Buffer Stock Consumption Model (Normalized)¶

The canonical consumption-savings model with permanent and transitory income shocks, normalized by permanent income.

This example presents the buffer stock saving model (Carroll, 2024) in its normalized (ratio) form. Normalizing by permanent income reduces the state from two variables \((\mathbf{m}, \mathbf{P})\) to one \((m = \mathbf{m}/\mathbf{P})\), yielding a stationary single-stage problem suitable for EGM.

Source

Based on Carroll (2024), "Theoretical Foundations of Buffer Stock Saving," Problem \(\mathscr{P}_N\).

Model Overview¶

Feature	Description
Stages	1 (consumption-savings with expectation and maximization)
Shocks	Permanent (\(\psi\)) and transitory (\(\xi\)), IID
Solution	Endogenous Grid Method (EGM)
State	Normalized market resources \(m\)
Horizon	Infinite (stationary)

Bellman Equation¶

\[ v(m) = \max_{0 < c < m} \left\{ \frac{c^{1-\rho}}{1-\rho} + \beta \,\mathbb{E}\!\left[(\Gamma \psi')^{1-\rho}\, v\!\left(\frac{R\,(m - c)}{\Gamma \psi'} + \xi'\right)\right] \right\} \]

subject to:

\[ \begin{aligned} a &= m - c \\ m' &= \frac{R}{\Gamma \psi'}\, a + \xi' \end{aligned} \]

Variables (all normalized by permanent income \(P_t\)):

\(m\) = market resources
\(c\) = consumption
\(a\) = end-of-period assets
\(\psi\) = permanent income shock (\(\mathbb{E}[\psi] = 1\), \(\psi \in [\underline{\psi}, \bar{\psi}]\))
\(\xi\) = transitory income shock (\(\mathbb{E}[\xi] = 1\); equals zero with probability \(\wp > 0\))
\(R\) = gross interest rate
\(\Gamma\) = deterministic permanent income growth factor
\(\beta\) = discount factor
\(\rho\) = coefficient of relative risk aversion (\(\rho > 1\))

The weighting factor \((\Gamma \psi)^{1-\rho}\) arises from normalizing the value function: \(v(m) \;=\; \mathbf{V}(\mathbf{m}, \mathbf{P})\,/\,\mathbf{P}^{1-\rho}\).

First-Order Conditions¶

Euler equation:

\[ c^{-\rho} = R\,\beta\,\mathbb{E}\!\left[(\Gamma \psi')^{-\rho}\, c'(m')^{-\rho}\right] \]

Envelope condition:

\[ v'(m) = c(m)^{-\rho} \]

State-Space Diagram¶

(a) ──shocks (ψ, ξ)──▶ (m) ──c──▶ (a_nxt)

A single stage per period: the prestate \(a\) (end-of-period assets from the previous period) is transformed by two shocks into market resources \(m\); consumption \(c\) determines end-of-period assets \(a_{\text{nxt}} = m - c\).

Stage YAML¶

name: BufferStockConsumption

symbols:
  spaces:
    Xm: "@def R+"       # market resources
    Xa: "@def R+"        # end-of-period assets

  prestate:
    a: "@in Xa"          # end-of-period assets (from previous period)

  exogenous:
    psi:
      - "@in R++"
      - "@dist LogNormal(1, sigma_psi)"     # permanent shock, E[psi] = 1
    xi:
      - "@in R+"
      - "@dist Mixture(p_zero, 0, LogNormal(1, sigma_xi) / (1 - p_zero))"
                                             # transitory shock, E[xi] = 1

  states:
    m: "@in Xm"          # market resources (after shocks)

  poststates:
    a_nxt: "@in Xa"      # end-of-period assets

  controls:
    c: "@in R+"

  values:
    V[<]: "@in R"
    V: "@in R"

  values_marginal:
    dV[<]: "@in R+"
    dV: "@in R+"

  parameters:
    beta: "@in (0,1)"
    rho: "@in (1, Inf)"
    R: "@in R++"
    G: "@in R++"
    sigma_psi: "@in R+"
    sigma_xi: "@in R+"
    p_zero: "@in [0,1)"

equations:
  arvl_to_dcsn_transition: |
    m = R * a[<] / (G * psi) + xi

  dcsn_to_cntn_transition: |
    a_nxt[>] = m - c

  cntn_to_dcsn_mover:
    Bellman: |
      V = max_c{c^(1-rho)/(1-rho) + beta * V[>]}
    InvEuler: |
      c = (beta * dV[>])^(-1/rho)
    ShadowBellman: |
      dV = c^(-rho)

  dcsn_to_arvl_mover:
    Bellman: |
      V[<] = E_{psi, xi}[(G * psi)^(1-rho) * V]
    ShadowBellman: |
      dV[<] = R * E_{psi, xi}[(G * psi)^(-rho) * dV]

