Slide Lecture3

June 24th, 2010 @ Gakushuin

E-mail: [email protected]://masaru.inaba.googlepages.com/

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I

() 2010/06/24 @ Gakushuin 3 / 78

I.1

maxct ;it ;kt+1

TXt=0

tu(ct)

subject toct + it = f (kt)kt+1 kt = it kt; t = 0; ;T;k0 = k0kT+1 0:

0 < < 1 (discount factor)u(0) = 0; u0() > 0; u00() < 0; u0(0) = 1; u0(1) = 0

() 2010/06/24 @ Gakushuin 4 / 78

it

maxct ;kt+1

TXt=0

tu(ct)

subject to

kt+1 kt = f (kt) ct kt t = 0; ; T; ( (transition equation))k0 = k0kT+1 0:

() 2010/06/24 @ Gakushuin 5 / 78

L =TX

t=0tu(ct)+

TXt=0

ttnf (kt)ctkt+1+(1)kt

o+1(k0k0)+kT+1:

(1)kt (state variable)ct (control variable)tt (costate variable)

() 2010/06/24 @ Gakushuin 6 / 78

@L@ct

= tu0(ct) tt = 0 (t = 0; 1; 2; ;T )@L@kt+1

= tt + t+1t+1nf 0(kt+1) + (1 )

o= 0 (t = 0; 1; 2; ;T 1)

@L@kT+1

= TT + = 0@L@t

= tnf (kt) ct kt+1 + (1 )kt

o= 0 (t = 0; 1; 2; ;T )

@L@1

= k0 k0 = 0:

kT+1 0; 0; kT+1 = 0:

() 2010/06/24 @ Gakushuin 7 / 78

u0(ct) = t (t = 0; 1; 2; ;T ) (2)t = t+1

nf 0(kt+1) + (1 )

o(t = 0; 1; 2; ;T 1) (3)

TT = (4)kt+1 kt = f (kt) ct kt (t = 0; 1; 2; ;T ) (5)k0 = k0 (6)kT+1 0; 0; kT+1 = 0: (7)

() 2010/06/24 @ Gakushuin 8 / 78

(2)(3)Euler

u0(ct) = u0(ct+1)nf 0(kt+1) + (1 )

o(8)

(7)

TT kT+1 = 0 (9)TT > 0

kT+1 = 0:

() 2010/06/24 @ Gakushuin 9 / 78

I.2t 1 (shadow price) (3)

t = t+1nf 0(kt+1) + (1 )

o(for t = 0; 1; 2; ;T 2) (10)

(forward-looking variable)

TT kT+1 = 0

T kT+1 kT+1

() 2010/06/24 @ Gakushuin 10 / 78

II

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II.1

maxct ;kt+1

1Xt=0

tu(ct)

s.t. kt+1 kt = f (kt) ct ktk0 = k0:

L =1X

t=0tu(ct)+

1Xt=0

ttnf (kt)ctkt+1+ (1)kt

o+1(k0k0) (11)

() 2010/06/24 @ Gakushuin 12 / 78

kt+1ct

@L@ct

= tu0(ct) tt = 0@L@kt+1

= tt + t+1t+1nf 0(kt+1) + (1 )

o= 0

@L@t

= tnf (kt) ct kt+1 + (1 )kt

o= 0

@L@1

= k0 k0 = 0:

() 2010/06/24 @ Gakushuin 13 / 78

u0(ct) = u0(ct+1)nf 0(kt+1) + (1 )

o(12)

kt+1 kt = f (kt) ct kt (13)k0 = k0 (14)

T ! 1limt!1

tu0(ct)kt+1 = 0: (15)

() 2010/06/24 @ Gakushuin 14 / 78

II.1 (steady state)(1/3) (optimal path)

u0(ct) = u0(ct+1)nf 0(kt+1) + (1 )

o(16)

kt+1 kt = f (kt) ct kt (17)k0 = k0

ck

u0(c) = u0(c)nf 0(k) + (1 )

ok k = f (k) c k

() 2010/06/24 @ Gakushuin 15 / 78

(steady state)(2/3)

