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Chiang_Ch11_missing pages

# Chiang_Ch11_missing pages - Ch. Variable 11. Conditions 11....

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1 Ch. 11 The Case of More than One Choice  Variable 11.1 The Differential Version of Optimization  Conditions 11.2 Extreme Values of a Function of Two  Variables 11.3 Quadratic Forms - An Excursion 11.4 Objective Functions with More than Two  Variables [11.5 Second-Order Conditions in Relation to  Concavity and Convexity] 11.6 Economic Applications 11.7 Comparative-Static Aspects of  Optimization

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2 11.1 The Differential Version of  Optimization Conditions • 11.1(a) First-order condition • 11.1(b) Second-order condition • 11.1(c) Differential conditions vs.  derivative conditions
3 11.2 Extreme Values of a  Function of Two Variables • 11.2(a) First-order condition • 11.2(b) Second-order partial derivatives • 11.2(c) Second-order total differential • 11.2(d) Second-order condition

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4 11.3 Quadratic Forms - An Excursion • 11.3(a) Second-order total differential as a  quadratic form • 11.3(b) Positive and negative definiteness • 11.3(c) Determinant test for sign  definiteness • 11.3(d) Three-variable quadratic forms • 11.3(e) n-variable quadratic forms • 11.3(f) Characteristic-root test for sign  definiteness
5 11.2(a) First-order condition yy xy xx y x yy xy xx y x f f f f f dy f dxdy f dx f z d dy f dx f dz y) f(x z s derivative partial Find 2 ) 3 : (SoC) conditions order - second the Find 0 ) 2 : (FoC) conditions order - first the Find , ) 1 function •Given the 2 + + = = + = =

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6 11.2(b) Second-order own & cross partial  derivatives 2 2 2 2 2 ) ( ) 5 ) ( ) 4 ) ( ) 3 ) 2 ) , ( ) 1 y z y z y f y f y x z y z x f x f x z x z x f x f y z f x z f y x f z y yy y xy x xx y x = = = = = = = = = = = =
7 11.2(c) Second-order total differential ( 29 ( 29 ( 29 2 2 2 2 2 ) ( ) ( ) 3 ) 2 ) , ( ) 1 dy f dxdy f dx f dy f dxdy f dxdy f dx f dy y dy f dx f dx x dy f dx f dy y dz dx x dz dz d dy f dx f dy y z dx x z dz y x f z yy xy xx yy xy xy xx y x y x y x + + = + + + = + + + = + = + = + = =

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8 11.3(c) Second derivative as determinant [ ] definite negative definite positive is then , 0 and 0 0 If ) 3 ) 2 ) 1 2 2 2 2 2 < = = + + + = z d H f f f f f f H dy dx f f f f dy dx z d dy f dydx f dxdy f dx f z d xx xx yy yx xy xx yy yx xy xx yy yx xy xx
9 11.2(d) Second-order condition Condition Maximum Minimum Saddlepoint 1 st order Necessary Sufficient 2 nd  Order Necessary Sufficient  Condition                          ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 , 0 , 0 , , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 2 2 2 2 2 2 * * * * * * * * * * * * * * * * * * * * * * * * < < < ± < = = = = = = H H H f f f f f f f f f y x f y x f y x f y x f y x f y x f y x f y x f y x f y x f y x f y x f xy yy xx xy yy xx xy yy xx yy yy yy xx xx xx y y y x x x

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10 11.4 n-variable soc principal minors test for max or min :) 0 ,..., 0 , 0 , 0 H : min ( : 0 ) 1 ,...( 0 , 0 , 0 H : max 317) (p. soc, case variable n :) 0 , 0 H : min ( : 0 , 0 H : max soc of test variable 2 :) , 0 H : min ( : , 0 H : max t minor tes principal soc,
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