Taxes, transfers, asset tests, floors, and adjustment rules can make liquid resources
piecewise smooth or discontinuous. Hiding those rules inside where, maximum, or
Python branching gives a general solver executable arithmetic, but it does not give a
structure-specific solver the boundaries and one-sided ownership it needs.
pylcm therefore separates the economic function from its structural declaration.
The decorated functions remain ordinary callables and still work under GridSearch.
Two declaration forms¶
Case pieces¶
Use a case boundary when one supported binary predicate selects between two smooth formulas for the same output. The declaration identifies:
the liquid variable;
the threshold parameter;
which side owns equality;
whether the boundary is a continuous kink, jump, or hard constraint;
the
whenandotherwiseformula for the output.
This form is intentionally narrow. NBEGM validates that the declared split matches a
supported route instead of guessing at arbitrary branch code.
Piecewise-affine schedules¶
Use piecewise_affine when one output is affine in the liquid variable between a
sequence of declared breakpoints. Each affine_breakpoint supplies the threshold and
kind. Indexed thresholds can describe schedules that vary with a categorical state.
This covers bracketed taxes, floors, and mixed kink/jump schedules without inventing a separate case function for every interval.
What the solver does with the structure¶
Within each smooth run, NBEGM performs a one-dimensional endogenous-grid solve. It
then augments candidates at declared boundaries, applies exact side ownership, and takes
a branch-aware upper envelope. A parent can read a jump one-sidedly or use a faster
bridged finite-grid representation according to jump_read.
The period is solved tile-locally: each block of ride-along cells reads its transition-aware continuation and runs its envelope solve in savings space before the next block, so the expected continuation over every cell never exists as one array. A next-period law that reads the current liquid state through declared cliffs makes the continuation piecewise-constant across the cliff intervals; the read is then bound per interval and each interval is solved as its own case. A discrete action that only shifts the budget and utility shares one continuation read across its branches; an action that reaches the continuation (through the regime transition, a law of motion, stochastic-state transition weights, a child’s resources, the discount factor, or a schedule variable on a per-interval read) is read once per class of branches agreeing on it.
The structural declaration does not make an unsupported model supported. The solver
still checks its scope: margin count, action structure, budget form, differentiability,
and which names are available at each candidate stage. A retained Condition has the
same rule: structure enables a proof or compiler only when one exists.
Correctness and brute-force comparisons¶
Grid search is a useful diagnostic because the same functions execute without the specialized metadata. It is not automatically an accuracy reference at a cliff: a fixed action grid can miss a threshold or interpolate across it. Validation should compare values and decisions away from and at the declared boundaries, use one-sided checks where appropriate, and refine the brute grid until remaining differences have an economic interpretation.
Exact decorator contracts are split into Case pieces and Piecewise-affine schedules. Feasibility boundaries belong to Constraints and structured Conditions, not to either budget declaration.
Related pages¶
Guide: Choosing a solver and Authoring for EGM-family solvers
Methods: Solver families and The endogenous-grid method
Example: Mahler & Yum (2024)
Reference: Case pieces, Piecewise-affine schedules, and Solvers and capabilities