Analyzing the Demand for Deductible Insurance


 Myrtle Kelly
 2 years ago
 Views:
Transcription
1 Journal of Risk and Uncertainty, 18: Kluwer Academic Publishers. Manufactured in The Netherlands. Analyzing the emand for eductible Insurance JACK MEYER epartment of Economics, Michigan State Uni ersity MICAEL B. ORMISTON epartment of Economics, Arizona State Uni ersity, Tempe, Arizona Abstract This paper investigates aspects of insurance demand related to deductible insurance. In particular, an important issue concerning analysis of the optimal deductible level is resolved. A simple sufficient restriction on the pricing of insurance is given which ensures that the second order condition for choosing the expected utility maximizing deductible level is met for any risk averse decision maker. This restriction is stated and its sufficiency is demonstrated using the level of expected indemnification rather than the level of the deductible as the choice variable in the decision model. Key words: deductible insurance, second order conditions JEL Classification: 81, G The demand for insurance has been the subject of extensive effort in recent years. One portion of this literature is concerned with the optimal form of indemnification for insurance policies, and has shown that, from the viewpoint of the insured, deductible policies are optimal under quite general circumstances. This has been accomplished both in the expected utility maximization model and in nonexpected utility decision models that preserve second degree stochastic dominance. 1 This being the case, research concerned with the quantity of insurance demanded has frequently focused on determining the optimal deductible level, and how this level changes in response to changes in wealth, the price of insurance, and the distribution of the random loss variable. While many interesting results concerning the optimal deductible level and its response to shifts in model parameters have been obtained, the analysis has typically been carried out assuming that second order conditions for the maximization are met, without precisely stating what this assumption implies about the decision model in question. An important unresolved question asks which assumptions on the decision model are sufficient to ensure that the second order condition for choosing the expected utility maximizing deductible level is met for all risk averse decision makers? The main contribution of this paper is an answer to this question. In particular, here it is shown that a simple assumption concerning the
2 4 MEYER AN ORMISTON pricing of insurance, that the price of deductible insurance is a convex function of expected indemnification, is sufficient to guarantee that the second order condition is satisfied globally when preferences exhibit risk aversion. 3 The paper proceeds as follows. Section 1 introduces the notation and assumptions used throughout the paper and reviews the standard specification of the deductible insurance demand model used and analyzed in the literature. Section proposes an alternative methodology for examining the demand for deductible insurance. The primary change from the standard model that is reviewed in Section 1 is that a different metric or parameter is used to identify a particular deductible insurance policy and to quantify a particular insurance level. The usual measure, the deductible level, is replaced by the level of expected indemnification. For deductible insurance these two measures are related to one another in a one to one fashion; thus, finding the optimal level for one is equivalent to finding the optimal level for the other. The main theorem of the section shows that if the price of deductible insurance is a convex function of the level of expected indemnification and the decision maker is risk averse, then maximizing expected utility by solving the first order condition yields a global maximum. 4 Finally, Section 3 summarizes the results and concludes with remarks concerning linear pricing models. 1. Assumptions and notation The decision maker is assumed to be endowed with initial wealth w and a risky asset whose value is M when no loss occurs. This risky asset is subject to random loss of size x, where x has support in, M and is distributed continuously. The cumulative distribution function describing x is denoted by FŽ x. and the density function by fž x.. An insurance policy, IŽ x., P 4, against this loss is composed of an indemnification function IŽ x. providing reimbursement of size IŽ x. when loss x occurs and a price or premium for insurance, P. IŽ x. is assumed to satisfy IŽ x. x. With insurance, random final wealth, z, is given by: z w M x IŽ x. P Ž 1. The deductible form of insurance is characterized by an indemnification function IŽ x. satisfying: if x IŽ x. ½ Ž Ž. x. if x where M is the level of the deductible. In this standard specification, the price for obtaining a policy with deductible is a function of and is denoted by the premium function, P Ž., which is assumed to be continuous, twice differentiable, and decreasing in. Thus, final wealth when insurance is of the
3 ANALYZING TE EMAN FOR EUCTIBLE INSURANCE 5 deductible form is given by: w M x Ž. if x z ½ Ž 3. w M Ž. if x Let U denote expected utility from final wealth; that is, U Euz, where už. is the decision maker s twice continuously differentiable von Neumann Morgenstern utility function. When is selected to maximize UŽ,. the first and second order conditions for an interior solution are U Ž *. and U Ž *., respectively, where * is the optimal level of deductible and U and U are given by: U U u Ž w M x. dfž x. M u Ž w M.Ž 1. dfž x. u Ž w M x. dfž x. u Ž w M.Ž 1. Ž 1 FŽ.. Ž 4. u Ž w M. fž. u Ž w M x.ž. u Ž w M x. dfž x. u w M 1 1 F u Ž w M. Ž 1 FŽ.. Ž 5. It has proven difficult to determine reasonable, interpretable, conditions on the various components of the decision model under which U is indeed negative. Schlesinger Ž presents the most extensive and complete analysis of this issue and gives a rather complicated necessary and sufficient condition for determining the sign of U locally. Among other things, this condition involves the first and second derivatives of the price function, the first and second derivatives of the utility function, and the expectation of the derivatives of the utility function. Most research since 1981 simply assumes U holds locally at the stationary points and proceeds, referring the reader to Schlesinger s analysis. Eeckhoudt et al. Ž 1991., for instance, state that, The second order condition is far from trivial and assumed to hold. and refer the reader to Schlesinger.
