# Economics of Insurance

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1 Economics of Insurance In this last lecture, we cover most topics of Economics of Information within a single application. Through this, you will see how the differential informational assumptions allow us to capture various features of existing markets. Buying Insurance A consumer i faces a risk: with probability p i she suffers a monetary loss or accident of size l i. In the absence of any losses, her wealth is w i and her monetary Bernoulli utility function is an increasing, strictly concave, twice differentiable function u i ( ). These properties of the utility function are assumed to hold throughout this lecture. An insurance contract consists of an up-front fee f and a compensation b if a loss takes place. If the consumer buys contract (f, b), then her wealth across the two states (no loss vs. loss) is given by: (1 p i ) u i (w i f) + p i (w i l i f + b). Monopoly Insurance company Suppose there is a single insurance company with the power of making a takeit-or-leave-it offer to the consumer. Suppose also that the company knows all the relevant details of the consumer, i.e. p i, w i and l i. The consumer accepts contract (f i, b i ) if and only if (1 p i ) u i (w i f i ) + p i (w i l i f i + b i ) (1 p i ) u i (w i ) + p i (w i l i ) := u 0. 1

2 The expected profit of the insurance company is given by f i p i b i. ence the optimization problem of the insurance company is : max f i p i b i f i,b i subject to (1 p i ) u i (w i f i ) + p i (w i l i f i + b i ) u 0 i. The Lagrangean of the problem is: L (f i, b i λ) = f i p i b i + λ ( (1 p i ) u i (w i f i ) + p i u i (w i l i f i + b i ) u 0 i ). The first order conditions for the problem are: 1 = λ ((1 p i ) u i (w i f i ) + p i u i (w i l i f i + b i )), p i = λp i u i (w i l i f i + b i ), (1 p i ) u i (w i f i ) + p i (w i l i f i + b i ) = u 0 i. From the first two lines, we get: 1 = λu i (w i f i ) = λu i (w i l i f i + b i ). By the strict concavity of u i, we conclude that b i = l i. ence the monopolist insurance company offers full insurance. The fee f i is set so that w i f i is the certainty equivalent of the lottery without insurance: u i (w i f i ) = (1 p i ) u i (w i ) + p i (w i l i ) = u 0 i. Notice that the fee depends on the risk type p i, the wealth level w i, and the exact properties of the Bernoulli utility function u i. It is a good exercise to see how the optimal f i depends on w i. Since the buyer is kept to her outside option utility level (expected utility without any insurance), all gains from obtaining insurance go to the monopoly insurer. 2

3 Competitive insurance industry with observable risk types Suppose that the insurance industry offers a number of different contracts. For any fixed b, all consumers with an increasing utility function u i choose the contract with the minimal f. ence we may write the available contracts as (f (b), b). Suppose that insurance companies are risk-neutral and there are at least two companies. If the companies observe the risk type of each buyer and compete by offering insurance contracts simultaneously, then the contracts offered in equilibrium to consumers of type p i will be actuarially fair: f = p i b. Substituting into the objective function of the consumer, we get the optimal choice problem for coverage by the consumer as follows: max (1 p i ) u i (w i p i b) + p i u i (w i l i + (1 p i ) b). b By the concavity of u i, first order conditions are sufficient for the optimal choice of b and hence we have: p i (1 p i ) u (w i p i b) + p i (1 p i ) i u i (w i l i + (1 p i ) b) = 0. By the strict concavity of u i we have then: w i p i b = w i l i + (1 p i ) b, or b = l i. ence all buyers buy full insurance where the damages b coincide with the loss l i. The fee does not depend on initial wealth or u i. 3

4 Insurance with Incomplete Information Assume next that the buyer s risk type is private information, but w i = w and l i = l, and u i ( ) = u ( ) for all insurance buyers. As examples, consider life insurance with a privately known family history, unemployment insurance with private information on productivity etc. Notice by the way that for many of such cases, the institutional arrangements require transparency with respect to past accident history (e.g. traffic insurance). For simplicity, assume also that the risk type is either high or low with associated loss probabilities p and p L respectively with p > p L. Start with the case of a monopoly insurance seller. As seen in the first part of the lecture, first-best contracts offer full insurance: b = l. ence incentive compatibility of the contracts requires that only one fee can be charged if efficient level of insurance is provided. If both types are to accept the contract, then the fee must satisfy the participation constraint of the low risk type. This leaves a huge information rent for the high risk type. Denote the equilibrium expected payoff to the type with risk p at contract (f, b) by U (f, b, p). To ease notation, let w j N = w f j and w j A = w l f j +b j, the final wealth levels in the case of no accident and accident respectively when j {, L}. It will be useful to draw indifference curves for wealth levels over these two states. We can calculate the marginal rate of substitution between the fee f and benefit b by totally differentiating U (f, b, p) : MRS f,b = pu (w A ) + (1 p) u (w N ). pu (w A ) This is a decreasing function of p and hence the Spence-Mirrlees single crossing condition holds. This raises the possibility that a solution where high risk types are offered more coverage in exchange of a higher fee might be possible in this case. 4

