Discussion Paper No.397
A Dynamic Optimization Approach to MMT-type CoordinatedFiscal and Monetary Stabilization Policy

Ken Matsumoto
Chuo University

January 2024

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, JapanA Dynamic Optimization Approach to MMT-type CoordinatedFiscal and Monetary Stabilization Policy
Ken Matsumoto
Chuo University
November 2023

Abstract

In this paper, we aim to construct a dynamic Keynesian model incorporating the concepts ofModern Monetary Theory (MMT), a fiscal-dominant economic theory, and to use dynamic
optimization to derive an optimal fiscal and monetary policy path. MMT is characterized by
the dominance of fiscal policy over monetary policy and consider macroeconomic
environment (inflation) being the limit of fiscal expansion, rather than a government debt
ceiling, or a “sound” budget balance. We incorporate these MMT elements into the model,
referring to previous studies. Ultimately, we confirm stability/instability at the equilibrium
point and consider the validity of application to the fiscal and monetary policies in the realworld. The conclusion suggests 1.) forward-looking environment where economic agents trust
their central bank policy 2.) inflation target is sufficiently high, and 3.) nominal interest is
sufficiently low. Under these economic conditions, the dynamic system converges to an
equilibrium point, while confirming the point is “stable” and “determinate”. Stability implies
that the economy settles at the equilibrium point without experiencing overheating or
stagnation, and money supply and inflation expectations do not diverge upward or downward.
This argument serves as a counterpoint to frequent criticisms of MMT, asserting that fiscaldominant economic policies envisioned by MMT cannot stabilize the economy.

Keywords: MMT, Dynamic (in)stability, Dynamic Keynesian Model, Dynamic Optimization
JEL Classification Number: C61, E12, E31, E52, E62, E631

I.

Introduction

Since the Global Financial Crisis, advanced economies had persisted in prolonged
accommodative monetary policies, yet they struggled with continued low growth and unmet
inflation targets. In the midst of these challenges, what gained prominence around 2018 wasModern Monetary Theory (MMT), which places a primary focus on fiscal policy. MMT traces
its roots to Keynesian theory, but it has developed unique propositions regarding monetary
perspectives and policy discussions. While MMT has faced criticism from mainstream
economists and others 1 , there appears to be limited agreement with its arguments, andsupport for its assertions is relatively scarce2.
First of all, MMT, as the name suggests, begins with the question of what money is and how
it is supplied within the private sector and how it collects back afterwards as tax payments. To
explain briefly, according to Wray (1998), Wray (2015) and Kelton (2020), what sets MMT
apart from conventional mainstream economics is the fundamental belief that the source ofmoney circulation lies in government currency issuance through fiscal policy. This means thattax revenue is not a source of funding for the government budget in a nation that issues its
own currency. While this implies that the government can execute fiscal policy freely, there
are constraints in the form of limits on labor and resources, or supply constraints. If demandon goods exceeds supply constraints, in general, the economy enters an inflationary
environment. MMT proponents acknowledge the need to set limits to prevent excessive
inflation. Specifically, they propose Job Guarantee Program (JGP), which aims to continue
government employment until full employment is achieved during times of insufficientdemand. Conversely, if demand becomes excessive, the government controls private-sector
employment needs and inflation rates by setting wages lower than those in the private sector.
These are purely fiscal policy methodologies, and monetary policy plays no role here. The
basic premise of MMT is that fiscal policy is the driver while monetary policy is passive. In
the real world, monetary policy often seems dominant, which is the opposite of MMT's
perspective.
Regarding the feasibility of the JGP, this paper does not delve into micro-level discussions.
However, critics from both the public and mainstream economics often raise concerns about
MMT, primarily related to fears of excessive inflation or the diversion of public debt due toan unlimited expansion of government deficits, even before assessing the viability of JGP. For
instance, Globerman (2020) and Mankiw (2019) claim that controlling inflation is1

For instance, Krugman (2019) and Summers (2019).

2

For instance, Mackintosh (2021).

2

challenging under the MMT regime.
As a response to these criticisms, investment-savings (IS) balance is considered useful, which
is another fundamental aspect of MMT. This is because when considering the IS balance, as
the government deficit increases, private savings such as households and businesses, should
also increase by an equivalent amount (assuming we ignore foreign factors). It is unlikely thatthe private sector would hoard vast savings without engaging in consumption or investment.
Abba Lerner, a key figure in MMT, also discussed this in Lerner (1943), stating that "as thenational debt increases, it acts as a self-equilibrating force, gradually diminishing the further
need for its growth and finally reaching an equilibrium level where its tendency to grow comesto a complete end. The greater the national debt, the greater the quantity of private wealth."
He also pointed out that "an increase in private spending makes it less necessary for thegovernment to undertake deficit financing to maintain total spending at a level that ensures
full employment." Simply put, from these observations, whether or not we adopt the idea of
MMT, even if fiscal deficits expand to address insufficient demand, there are limits to howmuch they can expand, and demand cannot keep increasing indefinitely, nor can inflation
continue rising.
This paper, using mathematical equations, demonstrates that even in a macroeconomic
model incorporating MMT principles, it does not lead to the divergence of inflation or fiscal
deficits that mainstream critics often fear, but instead, reaches a stable equilibrium. Past
studies with similar objectives that utilize MMT, such as Asada (2020) and Matsumoto(2023c), have also attempted to achieve similar outcomes. In addition to these past findings,
this paper seeks to confirm whether the same stability at equilibrium can be achieved using
dynamic optimization methods, which enables to draw an optimal fiscal path of theconsolidated government.
This paper is particularly inspired by Matsumoto (2023c), which examine thestability/instability of a dynamic Keynesian model with a focus on MMT. This paper hascommon feature in a sense that dynamic equations for money supply, government
expenditure, and the expected inflation rate are formulated. Here, we attempt to move a little
step forward by incorporating a dynamic optimization analysis to investigate whether similar
conclusions can be obtained.
As for other MMT-related papers, examples include Tanaka (2021), in which the author
analyzes models incorporating functional fiscal policies of MMT while utilizing
microeconomic and neoclassical frameworks to examine consumer utility maximization withutility functions and budget constraints and profit maximization for firms under monopolisticcompetition. The idea of functional fiscal policies, or “functional finance”, the term firstdescribed by Abba Lerner, is considered as the root of MMT. Lerner (1943) explains it “The3

