Macroeconomics Exam Review 70

Macroeconomics Exam Review 70 - Solutions for Foundations...

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2.54 Assume ( 𝑁, 𝑤 ) is convex, that is 𝑤 ( 𝑆 𝑇 ) + 𝑤 ( 𝑆 𝑇 ) 𝑤 ( 𝑆 ) + 𝑤 ( 𝑇 ) for every 𝑆, 𝑇 𝑁 For all disjoint coalitions 𝑆 𝑇 = 𝑤 ( 𝑆 𝑇 ) 𝑤 ( 𝑆 ) + 𝑤 ( 𝑇 ) 𝑤 is superadditive. 2.55 Rewriting (2.18), this implies 𝑤 ( 𝑆 𝑇 ) 𝑤 ( 𝑇 ) 𝑤 ( 𝑆 ) 𝑤 ( 𝑆 𝑇 ) for every 𝑆, 𝑇 𝑁 (2.34) Let 𝑆 𝑇 𝑁 ∖ { 𝑖 } and let 𝑆 = 𝑆 ∪ { 𝑖 } . Substituting in (2.34) 𝑤 ( 𝑆 𝑇 ) 𝑤 ( 𝑇 ) 𝑤 ( 𝑆 ) 𝑤 ( 𝑆 𝑇 ) Since 𝑆 𝑇 𝑆 𝑇 = ( 𝑆 ∪ { 𝑖 } ) 𝑇 = 𝑇 ∪ { 𝑖 } 𝑆 𝑇 = ( 𝑆 ∪ { 𝑖 } ) 𝑇 = 𝑆 Substituting in the previous equation gives the required result, namely 𝑤 ( 𝑇 ∪ { 𝑖 } ) 𝑤 ( 𝑇 ) 𝑤 ( 𝑆 ∪ { 𝑖 } ) 𝑤 ( 𝑆 ) Conversely, assume that 𝑤 ( 𝑇 ∪ { 𝑖 } ) 𝑤 ( 𝑇 ) 𝑤 ( 𝑆 ∪ { 𝑖 } ) 𝑤 ( 𝑆 ) (2.35) for every 𝑆 𝑇 𝑁 ∖ { 𝑖 } . Let 𝑆 and 𝑇 be arbitrary coalitions. Assume 𝑆 𝑇 𝑆 and 𝑆 𝑇 𝑇 (otherwise (2.18) is trivially satisfied). This implies that 𝑇 𝑆 = . Assume these players are labelled 1 , 2 , . . . , 𝑚 , that is 𝑇
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