--- # Page-specific frontmatter (inherits bibliography from myst.yml) title: Appendix to "The Method of Moderation" short_title: Appendix abstract: | This appendix provides detailed mathematical derivations and technical results supporting the Method of Moderation. Topics include: value function transformations and their relationship to the inverse value function; explicit formulas for minimal and maximal marginal propensities to consume; cusp point calculations for tighter upper bounds; Hermite interpolation slope formulas and MPC derivations; patience conditions ensuring well-defined solutions; and extensions to stochastic returns with explicit formulas for portfolio problems. keywords: - Dynamic Stochastic Optimization - Consumption-Saving Models - Numerical Methods parts: jel_codes: C63; D81; E21 exports: - format: tex+pdf template: arxiv_two_column output: ../exports/appendix_letters.pdf --- # 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) {cite:p}`Carroll1997,SolvingMicroDSOPs,CarrollShanker2024`. ## 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 {cite:p}`Carroll2001MPCBound`: 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 {cite:p}`CarrollToche2009` 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 ```{math} :label: eq:mNrmCuspFull \begin{array}{rclcll} \bigl(\mNrmCuspEx + \hNrmEx\bigr)\,\MPCmin &= & \MPCmax\,\mNrmCuspEx & & \\ \mNrmCuspEx &= & \dfrac{\MPCmin\,\hNrmEx}{\MPCmax-\MPCmin} & & \\ \mNrmCusp &= & -\hNrmPes + \dfrac{\MPCmin\,\bigl(\hNrmOpt-\hNrmPes\bigr)}{\MPCmax-\MPCmin}, \end{array} ``` where $\mNrmCuspEx\equiv\mNrmCusp-\mNrmMin > 0$ since $\MPCmax > \MPCmin$. For $\mNrm \in (\mNrmMin, \mNrmCusp]$, the tighter upper bound yields $$ \begin{array}{rcl} \mNrmEx \MPCmin < & \cFuncReal(\mNrmMin+\mNrmEx) & < \MPCmax \mNrmEx \\ 0 < & \cFuncReal(\mNrmMin+\mNrmEx) - \mNrmEx \MPCmin & < \mNrmEx(\MPCmax- \MPCmin) \\ 0 < & \left(\frac{\cFuncReal(\mNrmMin+\mNrmEx) - \mNrmEx \MPCmin}{\mNrmEx(\MPCmax- \MPCmin)}\right) & < 1. \end{array} $$ This motivates the definition of the low-resource moderation ratio as in {eq}`eq:modRteLoTightUpBd`. ## 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 $$ \begin{aligned} \vFuncOpt_{T-1}(\mNrm_{T-1}) &\equiv \uFunc(\cNrm_{T-1})+\DiscFac \uFunc(\cNrm_{T}) \\ &= \uFunc(\cNrm_{T-1})\left(1+\DiscFac \AbsPatFac^{1-\CRRA}\right) \\ &= \uFunc(\cNrm_{T-1})\left(1+\AbsPatFac/\Rfree\right) \\ &= \uFunc(\cNrm_{T-1})\PDVCoverc_{T-1}^{T} \end{aligned} $$ The infinite horizon expression becomes ```{math} :label: eq:vFuncPF \begin{aligned} \vFuncOpt(\mNrm) &= \uFunc(\cFuncOpt(\mNrm))\PDVCoverc \\ &= \uFunc(\cFuncOpt(\mNrm))\MPCmin^{-1} \\ &= \uFunc((\mNrmEx+\hNrmEx)\MPCmin) \MPCmin^{-1} \\ &= \uFunc(\mNrmEx+\hNrmEx)\MPCmin^{-\CRRA}. \end{aligned} ``` This can be transformed as $$ \begin{aligned} \vInvOpt &\equiv \left((1-\CRRA)\vFuncOpt\right)^{1/(1-\CRRA)} \\ &= \cNrm\,\PDVCoverc^{1/(1-\CRRA)} \\ &= (\mNrmEx+\hNrmEx)\MPCmin^{-\CRRA/(1-\CRRA)}. \end{aligned} $$ 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 ```{math} :label: eq:valModRteReal \valModRteReal(\logmNrmEx) = \left(\frac{\vInvReal(\mNrmMin+e^{\logmNrmEx})-\vInvPes(\mNrmMin+e^{\logmNrmEx})}{\hNrmEx \MPCmin \,\PDVCoverc^{1/(1-\CRRA)}}\right) ``` and the logit-transformed counterpart: ```{math} :label: eq:ChiUpper \begin{aligned} \logitValModRteReal(\logmNrmEx) &= \log \left(\frac{\valModRteReal(\logmNrmEx)}{1-\valModRteReal(\logmNrmEx)}\right) \\ &= \log(\valModRteReal(\logmNrmEx)) - \log(1-\valModRteReal(\logmNrmEx)) \end{aligned} ``` Inverting these approximations yields ```{math} :label: eq:vInvHi \vInvReal = \vInvPes+\overbrace{\left(\frac{1}{1+\exp(-\logitValModRteReal)}\right)}^{=\valModRteReal} \hNrmEx \MPCmin \,\PDVCoverc^{1/(1-\CRRA) } ``` from which the value function approximation is $\vFuncReal = \uFunc(\vInvReal)$. (stochastic-returns-mgf-derivation)= ## 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 ```{math} :label: eq:stochReturnsEulerEqn \begin{aligned} 1 &= \DiscFac \Ex_t\left[\Risky_{t+1} \left(\frac{\cNrm_{t+1}}{\cNrm_t}\right)^{-\CRRA}\right]\\ &= \DiscFac \Ex_t\left[\Risky_{t+1} \left(\frac{\mNrm_{t+1}}{\mNrm_t}\right)^{-\CRRA}\right] \end{aligned} ``` 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 ```{math} :label: eq:stochReturnsEulerEqnContd 1 = \DiscFac \Ex_t\left[\Risky_{t+1} \left(\Risky_{t+1}(1-\MPCmin)\right)^{-\CRRA}\right] ``` 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[^lognormal-returns-intuition] $$ \Ex[\Risky^{1-\CRRA}] = \exp\left((1-\CRRA)\left(r+\equityPrem - \frac{\std_{\risky}^2}{2}\right) + \frac{(1-\CRRA)^2\std_{\risky}^2}{2}\right). $$ 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 $$ \Ex[\Risky^{1-\CRRA}] = \exp\left((1-\CRRA)\left(r+\equityPrem - \CRRA\std_{\risky}^2/2\right)\right). $$ [^lognormal-returns-intuition]: Here we can interpret $\equityPrem$ as the risk premium, that is, the additional average return from holding a risky asset compared to the risk-free rate $r$. Adjusting the average log return by the asset volatility ensures that increasing $\std_{\risky}^2$ constitutes a mean-preserving spread of the level of return.