lecture note information friction

8/13/2019 Lecture Note Information Friction

1/8

i [0, 1] u (a,A,) = a (1A> c)

BR (A, ) =1 A >

0 A

0 ai = 1i 1 ai = 0i

0< c)

xi = + i i N

0, 1x

z= + N

0, 1z

a (x, z)

A (, z) a (x, z) arg maxE[u (a, A (, z) , ) |x, z]

A (, z) = Ex[a (x, z) |z] =

a (x, z)

x (

x(x )) dx

z

a (x, z) = 1(,x(z)](x) a (x, z) 0 A (z) A ((z) , z) = (z)

u (1, A , ) u (0, A , ) =

1 c A > c A A

x (

x(x )) x f(x) = x (x(x ))


2/8

x(z) (z)

A (, z) = (

x(x(z) ))

A ((z) , z) = (z) (x(x(z) (z))) =(z)

x(z) = (z) +

1

x1 ((z))

(z) x(z)

E[u (a, A (, z) , ) |x, z] = a

P|x,z( (z)) c

= a

(z) x

x +

z

z

c

x a (1 c) ac a (x, z) = 1(,x(z)](x)

(z) x

x(z) +

z

z

c = 0

(z) x

x(z) +

z

z

= 1

1 (c)

((z) , x(z))

(z) x

(z) +

1x

1 ((z))

= z

z +

1

1 (c)

zx

(z) 1 ((z)) = zx

z+

x1 (c) .

G () := zx

1 () lim0 G () = lim1 G () = ((z) , x(z))

x

f1 (x) = 1f(f1(x)) G

() = zx

1(1())

zx

2, zx

= 12

zx

2 x 22z z, ! ((z) , x(z))

x ( x 0) z z 0 ( z ) x (z) 1 c z H((z, z, x) , z , z, x) = 0 H(1 c,z, 0, x) =

H(1 c ,z,z, 0) = 0 (z, z, x) (0, x) (z, 0) 1 () = 1 (c) = 1 c.

H1(1 c,z, 0, x) = H1(1 c ,z,z, 0)= 0

z

c


3/8

t= 0, 1

t= 0 r d Nd, 1d p

s

i [0, 1]

p z = d+ N

0, 1z

U= {p} , I= {z, p} U(c|) = E [ec |] w0

p (z) xI(z, p) , xU(p)

x () = arg maxc,x

U(c|) s.t. c= (w0 px) (1 + r) + dx

xI(z, p (z)) + (1

) xU(p (z)) = s

z.

x () = E[d|](1+r)p

V[d|]

p (z) = 0 +1z 1= 0 p z d|p, z = d|z N

dd+zzd+z

, 1d+z

1= 0 z p

xI(z, p) =

dd+zzd+z

(1 + r)p

1d+z

xU(p) =

dd+zp01

d+z

(1 + r)p

1d+z

(=xI(z, p) = xI(z, p) + (1 ) xU(p)) . xI(z, p) + (1 ) xU(p) = s 0 1= 0

p (z) = 1

(1 + r) (d+ z)

dd + zz s

=

dd s(1 + r) (d+ z)

+ z

(1 + r) (d+ z)z

0 =

dd s(1 + r) (d+ z)

, 1 = z

(1 + r) (d+ z)

1 xU(p) =

zd+z

11

(1+r)

1d+z

= 0

xI(z, p) = xU(p) = s

U(c|) E [c|] + 2

V [c|] = (w0 px) (1 +r) + E [d|] x+ 2 V [d|] x2

0 1

p (d|z) p (z|d)p (d) e z(zd)2

2 ed(dd)

2

2 e 12 [(d+z)d22(zz+dd)d] e 1

211

d+z

d zz+d

dd+z

2


4/8

u s s + u u N0, 2u p (z, u) xI(z, p) , xU(p)

x () = arg maxc,x U(c|) s.t. c= (w0 p (z, u) x) (1 + r) + dxxI(z, p (z, u)) + (1 ) xU(p (z, u)) = s + u z, u

p (z) =0+z(z+ uu) z= 0 u= 0 x () = E[d|](1+r)pV[d|] z

u

d

xI(z, p) =

dd+zzd+z

(1 + r)p

1d+z

. p (p) := p0

z=z + uu= d + + uu p p

Nd, 1

p 1

p=2z+

2u

2u

xU(p) =

dd+pp(p)d+p

(1 + r)p

1d+p

. xI(z, p (z, u)) + (1 ) xU(p (z, u)) = s + u

dd + zz (d+ z) (1 + r)p

+

1

dd + pp (d+ p) (1 + r)p

= s + u

p p= p0z

=z + uu

dd (1 ) p 0z

+ zz+ (1 ) p1

z (1 + r) (d+ z+ (1 ) p)p = (s + u)

dd + (z+ (1 ) p) z (1 + r) (d+ z+ (1 ) p)p + ((1 ) pu ) u = s. (0, z, u)

0 = dd(1)p 0zs

(1+r)(d+z+(1)p)(1)p 1zz =

z(1+r)(d+z+(1)p)(1)p 1z

uz =

(1+r)(d+z+(1)p)(1)p 1zp =

12z+

2u

2u

0 = dds

(1+r)(d+z+(1)p)z =

z+(1)p(1+r)(d+z+(1)p)

uz = (1)pu

(1+r)(d+z+(1)p)p =

12z+

2u

2u

.

u= z

p z

|u

|

2z

xU(p)< 0

p = 1

2z+2u

2u

< z z xU(p) =

d+p

pd+p

1z (1 +r)

p1

z (1 + r) (d+ p) = p

(1 + r) (d+ z+ (1 ) p)z+ (1 ) p

(1 +r) (d+ p)

= (1 +r)p(d+z+ (1 ) p) (z+ (1 ) p) (d+p)

z+ (1 ) p= (1 +r)

pd (z+ (1 ) p) dz+ (1 ) p


5/8

EecIEecU =

d+ pd+ z

c) p (, ) k (x, p) a (x, p)

A (, p)

NREE stage 1 :

k (x, p) arg maxE[V (w0 pk+ k) |xi, p] =

k (x, p (, )) f(x) dx

P BE stage 2 :

a (x, p) arg maxE[u (a, A (, p) , ) |xi, p]A (, p) = a (x, p) f(x) dx

.

p (, ) = + p k (xi, p) = E[|xi,p]pV[|x,p]

|xi,p Npp+xxip+x

, 1x+p

k (xi, p) =

pp+xxip+x

p 1x+p

= x

(xi p) K(, p) =

k (x, p) f(x) dx=

x

( p)

= x

( p (, )) p (, ) =

x, p = 2x

i

x >

22p 3x2 2 < 1

2.

x 0

x

2x

x

p= +p


6/8

i [0, 1] U(ai, a, ) = (1 r) (ai )2 r (ai a) a=

[0,1]\{i} ajdj =

ajdj

Uai = 2 [ (1 r) (ai ) r (ai a)] = 0 ai(a, ) = (1 r) + ra. ai= aji, j r= 1 ai = a=

i [0, 1] U(ai, a, ) = (1 r) (ai )2 r (ai a)2 a=

[0,1]\{i} ajdj =

ajdj

xi= + i i N

0, 1x

y = + N

, 1y

a (x, y) a (, y)

a (xi, y) maxa

E

(1 r) (a )2 r (a a (, y))2 |xi, y

a (, y) =

a (xi, y) dj =

a (xi, y) f(x) dx

ai = (1 r) E [|xi, y] + rE [a|xi, y] r

ai = xxi + yy a =[0,1]\{i} aidi =

aidi = x + yy

ai =

(1

r+ rx) E [

|xi, y] + ryy

|xi,y

Nxxi+yy

x+

y

, 1x+

y

ai = (1 r+ rx) xxi+ yyx+ y

+ ryy =(1 r+ rx) x

x+ yxi+

(1 r+ rx) y

x+ y+ ry

y.

(x, y) x = (1r+rx)x

x+y, y =

(1r+rx)yx+y

+ ry

x = (1 r) x

(1 r) x+ y , y = y

(1 r) x+ y .

r

a= x+ yy= (x+ y) + y= + y

(1 r) x+ y .

Ei[] = E [|xi, y] E(0) [] :=

E(1) [] :=

Ej[] dj

E(k) [] := EE(k1) [] =

Ej

E(k1) []

dj.

a

ai = ai(a, )

a= ai, r >1

xidi=

xf(x) dx

r= 0 ai


7/8

a

ai = (1 r)Kk=0

rkEi

E(k)

+ rK (1 r) EiE(K) [a]

EiE(k) []

= xx+y

E(k) [] =

1 k y+ k,

EiE(k) [] =

1 k+1 y+ k+1xi

limK rK (1 r) EiE(K) [a] = 0

ai = (1 r)k=0

rkEi

E(k)

= (1 r)

k=0

rk

1 k+1 y+ k+1xi

= (1 r)

y

1 r + xi y1 r

=

(1 r) 1 r xi+

1 (1 r)

1 r

y= (1 r) x

(1 r) x+ y xi+ y

(1 r) x+ y y.

i [0, 1] U(ai, a, ) = (1 r) (ai )2 r

Li L

Li=[0,1]\{i}(ai aj)2 dj L=

Ljdj

y N

0, 1y

ai = (1 r) Ei+rEia= (1 r) Ei+rEi

ajdj

= (1 r) Ei+rEi

(1 r) Ej+rEj

akdk

dj

= (1 r) Ei+r (1 r) EiE(1)+r2EiE(1)

ajdj

= (1 r) Ei+r (1 r) EiE(1)+r2EiE(1)

(1 r) Ej[] +rEj

akdk

dj

= (1 r) Ei+r (1 r) EiE(1)+r2 (1 r) EiE(2)+r2 (1 r) EiE(2)

ajdj

= (1 r)Kk=0

rkEi

E(k)

+rK (1 r) EiE(K)a.

|xi,y Nxxi+yyx+y

, 1x+y

Ei= xi+ (1 ) y k= 1

E(1) =

Ej[] dj = + (1 ) y

EiE(1)

= (xi+ (1 ) y) + (1 ) y= 2xi+

1 2 y

k

E(k+1) = EE(k)=

Ej

E(k) []

dj=

Ej

1 k

y+k

dj =

1 k

y+k (+ (1 ) y)

= k+1+

1 k+1

y

EiE(k+1) = k+1 (xi+ (1 ) y) +

1 k+1

y.

=

1 k+2

y+k+2xi

L


8/8

W(a, ) = 11r

u (ai, a, ) di=

(ai )2 di

ai = (1 r) E [|y] + rE [a|y] ai = aj = a= E [|y] =y

lecture note information friction

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