Revealed Preference Theory and the Slutsky Matrx Yuhk Hosoya Abstract: In ths paper, we prove that for any contnuous

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1 Powered by TCPDF ( Ttle 顕示選好理論とスルツキー行列 Sub Ttle Revealed preference theory and the Slutsky matrx Author 細矢, 祐誉 (Hosoya, Yuk) Publsher 慶應義塾経済学会 Publcaton year 2015 Jttle 三田学会雑誌 (Mta ournal of economcs). Vol.108, No.3 ( ),p.521(45)- 536(60) Abstract 本稿では, 連続微分可能な需要関数について, スルツキー行列の半負値定符号性および対称性と, 顕示選好の強公理が同値であることを示すこのために, まずはシェパードの補題と呼ばれる偏微分方程式の解の存在定理を示し, そこから具体的に効用関数を導く In ths paper, we prove that for any contnuously dfferentable demand functon, the strong axom of the revealed preference s equvalent to the negatve sem-defnteness and symmetry of the Slutsly matrx. To show ths, we frst prove the exstence theorem on a partal dfferental equaton called Shephard's lemma, and then lead a utlty functon concretely. Notes 特集 : 経済の数理解析 : 数理経済学の新展開 Genre Journal Artcle URL

2 Revealed Preference Theory and the Slutsky Matrx Yuhk Hosoya Abstract: In ths paper, we prove that for any contnuously dfferentable demand functon, the strong axom of the revealed preference s equvalent to the negatve semdefnteness and symmetry of the Slutsly matrx. To show ths, we frst prove the exstence theorem on a partal dfferental equaton called Shephard s lemma, and then lead a utlty functon concretely Khlstrom, Mas-Colell and Sonnenschen 1976 Hurwcz and Rchter 1979 Hurwcz and Uzawa 1971 Uzawa 1960 Mas-Colell 1977 Mas-Colell Hosoya 2013 College of Economcs, Kanto Gakun Unversty

3 2 Du(p) = f(p, u(p)) 2 f % u(p ) = m f(p, m ) = x u(p) = E(p; x) = nf{p y y % x} expendture functon n R n x x n R n f(p) f (p) f(p) f R m f p f (p) x x k (x k ) k

4 R n x y x y x y x y x > y x 0 R n + x 0 R n ++ Ω R n + = {x R n, x 0} R n + R n ++ % Ω % Ω 2 (x, y) % x y % 2 x, y Ω (x, y) % (y, x) % (x, y) % (y, z) % (x, z) % 2 % preference (x, y) % x % y Ω u % x % y u (x) u (y) u % represent % utlty functon 1 % Ω 2 2 f : R n ++ R ++ Ω R n ++ R ++ p m f(p, m) f demand functon f Kreps Debreu

5 0 a > 0 f(ap, am) = f(p, m) p f(p, m) = m % f % (p, m) 2 x Ω p x m p y m y Ω x % y % u f % (p, m) max u (x) subect to. x Ω, p x m f % f u f Ω 2 4 x r y x y, (p, m), x = f(p, m) and p y m, x r y x 0,..., x k Ω, x 0 = x, x k = y, and x r x +1 for any = 0,..., k 1. 4 f % f % f % (p, m) f % f % (p, m) f % (p, m) f % (p, m) 2 2 f % (p, m) 0 % Ω = R n + u f % Ω = R n ++ f %

6 r asymmetrc x r y y r x f weak axom r f strong axom f 5 f = f % % f C 1 s (p, m) = f (p, m) + f n+1(p, m)f (p, m) (, )- S f (p, m) S f Slutsky matrx S f (p, m) = D p f(p, m) + D m f(p, m)f T (p, m) D p D m p, m T f (NSD) S f (p, m) f (S) S f (p, m) f C 1 (NSD) (S) (p, m ) R n ++ R ++ Du(p) = f(p, u(p)) 1 u(p ) = m u : R n ++ R ++ 1 f 1 p R n ++ x Ω x = f(p, m) (p, m) u f, p (x) = 0 x = f(p, m) (p, m) ċ = f((1 t)p + t p, c) ( p p), c(0) = m [0, 1] c(1) (p, m) u f, p (x) = c(1) f = f u f, p 5 Rchter 1966 Mas-Colell, Whnston and Green

