(1) (2) 27 7 15 (1) (2), E-mail: bessho@econ.keio.ac.jp

1 2 1.1......................................... 2 1.2............................... 2 1.3............................... 3 1.4............................ 4 2 9 2.1...................................... 9 2.2................................ 15 3 18 3.1.......................................... 18 3.2.......................................... 23 4 27 4.1.................................. 27 4.2....................................... 30 5 33 5.1................................ 33 5.2 First-best..................... 34 5.3 Second-best................... 40 6 44 6.1 First-best......................................... 45 6.2 Second-best....................................... 48 1

7 54 7.1................................. 54 7.2................................ 55 7.3......................................... 62 8 68 8.1..................................... 68 8.2......................................... 74 2

1 1.1 2005 cost benefit B C NB NB = B C (1.1) NB 0 NB < 0 with/without 1.2 3

1.3 2 1 market failure 2 1 4

2 1 1 1 1 1 1 1.4 [1] [2] [3] [4] [5] [6] [7] Net Present Value 5

[8] [9] 2004 2 2 40 8 1 6 4 6

1 1 1 12 1 200 7

CEA: Cost Effective Analysis t B C s (1 + s) t s T P V (B) = T t=0 B T t (1 + s) t, P V (C) = C t (1 + s) t t=0 NP V = P V (B) P V (C) NPV IRR: Internal Rate of Return 8

9

2 2.1 Vilfredo Pareto, 1848-1923 Pareto efficient allocation Pareto improving Pareto dominant (1) 3 (2) (1) (2) 10

2.1.1 MRS: Marginal Rate of Substitution A B 2 X Y 2 X 1 Y (3) X Y A B X Y A B A 1 B 2 A B X 1 B A Y 1 A 1 X 1 Y 1 A B 2 X 1/2 Y 1 B 2 2 2.1.2 X Y (3) X Y 11

X Y X 1 Y MRT: Marginal Rate of Transformation 2.1.3 2 X 2.1: Y 2.1 e e E E e 12

2.1 4 1 4 2 X 1 Y 4 1 X 1 Y 1 Y 4 X 4 2 2.1 (4) E X 1 Y MRS F MRT MRS F F O e f f e f 2 e F G 2 G 2 E e 2.1.4 2 2 2.1 1 2 2.2 2.2 c abc (4) 13

2.2: c d d A B compensation a b c d (5) 2.1.5 2 2.2 a, b,d (5) B A B 14

3 social welfare function Bergson-Samuelson (6) A B 2 Bergson-Samuelson W = W (U A, U B ) (2.1) 2.2 5 (6) 15

W/ U i > 0 2.2 W (U A, U B ) = W (U B, U A ) 2.2 2 45 Bergson-Samuelson 2.2 2 W = U A + U B W = min(u A, U B ) 2.2 16

price taker 17

1 3 18

3 market failure 3.1 2 3.1.1 2 A B 2 x A, x B 2 19

G 2 M A, M B p 1 A B g A, g B G = g A + g B A u A = u A (x A, G), x A + pg A = M A (3.1) B u B = u B (x B, G), x B + pg B = M B (3.2) u A = u A (x A, g A + g B ) = u A (M A pg A, g A + g B ) (3.3) u B = u B (x B, g A + g B ) = u B (M B pg B, g A + g B ) (3.4) B u A max u A, s.t. u B u (3.5) L = u A (M A pg A, g A + g B ) + λ (u u B (M B pg B, g A + g B )) (3.6) FONC: First Order Necessary Conditions L = u A ( p) + u A g A x A G λ u B G = 0 (3.7) L = u ( A ub λ ( p) + u ) B = 0 (3.8) g B x A x B G L λ = u u B(M B pg B, g A + g B ) = 0 (3.9) (3.7) λ = 1 u B / G ( p u A + u A x A G ) (3.10) 20

(3.8) u A x A 1 u B / G ( p u A + u A x A G u A / G ( 1 = p u ) ( A/ x A u A / G + 1 p u ) B/ x B u B / G + 1 MRS A = u A/ G u A / x A ( 1 = p ) ( + 1 p ) + 1 MRS A MRS B ) ( ub ( p) + u ) B = 0 (3.11) x B G (3.12) (3.13) (3.14) p = MRS A + MRS B (3.15) 1 A B 2 3.1.2 A B A B A B 2 2 A B Nash A B g B A max u A = u A (x A, g A + g B ) s.t. x A + pg A = M A (3.16) 21

