From 35b823dbf3761d34bd4f65ad63e9c88dd324fd51 Mon Sep 17 00:00:00 2001 From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com> Date: Wed, 6 Aug 2025 02:35:58 +0000 Subject: [PATCH 1/3] Initial plan From 36f3fdeb76330d7bd76528a61b1e191a9c208e74 Mon Sep 17 00:00:00 2001 From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com> Date: Wed, 6 Aug 2025 02:52:21 +0000 Subject: [PATCH 2/3] Fix grammar and spelling issues in amss.md lecture Co-authored-by: mmcky <8263752+mmcky@users.noreply.github.com> --- lectures/amss.md | 46 +++++++++++++++++++++++----------------------- 1 file changed, 23 insertions(+), 23 deletions(-) diff --git a/lectures/amss.md b/lectures/amss.md index 0bb65d74..8ed72e24 100644 --- a/lectures/amss.md +++ b/lectures/amss.md @@ -32,7 +32,7 @@ tags: [hide-output] ## Overview -Let's start with following imports: +Let's start with the following imports: ```{code-cell} ipython import numpy as np @@ -46,7 +46,7 @@ from numba.experimental import jitclass In {doc}`an earlier lecture `, we described a model of optimal taxation with state-contingent debt due to -Robert E. Lucas, Jr., and Nancy Stokey {cite}`LucasStokey1983`. +Robert E. Lucas, Jr., and Nancy Stokey {cite}`LucasStokey1983`. Aiyagari, Marcet, Sargent, and Seppälä {cite}`aiyagari2002optimal` (hereafter, AMSS) studied optimal taxation in a model without state-contingent debt. @@ -65,7 +65,7 @@ We begin with an introduction to the model. Many but not all features of the economy are identical to those of {doc}`the Lucas-Stokey economy `. -Let's start with things that are identical. +We start with things that are identical. For $t \geq 0$, a history of the state is represented by $s^t = [s_t, s_{t-1}, \ldots, s_0]$. @@ -74,7 +74,7 @@ Government purchases $g(s)$ are an exact time-invariant function of $s$. Let $c_t(s^t)$, $\ell_t(s^t)$, and $n_t(s^t)$ denote consumption, leisure, and labor supply, respectively, at history $s^t$ at time $t$. -Each period a representative household is endowed with one unit of time that can be divided between leisure +Each period, a representative household is endowed with one unit of time that can be divided between leisure $\ell_t$ and labor $n_t$: ```{math} @@ -93,9 +93,9 @@ c_t(s^t) + g(s_t) = n_t(s^t) Output is not storable. -The technology pins down a pre-tax wage rate to unity for all $t, s^t$. +The technology pins down the pre-tax wage rate to unity for all $t, s^t$. -A representative household’s preferences over $\{c_t(s^t), \ell_t(s^t)\}_{t=0}^\infty$ are ordered by +The representative household's preferences over $\{c_t(s^t), \ell_t(s^t)\}_{t=0}^\infty$ are ordered by ```{math} :label: TS_prefr_amss @@ -108,13 +108,13 @@ where * $\pi_t(s^t)$ is a joint probability distribution over the sequence $s^t$, and * the utility function $u$ is increasing, strictly concave, and three times continuously differentiable in both arguments. -The government imposes a flat rate tax $\tau_t(s^t)$ on labor income at time $t$, history $s^t$. +The government imposes a flat-rate tax $\tau_t(s^t)$ on labor income at time $t$, history $s^t$. Lucas and Stokey assumed that there are complete markets in one-period Arrow securities; also see {doc}`smoothing models `. It is at this point that AMSS {cite}`aiyagari2002optimal` modify the Lucas and Stokey economy. -AMSS allow the government to issue only one-period risk-free debt each period. +AMSS allows the government to issue only one-period risk-free debt each period. Ruling out complete markets in this way is a step in the direction of making total tax collections behave more like that prescribed in Robert Barro (1979) {cite}`Barro1979` than they do in Lucas and Stokey (1983) {cite}`LucasStokey1983`. @@ -122,8 +122,8 @@ Ruling out complete markets in this way is a step in the direction of making tot In period $t$ and history $s^t$, let -* $b_{t+1}(s^t)$ be the amount of the time $t+1$ consumption good that at time $t$, history $s^t$ the government promised to pay -* $R_t(s^t)$ be the gross interest rate on risk-free one-period debt between periods $t$ and $t+1$ +* $b_{t+1}(s^t)$ be the amount of the time $t+1$ consumption good that at time $t$ and history $s^t$ the government promised to pay +* $R_t(s^t)$ be the gross interest rate on risk-free one-period debt between periods $t$ and $t+1$ * $T_t(s^t)$ be a non-negative lump-sum *transfer* to the representative household [^fn_a] That $b_{t+1}(s^t)$ is the same for all