Moral Hazard in Insurance Markets Optimality, Risk ...

Moral Hazard in Insurance Markets Optimality, Risk and Preferences in the First Best Case Kristian Sundström Department of Economics Lund University P.O. Box 7082 SE-220 07 Lund Sweden

Licentiate Dissertation May, 2003

CONTENTS

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Contents Acknowledgements 1 Introduction 1.1 Overview . . . . . . . . . . . . . . . . . . . 1.2 Areas of Speci…c Interest . . . . . . . . . . Fundamentals of Risk and Uncertainty . . Insurance with Perfect Information . . . . Risk and Informational Imperfections . . . 1.3 Environment and Objectives . . . . . . . . The General Environment . . . . . . . . . Objectives and Organization of the Paper

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Economy A Traditional Moral Hazard Model for Insurance Agents and Institutions . . . . . . . . . . . . . . . Action and Consumption . . . . . . . . . . . . . . Action . . . . . . . . . . . . . . . . . . . . . . . . Net consumption . . . . . . . . . . . . . . . . . . Initial endowment . . . . . . . . . . . . . . . . . . Gross consumption and insurance contract . . . . Gross consumption and action . . . . . . . . . . . 2.4 The Prevention Function . . . . . . . . . . . . . . 2.5 Utility and Preferences . . . . . . . . . . . . . . . Utility function . . . . . . . . . . . . . . . . . . . Monotonicity in care . . . . . . . . . . . . . . . . The uniform consumption set . . . . . . . . . . . U is not quasi-concave . . . . . . . . . . . . . . . 2.6 The Economy . . . . . . . . . . . . . . . . . . . .

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3 Market Balance and First Best Action 3.1 Feasible Actions . . . . . . . . . . . . . . . . . . Market balance . . . . . . . . . . . . . . . . . . The feasible set . . . . . . . . . . . . . . . . . . 3.2 Interpretation of the Economy . . . . . . . . . . An example illustrating this interpretation . . . 3.3 The First Best Action . . . . . . . . . . . . . . Existence and uniqueness of the …rst best action Properties of the …rst best action . . . . . . . .

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4 Risk and Expected Utility 4.1 The General Concepts . . . . . . . . . . . . . . . A simple example . . . . . . . . . . . . . . . . . . 4.2 The Concepts in the Moral Hazard Environment The function Fc¹ . . . . . . . . . . . . . . . . . . . A curve for a …xed x . . . . . . . . . . . . . . . . The family of curves of Fc¹ . . . . . . . . . . . . . The fold curve . . . . . . . . . . . . . . . . . . . . The envelope . . . . . . . . . . . . . . . . . . . . 4.3 Analysis: Results, Discussion and Example . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . Example of the envelope and of a …rst best action 5 Conclusions

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ACKNOWLEDGEMENTS

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Acknowledgements A number of people should get credit for contributing to this dissertation. The greatest gratitude goes to my supervisor, Anders Borglin, for his support and for sharing his extensive knowledge of economics for the great bene…t of the entire dissertation. Also, I would like to express my appreciation of the many constructive suggestions made by the participants of the departmental weekly seminars: Anders Bäckstrand, Joakim Ekstrand, Helen Forslind and Ola Jönsson. The …nancial support received from Trygg-Hansas Forskningsstiftelse, Svenska Försäkringsföreningen and the Crafoord Foundation is also gratefully acknowledged. Lund, May 7, 2003 Kristian Sundström

1 INTRODUCTION

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Introduction

The objective of this section is to provide an introduction to the analysis carried out in this paper. To this end we …rst present a cursory outline of those areas of economics which are considered germane to this analysis, with an emphasis on elucidating the interconnection between them. Secondly, we single out those areas which are regarded to be particularly relevant to this paper, areas which will accordingly be accounted for in greater detail, including an overview of the pertinent literature. Thirdly, with the title of the dissertation as a frame of reference, the general environment as well as the objectives will be comprehensively reviewed, winding up with an overview of how the paper has been organized.

1.1

Overview

The title of this dissertation contains the concepts of risk, moral hazard and insurance, three terms that indicate three diverse but interconnected areas of economics with a relevance to this paper: the economics of risk and uncertainty, the economics of information and insurance economics. The economics of risk and uncertainty analyzes the behaviour of economic actors who face a risky or uncertain prospect.1 The uncertainty occurs, since the future cannot be predicted with perfect accuracy. There might be several possible future states of the world, and it is assumed that an agent does not know for certain which of these states that will actually occur. However, it is a common supposition that the fundamental structure of the uncertainty is known, so that there is a generally prevailing cognizance among the economic agents as to the possible outcomes, as well as (individually or collectively formed) estimations of the probability with which each one of these outcomes 1

There have been attempts to distinguish between the terms risk and uncertainty in the economic literature. One illustrious, if disputed, distinction was made by Knight (1921), p 20, Ch 7, who argued that in a risky prospect the probabilities of the di¤erent states can be mathematically speci…ed by the economic actors, which is not the case when the prospect is uncertain. For the purpose of this paper, however, the two concepts will be used interchangeably.

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will take place.2 In traditional economic theory, correct and full information has been presumed to be unrestrainedly accessible, so that economic agents have been able to become perfectly informed at no cost. To appreciate and to be able to analyze the value of information, one has to compromise with this assumption, a compromise which lays the foundation of the economics of information. The research carried out within this discipline thus conveys economic behaviour when various deviations from the case of perfect knowledge are considered, including scarce and/or asymmetrically allocated information regarding economic agents and their properties.3 The economics of insurance, …nally, directs the attention towards means (including general requirements and institutional arrangements) of obtaining Pareto improvements by transferring risk from economic agents who do not appreciate risk, to parties that for some reason are relatively more indi¤erent to risk.4;5 In Figure 1 the relations between these areas of research are illustrated by three circles.6 Since a transfer of risk presupposes a presence of risk, the small circle, symbolizing insurance economics, is completely included in the circle representing the economics of risk and uncertainty, while it intersects only partially with the circle illustrating the economics of information. The reason for the last relation is that informational de…ciencies are complications that might (but do not have to) be added in order to analyze insurance. 2

Surveys of the economics of risk and uncertainty are provided by, among others, Karni and Schmeidler (1991), Hirschleifer and Riley (1992), La¤ont (1989) and Machina (1987) at di¤erent levels of technical di¢culty. 3 For a general treatment of the economics of information, see for example Hirschleifer and Riley (1992). 4 Extensive coverages of the economic aspects of insurance have been carried out by Borch (1990), Dionne and Harrington (1992a), Dionne (1992 ) and Dionne (2000). 5 A more detailed discussion of the possible occasions for risk indi¤erence is presented in Section 1.3 6 It should be emphasized that this introductory discussion springs directly from the contents of this paper, so that the relations discussed here should not be considered exhaustive in any respect.

1 INTRODUCTION

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economics of risk and uncertainty

economics of information

insurance economics

Figure 1: The relation between the three relevant areas of economic research

1.2

Areas of Speci…c Interest

In Figure 2 we have used the elementary structure given in Figure 1 in order to select four relevant areas of research which will be further elaborated on below. Firstly, since the economics of risk and uncertainty plays such a predominant role in insurance, there will be a brief historical overview of the literature within this …eld, leading up to its formal incorporation of insurance. This is represented by intersection 1 in Figure 2, and will be discussed below under the heading Fundamentals of Risk and Uncertainty. Secondly, because the focal point of this paper is insurance economics, we will examine this discipline both with and without informational imperfections, which is illustrated, respectively, by intersections 2 and 3. Intersection 2 will be reviewed under the heading Insurance with Perfect Information below. Finally, as depicted by intersection 4, we want to pay some heed also to the observation that the concepts used to analyze insurance under informational imperfections generally carry over to other areas of economics as

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well. We will thus give a few examples of some of these other areas, although any extensive overview is far beyond the scope of this introduction. On account of this relatively limited attention, and also since intersections 3 and 4 produce analogous interpretations, they will be reviewed concurrently in the subsection Risk and Informational Imperfections. Fundamentals of Risk and Uncertainty

Insurance with Perfect Information

Intersection 1

Intersection 2

Risk and Informational Imperfections

Intersection 3

Intersection 4

Figure 2: Four areas of speci…c interest

Fundamentals of Risk and Uncertainty Early economists generally accepted the idea that random ventures were to be valued according to the expected value generated by them. By the St Petersburg paradox, which was posed in 1713, some of the unintuitive features of such an approach were displayed, and the attempt in Bernouilli (1738) to …nd a solution to the paradox introduced two revolutionary concepts in this area of research: the notion of a diminishing marginal utility and that of expected utility. Surprisingly, these notions did not …nd their way into formal economic theory until von Neumann and Morgenstern (1944) managed, by means of axiomatization, to lay a rational foundation for choice under uncertainty in accordance with expected utility theory. Although heavily

1 INTRODUCTION

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criticized on several points, not least for the plausibility of the underlying axioms,7 this formalization led to an outburst of research in di¤erent areas of economics.8 Arguably the two most in‡uential subsequent developments in this …eld were led by Savage (1954), who derived the expected utility hypothesis using subjective probabilities,9;10 and by Arrow (1953) and Debreu (1959) who introduced the state-preference approach to uncertainty, which enabled a connection with general equilibrium theory without risk. Arrow (1953) considered the simple case of two dates with the choice of Nature revealed at the second date, an analysis which was extended by Debreu (1959) to the case of many dates and uncertainty unravelling gradually over time. The basic idea in these two contributions was inspired by insurance contracts. In a simple interpretation of the model each commodity could be traded at the …rst date for delivery at the same or later dates. The di¤erence from earlier models was that the commodity concept was reinterpreted, now including not only the type of good and the place of deliverance, but also the state of nature. Thus every good could be traded at the current date for delivery contingent upon each of the events at a later date. As is the case with an insurance contract, the price is paid at the current date, but delivery takes place only if a speci…c event occurs. To secure the sure delivery of a unit of a good at some date the agent has to buy, for delivery at that date, 7

The most illustrious challenge came from Allais (1953) who showed that the independence axiom, which was a consequence of the axioms laid out by von Neumann and Morgenstern, was a very bad indicator of how people actually made choices. Allais posed the poblem in such a way that very few would agree that people would act in accordance with the axiom. Thus the term Allais paradox. 8 On account of the sometimes rather heavy criticism of the axiomatization made by von Neumann and Morgenstern and their followers, alternatives have been put forward, including weighted expected utility (Chew and MacCrimmon (1979)), non-linear expected utility (Machina (1982)) and regret theory (Loomes and Sugden (1982)). 9 In von Neumann and Morgenstern (1944) it was assumed, in accordance with the classical view, that probabilities were given exogenously (or objectively) by Nature. 10 The result of Savage (1954) was improved and generalized upon by Anscombe and Aumann (1963) who simpli…ed the axiomatization, as well as allowing for both subjective and objective probabilities to be present in the same model.

