Reconsideration-Proofness: A Refinement for Infinite Horizon Time Inconsistency

GAMES AND ECONOMIC BEHAVIOR ARTICLE NO.

15, 33–54 (1996)

0058

Reconsideration-Proofness: A Refinement for Infinite Horizon Time Inconsistency Narayana R. Kocherlakota∗ The Federal Reserve Bank of Minneapolis, Minneapolis, Minnesota 55480 Received December 22, 1992

This paper examines a game in which an infinitely lived decision maker with inconsistent preferences makes choices over time. There are typically many subgame perfect equilibria to this game; I describe a refinement of subgame perfection that I term reconsideration-proofness. There is always a reconsideration-proof equilibrium and all reconsideration-proof equilibrium paths provide the same utility to the decision maker. I show that reconsideration-proofness is relevant for macroeconomic policy games by proving an equivalence result between sustainable equilibria in policy games (as defined by Chari and Kehoe, 1990) and subgame perfect equilibria in intertemporally inconsistent choice problems. Journal of Economic Literature Classification Numbers: C73, D91, E61. © 1996 Academic Press, Inc.

Suppose an individual can choose a sequence of current and future actions today. If tomorrow he wishes to deviate from his plan for the future without having received any new information, his choice problem is said to be temporally inconsistent (Strotz, 1956). This paper poses a positive question: If the individual cannot commit to a sequence of actions, what will he end up choosing? It is well known how to solve such a choice problem, consistent or not, with a finite horizon: simply begin with the last period and solve backwards. More technically, one treats the decision maker at different points in time as being different people, models these different people as playing a noncooperative game, and looks for a subgame perfect equilibrium. If the objective function is strictly quasiconcave, subgame perfection picks out a unique solution to the temporally inconsistent optimal control problem. ∗ The views expressed in this paper do not represent the views of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. I thank Geir Asheim, Andreas Blume, Yong-Gwan Kim, Chuck Whiteman, the participants of a seminar at the Federal Reserve Bank of Minneapolis, the referees, and especially Barbara McCutcheon for their comments. They are not responsible for any remaining errors.

33 0899-8256/96 $18.00 Copyright © 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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The purpose of this paper is to provide solutions for all infinite horizon stationary and history-independent choice problems under the assumption that the decision maker cannot commit today to future choices. Subgame perfection retains some of its appealing properties as a solution approach in stationary infinite horizon optimal control problems. For example, if the objective function is quasiconcave, there always exists a subgame perfect equilibrium. Generally, though, subgame perfection is troublesome because it does not select a unique value for the decision maker’s choice problem. In fact, I provide an example of a temporally consistent optimal control problem in which subgame perfection chooses all elements of the choice set as possible solutions. It is therefore necessary to refine subgame perfection in order to select a plausible solution for infinite horizon problems. I look at what I term reconsideration-proof equilibria. I think of the various subgame perfect equilibria as being descriptions of self-enforcing plans. If the individual can choose among such plans at time zero, he will pick the plan or subgame perfect equilibrium with the highest value. Following Farrell and Maskin’s (1989) approach to renegotiation in repeated games, I instead allow the individual to “reconsider” his plan at every node. The ability to reconsider imposes strong requirements on what subgame perfect equilibria are plausible. The basic logic of perfection dictates that every continuation equilibrium path should be “credible.” If the individual can reconsider at every node and thus choose among these credible paths, then no continuation equilibrium path should be dominated by some credible one. These two arguments together imply that we should restrict attention to equilibria which provide the same value to the individual at every node in the plan. I term such equilibria symmetric. If we view all continuation equilibrium paths to symmetric equilibria as being “credible,” then a reconsideration-proof equilibrium should provide higher value than any other symmetric equilibrium. I prove that a reconsideration-proof equilibrium always exists for stationary and history-independent optimal control problems. In general, many sequences of choices are reconsideration-proof equilibrium paths. However, by construction, all of these reconsideration-proof equilibrium paths have the same value. Thus, my solution approach associates a unique value with all stationary and history-independent optimal control problems. Many economists believe that temporal inconsistency is not widespread among individual decision makers. In the Appendix, I prove a proposition that broadens the scope of my results to include macroeconomic policy games in which a single decision maker plays against a continuum of agents. In these contexts, Chari and Kehoe (1990) and Stokey (1989, 1991) characterize the set of what the former call sustainable equilibria and show that the set of equilibrium outcomes may be quite large. The Appendix proves that for each policy game in a large set (including the ones studied by Chari and Kehoe, 1990, Stokey, 1989, and Whiteman, 1986), there exists an individual intertemporal choice problem

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such that the set of sustainable equilibrium strategies for the government in the policy game is equivalent to the set of subgame perfect equilibrium strategies for an individual facing the choice problem. Hence, reconsideration-proofness (or indeed any refinement developed for intertemporal choice problems) can also be applied to macroeconomic policy games. The last section of the paper provides an example to demonstrate that reconsideration-proofness does pick out reasonable equilibria in the context of a simple policy game. This paper is related to others in the literature. As indicated above, the reconsideration-proof refinement in this paper is analogous to the strong renegotiationproof refinement developed by Farrell and Maskin (1989) for repeated games (which is a further refinement of Farrell and Maskin’s notion of weak renegotiationproofness and of Bernheim and Ray’s (1989) concept of internal consistency). Asheim (1991) discusses an alternative refinement which he terms “revisionproofness” which is based upon the theory of social situations of Greenberg (1990). Asheim also briefly mentions yet another refinement which is based upon Pearce’s (1987) notion of renegotiation-proofness. The rest of the paper is organized as follows. I first lay out the decision problem and discuss the subgame perfection approach to solving it. I demonstrate the existence of a subgame perfect equilibrium but then show that there are in general multiple values associated with the set of subgame perfect equilibria. I define the reconsideration-proof refinement and prove the existence of a reconsiderationproof equilibrium; I show that all reconsideration-proof equilibrium paths give the decision maker the same utility at every point in time. I discuss the relationship between revision-proofness and reconsideration-proofness. Finally, I discuss how reconsideration-proofness can be applied to macroeconomic policy games; in the Appendix, I demonstrate the general equivalence between the subgame perfect equilibria of individual intertemporal choice problems and the sustainable equilibria of infinite horizon macroeconomic policy games.

