sgz macro week 3, lecture 2 - boston...

SGZ Macro Week 3, Lecture 2:Suboptimal Equilibria

1SGZ 2008 Macro Week 3, Day 1 Lecture 2

Basic PointsBasic Points

• Effects of shocks can be magnifiedEffects of shocks can be magnified (damped) in suboptimal economies

• Multiple equilibria (stationary states• Multiple equilibria (stationary states, dynamic paths) in suboptimal economiesG t li i i lf• Government policies can improve welfare in these economies (or can be the source f b ti lit )of suboptimality)

SGZ 2008 Macro Week 3, Day 1 Lecture 2

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OutlineOutline

A Static example: production externalityA. Static example: production externality1. Positive effect of one individual’s effort on

level and marginal product of another’s effortlevel and marginal product of another s effort2. Optimal allocation3 Decentralized allocation (Nash equilibrium)3. Decentralized allocation (Nash equilibrium)

Output “too small”, if positiveOutput can be zero in equilibrium or “muchOutput can be zero in equilibrium or much too small”

4 Subsidy can raise output to efficient levelSGZ 2008 Macro Week 3, Day 1

Lecture 23

4. Subsidy can raise output to efficient level.

Outline (cont’d)Outline (cont d)B. Dynamic example: financing a given amount of y p g g

government purchases with an output tax, whose rate is endogenously set to raise specified revenuespecified revenue

1. Formulation as a “sequence” problem(later: formulation as a “dynamic program”)(later: formulation as a dynamic program )

2. Steady state outcomes:Accumulation too lowMultiple equilibria can occur with endogenous taxes

3. Dynamic outcomes: multiple equilibria can arise with endogenous taxation


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g

A. Static Example

• Consider an economy populated by manyConsider an economy populated by many identical agents

• All agents have a preferences over goods and g p gleisure which take the form

1( , ) log( ) (1 ) with 01

u c l c l γχ γγ

+= − − >+


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A1: Externality Model:I di id l d i f iIndividual production function

• Output for individual depends positively on his own labor p p p yinput (n) and the average amount of input (N) chosen by individuals in the economy.If th J i di id l th• If there are J individuals, then

1 1so thatJ

jNN n

J J∂

= =∂∑

where ni is the effort of the agent under examination.

1j

j iJ n J= ∂∑

i gWe’ll assume that δN/δni = 0, which requires that there are an infinite number of individuals, but is approximately true for large numbers of individuals


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true for large numbers of individuals.

Production FunctionProduction Function

• Cobb DouglasCobb Douglas– Marginal product if everyone works harder is higher

than if just the individual under study works harder1[ ] 0 1y a n Nα α α−= < <

| | N N ny y y yn n n n

α =

∂ ∂= =

∂ ∂

– But note social production function is y=an if n=N)


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Production Function (cont’d)Production Function (cont d)• If no one else works, there is no point to my , p y

working

1[ ] 0 1y a n Nα α α−= < <

0| 0 Nyn =

∂=

∂


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A2. Social production function d i l ll i iand optimal allocations in economy

• Production function and time constraintProduction function and time constraint

*c y a n= =

• Lagrangian for optimal allocations

1y

n l+ =

• Lagrangian for optimal allocations

( , ) [ ] [1 ]L u c l an c n lλ ω= + − + − −( , ) [ ] [ ]


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Social optimump• FOCs for planner

10 ( )l λ λ: 0 ( , )

: 0 ( ) (1 )

c

l

c u c lc

l u c l l γ

λ λ

ω χ ω

= − = −

= − = − −: 0 ( , ) (1 ): 0l t i t

ll u c l ln a

ω χ ωλ ω= −

• Optimal consumption and work

plus constraints

• Optimal consumption and work1

11( )n c anγ+= =


10

( )χ

A3. Decentralized (N h) E ilib i(Nash) Equilibrium

• Each agent chooses n taking as given NEach agent chooses n taking as given N• Symmetric equilibrium: because all agents

are same it must be that equilibrium N=nare same, it must be that equilibrium N=n

• Zero activity equilibrium (N=n=0) is transparent, so focus on positive activity equilibrium


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Individual optimizationIndividual optimization

• ProblemProblem

, ,max ( , )c l n u c l

s.t. ( , ) and 1c f n N n l= + =


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Lagrangian and FOCsLagrangian and FOCs

