Macroeconomics Exam Review 112

# Macroeconomics Exam Review 112 - c 2001 Michael Carter All...

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where 𝜒 { 𝑠 } is the indicator function of the set { 𝑠 } . Since 𝐸 is linear 𝐸 ( 𝑋 ) = 𝑠 𝑆 𝑋 ( 𝑠 ) 𝐸 ( 𝜒 { 𝑠 } ) = 𝑠 𝑆 𝑝 𝑆 𝑋 ( 𝑠 ) since 𝐸 ( 𝜒 { 𝑠 } = 𝑃 ( { 𝑠 } ) = 𝑝 𝑠 0. For the random variable 𝑋 = 1 , 𝑋 ( 𝑠 ) = 1 for every 𝑠 𝑆 and 𝐸 ( 1 ) = 𝑠 𝑆 𝑝 𝑆 = 1 3.44 Let 𝑥 1 , 𝑥 2 𝐶 [0 , 1]. Recall that addition in C[0,1] is defined by ( 𝑥 1 + 𝑥 2 )( 𝑡 ) = 𝑥 1 ( 𝑡 ) + 𝑥 2 ( 𝑡 ) Therefore 𝑓 ( 𝑥 1 + 𝑥 2 ) = ( 𝑥 1 + 𝑥 2 )(1 / 2) = 𝑥 1 (1 / 2) + 𝑥 2 (1 / 2) = 𝑓 ( 𝑥 1 ) + 𝑓 ( 𝑥 2 ) Similarly 𝑓 ( 𝛼𝑥 1 ) = ( 𝛼𝑥 1 )(1 / 2) = 𝛼𝑥 1 (1 / 2) = 𝛼𝑓 ( 𝑥 1 ) 3.45 Assume that x = x 1 + x 2 + ⋅ ⋅ ⋅ + x 𝑛 maximizes 𝑓 over 𝑆 . Suppose to the contrary that there exists y 𝑗 𝑆 𝑗 such that 𝑓 ( y 𝑗 ) > 𝑓 ( x 𝑗 ). Then y = x 1 + x 2 + ⋅ ⋅ ⋅ + y 𝑗 + ⋅ ⋅ ⋅ + x 𝑛 𝑆 and 𝑓 ( y ) = 𝑖 = 𝑗 𝑓 ( x 𝑖 ) + 𝑓 ( y 𝑖 ) > 𝑖 𝑓 ( x 𝑖 ) = 𝑓 ( x ) contradicting the assumption at 𝑓 is maximized at x . Conversely, assume 𝑓 ( x 𝑖 ) 𝑓 ( x 𝑖 ) for every x
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