Consider an economy that is composed of identical individuals who live for two periods. Individuals maximize: U = 3/5 ln(c1) + 2/5 ln(c2)

where c1 is consumption in each period i. In each period there are N young individuals and N old individuals. Each individual receives an income of $300 in Period 1 (their youth) and no income in Period 2 (their retirement). Individuals do not want to leave behind any money. They can save (or borrow) from Period 1 to Period 2 at interest rate r, and the price of consumption in each period is one.

1. Given the interest rate r, how much can an individual consume in Period 2 if he saves an amount s in Period 1? How much should the individual save if he wants to consume an amount c in Period 2?

2. In Period 1, individuals decide how much to save by maximising their lifetime utility subject to their lifetime budget constraint. The lifetime budget constraint states that the sum of individual Period-1 consumption and Period-1 savings cannot exceed Period-1 income. Write down the individuals maximization problem, and solve to find the optimal personal Period-1 savings (s*). What is the optimal level of consumption in each period?

Suppose the Federal government institutes a social security system. The government takes $50 from each agent in Period 1, saves it, and gives them back to each agent with interest r in Period 2. In Period 1, individuals are still able to privately save (and borrow) at interest rate r.

3. What is e terminology for this social security system? 4. What is the new amount of equilibrium individual savings? 5. What is the effect of this social security system on social welfare?

Suppose that some individuals do not care about their retirement very much, and have utility functions:

Uimpatient = 9/10 ln(c1) + 1/10 ln(c2).

As he subscript indicates, we call these individuals impatient, relative to the patient individuals, whose preferences are represented by the utility function in Equation (1).

Suppose the social security system still forces everyone to save $50.

6. In words, justify the use of the terms patient and impatient to characterise the two types of individuals. 7. In Period 1, how much will these impatient individuals save (or borrow) on top of the required saving of $50 under this social security scheme? In this case, what is the effect of this social security system on social welfare? 8. Suppose that these impatient agents cannot borrow because credit markets are imperfect. In that case, what is the effect of this social security system on the welfare of impatient individuals?