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Abstract:
This paper views uncertainty and economic fluctuations as being primarily endogenous and internally propagated phenomena. The most important Endogenous Uncertainty examined in this paper is price uncertainty which arises when agents do not have structural knowledge and are compelled to make decisions on the basis of their beliefs. We assume that agents adopt Rational Beliefs as in Kurz [1994a]. The trading of endogenous uncertainty is accomplished by using Price Contingent Contracts (PCC) rather than the Arrow-Debreu state contingent contracts. The paper provides a full construction of the "price state space" which requires the expansion of the exogenous state space to include the "state of beliefs." This construction is central to the analysis of equilibrium with endogenous uncertainty and the paper provides an existence theorem for a Rational Belief Equilibrium with PCC. It shows how the PCC completes the markets for trading endogenous uncertainty and lead to an allocation which is Pareto optimal. The paper also demonstrates that endogenous uncertainty is generically present in this new equilibrium.

Abstract:
This paper views uncertainty and economic fluctuations as being primarily endogenous and internally propagated phenomena. The most important Endogenous Uncertainty examined in this paper is price uncertainty which arises when agents do not have structural knowledge and are compelled to make decisions on the basis of their beliefs. We assume that agents adopt Rational Beliefs as in Kurz [1994a]. The trading of endogenous uncertainty is accomplished by using Price Contingent Contracts (PCC) rather than the Arrow-Debreu state contingent contracts. The paper provides a full construction of the "price state space" which requires the expansion of the exogenous state space to include the "state of beliefs." This construction is central to the analysis of equilibrium with endogenous uncertainty and the paper provides an existence theorem for a Rational Belief Equilibrium with PCC. It shows how the PCC completes the markets for trading endogenous uncertainty and lead to an allocation which is Pareto optimal. The paper also demonstrates that endogenous uncertainty is generically present in this new equilibrium.

Abstract:
We examine the equity premium puzzle with the perspective of the theory of Rational Beliefs Equilibrium (RBE) and show that from the perspective of this theory there is no puzzle. In an RBE agents need to be compensated for the endogenously propagated price uncertainty which is not permitted under rational expectations. It is then argued that endogenous uncertainty is the predominant uncertainty of asset returns and its presence provides a natural explanation of the observed premium. Utilizing data on the asset allocation of 63 U.S. mutual funds, we test some empirical implications of the theory of rational beliefs as well as estimate the parameters of risk aversion of mutual fund managers. Our tests show that the predictions of the theory are consistent with the empirical evidence. We then construct a simple two agent model of the U.S. economy in which the agents hold rational beliefs and calibrate it to the empirical experience in accord with the parameters of the Mehra and Prescott (1985) paper. The results of our calculations show that for a large set of parameter values the model predictions fit closely the historical record.

