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JEL Classification

The multiple equilibrium literature seeks explanations for excessive economic volatility, persistent poverty, market fads and fashions, and related macroeconomic phenomena that appear to be anomalies in standard models of rational economic behaviour. Terms like animal spirits, sunspots, irrational exuberance, indeterminacy, and bubbles describe situations of multiple equilibrium. All of these ideas assert that future values of macroeconomic states cannot be predicted accurately from current values of these states or from knowledge of economic fundamentals, even if households and firms behave with complete rationality.

Most of the economics research community has been sceptical of multiple equilibrium (cf. McCallum 1990), believing that it undermines the comparative statics and comparative dynamics exercises that are essential for policy evaluation and econometric prediction. Is it unreasonable, ask the sceptics, to know how the economy selects one equilibrium when many are possible, and how the expectations of economic actors settle on that particular outcome?

Economists have to weigh these legitimate reservations against direct evidence from laboratory experiments that beliefs do matter (Duffy and Fisher 2005) as well as against the continuing difficulties of unique equilibrium models to come to grips with an expanding array of empirical anomalies in many sub-fields of macroeconomics, from excessively volatile asset prices and exchange rates to persistent underdevelopment. This article describes briefly four types of multiple equilibria common in macroeconomics, discusses what causes them, and reviews briefly what they teach us about economic policy.

Typology and Examples

Multiple equilibria occur in dynamic economies whenever the laws of motion that describe macroeconomic states over time admit more than one solution sequence or, more broadly, several asymptotic states. The simplest mathematical example is a set valued, piecewise linear, deterministic law of motion for a scalar state variable x(t), expressed in terms of a vector v = (A, B, m, a,b) of fundamental parameters:

$$ x\left(t+1\right)=f\left(x(t),v\right)= mx(t)+a\ \ \ \ \ \mathrm{if}\ \ \ \ \ 0<x(t)<A=g\left(x(t),v\right)= mx(t)+b\ \ \ \ \ \mathrm{if}\ \ \ \ \ B<x(t) $$
(1)

for all t = 0, 1,…, with 0 < m < 1, 0 < A, 0 < B, 0 < a < b, and possibly some initial condition x(0) > 0 fixed by history.

For different values of the parameter vector v, Eq. (1) illustrates explicitly three major types of multiple equilibria: indeterminacy from missing initial conditions, indeterminacy from multiple laws of motion, and multiple attractors. A fourth type, non-fundamental state variables or sunspots, occurs when we randomly combine the two laws of motion f and g. All four types are associated with excessively volatile behaviour, that is, with macroeconomic states exhibiting abnormal sensitivity to small changes in fundamentals.

Missing initial conditions is the simplest and best-known type of indeterminacy. Suppose, for example, that there is a unique law of motion f, that is, the parameters A and B are infinitely large. If x(0) is an initial price or, more generally, a jump variable that is not predetermined by history but emerges instead from forward-looking markets, then there is a one-dimensional continuum of solutions x(t,a) to Eq. indexed on the indeterminate initial condition x(0):

$$ \log \left(x\left(t,a\right)-a=\left(1-m\right)\right)=t\;\log\;m\Big)+\log \left(x(0)-a/\left(1-m\right)\right) $$
(2)

More generally, an indeterminacy with S−I degrees of freedom appears in any dynamic economy when: (a) history predetermines I initial conditions; (b) the law of motion has S stable eigenvalues; and (c) I < S. Equation (2) illustrates the case (S, I) = (1, 0). A major set of economic examples for this kind of multiplicity comes from overlapping generations models. Fiat money in a dynamically inefficient exchange economy (Wallace 1980) has an indeterminate steady state with worthless money at which (S, I) = (1, 0) because history does not fix the initial price of money. Public debt in a dynamically inefficient production economy (Diamond 1965) leads to an indeterminate steady state, with worthless public debt and (S, I) = (2, 1) because the price of debt is also a jump variable. Finally, two-sector growth environments (Galor 1992), in which the distribution of capital between sectors is again a jump variable, exhibit indeterminacy with (S, I) = (2, 1) whenever the consumption good is more capital-intensive than the investment good.

