Beta-Delta (Quasi-Hyperbolic) Discounting

This notebook shows how to implement beta-delta discounting in PyLCM. The naive agent needs a phase-variant Koopmans aggregator, a custom $W$ declared via Phased(solve=..., simulate=...); the sophisticated and partially naive agents choose with one aggregator and store the value of another, declared via Decoupled(objective=..., value=...). It covers:

The beta-delta discount function
Why naive agents need different $W$ for solve vs. simulate
A 3-period consumption-savings model with analytical solutions
Verifying numerical results against closed-form solutions
Sophisticated and partially naive agents with Decoupled

Background: Beta-Delta Discounting¶

Standard exponential discounting uses a single discount factor $\delta$ to weight future utility:

U_t = u_t + \delta\, u_{t+1} + \delta^2\, u_{t+2} + \cdots

(1)

Quasi-hyperbolic (beta-delta) discounting (Laibson, 1997) introduces a present-bias parameter $\beta \in (0, 1]$ that discounts all future periods relative to the present:

U_t = u_t + \beta\delta\, u_{t+1} + \beta\delta^2\, u_{t+2} + \cdots

(2)

The discount weights are $\{1,\; \beta\delta,\; \beta\delta^2,\; \ldots\}$ . When $\beta = 1$ this reduces to exponential discounting. When $\beta < 1$ the agent is present-biased — they overweight current utility relative to any future period.

Note the key structure: $\beta$ appears once (not compounded). The weight $n$ periods ahead is $\beta\delta^n$ , not $(\beta\delta)^n$ .

PyLCM’s Koopmans Aggregator and the Naive Agent¶

PyLCM defines the value function recursively via an aggregation function $W$ :

V_t(s) = \max_a \; W\bigl(u(s, a),\; \mathbb{E}_t[V_{t+1}(s')]\bigr)

(3)

The default $W$ is $W(u, \mathbb{E}[V']) = u + \delta \cdot \mathbb{E}[V']$ .

A naive beta-delta agent believes their future selves will be exponential discounters, but at each decision point applies present bias $\beta\delta$ to the continuation value. This requires two different $W$ implementations:

Solve phase (backward induction): use exponential discounting with $\delta$ to compute the perceived continuation values $V^E$
Simulate phase (action selection): use $\beta\delta$ to choose actions, applying present bias to the perceived $V^E$

Phase is a broadcast dimension of the regime spec: a bare function applies to both phases, while Phased(solve=..., simulate=...) gives each phase its own implementation. Declaring koopmans_aggregator as Phased(solve=LinearAggregator(), simulate=linear_beta_delta) is all the naive agent needs — the params template becomes the union of both variants’ parameters, so one params dict serves both phases.

Why not just use $W(u, V') = u + \beta\delta V'$ for both phases? Because the recursion would compound $\beta$ at every step, giving effective weights $(\beta\delta)^n$ instead of the correct $\beta\delta^n$ . Splitting the phases ensures $\beta$ is applied exactly once — at the action selection step.

Setup: A 3-Period Model¶

We use a minimal consumption-savings model:

3 periods: $t = 0, 1$ (decisions), $t = 2$ (terminal)
1 state: wealth $w$
1 action: consumption $c$
Utility: $u(c) = \log(c)$
Budget: $w' = w - c$ (interest rate $R = 1$ )
Constraint: $c \le w$
Terminal: $V_2(w) = \log(w)$ (consume everything)

import jax.numpy as jnp

from lcm import (
    AgeGrid,
    LinSpacedGrid,
    Model,
    Phased,
    Regime,
    categorical,
)
from lcm.koopmans_aggregation import LinearAggregator
from lcm.typing import BoolND, ContinuousAction, ContinuousState, FloatND, ScalarInt

@categorical(ordered=False)
class RegimeId:
    working: ScalarInt
    dead: ScalarInt


def utility(consumption: ContinuousAction) -> FloatND:
    return jnp.log(consumption)


def terminal_utility(wealth: ContinuousState) -> FloatND:
    return jnp.log(wealth)


def next_wealth(
    *, wealth: ContinuousState, consumption: ContinuousAction
) -> ContinuousState:
    return wealth - consumption


def borrowing_constraint(
    *, consumption: ContinuousAction, wealth: ContinuousState
) -> BoolND:
    return consumption <= wealth


def linear_beta_delta(
    *,
    utility: float,
    CE: float,
    beta: float,
    delta: float,
) -> float:
    return utility + beta * delta * CE

We build two models:

One with the shipped LinearAggregator for the standard exponential agent
One with a phase-variant $W$ for the naive beta-delta agent — solving with exponential $W$ and simulating with beta-delta $W$

WEALTH_GRID = LinSpacedGrid(start=0.5, stop=50, n_points=200)
CONSUMPTION_GRID = LinSpacedGrid(start=0.001, stop=50, n_points=500)

dead = Regime(
    states={"wealth": WEALTH_GRID},
    functions={"utility": terminal_utility},
)

# Standard exponential model
working_exp = Regime(
    actions={"consumption": CONSUMPTION_GRID},
    states={"wealth": WEALTH_GRID},
    state_transitions={"wealth": next_wealth},
    constraints={"borrowing_constraint": borrowing_constraint},
    functions={"utility": utility},
    koopmans_aggregator=LinearAggregator(),
)

# Work at ages 0 and 1, then die. One edge per age, so the graph is the whole
# transition law; `dead` has no outgoing edges and is terminal.
edges = {"working": {"working": 0, "dead": 1}}

model_exp = Model(
    regimes={"working": working_exp, "dead": dead},
    ages=AgeGrid(start=0, inclusive_stop=2, step="Y"),
    edges=edges,
    initial_nodes=((0, "working"),),
    regime_id_class=RegimeId,
)

# Naive beta-delta model (a different aggregator per phase)
working_naive = Regime(
    actions={"consumption": CONSUMPTION_GRID},
    states={"wealth": WEALTH_GRID},
    state_transitions={"wealth": next_wealth},
    constraints={"borrowing_constraint": borrowing_constraint},
    functions={"utility": utility},
    koopmans_aggregator=Phased(
        solve=LinearAggregator(),
        simulate=linear_beta_delta,
    ),
)

model_naive = Model(
    regimes={"working": working_naive, "dead": dead},
    ages=AgeGrid(start=0, inclusive_stop=2, step="Y"),
    edges=edges,
    initial_nodes=((0, "working"),),
    regime_id_class=RegimeId,
)

Analytical Solution¶

With $\log$ utility, the optimal consumption rule is $c_t = w_t / D_t$ , where $D_t$ is a “marginal propensity to save” denominator.

At $t = 1$ (one period before terminal), the agent maximizes:

\max_c \;\bigl[\log(c) + \beta\delta \cdot \log(w - c)\bigr]

(4)

First-order condition: $1/c = \beta\delta / (w - c)$ , giving $c_1 = w / (1 + \beta\delta)$ .

