Analytical question

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.
a) Following the hint:

C3=Y3+(1+r₂)S

and

S=Y2-C2+(Y1-C1)(1+r₁)

so the intertemporal budget constraint of the agent is:

C3=Y3+(1+r₂)(Y2-C2)-(1+r₁)(1+r₂)(Y1-C1)

b) The agent's problem is then:

$$max_{\{C1,C2,C3\}} \ln(C1)+\ln(C2)+\ln(C3)$$

subject to:

C3=Y3+(1+r₂)(Y2-C2)-(1+r₁)(1+r₂)(Y1-C1)

or

$$max_{\{C1,C2,C3\}} \ln(C1)+\ln(C2)+\ln(Y3+(1+r)(Y2-C2)-(1+r)^2(Y1-C1))$$

FOCs are:

$$\frac1{C1}=\frac1{C3}(1+r_1)(1+r_2)$$

$$\frac1{C2}=\frac1{C3}(1+r_2)$$

c) Since r₁=r₂=0, the FOCs imply:

C1=C2=C3=C

(permanent income!) so from the intertemporal budget constraint:

C=1+(2-C)+(1-C)

or

$$C=\frac43$$

d) Permanent income implies that when the individual is young income is low (below permanent), and savings are negative. In fact: $Y1-C1=-\frac13$. When the individual is ``mature", income is high (above permanent), and savings are positive: $Y2-C2=\frac23$. When the individual is retired income is again low (below permanent), and savings are negative: $Y3-C3=-\frac13$.
e) In a closed economy people have to consume their resources. So C1=Y1=1, C2=Y2=2, C3=Y3=1. From the FOCs:

$$\frac1{Y1}=\frac1{Y3}(1+r_1)(1+r_2)$$

$$\frac1{Y2}=\frac1{Y3}(1+r_2)$$

so

$$1+r_2=\frac12,$$

and

1+r₁=2.