Reference no: EM133706578

The United States has one of the world's lowest savings rates, often near zero in recent years. 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 retirement income. Leaving the first question to macroeconomics, we will examine the second, rendered all the more important by a large wave of impending retirements as baby boomers reach retirement age. How might we ensure people have adequate retirement incomes? Consider the utility function U (x1, x2) = xα 1 + xβ 2 , where we now think x1 as the amount of money spent on current consumption and x2 the amount of money spent on future consumption. A more realistic model would require more periods, but we will think of period 1 as this person's working period and period 2 as retirement. Let the person's income be I1 in period 1 and I2 in period 2. Under ordinary circumstances we would expect I2 to be quite a bit smaller than I1, possibly with I2 = 0. Let the interest rate be r. (a) Write the budget constraint for this person, find the first-order conditions for utility maxi-mization, and find the associated demand functions for x1 and x2.

(b) Under what conditions (i.e., for what values of α, β, I1, I2 and r) is this person a saver? (Your answer will be in the form of an inequality that these variables together must satisfy. If this inequality is complicated, do not worry about simplifying it.) In particular, show that if I2 = 0, this person will compensate for the lack of retirement income by saving for retirement. Let us hereafter simplify by setting I2 = 0. Then, to compute an example, let I1 = 1000 and r = 0.1. Find this person's consumption of goods 1 and 2.

(c) Now let us model a simple social security program. In period 1, the person faces an income tax (in practice made via payroll deductions), with a tax rate of t and hence a total tax of tI1. The proceeds of this tax are invested, and the person receives an income of (1 + r)tI1 in the second period. Let us refer to this as an "investment" social security plan. The original United States' social security plan was an investment plan. Write the new budget constraint facing this consumer and find the new demand functions, and then let t = 0.2 (keeping the other values from (b)) and find this person's optimal consumption of goods 1 and 2.

(d) Compare your answers to (b) and (c). In light of this comparison, what good is an investment social security plan? There have recently been suggestions that people be able to invest their social security contributions in a private investment plan like that modeled in (c). One possible conclusion of your analysis is that such investment social security plans (whether private of public) are of no value. A more nuanced conclusion would be that if an investment social security plan is to be of value, something must be missing from our model (and the model's primary contribution is to tell us something is missing). What do you think it is?

(e) The current social security system is not fully funded, i.e., it is not an investment plan. Instead, it is a "pay-as-you-go" plan, in which taxes from current workers are used to pay benefits to current retirees. Suppose every consumer pays a tax of tI1 in period 1, and receives a benefit of tI1 in period 2. For this to happen, of course, we need to have an economy that always has some people in period 1 of their life and some in period 2. Notice that the period-2 payment is not augmented by investment income, since proceeds are paid out as they come in. Write the utility maximization problem facing an individual and then find this person's consumption of goods 1 and 2.

(f) Compare your answers to (c) and (e). In light of these answers, when might a pay-as-you-go social security system be beneficial? (For example, would it make a difference if the population was growing, so that there are always more period-1 people than period-2 people? This helps explain why countries whose populations are declining face serious challenges.)

(g) Now suppose that we manipulate the interest rate instead of incomes. In particular, suppose we increase r from 0.1 to 0.2. For example, one way of increasing the interest rate (though the magnitude is not as large as in our example) is to provide tax incentives for investing in 401k accounts. Compute this person's optimal consumption of x1 and x2 when r = 0.2, and compare them to the original values. Does this person save more? Do they consume more in period 2? How about period 1?

(h) Could an increase in the interest rate prompt a person to save less for retirement? Use the ideas of income and substitution effects to explain your answer. Given your answers to all of these questions, would you be more inclined to address the issue of retirement savings by a policy that targets incomes or a policy that targets the return to savings?