Problem Set 5 - Solution Hints

ETH Zurich D-MTEC Chair of Risk & Insurance Economics (Prof. Mimra) Exercise Class Spring 06 Anastasia Sycheva Contact: asycheva@ethz.ch Office Hour: on appointment Zürichbergstrasse 8 / ZUE, Room F Problem Set 5 - Solution Hints. Altruistic Externality/ Minimum Income a.) To infer the preferences of the individual, we compare the expected utilities of both alternatives. The expected utility of an individual without insurance: EU w/o = 00 + 64 = 9. To calculate the expected utility with insurance we first need to calculate the premium amount. Since the premium is fair, it is equal to the expected loss. The expected loss is L = 7 = 36 = P. Thus the expected utility with insurance is given by: EU w = 00 36 + 00 36 7 + 7 = 8 < EUw/o. One can conclude that the individual does not purchase the insurance policy. The situation is depicted in a two-state-of-the-world diagram in Figure. b.) The society incurs costs in the loss state, when the minimum income is guaranteed. In this case, society compensates the difference between the minimum income and actual income. Thus, the expected costs are: EC S = 0 + (64 (00 7)) = 8. c.) The individual buys insurance if its expected utility with insurance and the premium subsidy s is at least as high as the expected utility from not insuring. In case when the individual buys the insurance it reaches a wealth level of 64

Introduction W 45 W min W 0 L π π EU w/o EU w W 0 W Figure Figure : Indifference : The curves effect of individual of a minimum if he purchases income on insurance insurance and if demand he abstains. in both states of the world (we have full insurance). Thus, the question boils down to finding the certainty equivalent to the non-insurance case: EC w (s) = (64 + s)! = 9 s = 7. The premium subsidy has to be (at least) 7. However paying the premium subsidy is cheaper for society than to guarantee the minimum income at the expected costs EC S = 8. The situation is depicted in Figure d.) Mandatory health insurance is another possible solution to the problem (the European approach). In our example, mandatory health insurance is only paretoimproving, however, if the individual gets paid at least 7 as a compensation for the lower utility with insurance. This leads exactly to the same outcome as in c.). The individual is as good off as before, and society is even better off than before.. Rothschild-Stiglitz a.) In our setting the non-existence of a pooling equilibrium is deduced from the single crossing property Single crossing property: For every contract w, the slope of the indifference curve of the low risks in a two-state-of-the-world diagram is steeper than the

Introduction W EU w EUw (s), EU w/o, 45 W min π π W 0 L s s W 0 W Figure :... Figure : The comparison of indifference curves of an individual without insurance, with and without a premium subsidy s slope of the indifference curve of the high risks. (which implies that every two curves for high and low risks cross only once). The assumption holds in current setting since W, W : ( π h )u (W ) π h u (W ) < ( π l)u (W ) π l u (W ) NB if individuals of different types differ in more than one characteristics, than the property might be violated: Advantegeous selection Every pooling contract (w pool ) has to lie below or on the pooling-zero-profit line in order to be a feasible outcome. If the single crossing property holds there is an incentive to offer cream-skimming contracts w CS, that yield higher utility for the low-risk types and lower for the high risk types. Thus all the low risk types will prefer w CS to w pool, the original pooling contract would be chosen only by high risk types and becomes unprofitable. The Rothschild-Stiglitz equilibrium has the following properties:.) separating equilibrium.) full insurance for hight risks at their fair premium rate π h 3.) partial insurance for low risks at their fair premium rate π l 4.) high risks are just indifferent between both offered contracts and the low risks strictly prefer the partial-insurance-contract. 3

5.) every contract makes zero profit 6.) there is no cross-subsidization between contracts b.) An equilibrium in pure strategies does not exist if the share of high risks γ h is sufficiently low. In that case there would be a pooling contract on or below the fair-insurance line that would be preferred by both types and still be profitable. But as explained in task a.), a pooling equilibrium does not exist. No Risk, No Fun c.) The Rothschild-Stiglitz contracts are showed on the two-state-of-the-world diagram in Figure 3 π l π l w π π 45 Contract A: the contract chosen by the high risk types Contract B: the contract chosen by the low risk types Utility of the high risk types with contract A w 0 L π h π h Utility of the low risk types with contract B The average-risk-line. w 0 w Figure 3: Task c): Rothschild-Stiglitz contracts d.) The WMS equilibrium is obtained by a change of the assumptions on the actions of the insurers. Insurance companies can withdraw contracts after they have become unprofitable. Furthermore they offer two contracts, whereas in R/S framework insurance companies could only offer one contract at a time. The WMS contracts are obtained by maximizing the low risk type utility under the incentive constraint (IC) and participation constraint (PC). IC: high risk types must weakly prefer their intended contract wh W MS make nonnegative profit overall. More formally: over wl W MS. PC: insurers max Pl,C l,p h,c h [( π l )u(w 0 P l ) + π l u(w 0 P l L + C l )] PC: γ h (P h π h L) + ( γ h )(P l π l C l ) 0 IC: ( π h )u(w 0 P h ) + π h u(w 0 P h L + C h ) 4