principle of judging fiscal measures by the way they work or function in the economy”, “notto any established traditional doctrine about what is sound or unsound”. In contrast to Tanaka(2021), Asada (2020) investigates stability and instability of equilibrium points based on
coordinated financial and fiscal policies by using dynamic Keynesian model.
Also, Asada, Demetrian, Zimka and Zimková (2023) use a four dynamic equation model
consisting of fiscal policy, inflation expectations, money growth rate, and employment rate,
in a dynamic Keynesian model incorporating MMT ideas, with active fiscal policy and passive
monetary policy in financing the government debt. The study concluded that if the policy
authorities pursue proactive fiscal and monetary policies and can garner credibility in theirinflation target, it is possible to stabilize the economy.
The paper is structured as follows: Section II defines the variables and presents the model'sconstruction of the goods and money markets. In Section III, the induced system of dynamic
equations is presented. In Section IV, we present dynamic optimization method to analyze
the dynamic equations. Section V examines the stability and instability of equilibrium points.
Section VI considers the economic and policy implications of the previous analysis. Finally,
Section VII gives concluding remarks.
II.

Formulation of the IS-LM Part of the ModelIn this chapter, we will explain the framework and nature of this paper. The symbols used
are as follows, and the dot above a symbol denotes a derivative with respect to time.
𝑌 = real national income (real output). 𝐾 = real capital stock. 𝑦 =𝑌𝐾= output-capitalratio. 𝐶 = consumption expenditure. 𝑐 = marginal propensity to consume. 𝐼 = real private
investment expenditure. 𝑖 =
𝑔=

𝐺
𝐾

𝐼

= rate of investment. 𝐺 = real government expenditure.

𝐾= government expenditure-capital ratio. 𝜌 = nominal rate of interest of public bonds.

𝑝 = price level. 𝑇 = real income tax. 𝜏 =𝑇𝐾= real income tax-capital ratio. 𝐵 = nominal𝑀

stock of public debt (public bond). 𝑀 = nominal money supply. 𝑚 = 𝑃𝐾 = money-capitalratio.
Here, we start with the equilibrium condition of the goods market.
(1)

𝑌 = 𝐶+𝐼+𝐺
where,𝐵

𝐶 = 𝑐 [𝑌 + 𝜌 (𝑃 ) − 𝑇] ; 0 < 𝑐 < 1(2)

By substituting Eq(2) to Eq(1), we have Eq(3).
1𝐵

𝑌 = 1−𝑐 {𝑐 [𝜌 (𝑃 ) − 𝑇] + 𝐼 + 𝐺}
4(3)

We divide both sides of Eq(3) by 𝐾 to derive the equilibrium condition of the goods
𝐵market, which we call as an IS equation. Here, 𝑏 = 𝑃𝐾 = public debt-capital ratio. Rate of𝑑𝑖investment is a function of real expected rate of interest, where 𝑖 = 𝑖(𝜌 − 𝜋 𝑒 ), and 𝑑(𝜌−𝜋𝑒) <0.
1

𝑦 = 1−𝑐 {𝑐[𝜌(𝑦, 𝑚)𝑏 − 𝜏] + 𝑖(𝜌(𝑦, 𝑚) − 𝜋 𝑒 ) + 𝑔}

(4a)

We totally differentiate above equation with respect to 𝑦 and 𝑔 (We assume that 𝜏 is
fixed for simplicity following Asada (2020) and Asada, Chiarella, Flaschel and Franke(2003)).
𝑑𝑦 =
{1 −

𝑐
1
(𝜌𝑦 𝑑𝑦 + 𝑖𝜌 𝜌𝑦 ) +
𝑑
1−𝑐
1−𝑐 𝑔

𝑐
1
(1 + 𝑖𝜌 )𝜌𝑦 } 𝑑𝑦 =
𝑑𝑔
1−𝑐
1−𝑐

𝜕𝑦
𝑐
=(
)(
𝜕𝑔
1−𝑐 1−

1
)
𝑐
(1
+
𝑖
)𝜌
𝜌
𝑦
1−𝑐

Solving Eq(4a) with respect to 𝑦, we have Eq(4b).
𝑦 = 𝑦(𝑔, 𝑚, 𝑏, 𝜋 𝑒 )(4b)Nominal interest rate above is defined by LM equation.
𝑀= 𝑙(𝜌)𝑌 ; 𝑙𝜌 < 0𝑝𝑚 = 𝑙(𝜌)𝑦𝜌 = 𝜌(𝑦, 𝑚)
𝜌𝑦 =

𝜕𝜌
𝑙
=−
>0
𝜕𝑦
𝑙𝜌 𝑦

𝜌𝑚 =

𝜕𝜌
1
=
<0
𝜕𝑚 𝑙𝜌 𝑦(5)Here, we assume the economy is facing a “liquidity trap”, in which the economy is stagnant
and interest rate is too low so that the increase in money supply is ineffective to nominalinterest rate and output.
If |𝑙𝜌 | ≈ ∞ ,5𝜌𝑚 ≈ 0, 𝑦𝑚 ≈ 0 and 𝜌𝑦 ≈ 0.
(6)
We determine the characteristics of each variable through total differentiation of Eq (4).
Partial derivatives are given below.
𝜕𝑦

1

(7)

≈0

(8)

𝑦𝑔 = 𝜕𝑔 ≈ 1−𝑐 > 0
𝑦𝑚 =
𝑦𝜋𝑒 =

III.

