Jakub Muck - Analiza reakcji wybranych modeli długookresowego wzrostu gospodarczego na szoki stochastyczne

Analiza reakcji wybranych modeli

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

1/+),%&2+)-%03/)-40,1(

!"#$%&'$(#)

*"+,-."/01&2&34-567"+"&89::&;<&&=""#>,756?"/01&@&>A,;?>"&89:8&;<

5-'(),0,(12(

B,3,C& ";50#$.$& D,-5& ?.E>A,/4,& -A6#F?& -56>G"-50>A/0>G& +6& ?0%;"/0>G& C6+,34& +.$H66#;,-6?,H6& ?A;6-5$& H6-76+";>A,H6& 6;"A& 4+,/50I4#">D"& #4,;$/#$& 4& -4.0& ;,"#>D4& /"& 5,& -A6#4& /"D?"J/4,D-A0>G& AC4,//0>G&50>G&C6+,34<&K"+"/4,&6%DL.6&C6+,3,&M"C-,0"N&B"--"&4&O667C"/-"N&PA"?0&4&Q$>"-"&6;"A& !6/,-"<&P?AH3L+/46/,&A6-5".0&7;A0&50C&-A6#4&-56>G"-50>A/,&+?6D"#4,H6&;6+A"D$<&R4,;?-A,&A&/4>G&56& A"%$;A,/4"&A?4L#-A"DE>,&>".#6?45E&7;6+$#50?/6S(&>A0//4#F?&7;6+$#>D4&TUVRWN&>A034&-5"/+";+6?,& -A6#4& 5,>G/636H4>A/,<& X& 7;A07"+#$& C6+,34& PA"?0& 4& Q$>"-"& 6;"A& !6/,-"& ?7;6?"+A6/,& A6-5".0& 76/"+56&-A6#4&-56>G"-50>A/,&6++A4".$DE>,N&6+76?4,+/46N&/"&"#$C$3">DL&#"745".$&3$+A#4,H6&6;"A& 5,C76& 76-5L7$& 5,>G/636H4>A/,H6<& B,3& %"+"/4"& A6-5".& A;,"34A6?"/0& +?$,5"76?6<& *"& 74,;?-A0C& ,5"74,&+3"&76-5">4&>,/5;"3/,H6&73"/4-50&#"J+,H6&A&"/"34A6?"/0>G&C6+,34&A6-5".0&6%34>A6/,&?";56S>4& I$/#>D4&;,"#>D4&/"&4C7$3-<&Y;$H"&>ALS(&%"+"/4"&56&76;F?/"/4,&65;A0C"/0>G&-0C$3">D4&A&+"/0C4& ,C74;0>A/0C4&674-$DE>0C4&Z5"/0&=D,+/6>A6/,<

5"%*/+ &6#0,%*(7+ YZ[\N& ,/+6H,/4>A/0& ?A;6-5& H6-76+";>A0N& -,C4,/+6H,/4>A/0& ?A;6-5&

H6-76+";>A0 89:7+B@]N&^_: )&&*";6+6?0&K"/#&R63-#4N&`/-505$5&\#6/6C4>A/0a&ZA#6."&[.F?/"&b"/+36?"&?&X";-A"?4,N&`/-505$5&\#6/6C,5;44a& ,cC"431&D"#$%<C$>#d/%7<73<

;<+=)->.

R6>AE?-A0&6+&7;">0&Z636?"&T:ef@W&?&%"+"/4">G&C"#;6,#6/6C4>A/0>G&A">AL56&-#$74"(&-4L&/"&#6/ --5;$6?"/4$& C6+,34& +.$H66#;,-6?,H6& ?A;6-5$& H6-76+";>A,H6<& Y,5,;C4/4-50>A/0& >G";"#5,;& 50>G& #6/-5;$#>D4&$C6J34?4"&"/"3450>A/,&6#;,S3,/4,&?."-/6S>4&+0/"C4>A/0>G&6;"A&4+,/50I4#">DL&g;F+,.& ?A;6-5$&H6-76+";>A,H6&?&A"3,J/6S>4&6+&7;A0DL50>G&A".6J,h&4&76-5">4&A"H;,H6?"/,D&I$/#>D4&7;6+$# ->D4<&X&345,;"5$;A,&C6J/"&-765#"(&#43#"&#3"-&C6+,34N&#5F;,&?&;FJ/0&-76-F%&76A?"3"DE&?05.$C">A0(& AD"?4-#6&+.$H66#;,-6?,H6&?A;6-5$&H6-76+";>A,H6<&*"&76>AE5#$&;6A?6D$&5,D&+A4,+A4/0&%0.&6/&;6A -?"J"/0&D"#6&stricte&,HA6H,/4>A/,&AD"?4-#6&A?4EA"/,&A&"#$C$3">DE&76-5L7$&5,>G/636H4>A/,H6N&#5F -;0&;F?/4,J&$A/"?"/6&A"&AC4,//E&,HA6H,/4>A/E<&R$/#5,C&?0DS>4"&D,-5&5$5"D&C6+,3&M"C-,0"&T:e8iWN& #5F;0&A6-5".&;6A-A,;A6/0&7;A,A&B"--"&T:e@fW&4&O667C"/-"&T:e@fW<&Y;$H4C&76+,DS>4,C&-E&C6+,3,&,/ -+6H,/4>A/,H6&?A;6-5$N&#5F;,&g;F+,.&?A;6-5$&H6-76+";>A,H6&$7"5;$DE&?&?0-5L76?"/4$&"#$C$3">D4& -7,>0I4>A/0>G&>A0//4#F?&?05?F;>A0>G<&[.F?/,&#6/-5;$#>D,&$?AH3L+/4"DE&#"745".&3$+A#4&TPA"?"& :e@_a&&Q$>"-&:eiiW&&4&"#$C$3">DL&?4,+A0N&>A034&,/+6H,/4>A/0&76-5L7&5,>G/636H4>A/0&TM6C,;&:ee9a& jHG46/N&b6?455&:ee8W<&^-5"5/4E&#3"-E&C6+,34&-E&5,N&#5F;,&$?AH3L+/4"DE&-,C4,/+6H,/4>A/0&>G";"# -5,;&5,H6&AD"?4-#"<&X&50C&7;A07"+#$&+.$H66#;,-6?0&?A;6-5&H6-76+";>A0N&;6A$C4"/0&D"#6&?A;6-5& AC4,//0>G&C6+,3$&?&$DL>4$&per capitaN&A"3,J0&6+&A?4L#-A"/4"&-4L&&,HA6H,/4>A/0>G&?4,3#6S>4N&/7<& A"-6%$&7;">0<&X"J/"&#6/-5;$#>D"&+3"&5,D&#3"-0N&A"7;676/6?"/"&7;A,A&!6/,-"&T:eefWN&?&-7,>0I4#">D4& $?AH3L+/4"&-,#56;&%"+"h&4&;6A?6D$&TMkYN&"/H<&research and developmentW<&

K";+A6&767$3";/E&S>4,J#E&%"+"?>AE&?&%"+"/4">G&C"#;6,#6/6C4>A/0>G&D,-5&C,56+636H4"&YZ[\& T"/H<&dynamic stochastic general equilibriumW<&^+&>A"-F?&7$%34#">D4&O0+3"/+"&4&R;,->655"&T:ei8W&A" ->AL56&-4L&-#$74"(&/"&A"H"+/4,/4$&?"G"h&H6-76+";>A0>G<&',56+636H4"&5"&;FJ/4&-4L&6+&C6+,34&?A;6-5$& H6-76+";>A,H6&50CN&J,&$?AH3L+/4"&36-6?,&A"%$;A,/4"&AC4,//0>G&C"#;6,#6/6C4>A/0>G&4&76A?"3"& /"&C6+,36?"/4,&#;F5#66#;,-6?0>G&6->03">D4&H6-76+";>A0>GN&"&7$/#5,C&?0DS>4"&?&#6/-5;$6?"/4$& C6+,34&5,6;,50>A/0>G&D,-5&C6+,3&M"C-,0"&Tb,,;N&'"$--/,;&899fW<&X4L#-A6S(&%"+"h&A&5,H6&A"#;,-$& 76C4D"&D,+/"#&C,>G"/4AC0&?A;6-5$&H6-76+";>A,H6&4&-#$74"&-4L&/"&"/"34A4,&+,5,;C4/"/5&#;F5#66#;, --6?0>G&I3$#5$">D4<&'6J/"&?0;FJ/4(&+?"&H.F?/,&/$;50&50>G&"/"34A&T*6l"3,AN&V,;/"/+,AN&M$4A&899eW<& R4,;?-A0&A&/4>GN&C6+,36?"/4,&;,"3/,H6&>0#3$&#6/4$/#5$;"3/,H6&TMKBN&"/H< real business cycleWN&>G" -;"#5,;0A$D,&-4L&?0#6;A0-50?"/4,C&A".6J,h&;F?/6?"H4&6HF3/,D<&Y;$H4&/$;5&56&76+,DS>4,&/,6#,0/, --6?-#4,N&#5F;,&$?AH3L+/4"&;F?/4,J&-A50?/6S(&>,/&4&7."><&&R67$3";/6S(&YZ[\&?0/4#"&H.F?/4,&A&5;"I -/,H6&6+?A6;6?"/4"&#6/4$/#5$;"3/0>G&?"G"h&76+-5"?6?0>G&AC4,//0>G&,#6/6C4>A/0>G<