1 INTRODUCTION

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as many contingent commodities as there are (elementary) events. The formal theory, as given by Debreu (1959), is relevant also to economies with uncertainty, since the generalization relates only to the interpretation of the commodity concept. Radner (1968) further extended these results and showed that it was possible to allow for di¤erences in information. The reinterpretation of the model made it clear that the assumption of a complete market for all commodities at the …rst date and the corresponding equilibrium concept, that of a complete contingency market (CCM), is far removed from reality. But it is useful as a starting point in order to study market institutions closer to reality. The further development has taken place in three directions. Firstly, general equilibrium theory was extended to the case where the possibilities to transfer income over time or between states are limited, which gave rise to a general equilibrium theory with incomplete markets (GEI). Secondly, the contributions of Arrow (1953) and Debreu (1959) had a large impact on the theory of …nance11 (Cf. Du¢e (1991) who considers Arrow (1953) as the starting point of the modern theory of …nance). Thirdly, the introduction of uncertainty pointed to many problems related to information and incentives in insurance markets. Underlying the economics of …nance and insurance is naturally an assumption that individual agents have an aversion towards risk. The concept of risk aversion has been discussed extensively since the 1950’s.12 Theories of how to estimate risk aversion were put forward by Arrow (1965) and Pratt (1964), and were improved and simpli…ed by Ross (1981). The measurement of risk aversion also requires methods of comparing levels of risk. Theories of methods for such comparisons were advanced in some famous articles by Rotschild and Stiglitz in the early 1970’s.13 11

See Hirschleifer (1965) for a famous application to investment theory, and Diamond (1967) and Radner (1968, 1972) for extensions to …nance and general equilibrium. 12 See for example Friedman and Savage (1948) and Markowitz (1952). 13 Rotschild and Stiglitz (1970, 1971)

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Insurance with Perfect Information Until some forty years ago, insurance was very rarely considered an economic discipline on its own, but was rather viewed either as a contingent good or discussed in relation to gambling.14;15 The evolution of insurance economics as a distinct area of research was initiated primarily by Borch and Arrow in the early 1960’s.16 In his most in‡uential article, using Arrow’s (1953) model of equilibrium under uncertainty, Borch (1962) shed light on both similarities and di¤erences between …nancial markets and insurance markets,17 by placing them both in a larger context. He showed that by applying certain sharing rules of the collective risks, where these shares were to correspond with the absolute risk-tolerance of each individual agent,18 a Pareto optimal risk allocation could be obtained by means of the …nancial markets.19 On the other hand, by using the insurance markets, the individual risks were possible to diver14

Illustrious examples are provided by Friedman and Savage (1948), Allais (1953) and Arrow (1953). 15 This overview of insurance economics has bene…tted extensively from the excellent surveys provided by Dionne and Harrington (1992b) and Loubergé (2000). 16 In Loubergé (2000), p 4, three additional articles with a durable bearing upon the evolution of insurance as an economic discipline are mentioned. First among these is Mossin (1968) in which the theory of demand for insurance is analyzed successfully for the …rst time. The second important contribution is Ehrlich and Becker (1972), in which risk prevention mechanisms are studied. Among other things, the authors coin the two now well-established concepts of self-protection and self-insurance. Self-protection refers to measures carried out by the potential insuree in order to decrease the probability of an accident, while self-insurance bears upon means to temper the economic e¤ects of an accident. Both concepts have implications when introducing informational asymmetries, although the authors did not take these problems under consideration. Thirdly, and …nally, Loubergé (2000) mentions Joskow (1973), who laid the foundation to analyzing insurance and insurance markets empirically. 17 As the title of his article suggests, the analysis carried out in Borch (1962) concerns reinsurance markets, but the framework is immediately applicable to the ordinary insurance industry. See Mo¤et (1979) for details. 18 An agent’s risk tolerance is the inverse of his risk aversion. 19 If utility functions meet certain criteria, the sharing rule becomes linear, a general result of which the Capital Asset Pricing Model (CAPM) is a special case.

1 INTRODUCTION

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sify away. This distinction between …nance and insurance, which was made for e¤ectively the …rst time, and which inspired research in many areas of both these disciplines, has induced Loubergé to acknowledge Borch’s (1962) contribution as providing ”the cornerstone of insurance economics”.20 In his seminal article ”Uncertainty and the Welfare Economics of Medical Care”,21 Arrow provided a second cornerstone of the subsequent research in the …eld of insurance economics. He showed, for the …rst time, that incomplete coverage is optimal given an actuarially underfair premium. Furthermore he proved the Pareto optimality of introducing coinsurance above a deductible, when both parties to the insurance contract exhibit risk aversion. A third and extensively in‡uential contribution of this paper was its inauguration of informational problems into the realm of insurance. Since this hybrid of economic disciplines is the subject of the next subsection, this contribution will be further considered in that context. The advancements subsequent to these pioneering achievements can be roughly classi…ed into three subcategories.22 In the …rst of these, optimal insurance and protection, there have been numerous attempts to resolve incongruences between theory and real world observations. For example it has been noted that insurees do not in general require partial coverage, although this is the theoretical result that would be expected.23 The undertaking to dispel this anomaly has included an alleged non-applicability of the theoretical assumptions on real world insurance markets.24 Also it has been noted, as a result of the growing interest in portfolio theory, that no risks should be analyzed in isolation, and that therefore, by taking this into consideration, the con‡ict might be resolved without altering the analytical framework.25 20

Loubergé (2000), p 5 Arrow (1963) 22 See Loubergé (2000), p 8 23 See for example Arrow (1963) and Mossin (1968). 24 For disparities concerning behaviour, and consequences of these for the optimality of partial coverage, see Razin (1976). 25 See Schlesinger and Doherty (1985) for an analysis using the correlation concept to measure the dependence between risks, and Dionne and Gollier (1992) for the use of more sophisticated measures of dependence. 21

1 INTRODUCTION

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A third development within the subcategory of optimal insurance has been to analyze choices of insurance using state-dependent expected utility functions,26 as well as explicitly non-expected ones.27 A second path of academic research has been devoted to empirical analyses of the insurance market structure. Building on the result of Joskow (1973), and bene…tting from increasingly more accurate and extensive data availability, these include the organizational structure of the insurance market,28 returns to scale in the insurance industry29 and insurance distribution systems.30 The third development in insurance economics, being also the focal point of this paper, is its formal incorporation of information problems. To this subject the entire next subsection will be devoted. Risk and Informational Imperfections As referred to above, the idea of individual risks being completely shifted was challenged by Arrow (1963). According to this article, one ground for the risk-shifting de…ciency, and theoretically perhaps the least intriguing, was the prevalence of transaction costs. The administration necessary to complete an insurance contract was not for free, and once this cost was transferred to the risk averse agents, these were no longer willing to insure completely. The other two reasons for incomplete risk-sharing, adverse selection and moral hazard, were due to prevailing informational asymmetries. To categorize these asymmetries one may use the time of contracting as a guideline. Either the asymmetry existed before the time the contract was agreed upon, or else it is anticipated to occur subsequently. By this classi…cation, adverse selection falls into the …rst category, while moral hazard belongs to the second. The two concepts will now be examined in turn. 26

See for example Arrow (1974) and Schlesinger (1984). See Machina (1995) 28 See for example Mayers and Smith (1988) 29 See Doherty (1981) 30 Cummins and VanDerhei (1979) 27

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Adverse Selection and Insurance Adverse selection in insurance markets occurs when the population is heterogenous as to the probability of having an accident. If the risk level of an agent is known to the insurer, we have a …rst best case of adverse selection,31 while with the risk level as a piece of private information32 we turn out with a second best case. In order to at least partially overcome the inimical e¤ects of adverse selection, and since a direct revelation is not possible in the second best case, the (unfavoured) parties naturally reside to indirect ways of unveiling as much as possible of the concealed information. Generally, measures taken in order to indirectly reveal information are called signals if they are initiated by the informed party, and screening if inaugurated by the relatively uninformed agents. It has proved di¢cult to interpret the signaling device in an insurance market context, and the research within this area has thus essentially concentrated on screening as the mitigating instrument. In order to screen the risk level of the insurees, the insurer o¤ers a menu of contracts, from which he lets the agents choose. If constructed properly, the high risks and the low risks will opt for di¤erent contracts, thereby indirectly revealing their level of risk by self-selection. Other means of alleviating the adverse selection problem have included experience rating and risk categorization. Experience rating implies, in a multi-period context, that whenever a claim is presented, there will be increases in subsequent premiums. The idea is that the more claims presented by an insured, the more likely he is a high risk individual, and so the insurance company can use this information to set the premium higher. Risk categorization refers to the use of mainly exogenous variables which have been proved to signi…cantly relate to the level of risk. One example is the use of 31

In the …rst best case where the risk levels are monitorable by the insurer, we have a situation where the contracts can be made contingent upon the risk types. This, in turn, implies that both types of agents can receive full insurance at the prices induced by their respective risk levels. 32 In this case adverse selection becomes synonymous with the concept of hidden knowledge.

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age and gender as classi…ers of risk for insurees in automobile insurance. There has been an extensive debate regarding the existence of an equilibrium in insurance markets under adverse selection. The polemic was initiated by Rotschild and Stiglitz (1976) who used a model with two types of insurees with di¤erent risk levels, and with an insurance company o¤ering as a screening device a menu of two di¤erent price-and-quantity policies.33 The …rst of these policies, intended for the high risk agents, provided full insurance at a high price, while the other policy, designed to appeal to the low risk agents, was characterized by a partial coverage at a low premium. By adopting an equilibrium concept according to which an insurance company’s menu of contracts is independent of the menus o¤ered by other …rms, Rotschild and Stiglitz (1976) proved that a pooling equilibrium, in which all insurees are offered the same contract, cannot exist. Furthermore, a separating equilibrium, in which di¤erent risk types are provided with di¤erent contracts, did not always exist either, depending on the share of high risk agents in the economy.34 If a separating equilibrium is indeed obtained, the authors showed that the low risks will su¤er a welfare loss, while the high risks will receive the same expected utility as in the full information case. Accordingly, the high risks, by their very existence in the market, always exert an externality upon the entire economy. To overcome the serious existence problem induced by adverse selection, disputers have tried to modify the equilibrium concept. Both Wilson (1977) and Riley (1979) introduced alternative de…nitions, in which the di¤erent menus of contracts on the market are interdependent.35 Thus an insurance company cannot present a menu of contracts without taking into consideration the possible reactions from other companies, which may withdraw unpro…table contracts or provide new, pro…table ones, depending on the equilibrium concept at hand. By either of these alterations, there will always 33

For a case with price-only policies, see for example Pauly (1974). More speci…cally, when there exists only a small share of high risk persons, a separating equilibrium may not exist. 35 Various other equilibrium concepts have also been proposed. For a survey of these, consult Cooper and Hayes (1987). 34

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exist an equilibrium36 independent of the share of high risks. In the realm of non-insurance economics, the adverse selection concept has found its way to various applications, including Akerlof’s (1970) famous analysis of the second hand car market, the manufacturing industry37 , the labour market,38 monopolist behaviour39 and bargaining theory.40 Moral Hazard and Insurance A second reason for only partial risk transfer in insurance markets is a phenomenon known as moral hazard. It occurs when an insuree has the option to choose an action, and when the level of this action somehow a¤ects the insurer’s welfare. If the action is possible to monitor by the insurer we have a …rst best case of moral hazard, while the second best case requires the level of precaution to be a piece of private information.41;42 Being a postcontractual informational asymmetry according to the above categorization, moral hazard research has generally focused on constructing optimal contracts, designed as to provide the privately informed individuals with incentives to act in accordance with the interest of the less informed ones. For this reason the moral hazard problem has often been posed in a more general principal-agent framework,43 a setting in which the welfare of a principal relies partially on the often unobservable behaviour of an agent. On account of its conformity with this principal-agent framework, moral hazard applications have burgeoned in areas far beyond insurance economics, including labour economics (the possibility of job shirking)44 and banking 36

In Wilson’s (1977) case there always exists a pooling equilibrium, while in the case of Riley (1979) the equilibrium is a separating one. 37 Grossman (1981) 38 Spence (1974) 39 Milgrom and Roberts (1982) 40 Rubinstein (1985) 41 A much more precise de…nition of moral hazard (including possible recurrences of what has just been presented), will be provided in Section 1.3, where the environment of the paper is presented. 42 The concept of hidden action is also used to describe this case. 43 See Ross (1973) and Grossman and Hart (1983) for famous references on the principalagent theory. 44 See for example the famous analysis of sharecropping in Stiglitz (1974)

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(receivers of funds may select di¤erent, unobservable, levels of investment).45 Most academic research on moral hazard has been devoted to the case where the action is taken before the state of the world is revealed, which is known as ex ante moral hazard.46 This type of model was mathematically formulated by Pauly (1974), Marshall (1976) and Shavell (1979). They used a simple two-period model, and proved that in the second best case, insurance reduces the incentives to take care. The optimal solution, a partial coverage, is thus a compromise between risk-sharing and incentives: in order to induce the insurees to take su¢cient care, they must bear some of the risk themselves. Shavell (1979) also proved that in spite of its detrimental e¤ects on the market, the presence of moral hazard cannot completely eliminate the gains from trade: in his model it is always better for a risk averse individual to enter into an insurance contract than to remain uninsured. In a number of articles, Stiglitz and Arnott have analyzed the prejudicial e¤ects of moral hazard upon market equilibrium.47 Building on the pioneering work of Helpman and La¤ont (1975), who were the …rst to identify non-convexities of indi¤erence curves under moral hazard, Stiglitz and Arnott proved that a competitive equilibrium will not exist in general. Also, neither of the two theorems of welfare economics apply in the presence of moral hazard, suggesting potential welfare improvements by governmental intervention. Research which has concentrated upon multi-period moral hazard models has not been able to conclude that the moral hazard problem can be resolved in general despite the use of experience rating schemes.48 One reason might be that in competitive insurance markets, the insured can switch to another 45

See for example Boyd, Chang and Bruce (1998) for a discussion of di¤erent moral hazard problems in banking. 46 For a reference on ex post moral hazard, where the action is taken after the realization of the state, see Townsend (1979). The most important application of ex post moral hazard is in medical insurance, where decisions made after the occurrence of the illness a¤ect the requested compensation. 47 Arnott and Stiglitz (1988, 1990, 1991) 48 See Rubinstein and Yaari (1983) for a treatment of the in…nite period case, and Winter (1992) for an analysis of moral hazard in a …nite period model.

