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Abstract:
The rational expectations hypothesis swept through macroeconomics during the 1970's and permanently altered the landscape. It remains the prevailing paradigm in macroeconomics, and rational expectations is routinely used as the standard solution concept in both theoretical and applied macroeconomic modelling. The rational expectations hypothesis was initially formulated by John F. Muth Jr. in the early 1960s. Together with Robert Lucas Jr., Thomas (Tom) Sargent pioneered the rational expectations revolution in macroeconomics in the 1970s. We interviewed Tom Sargent for Macroeconomic Dynamics.

Abstract:
Using New Keynesian models, we compare Friedman's k-percent money supply rule to optimal interest rate setting, with respect to determinacy, stability under learning and optimality. We first review the recent literature. Open-loop interest rate rules are subject to indeterminacy and instability problems, but a properly chosen expectations-based rule yields determinacy and stability under learning, and implements optimal policy. We then show that Friedman's rule also can generate equilibria that are determinate and stable under learning. However, in computing the mean quadratic welfare loss, we find that for calibrated models Friedman's rule performs poorly compared to the optimal interest rate rule.

Monetary policy, determinacy, stability under learning

Abstract:
A fundamentals based monetary policy rule, which would be the optimal monetary policy without commitment when private agents have perfectly rational expectations, is unstable if in fact these agents follow standard adaptive learning rules. This problem can be overcome if private expectations are observed and suitable incorporated into the policy maker's optimal rule. These strong results extend to the case in which there is simultaneous learning by the policy maker and the private agents. Our findings show the importance of conditioning policy appropriately, not just on fundamentals, but also directly on observed household and firm expectations.

Abstract:
A fundamentals based monetary policy rule, which would be the optimal monetary policy without commitment when private agents have perfectly rational expectations, is unstable if in fact these agents follow standard adaptive learning rules. This problem can be overcome if private expectations are observed and suitably incorporated into the policy maker's optimal rule. These strong results extend to the case in which there is simultaneous learning by the policy maker and the private agents. Our findings show the importance of conditioning policy appropriately, not just on fundamentals, but also directly on observed household and firm expectations.

monetary policy, learning, expectational stability, optimal discretionary policy

Abstract:
A fundamentals-based monetary policy rule, which would be the optimal monetary policy without commitment when private agents have perfectly rational expectations, is unstable if in fact these agents follow standard adaptive learning rules. This problem can be overcome if private expectations are observed and suitably incorporated into the policy maker's optimal rule. These strong results extend to the case in which there is simultaneous learning by the policy maker and the private agents. Our findings show the importance of conditioning policy appropriately, not just on fundamentals, but also directly on observed household and firm expectations.

Adaptive learning, instability, stability, private expectations

Abstract:
A fundamentals based monetary policy rule, which would be the optimal monetary policy without commitment when private agents have perfectly rational expectations, is unstable if in fact these agents follow standard adaptive learning rules. This problem can be overcome if private expectations are observed and suitably incorporated into the policy maker's optimal rule. These strong results extend to the case in which there is simultaneous learning by the policy maker and the private agents. Our findings show the importance of conditioning policy appropriately, not just on fundamentals, but also directly on observed household and firm expectations.

expectations, stability, monetary policy, least squares learning

Abstract:
An economy exhibits structural heterogeneity when the forecasts of different agents have different effects on the determination of aggregate variables. Various forms of structural heterogeneity can arise and we study the important case of economies in which agents' behavior depends on forecasts of aggregate variables and show how different forms of heterogeneity in structure, forecasts, and adaptive learning rules affect the conditions for convergence of adaptive learning towards rational expectations equilibrium. Results are applied to the market model with supply lags and a New Keynesian model of interest rate setting.

