تعادل عمومی با عدم قطعیت درون زا و پیش فرض

We study the introduction of new assets that are defined in expected values rather than state by state. Individual default emerges naturally in an economy where such assets are introduced without completing all contingency markets. We further provide conditions under which individual default is propagated endogenously into a collective risk of widespread default in general equilibrium. We prove existence of a general equilibrium with endogenous uncertainty.

New financial instruments are introduced every day including indices, derivatives and innovative forms of government debt. They help manage risk and improve economic welfare. However, they can also increase macroeconomic volatility. The complexity of contractual obligation within a market can transmit individual risks and amplify them into correlated or collective risks. There are trade-offs arising from the gains and the losses created by financial innovation. This article shows the connection between financial innovation and default, and it focuses on the propagation of default in complex markets. Markets can magnify risk. As new assets are introduced, a creditor who is a victim of default in one transaction is unable to deliver in another, thereby causing default elsewhere. In this manner default by one individual leads, through a web of obligations, to a large number of defaults. Since new instruments create new webs of obligations, financial innovation is the precipitating factor. The transmission of default from one trader to another and from one market to another transmits individual risk and magnifies it into collective risk. Default by one individual leads to a collective risk of widespread default. We introduce a formal framework based on individual and collective risk. We show how financial instruments that are introduced to manage individual risk often increase collective risk. The newly created uncertainty does not originate in nature, but from market forces. It is endogenous uncertainty, and is best formalized by a set of simultaneous decisions that affect market behavior as in general equilibrium analysis. 1 Precisely how does financial innovation lead to collective default? We start from a large economy with an incomplete set of assets, where agents face individual risks. A new asset is introduced, whose payoffs are defined in terms of expected values rather than state by state. We call these ‘statistical assets’; similar assets have been studied in Arrow and Lind (1970) and Malinvaud, 1972 and Malinvaud, 1973, and we denote them Arrow–Lind–Malinvaud (ALM) assets. A typical example is provided by insurance contracts, which are valued based on their expected value. Such assets exist in large societies because of the inherent difficulties of dealing with contracts whose payoffs are contingent on each individual’s state,2 such as those in the Arrow–Debreu model. The next step is to show in Section 3 how individual default emerges with such ALM assets, and how individual default is propagated and magnified into a collective risk of widespread default once the new asset is introduced. Since the value of a newly introduced ALM asset is determined in terms of statistics this creates states of default. For example, in Malinvaud (1972), the statistic is the expected number of people who are ill, and the random variable is the number of sick people. As the population size increases, the law of large numbers predicts that the random variable representing the number of sick people converges to a fixed proportion almost surely. Therefore in the limit, but only in the limit, insurance that is provided at actuarially fair prices – expected value – matches premium precisely to the insurance payments. However, when the economy is large but finite, no matter how close we are to the limit, the law of large numbers does not operate exactly. Therefore insurance contracts designed to deal with an exact proportion of sick people will not be able to cope with actual payments in those cases where the realized numbers exceed the limiting proportions. Insurance contracts offered at actuarially fair values (even with a premium) promise payments that exceed physical endowments, with small but positive probability. This is how default arises when ALM assets are introduced.3 Default is a typical problem in large economies with individual risks, since in such economies it is standard to use statistics to describe the characteristics of a group. The next step is to show how individual risk, which in the limit is a statistically insignificant event, can be propagated and magnified into a major widespread default. Once default occurs the complexity of the web of trades within the economy determines how widely it spreads, and the total amount defaulted. The main result is Proposition 1 in Section 3, which proves the existence of a general equilibrium with default when agents recontract trades in the default states. The ‘default states’ are collective states that are defined following the introduction of the ALM asset; each represents endogenous uncertainty, namely uncertainty that is generated by the functioning of the economy. In Section 4, we show that in an open set of economies called ‘complex economies’, individual default leads to a widespread default no matter how large the economy is, i.e. no matter how close we are to the limiting economy. Proposition 2 shows that in a ‘robust’ set of large but finite complex economies, there exists a set of collective states with positive probability each, where an overwhelming majority of the households in the economy default. Following the main results we show examples of economies where the expected level of default increases with the size of each (finite) economy, although at the limit, by the law of large numbers, there is no default. Proposition 3 and Proposition 4 and Example 4 in Section 5 establish that the expected value of default may exceed any bound as the population size increases, no matter how close the economy is to its limit, and that the probability of default may decrease as more financial reserves are required. Appendix A provides the mathematical proofs, and Section 6 concludes the paper.

We have analyzed the effect of introducing ALM assets in a large but finite in complete market economy. Examples of ALM assets are actuarially fair insurance contracts, or shares in a firm which maximizes expected profits. States of default emerge because ALM assets promise deliveries that sometimes exceed physical endowments. Proposition 1 proved the existence of an equilibrium with recontracting and endogenous uncertainty of default. The new states of default in E are collective states because they depend on collective events, such as the proportion of the population in each individual state. We showed that in complex economies many individuals default at once at each state of default, emphasizing the collective nature of default. Proposition 2 shows that there is an open set of complex economies of complexity k , for any k≥2k≥2. In such economies default by one individual leads to default by kk individuals, due to the complex pattern of trading. The introduction of ALM assets to hedge against individual risk therefore may increase collective risk of default. We studied expected default as the population size increases without bounds. At the limit, and only at the limit, the economies have no default. This is because the distribution of risks across the population converges with certainty to a known one. Although the introduction of new assets, such as health insurance policies, may enhance the welfare of individuals, our results illustrate a familiar concern about financial innovation. The concern is that the introduction of new financial assets could in some circumstances lead to more instability, namely to new states of collective default. The results we presented formalize this concern. They offer a way to measure the benefits of financial innovation, as well as its drawbacks. The key is to understand the two circumstances under which collective risk increases with introduction of new financial assets. These circumstances are: the assets are ALM, and the economies are complex. The first feature of the problem, that the assets be ALM, is almost inevitable in large economies with individual risk, because of the difficulty of creating assets that depend on long list of individual characteristics, most of which are difficult to observe. So this first condition cannot be avoided. The second feature refers to the complexity of the economy. The introduction of an asset typically increases the web of trading in an economy and thus its complexity. Certain assets increase the collective risk of default more than others. They create “correlated” risks which cannot be properly insured. The computation of the collective risk of default and the total expected default from different assets, as defined in Section 5, could help to determine the extent of collective risk introduced by the asset. An interesting area of research would include the computation of the costs and benefits from the introduction of new securities. The benefits can be measured in terms of Pareto improvements in welfare, and the costs could be measured in terms of the increase is collective risks and complexity of the economy, which make it more vulnerable to financial instability. The other implication of our results is that they help to formalize a “multiplier effect” for policy. In a complex economy, financial policies which succeed in preventing default by one agent also prevent, by a chain reaction, a large number of other defaults at no additional cost. Therefore the benefits have a “multiplier effect”. Our results provide support for the policy of requiring reserves to enhance financial stability. The complexity of the economy is not a problem in itself, unless it leads to large correlated risks. However, if following the introduction of an ALM asset a second layer of securities is introduced to deal with the endogenous risk created by the first, and the latter securities are also of the ALM type, then the process is replicated. More endogenous uncertainty may be created, piling up the risk of default of an asset which was introduced to hedge against the risk of default of another a never-ending process. The results therefore suggest that large and complex market economies with individual and collective risks are likely to be incomplete.