the time consistency problem - theory and...

The Time Consistency Problem - Theory andApplications

Nils Adler and Jan Storger

Seminar on Dynamic Fiscal PolicyDr. Alexander Ludwig

November 30, 2006Universitat Mannheim

The Time Consistency Problem - Theory and Applications

Outline

1. Introduction1.1 The Dam Example

2. Theory2.1 A Stylized Model2.2 Proposed Remedies

3. Applications3.1 The Barro-Gordon Model 19833.2 Repeated Barro and Gordon with Tit for Tat3.3 Repeated Barro and Gordon with Grim Trigger

4. Summary

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1. Introduction

What is Time-Inconsistency?

I Optimal decision taken yesterday no longer optimal fromtodays point of view.

I If we could, we would choose differently today.

I Optimal decision is inconsistent.

Why it Matters...

I The Theory of Time-Inconsistency has a wide range ofapplications.

I Fiscal Policy.

I Monetary Policy.

See Part 2.

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1. Introduction

What is Time-Inconsistency?

I Optimal decision taken yesterday no longer optimal fromtodays point of view.

I If we could, we would choose differently today.

I Optimal decision is inconsistent.

Why it Matters...

I The Theory of Time-Inconsistency has a wide range ofapplications.

I Fiscal Policy.

I Monetary Policy. See Part 2.

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1. Introduction

Different Types of Time-Inconsistency

(i) Time-Inconsistency due to changes in preference over time.(Strotz (1956))

(ii) Time-Inconsistency of government plans when agents haverational expectations.(Lucas (1976), Kydland & Prescott (1977), Barro & Gordon(1983))

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1. Introduction

Different Types of Time-Inconsistency

(i) Time-Inconsistency due to changes in preference over time.(Strotz (1956))

(ii) Time-Inconsistency of government plans when agents haverational expectations.(Lucas (1976), Kydland & Prescott (1977), Barro & Gordon(1983))

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1. Introduction

1.1 The Dam Example

Intuitive Application: The Dam ExampleKydland & Prescott (1977)

I An potential housing area is prone to flooding.

I The government is benevolent.

I Flood protection or not?

Optimal Response

I No houses in area ⇒ No flood protection.

I Houses in area ⇒ Dams are built.

⇒ Optimal policy is inconsistent.

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1. Introduction

1.1 The Dam Example

Intuitive Application: The Dam ExampleKydland & Prescott (1977)

I An potential housing area is prone to flooding.

I The government is benevolent.

I Flood protection or not?

Optimal Response

I No houses in area ⇒ No flood protection.

I Houses in area ⇒ Dams are built.

⇒ Optimal policy is inconsistent.

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2. Theory

2.1 A Stylized Model

Consider the Following Stylized Model.Kydland & Prescott (1977)

Let

I {πt}Tt=1 be the policies,

I {x}Tt=1 the private agents’ decisions,

I S(xt , πt) the social welfare,

I and xt = X (x1, ..., xt−1, π1, ..., πt) ∀t = 1, ...,T the privateagents’ decision rule.

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2. Theory


Optimal vs. Time-Consistent Policy

Definition: Optimal Policy

A policy sequence {πt}Tt=1 is optimal if, for each time period t, πt

maximizes the social welfare subject to the decision rule of privatehouseholds xt = X (., .).

Definition: Time-Consistent Policy

A policy sequence {πt}Tt=1 is consistent if, for each time period t,

πt maximizes the social welfare, taking as given the history ofprivate decisions {xt}t−1

t=1 and having all future policy choices(πs |s > t) obey the same rule.

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2. Theory


Optimal vs. Time-Consistent Policy

Definition: Optimal Policy

A policy sequence {πt}Tt=1 is optimal if, for each time period t, πt

maximizes the social welfare subject to the decision rule of privatehouseholds xt = X (., .).

Definition: Time-Consistent Policy

A policy sequence {πt}Tt=1 is consistent if, for each time period t,

πt maximizes the social welfare, taking as given the history ofprivate decisions {xt}t−1

t=1 and having all future policy choices(πs |s > t) obey the same rule.

