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1 ( ) (i) ( ) (ii) ( ) (iii) ( ) ( ) 1) 1 x x X R X =[0, x] x b(x) b( ) 2 b(0) = 0 x<x b (x) > 0 b (0) = + b (x) =0 x 0 b (x) < 0 x 2 ( ) Θ Θ ={θ 0,θ 1 } 0 <θ 0 <θ 1 θ 0 θ 0 θ 1 θ 0 p p θ 0 1 p θ 1 p θ i x c i (x) c i (x) =θ i x (i =0, 1) 1) x

2 8 0 <θ i x>0 c 0 (x) <c 1 (x) c 0 (x) <c 1 (x) θ 0 θ 0 θ 1 w x w θ i U i (x, w) =w c i (x) b(x) w x x x x x w w w w M ( ) m M m µ M X R µ(m) m µ(m) =(δ(m),ρ(m)) m δ(m) X ρ(m) R 2 (i) M (ii) µ( ) =(δ( ),ρ( )) 2 (M,µ) (mechanism) (takeit-or-leave-it offer) 2 ( ) ( ) 3

3 9 (x, w) =(0, 0) b(0) c i (0) (reservation utility) θ i (x i,w i ) {(x fb 0,wfb 0 ), (xfb 1,wfb 1 )} (first-best) θ i b(x i ) c i (x i ) max b(x i ) c i (x i ) b (x fb x i )=θ i i w i w i c i (x fb i )=0 w fb i = θ i x fb i x fb i 2) Π fb = p[b(x fb 0 ) θ 0x fb 0 ]+(1 p)[b(xfb 1 ) θ 1x fb 1 ] Π(x 0,x 1 )=p[b(x 0 ) θ 0 x 0 ]+(1 p)[b(x 1 ) θ 1 x 1 ] Π(x) =Π(x fb 0,x)=p[b(xfb 0 ) θ 0x fb 0 ]+(1 p)[b(x) θ 1x] Π(x 0,x 1 ) θ i x i Π(x) θ 0 θ 1 x Π(x fb 1 )=Π(xfb 0,xfb 1 )=Πfb 1.1 x x θ 0 w fb 0 A B θ 1 w fb 1 A + C (M,µ) θ i m i i =0, 1 m i M σ =(m 0,m 1 ) σ (strategy) 2) w fb i

4 b'(x) θ 1 C θ 0 A B 0 x 1 fb x fb 0 x (M,µ) σ =(m 0,m 1) U i (δ(m i ),ρ(m i )) U i (δ(m),ρ(m)), m M,i =0, 1 (1.1) σ (M,µ) (1.1) θ i m i m = m j j i U i (δ(m i ),ρ(m i )) U i(δ(m j ),ρ(m j )). (1.2) (1.2) θ i θ j ( ) M (revelation principle) 3) (direct revelation mechanism) (M =Θ) θ 0 θ 1 (Θ,ν) ν ν = {(x 0,w 0 ), (x 1,w 1 )} θ i x i w i 4) σ =(ˆθ 0, ˆθ 1 ) (ˆθ i Θ) θ i ˆθ i ˆθ i = θ i ˆθ i = θ j (i, j =0, 1 i j) 3) 4) θ 0 θ 1 (x 0,w 0 ) (x 1,w 1 ) (x 0,w 0 ) (x 1,w 1 )

5 11 (Θ,ν) ν (M,µ) ( µ =(δ, ρ)) σ = (m 0,m 1 ) ν (i) (ii) ( ) i =0, 1 ˆθ i = θ i (ˆθ 0, ˆθ 1 ) ν (θ 0,θ 1 ) (M,µ) ν = {(x 0,w 0 ), (x 1,w 1 )} i =0, 1 θ i x i = δ(m i ) w i = ρ(m i ) ν = {(x 0,w 0 ), (x 1,w 1 )} x i = δ(m i ), w i = ρ(m i ), i =0, 1 (1.2) U i (x i,w i )=U i (δ(m i ),ρ(m i )) U i (δ(m j ),ρ(m j )) = U i (x j,w j ), j i ν θ i θ i i =0, 1 ˆθ i = θ i (i) i =0, 1 θ i ν θ i (x i,w i )=(δ(m i ),ρ(m i )) (ii) {(x 0,w 0 ), (x 1,w 1 )} ν = {(x i,w i )} (p) max p[b(x 0 ) w 0 ]+(1 p)[b(x 1 ) w 1 ] ν (1.3) subject to w 0 θ 0 x 0 0 (pc 0 ) w 1 θ 1 x 1 0 (pc 1 ) w 0 θ 0 x 0 w 1 θ 0 x 1 (ic 0 ) w 1 θ 1 x 1 w 0 θ 1 x 0 (ic 1 ) (1.3) (pc 0 ) (pc 1 ) (participation constraints) 5) θ 1 (ic 0 ) (ic 1 ) (incentive compatibility constraints) 6) 5) (individual rationality constraints) 6) (self-selection constraints) (truth-telling constraints)