Arrival Transition¶

The arrival transition \(m = R\,a^{<}\!/(\Gamma\psi) + \xi\) combines two steps from the levels problem:

Bank balances: \(b = R\,a\,/(\Gamma\psi)\), where the denominator \(\Gamma\psi\) is the realized permanent income growth factor.
Market resources: \(m = b + \xi\), adding transitory income.

Movers¶

cntn_to_dcsn_mover (\(\mathbb{B}\)): The decision mover contains the maximization problem and its EGM sub-equations:

Bellman: the Bellman equation at the decision perch
InvEuler: inverted Euler equation for EGM (consumption from marginal continuation value)
ShadowBellman: envelope condition (marginal value from consumption)

dcsn_to_arvl_mover (\(\mathbb{I}\)): The arrival mover integrates over both shocks with the normalization weight:

Bellman: \(V^{<}(a) = \mathbb{E}_{\psi,\xi}[(\Gamma\psi)^{1-\rho}\, V(m)]\) where \(m = Ra/(\Gamma\psi) + \xi\)
ShadowBellman: \(dV^{<}(a) = R\,\mathbb{E}_{\psi,\xi}[(\Gamma\psi)^{-\rho}\, dV(m)]\)

The \(R\) factor in the marginal-value propagation arises from differentiating the arrival transition with respect to \(a\).

Period and Nest¶

Period Template¶

name: BufferStockPeriod

stages:
  - consumption: !stage

A single non-branching stage per period.

Stationary Nest¶

name: BufferStockLifecycle

periods:
  - buffer_stock_period: !period    # repeated identically (stationary)

twisters:
  - rename: {a_nxt: a}             # end-of-period assets → next period's prestate

The model is stationary: every period uses the same stage template with the same parameters. The twister maps the poststate a_nxt to the next period's prestate a.

Solution Method: EGM¶

The Endogenous Grid Method solves this model by inverting the Euler equation:

Start with continuation marginal value \(dV^{>}(a)\) on an exogenous grid of end-of-period assets \(\{a_1, \dots, a_n\}\).
Expectation (dcsn_to_arvl_mover): \(dV^{<}(a_i) = R\,\mathbb{E}_{\psi,\xi}[(\Gamma\psi)^{-\rho}\, dV(Ra_i/(\Gamma\psi) + \xi)]\).
InvEuler: \(c_i = (\beta \cdot dV^{<}(a_i))^{-1/\rho}\).
Endogenous grid: \(m_i = a_i + c_i\).
Envelope: \(dV_i = c_i^{-\rho}\).
Iterate until convergence.

The normalization weight \((\Gamma\psi)^{-\rho}\) in the marginal-value expectation (step 2) is the key difference from a model without permanent income growth.

Calibration (Baseline)¶

calibration:
  R: 1.03              # gross interest rate
  beta: 0.96           # discount factor
  G: 1.03              # permanent income growth factor
  rho: 2               # CRRA coefficient
  sigma_psi: 0.1       # std dev of log permanent shock
  sigma_xi: 0.1        # std dev of log transitory shock (conditional on employment)
  p_zero: 0.005        # unemployment probability

settings:
  grid_max_a: 50        # maximum assets for grid
  grid_size: 100        # number of grid points

Key Features¶

Permanent Income Normalization¶

Dividing by \(P_t\) (for quantities) and \(P_t^{1-\rho}\) (for the value function) eliminates the permanent income state. The cost is a growth-adjusted arrival transition and a normalization weight \((\Gamma\psi)^{1-\rho}\) inside the expectation operator.

Natural Borrowing Constraint¶

The transitory shock \(\xi\) has a point mass at zero (probability \(\wp > 0\)): the agent may receive zero income. This prevents the agent from choosing \(c = m\) (which would risk zero consumption next period), generating a natural borrowing constraint without an explicit lower bound on assets.

Weighted Expectations in the Arrival Mover¶

The dcsn_to_arvl_mover propagates both value and marginal value through a \((\Gamma\psi)\)-weighted expectation. This weight is an artifact of the normalization, not an economic primitive: in the levels problem, the arrival mover would be a plain expectation \(V^{<} = \mathbb{E}[V]\).