advanced macroeconomics- 3 question-with reference books and slide

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Advanced Macroeconomics

Lecture 1. Quick Review of Undergraduate Macroeconomics:

Simple two-period models of consumption

Andrzej Cieślik

Spring 2013

Main assumptions:

- 2 periods of time: t, t+1 - Utility function additively separable:

UUU tt tt

CC CC

)()( ,

1 1

+ +

+= β

- constant discount factor: θ

β +

= 1

1 where: 10 ≤≤ β

- constant discount rate: where 0≥θ

Consumer utility maximization problem:

UUU tt tt

CC CC

Max )()( ,

1 1

+ +

+= β

s.t : o) no storage (benchmark) i) physical storage, no financial market, no production ii) financial market iii) production iv) financial market and production

Problem: UUU tt

tt CC

CC Max

)()( ,

1 1

+ +

+= β

CASE 0. (Benchmark): No Physical Storage

s.t. (1) YC tt ≤ (2) YC tt 11 ++ ≤

CASE 0. (Benchmark): No Physical Storage Budget Constraint = Endowment Point

(1) YC tt ≤ (2) YC tt 11 ++ ≤

E 1+tY

•

1+tC

Equilibrium with binding constraints

•11* ++ = tt YC

tt YC =*

1+tC

U C = E

Problem: UUU tt

tt CC

CC Max

)()( ,

1 1

+ +

+= β

s.t. (1) YSC ttt =+ (2) SYC ttt += ++ 11

(3) 0≥S t Combine (1) &(2) into the intertemporal budget constraint:

11 ++ +=+⇒ tttt YYCC 11 ++ +=+⇒ tttt dYdYdCdC ( )0,0 1 == +tt dYdY 11 −=⇒ +

dC dC (slope)

CASE 1. Physical Storage

1+tC

tCYt

Yt+1 •

CASE 1. Physical Storage Kinked Budget Constraint

1+tC

tCYt

Yt+1 •

CASE 1. Physical Storage Equilibrium with not binding saving constraint

C*t+1 •

C*t

CASE 1. Physical Storage Equilibrium with not binding saving constraint Equate the slope of the indifference curve to the slope of the budget constraint: 1

)(

)(1

−= ′ ′−

=⇒ +

U U

dC dC

β , CU c ln( )( =

01)()( , 11

=′+′= ++ +

tCtC CC

dCUdCUUd tt

1 1

=⇒

β )

)()( 1+ ′=′⇒

tt CC UU β

CfC

β=

=⇒

* 1

1 * )(

( )

( )1* 1

1 *

1 1

+⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ +

+⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ +

=⇒

ttt

YYC

β β

11 ++ +=+ tttt YYCC

0 1

1 1 1

** >⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ +

−⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ +

=−= +ttttt YYCYS ββ β

1+tC

•

CASE 1. Physical storage Equilibrium with binding saving constraint

11* ++ = tt YC

tt YC =*

C = E

CASE 2. Financial Market Problem: UUU tt

tt CC

CC Max

)()( ,

1 1

+ +

+= β

s.t. (1) YSC ttt =+ (2) SYC ttt r)1(11 ++= ++

Combine (1) &(2) into the intertemporal budget constraint:

( )

1 1

+ +=

+ +⇒

− +

=⇒

tttt

ttt

Y r

YC r

)(

)(1

)1( +

′ ′

−=+−=⇒ +

U U

r dC

dC β

PDV of consumption = PDV of income

CASE 2. Financial Market Numerical Example

Two Period Utility Function (additivelly separable)

ln lnt tU c cβ += +1442443

Logarithmic Utility Function

( ) lnU c c= 14243

1 1

Inter-temporal (between periods) Budget Constraint

1 1 1 1t t t t

C C Y Y r r+ +

+ = + + +1444442444443

CASE 2. Financial Market Numerical Example

Slope of the Indifference 1 Curve

1 ( )

1 1( )

t t

U c c r

U c c

β β+ +

′ = = +

′ 14243

{1 Consumption Policy Function

( ) (1 )t t tC f c r Cβ+ = = +

CASE 2. Financial Market

( )

