publications

2023

1. Time consistent policies and quasi-hyperbolic discounting Balbus, Łukasz, Reffett, Kevin, and Wozny, Łukasz Oxford Research Encyclopedia of Economics and Finance, Nov 2023, Oxford University Press
   [ed. Hamilton, J.H.] [Abs] [HTML] [PDF]

   In dynamic choice models, dynamic inconsistency of preferences is a situation in which a decision-maker’s preferences change over time. Optimal plans under such preferences are time inconsistent if a decision-maker has no incentive to follow in the future the (previously chosen) optimal plan. A typical example of dynamic inconsistency is the case of present bias preferences, where there is a repeated preference toward smaller present rewards versus larger future rewards. The study of dynamic choice of decision-makers who possess dynamically inconsistent preferences has long been the focal point of much work in behavioral economics. Experimental and empirical literatures both point to the importance of various forms of present-bias. The canonical model of dynamically inconsistent preferences exhibiting present-bias is a model of quasi-hyperbolic discounting. A quasi-hyperbolic discounting model is a dynamic choice model, in which the standard exponential discounting is modified by adding an impatience parameter that additionally discounts the immediately succeeding period. A central problem with the analytical study of decision-makers who possess dynamically inconsistent preferences is how to model their choices in sequential decision problems. One general answer to this problem is to characterize and compute (if they exist) constrained optimal plans that are optimal among the set of time consistent sequential plans. Time consistent plans are those among the set of feasible plans that will actually be followed, or not reoptimized, by agents whose preferences change over time. These are called time consistent plans or policies (TCPs). Many results of the existence, uniqueness, and characterization of stationary, or time invariant, TCPs in a class of consumption-savings problems with quasi-hyperbolic discounting, as well as provide some discussion of how to compute TCPs in some extensions of the model are presented, and the role of the generalized Bellman equation operator approach is central. This approach provides sufficient conditions for the existence of time consistent solutions and facilitates their computation. Importantly, the generalized Bellman approach can also be related to a common first-order approach in the literature known as the generalized Euler equation approach. By constructing sufficient conditions for continuously differentiable TCPs on the primitives of the model, sufficient conditions under which a generalized Euler equation approach is valid can be provided. There are other important facets of TCP, including sufficient conditions for the existence of monotone comparative statics in interesting parameters of the decision environment, as well as generalizations of the generalized Bellman approach to allow for unbounded returns and general certainty equivalents. In addition, the case of multidimensional state space, as well as a general self generation method for characterizing nonstationary TCPs must be considered as well..

2022

1. Time consistent equilibria in dynamic models with recursive payoffs and behavioral discounting Balbus, Łukasz, Reffett, Kevin, and Wozny, Łukasz Journal of Economic Theory vol. 204, p. 105493, 2022.
   [IF2022 1.6, 200p MEiN] [Abs] [HTML] [PDF]

   We prove existence of time consistent equilibria in a wide class of dynamic models with recursive payoffs and generalized discounting involving both behavioral and normative applications. Our generalized Bellman equation method identifies and separates both: recursive and strategic aspects of the equilibrium problem and allows to precisely determine the sufficient assumptions on a stochastic transition to establish existence. In particular we show existence of minimal state space stationary Markov equilibrium (a time-consistent solution) in a deterministic model of consumption-saving with beta-delta discounting and its generalized versions involving magnitude effects, non-additive payoffs, semi-hyperbolic or hyperbolic discounting (over possibly unbounded state and unbounded above reward space). We also provide an equilibrium approximation method for a hyperbolic discounting model.

2. On reference dependence and complementary symmetry Lewandowski, Michał, and Wozny, Łukasz Journal of Mathematical Psychology vol. 108, p. 102653, 2022
   [IF2022 1.8, 100p MEiN2021] [Abs] [HTML] [PDF]

   We reevaluate the complementary symmetry property and the corresponding experimental evidence. Originally the property was stated for binary risky prospects. We generalize it to arbitrary state-contingent real-valued outcomes, thus extending the domain of choice from risk to uncertainty/ambiguity and allowing for multiple outcomes. We consider various observable tasks related to the elicitation of buying and selling prices. In particular, for selected reference point models, we derive relevant definitions of gains and losses, and identify pairs of prices satisfying the complementary symmetry property. We then run an experiment to test these new predictions, and find that while some reference point models can be refuted based on our data, others perform reasonably well.

