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ff_l7&7$Fd[rFh\\nF_g_l7$FajqF^`p7$FajqFh\\n7&F\\g_lF\\g_lFag_lF_g_l7&F
_g_lF_g_lF`g_lFie_l7&Fie_lFie_lFje_lF`g_l7&7$Fd[rFatFfg_l7$FajqF^s7$Fd
[rF^s7&Ffg_lFfg_lFgg_l7$FajqFat7&F\\g_lF\\g_lFag_lF[g_l7&Fhg_lFhg_lFif
_lFgg_l7&Fhg_lFhg_lFif_lFef_l7&7$FajqF_x7$Fd[rF(F`[uF][u7&F`h_lF`h_lFd
[uF`[u7&F`h_lF`h_lF_h_l7$Fd[rF_x7&Fch_lFch_lFjg_lFfg_l7&Fch_lFch_lFjg_
lF_h_l7&F]f_lF]f_l7$F[[pFd[r7$F[[pFajq7&Fje_lFje_lFhh_lF]f_l7&F]f_lF]f
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g_lF]i_lFfe_l7&F`g_lF`g_l7$F[[pF[[pFje_l7&F`g_lF`g_lF`i_lF]i_l7&Fje_lF
je_lFhh_lF`i_l7&Ffe_lFfe_l7$F^`pF^`pF]i_l7&Ffe_lFfe_lFdi_lFce_l7&F]i_l
F]i_l7$F^`pF[[pF`i_l7&F]i_lF]i_lFgi_lFdi_l7&F`i_lF`i_l7$F^`pFajqFgi_l7
&Fhh_lFhh_l7$F^`pFd[rFji_l7&F`i_lF`i_lFji_lFhh_l7&F_h_lF_h_lF][uF[\\u7
&Fhh_lFhh_lF\\j_lFgh_l7&F_e^lF_h_lF[\\uFh[u7&F_h_lF_h_lF_e^lFjg_l7&Fjg
_lFjg_l7$F[[pF^sF_e^l7&Fji_lFji_lFce^lFae^l7&Fji_lFji_lFce^lF\\j_l7&F_
e^lF_e^lFh[uFc\\u7&F_e^lF_e^lFdd^lFcj_l7&Fcj_lFcj_l7$F^`pF\\glFdd^l7&F
cj_lFcj_lFij_l7$F[[pF\\gl7&F[[`lF[[`l7$F^`pFhcl7$F[[pFhcl7&F^[`lF^[`lF
be_lF][`l7&F^[`lF^[`lFbe_lFee_l7&Fce_lFce_lF[e^lFbf^l7&Fce_lFce_lF[e^l
Fdi_l7&Fdi_lFdi_lFgd^lF[e^l7&Fdi_lFdi_lFgd^lFgi_l7&Fgi_lFgi_lFae^lFji_
l7&Fgi_lFgi_lFae^lFgd^l7&F[[`lF[[`lF][`lFij_l7&Fbe_lFbe_lFbf^lF_f^l7&F
][`lF][`lF_f^lFhe^l7&Fij_lFij_lFhe^lF][`l7&F][`lF][`lF_f^lFbe_l7&Fij_l
Fij_lFhe^lFcd^l7&Fdd^lFdd^lFcd^lFij_l7&Fgg_lFgg_lF[[`lFif_l7&Fgg_lFgg_
lF[[`lFcj_l7&Fif_lFif_lF^[`lFff_l7&Fif_lFif_lF^[`lF[[`l7&Fff_lFff_lFee
_lF^[`l7&Fff_lFff_lFee_lF[g_l7&F[g_lF[g_lFfe_lFee_l7&Fjg_lFjg_lFcj_lFg
g_l-F[av6&F]avFd[r$\"*++++\"!\")Fd[rFaav-F$6U7$F^b\\lF`b\\l7$F^b\\lF^b
\\l7$Fb\\zF^b\\l7$Fb\\zFb\\z7$Fg]zFb\\z7$Fg]zFg]z7$F]dxFg]z7$F]dxF]dx7
$FcexF]dx7$FcexFcex7$Fh_wFcex7$Fh_wFh_w7$FacwFh_w7$FacwFacw7$F]\\wFacw
7$F]\\wF]\\w7$F`bwF]\\w7$F`bwF`bw7$F]ewF`bw7$F]ewF]ew7$F]_wF]ew7$F]_wF
]_w7$Fc]wF]_w7$Fc]wFc]w7$Fa\\wFc]w7$Fa\\wFa\\w7$Fd\\yFa\\w7$Fd\\yFd\\y
7$Fa]yFd\\y7$Fa]yFa]y7$F\\fyFa]y7$F\\fyF\\fy7$FgiyF\\fy7$FgiyFgiy7$Fbb
[lFgiy7$Fbb[lFbb[l7$F]e[lFbb[l7$F]e[lF]e[l7$Ffg[lF]e[l7$Ffg[lFfg[l7$Fi
j\\lFfg[l7$Fij\\lFij\\l7$F]\\^lFij\\l7$F]\\^lF]\\^l7$F\\b_lF]\\^l7$F\\
b_lF\\b_l7$Fja_lF\\b_l7$Fja_lFja_l7$Fgd_lFja_l7$Fgd_lFgd_l-F[av6&F]av
\"\"\"F*F*-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;Fd[
rFffnF^a`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "C
urve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 1
4" "Curve 15" "Curve 16" "Curve 17" "Curve 18" }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 71 "#observe that the absolute max of P lies outsi
de the constrained region" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
92 "#note that at the max level curves of the objective function and t
he constraint are parallel" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "Pl:=P-lambda*(x+y-10000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Pl
G,.*&,(\"$R$\"\"\"*&$F)!\"#F)%\"xGF)!\"\"*&$\"\"$!\"$F)%\"yGF)F.F)F-F)
F)*&,(\"$*RF)*&$\"\"%F2F)F-F)F.*&$F)F,F)F3F)F.F)F3F)F)\"'++SF.*&\"$&>F
)F-F)F.*&\"$D#F)F3F)F.*&%'lambdaGF),(F-F)F3F)\"&++\"F.F)F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dPldx:=diff(Pl,x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&dPldxG,**&$\"\"#!\"#\"\"\"%\"xGF*!\"\"$\"$W\"\"