6 12 (p) 1.1 x fb 0 xfb 1 θ 0 x fb 0 wfb 0 = A + B c 0(x fb 0 )=θ 0x fb 0 θ 0 1 = A θ 1 x fb 1 c 0(x fb 1 )=θ 0x fb w fb 1 = A + C C>0 θ 0 (p) (p) (second-best) 1 x 0 x 1 ( ) (ic 0 ) (ic 1 ) θ 1 (x 0 x 1 ) w 0 w 1 θ 0 (x 0 x 1 ). θ 1 >θ 0 x 0 x 1 2 (pc 1 ) (ic 0 ) θ 0 (pc 0 ) ( (pc 0 ) ) ( ) (ic 0 ) (pc 1 ) (pc 0 ) w 0 θ 0 x 0 w 1 θ 0 x 1 w 1 θ 1 x θ 0 (ic 0 ) ( ) (ic 0 ) (pc 0 ) 0=w 0 θ 0 x 0 >w 1 θ 0 x 1 w 1 θ 1 x 1. θ 1 (pc 1 ) (pc 0 ) (pc 0 ) (ic 0 ) w 0 (pc 1 ) (ic 1 ) (ic 1 ) w 0 w 0 (ic 0 ) 4 x 0 x 1 (ic 0 ) θ 1 (ic 1 ) ( ) (ic 0 ) θ 1 (x 0 x 1 ) (w 0 w 1 )=θ 1 (x 0 x 1 ) θ 0 (x 0 x 1 ). θ 1 >θ 0 (ic 1 ) w 1 θ 1 x 1 (pc 1) w 0 = w 1 + θ 0 (x 0 x 1 ) (ic 0 ) x 0 x 1 (m) 3 (pc 1 ) w 1 = θ 1 x 1, (pc 1 ) w 0 = θ 0 x 0 + θx 1. (ic 0) θ = θ 1 θ 0 (pc 1) (ic 0) (1.3)

7 13 (p ) max p[b(x 0 ) θ 0 x 0 θx 1 ]+(1 p)[b(x 1 ) θ 1 x 1 ] (1.4) x 0,x 1 subject to (m). 6 (m) (p ) b( ) (1.4) (b (0) = + b (x) =0) X (x 0,x 1) b (x 0 )=θ 0, (1.5) b (x 1 )=θ 1 + p θ, 1 p (1.6) (w0,w 1) (pc 1) (ic 0) x 0 = x 0 x 1 = x 1 7 (m) (1.6) (1.5) θ 1 >θ 0 b (x 0 ) <b (x 1 ) b( ) x 0 >x 1 1 (1.5) θ 0 (x 0 = xfb 0 ) (1.6) θ 1 ( ) (x 1 <xfb 1 ) 2 (pc 1) (ic 0) w0 θ 0x 0 = θx 1, (1.7) w1 θ 1 x 1 =0, (1.8) θ 0 θ 1 θ 0 θ 0 θ 0 (information rent) (1.7) θx θ 0 w fb 0 = A + B w 0 = A + B + C θ 0 w 0 θ 0 x fb 0 = C θ θ 1 x fb 1 1 θ 1 b (x fb 1 ) θ 1 ( ) θ 0 θ = θ 1 θ 0 ( ) 1 p p (1 p)(b (x 1 ) θ 1 )=p θ x 1 (1.6) Π Π = p[b(x fb 0 ) θ 0x fb 0 θx 1 ]+(1 p)[b(x 1 ) θ 1x 1 ] =Π(x 1) p θx 1 Π fb =Π(x fb 1 ) 2 1 θ 1 2 θ 0 θx 1

