Heterogeneous beliefs and trading inefficiencies

Abstract

Heterogeneous beliefs are introduced into the monetary economy of Lagos and Wright (2005) and the implications for monetary equilibria are considered. An endogenous fraction of agents hold rational expectations and the remaining agents employ an adaptive learning rule similar to Evans and Honkapohja (2001) and Brock and Hommes (1997). Three primary results follow from the finding that heterogeneous beliefs can destabilize a stationary monetary equilibrium and lead to non-linear dynamics bounded around the monetary steady state. First, heterogeneous beliefs can lead to equilibria that are welfare reducing due, in part, to a lower acceptance rate in decentralized meetings. Second, when buyers, who are uncertain about their beliefs, behave like Bayesians by placing a prior on sellers' beliefs, uncertainty impacts dynamic stability and welfare. Third, the model's unique predictions provide an explanation of new findings about the acceptance rate in monetary laboratory experiments.

Introduction

This paper introduces heterogeneous beliefs into the workhorse Lagos and Wright (2005) model of monetary theory. The main finding is that heterogeneous beliefs can alter the nature of trade in bilateral markets by generating dynamic monetary equilibria with distinct implications for the intensive and extensive margins of trade, welfare, and for reconciling key experimental findings.

Models of monetary economies take into account the role assets play in facilitating exchange in decentralized markets.¹ For example, in the Lagos and Wright (2005) model agents trade in bilateral meetings, while limited commitment and imperfect monitoring prevent buyers from using credit in exchange for goods. A demand for fiat money can arise in equilibrium because it facilitates exchange in these bilateral meetings.

The set of monetary equilibria depends on the fundamentals of the economy such as the available technology, preferences, search frictions, and the trading protocol in bilateral matches. The beliefs of agents are also fundamental to the model. Sellers are only willing to exchange goods for fiat money if they believe that they will be able to exchange that money for goods in later periods. Similarly, buyers will only demand fiat money if they believe that it has value in the future. The monetary equilibria commonly examined in the literature feature a (potentially) delicate coordination of beliefs, and the principal theory of how beliefs are formed is the rational expectations hypothesis. Rational expectations, while a natural benchmark, is a strong assumption that requires agents to form their expectations optimally with respect to the economy's true underlying conditional distributions that, in turn, depend upon agents' beliefs in a self-referential way.² We relax the rational expectations assumption by introducing heterogeneous beliefs – where buyers and sellers are distributed across a variety of forecasting models, including rational expectations – into the monetary search model.

We build on the framework in Lagos and Wright (2005).³ Each period is divided into two sub periods. In the first sub period, there is a decentralized market distinguished by bilateral exchange where buyers, who can consume but cannot produce, are matched with sellers, who can produce but cannot consume. When buyers and sellers are matched, the terms of trade are determined by take-it-or-leave-it offers made by buyers. In the second sub period there is a centralized market where buyers and sellers can both consume, trade assets and produce.

We introduce heterogeneity in beliefs by assuming that there is a fraction of agents who hold rational expectations and the remaining agents are distributed across other “boundedly rational” forecasting models. As a baseline example, we assume that agents are either rational or form forecasts using an adaptive learning rule. The adaptive learning rule is in the spirit of Marcet and Sargent (1989) and Evans and Honkapohja (2001), who argue that an economic agent should behave like a good econometrician and form forecasts from a well-specified forecasting model that adapts to account for recent data.

The structure of heterogeneous beliefs in our model is specified so that in a steady-state equilibrium all beliefs coincide and the perfect-foresight steady-state rational expectations equilibrium obtains. The range of equilibrium outcomes and of economic dynamics depends on the distribution of individuals across forecasting models. An important aspect to our model is that the distribution of heterogeneity is an endogenous object and perfect foresight remains a choice available to all agents, though they must pay a computational cost for its use. Following Brock and Hommes (1997) and Branch and Evans (2006), the fraction of agents with perfect foresight is increasing in its forecast accuracy, net of the computational cost, relative to the accuracy of the adaptive learning rule. Empirical evidence in favor of this framework for expectation formation is provided by Branch (2004) and Hommes (2013).

The main results of this paper are as follows. First, heterogeneous beliefs can lead to inefficient trading outcomes with lower welfare than would arise in a rational expectations equilibrium. In a steady state, all individuals hold identical beliefs. Thus, the inefficiencies that stem from heterogeneous beliefs arise along dynamic equilibrium paths. The novelty in this paper is that we allow beliefs, and hence trading inefficiencies, to be determined as an endogenous object of the model.

