2. Information Economics

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1 2. Information Economics In General Equilibrium Theory all agents had full information regarding any variable of interest (prices, commodities, state of nature, cost function, preferences, etc.) In many situations, though, information is not available to all the agents. The most important case is when some sort of asymmetric information prevails. In such cases, different agents have access to different information Examples of the above are: labor markets, used goods markets (lemons), insurance markets, and contractual relationships. The main phenomena that arise from these situations (and that lead to market failures) are: Adverse Selection: the actions of informed agents affect uninformed agents in an adverse way Moral Hazard: the actions of informed agents are unobservable to the uninformed agents (maybe through noisy signals only)

2 2. Information Economics Plan 2.1 Adverse Selection Information and Efficiency Signaling Screening 2.2 Moral Hazard Symmetric Information Asymmetric Information

3 2. Information Economics In all cases Market for car insurance Firms F = {1, 2,...,F } all identical All firms sell the same insurance policy at the price p per monetary unit insured All firms produce insurance policies at zero cost Individuals I = {1, 2,...,I} All individuals have the same initial wealth ω All individuals have a continuous, strictly increasing, and strictly concave utility function over wealth, u(.). When facing a risky situation individuals behave as VonNeumann- Morgenstern expected utility maximizers Individuals differentiate from each other in the probabilities they get involved in a car accident, π i [0,1] π i is the probability that individual i I has a car accident.

4 2. Information Economics 2.1 Adverse Selection Roughly speaking, adverse selection is a phenomena that occurs when the actions of informed agents affect the market outcome in such a way that the bad products (or customers) are more likely to be selected as the result of the actions of the uninformed agents To study this phenomena we use the market for car insurance with the following simplifying assumptions: All car accidents are equal: the monetary loss is of L monetary units in any case All insurance companies offer the same policy that pays L in case of an accident and 0 otherwise. Since p is the price per monetary unit insured, the cost of buying one unit of such policy is of pl

5 The symmetric case Suppose first (for comparison purposes) that all firms know the accident risks of all individuals (π 1,...,π I ) Firms seek to maximize expected profits by choosing the level of coverage offered at the unit price p i to each individual i I. (Since firms are able to distinguish between individuals, they can set different prices across individuals) Let x f i denote the amount of monetary units that firm f F offers to individual i I. Then, expected profits for firm f are given by E(Π(x f i )) = p ix f i π ix f i = (p i π i )x f i i I Clearly, p i > π i p i < π i p i = π i E(Π) > 0 x f i = E(Π) < 0 x f i = 0 E(Π) = 0 x f i = any quantity Hence, the firm f s coverage supply function for individual i I is

6 p i π i Individuals seek to maximize their expected utility by choosing the level of coverage to buy at the unit price p i Let x i denote the amount of monetary units that individual i I wants to buy Then, expected utility for individual i I is given by E(u(x)) = π i u i (ω L + x i p i x i ) + (1 π i )u i (ω p i x i ) x f i Expected utility maximization ( E(u(x))/ x = 0) yields π i u i(ω L + x i p i x i )(1 p i ) + (1 π i )u i(ω p i x i )( p i ) = 0

7 Rearranging, π i (1 p i )u i(ω L + x i p i x i ) = p i (1 π i )u i(ω p i x i ) (1) From here, (i) If p i = π i Then 1 becomes u i(ω L + x i p i x i ) = u i(ω p i x i ) and since u() is strictly concave we have that ω L + x i p i x i = ω p i x i So, p i = π i x i = L

8 (ii) If p i > π i Then π i (1 p i ) < p i (1 π i ) In such case, the maximization condition 1 holds if and only if u i(ω L + x i p i x i ) > u i(ω p i x i ) and since u() is strictly concave we have that ω L + x i p i x i < ω p i x i So, p i > π i x i < L (iii) If p i < π i Then π i (1 p i ) > p i (1 π i ) In such case, the maximization condition 1 holds if and only if u i(ω L + x i p i x i ) < u i(ω p i x i )

