Dissertation Roitman Final Version

8/3/2019 Dissertation Roitman Final Version

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Extreme events in economics: a suggested interpretation

Mauricio Ezequiel Roitman

This dissertation is being submitted in part-fulfillment of the requirements for the

degree of MSc in Economics

University of Surrey

2011



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‘I hereby declare that this dissertation has been composed by myself and no other and

to the best of my knowledge has not been submitted in any previous application for a

degree’.

‘I also give my permission for this dissertation to be available for photocopying and

inter-library loan’.

Mauricio Ezequiel Roitman

Guildford, Surrey, England, 06/09/2011



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Acknowledgements

First of all, I would like to express my gratitude to my supervisor, Professor Neil

Rickman, for his stimulus, helpful comments, suggestions and guide. Of course, all

remaining errors are exclusively responsibility of the author.

Secondly, I am really glad to have worked with valuable colleagues and fellow co-

workers at the “Instituto de Economía - UADE”, where I learnt to be a professional

economist and a better person. Particularly, I will always be grateful to Gustavo Ferro

for giving me that possibility almost twelve years ago.

Finally, I would like to thank those people who taught my first lessons of “Value

Theory”: my parents, my brothers and the rest of my family. They taught me the value

of friendship, integrity, responsibility and optimism, amongst others.



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To all people working for the truth,

the beauty and the goodness

To my parents and my brothers

To Silvana



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Contents

Abstract ........................................................................................................................... 8

1. Introduction ................................................................................................................. 9

2. Literature review I: extreme events and expected utility theory ............................... 12

2.1 Extreme events: definition and insurability ......................................................... 12

2.2 Extreme events and expected utility theory ......................................................... 13

3. A model of the extreme events with exogenous probability ..................................... 15

3.1 The model I .......................................................................................................... 15

3.2 The model I: comments and limitations .............................................................. 17

4. Literature review II: the toolkit to build a model of extreme events with endogenous

probability (financial contagion model) .................................................................... 204.1 The applications of the theory of networks to financial contagion models ......... 20

4.2 The financial theory: a short review of Mean-Variance and the Optimal Financial

Structure theories ....................................................................................................... 24

5. A model of the extreme events with endogenous probability: a rational explanation

of a financial contagion ............................................................................................. 26

5.1 The model II ........................................................................................................ 27

5.2 The model II: comments and limitations ............................................................. 32

6. Conclusion and further research ............................................................................... 33

7. Appendix ................................................................................................................... 36

8. References ................................................................................................................. 42

9. Bibliography ............................................................................................................. 47



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Figures

Figure N°1. The dynamic mechanism of the contagion model (Lemma 2) .................. 30 Figure N°2. A summary of the contagion model`s main building aspects.Figure N°3. A

suggested contagion model ........................................................................................... 31 Figure N°3. A suggested contagion model ................................................................... 31 Figure N°4. The sequential game for insurance demand against extreme events ........ 37



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“Before we can explain why people commit mistakes, we must first explain why they

should ever be right.”

Von Hayek, F. A., “Economics and knowledge”, Economica 4 (1937), 33-54

“Without effort, he had learned English, French, Portuguese, Latin. I suspect,

nevertheless, that he was not very capable of thought. To think is to forget difference,

to generalize, to abstract. In the overly replete world of Funes there were nothing but

details, almost contiguous details.”

In Ficciones by Jorge Luis Borges, edited by John Sturrock (original

publication 1942; English translation, Grove Press, 1962; rpt. by Alfred A.

Knopf/Everyman, 1993), 83-91.



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Abstract

Extreme events have been usually excluded from the economic analysis. This

dissertation will study some aspects of the economics of extreme events in the context

of the theory of risk and uncertainty, using common tools of the rational economic

science methodological approach.

A definition and a suggested classification of extreme events are made. Natural

disasters appear having an exogenous probability process and financial crises an

endogenous probability process.

The first hypothesis studied is that the agents do not buy insurance against extreme

events with exogenous probability because they know –by assumption- that the

government will act as an Insurer-of-Last-Resort in that contingency. Thus, the first

model and proposition explain the conditions under which a total crowding-out effect

of private insurance against extreme events is produced when the government ex-post

relief is common knowledge.

The second hypothesis regarding extreme events with endogenous probability is that

while the financial contagion depends on structural parameters of a network economy,

the individual rationality will carry the economy to a non-Pareto optimum Nash

equilibrium. The model not only shows the relevance of some costs of the government

IOLR role but principally describes the individual and rational mechanism that carried

the economy to the worst financial fragility position.



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1. Introduction

This dissertation will study anomalous or abnormal circumstances in economic

decisions under risk and uncertainty. Those circumstances are underlined by Bak and

Paczuski (1995) who have said that “Contingency was used as an argument for

statistical exclusion. Once the large events are discarded, the remaining events

trivially obey Gaussian statistics” (Ibid., p. 6690). Although this paper´s

methodological approach shares those research concerns, they will be treated with

ordinary tools from the rational paradigm of the economics science.

Two particular puzzles will be treated in this work taking into account worries about

extreme events (abnormal situations).

Firstly, the agent’s behaviour in presence of extreme events -also called catastrophes

or low probability-high losses events- is a theoretical puzzle for the economic science.

Even though the Von Neumann-Morgenstern Expected Utility theory approach has

constituted a mainstream methodology to analyse risk situations in economics, some

behavioral psychology and economic studies have shown that people systematically

ignore extreme events in their insurance decisions. This puzzle has been presented as a

contradiction that would undermine the rational choice behaviour assumption implicit

in the Expected Utility theory.

Secondly, the increasingly frequent government`s role as Lender-of-Last-Resort

(hereafter LOLR) or Insurer-of-Last-Resort (hereafter IOLR) over the last decades,

their determinants and the nature of the crises that have motivated that role –

sometimes called financial contagion- is not clear for the economic science yet. The

second part of this dissertation will be dedicated to study this issue because financial

contagion may be considered as an extreme event generated by an endogenous

probability process.

The extreme events will be considered the link between those two research aims.

It can be useful to begin this dissertation defining what an extreme event is and what

kind of extreme events will be considered here. At this respect, an extreme event can



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be defined, based on arguments that will be exposed at the Literature review I section,

as: rare, severe, nonlinear and outside the normal range of experience of the system in

question. In addition, it will be assumed that they can be described in terms of a

threshold with respect to one or more parameters.

The first hypothesis to be studied in this research will be that the economic agents do

not buy insurance against extreme events because they know –by assumption- the

government’s role as a public good provider of relief or an IOLR in that situation with

certainty. This hypothesis might be also thought by analogy with the Good Samaritan’s

dilemma (Buchanan, 1975).

At this regard, a particular feature of the extreme events is that they may force the

government to play the IOLR role to support their social cost, providing public

insurance under certain conditions.

The first model to be exposed is about insurance against extreme events with

exogenous probability exploring one ex-ante and one ex-post situation. At the ex-ante

moment, all private agents decide the level of private insurance that they are going to

buy in the market. At the ex-post situation, the extreme event might or might not

occur. If the extreme event takes place, the government will play the IOLR role for

certain because the government’s willingness to be an IOLR will be assumed

exogenously determined and common knowledge1. However, some doubts may remain

about why the government should guarantee the agents against extreme events and

when this situation is more probable.

