Elsevier

Journal of Economic Theory

Volume 178, November 2018, Pages 191-221
Journal of Economic Theory

A case for incomplete markets

https://doi.org/10.1016/j.jet.2018.09.004Get rights and content

Abstract

If two rational agents want to trade and there are no externalities, then trade is Pareto improving. Economists generally oppose restrictions on such trade. Complete markets allocations are Pareto optimal and thus complete markets are generally viewed as good. But when individuals want to trade because of heterogeneous beliefs, this standard argument is less compelling. We illustrate this in a standard general equilibrium setting and explore potential social benefits from restrictions on trade that make markets incomplete.

Introduction

A conventional wisdom in the economics profession is that complete markets are good. The welfare theorems state that complete markets outcomes are Pareto optimal and that any optimal allocation can be realized by trade in complete markets with an appropriate lump-sum transfer scheme. So limits that close trading opportunities leave potential mutual gains unrealized. This “wisdom” has practical consequences. Arguments for the privatization of social security and against the regulation of financial markets rely in part on the assertion that barriers to trade are bad things.

Complete markets have their critics. Some say that traders have market power and that their exploitation can be limited only by constraining trade. Others argue that lump-sum transfers are impossible. These critiques are empirical. The degree of market power could be large or small. Lump-sum transfers are not so much impossible as they are costly to execute. Consequently, these concerns are typically considered to be second-order.

We offer here a different and perhaps more fundamental critique of complete markets. When markets allocate contingent claims among expected-utility-maximizing agents who have heterogeneous beliefs, a market designer who cares about the realized discounted utility of consumption paths may prefer restrictions on trade to complete markets. As complete markets allocations are Pareto optimal (for the classical economies we study) we necessarily reject Pareto optimality as an appropriate welfare criterion for these economies. Our critique is motivated by two observations in the literature. First, Blume and Easley (2006) show that if traders have heterogeneous beliefs, a common discount factor and access to complete markets, only those traders whose beliefs are most nearly correct survive. The consumptions of all other traders converge to zero almost surely. Although the resulting allocation is Pareto optimal the objective impoverishment of traders with more incorrect beliefs seems undesirable. Second, Mongin (2005) shows in his discussion of “spurious unanimity” that, in the presence of heterogeneous beliefs, Pareto optimality is not a compelling welfare criterion. Other more recent literature (Brunnermeier et al., 2014 and Gilboa et al., 2014) proposes alternative welfare criteria for economies with heterogeneous beliefs. We do not advocate any of these welfare criteria; rather we argue that regardless of how a market designer views welfare, as long as he cares about the realized discounted utility of consumption paths, for some economies he should prefer markets in which trade is restricted to complete markets.

This critique is detailed in section 3, after an infinite-horizon model of trade in a single consumption good with complete markets is developed in section 2. If the actual data generating process were known to an omniscient social planner, Pareto calculations with correct beliefs is an obvious fix. Omniscient social planners do not exist, however, and without them there is no alternative welfare requirement that obviously ameliorates the issues raised in section 3. Section 3 also includes a discussion of alternative welfare criteria offered in the recent literature. We investigate the market design problem through computation of competitive equilibria for simple, classical economies. In sections 4 and 5 we describe the set of market restrictions we consider and how we propose that the market designer views welfare when traders have heterogeneous beliefs. Sections 6, 7 and 8 examine several policy alternatives to complete markets in Markovian instances of the model of financial restrictions developed in section 4, and there we explore the size and location of the set of data generating processes and beliefs for which these policies would lead to a true welfare improvement. We conclude in section 9 with a discussion of the theoretical and the policy implications of our findings. All proofs are provided in the Appendix A.1.

Section snippets

The model

We assume that time is discrete and begins at date 0. At each date a state is drawn from the set S={1,,S}. The set of all sequences of states is Σ with representative sequence σ=(s0,s1,...) called a path. Let σt=(s0,...,st) denote the partial history through date t. We use σ˜|σt to indicate that a path σ˜ coincides with a path σ up through period t.

The set Σ together with its product sigma-field is the measurable space on which everything is built. Let P0 denote the “true” probability measure

Welfare economics of heterogeneous beliefs

The welfare analysis of market outcomes begins with the Pareto order, taking preferences as given. “Tastes,” say Stigler and Becker (1977, p. 76), “are the unchallengeable axioms of a man's behavior: he may properly (usefully) be criticized for inefficiency in satisfying his desires, but the desires themselves are data.” Tastes, they say, “are not capable of being changed by persuasion.”

In contingent-claims markets, Pareto optimality is taken to be with respect to ex ante preferences (tastes).

Financial markets, competitive equilibria

In this section, we describe optimization problems of an agent under different financial market designs.

Welfare

A designer chooses a market structure M that, given beliefs P=(P1,...,PI), induces a competitive equilibrium allocation (c1(P|M),...,cI(P|M)).5 Individuals evaluate their welfare according to their own beliefs, but as we have argued, when beliefs are heterogeneous social welfare should not be based on individual perception of welfare. Instead the market designer should care about realized

Simple economies

We present a simple economy that we use to illustrate economic forces operating in economies with heterogeneous beliefs. In this section, we investigate social welfare using the Rawlsian utility aggregator (9). In section 7, we consider the Pareto criterion using market weights. The two criteria lead to remarkably similar results.

Agents share a common utility functionu(c)=c1γ/(1γ), where γ=2.8 There

Bergson–Samuelson criterion

Next, we examine the market design problem using the alternative utility aggregator that makes use of individuals' “market weights.” For each belief assignment P we first solve for the competitive equilibrium. Then we compute the vector of Pareto weights for which the competitive and the Pareto allocations coincide. Let θi(P) denote the implied Pareto weight, which we also call the market weight, of type-i agent.16

Dependence on P0

To this point we have analyzed the market design problem for a singleton B0. In this section, we confront our designer with multiple data-generating processes. Recall that our theoretical results hold for any P0 and, hence, for any B0. Our computed equilibria also show that welfare varies more under the complete markets design than under the designs with financial restrictions. In this section, we show that introducing ambiguity about P0 via expansion of B0 causes welfare gains from financial

Concluding remarks

We propose a framework to evaluate financial market designs for exchange economies in which agents have heterogeneous beliefs. Our analysis illustrates trade-offs between welfare-reducing speculation and welfare-improving insurance possibilities. Complete financial markets allow maximal insurance possibilities, but for economies with heterogeneous beliefs they also allow social welfare reducing speculation. In the economies that we study, financial market designs with simple restrictions like

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We thank Jonathan Parker for a helpful and detailed discussion. We also thank the referees for this journal, V.V. Chari, Vincenzo Quadrini, and seminar participants of the 2014 SITE conference, the 2014 ITAM Macroeconomics Workshop, the 2013 NBER Summer Institute, 2013 Econometric Society Australian meetings, 2012 North American Econometric Society meetings, 2012 Cornell/PennState Macroeconomics workshop.

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