Journal of Economic Dynamics and Control 18 (1994) 397-410. North-Holland

Big shocks versus small shocks in a dynamic stochastic economy with many interacting agents

Morgan Kelly* Cornell University, Ithaca, NY 14853. USA

Received December 1991, final version received January 1993

This paper investigates the conditions under which a dynamic, stochastic macroeconomic model with many interacting agents will exhibit the ‘small shocks, large shocks’ property that is often said to characterize observed time series: small shocks have a transient impact on the system, whereas a sufficiently large shock will change its trajectory forever. An example is analyzed where the steady state distribution of output has a unique mode if the variance of labor productivity is sufficiently high; but has two modes, each with its own basin of attraction, if the variance falls below a critical value. An unanticipated exogenous shock can have a long lasting impact if it moves the economy into the basin of attraction of a different output mode. A general result is derived showing that the invariant measure of output can have multiple modes if each firm’s probability of producing increases appropriately as the number of other firms producing rises.

Key words: Coordination failure; Production shocks

JEL class$cation: E32

1. Introduction

The purpose of this paper is to formulate a simple macroeconomic model with the ‘big shocks, small shocks’ dichotomy: small shocks have only a transitory impact on an economy’s output, whereas sufficiently large shocks can move the economy onto a completely different trajectory. Such behavior is in contrast to the natural rate hypothesis, where the effect of all shocks is transient, and the unit root hypothesis, where all shocks have a permanent impact. It is frequently

Correspondence to: Morgan Kelly, Department of Economics, Uris Hall, Cornell University, Ithaca, NY 14853, USA.

*I would like to thank William Brainard, John Geanakoplos, seminar participants at Yale and Cornell, and, in particular, two anonymous referees for helpful comments. Any errors are mine.

Ol65-1889/94/$07.00 0 1994-Elsevier Science BV. All rights reserved

398 M. Kelly. Big shocks versus small shocks

argued in empirical studies that such a dichotomy characterizes United States time series: large shocks such as the Great Crash and the Oil Shocks had irreversible effects on macroeconomic variables whereas smaller shocks had less lasting effects. This is the conclusion of the careful study of Blanchard and Watson (1985). More recently Perron (1989) has argued that the unit root which some maintain to be a property of United States output series is a consequence of shifts between different trend paths caused by large shocks.

To model this behavior we consider a real business cycle model with imper- fectly competitive firms or, equivalently, a dynamic stochastic coordination failure model.’ Firms move sequentially, each observing its own productivity realization before deciding whether to produce or not. For each firm there is a dynamic reaction function X(X) giving the probability that a firm will decide to produce when a fraction x of firms are currently producing. In the example considered in this paper, the reaction function is the outcome of the boundedly rational behavior of firms with access to a central market, but it can also be the result of a random matching process in the manner of Diamond (1982), where a firm will undertake more costly projects as the fraction of producing firms increases.

Extremal points of the steady state distribution of output are shown to correspond to fixed points of the dynamic reaction function rc(x). As a result, the economy can have multiple output modes if n(x) increases at an appropriate rate with x.

Each mode has its own basin of attraction. As a result, small exogenous shocks which do not move the economy from its original basin of attraction have a temporary effect on output, whereas a sufficiently large shock, which moves the economy into the orbit of a different output mode, will have a lasting impact.

It is notable that this result is exactly isomorphic to the results of Cooper and John (1988) on multiple symmetric Nash equilibria (SNE) in deterministic one-shot games. SNE correspond to fixed points of each player’s (conjectural) reaction function. Multiple SNE can then exist if this reaction function exhibits an appropriate slope.

To illustrate this general result we analyze a parametric example where a reduction in the variance of labor productivity below a critical value leads to a bifurcation of the set of output modes for the economy. Simulations of the impact of exogenous shocks on the path of output are included.

‘Classic references on real business cycle models are Kydland and Prescott (1982) and Long and Plosser (1983). Among papers in the coordination failure literature, this paper has borrowed most heavily from Blanchard and Kiyotaki (1987), Cooper and John (1988), Diamond (1982), Durlauf (1991a, b), Hart (1982) Heller (1986), Kiyotaki (1988) and Murphy et al. (1989).

