Solving a finite-horizon consumption problem via Stachurski's Slider¶

This note works through a concrete example — the portfolio-choice and consumption-savings problem over a finite horizon — to show how "order semiconjugate transform" connects a particular ordering of operators within a stage to the dolo solver -- which uses an alternative ordering of operators.

We begin from the terminal period, build the object we need to iterate on, and show that two natural orderings of the same Bellman operator are semiconjugate. We use this duality relationship to transform one ordering to the other, send it to dolo, and bring it back via the "slider".

1. The problem¶

Let's start with two equivalent formulations of the same finite-horizon consumption–savings problem.

Problem A: a single operator on cash-on-hand (dolo’s EGM “one-stage” view) where the decision problem computes the expectation.
Problem B: a two-stage factorization of the same period operator into a noport stage (pure expectation) and a cons stage (pure maximisation).

Suppose we have technology (code) to solve problem A, but we actually want to solve problem B. On the surface these two views look irreconcilable, and one may retort, "but problem B is totally different and cannot be solved using machinery used to solve problem A".

However, these two dynamical systems are related by order semiconjugacy (due to Stachurski and Sargent (2026) and other references): they use the same sub-operators in opposite order.

Thankfully, Bellman calculus stages can represent both problem A and problem B as stages, and we can see their duality clearly.

A. The primal sequence problem with shocks after decision¶

Let's start with the sequence problem that problem A tries to tackle. An agent lives for $H$ periods. In each period $t = 0, \ldots, H-1$ the agent:

enters with assets $a_{t-1}$,
receives a stochastic income shock $\theta_t \sim \text{LogNormal}(\mu_\theta, \sigma_\theta)$,
observes cash-on-hand $m_t = a_{t-1} R + \theta_t$,
chooses consumption $c_t \in (0, m_t)$,
exits with assets $a_t = m_t - c_t$.

The objective is to maximise

\[ \mathbb{E}_0 \sum_{t=0}^{H-1} \beta^t \, u(c_t), \qquad u(c) = \frac{c^{1-\rho}}{1-\rho}. \]

The continuation value at the horizon is an asset-space function $w_H(a)$ (e.g. $w_H(a)=0$ for no bequest).

A1. The Bellman equation¶

Working backwards from the terminal period, the Bellman equation at period $t$ is

\[ V_t(m) = \max_{c \in (0,m)} \Bigl\{ u(c) + \beta \, \mathbb{E}_{\theta'}\bigl[ V_{t+1}\bigl( (m - c)\,R + \theta' \bigr) \bigr] \Bigr\}. \tag{B} \]

Write $a = m - c$ for the post-consumption assets. The operator that maps the period $t+1$ value function to the period-$t$ value function is

\[ (\hat{\mathbf{S}}\, V)(m) = \max_{c} \Bigl\{ u(c) + \beta \, \mathbb{E}_{\theta'}\bigl[ V(aR + \theta') \bigr] \Bigr\}. \]

Solving the finite-horizon problem amounts to computing the iterates $\hat{\mathbf{S}}^n V_H$ for $n = 1, \ldots, H$.

The Bellman equation in ADC form¶

Map the period into a single ADC stage. The shock $\theta'$ occurs after the decision — it is a post-decision shock in $\mathrm{g}_{\sim\succ}$. The three perches carry:

Perch	Variable	Description
$\prec$ (arrival)	$m$	cash-on-hand (shock already resolved)
$\cdot$ (decision)	$m,\; c$	cash-on-hand and consumption choice
$\succ$ (continuation)	$m'$	next period's cash-on-hand (after post-decision shock)

Forward transitions:

\[ \mathrm{g}_{\prec\sim} \colon m \mapsto m \qquad\text{(identity)}, \qquad \mathrm{g}_{\sim\succ} \colon (m, c, \theta') \mapsto m' = (m - c)\,R + \theta', \]

where $\theta' \sim \text{LogNormal}(\mu_\theta, \sigma_\theta)$ is the post-decision shock and $\mathrm{g}_{\prec\sim}$ is trivial. The backward movers are:

$\mathbb{B}$ (continuation → decision): combines the max and the expectation, because the shock sits inside $\mathrm{g}_{\sim\succ}$:

\[ (\mathbb{B}\, \mathrm{v}_\succ)(m) = \max_{c \in (0,m)} \bigl\{ u(c) + \beta\, \mathbb{E}_{\theta'}\bigl[\mathrm{v}_\succ\bigl((m - c)\,R + \theta'\bigr)\bigr] \bigr\}. \]

$\mathbb{I}$ (decision → arrival): identity, $\mathrm{v}_\prec(m) = \mathrm{v}_\cdot(m)$.

