A household lives for two periods, has a natural logarithm utility function, and discounts the future at the rate of . Borrowing is not allowed. In period 1, the young household consumes c1, saves s1, and works for wage w1. In period 2, the old household consumes c2, receives gross return from saving (interest rate is r), and faces unemployment risk. With a probability p, the household keeps his job and earns a wage w2, and with a probability of 1 p, the household loses his job. The government taxes all employed workers (both young and old) at a proportional rate of and evenly transfers to all unemployed workers. The population is fixed.

Write down the maximization problem of the household.