Welfare cost of inflation, when credit card transaction services are included among monetary services

William A. Barnett; Sohee Park

doi:10.1515/snde-2022-0092

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Welfare cost of inflation, when credit card transaction services are included among monetary services

William A. Barnett and Sohee Park

Published/Copyright: May 29, 2023

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 28 Issue 3

Abstract

We investigate the welfare cost of anticipated inflation, when the volume of credit card transactions is included in measured monetary service flows. We use the credit-card-augmented Divisia monetary aggregates in a nonlinear dynamic stochastic general equilibrium (DSGE) New Keynesian model and calculate the welfare costs of inflation. The welfare costs of inflation with credit card services included are greater than without them in the New Keynesian DSGE model. Because of the complexity of the model’s dynamical structure, we are not aware of a simple explanation for the increased welfare sensitivity to inflation.

Keywords: credit-card-augmented Divisia; Divisia; inflation; monetary aggregates; nonlinear dynamics; welfare cost

JEL Codes: E31; E41; E51; E52

Corresponding author: Sohee Park, Department of Economics, Valparaiso University, Valparaiso, IN 46383, USA, E-mail: shpark0116@gmail.com

We thank the participants at the Sixth International Workshop on Financial Markets and Nonlinear Dynamics, Paris, June 2022.

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Household optimization

In Section 2.1, the representative household maximizes the utility function (2) subject to the constraints (3), (5), and (7). The Lagrangian for the problem can be written as

L t = E t ∑ t = 0 ∞ β t ζ ⁡ ln C t + η ⁡ ln ℓ t + θ ⁡ ln M t cc P t + Λ 1 , t ν 1 ω N t P t ω − 1 ω + ϕ 1 ω D t P t ω − 1 ω + ( 1 − ν − ϕ ) 1 ω C C t P t ω − 1 ω ω ω − 1 − M t cc P t + Λ 2 , t M t − 1 + B t − 1 + τ t − B t / r t − N t + L t + C C t P t + ∫ 0 1 Q i , t P t ( s i , t − 1 − s i , t ) d i − D t P t + Λ 3 , t N t + r t D D t + W t ( 1 − ℓ t ) P t + ∫ 0 1 F i , t P t s i , t d i − C t − r t L L t + e t C C t + M t P t .

The first-order conditions with respect to B _t, C _t, L _t, M _t, ℓ _t, M t cc , s _i,t, N _t, D _t, and CC_t are as follows, where Λ_1,t, Λ_2,t, and Λ_3,t denote the Lagrange multipliers on the constraints).

(40) Λ 2 , t P t r t = β E t Λ 2 , t + 1 P t + 1

(41) Λ 3 , t = ζ C t

(42) Λ 2 , t = r t L Λ 3 , t

(43) Λ 3 , t P t = β E t Λ 2 , t + 1 P t + 1

(44) Λ 3 , t W t P t = η ℓ t

(45) Λ 1 , t M t cc P t = θ

(46) Λ 2 , t Q i , t P t − Λ 3 , t F i , t P t = β E t Λ 2 , t + 1 Q i , t + 1 P t + 1

(47) Λ 2 , t − Λ 3 , t = Λ 1 , t ν 1 ω N t P t ω − 1 ω + ϕ 1 ω D t P t ω − 1 ω + ( 1 − ν − ϕ ) 1 ω C C t P t ω − 1 ω 1 ω − 1 ν 1 ω N t P t − 1 ω

(48) Λ 2 , t − r t D Λ 3 , t = Λ 1 , t ν 1 ω N t P t ω − 1 ω + ϕ 1 ω D t P t ω − 1 ω + ( 1 − ν − ϕ ) 1 ω C C t P t ω − 1 ω 1 ω − 1 ϕ 1 ω D t P t − 1 ω

(49) e t Λ 3 , t − Λ 2 , t = Λ 1 , t ν 1 ω N t P t ω − 1 ω + ϕ 1 ω D t P t ω − 1 ω + ( 1 − ν − ϕ ) 1 ω C C t P t ω − 1 ω 1 ω − 1 × ( 1 − ν − ϕ ) 1 ω C C t P t − 1 ω .

