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The Pitfall of Interest on Excess Reserves: A Perspective of Fiscal-Monetary Interaction

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Some central banks pay interest on excess reserves (IOER). Under IOER policy, monetary policy changes the budget balance of the fiscal authority, since the IOER decreases the seigniorage from the central bank. The IOER not only reduces the surplus of the fiscal authority but also increases the liabilities of the consolidated government. These changes caused by the IOER are crucial for price determination in joint fiscal-monetary policy. Here, I investigate the role of the IOER under Ricardian and non-Ricardian regimes and show that the IOER rate must be lower than the market interest rate to attain a finite inflation rate satisfying the transversality condition. Policy coordination between the fiscal authority and the central bank is needed to achieve the steady-state inflation rate and rule out real explosion.
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The Pitfall of Interest on Excess Reserves: A
Perspective of Fiscal–Monetary Interaction
Yuto Kajita
Graduate School of Economics, Waseda University, 1-6-1 Nishiwaseda
Shinjuku-ku, Tokyo 169-8050, Japan
December 24, 2018
Abstract
Some central banks pay interest on excess reserves (IOER). Under IOER policy, mone-
tary policy changes the budget balance of the fiscal authority, since the IOER decreases
the seigniorage from the central bank. The IOER not only reduces the surplus of the
fiscal authority but also increases the liabilities of the consolidated government. These
changes caused by the IOER are crucial for price determination in joint fiscal–monetary
policy. Here, I investigate the role of the IOER under Ricardian and non-Ricardian
regimes and show that the IOER rate must be lower than the market interest rate to
attain a finite inflation rate satisfying the transversality condition. Policy coordination
between the fiscal authority and the central bank is needed to achieve the steady-state
inflation rate and rule out real explosion.
Keywords: joint fiscal–monetary policy, interest on excess reserves, quantitative eas-
ing
JEL Codes: E63, E58
I would like to thank Kozo Ueda, Hajime Tomura, and other conference and seminar participants at the
2018 Asian Meeting of the Econometric Society and the 2018 Annual Meeting of the Central Bank Research
Association for their useful comments. All remaining errors are mine. E-mail address: kajita.yuto@gmail.com
1 Introduction
In recent years, interest on excess reserves (IOER) has served as a useful policy instrument
by central banks in an environment in which central banks hold huge balance sheets as a
result of massive quantitative easing (QE) in the aftermath of the global financial crisis.
From December 2015, the Federal Reserve in the US began to gradually increase the IOER
rate from 0.5%. The European Central Bank and the Bank of Japan introduced a negative
interest rate policy with the aim of strengthening monetary easing.
While IOER serves as a standard monetary policy instrument, as is often described in
new Keynesian models, there is another important fiscal channel through which the IOER
influences price stability. When combined with huge balance sheets of central banks, the
IOER changes the budget balance of the consolidated government: the fiscal authority and
the central bank. This is because interest paid to private financial institutions through the
IOER are deposited again in the central bank as reserve deposits. In other words, a rise in
the IOER rate causes an increase in central bank debt. In addition, because the IOER is
a liability cost, a rise in the IOER rate causes a decline in the profits of the central bank.
Thus, the IOER simultaneously changes the consolidated government liabilities and surplus.
These changes affect price level through the mechanism of the fiscal theory of price level.
In this study, I analyze price dynamics on a balanced growth path under the IOER policy
from the perspective of joint fiscal–monetary policy and discuss how the fiscal authority
and the central bank should cooperate to attain a steady-state inflation rate satisfying the
transversality condition (TVC) and stability under Ricardian and non-Ricardian regimes.
The main findings are as follows. First, there is no balanced growth path for inflation
rates when the IOER rate equals the market interest rate consistent with the Euler equation,
unless there is no excess reserve in the steady state. Second, under a Ricardian regime,
to satisfy the TVC, appropriate fiscal–monetary coordination is needed: the central bank
sets the IOER rate lower than market interest rate while the fiscal authority continues to
increase the total amount of its liabilities at a certain rate. The setting of the appropriate
1
IOER rate by the central bank is a necessary condition for stabilizing inflation rate under a
non-Ricardian regime.
