Analytical question

[50 points]
The setup of the problem is as follows:

• Consider an endowment economy that lasts three periods. Income is given and is equal to Y1=1, Y2=2, Y3=1.
• The agent has logarithmic utility over consumption in periods 1, 2 , and 3:
  
  $$U(C1,C2,C3)=\ln(C1)+\ln(C2)+\ln(C3)$$
  
  

• The economy is open (the agent can borrow and lend from abroad) and interest rate in both periods is given and equal to 0 (r₁=r₂=0) [Note: r₁ is the interest rate between periods 1 and 2. r₂ is the interest rate between periods 2 and 3.].

a) Write the intertemporal budget constraint of the agent. [Hint: remember that in the two period case the intertemporal budget constraint of the agent was:

(1+r)(Y1-C1)=C2-Y2

or, in words, ``what I save today I can consume tomorrow". With three periods, your reasoning may be as follows. In period 3 I can consume my income plus what I saved from the previous two periods, so:

C3=Y3+(1+r₂)S

where S is what I saved from the previous two periods. Now, what I saved from the previous two periods is equal to what I saved in period 2, that is Y2-C2, plus what I saved in period 1 times the interest rate, that is, (Y1-C1)(1+r₁).]
b) Find the first order conditions (note, they are two) of the problem. [Hint: use the intertemporal budget constraint to express, say, C3 as a function of C1 and C2, and then take the first order conditions with respect to C1 and C2.]
c) Find the equilibrium consumption.
d) Interpret the three periods as describing the life cycle of the individual (period 1=`young -low income', period 2=`mature -high income', period 3=` retired-low income'). What does the result you just found say about savings during the life cycle?
e) Now imagine that the economy is closed: the agent cannot borrow or lend. Find the equilibrium interest rates r₁ and r₂.