movers are made better oﬀ by increased nominalinterest rates. Thus, the function in (13) indicates that the nominal interest rate has two
opposing eﬀects on the expected utility, but it turns out that the net eﬀect is negative;the expected utility decreases as the nominal interest rate increases: ∂V (It , kt )/∂It < 0(see Appendix A.1 for the proof). This result is used in the welfare analysis in Section 4.

3

Equilibrium

In equilibrium, the factor pricing relationships in (4) and the no-arbitrage condition in(5) hold. In addition, the demand for reserves, γ (It ) (1 − τ )wt , must be equal to themonetary base, mt , in every period:
mt = γ (It ) (1 − τ ) (1 − α) Akt , t ≥ 0.

(14)This is the money market equilibrium condition. Similarly, period-t+1 capital stock, kt+1 ,must be equal to investment, it : kt+1 = it . From (1), (6), and (14), kt+1 is equivalent to
kt+1 = (1 − τ ) (1 − α) Akt − bt .

(15)This is the capital market equilibrium condition.
The treasury’s budget constraint in (3) is reformulated by using the factor pricing
relationships in (4), the no-arbitrage condition in (5), the definition of the nominal interest
rate in (10), and the money market equilibrium condition in (14), as follows:()
 bct−1 r 1 − 1 ,t = 0,It−1()
ξAkt = τ (1 − α) Akt + bt − rbt−1 + γ (It−1 ) (1 − τ ) (1 − α) Akt−1 r 1 − 1 , t ≥ 1,
It−1
(16)
where ξ ∈ (0, 1) denotes a constant fraction of government expenditure to GDP, Gt = ξYt ,which is aimed at controlling government expenditure.
The present study also assumes the following debt rule:bt = µyt ,(17)where µ ∈ (0, 1). This rule is aimed at controlling government bond issues. Thus, inthe present framework, there are two fiscal rules, represented by the parameters, ξ and10

µ. Finally, the central bank’s holdings of government bonds are limited by the stock ofgovernment bonds outstanding:bct ≤ bt .

(18)

We can now define a competitive equilibrium in the present model as follows.
nDefinition 1. A competitive equilibrium is a sequence of allocations, {bct , bpt , bt , kt , dmt , dt , it , mt , γt },and prices, {wt , rt , Rt , It }, such that given the initial conditions, k0 > 0, M−1 > 0,b−1 ≥ 0, bc−1 ≥ 0, and bp−1 ≥ 0, and the fiscal variables and rules, τ ∈ (0, 1),µ ∈ (0, 1), and ξ ∈ (0, 1), (i) γt solves the private bank’s optimization problem tonsatisfy (9), and the bank’s payments to movers and non-movers, {dmt , dt }, satisfy(11) and (12), respectively; (ii) wt and rt satisfy the factor pricing relationshipsin (4), and the no-arbitrage condition in (5) holds; (iii) the central bank’s balance
sheet in (2) holds; (iv) the private bank’s balance sheet in (6) holds; (v) the money
market equilibrium condition in (14) holds; (vi) the capital market equilibrium condition in (15) holds; (vii) the treasury’s budget constraint in (16) holds; (viii) the
restriction on the central bank’s holdings of government bonds in (18) holds; and(ix) the treasury authority follows the fiscal rules, Gt = ξyt and bt = µyt .
Summarizing the conditions described in Definition 1, a competitive equilibrium path
is characterized by a sequence of capital, monetary base, government bonds, and nominalinterest rate, {kt+1 , mt , bt , It }∞t=−1 . Given k0 (> 0), b−1 (≥ 0) , and m−1 (> 0), the sequencesatisfies the money market equilibrium condition in (14), the capital market equilibrium
condition in (15), the treasury’s budget constraint in (16), and the debt rule in (17) witha restriction on the central bank’s holding of government bonds in (18).
The capital market-clearing condition in (15) with the debt rule in (14) determines
the evolution of capital as
kt+1= [(1 − τ ) (1 − α) − µ] A.
kt

(19)

The growth rate of capital is constant across periods, since the production function exhibits a constant interest rate. In particular, a higher debt-to-GDP ratio lowers the growthrate, since government debt crowds out investment in capital. Given kt , the debt level bt
is determined by the debt rule, bt = µAkt .
The treasury’s budget constraint in (16) is reformulated by using the equation ofcapital in (19) and the debt rule in (17), as follows:11{[ξ − τ (1 − α) − µ] k0 + αb−1 = m−1 α (1 − 1/I−1 ) ,ϕ (·) /γ (It ) = α (1 − τ ) (1 − α) (1 − 1/It ) ,t ≥ 0,(20)

where ϕ (·) is defined byϕ (·) ≡ [ξ − τ (1 − α) − µ] [(1 − τ ) (1 − α) − µ] + αµ.
The nominal interest rate is constant across periods, except for the initial rate, I−1 . Givenkt and It , the monetary base, mt , is determined by the money market-clearing conditionin (14).
To determine the equilibrium nominal interest rate for t ≥ 0, recall the second expression in (20), which is rewritten as follows:

[

]
1−π
(1−σ)/σα (1 − τ ) (1 − α) (1 − 1/I) = ϕ (·)
(I)
+1 .
|
{z
}
π
{z
}
|
LHS

(21)

RHS

We assume ϕ (·) > 0 to find a nominal interest rate that is greater than one, I > 1. Under
this assumption, the left-hand side and the right-hand side of (21), denoted by LHS and
RHS, respectively, have the following properties:∂LHS∂ 2 LHS> 0,< 0, LHS|I=1 = 0, lim LHS = α (1 − τ ) (1 − α) ,
I→∞
∂I
∂I 2
∂LHS
1−σ1−π= ϕ (·)(I)(1−2σ)/σ > 0,∂Iσπ
 > 0 if σ ∈ (0, 1/2) ,2∂ LHS
1 − σ 1 − π 1 − 2σ
(1−3σ)/σ=0
if σ = 1/2,
ϕ
(·)
(I)

∂I 2
σ
π
σ
<0
if σ > 1/2,
1
RHS|I=1 = ϕ (·) , πif σ ∈ (0, 1), ∞ϕ (·) /πif σ = 1,
lim RHS =
I→∞

