Minor edits to opt savings

lectures/opt_savings_1.md

@@ -56,34 +56,52 @@ $$
 W_{t+1} + C_t \leq R W_t + Y_t 
 $$
 
-We assume that labor income $(Y_t)$ is a discretized AR(1) process.
+where
+
+* $C_t$ is consumption and $C_t \geq 0$,
+* $W_t$ is wealth and $W_t \geq 0$,
+* $R > 0$ is a gross rate of return, and
+* $(Y_t)$ is labor income.
+
+We assume below that labor income is a discretized AR(1) process.
 
-The right-hand side of the Bellman equation is
+The Bellman equation is
 
 $$   
-B((w, y), w', v) = u(Rw + y - w') + β \sum_{y'} v(w', y') Q(y, y'). 
+    v(w) = \max_{0 \leq w' \leq Rw + y}
+    \left\{
+        u(Rw + y - w') + β \sum_{y'} v(w', y') Q(y, y') 
+    \right\}
 $$
 
 where
 
 $$
-u(c) = \frac{c^{1-\gamma}}{1-\gamma} 
+    u(c) = \frac{c^{1-\gamma}}{1-\gamma} 
 $$
 
-## Starting with NumPy
+In the code we use the function
+
+$$   
+    B((w, y), w', v) = u(Rw + y - w') + β \sum_{y'} v(w', y') Q(y, y'). 
+$$
+
+the encapsulate the right hand side of the Bellman equation.
+
 
-Let's start with a standard NumPy version, running on the CPU.
 
-This is a traditional approach using relatively old technologies.
+## Starting with NumPy
+
+Let's start with a standard NumPy version running on the CPU.
 
-Starting with NumPy will allow us to record the speed gain associated with switching to JAX.
+Starting with this traditional approach will allow us to record the speed gain
+associated with switching to JAX.
 
 (NumPy operations are similar to MATLAB operations, so this also serves as a
 rough comparison with MATLAB.)
 
 
 
-
 ### Functions and operators
 
 The following function contains default parameters and returns tuples that
@@ -218,6 +236,8 @@ ax.legend()
 plt.show()
 ```
 
+
+
 ## Switching to JAX
 
 To switch over to JAX, we change `np` to `jnp` throughout and add some
@@ -284,7 +304,6 @@ def B(v, constants, sizes, arrays):
     return jnp.where(c > 0, c**(1-γ)/(1-γ) + β * EV, -jnp.inf)
 
 
-B = jax.jit(B, static_argnums=(2,))
 ```
 
 Some readers might be concerned that we are creating high dimensional arrays,
@@ -295,6 +314,12 @@ Could they be avoided by more careful vectorization?
 In fact this is not necessary: this function will be JIT-compiled by JAX, and
 the JIT compiler will optimize compiled code to minimize memory use.
 
+```{code-cell} ipython3
+B = jax.jit(B, static_argnums=(2,))
+```
+
+In the call above, we indicate to the compiler that `sizes` is static, so the
+compiler can parallelize optimally while taking array sizes as fixed.
 
 The Bellman operator $T$ can be implemented by 
 
@@ -505,14 +530,19 @@ print(jnp.allclose(v_star_vmap, v_star_jax))
 print(jnp.allclose(σ_star_vmap, σ_star_jax))
 ```
 
-Here's how long the `vmap` code takes relative to the first JAX implementation
-(which used direct vectorization).
+Here's the speed gain associated with switching from the NumPy version to JAX with `vmap`:
+
+```{code-cell} ipython3
+print(f"Relative speed = {numpy_elapsed / jax_vmap_elapsed}")
+```
+
+And here's the comparison with the first JAX implementation (which used direct vectorization).
 
 ```{code-cell} ipython3
-print(f"Relative speed = {jax_vmap_elapsed / jax_elapsed}")
+print(f"Relative speed = {jax_elapsed / jax_vmap_elapsed}")
 ```
 
-The execution times are relatively similar.
+The execution times for the two JAX versions are relatively similar.
 
 However, as emphasized above, having a second method up our sleeves (i.e, the
 `vmap` approach) will be helpful when confronting dynamic programs with more