Advanced Macro and Money Lecture notes (WS 09/10)

Prof. Dr. Gerhard Illing,

LMU University of Munich 1.1 Basic Tools: The Problem of Optimal Control – An intuitive Guide 1.1.1 Dynamic optimisation problem with inequality constraints – Discrete time ∑ == Tt t

t

tccUV 0)(

][)0(max β

)1/(1 ρβ += discount factor with ρ as time preference. Constraints: Law of motion tttt ckfkk −=−+ )(1

tc control variable; tk state variable (capital). Boundary constraints: Initial capital stock 0k given. Terminal constraint: 11 ++ ≥ TT kk . Furthermore, 0;0 ≥≥ tt ck . Characterise the optimal time path { } { } Ttkc tt ,...,0*;* = Necessary / sufficient properties of U(.) and f(.) to guarantee existence of optimum? Assume and ][ tcU )( tkf to be strictly concave functions.

For optimisation problems, we use Lagrangian method.

∑∑ = += −−−+= Tt tttttTt tt

tckkckfcUL 0 10)(

)](][[][max λβ

Recipe: Construct the Hamiltonian:

, ]][[][),,( ttttt

tttt ckfcUckH −+= λβλ Using Ht, reformulate the Lagrangian:

∑ −∑ −= = +=Tt ttt

Tt tttttc

kkckHL 0 10)()(],,[max λλ

∑ −+−∑ += = +−= Tt TTtttTt tttt kkkckH 1 10010 )(],,[ λλλλλ 1.1.2 How to handle inequality constraints? Kuhn Tucker conditions! Kuhn Tucker: Static optimisation with inequality constraints. Introduce Lagrange multiplier with complementary slackness

],...,[max 1)( ntcccU s. t. mjaccg jnj ,...,1),...,( 1 =≤

Lagrangian: += ],...,[ 1 nccUL [ ]),...,( 11 njjmj j ccga −∑ = ν with optimality conditions:

(1) i

jmj j

i cg

cU

∂∂

=∂∂

∑ =1ν

(2) mjga jjj ,...,1;0)( ==−ν complementary slackness: mjccga njj ;0),...,( 1 j ,...,1;0 =≥≥− ν

This way, Lagrange multipliers are non-negative!

2

Examples: Characterize optimal consumption bundle with inequality constraints: (1) Economic intuition for shadow prices:

],[max 21 ccUc s.t. 2211 ; cccc ≤≤

][][)(: 222111 ccccUL −+−+ νν FOC: 2211 /;/ νν =∂∂=∂∂ cUcU ; 0;0 ≥≥− iii cc ν ;

2,10] =[ =− iciciiν

→ 2

1

2

1

2

1

1

2

//

//

cUcU

cUcU

cdcd

∂∂∂∂

==∂∂∂∂

=−νν

Relative shadow price for resource constraints (2) Economic intuition for complementary slackness

],[max 21 ccUc=U(c1)+ß U(c2) s.t. 11221 ; ccYcpc ≤≤+

][][)()(: 1122121 cccpcYcUcUL −+−−++ νλβ ν: Shadow value for the constraint 11 cc ≤

(1) 221

; pcU

cU λβνλ =

∂∂

+=∂∂ ;

(2) 0;;0][ 221221 ≥≤+=−− λλ YcpccpcY ; 0;;0][ 1111 ≥≤=− νν cccc

→

( )2222

1

1

2 1/111//)1(

pppcUcU

cdcd

≥+=+

=∂∂∂∂

+=− λνλ

νλρ

If the shadow value ν is positive, the constraint 11 cc ≤ is binding. If the constraint 11 cc ≤ is not binding, ν=0

3

1.1.3 The Maximum Principle: Hamiltonian conditions Dynamic control problem We concentrate on the terminal value constraint 11 ++ ≥ TT kk And ignore the other inequality constraints 0;0 ≥≥ tt ckMaximise the Lagrangian:

∑∑ = +−= −+−+= Tt TTtttTt tttttckkkckHL 1 10010)(

)(],,[max λλλλλ +

)( 11 ++ − TT kkν This gives the FOC Hamiltonian conditions (Maximum Principle):

A) 0=∂∂

t

t

cH t=1,…,T

B) 0)( 1 =−+∂∂

−ttt

t

kH λλ t=1,…,T

C) ttt

t kkH −=∂∂

+1λ t=1,…,T

D) Transversality condition (note: νλ =T )

0)( 11 =− ++ TTT kkλ with complementary slackness: If 0011 =→>− ++ TTT kk λ ; if 110 ++ =→> TTT kkλ

Interpretation: tλ as shadow price (in present value terms)

of the optimised path: t

t kL∂

∂=

(.)*λ ; 0(.)*1

≤−=∂

∂

+T

TkL λ

Lagrangian approach gives us – as additional information – the shadow price of the state variable at each time t, once 0λ is determined.

00

(.)* λ=∂

∂k

L . For nontrivial problems, 00

(.)* λ=∂

∂k

L >0

4

Economic intuition behind Hamiltonian: is the sum of ])([][],,[ tttt

ttttt ckfcUckH −+= λβλ

1) The direct impact of the state variable kt on the payoff function at t plus

2) The change in the state variable (evaluated at present value λt) according to the law of motion: Try to maximise this sum, but take also into account that value of state variable may change:

][ 1−−+= ttttt kHL λλ Ht: at t realised returns;

][ 1−− tttk λλ : Indirect impact via capital gains of state variable Maximise, at each t, the sum of returns and capital gains → arbitrage equation:

B) )( 1−−−=∂∂

ttt

t

kH λλ

KEY economic insights: C): Characterises the law of motion for the state variable

(necessary condition, it is part of the basic constraints) B) Arbitrage condition: Characterizes the path (law of

motion) for shadow value D) Transversality condition: Either the value of the state

variable is positive, then the endpoint constraint is binding for the state variable; if endpoint constraint not binding, shadow value must be zero.

5

1.1.4 Note: Hamiltonian in current value vs. present value Shadow price in terms of present value (evaluated at initial stage): Alternative: Hamiltonian in current value terms:

])([][ ttttt

t ckfcUH −+= λβ

]))([][~ tttttt ckfcUH −+= μ Shadow price μ in terms of current value (price of capital in terms of current utility)!

tt

t HH~β= with ] t

tt μβλ =)([][ ttt

tt

t ckfcU −+= μββFOC:

A*) 0~

=∂∂

t

t

cH since t

t

t

t

t

cH

cH −

∂∂

=∂∂ β

~

B*) )(~

11 −− −−=∂∂

tttt

t

kH μμρμ

t

tt

t

t

kH

kH

∂∂=

∂∂ ~β = )()( 1

11 −

−− −−=−− t

tt

ttt μβμβλλ

( ) Inserting in B) gives tt

t μβλ = )(~

11

−−−−=

∂∂

ttt

t

kH μβμ

C*) ttt

t kkH −=∂∂

+1

~

μD*) Transversality condition 0)( 11 =− ++ TT

TT kkβμ

Straightforward interpretation of B*) as arbitrage condition:

ρμμ

μ=

Δ+

∂∂ kH /~

B*): → Value1 of capital kt: TtTtT

iit

itit k

Hμββμ −−=

+

+ +∂∂

= ∑ 1~

in present value terms: TtTiit

itt k

H λλ +∑∂∂

= −=+

+1

1 μt Value of state variable evaluated at t; λt value of state variable evaluated at initial stage 6

1.1.5 Continuous time analysis Apply the same method to continuous time problem

dtettcuV tTtc

ρ−∫= 0)(

]),([)0(max s.t. constraints:

a) Law of motion: transition equation for state variable: (Intertemporal resource constraint) )),(),(()( ttctkgtk =&b) 0)0( 0 >= kk initial value of state variable given c) Bounds: k(t) ≥ 0; c(t) ≥ 0. Fixed endpoint T given. We ignore bounds other than bound at endpoint: 0)( ≥Tk c1) Sometimes impose (in market economies) No Ponzi game condition: 0)( )( ≥⋅ − TTreTk Optimal solution: (continuous) path for control c*(t) and state variable k*(t). Necessary/sufficient conditions for existence of optimum: Strict concavity of u (.) and g (.). Technique: use generalized Lagrangian:

dttkttctkgtettcuL tT )]}()),(),(()[(]),([{0 &−+=−

∫ λρ Partial integration by parts

dttFtdtGdaGaFbGbFdttG

tdtFd b

aba )(

)()()()()()()( ∫−−=∫

gives:{ }

)()()0()0(

)()()),(),(()(]),([0TkTk

dttktttctkgtettcuL T t

λλ

λλρ

−+

++= ∫ − &

7

Construct Hamiltonian: )),(),(()(]),([ ttctkgtettcuH t λρ += −

Maximum Principle: First order conditions for the Hamiltonian (Necessary conditions):

A) 0)()(

=∂∂

tctH

B) )()()( t

tktH λ&−=

∂∂ Arbitrage equation:

C) )()()( tk

ttH &=

∂∂

λ Law of motion

D) k(0)=k0; Transversality condition 0)()( =TkTλ or 0)()(lim =

∞→tkt

tλ

Complimentary slackness: If 0)(0) =→>( TTk λ ; if 0)(0) =→>( TkTλ

Sufficient condition for Maximum: Concavity conditions:

