¹⁰Of course, if a starting population of such consumers were not refreshed by an inflow of new employed consumers, the population unemployment rate would asymptote to 100 percent. This problem can easily be addressed by introducing explicit demographics (which do not affect the optimization problem of the employed): Each period new employed consumers are born and a fraction of existing households dies, as in Carroll and Jeanne (2009). Because demographic effects are very gradual, the implications of the more complicated model are well captured by the simpler model presented here that ignores demographics and the behavior of the unemployed population.