Accordingly, a second model will be proposed at the subsequent part of the current

dissertation. The purpose of this model is to explain the effects that can be produced by

ex-ante independent and identically distributed (hereafter iid) shocks over a network

economy, which under some assumptions, can be transformed in ex-post large-effect

shocks (extreme events) by financial contagion.

1“Common knowledge” is a different concept to “belief”. Belief has involved that all known that all

known and ad infinitum. Common knowledge means that all known that all known and ad infinitum,

that a fact is true. The belief can be false but common knowledge to entail the veracity of the fact. See

Werlang (1989).



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The second hypothesis of this study is that while the financial contagion depends on

structural parameters of a network economy, the economic rationality to make the best

financial decisions in terms of individual welfare, e.g. full diversification incentives

and optimal financial structure decisions, will carry the agents to a non-Pareto

optimum Nash equilibrium (maximum financial fragility).

The most important characteristic of the second model is that it underlines the effects

of a financial contagion on the social welfare as a function of the exogenously

determined economic constraints (for example, the degree of financial liberalization,

laxness in the credit policy) and the endogenous process motivated by the interaction

between individual economic incentives and their knowledge about the government

action as an IOLR with certainty when an extreme event occurs.

This second model might be related to the general ideas of the non-micro-founded

Minsky’s Instability Financial Hypothesis (hereafter IFH) model (Minsky, 1975, 1992)

but considering exogenous shocks.

The paper is structured as follows. The general methodological approach, aims,

definitions, some brief comments about the structure of the first and the second models

and the two hypotheses, are offered here in Chapter 1 as an Introduction. In Chapter 2,

a survey of the literature about extreme events and the relationship between extreme

events and expected utility theory will be made. In Chapter 3, the first model, the first

proposition and some comments and limitations will be presented. In Chapter 4, a

second literature review on theory of networks, financial contagion and some aspects

of the financial theory will be carried out. In Chapter 5, the second model and somecomments and limitations will be exposed. Finally, a conclusion and further research

will be elaborated in Chapter 6.



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2. Literature review I: extreme events and expected utility theory

2.1 Extreme events: definition and insurability

There are several definitions of extreme events. In particular, Bier et al. (1999) said

that generally they are: “…rare, severe, and outside the normal range of experience of

the system in question.”(Ibid., p. 83). Moreover, they have added that:

Extreme events are often caused by non-linear phenomena, such that a relatively small

change in some causal factor can lead to a large increase in the severity of the event…

Additive or synergistic effects can also affect the frequency of extreme events,

especially when system behavior is nonlinear.

…Often, the event of interest can be described in terms of a threshold with respect to

one or more parameters, in which case its probability can be defined as a tail

probability of the distribution for the parameter(s). (Ibid., p. 84)

Other underlying features, underlined by Hirshleifer and Riley (1979), could be

presented in the case of extreme events. They have affirmed that some type of events

may not be actuarially fair and, hence, non-full-insurable by two reasons: “(1) if the

number of risks in the insurance pool is small, so that the Law of Large Numberscannot fully work, or (2) even with large numbers, if risks are on average correlated.”

(Ibid., p. 1387). It could be supported that these features are usually a constant when

an extreme event occurs.

A proposal of a classification methodology of extreme events will be made and used in

this paper.

At this regard, it can be suggested that there are two different types of extreme events,

one having an exogenous probability (e.g. hurricanes, earthquakes, etc.) and the other

having endogenous probability (e.g. financial contagion). However, both types of

extreme events share the same ex-post common feature: agents will be affected in a

positive correlated way by them (Embrechts et al., 1997; Bogachev et al., 2008;

Altmann and Kantz, 2005; Bunde et al., 2005).



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Consequently, the risks corresponding to those phenomenons, indentified by the

positive value of the correlation parameter, under some circumstances are considered

uninsurable (Hirshleifer and Riley, 1979)2.

2.2 Extreme events and Expected Utility theory

It was commented in the introduction that some behavioral psychology and economic

studies have shown that agents systematically ignore low probability-high losses

events in their insurance demand decisions and that puzzle has been presented as a

contradiction that would undermine the rationality assumption implicit in the Von

Neumann-Morgenstern Expected Utility theory approach.

One of the first studies that emphasized this fact as an anomaly of the expected utility

theory was Slovic et al. (1977), who underlined that, “people buy more insurance

against events having a moderately high probability of inflicting a relatively small loss

than against low-probability, high-loss events. Two explanations are discussed, both

contrary to traditional utility theory.” (Ibid., p. 237).

Likewise, one of the most recognized studies that support the same vision is

Kahneman and Tversky (1979), who have affirmed that:

…the simplification of prospects in the editing phase can lead the individual to discard

events of extremely low probability and to treat events of extremely high probability as

if they were certain. Because people are limited in their ability to comprehend and

evaluate extreme probabilities, highly unlikely events are either ignored or

overweighted, and the difference between high probability and certainty is either

neglected or exaggerated. (Ibid., p. 283)

That is, individuals ignore extreme events and, in their opinion, the reason is “because

people are limited in their ability to comprehend and evaluate extreme probabilities”.

Hence, this anomaly has been presented as evidence against the power of the

2For example, Lewis and Murdock (1996) have shown that in the natural disasters case -extreme events

with exogenous probability- insurance markets have not sufficient capability to diversify that type of

risks in an intertemporal way and have also underlined the necessity of federal governmental assistance.



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rationality assumption implicit in the Expected Utility theory to explain decisions

under risks when extreme events are possible.

On the contrary, Doherty and Schlesinger (1983) said that explanations including miss-

estimation of probabilities and simple irrational behaviour do not facilitate the

construction of alternative testable hypotheses.

At this respect, there are alternative explanations supporting rational behavior and the

Expected Utility Theory. Some of them will be cited subsequently.

Kunreuther and Pauly (2004) have highlighted the problem of agent's transaction costs

to collect information about the probability of this type of risk. Michel-Kerjan and

Mercellis-Warin (2006) have affirmed that the government could be forced to help

people without coverage when an extreme event happens. It has caused that

policymakers sometimes have been motivated to impose ex-ante contributions to

citizens. Furthermore, they have asserted some strengths of the government as an

insurer or a re-insurer, being: the ability to redistribute wealth and diversify risk

between present and future generations, the aptitude to be a reinsurer of last resort and

the capability to avoid adverse selection problems with mandatory coverage. They

have concluded that government action is necessary to solve the problem of the

insurability of extreme events.

In a more theoretical-oriented paper, Krueger and Perri (2010) have asked if public

income insurance through progressive income taxation could improve the allocation of

risk in an economy with incomplete private risk-sharing. They have argued that to

answer that question is necessary to understand the cause of private risk sharing

limitations. On one hand, if private risk sharing is limited by exogenous reasons,

government insurance –and redistributive taxation- improves the welfare. On the other

hand, if the limitations have been originated from enforcement problems of private

insurance markets, the government’s insurance provision might crowd out private

providers against idiosyncratic risk, potentially in a more than proportional way.

