Job Search: The McCall Search Model

Contents

The McCall Search Model

An unemployed worker receives in each period a job offer at wage $ W_t $.

At time $ t $, our worker has two choices:

  1. Accept the offer and work permanently at constant wage $ W_t $.
  2. Reject the offer, receive unemployment compensation $ c $, and reconsider next period.

The wage sequence $ \{W_t\} $ is assumed to be iid with probability mass function $ p_1, \ldots, p_n $.

Here $ p_i $ is the probability of observing wage offer $ W_t = w_i $ in the set $ w_1, \ldots, w_n $.

The worker is infinitely lived and aims to maximize the expected discounted sum of earnings.

$$ \mathbb{E} \sum_{t=0}^{\infty} \beta^t Y_t $$

The discount factor is $ \beta $ lies in $ (0, 1) $. The smaller is $ \beta $, the more the worker discounts future utility relative to current utility.

The variable $ Y_t $ is income, equal to

he worker faces a trade-off:

The Value Function

Let $ V $ be a function that assigns to each possible wage $ w $ the maximal lifetime value that can be obtained with that offer in hand.

$ V $ must satisfy the Bellman Equation:

$$ V(w) = \max \left\{ \frac{w}{1 - \beta}, \, c + \beta \sum_{i=1}^n V(w_i) p_i \right\} \tag{1} $$

for every possible $ w_i $ in $ w_1, \ldots, w_n $, where:

The Optimal Policy

Given any $ w $, we can read off the corresponding best choice (accept or reject) by picking the max on the r.h.s. of (1).

Thus, we have a map from $ \mathbb{R} $ to $ \{0, 1\} $, with 1 meaning accept and zero meaning reject.

We can write the policy as follows

$$ \sigma(w) := \mathbf{1} \left\{ \frac{w}{1 - \beta} \geq c + \beta \sum_{i=1}^n V(w_i) p_i \right\} $$

Here $ \mathbf{1}\{ P \} = 1 $ if statement $ P $ is true and equals zero otherwise.

We can also write this as

$$ \sigma(w) := \mathbf{1} \{ w \geq \bar w \} $$

where

$$ \bar w := (1 - \beta) \left\{ c + \beta \sum_{i=1}^n V(w_i) p_i \right\} \tag{2} $$

Here $ \bar w $ is a constant depending on $ \beta, c $ and the wage distribution, called the reservation wage.

The agent should accept if and only if the current wage offer exceeds the reservation wage.

Clearly, we can compute this reservation wage if we can compute the value function.

Computing the Optimal Policy I

The Algorithm (Interation based on Contraction Mapping)

Define that $v_i:=V(w_i)$, we proceed as follows:

$$ v'_i = \max \left\{ \frac{w_i}{1 - \beta}, \, c + \beta \sum_{i=1}^n v_i p_i \right\} \quad \text{for } i = 1, \ldots, n \tag{4} $$

This algorithm returns an arbitrarily good approximation to the true solution to (3), which represents the value function.

(Arbitrarily good means here that the approximation converges to the true solution as the tolerance goes to zero)

The Theory of Fixed Point

We can show that the mapping $T$ from $\mathbb{R}^n$ to itself defined as: $$ Tv_i = \max \left\{ \frac{w_i}{1 - \beta}, \, c + \beta \sum_{i=1}^n v_i p_i \right\} \quad \text{for } i = 1, \ldots, n \tag{5} $$ is a contraction mapping.

Computing the Value Functions

We assume that the wage offer is drawn from a Beta-Binomial Distribution.

The beta-binomial distribution is the binomial distribution in which the probability of success at each of n trials is not fixed but randomly drawn from a beta distribution.

We can use the function expectation to calculate moments.

Our intial guess $v$ is the value of accepting at every given wage. Below we show the sequence of value functions after 5 iterations.

Now we implement the iteration untile the deviation is quite low.

We can also use the fixedpoint algorithm in the NLsolve package.

Note: The convergence is very senstive to the initial values.

Comparative Statics

We want to know how the reservation wage differs for different parameters $(\beta, c)$

Intuition:

Computing the Optimal Policy Directly

Let $ \psi $ denote the value of not accepting a job in this period but then behaving optimally in all subsequent periods.

That is,

$$ \psi = c + \beta \sum_{i=1}^n V(w_i) p_i \tag{6} $$

where $ V $ is the value function.

By the Bellman equation, we then have

$$ V(w_i) = \max \left\{ \frac{w_i}{1 - \beta}, \, \psi \right\} $$

Substituting this last equation into (6) gives

$$ \psi = c + \beta \sum_{i=1}^n \max \left\{ \frac{w_i}{1 - \beta}, \psi \right\} p_i \tag{7} $$

Which we could also write as $ \psi = T_{\psi}(\psi) $ for the appropriate operator.

This is a nonlinear equation that we can solve for $ \psi $.

One solution method for this kind of nonlinear equation is iterative.

That is,

$$ \psi' = c + \beta \sum_{i=1}^n \max \left\{ \frac{w_i}{1 - \beta}, \psi \right\} p_i \tag{8} $$

Once again, one can use the Banach contraction mapping theorem to show that this process always converges.

The big difference here, however, is that we’re iterating on a single number, rather than an $ n $-vector.

Here’s an implementation using two methods:

Duration of Unemployment

We are also interested in the duration of unemployment in the economy. Now we compute the average duration of unemployment for different $c$s, which in reality can be affected by the government.

Simulation Method:

Mean Duration of Unemployment

Variance of Duration

Distribution of the Duration of Unemployment

References