The Time Consistency Problem - Theory and Applications

The Time Consistency Problem - Theory and Applications Nils Adler and Jan Störger Seminar on Dynamic Fiscal Policy Dr. Alexander Ludwig November 30, 2006 Universität Mannheim

Outline 1. Introduction 1.1 The Dam Example 2. Theory 2.1 A Stylized Model 2.2 Proposed Remedies 3. Applications 3.1 The Barro-Gordon Model 1983 3.2 Repeated Barro and Gordon with Tit for Tat 3.3 Repeated Barro and Gordon with Grim Trigger 4. Summary 2 / 35

1. Introduction What is Time-Inconsistency? Optimal decision taken yesterday no longer optimal from todays point of view. If we could, we would choose differently today. Optimal decision is inconsistent. Why it Matters... The Theory of Time-Inconsistency has a wide range of applications. Fiscal Policy. Monetary Policy. 3 / 35

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 have rational expectations. (Lucas (1976), Kydland & Prescott (1977), Barro & Gordon (1983)) 5 / 35

1. Introduction 1.1 The Dam Example Intuitive Application: The Dam Example Kydland & Prescott (1977) An potential housing area is prone to flooding. The government is benevolent. Flood protection or not? Optimal Response No houses in area No flood protection. Houses in area Dams are built. Optimal policy is inconsistent. 8 / 35

2. Theory 2.1 A Stylized Model Consider the Following Stylized Model. Kydland & Prescott (1977) Let {π t } T t=1 be the policies, the private agents decisions, {x} T t=1 S(x t, π t ) the social welfare, and x t = X (x 1,..., x t 1, π 1,..., π t ) t = 1,..., T the private agents decision rule. 9 / 35

2. Theory 2.1 A Stylized Model Optimal vs. Time-Consistent Policy Definition: Optimal Policy A policy sequence {π t } T t=1 is optimal if, for each time period t, π t maximizes the social welfare subject to the decision rule of private households x t = X (.,.). Definition: Time-Consistent Policy A policy sequence {π t } T t=1 is consistent if, for each time period t, π t maximizes the social welfare, taking as given the history of private decisions {x t } t 1 t=1 and having all future policy choices (π s s > t) obey the same rule. 10 / 35

2. Theory 2.1 A Stylized Model A Two-Period Version of the Stylized Model The government s problem at t = 0 The government maximizes welfare subject to the decision rule of the household s: max π 1,π 2 S(x 1, x 2, π 1, π 2 ) subject to x 1 = X 1 (π 1, π 2 ) x 2 = X 2 (x 1, π 1, π 2 ) 12 / 35

2. Theory 2.1 A Stylized Model First-Order-Conditions for Optimal Policy S(x 1, x 2, π 1, π 2 ) π 1 S(x 1, x 2, π 1, π 2 ) π 2 = S + S X 1 + S [ X2 π 1 x 1 π 1 x 2 = S π 2 + S x 2 X 2 π 1 + X 1 π 2 + X ] 2 X 1 π 1 x 1 π 1 [ S + S ] X 2 x 1 x 2 x 1! = 0! = 0 13 / 35

2. Theory 2.1 A Stylized Model First-Order-Conditions for Consistent Policy S(x 1, x 2, π 1, π 2 ) π 1 S(x 1, x 2, π 1, π 2 ) π 2 = S + S X 1 + S [ X2 + X ] 2 X 1 π 1 x 1 π 1 x 2 π 1 x 1 π 1 = S π 2 + S x 2 X 2 π 1! = 0! = 0 FOCs for the first period are identical. FOCs for the second period are different Consistent policy ignores past decisions of private agents. 14 / 35

2. Theory 2.1 A Stylized Model The Optimal Policy is Inconsistent The Government s Problem in Period t = 2 which implies that S(x 1, x 2, π 1, π 2 ) π 2 max S(x 2, π 2 ) subject to π 2 x 2 = X 2 (x 1, π 1, π 2 ) = S + S X 2 + X 1 π 2 x 2 π 1 π 2 [ S + S ] X 2 x 1 x 2 x 1 }{{} =0! = 0 15 / 35

2. Theory 2.1 A Stylized Model The Optimal Policy is Inconsistent The Government s Problem in Period t = 2 which implies that max S(x 2, π 2 ) subject to π 2 x 2 = X 2 (x 1, π 1, π 2 ) S + S X 2! = 0 π 2 x 2 π 2 16 / 35

2. Theory 2.2 Proposed Remedies Commitment Kydland & Prescott (1977) Circumvents the Time-Inconsistency by excluding the possibility to revise an optimal plan in a latter period. The government binds itself to a decision. Commitment technologies are: laws, institutional arrangements etc.. Kydland and Prescott (1977) advise: (...) rules rather than discretion (...). 17 / 35

2. Theory 2.2 Proposed Remedies Commitment The Dam Example Revisited The government would have to pass a law that forbids the building of dams in areas prone to flooding in any future period. Rational Expectations of private agents Government will never build any dams No houses will be build. 18 / 35

3. Applications 3.1 The Barro-Gordon Model 1983 Players Preferences Government s utility function is Public has the following utility function U Gov t = θ b(π t π e t ) a 2 π2 t (1) U Pub t = (π t π e t ) 2 (2) In the following subsections we consider a one-stage prisoners dilemma with perfect information and the assumption of a wet government, that is θ = 1, in which both players, government and public, move simultaneously. 19 / 35