Similarly, the effects of government relief on ex-ante incentives and risk bearing in

insurance private markets have been studied by Kaplow (1989). He has shown that



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government relief distorts the optimal private insurance attributed to moral hazard. As

the Model I that will be presented in Chapter 3 of this paper, he has demonstrated that

if the bailout is complete, no insurance is required by agents. In addition, if that is less

than complete, insurance coverage is complete but not enough to full-compensation.

From an empirical point of view, Jaffe and Russell (1997) have argued that the

problems why private insurance companies in the United States are not willing to

provide insurance against catastrophic events rest in the capital market rather than in

the insurance market3. However, they also have underlined -in the same way assumed

by this paper- that the government will continue to be an essential player in that

market.

In the next section, a simple model of the economic agents’ behavior in presence of

extreme events in a way compatible with the underlying rationality assumption of the

Expected Utility theory will be suggested.

3. A model of the extreme events with exogenous probability

3.1 The Model I

It can be argued that, taking into account latter arguments and definitions, an extreme

event with exogenous probability may be an uninsurable risk because of either there

are missing private insurers or there is an insurance market with private insurers as

suppliers and an implicit (but common knowledge) complementary insurance (or re-

insurance) supplied by the government.

In the first case, the puzzle of the missing supply of private insurance against that sort

of extreme events can be explained in a simple way by the actuarial features of them

(e.g. positive correlation and large number of agents affected, such as natural

disasters). Thus, the moral hazard costs highlighted by Buchanan (1975) could be

3Specifically, Jaffe and Russell (1997) have found that institutional factors such as accounting, tax,

takeover risk and the significant liquid capital requirements in relation to catastrophic risks discourage

insurers from offering these types of insurance products.



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sufficiently low with respect to welfare improvements produced by the government

insurance provision underlined by Krueger and Perri (2010).

In the second case, where that sort of extreme events are insurable by private insurers

with an implicit government guarantee, will be showed that the government IOLR role

will crowd out the insurance demand, under given circumstances. This is demonstrated

in the seemingly trivial Proposition 1.

Proposition 1. An economy with risk-averse agents, where the government guarantees

public insurance to agents without full-private insurance covering them for certain

when an ex–post known extreme event occurs –being it common knowledge-, and

financing its budget with ex-ante tax collection from all agents, will produce that no

one buys ex-ante private insurance against extreme events.

Proof.

The following notation will be used:

• Variables with a `hat' denote that insurance has been purchased. n individuals

do this.

• Variables with a `tilde' denote that insurance has not been purchased. m

individuals do this.

• u is the (`well-behaved') utility function of each individual.

• L

w is the wealth level of an individual in the `loss' state.

• N

w is the wealth level of an individual in the `no loss' state.

• p is the probability of an adverse event.

• T is total tax revenue.

It was assumed by Proposition 1 that:

1. The Government charges each individual with a positive amount of the total revenue

tax, regardless of whether they purchase insurance; i.e. everyone pays:

T/(n + m).



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2. The Government operates an ex post policy to ensure that wealth levels are equal

across the insured and uninsured agents in the “loss” state; w L = w L.4

Essentially, the Government provides an implicit, ex-ante, insurance to those who don'thave it and this information is common knowledge.

Now compare the expected utility of the uninsured with insured agents. It is optimal to

remain uninsured if and only if:

p.u( w L) + (1 - p).u( w N) > p.u( w L) + (1 - p).u( w N)

Given assumption 2 above, we know that the first parts of the left- and right-hand-sides are the same so the inequality depends on whether

u( w N) > u( w N).

This must be true because the insured agents will have paid an insurance premium so

w .N > w .N

Hence, the required result was obtained.

3.2 The Model I: comments and limitations

The latter simple argument is enough to explain the puzzle given initially respect to

why people do not buy insurance against extreme events. However, a more complex

and complete model including the case of two possible shocks, one being an insurable

shock and another, being an uninsurable shock (an extreme event with perfect positivecorrelation), will be considered in the Appendix of this dissertation.

The strong assumption and, at the same time, the most important limitation of the

proposed explanation is that the government playing the IOLR role is assumed with

probability one when an extreme event occurs. Although the proof could be extended

4The rationality of this “time-inconsistency” policy has been explained by Acharya and Yorulmazer

(2007) and Farhi and Tirole (2009) in the context of systemic crises.



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to the case when that probability is between zero and one, it will be left to further

research. Kaplow (1989) has obtained similar results using a different proof strategy.

This explanation of the missing extreme event insurance demand could be simpler than

other explanations based on different kind of market failures, highlighting the

relationship between rational individual incentives and the knowledge about the

government action. The government action as an IOLR is assumed to have a time

inconsistency problem (Acharya and Yorulmazer, 2007 and Farhi and Tirole, 2009).

That is, the government avoids announcing ex-ante a future relief but its action is ex-

post necessary and welfare improving. In addition, this model may be better in terms

of generally accepted properties of useful economic models: parsimony, tractability,

conceptual insightfulness, generalizability and falsifiability (Gabaix and Laibson,

2008).

The proposition could be considered trivial and valid for all types of events and not

only for extreme events. However, while the assumption that the government will play

the IOLR role with certainty could be excessively strong in general, it may be more

acceptable in particular to ex-post extreme events with exogenous probability. For

instance, Anderson (1976) has argued that, referring to natural catastrophes5, "Society

has established the working assumption that compensation for catastrophe property

damages will be provided by government imposed programs to the extent that

compensation is not provided by private insurance market" (Ibid, p. 650)6.

There is a well-documented literature, theoretical and empirical, about this issue but in

other contexts. One theoretical example is Varian (1980), who has argued that “if

consumers were guaranteed a second period income of w/2, and a marginal tax rate of

1 on any income in excess of w/2, they would choose to save nothing. This is simply the

disincentive effect of social insurance – a complete welfare state may result in a

significant incentive loss.”(Ibid, p. 53).

5They are considered here as extreme events with exogenous probability.

6Additionally, he said that (public) programs for compensating individuals’ material damage have

increased in size and number. At this respect, he has pointed out that prior 1950’s US Federal

Government had not a catastrophe insurance system to compensate people without private insurance.

Nevertheless, that policy was changed with the Federal Disaster Act of 1950.



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From the empirical side, Cutler and Gruber (1996) have studied the possible crowding

out effects and the mechanism through which public insurance might has affected

private insurance in expanding public sector health programs in the United States over

the period 1987-1992. They have found that, considering Medicaid expansions had an

exclusionary structure (individuals were either entirely eligible to public insurance or

not), the crowding-out was 49% (reduction in private coverage) of the public coverage

increase.

This model is also associated with the idea of the Good Samaritan Dilemma

(Buchanan, 1975). In that case, the Good Samaritan, generally the government, helps

people in bad state of nature. This action will cause moral hazard incentives on people

prior to a possible negative event. This moral hazard may produce a lower level of

insurance taken by agents.

