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 474c5cade967 phase public_simulate begin

INFO:lcm:solve call 474c5cade967 phase public_simulate begin

solve call 474c5cade967 phase params_validation begin

INFO:lcm:solve call 474c5cade967 phase params_validation begin

solve call 474c5cade967 phase params_validation end status=ok seconds=0.061298

INFO:lcm:solve call 474c5cade967 phase params_validation end status=ok seconds=0.061298

solve call 474c5cade967 phase simulation_inputs begin

INFO:lcm:solve call 474c5cade967 phase simulation_inputs begin

solve call 474c5cade967 phase simulation_inputs end status=ok seconds=0.094705

INFO:lcm:solve call 474c5cade967 phase simulation_inputs end status=ok seconds=0.094705

solve call 474c5cade967 phase authority_fingerprint begin

INFO:lcm:solve call 474c5cade967 phase authority_fingerprint begin

solve call 474c5cade967 phase authority_fingerprint end status=ok seconds=0.003770

INFO:lcm:solve call 474c5cade967 phase authority_fingerprint end status=ok seconds=0.003770

solve call 474c5cade967 phase solver_param_checks begin

INFO:lcm:solve call 474c5cade967 phase solver_param_checks begin

solve call 474c5cade967 phase solver_param_checks end status=ok seconds=0.000030

INFO:lcm:solve call 474c5cade967 phase solver_param_checks end status=ok seconds=0.000030

solve call 474c5cade967 phase state_action_spaces begin

INFO:lcm:solve call 474c5cade967 phase state_action_spaces begin

solve call 474c5cade967 phase state_action_spaces end status=ok seconds=0.000188

INFO:lcm:solve call 474c5cade967 phase state_action_spaces end status=ok seconds=0.000188

solve call 474c5cade967 phase continuation_templates begin

INFO:lcm:solve call 474c5cade967 phase continuation_templates begin

solve call 474c5cade967 phase continuation_templates end status=ok seconds=0.001099

INFO:lcm:solve call 474c5cade967 phase continuation_templates end status=ok seconds=0.001099

solve call 474c5cade967 phase program_graphs begin

INFO:lcm:solve call 474c5cade967 phase program_graphs begin

solve call 474c5cade967 phase program_graphs end status=ok seconds=0.000544

INFO:lcm:solve call 474c5cade967 phase program_graphs end status=ok seconds=0.000544

solve call 474c5cade967 phase structural_resolution begin

INFO:lcm:solve call 474c5cade967 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 474c5cade967 phase structural_resolution end status=ok seconds=0.030813

INFO:lcm:solve call 474c5cade967 phase structural_resolution end status=ok seconds=0.030813

solve call 474c5cade967 phase residency_inventory begin

INFO:lcm:solve call 474c5cade967 phase residency_inventory begin

solve call 474c5cade967 phase residency_inventory end status=ok seconds=0.000007

INFO:lcm:solve call 474c5cade967 phase residency_inventory end status=ok seconds=0.000007

solve call 474c5cade967 phase compilation_waves begin

INFO:lcm:solve call 474c5cade967 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 409.8ms

INFO:lcm:  lowered in 409.8ms

  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 256.6ms

INFO:lcm:  lowered in 256.6ms

  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 84.7ms

INFO:lcm:  lowered in 84.7ms

  compiled  working main (age 0, widths={'action_product': 64, 'cell': 128})  731.7ms

INFO:lcm:  compiled  working main (age 0, widths={'action_product': 64, 'cell': 128})  731.7ms

  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})  265.1ms

INFO:lcm:  compiled  dead main (age 2, widths={'cell': 128})  265.1ms
  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  726.1ms

INFO:lcm:  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  726.1ms

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 474c5cade967 phase compilation_waves end status=ok seconds=1.427443

INFO:lcm:solve call 474c5cade967 phase compilation_waves end status=ok seconds=1.427443

solve call 474c5cade967 phase workspace_selection begin

INFO:lcm:solve call 474c5cade967 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 474c5cade967 phase workspace_selection end status=ok seconds=0.022440

INFO:lcm:solve call 474c5cade967 phase workspace_selection end status=ok seconds=0.022440

Starting solution

INFO:lcm:Starting solution

solve call 474c5cade967 phase backward_induction begin

INFO:lcm:solve call 474c5cade967 phase backward_induction begin

Age 2 (1 regimes):

INFO:lcm:Age 2 (1 regimes):

  finished in 171.7ms

INFO:lcm:  finished in 171.7ms

Age 1 (1 regimes):

INFO:lcm:Age 1 (1 regimes):

  finished in 25.7ms

INFO:lcm:  finished in 25.7ms

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 474c5cade967 phase backward_induction end status=ok seconds=0.222042

INFO:lcm:solve call 474c5cade967 phase backward_induction end status=ok seconds=0.222042