Adaptive Learning, Expectations Formation, Stability of Equilibrium, Market Model, Monetary Policy

Abstract:
The current financial crisis, which has lasted almost one and a half years, is the 19th such crisis in the post-war period in advanced economies. Recent literature classifies the Nordic crises in Norway, Sweden and Finland in late 1980's and early 1990’s among the Big Five crises that have happened before the current crisis, which is now of a global nature. This paper outlines the developments of the Nordic crises, reasons behind them and crisis management by the authorities. Relatively more emphasis is placed on the Finnish crisis, as it was the deepest one. The paper concludes by considering the lessons that can be drawn from the Nordic crises.

financial deregulation, bank lending, overheating, financial crisis

Abstract:
Expectations about the future are central for determination of current macroeconomic outcomes and the formulation of monetary policy. Recent literature has explored ways for supplementing the benchmark of rational expectations with explicit models of expectations formation that rely on econometric learning. Some apparently natural policy rules turn out to imply expectational instability of private agents' learning. We use the standard New Keynesian model to illustrate this problem and survey the key results for interest-rate rules that deliver both uniqueness and stability of equilibrium under econometric learning. We then consider some practical concerns such as measurement errors in private expectations, observability of variables and learning of structural parameters required for policy. We also discuss some recent applications, including policy design under perpetual learning, estimated models with learning, recurrent hyperinflation, and macroeconomic policy to combat liquidity traps and deflation.

imperfect knowledge, learning, interest-rate setting, fluctuations, stability, determinacy

Abstract:
Commitment in monetary policy leads to equilibria that are superior to those from optimal discretionary policies. A number of interest-rate reaction functions and instrument rules have been proposed to implement or approximate commitment policy. We assess these rules in terms of whether they lead to a rational expectations equilibrium that is both locally determinate and stable under adaptive learning by private agents. A reaction function that appropriately depends explicitly on private sector expectations performs particularly well on both counts.

Abstract:
Commitment in monetary policy leads to equilibria that are superior to those from optimal discretionary policies. A number of interest rate reaction functions and instrument rules have been proposed to implement or approximate commitment policy. We assess these optimal reaction functions and instrument rules in terms of whether they lead to an RE equilibrium that is both locally determinate and stable under adaptive learning by private agents. A reaction function that appropriately depends explicitly on private expectations performs well on both counts.

Commitment, interest rate setting, adaptive learning, stability determinacy

Abstract:
Full commitment in monetary policy leads to equilibria that are superior to those from optimal discretionary policies. Different types of reactions functions to implement and instrument rules to approximate full commitment have been proposed in the literature. We assess optimal reaction functions and instrument rules, in terms of whether they lead to an RE equilibrium that is both locally determinate and stable under adaptive learning by private agents. The reaction function that appropriately depends explicitly on private expectations performs best on both counts.

Commitment, Interest Rate Setting, Adaptive Learning, Stability, Determinacy

Abstract:
We review the recent work on interest rate setting, which emphasizes the desirability of designing policy to ensure stability under learning. Appropriately designed expectations-based rules can yield optimal rational expectations equilibria that are both determinate and stable under learning. Some simple instrument rules and approximate targeting rules also have these desirable properties. We discuss various complications in implementing optimal policy, including the observability of key variables and the required knowledge of structural parameters. An additional issue that we take up concerns the implications of expectation shocks not arising from transitional learning effects.

Commitment, interest rate setting, adaptive learning, stability, determinacy, expectations shocks

Abstract:
We review the recent work on interest rate setting, which emphasizes the desirability of designing policy to ensure stability under private agent learning. Appropriately designed expectations based rules can yield optimal rational expectations equilibria that are both determinate and stable under learning. Some simple instrument rules and approximate targeting rules also have these desirable properties. We take up various complications in implementing optimal policy, including the observability of key variables and the required knowledge of structural parameters. An additional issue that we take up concerns the implications of expectation shocks not arising from transitional learning effects.

