ARE155FA11HW3 - 1 Managerial Economics(ARE 155 University...

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1 Managerial Economics (ARE) 155 University of California, Davis Fall Quarter, 2011 Instructor: John H. Constantine Homework 3: Due Friday, October 21, 2011 (ONE WEEK AFTER MIDTERM 1) Problems 1 – 4 should be done prior to the midterm. Problem 5 should be done after the midterm. Problem 1 : A furniture firm produces three outputs chairs, tables, and desks. Production of the output requires inputs labor (x 1 ) and wood (x 2 ). The firm’s objective is to minimize the cost of meeting the three production constraints. min TC = 10x 1 + 40x 2 s.t. 2x 1 + 8x 2 40 (chairs) 3x 1 + 9x 2 36 (tables) 8x 1 32 (desks) x 1 , x 2 0 (a) How many primal slack variables are there? Why? (b) How many dual choice variables and dual slack variables are there? Explain briefly. (c) Solve and graph the primal problem given above. State the optimal values for x j and s i , as well as TC. (d) Write the dual LP. (e) Clearly provide an economic interpretation for dual constraint 1. (Remember, in this problem the primal variables (x j ) are inputs and primal constraints are inputs (b 1 ). This changes how to interpret the problem relative to a “max” problem where the x j are outputs and b i are inputs.) (f) Using your results from the Primal problem above, state the all Complementary-Slackness relationships. State the dual variables that are possibly non-zero. (g) Rewrite the dual LP of problem (d), but now only including the potentially non-zero variables as implied by your answers in parts (a) thru (F). (f) Are your dual constraints of part (g) written with equality or inequality constraints? Explain. (g) State the complete dual LP answer by using the answer in part (e). State the optimal values for all (i) dual choice variables, (ii) dual slack variables and (iii) TC. You do not need to graph this problem to solve it. (i) Given the optimal solution to this problem, by how much will the value of the objective function change if b 2 increases from 36 to 37? Explain this answer in the context of the problem of the factory hiring inputs to produce outputs.
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This note was uploaded on 03/14/2012 for the course ARE 155 taught by Professor Staff during the Fall '08 term at UC Davis.

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ARE155FA11HW3 - 1 Managerial Economics(ARE 155 University...

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