Macroeconomics Exam Review 175

# Macroeconomics Exam Review 175 - c 2001 Michael Carter All...

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Assume to the contrary that 𝑆 int 𝑙 = . That is, there exists some z 𝑆 with z < 0 . This implies that there exists some allocation ( x 1 , x 2 , . . . , x 𝑛 ) such that z = 𝑖 x 𝑖 x < 0 and x 𝑖 x 𝑖 for every 𝑖 𝑁 . Distribute z equally to all the consumers. That is, consider the allocation y 𝑖 = x 𝑖 + z /𝑛 By strict monotonicity, y 𝑖 x 𝑖 x 𝑖 for every 𝑖 𝑁 . Since 𝑖 y 𝑖 = 𝑖 x 𝑖 + z = x = 𝑖 x 𝑖 ( y 1 , y 2 , . . . , y 𝑛 ) is a reallocation of the original allocation x which is strictly preferred by all consumers. This contradicts the assumed Pareto eﬃciency of x . We conclude that 𝑆 int 𝑙 = 3. Applying Exercise 3.227, there exists a hyperplane with nonnegative normal p 0 such that p z 0 for every z 𝑆 That is p ( x x ) 0 or p x p x for every x ( x ) (3.65) 4. Consider any allocation which is strictly preferred to x by consumer 𝑗 , that is x 𝑗 ∈ ≻ 𝑗 ( x 𝑗 ). Construct another allocation y by taking 𝜖 > 0 of each commodity away from agent
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