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A mean field game approach to equilibrium consumption under external habit formation

Lijun Bo, Shihua Wang, Xiang Yu

Abstract

This paper studies the equilibrium consumption under external habit formation in a large population of agents. We first formulate problems under two types of conventional habit formation preferences, namely linear and multiplicative external habit formation, in a mean field game framework. In a log-normal market model with the asset specialization, we characterize one mean field equilibrium in analytical form in each problem, allowing us to understand some quantitative properties of the equilibrium strategy and conclude some financial implications caused by consumption habits from a mean-field perspective. In each problem with n agents, we construct an approximate Nash equilibrium for the n-player game using the obtained mean field equilibrium when n is sufficiently large. The explicit convergence order in each problem can also be obtained.

A mean field game approach to equilibrium consumption under external habit formation

Abstract

This paper studies the equilibrium consumption under external habit formation in a large population of agents. We first formulate problems under two types of conventional habit formation preferences, namely linear and multiplicative external habit formation, in a mean field game framework. In a log-normal market model with the asset specialization, we characterize one mean field equilibrium in analytical form in each problem, allowing us to understand some quantitative properties of the equilibrium strategy and conclude some financial implications caused by consumption habits from a mean-field perspective. In each problem with n agents, we construct an approximate Nash equilibrium for the n-player game using the obtained mean field equilibrium when n is sufficiently large. The explicit convergence order in each problem can also be obtained.
Paper Structure (10 sections, 10 theorems, 164 equations, 3 figures)

This paper contains 10 sections, 10 theorems, 164 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Market Model
  3. Mean Field Game Problems
  4. Mean field equilibrium under linear habit formation
  5. Mean field equilibrium under multiplicative habit formation
  6. Numerical Illustrations of Mean Field Equilibrium
  7. Approximate Nash Equilibrium in n-player Games
  8. Approximation under linear habit formation
  9. Approximation under multiplicative habit formation
  10. Conclusions

Key Result

Lemma 3.1

Let $\bar{Z}=(\bar{Z}_t)_{t\in[0,T]}\in{\cal C}_{T, x_0}$. The classical solution of the HJB equation eq:V-PDE-two on the effective domain $\{(t,x)\in [0,T]\times \mathbb{R}_+: x>\int_t^T \alpha \bar{Z}_sds\}$ admits the closed-form that where and $a:= \frac{p\mu^2}{2(1-p)^2\sigma^2}$. The feedback functions of the optimal investment and consumption to the problem eq:value-function-two from the

Figures (3)

  • Figure 1: Top panel: The MFE consumption rate $C^l(t,x)$, the MFE portfolio $\pi^{l}(t,x)$, and the habit formation process $\bar{Z}^l_t$ with risk aversion parameters $p=0.2,~0.5$ and $0.7$. Bottom panel: The MFE consumption rate $C^m(t,x)$, the MFE portfolio $\pi^{m}(t,x)$, and the habit formation process $\bar{Z}^m_t$ with risk aversion parameters $p=0.2$, $0.5$ and $0.7$.
  • Figure 2: Top panel: The MFE consumption rate $C^l(t,x)$, the MFE portfolio $\pi^{l}(t,x)$, and the habit formation process $\bar{Z}^l_t$ with competition parameters $\alpha=0.2,~0.5$ and $1$. Bottom panel: The MFE consumption rate $C^m(t,x)$, the MFE portfolios $\pi^{m}(t,x)$, and the habit formation processes $\bar{Z}^m_t$ with competition parameters $\alpha=0.2,~0.5$ and $1$.
  • Figure 3: Top panel: The MFE consumption rate $C^l(t,x)$, the MFE portfolios $\pi^{l}(t,x)$, and the habit formation processes $\bar{Z}^l_t$ with habit persistence parameters $\delta = 0.1,~0.2$ and $0.3$. Bottom panel: The MFE consumption rate $C^m(t,x)$, the MFE portfolios $\pi^{m}(t,x)$, and the habit formation processes $\bar{Z}^m_t$ with habit persistence parameters $\delta=0.1,~0.2$ and $0.3$.

Theorems & Definitions (25)

  • Remark 2.1
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Definition 3.2
  • Lemma 3.2
  • proof
  • Theorem 3.2
  • ...and 15 more