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2. Theory


A Two-Period Version of the Stylized Model

The government’s problem at t = 0

The government maximizes welfare subject to the decision rule ofthe household’s:

maxπ1,π2

S(x1, x2, π1, π2)

subject to

x1 = X1(π1, π2)

x2 = X2(x1, π1, π2)

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2. Theory


First-Order-Conditions for Optimal Policy

∂S(x1, x2, π1, π2)

∂π1=

∂S

∂π1+

∂S

∂x1

∂X1

∂π1+

∂S

∂x2

[∂X2

∂π1+

∂X2

∂x1

∂X1

∂π1

]!= 0

∂S(x1, x2, π1, π2)

∂π2=

∂S

∂π2+

∂S

∂x2

∂X2

∂π1+

∂X1

∂π2

[∂S

∂x1+

∂S

∂x2

∂X2

∂x1

]!= 0

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2. Theory


First-Order-Conditions for Consistent Policy

∂S(x1, x2, π1, π2)

∂π1=

∂S

∂π1+

∂S

∂x1

∂X1

∂π1+

∂S

∂x2

[∂X2

∂π1+

∂X2

∂x1

∂X1

∂π1

]!= 0

∂S(x1, x2, π1, π2)

∂π2=

∂S

∂π2+

∂S

∂x2

∂X2

∂π1

!= 0

I FOCs for the first period are identical.

I FOCs for the second period are different ⇒ Consistent policyignores past decisions of private agents.

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2. Theory


The Optimal Policy is Inconsistent

The Government’s Problem in Period t = 2

maxπ2

S(x2, π2) subject to

x2 = X2(x1, π1, π2)

which implies that

∂S(x1, x2, π1, π2)

∂π2=

∂S

∂π2+

∂S

∂x2

∂X2

∂π1+

∂X1

∂π2

[∂S

∂x1+

∂S

∂x2

∂X2

∂x1

]︸︷︷︸

=0

!= 0

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2. Theory


The Optimal Policy is Inconsistent

The Government’s Problem in Period t = 2

maxπ2

S(x2, π2) subject to

x2 = X2(x1, π1, π2)

which implies that

∂S

∂π2+

∂S

∂x2

∂X2

∂π2

!= 0

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2. Theory

2.2 Proposed Remedies

CommitmentKydland & Prescott (1977)

I Circumvents the Time-Inconsistency by excluding thepossibility to revise an optimal plan in a latter period.

I The government binds itself to a decision.

I Commitment technologies are: laws, institutionalarrangements etc..

I Kydland and Prescott (1977) advise: ”(...) rules rather thandiscretion (...)”.

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2. Theory

2.2 Proposed Remedies

CommitmentThe Dam Example Revisited

I The government would have to pass a law that forbids thebuilding of dams in areas prone to flooding in any futureperiod.

I Rational Expectations of private agents ⇒ Government willnever build any dams ⇒ No houses will be build.

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3. Applications

3.1 The Barro-Gordon Model 1983

Players’ Preferences

Government’s utility function is

UGovt = θb(πt − πe

t )−a

2π2

t (1)

Public has the following utility function

UPubt = −(πt − πe

t )2 (2)

In the following subsections we consider a one-stage prisonersdilemma with perfect information and the assumption of a ”wet”government, that is θ = 1, in which both players, government andpublic, move simultaneously.

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3. Applications


Discretionary Policy

The unconstrained optimization problem of government is

maxπt


t )−a

2π2

t (3)

FOC

πt =b

a(4)

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3. Applications


Rational Expectations

By solving the unconstrained optimization problem of public,

maxπe

t

UPubt = −(πt − πe

t )2 (5)

FOC:πt = πe

t (6)

, rational expectations can be assumed. Later we will see that thisleads to the inefficient but time-consistent outcome becausesystematic cheating cannot take place under this assumption.

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3. Applications


Rule Policy

Now government maximizes his utility given that the publicanticipates his choice of πt correctly:

maxπt


t )−a

2π2

t (7)

s.t.πt = πe

t (8)

FOC:π∗t = 0 (9)

Payoffs UGov∗t = UPub∗

t = 0 are pareto-optimal but

time-inconsistent because πt = ba is a profitable deviation for

government.