1. THE DECISION PROBLEM An infinitely lived individual can choose an element from a choice set X in every period of life. His objective function over current and future choices is affected by a state variable z which is an element of a state space Z . There are two subperiods in each period. In the first subperiod, the state variable assumes its current value according to the law of motion z t = ξ(xt , xt+1 , . . .), where xt is the period t choice of the individual and ξ maps X ∞ into Z . At the time z t assumes its value, the arguments of ξ have yet to be determined. Hence, it is best to think of z t as being a function of the individual’s expectations of his future choices.

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In the second subperiod, the individual makes his choice of xt . He has an objective function over Z × X ∞ U (z t , xt , xt+1 , xt+2 , . . .), where U maps Z × X ∞ into <. Critically, the value of z t has already been determined in the first subperiod; its value is not affected by the choice of the individual in period t. In this way, the individual’s utility at time t is determined not only by his current and future choices, but also by his period (t − 1) expectations of those choices (because those expectations determine z t ). In an individual decision maker’s context, this dependence on expectations as well as actual decisions serves to capture some elements of regret. Later, when I discuss macroeconomic policy games, the variable z t will represent capital accumulated before the second subperiod of period t—it will therefore depend on investors’ expectations of government actions in period t. I impose the following loose structure upon the problem. (S1) X is a convex compact subset of a locally convex topological linear space with topology τx . Z is compact with respect to a topology τz . (S2) For all z in Z , U (z, ·) is quasiconcave over X ∞ . (S3) U is continuous over Z × X ∞ with respect to the product topology τz × τx∞ . ξ is continuous over X ∞ with respect to the (domain) topology τx∞ and the (range) topology τz . These assumptions are maintained throughout the paper. There is a natural way to define temporal consistency in this setting. Suppose that instead of choosing sequentially, the individual were able to commit to a full sequence of actions before the beginning of period one. If he has the period zero objective function, he would then try to solve the optimization problem Max

a∈X ∞

U (ξ(a), a)

(P)

(The compactness of X and the continuity of U and ξ guarantee that there exists a solution to (P)). In general, we think of the individual’s decision problem as being temporally consistent if he does not want to deviate from a precommitment optimum in future periods. In this setting, this intuition translates into the following requirements. DEFINITION 1. Suppose (x0 , x1 , x2 , x3 , . . .) is a solution to (P). The decision problem of the individual is temporally consistent if:

(i) xt = x ∗ for all t (ii) x ∗ = argmaxx∈X U (ξ(x ∗ , x ∗ , . . .), x, x ∗ , x ∗ , x ∗ , . . .) for all solutions to (P).

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The first condition guarantees that the individual will not want to deviate from the solution to (P) at the beginning of any period. The second condition guarantees that the individual will not want to deviate from the solution to (P) in between the two subperiods of any period. It is simple to construct examples of temporally inconsistent decision problems. P s EXAMPLE 1. Let U (xt , xt+1 , . . .) = ∞ s=0 β (x t+s − x t+s+1 ) and X = [0, 1]. (The state space Z is degenerate, that is, a singleton.) This decision problem is temporally inconsistent. If the individual can precommit in period zero, then he will choose (1, 0, 0, 0, . . .). However, he would like to deviate from this choice in period one (because (0, 0, 0, . . .) is not optimal from the perspective of period one). Hence, his choice problem is dynamically inconsistent. There are really two sources of dynamic inconsistency in the general choice problem stated above. Consider an individual before the beginning of period zero who can commit to a sequence of choices. When choosing x1 , he thinks about its effect on the current state of the world, z 0 . Once he reaches the beginning of period one (or indeed the end of the first subperiod of period zero), this effect no longer means anything to him. Similarly, before period zero, he takes into account the direct effect of x1 on his utility in period zero. This effect is also unimportant to him once he reaches period one.

2. SUBGAME PERFECTION The solution to the above choice problem is clear if the individual can commit to a particular set of choices at time zero; he merely solves (P) and commits himself to following these choices. Supopse though that the individual cannot commit to a particular set of choices at time zero, but instead can choose freely at each point in time. What will he choose? The traditional (see for example Epstein, 1989, or Strotz, 1956) way to answer this question is to treat the individual at different time periods as different people involved in a noncooperative game. Accordingly, I will use the term agent t to refer to the individual at time t. He chooses an element of X knowing the choices of the previous t agents (actually himself at the different points in time). A history at time t is a sequence of choices (x 0 , x1 , x2 . . . xt−1 ); the set of histories at time t is hence X t . A strategy for agent t is a mapping from the set of histories X t to X ; agents take as given the strategies of the other agents. At this point, one must decide which equilibrium concept to use to solve this game. Traditionally, the concept of subgame perfection has been used (again see Epstein, 1989).

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DEFINITION 2. A subgame perfect equilibrium is a specification of a strategy for each agent such that agent t’s choice after any history of play in X t is optimal given all other agents’ strategies.

A collection of equilibrium strategies for the infinite number of agents gives rise to an equilibrium path in the natural way; the collection of strategies also gives rise to a continuation equilibrium path after any history (x0 , x1 , . . . xt−1 ). If x 0 , x1 , x2 , . . . is a subgame perfect equilibrium path, it has value U (ξ((xt )∞ t=0 ), x 0 , x 1 , x 2 , . . .); the value of a continuation equilibrium path is defined similarly. Recall from the first section that the continuity of U and ξ combined with the compactness of X is sufficient to guarantee the existence of a solution to the maximization problem Pt . The further assumptions of the quasiconcavity of U in its second argument and the convexity of X guarantee the existence of a subgame perfect equilibrium. PROPOSITION 1. There exists a subgame perfect equilibrium.

