Consider a hypothetical overlapping generations economy studies in class. Assume

Question

Consider a hypothetical overlapping generations economy studies in class. Assume n = 0 and r = 0.1. The government oversees a PAYG SS system. It taxes each young in the amount of b = 20 and redistributes the proceeds equally among the old so that each old gets b (1 + n) = 20. Suppose y = 50, y' = 0, U (c, c') = Inc + Inc'. Write down the maximization problem of a representative consumer. Report two conditions that determine the optimal consumption bundle of the consumer. Solve for the optimal consumption bundle. Find the level of utility enjoyed by this consumer. Next consider the same hypothetical economy but with FF SS system in place. The government taxes each you in the amount of b = 20 and invests the proceeds at the rate of r, the gross return in paid out to the same person when he is old. Report two conditions that determine the optimal consumption bundle of the consumer. Solve for the optimal consumption bundle. Find the level of utility enjoyed by this consumer. (You should find that consumers are better off under the FF SS system.) Consider the hypothetical economy with PAYG SS system in place (from part a). In time period 10, the government decides to switch to the FF SS system. It also decides to keep its promise to the old in period 10, i.e. pay them the promised benefits. It will do so by incurring the debt in period 10 and paying it off by taxing future generations in the first period of their lives. Finally, suppose the government makes it a goal to pay off the debt as soon as possible subject to not making any generation worse off under the new system. In which time period will the government pay off its debt (In other words, in which period will consumers become strictly better off under the new system)?

Explanation / Answer

A. According to the information presented, we can write down the maximization problem of a representative consumer as it is shown:

Max Ut = ln c1 + ln c2t+1 where t > 0 U(c, c')

Also, we need to add certain conditions in orden to determine the optimal consumption bundle of the consumer.

s.a. y =50, n=0, b=20, r=0.1

y= c1t + st + b

c2t+1= (1+r)* st - b(1+n)

Now, in order to solve the optimal consumption bundle and find the level of utility enjoyed by this consumer, we proceed to write dwon this ecuations.

Max Ut = ln (y -st -b) + ln ( (1+r)*st -b(1+n) ) =ln(1)

(y -st -b) + ( (1+r)* st - b(1+n) ) = 0

(50 -st - 20) + (1.1)* st - 20 = 0

2st = 10/1.1 ---> St= 4.5

In this sense, we have the following results:

C1t= 50- 4.5 - 20= 25.5

C2t+1= (1.1* 4.5) - 20 = -15.05

Ut = ln c1 + ln c2t+1 = 3.2387

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