Optimality of a Linear Decision Rule in Discrete Time AK Model

Myungkyu Shim

doi:10.1515/bejte-2021-0061

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Optimality of a Linear Decision Rule in Discrete Time AK Model

Myungkyu Shim

Published/Copyright: November 24, 2021

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From the journal The B.E. Journal of Theoretical Economics Volume 23 Issue 1

Abstract

Surprisingly, formal proof on the optimality of a linear decision rule in the discrete time AK model with a CRRA utility function has not been established in the growth literature while that in the continuous time counterpart is well-established. This note fills such a gap: I provide a formal proof that consumption being linearly related to investment is a sufficient and necessary condition for Pareto optimality in the discrete time AK model.

Keywords: AK model; neo-classical discrete time model; optimality

JEL Classification: E13; O41

Corresponding author: Myungkyu Shim, School of Economics, Yonsei University, Yonsei-ro 50, Seodaemun-gu, Seoul 03722, South Korea, Phone: +82 2123 2481, E-mail: myungkyushim@yonsei.ac.kr

Funding source: Yonsei University

Award Identifier / Grant number: 2021-11-0410

Acknowledgments

I thank the anonymous referee for helpful suggestions. I would like to appreciate Kyung-Woo Lee for his constructive comments for the earlier version of this paper. Seoyoon Jeong provided excellent research assistance. This research was supported by the Yonsei University and Yongwoon Scholarship Foundation (Yonsei-Yongwoon Research Grant No. 2021-11-0410). Any remaining errors are the author’s sole responsibility.

Appendix A. Proof for Necessary Condition

Suppose that { C ̂ t , K ̂ t + 1 } is the optimal rule that satisfies Proposition 1. Multiplying C ̂ t − σ to the feasibility condition leads to the following equation after rearranging the terms:

(A.1) K ̂ t + 1 C ̂ t − σ = ( A + 1 − δ ) K ̂ t C ̂ t − σ ︸ ≡ K ̂ t C ̂ t − 1 − σ C ̂ t C ̂ t − 1 − σ − C ̂ t 1 − σ

From the optimality condition (Eq. (3.1)), C ̂ t C ̂ t − 1 = β ( A + 1 − δ ) 1 σ . Hence the feasibility condition becomes

(A.2) K ̂ t + 1 C ̂ t − σ = 1 β K ̂ t C ̂ t − 1 − σ − C ̂ t 1 − σ

Notice that this equation describes the sequence of K ̂ t + 1 C ̂ t − σ .

The next lemma would be useful for the proof:

Lemma 2

(Optimal consumption as a function of initial consumption). Let C ̂ 0 > 0 be the initial consumption level optimally chosen by the planner. Then optimal consumption can be described as follows.

(A.3) C ̂ t = β ( A + 1 − δ ) t σ C ̂ 0

Proof

Recursive substitution of the optimality condition (3.1) yields the above expression. □

Then C ̂ t 1 − σ = ω t C ̂ 0 1 − σ with ω ≡ β ( A + 1 − δ ) 1 − σ σ . Substituting this expression into the equation (A.2):

(A.4) K ̂ t + 1 C t − σ = 1 β K ̂ t C ̂ t − 1 − σ ︸ = 1 β K ̂ t − 1 C ̂ t − 2 − σ − ω t − 1 C ̂ 0 1 − σ − ω t C ̂ 0 1 − σ = 1 β 2 K ̂ t − 1 C ̂ t − 2 − σ − C ̂ 0 1 − σ ω t + 1 β ω t − 1

One can substitute the expression for K ̂ t + 1 C ̂ t − σ recursively and obtain the following expression for K ̂ t + 1 C t − σ .

(A.5) K ̂ t + 1 C ̂ t − σ = 1 β t K ̂ 1 C ̂ 0 − σ − C ̂ 0 1 − σ ω t + 1 β ω t − 1 + ⋯ + 1 β t − 1 ω ︸ ≡ ω t 1 − 1 β ω t 1 − 1 β ω

The next step is to verify that K ̂ t + 1 C ̂ t − σ , the term in the TVC, converges toward zero if it is multiplied by β ^t.