M. Kelly, Big shocks wrsus smd shocks 399

The intuition behind this behavior is straightforward. Suppose that the

economy starts at a low output level. There is a low payoff to each firm for producing, so few will do so. However, if individual productivity is highly stochastic, some firms will experience highly favorable productivity draws and will start to produce, pulling the economy away from its initial low output level.2 The converse holds if the economy starts at a very high level of output. On the other hand, if there is little idiosyncratic variance in productivity, the economy can remain for very long periods at high or low output levels.

It is often assumed that if an economic system has a unique invariant measure, then it will be well-behaved. The results here show that this is not the case when the invariant distribution has more than one mode.

This paper therefore encompasses important aspects of real business cycle and

coordination failure models. If there is high variance of productivity, the model behaves like a real business cycle model, converging to a unique natural rate of output. For sufficiently strong interaction there are multiple output modes in the manner of coordination failure models.

Referring to Krugman’s (1991) dichotomy between history and expectations in selecting among multiple equilibria, this model is driven by history. Given its starting point, the mode to which the economy will converge is determined. However, the usefulness of this dichotomy is called into question when we consider that the transition probabilities n(x) are determined, in part, by the expectations of players about each other’s actions.

The immediate inspiration for this paper comes from pioneering work of Durlauf (1991a,b) on phase transition (multiple invariant measures) in Ising models.3 He considers an economy where firms are sited as points on the d-dimensional integer lattice Zd. Like the model here, each firm has a binary output choice. The payoff to the firm’s actions depends on the actions taken by its 2d nearest neighbors. It may be shown that the invariant measure for this system is of the form (4.1) below. The ergodic behavior of the system is determined by the parameter fi of this distribution. For /I less than a critical value the economy exhibits long-run dependence in aggregate time series and has two invariant measures, one with most firms in the high-action state, the other with most firms in the low-action state. For /I above a critical value there is a unique invariant measure with output distributed normally about a mean

‘The role of the variance of productivity in determining uniqueness or multiplicity of output modes is overlooked in the influential Diamond (1982) search model. The important thing to remember is that in a dynamic stochastic game there are two equilibrium concepts. First, there is economic equilibrium which determines the probability of each one of the player’s actions through the dynamic reaction function. These probabilities determine the statistical equilibrium or invariant measure to which the economy will converge.

3Models with similar dynamics to the one here are considered in an interesting survey paper by Arthur (1989) but the economic motivation for his examples is sometimes less than complete.

400 M. Kelly, Big shocks versus small shocks

value. Brock (1991) derives similar results in a model of stock market participa- tion.

Our results may.thus be seen as a finite analog of those of Durlauf and Brock. The advantages of the approach taken here is that the mathematics are very simple (basic Markov chains rather than random fields), the fact that each firm interacts with every other firm in a global market rather than with its immediate neighbors, and in its robustness to asymmetry in the transition probabilities

between producing and not producing.

2. A parametric example

In this section we present an example of an economy where the probability that each firm produces depends on the number of other firms producing. In section 4 we show how a fall in the variance of labor productivity in this economy leads to a bifurcation in output modes. An economy with a high variance of labor productivity behaves in a classical manner: there is a unique natural or modal level of output and the effect of unanticipated exogenous shocks is transitory. For low variance of productivity the economy has two output modes and exhibits the ‘small shocks, large shocks’ dichotomy: small exogenous shocks have a transitory impact on output, whereas a sufficiently large shock will drive the economy into the basin of attraction of a different output mode.

2.1. Preferences and technology

Firms decide to produce sequentially.4 There are no stores of value: all output is consumed instantaneously. To facilitate aggregation of demand, it is assumed that each individual in the economy has an instantaneous CES utility function,

(2.1)

where qi is consumption of commodity i = 1,. . , N and L is the quantity of labor supplied. It will be assumed that the parameter 0 exceeds unity and that the labor supply parameter v is not less than unity. By appropriate choice of 6, all households may be aggregated into a single individual.

4This makes the calculation of the invariant distribution very much simpler. For the case where firms decide simultaneously analogous results to the ones derived here may be proved using the formulae of Freidlin and Wentzell (1984) for invariant measures of Markov chains.

