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 a521f0b998cb phase public_simulate begin

INFO:lcm:solve call a521f0b998cb phase public_simulate begin

solve call a521f0b998cb phase params_validation begin

INFO:lcm:solve call a521f0b998cb phase params_validation begin

solve call a521f0b998cb phase params_validation end status=ok seconds=0.066252

INFO:lcm:solve call a521f0b998cb phase params_validation end status=ok seconds=0.066252

solve call a521f0b998cb phase simulation_inputs begin

INFO:lcm:solve call a521f0b998cb phase simulation_inputs begin

solve call a521f0b998cb phase simulation_inputs end status=ok seconds=0.112160

INFO:lcm:solve call a521f0b998cb phase simulation_inputs end status=ok seconds=0.112160

solve call a521f0b998cb phase authority_fingerprint begin

INFO:lcm:solve call a521f0b998cb phase authority_fingerprint begin

solve call a521f0b998cb phase authority_fingerprint end status=ok seconds=0.004193

INFO:lcm:solve call a521f0b998cb phase authority_fingerprint end status=ok seconds=0.004193

solve call a521f0b998cb phase solver_param_checks begin

INFO:lcm:solve call a521f0b998cb phase solver_param_checks begin

solve call a521f0b998cb phase solver_param_checks end status=ok seconds=0.000036

INFO:lcm:solve call a521f0b998cb phase solver_param_checks end status=ok seconds=0.000036

solve call a521f0b998cb phase state_action_spaces begin

INFO:lcm:solve call a521f0b998cb phase state_action_spaces begin

solve call a521f0b998cb phase state_action_spaces end status=ok seconds=0.000194

INFO:lcm:solve call a521f0b998cb phase state_action_spaces end status=ok seconds=0.000194

solve call a521f0b998cb phase continuation_templates begin

INFO:lcm:solve call a521f0b998cb phase continuation_templates begin

solve call a521f0b998cb phase continuation_templates end status=ok seconds=0.001171

INFO:lcm:solve call a521f0b998cb phase continuation_templates end status=ok seconds=0.001171

solve call a521f0b998cb phase program_graphs begin

INFO:lcm:solve call a521f0b998cb phase program_graphs begin

solve call a521f0b998cb phase program_graphs end status=ok seconds=0.000499

INFO:lcm:solve call a521f0b998cb phase program_graphs end status=ok seconds=0.000499

solve call a521f0b998cb phase structural_resolution begin

INFO:lcm:solve call a521f0b998cb 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 a521f0b998cb phase structural_resolution end status=ok seconds=0.032555

INFO:lcm:solve call a521f0b998cb phase structural_resolution end status=ok seconds=0.032555

solve call a521f0b998cb phase residency_inventory begin

INFO:lcm:solve call a521f0b998cb phase residency_inventory begin

solve call a521f0b998cb phase residency_inventory end status=ok seconds=0.000335

INFO:lcm:solve call a521f0b998cb phase residency_inventory end status=ok seconds=0.000335

solve call a521f0b998cb phase compilation_waves begin

INFO:lcm:solve call a521f0b998cb 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 448.5ms

INFO:lcm:  lowered in 448.5ms

  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 283.1ms

INFO:lcm:  lowered in 283.1ms

  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 48.4ms

INFO:lcm:  lowered in 48.4ms

  compiled  working main (age 0, widths={'action_product': 64, 'cell': 128})  828.0ms

INFO:lcm:  compiled  working main (age 0, widths={'action_product': 64, 'cell': 128})  828.0ms

  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})  185.5ms

INFO:lcm:  compiled  dead main (age 2, widths={'cell': 128})  185.5ms

  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  785.4ms

INFO:lcm:  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  785.4ms

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 a521f0b998cb phase compilation_waves end status=ok seconds=1.557574