7 2 f C 1 3 I f (NSD) (S) II f III 1 u : R n ++ R f (NSD) (S) x = f(p, m) E(q) = nf{q y u f, p (y) u f, p (x)} 1 c(t) = E((1 t)p + t p) 1 u f, p (x) = E( p) y = f( p, E( p)) u f, p (y) = E( p) x y u f, p u f, p (x) m f( p, m) x 2 Rchter Debreu 1972 dual Debreu 1972 (NSD) 2 Rchter u f, p 1 0 < α < 1 P α = 1 f (p, m) = α m p p = (1, 1,..., 1) 1 ċ(t) = P α (1 p ) c(t), c(0) = m p + t(1 p )

8 ċ = a(t)c c(0)e R t 0 a(s)ds c(1) = c(0)e R 10 P α (1 p ) p +t(1 p ) dt = c(0)e P α log p = m Y x R n ++ p = α x m = 1 x = f(p, m) C u f, p (x) = Cx α x α n n p α 2 CES 0 < α < 1 P α = 1 < ρ < 1 ρ 0 1 f (p, m) = α 1 p m P α 1 ρ p p = (1, 1,..., 1) 1 ċ(t) = P α 1 P (p + t(1 p )) 1 (1 p ) α 1 c(1) = c(0)e = c(0)e R 10 = mc[ P (p + t(1 p )) ρ P α 1 P α 1 ρ (log P α 1 p c(t), c(0) = m (p + t(1 p )) 1 (1 p ) (p + t(1 p )) ρ 1 ρ α p log P 1 α ) ρ ] 1 ρ 1 C > 0 x R n ++ P m x = f(p, m) p m = P α x ρ p m = α x ρ 1 dt m α 1 p ρ = 1 u f, p (x) = C[α 1 x ρ α n x ρ n] 1 ρ

9 3 f = f u u Ω = R n ++ f G(x) = {p R n ++ P p = 1, x = f(p, p x)} G 1 u f, p f = f % G f r 6 (f k ) 1 f C 1 u fk, p u f, p 7 S f (p, m) n 1 Hosoya 2013 G C 1 (NSD) (S) C 1 u 6 x, x y x > y x < y y r x G

10 u f, p C 1 f C k u f, p C k Ω = R n ++ Ω = R n + Ω = R n + f = f u Hosoya Ω = R n ++ f(p, m) {x Ω p x m} R n ++ R ++ R n ++ R ++ 0 G 2 f = f % f % C (1) ẇ(t; p) = f((1 t)p + tp, w(t; p)) (p p ), w(0, p) = m. 2 (1) u [p, p] w(t; p) = u((1 t)p + tp) 1 (2) u(p) = w(1; p) p u(p ) = m (1)

11 h (t, p) = w p (t; p) tf ((1 t)p + tp, w(t; p)) 8 2 w = 2 w p t t p 9 (S) ḣ (t, p) = (f (p p )) f t X [f + f p n+1f ](p p ) = f + X» tf + f w n+1 (p p ) f t X [f + f p n+1f ](p p ) = t X [f f fn+1f ](p p ) + X f w n+1 (p p ) p» w X = tf fn+1(p p ) p = h (t, p) X f n+1(p p ) ḣ (t, p) = a(t, p)h (t, p) a(t, p) h(t, p) = h(0, p)e R t 0 a(s,p)ds = 0 h(0, p) = 0 w p (t; p) = tf ((1 t)p + tp, w(t; p)) u (p) = w (1; p) = f (p, w(1; p)) = f (p, u(p)) p p (2) p R n ++ [0, 1] 2 w(t; p) (2) [0, t ] p(t) = (1 t)p + tp x 1 = f(p, m ), x 2 = f(p(t ), w(t ; p)) p(t ) x 1 w(t ; p) p x 2 m c(t) = p f(p(t), w(t; p)) ċ(t) = (p ) T S f (p(t), w(t; p))(p p ) T p(t) T S f (p(t), w(t; p))(p p ) = f((1 t)p + tp, w(t; p)) f

