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Ken Urai May 19, 2004 5 27 date-event uncertainty risk 51 ordering preordering X X X (preordering) reflexivity x X x x transitivity x, y, z X x y y z x z asymmetric x y y x x = y X (ordering) completeness x, y X x y y x (total ordering) 27 40

i R l X i X i i, X i x, y x y x i y i X i 28 non-ordered i X i x i y y i x x i y x y x i y (y i x) x i y x y i irreflexsive x X i, (x i x) transitive i reflexive, transitive, symmetric X i i x, y X i x i y, x i y, y i x X X A b X x A, x b b x x X x X x x x x x X ( ) maximal element x x x = x ( ) x X y X y x x X ( ) greatest element greatest element ( ) maximal element Zorn Lemma 51 Zorn s Lemma Preordering A (preordering) A A maximal element x y x y y x x = y (ordering) Preordering Zorn s Lemma Ordering Zorn s Lemma Lemma 52 Zorn s Lemma Ordering X (ordering) X X maximal element maximal element x X x X P(X) F F f : F F f(x) F F f 28 41

X G g : G G g(g) F, g(g) G x 1 X A 1 = {x 1 } A 1 X totally ordered subset g(a 1 ) A 2 = A 1 {f(g(a 1 ))} A 2 {A 1, A 2 } F A 1 f g F F A F A = A A f(g(a)) (QED) Definition 53 X i i i i x X i {y y X i, y i x} is closed, {y y X i, x i y} is closed, i Theorem 54 X i R l i i u i : X i R x i y u i (x) u i (y) (Debreu, 1959; Chapter 4 ) 52 521 market market (1) (2) (2) 29 29 42

522 i R l X i X i i i X i R l, X i i u i : X i R, ω i R l, j = 1, 2,, n share holdings θ i1, θ i2,, θ in 0 θ ij 1, j = 1, 2,, n (X i, i, ω i, θ i1,, θ in ) X i 523 i, X i R l, X i i X X, ω i R l, Y 1,, Y n (θ i1,, θ in ) [0, 1] n X i i X i i X X 30 R l Y j π j i p W i (p) = p ω i + n θ ij π j (p) (40) R l W i : R π j, j = 1, 2,, n W i i (wealth function) W i W i R l + \ {0} R X i R l u i p R l W R j=1 : UMP p = (p 1, p 2,, p l ) R l W R x = (x 1, x 2,, x l ) R l u i (x) x X i, p x W (i) (ii), p, W, 30 43

(iii), p, W, (i) (ii) (iii) 524 i X i X i R l i i W i : R i i u i Definition 55 w R p R l β i (p, w) = {x X i p x w} β i i (budget correspondence) W i w = W i (p) p β i (p, W i (p)) B i (p) B i Definition 56 p = (p 1,, p l ) {x x X i, p x W i (p), ˆx X i, (p ˆx W i (p) = u i (ˆx) u i (x)} ξ i : X i i (demand correspondence) Theorem 57 W i p, inf x Xi p x < W i (p), ξ : X i ξ X i (Proof:) ξ i X i ξ X i {p ν } ν=1 p {x ν } ν=1 ν x ν ξ i (p ν ) x X i x ξ i (p ) x / ξ i (p ) ν p ν x ν W i (p ν ), p x W i (p ) x p x / ξ i (p ) y X i p y W i (p ) u i (x ) < u i (y ) x U x y U y x U x, y U y, u i (x) < u i (y), z X i p z < W i (p ) {y ν } ν=1 y ν = 1 ν z + (1 1 ν )y, ν = 1, 2,, y ν y, 44

x ν x ν ν ν ν, x ν U x, and y ν U y (41) lim ν pν y ν = lim ν pν ( 1 ν z + (1 1 ν )y ) = 1 ν p z + (1 1 ν )p y = p y ν < W i (p ) δ = Wi(p ν ) p y 2 > 0, ˆν ν ˆν, p ν y ν < W i (p ) δ (42) W i W i (p ν ) W i (p ) ν ν ν, W i (p ) δ < W i (p ν ) (43) ν 0 ν 0 > ν ν 0 > ˆν ν 0 > ν (41), (42), (43) u i (x ν0 ) < u i (y ν ), and p ν0 y ν < W i (p ) δ, and W i (p ) δ < W i (p ν0 ) y ν x ν0 p ν0 x ν0 p ν0 x ν0 ξ i (p ν0 ) (QED) Corollary 58 p ξ i (p) ξ i : X i (Proof:) p n p {x n } = ξ i (p n ), n = 1, 2,, {x n } n=1 x X i, {x } = ξ i (p ) ɛ > 0 x ɛ B(ɛ, x ) n 1, n 2, x nt / B(ɛ, x ), t = 1, 2, X i {x n } n=1 {xnt } t=1 {x mt } t=1 X i p mt p {x mt } t=1 ξ i (p ) x {x mt } t=1 {x nt } t=1 B(ɛ, x ) QED 525 Maximum Theorem 57 Budget β : p {x X i p x W i (p)} 427 Maximum Theorem 417 β Debreu (1959) Maximum Theorem Maximum Thorem price 45

