Consider the savings club problem from Example 13.1. Suppose again that Guadalupe earns $30 each week but that the time period is only three weeks. In weeks 1 and 2, instantaneous utility of consumption (for a week of consumption) is given by u(c) = √c, so that marginal utility of instantaneous consumption is given by 0.5c-05. In week 3, Guadalupe has utility given by u3(X) = 0.371X, where X is the amount spent on Christmas gifts (there is no other consumption in week 3). The instantaneous marginal utility of Christmas is given by 0.371. Savings beyond the third period leads to no additional utility. The regular savings account offers an annual interest rate of 5 percent, compounded weekly, whereas placing money in a savings club offers an annual interest rate of r. For the following, it may be useful to use the formulas derived in Example 13.1.
1. Suppose that δ = 0.97. Solve for the optimal consumption and savings decision in each period, supposing that the decision maker has time-consistent preferences. To do this, solve for the amount of consumption in weeks 1 and 2 and the amount of gifts in week 3 that yield equal discounted marginal utilities.
2. Suppose that β = 0.5.. Solve for the optimal savings and consumption decision supposing the decision maker is a naïf.
3. Now suppose that the decision maker is a sophisticate. Solve for the optimal savings and consumption decisions.
4. Finally, suppose that the decision maker is a partial naïf, with β = 0.8.. Now solve for the optimal savings and consumption decisions.
5. Solve for the r that would be necessary to induce the time-consistent decision maker, the naïf, the sophisticate, and the partial naïf to commit to the Christmas club. What is the optimal savings and consumption profile in this case?