McCall Model
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