Appendix: Mathematical Details¶

Patience Conditions Details¶

The patience conditions listed in the main text have clear economic interpretations. The FVAC $0<\DiscFac\PermGroFac^{1-\CRRA}\Ex[\permShk^{1-\CRRA}]<1$ ensures autarky (consuming zero forever) has finite disutility, guaranteeing the consumer values resources. The AIC $\AbsPatFac<1$ prevents indefinite consumption deferral by ensuring the marginal utility of current consumption exceeds the discounted marginal utility of future consumption under certainty. The RIC $\AbsPatFac/\Rfree<1$ ensures asset growth is slower than the patience-adjusted discount rate, preventing unbounded wealth accumulation. The GIC $\AbsPatFac/\PermGroFac<1$ ensures consumption grows slower than permanent income, establishing a target wealth ratio. The FHWC $\PermGroFac/\Rfree<1$ ensures the present value of future labor income is finite. Together, these conditions partition parameter space into regions with qualitatively different behavior: buffer-stock saving with a target wealth ratio (all conditions hold), perpetual borrowing (AIC fails), or unbounded wealth growth (GIC fails but RIC holds) Carroll, 1997Carroll, 2020Carroll & Shanker, 2024.

Human Wealth Formulas¶

The optimist’s human wealth (assuming $\tranShk_{t+n}=1~\forall~n>0$) can be computed three ways: backward recursion $\hNrmOpt_{T} = 0$, $\hNrmOpt_{t} = (\PermGroFac/\Rfree)(1 + \hNrmOpt_{t+1})$; forward sum $\hNrmOpt_{t} = \sum_{n=1}^{T-t}(\PermGroFac/\Rfree)^{n}$; or infinite-horizon $\hNrmOpt = \PermGroFac/(\Rfree-\PermGroFac)$ when $\Rfree>\PermGroFac$. With $\PermGroFac=1$, $\hNrmOpt = 1/(\Rfree-1)$.

The pessimist’s human wealth (assuming $\tranShk_{t+n}=\tranShkMin~\forall~n>0$) follows similarly: backward recursion $\hNrmPes_{T}=0$, $\hNrmPes_{t}=(\PermGroFac/\Rfree)(\tranShkMin + \hNrmPes_{t+1})$; forward sum $\hNrmPes_{t}=\tranShkMin\sum_{n=1}^{T-t}(\PermGroFac/\Rfree)^{n}$; or infinite-horizon $\hNrmPes=\tranShkMin\PermGroFac/(\Rfree-\PermGroFac)$. When $\tranShkMin=0$ (unemployment), $\hNrmPes=0$.

Marginal Propensity to Consume Formulas¶

The minimal MPC (perfect foresight consumer with horizon $T-t$) has three forms Carroll, 2001: backward recursion $\MPCmin_{t}=\MPCmin_{t+1}/(\MPCmin_{t+1}+\AbsPatFac/\Rfree)$ with $\MPCmin_T=1$; forward sum $\MPCmin_{t}=(\sum_{n=0}^{T-t}(\AbsPatFac/\Rfree)^{n})^{-1}$; or infinite-horizon $\MPCmin=1-\AbsPatFac/\Rfree = 1-(\Rfree \DiscFac)^{1/\CRRA}/\Rfree$.

The maximal MPC Carroll & Toche, 2009 satisfies backward recursion $\MPCmax_{t} = 1 - \WorstProb^{1/\CRRA} (\AbsPatFac/\Rfree) (1 + \MPCmax_{t+1})$ with $\MPCmax_T = 1$; forward sum $\MPCmax_{t} = 1 - \WorstProb^{1/\CRRA} (\AbsPatFac/\Rfree) \sum_{n=0}^{T-t}\left(\WorstProb^{1/\CRRA} (\AbsPatFac/\Rfree)\right)^{n}$; or infinite-horizon $\MPCmax = 1 - \WorstProb^{1/\CRRA} (\AbsPatFac/\Rfree)$.

Moderation Ratio Slope Formula¶

The moderation ratio slope needed for Hermite interpolation is derived from differentiating (8) from the main text. Using the chain rule:

Since $\partial e^{\logmNrmEx}/\partial \logmNrmEx = e^{\logmNrmEx} = \mNrmEx$ and $\cFuncPes$ is linear with slope $\MPCmin$, this yields

Asymptotic Linearity and Extrapolation¶

Under the GIC, the logit-transformed moderation ratio $\logitModRte(\logmNrmEx)$ is asymptotically linear with slope $\asympSlope = \lim_{\logmNrmEx \to +\infty} \frac{\partial \logitModRte}{\partial \logmNrmEx} \geq 0$ as $\logmNrmEx \to +\infty$. This slope may equal zero in theory, but is strictly positive on finite grids. For practical implementation, extrapolate $\logitModRte$ linearly using the positive boundary slope computed at the highest gridpoint. This linear extrapolation preserves $\modRte\in(0,1)$ and hence $\cFuncPes < \cFuncApprox < \cFuncOpt$ throughout the extrapolation domain, even if the theoretical limiting slope vanishes. The asymptotic linearity property is crucial because it allows the method to accurately represent the consumption function far beyond the range of gridpoints where the Euler equation was solved, without ever violating the theoretical bounds.

