Discussion Paper No.380

A Dynamic Optimization Model ofMonetary Stabilization Policy
Ken Matsumoto
Chuo University

January 2023

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, JapanA Dynamic Optimization Model of Monetary Stabilization Policy

Ken Matsumoto
Graduate School of Economics
Chuo University

January 2023

1

A Dynamic Optimization Model of Monetary Stabilization Policy

Abstract
By solving a dynamic optimization structure, this paper aims to achieve an optimal monetary
policy which enables the central bank to minimize a Taylor-rule embedded social loss functionof inflation and employment. Three different types of inflation expectations are tested:
forward-looking, backward-looking, and mixed expectation formation. As a result, under
several assumptions, we found stable and optimal policy paths with all of the three scenarios.
The results suggest that when the expected inflation rate is below the target, the central
bankers need to initially set the interest rate at a sufficiently low level but gradually raise it,
until they reach the equilibrium. Opposite solution stands if the inflation expectation is above
the target. These may be counterfactual to a commonly seen monetary policy, though we
believe this analysis is meaningful, in a way that our solution can be regarded as a “reference”
or “yardstick”, and indicate how the real-life monetary policy is deviating from the optimal
path.

Keywords: Optimal monetary policy, Taylor Rule, (in)stability, dynamic optimization,
inflation expectation
JEL Classification: E5, E12, E52

2

1. Introduction
At this moment in 2022, most of the major central banks have been struggling from the 40year high inflation which was triggered by the combination of supply bottlenecks andexpansive fiscal policy as a response to the Covid-19 pandemic, which has been spread from2020. As a response, a rapid pace of tightening has been taken which should eventually lower
the inflation at some point, but a trade-off between a loss of employment seems to be
inevitable. If the economy is overkilled, a deep recession is the possible outcome, anddisinflation risk could even emerge at some point. However, forecasting these scenarios arealways a big challenge. For example, the Federal Reserve System (Fed) has already admitted
that they overlooked the inflation outlook after the pandemic. Their outlooks on policy rate
(dot plot) have been revised upward repeatedly, adding uncertainty to the financial market.
Their stance of data-by-data policy decisions is widely known as a discretionary policy. The
opposite method is a rule-based policy, for example, a Taylor Rule. This comparison isoriginally termed by Kydland and Prescott (1977).
In this paper, a welfare loss function is formulated by inducing a simple Taylor-rule functionin the model. Taylor-rule, which was elaborated by Taylor (1993) is not explicitly adopted asa policy instrument to determine a policy rate, yet it is still a very influential notion widely
among policy makers. For example, in the Federal Reserve Bank (Fed) semiannual Monetary
Policy Report 1 , they provide Taylor Rule with several different versions since 2017 as areference to theoretically show the appropriate policy rate. The current Fed chairman Jerome
Powell pointed out on Taylor Rule that it never has been used as a strict policy instrument,
though acknowledged it is still a useful reference to signal an appropriate interest rate2.
In the Monetary Policy Report which was published in June 2022, all the versions of Taylor
Rule are implying that the Federal Funds rate should have been higher than what it was atthat moment3. The personal consumption expenditure (PCE) deflator as of Oct 2022 was
6.0% year on year, which is substantially higher than the Fed’s target of 2%. The
unemployment rate as of Nov 2022 was 3.7%, which is well below the Fed’s longer-run
projection of 4%4. The Fed intends to raise the policy rate expeditiously until the inflation

1

For example, the Monetary Policy Report (June 17, 2022)

https://www.federalreserve.gov/monetarypolicy/2022-06-mpr-summary.htm
2

Chair Powell’s quote on Taylor Rule mentioned in John Taylor’s blog

https://economicsone.com/2022/06/25/play-by-the-rules/
3

Monetary Policy Report (June 17, 2022)

4

December 14, 2022 FOMC Projection materials,
https://www.federalreserve.gov/monetarypolicy/fomcprojtabl20221214.htm

3rate heads back to the target, acknowledging the risk of facing higher unemployment rate.
However, from its significant gap from the target, a concern of over-killing the economy has
been arising and the Fed’s handling of the policy rate setting seems more and morechallenging. It would be beneficial if the Fed owns a certain instrument which guides anappropriate policy path, or an “optimal path” to achieve their policy goals.
There are several different approaches on the study of optimal policy. Rotemberg andWoodford (1997) analyzed monetary policy optimization through econometric approach,using vector autoregression models. As a variation of optimal monetary policy, for instance,
Acharya et al. (2020) takes into account consumption inequality in their model, by assuminga heterogeneous agent, instead of a representative agent model. Also, Brayton et al. (2014)
tackled this problem using the FRB/US Model to figure what they call “optimal-control
policy”. We mention the comparison with this paper in the end as well, to highlight thispaper’s policy implication more effectively.
We analyze central bank’s optimal policy path that minimizes the social loss function using
a dynamic optimization structure and test its stability at the equilibrium. The social loss
function here is defined by the Taylor Rule-like output/inflation gap combination, and the
dynamic optimization is solved subject to the inflation expectation function constraint. This
paper inherits the work by Matsumoto (2022), which assumed consolidated government(central bank and government) could control its real output to minimize the social loss
function of inflation and unemployment gap. Here, we change the control variable to nominalinterest rate and assess its policy implications. Similar approach was taken by Asada (2010)
and Semmler and Zhang (2004) to derive the appropriate monetary policy and its
characteristics at the equilibrium point, but this paper takes further steps, by switching theparameter of the inflation expectation function, which represents people’s stance on theirinflation expectation. Three different cases of optimization are tested: forward-looking,backward-looking, and mixed expectation formation scenarios.
2. Formulation of the Model
Basic formulation of the model used in this paper is based on Matsumoto (2022). Eq.(1) isa linear “expectations-augmented Phillips curve”, which variable 𝜋 and 𝑟 represents
inflation and nominal interest rate, respectively. As such, 𝑟 − 𝜋 𝑒 is the expected real rate ofinterest. Parameterεis a reaction parameter from the output gap, 𝜋 𝑒 is inflation expectation,
𝑌 is the real output, and 𝑌̅ represents natural output, or real output target. 𝑌(𝑟 − 𝜋 𝑒 ) below
represents a reduced form of the equilibrium condition of the goods market. Real output is adecreasing function of the expected real rate of interest, from a decrease in investment
4expenditure. What is different from Matsumoto (2022) is the control variable, which thispaper uses nominal interest rate 𝑟 to indirectly control the real output.

