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
1. WAand demand law Slutsky matrix(substitution matrix) s(p, w)=(SIK(P, w) Substitution effects SI(, w) ax, (p, w),ax, (p x,(p, w) s(p, w)Is n.s. d Giffen good is necessary inferior good S,,(p, w) ax,(p,w) ax, (p, w) x/(p,w)≤0
1.WA and demand law • Slutsky matrix (substitution matrix) • Substitution effects • is n.s.d • Giffen good is necessary inferior good. ( , ) ( ( , )) lk n n S w s w p p = ( , ) ( , ) ( , ) ( , ) l l lk k k x w x w s w x w p w = + p p p p S w ( , ) p ( , ) ( , ) ( , ) ( , ) 0 l l ll l l x w x w s w x w p w = + p p p p
2. From preferences to utility Definition: u: X>R is a utility function of preference % if x% yeu(x)2u(y),Vx,yeX Can we always find a utility function of Maybe If X is finite, there always exist utility function Proposition: only rational can be represented by a utility function (N c not s c) Lexicographic preference: rational but no utility function exist
2.From preferences to utility • Definition: is a utility function of preference , if • Can we always find a utility function of ? – Maybe – If X is finite, there always exist utility function. • Proposition1: only rational can be represented by a utility function. (N.C not S.C) • Lexicographic preference: rational but no utility function exist. u :X→ % x y u x u y x y % ( ) ( ), , X % %
2. From preferences to utility Continuity: V(r",yin=,x% y, and x=limx,y=limy thenx% y, or x upper contour sets x: y% x) and lower contour sets (yeX: x% y, are closure Proposition2: If is continuous, then exist a continuous utility function representing
2.From preferences to utility • Continuity: then ,or x upper contour sets and lower contour sets are closure. • Proposition2:If is continuous, then exist a continuous utility function representing . 1 { , } , ,and lim , lim n n n n n n n n n x y x y x x y y = → → = = % x y % {:} y y x X % {:} y x y X % % %
2. From preferences to utility Desirability: preference is desirable if is monotone: x,yEX, and v>>x.thenyx ify2x, then y>x, it's strongly monotone is local non-satiation: WxeX, and e >0 there is a y, that ly-xc, andyrx proposition: 9 is strong monotone, then it's monotone is monotone, it's local non-satiation
2.From preferences to utility • Desirability: preference is desirable if – is monotone: – is local non-satiation: there is a • proposition3: is strong monotone, then it’s monotone; is monotone, it’s local non-satiation. % % x y y x y x , ,and , then X if , then , it's strongly monotone y x y x % x X,and >0 y y x ,that y-x ,and % %
2. From preferences to utility Convexity: x upper contour sets are convex Decreasing in marginal rate of substitution people like variety Preferences are convex means utility function is quasi-concave
2.From preferences to utility • Convexity: x upper contour sets are convex. – Decreasing in marginal rate of substitution. – people like variety. • Preferences are convex means utility function is quasi-concave
