A significant fraction of the labor force consists of employed workers who are part-timeunemployed (underemployed) in the sense that they are unable to work as much as theyprefer. This paper develops a search and matching model to study the design of optimalunemployment insurance in an economy with unemployment as well as part-timeunemployment. Part-time unemployment provides income insurance and serves as astepping stone to full-time jobs. Unemployment benefits for part-timers increase the outflowfrom unemployment to part-time work but reduce the outflow from part-time work to full-timeemployment. We examine the optimal structure of benefits for unemployed andunderemployed workers. The results indicate non-negligible welfare gains associated withtime limits for unemployment benefits as well as for part-time benefits. The welfare gainsfrom optimal UI are larger when wages are fixed than when they are flexible....