It might also be valid to ask why the government should play the IOLR role. On the

one hand, there are two possible answers to this question in the case of extreme events

with exogenous probability. Firstly, the action may be determined by the nature of an

extreme event, that is, the large and positive correlated losses with an uninsurable

feature (by private companies). Secondly, it could appear “public pressures in the

aftermath of a catastrophe” (Miche-Kerjan and Marcellis-Warin, 2006, p. 25) to help

people without private coverage. On the other hand, an important reason for the

governmental bailout in the case of endogenous probability is to avoid that an ex-ante

small and iid exogenous shock become an ex-post large-effect positively correlated

shock in the way of a systemic crisis.

Considering mainly the latter case, an explanation taking into account the probability

of the government to play the IOLR role as endogenously determined will be

suggested in the next two chapters of this dissertation.



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4. Literature review II: the toolkit to build a model of extreme events with

endogenous probability (financial contagion model)

The use of the previous definition and categorization of the extreme events, the

influence on individual incentives of government`s action as an IOLR, together with

concepts of networks economic theory, mean-variance and optimal financial structure

theories, can be also useful to analyze the type of process usually characterized as

financial contagion. Main aspects of this concepts and theories will be reviewed in the

next two subsections.

Firstly, a literature review about network theory, financial contagion models and some

aspects of corporate finance theory will be made. Secondly, a model of extreme events

with endogenous probability offering a rational theory for financial contagion will be

proposed.

4.1 The applications of the theory of networks to financial contagion models.

Financial crises have been a focus of economist`s studies from the beginning of the

20th century. The classical works in this field are the debt-deflation theory of Fisher

(1933), the ideas exposed by Keynes (1936), Kindleberger’s (1978) history of financial

crisis and the FIH of Minsky (1975, 1992), amongst others.

In a broad sense, Minsky’s hypothesis is closest to this paper’s second model. The

main role given by that theory to the impact of debt (as a stock) on the system stability

and its relationship with profits (flows), related to what is called its “first theorem”, is

used in the model to define financial fragility through leverage ratios and wealth

effects. The difference is that in this case equilibrium situations and the motivations

are strictly explained by microeconomic incentives.

Economists have applied new methodologies to try to understand the microeconomic

foundations of financial crisis since some years ago. Nowadays, a popular and

increasingly used methodology is the application of the theory of network’s concepts

to study financial contagion.



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A complete survey about network theory in a strict sense, including references to

networks in the real world (social, information, technological and biological networks)

and theoretical developments can be founded in the work of Newman (2003)7.

From the economic point of view, a brief summary of the core concepts of networks

theory have been offered by Gale and Kariv (2007). First, they have characterized

networks as "natural tools" (Ibid., p.99) to explain complex phenomena and have

considered them an “alternative model of decentralized financial exchange" (Ibid.,

p.99). Second, they have clarified the key differences between the assumptions of

completeness and incompleteness, while a network is “complete” when “every

possible trading opportunity is present” (Ibid., p.99) and, on the contrary, it is

incomplete “if some pairs of traders cannot trade directly with each other.”(Ibid.,

p.99). The authors said that reasons for incompleteness –e.g. market failures- might

produce fragility when a sudden shock occurs. They have underlined one feature of

financial markets in the literature: their fragility, even though different models have

attributed that to diverse sources.

Specifically, among the most important papers in the financial contagion field are

Lagunoff and Schreft (1998), Scinasi and Smith (1999), Allen and Gale (2000),

Kodres and Pritsker (2002), Kiyotaki and Moore (2002), Goldstein and Pauzner (2004)

and Leitner (2005). Some characteristics of them, their similarities and differences in

contrast with the approach of the current research, will be explained.

For instance, Lagunoff and Schreft (1998) have constructed a model of financial

fragility, in which the agents’ financial position is given by interrelated and diversified

portfolios and payment commitments emerging by the credit market activity. These

7More precisely, Newman’s survey has analysed the most recent theoretical advances on networked

systems. For instance, he has reviewed: the main statistical properties of networks, such as resilience,

clustering, degree distributions and path lengths; the most used quantitative measurements; different

models to explain different network structures, such as: graph models, Markov graphs, small-world

model and model of network growth; and the behaviour of processes on networks, e.g. spread of

infections and failures of nodes. He has underlined that the most important remark is that networks are

not randomly formed while they have statistical common features shared between some easily

distinguished types of them. The author has concluded saying that complex networks study is still in its

infancy.



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payment commitments could be disturbed by iid shocks that can produce a default

spreading through portfolio linkages. They have established that if an economy is more

fragile, a financial crisis will be more severe ex post and may be accelerated by

preemptively portfolio re-optimization of forward-looking agents. Correspondingly,

the model to be proposed in this paper will consider in a similar way that small and iid

exogenous shocks can become uninsurable-positive-perfect-correlated-large-shocks,

when a complete interconnection situation is given but by a different propagation

mechanism (diversification). This idea of shock is harmonizing perfectly with the latter

authors` financial fragility definition. However, some doubts remain about the

mechanism by which “forward-looking and rational agents who anticipate the spread

of defaults” (Ibid., p. 5) while they “…do not know the chain structure of the economy,

which type of chain— closed or open— they are in, or their location within their chain

without knowledge about the chain structure of the economy.” (Ibid, p. 14).

In a similar way to the current dissertation model, Schinasi and Smith (1999) have

demonstrated that contagion can be explained using the basic portfolio theory without

market imperfection arguments. Equally to this work, they have concluded that

portfolio diversification and leverage could be key parameters to explain the investors’

rational behaviour in a financial contagion. Its slightly different contagion mechanism,

explained by contagious selling of higher-risk assets, has highlighted the positive

correlation between assets returns crossing markets.

Similarly to Lagunoff and Schreft (1998), financial contagion as an equilibrium

phenomenon providing microeconomic foundations has been modeled by Allen and

Gale (2000). The heart of this approach is the focus on the overlapping claims that

different regions/sectors of the banking system have between them as one channel of

contagion (shocks can affect the value of their claims). One of the important

characteristics that define contagion is the strength of the spillover effects affecting

other regions, defined by the network degree of completeness and connectedness. They

have argued that a greater completeness reduces contagion probability. Nevertheless,

they have not taken account of the case when the probability of a bank crisis is positive

(Ibid., p. 28). The latter not treated particular situation is the main connection with the

model proposed by this paper because it will show that a positive probability of a

systemic crisis and the assumed government bailout action to avoid that situation are



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key issues to understand the dynamics of the process and the welfare effects of

financial contagion.

In the case of Kodres and Pritsker (2002), they have used a multiple asset rational

expectations model of asset prices allowing contagion through several channels,

mainly portfolio rebalancing and information asymmetries. The major disagreement

with the model proposed in this dissertation, in addition to the modeling strategy based

on market failures, is that they have considered “re-optimization of portfolios” risk

exposures as a cause rather than an effect of contagion.

The contagion's propagation mechanisms through balance-sheet effects have been

examined by Kiyotaki and Moore (2002). They have defined two classes; the first one,

when fluctuations in assets prices have indirect effects on collateral values, and the

second is given by the direct effect of default or postponement debt repayments in an

existing credit chain. They have highlighted three main factors producing balance-

sheet effects: history dependence in the behavior of constrained firms (in greater depth

in recessions), a forward-looking asset market and the effects of leverage. The

contagion model to be presented in the next section takes into account these balance-

sheet contagion linkages.