( ) [ ( ) ] [1 ]L l f N lλ ω+ +( , ) [ ( , ) ] [1 ]

1

L u c l f n N c n lλ ω= + − + − −

1: 0 ( , )

: 0 ( , ) (1 )

c

l

c u c lc

l u c l l γ

λ λ

ω χ ω

= − = −

= − = − −

: 0 ( , )

l

nyn f n Nn

λ ω λα ω= − = −

plus constraints


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Equilibrium (with n=N)Equilibrium (with n N)

• Using above equations can work out that• Using above equations, can work out that

11α 1( ) c=a*nn n Nγα

χ+= =

• Contrast to social optimum which sets α=1 in above so that equilibrium n isin above, so that equilibrium n is inefficiently low (and consumption too because c=an)


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because c an)

ConceptsConcepts

• Output and work is inefficiently low becauseOutput and work is inefficiently low because people do not take into account the social benefit – raising other individual’s productivity –of their own work effort. This is a direct result of the interaction of the externality and competitive b h i t ti t ditibehavior, treating aggregate conditions as uninfluenced by one’s own actions.

• If all agents were to agree on a jointly optimal• If all agents were to agree on a jointly optimal level of work effort, it would be higher.


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Coordination failureCoordination failure• Low level of activity equilibrium is sometimes called a

“coordination failure” equilibrium, because individuals are unable to work together to bring about the higher eqbm.

• Zero is extreme case: think of an economy in which people must choose to work on one of two production methods (individual or joint), but do so knowing choices of others . Could have low level equilibrium just on relatively unproductive process 2. [Example is rigged so that n is hi h i 2 b t th t b h ff t th l l fhigher in process 2, but that b has no effect on the level of n, so that c can be made much lower on 2 than on 1. Hence, with higher work and lower consumption, 2 is evidently worse than 1]evidently worse than 1].

11 1Process 1: [ ] 0 1

Process 2 : b<<ay a n Ny bn

α α α−= < <=


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2Process 2 : b<<ay bn=

A4 Corrective subsidyA4. Corrective subsidy

• A government which can impose lumpA government which can impose lump sum taxes can subsidize employment and

Eliminate the zero equilibrium (by paying– Eliminate the zero equilibrium (by paying people to work even if others do not)

– Stimulate the equilibrium to an efficient levelStimulate the equilibrium to an efficient level, via a subsidy such that (1+s)α=1.

Question to think about: what if government must raise revenue via labor taxation?


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B Dynamic ExampleB. Dynamic Example

• Study economy with a large number ofStudy economy with a large number of agents concerned with maximizing

( ) jtU u cβ

∞

= ∑0

and sometimes ( ) log( )j

t tu c c=

=


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“Output taxation”Output taxation• Private agents have budget constraint

1( (1 ) ) (1 ) ( )t t t t t tc k k f k Tδ τ++ − − = − +

where i ti– c is consumption

– k is capital– f(k) is output( ) s ou pu– T is lump-sum transfer– τ is tax rate


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Agents take as givenAgents take as given

• The sequence of tax ratesThe sequence of tax rates

• The sequence of transfers• The sequence of transfers

B t th i th t th d d• But they recognize that these may depend on the actions of others

e g : If others accumulate more k they will– e.g.: If others accumulate more k, they will pay higher taxes and my transfers will go up…


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p

Dynamic LagrangianDynamic Lagrangian

( ) ttL u cβ

∞

= ∑0

[ (1 ) ( )

t

tt t t tf k Tβ λ τ

=

∞

+ − +

∑

∑0

1

[ (1 ) ( )

( (1 ) )]

t t t tt

t t t

f k T

c k k

β λ τ

δ=

+

+ +

− − − −

∑1( ( ) )t t t+


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FOCsFOCs

0 ( ) λ

1 1 1 1

: 0 ( ): 0 [(1 ) ( ) (1 )]

t c t t

t t t t k t

c u ck f k

λλ βλ τ δ+ + + +

= −

= − + − + −

1: 0 (1 ) ( ) ( (1 ) )t t t t t t tf k T c k kλ τ δ+= − + − − − −


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Equilibrium conditions: wordsEquilibrium conditions: words

