A couple more botched references fixed.

One figure and one equation.

Author: kchipeniuk

content/exports/appendix_letters.pdf

@@ -190,7 +190,7 @@ \subsection{Cusp Point Calculation}
 \end{array}
 \end{equation}
 
-This motivates the definition of the low-resource moderation ratio as in \{eq\} \texttt{eq:modRteLoTightUpBd}.
+This motivates the definition of the low-resource moderation ratio as in (\ref{eq:modRteLoTightUpBd}).
 
 \subsection{Value Function Derivation}
 

content/exports/appendix_letters_pdf_tex/appendix_letters.tex

@@ -407,7 +407,7 @@ \subsection{Hermite Interpolation}
 
 \subsection{Stochastic Rate of Return}
 
-For i.i.d. returns with $\log \Risky \sim \Nrml(r + \equityPrem - \std^{2}_{\risky}/2,\std^{2}_{\risky})$, \citet{Samuelson1969, Merton1969, Merton1971} showed that for a consumer without labor income (or with perfectly forecastable labor income) the consumption function is linear, with an MPC $= 1 - (\DiscFac \Ex[\Risky^{1 -\CRRA}])^{1/\CRRA}$. See \citet{CRRA-RateRisk, BBZ2016SkewedWealth, CKW2021Aggregation} for extensions. Simply substitute this stochastic MPC for $\MPCmin$ throughout our formulas. The pessimist and optimist still perceive their income streams with certainty, but both face the same stochastic return; thus the Merton-Samuelson result applies to them and their consumption functions remain linear. The realist, however, faces both labor income uncertainty and rate-of-return risk, so the moderation ratio captures the combined precautionary response to both sources of uncertainty. All moderation ratio calculations proceed identically. Extensions to serially correlated returns require tracking the return state as an additional state variable, complicating the analysis but not fundamentally altering the approach. As \{ref\} \texttt{fig:StochasticBounds} shows, consumption remains bounded between the pessimist and the optimist, each of which consume slightly less in the face of return uncertainty.
+For i.i.d. returns with $\log \Risky \sim \Nrml(r + \equityPrem - \std^{2}_{\risky}/2,\std^{2}_{\risky})$, \citet{Samuelson1969, Merton1969, Merton1971} showed that for a consumer without labor income (or with perfectly forecastable labor income) the consumption function is linear, with an MPC $= 1 - (\DiscFac \Ex[\Risky^{1 -\CRRA}])^{1/\CRRA}$. See \citet{CRRA-RateRisk, BBZ2016SkewedWealth, CKW2021Aggregation} for extensions. Simply substitute this stochastic MPC for $\MPCmin$ throughout our formulas. The pessimist and optimist still perceive their income streams with certainty, but both face the same stochastic return; thus the Merton-Samuelson result applies to them and their consumption functions remain linear. The realist, however, faces both labor income uncertainty and rate-of-return risk, so the moderation ratio captures the combined precautionary response to both sources of uncertainty. All moderation ratio calculations proceed identically. Extensions to serially correlated returns require tracking the return state as an additional state variable, complicating the analysis but not fundamentally altering the approach. As Figure~\ref{fig:StochasticBounds} shows, consumption remains bounded between the pessimist and the optimist, each of which consume slightly less in the face of return uncertainty.
 
 \begin{figure}[!htbp]
 \centering

content/exports/moderation_letters.pdf

@@ -414,7 +414,7 @@ \subsection{Hermite Interpolation}
 
 \subsection{Stochastic Rate of Return}
 
-For i.i.d. returns with $\log \Risky \sim \Nrml(r + \equityPrem - \std^{2}_{\risky}/2,\std^{2}_{\risky})$, \citet{Samuelson1969, Merton1969, Merton1971} showed that for a consumer without labor income (or with perfectly forecastable labor income) the consumption function is linear, with an MPC $= 1 - (\DiscFac \Ex[\Risky^{1 -\CRRA}])^{1/\CRRA}$. See \citet{CRRA-RateRisk, BBZ2016SkewedWealth, CKW2021Aggregation} for extensions. Simply substitute this stochastic MPC for $\MPCmin$ throughout our formulas. The pessimist and optimist still perceive their income streams with certainty, but both face the same stochastic return; thus the Merton-Samuelson result applies to them and their consumption functions remain linear. The realist, however, faces both labor income uncertainty and rate-of-return risk, so the moderation ratio captures the combined precautionary response to both sources of uncertainty. All moderation ratio calculations proceed identically. Extensions to serially correlated returns require tracking the return state as an additional state variable, complicating the analysis but not fundamentally altering the approach. As \{ref\} \texttt{fig:StochasticBounds} shows, consumption remains bounded between the pessimist and the optimist, each of which consume slightly less in the face of return uncertainty.
+For i.i.d. returns with $\log \Risky \sim \Nrml(r + \equityPrem - \std^{2}_{\risky}/2,\std^{2}_{\risky})$, \citet{Samuelson1969, Merton1969, Merton1971} showed that for a consumer without labor income (or with perfectly forecastable labor income) the consumption function is linear, with an MPC $= 1 - (\DiscFac \Ex[\Risky^{1 -\CRRA}])^{1/\CRRA}$. See \citet{CRRA-RateRisk, BBZ2016SkewedWealth, CKW2021Aggregation} for extensions. Simply substitute this stochastic MPC for $\MPCmin$ throughout our formulas. The pessimist and optimist still perceive their income streams with certainty, but both face the same stochastic return; thus the Merton-Samuelson result applies to them and their consumption functions remain linear. The realist, however, faces both labor income uncertainty and rate-of-return risk, so the moderation ratio captures the combined precautionary response to both sources of uncertainty. All moderation ratio calculations proceed identically. Extensions to serially correlated returns require tracking the return state as an additional state variable, complicating the analysis but not fundamentally altering the approach. As Figure~\ref{fig:StochasticBounds} shows, consumption remains bounded between the pessimist and the optimist, each of which consume slightly less in the face of return uncertainty.
 
