Laibson Notes 2013 0

7/25/2019 Laibson Notes 2013 0

1/54

v(x0)

v (x0) = sup{xt+1}t=0

t=0

tF(xt, xt+1)

xt+1 (x) x0

xt t

F(xt, xt+1) t

F

t


2/54

t

ln = dt

dt

t

e =

v(x) = supx+1(x)

{F(x, x+1) + v(x+1)}

F(x, x+1) v(x) v(x+1)

x v()

x v(x) x


3/54

sup{ct}t=0

t=0

t ln(ct)

c, k 0, k =c + k+1, k0

v(k0) = sup{kt+1}t=0

t=0

t ln(kt kt+1)

kt+1 [0, k] (k) k0

v(k) = supk+1[0,k]

{ln (k k+1) + v(k+1)} k


4/54

0 =F(x, x+1)

x+1+ v (x+1)

v(x) =F(x, x+1)

x

v() x

x

v(x) = max {x,Ev(x+1)}

x

Accept if x xReject if x < x

v

v(x) = x if x x

v if x < x

v() v= x

v(x) =

x if x xx if x < x


5/54

x

v(x) = x = E v(x+1)

x= x=x

x=0

xf(x)dx + x=1

x=xxf(x)dx=

1

2(x)2 +

1

2

x= 1

1

1 2

B w (Bw) (x) sup

x+1(x){F(x, x+1) + w(x+1)} x

Bw w sup Bw(x)= w(x) Bw(x) = w(x)

x x B w Bw

B

v Bv = v (Bv)(x) = B(x)x v B B v v

Bnw n

Bnw n B

(S, d) B : S S S B

(0, 1), d (Bf, Bg) d(f, g) f g


6/54

B B Bf Bg f g

(S, d) B : S S B v S v0 S lim Bnv0 = v

Bnv0 ln

X Rl C(X) f : X R B: C(X) C(X)

f, g C(X) f(x) g(x)x X (Bf)(x) (Bg)(x)x X (0, 1)

[B(f+ a)] (x) (Bf)(x) + a f C(X), a 0, x X B a f+ a f

f, g C(X) f g+ d(f, g)

Bf B (g+ d(f, g)) Bg + d(f, g)

Bg B (f+ d(f, g)) Bf+ d(f, g)

Bf Bg d(f, g)

Bg Bf d(f, g)

|(Bf)(x) (Bg)(x)| d(f, g) x

supx

|(Bf)(x) (Bg)(x)| d(f, g)

d(Bf,Bg) d(f, g)


7/54

(Bf)(x) = supc[0,x]

u(c) + Ef

R+1(x c) + y+1

x

f(x) g(x)x cf f

(Bf)(x) = supc[0,x]

u(c) + Ef

R+1(x c) + y+1

= u(cf) + Ef

R+1(x cf) + y+1

u(cf) + Eg

R+1(x cf) + y+1

sup

c[0,x]

u(c) + Eg

R+1(x c) + y+1

= (Bg)(x)

cf

()

[B(f+ )] (x) = supc[0,x]

u(c) + E

f

R+1(x c) + y+1

+

= supc[0,x]

u(c) + Ef

R+1(x c) + y+1

+ = (Bf)(x) +

v0(x) = 0

B

vn(x) = (Bnv0)(x) = B(B

n1v0)(x) = max

x, E(Bn1v0)(x)

xn E(Bn1v0)(x+1) vn(x) xn vn(x) = (B

nv0)(x)xn vn(x) = (B

nv0)(x)

(Bw)(x) max {x,Ew(x+1)} w v0(x) = 0

v1(x) = (Bv0)(x) = max {x,Ev0(x+1)} = max {x, 0} =x v1(x) = x

v2(x) =

(B2v0)(x)

= (Bv1)(x) = max {x,Ev1(x+1)} = max {x,Ex+1}x2 = E x+1 =

2


8/54

v2 =

x2 if x x2

x if x x2

(Bn1v0)(x) =

xn1 if x xn1x if x xn1

xn = E(Bn1v0)(x) =

x=xn1x=0

xn1f(x)dx + x=1

x=xn1

xf(x)dx

=

2

x2n1+ 1

xn = xn1

limnxn =

1

1

1 2

v(x)

v(x0) = sup{ct}0

E0

t=0

tu(ct)

ct C(x) xt+1 X

xt, ct, Rt+1,yt+1, . . .

x c R y

x

ct C(xt) [0, xt]; xt+1 X

xt, ct, Rt+1,yt+1, . . .

