Advanced economics (lecture 5: consumption theory l) Ye Jianliang
Advanced Economics (lecture 5: consumption theory II) Ye Jianliang
CONTENT · Wa and demand law From preferences to utility Utility maximization Expenditure minimization
CONTENT • WA and demand law • From preferences to utility • Utility maximization • Expenditure minimization
1. WAand demand law Walrasian demand function x(p, w) satisfied WA if for any (p, w)and(p, w,)we have p'x(p,w)>w',ifp·x(p',v)≤ w and x(p,w)≠X(p,)
1.WA and demand law • Walrasian demand function x(p,w) satisfied WA if for any we have: See the fig. ( , ) and ( , ) p p w w p x p p x p x p x p ( , ) ,if ( , ) and ( , ) ( , ) w w w w w w
1. WA and demand law Changing in price will change wealth too But how can we tell the demand changing by price changing from wealth changing? Given a changing from(p, w)to(p, w,), and people will not get worse that is w'>px(p, w) here wealth changing(compensation △=△pX(P,1) was called“ Slutsky wealth compensation"and Ap=p-p Slutsky) compensated price changing
1.WA and demand law • Changing in price will change wealth too. But how can we tell the demand changing by price changing from wealth changing? • Given a changing from ,and people will not get worse that is here wealth changing (compensation ) was called “Slutsky wealth compensation” and “(Slutsky) compensated price changing”. ( , ) to ( , ) p p w w w w p x p( , ) = w w p x p( , ) = − p p p
1. WAand demand law Proposition: x(p, w) satisfied WA if and only if: (p-p).[(p, w)-x(p, w)]<0 and when X(p,w)*x(p, w),(p-p).(p,w)-x(p, w)<0 Prop5 indicates,Ap·Ax≤0,orφ dx <o thats cal|ed" demand law”,or“ compensation demand|aw
1.WA and demand law • Proposition5: x(p,w) satisfied WA if and only if: and when , • Prop.5 indicates, , or that’s called “demand law ”, or “compensation demand law ”. ( ) [ ( , ) ( , )] 0 p p x p x p − − w w x p x p ( , ) ( , ) w w ( ) [ ( , ) ( , )] 0 p p x p x p − − w w p x 0 d d p x 0