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Update LaTeX notation and clarify economic expressions
Revised LaTeX math expressions for the consumption function, MPC, and value functions to use consistent variable names and notation (e.g., \cFunc(\mNrm), \vFunc(\mNrm)). Clarified economic interpretations and updated the description of deterministic and stochastic optimist MPC formulas for accuracy. Also updated Python version metadata in the notebook.
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‎code/notebook.ipynb‎

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"source": [
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"## Consumption Function Analysis\n",
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"\n",
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"The first set of figures will focus on the core of the consumption-saving problem: the consumption function, $c(m)$, which maps market resources, $m$, to a chosen level of 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",
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"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",
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"### Figure 1: The EGM Extrapolation Problem\n",
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"\n",
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"### Figure 8: MoM MPC Bounded by Theory\n",
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"\n",
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"The **MPC** ($\\partial c / \\partial m$) 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",
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"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",
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"\n",
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"```{tip} Policy Applications\n",
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"Bounded MPC estimates prevent nonsensical policy multipliers in DSGE models. MoM ensures economically meaningful MPCs for policy analysis.\n",
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"```{hint} MPC Economic Interpretation\n",
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"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",
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"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",
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"```\n",
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"\n",
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"### Figure 9: Value Functions Bounded by Theory\n",
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"\n",
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"The **value function** $v(m)$ is also bounded by optimist and pessimist solutions {cite:p}`Aiyagari1994,Huggett1993`. [](#fig:value-functions) compares truth, EGM, and MoM value functions."
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"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."
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"### Figure 10: Inverse Value Functions $\\vInv(\\mNrm)$\n",
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"The **inverse value function** $\\vInv(\\mNrm) = \\uFunc^{-1}(\\vFunc(\\mNrm))$ gives the consumption equivalent of lifetime utility. It is more linear than $v(m)$ near the borrowing constraint, making it better suited for interpolation. [](#fig:inverse-value-functions) compares the three solutions."
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"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."
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"```{hint} Stochastic Returns Interpretation\n",
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"Deterministic optimist uses $\\kappa = 1 - (\\beta R)^{1/\\rho}$; stochastic optimist uses $\\kappa_{stoch} = 1 - (\\beta \\mathbb{E}[R^{1-\\rho}])^{1/\\rho}$. Return uncertainty raises MPC and narrows the feasible region. See {ref}`stochastic-returns-mgf-derivation` for the MGF derivation.\n",
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"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",
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"```\n",
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"## Summary\n",
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"name": "python",
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"version": "3.11.13"
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"nbformat": 4,

‎code/notebook.md‎

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@@ -124,7 +124,7 @@ mNrmMax = IndShockMoMApproxSol.mNrmMin + IndShockMoMApprox.aXtraGrid.max()
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## Consumption Function Analysis
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The first set of figures will focus on the core of the consumption-saving problem: the consumption function, $c(m)$, which maps market resources, $m$, to a chosen level of 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.
127+
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.
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### Figure 1: The EGM Extrapolation Problem
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### Figure 8: MoM MPC Bounded by Theory
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The **MPC** ($\partial c / \partial m$) 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.
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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.
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```{tip} Policy Applications
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Bounded MPC estimates prevent nonsensical policy multipliers in DSGE models. MoM ensures economically meaningful MPCs for policy analysis.
@@ -344,12 +344,12 @@ plot_mom_mpc(
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```{hint} MPC Economic Interpretation
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MoM MPC declines with $\mNrm$: poor consumers spend windfalls immediately ($\MPC \to \MPCmax$), wealthy consumers save them ($\MPC \to \MPCmin$), reflecting diminishing marginal utility.
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MoM MPC declines with $\mNrm$: poor consumers spend windfalls immediately (MPC $\to \MPCmax$), wealthy consumers save them (MPC $\to \MPCmin$), reflecting diminishing marginal utility.
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```
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### Figure 9: Value Functions Bounded by Theory
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The **value function** $v(m)$ is also bounded by optimist and pessimist solutions {cite:p}`Aiyagari1994,Huggett1993`. [](#fig:value-functions) compares truth, EGM, and MoM value functions.
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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.
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```python
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# | label: fig:value-functions
@@ -374,7 +374,7 @@ Uncertainty matters most at low wealth where buffers are small; the optimist-pes
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### Figure 10: Inverse Value Functions $\vInv(\mNrm)$
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The **inverse value function** $\vInv(\mNrm) = \uFunc^{-1}(\vFunc(\mNrm))$ gives the consumption equivalent of lifetime utility. It is more linear than $v(m)$ near the borrowing constraint, making it better suited for interpolation. [](#fig:inverse-value-functions) compares the three solutions.
377+
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.
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```python
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# | label: fig:inverse-value-functions
@@ -477,7 +477,7 @@ plot_stochastic_bounds(
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```
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```{hint} Stochastic Returns Interpretation
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Deterministic optimist uses $\kappa = 1 - (\beta R)^{1/\rho}$; stochastic optimist uses $\kappa_{stoch} = 1 - (\beta \mathbb{E}[R^{1-\rho}])^{1/\rho}$. Return uncertainty raises MPC and narrows the feasible region. See {ref}`stochastic-returns-mgf-derivation` for the MGF derivation.
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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.
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```
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## Summary

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