ϕ (·)
if σ > 1.
Figure 1 graphically illustrates (21), using I on the horizontal axis. As illustrated in
Figure 1, there could be two solutions to (21). However, either or both of them might notconstitute an equilibrium, because the central bank’s holdings of government bonds arelimited by the stock of government bonds outstanding, bct ≤ bt , or:γ (It ) (1 − τ ) (1 − α) Akt ≤ bt .
With the debt rule in (17), this is rewritten as follows:γ (It ) ≤θµ.
(1 − τ ) (1 − α)Thus, the equilibrium nominal interest rate must satisfy (22).
12

(22)

[Figure 1 is here.]
Let Iˆ denote the nominal interest rate that satisfies (22) with an equality. Since γ (·)
is decreasing in its argument, the nominal interest rate that satisfies (21) constitutes anˆ Using the definition of γ (It ) in (9), we
equilibrium if it is greater than or equal to I.
derive Iˆ as follows:
(
Iˆ =

π
1−π

)σ/(1−σ) []σ/(1−σ)(1 − τ ) (1 − α)
−1
.
θµ

(23)

Thus, the restriction in (22) is equivalent toIˆ ≤ I.

(24)

The following proposition establishes the conditions for the existence of an equilibrium
nominal interest rate.
Proposition 1.
(i) There are two equilibria of the nominal interest rates if[]σ1−πϕ (·) 1−σ
ϕ (·)
1
σ
π
+
<1
α (1 − τ ) (1 − α) 1 − σ α (1 − τ ) (1 − α)(25)holds and either of the following is satisfied: (a) 0 < µ < π (1 − τ ) (1 − α) and()
1
α (1 − τ ) (1 − α) 1
αµ 1 −< ϕ (·) <(26)( )1/σ ,1−σ 1−πIˆˆσπIor (b) π (1 − τ ) (1 − α) ≤ µ < (1 − τ ) (1 − α) and0 ≤ ϕ (·) <α (1 − τ ) (1 − α)1−σ 1−π
σ
π

.

(27)

(ii) There is a unique equilibrium of the nominal interest rate if the following conditions
hold:

(
)
1
0 < µ < π (1 − τ ) (1 − α) and 0 ≤ ϕ (·) < αµ 1 −
.
Iˆ

(28)

Proof. See Appendix A.2.
The conditions presented in Proposition 1 are interpreted as follows. First, condition( )()(25) implies that there is some I, denoted by I˜ ∈ (1, ∞), such that α (1 − τ ) (1 − α) γ I˜ 1 − 1/I˜ >˜ the reϕ (·) holds. In other words, for some range of the nominal interest rate around I,
bate from the central bank to the treasury outweighs the net expenditure of the treasury
13

(i.e., government expenditure plus debt repayment minus tax revenue and revenue fromissuing new government bonds). Therefore, given the Laﬀer curve property of the rebate,
˜ and the other is above I—that˜
there are two nominal interest rates—one is below I,
equate the rebate from the central bank and the net expenditure of the treasury, andthereby balance the treasury’s budget constraint.
Given condition (25), the characterization of two equilibria are classified into two cases:(a) 0 < µ < π (1 − τ ) (1 − α) and (b) π (1 − τ ) (1 − α) ≤ µ < (1 − τ ) (1 − α). Case (a)corresponds to Iˆ ≥ 1, as depicted in Panel (a) of Figure 1, and case (b) corresponds toIˆ < 1, as depicted in Panel (b) of Figure 1. In the latter case, µ must be bounded abovefrom (1 − τ ) (1 − α) to ensure kt+1 /kt > 0.
Based on the classification above, we next consider the condition in (26). The firstˆ < RHS(I),ˆ implying that at I = I,ˆ the rebate of
inequality is equivalent to LHS(I)interest from the central bank to the treasury is outweighed by the net expenditure ofˆ In other words, revenue shortage occurs. The second inequality
the treasury at I = I.
is equivalent to ∂LHS/∂I|I=Iˆ > ∂RHS/∂I|I=Iˆ. This implies that a marginal increase inˆ Thus,the rebate of interest outweighs a marginal increase in the net expenditure at I = I.
these two conditions together imply that there are some nominal interest rates, satisfying
the restriction in (22), that balance the treasury’s budget constraint, as illustrated inPanel (a) of Figure 1.
The condition in (27) is relevant to the case depicted in Panel (b) of Figure 1. The
first inequality in (27) implies that the nominal interest rate that satisfies the treasury’sbudget in (21) is greater than one, and the second inequality in (27) implies that 1 < I˜
holds. Thus, if either of them fails to hold, at least one of the equilibrium interest rates
falls below one. This result implies that non-movers withdraw their deposits from theprivate bank before maturity, and thus, there is no investment in capital. Condition (27)prevents such an undesirable situation.
Finally, consider the condition in (28). The implications of 0 < µ < π (1 − τ ) (1 − α)()
and 0 ≤ ϕ (·) are already described above. The condition ϕ (·) < αµ 1 − 1/Iˆ is opposite
to the first inequality in (26). Thus, the revenue stemming from the rebate of interest
ˆ The treasury’srate is suﬃcient to cover the net expenditure of the treasury at I = I.
budget constraint could be balanced by lowering or raising the nominal interest rate fromˆ In other words, there are two candidates for the equilibrium nominal interest
I = I.
rates, but the lower one fails to satisfy the restriction of the central bank’s holdings ofgovernment bonds in (22). The higher one remains as an equilibrium nominal interest
14

