Create two notebook versions: portable and MyST

- method-of-moderation.ipynb: Expanded LaTeX, displays correctly in
  any Jupyter/KaTeX environment (macros like \mNrm → m)
- method-of-moderation-myst.ipynb: Original with MyST macros, for paper
  building where macros.yml is available

This enables the notebook to display properly in standard Jupyter
while preserving the MyST-compatible source for document builds.

Author: llorracc

code/method-of-moderation-myst.ipynb

@@ -325,7 +325,7 @@
    "source": [
     "## Consumption Function Analysis\n",
     "\n",
-    "The first set of figures will focus on the core of the consumption-saving problem: the consumption function $\\cFunc(\\mNrm)$, which maps market resources $\\mNrm$ to consumption. We will demonstrate the extrapolation problem inherent in the standard EGM and show how the Method of Moderation resolves it by respecting theoretical bounds.\n",
+    "The first set of figures will focus on the core of the consumption-saving problem: the consumption function $\\mathbf{c}(m)$, which maps market resources $m$ to consumption. We will demonstrate the extrapolation problem inherent in the standard EGM and show how the Method of Moderation resolves it by respecting theoretical bounds.\n",
     "\n",
     "### Figure 1: The EGM Extrapolation Problem\n",
     "\n",
@@ -432,7 +432,7 @@
    "metadata": {},
    "source": [
     "```{important} Key Theoretical Insight\n",
-    "The constraint $\\cFuncPes(\\mNrm) \\leq \\cFuncReal(\\mNrm) \\leq \\cFuncOpt(\\mNrm)$ is the foundation for MoM {cite:p}`CarrollShanker2024`. The lower bound arises from the natural borrowing constraint in incomplete markets models {cite:p}`Aiyagari1994,Huggett1993,Zeldes1989,Deaton1991`.\n",
+    "The constraint $\\underline{\\mathbf{c}}(m) \\leq \\hat{\\mathbf{c}}(m) \\leq \\bar{\\mathbf{c}}(m)$ is the foundation for MoM {cite:p}`CarrollShanker2024`. The lower bound arises from the natural borrowing constraint in incomplete markets models {cite:p}`Aiyagari1994,Huggett1993,Zeldes1989,Deaton1991`.\n",
     "```\n",
     "\n",
     "### Figure 3: Method of Moderation Solution\n",
@@ -494,22 +494,22 @@
     "\n",
     "MoM steps (notation matches the paper):\n",
     "1. Solve standard EGM for realist consumption at gridpoints\n",
-    "2. Transform to $\\logmNrmEx = \\log(\\mNrm - \\mNrmMin)$\n",
-    "3. Compute $\\modRte(\\logmNrmEx) = (\\cFuncReal - \\cFuncPes)/(\\cFuncOpt - \\cFuncPes) \\in [0,1]$ (Eq. {eq}`eq:modRte`)\n",
-    "4. Apply logit: $\\logitModRte = \\log(\\modRte/(1-\\modRte))$\n",
-    "5. Interpolate $\\logitModRte(\\logmNrmEx)$ with derivatives\n",
-    "6. Reconstruct: $\\cFuncReal = \\cFuncPes + \\modRte \\cdot (\\cFuncOpt - \\cFuncPes)$\n",
+    "2. Transform to $\\boldsymbol{\\mu} = \\log(m - \\underline{m})$\n",
+    "3. Compute $\\boldsymbol{\\omega}(\\boldsymbol{\\mu}) = (\\hat{\\mathbf{c}} - \\underline{\\mathbf{c}})/(\\bar{\\mathbf{c}} - \\underline{\\mathbf{c}}) \\in [0,1]$ (Eq. {eq}`eq:modRte`)\n",
+    "4. Apply logit: $\\boldsymbol{\\chi} = \\log(\\boldsymbol{\\omega}/(1-\\boldsymbol{\\omega}))$\n",
+    "5. Interpolate $\\boldsymbol{\\chi}(\\boldsymbol{\\mu})$ with derivatives\n",
+    "6. Reconstruct: $\\hat{\\mathbf{c}} = \\underline{\\mathbf{c}} + \\boldsymbol{\\omega} \\cdot (\\bar{\\mathbf{c}} - \\underline{\\mathbf{c}})$\n",
     "\n",
