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Dynamic Games and Applications
Erschienen in: Dynamic Games and Applications 2/2016

01.06.2016

Ruling Out Multiplicity of Smooth Equilibria in Dynamic Games: A Hyperbolic Discounting Example

verfasst von: Lilia Maliar, Serguei Maliar

Erschienen in: Dynamic Games and Applications | Ausgabe 2/2016

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Abstract

The literature that conducts numerical analysis of equilibrium in models with hyperbolic (quasi-geometric) discounting reports difficulties in achieving convergence. Surprisingly, numerical methods fail to converge even in a simple, deterministic optimal growth problem that has a well-behaved, smooth closed-form solution. We argue that the reason for nonconvergence is that the generalized Euler equation has a continuum of smooth solutions, each of which is characterized by a different integration constant. We propose two types of restrictions that can rule out the multiplicity: boundary conditions and shape restrictions on equilibrium policy functions. With these additional restrictions, the studied numerical methods deliver a unique smooth solution for both the deterministic and stochastic problems in a wide range of the model’s parameters.
Vorheriger Artikel The Time-Inconsistency Problem of Labor Taxes and Constitutional Constraints

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Fußnoten
1
The related literature includes Laibson et al. [35], Barro [9], O’Donoghue and Rabin [55], Harris and Laibson [24], Angeletos et al. [3], Krusell and Smith [29‐31], Krusell et al. [32], Luttmer and Mariotti [37], Maliar and Maliar [39‐44], Judd [25], Sorger [61], Gong et al. [23], Chatterjee and Eyigungor [16], Balbus et al. [6, 7], Bernheim et al. [11], among others.
 
2
Other methods can be used in the context of models with quasi-geometric discounting. One of them is a “recursive optimization” approach suggested in Strotz [65], and Caplin and Leahy [14]. A possible implementation of this approach is found in the “pseudo-state space/enlarged state space” analysis of Kydland and Prescott [33] and Feng et al. [22]. Recently, in the context of the game theoretic approach, some literature have suggested turning the problem into a stochastic game (e.g., [24], Balbus et al. [6]). Also, in this latter tradition, one could also attempt to apply incentive-constrained dynamic programming methods [60], recursive dual approaches [49, 52, 56], and [17], or set-value dynamic programming methods proposed in Abreu, Pearce and Stachetti [1] (e.g., Balbus and Wozny [6]).
 
3
For convergence properties of the envelope condition method (ECM) and its applications, see Maliar and Maliar [46], and Arellano et al. [4].
 
4
For the case of the standard geometric discounting, there is a general turnpike theorem that shows that an optimal program of a finite horizon economy asymptotically converges to an optimal program of the corresponding infinite horizon economy under very general assumptions; see Brock and Mirman [13], McKenzie [51], Joshi [26], Majumdar and Zilcha [38], Mitra and Nyarko [54], Becker [10], and Maliar et al. [47]. Turnpike theorems are also known for some dynamic games (see [28] for a survey), but they are not yet established for the economy with quasi-geometric discounting like ours.
 
5
In the standard geometric discounting case, there are monotone operators that converge to a limiting stationary solution by iteration on the finite horizon dynamic program; see Coleman [18, 19], Mirman et al. [53], Datta et al. [20], and Feng et al. [22].
 
6
See also Barillas and Fernandez-Villaverde [8], Maliar and Maliar [45], Fella [21] and White [67] for extensions and applications of EGM.
 
7
Maliar and Maliar [40] shows another example of the model with quasi-geometric discounting that admits a closed-form solution under the assumption of the exponential utility function.
 
8
Balbus, Reffett and Wozny [6, 7] suggest a different value-based recursion for the model with qusi-geometric discounting, which under appropriate assumptions delivers a unique SMNE equilibrium using a simple successive approximation scheme.
 
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Metadaten
Titel
Ruling Out Multiplicity of Smooth Equilibria in Dynamic Games: A Hyperbolic Discounting Example
verfasst von
Lilia Maliar
Serguei Maliar
Publikationsdatum
01.06.2016
Verlag
Springer US
Erschienen in
Dynamic Games and Applications / Ausgabe 2/2016
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-015-0177-8

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