8 b'(x) θ 1 θ 0 C A B 0 fb x x0 fb 1 x θ 0 p Π p dπ dp = Π p + Π x 1 x 1 p. [b(x fb 0 ) θ 0x fb 0 θx 1] [b(x 1) θ 1 x 1] 2 x 1 / p (1.6) x 1 (1.4) Π / x 1 =0 p ( ) x 1 (1.6) x 1 p p θ 0 θ 1 θ 1 p[b(x 0 ) w 0 ] (1.3) x 1 = w 1 =0 θ 1 θ 1 θ 0 (x 0,w 0 ) w 0 θ 0 x 0 0 (ic 0 ) θ 0 (x 0,w 0 )=(x fb 0,wfb 0 ) θ 1 θ 0 w fb 0 θ 1x fb 0 = θxfb 0 < 0 θ 1 (x fb 0,wfb 0 ) (p) (x 1,w 1 )=(0, 0) (p) (1.6) (1.8) (x 1,w 1 ) (0, 0) 1.1 ( )

9 15 3. ( ) ( ) Θ θ y (allocation) y =(x, w) x w Y U(y, θ) V (y, θ) ( ) M µ : M Y 7) (game with incomplete information) (game with imperfect information) p( ) (common knowledge) p( ) ( ) (M,µ) σ :Θ M 8) σ(θ) θ (M,µ) σ 1.3 Θ Y M Y µ (M,µ) σ σ (M,µ) U(µ(σ (θ)),θ) U(µ(m),θ), m M, θ Θ (1.9) 7) 8)

10 16 σ (θ) arg max m M U(µ(m),θ), θ Θ θ σ (θ) 1.3 Θ Y σ µ M (Θ,ν) ν Θ Y σ ν Θ 1.1 ( ) (M,µ) σ ν (i) (ii) ( ) θ Θ σ(θ) =θ σ ν (M,µ) σ θ Θ ν(σ(θ)) = µ(σ (θ)) (1.9) θ Θ U(µ(σ (θ)),θ) U(µ(σ (θ )),θ), θ Θ. ν( ) θ Θ ν(θ) =µ(σ (θ)) U(ν(θ),θ) U(ν(θ ),θ), θ Θ. σ(θ) =θ ν(σ(θ)) = ν(θ) =µ(σ (θ)) (ii) Θ ={θ 0,θ 1,θ 2 } M = {m 0,m 1 } (M,µ) σ σ (θ 0 )=m 0, σ (θ 1 )=m 1, and σ (θ 2 )=m 0. U(µ(m 0 ),θ 0 ) U(µ(m 1 ),θ 0 ) U(µ(m 1 ),θ 1 ) U(µ(m 0 ),θ 1 ) U(µ(m 0 ),θ 2 ) U(µ(m 1 ),θ 2 ) ν ν(θ) =µ(σ (θ)) ν(θ 0 )=µ(m 0 ), ν(θ 1 )=µ(m 1 ), and ν(θ 2 )=µ(m 0 )

11 17 σ (incentive compatible) y x X R w R y =(x, w) w X 1 U(y, θ) V (y, θ) 1.1 U = u(x, θ)+w V = v(x, θ) w u( ) v( ) x 2 ( ) S(x, θ) =u(x, θ)+v(x, θ) U θ x t = w U = u(x, θ) t V = t c(x) c(x) 1.1 ( ) ( ) x w U = w c(x, θ) V = v(x, θ) w θ 1.1 c(x, θ) =θx v(x, θ) =b(x) θ θ θ x w U = w c(x, θ) V = v(x) w + αu v w α θ x t U = u(x, θ) t

12 18 1 V = E θ [U] θ I A x w a = I A x + w n = I x U = θz(a)+(1 θ)z(n) z( ) ( ) V = x θw ( ) y( ) =(x( ),w( )) x( ) w( ) θ Θ S(x, θ) x x fb ( ) S(x, θ) x S x (x, θ) =0 x = x fb (θ), θ Θ u(x fb (θ),θ)+w = U w = w fb (θ) =U u(x fb (θ),θ) x( ) w( ) x( ) 2 Spence- Mirrlees (Spence-Mirrlees single crossing property) 9) 1.2 (SCP) u x θ x X θ, θ Θ θ >θ u x (x, θ) >u x (x, θ ) 1.5 x w dw = U x = u x dx U:const U w (SCP) θ θ θ θ <θ θ U = u(x, θ) t = θx t θ>0 θ 1 u x (x, θ) =θ (SCP) u x θ θ U = x/θ t x θ ( ) u x = θ 1 u x θ (SCP) ( ) 9) u( ) (x, θ) (strict increasing differences) ( A.2 )