𝜕𝑦
𝜕𝜋𝑒

=−

𝜕𝑦
𝜕𝑚
1

𝑖

1−𝑐 𝜌−𝜋

𝑒

(9)

>0

Dynamic Equations

In this section, we derive the dynamic equations for each variable. Below are some of thedefinitional equations.
𝐾̇ = 𝐼
𝐾̇
𝐾

(10)

𝐼

= 𝐾 = 𝑖(𝜌 − 𝜋 𝑒 )
𝑃̇
𝑃

(12)

=𝜋

𝜋 = 𝜀(𝑦 − 𝑦̅) + 𝜋 𝑒 ; 𝜀 > 0
𝑀̇
𝑀

(11)

(13)
(14)

=𝜇

Eq(10) is the rate of capital accumulation that is equivalent to investment expenditure.
Eq(11) is the investment function of firms, which is based on the standard Keynesian
theory3. The equation shows that investment is the decreasing function of the expected real
rate of interest. Eq(12) represents the growth rate of price equals inflation rate. Eq(13) isthe conventional linear “expectations-augmented Phillips curve”, which 𝜀 is the reaction
parameter from the output gap. 𝑦̅ is the natural output-capital ratio, and 𝜋 𝑒 is inflation
expectation variable. Eq(14) shows the growth rate of money.
Using Eqs(11) , (12) ,(14) and money-capital ratio 𝑚, dynamic law of the motion of the3

See Keynes (1936), Asada and Ouchi (2009).

6money-capital ratio can be described as below.
𝑚̇
𝑚

𝑀̇

𝑃̇

𝐾̇

= 𝑀 − 𝑃 − 𝐾 = 𝜇 − 𝜋 − 𝑖(𝜌 − 𝜋 𝑒 )𝑚̇ = 𝑚[𝜇 − 𝜋 − 𝑖(𝜌 − 𝜋 𝑒 )](15)

We can substitute each variable with equations we previously derived.
𝑚̇ = [𝜇 + 𝜀(𝑦̅ − 𝑦(𝑔, 𝑚, 𝑏, 𝜋 𝑒 )) − 𝜋 𝑒 − 𝑖(𝜌(𝑦(𝑔, 𝑚, 𝑏, 𝜋 𝑒 ), 𝑚) − 𝜋 𝑒 )]𝑚(16)
The dynamic equation of inflation expectation is described below. This equation represents
the dynamics of inflation expectations by capturing the distinction between forward-looking
and backward-looking expectation formations using θ. If θ is close to 1, it signifies aforward-looking expectation formation with a strong influence from the inflation target,
whereas if it is closer to zero, it indicates a backward-looking expectation formationinfluenced by the actual inflation rate. Therefore, θ can be referred to as the "credibility
parameter" with respect to the policy authority's inflation target. This formulation originatesfrom Asada, Chiarella, Flaschel and Franke (2003) and Asada (2020). 𝛾 can be interpreted
as a reaction parameter to inflation expectation from both of the gaps of inflation target andoutput.

𝜋̇ 𝑒 = 𝛾[𝜃(𝜋̅ − 𝜋 𝑒 ) + (1 − 𝜃)(𝜋 − 𝜋 𝑒 )] ; (𝛾 > 0, 0 ≤ 𝜃 ≤ 1)𝜋̇ 𝑒 = 𝛾{𝜃(𝜋̅ − 𝜋 𝑒 ) + (1 − 𝜃)[𝜀[𝑦(𝑔, 𝑚, 𝑏, 𝜋 𝑒 ) − 𝑦̅]]}(17)

From these dynamic equations, we can rewrite as below.
𝑚̇ = 𝐹1 (𝑔, 𝑚, 𝑏, 𝜋 𝑒 )𝜋̇ 𝑒 = 𝐹2 (𝑔, 𝑚, 𝑏, 𝜋 𝑒 )(18)
Next, we formulate the equation that also symbolizes the characteristic of MMT. Eq(19)below is the “budget constraint” of the consolidated government, including the central bank.
implying that the government deficit must be financed through the issuance of new highpowered money (monetary base) or government bonds. We acknowledge that the MMT
proponents do not have the idea of “budget constraint” of government since they can always
print their own money to finance the debt, but we still think this equation meaningfully
describe their basic idea. The formulation is based on Asada (2020) and Mitchell, Wray, and7

Watts (2019).
𝐻̇ + 𝐵̇ = 𝑃𝐺 + 𝜌𝐵 − 𝑃𝑇

(19)

Where, 𝐻 = high-powered money (monetary base). 𝐵 = nominal stock of public debt
(public bond). This simply tells that the government expenditure including the interest
payment of government bond is financed with the issuance of government bonds, highpowered money, or tax.
Money supply is described below, as the product of high-powered money and 𝜈, which is
the money multiplier (constant). These are differentiated with respect to time.
(20)

𝑀 = 𝜈𝐻 ; 𝜈 > 1
𝑀̇ = 𝜈𝐻̇
1
𝐻̇ = 𝑀̇
𝜈1(21)= 𝜈 𝜇𝑀
Below is the growth rate of nominal public debt stock.

𝐵̇𝜇𝐵 = 𝐵 , 𝐵̇ = 𝜇𝐵 𝐵

(22)Thus, it can be rewritten as below.
1𝜈𝜇𝑀 + 𝜇𝐵 𝐵 = 𝑃𝐺 + 𝜌𝐵 − 𝑃𝑇(23)

We divide this by price and capital to transform into a real term with respect to capital.

1
𝜈

𝑀

1
𝜈

We assume that 𝜏 =

𝑇
𝐾

𝐵

𝐺

𝐵

𝜇 (𝑃𝐾) + 𝜇𝐵 (𝑃𝐾) = 𝐾 + 𝜌 (𝑃𝐾) − 𝜏
𝜇𝑚 + 𝜇𝐵 𝑏 = 𝑔 + 𝜌𝑏 − 𝜏(24)is constant for simplicity following Asada (2020) and Asada,

Chiarella, Flaschel and Franke (2003). We also assume that the government controls theissuance of public debt, which means that they could keep its level constant, as described inEq(25). Eq(26) is the equation of growth rate of nominal public debt stock.

8

𝐵

𝑏 = 𝑃𝐾 = 𝑏̅
𝑏̇

𝐵̇

𝑃̇

𝐾̇

𝑏

𝐵

𝑃

𝐾

0= = − −(25)= 𝜇𝐵 − 𝜋 − 𝑖(𝜌 − 𝜋 𝑒 ) = 𝜇𝐵 − 𝜀(𝑦 − 𝑦̅) − 𝜋 𝑒 − 𝑖(𝜌 − 𝜋 𝑒 )𝜇𝐵 = 𝜀(𝑦 − 𝑦̅) + 𝜋 𝑒 + 𝑖(𝜌 − 𝜋 𝑒 )= 𝜀(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒 + 𝑖(𝜌(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝜋 𝑒 )(26)
We can insert Eq(26) into Eq(24).

