Discussion Paper No.374

Analysis of Optimal Macroeconomic Policy
Using Dynamic Optimization
Ken Matsumoto
Chuo University

August 2022

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Analysis of Optimal Macroeconomic Policy Using Dynamic Optimization

Ken Matsumoto
Chuo University
August 2022

1

Analysis of Optimal Macroeconomic Policy
Using Dynamic Optimization

Abstract
This paper analyzes policy maker’s optimal macroeconomic policy path that minimizes
Taylor-rule like social loss function of inflation gap and output gap, using a dynamic
optimization structure. Under the assumption that policy maker can dynamically control thereal output, this paper solves dynamic optimization equations and draw optimal path of output
and inflation expectation variables, with the analysis of its stability. Three different cases of
optimization are tested (forward-looking, backward-looking, and mixed expectation
formation scenario), and the phase diagrams show that each system obtains a converging path,
namely a saddle path, which ultimately reaches a stable and unique equilibrium point. The
policy implications may seem to be quite paradoxical, although it could still stand and avoid
inflation / deflation divergence under certain assumptions.
Keywords: Optimal macroeconomic policy, Taylor Rule, (in)stability, dynamic optimization
JEL Classification: E5, E12, E52

2

1. Introduction
For central bank/government policy makers, what is their definition of an optimal
macroeconomic policy? The Fed economists Khan et al. (2003) described from a central bankstandpoint: “Optimal monetary policy maximizes the welfare of a representative agent, given
frictions in the economic environment.”1
Monetary policy wouldn’t have to be a sole channel of optimizing social welfare. Davig andGürkaynak (2015) point out that many of the past literatures focusing on optimal policy referthem as a monetary policy, instead of mixing fiscal policy into their welfare function. They
point out that monetary policy which single-handedly maneuvers interest rates cannot address
all inefficiencies in the economy, especially in a world of multiple policymakers (such as fiscal
policymakers) with different objectives.
Approaches on its study of optimal policy differs among literatures. 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”.
In this paper, as many literatures do, a welfare loss function is formulated by inducing asimple Taylor-rule function in the model. Taylor-rule, which was elaborated by Taylor (1993)
is not explicitly adopted as a 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 Report2, they provide Taylor Rule with several different
versions since 2017 as a reference 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 anappropriate interest rate3.
In the Monetary Policy Report which was published in June 2022, all the versions of Taylor

1They challenged to incorporate each of the mainstream views of Fisher, Keynes, and Friedman, and cameto one of their conclusions that economy under a small deflation is a partial factor of an optimal monetary
policy to stabilize the price level.
2

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

https://www.federalreserve.gov/monetarypolicy/2022-06-mpr-summary.htm
3

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

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

3Rule are implying that the Federal Funds rate should have been higher than what it was atthat moment4. The core personal consumption expenditure inflation as of May 2022 was 4.7%
year on year, which is substantially higher than the Fed’s target of 2%. The unemployment
rate as of May 2022 was 3.6%, which is well below the Fed’s longer-run projection of 4%5.
The Fed intends to raise the policy rate expeditiously until the inflation rate heads back to thetarget, 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 andthe Fed’s handling of the policy rate setting seems more and more challenging. It would be
beneficial if the Fed / government owns a certain instrument which guides an appropriate
policy path to achieve the policy goals.
We analyze consolidated government (central bank and government)’s optimal policy paththat minimizes social loss function using a dynamic optimization structure. The 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. Under theassumption that policy maker can dynamically control the real output, this paper solves the
dynamic optimization equation and draw an optimal path of the output and inflation
expectation variable, with the analysis of its stability. Similar approach were taken by Asada(2010) and Semmler and Zhang (2004) to derive the optimal 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
Eq(1) is the conventional linear “expectations-augmented Phillips curve”. Variable 𝜋
represents inflation, parameterεis a reaction parameter from the output gap, 𝑌 is the real
output, and 𝑌̅ represents natural output, or real output target. Eq(2) is the dynamic inflation
expectation formation, which combines forward-looking and backward-looking expectations6.

4

Monetary Policy Report (June 17, 2022)

5

June 15, 2022 FOMC Projection materials,
https://www.federalreserve.gov/monetarypolicy/fomcprojtabl20220615.htm

6This paper sets real output Y as a control variable, whereas Asada(2010) puts nominal interest rate as acontrol variable.

