From e3af47624848338f5dc6b9999cfcdc461b55019c Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:11:49 +1100 Subject: [PATCH 01/28] Fix emphasis vs definitions in linear_algebra.md - Changed all first-use technical terms from italic to bold - Complies with QuantEcon style guide: bold for definitions, italic for emphasis - Related to issue #721 --- lectures/linear_algebra.md | 64 +++++++++++++++++++------------------- 1 file changed, 32 insertions(+), 32 deletions(-) diff --git a/lectures/linear_algebra.md b/lectures/linear_algebra.md index 9e89b2a33..36061bcb2 100644 --- a/lectures/linear_algebra.md +++ b/lectures/linear_algebra.md @@ -85,7 +85,7 @@ from scipy.linalg import inv, solve, det, eig ```{index} single: Linear Algebra; Vectors ``` -A *vector* of length $n$ is just a sequence (or array, or tuple) of $n$ numbers, which we write as $x = (x_1, \ldots, x_n)$ or $x = [x_1, \ldots, x_n]$. +A **vector** of length $n$ is just a sequence (or array, or tuple) of $n$ numbers, which we write as $x = (x_1, \ldots, x_n)$ or $x = [x_1, \ldots, x_n]$. We will write these sequences either horizontally or vertically as we please. @@ -225,15 +225,15 @@ x + y ```{index} single: Vectors; Norm ``` -The *inner product* of vectors $x,y \in \mathbb R ^n$ is defined as +The **inner product** of vectors $x,y \in \mathbb R ^n$ is defined as $$ x' y := \sum_{i=1}^n x_i y_i $$ -Two vectors are called *orthogonal* if their inner product is zero. +Two vectors are called **orthogonal** if their inner product is zero. -The *norm* of a vector $x$ represents its "length" (i.e., its distance from the zero vector) and is defined as +The **norm** of a vector $x$ represents its "length" (i.e., its distance from the zero vector) and is defined as $$ \| x \| := \sqrt{x' x} := \left( \sum_{i=1}^n x_i^2 \right)^{1/2} @@ -273,7 +273,7 @@ np.linalg.norm(x) # Norm of x, take three Given a set of vectors $A := \{a_1, \ldots, a_k\}$ in $\mathbb R ^n$, it's natural to think about the new vectors we can create by performing linear operations. -New vectors created in this manner are called *linear combinations* of $A$. +New vectors created in this manner are called **linear combinations** of $A$. In particular, $y \in \mathbb R ^n$ is a linear combination of $A := \{a_1, \ldots, a_k\}$ if @@ -282,9 +282,9 @@ y = \beta_1 a_1 + \cdots + \beta_k a_k \text{ for some scalars } \beta_1, \ldots, \beta_k $$ -In this context, the values $\beta_1, \ldots, \beta_k$ are called the *coefficients* of the linear combination. +In this context, the values $\beta_1, \ldots, \beta_k$ are called the **coefficients** of the linear combination. -The set of linear combinations of $A$ is called the *span* of $A$. +The set of linear combinations of $A$ is called the **span** of $A$. The next figure shows the span of $A = \{a_1, a_2\}$ in $\mathbb R ^3$. @@ -349,7 +349,7 @@ plt.show() If $A$ contains only one vector $a_1 \in \mathbb R ^2$, then its span is just the scalar multiples of $a_1$, which is the unique line passing through both $a_1$ and the origin. -If $A = \{e_1, e_2, e_3\}$ consists of the *canonical basis vectors* of $\mathbb R ^3$, that is +If $A = \{e_1, e_2, e_3\}$ consists of the **canonical basis vectors** of $\mathbb R ^3$, that is $$ e_1 := @@ -399,8 +399,8 @@ The condition we need for a set of vectors to have a large span is what's called In particular, a collection of vectors $A := \{a_1, \ldots, a_k\}$ in $\mathbb R ^n$ is said to be -* *linearly dependent* if some strict subset of $A$ has the same span as $A$. -* *linearly independent* if it is not linearly dependent. +* **linearly dependent** if some strict subset of $A$ has the same span as $A$. +* **linearly independent** if it is not linearly dependent. Put differently, a set of vectors is linearly independent if no vector is redundant to the span and linearly dependent otherwise. @@ -469,19 +469,19 @@ Often, the numbers in the matrix represent coefficients in a system of linear eq For obvious reasons, the matrix $A$ is also called a vector if either $n = 1$ or $k = 1$. -In the former case, $A$ is called a *row vector*, while in the latter it is called a *column vector*. +In the former case, $A$ is called a **row vector**, while in the latter it is called a **column vector**. -If $n = k$, then $A$ is called *square*. +If $n = k$, then $A$ is called **square**.*. -The matrix formed by replacing $a_{ij}$ by $a_{ji}$ for every $i$ and $j$ is called the *transpose* of $A$ and denoted $A'$ or $A^{\top}$. +The matrix formed by replacing $a_{ij}$ by $a_{ji}$ for every $i$ and $j$ is called the **transpose** of $A$ and denoted $A'$ or $A^{\top}$. -If $A = A'$, then $A$ is called *symmetric*. +If $A = A'$, then $A$ is called **symmetric**.*. -For a square matrix $A$, the $i$ elements of the form $a_{ii}$ for $i=1,\ldots,n$ are called the *principal diagonal*. +For a square matrix $A$, the $i$ elements of the form $a_{ii}$ for $i=1,\ldots,n$ are called the **principal diagonal**. -$A$ is called *diagonal* if the only nonzero entries are on the principal diagonal. +$A$ is called **diagonal** if the only nonzero entries are on the principal diagonal.l. -If, in addition to being diagonal, each element along the principal diagonal is equal to 1, then $A$ is called the *identity matrix* and denoted by $I$. +If, in addition to being diagonal, each element along the principal diagonal is equal to 1, then $A$ is called the **identity matrix** and denoted by $I$. ### Matrix Operations @@ -641,9 +641,9 @@ See [here](https://python-programming.quantecon.org/numpy.html#matrix-multiplica Each $n \times k$ matrix $A$ can be identified with a function $f(x) = Ax$ that maps $x \in \mathbb R ^k$ into $y = Ax \in \mathbb R ^n$. -These kinds of functions have a special property: they are *linear*. +These kinds of functions have a special property: they are **linear**. -A function $f \colon \mathbb R ^k \to \mathbb R ^n$ is called *linear* if, for all $x, y \in \mathbb R ^k$ and all scalars $\alpha, \beta$, we have +A function $f \colon \mathbb R ^k \to \mathbb R ^n$ is called **linear** if, for all $x, y \in \mathbb R ^k$ and all scalars $\alpha, \beta$, we have $$ f(\alpha x + \beta y) = \alpha f(x) + \beta f(y) @@ -773,7 +773,7 @@ In particular, the following are equivalent 1. The columns of $A$ are linearly independent. 1. For any $y \in \mathbb R ^n$, the equation $y = Ax$ has a unique solution. -The property of having linearly independent columns is sometimes expressed as having *full column rank*. +The property of having linearly independent columns is sometimes expressed as having **full column rank**. #### Inverse Matrices @@ -788,7 +788,7 @@ solution is $x = A^{-1} y$. A similar expression is available in the matrix case. In particular, if square matrix $A$ has full column rank, then it possesses a multiplicative -*inverse matrix* $A^{-1}$, with the property that $A A^{-1} = A^{-1} A = I$. +**inverse matrix** $A^{-1}$, with the property that $A A^{-1} = A^{-1} A = I$. As a consequence, if we pre-multiply both sides of $y = Ax$ by $A^{-1}$, we get $x = A^{-1} y$. @@ -800,11 +800,11 @@ This is the solution that we're looking for. ``` Another quick comment about square matrices is that to every such matrix we -assign a unique number called the *determinant* of the matrix --- you can find +assign a unique number called the **determinant** of the matrix --- you can find the expression for it [here](https://en.wikipedia.org/wiki/Determinant). If the determinant of $A$ is not zero, then we say that $A$ is -*nonsingular*. +**nonsingular**. Perhaps the most important fact about determinants is that $A$ is nonsingular if and only if $A$ is of full column rank. @@ -929,8 +929,8 @@ $$ A v = \lambda v $$ -then we say that $\lambda$ is an *eigenvalue* of $A$, and -$v$ is an *eigenvector*. +then we say that $\lambda$ is an **eigenvalue** of $A$, and +$v$ is an **eigenvector**. Thus, an eigenvector of $A$ is a vector such that when the map $f(x) = Ax$ is applied, $v$ is merely scaled. @@ -1034,7 +1034,7 @@ to one. ### Generalized Eigenvalues -It is sometimes useful to consider the *generalized eigenvalue problem*, which, for given +It is sometimes useful to consider the **generalized eigenvalue problem**, which, for given matrices $A$ and $B$, seeks generalized eigenvalues $\lambda$ and eigenvectors $v$ such that @@ -1076,10 +1076,10 @@ $$ $$ The norms on the right-hand side are ordinary vector norms, while the norm on -the left-hand side is a *matrix norm* --- in this case, the so-called -*spectral norm*. +the left-hand side is a **matrix norm** --- in this case, the so-called +**spectral norm**. -For example, for a square matrix $S$, the condition $\| S \| < 1$ means that $S$ is *contractive*, in the sense that it pulls all vectors towards the origin [^cfn]. +For example, for a square matrix $S$, the condition $\| S \| < 1$ means that $S$ is **contractive**, in the sense that it pulls all vectors towards the origin [^cfn]. (la_neumann)= #### {index}`Neumann's Theorem ` @@ -1112,7 +1112,7 @@ $$ \rho(A) = \lim_{k \to \infty} \| A^k \|^{1/k} $$ -Here $\rho(A)$ is the *spectral radius*, defined as $\max_i |\lambda_i|$, where $\{\lambda_i\}_i$ is the set of eigenvalues of $A$. +Here $\rho(A)$ is the **spectral radius**, defined as $\max_i |\lambda_i|$, where $\{\lambda_i\}_i$ is the set of eigenvalues of $A$. As a consequence of Gelfand's formula, if all eigenvalues are strictly less than one in modulus, there exists a $k$ with $\| A^k \| < 1$. @@ -1128,8 +1128,8 @@ Let $A$ be a symmetric $n \times n$ matrix. We say that $A$ is -1. *positive definite* if $x' A x > 0$ for every $x \in \mathbb R ^n \setminus \{0\}$ -1. *positive semi-definite* or *nonnegative definite* if $x' A x \geq 0$ for every $x \in \mathbb R ^n$ +1. **positive definite** if $x' A x > 0$ for every $x \in \mathbb R ^n \setminus \{0\}$ +1. **positive semi-definite** or **nonnegative definite** if $x' A x \geq 0$ for every $x \in \mathbb R ^n$ Analogous definitions exist for negative definite and negative semi-definite matrices. From 1b86b5ba8dc03cd1d1c9ade7a131b54d9131466e Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:13:58 +1100 Subject: [PATCH 02/28] Fix emphasis vs definitions in linear_models.md - Changed technical terms from italic to bold on first definition - Fixed: probability distributions, companion matrix, vector autoregression, deterministic/indeterministic seasonal, linear time trend, moving average, martingale with drift, unconditional mean/variance-covariance matrix, ensemble, cross-sectional average, autocovariance function, stationary distribution, covariance stationary, ergodicity, Markov property, conditional covariance matrix, discrete Lyapunov - Related to issue #721 --- lectures/linear_models.md | 40 +++++++++++++++++++-------------------- 1 file changed, 20 insertions(+), 20 deletions(-) diff --git a/lectures/linear_models.md b/lectures/linear_models.md index 43ae68038..4bfe67dca 100644 --- a/lectures/linear_models.md +++ b/lectures/linear_models.md @@ -112,7 +112,7 @@ The primitives of the model are Given $A, C, G$ and draws of $x_0$ and $w_1, w_2, \ldots$, the model {eq}`st_space_rep` pins down the values of the sequences $\{x_t\}$ and $\{y_t\}$. -Even without these draws, the primitives 1--3 pin down the *probability distributions* of $\{x_t\}$ and $\{y_t\}$. +Even without these draws, the primitives 1--3 pin down the **probability distributions** of $\{x_t\}$ and $\{y_t\}$. Later we'll see how to compute these distributions and their moments. @@ -259,7 +259,7 @@ C = \begin{bmatrix} \end{bmatrix} $$ -The matrix $A$ has the form of the *companion matrix* to the vector +The matrix $A$ has the form of the **companion matrix** to the vector $\begin{bmatrix}\phi_1 & \phi_2 & \phi_3 & \phi_4 \end{bmatrix}$. The next figure shows the dynamics of this process when @@ -301,7 +301,7 @@ Now suppose that * $\phi_j$ is a $k \times k$ matrix and * $w_t$ is $k \times 1$ -Then {eq}`eq_ar_rep` is termed a *vector autoregression*. +Then {eq}`eq_ar_rep` is termed a **vector autoregression**. To map this into {eq}`st_space_rep`, we set @@ -345,8 +345,8 @@ where $I$ is the $k \times k$ identity matrix and $\sigma$ is a $k \times k$ mat We can use {eq}`st_space_rep` to represent -1. the *deterministic seasonal* $y_t = y_{t-4}$ -1. the *indeterministic seasonal* $y_t = \phi_4 y_{t-4} + w_t$ +1. the **deterministic seasonal** $y_t = y_{t-4}$ +1. the **indeterministic seasonal** $y_t = \phi_4 y_{t-4} + w_t$ In fact, both are special cases of {eq}`eq_ar_rep`. @@ -376,7 +376,7 @@ The *indeterministic* seasonal produces recurrent, but aperiodic, seasonal fluct ```{index} single: Linear State Space Models; Time Trends ``` -The model $y_t = a t + b$ is known as a *linear time trend*. +The model $y_t = a t + b$ is known as a **linear time trend**. We can represent this model in the linear state space form by taking @@ -462,7 +462,7 @@ $x_0, w_1, w_2, \ldots, w_t$ can be found by using {eq}`st_space_rep` repeatedl \end{aligned} ``` -Representation {eq}`eqob5` is a *moving average* representation. +Representation {eq}`eqob5` is a **moving average** representation. It expresses $\{x_t\}$ as a linear function of @@ -503,7 +503,7 @@ The first term on the right is a cumulated sum of martingale differences and is The second term is a translated linear function of time. -For this reason, $x_{1t}$ is called a *martingale with drift*. +For this reason, $x_{1t}$ is called a **martingale with drift**. ## Distributions and Moments @@ -548,8 +548,8 @@ As with $\mu_0$, the matrix $\Sigma_0$ is a primitive given in {eq}`st_space_rep As a matter of terminology, we will sometimes call -* $\mu_t$ the *unconditional mean* of $x_t$ -* $\Sigma_t$ the *unconditional variance-covariance matrix* of $x_t$ +* $\mu_t$ the **unconditional mean** of $x_t$ +* $\Sigma_t$ the **unconditional variance-covariance matrix** of $x_t$ This is to distinguish $\mu_t$ and $\Sigma_t$ from related objects that use conditioning information, to be defined below. @@ -763,8 +763,8 @@ In the preceding figure, we approximated the population distribution of $y_T$ by 1. recording each observation $y^i_T$ 1. histogramming this sample -Just as the histogram approximates the population distribution, the *ensemble* or -*cross-sectional average* +Just as the histogram approximates the population distribution, the **ensemble** or +**cross-sectional average** $$ \bar y_T := \frac{1}{I} \sum_{i=1}^I y_T^i @@ -870,7 +870,7 @@ $$ #### Autocovariance Functions -An important object related to the joint distribution is the *autocovariance function* +An important object related to the joint distribution is the **autocovariance function** ```{math} :label: eqnautodeff @@ -958,11 +958,11 @@ the distribution at $T$. Apparently, the distributions of $y_t$ converge to a fixed long-run distribution as $t \to \infty$. -When such a distribution exists it is called a *stationary distribution*. +When such a distribution exists it is called a **stationary distribution**. ### Stationary Distributions -In our setting, a distribution $\psi_{\infty}$ is said to be *stationary* for $x_t$ if +In our setting, a distribution $\psi_{\infty}$ is said to be **stationary** for $x_t$ if $$ x_t \sim \psi_{\infty} @@ -1016,7 +1016,7 @@ Moreover, in view of {eq}`eqnautocov`, the autocovariance function takes the for This motivates the following definition. -A process $\{x_t\}$ is said to be *covariance stationary* if +A process $\{x_t\}$ is said to be **covariance stationary** if * both $\mu_t$ and $\Sigma_t$ are constant in $t$ * $\Sigma_{t+j,t}$ depends on the time gap $j$ but not on time $t$ @@ -1156,7 +1156,7 @@ $$ Do these time series averages converge to something interpretable in terms of our basic state-space representation? -The answer depends on something called *ergodicity*. +The answer depends on something called **ergodicity**. Ergodicity is the property that time series and ensemble averages coincide. @@ -1246,7 +1246,7 @@ $$ The right-hand side follows from $x_{t+1} = A x_t + C w_{t+1}$ and the fact that $w_{t+1}$ is zero mean and independent of $x_t, x_{t-1}, \ldots, x_0$. -That $\mathbb{E}_t [x_{t+1}] = \mathbb{E}[x_{t+1} \mid x_t]$ is an implication of $\{x_t\}$ having the *Markov property*. +That $\mathbb{E}_t [x_{t+1}] = \mathbb{E}[x_{t+1} \mid x_t]$ is an implication of $\{x_t\}$ having the **Markov property**. The one-step-ahead forecast error is @@ -1313,7 +1313,7 @@ $V_j$ defined in {eq}`eqob9a` can be calculated recursively via $V_1 = CC'$ and V_j = CC^\prime + A V_{j-1} A^\prime, \quad j \geq 2 ``` -$V_j$ is the *conditional covariance matrix* of the errors in forecasting +$V_j$ is the **conditional covariance matrix** of the errors in forecasting $x_{t+j}$, conditioned on time $t$ information $x_t$. Under particular conditions, $V_j$ converges to @@ -1324,7 +1324,7 @@ Under particular conditions, $V_j$ converges to V_\infty = CC' + A V_\infty A' ``` -Equation {eq}`eqob10` is an example of a *discrete Lyapunov* equation in the covariance matrix $V_\infty$. +Equation {eq}`eqob10` is an example of a **discrete Lyapunov** equation in the covariance matrix $V_\infty$. A sufficient condition for $V_j$ to converge is that the eigenvalues of $A$ be strictly less than one in modulus. From e4936b3f07efd40109de7b76c7ce410b226bca64 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:14:25 +1100 Subject: [PATCH 03/28] Fix emphasis vs definitions in lln_clt.md - Changed Kolmogorov's strong law from italic to bold (named theorem) - Changed variance-covariance matrix from italic to bold (definition) - Related to issue #721 --- lectures/lln_clt.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/lectures/lln_clt.md b/lectures/lln_clt.md index 7aa6954ae..d2839bede 100644 --- a/lectures/lln_clt.md +++ b/lectures/lln_clt.md @@ -84,7 +84,7 @@ will converge to their population means. The classical law of large numbers concerns independent and identically distributed (IID) random variables. -Here is the strongest version of the classical LLN, known as *Kolmogorov's strong law*. +Here is the strongest version of the classical LLN, known as **Kolmogorov's strong law**. Let $X_1, \ldots, X_n$ be independent and identically distributed scalar random variables, with common distribution $F$. @@ -563,7 +563,7 @@ $$ \right) =: \boldsymbol \mu $$ -The *variance-covariance matrix* of random vector $\mathbf X$ is defined as +The **variance-covariance matrix** of random vector $\mathbf X$ is defined as $$ \mathop{\mathrm{Var}}[\mathbf X] From c2e6432ebfed7d0276129f7d09f18fc25ce3c8ae Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:15:08 +1100 Subject: [PATCH 04/28] Fix emphasis vs definitions in markov_asset.md - Changed Gordon formula from italic to bold (named formula) - Changed Lucas tree model terms (tree, fruit, shares, dividend) to bold - Changed infinite horizon, call option, strike price to bold (definitions) - Note: kept 'exercises' and 'not to exercise' as italic (emphasis of choice) - Related to issue #721 --- lectures/markov_asset.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/lectures/markov_asset.md b/lectures/markov_asset.md index 1739dde0f..3ad757bda 100644 --- a/lectures/markov_asset.md +++ b/lectures/markov_asset.md @@ -264,7 +264,7 @@ $$ p_t = \frac{1 + \kappa}{ \rho - \kappa} d_t $$ -This is called the *Gordon formula*. +This is called the **Gordon formula**. (mass_mg)= ### Example 3: Markov Growth, Risk-Neutral Pricing @@ -451,8 +451,8 @@ Lucas considered an abstract pure exchange economy with these features: * a single non-storable consumption good * a Markov process that governs the total amount of the consumption good available each period -* a single *tree* that each period yields *fruit* that equals the total amount of consumption available to the economy -* a competitive market in *shares* in the tree that entitles their owners to corresponding shares of the *dividend* stream, i.e., the *fruit* stream, yielded by the tree +* a single **tree** that each period yields **fruit** that equals the total amount of consumption available to the economy +* a competitive market in **shares** in the tree that entitles their owners to corresponding shares of the **dividend** stream, i.e., the **fruit** stream, yielded by the tree * a representative consumer who in a competitive equilibrium @@ -473,7 +473,7 @@ where $u$ is a concave utility function and $c_t$ is time $t$ consumption of a r Assume the existence of an endowment that follows growth process {eq}`mass_fmce`. -The asset being priced is a claim on the endowment process, i.e., the *Lucas tree* described above. +The asset being priced is a claim on the endowment process, i.e., the **Lucas tree** described above. Following {cite}`Lucas1978`, we suppose that in equilibrium the representative consumer's consumption equals the aggregate endowment, so that $d_t = c_t$ for all $t$. @@ -748,7 +748,7 @@ We'll study an option that gives the owner the right to purchase a consol at a #### An Infinite Horizon Call Option -We want to price an *infinite horizon* option to purchase a consol at a price $p_S$. +We want to price an **infinite horizon** option to purchase a consol at a price $p_S$. The option entitles the owner at the beginning of a period either @@ -757,7 +757,7 @@ The option entitles the owner at the beginning of a period either Thus, the owner either *exercises* the option now or chooses *not to exercise* and wait until next period. -This is termed an infinite-horizon *call option* with *strike price* $p_S$. +This is termed an infinite-horizon **call option** with **strike price** $p_S$. The owner of the option is entitled to purchase the consol at price $p_S$ at the beginning of any period, after the coupon has been paid to the previous owner of the bond. From a28806b1c0de54504cdfaed5a6b3448d8f8c46d9 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:16:10 +1100 Subject: [PATCH 05/28] Fix emphasis vs definitions in markov_perf.md - Changed 'Markov perfect equilibrium' from italic to bold in formal definition - Related to issue #721 --- lectures/markov_perf.