Studying the Ramsey Model

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1Ramsey Model

OLG

1

1 t1 max J = I(x, u, t)dt + F (x1 , t1 ), (1.0.1){u(t)} t0

x = f (x, u, t), x(t0 ) = x(t1 ) = {u(t)} x0 , x1 , U.

(1.0.2) (1.0.3) (1.0.4) (1.0.5)

x state u control x u t x y co-state H

H(x, u, y, t) = I(x, u, t) + yf (x, u, t).

(1.0.6)

H u

= 0,

(1.0.7) (1.0.8) (1.0.9) (1.0.10) (1.0.11) (1.0.12)

H , y H , y = x x(t0 ) = x0 , x = x(t1 ) = y(t1 ) = x1 F . x1

1

Intriligator 1971

1

2

Social Planner :

Blanchard and Fisher (1989) 2 Social Planner

2.1

Kt t Nt t Yt

Yt = F (Kt , Nt ) Yt Ct dKt /dt dKt . (2.1.1) dt Yt = Ct + n n Constant Returns to Scale:CRS f a x af (x) = f (ax)

Yt F (Kt , Nt ) Kt N t Kt = = F( , ) = F ( , 1). Nt Nt Nt Nt Ntdt t Nt = K dKt

(2.1.2)

yt = Yt /Nt kt = Kt /Nt yt = f (kt ) = F (kt , 1)dkt dt +nkt

Ct /Nt = ct

2

yt = f (kt ) = ct +

dk + nkt . dt

(2.1.3)

dk = f (kt ) ct nkt . (2.1.4) dt f kt kt f (kt ) 0kt 0 f (kt )

2k

t

=

Kt Nt

kt Nt = Kt t

dkt Nt dt

+ k dNt = dt

dKt /dt dKt N dt t

=

dkt dt

+ nkt

2

2.2

k0 (> 0) T

lim kT eT 0

(2.2.1)

T kT +1

2.3

Social Planner

Social Planner kt ct kt state ct control > 0 U 3

U=0

u(ct )et dt.

(2.3.1)

U {ct }

U =0

u(ct )et dt

(2.3.2) (2.3.3) (2.3.4)

kt = f (kt ) ct nkt k0 > 0t t

lim kt e

0

(2.3.5)

state kt co-state qt

H(kt , ct , qt ) = u(ct )et qt (nkt + ct f (kt )). t = et qt 4

(2.3.6)

H(kt , ct , t (qt )) = et {u(ct ) t (nkt + ct f (kt ))}.ct , kt , qt H qt

(2.3.7)

H k(t) = = f (kt ) nkt ct . qtct

(2.3.8)

H = et (u (ct ) t ) = 0. ct

(2.3.9)

3 et u(c ) U < t 4 q t t

3

u (ct ) = t .kt

(2.3.10)

q= t

H = qt (n f (kt )) = et t (n f (kt )). kt d t (e qt ) = et qt + et qt . dt

(2.3.11)

t =

(2.3.12)

t

= et qt + et et t (n f (kt )) = t + t (n f (kt )) = t ( + n f (kt )).

(2.3.13) (2.3.14) (2.3.15)

kt , ct , t 1 c implicit

kt

=

f (kt ) nkt ct t t ( + n f (kt )).

(2.3.16) (2.3.17) (2.3.18)

u (ct ) = t =

2.4

lim kt et = 0 F (xT , T ) kT eT

F = T = T et kT complementary slackness condition

T kT eT = 05 T = u (cT )eT

kT u (cT )eT = 0 T T

lim kT u (cT )eT = 0

6

5 6

U

4

2.5

2 t

t = u (ct )ct . 3

(2.5.1)

u (ct )ct = u (ct )( + n f (kt )), kt ct ct kt = u (ct ) ( + n f (kt )), u (ct )

(2.5.2)

(2.5.3) (2.5.4)

= f (kt ) nkt ct ,

(ct , kt )

33.1

Nt wt kt rt t = F (Kt , Nt ) rt Kt wt Nt

t Kt t Nt

= FKt rt = 0 = FNt wt = 0.

