Two Tales of Time Consistency: The Generalized Euler Equation and the Bankruptcy–Sovereign Default Problem Organizational Equilibrium with Capital joint work with Marco Bassetto, Zhen Huo and Xavier Mateos-Planas Jos´ e-V´ ıctor R´ ıos-Rull March 28, 2016 Ridiculously Preliminary Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 1 / 89
Time Inconsistency is a Pervasive Issue It shows up very frequently, (policy making, nonstandard preferences, borrowing, being faithful). It is insufficiently well characterized. Benchmark outcome: Markov Perfect Equilibria. Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 2 / 89
Two Projects Today 1 Can we do better than the Markov Equilibrium without resorting to trigger strategies in standard time inconsistent environments such as hyperbolic discounting and fiscal policy? ◮ Yes, by extending the notion of Organizational Equilibrium (Prescott and R´ ıos-Rull (2000)) to Economies with state variables. We achieve notable gains. 2 Extension of the Characterization via Generalized Euler Equations (GEE) of the Markov equilibria of the most important versions of the Sovereign Default problem (Gomes, Jermann, and Schmid (2014)). ◮ Equilibrium is the solution to a pair of functional equations (without the max operator) using some auxiliary functions. It gets rid of the endogenous pricing functions. Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 3 / 89
I. Organizational Equilibrium with Marco Bassetto and Zhen Huo • Talk will concentrate on a Hyperbolic Discounting Example with Full Depreciation and log Utility Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 4 / 89
The Environment Preferences ∞ � β τ u ( c t + τ ) U t = u ( c t ) = log c t + δ τ =1 Technology f ( k t ) = k α t , k t +1 = f ( k t ) − c t . The differentiable Markov Perfect Equilibrium (Krusell, Kuruscu, and Smith (2010)) with closed form solution: δβ k ′ = α 1 − αβ + δαβ k α = α µ M k α Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 5 / 89
Aux Object: λ -sacrifice decision rule φ λ 1 β Discounted value of decision rule g : Γ g ( k ) = u [ f ( k ) − g ( k )] + β Γ g [ g ( k )] 2 λ -choice of hyperbolic agent given g : u [ f ( k ) − k ′ ] + λ δ β Γ g ( k ′ ) φ ( k , λ ; g ) = argmax k ′ Consider the following fixed point abusing notation φ λ ( k ) = φ ( k , λ ; φ λ ) We call φ λ ( k ) the λ -sacrifice decision rule. Its value is V λ ( k ) = u [ f ( k ) − φ λ ( k )] + λ δ β Γ g ( φ λ ( k )) Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 6 / 89
Characterization of φ λ 1 Clearly λ = 1 is the Markov φ 1 ( k ) = α µ M ( k ). 2 λ = 1 δ is the time consistent solution with discount rate β φ 1 /δ ( k ) = α β ( k ). 3 For some environments, there is a � λ < 1 we have � λ ( k ) = α δ β ( k ). φ Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 7 / 89
How is V λ ?: The best is λ ∗ < 1 δ Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 8 / 89
Can λ ∗ be achieved?: Logic of Organizational Equilibrium • Imagine that the current agent makes a proposal to all future selves Follow the λ ∗ sacrifice, by means of some proposal. All future selves can say yes or no within the same proposal Or they can propose something else. The best such Proposal will be issued. In this example it is the λ ∗ that we saw. But the proposals have to be large enough to accomodate various options. In particular, a non-constant λ . Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 9 / 89
Objects Needed: We need to think of proposals that Induce agents to go along. Do not want to restart the process by 1 making the same proposal themselves. Future Agents do not want to make a different proposal. 2 For this we need a function: λ = q ( k , λ − ) It is the proposal received that has to be accepted, and it is expected to be followed in the future. Note that it includes what the previous agent did (non-Markov). ◮ This function can be interpreted in many ways that look that renegotiation proof or “thank you for the idea, I will do it my self since I do not need you” (Kocherlakota (1996), Prescott and R´ ıos-Rull (2000), Nozawa (2014)). It also has to specify the starting outcome: we write λ = q ( k , ∅ ). Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 10 / 89
What are those functions? First, the proposal function q Consider the following extensions of the above objects 1 β Discounted value of proposal q : � Γ q ( k , λ − ) u [ f ( k ) − φ λ ( k )] + β � Γ q [ φ λ ( k ), q ( k , λ − )] = q ( k , λ − ) = λ 2 λ -modified choice of hyperbolic agent given q : φ ( k , λ − ; q ) = argmax � u [ f ( k ) − φ λ ( k )] + δ β � Γ q [ φ λ ( k ), λ ] λ A fixed point, q ∗ ( k , λ − ) ∈ � φ ( k , λ − ; q ∗ ), is a plausible proposal (it will be followed) Let Q be the set of plaussible proposal functions. Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 11 / 89
The Domain and Range of q Function q has to incorporate the behavior of the starter (the first agent). What is the λ − imputed to the starter depends on the environment? ◮ We assume it is the empty set. ◮ This is also available for any agent that wants to restart the process. Therefore the domain and the range of q will have to be extended to the empty set: q : [0, K ] × [ λ , λ ] ∪ ∅ → [ λ , λ ] ∪ ∅ Altough the range maynot be strictly necessary. Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 12 / 89
Second, the starting function h Any agent can make an initial proposal q ∈ Q and choose an initial λ − . It maximizes u [ f ( k ) − φ λ ( k )] + δ β � Γ q [ φ λ ( k ), q ( k , λ )] argmax λ , q s.t. q ∈ Q It is crucial that the same q is chosen for all k . This guarantees that no agent will later want to deviate. Denote the solution h ∗ ( k ), q ∗ . V: We have to make sure that we redefine the set Q so that it excludes blockable q ′ s like the ones that I talked about with Zhen Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 13 / 89
Organizational Equilibrium We have the elements that we need. Definition: An organizational equilibrium is a a pair h ∗ , q ∗ ∈ Q such • that { h ∗ ( k ), q ∗ } that solve the above problem (the same q ∗ ∈ Q , ∀ k ). Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 14 / 89
Characterization • There are a few steady state properties V: Zhen there are a few that have to be characterized 1 In the steady state, λ ∗ = h ∗ ( k ∗ ) = q ∗ ( k ∗ , λ ∗ ) 2 h ( k ) = q ( k , λ ∗ ) V: (This one I am not sure of) Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 15 / 89
In the full depreciation, log economy Theorem In a neighbourhood of the steady state k ∗ , the Organizational Equilibrium involves λ ∗ = h ( k ) and λ ∗ = q ( k , λ ∗ ): The sacrifice is constant and equal to λ ∗ So we can do much better than Markov even without triggers. Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 16 / 89
II. Characterization of Markov Perfect Equilibrium in Sovereign Default Environments Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 17 / 89
Main features of this environment 1 Borrowers have no commitment to return a loan. They sometimes default in circumstances that are different to those that they would have liked to have committed to. 2 If long term debt exists, the borrower cannot commit to limit additional borrowing in the future and there are no well defined seniority rules for debt. 3 There are multiple lenders and new lenders are always available. Past lenders cannot limit the activities of future lenders, at least in the absence of default. 4 Some form of punishment follows default. Typically, it is either output (or utility) reduction, or limited access to future borrowing, or both. Bassetto, Huo, Mateos-Planas, R´ ıos-Rull Two Tales of Time Consistency Wharton March 28, 2016 18 / 89
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