16-AUG-2025

We formulate a Huggett-Moll mean field game in which agents' labor productivity follows a Kou double-exponential jump-diffusion, capturing asymmetric income shocks: sudden collapses and rare windfalls superimposed on geometric Brownian motion. Working in log-productivity space $x = \log z$, the multiplicative jump structure becomes additive, and both the Hamilton-Jacobi-Bellman and Fokker-Planck partial integro-differential equations admit a complete reduction to coupled PDE systems via auxiliary first-order equations derived from the exponential jump kernel. General equilibrium is closed by a Cobb-Douglas production sector, with aggregate capital $K[m] = \iint a \, m(a,x) \, da \, dx$ determining wages and returns endogenously. We state the MFG fixed-point problem, derive the auxiliary PDE systems explicitly, and document a Method of Lines numerical algorithm combining upwind finite differences in wealth with centred fourth-order stencils in log-productivity and a stiff implicit integrator.

⇓ Download PDF

08-AUG-2024

We study a stationary discounted mean field game on the one-dimensional torus $\mathbb{T}=[0,1]$, in which a continuum of identical agents controls a Brownian drift subject to a local congestion cost and a sinusoidal external potential $f(x)=A\sin(2\pi x)$. The equilibrium pair $(V^*,m^*)$ satisfies a coupled Hamilton–Jacobi–Bellman and Kolmogorov–Fokker–Planck system. Lasry–Lions monotonicity guarantees a unique Nash mean field equilibrium. Numerically, Howard policy iteration solves the HJB at each Picard step, while the stationary KFP is handled by a direct adjoint-matrix solve; the outer loop converges in 33 iterations to an equilibrium featuring a bang-bang optimal policy and a unimodal density concentrated in the low-cost region.

⇓ Download PDF