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Robustly Optimal Monetary Policy in a Behavioral Environment

  • Lahcen Bounader EMAIL logo and Guido Traficante
Published/Copyright: September 13, 2022
The B.E. Journal of Macroeconomics
From the journal The B.E. Journal of Macroeconomics Volume 23 Issue 1

Abstract

This paper studies robustly optimal monetary policy with myopic agents in a behavioral New Keynesian model. The central bank is assumed to have Knightian uncertainty about the degree of myopia and the degree of price stickiness. We show that, under uncertainty about myopia, the Brainard’s attenuation principle holds under commitment and discretion, while, under uncertainty on price stickiness, alone or in addition to myopia, monetary policy becomes more aggressive. Welfare evaluation shows significant gains from a robust conduct of monetary policy when the model is distorted.

Keywords: optimal monetary policy; bounded rationality; min-max; parameter uncertainty
JEL Classification: E31; E52
Corresponding author: Lahcen Bounader, International Monetary Fund, Washington D.C., USA, E-mail:

Acknowledgements

We thank the editor Arpad Abraham, two anonymous referees, Jonathan Benchimol, Pompeo Della Posta, Domenico Delli Gatti, Thomas Lubik, Andrea Venegoni, all the participants at the WEAI Conference, the WEHIA October 2021 workshop, the 62nd Annual Conference of the Italian Economic Association, and at the ASSA 2022 Virtual Annual Meeting. Financial support from the European University of Rome is gratefully acknowledged. This paper does not represent the views of the IMF, its executive Board, or its Management. Any remaining errors are our own.

  1. Research funding: This work was supported by European University of Rome.

Appendix A: Phillips Curve Derivations

The problem of the behavioral firm is then to maximize

(32) ∑ k = 0 ∞ θ k E t BR Λ t , t + k P t * Y t + k | t − Ψ t + k Y t + k | t

subject to the sequence of demand constraints

(33) Y t + k | t = P t * P t + k − ε Y t + k

where Λ t , t + k = β k C t + k / C t − γ P t + k / P t is the stochastic discount factor in nominal terms, Ψ t + k . is the cost function, and Yt+k|t denotes the output in period t + k for a firm that last reset its price in period t.

The FOC of the problem is the following

(34) ∑ k = 0 ∞ θ k E t BR Λ t , t + k Y t + k | t P t * − M ψ t + k | t = 0

where M = ε ε − 1 is the desired or frictionless markup.

By dividing Eq. (34) by Pt−1 and defining Π t , t + k = P t + k P t and M C t + k | t = ψ t + k | t P t + k , we obtain the following

(35) ∑ k = 0 ∞ θ k E t BR Λ t , t + k Y t + k | t P t * P t − 1 − M M C t + k | t Π t − 1 , t + k = 0

We define the steady-state of Λt,t+k as β k , Yt+k|t as Y, P t * P t − 1 as 1, MCt+k|t as 1 M , and Πt−1,t+k as 1. These defined steady-states allow us to expand the FOC Eq. (35) as follows

(36) ∑ k = 0 ∞ β θ k E t BR p t * − p t − 1 − m c ̂ t + k | t + p t + k − p t − 1 = 0

with small letters denoting the logarithm of capital letters p t = ln P t and hat indicating the deviation with respect to the steady-state m c ̂ t + k | t = m c t + k | t − m c , where mct+k|t = ln MCt+k|t, and mc = −μ, where μ = ln M .

By simplifying Eq. (36) we obtain

(37) p t * − p t − 1 = 1 − β θ ∑ k = 0 ∞ β θ k E t BR m c ̂ t + k | t + p t + k − p t − 1

By rearranging the terms of Eq. (37), we obtain

(38) p t * = μ + 1 − β θ ∑ k = 0 ∞ β θ k E t BR m c t + k | t + p t + k

The (log) marginal cost can be expressed as

(39) m c t + k | t = m c t + k

We replace Eq. (39) in Eq. (37) and find

(40) p t * − p t − 1 = 1 − β θ ∑ k = 0 ∞ β θ k E t BR m c ̂ t + k + p t + k − p t − 1

