Multiple equilibria and hysteresis in simple exchange models

This paper presents a simple two-good general equilibrium model of exchange in which demands are linear functions of relative prices. The solution of the model is represented by a cubic equation which can have either a single root which corresponds to a unique equilibrium price or three roots corresponding to multiple equilibria. The major properties of the model are that the relative price can make large discrete jumps in response to small parameter changes, and can display hysteresis. The paper shows how recent developments in non-linear methods can be used to examine the equilibrium properties of the model and establish conditions under which multiple equilibria can arise. The model is compared with that used by Shapley and Shubik.     

Money illusion and coordination failure

Economists long considered money illusion to be largely irrelevant. Here we show, however, that money illusion has powerful effects on equilibrium selection. If we represent payoffs in nominal terms, choices converge to the Pareto inferior equilibrium; however, if we lift the veil of money by representing payoffs in real terms, the Pareto efficient equilibrium is selected. We also show that strategic uncertainty about the other players' behavior is key for the equilibrium selection effects of money illusion: even though money illusion vanishes over time if subjects are given learning opportunities in the context of an individual optimization problem, powerful and persistent effects of money illusion are found when strategic uncertainty prevails.     

Growth and multiple equilibria: A unique local dynamics

The paper investigates the local dynamics of a simple growth model with a public health factor. A notable feature of the model is that it employs a non-separable utility function between consumption and the level of public health infrastructure (publicly provided health environment). In addition to the effects on utility, an increase in the public health level contributes to labor productivity growth. From a three-dimensional dynamic system that includes physical capital, public health infrastructure and consumption, our model is found to have multiple equilibria (dual steady states) if the deep parameters of the utility function satisfy certain conditions. Then, numerical computations demonstrate that each equilibrium exhibits local saddle-path stability. This result deserves careful attention, as it indicates that the economy experiences a relatively undiversified growth pattern when converging on high- and low-growth equilibria. Our striking result has important implications in the study of growth and development.     

Endogenous public policy and multiple equilibria

One of the well known roles of public policy in models with indeterminacy is to reduce the set of equilibria. However, agents' expectations regarding future policy may be self fulfilling when public policy is endogenized. We show that a simple overlapping generations economy with public education may yield multiple equilibria. Under a laissez-faire system, our model has a unique equilibrium. Thus, the presence of public policy may generate, rather than eliminate, multiple equilibria.     

The replicator dynamics does not lead to correlated equilibria

It is shown that, under the replicator dynamics, all strategies played in correlated equilibrium may be eliminated, so that only strategies with zero marginal probability in all correlated equilibria survive. This occurs in particular in a family of 4×4 games built by adding a strategy to a Rock-Paper-Scissors game.     

Public health infrastructure and growth: Ways to improve the inferior equilibrium under multiple equilibria

This paper develops an endogenous growth model in which public health infrastructure, specified as a stock, plays an important role in economic growth. A notable feature of the model is that it employs a non-separable utility function for consumption, leisure, and the level of public health. In addition, increasing the level of health infrastructure contributes to the production of goods through labor augmentation. With these settings, our model is found to have a unique equilibrium or multiple equilibria, depending on the magnitude of the intertemporal elasticity of substitution. For the case of multiple equilibria, we numerically study the ways to avoid the low-growth state in developing countries. From this, we identify two feasible policy implications. The results indicate that public health infrastructure has a vital role in the development policies of low-income countries. Lastly, we show that there are two possibilities in regard to the local dynamics of the model.     

When and why? A critical survey on coordination failure in the laboratory

Coordination games with Pareto-ranked equilibria have attracted major attention over the past two decades. Two early path-breaking sets of experimental studies were widely interpreted as suggesting that coordination failure is a common phenomenon in the laboratory. We identify the major determinants that seem to affect the incidence, and/or emergence, of coordination failure in the lab and review critically the existing experimental studies on coordination games with Pareto-ranked equilibria since that early evidence emerged. We conclude that there are many ways to engineer coordination successes.      

Public capital, endogenous growth, and endogenous fluctuations

Cazzavillan [Cazzavillan, G., 1996. Public spending, endogenous growth, and endogenous fluctuations. Journal of Economic Theory 71, 394–415] studies a discrete-time, one-sector endogenous growth model with a flow of publicly enjoyable goods and productive services financed through income taxation. He demonstrates how equilibrium paths are indeterminate, for a large range of the consumption externality of public spending. This study extends [Cazzavillan, 1996] by considering an otherwise identical production function, except with public capital stock as an input. The results support the robustness of multiple growth paths even in a one-sector growth model with public capital stock, and modify the set of the consumption externality of public spending, in determining growth dynamics in a similar model with non-accumulated public spending.     

