QUESTION 4

Consider an economy that is composed of identical individuals who live for two periods.

These individuals have preferences over consumption in periods 1 and 2 given by

U = C C21 .

They receive an income of 100 in period 1 and an income of 50 in period 2. They can save as

much of their income as they like in bank accounts, earning an interest rate r of 10% per

period. They do not leave bequests so they spend all their money before the end of period 2.

Individuals choose consumption in each period by maximizing their lifetime utility subject to

their lifetime budget constraint.

(a) What is the individual's optimal consumption in each period? How much saving does he

do in the first period? Provide a diagram. (4 points)

(b) The government decides to set up a social security system. This system will take $10 from

each individual in period 1, and transfer this money back with 10% interest in period 2.

How does the system affect the amount of private savings? Show this situation in the

diagram carefully labeling all relevant points. (2 points)

(c) Assume now that the social security has a lower implicit rate of return (5%) than the

private return (10%). How would the introduction of the social security system affect the

budget constraint? Provide the amounts for consumption in periods 1 and 2, private

savings and total savings for this individual. Show this new allocation in the diagram. (4

points)

QUESTION 4 Consider an economy that is composed of identical individuals who live for two periods. These individuals have preferences over consumption in periods 1 and 2 given by U = BIC}. They receive an income of 100 in period 1 and an income of Si] in period 2. They can save as much of their income as they like in bank accounts, earning an interest rate r of 10% per period. They do not leave bequests so they spend all their money before the end of period 2. Individuals choose consumption in each period by maximizing their lifetime utility subject to their lifetime budget constraint. (a) What is the individual's optimal consumption in each period? How much saving does he do in the rst period? Provide a diagram. (4 points} (b) The government decides to set up a social security system. This system will take $10 from each individual in period t, and transfer this money back with 10% interest in period 2. How does the system affect the amount of private savings? Show this situation in the diagram carefully labeling all relevant points. {2 points] (c) Assume now that the social security has a lower implicit rate of return {5%} than the private return (10%]. How would the introduction of the social security system affect the budget constraint? Provide the amounts for consumption in periods 1 and 2. private savings and total savings for this individual. Show this new allocation in the diagram. {4 points}