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 (t):

At the beginning of each period, the worker has a certain career level denoted by θt and job quality denoted by t. 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;t) stays unchanged. This option is referred to as “stay put”

• Option B: retain current career, θt, but redraw a new job, t. This option is referred to as “a new job”

• Option C: redraw both a career θt and a job t. This option is referred to as “new life”

The draw of θ and  are independent of each other and their past values. θt is drawn from a distribution denoted by F and t 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

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;) denote the value function, which is the maximum of expected sum of discounted wages given initial state (θ;). The value function is represented as

v (θ;) = maxfA;B;Cg

where

Evidently A, B and C correspond to three possible options individual can choose in every period:
“stay put”, “new job” and “new life”, respectively.