4 6 MEYER AN ORMISTON. An alternate method for determining the optimal deductible level In this section an alternate method of finding the EU maximizing level of is presented. The change involves identifying a particular deductible insurance policy, and hence a quantity demanded, using Q, the expected level of indemnification, rather than. For deductible insurance, expected indemnification, Q, is given by M Q E IŽ x. Ž x. fž x. dx. Ž 6. This function is strictly decreasing in for all values of such that FŽ. 1 and, hence, has an inverse which we denote by Ž Q.. Because Q and are one to one, finding the EU maximizing level for Q and then using to determine, is an indirect and alternate method for determining the optimal level for. Later in the analysis, the following properties of are used. First, because dq d FŽ. 1 where 1 FŽ. 1, Ž Q. 1 FŽ Second, differentiating one more time implies that fž. FŽ Thus, is a decreasing and convex function. To restate the decision problem so that the level of expected indemnification, Q, rather than the level of the deductible is the choice variable, the price of this deductible insurance must be written as a function of Q rather than. efine to be price for deductible insurance policies expressed as a function of the level of expected indemnification Q. 5 Clearly, the two functions used to represent the price of deductible insurance, and Ž., are such that Q Ž.. This implies that and Ž.. As is typical in insurance demand models, it is assumed that Ž. and that 1. The latter restriction implies that the premium increases by at least as much as the increase in expected indemnification when the level of insurance is altered. This condition is necessary for an interior solution to the optimization problem. 6 Using this notation, final wealth can be written as a function of expected indemnification Q and is given by: w M x if x z ½ Ž 7. w M if x Proceeding as in the standard model, let VQ denote expected utility from final wealth; that is, VQ Euz, where z is now a function of expected indemnification. Note that VQ UŽ Ž Q... It is assumed that the decision maker chooses Q to maximize VQ. For this optimization problem the first and second order
5 ANALYZING TE EMAN FOR EUCTIBLE INSURANCE 7 conditions are V Ž Q*. and V Ž Q*. Q QQ, respectively, where Q* is the optimal level of expected indemnification and VQ and VQQ are given by: V Q M u Ž w M x. dfž x. u Ž w M.Ž. dfž x. u Ž w M x. dfž x. u Ž w M.Ž. 1 F Ž. Ž 8. V u Ž w M x.ž. u Ž w M x. dfž x. QQ u Ž w M.Ž. 1 F Ž. u Ž w M. f Ž Q. 1 F Ž Q. 9 Clearly, V U Ž Q., and because Ž Q. Q, VQ if and only if U. Thus, the first order equations resulting from maximizing U or VQ yield the same solutions; that is VQ and U, respectively, identify the same deductible insurance alternatives as potential maxima. Turning now to the second order condition for each optimization problem, observe that V QQ U Q U Ž Q..Atlocal optima, the first order condition U is satisfied, and hence it is the case that U if and only if VQQ at that point. Thus, at the values for or Q identified by the first order conditions, these second order restrictions are identical. What is less apparent, and is the basis for Theorem 1 which follows, is that under simple and reasonable assumptions, VQQ for all alues for Q and for all risk a erse decision makers; that is, V is globally concave in Q for all concave uz. This is the case even though U is not globally concave in. Obviously, when VQQ for all Q, then this ensures that solving VQ yields a global maximum for Q, and hence the associated value for the deductible given by Ž Q.,isa global maximum for U as well. Thus, even though U is not concave in, the first order condition for maximizing U does indeed yield a global maximum. The global concavity of V is formally demonstrated in the following theorem.