FONC u A x A ( p) + u A G = 0 (3.17) u A / G u A / x A = MRS A = p (3.18) B MRS B = p FONC MRS A + MRS B = 2p > p (3.19) G u A / G G 3.1.3 (1) A B 2MRS A = MRS B 1 A 1 3 p B 2 3p A B MRS A = 1 3 p MRS B = 2 3 p A B (1) 22

1 A B C A C 3.1 A B C 3.1: p 3 (1/3)p 23

A a B b A 3.2 externality 3.2 3.2.1 4 Coase 24

3.2: 25

2 26

3.2.2 1 p 1 27

4 4.1 p u 2 E(p 1, p 2, u) 1 p 1 p 0 1 p1 1 M 0 M 1 u 0 1 u 1 E(p 0 1, p 2, u 0 ) E(p 1 1, p 2, u 1 ) (4.1) p 1 u u 0 = u 1 EV: Equivalent Variation CV: Compensated Variation 28

EV = E(p 0 1, p 2, u 1 ) E(p 0 1, p 2, u 0 ) (4.2) CV = E(p 1 1, p 2, u 1 ) E(p 1 1, p 2, u 0 ) (4.3) WTA: Willingness to Accept WTP: Willingness to Pay 1 M 1 = E(p 1 1, p 2, u 1 ) EV = E(p 0 1, p 2, u 1 ) E(p 0 1, p 2, u 0 ) = E(p 0 1, p 2, u 1 ) E(p 1 1, p 2, u 1 ) + M 1 E(p 0 1, p 2, u 0 ) = E(p 0 1, p 2, u 1 ) E(p 1 1, p 2, u 1 ) + M 1 M 0 (4.4) 2 p 2 u 1 p 1 E E EV = E(p 0 1, p 2, u 1 ) E(p 1 1, p 2, u 1 ) + M 1 M 0 = p 0 1 p 1 1 p 1 E(p 1, p 2, u 1 )dp 1 + M 1 M 0 (4.5) Shephard s lemma x c EV = = p 0 1 p 1 1 p 0 1 p 1 1 p 1 E(p 1, p 2, u 1 )dp 1 + M 1 M 0 x c 1(p 1, p 2, u 1 )dp 1 + M 1 M 0 (4.6) 2 29

M 0 = E(p 0 1, p 2, u 0 ) CV = E(p 1 1, p 2, u 1 ) E(p 1 1, p 2, u 0 ) = E(p 1 1, p 2, u 1 ) M 0 + E(p 0 1, p 2, u 0 ) E(p 1 1, p 2, u 0 ) = M 1 M 0 + E(p 0 1, p 2, u 0 ) E(p 1 1, p 2, u 0 ) (4.7) 2 p 2 u 0 p 1 E E CV = M 1 M 0 + E(p 0 1, p 2, u 0 ) E(p 1 1, p 2, u 0 ) = M 1 M 0 + p 0 1 p 1 1 p 0 1 p 1 E(p 1, p 2, u 0 )dp 1 (4.8) CV = M 1 M 0 + E(p 1, p 2, u 0 )dp 1 p 1 p 1 1 p 0 1 = M 1 M 0 + x c 1(p 1, p 2, u 0 )dp 1 (4.9) p 1 1 2 p 0 1 > p1 1 1 4.1 1 p 0 1 p1 1 1 4.1 30

4.1: 4.2 31

1 2 2 1 1 p 1 x 1 2 v(x 1 ) 1 2 (x 1, x 2 ) = (0, 0) 1 x 1 2 v(x 1 ) v(0) = 0 2 quasi-linear f u(x 1, x 2 ) = f(v(x 1 ) + x 2 ) (4.10) (1) u(x 1, x 2 ) = v(x 1 ) + x 2 (4.11) MRS = u/ x 1 u/ x 2 = v (x 1 ) (4.12) 2 1 v (x 1 ) = p (4.13) 1 1 1 1 x 1 x 1 2 v(x 1 ) v (x 1 ) x 1 x 1 2 p x 1 v (x 1 ) x 1 p x 1 (4.14) 1 x 1 = 0 1 x 1 x1 (v (x 1 ) p)dx 1 0 (1) 2007 6 (4.15) 32

v (x 1 ) v(x 1 ) v(0) = 0 x1 0 (v (x 1 ) p)dx 1 = v(x 1 ) px 1 (4.16) FONC v (x 1 ) = p 1 2 1 consumer surplus 4.2: 4.2 33