The rest of this paper is organized as follows. In section 2, I review the literature about
the IOER. In section 3, I postulate the balance equation and nominal surplus of the central
bank and derive the government debt-value equation with QE and the IOER. I then discuss
how QE and the IOER affect prices in the model. In section 4, I show the condition for
the existence of a balanced growth path for the inflation rate in the model and examine the
relationship between the IOER and the inflation rate under Ricardian and non-Ricardian
regimes. I then discuss how the central bank should set the IOER rate to achieve the
steady-state inflation rate. Section 5 concludes.
2 Literature
The literature on the IOER begins with Friedman (1960), who proposes paying a market
interest rate on bank reserves. Since then, many researchers have studied policy about
interest on reserves from theoretical aspects. The first seminal work of this strand of the
literature is Sargent and Wallace (1985), who examine the determinacy condition of price
level under an interest-on-reserves regime. Smith (1991) and more recently Ennis (2018) also
discuss the determinacy of equilibrium in a different model.1
The two studies closes to mine are Cochrane (2014) and Del Negro and Sims (2015).
They incorporate interest on reserves into models of fiscal theory.2Cochrane (2014) discusses
monetary policy with interest on reserves under a non-Ricardian regime and analyzes the
interaction between a new Keynesian model and fiscal theory. Del Negro and Sims (2015)
discuss the problem behind the remittance from the central bank to the fiscal authority and
1Many studies analyze interest on reserves from other perspectives. Benigno and Nistic´o (2017) and
Ireland (2014) investigate the role of interest on reserves in dynamic stochastic general equilibrium models.
Kashyap and Stein (2012) consider optimal monetary policy with interest on reserves.
2To the best of my knowledge, Leeper (1991), Sims (1994), and Woodford (1994) are the seminal contribu-
tions in the literature. Leeper and Leith (2016) is a useful summary to aid understanding of fiscal–monetary
interaction for price-level determination.
2
show that if the fiscal authority does not provide appropriate financial support with the
central bank, extremely high inflation might occur when the central bank faces a huge loss.
There is a crucial difference in the monetary policy stance between Cochrane (2014) and
Del Negro and Sims (2015). Del Negro and Sims (2015) assume “active” monetary policy in
Leeper’s (1991) terminology3whereas Cochrane (2014) assume “passive” monetary policy,
in which the fiscal authority sets the nominal interest rate by controlling the total amount
of its liabilities given the real surplus.
This study complements Cochrane (2014) and Del Negro and Sims (2015); I explicitly
separate the balance equation of the consolidated government into that of the fiscal authority
and the central bank and discuss how QE and the IOER change the central bank’s balance
sheet and nominal surplus. I then examine whether the fiscal authority can pin down a
balanced growth path for the inflation rate alone under a “passive” fiscal policy (Ricardian
regime), and an “active” fiscal policy (non-Ricardian regime) with the IOER, and investigate
the role of the central bank in achieving a steady-state inflation rate.
3 The Government Debt-Value Equation with Quanti-
tative Easing and Interest on Excess Reserves
In this section, I postulate the budget balances of the fiscal authority and the central bank
and derive the government debt-value equation with QE and the IOER. I then discuss how
QE and the IOER affect prices in the model.
I follow the work of Cochrane (2001), who develops the government debt-value equation
with long-term bonds. For ease of interpretation, I assume there are only two types of bonds,
a short-term bond and a long-term bond, and households do not face any nominal friction.
3Leeper (1991) defines active and passive policy in terms of the constraints that a policy authority faces.
An active authority does not respond to a government debt shock and is free to set its control variable. A
passive authority responds to a government debt shock and is constrained by the active authority’s behavior.
3
3.1 Fiscal Authority
Let BF
t(t+j) denote the face value of zero-coupon government bonds outstanding at the
end of period t, which mature in t+j,j= 1 or 2. Thus, BF
t1(t) and BF
t1(t+ 1) represent a
short-term bond and a long-term bond, respectively, outstanding at the end of period t1.
