McCall Model

Author

Pablo Winant

Job-Search Model

  • When unemployed in date, a job-seeker
    • consumes unemployment benefit \(c_t = \underline{c}\)
    • receives in every date \(t\) a job offer \(w_t\)
      • \(w_t\) is i.i.d.,
      • takes values \(w_1, w_2, w_3\) with probabilities \(p_1, p_2, p_3\)
    • if job-seeker accepts, becomes employed at rate \(w_t\) in the next period
    • else he stays unemployed
  • When employed at rate \(w\)
    • worker consumes salary \(c_t = w\)
    • with small probability \(\lambda>0\) looses his job:
      • starts next period unemployed
    • otherwise stays employed at same rate
  • Objective: \(\max E_0 \left\{ \sum \beta^t \log(w_t) \right\}\)

What are the states, the controls, the reward of this problem ? Write down the Bellman equation.

Define a parameter structure for the model.

Define a function value_update(V_U::Vector{Float64}, V_E::Vector{Float64}, x::Vector{Bool}, p::Parameters)::Tuple{Vector, Vector}, which takes in value functions tomorrow and a policy vector and return updated values for today.

Define a function policy_eval(x::Vector{Bool}, p::Parameter)::Tuple{Vector, Vector} which takes in a policy vector and returns the value(s) of following this policies forever. You can add relevant arguments to the function.

Define a function bellman_step(V_E::Vector, V_U::Vector, p::Parameters)::Tuple{Vector, Vector, Vector} which returns updated values, together with improved policy rules.

Implement Value Function

Implement Policy Iteration and compare rates of convergence.

Discuss the Effects of the Parameters