 \begin{figure}[!htbp]
 \centering
@@ -474,7 +474,7 @@ \subsection{Cusp Point Calculation}
 \end{array}
 \end{equation}
 
-This motivates the definition of the low-resource moderation ratio as in \{eq\} \texttt{eq:modRteLoTightUpBd}.
+This motivates the definition of the low-resource moderation ratio as in (\ref{eq:modRteLoTightUpBd}).
 
 \subsection{Value Function Derivation}
 

content/exports/moderation_letters_pdf_tex/moderation_letters.tex

@@ -62,7 +62,7 @@ $$
 \end{array}
 $$
 
-This motivates the definition of the low-resource moderation ratio as in {eq} `eq:modRteLoTightUpBd`.
+This motivates the definition of the low-resource moderation ratio as in {eq}`eq:modRteLoTightUpBd`.
 
 ## Value Function Derivation
 

content/exports/moderation_with_appendix.pdf

@@ -268,7 +268,7 @@ For Hermite interpolation, compute $\modRteMu$ at gridpoints, then derive $\logi
 
 ## Stochastic Rate of Return
 
-For i.i.d. returns with $\log \Risky \sim \Nrml(r + \equityPrem - \std^{2}_{\risky}/2,\std^{2}_{\risky})$, {cite:t}`Samuelson1969,Merton1969,Merton1971` showed that for a consumer without labor income (or with perfectly forecastable labor income) the consumption function is linear, with an MPC $= 1- (\DiscFac \Ex[\Risky^{1-\CRRA}])^{1/\CRRA}$. See {cite:t}`CRRA-RateRisk,BBZ2016SkewedWealth,CKW2021Aggregation` for extensions. Simply substitute this stochastic MPC for $\MPCmin$ throughout our formulas. The pessimist and optimist still perceive their income streams with certainty, but both face the same stochastic return; thus the Merton-Samuelson result applies to them and their consumption functions remain linear. The realist, however, faces both labor income uncertainty and rate-of-return risk, so the moderation ratio captures the combined precautionary response to both sources of uncertainty. All moderation ratio calculations proceed identically. Extensions to serially correlated returns require tracking the return state as an additional state variable, complicating the analysis but not fundamentally altering the approach. As {ref} `fig:StochasticBounds` shows, consumption remains bounded between the pessimist and the optimist, each of which consume slightly less in the face of return uncertainty.  
+For i.i.d. returns with $\log \Risky \sim \Nrml(r + \equityPrem - \std^{2}_{\risky}/2,\std^{2}_{\risky})$, {cite:t}`Samuelson1969,Merton1969,Merton1971` showed that for a consumer without labor income (or with perfectly forecastable labor income) the consumption function is linear, with an MPC $= 1- (\DiscFac \Ex[\Risky^{1-\CRRA}])^{1/\CRRA}$. See {cite:t}`CRRA-RateRisk,BBZ2016SkewedWealth,CKW2021Aggregation` for extensions. Simply substitute this stochastic MPC for $\MPCmin$ throughout our formulas. The pessimist and optimist still perceive their income streams with certainty, but both face the same stochastic return; thus the Merton-Samuelson result applies to them and their consumption functions remain linear. The realist, however, faces both labor income uncertainty and rate-of-return risk, so the moderation ratio captures the combined precautionary response to both sources of uncertainty. All moderation ratio calculations proceed identically. Extensions to serially correlated returns require tracking the return state as an additional state variable, complicating the analysis but not fundamentally altering the approach. As {ref}`fig:StochasticBounds` shows, consumption remains bounded between the pessimist and the optimist, each of which consume slightly less in the face of return uncertainty.  
 
 :::{figure} #fig:stochastic-bounds
 :label: fig:StochasticBounds