526 UMP EMP Expenditure Minimization Problem X i u i EMP Ū p = (p 1, p 2,, p l ) R l u i (x) = Ū x = (x 1, x 2,, x l ) R l l p k x k k=1 (x 1, x 2,, x l ) X i UMP EMP EMP UMP ( EMP ) p R l Ū Definition 59 : R e i e i : p min x X i u i (x)=ū l p k x k e i i u i (x) = Ū e i Ū k=1 E i (Ū, p) = e i(p) e i Ū E i i ( expenditure function ) e i E i Ū e i α R, p, e i (αp) = αe i (p) p p, p q = αp + (1 α)p, (0 α 1), 3 e i (q) αe i (p) + (1 α)e i (p ) e i : R Theorem 510 u i (x) = Ū e i : R (i) p, α R +, αp and e i (αp) = αe i (p) (ii) p, p, α R, if αp + (1 α)p is an element of, then e i (αp + (1 α)p ) αe i (p) + (1 α)e i (p ) 46

Ū e i p e i (p ) = l k=1 q kx k (x 1, x 2,, x l ) X i Ū p x 1,, x l R l R H = {(p 1, p 2,, p l, l k=1 p k x k ) Rl R l k=1 p kx k = e i(p )} H (p, e i (p )) e i p = p p x e i (p) p e i (p) l k=1 p k(x k ) e i (p 1,, p l ) p p k, k = 1,, l ei p k (p ) = x k, k = 1,, l Shephard s Lemma Theorem 511 u i (x) = Ū e i : p = (p 1,, p l ) e i (p) R p = p p k (x 1,, y l ) X i p k = 1,, l e i p k (p ) = x k u i (x) = Ū e i Maximum Theorem 47

53 i convex X i int X i u : X i R locally non-satiated strictly convex locally non-satiated strictly convex 531 p = (p 1,, p l ) R l \ {0} W R i Max u(x 1,, x l ) (CP) subject to p 1 x 1 + + p l x l = w (x 1,, x l ) X i x(p, w) = (x 1 (p, w),, x l (p, w)) strictly convex unique (R l \ {0}) R O (p, w) O x(p, w) Marshallian Demand Function Marshallian demand function Assumption 512 Marshallian Demand Function int X i Marshallian demand function x(p, w) (p, w) O (CP) X i x 1,, x l λ Lagrangean L(x 1,, x l, λ) = u(x 1,, x l ) λ(p 1 x 1 + + p l x l w) (x 1 (p, w),, x l (p, w)) = x(p, w) l + 1 p 1 x 1 p l x l + w = 0 u λp 1 x 1 = 0 (44) u λp l x l = 0 λ II 2 u x i x j (x(p, w)) u ij 0 p 1 p l p 1 u 11 u l1 det 0 (45) p l u 1l u ll 48

(p, w) U O (ˆp, ŵ) U (44) x 1,, x l λ = λ(ˆp, ŵ) x 1 = ˆx 1 (ˆp, ŵ) (46) x l = ˆx l (ˆp, ŵ) U C 1 Martialian Demand Function x 1 (ˆp, ŵ),, x l (ˆp, ŵ) (44) x i (ˆp, ŵ) = ˆx i (ˆp, ŵ), i = 1,, l Demand Function U C 1 Marshallian (p, w) O O (45) Marshallian Demand Function O C 1 31 532 (CP) O (R l \ {0}) R Marshallian demand function x(p, w) v(p, w) = u(x(p, w)) (47) O v (Indirect Utility Function) Marshallian Demand Function C 1 v O C 1 Theorem 513 λ(p, w) (46) demand function O (ˆp, ŵ) = λ(ˆp, ŵ) w (Proof) w (ˆp, ŵ) = l i=1 u x i (x(ˆp, ŵ)) xi w (ˆp, ŵ) (44) u x i (x(ˆp, ŵ)) = ˆp i λ(ˆp, ŵ), i = 1,, l λ(ˆp, ŵ) l i=1 x i ˆp i (ˆp, ŵ) w x(p, w) (CP) p 1 x 1 (p, w) + + p l x l (p, w) = w w (ˆp, ŵ) l i=1 x i ˆp i (ˆp, ŵ) = 1 w QED λ 31 (45) 26 49