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insurer, rendering any scheme di¢cult to e¤ect.49

1.3

Environment and Objectives

This section will commence with a delineation of the general environment of the paper, as speci…ed by the three terms of the title: insurance markets with moral hazard in the …rst best case. A description and an integration of these terms (and discussing possible implications stemming from this integration) will hence be the sole objective of this …rst part. The section will then conclude with a presentation of the aim and scope of the paper, which should be considered a natural exposition of the remaining three concepts of the title: optimality, risk and preferences. The General Environment We de…ne moral hazard as being present whenever one party to a transaction, or to a contract, has the opportunity of altering the probability of a speci…c outcome, a probability which matters to the other party, by choosing a certain amount of some activity.50 Tacitly understood is that, under particular conditions, there will be a tendency of the former to choose a level of his activity which is not optimal for the latter, given the contract or transaction agreed upon; hence the term moral hazard. Now we add three assumptions to this de…nition: the presence of risk, risk aversion of the party that carries out the activity and the fact that the activity itself is costly. If all these conditions, but not the following, are met, we have a …rst best case of moral hazard. If, however, we add the criterion that the level of the action is a piece of private information to the party e¤ecting it, we arrive at a second best case of moral hazard. As the title indicates, the analysis will be con…ned to an insurance market 49

See Loubergé (2000), p 13 The reader should be reminded that this is not a universally accepted de…nition of moral hazard. Some may dispute, for example, that there has to be informational asymmetries present in order to use the concept. For a good reference on what line to take to this disunity, see Kreps (1990), p 578 (footnote). 50

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environment. An insurance market should be perceived as a market in which two types of actors with contrasting relations to risk exist synchronously (see Figure 3). The …rst type, which is conventionally interpreted as a single agent, will be simultaneously risk averse and exposed to a risk, whereas the second type is alternatively risk neutral or not exposed to any risk. For simplicity we shall refer to a party which satis…es either of the last two requirements (risk neutrality or non-exposure) as being risk indi¤erent.

Type of actor

Relation to risk risk neutrality

risk indifferent party contract opportunity

risk avert party

or no risk exposure

exposed to risk

Figure 3: The insurance market environment The interpretation of the risk indi¤erent actor is not entirely evident. Either we could think of this actor as a risk neutral insurance company which is exposed to a risk, or as a continuum of agents for which the …nal outcome, if the probabilities of the states are given, will be known for a certainty. In either case the two parties have incentives to enter into a contract (or into some sort of agreement), with the purpose of directing risk away from the risk averse agents to the party which, for either of the two reasons above, is indi¤erent to the level of risk. The potential contents of such a contract will be discussed in a few paragraphs. First, however, we need to distinctly nail down the environment in which the parties reside. The implication of integrating moral hazard into the insurance market

1 INTRODUCTION

19

described above is that a representative agent will now select his level of care, a level which will a¤ect the probability of the occurrence of some future ’good’ and ’bad’ states. Speci…cally a low level of precaution is supposed to induce a high probability of a bad state to come about in the future, whereas a high level will decrease this probability. This will result in con‡icting interests between the parties, given an agreed-upon contract. To elucidate this actuality, we …rst need to pinpoint the contents of such a contract. An insurance contract under moral hazard has to detail at least three particulars: an income transfer (from the agent to the risk indi¤erent party to be e¤ected in all circumstances), a contingent payout (from the risk neutral party to the agent), and conditions under which last payout will be realized. Normally two such conditions are included in the contract. Firstly a speci…c state of the world must have occurred, and secondly the agent must have chosen the amount of the risk reducing activity contracted upon. The con‡icting interests now emerge because with a given contract the agent wants to minimize his expenditures (and taking care is costly), whereas a high (and costly) level of precaution will be favoured by the risk indi¤erent party. Our eventual measure to delimit the setting described above by synthesizing it with the …rst best case criterion will be completely decisive in resolving this con‡ict, as it makes the requisite of taking su¢cient care practicable to incorporate into the contract, the level of preventive action being accordingly supervisable. Thus if an agent has not taken an adequate amount of care, the contract will simply be rendered useless, regardless of the relevant state having occurred or not. Objectives and Organization of the Paper From the very outset, models analyzing moral hazard in insurance markets have been notably analogous. The general design has been to comprise an individual agent’s prospective consumption, as well as all those elements which could have any pertinence to this consumption. Generally, therefore, the models have contained a host of variables including the payout level, the loss of income in the bad state(s), the endowments, initially as well as in the

1 INTRODUCTION

20

di¤erent states, and the level of care.51 A primary objective of this paper is to simplify this well-established framework as far as possible, thereby creating a foundation on which the additional objectives can be pursued. This simpli…cation will be e¤ected partly by reducing the number of variables included in the model to a minimum, and partly by giving an alternative interpretation of the institutional framework. A crucial requisite in this context is that this be made without any loss of generality, in the sense that all variables of previously employed models must be possible to map to the variables of this model. A second aim of the paper is to use this simpli…ed model in order to analyze the characteristics of the agents’ preferences. In this context we will discuss monotonicity properties of the utility function with regard to the level of care taken. The results of this section are then used to formulate a useful proposition regarding indi¤erence classes when the agents have a uniform consumption over states. Finally, we will analyze convexity properties of the preferences, and in this context we shall prove, given some standard assumptions, that the expected utility function cannot be quasi-concave, a result which has been shown to typically prevent the existence of a price equilibrium. Thirdly we will use our model to establish conditions for and properties of an individual optimum, which shall be referred to as the individual’s …rst best action. In short we will show existence and uniqueness requirements for a …rst best action. We also show …rst order conditions and the well-known result that in the individual optimum case full insurance is optimal. The last objective of the paper is to analyze the relation between risk level and expected utility. For this purpose we introduce some tools from di¤erential geometry, with a particular emphasis on the envelope concept. Using these tools, we come up with the rather counter-intuitive result that an increase in the risk of the economy can actually induce a higher utility level for the average agent (and thus for the economy as a whole). The section ends with a discussion of possible explanations of this predicament, 51

See for example Pauly (1974), Helpman and La¤ont (1975), Shavell (1979) and Arnott and Stiglitz (1988)

2 THE ECONOMY

21

and concludes that there might be a shortcoming in the way the probability function has traditionally been de…ned. It also suggests a possible resolution to this problem. The organization of the paper will strictly follow these objectives. Thus Section 2 will deal with the economy and the agent’s preferences, while Section 3 and Section 4 will discuss the …rst best action and the relation between risk and utility, respectively. Finally, Section 5 will conclude the results arrived at in the paper.

2

The Economy

The objective of this section is to introduce a simpli…ed framework for analyzing the moral hazard problem. With already established models as our point of departure, the ambition will be to reduce the number of variables only to those which are absolutely fundamental for comprehending the complexity stemming from moral hazard. By this approach we end up with only three variables; the net consumption in the two states of the world and the level of precaution chosen by the agent(s). As discussed in the introduction, there will be a substantial emphasis on manifesting a mapping between the variables used in most preceding analyses and those to be used in this model. In this sense the generality will be kept integral.

2.1

A Traditional Moral Hazard Model for Insurance

As a starting point we will, for comparative purposes, begin this section by describing a traditionally designed model of an insurance market under moral hazard.52 In this setting there is a pro…t maximizing and risk neutral insurance company in a competitive market on the one hand, and some unde…ned number of risk averse agents in want of insurance on the other. An agent’s choice of preventive action is assumed to directly a¤ect the probability of having a future income shortfall. The model thus complies with the 52

This particular example is taken from Pauly (1974)

2 THE ECONOMY

22

de…nitions of moral hazard and of an insurance market as presented in the introduction. More speci…cally, by signing a contract with the insurance company, agent i will receive the following income, wi1 and wi2 , in the two possible states: wi1 = Si0 ¡ zi ¡ Pi

(no income shortfall)

wi2 = Si0 ¡ zi ¡ Pi ¡ Li + Xi

(an income shortfall)

where Si0 zi Pi Li Xi

is the initial endowment in the present state for agent i is the level of precautionary measures chosen by agent i is the insurance premium paid to the insurance company by agent i is the potential income shortfall for agent i are the damages which are paid out in case of an income shortfall

The design now is to introduce another, simpli…ed, model which also conforms to the de…nitions made, and which is directly translatable from the model just described. In the following sections we will thus discuss the structure of this new model, and also general implications from introducing it.

2.2

Agents and Institutions

Our model includes a continuum of individuals with identical characteristics, implying, amongst other things, that there will be no complication stemming from adverse selection present in the economy. This could be the situation either if the risk levels of the agents are altogether possible to monitor by the risk indi¤erent party, or if these dissimilarities for some reason are considered negligible. We also assume that the agents have access to the same information regarding factors that might a¤ect the probability of the occurrence of either state. E¤ectively, these two assumed features imply that the agents’ subjective beliefs as to the probability of an accident will coincide. As indicated in the introduction, the idea of simplifying the moral hazard model will not only be realized by simplifying the model, but will also be

2 THE ECONOMY

23

applied to the interpretation of the institutional framework. For this reason we will not make the otherwise well-nigh catholic presumption that the economy accommodates an insurance company. By means of the abstract de…nitions made in the introduction, it has namely become practicable to interpret all de…nitions made and results concluded in this paper in terms of a pure exchange economy. The precise contents of such an interpretation will be presented in Section 3.2 as we discuss the concept of market balance.

2.3

Action and Consumption

Action An agent’s action is given by a vector (cu ; cd ; x) 2 R3 , where cu corresponds to the …nal consumption of the consumer good if there is no income shortfall, whereas cd denotes the consumption if there is. The variable x speci…es the amount spent upon a precautionary good, which has the feature that the more of it you buy, the less the risk you will have the income shortfall mentioned above. This matter is more comprehensively discussed in Section 2.4. A possible action is a vector (cu ; cd ; x) which belongs to the set © ª A = (cu ; cd ; x) 2 R3 j cu ¸ cd ; x ¸ 0; cd ¸ 0

Net consumption

An agent’s net consumption is a vector (cu ; cd ). A net consumption is possible if it belongs to the set © ª C = (cu ; cd ) 2 R2 j cu ¸ cd ; cd ¸ 0

Initial endowment

The fact that we have a potential loss of income implies there is an uncertain, state-dependent initial endowment confronting the agent in the future period. This endowment we denote by e = (eu ; ed ), a vector which describes the income when there is a shortfall, and where there is not, respectively. We

2 THE ECONOMY

24

further assume that e 2 C; so that (eu ; ed ) is a possible net consumption which is always obtainable by choosing the (possible) action (eu ; ed ; 0). Gross consumption and insurance contract An agent’s gross consumption is a vector (Cu ; Cd ) 2 R2 , which is possible if Cu ¸ Cd and Cd ¸ 0. There is an immediate connection between this gross consumption and the actual insurance contract (as described in the traditional model accounted for above), a contract that details a premium, P , payable under all circumstances, and an indemnity, X, which is e¤ected only in case of an accident. Given the initial endowment (eu ; ed ) (which is always known) there is namely a mapping between (P; X) and (Cd ; Cu ) speci…ed by Cu = (eu ¡ P ) Cd =

(ed ¡ P + X)

This means that the gross consumption is directly de…ned by the insurance contract. For reasons of simplicity, we shall therefore subsequently omit the pair (P; X) from the analysis. Gross consumption and action Since we want to use an agent’s action (which includes his net consumption) exclusively in the forthcoming analysis, we must, …nally, establish the link between an action and a gross consumption. This is given by (cu ; cd ; x) = (Cu ¡ x; Cd ¡ x; x) We thus have a mapping directly between the underlying insurance contract and an agent’s action. This means that we can exclude from the analysis all variables but those three associated with the action.