     "This ensures bound compliance via asymptotically linear extrapolation, as derived in the paper. HARK uses **cubic Hermite interpolation** {cite:p}`Fritsch1980,FritschButland1984` for accuracy; see {cite:p}`Santos2000,JuddMaliarMaliar2017` on function approximation and error bounding.\n",
     "```\n",
     "\n",
     "```{note} The Transformation\n",
-    "The logit maps $\\modRte \\in (0,1)$ to $\\logitModRte \\in (-\\infty, +\\infty)$ and becomes asymptotically linear with positive slope $\\logitModRteMu > 0$ as wealth increases.\n",
+    "The logit maps $\\boldsymbol{\\omega} \\in (0,1)$ to $\\boldsymbol{\\chi} \\in (-\\infty, +\\infty)$ and becomes asymptotically linear with positive slope $\\partial \\boldsymbol{\\chi} / \\partial \\boldsymbol{\\mu} > 0$ as wealth increases.\n",
     "```\n",
     "\n",
     "### Figure 4: MoM Consumption Function\n",
     "\n",
-    "[](#fig:mom-consumption-function) ({ref}`Figure 4 <fig:IntExpFOCInvPesReaOptNeed45>` in the paper) shows MoM consumption between optimist and pessimist bounds, plus a **tighter upper bound** derived from $\\MPCmax$ near the borrowing constraint {cite:p}`Carroll2001MPCBound,MaToda2021SavingRateRich,CarrollToche2009`. The cusp intersection is given by {eq}`eq:mNrmCusp`."
+    "[](#fig:mom-consumption-function) ({ref}`Figure 4 <fig:IntExpFOCInvPesReaOptNeed45>` in the paper) shows MoM consumption between optimist and pessimist bounds, plus a **tighter upper bound** derived from $\\bar{\\boldsymbol{\\kappa}}$ near the borrowing constraint {cite:p}`Carroll2001MPCBound,MaToda2021SavingRateRich,CarrollToche2009`. The cusp intersection is given by {eq}`eq:mNrmCusp`."
    ]
   },
   {
@@ -608,18 +608,18 @@
     "\n",
     "## Method of Moderation Framework\n",
     "\n",
-    "### Figure 6: Moderation Ratio Function $\\modRte(\\mNrm)$\n",
+    "### Figure 6: Moderation Ratio Function $\\boldsymbol{\\omega}(m)$\n",
     "\n",
     "::::{admonition} Definition: The Moderation Ratio\n",
     ":class: note\n",
     "\n",
     "The **moderation ratio** (Eq. {eq}`eq:modRte`):\n",
     "\n",
     "$$\n",
-    "\\modRte(\\mNrm) = \\frac{\\cFuncReal(\\mNrm) - \\cFuncPes(\\mNrm)}{\\cFuncOpt(\\mNrm) - \\cFuncPes(\\mNrm)} \\in (0,1)\n",
+    "\\boldsymbol{\\omega}(m) = \\frac{\\hat{\\mathbf{c}}(m) - \\underline{\\mathbf{c}}(m)}{\\bar{\\mathbf{c}}(m) - \\underline{\\mathbf{c}}(m)} \\in (0,1)\n",
     "$$\n",
     "\n",
-    "This ratio is strictly between 0 and 1 due to prudence {cite:p}`CarrollKimball1996`. At $\\modRte = 0$ the realist behaves like the pessimist (maximum precautionary saving); at $\\modRte = 1$ like the optimist (no precautionary saving). [](#fig:moderation-ratio) plots this ratio across wealth levels.\n",
+    "This ratio is strictly between 0 and 1 due to prudence {cite:p}`CarrollKimball1996`. At $\\boldsymbol{\\omega} = 0$ the realist behaves like the pessimist (maximum precautionary saving); at $\\boldsymbol{\\omega} = 1$ like the optimist (no precautionary saving). [](#fig:moderation-ratio) plots this ratio across wealth levels.\n",
     "::::"
    ]
   },
@@ -665,8 +665,8 @@
    "id": "587c8ea3",
    "metadata": {},
    "source": [
-    "```{note} Economic Interpretation of $\\modRte(\\mNrm)$\n",
-    "$\\modRte \\to 1$ at high wealth (approaches optimist), $\\modRte \\to 0$ at low wealth (approaches pessimist). This monotonic pattern ensures proper economic behavior across the wealth distribution.\n",
+    "```{note} Economic Interpretation of $\\boldsymbol{\\omega}(m)$\n",
