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1 6 3 JEL Classification: D4; K39; L86,,., JSPS 34304,
2 ,398 4% 017,
3 Novos and Waldman (1984) Johnson (1985) Chen and Png (003) Arai (011)
4 [0, 1] x [0, 1] 0 p 0 u x = v p 0 tx v tx x 1 p 1 u x = v p 1 t(1 x) x I x I = 1 + p 1 p 0 (1) t 0 x 0 1 x 1 x 0 = v p 0, x 1 = 1 v p 1 t t () (1) () x 1 x I x 0 v p 0 + p 1 + t (3) v (3) 3 3
5 Stage {r 0, r 1 } Stage Stage 3 0 ( 1 max π 0 = (p 0 r 0 ) p 0 + p ) 1 p 0 t subject to v p 1 + p 0 + t 1 ( 1 max π 1 = (p 1 r 1 ) p 1 p ) 1 p 0 t subject to v p 1 + p 0 + t 3 v (3t + r 0 + r 1 )/ p 0 = t + r 0 + r 1 3, p 1 = t + r 0 + r 1. 3 v r i / 3 4
6 0 1 1: ( 1 max π A = r 0 r 0,r 1 + r ) ( 1 r r 1 6t r ) 1 r 0 r 0 6t r 1 subject to v 3t + r 0 + r 1 v (3t + 1)/ r 0 = r 1 = 1 r0, r1 p 0, p 1 D 0, D1 π A π0, π1 CW SW 1 v (3t + 1)/ r 0 = r 1 = 1, p 0 = p 1 = t + 1, D 0 = D 1 = 1, π A = 1 4, π 0 = π 1 = t CS = v 1 5t 4, SW = v 1 4 t 4 5
7 0 ( ) v p0 max π 0 =(p 0 r 0 ) p 0 t subject to v < p 1 + p 0 + t ( ) v p1 max π 1 =(p 1 r 1 ) p 1 t subject to v < p 1 + p 0 + t v < (r 0 + r 1 + t)/ p 0 = v + r 0 p 1 = v + r 1 v max r 0,r 1 π A = r 0 ( ) v r0 t + r 1 ( v r1 t subject to v < r 0 + r 1 ) + t r 0 r 1 v < t(1 + t)/(t + 1) r 0 = r 1 = v (1 + t) v < t(1 + t)/(t + 1) D 0 = D 1 = r 0 = r 1 = (t + 1)v 4t(1 + t), π A = v (1 + t), p 0 = p 1 = (t + 3)v 4(1 + t), v 4t(1 + t), π 0 = π1 = (t + 1) v 16t(1 + t) CS = (t + 1) v 16t(1 + t), SW = (1t + 16t + 7) v 16t(1 + t) 6
8 F > 0 Stage 1 0 {r 0, r 1 } Stage 0 {p 0, p 1 } Stage 3 0 x 1 < x I < x 0 v x I v r 0 + t/ p 0 = p 1 = v t 0 max π A = r 0 r 0,r 1 + r 1 r 0 r 1 subject to v r 0 + t v (1 + t)/ r0 = r1 = 1 3 v (1 + t)/ r0 = r1 = 1, p 0 = p 1 = v t, D 0 = D 1 = 1, π A = 1 4, π 0 = v 1 t F CS = t 4, SW = v 1 4 t 4 F 7
9 0 : ( ) ( ) v p0 v p1 max π 0 = (p 0 r 0 ) + (p 1 r 1 ) p 0,p 1 t t subject to v < p 1 + p 0 + t v < (r 0 + r 1 + t)/ p 0 = v + r 0 p 1 = v + r 1 ( ) ( ) v r0 v r1 max π A = r 0 + r 1 r 0 r 0,r 1 t t r 1 subject to v < r 0 + r 1 v < t(1 + t)/(t + 1) r 0 = r 1 = v (1 + t) + t 8