𝜕𝑌

𝜋 = 𝜀(𝑌(𝑟 − 𝜋 𝑒 ) − 𝑌̅) + 𝜋 𝑒 ; 𝜀 > 0 , 𝑌(𝑟−𝜋𝑒) = 𝜕(𝑟−𝜋𝑒) < 0(1)

Eq.(2) is the dynamic inflation expectation formation that is based on Asada (2010),
which combines forward-looking and backward-looking expectations. Parameter ξ
represents the weight of people’s stance on their inflation expectation. If ξ=1, the equationof motion for 𝜋̇ 𝑒 is dependent on the inflation gap, which here is represented by thedifference between actual inflation 𝜋 and inflation expectation 𝜋 𝑒 . This shows that people
have adaptive behavior on inflation. On the other hand, if ξ=0 , the equation expresses
people’s behavior as forward-looking, since 𝜋̇ 𝑒 is now dependent on the gap between
government’s inflation target 𝜋̅ and 𝜋 𝑒 . If 0 < 𝜉 < 1, then people’s behavior on inflation ismixed between forward and backward-looking. The parameter α is the reaction parameterto the inflation expectation dynamics.
𝜋̇ 𝑒 = 𝛼{𝜉(𝜋 − 𝜋 𝑒 ) + (1 − 𝜉)(𝜋̅ − 𝜋 𝑒 )} ; 𝛼 > 0, 0 ≦ 𝜉 ≦ 1(2)

Eq.(3) is derived by substituting Eq.(1) into Eq.(2), and now the equation of motion for𝜋̇ 𝑒 has real output as one of the variables.
𝜋̇ 𝑒 = 𝛼{𝜉𝜀(𝑌(𝑟 − 𝜋 𝑒 ) − 𝑌̅) + (1 − 𝜉)(𝜋̅ − 𝜋 𝑒 )}(3)

For simplicity on solving the upcoming equations, linear approximation of 𝑌(𝑟 − 𝜋 𝑒 )below is used in the further analyses.
𝜕𝑌𝜕𝑌

𝑌 = −𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏 = 𝑌(𝜋 𝑒 , 𝑟) ; 𝜕𝑟 = −𝑎 < 0 , 𝜕𝜋𝑒 = 𝑎 > 0, 𝑏 > 0(4)

2.1 Case of ξ=0
Based on the model mentioned above, in this section, we work on the case of ξ=0. As
previously mentioned, this transforms the equation of 𝜋̇ 𝑒 into a forward-looking system.
Eq.(3) now becomes Eq.(5).
𝜋̇ 𝑒 = 𝛼(𝜋̅ − 𝜋 𝑒 ) ; 𝛼 > 05(5)The social loss function is defined as below. This transformation is similar to the method
used in Taylor(1989), Chiang(1992), Woodford(2001) and Asada(2010)5.
𝑉 = 𝜃(𝑌(𝑟 − 𝜋 𝑒 ) − 𝑌̅)2 + (1 − 𝜃)(𝜋 − 𝜋̅)2 = 𝑉(𝑌(𝑟 − 𝜋 𝑒 ), 𝜋) ; 0 < 𝜃 < 1(6)

The parameter 𝜃 is the positive parameter which represents the weight of how much thepolicy maker prioritize between real output and inflation gap. Eq.(1) is now substituted into
Eq.(6), and in order to minimize the loss function, Eq.(6) is turned into negative.
−𝑉 = −[𝜃(𝑌(𝑟 − 𝜋 𝑒 ) − 𝑌̅)2 + (1 − 𝜃){𝜀(𝑌(𝑟 − 𝜋 𝑒 ) − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}2 ] = 𝑊(𝑌(𝑟 − 𝜋 𝑒 ), 𝜋 𝑒 )

0<𝜃<1

(7)

Combining Eq.(5) and Eq.(7) gives the following dynamic optimization problem subject
to the constraint equation.