1𝜇𝑚 + [𝜀(𝑦 − 𝑦̅) + 𝜋 𝑒 + 𝑖(𝜌 − 𝜋 𝑒 )]𝑏̅ = 𝑔 + 𝜌𝑏̅ − 𝜏𝜈𝜇𝑚 = 𝜈 {𝑔 + 𝜌(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ), 𝑚)𝑏̅ − 𝜏− [[𝜀(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒 + 𝑖(𝜌(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 )) − 𝜋 𝑒 )]𝑏]}(27)
This is inserted into Eq(16).
We now have two dynamic equations, which are real money-capital ratio and inflation
expectation, respectively from Eq(16) and (17),𝑚̇ = 𝜈 {𝑔 + 𝜌[𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ), 𝑚]𝑏̅ − 𝜏− [[𝜀(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒 + 𝑖(𝜌(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 )) − 𝜋 𝑒 )]𝑏]}− 𝑚{𝜀(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒 + 𝑖(𝜌(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 )) − 𝜋 𝑒 )}= 𝑓1 (𝑔(𝑡), 𝑚(𝑡), 𝜋 𝑒 (𝑡))𝜋̇ 𝑒 = 𝛾 {𝜃(𝜋̅ − 𝜋 𝑒 ) + (1 − 𝜃) [𝜀[𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅]]}=𝑓2 (𝑔(𝑡), 𝑚(𝑡), 𝜋 𝑒 (𝑡))
(28)
IV.

Dynamic Optimization

From here, we use the dynamic equations formalized earlier to solve for the maximization(minimization) of the objective functional, or a social loss functional, through dynamic
optimization. Our objective functional is aimed at minimizing the inflation gap and output

9

gap4. Within the dynamic system of given money and inflation expectations, government
expenditure serves as the control variable.
∞max ∫ −{𝜉(𝜋 − 𝜋̅)2 + (1 − 𝜉)(𝑦 − 𝑦̅)2 }𝑒 −𝑟𝑡 𝑑𝑡
𝑔(𝑡)

0

2
2
∞
= max ∫0 − {𝜉[𝜀(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒 − 𝜋̅] − (1 − 𝜉)(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) } 𝑒 −𝑟𝑡 𝑑𝑡 ;
𝑔(𝑡)

0<𝜉<1
𝑠. 𝑡.
𝑚̇ = 𝜈 {𝑔 + 𝜌[𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ), 𝑚]𝑏̅ − 𝜏− [[𝜀(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒 + 𝑖(𝜌(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 )) − 𝜋 𝑒 )]𝑏]}− 𝑚 {𝜀(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒 + 𝑖 (𝜌 (𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 )) − 𝜋 𝑒 )}= 𝑓1 (𝑔(𝑡), 𝑚(𝑡), 𝜋 𝑒 (𝑡))𝜋̇ 𝑒 = 𝛾 {𝜃(𝜋̅ − 𝜋 𝑒 ) + (1 − 𝜃) [𝜀[𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅]]} = 𝑓2 (𝑔(𝑡), 𝑚(𝑡), 𝜋 𝑒 (𝑡)) ,(29)
where 𝑟 is the discount rate that is treated as a positive parameter.
Current value Hamiltonian and Pontryagin’s maximum-principle conditions5 are described
below.
2
2
𝐻 = −𝜉[𝜀(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒 − 𝜋̅] − (1 − 𝜉)(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅)+ 𝜙1 (𝑡)𝑓1 (𝑔(𝑡), 𝑚(𝑡), 𝜋 𝑒 (𝑡)) + 𝜙2 (𝑡)𝑓2 (𝑔(𝑡), 𝑚(𝑡), 𝜋 𝑒 (𝑡))Max 𝐻(𝑔, 𝑚, 𝜋 𝑒 ) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ∈ [0, ∞]𝑔(𝑡)𝑚̇(𝑡) =𝜋̇ 𝑒 =

𝜕𝐻(𝑡)= 𝑓1 (𝑔(𝑡), 𝑚(𝑡), 𝜋 𝑒 (𝑡))𝜕𝜙1 (𝑡)𝜕𝐻(𝑡)= 𝑓2 (𝑔(𝑡), 𝑚(𝑡), 𝜋 𝑒 (𝑡))𝜕𝜙2 (𝑡)𝜙1̇ (𝑡) = −𝜕𝐻(𝑡)+ 𝑟𝜙1 (𝑡)𝜕𝑚(𝑡)[𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑚̇][𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝜋̇ 𝑒 ][𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝜙1 ]4

Similar approach was taken by Matsumoto (2023a) and Matsumoto (2023b), and Taylor (1989).

5

Explanation on Pontryagin’s maximum principle conditions and dynamic optimization are available from

Chiang (1992), Chiang and Wainwright (2005).

10𝜙2̇ (𝑡) = −𝜕𝐻(𝑡)+ 𝑟𝜙2 (𝑡)𝜕𝜋 𝑒 (𝑡)[𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝜙2 ]lim 𝜙1 𝑒 −𝑟𝑡 = 0 , lim 𝜙2 𝑒 −𝑟𝑡 = 0

𝑡→∞

𝑡→∞

[𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑎𝑙𝑖𝑡𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠](30)

A first order condition is required to show that the control of 𝑔 will be an interior
solution.
𝜕𝐻= −2𝜉𝜀𝑦𝑔 [𝜀(𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒 − 𝜋̅] − 2(1 − 𝜉)𝑦𝑔 (𝑦(𝑔, 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅)𝜕𝑔(𝑡)+ 𝜙1 [𝑣{1 − 𝑏̅[𝜌𝑦 𝑦𝑔 + 𝜀𝑦𝑔 + 𝑖𝜌−𝜋𝑒 𝜌𝑦 𝑦𝑔 ]} − 𝑚{𝜀𝑦𝑔 + 𝑖𝜌−𝜋𝑒 𝜌𝑦 𝑦𝑔 }] + 𝜙2 𝛾(1− 𝜃)𝜀𝑦𝑔 = 0(31)Here, we figure the characteristics of the above equation of the first order condition, which
we derived in Eq(31).
𝐺(𝑔, 𝑚, 𝜋 𝑒 , 𝜙1 , 𝜙2 ) = 0(32)

We now solve the total derivation, as shown below. 𝐺𝑔 is equivalent to Eq (31).
𝐺𝑚 = −𝜙1 𝜀𝑦𝑔
𝐺𝜋𝑒 = −2𝜉𝜀𝑦𝑔 (𝜀𝑦𝜋𝑒 + 1) − 2(1 − 𝜉)𝑦𝑔 𝑦𝜋𝑒 < 0
𝐺𝜙1 = 𝑣(1 − 𝜀𝑦𝑔 𝑏̅) − 𝑚𝜀𝑦𝑔
𝐺𝜙2 = 𝛾(1 − 𝜃)𝜀𝑦𝑔 > 0(33)
By combining above equations, we can derive the following, under equilibrium.