4𝜋 = 𝜀(𝑌 − 𝑌̅) + 𝜋 𝑒 ; 𝜀 > 0(1)

𝜋̇ 𝑒 = 𝛼{𝜉(𝜋 − 𝜋 𝑒 ) + (1 − 𝜉)(𝜋̅ − 𝜋 𝑒 )} ; 𝛼 > 0, 0 ≦ 𝜉 ≦ 1(2)Parameter ξ represents the weight of people’s stance on their inflation expectation. If ξ
=1, the equation of motion for 𝜋̇ 𝑒 is dependent on the inflation gap, which here isrepresented by the difference between actual inflation 𝜋 and inflation expectation 𝜋 𝑒 . Thisshows that people have adaptive behavior on inflation. On the other hand, if ξ=0 , theequation turns out to express 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, thenpeople’s behavior on inflation is mixed between forward and backward-looking. The
parameter αis the reaction parameter to the inflation expectation dynamics.
Eq(3) is derived by substituting Eq(1) into Eq(2), and now the equation of motion for 𝜋̇ 𝑒
has real output as one of the variables.
𝜋̇ 𝑒 = 𝛼{𝜉𝜀(𝑌 − 𝑌̅) + (1 − 𝜉)(𝜋̅ − 𝜋 𝑒 )}(3)
Case of ξ=0

Based on the model mentioned above, here, the case of ξ=0 is considered. As previously
explained, this transforms the 𝜋̇ 𝑒 equation into a forward-looking system. Eq(3) now
becomes Eq(4).

𝜋̇ 𝑒 = 𝛼(𝜋̅ − 𝜋 𝑒 ) ; 𝛼 > 0(4)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)7.

(5)

𝑉 = 𝜃(𝑌 − 𝑌̅)2 + (1 − 𝜃)(𝜋 − 𝜋̅)2 = 𝑉(𝑌, 𝜋) ; 0 < 𝜃 < 1

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(5), and in order to minimize the loss function, Eq(5) isturned into negative.

7

Consideration in monetary policy tradeoffs, welfare loss function, and problem solving of optimalmonetary policy are also conducted by Galí (2015) and Woodford(2002).

5

−𝑉 = −[𝜃(𝑌 − 𝑌̅)2 + (1 − 𝜃){𝜀(𝑌 − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}2 ] = 𝑊(𝑌, 𝜋 𝑒 )

(6)

0<𝜃<1
Combining Eq(4) and Eq(6) gives the following dynamic optimization problem subject
to the constraint equation.
∞

max ∫0 𝑊(𝑌, 𝜋 𝑒 )𝑒 −𝜌𝑡 𝑑𝑡(7)

𝑌𝑠. 𝑡. 𝜋̇ 𝑒 = 𝛼(𝜋̅ − 𝜋 𝑒 )In order to derive the optimal macroeconomic policy path, a current-value Hamiltonian
equation is formalized as below, where 𝜆 is the co-state variable.
𝐻 = −[𝜃(𝑌 − 𝑌̅)2 + (1 − 𝜃){𝜀(𝑌 − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}2 ] + 𝜆𝛼(𝜋̅ − 𝜋 𝑒 )(8)As explained in Chiang (1992), Pontryagin’s maximum principle conditions are given as
follows8.

(9)

Max 𝐻(𝑌, 𝜋 𝑒 , 𝜆) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ∈ [0, ∞]𝑌𝜋̇ 𝑒 =

𝜕𝐻𝜕𝜆[𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝜋̇ 𝑒 ]

𝜕𝐻
+ 𝜌𝜆
𝜕𝜋 𝑒

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

𝜆̇ = −𝑡→∞This Hamiltonian system here is solved with respect to the real output, Y. This implies thatpolicy maker is controlling real output to minimize (optimize) the social loss function
previously proposed.
A first order condition is required to show that the optimal control of Y will be an interior
solution.
(10)

8

𝜕𝐻
𝜕𝑌

= −[2𝜃(𝑌 − 𝑌̅) + 2𝜀(1 − 𝜃){𝜀(𝑌 − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}] = 0

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

from Chiang and Wainwright (2010).