Solution complete  (224.0ms)

INFO:lcm:Solution complete  (224.0ms)

solve call 474c5cade967 phase result_assembly begin

INFO:lcm:solve call 474c5cade967 phase result_assembly begin

solve call 474c5cade967 phase result_assembly end status=ok seconds=0.004164

INFO:lcm:solve call 474c5cade967 phase result_assembly end status=ok seconds=0.004164

solve call 474c5cade967 phase solution_resolution begin

INFO:lcm:solve call 474c5cade967 phase solution_resolution begin

solve call 474c5cade967 phase solution_resolution end status=ok seconds=0.001751

INFO:lcm:solve call 474c5cade967 phase solution_resolution end status=ok seconds=0.001751

solve call 474c5cade967 phase solution_handover begin

INFO:lcm:solve call 474c5cade967 phase solution_handover begin

solve call 474c5cade967 phase solution_handover end status=ok seconds=0.000004

INFO:lcm:solve call 474c5cade967 phase solution_handover end status=ok seconds=0.000004

solve call 474c5cade967 phase chunk_planning begin

INFO:lcm:solve call 474c5cade967 phase chunk_planning begin

solve call 474c5cade967 phase chunk_planning end status=ok seconds=0.000311

INFO:lcm:solve call 474c5cade967 phase chunk_planning end status=ok seconds=0.000311

solve call 474c5cade967 phase simulation_setup begin

INFO:lcm:solve call 474c5cade967 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 474c5cade967 phase simulation_setup end status=ok seconds=0.086572

INFO:lcm:solve call 474c5cade967 phase simulation_setup end status=ok seconds=0.086572

solve call 474c5cade967 phase simulation_chunk begin

INFO:lcm:solve call 474c5cade967 phase simulation_chunk begin

Age 0 (1 regimes):

INFO:lcm:Age 0 (1 regimes):

solve call 474c5cade967 phase simulation_compilation begin

INFO:lcm:solve call 474c5cade967 phase simulation_compilation begin

solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.373761

INFO:lcm:solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.373761

solve call 474c5cade967 phase simulation_compilation begin

INFO:lcm:solve call 474c5cade967 phase simulation_compilation begin

solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.021191

INFO:lcm:solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.021191

solve call 474c5cade967 phase simulation_compilation begin

INFO:lcm:solve call 474c5cade967 phase simulation_compilation begin

solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.030133

INFO:lcm:solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.030133

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 783.0ms

INFO:lcm:  finished in 783.0ms

Age 1 (1 regimes):

INFO:lcm:Age 1 (1 regimes):

solve call 474c5cade967 phase simulation_compilation begin

INFO:lcm:solve call 474c5cade967 phase simulation_compilation begin

solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.290854

INFO:lcm:solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.290854

solve call 474c5cade967 phase simulation_compilation begin

INFO:lcm:solve call 474c5cade967 phase simulation_compilation begin

solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.023324

INFO:lcm:solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.023324

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 398.8ms

INFO:lcm:  finished in 398.8ms

Age 2 (1 regimes):

INFO:lcm:Age 2 (1 regimes):

solve call 474c5cade967 phase simulation_compilation begin

INFO:lcm:solve call 474c5cade967 phase simulation_compilation begin

solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.027560

INFO:lcm:solve call 474c5cade967 phase simulation_compilation end status=ok seconds=0.027560

  transitions:
  - dead → dead = 1

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

  finished in 37.4ms

INFO:lcm:  finished in 37.4ms

solve call 474c5cade967 phase simulation_chunk end status=ok seconds=1.230116

INFO:lcm:solve call 474c5cade967 phase simulation_chunk end status=ok seconds=1.230116

solve call 474c5cade967 phase simulation_completion begin

INFO:lcm:solve call 474c5cade967 phase simulation_completion begin

Simulation complete  (1.3s)

INFO:lcm:Simulation complete  (1.3s)

solve call 474c5cade967 phase simulation_completion end status=ok seconds=0.000569

INFO:lcm:solve call 474c5cade967 phase simulation_completion end status=ok seconds=0.000569

solve call 474c5cade967 phase simulation_result begin

INFO:lcm:solve call 474c5cade967 phase simulation_result begin

solve call 474c5cade967 phase simulation_result end status=ok seconds=0.005766

INFO:lcm:solve call 474c5cade967 phase simulation_result end status=ok seconds=0.005766

solve call 474c5cade967 phase result_finalization begin

INFO:lcm:solve call 474c5cade967 phase result_finalization begin

solve call 474c5cade967 phase result_finalization end status=ok seconds=0.000007

INFO:lcm:solve call 474c5cade967 phase result_finalization end status=ok seconds=0.000007

solve call 474c5cade967 phase public_simulate end status=ok seconds=3.237599

INFO:lcm:solve call 474c5cade967 phase public_simulate end status=ok seconds=3.237599