∞

max ∫0 𝑊(𝑌(𝑟 − 𝜋 𝑒 ), 𝜋 𝑒 )𝑒 −𝜌𝑡 𝑑𝑡 ; 𝜌 > 0𝑟≥0𝑠. 𝑡. 𝜋̇ 𝑒 = 𝛼(𝜋̅ − 𝜋 𝑒 )(8)In order to derive the optimal macroeconomic policy path, a current-value Hamiltonian
is formalized as below, where 𝜆 is the co-state variable.
𝐻 = −[𝜃((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅)2 + (1 − 𝜃){𝜀((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}2 ] + 𝜆𝛼(𝜋̅ − 𝜋 𝑒 )(9)As explained in Chiang (1992), Pontryagin’s maximum principle conditions are given as
follows6.
Max 𝐻(𝑌(𝑟 − 𝜋 𝑒 ), 𝜋 𝑒 , 𝜆) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ∈ [0, ∞)𝑟≥0𝜋̇ 𝑒 =

5𝜕𝐻
𝜕𝜆[𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝜋̇ 𝑒 ]Consideration in monetary policy tradeoffs, welfare loss function, and problem solving of optimalmonetary policy are also conducted by Galí (2015) and Woodford(2002).
6

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

from Chiang and Wainwright (2010).

6

𝜕𝐻
+ 𝜌𝜆
𝜕𝜋 𝑒

[𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝜆]lim 𝜆𝑒 −𝜌𝑡 = 0[𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑎𝑙𝑖𝑡𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛]

𝜆̇ = −𝑡→∞(10)
The Hamiltonian system here is solved with respect to the nominal interest rate, r. Thisimplies that policy maker is controlling nominal interest rate 𝑟 to minimize (optimize) thesocial loss function previously proposed.
A first order condition is required to show that the optimal control of 𝑟 will be aninterior solution.
𝜕𝐻̅) + 𝜋𝑒 − 𝜋= 2𝑎[𝜃((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅) + 𝜀 2 (1 − 𝜃){((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅ }] = 0𝜕𝑟(11)
Further differentiation of Eq.(11) with the result of negative shows that the control
variable 𝑟 does maximize the Hamiltonian system.
𝜕2 𝐻
𝜕𝑟 2

= 2𝑎[−𝜃𝑎 − 𝜀 2 𝑎(1 − 𝜃)] < 0(12)Equation of motion for 𝜋̇ 𝑒 is defined as below.
𝜋̇ 𝑒 =𝜕𝐻𝜕𝜆= 𝛼(𝜋̅ − 𝜋 𝑒 ) = 𝐹11 (𝜋 𝑒 )(13)Equation of motion for 𝜆, which is a costate variable, is described below.

𝜆̇ = −

𝜕𝐻+ 𝜌𝜆 = 2[𝜃𝑎((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅) + (1 − 𝜃)(𝜀𝑎𝜕𝜋 𝑒+ 1){𝜀((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}] + 𝜆(𝛼 + 𝜌)
(14)

Real output Y of Eq.(11) can be transformed as below.
𝑒̅ −𝜋 )
𝜀(1−𝜃)(𝜋((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − 𝑌̅) == 𝑓(𝜋 𝑒 )2
𝜃+𝜀 (1−𝜃)(15)

Eq.(15) can be substituted into Eq.(14) as below.
𝜆̇ = 2[𝜃𝑎𝑓(𝜋 𝑒 ) + (1 − 𝜃)(𝜀𝑎 + 1){𝜀𝑓(𝜋 𝑒 ) + 𝜋 𝑒 − 𝜋̅}] + 𝜆(𝛼 + 𝜌) = 𝐹21 (𝜋 𝑒 , 𝜆)7(16)

𝒹𝑓(𝜋𝑒 )

Where

𝒹𝜋𝑒

=−

𝜀(1−𝜃)
𝜃+𝜀2 (1−𝜃)<0

2.2 Phase diagram in case of 𝝃 = 𝟎
From the equilibrium point of 𝜋̇ 𝑒 in Eq.(13), the solution 𝜋̅ = 𝜋 𝑒 can be derived.
𝜋 𝑒 = 0 line is shown in Figure 1. It is assumed here that the policy maker would indirectly
maneuver the costate variable of 𝜆 by controlling the interest rate 𝑟, which can be termed
as a marginal increase (or decrease) in inflation expectation.

̇ <𝟎

̇ =𝟎

̅

̇ >𝟎

𝟎
Figure1
Total differentiation of Eq.(16) is described below.
∂𝐹21𝑑𝜋= − ∂𝜆1|𝑑𝜆 𝜆̇=0∂𝐹2∂𝜋 𝑒𝑒

We obtain Figure 2 by solving the equations below.
∂𝐹21
𝜀𝜃𝑎(1 − 𝜃) + 𝜃(1 − 𝜃)(𝜀𝑎 + 1)
= 2 [−] = 𝐹21 < 0𝑒∂𝜋
𝜃 + 𝜀 2 (1 − 𝜃)

