Title 最適年金の理論 Author(s) 藤井, 隆雄 ; 林, 史明 ; 入谷, 純 ; 小黒, 一正 Citation Issue 2012-06 Date Type Technical Report Text Version publisher URL http://hdl.handle.net/10086/23085 Right Hitotsubashi University Repository
2012 6 15 2 2 2 m (1) (2) m 0, m < 1 1, OLG Overlapping Generation Model,,,,, Feldstein [7], [8], ( ) ( ) ( (C) 24530349) E-mail:fujii@econ.kobe-u.ac.jp E-mail:082e527e@stu.kobe-u.ac.jp E-mail:iritani@fuec.fukuyama-u.ac.jp E-mail:k-oguro@ier.hit-u.ac.jp
[13],, 2, OLG, Barro [3], [4], Bohn [5] Judd [10],,, Samuelson [11] Diamond [6] OLG,,,,,, Auerbach and Kotlikoff [1], OLG, OLG (i) (ii) 2
(iii) m (i) (ii) Barro [3], [4], Bohn [5] Judd [10] (iii) 1 2 (1) (2) m 0, m < 1 (2-1) (2-2) (2-3) i (2-4) resp. 100 resp. (2-5) (3) m 1 1 100 0 (1) (2-3) 3
(2-2),, Barro [2],,,, OLG, [14] Horioka [9],,,, 2 3 4 2 2.1 t u(c 1t, c 2t+1, h t ), u : R 3 + R (1) c 1t, c 2t+1 t t + 1 h t 1 u (c 1t, c 2t+1 ) m m m 0 0 < m < 1 c 1t α c 2t+1 β h t γ, α + β + γ 1 4
τ t ξ t+1 r c 1t + s t = (1 τ t )w t (1 h t ), c 2t+1 = ()s t + ξ t+1 τ t w t (1 h t ) c 1t + c 2t+1 + (1 θ t)w t h t = (1 θ t )w t (2) θ t θ t = τ t ξ t+1 τ t /() ξ t+1 τ t = (τ t θ t )() r, w t, θ t c 1t, c 2t+1, h t t max u(c 1, c 2, h) subject to c 1 + c 2 + (1 θ)wh = (1 θ)w (3) c 1 ((1 θ)w, r), c 2 ((1 θ)w, r), h((1 θ)w, r) c 1t ((1 θ)w, r), c 2,t+1 ((1 θ)w, r), h t ((1 θ)w, r), c 1t ((1 θ)w, r) = c 1t ((1 θ)w, r), t, t c 2,t+1 ((1 θ)w, r) = c 2,t +1((1 θ)w, r), (4) h t ((1 θ)w, r) = h t ((1 θ)w, r), [13] [12] m = 1 h t w t c 1t c 2t+1 w t m 0 1 h t r c 1t c 2t+1 (1 θ t )w t 5
h t ((1 θ t )w t, r) = h t (1, r), c 1t ((1 θ t )w t, r) = (1 θ t )w t c 1t (1, r), c 2t+1 ((1 θ t )w t, r) = (1 θ t )w t c 2t+1 (1, r) (5) (3) max u(φ 1, φ 2, h) subject to φ 1 + φ 2 φ 1,φ 2,h + h = 1 (6) (3), (6) (c 1, c 2, h ), ( ˆφ 1, ˆφ 2, ĥ) (c 1/(1 θ)w, c 2/(1 θ)w, h ) (6) u( ˆφ 1, ˆφ 2, ĥ) u(c 1/(1 θ)w, c 2/(1 θ)w, h ) = u(c 1, c 2, h )/{(1 θ)w} m ((1 θ)w ˆφ 1, (1 θ)w ˆφ 2, ĥ) (3) u(c 1, c 2, h ) u((1 θ)w ˆφ 1, (1 θ)w ˆφ 2, ĥ) = {(1 θ)w}m u( ˆφ 1, ˆφ 2, ĥ) u(c 1, c 2, h ) = u((1 θ)w ˆφ 1, (1 θ)w ˆφ 2, ĥ) c 1 = (1 θ)w ˆφ 1, c 2 = (1 θ)w ˆφ 2, h = ĥ (6) w(1 θ) c i ((1 θ)w, r) = c i = (1 θ)w ˆφ i = (1 θ)wc i (1, r), i = 1, 2 h((1 θ)w, r) = h = ĥ = h(1, r) (5) 1 V t (θ t, r) = u(c 1t ((1 θ t )w t, r), c 2t+1 ((1 θ t )w t, r), h t ((1 θ t )w t, r)) = {(1 θ t )w t } m µ, µ := u(c 1 (1, r), c 2 (1, r), h t (1, r)) (7) m < 1 1 θ t r 6