9 and since u() is strictly concave we have that ω L + x i p i x i > ω p i x i So, p i < π i x i > L Therefore, the demand for coverage is decreasing as a function of the unit price p i p i π i L x i

10 Therefore, the unique equilibrium exists when p i = π i ( i I) p i π i L x i In this equilibrium: All individuals buy full insurance (x i = L i I) Any firm is willing to supply such coverage at the price p i = π i

11 We know (see Jehle-Reny for details) that it is efficient.

12 Asymmetric Information Suppose now that firms do not know (π 1,...,π I ) Firms know the maximum and minimum probabilities, [π, π] and a probability distribution G so that G(π) = p(π i π) Since firms can not distinguish among individuals they must set a unique price p At the equilibrium price p (if any), E(Π(p )) = 0 where E(Π(p)) = (p E(π))x For simplicity, we will assume that x = L or x = 0 If price is set as p = E(π) = ˆ π π πdg(π)

13 then, individuals i I such that π i < E(π) will not buy and, therefore, E(Π) < 0 Expectation must be conditional on individuals that buy. The incentives condition for individuals to buy the insurance is u(ω pl) π i u(ω L) + (1 π i )u(ω) That is, an individual with accident probability π i will buy the insurance at price p if and only if π i u(ω) u(ω pl) u(ω) u(ω L) h(p) Thus, p is an equilibrium price if p = E(π π h(p )) that is, p = ˆ π h(p ) πdg(π)

14 Then, E(Π) = (p ˆ π h(p ) πdg(π))l Does such p exist? Consider g(p) = E(π π h(p))

15 Notice the g(p) is well defined for p π, that is, h( π) π. To see this suppose, on the contrary, that h( π) > π, that is, u(ω) u(ω πl) u(ω) u(ω L) > π Then, u(ω) u(ω πl) > π(u(ω) u(ω L)) or πu(ω L) + (1 π)u(ω) > u(ω πl) (2) Notice now that (ω πl) = π(ω L) + (1 π)(ω). Replacing on the right-hand side of (2) yields πu(ω L) + (1 π)u(ω) > u( π(ω L) + (1 π)(ω)) which occurs whenever u( )is a convex function, thus contradicting the assumption of risk aversion (u( ) strictly concave ) Clearly, if p [π, π]then g(p) [π, π]. Thus, g(p) : [π, π] [π, π]

16 Also, since h(p) is increasing, g(p) must be non-decreasing. Hence, we can conclude that g(p) has a fixed point p such that p = g(p ), that is, p = E(π π h(p )) (such p might not be unique) We will show that this is not efficient (contrary to the symmetric information case) by means of one example

17 Example Let G be the uniform distribution over the whole unit interval, that is, G U[π,π] = U[0,1] In this case, we have that Hence, in equilibrium Notice that h(1) = 1. Hence, That is, g(p) = 1 + h(p) 2 p = 1 + h(p ) 2 1 = p = 1

18 is an (the) equilibrium. In this equilibrium, only individuals such that π i h(p ) will buy the insurance. Since p = 1 h(p ) = 1 we have that only individuals with π i = 1 will buy the insurance. All the other individuals (π i < 1) will remain uninsured! This is clearly inefficient as we know (from the symmetric case) that individuals are strictly better off when the are fully covered (at a fair price) than uninsured while, companies make zero profits in any case

19 Partial Solutions Although the inefficiencies brought by information asymmetries are difficult to solve completely, there are, in some cases, ways to mitigate the effect. The main point is that both, individuals and companies, will benefit if information about individuals risk were available to both sides of the market In practice, the strategic behavior of firms and individuals works towards the reduction of such asymmetry. The main problem, then, will be credibility. Signaling First we will study the situation in which low risk individuals try to differentiate themselves from high risk individuals by proposing a pair coverage-premium to the company Screening Second, we will analyze the case in which companies try to tell apart low risk individuals from high risk by offering a menu of policies We will see that, in both situations, individuals are correctly identified in some cases while in other cases they are not.