YA4L#4&76.E>A,/4$&6%$&76+,DS(&C6J/"&D,+/6>A,S/4,&$?AH3L+/4(&+.$H66#;,-6?,&C,>G"/4AC0& ?A;6-5$& H6-76+";>A,H6& 6;"A& #;F5#66#;,-6?,& ?"G"/4"& H6-76+";>A,<& & R6A?634& 56& C<4/<& ?05.$C" ->A0(& AC4,//6S(& ?4,3#6S>4& C"#;6,#6/6C4>A/0>G& 6& >AL-56534?6S>4& A";F?/6& #6/4$/#5$;"3/,DN& D"#& 4&S;,+/466#;,-6?,D&TB6C4/N&[,;53,;&899@a&[;"+A,?4>A&899iW<&mE>AE>&5,&6%-A";0N&C6J/"&65;A0C"(& >4,#"?0&A%4F;&/";AL+A4N&#5F;0&$."5?4&6>,/L&AH6+/6S>4&?-#"A0?"/0>G&?&345,;"5$;A,&g;F+,.&+.$H6 -6#;,-6?,H6&?A;6-5$&H6-76+";>A,H6&A&6%-,;?">D"C4&,C74;0>A/0C4<&X&";50#$3,&6CF?46/,&A6-5"/4,& ?&-A>A,HF3/6S>4N&#5F;,&A&#;F5#66#;,-6?0>G&?"G"h&H.F?/0>G&AC4,//0>G&C6+,3$&-E&?0?6.0?"/,& ,/+6H,/4>A/4,&7;A,A&C,>G"/4AC0&+0/"C4>A/,&?&?0%;"/0>G&C6+,3">G&?A;6-5$< B,3,C&";50#$.$&D,-5&4+,/50I4#">D"&#4,;$/#$&4&-4.0&;,"#>D4&/"D?"J/4,D-A0>G&AC4,//0>G&C6+,34& ?A;6-5$&H6-76+";>A,H6&/"&-A6#4&-56>G"-50>A/,&4&;FJ/4>&C4L+A0&50C4&;,"#>D"C4<&U"#&76-5"?46/0&>,3& %L+A4,&6-4EH/4L50&7;A,A&?7;6?"+A,/4,&+6&?0%;"/0>G&C6+,34&A"%$;A,h&-56>G"-50>A/0>G<&*"&76+ --5"?4,&+"/0>G&,C74;0>A/0>G&7;A,"/"34A6?"/,&A6-5"/4,&;F?/4,JN&#5F;,&A&;6A?"J"/0>G&#6/-5;$#>D4&

76A?"3"DE&/"&+6#."+/4,D-A,&6+?A6;6?"/4,&7;A,%4,H$&?"G"h&>0#34>A/0>G&?&Z5"/">G&=D,+/6>A6 -/0>G<&X&50C&>,3$&A6-5"/4,&?0#6;A0-5"/4,&-5"/+";+6?,&76+,DS>4,&-56-6?"/,&?&YZ[\N&"&+6#."+/4,D& MKBN&A/"/,&A&5,6;44&?A;6-5$<&X/46-#4&7.0/E>,&A&%"+"/4"&76-.$JE&+6&6+76?4,+A4&/"&705"/4,N&#5F -;0&A&;6A?"J"/0>G&C6+,34&3,74,D&6+?A6;6?$D,&?"G"/4"&H6-76+";>A,&?&H6-76+";>,&"C,;0#"h-#4,D<& BA0&D,-5&56&-5"/+";+6?0&C6+,3&MKB&67";50&/"&C6+,3$&M"C-,0"n&BA0&C6J,&D,H6&?,;-D"&;6A-A,;A6/"& 6&-,#56;&%"+"?>A0&3$%&,+$#">0D/0n&Y6+"5#6?,&C,>G"/4AC0&I3$#5$">D4&-56>G"-50>A/0>GN&$?AH3L+ -/46/,&?&C6+,3">G&+.$H66#;,-6?,H6&?A;6-5$&67";5,H6&/"&#;,">D4&?4,+A0&?&-,#56;A,&%"+"h&4&;6A -?6D$&%E+g&"#$C$3">DL&?4,+A0N&-E&%6?4,C&$A$7,./4,/4,C&76+-5"?6?0>G&C,>G"/4ACF?&A/"/0>G& A&C6+,34&;,"3/,H6&>0#3$&#6/4$/#5$;"3/,H6< K"+"/4,C&6%DL5,&A6-5"/E&H.F?/,&C6+,3,&76-A>A,HF3/0>G&#3"-&+.$H66#;,-6?,H6&?A;6-5$1&M"C --,0"&T:e8iWN&B"--"&T:e@fW&4&O667C"/-"&T:e@fWN&PA"?0&T:e@fW&4&Q$>"-"&T:eiiW&6;"A&!6/,-"&T:eefW<&B,3& %"+"/4"&A6-5"/4,&6-4EH/4L50&+?$,5"76?6<&*"&74,;?-A0C&,5"74,&+3"&76-5">4&o>,/5;"3/,H6&73"/4-50p& ?&#"J+0C&A&;6A?"J"/0>G&C6+,34&A6-5"/E&6%34>A6/,&?";56S>4&I$/#>D4&;,"#>D4&/"&4C7$3-<&R;A,+C46 -5,C& "/"34A0& %L+A4,& ?7.0?& %6+g>"& %L+E>,H6& A"%$;A,/4,C& -56>G"-50>A/0C& A?4L#-A"DE>,H6& .E>A/E& 7;6+$#50?/6S(&>A0//4#F?&UVR&T"/H<&total factor productivityW&6;"A&?7.0?"DE>,H6&/"&5,>G/636H4L&?& C6+,3$&!6/,-"&4&#"745".&3$+A#4&?&C6+,3$&PA"?0&4&Q$>"-"<&KL+A4,&56&7$/#5,C&?0DS>4"&+;$H4,D&>ALS>4& %"+"/4"N&>A034&76;F?/"/4"&65;A0C"/0>G&-0C$3">D4&A&+"/0C4&,C74;0>A/0C4&674-$DE>0C4&Z5"/0&=D,+ -/6>A6/,<&ZA,;,H4&>A"-6?,N&A";F?/6&6%-,;?">D4&C"#;6,#6/6C4>A/0>G&H6-76+";#4&"C,;0#"h-#4,DN&D"#& 4&?0/4#F?&-0C$3">D4&+3"&C6+,34&M"C-,0"N&PA"?0&4&Q$>"-"&6;"A&!6/,-"&76&?0-5E74,/4$&-A6#$&5,>G/6 -36H4>A/,H6N&A6-5"/E&A+,#6C76/6?"/,&A"&76C6>E&I435;"&b6+;4>#"&4&R;,->655"&T:ee2W<&*"&#6/4,>&&"/" -34A4,&76;F?/"?>A,D&A6-5"/4,&76++"/"&"$56#6;,3">D"&4&#6;,3">D,&#;A0J6?,&-#."+/4#"&>0#34>A/,H6&+3"& 76+-5"?6?0>G&AC4,//0>G&C6+,34N&>A034&7;6+$#5$N&#6/-$C7>D4&6;"A&#"745".$&I4A0>A/,H6< X&345,;"5$;A,&7;A,+C465$&%;"#$D,&-5$+4$C&76;F?/$DE>,H6&?."-/6S>4&>0#34>A/,&-5"/+";+6?0>G& C6+,34& +.$H66#;,-6?,H6& ?A;6-5$& ?& 7"SC4,& ?"G"h& #6/4$/#5$;"3/0>G<& X0#6;A0-5"/,& ?& /4/4,D --A0C& ";50#$3,& C,>G"/4AC0& 7;67"H">D4& -A6#F?& -56>G"-50>A/0>G& %L+E& AH6+/,& A,& -5"/+";+6?0C& 76+,DS>4,C& ?& C,56+636H44& YZ[\<& X7;6?"+A6/,& A"%$;A,/4"& %L+E& +650>A0(& A";F?/6& ?& .E>A/,D& 7;6+$#50?/6S>4&>A0//4#F?&7;6+$#>D4&UVRN&D"#&4&/4,?05?F;>A0>G&H".LA4&H6-76+";#4N&>A034&-,#56;"& ,+$#">0D/,H6&?&C6+,3$&PA"?0&4&Q$>"-"&6;"A&%"+"h&4&;6A?6D$&?&C6+,3$&!6/,-"<&X/46-#4&A&7;A, -7;6?"+A6/,H6&%"+"/4"&76-.$JE&5"#J,&+6&-#6/-5;$6?"/4"&C6+,34&H6-76+";,#&/"&765;A,%0&"/"34A0& I3$#5$">D4&?&S;,+/4C&6#;,-4,N&76/4,?"J&?"G"/4"&#6/4$/#5$;"3/,&-5"/6?4E&-56-$/#6?6&+$JE&>ALS(& AC4,//6S>4&?4,3#6S>4&C"#;6,#6/6C4>A/0>G&?&S;,+/4C&6#;,-4,&TB6C4/N&[,;53,;&899@W<& Z5;$#5$;"&";50#$.$&D,-5&/"-5L7$DE>"<&X&74,;?-A,D&>ALS>4&6CF?46/,&A6-5"/E&C6+,3,&?A;6-5$&H6-76 -+";>A,H6&M"C-,0"N&PA"?0&4&Q$>"-"&6;"A&!6/,-"<&Y;$H"&>ALS(&%L+A4,&A"?4,;"(&674-&/";AL+A4&+0/"C4>A -/,D&6750C"34A">D4&4&C,56+&/$C,;0>A/0>GN&?0#6;A0-5"/0>G&?&7;A,7;6?"+A6/0C&%"+"/4$<&X&5;A,>4C& ;6A+A4"3,&A6-5"/E&-A>A,HF.6?6&674-"/,&A".6J,/4"&#"34%;">D4&6;"A&?0/4#4&7;A,7;6?"+A6/,H6&%"+"/4"<& R6/"+56&?."-/6S>4&>0#34>A/,&65;A0C"/0>G&-A,;,HF?&>A"-6?0>G&#6/-$C7>D4N&#"745".$&I4A0>A/,H6&6;"A& 7;6+$#5$&-#6/I;6/56?"/,&%L+E&A&6%-,;?">D"C4&,C74;0>A/0C4&+3"&Z5"/F?&=D,+/6>A6/0>G<&

