Improve endogenous_lake.md: Split code blocks and add descriptions

Reorganized the endogenous lake model lecture for better readability and comprehension.

**Changes:**
- Split large monolithic code blocks into 12 smaller, focused blocks
- Added descriptive sentences before each code block explaining its purpose
- Separated function definitions from their usage and plotting code
- Added new subsection "Computing optimal unemployment insurance"

**Code block breakdown:**
1. Wage distribution function definition
2. Wage distribution creation and visualization
3. Utility function and McCall model data structure
4. Bellman equation operator
5. Value function iteration solver
6. Lake model functions (from previous lecture)
7. Economy parameters container
8. Function to compute optimal worker quantities
9. Function to compute steady state outcomes
10. Function to find balanced budget tax rate
11. Computation loop across policy range
12. Results visualization

Each block now has a clear explanation of what it does and how it fits into the overall analysis, making the lecture more pedagogically effective.

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

Co-Authored-By: Claude <[email protected]>

Author: jstac

lectures/endogenous_lake.md

@@ -144,7 +144,9 @@ $$
 
 where the notation $V$ and $U$ is as defined in the {doc}`McCall search model lecture <mccall_model>`.
 
-The wage offer distribution will be a discretized version of the lognormal distribution $LN(\log(20),1)$, as shown in the next figure
+The wage offer distribution will be a discretized version of the lognormal distribution $LN(\log(20),1)$.
+
+We first define a function to create a discretized wage distribution:
 
 ```{code-cell} ipython3
 def create_wage_distribution(max_wage: float,
@@ -159,29 +161,28 @@ def create_wage_distribution(max_wage: float,
     p_vec = pdf / pdf.sum()
     w_vec = (w_vec_temp[1:] + w_vec_temp[:-1]) / 2
     return w_vec, p_vec
+```
+
+Let's create a wage distribution and visualize it:
 
+```{code-cell} ipython3
 w_vec, p_vec = create_wage_distribution(170, 200, 20)
 
-# Plot the wage distribution
 fig, ax = plt.subplots()
-
 ax.plot(w_vec, p_vec)
 ax.set_xlabel('wages')
 ax.set_ylabel('probability')
-
 plt.tight_layout()
 plt.show()
 ```
 
 ### Fiscal policy code
 
-We will make use of techniques from the {doc}`McCall model lecture <mccall_model>`
+We will make use of techniques from the {doc}`McCall model lecture <mccall_model>`.
 
-The first piece of code implements value function iteration
+First, we define the utility function and the McCall model data structure:
 
 ```{code-cell} ipython3
-:tags: [output_scroll]
-
 @jax.jit
 def u(c, σ=2.0):
     return jnp.where(c > 0, (c**(1 - σ) - 1) / (1 - σ), -10e6)
@@ -214,8 +215,11 @@ def create_mccall_model(α=0.2, β=0.98, γ=0.7, c=6.0, σ=2.0,
         dist = BetaBinomial(n-1, a, b)
         p_vec = jnp.array(dist.pdf())
     return McCallModel(α=α, β=β, γ=γ, c=c, σ=σ, w_vec=w_vec, p_vec=p_vec)
+```
 
+Next, we implement the Bellman equation operator:
 
+```{code-cell} ipython3
 @jax.jit
 def bellman(mcm: McCallModel, V, U):
     """
@@ -228,8 +232,11 @@ def bellman(mcm: McCallModel, V, U):
     U_new = u(c, σ) + β * (1 - γ) * U + β * γ * (jnp.maximum(U, V) @ p_vec)
 
     return V_new, U_new
+```
 
+Now we define the value function iteration solver:
 
+```{code-cell} ipython3
 @jax.jit
 def solve_mccall_model(mcm: McCallModel, tol=1e-5, max_iter=2000):
     """
@@ -260,7 +267,7 @@ def solve_mccall_model(mcm: McCallModel, tol=1e-5, max_iter=2000):
     return V_final, U_final
 ```
 
-We also need to import the lake model functions from the previous lecture.
+We also need the lake model functions from the previous lecture to compute steady state unemployment rates:
 
 ```{code-cell} ipython3
 class LakeModel(NamedTuple):
@@ -331,8 +338,11 @@ def rate_steady_state(model: LakeModel) -> jnp.ndarray:
     return steady_state
 ```
 
-Now let's compute and plot welfare, employment, unemployment, and tax revenue as a
-function of the unemployment compensation rate
+### Computing optimal unemployment insurance
+
+Now we set up the infrastructure to compute optimal unemployment insurance levels.
+
+First, we define a container for the economy's parameters:
 
 ```{code-cell} ipython3
 class EconomyParameters(NamedTuple):
@@ -359,8 +369,11 @@ def create_economy_params(α=0.013, b=0.0124, d=0.00822,
                            log_wage_mean=log_wage_mean,
                            wage_grid_size=wage_grid_size,
                            max_wage=max_wage)
+```
 
+Next, we define a function that computes optimal worker behavior given policy parameters:
 
+```{code-cell} ipython3
 @jax.jit
 def compute_optimal_quantities(c, τ,
                     params: EconomyParameters, w_vec, p_vec):
@@ -384,8 +397,11 @@ def compute_optimal_quantities(c, τ,
 
     λ = params.γ * jnp.sum(p_vec * (w_vec - τ > w_bar))
     return w_bar, λ, V, U
+```
 
+This function computes the steady state outcomes given unemployment insurance and tax levels:
 
+```{code-cell} ipython3
 @jax.jit
 def compute_steady_state_quantities(c, τ,
                     params: EconomyParameters, w_vec, p_vec):
@@ -407,8 +423,11 @@ def compute_steady_state_quantities(c, τ,
     welfare = e * w + u * U
 
     return e, u, welfare
+```
 
+We need a function to find the tax rate that balances the government budget:
 
+```{code-cell} ipython3
 def find_balanced_budget_tax(c, params: EconomyParameters,
                              w_vec, p_vec):
     """
@@ -436,8 +455,11 @@ def find_balanced_budget_tax(c, params: EconomyParameters,
             t_high = t_mid
 
     return t_mid
+```
 
+Now we compute how employment, unemployment, taxes, and welfare vary with the unemployment compensation rate:
 
+```{code-cell} ipython3
 # Create economy parameters and wage distribution
 params = create_economy_params()
 w_vec, p_vec = create_wage_distribution(params.max_wage,
@@ -460,7 +482,11 @@ for c in c_vec:
     unempl_vec.append(u_rate)
     empl_vec.append(e_rate)
     welfare_vec.append(welfare)
+```
+
+Let's visualize the results:
 
+```{code-cell} ipython3
 fig, axes = plt.subplots(2, 2, figsize=(12, 10))
 
 plots = [unempl_vec, empl_vec, tax_vec, welfare_vec]