A consumer has preferences described by
where xt denotes consumption in period t and 0 < d < 1. Assume that the price of consumption is 1 in both periods and that the interest rate is r. If the consumer has income M > 0 in period 1 and no income on period 2, find her optimal level of savings and consumption plan. How is savings affected by changes in the interest rate and the discount factor d? Explain your results.