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/lectures/markov_perf.md b/lectures/markov_perf.md index c7b99c7ad..923cd07af 100644 --- a/lectures/markov_perf.md +++ b/lectures/markov_perf.md @@ -140,7 +140,7 @@ v_i(q_i, q_{-i}) = \max_{\hat q_i} \left\{\pi_i (q_i, q_{-i}, \hat q_i) + \beta v_i(\hat q_i, f_{-i}(q_{-i}, q_i)) \right\} ``` -**Definition** A *Markov perfect equilibrium* of the duopoly model is a pair of value functions $(v_1, v_2)$ and a pair of policy functions $(f_1, f_2)$ such that, for each $i \in \{1, 2\}$ and each possible state, +**Definition** A **Markov perfect equilibrium** of the duopoly model is a pair of value functions $(v_1, v_2)$ and a pair of policy functions $(f_1, f_2)$ such that, for each $i \in \{1, 2\}$ and each possible state, * The value function $v_i$ satisfies Bellman equation {eq}`game4`. * The maximizer on the right side of {eq}`game4` equals $f_i(q_i, q_{-i})$. From 123fd9f2bf22325e3a310f69d8d5827004bc8535 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:18:12 +1100 Subject: [PATCH 06/28] Fix emphasis vs definitions in mccall_model.md - Changed 'values' from italic to bold when introducing value functions concept - Note: 'Bellman equation', 'policy', 'reservation wage' already use bold - Related to issue #721 --- lectures/mccall_model.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/lectures/mccall_model.md b/lectures/mccall_model.md index 54f264c3e..f4cb83ceb 100644 --- a/lectures/mccall_model.md +++ b/lectures/mccall_model.md @@ -134,7 +134,7 @@ In order to optimally trade-off current and future rewards, we need to think abo 1. the current payoffs we get from different choices 1. the different states that those choices will lead to in next period -To weigh these two aspects of the decision problem, we need to assign *values* +To weigh these two aspects of the decision problem, we need to assign **values** to states. To this end, let $v^*(w)$ be the total lifetime value accruing to an From fa73e46cc8095f096ee454b6f6cc29bc85eecc01 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:18:49 +1100 Subject: [PATCH 07/28] Fix emphasis vs definitions in mle.md - Changed 'parametric class' from italic to bold (technical concept) - Changed 'Poisson regression' from italic to bold (named model) - Changed 'cumulative normal distribution' from italic to bold (technical term) - Note: 'maximum likelihood estimates' already uses bold - Related to issue #721 --- lectures/mle.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/lectures/mle.md b/lectures/mle.md index 929eed27c..4b3a2f047 100644 --- a/lectures/mle.md +++ b/lectures/mle.md @@ -74,7 +74,7 @@ Let's consider the steps we need to go through in maximum likelihood estimation The first step with maximum likelihood estimation is to choose the probability distribution believed to be generating the data. -More precisely, we need to make an assumption as to which *parametric class* of distributions is generating the data. +More precisely, we need to make an assumption as to which **parametric class** of distributions is generating the data. * e.g., the class of all normal distributions, or the class of all gamma distributions. @@ -183,7 +183,7 @@ In Treisman's paper, the dependent variable --- the number of billionaires $y_i$ Hence, the distribution of $y_i$ needs to be conditioned on the vector of explanatory variables $\mathbf{x}_i$. -The standard formulation --- the so-called *Poisson regression* model --- is as follows: +The standard formulation --- the so-called **Poisson regression** model --- is as follows: ```{math} :label: poissonreg @@ -861,7 +861,7 @@ f(y_i; \boldsymbol{\beta}) = \mu_i^{y_i} (1-\mu_i)^{1-y_i}, \quad y_i = 0,1 \\ \end{aligned} $$ -$\Phi$ represents the *cumulative normal distribution* and +$\Phi$ represents the **cumulative normal distribution** and constrains the predicted $y_i$ to be between 0 and 1 (as required for a probability). From 70115cafb1f9dad8ee91d3ceac8ce786a9e6e72a Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:19:17 +1100 Subject: [PATCH 08/28] Fix emphasis vs definitions in ols.md - Changed 'exogenous' from italic to bold (key econometric term) - Changed 'marginal effect' from italic to bold (technical definition) - Changed 'the sum of squared residuals' from italic to bold (OLS definition) - Note: 'omitted variable bias', 'multivariate regression model', 'endogeneity', 'two-stage least squares', 'instrument' already use bold - Related to issue #721 --- lectures/ols.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/lectures/ols.md b/lectures/ols.md index 22b5a0844..2416bf2b7 100644 --- a/lectures/ols.md +++ b/lectures/ols.md @@ -51,7 +51,7 @@ As an example, we will replicate results from Acemoglu, Johnson and Robinson's s In the paper, the authors emphasize the importance of institutions in economic development. -The main contribution is the use of settler mortality rates as a source of *exogenous* variation in institutional differences. +The main contribution is the use of settler mortality rates as a source of **exogenous** variation in institutional differences. Such variation is needed to determine whether it is institutions that give rise to greater economic growth, rather than the other way around. @@ -125,7 +125,7 @@ where: - $\beta_0$ is the intercept of the linear trend line on the y-axis - $\beta_1$ is the slope of the linear trend line, representing - the *marginal effect* of protection against risk on log GDP per + the **marginal effect** of protection against risk on log GDP per capita - $u_i$ is a random error term (deviations of observations from the linear trend due to factors not included in the model) @@ -169,7 +169,7 @@ The most common technique to estimate the parameters ($\beta$'s) of the linear model is Ordinary Least Squares (OLS). As the name implies, an OLS model is solved by finding the parameters -that minimize *the sum of squared residuals*, i.e. +that minimize **the sum of squared residuals**, i.e. $$ \underset{\hat{\beta}}{\min} \sum^N_{i=1}{\hat{u}^2_i} From dba04451b9531133f487a7c6b14f936d29ddaf6e Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:21:22 +1100 Subject: [PATCH 09/28] Fix emphasis vs definitions in rational_expectations.md Changes per #721: - rational expectations equilibrium (first introduction) - perceived law of motion - actual law of motion - belief function - Euler equation - transversality condition - recursive competitive equilibrium - planning problem All terms changed from italic to bold as they are definitions per style guide. --- lectures/rational_expectations.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/lectures/rational_expectations.md b/lectures/rational_expectations.md index dbc6e628a..2e4b51862 100644 --- a/lectures/rational_expectations.md +++ b/lectures/rational_expectations.md @@ -39,7 +39,7 @@ tags: [hide-output] ## Overview -This lecture introduces the concept of a *rational expectations equilibrium*. +This lecture introduces the concept of a **rational expectations equilibrium**. To illustrate it, we describe a linear quadratic version of a model due to Lucas and Prescott {cite}`LucasPrescott1971`. @@ -53,7 +53,7 @@ Because we use linear quadratic setups for demand and costs, we can deploy the L We will learn about how a representative agent's problem differs from a planner's, and how a planning problem can be used to compute quantities and prices in a rational expectations equilibrium. -We will also learn about how a rational expectations equilibrium can be characterized as a [fixed point](https://en.wikipedia.org/wiki/Fixed_point_%28mathematics%29) of a mapping from a *perceived law of motion* to an *actual law of motion*. +We will also learn about how a rational expectations equilibrium can be characterized as a [fixed point](https://en.wikipedia.org/wiki/Fixed_point_%28mathematics%29) of a mapping from a **perceived law of motion** to an **actual law of motion**. Equality between a perceived and an actual law of motion for endogenous market-wide objects captures in a nutshell what the rational expectations equilibrium concept is all about. @@ -309,7 +309,7 @@ Y_{t+1} = H(Y_t) where $Y_0$ is a known initial condition. -The *belief function* $H$ is an equilibrium object, and hence remains to be determined. +The **belief function** $H$ is an equilibrium object, and hence remains to be determined. #### Optimal Behavior Given Beliefs @@ -364,7 +364,7 @@ $$ v_y(y,Y) = a_0 - a_1 Y + \gamma (y' - y) $$ -Substituting this equation into {eq}`comp5` gives the *Euler equation* +Substituting this equation into {eq}`comp5` gives the **Euler equation** ```{math} :label: ree_comp7 @@ -377,7 +377,7 @@ The firm optimally sets an output path that satisfies {eq}`ree_comp7`, taking { * the initial conditions for $(y_0, Y_0)$. * the terminal condition $\lim_{t \rightarrow \infty } \beta^t y_t v_y(y_{t}, Y_t) = 0$. -This last condition is called the *transversality condition*, and acts as a first-order necessary condition "at infinity". +This last condition is called the **transversality condition**, and acts as a first-order necessary condition "at infinity". A representative firm's decision rule solves the difference equation {eq}`ree_comp7` subject to the given initial condition $y_0$ and the transversality condition. @@ -388,7 +388,7 @@ a decision rule that automatically imposes both the Euler equation {eq}`ree_comp As we've seen, a given belief translates into a particular decision rule $h$. -Recalling that in equilbrium $Y_t = y_t$, the *actual law of motion* for market-wide output is then +Recalling that in equilbrium $Y_t = y_t$, the **actual law of motion** for market-wide output is then ```{math} :label: ree_comp9a @@ -401,7 +401,7 @@ Thus, when firms believe that the law of motion for market-wide output is {eq}`r (ree_def)= ### Definition of Rational Expectations Equilibrium -A *rational expectations equilibrium* or *recursive competitive equilibrium* of the model with adjustment costs is a decision rule $h$ and an aggregate law of motion $H$ such that +A **rational expectations equilibrium** or **recursive competitive equilibrium** of the model with adjustment costs is a decision rule $h$ and an aggregate law of motion $H$ such that 1. Given belief $H$, the map $h$ is the firm's optimal policy function. 