5 The algebra to show this is not completely straightforward in this case, and you are strongly advised to draw indifference curves of the two risk types in the coordinate system where you put consumption in the two states (no accident, accident) on the axis. Incentive compatibility and IR constraints can be written as ( ) 1 p + p ( w ) A 1 p L + p wa L, ( ) 1 p L L + p L ( w ) A L 1 p L + p L wa, ( ) 1 p + p wa u,0, ( 1 p L ) u ( w L N ) + p L wa L u L,0. It is clear that the IR constraint of at least one risk type must bind. For example, one could reduce all ( w j N and u w j A) by reducing the b j and increasing the f j. It is a good exercise to show that this is indeed possible (what properties of u are needed here?). Denote the no insurance point where (f, b) = (0, 0) by (u 0 N, u0 A ). By the single crossing property, we get immediately that MRSf,b L (u 0 N,u 0 A) > MRSf,b (u 0 N,u and hence also all points that are preferable to no insurance 0 A) for the low risk type are also preferred to the no insurance point by the high risk type. ence the IR constraint of the low risk type must bind. Consider then the indifference curve of the low risk type through the point of no insurance. It intersects the 45-degree line at the point (( w f L), ( w f L)) and f L is determined from u ( w f L) = u,0. The optimal contract for the low risk type is then on the indifference curve of the low type that passes through (u 0 N, u0 A ) and (( w f L), ( w f L)) and on the segment between these two points. Consider a parametrized equation for this segment given by w (α) = (w N (α), w A (α)) with the property that α [0, 1], w (0) = (u 0 N, u0 A ), w (1) = (( w f L), ( w f L)), 5

6 ( ) 1 p L u (w N (α)) + p L u (w A (α)) = u L,0 for all α, and w A (α) > 0 For each w (α) draw the indifference curve of the high risk type through w (α). Let f (α) solve: From u ( w f (α) ) = ( 1 p ) u (w N (α)) + p u (w A (α)). the first section, we know that the contract ( f (α), l ) maximizes the insurance company s profit from the high risk buyers subject to the constraint: ( ) 1 p + p ( w ) A 1 p u (w N (α)) + p u (w A (α)). Notice that f (α) is decreasing and f (1) = f L. ence the insurance company gains on the low risk type buyers when α is increased and loses on df the high risk types. Since lim (α) a 1 > 0, we see that it is not optimal dα to insure the low risk types fully. You can also show that if the proportion of high risks is sufficiently high, then only high risks will be insured. This is an extreme form of adverse selection. ence the general lesson is that the optimal insurance contracts allow different risk types to self select into appropriate coverage through the use of deductibles. Insurance with Moral azard The problem is now that the buyer of insurance can be cautious at a cost and reduce her risk probability. Alternatively, she can be careless and have a higher probability of accident. The problem is actually quite is bit simpler this time since the buyers are all alike at the moment of buying the insurance. Examples here are reckless driving, unhealthy life habits, leaving oily rags in the basement etc. By exposing the agent to some residual risk, the insurer will reward risk avoidance. 6

7 Timing is as follows. In the first instance, the monopolist insurance company proposes a contract (f, b). the buyer either accepts or not. We say that the IR constraint is satisfied, if the buyer accepts. Then the buyer chooses an action x {x, x L }. Action x j results in probability of accident p j. The losses and the wealth levels are as above at w and l. Each action comes with an additively separable cost c (x). The incentive compatibility and IR constraints for this problem are thus: (IC) ( 1 p ) u (w f) + p u (w l f + b) c ( x ) ( 1 p L) u (w f) + p L u (w l f + b) c ( x L), ( (IR) ) 1 p u (w f) + p u (w l f + b) c ( x ) max{u,0 c ( x ), u L,0 c ( x L) } := u 0. Once we have written the problem in this form, we recognize that we are in the standard two action, two outcome framework for the Moral azard problem and we can simply refer to the lecture notes for the solution. To recap the main properties: both IC and IR must bind at the optimal insurance contract. Full insurance is not possible if the insurance company wants the agent to exert effort x L leading to the low probability of accident p L. On the other hand, if the insurance company is satisfied with x, then full insurance can be offered. Note that here we assume that c ( x L) > c ( x ). An obvious direction for extending the model would be to allow for insurance over a longer horizon. Then a very good question for further thought is the following: suppose that the effort determining the accident probabilities is chosen in each period. Suppose also that risks are independent across period (conditional on the effort chosen). When is it optimal for the insurance company to offer the same static optimal single period contract to the buyer in all periods? What are the key features in the model that affect the answer to this question? 7