?<+=4@'/1(+A%!(6(+*,'%)-#+$%).%!/'0,($%

X&7;,A,/5">D4&C6+,34&?A;6-5$&H6-76+";>A,H6&?0#6;A0-5"/6&A%4F;&A".6J,hN&#5F;0&D,-5&?-7F3/0&+3"& ?-A0-5#4>G&;6A?"J"/0>G&#6/-5;$#>D4<&R6&74,;?-A,N&>A"-&?&#6/-5;$#>D">G&D,-5&AC4,//E&+0-#;,5/E<&

R6&+;$H4,N&C6+,3,&7;A,+-5"?46/6&A&7$/#5$&?4+A,/4"&5A?<&>,/5;"3/,H6&73"/4-50&T"/H<&social plannerW<& X&C6+,3">G&/4,&$?AH3L+/46/6&76+A4".$&H6-76+";#4&/"&H6-76+";-5?"&+6C6?,&4&-,#56;&?05?F; ->A0<&R6/"+56&A&"/"34A0&?0.E>A6/6&+A4"."3/6S(&7"h-5?"&6;"A&C4L+A0/";6+6?E&?0C4"/L&G"/+36 -?E&T*6l"3,-N&V,;/"/+,AN&M$4A&899eW<&^H;"/4>A,/4,&-4L&+6&76+-5"?6?0>G&AC4,//0>G&C"#;6,#6/6 -C4>A/0>G&76A?"3"&/"&A,-5"?4,/4,&C6+,34&6&A$7,./4,&6+C4,//0>G&C,>G"/4AC">G&?A;6-5$&H6-76 -+";>A,H6<&R6&5;A,>4,N&?&#"J+0C&C6+,3$&A6-5".&$?AH3L+/46/0&?A;6-5&A"-6%$&7;">0&+6-5L7/,H6&?& H6-76+";>,<&Q4>A%"&6-F%&7;">$DE>0>GN&>A034&A"-F%&7;">0N&;6S/4,&?0#."+/4>A6&?,+.$H&-5".,D&,HA6H, -/4>A/,D&-5670&?A;6-5$&n<&Y0/"C4#L&5L&674-$D,&;F?/"/4,&T:W1&

&& & & & & & & & & & & &&&&&&&T:W

t

t

n

L

L

+₁=

(

1

+

)

H+A4,&L_t6A/">A"&6HF3/0&A"-F%&7;">0&?&C6C,/>4,&t TK";;6N&Z"3"c4c'";54/&:eefW<

O63,D/0&A,-5"?&A".6J,h&+650>A0&$J05,>A/6S>4<&X&C6+,3">G&$J056&I$/#>D4&6&-5".,D&;,3"50?/,D& "?,;-D4&+6&;0A0#"&BMMj&T"/H<&constant relative risk aversionW1

&& & & & & & & & & & & &&&&&&&T8W =

1

1

)

(

1 t t

c

θ

c

u

– – – &V$/#>D"&$J05,>A/6S>4&D,-5&;6-/E>"<&O6C,/5";A"&?0C"H"&7;A07"+,#N&H+0&?-7F.>A0//4# ! D,-5& %34-#4&D,+/6S>4<&X5,+0N&AH6+/4,&A&;,H$.E&+,qb6-745"3"N&I$/#>D"&$J05,>A/6S>4&"-0C75650>A/4,&+EJ0& +6&76-5">4&36H";05C$&/"5$;"3/,H6<&`/5,;7;,5">D"&7";"C,5;$&!"D,-5&A?4EA"/"&A&76C4";,C&"?,;-D4&+6& ;0A0#"<&O6;A0-5"DE>&A&C4";0&;,3"50?/,H6&;0A0#"&j;;6?"&4&R;"55"&MMj&T"/H<&relative risk aversionW& TR;"55&:e@_WN&#5F;"&D,-5&436;"A,C&+;$H4,D&76>G6+/,D&7;A,A&74,;?-AE&76>G6+/EN&?A4L5E&A,&A/"#4,C& C4/$-N&C6J/"&-5?4,;+A4(N&J,&?4,3#6S(&5"&/4,&A"3,J0&6+&?4,3#6S>4&#6/-$C7>D4N&"&?4L>&5,J&6+&>A"-$N& 4&D,-5&;F?/"&7";"C,5;6?4&!<&R6/"+56&C4";"&;,3"50?/,H6&;0A0#"&j;;6?"&4&R;"55"&D,-5&?&50C&7;A07"+ -#$& ;F?/"& ,3"-50>A/6S>4& #;"h>6?,D& $J05,>A/6S>4<& B6& ?4L>,DN& 6+?;65/6S(& 76?0J-A,H6& ?0;"J,/4"& D,-5&,#?4?"3,/5/"&?6%,>&C4L+A06#;,-6?,D&,3"-50>A/6S>4&-$%-505$>D4&TK";;6N&Z"3"c4c'";54/&:eefW<

?<;<+B%!(6+($,%$(120,1($%+*,'%)-#+C/A)(4/D+E/))/+2+F%%.A/1)/

R;A,H3E+& C6+,34& +.$H66#;,-6?,H6& ?A;6-5$& H6-76+";>A,H6& A6-5"/4,& ;6A76>AL50& 6+& #6/-5;$#>D4& A"7;676/6?"/,D& 7;A,A& M"C-,0"& T:e8iWN& "& /"-5L7/4,& ;6A?4/4L5,D& 7;A,A& B"--"& T:e@fW& 4& O667C"/-"& T:e@fW:_{<&X&C6+,3$&50C&76+-5"?6?0C&A"H"+/4,/4,C&D,-5&C"#-0C"34A">D"&$J05,>A/6S>4&#6/-$C7} ->D4&?&/4,-#6h>A6/0C&G6;0A6/>4,&>A"-6?0C1

β

Σ

∞ =0

)

(

max

t t t ct

u

c

T]W R";"C,5;&#"D,-5&$56J-"C4"/0&A,&-567E&+0-#6/56?EN&"&?4L>&>,/EN&A"&D"#E&;,7;,A,/5"50?/"&D,+ -/6-5#"&D,-5&?&-5"/4,&7;A,/4,S(&o+A4-4,D-AEp&#6/-$C7>DL&/"&oD$5;6p<&*"&76+-5"?4,&?A6;$&T]W&C6J/"& -5?4,;+A4(N&J,&7;6%3,C&6750C"34A">0D/0&D,-5&;6A?"J"/0&?&/4,-#6h>A6/0C&G6;0A6/>4,&>A"-6?0C<& :_{& &&X&+"3-A,D&>ALS>4&";50#$.$&C6+,3&5,/&%L+A4,&/"A0?"/0&C6+,3,C&M"C-,0"<}

X& 76+-5"?6?0C& C6+,3$& 7;A0DC$D,& -4LN& J,& -567"& ?A;6-5$& 3$+/6S>4 n D,-5& ,HA6H,/4>A/"<& R6A46C& ?4,+A0&A D,-5& 7,?/E& ?";56S>4E& -5".E& ?& >A"-4,N& #5F;"& 6#;,S3"& I$/#>DL& 7;6+$#>D4& +"/,D& H6-76+"; -#4<&j35,;/"50?/4,&C6J/"&7;A0DE(N&J,&76A46C&5,>G/636H4>A/0&?A;"-5"&AH6+/4,&A&,HA6H,/4>A/E&-56 -7E&g TK";;6N&Z"3"c4c'";54/&:eefW<&=C4,//"&5"&%L+A4,&674-0?"(&.E>A/E&7;6+$#50?/6S(&>A0//4#F?& ?05?F;>A0>G&UVR1

A

_t+1=

A

_t

(

1

+

g

)

&T_W X&7;A07"+#$ η 0 > g N&-A6#&5,>G/636H4>A/0N&674-"/0&+"3,D&D"#6 η1,tN&?7;6?"+A".%0&/4,6>A,#4?" -/E&AC4"/L&5,C7"&76-5L7$&5,>G/4>A/,H6&?&H6-76+";>,N&"&?&#6/-,#?,/>D4&5,C7"&?A;6-5$&7;6+$#5$N& 6+>G03"DE>&H6&6+&76A46C$&$-5"36/,H6&/"&S>4,J>,&A;F?/6?"J6/,H6&?A;6-5$< R;6>,-&7;6+$#>D4&A6-5"/4,&674-"/0&A"&76C6>E&I$/#>D4&B6%%"&4&Y6$H3"-"<&="H;,H6?"/0&7;6+$#5& Y_t&?&C6C,/>4,&t D,-5&I$/#>DE&#"745".$&I4A0>A/,H6&K_t6;"A&A"-6%$&7;">0&L_t 4&76A46C$&5,>G/636H4>A -/,H6&A_t TfW<&Z#$5#4,C&&7;6>,-$&?05?F;>A,H6&D,-5&7,?/"&436S(&G6C6H,/4>A/,H6&+6%;"<&X&A"7;676/6 -?"/,D&-7,>0I4#">D4&?0-5L7$D,&5,J&-#."+/4#&6+76?4,+A4"3/0&A"&-A6#4&-56>G"-50>A/,N&>A034&η1,tN&#5F;0& ?&5,D&>ALS>4&";50#$.$&A6-5".&$A/"/0&A"&-5".0&4&;F?/0&D,+/6S>48_<

η

α

α

γ

γ

)