(3.1.1) (3.1.2)

FK

= = = =

F (K + , N ) F (K, N ) = lim 0 0 lim0

1 N (F (K

+ , N ) F (K, N )) 1 N F ( K , 1) N

(3.1.3) (3.1.4) (3.1.5) (3.1.6)

lim

F ( K+ , N ) F ( K , N ) N N N N N

= lim

F(K + N

0

N , 1) N

0

lim

F (k +

N , 1) N

F (k, 1)

= lim

f (k +

0

N) N

f (k)

f (k + ) f (k) = f (k). 0 lim

= /N 0 0 F 7

F (K, N ) = KFK + N FN . N F (K, N )/N = f (k)

(3.1.7)

K FK + FN = kf (k) + FN . N7g

(3.1.8)g x

k g(x) = kx

g . x

k = 1 g(x) = x

5

FN = f (k) kf (k).

(3.1.9)

f (kt ) = rt f (kt ) kt f (kt )

(3.1.10) (3.1.11)

= wt

3.2

K0

r w C = A dA/dt

dAt = wt Nt + rt At . dt Ct + ct +

(3.2.1)

dat + nat = wt + rt at , dt

(3.2.2)

3.3

NPG

at

ct +

dat = wt + (rt n)at , dt

(3.3.1)

rt n 8 aT 9 RT0

aT eT

(rs n)ds 0

(3.3.2)

lim aT e

RT0

(rs n)ds 0

(3.3.3)

No Ponzi Game 10 Rt

8

rt a0 e(rn)t a0 e RT 9 a = a e 0 (rs n+)ds , > 0 t 0 a0 eRT0

0 (rn)ds

(rs n + )ds e

RT0

(rs n)ds = a e 0

RT0

ds

= a0 eT ,

T .

10

6

NPG NPG at (rt n)at = wt ct . eRt0

(3.3.4)(rs n)ds

0

at e (at eRt0

Rt0

(rs n)ds

(rt n)at eRt0

Rt0

(rs n)ds

= (wt ct )eRt0

Rt

(rs n)ds

.

(3.3.5)

(rs n)ds

) = at e

(rs n)ds

(rt n)at eT 0

(rs n)ds

0 T

[ ]T Rt 0 (rs n)ds at e =0

(wt ct )e

Rt0

(rs n)ds

dt,

(3.3.6)

t = 0 a0 T RT Rt aT e 0 (rs n)ds a0 = (wt ct )e 0 (rs n)ds dt0

(3.3.7)

NPG T RT RT 0 (rs n)ds ct e dt = wt e 0 (rs n)ds dt + a0 , (3.3.8)0 0

3.4

U =0

u(ct )et dt

(3.4.1) (3.4.2) (3.4.3)

at = wt ct + (rt n)at a0 0t Rt

lim at e

(rs n)ds 0 0

(3.4.4)

at co-state qt

H(ct , at , qt ) = u(ct )et + qt (wt ct + (rt n)at ).

(3.4.5)

H ct at qt

= u (ct )et qt = 0, H = wt ct + (rt n)at , qt H = = qt (rt n). at =

(3.4.6) (3.4.7) (3.4.8)

ct qt qt = u (ct )ct et u (ct )et = (u (ct )ct u (ct ))et . 3

(3.4.9)

u (ct )ct = u (ct )( rt + n). 7

(3.4.10)

3.5

t

rt kt

= f (kt ), = at ,

(3.5.1) (3.5.2)

1

wt = f (kt ) kt f (kt ),

(3.5.3)

kt

= f (kt ) kt f (kt ) ct + (f (kt ) n)kt , = f (kt ) ct nkt , u (ct ) ( f (kt ) + n). = u (ct )

(3.5.4) (3.5.5) (3.5.6)

ct

Social Planner

c

E c=0

E k=0O 1:

k

8

44.1

Degree of Relatvie Risk Aversion: RRA(ct )

(ct ) = ctct

u (ct ) . u (ct )

(4.1.1)

ct =

ct (f (kt ) n). (ct )

(4.1.2)

ct = 0 ct = 0 f (kt ) = + n 11 ct kt + n kt = 0 ct = f (kt ) nkt 12

4.1.1

CRRA

u(c) =

c1 , 1

(0 < < 1)

(4.1.3)

u (c) = c , u (c) = c1

(c) = c

c1 = . c

Constant Relative Risk Averse CRRA

f (k) = k

(0 < < 1)

f (k) = k 1

ct kt

=

ct (k 1 n),

(4.1.4) (4.1.5)

= kt ct nkt .