0)()(

2

2<

∂∂

tktH ; 0

)()(

2

2<

∂∂

tctH

Necessary conditions are sufficient (unique global maximum) if functions f(.) and g(.) are strictly concave and tt ∀≥ 0)(λ ~~ Hamiltonian is strictly concave

8

Again: Current value vs. present value Hamiltonian Up to now: Shadow price (λ) defined in terms of present value (evaluated at initial stage) (.))((.) gteuH t λρ += − Alternatively: construct Hamiltonian in current value terms: Shadow price (μ(t)) – price of capital in terms of current utility:

(.)(.)~ guH μ+= HeH t ~ρ−= = (.))((.) gteue tt μρρ −− + )()( tet t μλ ρ−=

FOC of the optimal control problem:

A*) 0)()(~

=∂∂

tctH since te

tctH

tctH ρ

)()(

)()(~

∂∂

=∂∂

B*) μρμ &−=∂∂

)()(~

tktH

Because: tetctH

tctH ρ−

∂∂

=∂∂

)()(~

)()( ;

μλ ρ te−= → μρμλ ρρ tt ee −− −= && →B)

C*) )()()( tk

ttH &=

∂∂

μ

D*) Transversality condition: 0)()(lim =−∞→

tket tt

ρμ

Arbitrage condition:

ρμμ

μ=+

∂∂ &kH /~

Marginal contribution of capital to utility + capital gain = marginal return on current consumption

9

1.1.6 Proof of Optimality: Barro/Sala-i-Martin or Chiang Dynamic problem with endpoint constraint 0)( ≥Tk Let )}(*{)},(*{ tctk be the optimal path for state and control variable. Then, any small perturbation of the optimal control path )}()(*{ tptc ε+ with the corresponding perturbation of the path for the state variable )}()(*{ tqtk ε+ and

)(( )(*) TkdTkTk ε+= should leave the maximum value L

unchanged: 0=∂∂

εL (given the endpoint constraint 0)( ≥Tk )

])}(.,(.))(.,),(.,[{)(., 0 dtkckHLT ελεεε &+∫=

),()()0()0( ελλ TkTk −+ + ),( εν Tk Take derivative wrt ε :

0)())((0 =∂∂

−+∫⎭⎬⎫

⎩⎨⎧

∂∂

+∂∂

=∂∂

ελν

ελ

εεTkTtdkHL T &

Since )()();()( TkdTktqkHtp

cHH =

∂∂

∂∂+

∂∂=

∂∂

εε, we get:

0)()]()([)()(0 =−+∫⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

∂∂

+∂∂

=∂∂ TkdTTtdtq

kHtp

cHL T λνλ

ε&

Must hold for arbitrary small perturbations paths );();( tqtp )(Tkd , so for each component, the Hamilton conditions have to hold:

cH

∂∂ =0

0=+∂∂ λ&

cH

)()( TT λν = resp. CS: 0)()( =TkTλ or 0)()(lim =∞→

tktt

λ

10

1.1.7 Example: Optimal extraction path for exhaustible resources

dtetCuV tρ−∞∫= 0 )]([ s.t. constraints:

a) Intertemporal resource constraint )()( tCtK −=& b) 0)0( 0 >= KK initial capital stock; c) ttK ∀≥ 0)( So: 00 )( KdttC ≤∫ ∞ Construct Hamiltonian: )()()]([ tCtetCuH t λρ −= − First order conditions:

A) 0)()(

=∂∂

tCtH → tetCu ρλ −= ))(('

B) )()()( t

tKtH λ&−=

∂∂ → 0)( =tλ&

C) )()()( tK

ttH &=

∂∂

λ → )()( tCtK −=&

D) K(0)=K0 ; Transversality condition: 0)()(lim =

∞→tKt

tλ

B): Arbitrage equation: 0)( =tλ& present value constant → Hotelling rule: current value must increase at discount rate:

ρμμμμ ρ=

)()(;)0()(

t=

tet t &

Reformulate conditions using the current value Hamiltonian:

)()()]([~ tCttCuH μ−= !

11

Example: Calculate optimal path for CES payoff function:

σσ

11

/11

1 −−

= tt cU ;

Example (σ=1):

)]([ln)]([ tCtCu = → tetC ρλ

−=1)( ;

λ1)0( =C

000111)( KdtedttC t ==∫=∫ ∞ −∞ ρλλ

ρ

(using the resource constraint)

Thus 0)0( KC ρ= and teKtC ρρ −= 0)(

][0Tt eKtd

; so that 1[ 00K)( TeTK

ρρρ −− =−= ∫ Transversality condition holds: 0)( >= λλ t

=∞→

)()(lim tKtt

λ

∞→t

lim ][lim1[ 000T

tT t eKtdeK ρρ λρλ −

∞→

− =− ∫ =0

Hotelling rule: Current value shadow price: tet ρμμ 0)( = ;

tett ρμλ −= )()(

12

1.2 The Basic Framework Modern Macro is about dynamic stochastic general equilibrium in a set up with representative, optimising agents in an economy with imperfect competition and possibly incomplete asset markets. Here, in order to gain the basic intuition for intertemporal optimisation, we start by reducing the problem to the simplest case: We consider first a pure endowment economy (1.2.1) with one good (consumption) and no uncertainty. We allow for intertemporal trade on perfect capital markets via real bonds with real interest rate rt. In this setup, we analyse incentives for consumption smoothing. Later, we extend this setup by introducing capital accumulation, uncertainty, government activity and finally money (section 1.3). In the next step (1.2.2), we look at the Ramsey growth model. It characterises the optimal intertemporal path for consumption and capital formation {Ct*} {kt*} (t=0,1,…,T) for a representative agent, starting with an initial capital stock k0. We solve it as optimal control problem. With a complete set of markets, a decentralised market economy can implement the efficient allocation. Section 1.2.3 looks at some applications, such as introducing government spending and taxes and international trade of small countries. In both cases, we concentrate on wealth constraints and try to figure out the implications for sustainability of (public or international) debt. Section 1.2.4 then looks at some basics of dynamic stochastic optimisation. A promising route for modern macro is to integrate macro and finance. Stochastic optimisation is the starting point for this research area.

13

1.2.1 Consumption Smoothing: The Euler Equation The representative agent maximizes

∑ == Ttt

tCUU 0 )( β s. t. per period budget constraint: tttttt CBrYBB −+=− −+ 11

with B0 and r-1 given and the end point constraint BT+1≥ 0 (No Ponzi Game constraint). rt-1: real return of asset Bt invested during the last period t-1. Pure endowment economy: Path {Yt} is given.

Define gt as the growth rate of endowment t

ttt Y

YYg −= +1 .

Start with a two period model U = U (C1) +ß U (C2). Budget constraints per period: tttttt CBrYBB −+=− −+ 11 t=1,2 Two Periods: Starting point (t=0) is 1; end point T is T=2! Choose {C1; C2} and {B2; B3} optimally! Constraints: (1+ r0) B1 given; terminal constraint B3≥0. A straightforward way to solve is to merge both per period budget constraints into a life time constraint (eliminating B2):

101

21

1

3

1

21 )1(111

Brr

YYr

Br

CC +++

+≤+

++

+ ,

and take also the terminal constraint B3≥ 0 into account.

14

But let us instead use the optimal control approach. First, define the Hamiltonian in present value:

)()( 1 ttttttt

t CBrYCUH −++= −λβ Key FOC for intertemporal optimality:

A*) 0=∂∂

t

t

cH → t

t

t

cU λβ =

∂∂

B*) )( 111

1tttt

t

t rBH λλλ −−==

∂∂

+++

+ → tt

t

r+=+

111

λλ

C*) ttt

t BBH −=∂∂

+1λ → tttttt CBrYBB −+=− −+ 11

D*) Transversality condition: 01 =+TT Bλ → 032 =Bλ By combining A*) and B*), we derive the Euler equation:

E*) ttc

tc

rCUCU

+=

++

11

)()(

11 1

ρ

Define the MRS as internal discount factor and 1/1+rt as

market discount factor t

tt rq

+=+ 1

11, . Then, we can rewrite E*

ttt

t

t

tc

tc

rq

CUCU

+===

+ +++

11

)()(

11

1,11

λλ

ρ

Using t

tt r+=+ 1

11 λλ , D* gives 01

13

1132,11 =+

= Br

Bq λλ .

Because 01

1 >∂∂

=cUλ this simplifies to: 0

11

31

32,1 =+= B

rBq

15

An equivalent formulation of the budget constraint is in terms of wealth: If I plan to have wealth Wt+1 at the beginning of next period t+1, I must spend today 11, ++ ttt Wq (Think of buying zero bonds at the price ). Note: Wt= (1+rt-1) Bt . 1, +ttq

This gives: ttt

tt WYr

WC +≤+

+ +1

1

with the end point constraints 0; 311 ≥= WWW . The wealth constraint per period can be formulated as:

ttttttttt WqCYqWW )1/1()(/1 1,1,1 −+−≤− +++ since tttttt WCYWq +−≤++ 11, .

