Department of Economics Microeconomic TheoryUniversity of California, Berkeley Economics 101AApril 11, 2007 Spring 2007Economics 101A: Microeconomic TheorySection Notes for Week 121 Efficient InsuranceWe will show that if actuarially fair insurance is offered, then it will be optimal for consumersto fully insure. Actuarially fair insurance is an insurance contract where the expected costto the consumer and the expected profit of the firm selling the contract are both zero. Themodel set up is:- The consumer begins with income of y0.- The consumer has probability, p > 0 of getting into an accident which will cause theconsumer to incur a loss of income of L.- With probability 1 − p there won’t be an accident.- An insurance contract is available which has a payout of −πC if no accident occurs anda payout of C − πC if an accident happens.- We assume that the consumer is an expected utility maximizer and thus picks the levelof coverage C to solvemaxCE[u(˜y)] = maxC{pu(y0− L + C − πC) + (1 − p)u(y0− πC)}where ˜y is a random variable representing the consumer’s income.- We can think of the amount of income in the accident state, s1and the no accidentstate, s2as state contingent goods. If s1occurs then the consumer has income x1=y0− L + C − πC and if s2occurs then the consumer has income x2= y0− πC.To solve the consumer’s problem we find the first order condition∂E[u(˜y)]∂C= p(1 − π)u0(y0− L + C − πC) − (1 − p)πu0(y0− πC) = 0u0(y0− πC)u0(y0− L + C − πC)=p1−pπ1−πThis relationship provides the same intuition that we saw graphically in lecture yesterday.1. If firm’s offer actuarially fair insurance then π = p and the odds ratio on the right handside equals one. Since u0(·) > 0 and u00(·) < 0 everywhere marginal utility is alwaysdecreasing in income, thus u0(x2) = u0(x1) implies that x1= x2, which means thatincome is the same no matter which state of the world occurs. The consumer is fullyinsured.12. If less than actuarially fair insurance is offered, then π > p hence the right hand sidewill be less than one. This implies that u0(x2) < u0(x1) thus x2> x1meaning thatincome is higher if no accident occurs or that the consumer is less than fully insured.3. Similarly, if more than actuarially fair insurance is offered, then π < p hence the righthand side will be greater than one. This implies that u0(x2) > u0(x1) thus x2< x1meaning that income is higher if the accident occurs or that the consumer is more thanfully insured.2 Adverse SelectionThe setup for adverse selection is the same as the previous model, except now there are twotypes of individuals with different probabilities of having an accident. We assume that theindividuals have the exact same expected utility function which implies that they have thesame degree of risk aversion. The low type has an accident with probability pLand the hightype has an accident with probaility pHwhere pL< pH.2.1 Efficient Solution: Perfect MonitoringFirst we consider the c ase where insurance firms can tell which type of consumer they arefacing (this assumption is sometimes referred to as perfect monitoring). In this case theinsurance agency will set a premium for the low type πL= pLand a premium for the hightype πH= pH. Note that this implies that for the low typeu0(y0− πLC)u0(y0− L + C − πLC)=pL1−pLπL1−πL= 1which means that x1= x2for the low type (they insure fully). Similarly for the high typeu0(y0− πHC)u0(y0− L + C − πHC)=pH1−pHπH1−πH= 1which implies that the high type also fully insures. Note, however, that income in bothstates is lower for the high type than it is for the low type since πHC > πLC.2.2 Failure of EfficiencyNext we consider the case where firms can’t tell whether they are dealing with a consumerwho has a low probability of getting into an accident or a high probability of getting intoan accident. The model is the same except now we assume that λ is the proportion of thepopulation that is low risk and the 1 − λ is the proportion of the population that is highrisk.The efficient contract where the firm offers two premiums πL= pLand πH= pHwillno longer work. To see this, note that all high risk type consumers will pretend to be lowrisk since in order to get the lower premium, thus the firm’s profits will beE[profits] = λ[pL((πL− 1)C + (1 − pL)πLC] + (1 − λ)[pH((πL− 1)C + (1 − pH)πLC]2since πL= pLthe first term equals zero. Thus,E[profits] = (1 − λ)[pH((πL− 1)C + (1 − pH)πLC= C(1 − λ)[pHπL− pH+ πL− pHπL]= C(1 − λ)[−pH+ πL]= C(1 − λ)[pL− pH] < 0meaning that the firms will choose to go out of business, hence the efficient contracts outcomethat we found under perfect monitoring is not an equilibrium.2.3 Other Pooling Equilibria?Are there any coverage level and premium combinations (πP, CP) that an insurance companycould offer to both types and still make a profit in a competitive industry? No.For the firm to be able to earn zero profits and stay in business, it must be able to subsidizeits losse s from the high risk types with gains from the low risk types. However, pH> pLimplies that the MRS of the high types is always greater than the MRS of the low types.Thus, for any possible pooling contract, a rival firm can offer another c ontract that is betterfor the low types but worse for the high types. This contract will pull low type business awayfrom the first firm, thus causing them to earn negative profits and go out of business. T hisargument can be made for every possible pooling contract, thus there are no equliibirumpooling contracts.Draw Graph.2.4 Separating EquilibriaEven though pooling equilibria do not exist, there are separating equilibria where one pre-mium / coverage combination is chosen by the low risk types (πSL, CSL) and a different combi-nation is chosen by the high risk types (πSH, CSH). For a separating equilibrium to be stable,it must be the case that the high risk types have no incentive to select the contract intendedfor the low risk types, and the firm offereing the contracts must earn expected profits ≥ 0so it will stay in business.Competition among firms for the high risk consumers will ensure that they are able tobuy their efficient contract, thus (πSH, CSH) = (pH, L).To ensure that the high risk types don’t want to switch to the contract intended for thelow risk types we must make sure that the expected utility of the high risk types is lowerwhen