rate, as illustrated in Panel (c) of Figure 1.
Figure 2 illustrates the conditions presented in Proposition 1, using µ on the horizontal
axis. Panels (a) and (b) depict the cases of ξ = 0.2 and 0.3, respectively. Other parametervalues are set as θ = 0.99, σ = 0.85, π = 0.4, α = 0.3, and τ = 0.3 When ξ = 0.2 inPanel (a), the model displays the two equilibria case for a certain range of µ, but not theunique equilibrium case for any value of µ. However, when ξ is slightly higher and is set
at ξ = 0.3, as in Panel (b), the model displays two equilibria for a high range of µ whereas
it displays a unique equilibrium for a low range of µ. Thus, the figure indicates that themodel economy is more likely to attain two equilibria when the government sets higher
ratios of µ and ξ.
[Figure 2 is here.]
In order to understand the abovementioned result, recall that there are two solutionsof the nominal interest rates I that satisfy the treasury’s budget constraint in (21), but thelower one may violate the restriction of the central bank’s holding of government bondsin (22). Given this feature, consider an increase in the ratio of government expenditure
to GDP, ξ, and an increase in the ratio of debt to GDP, µ. These increases imply thatthe government issues more public bonds to finance increased expenditure. This in turnmakes the restriction in (22) less severe, resulting in the lower solution of (21) being more
likely to satisfy the restriction in (22). Thus, the economy is more likely to attain two
equilibria as fiscal discipline is weakened. This is a noteworthy feature of the present
model.
Another noteworthy feature is that the equilibrium nominal interest rates could be
two, whereas the growth rate of physical capital is always unique, as shown in equation(19). This result is in sharp contrast to those of previous studies that show multiple equilibrium paths of physical capital (Schreft and Smith, 1997, 1998; Espinosa-Vega and Yip,
1999, 2002). Given this multiplicity of equilibrium paths, Schreft and Smith (1997, 1998)
argue that low capital equilibrium represents a development trap. In the present framework, such a trap never arises in equilibrium; any two economies attain the same growthrate. Nevertheless, two economies could be ranked in terms of expected utility owing to
diﬀerence of the nominal interest rates. Given that the nominal interest rates depend on
the two fiscal rules, µ and ξ, changes in these rules may have diﬀerent implications forthe two equilibria. This point is investigated further in the next section.

15

4

Welfare Analysis

The aim of this section is first to evaluate the two equilibria in terms of utility. Hereafter,
an equilibrium with a lower (higher) nominal interest rate is simply called a low (high)equilibrium.
To achieve this aim, first recall the expected utility function of generation t in (13),[r (1 − τ ) (1 − α) Akt ]1−σ
Vt = V (It , kt ) =Φ (It ),
| {z }
1 {z
−σ
|
}

(29)

(∗2)

(∗1)

where Φ (It ) is defined as{
Φ (It ) ≡

[
σ

(π)

γ (It )It}]1−σ
+ (1 − π) [1 − γ (It )]
σ

1−σ

.

The function Φ (I) is decreasing in I : Φ′ (I) < 0 (see Appendix A.1).
Term (*1) in (29) is associated with capital accumulation, and term (*2) in (29)is associated with nominal interest rates. Because the growth rates are the same inboth equilibria, the diﬀerence in utility between the two equilibria arises solely from thediﬀerence in the nominal interest rates. Given the property of ∂V (It , kt )/∂It < 0, weconclude that a low equilibrium is superior to a high equilibrium.
Our next aim is to analyze the eﬀects of the two fiscal rules, represented by ξ and µ,
on expected utility. In particular, we consider an unanticipated and permanent decreasein ξ and µ from period 0, which is aimed at tightening fiscal discipline, and its impact on
the expected utility of generation t(≥ 0). First, let us focus on the fiscal rule, ξ, which
controls the expenditure-to-GDP ratio. This rule has an eﬀect on the expected utility
only via term (*2) in (29), since the growth rate of capital in (19) is independent of ξ.
This feature leads to the following proposition.
Proposition 2. Suppose that there are two equilibria of the nominal interest rates inperiod t ≥ 0 as in Proposition 1(i). An unanticipated, permanent decrease in the
expenditure-to-GDP ratio, ξ, from period 0 lowers (raises) the nominal interest rate
and thereby raises (lowers) the expected utility of each generation t(≥ 0) in a low(high) equilibrium.
[Figure 3 here.]
Proof. See Appendix A.3.
16

Figure 3 illustrates the eﬀect of a decrease in ξ on the nominal interest rates. The
treasury’s budget constraint in (16) suggests that a decrease in the ratio of expenditure
to GDP, ξ, aﬀects the determination of the nominal interest rate via term γ(I)(1 − τ )(1 −α)Ar(1 − 1/I) representing the rebate from the central bank to the treasury. In otherwords, the rebate must be reduced through changes in the nominal interest rates to
respond to decreased government expenditure. In particular, the nominal interest rates
have two opposing eﬀects on the rebate, described as follows.
First, lower nominal interest rates imply a lower opportunity cost of holding money
and thus, a higher reserve-deposit ratio for the private bank. The central bank’s balancesheet constraint requires that the value of the central bank’s outstanding liabilities, m,equals the value of its holding of government debt, bc . Thus, a lower nominal interest rate
leads to a larger value of the central bank’s holding of government debt and thereby to alarger value of the rebate to the treasury. This is a positive eﬀect of the nominal interest
rate on the rebate.
Second, lower nominal interest rates imply a lower inflation rate. This in turn raises
the real value of the bond’s principal, bct−1 /(pt /pt−1 ), and thus, decreases the gross return
minus the principal, which equals the rebate. This is a negative eﬀect of the nominalinterest rates on the rebate. In a low (high) equilibrium, the first positive eﬀect is smaller(larger) than the second negative eﬀect. Therefore, a decrease in ξ leads to a decrease(an increase) in the nominal interest rate and thus, raises (lowers) the expected utility ofeach generation in a low (high) equilibrium, since ∂V /∂I < 0.
Next, consider the fiscal rule, µ, which controls the government debt-to-GDP ratio.
This rule has an eﬀect on the expected utility via the term (*1) in (29), since the growthrate of physical capital in (19) increases as µ decreases. The rule µ also has an eﬀect
on the expected utility via term (*2) in (29), since the determination of the equilibrium
nominal interest rate depends on this rule, as observed in equation (21). The following
proposition summarizes the net eﬀect of decreased µ on the expected utility of generationt ≥ 0.
Proposition 3. Suppose that there are two equilibria of the nominal interest rates
in period t ≥ 0 as in Proposition 1(i). Consider an unanticipated, permanent
decrease in the government debt-to-GDP ratio, µ, from period 0, and its impact on
the expected utility across generations.
(i) If 2µ < (1 − ξ)(1 − α) + ξ − τ (1 − α) − α holds, then a decrease in µ leads to (a)17a trade-oﬀ between generations in terms of utility in a low equilibrium; and (b) aPareto improvement in a high equilibrium.
(ii) If 2µ > (1 − ξ)(1 − α) + ξ − τ (1 − α) − α holds, then a decrease in µ leads to (a) aPareto improvement in a low equilibrium; and (b) a trade-oﬀ between generationsin terms of utility in a high equilibrium.
The proof and interpretation of the result in Proposition 3 are as follows. Recall theindirect utility function in (29). With the capital accumulation equation in (19), equation(29) is reformulated as follows:[r (1 − τ ) (1 − α) Ak0 ]1−σ
Vt ={[(1 − τ ) (1 − α) − µ] A}t(1−σ) Φ (It ).
|{z}| {z }
1−σ
(∗3)