2.4

The Prevention Function

The probability of not having an accident is described by a function q : R+ ¡! [0; 1] with x ¡! q(x). The probability of actually having the accident

2 THE ECONOMY

25

will, accordingly, be (1 ¡ q(x)). It is thus assumed that an agent can alter his probability of meeting with an impending accident by investing some of his future income in precautionary measures, the level of which is given by x. For example, to abate the probability of having a detrimental …re in his house, the agent could purchase a …re alarm system, thereby increasing the practicality of detecting the …re and preventing it from extending. We now impose a number of maintained assumptions on q, numbered from Q0 to Q6 , which will be su¢cient for all Propositions made in this paper. Q0 : q is twice continuously di¤erentiable for x 2 ]0; 1] Q1 : 0 · q(x) < 1

Q4 : q 0 (x) > 0 for x 2 ]0; 1]

Q2 :

Q5 :

lim q(x) = 1

x¡!1

Q3 : q(0) = 0

lim q 0 (x) = 1

x¡!0

Q6 : q 00 (x) < 0 for x 2 ]0; 1]

Assumptions Q0 , Q2 and Q5 are present mainly for technical reasons. In Q1 we assume that the occurrence of the accident will always have a strictly positive probability. Thus an agent cannot eliminate this probability altogether whatever the size of his choice of precaution. Assumption Q3 states that if no preventive measures are taken by the agent, then the income shortfall will occur for sure. In Q4 we make the normally made assumption that an increase in the level of precaution will increase the probability of the non-occurrence of an accident, while Q6 states that this takes place at a decreasing rate, so that the marginal e¤ect of x on q(x) decreases with the size of x. It should be emphasized that these assumptions are assumed to hold, unless otherwise stated. At places, however, in order to conclude an interesting result or to make an example, there might be small deviations from these assumptions. The changes made at these places, and the implications of making them, will always be accounted for. For the moment we assume that q is independent of the magnitude of the potential accident, which, in turn, means that there are actually increasing

2 THE ECONOMY

26

returns to scale of taking care; the larger the accident (the bigger the di¤erence between the endowments in the two states), the less x has to be as a percentage of the potential loss due to the accident. As a consequence of the results of Proposition 13 and Corollary 15, this predicament will be brought up again in that context.

2.5

Utility and Preferences

Utility function The state utility function, equal for all agents, is given by u : R+ ¡! R, with c ¡! u(c). As with q above, we impose some maintained assumptions on u, su¢cient for the results concluded in the propositions and corollaries of the paper: U0 : u is twice continuously di¤erentiable U1 : u0 (c) > 0 U2 : u00 (c) < 0 We assume that the utility of an agent can be represented by an expected utility function U : R3+ ¡! R, where U (cu ; cd ; x) = q(x)u(cu ) + (1 ¡ q(x))u(cd ) With the sole exception of Proposition 2 the domain of U is restricted to A ½ R3+ . The expected utility of an action can be reformulated in the following way: U (cu ; cd ; x) = u(cd ) + q(x) [u(cu ) ¡ u(cd )] With this interpretation the agent will obtain the utility u(cd ) regardless of the future state. However, if the good state occurs, which will happen with probability q(x), he will also receive an additional utility of [u(cu ) ¡ u(cd )]. This formulation will prove useful at several places in this paper. In the following two sections we will examine some interesting properties of the function U.

2 THE ECONOMY

27

Monotonicity in care In this subsection we will prove, and examine the economic implications of, two monotonicity properties of U in relation to the preventive action taken. For this reason we introduce two concepts, monotonicity and negative monotonicity. De…nition 1 For x¹ > x ¸ 0 U is monotone in x if U (cu ; cd ; x¹) > U (cu ; cd ; x) U is negatively monotone in x if U (cu ; cd ; x¹) < U (cu ; cd ; x) Given these de…nitions we can now conclude the following result. Proposition 2 (a) If cu > cd then U is monotone in x (b) If cu = cd , then U is neither monotone nor negatively monotone in x Proof: From our maintained assumptions we have @U (cu ; cd ; x) ´ q 0 (x)(u(cu ) ¡ u(cd )) > 0 if and only if cu > cd @x @U (ii) (cu ; cd ; x) ´ q 0 (x)(u(cu ) ¡ u(cd )) < 0 if and only if cu < cd @x (i)

where (i) proves (a), and (ii) and (i) combined prove (b).

¤

The economic interpretation of Proposition 2 becomes patent by using the formulation U (cu ; cd ; x) = u(cd ) +q(x) [u(cu ) ¡ u(cd )] introduced above. If cu > cd the additional part [u(cu ) ¡ u(cd )] is positive. There will therefore, for given levels of cu and cd , be a positive gain in expected utility from increasing the probability, q(x), of having this addition. However, when cu = cd the additional part is zero, and thus the size of q(x) (and hence x) will become inconsequential.

2 THE ECONOMY

28

The uniform consumption set We will now discuss implications of the result arrived at in Proposition 2 (b). This analysis, which is carried out both analytically and graphically, will thus highlight some properties of the expected utility function when the consumption is equal in the two states. There are mainly two reasons for taking an interest in this case. Firstly it is a highly relevant one to discuss, since an optimal …rst best solution always will imply an equal consumption over states. Corollary 11 (a) below will provide a proof of this fact. Secondly it is also fruitful to penetrate this issue as it will enable a reduction of the number of dimensions necessary for comparing utility levels from three (cu ; cd ; x) to one (cu = cd = c). First we introduce two necessary concepts, and then the main result of this subsection is stated. De…nition 3 The indi¤erence class of a possible action is de…ned by indi¤ (~ cu ; c~d ; x~) = f(cu ; cd ; x) 2 A j U(cu ; cd ; x) = U(~ cu ; c~d ; x~)g De…nition 4 The uniform consumption set L is de…ned as L = f(cu ; cd ; x) 2 A jcu = cd g These de…nitions now lead us to the following proposition: Proposition 5 Take any two vectors (~ cu ; c~d ; x) 2 L and (~ cu ; c~d ; x0 ) 2 L with x 6= x0 . Then indi¤ (~ cu ; c~d ; x) = indi¤ (~ cu ; c~d ; x0 ), i e the two vectors belong to the same indi¤erence class Proof: From Proposition 2 (b) we see that the function U (~ cu ; c~d ; ¢), taking x to U(~ cu ; c~d ; x), is neither monotone nor negatively monotone in x. Thus U(~ cu ; c~d ; x) = U(~ cu ; c~d ; x0 ). ¤

2 THE ECONOMY

29

cd L

(a,a,x**) (a,a,x*) (a,a,x)

x

cu

Figure 4: The uniform income set for cu = cd = a In order to gain a further insight into the matter, we consider two …gures which visualize the result of Proposition 5. Figure 4 depicts three vectors which belong to the subset of L given by f(cu ; cd ; x) 2 L j cu = cd = ag This subset is indicated by a white line, and what is proved in Proposition 5 is that all actions along such a line, that is actions with the same uniform consumption, must belong to the same indi¤erence class. In other words, the level of x will have no consequence for the utility of such an action. That this feature is not true in general, that is for arbitrary actions with the same53 consumption, is depicted in Figure 5, where the indi¤erence surface given by indi¤ (a; a; 0) is displayed. As a consequence of Proposition 5 the three vectors in Figure 4 will all belong to this surface. 53

To eliminate confusion, two actions (¹ cu ; c¹d ; x ¹) and (~ cu ; c~d ; x ~) have the same consumption if c¹u = c~u and c¹d = c~d . Thus two actions with the same consumption do not, generally, have a uniform consumption over states, which would further require c¹u = c¹d and c~u = c~d .

2 THE ECONOMY

30

The …gure also visualizes indi¤erence curves for the three di¤erent levels of x, and we see that when x = 0, resulting in a probability of having the accident of 1, the level of cu will become inconsequential, since it represents consumption which is contingent upon a state that cannot occur. We thus have a horizontal indi¤erence curve in this case. At the next cross section,

cd

(a,a,x**)

cd

x (a,a,x*)

cd

cu (a,a,x)

cu

cu

Figure 5: Actions that belong to indi¤ (a; a; 0) where x = x¤ > 0, we have a more conventionally shaped indi¤erence curve, whereas it becomes close to vertical with x = x¤¤ > x¤ , where the probability of having the accident is close to zero. We thus observe that, generally, actions with the same consumption do not yield the same utility. This is because, apart from the case where an action belongs to the uniform consumption set, the size of x does have an e¤ect on the expected utility, since the probability of an accident decreases with x. An interesting implication of this fact is that when we compare actions with the same uniform consumption (but only in this case), the utility level can be measured simply by means of the state utility function. This is because

2 THE ECONOMY

31

for any action (c; c; x) we then have that U (c; c; x) = q(x)u(c) + (1 ¡ q(x))u(c) = u(c) This implies that the problem of comparing utility levels in the …rst best case, where we always have a uniform consumption,54 is considerably simpli…ed. U is not quasi-concave Helpman and La¤ont (1975) have emphasized the importance of the expected utility function to be quasi-concave, since, otherwise, there typically does not exist an equilibrium. The authors conclude, however, that ”...we were not able to …nd economically meaningful conditions on the probability function which assure quasi-concavity of V (¢) [the expected utility function]”.55 In the next proposition we will go one step further by suggesting that under our maintained assumptions, the expected utility function can never be quasiconcave. Proposition 6 Under the maintained assumptions U is not a quasi-concave function Proof: Let G = f(cu ; cd ; x) 2 A j cu > cd ; x > 0g

and let V : G ¡! R with (cu ; cd ; x) ¡! q(x)u(cu ) + (1 ¡ q(x))u(cd ) so that V is the restriction of U to G. Thus the function U cannot be quasi-concave unless V is. We now construct the Bordered Hessian of V , BV : 2

0

u0 (cu )q(x)

u0 (cd )(1 ¡ q(x))

6 6 6 u0 (cu )q(x) q(x)u00 (cu ) 0 6 6 6 u0 (cd )(1 ¡ q(x)) 0 u00 (cd )(1 ¡ q(x)) 6 6 4 q 0 (x) [u(cu ) ¡ u(cd )] u0 (cu )q0 (x) ¡q 0 (x)u0 (cd ) 54 55

q0 (x) [u(cu ) ¡ u(cd )]

3

7 7 7 u (cu )q (x) 7 7 0 0 7 ¡q (x)u (cd ) 7 7 q 00 (x) [u(cu ) ¡ u(cd )] 5

As we have indicated this will be proved in Corollary 11 (a) below. Helpman and La¤ont (1975), p 13

0

0

2 THE ECONOMY

32

Since the domain of V is an open, convex set, a necessary condition for V to be quasi-concave is that jBV j < 0 for each action (cu ; cd ; x) 2 G, where jBV j is the determinant of BV .56 Now let bij denote the element in the i:th row and the j :th column of BV . Also, let Bmn be the matrix obtained by deleting the m:th row and the n:th column from BV . Now assume that cu = cd , so that b11 = b14 = 0. By expanding jBV j in terms of the elements of the …rst row we thus have: jBV j = ¡b12 jB12 j + b13 jB13 j where b12 = u0 (cd )q(x) b13 = u0 (cd )(1 ¡ q(x) ¯ ¯ u0 (c )q(x) 0 u0 (cd )q0 (x) ¯ d ¯ ¯ 0 00 0 0 jB12 j = ¯¯ u (cd )(1 ¡ q(x)) u (cd )(1 ¡ q(x)) ¡q (x)u (cd ) ¯ ¯ 0 ¡q 0 (x)u0 (cd ) 0 ¯ = ¡(u0 (cd ))3 (q 0 (x))2

¯ ¯ u0 (c )q(x) q(x)u00 (cd ) u0 (cd )q 0 (x) ¯ d ¯ ¯ 0 ¡q 0 (x)u0 (cd ) jB13 j = ¯¯ u (cd )(1 ¡ q(x)) 0 ¯ ¯ 0 u0 (cu )q 0 (x) 0 ¯ = (u0 (cd ))3 (q0 (x))2

56

Sydsaeter and Hammond (1985), p 647

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

2 THE ECONOMY

33

Thus jBV j = ¡b12 jB12 j + b13 jB13 j

= u0 (cd )q(x)(u0 (cd ))3 (q0 (x))2 + u0 (cd )(1 ¡ q(x)(u0 (cd ))3 (q0 (x))2 = u0 (cd )4 (q 0 (x))2

> 0 Since the inequality is strict, and by the continuity of jBV j there will exist actions (cu ; cd ; x) 2 G for which jBV j > 0. Thus V and U are not quasiconcave functions. ¤

2.6

The Economy

Now that all relevant de…nitions have been itemized, we can formally express the moral hazard economy by E = (u(¢); e; q(¢)). For easy reference we now summarize the Maintained Assumptions we have made regarding this economy. Maintained Assumptions of the economy on q Q0 : q is twice continuously di¤erentiable for x 2 ]0; 1]

Q1 : 0 · q(x) < 1

Q4 :

Q2 :

Q5 :

lim q(x) = 1

x¡!1

Q3 : q(0) = 0

Q6 :

q 0 (x) > 0 for x 2 ]0; 1] lim q 0 (x) = 1

x¡!0

q 00 (x) < 0 for x 2 ]0; 1]

on u U0 : U1 : U2 :

u is twice continuously di¤erentiable u0 (c) > 0 u00 (c) < 0 on e e 2 C, so that eu ¸ ed ¸ 0

3 MARKET BALANCE AND FIRST BEST ACTION

34

As was discussed initially, we also assume that there is a continuum of agents, an assumption that enables a re-interpretation of the economy, a subject that will be penetrated more thoroughly in the next section.

