Graduate Microeconomics II Lecture 8: Insurance Markets

Transcription

1 Graduate Microeconomics II Lecture 8: Insurance Markets Patrick Legros 1 / 31

2 Outline Introduction 2 / 31

3 Outline Introduction Contingent Markets 3 / 31

4 Outline Introduction Contingent Markets Insurance firms 4 / 31

5 Outline Introduction Contingent Markets Insurance firms Moral Hazard 5 / 31

6 Outline Introduction Contingent Markets Insurance firms Moral Hazard Adverse Selection 6 / 31

7 Outline Introduction Contingent Markets Insurance firms Moral Hazard Adverse Selection Akerlof; Outside Options 7 / 31

8 Introduction Insurance is a way for society to share risk Independent versus aggregate risks Parallel with contingent markets 8 / 31

9 Introduction Insurance is a way for society to share risk Independent versus aggregate risks Parallel with contingent markets 9 / 31

10 Contingent Markets Basic Model I (identical) agents and a single good. Endowment of an agent is stochastic: w with probability 1 π and w L with probability π. Probability are independent across agents Interpretations: loss of an asset, risk of being laid-off, risk of being sick. vnm utility function: u increasing concave 10 / 31

11 Q: What is a state of nature? 11 / 31

12 Q: What is a state of nature? A: Who are the agents suffering a loss. There are 2 I states of the world, and from Arrow-Debreu, need to organize 2 I markets. x i s is agent i s consumption of the good in state s, s = 1, 2,, S with S = 2 I. The probability of state s is π s = π ns (1 π) I ns We can write state-contingent endowments: w i s = w if i did not suffer a loss and w i s = w L if s/he suffered a loss. Note that only the i-th component of s matters. 12 / 31

13 Let {p s } be a vector of contingent prices. An Arrow-Debreu competitive equilibrium is a vector of prices p s and quantities consumed x s, s = 1, 2,, S such that: Each agent finds x to be the optimal contingent consumption plan given prices p : s.t. max x i s S π s u(xs) i s=1 S p s xs i = s=1 S p s ws i s=1 The aggregate budget constraint is satisfied: S s=1 x i s = S s=1 w i s An equilibrium exists and is Pareto Optimal. (Note necessarily symmetric, but focus on symmetric allocations later on.) 13 / 31

14 Contingent Markets I large When I is large, the law a large numbers implies that the number of agents who suffer a loss is equal to πi almost surely. Hence, almost surely, the per-capita endowment in the economy is π(w L) + (1 π)w = w πl Because agents are risk-averse, it is Pareto optimal to have each agent consume the quantity in each state of nature; that is x i s = w πl, for all agent i and for all state s Hence when I goes to infinity the egalitarian optimum tends toward the constant allocation w πl. However, rather than using incomplete markets, this allocation can be implemented if insurance is provided by firms on a competitive market. 14 / 31

15 Insurance firms Firms serve as intermediaries and provide insurance - smoothing of consumption - on an individual basis. A typical contract specifies a premium P that the insuree pays - independently of the state of the world - and a reimbursement R that is paid to the insuree if there is a loss. When contracts are individual (do not depend on what happens to other agents), the total premium paid is PI and the expected reimbursement is. πir 15 / 31

16 The expected utility from an individual contract is πu(w L + R P) + (1 π)u(w P) Competition between insurers zero profit condition P = πr To recover the Pareto optimal egalitarian allocation we need w L + R P = w P = w πl Hence, P = πl & R = L 16 / 31

17 Insurance firms Graphical representation: state space diagram state 2 ( loss ) full insurance w L 1 π π w πl w state 1 ( no loss ) 17 / 31

18 Insurance firms The insurance firm can allow agents to purchase partial insurance: if the agent insures a loss z then the price s/he pays is pz. In this case an agent will solve max x 1,x 2,z (1 π)u(x 1 ) + πu(x 2 ) x 1 = w pz x 2 = w L + z pz (Note the change of variable: a contract gives an allocation x s which is equivalent to paying a premium of pz and getting reimbursed z. Hence max(1 π)u(w pz) + πu(w L + z pz) z The FOC is (1 π)pu (w pz) + π(1 p)u (w L + z pz) = 0 and when p = π (that is when there is zero profit) z = L, that is, the agent chooses to be fully covered. 18 / 31

19 Moral hazard Suppose that the probability π of a loss is a function of the care a taken by the agent: drive carefully to avoid a car accident close doors to avoid theft eliminate some foods to decrease the risk of heart failure If the a has cost a, the income per capita is now w π(a)l a. The social optimum: full insurance and maximize u(w π(a)l a), hence (1 + π (a)l)u (w π(a)l a) = 0 1 = π (a)l The optimum cannot be incentive compatible since full insurance would lead the agent to choose a = / 31