(

, 1t t t t t

K

L

A

Y

=

1

& & & & &&&&&&TfW X";56S>4&7";"C,5;F?&$&&4&%"+,5,;C4/$DE&>G";"#5,;&7;6>,-$&7;6+$#>0D/,H6<&X&7;A07"+#$&H+0& 4>G&-$C"&D,-5&?4L#-A"&6+&D,+/6S>4N&I$/#>D"&7;6+$#>D4&>G";"#5,;0A$D,&-4L&#6;A0S>4"C4&-#"34<&=&#63,4& ?&C6+,3$&A&C"3,DE>0C4&7;A0>G6+"C4&?AH3L+,C&-#"34&A".6J6/6N&J,&α γ 1+ < <&R;A07"+,#&H;"/4>A/0N& H+0& η α γ 1+ = N&D,-5&56J-"C0&A&?0-5L76?"/4,C&-5".0>G&#6;A0S>4&?AH3L+,C&?63$C,/$&7;6+$#5$&4&5,/& ?";4"/5&A6-5"/4,&?0#6;A0-5"/0&?&+"3-A,D&"/"34A4,<& O63,D/0C&#;6#4,C&D,-5&?7;6?"+A,/4,&AC4,//0>G&C6+,3$&?&$DL>4$&per capitaN&>6&76A?634&/"& -I6;C$.6?"/4,& -#6/>,/5;6?"/,D& I$/#>D4& 7;6+$#>D4<& X& I6;C$3,& 674-$DE>,D& 7;6>,-& ?05?F;>A0& TfW& A6-5"."&?0#6;A0-5"/"&/,$5;"3/6S(&?&-,/-4,&b4>#-"<&YA4L#4&5,C$&7;6+$#5&4&#"745".N&6+76?4,+/46&y_t 6;"A&&k_t N&C6HE&A6-5"(&674-"/,&D"#6&&?";56S>4&AC4,//0>G&A"H;,H6?"/0>G&7;A,34>A6/0>G&/"&D,+/6-5#L& ,I,#50?/,D&7;">0]_< X&7;A0DL5,D&-7,>0I4#">D4&-#6/>,/5;6?"/,D&I$/#>D4&7;6+$#>D4&7;6+$#5&y_t A"3,J0&D,+0/4,&6+&?4,3 -#6S>4&#"745".$&I4A0>A/,H6&7;A07"+"DE>,H6&/"&D,+/6-5#L&7;">0&k_t 6;"A&>A0//4#"&-56>G"-50>A/,H6η1,t1

η

η

α

α γ

t t t

k

y

= _{1, & & & & & &&&&&&T@W} X05?6;A6/0& ?& 6#;,-4,&t 7;6+$#5& C6J,& A6-5"(& 7;A,A/">A6/0& /"& #6/-$C7>DL&c_t 3$%& 6-A>AL+ -/6S>4rs_t .&=&#63,4&>A0//4# η1,tN&#?"/50I4#$DE>0&A"%$;A,/4"&5,>G/636H4>A/,N&D,-5&#-A5".56?"/0&7;A,A& &7;6>,-& "$56;,H;,-0D/0& 74,;?-A,H6& -567/4"& 6& 7";"C,5;A,&&₁_<& X& 76+-5"?6?0C& C6+,3$& M"C-,0"& &A"#."+"&-4LN&J,&>".0&?63$C,/&6-A>AL+/6S>4&7;A,A/">A"/0&D,-5&/"&4/?,-50>D,&i_t1 t t t t t

s

c

i

c

y

= + = + T2W

8_{& &=".6J,/4,& 56& +650>A0& ?-A0-5#4>G& C6+,34& %L+E>0>G& 7;A,+C465,C& "/"34A0<& R;A,H3E+& 345,;"5$;0& 674,;"& -4L& /"& stricte}

+,5,;C4/4-50>A/0>G&#6/-5;$#>D">G<&

]_{& &X&7;A07"+#$N&H+0}

'"("0N&76?0J-A0&A"%4,H&-7;6?"+A"&-4L&+6&?0;"J,/4"&AC4,//0>G&C6+,3$&?&$DL>4$&per capita<&

PA$7,./4,/4"&?0C"H"&674-&+0/"C4#4&#"745".$&I4A0>A/,H6&7;A07"+"DE>,H6&/"&,I,#50?/E&D,+ -/6-5#L&7;">0<&X";56S(&5,D&AC4,//,D&?&C6C,/>4,&t+1&D,-5&;F?/"&4/?,-50>D6C&?&767;A,+/4C&6#;, --4,&i_tN&76C/4,D-A6/0C&6&+,7;,>D">DL<&jC6;50A">DL&#"745".$&I4A0>A/,H6&+,5,;C4/$D,&-5"."&?&>A"-4,& 4& ,HA6H,/4>A/"& -567"&)N& #5F;"& AC/4,D-A"& +650>G>A"-& A"#$C$36?"/E& ?4,3#6S(& 5,H6& >A0//4#"& &?05?F;>A,H6<&^-5"5,>A/4,&+0/"C4#"&5,D&AC4,//,D&D,-5&-#6;0H6?"/"&6&?A;6-5&A"-6%$&7;">0&n 6;"A& ,HA6H,/4>A/0&76-5L7&5,>G/636H4>A/0&g1&

_δ

_δ

_η

α t t t t t t t

i

k

k

Ak

c

k

g

n

+ = – = – + – +

)

₊₁

(

1

)

_1,

1

(

&& & & & & & & & & & & &&&&&&TiW Z5"/&$-5"36/0&C6+,3$f_{N&"&?4L>&#6C%4/">D"&?";56S>4&AC4,//0>G&C6+,3$&76?6+$DE>"N&J,&?";}

-56S>4&AC4,//0>G&C6+,3$&?&#63,D/0>G&6#;,-">G&-4L&/4,&AC4,/4"DEN&C6J,&A6-5"(&6#;,S36/0&+3"&#" -745".$&6;"A&#6/-$C7>D4&?&7;A,34>A,/4$&/"&,I,#50?/E&D,+/6-5#L&7;">0&TeW<&^A/">A"&56N&J,&76A46C& #6/-$C7>D4& ?& #"J+0C& 6#;,-4,& D,-5& 5"#4& -"C& 4& ?0/6-4&c

_

<& XA;6-5& H6-76+";>A0N& ;6A$C4"/0& D"#6& A?4L#-A"/4,&-4L&AC4,//0>G&?&$DL>4$&per capitaN&D,-5&/"56C4"-5&C6J34?0&D,+0/4,&?&7;A07"+#$N&H+0& ,HA6H,/4>A/"&-567"&?A;6-5$&g D,-5&+6+"5/4"<&X";56S>4&#6/-$C7>D4&c _ N&#"745".$&I4A0>A/,H6&k _ 6;"A& 7;6+$#5$&y _ &-E&+,5,;C4/6?"/,&?0.E>A/4,&7;A,A&7";"C,5;0&C6+,3$<&B".#6?45,&4/?,-50>D,&-E&;F? -/,&-#$5#6C&7;6>,-F?&AC/4,D-A"DE>0>G&#"745".&I4A0>A/0N&"&?4L>&+,7;,>D">DLN&76-5L7&5,>G/636H4>A/0& 6;"A&7;A0;6-5&A"-6%$&7;">0<

η

β

β

β

α

α

β

δ

δ

δ

δ

α – – – –

β

– – – – –

(

(

(

)

)

)

= + + + + + + = + + =

1

)

1

(

1

)

(

)

1

(

1

)

1

(

1

1 1 α –1 1 α α –1

_n

_g

g

n

A

α A

β

g

n

c

A

g

n

k

&& & & & & & & & & & & &&&&&&TeW

R6?0J,D&674-"/0&C6+,3&-5"/6?4&A"3,+?4,&7$/#5&?0DS>4"&+6&-#6/-5;$6?"/4"&%";+A4,D&A""?"/ --6?"/0>G&C,>G"/4ACF?&6%D"S/4"DE>0>G&?A;6-5&H6-76+";>A0<&

?<?<+B%!(6+(1!%$(120,1($%+*,'%)-#+G,/*4+2+:#0/)/

PA"?"&T:e@_W&?7;6?"+A4.&;6A-A,;A,/4,&-5"/+";+6?,H6&C6+,3$&6&#"745".&3$+A#4&h_t<&M,7;,A,/5"50? -/0&"H,/5&67;F>A&76A46C$&#6/-$C7>D4&?0%4,;"&u_tN&>A034&>A"-&76S?4L>6/0&/"&7;">L<&=C4,//"&u_t&6A/" ->A"&6+-,5,#&>A"-$&76S?4L>"/,H6&7;">0&4&7;A0DC$D,&?";56S>4&A&7;A,+A4".$&s9N&:t<&X0;"J,/4,&:&u&u_t 6A/">A"&/"56C4"-5&6+-,5,#&>A"-$&7;A,A/">A"/,H6&7;A,A&;,7;,A,/5"50?/,H6&"H,/5"&/"&/"%0?"/4,& $C4,DL5/6S>4&4&/"$#LN&>A034&"#$C$3">DL&#"745".$&3$+A#4,H6<&Y3"&$7;6-A>A,/4"&7;A0DL56N&J,&>A"-&76 -S?4L>6/0&/"&7;">L&/4,&?7.0?"&/"&76A46C&$J05,>A/6S>4&6-4EH"/,D&7;A,A&;,7;,A,/5"50?/E&D,+/6-5

#L<&R6+-5"?6?0&7;6%3,C&?&$DL>4$&C"5,C"50>A/0C&D,-5&5"#4&-"C&D"#&?&7;A07"+#$&C6+,3$&M"C-,0"& 4&-7;6?"+A"&-4L&+6&C"#-0C"34A">D4&%4,JE>,D&?";56S>4&#6/-$C7>D4&6-4EH"/,D&?&G6;0A6/>4,&/4,-#6h ->A6/0C&A&5E&;FJ/4>EN&J,&AC4,//0C4&+,>0A0D/0C4&-E&c_t 6;"A&u_t1

β

,

(

)

max

t _t u ct t

Σ

u

c

∞ =0 t T:9W `-565/0C&;6A-A,;A,/4,CN&?&76;F?/"/4$&A&C6+,3,C&M"C-,0"N&D,-5&?7;6?"+A,/4,&+6+"5#6?,H6& >A0//4#"&7;6+$#>D4&h_tN&#5F;,H6&"#$C$3">D"&>G";"#5,;0A$D,&-4L&6+76?4,+/4E&#6/-5;$#>DE<&O"745".& 3$+A#4&D,-5&%6?4,C&;6A$C4"/0&D"#6&6HF3/0&76A46C&$C4,DL5/6S>4&4&#?"34I4#">D4&7;">6?/4#F?&A" -5;$+/46/0>G&?&7;6>,-4,&7;6+$#>0D/0C<&*"&+0/"C4#L&h_t&%L+E&?7.0?"(&+?4,&?4,3#6S>4&T::W<&R4,;? --A"&A&/4>G&56&+650>G>A"-&A"#$C$36?"/"&?";56S(&#"745".$&3$+A#4,H6a&+;$H"&56&>A"-&76S?4L>"/0&/"& +A4"."3/6S(&7;6+$#>0D/E<

