Stackelberg solution for a two-agent rational expectations model

This paper is concerned with the optimal control of rational expectations models in the general case of longer forward-looking time (d ≥ 2). The main contribution is the necessary and sufficient condition for the solvability of the finite-horizon problem. In particular, explicit characterizations of the optimal solution and the optimal cost are given in terms of difference equations. The key technique is to solve the forward and backward stochastic difference equations (FBSDEs) obtained by the stochastic maximum principle.

The decentralized linear–quadratic–Gaussian (LQG) control problem for networked control systems (NCSs) with asymmetric information is investigated, where controller 1 shares its historical information with controller 2, and not vice versa. The asymmetry of the information structure leads to the coupling between controller 2 and estimator 1, and hence the classical separation principle fails. Through the assumption of linear control strategy, the coupling between controller 2 and estimator 1 (CCE) is decoupled, but the estimation gain is still coupled with the control gain. It is noted that the control gain conforms to the backward Riccati equation while estimation gain abides by the forward equation, which is computationally challenging. Applying the stochastic maximum principle, the solvability of the decentralized LQG control problem is reduced to that of corresponding forward and backward stochastic difference equations (FBSDEs). Further, necessary and sufficient conditions for the solvability of optimal control problem are presented by two Riccati equations, one of which is nonsymmetric. Moreover, a novel iterative forward method is proposed to calculate the coupled backward control gain and forward estimation gain.

We provide existence results and comparison principles for solutions of backward stochastic difference equations (BS$\Delta$Es) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BS$\Delta$Es and BSDEs are governed by drivers $f^N(t,\omega,y,z)$ and $f(t,\omega,y,z),$ respectively. The new feature of this paper is that they may be non-Lipschitz in z. For the convergence results it is assumed that the BS$\Delta$Es are based on d-dimensional random walks $W^N$ approximating the d-dimensional Brownian motion W underlying the BSDE and that $f^N$ converges to f. Conditions are given under which for any bounded terminal condition $\xi$ for the BSDE, there exist bounded terminal conditions $\xi^N$ for the sequence of BS$\Delta$Es converging to $\xi$, such that the corresponding solutions converge to the solution of the limiting BSDE. An important special case is when $f^N$ and f are convex in z. We show that in this situation, the solutions of the BS$\Delta$Es converge to the solution of the BSDE for every uniformly bounded sequence $\xi^N$ converging to $\xi$. As a consequence, one obtains that the BSDE is robust in the sense that if $(W^N,\xi^N)$ is close to $(W,\xi)$ in distribution, then the solution of the Nth BS$\Delta$E is close to the solution of the BSDE in distribution too.

This article considers the decentralized linear–quadratic (LQ) optimal control problem for the multiplicative-noise stochastic systems in discrete time with asymmetric information. There exist two controllers in the system, and each controller has its own information in general structure. The adopted information structure is adapted open loop for both controllers. The main contribution is to derive the necessary and sufficient condition for the unique solvability of the LQ control problem with a general asymmetric information structure and present the explicitly optimal controllers in terms of the proposed Riccati equations. The key is to explicitly solve the forward and backward stochastic difference equations, which are derived from the maximum principle.

This article considers the decentralized control for networked control systems (NCSs) with asymmetric information. In this NCSs model, the controller 2 (C2) shares its observations and part of its historical control inputs with the controller 1 (C1), whereas C2 cannot obtain the information of C1 due to network constraints. Under the linear control strategies assumption, we present the optimal estimators for C1 and C2 respectively based on asymmetric observations. Since the information for C1 and C2 are asymmetric, the estimation error covariance (EEC) is coupled with the controller which means that the classical separation principle fails. By applying the Pontryagin's maximum principle, we obtain a solution to the forward and backward stochastic difference equations. Based on this solution, we derive the optimal controllers to minimize a quadratic cost function. Combining the linear optimal controllers with the EEC, the controller C1 is decoupled from the EEC. It should be emphasized that the control gain is dependent on the estimation gain. What is more, the estimation gain satisfies the forward Riccati equation and the control gain satisfies the backward Riccati equation which makes the problem more challenging. We propose iterative solutions to the Riccati equations and give a suboptimal solution to the optimal decentralized control problem.

In this paper, the open-loop and closed-loop local and remote stochastic nonzero-sum game (LRSNG) problem is investigated. Different from previous works, the stochastic nonzero-sum game problem under consideration is essentially a special class of two-person nonzero-sum game problem, in which the information sets accessed by the two players are inconsistent. More specifically, both the local player and the remote player are involved in the system dynamics, and the information sets obtained by the two players are different, and each player is designed to minimise its own cost function. For the considered LRSNG problem, both the open-loop and closed-loop Nash equilibrium are derived. The contributions of this paper are given as follows. Firstly, the open-loop optimal Nash equilibrium is derived, which is determined in terms of the solution to the forward and backward stochastic difference equations (FBSDEs). Furthermore, by using the orthogonal decomposition method and the completing square method, the feedback representation of the optimal Nash equilibrium is derived for the first time. Finally, the effectiveness of our results is verified by a numerical example.

