Chapter 9. Optimization: One Choice Variable. 9.1 Optimum Values and Extreme Values

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1 RS - Ch 9 - Optimization: On Vaiabl Chapt 9 Optimization: On Choic Vaiabl Léon Walas 8-9 Vildo Fdico D. Pato Optimum Valus and Etm Valus Goal vs. non-goal quilibium In th optimization pocss, w nd to idntiy th objctiv unction to optimiz. In th objctiv unction th dpndnt vaiabl psnts th objct o maimization o minimization Eampl: - Din poit unction: = PQ - CQ - Objctiv: Maimiz - Tool: Q

2 RS - Ch 9 - Optimization: On Vaiabl 9. Rlativ Maimum and Minimum: Fist- Divativ Tst Citical Valu Th citical valu o is th valu i = A stationay valu o y is A stationay point is th point with coodinats and A stationay point is coodinat o th tmum Thom Wistass Lt : S R b a al-valud unction dind on a compact boundd and closd st S R n. I is continuous on S, thn attains its maimum and minimum valus on S. That is, th ists a point c and c such that c c o all S. 9. Fist-divativ tst Th ist-od condition.o.c. o ncssay condition o tma is that ' = and th valu o is: A lativ minimum i ' changs its sign om ngativ to positiv om th immdiat lt o to its immdiat ight. ist divativ tst o min. y B '= A lativ maimum i th divativ ' changs its sign om positiv to ngativ om th immdiat lt o th point to its immdiat ight. ist divativ tst o a ma. y A ' =

3 RS - Ch 9 - Optimization: On Vaiabl 9. Fist-divativ tst Th ist-od condition o ncssay condition o tma is that ' = and th valu o is: Nith a lativ maima no a lativ minima i ' has th sam sign on both th immdiat lt and ight o point ist divativ tst o point o inlction. y D ' = 5 9. Eampl: Avag Cost Function AC Q Q Q 5Q 8 Q 5 Q 5 Q 5.5 Objctiv unction st divativ unction.o.c. tma lt '...5 '.6., AC Q ight lativ min 6

4 RS - Ch 9 - Optimization: On Vaiabl 9. Th smil tst Convity and Concavity: Th smil tst o aimumminimum I " < o all, thn stictly concav. => citical points a global maima I " > o all, thn stictly conv. => citical points a global minima I a concav utility unction typical o isk avsion is assumd o a utility maimizing psntativ agnt, th is no nd to chck o s.o.c. Simila situation o a concav poduction unction 7 9. Th smil tst: Eampls: Eampl : Rvnu Function TR Q Q MR Q Q MR 5 MR Rvnu unction tma nd divativ Q.o.c. is a maimum Eampl : Avag Cost Function AC Q Q Q 5Q 8 Q 5 Q 5.5tma Objctiv unction.o.c. Q 5.5 is a minimum 8

5 RS - Ch 9 - Optimization: On Vaiabl Inlction Point Dinition A twic dintiabl unction has an inlction point at i th scond divativ o. changs om ngativ positiv in som intval m,~ to positiv ngativ in som intval ~,m, wh ~ m,n. Altnativ Dinition An inlction point is a point, y on a unction,, at which th ist divativ,, is at an tmun, -i.. a minimum o maimum. Not: = is ncssay, but not suicint condition. Eampl: Uw = w - w + w U w=w + w U w=+6w => +6w~ = w~= is an inlction point 9. Eampls: Optimal Signoag maimum a is : S.o.c. point Citical : assum F.o.c. Signoag mony o Dmand d S d d S d d ds P M S P M

6 RS - Ch 9 - Optimization: On Vaiabl 9. Eampls: Optimal Timing - win stoag A t V V k At k lna t da dt A F.o.c. : t t t t lnk ln lnk ln k k t t ½ ½ t t ½ t t t ln t t da A t dt t Psnt valu Gowth in valu Monotonic tansomation o objctiv unction t t 9. Eampls: Optimal Timing - win stoag Optimal tim : Lt % t. Dtmin optimal valus o At and V : At k lt k At V A V $ bottl 5. 5 t 5 yas $8.8 bottl t ½ t t. 5 V $. 8bottl. 5 $.8 bottl $8.8 bottl 6

7 RS - Ch 9 - Optimization: On Vaiabl 9. Eampls: Optimal Timing - win stoag Plot o Optimal Tim with =. => t=5 9. Fomal Scond-Divativ Tst: Ncssay and Suicint Conditions Th zo slop condition is a ncssay condition and sinc it is ound with th ist divativ, w to it as a st od condition. Th sign o th scond divativ is suicint to stablish th stationay valu in qustion as a lativ minimum i " >, th nd od condition o lativ maimum i " <. 7

8 RS - Ch 9 - Optimization: On Vaiabl 9. Eampl : Poit unction Rvnu and Cost unctions TR Q Q TC Q 6.5Q 58.5Q Poit unction π TR-TC Q 59. 5Q 8. 5Q st divativ o 5 Q Q poit unction 8.5Q 8.5 Q 6.5 nd divativ o poit unction 6 6Q applying th smil tst 8 Q min Q ma 5 9. Eampl : Impct Comptition Total vnu unction TR AR Q 8 Q.Q.8 Q Q 6 8

9 RS - Ch 9 - Optimization: On Vaiabl 9. Eampl : TR and st, nd & d divativs Avag and total v nu unctions nd divativ : smil tst st divativ - Maginal vnu MR 8 6Q.Q MR 6 6.6Q.6Q.76 d divitiv : 5 MR Q 6.6.Q 6 MR MR" TR Q Q Q MR Q min. 8Q Q : Q 9.79 :..7Q.Q.8Q MR" Q ma Taylo Sis o a polynomial unction: Rvisitd Taylo sis o an abitay by th wightd Wh I n - At Thn,, - R n, thn sum o its divativ s. Thn th n n!! n! n p n n!!! - i.., is a unction : Any unction! chang is givn ma., min., o inlction. th sign o can b appoimat d R by dtmins what is. 8 R n 9

10 RS - Ch 9 - Optimization: On Vaiabl 9.5 Taylo pansion and lativ tmum A unction attains a o valuso in th immdiat nighbohood o lt and ight Taylo sis appoimation o a small chang in : - At th ma., min., o inlction, and i thn -, and i lativ ma min valu at! R n! What is th sign o R o th ist nonzo divativ?, i - is ng.pos. th citical valu both to its... R n N th -divativ tst I th ist divativ o a unction at i th ist nonzo divativ valu at succssiv divation is that o th N thn th stationay valu a lativ ma i a lativ min i an inlction point i will b : N is vn and N is vn and N is odd th is divativ, ncountd in n n n and,

11 RS - Ch 9 - Optimization: On Vaiabl 9.5 N th -divativ tst Bcaus ist nonzo divativ and at 7 th citical valu pimitiv unction divativ divativ n, citical valu is a min. st nd d th diativ diviativ is vn 9.5 N th -divativ tst 7- dcision ul : n is vn and tho a minimum pimitiv unction th divativ