10 4 v < t(1 + t)/(t + 1) D 0 = D 1 = r 0 = r 1 = (t + 1)v 4t(1 + t), π A = v (1 + t), p 0 = p 1 = (t + 3)v 4(1 + t), v 4t(1 + t), π 0 = (t + 1) v 8t(1 + t) F CS = (t + 1) v 16t(1 + t), SW = (1t + 16t + 7) v 16t(1 + t) F re / 3 Stage 1 0 {r 0, r 1 } Stage 0 1 {r E } Stage 3 0 {p 0, p 1 } Stage 4 0 ( 1 max Π 0 = (p 0 r 0 ) p 0 + p ) 1 p 0 + (r E r 1 ) t subject to v p 1 + p 0 + t 1 ( 1 maxπ 1 = (p 1 r E ) p 1 p ) 1 p 0 t subject to v p 1 + p 0 + t ( 1 p ) 1 p 0 r E t v r E + (r 0 r 1 + 3t)/ p 0 = t + r E + (r 0 r 1 ) 3 p 1 = t + r E + r 0 r 1 3 9
11 0 1 3: 0 r E 0 ( max Π 0 = t + r E r ) ( 0 + r 1 1 r E 3 + r ) ( 1 r (r E r 1 ) 6t r ) 1 r 0 r E 5t subject to v r E + r 0 r 1 + 3t 0 r E = 1 ( 1 max π A = r 0 r 0,r 1 + r ) ( 1 r r 1 6t r ) 1 r 0 r 0 6t r 1 subject to v 3t + r 0 + r 1 v 3t/ r 0 = r 1 = 1 10
12 5 v 3t/ + 1 r 0 = r 1 = 1, r E = 1, p 0 = p 1 = t + 1, D 0 = D 1 = 1, π A = 1 4, π 0 = π 1 = t CS = v 1 5t 4, SW = v 3 4 t 4 ( ) ( ) v p0 v p1 max Π 0 = (p 0 r 0 ) + (r E r 1 ) p 0 t t subject to v < p 1 + p 0 + t 1 ( ) v p1 max Π 0 = (p 1 r E ) p 1 t subject to v < p 1 + p 0 + t r E v < t + (r 0 + r E )/ p 0 = v + r 0 p 1 = v + r E 0 0 ( ) ( ) ( ) v r0 v r0 v re max Π 0 = + (r E r 1 ) r E r E t t subject to v < t + r 0 + r E 0 r E = v + r 1 (1 + t) ( ) ( ) v r0 v + tv r1 max π A =r 0 + r 1 r 0 r 0,r 1 t 4t(1 + t) r 1 subject to v < (1 + t)(t + r 0) + r 1 4t
13 v < 8t(1 + t)(t + t + 1)/(t + 1)(8t 8 + 8t + 3) r0 v = (1 + t), v(t + 1) r 1 = (t + t + 1) 6 v < 8t(1 + t)(t + t + 1)/(t + 1)(8t 8 + 8t + 3) r0 v = (1 + t), v r 1 = (1 + t) vt (1 + t)(t + t + 1), v(4t + 6t + 3) r E = 4(1 + t)(t + t + 1), p v(t + 3) 0 = 4(1 + t), v(t + 3) p 1 = 4(1 + t) + v(t + 1) 8(1 + t)(t + t + 1), D 0 = v(t + 1) 4(1 + t), D 1 = π A = v(t + 1) 4(1 + t) v(t + 1) 8(1 + t)(t + t + 1), v 4t(1 + t) v 16t(1 + t)(t + t + 1), π 0 = v (t + 1) 16t(1 + t) v (16t 4 + 3t 3 + 0t + 4t 1) 3t(1 + t)(t + t + 1), π 1 = CS = v (t + 1) 16t(1 + t) v (t + 1) (8t + 8t + 3) 18t(1 + t) (t + t + 1), v (t + 1) 6 64t(1 + t) (t + t + 1) SW = v (1t + 16t + 7) v (t + 1) (16t t + 44t + 13) 16t(1 + t) 18t(1 + t) (t + t + 1).4 4 Stage {r 0, r 1 } Stage 0 1 {r E } Stage 3 Stage 4 1