• Private agent efficiency conditionsPrivate agent efficiency conditions• Government budget constraint

C i t (k K)• Consistency (k=K) • Government rules for g, τ and T


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Equilibrium Conditions: EquationsEquilibrium Conditions: Equations: 0 ( )t c t tc u c λ= −

1 1 1 1: 0 [(1 ) ( ) (1 )]: 0 (1 ) ( ) ( (1 ) )

t t t t k tk f kf k T c k k

λ βλ τ δλ τ δ+ + + += − + − + −

= − + − − − −1: 0 (1 ) ( ) ( (1 ) )

( )

t t t t t t tf k T c k k

T f K

λ τ δ+= +

( )t t t t

t t

T f K gk K

τ= −

=

plus appropriate rule (A,B,C) for g and τ


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Equilibrium Conditions: Equations simplifiedEquations simplified

0 ( )u c λ= −

1 1 1

0 ( )0 [(1 ) ( ) (1 )]0 ( ) ( (1 ) )

c t t

t t t k t

u cf k

f k c k k g

λλ βλ τ δ

δ+ + += − + − + −

= 10 ( ) ( (1 ) )t t t t t

t t

f k c k k gk K

δ+= − − − − −

=


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What does governmentd i h ?do with revenue?

• Case A (Redistribution/Endogenous Transfers): ( g )Refunds lump sum to individuals Tt=τt yt

• Case B (Endogenous purchases): Spends on a• Case B (Endogenous purchases): Spends on a time-varying quantity of government purchases, depending on the exogenous tax rate and

d t bendogenous tax base: gt=τt yt

• Case C: (Endogenous Tax Rates) Finances a• Case C: (Endogenous Tax Rates) Finances a given amount of goods, allow tax rate to vary endogenously with the tax base: τt =g/yt


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Implications for Stationary States• Case A and B

1 [(1 ) ( ) (1 )]f kβ δ1 [(1 ) ( ) (1 )]

(1 ) ( ) 1/(1 )

kf k

f k r with r

β τ δ

τ δ β

= − + −

= + = +

Case C

(1 ) ( ) 1/(1 )kf k r with rτ δ β− = + = +

• Case C

1 [(1 ( )) ( ) (1 )] ( ) / ( )kk f k with k g f kβ τ δ τ= − + − =[( ( )) ( ) ( )] ( ) ( )

(1 ) ( )

kf g f

g f k r

β

δ= +


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(1 ) ( )( ) kf k r

f kδ− = +

CD exampleCD example

• Case A and B:Case A and B:

1( ) ( )f k k f k kα αα −⇒( ) ( )kf k k f k kα= ⇒ =

11(1 )(1 ) ( ) ( ) [ ]k ssf k r k

rαα ττ δ

δ−−

− = + ⇒ =+


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Cases A and BCases A and B

0 24Stationary capital stock in cases A and B

0.2

0.22

0.24r+δfk(k)

0.16

0.18

of re

turn

0.1

0.12

0.14

rate

0 0.2 0.4 0.6 0.8 1 1.20.06

0.08

f ti f it l


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fraction of ss capital

Tax Rate and Stationary StateTax Rate and Stationary State10 2.5

5

capi

tal

1.5

2

Out

put

0 0.2 0.4 0.6 0.80

tax rate(τ)0 0.2 0.4 0.6 0.8

0.5

1

tax rate(τ)

0 2

0.3

0.4

0.5

nal p

rodu

ct

0.4

0.6

0.8ve

nue

0 0.2 0.4 0.6 0.80

0.1

0.2

tax rate(τ)

mar

gin

0 0.2 0.4 0.6 0.80

0.2

tax rate(τ)

rev


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tax rate(τ) tax rate(τ)

Key pointsKey points

• Higher tax rate yieldsHigher tax rate yields– lower capital;

lower output;– lower output;– higher pre-tax marginal product

L ff• Laffer curve:– high tax rates can produce less revenue– revenue maximizing tax rate


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Stationary State in Case CStationary State in Case C

0 1Stationary capital stocks in case C

0 08

0.09

0.1e

of re

turn

r+δ(1-τ)fk(k) with τ(k)=g/f(k)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0.08


rate

0.7

0.8

0.9

rate

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.4

0.5

0.6tax

f ti f it l


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Key featuresKey features

• Two stationary points, both of whichTwo stationary points, both of which produce the required revenue g– One with high tax, low capitalg , p– One with low tax, high capital

• Tax rates are sharply different across two p ystationary points

• From earlier figure, know that revenue is g ,rising with taxes near low tax point and falling with taxes near high tax point


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Topic C2: Analyzing dynamicsTopic C2: Analyzing dynamics• Case A: Exogenous movements in tax rates g

highlighting substitution effects, because tax revenues rebated.

