Consider an agent who lives for two periods. His income is Y1 in period 1 and Y2

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Question:

Consider an agent who lives for two periods. His income is Y1 in period 1 and Y2 in period 2. In period 1 he decides how much to consume C1, and how much to save, S. His second period consumption is denoted by C2. Savings are invested and earn interest rate R. The agent’s utility is U(C1, C2) = ln(C1) + β ln(C2) where ln(x) is the natural logarithm of x and β is the agent’s discount factor which measures his impatience (β ∈ (0, 1)).

1. Use the first and second period constraints to derive the agent’s lifetime budget constraint. Maximize the agent’s utility subject to his lifetime budget constraint to find his optimal consumption for both periods and optimal savings as a function of his lifetime income, interest rate and discount factor.

2. The agent’s income is given by Y1 = 160, Y2 = 165. Furthermore, β = 0.9 and R = 0.3. Calculate the agent’s lifetime income, his optimal consumption for both periods C ∗ 1 and C ∗ 2 , his optimal savings S ∗ and his utility level.

3. Draw a graph with period 1 consumption on the horizontal axis and period 2 consumption on the vertical axis. Draw the agent’s lifetime budget constraint on the graph, show his endowment point E and his optimal consumption point C ∗ . Draw indifference curves through both both points.

4. Suppose that the agent loses his job and, as a result, his first period income drops to Y 0 1 = 80. His second period income is not affected because he eventually finds another job (his discount factor β and the interest rate R are also unaffected). Calculate the agent’s new lifetime income W0 and his optimal consumption and savings C ∗ 1 0, C ∗ 2 0, S ∗ 0.

5. In a new graph with period 1 consumption on the horizontal axis and period 2 consumption on the vertical axis, draw the agent’s lifetime budget constraint before and after the change in income. Show his endowment point before and after the income drop (E and E 0 ) and his optimal consumption point before and after the income drop (C ∗ and C ∗ 0).