3

Market Balance and First Best Action

In this section we aim at introducing the concept of a …rst best action, which is the full information individual optimum in this economy. Furthermore we will analyze the existence of such an action, and also under what circumstances it is uniquely determined. Finally, after having determined its existence and uniqueness properties, we will discuss some of its general features. To enable such an analysis, the section will commence with the de…nition of a feasible action.

3.1

Feasible Actions

In order to de…ne a feasible action, we …rst need to know what is meant by a market balanced action in this economy. Market balance The limitations dictating a market balanced action are de…ned in the following manner. De…nition 7 An action (cu ; cd ; x) is market balanced if q(x) (cu + x) + (1 ¡ q(x)) (cd + x) = q(x)eu + (1 ¡ q(x))ed Section 3.2 below will provide an interpretation of this market balance criterion in terms of a continuum of agents. Now we want to use this formalization in order to de…ne the feasible set.


35

The feasible set In the de…nition of the possible action set, A, there was no essential requiring that an action be market balanced. If we add this requirement to the criteria de…ning A, we end up with the feasible set of actions. De…nition 8 The set F = f(cu ; cd ; x) 2 A j (cu ; cd ; x) is market balancedg is the feasible set of actions. We say that an action that belongs to the feasible set is a feasible action.

3.2

Interpretation of the Economy

As was discussed in the introduction to this paper, an insurance market has to accommodate two types of actors - one which is simultaneously risk avert and exposed to a risk (and thus in want of risk reduction), and one which is risk indi¤erent (and thereby willing to take on risk). The issue towards which we now intend to direct the attention is the institutional foundation required to ensure the existence of the risk indi¤erent party. Traditional models dealing with moral hazard in insurance markets have frequently assumed the existence of a risk neutral insurance company in order to satisfy this prerequisite. However, by regarding the economy in a somewhat modi…ed manner, the risk neutral insurance company, which in contrast to our aims mainly adds to the complexity of the model, can be exchanged for the continuum of agents. This is because if we have identical agents facing the same prospective endowment and the same information regarding factors that might a¤ect the probability of an accident, then they will all choose the same amount of prevention and hence they will all face the same probability of an income shortfall. For the continuum of agents this probability will correspond with the share of the population that will have the accident, a share which is thus known for a certainty. The continuum of agents will accordingly be risk indi¤erent in accordance with the de…nition, since it is in no way exposed to any risk. We thus have individual risk but no collective risk.


36

As a consequence of the above, we can now remove the insurance company altogether from the economy. This removal is directly in line with our intentions of simplifying the model as far as possible, since the insurance company in itself adds to the complexity of the model without contributing much (if anything) to the understanding of any complications stemming from moral hazard. To relate this new interpretation of the economy to De…nition 7 of a market balanced action, we assume that the agents can insure themselves completely (the rationale for which is given in Corollary 11 (a)). Under these premises the agents could be thought of as putting together their endowments before the state of the world is disclosed, and then this pool of endowments is evenly distributed among them after the true state is realized. The total resources needed for acquiring the total amount of the consumer good must then equal the total initial endowments net precautionary measures. But then this must also hold for the average endowment and consumption, which is exactly what is described by the market balance condition as given by De…nition 7. An example illustrating this interpretation Assume there are only three agents, A, B and C, and that we know for certain, as in the case of a continuum of agents, the share of this population which will have an accident, a share given by the probability q(x). The agents are identical, and they therefore choose the same level of x, a level which we denote by x¹. Then q(¹ x) is determined, and we assume this probability (and thus the total share of the population not having the accident) is 32 . Hence we will have three di¤erent possible states of the world (’d’ implying the agent has an accident, and ’u’ indicating he has not): Agent A Agent B Agent C State 1

u

u

d

State 2

d

u

u

State 3

u

d

u

In all three cases we know that the total net initial endowment is given by


37

2eu + ed ¡ 3¹ x, and it is thus unambiguously determined once the level of the preventive action is decided upon. The average net initial endowment (corresponding with the average expected net initial endowment as given by the market balance condition) will thus be 32 eu + 13 ed ¡ x¹.

3.3

The First Best Action

By the de…nition of a market balanced action, we now have what is required for introducing the concept of a …rst best action. De…nition 9 An action (¹ cu ; c¹d ; x¹) 2 A is a …rst best action if it is a solution to Max

(cu ;cd ;x)2A

q(x)u(cu ) + (1 ¡ q(x))u(cd )

s.to

q(x) (cu + x) + (1 ¡ q(x)) (cd + x) = q(x)eu + (1 ¡ q(x))ed

(1)

In order to acquire an intuition of what the …rst best action would entail assuming a continuum of agents, we …rst rewrite the above maximization problem under the condition that the agents can insure themselves unrestrainedly,57 so that cu = cd = c. Max

x¸0 u(c)

s.to

c + x = q(x)eu + (1 ¡ q(x))ed Given our maintained assumptions on u, the same solution is yielded by Max

x¸0 q(x)eu

+ (1 ¡ q(x))ed ¡ x

This means that in order to select a …rst best action, the average agent must ascertain that his choice of x maximizes the average initial endowment net the average level of precaution. This criterion is readily translatable also to the continuum of agents, for which the share having the accident, as well as the total initial endowment, is known for certain as demonstrated above. Thus the collective of agents must ensure that the level of x maximizes the total initial endowment net the total level of preventive action. 57

See Corollary 11 (a) for a justi…cation of this supposition.


38

Existence and uniqueness of the …rst best action Now we are interested in when the maximization problem (1) has a solution generally, and under what pre-conditions it can be concluded that this solution be unique. To elucidate these queries is the objective of the following proposition. Proposition 10 Let E = fu(¢); q(¢); eg be an economy with moral hazard. Then there exists a unique …rst best action (¹ cu ; c¹d ; x¹) 2 A Proof: First consider the function h de…ned by x ! h(x) = ed + q(x) [eu ¡ ed ] ¡ x, de…ned for x 2 R+ . Using Q6 we have that h00 (x) < 0. We now have two cases, eu = ed and eu > ed . In the …rst case h0 (x) < 0, and thus h(0) ¸ h(x). In the second case, which is hereafter considered, we let x 2 I, where I is the interval [0; eu ], which in this case is both compact and non-empty. Thus there is x¹ 2 I such that h(¹ x) > h(x) for 8x 2 I. Since h(x) · ed + (eu ¡ ed ) ¡ x = eu ¡ x · 0 for x 2 R+ ÂI and h(¹ x) ¸ 0 (as h(0) = ed + q(0) [eu ¡ ed ] ¸ 0), this implies that h(¹ x) ¸ h(x) for x 2 R+ . Now let c = ed +q(x) [eu ¡ ed ]¡x, and, in particular, let c¹ = ed +q(¹ x) [eu ¡ ed ]¡ x¹. Then (¹ c; c¹; x¹) 2 F. The last step is to prove that (¹ c; c¹; x¹) 2 F is actually a …rst best action. Therefore consider another action, (cu ; cd ; x) 2 F. Then U(cu ; cd ; x) = q(x)u(cu ) + (1 ¡ q(x))u(cd ) < u(q(x)cu ) + (1 ¡ q(x))cd )

(*)

= u(c) = u(h(x)) < u(h(¹ x))

(**)

= u(¹ c) = q(¹ x)u(¹ c) + (1 ¡ q(¹ x))u(¹ c) = U(¹ c; c¹; x¹) Thus (¹ c; c¹; x¹) 2 F is a unique …rst best action.

¤


39

Properties of the …rst best action In the following corollary, we will discuss some general properties of the …rst best action. They are all more or less evident from the way the existence problem was solved. Corollary 11 Let E = fu(¢); q(¢); eg be an economy with moral hazard with the …rst best action (¹ cu ; c¹d ; x¹) 2 F. Then (a) c¹u = c¹d = c¹ (b) c¹ = M axx¸0 q(x)eu + (1 ¡ q(x))ed ¡ x (c) If (¹ cu ; c¹d ; x¹) is such that x¹ > 0, then q 0 (¹ x) [eu ¡ ed ] = 1 (d) x¹ > 0 if and only if eu 6= ed Proof: (a) If (¹ cu ; c¹d ; x¹) 6= (¹ c; c¹; x¹), then from the proof of Proposition 10 we know that U(¹ cu ; c¹d ; x¹) < U (¹ c; c¹; x¹). Thus (¹ cu ; c¹d ; x¹) cannot be a …rst best action unless (¹ cu ; c¹d ; x¹) = (¹ c; c¹; x¹). (b) This follows from the fact that c¹ = h(¹ x) where x¹ = arg max h(x) x¸0

(c) As we know that x¹ = arg max h(x), we get this …rst order condition by x¸0

di¤erentiating h with respect to x: dh = 0; q 0 (¹ x)[eu ¡ ed ] ¡ 1 = 0 dx (d) First recall from the proof of Proposition 10 that h(x) = ed + q(x)(eu ¡ ed ) ¡ x, for x ¸ 0, and that x¹ = arg max h(x). In the …rst direction we x¸0

prove that x¹ > 0 =) eu 6= ed . In order to get a contradiction, assume x¹ > 0 and eu = ed . Then h(¹ x) = ed ¡ x¹, so that x¹ = 0 uniquely maximizes h(x), yielding the contradiction. To make the proof in the other direction, assume that eu 6= ed , so that eu ¡ ed > 0. Then h0 (x) > 0 for x > 0 and x su¢ciently small, and so x¹ > 0 is the unique maximizer of h(x). The result yielded from Corollary 11 (a) is that the …rst best action entails a full insurance. This is a result which is in full accordance with intuition.


40

Despite the presence of moral hazard, we do have a situation with complete information, since the level of care chosen by the agents is observable and thus possible to contract upon This is a standard result in the literature, a result which is here restated in the context of our simpli…ed model. In (b) we have indicated a simple way of calculating a …rst best action. We merely need to maximize the average expected initial endowment net the precautionary action, and set c¹ equal to this value.

cd L

(h(x), h(x), x) c

x

x

c

cu

Figure 6: The …rst best action The interpretation of (c) is that given the …rst best action, under those presumptions made above, the marginal cost of one additional x should equal the marginal reduction of the expected loss provided by that extra spending. This is the standard …rst order condition of problems with moral hazard in the …rst best case, but in a version induced by our simpli…ed model. In (d) we have proved a property connected with maintained assumption Q5 , a property that will be used later in this paper. It simply says that given that we have individual risk (eu 6= ed ) it is optimal to use some of the expected initial endowment for precautionary measures. It also says that if there is no risk, then an agent will not use any of his endowment for precaution.