The Hamiltonian gives us as FOC:

11,11, ~

~)~~()1/1(~

−+−+ =→−−=−=∂

∂

t

tttttttt

t

t qqWH

λλλλλ

Note: ttttt q λλλ 1,1~

++ ==Transversality condition: 011, =++++ TtTtt Wq or 033,1 =Wq The Euler equation E* characterises intertemporal demand. Aggregate Equilibrium: For the whole economy, demand must be equal to endowment. Equilibrium path depends on:

a) Preferences (discount rate, intertemporal elasticity of substitution)

b) Technology (endowment; productivity growth → real intertemporal exchange rate)

Let us concentrate on Euler equation )(11)( 1++

+= tc

ttc CU

rCUρ

Marginal utility out of consumption, adjusted for the relevant rate of return, has to be equal across time Due to concavity (u”(c)

Elasticity of marginal utility reflects the concavity of the utility function: How does the slope of the indifference curve change with a change in the consumption ratio ? 1/ +tt cc

)('/)('/

}/{)}('/)('{)(

1

1

1

1

+

+

+

+−=tt

tt

tt

tt

cucucc

ccdcucudcε

Inverse to the intertemporal elasticity of substitution σ

)(')("

)(1)(

cuccu

cc −==

σε (for continuous case)

Example: CES payoff function:

σσ

11

/11

1 −−

= tt cU ;

Euler equation:

tt cU ln:1For0 ==> σσ

tt

t

rC +=⎟⎟

C⎠

⎞⎜⎜⎝

⎛+11

ρ

−+

11

1

1 σ or ttt CrC

σ

ρ⎟⎠

⎞⎜⎝

⎛++

=+ 11

1 ;

tC is increasing [decreasing] for ρ>tr [ ρ

Equilibrium: Yt+1=(1+g) Yt;→ grnt σρ

1+= → Ct+1=(1+g) Ct =Yt+1 The natural rate of interest: Equilibrium real interest rate is characterised by equilibrium condition tYC tt ∀= (aggregate consumption = endowment). Using the growth rate: ttt YgY )1(1 +=+ , we can write E* as:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛++

=+ρ

σ111

1 nt

trg

Log-linear approximation: or )( ρσ −= ntt rg tn

t gr σρ1+= .

The natural rate of interest is higher ntra) The higher ρ: impatience requires stronger incentives to

save! b) The higher the growth rate g (because of incentives for

consumption smoothing, high growth has to be balanced by stronger incentives to save!

c) The lower the elasticity of substitution (the more concave the utility, the stronger incentives to smooth consumption for a given growth rate)!

18

1.2.2 The Ramsey Model: Optimal Growth – Continuous Time Version New twist: Include production and capital formation! Neoclassical linear homogenous production function F(K,N) (CRS)

N(t): Labour supply at t; K(t): capital stock at t Law of motion: )()())(),(()( tCtKtNtKFtK −−= δ&

)1,)()(())(),((

)(1))((

tNtKFtNtKF

tNtkf ==

0)(";0)(');( = kfkfkfy ;

kkfkfNkkfNkfNKF

NKFkfNNKF

kN

K

)()(),(

,()(),(

∂∂

+=

→= kfKkkfN k

)(')(

);(')()

−=

=∂∂

=

Inada conditions: ∞=

→)('lim

0kf

k; 0)('lim =

∞→kf

k

tneNtN 0)( = ; δ rate of depreciation Representative agent maximises per capita consumption:

dtetcuV tρ−∞∫= 0 ))(( with )(/)()( tNtCtc =

Strictly concave payoff function: ; 0)(";0)(' cucuρ discount factor s. t. constraints:

a) Intertemporal resource constraint: Capital stock per head

evolves according to: )()())(()( tckntkftk −+−= δ& , because: NNKKkk /// &&& −=

nNKknk or

kKKk −=⋅−⋅= /&& /& ; ckkfNK −−= δ)(/& b) Initial capital stock )0(k given; c) Endpoint constraint 0)(lim ≥

∞→Tk

T

19

1.2.2.1 Hamiltonian: )]()()([)()]([: tcknkftetcuH t −+−+= − δλρ

FOC:

A) 0)()(

=∂∂

tctH → tecu ρλ −= )('

B) )()()( t

tktH λ&−=

∂∂ → λδλ &−=+− )]()('[ nkf

C) )()()( tk

ttH &=

∂∂

λ → )()()()( tcknkftk −+−= δ&

D) 0)()(lim =∞→

tktt

λ → 0)())((('lim =−∞→

tketcu tt

ρ

FOC give 2 differential equations (characterising the evolution of the state variable kk /& and of the shadow price λλ /& ); B) λδλ &−=+− )]()('[ nkf C) )()()()( tcknkftk −+−= δ& Using A), B) can be transformed into a differential equation for the evolution of control variable : cc /&

ttt ecucu

cu⎢

cecueccu ρρρ ρρλ −−− ⎥⎦⎤

⎣

⎡('

("−=−= )('

)))(')(" &&&

Because ccu

cuc)(")(')( −=σ , we get: λρ

σλ ⎥⎦

⎤⎢⎣⎡ +=−

cc&& 1 or

AB) )])('[)( nkfccc

−−−= δρσ&

C) )()()()( tcknkftk −+−= δ& 2 boundary conditions give a determined solution for the 2 differential equations: (1) )0(k and (2) the transversality condition: 0)())((('lim =−

∞→tketcu t

t

ρ

20

1.2.2.2 Steady State and Transition Dynamics Digression: Stability analysis

How can we solve a differential equation )(yfy =& ? Is there convergence to a steady state? Intuition by graphical analysis: Two steady states. One stable, one unstable

Explicit analytical solution for linear differential equations:

byay =−&

Steady state 0=y& : aby /−=

21

Solution procedure: a) Solve for the homogeneous differential equation 0=− yay&

with ayy

=& stable system: a0

General solution: with taeAty =)( )()( tyaeAaty ta ==& Particular solution: taeAty )0()( = b) Take some particular solution for the non-homogenous

differential equation: For instance, take steady state 0=y& with aby /−=

General solution for the complete system: abeAty ta −=)(

The definite solution is determined by a specific boundary

condition: y(0) → abyA += )0(

→ Determines the unique solution: abe

abyty ta −+= ])0([)(

22

Steady State in the Ramsey Model: The equilibrium path is characterised by the 3 conditions AB), C) and D). In a Steady State:

0)])('[)( =−−−= nkfccc δρσ&

Modified golden rule: nkf ++= δρ*)(' 0)()()()( =−+−= tcknkftk δ& or **)(*)( cknkf ++= δ

For the steady state {c*; k*}, the transversality condition is fulfilled since k=k* and 0*)('))((('

∞→

−− →→T

TT ecueTcu ρρ .

Is the steady state stable? Graphical analysis:

23

Saddle point stability:

*)('0 kknkfcc

>≤

⇔++<≥

⇔<≥ δρ&

knkftckk )()()(0 +−

>≤

⇔<≥ δ

&

For any initial k(0), exactly one unique path of convergence to the steady state {c*; k*} starting with initial consumption

))0((ˆ kc - all other paths are diverging. For ))0((ˆ)0( kcc > : Savings too low → eventually cross →

at some finite T: k(T)=0 → jump in consumption to c=0 0)( =tk&

If ))0((ˆ)0( kcc < : Excessive Savings → Overaccumulation → in finite time: c=0 → violation of transversality condition.

Comparative statics: What happens, if some exogenous parameter changes? What is the impact of such a shock?

↓→↑→↑ **)(' kkfρ Investment reduced; consumption goes up

(consumption path tilted more towards present consumption).

24

No further adjustment in steady state: gknkftc −+−= *)(*)()(* δ

Increase in real rate of interest: consumption tilted more towards future consumption

25

1.2.2.3 The Decentralised Market Economy The competitive equilibrium is efficient in an economy with infinitely lived agents. Before reading the next pages, try first to show this yourself. Assume that some firms own the capital stock k(0), financed by issuing bonds to households. Bonds b(t) pay a return r(t) at t. The initial wealth of households is the value of the firm b(0). Labour is supplied inelastically, household’s wealth per capita b(t) evolves according to:

)()()()()()( tbntctbtrtwtb −−+=& Formulate the firm’s and household’s optimisation problem. Then, characterize the intertemporal price path. Note that in a one good economy, only the intertemporal price (the rate of return on assets, the interest rate r(t)) has to be determined. Use the Hamiltonian approach to characterise the FOC Finally, show that the FOC for the optimum are equivalent to the conditions of the Ramsey problem. Hints: 1) Define })]([1)(

0νν dr

ttr

t

∫⋅= as average interest rate up to t giving

the present value factor: ttre ⋅− )( 2) Impose the No Ponzi-Game-constraint:

→ 0)(lim ])([ ≥⋅−−∞→

TnTr

TeTb

Transversality condition: this inequality must be binding for the optimal solution

3) The firm’s problem is not really dynamic. Value of the firm: b(0)= k(0)!

26

1) Firms In the absence of adjustment costs, firms face not really a dynamic problem. Firms rent labour and capital at each point in time. Assume some firms own the capital stock, financed by issuing bonds. Assume CRS (Constant returns to scale).