Rt+1(xt ct) + yt+1; x0 = y0

y u limc0 u(c) = c > 0 x > 0


9/54

c x, xt+1 t

v(xt) = supct[0,xt] {

u(ct) + Etv(xt+1)}

x

xt+1 = Rt+1(xt ct) + yt+1

x0 = y0

v(xt) = supct[0,xt]

u(ct) + Etv

Rt+1(xt ct) + yt+1

x

u(ct) = EtRt+1v(xt+1) if 0< ct < xt u(ct) EtRt+1v(xt+1) if ct = xt

F OCct : 0 = u(ct) + Etv(xt+1) (Rt+1)

v(xt) = u(ct) xt: v(x) = u(ct) ctxt =u

(ct)

xt+1 = Rt+1(xt ct) + yt+1 = ctRt+1 = Rt+1xt xt+1+ yt+1 = ct= xt xt+1+yt+1Rt+1

u(ct) = EtRt+1u(ct+1) if 0< ct < xt u(ct) EtRt+1u(ct+1) if ct = xt


10/54

=u(ct)

Rt+1

=EtRt+1u(ct+1)

u(ct)< EtRt+1u(ct+1)

u(ct)

ct ct+1 Rt+1

0 EtRt+1u(ct+1)

ct ct+1 Rt+1 u(ct)

u(ct) EtRt+1u(ct+1)

> 0

=

ct < xt

u(ct) EtRt+1u(ct+1) ct < xt


11/54

u(ct) = EtRt+1u(xt+1) if 0< ct < xt u(ct) EtRt+1u(xt+1) if ct= xt

ln ct+1

u

u(c) = c1 1

1

lim1c11

1 =ln c.

Rt+1 t

ct =EtRt+1ct+1

ct

1 = EtRt+1ct+1c

t

1 = Etexp

ln

Rt+1ct+1c

t

1 = Etexp [rt+1 + () ln ct+1/ct][ ln = ; ln Rt+1 = rt+1]

1 =Etexp [rt+1 ln ct+1][ ln(ct+1/ct) = ln ct+1

ln ct

ln ct+1]

ln ct+1

1 = exp

Etrt+1 ln ct+1+ 1

22Vt ln ct+1

Eea =eEa+12

V ara a

ln ct+1 N( ln ct+1, 2Vt ln ct+1)


12/54

0 = Etrt+1 ln ct+1+ 12

2Vt ln ct+1

() l n ct+1 = 1

(Etrt+1 ) +12

Vt ln ct+1

() ln ct+1 = Vt ln ct+1 =

ln ct+1 ln ct+1

ct+1 ctct

ln

1 +ct+1 ct

ct

= ln

1 +

ct+1ct

1

= ln

ct+1

ct

ln ct+1

x ln (1 + x)

Rt = R

R = 1 1

v(x) = supcx

{u(c) + Ev(x+1)} x

x+1 = R(x c)

x0 = E

t=0

Rtyt


13/54

r

r

u(ct) = Ru(ct+1) = 1 u(ct+1) = u(ct+1) =

t=0

Rtct = E0

t=0

Rtyt

t=0 R

t

c0 = E0

t=0 R

t

yt

t=0 R

t = 11 1

R

c0 =

1 1

R

E0

t=0

Rtyt t

01 1

R

Rt = R

R = 1 1

u(c) = c 2 c2 limc0 u(c) =

v(x) = supc

{u(c) + Ev(x+1)} x

x+1 = R(x c) + y+1

x0 = y0


14/54

ct = Etct+1 = Etct+nu(c) = c. u(ct) = EtRt+1; u(ct+1) = Etu(ct+1) R = 1= ct = Et[ ct+1] = Etct+1 = ct =Etct+1 = ct =