The balance equation of the fiscal authority in period tis
BF
t1(t) + Qt(t+ 1)BF
t1(t+ 1) = PtsF
t+PtτCB
t+Qt(t+ 1)BF
t(t+ 1) + Qt(t+ 2)BF
t(t+ 2).(1)
The left-hand side of equation (1) shows redemption of the government bonds and the right-
hand side the nominal surplus of the fiscal authority (PtsF
t), which is a remittance from the
central bank (PtτCB
t)4and new government bonds issued by the fiscal authority. Qt(t+j)
denotes the nominal price at tof a bond that matures in t+j(j= 1 or 2) and is endogenously
determined by the Euler equation:
Qt(t+j) = ρ
1 + gyjPt
Pt+j
,(2)
where I assume there is an endowment economy and logarithmic utility with respect to
consumption. Here, ρand gydenote the subjective discount factor and the growth rate of
output, respectively. Equation (2) pins down market interest rate it:
1
1 + it
=Qt(t+ 1).(3)
In this economy, equation (1) is a household budget constraint. Therefore, it must always
be satisfied in an equilibrium for all periods.
There are two ways in which equation (1) is satisfied. One is a Ricardian regime, in which
the consolidated government adjusts the policy variables according to the given prices {Pt}.
The other case is a non-Ricardian regime in which the consolidated government determines
4The nominal surplus of the central bank is transferred to the fiscal authority by a fixed proportion every
period, PtτC B
t=θPtsC B
t, where PtsCB
tdenotes the nominal surplus of the central bank and θ[0,1] is the
remittance rate that the fiscal authority sets.
4
the policy variables independently of equation (1) and it is satisfied by the adjustment of
the price level.
3.2 Central Bank
The central bank’s balance equation consists of the remittance to the fiscal authority, the
purchase and redemption of bonds by QE, changes in excess reserves caused by the IOER
and QE, and the money issued by the central bank. For simplicity, I assume there is no
required reserve. The balance equation of the central bank in period tis
Q
t(t+ 1)BCB
t(t+ 1) + Q
t(t+ 2)BCB
t(t+ 2) BCB
t1(t)Qt(t+ 1)BCB
t1(t+ 1)
=1
1 + iIOER
t
DEX
tDEX
t1+MtMt1PtτCB
t,(4)
where BCB
t(t+j) denotes the amount of bonds held by the central bank in period tthat
mature in t+j(j= 1 or 2), and Q
t(t+j) is the purchase price in tof bonds that mature
in t+j(j= 1 or 2). In contrast to Qt(t+j) being determined by the Euler equation, I
assume that Q
t(t+j) is an exogenous variable. Mtand iI OER
trepresent money and the
IOER rate, respectively. DEX
tdenotes excess reserves that pay IOER in the next period.
Hence, I discount DEX
tby the IOER rate.
The right-hand side in equation (4) represents a change in assets of the central bank and
the left-hand side represents a change in its liabilities. Equation (4) shows that the cost of
buying the government bonds must be financed from excess reserves, money, or transfers
from the fiscal authority. During the global financial crisis, central banks created excess
reserves in exchange for asset purchases. Thus, I assume the central bank finances the cost
of asset purchases by issuing excess reserves.
The nominal surplus of the central bank is
PtsCB
t= [Qt(t+ 1) Q
t(t+ 1)] BC B
t(t+1)+[Qt(t+ 2) Q
t(t+ 2)] BC B
t(t+ 2)
+1
1+iIOE R
tQt(t+ 1)DEX
t+ [1 Qt(t+ 1)] Mt.(5)
5
The first and second terms on the right-hand side represent the change in the consolidated
government surplus caused by the central bank’s purchase of bonds. When the central bank
buys the bonds at a higher price than the market price (Qt(t+ 1) or Qt(t+ 2)), interest
payment of bonds for the consolidated government increases owing to the intervention by the
central bank. In the real economy, the bond price Qt(t+j) can be interpreted as the market
price in the primary market or secondary market. When a crisis occurs, the central bank
tends to buy bonds by adding some premium to the market price. Therefore, in reality, QE
might result in slightly increased interest payment costs for the consolidated government.
