Economic Model
Overview
The model is a life-cycle problem where a household chooses consumption and portfolio allocation each period, facing risky labor income, time-varying equity returns, and asymmetric consumption adjustment costs.
State Variables (Scaled Mode)
| Variable | Description |
|---|---|
| age | Current age |
| \(x_t\) | Equity premium state (AR(1)) |
| \(m_t\) | Cash-on-hand / persistent earnings |
| \(cm_t\) | Lagged consumption ratio |
Controls
- \(c_t\): consumption (scaled)
- \(\theta_t\): stock share of liquid wealth, \(\theta \in [0, 1]\)
Key Equations
Equity premium transition: $\(x_{t+1} = \bar{x} + \phi_x (x_t - \bar{x}) + \xi_{t+1}\)$
Realized stock return: $\(R_{stock} = r_f + x_t + N_{CF} - N_{DR}\)$
Consumption adjustment cost (asymmetric, applies only when \(c_t < c_{t-1}\)): $\(\Phi_C = \frac{\phi_c}{2} \frac{(\max(0, cm_t - c_t))^2}{cm_t}\)$
Epstein-Zin utility recursion: $\(V_t = [(1-\beta) c_t^\rho + \beta \text{CE}_t^\rho]^{1/\rho}\)$
where \(\rho = 1 - 1/\psi\) and \(\text{CE}_t = (\mathbb{E}[V_{t+1}^{1-\gamma}])^{1/(1-\gamma)}\).
Paper vs. Package
| Aspect | Paper | Package |
|---|---|---|
| Income process params | Estimated from confidential tax data | Public approximation from qualitative targets |
| Crash mixture | Calibrated to data | Parametric approximation |
| Grid resolution | Not specified | Configurable via GridSpec |
| Quadrature | "Gaussian quadrature over 4 shocks" | Gauss-Hermite with configurable nodes |
Approximations
All income process slope parameters (lambda coefficients linking persistent
earnings shocks to return news) are derived from the paper's qualitative
targets rather than confidential estimation. These are marked with
source="public_approximation" in the calibration metadata.