diff --git a/lectures/markov_chains_II.md b/lectures/markov_chains_II.md index 59b8e23c..fe2ad9b3 100644 --- a/lectures/markov_chains_II.md +++ b/lectures/markov_chains_II.md @@ -71,6 +71,8 @@ that The stochastic matrix $P$ is called **irreducible** if all states communicate; that is, if $x$ and $y$ communicate for all $(x, y)$ in $S \times S$. +```{prf:example} +:label: mc2_ex_ir For example, consider the following transition probabilities for wealth of a fictitious set of households @@ -95,6 +97,7 @@ $$ It's clear from the graph that this stochastic matrix is irreducible: we can eventually reach any state from any other state. +``` We can also test this using [QuantEcon.py](http://quantecon.org/quantecon-py)'s MarkovChain class @@ -107,6 +110,9 @@ mc = qe.MarkovChain(P, ('poor', 'middle', 'rich')) mc.is_irreducible ``` +```{prf:example} +:label: mc2_ex_pf + Here's a more pessimistic scenario in which poor people remain poor forever ```{image} /_static/lecture_specific/markov_chains_II/Irre_2.png @@ -116,6 +122,7 @@ Here's a more pessimistic scenario in which poor people remain poor forever This stochastic matrix is not irreducible since, for example, rich is not accessible from poor. +``` Let's confirm this @@ -272,6 +279,9 @@ In any of these cases, ergodicity will hold. ### Example: a periodic chain +```{prf:example} +:label: mc2_ex_pc + Let's look at the following example with states 0 and 1: $$ @@ -291,7 +301,7 @@ The transition graph shows that this model is irreducible. ``` Notice that there is a periodic cycle --- the state cycles between the two states in a regular way. - +``` Not surprisingly, this property is called [periodicity](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/16%3A_Markov_Processes/16.05%3A_Periodicity_of_Discrete-Time_Chains). @@ -392,7 +402,7 @@ plt.show() ````{exercise} :label: mc_ex1 -Benhabib el al. {cite}`benhabib_wealth_2019` estimated that the transition matrix for social mobility as the following +Benhabib et al. {cite}`benhabib_wealth_2019` estimated that the transition matrix for social mobility as the following $$ P:= diff --git a/lectures/mle.md b/lectures/mle.md index ee00c399..8a15d6ac 100644 --- a/lectures/mle.md +++ b/lectures/mle.md @@ -39,6 +39,8 @@ $$ where $w$ is wealth. +```{prf:example} +:label: mle_ex_wt For example, if $a = 0.05$, $b = 0.1$, and $\bar w = 2.5$, this means @@ -46,7 +48,7 @@ For example, if $a = 0.05$, $b = 0.1$, and $\bar w = 2.5$, this means * a 10% tax on wealth in excess of 2.5. The unit is 100,000, so $w= 2.5$ means 250,000 dollars. - +``` Let's go ahead and define $h$: ```{code-cell} ipython3 @@ -242,7 +244,7 @@ num = (ln_sample - μ_hat)**2 σ_hat ``` -Let's plot the log-normal pdf using the estimated parameters against our sample data. +Let's plot the lognormal pdf using the estimated parameters against our sample data. ```{code-cell} ipython3 dist_lognorm = lognorm(σ_hat, scale = exp(μ_hat)) diff --git a/lectures/money_inflation.md b/lectures/money_inflation.md index 17ec7e0f..216a9a63 100644 --- a/lectures/money_inflation.md +++ b/lectures/money_inflation.md @@ -35,7 +35,7 @@ Our model equates the demand for money to the supply at each time $t \geq 0$. Equality between those demands and supply gives a *dynamic* model in which money supply and price level *sequences* are simultaneously determined by a set of simultaneous linear equations. -These equations take the form of what are often called vector linear **difference equations**. +These equations take the form of what is often called vector linear **difference equations**. In this lecture, we'll roll up our sleeves and solve those equations in two different ways. @@ -49,19 +49,19 @@ In this lecture we will encounter these concepts from macroeconomics: * perverse dynamics under rational expectations in which the system converges to the higher stationary inflation tax rate * a peculiar comparative stationary-state outcome connected with that stationary inflation rate: it asserts that inflation can be *reduced* by running *higher* government deficits, i.e., by raising more resources by printing