```{code-cell} ipython3

:tags: [hide-output]

!pip install quantecon

!pip install quantecon jax myst-nb

```

## Overview

from myst_nb import glue

```

## The Model

## The model

The model is similar to the {doc}`baseline McCall job search model <mccall_model>`.

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

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.

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

Once we have them in hand, we will be able to make optimal choices.

### The Reservation Wage

### The reservation wage

Let

{eq}`bell2_mccall` and {eq}`bell1_mccall`.

### Set Up

### Set up

The default utility function is a CRRA utility function

```{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.

### Computing the Reservation Wage

### Computing the reservation wage

Now that we can solve for both value functions, let's investigate the reservation wage.

(ast_mcm)=

## A Simplifying Transformation

## A simplifying transformation

The approach above works, but iterating over two vector-valued functions is computationally expensive.

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

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

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.

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.

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 $\alpha$.

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()

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()

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()