4 Rothschild-Stiglitz insurance market

Transcription

1 4 Rothschild-Stiglitz insurance market Firms simultaneously offer contracts in final wealth, ( 1 2 ), space. state 1 - no accident, and state 2 - accident Premiumpaidinallstates, 1 claim (payment from firm in state 2), b 2 is damages and is initial wealth 1 = 1 2 = 1 + b 2

2 probability of accident, high risk Firms: risk neutral, free entry low risk Assume Firms cannot identify the risk type of consumers. A firm can only offer one contract. Consumers can only purchase one contract. Consumers differ only in accident probabilities. Utilityisoftheform(1 ) ( 1 )+ ( 2 )

3 Equilibrium: A set if contracts s.t. when consumers choose contracts to max, (i) no contract makes negative expected profits (ii) no unoffered contract makes positive expected profits. (Nash equilibrium of game where firms select contracts)

4 Fair odds: expected profit is the premium minus the expected claim expected claim = b 2 payment(premium) = 1 fair odds if 1 = b 2

5 fair odds in terms of 1 and 2 : z} { 1 1 ] {z } b 2 {z 1 = [ } = =(1 ) One interpretation of this fair odds equation is that the expected endowment consumption (left side) equals the expected consumption after buying the contract (right side).

6 slopeoffairoddslinein 1 2 space is: 1 (ratio of probabilities) Utility as a function of 1 and 2 is (1 ) ( 1 )+ ( 2 ),so = (1 ) 0 ( 1 ) 0 ( 2 ) At the 45 line, = 1 Since high risks have a larger, their indifference curve is flatter at any point

7 +(1 ) where is the population fraction of high risks To be an equilibrium, any pooling contract must be on the fair-odds line based on accident probability. Above it, firms lose money. Below it, a firm could offer higher consumption, steal the customers, and still make profits. For any pooling contract,, there is always a offering the high risks a worse deal and the low risks a better deal. Since it is close to the pooled fair odds line, the firm will profit. Thus, there cannot be a pooling equilibrium.

8 Pooling Equilibrium is Blocked.

9 Separating Equilibrium

10 4.1 Separating Equilbrium exists if there are enough high types. does not exist if there are too many low types. In the "candidate equilibrium", high risks are given complete and fair insurance, because firms need not fear that low risks will take the contract. Low risks must be offered a contract that is on their fair odds line, and lies below the indifference curve of the high risk type running through, labeled. (If it is above, highs will take it and we don t have separation. If it isn t on fair odds line, either losses or profitable entry.) Low risks must be offered, because anything below allows profitable entry.

11 Equilibium, when it exists, is inefficient. When there are just a few bad apples, no equilibrium. More than one price is offered in equilibrium. If firms can offer multiple contracts, a pair in which are contract loses and the other gains might break the separating equilibrium. If consumers can privately accept as many contracts as they wish (i) unmodelled moral hazard problems might emerge (ii) there cannot be a separating equilibrium (iii) pooling equilibrium where overinsure and underinsure. Alternative equlibrium concepts

12 5 Lemons Market (Akerlof) Why does the value of your car drop by 10% or more as soon as you leave the dealership? Adverse selection if you try to sell it, it is more likely to be a lemon (bad car). Two-quality Example (Sketchy) New cars: good quality (sometimes called a cream puff) with prob., and a lemon with prob. (1 )

13 Owners know the quality of their car, but buyers do not. Therefore, good and bad cars must sell at the same price. Even without depreciation, a used car must have a lower price than a new car. Proof: If used new, all the lemon owners will sell, since they can buy a new car, getting a fresh chance at a good car with money left over. In this case, the average quality of used cars is weakly lower than the average quality of new cars, so no one will buy a used car instead of a new car.

14 If used = new, again all lemon owners will sell. No one with a good car will sell in order to buy a new (or used) car. This is because their good car is higher quality than the average new (or used) car. As long as someone with a good car wants to have a car, the average quality of used cars will be less than the average quality of new cars. But again, no one will buy a used car. So used new, and no one that wants to have a car will sell a good one. Like Gresham s law: the bad cars drive the good out of circulation.

15 Continuous Model average quality of used cars on the market. =1 Type 1 traders: 1 = + P money consumption automobiles owned quality of car Type 2 traders: 2 = + P 3 2 =1 Type 1 traders are endowed with cars (due to linearity, distribution across traders does not matter) Type 2 traders endowed with 0 cars and 2 units of money

16 Asymmetric Information Assume [0 2] for =1 Now average quality of cars offered for sale depends on the price. If the price is, anytype1 with a car of quality will sell. ( ) = 2 Given ( ), notype1 will ever buy for any 0. Type 2 s are willing to buy iff 3 2, but since = 2 3 2,notype2 will buy for any 0. Excess supply unless =0 Complete unraveling of the market.