20 Moral hazard Second best Cannot provide full insurance in a competitive market with moral hazard: the agent must bear some of the risk. How much? Let z the amount of insurance and p the price. The problem for the agent is now, max z,a π(a)u(x 2(z)) + (1 π(a))u(x 1 (z)) p = π(a) x 1 (z) = w a pz x 2 (z) = w L a + z pz π (a)[u(x 2 (z)) u(x 1 (z))] π(a)u (x 2 (z)) (1 π(a))u (x 1 (z)) = 0 (1) The incentive constraint (1) is compatible with a > 0 only if u(x 2 (z)) u(x 1 (z)) > 0 20 / 31

21 Adverse Selection Unique price of coverage Agents face different risks (Good and bad drivers). The population is composed of good risks (π G ) and bad risks (π B > π G ). Then, 1 π G π G > 1 π B π B. (2) The insurance market should discriminate between the two types. Suppose however that the price per unit of coverage is the same p for type t. Each type chooses full insurance By (2), we have u (w L pz t + z t ) u (w pz t ) z G < z B = 1 π t π t p 1 p Bad types choose to purchase more insurance than good types if the price of insurance is uniform. 21 / 31

22 Adverse selection State space state 2 ( loss ) full insurance w L 1 π G π G 1 π B π B B G c w πl w state 1 ( no loss ) 22 / 31

23 Adverse selection Uniform price - continued Zero profit condition given price q per unit of coverage: line going through the initial endowment with slope between (1 π G )/π G and (1 π B )/π B. If maximum price: sell at zero profit to B but G types prefer to bear the risk without insurance. If minimum price: zero profit with G but loses with B. Rather than uniform price: discrimination between types. Firm offers two contracts at most. If different: separation of types in equilibrium, in one only: pooling. Still problem with existence of an equilibrium 23 / 31

24 Adverse selection Existence - Pooling A special screening model, but must take into account the possibility of free entry in the insurance market. If firms offer a pooling equilibrium: they must make zero profit, and the G must subsidize the B. why can t a new firm offers contracts that are attractive to the G only? Always can find a contract on the iso-profit line going thru the pooling equilibrium that attracts only the G types: must then be consistent with positive profit! Pooling contracts can never be part of an equilibrium. 24 / 31

25 Adverse selection State space Candidate separating equilibrium: no profit from G and no profit from B state 2 ( loss ) w L B G O w πl w state 1 ( no loss )

26 Adverse selection State space Candidate separating equilibrium: no profit from G and no profit from B state 2 ( loss ) pooling contract w L B G O w πl w state 1 ( no loss ) 26 / 31

27 Therefore if the measure of type G is sufficiently high, a pooling contract will induce both B and G to deviate from the separating contracts: there is no equilibrium! However we know that there does not exist a pooling equilibrium, hence the firm that enters and offers a pooling contract will later face entry by a firm offering to cater only to G types. Refinements; Riley s reactive equilibrium. 27 / 31

28 Akerlof First best N agents. Productivity θ and outside option r(θ). For instance r(θ) is what the agent could produce by home production. θ F Θ = [θ 0, θ 1 ] Firms compete for workers by offering wages. Hence if the type of the agent is known the competitive wage is w (θ) = θ as long as the participation constraint holds. θ r(θ) 28 / 31

29 Akerlof Second best Since there is only one instrument (the wage), it is not possible to offer separating contracts: conditional on participating in the labor market, all agents prefer a higher wage! remember that separation works only if there are at least two instruments. However, can separate the agents by inducing some to participate and the others not to participate in the labor market. Let Θ Θ be the types who participate. Then the competitive wage must be w = µ = θdf (θ) Θ Now since Θ = {θ : r(θ) µ}, we have w = E(θ r(θ) µ). This equilibrium is clearly not Pareto optimal 29 / 31

30 Akerlof The Lemon s problem [Exemple] r(θ) = r for all theta. Either w r and all workers participate or w < r and no worker participates. Now, since w = µ = E(θ), if µ < r since no worker wants to participate. [Exemple] r(θ) θ for all θ and r increasing. Then all workers would work in firms. In a competitive equilibrium, only those workers with r(θ) w work. This is the adverse selection problem: only worse types work; impossible to attract the good types. 30 / 31

31 Akerlof The Lemon s problem [Exemple] r(θ) = r for all theta. Either w r and all workers participate or w < r and no worker participates. Now, since w = µ = E(θ), if µ < r since no worker wants to participate. [Exemple] r(θ) θ for all θ and r increasing. Then all workers would work in firms. In a competitive equilibrium, only those workers with r(θ) w work. This is the adverse selection problem: only worse types work; impossible to attract the good types. E.g., if θ uniform on [0, 1], and r(θ) = rθ, with r < 1. Then E(θ r(θ) w) = w 2r which is equal to w only if r = 1/2. For r 1/2 there is market failure. Possibility of multiple equilibria: in previous example, if r = 1/2, an infinity of equilibria. 31 / 31