4.1.2

CARA

1 u(c) = et , 11 k

( > 0)

0 > 0, u (c) (c 0) ct = 0 12

9

(Constant Absolute Risk Aversion) CARA u (c)/u (c) CARA

CARA ct ct kt = 1 (k 1 n), (4.1.6) (4.1.7)

= kt ct nkt .

4.2

n n < 0 (rt n) n 13

c

E c=0

E k=0

E k=0O 2:

k

13

10

55.1

t t gt gt

ct + at + nat = wt + rt at t .

(5.1.1)

at = wt + (rt n)at t ct , at co-state qt

H(ct , at , qt ) = u(ct )et + qt (wt + (rt n)at t ct )

H ct qt at

= u (ct )et qt = 0, H = qt (rt n), at = wt + (rt n)at t ct . =

(5.1.2) (5.1.3) (5.1.4)

kt = f (kt ) nkt t ct , kt = 0 ct = f (kt ) nkt t ,

(5.1.5)

(5.1.6)

t = k

NPG Rt 0 (rs n)ds ct e dt =0 0

wt e

RT0

(rs n)ds

dt 0 RT0

t e

RT0

(rs n)ds

dt + a0 .RT0

(5.1.7) dt

t = gt

0

t e

(rs n)ds

dt =

0

gt e

(rs n)ds

Rt RT RT 0 (rs n)ds 0 (rs n)ds ct e dt = wt e dt gt e 0 (rs n)ds dt + a0 .0 0 0

(5.1.8)

11

5.2

bt bt t bt rt

bt + nbt = gt t + rt bt , kt

(5.2.1)

NPG RT RT t e 0 (rs n)ds dt = gt e 0 (rs n)ds dt + b0 ,0 0

(5.2.2)

vt = at + bt ct + vt + nvt = wt + rt vt t RT 0 (rs n)ds dt = ct e0 0 RT0

wt e

(rs n)ds

dt 0

t e

RT0

(rs n)ds

dt + v0 ,

(5.2.3)

RT RT ct e 0 (rs n)ds dt = wt e 0 (rs n)ds dt 0 0 0

gt e

RT0

(rs n)ds

dt + a0

(5.2.4)

5.3

rt t zt

ct +

dat + nat = wt + (1 t )rt at + zt . dt

(5.3.1)

H(ct , at , qt ) = u(ct )et + qt (wt + ((1 t )rt n)at + zt ct ),

(5.3.2)

H ct qt at

= u (ct )et qt = 0, = H = qt (n (1 t )rt ), at = wt + ((1 t )rt n)at + zt ct . 12

(5.3.3) (5.3.4) (5.3.5)

u (ct )ct = + n (1 t ) u (ct ) ct = 0

(5.3.6)

rt =

n+ , 1 t

(5.3.7)

rt = f (kt ) t rt 1 1 f k k

k

1

kn+f (k)

n+ 1- 3:

kt (r = f (k), w = f (k) kf (k)) k = f (k) ( f (k) + n)z c, k = 0 c = f (k) + z nk f (k)k,

(5.3.8)

(5.3.9)

z f (k)k

13

c

c=0

z O k=0 4:

k

14

6

Blanchard, Olivier and Stanley Fischer, A Lecture Note on Macroeconomics, MIT Press, 1989. Intriligator, Michael D., Mathematical Optimization and Economic Theory, Prentice-Hall, 1971. Barro, Robert J and Xavier Sala-i-Martin, Economic Growth, MIT Press, 1995.

15