20 Signaling There are two types of risk averse individuals: High risk individuals, with probability π of having an accident Low risk individuals, with probability π of having an accident There is a proportion λ of high risk individuals, which is known to everybody Each individual proposes a policy (B,p) to the company, where: B is the proposed coverage p is the proposed price per monetary unit covered After each proposal (B, p) by an individual, the risk neutral insurance company chooses whether to Accept (A) or Reject (R) the proposal.

21 The Insurance Signaling Game Nature λ 1 λ High Risk Individual Low Risk Individual (B, p) (B, p) (B, p ) (B, p ) Insurance Company A R A R Insurance Company A R A R Let (B,p) and (B, p) be the policies proposed by the high and low risk individuals respectively Let σ(b, p) {A,R} be the reaction function of the insurance company let µ(b,p) be the insurance company s beliefs that the consumer who proposed (B,p) is a high risk individual.

22 Definition. Pure Strategy Sequential Equilibrium A pair of policy proposals (B,p) and (B,p) by the high and low risk individuals, together with a reaction function σ(b, p) and the beliefs µ(b, p) by the firm, constitute a pure strategy sequential equilibrium of the insurance signaling game if: (i) Given σ(b,p), (B, p) and (B,p) maximize the high and low risk individuals expected utility respectively, (ii) the insurance company s beliefs µ(b.p) satisfies the Bayes rules, that is, (a) µ(b,p) [0, 1] for any proposal (B,p), (b) (B,p) (B,p) µ(b,p) = 1, µ(b,p) = 0 (c) (B, p) = (B, p) µ(b,p) = µ(b, p) = λ, (iii) for any policy proposal (B, p), the insurance company s reaction function σ(b, p) maximizes its expected profits given its beliefs µ(b,p)

23 Definition. Separating and Pooling Signaling Equilibria A pure strategy sequential equilibrium of the insurance signaling game is Separating if (B, p) (B, p) while is Pooling otherwise Theorem 4.1 Separating Equilibrium Characterization The policies (B, p) and (B,p) are proposed by the high and low risk individuals respectively and accepted by the insurance company in some separating equilibrium if and only if (i) (L, π) = (B,p) (B,p) (ii) p π (iii) E(u(B, p)) max (B,p) E(u(B,p)) s.t. p = π (iv) E(u(B,p)) E(u(B, p))

24 Theorem 4.2 Pooling Equilibrium Characterization The policy (B,p) is proposed by both, the high and low risk individuals, and accepted by the insurance company in some pooling equilibrium if and only if (i) E(u(B,p)) max (B,p) E(u(B,p)) s.t. p = π (ii) E(u(B,p)) E(u(L,π)) (iii) p λπ + (1 λ)π

25 Screening There are two types of risk averse individuals: High risk individuals, with probability π of having an accident Low risk individuals, with probability π of having an accident There is a proportion λ of high risk individuals, which is known to everybody Each risk neutral insurance companies offer a finite menu of policies of the form (B,p) Nature moves second and determines, according to the probability λ, which individual is looking for insurance The selected individual moves last by choosing a policy from one of the two insurance companies lists

26 Graphical Representation Consider the following picture representing the situation of one individual with risk π ω accident Actuarilly Fair Line 45 ω Slope = 1 π π ω πl ω L ω L ω πl ω ω no accident (1 π)ω + π(ω L) 1 π

27 There are two states, {no accident, accident} The probability of having an accident is π The Expected Income E(ω) = (1 π)ω + π(ω L) = ω πl determines the Actuarially Fair line, with slope 1 π π Any point in the actuarilly fair line yields the same expected income, ω πl If the price per monetary unit insured is actuarilly fair (that is, p = π), then the actuarilly fair line acts as a Budget Constrain

