5. Consider an economy that s composed ofidentical individuals who live for two

Question

5. Consider an economy that s composed ofidentical individuals who live for two periods. Preterences over consumption in periods I and 2 that is given by of U(c1,Cz) = 1n(q) + In(ca). save as much ves an income of 100 in period 1 and an income of 50 in period 2. They come as they like in bank accounts that earn 10% interest. Individuals do so they spend all their money before the end of period 2. not care about their children spen Individuals choose consumption in each period subject to their lifetime budget a) What is an individual's optimal constraint. consumption in each period? How much saving does he or she do in the first period? b) Now the government decides to set up a social security system. This system will take $10 from each individual i her with interest in the second peri const od. Write out an individual's new lifetime budget raint. How does this system affect the amount of private saving that occurs in the economy? How does the system affect national savings (total saving throughout the economy)? Now suppose that the existence of the new social security program c) causes an individual to retire in period 2, so he or she receives no labor income in period 2. Solve for this individual's new optimal consumption in each period. d) What is the new level of private and national saving? Does this differ from the level of saving you found in part (b)? If so, why? (Explain intuitively.)

Explanation / Answer

U(c1,c2)=ln(c1)+ln(c2) [Given].

Each individual receives an income of 100 in period 1 and 50 in period 2.

Each individual chooses consumption in each period subject to their life time budget constraint.

(a) Each individual lifetime budget constraint is given by: c1+ c2/(1+r) = Y1+ Y2/(1+r).

Individuals choose consumption in each period by maximising lifetime utility subject to this lifetime budget constraint.

Max U = ln(c1) + ln(c2) subject to c1 + c2/(1+ 0.1) = 100+50/1.1 [interest rate is given -10%]

Rearrange the budget constraint will give : c2= 110 + 50-1.1 c1 . Now put this into utility maximisation function will give : Max U = ln(c1) + ln (160 - 1.1 c1).

Now take the derivative and set it equal to zero : 1/c1 = 1.1/(160 - 1.1 c1) or 2.2 c1 = 160.

So, c1 = 72.7 and savings= Y1 - c1= 100- 72.7 = 27.3.

c2 = 50+ 1.1(100 - c1) = 80.

Hence optimal consumption in each period is c1= 72.7 and c2= 80.

and savings in ist period is equal to 27.3.

(b) Now the government decides to set up a social security system . This system will take $10 take dollars from each person and put in a financial institution then tranfer it back with interest rate in the second period.

so their new lifetime budget constraint:

c1 + 10 + s = 100 = Y1 and

s= 90 - c1 and

c2 = 1.1(s+ 10) + Y2= 1.1(100 - c1) + 50

c2 = 160 - 1.1 c1

c2 + 1.1c1 =160 [ new budget constraint]

This social security system affect the amount of private savings negatively that occurs in the economy. Private savings in the economy decreases.

Private savings = s = 100 - 10 - c1 = 17.27

Social savings= s + 10 = 27.273.

(c) New optimal consumption in period 1

c1 + 10 + s = 100 = Y1

s= 90 - c1

c2 = 1.1(s + 10 ) + Y2 (and Y2 = 0)

now, c2 = 1.1 (100 - c1)

Now c2= 50 and c1= 45.45 (optimal consumption in each period)

(d) New level of private saving = 100 - 10 - c1 = 40

This is differs from earlier level of savings.

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