·L¤À»PLagrange-multiplier Method
¡@
·L¤À
(differentiation) ¬O¤@ºØ¼Æ¾Ç¹Bºâ¡A¥iÀ°§Ų́D¸Ñ¤@Ó¨ç¼Æªº·¥¤jÈ(©Î·¥¤pÈ)¡C
(A)The Case of a Function of One
Variable (³æ¤@ÅÜ¼Æ¨ç¼Æ)
¥OU¬°Xªº¨ç¼Æ¡A§YU=U(X)¡C
¥H¦¹¨ç¼Æ´NX¶i¦æ¤@¦¸·L¤À¡A¥i±o¸Ó¨ç¼Æ¤§¾É¨Ó¦¡(derivative)¡AU'(X) ©ÎUx¡C
±ý¨DU¤§·¥¤jÈ¡A»Ý¨D¸ÑU'(X) = 0 (©ÎUx = 0)¡AºÙ²Ä¤@¶¥±ø¥ó(first
order condition)¡C
µù¡G²Ä¤G¶¥±ø¥ó (second order condition) ¥i¥Î¨Ó¨M©w¬O·¥¤j©Î·¥¤pÈ¡C
¦¹¦b¡uÓ¸g¡v½Ò¤¤À³¦³¸Ô²Ó³B²z¡A¥»½Ò¼È¤£½Í²Ä¤G¶¥±ø¥ó¡C
(½m²ß¨Ò¡GU = X3¤@12X2¤Q36X¤Q8)
(B)The Case of a Function of Multiple
Variables (¦hÅÜ¼Æ¨ç¼Æ)
¥OU¬°X»PYªº¨ç¼Æ¡A§YU = U(X,Y)¡C
«h²Ä¤@¶¥±ø¥ó¬°Ux = 0»PUy
= 0¡C
Y¦³¤TÓÅܼơAU(X,Y,Z)¡A«h²Ä¤@¶¥±ø¥ó¬°Ux =
0, Uy = 0, Uz = 0¡C¥H¦¹Ãþ±À¡C
(C)ÂùÅÜ¼Æ¨ç¼Æ¥[¤WConstraint
¨Dmax U(X,Y)
s.t. PxX¤QPyY = I
¦¹®É¡A©w¸q¤@Ó·sªº¤TÅܼƤ§µêÀÀ¨ç¼Æ (hypothetical function)
L¡×U (X,Y)¡Ð£f[PxX¡ÏPyY¡ÐI]
¨ä¤¤£f¬°·s°²³]¤§²Ä¤TÓÅܼơA¥sLagrange multiplier¡AL¨ç¼Æ«hºÙLagrange function¡C
¨D L(X,Y,£f) ¤§·¥¤jÈ
¡A¨ä²Ä¤@¶¥±ø¥ó¡G:
Lx
= 0
Ux¡Ð£fPx
= 0
Ly
= 0
¥ç§Y
Uy¡Ð£fPy
= 0
L£f = 0
PxX¡ÏPyY¡ÐI
= 0
¦¹®É¥i±oU¤§·¥¤jÈ¡A¦P®Éº¡¨¬constraint¡C
(¥H¤W¥i°Ñ¦Ò A. Chiang
"Fundamental Method of Mathematical Economics" ùئ³Ãö Differentiation, Optimization, Constrained Optimizationµ¥¤§³¹¸`) ¡C
¡@