Multiple Equilibria and Persistence in Aggregate Fluctuations Author(S): Steven N

American Economic Association

Multiple Equilibria and Persistence in Aggregate Fluctuations Author(s): Steven N. Durlauf Source: The American Economic Review, Vol. 81, No. 2, Papers and Proceedings of the Hundred and Third Annual Meeting of the American Economic Association (May, 1991), pp. 70-74 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/2006828 Accessed: 22-08-2014 01:56 UTC

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This content downloaded from 128.104.46.206 on Fri, 22 Aug 2014 01:56:04 UTC All use subject to JSTOR Terms and Conditions PATHDEPENDENCE IN ECONOMICS:THE INVISIBLE HAND IN THEGRIP OF THEPASTt

Multiple Equilibriaand Persistence in Aggregate Fluctuations

BY STEVEN N. DURLAUF*

Recent developments in theoretical tains a unit root. Despite controversy over macroeconomics have emphasized the po- the exact magnitude of the permanent com- tential for multiple, Pareto-rankable equi- ponent, the effects of current events on real libria to exist for economies where various activity apparently persist over long hori- Arrow-Debreu assumptions are violated. zons. Authors such as Peter Diamond (1982) em- The purpose of the current paper is to phasized how incomplete markets can allow link the new multiplicity results in macroe- economies to become trapped in Pareto- conomic theory with the evidence on output inferior equilibria; Walter Heller (1986) ob- persistence. I do this by modeling coordina- tained similar results due to imperfect com- tion problems in an explicitly stochastic petition. These different approaches share framework. As developed in my earlier pa- the idea that strong complementarities in pers (1990;1991), the microeconomic speci- behavior can lead to multiplicity. Intuitively, fication of the economy is expressed as a set when technological or demand spillovers of conditional probability measures describ- make agents sufficiently interdependent, ing how individual agents behave given the high and low levels of activity can represent economy's history. An aggregate equilib- internally consistent equilibria in the ab- rium exists when one can find a joint proba- sence of complete, competitive markets. bility measure over all agents which is con- Most of these models describe multiple sistent with these conditional measures; steady states in economies rather than mul- multiplicity occurs when several such mea- tiple nondegenerate time-series paths, and sures exist. This approach permits one to consequently cannot address issues of ag- directly describe the time-series properties gregate fluctuations. Further, this literature of aggregate fluctuations along different has not shown how economies can shift equilibrium paths. across equilibria, inducing periods of boom Specifically, I examine the capital accu- and depression. mulation problems of a set of infinitely lived An independent literature has argued that industries. I deviate from standard analyses aggregate fluctuations are strongly persis- in two respects. First, each industry faces a tent. Researchers have concluded from a nonconvex production technology. Second, variety of perspectives that aggregate output industries experience technological comple- in advanced industrialized economies con- mentarities as past high production decisions by each industry increase the current productivity of several industries through dynamic learning by doing or other effects. tDiscussants: W. Brian Arthur, Stanford University; Industries do not coordinate production de- Paul A. David, Stanford University; Paul Romer, Uni- cisions because of incomplete markets. By versity of California-Berkeley; Robert M. Solow, MIT. describing how output levels and productiv- *Department of Economics, Stanford University, ity evolve as industries interact over time, Stanford, CA 94305. I thank Charles Bean, Suzanne Cooper, Paul David, Avner Greif, Robert Solow, Doug the model characterizes the impact of com- Steigerwald, and Jeroen Swinkels for helpful complementarities and incomplete markets on ments. the structure of aggregate fluctuations.