Abstract:
Endogenous Uncertainty is that component of economic risk and market volatility which is propagated within the economy by the beliefs and actions of agents. The theory of Rational Belief (see Kurz [1994]) permits rational agents to hold diverse beliefs and consequently, a Rational Belief Equilibrium (in short, RBE) may exhibit diverse patterns of Endogenous Uncertainty. This paper shows that most of the observed volatility in financial markets is generated by the beliefs of the agents and the diverse market puzzles which are examined in this paper, such as the equity premium puzzle, are all driven by the structure of market expectations. To make the case for this theory we present a single RBE model, which builds on developments in Kurz and Beltratti [1997] and Kurz and Schneider [1996], with which we study a list of phenomena that have been viewed as "anomalies" in financial markets. The model is able to predict the correct order of magnitude of: (i) the long term mean and standard deviation of the price\dividend ratio;(ii) the long term mean and standard deviation of the risky rate of return on equities;(iii) the long term mean and standard deviation of the riskless rate;(iv) the long term mean equity premium.In addition, the model predicts (v) the GARCH property of risky asset returns;(vi) the Forward Discount Bias in foreign exchange markets.We also conjecture that an adaptation of the same model to markets with derivative assets will predict the appearance of "smile curves" in derivative prices.The common economic explanation for these phenomena is the existence of heterogenous agents with diverse but correlated beliefs. Given such diversity, some agents are optimistic and some pessimistic. We develop a simple model which allows agents to be in these two states of belief but the identity of the optimists and the pessimists fluctuates over time since at any date any agent may be in these two states of belief. In this model there is a unique parameterization under which the model makes all the above predictions simultaneously. That is, although the parameter space of the RBE is large, all parameterizations outside a small neighborhood of the parameter space fail significantly to reproduce some subset of variables under consideration. Any parameter choice in this small neighborhood requires the optimists to be in the majority but the rationality of belief conditions of the RBE require the pessimists to have a higher intensity level. This higher intensity has a decisive effect on the market: it increases the demand for riskless assets, decreases the equilibrium riskless rate and increases the equity premium. In simple terms, the large equity premium and the lower equilibrium riskless rate are the result of the fact that at any moment of time there are agents who hold extreme pessimistic beliefs and they have a relatively stronger impact on the market. The relative impact of these two groups of agents who are, at any moment of time, in the two states of belief is a direct consequence of the rationality of belief conditions and in that sense it is unique to an RBE. As for the correlation among the beliefs of agents, the paper shows that the dynamics of asset prices are strongly affected by such correlation. The pattern of correlation which was used in the model can be explained intuitively in terms of its effect on the dynamics of prices. The model correlation causes periods of price rises (i.e. bull markets) to develop slower than periods of decline (i.e. bear markets) hence the model dynamics do not permit prices to shoot directly from the bottom to the top but the opposite is possible and takes the form of market crashes.

Abstract:
It is argued that the theory of Rational Belief Equilibria (RBE) provides a unified paradigm for explaining market volatility by the effect of "Endogenous Uncertainty" on financial markets. This uncertainty is propagated within the economy (hence "endogenous") by the beliefs of the agents who trade assets. The theory of RBE was developed in a sequence of papers assembled in a recently published book entitled Endogenous Economic Fluctuations: Studies in the Theory of Rational Beliefs, M. Kurz (Ed.), Springer Verlag, 1997. The present paper provides a non-mathematical exposition of both the main ideas of the theory of RBE as well as a summary of the main results of the book regarding market volatility.

The structure of the paper is as follows. Section I starts by reviewing the standard assumptions underlying models of rational expectations equilibria (REE) and their implications to the study of market volatility. The paper then reviews four basic problems which have constituted puzzles or anomalies in relation to the assumptions of REE: (i) Why are asset prices much more volatile than their underlying fundamentals? (ii) The equity premium puzzle: why under REE the predicted riskless rate is so high and the equity risk premium so low? (iii) Why do asset prices exhibit the "GARCH" behavior without exogenous fundamental variables to explain it? (iv) the "Forward Discount Bias" in foreign exchange markets: why are interest rate differentials such poor predictors of future changes in the exchange rates? Section II outlines the basic ideas and assumptions of the theory of RBE and the main propositions which it implies in relation to the problems of market volatility. Section III first develops the simulation models of RBE which are used in the analysis of the four problems above and explains that the domestic economy is calibrated, as in Mehra and Prescott [1985], to the U.S. economy. Then for each of the four problems the relevant simulation results of the RBE are presented and compared both to the results predicted by a corresponding RBE as well as to the actual empirical observations in the U.S.

The conclusion of the paper is that the main cause of market volatility is the distribution of beliefs and expectations of agents. The theory of RBE shows that if agents disagree then the state of belief of each agent, represented by his conditional probability, must fluctuate over time. Hence the nature of the distribution of the individual states of belief in the market is the root cause of all phenomena of market volatility. The paper shows that the GARCH phenomenon of time varying variance of asset prices is explained in the simulation model by the presence of both persistence in the states of beliefs of agents as well as correlation among the states of beliefs of the agents. Correlation makes beliefs either narrowly distributed (i.e. "consensus") or widely distributed (i.e. "non-consensus"). When a belief regime of consensus is established (and due to persistence it remains in place for a while) then agents seek to buy or sell the same portfolio leading to high volatility. On the other hand, the widespread disagreement in a belief regime of non-consensus entails a balance between sellers and buyers leading to low market volatility. In short, the theory proposes that the GARCH phenomenon is the result of shifts in the distribution of beliefs in the market and these shifts are caused by the dynamics of the states of beliefs of the agents.