)()()()()())(()()(),(

tNtwtKtrtKtkftNMaxtKtN

−−− δ

FOC: )())((' trtkf += δ or δ−= ))((')( tkftr ; )()())(('))(( twtktkftkf =− Zero Profits: Labour income: )()( tNtw ; Capital income per worker: )()( tktr . With CRS, the value of the firm is b(0)= k(0)! 2) Households: The representative household starts with initial financial wealth b0 and earns income from interest on bonds and labour income w(t): (→ human wealth). Assets per capita for the household evolve according to:

)(])([)()()()()()()()( tbntrtctwtctbntbtrtwtb −+−=−−+=&

Let us follow the standard Hamiltonian approach:

Max dtetcudtetNtCuV tt ρρ −∞−∞ ∫=∫ ⎟

⎠

⎞⎜⎝

⎛= 00 ))(()(

)(

s.t. (1) the law of motion

)(])([)()()()()()()()( tbntrtctwtctbntbtrtwtb −+−=−−+=& (2) The No Ponzi-Game-constraint

0})({lim}])([exp[)({lim ])([0 ≥=∫ −−⋅⋅−−

∞→∞→

tntr

t

t

tetbdnrtb νν

(2b) Initial financial wealth: 0)0( bb =

27

Hamiltonian)](])([)()([)()]([: tbntrtctwtetcuH t −+−+= − λρ

FOC:

mA) 0)()(

=∂∂

tctH → tecu ρλ −= )('

mB) )()()( t

tbtH λ&−=

∂∂ → λλ &−=− )])([ ntr

mC) )()()( tb

ttH &=

∂∂

λ → )(])([)()()( tbntrtctwtb −+−=&

mD) 0)()(lim =∞→

tbtt

λ → 0)())((('lim =−∞→

tbetcu tt

ρ

Compare to the planning solution: The planning problem was:

Max dtetcuV tρ−∞∫= 0 ))(( s.t. a) Law of motion: )()()())(()( tkntktctkftk −−−= δ& b) Endpoint constraint 0})({lim ≥

∞→tk

t

c) Initial capital stock: 0)0( kk = with Hamiltonian:

)]()()())(([)()]([: tkntktctkftetcuH t −−−+= − δλρ FOC:

A) 0)()(

=∂∂

tctH → tecu ρλ −= )('

B) )()()( t

tktH λ&−=

∂∂ → λδλ &−=−− ])('[ nkf

C) )()()( tk

ttH &=

∂∂

λ → )()()())(()( tkntktctkftk −−−= δ&

D) 0)()(lim =∞→

tktt

λ → 0)())((('lim =−∞→

tketcu tt

ρ

So ntkf −−=− δλλ ))(('&

c

c

UU&&

−=− ρλλ

28

Equivalence: mA) is the same as in the planning solution. mB) is equivalent to B) because in a market economy,

δ−= ))((')( tkftr mC) is equivalent to C) )()()()( tcknkftk −+−= δ& because

δ−= ))((')( tkftr(')())((and b(t)=k(t) and under CRS

)())( tktkftwtkf += . But more generally, even in the absence of CRS, the value of

the firm includes all profits, so we get the analogous result also in that case.

mD) Transversality condition 0)()(lim =

∞→tkt

tλ is equivalent

to the No Ponzi game constraint being binding: 0})({lim ])([ =⋅−−

∞→

tntr

tetb , because b(t)=k(t) and

δ−= ))((')( tkftr , so from B): λλ &−=− ]n)([ tr or tntrt r ⋅−− ])([0 )([ ν edn ⋅=− )0(}] λνt ∫−⋅= exp[)0()( λλ

29

No Ponzi Game Constraint and Transversality Condition From the (flow) budget constraint, integration gives evolution of wealth across time:

tddnrtctwdnrbTb TtTT ]])([[exp)]()([]])([[exp)0()( 00 νννν ∫∫∫ −−+−=

Notation: ttre )(− = νν drt∫− 0 )(exp[ with νν drttr t∫⋅= 0 )(

1)(

So tdetctwebTb tTntrTTnTr )(])([0])([ )]()([)0()( −⋅−⋅− −∫+=

The intertemporal wealth constraint is: tdetwbeTbtdetc tntrTTnTrtntrT ⋅−−⋅−−⋅−− ∫+=+∫ ])([0

])([])([0 )()0()()(

or for T→∞ tdetwbeTbtdetc tntrTnTr

Ttntr ⋅−−∞⋅−−

∞→

⋅−−∞ ∫∫ +=+ ])([0])([])([

0 )()0()(lim)(

No Ponzi-Game-constraint:

Finite Time: Impose condition 0)( ≥Tb for representative agent: Intuition: Otherwise (that is if for some j 0)( Tb at the end of the world. i

No Ponzi-Game-constraint with infinite horizon ( ∞→T ): 0})({lim}])([exp[)({lim ])([0 ≥=∫ −−⋅

⋅−−

∞→∞→

tntr

t

t

tetbdnrtb νν

Present value of assets must be asymptotically non-negative! Motivation: lack of counterparty - no one is willing to hold assets with positive present value forever. Otherwise (if constraint were not imposed): it would be optimal to go into infinite debt at the starting period (cannot be an equilibrium).

But from the Transversality condition (as part of the optimality conditions): we know that this inequality must be binding for the optimal path. 30

1.2.2.4 OLG Model: Dynamic Inefficiency Simplest case: Agents live two periods; income only in first period U = U (C1t) +ß U (C2t+1)

s. t. wealth constraint: C1t +1/(1+rt) C2t+1= wt

FOC : ttc

tc

rCUCU

++

=+11

)()(

1

12 ρ or ttt CrC 112 1

1 σ

ρ⎟⎠

⎞⎜⎝

⎛++

=+

Consider σ=1:

tt wC ρρ

++

=21

1

ttt wwrs ρ+=

21)(

; t

tt w

rCρ+

+=+ 2

112

Population Growth: tt NnN )1(1 +=+ Evolution of capital stock:

12111 ),( −−+ −−+=−=− ttttttttttttt NCNCKrNwCNKFKK = 1211 ][ −− −+− ttttttt NCKrNCw ; Consumption of old: ttttt KrKNC 112 −− +=

ttttttt wrsNCwNK )()( 11 =−=+ or in per capita terms: tttttt

t

tt wrsn

Cwn

CwNNk )(

11)(

11)( 11

11 +

=−+

=−=+

+

= [ ])(')()('(1

11 tttt kfkkfkfsn

−+ +

For σ=1 ρ+

=+ 21)('( 1tkfs ; for

αtt kkf =)( :

αα tt kw )1( −= so

ααρ tt

kn

k )1(2

11

11 −++

=+ → α

αρ

−⎟⎠

⎞⎜⎝

⎛ −++

=1

1

)1(2

11

1*n

k

)2()1(1

*)(*)(' 1 ρα

αα α ++−

== − nkfkf

Compare with golden rule path: nkf g =)(' ! Overaccumulation if α is small: nkf

Dynamic Inefficiency ~ Reason? Market failure: missing markets Future generations do not participate in active markets today Incomplete markets Mathematical joke: Hotel infinity: one missing bed nor problem, just ask all to move one bed ahead Paul Samuelson: OLG model as basis for: a) social security system Intergenerational Transfer as free lunch b) Introduce Money (Sargent/ Wallace): OLG model as microfoundation for monetary theory (fad) no room for money as soon as there is storage; dominated asset! (You need transaction role for money)

32

1.2.2.5 Financial Wealth/ Asset Pricing: Value of firms (shares): Basics of asset pricing (see Obstfeld/Rogoff 100ff; Appendix 2B, 121) The value of a firm { } depends on the dividend stream { } and the terminal value. Each period, the household decides about the number of shares held until next period. She purchases shares of the firm at the value Vt-1. At time t, these shares yield a dividend dt and (after dividend payments, before consumption) have the value Vt.

tV td

ts

At t, additional shares can be sold or bought at the current value Vt. Given the dividend stream, how can we calculate the value of the firm? Intuition: Intertemporal evolution of the firm’s value should follow the intertemporal market prices.