Multiple laws of motion describe a less understood but more pernicious kind of indeterminacy that arises even if there are no jump variables. Examples of this phenomenon are growth models with private information or limited enforcement (Azariadis and Smith 1998; Azariadis and Kaas 2008) as well as Markov switching models in time-series econometrics and empirical finance (Hamilton 1994). To illustrate, let us choose the parameter vector v in Eq. (1) so that

$$ \left(1-m\right)B<a<b<\left(1-m\right)A $$
(3)

Then the two laws of motion, f and g, overlap in the interval (B, A); each of them has a steady state, a = (1−m) and b = (1−m) respectively, which is a suitable initial condition for the other law. If x(t,a) and x(t,b) are dynamic equilibria for the two laws in Eq. (2), then for any initial condition x(0) in the interval (B, A), we can write down a deterministic general solution z(t) that combines regimes f and g in any arbitrary time sequence, that is,

$$ z(t)=x\left(t,a\right)\ \ \ \ \ \mathrm{for}\;\mathrm{some}\;t=x\left(t,b\right)\ \ \ \ \ \mathrm{for}\;\mathrm{all}\;\mathrm{other}\;t $$
(4)

For each x(0), we may freely select either regime in each time period. In particular, choosing the same regime every period leads to the steady state of that regime; switching regimes periodically leads to deterministic periodic cycles, as in Grandmont (1985), and so on.

Sunspot equilibria are mixtures of multiple deterministic equilibria – static ones as in Cass and Shell (1983) or dynamic ones as in Azariadis (1981) – connected by a non-fundamental or extraneous random variable. Market sentiment, investor beliefs, and consensus forecasts are three examples of extraneous random variables which often take on more colourful names like ‘animal spirits’, ‘sunspots’ or ‘self-fulfilling prophecies’. A simple illustration of a non-fundamental state variable is a lottery s(t) played each period over the intercept, a or b, of the two laws of motion in Eq. (1). For instance, if s(t) is a two-state Markov process, then s(t) = s(t−1), with probability p(a) if s(t−1) = a, and with probability p(b) if s(t−1) = b. The general stochastic solution Z(t, s(t)) to Eq. (1) shows how outcomes depend on the non-fundamental macroeconomic state s(t). Specifically,

$$ {\displaystyle \begin{array}{l}\mathrm{If}\;s\left(t-1\right)=a,\ \ \ \mathrm{then}\ \ \ z\left(t,s(t)\right)=\hfill \\ {}x\left(t,a\right)\ \ \ w.p.p(a)=x\left(t,b\right)\ \ \ w.p.1\hfill \\ {}-p(a)\ \ \ \mathrm{If}\ \ \ s\left(t-1\right)=b,\ \ \ \mathrm{then}\ \ \ z\left(t,s(t)\right)=x\left(t,a\right)\ \ \ w.\hfill \\ {}p.1-p(b)=x\left(t,b\right)\ \ \ w.p.p(b)\hfill \end{array}} $$
(5)

The last type of non-uniqueness, multiple attractors, describes environments with several asymptotic states. Here long-run values of macroeconomic states depend on the corresponding initial values, as in Murphy et al. (1989), Azariadis and Drazen (1990), and Matsuyama (1991). We call these environments ‘non-ergodic’ or ones in which ‘history matters’. For example, suppose we pick the parameter vector v in Eq. (1) to eliminate the overlap between regimes f and g, and obtain one piecewise linear law of motion. Specifically, we replace (3) by

$$ a<\left(1-m\right)A<\left(1-m\right)B<b $$
(6)

Then, for each initial x(0), the general deterministic solution z(t) to Eq. (1) is a unique step function, which traces the law f up to x = A, and jumps to the other law g at that point. Mathematically,

$$ z(t)=x\left(t,a\right)\ \ if\ \ z\left(t-1\right)<A=x\left(t,b\right)\ \ \ if\ \ z\left(t-1\right)>A $$
(7)

Equilibrium here is completely determinate and utterly predictable if history fixes x(0), but the asymptotic state is a = (1−m) if x(0) < A, and b = (1−m) if x(0) > A. History matters in this situation because small or temporary shocks to the macroeconomic state z(t) can have substantial and long-lasting consequences if that state is anywhere near the critical value A.

