Credits & Acknowledgments

pylcm stands on a large body of methodological and applied research, and on a vibrant open-source ecosystem for solving discrete-continuous dynamic models. This page credits the methods, models, and software pylcm draws on. We are grateful to all of these authors and maintainers.

Authors & maintainers¶

pylcm is developed under the OpenSourceEconomics organization by Tim Mensinger, Maximilian Jahn, Janos Gabler, and Hans-Martin von Gaudecker.

Numerical methods¶

The endogenous grid method (EGM)¶

The endogenous grid method — inverting the Euler equation on an exogenous end-of-period grid instead of root-finding on a dense grid — originates with Carroll (2006), “The Method of Endogenous Gridpoints for Solving Dynamic Stochastic Optimization Problems,” Economics Letters 91(3), 312–320, Carroll (2006).

Discrete-continuous EGM (DC-EGM)¶

pylcm’s DCEGM solver implements the discrete-continuous endogenous grid method of Iskhakov, Jørgensen, Rust & Schjerning (2017), “The endogenous grid method for discrete-continuous dynamic choice models with (or without) taste shocks,” Quantitative Economics 8(2), 317–365, Iskhakov et al. (2017). The deterministic retirement model from that paper is shipped as a closed-form test oracle (see below).

• Original MATLAB implementation: fediskhakov/dcegm.

• A fully JAX-compatible Python implementation in the same organization, whose design pylcm draws on: OpenSourceEconomics/dcegm.

Upper-envelope refinement¶

pylcm refines the (non-monotone) EGM candidate correspondence into the upper envelope with the Fast Upper-Envelope Scan (FUES) of Dobrescu & Shanker (2022), “Fast Upper-Envelope Scan for Discrete-Continuous Dynamic Programming,” SSRN working paper 4181302. The treatment of non-smooth, non-concave value functions traces to Fella (2014), “A Generalized Endogenous Grid Method for Non-Smooth and Non-Concave Problems,” Review of Economic Dynamics 17(2), 329–344, Fella (2014).

pylcm’s FUES backend is adapted from OpenSourceEconomics/upper-envelope (Apache-2.0, © The Upper-Envelope Authors), then substantially modified for the JAX kernel. The reference implementation by the method’s authors is akshayshanker/FUES.

Multidimensional & nested EGM (reserved / planned backends)¶

pylcm reserves an upper-envelope backend slot for multidimensional envelopes and is designed to grow toward solving models with more than one continuous (Euler) state. That roadmap builds on:

• Druedahl (2021), “A Guide on Solving Non-convex Consumption-Saving Models,” Computational Economics 58(3), 747–775, Druedahl (2021) — nested EGM (NEGM). Code: NumEconCopenhagen/ConsumptionSaving.

• Druedahl & Jørgensen (2017), “A General Endogenous Grid Method for Multi-Dimensional Models with Non-Convexities and Constraints,” Journal of Economic Dynamics and Control 74, 87–107, Druedahl & Jørgensen (2017) — G2EGM. Code: JeppeDruedahl/G2EGM.

• Dobrescu & Shanker (2024), “Using Inverse Euler Equations to Solve Multidimensional Discrete-Continuous Dynamic Models: A General Method,” SSRN working paper 4850746 — the inverse-Euler / Rooftop-Cut (RFC) multidimensional envelope. Code: akshayshanker/InverseDCDP.

Stochastic-process discretization¶

pylcm’s AR(1) / Markov discretization follows Tauchen (1986), Rouwenhorst (1995), and Kopecky & Suen (2010), with the Tauchen implementation following QuantEcon. The choice of Rouwenhorst as the default for non-stationary lifecycle processes follows Fella, Gallipoli & Pan (2019), “Markov-Chain Approximations for Life-Cycle Models,” Review of Economic Dynamics 34, 183–201, Fella et al. (2019), against whose results pylcm’s discretization accuracy is validated. Code & data: RePEc red/ccodes/17-149.

Replicated example models¶

Models shipped with pylcm (src/lcm_examples/) that replicate published work:

• Iskhakov, Jørgensen, Rust & Schjerning (2017) — the deterministic retirement model (lcm_examples.iskhakov_et_al_2017), used as a closed-form test oracle.

• Mahler & Yum (2024), “Lifestyle Behaviors and Wealth-Health Gaps in Germany,” Econometrica 92(5), 1697–1733, Mahler & Yum (2024) — a finite-horizon discrete-continuous lifecycle model (lcm_examples.mahler_yum_2024). The replication package is available via the Econometrica supplemental materials.

Pedagogy¶

pylcm’s design and documentation are informed by Sargent & Stachurski’s Dynamic Programming (QuantEcon, 2024).

The open-source ecosystem¶

pylcm interoperates with, learns from, and is grateful to the broader ecosystem for solving discrete-continuous dynamic models:

• OpenSourceEconomics — dcegm, upper-envelope.

• NumEconCopenhagen — ConsumptionSaving (ConSav), EconModel.

• akshayshanker — FUES, InverseDCDP.

• JeppeDruedahl — G2EGM.

• fediskhakov — dcegm (original DC-EGM).

• QuantEcon — dynamic-programming pedagogy and reference code.