E517 Computational Economics Midterm Exam
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E517 Computational Economics
Midterm Exam
Consider a simple model of career and job search. We assume a worker’s wage, wt, can be decomposed into two components: career level (θt) and job quality (et):
wt = θt + et
At the beginning of each period, the worker has a certain career level denoted by θt and job quality denoted by et. The only choice the individual has to make in very period is one of the following options:
• Option A: retain the current career and job; (θt, et) stays unchanged. This option is referred to as “stay put”
• Option B: retain current career, θt, but redraw a new job, et. This option is referred to as “a new job”
• Option C: redraw both a career θt and a job et. This option is referred to as “new life”
The draw of θ and e are independent of each other and their past values. θt is drawn from a distribution denoted by F and et is drawn from a distribution denoted by G. Notice that the worker does not have the option to retain a job but redraw a career – starting a new career always requires starting a new job.
A worker aims to maximize the expected sum of discounted wages
{
E βtwt
=0
subject to the choice restrictions specified above. E is the expectation operator, β is the discount factor, and wt is the wage at time t.
Let v (θ, e) denote the value function, which is the maximum of expected sum of discounted wages given initial state (θ, e). The value function is represented as
v (θ, e) = max）A, B, C_
where
A = θ + e + βv (θ, e)
B = θ + (eo + βv (θ, eo)) G (eo)
o
C = (θo + eo + βv (θo, eo))F (θo) G (eo)
eo θ o
Evidently A, B and C correspond to three possible options individual can choose in every period: “stay put”, “new job” and “new life”, respectively.
Set β = 0.95. θ and e can take values in the set H which consists of 50 evenly distributed grid points between 0 and 5, both inclusive.1 Set the probability of each observation to 
. That is F (θ) = G (e) = 
for every θ and e in the set H .
1. Solve the value function v (θ, e) for every grid point θ and e in the set H. This is a similar problem to the job search model we solved in the class. Here, the value function depends on two objects, θ and e, as opposed to the job search model, which only depends on the current wage.
2. Solving the value function above should give you the policy function: the choice of A, B or C for every pair θ and e. Using this policy function, simulate the economy for 10,000 individuals for 100 periods. Start each individual with state (θ, e) = (0, 0). Report the average wage at the end of 100 periods.
3. Report the average number of periods it takes for each individual to end up with a permanent job, i.e. realization of a pair (θ, e) upon which the individual always chooses to stay put.
2021-12-15