Aggregation of heterogenous beliefs,asset pricing,and risk sharing in complete financial markets

Given a competitive equilibrium in complete asset markets, we propose a method that aggregates heterogeneous individual beliefs into a single “market probability,” which, if commonly shared by investors, generates the same marginal valuation of assets by the market as well as by each individual investor. As a result of the aggregation process, the market portfolio may have to be scalarly adjusted, upward or downward, a reflection of an aggregation bias due to the diversity of beliefs. From a dual viewpoint, the standard construction of an expected utility-maximizing aggregate investor designed to represent the economy in equilibrium, is shown to be also valid in the case of heterogeneous beliefs, modulo the above scalar adjustment of the market portfolio, thereby generating an Adjusted version of the Consumption based Capital Asset Pricing Model (ACCAPM). We analyze how the allocation of aggregate and individual risks relates to deviations of individual beliefs from the aggregate market probability. Finally, we identify the channels through which the distribution of beliefs and other microeconomic characteristics (incomes, attitudes toward risk) across investors impact the pricing of risky assets an may contribute to explaining the equity premium puzzle.     

Gurus and belief manipulation

We analyze a model with two types of agents: standard agents and gurus, i.e. agents who have the ability to influence the other investors. Gurus announce their beliefs and act accordingly. Gurus are strategic: they take into account the impact of their announced beliefs on the other agents, hence on prices. Standard agents observe gurus' performances, choose a guru and follow her/his recommendations. Prices are determined through a classical Walras mechanism. The competition among gurus for attracting followers among standard agents is governed by the level of accuracy of their predictions. The strategic behavior leads to belief subjectivity and heterogeneity among the gurus even when gurus' initial beliefs coincide with the objective belief. Optimism as well as pessimism can both emerge. We find a positive correlation across the agents between pessimism and risk tolerance. The representative agent belief, or the consensus belief is pessimistic. As a consequence of the pessimistic bias at the aggregate level, the risk premium is greater than in the standard rational expectations equilibrium. We show on an example that this impact is significative. In a multi-asset framework, the impact is stronger on the riskier assets.     

Optimal asset taxes in financial markets with aggregate uncertainty

This paper studies Pareto-optimal risk-sharing arrangements in a private information economy with aggregate uncertainty and ex ante heterogeneous agents. I show how to implement Pareto optima as equilibria when agents can trade claims to consumption contingent on aggregate shocks in financial markets. The first result is that if aggregate and idiosyncratic shocks are independent, the implementation of optimal allocations does not require any interventions in financial markets. This result can be extended to dynamic settings in the sense that, in this case, only savings need to be distorted, but not trades in financial markets. Second, I characterize optimal trading distortions in financial markets when aggregate and idiosyncratic shocks are not independent. In this case, optimal asset taxes must be higher for those securities that pay out in aggregate states in which consumption is more volatile. For instance, this can provide an efficiency justification for the frequently observed differential tax treatment of different asset classes, such as debt and equity claims.     

Efficient allocations under ambiguity

Important implications of the expected utility hypothesis and risk aversion are that if agents have the same probability belief, then consumption plans in every efficient allocation of resources under uncertainty are comonotone with the aggregate endowment, and if their beliefs are concordant, then the consumption plans are measurable with respect to the aggregate endowment. We study these two properties of efficient allocations for models of preferences that exhibit ambiguity aversion using the concept of conditional beliefs, which we introduce in this paper. We provide characterizations of such conditional beliefs for the standard models of preferences used in applications.     

Consensus Consumer and Intertemporal Asset Pricing with Heterogeneous Beliefs

The aim of this paper is to analyse the impact of heterogeneous beliefs in an otherwise standard competitive complete market economy. The construction of a consensus probability belief, as well as a consensus consumer, is shown to be valid modulo an aggregation bias, which takes the form of a discount factor. In classical cases, the consensus probability belief is a risk tolerance weighted average of the individual beliefs, and the discount factor is proportional to beliefs dispersion. This discount factor makes the heterogeneous beliefs setting fundamentally different from the homogeneous beliefs setting, and it is consistent with the interpretation of beliefs heterogeneity as a source of risk.
We then use our construction to rewrite in a simple way the equilibrium characteristics (market price of risk, risk premium, risk-free rate) in a heterogeneous beliefs framework and to analyse the impact of beliefs heterogeneity. Finally, we show that it is possible to construct specific parametrizations of the heterogeneous beliefs model that lead to globally higher risk premia and lower risk-free rates.     