5 5.1 (1) 2 CVM (1) 1996 A-201 http://www3.grips.ac.jp/ kanemoto/bennkk.pdf 1999 4 http://www3.grips.ac.jp/ kanemoto/bc/sec4.pdf 34

5.2 First-best 5.2.1 5.2.2 3 1 1 2 1 2 2 3 3 p 3 = 1 1 2 3 u(x 1, x 2, x 3 ) = v(x 1, x 2 ) + x 3 (5.1) 35

M M = p 1 x 1 + p 2 x 2 + x 3 (5.2) v (x 1, x 2 ) = p 1 x 1 v (x 1, x 2 ) = p 2 x 2 (5.3) (5.4) x 1 = x 1(p 1, p 2 ) x 2 = x 2(p 1, p 2 ) (5.5) (5.6) 3 1 2 1 2 1 2 1 C 1 (x 1 ) = p 1 x 1 1 2 C 2 (x 2 ) = c(x 2 ) (2) 1 2 3 first-best (3) first-best p 1 = c 1 p 2 = c 2(x 2 ) (5.7) (5.8) 1 c 1 c 1 (2) 3 1 (3) First-best 36

2 x 2 = x 2 (p 1, p 2 ) p 2 = c 2 (x 2) p 2 = c 2(x 2(p 1, p 2 )) (5.9) p 1 p 2 implicit p 2 explicit p 2 = p 2(p 1 ) (5.10) 1 p 1 = c 1 p 2 = p 2(c 1 ) (5.11) 1 2 x 1 = x 1(c 1, p 2(c 1 )) x 2 = x 2(c 1, p 2(c 1 )) (5.12) (5.13) 1 2 c 1 1 2 (5.10) x 1 = x 1(p 1, p 2(p 1 )) (5.14) x 2 = x 2((p 2) 1 (p 2 ), p 2 ) (5.15) (4) 1 2 c 1 c 1 1 2 u(x 1, x 2, x 3 ) = v(x 1, x 2 ) + x 3 = v(x 1, x 2 ) + M p 1 x 1 p 2 x 2 = v(x 1(c 1, p 2(c 1 )), x 2(c 1, p 2(c 1 ))) + M c 1 x 1(c 1, p 2(c 1 )) p 2(c 1 )x 2(c 1, p 2(c 1 )) (5.16) (4) (p 2) 1 p 2 37

c 1 c 1 du = v dx 1 dc 1 x 1 + v dx2 dc 1 x 2 dc ( 1 x 1(c 1, p dx 1 2(c 1 )) + c 1 dc 1 ) ( ) dp 2 x dc 2(c 1, p 2(c 1 )) + p 2(c 1 ) dx 2 1 dc 1 (5.17) 1 v/ x 1 = p 1 = c 1 v/ x 2 = p 2 (c 1) du dc 1 = c 1 dx 1 + p dc 2(c 1 ) dx 2 1 dc ( 1 x 1(c 1, p dx 1 2(c 1 )) + c 1 dc 1 ) ( ) dp 2 x dc 2(c 1, p 2(c 1 )) + p 2(c 1 ) dx 2 1 dc 1 = x 1(c 1, p 2(c 1 )) dp 2 dc 1 x 2(c 1, p 2(c 1 )) (5.18) 1 1 (5) 2 π 2 = p 2 x 2 C 2 (x 2 ) (5.19) c 1 1 2 π 2 = p 2(c 1 )x 2(c 1, p 2(c 1 )) c 2 (x 2(c 1, p 2(c 1 ))) (5.20) c 1 c 1 dπ 2 dc 1 = dp 2 dc 1 x 2(c 1, p 2(c 1 )) + p 2(c 1 ) dx 2 dc 1 c 2(x 2 ) dx 2 dc 1 (5.21) p 2 = c 2 (x 2) dπ 2 dc 1 = dp 2 dc 1 x 2(c 1, p 2(c 1 )) + p 2(c 1 ) dx 2 dc 1 p 2 dx 2 dc 1 = dp 2 dc 1 x 2(c 1, p 2(c 1 )) (5.22) S ds = du + dπ 2 = x dc 1 dc 1 dc 1(c 1, p 2(c 1 )) (5.23) 1 1 1 2 2 (5) 38

