Numerical Simulation of Nonoptimal Dynamic Equilibrium Models

In this paper we present a recursive method for the computation of dynamic competitive equilibria in models with heterogeneous agents and market frictions. This method is based upon a convergent operator over an expanded set of state variables. The fixed point of this operator defines the set of all Markovian equilibria. We study approximation properties of the operator as well as the convergence of the moments of simulated sample paths. We apply our numerical algorithm to two growth models, an overlapping generations economy with money, and an asset pricing model with financial frictions.Heterogeneous agents, taxes, externalities, financial frictions, competitive equilibrium, computation, simulation

Problems in the numerical simulation of models with heterogeneous agents and economic distortions

Our work has been concerned with the numerical simulation of dynamic economies with heterogeneous agents and economic distortions. Recent research has drawn attention to inherent difficulties in the computation of competitive equilibria for these economies: A continuous Markovian solution may fail to exist, and some commonly used numerical algorithms may not deliver accurate approximations. We consider a reliable algorithm set forth in Feng et al. (2009), and discuss problems related to the existence and computation of Markovian equilibria, as well as convergence and accuracy properties. We offer new insights into numerical simulation.Econometric models

Numerical simulation of nonoptimal dynamic equilibrium models

In this paper we present a recursive method for the computation of dynamic competitive equilibria in models with heterogeneous agents and market frictions. This method is based on a convergent operator over an expanded set of state variables. The fixed point of this operator defines the set of all Markovian equilibria. We study approximation properties of the operator as well as the convergence of the moments of simulated sample paths. We apply our numerical algorithm to two growth models, an overlapping generations economy with money, and an asset pricing model with financial frictions.Econometric models

Endogenous Market Incompleteness Without Market Frictions: Dynamic Suboptimality of Competitive Equilibrium in Multiperiod Overlapping Generations Economies

In this paper, we show that within the set of stochastic three-period-lived OLG economies with productive assets (such as land), markets are necessarily sequentially incomplete, and agents in the model do not share risk optimally. We start by characterizing perfect risk sharing and find that it requires a state-dependent consumption claims which depend only on the exogenous shock realizations. We show then that the recursive competitive equilibrium of any overlapping generations economy with weakly more than three generations is not strongly stationary. This then allows us to show directly that there are short-run Pareto improvements possible in terms of risk-sharing and hence, that the recursive competitive equilibrium is not Pareto optimal. We then show that a financial reform which eliminates the equity asset and replaces it with zero net supply insurance contracts (Arrow securities) will implement to Pareto optimal stochastic steady-state known to exist in the model. Finally, we also show via numerical simulations that a system of government taxes and transfers can lead to a Pareto improvement over the competitive equilibrium in the model.

Equilibria, Fixed Points, and Complexity Classes

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in 2-player normal form games, and (mixed) Nash equilibria in normal form games with 3 (or more) players. This paper reviews the underlying computational principles and the corresponding classes

THE UNIT ROOT PROPERTY WHEN MARKETS ARE SEQUENTIALLY INCOMPLETE

We consider pure exchange, one good OLG economies under stationary Markov uncertainty. It is known that when markets are sequentially complete, a stationary equilibrium at which the agents common matrix of intertemporal rates of substitution has a Perron root which is less than or equal to one is conditionally Pareto optimal (CPO). We assume that there exists a long-lived dividend paying asset and show that if dividends are strictly positive then the relation between the unit root condition and optimality holds even if markets are not sequentially complete. However, every equilibrium allocation is shown to be constrained CPO under the additional requirement that assets be freely disposable, which seems reasonable when dividends are positive and whose importance was pointed out by Santos and Woodford (1997) in their work on bubbles; this fact undermines the relation between the unit root property and optimality. The relation is less clear when dividends and asset prices are allowed to be negative in some states.Stochastic Overlapping Generations Models, IncompleteMarkets

LONG-LIVED ASSETS, INCOMPLETE MARKETS, AND OPTIMALITY

We consider general OLG economies under uncertainty, with dividend paying assets of infinite maturity and money, and in which one good is available for consumption. We study the optimality properties of equilibria when asset markets are allowed to be sequentially incomplete. We show that if equilibrium in asset markets has to be restored once an intervention has been made, then all non-monetary competitive equilibria are locally constrained optimal. We proceed to a notion of optimality which allows asset markets to not clear and provide a complete characterization of those equilibria that are optimal in terms of the prices and dividends of assets of infinite maturity and feasible portfolio reassignments. Results for various special cases follow; in particular, we show that if dividends are non-negative and assets are freely disposable then every non-monetary equilibrium allocation is optimal. Other results shed light on the role played by money vis-a-vis other assets of infinite maturity in determining the optimality properties of equilibria when markets are sequentially complete/incomplete and free disposal of assets is or is not allowed.Stochastic Overlapping Generations Models; Sequentially Incomplete Markets