6

Further differentiation of Eq(10) with the result of negative shows that the control variable
Y does maximize the Hamiltonian system.
𝜕2 𝐻

(11)

𝜕𝑌 2

= −[2𝜃 + 2𝜀 2 (1 − 𝜃)]= −2[𝜃 + 𝜀 2 (1 − 𝜃)] < 0Equation of motion for 𝜋̇ 𝑒 is defined as below.
𝜋̇ 𝑒 =

(12)

𝜕𝐻
𝜕𝜆

= 𝛼(𝜋̅ − 𝜋 𝑒 ) = 𝐹11 (𝜋 𝑒 )Equation of motion for 𝜆, which is a costate variable, is described below.

(13)

𝜕𝐻
𝜆̇ = − 𝑒 + 𝜌𝜆 = [2(1 − 𝜃){𝜀(𝑌 − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}] + 𝜆(𝛼 + 𝜌)𝜕𝜋Real output Y of Eq(10) can be transformed as below.

(14)

𝑌=

̅ −𝜋𝑒 )
𝑌̅{𝜀 2 (1−𝜃)+𝜃}+𝜀(1−𝜃)(𝜋= 𝑌(𝜋 𝑒 )𝜃+𝜀 2 (1−𝜃)Eq(14) can be substituted into Eq(13) as below.
(15)𝜆̇ = [2(1 − 𝜃){𝜀(𝑌(𝜋 𝑒 ) − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}] + 𝜆(𝛼 + 𝜌) = 𝐹21 (𝜋 𝑒 , 𝜆)Phase diagram in case of 𝝃 = 𝟎
From the equilibrium point of 𝜋̇ 𝑒 in Eq(12), the solution 𝜋̅ = 𝜋 𝑒 can be derived. Then,
we can derive Figure 1. It is assumed here that the policy maker would indirectly maneuver
the costate variable of 𝜆 by controlling the real output Y, which can be termed as a marginalincrease (or decrease) in inflation expectation,

7

̇ =𝟎

̇ <𝟎

̇ >𝟎

𝟎

̅

Figure 1̇Assuming equilibrium 𝜆 = 0 in Eq(13), equation of 𝜆 can be derived as below.
(16)
𝜆=

Since

𝑑𝑌
𝑑𝜋 𝑒

−2(1 − 𝜃){𝜀(𝑌(𝜋 𝑒 ) − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}
𝛼+𝜌=−𝜀(1−𝜃)
𝜃+𝜀2 (1−𝜃)<0 ,We have the following relationship from Eq(16).
𝑑𝑌−2(1 − 𝜃) {𝜀 𝑒 + 1}
𝑑𝜆
𝑑𝜋
=
<0
|
𝑑𝜋 𝑒 𝜆=0𝛼+𝜌̇Therefore, we obtain Figure 2, since

𝜕𝜆̇
𝜕𝜋𝑒

> 0 from Eq(15).

8

̇ >𝟎
̇ <𝟎

̇ =𝟎
𝟎
Figure 2
By combining Figure 1 and Figure 2, the phase diagram is complete.

̇ =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 which enables policy maker to achieve full employment and the inflation target
goal, and this unique path toward the equilibrium is the saddle path. Only the saddle pathsatisfies the transversality condition in Eq(9). From a given initial state variable 𝜋 𝑒 (0), an
inflation expectation, policy maker needs to select the initial point of 𝜆 and control it
following the saddle path, by indirectly maneuvering Y. Or else, the inflation expectation
9and costate variable 𝜆 would not end up at the equilibrium point of E.
If the starting point of inflation expectation exceeds the inflation target, the policy maker
would have to raise Y to lower inflation expectation while increasing the costate variable, 𝜆,𝑑𝑌

since 𝑑𝜋𝑒 < 0. On the other hand, if the initial point of inflation expectation is below thetarget, Y would have to decrease in order to lower 𝜆. This is quite a paradoxical result, sincein general, output and inflation expectation have a positive correlation, and increase in
output would lead to a further rise in inflation expectation. The only way to achieve thepolicy target is to follow the saddle path, and if any other path is to be chosen, the output
and inflation expectation would spread out indefinitely.
By taking into account Eq(12) and Eq(15), the characteristic equation of this system at theequilibrium point is described below, where 𝜆1 .and 𝜆2 are two characteristic roots9.
(17)|𝜆𝐼 − 𝐽1 | = 𝜆2 − (𝑡𝑟𝑎𝑐𝑒𝐽)𝜆 + (𝑑𝑒𝑡𝐽)= (𝜆 − 𝜆1 )(𝜆 − 𝜆2 ) = 𝜆2 − (𝜆1 + 𝜆2 )𝜆 + 𝜆1 𝜆2 = 0
Where,−𝛼𝐽1 = [ 1
𝐹21
𝐹21 =

