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Do Sunspots Matter? Do Sunspots Matter? David Cass and KarlShell Univer8tssitT,t# Penn~qdlvania~ Can extrinsic uncertainty (aniniral spirits," markett psychology," "sunspots,". .) play a signihicant role in rational expectations e(JIi- librium models? We establish that extrinsic unce tainty cannot niat- ter in the static Arrow-Debreu economy with complete markets. Bt we also establish that extrinsic uncertainty can matter in the overlap- ping-generations economy with complete i marketsbut where nanrket participation is limited to those consumers alive when the nvurkets are open. Equilibrium allocations in w hic h extrinsic unce rtainty plays n() role are Pareto optimal in the traditional sense. Equmilibritum allocations in which extrinsic uncertaintv does play a role aIe Pareto optimal in a (weaker) sensse which is appropriate to dynamic analysis. I. Introduction What is the best strategy for playing the stock market Sholtkh one concentrate on ftundarentals' or shotild one instead f(tcus on the psychology of the market'? These are interesting qtiestiolls for those who would like to get rich on Wall Street. They are also interesting for macroeconomists. For Keynes and many Keynesians, the volatility of investment iS at least in part based on the volatility of inarket psychology or the aninml spirits of capitalists, or, more generally, extrinsic uncertainty. 1 his alleged unreliability of the intertemporal allocation of resources under laissez-faire capitalism is an important basis of the Keynesian Research support Und'er NSF grant S0-07()012to the (Center for Andaltic Research in Economics anld the Social Sciences at the Ulinersity of' Pennsylvinia is gratefully ac- kitnocwledged.We think the membersrofthe (NRS Seniinaire c'Economrir ie, espec ially jeaFn-Michel (Grandniont, for helpfI l cniticisn. We have also benefited frommi traction with our colleames (Costas Azarialis andl Yves alasko. 1fu l Politil Ecl Ii I. 1 . f,#'' onomry 98s v) no, 21 , tI9) it CIi' i i ( ( hicago All ligries 0(s1122-3808/8:1/1(1-1(20111er d. $15(11 )(1 193 194 JOURNAL OF POLITICAL ECONOMY claim f'or the desirability of' active fiscal and monetary policies. On the other han-d, devotees of' the rational expectations school of mac- roeconomics hold that fundamentals play the determining role in the allocation of resources. Therefore, they claim that if' the environment is stable (in particular, if' the government does not behave capri- ciously), then private investment will also be appropriately stable. In this paper, we investigate whether or not rational expectations equilibria are immune to influence from extrinsic uncertainty. That is, Can self-fulfilling prophecies have a real impact? Or, in short, Do sunspots matter? We consider a simple exchange economy with a finite time horizon, two commodities, two states of' nature, and a finite number of' con- sumers. In Section II, consumers are introduced. They are assumed to have von Neumann-Morgenstern utility functions and commodity endowments which are independent of' the state of' nature. Since un- certainty does not then have any effect on the fundamentals of' the economy, it is purely extrinsic uncertainty; we refer to it simply as sunspot activity. Throughout most of' the paper, we adopt a very strong version of the rational expectations hypothesis: Consumers share the same beliefs about sunspot activity. This allows the interpre- tation that subjective probabilities are equal to objective probabilities. The usual extensions of general equilibrium theory flail to repre- sent an essential and significant aspect of actual intertemporal econo- mies: The trading process must take place in time. Individuals may be able to make complicated trades involving future deliveries contin- gent upon the revealed state of' nature, but, these trades must be struck when each of' the parties is alive. Markets in all commodities may be complete, but consumers are restricted from participating in markets which meet (say) before they are born. We assume that markets are complete. In Section III, we describe s IThrotighOLIt this essay, 'sLUIspot is meant to represent extrinsic Uncertainitv, that is, random phenimenia that dlo not affect tastes, end(wmenits, o- pro action pos- siI)ilities.