8 Insurance with Signaling The order of moves is reversed now. Nature chooses the type of the buyer. Let the fraction of low risk types be µ > 0. In the next step, the buyer proposes (f, b) to the insurance company. We consider pure weak PBE of the game. Let { ( f, b ), ( f L, b L) } denote the contracts that are proposed by the high and low risk types respectively. The insurance company either accepts or rejects the offer. If the offer is accepted, then the final wealth level of the buyer is across the two states of no accident and accident (w f, w l f + b) as before. Let π (f, b) denote the insurance company s posterior belief that the buyer is the low risk type given that (f, b) was proposed. Furthermore, let σ (f, b) {0, 1} denote the probability that the insurance company accepts the contract (f, b). Definition 1 A weak PBE of the game is a collection { ( f, b ), ( f L, b L) }, π (f, b), σ (f, b) such that 1. Given σ (f, b), (f j, b j ) maximizes for j {, L} σ (f, b) (( 1 p j) u (w f) + p j u (w l f + b) ) + (1 σ (f, b)) u j,0. 2. Given π (f, b), σ (f, b) maximizes σ(f ( π (f, b) p L + (1 π (f, b)) p ) b). 3. π (f, b) is derived by Bayes rule for (f, b) { ( f, b ), ( f L, b L) }. Otherwise π (f, b) [0, 1]. As usual, there are two types of pure strategy equilibria: separating and pooling. In separating equilibria, the two different risk types propose different contracts in equilibrium and hence the resulting beliefs on equilibrium path are degenerate. If we want to find the maximal set of separating equilibria, 8

9 then we should concentrate on beliefs of the insurance company that reject contract proposals that are not on the equilibrium path as often as possible. Let s assume that the insurance company is risk-neutral and has an outside option profit of 0. Then it will reject all offers with f < p b if p (f, b) = 0. It is never rational to reject a contract with f > p b. Since the high risk type reveals itself on the equilibrium path of a separating equilibrium, the above reasoning already gives our first result: ( f, b ) = ( p l, l ). This follows from the first section of this lecture since full insurance is optimal at actuarially fair prices. The remaining properties just state that the insurance company must break at least even with the low risk types, the low risk types must prefer insurance to no insurance, and the high risk type must not prefer the contract of the low risk type to its contract. Proposition 2 { ( f, b ), ( f L, b L) } is part of a separating weak PBE of the insurance game with signaling where the insurance company accepts both offers if and only if 1. ( f, b ) = ( p l, l ), 2. f L p L l, 3. U ( f L, b L ; p L) U ( f, b; p L) for f p l, 4. U ( p l, l; p ) U ( f L, b L ; p ). The second part assures acceptance of the low offer. The third states that there is no profitable deviation for the low risk type when a deviation is interpreted as coming from a high risk type. The fourth requirement is just the incentive compatibility requirement for the high risk type (the third part takes care of the other IC as well). 9

10 It is instructive to draw the picture of this situation in the (w N, w A ) coordinate system. The other equilibrium type is a pooling equilibrium. In this case, (f j, b j ) = ( f P, b P ) for j {, L}. The requirements for pooling can now be summarized as: Proposition 3 ( f P, b P ) is part of a pooling weak PBE of the insurance game with signaling where the insurance company accepts both offers if and only if 1. f P ( µp L + (1 µ) p ) l, 2. U ( f P, b P ; p L) U ( f, b; p L) for f p l, 3. U ( f P, b P ; p ) U ( p l, l; p ). The first part just requires that the insurance company must break even. The second and third require that pooling must be better for the two risk types than any contract that the insurance company would accept from the high risk type. Again, you should draw the pictures with indifference curves to see what is going on. The basic lesson is not too different from the monopoly insurance case with adverse selection (except of course how the surplus is divided between the firm and the buyers). In separating equilibria, the low risk send a costly signal by proposing a contract that leaves part of the loss uninsured. In practice, this corresponds to the deductible in the insurance contract. The pooling equilibria do not really have a counterpart in the adverse selection model. As is usually the case with models of signaling, the set of equilibria is large. Partially this results from the freedom of choosing beliefs at off equilibrium path contract offers. To see how much can be explained by this, you may want to ask yourself which of the equilibria survive the Intuitive Criterion. 10

11 In particular, you may want to show that pooling equilibria are in trouble. Another excellent question is whether it is possible that the best pooling equilibrium Pareto-dominates all separating equilibria. If this is the case, then you will notice a tension between efficiency and Intuitive Criterion as in the Spence job market signaling model. In general, this signaling approach to the contracting problem leads in the direction of models with Informed Principal. These are mechanism design models where a principal with private information starts the game by proposing a set of potential contracts. The agent either accepts to be in the game or not. If she agrees, then the principal picks one contract from the set proposed and then the contract is executed (typically then this is a direct revelation mechanism). The analysis of such models is beyond the scope of this course. Competition in Insurance with Private Information See the textbooks (Jehle and Reny, Salanie 3.2.1). 11

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