Option smiling when investors’ estimates of asset volatility disagree

This study sheds light on why heterogeneous beliefs in volatility manifest the smile effect of options and how the degree of belief disagreement influences option-implied volatility. It is found that when investors’ level of heterogeneous beliefs increases, agents who over estimate volatility raise their subjective probability about those outer dividend states and hence increase more consumption to those states. As a result, an increase of consumption causes a decrease of their marginal rate of transformation in outer states. The raising subjective probability increases the prices of outer states, while the lower marginal rate of transformation decreases the prices of that. However, the influence of the former dominates the latter, causing a fatter tail of the state-price density. As a consequence, the values increase for those call options with high strike prices and put options with low strike prices, leading to a U-shaped implied volatility and hence causes a smell effect.     

Resolving Indeterminacy in Coordination Games: A New Approach Applied to a Pay-as-you-go Pension Scheme

This paper shows that when players ignore what outcome will emerge because of the presence of multiple equilibria, they can coordinate their expectations by forming an initial belief based on the principle of indifference followed by a process of reasoning that updates this belief. Since this procedure describes a natural way to form beliefs under indeterminacy, it is reasonable for every agent to conjecture that all the others form their beliefs according to the same logic. Exactly the fact that agents are aware that they form their beliefs following the same procedure allows them to successfully coordinate their expectations.      

How does the diversity of investors’ beliefs affect stock price informativeness?

A growing number of studies have investigated the role of stock prices in aggregating private information and guiding resource reallocation. However, this article may be the first attempt to study how the diversity of beliefs affects stock price informativeness. The framework of the noisy rational expectations model shows that stock informativeness is determined by both the precision and use of private information in trading. If private beliefs about the value are highly diverse, the aggregate average opinion revealed in a stock’s price will be more accurate and, thus, more informative. As the price becomes more informative, however, individual investors will rely less on their private information. When this occurs, less private information will be absorbed in price, which, in turn, reduces price informativeness. Our model shows that the relationship between belief diversity and price informativeness is U-shaped in equilibrium.     

Non-Laplacian beliefs in a global game with noisy signaling

In standard global games, individual behavior is optimal if it constitutes a best response to agnostic—Laplacian—beliefs about the aggregate behavior of other agents. This paper considers a standard binary action global game augmented with noisy signaling by an informed policy-maker and shows that in this game, equilibrium beliefs depart in quite stark ways from the Laplacian benchmark. In the limit as signals become arbitrarily precise, so that all fundamental uncertainty is removed (leaving only strategic uncertainty), the equilibrium beliefs of the marginal individual concerning the aggregate action collapse to a discrete Bernoulli distribution, giving probability mass only to the polar extreme outcomes. By contrast in the underlying standard global game the marginal individual believes the aggregate action has a continuous uniform distribution, giving equal likelihood to all possible outcomes.     

Coordination and correlation in Markov rational belief equilibria

Summary This paper studies the effect of correlation in the rational beliefs of agents on the volatility of asset prices. We use the technique of generating variables to study stable and non-stationary processes needed to characterize rational beliefs. We then examine how the stochastic interaction among such variables affects the behavior of a wide class of Rational Belief Equilibria (RBE). The paper demonstrates how to construct a consistent price state space and then shows the existence of RBE for any economy for which such price state space is constructed. Next, the results are used to study the volatility of asset prices via numerical simulation of a two agents model. If beliefs of agents are uniformly dispersed and independent, we would expect heterogeneity of beliefs to have a limited impact on the fluctuations of asset prices. On the other hand, our results show that correlation across agents can have a complex and dramatic effect on the volatility of prices and thus can be the dominant factor in the fluctuation of asset prices. The mechanism generating this effect works through the clustering of beliefs in states of different levels of agreement. In states of agreement the conditional forecasts of the agents tend to fluctuatetogether inducing more volatile asset prices. In states of disagreement the conditional forecasts fluctuatein diverse directions tending to cancel each other's effect on market demand and resulting in reduced price volatility.This research was supported, in part, by the Fondazione Eni Enrico Mattei of Milan, Italy, and by the Research Incentive Fund of Stanford University. The authors thank Carsten K. Nielsen and Ho-Mou Wu for valuable discussions on an earlier draft. Carsten K. Nielsen also made an important contribution to the development of Section 3.     

