THEORETICAL PROPOSITIONS

For each stage kind κ ∈ Κ, we define two shock spaces corresponding to the two within-stage transitions: - 𝒵ₐᵥ(κ): Shock space for the arvl→dcsn transition, with probability measure Pₐᵥ(ζₐᵥ) - 𝒵ᵥₑ(κ): Shock space for the dcsn→cntn transition, with probability measure Pᵥₑ(ζᵥₑ)

Either or both shock spaces may be trivial (containing only ∅), in which case the corresponding transition is deterministic. The shock spaces may be discrete or continuous: - For discrete shock spaces, P(ζ) is a probability mass function - For continuous shock spaces, P(ζ) is a probability density function

Let 𝔭 denote the vector of all parameters that define the model. These parameters 𝔭 are fixed during the solution of any specific instance of the model, though they may vary across different model specifications or calibrations.

The state spaces 𝓧ₐ(κ), 𝓧ᵥ(κ), 𝓧ₑ(κ) and all functions between them are assumed to be measurable in the standard ways that allow: - Integration with respect to shock distributions - Composition of functions - Well-defined population dynamics

To reduce clutter, throughout the remainder of this section we will drop the stage kind κ from the notation, because it should be clear from the context which stage kind is being considered.
The transition functions map between states as follows:

Within-Stage Perch Transitions: 1. Arrival-to-Decision (gₐᵥ): - General signature: gₐᵥ: 𝓧ₐ × 𝒵ₐᵥ → 𝓧ᵥ - For each fixed shock ζₐᵥ, the map xₐ ↦ xᵥ is bijective - Reverse mapping function: gᵥₐ: 𝓧ᵥ × 𝒵ₐᵥ → 𝓧ₐ - When 𝒵ₐᵥ is trivial, simplifies to gₐᵥ: 𝓧ₐ → 𝓧ᵥ - Defined at stage level as part of stage's internal structure

1. Decision-to-Continuation (gᵥₑ):
2. General signature: gᵥₑ: 𝓧ᵥ × Π × 𝒵ᵥₑ → 𝓧ₑ
3. For each fixed choice 𝜋 and shock ζᵥₑ, the map xᵥ ↦ xₑ is bijective
4. Reverse mapping function: gₑᵥ: 𝓧ₑ × Π × 𝒵ᵥₑ → 𝓧ᵥ
5. When 𝒵ᵥₑ is trivial, simplifies to gᵥₑ: 𝓧ᵥ × Π → 𝓧ₑ
6. Defined at stage level as part of stage's internal structure

The bijectivity properties hold for each fixed realization of shocks. When both shock spaces are non-trivial, expectations are nested: the outer expectation is over ζₐᵥ (before decision), the inner over ζᵥₑ (after decision). See "Shock Spaces and Timing" in Recursive Problem Definition for the full taxonomy.

Connector Functions: These functions are defined at the period level and connect stages to their successors:

1. Sequential Stage Connector (gₑₐ₊):
2. Function signature: gₑₐ₊: 𝓧ₑ → 𝓧ₐ₊
3. Maps continuation states to next stage's arrival states
4. Bijective: Each xₑ maps to unique xₐ₊
5. Reverse mapping function: gₐ₊ₑ: 𝓧ₐ₊ → 𝓧ₑ

6. Used for deterministic transitions to next stage

7. Branching Stage Connector (gᵥₐ₊):

8. Function signature: gᵥₐ₊: 𝓧ᵥ × Π → 𝓧ₐ₊
9. Maps decision states directly to next stage's arrival states
10. For each branch j ∈ N₊ and fixed choice 𝜋:
    • The map xᵥ ↦ xₐ₊ is bijective when restricted to that branch
    • Reverse mapping function: gₐ₊ᵥ: 𝓧ₐ₊ × Π → 𝓧ᵥ
11. Used with branching probabilities p(j|xᵥ,𝜋) to determine transitions

These bijectivity properties ensure that: 1. Population measures can be pushed forward uniquely: - Through within-stage perch transitions via gₐᵥ and gᵥₑ - Through sequential transitions via gₑₐ₊ - Through each branch of branching transitions via gᵥₐ₊ 2. Individual state trajectories can be tracked both forward and backward 3. For branching stages: - Population splits correctly according to branching probabilities - Each branch preserves bijectivity for proper measure evolution

Use of Transition Functions in Simulation and Backward Induction

The transition functions serve dual purposes in our framework:

• Forward Transition Functions (gₑₐ₊ or gᵥₐ₊): These are used during simulation to map the continuation state of the final perch in a stage to the arrival perch in the successor stage.
• for a sequential stage, the forward transition function is gₑₐ₊
• for a branching stage, the forward transition function is gᵥₐ₊

They are required to simulate the evolution of states over time, allowing us to observe system behavior under different scenarios. Also, when integrated with a population measure, they allow us to calculate the expected number of individuals in each state at the next period.