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3. Applications


Cheating

Assuming that public expects the optimal rule zero inflationπe

t = π∗t = 0, the maximization problem of government becomes:

maxπt


t )−a

2π2

t (10)

s.t.πe

t = π∗t = 0 (11)

FOC

πt =b

a(12)

The resulting payoffs UGovt = b2

2a > UPubt = −

(ba

)2are

sub-optimal but time-consistent and considered the first bestsolution from government’s perspective.

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3. Applications


Wet Government θ = 1

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3. Applications


Hard-Nosed Government θ = 0

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3. Applications

3.2 Repeated Barro and Gordon with Tit for Tat

Public plays Tit For Tat

Barro and Gordon consider a ”Tit For Tat” strategy played by thepublic:

πet = π∗t = 0 if πt−1 = πe

t−1

πet = πt = b

a if πt−1 6= πet−1

Public expects zero inflation if government met public’sexpectations in the preceding period. Otherwise public expects thediscretionary inflation rate. Since it takes public only one period tofully update their expectations, government gets punished for oneperiod only and its credibility is completely restored thereafter.

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3. Applications


Government’s Incentive Constraint

A mechanism to enforce cooperation, that is here the ideal ruleπ∗t = 0, has to satisfy the following incentive constraint:

q ·[UGov∗

t+1 − UGovt+1

]︸︷︷︸

Enforcement

≥ UGovt − UGov∗

t︸︷︷︸Temptation

(13)

q ≥ 1 (14)

, which cannot be true for plausible values of q satisfying0 < q < 1.

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3. Applications


Best Enforceable Rule (BER)

If not not the ideal rule, what rule can be enforced here? Let π besome positive inflation rule. Then solving for π we get[

1− q

1 + q

]b

a︸︷︷︸πBER

≤ π ≤ b

a︸︷︷︸π

(15)

, where πBER is the Best Enforceable Rule (BER), that is thelowest enforceable inflation rate using the above mechanism. ThisBER is in fact a weighted average of π∗ = 0 and π = ba. Theweights are determined by the discount factor q. For q = 1 theBER would be the ideal rule (πBER = π∗ = 0) and for q = 0 it

would be the discretionary inflation rate (πBER = π = ba ).

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3. Applications


Enforcement vs. Temptation - Graph

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3. Applications

3.3 Repeated Barro and Gordon with Grim Trigger

Public plays Grim Trigger

Public now plays the following punishment mechanism called grimtrigger:

πet = π∗t = 0 if πs = πe

2 ∀s < t

πet = πt = b

a otherwise

As we see the length of the punishment interval is now extendedfrom one period to eternal punishment.

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3. Applications


Government’s Present Value - Enforcement

The present value of government’s expected payoff if it alwaysplays the ideal rule π∗t = 0 is

PV GovEnforcement =

T∑t=0

qt · 0︸︷︷︸π∗=0

= 0 (16)

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3. Applications


Government’s Present Value - Temptation

Government’s present value when surprising once in the first periodand being punished forever afterwards is

PV GovTemptation =

1

2· b2

a︸︷︷︸πt=0>πe

t=0=0

+T∑

t=1

qt ·(−1

2· b2

a

)︸︷︷︸

πt>0=πet>0=

ba

(17)

Assuming T = ∞ this becomes

PV GovTemptation =

1

2· b2

a︸︷︷︸πt=0>πe

t=0=0

+q

1− q·(−1

2· b2

a

)︸︷︷︸

πt>0=πet>0=

ba

(18)

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3. Applications


Government’s Incentive Constraint

Government will comply with the announced zero inflation withoutan explicit agreement over the ideal rule whenever

PV GovEnforcement ≥ PV Gov

Temptation (19)

, which is

q ≥ 1

2(20)

For q ≥ 12 there is no time-inconsistent behaviour and the

policymaker always follows the zero inflation rule. We thereforehere get an optimal and time-consistent solution.

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4. Summary

Summary

I Optimal policy plans are subject to Time-Inconsistency ifpeople have rational expectations.

I Commitment can help to circumvent this problem.

I If people are too myopic, commitment to the optimal outcomeis not credible.

I Whether cooperation can be induced by a trigger mechanism,depends on the type of mechanism and specifically on thelength of the punishment interval.

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4. Summary

Thank You!

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