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(The second equation for the value function of the unemployed should be $$v(w,U)= \max \{v(w,E); \,u[c,1]+\beta\int v(w', U) dF(w')\}. \quad (*)$$ ) ...how do you know when your problem solution require more than one Bellman equation? Whenever the state space of the problem contains discrete coordinates, there would be "multiple" value ...