Proof. Define ψ(y) = argmaxx∈X U (ξ(y, y, y, . . .), x, y, y, . . .). From the Theorem of the Maximum, ψ is an upper semicontinuous correspondence. Because U is quasiconcave in its second argument, ψ(y) is convex for all y in X . It follows from Fan’s (1952) extension of Kakutani’s fixed point theorem that there exists x ∗ in X such that x ∗ ∈ ψ(x ∗ ). The strategies xt (h t ) = x ∗ for any history h t form a subgame perfect equilibrium. As long as U is quasiconcave in its second argument, there exists a subgame perfect equilibrium. Indeed, the proposition shows that (S1)–(S3) imply the existence of a special type of subgame perfect equilibrium: a recursive or stationary subgame perfect equilibrium. DEFINITION 3. A stationary subgame perfect equilibrium is a subgame perfect equilibrium in which every agent’s strategy is independent of the history. COROLLARY 1. There exists a stationary subgame perfect equilibrium.

Proof. See the proof of Proposition 1. Thus, there is always at least one subgame perfect equilibrium to a given choice problem. Unfortunately, there will in general be many more with different values. In fact, this nonuniqueness problem with subgame perfection can be present even if the individual’s choice problem is temporally consistent.1 1

Asilis et al. (1990) present a similar example.

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P∞ t 1/2 1/2 EXAMPLE 2. Let U (x 0 , x1 , . . .) = t=0 β x t x t+1 and X = [0, 1] (again the state space is degenerate). This decision problem is temporally consistent. The solution to (P) is (1, 1, 1, . . .) and the individual does not want to deviate from this solution in future periods. Yet any path in X ∞ is a subgame perfect equilibrium path. To see this, let x˜0 , x˜1 , x˜2 , . . . be a path in X ∞ . The strategies xt (h t ) = x˜t =0

if h t = (x˜0 , x˜1 , . . . , x˜t−1 ) if h t 6= (x˜0 , x˜1 , . . . , x˜t−1 )

form a subgame perfect equilibrium.2 This example is troubling. It is intuitively obvious what the solution to the choice problem should be because the individual’s preferences do not change over time. Subgame perfection does admit this obvious solution. On the other hand, it also identifies any element of X ∞ as a potential solution. Moreover, these solutions yield any value from 0 to 1/(1 − β). This example also points out this same difficulty plagues stationarity as a potential refinement. The subgame perfect equilibria xt = 1 and xt = 0 are both stationary; they have different values. Thus, while it is a powerful solution device in finite horizon settings, subgame perfection loses much of its punch when the horizon is infinite. This observation is familiar to any student of the repeated game literature. In the next section, I modify a refinement from that literature (Farrell and Maskin, 1989; Bernheim and Ray, 1989) to eliminate the “unreasonable” subgame perfect equilibria.

3. RECONSIDERATION-PROOFNESS Which of the many subgame perfect equilibria in an intertemporal choice problem will actually be played? In the repeated game literature, especially in its applications to the study of bilateral oligopoly, it is commonly assumed that the two players can meet before the beginning of play and pick a nonbinding but self-enforcing contract to guide their play. This contract shapes a given player’s beliefs about the other player’s response to his actions. Under this assumption, and abstracting from bargaining problems, they end up playing an equilibrium path which is Pareto undominated by any other. This same logic can be employed in the case of an infinite horizon single player decision problem. Before period zero, an individual or institution chooses 2 As Andreas Blume has indicated to me, the multiplicity of subgame perfect equilibria disappears in this example if one models the choice problem as a single player game as opposed to an infinite agent game. This is because the choice problem is consistent. However, if the choice problem is temporally inconsistent, it is no longer sensible to model it as a single player game. My goal in this paper is to develop a solution method which is applicable to both consistent and inconsistent choice problems.

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a nonbinding but self-enforcing plan (that is, a subgame perfect equilibrium) that will guide his choices in the future. Again, the role of this plan is to dictate the individual’s beliefs about how he will respond in the future to his choices today. Given that he can choose any such plan, he will pick the one that maximizes his period zero utility. The problem with this approach is clear in the case of bilateral oligopoly. Suppose one of the firms cheats on a collusive arrangement. The arrangement may then require that two firms enter a “punishment” phase in which both firms receive low payoffs. It may well seem to the two firms at this point that there are gains to getting together and choosing a new nonbinding but self-enforcing contract. Of course, if they can do so, then the threats that supported the original Pareto efficient arrangement are no longer credible. A similar argument can be made for infinite horizon decision problems. The subgame perfect equilibrium that maximizes the individual’s utility in period zero may involve continuation equilibrium paths (punishments) that have very low value. Once the individual starts playing such a path, he may think about choosing another, more rewarding, self-enforcing plan. For example, consider an individual with objective function U (xt , xt+1 , xt+2 , . . .) =

∞ X

β s (xt+s − 2xt+s+1 )

s=0

and constraint set X = [0, 1] (0 < β < 1). We can think of xt as measuring the number of gallons of alcohol consumed in evening t. The individual likes to drink tonight, but does not like the idea of his drinking in the future. Hence, he draws up a plan that says, “I’ll drink one gallon of alcohol tonight. But if I ever drink again, then I’ll drink one gallon of alcohol every night thereafter.” This plan (which is a description of a subgame perfect equilibrium) supports the optimal pattern of drinking (1, 0, 0, 0, . . .). However, suppose the individual cheats and drinks a mug of beer one evening. His plan now says: drink one gallon of alcohol every night thereafter. While this is a subgame perfect equilibrium, it is not a particularly desirable one. Intuitively, it seems that this individual will say at some point, “Look, I’m killing myself. Let me try to choose a better way to spend my life.” If the individual does say this, then the original equilibrium (that supports the optimal drinking pattern) does not really make sense. There is another and perhaps more compelling way to summarize the difficulties with subgame perfection as a solution concept. A key feature of subgame perfection is that a player’s beliefs about the other players’ future responses to current and future actions are fixed at the beginning of time and can never be altered. Implicitly, subgame perfection regards communication and contact among the players as being extremely costly, and so coordinated deviations from a proposed equilibrium are impossible.

