Ghpdqg iru Vriw Gulqnv= Fkdudfwhulvwlfv/Fruqhuv/ dqg Frqwlqxrxv Fkrlfh

Wdw \1 Fkdq�

Ghsduwphqw ri Hfrqrplfv/ \doh Xqlyhuvlw|�

Qryhpehu/ 5333

� � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! " # $ " % & ' ( ) * + , - . / / . / 0 1 2 3 . 45 6 7 8 9 : ; ; < 6 : 9 = > 6 : 7 : ? @ A B C D B E F G C H I G B I J K L F M C N O P Q R S T Q U P V S R T P W X Y Z [ \S V ] ^ _ ` a b c d e f g h i j k l m n o h n p q r l m o m j l h n s r o r t h p r m i h n m u q v o t p q i h n m ut k q k w k n x k w o k y m h z { | | } ~ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � �   ¡ ¢ £ ¤ ¢ ¥ ¡ ¢ ¦ § ¨ ¥ § © § ¢ ¥ § ¢ ª § « ¬ © ¨ « ¥ ­ ª ¦ ª ® « ¡ ª § ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º ´ »· ´ ¼ ² ¹ ³ ½ ¼ ² · ´ ² ¾ ¼ ¾ ¿ º ¶ À ¹ ² º À ¼ º ¶ ³ ¿ Á  ´ ½  ¸ ¶ ¿ · ¸ ´ ½ · ¸ ¶ ¿ ¶ ¹ ¸ ½  ¶ ¸ ¿ · ¼ ¾ ¿ ´ ² ÃÀ ¹ ¸ µ ¶ · ³ ¶ Ä ¶ ³ º ¹ · ¹ Å ´ · ½ ¹ ² Æ ¶ ¹ Ç Ç ³ ´ ¶ º · ¼ À ´ ½ ¸ ¼ ³ ¶ Ä ¶ ³ º ¹ · ¹ Á ´ ·  ½ ¼ ¸ ² ¶ ¸¿ ¼ ³ ¾ · ´ ¼ ² ¿ È É Ê Ë Ì Í Î Ì Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Ü Þ Ø Ü ß Ù à Ú á Ú Ü Ù á Ú â ß ã Ú Ù Ú Û Ùß Ü Ý Ù ä å Û Ù à æ Ù á Ü ä ç Ù Ú Û Ù è é Ü ê ä Ù ß Ü Þ Û â ë Û Ü é Ý Ù é â à Ú Ù ë é ã Ú â Ü à ì Û Ù à Ú Û Ù é Ùã é Ù à í ß Ù é Ü í á î Ù é Ü è í é æ Û ã á Ù á ï

ð ã è è ä ñ Ú Û â á ß Ù Ú Û Ü Ý Ü ä Ü ë ñ Ú Ü Ù á Ú â ß ã Ú Ù Ú Û Ù Ý Ù ß ã à Ý Þ Ü é á Ü Þ Ú Ý é â à ò á ïó Û Ù é Ù á í ä Ú á á Û Ü ì Ú Û ã Ú æ Ü à á í ß Ù é á á Ù Ù ò ç ã é â Ù Ú ñ Ü ç Ù é è é Ü Ý í æ Ú æ Û ã é ã æ ôÚ Ù é â á Ú â æ á á í æ Û ã á õ ã ç Ü é å è ã æ ò ã ë â à ë å ã à Ý æ Ü à Ú ã â à Ù é á â î Ù å ã à Ý Ý â ö ÷ ø ÷ ù úû ÷ ü ø ÷ ÷ ý þ ÿ ý � � ý ú � ú � ú � � � � � ú � � ù û � þ � � � ÷ � ÷ ù ú � ø � ú � ÷ � ý ú � � þ ù ü û � ö ÷ ø ÷ ù ú � ø þ û � � ú ý � � ÷ � ø � � � � � � � ú � þ ù ý � ø ÷ � ý ÷ ÿ � � ú þ � ù û ÷ ø ý ú � ù û � ù û � ý ú ø � � �� ø � � � ù ü � ù û � ø þ û � � ú û � ý � � � � � þ � � � � ÷ ý � ÷ ù � � � ù � ÿ � � ú � ø ÷ ø � ø þ û � � ÷ ý� � � ú � � � ÷ � ø þ û � � ú ý �

� � � � � � � � � � � � � � � � � � ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 7 9 : ; ; < = > ? @ A B C 9 ; ? D @ A @ C E F G HI J K L M N O P Q R S O T U V W X Y Z [ \ ] ^ _ ` a b c a d _ a e f g h _ h i c j f k l m n l o p q r s t u v w x y z { z x | } u ~ z u t x | � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �   � � � ¡ � ¢ ¢ � £ ¤ �   ¥ � � ¦ ¢ ¢ §   ¨ © ª � « ¢ � � ¤ � � ¬

­ ® ¯ ° ± ² ³ ´ ¯ µ ³ ¶ · ¸ ¹ º » º ¼ ½ ¹ ¾ ¿ À Á Â Ã Ä Å Æ Ç È É Ê Æ Ë Ì Í Î Ï Ð Ñ Ò Ò Ó Ô Õ Ö × Ø Ù × Ú Õ × Û Ü Ý Þ ß à á Ý â ã ä åæ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ � � � � � þ � � � � � � � � � � � � � � � � �

4

4 Lqwurgxfwlrq

Lq wklv sdshu L frqvlghu d ghpdqg prgho zkhuh +4, frqvxphuv fduh derxwsurgxfw fkdudfwhulvwlfv/ +5, frqvxphuv fdq fkrrvh pxowlsoh surgxfwv dqg frq0wlqxrxv txdqwlw|/ dqg +6, wkh h{lvwhqfh ri fruqhu vroxwlrqv +wkdw lv/ }hur sxu0fkdvhv ri vrph surgxfwv, lv wkh qrup1 L dsso| wklv prghov hpslulfdoo| wrsxufkdvlqj ghflvlrqv ri vriw gulqnv1

Pxowlsoh surgxfw dqg pxowlsoh xqlw vkrsslqj ehkdylru iru vriw gulqnvlv vljql�fdqw lq vxshupdunhwv ehfdxvh frqvxphuv xvxdoo| ex| iru krxvhkrogfrqvxpswlrq zlwklq d fhuwdlq shulrg1 Wkh| pd| ex| pdq| xqlwv/ hyhq wkrxjkwkh xqlw lv d 450sdfn/ li wkh| eholhyh wkdw krxvhkrog qhhgv zloo eh odujh1Pruhryhu/ gl�huhqw surgxfwv kdyh gl�huhqw frpelqdwlrqv ri fkdudfwhulvwlfv/vxfk dv dyru/ sdfndjlqj/ dqg frqwdlqhu vl}h1 Frqvxphuv pd| vhhn ydulhw|lq wdvwhv e| ex|lqj pruh wkdq rqh w|sh ri gulqn1 Dw wkh vdph wlph }hursxufkdvhv ri pdq| gulqnv duh dovr frpprq= wkhuh duh qxphurxv vriw gulqnvfr0h{lvwlqj lq vxshupdunhwv/ dqg frqvxphuv xvxdoo| fkrrvh rqo| d ihz riwkhp

�1

Glvfuhwh Surgxfw Fkrlfh Prghov Glvfuhwh fkrlfh prghov +GFPv, dv0vxph wkdw frqvxphuv ex| dw prvw rqh xqlw +dqg khqfh rqo| rqh surgxfw, rijrrgv1 Wr ghdo zlwk wkh pxowlsoh surgxfw/ pxowlsoh xqlw sxufkdvlqj ehkdylru/d qhz prgho zdv ghyhorshg e| Khqgho +4<<<, iru �upv* kroglqjv ri frpsxw0huv dqg h{whqghg e| Gxeh +4<<<, wr wkh ghpdqg iru fduerqdwhg vriw gulqnv1Edvhg rq Kdqhpdqq*v glvfuhwh2frqwlqxrxv dssurdfk +4<;7,/ wklv prgho uh0vwulfwv frqvxphuv wr fkrrvlqj rqo| rqh surgxfw iru hdfk frqvxpswlrq qhhgexw doorzv pxowlsoh xqlw frqvxpswlrq1 Wkhuh pd| eh pxowlsoh frqvxpswlrqqhhgv gxulqj wkh shulrg/ dqg frqvxphuv pd| fkdqjh wkhlu surgxfw fkrlfhviru gl�huhqw qhhgv1 Dv d uhvxow/ wkh| pd| fkrrvh pruh wkdq rqh surgxfw dwwkh wlph ri sxufkdvh1

Wkh deryh fdq eh fdoohg �glvfuhwh surgxfw fkrlfh� prghov ehfdxvh frq0vxphuv duh uhvwulfwhg wr fkrrvlqj rqo| rqh surgxfw iru hdfk frqvxpswlrqqhhg1 Fkrlfhv pdgh iru hdfk frqvxpswlrq qhhg duh lqghshqghqw ri hdfkrwkhu1 Wkrxjk vxffhvvixoo| surylglqj dq hfrqrphwulf prgho iru wkh pxowl0soh surgxfw/ pxowlsoh xqlw sxufkdvh ehkdylru/ glvfuhwh surgxfw fkrlfh prghov

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � " # $ % � % & � � � � � & � � � ' % � � ' ( ! % ( � % ) $ � & $

* � � � � & � � � � � � � ( � $ � ) ' � & $ � � � � � "

5

odujho| ljqruh wkh lqwhughshqghqfh dprqj fkrlfhv pdgh iru gl�huhqw surgxfwv+vhh Khqgho +4<<<,/ s1776/ iru d glvfxvvlrq ri wkh sureohp1, Lw lv srvvleoh wkdw/iru h{dpsoh/ li d frqvxphu kdv erxjkw d 50olwhu sulydwh odeho frod/ wkh pdu0jlqdo ehqh�w ri ex|lqj rqh fdq ri Shsvl ru Frnh zloo eh orzhu/ wkrxjk wkdwri ex|lqj :0Xs pd| qrw eh d�hfwhg1 Wkhuh lv qr reylrxv zd| wr lqfrusrudwhwklv uhodwlrqvkls lq wkh glvfuhwh surgxfw fkrlfh prghov1

Wudglwlrqdo Frqwlqxrxv Ghpdqg Prghov Wudglwlrqdo frqwlqxrxv gh0pdqg prghov dvvxph wkh h{lvwhqfh ri d uhsuhvhqwdwlyh frqvxphu dqg vrph h{leoh sdudphwulf lqgluhfw xwlolw| ixqfwlrqv1 XvlqjUr|*v lghqwlw|/ uhvhdufkhuvkdyh ghulyhg pdunhw ghpdqg iru hdfk surgxfw +vhh Ghdwrq dqg Pxhooedxhu+4<;3,,1 Wkh qhz sdudphwulf ghpdqg v|vwhpv sursrvhg e| Gdylv +5333, fdqeh hvwlpdwhg hyhq zkhq surgxfwv hqwhu ru h{lw wkh pdunhw gxulqj wkh vdp0soh shulrg1 Krzhyhu/ wkhvh prghov uhvwulfw uhvhdufkhuv wr xvlqj pdunhw ohyhogdwd1 Rqh ri wkh sureohpv iroorz wklv uhvwulfwlrq lv ryhu0sdudphwhul}dwlrq=uhvhdufkhuv pd| qrw kdyh hqrxjk gdwd wr vxssruw wkh hvwlpdwlrq ri rzq0sulfhdqg furvv0sulfh ghulydwlyhv zkhq wkhuh duh qxphurxv surgxfwv1 Plfur ohyhogdwd surylghv pruh lqirupdwlrq/ |hw zkhq fruqhu vroxwlrqv h{lvw/ uhvhdufkhuvfdqqrw fduu| wkh wudglwlrqdo frqwlqxrxv ghpdqg dssurdfk lqwr wklv gdwd1Wkhuh fdq eh pdq| �nlqnv� lq wkh lqgluhfw xwlolw| ixqfwlrq/ dqg lw lv gl!fxowli qrw lpsrvvleoh wr xvh d uhodwlyho| orz rughu sro|qrpldo dssur{lpdwlrq iruwklv ixqfwlrq1 Zh fdqqrw ljqruh wkh h{lvwhqfh ri fruqhu vroxwlrqv/ rwkhuzlvhhvwlpdwruv duh eldvhg1

Frqwlqxrxv Fkrlfh Prgho xqghu wkh Khgrqlf Dssurdfk L sursrvhwr xvh wkh fodvvlfdo frqwlqxrxv fkrlfh prgho +FFP, doorzlqj wkh fdvhv zkhuhsurgxfwv duh qrw shuihfw vxevwlwxwhv zlwk hdfk rwkhu vr wkdw frqvxphuv pd|fkrrvh pxowlsoh surgxfwv exw/ dw wkh vdph wlph/ pdq| surgxfwv duh qrwsxufkdvhg e| hdfk frqvxphu dw hdfk wlph ri sxufkdvh1 Lqvwhdg ri dvvxplqjd sdudphwulf lqgluhfw xwlolw| ixqfwlrq/ L vwduw iurp d gluhfw xwlolw| ixqfwlrq=frqvxphuv duh dvvxphg wr kdyh d vwdwlf krxvhkrog xwlolw| ixqfwlrq dw wkhwlph ri sxufkdvh/ dqg wkh| duh pd{lpl}lqj xwlolw| xqghu lqfrph dqg qrq0qhjdwlylw| frqvwudlqwv1

Wkh FFP lv ulfk hqrxjk wr lqfrusrudwh d frpsolfdwhg uhodwlrqvkls rifkrlfhv dprqj gl�huhqw surgxfwv1 Li ydulhw| vhhnlqj whqghqf| lv vwurqj/ frq0vxphuv duh olnho| wr doorfdwh wkhlu exgjhw ryhu gl�huhqw surgxfwv1 Wklv lvvkrzq e| d udslgo| ghfolqlqj pdujlqdo xwlolw| ri frqvxplqj dq| vshfl�f surg0

6

xfw1 Li frqvxphuv whqg wr vwlfn wr rqh ru d ihz wdvwhv/ wkh| zrxog doorfdwhwkhlu exgjhw ryhu rqo| rqh ru d ihz surgxfwv1 Wklv lv vkrzq e| d vorzo|ghfolqlqj pdujlqdo xwlolw| dqg orz vxevwlwxwdelolw| dprqj surgxfwv1

L dgrsw wkh khgrqlf prgho dssurdfk e| dvvxplqj xwlolw| ghulyhg iurp wkhfrqvxpswlrq ri surgxfw fkdudfwhulvwlfv ru dwwulexwhv1 Wkhuh duh wzr dgydq0wdjhv wr xvlqj vxfk dq dssurdfk1 Iluvw/ lw uhgxfhv wkh frpsxwdwlrqdo exughqgxh wr ryhu0sdudphwhul}dwlrq li wkh glphqvlrq ri wkh fkdudfwhulvwlfv vsdfh lvohvv wkdq wkh glphqvlrq ri wkh surgxfw vsdfh1 Vhfrqg/ uhodwlrqvklsv ehwzhhqgl�huhqw surgxfwv zloo eh ehwwhu xqghuvwrrg e| dqdo|}lqj wkh vxevwlwxwdelolw|dqg frpsohphqwdulw| ri gl�huhqw fkdudfwhulvwlfv lq wkh xwlolw| ixqfwlrq1

Wkh khgrqlf dssurdfk kdv ehhq zlgho| xvhg lq wkh hpslulfdo olwhudwxuh/hvshfldoo| lq wkh GFP dqdo|vlv1 Krzhyhu/ wkhuh lv dq lpsruwdqw gl�huhqfhehwzhhq p| dssurdfk dqg wkh GFP dssurdfk1 Xqghu GFP/ surgxfw fkdu0dfwhulvwlfv duh ixuwkhu uhgxfhg wr rqh fkdudfwhulvwlf fdoohg �txdolw|�1 +Qrwhwkdw sulfh lv dovr rqh ri wkh surgxfw fkdudfwhulvwlfv1, Surgxfwv duh shuihfwvxevwlwxwhv iru hdfk rwkhu ehfdxvh frqvxphuv zloo rqo| fkrrvh wkh rqh zlwkwkh kljkhvw txdolw|1 Glvfuhwh surgxfw fkrlfh prghov L glvfxvvhg deryh kdyhwklv shuihfw vxevwlwxwdelolw| surshuw|1 Lq FFP hdfk frqvxphu kdv gl�huhqwghjuhhv ri vxevwlwxwdelolw|/ dqg hyhq frpsohphqwdulw|/ dprqj surgxfwv/ gh0shqglqj rq wkh dprxqw ri fkdudfwhulvwlfv suhvhqwhg1 Wzr vriw gulqnv zrxogeh forvh vxevwlwxwhv li wkh| kdyh vlplodu fkdudfwhulvwlfv/ exw wkh| fdq dovr ehfrpsohphqwv li wkh| duh yhu| gl�huhqw1 L dp deoh wr lqfoxgh wklv frpsolfdwhguhodwlrqvkls lq d sduvlprqlrxv zd| e| xvlqj wkh khgrqlf dssurdfk lq wkhiudphzrun ri FFP1

Vlpxodwhg Phwkrg ri Prphqwv Wkh h{lelolw| dqg frqflvhqhvv ri FFPduh zhoo nqrzq1 Krzhyhu/ wkhuh lv d elj whfkqlfdo sureohp wr lwv ehlqj dssolhglq wkh hpslulfdo olwhudwxuh= wkh h{lvwhqfh ri }hur sxufkdvhv eulqjv d w|slfdofhqvruhg gdwd sureohp1 Frqyhqwlrqdo dssurdfkhv/ vxfk dv wkh pd{lpxp olnh0olkrrg dssurdfk ru wkh ohdvw vtxduhv dssurdfk/ fdq eh xvhg li wkhuh duh rqo|d ihz dowhuqdwlyhv +vhh Dphpl|d +4<;7,1, Zkhq frqvxphuv* fkrlfh vhw lvodujh/ }hur sxufkdvhv ri pdq| surgxfwv h{lvw/ dqg frqyhqwlrqdo dssurdfkhvduh ghhphg wr eh lpsudfwlfdo gxh wr wkh frpsxwdwlrqdo frpsoh{lw|1 Lw lvqhfhvvdu| wr lqwhjudwh rxw wkh fruqhu vroxwlrqv/ exw wkh rughu ri lqwhjudwlrqlv xvxdoo| wrr kljk +wklv lv wkh zhoo0nqrzq �fxuvh ri glphqvlrqdolw|�, dqg wkhfruuhvsrqglqj Mdfreldq pdwul{ wrr frpsolfdwhg zkhq wkh qxpehu ri dowhuqd0wlyhv lv odujh1 Wklv sdshu lv ghdolqj zlwk d sureohp zlwk d yhu| odujh fkrlfh

7

vhw vlqfh wkhuh duh qxphurxv vriw gulqnv fr0h{lvwlqj lq vxshupdunhwv1Wr vroyh wklv sureohp L xvh wkh vlpxodwhg phwkrg ri prphqwv +VPP,/

ghyhorshg e| Sdnhv +4<;9,/ PfIdgghq +4<;<,/ dqg Sdnhv dqg Sroodug +4<;<,1L gluhfwo| vlpxodwh wkh rswlpdo txdqwlw| sxufkdvhg ri hdfk surgxfw/ dqg wkhqxvh prphqw frqglwlrqv wr vhdufk sdudphwhuv1 Li wkh vdpsoh vl}h lv odujh/ wkhVPP fdq lqwhjudwh rxw }hur sxufkdvhv zlwk d �qlwh qxpehu ri vlpxodwlrqgudzv zlwkrxw zruu|lqj wkh h!flhqf| ri hvwlpdwruv1 L zloo vkrz lq Dsshqgl{D wkdw wkh VPP hvwlpdwruv dssolhg wr FFP kdyh wkh surshuwlhv ri frqvlv0whqf| dqg dv|pswrwlf qrupdolw| xqghu vrph jhqhudo frqglwlrqv

�1

Vrph Uhvxowv dqg d Urdg Pds L frpsxwh wkh hvwlpdwhg pdunhw gh0pdqg/ dqg xvh wkh uhvxow wr hvwlpdwh rzq0sulfh dqg furvv0sulfh ghpdqg hodv0wlflwlhv dqg surgxfw glvsod| h�hfwv1 L �qg gl�huhqw ghjuhhv ri vxevwlwxwdelolw|dprqj surgxfwv/ dqg wzr surgxfwv duh forvh vxevwlwxwhv zkhq wkh| kdyh wkhvdph dyruv exw gl�hu lq sdfndjlqj dqg frqwdlqhu vl}h1 Rqh lqwhuhvwlqj uhvxowlv wkdw frpsohphqwdulw| dovr h{lvwv dprqj surgxfwv1 Iru h{dpsoh/ d 450sdfnFrfd Frod Fodvvlf dqg d 50olwhu Vsulwh duh frpsohphqwv1 D jhqhudo sdwwhuqlv wkdw wzr surgxfwv duh frpsohphqwv zkhq wkh| kdyh gl�huhqw dyruv dqgsdfndjlqj ru frqwdlqhu vl}h1 Fkhfnlqj iurp d vlpsoh uhjuhvvlrq prgho L �qgwkdw frpsohphqwdulw| grhv h{lvw lq wkh gdwd1 Wklv uhodwlrqvkls lv jhqhudwhge| frqvxphuv* ehkdylru ri ydulhw| vhhnlqj dqg lqwhughshqghqfh ri surgxfwfkrlfhv/ dqg h{sodlqhg lq d vwudljkwiruzdug zd| e| xvlqj wkh khgrqlf ds0surdfk1

Surgxfw glvsod| h�hfwv dovr �qg vlplodu vxevwlwxwdelolw| dqg frpsohphq0wdulw| sdwwhuqv1 L dovr �qg wkdw uhyhqxh zloo eh kljkhu zkhq wzr frpsohphqwvduh glvsod|hg vlpxowdqhrxvo| wkdq zkhq wkhvh wzr surgxfwv duh glvsod|hgvhsdudwho|1 Uhyhqxh zloo dovr eh kljkhu zkhq surgxfwv zlwk odujh pdunhwvkduh duh glvsod|hg1 Wkhvh uhvxowv h{sodlq zk| zh �qg Frfd Frod Fodvvlf ruShsvl duh glvsod|hg pruh iuhtxhqwo|/ dqg zk| zh vhogrp �qg 450sdfn Frnhglvsod|hg wrjhwkhu zlwk 50olwhu Frnh1

Wklv prgho fdq eh h{whqghg wr dqdo|}h frqvxphu zhoiduh fkdqjh zkhqsulfhv fkdqjh/ ru zkhq qhz surgxfwv hqwhu ru h{lvwlqj surgxfwv h{lw wkh pdu0nhw1 Uhvxowv lq wklv sdshu duh khosixo lq xqghuvwdqglqj sulflqj dqg surgxfw

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " � � � � � � � � � � � � # $ % $ & ' ( % ) * ' + $ , - - . / 0 $ 1 2 $ 0 3 4 $ 0 5 ' + $ 6 7 0 0 / * $$ 7 8 + 9 ) ) 4 7 0 7 & 3 & 4 $ : $ & 4 $ & ' 3 4 $ & ' 3 ' 6 3 & ' + $ / ' 3 ; 3 ' 6 ( / & 8 ' 3 ) & 1 < $ & 8 $ $ 0 ' 3 * 7 ' 3 ) & % $ 0 / ; ' 0 3 &* 6 * ) 4 $ ; 8 ) & ' 7 3 & * 7 & 6 3 * : ; 3 8 7 ' 3 ) & 0 4 3 # $ % $ & ' ( % ) * ' + $ 3 % 0 1

8

glvsod| vwudwhjlhv zkhq d pdqxidfwxuhu surgxfhv pxowlsoh surgxfwv1 Wkhprgho fdq dovr eh dssolhg wr rwkhu hpslulfdo vwxglhv vxfk dv vwruh surgxfwglvsod| ghflvlrqv/ lqglylgxdo lqyhvwphqw ghflvlrqv/ dqg pdqxidfwxuhuv* dgyhu0wlvlqj ghflvlrqv/ lq zklfk fdvhv ghflvlrq pdnlqj lqyroyhv pxowlsoh dowhuqdwlyhvdqg pxowlsoh xqlwv zlwk wkh h{lvwhqfh ri pdq| fruqhu vroxwlrqv1 Lw lv prvwxvhixo wr wkh w|sh ri ghflvlrq pdnlqj zkhq fkrlfhv pdgh iru rqh dowhuqdwlyhkdyh h{whuqdo h�hfwv rq fkrlfhv pdgh iru rwkhu dowhuqdwlyhv1

Wkh uhvw ri wkh sdshu lv vwuxfwxuhg dv iroorzv1 Vhfwlrq 5 glvfxvvhv wkhprgho dqg surylghv vrph ghwdlov ri wkh pxowlsoh surgxfw dqg frqwlqxrxvtxdqwlw| sxufkdvlqj ehkdylru ri vriw gulqnv1 Vhfwlrq 6 suhvhqwv wkh hvwlpdwlrqdssurdfk dqg jlyhv uhvxowv frqfhuqlqj frqvlvwhqf| dqg dv|pswrwlf qrupdolw|ri hvwlpdwruv1 Wkh gdwd dqg hvwlpdwlrq uhvxowv duh glvfxvvhg lq Vhfwlrq 71Vrph lpsolfdwlrqv/ dssolfdwlrqv/ h{whqvlrqv dqg olplwdwlrqv ri wkh prgho duhdovr glvfxvvhg1 Ilqdoo| Vhfwlrq 8 frqfoxghv1