2.2 t = 0 G (τ t, ξ t+1 ), t = 0, 1,... τ t t ξ t+1 t t + 1 (τ t, ξ t+1 ), t = 0, 1, t = 0 G = τ 0 w 0 (1 h 0 )L 0 + B 0 (8) B 0 t 1 ξ t τ t 1 w t 1 (1 h t 1 )L t 1 + ()B t 1 = τ t w t (1 h t )L t + B t (9) (8), (9) B t τ t, ξ t+1 2 r t g t, n t w 0, L 0 t 1 t 1 w t = w 0 (1 + g j ), L t = L 0 (1 + n j ) 0 < q < 1 q() > (1 + g t )(1 + n t ), t = 1, 2,... 1 () < (1 + g t )(1 + n t ) 1 () > (1 + g t )(1 + n t ), t = 1, 2,... (1 + g t)(1 + n t )/() = 0 ( ) 7
w t lim w t = w > 0 t θ t > 0 1 > ξ t+1, t = 0, 1,... θ t 0 ξ t+1 h t h t ((1 θ t )w t, r) 2 τ t, t = 0, 1,... (8) (9) lim t B t+1 = 0 (10) () t+1 θ t I t () t = G (11) θ t = τ t ξ t+1 τ t /(1+r), I t = w t (1 h t )L t, t = 0, 1,... θ t 1, t = 0, 1,... (11) τ t, B t, t = 0, 1,... (8), (9) (10) (9) (τ t 1 θ t 1 )w t 1 (1 h t 1 )L t 1 + B t 1 = τ tw t (1 h t )L t + B t 8
t I t = w t (1 h t )L t θ t 1 I t 1 = τ t 1 I t 1 + B t 1 τ ti t B t t = 1,..., n (8) 0 = G τ 0 I 0 B 0 θ 0 I 0 = τ 0 I 0 + B 0 τ 1I 1 B 1 θ 1 I 1 = τ 1I 1 + B 1 τ 2I 2 () B 2 2 () 2 θ 2 I 2 () = τ 2I 2 2 () + B 2 2 () τ 3I 3 2 () B 3 3 () 3... θ n I n () = τ ni n n () + B n n () τ n+1i n+1 n () B n+1 n+1 () n+1 (12) n θ t I t () t = G τ n+1i n+1 () n+1 B n+1 () n+1 τ n+1 I n+1 () τ n+1w n+1 L n+1 n+1 () n+1 = τ n n+1w 0 (1 + g n t)l 0 (1 + n t) () n+1 n (1 + g t )(1 + n t ) = τ n+1 w 0 L 0 () (1 + g t )(1 + n t )/() < q < 1 (10) θ t I t () t = G 9
G θ t 1, t = 0, 1,... (11) θ t = 1 ξ t+1 = 0, τ t = 1 θ t = τ t τ t ξ t+1 /() θ t < 1 ξ t+1 τ t 1 θ t = τ t τ t ξ t+1 /() τ t θ t θ t = τ t τ t ξ t+1 /() τ t, ξ t+1 B 0 = G τ 0 w 0 (1 h 0 )L 0 B t+1 = (τ t θ t )I t () τ t+1 I t+1 + ()B t, t = 0, 1,... B t, t = 0, 1, G = τ 0 w 0 (1 h 0 )L 0 + B 0 θ t I t = B t B t+1 + τ ti t τ t+1i t+1, t = 0, 1,... θ 0 I 0 + + θ ni n () n = B 0 + τ 0 I 0 τ n+1i n+1 () n+1 B n+1 () n+1 = G τ n+1i n+1 () n+1 B n+1 () n+1 n τ n+1 I n+1 /() n+1 0 θ ti t /() t = G lim t B t () t = 0 (11) (θ t ) 3 I t/() t > G I t/() t = (1 + g t )(1 + n t ) 2 lim t B t /() t = b lim t τ t I t /() t = g (11) θ t I t () = G g b t 10
3 W (θ 0, θ 1,... ) = β t V t (θ t, r), β 0 = L 0, β t = L 0 t 1 (1 + n j) (1 + R) t, t = 1, 2,... R W (θ 0, θ 1,... ) = µ β t {(1 θ t )w t } m, µ = u(c 1 (1, r), c 2 (1, r), h t (1, r)) I t = (1 h t ((1 θ t )w t, r))w t L t = (1 h(1, r))w t L t, t = 1, 2,... I t max W (θ 0, θ 1,... ) subject to θ 0,θ 1,... θ t I t () t = G γ t = I t /() t, t = 0, 1, 2,... max µ β t {(1 θ t )w t } m subject to θ 0,θ 1,... γ t θ t = G, θ t 1, t = 0, 1,... (13) θ t < 0 4 (13) (θ t ) R 3.1 m < 1, m 0 3.2 1 m 11