Commitment, interest rate setting, adaptive learning, stability, determinacy, expectations shocks

Abstract:
We present a new application of Samuelson's Correspondence Principle to the analysis of comparative dynamics in stochastic rational expectations models. Our version, which we call the E-correspondence principle, applies to rational expectations equilibria that are stable under least squares and closely related learning rules. With this technique it is sometimes possible to study, without explicitly solving for the equilibrium, how qualitative properties of the equilibrium are affected by changes in the model parameters. Applications to overlapping generations and New Keynesian models illustrate the potential of the technique.

Abstract:
We introduce the E-correspondence principle for stochastic dynamic expectations models as a tool for comparative dynamics analysis. The principle is applicable to equilibria that are stable under least squares and closely related learning rules. With this technique it is possible to study, without explicit solving for the equilibrium, how properties of the equilibrium are affected by changes in the structural parameters of the model. Even when qualitative comparative dynamics results are not obtainable, a quantitative version of the principle can be applied.

comparative dynamics, rational expectations, stability of equilibrium, adaptive

Abstract:
We develop a monetary model with flexible supply of labor, cash in advance constraints and government spending financed by seignorage. This model has two regimes. One regime is conventional with two steady states. The other regime has a unique steady state which can be determinate or indeterminate. In the latter case there exist sunspot equilibria which are stable under adaptive learning, taking the form of noisy finite state Markov processes at resonant frequencies. For a range of parameter values, a sufficient reduction in government purchases will eliminate these equilibria.

Indeterminacy, Learnability, Expectational Stability, Endogenous Fluctuations, Seignorage

Abstract:
We study how the use of judgement or "add-factors" in macroeconomic forecasting may disturb the set of equilibrium outcomes when agents learn using recursive methods. We isolate conditions under which new phenomena, which we call exuberance equilibria, can exist in standard macroeconomic environments. Examples include a simple asset pricing model and the New Keynesian monetary policy framework. Inclusion of judgement in forecasts can lead to self-fulfilling fluctuations, but without the requirement that the underlying rational expectations equilibrium is locally indeterminate. We suggest ways in which policymakers might avoid unintended outcomes by adjusting policy to minimize the risk of exuberance equilibria.

Learning, expectations, excess volatility, bounded rationality, monetary policy

Abstract:
Monetary policy is sometimes formulated in terms of a target level of inflation, a fixed time horizon and a constant interest rate that is anticipated to achieve the target at the specified horizon. These requirements lead to constant interest rate (CIR) instrument rules. Using the standard New Keynesian model, it is shown that some forms of CIR policy lead to both indeterminacy of equilibria and instability under adaptive learning. Some other forms of CIR policy perform better, however. We also examine the properties of the different policy rules in the presence of inertial demand and price behaviour.

Indeterminacy, instability under learning, inflation targeting, inertia in demand, inflation inertia

Abstract:
Monetary policy is sometimes formulated in terms of a target level of inflation, a fixed time horizon and a constant interest rate that is anticipated to achieve the target at the specified horizon. These requirements lead to constant interest rate (CIR) instrument rules. Using the standard New Keynesian model, it is shown that some forms of CIR policy lead to both indeterminacy of equilibria and instability under adaptive learning. However, some other forms of CIR policy perform better. We also examine the properties of the different policy rules in the presence of inertial demand and price behaviour.

Indeterminacy, instability under learning, inflation targeting, inertia in demand, inflation inertia

Abstract:
We consider the stability under adaptive learning of the complete set of solutions to the basic linear forward looking model in which the current value of the state variable depends linearly on the (subjectively) expected value of the state next period and the coefficient of the expected state is bigger than one in absolute value. In addition to the fundamentals solution, the literature describes both finite-state Markov sunspot solutions, satisfying a resonant frequency condition, and autoregressive solutions depending on an arbitrary martingale difference sequence. We clarify the relationships between these solutions and show that the stability properties of equilibria may depend crucially on the representation used by agents in the learning process. Only the finite-state Markov sunspot solutions can be stable under learning.