INFO:lcm:solve call a521f0b998cb phase compilation_waves end status=ok seconds=1.557574

solve call a521f0b998cb phase workspace_selection begin

INFO:lcm:solve call a521f0b998cb phase workspace_selection begin

core plan record {"active_replica_bytes":0,"axis_extents":{"action_product":500,"cell":200},"compiled_collectives":{},"compiler_peak_bytes":4523,"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":4523,"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":4523,"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":4523,"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 a521f0b998cb phase workspace_selection end status=ok seconds=0.024925

INFO:lcm:solve call a521f0b998cb phase workspace_selection end status=ok seconds=0.024925

Starting solution

INFO:lcm:Starting solution

solve call a521f0b998cb phase backward_induction begin

INFO:lcm:solve call a521f0b998cb phase backward_induction begin

Age 2 (1 regimes):

INFO:lcm:Age 2 (1 regimes):

  finished in 204.0ms

INFO:lcm:  finished in 204.0ms

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 5.9ms

INFO:lcm:  finished in 5.9ms

  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 a521f0b998cb phase backward_induction end status=ok seconds=0.259792

INFO:lcm:solve call a521f0b998cb phase backward_induction end status=ok seconds=0.259792

Solution complete  (261.8ms)

INFO:lcm:Solution complete  (261.8ms)

solve call a521f0b998cb phase result_assembly begin

INFO:lcm:solve call a521f0b998cb phase result_assembly begin

solve call a521f0b998cb phase result_assembly end status=ok seconds=0.004334

INFO:lcm:solve call a521f0b998cb phase result_assembly end status=ok seconds=0.004334

solve call a521f0b998cb phase solution_resolution begin

INFO:lcm:solve call a521f0b998cb phase solution_resolution begin

solve call a521f0b998cb phase solution_resolution end status=ok seconds=0.001874

INFO:lcm:solve call a521f0b998cb phase solution_resolution end status=ok seconds=0.001874

solve call a521f0b998cb phase solution_handover begin

INFO:lcm:solve call a521f0b998cb phase solution_handover begin

solve call a521f0b998cb phase solution_handover end status=ok seconds=0.000004

INFO:lcm:solve call a521f0b998cb phase solution_handover end status=ok seconds=0.000004

solve call a521f0b998cb phase chunk_planning begin

INFO:lcm:solve call a521f0b998cb phase chunk_planning begin

solve call a521f0b998cb phase chunk_planning end status=ok seconds=0.000383

INFO:lcm:solve call a521f0b998cb phase chunk_planning end status=ok seconds=0.000383

solve call a521f0b998cb phase simulation_setup begin

INFO:lcm:solve call a521f0b998cb 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 a521f0b998cb phase simulation_setup end status=ok seconds=0.104077

INFO:lcm:solve call a521f0b998cb phase simulation_setup end status=ok seconds=0.104077

solve call a521f0b998cb phase simulation_chunk begin

INFO:lcm:solve call a521f0b998cb phase simulation_chunk begin

Age 0 (1 regimes):

INFO:lcm:Age 0 (1 regimes):

solve call a521f0b998cb phase simulation_compilation begin

INFO:lcm:solve call a521f0b998cb phase simulation_compilation begin

solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.389969

INFO:lcm:solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.389969

solve call a521f0b998cb phase simulation_compilation begin

INFO:lcm:solve call a521f0b998cb phase simulation_compilation begin

solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.025406

INFO:lcm:solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.025406

solve call a521f0b998cb phase simulation_compilation begin

INFO:lcm:solve call a521f0b998cb phase simulation_compilation begin

solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.034788

INFO:lcm:solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.034788

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 862.6ms

INFO:lcm:  finished in 862.6ms

Age 1 (1 regimes):

INFO:lcm:Age 1 (1 regimes):

solve call a521f0b998cb phase simulation_compilation begin

INFO:lcm:solve call a521f0b998cb phase simulation_compilation begin

solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.302570

INFO:lcm:solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.302570

solve call a521f0b998cb phase simulation_compilation begin

INFO:lcm:solve call a521f0b998cb phase simulation_compilation begin

solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.026522

INFO:lcm:solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.026522

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 429.3ms

INFO:lcm:  finished in 429.3ms

Age 2 (1 regimes):

