Consider an economy where each representative agent lives two periods. In each p

Question

Consider an economy where each representative agent lives two periods. In each period t, there are N_t young individuals and N_t-1 old individuals, where Nt/Nt 1 = 1. Each young individual obtains a quantity I_t of the unique good, which can be used for consumption in period 1 c_yt or saved s_t. In the second period, when the individual is old, h/she obtains a quantity I_t+1 of the good and obtains (1 + r_t) s_t. The old individual uses the proceeds for consumption c_ot+1. Assume the individual maximizes lifetime utility U = ln (c_yt) + ln (c_ot+1), where 0 < < 1. (a) State mathematically the individual’s optimization problem. (b) Solve for the individual’s utility-maximizing path of consumption c_yt and c_ot+1 and of saving st. (c) Will intergenerational transactions take place? Why/why not? Will intragenerational transactions take place? Why/why not?

Explanation / Answer

a)

Consider this problem for each young individual in period t and each old individual in peiord t + 1 separately.

The budget for period t is

It = Cyt

Or

It = St

We assume that this good can either be consumed or saved. If consumed, then the budget for the second period t + 1, when individuals are old, will be:

It+1 = Cot+1

But when they save, this budget becomes:

It+1 = Cot+1 + (1 + rt) St

Nothing is saved                                                                    Everthing is saved

Lifetime budget constraint in this case is:                Lifetime budget constraint in this case is

It + It+1 = Cyt + Cot+1                                                    It+1 + (1 + rt) St = Cot+1

                                                                                    It+1 + (1 + rt) It = Cot+1

Setting Lagrangian for the first condition we have:

Max U = ln (Cyt) + ln (Cot+1) + (It + It+1 – Cyt – Cot+1) when there are no savings

and

Max U = ln (Cyt) + ln (Cot+1) + (It+1 + (1 + rt) It - Cot+1) when everthing is saved.

b)

To solve this equation, set the first order partial derivatives of this equation with respect to X, Y and equal to zero. This implies:

1/ Cyt =

/ Cot+1 =

It + It+1 = Cyt + Cot+1

The first two equations when solved gives:

Cot+1 = Cyt

Substituting this relation in the third and solving that equation gives the optimal solution for intertemporal choice when nothing is saved:

Cyt = (1/1+ )( It + It+1)

Cot+1 = (/1 + )( It + It+1)        

When everything is saved, the optimal values are:

Cyt = (1/)( It+1 + (1 + rt) It)

Cot+1 = It+1 + (1 + rt) It

c)

Intergeneration transfers will not occure given the fact that number of young and old inviduals are same and old consumes everyting they save.

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