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 that autarky (saving nothing, consuming all income each period) yields finite expected discounted utility, 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, 2009: 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 $\MPCmax_{t}=\MPCmax_{t+1}/(\MPCmax_{t+1}+\WorstProb^{1/\CRRA}\AbsPatFac/\Rfree)$ with $\MPCmax_T=1$; forward sum $\MPCmax_{t}=(\sum_{n=0}^{T-t}(\WorstProb^{1/\CRRA}\AbsPatFac/\Rfree)^{n})^{-1}$; or infinite-horizon $\MPCmax = 1 - \WorstProb^{1/\CRRA} (\AbsPatFac/\Rfree)$.

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

This motivates the definition of the low-resource moderation ratio as in (15).

Value Function Derivation¶

Under perfect foresight, consumption grows at constant rate equal to the absolute patience factor $\AbsPatFac$: $\cLvl_{t+n}=\cLvl_{t}\AbsPatFac^{n}$. 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. Defining $\PDVCoverc \equiv \lim_{T\to\infty} \PDVCoverc_{t}^{T}$, this yields the key identity $\PDVCoverc = \MPCmin^{-1}$, connecting the infinite-horizon PDV-to-consumption ratio to the minimal MPC.

The optimist’s value function satisfies

The infinite horizon 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 $\vFuncReal = \uFunc(\vInvReal)$.

I.I.D. Stochastic Returns: MPC Derivation¶

The fact that a linear consumption function with an MPC $= 1- (\DiscFac \Ex[\Risky^{1-\CRRA}])^{1/\CRRA}$ satisfies the Euler equation with i.i.d. returns and no labor income can be derived by the method of undetermined coefficients. In particular, assume that $\cFuncOpt(\mNrm) = \mNrm\MPCmin$, with a time-independent MPC $\MPCmin$ to be determined. Substituting this into the Euler equation, we have

where the second equality uses the assumed form of the consumption function. Since there is no labor income, $\mNrm_{t+1} = \Risky_{t+1}(\mNrm_t - \cNrm_t)$. Substituting this into the above we obtain

Solving for $\MPCmin$ and recalling that returns are i.i.d. gives $\MPCmin=1- (\DiscFac \Ex[\Risky^{1-\CRRA}])^{1/\CRRA}$.

In the particular case of lognormal returns, the MPC can be written in closed form. 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^[1]

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