3. Markov distributional equilibrium dynamics in games with complementarities and no aggregate risk Balbus, Łukasz, Dziewulski, Paweł, Reffett, Kevin, and Wozny, Łukasz Theoretical Economics vol. 17 No. 2, p. 725–762, 2022.
   [IF2022 1.671, 140p MEiN2021] [Abs] [HTML] [PDF]

   We present a new approach to studying equilibrium dynamics in a class of stochastic games with a continuum of players with private types and strategic complementarities. We introduce a suitable equilibrium concept, called Markov Stationary Nash Distributional Equilibrium (MSNDE), prove its existence, and determine comparative statics of equilibrium paths and the steady state invariant distributions to which they converge. Finally, we provide numerous applications of our results including: dynamic models of growth with status concerns, social distance, and paternalistic bequest with endogenous preference for consumption.

2019

1. Lipschitz recursive equilibrium with a minimal state space and heterogeneous agents Raad, Rodrigo, and Wozny, Łukasz Journal of Mathematical Economics vol. 82 p. 98-111, 2019
   [IF2019 0.725, 70p MNiSW2019] [Abs] [HTML] [PDF]

   This paper analyzes the Lucas tree model with heterogeneous agents and one asset. We show the existence of a minimal state space Lipschitz continuous recursive equilibrium using Montrucchio (1987) results. The recursive equilibrium implements a sequential equilibrium through an explicit functional equation derived from the Bellman Equation. Our method also allows to prove existence of a recursive equilibrium in a general class of deterministic or stochastic models with several assets provided there exists a Lipschitz selection on the demand correspondence. We provide examples showing applicability of our results.

2. A qualitative theory of large games with strategic complementarities Balbus, Łukasz, Dziewulski, Paweł, Reffett, Kevin, and Wozny, Łukasz Economic Theory vol. 67 no. 3 p. 497-523, 2019
   [IF2019 1.051, 140p MNiSW2019] [Abs] [HTML] [PDF]

   We study the existence and computation of equilibrium in large games with strategic complementarities. Using monotone operators defined on the space of distributions partially ordered with respect to the first-order stochastic dominance, we prove existence of a greatest and least distributional Nash equilibrium. In particular, we obtain our results under a different set of conditions than those in the existing literature. Moreover, we provide computable monotone distributional equilibrium comparative statics with respect to the parameters of the game. Finally, we apply our results to models of social distance, large stopping games, keeping up with the Joneses, as well as a general class of linear non-atomic games.

2018

1. Comparing recursive equilibrium in economies with dynamic complementarities and indeterminacy Datta, Manjira, Reffett, Kevin, and Wozny, Łukasz Economic Theory vol. 66 no. 3 p. 593-626, 2018
   [IF2018 0.961, 25p MNiSW] [Abs] [HTML] [PDF]

   We prove the existence of a (minimal state space) recursive equilibrium (RE) for a broad class of infinite horizon dynamic general equilibrium models with positive externalities, dynamic complementarities, public policy, equilibrium indeterminacy and sunspots. These are new multistep monotone map methods, and apply to economies for which results on the existence of dynamic equilibria are unknown, and where existing methods are difficult to apply. Our methods are global based on monotone operators defined on Euler equations, and we do not appeal to the theory of smooth dynamical systems that are commonly applied in the literature. Rather, using partial ordering methods, we are able to provide a qualitative theory of equilibrium comparative statics for some equilibria even in the presence of multiple equilibrium, and these comparison results are computable via successive approximations from upper and lower bounds of particular functions sets. We provide applications of our results to an extensive literature on local indeterminacy of dynamic equilibrium.

2. Dynamic Games in Macroeconomics Balbus, Łukasz, Reffett, Kevin, and Wozny, Łukasz In Handbook of Dynamic Game Theory p. 1–50, Springer International Publishing 2018
   [ed. Basar, Tamer and Zaccour, Georges] [Abs] [HTML] [PDF]

   In this chapter, we survey how the methods of dynamic and stochastic games have been applied in macroeconomic research. In our discussion of methods for constructing dynamic equilibria in such models, we focus on strategic dynamic programming, which has found extensive application for solving macroeconomic models. We first start by presenting some prototypes of dynamic and stochastic games that have arisen in macroeconomics and their main challenges related to both their theoretical and numerical analysis. Then, we discuss the strategic dynamic programming method with states, which is useful for proving existence of sequential or subgame perfect equilibrium of a dynamic game. We then discuss how these methods have been applied to some canonical examples in macroeconomics, varying from sequential equilibria of dynamic nonoptimal economies to time-consistent policies or policy games. We conclude with a brief discussion and survey of alternative methods that are useful for some macroeconomic problems.