1 = nf 0(k) + (1 )

o(18)

c + k = f (k) (19)

() 2010/06/24 @ Gakushuin 16 / 78

(steady state)(3/3)c

kk

ct+1 = ct

kt+1 = ktE

kg

c

() 2010/06/24 @ Gakushuin 17 / 78

(modified golden rule) kg(19)

f 0(kg) =

k > kg (dynamic inefficiency).k kg (dynamic efficiency).

k (18)

f 0(k) = 1 1 +

. f 0()k < kg =) k

() 2010/06/24 @ Gakushuin 18 / 78

II.3.(i): (phasediagram)(1/6)

(optimal path)

u0(ct) = u0(ct+1)nf 0(kt+1) + (1 )

o(20)

kt+1 kt = f (kt) ct kt (21)k0 = k0

() 2010/06/24 @ Gakushuin 19 / 78

(phase diagram)(2/6)(ct+1 = ct)ct+1 = ct k < k f 0(k)

1 < f 0(k) + (1 )

(20)u0(ct)

u0(ct+1) = nf 0(kt+1) + (1 )

o> 1

()u0(ct) > u0(ct+1)()ct+1 > ct

u0()ct+1 = ctct+1 = ct

() 2010/06/24 @ Gakushuin 20 / 78

(phase diagram)(3/6)

k < k

.

.

.

1 f 0(k) + (1 )

.

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2

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3 ct+1 > ct.

k > k

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1 f 0(k) + (1 )

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2

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3 ct+1 < ct

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(phase diagram)(4/6)c

kk

ct+1 = ct

kgFigure: ct+1 = ct

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(phase diagram)(5/6)(kt+1 = kt)kt+1 = kt kt+1 = kt c + k = f (k) kc

c + k > f (k)(21)

kt+1 kt = f (kt) ct kt < 0()kt+1 < kt

kt+1 = kt

() 2010/06/24 @ Gakushuin 23 / 78

(phase diagram)(6/6)c

k

kt+1 = kt

kgFigure: kt+1 = kt

() 2010/06/24 @ Gakushuin 24 / 78

(ii): (optimalpath)(1/7)c

kk

ct+1 = ct

kt+1 = ktE

kg

c

IV

III

III

Figure: () 2010/06/24 @ Gakushuin 25 / 78

(optimal path)(2/7)

k0

u0(ct) = u0(ct+1)nf 0(kt+1) + (1 )

okt+1 kt = f (kt) ct ktk0 = k0

c0

() 2010/06/24 @ Gakushuin 26 / 78

(optimal path)(3/7)c

kk

ct+1 = ct

kt+1 = ktE

kgk0

cl0

c0cu0

O E0

Figure: II () 2010/06/24 @ Gakushuin 27 / 78

(optimal path)(4/7)c0cu0

.

. .1 I

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.

2 kt+1 = kt IV

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3 IV

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4 (c; k) = (0; 0)

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1

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2 limc!0 u0(c) = 1.

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3

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4 cu0 () 2010/06/24 @ Gakushuin 28 / 78

(optimal path)(5/7)

cl0

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1 I

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2 ct+1 = ct II

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3

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4 E0

() 2010/06/24 @ Gakushuin 29 / 78

(optimal path)(6/7)

.

.

.

1 limc!0 u0(c) = 1

.

.

.

2 E0

.

.. 3 E0 kg

f 0(k) <

.

.

.

4 (20)u0(ct+1)u0(ct) =

1 f 0(kt+1) + (1 ) > 1

u0(ct) 1t

.

.

.

5 limt!1 tu0(ct) = 1 limt!1 kt = kE0

limt!1

tu0(ct)kt+1 = 1

.

.

.

