Macroeconomics Exam Review 234

Macroeconomics Exam Review 234 - Solutions for Foundations...

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5.23 Let the dimensions of the vat be 𝑤 × 𝑙 × . We wish to min 𝑤,𝑙,ℎ Surface area = 𝐴 = 𝑤 × 𝑙 + 2 𝑤ℎ + 2 𝑙ℎ subject to 𝑤 × 𝑙 × = 32 The Lagrangean is 𝐿 ( 𝑤, 𝑙, ℎ, 𝜆 ) = 𝑤𝑙 + 2 𝑤ℎ + 2 𝑙ℎ 𝜆𝑤𝑙ℎ. The first-order conditions for a maximum are 𝐷 𝑤 𝐿 = 𝑙 + 2 𝜆𝑙ℎ = 0 (5.86) 𝐷 𝑙 𝐿 = 𝑤 + 2 𝜆𝑤ℎ = 0 (5.87) 𝐷 𝐿 = 2 𝑤 + 2 𝑙 𝜆𝑤𝑙 = 0 (5.88) 𝑤𝑙ℎ = 32 Subtracting (5.45) from (5.44) 𝑙 𝑤 = 𝜆 ( 𝑙 𝑤 ) This equation has two possible solutions. Either 𝜆 = 1 or 𝑙 = 𝑤 But if 𝜆 = 1 /ℎ , (5.44) implies that 𝑙 = 0 and the volume is zero. Therefore, we conclude
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