Etct+1 ct = Etct+1 = Etct+n

ct+1 = ct+ t+1

ct+1 t

ct ct+1

t=0

Rtct

t=0

Rtyt

t

s=0

Rsct+s = xt+

s=1

Rsyt+s

Et

s=0

Rsct+s= xt+ Et

s=1

Rsyt+s

[ct = Etct+s]

s=0

Rsct = xt+ Et

s=1

Rsyt+s

t=0 R

t = 11 1

R

ct=

1 1

R

xt+ Et

s=1

Rsyt+s

t

t

1 1

R

(xt+ Et

s=1 Rsyt+s) t


15/54

t t + 1

ln ct+1 = 1

(Etrt+1 ) +1

2Vt ln ct+1+ t+1

t+1 t

ln ct+1 = + 1

Etrt+1+ t+1

1

1

1

ln ct+1Etrt+1

ln ct+1 = + 1

Etrt+1+ t+1

tt+1 t

Xt

ln ct+1 = +1

Etrt+1+ Xt+ t+1

t Et ln Yt+1 t+ 1

ln ct+1 = +1

Etrt+1+ Et ln Yt+1+ t+1

1 [0, 0.2]


16/54

[0.1, 0.8]

= t t + 1

= Vt ln ct+1

cit = Yit

Et ln ct+1 = 1

(Etrt+1 ) +

2Vt ln ct+1

Vtct+1 = Et[ln ct+1 Et ln ct+1]2

Vtct+1 Et ln ct+1

u(ct) =u(Et(ct+1)) u(ct) = Etu(ct+1)


17/54

= Et(ct+1)ct

>1

=

ct xt

> r


18/54

W0= E0

t=0

Rtyt

W+1 = R(W c)

v(W) = supc[0,W]

{u(c) + Ev(R(W c))} x

v(W) =

W

1

1 [0, ], = 1 + ln W = 1

c= 1 W

1 = 1 (R1) 1


19/54

t= 1 t= T = 40

vt(x) t

vt(x) = Et Ts=t stu(cs)

vT(x) T

vT(x) = u(x)

vt1(x) = sup

c[0,x]{u(c) + Etvt(R(x c) + y)} x

vt1(x)

vt1(x) = (Bvt)(x) = supc[0,x]

{u(c) + Etvt(R(x c) + y)} x

T

vTn(x) = (BnvT)(x)


20/54


21/54

u(xt+1) W(ct+1)

=REt

u(xt+1) (1 )u(ct+1) dCt+1

dxt+1

=REt

dCt+1dxt+1

+

1 dCt+1

dxt+1

u(ct+1) ()

() dCt+1

dxt+1

=REt

dCt+1dxt+1

+

1 dCt+1

dxt+1

u(ct+1)

e


22/54

Vet+1(x) = maxcet+1[0,x]u cet+1 + Et+1Vet+2 R(x c

et+1) + y

Wnt (x) = maxcet[0,x]

u (cn) + EtV

et+1(R(x cet ) + y)

{cn(x)}T=t t

u(cet+1) = REt+1u

cet+2

u(cnt) =REtV

e(xt+1) = REtu cet+1

ij =ic jc


23/54

ij =ri rj i j

ic = C ov(ii, ln c)

Rft t1 Rft i= j =

equity,f =equity,c

equity,f equity,c

R1t+1, R2t+1, . . . , R

it+1, . . . , R

It+1

Rit+1 = e

rit+1+

iit+1 12 [i]2

it+1

it+1 = it+1 it+1 N(0, 1)x N(, 2) Ax + B N(A + B, A22)

E[exp(Ax + B)] = exp A + B+ 12A22 x N(, 2) E( (x)) = (E(x) + 12V ar(x)) =exp( + 12

2)

iit+1+ rit+1 12 [i]2 N(i 0 + rit+1 12 [i]2, i2 12) = N(rit+1 12 [i]2, i2)

E[Rti] =Eexp(rit+1+ iit+1 12 [i]2)

= exp(mean + 12V ar) = exp(rit+1 12 [i]2 + 12i2) = exp(rit+1)

x ln(1 + x) x = 1 + x exexp(rit) 1 + rit+1

ij =ic jc

u(ct) = Et

Rit+1u(ct+1)

u

u(c) = c1 1

1


24/54

= exp( Rit+1 = e

rit+1+

iit+1 12 [i]2

it+1

u(c) = c

c =Et

Rit+1ct+1

c =Et

exp

+ rit+1+ iit+1

1

2

i2

ct+1

1 = Et

exp

+ rit+1+ iit+1

1

2

i2ct+1

ct

1 = Et

exp

+ rit+1+ iit+1

1

2

i2

exp

ln

ct+1

ct

1 = Et

exp

+ rit+1+ iit+1

1

2

i2 l n (ct+1)

