Apped III APPENDIX III THE ENVELOPE PROPERTY Optmzato mposes a very strog structure o the problem cosdered Ths s the reaso why eoclasscal ecoomcs whch assumes optmzg behavour has bee the most successful of socal sceces Oe of ts mportat aspects s the evelope property dscussed ths Apped The evelope property s cocered wth the rate of chage of the mamum (or mmum) value of a crtero fucto caused by a chage some parameter; for eample a chage the mamum utlty level of a household caused by a chage come a chage the mmum cost of producto caused by a chage the output level ad so o A chage a parameter geeral duces a chage the optmum levels of choce varables Accordg to the evelope property however the duced chage the choce varables may be gored calculatg the effect of a chage a parameter o the mamum value f the chage s very small I other words a chage the mamum value caused by a margal chage a parameter whch also duces a chage the choce varables s equal to a chage crtero fucto wth choce varables fed I secto the evelope property s eplaed the smplest possble case The Evelope Theorem s stated ad proved secto 2 I secto 3 propertes of the drect utlty fucto ad the epedture fucto are derved as applcatos of the Evelope Theorem The Smplest Case The essece of the evelope property may be eplaed usg the followg smple mamzato problem Cosder the problem of mamzg the crtero fucto V ( wth respect to for a gve parameter b A teror mamum s obtaed at the pot where the dervatve of the crtero fucto wth respect to s zero V ( = 0 The mamzg value of chages as the parameter b chages: from to ' as b moves to b' The evelope property states that the total effect of a ftesmal chage the parameter o the mamzed value of the crtero fucto (cludg the effect of a duced chage the optmum value of ) equals the partal effect o the crtero fucto wth the level of fed I Fgure the former s the movemet from V to V ; ad the latter from V to V ~ Sce the crtero fucto s 80
Apped III ~ appromately flat ear the optmum pot the dfferece betwee the two V V s very small compared wth V V ~ As the chage the parameter approaches zero the dfferece becomes eglgble ad the evelope property ca be voed The evelope property ca be derved by mechacally dfferetatg the crtero fucto at the mamum Sce the optmum value of depeds o b t ca be descrbed as a fucto ( of b The the total effect cludg a chage s dv ( ( V = db d V + db b ad the partal effect s V ( b The two are equal sce V / = 0 at the optmum Fgure 2 llustrates why ths property s called the evelope property The heavy curve represets the mamum value V ( = V ( ( of the crtero fucto correspodg to dfferet values of the parameter The lghter curves descrbe the value of the crtero acheved wth fed values (ad ) of as b s vared The values ad the slopes of the two types of curves V ( ( ad V ( are equal at the value of b for whch s optmal that s where = ( The two curves are taget at that pot ad V ( s below V ( everywhere else sce 8
Apped III V ( s the mamum: V ( > V ( f = ( Ths holds for ay ad the curve V ( s the evelope of the curves V ( Fgure 2 suggests aother way of provg the evelope property Sce V ( s mamum V ( V ( for ay b ad V ( = V ( f = ( Ths mples that V ( les below V ( everywhere ad the two cocde at the value of b for whch s optmal If the two curves are smooth ths s possble oly whe the two curves are taget at ths pot whch proves the evelope property: dv ( / db = V ( f = ( The evelope property appears may areas of ecoomcs Probably the most famous applcato s the relatoshp betwee the log-ru cost curve ad the short-ru cost curve The short-ru cost curve s obtaed whe oly a subset