Computing minimal state space recursive equilibrium in OLG models with stochastic production

Using order-theoretic methods, we derive sufficient conditions for the existence, characterization, and computation of minimal state space recursive equilibrium (RE), as well as Stationary Markov equilibrium (SME) for various classes of stochastic overlapping generations models. In contrast to previous work, our methods focus on constructive methods. Our existence results are obtained for models that include public policy (e.g., social security policies, transfers, taxes, etc), production nonconvexities, elastic labor supply, non-monotone income processes, and long-lived agents. We distinguish conditions under which there exist various subclasses of minimal state space RE, including bounded, monotone, non-monotone, semicontinuous, Lipschitz continuous RE. Finally, we provide monotone equilibrium comparative statics results on the space of economies for some RE.     

Construction of a state space for interrelated securities with an application to temporary equilibrium theory

Summary We construct an endogenous state space in an exchange economy with possibly infinite horizon. Every period agents trade securities whose payoffs depend on future dividends and asset prices. We reject the perfect foresight assumption on the ground that agents have not only limited knowledge of other individuals' endowments and preferences, but also limited capacity to compute equilibria. We choose instead absence of arbitrage as the principle which allows agents to determine if a system of future prices is possible. We give an alogrithm to compute the set of nonarbitrage prices every period, with both finite and infinite horizon. We then apply this endogenous structure of uncertainty to an infinite horizon temporary equilibrium model.I would like to thank Professor Donald Brown for his constant help and guidance. I have also greatly benefited from helpful discussions with Professors Jacques Drèze, Bernard Dumas, Mordecai Kurz, Carsten Nielsen, Jan Werner, and Ho-Mou Wu.     

p-Dominance and Equilibrium Selection under Perfect Foresight Dynamics

This paper studies equilibrium selection based on a class of perfect foresight dynamics and relates it to the notion of p-dominance. A continuum of rational players is repeatedly and randomly matched to play a symmetric n×n game. There are frictions: opportunities to revise actions follow independent Poisson processes. The dynamics has stationary states, each of which corresponds to a Nash equilibrium of the static game. A strict Nash equilibrium is linearly stable under the perfect foresight dynamics if, independent of the current action distribution, there exists a consistent belief that any player necessarily plays the Nash equilibrium action at every revision opportunity. It is shown that a strict Nash equilibrium is linearly stable under the perfect foresight dynamics with a small degree of friction if and only if it is the p-dominant equilibrium with p<1/2. It is also shown that if a strict Nash equilibrium is the p-dominant equilibrium with p<1/2, then it is uniquely absorbing (and globally accessible) for a small friction (but not vice versa). Set-valued stability concepts are introduced and their existence is shown. Journal of Economic Literature Classification Numbers: C72, C73.     

Recursive equilibrium in endogenous growth models with incomplete markets

This paper analyzes the existence of recursive equilibria in a class of convex growth models with incomplete markets. Households have identical CRRA-preferences, production displays constant returns to scale with respect to physical and human capital, and all markets are competitive. There are aggregate productivity shocks that affect aggregate returns to physical and human capital investment (stock returns and wages), and there are idiosyncratic shocks to human capital (idiosyncratic depreciation shocks) that only affect individual human capital returns. Aggregate and idiosyncratic shocks follow a joint Markov process. Conditional on the aggregate state, idiosyncratic shocks are independently distributed over time and identically distributed across households. Finally, households have the opportunity to trade assets in zero net supply with payoffs that depend on the aggregate shock, but markets are incomplete in the sense that there are no assets with payoffs depending on idiosyncratic shocks. It is shown that there exists a recursive equilibrium for which equilibrium prices (returns) only depend on the exogenous aggregate shock variable (the wealth distribution is not a relevant state variable). Moreover, the allocation associated with this recursive equilibrium is identical to the equilibrium allocation of an economy in which households live in autarky and face both aggregate and idiosyncratic risk.I would like to thank for helpful comments Peter Howitt, Bob Lucas, Michael Magill, Tomo Nakajima, Herakles Polemarchakis, Martine Quinzii, Kevin Reffett, an anonymous referee, and seminar participants at various universities and conferences.     

Lexicographic expected utility with a subjective state space

This paper provides a model that allows for a criterion of admissibility based on a subjective state space. For this purpose, we build a non-Archimedean model of preference with subjective states, generalizing Blume et al. (Econometrica 59:61–79, 1991), who present a non-Archimedean model with exogenous states; and Dekel et al. (Econometrica 69:891–934, 2001), who present an Archimedean model with an endogenous state space. We interpret the representation as modeling an agent who has several “hypotheses” about her state space, and who views some as “infinitely less relevant” than others.     