12 ċ(t) = t(p p ) T S f (p(t), w(t; p))(p p ) 0 (NSD) p x 2 = c(t ) c(0) = p x 1 = m 2 p [0, 1] p t [0, t] t [0, 1] [0, t ] w( ; p) [0, t [ p(t) = (1 t)p + tp 10 R n ++ R ++ C t t (p(t), w(t; p)) C p(t) [p, p] R n ++ nf t [0,t [ w(t; p) = 0 sup t [0,t [ w(t; p) = + 2 p(t) f(p, m ) w(t; p) w(t; p) p(t) f(p, m ) = tp f(p, m ) + (1 t)m p f(p, m ) + m (t k ) t k t w(t k ; p) 0 x k = f(p(t k ), w(t k ; p)) 2 p x k m t k p f(p, m ) + (1 t k )m = p(t k ) f(p, m ) p(t k ) x k = t k p x k + (1 t k )p x k p f(p, m ) p x k p(t ) f(p, m ) p(t ) x k x k {x R n + p(t ) x p(t ) f(p, m )} x R n + p x m x 0 0 < p(t ) x = lm k p(t k) x k = lm k w(t k, p) =

13 u 1, u 2 : R n ++ R ++ p q R n ++ c (t) = u ((1 t)p + tq) ċ = f((1 t)p + tq, c ) (q p), c (0) = u 1 (p) = u 2 (p) 2 2 u 1 (q) = c 1 (1) = c 2 (1) = u 2 (q) u(p) = m u : R n ++ R ++ x = f(p, m) q x u(q) 4 x y x = f(p, m), y = f(q, w) u : R n ++ R ++ u(p) = m 1 w u(q) p y > m w > u(q) w = u(q) w > u(q) v : R n ++ R ++ v(q) = w 1 v(q) > u(q) v(p) u(p) [p, q] v(r) = u(r) r 3 u v u(q) v(q) v(p) > u(p) 2 p y v(p) p y > u(p) w = u(q) 2 p y u(p) = m p y = m 2 p(t) = (1 t)p + tq c(t) = p f(p(t), u(p(t))) ċ(t) = t(q p) T S f (p(t), u(p(t)))(q p) 0 c(0) = m c(1) = p y ċ(t) 0 S f (p(t), u(p(t))) S t c t 1,..., c t n (NSD) 0 (S) P t

14 0 1 c t P T 0 c t t S t P t = B A c t n 0p 1 c t p 0 c t A t = P t B A c t n A 2 t = S t A t P T t ċ(t) = t (A t (q p)) 2 0 t A t (q p) = 0 S t (q p) = 0 x(t) = f(p(t), u(p(t))) ẋ(t) = S t (q p) = 0 x = x(0) = x(1) = y x y 4 x = f(p, m), y = f(q, w), x y p y m 4 u : R n ++ R ++ 1 u(p) = m u(q) > w v : R n ++ R ++ 1 v(q) = w 4 m = u(p) > v(p) 4 q x > w x = f(p, m) = f(q, w) u 1 u(p) = m v 1 v(q) = w u( p) = v( p) 3 u(q) = w t [0, 1] p(t) = (1 t)p + tq, d(t) = (1 t)m + tw f(p(t), d(t)) = x f(p(t), d(t)) = y x 0 t 1 d(t) = p(t) y = p(t) x p x < p y q x < q y p(t) y > d(t) d(t) = p(t) x d(t) = x (q p) = f(p(t), d(t)) (q p), d(0) = m