1. The law of motion $H$ satisfies $H(Y)= h(Y,Y)$ for all @@ -469,7 +469,7 @@ s(Y_t, Y_{t+1}) The first term is the area under the demand curve, while the second measures the social costs of changing output. -The *planning problem* is to choose a production plan $\{Y_t\}$ to maximize +The **planning problem** is to choose a production plan $\{Y_t\}$ to maximize $$ \sum_{t=0}^\infty \beta^t s(Y_t, Y_{t+1}) From ceb29ae5c5c8751e149840b99972cacc366acf8a Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:21:56 +1100 Subject: [PATCH 10/28] Fix emphasis vs definitions in re_with_feedback.md Changes per #721: - backward shift (operator definition) - lag (operator definition) - forward shift (operator definition) - explosive solution All terms changed from italic to bold as they are definitions per style guide. --- lectures/re_with_feedback.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/lectures/re_with_feedback.md b/lectures/re_with_feedback.md index 48a0aae94..da5a647dc 100644 --- a/lectures/re_with_feedback.md +++ b/lectures/re_with_feedback.md @@ -78,14 +78,14 @@ first-order and second-order linear difference equations. ## Linear Difference Equations -We'll use the *backward shift* or *lag* operator $L$. +We'll use the **backward shift** or **lag** operator $L$. The lag operator $L$ maps a sequence $\{x_t\}_{t=0}^\infty$ into the sequence $\{x_{t-1}\}_{t=0}^\infty$ We'll deploy $L$ by using the equality $L x_t \equiv x_{t-1}$ in algebraic expressions. -Further, the inverse $L^{-1}$ of the lag operator is the *forward shift* +Further, the inverse $L^{-1}$ of the lag operator is the **forward shift** operator. We'll often use the equality $L^{-1} x_t \equiv x_{t+1}$ below. @@ -345,7 +345,7 @@ F = (1-\lambda) G (I - \lambda A)^{-1} ``` ```{note} -As mentioned above, an *explosive solution* of difference +As mentioned above, an **explosive solution** of difference equation {eq}`equation_1` can be constructed by adding to the right hand of {eq}`equation_4` a sequence $c \lambda^{-t}$ where $c$ is an arbitrary positive constant. From 80c6cdbcb1a9745fe9e959fac9a2ec14c77ac5e9 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:23:52 +1100 Subject: [PATCH 11/28] Fix emphasis vs definitions in samuelson.md Changes per #721: - second-order linear difference equation - national output identity - consumption function - accelerator - accelerator coefficient - aggregate demand - aggregate supply - business cycles - stochastic linear difference equation - marginal propensity to consume - steady state - random - stochastic - shocks - disturbances - second-order scalar linear stochastic difference equation - characteristic polynomial - zeros - roots All terms changed from italic to bold as they are definitions per style guide. --- lectures/samuelson.md | 32 ++++++++++++++++---------------- 1 file changed, 16 insertions(+), 16 deletions(-) diff --git a/lectures/samuelson.md b/lectures/samuelson.md index 44aaf3b91..f5f1ad534 100644 --- a/lectures/samuelson.md +++ b/lectures/samuelson.md @@ -66,27 +66,27 @@ from cmath import sqrt ### Samuelson's model -Samuelson used a *second-order linear difference equation* to +Samuelson used a **second-order linear difference equation** to represent a model of national output based on three components: -- a *national output identity* asserting that national output or national income is the +- a **national output identity** asserting that national output or national income is the sum of consumption plus investment plus government purchases. -- a Keynesian *consumption function* asserting that consumption at +- a Keynesian **consumption function** asserting that consumption at time $t$ is equal to a constant times national output at time $t-1$. -- an investment *accelerator* asserting that investment at time - $t$ equals a constant called the *accelerator coefficient* +- an investment **accelerator** asserting that investment at time + $t$ equals a constant called the **accelerator coefficient** times the difference in output between period $t-1$ and $t-2$. Consumption plus investment plus government purchases -constitute *aggregate demand,* which automatically calls forth an -equal amount of *aggregate supply*. +constitute **aggregate demand,** which automatically calls forth an +equal amount of **aggregate supply**. (To read about linear difference equations see [here](https://en.wikipedia.org/wiki/Linear_difference_equation) or chapter IX of {cite}`Sargent1987`.) Samuelson used the model to analyze how particular values of the marginal propensity to consume and the accelerator coefficient might -give rise to transient *business cycles* in national output. +give rise to transient **business cycles** in national output. Possible dynamic properties include @@ -100,7 +100,7 @@ adds a random shock to the right side of the national income identity representing random fluctuations in aggregate demand. This modification makes national output become governed by a second-order -*stochastic linear difference equation* that, with appropriate parameter values, +**stochastic linear difference equation** that, with appropriate parameter values, gives rise to recurrent irregular business cycles. (To read about stochastic linear difference equations see chapter XI of @@ -152,7 +152,7 @@ and the national income identity Y_t = C_t + I_t + G_t ``` -- The parameter $\alpha$ is peoples' *marginal propensity to consume* +- The parameter $\alpha$ is peoples' **marginal propensity to consume** out of income - equation {eq}`consumption` asserts that people consume a fraction of $\alpha \in (0,1)$ of each additional dollar of income. - The parameter $\beta > 0$ is the investment accelerator coefficient - equation @@ -193,7 +193,7 @@ a constant value as $t$ becomes large. We are interested in studying - the transient fluctuations in $Y_t$ as it converges to its - *steady state* level + **steady state** level - the *rate* at which it converges to a steady state level The deterministic version of the model described so far --- meaning that @@ -204,10 +204,10 @@ fluctuations by adding a random shock to aggregate demand. ### Stochastic version of the model -We create a *random* or *stochastic* version of the model by adding -a random process of *shocks* or *disturbances* +We create a **random** or **stochastic** version of the model by adding +a random process of **shocks** or **disturbances** $\{\sigma \epsilon_t \}$ to the right side of equation {eq}`second_order`, -leading to the *second-order scalar linear stochastic difference equation*: +leading to the **second-order scalar linear stochastic difference equation**: ```{math} :label: second_stochastic @@ -235,7 +235,7 @@ Y_{t+2} - \rho_1 Y_{t+1} - \rho_2 Y_t = 0 ``` To discover the properties of the solution of {eq}`second_stochastic2`, -it is useful first to form the *characteristic polynomial* +it is useful first to form the **characteristic polynomial** for {eq}`second_stochastic2`: ```{math} @@ -246,7 +246,7 @@ z^2 - \rho_1 z - \rho_2 where $z$ is possibly a complex number. -We want to find the two *zeros* (a.k.a. *roots*) -- namely +We want to find the two **zeros** (a.k.a. **roots**) -- namely $\lambda_1, \lambda_2$ -- of the characteristic polynomial. These are two special values of $z$, say $z= \lambda_1$ and From 567ea1f47f69bbdffe1e31ec0959d2a011877533 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:24:19 +1100 Subject: [PATCH 12/28] Fix emphasis vs definitions in sir_model.md Changes per #721: - transmission rate - infection rate - recovery rate - effective reproduction number All terms changed from italic to bold as they are definitions per style guide. --- lectures/sir_model.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/lectures/sir_model.md b/lectures/sir_model.md index 24682939e..cbd82c45b 100644 --- a/lectures/sir_model.md +++ b/lectures/sir_model.md @@ -108,9 +108,9 @@ dynamics are In these equations, -* $\beta(t)$ is called the *transmission rate* (the rate at which individuals bump into others and expose them to the virus). -* $\sigma$ is called the *infection rate* (the rate at which those who are exposed become infected) -* $\gamma$ is called the *recovery rate* (the rate at which infected people recover or die). +* $\beta(t)$ is called the **transmission rate** (the rate at which individuals bump into others and expose them to the virus). +* $\sigma$ is called the **infection rate** (the rate at which those who are exposed become infected)d) +* $\gamma$ is called the **recovery rate** (the rate at which infected people recover or die). * the dot symbol $\dot y$ represents the time derivative $dy/dt$. We do not need to model the fraction $r$ of the population in state $R$ separately because the states form a partition. @@ -141,7 +141,7 @@ As in Atkeson's note, we set The transmission rate is modeled as -* $\beta(t) := R(t) \gamma$ where $R(t)$ is the *effective reproduction number* at time $t$. +* $\beta(t) := R(t) \gamma$ where $R(t)$ is the **effective reproduction number** at time $t$. (The notation is slightly confusing, since $R(t)$ is different to $R$, the symbol that represents the removed state.) From 6c176f19d4780eccb139c99646ea3f988b5a33c6 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:25:02 +1100 Subject: [PATCH 13/28] Fix emphasis vs definitions in uncertainty_traps.md Changes per #721: - propagation mechanism Term changed from italic to bold as it is a definition per style guide. --- lectures/uncertainty_traps.