14 0 1 4: 0 ( 1 max π 0 = (p 0 r 0 ) p 0 + p ) ( 1 p r E t p 1 p 0 t subject to v p 1 + p 0 + t 1 ( 1 max π 1 = (p 1 r 1 r E ) p 1 p ) 1 p 0 t subject to v p 1 + p 0 + t ) r E v r E + (3t + r 0 + r 1 )/ p 0 = t + r E + r 1 + r 0 3 p 1 = t + r E + r 1 + r ( max π 0 = t + r E + r ) ( 1 r 0 1 r E 3 + r ) ( 1 r r E 6t r ) 1 r 0 r E 6t subject to v r E + 3t + r 0 + r 1 v 1 + (3t + r 0 + r 1 )/ 0 r E = 1. 13
15 ( 1 max π A = r 0 r 0,r 1 + r ) ( 1 r r 1 6t r ) 1 r 0 r 0 6t r 1 subject to v 1 + 3t + r 0 + r 1 v 3(1 + t)/ r 0 = r 1 = 1. 7 v 3(1 + t)/ r 0 = r 1 = 1, r E = 1, p 0 = p 1 = t + 3, D 0 = D 1 = 1, π A = 1 4, π 0 = t + 1, π 1 = t CS = v 3 5t 4, SW = v 1 4 t 4 ( ) ( v p0 v p1 max Π 0 =(p 0 r 0 ) + r E p 0 t t subject to v < p 1 + p 0 + t ) r E ( ) v p1 max Π 1 =(p 1 r 1 r E ) p 1 t subject to v < p 1 + p 0 + t v < (r 0 + r 1 + t + r E )/ p 0 = v + r 0 p 1 = v + r 1 + r E. 14
16 0 1 0 max π 0 = v r ( ) ( ) 0 v r0 v r1 r E + r E r E r E t t subject to v < r 0 + r 1 + t + r E v < {(1 + t)(r 0 + r 1 + t) r 1 }/(4t + 3) r E = v r 1 (1 + t). ( ) v r0 max π A =r 0 r 0,r 1 t + r 1(t + 1)(v r 1 ) 4t(1 + t) subject to v < (1 + t)(r 0 + r 1 + t) r 1 4t + 3 r 0 r 1 v < 8t(1 + t)(t + 4t + 1)/(4t + 1)(t + 1)(t + 3) r0 v v(t + 1) = (1 + t) r 1 = (t + 4t + 1). 8 v < 8t(1 + t)(t + 4t + 1)/(4t + 1)(t + 1)(t + 3) r0 v = (1 + t), v r 1 = (1 + t) vt (1 + t)(t + 4t + 1), v(4t + 6t + 1) r E = 4(1 + t)(t + 4t + 1) p v(t + 3) 0 = 4(1 + t), v(t + 3) p 1 = 4(1 + t) + v(t + 1) 8(1 + t)(t + 4t + 1), D0 v(t + 1) = 4t(1 + t), v(t + 1) D 1 = 4t(1 + t) v(t + 1) 8t(1 + t)(t + 4t + 1), πa v = 4t(1 + t) v (4t + 1) 16t(1 + t)(t + 4t + 1), π 0 = (1 + t) v 16t(1 + t) + v (4t + 6t + 1) 3t(1 + t)(t + 4t + 1), π1 = v (4t + 6t + 1) (t + 1) 64t(1 + t)(t + 4t + 1), CS = (t + 1) v 16t(1 + t) v (t + 1) 3 (4t + 1)(t + 3) 18t(1 + t) (t + 4t + 1) SW = v (t + 1) 16t(t + 1) v (t + 1) (16t t + 44t + 13) 18t(1 + t) (t + 4t + 1) 15
17 v 3(1 + t)/ v < t(1 + t)/(t + 1) v 3(1 + t)/ F re / v < t(1 + t)/(t + 1) r E r 1 p v 3(1 + t)/ 1. F < v t 1. F v t 1 F F F 16
18 r E p 1 p 0 p v < t(1 + t)/(t + 1) 1. 4t(1 + t)(t + 1) > 1 t F < α F α. 4t(1 + t)(t + 1) 1 t F < β F β α v (t+1), β v (t+1) v (1 16t 4 3t 3 0t 4t) 16t(1+t) 16t(1+t) 3t(1+t)(t +t+1) F 9 v < t(1+t)/(t+ 1) v 3(1 + t)/ t r 1 r E 0 1 r E t F t
