amss and arellano

Author: longye-tian

lectures/amss3.md

@@ -73,7 +73,7 @@ import matplotlib.pyplot as plt
 from scipy.optimize import minimize
 ```
 
-## The Economy
+## The economy
 
 As in {doc}`Optimal Taxation without State-Contingent Debt <amss>` and {doc}`Optimal Taxation with State-Contingent Debt <opt_tax_recur>`,
 we assume that the  representative agent has  utility function
@@ -115,7 +115,7 @@ The following Python code sets up the economy
 :load: _static/lecture_specific/amss2/crra_utility.py
 ```
 
-### First and Second Moments
+### First and second moments
 
 We'll want  first and second moments of some key random variables below.
 
@@ -136,7 +136,7 @@ def covariance(x, y, s):
     return x * y @ u.π[s] - mean(x, s) * mean(y, s)
 ```
 
-## Long Simulation
+## Long simulation
 
 To generate a long simulation we use the following code.
 
@@ -263,7 +263,7 @@ these early observations
 * early observations are more influenced by the initial value of the par value of government debt than by the ergodic mean of the par value of government debt
 * much later observations are more influenced by the ergodic mean and are independent of the par value of initial government debt
 
-## Asymptotic Mean and Rate of Convergence
+## Asymptotic mean and rate of convergence
 
 We apply the results of BEGS {cite}`BEGS1` to interpret
 
@@ -310,7 +310,7 @@ BEGS interpret random variations in the right side of {eq}`eq_fiscal_risk_1`  as
   ${\mathcal R}_\tau(s, s_{-}) {\mathcal B}_{-}$,  and
 - fluctuations in the effective government deficit ${\mathcal X}_t$
 
-### Asymptotic Mean
+### Asymptotic mean
 
 BEGS give conditions under which the ergodic mean of ${\mathcal B}_t$ is approximated by
 
@@ -347,7 +347,7 @@ AMSS model should be approximately
 
 where mathematical expectations are taken with respect to the ergodic distribution.
 
-### Rate of Convergence
+### Rate of convergence
 
 BEGS also derive the following  approximation to the rate of convergence to ${\mathcal B}^{*}$ from an arbitrary initial condition.
 
@@ -361,13 +361,13 @@ BEGS also derive the following  approximation to the rate of convergence to ${\m
 
 (See the equation above equation (47) in BEGS {cite}`BEGS1`)
 
-### More Advanced Topic
+### More advanced topic
 
 The remainder of this lecture is about  technical material based on  formulas from BEGS {cite}`BEGS1`.
 
 The topic involves interpreting  and extending formula {eq}`eq_criterion_fiscal_1` for the ergodic mean ${\mathcal B}^*$.
 
-### Chicken and Egg
+### Chicken and egg
 
 Notice how attributes of the ergodic distribution for ${\mathcal B}_t$  appear
 on the right side of  formula {eq}`eq_criterion_fiscal_1` for approximating  the ergodic mean via ${\mathcal B}^*$.
@@ -385,7 +385,7 @@ As an example, notice how we used the formula for the mean of ${\mathcal B}$ in
 * **first** we computed the ergodic distribution using a reverse-engineering construction
 * **then** we verified that ${\mathcal B}^*$  agrees with the mean of that distribution
 
-### Approximating the Ergodic Mean
+### Approximating the ergodic mean
 
 BEGS also {cite}`BEGS1` propose  an approximation to  ${\mathcal B}^*$ that can be computed **without** first approximating  the
 ergodic distribution.
@@ -400,7 +400,7 @@ To  construct the BEGS  approximation to ${\mathcal B}^*$, we just follow steps
 
 Here is a step-by-step description of the BEGS {cite}`BEGS1` approximation procedure.
 
-### Step by Step
+### Step by step
 
 **Step 1:** For a given $\tau$ we  compute a vector of
 values $c_\tau(s), s= 1, 2, \ldots, S$ that satisfy
@@ -559,7 +559,7 @@ root(solve_c, np.ones(S), args=(τ, u))
 n = c + u.G   # Compute labor supply
 ```
 
-### Note about Code
+### Note about code
 
 Remember that in our code $\pi$ is a $3 \times 3$ transition
 matrix.

lectures/arellano.md

@@ -81,7 +81,7 @@ from numba import njit, prange
 
 In this section we describe the main features of the model.
 
-### Output, Consumption and Debt
+### Output, consumption and debt
 
 A small open economy is endowed with an exogenous stochastically fluctuating potential output
 stream $\{y_t\}$.
@@ -116,7 +116,7 @@ The government is the only domestic actor with access to foreign credit.
 Because household are averse to consumption fluctuations, the government will try to smooth
 consumption by borrowing from (and lending to) foreign creditors.
 
-### Asset Markets
+### Asset markets
 
 The only credit instrument available to the government is a one-period bond traded in international credit markets.
 
@@ -151,7 +151,7 @@ To rule out Ponzi schemes, we also require that $B \geq -Z$ in every period.
 
 * $Z$ is chosen to be sufficiently large that the constraint never binds in equilibrium.
 
-### Financial Markets
+### Financial markets
 
 Foreign creditors
 
@@ -176,7 +176,7 @@ q = \frac{1 - \delta}{1 + r}
 
 Next we turn to how the government in effect chooses the default probability $\delta$.
 
-### Government's Decisions
+### Government's decisions
 
 At each point in time $t$, the government chooses between
 
@@ -195,7 +195,7 @@ But a sovereign default has two consequences:
       markets.
 1. The country loses access to foreign credit markets.
 
-### Reentering International Credit Market
+### Reentering international credit market
 
 While in a state of default, the economy regains access to foreign credit in each subsequent
 period with probability $\theta$.
@@ -283,7 +283,7 @@ to pin down the bond price function:
 q(B', y) = \frac{1 - \delta(B', y)}{1 + r}
 ```
 
-### Definition of Equilibrium
+### Definition of equilibrium
 
 An *equilibrium* is