From a different methodological approach, Goldstein and Pauzner (2004) have

ascribed diversification of financial portfolio as an explanation of contagion likewise

this paper. They have emphasized that diversification -as a negative externality or a

social cost- generate positive correlation between returns and have facilitated the

transmission of negative shocks –through wealth effect- across countries. They have

concluded that the overall welfare effect of diversification is ambiguous instead of an

unambiguous positive welfare outcomes commonly obtained in the finance literature.

Finally, the main difference with the model proposed in this paper is the nature of

shocks (based on a self-fulfilling origin).

The model of financial networks proposed by Leitner (2005) is characterized by

linkages that spread contagion and private bailouts induced from lucky agents to shock

affected agents. He has argued that those bailouts` motivations are the benefits from

the mutual insurance. On the other hand, linkages may collapse in a contagion



Page 24 of 48

situation. From a critical point of view, it could be affirmed that the optimal risk-

sharing and insurance between banks is not possible owing to the fact that contagion

crises could produce a privately uninsurable extreme event.

From an empirical point of view, Nier et al. (2008) also has used network models to

investigate how systemic risk is affected by the structure of the financial system. They

have built a banking system using nodes (banks) connected by directed linkages

(interbank exposures) with a certain predefined probability. The ability to resist losses

is determined by the weight of these linkages to flow from one bank to another. The

authors have found, amongst other conclusions, that: a) the better capitalised banks

are, the more flexible the system is against contagious crisis in a non-linear way, b) the

degree of connectivity is initially increasing related with the contagion effect but, after

a certain value, it improves the ability of the banking system to absorb shocks, and c)

the quantity of banking debts tends to increase the risk of contagion default.

Nonetheless, that model is not taking into account possible negative systemic effect of

endogenous connectivity, the Central Bank strategic influence on individual incentives

and its probable endogenous effect on leverage, as the current paper does.

4.2 The financial theory: a short review of Mean-Variance and the Optimal

Financial Structure theories.

A short refresh of some main aspects of the Mean-Variance and Optimal Financial

Structure theories will be made in this section to be used in the construction of the next

model.

On the one hand, the starting point of the modern Corporate Finance theory could be

indicated by the publication of the Modigliani and Miller’s (1958) paper that has

established that: a) the firm valuation is independent of its financial structure

(Proposition I), b) leveraged firms are not more economically valuable than their

unlevered counterparts (Proposition II) and c) investment decisions of firms are not

affected by the instruments (equity or debt ) used to finance those (Proposition III).

However, Modigliani and Miller (1963) have introduced an important correction to

their previous paper. The corrected version has shown that leverage has increased the



Page 25 of 48

after-tax income and, then debt is more desirable than equity. This advantage has been

explained by the deductibility of interest payments of debt. Some years later, Hamada

(1969) has re-derived the three Modigliani and Miller’s propositions using a mean-

standard deviation approach in a market equilibrium context. Hamada has established

that the rate of return required by risk-averse investors increases linearly with firm's

leverage (debt-equity ratio).

Furthermore, Modigliani and Miller (1963) have indicated that the lenders' reluctance

to borrow after a determined leverage level of the firms and the firm's decision to have

a reserve to “ preserve flexibility” (Modigliani and Miller, 1963, p. 442) were the

reasons why there was a factual limit for the previously commented advantage of

financing investments with debt. However, the theoretical issue remained unclear and

open. Several authors have treated this puzzle, such as Baxter (1967), Stiglitz (1972)

and Scott (1976), who have attributed that empirical limit to the risk of ruin or

bankruptcy costs resulting as a counterbalance danger of increasing debt.

Summarizing, considering the absolute advantage of debt by deductible tax interests

and, alternatively, the disadvantage of debt endorsed to bankruptcy costs and other

costs, the modern Optimal Financial Structure theory appear to be dominated by the

called “Trade-Off” theory. This latter theory was tested and considered empirically

successful by Ju et al. (2005).

On the other hand, Markowitz (1952, 1959) has shown that the Mean-Variance

approach to choose the best financial portfolio has implied both, diversification

behaviour and the maximization of the expected return. That is, using Mean-Variance

decision rule, in any case a non-diversified portfolio will be preferred to a diversified

one by a rational agent. Mean-Variance decision rule will be assumed to be used by

decision makers in the following contagion model.

Several papers have improved the technicalities and the testable results of Markowitz’s

model without changing substantially its most important lessons. Tobin (1958) has

demonstrated that the risk-averse investor’s indifference curves are concave upward,

when expected return is plotted in the vertical axis and standard deviation on the

horizontal axis. Whitmore (1970) has presented an exact mathematical relationship



Page 26 of 48

between the number of securities in a portfolio and the reduction in portfolio

dispersion. Elton and Gruber (1977) have obtained an analytical solution (exact and

approximated formulas) to determine the optimal portfolio size with respect to risk

reduction. Epps (1981) has shown that the mean-variance model with constant risk

aversion and other similar models of market equilibrium derived from that, are

consistent with rational choice axioms only if both preferences and the stochastic

behaviour of asset returns are very narrowly specified (negative exponential and

normal).

Some conceptual and qualitative aspects were added by Sharpe (1964) and Lintner

(1965), who have developed a model to quantify risk in asset pricing, commonly called

Capital Asset Pricing Model (CAPM). The central idea is that investors compare the

assets’ expected returns to their inherent risks through a concept of risk-adjusted return

in a general equilibrium model. With respect to risk, two type of risk are underlying in

CAPM, the systematic risk and unsystematic risk. While the first one can be

diversified, the second is not possible to be eliminated by diversification.

The contagion model that will be presented in the next section uses an assumption

based on Markowitz’s explanation of microeconomic motivations to choose a full-

degree portfolio diversification. At the same time, it will be assumed that the

diversification, financial integration and interconnectedness are synonyms. It view is

closely related to Wagner (2010), who has shown that the risk diversification of

financial institutions can be undesirable because it increases the probability of a

systemic crisis. Particularly, this adverse effect of diversification is produced not by

contagion but because it makes a joint failure more likely. Moreover, the author has

said that diversification would amplify that problem when contagion is permitted.

5 A model of the extreme events with endogenous probability: a rational

explanation of a financial contagion

It has been argued that the positive correlation can produce the uninsurable feature of

extreme events (see Literature Review I). At this regard, in some circumstances the

government will play the IOLR role trying to avoid that an ex-ante insurable small iid

shock becomes an endogenously produced, ex-post positively correlated and



Page 27 of 48

uninsurable extreme event. However, it also was considered before that the certainty of

the government playing the IOLR role is a strong assumption.

For this reason, a situation where the probability of IOLR is positive but lower than

one and influenced by an endogenous process will be analysed.

5.1 The Model II

This model has assumed that the government’s decision to bail out uninsured agents

will be determined by the probability that a small, exogenous and ex-ante insurable iid

shock become an ex-post large-effect uninsurable shock (an extreme event), affecting

the vast majority of agents in a positive correlated way. Specifically, the government’s

decision rule to be an IOLR is given by a probability threshold parameter8. That is,

when that parameter is lower, the probability of the government to play the IOLR role

will be higher.