Causes

Dynamic inefficiency and dynamic complementarities are the two most common proximate causes of multiple equilibrium in macroeconomic models. Dynamic inefficiency is a property of economies with very patient consumers who are energetic savers at low interest rates. For example, holders of short-term US Treasury bills in the last 50 years seem content with an average real pre-tax annual yield of about 1%. Very patient savers are willing to invest in bubbles, paying top dollar for assets with low dividends. Bubbles themselves (Tirole 1985; Shiller 1989) are notoriously indeterminate objects in their initial conditions and laws of motion; they may deflate now, later or not at all, depending on investor sentiment.

Economies with externalities, increasing returns and, most notably, imperfect asset markets often exhibit complementarities in production or consumption which cause excess demands for consumption goods and productive factors to bend backward instead of sloping downward. The typical outcome is several steady states and several laws of motion or stable manifolds, each one leading to a distinct asymptotic state. In particular, multiple equilibria occur when externalities or increasing returns link the payoffs of each agent with the actions of others, both in strategic environments (Cooper and John 1988) and in competitive ones (Benhabib and Farmer 1994). Producers, for example, find it advantageous to raise, hold steady, or lower output in tandem with their industry or the whole national economy.

Imperfect asset markets, especially restrictions on debt and short sales (Bewley 1986; Kehoe and Levine 1993; Kiyotaki and Moore 1997) are an intellectually bountiful and empirically compelling source of complementarities in consumption. This literature motivates restrictions on short sales by the collateral requirements of creditors and, more generally, as a deterrent to debtor default. Short-sales constraints depend on the excess payoff of solvency (which guarantees unfettered participation in future asset markets) over default (which restricts trading in future asset markets). Constraints on short sales are tighter the smaller this excess payoff is because smaller excess payoffs strengthen the temptation to default.

Debt constraints cause two dynamic complementarities in consumption, one through prices and the other through quantities (Azariadis and Kaas 2007). Either one may be sufficient to overcome the intertemporal substitution effect embedded in the consumer’s utility function. Specifically, price changes create a dynamic complementarity when the ordinary income effect is amplified by a relaxation of binding short-sale restrictions. The same outcome is achieved by quantity changes when an anticipated relaxation of future constraints increases the current payoff to solvency, and to continued market participation, thus slackening today’s constraints.

Lessons for Policy

What is the function of economic policy in a deterministic world of many steady states like the one described in Eq. (7)? What should policy do in the stochastic world of Eq. (5) where non-fundamental variables like beliefs, forecasts, consumer sentiment, ‘sunspots’, or ‘animal spirits’ could be every bit as important as fundamentals? Dynamic economies with several asymptotic states have two special properties: long-run performance depends on the starting state x(0); and temporary shocks may have permanent consequences. Any economy that is headed towards an inferior or undesirable steady state may be shocked temporarily until it finds a path leading to a more desirable state. In growth models with many asymptotic states, these shocks are easy to achieve in principle via short-lasting gifts of physical or human capital, by forgiving international debt, and so on. The US-supported Marshall Plan for Europe did exactly that in the 1940s and 1950s. Africa seems in need of a similar plan now but the internal situation in that continent is more problematic than Europe’s was at the end of the Second World War.

A bigger conceptual, as distinct from political, challenge is to formulate policies appropriate for environments swayed by non-fundamental variables and vulnerable to spurious volatility. If equilibria were well described by the stochastic process of Eq. (5), could we find an economic policy to eliminate the unnecessary randomness, and bolster among consumers the belief that the economy is headed toward the more desirable of the two steady states, say, b/(1−m)? Viewing economic policy as equilibrium selection is fairly widespread in the monetary policy literature (Woodford 2003), and broadly consistent with monetary neutrality. On this view, credible monetary policy may be unable to influence the set of possible long-run equilibria, but it does bear on which one the economy selects. In Eq. (5), for example, reactive policy rules may be unable to change the laws of motion f and g but they can still deliver the long-run state b/(1−m) if they influence the public’s beliefs about the long-run likelihood of each state. All it takes to achieve the high state is nudging the two mixing probabilities, p(a) towards zero and p(b) towards 1.

See Also