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/lectures/uncertainty_traps.md b/lectures/uncertainty_traps.md index 2cef53ace..52aa5b4dd 100644 --- a/lectures/uncertainty_traps.md +++ b/lectures/uncertainty_traps.md @@ -323,7 +323,7 @@ at once, for a given set of shocks Notice how the traps only take hold after a sequence of bad draws for the fundamental. -Thus, the model gives us a *propagation mechanism* that maps bad random draws into long downturns in economic activity. +Thus, the model gives us a **propagation mechanism** that maps bad random draws into long downturns in economic activity. ## Exercises From 94961bb095324124e595e22524d6205801daac56 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:25:39 +1100 Subject: [PATCH 14/28] Fix emphasis vs definitions in von_neumann_model.md Changes per #721: - activities - goods - input matrix - output matrix - intensity - goods used in production - total outputs - productive - cost - revenue - costs - revenues - irreducibility All terms changed from italic to bold as they are definitions per style guide. --- lectures/von_neumann_model.md | 26 +++++++++++++------------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/lectures/von_neumann_model.md b/lectures/von_neumann_model.md index b26016286..7532797fa 100644 --- a/lectures/von_neumann_model.md +++ b/lectures/von_neumann_model.md @@ -364,11 +364,11 @@ respectively. A pair $(A,B)$ of $m\times n$ non-negative matrices defines an economy. -- $m$ is the number of *activities* (or sectors) -- $n$ is the number of *goods* (produced and/or consumed). -- $A$ is called the *input matrix*; $a_{i,j}$ denotes the +- $m$ is the number of **activities** (or sectors) +- $n$ is the number of **goods** (produced and/or consumed).). +- $A$ is called the **input matrix**; $a_{i,j}$ denotes the amount of good $j$ consumed by activity $i$ -- $B$ is called the *output matrix*; $b_{i,j}$ represents +- $B$ is called the **output matrix**; $b_{i,j}$ represents the amount of good $j$ produced by activity $i$ Two key assumptions restrict economy $(A,B)$: @@ -388,28 +388,28 @@ Two key assumptions restrict economy $(A,B)$: ``` ```` -A semi-positive *intensity* $m$-vector $x$ denotes levels at which +A semi-positive **intensity** $m$-vector $x$ denotes levels at which activities are operated. Therefore, -- vector $x^\top A$ gives the total amount of *goods used in - production* -- vector $x^\top B$ gives *total outputs* +- vector $x^\top A$ gives the total amount of **goods used in + production** +- vector $x^\top B$ gives **total outputs**** -An economy $(A,B)$ is said to be *productive*, if there exists a +An economy $(A,B)$ is said to be **productive**, if there exists a non-negative intensity vector $x \geq 0$ such that $x^\top B > x^\top A$. The semi-positive $n$-vector $p$ contains prices assigned to the $n$ goods. -The $p$ vector implies *cost* and *revenue* vectors +The $p$ vector implies **cost** and **revenue** vectors -- the vector $Ap$ tells *costs* of the vector of activities -- the vector $Bp$ tells *revenues* from the vector of activities +- the vector $Ap$ tells **costs** of the vector of activities +- the vector $Bp$ tells **revenues** from the vector of activitieses -Satisfaction of a property of an input-output pair $(A,B)$ called *irreducibility* +Satisfaction of a property of an input-output pair $(A,B)$ called **irreducibility** (or indecomposability) determines whether an economy can be decomposed into multiple "sub-economies". From 3b8d5e7873a8f390e56ac60bdac300c9e4790b6b Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:26:46 +1100 Subject: [PATCH 15/28] Fix emphasis vs definitions in ak_aiyagari.md Changes per #721: - Lifecycle patterns - Within-cohort heterogeneity - Cross-cohort interactions All terms changed from italic to bold as they are definitions per style guide. --- lectures/ak_aiyagari.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/lectures/ak_aiyagari.md b/lectures/ak_aiyagari.md index 4df268ed5..041e5428f 100644 --- a/lectures/ak_aiyagari.md +++ b/lectures/ak_aiyagari.md @@ -151,17 +151,17 @@ Summarizing activities in the asset market, all agents, regardless of age $j \in ### Key features -*Lifecycle patterns* shape economic behavior across ages: +**Lifecycle patterns** shape economic behavior across ages: - Labor productivity varies systematically with age according to the profile $l(j)$, while asset holdings typically follow a lifecycle pattern of accumulation during working years and decumulation during retirement. - Age-specific fiscal transfers $\delta_{j,t}$ redistribute resources across generations. -*Within-cohort heterogeneity* creates dispersion among agents of the same age: +**Within-cohort heterogeneity** creates dispersion among agents of the same age: - Agents of the same age differ in their asset holdings $a_{i,j,t}$ due to different histories of idiosyncratic productivity shocks, their current productivities $\gamma_{i,j,t}$, and consequently their labor incomes and financial wealth. -*Cross-cohort interactions* determine equilibrium outcomes through market aggregation: +**Cross-cohort interactions** determine equilibrium outcomes through market aggregation: - All cohorts participate together in factor markets, with asset supplies from all cohorts determining aggregate capital and effective labor supplies from all cohorts determining aggregate labor. From 6726aae7bd4057b0126afa7fee5f5cf41648ba6a Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:27:16 +1100 Subject: [PATCH 16/28] Fix emphasis vs definitions in ak2.md Changes per #721: - numeraire Term changed from italic to bold as it is a definition per style guide. --- lectures/ak2.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/lectures/ak2.md b/lectures/ak2.md index e1a516725..cdcf78c0b 100644 --- a/lectures/ak2.md +++ b/lectures/ak2.md @@ -209,7 +209,7 @@ Units of the rental rates are: * for $r_t$, output at time $t$ per unit of capital at time $t$ -We take output at time $t$ as *numeraire*, so the price of output at time $t$ is one. +We take output at time $t$ as **numeraire**, so the price of output at time $t$ is one. The firm's profits at time $t$ are From 0971d6306a57a5e2d5e69b0184ef7afdc741078b Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:29:25 +1100 Subject: [PATCH 17/28] Fix emphasis vs definitions in cake eating lectures Changes per #721: - exogenous (cake_eating_egm.md) - adapted (cake_eating_stochastic.md) - state (cake_eating_stochastic.md) - control (cake_eating_stochastic.md) - topologically conjugate (cake_eating_time_iter.md) All terms changed from italic to bold as they are definitions per style guide. --- lectures/cake_eating_egm.md | 2 +- lectures/cake_eating_stochastic.md | 6 +++--- lectures/cake_eating_time_iter.md | 2 +- 3 files changed, 5 insertions(+), 5 deletions(-) diff --git a/lectures/cake_eating_egm.md b/lectures/cake_eating_egm.md index e9f7e272d..e12444eec 100644 --- a/lectures/cake_eating_egm.md +++ b/lectures/cake_eating_egm.md @@ -114,7 +114,7 @@ Let $(u')^{-1}$ be the inverse function of $u'$. The idea is this: -* First, we fix an *exogenous* grid $\{k_i\}$ for capital ($k = x - c$). +* First, we fix an **exogenous** grid $\{k_i\}$ for capital ($k = x - c$). * Then we obtain $c_i$ via ```{math} diff --git a/lectures/cake_eating_stochastic.md b/lectures/cake_eating_stochastic.md index ff1187f0b..246b69b54 100644 --- a/lectures/cake_eating_stochastic.md +++ b/lectures/cake_eating_stochastic.md @@ -164,13 +164,13 @@ In summary, the agent's aim is to select a path $c_0, c_1, c_2, \ldots$ for cons 1. nonnegative, 1. feasible in the sense of {eq}`outcsdp0`, 1. optimal, in the sense that it maximizes {eq}`texs0_og2` relative to all other feasible consumption sequences, and -1. *adapted*, in the sense that the action $c_t$ depends only on +1. **adapted**, in the sense that the action $c_t$ depends only on observable outcomes, not on future outcomes such as $\xi_{t+1}$. In the present context -* $x_t$ is called the *state* variable --- it summarizes the "state of the world" at the start of each period. -* $c_t$ is called the *control* variable --- a value chosen by the agent each period after observing the state. +* $x_t$ is called the **state** variable --- it summarizes the "state of the world" at the start of each period. +* $c_t$ is called the **control** variable --- a value chosen by the agent each period after observing the state. ### The Policy Function Approach diff --git a/lectures/cake_eating_time_iter.md b/lectures/cake_eating_time_iter.md index 21f30141f..9fe5d4ad9 100644 --- a/lectures/cake_eating_time_iter.md +++ b/lectures/cake_eating_time_iter.md @@ -237,7 +237,7 @@ whenever $\sigma \in \mathscr P$. It is possible to prove that there is a tight relationship between iterates of $K$ and iterates of the Bellman operator. -Mathematically, the two operators are *topologically conjugate*. +Mathematically, the two operators are **topologically conjugate**. Loosely speaking, this means that if iterates of one operator converge then so do iterates of the other, and vice versa. From 14d4f64a16f3e5406ea3ef3b59b0ea85588603c3 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:29:55 +1100 Subject: [PATCH 18/28] Fix emphasis vs definitions in career and cass_koopmans_1 Changes per #721: - career (career.md) - job (career.md) - aggregation theory (cass_koopmans_1.md) All terms changed from italic to bold as they are definitions per style guide. --- lectures/career.md | 4 ++-- lectures/cass_koopmans_1.md | 2 +- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/lectures/career.md b/lectures/career.md index 63cb10626..c8fed1268 100644 --- a/lectures/career.md +++ b/lectures/career.md @@ -66,8 +66,8 @@ from matplotlib import cm In what follows we distinguish between a career and a job, where -* a *career* is understood to be a general field encompassing many possible jobs, and -* a *job* is understood to be a position with a particular firm +* a **career** is understood to be a general field encompassing many possible jobs, and +* a **job** is understood to be a position with a particular firm For workers, wages can be decomposed into the contribution of job and career diff --git a/lectures/cass_koopmans_1.md b/lectures/cass_koopmans_1.md index 30b9cbdb0..7525ff0fc 100644 --- a/lectures/cass_koopmans_1.md +++ b/lectures/cass_koopmans_1.md @@ -143,7 +143,7 @@ $$ c(\omega) = c = C. $$ -This line of argument indicates the special *aggregation theory* that lies beneath outcomes in which a representative consumer +This line of argument indicates the special **aggregation theory** that lies beneath outcomes in which a representative consumer consumes amount $C$. It appears often in aggregate economics. From d0fe1d51a6614e55cb3871c75f692a69b26c9f1e Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:30:49 +1100 Subject: [PATCH 19/28] Fix emphasis vs definitions in likelihood_bayes.md Changes per #721: - recursion - multiplicative decomposition Terms changed from italic to bold as they are definitions per style guide. --- lectures/likelihood_bayes.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/lectures/likelihood_bayes.md b/lectures/likelihood_bayes.md index c9b203f11..da81a560f 100644 --- a/lectures/likelihood_bayes.md +++ b/lectures/likelihood_bayes.md @@ -129,8 +129,8 @@ $$ where we use the conventions that $f(w^t) = f(w_1) f(w_2) \ldots f(w_t)$ and $g(w^t) = g(w_1) g(w_2) \ldots g(w_t)$. -Notice that the likelihood process satisfies the *recursion* or -*multiplicative decomposition* +Notice that the likelihood process satisfies the **recursion** or +**multiplicative decomposition** $$ L(w^t) = \ell (w_t) L (w^{t-1}) . From 4971ec3f9f1cbbc40317a6e3546529e4ad7d7dc2 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:31:21 +1100 Subject: [PATCH 20/28] Fix emphasis vs definitions in morris_learn.md Changes per #721: - prior distributions - posterior distributions - speculative behavior - ex dividend - Short sales are prohibited - Harsanyi Common Priors Doctrine All terms changed from italic to bold as they are definitions per style guide. --- lectures/morris_learn.