Macroeconomic coordination failure under oligempory

Summary. Each sector of a multi-sector overlapping generations model is an oligempory with a given number of firms, oligopsonists in the sectoral (spatially differentiated) labour market and oligopolists in the sectoral (homogeneous) output market. When there is aggregate unemployment, and a firm raises wages beyond the local full employment level acquiring labour from neighbours, sectoral output supply becomes constant and the firm faces a flat output demand curve under constant returns to labour (upward sloping under decreasing returns). Multiple temporary equilibria and Pareto-ranked steady-state equilibria emerge; the associated sunspot equilibria exhibit counter-cyclical markups, inter alia. Received: February 28, 2000; revised version: March 16, 2001     

Unique and multiple equilibria in a macroeconomic model with environmental quality: An analysis of local stability

This paper develops and analyzes an economic growth model which incorporates environmental quality into the production and utility functions. We solve our model for the balanced growth path and find that a unique low growth equilibrium is attained when environmental quality is given less weight in the utility function. Multiple equilibria exist if environment quality is given greater weight in the utility function. We also perform local stability analysis of our model. We conclude that an economy in which the environmental quality is given relatively less importance by the agents will be caught in low growth, high consumption poverty traps as is the case for many developing countries while other economies can potentially reach a relatively low consumption, high growth steady state if they place greater weight on environmental quality. Finally we look at how the gap between low and high growth equilibria shrinks when individuals place greater weight on environmental quality and how governmental policies can promote growth when societies give less weight to the environment.     

Concatenate coordination and mutual coordination

We tell of the evolving meaning of the term coordination as used by economists. The paper is based on systematic electronic searches (on “coord,” etc.) of major works and leading journals. The term coordination first emerged in professional economics around 1880, to describe the directed productive concatenation of factors or activities within a firm. Also, transportation economists used the term to describe the concatenation of routes and trips of a transportation system. These usages represent what we term concatenate coordination. The next major development came in the 1930s from several LSE economists (Hayek, Plant, Hutt, and Coase), who extended that concept beyond the eye of any actual coordinator. That is, they wrote of the concatenate coordination of a system of polycentric or spontaneous activities. These various applications of concatenate coordination prevailed until the next major development, namely, Thomas Schelling and game models. Here coordination referred to a mutual meshing of actions. Game theorists developed crisp ideas of coordination games (like “battle of the sexes”), coordination equilibria, convention, and path dependence. This “coordination” was not a refashioning, but rather a distinct concept, one we distinguish as mutual coordination. As game models became more familiar to economists, it was mutual coordination that economists increasingly had in mind when they spoke of “coordination.” Economists switched, so to speak, to a new semantic equilibrium. Now, mutual coordination overshadows the older notion of concatenate coordination. The two senses of coordination are conceptually distinct and correspond neatly to the two dictionary definitions of the verb to coordinate. Both are crucial to economics. We suggest that distinguishing between the two senses can help to clarify “coordination” talk. Also, compared to talk of “efficiency” and “optimality,” concatenate coordination allows for a richer, more humanistic, and more openly aesthetic discussion of social affairs. The narrative is backed up by Excel worksheets that report on systematic content searches of the writings of economics using the worldwide web and, using JSTOR, of Quarterly Journal of Economics, Economic Journal, Journal of Political Economy, American Economic Review, and Economica.     

Evidence on learning in coordination games

This paper reports an experiment designed to detect the influence of strategic uncertainty on behavior in order statistic coordination games, which arise when a player’s best response is an order statistic of the cohort’s action combination. Unlike previous experiments using order statistic coordination games, the new experiment holds the payoff function constant and only changes cohort size and order statistic.

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Rich Nations, Poor Nations: How Much Can Multiple Equilibria Explain?

This paper asks whether the income gap between rich and poor nations can be explained by multiple equilibria. We explore the quantitative implications of a simple two-sector general equilibrium model that gives rise to multiplicity, and calibrate the model for 127 countries. Under the assumptions of the model, around a quarter of the world’s economies are found to be in a low output equilibrium. We also find that, since the output gains associated with an equilibrium switch are sizeable, the model can explain between 15 and 25% of the variation in the logarithm of GDP per worker across countries.     