At $t = 0$ , the naive agent uses the perceived exponential continuation value $V^E_1$ , which has coefficient $(1 + \delta)$ on $\log w$ . So the agent maximizes:

\max_c \;\bigl[\log(c) + \beta\delta (1 + \delta) \log(w - c) + \text{const}\bigr]

(5)

giving $c_0 = w / (1 + \beta\delta(1 + \delta))$ .

	$D_1$	$D_0$
Exponential ( $\beta = 1$ )	$1 + \delta$	$1 + \delta(1 + \delta)$
Naive ( $\beta < 1$ )	$1 + \beta\delta$	$1 + \beta\delta(1 + \delta)$

BETA = 0.7
DELTA = 0.95
W0 = 20.0


def analytical_consumption(*, beta, delta, w0):
    """Return (c0, c1) from the closed-form solution."""
    bd = beta * delta
    d1 = 1 + bd
    d0 = 1 + bd * (1 + delta)
    c0 = w0 / d0
    c1 = (w0 - c0) / d1
    return c0, c1


c0_exp, c1_exp = analytical_consumption(beta=1.0, delta=DELTA, w0=W0)
c0_naive, c1_naive = analytical_consumption(beta=BETA, delta=DELTA, w0=W0)

print(f"{'Type':<15} {'c_0':>8} {'c_1':>8}")
print("-" * 33)
print(f"{'Exponential':<15} {c0_exp:8.4f} {c1_exp:8.4f}")
print(f"{'Naive':<15} {c0_naive:8.4f} {c1_naive:8.4f}")

Type                 c_0      c_1
---------------------------------
Exponential       7.0114   6.6608
Naive             8.7080   6.7820

The naive agent consumes more at $t = 0$ than the exponential agent — present bias pulls consumption forward. At $t = 1$ , the naive agent also consumes more because $\beta\delta < \delta$ .

Numerical Results¶

Exponential Agent ( $\beta = 1$ )¶

initial_conditions = {
    "age": jnp.array([0.0]),
    "wealth": jnp.array([W0]),
    "regime_id": jnp.array([RegimeId.working]),
}

result_exp = model_exp.simulate(
    params={"working": {"koopmans_aggregator": {"discount_factor": DELTA}}},
    initial_conditions=initial_conditions,
    log_level="debug",
)

df_exp = result_exp.to_dataframe().query('regime_name == "working"')
print("Exponential agent:")
print(
    f"  c_0 = {df_exp.loc[df_exp['age'] == 0, 'consumption'].iloc[0]:.4f}  "
    f"(analytical: {c0_exp:.4f})"
)
print(
    f"  c_1 = {df_exp.loc[df_exp['age'] == 1, 'consumption'].iloc[0]:.4f}  "
    f"(analytical: {c1_exp:.4f})"
)

solve call 322764a622de phase public_simulate begin

INFO:lcm:solve call 322764a622de phase public_simulate begin

solve call 322764a622de phase params_validation begin

INFO:lcm:solve call 322764a622de phase params_validation begin

solve call 322764a622de phase params_validation end status=ok seconds=0.064228

INFO:lcm:solve call 322764a622de phase params_validation end status=ok seconds=0.064228

solve call 322764a622de phase simulation_inputs begin

INFO:lcm:solve call 322764a622de phase simulation_inputs begin

solve call 322764a622de phase simulation_inputs end status=ok seconds=0.099455

INFO:lcm:solve call 322764a622de phase simulation_inputs end status=ok seconds=0.099455

solve call 322764a622de phase authority_fingerprint begin

INFO:lcm:solve call 322764a622de phase authority_fingerprint begin

solve call 322764a622de phase authority_fingerprint end status=ok seconds=0.004101

INFO:lcm:solve call 322764a622de phase authority_fingerprint end status=ok seconds=0.004101

solve call 322764a622de phase solver_param_checks begin

INFO:lcm:solve call 322764a622de phase solver_param_checks begin

solve call 322764a622de phase solver_param_checks end status=ok seconds=0.000035

INFO:lcm:solve call 322764a622de phase solver_param_checks end status=ok seconds=0.000035

solve call 322764a622de phase state_action_spaces begin

INFO:lcm:solve call 322764a622de phase state_action_spaces begin

solve call 322764a622de phase state_action_spaces end status=ok seconds=0.000152

INFO:lcm:solve call 322764a622de phase state_action_spaces end status=ok seconds=0.000152

solve call 322764a622de phase continuation_templates begin

INFO:lcm:solve call 322764a622de phase continuation_templates begin

solve call 322764a622de phase continuation_templates end status=ok seconds=0.000889

INFO:lcm:solve call 322764a622de phase continuation_templates end status=ok seconds=0.000889

solve call 322764a622de phase program_graphs begin

INFO:lcm:solve call 322764a622de phase program_graphs begin

solve call 322764a622de phase program_graphs end status=ok seconds=0.000603

INFO:lcm:solve call 322764a622de phase program_graphs end status=ok seconds=0.000603

solve call 322764a622de phase structural_resolution begin

INFO:lcm:solve call 322764a622de phase structural_resolution begin

structural blueprint: miss (hits=0, misses=1, entries=1)

DEBUG:lcm:structural blueprint: miss (hits=0, misses=1, entries=1)

solve call 322764a622de phase structural_resolution end status=ok seconds=0.032047

INFO:lcm:solve call 322764a622de phase structural_resolution end status=ok seconds=0.032047

solve call 322764a622de phase residency_inventory begin

INFO:lcm:solve call 322764a622de phase residency_inventory begin

solve call 322764a622de phase residency_inventory end status=ok seconds=0.000007

INFO:lcm:solve call 322764a622de phase residency_inventory end status=ok seconds=0.000007

solve call 322764a622de phase compilation_waves begin

INFO:lcm:solve call 322764a622de phase compilation_waves begin

candidate evaluation ('working', 0, 'main') {'action_product': 64, 'cell': 128}

DEBUG:lcm:candidate evaluation ('working', 0, 'main') {'action_product': 64, 'cell': 128}

candidate evaluation ('working', 1, 'main') {'action_product': 64, 'cell': 128}

DEBUG:lcm:candidate evaluation ('working', 1, 'main') {'action_product': 64, 'cell': 128}

candidate evaluation ('dead', 2, 'main') {'cell': 128}

DEBUG:lcm:candidate evaluation ('dead', 2, 'main') {'cell': 128}

AOT compilation wave 0: 3 unique lowerings for 3 regime-period-core triples (2 workers)

INFO:lcm:AOT compilation wave 0: 3 unique lowerings for 3 regime-period-core triples (2 workers)

1/3  working main (age 0, widths={'action_product': 64, 'cell': 128})

INFO:lcm:1/3  working main (age 0, widths={'action_product': 64, 'cell': 128})

  lowering working main (age 0, widths={'action_product': 64, 'cell': 128}) ...