c 1 c 1 1 5.2.3 First-best 3 1 2 1 p 0 1 p1 1 5.1: First-best 5.1 0 1 1 2 39

1 p 0 1 2 p0 2 x 1 (p 1, p 0 2 ), x 2 (p0 1, p 2) 1 1 p 1 1 1 1 2 2 2 5.1 2 2 2 1 5.1 1 2 1 2 1 p 1 1 2 p1 2 x 1 (p 1, p 1 2 ), x 2 (p1 1, p 2) 1 2 p 2 2 p 2 1 p 1 2 1 1 x 1 (p 1, p 2 (p 1)) 5.1 p 1 = p 0 1 x 1 (p0 1, p0 2 ) p 1 = p 1 1 x 1 (p1 1, p1 2 ) 2 2 p 2 = p 0 2 x 2 (p0 1, p0 2 ) p 2 = p 1 2 x 2 (p1 1, p1 2 ) 2 c (x 2 ) 5.1 1 2 2 1 1 2 40

5.3 Second-best 1 2 1 2 first-best 1 2 2 second-best 5.3.1 First-best 2 2 wedge t p 2 p 2 t c 2(x 2 ) = p 2 + t (5.24) t t > 0 t > 0 t first-best 2 (5.10) p 2 = p 2 (p 1, t) du dc 1 = x 1(c 1, p 2(c 1, t)) dp 2 dc 1 x 2(c 1, p 2(c 1, t)) (5.25) dπ 2 dc 1 = dp 2 dc 1 x 2(c 1, p 2(c 1, t)) (5.26) First-best 2 (6) (6) 41

(7) T = tx 2 (8) c 1 dt dc 1 = t dx 2 dc 1 (5.27) ds = du + dπ 2 + dt = x dc 1 dc 1 dc 1 dc 1(c 1, p 2(c 1 )) t dx 2 (5.28) 1 dc 1 First-best 2 t dx 2 /dc 1 2 c 1 dx 2 /dc 1 > 0 t > 0 c 1 1 1 t(dx 2 /dc 1) (9) 2 2 t dx 2 /dc 1 dx 2 /dc 1 < 0 5.1 5.1: Second-best t < 0 t = 0 t > 0 First best dx 2 dx 2 > 0 < 0 (7) 1 2 (8) t > 0 tx 2 (9) c 1 42

5.3.2 5.2: Second-best First-best 1 2 5.2 first-best first-best c 2 (x 2) t > 0 p 2 t p 2 p 2 + t first-best t c 2 (x 2) first-best first-best 2 2 43

2 2 1 2 2 first-best 44

6 (1) sunk cost 1 shadow price first-best secondbest first-best second-best Second-best (1) 112 124 45

6.1 First-best 6.1.1 first-best = 6.1: 6.1 x 1 (p 1, g = q 0 ) x 1 (p 1, g = q 1 ) 46

6.1.2 6.2: 6.2 6.1.3 6.3 47

6.3: x 1 (p 1 ; g = q 0 ) q 0 q 1 x 1 (p 1 ; g = q 1 ) p 0 1 p1 1 p 1 1 (q 1 q 0 ) 6.3 x 1 (p 1 ; g = q 0 ) 6.3 A B 6.3 A B C C 48

C 6.3 6.3 q 1 q 0 1 shadow price 6.2 Second-best wedge 2 6.2.1 6.4 / S S + t g = q 0 x 1 (p 1, g = q 0 ) S + t p 0 1 + t p0 1 g = q 1 x 1 (p 1, g = q 0 ) 49

6.4: x 1 (p 1, g = q 1 ) x 1 (p 1, g = q 1 ) S + t p 1 1 + t p1 1 (p 1 1 + t) (q 1 q 0 ) (2) A B 6.4 A B C S + t S A A C (2) 50