(30)

(∗2)

Equation (30) indicates that a decrease in µ has eﬀects on the expected utility of generation t via term (*3) representing the capital accumulation eﬀect as well as term (*2)
representing the nominal interest rate eﬀect.
Consider first term (*2), showing the eﬀect of a decrease in µ through the nominalinterest rate. As we observe below, this eﬀect is either positive or negative depending on
the initial value of µ as well as the state of the equilibrium (i.e., low or high). To investigate
the eﬀect precisely, recall the treasury’s budget constraint in (21), which determines theequilibrium nominal interest rate I. This is rewritten asξ − τ (1 − α) − µ  (1 − τ ) (1 − α) − µ  A+ αµ = γ (I) (1 − τ ) (1 − α) A(1−1/I),
|{z}
|{z}
|{z}
(∗4)

(∗5)

(∗6)

(31)
where the right-hand side presents the rebate from the central bank to the treasury.
Equation (31) suggests that a decrease in µ aﬀects the nominal interest rate and therebate in the following three ways. First, revenue from new bond issuance, represented
by term (*4), decreases as µ is reduced. Second, the crowding-out eﬀect on capital, represented by term (*5), also decreases as µ is reduced, which in turn increases government
expenditure. Finally, the debt outstanding, represented by term (*6), decreases as µ is
reduced.
The first two eﬀects imply that the rebate must increase to balance the treasury’sbudget, whereas the last eﬀect implies the opposite. The net eﬀect depends on the initialvalue of µ. Diﬀerentiation of the left-hand side of (31), denoted by LHS, with respect to18

µ yields∂LHS
1≷ 0 ⇔ µ ≷ µ̃ ≡ [(1 − τ ) (1 − α) + ξ − τ (1 − α) − α] .
∂µ
2If µ is high such that µ > µ̃ holds, a decrease in µ results in a decrease in LHS. Thisimplies there is a decrease in the rebate, which is realized by a decrease (an increase) in
the nominal interest rate in a low (high) equilibrium, as illustrated in Figure 4. Therefore,a decrease in µ works to raise (lower) the expected utility through term (*2) in (30) ina low (high) equilibrium if µ < µ̃ holds. The opposite result holds if µ is high such thatµ > µ̃.
[Figure 4 is here.]
Next, consider term (*3) in (30), showing a positive eﬀect on expected utility throughcapital accumulation. A decrease in µ lowers the debt-to-GDP ratio, reduces the crowdingout eﬀect of government debt, and thus, enhances capital accumulation. When the eﬀect
via term (*2) is positive, as in a high equilibrium in Proposition 3(i) and a low equilibrium in Proposition 3(ii), a decrease in µ definitely improves the expected utility of all
generations from generation 0.
When the eﬀect via term (*2) is negative, as in a low equilibrium in Proposition
3(i) and a high equilibrium in Proposition 3(ii), generation 0 becomes worse oﬀ, because
there is no positive eﬀect via term (*3) in period 0. However, all generations from some
generation, say generation T , onward benefit from a decrease in µ, because the positiveeﬀect via term (*3) accumulates over time and starts to dominate the negative eﬀect via
term (*2) in some future period T . Therefore, there is a trade-oﬀ in terms of the expectedutility between current and future generations.
Thus far, we have assumed two equilibria of the nominal interest rates. However, aspresented in Proposition 1(ii), the low equilibrium may fail to satisfy the restriction on
the central bank’s holding of government bonds. When this is the case, we obtain thefollowing corollary of Proposition 3.
Corollary 1. Suppose that there is a unique equilibrium of the nominal interest rate, as
in Proposition 1(ii). Consider an unanticipated, permanent change in fiscal rules
from period 0.
(i) A decrease in ξ raises the nominal interest rate and thus, lowers the expected utilityof generation 0.
19

(ii) If 2µ < (>)(1 − τ )(1 − α) + ξ − τ (1 − α) − α holds, a decrease in µ lowers (raises)
the nominal interest rate and thus, raises (lowers) the expected utility of generation0.
The first result shows that the current generation has little incentive to cut theexpenditure-to-GDP ratio, and thus, suggests the diﬃculty of fiscal reform that is aimed
at cutting public expenditure. The second result implies that the eﬀect of a decrease in
the debt-to-GDP ratio, µ, depends on the initial value of µ. When µ is low such that
2µ < (1 − ξ)(1 − α) + ξ − τ (1 − α) − α, a decrease in µ benefits the current generation, andthus, incentivizes the current generation to decrease the ratio further. However, when µ,
is high such that 2µ > (1 − ξ)(1 − α) + ξ − τ (1 − α) − α, the current generation opposesa reduction of µ, because it lowers their expected utility. Thus, the result in Corollary 1
provides a possible explanation for an observed diﬀerence in debt-to-GDP ratios amongdeveloped countries sharing similar economic characteristics (OECD, 2018). The result
also explains why fiscal reform is diﬃcult in countries with already high debt-to-GDP
ratios, such as Greece, Italy, and Japan.