INFO:lcm:Age 2 (1 regimes):

solve call a521f0b998cb phase simulation_compilation begin

INFO:lcm:solve call a521f0b998cb phase simulation_compilation begin

solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.031791

INFO:lcm:solve call a521f0b998cb phase simulation_compilation end status=ok seconds=0.031791

  transitions:
  - dead → dead = 1

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

  finished in 42.3ms

INFO:lcm:  finished in 42.3ms

solve call a521f0b998cb phase simulation_chunk end status=ok seconds=1.346742

INFO:lcm:solve call a521f0b998cb phase simulation_chunk end status=ok seconds=1.346742

solve call a521f0b998cb phase simulation_completion begin

INFO:lcm:solve call a521f0b998cb phase simulation_completion begin

Simulation complete  (1.5s)

INFO:lcm:Simulation complete  (1.5s)

solve call a521f0b998cb phase simulation_completion end status=ok seconds=0.000798

INFO:lcm:solve call a521f0b998cb phase simulation_completion end status=ok seconds=0.000798

solve call a521f0b998cb phase simulation_result begin

INFO:lcm:solve call a521f0b998cb phase simulation_result begin

solve call a521f0b998cb phase simulation_result end status=ok seconds=0.005843

INFO:lcm:solve call a521f0b998cb phase simulation_result end status=ok seconds=0.005843

solve call a521f0b998cb phase result_finalization begin

INFO:lcm:solve call a521f0b998cb phase result_finalization begin

solve call a521f0b998cb phase result_finalization end status=ok seconds=0.000033

INFO:lcm:solve call a521f0b998cb phase result_finalization end status=ok seconds=0.000033

solve call a521f0b998cb phase public_simulate end status=ok seconds=3.572377

INFO:lcm:solve call a521f0b998cb phase public_simulate end status=ok seconds=3.572377

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 5fc1a637d041 phase public_simulate begin

INFO:lcm:solve call 5fc1a637d041 phase public_simulate begin

solve call 5fc1a637d041 phase params_validation begin

INFO:lcm:solve call 5fc1a637d041 phase params_validation begin

solve call 5fc1a637d041 phase params_validation end status=ok seconds=0.015690

INFO:lcm:solve call 5fc1a637d041 phase params_validation end status=ok seconds=0.015690

solve call 5fc1a637d041 phase simulation_inputs begin

INFO:lcm:solve call 5fc1a637d041 phase simulation_inputs begin

solve call 5fc1a637d041 phase simulation_inputs end status=ok seconds=0.005430

INFO:lcm:solve call 5fc1a637d041 phase simulation_inputs end status=ok seconds=0.005430

solve call 5fc1a637d041 phase authority_fingerprint begin

INFO:lcm:solve call 5fc1a637d041 phase authority_fingerprint begin

solve call 5fc1a637d041 phase authority_fingerprint end status=ok seconds=0.001785

INFO:lcm:solve call 5fc1a637d041 phase authority_fingerprint end status=ok seconds=0.001785

solve call 5fc1a637d041 phase solver_param_checks begin

INFO:lcm:solve call 5fc1a637d041 phase solver_param_checks begin

solve call 5fc1a637d041 phase solver_param_checks end status=ok seconds=0.000036

INFO:lcm:solve call 5fc1a637d041 phase solver_param_checks end status=ok seconds=0.000036

solve call 5fc1a637d041 phase state_action_spaces begin

INFO:lcm:solve call 5fc1a637d041 phase state_action_spaces begin

solve call 5fc1a637d041 phase state_action_spaces end status=ok seconds=0.000196

INFO:lcm:solve call 5fc1a637d041 phase state_action_spaces end status=ok seconds=0.000196

solve call 5fc1a637d041 phase continuation_templates begin

INFO:lcm:solve call 5fc1a637d041 phase continuation_templates begin

solve call 5fc1a637d041 phase continuation_templates end status=ok seconds=0.000327