+    "$\\boldsymbol{\\omega} \\to 1$ at high wealth (approaches optimist), $\\boldsymbol{\\omega} \\to 0$ at low wealth (approaches pessimist). This monotonic pattern ensures proper economic behavior across the wealth distribution.\n",
     "```\n",
     "\n",
     "### Figure 7: The Logit Transformation\n",
@@ -677,10 +677,10 @@
     "The **logit transformation** (Eq. {eq}`eq:chi`) maps the bounded ratio to an unbounded space:\n",
     "\n",
     "$$\n",
-    "\\logitModRte(\\logmNrmEx) = \\log\\left(\\frac{\\modRte(\\logmNrmEx)}{1 - \\modRte(\\logmNrmEx)}\\right)\n",
+    "\\boldsymbol{\\chi}(\\boldsymbol{\\mu}) = \\log\\left(\\frac{\\boldsymbol{\\omega}(\\boldsymbol{\\mu})}{1 - \\boldsymbol{\\omega}(\\boldsymbol{\\mu})}\\right)\n",
     "$$\n",
     "\n",
-    "where $\\logmNrmEx = \\log(\\mNrm - \\mNrmMin)$. As [](#fig:logit-transformation) shows, $\\logitModRte$ is nearly linear, making it well-suited for interpolation.\n",
+    "where $\\boldsymbol{\\mu} = \\log(m - \\underline{m})$. As [](#fig:logit-transformation) shows, $\\boldsymbol{\\chi}$ is nearly linear, making it well-suited for interpolation.\n",
     "::::"
    ]
   },
@@ -726,18 +726,18 @@
    "metadata": {},
    "source": [
     "```{important} Why Asymptotic Linearity Matters\n",
-    "As $\\logmNrmEx \\to \\infty$, $\\logitModRte$ becomes linear with slope $\\logitModRteMu > 0$. This prevents extrapolation errors, ensures smooth convergence to the optimist bound, and maintains numerical stability.\n",
+    "As $\\boldsymbol{\\mu} \\to \\infty$, $\\boldsymbol{\\chi}$ becomes linear with slope $\\partial \\boldsymbol{\\chi} / \\partial \\boldsymbol{\\mu} > 0$. This prevents extrapolation errors, ensures smooth convergence to the optimist bound, and maintains numerical stability.\n",
     "```\n",
     "\n",
-    "```{note} Properties of $\\logitModRte(\\logmNrmEx)$\n",
-    "Unbounded domain $(-\\infty, \\infty)$, monotonically increasing, asymptotically linear. $\\logitModRte > 0$ indicates behavior closer to optimist; $\\logitModRte < 0$ closer to pessimist.\n",
+    "```{note} Properties of $\\boldsymbol{\\chi}(\\boldsymbol{\\mu})$\n",
+    "Unbounded domain $(-\\infty, \\infty)$, monotonically increasing, asymptotically linear. $\\boldsymbol{\\chi} > 0$ indicates behavior closer to optimist; $\\boldsymbol{\\chi} < 0$ closer to pessimist.\n",
     "```\n",
     "\n",
     "## Function Properties and Bounds\n",
     "\n",
     "### Figure 8: MoM MPC Bounded by Theory\n",
     "\n",
-    "The **MPC** ($\\partial \\cNrm / \\partial \\mNrm$) is bounded between $\\MPCmin$ (optimist) and $\\MPCmax$ (at the borrowing constraint), as detailed in {ref}`the paper <a-tighter-upper-bound>` and Eq. {eq}`eq:MPCModeration` {cite:p}`Carroll2001MPCBound`. [](#fig:mpc-bounds) confirms MoM respects these bounds.\n",
+    "The **MPC** ($\\partial c / \\partial m$) is bounded between $\\underline{\\boldsymbol{\\kappa}}$ (optimist) and $\\bar{\\boldsymbol{\\kappa}}$ (at the borrowing constraint), as detailed in {ref}`the paper <a-tighter-upper-bound>` and Eq. {eq}`eq:MPCModeration` {cite:p}`Carroll2001MPCBound`. [](#fig:mpc-bounds) confirms MoM respects these bounds.\n",
     "\n",
     "```{tip} Policy Applications\n",
     "Bounded MPC estimates prevent nonsensical policy multipliers in DSGE models. MoM ensures economically meaningful MPCs for policy analysis.\n",
@@ -785,12 +785,12 @@
    "metadata": {},
    "source": [