η

_{t t} – _t –

δ

_{t t} t

h

B

u

h

h

₊₁= _2,

(

1

)

+

(

1

)

T::W U"#&-#6/-5;$6?"/0&&7;6>,-&"#$C$3">D4&h_t ;FJ/4&-4L&6+&A"7;676/6?"/,D&7;A,A&PA"?L&T:e@_W&>A0& Q$>"-"&T:eiiW<&P?AH3L+/46/6&?&/4,D&+,7;,>D">DL&#"745".$&3$+A#4,H6N&>6&76-5$36?"34&'"/#4?N&M6 -C,;&4&X,43&T:ee8W<&[+0&;,7;,A,/5"50?/0&"H,/5&>".0&A"-F%&>A"-$&76S?4L>"&7;">0N&>A034&u_t= 1N&#"745".& 3$+A#4&+,7;,>D6/$D,&-4L&6&7,?/E&-5".E&?&>A"-4,&4&,HA6H,/4>A/E&-567L&+,7;,>D">D4&!_h<&=&+;$H4,D&-5;6 -/0&H+0&;,7;,A,/5"50?/0&"H,/5&?&6HF3,&/4,&A"DC$D,&-4L&+A4"."3/6S>4E&?05?F;>AEN&>A034&u_t= 0N&"#$ -C$3">D"&#"745".$&3$+A#4,H6&D,-5&/"D-A0%-A"&4&6+76?4"+"&-5".,D&B&76C/4,D-A6/,D&6&)_h<&*"D?"J/4,D --AE&?."-/6S>4E&D,-5&5$5"D&34/46?6S(&I$/#>D4&#;,$DE>,D&#"745".&3$+A#4N&-A>A,HF3/4,&>ALS>4&6#;,S3"DE>,D& 7;6>,-&?05?F;>A0&5,H6&A"-6%$N&>A034&htB(1–ut)<&[+0&?4,3#6S(&5"&D,-5&?0J-A"&6+&+,7;,>D">D4&#"74 -5".$&3$+A#4,H6N&?&C6+,3$&PA"?0&4&Q$>"-"&76D"?4"&-4L&,/+6H,/4>A/0&?A;6-5N&#5F;,H6&g;F+.,C&D,-5& "#$C$3">D"& 5,H6& >A0//4#"& 7;6+$#>D4<& O6C,/5";A"& ?0C"H"& 5,J& -76-F%& ?7;6?"+A,/4"& -A6#$& -56 ->G"-50>A/,H6&+6&;F?/"/4"&;$>G$&#"745".$&3$+A#4,H6<&R6?6+$D,&6/6&/4,"/50>076?"/,&A?4L#-A, -/4,&,I,#50?/6S>4&-,#56;"&,+$#">0D/,H6<&',>G"/4AC&7;67"H">D4&-A6#$&A?4L#-A"DE>,H6&,I,#50?/6S(& "#$C$3">D4&#"745".$&3$+A#4,H6&D,-5&7;6>,-,C&jMT:W&6&?-7F.>A0//4#$&&2<

=C4,/4"&-4L&5,J&-7,>0I4#">D"&I$/#>D4&7;6+$#>D4<&=%4F;&>A0//4#F?&A6-5"D,&76?4L#-A6/0&6&#"745".& 3$+A#4&h_t 6;"A&>A"-&76S?4L>"/0&+A4"."3/6S>4&7;6+$#>0D/,D&u_t<&`>G&,3"-50>A/6S>4&?AH3L+,C&7;6+$#5$& ?0/6-AE&1 – $<&Y6+"5#6?6&+6&C6+,3$&C6J,&%0(&?7;6?"+A6/0&#"745".&3$+A#4N&>6&76-5$36?".&/7<& Q$>"-&T:eiiW<&X5,+0&?0#."+/4#&#"745".$&3$+A#4,H6&?&I$/#>D4&7;6+$#>D4&D,-5&76?4L#-A6/0&6&?";56S(& 7";"C,5;$&%<&X&5,D&-05$">D4&?7;6?"+A6/,&A6-5"DE&,I,#50&-#"34&"#$C$3">D4&#"745".$&3$+A#4,H6<&=" -%4,H&5,/&D,-5&$A"-"+/46/0&C<4/<&765;A,%E&$?AH3L+/4,/4"&7;6>,-F?&A?4L#-A"DE>0>G&#"745".&3$+A#4& 76+>A"-&+A4"."3/6S>4&?05?F;>A,DN&/7<&learning by doing&TO3,/6?N&M6+;4H$,AcB3";,&899_W1

η

– – +γ = _{1,t t t}1 _t1 t

Ak

u

h

y

α α α &T:8W '6+,3&D,-5&$A$7,./46/0&;F?/"/4,C&;$>G$&#"745".$&per capita<&*"&+0/"C4#L&5,D&AC4,//,D&,#6 -/6C4>A/,D&?7.0?"&?";56S(&4/?,-50>D4&6;"A&D,D&+,7;,>D">D"&4&5,C76&?A;6-5$&A"-6%$&7;">0&T:]W<&`/ -?,-50>D,N&76+6%/4,&D"#&?&C6+,3$&M"C-,0"N&-5"/6?4E&;FJ/4>L&76C4L+A0&7;6+$#5,C&"&#6/-$C7>DE<& jC6;50A">D"&#"745".$&I4A0>A/,H6&6;"A&5,C76&?A;6-5$&A"-6%$&7;">0&-E&,HA6H,/4>A/,< +

n

k

t+ = –

δ

k

k

t+

η

t

Ak

t

u

t

h

t –

c

t 1 , 1 1

(

1

)

)

1

(

– 1 α– +γ T:]W

R;A,+-5"?4,/4,&AC4,//0>G&A"H;,H6?"/0>G&C6+,3$&PA"?0&4&Q$>"-"&?&$DL>4$&per capita&/4,&?0 --5";>A"N&%0&6#;,S34(&-5"/&-5">D6/";/0&C6+,3$&6&76JE+"/0>G&?."-/6S>4">G<&KL+A4,&/4C&A"5,C&K[R& urS>4,J#"&A;F?/6?"J6/,H6&?A;6-5$&T"/H<&balanced growth pathW&+3"&#"745".$&I4A0>A/,H6N&3$+A#4, -H6N&#6/-$C7>D4&6;"A&-5"."&?";56S(&u_t<& j;%45;"3/"&-7,>0I4#">D"&I$/#>D4&7;6+$#>D4N&A"#."+"DE>"N&J,&7";"C,5;&%D,-5&;F?/0&A,;6N&>A034&7;A0 -7"+,#N&H+0&7;6>,-6?4&?05?F;>A,C$&/4,&56?";A0-A0&+6+"5#6?"&"#$C$3">D"&#"745".$&3$+A#4,H6N&76 -A?"3"&?0A/">A0(&-5"/&-5">D6/";/0&C6+,3$&76C4C6&%;"#$&?07$#.6S>4&I$/#>D4&7;6+$#>D4&Tb4;"H$>G4& 899eW<&U,C76&?A;6-5$&#6/-$C7>D4N&#"745".$&4&7;6+$#5$&A"3,J0&?F?>A"-&D,+0/4,&6+&7";"C,5;F?&674 --$DE>0>G&7;6>,-0&"#$C$3">D4&#"745".$&3$+A#4,H6&6;"A&C"#-0C"34A">D4&%4,JE>,D&?";56S>4&#6/-$C7>D41&

[

+ –

δ

]

1

)

1

(

B

_h

g

=

β

θ& & & & &&&&&T:_W

Z5"/&$-5"36/0&%L+A4,&-4L&A"5,C&>G";"#5,;0A6?".&-5".0C4&7;676;>D"C4&76C4L+A0&AC4,//0C4N& #5F;,&?0#"A$DE&5;,/+&+.$H66#;,-6?0&+,5,;C4/6?"/0&-567E&?A;6-5$&g<&U,C76&?A;6-5$&?4,3#6S>4& C6+,3$&g A"3,J0&6+&7";"C,5;F?&;F?/"/4"&;$>G$&#"745".$&3$+A#4,H6&T::W<&&O6;A0-5"DE>&A&7;A0DL5,D& 76?0J,D&76-5">4&C6+,3$&PA"?0&4&Q$#"-"N&C6J/"&6#;,S34(&+6#."+/,&?";56S>4&?&-5"/4,&-5">D6/";/0C& +3"&u_t&4&/6?0>G&+?F>G&AC4,//0>G1&x_t&6;"A&z_t<&R4,;?-A"&A&/4>G&6A/">A"&-56-$/,#&#"745".$&I4A0>A/,H6& +6&3$+A#4,H6N&"&+;$H"&v&#6/-$C7>D4&+6&#"745".$&I4A0>A/,H61& – _–

δ

B

g

u

₌

₁

+ k

δ

– – – – – + + = 1

)

1

(

)

1

)(

1

(

u

A

g

n

z

k θ

β

α

α

&& & & & & & & & & & & &&&&&T:fW –

δ

– – _– –

_A

_z

– 1

_u

–1

₍

₁

₎

₍

₁

_n

₎₍

₁

_g

₎

x

= α α+ _k + + U"#&D"#&?&C6+,3$&M"C-,0"&-5"/&-5">D6/";/0&C"&5$5"D&>G";"#5,;&7$/#5$&-46+.6?,H6<&^A/">A"& 56N& J,& 76& $?AH3L+/4,/4$& ?";$/#F?& 5;"/-?,;-"3/6S>4& 6750C"3/0C& A%46;,C& AC4,//0>G& +,>0A0D -/0>G& C6+,3$& D,-5& S>4,J#"& -46+.6?"N& #5F;"& 76A?"3"& /"& -567/46?E& A%4,J/6S(& +6& -5"/$& $-5"36/,H6& TK";;6N&Z"3"c4c'";54/&:eefW< X,+.$H&#6/>,7>D4&A"7;676/6?"/,D&7;A,A&PA"?L&4&;6A?4/4L5,D&7;A,A&Q$>"-"&g;F+.,C&?A;6-5$& H6-76+";>A,H6&D,-5&"#$C$3">D"&#"745".$&3$+A#4,H6<&X&C6+,3$&50C&5,C76&?A;6-5$&AC4,//0>G&,#6 -/6C4>A/0>G&A"3,J0&6+&7";"C,5;F?&A?4EA"/0>G&A&+0/"C4#E&#"745".$&3$+A#4,H6<&XA;6-5&H6-76+"; ->A0&D,-5&A"5,C&AD"?4-#4,C&,/+6H,/4>A/0C<