Price and quantity regulation in general equilibrium

We consider a general equilibrium model with a production externality (e.g. pollution), where the regulator does not observe firm productivity shocks. We examine quantity (permit) regulation and price (tax) regulation. The quantity of permits issued by the regulator are independent of the productivity shock, since shocks are unobserved. Price regulation implies use of the regulated input is an increasing function of the productivity shock because firms take advantage of a good productivity shock by increasing input use. Thus price regulation generates higher average, but more variable, production. Therefore, we show that in general equilibrium the relative advantage of quantity versus price regulation depends not only on the slopes of marginal benefits and costs, but on general equilibrium effects such as risk aversion. The general equilibrium effects are often more important than the slopes of the marginal benefits and cost curves. In the simplest model, a reasonable risk aversion coefficient implies quantity regulation generates higher welfare regardless of the benefit function.     

Contract renegotiation in a continuous state space

Summary. When players are unable to write complete state contingent contracts it is shown, within the context of a non-cooperative contracting-renegotiation game, that the only subgame perfect equilibrium allocations are those that correspond to the set of first-best allocations. Players are able to implement this set of allocations by signing an initial contract that is subsequently renegotiated in all states of the world. The contracting-renegotiation problem is complicated in an interesting way by assuming that the state space is continuous. The issue of the existence of an initial contract, that is subsequently renegotiated to the set of first-best allocations, must be resolved. Unlike Aghion, Dewatripont and Rey [1994], the results here do not require nor depend upon the comonotonicity of the objective functions. Received: January 27, 1995; revised version July 1, 1996     

Valuation of assets on a markov state space

In the absence of opportunities for arbitrage, a very general valuation equation is derived for risky assets. This derivation exploits the linear operators associated with the transition probabilities of the state process.     

Limited information minimal state variable learning in a medium-scale multi-country model

Rational expectations has been the dominant way to model expectations, but the literature has quickly moved to a more realistic assumption of boundedly rational learning where agents are assumed to use only a limited set of information to form their expectations. A standard assumption is that agents form expectations by using the correctly specified reduced form model of the economy, the minimal state variable solution (MSV), but they do not know the parameters. However, with medium-sized and large models the closed-form MSV solutions are difficult to attain given the large number of variables that could be included. Therefore, agents base expectations on a misspecified MSV solution. In contrast, we assume that agents know the deep parameters of their own optimising frameworks. However, they are not assumed to know the structure nor the parameterisation of the rest of the economy, nor do they know the stochastic processes generating shocks hitting the economy. In addition, agents are assumed to know that the changes (or the growth rates) of fundament variables can be modelled as stationary ARMA(p,q) processes, the exact form of which is not, however, known by agents. This approach avoids the complexities of dealing with a potential vast multitude of alternative misspecified MSVs.Using a new multi-country euro area model with boundedly estimated rationality we show that this approach is compatible with the same limited information assumption that was used in deriving and estimating the behavioural equations of different optimising agents. We find that there are strong differences in the adjustment path to the shocks to the economy when agents form expectations using our learning approach compared to expectations formed under the assumption of strong rationality. Furthermore, we find some variation in expansionary fiscal policy in periods of downturns compared to boom periods.     

Price taking equilibrium in economies with multiple memberships in clubs and unbounded club sizes

We model an economy with clubs (or jurisdictions) where individuals may belong to multiple clubs and where clubs sizes are arbitrary—clubs may be restricted to consist of only one or two persons, or as large as the entire economy, or anything in-between. Notions of price-taking equilibrium and the core, both with communication costs, are introduced. These notions take into account that there is a small communication cost of deviating from a given outcome. We demonstrate that, given communication costs, for all sufficiently large economies the core is nonempty and the set of price-taking equilibrium outcomes is equivalent to the core.     

Non-stationary,stable Markov processes on a continuous state space

We extend our previous result on simple stable Markov (SSM) processes to the case where the state space is continuous. As anapplication we show the existence of a competitive general equilibrium of a cobweb model where price volatility is generated both by exogenous shocks and by stochastic, so called generating variables (that may be interpreted as sunspots) that govern the correlation of the rational beliefs of individual agents. I would like to thank Danish Social Science Foundation, The Carlsberg Foundation, Stanford Institute for Theoretical Economics (SITE) and Universita Cattolica for financial help. I would also like to thank Trinidad Casasus, Mordecai Kurz, an anonymous referee as well as participants at SITE, ESAM (2004) and ESEM (2004) for helpful discussions and comments. Peter Harremoes provided me with an illuminating counter example and Hiro Nakata provided many comments that helped improve the exposition of the paper. Part of this work is from a paper previously circulated under the title: “Sunspot rational belief structures: anonymity and endogenous uncertainty”.     