5 Wkh Prgho

514 Vrph Prwlydwlqj Idfwv

� � � � � � � � � 4 5 6 7 8 9 : ; < 43. Wrwdo

4 7:49 4:37 789 859 44: 53< 73 458 45 594 ;4995 4<74 :<8 797 595 4;9 43; <6 7; 4<9 73<66 963 746 53< 494 :4 <3 6; 4;< 4;347 576 474 459 8: 8; 77 44; :;:8 434 :; 7< 85 5< ;: 6<99 6; 57 53 47 :4 49:: 47 4; 8 84 ;;; 8 ; 64 77< 59 5943. 4< 4<Wrwdo 7:49 6978 4;;4 4979 ;63 :<; 696 794 4<; 437< 488;:

Wdeoh 4= Mrlqw iuhtxhqf| ri qxpehu ri VNXv

dqg qxpehu ri xqlwv sxufkdvhgWr xqghuvwdqg wkh h{whqw ri wkh pxowlsoh surgxfw pxowlsoh xqlw sxufkdvh

ehkdylruv/ orrn dw wkh gdwd1 P| gdwd frqwdlqv lqirupdwlrq derxw jurfhu|vkrsslqj iru vriw gulqnv e| 87; krxvhkrogv dw vl{ vxshupdunhwv gxulqj wkh

9

shulrg iurp Mxqh 4<<4 wr Mxqh 4<<6/ dowrjhwkhu 437 zhhnv1 Wkhuh duh dowr0jhwkhu 488;: ylvlwv1 Wkh gdwd frqwdlq lqirupdwlrq derxw 43:; vwrfn nhhslqjxqlwv +VNXv,1 Wdeoh 4 uhsruwv wkh qxpehu ri ylvlwv zlwk wkh vshfl�hg qxpehuri VNXv dqg xqlwv sxufkdvhg1

Derxw :3 shufhqw ri ylvlwv kdyh pxowlsoh xqlw sxufkdvhv/ zklfk phdqvwkdw wkh wudglwlrqdo GFP fdq rqo| h{sodlq derxw 63 shufhqw ri vkrsslqjehkdylru1 Lqghhg derxw kdoi wkh ylvlwv kdyh sxufkdvhv ri wkuhh ru pruh xqlwv/dqg derxw <1: shufhqw ri ylvlwv kdyh sxufkdvhv ri pruh wkdq rqh xqlw risdfnhg gulqnv1 Pruhryhu/ derxw 7; shufhqw ri ylvlwv kdyh sxufkdvhv ri pxowl0soh VNXv/ lpso|lqj d vljql�fdqw ydulhw|0vhhnlqj whqghqf|1 Zh pd| vxvshfwwkdw frqvxphuv vwlfn wr rqh dwwulexwh dqg qhyhu vhhn ydulhw|1 Iru h{dpsoh/frqvxphuv pd| rqo| ex| glhw gulqnv1 Wdeoh 5 olvwv vhyhudo dwwulexwhv dqgvkrzv wkh shufhqwdjh ri ylvlwv zlwk sxufkdvhv ri �vlqjoh� ru �pxowlsoh� dw0wulexwhv= li frqvxphuv sxufkdvh/ vd|/ erwk glhw dqg qrq0glhw gulqnv lq d ylvlw/lw lv ghqrwhg e| �pxowlsoh�/ rwkhuzlvh lw lv ghqrwhg e| �vlqjoh�1

Dwwulexwhv Vlqjoh Pxowlsohfrod2rwkhu dyruv ;:19 4517glhw2qrq0glhw <315 <1;fd�2qrq0fd� ;913 5713fdq2erwwoh <518 :18vl}hv <319 <17eudqgv :914 561<Wdeoh 5= Shufhqwdjh ri sxufkdvhv

ri surgxfw dwwulexwhv

Lw lv fohdu wkdw ydulhw|0vhhnlqj lq surgxfw dwwulexwhv lv dovr vljql�fdqw1 Iruh{dpsoh/ derxw 57 shufhqw ri ylvlwv kdyh pxowlsoh0eudqg sxufkdvhv

�/ lpso|lqj

ydulhw|0vhhnlqj lq eudqg qdphv1 Frqvxphuv ex| erwk frod dqg rwkhu dyruvlq derxw 45 shufhqw ri ylvlwv/ dqg erwk fd�hlqdwhg dqg qrq0fd�hlqdwhg gulqnvlq derxw 57 shufhqw ri ylvlwv1 Wkhvh ehkdylruv lqglfdwh hlwkhu d glyhuvlw| risuhihuhqfhv dprqj krxvhkrog phpehuv/ ru ydulhw|0vhhnlqj ri wdvwhv ri d vlqjohfrqvxphu1

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " � � � � # $ � % & � # ' ( � � ) *� + , - . / - 0 1 2 3 0 , 4 5 6 2 / 7 2 8 , 2 0 9 : - - 0 ; 2 : 2 : 6 , / 3 < = > ? @ A

:

515 Prgho ri Xwlolw| Pd{lpl}dwlrq

Lw lv dvvxphg wkdw wkh vxsso| vlgh/ zklfk lqfoxghv sulfhv dqg surgxfw fkdu0dfwhulvwlfv +lqfoxglqj pdunhwlqj dfwlylwlhv vxfk dv surgxfw glvsod|v dqg ihd0wxuhv,/ lv h{rjhqrxv

�1 Frqvxphuv pd{lpl}h wkhlu xwlolw| xqghu lqfrph frq0

vwudlqwv rq hdfk sxufkdvh rffdvlrq1L dvvxph wkdw xwlolw| lv ghulyhg qrw iurp wkh frqvxpswlrq ri surgxfwv

exw iurp wkh frqvxpswlrq ri surgxfw fkdudfwhulvwlfv/ dqg wkh uhodwlrqvklsehwzhhq fkdudfwhulvwlfv dqg txdqwlw| sxufkdvhg lv ghwhuplqhg e| d fkdudf0whulvwlfv htxdwlrq=

5 ' ��+

zkhuh 5 lv d ��� yhfwru ri wrwdo frqvxpswlrq ri fkdudfwhulvwlfv +erwk revhuy0deoh dqg xqrevhuydeoh,/ + d a � � yhfwru ri txdqwlw| sxufkdvhg ri surgxfwv/dqg � d a � � pdwul{ ri dprxqw ri fkdudfwhulvwlfv lq surgxfwv1 Wkhuh duh� � � revhuydeoh surgxfw fkdudfwhulvwlfv dqg � xqrevhuydeoh surgxfw fkdu0dfwhulvwlf1 Fruuhvsrqglqjo| wkh fkdudfwhulvwlf pdwul{ � kdv wzr sduwv= da � E� � �� vxepdwul{ iru wkh revhuydeoh fkdudfwhulvwlfv/ dqg d a � � yhf0wru iru wkh xqrevhuydeoh fkdudfwhulvwlf1 Ixoo ghwdlov ri surgxfw fkdudfwhulvwlfvzloo eh glvfxvvhg odwhu1 Wkh fkdudfwhulvwlfv htxdwlrq lv dqdorjrxv wr wkh khgr0qlf prgho lq Odqfdvwhu +4<99, dqg Guh}h dqg Kdjdq +4<:;,/ lpso|lqj olqhduvxppdelolw| dqg glylvlelolw| ri surgxfw fkdudfwhulvwlfv1

Wkh xwlolw| ixqfwlrq ri d frqvxphu lv ghqrwhg e| T E+ � c 5(Mc 0( w�c zkhuh+ � lv wkh rxwvlgh jrrg/ M d yhfwru ri krxvhkrog fkdudfwhulvwlfv/ vxfk dv krxvh0krog lqfrph dqg idplo| vl}h/ dqg w d g�� yhfwru ri sdudphwhuv1 0 lv d (��yhfwru ri vwrfkdvwlf yduldeohv zklfk lqfoxghv wzr sduwv= +4, vwrfkdvwlf ghyl0dwlrqv iurp wkh phdq suhihuhqfh iru revhuydeoh surgxfw fkdudfwhulvwlfv dqg/+5, vwrfkdvwlf xqrevhuydeoh surgxfw fkdudfwhulvwlfv1 Wklv zloo eh h{sodlqhglq ghwdlo odwhu1 Frqvxphuv pd{lpl}h wkh xwlolw| ixqfwlrq xqghu wkh lqfrphfrqvwudlqw

R�+ n R � + � ' U

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! " # $ % & ' ( ) * + , - . / 0 - 1 2 * 3 4 5 4 6 - 7 8 4 0 . + 8 - , 8 2 6 , 08 9 : ; < = > ? @ A B C D C E F G H I J K L M N O J K P Q R S T T U V W W X R W V Y S Z [ \ ] \ ^ [ _ ] ` \ a b c d e f g h e i j g k l m n o pq r s t u v s w x y z z { | } ~ � � � z � ~ � � � � � � � � � ~ | � } � � � � z � � � � z � � � � � z � � y � � | z � � z � � � � � z � � z � ~ � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �   ¡ ¢ £ ¤ ¥ ¦ £ § ¥ ¤ ¨ © ª ¦ ¥ « ¤ £ ¢ © ¬ ¦ ­ £ ¢ ® ¢ £ ¯ ° § ­ ± ¦ °² ¥ ¢ ­ ¤ ² ¥ ¨ ª ¨ £ ¢ ¨ © ° ¢ © £ ¤ ¥ ² ¥ ¤ £ ° ¦ ª ¢ ³ ¤ ´ ° £ ¥ ¦ £ ¤ ¬ ¯ ¨ µ ¨ ¶ · ¸ ¹ º · » ¼ ½ ¹ º ¹ ¾ ¿ À º Á  · · Ã Ä Å Æ Ä Ç È É Ê Ë Ì Í ÎÏ Ð Ñ Ð Ò Ó Ô Õ Ð Ö × Ø Ù Ú Ú Û Ü Ý Þ ß Þ à á â ã ß ß Þ ä å æ ç è é ê ë ì ì í î è ï ð ç ñ ò ó ô é õ î ö ÷ ø ù ø ú û ü ø ý ù þ ÷ ÿ þ � � � � ý ü ÿ þ ø � �� ý þ � ý ý � û � ø � ý � û � � � � � þ � ø ù û ü ÿ ÿ � � ý ÿ þ � � ý � ÷ ÿ � � ý ù � ÿ � � þ ÷ ý � � � � ù ý � � ý � û � � � � � þ � ÷ ÿ � ÿ � þ ý � ø ù þ ø � ù ù ÷ � � ü � � ý ú ø � ø ú ÿ ü � ÿ � � þ ÷ ý ÿ ù ù � ú û þ ø � � ø ù � ÿ ü ø � �

;

zkhuh R lv d a�� yhfwru ri sulfhv ri surgxfwv/ dqg R � wkh sulfh ri wkh rxwvlghjrrg/ dqg U wkh krxvhkrog lqfrph1

Wkhuhiruh wkh remhfwlyh ixqfwlrq ri d frqvxphu fdq eh zulwwhq dv

4@ � � � � � � T E+ � c 5(Mc 0( w�

r�|� 5 ' �� +R � + n R + ' U

+c + � f +4,

Wkh xwlolw| ixqfwlrq T E�� lv gh�qhg lq wkh fkdudfwhulvwlfv vsdfh ] dqg wkhsurgxfw vsdfh ri wkh rxwvlgh jrrg1 L fdq vxevwlwxwh wkh fkdudfwhulvwlfv htxd0wlrq lqwr wkh xwlolw| ixqfwlrq lq +4,1 L dovr dvvxph wkdw wkh xwlolw| ixqfwlrq lvdgglwlyho| dqg olqhduo| vhsdudeoh lq + 1 Wkhuhiruh L fdq uhzulwh

T E+ c �� +c 0( w� ' LE�� +(Mc 0(�� n b+

zkhuh � lv d Eg � �� � � yhfwru ri sdudphwhuv/ dqg b lv d vfdodu1 Khuh zhkdyh w ' i�� c bj � �

Wkh iroorzlqj duh vhyhudo dvvxpswlrqv=

Dvvxpswlrq 4=

Wkh ixqfwlrq LE�� lv wzlfh0gl�huhqwldeoh lq 5 +dqg khqfh dovr lq+,/ dqg L� � E�� qhjdwlyh gh�qlwh zlwk suredelolw| rqh1 Pruhryhu/LE�� lv frqwlqxrxvo| gl�huhqwldeoh lq �1

Dvvxpswlrq 5=

Wkh glphqvlrq ri wkh fkdudfwhulvwlfv vsdfh � lv vpdoohu wkdq wkhglphqvlrq ri wkh surgxfw vsdfh a 1 Wkh fkdudfwhulvwlfv pdwul{ �lv ri ixoo froxpq udqn1

Dvvxpswlrq 6=

Dq lqwhulru vroxwlrq iru + � dozd|v h{lvwv1 Wkdw lv/ frqvxphuv grqrw vshqg wkhlu hqwluh exgjhwv mxvw rq gulqnv1

Dvvxpswlrq 7=

Wkh sulfh ri wkh rxwvlgh jrrg/ R � c lv qrupdol}hg wr rqh1

<

Dvvxpswlrq 8=

Vwrfkdvwlf yduldeohv 0*v duh ����_� ryhu gl�huhqw frqvxphuv�1

Ohw �b eh wkh Odjudqjldq pxowlsolhu ri lqfrph1 Iurp Dvvxpswlrq 6 dqgwkh �uvw rughu frqglwlrqv/ lw lv hdv| wr rewdlq b ' �b � R � � Iurp Dvvxpswlrq 7/zh kdyh b ' �b� Vxevwlwxwlqj wkh lqfrph frqvwudlqw lqwr wkh xwlolw| ixqfwlrq/frqvxphu*v remhfwlyh ixqfwlrq fdq eh zulwwhq dv

4@ � � � T EUc �� +c 0( w� ' LE�� +(Mc 0(��� bR � +r�|� + � f +4*,

Qrwh wkdw wkh whup bU lv uhpryhg iurp htxdwlrq +4, dqg uhvxowv zloo qrw ehd�hfwhg1

Surgxfw Fkdudfwhulvwlfv Wkhuh duh wzr vxepdwulfhv lq wkh fkdudfwhu0lvwlfv pdwul{ �= d a � E� � �� vxepdwul{ fruuhvsrqglqj wr wkh revhuydeohsurgxfw fkdudfwhulvwlfv/ dqg d a�� yhfwru/ �c fruuhvsrqglqj wr wkh xqrevhuy0deoh surgxfw fkdudfwhulvwlfv +zklfk duh lqglylgxdo vshfl�f dqg xqrevhuyhg wrhfrqrphwulfldqv1,

Li zh lqfoxgh pdunhwlqj dfwlylwlhv vxfk dv surgxfw glvsod|v dqg ihdwxuhvdv revhuydeoh surgxfw fkdudfwhulvwlfv � / wkh vxepdwul{ lq � fruuhvsrqglqj wrwkh revhuydeoh surgxfw fkdudfwhulvwlfv fdq eh ixuwkhu eurnhq lqwr wzr sduwv=+4, d a � �� vxepdwul{/ ��c fruuhvsrqglqj wr �lqwulqvlf� revhuydeoh surgxfwfkdudfwhulvwlfv/ vxfk dv dyru/ sdfndjlqj dqg frqwdlqhu vl}h ri vriw gulqnv/zkhuh �� lv wkh glphqvlrq ri wkh lqwulqvlf revhuydeoh surgxfw fkdudfwhulvwlfvdqg/ +5, d a � 2 vxepdwul{/ E� � c � � �/ fruuhvsrqglqj wr surgxfw glvsod|v dqgihdwxuhv1 L xvh lqglfdwruv wr uhsuhvhqw pdunhwlqj dfwlylwlhv/ zkhuh � lv d a��yhfwru ri lqglfdwruv zkrvh &0wk frpsrqhqw lv 4 li surgxfw & lv glvsod|hg/ dqg3 rwkhuzlvh1 � lv d vlplodu a � � yhfwru iru surgxfw ihdwxuhv1

Lq wklv sdshu L hvwlpdwh wzr w|shv ri prghov= wkh �uvw rqh/ prgho P40D/ ljqruhv wkh h�hfw ri surgxfw glvsod|v dqg ihdwxuhv zkloh wkh vhfrqg rqh/prgho P40E/ lqfoxghv wkhp lq wkh xwlolw| ixqfwlrq1 Vr zh kdyh � ' d ��c � o

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " � # � � � # $ % " & " ' ( � ( " & � � ) � & ) " � $ # � � ! # & ' ( * �� + & � � ! ( , � - � " & � # ! # & ' � � " # ! � % . � � � � � � � � � � ! " � /

0 1 2 3 4 3 5 6 7 3 8 4 9 2 : 9 5 : ; < 8 9 3 = > : ? 9 3 @ 3 9 3 8 4 ? ; 8 : 9 8 = 8 A B 3 5 8 = 4 3 C = 4 3 = 9 2 8 ? 2 : ; : ? 9 8 ; 3 4 9 3 ? 44 6 : ? 8 : = B B 3 ; 8 ? 9 7 D 8 = 9 8 ; ? C = 4 E 5 8 ; 4 F E 9 3 7 3 9 D G E = ? 9 3 C = H I B @ 8 ; 9 3 4 3 = > J : = C 9 2 8 ; 3 5 6 C ; 9 : = 9 5 : ; K

< 8 9 3 = > @ : ; 3 : L 7 8 3 = 9 2 8 4 C G 9 B ; 3 = < 3 = B E 4 9 ; D L E 9 = C 9 ? C = 4 3 B 8 ; 8 B 3 = 9 2 3 4 6 : 6 8 ; J ? : = L 8 9 ; 8 : 9 8 B3 = : 4 3 5 3 7 : ; A : D H

43

lq wkh �uvw prgho/ dqg � ' d ��c � � c � � c �o lq wkh vhfrqg1 Fruuhvsrqglqjo|/fkdudfwhulvwlfv sxufkdvhg 5 fdq eh zulwwhq dv

5 ' ��+ '

���� +� � + � dqg/ 5 ' �� + ' �����

�� +� � +� � +� +

�����lq P04D dqg P04E uhvshfwlyho|1

L dvvxph wkdw h�hfwv ri surgxfw glvsod|v dqg ihdwxuhv dv zhoo dv wkh xqre0vhuydeoh surgxfw fkdudfwhulvwlf hqwhu wkh xwlolw| ixqfwlrq T E�� olqhduo|1 Wkhuh0iruh L fdq uhzulwh htxdwlrq +4*, iru P40D dv

4@ � � � T EUc �� +c 0( w� ' �LE �� � +(Mc #( ���� bR� + n � � + r�|� + � f +P40D,

zkhuh �LE�� lv wkh sduw lq wkh xwlolw| ixqfwlrq fruuhvsrqglqj wr wkh �lqwulqvlf�revhuydeoh surgxfw fkdudfwhulvwlfv/ # wkh vwrfkdvwlf ghyldwlrqv iurp wkh phdqsuhihuhqfh iru wkhvh fkdudfwhulvwlfv/ dqg �� d yhfwru ri sdudphwhuv lq �LE���Vlploduo|/ L fdq uhzulwh htxdwlrq +4*, iru prgho P40E dv

4@ � � � T EUc �� +c 0( w� ' �LE ��� +(Mc #( ���n� � �� � +n� � � � � +�bR � +n� � + r�|� + � f

+P40E,zkhuh � � dqg � � duh wzr vfdoduv uhsuhvhqwlqj wkh h�hfwv ri surgxfw ihdwxuhv

dqg glvsod|v1

Krxvhkrog Fkdudfwhulvwlfv L zloo qrz dgguhvv krz wkh sdqho dvshfw rigdwd hqwhuv wkh prgho1 Ohw � ' �c ���c� ghqrwhv gl�huhqw frqvxphuv lq wkhgdwd/ dqg | ' �c ���cA� gl�huhqw ylvlwv ri frqvxphu � gxulqj wkh vdpsoh shulrg1H{rjhqrxv yduldeohv f� � c lqfoxglqj wkh fkdudfwhulvwlfv vxepdwul{ d ��c � � � c � � odqg iR! c M" jc duh revhuyhg/ zkhuh M" lv d yhfwru ri krxvhkrog gdwd iru frq0vxphu �/ vxfk dv krxvhkrog lqfrph dqg idplo| vl}h/ R ! d yhfwru ri surgxfwsulfhv/ dqg � # � dqg � � wzr yhfwruv ri lqglfdwruv iru surgxfw glvsod|v dqgihdwxuhv dw wlph |1

Lqwxlwlyho|/ wkh odujhu wkh idplo| vl}h wkh odujhu wrwdo txdqwlw| ghpdqghgiru vriw gulqnv zloo eh1 Wr lqfrusrudwh wkh idplo| vl}h h�hfw/ L kdyh 7+$ % dqg75$ % uhsuhvhqwlqj wkh shu fdslwd krxvhkrog txdqwlw| sxufkdvhg ri surgxfwv dqgsurgxfw fkdudfwhulvwlfv uhvshfwlyho|1 Wkh shu fdslwd txdqwlw| sxufkdvhg lvgh�qhg dv

7+$ % ' +$ % *E6& � '44

zkhuh6� lv wkh idplo| vl}h ri frqvxphu �/ dqg frh!flhqw k wkh h�hfw ri idplo|vl}h rq wrwdo krxvhkrog frqvxpswlrq1 k htxdo wr f lpsolhv wkdw wkh fkdqjhlq idplo| vl}h kdv qr h�hfw rq wrwdo frqvxpswlrq/ dqg k htxdo wr � lpsolhv/kroglqj rwkhu wklqjv frqvwdqw/ wkdw grxeolqj wkh idplo| vl}h zloo grxeoh wrwdofrqvxpswlrq1 Fruuhvsrqglqjo|/ wkh shu fdslwd txdqwlw| sxufkdvhg ri surgxfwfkdudfwhulvwlfv lv 75� � ' �

�7+� � �

L xvh b � wr uhsuhvhqw frqvxphu �*v pdujlqdo xwlolw| ri lqfrph1 L dvvxph

b� ' 7bn � *?EU � �

zkhuh � uhsuhvhqwv wkh h�hfw ri krxvhkrog lqfrph rq frqvxphu �*v pdujlqdoxwlolw|1

Wkh Vshfl�fdwlrq ri wkh Ixqfwlrq �LE�� L xvh wkh wudglwlrqdo udqgrpfrh!flhqw prgho dqg jlyh d vshfl�f dvvxpswlrq iru wkh ixqfwlrqdo irup ri�LE���

�LE ���7+� � � ' q

� d*?E ��

�7+ n����iq � � fjonq

� dEi TE ��

�7+� ������iq � � fjo +5,

zkhuh q � � lv frqvxphu �*v udqgrp frh!flhqwv iru wkh lqwulqvlf revhuydeohsurgxfw fkdudfwhulvwlfv dw wlph |> wkh ixqfwlrq *?E�� dq hohphqw0e|0hohphqw *?ixqfwlrq ri wkh yhfwru lqvlgh wkh eudfnhwv/ dqg i TE�� vlploduo| gh�qhg> �i�jd yhfwru ri lqglfdwru ixqfwlrqv zkhuh wkh &0wk hohphqw htxdov � li wkh orjlfdoh{suhvvlrq lqvlgh eudfnhwv lv wuxh/ dqg f rwkhuzlvh> ��� dq hohphqw0e|0hohphqwpxowlsolfdwlrq/ dqg �� �� dqg �� � hohphqw0e|0hohphqw orjlfdo h{suhvvlrqv1

Wkh vshfldo ixqfwlrqdo irup lq htxdwlrq +5, pd| vhhp vwudqjh exw lv lq0wxlwlyho| dsshdolqj1 Wr pdnh �LE�� vwulfwo| frqfdyh lq 75/ L qhhg Y

��LE��*Y75

��

f iru doo lqwulqvlf revhuydeoh surgxfw fkdudfwhulvwlfv S/ hyhq zkhq vrph fkdudf0whulvwlfv duh �edgv� wr lqglylgxdo frqvxphuv +lq zklfk fdvh wkh fruuhvsrqglqjq*v duh qhjdwlyh1, L gr qrw h{foxgh wkh srvvlelolw| wkdw vrph q*v duh qhjd0wlyh wr vrph frqvxphuv ehfdxvh/ iru h{dpsoh/ vrph frqvxphuv pd| wuhdw wkhfkdudfwhulvwlf ri �fd�hlqh� dv d edg zkloh rwkhuv wuhdw lw dv d jrrg1 Qrwhwkdw lw grhv qrw lpso| wkh surgxfw ehlqj d edg/ vlqfh pdujlqdo xwlolw| rifrqvxplqj wkh surgxfw lv qrw wkh ghulydwlyh Y �LE��*Y75 � c exw �� EYT E��*Y75�/zkhuh �� lv wkh �0wk urz lq pdwul{ �1

Wkh vshfl�fdwlrq lq htxdwlrq +5, jxdudqwhhv wkdw/ li q � � 9' f/ Y��LE��*Y75

��

f iru doo S/ uhjdugohvv wkh vljq ri q � � 1 Wklv lpsolhv d ghfuhdvlqj pdujlqdo xwlo0lw| ri frqvxplqj jrrgv/ dqg dq lqfuhdvlqj pdujlqdo glvxwlolw| ri frqvxplqj