The full stage operator is just $\mathbb{B}$ (since $\mathbb{I}$ is trivial). With the inter-period twister identifying $m' = m$, this gives exactly (B). The expectation is locked inside $\mathbb{B}$ — inseparable from the maximisation.

Factoring the operator¶

Although $\mathbb{B}$ is monolithic in the ADC, it decomposes analytically into two sub-operators:

\[ (\mathbf{E}\, v)(a) = \mathbb{E}_\theta\bigl[v(aR + \theta)\bigr], \qquad (\mathbf{M}\, w)(m) = \max_{c \in (0,m)}\bigl\{u(c) + \beta\, w(m - c)\bigr\}. \]

$\mathbf{E}$ maps a function on cash-on-hand to a function on assets (integration); $\mathbf{M}$ maps a function on assets to a function on cash-on-hand (optimisation). Then

\[ \hat{\mathbf{S}} = \mathbf{M} \circ \mathbf{E}, \]

since $(\mathbb{B}\,v)(m) = \max_c\bigl\{u(c) + \beta\, \underbrace{\mathbb{E}_\theta[v((m-c)R + \theta)]}_{(\mathbf{E}\,v)(m-c)}\bigr\} = (\mathbf{M} \circ \mathbf{E}\, v)(m)$. Since $\mathbb{I}$ is trivial, the full stage operator $\mathbb{I} \circ \mathbb{B}$ equals $\hat{\mathbf{S}} = \mathbf{M} \circ \mathbf{E}$ — the period operator on $m$-space.

B. The continuation value formulation¶

The reverse factoring $\mathbf{S} = \mathbf{E} \circ \mathbf{M}$ defines the period operator on asset-space values:

\[ (\mathbf{S}\, w)(a) = \mathbb{E}_\theta \Bigl[ \max_{c} \bigl\{ u(c) + \beta\, w\bigl( aR + \theta - c \bigr) \bigr\} \Bigr]. \tag{B$'$} \]

This operator splits into two self-contained stages:

cons stage ($\mathbf{M}$): $(\mathbf{M}\,w)(m) = \max_c\{u(c) + \beta\, w(m-c)\}$. Pure optimisation — no stochastic integration.
noport stage ($\mathbf{E}$): $(\mathbf{E}\,v)(a) = \mathbb{E}_\theta[v(aR + \theta)]$. Pure expectation — no decision.

Each stage is a complete ADC unit with its own perches, transitions, and backward movers. The stages compose via a connector identifying post-consumption assets $a$ with investable assets $k$. The period backward sweep reads right-to-left: $\mathbf{S} = \mathbf{E} \circ \mathbf{M}$.

2. Solving Problem B via semiconjugacy¶

We want to iterate Problem B's modular operator $\mathbf{S} = \mathbf{E} \circ \mathbf{M}$ on asset-space values. Problem A's monolithic operator $\hat{\mathbf{S}} = \mathbf{M} \circ \mathbf{E}$ acts on cash-on-hand values — the same sub-operators in reverse order. Stachurski's semiconjugate transform connects the two.

Definition (Stachurski 2009, 2022). Dynamical systems $(V, \mathbf{S})$ and $(\hat V, \hat{\mathbf{S}})$ on posets are order semiconjugate under $\mathbf{F}, \mathbf{G}$ when there exist order-preserving maps $\mathbf{F} \colon V \to \hat V$ and $\mathbf{G} \colon \hat V \to V$ satisfying

\[ \mathbf{S} = \mathbf{G} \circ \mathbf{F} \qquad\text{and}\qquad \hat{\mathbf{S}} = \mathbf{F} \circ \mathbf{G}. \]

From this, two commuting identities follow:

\[ \mathbf{F} \circ \mathbf{S} = \hat{\mathbf{S}} \circ \mathbf{F} \qquad\text{and}\qquad \mathbf{G} \circ \hat{\mathbf{S}} = \mathbf{S} \circ \mathbf{G}. \]

Iterates Lemma. For all $n \geq 1$,

\[ \mathbf{S}^n = \mathbf{G} \circ \hat{\mathbf{S}}^{\,n-1} \circ \mathbf{F} \qquad\text{and}\qquad \hat{\mathbf{S}}^n = \mathbf{F} \circ \mathbf{S}^{\,n-1} \circ \mathbf{G}. \]

The proof (by induction, see semiconjugacy-iterates) uses the key step $\mathbf{F} \circ \mathbf{G} = \hat{\mathbf{S}}$.