Define 1,11 11

)()(

+++ =

+== tt

tt

t

tc

tc qrCu

Cuλ

λβ as the market

discount factor. Then, the asset evaluation equation should be: 1,11 )( +++ += ttttt qdVV = TttTittTi it qVqd ++= + +∑ ,,1

Let us derive this value function explicitly. With other income and spending Nwt

−t Vs, available consumption in each period is:2

+1)( ++= ttttt sVdC Nwt So })({( 10 NwssVsduU ttttttt

t +−+= +∞=∑ β For choosing the number of shares held optimally, we must have as FOC:

ts

][][)(')('

)('])[('

111

111

ttt

ttt

t

tt

ttt

tttt

VdVdCu

CuV

VCuVdCu

+=+=→

=+

−−−

−−−

λλβ

ββ

2 Budget constraint: tttttt CsVdsV −+=+ )(1 Nwt+ . Use the standard Lagrangian:

]Nwt+)[( 1sVCsVd ttttttt −−+ +λ + ]111 NwsVC tttt −−−) 1st−[( 111 Vd ttt −−− +−−+λ . Alternatively, try Hamiltonian approach by reformulating: tt ss −+1

33

By iterated substitution, the value of the firm is the present value of the dividend stream (discounted by the marginal rate of substitution between t and today [at each state s]):

01

00 λ

λλλ T

TTt

tt VdV +∑= = = = TTtTt t qVqd ,0,01 +∑ =

)('

)(')('

)('

001 Cu

CuVCu

Cud TT

Tt

tTt t

ββ +∑ =

Transversality condition: 0)(')('lim

0=

∞→ CuCuV TTTT

β , so

∑= ∞=10

0 tt

tdV λλ

tTt t qd ,01∑= =

Extension: Introduce an additional international asset with constant interest rate r: With 2 assets and a constant rate of interest r, real wealth is:

tttt BsVW += −1

tttttttttttttt CNwsVVsdrBsVsVBB −+−++=−+− −−++ )( 1111

})()1{( 110 NwssVsdBBruU tttttttttt +−++−+= −+∞=∑ β

FOC: )(')()(' 111 +++ += ttttt CudVCuV β and

)(')1()(' 1++= tt CurCu β thus:

)(1

11 ttt Vdr

V ++

=− or

∑+

++

=++

= =Tt TTtt Vr

dr

Vdr

V 1110 )1(1

)1(1)(

11

Using the transversality condition: ∑+

→ ∞=10 )1(1

t tt drV

34

Remarks: 1) Under CRS: Value of capital stock ~ Value of Firm: tt VK =

)(),( 1 ttttt KKNwNKFd −−−= + , so firms maximise at t=0:

)}(),({)1(

11000 tttt tt KKNwNKFr

Vd −−−∑+

=+ +∞=

with rFwFtKtN == ; .

Under constant returns to scale tKN KFNFF t+=(.) , so

tt KrNwF =−(.) So there are zero profits and 00000 )1( rKKKrdV +=+=+

2) No Bubble condition: In equilibrium, it must be true that

. 0)('lim =∞→

TT

TTCuV β

with constant r, equivalent to 0)1(

1lim =+∞→ TTT r

V because

ttt

rCuCu

)1(1

)(')('

0 +=β and 0)(' 0 >Cu

Proof: Suppose . Then, agents would

be better off (increase present value of consumption) by short-selling shares (maintain forever a constant negative value of the asset). But as long as , no

one would be willing to hold the positive counterpart (all agents would like to do the same – so that cannot be an equilibrium). Nobody wants to be the victim of a Ponzi game!! So we impose the no bubble condition:

. Optimality with free disposal implies

that inequality is binding: Transversality condition:

0)('lim >∞→

TT

TTCuV β

lim∞→T

0) ≤

0) =

0)(' >TT

T CuV β

('lim∞→

TT

TTCuV β

('lim∞→

TT

TTCuV β

35

Value of capital in the case of adjustment costs: Tobin’s q See: Barro/Sala-i-Martin, section 3.6 and Romer, section 8.2 In the presence of adjustment costs, the value of capital may deviate from replacement cost. Solve: Max

dtetKtItItNtwtNtKFV ttr )(0 )()(1)()()()](),([ −∞∫

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛Φ+−−=

s.t. )()()( tKtItK δ−=& ; 0)(";0)(';0)0( ≥⋅Φ>⋅Φ=Φ Shadow price: q(t)=V/K is the price of the capital stock at period t in current value (Market value of a firm/ capital stock). (Check with present value Hamiltonian: ttretqt )()()( −⋅=λ !) Current value Hamiltonian:

)]()([)()()(1)()()()](),([)(~ tKtItqtKtItItNtwtNtKFtH δ−+⎥

⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛Φ+−−=

FOC:

0)()(~

=∂∂

tNtH or 0)](

)()([ =−

∂∂ tw

tNtF

0)()(~

=∂∂

tItH or

)()(

)()('

)()(1)(

tKtI

tKtI

tKtItq ⎟

⎠

⎞⎜⎝

⎛Φ+⎟

⎠

⎞⎜⎝

⎛Φ+=

)()()()()(~ tqtrtq

tKtH

+−=∂∂

& or

)()())(()()(

)()('

)()(

2

2tqtqtr

tKtI

tKtI

tKtF

&−=+−⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛Φ+

∂∂ δ

or: ⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛Φ+

∂∂

−+= 22

)()(

)()('

)()()())(()(

tKtI

tKtI

tKtFtqtrtq δ&

36

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛Φ+

∂∂

−+= 22

)()(

)()('

)()(

)(1))((

)()(

tKtI

tKtI

tKtF

tqtr

tqtq δ&

Return on existing capital, deflated by cost of capital q(t): F’ + marginal reduction in adjustment cost with rising K Without adjustment costs: δ+=== rFtqtq ';0)(;1)( & Transversality constraint (motivated by the no bubble constraint): 0)()(lim )( =−

∞→

ttrt

etKtq

Example: Linear adjustment costs )()(

2)()(

tKtIb

tKtI =⎟

⎠

⎞⎜⎝

⎛Φ

FOC: )()(1)(tKtIbtq +=

btq

tKtI 1)(

)()( −

=→

)(1)()( tKbtqtK ⎟

⎠⎞

⎜⎝⎛ −

−= δ&

⎥⎦

⎤⎢⎣

⎡ −+−+=

btqtKFtqtrtq2

]1)([))((')())(()(2

δ&

Convergence of shadow price of capital q(t) to q*=1+δ b? Saddle path stability:

37

1.2.3 Applications: 1.2.3.1 Permanent income hypothesis Milton Friedman argued that a representative agent consumes a constant proportion of his permanent income. Under what conditions will this be optimal? Using the transversality condition, the intertemporal wealth constraint for a representative agent is:

tdetwbtdetc tntrtntr ⋅−−∞⋅−−∞ ∫∫ += ])([0])([

0 )()0()( Using the optimality conditions derived in 1.2.1, we can characterise the optimal consumption path in a straightforward way:

Because of )][)( nrccc

−−= ρσ& , we get via integration:

tntrtcectc ⋅−−= ])([))(()0()( ρσ Given this path, c(0) is determined by the intertemporal wealth constraint. For constant σ we can solve explicitly for the consumption path:

dtectdec tntrtntrntr ⋅−−−∞⋅−−−−−∞ ∫∫ = }]1[])({[0)])(()([

0 )0()0(ρσσρσ

)0()0( HWb +=

A) Special case σ=1: )]0()0([)0( HWbc += ρ

B) If the real rate of interest r is constant, this gives as optimal initial consumption level:

)]0()0([])([)0( HWbnrnrc +−−−−= ρσ Obviously, c(t)= c(0) only if 0=−− nr ρ :

)0()0()( ][ cectc tnr == ⋅−−ρσ

)0()0()0()0( ][0 Htnr Wb

nrctdec +=

−=∫ ⋅−−

∞

or pH wbnrWbnrc +−=+−= )0()()]0()0()[()0( Note: !ρ=− nr (steady state analysis) Modern Version: Consumption growth follows random walk 38

1.2.3.2 Government Spending and Ricardian equivalence 1.2.3.2.1 Basic Idea: Two Period Framework What changes if we introduce government activities? Take the path of government spending and taxes as given. The government can issue bonds to finance spending via debt.

2121 ;;; TTGG

The government budget constraint per period is: 10211 )1( BrBTG +−+= ; 21322 )1( BrBTG +−+=

Consider a pure endowment economy. The initial government debt has to be equal to the initial wealth 10 )1( Br+ of private agents who initially bought bonds . Since for private agents terminal condition is binding, we have to impose the No Ponzi game constraint for the government:

[resulting in the transversality condition ].

1B03 ≥B

03 ≤B 03 =B

)1(1)1(

1 13

1

21́10

1

21 r

Br

TTBrr

GG+

++

+=+++

+

The budget constraint for private agents is now:

101

2211

1

3

1

21 )1(111

BrrTYTY

rB

rCC ++

+−

+−=+

++

+

Ricardian equivalence: Government debt is no net wealth for private agents! From the government budget constraint, we can substitute for taxes in the private agent constraint:

1

2211

1

21 11 r

GYGYr

CC+−

+−=+

+

So the modified equilibrium conditions are now: 222111 ; GYCGYC −=−= , and the natural real interest rate is

now determined by:

11,

1

11

22

11

)()(

11

rq

GYuGYu

ttt

t

c

c

+===

−−

+ ++

λλ

ρ

39

1.2.3.2.2 Infinite Horizon Growth Model Assume that the representative household owns the capital stock )(tk and also holds government bonds )(tb [both defined as per capita]. This defines wealth. She has to pay taxes and may appreciate g(t). The growth rate of labour is n. The household’s problem (with the capital stock rented out to the firms) is:

Max dtetgtcuV tρ−∞∫= 0 ))(),(( s.t. law of motion a) Total wealth Ω(t)=k(t)+b(t) evolves according to: with )()()( tbtkt &&& +=Ω

)()()()(])([))(()( ttkntctbntrtkft τ−−−−+=Ω& b) No Ponzi-Game-constraint 0})({lim ])([ ≥Ω ⋅−−

∞→

tntr

tet

c) Initial wealth out of government bonds: )0(bd) Initial capital stock: )0(k e) Assume: ))(),(( tgtcu is additive separable

Hamiltonian: )()()]([: ttetcuH t Ω+= − &λρ From Lagrangian: )()()]([:)( ttetcuHL t Ω+= − &λρ - )]([)( tt Ωλ& So:

)]()()(])([)())(([)()]([: ttkntbntrtctkftetcuH t τλρ −−−+−+= −

FOC:

A) 0)()(

=∂∂

tctH → tecu ρλ −= )('

B) )()()(

)()(

)()( t

tbtH

tktH

ttH λ&−=

∂∂

=∂∂

=Ω∂

∂

→ λλ &−=− ])([ ntr and λλ &−=− ])('[ nkf

40

C) )()()()()()( tktbtt

ttH &&& +=Ω==

∂∂ ω

λ

)()()(])([)())(()()()( ttkntbntrtctkftktbt τ−−−+−=+=Ω &&&

D1) 0)()(lim =∞→

tbtt

λ → 0)())((('lim =−∞→

tbetcu tt

ρ

D2) 0)()(lim =∞→

tktt

λ → 0)())((('lim =−∞→

tketcu tt

ρ

So ntrntkf −=−=− )())(('λλ&

Equal return on capital and government bonds: )())((' trtkf =

Rate of return on consumption:c

c

UU&&

−=− ρλλ

With fixed labour supply and utility additive separable in g(t) ( ) : 0=cgU cUU ccc && =

D1) is equivalent to: 0})({lim ])([ =⋅−−∞→

tntr

tetb

D2) is equivalent to: 0})({lim ])([ =⋅−−∞→

tntr

tetk

0)0(;)0(])([exp[)0()( ])([0 >=−−= ⋅−−∫ λλννλλ tntrt ednrt The path of spending and taxation cannot be arbitrary: The government has to take into account its budget constraint. Its flow constraint is:

)()(])([)()( ttbntrtgtb τ−−+=& Starting with some initial level of government debt b(0), debt evolves according to:

dtettgebTb tTntrTTnTr )(])([0])([ )]()([)0()( −−− ∫ −+= τ or:

TntrT

tntrT ebdtetgtb ])([])([0 )]()([)0(−−−− +−= ∫ τ

41

When the transversality condition for households holds, the government is subject to the “No Ponzi game condition:” 0})({lim ])([ ≤⋅−−

∞→

tntr

tetb

The transversality condition gives as intertemporal wealth constraint for the government: (GW) dtetgtb tntr ])([0 )]()([)0(

−−∞ −= ∫ τ Independent of the timing of taxes and government spending, the present value of net earnings (discounted tax earnings minus government spending) must equal present outstanding debt. Ricardian equivalence: How does government debt policy affect private wealth? Using the transversality condition, the household’s wealth constraint is:

tdettkfbktdetc tntrtntr ⋅−−∞⋅−−∞ ∫ −++=∫ ])([0])([

0 )]())(([)0()0()( τ Government bonds are not net wealth: Using (GW), we can transform the household’s wealth constraint into:

tdetgtkfktdetc tntrtntr ⋅−−∞⋅−−∞ ∫ −+=∫ ])([0])([

0 )]())(([)0()( So the household’s wealth constraint is independent of the tax and debt path. Intuition: Higher government debt results in an equivalent increase in private savings in order to offset future tax obligations. This result depends on strong assumptions: No imperfections of capital markets, private equal to social discount rate…!

Credit constrained agents: liquidity constraints OLG structure: Distortionary taxes: Incentives for tax smoothing Non-optimising agents (Keynesian type agents; rule of thumb ~ Akerlof: inclusion of those norms (AER presidential address 2007).

42

1.2.3.2.3 Sustainability of Government Debt In the last section, we characterised the government wealth constraint. Will a government with a constant deficit run into solvency problems? Does it make sense to impose some upper limit to the deficit ratio such as in the Maastricht criteria? We have to distinguish between primary deficit and total deficit. Consider as simplest case an economy with constant growth rate y and constant real rate of interest r. We assume the interest rate exceeds the growth rate: r>y (only this assumption is essential!) When will the government stay solvent, starting with some initial debt (outstanding bonds)

>0? (Note: here b(t)=B(t)/Y(t); issuing bonds b>0 is equivalent to debt!)

)0(b

With Pd)t

as primary deficit, the debt ratio evolves according to: b )()()()()()()(( tbyrtdtbyrttg P −+=−+−= τ& giving the differential equation:

dtettgebTb TtyrTTyr )()(0)(0 )]()([)(−−−− −+= ∫ τ or

TyrT

tyrT ebdtetgtb )()(00 )]()([−−−− +−= ∫ τ

0lim )( = Imposing the

transversality condition −−∞→

TyrTT

eb

dtetdb tyr

gives (GW)

P)(

00 )(−−∞∫ −= etgt0 )]()([ dttyr )( −−∞ −= ∫ τ = yr

g−−τ

The present value (discounted by the effective rate r-y) of ratio of permanent future primary surplus to GDP has to be equal to the current level of debt /GDP. Consider a path with a constant primary deficit rate:

τ−= gpd such that: )()()( tbyrpdtb −+=& . Solution to this linear differential equation: cbab =+& Homogenous part: taeAtb −=)( ; given 0b

Particular solution: acbp =

43

Definite solution: aceacbtb ta /]/[)( 0 +−=−

Since a=-(r-y)0, we get )/()]/([)( )(0 yrpdeyrpdbtb

tyr −−−+= − Obviously, the government debt ratio is unstable unless

)/(0 yrpdb −−= But: temporary deviations - ])([)( pdtpdtb −=& - do no harm if offset later by additional surplus:3

dtepdtpdyr

pdTb TtyrT )()(0´ ])([)(−−−−+

−−= ∫

The government stays solvent if minmaxmax gsppd −=≤− τ with 1max g as minimum sustainable spending rate. Since these rates are somewhat arbitrary, as pragmatic approach, calculate as sustainable tax rate the average tax rate which would be needed to finance annuity value of future expected spending and transfers, plus the real interest rate burden on current debt:

])()[( 0)(

0 bdtetgyr tyr +−= −−∞∫τ If ττ < , taxes have to be raised or spending has to be cut when debt should be sustainable. The magnitude ττ − gives a measure of the adjustment needed! How does delay of adjustment until time t affect the size of needed policy action? The timing of adjustment is given by

differentiating the equation above: )()( τττ −= − ttyre

tdd

Key ingredients: 1) The real interest rate exceeds growth rate: r>y 2) Transversality condition holds: 0lim )( =−−

∞→

TyrTT

eb 3 Transversality condition 0lim )( =−−

∞→

TyrTT

eb implies:

0])([ )(0´ =− −−∞∫ dtedtd tyrpp 44

2) Maastricht treaty: constant total budget deficit rate The Maastricht criteria impose a limit for the nominal debt ratio b≤0,6 and for the total nominal deficit ratio d≤ 0,03. So now we have to analyse an economy with nominal debt and inflation. pY is nominal GDP. We assume a constant rate of inflation π and the Fisher equation i= r+π. We also include seigniorage μk in the government wealth constraint (see section 1.3 for details). How will the debt ratio evolve when the total budget deficit ratio (including debt service) stays constant? Will the debt ratio converge to some steady state?

Let constant==+−−= dbikgYp

B μτ&

Then bydb )( π+−=& . Note: d is total deficit (including interest payments)!

Solution of the Differential equation bydb )( +−=•

π for any initial level : 0b

yde

ydbdtedebTb TyTtyTTy

++⎥

⎦

⎤⎢⎣

⎡+

−=+= +−−++− ∫ππ

πππ )(0

)()(0´

)(0 ][)(

So we have y

dTbT +

=∞→ π

)(lim , converging to the steady state:

ydb+

=π

*

Maastricht criteria: For d=0,03; y=0,03 and 02.0=π : b converges to b=0,6 The lower y and π , the higher the steady state ratio b. Implies: higher tax burden on economy (depending on real interest rate r-y). The debt limit is feasible as long as

yrgbb

−−

=< minmaxτ ( 0;1 minmax >< gτ )

45

1.2.3.3 Current Account Deficits: Optimising Countries? Frequently, economists analyse dynamics with constant interest rate to gain a better intuition about key issues, such as solvency constraints and sustainability of debt. In this section, we consider simple examples of a pure endowment economy with infinite horizon to get a feeling for solvency constraints and for the implications of transversality conditions. We pose the following questions:

1) Can we have an economy with optimising heterogeneous agents, but a constant real rate of interest without violation of the transversality condition? (Example: Countries with different time preferences in open economies)?

2) Sustainability of Current account deficits: Under what conditions is the debt dynamic of a small country sustainable? Assume the country has an exogenous constant growth rate y with: ytYtY =)(/)(& . Ignore capital formation and set n=0. Preferences are characterised by a constant elasticity of substitution σ. Let b(0)

We analyse the debt dynamic in a small country with preferences

dtetcuV tρ−∞∫= 0 ))(( with σσ

11)(1)(

/11

−

−

= tctU

Initial wealth: b(0)= B(0)/ Y(0) a) Characterise the law of motion for wealth/GDP:

Show that b(t= B(t)/Y(t) evolves according to: )()()(1)()(/)]()()([)( tbyrtctbytYtBrtCtYtb −+−=−+−=&

b) Characterise FOC using the Hamiltonian! Show that

Consumption path for constant σ is characterised by: treCtC ⋅−= ][)0()( ρσ Under what conditions is the

problem well-defined? c) Characterize the autarchy solution. d) Assume the country has initial debt b(0)= B(0)/ Y(0)n! For n−−−− nrnr ρσ or ρ

σ−−

−<

nrnr

See also next section on sustainability analysis!