S + t S C D (p 1 1 + t) (q 1 q 0 ) C 6.4 6.4 q 1 q 0 p 0 1 p1 1 p 1 p 1 + t D S q 1 q 0 = D + S 6.4 p p D = (p 1 + t) D + S + p S 1 D + S (6.1) Harberger s weighted average shadow price formula 6.2.2 2 51

(3) 6.5: 6.5 p m L (3) 52

A 2 1 2 2 B p m p m 6.5 p m C 6.5 p c p d p c p d p c p d D 6.5 p m p r C p m p r p m p r 53

p r E (1/2)p m D p r p m = 0 p m p r D (1/2)p m p m E B 54

7 1 q 2 r 3 i (1) (2) 7.1 2 7.1 1 1 + r (1) (2) 55

7.1: 1 + q 1 i 7.2 56

7.2.1 7.2: 7.2 D S S + t / E 0 I E 1 E 1 i 57

q D S I = D + S 7.2 i D I + q S I (7.1) D S 8% (3) 8% 1 2 3 4 5 8% (3) 2004 10 302 AAA 1947 1999 6.86% 38% 0.0686/(1 0.38) = 0.1106 11.06% 3% 8% 58

7.2.2 2% (4) 1 2 3 (4) 2004 10 304 10 1953 1999 6.67% 30% 0.0667 (1 0.30) = 0.0474 4.74% 3% 2% 59

7.2.3 7.2 = = 3 7.2.4 1 1 1 60

depreciate capital depreciation r δ (5) 1 + q t C t K t s 0 1 1 C 1 = r(1 s), K 1 = (1 d) + sr (7.2) r C 2 = r(1 s)(1 d + sr), K 2 = (1 d + sr)(1 d) + sr(1 d + sr) = (1 d + sr) 2 (7.3) C 3 = r(1 s)(1 d + sr) 2, K 3 = (1 d + sr) 3 (7.4) θ C 1, C 2, C 3,... θ = = = r(1 s) 1 + q r(1 s) 1 + q r(1 s) 1 + q r(1 s)(1 d + sr) r(1 s)(1 d + sr)2 + (1 + q) 2 + (1 + q) 3 + [ 1 + 1 d + sr ] (1 d + sr)2 + 1 + q (1 + q) 2 + [ 1 + 1 d + sr ] 1 1 + q θ = r(1 s) q + d sr (7.5) s = 0 d = 0 r θ = r q (7.6) (5) Lyon, Randolph M. 1990. Federal discount rate policy, the shadow price of capital, and challenges for reforms. Journal of Environmental Economics and Management 18(2-2), pp.s29 S50. Appendix 1 61

1 s d 1.3% 2.7% 7.2.5 10% 2 A 1 2 36 37 PV A 100 100 100 100 100 1,000 B 151 100 100 0 0 1,014 7.1: 2 100 B 151 2 36 100 10% A 1,000 B 1,014 B A B 1 51 37 100 A B 62

F. Ramsey 7.2.6 7% 4% 7.3 7.3.1 k T (k) T (k) = t=0 B t C t k (1 + r) t = B t C t + T (k) (7.7) (1 + r) t t=0 T (k) 5 63

k (1) (2) 64

7.3.2 B/C = t=0 B t/(1 + r) t t=0 C t/(1 + r) t (7.8) 1 net B/C = t=0 (B t C t )/(1 + r) t t=0 t=0 C t/(1 + r) t = B t/(1 + r) t t=0 C t/(1 + r) t 1 (7.9) IRR: Internal Rate of Return π t=0 B t (1 + π) t t=0 C t (1 + π) t = 0 (7.10) 65

t B t C t t (7.10) (6) 1.5 30 1500 2850 7.2 7.2: 0 1 2 30 15,000 1,500 1,500 1,500 0 2,850 2,850 2,850 6% NPV: Net Present Value NPV = 15, 000 + 30 t=1 2, 850 1, 500 (1 + 0.06) t 3, 583 (7.11) 7.3 8.1395 1 8.1395% (6) 2007 8 2007 8.1 66

7.3: B/C 4 40,938 49,282 8,344 1.204 6 35,647 39,230 3,583 1.100 8 31,887 32,085 198 1.006 8.1395 31,667 31,667 0 1.000 10 29,140 26,867-2,274 0.922 7.3.3 B/C B/C 7.4 4 I 0 NP V 7.4: I 0 NPV B/C A 1,000 100 1.10 B 500 60 1.12 C 400 80 1.20 D 100 15 1.15 B/C 1 1,000 I 0 1, 000 67

