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Notes for the Presentation in Public Economics a Summary on Adverse Selection

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Notes for the Presentation in Public Economics a Summary on Adverse Selection Notes for the presentation in Public Economics A summary on adverse selection Eduard K. B¨ohm/ Morten N. Støstad October 16, 2017 Question: Compare the efficiency of the competitive equilibrium in a market with moral hazard and a market with adverse selection. Is there a role for a government in both markets? Definitions H. Varian:\Adverse selection refers to situations where one side of the market can't observe the \type" or quality of the goods on other side of the market. For this reason it is sometimes called a hidden information problem." J. Gruber:\The fact that insured individuals know more about their risk level than does the insurer might cause those most likely to have the adverse outcome to select insurance, leading insurers to lose money if they offer insurance." The literature on adverse selection was especially popular after the publication of the paper \The Market for Lemons" by George Akerlof in 1970. Akerlof already suggests an application of the theory to insurance markets, however, the most famous model regarding this particular application goes back to Michael Rothschild and Joseph Stiglitz (1976). The Nobel prize awarded in 2001 to Akerlof, Stiglitz and Michael Spence was done so on the merit of these authors' work in the field of asymmetric information. The textbook model, the lemon market and a simple numerical example The textbook model This is the graphical model used illustratively in Chetty and Finkelstein (2013), which is based on the work of Einav and Finkelstein (2011). It represents a case of competitive insurance firms offering a single contract meant to cover a concrete probabilistic loss. Individuals will make a binary decision of buying the insurance depending on their own private information about their personal risk level. Demand for insurance will fall with price. Assuming individuals are risk averse, the demand (price they are willing to pay for being insured) will always be above the marginal cost (which is the cost of the loss). Since individuals self-select into insurance, the marginal cost curve will have a negative slope. The additional buyers of insurance will always be of lower risk than the previous ones and will thus be less expensive. As the marginal cost falls, so does the average cost. Inefficiency arises in the market, as a number of individuals who value insurance for more than its marginal cost opt out due to the high costs. The equilibrium quantity Qeqm lies below the efficient Qmax. The welfare effect can vary depending on the shape 1 Figure 1: Textbook model in Einav, Finkelstein (2011). Falling marginal cost curve reflects self selection, efficiency is not achieved. and relative position of the demand curve to the cost curves. Two extreme cases are given by the authors: Figure 2: Einav and Finkelstein (2011). To the left, a case where adverse selection does not cause any inefficiency. To the right, a case where the market breaks down (or converges towards a failure), as there is no contract that allows for the insurance company to break even. In insurance, the problem that arises feeds on a loop which ends with complete market failure. The offer of insurance at a sustainable price (zero profits) will need to set the expectation of the losses of the entire population. Alas, people knowing their own type (riskiness in this model) will react heterogeneously, with a chance for those with low risk not buying any insurance due to its price being prohibitely expensive given the overall risk associated with the other group. Thus, the composition of those buying insurance will be skewed towards the group which conveys higher losses. In turn, this will mean that insurance offered at that price will lead to losses for the insurance company. This, in turn, will raise prices and cause 2 a further wave of \less risky" individuals to refuse to buy insurance. The lemon market Assume in the market for used cars the quality is normalized to be in the interval [0; 1]. 1 There are two groups (or a 50% chance of either realization): \lemons" of quality q = 5 4 and \peaches" of quality q = 5 . The sum of the two groups makes for a continuum of 1. Sellers have private information about the quality of their own product. A similar continuum of buyers can be found. Buyers do not know the quality of each car, but they do know the overall distribution of quality. The decision to buy or sell a car is made as follows. A seller will only sell her car if the price exceeds the quality, that is if p ≥ q1. 