1 = Et

exp

+ rit+1

1

2 i

2

+ iit+1 l n (ct+1)

Sx N(S,S22) E[exp(Sx)] = exp S + 12S22

iit+1 l n (ct+1)

E

iit+1 l n (ct+1)

= expEt[ ln (ct+1)] + 12V iit+1 l n (ct+1)

1 =

exp

+ rit+1

1

2

i2 Et[ln (ct+1)] +1

2V

iit+1 l n (ct+1)

0 = + rit+1 1

2

i2 Et[ln (ct+1)] +1

2V

iit+1 l n (ct+1)

i j

0 = rit+1 rjt+1 1

2

i2 j2+1

2

V

iit+1 l n (ct+1) V j jt+1 l n (ct+1)


25/54

,Et[ ln (ct+1)]

rit+1 rjt+1 =1

2i2 j2 12 V iit+1 l n (ct+1) V j jt+1 l n (ct+1)

V(A + B)

[V(A + B) = V(A) + V(B) + 2Cov(A, B)]

2Cov(A, B) = 2Cov(A, B) V

iit+1 l n (ct+1)

= V

iit+1

+V ( l n (ct+1))+2Cov

iit+1, l n (ct+1)

=

i2

+ 2V l n (ct+1) 2ic

ri

t+1rj

t+1 =

1

2 i2 j2 i2 + 2V l n (ct+1) 2ic + j2 + 2V l n (ct+1) 2jc ritrjt = 12

(i)2 (i)2 + (j )2 (j )2 + 2V l n (ct+1) 2V l n (ct+1) + 2ic 2

ij =rit+1 rjt+1 = ic jc

equity,f =equity,c

=equity,f

equity,c

equity,f .06 equity,c .0003

= .06

.0003= 200


26/54

x x(t + t) x(t) =

+h ph q= 1 p

E[x(t) x(0)] =n ((p q)h) = tt (p q)h V [x(t) x(0)] =n 4pqh2= tt 4pqh2

h=

t

p= 12

1 +

t

(p q) =

t

E[x(t) x(0)] = tt (p q)h= tt

t

t

= t

V [x(t) x(0)] = 2t

z =

t N(0, 1) z N(0, t) t

z

dx= a(x, t)dt + b(x, t)dz dx= a(x, t)dt

+ b(x, t)dz

dx= dt + dz 2

dx = xdt+xdz 2

z(t) x(t) dx= a(x, t)dt + b(x, t)dz V =V(x, t) dV = V

tdt + V

xdx + 12

2Vx2

b(x, t)2dt

=

Vt

+ Vx

a(x, t) + 122Vx2

b(x, t)2

dt + Vx

b(x, t)dz

(V) (x) (z)

dV = a(x, t)dt + b(x, t)dz = Vt + Vxa(x, t) +12 2V

x2b(x, t)2

a

dt +

V

xb(x, t) b

dz

dV = V

tdt + 12

2Vt2

(dt)2 + Vx

dx + 122Vx2

(dx)2 + 2V

x2dxdt +

(dx)2

=b(x, t)2(dz)2 + = b(x, t)2dt +


27/54

t t

0

t x(t)

x x(t + t) x(t) =

+h ph q= 1 p

E(x) = ph + q(h) = (p q)h

V(x) = E[x Ex]2

V(x) = E[x Ex]2 =E(x)2 [Ex]2 = 4pqh2 E

(x)2

= ph2 + q(h)2 =h2

x(t) x(0) t n= tt x(t) x(0)

= x(t) x(0) x(t) x(0) h h

x(t)

x(0) = (k)(h) + (n

k)(

h)

nkpkqnk

nk

= n!

k!(nk)!

E[x(t) x(0)] =n ((p q)h) = tt (p q)h

n x n

V [x(t) x(0)] =n 4pqh2= tt 4pqh2

x t t n

n= tt

x(t) x(0)


28/54

tt (p q)h tt 4pqh2

h p t 0

h=

t

p= 12

1 +

t

q= 1 p= 12

1

t

(p q) =

t

t h,p,q

t

tt 4pqh

2

t p q .5 4 t

h t (pq) t E[x(t) x(0)] = tt (p q)h

(p q) t p, q (p q)