Mixed equilibria in games of strategic complementarities

Summary. The literature on games of strategic complementarities (GSC) has focused on pure strategies. I introduce mixed strategies and show that, when strategy spaces are one-dimensional, the complementarities framework extends to mixed strategies ordered by first-order stochastic dominance. In particular, the mixed extension of a GSC is a GSC, the full set of equilibria is a complete lattice and the extremal equilibria (smallest and largest) are in pure strategies. The framework does not extend when strategy spaces are multi-dimensional. I also update learning results for GSC using stochastic fictitious play. Received: October 16, 2000; revised version: March 7, 2002 RID="*" ID="*" I am very grateful to Robert Anderson, David Blackwell, Aaron Edlin, Peter De Marzo, Ted O'Donoghue, Matthew Rabin, Ilya Segal, Chris Shannon, Clara Wang and Federico Weinschelbaum for comments and advise.     

A computational analysis of core convergence in a multiple equilibria economy

We examine the structure of the core of a trading economy with three competitive equilibria as the number of traders (N) is varied. The core first splits into two pieces at N=5 and then splits a second time into three pieces at N=12. Both of these splits occur not at a point but as a contiguous gap. We find that the speed of convergence of the core toward the three competitive equilibria is not uniform but when N is large, the convergence rate is approximately of the order 1/N.     

On learning equilibria

Bullard (1994) and Schönhofer (1999) show that endogenous business cycles may emerge in an inflationary overlapping generations model where households predict future inflation rates by running a least squares regression on prices. We show that given the same beliefs but under an alternative, more natural, estimation procedure based upon inflation rates the monetary steady state will be globally stable for a large set of savings functions. We also study an evolutionary competition between the two estimation procedures. Although the dynamics are stabilized for a large set of parameter values, endogenous business cycles may still emerge in this heterogeneous beliefs framework.     

Knightian uncertainty and poverty trap in a model of economic growth

This paper explores how Knightian uncertainty affects dynamic properties in an economic growth model. The decision-making theory employed in the analysis is the theory of expected utility under a non-additive probability measure, i.e., the Choquet expected utility model of preference. We apply this decision-making theory to an overlapping generations model where producers face “uncertainty” in their technologies. When the producer is averse to uncertainty, the firm's profit function may not be differentiable. Therefore, the firm's decision to invest and hire labor becomes rigid for a certain measurable range of real interest rates. In dynamic equilibrium, the existence of firm-level rigidity causes discontinuity in the wage function; this makes multiple equilibria the more likely outcomes under the log utility and Cobb–Douglas production functions. In this paper, we show that even if aversion to uncertainty is small, the “poverty trap” can arise for a wide range of parameter values.     

Run equilibria in the Green–Lin model of financial intermediation

We study the Green–Lin model of financial intermediation [E.J. Green, P. Lin, Implementing efficient allocations in a model of financial intermediation, J. Econ. Theory 109 (2003) 1–23] under a more general specification of the distribution of types across agents. We derive the efficient allocation in closed form. We show that, in some cases, the intermediary cannot uniquely implement the efficient allocation using a direct revelation mechanism. In these cases, the mechanism also admits an equilibrium in which some (but not all) agents “run” on the intermediary and withdraw their funds regardless of their true liquidity needs. In other words, self-fulfilling runs can arise in a generalized Green–Lin model and these runs are necessarily partial, with only some agents participating.     

Undominated (and) perfect equilibria in Poisson games

In games with population uncertainty some perfect equilibria are in dominated strategies. We prove that every Poisson game has at least one perfect equilibrium in undominated strategies.     

Creditworthiness and thresholds in a credit market model with multiple equilibria

Summary. The paper studies creditworthiness in a model with endogenous credit cost and debt constraints. Such a model can give rise to multiple candidates for steady state equilibria. We use new analytical techniques such as dynamic programming (DP) with flexible grid size to find solutions and to locate thresholds that separate different domains of attraction. More specifically, we (1) compute present value borrowing constraints and thus creditworthiness, (2) locate thresholds where the dynamics separate to different domains of attraction, (3) show jumps in the decision variable, (4) distinguish between optimal and non-optimal steady states, (5) demonstrate how creditworthiness and thresholds change with change of the credit cost function of the debtor and (6) explore the impact of debt ceilings and consumption paths on creditworthiness.JEL Classification Numbers: C61, C63, D91, D92, E51, G12, G32.An earlier version of this paper has been prepared for the 1998 North American Winter Meeting of the Econometric Society, January 1998, Chicago. We want to thank Jess Benhabib, Buz Brock, Gustav Feichtinger, Franz Wirl, Michael Woodford, Wolf-Jürgen Beyn and Thorsten Pampel for helpful discussions and comments on various versions of the paper. We also want to thank participants in a workshop at the University of Technology, Vienna, the Macroeconomic Workshop at Columbia University, and the SCE conference, at Yale University, June 2001. We are also grateful for comments from a referee of the journal.