money. -The same qualitive outcomes prevail in this lecture {doc}`money_inflation_nonlinear` that studies a nonlinear version of the model in this lecture. +The same qualitative outcomes prevail in this lecture {doc}`money_inflation_nonlinear` that studies a nonlinear version of the model in this lecture. These outcomes set the stage for the analysis to be presented in this lecture {doc}`laffer_adaptive` that studies a nonlinear version of the present model; it assumes a version of "adaptive expectations" instead of rational expectations. That lecture will show that * replacing rational expectations with adaptive expectations leaves the two stationary inflation rates unchanged, but that $\ldots$ -* it reverse the pervese 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 -This outcome will be used to justify a selection of a stationary inflation rate that underlies the analysis of unpleasant monetarist arithmetic to be studies in this lecture {doc}`unpleasant`. +This outcome will be used to justify a selection of a stationary inflation rate that underlies the analysis of unpleasant monetarist arithmetic to be studied in this lecture {doc}`unpleasant`. -We'll use theses tools from linear algebra: +We'll use these tools from linear algebra: * matrix multiplication * matrix inversion @@ -349,7 +349,7 @@ g2 = seign(msm.R_l, msm) print(f'R_l, g_l = {msm.R_l:.4f}, {g2:.4f}') ``` -Now let's compute the maximum steady-state amount of seigniorage that could be gathered by printing money and the state state rate of return on money that attains it. +Now let's compute the maximum steady-state amount of seigniorage that could be gathered by printing money and the state-state rate of return on money that attains it. ## Two computation strategies @@ -950,7 +950,7 @@ Those dynamics are "perverse" not only in the sense that they imply that the mon ```{note} -The same qualitive outcomes prevail in this lecture {doc}`money_inflation_nonlinear` that studies a nonlinear version of the model in this lecture. +The same qualitative outcomes prevail in this lecture {doc}`money_inflation_nonlinear` that studies a nonlinear version of the model in this lecture. ``` diff --git a/lectures/money_inflation_nonlinear.md b/lectures/money_inflation_nonlinear.md index 7bd8306a..e120bdf1 100644 --- a/lectures/money_inflation_nonlinear.md +++ b/lectures/money_inflation_nonlinear.md @@ -35,14 +35,14 @@ As in that lecture, we discussed these topics: * an **inflation tax** that a government gathers by printing paper or electronic money * a dynamic **Laffer curve** in the inflation tax rate that has two stationary equilibria * perverse dynamics under rational expectations in which the system converges to the higher stationary inflation tax rate -* a peculiar comparative stationary-state analysis connected with that stationary inflation rate that assert that inflation can be *reduced* by running *higher* government deficits +* a peculiar comparative stationary-state analysis connected with that stationary inflation rate that asserts that inflation can be *reduced* by running *higher* government deficits These outcomes will set the stage for the analysis of {doc}`laffer_adaptive` that studies a version of the present model that uses a version of "adaptive expectations" instead of rational expectations. That lecture will show that * replacing rational expectations with adaptive expectations leaves the two stationary inflation rates unchanged, but that $\ldots$ -* it reverse the pervese 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 ## The model @@ -399,7 +399,7 @@ Those dynamics are "perverse" not only in the sense that they imply that the mon * the figure indicates that inflation can be *reduced* by running *higher* government deficits, i.e., by raising more resources through printing money. ```{note} -The same qualitive outcomes prevail in {doc}`money_inflation` that studies a linear version of the model in this lecture. +The same qualitative outcomes prevail in {doc}`money_inflation` that studies a linear version of the model in this lecture. ``` We discovered that