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/lectures/morris_learn.md b/lectures/morris_learn.md index b41d81da4..95a920443 100644 --- a/lectures/morris_learn.md +++ b/lectures/morris_learn.md @@ -50,12 +50,12 @@ Key features of the environment in Morris's model include: * A single parameter indexes the set of statistical models * All traders observe the same dividend history * All traders use Bayes' Law to update beliefs -* Traders have different initial *prior distributions* over the parameter -* Traders' *posterior distributions* over the parameter eventually merge +* Traders have different initial **prior distributions** over the parameter +* Traders' **posterior distributions** over the parameter eventually merge * Before their posterior distributions merge, traders disagree about the predictive density over prospective dividends * therefore they disagree about the value of the asset -Just as in the hard-wired beliefs model of Harrison and Kreps, those differences of opinion induce investors to engage in *speculative behavior* in the following sense: +Just as in the hard-wired beliefs model of Harrison and Kreps, those differences of opinion induce investors to engage in **speculative behavior** in the following sense: * sometimes they are willing to pay more for the asset than what they think is its "fundamental" value, i.e., the expected discounted value of its prospective dividend stream @@ -110,11 +110,11 @@ Traders buy and sell the risky asset in competitive markets each period $t = 0, As in Harrison-Kreps: -* The asset is traded *ex dividend* +* The asset is traded **ex dividend** * An owner of a share at the end of time $t$ is entitled to the dividend at time $t+1$ * An owner of a share at the end of period $t$ also has the right to sell the share at time $t+1$ after having received the dividend at time $t+1$. -*Short sales are prohibited*. +**Short sales are prohibited**. This matters because it limits how pessimists can express their opinions: @@ -151,7 +151,7 @@ reduce the set of models to a single model by imputing to all agents inside the A set of statistical models that has a particular geometric structure is called a [manifold](https://en.wikipedia.org/wiki/Manifold) of statistical models. Morris endows traders with a shared manifold of statistical models. ``` -Proceeding in this way adheres to the *Harsanyi Common Priors Doctrine*. +Proceeding in this way adheres to the **Harsanyi Common Priors Doctrine**. From 3325db673188fd0815c13f4c3ff63a01a10fe74d Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:31:55 +1100 Subject: [PATCH 21/28] Fix emphasis vs definitions in odu and opt_transport Changes per #721: - reservation wage (odu.md) - reservation wage functional equation (odu.md) - matrix (opt_transport.md) - vector (opt_transport.md) All terms changed from italic to bold as they are definitions per style guide. --- lectures/odu.md | 4 ++-- lectures/opt_transport.md | 6 +++--- 2 files changed, 5 insertions(+), 5 deletions(-) diff --git a/lectures/odu.md b/lectures/odu.md index 656b141db..c62519468 100644 --- a/lectures/odu.md +++ b/lectures/odu.md @@ -111,7 +111,7 @@ v(w) ``` The optimal policy has the form $\mathbf{1}\{w \geq \bar w\}$, where -$\bar w$ is a constant called the *reservation wage*. +$\bar w$ is a constant called the **reservation wage**. ### Offer Distribution Unknown @@ -545,7 +545,7 @@ and using $\circ$ for composition of functions yields Equation {eq}`odu_mvf4` can be understood as a functional equation, where $\bar w$ is the unknown function. -* Let's call it the *reservation wage functional equation* (RWFE). +* Let's call it the **reservation wage functional equation** (RWFE). * The solution $\bar w$ to the RWFE is the object that we wish to compute. ## Solving the RWFE diff --git a/lectures/opt_transport.md b/lectures/opt_transport.md index 9ce3215ef..8791494f0 100644 --- a/lectures/opt_transport.md +++ b/lectures/opt_transport.md @@ -136,12 +136,12 @@ tackle the optimal transport problem. ### Vectorizing a Matrix of Decision Variables -A *matrix* of decision variables $x_{ij}$ appears in problem {eq}`plannerproblem`. +A **matrix** of decision variables $x_{ij}$ appears in problem {eq}`plannerproblem`. -The SciPy function `linprog` expects to see a *vector* of decision variables. +The SciPy function `linprog` expects to see a **vector** of decision variables. This situation impels us to rewrite our problem in terms of a -*vector* of decision variables. +**vector** of decision variables. Let From 9f1b3ed25af6f43de3cedbfd2ede44f144ff11d1 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:33:03 +1100 Subject: [PATCH 22/28] Fix emphasis vs definitions in kalman and ifp_advanced Changes per #721: - prior (kalman.md) - filtering distribution (kalman.md) - predictive (kalman.md) - Kalman gain (kalman.md) - predictive distribution (kalman.md) - savings (ifp_advanced.md) All terms changed from italic to bold as they are definitions per style guide. --- lectures/ifp_advanced.md | 2 +- lectures/kalman.md | 10 +++++----- 2 files changed, 6 insertions(+), 6 deletions(-) diff --git a/lectures/ifp_advanced.md b/lectures/ifp_advanced.md index d0eb01c80..265e0f061 100644 --- a/lectures/ifp_advanced.md +++ b/lectures/ifp_advanced.md @@ -264,7 +264,7 @@ assets is low. #### Finding Optimal Consumption -The endogenous grid method (EGM) calls for us to take a grid of *savings* +The endogenous grid method (EGM) calls for us to take a grid of **savings** values $s_i$, where each such $s$ is interpreted as $s = a - c$. diff --git a/lectures/kalman.md b/lectures/kalman.md index a516a8eb2..fa089320f 100644 --- a/lectures/kalman.md +++ b/lectures/kalman.md @@ -85,7 +85,7 @@ One way to summarize our knowledge is a point prediction $\hat x$ * Then it is better to summarize our initial beliefs with a bivariate probability density $p$ * $\int_E p(x)dx$ indicates the probability that we attach to the missile being in region $E$. -The density $p$ is called our *prior* for the random variable $x$. +The density $p$ is called our **prior** for the random variable $x$. To keep things tractable in our example, we assume that our prior is Gaussian. @@ -317,7 +317,7 @@ We have obtained probabilities for the current location of the state (missile) g This is called "filtering" rather than forecasting because we are filtering out noise rather than looking into the future. -* $p(x \,|\, y) = N(\hat x^F, \Sigma^F)$ is called the *filtering distribution* +* $p(x \,|\, y) = N(\hat x^F, \Sigma^F)$ is called the **filtering distribution** But now let's suppose that we are given another task: to predict the location of the missile after one unit of time (whatever that may be) has elapsed. @@ -331,7 +331,7 @@ Let's suppose that we have one, and that it's linear and Gaussian. In particular x_{t+1} = A x_t + w_{t+1}, \quad \text{where} \quad w_t \sim N(0, Q) ``` -Our aim is to combine this law of motion and our current distribution $p(x \,|\, y) = N(\hat x^F, \Sigma^F)$ to come up with a new *predictive* distribution for the location in one unit of time. +Our aim is to combine this law of motion and our current distribution $p(x \,|\, y) = N(\hat x^F, \Sigma^F)$ to come up with a new **predictive** distribution for the location in one unit of time. In view of {eq}`kl_xdynam`, all we have to do is introduce a random vector $x^F \sim N(\hat x^F, \Sigma^F)$ and work out the distribution of $A x^F + w$ where $w$ is independent of $x^F$ and has distribution $N(0, Q)$. @@ -356,7 +356,7 @@ $$ $$ The matrix $A \Sigma G' (G \Sigma G' + R)^{-1}$ is often written as -$K_{\Sigma}$ and called the *Kalman gain*. +$K_{\Sigma}$ and called the **Kalman gain**. * The subscript $\Sigma$ has been added to remind us that $K_{\Sigma}$ depends on $\Sigma$, but not $y$ or $\hat x$. @@ -373,7 +373,7 @@ Our updated prediction is the density $N(\hat x_{new}, \Sigma_{new})$ where \end{aligned} ``` -* The density $p_{new}(x) = N(\hat x_{new}, \Sigma_{new})$ is called the *predictive distribution* +* The density $p_{new}(x) = N(\hat x_{new}, \Sigma_{new})$ is called the **predictive distribution** The predictive distribution is the new density shown in the following figure, where the update has used parameters. From 373620611acabd7eeb1400f02f08fec77aa3044e Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:33:52 +1100 Subject: [PATCH 23/28] Fix emphasis vs definitions in cass_fiscal.md Changes per #721: - Household (changed from italic to bold - also fixed typo 'Frim' to 'Firm') - Firm Terms changed from italic to bold as they are economic agents being defined per style guide. --- lectures/cass_fiscal.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/lectures/cass_fiscal.md b/lectures/cass_fiscal.md index fdf5c274d..b7e3646df 100644 --- a/lectures/cass_fiscal.md +++ b/lectures/cass_fiscal.md @@ -147,8 +147,8 @@ $$ (eq:gov_budget) Given a budget-feasible government policy $\{g_t\}_{t=0}^\infty$ and $\{\tau_{ct}, \tau_{kt}, \tau_{nt}, \tau_{ht}\}_{t=0}^\infty$ subject to {eq}`eq:gov_budget`, -- *Household* chooses $\{c_t\}_{t=0}^\infty$, $\{n_t\}_{t=0}^\infty$, and $\{k_{t+1}\}_{t=0}^\infty$ to maximize utility{eq}`eq:utility` subject to budget constraint{eq}`eq:house_budget`, and -- *Frim* chooses sequences of capital $\{k_t\}_{t=0}^\infty$ and $\{n_t\}_{t=0}^\infty$ to maximize profits +- **Household** chooses $\{c_t\}_{t=0}^\infty$, $\{n_t\}_{t=0}^\infty$, and $\{k_{t+1}\}_{t=0}^\infty$ to maximize utility{eq}`eq:utility` subject to budget constraint{eq}`eq:house_budget`, and +- **Firm** chooses sequences of capital $\{k_t\}_{t=0}^\infty$ and $\{n_t\}_{t=0}^\infty$ to maximize profits $$ \sum_{t=0}^\infty q_t [F(k_t, n_t) - \eta_t k_t - w_t n_t] From deb1944ad557694a06e3495a98d5acd555ff504f Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 20 Nov 2025 19:34:31 +1100 Subject: [PATCH 24/28] Fix emphasis vs definitions in exchangeable.md Changes per #721: - conditionally (as part of 'conditionally independently and identically distributed') Term changed from italic to bold as it is a definition per style guide. --- lectures/exchangeable.