2008 11 10update 2010 5 21update 2010 5 26

I 2

I.1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2I.2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3

II 5II.1 (Consumer Problem (CP)) : : : : : : : : : : : : : : : : : 5II.2 (Firm Problem (FP)) : : : : : : : : : : : : : : : : : : : : 9II.3 (Competitive Equilibrium) : : : : : : : : : : : : : : : : : : : : : : : : : 10

III (Social Planner Problem) 12III.1 The Welfare Theorem : : : : : : : : : : : : : : : : : : : : 13

(real business cycle: RBC)RBC (dynamic stochastic generalequilibrium model: DSGE model) Ramsey(1928) (s) (1 s)Gertler

1

I t = 0; 1 consumer (one representative household)1

1 12

()

(goods)

I.1

(1) u(c0) + u(c1)

0 < < 1 (discount factor)3u()

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

u() (concave function)4

(marginal utility)

1 (2006)Ljungqvist and Sargent(2004)

23 11+

u(c0) +1

1 + u(c1)

(discount rate)4 (1995) (1990)

2

I.2 KL

(2) Yt = Ft(Kt; Lt)

1. (Constant return to scale) z z z

zYt = Ft(zKt; zLt)

2. (increasing function with respect to each input)

MPK =@Y

@K> 0; MPL =

@Y

@L> 0

MPK: (marginal product of capital)MPL: (marginal product of labor)

3. (diminishing marginal product with respect to eachinput)

@MPK

@K=

@2Y

@K2< 0;

@MPL

@L=

@2Y

@L2< 0

4. (Inada condition)

limK!0

@Y

@K=1; lim

K!1@Y

@K= 0;

limL!0

@Y

@L=1; lim

L!1@Y

@L= 0:

z zYt = Ft(zKt; zLt) z = 1Lt

zYt = Ft(zKt; zLt)

() YtLt

= Ft(KtLt

; 1)

yt YtLt; kt Kt

Lt

3

yt Ft(kt; 1)

Ft(kt; 1) f(kt)

(3) yt = f(kt)

f(kt)

MPK = f 0(k) > 0

f 00(k) < 0

limk!0

f 0(k) =1limk!1

f 0(k) = 0

4

II II.1 (Consumer Problem (CP))

(CP) maxc0;i0;c1;i1

u(c0) + u(c1)

c0 + i0 = w0 + r0a0(4)

c1 + i1 = w1 + r1a1(5)

a1 a0 = i0 a0(6)a2 a1 = i1 a1(7)a0 = a0(8)

a2 0:(9)

(4) 0a a0 a0w (real wage) w 1 r (real rental price)5 0(5) 1(6)(7)12a2a2 0a2 = 0a2 > 0a2 > 0 (transversalitycondition)a2 < 0 (no-Ponzi-game condition)6 a2 0a2 a2 = 0 c0c1i0i1

(10) maxa1

uw0 a1 + (1 + r0 )a0

+ u

w1 + (1 + r1 )a1

a1 (rst order conditions, F.O.C.s)

(11) u0(c0) + u0(c1)(1 + r1 ) = 05w r6Ponzi

5

7

(12) u0(c0) = u0(c1)(1 + r1 )

(Euler equation)(14) 0 (opportunitycost) (marginal utility) 0 1

(4)(5)(6)(7)

(13) c0 +c1

1 + r1 = w0 +w1

1 + r1 + (1 + r0 )a0

0 1 0(13) (intertemporal budget constraint)c0c1

(14)u0(c0)u0(c1)

= (1 + r1 )

(marginal rate of substitution: MRS) 1Friedman (permanent income hypothesis)Modigliani (life cycle hypothesis)

(marginal propensity to consume)

7 (1995) (1990)

6

c0

c1

c0

c1 O

u(c0) + u(c1)

(11)

1:

Lucas(1976) (Lucas critique) (2006) 2 2.5.2

7

(CP)i0i1 (CP)

maxc0;c1;a1;a2

u(c0) + u(c1)

subject to

c0 + a1 (1 )a0 = w0 + r0a0c1 + a2 (1 )a1 = w1 + r1a1a0 = a0

a2 0:

L = u(c0) + u(c1) + 0hw0 + r0a0

c0 + a1 (1 )a0

i+ 1

hw1 + r1a1

c1 + a2 (1 )a1

i+ 1(a0 a0) + 2a2:

c0; c1; a1; 0; 1; a2; 2

@L@c0

= 0 () u0(c0) = 0(15)@L@c1

= 0 () u0(c1) = 1(16)@L@a1

= 0 () 0 = 1(1 + r1 )(17)@L@0

= 0 () c0 + a1 (1 )a0 = w0 + r0a0(18)@L@1

= 0 () c1 (1 )a1 = w1 + r1a1(19)@L@a2

= 0 () 1 = 2(20)a2 0; 2 02a2 = 0 ()(21)@L@1

= 0 () a0 = a0 ()(22)