Beliefs and Public Good Provision with Anonymous Contributors

We analyze a static game of public good contributions where finitely many anonymous players have heterogeneous preferences about the public good and heterogeneous beliefs about the distribution of preferences. In the unique symmetric equilibrium, the only individuals who make positive contributions are those who most value the public good and who are also the most pessimistic; that is, according to their beliefs, the proportion of players who most like the public good is smaller than it would be according to any other possible belief. We predict whether the aggregate contribution is larger or smaller than it would be in an analogous game with complete information and heterogeneous preferences, by comparing the beliefs of contributors with the true distribution of preferences. A trade‐off between preferences and beliefs arises if there is no individual who simultaneously has the highest preference type and the most pessimistic belief. In this case, there is a symmetric equilibrium, and multiple symmetric equilibria occur only if there are more than two preference types.     

Credit constraints,growth and inequality dynamics

This paper examines how credit constraints affect the dynamics of wealth and thereby the dynamics of capital and output growth. We develop standard Ak growth models that display transitional dynamics, contrary to general belief, once the complete credit markets assumption is relaxed. The mechanism is that credit constraints make individual productivity differences persist, which in turn leads to the persistence of income inequality. The dynamics of inequality is jointly determined with the dynamics of aggregate capital. The economy thus passes through a transitional period of inequality, individual and aggregate capital dynamics before it converges to a long-run balanced growth path. The application of the model to the analysis of intergenerational mobility and inequality dynamics suggests substantial economic and policy significance. In particular, introducing credit constraints to the Barro Ak model, public investment could have an indirect impact on growth via its effect on inequality and mobility.     

Belief and higher-order belief in the centipede games: An experimental investigation

This paper experimentally explores the epistemic conditions behind people's non-equilibrium behaviour in the centipede games. We propose a novel design of laboratory experiment to elicit people's first- and second-order beliefs regarding their opponents' choices and beliefs. The measured beliefs, together with the choice data, help us to estimate people's level of rationality, belief of rationality and second-order belief of rationality. To examine how these epistemic variables are affected by the social-efficiency property of the classic increasing-sum centipede game, we revisit the constant-sum centipede and compare the measured epistemic conditions from the constant-sum with those from the classic centipede. We find that people's non-backward induction behaviour may be attributed to the diffusion of beliefs and higher-order beliefs in the increasing-sum centipede. We consider a behavioural model in which people's preferences for social efficiency are incorporated into the extended utility maximization problem. Our analytical and estimation results indicate that the presence of efficiency-oriented players and people's belief towards the uncertain portion of such type of players may play a part in the non-backward-induction outcomes in experimental centipede games.     

Constructing Historical Euro-zone Data

Existing methods of reconstructing historical Euro-zone data by aggregation of the individual countries' aggregate data raises numerous difficulties, especially due to past exchange rate changes. The approach proposed here is designed to avoid such distortions, and aggregate exactly when exchange rates are fixed. We first compute growth rates within states, aggregate these, then cumulate this Euro-zone growth rate to obtain the aggregated levels variables. The aggregate of the implicit-deflator price index coincides with the implicit deflator of our aggregate nominal and real data. We apply the method to Euro-zone M3, GDP and prices over the previous two decades.     