INFO:lcm:solve call 5fc1a637d041 phase continuation_templates end status=ok seconds=0.000327

solve call 5fc1a637d041 phase program_graphs begin

INFO:lcm:solve call 5fc1a637d041 phase program_graphs begin

solve call 5fc1a637d041 phase program_graphs end status=ok seconds=0.000269

INFO:lcm:solve call 5fc1a637d041 phase program_graphs end status=ok seconds=0.000269

solve call 5fc1a637d041 phase structural_resolution begin

INFO:lcm:solve call 5fc1a637d041 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 5fc1a637d041 phase structural_resolution end status=ok seconds=0.005566

INFO:lcm:solve call 5fc1a637d041 phase structural_resolution end status=ok seconds=0.005566

solve call 5fc1a637d041 phase residency_inventory begin

INFO:lcm:solve call 5fc1a637d041 phase residency_inventory begin

solve call 5fc1a637d041 phase residency_inventory end status=ok seconds=0.000024

INFO:lcm:solve call 5fc1a637d041 phase residency_inventory end status=ok seconds=0.000024

solve call 5fc1a637d041 phase compilation_waves begin

INFO:lcm:solve call 5fc1a637d041 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 185.3ms

INFO:lcm:  lowered in 185.3ms

  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})  87.4ms

INFO:lcm:  compiled  working main (age 0, widths={'action_product': 64, 'cell': 128})  87.4ms

  lowered in 198.0ms

INFO:lcm:  lowered in 198.0ms

  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 27.9ms

INFO:lcm:  lowered in 27.9ms

  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})  5.4ms

INFO:lcm:  compiled  dead main (age 2, widths={'cell': 128})  5.4ms

  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  51.0ms

INFO:lcm:  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  51.0ms

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 5fc1a637d041 phase compilation_waves end status=ok seconds=0.452543

INFO:lcm:solve call 5fc1a637d041 phase compilation_waves end status=ok seconds=0.452543

solve call 5fc1a637d041 phase workspace_selection begin

INFO:lcm:solve call 5fc1a637d041 phase workspace_selection begin

core plan record {"active_replica_bytes":0,"axis_extents":{"action_product":500,"cell":200},"compiled_collectives":{},"compiler_peak_bytes":4523,"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":4523,"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":4523,"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":4523,"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 5fc1a637d041 phase workspace_selection end status=ok seconds=0.019376

INFO:lcm:solve call 5fc1a637d041 phase workspace_selection end status=ok seconds=0.019376

Starting solution

INFO:lcm:Starting solution

solve call 5fc1a637d041 phase backward_induction begin

INFO:lcm:solve call 5fc1a637d041 phase backward_induction begin

Age 2 (1 regimes):

INFO:lcm:Age 2 (1 regimes):

  finished in 4.1ms

INFO:lcm:  finished in 4.1ms

Age 1 (1 regimes):

INFO:lcm:Age 1 (1 regimes):

  finished in 5.9ms

INFO:lcm:  finished in 5.9ms

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 5fc1a637d041 phase backward_induction end status=ok seconds=0.020447

INFO:lcm:solve call 5fc1a637d041 phase backward_induction end status=ok seconds=0.020447

Solution complete  (21.4ms)

INFO:lcm:Solution complete  (21.4ms)

solve call 5fc1a637d041 phase result_assembly begin

INFO:lcm:solve call 5fc1a637d041 phase result_assembly begin

solve call 5fc1a637d041 phase result_assembly end status=ok seconds=0.001970

INFO:lcm:solve call 5fc1a637d041 phase result_assembly end status=ok seconds=0.001970

solve call 5fc1a637d041 phase solution_resolution begin

INFO:lcm:solve call 5fc1a637d041 phase solution_resolution begin

solve call 5fc1a637d041 phase solution_resolution end status=ok seconds=0.001045

INFO:lcm:solve call 5fc1a637d041 phase solution_resolution end status=ok seconds=0.001045