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This restriction may seem unduly strong in bilateral oligopoly models. It is utterly ridiculous in an infinite horizon individual choice problem, where the different “players” are merely the individual at different points in time. In such a situation, it is easily conceivable and almost inevitable that a player in period t can and will alter the beliefs of players (future incarnations of himself!) in succeeding periods about the appropriate way to respond to preceding choices. Thus, subgame perfection treats the “multiple player” fiction too seriously: it not only eliminates the ability of an individual to bind his future self (which seems reasonable in many circumstances), but also eliminates the ability of an individual in period t to coordinate with this future self (which seems implausible in all circumstances).3 I address this defect in subgame perfection in the same way as Farrell and Maskin (1989). They point out that all continuation equilibrium paths ought to be “credible.” Further, they argue that if the players in a repeated game have the ability to renegotiate, they can pick any credible equilibrium path. Hence, no continuation equilibrium path should be Pareto dominated by some credible one. There is an exact analogy to these notions in the context of an individual choice problem. Again, all continuation equilibrium paths should be “credible.” Suppose that at any node, the individual has the ability to reconsider his plan, that is, change his current and future beliefs (or, put another way, the beliefs of his future selves) about the appropriate response to previous actions. This ability to reconsider means that at any node, no continuation equilibrium path should be dominated by any other “credible” path. As long as the same paths are “credible” at every node, we are led to focus on symmetric subgame perfect equilibria. DEFINITION 4. A subgame perfect equilibrium is symmetric if after every history (including off-equilibrium histories), the continuation equilibrium paths have the same value.

Any stationary subgame perfect equilibrium is symmetric (although as we shall see, the converse is not true). Hence, if (S1)–(S3) are true, there always exists a symmetric subgame perfect equilibrium. Symmetry is analogous to Farrell and Maskin’s definition of weak renegotiation-proofness and Bernheim and Ray’s (1989) definition of internal consistency. An element of X ∞ can be a symmetric subgame perfect equilibrium path only if it provides the same utility to the decision maker at every point in time. ∞ ∞ | U (ξ((xs )∞ Let D(V ) = {(xs )∞ s=t ), (x s )s=t ) = V for all t} be the s=0 ∈ X set of paths that provide utility V to the decision maker at every point in time. Similarly, let Z (V ) = ξ(D(V )). The following proposition uses these sets to completely characterize the range of values associated with symmetric subgame perfect equlibria. 3

I thank a referee for making this point clear to me.

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PROPOSITION 3. There exists a symmetric subgame perfect equilibrium with value V if and only if there exists a subset D ∗ ⊆ D(V ) and a subset Z ∗ ⊆ Z (V ) such that:

(i) Z ∗ = ξ(D ∗ ). (ii) For all x in X and all z in Z ∗ , there exists d in D ∗ such that U (z, x, d) ≤ V. Proof. Suppose there exists a symmetric equilibrium with value V . Define D ∗ to be the set of all possible continuation equilibrium paths in this equilibrium and Z ∗ to be ξ(D ∗ ). Is (ii) satisfied? Suppose not. Then there exists (x, z) in X × Z ∗ such that U (z, x, d) > V for all d in D ∗ . Note that if z is in Z ∗ , there exists some history h t with continuation equilibrium path (xt+1 , xt+2 , . . .) such that z = ξ(x t+1 , xt+2 , . . .). At this history, the value of the game must be larger than V because U (z, x, d) > V for all possible continuation equilibrium paths. But this is a contradiction because the value of the game at every node is V . Now suppose there exist Z ∗ and D ∗ such that both (i) and (ii) are satisfied. Define S: Z ∗ × X → D ∗ to be a function such that U (z, x, S(z, x)) ≤ V for all x in X . I claim any element of D ∗ can be supported as a subgame perfect equilibrium path as follows. Let x + = (x0 , x1 , . . .) be any element of D ∗ . Define the equilibrium strategies to support this path as follows. If the agent at time t deviates from x + by choosing x instead of x t , the successive agents are supposed to punish him by following S(z t , x). If one of them (say agent s) deviates and chooses y instead of following the prescribed punishment path, the successive agents begin punishing him by following S(z s , y). This process continues forever. It is easy to check that this collection of strategies forms a symmetric subgame perfect equilibrium. Suppose at time t no previous agent has deviated. If agent t does not choose xt , then successive agents will choose to follow S(z t , x). Agent t gets value U (z t , x, S(z t , x)), which by construction means that he is at least as well off choosing xt . On the other hand, suppose agent s was the last to deviate from the described strategy. Agent t is supposed to play the (t − s)th component of S(z s , x). If he does, he gets value V . If he does not and plays y instead, he gets value U (z t , y, S(z t , y)), which is no larger than V . Proposition 3 makes two contributions. First, it provides a check as to whether there exists a symmetric equilibrium with value V . Second, its proof shows how to construct a symmetric equilibrium with value V if one does exist. P t 1/2 1/2 EXAMPLE 3. Let U (x0 , x1 , x2 , . . .) = ∞ t=0 β x t x t+1 and X = [0, 1]; Z is degenerate. Suppose I want to construct a symmetric subgame perfect solution with value a/(1 − β), 0 < a < 1. Define S(x) = 1 = a 2 /x

if x ≤ a 2 if x ≥ a 2 .

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The collection of strategies x t (x0 , x1 , . . . , xt−1 ) = S(xt−1 ) is a symmetric subgame perfect equilibrium with value a/(1 − β). The concept of symmetry is closely related to an equilibrium notion originally introduced by Dekel and Farrell (1990). They consider a repeated game in which a single long-lived “leader” faces a sequence of short-lived “followers.” They term any subgame perfect equilibrium “weakly reselection-proof” if no two of its continuation equilibria are strictly ranked according to the long-lived leader’s payoffs. However, while symmetry and weak reselection-proofness are certainly analogous, they have profoundly different implications. Example 3 demonstrates that there may be a continuum of symmetric equilibria. In contrast, Dekel and Farrell (1990) prove that in their context, only repetitions of equilibria to the one-shot game are weakly reselection-proof. There is a simple intuition underlying this difference in results. In Dekel and Farrell (1990), the leader’s intraperiod utility (profit) is affected only by current decisions. Hence, the leader can only be punished for a current action by future choices if the leader’s future utility is lowered by those future actions; this possibility is ruled out by reselection-proofness. In contrast, in the general decision problem studied in this paper, it is possible for future choices to affect current intraperiod utility. As a consequence, the utility of an agent in period t can be lowered by the choices of agents in future periods without their utility being lowered. Thus, there are typically many symmetric equilibria precisely because the decision problem is temporally inconsistent: future agents may be indifferent between two sequences of actions which the current agent can unambiguously rank. Any continuation equilibrium path in any symmetric subgame perfect equilibrium is “credible.” At any node, an individual with the ability to reconsider will choose the best possible equilibrium path out of the set of credible ones. This argument leads us to focus on the following equilibria. DEFINITION 5. A subgame perfect equilibrium is reconsideration-proof if it is symmetric and there exists no other symmetric subgame perfect equilibrium which provides higher value at every node.

