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# GENERAL EQUILIBRIUM AND ASSET PRICING: EXISTENCE AND SPECULA TIVE BUBBLES IN PURE EXCHANGE ECONOMIES.

MATTALIA, CLAUDIO

2000-02-14

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Contributor(s)

ZECCHIN, MARCO

Abstract

The theory of general equilibrium is the branch of economic theory that studies the interactions between demand and supply of the different goods in the different markets in order to determine the prices of these goods ( while the partial equilibrium analysis considers only the relations between demand and supply of a specific good and the price of the same good). In the study of general equilibrium some simplifications are usually introduced, in particular it is assumed that markets are competitive and individuals are optimizing, that there is no production (at least in first approximation) and that agent shave fixed endowments of the goods and must determine only the quantities to exchange(i. e. a pure exchange economy is considered).One of the central features of modern economics is then the introduction of time and uncertainty, and the consequent attempt to analyse an environment characterized by the presence of these elements. The main consequence for the behaviour of individuals is that they have only a limited ability to make decisions in such an environment; with reference to the theory of general equilibrium, in particular, this implies that, when agents ha velimited knowiedge and ability to face uncertainty, they trade sequentially (i.e. period by period) and use a system of contracts which involve only limited commitments into the future. The standard model for the analysis of general equilibrium is that developed by Arrow and Debreu; the principal objective of the Arrow-Debreu theory is to study the allocation of resources achievable through a system of markets, and the central result of this theory is that, when there are markets and associated prices for all goods and services in the economy, no externalities or public goods, and no informational asymmetries, then competitive markets allocate resources efficiently. This framework can be adapted in order to take into account the fact that economic activity (production, exchange and consumption) takes place over time and involves the presence of uncertainty; in this casethe Arrow-Debreu model assumes that at the initial date there is a market for each good produced or consumed in every possible future contingency, i.e. it assumes the presence of a complete set of contingent markets. Nevertheless, this structure is an idealization of the situation we can observe in the real world, since the individuals do not have full knowledge of all possible future events. The market structure that it is possible to observe in the real world, on the contrary, takes into account the fact that agents have limited capacity to face uncertainty, and it consists of a sequential system of spot markets for the exchange of goods and services and of contractual (financial} markets that involve limited commitments into the future. For this reason it is necessary to consider a general equilibrium model that constitutes an extension of the basic framework represented by the Arrow-Debreu theory. In such an extension the basic set of markets is represented by a sequence of spot markets, on which goods and services are exchanged, and by a sequence of contractual (financial)markets to make commitments for the future, commitments that typically involve either the promise to deliver goods and services ( real contracts) or the promise to deliver a certain amount of money (nominailontra.Cts). When there is only a limited set of such contracts, in particular, the economy is characterized by a system of incomplete markets( that is typical of the real world).The equilibrium solution of this kind of models (if it exists) gives the values of prices and quantities ( of the goods and of the financial activities) in correspondence of which the individuals solve their optimization problem and the markets (real and financial) clear(i. e. demand equals supply o n these markets). A first important problem is there for represented by the analysis of conditions that guarantee the existence of solutions in this kind of models. These models can then be used to analyse the issue of asset pricing, and in particular the relation between the equilibrium price of an asset and the stream of future dividends on which the asset represents a claim. What emerges is that, while in the finite-horizon case the equilibrium price equals the fundamental value of the asset (i.e. the discountedsum of future dividends), in the infinite-horizon case this is not necessarily true (in particular, it is possible for the price to be larger than the fundamental value). In this case the price of the asset is said to involve a speculative bubble. A second important question is therefore represented by the analysis of conditions that allow to exclude the presence of such bubble components, together with the study concerning the fragility of this phenomenon. The analysis presented in this thesis moves along these lines and is divided into three parts. In the first part (Chapter l) the problem of existence of solutions in temporal general equilibrium models is treated, and the main results are given. Since the literature on this argument is very extensive, and in the light also of the results discussed in the second part, this analysis is limited to the (relatively) simplest case, the one in