Rearranging terms leads to the following expression

(41) p t * − p t − 1 = 1 − β θ ∑ k = 0 ∞ β θ k E t BR m c ̂ t + k + p t + k − p t − 1

We apply the transition from bounded rationality expectations to objective expectations following Eq. (4)

(42) p t * − p t − 1 = 1 − β θ ∑ k = 0 ∞ β θ m ̄ k E t m c ̂ t + k + p t + k − p t − 1

By writing this equation as a difference equation, we find

(43) p t * − p t − 1 = β θ m ̄ E t p t + 1 * − p t + 1 − β θ m c ̂ t + π t

We combine Eq. (43) with π t = 1 − θ p t * − p t − 1 and

(44) m c ̂ t = γ + ϕ y t − y t n

to obtain the Phillips curve Eq. (9).

Appendix B: Proofs

B.1 Proposition 1

The evil agent’s problem is the following

(45) max m ̄ 1 + κ ϵ 1 + κ ϵ − β M f ρ u 2 σ u 2

We compute the FOC with respect to m ̄ obtaining:

(46) ∂ L ∂ m ̄ = − 2 ( 1 + κ ϵ ) − β ρ u ∂ M f ∂ m ̄ 1 + κ ϵ − β M f ρ u 3 σ u 2

where

(47) ∂ M f ∂ m ̄ = θ + ( 1 − β θ ) ( 1 − θ ) 1 − β θ m ̄ + β θ m ̄ ( 1 − β θ ) ( 1 − θ ) ( 1 − β θ m ̄ ) 2

It is easy to verify that ∂ M f ∂ m ̄ > 0 . Moreover, since the denominator is surely positive (βM f ρ u is the product of three positive parameters less than one) we can conclude ∂ L ∂ m ̄ > 0 .

B.2 Proposition 2

The evil agent’s problem in this case is the following

(48) max θ 1 + κ ϵ 1 + κ ϵ − β M f ρ u 2 σ u 2

The parameter θ enters in coefficient κ, hence the derivative of the loss function (17) with respect to θ will be

(49) ∂ L ∂ θ = − 2 1 + κ ϵ ϵ ∂ κ ∂ θ − β ρ u ∂ M f ∂ θ 1 − β M f ρ u + κ ϵ 3 + 1 + ϵ ∂ κ ∂ θ 1 − β M f ρ u + κ ϵ 2 σ u 2

which can be simplified as follows

(50) ∂ L ∂ θ = − 1 + β M f ρ u + κ ϵ ϵ ∂ κ ∂ θ + 2 1 + κ ϵ β ρ u ∂ M f ∂ θ + 1 − β M f ρ u + κ ϵ 1 − β M f ρ u + κ ϵ 3 σ u 2

Now we write down the expression for derivatives inside

(51) ∂ κ ∂ θ = β θ 2 − 1 θ 2 ( γ + ϕ ) < 0

and,

(52) ∂ M f ∂ θ = 1 + m ̄ β m ̄ − ( β θ ) 2 m ̄ + 2 β θ − β − 1 1 − β θ m ̄ 2

By working on (52), we can show that the derivative is positive if

β θ 2 m ̄ − 2 θ + 1 > 0

which occurs for values of θ < 2 − 4 − 4 β m ̄ 2 β m ̄ , while the other analytical condition that implies a positive sign for the inequation, i.e. θ > 2 + 4 − 4 β m ̄ 2 β m ̄ is ruled out because this second root is always larger than one.

Therefore, in this case, the sign of the loss function derivative is unclear. The whole denominator should be evaluated. By replacing (51) and (52) in (66) and simplifying, we obtain

∂ L ∂ θ ∝ − 1 + β M f ρ u + κ ϵ ϵ β θ 2 − 1 γ + ϕ 1 − β θ m ̄ 2 + 2 1 + κ ϵ β ρ u m ̄ θ 2 β m ̄ − ( β θ ) 2 m ̄ + 2 β θ − β − 1 + 1 − β M f ρ u + κ ϵ θ 2 1 − β θ m ̄ 2

Evaluating each term of the above derivative reveals that ∂ L ∂ θ > 0 . Thus, we obtain the proposition.

