Macroeconomics Exam Review 239

# Macroeconomics Exam Review 239 - Solutions for Foundations...

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The necessary condition for optimality can be expressed as 𝐷 𝑘 𝑓 ( 𝑘, 𝑙 ) 𝐷 𝑙 𝑓 ( 𝑘, 𝑙 ) = 𝑟 𝑤 𝜆 1 𝜆 𝑠 𝑟 𝑤 whereas the necessary condition for cost minimization is (Example 5.16) 𝐷 𝑘 𝑓 ( 𝑘, 𝑙 ) 𝐷 𝑙 𝑓 ( 𝑘, 𝑙 ) = 𝑟 𝑤 The regulated firm does not use the cost-minimizing combination of inputs. 5.32 The general constrained optimization problem max x 𝑓 ( x ) subject to g ( x ) 0 can be transformed into an equivalent equality constrained problem max x , s 𝑓 ( x ) subject to g ( x ) + s = 0 and s 0 through the addition of nonnegative slack variables s . Letting ˆ g ( x , s ) = g ( x ) + s , the first-order conditions a local optimum are (Exercise 5.16) 𝐷 x 𝑓 ( x ) = 𝜆 𝑗 𝐷 x ˆ 𝑔 𝑗 ( x , s ) = 𝜆 𝑗 𝐷 x 𝑔 𝑗 ( x ) 0 = 𝐷 s 𝑓 ( x ) 𝜆 𝑗 𝐷 s ˆ 𝑔 𝑗 ( x , s ) = 𝝀 (5.98) s 0 𝝀 𝑇 s = 0 (5.99) Condition (5.98) implies that 𝜆 𝑗 0 for every
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