Joh Rley Novembe ANSWERS O ODD NUMBERED EXERCISES IN CHAPER Seo Eese -: asvy (a) Se y ad y z follows fom asvy ha z Ehe z o z We suppose he lae ad seek a oado he z Se y follows by asvy ha z y Bu hs oads he al hypohess ha he osume sly pefes y o z (b) Yes s ue he poof s smla Eese -3: Suffe odo fo ove pefeees (a) We wsh o show ha fo ay o y, y all ove ombaos ae also pefeed ha s, f U( ) U( y) ad U( ) U( y) he U( ) U( y) Se u ( ) = fu ( ( )) s oave, ha s, u ( ) ( ) u ( ) + u ( ) f ( U( )) ( ) f( U( )) + f( U( )) Se f s easg, f ( U( )) f( U( y)) ad f ( U( )) f( U( y)) I follows ha f ( U( )) ( ) f( U( y)) + f( U( y)) = f( U( y)) Aga, se f s easg, U( ) U( y) (b) he poof s almos deal () No Suppose U( ) = a+ a he u ( ) = f( a ) Idffeee uves ae les Aswes o odd umbeed eeses
Joh Rley Novembe Eese -5: Sly Cove Pefeees (a) he logahm of U( ) s well defed ove + Defe u ( ) = l U ( ) = l( + ) + l( + ) Se eah em o he gh had sde s oave follows fom he pevous eese ha pefeees ae ove Ideed se l( + j ) s a sly oave fuo, u ( ) s sly oave ad so pefeees ae sly ove (b) Fo all > he logahm of U( ) s well defed ad sly oave he pefeees ae sly ove I follows ha fo all sly posve, ad ove ombao ( < < ), suh ha ouous follows ha fo all o-egave, U( ) U( ) hus pefeees ae quas-oave U( ) U( ), U( ) > U( ) Se U( ) s Noe ha f = (, a) ad = (, b), U( ) = U( ) = U( ) = so ha pefeees ae o sly ove Eese -7: Quas-lea pefeees (a) Se a lea fuo s oave follows ha f V( z) s oave he sum U( ) = y+ V( z) s oave ad hee quas-oave (b) Fo ay veos z, z, hoose y, y so ha U y z y V z y V z U y z (, ) = + ( ) = + ( ) = (, ) If U s quas-oave, he U( y, z ) U( y, z ) ad U( y, z ) U( y, z ) Mulply he fs equaly by ( ) ad he seod by U( ) = y + V( z ) ( )( y + V( z )) + ( y + V( z )) = y + ( ) V( z ) + V( z ) Subag y fom boh sdes follows ha V s oave Aswes o odd umbeed eeses
Joh Rley Novembe Cosume Choe wh ommodes Eese -: Cosume Choe () If U( ) = l( + ) + l( + ) he magal ae of subsuo If = + MRS(, ) = = U +, he I = ad p I p MRS( ) = + p p Reaagg hs epesso, p p I Gve he symmey of he uly fuo, f he osume oly osumes ommody he p p I, ha s p p+ I hus ase () oespods o he hd dagam Eese -3: Elasy Rules Fs we show ha he hee defos ae equvale θ l = Hee θ l = = E [, θ ] θ Also lθ = Hee θ l θ θ = = = E [, θ ] lθ θ Appealg o hese defos yeld he followg esuls l a (l a + l ) l () Eab [, θ ] = = = = E [, θ ] l bθ (l b+ l θ) lθ () E[ y, θ ] = θ l y = θ (l + l y) = θ l + θ l y = E[, θ] + E[ y, θ] l l (v) E[, ] = = = E[, θ ] θ lθ lθ Aswes o odd umbeed eeses 3
Joh Rley Novembe Eese -5: CES Pefeees ( > ) U = + Defe u = U he aoud a dffeee uve dffeee uve hough ( aa, ), u = + Hee o he + = a (a) If = hs beomes Noe ha as, 4a (b) MRS(, ) = / = ( ) / / / + = a / hus MRS(, ) alog he dffeee uve as 4a ad MRS(, ) alog he dffeee uve as () Aoud he dffeee uve hough ( aa, ), + = a heefoe a =, = a Noe ha l = ( ) l + l a Se he gh had sde eases wh, he eep s a easg fuo of (d) I he lm ( ) a hus he dffeee uve passes hough he pos ( a,), ( a, a ) ad (, a ) Se pefeees ae ove, hs mus be a le of slope - 3 Cosume Choe wh ommodes Eese 3-: he Epedue Fuo wh Cobb-Douglas pefeees Suppose a osume has a Cobb-Douglas uly fuo Aswes o odd umbeed eeses 4
