Recall the consumer utility maximization problem between consumption and saving as follows :

maxU = u(c₁) + βu(c₂) + β²u(c₃)

c1,c2

I1 = c1 + s1

I₂ + (1 + r)s₁ = c₂ + s₂

I₃ + (1 + r)s₂ = c₃

(a) Combine those three budget constraints into one life time budget constraint and give interpretation.

(b) Exploiting the optimization result that the marginal rate of intertemporal substitution between ci and cj for all i ̸= j should equal to the price, derive the optimality condition when u(c) = logc so that .

(c) Give interpretation the optimality condition above.

(d) Suppose the real interest rate r increases. What happens to c1, c2, c3, s1 and s2?

(e) Suppose both I1 and I3 increase at the same time, while I1 stays the same. What happens to c1, c2, c3, s1 and s2?