M. Keli~, Big shocks LWSUS small shocks 401

If Y is the value of output produced, the total demand for each commodity is

(2.2)

where pi is the price of good i. To induce a labor supply of L, workers must be paid a wage,

i/(0- 1) w= 6NL’-’ . (2.3)

It will be assumed that prices and wages are set by a central market clearing agency to clear markets: price inflexibility will play no role in generating low-output equilibria.5

Each of the N commodities is produced by a single firm. Each firm has a choice of two possible actions: it may produce a positive output of k units or it may produce nothing.6 The firm has no fixed costs and by producing nothing it is guaranteed to break even. There are no start-up or lay-off costs: the firm may alternate between production and shutdown costlessly.

To produce k units of output the firm requires pi units of labor. A is a random variable with support on the positive real line. It has distribution function F(A) and expectation /i. It is i.i.d. across firms and through time.

2.2. Prices and output

If n firms produce and the remaining (N - n) are idle, aggregate demand equals

Y(n) = p(n) n k , (2.4)

where p(n) is the price per unit of output if n firms are producing. Given the elastic demand and the ability to produce only k goods, no firm has an incentive to charge a price higher or lower than other firms. The price of a good which is not being produced is infinite.

If a firm produces, it must set its price such that demand equals k at every level of aggregate income. Consequently, demand when n firms produce must be

‘Since we are concerned with the impact of large shocks, any second-order losses from suboptimal price setting by firms in the spirit of Akerlof and Yellen (1985) will be very large.

‘This captures the empirical finding of Blanchard and Diamond (1990) amongst others that recessions correspond to periods where a small number of firms experience large reductions in output, rather than all firms experiencing small reductions in output.

402 M. Kelly, Big shocks versus small shocks

equal to demand when N firms produce: from (2.2)

Y(n) P(V’ = k = Y(N) p(N))’ np(n)rPe Np(N)‘-’ .

(2.5)

Given that the price level depends only on the number of firms producing (because output per firm is fixed at k, and expenditure is independent of the distribution of income between wages and profits resulting from different pro- ductivity realizations), we normalize p(N), the price level if all firms produce, to unity. Substituting for Y(n) and Y(N) from (2.4) and equating either the numer- ators and denominators in (2.5) implies that

n l/C@-1)

p(n) = jq 0 . (2.6)

No other solution is consistent with the normalization on prices. As aggregate demand rises, the firm must set a higher price to maintain demand constant at k.

Suppose that the firm expects (n - 1) other firms to produce. If its labor requirement is Ai, it expects to face a wage rate of

w(n) = ; 0 l/(0- 1)

6 N((n - 1) J(n) + ni) “-’ 3 (2.7)

if it produces. J(n) represents the expected labor requirement of each one of the firms that is producing when n firms are producing,

_ jp(“) 1 dF(i)

n(n) = F@,,,(n)) ’ (2.8)

where /1,,,(n) denotes the maximum labor input at which a firm can break even when II firms are producing. By the strong law of large numbers this expectation will be equal to the actual average labor requirement when n is large.

Firms behave in a monopolistically competitive manner: the firm assumes that its decision to produce or not to produce will have a negligible impact on prices and wages.

It is assumed that firms adhere to a simple form of bounded rationality: each assumes that the number of firms n currently producing will remain constant. We discuss later how the results change if learning is allowed.

As a result, if n firms are currently producing, the firm whose turn it is to decide will produce if its expected profit,

p(n) k - w(n) Ai , (2.9)

_.is positive. Let n(n) denote the probability that this is the case.

M. Kelly, Big shocks WTUS small shocks 403

3. Invariant measures

Given that firms decide to produce sequentially, each with probability n(n), we wish to determine the properties of the invariant measure or steady state distribution of output for this economy. A simple result in Markov chains [cf. Karlin and Taylor (1975, pp. 85-86)] establishes that the invariant measure Il is given by

n(n) = n(0) fi (N -jl”;;.y)- l)) n=l,...,N. i=l

where n(O) is chosen to satisfy the normalization constraint

nio n(n) = 1

(3.1)

(3.2)

To determine the extrema of this distribution let x = n/N define the fraction of firms which are producing. In a convenient abuse of notation, we will use X(X) interchangeably with n(n) to denote the probability that each firm will produce. As the number of firms in the economy N becomes large, x comes to approxi- mate the unit interval [0, 11.