45

edgv1 Lw lv reylrxv wkdw wkh Mdfreldq pdwul{ �L �� �� E�� lv qhjdwlyh gh�qlwh zlwksuredelolw| rqh/ dqg �LE�� lv frqwlqxrxvo| gl�huhqwldeoh lq wkh sdudphwhu vsdfhX h{fhsw dw q � � ' f1

Wr dyrlg wkh pdwkhpdwlfdo sureohp ri *?Ef� L kdyh d frqvwdqw � lq wkh *?ixqfwlrq lq htxdwlrq +5,1

Udqgrp frh!flhqwv q � � duh gh�qhg dv

q � � ' 7q n e� � c e� � ' l � #

zkhuh # lv d ���� yhfwru ri vwdqgdug qrupdo ghyldwhv/ dqg l � d ��� �� pdwul{vxfk wkdw l

�� l � lv d yduldqfh0fryduldqfh pdwul{/ dqg �� wkh glphqvlrq ri

wkh lqwulqvlf revhuydeoh surgxfw fkdudfwhulvwlfv1 Wr plqlpl}h frpsxwdwlrqdoexughq L dvvxph wkdw wkh glvwulexwlrq ri doo hohphqwv lq # duh lghqwlfdo dqglqghshqghqw ri hdfk rwkhu/ vr L fdq uhzulwh l � # dv j � # � c j � d vfdodu1

L rqo| qhhg wr hvwlpdwh d j � c lqvwhdg ri wkh pdwul{ l � / xqghu wkh ����_�dvvxpswlrq iru #1 Krzhyhu/ gldjrqdo hohphqwv lq l � / lpso|lqj wkh yduldqfhvri suhihuhqfhv iru surgxfw fkdudfwhulvwlfv/ fdq eh gl�huhqw/ dqg r�0gldjrqdohohphqwv lql � c lpso|lqj fruuhodwlrqv ri suhihuhqfhv iru gl�huhqw surgxfw fkdu0dfwhulvwlfv/ fdq dovr eh qrq0}hur1 Dv dq h{whqvlrq ri wklv sdshu/ L zloo hvwlpdwhwkh pdwul{ l � 1 Rqh zd| wr uhgxfh wkh frpsxwdwlrqdo exughq lv wr dvvxphwkdw wkh yduldqfhv ri suhihuhqfhv iru ydulrxv vriw gulqn dyruv duh krpr0jhqhrxv/ exw wkh| duh gl�huhqw iurp wkh yduldqfhv ri suhihuhqfhv iru rwkhufkdudfwhulvwlfv vxfk dv sdfndjlqj dqg frqwdlqhu vl}h1 Lw dovr vhhpv sodxvleohwkdw suhihuhqfhv iru dyruv lv lqghshqghqw iurp suhihuhqfhv iru sdfndjlqj dqgfrqwdlqhu vl}h/ exw fruuhodwlrqv h{lvw dprqj suhihuhqfhv iru gl�huhqw dyruv1

Rwkhu Ghwdlov Ilqdoo| ��+ lq prghov P40D dqg P40E wdnhv dffrxqw ri wkh

xqrevhuyhg surgxfw fkdudfwhulvwlfv/ zkhuh � lv d a � � yhfwru ri vwrfkdvwlffrh!flhqwv1 Lqfrusrudwlqj wkh sdqho dvshfw dqg krxvhkrog fkdudfwhulvwlfv lqwkh gdwd/ ohw*v gh�qh ��

�� � 7+� � ' �Eb

�R � � �

� � ��7+

� � 1 Wr dyrlg wkh sureohp ri�

� � � � f iru vrph surgxfw �/ xqghu zklfk �*v txdqwlw| ghpdqghg zrxog ehlq�qlwh li lw lv d �jrrg� dqg ihdwxuh dqg glvsod| gr qrw eulqj glvxwlolw|/ Ldvvxph b

�: f iru doo � dqg 0� � � orj0qrupdoo| glvwulexwhg1 Wkdw lv/

*L}E�� � � � ' l � D � � c D � � � �Efc U � �

zkhuh l � lv d a � a pdwul{ vxfk wkdw l� �l

�d yduldqfh0fryduldqfh pdwul{/

dqg D � d a � � yhfwru ri vwdqgdug qrupdo ghyldwhv1 L dovr dvvxph wkdw wkhghyldwhv ri gl�huhqw surgxfwv duh ����_1/ vr l ! D " # fdq eh uhzulwwhq dv j $ D % & �

46

Wkh sxusrvh ri dvvxplqj ����_� iru D � � lv djdlq wr plqlpl}h wkh frpsxwd0wlrqdo exughq1 Exw wklv dvvxpswlrq lv sodxvleoh ehfdxvh/ dv zloo eh h{sodlqhglq odwhu vhfwlrqv/ hdfk surgxfw lq wklv sdshu lv d surgxfw fdwhjru|/ dqg qrwd eudqg1 Surgxfwv duh gl�huhqwldwhg iurp hdfk rwkhu e| srvvhvvlqj vrphxqltxh frpelqdwlrq ri lqwulqvlf revhuydeoh surgxfw fkdudfwhulvwlfv1 Khqfh/doo khwhurjhqhlw| dqg fruuhodwlrqv dprqj surgxfwv vkrxog eh devruehg e|wkh udqgrp frh!flhqwv q*v/ dqg zh pd| dvvxph � ehlqj ����_�1

Qrwh wkdw wkh vwrfkdvwlf yduldeoh 0 ' i#�c D

�j

�/ dqg lwv glphqvlrq ( '

�� n a�

Ilqdoo|/ p| sdudphwhuv lq prgho P40D duh w ' i7q�c j � ckc 7bc �c j � j

�/ dqg

lq prgho P40E duh w ' i7q�c j c kc � c � � c 7bc � c j � j

Prgho Urexvwqhvv Prghov +P40D, dqg +P40E, dvvxph ixqfwlrqv *? dqg

i T lq �LE��1 Wklv vshfl�fdwlrq uhvwulfwv wkh fxuydwxuh ri wkh xwlolw| ixqfwlrq1D pruh h{leoh ixqfwlrqdo irup/ fruuhvsrqglqj wr htxdwlrq +5,/ lv

�LE ���7+� � � '

���� � � q � � � � E ��� � 7+� � � � �� � +6,

zkhuh �� lv wkh S0wk froxpq lq wkh vxepdwul{ ��1 Wkh srzhu whup 4 ! lvuhvwulfwhg wr olh zlwklq Efc �� li q " # $ % � f/ dqg juhdwhu wkdq � li q & ' ( ) f1 E| wklv

vshfl�fdwlrq �LE�� lv vwloo qhjdwlyh gh�qlwh dqg kdv doo wkh rwkhu surshuwlhv riP40D dqg P40E/ exw lw lv pruh h{leoh1 Li ydulhw| vhhnlqj whqghqf| lv vwurqj/4 * zloo eh vpdoo iru doo wkrvh S*v zlwk q + , - * � f/ dqg frqvxphuv zloo doorfdwhwkhlu exgjhw wr ex| pdq| gl�huhqw surgxfw fkdudfwhulvwlfv1 Rwkhu qrwdwlrqvlq +6,/ vshfl�fdwlrqv ri q . / / b. dqg doo vwrfkdvwlf yduldeohv duh gh�qhg dvehiruh1

Fruuhvsrqglqj wr wklv vshfl�fdwlrq wkhuh zloo eh dqrwkhu vhw ri prghov=frqvxphu �*v remhfwlyh ixqfwlrq dw wlph |/ zkhq surgxfw glvsod|v dqg ihdwxuhvduh ljqruhg/ lv

4@ 0 1 2 3 4 T EU5 c �c +5 6 c 0 5 6 ( w� '78

9: ; < q = > ? : E ��@ : 7+ = > ? : �A B � bRC + n � C +

r�|� +D E � f +P50D,

zkhuh 4 F lv uhvwulfwhg wr olh zlwklq Efc �� li q D E G F � f/ dqg juhdwhu wkdq � liq D E G F f1 Zkhq surgxfw glvsod|v dqg ihdwxuhv duh lqfoxghg lq wkh xwlolw|

47

ixqfwlrq zh kdyh

4@ � � � � � T EU� c �c +� � c 0 � � ( w� '�� � � q � � E ��� 7+ � � �� � n � � � � � + n � � � � � + � bR� + n � � +

r�|� +� � � f +P50E,

Qrwh wkdw p| sdudphwhuv lq prgho P50D duh w ' i7q � c 4c j � c kc 7bc �c j � j � /dqg lq prgho P50E duh w ' i7q � c 4c j � c kc � � c � � c 7bc � c j � j � �

+Hvwlpdwlrq uhvxowv ri wkhvh wzr prghov zloo eh glvfxvvhg lq d uhylvhgyhuvlrq1,

6 Prgho Hvwlpdwlrq

614 Sureohpv

Jlyhq h{rjhqrxv yduldeohv f � i ��c � � c � � c RcMjc dqg vwrfkdvwlf yduldeohv 0czh duh deoh wr �qg wkh rswlpdo ydoxh +Efc 0( w� e| vroylqj wkh xwlolw| pd{l0pl}dwlrq prgho/ wkdw lv/

+Efc 0( w� '@h}4@ � � � T E+(fc 0( w� r�|� �?SJ6e SJ?r|o@�?|r +7,

Vxssrvh zh nqrz wkh frqglwlrqdo glvwulexwlrq ri 0c � E�mf�1 Wkh h{shfwhgghpdqg ixqfwlrq/ .+Efc w�c lv/

.+Efc w� ' � i@h}4@ � � � T E+(fc 0( w� r�|� �?SJ6e SJ?r|o@�?|rj� E_0mf� +8,

Gh�qh w � dv wkh yhfwru ri wuxh sdudphwhuv1 Zh fdq pdnh xvh ri wkhprphqw frqglwlrq

.E+ � .+Efc w � �mf� ' f +9,

wr �qg rxu hvwlpdwhv1 Krzhyhu/ .+Efc w� lv qrq0wudfwdeoh1Ohw*v ghqrwh e| + ' +Efc 0( w�� Rqh sureohp lq prgho hvwlpdwlrq lv zkhq

+ grhv qrw kdyh lqwhulru vroxwlrqv1 Lq d vxshupdunhw zh �qg qxphurxv vriwgulqnv/ dqg frqvxphuv xvxdoo| kdyh }hur sxufkdvhv iru prvw surgxfwv1 Hfrqr0phwulfldqv rqo| revhuyh }hur ru srvlwlyh ex|lqj/ zkloh frqvxphuv* zloolqjqhvvwr ex| ohvv wkdq }hur dprxqw lv klgghq1 Wklv lv d w|slfdo fhqvruhg gdwdsureohp1

48

Frqyhqwlrqdo hvwlpdwlrq phwkrgv vxfk dv wkh pd{lpxp olnholkrrg ru ohdvwvtxduhv dssurdfkhv pdnh xvh ri wkh Nxkq0Wxfnhu frqglwlrqv1 Iru hdfk surg0xfw �/

_T

_+� E+(fc 0( w�� �bR� ' f �s + � : f

_T

_+� E+(fc 0( w�� �bR� � f �s + � ' f +:,

Zh qhhg wr lqwhjudwh rxw wkh fruqhu vroxwlrqv1 Lw lv srvvleoh wr gr vr liwkhuh duh rqo| d ihz ghshqghqw yduldeohv exw/ zkhq wkh qxpehu ri ghshqghqwyduldeohv lv odujh/ wkh rughu ri lqwhjudwlrq zloo eh wrr kljk +wkdw lv/ �fxuvhri glphqvlrqdolw|�, dqg wkh fruuhvsrqglqj Mdfreldq pdwul{ wrr frpsolfdwhgwr frpsxwh1 Frqyhqwlrqdo dssurdfkhv duh lpsudfwlfdo gxh wr frpsxwdwlrqdofrpsoh{lw|1

615 Vlpxodwhg Phwkrg ri Prphqwv

Iru qrwdwlrqdo vlpsolflw| L dvvxph wkdw wkh gdwd lv furvv0vhfwlrqdo/ dqg ghqrwhhdfk revhuydwlrq ri sxufkdvh dv � ' �c ���c ?1 L xvh wkh phwkrg ri vlpxodwlrq wrvroyh wkh deryh sureohp1 Vxssrvh wkh frqglwlrqdo glvwulexwlrq ri 0c � E�mf�c lvnqrzq1 L fdq vlpxodwh.+Efc w� e| gudzlqj 0 � c ���c 0� � c zkhuh ?r lv wkh qxpehuri vlpxodwlrq gudzv iru hdfk revhuydwlrq/ iurp � E�mf�� Grlqj vlpxodwlrqv iruhyhu| �c L jhqhudwh d vhw ri gdwd �+Ef� c w� dv

�+Ef� c w� ' �

?r

� �� �

i@oS4@ � � T E+(f� c 0� � � ( w� r�|� �?SJ6e SJ?r|o@�?|rj +;,

Iurp wkh prgho frqvwuxfwlrq L kdyh wkh iroorzlqj prphqw frqglwlrq=

.E+� � �+Ef� c w � �mf� � ' f +<,

Khqfh L kdyh

CEw � � � .6Ef� c w � � ' .dE+ � � �+Ef� c w � ��� �Ef� �o ' f +43,

zkhuh �Ef� � lv d frqirupdeoh pdwul{ ri phdvxudeoh ixqfwlrqv1 Wkh vdpsohdqdorjxh ri CEw� lv

C� Ew� ' �

?

��� � �

6Ef� c w� ' �

?

��� � �

E+ � � �+Ef� c w��� �Ef� � +44,

49

Hvwlpdwru w � lv rewdlqhg e| �qglqj wkh plqlpxp ri

'� Ew� ' nC� Ew�nc ru '� Ew� ' nC� Ew�n � � � � � +45,

zkhuh n�n lv d qrup/ dqg �̀ Ew� d srvlwlyh gh�qlwh pdwul{1 Hvwlpdwruv re0wdlqhg iurp vxfk d dssurdfk duh fdoohg VlpxodwhgPhwkrg ri Prphqwv +VPP,hvwlpdwruv1

Wkh VPP lv zlgho| xvhg lq wkh hpslulfdo olwhudwxuh iroorzlqj PfIdgghq+4<;<, dqg Sdnhv dqg Sroodug +4<;<,1 \hw prvw ri wkh dssolfdwlrqv duh rqGFPv1 Iru h{dpsoh/ vhh Sdnhv +4<;9, rq ghflvlrqv ri wkh uhqhzdo ri sdwhqwv/dqg Ehuu|/ Ohylqvrkq dqg Sdnhv +4<<8, rq ghpdqg ixqfwlrqv lq wkh dxwrpdunhw1 Ghshqghqw yduldeohv lq Od�rqw/ Rvvdug dqg Yxrqj*v uhvhdufk +4<<8,rq �uvw0sulfh dxfwlrqv duh ydoxhv ri zlqqlqj elgv/ zklfk duh frqwlqxrxv1 Wkh|duh deoh wr vlpsoli| wkh sureohp e| dvvxplqj qr fruuhodwlrqv h{lvw dprqjgl�huhqw elgglqjv1

L zdqw wr vkrz wkdw wkh VPP hvwlpdwruv lq p| prgho/ xqghu vrph jhq0hudo frqglwlrqv/ duh frqvlvwhqw dqg dv|pswrwlfdoo| qrupdo1 L frqfhqwudwh p|dqdo|vlv rq htxdwlrq +4*,1 Wkdw lv

4@ � T EUc �� +c 0( w� ' LE�� +(Mc 0(��� bR +

r�|� + � f +4*,

Wr dyrlg frqixvlrq lq qrwdwlrqv/ qrwh wkdw w ' i� c bj �Pdlq uhvxowv duh jlyhq lq wkh Dsshqgl{1 L zloo dovr glvfxvv krz wr hvwlpdwh

wkh yduldqfhv ri hvwlpdwruv zkhq sdqho gdwd dvshfw hqwhuv wkh prgho1 Fkrlfhvri lqvwuxphqwv dv zhoo dv wkh dojrulwkp ri wkh prgho hvwlpdwlrq zloo dovr ehfryhuhg wkhuh1

7 Gdwd dqg Uhvxowv

714 Wkh Gdwd

L xvh d pdunhw gdwdedvh froohfwhg e| Lqirupdwlrq Uhvrxufhv Lqf1 +LUL,1Wkh gdwdedvh frqwdlqv lqirupdwlrq derxw jurfhu| vkrsslqj ehkdylru iru vriwgulqnv e| 87; krxvhkrogv dw vl{ gl�huhqw vxshupdunhwv lq d odujh phwursrol0wdq duhd lq wkh Xqlwhg Vwdwhv1 Lw frqwdlqv wzr |hduv ri gdwd iurp Mxqh 4<<4

4:

wr Mxqh 4<<61 Wkhuh duh dowrjhwkhu 4/3:; VNXv1 Wkh gdwd lqfoxghv +4, phu0fkdqgl}lqj lqirupdwlrq/ vxfk dv uhwdlo sulfhv/ ihdwxuhv dqg glvsod|v/ ri hdfkVNX dw hdfk vxshupdunhw lq hdfk zhhn> +5, sxufkdvh lqirupdwlrq iru hdfkkrxvhkrog iru hdfk VNX dw hdfk vxshupdunhw zkhq wkhuh lv srvlwlyh ex|lqj>+6, ghprjudsklf gdwd vxfk dv idplo| vl}h dqg krxvhkrog lqfrph1

L fkrrvh wkh wkuhh vxshupdunhwv zklfk kdyh wkh pdmrulw| ri ylvlwv dqgfrqfhqwudwh rq wkrvh zhhnv zlwk 83 ru pruh ylvlwv1 Wkhuh duh dowrjhwkhu 4;8vxshupdunhw0zhhnv dqg 44/43: ylvlwv1 Zlwk wklv gdwd vhohfwlrq L dp deoh wrpdnh prgho frpsdulvrqv lq odwhu vhfwlrqv

�1

Surgxfw Fkdudfwhulvwlfv Lw lv lpsrvvleoh wr hvwlpdwh wkh ghpdqg iru4/3:; VNXv1 Lqvwhdg L jurxs gl�huhqw VNXv dffruglqj wr wkhlu surgxfw fkdu0dfwhulvwlfv1 Lqglfdwruv duh xvhg wr uhsuhvhqw wkh frqwhqw ri surgxfw fkdudf0whulvwlfv= �4� li wkh fkdudfwhulvwlf h{lvwv lq wklv eudqg/ dqg �3� rwkhuzlvh1

L lghqwli| vl{ surgxfw fkdudfwhulvwlfv iru doo vriw gulqnv=

41 frod dyru +FROD, yv1 rwkhu dyruv +RWKHU,>

51 glhw gulqnv +GLHW, yv1 qrq0glhw gulqnv +QGLHW,>

61 fd�hlqdwhg gulqnv +FDII, yv1 qrq0fd�hlqdwhg gulqnv +QFDII,>

71 odujh0vl}hg frqwdlqhu +ODUJ, +frqwdlqhu vl}h odujhu wkdq ru htxdo wr650r}1,>

81 fdq +FDQ,> dqg

91 sdfnhg +SDFN, +pruh wkdq 4 xqlw lq wkh sdfndjh,1

Dv d uhvxow/ wkhuh duh dowrjhwkhu 63 jurxsv ri VNXv lq wkh gdwd/ hdfk zlwkvrph xqltxh frpelqdwlrq ri surgxfw fkdudfwhulvwlfv1 Khuhdiwhu L xvh �sur0gxfw� wr uhsuhvhqw d jurxs ri VNXv vkdulqj wkh vdph surgxfw fkdudfwhulvwlfv1Lq rughu wr rewdlq d ehwwhu hvwlpdwlrq/ vrph VNXv duh ljqruhg li wkh| fdqqrweh sxw lqwr dq| ri wkhvh wkluw| surgxfwv1 Wklv vkrxog qrw d�hfw hvwlpdwlrquhvxowv vlqfh wkhvh VNXv kdyh d qhjoljleoh pdunhw vkduh1 L xvh dyhudjh sulfhvzhljkwhg e| wrwdo pdunhw vkduh ri VNXv lq hdfk surgxfw dv surgxfw sulfhv1

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " # $ % & ' ( ) * + , - . - , - / 0 1 2 3 4 5 6 7 8 9 : ; < => ? @ A B > C D E @ E @ ? = C F G E H I J K L M N O P Q R S T T U V W X T T Y Z [ \ W ] X [ X S ^ _ Y ] W T W V [ S _ X [ S Y ^ V X V X Z U [ U ` WW a b c d e f g d h i d j k g l m i n c b o c c e b f l i b g n e p f b o k q n n c d b n c e q r b e s

4;

Lea

din

gM

arke

t S

har

eF

eatu

reD

isp

lay

Pro

du

ct C

har

acte

rist

ics

Exa

mp

le(%

)M

ea

nM

ax

.M

in.

Std

. Dev

.M

ea

nM

ea

n

1. O

TH

ER

, NC

AF

F, N

DIE

TS

prite

16-

oz.

Bot

tle0.

281.

755

3.12

20.

80.

504

0.15

10.

071

2. O

TH

ER

, NC

AF

F, N

DIE

T,

PA

CK

Spr

ite 1

6-oz

. 6

Pac

k1.

470.

968

1.15

80.

860.

057

0.35

90.

285

3. O

TH

ER

, NC

AF

F, N

DIE

T,

LAR

G2-

lit.

Bot

tle S

prite

130.

611

0.72

50.

447

0.05

0.91

0.97

1

4. O

TH

ER

, NC

AF

F, N

DIE

T,

CA

NS

prite

12-

oz.

Can

2.21

0.52

30.

597

0.18

20.

056

0.10

90.

083

5. O

TH

ER

, NC

AF

F, N

DIE

T,

CA

N P

AC

KS

prite

12-

oz. C

an 1

2 P

ack

8.08

0.77

1.05

0.52

30.

114

0.95

80.

916.

OT

HE

R, C

AF

F, N

DIE

T,

PA

CK

Dr.

Pep

per

16-oz

. 6

Pac

k 0.

080.

81.

163

0.64

50.

066

0.26

0.05

87.

OT

HE

R, C

AF

F, N

DIE

T,

LAR

G2-

lit. B

ottle

Dr.

Pep

per

1.58

0.49

60.

697

0.33

60.

062

0.75

0.64

78.

OT

HE

R, C

AF

F, N

DIE

T,

CA

N, P

AC

KD

r. P

eppe

r 12

- oz. C

an 1

2 P

ack

3.25

0.77

71.

649

0.47

30.

123

0.77

20.

724

9. O

TH

ER

, NC

AF

F, D

IET

, P

AC

KD

iet 7

-Up

16- oz

. 6

Pac

k0.

620.

956

1.05

80.

716

0.05

10.

196

0.08

710

. OT

HE

R, N

CA

FF

, DIE

T,

LAR

G2-

lit. B

ottle

Die

t 7-U

p5.

040.

568

0.74

10.

428

0.05

40.

843

0.76

11. O

TH

ER

, NC

AF

F, D

IET

, C

AN

Die

t 7-U

p 12

- oz.

Can

1.02

0.72

81.

043

0.45

30.

110.

147

0.13

112

. OT

HE

R, N

CA

FF

, DIE

T,

CA

N, P

AC

KD

iet 7

-Up

12- o

z . C

an 1

2 P

ack

4.64

0.78

61.

064

0.56

70.

112

0.92

30.

7413

. OT

HE

R, C

AF

F, D

IET

, P

AC

KD

iet R

ite 1

6-oz

. 6

Pac

k0.

020.

789

1.16

30.

297

0.10

20.

138

0.03

214

. OT

HE

R, C

AF

F, D

IET

, LA

RG

2-lit

. Bot

tle D

iet R

ite2.

190.

481

0.64

20.

356

0.04

80.

462

0.45

515

. OT

HE

R, C

AF

F, D

IET

, C

AN

, PA

CK

Die

t Rite

12-

oz. C

an 1

2 p

ac

k1.

610.

826

1.13

50.

599

0.09

80.

606

0.57

4

Pri

ce (

32 o

z.)

Tab

le 3

: D

escr

ipti

ve S

tati

stic

s: M

arke

t S

har

e, P

rice

s, D

isp

lays

an

d F

eatu

res

of

Pro

du

cts

Iljxuh 4=

4<

Lea

din

gM

arke

t S

har

eF

eatu

reD

isp

lay

Pro

du

ct C

har

acte

rist

ics

Exa

mp

le(%

)M

ea

nM

ax

.M

in.

Std

. Dev

.M

ea

nM

ea

n

16. C

OLA

, NC

AF

F, N

DIE

T,

PA

CK

Caf

fein

e F

ree

Pep

si 1

6-oz

. 6

Pa

ck0.

080.

801

1.16

0.41

70.

090.

282

0.04

817

. CO

LA, N

CA

FF

, ND

IET

, LA

RG

2-lit

. Bot

tle C

affe

ine

Fre

e P

epsi

0.43

0.61

20.

847

0.32

20.

096

0.68

90.

103

18. C

OLA

, NC

AF

F, N

DIE

T,

CA

N, P

AC

KC

affe

ine

Fre

e P

epsi

12-

oz.