3.1 m < 1 m 0 V/ θ t (θ t, r) = µm(1 θ t ) m 1 w t m lim θt 1 V/ θ t (θ t, r) = θ t = 1 t 2 (13) µmw t β t {(1 θ t )w t } m 1 = δγ t, t = 0, 1, 2,... µ, δ 3 w t β t {(1 θ t )w t } m 1 w t+1 β t+1 {(1 θ t+1 )w t+1 } = γ t m 1 γ t+1 ( ) 1 θt 1/(1 m) 1 + R = (1 + g 1 θt+1 t ), t = 0, 1,... θ t > θ t+1, iff (1 + g t ) ( ) 1/(1 m) 1 + R < 1 (14) t, t (t < t ) 1 θ t 1 θ t = ( ) (t 1 + R t)/(1 m) t 1 j=t (1 + g j ) (15) 1 m < 1, m 0 2 m < 0 µ < 0 µm > 0 m < 0 n (1 + δ) (i s) c s 1 1/ν 1 1/ν, ν < 1 i=s m > 0 µ > 0 mµ > 0 3 w t = 1, w t+1 1 + g t γ t γ t+1 = (1 + g t )(1 + n t ), β t β t+1 = 1 + R 1 + n t. 12
(i) resp. R r (r R ) t t (ii) (iii) (iii) (ii) 1 (i) r > R (i) (15) t = 1, 2,... 1 θ t = α t (1 θ 0), α t def = 1 t 1 (1 + g j) ( 1 + R ) t/(1 m) = w ( ) t/(1 m) 0, w t 1 + R α 0 def = 1 α α def = γ t α t. 13
γ t G = θ 0 = 1 γ t (1 θt ) = γ j G α γ t α t (1 θ0) (16) θt = 1 α γ j G t, t = 0, 1,... (17) α α, γ = γ j α, γ 3.1.1 3.1.2 x = (()/(1 + R)) 1/(1 m) α n i = I 0 1 + n i γ n i = I 0 1 + n i ( i ( i 1 + n j i+1 1 + n j xi+1 + xi+2 + (1 + g j )(1 + n j ) + i+1 t 1 1 + n j t 1 (1 + g j )(1 + n j ) xt = t 1 ) (1 + g j )(1 + n j ) 1 + n j + ) ( ) t 1 x t (1 + g j ) t = i + 1, i + 2,... x ( t 1 1/t (1 + g j )) = 1 + ḡ t ḡ t t 0 < α γ n i n i x i + 1 n i γ α (γ G)/α 14
2 m < 1, m 0 (iv) (16) (17) (v) x = (()/(1 + R)) 1/(1 m) i ḡ i+1 x < 1 + ḡ i+j, j = 1, 2,... i 3.1.1 j γ j (16), (17) γ j 2 γ j = I 0 + I 1 + I 2 () 2 + γ j G < w 0 L 0 (1 + q + q 2 + ) = w 0L 0 1 q < 3.1.2 α α t (θ t ) budget (11) γ j G = lim t (1 θ 0) t γ j α j θ0 1 θ0 = 1 θ0 < 1 α = t γ t α t = 2 α < 15
α < α = α t γ t = I 0 (1 + t=1 ( ) t/(1 m) t 1 1 + R 1 + n j x = ((1+r)/(1+R)) 1/(1 m), a 0 = 1, a t = t 1 (1+n j)/(1+ r), t = 1, 2,... α I 0 = a 0 + a 1 x + a 2 x 2 + x Cauchy-Hadamard k 1 k = lim sup a t 1/t t 4 t ( ) 1/t (1 + a 1/t n0 ) (1 + n t 1 ) t = q () t 5 k 1/q(> 1) (A) x > k α = (B) x < k 0 < α < 2 α = (A) α t α = (w 0 /w t )x t I 0 (a 0 + a 1 x + a 2 x 2 + ) w t w (B) ( ) k > x > 1 α t /α 4 [15], 15 page 5 t ( ) 16 )
{ ( ) 1 = x α t /α (w(1 h) } 1 a j ( ) 1 > x > 0 α t /α 0 0 t 1 t n t ((1 + n 0 ) (1 + n t 1 )) 1/t = 1 + n t, t = 1, 2,... n t, t = 1, 2,... n k = 1 + n > 1 q 3.1.3 θ t ( ) ( ) r x ( ) k > x > 1 θ 0 > 0, t : t t, θ t < 0, lim t θ t = x 1 m = ()/(1 + R) x > 1 r > R ( ) x = 1 θt i 1 (1+n j )(1+g j ) θ0 i=1 G/I (1+r) 0 = 1 i 1 (1+n j ) i=1 (1+r) i 1 (1+n j )(1+g j ) θt 1 i=1 G/I (1+r) 0 = 1 t 1 (1 + g i 1 (1+n j) j ) i=1 (1+r) 17