Heterogeneous beliefs alter the set of monetary equilibria. Unlike rational expectations monetary equilibria which require full coordination of beliefs, we show the existence of monetary equilibria with heterogeneous, boundedly rational beliefs. Although the steady state of the monetary equilibrium we consider coincides with the steady state under rational expectations, heterogeneous beliefs can destabilize the steady state and lead to a variety of periodic and aperiodic cycles, including complicated dynamics, that remain bounded around the steady-state monetary equilibrium. Along these dynamic equilibria, the distribution of money holdings across agents is non-degenerate and time-varying.

An interesting implication of the framework in this paper is the role that within-match common knowledge assumptions play in bilateral trade. In random, anonymous decentralized meetings it is natural to assume that buyer and seller beliefs are not common knowledge within a match. To see how this impacts trading behavior, consider a take-it-or-leave-it offer made by a buyer to a seller. Under common knowledge, the buyer would select an offer that just meets the seller's participation constraint. Without common knowledge, the buyer must form expectations about the seller's participation constraint, and this constraint depends on the seller's beliefs about the future value of money. Thus, when making a take-it-or-leave-it offer, a buyer must allow for the possibility that his beliefs do not align with the seller's. Consequently, we model buyers as Bayesians who holds priors over sellers' beliefs and, therefore, who explicitly acknowledge their uncertainty about the sellers' participation constraints. We call the bargaining offers that arise in this setting Bayesian offers.

Along a dynamic path, when a buyer and seller with different expectations about the value of money are matched, they may be unable to reach agreement on the terms of trade. Thus, heterogeneous beliefs can lead to an endogenous extensive margin of trade, which we measure as an acceptance rate, that is, the proportion of take-it-or-leave-it offers accepted by sellers. Naturally, the failure of some offers to be accepted in some matches, that is, an acceptance rate lower than one, magnifies search frictions.

We present results that demonstrate how heterogeneous beliefs reduce welfare and, in particular, we characterize how welfare is impacted by variation in buyers' uncertainty, parameterized in terms of the spread of their prior distribution. We identify three ways uncertainty impacts welfare. First, uncertainty affects the intensive margin of trade as buyers make “cautious” offers whereby they offer a higher payment in exchange for a smaller quantity. By altering the demand for money, this affects the equilibrium price. Second, these cautious offers, all else equal, increase the acceptance rate of sellers, i.e. the extensive margin of trade. Third, small changes in uncertainty can bifurcate the equilibrium leading to qualitatively distinct dynamics that can feature higher or lower welfare depending on the resulting dynamics for the acceptance rate.

Another main result of this paper offers the unique theoretical predictions from the model with heterogeneous beliefs as an explanation for the results from a Lagos–Wright laboratory environment in Duffy and Puzzello (2014), who provide evidence in favor of monetary equilibria. They also find, however, that

• a large fraction of offers made by buyers are not accepted by sellers;

• the accepted offers are different from the theoretical stationary equilibrium price and quantity;

• there is a non-degenerate distribution of money holdings despite evidence of portfolio rebalancing in the centralized market;

• higher-quantity (buyer) offers are more likely to be rejected.

These experimental results are inconsistent with the rational expectations equilibrium but are consistent with the theoretical predictions of the monetary search model with heterogeneous expectations. Furthermore, the model presented in this paper makes sharp predictions regarding the relationship between prices and the acceptance rate, as well as between prices and sellers' realized surpluses, which should be invariant to the price of money under rational expectations.

What is the intuition for why heterogeneous beliefs can destabilize the stationary monetary equilibrium? In a stationary equilibrium, rational expectations (perfect foresight) and the adaptive predictor deliver the same forecast. Depending on the parameterization, the perfect-foresight steady state corresponding to the monetary equilibrium may be determinate or indeterminate. When the steady state is indeterminate, the monetary equilibrium is a sink under perfect foresight dynamics and a source under adaptive expectations. In a neighborhood of the stationary equilibrium, rational expectations and the adaptive predictor will both forecast similarly and agents will be unwilling to pay the computational costs necessary for perfect foresight. As the fraction of adaptive agents increases, the dynamics push the economy away from the stationary equilibrium until it reaches a point at which the accuracy gains associated to rational expectations outweigh the costs required for its use; hence, the fraction of rational agents increases and the economy moves towards the stationary equilibrium. The resulting tension between the stabilizing (or, attracting) and repelling dynamics can yield periodic orbits and complex dynamics.