0
1 ]
𝐹22

𝑑𝐹2
𝜃
= 2(1 − 𝜃) {
}>0
𝑒
2
𝑑𝜋
𝜃 + 𝜀 (1 − 𝜃)
𝐹22 =

𝑑𝐹2
=𝛼+𝜌 >0𝑑𝜆

𝑡𝑟𝑎𝑐𝑒 𝐽 = 𝜆1 + 𝜆2 = −𝛼 + 𝐹22
𝑑𝑒𝑡𝐽 = 𝜆1 𝜆2 = −𝛼𝐹22 < 0.
This shows that the equilibrium point is a saddle point because the characteristic equation
Eq(17) has one positive real root and one negative real root..
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 real output Y.
9Evaluating the dynamical properties of interest rate and policy tool is also done by Drumon et al.(2021)10𝜋̇ 𝑒 = 𝛼𝜀(𝑌 − 𝑌̅) ; 𝛼 > 0(18)

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

0𝑠. 𝑡. 𝜋̇ 𝑒 = 𝛼𝜀(𝑌 − 𝑌̅)
As similar to Eq(8), a current value Hamiltonian becomes as follows.
(19)

𝐻 = −[𝜃(𝑌 − 𝑌̅)2 + (1 − 𝜃){𝜀(𝑌 − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}2 ] + 𝜆𝛼𝜀(𝑌 − 𝑌̅)

where 𝜆 is the co-state variable.
First order condition and second derivative are described below.

(20)

𝜕𝐻
𝜕𝑌

= −[2𝜃(𝑌 − 𝑌̅) + 2𝜀(1 − 𝜃){𝜀(𝑌 − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}] + 𝜆𝛼𝜀 = 0
𝜕2𝐻= −[2𝜃 + 2𝜀 2 (1 − 𝜃)]𝜕𝑌 2= −2[𝜃 + 𝜀 2 (1 − 𝜃)] < 0Equation of motion for the inflation expectation 𝜋̇ 𝑒 is given below.

(21)𝜋̇ 𝑒 =𝜕𝐻𝜕𝜆= 𝛼𝜀(𝑌 − 𝑌̅) = 𝐹12 (𝑌)From the first order condition in Eq(20), the equation of 𝜆 can be obtained.

(22)

2
1
𝜆 = 𝛼 [(𝑌 − 𝑌̅) {𝜀 𝜃 + 𝜀(1 − 𝜃)} + (1 − 𝜃)(𝜋 𝑒 − 𝜋̅)]Differentiation of Eq(22) with respect to time t gives Eq(23).

(23)

2
1
𝜆̇ = 𝛼 [𝑌̇ {𝜀 𝜃 + 𝜀(1 − 𝜃)} + 𝛼𝜀(1 − 𝜃)(𝑌 − 𝑌̅)]

11

Additionally, from the one of the maximum principal conditions, the motion for 𝜆 is
obtained below.

(24)

𝜕𝐻
𝜆̇ = − 𝜕𝜋𝑒 + 𝜌𝜆 = 2(1 − 𝜃){𝜀(𝑌 − 𝑌̅) + 𝜋 𝑒 − 𝜋̅} + 𝜌𝜆(𝑌, 𝜋 𝑒 )Transversality condition is termed as follows.

lim 𝜆 𝑒 −𝜌𝑡 = 0

(25)

𝑡→∞

By combining Eq(23) and (24) and bringing 𝑌̇ to the left side, the equation of motion forreal output Y can be described as below.
(26)1𝑌̇ = 𝐴 [2(1 − 𝜃)(𝜋 𝑒 − 𝜋̅) + 𝜌𝜆(𝑌, 𝜋 𝑒 )] = 𝐹2 (𝑌, 𝜋 𝑒 )Phase diagram in case of 𝝃 = 𝟏
The locus for 𝜋̇ 𝑒 = 0 is derived from Eq(21), where0 = 𝛼𝜀(𝑌 − 𝑌̅)
The 𝜋̇ 𝑒 = 0 curve becomes horizontal, as sketched below. The level of Y at theequilibrium point, 𝑌̅ is given below.
𝑌 = 𝑌̅

12

̇ >𝟎

̅

̇ =𝟎
̇ <𝟎

𝟎
Figure 4
The 𝑌̇ = 0 line is depicted by solving the total derivation of Eq(26), as shown below.

(27)

𝑑𝑌
𝑑𝜋

∂𝐹2
𝑒

= − ∂𝜋∂𝐹2 < 0𝑒∂𝑌

Each partial derivations are solved as below.