Identifying $\mathbf{F}$, $\mathbf{G}$, $\mathbf{S}$, $\hat{\mathbf{S}}$¶

Object	Definition	Description
$\mathbf{E}$	$(\mathbf{E}\,v)(a) = \mathbb{E}_\theta[v(aR + \theta)]$	Integration over income shock
$\mathbf{M}$	$(\mathbf{M}\,w)(m) = \max_{c \in (0,m)}\{u(c) + \beta\, w(m-c)\}$	Bellman maximisation
$\mathbf{S} = \mathbf{E} \circ \mathbf{M}$	Problem B's period operator (on $a$-space)	optimise, then integrate
$\hat{\mathbf{S}} = \mathbf{M} \circ \mathbf{E}$	Problem A's operator (on $m$-space)	integrate, then optimise
$\mathbf{F} = \mathbf{M}$	$V \to \hat V$ (a-space → m-space)
$\mathbf{G} = \mathbf{E}$	$\hat V \to V$ (m-space → a-space)

Check: $\mathbf{S} = \mathbf{G} \circ \mathbf{F} = \mathbf{E} \circ \mathbf{M}$ ✓ and $\hat{\mathbf{S}} = \mathbf{F} \circ \mathbf{G} = \mathbf{M} \circ \mathbf{E}$ ✓. The iterates lemma gives, for the finite-horizon problem:

\[ \mathbf{S}^n = \mathbf{E} \circ (\mathbf{M} \circ \mathbf{E})^{n-1} \circ \mathbf{M}. \]

Reading right-to-left: apply $\mathbf{M}$ once to the terminal value, iterate $\hat{\mathbf{S}} = \mathbf{M} \circ \mathbf{E}$ for $n-1$ steps, then apply one final $\mathbf{E}$. Each application of $\hat{\mathbf{S}}$ first integrates ($\mathbf{E}$), then optimises ($\mathbf{M}$) — EGM-compatible, since $\mathbf{M}$ maximises over a smooth expected continuation value.

3. The syntax¶

3.1 Problem A: single-stage representation¶

Existing Dolo syntax. The Dolo EGM model represents the full period as a single monolithic step. The state is cash-on-hand $w$ and the transition folds both the decision and the next-period shock into one equation — exactly $\hat{\mathbf{S}} = \mathbf{M} \circ \mathbf{E}$:

# consumption_savings_iid_egm.yaml (existing Dolo format)
symbols:
    exogenous: [y]
    states: [w]
    expectations: [mr]
    poststates: [a]
    controls: [c]
    parameters: [β, γ, σ, ρ, r]

equations:
    transition:
        - w[t] = exp(y[t]) + (w[t-1]-c[t-1])*r

    arbitrage:
        - β*( c[t+1]/c[t] )^(-γ)*r - 1  | 0.0<=c[t]<=w[t]

    half_transition: |
        w[t] = exp(y[t]) + a[t-1]*r

    auxiliary_direct_egm: |
        a[t] = w[t] - c[t]

    reverse_state: |
        w[t] = a[t] + c[t]

    expectation: |
        mr[t] = β*( c[t+1] )^(-γ)*r

    direct_response_egm: |
        c[t] = (mr[t])^(-1/γ)

The transition equation w[t] = exp(y[t]) + (w[t-1]-c[t-1])*r composes the decision ($a = w - c$) with the next-period shock realisation ($w' = \exp(y') + a \cdot r$) into a single step — the hallmark of $\hat{\mathbf{S}} = \mathbf{M} \circ \mathbf{E}$. The Dolo solver internally separates the expectation (expectation:) from the policy update (direct_response_egm:, reverse_state:) to implement EGM, but the YAML presents the problem monolithically.

dolo-plus ADC form (one-stage). A dolo-plus adc-stage version of the same one-stage dtcc/EGM model lives at:

packages/dolo/examples/models/doloplus/consumption_savings_iid_egm_doloplus/stage.yaml

with split-file bindings:

calibration.yaml (parameters), settings.yaml (numerical settings), methodization.yml (schemes)

An abbreviated excerpt (using Dolo’s parameter name γ for CRRA) is:

