Consider a consumer with a lifetime utility function U(c, c′) = u(c) + βu(c′)
that satisfies all the standard assumptions listed in the book. The period 1 and 2 budget constraints are
c+s=y
c′ +s′ =y′ +(1+r)s
(a) What is the optimal value of s′? 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 point, (c∗, c∗′ )?
(d) Graphically depict the effects of an increase in y′. Carefully label the intercepts of the budget constraint. Is the slope of the indifference curve at the optimal, (c∗, c∗′ ), different from before?