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I. A Model of Interacting Industries shock and Fi is a fixed overhead capital cost; iT,t-1 7iht-1. and (,- are elements Consider a countable infinity of industries of at-l. Recalling that firms within an in- indexed by i.1 Each industry consists of dustry are identical, let us define wi,t which many small, identical firms. All firms pro- equals 1 if technique 1 is used by industry duce a homogeneous good; industries are i at t, 0 otherwise and wt ={I... )i-I,t i t distinguished by distinct production func- )i+I t ... } which equals the joint set of tech- tions rather than distinct outputs. The ho- niques employed at t. mogeneous final good may be consumed by I make the following assumptions. First, the owners of the firms, or converted to a each technique fulfills standard curvature capital good that fully depreciates after one conditions. Further, I associate technique 1 period. Industry i's behavior is proportional with high production. Specifically, net capi- to the behavior of a representative firm that tal NKi ,, that equals Ki, - Fi for tech- chooses a capital stock sequence {Ki ,1 to nique 1 and Ki, t for technique 2, has a maximize the present discounted value of strictly higher marginal (and by implication profits Hi't total) product when used with technique 1 than technique 2. A firm chooses technique 1 if it is willing to pay fixed capital costs in = (1) Hi,t E( E-(Yi t+j-Kj t+j)jat exchange for higher output. where Y, t equals the output of the ith ASSUMPTION 1: Restrictions on tech- industry'srepresentative firm at t; at equals nique-specificproduction functions. all available information at t. Initial endow- For all realizationsof Si t, 7iqt, and NK, ments Y1,0provide starting capital. = Aggregate behavior is determined by the A. M0(,;i,t,61) =fO 77i,t, 0 0. interactions of many heterogeneous industries employing nonconvex technologies. afl(?f vi,t, 0t af2 (? 'qi, t t) B. - oo; Production occurs with a one-period lag; dNK aNK firms at t -1 employ one of two production techniques and a level of capital to deter- dfl (00 i., t) df2( 00,ni., t) mine output at t. Only one technique may be used at a time. Russell Cooper (1987) aNK dNK and Kevin Murphy et al. (1989) exploit similar technologies to analyze multiple c f, (NK, 4i t ,) fAfNK, 7i,t) equilibria; Paul Milgrom and John Roberts aNK aNK (1990) discuss how this type of nonconvexity can arise as firms internally coordinate many Both techniques are assumed to exhibit complementary activities. The technique- technological complementarities, as the his- specific production functions produce Yl,iNt tory of realized activity determines the pa- and Y2, ,t through rameters of the production function at t. Paul Romer's (1986) model of social in- (2) Yliit= f(Kt -l-Fi;i t-1,?t-1) creasing returns shares this feature. My complementarities differ from Romer's in Y2, i,t = f2( Ki t - Ijq, it - I t - I), two respects. First, all complementarities are local as the production function of each i, t and 7i t are industry-specific productiv- firm is affected by the production decisions ity shocks; 6t is an aggregate productivity of a finite number of industries. The index i orders industries by similarity in technology; spillovers occur only between similar technologies. Paul David (1988) describes the IMy 1990 paper derives a general equilibrium ver- sion of this economy. historical importance of local complemen-

This content downloaded from 128.104.46.206 on Fri, 22 Aug 2014 01:56:04 UTC All use subject to JSTOR Terms and Conditions 72 AEA PAPERS AND PROCEEDINGS MAY 1991 tarities in the evolutionof technicalinnova- rium industry technique choices obey conditions. Second, my complementaritiesare ex- tional probabilities of the form plicitly dynamic. Past production decisions affect current productivity,which captures (3) Prob(wci,I k)1) V the idea of learningby doing. Specifically,I model the complementari- =Prob((i, tI ojj t - 1 Vfi EQ= ties throughthe dependence of the productivity shocks ai,, and 'ri,, on the history of Once technique choices are determined, one technique choices (see my 1990 paper for a can solve for the optimal levels of capital justification). Complementarities are as- and output for each firm. In fact, a suffi- sumed to be the only source of dependence cient condition for the existence of equilib- across shocks. Prob(xly) denotes the condi- rium capital and output sequences for all tional probabilitymeasure of x given infor- firms is the existence of a joint probabil- mation y; x(y) denotes the random vari- ity measure over all technique choices which able associated with this measure. Ak, = is consistent with the conditional measures {i - k ... i ... i + 1) indexes the industries that (3). My 1990 paper verifies that such a joint affect industryi's productivity. measure exists for any initial conditions wo. Let us now restrict the conditional proba- ASSUMPTION 2: Conditional probability bilities in order to discuss multiplicity and structure of productivity shocks. dynamics. Past choices of technique 1 are assumed to improve the current relative A. ProW(i,tiat_ 1) productivity of the technique. As a result, technique 1 choices will propagate over time. = Prob(Wit,)w,t1I V j E Ak,l)- Further, it is assumed that wt = 1 is a steady state, which means that when all productiv- B. l Prob(Y7i,lt1- ) ity spillovers are active, the effects are so strong that high production is always opti- = V E Ak,l) Prob(yJ,t,I1 t-i i mal. C. The random pairs (L, - hi,t - 71i, t(9t 1)) are mutually independent ASSUMPTION 3: Impact of past technique choices on current of each other and of 6t - 6t(at -1) V i. techniqueprobabilities.2 Let X and &' denote two realizations of No marketsexist wherebyindividual firms )t-l If (w? v EAkl1 then can coordinate to exploit complementarities. Consequently,no industrymay be com- pensated for choosing technique 1 in order A. Prob(w1 w) ]/j k,l) to expand the productionsets of other in- A=k,= dustries;nor, given my conceptualizationof >Prob(',t llWI= 1 = w;Viek,l) industriesas aggregatesof many small pro- B. Prob(oi ,=lj,ll j Ak,l)= ducers,can firmswithin an industrystrategi- B. =1V i EE = 1. cally choose a techniquein order to induce Prob(wi't= 1I1Wit1 Ak,l) higher future productivitythrough complementarities. Market incompleteness com- Whenever some industry chooses witt = 0 bines with the production nonconvexityto a positive productivity feedback is lost. Dif- fundamentallyaffect aggregatedynamics. ferent configurations of choices at t - 1 de- termine different production sets and condi- II. Local Complementarities and tional technique choice probabilities for Multiple Equilibria

I initially analyze the economy without 2This assumption can be reformulated in terms of aggregateshocks, by setting 6t = 0 V t. From restrictions on the technique-specific production func- my assumptions,one may show that equilib- tions.