8

∂𝐹21
= 𝛼 + 𝜌 = 𝐹22 > 0
∂𝜆

̇ =𝟎

̇ <𝟎

̇ >𝟎

̅

𝟎
Figure 2
We now have the phase diagram of this analysis below in Figure 3, by combining Figure 1
and 2.

̇ =𝟎

E

̅

̇ =𝟎

𝟎
Figure 3
As depicted in Figure 3, an implication drawn under the rule ξ=0 is that there is only one
policy path (from both sides) which enables policy maker to achieve full employment and9the inflation target goal, and this unique path toward the equilibrium is the saddle path.
Only the saddle path satisfies the transversality condition in Eq.(10). From a given initialstate variable 𝜋 𝑒 (0) of inflation expectation, policy maker needs to select the initial point of
𝜆 and control it following the saddle path, by indirectly maneuvering the interest rate 𝑟. Or
else, the inflation expectation and co-state variable 𝜆 would not end up at the equilibrium
point of E.
If the starting point of inflation expectation is exceeding the inflation target, the policy
maker would have to set the initial interest rate 𝑟 at a sufficiently high point but lower it
gradually afterwards in order to reach the equilibrium. On the other hand, if the initial pointof inflation expectation is below the target, the starting point of 𝑟 would be set sufficiently
low, and the optimal path is to raise it upward until the target is achieved. This is quite aparadoxical result, since in general, output and inflation expectation have a positive
correlation, which the increase in output would lead to a further rise in inflation expectation.
The only way to achieve the policy target is to follow the saddle path, and if any other path isto be chosen, the output and inflation expectation would then spread out indefinitely.
By taking into account Eq.(13) and Eq.(16), the characteristic equation of this system at theequilibrium point is described below, where 𝜆1 and 𝜆2 are two characteristic roots7.
|𝜆𝐼 − 𝐽1 | = 𝜆2 − (𝑡𝑟𝑎𝑐𝑒𝐽)𝜆 + (𝑑𝑒𝑡𝐽)= (𝜆 − 𝜆1 )(𝜆 − 𝜆2 ) = 𝜆2 − (𝜆1 + 𝜆2 )𝜆 + 𝜆1 𝜆2 = 0(17)
Where,−𝛼𝐽1 = [ 1
𝐹21

0
1 ]
𝐹22

1
𝑡𝑟𝑎𝑐𝑒 𝐽 = 𝜆1 + 𝜆2 = −𝛼 + 𝐹22
1
𝑑𝑒𝑡𝐽 = 𝜆1 𝜆2 = −𝛼𝐹22<02.3 Case of ξ=1
Now, this time, a backward-looking type of inflation expectation, is considered. This is thecase of ξ=1 in Eq.(3).
The equation of motion for 𝜋̇ 𝑒 now becomes a function of the interest rate 𝑟.
7

Evaluating the dynamical properties of interest rate and policy tool is also done by Drumond et al.(2022)10𝜋̇ 𝑒 = 𝛼{𝜀(𝑌(𝑟 − 𝜋 𝑒 ) − 𝑌̅)} ; 𝛼 > 0(18)

Eq.(18) becomes a constraint function, and the dynamic optimization problem is formulated
as below.
∞max ∫ 𝑊(𝑌(𝑟 − 𝜋 𝑒 ), 𝜋 𝑒 )𝑒 −𝜌𝑡 𝑑𝑡
𝑟≥00

𝑠. 𝑡. 𝜋̇ 𝑒 = 𝛼𝜀(𝑌(𝑟 − 𝜋 𝑒 ) − 𝑌̅)
As similar to Eq.(9), a current value Hamiltonian is described below.
𝐻 = −[𝜃((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅)2 + (1 − 𝜃){𝜀((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}2 ] + 𝜆𝛼𝜀(𝑌(𝑟 − 𝜋 𝑒 ) − 𝑌̅)(19)First order condition and second derivative are described below.

𝜕𝐻= 2𝑎[𝜃((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅) + 𝜀 2 (1 − 𝜃){((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}] − 𝜆𝛼𝜀𝑎 = 0
𝜕𝑟

𝜕2𝐻= −2𝑎2 {𝜃 + 𝜀 2 (1 − 𝜃)} < 0
𝜕𝑟 2
(20)

Equation of motion for the inflation expectation 𝜋̇ 𝑒 is given below.
𝜋̇ 𝑒 =𝜕𝐻𝜕𝜆= 𝛼𝜀((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − ̅𝑌) = 𝐹12 (𝜋 𝑒 , 𝑟)

(21)

From the first order condition in Eq.(20), the equation of 𝜆 can be obtained.

21̅) { 𝜃 + 𝜀(1 − 𝜃)} + (1 − 𝜃)(𝜋 𝑒 − 𝜋̅)]𝜆 = 𝛼 [((−𝑎(𝑟−𝜋𝑒 ) + 𝑏) − 𝑌𝜀(22)

Differentiation of Eq.(22) with respect to time t gives Eq.(23).
21𝜆̇ = 𝛼 [{−𝑎(𝑟̇ − 𝜋̇ 𝑒 )} {𝜀 𝜃 + 𝜀(1 − 𝜃)} + 𝜋̇ 𝑒 (1 − 𝜃)]Additionally, from the one of the maximum principle conditions, the motion for 𝜆 is
obtained below.
11

(23)

𝜆̇ = −

𝜕𝐻̅)}{𝜃𝑎 + 𝜀(1 − 𝜃)(𝜀𝑎 + 1)}+ 𝜌𝜆 = 2[{((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − 𝑌𝜕𝜋 𝑒+ (1 − 𝜃)(𝜀𝑎 + 1)(𝜋 𝑒 − 𝜋̅)] − 𝜆(𝛼𝜀𝑎 − 𝜌)(24)Transversality condition is termed as follows.
lim 𝜆 𝑒 −𝜌𝑡 = 0

𝑡→∞

(25)

