Advanced microeconomics lecture 7: consumption theory IV
Advanced Microeconomics lecture 7:consumption theory IV
Aggregation and Welfare Content Consumer's surplus Aggregation Representation consumer
Aggregation and Welfare • Content: – Consumer’s surplus – Aggregation – Representation consumer
1. Consumer's surplus Money metric utility function e(p, u)measure the welfare of consumer in economy changing Definition: equivalent variation(En and compensating Variation( CV): see the fig Letu=v(p,w),u=v(p, w),e(po,u)=e(p, u)=w EV(p,p,w)=e(p,u-e(p,u=e(p, u)-w Cv(p,,p,w)=e(p,u-e(p, u)=w-e(p,u)
1.Consumer’s surplus • Money metric utility function measure the welfare of consumer in economy changing. • Definition: equivalent variation (EV) and compensating Variation (CV): see the fig. Let e u ( , ) p 0 1 0 1 0 0 0 1 EV w e u e u e u w ( , , ) ( , ) ( , ) ( , ) p p p p p = − = − 0 1 1 1 1 0 1 0 CV w e u e u w e u ( , , ) ( , ) ( , ) ( , ) p p p p p = − = − 0 0 1 1 0 0 1 1 u v w u v w e u e u w = = = = ( , ), ( , ), ( , ) ( , ) p p p p
1.Consumer's surplus Consider a changing only accurse in price of commodity 1 ev(p,p,w)=e(p,u)-w (p,)-e(p h,(pu, p_,u ) dp (p,p,w)=h,(pi AV(P,P,1)=x(2p1,)
1.Consumer’s surplus • Consider a changing only accurse in price of commodity 1. 0 1 1 1 0 1 0 1 0 1 1 1 1 1 1 1 1 ( , , ) ( , ) ( , ) ( , ) ( , , ) p p EV w e u w e u e u h p u dp − = − = − = p p p p p p 0 1 1 1 0 1 0 1 1 1 1 ( , , ) ( , , ) p p CV w h p u dp p p p = − 0 1 1 1 0 1 0 1 1 1 1 ( , , ) ( , , ) p p AV w x p u dp p p p = −
1.Consumer's surplus pI h2(P12P12°) h1(P1,p1,) EV P1 A x,(pi, p x(P,1)
EV AV 1.Consumer’s surplus p1 x1 0 1 1 1 h p u ( , , ) p− 1 1 1 x p w ( , , ) p− 1 1 1 1 h p u ( , , ) p− 1 1 x p w ( , ) 0 1 x p w ( , ) 1 1 p 0 1 p cV
1.Consumer's surplus Commodity taxation p=pI+t, T=tx,(p, w) The loss of the taxation 1.D=-E(p,p2W)-7=e(p,)-e(p,)-T P1 =(P,n)(p+1p,n) 2. L2=-Cv(po,p, w)-T=e(p,u)-e(po, u)-T =[(P1p2n)(P+p,n Pi +t
1.Consumer’s surplus • Commodity taxation: • The loss of the taxation: – 1. – 2. 1 0 1 1 1 1 p p t T t x p w = + = , ( , ) 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 ( , , ) ( , ) ( , ) [ ( , , ) ( , , )] p t p L EV w T e u e u T h p u h p t u dp + − − = − − = − − = − + p p p p p p 0 1 0 1 2 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 ( , , ) ( , ) ( , ) [ ( , , ) ( , , )] p t p L CV w T e u e u T h p u h p t u dp + − − = − − = − − = − + p p p p p p
1.Consumer's surplus Ph(P1,P1,) h1(P12p12) Is here a mistake? +t x1(1,p12) h(p8+1,p1,l)h(p?+t,p1,l)
1.Consumer’s surplus p1 x1 0 1 1 1 h p u ( , , ) p− 1 1 1 x p w ( , , ) p− 1 1 1 1 h p u ( , , ) p− 0 1 p 0 1 p t + 0 1 1 1 1 h p t u ( , , ) + p− 0 0 1 1 1 h p t u ( , , ) + p− Is here a mistake?
1.Consumer's surplus
1.Consumer’s surplus x2 x(p 1 ,w) x(p 0 ,w) u 1 u 0 L 1 x1 L 2
2. Aggregation across consumers Can a society behave like a ration man? If a society behave like a ration man, can the utility of this"man" represented the welfare of society? Aggregate demand:x(p…m)=∑x() Wealth distribution w=(w,.w,) And total wealth w=>w
2.Aggregation across consumers • Can a society behave like a ration man? • If a society behave like a ration man, can the utility of this “man” represented the welfare of society? • Aggregate demand: • Wealth distribution • And total wealth 1 1 ( , ) ( , ) I I i i i w w x w = x p p = 1 ( ) w = w wI 1 I i i w w = =
2. Aggregation across consumers Step1: x(p, w, w,)=x(p, w), that means aggregate demand are independent on the distribution of the wealth Proposition 1: if and only if each consumer has a indirect utility function of Gorman's v(P)=a1()+b(p) aggregate demands are independent on the distribution of the wealth
2.Aggregation across consumers • Step1: , that means aggregate demand are independent on the distribution of the wealth. • Proposition1:if and only if each consumer has a indirect utility function of Gorman’s aggregate demands are independent on the distribution of the wealth. 1 ( , ) ( , ) x p x p w w w I = ( , ) ( ) ( ) i i i v w a p b p w p = +