Reconsideration-proofness is analogous to “strong renegotiation-proofness” in the sense of Farrell and Maskin (1989). As we shall see, there may be many reconsideration-proof equilibrium paths. However, by construction, reconsiderationproofness provides a unique value for the individual’s choice problem. Given Proposition 3, it is conceptually straightforward to find a reconsiderationproof equilibrium. Find the set VAL of all possible values of symmetric equilibria; Corollary 1 guarantees that it is nonempty. Look for its largest element. Construct a symmetric equilibrium with that value using the proof of Proposition 3. The following proposition guarantees that this method always works. PROPOSITION 4. There exists a reconsideration-proof equilibrium.

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Proof. Let V ∗ denote the supremum of VAL. Suppose {Vn }∞ 1 is a sequence of elements of VAL that converge to V ∗ . Let Z n∗ and Dn∗ be subsets of Z (Vn ) and D(Vn ), respectively, that satisfy conditions (i) and (ii) of Proposition 3. Define ∗ ∗ so that d is in D∞ if and only if it is a limit point of a collection of points D∞ ∞ {dn }n=1 such that dn is in Dn∗ for all n. The Bolzano–Weierstrass theorem implies ∗ ∗ ∗ is nonempty. Define Z ∞ = ξ(D∞ ). that D∞ ∗ Because U and ξ are continuous, D∞ is a subset of D(V ∗ ). It follows that ∗ ∗ ∗ is a subset of Z (V ∗ ). Thus, it is only left to prove that D∞ and Z ∞ satisfy Z∞ condition (ii) of Proposition 3. ∗ × X . There exists some element d in Let (z, x) be an arbitrary element of Z ∞ ∗ D∞ such that z = ξ(d). Because the element d is a limit point of some collection ∗ of points {dn }∞ n=1 such that dn is in Dn , the element z is a limit point of some ∞ collection of points {z n }n=1 such that z n is in Z n . From Proposition 3, we know that for all n, there exists dn0 in Dn such that U (z n , x, dn0 ) ≤ V. ∞ Let {z n k }∞ k=1 be a subsequence of {z n }n=1 that converges to the limit point z. Now think about the corresponding subsequence {dn0 k }∞ k=1 . This subsequence is not guaranteed to converge. However, the Bolzano–Weierstrass theorem implies that there exists some subsequence of the subsequence {dn0 k }∞ k=1 that converges , we know that d 0 is to an element d 0 of X ∞ . Since d 0 is a limit point of {dn0 }∞ n=1 ∗ in D∞ . Because U and ξ are continuous, it follows that U (z, x, d 0 ) ≤ V . This logic implies that the two conditions of Proposition 3 are satisfied for V ∗ . Hence, there exists a symmetric subgame perfect equilibrium with value V ∗.

Thus, there is always a reconsideration-proof equilibrium.4 Example 2 demonstrated that subgame perfection did not “correctly” solve consistent decision problems in general. The following proposition shows that reconsideration-proofness has an attractive feature that subgame perfection does not possess: it uniquely picks out the “correct” value for consistent decision problems. PROPOSITION 5. If the individual’s choice problem is temporally consistent, any solution to (P) is a reconsideration-proof equilibrium path.

Proof. If the choice problem is dynamically consistent, any solution to (P) must be constant over time: (x ∗ , x ∗ , x ∗ , . . .). Define xt (h t ) = x ∗ . From the 4 In contrast to strong renegotiation-proofness, it is possible to prove the existence of a reconsideration-proof equilibrium in very general contexts. This is because there is only one player at every history. Hence, Pareto dominance is a one-dimensional, not multidimensional, concept. Dekel and Farrell (1990) make use of a similar property when they assert that “reselection-proof” equilibria exist in a wide class of games.

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second condition in the definition of temporal consistency, this collection of strategies is a stationary subgame perfect equilibrium; hence, it is a symmetric equilibrium. It is reconsideration-proof because there is no symmetric subgame perfect equilibrium with greater value (or its equilibrium path would be the solution to (P)). In the context of Proposition 5, reconsideration-proofness merely serves to pick out the best (highest value) stationary subgame perfect equilibria. More generally, the reconsideration-proof equilibrium will not be stationary and its value will lie between that of the solution to (P) and the best stationary subgame perfect equilibrium. The following example makes this point explicit. P t EXAMPLE 4. Let U (x0 , x1 , x2 , . . .) = ∞ t=0 β (x t − 2x t+1 ) and X = [0, 1]; Z is degenerate. There is a unqiue stationary subgame perfect solution described by xt (x0 , x1 , . . . , xt−1 ) = 1; it has value −1/(1 − β). The solution to (P), on the other hand, is (1, 0, 0, . . .); it has value 1. (It can be shown that the solution to (P) is the equilibrium path of a subgame perfect equilibrium.) What is a reconsideration-proof equilibrium? If (x0 , x1 , . . .) is in D(V ), xt − 2x t+1 is constant for all t. It is straightforward to see that D(V ) is empty for all V > 0. Furthermore, there exists a symmetric subgame perfect equilibrium with value zero because for all x in X , there exists (x/2, x/4, x/8 . . .) in X ∞ such that U (x, x/2, . . .) = 0. This symmetric subgame perfect equilibrium is reconsideration-proof. In the stationary subgame perfect equilibrium, the individual at time t realizes he cannot control what the individual next period is going to do. Hence, he does the best he can: he sets xt = 1. In the reconsideration-proof equlibrium, the individual at time t realizes that he cannot choose any subgame perfect equilibrium path with positive value (or there would not be any equilibrium at all). He is indifferent among all the equilibrium paths that have zero value; these paths can start with any play on his part. He chooses xt−1 /2 because it is weakly optimal. Any path of the form (x0 , x0 /2, x0 /4, . . .) is a reconsideration-proof equilibrium path. (Section 5 presents a macroeconomic example in which the policy game reduces to this individual choice problem.)

