Part IV The savings problem and Bewley models Chapter 16 Self-Insurance . Introduction This chapter describes a version of what is sometimes called a savings problem (., Chamberlain and Wilson, 2000). A consumer wants to maximize the expected discounted sum of a concave function of one-period consumption | Part IV The savings problem and Bewley models Chapter 16 Self-Insurance . Introduction This chapter describes a version of what is sometimes called a savings problem . Chamberlain and Wilson 2000 . A consumer wants to maximize the expected discounted sum of a concave function of one-period consumption rates as in chapter 8. However the consumer is cut off from all insurance markets and almost all asset markets. The consumer can only purchase nonnegative amounts of a single risk-free asset. The absence of insurance opportunities induces the consumer to adjust his asset holdings to acquire self-insurance. This model is interesting to us partly as a benchmark to compare with the complete markets model of chapter 8 and some of the recursive contracts models of chapter 19 where information and enforcement problems restrict allocations relative to chapter 8 but nevertheless permit more insurance than is allowed in this chapter. A generalization of the single-agent model of this chapter will also be an important component of the incomplete markets models of chapter 17. Finally the chapter provides our first brush with the powerful supermartingale convergence theorem. To highlight the effects of uncertainty and borrowing constraints we shall study versions of the savings problem under alternative assumptions about the stringency of the borrowing constraint and alternative assumptions about whether the household s endowment stream is known or uncertain. - 540 - The consumer s environment 541 . The consumer s environment An agent orders consumption streams according to E Ptu Ci t 0 where p G 0 1 and u c is a strictly increasing strictly concave twice continuously differentiable function of the consumption of a single good c. The agent is endowed with an infinite random sequence yt 0 of the good. Each period the endowment takes one of a finite number of values indexed by s G S. In particular the set of possible endowments is y1 y2 yS. Elements of the .