Indeterminacy, Representations of Solutions, Learnability, Expectational Stability, Endogenous Fluctuations

Abstract:
In this paper we consider inflation and government debt dynamics when monetary policy employs a global interest rate rule and private agents' forecasts using adaptive learning. Because of the zero lower bound on interest rates, active interest rate rules are known to imply the existence of a second, low inflation steady state, below the target inflation rate. Under adaptive learning dynamics we find the additional possibility of a liquidity trap, in which the economy slips below this low inflation steady state and is driven to an even lower inflation floor which, in turn, is supported by a switch to an aggressive money supply rule. Fiscal policy alone cannot push the economy out of the liquidity trap. However, raising the threshold at which the money supply rule is employed can dislodge the economy from the liquidity trap and ensure a return to the target equilibrium.

stability of equilibria, fiscal and monetary policy, interest rate and

Abstract:
Recent models of monetary policy have analyzed the desirability of different optimal and ad hoc interest rules under the restrictive assumption that forecasts of the private sector and the central bank are homogenous. In this paper, we study the implications of heterogeneity in forecasts of the central bank and private agents for the performance of interest rules from the learning viewpoint.

Adaptive Learning, Stability, Heterogeneity, Monetary Policy

Abstract:
We examine the nonlinear one-step forward-looking model, in which the current state is a function of the (subjective) expected value of a nonlinear function of the state next period. Stationary Markov Sunspot Equilibria (SSEs) are known to exist near an indeterminate steady state, i.e. when the derivative of the function at the steady state is bigger than one in absolute value. We show that there exist Markov SSEs that are E- stable, and therefore locally stable under adaptive learning, if the value of this derivative is less than minus one.

Indeterminacy, Learnability, Expectational Stability, Endogenous Fluctuations

Abstract:
We investigate both the rational explosive inflation paths studied by McCallum (2001) and the classification of fiscal and monetary policies proposed by Leeper (1991) for stability under learning of rational expectations equilibria (REE). Our first result is that the fiscalist REE in the model of McCallum (2001) is not locally stable under learning. By contrast, in the setting of Leeper (1991), different possibilities can obtain. We find, in particular, that there are parameter domains for which the fiscal theory solution - in which fiscal variables affect the price level - can be a stable outcome under learning. For other parameter domains, the monetarist solution is the stable equilibrium.

Inflation, expectations, fiscal and monetary policy, explosive price paths

Abstract:
We investigate both the rational explosive inflation paths studied by McCallum (2001) and the classification of fiscal and monetary policies proposed by Leeper (1991) for stability under learning of rational expectations equilibria (REE). Our first result is that the fiscalist REE in the model of McCallum (2001) is not locally stable under learning. By contrast, in the setting of Leeper (1991), different possibilities can obtain. We find, in particular, that there are parameter domains for which the fiscal theory solution - in which fiscal variables affect the price level - can be a stable outcome under learning. For other parameter domains, the monetarist solution is the stable equilibrium.

Inflation, expectations, fiscal and monetary policy, explosive price paths

Abstract:
We study the properties of generalized stochastic gradient (GSG) learning in forward-looking models. We examine how the conditions for stability of standard stochastic gradient (SG) learning both differ from and are related to E-stability, which governs stability under least squares learning. SG algorithms are sensitive to units of measurement and we show that there is a transformation of variables for which E-stability governs SG stability. GSG algorithms with constant gain have a deeper justification in terms of parameter drift, robustness and risk sensitivity.

adaptive learning, E-stability, recursive least squares, robust estimation

Abstract:
Earlier studies of the seigniorage inflation model have found that the high-inflation steady state is not stable under adaptive learning. We reconsider this issue and analyze the full set of solutions for the linearized model. Our main focus is on stationary hyperinflationary paths near the high-inflation steady state. The hyperinflationary paths are stable under learning if agents can utilize contemporaneous data. However, in an economy populated by a mixture of agents, some of whom only have access to lagged data, stable inflationary paths emerge only if the proportion of agents with access to contemporaneous data is sufficiently high.

