1 27 4 10 1 toyotaka.sakai@gmail.com
Welfare Economics (1920) The main motive of economic study is to help social improvement help social improvement society society improvement help 1885 cool heads but warm hearts warm hearts cool heads hearts warm cool heads warm hearts 2
..................................... 2 5..................................... 5.......................... 5................................. 6.......................... 7 1 9 1.1............................. 9 1.2......................... 16 1.3.............................. 16 2 21 2.1............................. 21 2.2.................................. 21 2.3.............................. 25...................... 28 3 31 3.1.................................. 31 3.2.............................. 33 3.3........................... 36 4 43 4.1............................. 43 4.2........................... 43 4.3................................ 45 4.4................................ 46 4.5......................... 48 4.6......................... 51 3
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1 1 1 6
(for all) R = (, ) R + = [0, ) x X x X x [0, ) x R + 0 x < x R + 0 x < (1) x X y Y (x, y) X Y (2) 5 R + 2 R (5, 2) (5, 2) R + R 1789 1770 7
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1 1.1 1 x R + m R 1 x m 2 m 3 (x, m) R + R (x, m) (x, m ) (x, m) (x, m ) (x, m) (x, m ) (x, m) (x, m ) (x, m) (x, m ) (x, m) (x, m ) (x, m) (x, m ) (x, m) (x, m ) x p 1 1 M px + m = M (1.1) (x, m) R + R B(p, M) = {(x, m) R + R : px + m = M} (1.2) 1 m 2 3 9
1 ( ). B(p, M) (x, m ) (x, m ) (x, m ) B(p, M) (1.3) (x, m ) (x, m) (x, m) B(p, M) (1.4) (p, M) (p, M) (x, m ) (x (p, M), m (p, M)) (1.5) x (p, M) m (p, M) x, m m (p, M) px (p, M) + m (p, M) = M (1.6) m (p, M) = M px (p, M) (1.7) x (p, M) m (p, M) x x (p, M) 4 4 10
x R + (x, 0) (0, V (x)) (1.8) V (x) (1.8) x V (x) x V V (0) = 0 (1.9) V > 0 (1.10) V < 0 (1.11) (1.9) (1.10) (1.11) 2 ( ). R + R (x, m), (x, m ) R + R (x, m) (x, m ) V (x) + m V (x ) + m (1.12) U(x, m) = V (x) + m 11
V x 1.1: V 12
V x 1.2: V 13
(x, m ) (x, m ) B(p, M) (1.13) (x, m ) (x, m) (x, m) B(p, M) (1.14) (x, m ) B(p, M) (1.15) V (x ) + m V (x) + m (x, m) B(p, M) (1.16) (x, m ) px + m = M V (x) + m m = M px x V (x) + M px x V (x) + M px V (x) p = 0 (1.17) (1.17) M V (x) + M px x M (1.17) V 1 V (x) = p (1.18) x = V 1 (V (x)) = V 1 (p) (1.19) x (p, M) = V 1 (p) (1.20) (1.20) M x (p, M) M M x (p) = V 1 (p) (1.21) 14
x. U(x, m) = V (x) + m (1.22) (1.22) 20 15