13 (θ >θ ) w θ' θ x 1.3 u θ x X θ, θ Θ θ >θ u(x, θ) >u(x, θ ) u(x, θ) =θx u(x, θ) =x/θ u θ θ = θ 1 u u x θ θ u u x Θ Θ 1.3 Θ Θ ={θ 0,...,θ N } 1.1 N =1 1.1 N +1 θ i θ 0 < <θ N θ i p i i =0,...,N p i > 0 F i i F i =Pr{θ θ i } = i j=0 p j F N = y( ) u(x(θ),θ)+w(θ) u(x(θ ),θ)+w(θ ), θ, θ Θ. (IC)

14 20 U(θ θ) =u(x(θ ),θ)+w(θ ) θ θ U(θ) =U(θ θ) (IC) U(θ) U(θ θ), θ, θ Θ (IC ) θ i y i =(x i,w i ) (i =0,...,N) U(θ i )=u(x i,θ i )+w i u(x j,θ i )+w j = U(θ j θ i ), i,j =0,...,N (IC N ) y =(y 0,...,y N ) x =(x 0,...,x N ) w =(w 0,...,w N ) ) 1.1 y =(x, w) (IC N ) i =1,...,N U(θ i ) >U(θ i 1 ) ( ) (IC N ) 1.3 U(θ i ) U(θ i 1 θ i ) >U(θ i 1 ) (SCP) 1.2 y =(x, w) (IC N ) U(θ i ) U(θ i 1 θ i ), i 1 (LICD) U(θ i ) U(θ i+1 θ i ), i N 1 (LICU) ( ) (IC N ) (LICD) (LICU) (LICD) (LICU) U(θ i ) U(θ i 1 θ i ) 0 U(θ i θ i 1 ) U(θ i 1 ), i =1,...,N U u(x i,θ i ) u(x i 1,θ i ) u(x i,θ i 1 ) u(x i 1,θ i 1 ), i =1,...,N. xi x i 1 [u x (x, θ i ) u x (x, θ i 1 )]dx 0, i =1,...,N. (SCP) x 0 x N (IC N ) N =1 (LICD) (LICU) N = M N = M +1 u(x M+1,θ M+1 )+w M+1 u(x i,θ M+1 )+w i, i =0,...,M, (1.10) u(x i,θ i )+w i u(x M+1,θ i )+w M+1, i =0,...,M, (1.11) 2 (1.10) (1.11) (LICD) u(x M+1,θ M+1 ) u(x M,θ M+1 ) w M w M+1. 10) u θ 1.1 U(θ i ) <U(θ i 1 )

15 21 u(x M,θ M ) u(x i,θ M ) w i w M, i M 1. u(x M+1,θ M+1 ) u(x M,θ M+1 )+u(x M,θ M ) u(x i,θ M ) w i w M+1, i =0,...,M. (1.12) (SCP) x M x i (i =0,...,M 1) u(x M,θ M ) u(x i,θ M ) u(x M,θ M+1 ) u(x i,θ M+1 ). (1.12) u(x M+1,θ M+1 ) u(x i,θ M+1 ) w i w M+1, i =0,...,M (1.10) 1.2 (IC N ) (LICD) (LICU) (LICD) (LICU) 1.6 θ i θ i (x i,w i ) θ i+1 θ i+1 θ i+1 (x i+1,w i+1 ) θ i (x i,w i ) (x i+1,w i+1 ) ( ) (x i+1,w i+1 ) θ i θ i+1 (x i+1,w i+1 ) (x i,w i ) (x i+1,w i+1 ) θ i+1 x i x i w θ i θ i+1 w i xx xxxxxx xxxx xxxxxx xxxxx xxx xx xxxxx xxx x xxxxx xxx xxxxx x x i x y =(x, w) (IC N ) x 0 x N

