[mccall_model_with_separation] Update Exercises into Discussions (#701)

* update exercises

* Update lectures/mccall_model_with_separation.md

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>

* revert changes but add stylesheet compliance enforcement

---------

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>

Author: HumphreyYang

lectures/mccall_model_with_separation.md

@@ -34,7 +34,7 @@ In addition to what's in Anaconda, this lecture will need the following librarie
 ```{code-cell} ipython3
 :tags: [hide-output]
 
-!pip install quantecon
+!pip install quantecon jax myst-nb
 ```
 
 ## Overview
@@ -65,7 +65,7 @@ from quantecon.distributions import BetaBinomial
 from myst_nb import glue
 ```
 
-## The Model
+## The model
 
 The model is similar to the {doc}`baseline McCall job search model <mccall_model>`.
 
@@ -94,7 +94,7 @@ Wage offers $\{ W_t \}$ are IID with common distribution $q$.
 
 The set of possible wage values is denoted by $\mathbb W$.
 
-### Timing and Decisions
+### Timing and decisions
 
 At the start of each period, the agent can be either
 
@@ -118,7 +118,7 @@ The process then repeats.
 We do not allow for job search while employed---this topic is taken up in a {doc}`later lecture <jv>`.
 ```
 
-## Solving the Model
+## Solving the model
 
 We drop time subscripts in what follows and primes denote next period values.
 
@@ -135,7 +135,7 @@ the worker makes optimal decisions at all future points in time.
 As we now show, obtaining these functions is key to solving the model.
 
 
-### The Bellman Equations
+### The Bellman equations
 
 We recall that, in {doc}`the original job search model <mccall_model>`, the
 value function (the value of being unemployed with a given wage offer) satisfied
@@ -200,7 +200,7 @@ enough information to solve for both $v_e$ and $v_u$.
 Once we have them in hand, we will be able to make optimal choices.
 
 
-### The Reservation Wage
+### The reservation wage
 
 
 Let
@@ -234,7 +234,7 @@ Let's now implement a solution method based on the two Bellman equations
 {eq}`bell2_mccall` and {eq}`bell1_mccall`.
 
 
-### Set Up
+### Set up
 
 The default utility function is a CRRA utility function
 
@@ -310,8 +310,8 @@ than some small tolerance level.
 ```{code-cell} ipython3
 def solve_full_model(
         model,
-        tol: float=1e-6,
-        max_iter: int=1_000,
+        tol: float = 1e-6,
+        max_iter: int = 1_000,
     ):
     """
     Solves for both value functions v_u and v_e iteratively.
@@ -337,7 +337,7 @@ def solve_full_model(
 
 
 
-### Computing the Reservation Wage
+### Computing the reservation wage
 
 Now that we can solve for both value functions, let's investigate the reservation wage.
 
@@ -390,7 +390,7 @@ This value seems close to where the two lines meet.
 
 
 (ast_mcm)=
-## A Simplifying Transformation
+## A simplifying transformation
 
 The approach above works, but iterating over two vector-valued functions is computationally expensive.
 
@@ -428,7 +428,7 @@ useful.
 But we can go further, but eliminating $v_e$ from the above equation.
 
 
-### Simplifying to a Single Equation
+### Simplifying to a single equation
 
 As a first step, we rearrange the expression defining $h$ (see {eq}`defh_mm`) to obtain
 
@@ -485,7 +485,7 @@ If we can solve this for $h$, we can easily recover $v_e$ using
 Then we have enough information to compute the reservation wage.
 
 
-### Solving the Bellman Equations
+### Solving the Bellman equations
 
 To solve {eq}`bell_scalar`, we use the iteration rule 
 
@@ -595,11 +595,11 @@ However, the simplified method is far more efficient.
 Next we will investigate how the reservation wage varies with parameters.
 
 
-## Impact of Parameters
+## Impact of parameters
 
 In each instance below, we'll show you a figure and then ask you to reproduce it in the exercises.
 
-### The Reservation Wage and Unemployment Compensation
+### The reservation wage and unemployment compensation
 
 First, let's look at how $\bar w$ varies with unemployment compensation.
 
@@ -615,7 +615,7 @@ As expected, higher unemployment compensation causes the worker to hold out for
 
 In effect, the cost of continuing job search is reduced.
 
-### The Reservation Wage and Discounting
+### The reservation wage and discounting
 
 Next, let's investigate how $\bar w$ varies with the discount factor.
 
@@ -629,7 +629,7 @@ $\beta$
 
 Again, the results are intuitive: More patient workers will hold out for higher wages.
 
-### The Reservation Wage and Job Destruction
+### The reservation wage and job destruction
 
 Finally, let's look at how $\bar w$ varies with the job separation rate $\alpha$.
 
@@ -682,7 +682,7 @@ w_bar_vals = jax.vmap(compute_res_wage_given_c)(c_vals)
 
 fig, ax = plt.subplots()
 ax.set(xlabel='unemployment compensation', ylabel='reservation wage')
-ax.plot(c_vals, w_bar_vals, label=r'$\bar w$ as a function of $c$')
+ax.plot(c_vals, w_bar_vals, lw=2, label=r'$\bar w$ as a function of $c$')
 ax.legend()
 glue("mccall_resw_c", fig, display=False)
 plt.show()
@@ -700,7 +700,7 @@ w_bar_vals = jax.vmap(compute_res_wage_given_beta)(β_vals)
 
 fig, ax = plt.subplots()
 ax.set(xlabel='discount factor', ylabel='reservation wage')
-ax.plot(β_vals, w_bar_vals, label=r'$\bar w$ as a function of $\beta$')
+ax.plot(β_vals, w_bar_vals, lw=2, label=r'$\bar w$ as a function of $\beta$')
 ax.legend()
 glue("mccall_resw_beta", fig, display=False)
 plt.show()
@@ -718,7 +718,7 @@ w_bar_vals = jax.vmap(compute_res_wage_given_alpha)(α_vals)
 
 fig, ax = plt.subplots()
 ax.set(xlabel='separation rate', ylabel='reservation wage')
-ax.plot(α_vals, w_bar_vals, label=r'$\bar w$ as a function of $\alpha$')
+ax.plot(α_vals, w_bar_vals, lw=2, label=r'$\bar w$ as a function of $\alpha$')
 ax.legend()
 glue("mccall_resw_alpha", fig, display=False)
 plt.show()