?<H<+B%!(6+)(A2(1!%$(120,1($%+*,'%)-#+8%1()/

X&C6+,3$&!6/,-"&T:eefW&?7;6?"+A6/6&G,5,;6H,/4>A/0&A"-F%&7;">0&T:@W<&MFJ/6;6+/6S(&5"&?4EJ,&-4L& A&76+A4".,C&-4.0&;6%6>A,D&/"&A"5;$+/46/0>G&?&-,#56;A,&%"+"h&4&;6A?6D$&L_A,t&6;"A&A""/H"J6?"/0>G& ?&+A4"."3/6S(&?05?F;>AE&L_y,t1&

t yt At

L

L

L

= , + , &T:@W X&+"3-A,D&"/"34A4,&C6+,3$&?7;6?"+A6/6&6+-,5,#&>A"-$&76S?4L>"/,H6&/"&+A4"."3/6S(&?&D,+ -/0C&A&+?F>G&-,#56;F?&H6-76+";#4&T:2W<&X&5,/&-76-F%&;,7;,A,/5"50?/"&D,+/6-5#"&%L+A4,&+,>0+6 -?"(N&43,&>A"-$&7;A,A/">A0&/"&+A4"."3/6S(&%"+"?>AEN&"&43,&/"&?05?F;>AE<&*4,>G&l_A,t&6A/">A"&>ALS(& A"-6%$&7;">0&7;A,A/">A6/E&/"&%"+"/4"&4&;6A?FDN&"&l_y,tv&/"&7;6+$#>DL@1

1

, ,t+ At= y

l

l

&T:2W X05?";A"/4,& 7;6+$#5$& ?& H6-76+";>,&y_t& D,-5& 674-"/,& I$/#>DE& 7;6+$#>D4& B6%%"& 4& Y6$H3"-"<& ^7;F>A&#"745".$&k >A0//4#"C4&7;6+$#>D4&-E1&76A46C&5,>G/636H4>A/0&A_t6;"A&6+-,5,#&>A"-$&7;A,A/" ->A"/,H6&/"&+A4"."3/6S(&?05?F;>AE&&l_y,t AH6+/4,&A&76/4J-AE&I6;C$.E1

= – – = 1, , 1 1 , , 1t t

(

t yt

)

t t

(

t

(

1

At

))

t

η

Bk

A

l

Bk

A

l

α α

_η

α α

y

_T:iW Z7,>0I4#">D"&7;6>,-$&7;6+$#>0D/,H6&$C6J34?4"&A"74-"/4,&;F?/"/4"&;$>G$&#"745".$&I4A0>A/, -H6&per capita<&U"#&D"#&?&C6+,3">G&M"C-,0"&6;"A&PA"?0&4&Q$>"-"&?4,3#6S(&5,H6&>A0//4#"&?&6#;,-4,& t + 1 6+76?4"+"&?";56S>4&?&C6C,/>4,&tN&76C/4,D-A6/,D&6&+,7;,>D">DL&4&?A;6-5&A"-6%$&7;">0&6;"A& 76?4L#-A6/,D&6&4/?,-50>D,1 – – – –

η

δ

_{t t t}α _{t At} α _t t

k

Bk

A

l

c

k

n

= + +

)

₊₁

(

1

)

_1,

(

(

1

_,

))

1

1

(

&& & &&&&&T:eW

^-5"5/4C&,3,C,/5,C&$#."+$&+0/"C4>A/,H6&43$-5;$DE>,H6&C6+,3&D,-5&;F?/"/4,&674-$DE>,&AC4" -/L&76A46C$&5,>G/636H4>A/,H6&T89W<&*"&?4,3#6S(&76A46C$&5,>G/636H4>A/,H6&?7.0?"DE1&+650>G>A"-& 6-4EH/4L5"&?4,+A"&A_tN&?63$C,/&-4.0&;6%6>A,D&A"5;$+/46/,D&?&-,#56;A,&%"+"h&L_A,t 6;"A&,3,C,/5&6+76 -?4,+A4"3/0&A"&7;67"H">DL&-A6#F?&-56>G"-50>A/0>G&*3,tN&#5F;,H6&#-A5".56?"/4,&%L+A4,&76+3,H"(&7;6

->,-6?4&jMT:W&6&?-7F.>A0//4#$&&31&

η

η

φ φ t t t A t t t t A t t t DA L A DA l L A

A

+ = + = + μ μ μ , , 3 , , 3

1 & & & &&&&&T89W

MF?/"/4,& ;$>G$& 5,>G/636H44& D,-5& /4,34/46?,<& O6C,/5";A"& ?0C"H"DE& ?";56S>4& 7";"C,5;F?&&" 6;"A&+<& [+0& ?";56S>4& 6%$& 50>G& 7";"C,5;F?& -E& ;F?/,& D,+/6S>4N& C6+,3& -7;6?"+A"& -4L& +6& 76-5">4& 76-5$36?"/,D&7;A,A&M6C,;"&T:ee9WN&[;6--C"/"&4&b,37C"/"&TO3,/6?N&M6+;4H$,AcB3";,&899_W&6;"A& jHG46/&4&b6?455"&T:ee8W<&'6+,3,&5,&/4,&674-$DE&D,+/"#&?."S>4?4,&-,#56;"&%"+"h&4&;6A?6D$&?&$DL>4$& ,C74;0>A/0C&T!6/,-&899fW<

X7.0?& %4,JE>,H6& 76A46C$& 76-5L7$& 5,>G/636H4>A/,H6& /"& +0/"C4#L& 5,D& AC4,//,D& A"3,J0& 6+& ?";56S>4& 7";"C,5;$&&<& !,S34& ?-7F.>A0//4#&&"D,-5& ;F?/0& A,;6N& 5,C76& ?A;6-5$& ?4,+A0& /4,& A"3,J0& 6+&76A46C$&+650>G>A"-&6-4EH/4L5,D&5,>G/636H44<&=&#63,4N&H+0&7";"C,5;&&"D,-5&$D,C/0N&56&?&76-5" ->4&;F?/"/4"&76-5L7$&5,>G/636H4>A/,H6&7;A0DC$D,C0N&J,&-,#56;&%"+"h&4&;6A?6D$&>G";"#5,;0A$D,&-4L& C"3,DE>0C4&7;A0>G6+"C4&?AH3L+,C&-#"34<&'6J/"&56&.E>A0(&A,&AD"?4-#4,C&fishing-outN&#5F;,&763, -H"&/"&-7"+#$&-5670&?A;6-5$&7;6+$#5$&7;A0&D,+/6>A,-/0C&?A;6S>4,&7;6+$#50?/6S>4&>A0//4#F?& ?05?F;>A0>G<&!,S34&?";56S(&?-7F.>A0//4#"&&"D,-5&?4L#-A"&6+&A,;"N&7;6>,-6?4&"#$C$3">D4&?4,+A0& 56?";A0-AE&76A050?/,&,I,#50&A,?/L5;A/,&T!6/,-&899fW<&

Y;$H4C& >A0//4#4,C& ?7.0?"DE>0C& /"& 76-5L7& 5,>G/636H4>A/0& D,-5& A"-F%& 7;">0& 7;A,A/">A6/0& /"&+A4"."3/6S(&%"+"?>A6c;6A?6D6?E<&O4,;$/,#&?7.0?$&D,-5&6#;,S3"/0&7;A,A&7";"C,5;&+<&K;"#&A" -3,J/6S>4&5,C7"&"#$C$3">D4&76-5L7$&5,>G/636H4>A/,H6&6+&A"-6%$&7;">0&?&-,#56;A,&%"+"h&4&;6A?6D$& $A0-#$D,C0N&H+0&+"D,-5&;F?/,&A,;6<&Y3"5,H6&?&C6+,3$&A"#."+"&-4LN&J,&?";56S>4&+&/"3,JE&+6&7;A, -+A4".$&T9N:t<&Y6+"5/4"&?";56S(&%L+A4,&S?4"+>A0(&6&76A050?/0C&?7.0?4,&?63$C,/$&A"5;$+/4,/4"& ?&-,#56;A,&MkY&/"&+0/"C4#L&"#$C$3">D4&?4,+A0<&[;"/4>A/0&7;A07"+,#N&?&#5F;0C&7";"C,5;&D,-5& ;F?/0&D,+/6S>4N&6A/">A"&-05$">DLN&?&#5F;,D&A?4L#-A,/4,&A"-6%$&7;">0&?&-,#56;A,&%"+"h&4&;6A?6D$& D,-5&?7;6-5&7;676;>D6/"3/,&+6&+0/"C4#4&76A46C$&76-5L7$&5,>G/636H4>A/,H6&T!6/,-&:eefW<& X&C6+,3$&7;A0DC$D,&-4L&D,+/"#N&J,&?-7F.>A0//4#&+&C"&?";56S(&/4J-AE&6+&D,+/6S>4<&Y"/,&,C -74;0>A/,&A&6-5"5/4>G&8fN&"&/"?,5&:99&3"5&76#"A$DEN&J,&34>A%"&6-F%&A"5;$+/46/0>G&?&-,#56;A,&%"+"? ->A0C&A/">A/4,&?A;6-."&6+&5,H6&>A"-$<&=&#63,4&-567"&?A;6-5$&7;6+$#5$&%0."&?";56S>4E&-5".E<&