6 8 MEYER AN ORMISTON Theorem 1. VŽ Q. is strictly conca e ifu Ž z., u Ž z.,, Ž Q. 1, and. Proof: Recall that Ž Q. 1 FŽ. 1 and fž. FŽ Substituting these into Ž. 9 in the last term gives V u Ž w M x.ž. u Ž w M x. dfž x. QQ u Ž w M.Ž. 1 F Ž. u Ž w M. 1 F Ž. Ž 1. which is strictly negative under the conditions of the theorem. QE Theorem 1 indicates that for deductible insurance, any Q* which solves the first order condition for EU maximization is a global maximum. Because is a strictly decreasing function of Q, the corresponding * Ž Q*. also globally maximizes UŽ.. That is, when u is strictly concave and the price of deductible insurance is an increasing and convex function of expected indemnification, both U and VQ have a global maximum, and at the maximum, the objective function is locally concave. VQ displays the additional property that it is globally concave, while U need not display this property. It is the case, however, that U is strictly quasiconcave in the level of the deductible. The following Corollary makes this point formally. Corollary 1. UŽ. is strictly quasiconca e ifu Ž z., u Ž z.,, Ž Q. 1, and. Proof: Recall that U is strictly quasiconcave if U whenever U. This follows directly from Theorem 1 and the fact that V QQ U Q U Ž Q.. 3. Concluding remarks Researchers frequently assume that the price of all insurance policies offered by sellers, whether of the deductible form or not, is proportional to expected indemnification; that is, Q, where Ž 1. is the loading factor and this price function applies to insurance of any form. Clearly this is a pricing assumption with premium schedules which are convex in expected indemnification and hence satisfy the conditions in Theorem 1. This fact has not been recognized by the many researchers in that they assume risk aversion and that price is proportional to expected indemnification, but then proceed to add to these assumptions the restriction that the second order condition must be satisfied. Of course, Theorem 1
7 ANALYZING TE EMAN FOR EUCTIBLE INSURANCE 9 indicates that the latter assumption is redundant. 7 That is, Theorem 1 demonstrates that the second order condition for choosing the EU maximizing deductible level is automatically satisfied whenever risk aversion and proportional pricing are assumed. This paper demonstrates an advantage of formulating the deductible insurance demand model using expected indemnification as the decision variable. With Q as the decision variable, it is easy to show that when the decision maker is risk averse and the insurance premium schedule is increasing and convex in Q, then the first order condition for choosing Q to maximize EU identifies a unique global maximum. Furthermore, once this fact is known, the same result holds in the standard formulation. That is, risk aversion and convexity in Q is sufficient for the first order condition for choosing to maximize EU to identify a unique global maximum. This fact can be demonstrated directly by converting the restrictions in Theorem 1 concerning how the premium varies with expected indemnification, Ž Q., into the corresponding restrictions on how the premium varies with the deductible level, Ž.. To see this note that Ž Q. 1 implies that Ž. FŽ. 1, and implies Ž. Ž. fž. ŽFŽ. 1.. It is easily verified that given these restrictions on Ž., the second order restriction, U, holds whenever the first order restriction, U, is satisfied; that is, at *. ence, this first order condition, if satisfied, identifies a global maximum. 8 To interpret these complex restrictions on Ž., however, involves returning to the relationship between price and expected indemnification. Thus, the primary reason for conducting the analysis using expected indemnification as the decision variable is to allow reasonable and interpretable conditions to be stated as sufficient for the maximization of EU. Acknowledgments The authors thank Ed Schlee, an anonymous referee, and the participants of the 8th International Conference on the Foundations and Applications of Utility, Risk, and ecision Theories for their valuable comments and suggestions. This research was funded in part by the Arizona State University, College of Business, Summer Research Grant Program. Notes 1. Examples of papers demonstrating the optimality of the deductible form for insurance in an expected utility setting include Arrow Ž 1971, and Raviv Ž Gollier and Schlesinger Ž demonstrate the optimality of the deductible form for insurance in a nonexpected utility setting.. Papers examining factors affecting the demand for deductible insurance include Pashigan et al. Ž 1966., Mossin Ž 1968., Gould Ž 1969., Schlesinger Ž 1981., Eeckhoudt et al. Ž 1991., Eeckhoudt and Gollier Ž and Schlee Ž to name a few.