28 The full insurance allocation is in the actuarilly fair line. At this allocation: The firm makes zero profits The individual maximizes the expected utility Hence, it corresponds to the symmetric information equilibrium

29 Pooling Equilibrium The two individuals have different risks. Therefore, the situation ω accident Actuarilly Fair Line 45 Slope = 1 π π ω L Slope = 1 π π ω ω no accident In a Pooling Equilibrium both risk types buy the same policy (B p, p p )

30 In equilibrium, firms must make zero expected profits. That is, if E(π) = λπ + (1 λ)π is the expected risk, then the pooling equilibrium price must be p p = E(π) Hence, in the pooling equilibrium, the equilibrium policy (B p, p p ) must lie in the expected actuarilly fair line

31 ω accident Slope = 1 E(π) E(π) 45 Slope = 1 π π ω L Slope = 1 π π ω ω no accident Notice now, that for any policy (B,p), we have that MRS(ω na,ω a ) < MRS(ω na,ω a )

32 where ω na = ω pb ω a = ω pb L + B This is so because, MRS(ω na, ω a ) = u (ω pb)(1 π) u (ω pb L+B)π Hence, MRS(ω na, ω a ) = u (ω pb)(1 π) u (ω pb L+B)π MRS(ω na, ω a ) MRS(ω na, ω a ) = 1 π π π 1 π < 1 which shows that the indifference curve of low risk individuals is more steep than that of high risk individuals. This is known as the single-cross property.

33 ω accident Slope = 1 E(π) E(π) 45 Slope = 1 π π ω L Slope = 1 π π A u B u ω ω no accident Notice now that only points (policies) like A can be pooling equilibria because Is on the expected actuarilly fair line and so firms make zero profits Both individuals are better off at A than not having any insurance (initial situation) But... Suppose that another insurance company offers a policy like point B

34 High risk individuals do still prefer A to B But low risk individuals will switch to the new company offering B! This is profitable for the company offering this policy as it attracts only low risk individuals and at a price that is above the risk π This is called cream skimming After this, no firm will remain offering A as only high risk individuals will buy it and then the expected actuarilly fair line will no longer be valid Consequently, the pooling equilibrium does never exist. It is always disrupted by a rival policy that skims the low risk individuals from the pool

35 Consider the following figure Separating Equilibrium ω accident Slope = 1 π π 45 A A ū ω L Slope = 1 π π ω ω no accident Ù Notice that, if A is offered to high risk individuals, then A is the best separating equilibrium policy that can be offered to low risk individuals because It is on the actuarilly fair line of the low risk individuals, so firms make zero profits

36 Other policies in the same actuarilly fair line which would be better for low risk individuals would also attract high risk individuals and, hence, would not be separating Therefore, A is the policy that determines the separating constraint Consequently, a separating equilibrium could be The menu of policies A, A is offered High risk individuals choose A while low risk individuals prefer A Both policies produce zero profits to the insurance companies as each lies on the actuarilly fair line of each group Can it be disrupted???

37 2. Information Economics 2.2 Moral Hazard Moral Hazard (danger because of immoral behavior) is a situation that appears because of the null incentives that insured individuals have to drive as carefully as without any insurance. The information problem is that insurance companies can not monitor all individuals to determine how much effort they put on driving. Incentives is the only way to induce individuals to behave properly. This is known as the principal-agent problem. To study this phenomena we use the same market for car insurance with the following simplifying assumptions: Car accidents may result in different losses, No accident corresponds to the case l = 0 l {0,1, 2,...,L}. There is only one consumer. His probability of having an accident with a loss of l depends on how much effort e he puts on driving, π l (e) > 0. For simplicity, effort can be either high (e = 1) or low (e = 0). For completeness, L l=0 π l(e) = 1 for any e {0,1} Individuals are assumed to dislike effort. In this sense, d(e) will measure the disutility of effort (d(1) > d(0))