𝑔𝑚 =

𝜕𝑔
𝐺𝑚
=−
𝜕𝑚
𝐺𝑔

𝜕𝑔
𝐺𝜋𝑒
=−
<0
𝑒
𝜕𝜋
𝐺𝑔
𝐺𝜙
𝜕𝑔
𝑔𝜙1 =
=− 1
𝜕 𝜙1
𝐺𝑔
𝐺𝜙
𝜕𝑔
𝑔𝜙2 =
=− 2>0
𝜕𝜙2
𝐺𝑔
𝑔𝜋𝑒 =

𝑔 = 𝑔(𝑚(𝑡), 𝜋𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡))
11

(34)
Further differentiation of Eq(31) with the result of negative shows that the control
variable 𝑔 does maximize the Hamiltonian system.
𝜕2𝐻= −2𝑦𝑔 2 {𝜉𝜀 2 − (1 − 𝜉)} < 0
𝜕𝑔2
The condition to fulfill above equation is shown below.
𝜀2 >

1−𝜉
;0 ≤ 𝜉 ≤ 1𝜉𝜉>1
1 + 𝜀2(35)
Solving the maximum-principle conditions give us equations below.
𝑚̇(𝑡) = 𝜈 {𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡)) + 𝜌[𝑦(𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡)), 𝑚, 𝑏̅, 𝜋 𝑒 ), 𝑚]𝑏̅ − 𝜏− [[𝜀(𝑦(𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡)), 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒+ 𝑖(𝜌(𝑦(𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡)), 𝑚, 𝑏̅, 𝜋 𝑒 )) − 𝜋 𝑒 )]𝑏]}− 𝑚{[𝜀(𝑦(𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡)), 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒 ]+ 𝑖[𝜌(𝑦(𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡)), 𝑚, 𝑏̅, 𝜋 𝑒 ), 𝑚) − 𝜋 𝑒 ]}= 𝐹1 (𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡))𝜋̇ 𝑒 (𝑡) = 𝛾{𝜃(𝜋̅ − 𝜋 𝑒 ) + (1 − 𝜃)[𝜀[𝑦(𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡)), 𝑚, 𝑏, 𝜋 𝑒 ) −𝑦̅]]}=𝐹2 (𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡))𝜙1̇ (𝑡) = 2𝜉𝜀𝑦𝑚 [𝜀(𝑦(𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡), 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒 − 𝜋̅)]+ 2(1 − 𝜉)𝑦𝑚 [𝑦(𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡)), 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅]− 𝜙1 {𝑣{(𝜌𝑦 𝑦𝑚 + 𝜌𝑚 )𝑏̅ − [𝜀𝑦𝑚 + 𝑖𝜌−𝜋𝑒 (𝜌𝑦 𝑦𝑚 + 𝜌𝑚 )]𝑏̅}− 𝑚[𝜀𝑦𝑚 + 𝑖𝜌−𝜋𝑒 (𝜌𝑦 𝑦𝑚 + 𝜌𝑚 )]− 𝜀(𝑦(𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡), 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅) + 𝜋 𝑒− 𝑖[𝜌(𝑦(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡), 𝑚, 𝑏̅, 𝜋 𝑒 ), 𝑚) − 𝜋 𝑒 ]} − 𝜙2 {𝛾[(1 − 𝜃)𝜀𝑦𝑚 ]}+ 𝑟𝜙1 (𝑡) = 𝐹3 (𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡); 𝑟)12

𝜙2̇ (𝑡) = 2𝜉(𝜀𝑦𝜋𝑒 + 1)[𝜀(𝑦(𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡)), 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅)]+ 2(1 − 𝜉)𝑦𝜋𝑒 [𝑦(𝑔(𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡)), 𝑚, 𝑏̅, 𝜋 𝑒 ) − 𝑦̅]− 𝜙1 {𝑣{𝜌𝑦 𝑦𝜋𝑒 𝑏̅ − [𝜀𝑦𝜋𝑒 + 1 + 𝑖𝜌−𝜋𝑒 (𝜌𝑦 𝑦𝜋𝑒 − 1)]𝑏̅}− 𝑚{𝜀𝑦𝜋𝑒 + 1 − 𝑖𝜌−𝜋𝑒 (𝜌𝑦 𝑦𝜋𝑒 − 1)}} − 𝜙2 {𝛾[−𝜃 + (1 − 𝜃)𝜀𝑦𝜋𝑒 ]} + 𝑟𝜙2 (𝑡)= 𝐹4 (𝑚(𝑡), 𝜋 𝑒 (𝑡), 𝜙1 (𝑡), 𝜙2 (𝑡); 𝑟)(36)
In the state of long-run equilibrium, we expect the output and inflation to equal each of thepolicy target. The costate variables should also become zero6. Applying this assumption to
the dynamic equations, we have the following results.
𝑦 ∗ = 𝑦̅𝜋 𝑒 ∗ = 𝜋̅
𝜙1 ∗ = 0𝜙2 ∗ = 0(37)
If we put figures below on each dynamic equations, we can analyze the characteristics ofthe equilibrium.
𝑚̇ = 𝜋̇ 𝑒 = 𝜙1̇ = 𝜙2̇ = 0 .

