From 20a3b6c7a8e7468b784b9de1535ac528bc132201 Mon Sep 17 00:00:00 2001 From: Longye Tian <133612246+longye-tian@users.noreply.github.com> Date: Wed, 7 Aug 2024 19:33:16 +1000 Subject: [PATCH] [laffer_adaptive] [lake_model] spelling and check example admonition (#545) * [laffer_adaptive] update style and spelling - change 'it reverse' to 'it reverses' - change the subtitle to lower case to match with the style - change 'pseudo code' to 'pseudo-code' to match with the title for consistency - change 'limiting values exists' to 'limiting values exist' - add hyphen to steady state when using is as adjective for consistency. * [lake_model] update spelling - change 'the below graph' to 'the graph below' for better word ordering - change 'long run growth' to 'long-run growth' for consistency --- lectures/laffer_adaptive.md | 28 ++++++++++++++-------------- lectures/lake_model.md | 4 ++-- 2 files changed, 16 insertions(+), 16 deletions(-) diff --git a/lectures/laffer_adaptive.md b/lectures/laffer_adaptive.md index fd7b7f37..684f2e6f 100644 --- a/lectures/laffer_adaptive.md +++ b/lectures/laffer_adaptive.md @@ -33,7 +33,7 @@ that we adopted in lectures {doc}`money_inflation` and lectures {doc}`money_infl We shall discover that changing our hypothesis about expectations formation in this way will change some our findings and leave others intact. In particular, we shall discover that * replacing rational expectations with adaptive expectations leaves the two stationary inflation rates unchanged, but that $\ldots$ -* it reverse the perverse dynamics by making the **lower** stationary inflation rate the one to which the system typically converges +* it reverses the perverse dynamics by making the **lower** stationary inflation rate the one to which the system typically converges * a more plausible comparative dynamic outcome emerges in which now inflation can be **reduced** by running **lower** government deficits These more plausible comparative dynamics underlie the "old time religion" that states that @@ -50,7 +50,7 @@ by dropping rational expectations and instead assuming that people form expecta {cite}`marcet2003recurrent` and {cite}`sargent2009conquest` extended that work and applied it to study recurrent high-inflation episodes in Latin America. ``` -## The Model +## The model Let @@ -88,9 +88,9 @@ $$ (eq:adaptex) where $\delta \in (0,1)$ -## Computing An Equilibrium Sequence +## Computing an equilibrium sequence -Equation the expressions for $m_{t+1}$ promided by {eq}`eq:ada_mdemand` and {eq}`eq:ada_msupply2` and use equation {eq}`eq:adaptex` to eliminate $\pi_t^*$ to obtain +Equation the expressions for $m_{t+1}$ provided by {eq}`eq:ada_mdemand` and {eq}`eq:ada_msupply2` and use equation {eq}`eq:adaptex` to eliminate $\pi_t^*$ to obtain the following equation for $p_t$: $$ @@ -99,7 +99,7 @@ $$ (eq:pequation) **Pseudo-code** -Here is pseudo code for our algorithm. +Here is the pseudo-code for our algorithm. Starting at time $0$ with initial conditions $(m_0, \pi_{-1}^*, p_{-1})$, for each $t \geq 0$ deploy the following steps in order: @@ -111,14 +111,14 @@ deploy the following steps in order: This completes the algorithm. -## Claims or Conjectures +## Claims or conjectures It will turn out that * if they exist, limiting values $\overline \pi$ and $\overline \mu$ will be equal -* if limiting values exists, there are two possible limiting values, one high, one low +* if limiting values exist, there are two possible limiting values, one high, one low * unlike the outcome in lecture {doc}`money_inflation_nonlinear`, for almost all initial log price levels and expected inflation rates $p_0, \pi_{t}^*$, the limiting $\overline \pi = \overline \mu$ is the **lower** steady state value @@ -128,7 +128,7 @@ It will turn out that * the preceding equation for $p_0$ comes from $m_1 - p_0 = - \alpha \bar \pi$ -## Limiting Values of Inflation Rate +## Limiting values of inflation rate As in our earlier lecture {doc}`money_inflation_nonlinear`, we can compute the two prospective limiting values for $\bar \pi$ by studying the steady-state Laffer curve. @@ -213,15 +213,15 @@ print(f'The two steady state of π are: {π_l, π_u}') We find two steady state $\bar \pi$ values -## Steady State Laffer Curve +## Steady-state Laffer curve -The following figure plots the steady state Laffer curve together with the two stationary inflation rates. +The following figure plots the steady-state Laffer curve together with the two stationary inflation rates. ```{code-cell} ipython3 --- mystnb: figure: - caption: Seigniorage as function of steady state inflation. The dashed brown lines + caption: Seigniorage as function of steady-state inflation. The dashed brown lines indicate $\pi_l$ and $\pi_u$. name: laffer_curve_adaptive width: 500px @@ -258,11 +258,11 @@ def plot_laffer(model, πs): plot_laffer(model, (π_l, π_u)) ``` -## Associated Initial Price Levels +## Associated initial price levels Now that we have our hands on the two possible steady states, we can compute two initial log price levels $p_{-1}$, which as initial conditions, imply that $\pi_t = \bar \pi $ for all $t \geq 0$. -In particular, to initiate a fixed point of the dynamic Laffer curve dynamics we set +In particular, to initiate a fixed point of the dynamic Laffer curve dynamics, we set $$ p_{-1} = m_0 + \alpha \pi^* @@ -348,7 +348,7 @@ eq_g = lambda x: np.exp(-model.α * x) - np.exp(-(1 + model.α) * x) print('eq_g == g:', np.isclose(eq_g(m_seq[-1] - m_seq[-2]), model.g)) ``` -## Slippery Side of Laffer Curve Dynamics +## Slippery side of Laffer curve dynamics We are now equipped to compute time series starting from different $p_{-1}, \pi_{-1}^*$ settings, analogous to those in this lecture {doc}`money_inflation` and this lecture {doc}`money_inflation_nonlinear`. diff --git a/lectures/lake_model.md b/lectures/lake_model.md index ba11f07f..f70da94f 100644 --- a/lectures/lake_model.md +++ b/lectures/lake_model.md @@ -36,7 +36,7 @@ The "flows" between the two lakes are as follows: 3. employed workers separate from their jobs at rate $\alpha$. 4. unemployed workers find jobs at rate $\lambda$. -The below graph illustrates the lake model. +The graph below illustrates the lake model. ```{figure} /_static/lecture_specific/lake_model/lake_model_worker.png :name: lake_model_graphviz @@ -216,7 +216,7 @@ Moreover, the times series of unemployment and employment seems to grow at some Since by intuition if we consider unemployment pool and employment pool as a closed system, the growth should be similar to the labor force. -We next ask whether the long run growth rates of $e_t$ and $u_t$ +We next ask whether the long-run growth rates of $e_t$ and $u_t$ also dominated by $1+b-d$ as labor force. The answer will be clearer if we appeal to {ref}`Perron-Frobenius theorem`.