Parent spec: [[spec_0.1h-periods-nests|spec 0.1h — periods and nests]]

Syntax rules: [[05-periods-models|0.4 Periods and models]], [[02-stage-structure-and-timing|0.2 Stage structure and timing]]

Connector spec: [[spec_0.1lA-connector-unification|spec 0.1lA — connector unification]] (discount stage connects via identity connector)

Theory: [[periods-nests-theory|Composition as twister (foundation)]], [[theoretical-propositions|Theoretical propositions (contraction conditions)]]

Examples:

[[port_stage|port-with-shocks: port_stage.yaml]] — currently β = 1
[[cons_stage|port-with-shocks: cons_stage.yaml]] — currently β < 1 (will become β = 1)
[[modular-examples-port-able_port-with-shocks|Unified: port-able stage examples]]
[[modular-examples-cons_port-with-shocks|Unified: cons stage examples]]

Spec 0.1hA — Discount stage as period-terminal stage¶

1. Problem¶

Time discounting (β) is currently bundled into whichever stage carries the reward function. In the cons-port architecture, the cons stage has β < 1 while the port stage has β = 1. This creates an asymmetry: a period with only a port stage cannot express inter-temporal discounting, because no stage carries β.

This limits composability. With three basic stage types (cons, port, discount), we should be able to form any combination:

Period type	Stages (proposed)	Currently possible?
cons only	cons → disc	Yes (β bundled in cons)
port → cons	port → cons → disc	Yes (β bundled in cons)
cons → port	cons → port → disc	Yes (β bundled in cons)
port only	port → disc	No — no stage carries β

2. Decision¶

Introduce a dedicated discount stage (disc) that appears as the terminal stage in every period. This stage:

Has no choice variable (control space is empty)
Has no shocks (exogenous is trivial)
Has zero stage reward: $r = 0$
Applies the period's time discount factor: $\beta < 1$
Has identity transitions: arrival state = decision state = continuation state

Its Bellman equation is:

\[ \mathrm{v}_{\mathrm{disc}}(x) = \beta \, \mathrm{v}_{\succ}(x) \]

The discount stage scales the continuation value by β and passes the state through unchanged. It produces no policy, no optimization, no expectation.

3. Impact on existing stages¶

With discounting extracted, all "economic" stages become β = 1:

Stage	Before	After
cons	β < 1, r = u(c)	β = 1, r = u(c)
port	β = 1, r = 0	β = 1, r = 0 (unchanged)
disc (new)	—	β < 1, r = 0

Economic stages focus purely on their content (optimization, shocks, rewards). The discount stage is the sole bearer of β.

4. Stage YAML¶

name: discount_stage
stage_type: adc-stage

symbols:
    # The discount stage inherits the period's state space.
    # The specific state variables are determined by the
    # preceding stage's continuation interface via the connector.
    parameters: [β]

equations:
    # Identity transitions (state passes through)
    arvl_to_dcsn_transition: |
        # identity — arrival state = decision state

    dcsn_to_cntn_transition: |
        # identity — decision state = continuation state

    # Backward mover: pure discounting
    cntn_to_dcsn_mover:
        Bellman: |
            V = β * V[>]

Note

The discount stage's state variables are not declared in the stage YAML itself — they are inherited from the preceding stage's continuation interface via the within-period connector (which is identity, since the discount stage doesn't rename anything). This is consistent with the modularity principle in [[spec_0.1lA-connector-unification|spec 0.1lA]]: the stage declares its equations, and the assembly determines the wiring.

5. Period composition (with `:=:` connectors)¶

Using the interleaved connector syntax from [[spec_0.1lA-connector-unification|spec 0.1lA]]:

# Port-cons period (the common case)
sequence:
  - port_stage
  - connector:
      - m :=: m        # port continuation → cons arrival (identity)
  - cons_stage
  - connector:
      - a :=: a        # cons continuation → disc arrival (identity)
  - discount_stage

Since all connectors to the discount stage are identity, they can be omitted:

# Equivalent (identity connectors implied)
sequence:
  - port_stage
  - cons_stage
  - discount_stage

The port-only period that was previously impossible:

sequence:
  - port_stage
  - discount_stage

6. Impact on value function composition¶

The discount factor moves from inside the cons stage's Bellman equation to a separate multiplicative step:

Before (β inside cons): $$ \mathrm{v}{\mathrm{cons}}(m) = \max_c \bigl{ u(c) + \beta \, \mathrm{v}(m - c) \bigr} $$

After (β in disc stage): $$ \mathrm{v}{\mathrm{cons}}(m) = \max_c \bigl{ u(c) + \mathrm{v}(m - c) \bigr}, \qquad \mathrm{v}}{\mathrm{disc}}(a) = \beta \, \mathrm{v}(a) $$

Composing these two stages recovers the same equation: $\mathrm{v}_{\mathrm{cons}}(m) = \max_c \{ u(c) + \beta \, \mathrm{v}_{\succ}(m - c) \}$.

7. Contraction conditions¶

The [[theoretical-propositions|theoretical propositions]] document already states that contraction conditions apply at the period level (the product of stage β values must satisfy $0 < \beta_{\mathrm{period}} < 1$). With the discount stage, this simplifies: only the disc stage has $\beta < 1$, so the period-level condition is equivalent to requiring the disc stage's β ∈ (0, 1).

8. Methodization¶

The discount stage requires minimal methodization — no expectation, no grid, no maximization. A bare entry suffices:

stage: discount_stage

methods:
  - on: cntn_to_dcsn_mover
    schemes:
      - scheme: bellman_backward
        method: !scale
  - on: cntn_to_dcsn_mover.Bellman
    schemes: []
  - on: arvl_to_dcsn_transition
    schemes: []
  - on: dcsn_to_cntn_transition
    schemes: []

The !scale method signals to the whisperer that this mover is a pure scalar multiplication — no optimization, no interpolation, no grid.

9. Acceptance criteria¶

[ ] discount_stage.yaml created in packages/dolo/examples/models/doloplus/port-with-shocks/library/
[ ] discount_stage_methods.yml created alongside it
[ ] Existing cons stage YAML updated: β removed from stage (moved to calibration of disc)
[ ] Period builder scripts updated to append discount stage as terminal
[ ] Port-only period demonstrated as a test case
[ ] [[05-periods-models]] updated with discount stage examples
[ ] [[20260122-discount-stage-as-period-terminal-stage|CDC #11]] marked resolved