\"!F**&$\"\"(!\"$F*%\"yGF*F,%'lambdaGF," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "dPldy:=diff(Pl,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%&dPldyG,**&$\"\"(!\"$\"\"\"%\"xGF*!\"\"*&$\"\"#!\"#F*%\"yGF*F,$\"$u
\"\"\"!F*%'lambdaGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dPl
dl:=diff(Pl,lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dPldlG,(%\"
xG!\"\"%\"yGF'\"&++\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "sols:=solve(\{dPldx,dPldy,dPldl\},\{x,y,lambda\});" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%solsG<%/%\"yG$\"+ah%Q:'!\"'/%\"xG$\"+YQ:YQF*/%'
lambdaG$\"#C\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "assign
(sols);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "P;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"*#p2B`!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "unassign('x');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "unassign('y');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "u
nassign('lambda');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#star
t over with a new constraint x=5000" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "#we need to solve: max P, subject to y<=8000, y<=1000
0-5000=5000 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "#this impli
es new constraint y<=5000" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "#form new function P1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "assign(x=5000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "P1:=
P-lambda*(y-5000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G,*$\"(+++(
!\"#\"\"\"*&$\"'++C!\"$F)%\"yGF)!\"\"*&,&$\"'+!z$F-F)*&$F)F(F)F.F)F/F)
F.F)F)*&%'lambdaGF),&F.F)\"%+]F/F)F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "dP1dy:=diff(P1,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%&dP1dyG,($\"'+!R\"!\"$\"\"\"*&$\"\"#!\"#F)%\"yGF)!\"\"%'lambdaGF/" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dP1dl:=diff(P1,lambda);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dP1dlG,&%\"yG!\"\"\"%+]\"\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sols1:=solve(\{dP1dy,dP1dl
\},\{y,lambda\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sols1G<$/%'lam
bdaG$\"#R\"\"!/%\"yG$\"%+]F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "assign(sols1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "P;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*+++:&!\"$" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 70 "#we get a smaller value than before; other cas
e can be done similarly " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"unassign('x');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign
('y');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "unassign('lambda'
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "#Sensitivity with res
pect to manufacturing capacity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 69 "#Introduce new parameter and new form of the constraint: g(x,y
)=x+y=c" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "#Set up constrai