∂𝐹22
1
2
= 𝐹21
= {2(1 − 𝜃) + 𝜌(1 − 𝜃)} > 0
𝑒
∂𝜋
𝐴
∂𝐹22
1 2 1
2
= 𝐹22
= 𝜌 { 𝜃 + 𝜀(1 − 𝜃)} > 0∂𝑌𝐴 𝛼 𝜀From the result of the total derivation, it is clear that 𝑌̇ = 0 line is negatively inclined.

13

̇ >𝟎

̇ <𝟎
̇ =𝟎
𝟎
Figure 5
By combining both lines, the phase diagram is depicted as below. It is now clear that thesystem with ξ=1 also has an unique path which converges into a saddle 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 decrease the output,
𝑌. On the other hand, if the inflation expectation is exceeding the target, the policy maker
now needs to increase the output to suppress the inflation expectation.

Y

E

̅

̇ =𝟎

̇ =𝟎
𝟎

̅
Figure 6

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.
14

0
𝐽2 = [ 2
𝐹21

(28)
2
𝐹21
=

𝛼𝜀
2 ]
𝐹22

1
{2(1 − 𝜃) + 𝜌(1 − 𝜃)} > 0
𝐴

2
𝐹22
=

1 2 1
𝜌 { 𝜃 + 𝜀(1 − 𝜃)} > 0𝐴 𝛼 𝜀𝑑𝑒𝑡𝐽 = 𝜆1 𝜆2 = 0 − 𝛼𝜀𝐹21 < 0
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 ∫0 𝑊(𝑌, 𝜋 𝑒 )𝑒 −𝜌𝑡 𝑑𝑡

(29)

𝑌

𝑠. 𝑡. 𝜋̇ 𝑒 = 𝛼{𝜉𝜀(𝑌 − 𝑌̅) + (1 − 𝜉)(𝜋̅ − 𝜋 𝑒 )}Thus, current-value Hamiltonian is formatted using the new constraint function.
(30)

𝐻 = −[𝜃(𝑌 − 𝑌̅)2 + (1 − 𝜃){𝜀(𝑌 − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}2 ] + 𝜆𝛼{𝜉𝜀(𝑌 − 𝑌̅) + (1 − 𝜉)(𝜋̅ − 𝜋 𝑒 )}First order condition and second derivative become as follows.

(31)

𝜕𝐻
𝜕𝑌

= −[2𝜃(𝑌 − 𝑌̅) + 2𝜀(1 − 𝜃){𝜀(𝑌 − 𝑌̅) + 𝜋 𝑒 − 𝜋̅}] + 𝜆𝛼𝜉𝜀 = 0
𝜕2𝐻= −[2𝜃 + 2𝜀 2 (1 − 𝜃)] < 0
𝜕𝑌 2

We can derive the following expression of 𝜆 by transforming the result from the first
derivative equation above.

(32)

2
1
𝜆 = 𝛼𝜀 [(𝑌 − 𝑌̅) {𝜀 𝜃 + 𝜀(1 − 𝜃)} + (1 − 𝜃)(𝜋 𝑒 − 𝜋̅)] = 𝜆(𝑌, 𝜋 𝑒 )Equation of motion for 𝜋̇ 𝑒 is obtained below.
15

(33)

𝜕𝐻
𝜋̇ 𝑒 = 𝜕𝜆 = 𝛼{𝜉𝜀(𝑌 − 𝑌̅) + (1 − 𝜉)(𝜋̅ − 𝜋 𝑒 )} = 𝐹13 (𝑌, 𝜋 𝑒 )Differentiation of Eq(32) with respect to time gives an equations of 𝜆̇.
(34)

2
1
𝜆̇ = 𝛼𝜉 [𝑌̇ {𝜀 𝜃 + 𝜀(1 − 𝜃)} + 𝛼(1 − 𝜃){𝜉𝜀(𝑌 − 𝑌̅) + (1 − 𝜉)(𝜋̅ − 𝜋 𝑒 )}]Another equation of 𝜆̇, which is part of the maximum principle, is described below.