Preference shocks from aggregation: time series data evidence

Abstract.  We investigate the impact of preference shocks on the aggregate dynamics of the U.S. economy in the context of a neoclassical growth model derived from aggregation. The aggregation result we use is as follows: if markets are complete and if agents have identical preferences of the addilog type, then the heterogeneous‐agent economy where agents are subject to idiosyncratic productivity shocks behaves as if there was a representative consumer who faces shocks to preferences and technology. We estimate the parameters in the aggregation‐based model from the aggregate time‐series data and compute the numerical solution. We find that the preference shocks play an important role in the aggregate labour‐market fluctuations. JEL classification: C73, D90, E21     

MODELING THE EVOLUTION OF EXPECTATIONS AND UNCERTAINTY IN GENERAL EQUILIBRIUM

We develop methods to solve general equilibrium models in which forward‐looking agents are subject to waves of pessimism, optimism, and uncertainty that turn out to critically affect macroeconomic outcomes. Agents in the model are fully rational and conduct Bayesian learning, and they know that they do not know. Therefore, agents take into account that their beliefs will evolve according to what they will observe. This framework accommodates both gradual and abrupt changes in beliefs and allows for an analytical characterization of uncertainty. We use a prototypical Real Business Cycle model to illustrate the methods.     

The anatomy of the aggregate labor supply elasticity

We show that the aggregate Frisch elasticity of labor supply can greatly exceed the corresponding individual-level parameter, and we illustrate the “anatomy” of the former in terms of intensive and extensive margins. The methodology consists of using micro data from the PSID to construct a panel of individuals and an aggregate time series obtained by aggregating these individuals each year. These two data sets represent exactly the same sample at different levels of aggregation, and we use them to identify the parameters of two distinct MaCurdy-type micro and macro equations. We find a micro elasticity of about 0.1 and a much larger macro elasticity that ranges from 1.1 to 1.7. There is no conflict between the two estimates: the micro one reflects only the intensive margin while the macro one reflects, in addition, the much more volatile extensive margin. Furthermore, aggregation of only continuously employed individuals allows us to provide a reliable estimate of the intensive margin elasticity in the range 0.3–0.4. This implies an extensive margin elasticity in the range 0.8–1.4. These findings suggest that micro evidence is not a benchmark for assessing how large the Frisch elasticity of labor supply should be in a model of the aggregate economy.     

Out of your mind: Eliciting individual reasoning in one shot games

We experimentally investigate the fundamental element of the level-k model of reasoning, the level-0 actions and beliefs. We use data from a novel experimental design that allows us to obtain incentivised written accounts of individuals' reasoning. In particular, these accounts allow to infer level-0 beliefs. Level-0 beliefs are not significantly different from 50, and almost 60% of higher level players start their reasoning from a level-0 belief of exactly 50. We also estimate that around one third of the participants play non-strategically. The non-strategic level-0 actions are not uniformly distributed.     

Municipal aggregation and retail competition in the Ohio energy sector

Ohio allows communities to vote to aggregate the loads of individual consumers (unless they opt out) in order to seek a competitive energy supplier. Over 200 communities have voted to do this for electricity. By 2004 residential switching reached 69% in Cleveland territory (95% from municipal aggregation) but by 2006 had fallen to 8%. Savings are now small, but customer acquisition costs are low and the cost to consumers is negligible. Aggregation and retail competition have been thwarted by Rate Stabilization Plans holding incumbent utility prices below cost since 2006. In the Ohio gas sector, rate regulation has not discouraged aggregation and competition, but market prices falling below municipally negotiated rates can be politically embarrassing. Municipal aggregation thus works when conditions allow it, and enhances competition. How it would fare against individual choice in a market conducive to retail competition remains an open question.      

Markets and relationships in a learning economy

It is generally agreed that within long-term relationships agents learn the characteristics of their market partners better than through spot transactions. In contrast, little is known on how relationship-based and transaction-based markets compare when agents learn about the aggregate economy from market exchanges. In this paper, we study the market structure that arises in an economy where agents learn the aggregate productivity from market exchanges. The model allows to relate the market structure to macroeconomic fundamentals such as the persistence and volatility of the aggregate productivity and the cross-sectional volatility of productivity.