5

Case of High Risk Aversion

Thus far, we have conducted the analysis by assuming that σ of the utility function(c)1−σ /(1 − σ) is set within the range, (0, 1), as in Schreft and Smith (2002). Under
this assumption, higher nominal interest rates imply higher opportunity costs of holding
reserves, and thereby lead to a lower reserve-deposit ratio for the private bank. However,in their companion paper, Schreft and Smith (1998) argued that the demand for reserves
is independent of the nominal rate of interest in their framework, and assumed σ > 1.
Given these conflicting arguments in the literature, this section provides the characterization of the competitive equilibrium when σ > 1. This assumption implies that agentsare highly risk averse. We show that when σ > 1, the equilibrium, if it exists, is always
unique, and that the welfare eﬀects of changes in fiscal rules are totally diﬀerent from theprevious welfare eﬀects.
Recall equation (21), which characterizes the equilibrium nominal interest rate fort ≥ 0. When σ > 1, LHS and RHS in (21) have the following properties:∂LHS> 0, LHS|I=1 = 0, lim LHS = α (1 − τ ) (1 − α) ,
I→∞
∂I
∂RHS
1
< 0, RHS|I=1 = ϕ (·) , lim RHS = ϕ (·) .
∂I
π I→∞
20

These properties imply that the solution of (21), if it exists, is unique as illustrated in
Figure 5. In other words, there is no multiplicity of equilibria when σ > 1. Thus, whenσ > 1, there is a unique I that satisfies (21) if the following condition holds:
lim LHS > lim RHS ⇔ α (1 − τ ) (1 − α) > ϕ (·) .

I→∞

I→∞

(32)

[Figure 5 is here.]
We reformulate the condition in (32) in terms of µ, as follows:f (µ) < 0,where f (·) is defined by[]1
f (µ) ≡ (µ) − 2 (1 − τ ) (1 − α) − (1 − ξ) µ2
+ (1 − τ ) (1 − α) [(1 − τ ) (1 − α) − (1 − ξ)] .
2

f (µ) = 0 has two solutions, µ = (1 − τ ) (1−α) and (1 − τ ) (1−α)−(1 − ξ), and f (µ) < 0
holds for µ ∈ ((1 − τ ) (1 − α) − (1 − ξ) , (1 − τ ) (1 − α)). With the non-negative constraint of µ, we reformulate the condition in (32) as follows:
max {0, (1 − τ ) (1 − α) − (1 − ξ)} < µ < (1 − τ ) (1 − α).

(33)

Next, consider the restriction on the central bank’s holding of government bonds in(22). Using the definition of γ(·) in (9), we reformulate (22) as follows:(1 − τ ) (1 − α) − θµ1−π≤(I)(1−σ)/σ ,
θµ
π

(34)

ˆ
I ≤ I,(35)

or

where Iˆ is defined in (23). Thus, when σ > 1, the restriction is satisfied if the nominalˆ This result is opposite to (24) for the case ofinterest rate I is below the critical level I.
σ ∈ (0, 1).
The analysis thus far suggests that there is a unique equilibrium nominal interest
rate I(> 1) if (33) and (35) hold. These conditions are summarized in the following
proposition.
Proposition 4. Suppose that σ > 1 holds. There is a unique equilibrium of the nominalinterest rate if the following condition holds:
21

(
)
π
ˆ
max {0, (1 − τ ) (1 − α) − (1 − ξ)} < µ < (1 − τ ) (1−α) and ϕ (·) ≤ αθµ 1 − 1/I .
θ(36)
Proof. See Appendix A.4.
The lower bound of µ corresponds to the condition in (32). The upper bound ofˆ is greater than 1 if µ <µ, π (1 − τ ) (1 − α)/θ, implies that the critical value of I, I,
π (1 − τ ) (1−α)/θ. Otherwise, Iˆ is below 1, so that there is no equilibrium nominal interest()
rate I(> 1) that satisfies the restriction in (35). The condition ϕ (·) ≤ αθµ 1 − 1/Iˆ
ensures that the solution of (21) satisfies the restriction in (35).
Given the characterization of the equilibrium in Proposition 4, we investigate theeﬀects of fiscal rules, ξ and µ, on the equilibrium nominal interest rate as well as the
expected utility of agents.
Proposition 5. Suppose that σ > 1 and (36) hold. Consider an unanticipated, permanent decrease in ξ and µ from period 0.
(i) A decrease in ξ lowers the nominal interest rate and thereby raises the expected utilityof generation 0.
(ii) A decrease in µ raises (lowers) the nominal interest rate and thereby lowers (raises)
the expected utility of generation 0 if 2µ < (>) (1 − τ ) (1 − α) + ξ − τ (1 − α) − α.
Proof. See Appendix A.5.
Given that ∂V /∂I < 0, the welfare of generation 0 is raised (lowered) by decreasesin ξ and µ if the nominal interest rate decreases (increases). The result in Proposition
5 resembles (opposes) that obtained in a low (high) equilibrium when σ ∈ (0, 1) (seePropositions 2 and 3). Thus, if the low equilibrium when σ ∈ (0, 1) fails to satisfy the
restriction on the central bank’s holding of government bonds, as in Corollary 1, the eﬀects
on the expected utility are diﬀerent in the two cases, σ ∈ (0, 1) and σ > 1. In other words,
the welfare eﬀects of changes in the fiscal rules depend heavily on the curvature of theutility function related to risk aversion. This result suggests that agents’ attitudes towardrisks shape the political feasibility of fiscal reforms.

6

Concluding Remarks

This study presented a monetary growth model with financial intermediation, in which
the monetary base of the central bank is constrained by its holding of government debt,
22

