Refactor f and f_prime to use explicit alpha parameter

Changed from closure-based approach to explicit parameter passing:
- f = lambda k, α: k**α (instead of f = lambda k: k**α with global α)
- f_prime = lambda k, α: α * k**(α - 1)
- Updated K operator to call f(s, α) and f_prime(s, α)

This makes the NumPy version consistent with the JAX implementation and
ensures the α stored in the Model is actually used in the K operator
(previously it was unpacked but unused).

Benefits:
- Consistency between NumPy and JAX versions
- Clearer function dependencies (α is an explicit parameter)
- Actually uses model.α instead of relying on closure

Tested successfully with same convergence behavior.

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

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

Author: jstac

lectures/cake_eating_egm.md

@@ -243,7 +243,7 @@ def K(
 
     # Solve for updated consumption value
     for i, s in enumerate(s_grid):
-        vals = u_prime(σ(f(s) * shocks)) * f_prime(s) * shocks
+        vals = u_prime(σ(f(s, α) * shocks)) * f_prime(s, α) * shocks
         c_out[i] = u_prime_inv(β * np.mean(vals))
 
     # Determine corresponding endogenous grid
@@ -266,12 +266,11 @@ First we create an instance.
 
 ```{code-cell} python3
 # Define utility and production functions with derivatives
-α = 0.4
 u = lambda c: np.log(c)
 u_prime = lambda c: 1 / c
 u_prime_inv = lambda x: 1 / x
-f = lambda k: k**α
-f_prime = lambda k: α * k**(α - 1)
+f = lambda k, α: k**α
+f_prime = lambda k, α: α * k**(α - 1)
 
 model = create_model(u=u, f=f, u_prime=u_prime,
                      f_prime=f_prime, u_prime_inv=u_prime_inv)