1 1

1 1 1 1

1 1 1

1 1

t t t t

t t t

C C Y Y r r

C r C Y Y r r

C Y Y r

+ +

⎛ ⎞ ⎛ ⎞ + = +⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞

+ + = +⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠ ⎛ ⎞

+ = + ⎜ ⎟+⎝ ⎠

{ 1 Optimal Amount of Consumption in period (t)

1 1 1 1t t t

C Y Y rβ

∗ +

⎛ ⎞⎛ ⎞ = +⎜ ⎟⎜ ⎟+ +⎝ ⎠⎝ ⎠

We have to solve a simple two-period consumption = income equality

( )* 1 1

1 1 1t t t

r C Y Y

β β β β+ + +⎧ ⎫ ⎧ ⎫

= +⎨ ⎬ ⎨ ⎬ + +⎩ ⎭⎩ ⎭

( )* 1 1 1 1

1 1 1t t t

C r Y Y r

β β+ +

⎧ ⎫⎛ ⎞⎛ ⎞⎛ ⎞ = + +⎨ ⎬⎜ ⎟⎜ ⎟⎜ ⎟+ +⎝ ⎠⎝ ⎠⎝ ⎠⎩ ⎭

CASE 2. Financial Market Numerical Example

( )* *1 1t tC r Cβ+ = +

CASE 2. Financial Market Optimal Savings

( )( ) 1 1 1

1 1 1t t t t t t S Y C Y Y Y

rβ β ∗ ∗

⎛ ⎞ = − = − −⎜ ⎟⎜ ⎟+ + +⎝ ⎠

( )( ) 1 1 1 1 1 1 1 1t t t

S Y Y r

β β β β

∗ +

⎛ ⎞+ = − −⎜ ⎟+ + + +⎝ ⎠

( )

{ ( )( ) 1 Optimal Amount of Savings in period

1 1 1 1t t t

S Y Y r

β β β

∗ += −+ + +

1+tC

tCYt

Yt+1 •

CASE 2. Financial Market Equilibrium with positive savings (lending)

C*t+1 •

C*t

1+tC

tCYt

Yt+1 •

CASE 2. Financial Market Equilibrium with negative savings (borrowing)

C*t+1 • C*t

C E

Problem: UUU tt tt

CC CC

Max )()(

, 1

1 +

+= β

s.t : (1) tttt KYKC )1(1 δ−+=+ + (2) 1121 )1( ++++ −+=+ tttt KYKC δ (3) )(1 1+=+ tKt FY

CASE 3. Production

Solution: (assume 02 =+tK ) Substitute (3) into (2)

1)(1 )1(1 ++ −+=⇒ + tKt KFC t δ

(1) tttt KCYK )1()(1 δ−+−=⇒ +

CASE 3. Production Intertemporal budget constraint

( ) ( )[ ] ( )[ ]ttttttt KCYKCYFC )1(111 δδδ −+−−+−+−=⇒ +

( ) ( )[ ] =−−+−

∂ ∂

∂ −+−∂

== ′

′ − +

)1)(1()1( 1 1

)(

(

) δ δ

β t t

ttt

C K

K KCYF

dC dC

[ ])1( δ−+−= kF

Equate slopes of indifference curve and intertemporal budget constraint

1+tC

tCYt

Yt+1

•

CASE 3. Production Equilibrium with positive investment

C*t+1

• C*t

CASE 3. Production Numerical Example:

( ) 1 21 1 1( )t t tY F K K+ + += =

( ) 1 21

2K t F K

− +=

1 δ =

( ) 1 2

1 1 1t t tC Y K+ + += =

( )( ) 1 2

1 1t t t tC Y K Cδ+ = + − −

From the budget constraint we know that

CASE 3. Production Numerical Example:

( ) 1 1

t t

C C

Y C

β∗ + =

−

From the utility maximization we know that

2t t C Y

β ∗ =

Equation we have our solutions:

1 2

2 2t t t t

K Y Y Y β

β β ∗

+ = − =+ + 1 2

1 1t t C Y

β β

∗ +

⎛ ⎞ = ⎜ ⎟+⎝ ⎠

Problem: UUU tt tt

CC CC

Max )()(

, 1

1 +

+= β

s.t : (1) tttttt BrKYKBC )1()1(11 ++−+=++ ++ δ , [ 0,0 == tt BK ] (2) 1111 )1()1( ++++ ++−+= tttt BrKYC δ (3) )(1 1+=+ tKt FY , 01≥+tK

CASE 4. Financial Market and Production

Note that S splits into B & K

CASE 4. Financial Market and Production

Solution: (1)

11 ++ −−=⇒ tttt KCYB

[ ]11)(1 )1()1(1 1

1 1

1 +++ +−−+

+ +=

+ +

+ ttKttt KrKF

r YC

r C

t δ

0)1()1()( 1

1 =+−−+′=

+ +

rF dK dPDV

tK t

)1()1()( 1 rF tK +=−+′ + δ

CASE 4. Financial Market and Production Numerical Example:

1ln lnt tMaxU C Cβ += +

1 1t t t tC B K Y+ ++ + =

1 1 1 1(1 ) (1 )t t t tC Y K r Bδ+ + + += + − + +

1 1 1( )t t tY F K AK α

+ + += =

CASE 4. Financial Market and Production Numerical Example:

1 1 1 1 1 1

(1 ) (1 ) 1 1t t t t t t

C C Y AK K r K r r

α δ+ + + −⎡ ⎤+ = + + − − +⎣ ⎦+ +

Maximize the value of recourses 1 1 1 (1 ) (1 )t t tAK K r K α δ+ + ++ − − +

{ 1

1 1 Re

(1 ) (1 ) (1 ) (1 )K t t GrossRateOf turnInFinancialMarkGrossRateOnCapital

dPDV F r AK r

dK αδ α δ−+

= + − − + = + − = + 1442443

CASE 4. Financial Market and Production Numerical Example:

}

Net Rate of Retrun in Financial

Net Rate of Return on capital Markets 1

NetRateOf turn

AK rαα δ−+ − = 64748

144444424444443

{ { {

1 1 1 1

1 Optimal

Opportunity Cost Improvements Capital of Holding Capital in TechnologyStock decreases Capital Stock increases Capital

Stock

K 0 0t tt dK dKA

r dr dA αα

δ −∗ + +

+ ⎛ ⎞=⎜ ⎟+⎝ ⎠

p f

CASE 4. Financial Market and Production Numerical Example:

1 1 1 1 1 1

Maximum Resources that we can have in the next period

Intertemporal Budget Co

1 1 (1 ) (1 )

1 1t t t A A A

C C Y A r r r r r r

α α α αα α α

δ δ δ δ

− − − +

⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥+ = + + − − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥+ + + + +⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎣ ⎦14444444444244444444443

nstraint 14444444444444444244444444444444443

{

1 1 1 1 1 1

1 Maximum Resources Available in Period (t+1)

(1 ) (1 ) t A A A

A r X r r r

α α α αα α α

δ δ δ δ

− − − +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − − + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠ ⎝ ⎠

CASE 4. Financial Market and Production Numerical Example:

{

Optimality Condition (Utility Maximization)

Slope of the Budget Constraint

Slope of the Indifference Curve

1 1 t

C r

C β

= +

64444744448

123

( )1 Consumption Policy function

1t tC r Cβ ∗

+ = +144424443

CASE 4. Financial Market and Production Numerical Example:

( ) 1 1 1

1 1 (1 )t t t C Y X

rβ β ∗

+= ++ + +

1 1 t t t tB Y C K ∗ ∗ ∗

+ += − −

( ) 1 1 1

1 1 (1 )t t t C Y X

rβ β ∗

+= ++ + +

CASE 4. Financial Market and Production Numerical Example:

1 1 0 0.25 0.5A rβ δ α= = = = =

1 1 0.5 0.5(2 1) 2 0.5( -5) 0 5t t t tB Y Y B if Y ∗ ∗

+ += − − − = f f

Advanced Macroeconomics
Main assumptions:�
Consumer utility maximization problem:
CASE 0. (Benchmark): No Physical Storage�Budget Constraint = Endowment Point