这个是我们学校的期末考题。就是死在这道题上了。不知道那位高人能为我指点一下迷津！万分的感谢。



Consider an economy that consists of workers, who can be employed (E) or
unemployed (U), and jobs, which can be either filled (F) or vacant (V). Time is
continuous, and we consider a steady state. Each job can have at most one worker.
Vacancies can be created or eliminated freely. There is a cost C per unit of time of
maintaining a job. When a worker is employed, she produces output at rate A > C per
unit of time, and is paid w per unit of time. The labor force is constant at L, L = E + U.
A worker’s utility per unit of time is 0 if unemployed and w if employed. Jobs end at
exogenous rate b per unit of time. Labour market frictions are proxied by the so-called
matching function: unemployment and vacancies yield a flow of new jobs at rate per
unit of time M = M(U,V).
When an unemployed worker and a firm with a vacancy meet, they choose w so that
each gets the same gain. Workers maximize the expected present discounted value of
their lifetime utility. Firms maximize the expected present discounted value of lifetime
profits. The time horizon is infinite. The discount rate for both firms and workers is r. 
(a)  [25 marks] Derive the Bellman equations that characterize the optimal behaviour
of both workers and firms.
(b) [25 marks] What would be the competitive labour market equilibrium in the
absence of frictions? Why does the search/matching model predict unemployment? 
Why does a fall in productivity raise unemployment? [There is no need to solve for
the model’s equilibrium to answer this part.]