INFO:lcm:  lowering working main (age 0, widths={'action_product': 64, 'cell': 128}) ...

  lowered in 396.0ms

INFO:lcm:  lowered in 396.0ms

  compiling working main (age 0, widths={'action_product': 64, 'cell': 128}) ...

INFO:lcm:  compiling working main (age 0, widths={'action_product': 64, 'cell': 128}) ...

2/3  working main (age 1, widths={'action_product': 64, 'cell': 128})

INFO:lcm:2/3  working main (age 1, widths={'action_product': 64, 'cell': 128})

  lowering working main (age 1, widths={'action_product': 64, 'cell': 128}) ...

INFO:lcm:  lowering working main (age 1, widths={'action_product': 64, 'cell': 128}) ...

  lowered in 260.3ms

INFO:lcm:  lowered in 260.3ms

  compiling working main (age 1, widths={'action_product': 64, 'cell': 128}) ...

INFO:lcm:  compiling working main (age 1, widths={'action_product': 64, 'cell': 128}) ...

3/3  dead main (age 2, widths={'cell': 128})

INFO:lcm:3/3  dead main (age 2, widths={'cell': 128})

  lowering dead main (age 2, widths={'cell': 128}) ...

INFO:lcm:  lowering dead main (age 2, widths={'cell': 128}) ...

  lowered in 72.6ms

INFO:lcm:  lowered in 72.6ms

  compiled  working main (age 0, widths={'action_product': 64, 'cell': 128})  518.3ms

INFO:lcm:  compiled  working main (age 0, widths={'action_product': 64, 'cell': 128})  518.3ms

  compiling dead main (age 2, widths={'cell': 128}) ...

INFO:lcm:  compiling dead main (age 2, widths={'cell': 128}) ...

  compiled  dead main (age 2, widths={'cell': 128})  166.0ms

INFO:lcm:  compiled  dead main (age 2, widths={'cell': 128})  166.0ms

  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  556.6ms

INFO:lcm:  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  556.6ms

AOT compilation wave 0: 3 unique variants compiled, 0 fallback evaluations skipped, 0 fallback requests avoided

INFO:lcm:AOT compilation wave 0: 3 unique variants compiled, 0 fallback evaluations skipped, 0 fallback requests avoided

solve call 322764a622de phase compilation_waves end status=ok seconds=1.243935

INFO:lcm:solve call 322764a622de phase compilation_waves end status=ok seconds=1.243935

solve call 322764a622de phase workspace_selection begin

INFO:lcm:solve call 322764a622de phase workspace_selection begin

core plan record {"active_replica_bytes":0,"axis_extents":{"action_product":500,"cell":200},"compiled_collectives":{},"compiler_peak_bytes":4491,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"action_product":"streamed","cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":0,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"working","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"action_product":64,"cell":128}}

DEBUG:lcm:core plan record {"active_replica_bytes":0,"axis_extents":{"action_product":500,"cell":200},"compiled_collectives":{},"compiler_peak_bytes":4491,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"action_product":"streamed","cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":0,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"working","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"action_product":64,"cell":128}}

core plan record {"active_replica_bytes":0,"axis_extents":{"action_product":500,"cell":200},"compiled_collectives":{},"compiler_peak_bytes":4491,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"action_product":"streamed","cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":1,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"working","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"action_product":64,"cell":128}}

DEBUG:lcm:core plan record {"active_replica_bytes":0,"axis_extents":{"action_product":500,"cell":200},"compiled_collectives":{},"compiler_peak_bytes":4491,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"action_product":"streamed","cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":1,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"working","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"action_product":64,"cell":128}}

core plan record {"active_replica_bytes":0,"axis_extents":{"cell":200},"compiled_collectives":{},"compiler_peak_bytes":1636,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":2,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"dead","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"cell":128}}

DEBUG:lcm:core plan record {"active_replica_bytes":0,"axis_extents":{"cell":200},"compiled_collectives":{},"compiler_peak_bytes":1636,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":2,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"dead","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"cell":128}}

solve call 322764a622de phase workspace_selection end status=ok seconds=0.017485

INFO:lcm:solve call 322764a622de phase workspace_selection end status=ok seconds=0.017485

Starting solution

INFO:lcm:Starting solution

solve call 322764a622de phase backward_induction begin

INFO:lcm:solve call 322764a622de phase backward_induction begin

Age 2 (1 regimes):

INFO:lcm:Age 2 (1 regimes):

  finished in 184.0ms

INFO:lcm:  finished in 184.0ms

Age 1 (1 regimes):

INFO:lcm:Age 1 (1 regimes):

  finished in 26.0ms

INFO:lcm:  finished in 26.0ms

Age 0 (1 regimes):

INFO:lcm:Age 0 (1 regimes):

  finished in 5.8ms

INFO:lcm:  finished in 5.8ms

  dead  age 2.0   V min=-0.693  max=3.91  mean=2.95

DEBUG:lcm:  dead  age 2.0   V min=-0.693  max=3.91  mean=2.95

  working  age 1.0   V min=-2.19  max=6.28  mean=4.41

DEBUG:lcm:  working  age 1.0   V min=-2.19  max=6.28  mean=4.41

  working  age 0.0   V min=-3.73  max=8.03  mean=5.3

DEBUG:lcm:  working  age 0.0   V min=-3.73  max=8.03  mean=5.3

solve call 322764a622de phase backward_induction end status=ok seconds=0.237576

INFO:lcm:solve call 322764a622de phase backward_induction end status=ok seconds=0.237576

Solution complete  (239.3ms)

INFO:lcm:Solution complete  (239.3ms)

solve call 322764a622de phase result_assembly begin

INFO:lcm:solve call 322764a622de phase result_assembly begin

solve call 322764a622de phase result_assembly end status=ok seconds=0.004183

INFO:lcm:solve call 322764a622de phase result_assembly end status=ok seconds=0.004183

solve call 322764a622de phase solution_resolution begin

INFO:lcm:solve call 322764a622de phase solution_resolution begin

solve call 322764a622de phase solution_resolution end status=ok seconds=0.001598

INFO:lcm:solve call 322764a622de phase solution_resolution end status=ok seconds=0.001598

solve call 322764a622de phase solution_handover begin

INFO:lcm:solve call 322764a622de phase solution_handover begin

solve call 322764a622de phase solution_handover end status=ok seconds=0.000004

INFO:lcm:solve call 322764a622de phase solution_handover end status=ok seconds=0.000004

solve call 322764a622de phase chunk_planning begin

INFO:lcm:solve call 322764a622de phase chunk_planning begin

solve call 322764a622de phase chunk_planning end status=ok seconds=0.000333

INFO:lcm:solve call 322764a622de phase chunk_planning end status=ok seconds=0.000333

solve call 322764a622de phase simulation_setup begin

INFO:lcm:solve call 322764a622de phase simulation_setup begin

Starting simulation

INFO:lcm:Starting simulation

Simulation plan: route=legacy; 1 cpu device(s) (0,); outer chunks=1 (widths=(1,)); unbudgeted.