and introduced two fiscal rules into the model, that is, the debt rule, which keeps thedebt-to-GDP ratio constant, and the expenditure rule, which keeps the expenditure-toGDP ratio constant. Within this framework, the study investigated the eﬀects of these
two fiscal rules on the nominal interest rate, growth rate of physical capital, and welfare
across generations. The analysis showed that strengthening the fiscal rules reduces thecrowding-out eﬀects of government debt, promotes physical capital accumulation, andbenefits future generations. However, there could be multiple equilibria of the nominalinterest rates, and thus, the welfare eﬀects of the fiscal restraints through the nominalinterest rates are diﬀerent between the two equilibria. Thus, the net welfare consequences
are also diﬀerent between the two equilibria.
The present study focused on the debt rule to provide a mechanism for explaining
the cross-country diﬀerence in government debt accumulation. It was shown that undera certain condition, the equilibrium nominal interest rate is unique and distinguished by
the higher nominal interest rate, because the lower rate fails to satisfy the central bank’sbalance sheet constraint. In this situation, a reduction in the debt-to-GDP ratio raises(lowers) the expected utility of the current generation when the ratio is initially low (high).
This result suggests that the current generation supports (opposes) a reduction of the ratio
from the viewpoint of its utility when the ratio is initially low (high). Thus, low-ratio
countries have an incentive to reduce the ratios further, while high-ratio countries haveno such incentive. This result provides a mechanism for the cross-country diﬀerencesin government debt accumulation. In addition, this mechanism could be viewed as analternative to that proposed by Song, Storesletten, and Zilibotti (2012), who showed thatthe variation in countries’ preferences for public goods is key to explaining the crosscountry diﬀerences in government debt accumulation.
As a caveat to the analysis, it should be noted that the present analysis is based on theassumption that the treasury acts first and the central banks follow, because the present
study focuses on the fiscal rules eﬀects. An alternative assumption is that the central
bank acts first and sets a sequence of the nominal interest rates, as in Schreft and Smith(2002, Section 3). In addition, regarding the fiscal rules, there are alternatives, such as thebalanced-budget rule (Azzimonti, Battaglini, and Coate, 2016), the golden rule of publicfinance (Greiner and Semmler, 2000; Ghosh and Mourmouras, 2004a, 2004b; Minea andVillieu, 2009; Ueshina, 2018); and a fixed ratio of primary surplus to GDP (Bohn, 1995,
1998; Greiner, 2008). These alternatives are expected to provide interesting implications,and are left for future study.
23

A

ProofsA.1

Proof of ∂V (It , kt )/∂It < 0

Define Ṽ (It ) as follows:
V (It , kt )[r (1 − τ ) (1 − α) Akt ]1−σ
{
}
[
]1−σ
1
γ
(I
)
t
=
(π)σ
+ (1 − π)σ [1 − γ (It )]1−σ .
1−σItṼ (It ) ≡Diﬀerentiation of Ṽ (It ) with respect to It leads to[]−σ ( )2 1−π 1(It )(1−σ)/σ + 1∂ Ṽ (It )1
σ γ (It )π σ= − (π)
[
]2
∂It
It
It
(1−σ)/σ
1−π
(I
)
+
1
t
π
1−π 1−σ(It )[(1−σ)/σ]−1σ]2 .
(1−σ)/σ
1−π
(It )
+1
π

+ (1 − π)σ [1 − γ (It )]−σ [ πThe first term on the right-hand side is negative; the second term on the right-hand sideis positive as long as σ ∈ (0, 1). To determine the net eﬀect, we reformulate the above
expression as follows:[]σ−2∂ Ṽ (It )σ−2 1 − π(1−σ)/σ= (It )
(It )
+1
∂It
π
]
}
{
[
(1−σ)/σσ 1−π1−σ(1−σ)/σσ 1−π 1(It )+ 1 + (π)(It )× − (π)
π σ
π
σ
[
]σ−2
[
]
1−π
σ−2 1 − π(1−σ)/σσ(1−σ)/σ= (It )
(It )
+1
(π) −
(It )
−1
π
π
< 0,
where the last inequality holds because the sign of term − 1−π(It )(1−σ)/σ − 1 is negative.
π
■

A.2

Proof of Proposition 1

To show the existence of an equilibrium nominal interest rate, recall equation (21), which
determines the equilibrium nominal interest rate. Figure 1 graphically illustrates equation(21), using I on the horizontal axis. Let I˜ denote the value of I that satisfies ∂LHS/∂I =∂RHS/∂I. Solving ∂LHS/∂I = ∂RHS/∂I for I, we obtain
[
]σ
α
(1
−
τ
)
(1
−
α)
I˜ ≡
.
1−π
ϕ (·) 1−σσπ24As observed in Figure 1, there are two solutions of equation (21) distinguished by I > 1
if the following conditions hold:˜ > RHS(I).
˜
I˜ > 1 and LHS(I)(37)

The two conditions in (37) are reformulated as follows:α (1 − τ ) (1 − α)I˜ > 1 ⇔ ϕ (·) <,1−σ 1−πσand˜ > RHS(I)˜ ⇔LHS(I)

(38)

π

[
]σ
1−π
ϕ (·) 1−σ
ϕ (·)
1
σ
π
< 1.
+
α (1 − τ ) (1 − α) 1 − σ α (1 − τ ) (1 − α)(39)Given conditions in (38) and (39), either of the following two cases occur: (i) both
solutions satisfy the restriction of the central bank’s holdings of government bonds in(22); and (ii) only one of the solutions satisfies the restriction in (22). In what follows,
we consider these two cases in turn.
(i) The two solutions satisfy the restriction of the central bank’s holdings of government
bonds in (22) if either of the following holds:Iˆ ≤ 1,

(40)

∂LHS
∂RHS
ˆ < RHS(I).
ˆIˆ > 1,>, and LHS(I)∂I I=Iˆ∂I I=Iˆ

(41)

or

Panels (a) and (b) of Figure 1 illustrate cases satisfying (40) and (41), respectively.
The condition in (40) is reformulated as follows:Iˆ ≤ 1 ⇔ π (1 − τ ) (1 − α) ≤ µ.

(42)

The three conditions in (41) are reformulated as follows:Iˆ > 1 ⇔ π (1 − τ ) (1 − α) > µ,

(43)

1−π
ϕ (·) 1−σ
σ
π

∂RHS
1
∂LHS
>
⇔
<(
[
)
∂I I=Iˆ∂I I=Iˆα (1 − τ ) (1 − α)1/(1−σ) (1−τ )(1−α)
π
1−π

(
)
1
ˆ < RHS(I)ˆ ⇔ αµ 1 −LHS(I)< ϕ (·)
Iˆµ]1/(1−σ) ,
−1
(44)
(45)

Thus, there are multiple equilibria of nominal interest rates if (38), (39), and either (42)or the combination of (43), (44), and (45) hold. These conditions are summarized in thefirst part of Proposition 1.
25

(ii) As illustrated in Panel (c) of Figure 1, a lower nominal interest rate fails to satisfy
the restriction in (22) if the following conditions hold:ˆ > RHS(I).
ˆIˆ > 1 and LHS(I)
These conditions are reformulated as follows:Iˆ > 1 ⇔ π (1 − τ ) (1 − α) > µ,()
1
ˆ > RHS(I)ˆ ⇔ ϕ (·) < αµ 1 −LHS(I)
.
Iˆ