8 3 MEYER AN ORMISTON 3. Meyer and Ormiston Ž show that these same pricing assumptions are necessary for the optimality of the deductible form of indemnification. 4. The assertions made by Schlee Ž in footnotes 8 and 15 yield a similar conclusion concerning a local maximum; however, no proof is given. 5. It is important to recognize that representing the price of deductible insurance as a function of Q in no way assumes that all insurance contracts are priced by this function of Q. Other indemnification functions could display a different relationship between price and Q than does the set of deductible indemnification functions. 6. It is also the case that Ž. and 1 together imply that expected profit from providing insurance is nonnegative. This is necessary to guarantee that a risk neutral insurer will, in fact, be willing to offer the insurance. 7. See, for example, Mossin Ž 1968., oherty and Schlesinger Ž 1983., Eeckhoudt, Gollier and Schlesinger Ž 1991., and Eeckhoudt and Gollier Ž Each assumes risk aversion and linear pricing. In addition, each goes on to assume that the second order conditions hold, which in light of Theorem 1 is redundant. 8. Substituting these inequalities into Ž. 5 and rearranging gives the desired result. References Arrow, K. Ž Essays in the Theory of Risk Bearing. Markham. Arrow, K. Ž Optimal Insurance and Generalized eductibles, Scandina ian Actuarial Journal, 1 4. oherty, N. A., and. Schlesinger. Ž The Optimal eductible for an Insurance Policy When Initial Wealth Is Random, Journal of Business 56, Eeckhoudt, L., and C. Gollier. Ž Risk. New York: arvesterwheatsheaf. Eeckhoudt, L., C. Gollier, and. Schlesinger. Ž Increases in Risk and eductible Insurance, Journal of Economic Theory 55, Gollier, C., and. Schlesinger. Ž Arrow s Theorem on the Optimality of eductibles: A Stochastic ominance Approach, Economic Theory. Gould, J. P. Ž The Expected Utility ypothesis and the Selection of Optimal eductibles for a Given Insurance Policy, Journal of Business 4, Meyer, J., and M. B. Ormiston. Ž The Pricing of Optimal Insurance Contracts, Proceedings from FUR 8, Mons Belgium, July 1997 Ž forthcoming.. Mossin, J. Ž Aspects of Rational Insurance Purchasing, Journal of Political Economy 76, Pashigan, B. P., L. Schkade, and G. Menefee. Ž The Selection of an Optimal eductible for a Given Insurance Policy, Journal of Business 39, Raviv, A. Ž The esign of an Optimal Insurance Policy, American Economic Re iew 69, Schlee, E. Ž The Comparative Statics of eductible Insurance in Expected and NonExpected Utility Theories, The Gene a Papers on Risk and Insurance Theory, Schlesinger,. Ž The Optimal Level of eductibility in Insurance Contracts, Journal of Risk and Insurance 48,
A Portfolio Model of Insurance Demand. April 2005. Kalamazoo, MI 49008 East Lansing, MI 48824
A Portfolio Model of Insurance Demand April 2005 Donald J. Meyer Jack Meyer Department of Economics Department of Economics Western Michigan University Michigan State University Kalamazoo, MI 49008 East
More informationOn Compulsory PerClaim Deductibles in Automobile Insurance
The Geneva Papers on Risk and Insurance Theory, 28: 25 32, 2003 c 2003 The Geneva Association On Compulsory PerClaim Deductibles in Automobile Insurance CHUSHIU LI Department of Economics, Feng Chia
More informationRegret and Rejoicing Effects on Mixed Insurance *
Regret and Rejoicing Effects on Mixed Insurance * Yoichiro Fujii, Osaka Sangyo University Mahito Okura, Doshisha Women s College of Liberal Arts Yusuke Osaki, Osaka Sangyo University + Abstract This papers
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationBasic Utility Theory for Portfolio Selection
Basic Utility Theory for Portfolio Selection In economics and finance, the most popular approach to the problem of choice under uncertainty is the expected utility (EU) hypothesis. The reason for this
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationThe Effect of Ambiguity Aversion on Insurance and Selfprotection
The Effect of Ambiguity Aversion on Insurance and Selfprotection David Alary Toulouse School of Economics (LERNA) Christian Gollier Toulouse School of Economics (LERNA and IDEI) Nicolas Treich Toulouse
More informationLecture notes for Choice Under Uncertainty
Lecture notes for Choice Under Uncertainty 1. Introduction In this lecture we examine the theory of decisionmaking under uncertainty and its application to the demand for insurance. The undergraduate
More informationPortfolio Allocation and Asset Demand with MeanVariance Preferences
Portfolio Allocation and Asset Demand with MeanVariance Preferences Thomas Eichner a and Andreas Wagener b a) Department of Economics, University of Bielefeld, Universitätsstr. 25, 33615 Bielefeld, Germany.
