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PS8Solutions - Economics 1011b Problem Set 8 Professor Aleh...

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Economics 1011b Problem Set 8 Professor Aleh Tsyvinski The problem set is due next Thursday, May 3rd, by 5 pm in your TF’s mailbox in Littauer. Late problem sets will not be accepted. You can work in groups (discuss the solutions, etc). However, you must write the solutions by yourself. Please write down all derivations/explanations. For clarifications (not answers) please contact Leon Berkelmans ([email protected]). Exercise 1. Asset Pricing Consider a two period economy with a representative household with CES preferences who faces some uncertainty about their period-two income y2: In particular, suppose that there are two equally-likely states of the world s in period 2, s h ; s l . Suppose that income in the high state, s h is 2, and in the low state, s 0 1. Suppose that income in the first period is 1 and that there is no storage technology in the economy. Suppose preferences are: c 1 - σ 1 1 - σ + Ec 1 - σ 2 1 - σ a. What will be the amount of money saved in the first period? Solution: There will be no money saved in the first period because there can be nothing saved in the aggregate (there is no storage technology) b. Suppose there are two assets, a risk free asset and a risky asset. What must the risk free interest rate be? (For this question and the next you need to think very carefully about what the budget constraint(s) must be.) Solution: Suppose that the amount of the risky asset bought by the representative agent is a r and the amount of the riskless asset bought be a b . Let the return on the riskless asset be R b . Let the price of the risky asset be p r , and let the amount the risky asset pays in each state be π ( s )Consumption in the second period must be: c 2 ( s ) = a r π ( s ) + a b R b + Y 2 ( s ) Consumption in the first period must be: c 1 = y 1 - p r a r - a b Inserting this into the utility function gives: ( y 1 - p r a r - a b ) 1 - σ 1 - σ + β E ( a r π ( s ) + a b R b + Y 2 ( s )) 1 - σ 1 - σ 1
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Maximizing this with respect to a b gives: ( y 1 - p r a r - a b ) - σ = R b E ( a r π ( s ) + a b R b + Y 2 ( s )) - σ R b = ( c 1 ) - σ E ( c 2 ( s )) - σ We know that c 1 = y 1 and c 2 ( s ) = y 2 ( s ). Therefore: E ( c 2 ( s )) - σ = 0 . 5 * 1 - σ + 0 . 5 * 2 - σ = 0 . 5 + 0 . 5 * 2 - σ Therefore: R b = 1 - σ 0 . 5 + 0 . 5 * 2 - σ = 1 0 . 5 + 0 . 5 * 2 - σ This means that the risk free interest rate is: r b = R b - 1 = 1 0 . 5 + 0 . 5(2) - σ - 1 = 0 . 5 - 0 . 5(2) - σ 0 . 5 + 0 . 5(2) - σ c. Suppose that one unit of the risky asset pays 0.5 in period s l , and 1 in s h . What is the equity premium?
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