Assume that rc is a continuous function of x.’ The next result now follows immediately from eq. (3.1).

Proposition 1. Extrema x, of the inuariant distribution II correspond to fixed points of the dynamic reaction function z(x).

Note that this result is closely analogous to a one-shot game where Nash equilibria correspond to fixed points of the (conjectural) reaction function. As a result we can use the result of Cooper and John (1988) that a necessary condition for multiple symmetric Nash equilibria (SNE) is that the slope of the reaction function is positive, and that a sufficient condition is that this slope exceeds unity at some fixed point. In words, multiple SNE occur if the degree of strategic complementarity, as measured by the slope of the reaction function, is sufficiently strong. Similarly, in the model here, the steady state distribution of output has multiple modes if, at some fixed point, the probability that a firm produces X(X) depends appropriately on the fraction of firms producing x. This is illustrated in fig. 1.

‘A sufficient condition for this is that the instantaneous payoff correspondence of each player is continuous and strictly concave in the state variables x and E.,. See, for example. Stokey and Lucas (1989, p. 63).

404 M. Kelly, Big shocks versus small shocks

Fig. 1. Extrema of the invariant measure correspond to fixed points of the dynamic reaction Function n(x).

3.1. Convergence to the steady state

Convergence to the invariant distribution is illustrated in fig. 2, where x _ and x+ represent modes and x0 is a minimum. For starting points other than the three extrema the system converges (in expectation) at an exponential rate equal in absolute value to In(x) - xl. Points below x0 will converge to the neighbor- hood of x_ and points above it will converge to the neighborhood of x+ . For a process starting at the unstable point x0 where this drift is zero, initial stochastic fluctuations around x0 are enhanced exponentially until the previous exponential convergence occurs. For a system starting at one mode, the time it takes to flip to the other mode is extremely long if the invariant distribution is not too asymmetric. For further discussion and proofs see Weidlich and Haag (1983, sect. 2.3.2).

To summarize. This section has shown that an economy can have an invari- ant distribution of output with several modes if the probability of each firm producing Z(X ) depends appropriately on the fraction of firms producing x. Each mode has its own basin of attraction. Small shocks which do not perturb the system from its current basin will have a temporary impact on the economy, whereas large shocks which move the economy into the orbit of another output mode have a lasting impact.

M. Kelly, Big shocks versw small shocks

Fig. 2. Deterministic trajectories in the t, x(t) plane.

4. Steady state distribution of output

To calculate the invariant measure of output for the economy of section 2 it will be assumed that the distribution function of the random variable A is

(4.1)

There are two reasons for choosing this functional form. First its exponential form makes the calculation of invariant measures very simple. More impor- tantly, this is the same form (the so-called Gibbs distribution) as the invariant distribution used in Durlaufs and Brock’s random field models. The magnitude of the parameter /? will be critical in what follows. Increases in the value of the parameter p correspond to reductions in the variance of labor productivity.8

*E(n) = In(1 + exp(2?))/2P and E(n*) = S(1 + exp(2y))/2p2. S(X) = ji Int/(t - 1)dt is the nega- tive of Spence’s Integral. See Abramowitz and Stegum (1970, p. 1004).

406 M. Kelly, Big shocks versus small shocks

As a consequence of the monopolistically competitive behavior of firms we can approximate the wage rate (2.7) as

w(n) = ; 0

l/(Bp 1) 6 N(n /i(n))V- l. (2.7’)

If n firms are producing, from (2.6), (2.7’) and (2.9) the probability that any one firm will produce is

7c(n) = Pr i I

where q = (1 - v + l/(0 - 1)). Substituting into (2.8) and defining /I’ = jl k/6,

( ’ ntl exp N/qn)‘Fl - ‘/

n(n) =

exp (

Fnq i--y]:exp-[

1 /Ynq ’

>

(4.2)

N A(n)“- N/l(n)“v’ - 7

It will be assumed that there are N = 100 firms in the economy and that the elasticity of substitution parameter 6 = 2.9

Fig. 3 shows the steady state distribution of output for the empirically important case of a constant real wage. In this simple model this corresponds to a perfectly elastic labor supply (v = l), but in a more complex model this could result from efficiency wage considerations or the bargaining power of insiders. If the wage rate increases with employment, more weight shifts to the lower part of the distribution. It is assumed that y = 0.99/Y/2, with values of p ranging from 0.2 to 3.0. This generates an economy where high-output states are more likely than low-output ones: for y = /I’/2 the distribution would be symmetric.

