浙江大学：《高级微观经济学》课程PPT教学课件（英文版）lecture 7 Advanced microeconomics

Advanced microeconomics lecture 7 consumption theory V

Advanced Microeconomics lecture 7:consumption theory IV

Aggregation and Welfare Content Consumers 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. Let u=v(po, w),u'=v(p, w),e(po,u)=e(p,u)=w E(p,p,w)=e(p,)-e(p,)=e(p,) Cy(p,p, w)=e(p,u)-elp,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 e(p,u-e(p, u) h1(P1,p12u) CV(P,P,)=h(P2,F122)k AV(P,p,w)=x(2p1)如

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 h2(P12p12°) h1(P1,p1,) P1 x(n1p12) x(p,) x,(p, w)

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

1.Consumer's surplus Commodity taxation: pi=PI+t, T=tx,(p, w) The loss of the taxation 1.=-E(p,p,1)-T=e(p,l)-e(p,l)-T P1 t Je [h(P1, P-l,u)=(P0+t, p-1, u)ldp. 2.D2=-CW(p°,p,w)-T=e(p,l)-e(p,u°)-T [(np,2)-(P+p,n2)n

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.Consumers surplus P h1(P1,p1,a2) h1(P12p12) +t PI x(1,p1,w) h,(pi+t, p-l,u) h,(pi+t, p_,u)

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−

1.Consumers surplus L x(pI, w)

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""represented the welfare of society? Aggregate demand:x(pw…w)=∑x(p) Wealth distribution w=(w,.w,) And total wealth w=∑m i=1

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 Gormans v, (p, w)=a, (p)+b(p)w 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 = +