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Ch.1-Solution - Microeconomics Answer Key Ch.1 Kohsuke...

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Microeconomics Answer Key Ch.1 Kohsuke Kakami 2011 6 1 1.2 The elasticity of substitution σ is given by 𝜎 = 𝑑 ?? ? 2 ? 1 𝑑 ?? TRS . Note that TRS = 𝜕? 𝜕? 1 𝜕? 𝜕? 2 = ? 1 ? 2 ∙ ( ? 1 ? 2 ) 𝜌−1 ? 2 ? 1 = ( ? 2 ? 1 ∙ TRS) 1 1−𝜌 . Taking natural log of both sides, this equation can be written as ?? ? 2 ? 1 = 1 1 − 𝜌 ?? ? 2 ? 1 + 1 1 − 𝜌 ?? TRS , then we have 𝜎 = 𝑑 ?? ? 2 ? 1 𝑑 ?? TRS = 1 1 − 𝜌 . 1.3 𝜀 1 (𝐱) = 𝜕?(𝐱) 𝜕? 1 ? 1 ?(𝐱) = ?? 1 ?−1 ? 2 ? ? 1 ? 1 ? ? 2 ? = ? 𝜀 2 (𝐱) = 𝜕?(𝐱) 𝜕? 2 ? 2 ?(𝐱) = ?? 1 ? ? 2 ?−1 ? 2 ? 1 ? ? 2 ? = ?
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1.5 Note that ?(𝑡? 1 , 𝑡? 2 ) = [(𝑡? 1 ) 𝜌 + (𝑡? 2 ) 𝜌 ] 1 𝜌 = 𝑡(? 1 𝜌 + ? 2 𝜌 ) 1 𝜌 . Then the elasticity of scale is 𝑑?(𝑡? 1 , 𝑡? 2 ) 𝑑𝑡 𝑡 ?(𝑡? 1 , 𝑡? 2 ) | 𝑡=1 = (? 1 𝜌 + ? 2 𝜌 ) 1 𝜌 𝑡 𝑡(? 1 𝜌 + ? 2 𝜌 ) 1 𝜌 | 𝑡=1 = 1 . 1.8 Theorem1. If ?(𝒙): ℛ + 𝑛 → ℛ is a differentiable function that is homogeneous of degree ? ≥ 1 , then 𝜕𝑓 𝜕𝑥 𝑖 (𝐱) is homogeneous of degree k-1. (?𝑟???) Since ? is homogeneous of degree ? , ?(𝑡𝒙) = 𝑡 𝑘 ?(𝒙) holds. Differentiating both sides of this identity with respect to ? 𝑖 , we have 𝜕? 𝜕? 𝑖 (𝑡𝒙) 𝜕𝑡? 𝑖 𝜕? 𝑖 = 𝑡 𝑘 𝜕? 𝜕? 𝑖 𝜕? 𝜕? 𝑖 (𝑡𝒙) = 𝑡 𝑘−1 𝜕? 𝜕? 𝑖 (𝒙). (?. ?. ?) (Proof) Let ?: ℛ → ℛ be a strictly increasing function, ?: ℛ → ℛ be a homogeneous of degree
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