Can

12

Pac

k1.

220.

752

1.26

60.

543

0.12

0.87

50.

292

19. C

OLA

, CA

FF

, ND

IET

, P

AC

KC

oca

Col

a C

lass

ic 1

6-oz

. 6

Pa

ck0.

760.

694

1.16

60.

150.

074

0.44

60.

263

20. C

OLA

, CA

FF

, ND

IET

, LA

RG

2-lit

. Bot

tle C

oca

Col

a C

lass

ic10

.25

0.51

80.

740.

345

0.06

70.

958

0.83

7

21. C

OLA

, CA

FF

, ND

IET

, CA

NC

oca

Col

a C

lass

ic 1

2-oz

. C

an0.

730.

512

0.59

40.

160.

055

0.11

20.

067

22. C

OLA

, CA

FF

, ND

IET

, C

AN

, PA

CK

Coc

a C

ola

Cla

ssic

12-

oz.

Can

12

Pac

k17

.90.

746

1.05

20.

548

0.09

80.

984

0.92

323

. CO

LA, N

CA

FF

, DIE

T,

PA

CK

Caf

fein

e F

ree

Die

t Cok

e 16

-oz

. 6 P

ack

0.7

0.68

20.

813

0.28

60.

065

0.33

0.10

624

. CO

LA, N

CA

FF

, DIE

T,

LAR

G2-

lit. B

ottle

Caf

fein

e F

ree

Die

t Cok

e 3.

190.

550.

773

0.36

40.

068

0.81

40.

58

25. C

OLA

, NC

AF

F, D

IET

, CA

NCaf

fein

e F

ree

Die

t Cok

e 12

-oz

. Can

0.08

0.51

60.

533

0.16

0.04

70.

035

0.03

826

. CO

LA, N

CA

FF

, DIE

T,

CA

N, P

AC

KC

affe

ine

Fre

e D

iet C

oke

12-

oz. C

an 1

2 P

ack

6.59

0.75

71.

085

0.55

30.

103

0.95

20.

699

27. C

OLA

, CA

FF

, DIE

T, P

AC

KD

iet C

oke

16-o

z. 6

Pac

k 0.

740.

630.

848

0.26

60.

090.

420.

202

28. C

OLA

, CA

FF

, DIE

T, L

AR

G2-

lit. B

ottle

Die

t Cok

e3.

460.

548

0.77

20.

329

0.07

40.

904

0.51

29. C

OLA

, CA

FF

, DIE

T, C

AN

Die

t Cok

e 12

-oz.

Can

0.3

0.62

71.

093

0.4

0.17

0.08

0.01

6

30. C

OLA

, CA

FF

, DIE

T, C

AN

, P

AC

KD

iet C

oke

12-o

z. C

an 1

2 P

ack

8.48

0.77

51.

127

0.53

90.

112

0.95

80.

782

Pri

ce (

32 o

z.)

Tab

le 3

(C

on

t.):

Des

crip

tive

Sta

tist

ics:

Mar

ket

Sh

are,

Pri

ces,

Dis

pla

ys a

nd

Fea

ture

s o

f P

rod

uct

s

Iljxuh 5=

53

Fkdudfwhulvwlfv/ wrwdo txdqwlw| vrog dqg pdunhw vkduh ri hdfk surgxfw duhsurylghg lq Wdeoh 61 +FROD/ FDII/ QGLHW/ FDQ/ SDFN, kdv wkh odujhvwpdunhw vkduh/ dqg +FROD/ FDII/ QGLHW/ ODUJ, wkh wklug1 Lqghhg Frfd0Frod Fodvvlf dqg Shsvl/ wzr prvw srsxodu gulqnv lq wkh lqgxvwu|/ ehorqj wrwkhvh surgxfwv1 +RWKHU/ QFDII/ QGLHW/ ODUJ, dqg +RWKHU/ QFDII/QGLHW/ FDQ/ SDFN, duh wkh vhfrqg dqg �iwk odujhvw surgxfw1 Vrph zhoo0nqrzq gulqnv vxfk dv Vsulwh dqg Vhyhq0Xs ehorqj wr wkhp1 +FROD/ FDII/GLHW/ FDQ/ SDFN,/ zklfk lqfoxghv gulqnv vxfk dv Glhw Frnh dqg GlhwShsvl/ lv wkh irxuwk odujhvw1 Lw lv zruwk| ri srlqwlqj rxw/ hyhq wkrxjk wkhlqirupdwlrq lv qrw lq wkh wdeoh/ wkdw sulydwh odehov kdyh d odujh pdunhw vkduhlq wkh gdwd1 Wkhlu sulfhv duh jhqhudoo| pxfk orzhu wkdq qdwlrqdo eudqgv1

Qrwh wkdw wkh +FDQ/ SDFN, w|sh dqg +ODUJ, w|sh gulqnv kdyh d odujhupdunhw vkduh wkdq rwkhu gulqnv1 Lw lv w|slfdo iru vxshupdunhw vkrsslqj ehkdy0lruv= frqvxphuv whqg wr ex| sdfndjhv ru odujh0vl}hg erwwohv lq vxshupdunhwviru ixwxuh krxvhkrog frqvxpswlrq1

Suredeo| wkh xvh ri jurxsv ri VNXv lqvwhdg ri eudqgv lv wkh prvw frq0whqwlrxv lvvxh lq p| dssolfdwlrq1 Lw lpsolhv/ iru h{dpsoh/ wkdw d 450sdfnFrfd0Frod Fodvvlf lv htxlydohqw wr d 450sdfn sulydwh odeho frod/ exw gl�huhqwiurp d 50olwhu erwwoh Frfd0Frod Fodvvlf1 Zh pd| dujxh wkdw eudqg qdph lvpruh lpsruwdqw wkdq frqwdlqhu vl}h1 Krzhyhu/ p| prgho kdv pxfk h{lelo0lw|1 Iru h{dpsoh/ L fdq lqfoxgh eudqg qdphv �Frfd0Frod�/ �Shsvl�/ �Vsulwh�/ru �sulydwh odehov� dv surgxfw fkdudfwhulvwlfv dqg rewdlq d �qhu fodvvl�fd0wlrq1 Dw wklv vwdjh L �qg wkdw wklv urxjk jurxslqj kdv douhdg| h{sodlqhgd orw ri frqvxphu sxufkdvh ehkdylru1 Ri frxuvh/ li p| phwkrg lv dssolhgwr rwkhu hpslulfdo vwxglhv/ iru h{dpsoh wr suhglfw pdunhw ghpdqg iru qhzsurgxfwv2eudqgv/ eudqg qdphv dqg rwkhu surgxfw fkdudfwhulvwlfv vkrxog ehlqfrusrudwhg1

Prgho Urexvwqhvv Wkh vl{ lqwulqvlf revhuydeoh surgxfw fkdudfwhulv0wlfv olvwhg deryh qhhg qrw eh frpsohwh iru vriw gulqnv1 Rqh ri wkh sureohpv lvwkdw zh kdyh vrph surgxfwv frpsohwho| gl�huhqw lq wkh fkdudfwhulvwlfv vsdfh1Iru h{dpsoh/ d fdq ri Shsvl +zklfk ehorqjv wr +FROD/ FDII/ QGLHW/FDQ,, vkduhv qr revhuydeoh fkdudfwhulvwlfv zlwk d 50olwhu erwwoh ri Glhw :0Xs+zklfk ehorqjv wr +RWKHU/ QFDII/ GLHW/ ODUJ,,1 Dvvxplqj dq dggl0wlyho| vhsdudeoh xwlolw| ixqfwlrq vxfk dv wkrvh lq prghov P04D/ P04E/ P50Dru P50E/ pdujlqdo xwlolw| ri frqvxplqj wkh iruphu lv lqghshqghqw iurp frq0vxplqj wkh odwwhu1 Wkhvh wzr surgxfwv duh qrw vxevwlwxwhv lq rxu prgho1 \hw

54

wkh| duh erwk vriw gulqnv/ dqg lqwxlwlyho| ex|lqj d orw ri 50olwhu erwwoh Glhw:0Xs fdq pdnh frqvxphuv ex|lqj ohvv ri Shsvl lq fdq1

Wr fkhfn wkh urexvwqhvv ri wkh vshfl�fdwlrqv lq prghov P40D/ P40E/P50D ru P50E/ L hvwlpdwh dqrwkhu vhw ri prghov zklfk kdyh dq dgglwlrqdolqwulqvlf surgxfw fkdudfwhulvwlf/ �Vriw Gulqnv� +GULQN,1 Wklv fkdudfwhulvwlflv hpehgghg lq hyhu| surgxfw/ vr wkhuh zloo eh qr wzr surgxfwv frpsohwho|vhsdudwh iurp hdfk rwkhu lq wkh fkdudfwhulvwlfv vsdfh1 Li frqvxphuv ex| vriwgulqnv rqo| iru wkluvw| uhdvrq dqg gr qrw frqfhuq derxw rwkhu fkdudfwhulvwlfv/zh zloo kdyh d uhodwlyho| kljk phdq suhihuhqfh dqg d uhodwlyho| vorz pdujlqdoxwlolw| ghfolqh iru +GULQN,1 Gl�huhqw surgxfwv zloo eh forvh vxevwlwxwhv1 Wkhrssrvlwh lv wuxh li rwkhu fkdudfwhulvwlfv duh pruh lpsruwdqw lq wkh xwlolw|ixqfwlrq1 Wkh vxevwlwxwlrq sdwwhuq dprqj surgxfwv zloo eh pruh frpsolfdwhgdqg/ dv zloo eh h{sodlqhg lq odwhu vhfwlrqv/ frpsohphqwdulw| pd| dulvh1

L hvwlpdwh wzr prghov/ P60D dqg P60E/ iru wklv sduw1 P60D dqg P60Eduh dqdorjrxv wr prghov P50D dqg P50E uhvshfwlyho|/ h{fhsw wkdw wkhuh lvdq dgglwlrqdo fkdudfwhulvwlf/ +GULQN,/ lq wkh xwlolw| ixqfwlrq1

+Hvwlpdwlrq uhvxowv ri wkhvh wzr prghov zloo eh glvfxvvhg lq d uhylvhggudiw1,

Surgxfw Fkdudfwhulvwlfv lq wkh Xwlolw| Ixqfwlrq Lq wklv vhfwlrq L zlooglvfxvv krz wkh deryh lqwulqvlf revhuydeoh surgxfw fkdudfwhulvwlfv hqwhu wkhxwlolw| ixqfwlrq1 Wkh hdvlhvw zd| zrxog eh wr dvvxph d xwlolw| ixqfwlrq dg0glwlyho| vhsdudeoh lq vhyhq fkdudfwhulvwlfv/ wkdw lv/ FROD/ RWKHU/ GLHW/FDII/ ODUJ/ FDQ/ dqg SDFN1 Wklv lpsolhv wkh olqhdu vxppdelolw| dqgvhsdudelolw| ri fkdudfwhulvwlfv1 Krzhyhu/ wklv dvvxpswlrq grhv qrw frqvlghu lq0whudfwlqj uhodwlrqvklsv ehwzhhq wkhvh fkdudfwhulvwlfv/ hvshfldoo| dyruv1 Iurpwkh gdwd L �qg wkdw +FROD/ FDII/ QGLHW, kdv wkh odujhvw pdunhw vkduhdprqj frod gulqnv/ zkloh +RWKHU/ QFDII/ QGLHW, kdv wkh odujhvw pdunhwvkduh dprqj gulqnv zlwk rwkhu dyruv1 +RWKHU/ QFDII/ QGLHW, kdv dodujhu pdunhw vkduh wkdq +RWKHU/ QFDII/ GLHW,1 Li wkh deryh dvvxps0wlrq lv fruuhfw/ zh vkrxog h{shfw +FROD/ QFDII/ QGLHW, wr kdyh d odujhupdunhw vkduh wkdq +FROD/ QFDII/ GLHW,1 \hw wkh rssrvlwh lv wuxh1 Lqghhg+FROD/ QFDII/ QGLHW, kdv wkh vpdoohvw pdunhw vkduh/ zkloh +RWKHU/QFDII/ QGLHW, kdv wkh vhfrqg odujhvw pdunhw vkduh1 Wkhvh skhqrphqdfdqqrw eh h{sodlqhg e| sulfh gl�huhqfhv1 Wkh| lpso| wkdw � dyru pl{wxuhv�/wkh lqwhudfwlrqv ri dyruv/ duh qrw mxvw wkh vxp ri lqglylgxdo dyruv1

55

Zkhq d surgxfw kdv fkdudfwhulvwlfv ri +FDQ, dqg +SDFN,/ lw kdv d iduodujhu pdunhw vkduh wkdq wkh frpelqhg pdunhw vkduh ri surgxfwv zklfk kdyhfkdudfwhulvwlfv ri hlwkhu +FDQ, ru +SDFN,1 Wklv lpsolhv wkdw wkh olqhduo|vxppdelolw| dqg vhsdudelolw| dvvxpswlrq ri +FDQ, dqg +SDFN, lv dovr lq0ydolg1

L �qg wkdw hyhq wkh olqhdu vxppdelolw| dqg vhsdudelolw| dvvxpswlrq rifrqwdlqhu vl}h dqg sdfndjlqj zlwk dyru pl{wxuhv lv qrw mxvwl�hg lq vrphfdvhv1 Iru h{dpsoh/ +RWKHU/ QFDII/ QGLHW/ ODUJ, kdv d odujhu pdunhwvkduh wkdq +RWKHU/ QFDII/ QGLHW/ FDQ/ SDFN,/ lpso|lqj wkdw fkdudf0whulvwlf +ODUJ, lv suhihuuhg wr fkdudfwhulvwlf +FDQ/ SDFN, rq dyhudjh1 E|wkh deryh dvvxpswlrq/ +FROD/ FDII/ QGLHW/ ODUJ, vkrxog eh suhihuuhgwr +FROD/ FDII/ QGLHW/ FDQ/ SDFN,1 Krzhyhu/ wkh odwwhu surgxfw kdvd pxfk odujhu pdunhw vkduh wkdq wkh iruphu1

Ehfdxvh ri wkh deryh sureohpv/ L sxw gl�huhqw lqwhudfwlrqv ri surgxfwfkdudfwhulvwlfv lqwr wkh xwlolw| ixqfwlrq lqvwhdg1 Wkhuh duh dowrjhwkhu 44 lq0whudfwlrqv L xvh lq wkh xwlolw| ixqfwlrq=

41 frod dyru/ fd�hlqdwhg/ glhw +FROD ) FDII ) GLHW,>

51 frod dyru/ fd�hlqdwhg/ qrq0glhw +FROD ) FDII ) QGLHW,>

61 frod dyru/ qrq0fd�hlqdwhg/ glhw +FROD ) QFDII ) GLHW,>

71 frod dyru/ qrq0fd�hlqdwhg/ qrq0glhw +FROD ) QFDII ) QGLHW,>

81 rwkhu dyruv/ fd�hlqdwhg/ glhw +RWKHU ) FDII ) GLHW,>

91 rwkhu dyruv/ fd�hlqdwhg/ qrq0glhw +RWKHU ) FDII ) QGLHW,>

:1 rwkhu dyruv/ qrq0fd�hlqdwhg/ glhw +RWKHU ) QFDII ) GLHW,>

;1 rwkhu dyruv/ qrq0fd�hlqdwhg/ qrq0glhw +RWKHU)QFDII)QGLHW,>

<1 rwkhu dyruv/ odujh vl}hg frqwdlqhu +RWKHU ) ODUJ,>

431 odujh vl}hg frqwdlqhu +ODUJ,>

441 fdq dqg sdfnhg +FDQ ) SDFN,1

56

Wkh �uvw hljkw fkdudfwhulvwlfv duh � dyru pl{wxuhv� +zklfk lqfoxgh frodru rwkhu dyruv/ glhw ru qrq0glhw/ dqg fd�hlqdwhg ru qrq0fd�hlqdwhg,1 Wkhqlqwk whup/ +RWKHU ) ODUJ,/ wdnhv fduh ri wkh qrq0vxppdelolw| dqg qrq0vhsdudelolw| ri frqwdlqhu vl}h dqg dyruv/ dqg wkh hohyhqwk whup/ +FDQ )SDFN,/ wdnhv fduh ri wkh qrq0vxppdelolw| dqg qrq0vhsdudelolw| ri frqwdlqhudqg sdfndjlqj1

Wkhvh lqwhudfwlrqv pd| vhhp vwudqjh dqg vrphwlphv dg krf/ dqg pd|qrw eh wkh prvw frpsohwh ru vxfflqfw zd| ri vshfli|lqj wkh xwlolw| ixqfwlrquhvhdufkhuv fdq xvh1 Krzhyhu/ wklv lv d frpsurplvh ehwzhhq wkh gdwd dqg wkhqhhg ri prgho sduvlprq|1 Wr uhgxfh wkh frpsxwdwlrqdo exughq L gr qrw xvhdoo lqwhudfwlrqv ri surgxfw fkdudfwhulvwlfv/ exw wkh vlpsohvw zd| ri dvvxplqjolqhdu vxppdelolw| dqg vhsdudelolw| ri gl�huhqw fkdudfwhulvwlfv lq wkh xwlolw|ixqfwlrq zloo/ dv glvfxvvhg deryh/ eulqj vhulrxv glvwruwlrqv wr wkh prgho1

Sulfhv/ Glvsod|v/ Ihdwxuhv dqg Ghprjudsklfv Wdeoh 6 dovr surylghvvxppdu| ghvfulswlyh vwdwlvwlfv ri phufkdqgl}lqj lqirupdwlrq lqfoxglqj sulfhv+lq vwdqgdugl}hg xqlwv ri 65 r}1,/ ihdwxuhv/ dqg glvsod|v1 L �qg odujh sulfh xfwxdwlrqv1 Iru h{dpsoh/ +RWKHU/ QFDII/ QGLHW, kdv d plqlpxp sulfhri ;3 fhqwv dqg d pd{lpxp sulfh ri '6145/ dqg vwdqgdug ghyldwlrq ri 31831+FROD/ FDII/ QGLHW/ FDQ/ SDFN,/ zklfk lv wkh surgxfw zlwk wkh odujhvwpdunhw vkduh/ kdv d plqlpxp sulfh ri 88 fhqwv dqg d pd{lpxp sulfh ri '4138/dqg vwdqgdug ghyldwlrq ri 31431 Phdq sulfhv duh dovr yhu| gl�huhqw dprqjsurgxfwv/ zlwk d plqlpxp ri 7; fhqwv iru +RWKHU/ FDII/ GLHW/ ODUJ,/dqg d pd{lpxp ri '41:9 iru +RWKHU/ QFDII/ QGLHW,1

Odujh surgxfwv +surgxfwv zlwk odujh pdunhw vkduh, duh iuhtxhqwo| ihdwxuhgdqg glvsod|hg lq vxshupdunhwv1 Iru h{dpsoh/ +FROD/ FDII/ QGLHW/ FDQ/SDFN,/ wkh odujhvw surgxfw/ lv wkh surgxfw prvw iuhtxhqwo| ihdwxuhg dqg rqhri wkh surgxfwv prvw iuhtxhqwo| glvsod|hg1 Wklv lv dovr wuxh iru rwkhu odujhsurgxfwv1

Yduldeohv Phdq Pd{1 Plq1 Vwg1 Ghy1Idplo| Vl}h 2�SH2 S � ��ee�Krxvhkrog Lqfrph +', �ec bef Hfc fff Dc fff 2ec ��2Wdeoh 7= Ghvfulswlyh Vwdwlvwlfv ri Krxvhkrog Ghprjudsklfv

Wdeoh 7 surylghv vrph vxppdu| ghvfulswlyh vwdwlvwlfv ri ghprjudsklf ydul0deohv1 Krxvhkrogv zlwk idplo| vl}h odujhu ru htxdo wr vl{ duh uhfrughg dv �vl{

57

ru pruh shrsoh�/ dqg L wuhdw wkhp dv vl{1 Krxvhkrog lqfrphv duh fdwhjrul}hglqwr whq jurxsv/ iurp �ohvv wkdq 43/333� wr �:8/333 dqg ryhu�1 L wdnh wkhphdq lqfrph lq hdfk jurxs/ dqg dvvhuw krxvhkrog lqfrph ri wkh odvw jurxshtxdo wr ;3/3331

715 Uhvxowv

71514 Vrph Hvwlpdwhv

L hvwlpdwh wzr prghov= Prgho P40E lqfoxghv ihdwxuhv dqg surgxfw glvsod|vlq wkh xwlolw| ixqfwlrq zkloh prgho P40D grhv qrw1 Lq rughu wr lghqwli|sdudphwhuv lq prghov L qrupdol}h 7b/ wkh phdq pdujlqdo xwlolw| ri lqfrph/ wr�1

Hvwlpdwhv ri wkh wzr prghov duh uhsruwhg lq Wdeoh 81Wkh srlqw hvwlpdwh ri wkh frh!flhqw iru +Sulfh� *?E�?SJ6e�, lq erwk prg0

hov lv qhjdwlyh/ lpso|lqj wkdw wkh kljkhu wkh krxvhkrog lqfrph/ wkh orzhu wkhpdujlqdo xwlolw| wr lqfrph zloo eh1 Lw fdq dovr eh lqwhusuhwhg dv wkh ulfk ehlqjohvv frqfhuqhg derxw sulfh1 Lqfoxglqj ihdwxuhv dqg glvsod|v/ wkh hvwlpdwh riprgho P04E kdv d orzhu devroxwh ydoxh/ dqg qrw vwdwlvwlfdo vljql�fdqw1

Hvwlpdwhv ri phdq ydoxhv ri frh!flhqw iru dyru pl{wxuhv/ 7qc duh uhsruwhgiurp urz wzr wr urz qlqh1 Rqo| wkh frh!flhqw iru +FROD ) FDII )QGLHW,/ wkh dyru pl{wxuh zlwk wkh odujhvw pdunhw vkduh/ lv vljql�fdqwo|srvlwlyh1 Wkh frh!flhqw iru +RWKHU ) QFDII ) QGLHW,/ wkh dyrupl{wxuh zlwk wkh vhfrqg odujhvw pdunhw vkduh/ lv srvlwlyh exw qrw vljql�fdqwlq erwk prghov1 Hvwlpdwhv ri frh!flhqwv iru rwkhu dyru pl{wxuhv duh doovljql�fdqwo| qhjdwlyh1 Wkhvh hvwlpdwhv qhhg qrw lpso| wkdw prvw frqvxphuvwuhdw wkhvh dyru pl{wxuhv dv edgv1 Wkhlu ehlqj qhjdwlyh lv dfwxdoo| fdxvhg e|wkh dvvxpswlrq wkdw #c wkh vwrfkdvwlf ghyldwhv ri q/ duh qrupdoo| glvwulexwhg1Li zh duh zloolqj wr dvvxph wkdw # kdyh d wuxqfdwhg glvwulexwlrq dqg uhvwulfw7q n j � # wr ehlqj srvlwlyh/ wkhq qr dyru pl{wxuh lv d edg1 L pdlqwdlq wkhruljlqdo dvvxpswlrq ehfdxvh +4, lw lv sodxvleoh wkdw vrph frqvxphuv wuhdwvrph dyru pl{wxuhv dv edgv/ dqg +5, prgli|lqj wkh dvvxpswlrq ri # rqo|frpsolfdwhv hvwlpdwlrqv exw zrxog qrw fkdqjh uhvxowv1

58

P40D P40EYduldeohv Hvwlpdwhv V1H1 Hvwlpdwhv V1H1Sulfh� *?E�?SJ6e� 0313355 313345 0313349 313344RWKHU ) QFDII ) QGLHW 31334< 3133<4 313354 3133;4RWKHU ) FDII ) QGLHW 031:8;5 313477 031:37: 313445RWKHU ) FDII ) GLHW 031;;;3 3134:; 031;387 313477RWKHU ) QFDII ) GLHW 03174:: 313443 031737; 3133;<FROD ) QFDII ) QGLHW 031:<69 313535 031:877 313534FROD ) FDII ) QGLHW 315739 313439 31543; 3133<8FROD ) FDII ) GLHW 031493< 3133:; 03147<6 3133;6FROD ) QFDII ) GLHW 0315948 313438 0315676 3133<5ODUJ 31:68< 3134<7 31:8<9 31347<FDQ ) SDFN 41:<6< 3135<4 41:<46 31355:RWKHU ) ODUJ 317447 3133<3 316<75 3133:4Ihdwxuh 31333: 313338Glvsod| 313466 313349j � � � � � � � � � � � � � � � 3189:5 3133<4 318434 313343j � � � � � � � � � � � � � � 313736 313355 313736 31333:Ip|1 Vl}h 3187<6 313454 318746 31344:

Wdeoh 8= Sdudphwhu Hvwlpdwhv

Hvwlpdwhv ri phdq ydoxhv ri wkh frh!flhqwv iru +ODUJ, dqg +FDQ )SDFN, duh vljql�fdqwo| srvlwlyh lq erwk prghov1 Wkh hvwlpdwh ri wkh frh!0flhqw iru +ODUJ,/ 7q � c lv f�.�Db lq prgho P04D dqg f�.DbS lq prgho P04E/zkloh wkh hvwlpdwh ri wkh frh!flhqw iru +FDQ ) SDFN,/ 7q � ! c lv ��.b�blq prgho P40D dqg ��.b�� lq prgho P40E1 Hvwlpdwhv ri wkh phdq ydoxh riwkh frh!flhqw iru +RWKHU ) ODUJ,/ q " # $ c duh f�e��e lq prgho P40D dqgf��be2 lq prgho P40E/ erwk vwdwlvwlfdoo| vljql�fdqw1 Hyhq wkrxjk L fdqqrwh{sodlq zk|/ wkh uhvxow vkrzv d frpsohphqwdu| h�hfw ehwzhhq rwkhu dyruvdqg odujh0vl}hg erwwohv1 Wklv phdqv wkdw zkloh 7q % # & lv htxdo iru erwk frod dyru dqg rwkhu dyruv +��.b�b lq prgho P40D dqg ��.b�� lq prgho P40E,/wkh sdudphwhu iru +ODUJ, lv 7q $ iru frod dyru +f�.�Db lq prgho P40D dqgf�.DbS lq prgho P40E,/ dqg 7q $ n 7q " # $ iru rwkhu dyruv +���e.� lq prghoP40D dqg ���D�H lq prgho P40E,1

Frh!flhqwv ri ihdwxuhv dqg glvsod|v duh irxqg wr eh srvlwlyh lq prgho P40E1 Krzhyhu/ wkh frh!flhqw iru ihdwxuhv lv vpdoo/ lpso|lqj wkdw ihdwxuhv kdyhd qhjoljleoh pdunhwlqj h�hfw 1 Surgxfw glvsod|v kdyh d pxfk odujhu h�hfw

59

rq wkh xwlolw| ixqfwlrq +f�f���,/ dqg lv vwdwlvwlfdoo| vljql�fdqw1 Wklv h{sodlqvzk| pdqxidfwxuhuv frpshwh zlwk hdfk rwkhu wr glvsod| wkhlu surgxfwv lqvxshupdunhwv/ exw whqg wr ljqruh surgxfw ihdwxulqj1

Ilqdoo|/ wkh hvwlpdwh ri wkh h�hfw ri idplo| vl}h rq wrwdo krxvhkrog frq0vxpswlrq lv 3187<6 lq prgho P40D dqg 318746 lq prgho P40E1 Lw lv vwd0wlvwlfdoo| vljql�fdqw lq erwk prghov1 Wkh frh!flhqw ehlqj srvlwlyh lpsolhvd odujhu frqvxpswlrq qhhg iru vriw gulqnv zkhq wkhuh duh pruh krxvhkrogphpehuv1 Krzhyhu/ lw lv vpdoohu wkdq rqh/ lpso|lqj wkdw wkh jurzwk ri wrwdokrxvhkrog frqvxpswlrq odjv ehklqg wkh jurzwk ri idplo| vl}h1 Wklv frxogeh h{sodlqhg e| wkh idfw wkdw odujh idplolhv xvxdoo| kdyh pdq| fkloguhq/ dqgfkloguhq jhqhudoo| kdyh d orzhu ghpdqg iru gulqnv wkdq dgxowv1

Iljxuhv 4 dqg 5 vkrz wkh hvwlpdwhg pdunhw ghpdqg dqg wkh revhuyhg pdu0nhw ghpdqg iru hdfk surgxfw1 Surgxfw qxpehu lv ghvfulehg lq Wdeoh 6 ehiruh1Prghov P40D dqg P40E jhqhudoo| jlyh d jrrg hvwlpdwlrq ri pdunhw ghpdqgiru surgxfwv zlwk odujh pdunhw vkduh1 Krzhyhu/ erwk prghov xqghuhvwlpdwhpdunhw ghpdqg iru surgxfwv zlwk vpdoo pdunhw vkduh1 D frqmhfwxuh lv wkdwwkhvh uhvxowv vlpso| uh hfw wkh h{lvwhqfh ri pdq| }hur sxufkdvhv1 Zkhq }hursxufkdvh lv wkh qrup iru d surgxfw/ wkh hvwlpdwlrq dojrulwkp whqgv wr sur0gxfh }hurv iru vlpxodwhg ghpdqg lq rughu wr plqlpl}h wkh remhfwlyh ixqfwlrq/dqg wkhuhiruh jhqhudwhv d orzhu hvwlpdwhg pdunhw ghpdqg1

71515 Lpsolfdwlrqv

Lq wklv vhfwlrq L xvh wkh hvwlpdwlrq uhvxowv wr qxphulfdoo| frpsxwh sulfhhodvwlflwlhv dqg hydoxdwh wkh h�hfwv ri surgxfw glvsod|v lq vxshupdunhwv1

Rzq0Sulfh dqg Furvv0Sulfh Hodvwlflwlhv Wr jhqhudwh sulfh hodvwlflwlhvL dvvxph wkdw frqvxphuv dw hyhu| vxshupdunhw lq hyhu| zhhn idfh wkh phdqsulfhv +vhh Wdeoh 6, dqg frpsxwh wkh wrwdo ghpdqg iru hdfk surgxfw xvlqjsdudphwhuv hvwlpdwhg lq prgho P40E1 Wkhq L dvvxph frqvxphuv idfh d sulfhfkdqjh ri gl�huhqw surgxfwv/ dqg frpsxwh wkh qhz wrwdo ghpdqg1 Ohw 0� � � c�c & ' �c ���c a / eh wkh sulfh hodvwlflw|/ + � wkh pdunhw ghpdqg xqghu wkh phdqsulfhv 7R/ dqg +

�� wkh qhz pdunhw ghpdqg zkhq sulfh ri surgxfw & fkdqjhv

iurp 7R � wr R�� 1 L gh�qh

0� � � '+

�� � +�R

�� � 7R �

�7R �+�

5:

Fig

1:

Mo

del

M1-

A:

Ob

serv

ed a

nd

Est

imat

ed D

eman

d

...

0

2000

4000

6000

8000

1000

0

1200

0

1400

0

1600

0

1800

0

2000

0

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

2930

Pro

du

ct...

Quantity Demanded...

Ob

serv

ed

Ma

rke

t D

em

an

d

Est

ima

ted

Ma

rke

t D

em

an

d

Iljxuh 6=

5;

Fig

2:

Mo

del

M1-

B:

Ob

serv

ed a

nd

Est

imat

ed D

eman

d

.

0

2000

4000

6000

8000

1000

0

1200

0

1400

0

1600

0

1800

0

2000

0

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

2930

Pro

du

ct

.

Quantity Demanded .

Obs

erve

d M

arke

t D

eman

d

Est

imat

ed M

arke

t D

eman

d

Iljxuh 7=

5<

Lw fdq eh vkrzq wkdw 0 � � � lv mxvw wkh vxp ri sulfh hodvwlflwlhv ri lqglylgxdofrqvxphuv zhljkwhg e| sxufkdvh vkduh +vhh Gxeh +4<<<,/ vhfwlrq 915,1 0� � � lvdq rzq0sulfh hodvwlflw| li � ' &/ dqg d furvv0sulfh hodvwlflw| li � 9' &1 Surgxfwv� dqg & duh vxevwlwxwhv li 0� � � lv srvlwlyh/ dqg frpsohphqwv li 0 � � � lv qhjdwlyh1L fdqqrw ghulyh dq dqdo|wlfdo h{suhvvlrq iru 0� � � ehfdxvh pdunhw ghpdqg +

fdq rqo| eh frpsxwhg qxphulfdoo|1Wr vlpsoli| wkh dqdo|vlv L rqo| uhsruw sulfh hodvwlflwlhv ri hljkw surgxfwv

zlwk d odujh pdunhw vkduh +vhh Wdeohv 9 dqg :1, Wkh| fdq eh frqvlghuhg dvsdluv ri gulqnv/ hdfk sdlu kdv wkh vdph dyru pl{wxuhv exw gl�huhqw frqwdlqhuvl}hv dqg sdfndjlqj1 Wkh �uvw lq hdfk sdlu lv lq odujh0vl}hg erwwohv/ zkloh wkhvhfrqg lq sdfndjhg fdqv1 0� � � lq wkh htxdwlrq deryh lv uhfrughg e| wkh �0wkurz dqg &0wk froxpq lq erwk wdeohv1 Wdeoh 9 uhsruwv sulfh hodvwlflwlhv zkhqwkh sulfh ri d surgxfw idoov e| 43 shufhqw/ dqg Wdeoh : hodvwlflwlhv zkhq sulfhulvhv e| 43 shufhqw1 Wkh| duh gl�huhqw lq ydoxh/ lpso|lqj wkdw hodvwlflwlhv righpdqg duh qrw frqvwdqw1

Doo gldjrqdo hohphqwv/ wkh rzq0sulfh hodvwlflwlhv/ duh qhjdwlyh1 Wkhlu de0vroxwh ydoxhv duh juhdwhu wkdq rqh/ vkrzlqj wkdw dw phdq sulfhv doo surgxfwvduh lq wkh udqjh ri hodvwlf ghpdqg

�1

Frpsohphqwdulw| dprqj Surgxfwv Shukdsv wkh prvw vxusulvlqj uh0vxow lv wkh h{lvwhqfh ri pdq| frpsohphqwv/ zklfk lv lpsolhg e| qhjdwlyh r�0gldjrqdo hohphqwv lq erwk wdeohv1 D jhqhudo sdwwhuq lv wkdw wzr surgxfwvduh frpsohphqwv zkhq wkh| kdyh yhu| gl�huhqw surgxfw fkdudfwhulvwlfv/ dqgduh vxevwlwxwhv zkhq wkh| kdyh vrph frpprq surgxfw fkdudfwhulvwlfv1 Iruh{dpsoh/ +RWKHU/ QFDII/ QGLHW/ ODUJ, dqg +FROD/ FDII/ QGLHW/FDQ/ SDFN, duh frpsohphqwv1 Wkh| duh gl�huhqw lq erwk dyru pl{wxuhvdqg frqwdlqhu vl}h dqg sdfndjlqj1 +RWKHU/ QFDII/ QGLHW/ ODUJ, dqg+FROD/ FDII/ QGLHW/ ODUJ, duh vxevwlwxwhv1 Wkh| kdyh gl�huhqw dyrupl{wxuhv exw erwk duh lq odujh0vl}hg erwwohv1 Wkh uhdvrq ehklqg wklv sdw0whuq lv wkdw frqvxphuv sxufkdvh pxowlsoh surgxfwv dqg pxowlsoh xqlwv/ dqgwkh| vhhn ydulhw| ryhu surgxfw fkdudfwhulvwlfv1 Zkhq wkhuh lv d sulfh fxw lq/vd|/ 50olwhu Vsulwh/ lwv txdqwlw| ghpdqghg ulvhv/ zklfk phdqv frqvxphuv kdyhpdq| odujh0vl}hg erwwohv qrz1 Li wkh| dovr kdyh d ghpdqg iru dyru pl{wxuh

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " # � $ % � � & ' ( ) *+ � ' , - # & ' " , . - � & # * � & $ � � ! # & / # & � � � � * + � ' , - / 0 � � 1 � # # % # & 0 � # & # � ' $ / & � + # + , 0 2' � � & 0 � 0 # & - + � $ ! # � - $ & 3 4 5 6 4 7 8 9 6 : ; < 8 = > 4 ? @ A 5 < B 9 > C 5 4 = 7 D E 8 9 > 4 7 8 F @ 4 7 A D E ; 7 8 < < 4 = GH I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ [ _ ` a ` b c d e f c ` g h i j k k l m n o i j i o i k n p

63

Pro

du

ctO

TH

ER

, NC

AF

F,

ND

IET

, LA

RG

OT

HE

R, N

CA

FF

, N

DIE

T,

CA

N,

PA

CK

OT

HE

R, N

CA

FF

, D

IET

, LA

RG

OT

HE

R, N

CA

FF

, D

IET

, C

AN

, P

AC

KC

OLA

, C

AF

F,

ND

IET

, LA

RG

CO

LA,

CA

FF

, N

DIE

T,

CA

N,

PA

CK

CO

LA,

CA

FF

, D

IET

, LA

RG

CO

LA,

CA

FF

, D

IET

, C

AN

, P

AC

K

OT

HE

R, N

CA

FF

, N

DIE

T, L

AR

G-3

.05

0.95

0.96

-0.7

0.5

-0.1

60.

56-0

.19

OT

HE

R, N

CA

FF

, N

DIE

T, C

AN

PA

CK

0.84

-5.8

8-0

.32

1.09

-0.4

60.

78-0

.51

0.9

OT

HE

R,

NC

AF

F,

DIE

T,

LAR

G0.

34-0

.15

-3.3

1.17

0.21

-0.0

30.

23-0

.03

OT

HE

R,

NC

AF

F,

DIE

T,

CA

N,

PA

CK

-0.1

60.

440.

86-9

.16

-0.1

20.

24-0

.18

0.28

CO

LA,

CA

FF

, N

DIE

T,

LAR

G0.

37-0

.30.

35-0

.33

-4.6

70.

761.

68-0

.66

CO

LA,

CA

FF

, N

DIE

T,

CA

N,

PA

CK

-0.3

11.

44-0

.15

1.69

2.13

-4.0

5-1

.81.

73C

OLA

, C

AF

F,

DIE

T,

LAR

G0.

15-0

.12

0.17

-0.1

80.

680.

23-6

.69

1.24

CO

LA,

CA

FF

, D

IET

, C

AN

, P

AC

K-0

.14

0.69

-0.0

50.

72-0

.68

0.69

3.06

-6.1

7

Pro

du

ctO

TH

ER

, NC

AF

F,

ND

IET

, LA

RG

OT

HE

R, N

CA

FF

, N

DIE

T,

CA

N,

PA

CK

OT

HE

R, N

CA

FF

, D

IET

, LA

RG

OT

HE

R, N

CA

FF

, D

IET

, C

AN

, P

AC

KC

OLA

, C

AF

F,

ND

IET

, LA

RG

CO

LA,

CA

FF

, N

DIE

T,

CA

N,

PA

CK

CO

LA,

CA

FF

, D

IET

, LA

RG

CO

LA,

CA

FF

, D

IET

, C

AN

, P

AC

K

OT

HE

R, N

CA

FF

, N

DIE

T, L

AR

G-2

.35

0.84

0.72

-0.2

90.

49-0

.16

0.52

-0.1

8O

TH

ER

, NC

AF

F,

ND

IET

, CA

N P

AC

K0.

73-4

.04

-0.3

81.

44-0

.28

0.53

-0.3

0.66

OT

HE

R,

NC

AF

F,

DIE

T,

LAR

G0.

35-0

.18

-2.6

71.

440.

19-0

.03

0.25

-0.0

6O

TH

ER

, N

CA

FF

, D

IET

, C

AN

, P

AC

K-0

.05

0.19

0.47

-4.8

2-0

.06

0.13

-0.0

50.

15C

OLA

, C

AF

F,

ND

IET

, LA

RG

0.28

-0.2

30.

26-0

.25

-3.5

30.

681.

61-0

.51

CO

LA,

CA

FF

, N

DIE

T,

CA

N,

PA

CK

-0.3

41.

46-0

.21

1.77

2.18

-3.3

-1.4

51.

68C

OLA

, C

AF

F,

DIE

T,

LAR

G0.

11-0

.09

0.1

-0.0

80.

52-0

.17

-4.3

50.

98C

OLA

, C

AF

F,

DIE

T,

CA

N,

PA

CK

-0.1

30.

57-0

.10.

66-0

.50.

523.

19-4

.35

Tab

le 6

: Pric

e D

eman

d E

last

iciti

es w

hen

Pric

es F

all

Tab

le 7

: Pric

e D

eman

d E

last

iciti

es w

hen

Pric

es R

ise

Iljxuh 8=

64

+FROD/ FDII/ QGLHW,/ frqvxphuv zrxog/ kroglqj rwkhu wklqjv frqvwdqw/wu| wr ex| pruh gulqnv zlwk wklv dyru pl{wxuh exw lq sdfndjhg fdqv1 Khqfhzh pd| revhuyh pdunhw ghpdqg iru 450sdfn Frfd Frod Fodvvlf ulvhv1

Wr vhh zkhwkhu ru qrw p| prgho suhglfwv wrr pdq| frpsohphqwv/ L gr dvlpsoh h{huflvh e| uhjuhvvlqj olqhduo| wkh wrwdo ghpdqg ri hdfk surgxfw/ dwhdfk vxshupdunhw lq hdfk zhhn/ zlwk uhvshfw wr lwv rzq sulfh dqg sulfhv rirwkhu surgxfwv1 Wkdw lv/ iru hdfk vxshupdunhw V dqg zhhn |/

+ � � � ' kn EU � R � � � ��q n 1 �

zkhuh + � lv d a � � yhfwru ri txdqwlwlhv ghpdqghg/ R � � d a � � yhfwru risulfhv/ U � d a � a lghqwlw| pdwul{/ wkh Nurqhfnhu surgxfw/ dqg 1 � � � wkhlglrv|qfudwlf huuruv1 Sdudphwhuv duh d a � � yhfwru ri k dqg d a

�� � yhfwru

ri q1 Wkh sdudphwhuv duh sulfh ghulydwlyhv/ zkhuh d srvlwlyh vljq lpsolhvvxevwlwxwdelolw| dqg d qhjdwlyh vljq lpsolhv frpsohphqwdulw|1 Wr vlpsoli|qrwdwlrq L xvh wkh surgxfw qxpehu lq Wdeoh 6 iru wkhvh hljkw surgxfwv/ zklfkduh surgxfw 6/ 8/ 43/ 45/ 53/ 55/ 5;/ dqg 631 Wkh uhvxowv duh uhsruwhg lq Wdeoh; ehorz1

Glvuhjduglqj frqvwdqwv/ wkhuh duh 53 vwdwlvwlfdoo| vljql�fdqw hvwlpdwhv1Ilyh duh rzq0sulfh ghulydwlyhv/ doo qhjdwlyh1 Wklv lv frqvlvwhqw zlwk wkh suh0glfwlrqv ri p| prgho1 Vhyhq furvv0sulfh ghulydwlyhv duh srvlwlyh/ lpso|lqjd vxevwlwxwdelolw| uhodwlrqvkls1 Ilyh ri wkhp duh fruuhfwo| suhglfwhg lq p|prgho1 Wkhuh duh hljkw qhjdwlyh furvv0sulfh ghulydwlyhv/ lpso|lqj d uhodwlrq0vkls ri frpsohphqwdulw|1 Rqo| wkuhh duh fruuhfwo| suhglfwhg e| p| prgho1P| prgho suhglfwv wrr ihz frpsohphqwv/ dqg qrw wrr pdq|1

Frpsohphqwdulw| fdqqrw eh h{sodlqhg e| glvfuhwh surgxfw fkrlfh prghov1Dvvxplqj frqvxphuv rqo| fkrrvh rqh surgxfw iru hdfk frqvxpswlrq qhhg/surgxfwv duh shuihfw vxevwlwxwhv iru hdfk rwkhu1 Zkhq wkhuh lv d sulfh fxw lqrqh surgxfw/ wkh suredelolw| ri fkrrvlqj dqrwkhu surgxfw zloo ghfuhdvh1 Vrwkhuh lv rqo| vxevwlwxwdelolw| dprqj surgxfwv1 D pruh h{leoh prgho doorzvgl�huhqw ghjuhhv ri vxevwlwxwdelolw| zkhq frqvxphu khwhurjhqhlw| lv dvvxphg+vhh Ehuu|/ Ohylqvrkq dqg Sdnhv +4<<8,,1 Exw wkh idfw wkdw surgxfwv duhshuihfw vxevwlwxwhv wr hdfk frqvxphu grhv qrw fkdqjh1 P| prgho doorzvsurgxfwv wr eh lpshuihfw vxevwlwxwhv dqg hyhq frpsohphqwv zlwk hdfk rwkhu/dqg khqfh lpsolhv d ulfkhu vxevwlwxwlrq sdwwhuq1

Frpsohphqwdulw| dovr fdqqrw eh h{sodlqhg e| FFP zkhq wkh khgrqlfdssurdfk lv qrw xvhg dqg dq dgglwlyh vhsdudeoh xwlolw| ixqfwlrq lv dvvxphg

�1

� � � � � � � � � � � � � � ! � " # $ $ � % & " & ' # ( # ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < < 8 6 5 = > ?@ A B C D E F G H I J

65

Qr frpsohphqwdu| h{lvwv dprqj surgxfwv xqghu wkh dgglwlyh vhsdudelolw| dv0vxpswlrq/ dqg gl�huhqw surgxfwv frpshwh gluhfwo| iru exgjhw vkduh1 Wkhuh0iruh wkh| duh rqo| vxevwlwxwhv iru hdfk rwkhu1 Lqghhg/ li zh ixuwkhu dvvxphtxdvlolqhdu suhihuhqfh zlwk uhvshfw wr + � / wkh rxwvlgh jrrg/ dqg lwv lqwhulruvroxwlrq/ wkhq sulfh fkdqjh lq rqh surgxfw kdv qr h�hfw rq wkh ghpdqg irurwkhu surgxfwv1 Wkhuhiruh wkhuh lv qhlwkhu vxevwlwxwdelolw| qru frpsohphq0wdulw|1

Yduldeohv �J?r| R � R � R � � R � � R � � R � � R � � R � �+ � 594 05:4 6< 73 045 054: 089 48< 043 � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � " # $+ % 48< 0438 04;; 433 5; 0746 498 5<: 0:9& ' ( $ & ) * $ & ( + $ & ( , $ & + ) $ & , - . $ & + , $ & . + $ & ( # $+ / 0 434 089 < 043; 78 95 086 075 45& ' ( $ & ) * $ & ( + $ & ( , $ & + ) $ & , - . $ & + , $ & . + $ & ( # $+ / 1 83 05: 054 :< 045 09: 08 45 5

E2S� 2 3 4 5 2 6 7 5 2 6 8 5 2 7 3 5 2 8 9 : 5 2 7 8 5 2 : 7 5 2 6 ; 5+ < = 4<6 0449 056 <4 4; 0738 <7 495 0;;

2 > 6 5 2 3 4 5 2 6 7 5 2 6 8 5 2 7 ? @ A B C D @ A E B @ A D E @ A F G @+ H H 7;; 05;8 0444 :: 485 0478 0597 46; 046<

A I F @ A J K @ A F E @ A F C @ A E J @ A B C D @ A E B @ A D E @ A F G @+ H L 8< 073 3 83 046 054 55 0:; 5

A I F @ A J K @ A F E @ A F B @ A E J @ A B C D @ A E B @ A D E @ A F G @+ M N 44: 55 088 07 435 047< 04<9 4:4 <

A I F @ A J K @ A F E @ A F B @ A E J @ A B C D @ A E B @ A D E @ A F G @O P Q R S T U V W X Y Y Z Y [ \ ] ^ _ ` a b c d e f g h i h iWdeoh ;= Sdudphwhu Hvwlpdwhv ri d Vlpsoh Uhjuhvvlrq

Xvxdoo| wzr surgxfwv duh forvh vxevwlwxwhv zkhq wkh| kdyh wkh vdph d0yru pl{wxuh dqg rqo| gl�hu lq sdfndjlqj dqg frqwdlqhu vl}h1 Iru h{dpsoh/+FROD/ FDII/ GLHW/ ODUJ, lv wkh forvhvw vxevwlwxwh ri +FROD/ FDII/GLHW/ FDQ/ SDFN,1 Wklv uhvxow lv lqwxlwlyh= zh jhqhudoo| h{shfw wkh forvhvwvxevwlwxwh ri d 450sdfn Frfd Frod Fodvvlf lv d 50olwhu Frfd Frod Fodvvlf1

Qhjdwlyh furvv0sulfh hodvwlflwlhv duh jhqhudoo| vpdoohu lq devroxwh ydoxhwkdq srvlwlyh furvv0sulfh hodvwlflwlhv1 Wklv lpsolhv wkdw zkhq wkh sulfh ri dsurgxfw idoov/ wkh wrwdo fkdqjh lq ghpdqg iru rwkhu surgxfwv lv qhjdwlyh1 Vrzkhq wkh sulfh ri 450sdfn Frfd0Frod Fodvvlf idoov/ wkh frpelqhg ghpdqg ridoo Shsvl Fr1 surgxfwv zloo idoo/ hyhq wkrxjk zh pd| revhuyh wkdw wkh ghpdqg

j k l m k n o p k l l o q m o r s t u vw x y z { | } ~ � � � �� � �

66

iru 50olwhu Prxqwdlq Ghz ulvhv1

Surgxfw Glvsod| H�hfwv Wkh h�hfwv ri surgxfw glvsod|v lq vxshupdunhwvduh uhsruwhg lq Wdeoh <1 Iluvw L frpsxwh wkh pdunhw ghpdqg iru hdfk surgxfwzkhq wkhuh duh qr surgxfw glvsod|v iru doo surgxfwv/ dqg wkhq wkh pdunhwghpdqg zkhq wkhuh duh surgxfw glvsod|v1 Froxpqv 5 wr < uhsruw wkh fkdqjhlq groodu uhyhqxhv ri gl�huhqw surgxfwv zkhq rqh ri wkh surgxfwv lv glvsod|hg1Froxpqv 43 wr 48 uhsruw wkh fkdqjh lq groodu uhyhqxhv ri gl�huhqw surgxfwvzkhq wzr surgxfwv duh glvsod|hg vlpxowdqhrxvo|1

