Abstract
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.
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Notes
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Strategic dynamic programming methods were first described in the seminal papers of Abreu (1988) and Abreu et al. (1986, 1990), and they were used to construct the entire set of sequential equilibrium values for repeated games with discounting. These methods have been subsequently extended in the work of Atkeson (1991), Judd et al. (2003), and Sleet and Yeltekin (2016), among others.
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- 5.
In some cases, researchers also seek further restrictions of the set of dynamic equilibria studied in these models, and they focus on Markov perfect equilibria. Hence, the question of memory in strategic dynamic programming methods has also been brought up. To answer this question, researchers have sought to generate the value correspondence in APS type methods using nonstationary Markov perfect equilibria. See Doraszelski and Escobar (2012) and Balbus and Woźny (2016) for discussion of these methods.
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The key difference between the standard APS methods and those using dual variables such as in Kydland and Prescott (1980), and Feng et al. (2014) is that in the former literature, value functions are used as the new state variables; hence, APS methods are closely related to “primal” methods, not dual methods.
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It bears mentioning that this continuity problem is related to difficulties that one finds in looking for continuity in best reply maps of the stage game given a continuation value function. It was explained nicely in the survey by Mirman (1979) for a related dynamic game in the context of equilibrium economic growth without commitment. See also the non-paternalistic altruism model first discussed in Ray (1987).
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For example, models of economic growth with strategic altruism under perfect commitment have also been studied extensively in the literature. For example, see Laitner (1979a,b, 1980, 2002), Loury (1981), and including more recent work of Alvarez (1999). Models of infinite-horizon growth with strategic interaction (e.g., “fishwars”) are essentially versions of the seminal models of Cass (1965) and Brock and Mirman (1972) , but without commitment.
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Also see Balbus et al. (2012) section 5 for a discussion of these methods for this class of models.
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For competitive economies, progress has been made. See Peralta-Alva and Santos (2010).
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See also the correction in Sundaram (1989b).
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See Dominguez (2005) for an application to models with public dept and time-consistency issues, for example.
- 15.
See Baldauf et al. (2015) for a discussion of this fact.
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However, Berg and Kitti (2014) show that this characterization is satisfied for (elementary) paths of action profiles.
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Both of these early proposals suffer from some well-known issues, including curse of dimensionality or lack of convergence.
- 19.
See Rockafellar and Wets (2009), chaps. 4 and 5, for theory of approximating sets and correspondences.
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Phelan and Stacchetti (2001) prove that a sequential equilibrium exists in this economy for each feasible sequence of tax rates and expenditures.
- 22.
It bears mentioning that in Phelan and Stacchetti (2001) and Feng et al. (2014) the authors actually used envelope theorems essentially as the new state variables. But, of course, assuming a dual representation of the sequential primal problem, this will then be summarized essentially by the KKT/Lagrange multipliers.
- 23.
It is worth mentioning that Messner et al. (2012, 2014) often do not have sufficient conditions on primitives to guarantee that dynamic games studied using their recursive dual approaches have recursive saddle point solutions for models with state variables. Most interesting applications of game theory in macroeconomics involve states variable (i.e., they are dynamic or stochastic games).
- 24.
For example, it is not a contraction in a the “sup” or “weighted sup” metric. It is a contraction (or a local contraction) under some reasonable conditions in the Thompson metric. See Messner et al. (2014) for details.
- 25.
By “most general”, we mean has the weakest assumptions on the assumed structure of Markov perfect stationary equilibria. That is, in other implementations of the generalized Euler equation method, authors often assume smooth Markov perfect stationary equilibria exist. In none of these cases do the authors actually appear to prove the existence of Markov perfect stationary equilibria within the class postulated.
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Balbus, Ł., Reffett, K., Woźny, Ł. (2018). Dynamic Games in Macroeconomics. In: Başar, T., Zaccour, G. (eds) Handbook of Dynamic Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-44374-4_18
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