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 706b16cfd7cc phase public_simulate begin

INFO:lcm:solve call 706b16cfd7cc phase public_simulate begin

solve call 706b16cfd7cc phase params_validation begin

INFO:lcm:solve call 706b16cfd7cc phase params_validation begin

solve call 706b16cfd7cc phase params_validation end status=ok seconds=0.014667

INFO:lcm:solve call 706b16cfd7cc phase params_validation end status=ok seconds=0.014667

solve call 706b16cfd7cc phase simulation_inputs begin

INFO:lcm:solve call 706b16cfd7cc phase simulation_inputs begin

solve call 706b16cfd7cc phase simulation_inputs end status=ok seconds=0.004929

INFO:lcm:solve call 706b16cfd7cc phase simulation_inputs end status=ok seconds=0.004929

solve call 706b16cfd7cc phase authority_fingerprint begin

INFO:lcm:solve call 706b16cfd7cc phase authority_fingerprint begin

solve call 706b16cfd7cc phase authority_fingerprint end status=ok seconds=0.001740

INFO:lcm:solve call 706b16cfd7cc phase authority_fingerprint end status=ok seconds=0.001740

solve call 706b16cfd7cc phase solver_param_checks begin

INFO:lcm:solve call 706b16cfd7cc phase solver_param_checks begin

solve call 706b16cfd7cc phase solver_param_checks end status=ok seconds=0.000036

INFO:lcm:solve call 706b16cfd7cc phase solver_param_checks end status=ok seconds=0.000036

solve call 706b16cfd7cc phase state_action_spaces begin

INFO:lcm:solve call 706b16cfd7cc phase state_action_spaces begin

solve call 706b16cfd7cc phase state_action_spaces end status=ok seconds=0.000207

INFO:lcm:solve call 706b16cfd7cc phase state_action_spaces end status=ok seconds=0.000207

solve call 706b16cfd7cc phase continuation_templates begin

INFO:lcm:solve call 706b16cfd7cc phase continuation_templates begin

solve call 706b16cfd7cc phase continuation_templates end status=ok seconds=0.000343

INFO:lcm:solve call 706b16cfd7cc phase continuation_templates end status=ok seconds=0.000343

solve call 706b16cfd7cc phase program_graphs begin

INFO:lcm:solve call 706b16cfd7cc phase program_graphs begin

solve call 706b16cfd7cc phase program_graphs end status=ok seconds=0.000243

INFO:lcm:solve call 706b16cfd7cc phase program_graphs end status=ok seconds=0.000243

solve call 706b16cfd7cc phase structural_resolution begin

INFO:lcm:solve call 706b16cfd7cc 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 706b16cfd7cc phase structural_resolution end status=ok seconds=0.005531

INFO:lcm:solve call 706b16cfd7cc phase structural_resolution end status=ok seconds=0.005531

solve call 706b16cfd7cc phase residency_inventory begin

INFO:lcm:solve call 706b16cfd7cc phase residency_inventory begin

solve call 706b16cfd7cc phase residency_inventory end status=ok seconds=0.000008

INFO:lcm:solve call 706b16cfd7cc phase residency_inventory end status=ok seconds=0.000008

solve call 706b16cfd7cc phase compilation_waves begin

INFO:lcm:solve call 706b16cfd7cc 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 179.4ms

INFO:lcm:  lowered in 179.4ms

  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})  69.9ms

INFO:lcm:  compiled  working main (age 0, widths={'action_product': 64, 'cell': 128})  69.9ms

  lowered in 185.5ms

INFO:lcm:  lowered in 185.5ms

  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 24.6ms

INFO:lcm:  lowered in 24.6ms

  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})  6.3ms

INFO:lcm:  compiled  dead main (age 2, widths={'cell': 128})  6.3ms

  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  46.3ms

INFO:lcm:  compiled  working main (age 1, widths={'action_product': 64, 'cell': 128})  46.3ms

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 706b16cfd7cc phase compilation_waves end status=ok seconds=0.430103

INFO:lcm:solve call 706b16cfd7cc phase compilation_waves end status=ok seconds=0.430103

solve call 706b16cfd7cc phase workspace_selection begin

INFO:lcm:solve call 706b16cfd7cc 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 706b16cfd7cc phase workspace_selection end status=ok seconds=0.018311

INFO:lcm:solve call 706b16cfd7cc phase workspace_selection end status=ok seconds=0.018311

Starting solution

INFO:lcm:Starting solution

solve call 706b16cfd7cc phase backward_induction begin

INFO:lcm:solve call 706b16cfd7cc 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 6.2ms

INFO:lcm:  finished in 6.2ms

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 706b16cfd7cc phase backward_induction end status=ok seconds=0.020217