of factors are optmally chose ad the log-ru cost curve whe all factors are chose optmally I the short ru some factor puts are fed whereas the log ru they become varable ad ca be chose optmally Cost curves descrbe the mmzed cost as fuctos of the output The argumet the last proof of the evelope property ca be appled to show that the log-ru cost curve s a evelope of short-ru cost curves Aother mportat eample s cocered wth beefts of a publc good Cosder a household wth the utlty fucto u ( z h X ) where z s the composte cosumer good ad the umerare h s the lot sze ad X s the supply of a publc good For a gve cosumpto budle the margal beeft of the publc good s u( z h X )/ X Whe the cosumpto budle s optmally chose the mamum utlty level depeds o the come I the lad ret R ad the level of the publc good See Dt (976) 82
Apped III X ad t ca be descrbed by the drect utlty fucto v ( R I X ) For the optmum cosumpto budle the margal beeft of the publc good s v( R I X ) / X The evelope property mples that u( z h X )/ X = v( R I X )/ X f z ad h are optmal gve R I ad X Ths result s used Chapter III 2 The Evelope Theorem Cosder the problem of mamzg the crtero fucto f ( subect to the costrats ( = 0 = 2 K m wth respect to the vector = K ) for g ( 2 a fed vector of parameters b = b b K b ) Let ( be the optmal choce for ( 2 q ths problem The grated a certa regularty codto 2 there ests the vector of Lagrage multplers λ = λ λ K λ ) such that ( mamzes the Lagraga ( 2 m Φ( λ = f ( + λ g( (2) wthout ay costrat where g( = ( g( g2( K gm ( ) ad the dot betwee ad g ( deote the er product so that λ g( = λ h = g ( If f ( ad g ( are dfferetable wth respect to the optmal choce = ( satsfes the frst order ecessary codtos f ( / + λ ( / = 0 = 2 K (22) g The Evelope Theorem descrbes a relatoshp betwee the mamum value fucto f ( = f ( ( ad the Lagraga Φ ( λ 2 The codto s called the Jacoba codto ad requres that the Jacoba matr of frst order partal dervatves of costrat fuctos g g2 KKK gm g 2 g2 2 gm 2 g g2 g m be of full row ra m at the optmum I olear programmg whch deals wth the more geeral case whch cludes equalty costrats a smlar codto called the costrat qualfcato must be satsfed 83
Apped III The Evelope Theorem: Assume that f ( ad Φ ( λ are cotuously dfferetable b The at = ( f ( / b = φ ( λ / b = 2 K q (23) Proof Sce ( satsfes the costrat g ( ( = 0 for ay b we have ( g / )( ( ) + g = 0 = 2 K q = (24) By the defto of the mamum value fucto ad the frst order codto (22) we obta f ( = = = λ ( f / )( = ( g / )( ) + f ) + f (25) (24) ow yelds the desred result: f ( = λ g + f = φ ( λ QED 3 Applcatos: Propertes of Idrect Utlty Fucto ad the Epedture Fucto Cosder a cosumer wth a utlty fucto u () where s the cosumpto vector ( 2 ) The cosumer mamzes the utlty fucto subect to the budget costrat p = I (3) where p s the prce vector p p p p ) I the moey come ad ( 2 p = = p The Lagraga for ths mamzato problem s 84
Apped III Ψ = u( ) + δ [ I p ] (32) where s the Lagrage multpler assocated wth the budget costrat (3) The frst order codtos are u / = δ p = 2 K (33) The optmal cosumpto depeds o come ad prces ad ca be wrtte as Substtutg = to the utlty fucto u () yelds the mamum utlty level v u( ( I )) whch ca be acheved at the gve values of come ad prces v s called the drect utlty fucto The Evelope Theorem (23) may the be appled to eame the effect of a chage prces ad the come o the mamzed utlty level: v/ p = δ = 2 K (34) v/ I =δ (35) The latter equalty shows that the Lagrage multpler equals the margal cotrbuto to the mamum utlty level made by a crease come or the margal utlty of come The multpler s therefore