*4,34/46?6S(& I$/#>D4& #;,$DE>,D& ?4,+AL& ?AH3L+,C&A_t -7;"?4"N& J,& ?& C6+,3$& /4,& ?0-5L7$D,& ?&7,./4&,/+6H,/4>A/0&?A;6-5N&#5F;,H6&6-5"5,>A/0C&g;F+.,C&D,-5&"#$C$3">D"&A_tN&3,>A&+6&$A0-#"/4"& 5;?".,H6&?A;6-5$&y_t&/4,A%L+/0&D,-5&7;A0;6-5&/"5$;"3/0N&"&?4L>&+6+"5/4"&?";56S(&n<&U,/&A"SN&+A4L#4& -,#56;6?4&KkMN&7;A,#."+"&-4L&/"&?A;6-5N&H+0J&?4,+A"&/4,&D,-5&>A0//4#4,C&7;6+$#>D4N&#5F;0&76+3,H"& -5"/+";+6?0C&C,>G"/4AC6C&;0/#6?0C<&YA4L#4&/4,D&C6J34?,&D,-5&?0#6;A0-5"/4,&;6-/E>0>G&#6 -;A0S>4&-#"34&TM6C,;&:ee9W<&ZA6#&5,>G/636H4>A/0&*_3,t &?&;F?/"/4$&T89W&%L+A4,&6+/6-4.&-4L&+6&/4,"/ -50>076?"/,D&AC4"/0&5,C7"&76-5L7$&5,>G/636H4>A/,H6&T,I,#50?/6S>4&-,#56;"&KkMW&?&H6-76+";>,N& "&?&#6/-,#?,/>D4r+6&5,C7"&?A;6-5$&7;6+$#5$N&6+>G03"DE>&H6&6+&76A46C$&674-$DE>,H6&K[R< X&C6+,3$&!6/,-"&-5"/&$-5"36/0&-5"/6?4&S>4,J#L&A;F?/6?"J6/,H6&?A;6-5$&K[R&+3"&AC4,//0>G& ?0-5L7$DE>0>G&?&C6+,3$<&U,C76&?A;6-5$&?-A0-5#4>G&AC4,//0>G&D,-5&D,+/"#6?,&T8:W<&B6&?4L>,DN& -567"&?A;6-5$&D,-5&I$/#>DE&,HA6H,/4>A/,D&-5670&?A;6-5$&-4.0&;6%6>A,D&6;"A&7";"C,5;F?&&"6;"A&+<&

₌

(

1

₊

)

1φ –

1

μ

n

g

– && & & & &&&&&T8:W

Z567L&?A;6-5$&?&H6-76+";>,&C6J/"&674-"(&D"#6&;6-/E>E&I$/#>DL&?-7F.>A0//4#F?&+&4&&&6;"A& 5,C7"&?A;6-5$&A"-6%$&7;">0<& R6A%"?4,/4,&5;,/+$&AC4,//0>G&?&C6+,3$&76A?"3"&6#;,S34(N&D"#4&6+-,5,#&>A"-$&76S?4L>"/0&D,-5& /"&+A4"."3/6S(&%"+"?>A6c;6A?6D6?EN&>A034&l _ A&T8:W<&Y6+"5#6?6&C6J/"&?0A/">A0(&?4,3#6S>4&76A6 --5".0>G&AC4,//0>G&$#."+$N&5D<&436;"A&#6/-$C7>D4&+6&#"745".$&I4A0>A/,H6&z_t&6;"A&#"745".$&I4A0>A/,H6& +6&76A46C$&5,>G/636H4>A/,H6&x_t &4&AC4,//,D&w_t 6+76?4"+"DE>,D&?0;"J,/4$& φ μ

t A t L A –1 _, 1&

_D

g

w

= – – 1

)

1

(

1

1

+ + =

g

g

l

_A

( (

β

))

β

θ

φ

μ

– – – – –

(

)

+ + + = 1 1 1

)

1

(

1

)

1

)(

1

(

A

l

g

n

x

β

δ

α

α α θ – – –

& & & & &&&&&T88W

– – – –

(

1

)

(

1

)

1 1

)

1

)(

1

(

+ + + + =

g

n

B

l

x

z

δ

_A α α

R6+6%/4,&D"#&?&7;A07"+#$&767;A,+/4>G&C6+,34&-5"/&$-5"36/0&D,-5&5$&7$/#5,C&-46+.6?0C&$#." -+$&+0/"C4>A/,H6<&=?"J0?-A0&/"&?";$/#4&5;"/-?,;-"3/6S>4&;6A?4EA"/4,C&6750C"3/0C&D,-5&A" -5,C&S>4,J#"&-46+.6?"&A"7,?/4"DE>"&-567/46?E&A%4,J/6S(&+6&-5"/$&-5">D6/";/,H6&C6+,3$&Tj;/63+& 899@W< O6/-5;$#>D"&A"7;676/6?"/"&7;A,A&!6/,-"&>AL-56&D,-5&674-0?"/"&D"#6&C6+,3&-,C4,/+6H,/4>A -/,H6&?A;6-5$<&X0/4#"&56&A&I"#5$N&J,&?A;6-5&H6-76+";>A0&A"3,J0&6+&+0/"C4#4&A"-6%$&7;">0N&#5F -;"&D,-5&,HA6H,/4>A/"<&=&+;$H4,D&-5;6/0&?&C6+,3$&?0-5L7$D,&?A;6-5&?";56S>4&#6/-$C7>D4N&#"745".$& I4A0>A/,H6&6;"A&7;6+$#5$&T!6/,-&899fW<&

H<+B(-%!4&/+@/!/12/

X&5,D&>ALS>4&";50#$.$&A6-5"/4,&6CF?46/"&C,56+0#"&%"+"/4"<&X0#6;A0-5"/,&/";AL+A4"&A6-5".0&A" -76J0>A6/,&H.F?/4,&A&345,;"5$;0&76S?4L>6/,D&YZ[\<&R6+-5"?6?0C&/";AL+A4,C&?0A/">A"/4"&,#- -5;,C$C& 36#"3/,H6& I$/#>D4& D,-5& C,56+"& C/6J/4#F?& Q"H;"/H,q"<& Z56-$D,& -4L& DE& ?& C6+,3">G& #3"-0& YZ[\N&">A#63?4,#&D,-5&5,J&7;A0+"5/"&?&#6/-5;$#>D">G&?0?6+AE>0>G&-4L&A&5,6;44&?A;6-5$N&A""+"7 -56?"/0>G&C<4/<&+6&"/"34A0&I3$#5$">D4<

j/"34A6?"/,&?&";50#$3,&C6+,3,&A6-5".0&A"74-"/,&?&6HF3/,D&76-5">4&7;6%3,C$&C"#-0C"34A" ->0D/,H6N&?&#5F;0C&6750C"34A6?"/"&D,-5&I$/#>D"&6&L AC4,//0>G&-5,;$DE>0>G&x_1,t6;"A&K AC4,//0>G& -5"/$&TX4>#,/-&899iW<&*"-5L7/4,&-#6/-5;$6?"/6&I$/#>DL&Q"H;"/H,q"<&j;H$C,/5"C4&5,D&I$/#>D4&-E& AC4,//,&-5"/$&y_j,t N&AC4,//,&-5,;$DE>,&x_1,t 6;"A&C/6J/4#&"_j,t&+3"&#"J+,H6&6#;,-$&t ?&/4,-#6h>A6/0C& G6;0A6/>4,&/4,-#6h>A6/0C<&Q4>A%"&C/6J/4#F?&Q"H;"/H,q"&?&#"J+0C&6#;,-4,&t&D,-5&;F?/"&34>A%4,& 6H;"/4>A,hN&>A034&34>A%4,&;F?/"h&;$>G$&AC4,//0>G&T*6l"3,-N&V,;/"/+,AN&M$4A&899eW1 – ( ( ( ))+ + (

(

)

)

)

= = 1 1 1, , 1, , 1, , 1, , 0 , , 1 0 ( ,..., ) t,..., Kt, t,..., Lt ... l l t,..., Kt, t,..., Lt t t K t

y

G

y

y

x

x

G

y

y

x

x

y

F

E

∑

L

λ

λ

∞ =0 t

∑

∞ _T8]W *"-5L7/4,&6%34>A6/6&5A?<&?";$/#4&74,;?-A,H6&;AL+$&TX4>#,/-&899iW<&R63,H"&56&/"&6%34>A,/4$& 76>G6+/0>G&>AE-5#6?0>G&I$/#>D4&Q"H;"/H,q"&?AH3L+,C&?-A0-5#4>G&AC4,//0>G&-5"/$N&-5,;$DE>0>G& 6;"A&C/6J/4#F?&Q"H;"/H,q"&?&C6C,/>4,&t<&RFg/4,D&#"J+,&A&6%34>A6/0>G&?&5,/&-76-F%&?0;"J,h&D,-5& 7;A0;F?/0?"/,&+6&A,;"&TO34C"&899fW1

∀

∀

∀

= = =

0

0

0

, } ,.., 1 { , } ,.., 1 { , } ,.., 1 { t l L l t l L l t k K k

L

x

L

y

L

λ

∂

∂

∂

∂

∂

∂

∈ ∈ ∈

&& & & & & & & & & & & &&&&&T8_W

R4,;?-A,& ;F?/"/4,& $#."+$& T8_W& 6+/6-4& -4L& +6& AC4,//0>G& C"#-0C"34A6?"/0>GN& >A034& 5A?<& AC4,//0>G& -5"/$<& Y;$H4,& D,-5& A?4EA"/,& A,& AC4,//0C4& -5,;$DE>0C4N& "& 6-5"5/4,& +650>A0& I$/#>D4& &Q"H;"/H,q"&?0#6;A0-50?"/,D&?&A"+"/4$&6750C"34A">0D/0C&T8_W<