However, if the crisis causes the bond price to decrease below some fundamentals, the bond
price might increase when the market is normalized. If so, the central bank might earn a
positive profit from the market intervention.
The third and fourth terms on the right-hand side are seigniorage created by high-powered
money. As this third term shows, the IOER offsets the benefits arising from seigniorage. If
the IOER rate equals the yield on a short-term bond, the profit from issuing excess reserves
is zero.
When the IOER rate equals the market interest rate it, seigniorage generated by issuing
excess reserves disappears. Therefore, the impact of the IOER on prices seems to be neutral.
However, the IOER changes not only the nominal surplus of the central bank but also the
total amount of excess reserves, because the excess reserves evolve as
DEX
t= (1 + iI OE R
t)DEX
t1+Q
t(t+ 1)BCB
t(+1) + Q
t(t+ 2)BCB
t(t+ 2) BCB
t1(t)Qt(t+ 1)BCB
t1(t+ 1).(6)
The first term in euqation (6) represents an increment of excess reserves by paying IOER
in the previous period. The second and third terms indicate that excess reserves increase
as a consequence of buying bonds. The fourth and fifth terms indicate that excess reserves
decrease owing to the redemption of bonds held by the central bank.
The crucial point that equation (6) shows is that there is no guarantee that the reserve
will be zero when QE ends. When the central bank initiates raising the IOER after QE, the
consolidated government debt increases through the increase in excess reserves.
6
3.3 Consolidated Government
Summing up (1) and (4) for all periods and using the definition of the central bank’s nominal
surplus (5), I obtain
BF
t1(t) + DEX
t1+Mt1BCB
t1(t)
Pt
+βBF
t1(t+ 1) BCB
t1(t+ 1)
Pt+1
=
i=0
βisF
t+i+sCB
t+i,(7)
with βρ
1+gyif the TVC holds:
lim
J→∞
βJβBF
t+J(t+J+ 1) + DEX
t+J+Mt+J
Pt+J+1
+β2BF
t+J(t+J+ 2)
Pt+J+2 = 0.(8)
If the fiscal authority follows the “active” fiscal policy, not constrained by the balance
equation (1), the equilibrium price level must satisfy government debt-value equation (7).
Equation (7) implies that the central bank’s debt (DEX
tand Mt) plays an important role in
determining price level. When the debt of the central bank is massive, and the shock occurs
on the present discounted value of real surplus, the price level must respond more strongly
to satisfy equation (7). Furthermore, the IOER affects the surplus on the right-hand side of
equation (7). Thus, a rise in the IOER simultaneously causes an increase in the consolidated
government debt (DEX
t) and a decrease in the real surplus (sCB
t).
Since equation (7) pins down price level rather than inflation rate, it is not immediately
clear how the IOER affects the inflation rate. In addition, the IOER increases the consoli-
dated government debt through an increase in excess reserves. To discuss this point in detail,
in the subsequent section, I examine the relationship between the IOER and the inflation
rate after QE.
4 How Should the Central Bank Set the IOER Rate?
In this section, I investigate the role of the IOER in the two Ricardian and non-Ricardian
regimes. The government debt-value equation derived in the previous section is equivalent to
7
the present-value budget constraint of households. Hence, it should be satisfied by adjusting
either policy variables or prices. A Ricardian regime means that the consolidated government
sets its policy variables to satisfy equation (7) given prices {Pt}. Meanwhile, when the
consolidated government policy variables are not calibrated to satisfy the government debt-
value equation for all value of prices, prices should satisfy equation (7) in equilibrium, which
implies that the economy is under a non-Ricardian regime.
One important question is how the IOER and inflation rate are related in the steady state
under each regime. To answer the question, I first examine how the consolidated government
policy variables influence the inflation rate on a balanced growth path.