It is also assumed that the probability threshold parameter is determined by the average

degree of diversification of the agents and by the average degree of agents’ financial

leverage or financial fragility (debt-equity ratio) in the economy. The average degree

of diversification and the average financial fragility ratio could be interpreted as the

speed of spreading and the degree of susceptibility to infections of an epidemic,

respectively, by analogy with the epidemiology theory.

The first component of the latter proposed function, the average degree of

diversification, will be determined by the individual`s rational financial incentives to

full-diversification. The diversification is carried out by agents buying other agents`

debt, being it the contagion's propagation mechanism called “balance-sheet effects”

(Kiyotaki and Moore, 2002).

It is also understood that the rational full-diversification will produce the maximum

financial “connection” and potential positive correlation between agents. The

maximum average degree of diversification will be determined exogenously by

8This threshold parameter should be differently determined if the extreme event has an exogenous

probability.



Page 28 of 48

financial architecture or financial regulation (e.g. degree of financial liberalization of

the economy, tightness of credit regulations). This first assumption about the dynamics

of the agents` behavior is set out in the following Lemma 1.

Lemma 1. Rational economic agents will achieve financial full-diversification, given

exogenous constraints.

Proof. It has been proven by Markowitz (1959, p. 105).

The second component to determine the threshold probability parameter is the average

debt(stock)/equity(flow) ratio that indicates the average degree of financial fragility of

agents to shocks on their equity (flow)9. It follows that individual decision makers will

choose the optimal capital structure (relationship between debt and equity) taking into

consideration the tax benefits of debt (Modigliani and Miller, 1963) and the costs of

debt, which is represented by the probability of bankruptcy (Baxter, 1967; Stiglitz,

1972). The usually called “Trade-Off theory” has determined that debt-equity

equilibrium will be given by both counterbalance factors.

The latter assumed agents’ behaviour has appeared coincident with the mainstream

financial theory. However, the equilibrium produced by a stylised dynamic version of

the model might be surprising in comparison with the most common results in the

financial contagion literature.

At this respect, the latter commented optimal capital structure decision will be affected

by the assumed agents’ knowledge on the positive probability (different from zero)

that government will play the IOLR role. Given that diversification will be

increasingly higher each period of time, the systemic risk will be also higher (Wagner,

2010). If the systemic risk is increasing, the probability that the government play the

IOLR role will increase, decreasing the individual bankruptcy risk, which will give

incentives to agents to take a higher leverage (debt-equity ratio). The latter argument is

9This mean of financial fragility is close to Minsky’s FIH. He has argued that “For each economic unit,

the liabilities on its balance sheet determine a time series of prior payment commitments, even as the

assets generate a time series of conjectured cash receipts.”(Minsky, 1992, p.2). In the proposed model,

the conjectured cash receipts are assumed to be affected by iid exogenous shocks.



Page 29 of 48

close to the idea of time inconsistency10

policy of “too many to fail”, where a regulator

(Central Bank) is induced by a failure of a large number of banks together to bail out

as an ex-post optimal remedy (Acharya and Yorulmazer, 2007). Additionally, a similar

idea is developed by Farhi and Tirole (2009).

Thus, the government increasingly higher probability to be an IOLR will decrease the

individual risk of bankruptcy, affecting the rational incentives to choose an optimal

capital structure, which will increase the benefits of a higher debt-equity ratio. Thus,

agents will have a higher debt-equity ratio -a more financially fragile situation- and

consequently, the probability threshold parameter of the economy will be lower again,

increasing the probability of contagion. That is, a smaller iid exogenous shock will be

necessary to generate a financial contagion under the new parameter values. At the

same time, it will increase the probability that the government plays the IOLR role,

raising the benefits of a higher debt-equity ratio. This endogenous process will

continue on and on until the Nash equilibrium, characterized by the maximum debt-

equity ratio and the maximum achievable average degree of diversification. This

second assumption about the dynamics of the agents’ behavior is set out in the

following Lemma 2 and Figure N°1.

Lemma 2. Rational economic agents will achieve an optimal debt-equity ratio, given a

deductible tax on debt interests, a positive bankruptcy probability and positive

bankruptcy costs (*). It has been assumed that a higher probability of the government

playing the IOLR role when the agents` portfolio diversification is higher will produce

a lower risk of individual bankruptcy (**). This lower risk of individual bankruptcy

will affect the rational incentives to choose an optimal capital structure, resulting in a

debt-equity ratio biased towards a higher optimum level (***).

Proof. (*) It has been proven by Modigliani and Miller (1963), Baxter (1967), Stiglitz

(1972) and Scott (1976). (**) Similar idea has been proven by Wagner (2010),

Acharya and Yorulmazer (2007) and Farhi and Tirole (2009). (***) It has been proven

by Baxter (1967), Stiglitz (1972) and Scott (1976).

10It is because the ex-post optimal bailout is not ex-ante optimal.





Page 31 of 48

Figure N°2. A summary of the contagion model`s main building aspects.

Figure N°3. A suggested contagion model

Max. Diversification***

Systemic crisis more probable**

Government IOLR’s action

more probable

The unsystematic risk decreases

and then, the prob. of individual

bankruptcy decreases

D/E increases (by “trade-

off” optimal capital

structure theory)

p* decreases

#

Threshold prob. parameter (p*)

is determined by:

1. Degreeof diversification

2. Financialfragility(D/E)

Nash Equilibriumis:

1’. Full diversification

2’. Max. FinancialFragility

Lowest threshold prob.

parameter (p*) and largest

prob. of financial contagion

** By Wagner (2010). ***By Lemma 1. # The referedred circleis the dynamicprocess described by Lemma 2.

Shocks

1) There is a threshold parameter under which government will be an IOLR trying to

avoid that an ex-ante iid insurable shock becomes an endogenously produced,positively correlated and uninsurable extreme event.

2) Threshold parameter determinants:

a. The average degree of diversification of the agents (“speed of spreading of

an infection”). More diversification involves a lower threshold parameter

(See Fig. N°1).

b. The average degree of agents’ financial leverage or financial fragility, that

is, the debt-equity ratio (“degree of susceptibility to infections”). More

leverage implies a lower threshold parameter (See Fig. N°1).

3) Contagion’s propagation mechanism:

a. It is given by the degree of diversification -“connection”- that has permitted

the positive correlation of the shock between agents.

i. The diversification is carried out by agents buying other agents`

debt.

ii. The agents` equity is hit by iid shocks and “balance sheet effects”

are produced.

4) The dynamics and the equilibrium: Lemma 1, Lemma 2 and Conjecture.



Page 32 of 48

5.2 The Model II: comments and limitations

Generally speaking, the dynamics of the model could be thought by analogy to an

epidemic contagion model. However, it should be underlined that the idea has been to

make a model based on rational behavior.

The contagion model described before has shown that the physical feature of the

network economy, constraining the degree of potential financial diversification ability,

is a key aspect to determine the probability of the government to play the IOLR role

and the financial fragility of agents11

.

Broadly speaking, the current model could be understood as a micro-founded and

extended version of Minsky’s FIH model (Minsky, 1975, 1992), considering

exogenous shocks. In fact, the similarity is that in both models an endogenous process

carries the economy to a Nash Equilibrium that is socially non-Pareto optimum. The

main difference is that at the Minsky’s model the crisis is not exogenously motivated

as it is in the present model.