INFO:lcm:Simulation plan: route=legacy; 1 cpu device(s) (0,); outer chunks=1 (widths=(1,)); unbudgeted.

Simulation plan detail: route=legacy; subject_devices=(cpu) (0,); axis_widths_by_regime=[working: {}; dead: {}]; outer_chunk_count=1; admitted_chunk_widths=(1,); budget_mode=unbudgeted; effective_device_memory_bytes=None; subject_grouping=None.

DEBUG:lcm:Simulation plan detail: route=legacy; subject_devices=(cpu) (0,); axis_widths_by_regime=[working: {}; dead: {}]; outer_chunk_count=1; admitted_chunk_widths=(1,); budget_mode=unbudgeted; effective_device_memory_bytes=None; subject_grouping=None.

solve call 322764a622de phase simulation_setup end status=ok seconds=0.090848

INFO:lcm:solve call 322764a622de phase simulation_setup end status=ok seconds=0.090848

solve call 322764a622de phase simulation_chunk begin

INFO:lcm:solve call 322764a622de phase simulation_chunk begin

Age 0 (1 regimes):

INFO:lcm:Age 0 (1 regimes):

solve call 322764a622de phase simulation_compilation begin

INFO:lcm:solve call 322764a622de phase simulation_compilation begin

solve call 322764a622de phase simulation_compilation end status=ok seconds=0.300657

INFO:lcm:solve call 322764a622de phase simulation_compilation end status=ok seconds=0.300657

solve call 322764a622de phase simulation_compilation begin

INFO:lcm:solve call 322764a622de phase simulation_compilation begin

solve call 322764a622de phase simulation_compilation end status=ok seconds=0.022588

INFO:lcm:solve call 322764a622de phase simulation_compilation end status=ok seconds=0.022588

solve call 322764a622de phase simulation_compilation begin

INFO:lcm:solve call 322764a622de phase simulation_compilation begin

solve call 322764a622de phase simulation_compilation end status=ok seconds=0.031964

INFO:lcm:solve call 322764a622de phase simulation_compilation end status=ok seconds=0.031964

Off-grid next states: regime='working', target='working', period=0, state='wealth', phase='realized', count=0

INFO:lcm:Off-grid next states: regime='working', target='working', period=0, state='wealth', phase='realized', count=0

  transitions:
  - working → working = 1

DEBUG:lcm:  transitions:
  - working → working = 1

  finished in 735.6ms

INFO:lcm:  finished in 735.6ms

Age 1 (1 regimes):

INFO:lcm:Age 1 (1 regimes):

solve call 322764a622de phase simulation_compilation begin

INFO:lcm:solve call 322764a622de phase simulation_compilation begin

solve call 322764a622de phase simulation_compilation end status=ok seconds=0.223009

INFO:lcm:solve call 322764a622de phase simulation_compilation end status=ok seconds=0.223009

solve call 322764a622de phase simulation_compilation begin

INFO:lcm:solve call 322764a622de phase simulation_compilation begin

solve call 322764a622de phase simulation_compilation end status=ok seconds=0.025291

INFO:lcm:solve call 322764a622de phase simulation_compilation end status=ok seconds=0.025291

Off-grid next states: regime='working', target='dead', period=1, state='wealth', phase='realized', count=0

INFO:lcm:Off-grid next states: regime='working', target='dead', period=1, state='wealth', phase='realized', count=0

  transitions:
  - working → dead = 1

DEBUG:lcm:  transitions:
  - working → dead = 1

  finished in 336.7ms

INFO:lcm:  finished in 336.7ms

Age 2 (1 regimes):

INFO:lcm:Age 2 (1 regimes):

solve call 322764a622de phase simulation_compilation begin

INFO:lcm:solve call 322764a622de phase simulation_compilation begin

solve call 322764a622de phase simulation_compilation end status=ok seconds=0.029320

INFO:lcm:solve call 322764a622de phase simulation_compilation end status=ok seconds=0.029320

  transitions:
  - dead → dead = 1

DEBUG:lcm:  transitions:
  - dead → dead = 1

  finished in 40.3ms

INFO:lcm:  finished in 40.3ms

solve call 322764a622de phase simulation_chunk end status=ok seconds=1.124300

INFO:lcm:solve call 322764a622de phase simulation_chunk end status=ok seconds=1.124300

solve call 322764a622de phase simulation_completion begin

INFO:lcm:solve call 322764a622de phase simulation_completion begin

Simulation complete  (1.2s)

INFO:lcm:Simulation complete  (1.2s)

solve call 322764a622de phase simulation_completion end status=ok seconds=0.000828

INFO:lcm:solve call 322764a622de phase simulation_completion end status=ok seconds=0.000828

solve call 322764a622de phase simulation_result begin

INFO:lcm:solve call 322764a622de phase simulation_result begin

solve call 322764a622de phase simulation_result end status=ok seconds=0.006066

INFO:lcm:solve call 322764a622de phase simulation_result end status=ok seconds=0.006066

solve call 322764a622de phase result_finalization begin

INFO:lcm:solve call 322764a622de phase result_finalization begin

solve call 322764a622de phase result_finalization end status=ok seconds=0.000009

INFO:lcm:solve call 322764a622de phase result_finalization end status=ok seconds=0.000009

solve call 322764a622de phase public_simulate end status=ok seconds=2.974445

INFO:lcm:solve call 322764a622de phase public_simulate end status=ok seconds=2.974445

Exponential agent:
  c_0 = 7.0149  (analytical: 7.0114)
  c_1 = 6.7143  (analytical: 6.6608)

Naive Agent¶

The naive agent uses model_naive, whose $W$ is phase-variant. During backward induction, the solve variant (LinearAggregator) computes the perceived value function $V^E$ ; during simulation, the simulate variant (linear_beta_delta) applies present bias for action selection.