(46)
(47)

Thus, there is a unique equilibrium of the nominal interest rate if (38), (39), (46), and(47) hold. These are summarized in the second part of Proposition 1.
■

A.3

Proof of Proposition 2

Recall (21), which determines the equilibrium nominal interest rates. LHS of (21) is
independent of ξ, whereas RHS of (21) decreases as ξ decreases. Thus, as illustrated in
Figure 3, the nominal interest rate decreases (increases) in a low (high) equilibrium as ξ
decreases.
Recall equation (29), which presents the expected utility of generation t, Vt . The
equation indicates that Vt depends on kt and It , but kt is independent of ξ, as demonstratedin (19). Thus, given the property of ∂V /∂I < 0, we obtain the result in Proposition 2.
■

A.4

Proof of Proposition 4

If µ ≥ (1 − τ ) (1 − α)/θ, (34) always holds for I > 1 because the left-hand side of (34)
is non-positive. Thus, the solution of equation (21) always satisfies the restriction inˆ Since the(35). If µ < (1 − τ ) (1 − α)/θ, the solution of equation (21) must be below I.
equilibrium I must be also greater than 1, the critical value Iˆ must be greater than 1:πIˆ > 1 ⇔ (1 − τ ) (1 − α) > µ.
θ
As depicted in Figure 5, the solution of (21) satisfies the restriction in (35) if LHS|I=Iˆ ≥()ˆ
RHS| ˆ holds, that is, if αθµ 1 − 1/I ≥ ϕ (·) holds. Thus, the solution of equation
I=I(21) satisfies the restriction in (35) if either of the following conditions holds:µ≥1(1 − τ ) (1 − α),
θ
26

(48)

or
µ<

(
)
π
(1 − τ ) (1 − α) and ϕ (·) ≤ αθµ 1 − 1/Iˆ .
θ

(49)

There is a nominal interest rate I (≥ 1) that satisfies (21) and (35) if either (i) (33) and(48) hold, or (ii) (33) and (49) hold. There is no µ(> 0) that satisfies the first suﬃcientcondition. The second suﬃcient condition is summarized as follows:
max {0, (1 − τ ) (1 − α) − (1 − ξ)} < µ <()π(1 − τ ) (1 − α) and ϕ (·) ≤ αθµ 1 − 1/Iˆ .
θ(50)

Thus, there is a unique equilibrium of the nominal interest rate I (≥ 1) if (50) holds.
■

A.5

Proof of Proposition 5

Recall equation (21), which determines the equilibrium nominal interest rate for t ≥ 0,
which is rewritten asα (1 − τ ) (1 − α) (1 − 1/I)
1−π(I)(1−σ)/σ + 1π= ϕ (·) .

(51)

The left-hand side of (51) is increasing in I when σ > 1 and is independent of ξ and µ.
The right-hand side is independent of I but is aﬀected by ξ and µ in the following way:∂ϕ (·) /∂ξ > 0,∂ϕ (·) /∂µ ≷ 0 ⇔ 2µ ≷ (1 − τ ) (1 − α) + ξ − τ (1 − α) − α.
Thus, we obtain∂I/∂ξ > 0,∂I/∂µ ≷ 0 ⇔ 2µ ≷ (1 − τ ) (1 − α) + ξ − τ (1 − α) − α.
From (29), the expected utility of generation 0 is
V0 = V (I0 , k0 ) =[r (1 − τ ) (1 − α) Ak0 ]1−σΦ (I0 ) ,1−σ

where Φ (I0 ) is decreasing in I0 : Φ′ (I0 ) < 0. Since k0 is a given initial condition, a higher(lower) I0 is associated with a lower (higher) V0 .
■27

References[1] Antinolfi, G., and Keister, T., 2006. Discount window policy, banking crises, and indeterminacy of equilibrium. Macroeconomic Dynamics 10, 1–19.
[2] Antinolfi, G., Landeo, C.M., and Nikitin, M., 2007. Dollarization and the inflation threshold. Canadian Journal of Economics 40, 628–649.
[3] Azzimonti, M., Battaglini, M., and Coate, S., 2016. The costs and benefits of balanced
budget rules: Lessons from a political economy model of fiscal policy. Journal of Public
Economics 136, 45–61.
[4] Barro, R.J., 1990. Government spending in a simple model of endogenous growth. Journalof Political Economy 98, S103–S125.
[5] Bhattacharya, J., Guzman, M.G., Huybens, E., and Smith, B.D., 1997. Monetary, fiscal,
and reserve requirement policy in a simple monetary growth model. International Economic
Review 38, 321–350.
[6] Bhattacharya, J., Haslag, J., and Martin, A., 2009. Optimal monetary policy and economic
growth. European Economic Review 53, 210–221.
[7] Bohn, H., 1995. The sustainability of budget deficit in a stochastic economy. Journal of
Money, Credit and Banking 27, 257–271.
[8] Bohn, H., 1998. The behavior of US public debt and deficits. Quarterly Journal of Economics 113, 949–963.
[9] Bom, P.R.D, and Ligthart, J.E., 2014. Public infrastructure investment, output dynamics,
and balanced budget fiscal rules. Journal of Economic Dynamics and Control 40, 334–354.
[10] Budina, N., Kinda, T., Schaechter, A., and Weber, A., 2012. Fiscal rules at a glance:
Country details from a new dataset. IMF Working Paper 12/273, International MonetaryFund.
[11] Diamond, D.W., and Dybvig, P.H., 1983. Bank runs, deposit insurance, and liquidity.
Journal of Political Economy 91, 401–419.
[12] Diamond, P.A., 1965. National debt in a neoclassical growth model. American Economic
Review 55, 1126–1150.
[13] Espinosa-Vega, M.A., and Yip, C.K., 1999. Fiscal and monetary policy interactions in an
endogenous growth model with financial intermediaries. International Economic Review 40,
595–616.
[14] Espinosa-Vega, M.A., and Yip, C.K., 2002. Government financing in an endogenous growthmodel with financial market restrictions. Economic Theory 20, 237–257.
[15] Fernandez-Huertas Moraga, J., and Vidal, J.-P., 2010. Fiscal sustainability and public debt
in an endogenous growth model. Journal of Pension and Economic Finance 9, 277–302.
[16] Ghosh, S., and Mourmouras, I.A., 2004a. Endogenous growth, welfare and budgetary
regimes. Journal of Macroeconomics 26, 623–635.
[17] Ghosh, S., and Mourmouras, I.A., 2004b. Debt, growth and budgetary regimes. Bulletin of
Economic Research 56, 241–250.
[18] Ghossoub, E.A., and Reed, R.R., 2010. Liquidity risk, economic development, and theeﬀects of monetary policy. European Economic Review 54, 252–268.
[19] Ghossoub, E.A., and Reed, R.R., 2014. The cost of capital, asset prices, and the eﬀects ofmonetary policy. Journal of Macroeconomics 42, 211–228.