Note how the set of modes of the steady state distribution of output bifurcates when /Y = 2. For /I’ I 2, variations in productivity dominate firms’ actions producing a unique mode where half of the firms produce at any instant. For /Y > 2, two symmetric equilibria appear which tend to the opposite extremes of all firms producing or no firm producing as interaction strength /Y rises in value.

4.1. Time series properties of output

From the previous section we know that an exogenous shock can have a permanent impact on output if there is more than one output mode and the

‘As N increases it may be shown that the modes of the steady state distributions become more sharply peaked. See Weidlich and Haag (1983, ch. 2).

M. Kelly, Big shocks versus small shocks 407

Fig. 3. Bifurcation of modes of the steady state distribution of output as the variance of labor productivity falls.

economy is moved into a new basin of attraction. In an economy whose steady state distribution of output has a unique mode, observed realizations of output will cluster around the modal value, sometimes above or below it depending on whether firms had lucky or unlucky productivity draws on average. Large shocks have only a transitory impact on output. We now illustrate this.

Fig. 4a is drawn for an economy with a high variance of labor productivity /3’ = 0.1 and 200 firms. (By increasing the number of firms, the variance about the modal output level can be reduced: each firm has less impact on the aggregate behavior of the economy.)

The economy starts at 150, but rapidly converges to its modal level of 100. Halfway through, a shock reduces output to 50. Again, output returns exponen- tially fast to its modal value. This strong mean reversion precludes long periods of low output in the aftermath of a severe exogenous shock.

Fig. 4b depicts the contrasting behavior of an economy with lower variance of labor productivity fi’ = 2.4. Here the economy stays initially in the neighbor- hood of the high output mode. However, halfway through, a large shock drives the economy to a situation where only 90 firms produce, below its unstable mode of 100. This causes the economy to converge rapidly to the neighborhood

408 M. Kelly, Big shocks

u4 Fig. 4. Impact of an exogenous shock on an economy with (a) a unique output mode and (b) two

output modes.

of its low output equilibrium. Note that, although the low-output mode is less likely than the high-output one, the economy shows no tendency to leave it even after 10,000 iterations.

Perron (1990) has demonstrated that such an economy will seem to exhibit a unit root in output, and argues that this seems to be the reason for the unit root in U.S. output data. lo After 1929 and 1973 the economy moves to different

equilibria, but between these large shocks output is trend-reverting.

4.2. Learning

In common with the growing literature on evolutionary games [such as Kandori, Mailath, and Rob (forthcoming)] which address issues similar to the ones considered here, we have considered an economy with myopic agents. This behavior is simpler to analyse but is not optimal: firms can increase their payoffs by learning about the dynamics of the economy and modifying their behavior accordingly.

Specifically, at points where Z(X) - x, the deterministic drift in the proportion of firms producing, is positive, static expectations will systematically underesti- mate demand, and conversely where the deterministic drift is negative. If we observe an economy where firms learn about its dynamics from its past behav- ior, we will observe output levels growing faster along expanding trajectories than in the static expectations economy, and shrinking faster along declining trajectories. However, the extrema of the invariant distribution of output will be

“Banerjee et al. (1990) and others whom they cite, dispute Perron’s findings on the grounds that the structural breaks are imposed a priori. To the extent that the econometrician should use all available information this criticism is not opposite.

M. Kelly, Big shocks versus small shocks 409

unaffected by this learning process because there is zero deterministic drift at these points by definition, so static expectations will be rational.

To summarize, as agents start from static expectations and learn about the trajectory of the economy, the absolute value of the deterministic drift 1 n(x) - x 1 increases. As a result, the extrema of the invariant distribution of output will be unaffected, but convergence to these extrema will be faster.