The interpretation of financial fragility in the suggested model must be clearly

understood because it is particularly different than several financial contagion models

in the literature. One of them, Lagunoff and Schreft (1998), has characterized the

financial fragility in terms of severity of the resulting crisis. That interpretation has

shown the financial fragility as an ex-post concept or an effect more than a cause of

contagion crisis. That is, those types of models have considered the financial fragility

not as a determinant of crisis but a consequence of that. On the contrary, the proposed

model has considered an ex-ante approach of the “financial fragility” concept. That is,

financial fragility is a determinant of a contagion crisis not a consequence of it.

In addition, Ibid. have pointed out that “Economies with an insurance equilibrium

should be less fragile than those without because the insurance essentially allows

complete diversification” (Ibid., p. 39). It can be affirmed that the latter “insurance

11Many empirical papers of network theory have studied the relationship between different structures of

networks. For example, Pautasso and Jeger (2007) have shown that there are key parameters influencing

the probability threshold point, in the context of small-size directed networks.



Page 33 of 48

equilibrium” is not defining fragility because the causality direction may be wrong.

From the point of view of the current model implications, financial fragility defines the

Nash Equilibrium. In addition, the same definition of financial fragility used by Ibid.

has shown that an ex-ante small shock generating financial contagion produces an ex-

post large-scale, positively correlated effect (extreme event). Thus, it is usually

uninsurable by private firms.

Another difference with several papers in the literature is that the vast majority of them

have interpreted the degree of interconnectedness as given exogenously and not

influenced by the degree of financial diversification. The current dissertation’s

contagion model has assumed that both concepts are synonyms. Paradoxically,

although the model has assumed that diversification is a rational and optimal strategy

for individual agents, it has carried the economy to a non-Pareto optimum Nash

Equilibrium. The reason is that while diversification improves individual sharing risk,

it also increases the degree of interconnection and positive correlation of shocks effects

between the agents’ financial portfolios.

Last but not least, it could be useful to underline that the most important point of the

endogenous process determining optimal financial structure, and then, the financial

fragility, is the initial positive probability that the government will play the IOLR role

to avoid the large-effects of an extreme event by assumption. It can be considered a

logic assumption –and not excessively strong- if an extreme event is possible, with

small but positive probability.

6. Conclusion and further research

This dissertation has studied anomalous or abnormal circumstances –extreme events-

in decisions under risk and uncertainty using common methodological tools from the

rational paradigm of the economic science.

Two particular puzzles were treated by this dissertation about extreme events.

It was argued that when an extreme event with exogenous probability happens, the

government bailout to uninsured agents could be sometimes strictly necessary.



Page 34 of 48

Essentially, the first model has shown that rational agents do not buy private insurance

against extreme events (with exogenous probability) when is common knowledge that

the government will play the IOLR role for certain in the case of large-effects and

positive perfect correlated shocks.

The contagion model suggested in this paper not only has shown the relevance of some

moral hazard costs of the government IOLR role in the presence of small iid

exogenous shocks potentially becoming in an uninsurable extreme event (systemic

crisis) but it principally has described the rational individual incentives mechanism

that carried to a Nash equilibrium that is not Pareto optimum. The model’s main

aspects are two. Firstly, the endogenous process by which individuals with rational

incentives, looking for an optimal financial structure, performs the economy to the

worst financially fragile situation. Secondly, the paradoxical contraindication of full-

diversification, while it is individually rational, it also increases the systemic risk.

Summarizing, the main contributions of this dissertations are several. Firstly, it has

offered a simple explanation, not based on market failures, of the “extreme event (with

exogenous probability) puzzle”. Secondly, an original extreme events’ classification

methodology was presented. Thirdly, a new financial contagion model was built based

on the idea that an extreme event could be produced endogenously by the interaction

of the economic structure, individual and rational incentives, the government action as

an IOLR and ex-ante iid shocks. Finally, the latter model has underlined the

paradoxical negative welfare effect of individual financial full-diversification that

together with financial fragility and Government playing the IOLR role, get worse the

systemic risk.

There are several topics for further research in this work. With respect to the model of

extreme events with exogenous probability, one improvement could be achieved

through the analysis of the case when the Government's IOLR role is not assumed for

certain but the shock is still an extreme event.

In relation to the financial contagion model proposed, more mathematical

technicalities, for instance a better mathematical dynamic modeling, could be

necessary to enrich this schematic and stylized model.



Page 35 of 48

Particularly, some aspects of the suggested model should be analysed in depth.

First, the role of diversification on the systemic risk remains controversial in the

literature.

Second, the trade-off between moral hazard costs and systemic crisis costs and the

decision that motivates an IOLR action by the government is also unclear. At this

respect, an evaluation of the importance of those costs is still necessary. For instance,

it may be suggested that the stability versus efficiency trade-off could be expressed in

terms of the statistical analogy of the Type I error and the Type II error.

Third, a more focused discussion about the causes and effects of financial contagion

might be also important.

To conclude, it might be affirmed that these abnormal situations should be studied in

depth in economics. An analogy supporting the latter statement may be suggestive.

Would you take a flight that in the ticket said: “this airplane is only prepared to normal

weather”?



Page 36 of 48

7. Appendix

In this Appendix, an alternative more complete and complex model of extreme events

with exogenous probability will be offered. The main difference between the

simplified model exposed in the Chapter 3.1 and the present model is that the former is

only considering two scenarios, the extreme event occurs or nothing happens. In the

present case, there are three scenarios. They are: the shock occurs or not, and when it

occurs that could be an insurable (iid shocks) or an uninsurable shock (an extreme

event with exogenous probability).

This alternative model is a finite, two period’s sequential game, with three players:

Customers or Firms, Nature and Government. There is no transaction cost. The

information is symmetric but incomplete. All agents are rational. No discount factor is

assumed.

The action of the Nature is random and exogenously determined, with three options in

the second period: a negative and uninsurable extreme event (Li and Lj are positive

perfectly correlated) or an insurable shock (iid); the shock does not occur.

The economy is hit by ex-ante iid shocks. By definition, if an ex-ante iid shock

becomes an ex-post extreme event, all agents in the model know by common

knowledge that the Government will play the IOLR role assisting agents with not

enough or without insurance against that. The government budget is in equilibrium

while it collects taxes in the first period to finance the contingency action as an IOLR

when the Nature to reveal the bad state of the world in the second period.

Customers or firms are risk-averse and choose to buy or not full-private insurance

against extreme events in the first period. They have three options at the first period:

anybody buys full-private-insurance ( 0δ = ), some agents buy full-private-insurance

( 0 1δ ⟨ ⟨ ) and all agents buy full-private-insurance ( 1δ = )12.

The sequence of the game is as follows:

12This problem is analogous that the case of only one representative agent choosing between different

percentages of insurance.



Page 37 of 48

I. At the first period, customers or firms decide if: anybody buys full-private-

insurance ( 0δ = ), some agents buy full-private-insurance ( 0 1δ ⟨ ⟨ ) and all

agents buy full-private-insurance ( 1δ =

).

