Chapter 10. Optimization: More than One Choice Variable

Transcription

1 Chaper Opimiaion: More han One Choice Variable William Sanle Jevons Carl Menger 8 9. Opimiaion Problems Chaper 9: ma u one choice variable: consumpion Chaper : ma u o choice variables: leisure Chaper : ma u s. p + p = I here p p are eogenous prices an I is eogenous income. Wih one choice variable he.o.c. is eine b seing u equal o ero. Noe ha u=u u = u. hen u = => u=. Wih o or more choice variables he.o.c. is eine b a ierenial hen u equals ero.

2 Figure. Saionar Poins an he angen Planes o Bivariae Funcions. Firs-orer coniion Given he uncion = Fin he oal ierenial = + Se =. hen = =.o.c. = is a necessar coniion or an ereme poin. Bu like in he one variable case i is no suicien. See Figure.. Fin he secon- orer ierenial: Fin parialerivaives :

3 Figure. A Sale Poin 5. Secon-orer on & cross parial erivaives 5

4 . Secon-orer oal ierenial Noe: is a quaraic orm. 7. Secon-orer oal ierenial Eample: = + + Firs orer coniion ; / ; Quesion ; ; ; / : ha kin.o.c. o criical poin is? 8

5 . Secon-orer oal ierenial I < evaluae a in ever an irecion hen is a maimum. On he oher han i > evaluae a in ever an irecion hen is a minmum. We can use o sae he secon orer coniion: - < or an o be a maimum - > or an o be a minimum When can e kno he sign o? - When = sign is eermine b he sign o an : -I < an < hen sign < -I > an > hen sign > - Bu in general hen geing he sign is no ha simple. 9. Quaraic orms A orm is a polnomial epression in hich each componen erm has a uniorm egree. A quaraic orm has a uniorm secon egree. Eamples: irs egree orm. -secon egree quaraic orm. is a quaraic orm in an. 5

6 . Quaraic orms Le q be a quaraic orm. We sa q is: Posiive einie i q is invariabl posiive q> Posiive semi-einie i q is invariabl non-negaive q Negaive semi-einie i q is invariabl non-posiive q Negaive einie i q is invariabl negaive q< A quaraic orm is sai o be ineinie i q changes signs.. Quaraic orms is a quaraic orm in an : Le u ; Le a Reriing q au ; v b : huv bv ; h rick o eermine sign : complee he square : h h q au huv bv v v a a h h h q a u uv v bv v a a a a u h a v h b v a

7 . Quaraic orms No e are rea o eermine sign : q a u sign q epens h a v on a h b a v sign We can rie in a more compac orm using linear algebra. u u here u is he vecor o s erivaives an is he mari o secon erivaives he essian.. Sign as a Deerminan Rerie using linear algebra: I Le is he eerminan o he mari above. an hen posiive einie is negaiveeinie 7

8 8. Sign as a Deerminan: Eample minimum einie posiive Q is 8 8 einie negaive einie posiive is hen an I 8 ; ; 8; ; 8 Q v u v u Q v u v u v uv u Q uu vv vu v uv uu u 5 & & FOC i i poin i sale i min i ma or solve parameers eog. enog. i here. Opimiaion coniions: Summar

9 9. Opimiaion coniions: Summar Coniion Maimum Minimum Salepoin s orer Necessar n Orer Necessar Suicien Coniion 7 8 minimum is p is Secon - orer oal ierenial F.o.c. ierenial - orer oal Firs maima or minima heher he are Deermine. o values Fin he ereme 7. Opimiaion: Eample

10 L L L ' ' ' ' ln ln ln ln... log Soluion : F.o.c. s erivaives : is :.r.. an o maimie e he log likelihoo uncion We assume eplanaor variables.. Opimiaion: ML Eample 9. eermine o be S.o.c.? '. Opimiaion: ML Eample

11 . Sign o a quaraic orm: Eigenvalue ess From Chaper 5: Orhogonall iagonalie a ssem o equaions: A = Pre-mulipl boh sies b - : - A = - = ν - A - = ν Le υ = - => υ = ν here is he eigenvalue iagonal mari. Le s re-rie he quaraic orm: q q u ' u. Sign o a quaraic orm: Eigenvalue ess.quaraic orm: q = u u noe: he essian is a smmeric mari Le u= here is he mari o eigenvecors o such ha = I. hen q = = Λ = Λ q = λ + λ λ i i λ n n => he signq epens on he λ i s onl. We sa: q is posiive einie i λ i > or all i. q is posiive semi-einie i λ i or all i. q is negaive semi-einie i λ i or all i. q is negaive einie i λ i < or all i. q is ineinie i some λ i > an some λ i <.

12 . Sign o a quaraic orm: Eigenvalue ess Eample coninuaion o.: F.o.c. Calculae mari o secon erivaive s an are is minimum.5858 ; posiive q is posiive einie.. n-variable soc principal minors es or ma or min - variable es o soc principal ma : min : variable es o soc ma : min : n variable ma : min : case soc : : minor es : : n n n : :

13 5 or Solve 5. N-variable case: Eample ih n= Eigenvalue check : minimum posiive einie minors Principal. N-variable case: Eample ih n=

14 .5 Economic Applicaions: Eample Muli-prouc irm. Assume a compeiive irm has revenue an cos uncions belo. Deermine Q Q P π π R P R - C Q P Q P P P 8 Q 5 5 P P 8 Q 5 5 π P Q 8 P Q P Q P Q Q Q P 8 C Q Q Q Q Q 5 5 Q Q π P Q Q Q Q negaive Q P P Q Q einie ma. 7.5 Economic Applicaions: Eample Assumpions behin classical linear regression CLM moel: A = + is correcl speciie. A E[ ] = A Var[ ] = σ I A has ull column rank rank=k here k. Objecive uncion: min {Σ i i = = - - } F.o.c. normal equaions: - Σ i [ i - Σ k ik b k ] ik = i=... => - - b = Solving or b => b = - 8

15 .5 Economic Applicaions: Eample Q: Is b is a minimum? We nee o check he s.o.c. - b' - b ' - b b - b' - b - b' - b = b bb b column vecor = ro vecor = ' I here ere a single b e oul require his o be posiive hich i oul be: = Σ k i >. he mari counerpar o a posiive number is a posiive einie mari. 9.5 Economic Applicaions: Eample n n n... i i i i i i i ik i i i... i ik n n n iii i i... i iik... = n i i i i ik ' = i n n n iiki i iki... i ik iki iki... ik i = n i i i i ik ik n = i i i Deiniion: A mari A is posiive einie p i A > or an. In general e nee eigenvalues o check his all shoul be posiive. For some marices i is eas o check. Le A =. hen A = = v v >. => is p => b is a min! 5