Consider an equilibrium search model. Let y be the productivity of a job when it is filled, w be the wage paid to the worker, h be the cost of a vacant job per unit of time, q be the probability of job destruction, θ = V/U be labour market tightness - the ratio of the number of vacant jobs to the number of unemployed persons, M(V,U) be a constant-return-to-scale matching function, and m(θ) = M(1, V/U).
(a) Express the probability of filling a vacant job and the exit rate from unemployment as a function of M, V, and U.