8 0 if p < 1 > 5 > < 1 1 4 S(p) = 2 if p 2 [ 5 ; 5 ) > > :> 4 1 if p ≥ 5 A buyer will buy a car depending on the quality he can expect to find in the market. The decision of each buyer can be characterized as: 8 1 if p ≤ 3 qE < 2 D(p) = : 3 E 0 if p > 2 q Since this example is simple, we can solve for the expected quality qE and write the demand function only in terms of p. In fact: 8 0 if p < 1 > 5 > E < 1 1 4 q = 5 if p 2 [ 5 ; 5 ) > > :> 1 4 2 if p ≥ 5 Now we check with the demand function, excluding the case of p = 0. Whenever the expected quality is zero, there will be no demand. When there are only \lemons" in the market, there will be demand as long as the price is not too high: 1 3 3 qE = =) D(p) = 1 iff p ≤ qE = 5 2 10 When every seller is active in the market: 1 3 qE = =) D(p) = 1 iff p ≤ 2 4 but this does not turn out to be possible, as the \peach" sellers will only enter the 4 market at the price of 5 . Thus, 8 0 if p < 1 > 5 > < 1 3 D(p) = 1 if p 2 [ 5 ; 10 ] > > :> 3 0 if p > 10 1Assume she is indifferent when equality holds. 3 Plotting this, we get our equilibrium2: p D(p) S(p) ∗ 3 p = 10 ∗ 1 Q Q = 2 Note that if we make the condition for the buyers stricter (say, only demand whenever 1 p ≤ 2 q), then there might very well not be an equilibrium in the market at all: 1 1 1 qE = =) D(p) = 1 iff p ≤ qE = 5 2 10 4 1 4 qE = =) D(p) = 1 iff p ≤ qE = 5 2 10 p D(p) S(p) Q Asymmetry of information (together with heterogeneity of agents) results in a market breakdown: supply will be cut down due to the price resulting from the beliefs about the market composition leads to suppliers of how quality exiting the market. A numerical example on insurance (Gruber) This is a brief example (outlined in Chapter 12 of Gruber's textbook) of the aforemen- tioned loop which leads to the market failure. 2I ignore the first part of the curve for clarity of presenting only one equilibrium. The way the problem 1 was set up, there would be another possibility at price p = 5 4 Suppose there are two groups, each one with 100 persons. They have different prob- abilities of suffering an accident due to one group being more careful. Careful people get hit by a car with 0.5% probability, careless ones with a probability of 5%. The insurance company is unable to distinguish between both groups. The costs of suffering an accident when insured are carried by the company and are of 30000. When attempting to offer contracts, the company could try to provide full insurance for both groups, offering a premium of 150 for careful and of 1500 for careless consumers. If people were to select the one designed for their own type, the net profit for the insurance company would be zero. Yet, the careless group has private information on their own type and thus an incentive to buy the insurance intended for the other group. This will mean that the company will make a net loss when offering the set of contracts detailed above: every policyholder will buy the contract for 150. If they offer only a single contract with which they would (aritmetically) break even, a single contract with a premium of 825, the careful group will not buy insurance and once again the company will make a net loss. Thus, the model once again predicts market failure. The model by Rothschild and Stiglitz (1976) offers a more detailed characterization of the possibilities of contract specification to an insurance firm. In a setup in which a firm chooses price and quantity of insurance contracts offered, they prove that the only possible equilibrium will be a separating one, and that this will only exist under certain conditions. Thus, market failure is certainly one outcome but not a direct prediction of the model. Key elements of markets with adverse selection Summarizing a couple of important aspects: • Adverse selection requires heterogeneity in one side of the market. • The heterogeneity must be unobservable for the other market participant. • The informational advantage will lead to certain transactions which are possible under full information to fail, leading to inefficient outcomes or even to market failure. Model predictions tested in markets There is a somehow vast literature which has been concerned with testing the predictions of the adverse selection models. Examples for several insurance markets are summarized an extensively discussed in the handbook of Public Economics article by Chetty and Finkelstein (2013), a very extensive review of the empirical work is offered by Cohen and Siegelman (2009).
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