4. RECONSIDERATION-PROOFNESS AND REVISION-PROOFNESS As mentioned in the Introduction, Asheim (1991) presents an alternative refinement of subgame perfection for intertemporally inconsistent choice problem. His refinement, which he terms revision-proofness, is based upon Greenberg’s (1990) theory of social situations. He does not prove the existence of a revisionproof equilibrium in the infinite horizon case, but he does prove a version of Proposition 5 for his concept.

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Proposition 7.1 of Asheim (1991) shows that a revision-proof equilibrium is reconsideration-proof if and only if it is symmetric. In this sense, the main difference between revision-proofness and reconsideration-proofness is that the latter imposes symmetry. If I want to claim that reconsideration-proofness is reasonable, I had better be able to argue that symmetry is reasonable. I find symmetry appealing for two reasons. First, like Farrell and Maskin (1989), I find it implausible that the history of play should affect the set of plans from which a reconsidering individual chooses. Any “credible” plan should be available to him; why should this set be affected by the history of play? My second reason is more mechanical. The individual faces the same intertemporal choice problem at every node; history does not change the constraint set or the objective function. In general, we think that choice problems may admit many solutions but they should admit only a single value. Hence, the individual should receive the same value at every node, even if he does not follow the same policy. This kind of reasoning leads immediately to my focus on symmetric equilibria. Asheim points out that in Example 1 of this paper, there exists a revision-proof equilibrium which has a higher value at every node than the reconsideration-proof equilibrium. I do not regard this possibility as a serious defect of reconsiderationproofness. There is no sense in which an individual can bargain with his future selves (what kind of bargaining power does my 60-year-old self have with me when I am 30?). Why then should we expect the outcome of this infinite individual game to be “Pareto optimal”?

5. RELATIONSHIP TO MACROECONOMIC POLICY GAMES Stokey (1989) and Chari and Kehoe (1990) discuss the following set of macroeconomic policy games. There are an infinite number of periods. In each period, there are two subperiods. In the first subperiod, a continuum of identical households each takes an action. Then, in the second subperiod, an infinitely lived government makes some choice. There are no intertemporal linkages beyond the memory of past choices. To solve these games, Chari and Kehoe (1990) define what they term “sustainable equilibria.” In a sustainable equilibrium, a history describes the sequence of past choices by the government; households cannot influence the history of play, because each one is atomistic and they cannot engage in simultaneous group deviations (see Section V of Chari and Kehoe for a more complete discussion of this point). Then, a strategy for households prescribes an action for a household after every possible history; a strategy for the government prescribes an action for the government after every possible history. A collection of strategies forms a sustainable equilibrium if each strategy prescribes actions that are optimal at each history, taking as given the other strategies. Stokey (1989) and Chari and

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Kehoe (1990) use the techniques of Abreu (1988) to completely characterize the set of sustainable equilibria. In the Appendix, I show how to construct an intertemporal optimization problem for the government that corresponds to a policy game of the form described by Stokey and Chari/Kehoe. I prove in Proposition A1 that the set of subgame perfect equilibrium paths to the (possibly inconsistent) optimization problem is the same as the set of sustainable equilibrium paths to the macroeconomic policy game. While I will not describe the details of the argument here, the intuition behind the result is straightforward. The history of play only influences consumer choices through their expectations of future government actions. Put another way, a consumer’s action at any point in the game tree can be expressed as a history-independent function of government actions along the continuation equilibrium path. Thus, it is possible to “substitute” the consumers’ choices out of the government’s objective function and replace them with a history-independent function of future government actions. Because of this equivalence result, reconsideration-proofness can be applied as a refinement concept for sustainable equilibria. Essentially, we can think of a sustainable equilibrium as a plan of governmental actions that is self-enforcing given that the government has no ability to revise the plan. Suppose though that the government can costlessly revise its plan of actions after any history and adopt another “credible” plan.5 Since the plan of actions after any history should be credible, at the minimum the government could switch to any other continuation equilibrium path. This kind of thinking implies that if the government can costlessly revise, only equilibria in which all continuation equilibrium paths have the same value can survive. Thus, as in Section 3, the government’s ability to costlessly revise leads us to focus on reconsideration-proof equilibria, that is, the best possible symmetric sustainable equilibria. Imposing reconsideration-proofness as a requirement has a drastic effect in the games described by Chari and Kehoe (1990) and Stokey (1989). In those games, there exists a stationary sustainable equilibrium. Other paths are supportable as equilibria only because the current government (and future households) expects its successors to revert to a worse equilibrium following any deviation. But in a reconsideration-proof equilibrium, no continuation equilibrium path can be worse than any other. Hence, the only reconsideration-proof equilibrium in the Stokey and Chari/Kehoe games is the stationary equilibrium.6 As seen in Example 4, this phenomenon is not a general defect of reconsideration-proofness. It occurs because Chari and Kehoe (1990) and Stokey (1989) 5 I argued earlier that being able to costlessly revise one’s plans is highly intuitive when one thinks about a single individual choosing over time. On the other hand, costless plan revision may not be as compelling in a macroeconomic policy game: the government has to be able to influence the belief structure of the continuum of agents in the economy. 6

This is equivalent to Dekel and Farrell’s (1990) argument (described in Section 3).

