diff --git a/lectures/amss.md b/lectures/amss.md index 0bb65d74..7514e2c2 100644 --- a/lectures/amss.md +++ b/lectures/amss.md @@ -63,9 +63,9 @@ We begin with an introduction to the model. ## Competitive Equilibrium with Distorting Taxes -Many but not all features of the economy are identical to those of {doc}`the Lucas-Stokey economy `. +Many features of the economy are identical to those of {doc}`the Lucas-Stokey economy `, though not all. -Let's start with things that are identical. +Let us 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 +In 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} @@ -93,7 +93,7 @@ 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 a pre-tax wage rate equal 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 @@ -110,13 +110,14 @@ where 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 `. +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. -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`. +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`. +This is in contrast to the behavior they exhibit in Lucas and Stokey (1983) {cite}`LucasStokey1983`. ### Risk-free One-Period Debt Only @@ -168,7 +169,7 @@ 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 +204,7 @@ 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 let us * 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 @@ -246,13 +247,15 @@ Equation {eq}`TS_gov_wo4a` must hold for each $s^t$ for each $t \geq 1$. ### Comparison with Lucas-Stokey Economy -The expression on the right side of {eq}`TS_gov_wo4a` in the Lucas-Stokey (1983) economy would equal the present value of a continuation stream of government net-of-interest surpluses evaluated at what would be competitive equilibrium Arrow-Debreu prices at date $t$. +The expression on the right side of {eq}`TS_gov_wo4a` in the Lucas-Stokey (1983) economy would equal the present value of a continuation stream of government net-of-interest surpluses. +This present value would be evaluated at what would be competitive equilibrium Arrow-Debreu prices at date $t$. In the Lucas-Stokey economy, that present value is measurable with respect to $s^t$. -In the AMSS economy, the restriction that government debt be risk-free imposes that that same present value must be measurable with respect to $s^{t-1}$. +In the AMSS economy, the restriction that government debt be risk-free imposes that the same present value must be measurable with respect to $s^{t-1}$. -In a language used in the literature on incomplete markets models, it can be said that the AMSS model requires that at each $(t, s^t)$ what would be the present value of continuation government net-of-interest surpluses in the Lucas-Stokey model must belong to the **marketable subspace** of the AMSS model. +In the language used in the literature on incomplete markets models, the AMSS model requires the following. +At each $(t, s^t)$, what would be the present value of continuation government net-of-interest surpluses in the Lucas-Stokey model must belong to the **marketable subspace** of the AMSS model. ### Ramsey Problem Without State-contingent Debt @@ -312,7 +315,7 @@ A negative multiplier $\gamma_t(s^t)<0$ means that if we could relax constraint {eq}`AMSS_46`, we would like to *increase* the beginning-of-period indebtedness for that particular realization of history $s^t$. -That would let us reduce the beginning-of-period indebtedness for some other history [^fn_b]. +This would allow us to reduce the beginning-of-period indebtedness for some other history [^fn_b]. These features flow from the fact that the government cannot use state-contingent debt and therefore cannot allocate its indebtedness efficiently across future states. @@ -322,7 +325,7 @@ 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 @@ -435,8 +438,8 @@ $$ where $R_t(s^t)$ is the gross risk-free rate of interest between $t$ and $t+1$ at history $s^t$ and $T_t(s^t)$ are non-negative transfers. -Throughout this lecture, we shall set transfers to zero (for some issues about the limiting behavior of debt, this is possibly an important difference from AMSS {cite}`aiyagari2002optimal`, who restricted transfers -to be non-negative). +Throughout this lecture, we shall set transfers to zero. +(For some issues about the limiting behavior of debt, this is possibly an important difference from AMSS {cite}`aiyagari2002optimal`, who restricted transfers to be non-negative). In this case, the household faces a sequence of budget constraints @@ -512,7 +515,7 @@ constraint imposed in the Lucas and Stokey model. ### Two Bellman Equations -Let $\Pi(s|s_-)$ be a Markov transition matrix whose entries tell probabilities of moving from state $s_-$ to state $s$ in one period. +Let $\Pi(s|s_-)$ be a Markov transition matrix whose entries give probabilities of moving from state $s_-$ to state $s$ in one period. Let @@ -544,8 +547,8 @@ for each $s \in {\cal S}$: ``` A continuation Ramsey planner at $t \geq 1$ takes -$(x_{t-1}, s_{t-1}) = (x_-, s_-)$ as given and before -$s$ is realized chooses +$(x_{t-1}, s_{t-1}) = (x_-, s_-)$ as given. +Before $s$ is realized, the planner chooses $(n_t(s_t), x_t(s_t)) = (n(s), x(s))$ for $s \in {\cal S}$. The **Ramsey planner** takes $(b_0, s_0)$ as given and chooses $(n_0, x_0)$. @@ -669,9 +672,8 @@ $$ When $\mu(s|s_-) = \beta V_x(x(s),x)$ converges to zero, in the limit $u_l(s)= 1 =u_c(s)$, so that $\tau(x(s),s) =0$. -Thus, in the limit, if $g_t$ is perpetually random, the government -accumulates sufficient assets to finance all expenditures from earnings on those -assets, returning any excess revenues to the household as non-negative lump-sum transfers. +Thus, in the limit, if $g_t$ is perpetually random, the government accumulates sufficient assets to finance all expenditures from earnings on those assets. +It returns any excess revenues to the household as non-negative lump-sum transfers. ### Code @@ -705,7 +707,7 @@ 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 {doc}`optimal taxation with state-contingent debt `. -Government expenditures are known for sure in all periods except one. +Government expenditures are known with certainty in all periods except one. * For $t<3$ or $t > 3$ we assume that $g_t = g_l = 0.1$. * At $t = 3$ a war occurs with probability 0.5. @@ -844,13 +846,13 @@ If it is able to trade state-contingent debt, then at time $t=2$ * the government **purchases** an Arrow security that pays off when $g_3 = g_h$ * 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 +* the Ramsey planner designs these purchases and sales 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. -Instead, it must enter time $t=3$ with the same level of debt falling due whether there is peace or war at $t=3$. +Instead, the government must enter time $t=3$ with the same level of debt falling due, regardless of whether there is peace or war at $t=3$. The risk-free rate between time $2$ and time $3$ is unusually **low** because at time $2$ consumption at time $3$ is expected to be unusually **low**. @@ -877,7 +879,7 @@ Without state-contingent debt, the optimal tax rate is history dependent. #### Perpetual War Alert History dependence occurs more dramatically in a case in which the government -perpetually faces the prospect of war. +perpetually faces the possibility of war. This case was studied in the final example of the lecture on {doc}`optimal taxation with state-contingent debt `. @@ -973,7 +975,7 @@ 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, when faced with a long war, the government finances it by borrowing and raising taxes. This results in a drift away from policies with state-contingent debt that depends on the history of shocks. @@ -1003,8 +1005,7 @@ titles = ['Consumption', 'Labor Supply', 'Government Debt', fig, axes = plt.subplots(3, 2, figsize=(14, 10)) -for ax, title, ls, amss in zip(axes.flatten(), titles, sim_ls, \ - sim_amss): +for ax, title, ls, amss in zip(axes.flatten(), titles, sim_ls, sim_amss): ax.plot(ls, '-k', amss, '-.b', alpha=0.5) ax.set(title=title) ax.grid()