Macroeconomics Exam Review 214

Macroeconomics Exam - c 2001 Michael Carter All rights reserved Solutions for Foundations of Mathematical Economics 4.52 The partial derivatives of

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4.52 The partial derivatives of the quadratic function are ? 1 ± =2 ²³ 1 +2 ´³ 2 ? 2 ± ´³ 1 µ³ 2 The second-order partial derivatives are ? 11 ± ²? 21 ± ´ ? 12 ± ´? 22 ± µ 4.53 Apply Exercise 4.37 to each partial derivative ? ? ± [ x ]. 4.54 ( x 0 )= ( ? 11 ±± [ x 0 ] ? 12 [ x 0 ] ? 21 [ x 0 ] ? 22 [ x 0 ] ) ( ²´ µ· ) 4.55 4.56 For any ³ 1 ¸ , de±ne ¹ : ¸ →ℜ by ¹ ( º ± ( º )+ ± [ º ]( ³ 1 º ² 2 ( ³ 1 º ) 2 ¹ is di²erentiable on ¸ with » ( º ± [ º ] ± [ º ]+ ± ′′ [ º ]( ³ 1 º ) 2 ² 2 ( ³ 1 º ± ′′ [ º ]( ³ 1 º ) 2 ² 2 ( ³ 1 º ) Note that ¹ ( ³ 1 ± ( ³ 1 )and ¹ ( ³ 0 ± ( ³ 0 ± ( ³ 0 )( ³ 1 ³ 0 ² 2 ( ³ 1 ³ 0 ) 2 (4.50) is a quadratic approximation for ± near ³ 0 . If we require that this be exact at ³ 1 = ³ 0 , then ¹ ( ³ 0 ± ( ³ 1 ¹ ( ³ 1 ). By the mean value theorem (Theorem 4.1), there exists some ¯ ³ between ³ 0 and ³ 1 such that ¹ ( ³ 1 ) ¹ ( ³ 0 » ³ )( ³ 1 ³ 0 ± ′′ ³ )( ³ 1 ³ 0 ) 2 ² 2 ( ³ 1 º )=0 which implies that ² 2 = 1 2 ± ′′ ³ ) Setting ³ = ³ 1 ³ 0 in (4.50) gives the required result. 4.57 ³ 1 ¸ , de±ne ¹ : ¸ ¹ ( º ± ( º ± [ º ]( ³ 1 º 1 2 ± ′′ [ º ]( ³ 1 º ) 2 + 1 3! ± (3) [ º ]( ³ 1 º ) 3 + ... + 1 ¼ ! ± ( ± ) [ º ]( ³ 1 º ) ± + ² ± +1 ( ³ 1 º ) ± +1 ¹ is di²erentiable on
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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