C 2 D 2 1 B C D A (7) 7.5 2 A 75 3,000 B 15 2,400 2 7.5: NPV A 3,000 75 B 2,400 15 A B 5 15 75 4 5 8% B 5 = 2, 400 + 2, 400 2, 400 2, 400 2, 400 + + + 3, 494 1.0815 1.0830 1.0845 1.0860 (8) (7) 2004 pp.171-172 (8) 68

8 (1) 8.1 8.1.1 1 1km 0.001 0.1% 20m 1km 0.004 0.4% 20m 0.995 99.5% 1km 30 20m 1km 6 600 1km 5 20m 1km 1 1km 5 600 8.1 (1) 8 9 69

8.1: 1km 20m-1km 1 0.001 0.004 0.995 10 5,060 1,060 60 69 30,000 6,000 0 54 5.06 0.001 + 1.06 0.004 + 0.06 0.995 = 0.069 69 54 8.1 1000 600 1000 600 400 150 550 8.1.2 (2) Y U(Y ) du/dy > 0 Y α 1, α 2,..., α K K π 1, π 2,..., π K (2) 2010 8 70

K E[U(Y )] = π 1 U(α 1 ) + π 2 U(α 2 ) + + π K U(α K ) = π i U(α i ) (8.1) Y U(Y ) E[U(Y )] Y Y i=1 E[U(Y )] = U(Y ) (8.2) Y certainty equivalence U risk averse U < 0 Y E[Y ] Y < E[Y ] risk premium = E[Y ] Y (8.3) 8.1: K = 2 Y β, γ β < γ β π γ 1 π 71

β, γ E[Y ] = πβ + (1 π)γ β γ 1 π : π E[Y ] U(E[Y ]) U(β), U(γ) 2 1 π : π E[U(Y )] = πu(β) + (1 π)u(γ) E[U(Y )] Y Y < E[Y ] 8.1.3 certainty equivalence 2 (3) q 1 q 2 = 1 q 1 G G = G G = 0 X r 1 (> q 1 ) r 2 (< q 2 ) (3) 2002 5(3), 2-11 Freeman III, A.M. 1986. Uncertainty and option value in environmental policy. in E. Miles, R. Pealy and R. Stokes eds. Natural Resources Economics and Policy Applications Essay in honor of James A. Crutchfield, Seattle and London, University of Washington Press, pp.251-271. 72

i = D p 1 p 2 = 1 p 1 i = N U D (Y D, P D, G) U N (Y N, P N ) Y P U D (Y D, P D, G ) > U D (Y D, P D, 0) CS i CS D : U D (Y D CS D, P D, G ) = U D (Y D, P D, 0) (8.4) CS N : CS N = 0 (8.5) U N (Y N, P N ) p 2 U D (Y D, P D, G ) p 1 q 1 U D (Y D, P D, 0) p 1 q 2 EU SQ EU SQ = p 1 q 1 U D (Y D, P D, G ) + p 1 q 2 U D (Y D, P D, 0) + p 2 U N (Y N, P N ) (8.6) X r 1 > q 1 X EU OP EU OP = p 1 r 1 U D (Y D X, P D, G ) + p 1 r 2 U D (Y D X, P D, 0) + p 2 U N (Y N X, P N ) (8.7) p 1, q 1, r 1 X X option price X EU OP EU OP = EU SQ X 73

ECS CS N = 0 CS i q 1 r 1 ECS = p 1 (r 1 q 1 )CS D + p 2 0 CS N = p 1 (r 1 q 1 )CS D (8.8) option value Option Value = Option Price ECS (8.9) (4) p 2 = 0 X r 2 = 0 EU SQ = q 1 U D (Y D, P D, G ) + q 2 U D (Y D, P D, 0) (8.10) EU OP = U D (Y D X, P D, G ) (8.11) EU OP = EU SQ X 8.1.4 U D (Y D CS D, P D, G ) = U D (Y D, P D, 0) (8.12) (4) 2002 74

CS D G 1 2 individual risk, idiosyncratic risk collective 8.2 75

8.2.1 1 2 3 4 76

8.2.2 2 DVD 1 2 3 77