Question

Beta lives for two periods. In period 1, Beta works and earns a total income of $2, 000. If she consumes $c1 in period 1, then she deposits her savings of 2, 000 − c1 dollars in a bank account that gives her an interest rate of 10% per period. (Notice that Beta is not able to borrow in period 1, so c1 ≤ 2, 000.) In period 2, Beta leads a retired life and receives $110 in social-security income. Beta consumes all her wealth in period 2. Thus, her period 2 consumption c2 = s(1.1) + 110, where s are her period 1 savings.

1. Write down Beta’s budget equation which gives a relation between her period 1 and period 2 consumption.

2. Beta’s lifetime utility u(c1, c2) = √ c1 + 1 1.1 √ c2. Find the (c1, c2) that maximizes Beta’s lifetime utility. How much does Beta save in period 1?

3. Now suppose that Beta does not take into account her period 2 social-security income while making her period 1 consumption and saving decision. The reason for this is that in her mental accounting framework, social-security income is future unearned income. Let’s see how this affects her savings and lifetime utility.

(a) First, write down Beta’s budget equation which gives a relation between her period 1 and period 2 consumption. Hint: In contrast to Question 1, now Beta will not include social-security income in her budget equation.

(b) Now find the (c1, c2) that maximizes Beta’s lifetime utility u(c1, c2) = √ c1 + 1 1.1 √ c2 and satisfies the budget equation that you found in part (a). How much does Beta save in period 1?

(c) What is the impact of not taking social-security income into account on her lifetime utility?

(d) Suppose the government imposes a tax rate of t on savings. Thus, if Beta saves s in period 1, then she can only deposit s(1 − t) in her bank account. Can the government increase Beta’s utility by imposing such a tax on savings?

Answer 1

d)

The post-tax budget will be

........(4)

At equilibrium,

............... (5)

Substituting (5) in (4)

Therefore, for t<1, the utility ut<U1. That is the government can not increase utility by imposing a tax on savings.