This content downloaded from 128.104.46.206 on Fri, 22 Aug 2014 01:56:04 UTC All use subject to JSTOR Terms and Conditions VOL.81 NO. 2 PATH DEPENDENCE IN ECONOMICS 73 each industry.I bound the techniquechoice For every nonnull index set Ak 1, there exist probabilitiesfrom below and above by 0,in numbers0 < QAk,I < 0Ak,l <1 suchthat and 0,a,X, respectively. A. If 0 l k, then produce using technique 1. The choice of technique 1 by one industry, through the lim Prob(woit = lko = 0) = 1. complementarities,increases the probability t =:00 that the techniqueis subsequentlychosen in several industries. With strong spillovers, If complementarities are sufficiently weak, these effects may build up, allowing w, = 1 each industry converges to the high produc- to emerge from any initial conditions. The tion technique almost surely from economy- model therefore allows us to analyze the wide low production technique initial condi- stabilityof a high aggregateoutput equilib- tions. rium from arbitraryinitial conditions. In fact, the limitingbehavior of the econ- One can associate t = 1 with the equi- omy is determinedby the bounds omin and libriumthat would emerge if all firmschose Ok, X If the probabilityof high production their productionlevels cooperatively.If pro- by an industry is sufficientlylarge for all duction through technique 1 is sufficiently production histories, then the spillover ef- large for t = 1 versus any other configura- fects induced by spontaneous technique 1 tion, then t = 1 emerges as the coopera- choices cause the economy to iterate to- tive equilibriumafter one period. Conse- wards high production. Alternatively, if quently, incompletenessof markets lowers technique 1 probabilitiesare too low in the the mean and increases the varianceof in- absence of active spillovers, spontaneous dustry and aggregate output along the in- technique 1 choices will not generate suffi- efficient equilibrium path, as technique cient momentumto achievethe t = 1 equi- choices fluctuate over time. When indus- librium.0min and 0mX bound the degree of tries fail to coordinate,production decisions complementarityin the economy.Large val- become dependenton idiosyncraticproduc- ues of omin, imply complementarities are tivity shocks. Observe that the volatilityas- weak as technique 1 is chosen relatively sociated with the inefficient equilibriumis frequently regardless of the past. Con- causedby fundamentals.Simulations in both versely, small values of 0mX imply strong of my earlier papers show that aggregate complementarities;the probabilityof cur- output can obey a wide range of AR pro- rent high productionis very sensitiveto past cesses, dependingon 4k technique choices. Theorem 1 (proven in my 1990 paper) shows how long-runindus- III. Path Dependence and Aggregate Shocks try behavioris jointly determinedby initial conditionsand conditionaltechnique proba- Now consider the role of the aggregate bilities. shocks ft. By affecting many industries si- multaneously, these shocks act in a way THEOREM1: Conditionsfor uniquenessvs. analogousto changingthe initial conditions multiplicityof long-runequilibrium. of the economy.Path dependenceoccurs as

This content downloaded from 128.104.46.206 on Fri, 22 Aug 2014 01:56:04 UTC All use subject to JSTOR Terms and Conditions 74 AEA PAPERS AND PROCEEDINGS MAY 1991 one realization of (t permanently changes Several interpretations beyond productiv- the equilibrium in the absence of future ity can be applied to the aggregate shocks. offsetting shocks. I assume that sufficiently Interpreting (t as a proxy for the financial unfavorable aggregate productivity draws sector, the model indicates how the break- make technique 1 unlikely whereas suffi- down of financial institutions, such as oc- ciently favorable draws ensure the use of curred during the Great Depression, can the technique. cause indefinite output loss. Alternatively, my 1990 paper shows how (t can represent ASSUMPTION 4: Impact of aggregate the cost of production inputs provided by shocks on technique choice. leading sectors such as transportation or There exist numbers a and b, with steel. In this case, the growth of leading Prob(4t < a) and Prob(4t 2 b) both nonzero, sectors improves the relative profitability of such that3 high production, which can lead to a takeoff in growth as the economy shifts across equi- A . Prob(t, t= llft 'a,wXJ,t- I= 1vi6k,l) libria.

< OAk,/. REFERENCES

B. Prob(w.)it=1llt4b Xj,t- I =OViEAk,l) Cooper, Russell, "Dynamic Behavior of Im- =1. perfectly Competitive Economies with Multiple Equilibria," NBER Working Pa- When this assumption holds, aggregate per No. 2388, 1987. shocks can have an indefinite effect on real David, Paul A., "Path-Dependence: Putting activity. My 1991 paper verifies the Past in the Future of Economics," Stanford University, 1988. THEOREM 2: Path dependence due to ag- Diamond, Peter A., "Aggregate Demand In gregate shocks. Search Equilibrium," Journal of Political Let t = 0 V t> T and 0max < k*, The Economy, October 1982, 90, 881-94. economy exhibitspath dependence as the real- Durlauf, Steven N., "Nonergodic Economic ization of 4T affects the limiting technique Growth," Stanford University, 1990. choice probabilitiesfor all industries. , "Path Dependence in Aggregate Output," Stanford University, 1991. Heller, WalterP., "Coordination Failure Un- A. lim 11T

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