Problem
Suppose that in the Shapiro-Stieglitz model, unemployed workers are hired according to how long they have been unemployed rather than at random; specifically, suppose that workers who have been unemployed the longest are hired first.
(a) Consider a steady state where there is no shirking. Derive an expression for how long it takes a worker who becomes unemployed to get a job as a function of b, L, N, and L.
(b) Let VU be the value of being a worker who is newly unemployed. Derive an expression for VU as a function of the time it takes to get a job, workers' discount rate (ρ), and the value of being employed (VE).
(c) Using your answers to parts (a) and (b), find the no-shirking condition for this version of the model. (d) How, if at all, does the assumption that the longer-term unemployed get priority affect the equilibrium unemployment rate?