• Case B: Just like a productivity shift because revenues are “used up”: shift to output

il bl f i t ti davailable for private consumption and investment

( )( ) (1 ) ( )

t t t t

t t t t t t

c g i f kc i f k g f kτ

+ + =

⇒ + = − = −


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Analyzing DynamicsAnalyzing Dynamics

• To consider the effects of endogenous• To consider the effects of endogenous government policy on local dynamics, let’s use general policy rules that allow for rich g p ydependence of taxes and purchases on capital, then we can specialize to A,B, C or cover other cases

( )Kτ τ ( )( )

( ) ( ) ( )

t t

t t

Kg g KT K f K g K

τ τ

τ

=

=

= −


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( ) ( ) ( )t t t tT K f K g Kτ= −

Recalling equilibrium conditions( h k )(note no shocks)

0 ( )c t tu c λ= −

1 1 1

1

0 [(1 ( )) ( ) (1 )]0 ( ) ( (1 ) ) ( )

t t t k t

t t t t t

k f kf k c k k g kλ βλ τ δ

δ+ + +

+

= − + − + −

= − − − − −

t tk K=


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Linearization (near stationary point)Linearization (near stationary point)

0 *( )t tc cc

λ λσλ

− −= − −

1 10 ( ) ( ) ( )t t tk kk

λ λ λ λ ηλ λ

+ +− − −= − + −

10 ( ) ( ) ( )t t tk k c c k klk c k

φ +− − −= − −


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Dynamic systemDynamic system

λ λ λ λ− −⎡ ⎤ ⎡ ⎤1

1

1 1 00 1 /

t t

t tk k k k

λ λ λ λη λ λ

φ σ χ

+

+

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦10 1 /t tk k k kk k

φ σ χ+⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦


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CoefficientsCoefficients

Standard With endogenous fiscalStandard With endogenous fiscal policy

χ =fk+(1-δ)=1/β=(1+r)>1 χ=fk+(1-δ)- gk =(1+r) – gkχ k ( δ) /β ( ) χ k ( δ) gk ( ) gk

φ = c/k Sameφ

σ= -(c* ucc )/ uc Same

η =- βkfkk>0 η = -βk(1-τ)fkk+βkfkτk


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Characteristic polynomialCharacteristic polynomial

Write in con enient form to identif s m of• Write in convenient form to identify sum of roots [1+χ+ϕ ] and product (χ) of roots

21 20 [1 ] ( )( ) with /z z z zχ ϕ χ μ μ ϕ ησ φ= − + + + = − − =

• Write in convenient form for graphical analysisanalysis

( 1)( )z z zχ ϕ− − =


40


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Real root casesReal root cases

0 1 10 A: 2 stable B: 2 unstable

χ χϕ

< < ><

0 C: 1 of each D: 1 of each ϕϕ >


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Complex root casesComplex root cases

• Stability is dictated entirely by χ sinceStability is dictated entirely by χ since complex roots have common modulus equal to square root of χ equal to square root of χ

C l t bt i if di i i t i• Complex roots obtain if discriminant is negative, i.e., if [1+χ+ϕ]2 - 4* χ<0.


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What’s the point?What s the point?

• If we change values of χ and η in criticalIf we change values of χ and η in critical ways, then multiplicity obtains.

• Critical values are:• Critical values are:– Resources: χ<1 versus χ>1, which means

that a higher capital stock reduces ratherthat a higher capital stock reduces rather than increases future k, given the c level.Incentives: η <0 versus η >0 which means– Incentives: η <0 versus η >0, which means that return to investing rises with capital rather than fall. (η dictates sign of ϕ)


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(η g ϕ)

How does extra stable root change RE l i ?RE solutions?

