A consumer lives for three periods, 0, 1, and 2. She derives utility from consumption in periods 1 and 2 (we are ignoring her consumption in period 0 for the purposes of this problem). Her instantaneous utility from consuming a dollar amount ?1 in period 1 is ?1, and her instantaneous utility from consuming ?2 in period 2 is ?2. The consumer is a naive hyperbolic discounter with ? = 0.5 and ? = 1. The consumer starts off with a wealth of $60 in period 0. She can keep her wealth in a checking account, in which case it earns no interest, and she arrives in period 1 with $60 in cash. Alternatively, she can put $10, $20, or $30 in a retirement account that earns 10 percent interest until period 2. (That is, if she puts $10 into the account in period 0, she gets back $11 in period 2, and so on.) If she puts money in her retirement account in period 0, that money cannot be withdrawn in period 1. Whatever dollar amount she has access to in period 1, she consumes in period 1.

In addition to being able to put money in the retirement account in period 0, the consumer can borrow $10 on her credit card in period 1. If she decides to borrow the $10, she has to pay back $15 in period 2.

(a) How much money does the consumer put in her retirement account in period 0? Does she plan to use her credit card in period 1? Show your work.

(b) Does the consumer borrow on her credit card in period 1? What is her actual consumption pattern ?1 and ?2? Show your work.

(c) Given her actual consumption pattern determined in part (b), if she could commit, what is a different choice she could have made that would give her at least as much consumption in each period?

(d) Discuss how this question relates to the empirical findings of Laibson, Repetto, and Tobacman discussed at the end of Lecture 6.