1.2 M x px U(x, M px) = V (x) + M px = V (0) +M + V (x) px = U(0, M + V (x) px) }{{} =0 (1.23) (x, M px) }{{} x px (0, M + V (x) px) }{{} v(x) px (1.24) x px V (x) px V (x) px (1.25) V (x (p)) px (p) x (p) 0 V (x)dx = V (x (p)) V (0) = V (x (p)) (1.26) }{{} =0 V (x (p)) px (p) = x (p) 0 V (x)dx px (p) (1.27) 1.3 i = 1, 2,..., I i i x i R + V i 5 p 5 x 2 2 2 1 16
V p x (p) 1.3: V (x (p)) px (p) 17
(x i (p)) I (x 1(p), x 2(p),..., x I (p)) D(p) I x i (p) = I V i 1 (p) (1.28) D i x i (p) px i (p) CS(p) = I ( ) V i (x i (p)) px i (p) (1.29) 1 18
x 2 x 1 p CS(p) D 1.4: 19
2 2.1 y C(y) C C(0) = 0 (2.1) C (y) > 0 (2.2) C (y) > 0 (2.3) (2.1) (2.2) (2.3) C (y) y 2 2 1 2.2 y π(y) = py }{{} C(y) }{{} (2.4) π 1 21
C y 2.1: C 22
2 p π (y) = p C (y) = 0 (2.5) p = C (y) (2.6) 1 1 (2.6) 2 23
C p y 2.2: 24
(2.6) C C 1 C 1 (p) = C 1 (C (y)) = y (2.7) y (p) C 1 (p) (2.8) y (p) p y y (p) π(y (p)) = py (p) C(y (p)) (2.9) C(y (p)) = C(y (p)) C(0) = }{{} =0 π(y (p)) = py (p) y (p) 0 y (p) 0 C (y)dy (2.10) C (y)dy (2.11) 2.3 j = 1, 2,..., J j j y j R + C j π j p (yj (p)) J j=1 (y1(p), y2(p),..., yj (p)) (2.8) J J Y (p) yj (p) = C j 1 (p) (2.12) j=1 Y p j y j (p) P S(p) J j=1 j=1 ( ) pyj (p) C j (yj (p)) (2.13) 3 3 25
C p y (p) 2.3: py (p) C(y (p)) 26
y 2 y 1 S p P S(p) 2.4: 27
z y = F (z) F F (0) = 0 (2.14) F > 0 (2.15) F < 0 (2.16) (2.14) z y (2.15) (2.16) z c y p c c z y y = F (z) (2.17) 28
y z F F 1 (y) = z (2.18) y z F 1 (y) z c cz (2.18) cf 1 (y) (2.19) C(y) cf 1 (y) (2.20) C y 4 C (y) = cf 1 (y) > 0 (2.21) 4 29
3 3.1 p I x i (p ) = J yj (p ) (3.1) j=1 D S D(p) S(p) I x i (p) (3.2) J yj (p) (3.3) j=1 p p ((x i (p )) I, (y j (p )) Jj=1, p ) (3.4) V i (x i (p )) = p i = 1, 2,..., I (3.5) C j(y j (p )) = p j = 1, 2,..., J (3.6) 31
x 2 y 2 y 1 S x 1 p D 3.1: 32
3.2 ( ) (x i ) I, (y j ) J j=1 (3.7) I x i J y j (3.8) j=1 ( ) (x i ) I, (y j ) J j=1 I V i(x i ) J j=1 C j(y j ) SS((x i ) I, (y j ) J j=1) I J V i (x i ) C j (y j ) (3.9) j=1 ( ) (x i ) I, (y j ) J j=1 I x i < J y j (3.10) j=1 max I V i (x i ) J C j (y j ) (3.11) j=1 sub to I J x i = y j (3.12) j=1 33
L((x i ) I, (y j ) J j=1, λ) = I J J V i (x i ) C j (y j ) + λ( y j j=1 j=1 I x i ) (3.13) L((x i ) I, (y j ) J j=1, λ) x i = V i (x i ) λ = 0 i = 1, 2,..., I (3.14) L((x i ) I, (y j ) J j=1, λ) = C y j(y j ) + λ = 0 j = 1, 2,..., J j (3.15) L((x i ) I, (y j ) J j=1, λ) J I = y j x i = 0 λ (3.16) j=1 V i (x i ) = λ i = 1, 2,..., I (3.17) C j(y j ) = λ j = 1, 2,..., J (3.18) I J x i = y j (3.19) j=1 ( ) (x i ) I, (y j ) J j=1, λ 34