g t > 0, g t = 0, g t < 0 θ t < θ t+1, θ t = θ t+1, θ t > θ t+1 θ t θ i 1 (1+n j )(1+g j ) θ 1 i=1 G/I (1+r) 0 = 1 (1 + g j) i 1 (1+n j ) i=1 (1+r) ( ) x < 1 θ t θ t 1 as t 3 m < 1, m 0 (vi) (vii) (viii) 100 (vii) Barro [3], [4], Bohn [5] R = r (vii) x = 1 G 1 ( γ j G){(w(1 h) a j} 1 t θ t 18
3.2 m > 1 (13) θ t 1 0 θ t 1 t m > 1 c t = β tw t m γ t, b t = c t (1 θ t ) m 1 t = 1, 2,... θ t, t = 0, 1,... (13) θ t, θ t 0 < θ t, θ t < 1 b t b t > 0 ε > 0 θ t = θ t γ t γ t ε, θ t = θ t + ε, θ j = θ j, if j t, t (18) ε- ( γ t θt + γ t θt = γ t θ t γ ) t ε + γ t (θ t + ε) = γ t θ t + γ t θ t γ t θ j, j = 0, 1,... (13) 0 < η < 1 W ( θ 0, θ 1,... ) n=1 W (θ 0, θ 1,... ) n=1 { = mεγ t ct (1 θ t + ηεγ t /γ t ) m 1 c t (1 θ t ηε) m 1} m > 1, b t b t > 0 c t (1 θ t ) m 1 = b t b t = c t (1 θ t ) m 1 ε > 0 c t (1 θ t ηε) m 1 < b t b t < c t (1 θ t + ηεγ t /γ t ) m 1 19
W ( θ 0, θ 1,... ) n=1 W (θ 0, θ 1,... ) > 0 n=1 (13) θ t, t = 0, 1,... #{i 1 > θ i > 0} 1 1 0 4 m > 1 θ t, t = 0, 1,... 1 0 3.3 m = 1 r R r = R 3.3.1 r R β t w m t γ t = L 0 = Q t 1 (l+n j) (1+R) t w 0 t 1 (1 + g j) I t /() t ( ) t 1, t = 0, 1, 2, 1 + R 1 h d t def = β t w t /γ t, t = 0, 1, 2 γ t = γ t 1 (l + n t 1 )(1 + g t 1 ) < γ t 1 γ t, t = 1, 2,... 20
Assertion 1 θ t, t = 0, 1,... t=1 γ tθ t = G t, t d t = β tw t > d t = β t w t, 0 < θ t 1, 0 θ t < 1 γ t γ t θ t = 0 θ t = 1 ε- ε Assertion 1 1 0 N ā N(a) def = {t N d t a}, R(a) def = γ t. t N\N(a) ā def = inf {a R(a) > G }. Assertion 2 R(a) d t Assertion 3 ā = β t w t /γ t t Assertion 4 R(ā) G Assertion 5 (ˆθ j ) ˆθ j = 0 if j N(ā) \ {t} = 1 if j N \ N(ā) (ˆθ j ) t Assertion 3 t (ˆθ j ) (13) Appendix 21
3.3.2 r = R β t w t γ t = z, t = 1, 2, θ t, t = 0, 1, θ tγ t = G W (θ 0, θ 1, ) = β t w t (1 θ t ) = zγ t (1 θ t ) = zγ t zγ t θ t = z γ t zg 5 m = 1 R = r R r β t w t /γ t 100 4 c 1t + s t = (1 τ t )w t (1 h t ), c 2t+1 = ()s t + ( d t )τ t w t (1 h t ) τ t w t (1 h t ) d t τ t w t (1 h t ) c 1t + c 2t+1 + ( 1 τ td t ) ( w t h t = 1 τ ) td t w t 22
θ t θ t = τ t d t /() r, w t, θ t c 1t, c 2t+1, h t t max u(c 1, c 2, h) subject to c 1 + c 2 + (1 θ)wh = (1 θ)w (3) G 0 = B 0 ()B t 1 = B t + d t τ t 1 w(1 h t 1 )L t 1, t = 1, 2,... B 0 G 0 B t d t τ t 1 w(1 h t 1 )L t 1 ()B t 1 P 0 + A 0 = G 0 + τ 0 w 0 (1 h 0 )L 0, ()τ t 1 w t 1 (1 h t )L t 1 + A t = ()A t 1 + τ t w t (1 h t )L t, t = 1, 2,... G 0 τ 0 w 0 (1 h 0 ) P 0 A 0 ()A t 1 + τ t w t (1 h t ) ( a t )τ t 1 w t 1 (1 h t ) + a t τ t 1 w t 1 (1 h t ) + A t P 0 + A 0 = B 0 + τ 0 w 0 (1 h 0 )L 0 ( a t )τ t 1 w t 1 (1 h t 1 )L t 1 + A t + ()B t 1 = ()A t 1 + τ t w t (1 h t )L t + B t, t = 1, 2,... 23
