w06mt_a - GS/ECON 5300 PUBLIC ECONOMICS I Answers to...

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GS/ECON 5300 PUBLIC ECONOMICS I Answers to Midterm Exam February 2006 Q 1. What is the effect on a person’s saving of a tax on the return to saving to the following person? The person lives for two periods, earns all of her income in the first period and has preferences which can be represented by a utility function U ( C 1 , C 2 ) = C 1 + C 2 defined over consumption in the first and second period of her life. Here C 1 is her first–period consumption, and C 2 is her second–period consumption. A 1. First, find a person’s demand function for good 1, if she had an exogenous income of Y , and wanted to maximize U ( C 1 , C 2 ) with respect to C 1 and C 2 , subject to the budget constraint p 1 C 1 + p 2 C 2 = Y . Tangency of her indifference curve requires that her MRS equal the price ratio. Here, since MU 1 = 1 2 C 1 MU 2 = 1 2 C 2 this tangency condition is C 1 C 2 = p 2 p 1 (1 - 1) Equation (1 - 1) implies that C 2 = ( p 1 p 2 ) 2 C 1 so that the person’s budget constraint can be written p 1 C 1 + p 2 ( p 1 p 2 ) 2 C 1 = Y or C 1 = p 2 p 1 Y p 1 + p 2 (1 - 2) Now let period–1 consumption be the num´ eraire. So the present value of her lifetime income, expressed in terms of period–1 consumption is W , the income she earns in period 1. Setting p 1 = 1, since good 1 is the num´ eraire, (1 - 2) becomes C 1 = p 2 W 1 + p 2 (1 - 3) implying that ∂C 1 ∂p 2 = 1 (1 + p 2 ) 2 W > 0 (1 - 4)
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The price of second–period consumption, in terms of first–period consumption, is p 2 = 1 1 + r (1 - τ ) where r is the gross–of–tax rate of return on saving, and τ is the tax rate on the return to saving.
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