()f jrse, as Jevons noted, real-woritl stinspots maIy erN well provide i'otritoow Uncertainty to the economy, aff'ectilng, e.g., agriclt-otralplt - Oluctioll possibilities. Here we are interested only in otir highly stylized Vel-sioll of' SLnspot actix itv. 2 We have heen worry ing ahoult SLunIspotS for some time. A theoretical example of' a monetary overlapping-generations economy ill which SulinSpOtS Imattel is reported in Shell (1977). (O)nr colleague Costas Azariatlis (1981) has presented similar reSlItS ill a more familiar macloeconomic setting. These examples, however, are hased on x ery complicated (iyxnamic models which (ti not permit tis to separate the respective roles of' generational overlap, the infinite time horizon, andt government debt. In the present paper, we f'octls on the implicatiois of' generational overlap in a very siml)le e(lUilih- ritim model. In a coml)anion paper, Yves Balasko (f'orthcoming) inv estigates a more general f'Ormulation of' the notion of' extrinisic tincertcainty. Foi an overview of' the stibject of' ratiinal expectations eqUilihi'ia in which extrinsic uincertainty matters, see 0ir propagandia )iece, Cass and Shell (1980). DO SUNSPOTS MATTER? 195 the natural restrictions which must be imposed on market participa- tion-namely, that consumers born after sunspot activity is observed cannot possibly trade on the securities market meant to provide the possibility of hedging against the effects of' this uncertainty. Section III also provides a complete description of consumer behavior in the securities and commodities markets. Section IV defines and analyzes market equilibrium. This section might appear to be our mathematical ghetto. Indeed, there is a good deal of notation (which is the only way we know to make these ideas precise!), but the arguments themselves are not very complicated. Equilibrium in the economy with securities and commodities markets is shown to be equivalent to equilibrium in an economy with contill- gent-claims markets. (This equivalence is the one established by Ar- row [ 1964].) The latter formulation of market equilibrium is the more tractable and is exclusively employed in the succeeding analysis. The traditional general equilibrium model yields "certainty equilibria" in which sunspots cannot matter. It is shown that, in principle, our mar- ket equilibrium concept is broader than that of the traditional model; certainty equilibrium is a special case of market equilibrium. Section V is devoted to the degenerate-and contraf actual-case in which there are no restrictions on market participation. If consumers share the same probability beliefs, then sunspots do not matter; that is, sunspot activity does not affect the allocation of resources. In this case, market equilibrium is equivalent to certainty equilibrium. If', however, subjective probabilities differ across consumers, then sun- spots are bound to matter. These ideas are presented in a series of' three Edgeworth box diagrams in Section V. The reader should seek the return on his investment in Section VI, where the consequences of' restricted market participation are ana- lyzed. Attention is focused on the more difficult (not necessarily the more realistic) case of' shared probability beliefs. There is always a market equilibrium in which sunspots do not matter, that is, a market equilibrium equivalent to a certainty equilibrium. But we also demon- strate that in fact our market equilibrium concept is broader than the traditional certainty equilibrium concept. In Section VI, we give sim- ple examples in which sunspot equilibria are lotteries over certainty equilibria. Randomnization over certainty equilibria is not, however, the only possible source of sunspot equilibria. A more subtle example presented in the Appendix possesses a unique certainty equilibrium and (at least) one sunspot equilibrium. Section VII is devoted to the welfare analysis. Equilibrium alloca- tions in which sunspots do not matter are Pareto optimal in the tradi- tional sense. While equilibrium allocations in which sunspots do mat- ter are not Pareto optimal in this sense, they are Pareto optimal in a 196 JOURNAL OF POLITICAL ECONOMY natural, weaker sense, where each consumer is identified by his state of' birth (as well as by the other usual parameters-namely, tastes and endowments). The upshot of this analysis is that in a rational expectations, general equilibrium world, the presence of extrinsic uncertainty-sunspots, waves of pessimism/optimism, and so forth-may well have real ef'- fects. The lesson for macroeconomics is that, even if one assumes the most favorable informational and institutional conditions imaginable, there may be a role for the government to stabilize fluctuations arising from seemingly noneconomic disturbances. II. Extrinsic Uncertainty The basic parameters defining an economy-such as endowments, preferences, and production possibilities-are referred to as the fun- damentals of that economy. We make a sharp distinction between those things which influence the fundamentals and those which do not. If a variable has an effect on the fundamentals, we say that it is intrinsic (to the economy); otherwise, the variable is said to be extrinsic. Our basic question is: What effect can an extrinsic variable, say, sun- spot activity, have on the allocation
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