ˆB t = B t A t, t = 0, 1,... P 0 = ˆB 0 + τ 0 w 0 (1 h 0 )L 0 ( d t )τ t 1 w t 1 (1 h t 1 )L t 1 + () ˆB t 1 = τ t w t (1 h t )L t + ˆB t, t = 1, 2,... (τ t 1 θ t )w t 1 (1 h t 1 )L t 1 + ˆB t 1 = τ tw t (1 h t )L t + ˆB t, t = 1, 2,... I t = w t (1 h t )L t 0 = P 0 ˆB 0 τ 0 I 0 θ 0 I 0 = τ 0 I 0 τ 1I 1 + ˆB 0 ˆB 1 θ 1 I 1 = τ 1I 1 τ 2I 2 () + ˆB 1 2 ˆB2 () 2.. θ n I n () = τ ni n n () τ n+1i n+1 n () + ˆBn n+1 () ˆBn+1 n () n+1 n θ t I t () t = P 0 τ n+1i n+1 () n+1 ˆBn+1 () n+1 Theorem 2 [1] Auerbach, A.J. and L.J. Kotlikoff (1987) Dynamic Fiscal Policy, Cambridge: Cambridge University Press. [2] Barro, R.J. (1974) Are Government Bonds Net Wealth?, The Journal of Political Economy, Vol. 82(6), 1095-1117. 24
[3] Barro, R.J. (1979) On the Determination of the Public Debt, Journal of Political Economy, Vol.87, 940-971. [4] Barro, R.J. (1999) Notes on Optimal Debt Management, Journal of Applied Economics, Vol.2, 281-289. [5] Bohn, H. (1990) Tax Smoothing with Financial Instruments, American Economic Review, Vol.80, 1217-1230. [6] Diamond,P.A. (1965) National Debt in A Neo-classical Growth Model, American Economic Review, Vol.55, 1125-1150. [7] Feldstein, M. (1995) Would privatizing social security raise economic welfare?, NBER. WorkinPaper, No.5281. [8] Feldstein, M. (1998) The effect of privatizing social security on economic welfare: Appendix to the Introduction, in M. Feldstein ed., Privatizing Social Security, The University of Chicago Press, 1-29. [9] Horioka, C. Y. (2002) Are the Japanese Selfish, Altruistic, or Dynastic?, The Japanese Economic Review, Vol.53(1), 26-54. [10] Judd, K.L. (1999) Optimal taxation and sepending in general competitive growth models, Journal of Public Economics, Vol.71, 1-26. [11] Samuelson, P.A. (1958) An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money, Journal of Political Economy, Vol.66, 467-482. [12] (2007), Discussion Paper Series 07A-5 [13] (2008) 224-233 [14] (1996),,. 25