INFO:lcm:solve call 706b16cfd7cc phase backward_induction end status=ok seconds=0.020217

Solution complete  (21.2ms)

INFO:lcm:Solution complete  (21.2ms)

solve call 706b16cfd7cc phase result_assembly begin

INFO:lcm:solve call 706b16cfd7cc phase result_assembly begin

solve call 706b16cfd7cc phase result_assembly end status=ok seconds=0.001892

INFO:lcm:solve call 706b16cfd7cc phase result_assembly end status=ok seconds=0.001892

solve call 706b16cfd7cc phase solution_resolution begin

INFO:lcm:solve call 706b16cfd7cc phase solution_resolution begin

solve call 706b16cfd7cc phase solution_resolution end status=ok seconds=0.001174

INFO:lcm:solve call 706b16cfd7cc phase solution_resolution end status=ok seconds=0.001174

solve call 706b16cfd7cc phase solution_handover begin

INFO:lcm:solve call 706b16cfd7cc phase solution_handover begin

solve call 706b16cfd7cc phase solution_handover end status=ok seconds=0.000004

INFO:lcm:solve call 706b16cfd7cc phase solution_handover end status=ok seconds=0.000004

solve call 706b16cfd7cc phase chunk_planning begin

INFO:lcm:solve call 706b16cfd7cc phase chunk_planning begin

solve call 706b16cfd7cc phase chunk_planning end status=ok seconds=0.000349

INFO:lcm:solve call 706b16cfd7cc phase chunk_planning end status=ok seconds=0.000349

solve call 706b16cfd7cc phase simulation_setup begin

INFO:lcm:solve call 706b16cfd7cc 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 706b16cfd7cc phase simulation_setup end status=ok seconds=0.003584

INFO:lcm:solve call 706b16cfd7cc phase simulation_setup end status=ok seconds=0.003584

solve call 706b16cfd7cc phase simulation_chunk begin

INFO:lcm:solve call 706b16cfd7cc phase simulation_chunk begin

Age 0 (1 regimes):

INFO:lcm:Age 0 (1 regimes):

solve call 706b16cfd7cc phase simulation_compilation begin

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation begin

solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.303609

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.303609

solve call 706b16cfd7cc phase simulation_compilation begin

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation begin

solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.010285

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.010285

solve call 706b16cfd7cc phase simulation_compilation begin

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation begin

solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.011313

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.011313

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 352.3ms

INFO:lcm:  finished in 352.3ms

Age 1 (1 regimes):

INFO:lcm:Age 1 (1 regimes):

solve call 706b16cfd7cc phase simulation_compilation begin

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation begin

solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.304829

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.304829

solve call 706b16cfd7cc phase simulation_compilation begin

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation begin

solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.011403

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.011403

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 339.6ms

INFO:lcm:  finished in 339.6ms

Age 2 (1 regimes):

INFO:lcm:Age 2 (1 regimes):

solve call 706b16cfd7cc phase simulation_compilation begin

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation begin

solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.011012

INFO:lcm:solve call 706b16cfd7cc phase simulation_compilation end status=ok seconds=0.011012

  transitions:
  - dead → dead = 1

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

  finished in 19.3ms

INFO:lcm:  finished in 19.3ms

solve call 706b16cfd7cc phase simulation_chunk end status=ok seconds=0.715691

INFO:lcm:solve call 706b16cfd7cc phase simulation_chunk end status=ok seconds=0.715691

solve call 706b16cfd7cc phase simulation_completion begin

INFO:lcm:solve call 706b16cfd7cc phase simulation_completion begin

Simulation complete  (720.8ms)

INFO:lcm:Simulation complete  (720.8ms)

solve call 706b16cfd7cc phase simulation_completion end status=ok seconds=0.000824

INFO:lcm:solve call 706b16cfd7cc phase simulation_completion end status=ok seconds=0.000824

solve call 706b16cfd7cc phase simulation_result begin

INFO:lcm:solve call 706b16cfd7cc phase simulation_result begin

solve call 706b16cfd7cc phase simulation_result end status=ok seconds=0.000786

INFO:lcm:solve call 706b16cfd7cc phase simulation_result end status=ok seconds=0.000786

solve call 706b16cfd7cc phase result_finalization begin

INFO:lcm:solve call 706b16cfd7cc phase result_finalization begin

solve call 706b16cfd7cc phase result_finalization end status=ok seconds=0.000008

INFO:lcm:solve call 706b16cfd7cc phase result_finalization end status=ok seconds=0.000008

solve call 706b16cfd7cc phase public_simulate end status=ok seconds=1.250060

INFO:lcm:solve call 706b16cfd7cc phase public_simulate end status=ok seconds=1.250060

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 ( $\beta = 1$ )¶