6This interpretation is explained by Intriligator (1971) and Chiang (1992). According to Intriligator(1971) chapter 14 and Chiang (1992) chapter 8, we have[𝜙1 (𝑡)𝑒 −𝑟𝑡 ]𝑡=0 = 𝜙1 (0) =

𝜕𝑊 ∗
𝜕𝑚(0)

,

(F1)

𝜕𝑊 ∗
[𝜙2 (𝑡)𝑒 −𝑟𝑡 ]𝑡=0 = 𝜙2 (0) = 𝑒 ,𝜕𝜋 (0)(F2)

𝑊 = ∫0 −{𝜉(𝜋(𝑡) − 𝜋̅)2 + (1 − 𝜉)(𝑦(𝑡) − 𝑦̅)2 }𝑒 −𝑟𝑡 𝑑𝑡.

(F3)

where 𝑊 ∗ is the optimal value of∞In this case, we have𝜕𝑊 ∗∞= ∫0 −2 {𝜉(𝜋 ∗ (𝑡) − 𝜋̅)𝜕𝜋∗ (𝑡)+ (1 − 𝜉)(𝑦 ∗ (𝑡) − 𝑦̅)𝜕𝑦 ∗ (𝑡)} 𝑒 −𝑟𝑡 𝑑𝑡,(F4)∞= ∫0 −2 {𝜉(𝜋 ∗ (𝑡) − 𝜋̅) 𝑒 + (1 − 𝜉)(𝑦 ∗ (𝑡) − 𝑦̅) 𝑒 } 𝑒 −𝑟𝑡 𝑑𝑡,
𝜕𝜋𝑒 (0)𝜕𝜋 (0)𝜕𝜋 (0)

(F5)

𝜕𝑚(0)
𝜕𝑊 ∗

𝜕𝑚(0)
𝜕𝜋∗ (𝑡)𝜕𝑚(0)𝜕𝑦 ∗ (𝑡)Where 𝜋 ∗ (t) and 𝑚∗ (𝑡) are 𝜋(𝑡) and 𝑚(𝑡) at the optimal path. Consider that we are in the equilibrium
point such that 𝑚(𝑡) = 𝑚∗ , 𝜋(𝑡) = 𝜋 ∗ = 𝜋̅, 𝜋 𝑒 (𝑡) = 𝜋 𝑒 ∗ = 𝜋̅, 𝑦(𝑡) = 𝑦 ∗ = 𝑦̅, 𝜙1 (𝑡) = 𝜙 ∗ , 𝜙2 (𝑡) = 𝜙 ∗ forall 𝑡 ≧ 0. Substituting these conditions to equations (F1), (F2), (F4) and (F5), we obtain𝜙1 ∗ = 𝜙2 ∗ = 0.

13

(F6)

𝑚̇ = 𝑣{𝑔(𝑚∗ , 𝜋̅, 0, 0) − 𝜏 − 𝑏[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅ − 𝑖[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅]]} − 𝑚∗ {𝜋̅ − 𝑖[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅]}= 𝐹(𝑚∗ ) = 0
𝜋̇ 𝑒 = 0𝜙1̇ = 0𝜙2̇ = 0(38)
Below is the partial derivative of 𝑚̇ with respect to 𝑚.
𝐹 ′ (𝑚) = 𝑣{𝑔𝑚∗ + 𝑏̅[𝜌𝑚∗ + 𝑖𝜌−𝜋𝑒 𝜌𝑚∗ ]} + 𝑖𝜌−𝜋𝑒 𝜌𝑚∗ 𝑚∗ − 𝜋̅ + 𝑖[𝜌(𝑦̅, 𝑚∗) − 𝜋̅] > 0(39)This gives the characteristic of optimal level of money capital ratio, 𝑚∗ , described below.

( )∗Figure 1

V.

Local Stability/Instability of the Long-run Equilibrium Point

In this section, we identify the state of local stability/instability of the long-run equilibrium
point of our dynamic system. Jacobian matrix of the four-dimensional dynamic system at theequilibrium point is described below. Here, the liquidity trap is assumed again. In reality,
the response of interest rates to increases in money or output is not entirely zero, but for thesake of simplification, it is treated as 𝜌𝑚 ≈ 0, 𝑦𝑚 ≈ 0 and 𝜌𝑦 ≈ 0.
14

𝐹13
𝐹14
𝐹11 𝐹12𝐹23𝐹24𝐹21 𝐹22
𝐽=[]
0 𝐹33 (𝑟) 000 𝐹42 𝐹43 𝐹44 (𝑟)(40)
In Appendix, the detailed expressions of the partial derivatives of Eq (40) are presented.
We can write the characteristic equation of the Jacobian matrix as follows.
Γ(𝜆) ≡ |𝜆𝐼 − 𝐽| = 𝜆4 + 𝑎1 𝜆3 + 𝑎2 𝜆2 + 𝑎3 𝜆 + 𝑎4 = 0 ,
where
𝑎1 = −𝑡𝑟𝑎𝑐𝑒𝐽4 = −𝐹11 (𝜀) − 𝐹22 − 𝐹33 − 𝐹44 ,𝑎2 = 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑠𝑒𝑐𝑜𝑛𝑑 𝑜𝑟𝑑𝑒𝑟 𝑚𝑖𝑛𝑜𝑟𝑠 𝑜𝑓 𝐽 ,𝑎3 = −(𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑡ℎ𝑖𝑟𝑑 𝑜𝑟𝑑𝑒𝑟 𝑚𝑖𝑛𝑜𝑟𝑠 𝑜𝑓 𝐽) ,𝑎4 = 𝑑𝑒𝑡𝐽 .
(41)
In the conventional dynamic system in which all initial values of the endogenous variables
are given, the equilibrium point of the dynamic system is considered to be locally stable if
and only if all of the roots of the characteristic equation have negative real parts, which isdefined as “Routh-Hurwitz conditions”, shown below. The system is considered as unstable
if any one of the roots has a positive real part7.
𝑎𝑗 > 0 (𝑗 = 1,2,3,4), 𝑎1 𝑎2 𝑎3 − 𝑎1 2 𝑎4 − 𝑎3 2 > 0 .
(42)
Using the results shown in the appendix, we can find that the above condition cannot be
attained because of the following inequality.
𝑎1 = −𝑡𝑟𝑎𝑐𝑒𝐽4 = −2𝑟 < 0 .
(43)This tells that at least one root has positive real part, and the dynamic system is unstable inthe conventional sense. However, in this 4-dimensional dynamic system derived from theconditions of dynamic optimization, it is not assumed that the initial values of all
endogenous variables are given. What is assumed as given in this system are the initialvalues of two 'state variables,' m and π, and it is anticipated that the initial values of the7

We follow steps taken by Asada (2024).

