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 = ln(C1) + ln(C2). 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 of 10% per period. They do not care about their children, so they spend all their money before the end of period 2. Each individual’s lifetime budget constraint is given by C1 + C2/(1 + r) = Y1 + Y2/(1 + r). Individuals choose consumption in each period by maximizing lifetime utility subject to this lifetime budget constraint.
What is the individual’s optimal consumption in each period? How much saving does he or she do in the first period?
Now the government decides to set up a social security system. This system will take $10 from each individual in the first period, put it in the bank, and transfer it to him or her with interest in the second period. Write out the new lifetime budget constraint. How does the system affect the amount of private savings? How does the system affect national savings (total savings in society)? What is the name for this type of social security system?

Explanation:

we calculate lifetime income,

= 100 + 50 = 150

individual consumes income in periods c1 and c2 and interest rate on savings is 10%,

we define the consumption basket;

c1 = 100

c2 = 50

i = 10% = 0.1

c1 + c2/(1+i) = y1 + y2/(1+i) = 100 + 50/1.1 = 100+45.45 = 145.45

c2 = (145.45 - c1) x 1.1

we have MUc1 = 1/c1

and MUc2 = 1/c2

we have to equate the ratio of marginal utilities with the prices of consumption of periods c1 and c2

c2/c1 = 1+i = 1.1

c2 = 1.1 x c1 (2)

we have to put  c2 into c1, we get

1.1 x c1 = (145.45 - c1) x 1.1

1.1c1 = (145.45- c1)1.1

divide through by 1.1

c1 = 145.45 - c1

c1+c1 = 145.45

2c1 = 145.45

to get value of c1;

c1 = 145.45/2 = 72.73

since c1 is known

c2 = 1.1 x 72.73 = 80

b. if government takes $10 from period 1 and add it to the income of a consumer in period 2,  we then have  income of individual to be;

Y1 = 100-10 = 90

Y2 = 50 + 10 + (10% of 10)

10%x10 = 1,

y2 = 60+1 = 61

c1 + c2/1.1 = 90 + (61/1.1)

= 90 + 55.45

= 145.45

c2 = (145.45 - c1) x 1.1

equilibrium values will be unchanged, apart fro the fact that $10 is a compulsry savings of an individual therefore private savings falls

100 - 72.73-10 = 17.27

This type of savings is called as Providend fund

answer; (c) run the name by everyone you can;

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