Some thoughts on rational expectations models, and alternate formulations

Indeterminate equilibrium rational expectations (RE) models are ubiquitous in both theoretical and applied work in dynamic macroeconomics. The issue of characterizing the exact dimension of indeterminacy—i.e. of deriving the full set of causal and stable solutions to linear RE models—has only recently been addressed in the context of general and multivariate settings. This paper complements existing results by identifying bounds on the observable dimension of indeterminacy of linear RE models in the presence of arbitrary initial conditions. Implications for the estimation of indeterminate equilibrium RE models are discussed.

Many macroeconomic models involving rational expectations give rise to an infinity of solution paths, even when the models are linear in all variables. Some writers have suggested that this non-uniqueness constitutes a serious weakness for the rational expectations hypothesis. One purpose of the present paper is to argue that the non-uniqueness in question is not properly attributable to the rationality hypothesis but, instead, is a general feature of dynamic models involving expectations. It is also argued that there typically exists, in a very wide class of linear rational expectations models, a single solution that excludes "bubbl&' or "bootstrap" effects--ones that occur only because they are arbitrarily expected to occur. A systematic procedure for obtaining solutions free from such effects is introduced and discussed. In addition, this procedure is used to interpret and reconsider several prominent examples with solution multiplicities, including ones developed by Fischer Black and

This note presents a model in which the rational expectations hypothesis does not restrict the number of equilibrium solutions in any meaningful way.1 The techniques and assumptions used to obtain this result, though applied to a very simple duopoly model, are likely to generalize to a variety of rational expectations models. This result suggests that the rationality assumptions often used in economic models may not narrow the range of predicted economic behavior by very much. Such a result is troubling insofar as models that are able to narrow the range of predicted behavior significantly are often the most useful in empirical work. There is a more positive way to state this result, for it suggests a potentially fruitful collaboration between economists and decision psychologists. From the viewpoint of economics, such collaboration may be a source of insight that can help to reduce the number of solutions that rational expectations models of the type presented here seem to generate. From the viewpoint of decision psychology, such collaborative research would provide a framework of equilibrium analysis (or even general equilibrium analysis) that might suggest new forms of experimentation and a broader context for the interpretation of experimental findings. It is worth mentioning in this respect that the feedback implied by the equilibrium framework of economics does not, in and of itself, depend on any rationality assumption but does recognize the effect of the interaction of an aggregation of numerous individual decisions on the individual decision makers themselves. So far as I can tell from my reading of the literature, such feedback from what might be described as the macro implications of the equilibrium process in economics seems to play little or no role in decision psychology. A formal model is presented and discussed in the next section, and

This paper discusses inference for rational expectations models estimated via minimum distance methods by characterizing the probability beliefs regarding the data generating process (DGP) that are compatible with given moment conditions. The null hypothesis is taken to be rational expectations and the alternative hypothesis to be distorted beliefs. This distorted beliefs alternative is analyzed from the perspective of a hypothetical semiparametric Bayesian who believes the model and uses it to learn about the DGP. This interpretation provides a different perspective on estimates, test statistics, and confidence regions in large samples, particularly regarding the economic significance of rejections in rational expectations models.

ABSTRACTA rational expectation model with lagged endogenous variables is used to describe how the current price level is influenced by the expectation and historic price level. The time domain of the rational expectation model is extended to a complex discrete time domain which is a collection of points along the real number line. The rational expectation model with lagged endogenous variables is solved in multi-dimensional cases where the agents possess multiple assets, and the current price of each asset is related to the expected price and historical prices. An example about price determination process of storable commodities is given to illustrate the advantages of the rational expectation model on isolate time domain.

The paper is concerned with the role of monetary policy and argues that activist monetary policy can affect the behavior of real output, rational expectations notwithstanding. A rational expectations model with overlapping labor contracts is constructed, with each labor contract being made for two periods. These contracts inject an element of short-run wage stickiness into the model. Because the money stock is changed by the monetary authority more frequently than labor contracts are renegotiated, and, given the assumed form of the labor contracts, monetary policy has the ability to affect the short-run behavior of output, though it has no effects on long-run output behavior.

This survey provides a detailed description of some of the recent theoretical and empirical literature on rational expectations econometrics. The survey pays special attention to non–stationarity of the data, and to the various methods for evaluating rational expectations models that have been developed as alternatives to the classical statistical approach of testing overidentifying restrictions. These methods have become very popular and widely used in empirical research. We provide an illustration using Danish stock market data, and we summarize the many results obtained recently using these measures in areas as diverse as stock prices, the term structure of interest rates, exchange rates, consumption and saving, the balance of payments, tax–smoothing, hyperinflation, and linear quadratic adjustment cost models for inventories, labour demand, and money demand.

Efficient solution techniques for dynamic non-linear rational expectations models