Forecasting with a state space time-varying parameter VAR model: Evidence from the Euro area

Standard VAR and Bayesian VAR models are proven to be reliable tools for modeling and forecasting, yet they are still linear and they do not consider time-variation in parameters. VAR modeling is subject to the Lucas critique and fails to take into account the inherent nonlinearities of the economy, while it can only be utilized in the analysis of stationary series and in many cases stationarity assumptions are too restrictive. A novel time-varying multivariate state-space estimation method for vector autoregression models is introduced. For the time-varying parameter model (TVP-VAR), the parameters are estimated using a multivariate specification of the standard Kalman filter (Harvey, 1990) combined with a suitable extension of the univariate methodology framework of Kim and Nelson (1999). The TVP-VAR model as well as standard VARs and Bayesian VARs, are used in a comparative investigation of their predicting performance for the monthly IP, CPI and Euribor rate of the EU economy. The total period covers 1999:1–2011:2 with an out-of-sample testing period of 2007:2 to 2011:2, which included the US sub-prime and the EU debt crisis sub-periods. The results varied across the investigated time series and indicated that the TVP-VAR model consistently outperforms the other models in case of the EU monthly CPI, while different specifications of the VAR and BVAR models for the IP and Euribor series provide with better forecasting performance. Interestingly, the robustness analysis showed that the TVP-VAR model provided with enhanced predictability in particular during “crisis times”.     

Price discrimination with a resource constraint

Introducing an aggregate resource constraint into Mussa and Rosen's [Mussa, M., Rosen, S., 1978. Monopoly and product quality. Journal of Economic Theory 18 (2), 301–317] model of monopoly price discrimination induces an upwards distortion in product quality for high valuation consumers because the standard downward distortions in product quality for other consumers relax the resource constraint.     

Identification of house price bubbles using user cost in a state space model

This article studies how much variation in house prices results from nonfundamental factors. We propose a relative valuation approach to quantifying a bubble in housing by incorporating the housing User Cost into a state space model. We find that UK house prices were undervalued from January 1995 to May 2001 and subsequently moved into a bubble over the period to October 2012. Our results support the bounded rationality hypothesis in the long run. However, we also find that the irrational and the rational expectation hypotheses can coexist in the short run when explosive bubbles are driven by price dynamics.     

Hypothetical Knowledge and Games with Perfect Information

Standard structures of information, in particular partition structures, are inadequate for the modeling of strategic thinking. They fail to capture the inner structure of hypotheses players make about situations they know will not occur. An extension of the partition structure is proposed in which such hypotheses can be modeled in detail. Hypothetical knowledge operators are defined for extended structures and are axiomatically characterized. The use of extended structures to model games with complete information is demonstrated. A sufficient condition is derived for players to play the backward induction in such games.Journal of Economic LiteratureClassification Numbers: C72, D81, D82.     

Selecting a unique competitive equilibrium with default penalties

The enlargement of the general-equilibrium structure to allow for default subject to penalties results in a construction of a simple mechanism for selecting a unique competitive equilibrium. We consider economies for which a common credit money can be applied to uniquely select each of the competitive equilibria with suitable default penalties. We identify two classes of such economies.     

Conditions for a CAPM equilibrium with positive prices

This paper examines the conditions required to guarantee positive prices in the CAPM. Positive prices imply an upper bound on the equity premium. This upper bound depends on the degree of diversity of firms’ fundamentals, and it is independent of investors’ preferences. In economies with realistically diverse assets the only positive-price CAPM equilibrium theoretically possible is a degenerate one, with zero equity premium. Furthermore, when specific standard investors’ preferences are assumed, the CAPM equilibrium with positive prices may be altogether impossible. A possible solution to these fundamental problems may be offered by the segmented-market version of the model.     

Recursive structure and equilibria in games with private monitoring

Summary. In each stage of a repeated game with private monitoring, the players receive payoffs and privately observe signals which depend on the players' actions and the state of world. I show that, contrary to a widely held belief, such games admit a recursive structure. More precisely, I construct a representation of the original sequential problem as a sequence of static games with incomplete information. This establishes the ground for a characterization of strategies and, hence, of behavior in interactive-decision settings where private information is present. Finally, the representation is used to give a recursive characterization of the equilibrium payoff set, by means of a multi-player generalization of dynamic programming. Received: February 11, 2002; revised version: July 22, 2002 RID="*" ID="*" I am very grateful to In-Koo Cho, Larry Epstein, Denis Gromb, Stephen Morris, Paolo Siconolfi, Lones Smith and Max Stinchcombe for several insights and suggestions. A referee's comments helped improving the exposition. Finally, I wish to thank the participants to the seminars at MEDS, NYU, Columbia University, Caltech, UCLA, University of Rochester, University of Texas-Austin, Northwestern Summer Microeconomics Conference 98, Summer in Tel Aviv 98, and NASM98.