¹²Note that Deaton’s condition is stronger than the one required for the problem to define a contraction mapping. This reflects a subtle distinction: If the weaker condition (6) is imposed, but Deaton’s stronger condition (11) is not satisfied, then a consumer who begins the period with zero resources will choose to save some strictly positive amount. In this case, zero wealth is NOT an absorbing state, and the target asset-to-permanent-income ratio is actually positive.