# consumption_savings_iid_egm_doloplus/stage.yaml (abbreviated)
equations:
  arvl_to_dcsn_transition: |
    w = exp(y) + b*r

  dcsn_to_cntn_transition: |
    a = w - c

  cntn_to_dcsn_transition: |
    w = a + c

  cntn_to_dcsn_opr:
    InvEuler: |
      c[>] = (β*dV[>])^(-1/γ)
    MarginalBellman: |
      dV = (c)^(-γ)

  dcsn_to_arvl_opr:
    MarginalBellman: |
      dV[<] = r * E_{y}(dV)

Conceptually, this one-stage dolo-plus representation matches the vanilla Dolo dtcc blocks (half_transition, auxiliary_direct_egm, reverse_state, expectation, direct_response_egm) used by egm(). A fully “materialized” translator-ready instance (with dolo_plus.equation_symbols, calibration, domain, options) is docs/old/examples/old/example-dolo-plus-consind-egm.yml.

3.2 Problem B: noport–cons decomposition¶

The modular operator $\mathbf{S} = \mathbf{E} \circ \mathbf{M}$ factors the period into two stages:

Stage 1: noport_stage — pure expectation ($\mathbf{E}$), no decision.

# noport_stage (abbreviated)
equations:
  arvl_to_dcsn_transition: |
    k_d = k
  dcsn_to_cntn_transition: |
    m = k_d*R + θ

  cntn_to_dcsn_mover:
    Bellman: |
      V = E_{θ}(V[>])
  dcsn_to_arvl_mover:
    Bellman: |
      V[<] = V

Stage operator: $(\mathbf{E}\,v)(k) = \mathbb{E}_\theta[v(kR + \theta)]$.

Stage 2: cons_stage — pure optimisation ($\mathbf{M}$), no shocks.

# cons_stage (abbreviated)
equations:
  arvl_to_dcsn_transition: |
    m_d = m
  dcsn_to_cntn_transition: |
    a = m_d - c

  cntn_to_dcsn_mover:
    Bellman: |
      V = max_{c}(u + β*V[>])
    InvEuler: |
      c[>] = (β*dV[>])^(-1/ρ)
    cntn_to_dcsn_transition: |
      m_d[>] = c[>] + a
    MarginalBellman: |
      dV = (c)^(-ρ)
  dcsn_to_arvl_mover:
    Bellman: |
      V[<] = V

Stage operator: $(\mathbf{M}\,w)(m) = \max_{c \in (0,m)} \{u(c) + \beta\, w(m - c)\}$.

The composed period operator reads right-to-left: $\mathbf{S} = \mathbf{E} \circ \mathbf{M}$. The connector $\mathrm{c} \colon a \mapsto k = a$ identifies post-consumption assets with investable assets.

4. Solving the finite-horizon problem¶

The iterates formula¶

For $n$ steps back from the terminal period:

\[ \mathbf{S}^n \, w_H = \mathbf{E} \circ (\mathbf{M} \circ \mathbf{E})^{n-1} \circ \mathbf{M} \, w_H. \]

Reading right-to-left: apply $\mathbf{M}$ once to the terminal value, iterate $\hat{\mathbf{S}} = \mathbf{M} \circ \mathbf{E}$ for $n-1$ steps, then apply one final $\mathbf{E}$. The inner iterates $(\mathbf{M} \circ \mathbf{E})^{n-1}$ are the conjugate operator $\hat{\mathbf{S}}$ applied $n-1$ times. Each application of $\hat{\mathbf{S}}$:

$\mathbf{E}$: integrate the current $m$-space value function $v(\cdot)$ over the next-period shock to get an $a$-space continuation value $w(a)=\mathbb{E}_\theta[v(aR+\theta)]$,
$\mathbf{M}$: maximise $u(c) + \beta\, w(m - c)$ over $c$ (EGM-compatible since $w$ is a smooth expected continuation value).

Conceptual seam (level values vs EGM objects). The semiconjugacy identities above are stated for operators acting on level value functions ($v, w, V$). The dtcc/EGM “Dolo horse”, however, does not carry level values explicitly: it propagates a marginal object (dV in dolo-plus stage syntax, renamed to mr in vanilla Dolo) built from the envelope condition $u'(c)$ and used in the inverse-Euler update. Accordingly, the translator synthesizes the Dolo expectation: block from the stage’s marginal equations (e.g. ShadowBellman / MarginalBellman), and EGM’s inner loop accumulates mr (its internal z) rather than $V$. This is why the final “$\mathbf{E}$” step in the code below returns mr(a) on the asset grid, not a level value function.

Pseudocode¶

Step 1: Translate a one-stage dolo-plus model → vanilla Dolo (dtcc/EGM horse)¶

The translator doloplus_to_dolo (in packages/dolo/dolo/compiler/doloplus_translator/core.py) converts a dolo-plus adc-stage YAML file to a vanilla Dolo dtcc model using external mapping tables.