3. Applications 3.1 The Barro-Gordon Model 1983 Discretionary Policy The unconstrained optimization problem of government is FOC max U π t Gov = θ b(π t πt e ) a t 2 π2 t (3) ˆπ t = b a (4) 20 / 35

3. Applications 3.1 The Barro-Gordon Model 1983 Rational Expectations By solving the unconstrained optimization problem of public, FOC: max U Pub πt e t = (π t πt e ) 2 (5) π t = π e t (6), rational expectations can be assumed. Later we will see that this leads to the inefficient but time-consistent outcome because systematic cheating cannot take place under this assumption. 21 / 35

3. Applications 3.1 The Barro-Gordon Model 1983 Rule Policy Now government maximizes his utility given that the public anticipates his choice of π t correctly: s.t. FOC: max U π t Gov = θ b(π t πt e ) a t 2 π2 t (7) π t = π e t (8) π t = 0 (9) Payoffs Ut Gov = Ut Pub = 0 are pareto-optimal but time-inconsistent because π t = b a is a profitable deviation for government. 22 / 35

3. Applications 3.1 The Barro-Gordon Model 1983 Cheating Assuming that public expects the optimal rule zero inflation π e t = π t = 0, the maximization problem of government becomes: s.t. FOC max U π t Gov = θ b(π t πt e ) a t 2 π2 t (10) π e t = π t = 0 (11) π t = b a ( ) The resulting payoffs Ũt Gov = b 2 2 2a > ŨPub t = b a are sub-optimal but time-consistent and considered the first best solution from government s perspective. (12) 23 / 35

3. Applications 3.1 The Barro-Gordon Model 1983 Wet Government θ = 1 24 / 35

3. Applications 3.1 The Barro-Gordon Model 1983 Hard-Nosed Government θ = 0 25 / 35

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 the public: π e t = π t = 0 if π t 1 = π e t 1 π e t = ˆπ t = b a if π t 1 π e t 1 Public expects zero inflation if government met public s expectations in the preceding period. Otherwise public expects the discretionary inflation rate. Since it takes public only one period to fully update their expectations, government gets punished for one period only and its credibility is completely restored thereafter. 26 / 35

3. Applications 3.2 Repeated Barro and Gordon with Tit for Tat Government s Incentive Constraint A mechanism to enforce cooperation, that is here the ideal rule πt = 0, has to satisfy the following incentive constraint: [ ] q Ut+1 Gov Ût+1 Gov Ũt Gov Ut Gov (13) }{{}}{{} Temptation Enforcement q 1 (14), which cannot be true for plausible values of q satisfying 0 < q < 1. 27 / 35

3. Applications 3.2 Repeated Barro and Gordon with Tit for Tat Best Enforceable Rule (BER) If not not the ideal rule, what rule can be enforced here? Let π be some positive inflation rule. Then solving for π we get [ ] 1 q b 1 + q a }{{} π BER π b a }{{} ˆπ (15), where π BER is the Best Enforceable Rule (BER), that is the lowest enforceable inflation rate using the above mechanism. This BER is in fact a weighted average of π = 0 and ˆπ = ba. The weights are determined by the discount factor q. For q = 1 the BER would be the ideal rule (π BER = π = 0) and for q = 0 it would be the discretionary inflation rate (π BER = ˆπ = b a ). 28 / 35

3. Applications 3.2 Repeated Barro and Gordon with Tit for Tat Enforcement vs. Temptation - Graph 29 / 35

3. Applications 3.3 Repeated Barro and Gordon with Grim Trigger Public plays Grim Trigger Public now plays the following punishment mechanism called grim trigger: πt e = πt = 0 if π s = π2 e πt e = ˆπ t = b a otherwise s < t As we see the length of the punishment interval is now extended from one period to eternal punishment. 30 / 35

3. Applications 3.3 Repeated Barro and Gordon with Grim Trigger Government s Present Value - Enforcement The present value of government s expected payoff if it always plays the ideal rule π t = 0 is T Enforcement = q t 0 = 0 (16) PV Gov t=0 } {{ } π =0 31 / 35

3. Applications 3.3 Repeated Barro and Gordon with Grim Trigger Government s Present Value - Temptation Government s present value when surprising once in the first period and being punished forever afterwards is Temptation = 1 2 b 2 T ) + q t 12 b 2 ( }{{ a} a t=1 π t=0 >πt=0 e =0 }{{} PV Gov Assuming T = this becomes PV Gov ˆπ t>0 =π e t>0 = b a Temptation = 1 2 b 2 + q ) }{{ a} 1 q 12 b 2 ( a }{{} π t=0 >πt=0 e =0 ˆπ t>0 =π e t>0 = b a (17) (18) 32 / 35

3. Applications 3.3 Repeated Barro and Gordon with Grim Trigger Government s Incentive Constraint Government will comply with the announced zero inflation without an explicit agreement over the ideal rule whenever, which is PVEnforcement Gov Gov PVTemptation (19) q 1 2 (20) For q 1 2 there is no time-inconsistent behaviour and the policymaker always follows the zero inflation rule. We therefore here get an optimal and time-consistent solution. 33 / 35

4. Summary Summary Optimal policy plans are subject to Time-Inconsistency if people have rational expectations. Commitment can help to circumvent this problem. If people are too myopic, commitment to the optimal outcome is not credible. Whether cooperation can be induced by a trigger mechanism, depends on the type of mechanism and specifically on the length of the punishment interval. 34 / 35

4. Summary Thank You! 35 / 35