1. Consumption-Savings Consider a consumer with a lifetime utility function

U = u(Ct) + u(Ct+1)

that satisfies all the standard assumptions listed in the book. The period t

and t + 1 budget constraints are (t is a subscript).

Ct + St = Yt

Ct+1 + St+1 = Yt+1 + (1 + r)St

(a) What is the optimal value of St+1? Impose this optimal value and derive

the lifetime budget constraint.

(b) Derive the Euler equation. Explain the economic intuition of the equation.

(c) Graphically depict the optimality condition. Carefully label the intercepts of the budget constraint. What is the slope of the indifference

curve at the optimal consumption basket, (C*t ;C*t+1)?

(d) Graphically depict the effects of an increase in Yt+1. Carefully label the

intercepts of the budget constraint. Is the slope of the indifference curve

at the optimal consumption basket, (Ct ;Ct+1), different than in part c?

(e) Now suppose Ct is taxed at rate so consumers pay 1 + for one unit

of period t consumption. Redo parts a-c under these new assumptions.

(f) Suppose the tax rate increases from to '. Graphically depict this.

Carefully label the intercepts of the budget constraint. Is the slope of

the indifference curve at the optimal consumption basket, (C*t ;C*t+1),

different than in part e? Intuitively describe the roles played by the

substitution and income effects. Using this intuition, can you definitively

prove the sign of C*t/ and (C*t+1)/ (derivatives)? It is not necessary to use math for this. Describing it in words is fine.