realizations of $s_{t+1}$ captures its *risk-free* character. @@ -149,7 +149,7 @@ where $z_t(s^t)$ is the net-of-interest government surplus. To rule out Ponzi schemes, we assume that the government is subject to a **natural debt limit** (to be discussed in a forthcoming lecture). The consumption Euler equation for a representative household able to trade only one-period risk-free debt -with one-period gross interest rate $R_t(s^t)$ is +with a one-period gross interest rate $R_t(s^t)$ is $$ {1 \over R_t(s^t)} @@ -168,7 +168,7 @@ b_t(s^{t-1}) = z_t(s^t) + \beta \sum_{s^{t+1}\vert s^t} \pi_{t+1}(s^{t+1} | s ``` Components of $z_t(s^t)$ on the right side depend on $s^t$, but the left side is required to depend only -on $s^{t-1}$ . +on $s^{t-1}$. **This is what it means for one-period government debt to be risk-free**. @@ -203,7 +203,7 @@ b_t(s^{t-1}) Notice how the conditioning sets in equation {eq}`TS_gov_wo3` differ: they are $s^{t-1}$ on the left side and $s^t$ on the right side. -Now let's +Now we * substitute the resource constraint into the net-of-interest government surplus, and * use the household’s first-order condition $1-\tau^n_t(s^t)= u_{\ell}(s^t) /u_c(s^t)$ to eliminate the labor tax rate @@ -222,7 +222,7 @@ If we substitute appropriate versions of the right side of {eq}`AMSS_44_2` for we obtain a sequence of *implementability constraints* on a Ramsey allocation in an AMSS economy. Expression {eq}`TS_gov_wo3` at time $t=0$ and initial state $s^0$ -was also an *implementability constraint* on a Ramsey allocation in a Lucas-Stokey economy: +is also an *implementability constraint* on a Ramsey allocation in a Lucas-Stokey economy: ```{math} :label: TS_gov_wo4 @@ -322,7 +322,7 @@ It is helpful to apply two transformations to the Lagrangian. Multiply constraint {eq}`AMSS_44` by $u_c(s^0)$ and the constraints {eq}`AMSS_46` by $\beta^t u_c(s^{t})$. -Then a Lagrangian for the Ramsey problem can be represented as +Then a Lagrangian for the Ramsey problem can be represented as ```{math} :label: AMSS_lagr;a @@ -435,7 +435,7 @@ $$ where $R_t(s^t)$ is the gross risk-free rate of interest between $t$ and $t+1$ at history $s^t$ and $T_t(s^t)$ are non-negative transfers. -Throughout this lecture, we shall set transfers to zero (for some issues about the limiting behavior of debt, this is possibly an important difference from AMSS {cite}`aiyagari2002optimal`, who restricted transfers +Throughout this lecture, we shall set transfers to zero (for some issues about the limiting behavior of debt, this is possibly an important difference from AMSS {cite}`aiyagari2002optimal`, who restricted transfers to be non-negative). In this case, the household faces a sequence of budget constraints @@ -502,7 +502,7 @@ The right side of equation {eq}`eqn:AMSSapp2b` expresses the time $t$ value of g in terms of a linear combination of terms whose individual components are measurable with respect to $s^t$. -The sum of terms on the right side of equation {eq}`eqn:AMSSapp2b` must equal +The sum of terms on the right side of equation {eq}`eqn:AMSSapp2b` must equal $b_t(s^{t-1})$. That implies that it has to be *measurable* with respect to $s^{t-1}$. @@ -512,7 +512,7 @@ constraint imposed in the Lucas and Stokey model. ### Two Bellman Equations -Let $\Pi(s|s_-)$ be a Markov transition matrix whose entries tell probabilities of moving from state $s_-$ to state $s$ in one period. +Let $\Pi(s|s_-)$ be a Markov transition matrix whose entries give probabilities of moving from state $s_-$ to state $s$ in one period. Let @@ -639,12 +639,12 @@ means that there is no state-variable degeneracy in the AMSS model. In the AMSS model, both $x$ and $s$ are needed to describe the state. -This property of the AMSS model transmits a twisted martingale +This property of the AMSS model transmits a twisted martingale component to consumption, employment, and the tax rate. ### Digression on Non-negative Transfers -Throughout this lecture, we have imposed that transfers $T_t = 0$. +Throughout this lecture, we have set transfers to zero, i.e., $T_t = 0$. AMSS {cite}`aiyagari2002optimal` instead imposed a nonnegativity constraint $T_t\geq 0$ on transfers. @@ -846,7 +846,7 @@ If it is able to trade state-contingent debt, then at time $t=2$ * the government **sells** an Arrow security that pays off when $g_3 = g_l$ * the Ramsey planner designs these purchases and sales designed so that, regardless of whether or not there