47

1.2.3.3.2 Sustainability analysis of current account deficits. Using the transversality condition 0})({lim =⋅ ⋅−

∞→

Tr

TeTb , the

intertemporal wealth constraint is: =+= ⋅−∞⋅−∞ ∫∫ tdetYBtdetC trtr 00 )()0()([

yrYBtdeYB tyr

−+=+ ⋅−−∞∫

)0()0()0()0( )(0

The consumption path for constant σ is characterised by: treCtC ⋅−= ][)0()( ρσ ;

)()0()0( 0

)]([

ρσρσ

−−=∫∞ ⋅−−−

rrCtdeC trr

The problem is well-defined for ρ

σρσ−

−−r

rrr ;0)(

(see 1.2.3.3.1) Otherwise there would be no convergence of consumption. So we get:

=+−

−−= )]0()0([)]([)0( Byr

YrrC ρσ

)0()]([)0()( BrrYyr

rr ρσρσ −−+−

−−

tyrtyr eyr

rreYC

tYtC ⋅−−⋅−−

−−−

== }][{}][{)(

)0()0(

)()( ρσρσ ρσ

>∞

=−→

<

∞→yrforconst

tYtC

t][

)()(

0

ρσ

48

A) In the case of autarchy, we must have:

yrσ

ρ 1+= ; B(0)=0; tyty eCeYtYtC ⋅⋅ === )0()0()()(

For yrσ

ρ 1+= : consttYtC

=)()( =1+(r-y)b0; consttY

tB=

)()( ;

)0()0(

)()(

)()(lim

YB

tYtB

tYtB

t==

∞→= b0

B) If yrσ

ρ 1+> , the consumption path is tilted towards the

future. There is no convergence towards a steady state, but the value of consumption will be bounded for σ : ∞→)()(

tYtC ; ∞→

)()(

tYtB

49

In that case, consumption grows faster than income, but initially the country builds up assets. Ratio of net foreign

assets to GDP shoots off towards infinity: ∞=∞→ )(

)(limtytb

t.

Note: no stable steady state, but the transversality condition holds even in that case as long as )( ρσ −> rr (compare Obstfeld/Rogoff, exercise 2.7)! C) Consider now the case that the world interest rate is less

than the autarchy rate yrσ

ρ 1+< . That is the interesting case

for solvency issues, since in that case, the consumption path of a small open economy is tilted towards the present relative to the autarchy case (C(0)>Ca(0). So, initially, there is a current account deficit B(t)r C(t)- Y(t) CA(t)(t)B +==&

B(t)r ;

CA(t) C(t)- Y(t))( −==tTB . What will happen to a country borrowing at the world rate r across time? Where does the debt/GDP ratio converge to?

50

Steady state: B(t)g CA(t)(t)Blim ==∞→&

t; B(t)r)-(yTB(t)lim =

∞→t

In the long run: yrtY

tBt −

−=∞→

1)()(lim ; 1

)()(lim =

∞→ tYtTB

t; 0

)()(lim =

∞→ tYtC

t;

Limit of debt/ GDP ratio converges to the entire present value of current and future output (debt has to be repaid out of future production).

Proof:

)()(

)()(

)()()(;

)()(

)()(

)()(;

)()()(

tYtB

tYtY

tYtBtb

tYtY

tBtB

tbtb

tYtBtb

&&&

&&&−=−== ; Thus:

)()()(

)()(1

)()(

)()()()()(

tYtByr

tYtC

tYtBy

tYtBrtCtYtb −+−=−+−=& .

In steady state: or 0)( =tb& ⎟⎠

⎞⎜⎝

⎛−

−−=

∞→ )()(11

)()(lim

tYtC

yrtYtB

t;

For yrσ

ρ 1+< : 0)()(lim =

∞→ tYtC

t;

yrtYtB

t −−=

∞→

1)()(lim

Crucial issue: Is the limit 0)()(lim =

∞→ tYtC

t sustainable?

51

Notes: a) Consumption does not need to converge to 0 since Y

grows at rate y. But 0)()(lim =

∞→ tYtC

t for growth rate of

consumption: yr y must hold (otherwise

infinite consumption would be feasible), we must have: σ ρσ for a stable

solution, the country’s growth rate has to exceed world rate!

d) Unrealistic feature of this thought experiment: Assumption of a constant interest and growth rate despite high growth economy. More realistic: assume that growth rate converges to world growth rate! (Otherwise small country assumption will be violated at least in the long run)!

52

1.2.3.3.3 Mechanics of Accounting: Sustainability Analysis of Current Account Deficits - The case of nominal assets and inflation Current account: equivalent to an increase in foreign assets

)()( tBtCA &= with BiAYtCA +−=)( π;y growth rate of real output and inflation

Growth of nominal BIP: π+= yYY&

byY

CAbYY

YBb )( π+−=−=

&&&

Sustainability Simulation: what would happen if CA/Y is sustained for constant π;y ? Will b (ratio of net foreign assets to GDP) converge to a steady state ratio ( )? 0=b&

Differential equation: ππ

π

++

+−= +−

yYCAe

yYCAbtb ty /)/()( )(0

YCA

yb

π+=∞

1)(

YBb = is constant if BytB )()( π+=&

Example: 05,0/;05,0 −==+ YCAy π b* = - 100 % With constant b, real net external asset stock evolves according to:

PBy

PB

PB

tdPBd

=−= π&)/(

Steady state share of absorption A= C+I+G or a=A/Y: BiAYtBtCA +−== )()( &

PBy

PBr

PA

PY

tdPBd !)/(

=+−= ; byra )(1 −−=−

53

If there is a critical lower bound aa > , yrabb

−−

−=>1 (present

value of excess production over minimum absorption level)

Obstfeld/Rogoff: Perspectives on OECD Economic Integration: Implications for U.S. Current Account Adjustment What is the long run equilibrium? 0)( =tCA ? Balanced current account? Note: Nominally balanced current account implies (when inflation is positive and with net foreign debt) an inflation adjusted current account surplus! Rate of change of the economy’s real net foreign asset position with =0 is: )()( tBtCA &=

PB

tdPBd π−=)/( >0 for B

Assume that absorption (trade balance) is such that it repays a small fraction of the interest burden. The rest will be rolled over into the next period.

BrAYTB ξ=−= , so BrBrAYCAB )1( ξ−=+−==& Thus, treBtB )1(0)(

ξ−= Transversality condition holds:

000001)( Br

BrdteBrdtetTB trtr −=−=∫=∫ −∞−∞ ξξξ ξ

But notice that debt (and thus interest burden) grows at rate

)1( ξ−r In order for repayments not to exceed total production at some stage BrBrAY )1( ξ−=+−

)1(

, Y must grow at a rate above

ξ−> ry . (with y1ξ

55

1.2.4 Stochastic Optimisation 1.2.4.1 Bonds and market discount rates How to extend market discount factor to stochastic problems? Wealth constraint per period: If I plan to have wealth Wt+1 at the beginning of next period t+1, I must spend today .

is the market discount factor between t and t+1. 11, ++ ttt Wq

1, +ttqSince tttttt WCYWq +−≤++ 11, , the wealth constraint per period is:

ttttttttt WqCYqWW )1/1()(/1 1,1,1 −+−≤− +++ From the Hamiltonian, we get the FOC:

11,11, ~

~)~~()1/1(~

−+−+ =→−−=−=∂

∂

t

tttttttt

t

t qqWH

λλλλλ

and the transversality condition 011, =++++ TtTtt Wq Two periods: Maximise )()( 21 CUCUu β+= subject to the per period budget constraints

111

21 1

WYr

WC +=+

+ ; 222

32 1

WYr

WC +≤+

+ ;

with end point constraints 0; 311 ≥= WWW . FOC:

a) t

ttt

t

tc

tc

rq

CuCu

+===

+ +++

11

)()(

11

1,11

λλ

ρ

b) 033,1 =Wq Transversality condition Equilibrium condition: 2211 , YCYC == Imposing equilibrium conditions, qt is determined by:

2,111

2

11

)()(

11 q

rYuYu

c

c =+

=+ ρ

What is the impact of uncertainty on consumption/ equilibrium?