6 cl0 () 2010/06/24 @ Gakushuin 30 / 78

(optimal path)(7/7)

c0 4 cl0

(saddle path)

() 2010/06/24 @ Gakushuin 31 / 78

II.4

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1

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2

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(i)

u0(ct) = u0(ct+1)nf 0(kt+1) + (1 )

o(22)

kt+1 kt = f (kt) ct kt: (23) (constant relative risk aversion, CRRA)

u(c) = c1

1 1

f (k) = Ak

ct = ct+1

nAk1t+1 + (1 )

o(24)

kt+1 kt = Akt ct kt (25)1 Y = AK(L)1 y = Ak

() 2010/06/24 @ Gakushuin 33 / 78

II.5 (shooting algorithm)

k0(24)(25) c0 c0 c0 (shooting algorithm)

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(steady state)(24) (25)

1 = nAk1 + (1 )

o(26)

c + k = Ak (27)

k =8>>>:

1 (1 )A

9>>=>>;1

1

(28)

c = Ak k (29)

() 2010/06/24 @ Gakushuin 35 / 78

(i): (shootingalgorithm)

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1

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2 cl cu

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3 c0 =cl+cu

2

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4 k0 c0(24)(25)t = 1; 2; 3; ;

.

.

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5 t

.

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.

1 kt > kcl = c0step 3

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2 kt < kcu = c0step 3

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3 k kt <

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.

6 c0ct; kt

() 2010/06/24 @ Gakushuin 36 / 78

(ii) Matlab Code (1/10)% determinisitic model

% shooting algorithm for simple optimal growth model

% u(c)=c^(1-sigma)/(1-sigma)

% y=Ak^alpha

% 2008/11/22

% Masaru Inaba at YNU

% 6Matlab% shootingclear all;

close all;

set(0,'defaultAxesFontSize',12);

set(0,'defaultAxesFontName','century');

set(0,'defaultTextFontSize',12);

set(0,'defaultTextFontName','century');

() 2010/06/24 @ Gakushuin 37 / 78

Matlab Code (2/10)% ================

% parameters

% ================

alpha=0.3; % capital share

beta=0.98; % discount factor

delta=0.08; % depreciation rate

sigma=2; % relative risk aversion

T=100; % maximum time for simulation

A=1*ones(T,1); % productivity (which is constant for all t)

% ====================

% steady state values

% ====================

A_ss=1; %productivity in ss

% ss capital, equation (28)

k_ss=((1/beta-(1-delta))/(alpha*A_ss))^(1/(alpha-1));

% ss consumption, equation(29)

c_ss=A_ss*k_ss^alpha-delta*k_ss;

() 2010/06/24 @ Gakushuin 38 / 78

Matlab Code (3/10)

% =====================

% initialization

% =====================

k_0=0.1; % initial value of capital

% lower bound of initial value of consumption

c_under=0.000000001;

% upper bound of initial value of consumption

c_upper=A_ss*k_0^alpha-delta*k_0;

% initialize path vector of capital as zero vector

k=zeros(T,1);

% initialize path vector of consumption as zero vector

c=zeros(T,1);

() 2010/06/24 @ Gakushuin 39 / 78

Matlab Code (4/10)

% ======================

% shooting algorithm

% ======================

% start while iteration

while (abs(k(T)-k_ss)>1e-8)&(abs(c(T)-c_ss)>1e-8);

% initialize path vector of capital as zero vector k=zeros(T,1);% initialize path vector of consumption as zero vector c=zeros(T,1); c0temp=(c_upper+c_under)/2; % initialization of c_0

() 2010/06/24 @ Gakushuin 40 / 78

Matlab Code (5/10)

%-----------% at t=1%----------- t=1; % note that Matlab can not apply the number of index 0 k(t)=k_0; c(t)=c0temp;% equation (25) k(t+1)=A(t)*k(t)^alpha-c(t)-delta*k(t)+k(t);% equation (24) c(t+1)=(c(t)^(-sigma).../(beta*(alpha*A(t+1)*k(t+1)^(alpha-1)+(1-delta))))^(-1/sigma);

() 2010/06/24 @ Gakushuin 41 / 78

Matlab Code (6/10)

%---------------% for t=2 to T-1%--------------- for t=2:T-1;% equation (25) k(t+1)=A(t)*k(t)^alpha-c(t)-delta*k(t)+k(t);% flag 1 means k(t) is too big if k(t+1)>2*k_ss; flag=1; break; end;% flag 2 means k(t) is too small if k(t+1)

Matlab Code (7/10)%---------------% diagnosis%--------------- if t==T-1; if k(T)>k_ss; flag=1 % flag 1 means k(T) is too big else flag=2 % flag 2 means k(T) is too small end end if flag==1; c_under=c0temp elseif flag==2; c_upper=c0temp end clear flagend % end of while iteration loop

disp('We found!'); % the end of shooting algorithm.

