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# 4-9 - =-50 shadow price on constraint 4 Marginal cost MC 1...

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Do first homework problem in Solver and Min cost/Max profit, Lagrange multipliers Exercise 4-9 Want to minimize cost: C(q;P) = 300K + 250L 1 + 400L 2 +50B Subject to q 1 ≤√KL 1 q 2 ≤√KL 2 q 1 + q 2 ≤ B K ≤ 200 Given: Q 1 = 200 Q 2 = 100 Substitute L 1 = q 1 2 /K, L 2 = q 2 2 /K C(q;P) = 200(300) + 250q 1 2 /K + 400q 2 2 /K+50B Substitute K= 200 C(q;P) = 200(300) + 250q 1 2 /200 + 400q 2 2 /200+50B Substitute B = q 1 +q 2 = 60,000 + (5/4)q 1 2 + 2q 2 2 + 50(q 1 + q 2 ) Find MC 1 and MC 2 : For MC 1 MC 1 = dc/dq 1 = (5/2)q 1 + 50 Substitute q 1 = 200 MC 1 = (5/2)(200) + 50 = 550 For MC 2 MC 2 = dc/dq 2 = 4q 2 + 50 Substitute q 2 = 100 = 4(100) + 50 = 450 Shadow prices: Breakdown of MC 1 : (5/2)q 1 is from production function shadow price = 500 50 is from box constraint shadow price = 50 Breakdown of MC 2 : 4q 2 is from production function shadow price = 400 50 is from box constraint shadow price = 50 What about SP on K=200 constraint? Recall C(q;P) = 300 K +250q 1 2 / K + 400q 2 2 / K + 50(q 1 + q 2 ) Differentiate with respect to K dc/d K = 300 -250q 1 2 / K 2 – 400q 2 2 / K 2 Plug in q 1 = 200, q 2 = 100, K = 200 = 300 – 250 – 400 (100*100)/(200*200)

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Unformatted text preview: = -50 shadow price on constraint 4 Marginal cost MC 1 = 550 MC 2 = 450 C(q;P) = 60,000 + (5/4)q 1 2 + 2q 2 2 + 50(q 1 + q 2 ) K L 1 L 2 B Costs: [Indirect] [--Direct--] [Period] Total Cost, substituting C(q;P) = 60,000 + (5/4)200 2 + 2q 2 2 + 50(200 + q 2 ) q 1 = 200, q 2 = 100 = 60,000 + 50000 + 2*100 2 + 50(300) = 145,000 Allocate Indirect Cost (60,000) to products in proportion to direct labor: Product 1 60000 * 50000/( 50,000+20,000) = 42857 1/7 Product 2 60000 * 20000/( 50000+20000) = 17,142 6/7 Total 60,000 Calculate Unit Cost: Product 1 Product 2 Labor 50,000 20,000 K 42,857 1/7 17,142 6/7 92,857 1/7 37,142 6/7 Unit Cost (divide by q 1 = 200) (q 2 = 100) 464 2/7 371 3/7 Recall MC 550 450 If you’re not satisfied with this cost, you could have added the \$50/box to the unit cost. Product 1 Product 2 464 2/7 371 3/7 50 50 . 514 2/7 421 3/7 Closer to MC ~6% ~5%...
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