I assume that the central bank does not hold government bonds after QE ends (BC B
t1(t) =
BCB
t1(t+ 1) = 0) and denote the growth rate of the fiscal authority’s real surplus (sF
t), the
government bonds (BF
t1(t) and BF
t1(t+ 1)), and money (Mt) as gs,b, and m, respectively.
From equation (6), the steady-state excess reserves evolve as
DEX
t= (1 + iI OE R )DEX
t1.(9)
Suppose that there is a finite inflation rate πon a balanced growth path; then, we obtain
the following lemma.
Lemma 1 (The TVC).The TVC holds only if the growth rate of the consolidated government
debt is lower than the market interest rate,
|1 + gd|<1 + i, (10)
where
1 + gdmax 1 + b, 1 + iIOER ,1 + m.(11)
Proof. See Appendix.
8
Condition (10) states that the growth rate of the consolidated government debt must be
lower than the market interest rate to satisfy the TVC. Thus, to obtain an equilibrium
solution, the central bank should not set the IOER rate higher than or equal to the market
interest rate.
Using these conditions, including the TVC, and equations (2), (5), and (7) in the steady
state, I obtain the steady-state condition of the government debt-value equation,
BF
t1(t)
Pt
+XDEX
t1
Pt
+YMt1
Pt
+βBF
t1(t+ 1)
(1 + π)Pt
=sF
t
1β(1 + gs),(12)
where
X1iiIOER
(1 + i)1β(1+iIOE R)
1+π,(13)
Y1i(1 + m)
(1 + i)1β1+m
1+π,(14)
with |β1+iIOE R
1+π|<1, |β1+m
1+π|<1 and |β(1 + gs)|<1. Although BF
t1(t), BF
t1(t+ 1), DEX
t1,
and Mt1are non-stationary, their ratios to Ptneed to be stationary. From equation (12), I
immediately obtain the following lemma.
Lemma 2 (BGP condition).Suppose that DEX
t1, Mt1, BF
t1(t), and BF
t1(t+ 1) >0,X >
0, and Y > 0; then, the balanced growth path for inflation rate πexists only when the
consolidated government debt grows at the rate of 1 + gdmax[1 + b, 1 + iIOE R,1 + m] =
(1 + π)(1 + gs)on a balanced growth path.
Proof. The real value of liabilities, the left-hand side of equation (12), grows at (1+gd)/(1+π)
and the real surplus sF
tgrows at 1 + gs. The real value of liabilities must be backed by the
present-discounted value of real surplus for all periods. Thus, the growth rate of the real
value of liabilities and real surplus must be equal.
The BGP condition indicates two things. Under a non-Ricardian regime, the growth
9
rate of the consolidated government liabilities (1 + gd) and that of real surplus (1 + gs)
are arbitrary, and their relative size determines the steady-state level of inflation (1 + π).
Under a Ricardian regime, the consolidated government should set an appropriate ratio
(1 + gd)/(1 + gs) given the steady-state inflation rate to satisfy the government debt-value
equation.
In the model, the IOER rate is a factor determining the growth of consolidated govern-
ment debt and could influence the steady-state inflation rate and equilibrium condition. In
the next subsection, I investigate the property of the IOER rate by using the BGP condition.
4.1 Ricardian Regime
Suppose that the central bank actively sets nominal interest rate (i). The fiscal authority
passively chooses the growth rate of the government bonds (b) and real surplus (gs) to satisfy
the BGP condition given households’ intertemporal substitution. In other words, band gs
satisfy (1 + b)/(1 + gs) = 1 + π.
Under a Ricardian regime, how is the IOER rate related to the economy? What is
immediately observable from the BGP condition is that the steady-state inflation rate always
exists when the central bank sets the IOER rate lower than the growth rate of government
bonds, that is, iI OE R b. In this case, government bonds dominate the debt structure of
the consolidated government, that is, 1 +gdequals 1 + b. The IOER rate does not influence
the inflation rate.