() 2010/06/24 @ Gakushuin 43 / 78

Matlab Code (8/10)% =============================

% code for the graph plottings

% =============================

% data for equation (27);

k_Eprime=(delta/A_ss)^(1/(alpha-1)); % position of E'

% data for plotting equation (27)

k_27=linspace(0,k_Eprime,1000); % vector of k in (27)

c_27=A_ss.*k_27.^alpha-delta.*k_27; % vector of c in (27)

% data for plotting equation (26)

c_26=linspace(0,2*c_ss,100);

k_26=k_ss*ones(100,1);

% data for initial value plotting

k_initial=k_0*ones(100);

c_kinitial=linspace(0,(A_ss*k_0^alpha-delta*k_0),100);

% data for optimal initial consumption

coptini=c(1,1)*ones(100);

koptini=linspace(0,k_0,100);

() 2010/06/24 @ Gakushuin 44 / 78

Matlab Code (9/10)

figure(1);

% plot equation(27)

plot(k_27,c_27);axis([0,1.05*k_Eprime,0,1.3*c_ss]);hold on;

% plot equation(26)

plot(k_26,c_26);

plot(k,c,'m','LineWidth',2);% plot for optimal path

plot(k_initial,c_kinitial,':'); % plot for k_0

plot(koptini,coptini,':'); % plot for optimal c_0

text(k_0,-0.15,'k_{0}');

text(-0.8,c(1,1),'c_0');

text(k_ss,-0.15,'k_{ss}');

text(k_ss,1.03*c_ss,'E');

text(k_Eprime,-0.15,'E^\prime');

hold off;

() 2010/06/24 @ Gakushuin 45 / 78

Matlab Code (10/10)figure(2);

% plot equation(27)

plot(k_27,c_27);axis([0,1.3*k_ss,0,1.3*c_ss]);hold on;

% plot equation(26)

plot(k_26,c_26;

plot(k,c,'m','LineWidth',2);% plot for optimal path

plot(k_initial,c_kinitial,':'); % plot for k_0

plot(koptini,coptini,':'); % plot for optimal c_0

text(k_0,-0.15,'k_{0}');

text(-0.2,c(1,1),'c_0');

text(k_ss,-0.15,'k_{ss}');

text(k_ss,1.03*c_ss,'E');

text(k_Eprime,-0.15,'E^\prime');

hold off;

Matlab

() 2010/06/24 @ Gakushuin 46 / 78

1 by shooting algorithm

0 5 10 15 20 25 30 350

0.5

1

1.5

k0

c0

kss

E

E

Figure: Phase diagram for shooting 1 () 2010/06/24 @ Gakushuin 47 / 78

0 1 2 3 4 5 60

0.5

1

1.5

k0

c0

kss

E

Figure: Phase diagram for shooting 2 () 2010/06/24 @ Gakushuin 48 / 78

II.6

() 2010/06/24 @ Gakushuin 49 / 78

(i): (1/6): f (x) a

f (x) = f (a) + f 0(a)(x a) + 12!

f 00(a)(x a)2 + 13! f000(x a)2 +

() =1X

n=0

1n!

f n(a)(x a)n

f (x) f (a) + f 0(a)(x a) (30) () 2010/06/24 @ Gakushuin 50 / 78

(2/6) f (x) x = aax a

a

f (x)

x

f (x)

f (a) + f 0(a)(x a)

Figure: () 2010/06/24 @ Gakushuin 51 / 78

(3/6)

f (x; y) = f (a; b) + fx(a; b)(x a) + fy(a; b)(y b)+

12!

