Utility maximization. Conditions for utility maximization. Consumer theory: Utility maximization and expenditure minimization

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1 Consmer theory: Utlty mmzton nd ependtre mnmzton Lectres n Mcroeconomc Theory Fll 006 Prt GB Ashem ECON #7 Utlty mmzton Assme prce-tng ehvor n good mrets m p Bdget set : { X p m} where m s n ncome nd p p p ) re good prces m p Indrect tlty fncton : v m ) sch tht p m 0006 GB Ashem ECON #7 Condtons for tlty mmzton Ind tlty fn : v m ) sch tht p m Frst order condtons : ) λp 0 for ) ) p p 0006 GB Ashem ECON #7 3

2 Utlty fncton : A Ind tlty fn : m A st p + p m λp p p + p m λ p p m p m m p + p m p p m + p m p p m A + p p + Mrshlln demnd fncton m ) rg m ) st p m Propertes: Comp sttcs) tp t 0 for?? for?? Ind tlty fn n the Co-Dogls cse Utlty fncton : ) A Ind tlty fn : p m A st p + p m λp λp p + p m p p m p p m p + p m p p + p m p p A + ) p p 0006 GB Ashem ECON #7 4 Indrect tlty fn v m ) Propertes: sch tht p m ) Non - ncresng n non - decresng n m ) Homogeneos of degree 0 n t t for ll t 0 : p 3) Qsconve n p; tht s { p v } s conve for ll 4) Contnos n for postve p prces nd ncome Ind tlty fn n the Co-Dogls cse ) ) ) ) ) 0006 GB Ashem ECON #7 5 Roy s Identty ) m provded the rght-hnd sde s well-defned 0006 GB Ashem ECON #7 6

3 : Ependtre mnmzton Ependtre fn : mn p sch tht ) Condtons for ependtre mnmzton Frst order condtons : ) p λ 0 for ) p p ) ) 0006 GB Ashem ECON #7 7 Hcsn demnd fncton h rg mn p st ) Propertes: Comp sttcs) h tp h h 0 for h h for Ependtre fn n the Co-Dogls cse Utlty fncton ) A Epend fn : mn p + p st A p λ p λ A p p p + h p p A p ) A p + h p p A p ) e p p A + ) + ) p p ) ) 0006 GB Ashem ECON #7 8 Wht hppens when good prce ncreses? If p p ) 0 ) 0 p for ll then p ) ) 0 p > mples ) ) ) 0006 GB Ashem ECON #7 9 3

4 Ependtre fn: Proof of property ) Assme p for ll goods Let e ependtre mnmzng t so tht p nd ) Let e ependtr e mnmzng t so tht ) nd ) By ependtr e mnmzt on : p Snce p for ll goods p p Hence : Ependtre fn: Proof of property ) Let e ependtre mnmzng t so tht p nd ) However f e p p p then tp tp Hence s ependtre - mnmzng t t Ependtre fn: Proof of property 3) Let e ependtr e mnmzng t Let e ependtre mnmzng t Let e ependtre mnmzng t where tp + t) By ependtre mnmzton : tp tp t t) t) t) Addng these two neqltes together p t + t) Sppose s Defne the fncton g p ) p Then g p ) 0 nd g p ) p ependtre mnmzng good vector 0 for t p Utlty fncton : A Epend fn : mn p + p st A p λ p A p p + p h p p A A p p h p p A p e p p A p p Ependtre fncton Propertes: ) Non - decresng n p ) Homogeneos of 3) Concve n p degree n p : t t for ll t 0 p p p p mn p sch tht ) Slope : Ependtre fn n the Co-Dogls cse Utlty fncton : ) A Epend fn : p mn p + p st A p λ p λ A p p p + h p p A p ) A p + h p p A p ) e p p A + ) ) ) + p p ) h p p p p 4) Contnos n p p 0006 GB Ashem ECON #7 0 Shephrd s Lemm Proof of Shephrd s Lemm p p Slope : h p p p p p p ph Assme dfferentlty h h 0 Invoe concvty p Ependtre fn n the Co-Dogls cse ) λ ) ) ) ) ) ) 0006 GB Ashem ECON #7 More on Shephrd s Lemm Assme dfferentlty Slope : h p p p p p p h w y) Invoe Yong s theorem p h h ) 0006 GB Ashem ECON #7 4

5 For mportnt denttes If tlty mmzton s eqvlent to ependtre ) m mnmzton The mnmm ependtre necessry to rech tlty s m ) Is tlty mmzton eqvlent to ependtre mnmzton? Utlty mmzton mples ependtre mnmzton f The mmm tlty from ncome locl non-stton s Ependtre mnmzton mples tlty mmzton f preferences re contnos h ) The Mrshlln demnd t ncome m s the sme s the Hcsn demnd t tlty h ) The Hcsn demnd t tlty s the sme s the Mrshlln demnd t ncome 0006 GB Ashem ECON #7 3 Proof of Roy s Identty Sppose tht mml tlty p m ) yelds : p m ) h p ) m We lso hve tht )) m ) m ) ) 0 + m m ) ) p m ) h p ) m ) m 0006 GB Ashem ECON #7 4 Ependtre fn: Proof of property ) Assme for ll fctors Let p e ependtre mnmzng t so tht p nd ) Let e ependtre mnmzng t so tht nd ) By ependtre mnmzton : p Snce p for ll goods p p Hence : 0006 GB Ashem ECON #7 5 5

6 Ependtre fn: Proof of property ) Let e ependtre mnmzng t so tht p nd ) However f e p p p then tp tp Hence s ependtre - mnmzng t t 0006 GB Ashem ECON #7 6 Ependtre fn: Proof of property 3) Let e ependtre mnmzng t Let e ependtre mnmzng t Let e ependtre mnmzng By ependtre mnmzton : tp tp t t) t) t) Addng these two neqltes together p t + t) t p where tp + t) 0006 GB Ashem ECON #7 7 Proof of Shephrd s Lemm Sppose Defne the fncton s ependtre mnmzng g p ) p good vector t p Then g p ) 0 nd g p ) p 0 for 0006 GB Ashem ECON #7 8 6