(35)

𝜕𝐻
𝜆̇ = − 𝑒 + 𝜌𝜆 = 2(1 − 𝜃){𝜀(𝑌 − 𝑌̅) + 𝜋 𝑒 − 𝜋̅} + 𝜆(𝑌, 𝜋 𝑒 ){𝛼(1 − 𝜉) + 𝜌}𝜕𝜋

By combining Eq(34) and (35), and bringing 𝑌̇ to the left side, the equation of motion forreal output Y is available.

(36)

1
1
𝑌̇ = 𝐴 [2(1 − 𝜃) {𝜋 𝑒 − 𝜋̅ − 𝜉 (1 − 𝜉)(𝜋̅ − 𝜋 𝑒 )} + 𝜆(𝑌, 𝜋 𝑒 ){𝛼(1 − 𝜉) + 𝜌}]= 𝐹23 (𝑌, 𝜋 𝑒 )Phase Diagram in case of 𝟎 < 𝝃 < 𝟏
Phase diagram of this system can be depicted from Eq(33) and Eq(36).
It is clear that the slope of 𝜋̇ 𝑒 = 0 line is positive.
(38)

𝑑𝑌

|

𝑑𝜋 𝑒 𝜋̇ =0

=−

𝜕𝐹3
1
𝜕𝜋𝑒
𝜕𝐹3
1
𝜕𝑌

>0

Where,
𝜕𝐹13= 𝐹11 = −𝛼(1 − 𝜉) < 0
𝜕𝜋 𝑒
𝑑𝐹13
= 𝐹12 = 𝛼𝜉𝜀 > 0
𝑑𝑌

16

̇ =𝟎

̇ >𝟎
̇ <𝟎

𝟎
Figure 7
The slope of 𝑌̇ = 0 line is negative, as described below.
𝜕𝐹23
𝑑𝑌
𝑒
= − 𝜕𝜋 3 < 0|𝑒𝑑𝜋 𝑌̇=0
𝜕𝐹2
𝜕𝑌
Where,
𝜕𝐹23
1
1
2
3
= 𝐹21
= [2(1 − 𝜃) {1 − (1 − 𝜉)} (1 − 𝜃){𝛼(1 − 𝜉) + 𝜌}] > 0
𝑒
𝜕𝜋
𝐴
𝜀
𝛼𝜉
𝑑𝐹2
1 2 1
3
= 𝐹22
= [ { 𝜃 + 𝜀(1 − 𝜃)} {𝛼(1 − 𝜀) + 𝜌}] > 0
𝑑𝑌
𝐴 𝛼𝜉 𝜀

̇ >𝟎

̇ <𝟎
̇ =𝟎
𝟎
Figure 817

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.
−𝛼(1 − 𝜉) 𝛼𝜉𝜀𝑑𝑒𝑡𝐽3 = 𝑑𝑒𝑡 [
3
3 ]<0
𝐹21
𝐹22
Where again,
3
𝐹21
=

1
1
2
[2(1 − 𝜃) {1 − (1 − 𝜉)} (1 − 𝜃){𝛼(1 − 𝜉) + 𝜌}] > 0
𝐴
𝜀
𝛼𝜉
3
𝐹22
=

1 2 1
[ { 𝜃 + 𝜀(1 − 𝜃)} {𝛼(1 − 𝜀) + 𝜌}] > 0
𝐴 𝛼𝜉 𝜀As discovered previously in the other cases, the system with the case of 0 < 𝜉 < 1 also
owns a saddle path which converges to the equilibrium point of E.
3. Policy Implications
Here, we consider the policy implication under different scenarios. First is the case whichinflation (inflation expectation) is above the target. In this case, the initial point of real
output 𝑌 would have to be placed below the target, 𝑌̅. From this starting point, theappropriate path converging to the equilibrium point of 𝑌̅ and 𝜋̅ would be to increase thereal output. This initial condition of high inflation and low output is commonly called as
stagflation. The process of increasing the output while lowering inflation rate may be18possible through the increase in investment to enlarge its supply capacity.
The opposite case is when inflation rate is below the target, a situation which is called as a
deflationary economy. Under this scenario, the initial point of real output needs to be above
the target. From there, the appropriate path for the policy maker to reach the inflation target
is to continuously lower the real output. This is somewhat counterintuitive, though one
possible case is when policy maker stimulates the economy at the starting point and locatesthe initial point of real output substantially higher than the target. Inflation generally reacts
with a lag, so the inflation rate could possibly move higher while the real output is
contracting to the equilibrium point.
Our findings imply that the optimal policy solutions for the policy makers are more or lessthe same, regardless of people’s forward/backward-looking stance on their inflation
expectation. As a contrast, although Williams (2003) uses FRB/US macroeconomic model in
its analysis, 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
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.
The implications above are the policy paths which converge to the stable equilibrium point,
but we need to consider here the cases when the policy actions were unsuccessful. 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 the rampant hike of the policy rate, which consequently led to a deep recession
afterwards10. Since then, the Fed’s important objective upon achieving price stability is to
first stabilize people’s inflation expectation close their target, and it has seemingly beensuccessful until the Covid-19 inflation. The opposite case, –deflationary spiral – is relatively