nfxx(a; b)(x a)2 + 2 fxy(x a)(y b) + fyy(a; b)(y b)3

o+

fx = @ f (x;y)@x f (x; y) f (a; b) + fx(a; b)(x a) + fy(a; b)(y b) (31)

() 2010/06/24 @ Gakushuin 52 / 78

(4/6) (log-linearization) xln xln a2

x = eln x

3 f (x) = feln x

ln x

(30)

f (x) feln a

+@ f

eln x

@ ln x

(ln x ln a)() f (x) f (a) + a f 0(a)(ln x ln a)

2ln3e dexdx = ex

() 2010/06/24 @ Gakushuin 53 / 78

(5/6)

x = eln x

f (x) = feln x

ln x

@ feln x

@ ln x

=@ f (x)@x

@eln x

@ ln x= f 0(x)x

() 2010/06/24 @ Gakushuin 54 / 78

(6/6)

f (x) f (a) + a f 0(a)(ln x ln a)

ln(x) ln(a) = ln

x

a

= ln

x a

a+ 1

x a

a(32)

xaax a

apersentage deviationa

() 2010/06/24 @ Gakushuin 55 / 78

(ii)

(24) (25)(26) (27)

k =8>>>:

1 (1 )A

9>>=>>;1

1

c = Ak k:

() 2010/06/24 @ Gakushuin 56 / 78

(25)(25)

kt+1 k + 1 (kt+1 k)kt k + 1 (kt k)

Akt Ak + Ak1(kt k)ct c + 1 (ct c)kt k + (kt k)

(25)kt+1 kt = Akt ct kt

()(k + kt+1 k) (k + kt k) = Ak + Ak1(kt k) (c + ct c) fk + (kt k)g

()(kt+1 k) (kt k) = Ak1(kt k) (ct c) (kt k)( (27))

() 2010/06/24 @ Gakushuin 57 / 78

xt xt x (x)

kt+1 kt = Ak1 kt ct kt()kt+1 = Ak1 + 1 )kt ct()kt+1 = 1

kt ct ((26))

()kt+1 = 1 kt ct (33)1 = 1 > 0

() 2010/06/24 @ Gakushuin 58 / 78

(24)(24)

ct c c1(ct c)

ct+1nAk1t+1 + (1 )

o c

nAk1 + (1 )

o c1

nAk1 + (1 )

o(ct+1 c)

+ cn( 1)Ak2

o(kt+1 k)

(24)c1(ct c)

= c1(ct+1 c) + cn( 1)Ak2

o(kt+1 k)

() 2010/06/24 @ Gakushuin 59 / 78

xt xt x c1ct = c1ct+1 +

n( 1)Ak2

okt+1

, ct = ct+1 + 2 kt+1, ct+1 ct = 2 kt+1 (34)

2 =

n(1)Ak2

oc1 > 0

() 2010/06/24 @ Gakushuin 60 / 78

(33)(34)kt+1 = 1 kt ctct+1 ct = 2 kt+1

ct

kt+2 (1 + 1 + 2)kt+1 + 1 kt = 0, kt+2 3 kt+1 + 1 kt = 0

3 = 1 + 1 + 2

() 2010/06/24 @ Gakushuin 61 / 78

2 3 + 1 = 01;2 =

3p

23412

1 > 1 2 < 1=)

kt k = b1t1 + b2t2

1 > 1 2 < 1 0 kt = k b1 = 0

() 2010/06/24 @ Gakushuin 62 / 78

ckk

ct+1 = ct

kt+1 = ktE

kg

c

Figure: Linearized dynamics () 2010/06/24 @ Gakushuin 63 / 78

III

() 2010/06/24 @ Gakushuin 64 / 78

III

() 2010/06/24 @ Gakushuin 65 / 78

III.1

(one representative household) (goods)

() 2010/06/24 @ Gakushuin 66 / 78

(i)

1Xt=0

tu(ct): (35)

u()u(0) = 0; u0() > 0; u00() < 0; u0(0) = 1; u0(1) = 0

() 2010/06/24 @ Gakushuin 67 / 78

yt = f (kt) (36) f (kt)