Gldjrqdo hohphqwv lq wkh wdeoh iurp froxpqv 5 wr < uhsruw wkh �rzq0glvsod|v� h�hfwv1 Wkh odujhu wkh pdunhw vkduh wkh surgxfw kdv/ wkh odujhu wkhlqfuhdvh lq uhyhqxh zkhq wkh surgxfw lv glvsod|hg1 +Iru h{dpsoh/ +FROD/FDII/ QGLHW/ FDQ/ SDFN, kdv wkh odujhvw uhyhqxh jurzwk1, Li wkh pdu0jlqdo frvw ri surgxfwlrq lv frqvwdqw dqg wkh frvwv ri surgxfw glvsod|v duh�{hg/ odujh surgxfwv duh pruh zloolqj wr glvsod| wkhlu surgxfwv1 Wdeoh < dovrvkrzv wkdw wkh wrwdo uhyhqxh fkdqjh/ zklfk lv wkh vxp ri uhyhqxh fkdqjhv ridoo surgxfwv/ lv dozd|v srvlwlyh zkhq wkhuh duh surgxfw glvsod|v1 Wklv lv eh0fdxvh wkh prgho dvvxphv wkdw surgxfw glvsod|v gluhfwo| lqfuhdvh frqvxphu*vpdujlqdo xwlolw| ri frqvxplqj wkh glvsod|hg gulqnv1 L �qg wkdw wkh wrwdo uhy0hqxh lqfuhdvh lv odujh zkhq odujh surgxfwv duh glvsod|hg +vxfk dv +FROD/FDII/ QGLHW/ FDQ/ SDFN,,1 Vxssrvh vxshupdunhwv fkdujh d �{hg shu0fhqwdjh ri pdun0xs deryh zkrohvdoh sulfhv1 Lw zrxog eh pruh sur�wdeoh iruvxshupdunhwv wr glvsod| odujh surgxfwv1 Wkh deryh wzr �qglqjv h{sodlq zk|odujh surgxfwv duh glvsod|hg pruh iuhtxhqwo| lq vxshupdunhwv1

Frpsohphqwdulw| dovr h{lvwv lq surgxfw glvsod|v1 Wkh sdwwhuq lv vlploduwr zkdw L glvfxvvhg deryh= surgxfwv duh vxevwlwxwhv zkhq wkh| kdyh vrphfrpprq surgxfw fkdudfwhulvwlfv/ dqg duh frpsohphqwv zkhq wkh| duh yhu|gl�huhqw lq wkh fkdudfwhulvwlfv vsdfh1 Frpsdulqj Wdeoh < zlwk Wdeohv : dqg;/ surgxfwv duh frpsohphqwv lq surgxfw glvsod|v zkhq wkh| duh frpsohphqwvlq sulfhv/ dqg ylfh yhuvd iru vxevwlwxwhv1

Ilqdoo| wrwdo uhyhqxh fkdqjhv duh odujhuzkhq wkhuh duh vlpxowdqhrxv surg0xfw glvsod|v1 Wklv lv ehfdxvh rxu ixqfwlrqdo irup ri wkh xwlolw| ixqfwlrq grhvqrw dvvxph d ghfolqlqj h�hfw zkhq wkh qxpehu ri surgxfwv glvsod|hg lq0fuhdvhv1 Vxshupdunhwv zloo suhihu wr glvsod| vhyhudo surgxfwv vlpxowdqhrxvo|lq rughu wr pd{lpl}h wkh wrwdo h�hfw1 Zkhq wzr vxevwlwxwhv/ vxfk dv +FROD/FDII/ QGLHW/ ODUJ, dqg +FROD/ FDII/ QGLHW/ FDQ/ SDFN,/ duhglvsod|hg wrjhwkhu/ hdfk kdv d vpdoohu uhyhqxh lqfuhdvh wkdq zkhq wkh| duh

67

Prod

uct

OTH

ER, N

CA

FF, N

DIE

T,

LAR

G (3

)O

THER

, NC

AFF

, ND

IET,

C

AN

PA

CK

(5)

OTH

ER, N

CA

FF,

DIE

T, L

AR

G (1

0)O

THER

, NC

AFF

, DIE

T,

CA

N, P

AC

K (1

2)C

OLA

, CA

FF,

ND

IET,

LA

RG

(20)

CO

LA, C

AFF

, ND

IET,

C

AN

, PA

CK

(22)

CO

LA, C

AFF

, D

IET,

LA

RG

(28)

CO

LA, C

AFF

, D

IET,

CA

N, P

AC

K

(30)

OTH

ER, N

CA

FF, N

DIE

T,

LAR

G48

3-1

11-7

214

-82

51-2

617

OTH

ER, N

CA

FF, N

DIE

T,

CA

N P

AC

K-1

4362

930

-43

57-2

0715

-82

OTH

ER, N

CA

FF, D

IET,

LA

RG

-67

2226

1-4

0-3

512

-12

4O

THER

, NC

AFF

, DIE

T,

CA

N, P

AC

K18

-40

-51

215

16-6

24

-22

CO

LA, C

AFF

, ND

IET,

LA

RG

-67

38-3

011

514

-173

-67

43C

OLA

, CA

FF, N

DIE

T,

CA

N, P

AC

K64

-214

14-6

3-2

5110

1366

-163

CO

LA, C

AFF

, DIE

T,

LAR

G-2

311

-11

3-6

747

224

-80

CO

LA, C

AFF

, DIE

T,

CA

N, P

AC

K22

-83

5-2

264

-172

-120

478

Tota

l28

825

214

675

216

510

8419

5

Pro

du

ct3

& 5

3 &

22

22 &

20

22 &

520

& 3

20 &

5O

TH

ER

, NC

AF

F, N

DIE

T,

LA

RG

367

533

-27

-56

400

-194

OT

HE

R, N

CA

FF

, ND

IET

, C

AN

PA

CK

486

-342

-156

399

-88

691

OT

HE

R,

NC

AF

F,

DIE

T,

LA

RG

-46

-56

-21

30-1

01-1

4O

TH

ER

, N

CA

FF

, D

IET

, C

AN

, PA

CK

-24

-45

-49

-100

34-2

5

CO

LA

, CA

FF

, ND

IET

, LA

RG

-30

-236

336

-139

441

556

CO

LA

, CA

FF

, ND

IET

, CA

N,

PA

CK

-147

1068

759

802

-184

-468

CO

LA

, CA

FF

, DIE

T, L

AR

G-1

124

-23

58-9

0-5

7C

OL

A, C

AF

F, D

IET

, CA

N,

PA

CK

-63

-102

-105

-202

86-1

9To

tal

531

797

714

792

499

469

Tab

le 9

: E

ffec

ts o

f P

rod

uct

Dis

pla

ys o

n R

even

ues

No

te:

(1)

Ch

an

ge

s i

n r

ev

en

ue

are

in

do

lla

r;

(

2)

i &

j m

ea

ns

pro

du

ct i

& j

are

dis

pla

yed

to

ge

the

r, w

he

re p

rod

uct

nu

mb

er

is g

ive

n i

n t

he

pa

ren

the

ses

in t

he

fir

st r

ow

of

ea

ch p

rod

uct

.

Iljxuh 9=68

glvsod|hg vhsdudwho|1 H�hfw ri glvsod|v lv wkh odujhvw zkhq wzr frpsohphqwvduh glvsod|hg vlpxowdqhrxvo|1 Wkhvh h{sodlq zk| vxshupdunhwv dozd|v glv0sod| surgxfwv zlwk gl�huhqw surgxfw fkdudfwhulvwlfv/ vxfk dv 50olwhu FrfdFrod Fodvvlf dqg Vsulwh/ exw qrw/ iru h{dpsoh/ 50olwhu dqg 450sdfn Frfd FrodFodvvlf/ vlpxowdqhrxvo|1

71516 Olplwdwlrqv

Lq wklv vhfwlrq L zloo glvfxvv vhyhudo olplwdwlrqv dqg srwhqwldo eldvhv lq wkhprgho/ dqg vxjjhvw vrph vroxwlrqv1

Iluvw/ lpsruwdqw surgxfw fkdudfwhulvwlfv duh gl!fxow wr gh�qh1 Lq wkhfdvh ri vriw gulqnv/ dyruv/ fd�hlqdwhg ru qrw/ glhw ru qrw/ frqwdlqhu vl}hvdqg sdfndjlqj duh vrph fkdudfwhulvwlfv lghqwl�hg dv vljql�fdqw lq d�hfwlqjfrqvxphu*v sxufkdvh ghflvlrqv1 Rwkhu fkdudfwhulvwlfv/ hvshfldoo| wkrvh rq dfrqwlqxrxv vfdoh/ vxfk dv wkh dprxqw ri fdorulhv/ vxjdu/ vrglxp/ duh qrw hdv|wr sxw lqwr wkh xwlolw| ixqfwlrq

� �1 D pruh gl!fxow sureohp lv krz fkdudfwhu0

lvwlfv lqwhudfw zlwk hdfk rwkhu1 Vlpsoh dvvxpswlrq ri olqhdu vxppdelolw| dqgvhsdudelolw| ri fkdudfwhulvwlfv lq wkh xwlolw| ixqfwlrq eulqjv srwhqwldo eldvhv1L wu| wr vroyh wkh sureohp e| wdnlqj lqwhudfwlqj whupv vxfk dv gl�huhqw d0yru pl{wxuhv/ +RWKHU ) ODUJ,/ dqg +FDQ ) SDFN,1 Lq rughu wr gh�qhsrvlwlrqv ri surgxfwv lq wkh fkdudfwhulvwlfv vsdfh/ uhvhdufkhuv vkrxog xvh dpruh h{leoh xwlolw| ixqfwlrq/ vxfk dv wkrvh doorzlqj gl�huhqw lqwhudfwlrqv rifkdudfwhulvwlfv1 Rq wkh rwkhu kdqg/ dgglqj wkhvh lqwhudfwlrqv zloo kdup wkhprgho sduvlprq|1 Lw frxog eh d elj sureohp zkhq wkh frpsxwdwlrqdo exughqlv khdy|1

Vhfrqg/ hdfk surgxfw lq wklv prgho lv d jurxs ri VNXv zklfk vkduh doosurgxfw fkdudfwhulvwlfv1 Wkhlu sulfhv duh dyhudjh sulfhv zhljkwhg e| pdunhwvkduh1 Wklv zd| d fdq ri Frfd0Frod Fodvvlf/ d fdq ri Shsvl/ dqg d fdq risulydwh odeho frod duh jurxshg wrjhwkhu dv rqh surgxfw1 Wklv zloo eldv hvwlpd0wlrqv li frqvxphuv gr qrw wuhdw wkhp htxdoo|1 Rqh vroxwlrq lv wr lqfrusrudwheudqg qdphv/ vxfk dv �Frfd0Frod�/ �Shsvl� dqg �sulydwh odehov�/ dv surg0xfw fkdudfwhulvwlfv1 Wklv pd| eh hvvhqwldo lq wkh fdvh ri vriw gulqnv/ zkhuhlw lv eholhyhg wkdw or|dow| wr eudqg qdphv lv dq lpsruwdqw ghwhuplqdqw lqfrqvxphuv* sxufkdvh ghflvlrqv1 Wklv lv d wrslf L dp zrunlqj rq1

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " # $ % & $ ' $ % ( ) * + + , - . - , / . 0 / 1 2 0 / , 3 1 0 / 0 4 5 67 8 9 : : ; < 7 ; = > ? @ A B C B D E F E B C G F H I C H B I E F G J K F G L H M N O P Q Q R S T U V W X V Y Z W X [ \ ] ] ^ X _ \ Y X ` V ` V Y Z Wa W b \ V a c d e f d g h e i j

69

Odvwo|/ wklv prgho rqo| frqvlghuv vwdwlf xwlolw| pd{lpl}dwlrq/ dqg lqyhqwru|0kroglqj ehkdylru lv odujho| ljqruhg1 Li d frqvxphu*v ex|lqj pruh wrgd| lviroorzhg e| d orqj shulrg zlwkrxw sxufkdvh/ zh zloo revhuyh d orzhu ghpdqgiru vhyhudo shulrgv diwhu d vkruw whup vxujh lq ghpdqg zkhqhyhu wkhuh lv dsulfh fxw/ dqg wkh qhw h�hfw rq ghpdqg zloo qrw eh dv elj dv suhglfwhg lq wklvprgho1 Pdqxidfwxuhuv duh lqwhuhvwhg wr nqrz wkh orqj0whup h�hfw ri sulfhfkdqjh1 Wr dffrpprgdwh wklv frqfhuq d g|qdplf frqwlqxrxv fkrlfh prgho/wdnlqj dffrxqw ri lqyhqwru|0kroglqj ehkdylru/ vkrxog eh ghyhorshg1 Wklv lvwkh vhfrqg sduw ri p| uhvhdufk rq wkh ghpdqg iru vriw gulqnv1

71517 Dssolfdwlrqv

Wkrxjk lw kdv vhyhudo olplwdwlrqv/ wkh uhvxowv ri wklv sdshu fdq eh dssolhgwr pdq| uhodwhg duhdv vxfk dv zhoiduh dqdo|vhv ri sulfh fkdqjhv dqg qhzsurgxfwv hqwhulqj ru h{lvwlqj surgxfwv h{lwlqj wkh pdunhw1 Wkh fkdqjh lqfrqvxphu zhoiduh fdq eh frpsxwhg qxphulfdoo|1 Khqfh L fdq suhglfw krzfrqvxphuv zloo eh d�hfwhg zkhq wkhuh lv d sulfh frqwuro/ ru zkhq glhw gulqnvduh qhzo| lqwurgxfhg ru qrq0fd�hlqdwhg frod gulqnv h{lw iurp wkh pdunhw1

Uhvxowv lq wklv sdshu dovr khos xv wr xqghuvwdqg lqgxvwuldo sulflqj dqgsurgxfw glvsod|v vwudwhjlhv1 Iru h{dpsoh/ vriw gulqn pdqxidfwxuhuv jhqhudoo|surgxfh pxowlsoh eudqgv ri gulqnv/ hdfk eudqg zlwk pdq| frqwdlqhu vl}hv dqgsdfndjhv1 D pdqxidfwxuhu nqrzv wkdw sulfhv fxw dqg surgxfw glvsod|v ri rqhgulqn zloo kdyh h�hfwv rq rzq dqg ulydo gulqnv dv zhoo 1 L kdyh vkrzq wkdw wkhh�hfwv duh wkh odujhvw zkhq wzr frpsohphqwdu| surgxfwv duh glvsod|hg wr0jhwkhu/ dqg ohvv zkhq wzr vxevwlwxwh surgxfwv duh glvsod|hg wrjhwkhu1 Lqghhgwkh h�hfwv ri sulfh fkdqjh kdyh d vlplodu sdwwhuq1 Wkh h{lvwhqfh ri frpsoh0phqwdulw| fdq kdyh d odujh lpsdfw rq pdqxidfwxuhuv* ghflvlrqv pdnlqj1

Wkh phwkrgrorj| xvhg lq wklv sdshu fdq eh h{whqghg wr rwkhu nlqgv righflvlrq pdnlqj lqyroylqj pxowlsoh dowhuqdwlyh dqg pxowlsoh xqlw fkrlfhv zlwkqxphurxv fruqhu vroxwlrqv1 Wkh iroorzlqj duh vhyhudo h{dpsohv=

� Surgxfw glvsod|v lq vwruhv= vwruh pdqdjhuv kdyh wr ghflgh zklfk surg0xfwv wr glvsod| dqg krz pxfk vkhoi vsdfh wr jlyh wkhp dw iurqw0hqgru edfn0hqg dlvohv1 Xvxdoo| vhyhudo surgxfwv duh glvsod|hg dw wkh vdphwlph exw/ vlqfh dydlodeoh vsdfh lv olplwhg/ qrw doo surgxfwv zrxog eh fkr0vhq1 Iru dq rswlpdo surgxfw glvsod| srolf|/ wkh h�hfwv rq wkh ghpdqgiru rwkhu surgxfwv/ rqfh d surgxfw lv glvsod|hg/ vkrxog eh xqghuvwrrg1

6:

� Lqyhvwphqw ghflvlrqv= lq hyhu| shulrg rxu hfrqrplf djhqwv kdyh wr gh0flgh krz wr doorfdwh wkhlu prqh| rq gl�huhqw �qdqfldo dvvhwv vxfk dvvwrfnv/ erqgv/ edqn vdylqjv dqg uhdo hvwdwh1 Lq pdq| ohvv ghyhorshgfrxqwulhv mhzhou| dqg jrog duh dovr frpprq dvvhwv1 Wkhvh lqyhvwphqwghflvlrqv wr vrph h{whqw duh xqghu �qdqfldo frqvwudlqwv1 Fruqhu vrox0wlrqv h{lvw ehfdxvh shrsoh duh qrw lqyhvwlqj lq hyhu| dvvhw1 Ehvlghv/lqyhvwlqj lq rqh dvvhw zloo d�hfw ehqh�wv ru frvwv ri lqyhvwlqj lq rwkhudvvhwv1 Iru h{dpsoh/ lqyhvwlqj pruh lq grw frp vkduhv frxog eh kljko|ulvn|/ dqg lqyhvwruv pd| zdqw wr sxw vrph prqh| lqwr dvvhwv zlwk dvwdeohu uhwxuq/ vxfk dv erqgv1

� Dgyhuwlvlqj ghflvlrqv= d pdqxidfwxuhu pd| surgxfh pxowlsoh surgxfwv/dqg kdv gl�huhqw phgld wr fduu| lwv surgxfw dgyhuwlvlqj1 Iru h{dpsoh/Frfd Frod Fr1 surgxfhv Frfd Frod Fodvvlf/ Frfd Frod Fd�hlqh Iuhh/Glhw Frnh/ Fkhuu| Frnh/ Phoorz \hoorz/ Iuhvfd/ Plqxwh Pdlg/ Vsulwh/Wde dqg rwkhuv/ dqg wkhuh duh vhyhudo phgld ri dgyhuwlvlqj/ lqfoxglqjpdjd}lqhv/ qhzvsdshu/ rxwgrru/ qhwzrun WY/ fdeoh WY/ qhwzrun udglrdqg rwkhuv

� �1 Lw kdv wr ghflgh krz wr doorfdwh lwv exgjhw rq gl�huhqw

phgld iru gl�huhqw surgxfwv1 Prqh| vshqw rq rqh surgxfw fdq kdyhh�hfwv rq rwkhu surgxfwv1 Iru h{dpsoh/ pruh dgyhuwlvlqj iru Glhw Frnhpd| eh khosixo iru wkh vdoh ri Fkhuu| Frnh exw pd| gulyh frqvxphuvdzd| iurp Vsulwh1 Fruqhu vroxwlrqv duh dovr frpprq= L �qg iurp wkhgdwd pdq| vriw gulqn pdqxidfwxuhuv kdyh }hur h{shqglwxuh rq gl�huhqwphgld iru gl�huhqw surgxfwv1

8 Frqfoxvlrq

Lq wklv sdshu L frqvlghu ghpdqg prghov zkhuh +4, frqvxphuv fduh derxwsurgxfw fkdudfwhulvwlfv/ +5, frqvxphuv fdq fkrrvh pxowlsoh eudqgv dqg dfrqwlqxrxv txdqwlw| ri surgxfwv/ dqg +6, wkh h{lvwhqfh ri fruqhu vroxwlrqv lvwkh qrup1 L dsso| wkhvh prghov hpslulfdoo| wr sxufkdvlqj ghflvlrqv ri vriwgulqnv1

L sursrvh wr xvh d frqwlqxrxv fkrlfh prgho hpehgghg lq d khgrqlf ds0surdfk1 Wudglwlrqdo frqwlqxrxv ghpdqg prghov uhvwulfwv uhvhdufkhuv wr xvlqjpdunhw ohyho gdwd/ exw wklv prgho doorzv plfur ohyho gdwd zlwk qxphurxv fru0qhu vroxwlrqv1 Frpsdulqj zlwk glvfuhwh surgxfw fkrlfh prghov/ wklv prgho

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " � # $ � % & ' ' ( # ) * + , - . / 0 . / 1

6;

kdv dq dgydqwdjh ri doorzlqj d pruh h{leoh sdwwhuq ri vxevwlwxwdelolw| dqghyhq frpsohphqwdulw| frqvlvwhqw zlwk frqvxphu ehkdylruv ri ydulhw| vhhnlqjdqg lqwhughshqghqfh ri fkrlfhv dprqj surgxfwv1 L xvh wkh VPP wr hvwlpdwhwkh prgho wr vroyh wkh frpsxwdwlrqdo frpsoh{lw| sureohp iroorzlqj wkh h{0lvwhqfh ri pdq| fruqhu vroxwlrqv1

Wklv phwkrgrorj| lv dssolhg wr hvwlpdwh wkh ghpdqg iru vriw gulqnv xv0lqj plfur ohyho gdwd1 E| hvwlpdwlqj sdudphwhuv lq wkh xwlolw| ixqfwlrq rilqglylgxdo frqvxphuv/ L dp deoh wr jhqhudwh wkh pdunhw ghpdqg dqg wkh fru0uhvsrqglqj rzq0sulfh dqg furvv0sulfh hodvwlflwlhv ri hdfk surgxfw1 Gl�huhqwghjuhhv ri vxevwlwxwdelolw|/ dqg hyhq frpsohphqwdulw|/ duh lghqwl�hg dprqjgl�huhqw surgxfwv1 Wkh uhvxowv vkrz wkdw frqvxphuv vhhn ydulhw| ryhu surg0xfw fkdudfwhulvwlfv vxfk dv dyru/ sdfndjlqj/ dqg frqwdlqhu vl}h/ dqg |lhogfrpsohphqwdulw| zkhq wzr surgxfwv duh yhu| gl�huhqw lq surgxfw fkdudfwhu0lvwlfv1 L dovr frpsxwh h�hfwv ri surgxfw glvsod|v/ dqg �qg vlplodu sdwwhuqvri vxevwlwxwdelolw| dqg frpsohphqwdulw| dprqj surgxfwv1

Wkh uhvxowv lq wklv sdshu fdq eh xvhg iru zhoiduh dqdo|vlv zkhq sulfhvfkdqjh/ qhz surgxfwv hqwhu ru h{lvwlqj surgxfwv h{lw iurp wkh pdunhw1 Wkh|fdq dovr khos pdqxidfwxuhuv wr ghflgh rswlpdo sulflqj dqg surgxfw glvsod|vwudwhjlhv1 Wklv prgho srwhqwldoo| fdq eh dssolhg wr hvwlpdwh sdudphwhuv rirwkhu nlqgv ri ghflvlrq pdnlqj zkhq pxowlsoh dowhuqdwlyh dqg pxowlsoh xqlwfkrlfhv zlwk wkh h{lvwhqfh ri pdq| fruqhu vroxwlrqv duh wkh qrup1 Lw doorzv dfrpsolfdwhg uhodwlrqvkls dprqj gl�huhqw dowhuqdwlyhv xqghu zklfk fkrrvlqjrqh dowhuqdwlyh zloo d�hfw wkh ixqfwlrqlqj ri rwkhu dowhuqdwlyhv1

6<

Uhihuhqfhv

^4` Dphpl|d/ W1 +4<;7,= �Wrelw Prghov= D Vxuyh|/� Mrxuqdo ri Hfrqrphw0

ulfv/ 57/ 60941

^5` Ehuu|/ V1 +4<<7,= �Hvwlpdwlqj Glvfuhwh0Fkrlfh Prghov ri Surgxfw Gli0ihuhqwldwlrq/� Udqg Mrxuqdo ri Hfrqrplfv/ 58/ 57505951

^6` Ehuu|/ V1/ Ohylqvrkq/ M1/ dqg Sdnhv/ D1 +4<<8,= �Dxwrpreloh Sulfhv lqPdunhw Htxloleulxp/� Hfrqrphwulfd/ 96/ ;740;<31

^7` Ehoo/ G1U1/ Kr/ W1/ dqg Wdqj/ F1V1 +4<<;,= �GhwhuplqlqjZkhuh wr Vkrs=Il{hg dqg Yduldeoh Frvwv ri Vkrsslqj/� Mrxuqdo ri Pdunhwlqj Uhvhdufk/Dxjxvw1

^8` Eurzq/ G1M1/ dqg Pdw}nlq/ U1O1 +4<<;,= �Hvwlpdwlrq ri QrqsdudphwulfIxqfwlrqv lq Vlpxowdqhrxv Htxdwlrqv Prghov/ zlwk dq Dssolfdwlrq wrFrqvxphu Ghpdqg/� pdqxvfulsw/ \doh Xqlyhuvlw|1

^9` Fkdq/ W1\1 +4<<<,= �Lghqwl�fdwlrq lq Frqwlqxrxv Ghpdqg Prghov/�pdqxvfulsw/ \doh Xqlyhuvlw|1

^:` Fkldqj/ M1 +4<<4,= �D Vlpxowdqhrxv Dssurdfk wr wkh Zkhwkhu/ Zkdwdqg Krz Pxfk wr Ex| Txhvwlrqv/� Pdunhwlqj Vflhqfh/ Yro143/ Qr17/5<:06481

