We introduce a preference for wealth into the standard search and matching model to analyze the labor market when there is persistent demand shortage. We show that, under some conditions, a secular stagnation steady state exists in which the economy permanently operates below capacity due to both structural unemployment and underemployment. The latter is a direct consequence of the lack of aggregate demand. Our findings are as follows. In the absence of demand shortage, the preference for wealth creates a new transmission channel for shocks and policy measures due to induced changes in the real interest rate, in addition to the job creation channel of the standard matching model. Turning to the stagnation equilibrium, the effects of demand and supply shocks are opposite to those of the standard case and result in a co-movement of unemployment and underemployment. In contrast, the effects of wage and cost shocks depend on the degree of aggregate demand shortage, but they can explain movements of unemployment and underemployment in opposite directions. Finally, we show that fluctuations in the total employment gap under stagnation are primarily driven by fluctuations in underemployment instead of structural unemployment. Our analysis helps to understand why the unemployment rate in Japan has been surprisingly low during its lost decades and highlights the need for further policy interventions in support of aggregate demand despite a seemingly decent employment record.