     "```{hint} MPC Economic Interpretation\n",
-    "MoM MPC declines with $\\mNrm$: poor consumers spend windfalls immediately (MPC $\\to \\MPCmax$), wealthy consumers save them (MPC $\\to \\MPCmin$), reflecting diminishing marginal utility.\n",
+    "MoM MPC declines with $m$: poor consumers spend windfalls immediately (MPC $\\to \\bar{\\boldsymbol{\\kappa}}$), wealthy consumers save them (MPC $\\to \\underline{\\boldsymbol{\\kappa}}$), reflecting diminishing marginal utility.\n",
     "```\n",
     "\n",
     "### Figure 9: Value Functions Bounded by Theory\n",
     "\n",
-    "The **value function** $\\vFunc(\\mNrm)$ is also bounded by optimist and pessimist solutions {cite:p}`Aiyagari1994,Huggett1993`. [](#fig:value-functions) compares truth, EGM, and MoM value functions."
+    "The **value function** $\\mathbf{v}(m)$ is also bounded by optimist and pessimist solutions {cite:p}`Aiyagari1994,Huggett1993`. [](#fig:value-functions) compares truth, EGM, and MoM value functions."
    ]
   },
   {
@@ -843,9 +843,9 @@
     "Uncertainty matters most at low wealth where buffers are small; the optimist-pessimist gap narrows with wealth as assets provide natural insurance.\n",
     "```\n",
     "\n",
-    "### Figure 10: Inverse Value Functions $\\vInv(\\mNrm)$\n",
+    "### Figure 10: Inverse Value Functions ${\\scriptsize \\boldsymbol{\\Lambda}}(m)$\n",
     "\n",
-    "The **inverse value function** $\\vInv(\\mNrm) = \\uFunc^{-1}(\\vFunc(\\mNrm))$ gives the consumption equivalent of lifetime utility. It is more linear than $\\vFunc(\\mNrm)$ near the borrowing constraint, making it better suited for interpolation. [](#fig:inverse-value-functions) compares the three solutions."
+    "The **inverse value function** ${\\scriptsize \\boldsymbol{\\Lambda}}(m) = \\mathbf{u}^{-1}(\\mathbf{v}(m))$ gives the consumption equivalent of lifetime utility. It is more linear than $\\mathbf{v}(m)$ near the borrowing constraint, making it better suited for interpolation. [](#fig:inverse-value-functions) compares the three solutions."
    ]
   },
   {
@@ -892,15 +892,15 @@
    "metadata": {},
    "source": [
     "```{note} Inverse Value Function Interpretation\n",
-    "The inverse transformation converts utility to consumption units: $\\vInv(5) = 0.8$ means 5 units of wealth provides lifetime utility equivalent to consuming 0.8 forever. HARK uses this more linear representation for interpolation.\n",
+    "The inverse transformation converts utility to consumption units: ${\\scriptsize \\boldsymbol{\\Lambda}}(5) = 0.8$ means 5 units of wealth provides lifetime utility equivalent to consuming 0.8 forever. HARK uses this more linear representation for interpolation.\n",
     "```\n",
     "\n",
     "### Figure 11: Value Function Moderation Ratio\n",
     "\n",
     "MoM applies to any bounded function. The **inverse value function moderation ratio** (Eq. {eq}`eq:valModRteReal`):\n",
     "\n",
     "$$\n",
-    "\\valModRte(\\mNrm) = \\frac{\\vInvReal(\\mNrm) - \\vInvPes(\\mNrm)}{\\vInvOpt(\\mNrm) - \\vInvPes(\\mNrm)}\n",
+    "\\boldsymbol{\\Omega}(m) = \\frac{\\hat{{\\scriptsize \\boldsymbol{\\Lambda}}}(m) - \\underline{{\\scriptsize \\boldsymbol{\\Lambda}}}(m)}{\\bar{{\\scriptsize \\boldsymbol{\\Lambda}}}(m) - \\underline{{\\scriptsize \\boldsymbol{\\Lambda}}}(m)}\n",
     "$$\n",
     "\n",
     "follows the same pattern as the consumption ratio, as shown in [](#fig:value-moderation-ratio)."