More information2. Information Economics
2. Information Economics In General Equilibrium Theory all agents had full information regarding any variable of interest (prices, commodities, state of nature, cost function, preferences, etc.) In many
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 PrincipalAgent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationFinancial Markets. Itay Goldstein. Wharton School, University of Pennsylvania
Financial Markets Itay Goldstein Wharton School, University of Pennsylvania 1 Trading and Price Formation This line of the literature analyzes the formation of prices in financial markets in a setting
More informationMULTIPLE LOSSES, EX ANTE MORAL HAZARD, AND THE IMPLICATIONS FOR UMBRELLA POLICIES
C The Journal of Risk and Insurance, 2005, Vol. 72, No. 4, 525538 MULTIPLE LOSSES, EX ANTE MORAL HAZARD, AND THE IMPLICATIONS FOR UMBRELLA POLICIES Michael Breuer ABSTRACT Under certain cost conditions
More informationThe Values of Relative Risk Aversion Degrees
The Values of Relative Risk Aversion Degrees Johanna Etner CERSES CNRS and University of Paris Descartes 45 rue des SaintsPeres, F75006 Paris, France johanna.etner@parisdescartes.fr Abstract This article
More informationName. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.
Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price
More informationOptimal Reinsurance Arrangements Under Tail Risk Measures
Optimal Reinsurance Arrangements Under Tail Risk Measures Carole Bernard, and Weidong Tian 6th March 2007 Abstract Regulatory authorities demand insurance companies to control the risks by imposing stringent
More informationOn Prevention and Control of an Uncertain Biological Invasion H
On Prevention and Control of an Uncertain Biological Invasion H by Lars J. Olson Dept. of Agricultural and Resource Economics University of Maryland, College Park, MD 20742, USA and Santanu Roy Department
More informationOn the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information
Finance 400 A. Penati  G. Pennacchi Notes on On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information by Sanford Grossman This model shows how the heterogeneous information
More informationOptimal insurance contracts with adverse selection and comonotonic background risk
Optimal insurance contracts with adverse selection and comonotonic background risk Alary D. Bien F. TSE (LERNA) University Paris Dauphine Abstract In this note, we consider an adverse selection problem
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationLecture 13: Risk Aversion and Expected Utility
Lecture 13: Risk Aversion and Expected Utility Uncertainty over monetary outcomes Let x denote a monetary outcome. C is a subset of the real line, i.e. [a, b]. A lottery L is a cumulative distribution
More information3 Introduction to Assessing Risk
3 Introduction to Assessing Risk Important Question. How do we assess the risk an investor faces when choosing among assets? In this discussion we examine how an investor would assess the risk associated
More informationExplaining Insurance Policy Provisions via Adverse Selection
The Geneva Papers on Risk and Insurance Theory, 22: 121 134 (1997) c 1997 The Geneva Association Explaining Insurance Policy Provisions via Adverse Selection VIRGINIA R. YOUNG AND MARK J. BROWNE School
More information1 Uncertainty and Preferences
In this chapter, we present the theory of consumer preferences on risky outcomes. The theory is then applied to study the demand for insurance. Consider the following story. John wants to mail a package
More informationExistence of Sunspot Equilibria and Uniqueness of Spot Market Equilibria: The Case of Intrinsically Complete Markets
Existence of Sunspot Equilibria and Uniqueness of Spot Market Equilibria: The Case of Intrinsically Complete Markets Thorsten Hens 1, Janós Mayer 2 and Beate Pilgrim 3 1 IEW, Department of Economics, University
More informationCost Minimization and the Cost Function
Cost Minimization and the Cost Function Juan Manuel Puerta October 5, 2009 So far we focused on profit maximization, we could look at a different problem, that is the cost minimization problem. This is
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More informationMathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions
Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions
More informationGame Theory: Supermodular Games 1
Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike 4.0 1 / 22 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22 Motivation I several solution
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationMidterm Exam:Answer Sheet
Econ 497 Barry W. Ickes Spring 2007 Midterm Exam:Answer Sheet 1. (25%) Consider a portfolio, c, comprised of a riskfree and risky asset, with returns given by r f and E(r p ), respectively. Let y be the
More informationA Generalization of the MeanVariance Analysis
A Generalization of the MeanVariance Analysis Valeri Zakamouline and Steen Koekebakker This revision: May 30, 2008 Abstract In this paper we consider a decision maker whose utility function has a kink
More informationInsurance. Michael Peters. December 27, 2013
Insurance Michael Peters December 27, 2013 1 Introduction In this chapter, we study a very simple model of insurance using the ideas and concepts developed in the chapter on risk aversion. You may recall
More informationEconomics 401. Sample questions 1
Economics 40 Sample questions. Do each of the following choice structures satisfy WARP? (a) X = {a, b, c},b = {B, B 2, B 3 }, B = {a, b}, B 2 = {b, c}, B 3 = {c, a}, C (B ) = {a}, C (B 2 ) = {b}, C (B
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore
More informationOptimal Risk Sharing With Limited Liability
Optimal Risk Sharing With Limited Liability Semyon Malamud, Huaxia Rui, and Andrew Whinston Abstract We solve the general problem of optimal risk sharing among a finite number of agents with limited liability.
