HW5Sol - Fall 2010 IEOR 160 Industrial Engineering &...

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Unformatted text preview: Fall 2010 IEOR 160 Industrial Engineering & Operations Research October 2, 2010 Page 1 of 3 HOMEWORK 5 SOLUTIONS Chapter 12.4 7. Let T be the tax rate and e be the individual effort level. Given T , individual chooses e to maximize f ( e ) = 10 √ e (1- T )- e . f ( e ) = 5 √ e (1- T )- 1 = 0 for e = 25(1- T ) 2 . Let r ( t ) be the govt. revenue when tax rate is T . Then r ( t ) = 10 T p 25(1- T )2 = 50 T (1- T ) which is maximized for T = 0 . 5. Chapter 12.8 4. L ( x 1 ,x 2 ,λ ) = 30 x 1 / 2 1 + 20 x 1 / 2 2- x 1- x 2 + λ (100- x 1- x 2 ). Hence (1) ∂L ∂x 1 = 15 x- 1 / 2 1- 1- λ = 0 (2) ∂L ∂x 2 = 10 x- 1 / 2 2- 1- λ = 0 (3) ∂L ∂λ = 100- x 1- x 2 = 0 From (1) and (2), x 1 = 225 / ( λ +1) 2 and x 2 = 100 / ( λ +1) 2 . Then (3) yields 325 / ( λ +1) 2 = 100 or ( λ + 1) 2 = 3 . 25 and λ = 0 . 80. Now we find that x 1 = 225 / 3 . 25 = 900 / 13 = 69 . 23 and x 2 = 400 / 13 = 30 . 77. If an extra dollar can be spent on promotion the objective function (profit) increases by...
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This note was uploaded on 10/31/2010 for the course IEOR 41027 taught by Professor Glassey during the Fall '10 term at University of California, Berkeley.

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HW5Sol - Fall 2010 IEOR 160 Industrial Engineering &...

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