The endogenous-grid method

The endogenous-grid method (EGM) removes a continuous root search from a smooth consumption-saving problem by choosing tomorrow’s assets first and recovering today’s resources from the Euler equation Carroll (2006).

Write post-decision savings as

where $m$ is liquid resources and $c$ is consumption. For an exogenous savings node $s_i$, the Euler condition has the schematic form

When marginal utility is invertible,

The pairs $(m_i,c_i)$ form an endogenous policy grid. Interpolation evaluates that policy on the regime’s exogenous liquid-state grid.

Why it can be faster¶

A brute-force solve compares many consumption candidates for every current resource node. EGM evaluates the Euler right-hand side once per savings node and obtains the consumption choice by inversion. The benefit relies on three properties:

1. the relevant continuous choice is a one-dimensional liquid margin;

2. marginal utility can be inverted;

3. conditional on solver rows and smooth branches, the Euler right-hand side is a function of the post-decision state rather than every current liquid node.

Violating the third property can require one EGM construction per current state, losing the amortization that motivated the method.

EGM and DCEGM encode different budgets¶

pylcm’s plain EGM uses the liquid state itself as cash-on-hand. Its kernel constructs the endogenous location as consumption + savings, and model validation checks that the declared post-decision function implements the corresponding identity.

DCEGM binds a genuine resources node. Wealth, labor income, taxes, and transfers may feed that node before consumption is paid. It also handles a discrete choice and takes an upper envelope across the resulting branches.

This distinction affects model authoring. Both use a ConsumptionSavingsRegime and LiquidMargin, but the resources role is the liquid state for the plain identity case and a named function for the richer budget. See Consumption-saving regimes and margins.

Borrowing constraints¶

An EGM solver enforces the lower edge of its savings grid by construction. Declaring that economic restriction with post_decision_lower_bound lets pylcm verify that the constraint and grid agree. An arbitrary callable that happens to compute the same Boolean does not retain enough structure for that proof.

This does not mean every condition must be declarative. Read Constraints and structured Conditions for the precise boundary between ordinary callables and retained structure.

Corners, discrete choices, and non-convex budgets¶

Plain EGM is for a smooth concave margin. A borrowing corner, discrete branch, or institutional cliff changes how candidates are generated and compared:

• DCEGM handles a genuine resources node, supported passive or process states, and optional discrete-continuous non-concavity with an upper envelope.

• NBEGM handles declared budget runs, kinks, jumps, and supported hard constraints.

• NEGM and NNBEGM place one of those liquid solves inside an outer search.

Use Choosing a solver rather than treating these classes as interchangeable performance switches.

Related pages¶

• Guide: Authoring for EGM-family solvers

• Methods: Solver families and Nested endogenous-grid methods

• Example: Iskhakov et al. (2017)

• Reference: Consumption-saving regimes and margins and Solvers and capabilities