8

0 1

u0(c0) = u0(c1)(1 + r1 )c0 + a1 (1 )a0 = w0 + r0a0c1 (1 )a0 = w1 + r1a1

(14)8(21)

2a2 = 0

complementary slackness conditiona2 > 02 = 0 a2 = 02 > 01 > 0 1 = 22 > 0k2 = 0 (1995) (1990)

II.2 (Firm Problem (FP))

(23) maxKt;Lt

Ft(Kt; Lt) rtKt wtLt for t = 0; 1

Lt

(24) maxkt

f(kt) rtkt wt for t = 0; 1

kt9

(25) f 0(kt) = rt for t = 0; 1

wt10

(26) wt = f(kt) rtkt for t = 0; 1

9 (1995) (1990)

10

9

II.3 (Competitive Equilibrium) (1994)

(competitive equilibrium)

1.

2. t( (market clearing condition) (resource constraint)

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

1. w0; w1; r0; r1 a = 0c0; c1, i0, i1, a1, a2

2. w0; w1; r0; r1 k0; k1

3. (market clearing)

1 1 at kt

4.

c0 + i0 = y0

c1 + i1 = y1

10

1. w0; w1; r0; r1

(CP) maxc0;i0;c1;i1

u(c0) + u(c1)

c0 + i0 = w0 + r0a0

c1 + i1 = w1 + r1a1

a1 a0 = i0 a0a2 a1 = i1 a1a0 = a0

a2 0:

2. w0; w1; r0; r1

(FP) maxkt

f(kt) rtkt wt for t = 0; 1

3.

at = kt ct + it = yt

F.O.C.s

u0(c0) = u0(c1)(1 + r1 )(27)f 0(kt) = rt(28)

wt = f(kt) rtkt(29)yt = f(kt)(30)

ct + it = yt(31)

it = kt+1 (1 )kt(32)k0 = k0(33)

k2 = 0:(34)

t = 0; 1

11

III (Social Planner Problem)

(SP) maxc0;i0;c1;i1

u(c0) + u(c1)

s.t. c0 + i0 = f(k0)

c1 + i1 = f(k1)

k1 k0 = i0 k0k2 k1 = i1 k1k0 = k0

k2 0:

i k2

maxc0;c1;k1

u(c0) + u(c1)

s.t. c0 + k1 (1 )k0 = f(k0)c1 (1 )k1 = f(k1)k0 = k0

k2 0:

c0; c111

maxk1

uf(k0) k1 + (1 )k0

+ u

f(k1) + (1 )k1

12

u0(c0) + u0(c1)nf 0(k1) + (1 )

o= 0

() u0(c0) = u0(c1)nf 0(k0) + (1 )

o11k2 = 012

12

u0(c0) = u0(c1)nf 0(k1) + (1 )

o(35)

c0 + k1 (1 )k0 = f(k0)(36)c1 (1 )k1 = f(k1)(37)k0 = k0(38)

k2 = 0:(39)

III.1 The Welfare TheoremLjungqvist and Sargent (2004)

First Welfare Theorem fct; kt+1g1t=0 (socially op-timal)

Proof

fct; ktg1t=0 (27),(28)

(40) u0(c0) = u0(c1)nf 0(k1) + (1 )

o (35) fct; kt+1g1t=0Q.E.D.

Second Welfare Theorem fct; kt+1g1t=0fct; at+1; kt+1g1t=0 fwt; rtg1t=0.

Proof

(market clearing condition) ((36)(37))

rt = f0(kt)(41)

wt = f(kt) rtkt(42)

(35) (27)Q.E.D.

13

[1] , A., C., (1995),CAP

[2] Romer, David, (2005) \Advanced Macroeconomics", 3rd edition, McGraw-Hill.

[3] Ljungqvist, Lars and Thomas Sargent, (2004), \Recursive Macroeconomic Theory," 2nd

edition, MIT Press.

[4] Lucas, Robert, (1976), \ Econometric Policy Evaluation: A Critique," in K. Brunner and

H. Meltzer (eds.), The Phillips Curve and Labor Markets, Amsterdam: North-Holland,

19-46.

[5] , G., , N.,, ,1994.

[6] Ramsey, F. P., (1928), \A Mathematical Theory of Saving," Economic Journal, 38, 152,

543-559.

[7] (2006),

[8] (1990)

14