Then, what if the central bank sets the IOER rate higher than the growth rate of govern-
ment bonds (iIOER > b)? Because 1+gd= max[1 +iIOER ,1+ b] and 1+b= (1+g
s)(1+π),
1 + gdbecomes strictly larger than (1 + g
s)(1 + π), and the condition violates the BGP con-
dition. Thus, there is no finite inflation rate.
However, the condition does not necessarily mean an invalid equilibrium in real terms,
because the TVC is satisfied as long as 1 + iIOER <1 + ifrom inequality (10) and equation
(11). Thus, even if the IOER rate violates the BGP condition, the TVC is satisfied if it
10
is slightly lower than the market interest rate. In this case, nominal explosion occurs but
the government debt-value equation is satisfied and there is a valid equilibrium. Cochrane
(2011) carefully discusses this issue.
Here, it is important to review previous studies on how the central bank is supposed to
set the IOER rate. Cochrane (2014) and Del Negro and Sims (2015) assume a no-arbitrage
condition between the IOER rate and the market interest rate (i.e., iIOER =i). Actually,
introducing no arbitrage leads to a cumbersome problem. As Figure 1 shows, the BGP con-
dition and no-arbitrage condition are incompatible. Furthermore, the no-arbitrage condition
violates the TVC. This is because the no-arbitrage condition imposes restriction 1+gd1+i
on the growth rate of the consolidated government liabilities and therefore, the growth rate
of real value of the consolidated government liabilities is restricted as
1 + gd
1 + π1 + i
1 + π=1 + gy
ρ>Yt+1
Yt
.(15)
The condition implies that the real value of the consolidated government liabilities grows
faster than that of endowment. The real surplus cannot back the real value of its liabilities
given resource constraint. Real explosion occurs. To rule out real explosion, the central bank
should set the IOER rate lower than the market interest rate. However, previous studies do
not assume this. How do previous studies excludes real explosion? I discuss this point in
subsection 4.3.
4.2 Non-Ricardian Regime
Under a non-Ricardian regime, the fiscal authority determines the growth rate of government
bonds band real surplus g
sindependently of the BGP condition and the steady-state
inflation rate is endogenously determined by their ratio, that is, 1 + π= (1 + gd)/(1 + g
s).
Suppose that the fiscal authority pins down a steady-state inflation rate and the money
growth rate is lower than the growth rate of government bonds (b> m). In this case, the
11
Figure 1: The existence of πon a BGP and the TVC with the IOER rate
relationship between the steady-state inflation rate and the IOER rate is
1 + π=max[1 + b,1 + iIOER ]
1 + g
s
.(16)
12
Thus, if the IOER rate is higher than the growth rate of the government bonds, the propor-
tion of excess reserves in the consolidated government liabilities becomes dominant and the
IOER rate determines the steady-state inflation rate.
The TVC under a non-Ricardian Regime is
|max[1 + b,1 + iIOER ]|<1 + i=(1 + π)(1 + gy)
ρ.(17)
Notice that the left-hand side in the inequality depends on the IOER rate. The steady-state
inflation rate satisfying the BGP condition is always valid.
Then, what happens by introducing the no-arbitrage condition? Combining the no-
arbitrage condition, Euler equation, and BGP condition yields
β(1 + g
s) = 1.(18)
The condition implies that the present discounted value of real surplus explodes. As in the
case of a Ricardian regime, to satisfy the no-arbitrage condition, a real surplus exceeding
resource constraints is necessary. There is no valid equilibrium under a non-Ricardian regime
with the no-arbitrage condition.
4.3 Discussion of the No-arbitrage Condition
I show that no equilibrium exists under the no-arbitrage condition between iand iIOER
irrespective of the existence of a Ricardian or non-Ricardian regime, because the no-arbitrage
condition and the TVC are incompatible. This result is not consistent with previous studies,
Cochrane (2014) and Del Negro and Sims (2015), which introduce the no-arbitrage condition
without violating the TVC. How do they make their models compatible with the no-arbitrage
condition and the TVC? Actually, their models do not explicitly consider the law of motion
with respect to excess reserves, and thus, assume that excess reserves do not depend on the
13
IOER rate. Therefore, the no-arbitrage condition and the TVC can be compatible.5One
way to exclude real explosion is to specify an alternative law of motion of excess reserves
that does not evolve with the IOER rate.
