Consider two agents, A and B, who live for two periods. Their income is Y A 1 and Y B 1 in period 1 and Y A 2 and Y B 2 in period 2. In period 1 each agent decides how much to consume C A 1 and C B 1, and how much to save, S A and S B. Their second-period consumption is denoted by C A 2 and C B 2 . Savings earn interest rate R. The utility of agent A is given by U(C A 1, CA 2 ) = ln(C A 1 ) + β ln(C A 2 ) where ln(x) is the natural logarithm of x and β is the agent’s discount factor which measures his impatience. The utility of agent B is defined similarly. The income of the two agents for the two periods is given by: Y A 1 = 80, Y A 2 = 160, Y B 1 = 132, Y B 2 = 95. The interest rate is R = 0.25. The discount factor is β = 1.

1. Calculate agent A’s lifetime income, his optimal consumption for both periods (C A∗ 1 and C A∗ 2 ), his optimal savings (S A∗ ), and his utility level.

2. Calculate agent B’s lifetime income, his optimal consumption for both periods (C B∗ 1 and C B∗ 2 ), his optimal savings (S B∗ ), and his utility level.

3. Which agent saves more? Explain how the difference in the timing of income affects each agent’s saving/borrowing behavior.

4. Draw a graph with period 1 consumption on the horizontal axis and period 2 consumption on the vertical axis. Draw the lifetime budget constraints of each agent and denote their endowment points (E A and E B) and their optimal consumption points (C A∗ and C B∗ ). Draw indifference curves through the agents’ consumption points and describe which agent is better off.

5. Suppose that credit markets are subject to severe credit constraints and that agents cannot borrow at all: S ≥ 0. What is the maximum first-period consumption that is feasible for agent A? What is the maximum first-period consumption that is feasible for agent B?

6. Draw a new graph for each agent with period 1 consumption on the horizontal axis and period 2 consumption on the vertical axis. On the graph, show the new budget line (after credit constraints) and the old budget line (before credit constraints). Denote the consumption point that you derived in the previous question on the new graph for each agent. Is this consumption point feasible for agent A? Is it feasible for agent B?

7. Calculate the consumption and savings under credit constraints for agent A (CˆA 1 , CˆA 2 , SˆA) and agent B (CˆB 1 , CˆB 2 , SˆB). Calculate the utility of the two agents and describe which agent is better off. Compare with your response to question 4 and comment on any difference.

8. The government wants to stimulate consumption to help economic recovery. For this reason, it provides a 10-pound rebate to each agent. Calculate the change in the first-period consumption of agent A and agent B. Which agent’s consumption increases more?

9. The economy consists of 2,000 people: 1,000 whose income is like agent-A’s and 1,000 agents whose income is like agent Bs. The total cost of the tax rebate is 20,0000 pounds (10 pounds per person). Calculate the total increase in consumption as a result of the tax rebate. Could the government have generated a greater increase in consumption for the same cost? Explain.

10. OPTIONAL QUESTION (for the intrepid). As we discussed in class, an increase in the interest rate might make an agent better off or worse off. Provide a condition on the timing of the agent’s income such that the agent’s utility increases in response to an increase in R. [Hint: introduce the optimal consumption decision in the utility function, differentiate with respect to R and provide a condition such that the derivative is positive, i.e. utility increases in R. ]