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rule out any dynamic linkages between periods for expository purposes (see Stokey, 1991, for a more general treatment). In the Chari–Kehoe and Stokey models, future government actions have no impact on the intraperiod utility of the current government; the total utility of the current government can be lowered only if the total utility of its successors falls. Reconsideration-proofness eliminates this kind of “self-punishment” by the players of the game. Hence, a reconsideration-proof equilibrium can be different from a stationary equilibrium only if future governments can punish current governments without punishing themselves. Fortunately, it is possible in plausible macroeconomic settings for future governments to punish current governments without punishing themselves. The following example makes this point clear.7 Consider an overlapping generations economy in which a continuum of two-period-lived individuals are born at each date. The typical individual has preferences given by c1 + 0.5g2 , where c1 is his consumption level in the first period of life and g2 is the level of per capita public goods created by the government in the second period of his life. Note that the individual does not value consumption in the second period of life. The individual receives no income in the first period of life and y units of consumption in the second period of his life. Thus, the individual has to borrow against his future income in order to consume today; he is able to borrow from an international loan market at a fixed rate r , where 1 > 0.5(1 + r ). In period t, the government’s objective function takes the form ∞ X

β s−t {c1s + 0.5g2,s−1 },

0 < β < 1,

s=t

where c1s is the per capita consumption of agents born in period s and g2,s−1 is the per capita level of public goods created for agents born in period (s − 1). The government creates public goods in period s by taxing the income y of the current old at rate τs and transforming the proceeds into public goods. The order of events in period t is as follows. First, the old receive their income y. The government then taxes this income at rate τt (chosen from the interval [0, 1]) and converts it (one for one) into τt y (per capita) public goods. Using their post-tax income, the old pay back any loans they had from last period; they throw any remaining income into the ocean because it is worthless to them. 7 The stated version of Proposition A1 does not apply to this example. I conjecture though that some verison of the proposition applies in any game in which a single strategic player is playing against a continuum of identical players who cannot simultaneously deviate from the equilibrium path. In particular, the discussion that follows makes clear that there exists some more general version of Proposition A1 that is applicable to this example.

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The young then enter the loan market and borrow against the income they will receive when they are old. Implicit in the above description is a crucial feature of the environment: different causes of default generate different penalties. If an individual can afford to repay any fraction of a loan out of his after-tax income and does not, he is punished in a sufficiently painful fashion that this type of default provides him with negative utility. On the other hand, if an individual is forced into partial default because the government taxes away more of his income than is anticipated in equilibrium, then he is not punished. There is a fundamental tension in this environment. Current governments want to impose high tax rates in order to create as many public goods as possible; untaxed goods are lost because they either flow out of the country to foreigners or get thrown into the ocean. On the other hand, the current government wants next period’s tax rate to be low, so that the current young can borrow a lot. The goal of my analysis is to resolve this tension through an appropriate equilibrium concept. As mentioned above, a history in a sustainable equilibrium of this game is fully characterized by the past choices of the government. Hence, a strategy for the government in period (t + 1) describes its tax rate choice τt+1 as a function of past choices (τ0 , τ1 , . . . , τt ). Given a strategy for the government, and given a history of governmental choices (τ0 , τ1 , . . . , τt ), a young individual in period t will choose to borrow (1 − τt+1 (τ0 , . . . , τt ))y/(1 + r ) units of consumption. Thus, any strategy for the government translates directly into a strategy for individuals. Following Chari and Kehoe, a sustainable equilibrium is a strategy for the government that is optimal (in terms of the government’s objective function) after every history, taking as given what the strategy dictates about the choices of future governments and individuals. The class of sustainable equilibrium paths is very large, as is the set of values associated with the equilibrium paths. The worst equilibrium path is defined by the unique stationary sustainable equilibrium in which the government sets τt = 1 after every history. On the other hand, the best equilibrium path is the sequence of tax rates (1, 0, 0, . . .) (recall that (1 + r )−1 > 0.5). This path can be supported as an equilibrium by the individuals’ expecting the government to set τt = 1 after any deviation. How does reconsideration-proofness work in this environment? We can think of a symmetric sustainable equilibrium as being an equilibrium in which the value of the government’s objective function is the same after any history (including histories that exhibit deviations from the equilibrium path). Thus, a path of tax rates (τ0 , τ1 , τ2 , . . .) can be a symmetric equilibrium path only if the government

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receives the same intraperiod utility at every date. Hence, there exists v such that −τt+1 y/(1 + r ) + 0.5τt y = v for all t ≥ 0. If v is positive, it is easy to show that any paths that solve this difference equation feature some negative τt (again, recall that 0.5(1 + r ) < 1). Hence, v must be nonpositive in any symmetric equilibrium path. To find a reconsideration-proof equilibrium, we must characterize the best possible symmetric equilibrium. One such equilibrium is for a government to choose τt = 0.5(1 + r )τt−1 after any history (τ0 , τ1 , . . . , τt−1 ), t ≥ 1. It is easy to check that this strategy is sustainable (because it leads to any government being indifferent among all choices) and generates a symmetric equilibrium. The argument in the previous paragraph implies that this is the best possible symmetric equilibrium. Thus, the reconsideration-proof equilibrium paths consist of all sequences of tax rates of the form {0.5t (1 + r )t τ0 }∞ t=0 (τ0 arbitrary). Note that all reconsideration-proof paths share the same feature of converging to zero over time (see Example 4 for a similar result). We can therefore think of the individuals in the economy as viewing zero taxation as a long-run goal of the government. In a reconsideration-proof equilibrium, the agents expect the economy to partially adjust toward zero taxation in each period. This kind of continuous partial adjustment toward a long run goal, even after a deviation, seems more intuitively appealing than the sudden jump downward in the tax rate posited by the “best” sustainable equilibrium or the steady-state “bad” outcome posited by the stationary equilibrium. Thus, reconsideration-proofness can pick out a “reasonable” sustainable equilibrium in a macroeconomic policy game. I conjecture that it can serve this role in more elaborate settings. The key, as indicated above, is that future governments must have a policy lever that can harm current governments.

6. CONCLUSIONS This paper looks at a simple choice problem which is in general temporally inconsistent. If the individual cannot commit to a given sequence of choices at time zero, what will his choices look like? Typically, researchers have treated this choice problem as a noncooperative game between an infinite number of different individuals and looked for the subgame perfect equilibria of this game. This paper shows that in the infinite horizon case, subgame perfection cannot in general provide a unique value—even if the control problem is temporally consistent. It instead uses a refinement of subgame perfection called reconsideration-proofness

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to select a set of solutions which all have the same value at any given time. Mechanically, this refinement simply selects the symmetric subgame perfect equilibrium with the highest value. In the Appendix, I show that there is an equivalence theorem relating the class of individual choice problems considered in this paper and the class of macroeconomic policy games considered by Chari and Kehoe (1990) and Stokey (1989, 1991). Refinements of subgame perfection applicable to individual choice problems can also be used for macroeconomic policy games. I demonstrate in a particular macroeconomic example how reconsideration-proofness can select a plausible outcome.

