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# 420KSpring10HW2answer - Homework 2 Spring 08(Total 20...

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Homework 2, Spring 08 (Total 20 points) Q1 (2 × 4): (a) MU 1 ( x 1 , x 2 ) = ∂x 1 (3 ln x 1 +5 ln x 2 ) = 3 x 1 , MU 2 ( x 1 , x 2 ) = ∂x 2 (3 ln x 1 +5 ln x 2 ) = 5 x 2 . MRS ( x 1 , x 2 ) = MU 1 ( x 1 , x 2 ) MU 2 ( x 1 , x 2 ) = 3 x 1 5 x 2 = 3 x 2 5 x 1 (b) From the tangency condition MRS ( x 1 , x 2 ) = p 1 p 2 , we have 3 x 2 5 x 1 = p 1 p 2 , hence x 2 = 5 p 1 3 p 2 · x 1 . Plug this into the budget equation, then p 1 x 1 + p 2 x 2 = p 1 x 1 + p 2 · 5 p 1 3 p 2 · x 1 = 8 p 1 3 · x 1 = m, which yields x 1 = 3 m 8 p 1 . Plug this to the previous formula, then x 2 = 5 m 8 p 2 Summing up, x 1 ( p 1 , p 2 , m ) = 3 m 8 p 1 , x 2 ( p 1 , p 2 , m ) = 5 m 8 p 2 (c): (a) MU 1 ( x 1 , x 2 ) = ∂x 1 x 3 8 1 x 5 8 2 = 3 8 x - 5 8 1 x 5 8 2 , MU 2 ( x 1 , x 2 ) = ∂x 2 x 3 8 1 x 5 8 2 = 5 8 x 3 8 1 x - 3 8 2 . MRS ( x 1 , x 2 ) = MU 1 ( x 1 , x 2 ) MU 2 ( x 1 , x 2 ) = 3 8 x - 5 8 1 x 5 8 2 5 8 x 3 8 1 x - 3 8 2 = 3 x 2 5 x 1 (b): same as above. Warning : Many students calculated MRS by MU 2 MU 1 . Our MRS is defined by the local slope of indifference curve - Δ x 2 Δ x 1 where 1 comes in the denominator and 2 comes in the numerator. But in terms of marginal utilities it

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420KSpring10HW2answer - Homework 2 Spring 08(Total 20...

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