3. WMS equilibrium -Miyazaki-Spence contracts RS and WMS contracts S contracts: Utility of low risk maximized subject to incentive compatibili ( π h )u(w 0 P l ) + π h u(w 0 P l L + C l ) overall zero-profit constraint, i.e. cross-subsidization is possible The contracts are showed on Figure 4 w H L L H WMS contracts w 0 RS contracts w 0 w WMS contracts are second-best efficient Figure 4: Task d): WMS contracts e.) The proposal is not suited to benefit all insured agents. The reason is that one of the two groups will be worse off. Imagine for instance the pooling contract on the crossing point of the 45-degree-line and the overall-fair-insurance-line in figure in d). No matter if the consequence of a liberalization will lead to a R/S or a WMS-equilibrium, the high types will be worse off. 3. Adverse Selection a.) Since the insurance market is perfectly competitive, there are no administrative costs and the insurance companies can observe the risk type of the client, all the clients are offered contracts at their fair premium rates (π h, π l ). Under these conditions they prefer to purchase full insurance (see Problem Set 3). The high risks premium amount is π h 000 = 6000. The low risks premium amount is π l 000 = 00. Under this arrangement, the utility of high risks 5

is: u H (4000 6000) = 8000 34.640, and the utility of low risks is u L (4000 00) = 800 50.99 b.) Let w h = (P h, C h ) and w l = (P l, C l ) be the contracts for high and low risk types, respectively. From the previous task we know that they are characterized by: w h provides full coverage:c h = 000 w h and w l are actuarially fair: P l = 6000 P l = 0.C l High risk type s incentive compatibility constraint holds with equality: u h (w h ) = u h (w l ) u(8000) = 0.5u(4000 P l ) + 0.5u(000 + C l P l ) 8000 = 0.5 4000 0.Cl + 0.5 000 + 0.9C l ( 8000 0.5 4000 0.C l ) = (0.5 000 + 0.9C l ) 8000 8000 4000 0.C l +0.5(4000 0.C l ) = 0.5(000+0.9C l ) 000 0.5C l = 8000 4000 0.C l C l 04.870 Thus the optimal contracts are: w h = (6000, 000), w l (04.870, 04.870) c.) Now you are asked to check that the pooling contract does not dominate {w h, w l } menu; this is a necessary condition for this self-selection menu to be a competitive equilibrium. In the two-state-of-the-world diagram on Figure 3 we can see that high risks prefer a fair pooling contract to w h. Thus to answer the question we need to compare the utility of the low-risk types for two contracts. The premium amount of the pooling contract is: (γ h π h + ( γ h )π l )L = 0.5 0.5 000 + 0.5 0. 000 = 3600, thus the utility of the pooling contract is: u(4000 3600) = 0400 4.88 The utility of low risks with a contract w l is greater than: 0.9 4000 05 + 0. 000 + 04 05 50.49 We can conclude that the pooling contract is not preferred by low-risk individuals. 6

d.) The fair self-selection menu does not depend on the proportion of the low risks γ l, but whether the pooling contract dominates does. The actuarially fair premium for the pooling contract is γ l 0. 000 + ( γ l )0.5 000 = 00(5 4γ l ) = π pool (γ l ) Suppose that γ l is close to one. Then this premium is close to the premium the low risk types would pay for full insurance if there was no adverse selection at all, and so the utility from this contract would exceed the utility the low risk types get from the separating menu of contracts we found above. That means that the separating menu cannot be an equilibrium, because some insurance companies would come in and offer this pooling contract, which both types prefer to the menu {w h, w l } (certainly the high risk types really like this contract, since they get the same coverage at a lower premium). The cutoff level of γ l such that the low-risk type are just indifferent between the pooling contract and the fair, separating menu is the solution to: u(4000 π pool (γ l )) = 0.9u(4000 0.C l ) + 0.u(000 + 0.9C l ) Substituting in π pool (γ l ) and C l : 4000 00(5 4γl ) = 0.9 4000 0. 04 + 0. 000 + 0.9 04 8000 + 4800γl = 0.9 4000 0. 04+0. 000 + 0.9(04) 50.4988 649.88 8000 = 4649.88 = 4800γ l γ l 0.96875 Thus if the proportion of low risks is higher than 96, 875% the fair pooling contract attracts all the customers. 4. Moral Hazard a.) With high effort level e = e h the expected loss is: 900 = 00 9 With low effort level e = e l the expected loss is: 900 = 450 b.) A contract specifies the level e of effort, the reimbursement C in case of a loss, and the premium amount P : (e, C, P ). Zero profits imply fair premium rate p = π(e i ) and thus the premium amount P equals the expected loss. Thus in 7

the absence of moral hazard efficient contracts feature full insurance (C = 900). It remains to determine the effort level e. The expected utility of a best high effort contract (w w/omh h u(e h, w w/omh h ) = 500 00 3 = 45.9898 = (e h, 900, 00)) is: The expected utility of a best low effort contract (w w/omh l = (e l, 900, 450)) is: u(e L, w w/omh l ) = 500 450 = 45.769 Therefore, the high effort level contract is better. c.) These policies are not incentive compatible, since the clients are fully insured and end up with the same income regardless of their actions. d.) With moral hazard the optimal contract to implement low effort level is the same contract, as without moral hazard (wl wmh = w w/0mh l = (e L, 900, 450)), since it is not necessary to provide special incentives to take the low level of care. e.) The optimal contract that implements high effort level can be derived as a solution to the following maximization problem: max P,C [( π(e h ))u(e h, W 0 P ) + π(e h )u(e h, W 0 L + C P )] P C : ( π(e h ))P π(e h )(C P ) 0 IC : ( π(e h ))u(e h, W 0 P ) + π(e h )u(e h, W 0 L + C P ) ( π(e l ))u(e l, W 0 P ) + π(e l )u(e l, W 0 L + C P ) Due to competition the policy will be offered at a fair premium rate, thus the optimal contract can be written as w wmh h = (e h, C, C), where the level of 9 coverage C is the highest for which the incentive compatibility constraint holds: 9 u(e H, 500 900 + 8 9 C) + 8 9 u(e H, 500 9 C) u(e L, 500 900 + 8 9 C) + u(e L, 500 9 C) We find C that satisfies the constraint with equality, i.e. we solve the equation 9 [600 + 89 C 3 ] The solution is C 9.369539359 [500 9 C 3 ] + 8 9 600 + 8 9 C + 500 9 C = 8