Indeterminacy, Inflation, Stability of Equilibria, Seigniorage

Abstract:
We consider "robust stability" of a rational expectations equilibrium, which we define as stability under discounted (constant gain) least-squares learning, for a range of gain parameters. We find that for operational forms of policy rules, i.e. rules that do not depend on contemporaneous values of endogenous aggregate variables, many interest-rate rules do not exhibit robust stability. We consider a variety of interest-rate rules, including instrument rules, optimal reaction functions under discretion or commitment, and rules that approximate optimal policy under commitment. For some reaction functions we allow for an interest-rate stabilization motive in the policy objective. The expectations-based rules proposed in Evans and Honkapohja (2003, 2006) deliver robust learning stability. In contrast, many proposed alternatives become unstable under learning even at small values of the gain parameter.

Adaptive learning, Commitment, determinacy, interest-rate setting, stability

Abstract:
We consider the robust stability of a rational expectations equilibrium, which we define as stability under discounted (constant gain) least-squares learning, for a range of gain parameters. We find that for operational forms of policy rules, ie rules that do not depend on contemporaneous values of endogenous aggregate variables, many interest-rate rules do not exhibit robust stability. We consider a variety of interest-rate rules, including instrument rules, optimal reaction functions under discretion or commitment, and rules that approximate optimal policy under commitment. For some reaction functions we allow for an interest-rate stabilization motive in the policy objective. The expectations-based rules proposed in Evans and Honkapohja (2003, 2006) deliver robust learning stability. In contrast, many proposed alternatives become unstable under learning even at small values of the gain parameter.

Commitment, interest-rate setting, adaptive learning, stability, determinacy.

Abstract:
We study how the use of judgment or "add-factors" in forecasting may disturb the set of equilibrium outcomes when agents learn using recursive methods. We isolate conditions under which new phenomena, which we call exuberance equilibria, can exist in a standard self-referential environment. Local indeterminacy is not a requirement for existence. We construct a simple asset pricing example and find that exuberance equilibria, when they exist, can be extremely volatile relative to fundamental equilibria.

Learning, expectations, excess volatility, bounded rationality

Abstract:
We construct a rational expectations model in which aggregate growth alternates between a low growth and a high growth state. When all agents expect growth to be slow, the returns on investment are low, and little investment takes place. This slows growth and confirms the prediction that the returns on investment will be low. But if agents expect fast growth, investment is high, returns are high, and growth is rapid. This expectational indeterminacy is induced by complementarity between different types of capital goods. In a growth cycle there are stochastic shifts between high and low growth states and agents take full account of these transitions. The rules that agents need to form rational expectations in this equilibrium are simple. The equilibrium with growth cycles is stable under the dynamics implied by a correspondingly simple learning rule

Abstract:
We develop a monetary model with flexible supply of labor, cash in advance constraints and government spending financed by seignorage. This model has two regimes. One regime is conventional with two steady states. The other regime has a unique steady state which can be determinate or indeterminate. In the latter case there exist sunspot equilibria which are stable under adaptive learning, taking the form of noisy finite state Markov processes at resonant frequencies. For a range of parameter values, a sufficient reduction in government purchases will eliminate these equilibria.

Indeterminacy, learnability, expectational stability, endogenous fluctuations, seignorage

Abstract:
We study how the use of judgment or "add-factors" in macroeconomic forecasting may disturb the set of equilibrium outcomes when agents learn using recursive methods. We examine the possibility of a new phenomenon, which we call exuberance equilibria, in the New Keynesian monetary policy framework. Inclusion of judgment in forecasts can lead to self-fulfilling fluctuations in a subset of the determinacy region. We study how policymakers can minimize the risk of exuberance equilibria.