Appendix B: The stationary system

Since most variables will be nonstationary in Section 2, we transform the system to be stationary by defining the new set of variables c _t = C _t/Z _t−1, y _t = Y _t/Z _t−1, y t * = Y t * / Z t − 1 , f _t = (F _t/P _t)/Z _t−1, λ _1,t = Z _t−1Λ_1,t, λ _2,t = Z _t−1Λ_2,t, λ _3,t = Z _t−1Λ_3,t, m _t = (M _t/P _t)/Z _t−1, m t cc = M t cc / P t / Z t − 1 , n _t = (N _t/P _t)/Z _t−1, n t υ = N t υ / P t / Z t − 1 , l _t = (L _t/P _t)/Z _t−1, d _t = (D _t/P _t)/Z _t−1, cc_t = (CC _t/P _t)/Z _t−1, w _t = (W _t/P _t)/Z _t−1, z _t = Z _t/Z _t−1, π _t = P _t/P _t−1, q _t = (Q _t/P _t)/Z _t−1.

We get Equations (50)–(89) in terms of the transformed new variables:

(50) m t cc = ν 1 ω n t ω − 1 ω + ϕ 1 ω d t ω − 1 ω + ( 1 − ν − ϕ ) 1 ω cc t ω − 1 ω ω ω − 1

(51) m t − n t + l t + c c t = d t

(52) n t + r t D d t + w t h t + f t = c t + r t L l t + e t c c t + m t

(53) y t = z t h t

(54) ln z t = ln ⁡ z + ε z , t

(55) f t = t t − w t h t z t − γ 2 π t π − 1 2 y t

(56) ( 1 − σ ) λ 3 , t y t + σ λ 3 , t w t y t z t − γ λ 3 , t π t π − 1 y t π t π + β γ E t λ 3 , t + 1 π t + 1 π − 1 y t + 1 π t + 1 π = 0

(57) y t * = ζ η z t ( 1 − h t )

(58) g t * = η y t ζ z t ( 1 − h t )

(59) l t = ( 1 − k t ) d t

(60) ln k t = ( 1 − ρ k ) ln ⁡ k + ρ k ⁡ ln k t − 1 + ε k , t

(61) c c t = a t d t

(62) ln a t = ( 1 − ρ a ) ln ⁡ a + ρ a ⁡ ln a t − 1 + ε a , t

(63) ln x t = ( 1 − ρ x ) ln ⁡ x + ρ x ⁡ ln x t − 1 + ε x , t

(64) r t D = r t L − 1 ( 1 − k t ) + ( e t − x t − 1 ) a t − x t + 1

(65) m t = n t + n t υ

(66) n t υ = k t d t

(67) g t y y t − 1 = y t z t − 1

(68) r t r = r t − 1 r ρ r π t π ρ π g t * g * ρ g * g t y g y ρ g y ⁡ exp ( ε r , t )

(69) m t sim = n t + d t

(70) u t N = 1 − 1 r t

(71) u t D = 1 − r t D r t

(72) u t cc = e t r t − 1

(73) s t N = u t N n t u t N n t + u t D d t + u t cc c c t

(74) s t D = u t D d t u t N n t + u t D d t + u t cc c c t

(75) s t cc = u t cc c c t u t N n t + u t D d t + u t cc c c t

(76) μ t C Div = μ t N s t N + s t − 1 N / 2 μ t D s t D + s t − 1 D / 2 μ t c c s t cc + s t − 1 c c / 2

(77) n t − 1 μ t N = n t z t − 1 π t

(78) d t − 1 μ t D = d t z t − 1 π t

(79) c c t − 1 μ t cc = c c t z t − 1 π t

(80) z t λ 2 , t = β r t E t λ 2 , t + 1 π t + 1

(81) λ 3 , t = ζ c t

(82) λ 2 , t = r t L λ 3 , t

(83) z t λ 3 , t = β E t λ 2 , t + 1 π t + 1

(84) λ 3 , t w t = η ℓ t

(85) λ 1 , t m t cc = θ

(86) λ 2 , t q t − λ 3 , t f t = β E t λ 2 , t + 1 q t + 1

(87) λ 2 , t − λ 3 , t = λ 1 , t m t c c 1 ω ν 1 ω n t − 1 ω

(88) λ 2 , t − r t D λ 3 , t = λ 1 , t m t c c 1 ω ϕ 1 ω d t − 1 ω

(89) e t λ 3 , t − λ 2 , t = λ 1 , t m t c c 1 ω ( 1 − ν − ϕ ) 1 ω c c t − 1 ω

(90) ℓ t + h t = 1 .

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2022-0092).

Received: 2022-10-11

Accepted: 2023-05-05

Published Online: 2023-05-29

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Keywords for this article

credit-card-augmented Divisia; Divisia; inflation; monetary aggregates; nonlinear dynamics; welfare cost