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?

Now assume that Betty meets Andy, who is currently a student and has no current income at all, but who will be working tomorrow and is guaranteed to have income of w tomorrow. Assume that both Andy and Betty have the same intertemporal discount factor, B, and that they both have the same per-period indirect utility function, which is u(z) = ln(z + 1). Since there is no banking system, all by himself Andy cannot bring any of his future income into the present, but he suggests to Betty that if Betty would give him an amount of money today, s, then from his future income tomorrow he will return to her an amount (1 + r)s.

6. Find the first-order condition for Betty's optimal value of s at any given interest rate r, and find the first-order condition for Andy's optimal value of s at any given interest rate r.

7. Find the signs of the effect of an increase in r upon these two first-order conditions, and sketch them (the two first-order conditions, one for Betty and another for Andy) in a graph with s on the vertical axis and r on the horizontal.

8. A market equilibrium in this model is a level of savings, se, and a level of the interest rate, re, such that both Betty and Andy simultaneously satisfy their first-order conditions. Find the equilibrium interest rate and the equilibrium level of s, and show on your graph how the equilibrium changes with increases in (first) w and (second) B.

9. The market equilibrium that is arrived at above can be easily compared to the social welfare maximising equilibrium. So, now assume that there exists a social planner (i.e. a regulator) who sets r and s so as to maximise the sum of the utilities of Andy and Betty (which is the social welfare function). For this question, go back to the very general case in which the per-period utility function of both Andy and Betty is just u(z), which is increasing and concave in z, and both Andy and Betty continue to discount tomorrow's utility by the common discount factor B. Assume that the social planner sets r and s as an unconstrained maximisation of social welfare. Begin by checking that the social welfare function (the sum of the utilities of Andy and Betty) is concave in each of the two choice variables (r and s), and then find the unconstrained optimal values (those that maximise social welfare). Compare your answer with the market solution found earlier