*"&76+-5"?4,&65;A0C"/,H6&$#."+$&6%34>A6/6&;F?/"/4"&\$3,;"N&7;A,A&?0,34C4/6?"/4,&C/6J -/4#F?&Q"H;"/H,q"&A&+;$H4,H6&A,-5"?$&T8_W<&MF?/"/4"&\$3,;"&6#;,S3"DE&A"3,J/6S(&AC4,//0>G&-5"/$& 76C4L+A0&6#;,-"C4<&X&C6+,3">G&$?AH3L+/4"DE>0>G&-A6#4&-56>G"-50>A/,&;F?/"/4"&\$3,;"&674-$DE& ?";56S(&6>A,#4?"/E&?&6#;,-4,&t ?&A"3,J/6S>4&6+&?4,3#6S>4&AC4,//0>G&?&6#;,-4,&t + 1 TX4>#,/-& 899iW<&`-565/0C&,3,C,/5,C&C6+,34&67";50>G&/"&+0/"C4>A/,D&6750C"34A">D4&D,-5&-5"/&$-5"36/0&T"/H<&

steady stateW<& X& C6+,3$& ?0;"J6/0C& ?& >A"-4,& +0-#;,5/0C& 6& /4,-#6h>A6/0C& G6;0A6/>4,&

>A"-6-?0C&6+/6-4&-4L&6/&+6&-05$">D4N&?&#5F;,D&J"+/"&A,&AC4,//0>G&$#."+$&+0/"C4>A/,H6N&"&?4L>&AC4,/ -/0>G&-5,;$DE>0>G&6;"A&AC4,//0>G&-5"/$N&/4,&AC4,/4"&-4L&?&>A"-4,&T*6l"3,-N&V,;/"/+,AN&M$4A&899eW<& =C4,//,&36-6?,&C6+,3$N&5"#&D"#&?&7;A07"+#$&6#;,S3"/4"&?";$/#F?&74,;?-A,H6&;AL+$N&%L+E&-5".,& ?&>A"-4,N&"&4>G&?";56S(&%L+A4,&;F?/"&?";56S>4&6>A,#4?"/,D<& X&C6+,3">GN&?&#5F;0>G&-5"/&-5">D6/";/0&C"&76-5"(&5;,/+$&AC4,//0>G&-5"/$&3$%&-5,;$DE>0>GN& 76?0J-A0&$#."+&%L+A4,&?0;"J6/0&?&76-5">4&;,3">D4&76C4L+A0&AC4,//0C4<&X7;6?"+A6/,&A6-5"/E& /6?,&6A/">A,/4"N&?0;"J"DE>,&;,3">D,&C4L+A0&76-A>A,HF3/0C4&AC4,//0C4<&=C4,/4&-4L&5,J&;F?/"/4,& \$3,;"N& #5F;,& %L+A4,& ?0C"H"(& 6+5;,/+6?4,/4"& $#."+$N& >A034& ?7;6?"+A,/4"& /6?0>G& AC4,//0>G& 6#;,S36/0>G&D"#6&7;676;>D,&76C4L+A0&76+-5"?6?0C4&AC4,//0C4&C6+,3$2_<&

O63,D/0C&#;6#4,C&D,-5&A"-56-6?"/4,&C,56+0&"7;6#-0C">0D/,D<&X&%"+"/4$&?0#6;A0-5"/6&7;A0 -%34J,/4,&36H";05C4>A/6c34/46?,N&763,H"DE>,&/"&A"74-"/4$&$#."+$&;F?/"h&\$3,;"&6;"A&;F?/"h&;$ ->G$&AC4,//0>G&-5,;$DE>0>G&D"#6&$#."+$&34/46?,H6<&X&50C&>,3$&+3"&#"J+,D&AC4,//,D&$#."+$&?7;6 -?"+A6/6&AC4,//E&76C6>/4>AE&-5"/6?4E>E&;FJ/4>L&C4L+A0&36H";05C"C4&AC4,//,D&?&6#;,-4,&t 6;"A& ?&-5"/4,&-5">D6/";/0C<&!"#6&yˆ_k,t 6A/">A6/6&AC4,//E&76C6>/4>AE&?&C6C,/>4,&t +3"&kc5,D&AC4,//,D& -5"/$i<&^HF3/"&A"3,J/6S(&76C4L+A0&6%-,;?">D"C4&C6J,&A6-5"(&A"74-"/"&A"&76C6>E&/"-5L7$DE>,H6& ;F?/"/4"&T*6l"3,-N&V,;/"/+,AN&M$4A&899eW1 – – –

)

∀

(

_kt

)

(

_k

)

k t k t k K k

y

y

y

y

y

ˆ

_,

ln

,

ln

_,

ln

} ,.., 1 { =

(

= ∈

&& & & & & & & & & & & &&&&&T8fW

X0#6;A0-5$DE>&7;"?6&45,;6?"/0>G&6>A,#4?"h&6;"A&34/46?6S(&67,;"56;"&?";56S>4&6>A,#4?"/,DN& ?";56S>4&6+76?4"+"DE>,&>ALS>4&-56>G"-50>A/,D&?&6#;,-4,&t + 1 C6J/"&?0A/">A0(&A"&76C6>E&"$56 -;,H;,-0D/,D&76-5">4&;F?/"/4"&674-$DE>,H6&5L&>ALS(&C6+,3$&T*6l"3,-N&V,;/"/+,AN&M$4A&899eW<&="&76 -C6>E&?A6;$&T8fW&?07;6?"+A6/0&A6-5"/4,&$#."+&;F?/"h&T8@W&674-$DE>0&#-A5".56?"/4,&-4L&AC4,/ -/0>G&?6#F.&-5"/$&$-5"36/,H6 + = + + + + t L t t K t t L t t K t t L t

M

y

y

x

x

M

y

E

y

E

x

x

M

, , 1 1 , , 1 , , 1 1 1 , 1 , 1 1 , 1 , 1 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

η

η

&& & & & & & & & & & & &&&&&T8@W

2_{& &="%4,H&5,/&A6-5"/4,&?0#6;A0-5"/0&?&C6+,3$&PA"?0&4&Q$>"-"&6;"A&!6/,-"<}

i_{& &j/"36H4>A/,&?."-/6S>4&A?4EA"/,&A&7;A0DL5E&5,>G/4#E&"7;6#-0C">0D/E&C6J/"&76#"A"(&+3"&#"J+,D&AC4,//,D&-5,;$DE>,Dxˆ}

H+A4,&M₀ 6;"A&M₁ -E&C">4,;A"C4&#?"+;"56?0C4&6&?0C4";">G (K ₊L)×(K+L)N&"&,3,C,/50&50>G& C">4,;A0&-E&I$/#>D"C4&7";"C,5;F?&C6+,3$e<&'">4,;A&M₂N&674-$DE>"&?7.0?&,3,C,/5$&-56>G"-50>A -/,H6&C6+,3$N&C"&?0C4";0&(K + L) ! L<

P#."+&T8@W&C6J/"&-7;6?"+A4(&+6&76-5">4&;,#$;,/>0D/,D&T82W<&'">4,;A,&674-$DE>,&;FJ/4>,&C4L+A0& 36H";05C"C4&?&6#;,-4,&t 6;"A&-A6#4&?&6#;,-4,&t A6-5"/E&7;"?6-5;6//4,&7;A,C/6J6/,&7;A,A&6+?;65 -/6S(&C">4,;A0&?-7F.>A0//4#F?&+3"&;FJ/4>&36H";05CF?&?&6#;,-4,&t + 1<&X&;,A$35">4,&7;A,#-A5".>6 -/0&$#."+&%L+A4,&D,+/6A/">A/4,&?0A/">A".&;FJ/4>,&C4L+A0&36H";05C"C4&AC4,//0>G&C6+,3$1 + + + + + t L t t K t t L t t K t t L t

M

y

y

x

x

M

y

E

y

E

x

x

, , 1 4 , , 1 , , 1 3 1 , 1 , 1 1 , 1 , 1

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

=

η

η

&& & & & & & & & & & & &&&&&T82W

X&+"3-A,D&#63,D/6S>4&?0#6;A0-5"/6&;6A#."+&C">4,;A0&?AH3L+,C&?";56S>4&?."-/0>G<&m"5?6&A" -$?"J0(N&J,&C">4,;A&&M₃ C6J,&A6-5"(&A6;56H6/"34A6?"/"1& –1 3=

P

P

M

Δ

&& & & & & & & & & & & &&&&T8iW '">4,;A&P -5?6;A6/"&A6-5"."&A&?,#56;F?&?."-/0>G&C">4,;A0&M₃?&5,/&-76-F%N&J,&jc50&?,#56;& ?."-/0&C">4,;A0&M₃ D,-5&jc5E&#63$C/E&C">4,;A0&P<&=&#63,4&!"D,-5&C">4,;AE&+4"H6/"3/E<&\3,C,/5"C4& /"&7;A,#E5/,D&-E&?";56S>4&?."-/,&C">4,;A0 M₃1 ₌ +L K

0

0

0

0

0

0

1

λ

Δ

λ

T8eW

^A/">A"DE>&C">4,;A&6+?;65/E&+6&C">4,;A0&P D"#6&pN&>A034&#$ %$ &'-1N&;6A?"J"/0&$#."+&C6J/"& 674-"(&/"-5L7$DE>61

Q

₌

P

–1

M

_{4 =}

pM

₄& & & & &&&&&T]9W R;A0DL56N&J,&74,;?-A,&L T34>A%"&AC4,//0>G&-5"/$W&?";56S>4&?."-/0>G&D,-5&-5"%43/,N&>A034&C6+$.& 50>G&?";56S>4&%,A?AH3L+/0>G&D,-5&C/4,D-A0&6+&D,+/6S>41

∀

i∈{1,..,L}

λ

i <

1

& & & & & &&&&&T]:W

*"-5L7/4,& A+,I4/46?"/6& ?,#56;& 76C6>/4>A0&f_tN& %L+E>0& 436>A0/,C& C">4,;A0&p 6;"A& ?,#56;"& AC4,//0>G&?&C6C,/>4,&t1