There is a substantial literature that studies dynamic monetary equilibria in versions of the Lagos–Wright model. Lagos and Wright (2003) show that periodic and aperiodic equilibria can arise when bilateral trade is negotiated via Nash bargaining. Nosal and Rocheteau (2011) show the existence of periodic cycles in a model very close to ours with buyer-takes-all bargaining. The novelty in this paper is that these fluctuations arise for a different reason, namely, the endogenous distribution of heterogeneous beliefs.

It is also important to note that the framework employed here can be applied in more general settings such as any extension of the model that might include other assets, e.g. stocks or bonds, that can have a liquidity role in over-the-counter markets. Thus, the types of dynamics present in this paper might arise in other asset pricing settings with search frictions, and may provide realistic asset price dynamics such as bubbles and crashes.

The paper proceeds as follows. Section 2 details the general model environment and introduces take-it-or-leave it offers with belief uncertainty, Bayesian offers. Section 3 studies the steady-state properties with belief uncertainty, and presents a set of benchmark results on the dynamic stability properties of heterogeneous beliefs. Section 4 presents the implications of heterogeneous beliefs for welfare. Section 5 explores the unique theoretical predictions of the model in the Duffy–Puzzello experimental data, while Section 6 concludes.

The results in this paper relate to a literature in monetary theory which incorporates heterogeneous valuations in search models, see for instance Rocheteau (2011). In Jacquet and Tan (2011), an extension to Rocheteau (2011), buyers and sellers attach different values to money because of shocks to their disutility of effort in the centralized market. In their model, sellers value money more and so are willing to produce more in any given match. The results in this paper are closely related, but the difference in values arises because of endogenously heterogeneous beliefs. Moreover, we show that it is possible for output, within a match, to be inefficiently low or high. In Engineer and Shi, 1998, Engineer and Shi, 2001 and Berentsen and Rocheteau (2003), buyers and sellers have asymmetric demands for the other's goods and study whether monetary equilibria can exist even without the double coincidence problem. Berentsen and Rocheteau (2003) show in settings with asymmetric demands that money can have value in equilibrium because agents value money symmetrically. In our model, buyers and sellers value money asymmetrically when they have heterogeneous beliefs. It would be interesting to consider the choice of which assets to use as a medium of exchange in an environment where buyers and sellers disagree about the value of only a subset of assets.

The existence of cycles and non-linear dynamics in monetary models is well-known. For example, in addition to the aforementioned Lagos and Wright (2003), Gu et al. (2013) demonstrate non-linear perfect foresight dynamics in a model with endogenous credit constraints. Most of these perfect foresight models require the dynamic pricing equation to be non-monotonic much like in the extensive literature that studies non-linear dynamics in overlapping generations models. In the present paper, interesting dynamics can arise under more general conditions and rely on the tension between attracting and repelling dynamics that are inherent to heterogeneous expectations. The non-linear dynamics that arise in our model are very close to those in Brock and Hommes (1997) and Hommes (2013).

There is an extensive literature that studies bounded rationality and learning in macroeconomic models. Adaptive learning models formulated by Marcet and Sargent (1989) and Evans and Honkapohja (2001) are based on a “cognitive consistency principle” that states that economic agents should be modeled like economists and econometricians who specify models and revise their beliefs in light of data. Here, the choice of the forecasting model is an endogenous object. Moreover, most monetary models with adaptive learning do not feature bilateral bargaining. One contribution of this paper is to extend learning analysis to frameworks where higher-order expectations matter.

The results of this paper are also related to monetary search models with non-degenerate wealth distributions that arise from the idiosyncratic consumption and production possibilities in search markets. Shi (1997b) uses a large household model to construct a model of divisible money with a non-degenerate wealth distribution. Similarly, Berentsen et al. (2005) and Molico (2006) study the distributional consequences of monetary policy in monetary search models. Here, we work with a very tractable extension of the Lagos–Wright model, where the distribution of money holdings arises because of heterogeneous beliefs. Because of the non-degenerate monetary distribution, there will be an endogenous price dispersion reminiscent of Peterson and Shi (2004).⁴

There is also a literature on private information in payoffs to assets and the potential role of signaling and other strategic considerations in the decentralized market. See, for example, Nosal and Wallace (2007), Lester et al. (2012), Li et al. (2012), and Golosov et al. (2014). Relatedly, Berentsen et al. (2014) find that uncertainty and private information in the payoffs of an asset can account for the large number of rejected offers in the laboratory. This paper focuses on agents with heterogeneous beliefs – who make take-it-or-leave-it offers given their priors about the value of money – and abstracts from the strategic issues. These and other issues are discussed in greater detail below.