56

1.2.4.2 Stochastic Discount Rates Simple example: 2 Periods; uncertainty about the outcome of the second period with 2 possible states: high or low income. Arrow Debreu economy: Assume a complete set of risk markets – there is a price defined for each state of the world.

hp probability of high endowment; otherwise =1- lp hp Representative agents maximises expected utility:

[ ])()()( 221 llhh CUpCUpCUu ++= β Intertemporal buget constraint with Arrow Debreu securities: At present value price , buy assets in period 1 giving a real claim of 1 unit in state s next period:

sq2

With claims, wealth in state s is sD2 ss DW 22 = . So )( 222 sss DYC +=

122122 YYqCCq sss

sss

+=+ ∑∑ or )( 1122 CYDq sss

−=∑

FOC:

'2

2

'2'2'

22

/)(/)(

s

s

sss

sss

qq

CCUpCCUp

=∂∂

∂∂

sss

s qCCUCCU

p 211

22

/)(/)(

=∂∂

∂∂β

Define s

ss p

qQ 22 = as stochastic discount factor

),();(),( 11,111, +++++ = ttssttttss zzqzzprobzzQ

ss

sss Qpq

CCUCCU

22

11

22

/)(/)(

==∂∂

∂∂β

With complete markets the stochastic discount factor is unique. sQ2

57

How does uncertainty affect the price of a risk free asset (the current value price for one unit of consumption in all states)? The stochastic discount factor for each state is characterized by:

⎟⎟⎠

⎞⎜⎜⎝

⎛= ++++ );(

);(),( 1111,ttc

ttcttss cu

cuzzQξξβ

The risk free rate between some state s and s+1 is:

== ++

+++ ∑ 11

111, ),(( sss

sttss QpzzQE ⎟⎟⎠

⎞⎜⎜⎝

⎛=

+++

);();(

11 11

ttc

ttct

t cucuE

r ξξβ

Explicit analytical solutions can be obtained only for some examples with special preferences, combined with specific distributions.

58

Example 1) Assume (1) CRRA and (2) consumption growth is log normally distributed. In that case, we had without risk:

σ

ρ

1

1

11

11

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=+ t

t

t CC

ror log-linearized: tt gr σ

ρ 1+=

When future consumption is risky, the Euler equation in expected terms is:

σ

ρ

1

1

11

11

−

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

=+ t

t

t CCE

r

In the presence of consumption risk, a safe asset provides insurance for risky future consumption: risk averse agents want to do precautionary saving. From Jensen’s inequality, we guess that the safe return will be lower than a risky consumption profile with same expected value (incentive to save for precaution drives down the sure rate of interest). Define ttt CCC lnlnln 11 −=Δ ++ . Taking logs, we can rewrite

the FOC condition: ( )1ln1

11ln +Δ−=+

+t

tCE

r σρ .

If )),((~ 2xxENx σ , then ( )221)(exp))((exp xxExE σ+= . Assuming that consumption growth is log normally

distributed, here we have: 1ln1

+Δ−= tCx σ, so we get:

( ) ⎟⎠⎞

⎜⎝⎛ +Δ−=

++

+2

221

11ln1exp

11

Ctt

CEr

σσσ

ρ

Or, log linearising, we get:

( ) 222111ln1 Ctt CEr σσσ

ρ −Δ+= + = ( ) 222111

CCgE σσσρ −+

Precautionary saving With risky consumption people save more in order to insure against risk of being in a state with low consumption, driving down the risk-free interest rate. The more risk averse people are (the lower σ ), the stronger this effect!

59

Example: 2) Quadratric utility function (certainty equivalence):

221 )( tCBU −−= gives:

[ ]tt

t CBrCEB −

++

=− + 11)( 1

ρ

Br

CCEt

tt ⎥⎦

⎤⎢⎣

⎡−

++

−=+ 111)( 1

ρ ;

for ρ=tr : tt CCE =+ )( 1 or 11 ++ += ttt CC ε With 0)( 1 =+tE ε - consumption follows a random walk Example 3) Exponential utility function:

)(exp1)( tt CCU αα−−=

This gives the Euler equation:

( ))(exp11)(exp 1+−+

+=− t

tt CE

rC αρ

α

Note: For )),((~ 2xxENx σ ( )221)(exp))((exp xxExE σ+= . So we get:

⎥⎦

⎤⎢⎣

⎡+−

++

=− ++ )(2)(exp

11)(exp 1

2

1 ttt

t CVarCErC ααρ

α

or in logs: )(2

)()( 12

1 ++ +−−=− tttt CVarCErCααρα , so:

)(2

)(])([ 12

1 ++ +−=− tttt CVarrCCEαρα

Calculate equilibrium for the stochastic endowment process: 11 ++ += ttt YY ε (persistent effect of shocks); ),0(~

21 σε Nt+

60

1.2.4.3 Stochastic Returns Let us now extend the analysis to holding arbitrary assets with random return 2,~ir . Again, assume complete contingent prices. Using the stochastic discount factor sQ2

~ the MRS between consumption in state s and s+1 is characterised by:

11

2

((

CC 2

2 /)/)

CUCU

Q sss ∂∂∂∂

= β . So the Euler equation gives:

A) ( )sissi QrECCUCCU

rE 22,11

222,

~)~1(/)(/)(

)~1(1 +=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂+= β

Rewrite condition A) as the product of expected variables plus the covariance A+) ( ) ( )sisi QrCovQErE 22,22, ~;~1)~(~11 +++= . For a risk-less asset with safe return r, the covariance with the stochastic discount factor (and any other random variable) is zero. Thus:

B) t

ttss rzzQE

+=++ 1

1),(~( 11,

Using B) in A), we have for arbitrary risky assets: C) ( ) ( )( )sii QrCovrrE 22,12, ~;~11)1(~1 +−+=+ Equation C) is the key relation for the consumption CAPM Model: The expected return premium of an asset relative to risk-free assets is negatively [positively] proportional to its covariance with MRS [consumption]. Correlation (covariance) between the asset’s payoff and consumption as decisive criterion for holding the asset: An asset with high returns is attractive when consumption is low ( is high). → Demand for such an asset will be high; its return will be lower than a risk free asset.

sQ2

61

Applications: 1) Evaluation of risky shares (dividend streams)

Without risk, the value of the firm is ∑∞== 10

0 tt

tdV λλ

The dividend stream is evaluated with )(')('

0,0 Cu

Cuq tto

tt βλ

λ== .

With )1(

1...)1(

1)1(

1)1(

1

1,1,20,11

1,,0

−=

− +++=

+= ∏

tt

ti

iit rrrr

q

How to evaluate a risky dividend stream? In case of risk neutrality, utility is linear in consumption, so

1)(')('

0=

CuCu t . Thus t

o

tt Eq βλ

λ=⎟⎟

⎠

⎞⎜⎜⎝

⎛=,0

Evaluation equation: ( ) )(1,00 ∑∞=== t tttt dEdqEV β With risk aversion, expected market discount factor depends on the covariance between dividends and the discount factor. To see this write the Euler equation as:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+++⎟⎟

⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛+=

+++

+++

+++

][)(')('][

)(')('

][)(')('

111

111

111

ttt

ttt

t

t

ttt

tt

VdCu

CuCovVdECu

CuE

VdCu

CuEV

ββ

β

If dividends are positively correlated with the discount factor, the price of the claim, Vt, is higher (a positive correlation with the discount factor means a negative correlation with consumption growth). Hence, if the dividends are negatively correlated with the cycle, Vt, the price of the shares, is higher, since the claim on the firm provides a good hedge against the variability of consumption.

62

2) Lognormal distribution and homoskedasticity Characterise the exact solution for lognormal distributions of the random variable If )),((~ 2xxENx σ , then ( )221)(exp))((exp xxExE σ+= or

( )221)(log xx σ+))((exp ExE = So if Y is log normally distributed, (that is, X= log Y is normally distributed), ( ))(log)(log)(log 221 YYEYE σ+= If Y is homoskedastic, then

)](log[log)](log[log)(log 2 YEYVarYEYEYVar −=−= Logarithmic Version: Taking the log of A) ( )si QrE 22, )~1(1 += , we get:

Log A) ( )

)]log;~(2)(log)~([21

)(log~0

22,22,

22,

sisi

si

QrCovQVarrVar

QErE

+++

+=

For the riskless asset r with Var (r) =0 and )log;( 2 sQrCov =0, the log-version of B) is:

Log B) )](log[21)(log 22 ss QVarQEr −−=

For risky assets, the log-version of C) is:

Log C) ( ) ( ) ( )siii QrCovrVarrrE 22,2,12, log;~~21~ −−=−

63

Explicit solution for CRS preferences

With CRS preferences, σ1

1

2

11

22

/)(/)( −

⎟⎟⎠

⎞⎜⎜⎝

⎛=

∂∂

∂∂

CC

CCUCCU sss , the

stochastic discount factor is: σβ1

1

22

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

CC

Q ss . The inverse of σ is

the coefficient of relative risk aversion. The lower σ, the higher the risk aversion. Defining 1+g as the gross growth rate of consumption

, we get in A+): ttst CgC )1( 11 ++ +=

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+++++=

−−σσβ1

2,

1

2, )~1();~1()~1()~1(1 grCovgErE ii

Log B) gives for the risk-free rate: )]([1

21)(1 121 ++ −+= tt gVargEr σσ

ρ

Log C) gives [ ] ),(1 criskyfreeriskriskyt grCovrrE σ≅−−

For these preferences, the Euler-equation gives a linear relation between expected growth of consumption and the expected return of any asset:

)]([121)()( 11 ++ +−= tt gVarrgE σ

ρσ or

1,1)( ++ +=Δ tWtttt rEcE σμ With tμ determined by variance and covariance between consumption and the asset’s return. If both Wt rE and μ are constant across time, growth of consumption follows a random walk.

64

Advanced Macro and MoneyLecture notes (WS 09/10)Prof. Dr. Gerhard Illing, LMU University of Munich1.1 Basic Tools: The Problem of Optimal Control – An intuitive Guide resp. CS: or 1.1.7 Example: Optimal extraction path for exhaustible resources