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retakeprelim2010 - Preliminary Examination in...

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Preliminary Examination in Microeconomics August 25, 2010 Question 1 Harry consumes just one commmodity and he will live for T periods. His cur- rent preferences over consumption streams are represented by a utility function of the form U ( x 1 , . . . , x T ) = T X t =1 β t u ( x t ) where x t is the amount of the commodity that he will consume in year t and where the function u ( · ) is strictly concave and twice continuously differentiable. Harry knows that his income stream will be ( w 1 , . . . , w T ) where w t is the income that he will receive in period t . Harry is able to borrow or lend at the constant interest r . At time 1, Harry is able to commit himself to any time path of consumption that satisfies his budget constraint. His budget constraint is that the present value of his lifetime consumption must not exceed the present value of his lifetime income stream. Part 1) Suppose that for some α where 0 < α < 1, and for all t = 1 , . . . T , β t = α t - 1 . At what interest rate will Harry will choose to consume the same amount of goods in every period of his life? Explain why your answer is correct.
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