A Promised Value Approach to Optimal Monetary Policy*
| DOI | http://doi.org/10.1111/obes.12401 |
| Author | Taisuke Nakata,Takeki Sunakawa,Timothy Hills |
| Published date | 01 February 2021 |
| Date | 01 February 2021 |
176
©2020 TheAuthors. OxfordBulletin of Economics and Statistics published by Oxford University and John Wiley & Sons Ltd.
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OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 83, 1 (2021) 0305–9049
doi: 10.1111/obes.12401
A PromisedValue Approach to Optimal Monetary
Policy*
Timothy Hills,†Taisuke Nakata‡ and Takeki Sunakawa§
†Stern School of Business, New York University, New York, 10003, USA (e-mail: hills.
timoteo@gmail.com)
‡Faculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo City, Tokyo, 113-0033,
Japan (e-mail: taisuke.nakata@e.u-tokyo.ac.jp)
§Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan (e-mail: takeki.
sunakawa@gmail.com)
Abstract
This paper characterizes optimal commitment policy in the New Keynesian model using a
recursive formulation of the central bank’s infinite-horizon optimization problem in which
promised inflation and output gap – as opposed to lagged Lagrange multipliers – act as
pseudo-state variables. Our recursiveformulation is motivated by (Kydland,F. and Prescott,
E. C. (1980). Journal of Economic Dynamics and Control Vol. 2, pp. 79–91). Using three
well-known variantsof the model – one featuring inflation bias, one featuring stabilization
bias and one featuring a lowerbound constraint on nominal interest rates – we show that the
proposed formulation sheds new light on the nature of the intertemporal trade-off facing
the central bank.
I. Introduction
Optimal commitment policy is a widely adopted approach among economists and policy-
makers to studying the question of how to best conduct monetary policy. For example, at
the Federal Reserve, the results of optimal commitment policy analysis from the FRB/US
model have for some time been regularlypresented to the Federal Open Market Committee
to help inform its policy decisions (Brayton, Laubach and Reifschneider, 2014). Most re-
cently,in many advanced economies where the policy rate was constrained at the effective
lower bound (ELB), the insights from the optimal commitment policy in a stylized New
Keynesian model have played a key role in the inquiry on how long the policy rate should
be kept at the ELB (Bullard (2013), Evans (2013), Kocherlakota (2011), Plosser (2013),
Woodford (2012)).Accordingly, a deep understanding of optimal commitment policy is as
relevant as ever.
JEL Classification numbers: E32, E52, E61, E62, E63.
*Wethank Thomas Sargent for introducing us to the promised value approach to solving the Ramsey problem. We
also thank Editor, two anonymous referees, seminar participants atTohoku University, and Yuichiro Wakifor useful
comments. Satoshi Hoshino and Donna Lormand provided excellent research and editorial assistance respectively.
A promised value approach 177
In this paper, we contribute to a better understanding of optimal commitment policy
in the New Keynesian model–aworkhorse model for analysing monetary policy – by
characterizing it using a novel recursive method. Our method uses promised values of
inflation and output as pseudo-state variables in the spirit of Kydland and Prescott (1980)
instead of lagged Lagrange multipliers as in the standard method of Marcet and Marimon
(2019). We describe our recursive approach – which we will refer to as the promised value
approach – in three variants of the New Keynesian model that have been widely studied
in the literature: the model with inflation bias, the model with stabilization bias and the
model with an ELB constraint. In each model, we define the infinite-horizon problem of
the Ramsey planner, provide the recursive formulations of the Ramsey planner’s problem
via the promised value approach, and describe the trade-off facing the central bank in
determining the optimal commitment policy.1
The idea of using promised values as pseudo-state variables to recursify the infinite-
horizon problem of the Ramsey planner was first suggested byKydland and Prescott (1980)
in the context of an optimal capital taxation problem. Later, Chang (1998) and Phelan
and Stacchetti (2001) formally described, as an intermediate step towards characterizing
sustainable policies, the recursive formulation of the Ramsey planner’s problem using
promised marginal utility in models with money and with fiscal policy respectively. How-
ever, because their focus was on characterizing sustainable policies, they did not solve for
the Ramsey policy. To our knowledge, we are the first to formulate and solve the Ramsey
policy using the promised value approach.2
Our aim is not to argue that readers should use the promised valueapproach instead of the
Lagrange multiplier approach. Rather, our aim is to show that the promised value approach
can be a useful analytical tool to supplement the analysis based on the standard Lagrange
multiplier approach. Both approaches should be able to find the same allocation; we indeed
find that both approaches reliably compute the optimal commitment policies in the New
Keynesian model. However, the Ramsey policies are often history dependent in complex
ways, and it is not always straightforward for researchers to understand the trade-off facing
the central bank. Accordingly, it is useful for researchers to have an alternative way to
analyse the Ramsey policy, as it may provide new insights on the optimal commitment
policy.3
One difficulty associated with the promised valueapproach is that it requires researchers
to compute the set of feasible promised values (see discussion in Marcet and Marimon
(2019)). We find that the extent to which this computation poses a challenge depends on
the model. For the model with inflation bias and the model with stabilization bias, the
promised rate of inflation is the only pseudo-state variable, and we analytically show that
the set of feasible promised inflation rates are identical to the set of feasibleactual inflation
1In the online appendix, we also analyse the model with the ELB and quantitative easing.
2The only exception is a recent lecture note by Sargent and Stachurski (2018) which characterizes the Ramsey
policy in the linear-quadratic version of the model of Cagan (1956) using the promised value approach. Note that,
while the Lagrange multiplier approach is almost always used in solving the Ramsey problems in business cycle
models, the promised value approach is extensivelyused in the literature of dynamic contract.
3In simple models such as the model with inflation bias considered in the next section, one can easily reverse-
engineer policy functions associated with the promised value approach from those associated with the Lagrange
multiplier approach. However, in more complicated setup, such reverse engineering is either very complicated or
impossible without some numerical methods.
©2020 The Authors. Oxford Bulletin of Economics and Statistics published by Oxford University and JohnWiley & Sons Ltd.
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