4 RISK AND EXPECTED UTILITY

41

By momentarily deviating slightly from our maintained assumptions, we note an additional property connected with the …rst best action. By assuming that we exchange U1 and Q4 for the criteria q 0 (x) ¸ 0 and u0 (c) ¸ 0, the two strict inequalities at (*) and (**) respectively, in the proof of Proposition 10, will become non-strict. This implies that if neither q nor u is strictly increasing, there might be several …rst best actions. Figure 6 depicts a …rst best action with a strictly concave prevention function q. Here we note that since the …rst best action must be in the uniform consumption set, L, the expected utility function will not depend upon the level of x, as was proved in Proposition 5. This implies that the indi¤erence curves are made out of straight horizontal lines along L, one of which is indicated by the white line in Figure 6. The …rst best action is where the relevant indi¤erence curve and the curve given by (h(x); h(x); x) have their common tangency point in the set L. This point de…nes x¹ which, in turn, determines c¹, since c¹ = h(¹ x).

4

Risk and Expected Utility

In this section we will illustrate an oddity in traditionally particularized moral hazard insurance models. What we will discuss is the relation between the risk to which an average agent is exposed, and his corresponding expected utility. Intuitively, as in our maintained assumptions we have assumed risk aversion, one would expect that a higher risk, given a constant probability of having an income shortfall, should at least not induce a higher expected utility. The overarching objective of this section is to prove that it can. To attain this objective we will initially introduce some concepts, de…ned in general terms, which will be used in the proceedings. We then, secondly, continue by describing the counterparts of these concepts in the speci…c model used in this paper. Finally, we proceed to the analytical part of this section, which will conclude with a discussion of the result and its possible causes, as well as an example illustrating the outcome.


4.1

42

The General Concepts

We now brie‡y introduce some fundamental concepts in di¤erential geometry which relate to the envelope, a concept which will be used as a tool to obtain the results of this section. We will give mathematical de…nitions, and brief interpretations of these di¤erent concepts, as well as a concrete example with a graphical representation of each.58 In the analytical subsection, the envelope will actually be the only concept from di¤erential geometry that we apply to the problem discussed in this section. The remaining concepts introduced below should be appreciated solely by right of their adding to the comprehension of the characteristics of an envelope, both in general terms, and in this concept’s connection with the subject analyzed in this section. The reason for introducing the envelope is because it turns out to have properties which are particularly opportune when analyzing the relation between risk and expected utility. Firstly, the concept and its application to the problem at hand is very intuitive and easy to grasp. Secondly, these favourable properties seem to be considerably invariable to alterations in the model, so that variations in the relation between risk and utility stemming from changes in for example the assumptions on the probability function or on the information structure should be easy to incorporate.59 We commence by considering a function, F , de…ned by F : R3 ¡! R so that (t; x1 ; x2 ) ¡! F (t; x1 ; x2 ). We assume that, for each t, 0 is a regular value of F , so that at least one of the partial derivatives Dxi F (t; x1 ; x2 ) 6= 0 for i = 1; 2. We now de…ne the following concepts: 58

The essence of the material presented in this section emanates from Hsiung (1981) and Bruce and Ghiblin (1987). 59 This argument is based on research which is not presented in this paper, and which is not, as yet, entirely conclusive. The argument should thus be considered indicative at this stage.


43

De…nition

Interpretation

(0) F

a function

(1) F (t; ¢; ¢) = 0

for each t a curve in (x1 ; x2 )-space

(2) F (t; x1 ; x2 ) = 0

as t varies a family of curves in (t; x1 ; x2 )-space

8 ¯ 9 ¯ F (t; x1 ; x2 ) = 0 and > > < ¯ = ¯ (3) (t; x1 ; x2 ) ¯ > ¯ @F (t; x1 ; x2 ) > : ; ¯ =0 @t 8 ¯ ¯ there exists t such that > > ¯ > > ¯ > < ¯ ¯ F (t; x1 ; x2 ) = 0 and (4) (x1 ; x2 ) ¯ > ¯ > ¯ @F (t; x ; x ) > > 1 2 ¯ > : =0 ¯ @t

the fold curve of the family of curves given by F (t; x1 ; x2 ) = 0 9 > > > > > = > > > > > ;

the envelope of the family of curves given by F (t; x1 ; x2 ) = 0

The requirements that have to be met by the envelope in accordance with the above table, could either be regarded as an actual de…nition, or as conditions derived from a more primary de…nition. In the latter case we could de…ne the envelope of a one-parameter family of curves as a curve which satis…es (i) at every point it is tangent to at least one curve of the family, and (ii) it is tangent to every curve of the family at at least one point. From this de…nition it is possible, given some basic di¤erentiability assumptions, to derive the criterion given in (4) in the above table.60 In order to gain some more intuition of the concepts and their related de…nitions, we will now present a simple but illuminating example. For those readers already conversant with these concepts, the example could be passed over without any loss of coherence. 60

Hsiung (1981), p112


44

A simple example Let F be de…ned by F (t; x1 ; x2 ) = (x1 ¡ t)2 + x22 ¡ 1.61 Employing de…nition (1) from the table above, and setting the value of t to 2, we get a circle de…ned by (x1 ¡ 2)2 + x22 ¡ 1 = 0. This circle is centred at (x1 ; x2 ) = (2; 0) and has a radius of 1 as shown in Figure 7.

x2 2 1

1

2

3

4

x1

-1 -2

Figure 7: F (t; ¢; ¢) = 0; a circle given by t = 2 By applying de…nition (2) from the table to this function, we get a family of circles with radius one, centred at (x1 ; x2 ) = (t; 0). This family is speci…ed by (x1 ¡ t)2 + x22 ¡ 1 = 0, and is visualized for a number of values of t in Figure 8. This …gure also displays the projection of one of these circles onto the (x1 ; x2 )-space in the case where t = 2 (the special case shown in Figure 7). We should think of this as a family of functions of (x1 ; x2 ) parametrized by t. The fold curve, given by de…nition (3) in the table, is the locus of points (t; x1 ; x2 ) where Dt F = 0. In our example we have that 61

The core of this example is taken from Bruce and Ghiblin (1987).


3

45

t t=2

2 1

x2

1

x1

-1

1

2

3

4

-1 -2

Figure 8: F (t; x1 ; x2 ) = 0; as t varies, a family of circles

¯ ) ¯ (x1 ¡ t)2 + x2 ¡ 1 = 0 2 ¯ (t; x1 ; x2 ) 2 R3 ¯ ¯ and ¡ 2x1 + 2t = 0

(

which is equal to ª © (t; x1 ; x2 ) 2 R3 j x1 = t; x2 = §1

This set consists of two lines in R3 , as shown in Figure 9. In this …gure we have also indicated the projection of the fold curve onto the (x1 ; x2 )-plane, which is in fact the envelope. More speci…cally, the envelope is given by 8 > >
> ¯ : ¯ ¡2x1 + 2t = 0

and > > ;

which is equal to ¯ ) ¯ there exists t with 2¯ (x1 ; x2 ) 2 R ¯ ¯ x1 = t; x2 = §1

(

9 > > =


46

t 3 2 x2

1 1 -1

1

2

3

4

x1

-1

Figure 9: The fold curve tracing out the envelope This set is shown in Figure 10 and consists, as was indicated in Figure 9, of two lines in the (x1 ; x2 )-plane.

4.2

The Concepts in the Moral Hazard Environment

In this section we will apply the di¤erent concepts introduced above to an economy with moral hazard. The function Fc¹ We …rst consider a net consumption (¹ c; c¹) 2 C, and note that any action (¹ c; c¹; x) will belong to the uniform consumption set L. We now de…ne Fc¹ by Fc¹(eu ; ed ; x) = q(x)eu + (1 ¡ q(x))ed ¡ x ¡ c¹ This function describes the di¤erence between the expected average net initial endowment and the net consumption level (¹ c; c¹). Thus if Fc¹(eu ; ed ; x) > 0 the agent is not using all his endowment for consumption, whereas if Fc¹(eu ; ed ; x) < 0 the agent is using more resources for consumption than he is initially endowed with.


47

x2 2 1 1

2

3

4

x1

-1 -2

Figure 10: The envelope A curve for a …xed x We now require that Fc¹(eu ; ed ; x) = 0, so that the level of the net consumption equals the expected initial endowment. Furthermore we assume that the level of x is …xed at x¹, so that q(¹ x)eu + (1 ¡ q(¹ x))ed ¡ x¹ ¡ c¹ = 0

(2)

Relation (2) describes a curve for the given x¹ in (eu ; ed )-space. In economic terms, this curve traces out all initial endowments which make the action (¹ c; c¹; x¹) market balanced. Note here that we have no optimum requirement, i e x¹ need not be the preventive action that maximizes the expected utility given the endowment (eu ; ed ). In other words, (¹ c; c¹; x¹) will not generally be a …rst best action. The family of curves of Fc¹ The family of curves of Fc¹ is obtained by letting the level of x vary, while still requiring that Fc¹ = 0. Accordingly, it is given by q(x)eu + (1 ¡ q(x))ed ¡ x ¡ c¹ = 0

(3)


48

This means that, for each value of x, we end up with a curve like the one described in relation (2). In economic terms, relation (3) speci…es, for each value of x, those initial endowments for which (¹ c; c¹; x) is a market balanced action. Geometrically, we have a surface in (eu ; ed ; x)-space, as shown in Figure 11. Again it is essential to underline that, for any x considered, the action (¹ c; c¹; x) need not be a …rst best action given the endowment (eu ; ed ). We merely regard di¤erent levels of x, and for each of these levels we consider all the initial endowments for which (¹ c; c¹; x) is a market balanced action.

ed, cd

c, c x

eu, cu Figure 11: The family of curves for Fc¹

The fold curve To de…ne the fold curve, however, we have to impose an optimum criterion: for each level of x we only consider the unique initial endowment which maximizes Fc¹ with regard to x. The fact that, for each value of x, there will actually be a unique (eu ; ed ) for which (¹ c; c¹; x) is a …rst best action, will be proved in Corollary 14. The fold curve of the family of curves de…ned in relation (3), is thus


49

de…ned by the set ¯ ) ¯ q(x)eu + (1 ¡ q(x))ed ¡ x ¡ c¹ = 0 ¯ (eu ; ed ; x) ¯ ¯ q 0 (x) [eu ¡ ed ] ¡ 1 = 0

(

(4)

Geometrically, this set describes a curve in (eu ; ed ; x)-space, as shown in Figure 12.

ed, cd

c, c x

eu, cu Figure 12: The fold curve of the family of curves given by Fc¹

The envelope Finally we want to de…ne the envelope for the family of curves speci…ed by relation (3). In accordance with our general de…nition the envelope is given by the set 8 ¯ 9 ¯ there exists x with > > < ¯ = ¯ (5) (eu ; ed ) ¯ q(x)eu + (1 ¡ q(x))ed ¡ x ¡ c¹ = 0 and > ¯ > : ; 0 ¯ q (x) [eu ¡ ed ] ¡ 1 = 0 This set is the projection of the fold curve onto the (eu ; ed )-space (see Figure 13). It describes, as we vary x, all the initial endowments (eu ; ed ) for which


50

(¹ c; c¹; x) is a …rst best action. For convenience, we shall denote the envelope of a family of curves, given by the function Fc¹, by DFc¹ for short. To use the envelope DFc¹ in the formal analysis, we would like to modify it in two ways. Firstly we would like (eu ; ed ) = (¹ c; c¹) to be included, since our intention is to consider all endowments with the property that the …rst best action corresponding with this endowment entails the net consumption (¹ c; c¹). However, (eu ; ed ) = (¹ c; c¹) 2 = DFc¹ , since the marginal condition de…ning DFc¹ will not be met at this particular endowment. Secondly we would like to direct the attention only towards endowments which are possible, so that we require that (eu ; ed ) 2 C. For these reasons we de…ne a modi…ed envelope by DF¤ c¹ = f(eu ; ed ) 2 C j (eu ; ed ) 2 DFc¹ or (eu ; ed ) = (¹ c; c¹)g We will henceforth, somewhat improperly, relate to DF¤ c¹ as the envelope of the family of curves given by Fc¹.

