Macroeconomics Exam Review 143

Macroeconomics Exam Review 143 - Solutions for Foundations...

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3.126 Suppose that x 1 minimizes the cost of producing 𝑦 at input prices w 1 while x 2 minimizes cost at w 2 . For some 𝛼 [0 , 1], let ¯ w be the weighted average price, that is ¯ w = 𝛼 w 1 + (1 𝛼 ) w 2 and suppose that ¯ x minimizes cost at ¯ w . Then 𝑐 ( ¯ w , 𝑦 ) = ¯ w ¯ x = ( 𝛼 w 1 + (1 𝛼 ) w 2 x = 𝛼 w 1 ¯ x + (1 𝛼 ) w 2 ¯ x But since x 1 and x 2 minimize cost at w 1 and w 2 respectively 𝛼 w 1 ¯ x 𝛼 w 1 x 1 = 𝛼𝑐 ( w 1 , 𝑦 ) (1 𝛼 ) w 2 ¯ x (1 𝛼 ) w 2 x 2 = (1 𝛼 ) 𝑐 ( w 2 , 𝑦 ) so that 𝑐 ( ¯ w , 𝑦 ) = 𝑐 ( 𝛼 w 1 + (1 𝛼 ) w 2 , 𝑦 ) = 𝛼 w 1 ¯ x + (1 𝛼 ) w 2 ¯ x 𝛼𝑐 ( w 1 , 𝑦 ) + (1 𝛼 ) 𝑐 w 2 , 𝑦 ) This establishes that the cost function 𝑐 is concave in w . 3.127 Since 𝑢 is concave, Jensen’s inequality implies 𝑢 ( 𝑇 𝑡 =1 1 𝑇 𝑐 𝑡 ) 𝑇 𝑡 =1 1 𝑇 𝑢 ( 𝑐 𝑡 ) = 1 𝑇 𝑇 𝑡 =1 𝑢 ( 𝑐 𝑡 ) for any consumption stream 𝑐 1 , 𝑐 2 , . . . , 𝑐 𝑇 so that 𝑈 = 𝑇 𝑡 =1 𝑢 ( 𝑐 𝑡 ) 𝑇 𝑢 ( 𝑇 𝑡 =1 1 𝑇 𝑐 𝑡 ) = 𝑇 𝑢 𝑐 ) It is impossible to do better than consume a constant fraction ¯ 𝑐 = 𝑤/𝑇 of wealth in each period. 3.128 If 𝑥 1 = 𝑥 3 , the inequality is trivially satisfied.
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