At the moment, translation is driven entirely by the YAML payload (in particular dolo_plus.equation_symbols). Split-file functor attachments (methodize, configure, calibrate) are not consumed by the translator unless you first materialize them into the YAML.

A minimal dolo-plus input in this repo that is already “materialized” (has calibration, domain, options, exogenous, and dolo_plus.equation_symbols) is:

docs/old/examples/old/example-dolo-plus-consind-egm.yml

from dolo.compiler.model_import import yaml_import
from dolo.compiler.doloplus_translator import doloplus_to_dolo

model_plus = yaml_import(
    "docs/old/examples/old/example-dolo-plus-consind-egm.yml",
    compile_functions=False,
)

model = doloplus_to_dolo(
    model_plus,
    mapping_tables="explore/transformations/egm-ADC-to-dolo/mapping_tables.yaml",
    compile_functions=True,
)

The expected translated output is documented at explore/transformations/egm-ADC-to-dolo/transformation_example.yaml, and a concrete output instance lives at explore/transformations/egm-ADC-to-dolo/output_dolo.yaml.

Step 2: Provide the terminal/initial decision rule `dr0`¶

In Dolo, terminal/initial conditions for decision rules are not part of the YAML surface syntax — they are supplied as an argument to the solver. For finite-horizon iterates, dr0 plays the role of the rightmost $\mathbf{M}\,w_H$ boundary condition in the semiconjugacy formula.

For the simple “no bequest” horizon $w_H(a)=0$, the terminal maximisation is “consume all”, i.e. $c(m)=m$. We pass that as a decision rule dr0:

import numpy as np

H = 50
a_grid = np.linspace(0.01, 10.0, 100)  # poststate grid for EGM

# dr0(i, s) -> controls at (future) state s
# Here: c = w (consume all cash-on-hand).
dr0 = lambda i, s: s

Step 3: Iterate $\hat{\mathbf{S}}^{H-1}$ with Dolo's EGM solver¶

With dr0 from Step 2 as the starting point, iterate the conjugate operator $\hat{\mathbf{S}} = \mathbf{M} \circ \mathbf{E}$ for $H-1$ steps. To force exactly H-1 iterations (rather than stopping early on convergence), set η_tol=0:

from dolo.algos.egm import egm

sol = egm(model, dr0=dr0, a_grid=a_grid, maxit=H-1, η_tol=0.0, verbose=True)
# sol.dr is the final m-space decision rule iterate

After H-1 iterations, sol.dr contains the $m$-space policies from $\hat{\mathbf{S}}^{H-1} \circ \mathbf{M}\, w_H$.

Step 4: Extract the final expectation output `mr(a)` (the “E-step” of EGM)¶

Dolo’s egm() returns a decision rule sol.dr. The expectation output that feeds the inverse-Euler step (often called z internally, and mr in dtcc models) is formed by the accumulation loop $z \mathrel{+}= w \cdot h(M,S,X,p)$.

We can run that expectation-accumulation half once and stop, using the compiled gufunctions from the same translated model:

import numpy as np

funs = model.__original_gufunctions__
gt = funs["half_transition"]
h  = funs["expectation"]

dp = sol.dprocess
p  = model.calibration["parameters"]
a  = a_grid[:, None]

# For IID shocks, dp.n_nodes == 1 and i_m=0 is the only current node.
i_m = 0
m = dp.node(i_m)

mr = np.zeros((len(a_grid), 1))
for i_M in range(dp.n_inodes(i_m)):
    w = dp.iweight(i_m, i_M)
    M = dp.inode(i_m, i_M)
    S = gt(m, a, M, p)       # next-period cash-on-hand grid
    X = sol.dr(i_M, S)       # future consumption evaluated at S
    mr += w * h(M, S, X, p)  # accumulate into mr(a)

This is exactly the z += w * h(...) loop inside dolo.algos.egm.egm; we’re just stopping after forming mr(a) instead of proceeding to the policy update (direct_response_egm, reverse_state).

Convention note. Whether expectation: computes $mr = R\cdot\mathbb{E}[u'(c')]$ or $mr = \beta R\cdot\mathbb{E}[u'(c')]$ is a mapping-table convention (β can live in expectation: or be applied in direct_response_egm:). The loop above always returns whatever object your compiled h = funs["expectation"] encodes.

