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Macroeconomics Exam Review 176

Macroeconomics Exam Review 176 - Solutions for Foundations...

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5. Trivially, x is a feasible allocation with endowments w 𝑖 = x 𝑖 and 𝑚 𝑖 = p w 𝑖 = p x 𝑖 . To show that ( p , x ) is a competitive equilibrium, we have to show that x 𝑖 is the best allocation in the budget set 𝑋 𝑖 ( p , 𝑚 𝑖 ) for each consumer 𝑖 . Suppose to the contrary there exists some consumer 𝑗 and allocation y 𝑗 such that y 𝑗 x 𝑗 and py 𝑗 𝑚 𝑗 = px 𝑗 . By continuity, there exists some 𝛼 (0 , 1) such that 𝛼 y 𝑗 𝑖 x 𝑗 and p ( 𝛼 y 𝑗 ) = 𝛼 py 𝑗 < py 𝑗 px contradicting (3.66). We conclude that x 𝑖 𝑖 x 𝑖 for every x 𝑋 ( p , 𝑚 𝑖 ) for every consumer 𝑖 . ( p , x ) is a competitive equilibrium. 3.229 By the previous exercise, there exists a price system p such that x 𝑖 is optimal for each consumer 𝑖 in the budget set 𝑋 ( p , p x 𝑖 ), that is x 𝑖 𝑖 x 𝑖 for every x 𝑖 𝑋 ( p , p x 𝑖 ) (3.67)
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