The params dict is the union of both variants’ parameters — discount_factor for LinearAggregator, plus beta and delta for linear_beta_delta. Each variant receives only the kwargs it expects.

result_naive = model_naive.simulate(
    params={
        "working": {
            "koopmans_aggregator": {
                "discount_factor": DELTA,
                "beta": BETA,
                "delta": DELTA,
            },
        },
    },
    initial_conditions=initial_conditions,
    log_level="debug",
)

df_naive = result_naive.to_dataframe().query('regime_name == "working"')
print("Naive agent:")
print(
    f"  c_0 = {df_naive.loc[df_naive['age'] == 0, 'consumption'].iloc[0]:.4f}  "
    f"(analytical: {c0_naive:.4f})"
)
print(
    f"  c_1 = {df_naive.loc[df_naive['age'] == 1, 'consumption'].iloc[0]:.4f}  "
    f"(analytical: {c1_naive:.4f})"
)

solve call 2399503668bd phase public_simulate begin

INFO:lcm:solve call 2399503668bd phase public_simulate begin

solve call 2399503668bd phase params_validation begin

INFO:lcm:solve call 2399503668bd phase params_validation begin

solve call 2399503668bd phase params_validation end status=ok seconds=0.015865

INFO:lcm:solve call 2399503668bd phase params_validation end status=ok seconds=0.015865

solve call 2399503668bd phase simulation_inputs begin

INFO:lcm:solve call 2399503668bd phase simulation_inputs begin

solve call 2399503668bd phase simulation_inputs end status=ok seconds=0.004833

INFO:lcm:solve call 2399503668bd phase simulation_inputs end status=ok seconds=0.004833

solve call 2399503668bd phase authority_fingerprint begin

INFO:lcm:solve call 2399503668bd phase authority_fingerprint begin

solve call 2399503668bd phase authority_fingerprint end status=ok seconds=0.001838

INFO:lcm:solve call 2399503668bd phase authority_fingerprint end status=ok seconds=0.001838

solve call 2399503668bd phase solver_param_checks begin

INFO:lcm:solve call 2399503668bd phase solver_param_checks begin

solve call 2399503668bd phase solver_param_checks end status=ok seconds=0.000029

INFO:lcm:solve call 2399503668bd phase solver_param_checks end status=ok seconds=0.000029

solve call 2399503668bd phase state_action_spaces begin

INFO:lcm:solve call 2399503668bd phase state_action_spaces begin

solve call 2399503668bd phase state_action_spaces end status=ok seconds=0.000158

INFO:lcm:solve call 2399503668bd phase state_action_spaces end status=ok seconds=0.000158

solve call 2399503668bd phase continuation_templates begin

INFO:lcm:solve call 2399503668bd phase continuation_templates begin

solve call 2399503668bd phase continuation_templates end status=ok seconds=0.000364

INFO:lcm:solve call 2399503668bd phase continuation_templates end status=ok seconds=0.000364

solve call 2399503668bd phase program_graphs begin

INFO:lcm:solve call 2399503668bd phase program_graphs begin

solve call 2399503668bd phase program_graphs end status=ok seconds=0.000246

INFO:lcm:solve call 2399503668bd phase program_graphs end status=ok seconds=0.000246

solve call 2399503668bd phase structural_resolution begin

INFO:lcm:solve call 2399503668bd phase structural_resolution begin

structural blueprint: miss (hits=0, misses=1, entries=1)

DEBUG:lcm:structural blueprint: miss (hits=0, misses=1, entries=1)

solve call 2399503668bd phase structural_resolution end status=ok seconds=0.005939

INFO:lcm:solve call 2399503668bd phase structural_resolution end status=ok seconds=0.005939

solve call 2399503668bd phase residency_inventory begin

INFO:lcm:solve call 2399503668bd phase residency_inventory begin

solve call 2399503668bd phase residency_inventory end status=ok seconds=0.000007

INFO:lcm:solve call 2399503668bd phase residency_inventory end status=ok seconds=0.000007

solve call 2399503668bd phase compilation_waves begin

INFO:lcm:solve call 2399503668bd phase compilation_waves begin

candidate evaluation ('working', 0, 'main') {'action_product': 64, 'cell': 128}

DEBUG:lcm:candidate evaluation ('working', 0, 'main') {'action_product': 64, 'cell': 128}

candidate evaluation ('working', 1, 'main') {'action_product': 64, 'cell': 128}

DEBUG:lcm:candidate evaluation ('working', 1, 'main') {'action_product': 64, 'cell': 128}

candidate evaluation ('dead', 2, 'main') {'cell': 128}

DEBUG:lcm:candidate evaluation ('dead', 2, 'main') {'cell': 128}

AOT compilation wave 0: 3 unique lowerings for 3 regime-period-core triples (2 workers)

INFO:lcm:AOT compilation wave 0: 3 unique lowerings for 3 regime-period-core triples (2 workers)

1/3  working main (age 0, widths={'action_product': 64, 'cell': 128})

INFO:lcm:1/3  working main (age 0, widths={'action_product': 64, 'cell': 128})

  lowering working main (age 0, widths={'action_product': 64, 'cell': 128}) ...

INFO:lcm:  lowering working main (age 0, widths={'action_product': 64, 'cell': 128}) ...

  lowered in 152.0ms

INFO:lcm:  lowered in 152.0ms

  compiling working main (age 0, widths={'action_product': 64, 'cell': 128}) ...

INFO:lcm:  compiling working main (age 0, widths={'action_product': 64, 'cell': 128}) ...

2/3  working main (age 1, widths={'action_product': 64, 'cell': 128})

INFO:lcm:2/3  working main (age 1, widths={'action_product': 64, 'cell': 128})

  lowering working main (age 1, widths={'action_product': 64, 'cell': 128}) ...

INFO:lcm:  lowering working main (age 1, widths={'action_product': 64, 'cell': 128}) ...

  compiled  working main (age 0, widths={'action_product': 64, 'cell': 128})  43.6ms

INFO:lcm:  compiled  working main (age 0, widths={'action_product': 64, 'cell': 128})  43.6ms

  lowered in 154.3ms

INFO:lcm:  lowered in 154.3ms

  compiling working main (age 1, widths={'action_product': 64, 'cell': 128}) ...

INFO:lcm:  compiling working main (age 1, widths={'action_product': 64, 'cell': 128}) ...
3/3  dead main (age 2, widths={'cell': 128})

INFO:lcm:3/3  dead main (age 2, widths={'cell': 128})

  lowering dead main (age 2, widths={'cell': 128}) ...

INFO:lcm:  lowering dead main (age 2, widths={'cell': 128}) ...

  lowered in 25.2ms

INFO:lcm:  lowered in 25.2ms
  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  32.6ms

INFO:lcm:  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  32.6ms
  compiling dead main (age 2, widths={'cell': 128}) ...