@@ -949,18 +949,18 @@
    "metadata": {},
    "source": [
     "```{hint} Value Function Moderation Interpretation\n",
-    "$\\valModRte \\to 1$ at high wealth (low uncertainty cost), $\\valModRte \\to 0$ at low wealth (high uncertainty cost). The pattern confirms uncertainty's welfare cost diminishes with wealth.\n",
+    "$\\boldsymbol{\\Omega} \\to 1$ at high wealth (low uncertainty cost), $\\boldsymbol{\\Omega} \\to 0$ at low wealth (high uncertainty cost). The pattern confirms uncertainty's welfare cost diminishes with wealth.\n",
     "```\n",
     "\n",
     "### Figure 12: Cusp Point Visualization\n",
     "\n",
     "The **cusp point** (Eq. {eq}`eq:mNrmCusp`) is where optimist and tighter upper bounds intersect:\n",
     "\n",
     "$$\n",
-    "\\mNrmCusp = \\mNrmMin + \\frac{\\MPCmin \\cdot \\hNrmEx}{\\MPCmax - \\MPCmin}\n",
+    "m^* = \\underline{m} + \\frac{\\underline{\\boldsymbol{\\kappa}} \\cdot \\Delta h}{\\bar{\\boldsymbol{\\kappa}} - \\underline{\\boldsymbol{\\kappa}}}\n",
     "$$\n",
     "\n",
-    "Below the cusp, the tighter bound ($\\MPCmax$ slope) constrains; above, the optimist bound constrains. See `IndShockMoMCuspConsumerType` for the three-piece implementation."
+    "Below the cusp, the tighter bound ($\\bar{\\boldsymbol{\\kappa}}$ slope) constrains; above, the optimist bound constrains. See `IndShockMoMCuspConsumerType` for the three-piece implementation."
    ]
   },
   {
@@ -1005,7 +1005,7 @@
    "metadata": {},
    "source": [
     "```{hint} Cusp Point Interpretation\n",
-    "Below cusp: MPC near $\\MPCmax$, tighter bound constrains. Above cusp: behavior approaches optimist ($\\MPCmin$), optimist bound constrains. The envelope is the minimum of both bounds.\n",
+    "Below cusp: MPC near $\\bar{\\boldsymbol{\\kappa}}$, tighter bound constrains. Above cusp: behavior approaches optimist ($\\underline{\\boldsymbol{\\kappa}}$), optimist bound constrains. The envelope is the minimum of both bounds.\n",
     "```\n",
     "\n",
     "## Further Extensions: Stochastic Rate of Return\n",
@@ -1070,7 +1070,7 @@
    "metadata": {},
    "source": [
     "```{hint} Stochastic Returns Interpretation\n",
-    "Deterministic optimist uses $\\MPCmin = 1 - (\\DiscFac \\Rfree)^{1/\\CRRA}$; stochastic optimist uses $\\MPCmin = 1 - (\\DiscFac \\Ex[\\Risky^{1-\\CRRA}])^{1/\\CRRA}$. Return uncertainty raises MPC and narrows the feasible region. See {ref}`stochastic-returns-mgf-derivation` for the MGF derivation.\n",
+    "Deterministic optimist uses $\\underline{\\boldsymbol{\\kappa}} = 1 - (\\beta \\text{R})^{1/\\rho}$; stochastic optimist uses $\\underline{\\boldsymbol{\\kappa}} = 1 - (\\beta \\mathbf{\\mathbb{E}}[\\mathbf{R}^{1-\\rho}])^{1/\\rho}$. Return uncertainty raises MPC and narrows the feasible region. See {ref}`stochastic-returns-mgf-derivation` for the MGF derivation.\n",
     "```\n",
     "\n",
     "## Summary\n",