More informationCompulsory insurance and voluntary selfinsurance: substitutes or complements? A matter of risk attitudes
Compulsory insurance and voluntary selfinsurance: substitutes or complements? A matter of risk attitudes François Pannequin, Ecole Normale Supérieure de Cachan and Centre d Economie de la Sorbonne pannequin@ecogest.enscachan.fr
More informationSCREENING IN INSURANCE MARKETS WITH ADVERSE SELECTION AND BACKGROUND RISK. Keith J. Crocker
SCREENING IN INSURANCE MARKETS WIT ADVERSE SEECTION AND BACKGROUND RISK Keith J. Crocker University of Michigan Business School 70 Tappan Street Ann Arbor, MI 4809 and Arthur Snow Department of Economics
More informationC2922 Economics Utility Functions
C2922 Economics Utility Functions T.C. Johnson October 30, 2007 1 Introduction Utility refers to the perceived value of a good and utility theory spans mathematics, economics and psychology. For example,
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationGeneral Equilibrium Theory: Examples
General Equilibrium Theory: Examples 3 examples of GE: pure exchange (Edgeworth box) 1 producer  1 consumer several producers and an example illustrating the limits of the partial equilibrium approach
More informationThe Stewardship Role of Accounting
The Stewardship Role of Accounting by Richard A. Young 1. Introduction One important role of accounting is in the valuation of an asset or firm. When markets are perfect one can value assets at their market
More informationProspect Theory and the Demand for Insurance
Prospect Theory and the emand for Insurance avid L. Eckles and Jacqueline Volkman Wise ecember 211 Abstract We examine the effect of prospect theory preferences on the demand for insurance to determine
More informationA Simple Model of Price Dispersion *
Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 112 http://www.dallasfed.org/assets/documents/institute/wpapers/2012/0112.pdf A Simple Model of Price Dispersion
More informationAsset Pricing. Chapter IV. Measuring Risk and Risk Aversion. June 20, 2006
Chapter IV. Measuring Risk and Risk Aversion June 20, 2006 Measuring Risk Aversion Utility function Indifference Curves U(Y) tangent lines U(Y + h) U[0.5(Y + h) + 0.5(Y h)] 0.5U(Y + h) + 0.5U(Y h) U(Y
More informationOptimal demand management policies with probability weighting
Optimal demand management policies with probability weighting Ana B. Ania February 2006 Preliminary draft. Please, do not circulate. Abstract We review the optimality of partial insurance contracts in
More informationEffects of Subsidized Crop Insurance on Crop Choices
Effects of Subsidized Crop Insurance on Crop Choices Jisang Yu Department of Agricultural and Resource Economics, UC Davis UC Agricultural Issues Center 2015 AAEA&WAEA Annual Meeting, San Francisco, CA,
More informationThe Demand for Life Insurance: An Application of the Economics of Uncertainty: A Comment
THE JOlJKNAL OF FINANCE VOL. XXXVII, NO 5 UECEMREK 1982 The Demand for Life Insurance: An Application of the Economics of Uncertainty: A Comment NICHOLAS ECONOMIDES* IN HIS THEORETICAL STUDY on the demand
More informationMultivariable Calculus and Optimization
Multivariable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multivariable Calculus and Optimization 1 / 51 EC2040 Topic 3  Multivariable Calculus
More informationALMOST COMMON PRIORS 1. INTRODUCTION
ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type
More informationLecture 11 Uncertainty
Lecture 11 Uncertainty 1. Contingent Claims and the StatePreference Model 1) Contingent Commodities and Contingent Claims Using the simple twogood model we have developed throughout this course, think
More information1 Portfolio mean and variance
Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a oneperiod investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring
More informationThe Cumulative Distribution and Stochastic Dominance
The Cumulative Distribution and Stochastic Dominance L A T E X file: StochasticDominance Daniel A. Graham, September 1, 2011 A decision problem under uncertainty is frequently cast as the problem of choosing
More informationA FIRST COURSE IN OPTIMIZATION THEORY
A FIRST COURSE IN OPTIMIZATION THEORY RANGARAJAN K. SUNDARAM New York University CAMBRIDGE UNIVERSITY PRESS Contents Preface Acknowledgements page xiii xvii 1 Mathematical Preliminaries 1 1.1 Notation
More informationRevealed Preference. Ichiro Obara. October 8, 2012 UCLA. Obara (UCLA) Revealed Preference October 8, / 17
Ichiro Obara UCLA October 8, 2012 Obara (UCLA) October 8, 2012 1 / 17 Obara (UCLA) October 8, 2012 2 / 17 Suppose that we obtained data of price and consumption pairs D = { (p t, x t ) R L ++ R L +, t
More informationA Simple Insurance Model: Optimal Coverage and Deductible
A Simple Insurance Model: Optimal overage and eductible hristopher Gaffney Adi BenIsrael Received: date / Accepted: date Abstract An insurance model, with realistic assumptions about coverage, deductible
More informationFinancial Services [Applications]