(A.6) β t K ̂ t + 1 C ̂ t − σ = β t 1 β t K ̂ 1 C ̂ 0 − σ − C ̂ 0 1 − σ ω t 1 − 1 β ω t 1 − 1 β ω = K ̂ 1 C ̂ 0 − σ − C ̂ 0 1 − σ β ω β ω − 1 β ω t 1 − 1 β ω t = K ̂ 1 C ̂ 0 − σ − C ̂ 0 1 − σ β ω β ω − 1 β ω t − 1 ︸ = ( β ω − 1 ) 1 + β ω + ⋯ + β ω t − 1 = K ̂ 1 C ̂ 0 − σ − C ̂ 0 1 − σ β ω 1 + β ω + ⋯ + β ω t − 1 ︸ = 1 − β ω t 1 − β ω = K ̂ 1 C ̂ 0 − σ − C ̂ 0 1 − σ β ω 1 − β ω t 1 − β ω

Using the definition of ω, β ω = β β ( A + 1 − δ ) 1 − σ σ = β 1 σ A + 1 − δ 1 − σ σ ≡ 1 − ϕ . Then β ω 1 − β ω t 1 − β ω = ( 1 − ϕ ) ϕ 1 − 1 − ϕ t .

Thus

(A.7) lim t → ∞ β t K ̂ t + 1 C ̂ t − σ = lim t → ∞ C ̂ 0 − σ K ̂ 1 − ( 1 − ϕ ) ϕ 1 − 1 − ϕ t ︸ → 0 when t → ∞ C ̂ 0 = lim t → ∞ C ̂ 0 − σ K ̂ 1 − ( 1 − ϕ ) ϕ C ̂ 0

Since C ̂ 0 ∈ ( 0 , ∞ ) (if C ̂ 0 = 0 , the optimality condition (3.1) yields C ̂ t = 0 for all t, which is not optimal given that marginal utility of consumption diverges toward infinity when consumption is near zero), the TVC holds only when K ̂ 1 = ( 1 − ϕ ) ϕ C ̂ 0 , implying that investment and consumption chosen by the planner should be linearly related with each other.

To further show that the above property holds for any t, I will use mathematical induction. Suppose that K ̂ t = 1 − ϕ ϕ C ̂ t − 1 for some t ≥ 1. The following lemma is helpful for the proof:

Lemma 3

(Optimal rule for capital growth). Along the optimal path, the following should hold.

(A.8) K ̂ t + 1 = β 1 σ A + 1 − δ 1 σ K ̂ t

Proof

From Eq. (3.1), C ̂ t = β ( A + 1 − δ ) 1 σ C ̂ t − 1 . From the feasibility condition (3.2),

(A.9) K ̂ t + 1 = ( A + 1 − δ ) K ̂ t − C ̂ t = ( A + 1 − δ ) K ̂ t − β ( A + 1 − δ ) 1 σ C ̂ t − 1 = ( A + 1 − δ ) K ̂ t − β ( A + 1 − δ ) 1 σ ϕ 1 − ϕ K ̂ t = ( A + 1 − δ ) − [ β ( A + 1 − δ ) ] 1 σ 1 − β 1 σ ( A + 1 − δ ) 1 σ − 1 β 1 σ ( A + 1 − δ ) 1 σ − 1 K ̂ t = β 1 σ ( A + 1 − δ ) 1 σ K ̂ t

Hence, C ̂ t C ̂ t − 1 = K ̂ t + 1 K ̂ t = β 1 σ ( A + 1 − δ ) 1 σ , implying K ̂ t + 1 = 1 − ϕ ϕ C ̂ t for any t. As this relationship holds for t = 0, K ̂ t + 1 = 1 − ϕ ϕ C ̂ t for all t ≥ 0.

□

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Received: 2021-05-07

Accepted: 2021-11-03

Published Online: 2021-11-24

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Articles in the same Issue

https://doi.org/10.1515/bejte-2021-0061

Keywords for this article

AK model; neo-classical discrete time model; optimality