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econ11_08_hw2_sol

# econ11_08_hw2_sol - Eco11 Fall 2008 Simon Board Economics...

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Eco11, Fall 2008 Simon Board 2. Intertemporal Choice with Differential Interest Rates An agent allocates consumption across two periods. Let the consumption in period t be x t , and the income in period t be m t . The agent’s utility is u ( x 1 , x 2 ) = ln( x 1 ) + 3 4 ln( x 2 ) The agent is poor in period 1 but wealthy in period 2. In particular, she has income m 1 = 3 in period 1 and m 2 = 4 in period 2. (a) Suppose the agent can borrow and save at interest rate r = 1 / 3, so \$1 in period 1 is worth \$(1+1/3) in period 2. Sketch the agent’s budget constraint. Solve for her optimal consumption. (b) Suppose the agent can still save at r = 1 / 3, but can only borrow at r = 1 / 2. Sketch the agent’s budget constraint. Solve for her optimal consumption. (c) Suppose the agent can still save at r = 1 / 3, but can only borrow at r = 1. Sketch the agent’s budget constraint. Solve for her optimal consumption. [Hint: Beware of kinks.] Solution (a) Writing everything in terms of period 1 money, the agent’s budget constraint is m 1 + 3 4 m 2 = x 1 + 3 4 x 2 The tangency condition states that MRS = p 1 /p 2 . That is, 4 3 x 2 x 1 = 4 3 which implies x 1 = x 2 . The LHS of the budget equation is 6. Using this, we obtain x * 1 = x * 2 = 24 / 7. Since x * 1 > 3, the agent borrow in period 1.
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econ11_08_hw2_sol - Eco11 Fall 2008 Simon Board Economics...

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