Betty lives alone in an economy with no banking system. She has income today of w, but she must live for two periods of time, today and tomorrow. Tomorrow she will be retired, and will have no income at all. She can save s dollars, where 0 s w, from her current wealth for consumption in the next period, but since there is no banking system, savings generates no interest payment at all. Assume her indirect utility function for a given amount of wealth z is u(z), and it is the same in each period of time, it is increasing and strictly concave, and it satisfies u(0) = 0. Further, Betty discounts tomorrow's utility by a discount factor B which satisfies 0 < B < 1. Her overall utility is the sum of her present utility and her discounted future utility.

1. Write down Betty's optimal savings problem as a constrained optimisation problem, and find the first-order condition for her optimal amount of savings, s*.

2. Can it ever be optimal for Betty to save all her current wealth, s* = w? Explain why or why not.

3. Can it ever be optimal for Betty to save nothing, s* = 0? Explain why or why not.

4. How is Betty's optimal savings choice affected by an increase in B? How is her savings choice affected by an increase in w?

5. If Betty's utility function is u(z) = ln(z+ 1), what is her exact level of optimal savings?