MPK = f 0(k) > 0f 00(k) < 0limk!0

f 0(k) = 1limk!1

f 0(k) = 0

() 2010/06/24 @ Gakushuin 68 / 78

III.2.(i) (ConsumerProblem (CP))

(CP) maxct ;it ;at+1

1Xt=0

tu(ct):

s.t. ct + it = wt + rtat (37)at+1 at = it at (38)a0 = a0

(37)ta a0w (real wage) w 1 r (real rental price)

() 2010/06/24 @ Gakushuin 69 / 78

at+1ctt

u0(ct) = u0(ct+1)nrt+1 + 1

o

at+1 at = wt + rtat ct ata0 = a0.

T ! 1limt!1

tu0(ct)at+1 = 0: (39)

() 2010/06/24 @ Gakushuin 70 / 78

a1 = w0 + (r0 + 1 )a0 c0 (t = 0) (40)a2 = w1 + (r1 + 1 )a1 c1 (t = 1) (41)a3 = w1 + (r1 + 1 )a2 c2 (t = 2) (42):::

at+1 = wt + (rt + 1 )at ct (for all t) (43)(40) (41) a1

c0

1 + r0 +c1

(1 + r0 )(1 + r1 ) +a2

(1 + r0 )(1 + r1 ) =w0

1 + r0 +w1

(1 + r0 )(1 + r1 ) + a0

() 2010/06/24 @ Gakushuin 71 / 78

(42)a2c0

1 + r0 +c1

(1 + r0 )(1 + r1 ) +c2

(1 + r0 )(1 + r1 )(1 + r2 )+

a3

(1 + r0 )(1 + r1 )(1 + r2 )=

w0

1 + r0 +w1

(1 + r0 )(1 + r1 )+

w2

(1 + r0 )(1 + r1 )(1 + r2 ) + a0:

t (43)1X

t=0

ctQtj=0(1 + r j )

+ limt!1

at+1Qtj=0(1 + r j )

= a0 +

1Xt=0

wtQtj=0(1 + r j )Qt

j=1 x j = x1 x2 xt

() 2010/06/24 @ Gakushuin 72 / 78

limt!1

at+1Qtj=0(1 + r j )

at (no Ponzi gamecondition)

limt!1

at+1Qtj=0(1 + r j )

= 0 (44)

() 2010/06/24 @ Gakushuin 73 / 78

u0(ct) = u0(ct+1)

nrt+1 + 1

o() 1

1 + rt+1 = u0(ct+1)u0(ct)

(45)limt!1

tu0(ct)at+1u0(c0)(1 + r0 ) = 0 (45)

u0(c0)(1 + r0 ) = 1 (intertemporal budget constraint)

1Xt=0

ctQtj=0(1 + r j )

= a0 +

1Xt=0

wtQtj=0(1 + r j )

: (46)

.

() 2010/06/24 @ Gakushuin 74 / 78

III.3 (Firm Problem(FP))

maxKt ;Lt

Ft(Kt; Lt) rtKt wtLt (47)Lt

maxkt

f (kt) rtkt wt (48)

kt

f 0(kt) = rt (49) wt

wt = f (kt) rtkt (50) () 2010/06/24 @ Gakushuin 75 / 78

III.4 (Competitive Equilibrium)

market clearingcondition

() 2010/06/24 @ Gakushuin 76 / 78

(allocation)ct; it; at+1; kt+1; yt (prices)wt; rt

.

..

1 wt; rtct, it, at+1

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.

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2 wt; rt kt

.

.

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3 (market clearing)

I 1 1I at kt

.

.

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4

ct + it = yt

() 2010/06/24 @ Gakushuin 77 / 78

F.O.C.s

u0(ct) = u0(ct+1)(1 + rt+1 ) (51)f 0(kt) = rt (52)wt = f (kt) rtkt (53)yt = f (kt) (54)ct + it = yt (55)it = kt+1 (1 )kt (56)k0 = k0: (57)

() 2010/06/24 @ Gakushuin 78 / 78

(steady state)(1/3)(shooting algorithm)

(Firm Problem (FP))(Competitive Equilibrium)

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Slide Lecture3