ned problem" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Plc:=P-lambda*(x+y-c);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$PlcG,.*&,(\"$R$\"\"\"*&$F)!\"#F)%\"xGF)!\"\"
*&$\"\"$!\"$F)%\"yGF)F.F)F-F)F)*&,(\"$*RF)*&$\"\"%F2F)F-F)F.*&$F)F,F)F
3F)F.F)F3F)F)\"'++SF.*&\"$&>F)F-F)F.*&\"$D#F)F3F)F.*&%'lambdaGF),(F-F)
F3F)%\"cGF.F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "#Find pa
rtials with respect to x,y,lambda respectively" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "dPlcdx:=diff(Plc,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'dPlcdxG,**&$\"\"#!\"#\"\"\"%\"xGF*!\"\"$\"$W\"\"\"!F
**&$\"\"(!\"$F*%\"yGF*F,%'lambdaGF," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "dPlcdy:=diff(Plc,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%'dPlcdyG,**&$\"\"(!\"$\"\"\"%\"xGF*!\"\"*&$\"\"#!\"#F*%\"yGF*F,$
\"$u\"\"\"!F*%'lambdaGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 
"dPlcdl:=diff(Plc,lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dPlcd
lG,(%\"xG!\"\"%\"yGF'%\"cG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "solsc:=solve(\{dPlcdx,dPlcdy,dPlcdl\},\{x,y,lambda\})
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&solscG<%/%'lambdaG,&$\"$f\"\"
\"!\"\"\"*&$\"++++]8!#6F,%\"cGF,!\"\"/%\"yG,&*&$\"+++++]!#5F,F1F,F,$\"
+ah%Q:\"!\"'F,/%\"xG,&*&F7F,F1F,F,$\"+ah%Q:\"F<F2" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "dPdx:=diff(P,x);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "dPdy:=diff(P,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%%dPdxG,(*&$\"\"#!\"#\"\"\"%\"xGF*!\"\"$\"$W\"\"\"!F**&$\"\"(!\"$F*%\"
yGF*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dPdyG,(*&$\"\"(!\"$\"\"\"
%\"xGF*!\"\"*&$\"\"#!\"#F*%\"yGF*F,$\"$u\"\"\"!F*" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 14 "assign(solsc);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "dxdc:=diff(x,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%%dxdcG$\"+++++]!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dydc
:=diff(y,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dydcG$\"+++++]!#5" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "SPc:=dPdx*dxdc+dPdy*dydc;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SPcG,&*&$\"++++]8!#6\"\"\"%\"cG
F*!\"\"$\"++++!f\"!\"(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "
P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&,&$\"+I#p2Z$!\"(\"\"\"*&$\"+
++++l!#7F)%\"cGF)!\"\"F),&*&$\"+++++]!#5F)F.F)F)$\"+ah%Q:\"!\"'F/F)F)*
&,&$\"+J#p2#RF(F)*&$\"+++++qF-F)F.F)F/F),&*&F2F)F.F)F)$\"+ah%Q:\"F7F)F
)F)$\"+YQ:YV!\"%F/*&$\"+++++@F(F)F.F)F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "assign(c=10000);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*#p2B`!\"$" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "SPc;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"*+++S#!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 92 "#lambda can be interpreted as the shadow (dual) price of a resou
rce, in this case production" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "#capacity x+y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "#equ
ality Spc=lambda is a general fact " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "43 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }