100b_mt1_q258

100b_mt1_q258 - Solutions to Econ 110B Midterm 1 The...

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Unformatted text preview: Solutions to Econ 110B Midterm 1 October 29, 2010 The solutions and grading guidelines that will be used to assess this exam. 1. Question 2 - [15 pts] : We are given that total output Q = 100 and input price for each unit of x 1 is w 1 = 3 and for each unit of x 2 is w 2 = 7 . Thus, the Total Cost is given by: w 1 x 1 + w 2 x 2 . Our optimization problem can be written as, minimizing the cost function (by choosing x 1 and x 2 , such that the output produced is 100 units. min x 1 ,x 2 w 1 x 1 + w 2 x 2 (1) s.t.x 1 x 1 3 2 = Q (2) Setting up a lagrangian yields: L = w 1 x 1 + w 2 x 2- λ [ x 1 x 1 3 2- Q ] (3) Taking the derivatives with respect to x 1 , x 2 and λ yields the following three first order conditions: w 1- λ [ x 1 3 2 ] = 0 (4) w 2- λ [ 1 3 x 1 x- 2 3 2 ] = 0 (5) [ x 1 x 1 3 2- Q ] = 0 (6) By dividing up the first two first order conditions above, we get: x 1 3 2 1 3 x 1 x- 2 3 2 = w 1 w 2 (7) or 3 x 2 x 1 = w 1 w 2 (8) 1 Substituting the values of w 1 (= 3) and w 2 (= 7) yields: x 2 = 1 7 x 1 (9) Substituting this relationship in the third first order condition yields: [ x 1 ( 1 7 x 1 ) 1 3 ] = 100 (10) Calculating for x 1 yields x 1 = 51 . 43 (11) Calculating for x 2 yields x 2 = 1 7 * 51 . 43 = 7 . 34...
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100b_mt1_q258 - Solutions to Econ 110B Midterm 1 The...

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