Advanced microeconomics lecture 7: consumption theory IV
Advanced Microeconomics lecture 7:consumption theory IV
Aggregation and Welfare · Content Consumers surplus gregation 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(po, w), u'=vp, w),e(po,u)=e(p, u)=w E(p",p,w)=e(p,l)-e(p,u)=e(p,)- 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 E(p,p,w)=e(p,)- e(p,u-elp h(P1,p1,u地d CV(p,P, w)=h(P, P-, u) Av(P, P, w)=x(Pi, P-1, u dp
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 h1(P12p1,2 h1(P1,p1) E∨ PI CV PI x, (p, p. x(p,) x(p,)
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: pi=PI +t, T=tx(p, w) The loss of the taxation 1.L=-EV(p,p,w)-7=e(p,u)-e(p,u)-7 =。[(P,p12)+(3+,p1,) 2.=-CW(p,p,y)-T=e(p,l")-e(p,l2)-7 ∫(P1p,n2)-h(P+1p,n2
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 P1 h,(P,P-I,u) h,(pi,p_i,u) Is here a mistake? +t x(P12p1,w) h,(pi+t,p_, 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− 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(P) Wealth distribution w=(w,wn) And total wealth 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 Gormans (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 = +