By combining Eq.(23) and (24) and bringing 𝑟̇ to the left side, the equation of motion for
the interest rate 𝑟 can be described as below.
̅)}{𝜃𝑎 + 𝜀(1 − 𝜃)(𝜀𝑎 + 1)} + (1 − 𝜃)(𝜀𝑎 + 1)(𝜋 𝑒 − 𝜋̅)] − 𝜆(𝛼𝜀𝑎 − 𝜌)
2[{((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − 𝑌=21[{−𝑎(𝑟̇ − 𝜋̇ 𝑒 )} { 𝜃 + 𝜀(1 − 𝜃)} + 𝜋̇ 𝑒 (1 − 𝜃)]𝛼𝜀̅)}{𝜃𝑎 + 𝜀(1 − 𝜃)(𝜀𝑎 + 1)} + (1 − 𝜃)(𝜀𝑎 + 1)(𝜋 𝑒 − 𝜋̅)
2 [{((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − 𝑌−

1(1 − 𝜃)𝐹12 (𝜋 𝑒 , 𝑟)] − 𝜆(𝛼𝜀𝑎 − 𝜌) = −𝐴(𝑟̇ − 𝜋̇ 𝑒 )
𝛼
𝐴=

2 1
( 𝜃 + 𝜀(1 − 𝜃)) 𝑎𝛼 𝜀1̅)}{𝜃𝑎 + 𝜀(1 − 𝜃)(𝜀𝑎 + 1)} + (1 − 𝜃)(𝜀𝑎 + 1)(𝜋 𝑒 − 𝜋̅)𝑟̇ = − [2 [{((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − 𝑌𝐴−1(1 − 𝜃)𝐹12 (𝜋 𝑒 , 𝑟)] − 𝜆(𝛼𝜀𝑎 − 𝜌)] + 𝜋̇ 𝑒 = 𝐹22 (𝜋 𝑒 , 𝑟)𝛼(26)

2.4 Phase diagram in case of 𝝃 = 𝟏
The locus for 𝜋̇ 𝑒 = 0 is derived from Eq.(21) and shown in Figure 4.
𝑑𝜋𝑒|

𝑑𝑟 𝜋𝑒̇ =0

=−

2
∂𝐹1
∂𝑟
2
∂𝐹1
∂𝜋𝑒

Where numerator and denominator are solved as below.

12

(27)

∂𝐹12
= −𝛼𝜀𝑎 < 0∂𝑟∂𝐹12= 𝛼𝜀𝑎 > 0
∂𝜋 𝑒

̇ =𝟎

̅

̇ <𝟎

̇ >𝟎

𝟎

̅
Figure 4𝜋̇ 𝑒 = 𝛼𝜀((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − ̅𝑌) = 𝐹12 (𝜋 𝑒 , 𝑟)𝑟̅ =𝑏 − 𝑌̅
>0𝑎Similarly, the 𝑟̇ = 0 line is depicted by solving the total derivation of Eq.(26), as shown
below in Figure 5.
𝑑𝜋𝑒|𝑑𝑟 𝑟̇ =0

=−

2
∂𝐹2
∂𝑟
2
∂𝐹2
∂𝜋𝑒

Each of the partial derivations are solved as below.

∂𝐹22
=0
∂𝑟
1
−2[𝑎{𝜃𝑎 + 𝜀 2 𝑎(1 − 𝜃)} + (1 − 𝜃)(𝜀𝑎 + 1)] − 𝜀𝑎2 {𝜀 𝜃 + 𝜀(1 − 𝜃)}∂𝐹222= 𝐹11 =
<0
∂𝜋 𝑒
𝐴

13

(28)

𝐴=

2 1
{ 𝜃 + (1 − 𝜃)} 𝑎
𝛼 𝜀

̇ <𝟎

̇ =𝟎
̇ >𝟎

𝟎
Figure 5
By combining both lines, the phase diagram is depicted below in Figure 6. It is now clearthat the system with ξ=1 also has an unique path which converges into the equilibrium
point E. Again, as like the case with ξ=0, the conclusion is quite paradoxical. If the starting
point of inflation expectation is below the inflation target 𝜋̅, the policy maker needs to set
the initial interest rate 𝑟 at a low point and raise it later on. On the other hand, if theinflation expectation is exceeding the target, the policy maker now needs to start from a highinterest rate and lower it to suppress the inflation expectation.

̇ =𝟎

E

̅

̇ =𝟎

𝟎
Figure 6
14

The evaluation of the dynamical properties of the model using the Jacobian matrix is
described below, which shows that the equilibrium point becomes saddle point.

𝐹2
𝐽2 = [ 11
2
𝐹21

2
𝐹12
]
0

2
𝑡𝑟𝑎𝑐𝑒 𝐽 = 𝜆1 + 𝜆2 = 𝐹11>0

𝑑𝑒𝑡𝐽 = 𝜆1 𝜆2 < 0
2.5 Case of 𝟎 < 𝝃 < 𝟏
Thirdly, the system with 0 < 𝜉 < 1 is tested to figure out its dynamic behavior. The loss
function of policy maker is same as the previous cases. The dynamic optimization problem
in this model is described as below.
∞

max ∫ 𝑊(𝑌(𝑟 − 𝜋 𝑒 ), 𝜋 𝑒 )𝑒 −𝜌𝑡 𝑑𝑡
𝑟≥00

𝑠. 𝑡. 𝜋̇ = 𝛼{𝜉𝜀(𝑌(𝑟 − 𝜋 𝑒 ) − 𝑌̅) + (1 − 𝜉)(𝜋̅ − 𝜋 𝑒 )}𝑒

(29)Thus, current-value Hamiltonian is formatted using the new constraint function.