38 2. Information Economics There is only one insurance company that can observe the severity of the accident (l) but can not observe the level of effort (e). Consequently, the company can only link the benefit in case of an accident to its severity l (not to the effort), Thus, policies offered by the company are of the form (p,b 0,B 1...,B L ) where p is the premium paid by the individual to the company in exchange of the title to receive B l monetary units if an accident with a loss l occurs First, we will study the ideal case of symmetric information in which the company can tie each contingent benefit B l to the level of effort e

39 2. Information Economics 2.2 Moral Hazard Assumption π l (0) > π l (1) for any l {1,...,L} Assumption The Monotone Likelihood Ratio is strictly increasing in l π l (0) π l (1)

40 2. Information Economics 2.2 Moral Hazard Symmetric case If the company can verify the level of effort e {0,1} by the individual, the it can set up the policy so that it pays benefit B l only if some particular level of effort is observed. The maximization problem of the company illustrates how, in fact, the company may act as if it could choose the level of effort max e,p,b 0,B 1,...,B L p L l=0 π l (e)b l s.t. L l=0 π l(e)u(ω p l + B l ) d(e) u where u denotes the individual s reservation utility. Given a particular level of effort e, standard maximization techniques produce the following Lagrangian

41 2. Information Economics 2.2 Moral Hazard L = p [ L L ] π l (e)b l + λ π l (e)u(ω p l + B l ) d(e) u l=0 whose First Order Conditions are l=0 L [ p = 1 λ L ] l=0 π l(e)u (ω p l + B l ) = 0 (3) L = π l (e) + λπ l (e)u (ω p l + B l ) = 0 ( l) (4) B l L λ = L l=0 π l(e)u(ω p l + B l ) d(e) u 0 (5) where (5) holds with equality when λ 0

42 2. Information Economics 2.2 Moral Hazard Notice that: Condition (3) is implied by the (L + 1) conditions in (4) Hence, we have (L + 2) independent equations (at most) and (L + 3) unknowns. We can thus normalize B 0 = 0 Conditions (4) imply that λ 0 and, thus, u (ω p l + B l ) = 1 λ l Hence, it must be the case that (B l l) is constant for all l {0, 1,...,L} Since λ 0, condition (5) must hold with equality. That is, L π l (e)u(ω p l + B l ) d(e) u = 0 l=0

43 2. Information Economics 2.2 Moral Hazard but since (B l l) is constant, this can be written as u(ω p l + B l ) L π l (e) = d(e) + u l=0 that is, u(ω p l + B l ) = d(e) + u Notice now that, because of the normalization B 0 = 0, the equation above depends only on p and thus p = ω u 1 (d(e) + u) Finally, since B 0 0 = 0 and we know that B l l must be constant for any l, it must be the case that B l = l for any loss l {0, 1,...,L} and for any effort level e {0, 1} we have fixed. Thus, in (the symmetric) equilibrium, the individual is fully insured against any loss

44 2. Information Economics 2.2 Moral Hazard Asymmetric Case Now we assume (interesting case) that the company can not observe the individual s effort. The problem for the company is to set up a policy in such a way that not only the the individual prefers to exert a particular level of effort rather than receiving his reservation utility (Individual rationality constraint). Now, it must be in the interest of the individual to exert the level of effort that the company prefers, which is known as the incentive compatible constraint The company s problem becomes:

45 2. Information Economics 2.2 Moral Hazard max e,p,b 0,B 1,...,B L p L l=0 π l (e)b l s.t. and L l=0 π l(e)u(ω p l + B l ) d(e) u L l=0 π l(e)u(ω p l + B l ) d(e) L l=0 π l(e )u(ω p l + B l ) d(e ) where e, e {0, 1} and e e As before (symmetric case) we solve the maximization problem for each level of effort e {0,1} and then the company will set up the policy that induces its desired level of effort on the side of the individual