E[x(t) x(0)] = tt

(p q)h= tt

t

t

= t

V [x(t) x(0)] = tt

4pqh2 = t

t4

1

4

1

2t

2t

= t2

1

2t

2t

( t 0)

t 0 t 0 t,

t

2t

t


29/54

t t [x(t) x(0)] D N(t,2t) Binomial D Normal nh= 1t

t= 1

t

xt =

tt =

t

Edxdt

E(x)t =

(pq)ht =

(t)(

t)

t =

E(dx) = dt

t

V(x)t =

4( 14)1( )

2t

2t

t 2

V(dx) = 2dt

t 2

x z

z(t) z, z t

z =

t N(0, 1)


30/54

z N(0, t)

t

z

t1 t2 t3 t4 E[(z(t2) z(t1)) (z(t4) z(t3))] = 0

z(t)

y

z(t) N(z(0), t) z(t) t

a x t a(x, t) t x limt0 Ext a

z(t) x(t)

limt0 Ext =a (x, t) E(dx) = a (x, t) dt limt0 Vxt =b (x, t)

2

V (dx) = b (x, t)2 dt

dt

dx= a(x, t)dt + b(x, t)dz

dx= a(x, t)dt

+ b(x, t)dz


31/54

dz

dz z z

x

z

x b

dx= a(x, t)dt

+ b(x, t)dz

dx= dt + dz

2

dx= xdt + xdz

2

x dx

x =dt + dz dx

x

z(t) x(t) dx = a(x, t)dt+b(x, t)dz V =V(x, t)

dV =V

tdt +

V

xdx +

1

2

2V

x2b(x, t)2dt


32/54

=

V

t +

V

xa(x, t) +

1

2

2V

x2b(x, t)2

dt +

V

xb(x, t)dz


x x

V

V 2nd x x V

x

dx= a(x, t)dt+b(x, t)dz x

dx= xdt + xdz V(x, t) x

V(x, t) x

dV = a(V,x,t)dt+b(V,x,t)dz

a b V

V

dV = a(x, t)dt + b(x, t)dz

V (a, b)

(a, b)

dV = Vt

dt+ Vx

dx+ 12

2Vx2

b(x, t)2dt=V

t + V

xa(x, t) +1

22Vx2

b(x, t)2

dt+ Vx

b(x, t)dz

dV =

V

t +

V

xa(x, t) +

1

2

2V

x2b(x, t)2

a

dt + V

xb(x, t)

b

dz


33/54

dV =

V

t dt +

1

2

2V

t2 (dt)2

+

V

xdx +

1

2

2V

x2(dx)2

+

2V

x2dxdt +

(dt) 32 dt

(t)2 (t)32

(t)

(dz)2 = dt. z t z h

(dt)2 =

dxdt= a(x, t)(dt)2 + b(x, t)dzdt= (dx)2 =b(x, t)2(dz)2 + = b(x, t)2dt +

V

tdt +

V

xdx +

1

2

2V

x2b(x, t)2dt

dz t dx

x

t

V(x, t) dx t

V V

a(x, t) = 0 Vt

= 0 x V t

E(dV) = 12 2V

x2b(x, t)2dt = 0

V V x V(x) = ln x V= 1

x V = 1

x2

dx= xdt + xdz dV =

Vt

+ Vx

a(x, t) + 122Vx2

b(x, t)2

dt + Vx

b(x, t)dz

= 0 + 1x x

12x2(x)

2 dt + 1x xdz = 122 dt + dz = V V

(dx)2 b(x, t)2dt dt V

x


34/54

V

V(x, t)dt= maxu

{w(u,x,t)dt + E [dV]}

V(x, t)dt= maxu

w(u,x,t)dt +

V

t +

V

xa +

1

2

2V

x2b(x, t)2

dt

V

V(x, t)dt

= maxu

w(u,x,t)dt

+ E [dV]

w(x,u,t) = x u t

x = x + x t = t + t t 0


35/54

V(x, t) = maxu

w(x,u,t)t + (1 + t)

1 EV(x, t)

V(x, t) = maxu w(x,u,t)t

+ (1 + t)1

EV (x, t)

(1 + t)

(1 + t) V(x, t) = maxu

{(1 + t) w(x,u,t)t + EV (x, t)}

tV(x, t) = maxu

{(1 + t) w(x,u,t)t + EV(x, t) V(x, t)}

tV(x, t) = maxu

w(x,u,t)t + w(x,u,t) (t)