22̅) + (1 − 𝜃){𝜀((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − 𝑌̅) + 𝜋 𝑒 − 𝜋̅} ]𝐻 = − [𝜃((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − 𝑌̅) + (1 − 𝜉)(𝜋̅ − 𝜋 𝑒 )}+ 𝜆𝛼{𝜉𝜀((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − 𝑌(30)First order condition and second order derivative become as follows.

𝜕𝐻= 2𝑎[𝜃((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅) + 𝜀(1 − 𝜃){((−𝑎(𝑟 − 𝜋 𝑒 ) + 𝑏) − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}] − 𝜆𝛼𝜉𝜀𝑎 = 0
𝜕𝑟

𝜕2𝐻= −2𝑎{−𝜃𝑎 + 𝜀 2 𝑎(1 − 𝜃)} < 0
𝜕𝑟 2
(31)
We can derive the following expression of 𝜆 by transforming the result from the first
derivative equation above.
𝜆=21𝑌) { 𝜃 + 𝜀(1 − 𝜃)} + (1 − 𝜃)(𝜋 𝑒 − 𝜋̅)] = 𝜆(𝜋 𝑒 , 𝑟)[((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − ̅
𝛼𝜉
𝜀
(32)

Equation of motion for 𝜋̇ 𝑒 is obtained below.
15𝜕𝐻𝜋̇ 𝑒 = 𝜕𝜆 = 𝛼[𝜉𝜀((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − ̅𝑌) + (1 − 𝜉)(𝜋
̅ − 𝜋 𝑒 )] = 𝐹13 (𝜋 𝑒 , 𝑟)(33)

Differentiation of Eq.(32) with respect to time gives an equations of 𝜆̇.
21
𝜆̇ = 𝛼𝜉 [{−𝑎(𝑎̇ − 𝜋̇ 𝑒 ) (𝜀 𝜃 + 𝜀(1 − 𝜃))} + 𝜋̇ 𝑒 (1 − 𝜃)](34)

Another equation of 𝜆̇, which is part of the maximum principle, is described below.
𝜆̇ = −𝜕𝐻̅)+ 𝜌𝜆 = 2[𝜃𝑎((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − 𝑌𝜕𝜋 𝑒+ (1 − 𝜃)(𝜀𝑎 + 1){𝜀((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − ̅𝑌) + 𝜋 𝑒 − 𝜋̅}]− 𝜆[𝛼{𝜉𝜀𝑎 − (1 − 𝜉)} − 𝜌](35)
2[𝜃𝑎((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − ̅𝑌) + (1 − 𝜃)(𝜀𝑎 + 1){𝜀((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − ̅𝑌) + 𝜋 𝑒 − 𝜋̅}]− 𝜆[𝛼{𝜉𝜀𝑎 − (1 − 𝜉)} − 𝜌]=212{−𝑎(𝑎̇ − 𝜋̇ 𝑒 ) ( 𝜃 + 𝜀(1 − 𝜃))} + (1 − 𝜃)𝐹13 (𝜋 𝑒 , 𝑟) = −𝐴(𝑟̇ − 𝜋̇ 𝑒 )
𝛼𝜉
𝜀
𝛼𝜉

𝐴=

2𝑎 1
{ 𝜃 + 𝜀(1 − 𝜃)}
𝛼𝜉 𝜀
(36)

By combining Eq.(34) and (35), and bringing 𝑟̇ to the left side, the equation of motion forinterest rate 𝑟 is available.
1̅) + (1 − 𝜃)(𝜀𝑎 + 1){𝜀((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}𝑟̇ = − [2 [𝜃𝑎((−𝑎(𝑟 − 𝜋𝑒 ) + 𝑏) − 𝑌𝐴−1(1 − 𝜃)𝐹13 (𝜋 𝑒 , 𝑟)] − 𝜆{𝛼{𝜉𝜀𝑎 − (1 − 𝜉)} − 𝜌}] + 𝐹13 (𝜋 𝑒 , 𝑟) = 𝐹23 (𝜋 𝑒 , 𝑟)𝛼𝜉(37)

2.6 Phase Diagram in case of 𝟎 < 𝝃 < 𝟏
Phase diagram of this system can be depicted from Eq.(33) and Eq.(37).
It is clear that the slope of 𝜋̇ 𝑒 = 0 line is positive, as shown in Figure 7.

16

𝑑𝜋𝑒

|

𝑑𝑟 𝜋̇ =0

=−

𝜕𝐹3
1
𝜕𝑟
𝜕𝐹3
1
𝜕𝜋𝑒

>0

(38)

Where,
𝑑𝐹13= 𝐹11 = 𝛼{𝜉𝜀𝑎 − (1 − 𝜉)} > 0
𝑑 𝜋𝑒
𝜕𝐹13
= 𝐹12 = −𝛼𝜉𝜀𝑎 < 0
𝜕𝑟

̇ =𝟎
̇ >𝟎

̇ <𝟎

𝟎
Figure 7

The slope of 𝑟̇ = 0 line is horizontal, as described below in Figure 8.
𝜕𝐹23= 𝐹21 = −2[𝑎{(𝜀𝑎 + 1)(1 − 𝜃)(𝜀(−1 + 𝑎) + 1)}] −𝜕𝜋𝑒1(1 − 𝜉) {2(1 − 𝜃) + 𝛼𝐴}
𝜉

