?dmf =`mb`h\iiNjgpodjin km`k\m`_ ]t Qdi^`io Kjcg?`k\moh`io ja @^jijhd^nT\g` Pidq`mndot@^jijhd^n ,-,]5 Dio`mh`_d\o` Hd^mj`^jijhd^nKmj]g`h N`o .5 ?`h\i_ Api^odjin-*.*,+B`i`m\g >jhh`ion5 Ocdn kmj]g`h n`o r\n b`i`m\ggt q`mt r`gg _ji`) Hjno nop_`ion r`m` \]g` ojnjgq` oc` kmj]g`hn ^jmm`^ogt pndib oc` orj h`ocj_n ^jq`m`_ di ^g\nn #np]nodopodji \i_ G\bm\ib`$)Orj ^jhhji hdno\f`n \m` gdno`_ ]`gjr5ȸ <nnphdib oc\o a + b =1di kmj]g`h ,) <gocjpbc oc` hjijoji` om\inajmh\odji f(u)=u1a+brjpg_ h\f` oc` nph ja oc` kjr`mn `lp\g oj ,' ocdn c\n oj ]` h\_` `skgd^do) Pndib ocdnhjijoji` om\inajmh\odji g`\_n oj oc` ^jmm`^o m`npgo' ]po oc` m`npgodib _`h\i_ api^odji \m`_dz`m`io da a + b =1dn ndhkgt \nnph`_)ȸ Oc` di_dz`m`i^` ^pmq`n di kmj]g`h - r`m` _m\ri \n ijo dio`mn`^odib oc` \s`n),) >cjd^` \i_ ?`h\i_#\$ >j]](?jpbg\n podgdot) Oc` jkodhdu\odji kmj]g`h dnh\sx,yxaybn)o) pxx + pyy = m) #,$R` ^jind_`m orj \kkmj\^c`n oj njgq` ajm oc` jkodh\g ^jinphkodji ]pi_g`) Admno' r`pn` oc` \kkmj\^c ]t np]nodopodji) Ndi^` oc` >j]](?jpbg\n podgdot api^odji m`km`n`ionhjijoji` km`a`m`i^`n' oc` ]p_b`o ^jinom\dio cjg_n rdoc `lp\gdot \i_ r` ^\i `skm`nn do\n ajggjrn5y =m − pxxpy) #-$Kgpbbdib ocdn `skm`nndji dioj oc` podgdot api^odji td`g_n oc` ajggjrdib pi^jinom\di`_ndibg`(q\md\]g` jkodhdu\odji kmj]g`h5h\sxxa!m − pxxpy"b)Oc` AJ> dnaxa−1!m − pxxpy"b− bxa!m − pxxpy"b−1pxpy=0',rcd^c ^\i ]` ndhkgd{`_ oja(m − pxx)=bpxx'rcd^c di opmi ^\i ]` njgq`_ ajm oc` jkodh\g g`q`g ja bjj_ xx∗(px,py,m)=am(a + b)px) #.$Kgpbbdib ocdn ]\^f dioj #-$ bdq`n pn oc` jkodh\g g`q`g ja bjj_ yy∗(px,py,m)=bm(a + b)py) #/$Oc` n`^ji_ h`ocj_ pn`n oc` G\bm\ib` \kkmj\^c) Di b`i`m\g' oc` G\bm\ibd\i ajm \ podgdoth\sdhdu\odji kmj]g`h dn _`{i`_ \nL(x, y, λ)=u(x, y)+λ(m − pxx − pyy))Di \ {mno no`k' r` ^\g^pg\o` oc` {mno(jm_`m ^ji_dodjin \i_ n`o oc`h `lp\g oj u`mj5∂L∂x=∂u∂x− λpx=0∂L∂x=∂u∂y− λpy=0∂L∂λ= m −pxx − pyy =0jm λ =0)Oc` g\no ^ji_dodji ^\i \gnj ]` rmdoo`i \n λ(m − pxx − pyy)=0' rcd^c dn fijri \n oc`^jhkg`h`io\mt ng\^fi`nn ^ji_dodji) Cjr`q`m' rc`i oc` ]p_b`o ^jinom\dio dn ]di_dib'd)`)' \gg di^jh` dn nk`io' r` c\q` oc\o m − pxx − pyy =0\i_ λ "=0) Oc` G\bm\ibd\i\nnj^d\o`_ rdoc kmj]g`h #,$ dnL(x, y, λ)=xayb+ λ(m − pxx − pyy))Oc` AJ>n ajm ocdn G\bm\ibd\i \m`∂L∂x= axa−1yb− λpxx =0 #0$∂L∂y= bxayb−1− λpyy =0 #1$∂L∂λ= m −pxx − pyy =0) #2$C`i^` r` c\q` ocm`` `lp\odjin di ocm`` pifijrin #x, y, λ$) Ijr' di jm_`m oj njgq` ajmoc` pifijrin' r` o\f` oc` m\odj ja AJ>n #0$ \i_ #1$' rcd^c td`g_n oc` a\hdgd\m ȳHMN 8kmd^` m\odjȴ ^ji_dodji5axa−1ybbxayb−1=λpxxλpyy⇔aybx=pxpy' #3$-rcd^c ^\i ]` njgq`_ ajm y =bpxapyx \i_ kgpbb`_ dioj AJ> #2$' ijodib oc\o oc` ]p_b`o^jinom\dio cjg_n rdoc `lp\gdot5pxx + pybpxapyx = m)Njgqdib ocdn `lp\odji ajm x td`g_n oc` n\h` jkodh\g _`h\i_ ajm bjj_ x \n ]`ajm` di`lp\odji #.$) Kgpbbdib oc` jkodh\g x∗]\^f dioj `lp\odji #3$ td`g_n oc` n\h` jkodh\g_`h\i_ ajm bjj_ y \n ]`ajm` di `lp\odji #/$)#]$ G`jiod`z podgdot) Oc` jkodhdu\odji kmj]g`h dnh\sx,yhdi {ax, by} n)o) pxx + pyy = m) #4$Dino`\_ ja pndib AJ>n #ndi^` oc` di_dz`m`i^` ^pmq`n \m` ijo _dz`m`iod\]g` \o oc` fdif$'ijo` oc\o \o oc` jkodh\g ^jinphkodji ]pi_g` r` c\q` oc` ^ji_dodjiax = by) #,+$Joc`mrdn` m`njpm^`n rjpg_ ]` r\no`_ rdocjpo kmjqd_dib \__dodji\g podgdot) Njgqdib ocdn^ji_dodji ajm y \i_ kgpbbdib dioj oc` ]p_b`o ^jinom\dio td`g_npxx + pyabx = m'rcd^c ^\i ]` njgq`_ ajm oc` jkodh\g x5x∗(px,py,m)=bmbpx+ apy)Kgpbbdib ocdn `skm`nndji dioj `lp\odji #,+$ bdq`n pn oc` jkodh\g g`q`g ja bjj_ y5y∗(px,py,m)=ambpx+ apy)-) =p_b`o \i_ ?`h\i_#\$ N`` Adbpm` , \i_ oc` _`mdq\odjin ]`gjr)d) Oc` bjq`mijmȱn podgdot api^odji dnu(x, y) = 2√x +2√y)Ajm \it bdq`i podgdot g`q`g ¯u' r` ^\i rmdo`2√x +2√y =¯u'rcd^c ^\i ]` `skm`nn`_ di o`mhn ja y \n \ api^odji ja x5y(x)=!¯u −2√x2"2) #,,$.educationother goodsindifference curvebudget lineAdbpm` ,5 Lp`nodji -#\$Rc`i r` n`o `doc`m x jm y `lp\g oj u`mj r` n`` oc\o oc` di_dz`m`i^` ^pmq`n ajm ocdnapi^odji dio`mn`^o oc` \s`n \o x = y =#¯u2$2) Ijo` oc\o ocdn _dz`mn amjh >j]](?jpbg\n podgdot) Di oc` km`n`io ^\n`' \ kjndodq` podgdot g`q`g ^\i ]` m`\gdu`_ rc`i`doc`m x jm y dn `lp\g oj u`mj) Oc` ngjk` ja oc` di_dz`m`i^` ^pmq`n ^\i ]` _`mdq`_ ]t{i_dib oc` _`mdq\odq` ja api^odji #,,$5dydx= −2!