^;` Fklqwdjxqwd/ S1N1 +4<<6,= �Lqyhvwljdwlqj Sxufkdvh Lqflghqfh/ EudqgFkrlfh dqg Sxufkdvh Txdqwlw| Ghflvlrqv ri Krxvhkrogv/�Pdunhwlqj Vfl0

hqfh/ Yro145/ Qr15/ 4;7053;1

^<` Gdylv/ S1 +5333,= �Qhz Sdudphwulf Ghpdqg V|vwhpv iru Pdunhw OhyhoGhpdqg Vwxglhv/� pdqxvfulsw/ PLW Vordq Vfkrro ri Pdqdjhphqw1

^43` Ghdwrq/ D1/ dqg M1 Pxhooedxhu +4<;3,= Hfrqrplfv dqg Frqvxphu Eh0

kdylru1 Fdpeulgjh/ X1N1= Fdpeulgjh Xqlyhuvlw| Suhvv1

^44` Guh}h/ M1K1 dqg Kdjhq/ N1S1 +4<:;,= �Fkrlfh ri Surgxfw Txdolw|= Htxl0oleulxp dqg H!flhqf|/� Hfrqrphwulfd/ 79/ 7<608461

73

^45` Gxeh/ Mhdq0Slhuuh +4<<<,= �Surgxfw Gl�huhqwldwlrq dqg Phujhuv lq wkhFduerqdwhg Vriw Gulqn Lqgxvwu|/� Glvvhuwdwlrq/ Qruwkzhvwhuq Xqlyhu0vlw|1

^46` Hughp/ W1 +4<<9,= �D G|qdplf Dqdo|vlv ri Pdunhw Vwuxfwxuh Edvhg rqSdqho Gdwd/� Pdunhwlqj Vflhqfh/ Yro148/ Qr17/ 68<06:;1

^47` Jrxulhurx{/ F1/ dqg Prqiruw/ D1 +4<<9,= Vlpxodwlrq0Edvhg Hfrqrphwulf

Phwkrgv1 R{irug Xqlyhuvlw| Suhvv1

^48` Jxdgdjql/ S1P1 dqg Olwwoh/ M1G1F1 +4<;6,= �D Orjlw Prgho ri EudqgFkrlfh Fdoleudwhg rq Vfdqqhu Gdwd/� Pdunhwlqj Vflhqfh/ Yro15/ Qr16/536056;1

^49` Kdqhpdqq/ Z1P1 +4<;7,= �Glvfuhwh2Frqwlqxrxv Prghov ri FrqvxphuGhpdqg/� Hfrqrphwulfd/ 85/ 87408941

^4:` Khqgho/ L1 +4<<<,= �Hvwlpdwlqj Pxowlsoh0Glvfuhwh Fkrlfh Prghov= DqDssolfdwlrq wr Frpsxwhul}dwlrq Uhwxuqv/� Uhylhz ri Hfrqrplf Vwxglhv/

99/ 75607791

^4;` Nlp/ M1/ Doohqe| J1P1/ dqg Urvvl/ S1H1 +5333,= �Prgholqj FrqvxphuGhpdqg iru Ydulhw|/� pdqxvfulsw/ JVE/ Xqlyhuvlw| ri Fklfdjr1

^4<` Od�rqw/ M1M1/ Rvvdug/ K/ dqg Yxrqj/ T1 +4<<8,= �Hfrqrphwulfv ri Iluvw0Sulfh Dxfwlrqv/� Hfrqrphwulfd/ 96/ <860<;31

^53` Odo/ U1 dqg Yloodv0Erdv/ M1P1 +4<<;,= �Sulfh Surprwlrqv dqg WudghGhdov zlwk Pxowlsurgxfw Uhwdlohuv/� Pdqdjhphqw Vflhqfh/ Yro1 77/ Qr1:/ <680<7<1

^54` Odqfdvwhu/ N1M1 +4<99,= �D Qhz Dssurdfk wr Frqvxphu Wkhru|/� Mrxu0

qdo ri Srolwlfdo Hfrqrp|/ :7/ 46504891

^55` PfIdgghq/ G1 +4<;<,= �D Phwkrg ri Vlpxodwhg Prphqwv iru Hvwlpdwlrqri Glvfuhwh Uhvsrqvh Prghov Zlwkrxw Qxphulfdo Lqwhjudwlrq/� Hfrqr0

phwulfd/ 8:/ <<8043591

^56` Qdudvlpkdq/ F1 +4<;;,= �Frpshwlwlyh Surprwlrqdo Vwudwhjlhv/� Mrxuqdori Exvlqhvv/ 94/ 7/ 75:077<1

74

^57` Qhoghu/ M1D1/ dqg Phdg U1 +4<98,= �D Vlpsoh{ Phwkrg iru IxqfwlrqPlqlpl}dwlrq/� Frpsxwdwlrqdo Mrxuqdo/ :/ 63;06461

^58` Sdnhv/ S1 +4<;9,= �Sdwhqwv dv Rswlrqv= Vrph Hvwlpdwhv ri wkh Ydoxh riKroglqj Hxurshdq Sdwhqw Vwrfnv/� Hfrqrphwulfd/ 87/ :880:;81

^59` Sdnhv/ D1 dqg Sroodug/ G1 +4<;<,= �Vlpxodwlrq dqg wkh Dv|pswrwlfv riRswlpl}dwlrq Hvwlpdwruv/� Hfrqrphwulfd/ 8:/ 435:0438:1

^5:` Urvhq/ V1 +4<:7,= �Khgrqlf Sulfhv dqg Lpsolflw Pdunhwv= Surgxfw Gli0ihuhqwldwlrq lq Sxuh Frpshwlwlrq/� Mrxuqdo ri Srolwlfdo Hfrqrp|/ ;5/670881

^5;` Ur|/ U1/ Fklqwdjxqwd S1N1/ dqg Kdogdu/ V1 +4<<9,= �D Iudphzrun iruLqyhvwljdwlqj Kdelwv/ cWkh Kdqg ri wkh Sdvw*/ dqg Khwhurjhqhlw| lq G|0qdplf Eudqg Fkrlfh/� Pdunhwlqj Vflhqfh/ Yro148/ Qr16/ 5;305<<1

^5<` Vhhwkdudpdq/ S1E1 dqg Fklqwdjxqwd/ S1N1 +4<<;,= �D Prgho ri Lqhuwlddqg Ydulhw|0Vhhnlqj zlwk Pdunhwlqj Yduldeohv/� Lqwhuqdwlrqdo Mrxuqdo

ri Uhvhdufk lq Pdunhwlqj/ 48/ 404:1

^63` Yduldq/ K1 +4<;3,= �D Prgho ri Vdohv/� Dphulfdq Hfrqrplf Uhylhz/Vhswhpehu/ 984098<1

75

Dsshqgl{= Surriv ri Frqvlvwhqf| dqg Dv|pswrwlfQrupdolw| ri VPP Hvwlpdwruv

Lq wklv dsshqgl{ L zloo suryh wkdw wkh VPP hvwlpdwruv/ dssolhg lq wkhFPP hpehgghg lq d khgrqlf dssurdfk zlwk wkh h{lvwhqfh ri fruqhu vrox0wlrqv/ vdwlv�hg frqglwlrqv ri frqvlvwhqf| dqg dv|pswrwlf qrupdolw| xqghuvrph jhqhudo frqglwlrqv1

Sursrvlwlrqv

L vwduw zlwk vrph dgglwlrqdo dvvxpswlrqv zklfk zloo eh khosixo wr suryhvhyhudo sursrvlwlrqv ehorz1

Dvvxpswlrq 9=

Wkhuh h{lvw R dqg 7R vxfk wkdw f R� �?u � ER� � � t�T� ER� � �7R 4� Vr sulfhv duh uhvwulfwhg lq d forvhg dqg erxqghg vxevhwlq -

�� � �

Dvvxpswlrq :=

Wkhuh h{lvwv d qxpehu 7� vxfk wkdw n�Ef� �n � 7� 4 iru doo ��

Dvvxpswlrq ;=

Krxvhkrog lqfrphv duh erxqghg iurp deryh1 Wkdw lv/ wkhuh h{lvwvd qxpehu 7U vxfk wkdw f � U � � 7U 4/ iru doo �1

Gh�qh +Ef� c w� wkh rswlpdo ohyho ri sxufkdvh/ jlyhq w dqg vwrfkdvwlf ydul0deohv 0/ dqg �+Ef� c w� wkh fruuhvsrqglqj vlpxodwlrq sduw jlyhq lq htxdwlrq +;,1Edvhg rq Dvvxpswlrqv 4 wr ; L kdyh wkh iroorzlqj �yh sursrvlwlrqv=

Sursrvlwlrq 4=

Wkhuh h{lvwv d _Ef � vxfk wkdw n6Ef c w�n � nE+� ��+Ef� c w�� �Ef� �n �

_Ef� � iru doo w 5 X/ dqg .d_Ef� �o 4�

76

Surri= E| Dvvxpswlrq 9/ :/ ; dqg wkh lqfrph frqvwudlqw/ +� 7U*R 4�Khqfh/ iru doo �c

nE+ � � �+Ef� c w����Ef� �n � nE+ � � �+Ef� c w��n � n�Ef� �n � d 7U*Ro � 7� 4

Ohw _Ef� � ' d7U*Ro � 7� iru doo �� Wkhuhiruh Sursrvlwlrq 4 lv suryhg1

Sursrvlwlrq 5=

Li wkhuh h{lvwv d xqltxh vroxwlrq �+Ef� c w� iru hyhu| w 5 X/ wkhq�+Ef� c w� lv frqwlqxrxv lq w iru hyhu| w 5 X1

Surri= Doo L qhhg wr suryh lv wkh frqwlqxlw| dqg gl�huhqwldelolw| ri wkhghpdqg ixqfwlrq +Ef� c w� ehfdxvh lwv fruuhvsrqglqj vlpxodwhg sduw/ �+Ef� c w�ckdv wkh vdph surshuwlhv1 Wr vlpsoli| qrwdwlrq L zloo zulwh dv +Ew� li wkhuh lvqr frqixvlrq1

Vxssrvh wkhuh h{lvwv d xqltxh vroxwlrq +Ew�1 Dvvxph wkdw +Ew� lv qrwfrqwlqxrxv lq wc wkdw lv/ wkhuh h{lvwv d vhtxhqfh w

�vxfk wkdw w

�$ wc + � '

+Ew � � $ + 9' +Ew�/ dv ? $ 41 Vlqfh + lv qrw dq rswlpdo fkrlfh/ T E+c w� T E+Ew�c w�� E| wkh frqwlqxlw| dvvxpswlrq ri T E�� lq + lq Dvvxpswlrq 4/ wkhuhh{lvwv d ihdvleoh 7+ forvh hqrxjk wr +Ew� vxfk wkdw T E7+c w� : T E+c w�1 Exw irudoo ? zh pxvw kdyh T E+ � c w � � : T E7+c w � �� Ohw ? $ 4c frqwlqxlw| ri T E��lq erwk + dqg w lq Dvvxpswlrq 4 lpsolhv wkdw T E+c w� : T E7+c w�c zklfk lv dfrqwudglfwlrq1 Khqfh Sursrvlwlrq 5 lv suryhg1

Sursrvlwlrq 6=

D xqltxh ghpdqg ixqfwlrq +Ef� c w� h{lvwv zlwk suredelolw| rqh1

Surri= IurpDvvxpswlrq 4 wkh Mdfreldq pdwul{ L E�� lv vwulfwo| qhjdwlyhgh�qlwh +khqfh wkh vhw ri 5*v suhihuuhg wr dq| vshfl�f 75 5 ] lv frqyh{,1 IurpDvvxpswlrq 5 wkh glphqvlrq ri wkh surgxfw vsdfh a lv juhdwhu wkdq wkhglphqvlrq ri wkh fkdudfwhulvwlf vsdfh �c dqg wkh fkdudfwhulvwlf pdwul{ � kdvixoo froxpq udqn1 Wklv lpsolhv dq| duelwudu| yhfwru 5 kdv dw ohdvw rqh yhfwru+ vxfk wkdw �

+ ' 5� Odqfdvwhu +4<99, suryhg wkdw wkh frqyh{lw| ri wkh xwlolw|

ixqfwlrq lq wkh fkdudfwhulvwlf vsdfh zloo eh suhvhuyhg lq wkh surgxfw vsdfh1Qrz wkh rqo| fdvh ri pxowlsolflw| ri vroxwlrq +Ew� lv zkhq wkh vorsh ri wkhlqgl�huhqfh iurqwlhu lv olqhdu dqg htxdo wr wkh vorsh ri wkh exgjhw vhw1 Iruh{dpsoh/ li surgxfw � kdv kdoi ri wkh dprxqw ri hdfk fkdudfwhulvwlf lq surgxfw

77

& dqg surgxfw �*v sulfh lv h{dfwo| kdoi ri surgxfw &*v/ wzr xqlwv ri surgxfw� zrxog eh lqgl�huhqw wr rqh xqlw ri surgxfw & wr doo frqvxphuv1 Krzhyhu/iurp Dvvxpswlrq 8 wkh xqrevhuydeoh vwrfkdvwlf surgxfw fkdudfwhulvwlf/ Dch{lvwv dqg lv frqwlqxrxvo| glvwulexwhg/ vr wkh deryh vlwxdwlrq h{lvwv rqo| zlwk}hur suredelolw|1 Khqfh wkh xqltxhqhvv ri vroxwlrq +Ew� zrxog eh jxdudqwhhgzlwk suredelolw| rqh/ dqg Sursrvlwlrq 6 lv ydolg1

Sursrvlwlrq 7=

Ghpdqg ixqfwlrq +Ef� c w� lv gl�huhqwldeoh dw w � zlwk suredelolw|rqh1

Surri= Wkh Nxkq0Wxfnhu wkhruhp whoov xv wkdw/ iurp +46,/ wkh iroorzlqjfrqglwlrqv vkrxog krog=

�� L� E��+Ew�cMc 0(��� bR� ' f �s +� : f +l,

�� L� E��+Ew�cMc 0(��� bR � � f �s + � ' f +ll,

Ohw a � a eh wkh vhw ri surgxfwv vxfk wkdw + Ew� : f iru hdfk � 5 a � c dqga � � a eh wkh vhw ri surgxfwv vxfk wkdw + Ew� ' f iru hdfk � 5 a � � Lw lvreylrxv wkdw L fdq uhzulwh wkh deryh frqglwlrqv dv

� L� E�� �+

�Ew�c 0(��� bR ' f li + : f +l*,

� L� E�� �+

�Ew�c 0(��� bR � f li + ' f +ll*,

zkhuh ��lv wkh fkdudfwhulvwlfv pdwul{/ R

�d a � � � yhfwru ri sulfhc dqg +

�Ew�

wkh qrq0}hur ghpdqg/ ri doo � 5 a � �Iru hdfk � 5 a � / iurp Dvvxpswlrq 8 wkh vwrfkdvwlf huuru 0 lv vprrwko|

glvwulexwhg/ wkhuh lv }hur suredelolw| ri +ll, ehlqj dq htxdolw|1 Vr L fdq uhzulwh+ll,/ zlwk suredelolw| rqh/ dv

�� L� E��+Ew�c 0(��� bR � f li +� ' f +ll�,

Vxssrvh w fkdqjhv iurp w wr/ vd|/ w � ' i�� �c b

�j

�1 Li +� c � 5 a � c fkdqjhv

iurp }hur wr vrph ydoxhv juhdwhu wkdq }hur/ lw lv qhfhvvdu| wkdw +ll�, krogv dww ' w exw

�� L� E��

� � + � � Ew � �c 0(� � �� b � R� � f +lll,

78

zkhuh � � � lv wkh fkdudfwhulvwlfv pdwul{ ghohwlqj wkh �0wk urzc dqg + � � Ew � �wkh qhz ghpdqg dw w � iru doo surgxfwv h{foxglqj surgxfw �1

Iurp Sursrvlwlrq 4 dqg 5 +Ew� lv frqwlqxrxv lq w zlwk suredelolw|rqh/ vr dv w � $ w zh kdyh n+ � � Ew � � � + � � Ew�n $ f1 Iurp Dvvxpswlrq 4L� E�� lv frqwlqxrxv lq erwk � dqg 5/ vr zh kdyh m�� L� E� � � + � � Ew � �c 0(� � � ��� L� E� � � + � � Ew�c 0(��m $ f� Khqfh +lll, krogv zlwk suredelolw| }hur1 Lw lp0solhv wkdw zlwk suredelolw| rqh +� E�( w � � lv htxdo wr }hur zkhq nw � � wn lv forvhwr f/ dqg wkh ghulydwlyh h{lvwv1 Vlqfh wkh deryh dujxphqw dssolhv wr hyhu|w 5 Xc lw lv dovr wuxh wkdw +� Ew� lv gl�huhqwldeoh zlwk suredelolw| rqh dw w � /iru � 5 a 1

Ohw*v wxuq wr surgxfw � 5 a � / zkrvh �uvw rughu frqglwlrqv duh jlyhq lq+l*,1 Iurp Dvvxpswlrq 4 wkh ixqfwlrq LE�( w� lv frqwlqxrxvo| gl�huhqwldeoh dwhdfk w 5 X1 Vxssrvh wkh Mdfreldq pdwul{ ri +l*, dw wkh wuxh sdudphwhu ydoxhw � lv qrq0vlqjxodu1 Wkdw lv/

m��L � � E� �

+�Ew � �c 0(� � �� �

m 9' f +ly,

Lw lv vkrzq deryh wkdw zkhq wkh fkdqjh ri w lv vpdoo zlwk suredelolw| rqh +�uhpdlqv htxdo wr }hur iru doo � 5 a 1 Vr L pd| ljqruh wkh fkdqjh lq wkh vhw a eurxjkw rq e| wkh fkdqjh ri w1 Qrz L fdq xvh wkh Lpsolflw Ixqfwlrq Wkhruhp

wr vwdwh wkdw E+ � c ���c + � �Ew� lv gl�huhqwldeoh dw w � zlwk suredelolw| rqh1Qrwh wkdw wkh uhtxluhphqw ri wkh Mdfreldq pdwul{ ehlqj qrqvlqjxodu dw

wkh wuxh sdudphwhu ydoxh w � kdv wr krog iru hyhu| vxevhw �a � a� D qhfhv0vdu| frqglwlrq iru +ly, lv wkdw wkh qxpehu ri urzv ri �

�lv vpdoohu ru htxdo

wr wkh glphqvlrq ri fkdudfwhulvwlfv vsdfh/ �1 Iurp Dvvxpswlrq 4 L� � E�� lvqrqvlqjxodu dw w � � �

1 Vr ��zlwk ixoo urz udqn zrxog eh vx!flhqw iru +ly,1

Wklv lpsolhv wkdw frqvxphuv zrxog qrw ex| dq| surgxfw ri zklfk wkh dprxqwri fkdudfwhulvwlfv lv d olqhdu frpelqdwlrq ri wkh dprxqw ri fkdudfwhulvwlfv rirwkhu surgxfwv wkdw wkh| ex| vlpxowdqhrxvo|1 Vxssrvh wkhuh lv d vxevhw risurgxfwv �a � � a � vxfk wkdw � � � �� � k � �� ' f iru k 5 - �� �

qifj� Wkhq wkhuhh{lvwv wzr jurxsv ri surgxfwv zklfk duh shuihfw vxevwlwxwhv ri hdfk rwkhu lqwkh fkdudfwhulvwlfv vsdfh1 Exw wklv h{lvwv rqo| zlwk suredelolw| }hur vlqfh/iurp Dvvxpswlrq 8/ wkh xqrevhuydeoh surgxfw fkdudfwhulvwlf lv frqwlqxrxvo|glvwulexwhg1 Wkhuhiruh wkh dvvxpswlrq ri ixoo urz udqn ri � � krogv zlwksuredelolw| rqh� Sursrvlwlrq 7 khqfh lv dovr suryhg1

� ! " # # $ % & " ' " ( ( ) * ) + & , - # & . " / " 0 , & $ * ) , ) * - 1 $ ' 2 * ) 3 ' ) ' % 3 ( & , # 4 5 6 7 8 9 6 : ; < = > ? @ < A B C= > ? ; D E F G H I J G K G J L M I N O J G P N H Q J G H R S H J F S O P N T G J G P N H G N O S J F S U Q J V G W X Y Z [ [ \ ] ^ _ ` a b c d e f g h ij k l m n o p q r s t u v s w x w x y z { | } ~ � � � � � � � � �

79

Sursrvlwlrq 8=

Ohw wkh xwlolw| ixqfwlrq T E�� eh gh�qhg lq htxdwlrq +7,1 Wkh vr0oxwlrq +Ef� c w� rewdlqhg iurp pd{lpl}lqj T E�� lv Olswvfklw} frq0wlqxrxv lq w zlwk suredelolw| rqh1

Surri= D ixqfwlrq sE%c |� lv Olswvfklw} frqwlqxrxv lq | li iru doo |c |�lq

wkh grpdlq/ wkhuh h{lvwv d ixqfwlrq -E%� vxfk wkdw nsE%c |�� sE%c |��n �

-E%� � n|� |� n c dqg .-E%� h{lvwv1 L zdqw wr vkrz wkdw zlwk suredelolw| rqh

+Ew� lv Olswvfklw} frqwlqxrxv1 L zloo frqfhqwudwh rq wkh vshfldo fdvh xvhg lq rxuhpslulfdo vwxg| vlqfh wklv lv qrw d jhqhudo surshuw| iru doo xwlolw| ixqfwlrqv1Iru wkh vdnh ri vlpsolflw| L dvvxph wkdw q

�r duh srvlwlyh lq ydoxh dqg wkh

xwlolw| ixqfwlrq lv vshfl�hg dv

T E�( w� ' q�*?E�

�+Ew�n ��� bR

�+ � jD

�+ +y,

Sdudphwhuv duh w ' iqc bc jj1 Vwrfkdvwlf ghyldwhv D

r duh orjqrupdoo| glv0

wulexwhg= D ' i TE#�c dqg #�r duh vwdqgdug qrupdo1 L zdqw wr vkrz wkdw wkh

vroxwlrq +Ew� rewdlqhg iurp pd{lpl}lqj T E�( w� lv Olswvfklw} frqwlqxrxv lq wzlwk suredelolw| rqh1

Dqdorjrxv wr wkh htxdwlrq v|vwhp lq +l, dqg +ll,/ L kdyh wkh Nxkq0Wxfnhufrqglwlrqv

� � � �

@� � q �� �� � � E@ � � + � Ew� n ��� bR � n jD � ' f �s +� : f +yl,

��� � �

@ � q �! "# � � E@ # � + # Ew� n ��

� bR n jD � f �s + ' f +yll,

zkhuh @ � lv wkh �wk urz dqg Swk froxpq ri pdwul{ �� Dv glvfxvvhg deryh/htxdolw| krogv lq +yll, zlwk suredelolw| }hur1 Khqfh zlwk suredelolw| rqh zhfdq fkdqjh +yl, dqg +yll,/ dqdorjrxv wr +l*, dqg +ll�,/ wr eh

$%

& ' (@) & q &

* + ,- . / E@ - 0 + - Ew� n ��� bR 1 n jD 1 ' f �s � 5 a 2 +yl*,

34

5 6 2@7 5 q 5

8 9 :; 6 2 E@ ; 5 + ; Ew� n ��� bR 7 n jD 7 f �s � 5 a < +yll*,

zkhuh a = dqg a < duh gh�qhg dv deryh1

7:

Vxssrvh sdudphwhuv w fkdqjh iurp w � c wkh wuxh sdudphwhuv/ wr w � 1 Iurp+yll*, dqg sursrvlwlrq 5 wkhuh h{lvwv/ zlwk suredelolw| rqh/ d B : f vxfk wkdwiru nw � � w � n � B zh kdyh/ iru � 5 a � dw w � c

��� � �

@ � q � �� � � � � �� � E@ � � + � Ew � � n ��� b � R� n j � D � f +ylll,

Wr dyrlg frqixvlrq lq qrwdwlrq qrwh wkdw w � ' iq � � c b � c j � j dqg w � ' iq � � c b � c j � j�Wkh lqhtxdolw| lq +ylll, lpsolhv wkdw +� Ew � � ' f li � 5 a � dw w � dqg nw � � w � n �B1 Wkhq m+ � Ew � �� +� Ew � �m ' f�

Vxssrvh nw � � w � n : B� Zh nqrz wkdw 4@ � +Ew�� � 7U*R � 7+ 4 dqg+� � f iru doo �1 Khqfh m+� Ew �� +� Ew ! �m � "# $ � nw % � w & n iru � 5 a ' �

Ohw*v frqvlghu wkh fdvh ri � 5 a % � Vxssrvh nw % � w & n � B�Zlwk suredelolw|rqh + ( Ew % � ' f iru & 5 a ) dw w * c dqg khqfh L gr qrw qhhg wr zruu| wkh h�hfw rifkdqjh lq + + rq wkh txdqwlw| ghpdqghg ri surgxfwv lq a , � Iurp +yl*, L kdyh/iru � 5 a , c

-./ 0 , d

@1 2 q 3 4 25 6 78 9 3 E@ 8 2 + 8 Ew 3 � n ��� @1 2 q : 4 25 6 78 9 3 E@ 8 2 + 8 Ew : � n ��

o�Eb ; �b < �R = nEj ; �j < �D = ' f

, >?

