3. There is widespread concern that Americans save too little. At an aggregate level, a lack of savings can lead to a lack of investment, which can attenuate the growth of the economy. At an individual level, a lack of savings can lead to inadequate re- tirement income. Leaving the first question to macroeconomics, we will examine the second. How might we ensure people have adequate retirement incomes? The answer to this question depends partly on income and substitution effects. Let us start with 3.2 Under what conditions (i.e., for what values of mi, m2 and r) is this agent a saver? the utility function U(g1, (2) = 200c, - ci + 20c2, where we now think of q, as the (Your answer will be in the form of an inequality that these variables together amount of money spent on current consumption and c2 the amount of money spent must satisfy. If this inequality is complicated, do not worry about simplifying it.) on future consumption. A more realistic model would require more periods, but we Let us simplify by setting m2 = 0. Then, to compute an example, let m1 = 150 will think of period one as this person's working period and period two as retirement. and r = 0.1. Find this person's consumption in periods one and two. Let the person's income be m, in period one and my in period two. Under ordinary 3.3 Now let us model a simple social security program. In period one, the person faces circumstances we would expect m2 to be quite a bit smaller than mi, possibly with an income tax (in practice made via payroll deductions), with a tax rate of t and m2 = 0. Let the interest rate be r. hence a total tax of t . m1. The proceeds of this tax are invested, and the person receives an income of (1 + r) . t . mi in the second period (on top of m2). Let us 3.1 Write the budget constraint for this person, find the first-order conditions for refer to this as an "investment" social security plan. (The original United States' utility maximization, and find the associated demand functions for consumption social security plan was an investment plan.) Write the new budget constraint in period one and in period two. facing this consumer and find the new demand functions, and then let t = .2 (keeping the other values from 3.2) and find this person's optimal consumption in period one and two. 3.4 Compare your answers to 3.2 and 3.3. How does the investment social security plan affect this person's utility? Suppose the government could invest money at a rate r, that is different from the market rate r at which the individual can invest. Under what relationship between r and r, can the investment social security plan make the person better off? 3.5 Our current social security system is a mixture of an investment plan and a "pay- as-you-go" plan, in which taxes from current workers are used to pay benefits to current retirees. Let's look at a pure pay-as-you-go plan. Suppose every consumer pays a tax of to mi in period 1, and receives a benefit of B in period 2. Write the utility maximization problem facing an individual and then find this person's consumption in period one and in period two. 3.6 Compare your answers to parts 3.2 and 3.5. For what values of B does the pay-as-you-go plan increase this person's utility? Presumably, we must balance the budget, in the sense that taxes on period-one individuals must suffice to pay benefits to current period-two individuals. Suppose the population is of constant size, so that in each period there are the same number of young and old people. Can we balance the budget and make people better off via the pay-as-you-go social security scheme? What if the population is shrinking? Growing