Recursive Equilibrium Research Articles

Uniformly self-justified equilibria

We consider dynamic stochastic economies with heterogeneous agents and introduce the concept of uniformly self-justified equilibria (USJE)—temporary equilibria for which expectations satisfy the following rationality requirements: i) individuals' forecasting functions for the next period's endogenous variables are assumed to lie in a compact, finite-dimensional set of functions, and ii) the forecasts constitute the best uniform approximation to a selection of the equilibrium correspondence. We show that in contrast to rational expectations equilibria, USJE always exist, and we develop a simple algorithm to compute them. As an application, we discuss a stochastic overlapping generations exchange economy. We give an example where recursive (rational expectations) equilibria fail to exist and explain how to construct USJE for that example. In addition, we provide numerical examples to illustrate our computational method.

Recursive equilibria in dynamic economies with bounded rationality

This paper provides a general framework to model bounded rationality in dynamic stochastic general equilibrium models with infinitely lived heterogeneous agents. A boundedly rational agent is associated with an information set $I$ and an extra parameter $\epsilon$, which can be interpreted as the ``level of irrationality''. To make decisions, the boundedly rational agent forms a belief of a stationary joint distribution of the exogenous and endogenous variables and uses the marginal distribution (conditional on $I$) to form forecasts. If the equilibrium distribution stays within $\epsilon$ of the forecasted next-period distribution, the agent would consider it as $\epsilon$-stationary. In equilibrium, each agent maximizes utility with an $\epsilon$-stationary belief and markets clear. The main theorem of this paper shows that for any strictly positive $\epsilon$, a recursive equilibrium exists. With a quantifiable ``level of irrationality'', the model incorporates many behavioral economics models as well as rational-expectations models with computational approximations into a unified framework. An illustration example is presented to show that the boundedly rational recursive equilibrium may substantially enrich the asset pricing dynamics of the rational-expectations model even for small $\epsilon$.

Impact assessment and a fiscal recovery policy for tsunami risk: GIS and the general equilibrium approach in Hakodate city, Japan

Given Japan’s substantial exposure to many kinds of natural hazards, such as earthquakes, tsunamis, and typhoons, it has been a priority to invest in resilience, guided by evidence-based modeling. In 2011, the Great East Japan Earthquake and Tsunami became the costliest natural disaster ever recorded. This study applied a geographic information system using assumed tsunami-affected data calibrated in a recursive computable general equilibrium model to perform an economic impact assessment and an estimated recovery budget. We simulated 100 years of tsunamis and a 10-year sectoral recovery package for the sectors related to the ocean economy, such as kelp, net fishery, squid, other fisheries, food processing, and recreation, with a capital-use subsidy policy regarding investment strategy. We found that the aqua sector is incredibly vulnerable and would not recover with the capital-use subsidy within Hakodate City’s financial capability. Nevertheless, the recovery policy could still ease output price changes. On the other hand, the recreation sector could recover to pre-disaster conditions, but at huge fiscal and social costs. Meanwhile, the food processing sector’s recovery could generate social benefits and have a spillover effect on other fisheries sectors. The application of geographic information system in tsunami-prone areas could strengthen the precision of economic analysis. Such evidence-based modeling could visualize the economic impact to assist policymakers and stakeholders in foreseeing disaster risk and implementing more effective building resilience measures.

Recursive equilibrium in Krusell and Smith (1998)

This paper uses the tools developed in the literature on dynamically incomplete markets with finite agents to study the large economy with a continuum of agents and both aggregate and idiosyncratic shocks in Krusell and Smith (1998). It establishes the existence of sequential competitive equilibrium, generalized recursive equilibrium, recursive equilibrium with an extended-state space, and characterizes several important properties of the equilibrium variables. The equilibrium process admits an ergodic measure, which enables the application of the ergodic theorem for simulating and calibrating the model. Without aggregate shocks, the existence and some characterization results carry over to economies with only idiosyncratic shocks such as Huggett (1997)'s economy. However, the existence of recursive equilibrium with the natural minimal state space in Krusell and Smith's economy remains elusive, as in finite-agent incomplete markets economies.

Lipschitz recursive equilibrium with a minimal state space and heterogeneous agents

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.

Dynamic performance evaluation with deadlines: The role of commitment

We consider an environment in which a principal hires an agent and evaluates his productivity over time. The problem is embedded in a continuous‐time model with both hidden action and hidden information, where the principal must induce the agent to exert effort to facilitate her learning process. The value of committing to a deadline is examined in this environment, and factors which make the deadline more profitable are identified. Our framework generates a unique recursive equilibrium structure under no commitment which can be exploited to obtain a full characterization of the unique pure‐strategy equilibrium. The analysis allows us to evaluate the exact value of commitment for any given set of parameters and provides insight into when it is beneficial to commit to an evaluation deadline at the outset of a relationship.

The Impact of an Early Career Recession on Schooling and Lifetime Welfare

This paper evaluates the long-term welfare consequences from experiencing a recession as youths, taking into account the impact on schooling, future job mobility, human capital accumulation, labor supply and wages. The paper also explores the mechanisms that account for lifetime wage changes by decomposing those changes into different channels: changes from schooling, from work experience, and from job mobility. To achieve these goals, this paper develops and estimates a search equilibrium model with heterogeneous agents and aggregate shocks. The model is an extension of a directed search model, the Block Recursive Equilibrium framework of Menzio and Shi (2010), which remains tractable when it is solved outside of the steady state. The counterfactual analysis shows that experiencing the 1981-1982 recession at age 16-22 causes a 2.2% to 3.0% loss in lifetime welfare. Endogenizing schooling decision avoids overestimation of the welfare loss for two reasons: first, there exists the selection of labor market entry timing and high ability individuals are more likely to postpone entry into the labor market during a recession; second, the additional schooling caused by an early career recession increases future wages and employment rate. The wage decomposition shows that the loss from job mobility explains the majority of the wage loss during the recession, and the loss in experience and tenure persists long after the recession.

Existence and Uniqueness of Recursive Equilibria With Aggregate and Idiosyncratic Risk

In this paper, I study the existence and uniqueness of recursive equilibria in economies with aggregate and idiosyncratic risk. Rather than relying on compactness to establish existence, I exploit the monotonicity property of the equilibrium model and rely on arguments from convex analysis. This methodology does not only give rise to a convergent iterative procedure, but more strikingly, it also yields uniqueness. To illustrate my theoretical results, I establish sufficient conditions for the existence and uniqueness of solutions to the stochastic growth model as in Krusell and Smith (1998) and the heterogeneous-agent exchange economy as in Huggett (1993) with aggregate risk.

Bubbly Markov equilibria

Bubbly Markov equilibria (BME) are recursive equilibria on the natural state space which admit a non-trivial bubble. The present paper studies the existence and properties of BME in a general class of overlapping generations economies with capital accumulation and stochastic production shocks. Using monotone methods, we develop a general approach to construct Markov equilibria and provide necessary and sufficient conditions for these equilibria to be bubbly. Our main result shows that a BME exists whenever the bubbleless equilibrium is Pareto inefficient due to either overaccumulation of capital or inefficient risk sharing between generations.

Assessment of model validation outcomes of a new recursive spatial equilibrium model for the Greater Beijing

Robust calibration and validation of applied urban models are prerequisites for their successful, policy-cogent use. This is particularly important today when expert assessment is questioned and closely scrutinized. This paper proposes a new model calibration-validation strategy based on a spatial equilibrium model that incorporates multiple time horizons, such that the predictive capabilities of the model can be empirically tested. The model is implemented for the Greater Beijing city region and the model validation strategy is demonstrated over the Census years 2000 to 2010. Through forward/backward forecasting, the model validation helps to verify the stability of the model parameters as well as the predictive capabilities of the recursive equilibrium framework. The proposed modelling strategy sets a new standard for verifying and validating recursive equilibrium models. We also consider the wider implications of the approach.

Comparing recursive equilibrium in economies with dynamic complementarities and indeterminacy

We develop a new multistep monotone map approach to characterize minimal state-space recursive equilibrium for a broad class of infinite horizon dynamic general equilibrium models with positive externalities, dynamic complementarities, public policy, equilibrium indeterminacy, and sunspots. This new approach is global, defined in the equilibrium version of the household’s Euler equation, applies to economies for which there are no known existence results, and existing methods are inapplicable. Our methods are able to distinguish different structural properties of recursive equilibria. In stark contrast to the extensive body of existing work on these models, our methods make no appeal to the theory of smooth dynamical systems that are commonly applied in the literature. Actually, sufficient smoothness to apply such methods cannot be established relative to the set of recursive equilibria. Our partial ordering methods also provide a qualitative theory of equilibrium comparative statics in the presence of multiple equilibrium. These robust equilibrium comparison results are shown to be computable via successive approximations from subsolutions and supersolutions in sets of candidate equilibrium function spaces. We provide applications to an extensive literature on local indeterminacy of dynamic equilibrium.

Collateral, Liquidity and Debt Sustainability

We study the sustainability of public debt in a closed production economy where a benevolent government chooses fiscal policies, including haircuts on its outstanding debt, in a discretionary manner. Government bonds are held by domestic agents to smooth consumption over time and because they provide collateral and liquidity services. We characterize a recursive equilibrium where public debt amounts to a sizeable fraction of output in steady state and is nevertheless fully serviced by the government. In a calibrated economy, steady state debt amounts to around 84% of output, the government's default threshold is at around 94% of output, and the haircut on outstanding debt at this threshold is around 40%. Both reputational costs of default and contemporaneous costs due to lost collateral and liquidity are essential to generate these empirically plausible predictions.

Dynamic Performance Evaluation with Deadlines: The Role of Commitment

We consider an environment in which a principal hires an agent and evaluates his productivity over time in an ongoing relationship. The problem is embedded in a continuous-time model with both hidden action and hidden information, where the principal must induce the agent to exert effort to facilitate her learning process. The value of committing to a deadline is examined in this environment, and factors which make the deadline more profitable are identified. Our framework generates a unique recursive equilibrium structure under no commitment which can be exploited to obtain a full characterization of equilibrium. The analysis allows us to evaluate the exact value of commitment for any given set of parameters and provides insight into when it is beneficial to commit to an evaluation deadline at the outset of a relationship.