@ A B d@C @ � q B D @ � E F GH A B E@ H @ + H Ew I � n ��� q I D @ � E F GH A B E@ H @ + H Ew B � n ��

E F GH A B E@ H @ + H Ew B � n �� � E F GH A B E@ H @ + H Ew I � n ��o�

Eb B � b I �R C n Ej B � j I �D C ' f

,>

?@ A B d

@C @E F GH A B E@ H @ + H Ew B � n �� � E F GH A B E@ H @ + H Ew I � n ��

�

q B D @ � d F G?H A B E@ H @ + H Ew I �n��� F G?

H A B E@ H @ + H Ew B �n��o� Eq I D @ �q B D @ � � F G?H A B E@ H @ + H Ew B �n��o

' Eb B � b I �RC � Ej B � j I �D C, >

?@ A B i

@C @ � q B D @E F GH A B E@ H @ + H Ew B � n �� � E F GH A B E@ H @ + H Ew I � n ��

� F G?H A B d@ H @ E+ H Ew I �� + H Ew B ��oj

7;

'

��� � � d

@� � � � � � � � E@ � + Ew � � n ��

� � � � � E@ � + Ew � � n �� � � � � � � E@ � + Ew � n ���

Eq � � � q � � � �o n Eb � � b �R � � Ej � � j �D �,

� �� � �

��� � � i

q � � � � @� � � @ �� � � � � E@ � + Ew � � n �� � � � � � � E@ � + Ew � n ��

� E+ Ew �� + Ew � ��j

'

��� � � d

@� � � � � � � � E@ � + Ew � � n ��

� � � � � E@ � + Ew � � n �� � � � � � � E@ � + Ew � n ���

Eq � � � q � � � �o n Eb � � b �R � � Ej � � j �D �L fdq uhzulwh wkh zkroh v|vwhp ri htxdwlrqv lq pdwul{ irup dv=

(d+Ew � �� +Ew �o ' gEq � � q � � n Eb � � b � �R� Ej � � j � �D � �gEw � � w � � +l{,

zkhuh ( lv d a � � a � pdwul{ zlwk wkh �0wk urz ,0wk froxpq hohphqw dv

��� � � i

q � � � � @� � � @ � �� � �� ! E@ � � + � Ew ! � n �� � � � �� ! E@ � � + � Ew " � n ��j +{,

dqg g lv d a ! � � pdwul{ zlwk wkh �0wk urz S0wk froxpq hohphqw dv

@# $ � % & '( ) * E@ ( $ + ( Ew * � n ��

% & '( ) * E@ ( $ + ( Ew * � n �� � % & '( ) * E@ ( $ + ( Ew + � n ��+{l,

dqg wkh da * � E� n 2�o pdwul{ �g ' Egc Rc D�1Ohw _ eh wkh plqlpxp ri doo hohphqwv lq (1 Lw fdq eh fkhfnhg wkdw

f _ 4 +Qrwh wkdw hyhu| hohphqw lq � lv dvvxphg wr eh juhdwhu ru htxdowr f1, Iurp +l{, zh kdyh

(

_d+Ew , ��+Ew - �o ' g

_Eq , �q - �nEb - �b , �R

_�Ej - �j , �D

_�

�g

_Ew , �w - � +{ll,

Hyhu| hohphqw lq . / � �( lv juhdwhu ru htxdo wr rqh1 Wkhuhiruh zh kdyh wkhlqhtxdolw|

n+Ew 0 �� +Ew 1 �n � 2222(

_d+Ew 3 �� +Ew 4 �o 2222

' 22222

�g

_Ew 3 � w 4 � 22222

+{lll,

7<

Ohw 7@ eh wkh pd{lpxp ri doo hohphqwv lq wkh fkdudfwhulvwlfv pdwul{ �� Iurp+{l, lw fdq eh fkhfnhg wkdw hyhu| hohphqw lq g lv juhdwhu ru htxdo wr f exwlv vpdoohu ru htxdo wr 7& � aE7@

�7+ n 7@� 4� Xvlqj wkh Vfkzdu} lqhtxdolw|/ L

kdyh wkh iroorzlqj lqhtxdolw| iru hohphqw � lq wkh yhfwru �gEw � � w � �c ghqrwhge| �g� Ew � � w � �c � ' �c ���c a � /

��� �g� Ew � � w � ���� �

��� 7&c ���c 7&c R� c D ���� � nw � � w � n

Xvh wkh lqhtxdolw| ri n% � c ���c % � n � n% � nn ���n n%� n c dqg iurp +{lll, L kdyhwkh iroorzlqj lqhtxdolw|=

n+Ew � �� +Ew � �n � E�

_

� � � �

7&c ���c 7&c R� c D � � � nw � � w � n +{ly,

Ohw - � Ef� ' E�� � � � � �

7&c ���c 7&c R� c D � �1 Vxssrvh ERc D� duh vtxduh0lqwhjudeoh/

dqg wkdw .- � Ef� h{lvwv1 Lq p| fdvh L dvvxph wkdw R� r duh erxqghg deryhdqg D � r duh orj0qrupdoo| glvwulexwhg/ vr wkh vtxduh0lqwhjudelolw| frqglwlrq lvvdwlv�hg1

Odvwo|/ li nw � � w � n : Bc wkhq m+ � Ew � �� +� Ew � �m � 7+*B � nw � � w � n c zkhuh7+ ' U*R�

Iurp wkh deryh glvfxvvlrqv L frqfoxgh wkdw li nw � � w � n � Bc wkhq n+Ew � �� +Ew � �n �- � Ef��nw � � w � n ( li nw � � w � n : Bc wkhq n+Ew � �� +Ew � �n � - � Ef��nw � � w � n czkhuh- � Ef� � � ��� �Wkhuhiruh n+Ew � �� +Ew � �n � E- � Ef���inw � � w � n � B � jn- � Ef� � �inw � � w � n : B � j� � nw � � w � n c zkhuh �i�j lv dq lqglfdwru ixqfwlrqzklfk htxdov wr � li wkh orjlfdo h{suhvvlrq lqvlgh eudfnhwv lv wuxh/ dqg 3 rwkhu0zlvh1 Ohw -Ef� ' E- � Ef� � �inw � � w � n � B � jn- � Ef� � �inw � � w � n : B � j��L kdyh .-Ef� 4�

Wkhuhiruh Sursrvlwlrq 8 lv vkrzq wr eh ydolg1

Frqvlvwhqf|

Wkh iroorzlqj lv wkh pdlq uhvxow ri frqvlvwhqf|=Sursrvlwlrq 9 +Frqvlvwhqf|,=

Dvvxph wkdw

+l,1 wkh hvwlpdwru w � lv gh�qhg vxfk wkdw

nC� Ew � �n ' �?u� � � nC Ew�nn J! E��c

83

+ll,1 lghqwl�fdwlrq frqglwlrq krogv1 Wkdw lv/

iru doo B : fc �?u� � � � � � � � nCEw�n � 0EB� : f

+lll, X lv d frpsdfw vxevhw ri d Hxfolghdq vsdfh1

Wkhq zh kdyh w � � w ' J E���

Wr vkrz wkdw VPP hvwlpdwruv dssolhg lq wkh FFP duh frqvlvwhqw/ wkhiroorzlqj ohppd derxw wkh xqlirup odz ri odujh qxpehuv lv xvhixo1 Lwv surrifdq eh irxqg lq Qhzh| dqg PfIdgghq +4<<7,=

Ohppd 4=

Vxssrvh wkh gdwd duh l1l1g11 Li

+l,1 6Efc w� lv frqwlqxrxv dw hdfk w 5 X zlwk suredelolw| rqh>

+ll,1 wkhuh h{lvwv _Ef� � zlwk n6Ef� c w�n � _Ef� � iru doo w 5 X/dqg .d_Ef� �o 4(

+lll,1 X lv d frpsdfw vxevhw ri d Hxfolghdq vsdfh1

Wkhq .i6Ef� c w�j lv frqwlqxrxv dqg t�T � � nC� Ew� � CEw�n 'J� E���

Iurp wkh idfw wkdw wkh vlpxodwlrqv 0� � � duh gudzq lqghshqghqwo|/ L jhq0hudwh d vhw ri l1l1g1 gdwd1 Sursrvlwlrq 4 surylghv frqglwlrqv xqghu zklfkdvvxpswlrq +ll, lq wkh deryh sursrvlwlrq zloo krog1 Sursrvlwlrqv 5 dqg 6 sur0ylgh frqglwlrqv xqghu zklfk dvvxpswlrq +l, zloo krog1 Li wkh frpsdfwqhvv riX lv dvvxphg/ xqlirup frqyhujhqfh ri C� Ew� lv jxdudqwhhg1

Wkh iroorzlqj wkhruhp rq frqvlvwhqf| ri wkh VPP hvwlpdwruv lv zhoonqrzq1 Lwv surri fdq eh irxqg lq PfIdgghq +4<;<, ru Sdnhv dqg Sroodug+4<;<,1

Wkhruhp 4=

Dvvxph nCEw � �n ' f/ dqg

+l,1 wkh hvwlpdwru w � lv gh�qhg vxfk wkdw

nC� Ew � �n ' �?u� � � nC� Ew�nn J� E��c

84

+ll,1 lghqwl�fdwlrq lv krog vxfk wkdw

iru doo B : fc �?u� � � � � � � � nCEw�n � 0EB� : f

+lll,1 Xqlirup Odz ri Odujh Qxpehuv +XOOQ, lv krog vxfk wkdw

t�T� � � nC� Ew��CEw�n ' J E��

Wkhq zh kdyh w � � w ' J E���

Vlqfh 0� � c � ' �c ���c? dqg r ' �c ���c ?rc lv gudzq iurp wkh vdph glvwulex0wlrq dv 0 � / wkh frqglwlrq CEw � � ' f lv jxdudqwhhg� Frqglwlrq +ll, gh�qhv wkhhvwlpdwru w � 1 Lghqwl�delolw| lv d vwdqgdug dvvxpswlrq lq wkhruhpv ri frqvlv0whqf| exw lv gl!fxow wr yhuli| � � 1 Wkhvh wzr frqglwlrqv duh dvvxphg wr kroglq Sursrvlwlrq 9 deryh1 Iurp sursrvlwlrq 4/ 5/ dqg 6/ dqg frqglwlrq +lll, lqOhppd 4/ wkh XOOQ lv jxdudqwhhg1 Khqfh Sursrvlwlrq 9 lv suryhg1

Dv|pswrwlf Qrupdolw|

Wkh iroorzlqj lv wkh pdlq uhvxow ri dv|pswrwlf qrupdolw|1 L xvh , wrghqrwh frqyhujhqfh lq glvwulexwlrq1

Sursrvlwlrq : +Dv|pswrwlf Qrupdolw|,=

Vxssrvh wkdw w � $� w � � Li nC� Ew � �n ' �?u � nC� Ew�n n J� E�*s?�dqg w � lv lq wkh lqwhulru ri X/ wkhq

s?Ew � � w � �, �Efc EK� K� � �

K � E� n ?r � ��T KEK� K� � �

��

zkhuh T lv wkh yduldqfh0fryduldqfh pdwul{ vxfk wkdw ? � � � �d+ � �

.+Ef� c w �o! �Ef" � , �Efc T �/ K d ixoo udqn pdwul{ vxfk wkdw iruw qhdu w # c CEw� ' KEw � w # � n JEnw� w # n�> dqg ?r wkh qxpehu rivlpxodwlrq gudzv iru hdfk revhuydwlrq �1

$ % & ' ( ) * + , - . / / 0 1 2 / 3 4 2 5 6 7 8 9 : ; < : = > ? @ A B C D E F G H I I J K L M N N O P Q R S T U V W X T S V W Y S Z [\ ] T [ Y \ ^ _ ` a b c d c e d c d f b d g ` h i b j k f a ` d g c k i g l m n o p q r s t q u v w s x n r y v z q m r u p r { | } ~ � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � �   ¡ ¢ £ ¤ ¥ ¦ § § ¨ © ª ¤ « ¬ ­ ® ¯ ° ± ² ² ² ³ ´ µ ¶ · ¸ ¹ · ¹ ¶ º · » · ¼ ½ ¾ ¿ À

¿ Á Â À Ã Ä Å Æ Ç È É Ê Ë Ì Í Í Î Ê Ì Ë Ì Ï Ð Ñ Ò Ð Ó Ï Î Ê Ð Ò Ô Ì Ï Ð Ò Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â â à ã ä å æ ã ç è é ê ë ì í î ï ð î í ñ ò óô õ î í ö ó ÷ ø ð ù ù ú û ü ý þ ÿ � � � � � � � � � � � � � �

85

Wr suryh wkh Sursrvlwlrq : L xvh d zhoo nqrzq wkhruhp/ zklfk surri fdqeh irxqg lq PfIdgghq +4<;<, dqg Sdnhv dqg Sroodug +4<;<,=

Wkhruhp 5=

Vxssrvh wkdw CEw � � ' fc dqg w � $� w � � Li+l,1 nC� Ew � �n ' �?u � nC� Ew�nn J� E�*s?�(

+ll,1 iru w qhdu w � c CEw� ' KEw� w � n JEnw� w n�c zkhuh K lv riixoo udqn>

+lll,1 iru dq| vhtxhqfh ri srvlwlyh qxpehu B $ fc

t�T� � � � � � � � ns?dC� Ew��CEw�o�

s?dC� Ew � ��CEw � �on*d�ns?nw�w � no ' J� E��

+ly,1s?C� Ew � �, �EfcT �( dqg

+y,1 w � lv lq wkh lqwhulru ri X1 Wkhq

s?Ew � � w � �, �Efc EK� K� � �

K � T KEK � K� � ���

Frqglwlrq +lll, lv d frqglwlrq iurp Qhzh| dqg PfIdgghq+4<<7, / s154;91

L zdqw wr vkrz wkdw frqglwlrqv +ll, dqg +lll, duh vdwlv�hg lq p| prgho1Wkh iroorzlqj ohppd/ iurp Qhzh| dqg PfIdgghq +4<<7,/ s154;;/ suryhg wreh xvhixo=

Ohppd 5=

Vxssrvh wkdw .6Ef c w ! � ' f dqg wkhuh duh KEf � dqg B : f vxfkwkdw zlwk suredelolw| rqh/

oEf" c w� ' n6Ef" c w��6Ef" c w # �� KEf" �Ew � w # �n*nw � w # n $ f

dv w $ w # / dqg .dr�R $ % & % ' $ ( ) oEf* c w�o 4c dqg +, - ./ 0 1 KEf2 �$3.dKEf2 �o�Wkhq frqglwlrq +ll, dqg +lll, duh vdwlv�hg iru K ' .dKEf2 �o�

Iurp Sursrvlwlrqv 7 dqg 8 zh nqrz wkdw zlwk suredelolw| rqh 6Ef2 c w�lq rxu prgho lv gl�huhqwldeoh dqg Olsvfklw} frqwlqxrxv lq w1 Frqglwlrqv lqOhppd 6 duh vdwlv�hg/ dqg khqfh dvvxpswlrq +ll, dqg +lll, lq Wkhruhp 5 dovr

86

krog1 Dvvxpswlrq +l, lq Wkhruhp 5 mxvw gh�qhv hvwlpdwruv w � 1 Qrz wkdws?C� Ew � � ' ? � � � �

d+ � � .+� Ew � �o � �Ef � � ? �� d �� � � � �� � � �+� � � � .+� Ew � �o� �Ef� �1

Vlqfh L dvvxph vwrfkdvwlf yduldeohv duh l1l1g1/ L fdq xvh wkh pxowlyduldwh fhqwudoolplw wkhruhp wr rewdlq

? � � � � d+ � � .+ � Ew � �o �Ef� �, �Efc T �

zkhuh T lv wkh yduldqfh0fryduldqfh pdwul{ ri wkh JPP hvwlpdwruv zlwkrxwvlpxodwlrqv1 Lq p| fdvh wkh gdwd jhqhudwlrq surfhvv dqg vlpxodwlrq surfhvvhvduh lqghshqghqw/ wkhuhiruh

s?C! Ew " �, �Efc E� n ?r # $ �T �

Dvvxph wkdw w " lv lq wkh lqwhulru ri X� L fdq xvh Wkhruhp 5 wr rewdlq

s?Ew ! � w " �, �Efc EK% K� # $ K % E� n ?r # $ �T KEK% K� # $ �

Khqfh Sursrvlwlrq : lv suryhg1

Rwkhu Ghwdlov

Wkh qh{w ohppd lv derxw wkh rswlpdo fkrlfh ri wkh zhljkwhg pdwul{ &̀ Ew�/dqg frphv iurp Jrxulhurx{ dqg Prqiruw +4<<9,/ s1641

Ohppd 6=

Wkh rswlpdo fkrlfh ri '̀ Ew� lv vxfk wkdw '̀ Ew� '̀ Ew�( ' T ) * �Wkh dv|pswrwlf glvwulexwlrq fruuhvsrqglqj wr wklv fkrlfh lv

s?Ew ' � w + �, �Efc EK ( �T ) * K� ) * ��

zkhuh �T ' E� n ?r ) * �T�

Ohw*v jr edfn wr wkh sdqho gdwd L xvh lq prgho hvwlpdwlrqv1 Hdfk frqvxphu�c � ' �c ���c �c kdv d qxpehu ri sxufkdvhv/ A, c lq wkh gdwd1 L dvvxph 0*v duhlqghshqghqw ryhu lqglylgxdo frqvxphuv/ exw lqwhuwhpsrudo ghshqghqfh ryhugl�huhqw sxufkdvhv pd| vwloo h{lvw1 Iru h{dpsoh/ d frqvxphu zkr kdv d kljksuhihuhqfh iru frod dyru dw wlph | lv olnho| wr kdyh d kljk suhihuhqfh iru frod dwwlph �c � 9' |1 Khqfh wkh uhvxowv iurp Sursrvlwlrq : dqg Ohppd 6 vkrxog eh

87

fruuhfwhg iru wklv sdqho dvshfw1 Gxulqj hvwlpdwlrqv L gudz vwrfkdvwlf yduldeohv0� � � � ' E#

�� � � � c D � � �/ r ' �c ���c ?r/ dqg #

� � � �Efc U �� � dqg *?ED � � � � � � �Efc U � �/qrwdwlrqv gh�qhg dv ehiruh� Wkh sdqho gdwd zloo hqwhu wkh hvwlpdwh ri wkhyduldqfh0fryduldqfh pdwul{ �T c T � � c lq Ohppd 6 dv

T � � '�

�

��� � � i

�

A�� ��

! "# ��

$ % & dE+' ( � �+Ef' ( c w�� ) �Ef' ( �o) dE+' $ � �+Ef' $ c w��) �Ef' $ �oj+{y,

T * + zloo eh odujhu zkhq lqwhuwhpsrudo ghshqghqfh h{lvwv wkdq zkhq 0*v duh����_� ryhu frqvxphuv dqg gl�huhqw sxufkdvhv1 Dqrwkhu zd| ri vlpxodwlrqgudzlqj iru 0*v/ zklfk zloo lqfuhdvh wkh h!flhqf| ri hvwlpdwruv/ lv wr dvvxphd sdudphwulf ixqfwlrqdo irup iru wkh gdwd jhqhudwlqj surfhvv ri 0, - 1 Iruh{dpsoh/ ohw*v dvvxph

0 , - ' �, n 1 . /zkhuh �. lv ����_� ryhu frqvxphuv exw �{hg ryhu wlph iru hdfk frqvxphu �/dqg kdv d frqglwlrqdo glvwulexwlrq ixqfwlrq � 0 E�mf1 �/ zkhuh f1 lv vrph vsh0fl�f lqglylgxdo frqvxphu fkdudfwhulvwlfv> 1 1 2 ����_� ryhu erwk � dqg |/ dqg kdvd frqglwlrqdo glvwulexwlrq �3 E�mf4 �1 Wklv vshfl�fdwlrq lqfrusrudwhv wkh idfwwkdw prvw frqvxphuv kdyh d �{hg suhihuhqfh iru surgxfw fkdudfwhulvwlfv xq0fkdqjhg ryhu wlph1 Wkhq zh fdq gudz �*v iurp � 5 E�mf4 � iru hdfk frqvxphu�/ dqg 1*v iurp � 3 E�mf4 � iru hdfk � dqg |1

Vlqfh wkh vdpsoh vl}h lq p| gdwd lv vr odujh wkdw wkh h!flhqf| ri hvwlpdwruvlv qrw fuxfldo/ L vlpso| xvh wkh iruphu phwkrg ri ����_� gudzlqjv iru 0*v 6 7 1

Wkh rswlpdo lqvwuxphqwv �Ef8 9 � duh

� : Ef; < � ' EY�+Ef= > c w ? �

Yw�T @ A

Krzhyhu/ L rqo| xvh �fuxgh� lqvwuxphqwv �EfB C � ' fB C khuh1 Wkh| duh lqgh0shqghqw ri w1 Li fB C duh srru lqvwuxphqwv/ wklv zloo eh vkrzq e| odujh ydul0dqfhv iru hvwlpdwruv w D � Wkhq zh fdq pdnh ixuwkhu lpsuryhphqwv wkurxjklwhudwlyh hvwlpdwlrq phwkrgv +iru h{dpsoh/ vhh PfIdgghq +4<;<,/ s143381,

Iru hdfk ++E F G H � �+H EfI J c w��/ zkhuh � ' �c ���c ac wkh lqvwuxphqwv L xvhlqfoxgh sulfhv ri doo surgxfwv dw wkh vxshupdunhw dw wkh wlph ri sxufkdvh/

K L M N O P O Q R O S T U V W X Y Z [ Z \ Y ] \ [ W U ^ _ ` a b c d e f g f h g f b i j f i d k g l e k m n b i o a p q r s t u v w x y z{ s w s t u | y } y x z | ~ z { } � s � z t } u | v w s ~ w y v ~ � x � } u � � v x � { z s � s u � } � x s ~ z | z � r v u | ~ � } ~ z v u �{ v � w | ~ � � z { } � � s � � x � � } � � v r z { } t u v w x y z | ~ z { } y x u u } ~ z t } u | v w � � { | � | ~ z } u z } � t v u s �w } t } ~ w } ~ y } w v } � ~ v z } � | � z | ~ � � � v w } � � } y s x � } q s � � x � } y v ~ � x � } u � v ~ � � t x u y { s � } r v u

z { } | u y v ~ � x � t z | v ~ | ~ z { } r v � � v � | ~ � t } u | v w � z { s z | � � ~ v | ~ � } ~ z v u � { v � w | ~ � � s u } s � � v � } w �

88

wkdw lv/ d � � a urz yhfwru R�� > pdunhwlqj dfwlylwlhv ihdwxuh dqg glvsod| ri

surgxfw � dw wkh vxshupdunhw dw wkh wlph ri sxufkdvh/ wkdw lv/ d � � 2 urzyhfwru E�Es� � � �c �E_ � � � ��> frqvxphu �*v idplo| vl}h/ 6� > dqg dq lqwhudfwlqj whupri sulfhv dqg qdwxudo orj ri krxvhkrog lqfrph/ wkdw lv/ d �� a urz yhfwru riE*?EU� � � R �

� �� L dovr lqfoxgh � � a surgxfw lqglfdwruv e � c zklfk kdyh � dw wkh�wk sodfh dqg }hurv hovhzkhuh/ wr wdnh fduh ri fkdudfwhulvwlfv lq hdfk surgxfw�Khqfh p| lqvwuxphqwv duh Ee � c R

�c �Es

� �c �E_

� �c6� c *?EU � � � R�

��Ilqdoo|/ jlyhq f�

dqg dq| w 5 Xc L xvh wkh Vhtxhqwldo Txdgudwlf Sur0

judpplqj +VTS, phwkrg� wr �qg �+Ef� � c w�/ wkh vroxwlrq ri wkh frqvwudlqhg

xwlolw| pd{lpl}dwlrq sureohp� Wr vhdufk iru d w � zklfk plqlpl}hv wkh re0mhfwlyh ixqfwlrq/ L xvh wkh Qhoghu0Phdg +4<98, qrqghulydwlyh vlpsoh{ vhdufkdojrulwkp1

Frpsxwdwlrqdo exughq lq hvwlpdwlrq lv khdy|= frpsxwdwlrq wlph e| xv0lqj wkh VTS phwkrg lv orqjhu wkdq zkhq d forvhg0irup vroxwlrq h{lvwv iru�+Ef� � c w�/ dqg wkh qxpehu ri vhdufk lwhudwlrqv iru wkh qrqghulydwlyh vlpsoh{vhdufk dojrulwkp lv xvxdoo| odujhu wkdq zkhq ghulydwlyh vhdufk dojrulwkpv fdqeh xvhg1 Rq wkh rwkhu kdqg/ h!flhqf| lv qrw lpsruwdqw lq hvwlpdwlrqv eh0fdxvh L kdyh d odujh qxpehu ri revhuydwlrqv1 Wkhuhiruh L vhw wkh qxpehu rivlpxodwlrq gudzv iru hdfk revhuydwlrq wr eh rqh1 Iurp wkh uhvxow lq Ohppd6/ yduldqfhv lq p| hvwlpdwruv duh wzlfh dv odujh dv wkh JPP hvwlpdwruv zkhqqr vlpxodwlrq gudz lv qhhghg1

� � � � � � � � � � � � � � � � � � � � � � ! " # $ % & ' ( ) * + , - % . ( / , ) 0 , 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J J K LM N O P Q R S T U V W X Y Z [ \ ] ]

89