Two-sided altruism and time inconsistency

Takaaki Aoki; Kazuo Nishimura; Makoto Yano

doi:10.1515/snde-2019-0022

Article

Two-sided altruism and time inconsistency

Takaaki Aoki , Kazuo Nishimura and Makoto Yano

Published/Copyright: April 17, 2019

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 23 Issue 4

Abstract

In this study, we examine dynamic allocation of resources and consumption over an infinite time horizon in an overlapping-generations model with two-sided altruism. Each generation cares not only about its own consumption and the utility of its successor, but also about the utility of its parental generation. The consumption decisions of different generations could display time-inconsistency in a model of two-sided altruism. Consequently, the outcome solutions could differ depending on whether members of the current generation are naïve and falsely believe that the future generation will comply with their, or are sophisticated and know exactly how the future generation will choose to behave. We assume that the young generation is naïve and behaves as a central planner in allocating capital resources. The capital stocks inherited by each generation from the parental generation are actually realized and used in the joint production of capital stocks and consumption goods. We refer to a sequence of such capital stocks as a two-sided altruistic path. This study examines a more general production function than earlier models of two-sided altruism, and our results prove that any two-sided altruistic path will eventually converge to the two-sided altruistic stationary path.

Keywords: global convergence; joint production; time-inconsistency; two-sided altruism

JEL Classification: C61; D91; O41

Funding source: Japan Society for Promotion of Science

Award Identifier / Grant number: #15H05729, #16H0233598 and #23000001

Funding statement: This work was supported by the Japan Society for Promotion of Science, Funder Id: http://dx.doi.org/10.13039/501100001691, Grants-in-Aid for Research #15H05729, #16H0233598 and for Specially Promoted Research #23000001.

A Appendix

A.1 Proof of Lemma 3:

Proof.

Let (k0,k1) be in the interior of D={(k,y)|T(y,k)≥0}. Suppose that k0′>k0 and k1′<k1. Because (k0,k1)∈D, (k0,k1′) with k1′<k1 and (k0′,k1) with k0′>k0 both belong to D. This means that (k0,k1′) and (k0′,k1) are feasible if (k0,k1) is feasible.

Because we can show easily that an optimal solution of Problem (I) is unique,

(21)wα(T(k1,k0))+βW¯(k1)>wα(T(k1′,k0))+βW¯(k1′)

and

(22)wα(T(k1′,k0′))+βW¯(k1′)>wα(T(k1,k0′))+βW¯(k1)

must be true. Combining (21) and (22) gives

(23)wα(T(k1,k0))+wα(T(k1′,k0′))>wα(T(k1′,k0))+wα(T(k1,k0′)).

Therefore,

(24)∫k1′k1[wα′(T(s,k0))T1(s,k0)−wα′(T(s,k0′))T1(s,k0′)]ds>0.

Because k0′>k0, T12>0 and w(⋅) is increasing, T(s,k0′)>T(s,k0), T1(s,k0′)>T1(s,k0), and wα′(T(s,k0))<wα′(T(s,k0′)). The left-hand side of (24) is negative.

This contradicts the inequality in (24). Therefore, k1′≥k1 must be true. Suppose that k1′=k1. By the Euler equation,

(25)wα′(T(k1,k0))T1(k1,k0)+βw1/β′(T(k2,k1))T2(k2,k1)=0.

Because k0′>k0 and k1′=k1, both {kt}t=1∞ and {kt′}t=1∞ are optimal solutions of Problem (II) from k1′=k1. Furthermore, because the optimal solution of Problem (II) is unique, k2′=k2 holds, and wα′(T(k1,x))T1(k1,x)+βw1/β′(T(k2,k1))T2(k2,k1)=0 has two distinct solutions, x = k₀ and x=k0′. This contradicts the strictly increasing property of wα′(T(k1,x))T1(k1,x). Hence, k1′>k1.

Next, {kt}t=1∞ and {kt′}t=1∞ are the optimal solutions of Problem (II), given k₁ and k1′, respectively. By the principle of optimality, W¯(k1)=w1/β(T(k2,k1))+βW¯(k2) and W¯(k1′)=w1/β(T(k2′,k1′))+βW¯(k2′). Therefore, the argument above may be applied, and k1′>k1 implies k2′>k2. Thus, it is proven that k0′>k0 implies kt′>kt for t ≥ 1. □

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Article Note

The earlier version of the paper was presented at the special session of the 26th Annual Symposium of the Society for Nonlinear Dynamics and Econometrics, held at Keio University, Tokyo (Japan), March 19–20, 2018.

Published Online: 2019-04-17

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Articles in the same Issue

https://doi.org/10.1515/snde-2019-0022

Keywords for this article

global convergence; joint production; time-inconsistency; two-sided altruism