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Chiang_Ch12 - 1 Ch 12 Optimization with Equality...

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Unformatted text preview: 1 Ch. 12 Optimization with Equality Constraints • 12.1 Effects of a Constraint • 12.2 Finding the Stationary Values • 12.3 Second-Order Conditions • 12.4 Quasi-concavity and Quasi-convexity • 12.5 Utility Maximization and Consumer Demand • 12.6 Homogeneous Functions • 12.7 Least-Cost Combination of Inputs • 12.8 Some concluding remarks 2 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 128 8 2 14 8 ) 14 13 8 14 2 4 60 ) 12 11 14 2 2 1 1 4 60 ) 10 9 2 2 2 1 2 ) 8 7 2 1 ; 2 ) 6 5 2 4 ; 2 4)- 3 functions both of als differenti l order tota- first the Taking 128 ; 14 ; 8 ) 10 8 maximum 4 ; 4 32 7)- 6 2 32 ) 5 2 30 ; 2 2 30 4)- 3 function utility the into ng substituti and for x Solving 2 4 60 2 2)- 1 B constraint budget and function U utility Given ation differenti total and substition by solving : Constraint a of Effects 1 . 12 * 1 1 * 2 2 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 1 2 * * 2 * 1 2 1 2 1 1 2 1 1 1 2 1 1 1 2 2 1 1 2 1 = + =- = + =- = + + =- + =- = +--- = +- =- =-- = = + + = = = =-- = =- =- =- = +- =-- = + = * U U x x x x x x x x x dx dx x x dx dx dx dx dB dx dx x dx x dU U x x dx U d x dx dU x x U x x x x x U x x B x x x U 3 ( 29 ( 29 y." and x values critical the find which to from equations two provide will 2), (eq. constraint with the together 10), (eq. condition The " ; ) 10 9 ; ) 8 7 B ; U ) 6 5 ; ) 4 3 , ; , 2)- 1 B constraint budget and function U utility Given 352) p. W, & (C- Constraint a of Effects 1 . 12 x x y y x y x y x y x y y x y x y x y x g f g f g g f f g g dy dx f f dy dx dy dx g g d dy dx f f d dy g dx g dB dy f dx f dU y x g B y x f U approach ferential Total dif = =-- =- =- = =- = + = = + =- = = 4 [ ] [ ] ( 29 ( 29 ( 29 ( 29 ( 29 128 8 2 14 8 2 ) 18 14 2 32 60 ) 17 2 8 4 60 ) 16 8 ) 15 64 8 ; 2 2 2 4 60 ) 14 13 2 2 ; 2 2 ) 12 11 2 2 4 2 ) 10 9 2 4 ; 2 1 1 U ) 8 7 ; 2 4 ; 2 ) 6 5 60 2 4 ; 2 2 1 2 1 U 4)- 3 60 2 4 ; 2 2)- 1 B constraint budget and function U utility Given 352) p. W, & (C- Constraint a of Effects 1 . 12 1 2 1 * * 2 2 * 1 1 1 1 1 2 1 2 1 2 1 2 1 2 2 1 1 2 1 2 1 2 1 1 1 2 2 1 2 1 1 2 1 2 1 2 1 1 2 1 = + = + = =- = + = = =- + =-- =- =---- =- =-- =- = + =- = + = = + + =-- = + = =- + = + = x x x U x x x x x x x x x x dx dx x x dx dx dx dx dB dx dx dx x x d dx dx dB dx dx x dx x dU x x B x x x x x x x B x x x U approach ferential Total dif 5 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 4 ; 128 ; 8 2 14 8 ) 17 15 14 ; 2 8 4 60 ) 14 13 8 ; 2 2 2 4 60 ) 12 11 2 2 ; 2 1 2 1 4 1 ) 10 9 2 1 ; λ2 ) 8 7 2 1 4 1 ; λ4 2 ) 6 5 2 4 60 ) 4 2 4 60 2 3) function Lagrangian 2 4 60 B 2 2)- 1 B constraint budget and function U utility Given on substituti with Lagrangian : constraint a of effect The 12.1 * * * 2 2...
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Chiang_Ch12 - 1 Ch 12 Optimization with Equality...

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