INFO:lcm:  compiling dead main (age 2, widths={'cell': 128}) ...

  compiled  dead main (age 2, widths={'cell': 128})  3.7ms

INFO:lcm:  compiled  dead main (age 2, widths={'cell': 128})  3.7ms

AOT compilation wave 0: 3 unique variants compiled, 0 fallback evaluations skipped, 0 fallback requests avoided

INFO:lcm:AOT compilation wave 0: 3 unique variants compiled, 0 fallback evaluations skipped, 0 fallback requests avoided

solve call 2399503668bd phase compilation_waves end status=ok seconds=0.365408

INFO:lcm:solve call 2399503668bd phase compilation_waves end status=ok seconds=0.365408

solve call 2399503668bd phase workspace_selection begin

INFO:lcm:solve call 2399503668bd phase workspace_selection begin

core plan record {"active_replica_bytes":0,"axis_extents":{"action_product":500,"cell":200},"compiled_collectives":{},"compiler_peak_bytes":4491,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"action_product":"streamed","cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":0,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"working","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"action_product":64,"cell":128}}

DEBUG:lcm:core plan record {"active_replica_bytes":0,"axis_extents":{"action_product":500,"cell":200},"compiled_collectives":{},"compiler_peak_bytes":4491,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"action_product":"streamed","cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":0,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"working","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"action_product":64,"cell":128}}

core plan record {"active_replica_bytes":0,"axis_extents":{"action_product":500,"cell":200},"compiled_collectives":{},"compiler_peak_bytes":4491,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"action_product":"streamed","cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":1,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"working","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"action_product":64,"cell":128}}

DEBUG:lcm:core plan record {"active_replica_bytes":0,"axis_extents":{"action_product":500,"cell":200},"compiled_collectives":{},"compiler_peak_bytes":4491,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"action_product":"streamed","cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":1,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"working","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"action_product":64,"cell":128}}

core plan record {"active_replica_bytes":0,"axis_extents":{"cell":200},"compiled_collectives":{},"compiler_peak_bytes":1636,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":2,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"dead","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"cell":128}}

DEBUG:lcm:core plan record {"active_replica_bytes":0,"axis_extents":{"cell":200},"compiled_collectives":{},"compiler_peak_bytes":1636,"compiler_reservation_bytes":null,"core":"main","device_ids":[0],"dispatch":{"cell":"streamed"},"logical_gathered_bytes":0,"logical_value_shape":[200],"period":2,"period_transfer_scratch_bytes":{},"physical_value_shape":[200],"regime":"dead","resident_bytes":null,"selected_block":null,"state_extents":{"wealth":200},"stored_owner_bytes":{},"transfer_workspace_bytes":0,"widths":{"cell":128}}

solve call 2399503668bd phase workspace_selection end status=ok seconds=0.013347

INFO:lcm:solve call 2399503668bd phase workspace_selection end status=ok seconds=0.013347

Starting solution

INFO:lcm:Starting solution

solve call 2399503668bd phase backward_induction begin

INFO:lcm:solve call 2399503668bd phase backward_induction begin

Age 2 (1 regimes):

INFO:lcm:Age 2 (1 regimes):

  finished in 3.8ms

INFO:lcm:  finished in 3.8ms

Age 1 (1 regimes):

INFO:lcm:Age 1 (1 regimes):

  finished in 6.2ms

INFO:lcm:  finished in 6.2ms

Age 0 (1 regimes):

INFO:lcm:Age 0 (1 regimes):

  finished in 5.7ms

INFO:lcm:  finished in 5.7ms

  dead  age 2.0   V min=-0.693  max=3.91  mean=2.95

DEBUG:lcm:  dead  age 2.0   V min=-0.693  max=3.91  mean=2.95

  working  age 1.0   V min=-2.19  max=6.28  mean=4.41

DEBUG:lcm:  working  age 1.0   V min=-2.19  max=6.28  mean=4.41

  working  age 0.0   V min=-3.73  max=8.03  mean=5.3

DEBUG:lcm:  working  age 0.0   V min=-3.73  max=8.03  mean=5.3

solve call 2399503668bd phase backward_induction end status=ok seconds=0.020500

INFO:lcm:solve call 2399503668bd phase backward_induction end status=ok seconds=0.020500

Solution complete  (21.8ms)

INFO:lcm:Solution complete  (21.8ms)

solve call 2399503668bd phase result_assembly begin

INFO:lcm:solve call 2399503668bd phase result_assembly begin

solve call 2399503668bd phase result_assembly end status=ok seconds=0.001883

INFO:lcm:solve call 2399503668bd phase result_assembly end status=ok seconds=0.001883

solve call 2399503668bd phase solution_resolution begin

INFO:lcm:solve call 2399503668bd phase solution_resolution begin

solve call 2399503668bd phase solution_resolution end status=ok seconds=0.000998

INFO:lcm:solve call 2399503668bd phase solution_resolution end status=ok seconds=0.000998

solve call 2399503668bd phase solution_handover begin

INFO:lcm:solve call 2399503668bd phase solution_handover begin

solve call 2399503668bd phase solution_handover end status=ok seconds=0.000004

INFO:lcm:solve call 2399503668bd phase solution_handover end status=ok seconds=0.000004

solve call 2399503668bd phase chunk_planning begin

INFO:lcm:solve call 2399503668bd phase chunk_planning begin

solve call 2399503668bd phase chunk_planning end status=ok seconds=0.000362

INFO:lcm:solve call 2399503668bd phase chunk_planning end status=ok seconds=0.000362

solve call 2399503668bd phase simulation_setup begin

INFO:lcm:solve call 2399503668bd phase simulation_setup begin

Starting simulation

INFO:lcm:Starting simulation

Simulation plan: route=legacy; 1 cpu device(s) (0,); outer chunks=1 (widths=(1,)); unbudgeted.

INFO:lcm:Simulation plan: route=legacy; 1 cpu device(s) (0,); outer chunks=1 (widths=(1,)); unbudgeted.

Simulation plan detail: route=legacy; subject_devices=(cpu) (0,); axis_widths_by_regime=[working: {}; dead: {}]; outer_chunk_count=1; admitted_chunk_widths=(1,); budget_mode=unbudgeted; effective_device_memory_bytes=None; subject_grouping=None.