((x i (p )) I, (y j (p )) Jj=1, ) p V i (x i (p )) = p i = 1, 2,..., I (3.20) C j(y j (p )) = p j = 1, 2,..., J (3.21) I J x i (p ) = yj (p ) (3.22) j=1 ((x i (p )) I, (y j (p )) Jj=1, ) p ( ) (x i (p )) I, (yj (p )) J j=1 V i C j I x i (p ) = J yj (p ) (3.23) j=1 I p x i (p ) = p I J x i (p ) = p yj (p ) = j=1 J p yj (p ) (3.24) j=1 35
SS((x i (p )) I, (yj (p )) J j=1) (3.25) I J = V i (x i (p )) C j (yj (p )) (3.26) = = I V i (x i (p )) I j=1 I p x i (p ) + ( ) V i (x i (p )) p x i (p ) + J p yj (p ) j=1 J j=1 J C j (yj (p )) (3.27) j=1 ( ) p yj (p) C j (yj (p )) (3.28) =CS(p ) + P S(p ) (3.29) (3.9) p 3.3 T > 0 D(r 1 ) = S(r 2 ) (3.30) r 1 = r 2 + T (3.31) (r 1, r 2 ) 36
x 2 y 2 y 1 S x 1 p CS(p ) P S(p ) D 3.2: 37
r 1 S T r 2 D D(r 1 ) = S(r 2 ) 3.3: T r 1, r 2 38
t 1 + t 2 = T (t 1, t 2 ) (t 1, t 2 ) = (T, 0) (t 1, t 2 ) = (0, T ) 1 t 1 1 t 2 p p + t 1 p t 2 t 1 + t 2 = T D(p + t 1 ) = S(p t 2 ) (3.32) (p + t 1 ) = (p t 2 ) + T (3.33) (3.30, 3.31) (r 1, r 2 ) (3.32, 3.33) (p + t 1, p t 2 ) (3.30, 3.31) p + t 1 = r 1 (3.34) p t 2 = r 2 (3.35) T r 1, r 2 t 1 + t 2 = T (t 1, t 2 ) (t 1, t 2 ) p r 1 (= p + t 1 ) r 2 (= p t 2 ) t 1 + t 2 = T (t 1, t 2 ) p (3.34, 3.35) r 1 r 2 (T, 0) (0, T ) ( ) SS (x i (p + t 1 ) I, (yj (p t 2 )) J j=1 = CS(r 1 ) + T D(r 1 ) + P S(r 2 ) (3.36) 39
1 (3.36) (3.38) (3.41) (3.41) (3.42) 2 D(r 1 ) = S(r 2 ) (3.37) 2 1 2 40
( ) SS (x i (r 1 ) I, (yj (r 2 )) J j=1 (3.38) I J = V i (x i (r 1 )) C j (yj (r 2 )) (3.39) = = j=1 I V i (x i (r 1 )) r 1 I I x i (r 1 ) + r 1 J C j (yj (r 2 )) + r 2 j=1 I x i (r 1 ) J yj (r 2 ) r 2 j=1 ( ) V i (x i (r 1 )) r 1 x i (r 1 ) + r 1 D(r 1 ) r 2 S(r 2 ) J ( ) + r 2 yj (r 2 ) C j (yj (r 2 )) j=1 J yj (r 2 ) (3.40) j=1 (3.41) =CS(r 1 ) + (r 1 r 2 )D(r 1 ) + P S(r 2 ) (3.42) =CS(r 1 ) + T D(r 1 ) + P S(r 2 ) (3.43) 41
r 1 CS(r 1 ) S p t 1 T t 2 r 2 P S(r 2 ) D D(r 1 ) = S(r 2 ) 3.4: 42
4 4.1 Y P p = P (Y ) (4.1) D(p) I x i (p) (4.2) p D X = D(p) X = Y Y = D(p) Y p = P (Y ) Y = D(p) Y = D(P (Y )) (4.3) P D P P < 0 1 1. D(p) = a p a > 0 Y = a p p = a Y P (Y ) = a Y 4.2 y J j = 1, 2,..., J j C j j y j 1 P D (y 1, y 2,..., y J ) (4.4) 43
y Y = y 1 + y 2 + + y J (4.5) j J = 4 j Y j = Y x j (4.6) Y 2 = x 1 + x 3 + x 4 (4.7) π j (y j Y j ) = P (y j + Y j ) y }{{} j C j (y j ) }{{} (4.8) π j (y j Y j ) Y j y j j π j(y j Y j ) = dp (y j + Y j ) y j dy j } {{ } C j(y j ) = 0 (4.9) }{{} dp (y j + Y j ) y j } dy {{ j } = C j(y j ) }{{} (4.10) df(a)g(a) da = f (a)g(a) + f(a)g (a) (4.11) (4.10) dp (y j + Y j ) y j dy j =P (y j + Y j ) y j + P (y j + Y j ) 1 (4.12) =P (y j + Y j ) y j + P (y j + Y j ) (4.13) P (y j + Y j ) y j + P (y j + Y j ) = C }{{} j(y j ) }{{} (4.14) 44