Joh Rley Novembe α j = U j α α ( ) =, α = j= Gve a pe veo p ad a budge I, he osume hooses pus o solve V( p, I) = Ma{ U( ), p I} Se wll be useful wha follows, we oe ha lu = α l heefoe α lu = = ad so U αu = Se he magal uly of eah ommody eases whou boud as osumpo dele owads zeo, we kow ha he opmal osumpo of eah ommody s sly posve he Lagaga s L (, ) = U( ) + ( I p ) hus he FOC fo he opmal hoe of pus ae: αu( ) p = p = Reaagg, we oba α α = = = p p U( ) = Appealg o he Equal Rao Rule he follows mmedaely ha α α α = = = = = p p p I heefoe, α I =, =,, p Fally, subsug bak o he uly fuo, mamzed uly s α α α V( p, I) = ( ) j = I = = p Aswes o odd umbeed eeses 5
Joh Rley Novembe Se U = V( p, I) s he mamum uly ha a be aheved gve a budge I, heefoe follows ha I s he smalles budge ha wll yeld a uly of he soluo o he followg "dual" poblem M qu M p U U (, ) {, ( ) } z Iveg, we oba he epedue fuo p = j α (, ) U ( ) j j= α j M p U U ha s, I s Eese 33: Cosumg pas of ommodes Suppose ha he osume dedes o sped y o he fs pa ommodes ad z o he seod pa of ommodes he hoe of how muh o sped o ommody ad ommody s a sadad Cobb-Douglas uly mamzao poblem heefoe p = p = y Smlaly p = p = z Subsug bak o he uly fuo yelds he 3 3 4 4 followg deved uly fuo uyzp (,, ) = ( pp) y+ ( pp) z / / 3 4 Noe ha he magal uly of y ad z ae as follows u = ( pp) y / ad u = ( pp) z / 3 4 If hese ae equal, he osume s dffee as o he fao of hs ome he speds o he fs wo ommodes Howeve, f hey dffe, he mamzes uly by spedg oly o he pa of ommodes wh he hghe magal uly Eese 3-5: Compesaed pe elases he epedue fuo s M ( pu, ) = M{ p U( ) U} Coveg hs o a mamzao poblem, M ( pu, ) = Ma{ p U( ) U} Aswes o odd umbeed eeses 6
Joh Rley Novembe he Lagaga of hs mamzao poblem s L (, ) = p + ( U( ) U) Appealg o he Evelope heoem, M L = = p p Hee M M = = p p p p p j j j he epedue fuo s a oave fuo of p hus, o=fo ay pa of pes, he Hessa ma M p p j Mus be egave sem-defe heefoe M M M M p p p p p p p p j j j j j Appealg o ou eale esul, follows ha j p p p p j j j Coveg hs o elases yelds he desed esul Seo 5 Choe Ove me Eese 5-: Epedue Choe (a) Le { } =, =, be opmal wh epedue sequee { } = ha s U ( ) = U( ) Se p, p ( ) p p ( ) = + + = hus s feasble gve he epedue sequee { } Suppose = U ( ) U ( ) he U( ) U( ) ad by quas-oavy, U( ) U( ) Se s feasble, U ( ) U( ) U( ) Aswes o odd umbeed eeses 7
Joh Rley Novembe (b) Se he uly fuo s addvely sepaable, he hoe of how o alloae epedue peod s depede of osumpo ohe peods hus, peod he osume solved=s he followg poblem Ma{ α j l j p = } j= hs s a sadad Cobb-Douglas poblem Solvg, α = Subsug bak o j j p j he peod uly fuo, α u ( ) = l = l + ems depede of j α j j= pj ems depede of = = Hee U () = δ u ( ) = δ l + () Le W be he pese value of he edowme He lfe-me epedue osa s + + + W We fom he Lagaga + ( + ) FOC L = l + ( ) δ W = = ( + ) L = =, L δ = ( + ) = + + Elmag, ( ) + = + δ () Subsug bak o he lfe-me budge osa, ( δ δ ) W + + + = Usg he fomula fo a geome sum, δ = ( ) W δ Eese -3: Savg ad boowg wh a ouum of ypes: Eah osume has a wealh of ω + + + + ω + = ω( + ) = ω, Aswes o odd umbeed eeses 8
Joh Rley Novembe Gve hs logahm pefeees, a ype α osume has fs peod osumpo + ( α) = αω + + Hee hs savg s( α) = ω ( α) = ω( α) + (a) Solvg fo he ype wh zeo savg follows ha ype α s a save f + α < ˆ α + (b) he oal savg by hose who save s ˆ α ˆ α () ( α) α ω( α α ) + ω + S = s d = = + + () Fo all he ohe osumes, s( α ) < hus oal boowg s, ˆ α ˆ α ˆ α + + B () = s( α) dα = ω ( α) dα = ω( α α ) + + ω + ω = ( ) ) = + + ( + )( + ) B() S () ω 4 B, S Fg A--: Savg ad Boowg Aswes o odd umbeed eeses 9