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Macroeconomics Exam Review 106

# Macroeconomics Exam Review 106 - Solutions for Foundations...

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3.27 ( 100 010 ) 3.28 We must specify bases for each space. The most convenient basis for ? ? is the T-unanimity games. We adopt the standard basis for ± . With respect to these bases, the Shapley value ± is represented by the 2 ± 1 × ² matrix where each row is the Shapley value of the corresponding T-unanimity game. For three player games ( ² = 3), the matrix is 001 1 2 1 2 0 1 2 0 1 2 0 1 2 1 2 1 3 1 3 1 3 3.29 Clearly, if ³ is continuous, ³ is continuous at 0 . To show the converse, assume that ³ : ´ µ is continuous at 0 . Let ( x ± ) be a sequence which converges to x ´ . Then the sequence ( x ± x )convergesto 0 ² and therefore ³ ( x ± x ) 0 ³ by continuity (Exercise 2.68). By linearity, ³ ( x ± ) ³ ( x )= ³ ( x ± x ) 0 ³ and therefore ³ ( x ± ³ ( x ). We conclude that ³ is continuous at x . 3.30 Assume that ³ is bounded, that is ³ ( x ) ∥≤ x for every x
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