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1 (1) (2) (1) (2), bessho@econ.keio.ac.jp
2 First-best Second-best First-best Second-best
3
4 cost benefit B C NB NB = B C (1.1) NB 0 NB < 0 with/without 1.2 3
5 market failure 2 1 4
6 [1] [2] [3] [4] [5] [6] [7] Net Present Value 5
7 [8] [9]
8
9 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
10 9
11 2 2.1 Vilfredo Pareto, Pareto efficient allocation Pareto improving Pareto dominant (1) 3 (2) (1) (2) 10
12 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 X Y (3) X Y 11
13 X Y X 1 Y MRT: Marginal Rate of Transformation X 2.1: Y 2.1 e e E E e 12
14 X 1 Y 4 1 X 1 Y 1 Y 4 X (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 c abc (4) 13
15 2.2: c d d A B compensation a b c d (5) a, b,d (5) B A B 14
16 3 social welfare function Bergson-Samuelson (6) A B 2 Bergson-Samuelson W = W (U A, U B ) (2.1) (6) 15
17 W/ U i > W (U A, U B ) = W (U B, U A ) Bergson-Samuelson W = U A + U B W = min(u A, U B )
18 price taker 17
19 1 3 18
20 3 market failure A B 2 x A, x B 2 19
21 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
22 (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 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
23 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 (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
24 1 A B C A C 3.1 A B C 3.1: p 3 (1/3)p 23
25 A a B b A 3.2 externality Coase 24
26 3.2: 25
27 2 26
28 p 1 27
29 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
30 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
31 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 p 0 1 > p p 0 1 p
32 4.1:
33 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) 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) (4.15) 32
34 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 consumer surplus 4.2:
35 5 5.1 (1) 2 CVM (1) 1996 A kanemoto/bennkk.pdf kanemoto/bc/sec4.pdf 34
36 5.2 First-best p 3 = u(x 1, x 2, x 3 ) = v(x 1, x 2 ) + x 3 (5.1) 35
37 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) C 1 (x 1 ) = p 1 x C 2 (x 2 ) = c(x 2 ) (2) 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
38 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 (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 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
39 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 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) (5) 38
40 c 1 c First-best p 0 1 p : First-best
41 1 p p0 2 x 1 (p 1, p 0 2 ), x 2 (p0 1, p 2) 1 1 p p p1 2 x 1 (p 1, p 1 2 ), x 2 (p1 1, p 2) 1 2 p 2 2 p 2 1 p 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 )
42 5.3 Second-best first-best second-best 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
43 (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 t(dx 2 /dc 1) (9) 2 2 t dx 2 /dc 1 dx 2 /dc 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
44 : Second-best First-best 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
45 first-best 44
46 6 (1) sunk cost 1 shadow price first-best secondbest first-best second-best Second-best (1)
47 6.1 First-best first-best = 6.1: 6.1 x 1 (p 1, g = q 0 ) x 1 (p 1, g = q 1 ) 46
48 :
49 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
50 C q 1 q 0 1 shadow price 6.2 Second-best wedge / S S + t g = q 0 x 1 (p 1, g = q 0 ) S + t p t p0 1 g = q 1 x 1 (p 1, g = q 0 ) 49
51 6.4: x 1 (p 1, g = q 1 ) x 1 (p 1, g = q 1 ) S + t p t p1 1 (p t) (q 1 q 0 ) (2) A B 6.4 A B C S + t S A A C (2) 50
52 S + t S C D (p t) (q 1 q 0 ) C 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
53 (3) 6.5: 6.5 p m L (3) 52
54 A 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
55 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
56 7 1 q 2 r 3 i (1) (2) r (1) (2) 55
57 7.1: 1 + q 1 i
58 : 7.2 D S S + t / E 0 I E 1 E 1 i 57
59 q D S I = D + S 7.2 i D I + q S I (7.1) D S 8% (3) 8% % (3) AAA % 38% /(1 0.38) = % 3% 8% 58
60 % (4) (4) % 30% (1 0.30) = % 3% 2% 59
61 = =
62 depreciate capital depreciation r δ (5) 1 + q t C t K t s 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 + [ d + sr ] (1 d + sr) q (1 + q) 2 + [ d + sr ] q θ = r(1 s) q + d sr (7.5) s = 0 d = 0 r θ = r q (7.6) (5) Lyon, Randolph M 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
63 1 s d 1.3% 2.7% % 2 A PV A ,000 B , : B % A 1,000 B 1,014 B A B A B 62
64 F. Ramsey % 4% 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
65 k (1) (2) 64
66 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
67 t B t C t t (7.10) (6) : ,000 1,500 1,500 1, ,850 2,850 2,850 6% NPV: Net Present Value NPV = 15, t=1 2, 850 1, 500 ( ) t 3, 583 (7.11) % (6)
68 7.3: B/C 4 40,938 49,282 8, ,647 39,230 3, ,887 32, ,667 31, ,140 26,867-2, B/C B/C I 0 NP V 7.4: I 0 NPV B/C A 1, B C D B/C 1 1,000 I 0 1,
69 C 2 D 2 1 B C D A (7) A 75 3,000 B 15 2, : NPV A 3, B 2, A B % B 5 = 2, , 400 2, 400 2, 400 2, , (8) (7) 2004 pp (8) 68
70 8 (1) km % 20m 1km % 20m % 1km 30 20m 1km km 5 20m 1km 1 1km (1)
71 8.1: 1km 20m-1km ,060 1, ,000 6, = (2) Y U(Y ) du/dy > 0 Y α 1, α 2,..., α K K π 1, π 2,..., π K (2)
72 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
73 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 ] 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) (3), 2-11 Freeman III, A.M 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
74 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
75 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 U D (Y D CS D, P D, G ) = U D (Y D, P D, 0) (8.12) (4)
76 CS D G 1 2 individual risk, idiosyncratic risk collective
77
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II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ =
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3.2 [ ]< 86, 87 > ( ) T = U V,N,, du = TdS PdV + µdn +, (3) P = U V S,N,, µ = U N. (4) S,V,, ( ) ds = 1 T du + P T dv µ dn +, (5) T 1 T = P U V,N,, T
![3.2 [ ]< 86, 87 > ( ) T = U V,N,, du = TdS PdV + µdn +, (3) P = U V S,N,, µ = U N. (4) S,V,, ( ) ds = 1 T du + P T dv µ dn +, (5) T 1 T = P U V,N,, T](/thumbs/82/86676506.jpg "3.2 [ ]< 86, 87 > ( ) T = U V,N,, du = TdS PdV + µdn +, (3) P = U V S,N,, µ = U N. (4) S,V,, ( ) ds = 1 T du + P T dv µ dn +, (5) T 1 T = P U V,N,, T")
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