16 22 (P N ) N max p y i [S(x i,θ i ) U(θ i )] (1.13) i=0 subject to (LICD) (LICU) and U(θ i ) U, i =0,...,N (PC N ) (P N ) (PC N ) θ (LICU) ( ) 1.2 y (P N ) y (P N ) N max p y i [S(x i,θ i ) U(θ i )] (1.14) i=0 subject to (LICD) U(θ 0 ) U and x 0 x N (M N ) ( ) ( ) 1.1 ( ) (P N ) (PC N ) U(θ 0 ) U 1.1 (LICU) 2 1 (P N ) (LICD) y =(x, w) i u(x i,θ i )+w i >u(x i 1,θ i )+w i 1 ɛ>0 u(x i,θ i )+w i ɛ>u(x i 1,θ i )+w i 1 (x, w ) { w k = w k, if k =0,...,i 1 w k ɛ, if k = i,...,n (P N ) (x, w) (x, w) (P N ) (LICD) 2 (P N ) (LICU) 1 (x, w) u(x i,θ i ) u(x i 1,θ i )=w i 1 w i, i =1,...,N (M N ) (SCP) u(x i,θ i ) u(x i 1,θ i ) u(x i,θ i 1 ) u(x i 1,θ i 1 ), i =1,...,N. u(x i,θ i 1 ) u(x i 1,θ i 1 ) w i 1 w i, i =1,...,N. (LICU) x w x i w i i U i = U(θ i )=u(x i,θ i )+w i x U =(U 0,...,U N )

17 23 (P N ) N max p i [S(x i,θ i ) U i ] (1.15) x,u i=0 subject to (M N ) U 0 U and U i U i 1 u(x i 1,θ i ) u(x i 1,θ i 1 ), i =1,...,N (LICD ) (M N ) i (LICD ) λ i (i =1,...,N) U 0 U λ 0 L = N p i [S(x i,θ i ) U i ]+λ 0 [U 0 U] i=0 + N λ i [U i U i 1 u(x i 1,θ i )+u(x i 1,θ i 1 )] i=1 i =0,...,N L/ x i =0 L/ U i =0 p i S x (x i,θ i )=λ i+1 [u x (x i,θ i+1 ) u x (x i,θ i )], i =0,...,N 1 (1.16) p N S x (x N,θ N ) = 0 (1.17) p i + λ i λ i+1 =0, i =0,...,N 1 (1.18) p N + λ N = 0 (1.19) λ i = N p j =1 F i 1 > 0, i =0,...,N. (1.20) j=i ( F 1 =0 ) (1.20) (M N ) 11) Φ(x i,θ i )=S(x i,θ i ) 1 F i [u(x i,θ i+1 ) u(x i,θ i )], i =0,...,N (1.21) p i (1.16) (1.20) x i Φ x (x i,θ i )= Φ/ x i (x i,θ i )=0 1.4 θ, θ Θ (i) (ii) Φ( ) (quasi-concave) x Φ( ) (x, θ) θ >θ x X Φ x (x, θ) Φ x (x, θ ) 0 (i) u(x, θ) v(x, θ) (ii) x i i (M N ) ( A.2 ) 11) Φ i p i θ i F i θ i Φ θ i i Φ

18 y =(x, w) S x (x i,θ i )= 1 F i [u x (x i,θ i+1 ) u x (x i,θ i )], p i i =0,...,N 1 (1.22) S x (x N,θ N ) = 0 (1.23) U 0 = U (1.24) i U i = U + [u(x j 1,θ j ) u(x j 1,θ j 1 )], i =1,...,N (1.25) j=1 1.1 (1.23) θ N (1.22) 12) (1.24) θ 0 (1.25) θ i i [u(x j 1,θ j ) u(x j 1,θ j 1 )] j=1 (j =0,...,i 1) x j ( (SCP) ) x j 1.4 Θ=[θ 0,θ 1 ] Θ f(θ) θ Θ f(θ) > 0 F (θ) (SCP) 1.3 u θ > 0 u xθ > 0 (optimal control theory) ( ) (IC) (IC ) U(θ) =u(x(θ),θ)+w(θ) u(x(θ ),θ)+w(θ )=U(θ θ), θ, θ Θ (IC) 1.1 (IC) 1.3 θ >θ U(θ) U(θ θ) >U(θ ) 1.3 y( ) (IC) U(θ) 12) x 0 x N v(x, θ) θ 1.1 v(x, θ) θ (SCP)

19 25 (SCP) y( ) 2 U(θ θ) 2 U 1 (θ) =U 1 (θ θ) θ U U 2 (θ) =U 2 (θ θ) θ 13) U 1 (θ) =0 du(θ) dθ = U 1 (θ)+u 2 (θ) =U 2 (θ) =u θ (x(θ),θ) (EC) U(θ) θ U(θ) =U(θ 0 )+ u θ (x(s),s)ds, θ Θ (EC ) θ 0 (EC) (EC ) (LICD) (LICU) U 1 (θ) =0 θ U 11 (θ)+u 12 (θ) =0 (EC) U 11 (θ) 0 U 12 (θ) 0 u θx (x(θ),θ)x (θ) 0. (SCP) x (θ) U( ) θ (EC) 1.4 θ Θ=[θ 0,θ 1 ] (i) (ii) (iii) ( ) θ>θ 0 U( ) θ U (θ ) u θ (x(θ),θ) θ<θ 1 U( ) θ U (θ+) u θ (x(θ),θ) θ (θ 0,θ 1 ) U( ) θ U (θ) =u θ (x(θ),θ) θ, θ Θ U(θ ) U(θ θ )=U(θ)+[u(x(θ),θ ) u(x(θ),θ)]. U(θ) U(θ ) u(x(θ),θ) u(x(θ),θ ). (1.26) (i) θ>θ θ θ > 0 U(θ) U(θ ) θ θ u(x(θ),θ) u(x(θ),θ ) θ θ. U( ) θ θ θ (ii) θ<θ (1.26) θ θ>0 U(θ ) U(θ) θ θ u(x(θ),θ ) u(x(θ),θ) θ. θ U( ) θ θ θ+ (iii) U( ) θ θ (i) (ii) 13) U 1 (θ) = U(θ θ) fi fifiθ θ =θ U 2 (θ) = U(θ θ) fi fi fi θ θ =θ

20 ( ) y( ) =(x( ),w( )) x( ) θ, θ Θ U(θ) U(θ θ) =U(θ )+[u(x(θ ),θ) u(x(θ ),θ )]. U(θ) U(ˆθ) u(x(ˆθ),θ) u(x(ˆθ), ˆθ). (1.27) (1.26) (1.27) u(x(θ),θ) u(x(θ),θ ) U(θ) U(θ ) u(x(θ ),θ) u(x(θ ),θ ). (1.28) (SCP) x(θ) x(θ ) 1.4 L x X θ Θ u θ (x, θ) <L y( ) =(x( ),w( )) 2 (a) (the envelope condition) (EC ) (b) x( ) ( ) ( ) (b) 1.5 (EC ) x( ) x( ) ( ) 1.4 θ (EC) (EC) (EC ) K U(θ) U(θ ) <K θ θ, θ, θ Θ (1.29) 14) θ, θ Θ θ >θ U(θ) U(θ ) = sup U(t θ) sup U(t θ ) sup U(t θ) U(t θ ) t Θ t Θ t Θ =sup u(x(t),θ) u(x(t),θ ). t Θ u(x, θ) θ u(x(t),θ) u(x(t),θ )= θ θ u θ (x(t),s)ds sup t Θ u(x(t),θ) u(x(t),θ ) =sup t Θ θ θ θ θ θ θ u θ (x(t),s)ds sup u θ (x(t),s) ds t Θ u θ (x(θ 1 ),s)ds x( ) K = L (1.29) (EC) (EC ) ( ) θ, ˆθ U(ˆθ θ) >U(θ) u(x(ˆθ),θ) u(x(ˆθ), ˆθ) >U(θ) U(ˆθ). 14) U( ) [θ 0,θ 1 ] (absolutely continuous) U( ) (EC ) ( ) ( A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, New York, Dover, 1970, Chapter 9) H. L. Royden, Real Analysis, Second Edition, Macmillan 1968

21 27 (a) θ ˆθ u θ (x(ˆθ),s)ds > θ ˆθ u θ (x(s),s)ds. θ ˆθ [u θ (x(ˆθ),s) u θ (x(s),s)]ds > 0 (b) (SCP) u xθ > (P) θ1 max y( ) θ 0 [S(x(s),s) U(s)]f(s)ds (1.30) subject to (EC ), x(θ) x(θ ), θ, θ Θ,θ >θ, (M) U(θ) U, θ Θ (PC) 1.3 (PC) U(θ 0 ) U (EC ) θ1 θ 0 U(s)f(s)ds = U(s)[1 F (s)] θ1 = U(θ 0 )+ θ 0 θ1 θ 0 + θ1 θ 0 du(s) dθ u θ (x(s),s) 1 F (s) f(s)ds. f(s) 1 F (s) f(s)ds f(s) (1.31) f(s) f(s) ( f(s) ) F (s) (1 F (s)) U(s) 2 F (θ 0 )=0 F(θ 1 )=1 (EC) (1.31) U(θ 0 ) U 15) (1.31) (P) (P ) θ1 [ max S(x(s),s) u θ (x(s),s) 1 F (s) ] f(s)ds U(θ 0 ) (1.32) x( ) θ 0 f(s) subject to (M) and U(θ 0 ) U U(θ 0 )=U (1.32) Φ(x, θ) =S(x, θ) 1 F (θ) u θ (x, θ) (1.33) f(θ) (M) E θ [Φ(x, θ)] θ Θ Φ(,θ) 1.4 (i) x(θ) Φ x (x(θ),θ)=0, θ Θ (1.34) 15) (EC ) (1.31) U(s) (1.31) ( )

