Macroeconomics Exam Review 187

Macroeconomics Exam Review 187 - Solutions for Foundations...

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has no solution 𝑦 ∈ ℜ 𝑚 + . If 𝐴 is the 𝑚 × n matrix whose rows are a 𝑖 , the latter system can be written as 𝐴 𝑇 y = 0 3. By Gordan’s theorem, the system 𝐴 x > 0 (3.78) has a solution ¯ x = 0 . 4. Without loss of generality, we can take ¯ x = 1. (3.78) implies that a 𝑖 ¯ x = ¯ xa 𝑖 > 0 for every 𝑖 = 1 , 2 . . . , 𝑚 so that ¯ x 𝑆 a 𝑖 . Hence ¯ x 𝑚 𝑖 =1 𝑆 a 𝑖 5. We have shown that for every finite set { a 1 , a 2 , . . . , a 𝑚 } ⊆ 𝑆 , 𝑚 𝑖 =1 𝑆 a 𝑖 is non- empty closed subset of the compact set 𝐵 = { 𝑥 ∈ ℜ 𝑛 : x = 1 } . By the Finite intersection property (Exercise 1.116) a 𝑆 𝑆 a = 6. For every p a 𝑆 𝑆 a pa 0 for every a 𝑆 p defines a hyperplane 𝑓 ( a ) = pa which separates 𝑆 from 0 . 3.253 The expected outcome if player 1 adopts the mixed strategy p = ( 𝑝 1 , 𝑝 2 , . . . , 𝑝 𝑚 ) and player 2 plays her 𝑗 pure strategy is 𝑢 ( p , 𝑗 ) = 𝑚 𝑖 =1 𝑝 𝑖 𝑎 𝑖𝑗 = pa 𝑗 where a 𝑗 is the 𝑗 th column of 𝐴 . The expected payoff to 1 for all possible responses of player 2 is the vector ( p 𝐴 ) = 𝐴 𝑇 p . The mixed strategy
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