¯u −2√x2"12√x=1−¯u2√x< 0)Don n`^ji_ _`mdq\odq` dnd2ydx2=¯u4√x3> 0)C`i^` oc` di_dz`m`i^` ^pmq`n \m` _`^m`\ndib #i`b\odq` ngjk`$ \i_ ^jiq`s #kjndodq`n`^ji_ _`mdq\odq`$)dd) Oc` ]p_b`o ^jinom\dio dn2x + y = 100'nj oc\o oc` ]p_b`o gdi` ^\i ]` _`n^md]`_ ]t oc` ajggjrdib `lp\odjiy(x) = 100 − 2x)#]$ Oc` jkodhdu\odji kmj]g`h dnh\sx,y2√x +2√y n)o) 2x + y = 100)=`^\pn` oc` njgpodji dn dio`mdjm \i_ ^jiq`s' r` ^\i n`o HMN 8 ngjk` ja ]p_b`o ^jinom\diooj {i_ oc` jkodh\g ]pi_g`5MUxMUy=pxpy)/Pndib oc` api^odji\g ajmh ja oc` podgdot \i_ oc` kmd^`n' ocdn dn `lpdq\g`io oj1√x1√y=2'rcd^c ^\i ]` njgq`_ ajm y5y =4x #,-$\i_ kgpbb`_ dioj oc` ]p_b`o ^jinom\dio52x +4x = 100)C`i^` oc` jkodh\g g`q`g ja `_p^\odji dn x∗= 100/6 = 16.67 \i_ oc` jkodh\g g`q`g jajoc`m bjj_n dn y∗= 200/3 = 66.67)#^$ Oc` HMN \o (x, y) = (10, 80) dnMUxMUy=%yx=√8)Ocdn dn ijo \i jkodh\g ^jinphkodji ]pi_g` ]`^\pn` oc` HMN dn ijo `lp\g oj' ]po g\mb`moc\i oc` kmd^` m\odji' rcd^c dn `lp\g oj -) Oc` di_dz`m`i^` ^pmq` \o ocdn kjdio dn no``k`moc\i oc` ]p_b`o gdi`) <o ocdn kjdio' oc` bjq`mijm ^\i ncdao ^jinphkodji amjh joc`m bjj_noj `_p^\odji di jm_`m oj di^m`\n` podgdot) Ijo` oc\o r` ajpi_ oc` jkodh\gdot ^ji_dodjiy =4x di k\mo #]$' rcd^c dn ^g`\mgt qdjg\o`_ c`m`)#_$ Oc` ]p_b`o ^jinom\dio ]`^jh`n2x + y = 150ndi^` oc` bjq`mijm ^\i nk`i_ oc` iji(h\o^cdib bm\io ji \it bjj_)d) Oc` `z`^o ja oc` iji(h\o^cdib bm\io dn \ k\m\gg`g jpor\m_ ncdao ja oc` ]p_b`o gdi`)dd) Kgpbbdib jkodh\gdot ^ji_dodji #,-$ dioj oc` i`r ]p_b`o ^jinom\dio td`g_n2x +4y = 150'rcd^c r` ^\i njgq` ajm oc` jkodh\g ^jinphkodji ]pi_g`(x∗,y∗) = (25, 100))Oc` podgdot g`q`g amjh ocdn ^jinphkodji ]pi_g` dnu(25, 100) = 2√25 + 2√100 = 30)#`$ Di ocdn ^\n` oc` ]p_b`o ^jinom\dio ]`^jh`nx + y = 100]`^\pn` oc` kmd^` ja `_p^\odji `z`^odq`gt ]`^jh`n ,' ]po di^jh` m`h\din \o ,++)d) Oc` `z`^o ja oc` iji(h\o^cdib bm\io dn \i \iod(^gj^frdn` mjo\odji ja oc` ]p_b`o gdi`\mjpi_ don dio`mn`^odji rdoc oc` y(\sdn)0dd) @lp\odib HMN \i_ kmd^` m\odj td`g_nMUxMUy=%yx=1nj oc\o oc` jkodh\gdot ^ji_dodji dnx = y)Ocdn g`\_n oj oc` jkodh\g ^jinphkodji ]pi_g`(x∗,y∗) = (50, 50))Oc` podgdot g`q`g amjh ocdn ^jinphkodji ]pi_g` dnu(25, 100) = 2√50 + 2√50 =