2+ EV(x, t) V(x, t)

t 0 (dt)2 = 0

V(x, t)dt= maxu

{w(x,u,t)dt + E [dV]}

E [dV]

dV =

V

t +

V

xa(x,u,t) +

1

2

2V

x2b(x,u,t)2

dt +

V

xb(x,u,t)dz

E

V

xbdz

= 0

E [dV] = Vt + Vxa +12 2

Vx2b2 dt

V(x, t)dt= maxu

w(u,x,t)dt +

V

t +

V

xa +

1

2

2V

x2b(x, t)2

dt

x t

t et

w (x(t), u(t), t) dt

V(x, t)dt

= maxu

w(x,u,t)

+ E [dV]


36/54

=

V

t +

V

xa +

1

2

2V

x2b(x, t)2

r r+ 2

x

dx= [rx + x c] dt + xdz

rxdt x xdt cdt

xdz

a= [rx + x c] b= x

x

c

c 0

0 x

V(x, t)dt= maxc,

w(u,x,t)dt +

V

x [rx + x c] +1

2

2V

x2(x)2

dt

c


37/54

c

V

t

t dt

u(c) = c1

1

V(x) = x

1

1

u(c) = c11 V(x) = x1

1

V(x) = x11

V(x) = x11

c

V(x) = x1

1

V(x) = x11 u(c) = c1

1 Vx

2Vx2

x1

1 dt= maxc,

u(c)dt +

x[(r+ )x c] 2

x1(x)2

dt

x1

1 = maxc,

c1

1 +

x[(r+ )x c] 2

x1(x)2

c: F OC : xx 2

2

x1(x)2 = 0

F OCc : u(c) = c =x

c= 1

x=

2

2


38/54

1 =

+

1 1

r+

2

22

= 1 c= x

MP C .05

= 2

= 0.061(0.16)2 = 2.34

5 = 5 = 0.065(0.16)2 = 0.47

8%

u(c) = ln(c)

V(x) = ln(x) +

V(x, t)dt= maxc,

w(u,x,t)dt +

Vx

[rx + x c] + 12 2V

x2(x)2

dt

V(x) u(c)

[ + ln(x)] = maxc,

ln(c) + [(r+ )x c] x1 1

2(x)2x2

F OCc : c= 1

x

F OC : = 2

+ ln(x) = r ln() 1 + 222 + ln(x)

ln(x) ln(x) = r ln() 1 +2

22 = 1

x

= 1 0 = r ln() 1 + 222 x

= 1 = r ln() 1 + 222


39/54

= 1

= 1

r

+ ln() 1 + 222

c= x = 2

dx= a(x,u,t)dt + b(x,u,t)dz

dx

u

V(x, t) = w(x, u, t) + Vt

+ Vx

a(x, u, t) + 122Vx2

b(x, u, t)2

u = u(x, t) =

V x t

= =

T


40/54

T

V(x, T) = (x, T)x T (x, t)

t = ODE

V(x) = w(x, u) + a(x, u)V+1

2b(x, u)2V

V V

V(x, t) = max

w(x, t)t + (1 + t)1EV(x, t), (x, t)

(x, t)

x > x(t) ; x x(t)


V(x, t)dt= w(x, t)dt +

Vt

+ Vx

a(x, t) + 122Vx2

b(x, t)2

dt

V(x, t) = (x, t)

x, t x(t) x(t)


41/54

V

V

V x

x

Vx(x(t), t) = x(x(t), t)

V

x(t) x(t)

t t


42/54

V x

x(t)

dx= adt + bdz

x

w(x) = x

w(x) = x c

= 0

x > x

x x = 0


43/54

V(x) = max

xt + (1 + t)1EV(x), 0

x= x(t + t)

V(x) = 0

V(x)dt= xdt + E(dV)

V(x) = xt + (1 + t)1EV(x)= (1 + t)V(x) = (1 + t)xt + EV(x)= tV(x) = (1 + t)xt + EV(x) V(x)

t 0 (dt)2 = 0 E(dV)

V =x + aV+b2

2V

V

x

V

= x

+

aV +

b2

2V

V

b22V

x V(x) = = 0 x x


44/54

x

V(x) = 0

V(x) = 0 x

x

x

V

limx

V(x)