Financial Services [Applications] Tomáš Sedliačik Institute o Finance University o Vienna tomas.sedliacik@univie.ac.at 1 Organization Overall there will be 14 units (12 regular units + 2 exams) Course
More informationThe MarketClearing Model
Chapter 5 The MarketClearing Model Most of the models that we use in this book build on two common assumptions. First, we assume that there exist markets for all goods present in the economy, and that
More informationAbstract. A Study of the Interaction of Insurance and Financial Markets: Efficiency and Full Insurance Coverage. Abstract
Abstract A Study of the Interaction of Insurance and Financial Markets: Efficiency and Full Insurance Coverage Abstract This paper studies the interaction between insurance and capital markets within a
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More informationMonetary Theory of Inflation
The monetary theory of inflation asserts that money supply growth is the cause of inflation. Faster money supply growth causes faster inflation. In particular, 1% faster money supply growth causes 1% more
More informationChapter ML:IV. IV. Statistical Learning. Probability Basics Bayes Classification Maximum aposteriori Hypotheses
Chapter ML:IV IV. Statistical Learning Probability Basics Bayes Classification Maximum aposteriori Hypotheses ML:IV1 Statistical Learning STEIN 20052015 Area Overview Mathematics Statistics...... Stochastics
More informationNotes on weak convergence (MAT Spring 2006)
Notes on weak convergence (MAT4380  Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p
More information1.3 Induction and Other Proof Techniques
4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.
More informationIndifference Curves and the Marginal Rate of Substitution
Introduction Introduction to Microeconomics Indifference Curves and the Marginal Rate of Substitution In microeconomics we study the decisions and allocative outcomes of firms, consumers, households and
More informationHARVARD John M. Olin Center for Law, Economics, and Business
HRVRD John M. Olin Center for aw, Economics, and Business ISSN 0456333 THE DISDVNTGES OF GGREGTE DEDUCTIBES lma Cohen Discussion Paper No. 367 6/00 Harvard aw School Cambridge, M 038 This paper can be
More informationExam Introduction Mathematical Finance and Insurance
Exam Introduction Mathematical Finance and Insurance Date: January 8, 2013. Duration: 3 hours. This is a closedbook exam. The exam does not use scrap cards. Simple calculators are allowed. The questions
More informationNo Claim Bonus? by John D. Hey*
The Geneva Papers on Risk and Insurance, 10 (No 36, July 1985), 209228 No Claim Bonus? by John D. Hey* 1. Introduction No claim bonus schemes are a prominent feature of some insurance contracts, most
More informationSecond degree price discrimination
Bergals School of Economics Fall 1997/8 Tel Aviv University Second degree price discrimination Yossi Spiegel 1. Introduction Second degree price discrimination refers to cases where a firm does not have
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More informationSinglePeriod Balancing of Pay PerClick and PayPerView Online Display Advertisements
SinglePeriod Balancing of Pay PerClick and PayPerView Online Display Advertisements Changhyun Kwon Department of Industrial and Systems Engineering University at Buffalo, the State University of New
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationEconomics 200B Part 1 UCSD Winter 2015 Prof. R. Starr, Mr. John Rehbeck Final Exam 1
Economics 200B Part 1 UCSD Winter 2015 Prof. R. Starr, Mr. John Rehbeck Final Exam 1 Your Name: SUGGESTED ANSWERS Please answer all questions. Each of the six questions marked with a big number counts
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationEffects of Subsidized Crop Insurance on Crop Choices
Effects of Subsidized Crop Insurance on Crop Choices Jisang Yu Department of Agricultural and Resource Economics University of California, Davis jiyu@primal.ucdavis.edu Selected Paper prepared for presentation
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More informationA Simpli ed Axiomatic Approach to Ambiguity Aversion
A Simpli ed Axiomatic Approach to Ambiguity Aversion William S. Neilson Department of Economics University of Tennessee Knoxville, TN 379960550 wneilson@utk.edu March 2009 Abstract This paper takes the
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationA Farkastype theorem for interval linear inequalities. Jiri Rohn. Optimization Letters. ISSN Volume 8 Number 4
A Farkastype theorem for interval linear inequalities Jiri Rohn Optimization Letters ISSN 18624472 Volume 8 Number 4 Optim Lett (2014) 8:15911598 DOI 10.1007/s1159001306759 1 23 Your article is protected
More informationBy W.E. Diewert. July, Linear programming problems are important for a number of reasons:
APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)
More informationOptimal health insurance contract : can moral hazard incresase indemnity?