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/lectures/exchangeable.md b/lectures/exchangeable.md index 564cbb1cc..7ebb7278e 100644 --- a/lectures/exchangeable.md +++ b/lectures/exchangeable.md @@ -47,7 +47,7 @@ search model. Among other things, this lecture discusses connections between the statistical concepts of sequences of random variables that are - independently and identically distributed -- exchangeable (also known as *conditionally* independently and identically distributed) +- exchangeable (also known as **conditionally** independently and identically distributed) Understanding these concepts is essential for appreciating how Bayesian updating works. From f3758ed5112547bf2826aaf3f8b68ee23f451e70 Mon Sep 17 00:00:00 2001 From: mmcky Date: Fri, 21 Nov 2025 08:16:37 +1100 Subject: [PATCH 25/28] Revert incorrect emphasis changes back to italic Per review feedback on #721: - ifp_advanced.md: 'savings' - emphasis on grid type, not definition - exchangeable.md: 'conditionally' - alternate description, not definition - cass_koopmans_1.md: 'aggregation theory' - referencing theory, not defining it These should remain italic for emphasis, not bold for definitions. --- lectures/cass_koopmans_1.md | 2 +- lectures/exchangeable.md | 2 +- lectures/ifp_advanced.md | 2 +- 3 files changed, 3 insertions(+), 3 deletions(-) diff --git a/lectures/cass_koopmans_1.md b/lectures/cass_koopmans_1.md index 7525ff0fc..30b9cbdb0 100644 --- a/lectures/cass_koopmans_1.md +++ b/lectures/cass_koopmans_1.md @@ -143,7 +143,7 @@ $$ c(\omega) = c = C. $$ -This line of argument indicates the special **aggregation theory** that lies beneath outcomes in which a representative consumer +This line of argument indicates the special *aggregation theory* that lies beneath outcomes in which a representative consumer consumes amount $C$. It appears often in aggregate economics. diff --git a/lectures/exchangeable.md b/lectures/exchangeable.md index 7ebb7278e..564cbb1cc 100644 --- a/lectures/exchangeable.md +++ b/lectures/exchangeable.md @@ -47,7 +47,7 @@ search model. Among other things, this lecture discusses connections between the statistical concepts of sequences of random variables that are - independently and identically distributed -- exchangeable (also known as **conditionally** independently and identically distributed) +- exchangeable (also known as *conditionally* independently and identically distributed) Understanding these concepts is essential for appreciating how Bayesian updating works. diff --git a/lectures/ifp_advanced.md b/lectures/ifp_advanced.md index 265e0f061..d0eb01c80 100644 --- a/lectures/ifp_advanced.md +++ b/lectures/ifp_advanced.md @@ -264,7 +264,7 @@ assets is low. #### Finding Optimal Consumption -The endogenous grid method (EGM) calls for us to take a grid of **savings** +The endogenous grid method (EGM) calls for us to take a grid of *savings* values $s_i$, where each such $s$ is interpreted as $s = a - c$. From d9d78101a148baceb36e7ad88271b8877489626f Mon Sep 17 00:00:00 2001 From: mmcky Date: Fri, 21 Nov 2025 08:23:08 +1100 Subject: [PATCH 26/28] Revert incorrect emphasis-to-bold changes (batch 2) Reverted changes in 3 files where emphasis/contrast was incorrectly changed to bold: - opt_transport.md: matrix/vector (contrast between types) - cake_eating_egm.md: exogenous (contrast with endogenous) - ak_aiyagari.md: section headers (organizational emphasis) These are not formal definitions, so should remain italic per style guide. --- lectures/ak_aiyagari.md | 6 +++--- lectures/cake_eating_egm.md | 2 +- lectures/opt_transport.md | 6 +++--- 3 files changed, 7 insertions(+), 7 deletions(-) diff --git a/lectures/ak_aiyagari.md b/lectures/ak_aiyagari.md index 041e5428f..4df268ed5 100644 --- a/lectures/ak_aiyagari.md +++ b/lectures/ak_aiyagari.md @@ -151,17 +151,17 @@ Summarizing activities in the asset market, all agents, regardless of age $j \in ### Key features -**Lifecycle patterns** shape economic behavior across ages: +*Lifecycle patterns* shape economic behavior across ages: - Labor productivity varies systematically with age according to the profile $l(j)$, while asset holdings typically follow a lifecycle pattern of accumulation during working years and decumulation during retirement. - Age-specific fiscal transfers $\delta_{j,t}$ redistribute resources across generations. -**Within-cohort heterogeneity** creates dispersion among agents of the same age: +*Within-cohort heterogeneity* creates dispersion among agents of the same age: - Agents of the same age differ in their asset holdings $a_{i,j,t}$ due to different histories of idiosyncratic productivity shocks, their current productivities $\gamma_{i,j,t}$, and consequently their labor incomes and financial wealth. -**Cross-cohort interactions** determine equilibrium outcomes through market aggregation: +*Cross-cohort interactions* determine equilibrium outcomes through market aggregation: - All cohorts participate together in factor markets, with asset supplies from all cohorts determining aggregate capital and effective labor supplies from all cohorts determining aggregate labor. diff --git a/lectures/cake_eating_egm.md b/lectures/cake_eating_egm.md index e12444eec..e9f7e272d 100644 --- a/lectures/cake_eating_egm.md +++ b/lectures/cake_eating_egm.md @@ -114,7 +114,7 @@ Let $(u')^{-1}$ be the inverse function of $u'$. The idea is this: -* First, we fix an **exogenous** grid $\{k_i\}$ for capital ($k = x - c$). +* First, we fix an *exogenous* grid $\{k_i\}$ for capital ($k = x - c$). * Then we obtain $c_i$ via ```{math} diff --git a/lectures/opt_transport.md b/lectures/opt_transport.md index 8791494f0..9ce3215ef 100644 --- a/lectures/opt_transport.md +++ b/lectures/opt_transport.md @@ -136,12 +136,12 @@ tackle the optimal transport problem. ### Vectorizing a Matrix of Decision Variables -A **matrix** of decision variables $x_{ij}$ appears in problem {eq}`plannerproblem`. +A *matrix* of decision variables $x_{ij}$ appears in problem {eq}`plannerproblem`. -The SciPy function `linprog` expects to see a **vector** of decision variables. +The SciPy function `linprog` expects to see a *vector* of decision variables. This situation impels us to rewrite our problem in terms of a -**vector** of decision variables. +*vector* of decision variables. Let From 8d221fda202ddf029d33839468048758b1e2c470 Mon Sep 17 00:00:00 2001 From: mmcky Date: Fri, 21 Nov 2025 11:40:32 +1100 Subject: [PATCH 27/28] Revert checked emphasis comments back to italic (batch 3) Based on PR review feedback with checked [x] emphasis comments, reverted the following terms from bold back to italic (emphasis, not definitions): - linear_models.md: ergodicity (concept emphasis) - markov_asset.md: tree, fruit, shares, dividend (metaphorical emphasis) - mccall_model.md: values (concept emphasis) - mle.md: parametric class (emphasis not definition) - morris_learn.md: prior/posterior distributions, speculative behavior, ex dividend, Short sales, Harsanyi Common Priors Doctrine (emphasis) - ols.md: exogenous, marginal effect (emphasis not definitions) - rational_expectations.md: rational expectations equilibrium (intro emphasis), perceived/actual law of motion (intro emphasis, formal definitions come later) - samuelson.md: second-order linear difference equation, national output identity, consumption function, accelerator, accelerator coefficient, aggregate demand/supply, random, stochastic, shocks, disturbances (emphasis not definitions) These are emphasis on concepts or references, not formal definitions. --- lectures/linear_models.md | 2 +- lectures/markov_asset.md | 4 ++-- lectures/mccall_model.md | 2 +- lectures/mle.md | 2 +- lectures/morris_learn.md | 12 ++++++------ lectures/ols.md | 4 ++-- lectures/rational_expectations.md | 4 ++-- lectures/samuelson.md | 20 ++++++++++---------- 8 files changed, 25 insertions(+), 25 deletions(-) diff --git a/lectures/linear_models.md b/lectures/linear_models.md index 4bfe67dca..feb76976a 100644 --- a/lectures/linear_models.md +++ b/lectures/linear_models.md @@ -1156,7 +1156,7 @@ $$ Do these time series averages converge to something interpretable in terms of our basic state-space representation? -The answer depends on something called **ergodicity**. +The answer depends on something called *ergodicity*. Ergodicity is the property that time series and ensemble averages coincide. diff --git a/lectures/markov_asset.md b/lectures/markov_asset.md index 3ad757bda..4421b37b4 100644 --- a/lectures/markov_asset.md +++ b/lectures/markov_asset.md @@ -451,8 +451,8 @@ Lucas considered an abstract pure exchange economy with these features: * a single non-storable consumption good * a Markov process that governs the total amount of the consumption good available each period -* a single **tree** that each period yields **fruit** that equals the total amount of consumption available to the economy -* a competitive market in **shares** in the tree that entitles their owners to corresponding shares of the **dividend** stream, i.e., the **fruit** stream, yielded by the tree +* a single *tree* that each period yields *fruit* that equals the total amount of consumption available to the economy +* a competitive market in *shares* in the tree that entitles their owners to corresponding shares of the *dividend* stream, i.e., the *fruit* stream, yielded by the tree * a representative consumer who in a competitive equilibrium diff --git a/lectures/mccall_model.md b/lectures/mccall_model.md index f4cb83ceb..54f264c3e 100644 --- a/lectures/mccall_model.md +++ b/lectures/mccall_model.md @@ -134,7 +134,7 @@ In order to optimally trade-off current and future rewards, we need to think abo 1. the current payoffs we get from different choices 1. the different states that those choices will lead to in next period -To weigh these two aspects of the decision problem, we need to assign **values** +To weigh these two aspects of the decision problem, we need to assign *values* to states. To this end, let $v^*(w)$ be the total lifetime value accruing to an diff --git a/lectures/mle.md b/lectures/mle.md index 4b3a2f047..7a1942d42 100644 --- a/lectures/mle.md +++ b/lectures/mle.md @@ -74,7 +74,7 @@ Let's consider the steps we need to go through in maximum likelihood estimation The first step with maximum likelihood estimation is to choose the probability distribution believed to be generating the data. -More precisely, we need to make an assumption as to which **parametric class** of distributions is generating the data. +More precisely, we need to make an assumption as to which *parametric class* of distributions is generating the data. * e.g., the class of all normal distributions, or the class of all gamma distributions. diff --git a/lectures/morris_learn.md b/lectures/morris_learn.md index 95a920443..b41d81da4 100644 --- a/lectures/morris_learn.md +++ b/lectures/morris_learn.md @@ -50,12 +50,12 @@ Key features of the environment in Morris's model include: * A single parameter indexes the set of statistical models * All traders observe the same dividend history * All traders use Bayes' Law to update beliefs -* Traders have different initial **prior distributions** over the parameter -* Traders' **posterior distributions** over the parameter eventually merge +* Traders have different initial *prior distributions* over the parameter +* Traders' *posterior distributions* over the parameter eventually merge * Before their posterior distributions merge, traders disagree about the predictive density over prospective dividends * therefore they disagree about the value of the asset -Just as in the hard-wired beliefs model of Harrison and Kreps, those differences of opinion induce investors to engage in **speculative behavior** in the following sense: +Just as in the hard-wired beliefs model of Harrison and Kreps, those differences of opinion induce investors to engage in *speculative behavior* in the following sense: * sometimes they are willing to pay more for the asset than what they think is its "fundamental" value, i.e., the expected discounted value of its prospective dividend stream @@ -110,11 +110,11 @@ Traders buy and sell the risky asset in competitive markets each period $t = 0, As in Harrison-Kreps: -* The asset is traded **ex dividend** +* The asset is traded *ex dividend* * An owner of a share at the end of time $t$ is entitled to the dividend at time $t+1$ * An owner of a share at the end of period $t$ also has the right to sell the share at time $t+1$ after having received the dividend at time $t+1$. -**Short sales are prohibited**. +*Short sales are prohibited*. This matters because it limits how pessimists can express their opinions: @@ -151,7 +151,7 @@ reduce the set of models to a single model by imputing to all agents inside the A set of statistical models that has a particular geometric structure is called a [manifold](https://en.wikipedia.org/wiki/Manifold) of statistical models. Morris endows traders with a shared manifold of statistical models. ``` -Proceeding in this way adheres to the **Harsanyi Common Priors Doctrine**. +Proceeding in this way adheres to the *Harsanyi Common Priors Doctrine*. diff --git a/lectures/ols.md b/lectures/ols.md index 2416bf2b7..bb4cc68af 100644 --- a/lectures/ols.md +++ b/lectures/ols.md @@ -51,7 +51,7 @@ As an example, we will replicate results from Acemoglu, Johnson and Robinson's s In the paper, the authors emphasize the importance of institutions in economic development. -The main contribution is the use of settler mortality rates as a source of **exogenous** variation in institutional differences. +The main contribution is the use of settler mortality rates as a source of *exogenous* variation in institutional differences. Such variation is needed to determine whether it is institutions that give rise to greater economic growth, rather than the other way around. @@ -125,7 +125,7 @@ where: - $\beta_0$ is the intercept of the linear trend line on the y-axis - $\beta_1$ is the slope of the linear trend line, representing - the **marginal effect** of protection against risk on log GDP per + the *marginal effect* of protection against risk on log GDP per capita - $u_i$ is a random error term (deviations of observations from the linear trend due to factors not included in the model) diff --git a/lectures/rational_expectations.md b/lectures/rational_expectations.md index 2e4b51862..ec8be4bc8 100644 --- a/lectures/rational_expectations.md +++ b/lectures/rational_expectations.md @@ -39,7 +39,7 @@ tags: [hide-output] ## Overview -This lecture introduces the concept of a **rational expectations equilibrium**. +This lecture introduces the concept of a *rational expectations equilibrium*. To illustrate it, we describe a linear quadratic version of a model due to Lucas and Prescott {cite}`LucasPrescott1971`. @@ -53,7 +53,7 @@ Because we use linear quadratic setups for demand and costs, we can deploy the L We will learn about how a representative agent's problem differs from a planner's, and how a planning problem can be used to compute quantities and prices in a rational expectations equilibrium. -We will also learn about how a rational expectations equilibrium can be characterized as a [fixed point](https://en.wikipedia.org/wiki/Fixed_point_%28mathematics%29) of a mapping from a **perceived law of motion** to an **actual law of motion**. +We will also learn about how a rational expectations equilibrium can be characterized as a [fixed point](https://en.wikipedia.org/wiki/Fixed_point_%28mathematics%29) of a mapping from a *perceived law of motion* to an *actual law of motion*. Equality between a perceived and an actual law of motion for endogenous market-wide objects captures in a nutshell what the rational expectations equilibrium concept is all about. diff --git a/lectures/samuelson.md b/lectures/samuelson.md index f5f1ad534..8616328e9 100644 --- a/lectures/samuelson.md +++ b/lectures/samuelson.md @@ -66,21 +66,21 @@ from cmath import sqrt ### Samuelson's model -Samuelson used a **second-order linear difference equation** to +Samuelson used a *second-order linear difference equation* to represent a model of national output based on three components: -- a **national output identity** asserting that national output or national income is the +- a *national output identity* asserting that national output or national income is the sum of consumption plus investment plus government purchases. -- a Keynesian **consumption function** asserting that consumption at +- a Keynesian *consumption function* asserting that consumption at time $t$ is equal to a constant times national output at time $t-1$. -- an investment **accelerator** asserting that investment at time - $t$ equals a constant called the **accelerator coefficient** +- an investment *accelerator* asserting that investment at time + $t$ equals a constant called the *accelerator coefficient* times the difference in output between period $t-1$ and $t-2$. Consumption plus investment plus government purchases -constitute **aggregate demand,** which automatically calls forth an -equal amount of **aggregate supply**. +constitute *aggregate demand,* which automatically calls forth an +equal amount of *aggregate supply*. (To read about linear difference equations see [here](https://en.wikipedia.org/wiki/Linear_difference_equation) or chapter IX of {cite}`Sargent1987`.) @@ -204,10 +204,10 @@ fluctuations by adding a random shock to aggregate demand. ### Stochastic version of the model -We create a **random** or **stochastic** version of the model by adding -a random process of **shocks** or **disturbances** +We create a *random* or *stochastic* version of the model by adding +a random process of *shocks* or *disturbances* $\{\sigma \epsilon_t \}$ to the right side of equation {eq}`second_order`, -leading to the **second-order scalar linear stochastic difference equation**: +leading to the *second-order scalar linear stochastic difference equation*: ```{math} :label: second_stochastic From f0264ea5faf5e9892d988c25b1d41d01006ec64f Mon Sep 17 00:00:00 2001 From: mmcky Date: Fri, 21 Nov 2025 12:35:41 +1100 Subject: [PATCH 28/28] Fix typos introduced during formatting changes MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Fixed the following typos: - linear_algebra.md: removed extra '.*.' after 'square' and 'symmetric' - linear_algebra.md: removed extra '.l.' after 'diagonal' - sir_model.md: removed extra 'd)' after 'infected)' - von_neumann_model.md: removed extra '.).' after 'consumed)' - von_neumann_model.md: removed extra '****' after 'outputs' - von_neumann_model.md: removed extra 'es' from 'activitieses' → 'activities' --- lectures/linear_algebra.md | 6 +++--- lectures/sir_model.md | 2 +- lectures/von_neumann_model.md | 6 +++--- 3 files changed, 7 insertions(+), 7 deletions(-) diff --git a/lectures/linear_algebra.md b/lectures/linear_algebra.md index 36061bcb2..33f3dc53e 100644 --- a/lectures/linear_algebra.md +++ b/lectures/linear_algebra.md @@ -471,15 +471,15 @@ For obvious reasons, the matrix $A$ is also called a vector if either $n = 1$ or In the former case, $A$ is called a **row vector**, while in the latter it is called a **column vector**. -If $n = k$, then $A$ is called **square**.*. +If $n = k$, then $A$ is called **square**. The matrix formed by replacing $a_{ij}$ by $a_{ji}$ for every $i$ and $j$ is called the **transpose** of $A$ and denoted $A'$ or $A^{\top}$. -If $A = A'$, then $A$ is called **symmetric**.*. +If $A = A'$, then $A$ is called **symmetric**. For a square matrix $A$, the $i$ elements of the form $a_{ii}$ for $i=1,\ldots,n$ are called the **principal diagonal**. -$A$ is called **diagonal** if the only nonzero entries are on the principal diagonal.l. +$A$ is called **diagonal** if the only nonzero entries are on the principal diagonal. If, in addition to being diagonal, each element along the principal diagonal is equal to 1, then $A$ is called the **identity matrix** and denoted by $I$. diff --git a/lectures/sir_model.md b/lectures/sir_model.md index cbd82c45b..5b0c5305c 100644 --- a/lectures/sir_model.md +++ b/lectures/sir_model.md @@ -109,7 +109,7 @@ dynamics are In these equations, * $\beta(t)$ is called the **transmission rate** (the rate at which individuals bump into others and expose them to the virus). -* $\sigma$ is called the **infection rate** (the rate at which those who are exposed become infected)d) +* $\sigma$ is called the **infection rate** (the rate at which those who are exposed become infected) * $\gamma$ is called the **recovery rate** (the rate at which infected people recover or die). * the dot symbol $\dot y$ represents the time derivative $dy/dt$. diff --git a/lectures/von_neumann_model.md b/lectures/von_neumann_model.md index 7532797fa..70c7afd9a 100644 --- a/lectures/von_neumann_model.md +++ b/lectures/von_neumann_model.md @@ -365,7 +365,7 @@ A pair $(A,B)$ of $m\times n$ non-negative matrices defines an economy. - $m$ is the number of **activities** (or sectors) -- $n$ is the number of **goods** (produced and/or consumed).). +- $n$ is the number of **goods** (produced and/or consumed) - $A$ is called the **input matrix**; $a_{i,j}$ denotes the amount of good $j$ consumed by activity $i$ - $B$ is called the **output matrix**; $b_{i,j}$ represents @@ -395,7 +395,7 @@ Therefore, - vector $x^\top A$ gives the total amount of **goods used in production** -- vector $x^\top B$ gives **total outputs**** +- vector $x^\top B$ gives **total outputs** An economy $(A,B)$ is said to be **productive**, if there exists a non-negative intensity vector $x \geq 0$ such @@ -407,7 +407,7 @@ the $n$ goods. The $p$ vector implies **cost** and **revenue** vectors - the vector $Ap$ tells **costs** of the vector of activities -- the vector $Bp$ tells **revenues** from the vector of activitieses +- the vector $Bp$ tells **revenues** from the vector of activities Satisfaction of a property of an input-output pair $(A,B)$ called **irreducibility** (or indecomposability) determines whether an economy can be decomposed