533 (p, w) O v V R v V C 1 v(p, w) = v (48) (48) ( p, w) O 513 λ( p, w) 0, 32 ( p, w) 0 (49) w U (R l \ {0}) V w = w(p, v) 33 ( p, v) (R l \ {0}) V v C 1 w(p, v) U C 1 w(p, v) (Expenditure Function) E i w(p, v) U h(p, v) = x(p, w(p, v)), (h i (p, v) = x i (p, w(p, v)), i = 1,, l), (50) h (Hicksian Compensated Demand Function) Theorem 514 (Shepard s Lemma) w (ˆp, ˆv) = h i (ˆp, ˆv) (Proof) v(p, w) = v w(p, v) w p (ˆp, ˆv) = i (ˆp, w(ˆp, ˆv)) = (ˆp, w(ˆp, ˆv)) w v(p, w) = u(x(p, w)) (ˆp, w(ˆp, ˆv)) λ(ˆp, w(ˆp, ˆv)) (51) (ˆp, w(ˆp, ˆv)) = l j=1 u x j (x(ˆp, w(ˆp, ˆv))) x j (ˆp, w(ˆp, ˆv)) (52) (44) u x j (ˆp, w(ˆp, ˆv)) = ˆp j λ(ˆp, w(ˆp, ˆv)) (ˆp, w(ˆp, ˆv)) = l j=1 ˆp j λ(ˆp, w(ˆp, ˆv)) x j (ˆp, w(ˆp, ˆv)) = λ(ˆp, w(ˆp, ˆv)) l j=1 ˆp j x j (ˆp, w(ˆp, ˆv)) (53), Marshallian Demand l j=1 p jx j (p, w) = w p i (ˆp, w(ˆp, ˆv)) x i (p, w) + x i (ˆp, w(ˆp, ˆv)) + l j=1 l j=1 p j x j (p, w) = 0 ˆp j x j (ˆp, w(ˆp, ˆv)) = 0 32 p 0 u strictly increasing 33 w local non-satiation w 1 < w 2 p v(p, w 1 ) < v(p, w 2 ), p, v w (48) 50

(53) (51) (ˆp, w(ˆp, ˆv)) = λ(ˆp, w(ˆp, ˆv))( x i (ˆp, w(ˆp, ˆv))) (54) w (ˆp, ˆv) = λ(ˆp, w(ˆp, ˆv))( x i(ˆp, w(ˆp, ˆv))) = x i (ˆp, w(ˆp, ˆv)) = h i (ˆp, ˆv), (55) λ(ˆp, w(ˆp, ˆv))) QED Theorem 515 (Roy s Identity) (ˆp, ŵ) w (ˆp, ŵ) = x i(ˆp, ŵ) (Proof) (54) (QED) Theorem 516 (Slutsky Equation) x i p j (ˆp, w(ˆp, ˆv)) = h i p j (ˆp, ˆv) x i w (ˆp, w(ˆp, ˆv))h j(ˆp, ˆv) (Proof) h i (p, v) = x i (p, w(p, v)) p j 514 (QED) 54 R n i X i R l 51

55 1 CP (p48) X (Assumption 512) Lagrangean L(x 1,, x l, λ) = u(x 1,, x l ) λ(p 1 x 1 + + p l x l w) 1 X i X i (x 1,, x l ) X i X i {(x 1,, x l ) 0 x 1,, 0 x l } Max u(x 1,, x l ) (56) Subto g 1 (x 1,, x l ) = 0 g m (x 1,, x l ) = 0 h 1 (x 1,, x l ) 0 h n (x 1,, x l ) 0 x = (x 1,, x l ) x Karush-Kuhn-Tucker 551 Max u(x 1,, x l ) (57) Subto g 1 (x 1,, x l ) = 0 g m (x 1,, x l ) = 0 1 m = 1 432 f(y 1,, y l ) = 0 y 1 = g(y 2,, y l ) I m m II m Assumption 517 Constraint Qualification (57) x = (x 1,, x l ) m C 1 D 1 g 1 (x ) D l g 1 (x ) rank = m D 1 g m (x ) D l g m (x ) 52

m Theorem 518 (57) u C 1 x = (x 1,, x l ) x 517 λ 1,, λ m R m D i u(x ) = λ j D i g j (x ), i = 1,, l (58) j=1 Proof : F : R l R m+1 u(x 1,, x l ) g 1 (x 1,, x l ) F (x 1,, x l ) = g m (x 1,, x l ) (59) u (x ) D 1 u(x ) D l u(x ) rank F (x (g 1 ) (x ) ) = rank = rank D 1 g 1 (x ) D l u(x ) (g m ) (x ) 517 rank m m + 1 Case 1 : F m D 1 g m (x ) D l u(x ) m + 1 0 a 0, a 1,, a m R a 0 u (x ) + a 1 (g 1 ) (x ) + + (g m ) (x ) = 0 (60) a 0 = 0 Constraint Qualification m a 0 0 λ i = a i /a 0, i = 1,, m Case 2 : F m + 1 x 1,, x l δ u(x 1,, x l ) u(x 1,, x l ) δ = 0 (61) g 1 (x 1,, x l ) g 1 (x 1,, x l ) = 0 g m (x 1,, x l ) g m (x 1,, x l) = 0 II x 1 = x 1,, x l = x l, δ = 0 F m + 1 x 1,, x l m + 1 x 1,, x m+1 D 1 u(x ) D m+1 u(x ) D 1 g 1 (x ) D m+1 g 1 (x ) det 0 (62) D 1 g m (x ) D m+1 g m (x ) 53