𝑎>1
𝜕𝐹23
𝜕𝑟

= 𝐹22 = −2𝑎[−𝜀(1 − 𝜃){𝜀𝑎 − 𝜀𝑎}]
=0
𝐴

𝜕𝐹23
𝑑𝜋 𝑒
= − 𝜕𝑟3 = 0|
𝑑𝑟 𝑟̇ =0
𝜕𝐹2
𝜕𝜋 𝑒
(39)

17

̇ <𝟎

̇ =𝟎
̇ >𝟎

𝟎
Figure 8
The combination of Figure 7 and 8 gives the phase diagram, as depicted in Figure 9.

̇ =𝟎

E

̇ =𝟎

̅

𝟎
Figure 9
The system is stable as it converges to the equilibrium point, as proven in the Jacobianmatrix below.
𝐹3
𝐽3 = [ 11
3
𝐹21

3
𝐹12
]
0

3
𝑡𝑟𝑎𝑐𝑒 𝐽 = 𝜆1 + 𝜆2 = 𝐹11>0

𝑑𝑒𝑡𝐽 = 𝜆1 𝜆2 < 0
As we have seen in the previous cases, the system with 0<ξ<1 also have the same policy
18implications. If the inflation expectation is above the target, the policy maker initially sets
the interest rate high, but lower it gradually. If the inflation expectation is below the target,
the opposite policy is required to reach the equilibrium.
3. Policy Implications
From the results of each analysis, we consider here the policy implications. First, ourresults tell us that if inflation expectation is above the target, the starting point of policy rateneeds to be set sufficiently above the natural rate of interest, 𝑟 . Afterwards, the policy rateshould be lowered gradually until it reaches the inflation target. This is quite
counterintuitive in which central banks usually deal with inflation by raising its interest rate.
One way to interpret this paradox is the implication that it’s more appropriate to set theinitial interest rate at drastically high/low level, instead of a gradual rate hike/cut.
Historically, major central banks including the Fed have responded to inflation (or potential
inflation) by raising its policy rate consecutively in an orderly way, for instance, the increaseof 25 basis point on each occasion. In some cases, the periods of policy tightening havelasted for few years. These are probably because the policy makers want to avoid volatilefinancial markets reacting to an abrupt policy change. For instance, the ongoing policy fromearly 2022 tightening have started from a mere 25bp, which then ratcheted up to 50bp and75bp consecutively. The solution from our model is to set the initial rate to where theexpected terminal rate is, and to lower it gradually, instead of hiking it.
If the inflation expectation is below the target, the solution from our model is to set theinitial rate at a sufficiently low level, and then raise it over the optimal path until the
inflation target is achieved, which also is against a standard economic theory. Unlike theformer case, somewhat similar action had been taken by the Fed right after the Covid-19 in
2020. The Fed immediately lowered its policy rate from 1.75% (upper range of corridor) to
0.25% in March 2020. Disregarding the option to implement a negative interest rate policy,
the Fed lowered the rate down to the minimal level instantly8.
Regardless of people’s forward/backward-looking stance on their inflation expectation, ourfindings imply that the optimal policy solutions for the policy makers are the same. Although
Williams (2003) uses FRB/US macroeconomic model in its analysis as contrary to ouranalysis, the author concluded that the performance of efficient monetary policy differs
under rational and backward-looking models. Carlstrom and Fuerst (2000) argues that
aggressive and backward-looking stance are necessary for the monetary authority to ensure

8

The Fed also implemented a quantitative easing policy but we do not take into accountsuch policy in this paper.
19

determinacy. Benhabib et al. (2003) tested the stabilization behavior of the backwardlooking interest-rate rules, which they figured that parameters distinguish the stabilization,and not all backward-looking feed-back rules guarantee the uniqueness of equilibrium.
Our findings show policy paths which converge to the stable equilibrium point, but weneed to consider here the cases when the policy actions were unsuccessful. Those cases arelikely to occur when policy makers misplace the initial interest rate or mis-follow the saddle
path. The output/inflation expectation paths would then diverge, letting the two variables
increase/decrease indefinitely. One consequence is the upward divergence of real output
and inflation expectations, what is called as an inflation spiral, and the other is thedownward divergence, which is a deflation spiral. The former case is frequently observableespecially among the emerging economies. Developed economies also faced somewhat
similar phenomenon in the 1970s that are commonly attributed to oil embargo factor, but at
the same time to a wage-price spiral. The Fed’s response under then-chairman Paul Volcker
was a severe tightening, which consequently led to a deep recession afterwards9. The
opposite case, –deflationary spiral – is a relatively rare scenario. One of the few examples isJapan, particularly since the beginning of the 1990s. The sluggish economy of Japan has
faced near zero or even negative year-on-year core inflation rate for nearly 20 years, whichshould be long enough to categorize Japan as the country which faced deflationary spiral.
This phenomenon can also be described as a liquidity trap, or an absolute preference ofliquidity, which was originally termed by Keynes (1936). The necessary action the Bank ofJapan had to enforce was to drastically ease its policy at the onset of the crisis, but they haverepeatedly failed to do so, resulting the interest rate to face lower bound while the inflation
remains muted. At this juncture, the central bank’s remaining solution is to stimulate thepublic’s inflation expectation to lower the expected real rate of interest. Japan succeeded thistask for a short period of time by implementing a massive quantitative easing policy from2013, until the austere fiscal policy offset the rise of inflation expectation.
We explained that our analyses earn an optimal, unique path which converges to theequilibrium. This shows that those are the “optimal monetary policy paths”, although thepolicy makers need to perfectly understand the exact starting point and interest rate path,otherwise the inflation expectation diverges indefinitely. A study by Drumond et al. (2022)
provide us a meaningful contrast. Their model depicts a phase diagram which converges into
an equilibrium, without a saddle path. The differential equations they have are described
below.