Cusp Point Calculation¶

The two upper bounds intersect at the cusp point $\mNrmCusp$ where

where $\mNrmCuspEx\equiv\mNrmCusp-\mNrmMin > 0$ since $\MPCmax > \MPCmin$. For $\mNrm \in (\mNrmMin, \mNrmCusp]$, the tighter upper bound yields

We define the low-region moderation ratio as

which measures how far consumption per unit of wealth exceeds the optimist’s MPC relative to the maximum possible excess.

Hermite Interpolation: Slope Derivations¶

The logit transformation slope follows from the chain rule Santos, 2000Judd et al., 2017:

where $\modRteMu$ is from (2). For monotone cubic Hermite schemes Fritsch & Carlson, 1980Fritsch & Butland, 1984Boor, 2001, theoretical slopes may be adjusted to enforce monotonicity Hyman, 1983. The Fritsch-Carlson algorithm modifies slopes at local extrema, while Fritsch-Butland uses harmonic mean weighting. Both preserve the shape-preserving property essential for consumption functions that must be strictly increasing.

The MPC weight derivation starts from differentiating (10) from the main text with respect to $\mNrm$:

Since $\cFuncPes$ has constant MPC and $\hNrmEx$ is constant, $\partial \cFuncPes/\partial \mNrm = \MPCmin$ and $\partial \modRte/\partial \mNrm = (\partial \modRte/\partial \logmNrmEx) \cdot (\partial \logmNrmEx/\partial \mNrmEx) \cdot (\partial \mNrmEx/\partial \mNrm) = \modRteMu \cdot (1/\mNrmEx) \cdot 1$. This yields

Factoring as a weighted average between $\MPCmin$ and $\MPCmax$ gives (13) from the main text, with weight

Value Function Derivation¶

Under perfect foresight, consumption grows at constant rate $\AbsPatFac$: $\cLvl_{t+n}=\cLvl_{t}\AbsPatFac^{n}$. The present discounted value of consumption satisfies $\PDV_{t}^{T}(\cLvl)=\sum_{n=0}^{T-t}\DiscFac^{n}\cLvl_{t}\AbsPatFac^{n}=\cLvl_{t}\sum_{n=0}^{T-t}(\AbsPatFac/\Rfree)^{n}$, where we use $\DiscFac\AbsPatFac^{1-\CRRA}=\AbsPatFac/\Rfree$. Dividing by consumption yields the PDV-to-consumption ratio $\PDVCoverc_{t}^{T}=\PDV_{t}^{T}(\cLvl)/\cLvl_{t}=\sum_{n=0}^{T-t}(\AbsPatFac/\Rfree)^{n}=\MPCmin_{t}^{-1}$, which is unchanged for normalized variables. This yields the key identity $\PDVCoverc = \MPCmin^{-1}$ in the infinite-horizon limit, connecting the PDV-to-consumption ratio to the minimal MPC.

The optimist’s value function satisfies

The general expression becomes

This can be transformed as

For the realist’s problem, we define $\vInvReal = \left((1-\CRRA)\vFuncReal(\mNrm)\right)^{1/(1-\CRRA)}$. Using the bounds $\vInvPes < \vInvReal < \vInvOpt$, we define

and:

Inverting these approximations yields

from which the value function approximation is $\vFuncReal = \uFunc(\vInvReal)$.

Stochastic Returns: MGF Derivation¶

The moment generating function (MGF) for lognormal returns provides the key formula. For $\log \Risky \sim \Nrml(\mu, \sigma^2)$, the MGF is $\Ex[e^{sX}] = \exp(\mu s + \sigma^2 s^2/2)$ where $X = \log \Risky$. Setting $s = 1-\CRRA$ and $\mu = r + \equityPrem - \std_{\risky}^2/2$ yields

Simplifying the variance terms: $(1-\CRRA)^2\std_{\risky}^2/2 - (1-\CRRA)\std_{\risky}^2/2 = (1-\CRRA)[(1-\CRRA)-1]\std_{\risky}^2/2 = (1-\CRRA)(-\CRRA)\std_{\risky}^2/2$, giving the final form

For serially correlated returns, the return state becomes an additional state variable, requiring two-dimensional interpolation of the moderation ratio.

Shock Discretization¶

Continuous shock distributions require discretization. The Tauchen method Tauchen, 1986 constructs a Markov chain by dividing the state space into bins. The Tauchen-Hussey method Tauchen & Hussey, 1991 uses Gaussian quadrature, often requiring fewer states for comparable accuracy. For unemployment shocks, assign probability $\WorstProb$ to zero income and $(1-\WorstProb)$ across positive realizations. Choose gridpoints and shock points via convergence analysis. The method of moderation is efficient because the transformed moderation ratio is better-behaved than consumption, requiring fewer gridpoints.