Learning, expectations, excess volatility, bounded rationality, monetary policy

Abstract:
We consider inflation and government debt dynamics when monetary policy employs a global interest rate rule and private agents forecast using adaptive learning. Because of the zero lower bound on interest rates, active interest rate rules are known to imply the existence of a second, low inflation steady state, below the target inflation rate. Under adaptive learning dynamics we find the additional possibility of a liquidity trap, in which the economy slips below this low inflation steady state and is driven to an even lower inflation floor that is supported by a switch to an aggressive money supply rule. Fiscal policy alone cannot push the economy out of the liquidity trap. Raising the threshold at which the money supply rule is employed can, however, dislodge the economy from the liquidity trap and ensure a return to the target equilibrium.

Learning, deflation, economic policy

Abstract:
Recent models of monetary policy can have indeterminacy of equilibria. The indeterminacy property is often viewed as a difficulty of these models. We consider its significance using the learning approach to expectations formation by employing expectational stability as a robustness criterion for different equilibria. We derive the expectational stability and instability conditions for forward-looking multivariate models, both with and without lags, that cover a wide range of monetary policies proposed in the literature.

Adaptive learning, stability, sunspots, monetary policy

Abstract:
We consider an endogenous growth model with international trade in complementary capital goods. The model possesses several distinct, balanced growth solutions, which we classify using stability under adaptive learning. Some of the equilibria can involve growth rates much higher than others. We show that, in addition to a small (usually positive) effect on a given equilibrium, an expansion in trade may sometimes yield a much larger, sudden jump in growth. The small effect on the initial equilibrium may reduce growth if the opportunity cost of capital rises very fast as growth accelerates.

Abstract:
We study the properties of generalized stochastic gradient (GSG) learning in forward-looking models. We examine how the conditions for stability of standard stochastic gradient (SG) learning both differ from and are related to E-stability, which governs stability under least squares learning. SG algorithms are sensitive to units of measurement and we show that there is a transformation of variables for which E-stability governs SG stability. GSG algorithms with constant gain have a deeper justification in terms of parameter drift, robustness and risk sensitivity.

Abstract:
Stochastic rationing when the market does not clear draws attention because both Dreze (1975) and Benassy (1975) quantity-constrained equilibria have some undesirable features. Gale (1978)gave the existence proof of trade under uncertainty. His stochastic rationing depends on all the individual effective demands. It is too vague to characterize a rationing mechanism. Moreover, his assumption to ensure a non-trivial equilibrium is economically not clear. In this paper we extend Green (1978) to characterizing the rationing scheme as the individual effective demand times the rationing number which is a function of the aggregate quantity signals. We also construct an economy with money and overlapping generations. We show the existence of the non-trivial equilibrium and provide an example of a non-Wairasian equilibrium at the Walrasian equilibrium prices.

Abstract:
No abstract is available for this paper.

Abstract:
Expectations about the future are central for determination of current macroeconomic outcomes and the formulation of monetary policy. Recent literature has explored ways for supplementing the benchmark of rational expectations with explicit models of expectations formation that rely on econometric learning. Some apparently natural policy rules turn out to imply expectational instability of private agents' learning. We use the standard New Keynesian model to illustrate this problem and survey the key results about interest-rate rules that deliver both uniqueness and stability of equilibrium under econometric learning. We then consider some practical concerns such as measurement errors in private expectations, observability of variables and learning of structural parameters required for policy. We also discuss some recent applications including policy design under perpetual learning, estimated models with learning, recurrent hyperinflations, and macroeconomic policy to combat liquidity traps and deflation.

Determinacy, fluctuations, imperfect knowledge, interest-rate setting, learning, stability

Abstract:
We consider the impact of anticipated policy changes when agents form expectations using adaptive learning rather than rational expectations. To model this we assume that agents combine limited structural knowledge with a standard adaptive learning rule. We analyze these issues using two well-known set-ups, an endowment economy and the Ramsey model. In our set-up there are important deviations from both rational expectations and purely adaptive learning. Our approach could be applied to many macroeconomic frameworks.

