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 c1 in period 1 is c1, and her instantaneous utility from consuming c2 in period 2 is c2. In each time period, she seeks to maximize the sum of the discounted utility of consumption in the two time periods. The consumer is a naive hyperbolic discounter with = 0.5 and = 1