Handout #5


·L¤À»PLagrange-multiplier Method

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·L¤À (differentiation) ¬O¤@ºØ¼Æ¾Ç¹Bºâ¡A¥iÀ°§Ú­Ì¨D¸Ñ¤@­Ó¨ç¼Æªº·¥¤j­È(©Î·¥¤p­È)¡C

(A)The Case of a Function of One Variable (³æ¤@Åܼƨç¼Æ)

  ¥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 (¦hÅܼƨç¼Æ)

  ¥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)ÂùÅܼƨç¼Æ¥[¤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

    


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