3. On uniqueness of time-consistent Markov policies for quasi-hyperbolic consumers under uncertainty Balbus, Łukasz, Reffett, Kevin, and Wozny, Łukasz Journal of Economic Theory vol. 176 p. 293 - 310, 2018
   [IF2018 1.181, 30p MNiSW] [Abs] [HTML] [PDF]

   We give a set of sufficient conditions for uniqueness of a time-consistent stationary Markov consumption policy for a quasi-hyperbolic household under uncertainty. To the best of our knowledge, this uniqueness result is the first presented in the literature for general settings, i.e. under standard assumptions on preferences, as well as some new condition on a transition probability. This paper advocates a generalized Bellman equation method to overcome some predicaments of the known methods and also extends our recent existence result. Our method also works for returns unbounded from above. We provide a few natural extensions of optimal policy uniqueness: convergent and accurate computational algorithm, monotone comparative statics result and generalized Euler equation.

2017

1. A note on Markov perfect equilibria in a class of non-stationary stochastic bequest games Balbus, Łukasz, Jaskiewicz, Anna, Nowak, Andrzej S., and Wozny, Łukasz Journal of Mathematical Analysis and Applications vol. 456 no. 1 p. 394 - 401, 2017
   [IF2018 1.188, 40p MNiSW] [Abs] [HTML] [PDF]

   In this note, we prove the existence of a Markov perfect equilibrium in a non-stationary version of a paternalistic bequest game. The method we advocate is general and allows to study models with unbounded state space and unbounded utility functions. We cover both, the stochastic and deterministic cases. We provide a characterization of the set of all Markov perfect equilibria by means of a set-valued recursive equation involving the best response operator. In the stationary case, we show that there exists a set of strategies that is invariant under the best response mapping.

2016

1. A strategic dynamic programming method for studying short-memory equilibria of stochastic games with uncountable number of states Balbus, Łukasz, and Wozny, Łukasz Dynamic Games and Applications vol. 6 no. 2 p. 187–208, 2016
   [IF2016 1.647, 25p MNiSW2016] [Abs] [HTML] [PDF]

   We study a class of infinite horizon stochastic games with uncountable number of states. We first characterize the set of all (nonstationary) short-term (Markovian) equilibrium values by developing a new (Abreu et al. in Econometrica 58(5):1041–1063, 1990)-type procedure operating in function spaces. This (among others) proves Markov perfect Nash equilibrium (MPNE) existence. Moreover, we present techniques of MPNE value set approximation by a sequence of sets of discretized functions iterated on our approximated APS-type operator. This method is new and has some advantages as compared to Judd et al. (Econometrica 71(4):1239–1254, 2003), Feng et al. (Int Econ Rev 55(1):83–110, 2014), or Sleet and Yeltekin (Dyn Games Appl doi:10.1007/s13235-015-0139-1, 2015). We show applications of our approach to hyperbolic discounting games and dynamic games with strategic complementarities.

2. On the price of commitment assets in a general equilibrium model with credit constraints and tempted consumers Wozny, Łukasz The B.E. Journal of Theoretical Economics vol. 16 no. 2 p. 579-598, 2016
   [IF2016 0.229, 20p MNiSW] [Abs] [HTML] [PDF]

   We analyze a three period production economy, where households exhibit problems of self-control and face credit constraints. Apart from liquid assets, a single commitment (illiquid) asset is available that allows to commit to a planned consumption path. We compare general equilibrium allocations of the two models: one, where households choices are determined using Gul, Pesendorfer (2001) model and the other, where households choices come from a beta-delta quasi-hyperbolic discounting model. Contrary to the results of Kocherlakota (2001) or Gabrieli, Ghosal (2013), we show that, when a production sector is incorporated into the economy with commitment asset and credit constraints, we can restore the equilibrium existence (without recalling measure space of consumers (see Luttmer, Mariotti (2006)) and unlike Gul, Pesendorfer (2004), we show that the equilibrium allocations of both models (GP and beta-delta) imply positive consumption of the commitment asset and corner consumption of one of the liquid assets. We also provide an example showing, when equilibrium allocations of both models are different.