• Backward Mapping Functions (gₐ₊ₑ or gₐ₊ᵥ): These mapping functions allow us to trace an individual's path back from a known future state to determine the preceding states that could have produced that future state. They are defined for completeness, but are not used in the backward induction solution, which requires only the forward transition functions because each earlier (induced) value function depends only on expectations about the forward transition functions.

Transition Equations in Backward and Forward Processes

The transition functions g defined for the backward iterating solution are equally applicable during forward simulation. These equations ensure that the state transitions remain consistent across both processes. However, during forward simulation, it is possible to construct 'auxiliary' variables that are not required in the backward solution stage.

For instance, while the backward solution may focus on normalized variables, the forward simulation can incorporate the construction of levels for these auxiliary variables. This is particularly relevant in scenarios involving permanent shocks, where the level of a variable, such as permanent income, may not be a state variable in the backward solution but is important for accurately simulating the system's dynamics.

This flexibility allows the model to capture a broader range of dynamics during simulation, providing a more comprehensive understanding of the system's behavior.

Existence and Uniqueness of Value Functions

The Bellman operator T defined by the recursive problem structure is a contraction mapping under standard regularity conditions, guaranteeing existence and uniqueness of the value function.

Scope: These contraction conditions apply to periods as a whole, not to individual stages. There is no requirement that each stage independently satisfy the contraction conditions. For example, a portfolio stage may have β = 1 and r = 0 (no discounting and no stage reward), yet the period-level Bellman operator still contracts as long as the combined effect of all stages within a period satisfies the conditions below.

Sufficient Conditions for Contraction:

1. Discounting: The effective discount factor across a full period satisfies 0 < β_period < 1 (i.e., the product of stage-level discount factors ∏_{s ∈ 𝕊} β_s satisfies this bound; individual stages may have β_s = 1). In the current architecture, the dedicated disc (discounting) stage is the sole bearer of β < 1, appearing as the last stage in every period; all other ("economic") stages have β = 1. The period-level condition thus reduces to requiring the disc stage's β ∈ (0, 1)
2. Bounded Rewards: The reward function r(xᵥ, 𝜋) is bounded, or grows sufficiently slowly relative to discounting
3. Compact Choice Sets: Π(xᵥ) is compact for all xᵥ ∈ 𝓧ᵥ
4. Continuity:
5. r(xᵥ, 𝜋) is continuous in (xᵥ, 𝜋)
6. gᵥₑ(xᵥ, 𝜋, ζᵥₑ) is continuous in (xᵥ, 𝜋) for each ζᵥₑ
7. Π(xᵥ) is continuous (upper and lower hemicontinuous) in xᵥ

Contraction Property:

Under these conditions, the Bellman operator T satisfies:

‖T𝒱 - T𝒲‖ ≤ β ‖𝒱 - 𝒲‖

where ‖·‖ is the sup norm. By the Banach fixed-point theorem: - A unique fixed point 𝒱 exists satisfying T𝒱 = 𝒱 - Value function iteration converges: 𝒱ₙ → 𝒱 as n → ∞ - Convergence is geometric: ‖𝒱ₙ - 𝒱‖ ≤ βⁿ‖𝒱₀ - 𝒱‖

Special Cases:

CRRA Utility with Borrowing Constraint: For consumption problems with u(c) = c^(1-ρ)/(1-ρ) and constraint a ≥ 0: - Utility is unbounded below as c → 0 - Standard approach: work with transformed value function or impose c ≥ c_min > 0 - Natural borrowing constraint ensures feasibility

Infinite Horizon: For stationary infinite-horizon problems: - Terminal condition replaced by fixed-point condition - Iteration continues until ‖𝒱ₙ - 𝒱ₙ₋₁‖ < ε for tolerance ε

Policy Function Existence:

Under the contraction conditions, the optimal policy function π*(xᵥ) exists and satisfies:

π*(xᵥ) ∈ argmax_{𝜋 ∈ Π(xᵥ)} [r(xᵥ, 𝜋) + β E[ℰ(gᵥₑ(xᵥ, 𝜋, ζᵥₑ))]]

If r and ℰ are strictly concave in 𝜋, the policy is unique.

Christopher Carroll and Akshay Shanker — repository in development, please do not cite.