DEBUG:lcm:Simulation plan detail: route=legacy; subject_devices=(cpu) (0,); axis_widths_by_regime=[working: {}; dead: {}]; outer_chunk_count=1; admitted_chunk_widths=(1,); budget_mode=unbudgeted; effective_device_memory_bytes=None; subject_grouping=None.

solve call 2399503668bd phase simulation_setup end status=ok seconds=0.006132

INFO:lcm:solve call 2399503668bd phase simulation_setup end status=ok seconds=0.006132

solve call 2399503668bd phase simulation_chunk begin

INFO:lcm:solve call 2399503668bd phase simulation_chunk begin

Age 0 (1 regimes):

INFO:lcm:Age 0 (1 regimes):

solve call 2399503668bd phase simulation_compilation begin

INFO:lcm:solve call 2399503668bd phase simulation_compilation begin

solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.240583

INFO:lcm:solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.240583

solve call 2399503668bd phase simulation_compilation begin

INFO:lcm:solve call 2399503668bd phase simulation_compilation begin

solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.009206

INFO:lcm:solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.009206

solve call 2399503668bd phase simulation_compilation begin

INFO:lcm:solve call 2399503668bd phase simulation_compilation begin

solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.011567

INFO:lcm:solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.011567

Off-grid next states: regime='working', target='working', period=0, state='wealth', phase='realized', count=0

INFO:lcm:Off-grid next states: regime='working', target='working', period=0, state='wealth', phase='realized', count=0

  transitions:
  - working → working = 1

DEBUG:lcm:  transitions:
  - working → working = 1

  finished in 286.2ms

INFO:lcm:  finished in 286.2ms

Age 1 (1 regimes):

INFO:lcm:Age 1 (1 regimes):

solve call 2399503668bd phase simulation_compilation begin

INFO:lcm:solve call 2399503668bd phase simulation_compilation begin

solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.235358

INFO:lcm:solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.235358

solve call 2399503668bd phase simulation_compilation begin

INFO:lcm:solve call 2399503668bd phase simulation_compilation begin

solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.011655

INFO:lcm:solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.011655

Off-grid next states: regime='working', target='dead', period=1, state='wealth', phase='realized', count=0

INFO:lcm:Off-grid next states: regime='working', target='dead', period=1, state='wealth', phase='realized', count=0

  transitions:
  - working → dead = 1

DEBUG:lcm:  transitions:
  - working → dead = 1

  finished in 270.5ms

INFO:lcm:  finished in 270.5ms

Age 2 (1 regimes):

INFO:lcm:Age 2 (1 regimes):

solve call 2399503668bd phase simulation_compilation begin

INFO:lcm:solve call 2399503668bd phase simulation_compilation begin

solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.010726

INFO:lcm:solve call 2399503668bd phase simulation_compilation end status=ok seconds=0.010726

  transitions:
  - dead → dead = 1

DEBUG:lcm:  transitions:
  - dead → dead = 1

  finished in 19.4ms

INFO:lcm:  finished in 19.4ms

solve call 2399503668bd phase simulation_chunk end status=ok seconds=0.579226

INFO:lcm:solve call 2399503668bd phase simulation_chunk end status=ok seconds=0.579226

solve call 2399503668bd phase simulation_completion begin

INFO:lcm:solve call 2399503668bd phase simulation_completion begin

Simulation complete  (587.4ms)

INFO:lcm:Simulation complete  (587.4ms)

solve call 2399503668bd phase simulation_completion end status=ok seconds=0.000559

INFO:lcm:solve call 2399503668bd phase simulation_completion end status=ok seconds=0.000559

solve call 2399503668bd phase simulation_result begin

INFO:lcm:solve call 2399503668bd phase simulation_result begin

solve call 2399503668bd phase simulation_result end status=ok seconds=0.000694

INFO:lcm:solve call 2399503668bd phase simulation_result end status=ok seconds=0.000694

solve call 2399503668bd phase result_finalization begin

INFO:lcm:solve call 2399503668bd phase result_finalization begin

solve call 2399503668bd phase result_finalization end status=ok seconds=0.000008

INFO:lcm:solve call 2399503668bd phase result_finalization end status=ok seconds=0.000008

solve call 2399503668bd phase public_simulate end status=ok seconds=1.050539

INFO:lcm:solve call 2399503668bd phase public_simulate end status=ok seconds=1.050539

Naive agent:
  c_0 = 8.7183  (analytical: 8.7080)
  c_1 = 6.8145  (analytical: 6.7820)

Comparison¶

The small differences between numerical and analytical solutions are due to grid discretization.

import pandas as pd

comparison = pd.DataFrame(
    {
        "Agent": ["Exponential", "Naive"],
        "c_0 (numerical)": [
            df_exp.loc[df_exp["age"] == 0, "consumption"].iloc[0],
            df_naive.loc[df_naive["age"] == 0, "consumption"].iloc[0],
        ],
        "c_0 (analytical)": [c0_exp, c0_naive],
        "c_1 (numerical)": [
            df_exp.loc[df_exp["age"] == 1, "consumption"].iloc[0],
            df_naive.loc[df_naive["age"] == 1, "consumption"].iloc[0],
        ],
        "c_1 (analytical)": [c1_exp, c1_naive],
    }
)
comparison = comparison.set_index("Agent")
comparison.style.format("{:.4f}")

Sophisticated and Partially Naive Agents¶

The naive agent stores the value it chooses with: its solve phase maximizes the exponential $W$ , and the stored $V^E$ is that maximum. A sophisticated agent knows its future selves are present-biased. The solve then has two jobs that need different aggregators:

choose the action each future self takes, with $u + \beta\delta \cdot \mathrm{CE}$ ;
store the long-run value of that action, with $u + \delta \cdot \mathrm{CE}$ , which is what the current self uses to price the future.

Decoupled(objective=..., value=...) declares exactly this. The objective ranks the actions, the value is stored at the chosen action, and only the objective may be Phased. One parameter $\tilde\beta$ , the agent’s belief about its future selves’ present bias, spans all three agent types:

$\tilde\beta = 1$ : naive, it expects exponential future selves;
$\tilde\beta = \beta$ : sophisticated, it expects its true present bias;
$\beta < \tilde\beta < 1$ : partially naive.

The acting self always chooses with its true $\beta$ , so the simulate variant of the objective is the linear_beta_delta aggregator from above.

from lcm import Decoupled


def perceived_beta_delta(
    *,
    utility: float,
    CE: float,
    beta_tilde: float,
    delta: float,
) -> float:
    return utility + beta_tilde * delta * CE


def long_run(*, utility: float, CE: float, delta: float) -> float:
    return utility + delta * CE


# The stored value is read off at the action the objective chooses, away from
# the value's own maximum, so grid error in that action enters it at first order.
# Finer grids than for the exponential agent keep it small.
FINE_WEALTH_GRID = LinSpacedGrid(start=0.5, stop=50, n_points=1000)
FINE_CONSUMPTION_GRID = LinSpacedGrid(start=0.001, stop=50, n_points=2000)

working_decoupled = Regime(
    actions={"consumption": FINE_CONSUMPTION_GRID},
    states={"wealth": FINE_WEALTH_GRID},
    state_transitions={"wealth": next_wealth},
    constraints={"borrowing_constraint": borrowing_constraint},
    functions={"utility": utility},
    koopmans_aggregator=Decoupled(
        objective=Phased(solve=perceived_beta_delta, simulate=linear_beta_delta),
        value=long_run,
    ),
)

model_decoupled = Model(
    regimes={
        "working": working_decoupled,
        "dead": Regime(
            states={"wealth": FINE_WEALTH_GRID},
            functions={"utility": terminal_utility},
        ),
    },
    ages=AgeGrid(start=0, inclusive_stop=2, step="Y"),
    edges=edges,
    initial_nodes=((0, "working"),),
    regime_id_class=RegimeId,
)

# Parameters shared by name across the three aggregators are one parameter.
model_decoupled.get_params_template()["working"]["koopmans_aggregator"]

{'beta': 'float', 'beta_tilde': 'float', 'delta': 'float'}

Analytical Solution¶

With log utility all three agents consume the same at $t = 0$ (Pollak, 1968). At $t = 1$ the agent expects to consume $w_1 / (1 + \tilde\beta\delta)$ , so the long-run value it stores is

V_1(w) = (1 + \delta) \log w + k(\tilde\beta), \qquad k(b) = -\log(1 + b\delta) + \delta \log \frac{b\delta}{1 + b\delta}.

