Problems - MC = 8/7q Set MC = MR : 8/7q = 15 q = 7(15)/8...

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Region 1: Ignore K≤7 C(q;P) = MIN 9K + 4L s.t. q≤√KL Therefore: L = q 2 /K Plug into Cost function: C(q;P) = 9K + 4q 2 /K Find K to Minimize Cost: Diff w.r.t. K MC = 9 – 4q 2 /K 2 Set = 0 9 – 4q 2 /K 2 = 0 Solve: 9K 2 = 4q 2 K 2 = 4/9q 2 K = 2/3q Substitute: C(q;P) = 9(2/3q) + 4q 2 /(2/3q) = 6q + 6q = 12q Differentiate to find MC MC = 12 Figure out q where cost curve changes: K = 2/3 q until K = 7 7 = 2/3 q q = 10.5 Note: Because MC = 12, if we were to try to set MC = MR for profit optimization, it would not work (since MR = 15, this would be trying to set 12 = 15, which is impossible) Region 2: Cost function: C(q;P) = Min 9K + 4L s.t. q≤√KL K≤√7 L = q 2 /K and substituting K = 7 : L = q 2 /K L = q 2 /7 Plug L = q 2 /7 into C(q;P): C(q;P) = 9(7) + 4(q 2 /7) = 63 + 4/7q 2 Differentiate (will be used later):
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Unformatted text preview: MC = 8/7q Set MC = MR : 8/7q = 15 q = 7(15)/8 answer to 12b (optimal output) = 13.125 Find cost: C(q = 13.125) = 63 + 4/7(13.125 2 ) = 161.4735 Find profit: answer to 12b(profit) = 15q - C(q;P) =15 (105/8) (63 + 4/7[105/8] 2 ) = 35.4375 Now we can write entire cost function the answer to 12a Region 1 (q10.5) C(q;P) = 12q MC = 12 Region 2 (q10.5) C(q;P) = 63 + 4/7q 2 MC = 8/7q What is the shadow price on K 7? C(q;P) = Min qK + 4L s.t. qKL K k Humor me and pretend italicized k is k bar :o) Want dc/d k (shadow price) C(q;P) =9 k + 4q 2 / k Dc/d k = 9 4q 2 / k 2 At optimal production (q = 13.125, k = 7) Dc/d k = 9 4(13.125 2 /7 2 ) = -5.0625...
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Problems - MC = 8/7q Set MC = MR : 8/7q = 15 q = 7(15)/8...

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