19 11 v 3(1 + t)/ 1. F < v t 1. F v t r 1 r E r E r v < t(1 + t)/(t + 1) 1. F < γ. F γ γ v (t+1) 16t(1+t) v (4t +6t+1) 3t(1+t)(t +4t+1) r E r 1 r v 3(1 + t)/ 1. 18
20 . 3. F > v t 1 v t 1 F v t 1 v t 1 > F 4. F > v t 1 v t 1 F v t 1 v t 1 > F v 3(1 + t)/ 9 F v t 1/ r 1 r E p 1 p v r E v t 1/ > F v t 1 0 F v t 1/ > F v t 1 v t 1 > F 3 v < t(1 + t)/(t + 1) 1. 19
21 . F γ F < γ 3. 4t(1 + t)(t + 1) 1 F β r 1 r E 4t(1 + t)(t + 1) > 1 γ < F 4t(1 + t)(t + 1) < 1 γ < F < β r 1 4. F γ F < γ 5. 4t(1 + t)(t + 1) > 1 F α 4t(1 + t)(t + 1) > 1 > 4t(1 + t) α > F γ 4t(1 + t)(t + 1) 1 β > F γ 4t(1 + t)(t + 1) 1 F β 4t(1 + t) > 1 α > F γ v 3(1 + t)/ 10 4t(1 + t)(t + 1) > 1 F α r 1 r E r E r 1 r 1 4t(1 + t)(t + 1) 1 F β 0
22 r E r 1 r E r 1 r E r 1 r E r 1 r 1 r 1 r E r 1 1 r E r 1 r E 4t(1 + t)(t + 1) > 1 α > F γ 4t(1 + t)(t + 1) 1 β > F γ r E r 1 r 1 r 1 1 4t(1 + t)(1 + t) 1 F β t r 1 r E r E 1 v < t(1 + t)/(t + 1) t 4t(1 + t) > 1 F
23 3 Arai, 011; Novos and Waldman, 1984; Yoon, 00 v 4
24 v < t(1 + t)/(t + 1) SW 4 SW 3 16t 6 + 1t 5 78t 4 148t 3 106t 35t t(1 + t)(t + 1) < 1 t 16t 6 + 1t 5 78t 4 148t 3 106t 35t < 4 SW4 > SW3 SW SW 4 F v (1 + t) 16t(1 + t) v (1 + t)(16t t + 30t + 5)(4t + 4t 1) 18t(t + 4t + 1) (1 + t) (4) 4t(1 + t)(t + 1) < 1 (4) α F < α SW > SW4 4t(1 + t)(t + 1) < 1 t 4t(1 + t)(t + 1) 4t(1 + t) > 1 (4) γ F SW < SW4. (013) 013 R&D. (01) 01 R&D Arai, Y. (011) Civil and Criminal Penalties for Copyright Infringement. Information Economics & Policy, Vol.3, pp Chen, Y. and Png, I. (003) Information Goods Pricing and Copyright Enforcement: Welfare Analysis. Information Systems Research, Vol. 14, No.1, pp Johnson, W. (1985) The Economics of Copying. Journal of Political Economy, Vol.93, No.1, pp
25 Novos, I. and Waldman, M. (1984) The Effect of Increased Copyright Protections: An Analytic Approach. Journal of Political Economy, Vol.9, No., pp Yoon, K. (00) The Optimal Level of Copyright Protection. Information Economics and Policy, Vol.14, pp
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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
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