Turning to the equity risk premium, it is clear that the key question is what are the conditions on the distribution of beliefs which will ensure that the average riskless rate is low and hence the average equity risk premium is high. It turns out that the key condition requires that on average the majority of agents are relatively optimistic about the prospects of capital gains in the subsequent period.

Consequently, the rationality of belief conditions require the pessimists (who are in the minority) to be on the average more intensely pessimistic. In this narrow sense of having higher intensity of beliefs, the pessimists tend to have a stronger impact on the market a significant fraction of time. When this occurs they protect their endowment by shorting the stock and increasing their purchases of the safe riskless bill. This tends to bid up the price of the bill and lowers the price of the stock resulting in a lower riskless rate and a higher equity risk premium.

The "Forward Discount Bias" in foreign exchange markets is explained by the same model used to explain the other three phenomena. It is the result of the fact that in an RBE agents often make the wrong forecasts although they are right on the average. Hence, in an RBE the exchange rate fluctuates excessively due to the errors of the agents and hence at almost no date is the interest differential between two countries an unbiased estimate of the rate of depreciation of the exchange rate one period later. The bias is positive since agents who invest in foreign currency demand a risk premium on endogenous uncertainty which is above and beyond the risk which exists in an REE. Hence the size of the bias is equal to the added risk premium due to endogenous uncertainty.

Abstract:
We review the issues related to the formulation of endogenous uncertainty in rational belief equilibria (RBE). In all previous models of RBE, individual states of belief were the foundation for the construction of the endogenous state space where individual states of belief were described with the method of assessment variables. This approach leads to a lack of "anonymity" where the belief of each individual agent has an impact on equilibrium prices but as a competitor he ignores it. The solution is to study a replica economy with a finite number of types but with a large number of agents of each type. This gives rise to "type-states" which are distributions of beliefs within each type. The state space for this economy is then constructed as the set of products of the exogenous states and the social states of belief which are vectors of distributions of all the types. Such an economy leads to RBE which do indeed solve the problem of anonymity. We then study via simulations the implications of the model of RBE with social states for market volatility and for the determinants of the equity risk premium in an RBE. Under i.i.d. assessments one uses the law of large numbers to induce a single social state of belief and we show that the RBE of such economies have the same number of prices as in rational expectations equilibrium (REE). However, the RBE may exhibit large fluctuations if agents are allowed to hold extreme beliefs. Establishing 5% boundary restrictions on beliefs we show that the model with a single social state of belief cannot explain all the moments of the observed distribution of returns. We then introduce correlation among beliefs and this leads both to the creation of new social states as well as fluctuations of the social states of belief over time. We next show that under correlation among beliefs the model simulations reproduce the values of four key moments of the empirical distribution of returns. The observed equity premium is then explained as follows. Note first, as a background, that in an RBE the main cause of market volatility is the dynamics of beliefs of agents. The theory of RBE shows that if agents disagree then the state of belief of each agent, represented by his conditional probability, must fluctuate over time. Hence the distribution of the social states of belief in the market is the root cause of all phenomena of market volatility. More specifically, at any moment of time there are both rational optimists as well as rational pessimists in our financial markets. Hence, in the case of the equity risk premium, the key question is what are the conditions of the distribution of beliefs which will ensure that the average riskless rate is low and hence the average equity risk premium is high. It turns out that the key condition requires that on average the majority of agents are relatively optimistic about the prospects of capital gains in the subsequent period. Since the agents hold Rational Beliefs, the rationality of beliefs conditions require the pessimists (who are in the minority) to be, on the average, more intensely pessimistic. In this narrow sense of having a higher intensity of beliefs, the pessimists tend to have a stronger impact on the market in a significant fraction of time. When this occurs they protect their endowment by shorting the stock and increasing their purchases of the riskless bill. This tends to bid up the price of the bill and lowers the price of the stock resulting in a lower average riskless rate and a higher equity risk premium. In addition, as noted earlier, correlation among beliefs of agents is needed in order to generate fluctuations in the social states of belief over time. These fluctuations generate volatility of the rate of return and contribute to the explanation of the equilibrium equity risk premium.

