Lecture 5
1 Lecture 5
A general lesson-aggregate outcomes do not always resemble individual outcomes. Let's start with a discrete commodity,of which only one unit can be consumed. Normalize this to be commodity 1. Proposition:Suppose that the preference relation≥on X={x∈{0,1}，(x2,xz)∈R1-} is rational,continuous and locally non- satiated on the commodities other than commodity 1
2 • A general lesson– aggregate outcomes do not always resemble individual outcomes. • Let’s start with a discrete commodity, of which only one unit can be consumed. Normalize this to be commodity 1. • Proposition: Suppose that the preference relation on is rational, continuous and locally nonsatiated on the commodities other than commodity 1, 1 X { {0,1}, ( ,... ) } 1 2 L L x x x − = +
。and if x={x∈B:x>y for all∈Bo} ·where B。={x∈X:x=0and ∑，P,x≤wW ·and x(p)={x∈B(p):x>y for all y∈B} ·where B,p)={x∈X:x=1and∑i2p,≤w-p
3 • and if for all • where and • and for all • where and * 0 x x B x y = { : y B 0 } 0 1 B x X x = = { : 0 2 } L i i i p x w = ** 1 0 1 x p x B p x y ( ) { ( ) : = 0 y B } 0 1 1 B p x X x ( ) { : 1 = = 2 1 } L i i i p x w p = −
·lfx*>(1,0,0..0)then there exists1∈[0，w] for all p>p,xx"(p)and x"x"(p) for all px (p), at the least xx"(0)and x'x"(w)
4 • If then there exists a for all and for all . If preferences are continuous then where . • Proof: It is obviously true that there exis some values of in which and some values for , at the least and . * x (1,0,0 0) 1 p w [0, ] * ** 1 1 1 p p x x p , ( ) * ** 1 x x p( ) 1 1 p p * ** 1 x x p( ) p w 1 0, * ** 1 x x p( ) * ** 1 x x p( ) * ** x x (0) * ** x x w( )
Furthermore,non-satiation tells us that x'>x"(p)implies that xx"(p)for all p>p,and that xx"(p)implies that x<x"(p)for all p<p. Hence,since the set of p for which is bounded above there exists an upper bound p. And since the set of p for which xx"(p) is bounded below there exists a greatest lower bound p,B·
5 • Furthermore, non-satiation tells us that implies that for all , and that implies that for all . • Hence, since the set of p1 for which is bounded above there exists an upper bound . • And since the set of p1 for which is bounded below there exists a greatest lower bound . * ** 1 x x p( ) * ** 1 x x p( ) p p 1 1 * ** 1 x x p( ) * ** 1 x x p( ) 1 1 p p 1,A p * ** 1 x x p( ) 1,B p
As it would be impossible to have x"(p)xx(p),it must be that Puspu.If pu>pthen there exists a continuum of values of p at which xzx"(p）and x'"(p,) i.e.for whichx-x(P)but local non- satiation rules that out,so it must be that p1.=p=p and by construction for all p>p,and xx"(p)and"(p) for all p<p
6 • As it would be impossible to have , it must be that . If then there exists a continuum of values of p1 at which and • i.e. for which but local nonsatiation rules that out, so it must be that and by construction for all , and and for all . ** * ** 1 1 x p x x p ( ) ( ) 1, 1, A B p p 1, 1, A B p p * ** 1 x x p( ) * ** 1 x x p( ) * ** 1 x x p( ) ppp 1, 1, 1 A B = = 1 1 p p * ** 1 x x p( ) p p 1 1 * ** 1 x x p( )
Continuity gives you the existence of a price where consumers are indifferent. Consider,the sequence and the sequence x where x"(p for all n,continuity implies that x(p)x. Likewise consider the sequences x"(+)and x,where x"(p+x for all n,so x(px which together imply that x"(p)-x
7 • Continuity gives you the existence of a price where consumers are indifferent. Consider, the sequence and the sequence , where for all n, continuity implies that . • Likewise consider the sequences and , where for all n, so which together imply that . ** 1 1 x p( ) n − * x ** * 1 1 x p x ( ) n − ** * 1 x p x ( ) ** 1 1 x p( ) n + * x ** * 1 1 x p x ( ) n + ** * 1 x p x ( ) ** * 1 x p x ( )
Thus each person has a price at which he is indifferent between consuming commodity one and not doing so. In this case,individual demand is always downward sloping-no income effects or cross-partials to worry about
8 • Thus each person has a price at which he is indifferent between consuming commodity one and not doing so. • In this case, individual demand is always downward sloping– no income effects or cross-partials to worry about
·Why? What would happen if we dropped continuity or local non-satiation?
9 • Why? • What would happen if we dropped continuity or local non-satiation?
Assume that there are J consumers,each with preferences and wealth w,,facing prices for all goods i>1,p,which generates a cutoff price for each consumer. ·Let D(P1,P,w)= the number of consumers s.t.P<p} This function is certainly weakly downward sloping-as p falls the set of consmers for which p<p must rise. 10
10 • Assume that there are J consumers, each with preferences and wealth , facing prices for all goods i>1 , , which generates a cutoff price for each consumer. • Let { the number of consumers s.t. } • This function is certainly weakly downward sloping– as p1 falls the set of consmers for which must rise. j wi p 0 1 i p 1 D p p w ( , , ) = 1 1 i p p 1 1 i p p