which the economy is characterized by discrete time periods and a finite number of states of nature in each period (an d in which the economy is a pure exchange one, with financial structure that consists of real assets only). In such a framework, the case of a two-period economy is initially studied, and the results are then extended to the case of a T-period economy ( with T finite) an d finally to the case of an infinite-horizon economy. While in the model with complete contingent markets an equilibrium exists for every economy,this is no longer true when we deal with an economy characterized by spot and financial markets. What can be proved, in the case of finite horizon, is that the equilibrium exists for a generic set of economies (i.e. for a set of the parameter values that characterize the economy whose complement has zero measure); in the case of infinite horizon ( with general financial structure) the equilibrium exists only fora dense set of economies, and there fore the result is weaker. The main conclusion of this first part is that, even if in the passage from an economy with complete contingent markets to an economy with spot financial markets and, in this economy, from the finite-horizon to the infinite-horizon case, something is "lost" in the proof of existence of equilibrium, nevertheless a form of existence is always guaranteed. The model is therefore consistent, and it can be used to explain something of the economy we are dealing with. In particular, this kind of models can be used t o investigate the problem of asset pricing, and the relation between the price of the assets and the stream of dividends to which they give rise. This is the question considered in the second part (Chapter 2),with particular reference to the phenomenon of speculative bubbles that can emerge in infinite-horizon incomplete-market economies. The analysis of this question by means of the model introduced in the first part shows how (under the hypothesis of the model)the occurrence of bubbles is limited to a particular class of securities, those in zero netsupply, while on securities in positive net supply (such as equity contracts) speculative bubbles cannot arise. Since this second type of securities represents an important part of the capital market, this result reduces consistently the role of speculation in the class of models considered. The issue of speculative bubbles is then studied following a general approach, inspired by a recent contribution that tries to make arder in the field and to give a definitive theoretical settlement to the question. In this context ( that is based on considerations of no-arbitrage opportunities), in particular, the possibility of a new type of bubbles (theso-called "ambiguos bubbles") is outlined, and the results concerning the fragility of this phenomenon are discussed. These results show how, under quite general assumptions,speculative bubbles do not occur in intertemporal economies, and they can appear only under rather special circurmstances ( and, again, they are possible only on securities in zero net supply). As a consequence, well-known examples of economies in which bubble sappear as an equilibrium phenomenon (such as some kinds of monetary models or of overlapping generations models) are not robust. In the last part of the thesis a different approach (based on Euler equations) is used to study the issue of speculative bubbles, and to obtain results that, again, are in favour of a substantial fragility of this phenomenon. This approach is far more limited than the previous ones, but its interest is due to the fact that it confirms the results obtained with the general models and, first of all, it can be used to build and study specific examples of economies in which bubbles appear. This method is first used to deal with a model in which agents are homogeneous, and then it is extended to the more general situation of etherogeneous agents. With reference to this context, a series of examples is presented (Chapter 3), and in this case the use of the Euler equations' approach allows us to obtain some new results (for instance in terms of multiplicity of equilibria) and to show how the presence of bubbles on asset prices is linked to the violation of specific conditions. The thesis shows how, in the light of the recent results obtained, speculative bubbles can be considered definitively as a marginal phenomenon in the kind of models presented. On the other hand, the real world is often characterized by the occurrence of "speculative episodes" in which bubbles do appear, and therefore this contrasts with the conclusions obtained. The final considerations try to explain this contrast, by describing a possible mechanism that produces these results: the models considered are based on the hypothesis of rational expectations of the individuals, which are assumed to use all the available information to make their predictions and to base these predictions on the correct model of the economy. Nevertheless, there are periods in which, as a consequence of specific situations, the hypotesis of rational expectations "breaks down"; it is precisely in these periods that speculative episodes arise, and speculative bubbles on asset prices appear. In conclusion, therefore, by relying on a mechanism of this kind it becomes possible to reconcile the results of the theory with the episodes of the reality.

Insegnamento

Publisher

Università degli studi di Trieste

Languages

en

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