Skip to content

Consumption-Savings with IID Shocks

A single-stage lifecycle model solved via EGM.

This example demonstrates the simplest complete dolo-plus model: a single stage containing both expectation and maximization, designed for solution via the Endogenous Grid Method (EGM).

Model Overview

Feature Description
Stages 1 (self-contained stage with both expectation and maximization)
Shocks IID log-normal income shock
Solution Endogenous Grid Method (EGM)
State Wealth w (cash-on-hand)

Economic Problem

The agent solves: $$ V(b) = \mathbb{E}_y \left[ \max_c \left{ u(c) + \beta V(a) \right} \right] $$

where: - b = beginning-of-period assets (prestate) - y = income shock (exogenous) - w = exp(y) + b*r = cash-on-hand (state after shock) - c = consumption (control) - a = w - c = end-of-period assets (poststate)

ADC Stage Structure

[ARRIVAL] ───g_ad───▶ [DECISION] ───g_de───▶ [CONTINUATION]
    b          y           w        c            a
              shock       choice
  • Arrival → Decision: Income shock realized, w = exp(y) + b*r
  • Decision → Continuation: Consumption choice, a = w - c

Files

File Purpose
stage.yaml Stage syntax (symbols, equations)
calibration.yaml Parameter values
settings.yaml Grid sizes, bounds
methodization.yml Solution scheme (EGM recipe)

Stage YAML

name: consumption_savings_iid_egm_doloplus

symbols:
  spaces:
    Xb: "@def R+"     # prestate assets
    Xw: "@def R+"     # cash-on-hand
    Xa: "@def R+"     # end-of-period assets
    Y:  "@def R"      # income support

  prestate:
    b: "@in Xb"
  exogenous:
    y:
      - "@in Y"
      - "@dist Normal(μ_y, σ_y)"
  states:
    w: "@in Xw"
  poststates:
    a: "@in Xa"
  controls:
    c: "@in R+"

  values:
    V: "@in R"
  values_marginal:
    dV: "@in R+"

  parameters:
    β: "@in (0,1)"
    γ: "@in R+"
    r: "@in R++"
    μ_y: "@in R"
    σ_y: "@in R+"

equations:
  # Half-transition: shock realization
  arvl_to_dcsn_transition: |
    w = exp(y) + b*r

  # Budget constraint
  dcsn_to_cntn_transition: |
    a = w - c

  # Reverse state for EGM
  cntn_to_dcsn_transition: |
    w = a + c

  # Backward mover with EGM sub-equations
  cntn_to_dcsn_opr:
    Bellman: |
      V = max_{c}((c^(1-γ))/(1-γ) + β*V[>])
    InvEuler: |
      c[>] = (β*dV[>])^(-1/γ)
    MarginalBellman: |
      dV = (c)^(-γ)

  # Expectation operator
  dcsn_to_arvl_opr:
    Bellman: |
      V[<] = E_{y}(V)
    MarginalBellman: |
      dV[<] = r * E_{y}(dV)

Key Features

Single Stage with Both Operators

Unlike the port-with-shocks model (which separates expectation and maximization), this model contains both in one stage: - dcsn_to_arvl_opr: Expectation over income shock - cntn_to_dcsn_opr: Maximization via EGM

EGM Sub-Equations

The cntn_to_dcsn_opr block contains: 1. InvEuler: Compute consumption from marginal value 2. MarginalBellman: Envelope condition for marginal value 3. cntn_to_dcsn_transition: Reverse mapping w = a + c

Marginal Value Propagation

The MarginalBellman in dcsn_to_arvl_opr: 

dV[<] = r * E_{y}(dV)
propagates the expected marginal value, accounting for the return r.