(6)

The slope in $\log w$ does not depend on $\tilde\beta$ , so neither does the choice $c_0 = w_0 / (1 + \beta\delta(1 + \delta))$ . The beliefs show up in the acting self’s assessment at $t = 0$ , $V_0 = \log c_0 + \delta V_1(w_0 - c_0)$ , the value simulation publishes. The solve’s own $V_0$ chooses $c_0$ with $\tilde\beta$ instead.

def analytical_value_at_t0(*, beta, beta_tilde, delta, w0):
    """Return the long-run value at t=0 that an agent with belief beta_tilde stores."""
    c0 = w0 / (1 + beta * delta * (1 + delta))
    k = -jnp.log(1 + beta_tilde * delta) + delta * jnp.log(
        beta_tilde * delta / (1 + beta_tilde * delta)
    )
    return float(jnp.log(c0) + delta * ((1 + delta) * jnp.log(w0 - c0) + k))


BETA_TILDE = {"naive": 1.0, "partially naive": 0.85, "sophisticated": BETA}

rows = {}
for agent, beta_tilde in BETA_TILDE.items():
    result = model_decoupled.simulate(
        params={
            "working": {
                "koopmans_aggregator": {
                    "beta": BETA,
                    "beta_tilde": beta_tilde,
                    "delta": DELTA,
                },
            },
        },
        initial_conditions=initial_conditions,
        log_level="off",
    )
    first = result.to_dataframe().query('regime_name == "working" and age == 0')
    rows[agent] = {
        "c_0 (numerical)": first["consumption"].iloc[0],
        "c_0 (analytical)": c0_naive,
        "assessed V_0 (simulated)": first["value"].iloc[0],
        "assessed V_0 (analytical)": analytical_value_at_t0(
            beta=BETA, beta_tilde=beta_tilde, delta=DELTA, w0=W0
        ),
    }

pd.DataFrame(rows).T.style.format("{:.4f}")

Analytically all three agents consume the same, and the numbers agree up to grid error. The assessed value falls as the agent expects more present bias from its future selves. Each agent acts on the same rule at $t = 1$ , so the long-run value every one of them actually gets is the sophisticated agent’s.

Keep in mind what the stored value is:

It is a self-assessment. For naive and partially naive agents it is the long-run value of the policy the agent expects to follow. The policy it does follow has a lower long-run value here. Welfare comparisons need that realized value, which the next section computes.
It needs finer grids. The value is read off at the objective’s choice, away from its own maximum, so grid error in the chosen action enters it at first order.
Only GridSearch solves it, and collective regimes and regimes whose transitions carry value-dependent gates refuse it.
Taste shocks work. With ExtremeValueTasteShocks the objective sets the choice probabilities and the stored value is the expected value at the noisy choice.

Realized Welfare¶

The stored value is what the agent expects. The long-run value of the policy it actually follows is $\sum_t \delta^t u_t$ along its simulated path. PyLCM does not compute it by a second backward pass; simulating and discounting the realized utilities estimates it. With deterministic wealth one subject suffices here; with risk, average over many subjects.

def realized_long_run_value(*, beta_tilde):
    """Discount the utilities the agent realizes along its simulated path."""
    result = model_decoupled.simulate(
        params={
            "working": {
                "koopmans_aggregator": {
                    "beta": BETA,
                    "beta_tilde": beta_tilde,
                    "delta": DELTA,
                },
            },
        },
        initial_conditions=initial_conditions,
        log_level="off",
    )
    path = result.to_dataframe(additional_targets=["utility"]).sort_values("age")
    return float((DELTA ** path["age"] * path["utility"]).sum())


pd.DataFrame(
    {
        "assessed V_0": {
            agent: rows[agent]["assessed V_0 (simulated)"] for agent in BETA_TILDE
        },
        "realized long-run value": {
            agent: realized_long_run_value(beta_tilde=beta_tilde)
            for agent, beta_tilde in BETA_TILDE.items()
        },
    }
).style.format("{:.4f}")

The realized values agree across the three agents up to grid error, and they equal the sophisticated agent’s assessed value: only the sophisticated agent forecasts its own behavior correctly. The naive and partially naive agents overstate what they get.

Summary¶

Beta-delta discounting in PyLCM requires no core modifications:

Pair the shipped LinearAggregator with a present-biased variant — only the second one has to be written:

from lcm.koopmans_aggregation import LinearAggregator  # utility + discount_factor * CE

def linear_beta_delta(utility, CE, beta, delta):
    return utility + beta * delta * CE

Declare koopmans_aggregator as phase-variant so backward induction uses the exponential $W$ while action selection uses the beta-delta $W$ :
```
from lcm import Phased

koopmans_aggregator=Phased(
    solve=LinearAggregator(),
    simulate=linear_beta_delta,
)
```

Provide the union of both variants’ parameters:

params = {
    "working": {
        "koopmans_aggregator": {
            "discount_factor": delta,
            "beta": beta,
            "delta": delta,
        }
    }
}

For sophisticated and partially naive agents, decouple the objective from the value, with the agent’s belief $\tilde\beta$ as a parameter:

from lcm import Decoupled

koopmans_aggregator=Decoupled(
    objective=Phased(solve=perceived_beta_delta, simulate=linear_beta_delta),
    value=long_run,
)

The phase split is essential: using $W(u, V') = u + \beta\delta V'$ for both phases would compound $\beta$ at every recursive step, giving effective weights $(\beta\delta)^n$ instead of the correct $\beta\delta^n$ .

Beta-Delta (Quasi-Hyperbolic) Discounting

Background: Beta-Delta Discounting¶

PyLCM’s Koopmans Aggregator and the Naive Agent¶

Setup: A 3-Period Model¶

Analytical Solution¶

Numerical Results¶

Exponential Agent (β=1\beta = 1β=1)¶

Naive Agent¶

Comparison¶

Sophisticated and Partially Naive Agents¶

Analytical Solution¶

Realized Welfare¶

Summary¶

Exponential Agent ( $\beta = 1$ )¶