Expectations, Ramsey model, taxation

Abstract:
We examine global economic dynamics under infinite-horizon learning in a New Keynesian model in which the interest-rate rule is subject to the zero lower bound. As in Evans, Guse and Honkapohja (2008), we find that under normal monetary and fiscal policy the intended steady state is locally but not globally stable. Unstable deflationary paths can arise after large pessimistic shocks to expectations. For large expectation shocks pushing interest rates to the zero lower bound, temporary increases in government spending can be used to insulate the economy from deflation traps.

Adaptive Learning, Fiscal Policy, Monetary Policy, Zero Interest Rate Lower Bound

Abstract:
We examine global economic dynamics under learning in a New Keynesian model in which the interest-rate rule is subject to the zero lower bound. Under normal monetary and fiscal policy, the intended steady state is locally but not globally stable. Large pessimistic shocks to expectations can lead to deflationary spirals with falling prices and falling output. To avoid this outcome we recommend augmenting normal policies with aggressive monetary and fiscal policy that guarantee a lower bound on inflation. In contrast, policies geared toward ensuring an output lower bound are insufficient for avoiding deflationary spirals.

Adaptive learning, fiscal policy, indeterminacy, monetary policy, zero interest rate lower bound

Abstract:
Earlier studies of the seigniorage inflation model have found that the high-inflation steady state is not stable under learning. We reconsider this issue and analyze the full set of solutions for the linearized model. Our main focus is on stationary hyperinflationary paths near the high-inflation steady state. These paths are shown to be stable under least squares learning if agents can utilize contemporaneous data. In an economy with a mixture of agents, some of whom only have access to lagged data, stable hyperinflationary paths emerge only if the proportion of agents with access to contemporaneous data is sufficiently high.

Indeterminacy, inflation, stability of equilibria, seigniorage

Abstract:
We consider the stability under adaptive learning of the complete set of solutions to the model x(t)= b*Ex{t+1} when |b| > 1. In addition to the fundamentals solution, the literature describes both finite-state Markov sunspot solutions and autoregressive solutions depending on an arbitrary martingale difference sequence. We clarify the relationships between these solutions and show that the stability properties of equilibria may depend crucially on the representation used by agents in the learning process. Autoregressive forms of solutions are not learnable, but finite-state Markov sunspot solutions are stable under learning if b < -1.

indeterminacy, representations of solutions, learnability, expectational stability, endogenous fluctuations

Abstract:
We examine the nonlinear model x(t)=E(t)F(x(t+1)). Markov SSEs (stationary sunspot equilibria) exist near an indeterminate steady state, x=F(x), provided |F'(x)| >1. Despite the importance of indeterminacy in macroeconomics, earlier results have not provided conditions for the existence of adaptively stable SSEs near an indeterminate steady state. We show that there exist Markov SSEs near x that are E-stable, and therefore locally stable under adaptive learning, if F'(x)<-1.

Indeterminacy, Learnability, Expectational Stability, Endogenous Fluctuations

Abstract:
Drawing upon recent contributions in the statistical literature, we present new results on the convergence of recursive, stochastic algorithms which can be applied to economic models with learning and which generalize previous results. The formal results provide probability bounds for convergence which can be used to describe the local stability under learning of rational expectations equilibria in stochastic models. Economic examples include local stability in a multivariate linear model with multiple equilibria and global convergence in a model with a unique equilibrium.

Abstract:
Drawing upon recent contributions in the statistical literature, we present a new result on the convergence of recursive, stochastic algorithms which can be applied to economic models with learning. The formal result provides probability bounds for convergence which can be used to describe the local stability under learning of rational expectations equilibria. Our treatment also generalizes previous results by permitting the state variable to follow a nonlinear Markov process. Two economic applications are discussed.