15

remaining two 'auxiliary variables' can be freely chosen by the consolidated government
including the central bank, which is the subject of dynamic optimization. The state variables
mentioned here are referred to as pre-determined variables, while the auxiliary variables arecalled not-pre-determined variables or jump variables. The consolidated government, being
the planner, can achieve dynamic optimization only by selecting initial values for theauxiliary variables that satisfy 'transversality conditions' with respect to the given statevariables and converge to equilibrium points. The dynamic system described here isconsidered locally stable and determinate only when the initial values determining such apath are uniquely determined.
Therefore, in the dynamic system described here, it is considered locally “stable” and“determinate” only when the characteristic equation has two roots with negative real parts
and two roots with positive real parts. According to Dockner and Feichtinger (1991), if theinequalities represented by 𝐸 ≡ 𝑎2 − 𝑟 2 < 0 and 𝑑𝑒𝑡𝐽 > 0 simultaneously hold in thisdynamic optimization system with two state variables, then the characteristic equation's tworoots have negative real parts, and the remaining two roots have positive real parts, leading
to the equilibrium point being a locally saddle point. The following verifies whether thisdefinition holds true8.
Proposition
If the "credibility parameter", θ is sufficiently close to 1, and the inflation target 𝜋̅satisfies the inequality 𝜋̅ > 𝑖[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅], the dynamic system is locally “stable” and“determinant”.
Proof
We use the solutions which are provided in Appendix to see if the conditions of Docknerand Feichtinger (1991) are met under the condition θ=1.

𝑎2 = 𝑎2 (𝑟) ≡ 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑠𝑒𝑐𝑜𝑛𝑑 𝑜𝑟𝑑𝑒𝑟 𝑚𝑖𝑛𝑜𝑟𝑠 𝑜𝑓 𝐽
𝐹
= | 11𝐹21

𝐹11𝐹12| + |𝐹
𝐹22
31

𝐹13
𝐹11
+
|
|
𝐹33(𝑟)
𝐹41

𝐹22
𝐹14
+
|
|
𝐹44(𝑟)
𝐹32

𝐹23
𝐹22
+
|
|
𝐹33(𝑟)𝐹42

𝐹33(𝑟)𝐹24+||
𝐹44(𝑟)𝐹43= (𝐹11 + 𝐹22 ){𝐹33 (𝑟) + 𝐹44 (𝑟)} + 𝐹33 (𝑟)𝐹44 (𝑟) + 𝐴= {−𝜋̅ + 𝑖[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅] − 𝛾}{𝜋̅ − 𝑖[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅] + 𝛾[𝜃 − (1 − 𝜃)𝜀𝑦𝜋𝑒 ] + 2𝑟}+ {[𝜋̅ − 𝑖[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅] + 𝑟][𝛾[𝜃 − (1 − 𝜃)𝜀𝑦𝜋𝑒 ] + 𝑟]} + 𝐴

8Follows the method by Asada (2024).

16

𝐹34
|
𝐹44(𝑟)

= 𝑟 2 − 𝑟𝜋̅ + 𝑟𝑖[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅] − 𝑟𝛾 + 𝐵
𝑬(𝒓) ≡ 𝑎2 (𝑟) − 𝑟 2 = 𝑟{−𝜋̅ + 𝑖[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅] − 𝛾} + 𝐵 ,where A and B are independent of 𝑟.
𝐸(𝑟) is negative if the condition below is met and 𝑟 is sufficiently large.
𝑖[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅] < 𝜋̅ + 𝛾 .
(44)
In terms of 𝑎4 = 𝑑𝑒𝑡𝐽,
𝐹11
𝑎4 = 𝑑𝑒𝑡𝐽 = 𝐹44 (𝑟) |𝐹21
𝐹31

𝐹12
𝐹22
𝐹32

𝐹12
𝐹
+ 𝐹41 | 22
𝐹32

𝐹13
𝐹11
𝐹23 | + 𝐹43 |𝐹21
𝐹31
𝐹33 (𝑟)

𝐹13
𝐹23
𝐹33(𝑟)𝐹= 𝐹44 (𝑟)𝐹33 (𝑟) | 11
𝐹21

𝐹12
𝐹22
𝐹32

𝐹11
𝐹14
𝐹24 | + 𝐹42 |𝐹21
𝐹34
𝐹31

𝐹13
𝐹14
𝐹23
𝐹24 |
𝐹33 (𝑟) 𝐹34

𝐹14
𝐹24 |
𝐹34

𝐹12
𝐹
| − 𝐹44 (𝑟) | 11
𝐹22
𝐹21

𝐹13
𝐹
| + 𝐹31 | 12
𝐹23
𝐹22

𝐹13
|+𝐶
𝐹23

= 𝐹44 (𝑟)𝐹33 (𝑟)(𝐹11 𝐹22 − 𝐹12 𝐹21 ) + 𝐷= {𝑟 2 + 𝑟{𝜋̅ − 𝑖[𝜌(𝑦̅, 𝑚∗) − 𝜋̅] + 𝛾}}{𝛾{𝜋̅ − 𝑖[𝜌(𝑦̅, 𝑚∗) − 𝜋̅]}} + 𝐷 ,where C and D are independent of 𝑟.
𝑎4 = 𝑑𝑒𝑡𝐽 is positive for all sufficiently large values of 𝑟 > 0 if
𝜋̅ > 𝑖[𝜌(𝑦̅, 𝑚∗) − 𝜋̅].
(45)
From the two results above, we can summarize that 𝐸 ≡ 𝑎2 − 𝑟 2 < 0 and 𝑑𝑒𝑡𝐽 > 0 are
satisfied for all sufficiently large 𝑟 > 0 if 𝜋̅ > 𝑖[𝜌(𝑦̅, 𝑚∗) − 𝜋̅]. This result is derived under
the assumption that θ=1. However, this result also applies in case of 0<θ<1 as long as θis close to 1 because of continuity.
□
VI.

