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Prof. Kranton's Econ 604 Spring 2001: Moral Hazard, Insurance & Principal-Agent Problem - , Assignments of Microeconomics

University of MarylandMicroeconomics
Prof. Rachel E. Kranton

Problem set questions from professor rachel kranton's university of maryland economics 604 course in spring 2001. The problems deal with moral hazard and insurance, and a principal-agent problem. The first problem discusses an individual's decision to take reasonable care given the cost of care and the probability of an accident. The insurer's optimization problem to determine the optimal insurance contract is also set up. The second problem involves a principal-agent problem where the agent's effort is not observable, and the optimal wage schedule is determined. The document also includes references to additional problems in the text 'mwg'.

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Uploaded on 02/13/2009

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ProfessorRachelKrantonUniversityofMaryland Econ604 Spring 2001
ProblemSet4
1.(MoralHazardand Insurance)Consideranindividualwithinitialwealthwwhocares
onlyaboutmoneyand hasBernoulli utilityfunctionu();whereu0>0.Theindividual isalso
strictlyriskaverse,u00 <0:Theindividualfacesthepossibilityofanaccidentwhichwould
involveamonetaryloss d.Theprobabilityofanaccidentdependsonthelevelofcaretaken by
theindividual. Therearethree possiblelevelsofcare:“extreme care,” “reasonable care,” and
“nocare.”Theprobablityofanaccidentiszeroforextreme care,1/2 for reasonable care,and
1fornocare.Takingcareiscostlyfortheindividual. LetcE;cR;and cNdenotethemonetary
costsofextreme,reasonable,and nocare,respectively.WeassumethatcE>cR>cN=0.
(i)Iftheindividualbuysnoinsurance,underwhatconditionswill shestrictlyprefertotake
“reasonable care?”Forthefollowingpartsofthequestion,assumethiscondition holds.
Supposethereis singlerisk-neutral insurerwho o¤ersaninsurance contract(q;a)tothe
individual, whereq>0isthetotalpremiumpaidand a>0isthetotalamountpaidincaseof
anaccident.On being o¤eredsuchacontract,theindividual…rstdecideswhetherornot tobuy
insurance,then decidesonthelevelofcare.Thelevelofcareisnotobservabletotheinsurer.
(ii)Arguethatfortheinsurer(a)itisnotpossibleto o¤eracontractunderwhichthe
individualwill take“extreme care,” and (b)itisnotpro…tableto o¤eracontractunderwhich
theindividualwill take“nocare.”
(iii)Setup theinsurer’soptimization problemtodeterminetheoptimal insurance contract
(q¤;a¤)whichinducestheindividualtotake“reasonable care.”
(iv)Showthata¤<d.
2.Considerthefollowingprincipal-agentproblem.TheagenthasBernoulli utilityfunction
v() and is strictlyriskaverse.TheagentcantakeoneoftwopossibleactionsfeL;eHg,wherethe
costoftakingtheeLislowerthanthe costofeH:c(eL)<c(eH).Theprincipal isriskneutral
and therearetwopossiblerealizationsofpro…ts:¼1>¼2.LetpL´probf¼=¼1je=eLgand
pH´probf¼=¼1je=eHg:Thenprobf¼=¼2je=eLg=1¡pLand probf¼=¼2je=eHg=
1¡pH:
pf2

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Professor Rachel Kranton University of Maryland Econ 604 Spring 2001

Problem Set 4

  1. (Moral Hazard and Insurance) Consider an individual with initial wealth w who cares only about money and has Bernoulli utility function u(); where u^0 > 0. The individual is also strictly risk averse, u^00 < 0 : The individual faces the possibility of an accident which would involve a monetary loss d. The probability of an accident depends on the level of care taken by the individual. There are three possible levels of care: “extreme care,” “reasonable care,” and “no care.” The probablity of an accident is zero for extreme care, 1/2 for reasonable care, and 1 for no care. Taking care is costly for the individual. Let cE ; cR;and cN denote the monetary costs of extreme, reasonable, and no care, respectively. We assume that cE > cR > cN = 0. (i) If the individual buys no insurance, under what conditions will she strictly prefer to take “reasonable care?” For the following parts of the question, assume this condition holds. Suppose there is single risk-neutral insurer who o¤ers an insurance contract (q; a) to the individual, where q > 0 is the total premium paid and a > 0 is the total amount paid in case of an accident. On being o¤ered such a contract, the individual …rst decides whether or not to buy insurance, then decides on the level of care. The level of care is not observable to the insurer. (ii) Argue that for the insurer (a) it is not possible to o¤er a contract under which the individual will take “extreme care,” and (b) it is not pro…table to o¤er a contract under which the individual will take “no care.” (iii) Set up the insurer’s optimization problem to determine the optimal insurance contract (q¤; a¤) which induces the individual to take “reasonable care.” (iv) Show that a¤^ < d.
  2. Consider the following principal-agent problem. The agent has Bernoulli utility function v() and is strictly risk averse. The agent can take one of two possible actions feL; eH g, where the cost of taking the eL is lower than the cost of eH: c(eL) < c(eH ). The principal is risk neutral and there are two possible realizations of pro…ts: ¼ 1 > ¼ 2. Let pL ´ probf¼ = ¼ 1 j e = eLg and pH ´ probf¼ = ¼ 1 j e = eHg: Then probf¼ = ¼ 2 j e = eLg = 1 ¡ pL and probf¼ = ¼ 2 j e = eH g = 1 ¡ pH :

(i) An assumption that “higher e¤ort leads to higher expected pro…ts” in the sense of …rst- order stochastic dominance implies a simple relationship between pL and pH. What is it? (ii) Suppose e¤ort is not observable and not veri…able, but the realization of pro…ts is veri…able. Set up the minimization problem to determine the optimal wage schedule that implements the high e¤ort level eH. Let (w 10 ; w^02 ) denote this optimal schedule, where w^0 i is the payment to the manager when pro…t level ¼i is realized). Establish the …rst-order conditions that determine (w^01 ; w^02 ) : (iii) Establish that, in this model, the …rst-order stochastic dominance assumption implies w 10 > w^02 : The reason for this result is that with two e¤ort levels and pro…ts being a binary random variable, …rst order stochastic dominance does imply the monotone likelihood ratio property. Show this. (iv) Show that 0 < w 10 ¡ w 20 < ¼ 1 ¡ ¼ 2.

  1. MWG 14.B.
  2. MWG 14.B.