ed, cd

c, c

eu, cu

Figure 13: The envelope of the family of curves given by Fc¹


4.3

51

Analysis: Results, Discussion and Example

In this section we will analyze the relation between risk and expected utility by means of the envelope concept. We begin with some preliminary de…nitions, and continue by stating the main results of the section. We then proceed with an economic examination of these results, describing the implications of and conceivable occasions to them. To enhance the understanding of the results further, the section concludes with an illustrating example. Preliminaries As a preliminary to the analysis we de…ne some sets to be used in the forthcoming proofs. Given the uniform consumption (¹ c; c¹) 2 C, we de…ne the at-least-as-bad set of endowments by Ec¹ = f(eu ; ed ) 2 C j q(x)eu + (1 ¡ q(x)ed ¡ x ¡ c¹ · 0; 8xg This set, which is the lower contour set of (¹ c; c¹), consists of all endowments (eu ; ed ) which cannot generate a …nal uniform consumption that exceeds (¹ c; c¹), no matter the choice of the preventive action x. We also de…ne the sets Dc¹ and H¡ x;¹ c by: Dc¹ = f(eu ; ed ) 2 Ec¹ j 9x where q(x)eu + (1 ¡ q(x)ed ¡ x ¡ c¹ = 0g

H¡ ¹g for x ¸ 0 x;¹ c = f(eu ; ed ) 2 C j q(x)eu + (1 ¡ q(x)ed · x + c

The set Dc¹ ½ Ec¹ is the set of initial endowments which exactly realize (¹ c; c¹) as a …nal uniform consumption, when the preventive action is selected optimally. For a given x, the set H¡ x;¹ c , de…nes a subset of C, with a boundary given by the hyperplane with normal (q(x); (1 ¡ q(x)) and the number x + c¹. In economic terms, for a given x, the set includes all initial endowments for which the action (¹ c; c¹; x) is market balanced, as well as all endowments that are insu¢cient for the realization of the action (¹ c; c¹; x).


52

Results In Proposition 12 below we look at some properties of the envelope and the at-least-as-bad set. Proposition 12 For x ¸ 0, the sets Dc¹; Ec¹ and DF¤ c¹ satisfy: (a) Dc¹ = DF¤ c¹ (b) Ec¹ is a closed, convex set (c) if (eu ; ed ) 2 Ec¹ and 0 · ® · 1 then (®eu ; ®ed ) 2 Ec¹ (d) if (eu ; ed ) 2 Dc¹ and (®eu ; ®ed ) 2 Dc¹ then ® = 1 Proof: (a) First we prove that Dc¹ ½ DF¤ c¹ . Let (eu ; ed ) 2 Dc¹ so that there exists an x¹ such that Fc¹(eu ; ed ; x¹) = 0 and where x¹ = arg max Fc¹(eu ; ed ; x). If eu = ed x¸0

then, by Corollary 11, x¹ = 0, so that (eu ; ed ) = (¹ c; c¹) 2 DF¤ c¹ . If eu > ed then, @F by Corollary 11, x¹ > 0, and so (eu ; ed ; x¹) = q 0 (¹ x)(eu ¡ ed ) ¡ 1 = 0. Hence @x (eu ; ed ) 2 DF¤ c¹ . Next we prove that DF¤ c¹ ½ Dc¹. Let (eu ; ed ) 2 DF¤ c¹ . If (eu ; ed ) = (¹ c; c¹) then Fc¹(eu ; ed ; x) = ¡x, which has its maximum for x¹ = 0 with Fc¹(eu ; ed ; x¹) = 0, so that (eu ; ed ) 2 Dc¹. If (eu ; ed ) 6= (¹ c; c¹) then eu > ed and by Corollary 11 x¹ > 0. From our maintained assumptions, q 00 (x)(eu ¡ ed ) < 0, and so x¹ maximizes Fc¹. Hence (eu ; ed ) 2 Dc¹ also in this case. To prove (b) note that from the de…nitions of Ec¹ and H¡ x;¹ c we have Ec¹ = \x¸0 H¡ x;¹ c Since H¡ x;¹ c is a convex, closed set, the set Ec¹ is also a closed and convex set, ¡ 0 being the intersection of sets H¡ x;¹ c for x ¸ 0. Note that if x 6= x then Hx;¹ c 6= ¡ Hx0 ;¹c : (c) Since Dc¹ is a convex set including the points (0; 0) and (eu ; ed ) we must have ®(eu ; ed ) + (1 ¡ ®)(0; 0) = ®(eu ; ed ) 2 Dc¹


53

(d) To get a contradiction assume (eu ; ed ) 2 Ec¹ and (®eu ; ®ed ) 2 Ec¹ and ® 6= 1. Then, if ® > 1, there exists an x so that q(x)®eu + (1 ¡ q(x))®ed ¡ x ¡ c¹ > 0. Hence (®eu ; ®ed ) 2 = Dc¹ ¾ Ec¹. If ® < 1, then q(x)®eu + (1 ¡ q(x))®ed ¡ x ¡ c¹ < 0 for all x, and so (®eu ; ®ed ) 2 = Ec¹. ¤ Proposition 12 characterizes the envelope in an economy with moral hazard. In (a) we prove that the two sets Dc¹ and DF¤ c¹ are identical. The reason for proving this identity is twofold. Firstly, Dc¹ is easier to use in the characterization of the envelope as made in the rest of Proposition 12. Secondly, Dc¹ also turns out to be more convenient when proving the subsequent propositions and corollaries. In (b) we prove that Ec¹, the lower contour set of (¹ c; c¹), is a closed and convex set. Two consequences of this …nding should be noted. Firstly, since Dc¹ is the boundary of Ec¹, the envelope, in accordance with Proposition 12 (a), is the boundary of a closed and convex set, which is not true generally (see for instance the example with the circles given earlier). Secondly the fact that Ec¹, the lower contour set of (¹ c; c¹), is convex should be contrasted with the standard model of exchange in which the upper contour set is generally required to be convex. The reason for this anomaly is that in a comparable model of exchange with two commodities, the convexity of the upper contour set simply re‡ects a diminishing marginal rate of substitution between these two commodities. The complication in our model is that, apart from the two state-contingent commodities, there is also the precautionary good, the level of which is given by x, and which is only implicitly present in the two-dimensional space of the envelope.62 The di¤erence to classical demand theory thus arises because the level of x varies along the envelope (we regard endowments which can exactly attain (¹ c; c¹) when x is chosen optimally), in combination with the fact that the level of x has a direct e¤ect on an agent’s utility. 62

By implicitly present we mean that the normal of the hyperplane tangent to any (¹ eu ; e¹d ) 2 DF¤ c¹ is given by (q(¹ x); (1 ¡ q(¹ x))), where (¹ c; c¹; x ¹) is the …rst best action for (¹ eu ; e¹d ). Given our maintained assumptions on q the level of x can thus be uniquely determined at each point of the envelope, although it is not directly visible.


54

In (c) we note an additional feature of the envelope and of the lower contour set of (¹ c; c¹), namely that all endowments along a ray from the origin of coordinates (eu ; ed ) = (0; 0) to an endowment belonging to Ec¹ (and thus, in particular, to an endowment which belongs to the envelope) also belongs to the lower contour set Ec¹. Finally, in (d), we conclude that the envelope is not thick. Thus, for a given level of x, there cannot be two endowments (^ eu ; e^d ) and (~ eu ; e~d ) with e^u ¸ e~u and e^d ¸ e~d , with at least one of the inequalities strict, which belong to the same envelope. Thus we can rule out local satiability, in the sense that for a given level of x more of either state-contingent commodity is always preferred as an initial endowment. In the next proposition we analyze the intersection of Dc¹ and the set of endowments for which a given action is market balanced. Proposition 13 Let (¹ eu ; e¹d ) 2 Dc¹ and let (¹ c; c¹; x¹) be the …rst best action for (¹ eu ; e¹d ) and let © ª Hx¹;¹c = (eu ; ed ) 2 R2 j q(¹ x)eu + (1 ¡ q(¹ x)ed ¡ x¹ ¡ c¹ = 0 Then Ec¹ \ Hx¹;¹c = f(¹ eu ; e¹d )g

Proof: Clearly (¹ eu ; e¹d ) 2 Ec¹ \ Hx¹;¹c . Assume, in order to get a contradiction, that (eu ; ed ) 2 Ec¹ \ Hx¹;¹c and (eu ; ed ) 6= (¹ eu ; e¹d ). Then q(¹ x)eu + (1 ¡ q(¹ x)ed ¡ x¹ ¡ c¹ = 0

since (eu ; ed ) 2 Hx¹;¹c

q(x)eu + (1 ¡ q(x))ed ¡ x ¡ c¹ · 0 for x ¸ 0

since (eu ; ed ) 2 Ec¹

so that (¹ c; c¹; x¹) is the …rst best action for (eu ; ed ): If x¹ = 0 then e¹u = e¹d and eu = ed so that (¹ eu ; e¹d ) = (eu ; ed ) in contradiction to our assumptions. If x¹ > 0 then q0 (¹ x)[¹ eu ¡ e¹d ] ¡ 1 = 0 and

q0 (¹ x)[eu ¡ ed ] ¡ 1 = 0

which implies e¹u ¡ e¹d = eu ¡ ed . Since (eu ; ed ) and (¹ eu ; e¹d ) both belong to Hx¹;¹c we get again the contradiction (¹ eu ; e¹d ) = (eu ; ed ): ¤


55

In Proposition 13 we consider the relation between initial endowments belonging either to Dc¹ or to Hx¹;¹c or to both these sets. All initial endowments belonging to Dc¹ (and thus to the envelope DF¤ c¹ ) have in common the property that (¹ c; c¹) is attainable as a …nal net consumption when the level of the preventive action x is chosen optimally. Furthermore we know that for any initial endowment belonging to Hx¹;¹c , the action (¹ c; c¹; x¹) is market balanced (see Figure 14).

ed, cd

Hx,c

c+x

Dc

c ed

Ec c c+x

eu

eu, cu

Figure 14: Graphical representation of Proposition 13 The conclusion stated in Proposition 13 is that it is only (¹ eu ; e¹d ), the initial endowment having (¹ c; c¹; x¹) as a …rst best action, that belongs to both these sets. This inference could be divided into two separate implications. Firstly, except for (¹ eu ; e¹d ), any initial endowment for which (¹ c; c¹) is the optimal …nal net consumption, the action (¹ c; c¹; x¹) is not market balanced. Secondly, if (¹ c; c¹; x¹) is a …rst best action for an initial endowment, then, except for the endowment (¹ eu ; e¹d ), the net consumption (¹ c; c¹) is not the best possible. In Corollary 14 and Corollary 15 we discuss immediate consequences of these two implications.