5. Conclusions

This paper has shown that the fact that an economy has a unique invariant measure for output does not imply that it is ‘well-behaved’ in the sense that it cannot go through long periods of low output. It was shown that the invariant measure can have several modes each with its own domain of attraction depending on how each firm’s dynamic reaction function n(n) depends on the number of other firms producing n. Unanticipated exogenous shocks can move the economy between modes. An obvious extension is to model such shocks, and their transmission mechanisms explicitly. An appealing start might be, following Bernanke’s (1983) analysis of the role of collapse of the U.S. financial system in propagating the Great Depression, to incorporate a credit market into the model, and also to allow firms to have more than a binary output choice.

References

Abramowitz, Moses and Irene Stegum, 1970, Handbook of mathematical functions (Dover, New York, NY).

Akerlof, George and Janet Yellen, 1985, The macroeconomic consequences of near-rational rules- of-thumb behavior, Quarterly Journal of Economics 100, 8233838.

Arthur, W. Brian, 1988, Self reinforcing mechanisms in economics, in: P.W. Anderson, K.J. Arrow, and D. Pines, eds., The economy as an evolving complex system (Addison Wesley, New York, NY).

Banerjee, A., R.L. Lumsdaine, and J.H. Stock, 1990, Recursive and sequential tests of the unit root and trend break hypotheses: Theory and international evidence, NBER working paper no. 3510.

Bernanke, Ben, 1983, Non-monetary effects of the financial crisis in the propagation of the great depression, American Economic Review 76, 822109.

Blanchard, Olivier and Peter Diamond, 1990, The cyclical behavior of gross flows of US workers, Brookings Papers on Economic Activity 2, 855143.

Blanchard, Olivier and Nobuhiro Kiyotaki, 1987, Monopolistic competition and the effects of aggregate demand, American Economic Review 77, 647-666.

Blanchard, Olivier and Mark Watson, 1986, Are business cycles all alike?, in: R. Gordon, ed., The American business cycle: Continuity and change (University of Chicago Press, Chicago, IL).

Brock, William, 1991, The economics of emergent noise, Working paper (University of Wisconsin, Madison, WI).

Cooper, Russell and Andrew John, 1988, Coordinating coordination failure in Keynesian models, Quarterly Journal of Economics 103, 441463.

Diamond, Peter, 1982, Aggregate demand management in search equilibrium, Journal of Political Economy 80, 88 l-894.

Durlauf, Steven N., 1991a, Path dependence in aggregate output, NBER working paper no. 3718.

410 M. Kelly, Big shocks versus small shocks

Durlauf, Steven N., 1991b, Multiple equilibria and persistence in aggregate fluctuations, American Economic Review Papers and Proceedings 81, 70-74.

Freidlin, Mark and Alexander Wentzell, 1984, Random perturbations of dynamical systems (Springer Verlag, New York, NY).

Hart, Oliver, 1982, A model of imperfect competition with Keynesian features, Quarterly Journal of Economics 97, 1099138.

Heller, Walter, 1986, Coordination failure under complete markets with applications to effective demand, in: W.P. Heller, R.M. Starr, and D.A. Starrett eds., Equilibrium analysis: Essays in honor of Kenneth J. Arrow, Vol. 2 (Cambridge University Press, Cambridge).

Kandori, Michihiro, George Mailath, and Rafael Rob, forthcoming, Learning, mutation and long run equilibrium in games, Econometrica.

Kiyotaki, Nobuhiro, 1988, Multiple expectational equilibria under monopolistic competition, Quar- terly Journal of Economics 103, 695-713.

Krugman, Paul, 1991, History versus expectations, Quarterly Journal of Economics 106, 651-667. Kydland, Finn and Edward Prescott, 1982, Time to build and aggregate fluctuations, Econometrica

30, 134551370. Long, John and Charles Plosser, 1983, Real business cycles, Journal of Political Economy 91, 39-69. Murphy, Kevin, Andrei Shleifer, and Robert Vishny, 1989. Increasing returns, durables and eco-

nomic fluctuations, Working paper (University of Chicago, Chicago, IL). Perron, Pierre, 1989, The great crash, the oil price shock, and the unit root hypothesis, Econometrica

57, 1361-1402. Stokey, Nancy and Robert Lucas, 1989, Recursive methods in economic dynamics (Harvard

University Press, Cambridge, MA). Weidlich, W. and G. Haag, 1983, Concepts and models of a quantitative sociology (Springer-Verlag,

New York, NY).