Recursive Equilibria in Dynamic Economies With Stochastic Production

In this paper we prove the existence of recursive equilibria in stochastic production economies with infinitely lived agents and incomplete financial markets. We consider a general dynamic model with several commodities, which encompasses heterogeneous agent versions of both the Lucas asset pricing model and the stochastic neo-classical growth model as special cases. Our main assumption is that there are atomless shocks to fundamentals that have a purely transitory component and a component that does not depend on last period's shocks directly.

Existence and uniqueness of equilibrium in a distorted dynamic small open economy

We provide a set of sufficient conditions for existence and uniqueness of recursive equilibrium for a class of small open economies with distortion. The second welfare theorem is not available in this framework and usual approach for studying optimal neoclassical growth models is also not applicable.

Existence of Generalized Recursive Equilibrium in Krusell and Smith (1998)

In this paper, I define and show the existence of a generalized recursive equilibrium in Krusell and Smith (1998)’s economy with both aggregate and idiosyncratic shocks. The method to establish existence using finite horizon approximation applies equally well to economies with a finite number of agents or with a continuum of agents. With a continuum of agents, my proof relies on the compactness of the space of probability measures over a bounded interval endowed with weak topology and of the space of Lipschitz continuous value functions and weakly increasing policy functions. These properties enable the use of the Kakutani-Glicksberg-Fan Fixed Point Theorem for infinite dimensional spaces. The existence proof suggests a numerical algorithm to compute a global nonlinear recursive equilibrium. I implement the algorithm and analyze its numerical solutions for two-agent economies.

Comparing Recursive Equilibrium in Economies with Dynamic Complementarities and Indeterminacy

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 existing methods do not apply. Our methods are global, based on monotone operators defined on Euler equations, and do not appeal to the theory of smooth dynamical systems that are commonly applied in the literature. Rather, using partial ordering methods, we provide a qualitative theory of equilibrium comparative statics in the presence of multiple equilibrium. These comparison results are computable via successive approximations from upper and lower bounds of particular sets of functions. We provide applications of our results to an extensive literature on local indeterminacy of dynamic equilibrium.

Recursive equilibrium with Price Perfect Foresight and a minimal state space

This paper analyzes general equilibrium models with finite heterogeneous agents who anticipate future prices through a price expectation function with or without accuracy. I show the existence of a recursive equilibrium with a minimal state space through the Kakutani–Fan–Glicksberg fixed point theorem. Moreover, any such recursive equilibrium implements a sequential equilibrium and its uniqueness implies its continuity. Particularly, I prove that an agent making persistent errors in the price expectation function is driven out of the market in any sequential equilibrium implemented by a continuous recursive equilibrium. This result is established under the condition that exogenous variables converge in probability and assuming that the relative variability of all stochastic discount factors is low.

Housing Tax Reform and Foreclosure Rates

Since the housing market collapse of 2008 and the ensuing economic recession were amplified by rising foreclosure rates, we study the potential impact of housing taxation reform on foreclosures. Two ways in which housing receives preferential treatment are the tax deductibility of mortgage interest payments and the non-recognition of imputed rents. We employ a recursive equilibrium model in which heterogeneous agents make housing tenure choices and in which mortgage default occurs in equilibrium. We calibrate the model and use the calibration results to quantify the effects of reforming these parts of the tax regime. We find that elimination of the mortgage interest tax deduction reduces the foreclosure rate by 12.4 percent. Alternatively, taxing imputed rents would reduce it by 32.5 percent.

The competitiveness of Canadian softwood lumber: a disaggregated trade-flow analysis

A recursive dynamic spatial equilibrium model is used to examine the global competitiveness of Canadian softwood lumber. To address the restrictive assumption of softwood lumber homogeneity, this study disaggregates softwood lumber into two product groups: (i) higher grade lumber that includes appearance, select structural-grade lumber, and Japanese J-grade lumber; and (ii) lower grade lumber that includes the United States dimension lumber that is commonly used in construction and utility- and economy-grade lumber. Factors that may affect global softwood lumber markets are simulated in the model to project global softwood lumber trade flows from 2012 to 2021. The results indicate that the reduced lumber supply in western Canada caused by the mountain pine beetle (Dendroctonus ponderosae Hopkins) infestation combined with demand increases in several regions of the world will contribute to a global increase in softwood lumber prices. Our results suggest that the global price increase will be greater for lower grade softwood lumber than for higher grade lumber. The United States and China will continue to be the top two markets for lower grade Canadian softwood lumber. Although Canadian exports of lower grade softwood lumber to the United States are expected to increase marginally over time in response to the recovery of American housing starts, softwood lumber exports to China are expected to drop significantly, and it is forecasted that exports from the Russian Federation will fill that void. These findings provide strong market signals for both forest managers and the forest-products industry to assess supply chain profitability and adjust production planning accordingly.