5. Implementation agnosticism¶

The Bellman calculus representation is agnostic to how the sub-operators are executed. Nothing requires stage-by-stage solving:

A Dolo horse can swallow an entire period and compute $\hat{\mathbf{S}}$ as a single compiled step.
A stage-separated orchestrator (spec 0.1eA) can call the expectation and maximisation as separate compiled primitives.
A generic operator can replace $\mathbf{M}$ with policy iteration, marginal value iteration, or any other method sharing the same signature.

The mathematical content — the stages, their transitions, and the semiconjugate relationship between the two orderings — is invariant across all three approaches.

refs:¶

semiconjugacy-iterates #transform-stachurski
spec 0.1eA
Stachurski, J. (2009). Economic Dynamics. MIT Press.
Stachurski, J. (2022). Economic Dynamics, 2nd ed. MIT Press.
Sargent, T. J. and Stachurski, J. (2025). Dynamic Programming, vol. 2. Manuscript.

Object	Definition	Description
\(\mathbf{E}\)	\((\mathbf{E}\,v)(a) = \mathbb{E}_\theta[v(aR + \theta)]\)	Integration over income shock
\(\mathbf{M}\)	\((\mathbf{M}\,w)(m) = \max_{c \in (0,m)}\{u(c) + \beta\, w(m-c)\}\)	Bellman maximisation
\(\mathbf{S} = \mathbf{E} \circ \mathbf{M}\)	Problem B's period operator (on \(a\)-space)	optimise, then integrate
\(\hat{\mathbf{S}} = \mathbf{M} \circ \mathbf{E}\)	Problem A's operator (on \(m\)-space)	integrate, then optimise
\(\mathbf{F} = \mathbf{M}\)	\(V \to \hat V\) (a-space → m-space)
\(\mathbf{G} = \mathbf{E}\)	\(\hat V \to V\) (m-space → a-space)

Solving a finite-horizon consumption problem via Stachurski's Slider¶

1. The problem¶

A. The primal sequence problem with shocks after decision¶

A1. The Bellman equation¶

The Bellman equation in ADC form¶

Factoring the operator¶

B. The continuation value formulation¶

2. Solving Problem B via semiconjugacy¶

Identifying \(\mathbf{F}\), \(\mathbf{G}\), \(\mathbf{S}\), \(\hat{\mathbf{S}}\)¶

3. The syntax¶

3.1 Problem A: single-stage representation¶

3.2 Problem B: noport–cons decomposition¶

4. Solving the finite-horizon problem¶

The iterates formula¶

Pseudocode¶

Step 1: Translate a one-stage dolo-plus model → vanilla Dolo (dtcc/EGM horse)¶

Step 2: Provide the terminal/initial decision rule `dr0`¶

Step 3: Iterate \(\hat{\mathbf{S}}^{H-1}\) with Dolo's EGM solver¶

Step 4: Extract the final expectation output `mr(a)` (the “E-step” of EGM)¶

5. Implementation agnosticism¶

refs:¶

Perch	Variable	Description
\(\prec\) (arrival)	\(m\)	cash-on-hand (shock already resolved)
\(\cdot\) (decision)	\(m,\; c\)	cash-on-hand and consumption choice
\(\succ\) (continuation)	\(m'\)	next period's cash-on-hand (after post-decision shock)

Solving a finite-horizon consumption problem via Stachurski's Slider¶

1. The problem¶

A. The primal sequence problem with shocks after decision¶

A1. The Bellman equation¶

The Bellman equation in ADC form¶

Factoring the operator¶

B. The continuation value formulation¶

2. Solving Problem B via semiconjugacy¶

Identifying \(\mathbf{F}\), \(\mathbf{G}\), \(\mathbf{S}\), \(\hat{\mathbf{S}}\)¶

3. The syntax¶

3.1 Problem A: single-stage representation¶

3.2 Problem B: noport–cons decomposition¶

4. Solving the finite-horizon problem¶

The iterates formula¶

Pseudocode¶

Step 1: Translate a one-stage dolo-plus model → vanilla Dolo (dtcc/EGM horse)¶

Step 2: Provide the terminal/initial decision rule dr0¶

Step 3: Iterate \(\hat{\mathbf{S}}^{H-1}\) with Dolo's EGM solver¶

Step 4: Extract the final expectation output mr(a) (the “E-step” of EGM)¶

5. Implementation agnosticism¶

refs:¶

Step 2: Provide the terminal/initial decision rule `dr0`¶

Step 4: Extract the final expectation output `mr(a)` (the “E-step” of EGM)¶