10

Detail on the history of the Fed’s battle against great inflation: Federal Reserve History“The Great Inflation 1965-1982”, (https://www.federalreservehistory.org/essays/greatinflation)19

rare, and its movement of economic variables haven’t been as dramatic. One of the few
examples is Japan, particularly since the beginning of the 1990s. The sluggish economy ofJapan has experienced negative rate of year-on-year core inflation for nearly 20 years, whichshould be long enough to categorize Japan as the country which faced deflationary spiral.
We think that it is reasonable for the policy makers to aim the optimal policy path whichconverges to the target, considering its detrimental consequences of both cases ofinflationary/deflationary spiral. As Bernanke (2002) made an important argument on
Japan’s stagnating economy: “Japan’s deflation problem is real and serious; but, in my view,political constraints, rather than a lack of policy instruments, explain why its deflation has
persisted for as long as it has”, the responsibility is on policy maker’s hands, and they shouldhave the tools and policy space to deal with the problem.
4. Conclusion
In this paper, by applying Hamiltonian equations, several optimal policy paths were derived
under different scenarios of inflation expectation behavior. First was a forward-looking
inflation expectation scenario, second was the backward-looking scenario, and the third was
the mixed behavior of the two. The phase diagrams showed that each system obtains a uniqueconverging path, namely a saddle path, which ultimately reaches a stable and uniqueequilibrium point.
The dynamic relationship between the real output and inflation expectation derived here
implies that if inflation expectation is above the inflation target, the optimal option for policy
maker is to select an initial point of real output that is below the target to position itself at theappropriate trajectory (saddle path). Then the optimal policy is to increase the output andfollow the saddle path until the expected inflation converges to the equilibrium, or the
inflation target. On the other hand, if the expected inflation is below the target, the optimal
policy is to lower the real output by controlling it over the saddle path. In either case, if thepolicy 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 anti-cyclicalmeasures.
Similar antitheoretical observations are pointed by Asada (2010) and Mankiw (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 aparadoxical properties of the “New Keynesian” dynamic model that shows a counterfactual
relationship between inflation and real output. According to Asada(2010), the New Keynsian
20dynamic model implies that “the rate of inflation accelerates whenever the actual output levelis 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.
Lastly, we want to conclude by pointing out some caveats on this paper’s analysis. The model
assumes that policy maker is omniscient and have the capacity to start from wherever they
prefer, which is referred to as a “jump variable” by Asada (2010) and Asada(2013). Also, it
assumes that the real output is controllable so it can follow the optimal path without deviation.
However, this is quite an unrealistic assumption, which similar argument has been made by
Asada(2010). The dynamically optimal solution which is derived in this study should beconsidered as a “reference” or “yardstick” to observe how the real-life macroeconomic policy
is deviating from the optimal path.
In the real-life policy setting, governors can only impact the real output 𝑌 indirectly. A
derived form of the equilibrium condition for the goods market is usually formed as below.

𝑌 = 𝑌(𝑟 − 𝜋 𝑒 , 𝐺);𝜕𝑌<0,𝜕(𝑟 − 𝜋 𝑒 )

𝜕𝑌
>0
𝜕𝐺

Here, 𝑟 =nominal interest rate and 𝐺 =real government expenditure. Normally, nominalinterest rate is controlled by a central bank’s monetary policy, and real government
expenditure is decided by a government’s fiscal policy. As such, the real output 𝑌 is not avariable which policy maker can directly control, instead is an indirect control variable whichcomes from a monetary and fiscal mixed policy. Optimal macroeconomic policy model
which sets these policy tools as control variables may be the next step for the future study.

21

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24

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