Consider an economy where each representative agent lives two periods. In each p
ID: 1200067 • Letter: C
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.