solve call 5fc1a637d041 phase solution_handover begin

INFO:lcm:solve call 5fc1a637d041 phase solution_handover begin

solve call 5fc1a637d041 phase solution_handover end status=ok seconds=0.000004

INFO:lcm:solve call 5fc1a637d041 phase solution_handover end status=ok seconds=0.000004

solve call 5fc1a637d041 phase chunk_planning begin

INFO:lcm:solve call 5fc1a637d041 phase chunk_planning begin

solve call 5fc1a637d041 phase chunk_planning end status=ok seconds=0.000366

INFO:lcm:solve call 5fc1a637d041 phase chunk_planning end status=ok seconds=0.000366

solve call 5fc1a637d041 phase simulation_setup begin

INFO:lcm:solve call 5fc1a637d041 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 5fc1a637d041 phase simulation_setup end status=ok seconds=0.003990

INFO:lcm:solve call 5fc1a637d041 phase simulation_setup end status=ok seconds=0.003990

solve call 5fc1a637d041 phase simulation_chunk begin

INFO:lcm:solve call 5fc1a637d041 phase simulation_chunk begin

Age 0 (1 regimes):

INFO:lcm:Age 0 (1 regimes):

solve call 5fc1a637d041 phase simulation_compilation begin

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation begin

solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.314124

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.314124

solve call 5fc1a637d041 phase simulation_compilation begin

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation begin

solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.009612

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.009612

solve call 5fc1a637d041 phase simulation_compilation begin

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation begin

solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.012336

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.012336

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 362.4ms

INFO:lcm:  finished in 362.4ms

Age 1 (1 regimes):

INFO:lcm:Age 1 (1 regimes):

solve call 5fc1a637d041 phase simulation_compilation begin

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation begin

solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.312811

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.312811

solve call 5fc1a637d041 phase simulation_compilation begin

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation begin

solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.012101

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.012101

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 349.7ms

INFO:lcm:  finished in 349.7ms

Age 2 (1 regimes):

INFO:lcm:Age 2 (1 regimes):

solve call 5fc1a637d041 phase simulation_compilation begin

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation begin

solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.011453

INFO:lcm:solve call 5fc1a637d041 phase simulation_compilation end status=ok seconds=0.011453

  transitions:
  - dead → dead = 1

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

  finished in 20.9ms

INFO:lcm:  finished in 20.9ms

solve call 5fc1a637d041 phase simulation_chunk end status=ok seconds=0.737599

INFO:lcm:solve call 5fc1a637d041 phase simulation_chunk end status=ok seconds=0.737599

solve call 5fc1a637d041 phase simulation_completion begin

INFO:lcm:solve call 5fc1a637d041 phase simulation_completion begin

Simulation complete  (744.3ms)

INFO:lcm:Simulation complete  (744.3ms)

solve call 5fc1a637d041 phase simulation_completion end status=ok seconds=0.000772

INFO:lcm:solve call 5fc1a637d041 phase simulation_completion end status=ok seconds=0.000772

solve call 5fc1a637d041 phase simulation_result begin

INFO:lcm:solve call 5fc1a637d041 phase simulation_result begin

solve call 5fc1a637d041 phase simulation_result end status=ok seconds=0.000756

INFO:lcm:solve call 5fc1a637d041 phase simulation_result end status=ok seconds=0.000756

solve call 5fc1a637d041 phase result_finalization begin

INFO:lcm:solve call 5fc1a637d041 phase result_finalization begin

solve call 5fc1a637d041 phase result_finalization end status=ok seconds=0.000008

INFO:lcm:solve call 5fc1a637d041 phase result_finalization end status=ok seconds=0.000008

solve call 5fc1a637d041 phase public_simulate end status=ok seconds=1.299981

INFO:lcm:solve call 5fc1a637d041 phase public_simulate end status=ok seconds=1.299981

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 ( $\beta = 1$ )¶