2015

1. Monotone equilibria in nonatomic supermodular games. A comment Balbus, Łukasz, Reffett, Kevin, and Wozny, Łukasz Games and Economic Behavior vol. 94 p. 182-187, 2015
   [IF2015 0.882, 30p MNiSW2015] [Abs] [HTML] [PDF]

   Recently Yang and Qi (GEB 2013) stated an interesting theorem on the existence of complete lattice of equilibria in a particular class of large nonatomic supermodular games for general action and players spaces. Unfortunately, their result is incorrect. In this note, we detail the nature of the problem with the stated theorem, provide a counterexample, and then correct the result under additional assumptions.

2. Time consistent Markov policies in dynamic economies with quasi-hyperbolic consumers Balbus, Łukasz, Reffett, Kevin, and Wozny, Łukasz International Journal of Game Theory vol. 44 no. 1 p. 83–112, 2015
   [IF2015 0.577, 20p MNiSW] [Abs] [HTML] [PDF] [Corrigendum]

   We study the question of existence and computation of time-consistent Markov policies of quasi-hyperbolic consumers under a stochastic transition technology in a general class of economies with multidimensional action spaces and uncountable state spaces. Under standard complementarity assumptions on preferences, as well as a mild geometric condition on transition probabilities, we prove existence of time-consistent solutions in Markovian policies, and provide conditions for the existence of continuous and monotone equilibria. We present applications of our methods to habit formation models, environmental policies, and models of consumption under borrowing constraints, and hence show how our methods extend the results obtained by Harris and Laibson (Econometrica 2001) to a broad class of dynamic economies. We also present a simple successive approximation scheme for computing extremal equilibrium, and provide some results on the existence of monotone equilibrium comparative statics in the model’s deep parameters.

3. On incentives, temptation and self-control Wozny, Łukasz Mathematical Social Sciences vol. 74 p. 60 - 67, 2015
   [IF2015 0.344, 25p MNiSW] [Abs] [HTML] [PDF]

   We consider a principal-agent model, where a single agent exhibits problems of self control modelled using Gul, Pesendorfer (2001) type temptation preferences. For a general class of preferences, we characterize the optimal contract in such a setting using standard Grossman, Hart (1983) techniques. Our analysis shows that the incentive compatibility constraint is not necessarily binding at the optimal solution. As a result, the solution to the relaxed problem (without the incentive compatibility constraint) provides a variable pay, which contrasts with the standard results for the separable utility case. These observations result from the fact that in our setting the principal trade-offs incentives and insurance, but also reduction of self control cost for the agent. Our results shed some light on the justification of randomized contracts (see Holmstrom 1979), the literature on behavioral contracts, but also show that in the presence of strong self-control problems moral hazard cost can be mitigated.

4. Differential information in large games with strategic complementarities Balbus, Łukasz, Dziewulski, Paweł, Reffett, Kevin, and Wozny, Łukasz Economic Theory vol. 59 no. 1 p. 201-243, 2015
   [IF2015 1.137, 25p MNiSW] [Abs] [HTML] [PDF]

   We study equilibrium in large games of strategic complementarities (GSC) with differential information. We define an appropriate notion of distributional Bayesian Nash equilibrium and prove its existence. Furthermore, we characterize order-theoretic properties of the equilibrium set, provide monotone comparative statics for ordered perturbations of the space of games, and provide explicit algorithms for computing extremal equilibria. We complement the paper with new results on the existence of Bayesian Nash equilibrium in the sense of Balder and Rustichini (J Econ Theory 1994) or Kim and Yannelis (J Econ Theory 1997) for large GSC and provide an analogous characterization of the equilibrium set as in the case of distributional Bayesian Nash equilibrium. Finally, we apply our results to riot games, beauty contests, and common value auctions. In all cases, standard existence and comparative statics tools in the theory of supermodular games for finite numbers of agents do not apply in general, and new constructions are required.