In reality, the problem of the no-arbitrage condition may not matter much. The no-
arbitrage condition between the IOER rate and market interest rate means that households
have direct access to excess reserves. In fact, households cannot hold excess reserves and
then, we do not need to assume the no-arbitrage condition.
Figure 2 shows the IORE rate set by the Federal Reserves and the yield of 1-year treasury
bill. Indeed, we confirm a deviation of the IOER rate and market interest rate. Furthermore,
the IOER rate is lower than the market interest rate in recent years. This observation is
consistent with the implication of Ennis (2018), which suggests there is a spread between
the IOER rate and market interest rate.
Two important implications emerge about the IOER policy. First, the central bank
should not accept the no-arbitrage condition. If it did, there would be no equilibrium in
either a Ricardian or non-Ricardian regime. Second, in a Ricardian regime, to satisfy the
TVC and attain a finite inflation rate, the central bank should set the IOER rate lower than
the market interest rate by the subjective discount factor ρ. In the absence of the appropriate
setting of the IOER rate by the central bank, the fiscal authority cannot stabilize inflation
rate by itself. Appropriate cooperation between the fiscal authority and the central bank is
necessary.
5 Conclusion
I show that there is no balanced growth path for inflation rate when excess reserves pay the
market interest rate. To attain a valid equilibrium, the law of motion of excess reserves should
5Even in my model, if X in equation (12) is zero, the IOER rate becomes -100% and there are no excess
reserves in the steady state. Then, the law of motion of the consolidated government liabilities and the IOER
rate become irrelevant and therefore, the no-arbitrage condition and the TVC can be compatible. Since this
assumption is somewhat extreme, I exclude it in the analysis.
14
Figure 2: The IOER rate and yield on 1-year treasury bill
not be assumed, as it depends on the IOER rate and no-arbitrage condition simultaneously.
In this study, I mainly focus on the relationship between the IOER and the inflation rate in
the steady state. The government debt-valuation equation with QE and the IOER I derive
is a useful framework for analyzing how much actual QE and the IOER influence prices
15
through the fiscal channel. To quantify this, a model to project the central bank’s balance
sheet and surplus is needed. Future work should examine how much actual QE and the
IOER change the inflation rate.
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16
Appendix:Proof of Lemma 1.
Proof. On a balanced growth path, the TVC, equation (8), can be expressed as
lim
J→∞ β
1 + πJβ(1 + b)JBF
t(t+1)+(1+iIOE R)JDE X
t+ (1 + m)JMt
Pt+1
+β2(1 + b)JBF
t(t+ 2)
Pt+2 = 0.(19)
Thus, the TVC holds when all the following conditions are satisfied:
lim
J→∞ β(1 + b)
1 + πJ
= 0,(20)
lim
J→∞ β(1 + iIOER )
1 + πJ
= 0,(21)
lim
J→∞ β(1 + m)
1 + πJ
= 0.(22)
To satisfy these equations, the absolute values in parentheses must be less than 1. Using the
Euler equation (1 + i=(1+π)(1+gy)
ρ=1+π
β), the conditions become
|1 + b|<1 + i, (23)
|1 + iIOER |<1 + i, (24)
|1 + m|<1 + i. (25)
I denote the growth rate of the consolidated government debt as gd. Its growth rate is
1 + gdBF
t(t+ 1) + BF
t(t+ 2) + DEX
t+Mt
BF
t1(t) + BF
t1(t+ 1) + DEX
t1+Mt1
= max 1 + b, 1 + iIOER ,1 + m.(26)
Using equation (27), the conditions (23), (24), and (25) can be summarized as
|1 + gd|<1 + i. (27)
17
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