Abstract:
This is a chapter in a new series of Handbooks in Financial Economics entitled Dynamics and Evolution of Financial Markets. It explores works which contribute to explaining market dynamics and volatility by employing models of rational but diverse beliefs. This excludes models of Behavioral Economics. We explore models that permit endogenous amplification which can, in turn, explain such basic facts as excess volatility and high risk premia. The first part gives an exposition of the Noisy Rational Expectations (i.e. REE) Asset Pricing Theory where diversity of beliefs arises from diverse private information. Examination of these models enable us to evaluate the assumptions made and the ability of the models to explain the data on market dynamics. Our conclusion is that, although Noisy REE asset pricing theories have many attractive features, they do not offer a satisfactory paradigm for market dynamics. Indeed, for most models increased amount of idiosyncratic private information leads to a reduction in volatility.

We next turn to models of diverse beliefs without private information and explore work on Rational Belief Equilibria starting with modeling beliefs. Agents do not know the structure of the dynamically complex economy but have data over a long time to enable the computation of an empirical probability on sequences which is then commonly known. A belief is a model of deviations from the empirical probability. It is shown that in a wold of diverse beliefs, to be rational a belief must fluctuate, generating a mechanism of endogenous amplification of market dynamics. An important innovation of the theory is the treatment of individual beliefs as state variables and market belief is then derived as the distribution of individual beliefs. Market belief is observable via sampling data from several sources and we review both data sources as well as computations of the first two moments of market belief. We explore an illustrative model of asset pricing theory with diverse belief for which a closed form solution is available. This enables us to explain the central theoretical implications of the theory for market dynamics, the nature of uncertainty and risk premia. Next, we review the ability of the theory to explain via simulations the stylized facts known about market volatility. It is shown that the model's predictions are compatible with the data. It is then argued that the results offer a unified explanation for key features of market dynamics such as excess asset price volatility, the Equity Premium Puzzle, the predictability of asset returns and stochastic volatility.

Market Volatility, Diverse Beliefs, Noisy REE, Private Information, Rational Beliefs, Rational Overconfidence, Equity Risk Premium, Market States of Belief, Bayesian updating, Money Non Neutrality, Propagation Mechanism, Optimism, Pessimism, Non Stationarity, Endogenous Uncertainty

Abstract:
Why do risk premia vary over time? We examine this problem theoretically and empirically by studying the effect of market belief on risk premia. Individual belief is taken as a fundamental state variables. Market belief is observable, it is central to the empirical evaluation and we show how to extract it from the data. The asset pricing model we use is familiar from the noisy REE literature but we adapt it to an economy with diverse beliefs. We derive the equilibrium asset pricing and the implied risk premium. Our approach permits a closed form solution of prices hence we trace the exact effect of market belief on the time variability of asset prices and risk premia. We test empirically the theoretical conclusions.