[15] (1965) 2 Appendix 4.3.1 Assertions Assertion 1 θ t, t = 0, 1,... t=1 γ tθ t = G t, t d t = β tw t > d t = β t w t, 0 < θ t 1, 0 θ t < 1 γ t γ t d t > 0 (18) θ k, k = 0, 1,... β t w t (1 θ t ) + β t w t (1 θ t ) {β t w t (1 θ t ) + β t w t (1 θ t )} = β t w t (θ t θ t ) + β t w t (θ t θ t ) = εγ t (d t d t ) > 0 θ t = 0 θ t = 1 ε welfare Assertion 2 a < a N(a ) N(a) R(a) a a = d t d t, t = 1, 2,... d k, d k d k < a < d k {t} = {j N d k < d j < a k } b [a, d k ) R(b) = R(a) γ t b (d k, a] R(b) = R(a) a = f def = inf{d t t N} b [0, f] R(b) = 0 f a = s def = sup{d t t N} s = s < b [s, ) R(b) = j=1 γ j s R(a) Assertion 3 d j = β j w j /γ j, j = 0, 1, R(0) = 0 t a(t) > max{d 0,..., d t } a(t) R ++ R(a(t)) t γ j t γ t(> G) j R(a(j)) j γ t > G j N ā β j w j /γ j 3 d k d k < ā d k d k > ā 26
k, k d k < ā < d k d t, t = 0, 1, 2... ā > d 0 = max{d 0, d 1,... } ε ā ε > d 0 N(ā ε) = N ā ε {a R(a) > G} ã > ā ε R(ã) < G ã R( ) G < γ t = R(ā ε) R(ã) < G R(ā) = 0 < G d t, t = 1, 2, ā < d 0 < d i, i ε > 0 ā + ε < d 0 < d i, i R(ā + ε) = 0 < G ā d t, t = 1, 2, ā = 0 j γ t + t=j+1 γ t = γ t > G j G j t=j+1 γ t < G γ j > γ j+1 ã γ j > ã > γ j+1 > 0 R(ã) = t=j+1 γ t < G ã {a R(a) < G} ā ã > 0 δ > 0 {j N a j (ā δ, ā+δ) } = a, a ā δ < a < ā < a < ā+δ N(ā) = N(a ) = N(a ) R(a ) = R(ā) = R(a ) R(a ) > G ā Assertion 4 R(ā) < G Assertion 3 ā = d t = β t w t /γ t d j, j = 0, 1, t k ā < d k δ > 0 ā+δ < β k w k /γ k R(ā) = R(a+δ) < G ā Assertion 5 Step 1 R(ā) > G Assertion 3 β t w t /γ t = ā t N(ā) t ā R(ā) = γ j > G > j N\N(ā) j N\(N(ā) {t}) ˆθ t G = γ j + ˆθ t γ t, 0 < ˆθ t < 1 j N\(N(ā) {t}) ˆθ j = 0 if j N(ā) \ {t} = 1 if j N \ N(ā) γ j 27
(ˆθ j ) step 2 R(ā) = G β t w t /γ t = ā t ā R(ā) = ˆθ j = 0, (ˆθ j ) j N\N(ā) γ j = G if j N(ā) = 1, if j / N(ā) Step 3 θ t, t = 0, 1,... θ j 0 1 P = { t θ t = 0} t, t N t N(ā) \ P, t P \ N(ā) β t w t ā > β t w t, 1 θt > 0, θt γ t γ = 0 t t, t (18) θk, k = 0, 1,... θ k, k = 0, 1,... P N(ā) or N(ā) P Step 4 claim P N(ā). N(ā) P N \ N(ā) N \ P d t = ā G = j N\P θ j γ j < j N\N(ā) γ j + ˆθ t γ t = G Step 5 P = N(ā) θ j = ˆθ j, j = 1, 2, Step 6 P N(ā) (a) R(ā) > G. (b) t / P where ā = d t. 28
(c) P = N(ā) \ {t}, where ā = d t. (d) θ j = ˆθ j, j = 1, 2,. (a) Assertion 4 R(ā) G P N(ā) R(ā) = G k N(ā) \ P θ k > 0 j / N(ā) 1 > θ j > 0 a k > a j, θ k > 0, 1 > θ j > 0 Assertion 1 θi, i = 0, 1, j / N(ā) 1 = θj G = j N\P θ j γ j > j N\N(ā) γ j = G (b) t P k N k N(ā) \ P k t d k > d t = ā, θ k > 0, θ t = 0 Step 1 θj, j = 0, 1, (c) k N k N(ā) \ {t} k / P d k > d t = ā, θ k > 0, θ t > 0 θ k = θ t = 1 h N\N(ā) θ h = 1 G = θj γ j > γ j = G j N\P j N\N(ā) h N \ N(ā) 1 > θh > 0 d k > d h, θ k = 1, 0 < θ h < 1 Step 1 k, h (d) P N(ā) θ j = ˆθ j, j = 1, 2, 29