Refine optimal savings lecture series with improved clarity and code organization

- Enhance code formatting and comments for better readability across all OS lectures
- Improve mathematical notation and explanations in stochastic optimal savings
- Restructure function definitions in os_egm_jax for better logical flow
- Simplify utility function and Bellman operator implementations
- Add clearer documentation of marginal utility approximations in EGM
- Remove redundant code and improve variable naming throughout

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

Author: jstac

lectures/os_egm.md

@@ -241,10 +241,12 @@ def K(
     # Allocate memory for new consumption array
     c_out = np.empty_like(s_grid)
 
-    # Solve for updated consumption value
     for i, s in enumerate(s_grid):
+        # Approximate marginal utility ∫ u'(σ(f(s, α)z)) f'(s, α) z ϕ(z)dz
         vals = u_prime(σ(f(s, α) * shocks)) * f_prime(s, α) * shocks
-        c_out[i] = u_prime_inv(β * np.mean(vals))
+        mu = np.mean(vals)
+        # Compute consumption
+        c_out[i] = u_prime_inv(β * mu)
 
     # Determine corresponding endogenous grid
     x_out = s_grid + c_out        # x_i = s_i + c_i

lectures/os_egm_jax.md

@@ -90,14 +90,16 @@ class Model(NamedTuple):
     α: float              # production function parameter
 
 
-def create_model(β: float = 0.96,
-                 μ: float = 0.0,
-                 s: float = 0.1,
-                 grid_max: float = 4.0,
-                 grid_size: int = 120,
-                 shock_size: int = 250,
-                 seed: int = 1234,
-                 α: float = 0.4) -> Model:
+def create_model(
+        β: float = 0.96,
+        μ: float = 0.0,
+        s: float = 0.1,
+        grid_max: float = 4.0,
+        grid_size: int = 120,
+        shock_size: int = 250,
+        seed: int = 1234,
+        α: float = 0.4
+    ) -> Model:
     """
     Creates an instance of the optimal savings model.
     """
@@ -111,6 +113,17 @@ def create_model(β: float = 0.96,
     return Model(β=β, μ=μ, s=s, s_grid=s_grid, shocks=shocks, α=α)
 ```
 
+
+We define utility and production functions globally.
+
+```{code-cell} python3
+# Define utility and production functions with derivatives
+u = lambda c: jnp.log(c)
+u_prime = lambda c: 1 / c
+u_prime_inv = lambda x: 1 / x
+f = lambda k, α: k**α
+f_prime = lambda k, α: α * k**(α - 1)
+```
 Here's the Coleman-Reffett operator using EGM.
 
 The key JAX feature here is `vmap`, which vectorizes the computation over the grid points.
@@ -135,10 +148,13 @@ def K(
 
     # Define function to compute consumption at a single grid point
     def compute_c(s):
+        # Approximate marginal utility ∫ u'(σ(f(s, α)z)) f'(s, α) z ϕ(z)dz
         vals = u_prime(σ(f(s, α) * shocks)) * f_prime(s, α) * shocks
-        return u_prime_inv(β * jnp.mean(vals))
+        mu = jnp.mean(vals)
+        # Calculate consumption
+        return u_prime_inv(β * mu)
 
-    # Vectorize over grid using vmap
+    # Vectorize and calculate on all exogenous grid points
     compute_c_vectorized = jax.vmap(compute_c)
     c_out = compute_c_vectorized(s_grid)
 
@@ -148,18 +164,6 @@ def K(
     return c_out, x_out
 ```
 
-We define utility and production functions globally.
-
-Note that `f` and `f_prime` take `α` as an explicit argument, allowing them to work with JAX's functional programming model.
-
-```{code-cell} python3
-# Define utility and production functions with derivatives
-u = lambda c: jnp.log(c)
-u_prime = lambda c: 1 / c
-u_prime_inv = lambda x: 1 / x
-f = lambda k, α: k**α
-f_prime = lambda k, α: α * k**(α - 1)
-```
 
 Now we create a model instance.
 
@@ -172,11 +176,13 @@ The solver uses JAX's `jax.lax.while_loop` for the iteration and is JIT-compiled
 
 ```{code-cell} python3
 @jax.jit
-def solve_model_time_iter(model: Model,
-                          c_init: jnp.ndarray,
-                          x_init: jnp.ndarray,
-                          tol: float = 1e-5,
-                          max_iter: int = 1000):
+def solve_model_time_iter(
+        model: Model,
+        c_init: jnp.ndarray,
+        x_init: jnp.ndarray,
+        tol: float = 1e-5,
+        max_iter: int = 1000
+    ):
     """
     Solve the model using time iteration with EGM.
     """

lectures/os_numerical.md

@@ -92,14 +92,14 @@ This is a form of **successive approximation**, and was discussed in our {doc}`l
 The basic idea is:
 
 1. Take an arbitrary initial guess of $v$.
-1. Obtain an update $w$ defined by
+1. Obtain an update $\hat v$ defined by
 
    $$
-       w(x) = \max_{0\leq c \leq x} \{u(c) + \beta v(x-c)\}
+       \hat v(x) = \max_{0\leq c \leq x} \{u(c) + \beta v(x-c)\}
    $$
 
-1. Stop if $w$ is approximately equal to $v$, otherwise set
-   $v=w$ and go back to step 2.
+1. Stop if $\hat v$ is approximately equal to $v$, otherwise set
+   $v=\hat v$ and go back to step 2.
 
 Let's write this a bit more mathematically.
 
@@ -109,7 +109,7 @@ We introduce the **Bellman operator** $T$ that takes a function v as an
 argument and returns a new function $Tv$ defined by
 
 $$
-Tv(x) = \max_{0 \leq c \leq x} \{u(c) + \beta v(x - c)\}
+    Tv(x) = \max_{0 \leq c \leq x} \{u(c) + \beta v(x - c)\}
 $$
 
 From $v$ we get $Tv$, and applying $T$ to this yields
@@ -206,13 +206,7 @@ Here's the CRRA utility function.
 
 ```{code-cell} python3
 def u(c, γ):
-    """
-    Utility function.
-    """
-    if γ == 1:
-        return np.log(c)
-    else:
-        return (c ** (1 - γ)) / (1 - γ)
+    return (c ** (1 - γ)) / (1 - γ)
 ```
 
 To work with the Bellman equation, let's write it as
@@ -240,8 +234,8 @@ def B(
     Right hand side of the Bellman equation given x and c.
 
     """
-    # Unpack
-    β, γ, x_grid = model.β, model.γ, model.x_grid
+    # Unpack (simplify names)
+    β, γ, x_grid = model
 
     # Convert array v into a function by linear interpolation
     vf = lambda x: np.interp(x, x_grid, v)
@@ -250,7 +244,12 @@ def B(
     return u(c, γ) + β * vf(x - c)
 ```
 
-We now define the Bellman operation:
+We now define the Bellman operator acting on grid points:
+
+$$
+    Tv(x_i) = \max_{0 \leq c \leq x_i} B(x_i, c, v)
+    \qquad \text{for all } i
+$$
 
 ```{code-cell} python3
 def T(
@@ -280,7 +279,7 @@ model = create_cake_eating_model()
 β, γ, x_grid = model
 ```
 
-Now let's see the iteration of the value function in action.
+Now let's see iteration of the value function in action.
 
 We start from guess $v$ given by $v(x) = u(x)$ for every
 $x$ grid point.