y j j y j P (4.14) P < 0 P (y j + Y j ) y j + P (y j + Y j ) < P (y j + Y j ) (4.15) 4.3 P (y j + Y j ) y j y j P (y j + Y j ) y j P (y j + Y j ) P (y j + Y j ) = 0 P < 0 P P (y j + Y j ) = 0 j y j P (y j + Y j ) P (y j + Y j ) y j + P (y j + Y j ) = C j(y j ) (4.16) y j = y j P (y j + Y j ) = 0 P (y j + Y j ) = C j(y j ) (4.17) y j = y j 45
(y 1, y 2,..., y J ) Y = y 1 + y 2 + + y J P (Y ) = C j(y j ) (4.18) j 2. P (Y ) = a Y C j (y j ) = cy j Y p j C (y j ) = c p = C j(y j ) = c p = P (Y ) = a Y c = p = a Y Y = a c p = c (4.19) Y = a c (4.20) 4.4 1 P (y j + Y j ) y j + P (y j + Y j ) = C j(y j ) (4.21) 1 j Y P (Y ) Y + P (Y ) = C (Y ) (4.22) Y Y Y Y 46
a p = 1 2 a + 1 2 c CS c PS a c a 4.1: 47
3. P (Y ) = a Y C(Y ) = cy Y p (4.21) }{{} 1 Y + (a Y ) = c (4.23) =P (Y ) Y Y = a c 2 p = P (Y ) = a Y = a a c 2 = a + c 2 (4.24) p = a + c 2 Y = a c 2 (4.25) (4.26) 4.5 2 (yj ) J j=1 = (y1, y2,..., yj ) j Y j y j π j (y j Y j) = P (y j + Y j) y j C j (y j ) (4.27) (y j ) J j=1 = (y 1, y 2,..., y J ) j y j 2 J = 2 48
(y j ) J j=1 P (Y ) = a Y C j (y j ) = cy j (4.14) P (y j + Y j ) y j + P (y j + Y j ) = C j(y j ) (4.28) Y = y j + Y j = y 1 + y 2 + + y J da (y 1 + y 2 + + y J ) dy j y j + a (y 1 + y 2 + + y J ) = dcy j dy j (4.29) 1 y j + a (y 1 + y 2 + + y J ) = c (4.30) y j + a (y 1 + y 2 + + y J ) = c (4.31) a Y c = y j (4.32) (4.32) (y j ) J j=1 (y j ) J j=1 j y j = a Y c (4.33) a Y c a Y c Y j y j = Y J Y J = a Y c (4.34) J + 1 Y = Y + Y J J = a c (4.35) Y = J (a c) J + 1 (4.36) 49
p P (Y ) = a Y (4.37) = a J (a c) (4.38) J + 1 J = (1 J J + 1 ) a + = ( J + 1 J + 1 = 1 J + 1 a + J + 1 c (4.39) J J + 1 ) a + J J + 1 c (4.40) J J + 1 c (4.41) p = 1 J + 1 a + J J + 1 c (4.42) Y = J (a c) J + 1 (4.43) J Y p J = 1 p = 1 1 + 1 a + 1 1 + 1 c = a + c 2 Y = 1 a c (a c) = 1 + 1 2 (4.44) (4.45) J J p = 1 J + 1 Y = }{{} 0 J J + 1 }{{} 1 a + J J + 1 }{{} 1 c 0 a + 1 c = c (4.46) (a c) 1 (a c) = a c (4.47) 50
4.6 X(p) = a p 2 j = 1, 2 p j (p 1, p 2 ) 1 c c (p 1, p 2 ) p 1 < p 2 1 X(p 1 ) = a p 1 2 p 2 < p 1 2 X(p 2 ) = a p 2 1 p 1 = p 2 X(p 1) = a p 1 2 2 p 1 < p 2 p 1 = p 2 π 1 (p 1, p 2 ) = p 1 (a p 1 ) c (a p 1 ) (4.48) π 2 (p 1, p 2 ) = 0 (4.49) π 1 (p 1, p 2 ) = π 2 (p 1, p 2 ) = p 1 (a p 1 ) 2 (4.50) p 1 > p 2 π 1 (p 1, p 2 ) = 0 (4.51) π 2 (p 1, p 2 ) = p 2 (a p 2 ) c (a p 2 ) (4.52) 51
a CS p = 1 J+1 a + J J+1 c c PS a c a 4.2: 52
(p 1, p 2) j = 1, 2 p j < c p j c j = 1, 2 p 1 > p 2 > c 1 1 p 2 > p 1 > c 2 p 2 > p 1 > c 2 2 p 1 > p 2 > c 1 p 1 = p 2 > c 1 2 p 1 (a p 1 ) 2 j = 1, 2 ε > 0 p 2 > p 1 = p 1 ε > c (4.53) p 1 p 1 = p 2 = c 1 2 2 53
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