Economic Interpretation

The proposition we presented, the “credibility parameter” 𝜃 ≈ 1, signifies a forwardlooking expectation formation with a strong influence from the inflation target. This means,the economy can achieve stability of output and inflation at the equilibrium, when the public
17

believes the consolidated government is capable and committed to the inflation target if the
discount rate (𝑟) is sufficiently large. Also, the inflation target is required to be sufficiently
high, while the nominal interest rate 𝜌(𝑦̅, 𝑚∗ ) is sufficiently low. The idea that the target
needs to be somewhat high is consistent with MMT’s argument9 that setting inflation target
too low can come at the expense of employment, and also aligns with the views of so-called
mainstream economists, often considered fiscal doves, who argue for a more expansionary
fiscal policy10. Based on these two assumptions, it can be demonstrated that achieving
economic stabilization through a fiscal-policy-led approach, characteristic of MMT, isfeasible, which is the main objective of this paper. Conversely, if there is low credibility
towards the inflation target, which means the public places more emphasis on past pricemovements and exhibits backward-looking inflation expectations, and if the inflation target
is too low, economic trends are likely to become destabilized.
VII.

Concluding Remarks

In this paper, we approached the examination of stability/instability in a dynamic
Keynesian model considering MMT through a different perspective by incorporating
dynamic optimization, in contrast to the approaches taken by Asada (2020) and Matsumoto(2023c). The dynamic system constructed sets money and inflation expectations as statevariables, with an MMT-inspired (fiscally dominant) consolidated government adjusting
government expenditure to achieve economic stability. The conclusion suggests a forwardlooking environment where economic agents initially trust central bank policy. In the eventof a deviation of the inflation rate from the target, the central bank acts to bring it back to
the target, ultimately achieving the inflation goal. Additionally, if the consolidated
government’s inflation target is sufficiently high, the dynamic system converges to an
equilibrium point, satisfying stability under sufficiently high discount rate (𝑟). Stability
implies that the economy settles at the equilibrium point without experiencing overheatingor stagnation, and money supply and inflation expectations do not diverge upward or
downward. This argument serves as a counterpoint to frequent criticisms of MMT, assertingthat fiscal-dominant economic policies envisioned by MMT cannot stabilize the economy.
One of the common critiques of MMT is that theory lacks mathematical models, which thispaper and the aforementioned previous studies aim to address. However, it is important tonote that this paper may face criticism from MMT proponents for certain simplifications,
keeping central bank’s inflation target, and imposing budget constraints, which might be

9Papadimitriou and Wray (1996)

10

Stiglitz (2008)18

considered not entirely MMT-consistent. Consequently, enhancing the model'scompleteness as a more MMT-centric framework remains a future challenge. Nonetheless,demonstrating the potential for stabilizing the economic system through a dynamic
Keynesian model infused with MMT ideas and explaining the existence of optimal paths
using dynamic optimization are considered contributions to the discourse.
In addition, there are considerations regarding the dynamic optimization. This modelusing dynamic optimization assumes that the government is omniscient and has the ability
to follow the appropriate path of government expenditure towards the equilibrium point.
Therefore, our findings should be considered as a “reference” or “yardstick” to observe how
the real-life economic policy is deviating from the optimal path. As such, we may
paradoxically say that the real-life policy settings by the policy makers are not optimal.
On the other hand, as mentioned earlier, the analysis of MMT-inspired dynamic models
not relying on dynamic optimization are conducted in studies like Asada (2020), Asada,
Demetrian, Zimka and Zimková (2023) and Matsumoto (2023). The significance of theseanalysis lies in the fact that it involves policies of government without complete informationof the economic structure.
Appendix
𝐹11 = −𝜋̅ + 𝑖[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅]
𝐹12 = 𝑣{𝑔𝜋𝑒 − 𝑏̅[𝜀(𝑦𝑔 𝑔𝜋𝑒 + 𝑦𝜋𝑒 ) + 1 − 𝑖𝜌−𝜋𝑒 ]} − 𝑚[𝜀(𝑦𝑔 𝑔𝜋𝑒 + 𝑦𝜋𝑒 ) + 1 + 𝑖𝜌−𝜋𝑒 ]𝐹13 = 𝑣{𝑔𝜙1 − 𝜀𝑦𝑔 𝑔𝜙1 } − 𝑚𝜀𝑦𝑔 𝑔𝜙1
𝐹14 = 𝑣(𝑔𝜙 + 𝜀𝑦𝑔 𝑔𝜙 𝑏̅) − 𝑚𝜀𝑦𝑔 𝑔𝜙222

𝐹21 = 𝛾[(1 − 𝜃)𝜀(𝑦𝑔 𝑔𝑚 + 𝑦𝑚 )]𝐹22 = 𝛾[−𝜃 + (1 − 𝜃)𝜀(𝑦𝑔 𝑔𝜋𝑒 + 𝑦𝜋𝑒 )]𝐹23 = 𝛾[(1 − 𝜃)𝜀𝑦𝑔 𝑔𝜙1 ]𝐹24 = 𝛾[(1 − 𝜃)𝜀𝑦𝑔 𝑔𝜙2 ]𝐹31 = 0𝐹32 = 0𝐹33 (𝑟) = 𝜋̅ − 𝑖[𝜌(𝑦̅, 𝑚∗ ) − 𝜋̅] + 𝑟
𝐹34 = 0𝐹41 = 0𝐹42 = 2𝜉(𝜀𝑦𝜋𝑒 + 1)𝜀(𝑦𝑔 𝑔𝜋𝑒 + 𝑦𝜋𝑒 )𝐹43 = 2𝜉(𝜀𝑦𝜋𝑒 + 1)[𝜀(𝑦𝑔 𝑔𝜙1 )] + 2(1 − 𝜉)𝑦𝜋𝑒 𝑦𝑔 𝑔𝜙1− {𝑣{(𝜀𝑦𝜋𝑒 − 1 + 𝑖𝜌−𝜋𝑒 )𝑏̅} − 𝑚[𝜀𝑦𝜋𝑒 + 1 + 𝑖𝜌−𝜋𝑒 ]}𝐹44 (𝑟) = 𝛾[𝜃 − (1 − 𝜃)𝜀𝑦𝜋𝑒 ] + 𝑟

19

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