9

For more specific history of 1970’s inflation and policy actions of the Fed, see Bryan(2013).
20

① 𝑟̇ = 𝛾(𝜋 − 𝜋 𝑇 ) + 𝛽(𝑢 − 𝑢𝑇 )② 𝜋̇ 𝑒 = 𝑘(𝜋 − 𝜋 𝑒 )Here, 𝑟̇ is the time differentiated variable of real interest rate, 𝜋 is actual inflation rate,𝜋̇ 𝑒 is the time differentiated variable of expected inflation rate, 𝜋 𝑇 is the inflation target, 𝑢
is the rate of capacity utilization, and 𝑢𝑇 is its target. 𝑘 is a inflation memory parameter,and 𝛾, 𝛽 are positive parameters, representing the policy maker’s focus on each gap fromthe target. After some algebraic manipulation and evaluating the dynamical properties, thephase diagram they obtained is shown below in Figure 10.

̇ =𝟎

̇ =𝟎

𝟎
Figure 10
The analysis earns an equilibrium point with vertical 𝜋̇ 𝑒 = 0 line and positive slope 𝑟̇ = 0line, which shows that the system is stable. Their result tells that there is no necessity for thepolicy makers to know the exact starting point and path of the interest rate. Instead ofaiming for a unique, optimal policy path, this may be described as a more practical,“appropriate” policy rule.
Similar conclusion has been drawn by Asada (2010). The slope of the 𝜋̇ 𝑒 = 0 line differsfrom the previous paper, but its phase diagram shows that they have a globally stableequilibrium point, if the central bank is sufficiently credible.
Our findings provide a strict, optimal policy path, but it is rather a more theoreticalsolution, and may not be suitable in a real-life policy tool. However, we argue that ouranalysis is still meaningful since we have shown the difficulty of implementing an optimalmonetary policy, and paradoxically proven that policies taken by the real-life central bankers
are not ideal.

21

4. Conclusion
In this paper, by applying Hamiltonian equations, several optimal policy paths are derived
under different scenarios of inflation expectation behavior. First was the forward-looking
inflation expectation scenario, second was the backward-looking scenario, and the third was
the mixed behavior of the previous two scenarios. The phase diagrams showed that eachsystem obtains a unique converging path, namely a saddle path, which ultimately reaches astable and unique equilibrium point.
The dynamic relationship between interest rate and inflation expectation derived here
implies that if inflation expectation is below the inflation target, the optimal option for thepolicy maker is to select an initial point of nominal interest rate that is below the target toposition itself at the appropriate trajectory (saddle path). Then the optimal policy is to raise
the nominal interest rate and follow the saddle path until the inflation expectation converges
to the equilibrium, or the inflation target. On the other hand, if the inflation expectation isabove the target, the optimal policy is to place the initial nominal policy rate at a sufficiently
high level, and afterwards lower it following the saddle path. In either case, if the policy
outcome fails to follow the optimal path, inflation expectation and real output would dissipate
and not end up reaching the equilibrium point. These implications are somewhat paradoxical,
since an orthodox policy to stabilize the economy is to undertake counter-cyclical measures.
As a side note, another antitheoretical observations are pointed by Asada(2010) andMankiw(2001), against the New Keynsian dynamic model which can be found in the literature
by Galí (2015). They referred to their findings as a “sign-reversal” problem. “Sign-reversal”
problem is a paradoxical properties of the “New Keynesian” dynamic model that shows acounterfactual relationship between inflation and real output. According to Asada(2010), the
New Keynsian dynamic model implies that “the rate of inflation accelerates whenever theactual output level is below the natural output level, and it decelerates whenever the actual
output level is above the natural output level”. The author pointed out the same problem for
the New Keynsian IS model as well.
There are some caveats on this paper’s analysis. The model assumes that policy maker isomniscient and knows the best starting point and path to lead them to the equilibrium point.
This assumption is referred as a “jump variable” by Asada (2010) and Asada(2013). We
assume that they can control the nominal interest rate at any time, which allow them to follow
the optimal path without deviation. Also, as a simplification, we defined nominal interest rate
as if it’s the only one that exists, and we didn’t specify if it’s a short term or long-term rate.
These may be quite an unrealistic assumption, which similar argument has been made by
Asada(2010). As mentioned previously, our findings provide an “optimal” policy path, which
does not necessarily mean a realistic policy. There are some literatures which imply a more22

“appropriate”, or realistic policy approach. Our findings of optimal solution should beconsidered as a “reference” or “yardstick” to observe how the real-life monetary policy isdeviating from the optimal path. As such, we may paradoxically say that the real-life policy
settings by the central bankers are not optimal.
Finally, in the real-life policy setting, policy makers also have the capacity to control fiscal
measures. As such, for the next step of the future study, although it may complicate theequations, we may form a more developed model with interest rate and government
expenditure as the control variables.

23

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