15 c(t) = u(p(t)) ċ(t) = f(p(t), c(t)) (q p), c(0) = m u(q) = c(1) = d(1) = w u f, p (x) (p, m) x y, f(p, m) = x p y m y f u f, p (y) = 0 < u f, p (x) y = f(q, w) (q, w) u, v 1 u(p) = m, v(q) = w 4 u(q) > v(q) 4 u( p) > v( p) u f, p (x) > u f, p (y) x = f u f, p (p, m) f (NSD) (S) 1 f = f u f, p f % f = f % (p, m ) x = f(p, m ) E(p) = nf{p y y % x} 5 E(p) E(p ) = m 11 1 p 1, p 2 R n ++ t [0, 1] (1 t)p 1 + tp 2 = p ε > 0 y % x p y E(p) + ε y E(p) + ε p y = (1 t)p 1 y + tp 2 y (1 t)e(p 1 ) + te(p 2 ) ε 0 E E y % x y x x = f(p, m ) = f % (p, m ) p y > m p x = m E(p ) = m x(p) = f(p, E(p)) p x(p) = E(p) ε > 0 x ε (p) = f(p, E(p) + ε) E(p) y % x p y < E(p) + ε

16 y x ε (p) % y x ε (p) % x p, q R n ++ p x(p) = E(p) p x ε (q) ε 0 f p x(p) p x(q) e p(t) = p + te E(p(t)) E(p) = (p + te ) x(p + te ) p x(p) = p (x(p + te ) x(p)) + tx (p + te ) tf (p + te, E(p + te )) E(p(t)) E(p) lm f E(p(t)) E(p) (p, E(p)) lm t 0 t t 0 t E E (p) = f (p, E(p)) E 1 1 u(p ) = m (p, m ) 1 u u(p ) = m C 2 D 2 u(p ) = S f (p, m ) S f (p, m ) C 2 (p, m ) f (NSD) (S) [1] Debreu, G.: Representaton of a Preference Orderng by a Numercal Functon. In: R. M. Thrall, C. H. Coombs and R. L. Davs (Eds.), Decson Processes, , Wley (1954) [2] : Smooth Preferences. Econometrca, 40, (1972) [3] Hosoya, Y.: Measurng Utlty from Demand. Journal of Mathematcal Economcs, 49, (2013) [4] : A Theory for Estmatng Consumer s Preference from Demand. Advances n Mathematcal Economcs, 19, (2015) [5] Hurwcz, L., Rchter, M. K.: Vlle Axoms and Consumer Theory. Econometrca, 47, (1979)

17 [6] Hurwcz, L., Uzawa, H.: On the Integrablty of Demand Functons. In: J. S. Chpman, L. Hurwcz, M. K. Rchter and H. F. Sonnenschen (Eds.), Preferences, Utlty and Demand, , Harcourt Brace Jovanovch, Inc., New York (1971) [7] Khlstrom, R., Mas-Colell, A., Sonnenschen, H.: The Demand Theory of the Weak Axom of Revealed Preference. Econometrca, 44, (1976) [8] Kreps, D.: Notes on the Theory of Choce. Westvew Press (1988) [9] Mas-Colell, A.: The Recoverablty of Consumers Preferences from Market Demand Behavor. Econometrca, 45, (1977) [10] Mas-Colell, A., Whnston, M. D. and Green, J.: Mcroeconomc Theory. Oxford Unversty Press, Oxford (1995) [11] Rchter, M. K.: Revealed Preference Theory. Econometrca, 34, (1966) [12] : Dualty and Ratonalty. Journal of Economc Thoery, 20, (1979) [13] Uzawa, H.: Preference and Ratonal Choce n the Theory of Consumpton. In: K. J. Arrow, S. Karln and P. Suppes (Eds.), Mathematcal Methods n the Socal Scences, 1959, Proceedngs, , Stanford Unversty Press (1960) [14] 1968 : :

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.~:ril.~, ー.. ~ 諭 Powered by TCPDF (www.tcpdf.org) Title 歴史の裏面 Author 久米, 邦武 Publisher 三田学会 Jtitle 三田学会雑誌 (Keio journal of economics). Vol.3, No.6 (1910. 6),p.705(77)- 715(87) Abstract Genre Journal Article URL http://koara.lib.keio.ac.jp/xoonips/modules/xoonips/detail.php?koara_id=an00234610-19100615