Directed search on the job and the wage ladder

We model a labor market where employed workers search on the job and firms direct workers' search using wage offers and employment probabilities. Applicants observe all offers and face a trade-off between wage and employment probability. There is wage dispersion among workers, even though all workers and jobs are homogeneous. Equilibrium wages form a ladder, as workers optimally choose to climb the ladder one rung at a time. This is because low-wage applicants are relatively more sensitive to employment probability than to wage and thus forgo the opportunity to apply for a high wage, with a lower chance of success.