1

x + a

= 1

limx V(x) = 1

x + a

limx

V(x) =1

x x

etx(t)dt

et(x)(t)dt

=

0

et 1 dt= 1

1

F x


45/54

F(x) + A(x)F(x) + B(x)F(x) = C(x)

C(x)

0

F(x) + A(x)F(x) + B(x)F(x) = 0

F(x) F1 F2

F(x) = C1F1(x) + C2F2(x) F1, F2

A1 A2

A1F1(x) + A2F2(x) = 0 x

V =x + aV+

b2

2V

0 = V + aV+ b2

2V


46/54

erx

0 = erx + arerx + b2

2r2erx

erx

0 = + ar+ b2

2r2

r=a a2 + 2b2b2

r+ r

C+er+

+ Cer

V =x + aV+ b2

2V

E

0

etx(t)dt

E[x(t)]

E[x(t)] =E

x(0) +

t0

dx(t)

=E

x(0) +

t0

[adt + bdz(t)]

= E

x(0) + at +

t0

bdz(t)

t0

bdz(t) = 0

E[x(t)] =x(0) + at

V(x) =


47/54

udv= uv

vdu

E0 e

tx(t)dt

E

0

etx(t)dt= E0

et [x(0) + at] dt

=E

0

et [x(0)] dt +0

et [at] dt=x(0)

+ E

0

et [at] dt

dv= et u= at v= 1

et du= adt

udv=

0

et(at)dt uv vdu = [at1

et]0 0 1 etadt

1

et]0 = 1

= [at 1 et]0 [ 12 eta]0 = 0 + a 12

=1

x(0) +

a

V(x) = 1

x +

a

V =x + aV+ b22V

1 x + a= x + a =x + a1 + b2

2 0

C+er+x + Cer

x

r+, r=a

a2 + 2b2

b2

1

x +a

V(x) = 1

x +

a

+ C+er

+x + Cerx


48/54

V(x) = 0

V(x) = 0

limx V(x) = 1

C+ = 0 limx V(x) = 1

V(x) = 1

x+ a

+ Cer

x = 0

V(x) = 1

+ rCerx = 0

Cerx = 1

x+ a

1 r 1

x+ a

= 0

1 = r

x+ a

= 1

r =x+ a

x = 1r

a

r = aa2+2b2b2

x

x = b2

a +

a2 + 2b2 a


49/54

i i

118

14

CU+ cuI

CU cU CD+ cD(I)

CD+ cD |I| cD > 0 I


50/54

X

dX= dt + dz

dz

V (X) = maxE

=t

e(t)

b2

X2

d

n=1

e((n)t)A (n)

A (n) =

CU+ cUIn In > 0CD+ cD |In| In < 0

(n) = n A (n) = n In = n

x X=U X= u x X=D X=d

V (X) = b2

X2 + V(X) +1

22V(X)

E [dV] = Vt

+ Vx

a + 122Vx2

b2

dt

X U

V (X) = V (u) [CU+ cU(u X)]

V(X) = cU X U

X D

V (X) = V (d) [CD+ cD(X d)]

V(X) = cD X D


51/54

limXU V(X) = V (u) [CU+ cU(u U)] =V (U) = limXU V (X)

limXD

V(X) = V(d) [CD+ cD(D d)] = V (D) = limXD

V (X)

limXU

V(X) = cU =V(U) = limXU

V(X)

limXD

V(X) = cD = V(D) = limXD

V(X)

=

V(u) = cU

V(d) = cD


52/54

V(X)

V(X) = b2X2

+ +2X

2 + 2

2

3

V (X) = b2X2 + V(X) + 122V(X)

V(X) = b2

X2

+

+ 2X

2 +

22

3

V(X) V(X)

V(X) = b2

2X

+22

2

, V(X) = b

V(X) = 1

b2

X2 + V(X) + 122V(X)

V(X) =1

b2

X2 b2

2X

+

22

2

1

22

b

= b

2

X2 +

2X

+

2

+

2

2

2


53/54

= b2

X2

+

2 + 2X

2 +

2

3

u= d= X R

|x|


54/54

E[Rti] =E[exp(rti+

iit 12 [i]2)] =exp(mean + 12V ar) = exp(r

ti 12 [i]2 + 12i2)

=exp(rti )

V(A + B) = V(A) + V(B) + 2Cov(A, B)

xln(1 + x) x1 + x ex

Laibson Notes 2013 0

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Transcript of Laibson Notes 2013 0