is a war at $t=3$, the government begins period $t=4$ with the *same* government debt -This pattern facilities smoothing tax rates across states. +This pattern facilitates smoothing tax rates across states. The government without state-contingent debt cannot do this. @@ -973,9 +973,9 @@ plt.show() When the government experiences a prolonged period of peace, it is able to reduce government debt and set persistently lower tax rates. -However, the government finances a long war by borrowing and raising taxes. +However, the government finances a long war by borrowing and raising taxes. -This results in a drift away from policies with state-contingent debt that +This results in a drift away from policies with state-contingent debt that depends on the history of shocks. This is even more evident in the following figure that plots the evolution of From 51665fb15c9641833080eac4a974046f8f9b2235 Mon Sep 17 00:00:00 2001 From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com> Date: Wed, 6 Aug 2025 02:58:03 +0000 Subject: [PATCH 3/3] Final grammar and spacing fixes for amss.md Co-authored-by: mmcky <8263752+mmcky@users.noreply.github.com> --- lectures/amss.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/lectures/amss.md b/lectures/amss.md index 8ed72e24..1061aadb 100644 --- a/lectures/amss.md +++ b/lectures/amss.md @@ -48,7 +48,7 @@ In {doc}`an earlier lecture `, we described a model of optimal taxation with state-contingent debt due to Robert E. Lucas, Jr., and Nancy Stokey {cite}`LucasStokey1983`. -Aiyagari, Marcet, Sargent, and Seppälä {cite}`aiyagari2002optimal` (hereafter, AMSS) +Aiyagari, Marcet, Sargent, and Seppälä {cite}`aiyagari2002optimal` (hereafter, AMSS) studied optimal taxation in a model without state-contingent debt. In this lecture, we @@ -71,7 +71,7 @@ For $t \geq 0$, a history of the state is represented by $s^t = [s_t, s_{t-1}, \ Government purchases $g(s)$ are an exact time-invariant function of $s$. -Let $c_t(s^t)$, $\ell_t(s^t)$, and $n_t(s^t)$ denote consumption, +Let $c_t(s^t)$, $\ell_t(s^t)$, and $n_t(s^t)$ denote consumption, leisure, and labor supply, respectively, at history $s^t$ at time $t$. Each period, a representative household is endowed with one unit of time that can be divided between leisure @@ -106,7 +106,7 @@ The representative household's preferences over $\{c_t(s^t), \ell_t(s^t)\}_{t=0} where * $\pi_t(s^t)$ is a joint probability distribution over the sequence $s^t$, and -* the utility function $u$ is increasing, strictly concave, and three times continuously differentiable in both arguments. +* the utility function $u$ is increasing, strictly concave, and three times continuously differentiable in both arguments. The government imposes a flat-rate tax $\tau_t(s^t)$ on labor income at time $t$, history $s^t$. @@ -138,9 +138,9 @@ The government’s budget constraint in period $t$ at history $s^t$ is \begin{aligned} b_t(s^{t-1}) & = \tau^n_t(s^t) n_t(s^t) - g(s_t) - T_t(s^t) + - {b_{t+1}(s^t) \over R_t(s^t )} + {b_{t+1}(s^t) \over R_t(s^t)} \\ - & \equiv z_t(s^t) + {b_{t+1}(s^t) \over R_t(s^t )}, + & \equiv z_t(s^t) + {b_{t+1}(s^t) \over R_t(s^t)}, \end{aligned} ``` @@ -380,7 +380,7 @@ and with respect to $b_t(s^t)$ as ``` If we substitute $z_t(s^t)$ from {eq}`AMSS_44_2` and its derivative -$z_c(s^t)$ into the first-order condition {eq}`AMSS_foc;a`, we find two +$z_c(s^t)$ into the first-order condition {eq}`AMSS_foc;a`, we find two differences from the corresponding condition for the optimal allocation in a Lucas-Stokey economy with state-contingent government debt. @@ -663,7 +663,7 @@ random, $V_x(x, s)$ almost surely converges to zero. For quasi-linear preferences, the first-order condition for maximizing {eq}`eqn:AMSSapp5` subject to {eq}`eqn:AMSSapp6` with respect to $n(s)$ becomes $$ -(1-\mu(s|s_-) ) (1 - u_l(s)) + \mu(s|s_-) n(s) u_{ll}(s) =0 +(1-\mu(s|s_-)) (1 - u_l(s)) + \mu(s|s_-) n(s) u_{ll}(s) =0 $$ When $\mu(s|s_-) = \beta V_x(x(s),x)$ converges to zero, in the limit @@ -702,7 +702,7 @@ For convenience in matching our computer code, we have expressed utility as a function of $n$ rather than leisure $l$. ``` -We first consider a government expenditure process that we studied earlier in a lecture on +We first consider a government expenditure process that we studied earlier in a lecture on {doc}`optimal taxation with state-contingent debt `. Government expenditures are known for sure in all periods except one.