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611-3sol - 316611 Microeconomics Semester 1 2007 Seminar 3...

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316611 Microeconomics Semester 1, 2007 Seminar 3 Yuelan Chen [email protected] March 22, 2007 Please prepare the following questions before the class on 21 March, 2007. Q 1. Derive the Marshallian demand functions from the following utility functions. For all parts, assume positive prices and positive income. (a) u ( x 1 , x 2 ) = x 1 (b) u ( x 1 , x 2 ) = x α 1 x 1 - α 2 , α (0 , 1) (c) u ( x 1 , x 2 ) = min { x 1 , x 2 } (d) u ( x 1 , x 2 ) = max { ax 1 , ax 2 } + min { x 1 , x 2 } , a (0 , 1) Solution. (a) This problem has a corner solution x * = ( y p 1 , 0). To see this, simply draw the budget set and the indifference curves. (b) We solve max x R 2 + , λ 0 L = α ln x 1 + (1 - α ) ln x 2 + λ ( y - p 1 x 1 - p 2 x 2 ) First order conditions are L x 1 = α x 1 - λp 1 = 0 L x 2 = 1 - α x 2 - λp 2 = 0 L λ = y - p 1 x 1 - p 2 x 2 = 0 So the solution is x * = ( αy p 1 , 1 - αy p 2 ). 1
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(c) The solution is x * = ( y p 1 + p 2 , y p 1 + p 2 ). To see this, we first look at the case where x 1 < x 2 . Then u ( x 1 , x 2 ) = min { x 1 , x 2 } = x 1 . But part (a) shows that x * 1 = y p 1 > 0 = x * 2 . So we cannot have x 1 < x 2 in the maximum. By symmetry, we can exclude x 2 < x 1 as a potential solution. This leaves only x 1 = x 2
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