II. At the second period, the Nature decides the state of the world. That is, the

shock occurs and it could be ex-post a privately insurable shock or an

uninsurable extreme event –where the Government pay transfers to agents

with not enough or without insurance-; or the shock does not occur.

Figure N°4. The sequential game for insurance demand against extreme events

( . ( / ( ))1

( ))i

u w q T n mi ii E

b q Li i i

π

ε

− − +

+ + − +

1( .

( / ( ) )

i i iu w q E

T n m

π −

− +

( ( / ( ))1

( ))

u w T n m j j E

b L j j jε

− +

+ − +

1(

( / ( ) )

j ju w E

T n m

− +

p

p

1 p−

1 p−

[ ]1. ( / ( ) )

i iw q T n mπ − − +

1 δ −

o r

C

F

a n d

N

G

δ

1 / ( )

jw T n m − +

Where,

w : wealth of customers or firms.

iu : utility function of “i” agents. i: 1,…,m.

iu ’(.) ⟩ 0,

iu ’’(.) ⟨ 0,

iu ’’’(.) ⟨ 0.

ju : utility function of “j” agents. j: 1,…,m.

ju ’(.) ⟩ 0,

ju ’’(.) ⟨ 0,

ju ’’’(.) ⟨ 0.

iu ≡ j

u



Page 38 of 48

iqπ : cost of full-private-insurance.

iq : private insurer firms pay “q” unit of money if the event occurs.

π : premium per unit of money of coverage.

L : total damage of extreme event.

- i L : damage of “i” agents.

- j

L : damage of “j” agents.

ib or jb : Governmental bailout (transfer) that covers the losses produced by the

extreme event. For “i” agents it could be interpreted as a re-insurance and for “j”

agents it could be simply thought as a transfer by Government play the IOLR (similar

to the Good Samaritan`s case).

T : tax collection

i jT q q= + (Government covers transfers with tax collection).

The additional assumptions are:

1) If ( / ) 1is jt Cov ε ε = , t s= , j jb L= and i i ib q L+ =

If ( / ) 0is jt Cov ε ε = , t s≠ , 0 jb = and 0ib =

2) The agents have common knowledge about the relation between the value of

covariance parameter and the probability of Government being an IOLR. Particularly,

agents know that the probability of Government to play the IOLR role is one (certain

case) when the correlation of events is one and, zero otherwise.

3) It is also supposed that statistical properties of shocks have transformed the two

different expected utility equations in a seemingly unrelated regression model, which

is formally:

3.1) ( / ) ( ) 0l l l E L E ε ε = = where l is i or j

3.2),

( / ) ( / ) 0l l l l lVar L Var Lε ε ε = = = Σ where Σ is a positive

definite NxN matrix.

3.3) / (0, )l l L N ε Σ∼

3.4) ( / )i j ij nCov I ε ε σ = ,



Page 39 of 48

( / ) 0is jt Cov ε ε = , t s≠

( / )is jt ijCov ε ε σ = , t s=

The latter option of the assumption 3.4 has suggested that disturbance terms may be

contemporaneously correlated when the shock is affecting both types of agents

(insured and uninsured) simultaneously.

Proposition 1’. An economy with risk-averse agents, where the government guarantees

the insurance to agents without (or not enough) full-private insurance to cover with

certain an ex–post known extreme event (with covariance one), financing it with ex-

ante tax collection, will produce that no one buys insurance against extreme events. At

the same time, agents will buy full private insurance against iid negative shocks, if the

expected shock is large enough and the premium is sufficiently small.

Proof.

It will be “rational” to remain uninsured if and only if:

The expected utility of uninsured agents:

( )1

1 1

/ ( )

. ( / ( ) ( )) (1 ). ( / ( ))

j

j j j j j j j

w T m n

p E u w T m n b L p E u w T m nε

− + +

− + + − + + − − + (1)

is greater than the expected utility of insured agents:

( )

( ) ( )

1

1 1

/ ( )

. ( / ( ) ( ) (1 ). ( / ( ) )

i i

i i i i i i i i i i

w q T m n

p E u w q T m n q b L p E u w q T m n

π

π ε π

− − + +

− − + + + − + + − − − + (2)

Where 1 1i jw w=

a) In the particular case when the covariance is 1, the event becomes an extreme

event and, then, uninsurable by private firms. That is, p=1, j j j

b L ε = + and

i i i ib q L ε + = + ,

Proof that (1)>(2) when the case is a):



Page 40 of 48

At the 2º period:

[ ] [ ]1 1

1 1

. ( ( / ( ))) (1 ). ( ( / ( )))

. ( ( / ( ))) (1 ). ( ( / ( )))

i i i i i i

j j j j

p E u w q T m n p E u w q T m n

p E u w T m n p E u w T m n

π π − − + + − − − + ⟨

− + + − − +

(3)

At the 1º period:

( ) ( )1 1 / ( ) / ( )i i jw q T m n w T m nπ − − + ⟨ − + (4)

While (3) and (4) are true, no one agent will buy insurance against an ex-post extreme

event considered by a).

b) In other cases, when the covariance is 0, the shocks are ex-ante and ex-post iid

and, then, insurable by private firms. That is, p=1, 0 jb = , 0

ib = ,

i i iq L ε = +

Then, it will be “rational” to buy insurance against iid shocks if and only if (2)>(1) in

the case b).

Proof that (2)>(1) when the case is b):

At the 2º period:

[ ] [ ]1 1

1 1

. ( ( / ( ))) (1 ). ( ( / ( )))

. ( ( / ( )) ( )) (1 ). ( ( / ( )))

i i i i i i

j j j j j j

p E u w q T m n p E u w q T m n

p E u w T m n L p E u w T m n

π π

ε

− − + + − − − + ⟩

− + − + + − − + (5)

Then, the LHS and the RHS`s first terms difference is13

:

( ) j j i L qε π + −

While the LHS and the RHS`s second terms difference is:

iqπ −

At the 1º period:

( ) ( )1 1 / ( ) / ( )i i jw q T m n w T m nπ − − + ⟨ − + (6)

13LHS: Left-Hand-Side. RHS: Right-Hand-Side.



Page 41 of 48

The difference between LHS and RHS is:

iqπ −

Thus, (2) is greater than (1) when the case is b) if and only if:

( ) 2 j j i i L q qε π π + − ⟩ (7)

If condition (7) is true, all agents will buy insurance against iid shocks.

Summarizing, two principal lessons can be learned from the latter proofs:

a. No one risk-adverse agent will buy insurance against extreme events because

they know that if an ex-ante negative iid shock become ex-post an extreme

event (explained by contemporaneous correlation) they will be privately

uninsurable events and Government will play the IOLR role with probability

one.

b. However, when the negative shock is iid, the covariance parameter is zero14

, all

risk-adverse agents buy insurance if the expected shock is large enough and the

premium is sufficiently small.

Last but not least, it is necessary to underline that agents are using Expected Utility

theory and rationality to decide not to buy insurance against ex-post extreme events

(events which covariance between agents is 1) because they know (ex-ante by common

knowledge) that the government will play the IOLR role with certainty under that

circumstance.

14In this case, there is not contemporaneous correlation.



Page 42 of 48

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