• Recall that when we talked about the single gvariable case, we said that there could be multiple equilibria if the difference equation’s root was less than one in absolute value

t i ( 1) 0 1/E 1 root is ( 1) 0 or 1/

1( ) with 0

t t t ty aE y cx az z a

cy E y x Eξ ξ

+= + − = =

⇒ = − + =1 1 1 1( ) with 0t t t t t t ty E y x Ea a

ξ ξ+ + + +⇒ = + =


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Current setupCurrent setup

• Two variablesTwo variables. • One initial condition.

T t bl t• Two stable roots• Implication can shock λ just as could

shock y above, resetting initial conditions.


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Sample dynamics (shock to λ)f i IIfrom region II

2Consumption

0.6Government purchases

0

1

0

0.2

0.4

0 10 20 30 40-2

-1

0 10 20 30 40-0.4

-0.2

Tax Rate Capital

0

5Tax Rate

2

4Capital

0 10 20 30 40-10

-5

0 10 20 30 40-2

0


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• Note endogenous variation in tax ratesNote endogenous variation in tax rates due to fiscal rule

• Thought experiment on impact:• Thought experiment on impact: – if everybody else invests a lot, then tax rates

will fall making it efficient for me to investwill fall making it efficient for me to invest more


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• Thought experiment subsequently: howThought experiment subsequently: how will the attempts of others to increase or decrease on capital formation affect mydecrease on capital formation affect my incentives to invest.


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Application to model CApplication to model C

• Low activity ss is unstable in prior modelLow activity ss is unstable in prior model

Hi h ti it i t bl ( t t it l• High activity ss is stable (return to capital diminishes locally). Stability is in

ddl i t i l i lsaddlepoint sense, as in neoclassical growth model without distortions because

d t f t (l) i till 1/β (1 ) 1product of roots (l) is still 1/β = (1+r)>1


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Topic C. Recursive formulation f di d iof distorted economies

• Agents take as given the sequence of taxes and transfers above, now view as function of a state variable which isfunction of a state variable which is exogenous to them

( )( )K

T T Kτ τ=

=

' ( )K H K=


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Dynamic programDynamic program

• Bellman equationBellman equation

, '( , ) max { ( ) ( ', ')}c kv k K u c v k Kβ= +

subject to' (1 ( )) ( ) (1 ) ( )' ( )

k K f k k T K cK H K

τ δ= − + − + −= ( )

Decision rules: ( ) and ' ( )c c k K k h k K= =SGZ 2008 Macro Week 3, Day 1

Lecture 252

Decision rules: ( , ) and ( , )c c k K k h k K= =

EquilibriumEquilibrium

( ) ( ) ( ) ( )T K K f K g Kτ= −

( ) | ( )h k K H K=( , ) | ( )k Kh k K H K= =


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An efficient way to construct h(k K) H(K)h(k,K)=H(K)

• Take FOCs and use Envelope theoremTake FOCs and use Envelope theorem• Impose consistency conditions• Equilibrium policies C(K) H(K) satisfyEquilibrium policies C(K), H(K) satisfy

0 ( ( )) ( )cu C K K= − Λ0 ( ) ( ')[(1 ( ') ( ') (1 )]0 ( ) (1 ) ( ) ( )

c

kK K K f Kf K K C K g K

β τ δδ

= −Λ + Λ − + −

= − − − −' ( )K H K=


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Note restrictionNote restriction

• Made equilibrium just a function ofMade equilibrium just a function of “natural” state variable K

• Limit of finite horizon economies, each of hi h t k h f ti i thwhich takes such a function governing the

future as given


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Policy function viewpointPolicy function viewpoint

• Think of K’=H(K) as coming out of theThink of K H(K) as coming out of the problem above. Some old and new possibilities, suggested by this nonlinear p gg yperspective

• Old: Multiple steady states

• New: Multiple equilibria at point in time if “backward bending” policy function


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backward bending policy function

Multiple steady-states ( bili )(note stability)


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Multiple equilibria i i iat a point in time


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Later pictureLater picture

• Clearly rule out in standard concaveClearly rule out in standard concave (growth model problem) without distortions

• Similar pictures have arisen in recent work ti i i di ti ton optimizing, discretionary monetary

policy authority in New Keynesian models lik th t b di d t klike those to be discussed next week.


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sgz macro week 3, lecture 2 - boston...

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