1.4 Problem 4: Two-period model with depreciation Maxine lives for two periods. Each period, she receives an endowment of consumption goods: e1 in the first, e2 in the second. She does not have to work for this output. Her preferences for consumption in the two periods are given by: u(c1, c2) = ln c1 + ? ln c2, where c1 and c2 are her consumptions in periods 1 and 2, respectively, and ? is some discount factor between zero and one. She is able to save some of her endowment in period 1 for consumption in period 2. Call the amount she saves s. Maxine's savings get invaded by rats, so if she saves s units of consumption in period 1, she will have only (1 - ?)s units of consumption saved in period 2, where ? is some number between zero and one. (a) Write down Maxine's maximization problem. (You should show her choice variables, her objective, and her constraints.) (b) Solve Maxine's maximization problem. (This will give you her choices for given values of e1, e2, ?, and ?.) (c) How do Maxine's choices change if she finds a way reduce the damage done by the rats? (You should use calculus to do comparative statics for changes in ?.)
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Her choice variables are \(c_1\) and \(c_2\), or equivalently, \(c_1\) and \(s\) (since \(c_2\) is determined by how much she saves and the depreciation caused by rats). Maxine's constraints are: Show more…
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