Our main result is that, above and beyond the effect of business cycles on risk premia, fluctuations in market belief have significant independent effect on the time variability of risk premia. We study the premia on long positions in Federal Funds Futures, 3-month and 6-month Treasury Bills. The annualized mean risk premium on holding such assets for 1-12 months is about 40-60 basis points and, on average, we find that the component of market belief in the risk premium at a random date exceeds 50% of the mean. Since time variability of market belief is large, this component frequently exceeds 50% of the mean premium. This component is larger the shorter is the holding period of an asset and it dominates the premium for very short holding returns of less than 2 months. As to the structure of the premium we show that when the market holds abnormally favorable belief about the future payoff of an asset the market views the long position as less risky hence the risk premium on that asset declines. More generally, periods of market optimism (i.e. "bull" markets) are shown to be periods when the market risk premium declines while in periods of pessimism (i.e. "bear" markets) the market's risk premium rises. Hence, fluctuations in risk premia are inversely related to the degree of market optimism about future prospects of asset payoffs. This effect is strong and economically very significant.

Risk premium, heterogenous beliefs, market state of belief, asset pricing, Bayesian learning, updating beliefs, Rational Beliefs

Abstract:
This paper studies the dynamic volatility properties of a monetary economy in which agents hold Rational Beliefs (see Kurz (1994), (1997)) rather than Rational Expectations. Except for this feature the examined Rational Belief Equilibrium (in short, RBE) is entirely standard: markets are competitive, prices are flexible and all information is symmetric. The paper demonstrates

a) The RBE paradigm offers an integrated theory of real and financial volatility with a high volume of trade. Most volatility in an RBE is induced endogenously through the beliefs of agents.

b) Although our RBE assumes fully competitive markets in which prices are fully flexible,the diverse expectations of agents can explain most of the familiar features of monetary equilibria. This includes, money non-neutrality, Phillips curve and impulse response functions with respect to monetary shocks.

c) Agents with diverse but inconsistent beliefs may induce socially undesirable excess fluctuations even when the allocation is ex-ante Pareto optimal. Central bank policy should aim to reduce the endogenous component of this volatility.

Abstract:
We show diverse beliefs is an important propagation mechanism of fluctuations, money non neutrality and efficacy of monetary policy. Since expectations affect demand, our theory shows economic fluctuations are mostly driven by varying demand not supply shocks. Using a competitive model with flexible prices in which agents hold Rational Belief (see Kurz (1994)) we show that (i) our economy replicates well the empirical record of fluctuations in the U.S. (ii) Under monetary rules without discretion, monetary policy has a strong stabilization effect and an aggressive anti-inflationary policy can reduce inflation volatility to zero. (iii) The statistical Phillips Curve changes substantially with policy instruments and activist policy rules render it vertical. (iv) Although prices are flexible, money shocks result in less than proportional changes in inflation hence the aggregate price level appears "sticky" with respect to money shocks. (v) Discretion in monetary policy adds a random element to policy and increases volatility. The impact of discretion on the efficacy of policy depends upon the structure of market beliefs about future discretionary decisions. We study two rationalizable beliefs. In one case, market beliefs weaken the effect of policy and in the second, beliefs bolster policy outcomes and discretion could be a desirable attribute of the policy rule. Since the central bank does not know any more than the private sector, real social gain from discretion arise only in extraordinary cases. Hence, the weight of the argument leads us to conclude that bank's policy should be transparent and abandon discretion except for rare and unusual circumstances. (vi) An implication of our model suggests the current effective policy is only mildly activist and aims mostly to target inflation.

Monetary policy rules, Money non neutrality, Business cycles, Market volatility, Propagation mechanism, Capacity utilization, Heterogenous beliefs, Over confidence, Rational Belief, Optimism; Pessimism, Non stationarity, Empirical distribution

Abstract:
The paper contrasts theories that explain diverse belief by asymmetric private information (in short PI) with theories which postulate agents use subjective heterogenous beliefs (in short HB). We focus on problems where agents forecast aggregates such as profit rate of the S&P500 and our model is similar to the one used in the literature on asset pricing (e.g. Brown and Jennings (1989), Grundy and McNichols (1989), Allen, Morris and Shin (2006)).