(61) x 1,, x m+1 x 1,, x l, 0 (m + 1) (m + 1) D 1 u(x ) D m+1 u(x ) D 1 g 1 (x ) D m+1 u(x ) D 1 g m (x ) D m+1 u(x ) (62) 0 II p31, 420 x 1,, x m+1 U x m+2,, x l, 0 V V x 1 = x 1 (x m+2,, x l, δ) x m+1 = x m (x m+2,, x l, δ) (x m+2,, x l, δ) V, δ > 0 (61) x (57) QED (63) 552 Max u(x 1,, x l ) (64) Subto g 1 (x 1,, x l ) = 0 g m (x 1,, x l ) = 0 h 1 (x 1,, x l ) 0 h n (x 1,, x l ) 0 x = (x 1,, x l ) x Constraint Qualification x = (x 1,, x l ) Index I(x ) Assumption 519 Constraint Qualification 2 (64) x = (x 1,, x l ) m m C 1 I(x ) i 1,, i k D 1 g 1 (x ) D l g 1 (x ) D 1 g m (x ) D l g m (x ) rank D 1 h i1 (x ) D l h i1 (x = m + k ) D 1 h i k (x ) D l h i k (x ) 54

Theorem 520 Karush-Kuhn-Tucker (64) x = (x 1,, x l ) u C1 x Constraint Qualification (519) λ 1,, λ m, µ 1,, µ n R m D i u(x ) = λ j D i g j (x ) + j=1 n µ j D i h j (x ), i = 1,, l (65) j=1 0 µ j, j = 1,, n (66) µ j h j (x ) = 0, j = 1,, n (67) 56 2 p i R l w i R x i R l (x i, p i, w i ) i I {x i i I} X X (x i, p i, w i ) i I i I, x X, (p i x w i ) (x x i ) (68) Assumption 521 X X (x i, p i, w i ) i I x X x i X x x i p i x w i x r x i x i is directly revealed prefered to x x i x x i x i (x i, p i, w i ) i I 2 (x i1, p i1, w i1 ),, (x in, p in, w in ) p i1 x in > w i1 34 x i1 r r x in (69) Theorem 522 (x i, p i, w i ) i I (521) X {x i i I} X (68) 35 Proof : X r X X y 1,, y n x = y 1 r r y n = y x y transitive asymmetric irreflexive X X X irreflexive transitive X X transitive irreflexive -maximal Zorn s Lemma 34 n = 2 x i 1 x i 2 x i 1 x i 2 35 55

x, y X, x y x y y x (x, y) / (y, x) / = {(z, w) (z x z = x) (w = y y w)} transitive irreflexive, -maximal x x (x = x x x ) reflexive, transitive anti-symmetric complete r greatest element QED 57 3 p x C 1 f Max f(p, x) (70) Subto g(p, x) = 0 (ˆp, ˆx (ˆp)) Ŵ = Û ˆV p Û x (p) ˆV C 1 x : Û ˆV Û F (p) = f(x (p), p) (71) F F f Û R f f x (ˆp) ˆp F F f F DF (p) = D p f(p, x (p)) + D x f(p, x p)dx (p) g 0 p x f (, 1996; p308) REFERENCES Debreu, G (1959): Theory of Value Yale University Press, New Haven, CT Hicks, J (1939): Value and Capital Clarendon Press, Oxford :, Tokyo Mas-Colell, A, Whinston, M D, and Green, J R (1995): Microeconomic Theory Oxford University Press, New York 56

Exercise 51 2 24 LAWSON 24 24 24 0 Exercise 52 X (y x)) y x (i) = X X\ irreflexive, transitive (y x) (y x x = y) (ii) transitive (iii) x y y x x y x, y X x y, x y, y x Exercise 53 X R l X X xxx X X Exercise 54 X X X X X Exercise 55 Debreu (1959) (Debreu, 1959; Chapter 4) 3 D Q mapping u Q Q Exercise 56 X R l W R p R l B = {x R l p x W } {x R l p x = W } A A X X R l x X B x A Exercise 57 518 x (57) Exercise 58 522 (1) asymmetric x y (y x) (2) transitive 57

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