terpreted as the shadow value of the moetary come utlty terms by If a dollar crease come s all spet o good the crease utlty s gve u / P Ths s equal to the margal utlty of come whch s obtaed whe the crease come ca be optmally dstrbuted amog all goods sce by (33) a margal crease epedtures creases the utlty by the same amout whchever good s purchased Thus u / P = v/ I = 2 K (34) has the followg terpretato If the prce of the -th good s rased by a dollar per ut ad cosumpto of the -th good s fed epedture o that good must crease by dollars ad epedture o other goods must decrease by the same amout The utlty level would therefore decle by tmes the margal utlty of come By (33) t does ot matter f substtuto occurs: at the optmum all goods have the same margal utlty per dollar epedture Combg (34) ad (35) yelds Roy's Idetty: = ( v( / p ) /( v( I )/ ˆ I ) = 2 K (36) 85
Apped III whch s derved Chapter I wthout usg the Evelope Theorem ˆ s the ucompesated (or Marshalla) demad fucto Ths result s qute useful: demad fuctos ca be obtaed smply by dfferetatg the drect utlty fucto Net cosder the problem of mmzg the epedture ecessary to acheve a gve utlty level I ths problem p s mmzed uder the costrat u ( ) = u (37) for a gve u The mmum epedture level s a fucto of prces ad the utlty level E whch s called the epedture fucto ad If λ s the Lagrage multpler the Lagraga s Φ = p + λ [ u u( )] (38) p = λ ( u / ) = 2 K (39) By the Evelope Theorem (23) λ = E / u (30) = E / p (3) The latter equato s usually called Shephard's Lemma ad gves the compesated demad fucto It ca be easly show that the epedture fucto s cocave as a fucto of prces for ay fed utlty level Let p ad p' be two arbtrary prce vectors ad ad ' be correspodg optmal cosumpto vectors The ad Cosder a ew prce vector the correspodg cosumpto vector ad E = p E( = ' p ˆ = tp + ( t) for a arbtrary t betwee 0 ad ad ˆ The followg equaltes hold: p p ˆ ˆ 86
Apped III Multplyg the frst equalty by t ad the secod by yelds t ad addg them E( tp+ ( t) = ( tp+ ( t) ) tp ˆ + ( t) = te( + ( t) E( Thus E s cocave wth respect to p If E s twce dfferetable the cocavty mples 2 2 E / p = / p 0 (32) Ths shows that prce crease for ay good does ot crease the ucompesated demad for that good e the ow substtuto effect s opostve Ths s used Equato (I20) of Chapter Now we derve the Slutsy equato descrbg the relatoshp betwee the ucompesated ad compesated demad fuctos For gve prces ad come utlty mamzato yelds the drect utlty fucto v ad the ucompesated demad fucto ˆ I ) = 2 K Cosder the epedture mmzato gve the mamum utlty level u = v Uless some prces are zero whch case some techcal dffculty appears the optmal choces cocde ad I = E The ucompesated demad fucto therefore satsfes = ˆ E( ) = 2 K (33) ' Dfferetato of ths equato wth respect to P yelds / p = ˆ / p = ˆ / p + [ ˆ / I ][ E( / p + [ ˆ / I] ] (34) where the last term results from substtutg accordg to (30) Ths s the Slutsy equato P u= cost = P I = cost + I (35) used dervg (V227) Compesated ad ucompesated demad fuctos satsfy aother relatoshp whch s also used dervg (V227) Followg a argumet smlar to that whch led to (33) we obta v( I )) = ˆ I ) = K 87
Apped III Tag a partal dervatve wth respect to I we obta [ ( / u][ v( / I] = ˆ / I = (36) Notes Dscussos ths Apped owe very much to Dt (976) The Evelope Theorem secto 2 was proved by Afrat (97) ad ca also be foud Taayama (974) Refereces Afrat SN (97) "Theory of Mama ad the Method of Lagrage " SIAM Joural of Appled Mathematcs 20 Dt AK (976) Optmzato Ecoomc Theory (Oford Uversty Press Oford) Taayama A (974) Mathematcal Ecoomcs (Dryde Press Hsdale IL) 88