We first argue there is no a-priori conceptual basis to assuming PI about economic aggregates. Since PI is not observed, models with PI offer no testable hypotheses, making it possible to prove anything with PI. In contrast, agents with HB reveal their forecasts hence data on market belief is used to test hypotheses of HB. We show the common knowledge assumptions of the PI theory are implausible. The theories differ on four main analytical issues. (1) The pricing theory under PI implies prices have infinite memory and at each t depend upon unobservable variables. In contrast, under HB prices have finite memory and depend only upon observable variables. (2) The "Beauty Contest" implications of the two are different. Under PI today's price depends upon today's market belief about tomorrow's mean belief which is correct and depends upon supply shock and inference from prices. Under HB it depends upon today's market belief about tomorrow's market beliefs. Tomorrow's beliefs are, in part, beliefs about future beliefs and are often mistaken. Market forecast mistakes are key to Beauty Contests, and are a central cause of market uncertainty called "endogenous uncertainty." (3) Contrary to PI, theories with HB have wide empirical implications which are testable with available data. (4) PI theories assume unobserved data and hence do not restrict behavior, while rationality conditions impose restrictions on any HB theory. We explain the tight restrictions on the model's parameters imposed by the theory of Rational Beliefs.

private information, Bayesian learning, updating beliefs, heterogenous beliefs, asset pricing, Rational Beliefs.

Abstract:
A strategic mechanism of price adjustment is introduced to explain inflations in the U.S. during 1909-1974. The mechanism follows from our theory that when the profit rate is above a normal-target rate, competitive forces operate to lower prices while if the profit rate is below the target a correlated strategy among firms operates to generate a rise in prices as a strategy to improve profitability. The notion of "correlated strategy" is adopted from game theory. The mechanism may operate in harmony or against demand and the net effect is what we call the "basic inflation." Contrary to a-priori notions of positive association between inflation rates and profit rates, our theory proposes a critical test of a negative association between these variables. Such a relationship is in fact empirically established.

The analysis shows that large and persistent inflationary pressures are generated by low profitability and during 1971-1977 those accounted for some 20%-50% of total inflation. These pressures would be present even if no increase in cost occurs. This suggests that an important cause of the 1970's inflation is the low profit rate in the private sector and any public policy against inflation will fail if it does not aim at the same time to raise the profit rate on private capital.

Abstract:
This paper introduces the concept of Rational Belief Equilibrium (RBE) as a basis for a new theory of asset pricing. Rational Beliefs are probability beliefs about future economic variables which cannot be contradicted by the data generated by the economy. RBE is an equilibrium in which the diverse beliefs of all the agents induce an equilibrium stochastic process of prices and quantities and these beliefs are, in general, wrong in the sense that they are different from the true probability of the equilibrium process. These beliefs are, however, Rational. Consequently, in an RBE agents use the wrong forecasting functions and their forecasting mistakes play a crucial role in the analysis. First, we show that these mistakes are the reason why stock returns are explainable in retrospect and forecastable whenever the environment remains unchanged over a long enough time interval for agents to learn the forecasting function. Second, the aggregation of these mistakes generates Endogenous Uncertainty: it is that component of the variability of stock prices and returns which is endogenously induced by the beliefs and actions of the agents rather than by the standard exogenous state variables. The paper develops some basic propositions and empirical implications of the theory of RBE. Based on the historical background of the post world war II era, we formulate an econometric model of stock returns which allows non-stationarity in the form of changing environments ("regimes"). A sequence of econometric hypotheses are then formulated as implications of the theory of RBE and tested utilizing data on common stock returns in the post war period. Apart from confirming the validity of our theory, the empirical analysis shows that (i) common stock returns are forecastable within each environment but it takes time for agents to learn and approximate the forecasting functions. For some agents the time is too short so that it is too late to profit from such learning; (ii) the equilibrium forecasting functions change from one environment to the other in an unforecastable manner so that learning the parameters of one environment does not improve the ability to forecast in the subsequent environments. (iii) more than 2/3 of the variability of stock returns is due to endogenous uncertainty rather than exogenous causes. The paper analyzes one example of a gross market overvaluation which was induced in the 1960's by an aggregation of agent's Mistakes.