B.3 Proposition 4

The policy problem of the central bank in this setup is the following

(53) ∑ t = 0 ∞ β t π t 2 + ϑ x t 2

subject to the Phillips curve Eq. (9).

As in the main text, the FOCs of this problem could be expressed as a targeting rule in a single equation

(54) π t = − ϑ κ x t + ϑ M f κ x t − 1

We replace π t in the Phillips curve Eq. (9), which yields the following difference equation

(55) − ϑ κ + β ϑ κ ( M f ) 2 + κ x t = − ϑ κ M f x t − 1 − β M f ϑ κ E t x t + 1 + u t

By changing the notation to α = − M f ϑ 2 1 + β ( M f ) 2 + ϑ κ 2 and rearranging, this can be written as

(56) x t = α x t − 1 + α β E t x t + 1 − α κ ϑ u t

The stationary solution of this equation is

(57) x t = ψ x t − 1 − κ ψ ϑ ( 1 − β ψ ρ u ) u t

where ψ = 1 − 1 − 4 β α 2 2 α β .

Plaguing this expression to the Phillips curve yields

(58) π t = ψ π t − 1 + ψ 1 − β ψ ρ u ( u t − u t − 1 )

Solving the IS Eq. (5) for the interest rate, we obtain

i t = − 1 σ ( x t − M E t x t + 1 ) + E t π t + 1 + r t e

We now replace the expressions for x and π Eqs. (57) and (58) in the interest rate equation above, which completes the proof of this proposition.

Appendix C: Alternative Specification of the Phillips Curve

In this Appendix, we derive the paper’s results using an alternative specification of the Phillips Curve (PC) following Benchimol and Bounader (2019). The PC is specified as follows

(59) π t = β M f E t π t + 1 + κ x t + u t

with M f = m ̄ and κ is defined as before.

The loss function of the central bank is defined as earlier, with the only exception that M f = m ̄ . Applying the first-order conditions we arrive, as in the main text, at the following expression where M f takes the value defined in (59):

(60) L = 1 + κ ϵ 1 − β M f ρ u + κ ϵ 2 σ u 2

The robustly optimal monetary policy is derived by considering the worst-case scenario of the loss function with respect to uncertain parameters

Proposition 5

If the policy maker is uncertain about:

  1. m ̄ , a robust policy should be based on m ̄ = m ̄ max .

  2. θ, a robust policy should be based on θ =θmax.

  3. m ̄ and θ, jointly, a robust policy should be based on m ̄ = m ̄ max and θ =θmax.

Proof

The evil agent’s problem is the following when there is uncertainty about m ̄

(61) max m ̄ 1 + κ ϵ 1 + κ ϵ − β M f ρ u 2 σ u 2

We compute the FOC with respect to m ̄ obtaining:

(62) ∂ L ∂ m ̄ = − 2 ( 1 + κ ϵ ) − β ρ u ∂ M f ∂ m ̄ 1 + κ ϵ − β M f ρ u 3 σ u 2

where

(63) ∂ M f ∂ m ̄ = 1 > 0

Since the denominator is surely positive (βM f ρ u is the product of three positive parameters less than one), we can conclude ∂ L ∂ m ̄ > 0 .

The evil agent’s problem is the following when there is uncertainty about θ

(64) max θ 1 + κ ϵ 1 + κ ϵ − β M f ρ u 2 σ u 2
(65) ∂ L ∂ θ = − 2 1 + κ ϵ ϵ ∂ κ ∂ θ − β ρ u ∂ M f ∂ θ 1 − β M f ρ u + κ ϵ 3 + 1 + ϵ ∂ κ ∂ θ 1 − β M f ρ u + κ ϵ 2 σ u 2

which can be simplified, given that ∂ M f ∂ θ = 0 , as follows

(66) ∂ L ∂ θ = − 1 + β M f ρ u + κ ϵ ϵ ∂ κ ∂ θ + 1 − β M f ρ u + κ ϵ 1 − β M f ρ u + κ ϵ 3 σ u 2

Now we write down the expression for derivatives inside

(67) ∂ κ ∂ θ = β θ 2 − 1 θ 2 ( γ + ϕ ) < 0

Since ∂ κ ∂ θ < 0 , it is easy to check that ∂ L ∂ θ > 0 , an thus θ =θmax.