2014

1. A constructive study of Markov equilibria in stochastic games with strategic complementarities Balbus, Łukasz, Reffett, Kevin, and Wozny, Łukasz Journal of Economic Theory vol. 150 p. 815-840, 2014
   [IF2014 1.033, 30p MNiSW] [Abs] [HTML] [PDF]

   We study a class of infinite horizon, discounted stochastic games with strategic complementarities. In our class of games, we prove the existence of a stationary Markov Nash equilibrium, as well as provide methods for constructing this least and greatest equilibrium via a simple successive approximation schemes. We also provide results on computable equilibrium comparative statics relative to ordered perturbations of the space of games. Under stronger assumptions, we prove the stationary Markov Nash equilibrium values form a complete lattice, with least and greatest equilibrium value functions being the uniform limit of approximations starting from pointwise lower and upper bounds.

2013

1. A constructive geometrical approach to the uniqueness of Markov stationary equilibrium in stochastic games of intergenerational altruism Balbus, Łukasz, Reffett, Kevin, and Wozny, Łukasz Journal of Economic Dynamics and Control vol. 37 no. 5 p. 1019 - 1039, 2013
   [IF2013 1.057, 25p MNiSW] [Abs] [HTML] [PDF]

   We provide sufficient conditions for existence and uniqueness of a monotone, Lipschitz continuous Markov stationary Nash equilibrium (MSNE) and characterize its associated Stationary Markov equilibrium in a class of intergenerational paternalistic altruism models with stochastic production. Our methods are constructive, and emphasize both order-theoretic and geometrical properties of nonlinear fixed point operators, and relate our results to the construction of globally stable numerical schemes that construct approximate Markov equilibrium in our models. Our results provide a new catalog of tools for the rigorous analysis of MSNE on minimal state spaces for OLG economies with stochastic production and limited commitment.

2. Markov stationary equilibria in stochastic supermodular games with imperfect private and public information Balbus, Łukasz, Reffett, Kevin, and Wozny, Łukasz Dynamic Games and Applications vol. 3 no. 2 p. 187-206, 2013
   [IF2013 1.041, 25p MNiSW2013] [Abs] [HTML] [PDF]

   We study a class of discounted, infinite horizon stochastic games with public and private signals and strategic complementarities. Using monotone operators defined on the function space of values and strategies (equipped with a product order), we prove existence of a Stationary Markov Nash Equilibrium via constructive methods. In addition, we provide monotone comparative statics results for ordered perturbations of our space of games. We present examples from industrial organization literature and discuss possible extensions of our techniques for studying principal-agent models.

2012

1. Intergenerational Interactions in Human Capital Accumulation Wozny, Łukasz, and Growiec, Jakub The B.E. Journal of Theoretical Economics vol. 12 no. 1 p. 1-47, 2012
   [IF2012 0.419, 15p MNiSW] [Abs] [HTML] [PDF]

   This paper considers an economy populated by a sequence of generations who decide over their consumption and investment in human capital of their immediate descendants. In such a framework, we first identify the impact of strategic interactions between consecutive generations on the time path of human capital accumulation. To this end, we characterize the decentralized Markov stationary Nash equilibrium (MSNE) and derive the sufficient conditions for its existence and uniqueness. We then provide sufficient conditions under which human capital accumulation is unambiguously (pointwise) lower in the “strategic” equilibrium than under the optimal dynastic policy, and discuss an example where this ordering does not hold. Secondly, we also run a numerical sensitivity analysis to assess the magnitude of discrepancies between the two analyzed cases and discuss the potential implications of overestimation of the human capital role if intergenerational interactions are not accounted for.

2. Stationary Markovian equilibrium in altruistic stochastic OLG models with limited commitment Balbus, Łukasz, Reffett, Kevin, and Wozny, Łukasz Journal of Mathematical Economics vol. 48 no. 2 p. 115 - 132, 2012
   [IF2012 0.321, 15p MNiSW] [Abs] [HTML] [PDF]

   We introduce a new class of infinite horizon altruistic stochastic OLG models with capital and labor, but without commitment between the generations. Under mild regularity conditions, for economies with either bounded or unbounded state spaces, continuous monotone Markov perfect Nash equilibrium (henceforth MPNE) are shown to exist, and form an antichain. Further, for each such MPNE, we can also construct a corresponding stationary Markovian equilibrium invariant distribution. We then show for many versions of our economies found in applied work in macroeconomics, unique MPNE exist relative to the space of bounded measurable functions. We also relate all of our results to those obtained by promised utility/continuation methods based upon the work of Abreu, Pearce, Stacchetti (1990). As our results are constructive, we can provide characterizations of numerical methods for approximating MPNE, and we construct error bounds. Finally, we provide a series of examples to show the potential applications and limitations of our results.