Lecture 3
Lecture 3
Back to preferences.We want four attributes: ·(1)completeness and ·(2)transitivity
• Back to preferences. We want four attributes: • (1) completeness and • (2) transitivity
Definition 3.B.2: The preference relation on X is monotone ifx,y X,and y>>x implies x.It is strongly monotone ify>x andy≠ximplies y>x. In many cases,we won't naturally have monotonicity,but then a little redefinition of variables does the trick (turn a bad into a good by multiplying by-1)
Definition 3.B.2: • The preference relation on X is monotone if ,and implies . It is strongly monotone if and implies . • In many cases, we won’t naturally have monotonicity, but then a little redefinition of variables does the trick (turn a bad into a good by multiplying by -1). x, y X y x y x y x y x y x
Definition 3.B.3: ·The preference relation≥on X is locally non-satiated if for every x,y, and every s>0,there exists a ysuch thaty-x≤sand y>x This means that there exists a y vector that is arbitrarily close to x that is strictly preferred to x
Definition 3.B.3: • The preference relation on X is locally non-satiated if for every , and every , there exists a y such that and . • This means that there exists a y vector that is arbitrarily close to x that is strictly preferred to x. x, y X 0 y − x y x
One last property: Definition 3.C.1.:The preference relation on X is continuous if it is preserved under limits.That is,for any sequence of pairs 《x,y")}a,withx"-y”for all n, x=limX and y=lim we have X-Y The famous counterexample is lexicographic preferences,where more of good 2 is preferred to less,but unless the bundles have the same amount good 1, then the bundle with more good 1 is always preferred
One last property: • Definition 3.C.1.: The preference relation on X is continuous if it is preserved under limits. That is, for any sequence of pairs with for all n, and we have . • The famous counterexample is lexicographic preferences, where more of good 2 is preferred to less, but unless the bundles have the same amount good 1, then the bundle with more good 1 is always preferred. ( ) =1 , n n n x y n n x x = → lim n n y y = → lim n n x y xy
Consider sequence 1,with no units of good 2 and 1/nunits of good 1,and sequence 2 with no units of good 1,and 1+1/n units of good 2. For all finite n,sequence 1 is preferred to Sequence 1,but in the limit sequence 1 yields zero unit of either good and sequence 2 yields 1 unit of good 2 and is therefore preferred. Lexicographic preferences are a famous example,but hardly a mainstay of either theory or empirical work
• Consider sequence 1, with no units of good 2 and units of good 1, and sequence 2 with no units of good 1, and units of good 2. • For all finite n, sequence 1 is preferred to Sequence 1, but in the limit sequence 1 yields zero unit of either good and sequence 2 yields 1 unit of good 2 and is therefore preferred. • Lexicographic preferences are a famous example, but hardly a mainstay of either theory or empirical work. 1+1 n 1 n
MWG Definition 1.B.2:A function u:Xis a utility function representing preference relation≥if for allx,y∈X, x≥y if and only if z(x)≥u(y) Proposition 3.C.1:If the rational preference relation on is continuous then there is a continuous utility function (x)that represents
• MWG Definition 1.B.2: A function is a utility function representing preference relation if for all , if and only if • Proposition 3.C.1: If the rational preference relation on is continuous , then there is a continuous utility function that represents . u : X → x, y X xy u(x) u(y) L u(x)
The proof in MWG requires monotonicity -a slight variant is to take a probability measure on that has positive density everywhere,and then let u(x)=1-prob(yx) ·By construction,ifxythen u(x)≥u(y) and ifu(x)u(y)thenxy
• The proof in MWG requires monotonicity – a slight variant is to take a probability measure on that has positive density everywhere, and then let • By construction, if then and if then . L u(x) =1− prob(y yx) xy u(x) u(y) u(x) u(y) xy
MWG Definition 3.B.4: ·The preference relation≥is convex if for every xe the upper contour set yY:vx is convex,that is if yx and zx then a+(1-a)zxfor every a∈[0，l
MWG Definition 3.B.4: • The preference relation is convex if for every the upper contour set is convex, that is if and then for every x X yx zx y + (1−)zx 0,1 yY : yx
MWG Definition 3.B.5: The preference relation is strictly convex if for everyxe if yx and z≥x and≠Z implies that oy+(1-ox)z>x for every a∈(0，l)
MWG Definition 3.B.5: • The preference relation is strictly convex if for every if and and implies that for every . x X yx zx y z y + (1−)z x (0,1)