Intentional time inconsistency

For the insights established by Adam Smith and David Hume, see Palacios-Huerta (2003), and for a review of studies providing evidence of preference reversal, see Green and Myerson (2004, 2010).

Consider a person with a bad habit of staying up too late. Every morning, he promises to go to bed early, but, at night, he always goes to bed later than he intended. By knowing that he breaks his promises so many times, a rational agent may pre-commit his future behavior. For example, he can say his spouse that he feels tired and sluggish, so that he should go to bed early. If you are married, then you know that your spouse will make you go to bed early either by kindness or by force.

For a comprehensive review on psychological determinants of intertemporal preferences, see Urminsky and Zauberman (2015).

Note that, even if both players know that they have time-consistent preferences, we might have a subgame perfect equilibrium where the consumption path coincides with the one defined in Proposition 1: The agent 1 continues to play \(g_{S}(x)\) and agent 2 continues to play \(g_{A_{S}}(x)\) as long as no agent deviates from this strategy. If at least one player deviates from this strategy, they both play \(h\left( x\right) =x\) and the resources are exhausted. For a set of payoffs to be supportable in discounted dynamic programming, see Fudenberg and Tirole (1991).

The pooling equilibrium that we define resembles the well-known variant of the chain-store game in which there is a small probability p that the monopolist is “tough” and prefers fight rather than cooperate if there is an entry to the market. In the original chain-store game, a monopolist plays against a succession of K potential competitors. In each period, one of the potential competitors decide whether or not to compete with the monopolist. If it decides to enter, then the monopolist chooses either to cooperate or to fight. Each potential competitor prefers to stay out rather than entering and being fought, but prefers the most when it enters and the monopolist does not fight. If a competitor enters, the monopolist prefers to cooperate rather than fight, but it prefers the most if there is no entry. In the unique subgame perfect equilibrium of the game, each potential competitor chooses to enter and the monopolists always chooses to cooperate (Selten 1978). Kreps and Wilson (1982) shows that the regular monopolist turns the failure of correct common knowledge about its payoff into an advantage by acting like a tough one and preserves its reputation at least until the horizon gets close. Similarly, we show that agent 1 turns the failure of correct common knowledge about its preferences into an advantage and acts as if he might have problems with self-control.

With heterogeneous discount factors, we get \(c=\frac{\left( 1-\delta _{1}\alpha \right) \delta _{2}\alpha }{\delta _{2} +\beta \delta _{1}-\alpha \delta _{1}\delta _{2}},\)\(d=\frac{\left( 1-\delta _{2}\alpha \right) \beta \delta _{1}\alpha }{\delta _{2}+\beta \delta _{1} -\alpha \delta _{1}\delta _{2}}\)and \(1-c-d=\frac{\beta \alpha \delta _{1}\delta _{2} }{\delta _{2}+\beta \delta _{1}-\alpha \delta _{1}\delta _{2}}.\)

We restrict ourselves to linear strategies to obtain definite results. By relaxing the assumption on output elasticity, one can show numerically that the decision to pretend to have time−inconsistent preferences and the preference between naive and sophisticated behavior may depend on the available resource stock.

As we did in Sect. 4, one can solve the model for heterogeneous discount factors. While an MPNE in linear strategies does not exist when agent 1 pretend to be sophisticated, it still exists when both agents act with time-consistent preferences or when agent 1 pretend to be naive. For the naive player, we plot the optimal level of \(\beta \) for multiple cases by freeing modified discount factor of agents one at a time. Our analysis confirms the discussion in Sect. 4 that the optimal level of \(\beta \) depends on the nonlinear interaction of agent 1’s own discount rate and the discount rate of the agent 2.