The proof of the last bullet is straightforward given ∂ L ∂ θ > 0 and ∂ L ∂ m ̄ > 0 .

Determinacy region under robustness Under passive monetary policy, the equilibrium is determinate if the below condition is satisfied

(68) 1 − β M f ( 1 − M ) κ σ > 1

Under uncertainty with respect to θ, we differentiate the determinacy condition with respect to θ

(69) − 1 − β m ̄ κ 2 σ β κ ∂ M f ∂ θ ︸ = 0 + 1 − β M f ∂ κ ∂ θ ︸ < 0

Uncertainty on price rigidity increases the determinacy area for any policy that does not respond at all to inflation. This result is quite straightforward under the current specification than the one derived in the core of the paper, since the derivative is undoubtedly positive.

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Received: 2022-01-27
Accepted: 2022-08-28
Published Online: 2022-09-13

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Advances
  3. A New General Equilibrium Welfare Measure, with Application to Labor Income Taxes
  4. Labor Share Dynamics and Factor Complementarity
  5. Effect of Monetary Policy on Government Spending Multiplier
  6. News-Driven Housing Booms: Spain Versus Germany
  7. Sovereign Debt Crisis, Fiscal Consolidation, and Active Central Bankers in a Monetary Union
  8. Housing Taxation and Economic Growth: Analysis of a Balanced-Growth Model with Residential Capital
  9. Intergenerational Linkages, Uncertain Lifetime and Educational and Health Expenditures
  10. Contributions
  11. Tolerance of Informality and Occupational Choices in a Large Informal Sector Economy
  12. Uncertainty Shocks, Innovation, and Productivity
  13. Asymmetric Effects of Private Debt on Income Growth
  14. Interpreting Structural Shocks and Assessing Their Historical Importance
  15. Charge-offs, Defaults and the Financial Accelerator
  16. Filtering Persistent and Asymmetric Cycles
  17. Population Aging and Convergence of Household Credit
  18. Robustly Optimal Monetary Policy in a Behavioral Environment
  19. Forward Guidance Effectiveness in a New Keynesian Model with Housing Frictions
  20. The Welfare Effects of Social Insurance Reform in the Presence of Intergenerational Transfers
https://doi.org/10.1515/bejm-2022-0031
Keywords for this article
optimal monetary policy; bounded rationality; min-max; parameter uncertainty

Articles in the same Issue

  1. Frontmatter
  2. Advances
  3. A New General Equilibrium Welfare Measure, with Application to Labor Income Taxes
  4. Labor Share Dynamics and Factor Complementarity
  5. Effect of Monetary Policy on Government Spending Multiplier
  6. News-Driven Housing Booms: Spain Versus Germany
  7. Sovereign Debt Crisis, Fiscal Consolidation, and Active Central Bankers in a Monetary Union
  8. Housing Taxation and Economic Growth: Analysis of a Balanced-Growth Model with Residential Capital
  9. Intergenerational Linkages, Uncertain Lifetime and Educational and Health Expenditures
  10. Contributions
  11. Tolerance of Informality and Occupational Choices in a Large Informal Sector Economy
  12. Uncertainty Shocks, Innovation, and Productivity
  13. Asymmetric Effects of Private Debt on Income Growth
  14. Interpreting Structural Shocks and Assessing Their Historical Importance
  15. Charge-offs, Defaults and the Financial Accelerator
  16. Filtering Persistent and Asymmetric Cycles
  17. Population Aging and Convergence of Household Credit
  18. Robustly Optimal Monetary Policy in a Behavioral Environment
  19. Forward Guidance Effectiveness in a New Keynesian Model with Housing Frictions
  20. The Welfare Effects of Social Insurance Reform in the Presence of Intergenerational Transfers
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