56

Corollary 14 For (¹ c; c¹; x¹) 2 A there is a unique economy (¹ eu ; e¹d ) such that (¹ c; c¹; x¹) is the …rst best action for (¹ eu ; e¹d ): Proof: The action(¹ c; c¹; x¹) is a …rst best action for (¹ eu ; e¹d ) if and only if q(¹ x)¹ eu + (1 ¡ q(¹ x)¹ ed ¡ x¹ ¡ c¹ = 0

q(x)¹ eu + (1 ¡ q(x))¹ ed ¡ x ¡ c¹ · 0 for x ¸ 0 which is equivalent to Ec¹ \ Hx¹;¹c = f(¹ eu ; e¹d )g. By Proposition 13 there is an (¹ eu ; e¹d ) such that Ec¹ \ Hx¹;¹c = f(¹ eu ; e¹d )g : ¤ Corollary 14 states that for any given combination of c and x, there is a unique initial endowment (eu ; ed ) for which (c; c; x) is a …rst best action. This could be regarded as an extension of Proposition 1 where the reverse relation was proved: given an initial endowment (eu ; ed ) there was a unique combination of c and x so that (c; c; x) was a …rst best action for (eu ; ed ). The next corollary, also a direct consequence of the result of Proposition 13, in its extension states what should be regarded as the main …nding of this section. Corollary 15 Let (¹ c; c¹; x¹) and (^ c; c^; x^) be …rst best actions for (¹ eu ; e¹d ) and (^ eu ; e^d ) respectively. Furthermore let (¹ eu ; e¹d ) 2 Hx¹;¹c \ Ec¹ and (^ eu ; e^d ) 2 Hx¹;¹c nEc¹. Then c^ > c¹. Proof: From Proposition 13, Hx¹;¹c \ Ec¹ = f¹ eu ; e¹d g, so that (^ eu ; e^d ) 2 (Ec¹)c . This implies q(x)^ eu + (1 ¡ q(x))^ ed ¡ x > c¹ for some x Speci…cally, by choosing x^, we have that c^ = q(^ x)^ eu + (1 ¡ q(^ x))^ ed ¡ x^ > c¹ ¤ The gist of Corollary 15 is that in the set of initial endowments for which (¹ c; c¹; x¹) is a market balanced action, (¹ eu ; e¹d ) is the one endowment that yields the lowest …nal net consumption, and thus, as a consequence of the maintained assumptions on u, the lowest utility. To grasp the interesting implication of this …nding, we need to relate it to the risk associated with an initial endowment. In Figure 15 the situation


57

is visualized. Since the normal to the hyperplane Hx¹;¹c is non-negative in both its elements (there cannot be negative probabilities), the direction of the arrow in Figure 15 will describe initial endowments with an increasing di¤erence between the outcomes in the two states, that is (eu ¡ ed ), which for an agent will represent an increasing risk. Thus, since the risk increases along Hx¹;¹c in the direction of the arrow, and since (¹ eu ; e¹d ) will not generally be at any of the endpoints of Hx¹;¹c , (¹ eu ; e¹d ) will, generally, be surrounded by other initial endowments (eu ; ed ) 2 Hx¹;¹c which represent both higher and lower risks than the risk generated by (¹ eu ; e¹d ) itself. Now we relate the level of risk, as described above, to the conclusion given in Corollary 15, which stated that of all endowments included in Hx¹;¹c , (¹ eu ; e¹d ) is the one that yields the lowest utility. In particular, it is thus possible to increase the risk of the average agent (and thus of the economy as a whole) by moving from (¹ eu ; e¹d ) along Hx¹;¹c and in the direction of the arrow, and still get a higher average (and total) utility. Since we have kept the rest of the economy constant, in the sense that we are only considering endowments which can precisely attain (¹ c; c¹) when the probability of not having an income shortfall is …xed at q(¹ x), this is a highly counter-intuitive property. A higher risk should not induce a higher utility, since neither party to the insurance transaction exhibits any risk-loving properties (and someone has to carry the risk). It is beyond the scope of this paper to penetrate the issue of why this result arises. However, a hint at a potential explanation will be given. As was indicated at the beginning, the probability function exhibits what could be regarded as increasing returns to scale of taking care. By this was meant the fact that the cost of reducing an income shortfall to the level (1 ¡ q(¹ x)) is x¹ regardless of the magnitude of the potential accident insured against. As an example, the cost of reducing the probability of a ‡ood from 75 to 50 percent will be equal to the cost of reducing the probability of a bicycle theft from 75 to 50 percent, which is not an intuitively appealing property. The reason we have this peculiarity of increasing returns to scale of taking care thus seems to be the fact that the probability function does not consider the size of the potential accident, as given by the di¤erence in the outcomes


58

ed, cd

c+x c ed

Hx,c Dc

Ec c c+x

eu

eu, cu

Figure 15: The variation of risk along the set Hx¹;¹c of the two states. As a direct consequence, an interesting idea would be to let the probability level of an accident be increasing in risk, as given by (eu ¡ed ), in addition to letting it be decreasing in the precautionary action x. In this way, for a certain x, the probability of an accident would be reduced by a smaller amount if the accident is large than if it is a small one. In the example above, choosing x¹ would perhaps still reduce the probability of a bicycle theft from 75 to 50 percent, while x¹ may only be able to reduce the probability of a ‡ood to 74 percent. Economically, this is a much more appealing property. The precise details of such an approach, however, and the possible implications it would have on the result of Corollary 15, has to be postponed to another paper. In order to get a more pronounced intuition of what has been discussed in this section we provide the following example, where a …rst best action as well as the envelope of a family of curves will be explicitly calculated. Example of the envelope and of a …rst best action First consider the family of curves given by Fc¹(eu ; ed ; x) = q(x)eu + (1 ¡ q(x))ed ¡ x ¡ c¹ = 0


59

The envelope, DFc¹ , of this family of curves is de…ned by

DFc¹

8 > > > > >
> ¯ > > ¯ > = ¯ ¯ Fc¹(eu ; ed ; x) = 0 and = (eu ; ed ) ¯ > > ¯ > > > > ¯ @F (e ; e ; x) > > u d > > ¯ : ; = 0 ¯ @x

Since the envelope belongs to (eu ; ed )-space, we need to eliminate the parameter x from these two equations (note that, for simplicity, we use ± for (eu ¡ ed )): Fc¹(eu ; ed ; x) ´ ed + q(x)± ¡ x ¡ c¹ = 0

(6)

@F (eu ; ed ; x) ´ q 0 (x)± ¡ 1 = 0 @x

We now set q(x) = x® , so that, from the second equality, we have that 1 ± which implies ®x®¡1 =

1

®

x = [®±] 1¡® and q(x) = [®±] 1¡®

(7)

Inserting (7) into (6), we get ed = c¹ ¡ ® = c¹ ¡ ®

® 1¡® ® 1¡®

= c¹ ¡ (®

For ® =

1 2

± ±

® 1¡®

® 1¡® 1 1¡®

±

1¡® 1¡®

+®

¡®

1 1¡®

+® 1 1¡®

)±

±

1 1¡®

1 1¡®

we can solve explicitly: 1 1 ed = c¹ ¡ ( ¡ )± 2 2 4 1 2 = c¹ ¡ ± 4

Since ± = (eu ¡ ed ) we have

1 1¡®

±

1 1¡®


1 ed = c¹ ¡ (eu ¡ ed )2 4 which implies p ed = eu ¡ 2 § 4 ¡ 4eu + 4¹ c

60

(8) (9)

Equation 9 explicitly de…nes the envelope of the family of curves given 1 by Fc¹ when we set q(x) = x 2 . By rewriting equation (8) a notable symmetrical property of the envelope becomes apparent: 1 1 [(eu + ed ) ¡ (eu ¡ ed )] = c¹ ¡ (eu ¡ ed )2 2 4 Thus we have that ¡2(eu + ed ) = (eu ¡ ed )2 ¡ 2(eu ¡ ed ) ¡ 4¹ c completing the square yields

¡2(eu + ed ) = (eu ¡ ed )2 ¡ 2(eu ¡ ed ) + 1 ¡ 1 ¡ 4¹ c which implies 1 1 (eu + ed ) = ¡ ((eu ¡ ed ) ¡ 1)2 + + 2¹ c 2 2

(10)

Equation (10) describes the envelope from a somewhat new perspective. Thinking in terms of a coordinate system, we have exchanged ed and eu for (eu + ed ) and (eu ¡ ed ) on the y-axis and the x-axis respectively This implies that we use the same origin of coordinates as in the original case (since (eu + ed ) = (eu ¡ ed ) = 0 implies eu = ed = 0). However, we have now rotated the whole coordinate system 45 degrees counter-clockwise around the origin of coordinates (and changed the scale somewhat),63 so that the 45 degree line in the original (eu ; ed )-space is now the y-axis. Interestingly, by looking at equation (10), we can conclude the envelope has a symmetrical feature (since it is a quadratic function). For a visualization of the above calculations we consider a particular initial endowment given by (¹ eu ; e¹d ) = (3; 2). From Corollary 11 (c) we know 63

To have the same scale in the new coordinate system as in the original one would require using p12 (eu + ed ) on the y-axis and p12 (eu ¡ ed ) on the x-axis in accordance with Pythagorean theory.


61

that the …rst best action (¹ c; c¹; x¹) corresponding with this endowment must meet: 1 ¡1 x¹ 2 = 1 2 which implies 1 1 x¹ = and q(¹ x) = 4 2

(11)

Inserting (11) and the initial endowment (¹ eu ; e¹d ) = (3; 2) into the market balance condition (6) we obtain c¹ by c¹ =

1 1 1 9 ¢3+ ¢2¡ = 2 2 4 4

Thus the …rst best action for (¹ eu ; e¹d ) = (3; 2) is given by (¹ c; c¹; x¹) = ( 94 ; 94 ; 14 ). The envelope derived above now has the equation ed = eu ¡ 2 §

p (13 ¡ 4eu )

(12)

Those initial endowments which market balance given c¹ and x¹ are those that meet: 1 1 1 9 = eu + ed ¡ 4 2 2 4 which implies (13)

ed = 5 ¡ eu

Equations (12) and (13) de…ne the envelope DF 9 and H 1 ; 9 , the set of 4

4 4

endowments for which ( 94 ; 94 ; 14 ) is a market balanced action, respectively, and these sets are depicted in Figure 16 below. A few things should be noted regarding this …gure. Firstly, it is not the entire envelope which is shown; it extends both left of the 45 degree line and below the x-axis. Thus the envelope is only shown for initial endowments which belong to the possible net consumption set, C. In the terminology introduced in the paper, it is DF¤ 2:25 rather than DF2:25 that we see in Figure 16. Secondly we see that the envelope has a positive slope for some initial endowments. This is because in this example, maintained assumption Q5 is


62

ed, cd

5 4 3

c, c

2 1

eu, ed Hx,c

Ec 1

2

Dc 3

4

eu, cu 5

6

Figure 16: Visualization of the example 1

not satis…ed for q(x) = x 2 . If it was, the slope of the envelope could only have taken negative values. In Figure 16 we also note that the envelope and the market balance line Hx¹;¹c have only (¹ eu ; e¹d ) = (3; 2) as a common tangency point. This was the result proved in Proposition 13. Since Dc¹ is the boundary of Ec¹, the at-leastas-bad set, this implies that any initial endowment (eu ; ed ) 6= (¹ eu ; e¹d ) along Hx¹;¹c yields an optimal net consumption that is higher than for (¹ eu ; e¹d ). This was proved in Proposition 15. To depict the symmetry of the envelope, as discussed above, we now turn to Figure 17. In the left part of this …gure we have indicated the di¤erence between DF¤ 2:25 , which is represented by the solid line (depicted in Figure 16), and DF2:25 (the unmodi…ed envelope) which also includes the dotted part of the curve. In the right portion of Figure 17 we have rotated the coordinate system in accordance with the above discussion in order to graphically represent the actuality that the unmodi…ed envelope DF2:25 is in fact symmetrical around the vertical line (eu ¡ ed ) = 1.

5 CONCLUSIONS

63

eu + ed

ed, cd

5 4 3 2 -4 -3

eu, cu

-1 -1 -2 -3 -4

1 1 2 3 4 5

eu - ed

Figure 17: The symmetry of the envelope

5

Conclusions

We have been able to show in this paper that it is actually possible to reduce the number of variables in moral hazard models to three. This was made possible by focusing exclusively upon the net consumption, rather than including all variables that in‡uence it, which has been the case in conventional models. The results stemming from this simpli…ed model will therefore be considerably easier to calculate, and much less complicated to interpret. We have also discussed an alternative interpretation of the institutional framework with a continuum of agents, resulting in an economy with exclusively individual risk and with no insurance company. We have then used this model to demonstrate that the property of monotonicity in care disappears when the consumption is uniform over states, a result which was used when proving that in the uniform consumption set, all bundles with the same level of income belong to the same indi¤erence class. Finally we also proved that the utility function cannot be quasi-concave, a result which has previously been shown to have far-reaching implications for the existence of an equilibrium. We have, thirdly, examined the conditions for a …rst best action to exist.

5 CONCLUSIONS

64

It was shown that our maintained assumptions su¢ced to ensuring a unique …rst best action, while there could potentially be many …rst best actions if both the prevention function and the utility function were concave but not strictly concave. Finally we were able to manifest a patent shortcoming in the way moral hazard models in insurance markets have traditionally been designed. It was proved, using the envelope concept as a tool, that an increase in the total risk of an economy, with other relevant variables held constant, can actually increase the total and average utility. It was then suggested that this counterintuitive result might be due to the fact that the prevention function exhibits increasing returns to scale of taking care; no matter the size of an accident, the cost of reducing the probability of its occurrence by a certain percentage will be the same.

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65

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