22 28 16) x(θ) (EC ) U(θ 0 )=U w(θ) = u(x(θ),θ)+u + θ θ 0 u θ (x(s),s)ds (1.35) 1.4 (ii) x X θ Θ Φ xθ (x, θ) 0 ( A.2 ) Φ (1.33) S(x, θ) =u(x, θ)+v(x, θ) (ii) (a) (b) (c) v xθ (x, θ) 0 x X, θ Θ u ( xθθ (x, θ) ) 0 x X, θ Θ d 1 F (θ) /dθ 0 θ Θ f(θ) (a) v(x, θ) θ (b) u(x, θ) x + θ 1.4 (i) (c) (monotone hazard rate condition, MHRC) (hazard rate) λ(θ) =f(θ)/(1 F (θ)) θ f(θ)dt θ θ + dt θ dt λ(θ)dt ( λ(θ) θ ) 1.5 x(θ) (1.34) w(θ) (1.35) y( ) =(x( ),w( )) (EC ) θ 0 θ θ θ 0 u θ (x(s),s)ds (1.34) S x (x(θ),θ)f(θ) =(1 F (θ))u xθ (x(θ),θ) (1.36) θ = θ 1 (1.36) θ x θ 1 F (θ) θ x(θ) U = θx t V = t c(x) θ 0 > 0 X =[0, + ) Spence-Mirrlees ) (1.33) Φ(x, θ) Φ(x, θ) =θx c(x) 1 F (θ) x f(θ) c Φ x (MHRC) Φ xθ 0 x(θ) 16) ) x =0 1.3

23 29 f(θ)(θ c (x(θ))) = 1 F (θ), θ Θ 1.1 U = w c(x, θ) V = b(x) w X =[0, + ) θ Θ x>0 c θ (0,θ) 0 c θ (x, θ) > 0 c xθ (x, θ) > 0 c xx (x, θ) c(x, θ) =θx u(x, θ) = c(x, θ) u u x θ (a) θ1 U(θ) =U(θ 1 )+ c θ (x(s),s)ds, θ [θ 0,θ 1 ], θ (b) x(θ) U(θ) θ1 θ1 U(s)f(s)ds = U(θ 1 )+ c θ (x(s),s) F (s) θ 0 θ 0 f(s) f(s)ds Φ(x, θ) Φ(x, θ) =b(x) c(x, θ) c θ (x, θ) F (θ) f(θ) c θxx 0 Φ x x(θ) b (x(θ)) = c x (x(θ),θ)+c θx (x(θ),θ) F (θ) f(θ), θ Θ c xθθ 0 F/f θ Φ xθ 0 x( ) 18) 1.1 c(x, θ) =θx θ + F (θ)/f(θ) (1.34) x(θ) (M) ˆx(θ) ˆx(θ) x(θ) ˆx(θ) θ θ ˆx(θ) ˆx(θ) x(θ) Fudenberg and Tirole (1991, Chapter 7) Laffont (1989, Chapter 10) 2. u θ u θ x u θxx ) Laffont and Tirole (1993)

24 θ θ 1 1 Rochet and Stole (2000) Lars Stole Stole (1998) (optimal control theory) 1.4 Fudenberg and Tirole (1991, Chapter 7) Laffont (1989, Chapter 10) Fudenberg and Tirole (1991, Chapters 6, 7) Myerson (1985) Spence-Mirrlees 2 A. Michael Spence Spence (1974) Kreps (1990, Chapter 17) James A. Mirrlees Mirrlees (1971) Milgrom and Segal (2002) Spence-Mirrlees Milgrom and Shannon (1994) Edlin and Shannon (1998) Spence-Mirrlees Wilson (1993) Baron (1989) Laffont and Tirole (1993) 19) Guesnerie (1995) 19) Laffont and Tirole

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Untitled II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j