APPENDIX Consider the following policy game. There is a single infinitely lived government and a continuum of infinitely lived households. In period t, there are two subperiods. In the first subperiod, each household chooses an action z t from a set Z . In the second subperiod, the government chooses a policy xt out from a set X . In period t, a generic household has the objective function ∞ X

β s−t u h (z s , zˆ s , (xr )r∞=s ),

s=t

where z s is its choice in period s, zˆ s is the choice of all other households in period s, and xr is the choice of the policy maker in period r . In period t, the government has the objective function ∞ G((z s )∞ s=t , (x s )s=t ),

where z s is the choice in period s of the households. (Following the macroeconomic policy literature, I assume that all households always make the same choice out of the set Z . Note that the government may or may not be benevolent.) Thus, there is a single long-lived decision maker playing against a continuum of long-lived households; the main result is still valid if the households only live for one period. In this game, a history in period t is a sequence of past choices by the government, (x0 , x1 , . . . , xt−1 ). A strategy for the period t government is a mapping from the set of all possible histories X t into X . A strategy for the period t households is a mapping from the set of all possible histories X t into Z (as is common in the macroeconomic policy literature, I assume that in all histories, all households follow the same strategy and no nonnegligible set of households deviate simultaneously). Following Chari and Kehoe (1990), I use the following concept of equilibrium for this game.

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DEFINITION A1. A collection of strategies forms a sustainable equilibrium if after any history, the government’s strategy specifies the optimal choice out of the set X , taking as given the strategies of the households and the prescriptions of its own strategy at future histories. Similarly, the period t households’ strategies specify an optimal choice out of the set Z , taking as given the strategy of the government and households.

Define the function ξ : X ∞ → Z to be the solution to the equation ξ(x∞ ) = argmax u h (z 0 , ξ(x∞ ), x∞ ), z0

where x∞ is in X ∞ . (Following the literature on macroeconomic policy games, I assume that ξ is well-defined.) In words, ξ(x∞ ) is the optimal choice of a generic household, given that all other households choose ξ(x∞ ) and future governments choose x∞ . Now consider the following decision problem. All symbols stand for the same sets and elements of sets as in the above macroeconomic policy game. There is a single infinitely lived agent. The agent’s objective function is influenced by a state variable z t which lies in the state space Z . In each period, there are two subperiods. In the first subperiod, a state variable z t assumes its value according to the law of motion z t = ξ((xs )∞ s=t ), where xs is the period s choice of the agent and ξ is defined as above. In the second subperiod, the agent makes a choice xt out of the set X . His objective function is given by ∞ ∞ ∞ U (z t , (xs )∞ s=t ) ≡ G(z t , (ξ((xr )r =s ))s=t+1 , (x s )s=t ).

Note that given certain regularity conditions on G, ξ , X , and Z , this individual choice problem lies in the class of problems considered in this paper. Hence, the concept of subgame perfection can be applied just as it was earlier in the paper. The following proposition implies that the refinement concepts discussed above can be applied to the macroeconomic policy game. PROPOSITION A1. Suppose the collection of functions x t : X t → X , 0 ≤ t < ∞, is an equilibrium government strategy in the policy game (x0 is an element of X ). Then this collection of functions is a subgame perfect collection of individual strategies in the choice problem. The converse is also true.

Proof. Consider a sustainable equilibrium government strategy in the macroeconomic policy game. I claim that this is a subgame perfect equilibrium in the individual choice problem. Suppose that after history h t , the government’s strategy is to play xt . Suppose the individual deviates from this strategy in

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the choice problem and plays xt0 , given that all future individuals will follow the strategies described in the policy game. Then the period t individual receives utility U (ξ(xt ), xt0 , x c (h t , xt0 )), where x c (h t , xt0 ) is the continuation sustainable equilibrium path after the history (h t , xt0 ). This utility level is lower than U (ξ(xt ), xt , x c (h t , xt )), or the government would find it optimal to play xt0 instead of xt . Consider a subgame perfect equilibrium in the individual choice problem. Suppose it prescribes xt (h t ) as the individual’s choice after history h t . I claim that the collection of government strategies xt (h t ) and household strategies z t (h t ) = ξ(xt (h t ), xt+1 (h t , xt ), xt+2 (h t , xt , xt+1 ), . . .) form a sustainable equilibrium. It is clear from the definition of ξ that it is suboptimal for any household to deviate from its strategy given that no other players do. Now consider the government’s choice after history h t . It is supposed to play xt (h t ). Suppose instead it plays xt0 . This results in the government’s playing a continuation subgame perfect equilibrium path x c (h t , xt0 ). The utility level for the government generated by this sequence of choices is lower than that generated by the sequence of choices (xt , x c (h t , xt )), or the individual in the choice problem would choose xt0 instead of xt . This equivalence result underscores the intuitive relationship between time consistency in macroeconomic policy environments and individual choice problems. It also serves to point out that the strategic interaction in macroeconomic policy games can be interpreted as being between the current government and future governments.

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FARRELL, J., AND MASKIN, E. (1989). “Renegotiation in Repeated Games,” Games Econ. Behav. 1, 327–360. GREENBERG, J. (1990). The Theory of Social Situations: An Alternative Game Theoretic Approach. Cambridge: Cambridge Univ. Press. PEARCE, D. (1987). “Renegotiation-Proof Equilibria: Collective Rationality and Intertemporal Cooperation,” working paper, Yale University. STOKEY, N. (1989). “Reputation and Time Consistency,” Amer. Econ. Rev. 79, 134–145. STOKEY, N. (1991). “Credible Public Policy,” J. Econ. Dynam. Cont. 15, 627–657. STROTZ, R. (1956). “Myopia and Consistency in Dynamic Utility Maximization,” Rev. Econ. Stud. 23, 165–180. WHITEMAN, C. (1986). “Analytical Policy Design under Rational Expectations,” Econometrica 54, 1387–1406.