28

[20] Ghossoub, E.A., and Reed, R.R., 2017. Financial development, income inequality, and theredistributive eﬀects of monetary policy. Journal of Development Economics 126, 167–189.
[21] Glomm, G., Jung, J., and Tran, C., 2018. Fiscal austerity measures: Spending cuts vs. tax
increases. Macroeconomic Dynamics 22, 501–540.
[22] Gomis-Porqueras, P., 2000. Money, banks and endogenous volatility. Economic Theory 15,
735–745.
[23] Greiner, A., 2008. Human capital formation, public debt and economic growth. Journal of
Macroeconomics 30, 415–427.
[24] Greiner, A., and Semmler, W., 2000. Endogenous growth, government debt and budgetary
regimes. Journal of Macroeconomics 22, 363–384.
[25] Heijdra, B.J., and Ligthart, J.E., 2000. The dynamic macroeconomic eﬀects of tax policy
in an overlapping generations model. Oxford Economic Papers 52, 677–701.
[26] Hung, F.-S., 2005. Optimal composition of government public capital financing. Journal of
Macroeconomics 27, 704–723.
[27] International
Monetary
Fund,
2009.
Fiscalrules—Anchoring
expectations for sustainable public finances. Staﬀ Board Paper,
December 16.
https://www.imf.org/external/np/pp/eng/2009/121609.pdf (Accessed on July 21, 2018).
[28] Matsuoka, T., 2011. Monetary policy and banking structure. Journal of Money, Credit, andBanking 43, 1109–1120.
[29] Minea, A., and Villieu, P., 2009. Borrowing to finance public investment? The ‘golden ruleof public finance’ reconsidered in an endogenous growth setting. Fiscal Studies 30, 103–133.
[30] OECD, 2018. General government debt (indicator). doi: 10.1787/a0528cc2-en (Accessed on
August 2, 2018).
[31] Paal, B., Smith, B.D, and Wang, K., 2013. Monopoly versus competition in banking: Some
implications for growth and welfare. Annals of Economics and Finance 14, 853–892.
[32] Schaechter, A., Kinda, T., Budina, N., and Weber, A., 2012. Fiscal rules in response to thecrisis—Toward the “next generation” rules. A new dataset. IMF Working Paper 12/187,
International Monetary Fund.
[33] Schreft, S.L., and Smith, B.D., 1997. Money, banking, and capital formation. Journal of
Economic Theory 73, 157–182.
[34] Schreft, S.L., and Smith, B.D., 1998. The eﬀects of open market operations in a model ofintermediation and growth. Review of Economic Studies 65, 519–550.
[35] Schreft, S.L., and Smith, B.D., 2000. The evolution of cash transactions: Some implications
for monetary policy. Journal of Monetary Economics 46, 97–120.
[36] Schreft, S.L., and Smith, B.D., 2002. The conduct of monetary policy with a shrinking
stock of government debt. Journal of Money, Credit, and Banking 34, 848–882.
[37] Song, Z., Storesletten, K., and Zilibotti, F., 2012. Rotten parents and disciplined children:
A politico-economic theory of public expenditure and debt. Econometrica 80, 2785–2803.
[38] Ueshina, M., 2018. The eﬀect of public debt on growth and welfare under the golden ruleof public finance. Journal of Macroeconomics 55, 1–11.
[39] Yakita, A., 2008. Sustainability of public debt, public capital formation, and endogenous
growth in an overlapping generations setting. Journal of Public Economics 92, 897–914.

29

Panel (a)LHS of Eq.(21)
RHS of Eq.(21)I

1

Panel (b)
LHS of Eq.(21)
RHS of Eq.(21)I
1
Panel (c)
LHS of Eq.(21)
RHS of Eq.(21)I
130
Figure 1: Illustration of LHS and RHS of Eq. (21), using I on the horizontal axis.

Panel (a)

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2

multiple

0.1
0
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.3

0.35

0.4

0.45

Panel(b)

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2

uniquemultiple

0.1

0.2

0.1
0
0

0.05

0.15

0.25

Figure 2: Conditions (25)–(28) in Proposition 1 when ξ = 0.2 (Panel (a)) and ξ = 0.3(Panel (b)). The horizontal axis is µ. Other parameter values are set as θ = 0.99,
σ = 0.85, π = 0.4, α = 0.3, and τ = 0.3; these values are used in Figures 3 and 4.

31

0.2
LHS of Eq.(21)
RHS of Eq.(21) when
RHS of Eq.(21) when

0.18
0.16

high equilibrium

0.14
0.12

low equilibrium

0.1
0.08
0.06
0.04
0.02

I

0
0

5

10

15

20

25

30

Figure 3: Illustration of Eq. (21) when µ = 0.26 and ξ = 0.3 or 0.295, using I on thehorizontal axis.

32

0.2
LHS of Eq.(31)
RHS of Eq.(31) when
RHS of Eq.(31) when

0.18

high equilibrium

0.16
0.14
0.12

low equilibrium

0.1
0.08
0.06
0.04
0.02

I

0
5

10

15

20

25

30

35

40

45

50

Figure 4: Illustration of Eq. (31) when ξ = 0.3 and µ = 0.26 or 0.25, using I on thehorizontal axis.

33

LHS of Eq.(21)
RHS of Eq.(21)I

1Figure 5: Illustration of Eq. (21) when σ > 1, using I on the horizontal axis.

34