Optimal health insurance contract : can moral hazard incresase indemnity? David Alary SDFi, University Paris Dauphine Franck Bien LEGOS, University Paris Dauphine Abstract In this note, we generalize the
More information= Q H Q C Q H Q C Q H Q C. ω = Q C W =
I.D The Second Law The historical development of thermodynamics follows the industrial olution in the 19 th century, and the advent of heat engines. It is interesting to see how such practical considerations
More informationWorking Paper Series
RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series A Market Game Approach to Differential Information Economies Guadalupe Fugarolas, Carlos HervésBeloso, Emma Moreno García and
More informationLOOKING FOR A GOOD TIME TO BET
LOOKING FOR A GOOD TIME TO BET LAURENT SERLET Abstract. Suppose that the cards of a well shuffled deck of cards are turned up one after another. At any timebut once only you may bet that the next card
More informationMTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages
MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem
More informationEquilibria and Dynamics of. Supply Chain Network Competition with Information Asymmetry. and Minimum Quality Standards
Equilibria and Dynamics of Supply Chain Network Competition with Information Asymmetry in Quality and Minimum Quality Standards Anna Nagurney John F. Smith Memorial Professor and Dong (Michelle) Li Doctoral
More informationThe Reasonable Person Negligence Standard and Liability Insurance. Vickie Bajtelsmit * Colorado State University
\ins\liab\dlirpr.v3a 060607 The Reasonable Person Negligence Standard and Liability Insurance Vickie Bajtelsmit * Colorado State University Paul Thistle University of Nevada Las Vegas Thistle s research
More informationArticle from: ARCH 2014.1 Proceedings
Article from: ARCH 20141 Proceedings July 31August 3, 2013 1 Comparison of the Standard Rating Methods and the New General Rating Formula Muhamed Borogovac Boston Mutual Life Insurance Company*; muhamed_borogovac@bostonmutualcom
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More information5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0
a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More informationSchooling, Political Participation, and the Economy. (Online Supplementary Appendix: Not for Publication)
Schooling, Political Participation, and the Economy Online Supplementary Appendix: Not for Publication) Filipe R. Campante Davin Chor July 200 Abstract In this online appendix, we present the proofs for
More informationECO 317 Economics of Uncertainty Fall Term 2009 Week 5 Precepts October 21 Insurance, Portfolio Choice  Questions
ECO 37 Economics of Uncertainty Fall Term 2009 Week 5 Precepts October 2 Insurance, Portfolio Choice  Questions Important Note: To get the best value out of this precept, come with your calculator or
More informationProblem Set 9 Solutions
Problem Set 9 s 1. A monopoly insurance company provides accident insurance to two types of customers: low risk customers, for whom the probability of an accident is 0.25, and high risk customers, for
More informationFund Manager s Portfolio Choice
Fund Manager s Portfolio Choice Zhiqing Zhang Advised by: Gu Wang September 5, 2014 Abstract Fund manager is allowed to invest the fund s assets and his personal wealth in two separate risky assets, modeled
More informationSingle item inventory control under periodic review and a minimum order quantity
Single item inventory control under periodic review and a minimum order quantity G. P. Kiesmüller, A.G. de Kok, S. Dabia Faculty of Technology Management, Technische Universiteit Eindhoven, P.O. Box 513,
More informationMARKET STRUCTURE AND INSIDER TRADING. Keywords: Insider Trading, Stock prices, Correlated signals, Kyle model
MARKET STRUCTURE AND INSIDER TRADING WASSIM DAHER AND LEONARD J. MIRMAN Abstract. In this paper we examine the real and financial effects of two insiders trading in a static Jain Mirman model (Henceforth
More information