46 2. Information Economics 2.2 Moral Hazard Optimal policy for e = 0 Recall that in the symmetric case (without the Incentives Compatible Constraint), the optimal policy for e = 0 would be to choose (p,b 0,...,B L ) to satisfy u(ω p) = d(0) + u B l = l l = 0, 1,...,L (6) Notice that: 1. Adding the [IC] constraint the company s profits can not go up 2. Hence, if the solution of (6) satisfies the [IC] constraint then it must also be optimal in this asymmetric case 3. And... indeed it does satisfy the [IC] constraint because if e = 0 then the [IC] constraint boils down to d(0) d(1)

47 2. Information Economics 2.2 Moral Hazard which is true by assumption Hence, choosing a policy that maximized the company s profits and that induces the consumer to choose the low effort means choosing the same policy as in the symmetric case!

48 2. Information Economics 2.2 Moral Hazard In this case, the corresponding Lagrangian is Optimal policy for e = 1 L = p + β [ L L ] π l (1)B l + λ π l (1)u(ω p l + B l ) d(1) u l=0 l=0 [ L ( L )] π l (1)u(ω p l + B l ) d(1) π l (0)u(ω p l + B l ) d(1) l=0 l=0

49 2. Information Economics 2.2 Moral Hazard and the corresponding First Order Conditions, L [ p = 1 λ L ] l=0 (π l(1) + β(π l (1) π l (0))) u (ω p l + B l ) = 0 (7) L = B l π l (1) + (λπ l (1) + β(π l (1) π l (0))) u (ω p l + B l ) = 0 ( l) (8) L λ = L l=0 π l(1)u(ω p l + B l ) d(1) u 0 (9) L β = L l=0 (π l(1) π l (0)) u(ω p l + B l ) + d(0) d(1) 0 (10) where (9) and (10) hold with equality if λ 0 and β 0 respectively

50 2. Information Economics 2.2 Moral Hazard As before, condition (7) is implied by the L + 1 conditions in (8) so that we have one degree of freedom to set B 0 = 0. Also, (8) can be written as 1 u (ω p l + B l ) = λ + β [ 1 π ] l(0) π l (1) Notice that if β = 0 then (11) would imply that (ω p l + B l ) is constant and then (10) will simplify to d(0) d(1), which is not true!. Hence, β 0 (In fact, because of the Kuhn-Tucker conditions, β > 0) Also, the Monotone Likelihood property implies that l, l such that π l (0) > π l (1) and π l (0) < π l (1). This means that if λ = 0 then the right hand side of (11) would take positive and negative values (because we know that β 0). But, if we pay attention to the left hand side of (11), we see that it is always positive. Hence it must be the case that λ 0 Finally, because β > 0, the Monotone Likelihood Ratio property implies that the right hand side of (11) is strictly decreasing in l. Hence, u (ω p l + B l ) must be strictly increasing and (ω p l + B l ) strictly decreasing (if the individual is risk avers). (11)

51 2. Information Economics 2.2 Moral Hazard Therefore, the optimal policy for the company when e = 1 must have the following property l B l isstrictly increasing (12) Since B 0 = 0, (12) implies that B l < l l > 0 Intuitively, this is the incentive for the individual to put high effort

52 2. Information Economics 2.2 Moral Hazard Final Remarks In both cases (symmetric and asymmetric), we have found what is the best policy for the company at each level of effort. Which one is the optimal level of effort depends on each particular situation Suppose that in the symmetric case the best for the company is that the individual chooses e = 0. Then, the optimal policy is to sell full insurance to the individual. And this will also be the optimal policy in the asymmetric case because when e = 0 because the expected profits in this case can not be higher than the expected profits in the symmetric case since we have one more constraint. Hence, if such policy was efficient in the symmetric case, it is also efficient in the asymmetric case Suppose now that in the symmetric case the best for the company is that the individual chooses e = 1. Then, in order to induce the individual to actually exert a high effort, the company must set up a policy in which the real loss to the individual (l B l ) increases with the severity of the accident. This cannot be efficient since in the symmetric case we have seen that the company offers full coverage (B l = l) for any level l regardless the effort

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