Microeconomics. 3. Information Economics

Transcription

1 Microeconomics 3. Information Economics Alex Gershkov 9. Januar / 19

2 1.c The model (Rothschild and Stiglitz 77) strictly risk-averse individual initial wealth of W (s)he runs a risk of a loss of d (accident) probability of an accident is p The individual can insure himself against this accident by buying an insurance contract. An insurance contract α is a pair (α 1,α 2 ) where α 1 is the premium α 2 compensation in case of accident 2 / 19

3 1.c without insurance, the individual s income is (W,W d) with insurance, the income is (W α 1,W d α 1 + α 2 ) The utility which an individual with risk of accident of p obtains if (s)he gets the insurance contract α is V (p,α) = (1 p)u(w α 1 ) + pu(w d α 1 + α 2 ). 3 / 19

4 1.c Insurance companies many (at least two) insurance companies risk neutral, i.e. contract α if sold to an individual with probability of accident p worth π(p,α) = α 1 pα 2. 4 / 19

5 1.c Timing 1. Nature selects individual s type, p {p H,p L } where Pr(p = p I ) = µ I ; this type is privately observed 2. two insurance firms simultaneously announce insurance contracts α 3. individual selects one preferred contract (ties are broken randomly, 50:50). Notice that since the informed party moves last, there is no learning in the game. 5 / 19

6 1.c Definition A separating eq m in a competitive screening model is an e qm where each type of agent chooses a different contract in e qm: ie. α H α L. Definition A pooling eq m in a competitive screening model is an e qm where each type of agent chooses the same contract in e qm: ie. α H = α L. 6 / 19

7 1.c Observable types: 1. if there is the set A of the available contracts, our individual p I will choose α such that if V (p,α) > V (p,0) α argmax α A V (p,α). 2. firms earn zero profits (Bertrand competition) 3. α I = (dp I,d). 7 / 19

8 Insurance ωa certainty line The slopes of the indifference curves are ω d dω A deu=0 = (1 pi )u (ω N ) dω N p I u. (ω A ) ω 1 p I p I ωn How much insurance will be provided? (1 p I )u (W α 1 ) p I u (W α 1 + α 2 d) = 1 pi p I. 8 / 19

9 1.c But, of course, types are not observable: were the above contracts to be offered, then all individuals would choose α L. Lemma 1: In eq m, firms earn zero profits. Proof: Let α L and α H be the contract chosen by the low- and high-accident probability individuals, respectively, and suppose that Π = i π > 0. Then one firm must earn π i Π/2. But this firm i could earn close to all Π by offering (α L 1 ε,αl 2 ) and (αh 1 ε,αh 2 ) for ε > 0. Thus we must have Π 0. Because no firm can incur a loss in any equilibrium (it can always gets non-negative profits by offering (0,0)), both firms must in fact earn a profit of zero. 9 / 19

10 1.c Lemma 2: There is no pooling eq m. Proof: Every contract ( α 1, α 2 ) attracting only the low-probability types yields positive profits and thus breaks the zero-profit pooling contract (α 1,α 2 ). 10 / 19

11 1.c Lemma 3: In (separating) eq m, all contracts earn zero profits. Proof: 1. Suppose α H 1 > ph α H 2. Then a firm can earn positive profits by offering just contract (α H 1,pH α H 2 ) which would be accepted by (at least) the high-probability types of individuals. 2. Suppose α L 1 > pl α L 2. Hence a firm could make positive profits by offering ( α 1, α 2 ) SE of (α L 1,αL 2 ) which would attract only low-probability type individuals. Hence there is no cross-subsidy in eq m. 11 / 19

12 1.c Lemma 4: In a separating eq m, α H 2 = d. 12 / 19

13 1.c Lemma 5: In a separating eq m, (α L 1,αL 2 ) satisfies U(W p H d) = (1 p H )U(W α L 1) + p H U(W α 1 d + α 2 ). with α L 1 = pl α L 2. Question: But does the separating equilibrium always exist? 13 / 19

14 1.c : Summary The main insights which can be derived from the simple 2x2 (competitive) model are eq a may not exist (in pure strategies) if an eq m exists, it is separating if an eq m exists, it is second best all rents go to the agents the high type is better off without screening. 14 / 19

15 1.c Observable types. If p I is observable, then the monopolist s problem is s.t. max α α 1 p I α 2. (1 p I )U(W α 1 )+p I U(W d α 1 +α 2 ) (1 p I )U(W )+p I U(W d). 15 / 19

16 1.c Theorem The optimal insurance contract with observable types sets α 2 = d and α 1 as the solution to U(W α 1 ) = (1 p I )U(W ) + p I U(W d). 16 / 19

17 1.c Unobservable types. The monopolist s optimal pricing rule is given by max α H,α L I {H,L} s.t. for I,J {H,L} holds µ I (α I 1 p I α I 2). (1) (1 p I )U(W α I 1)+p I U(W d α I 1+α I 2) (1 p I )U(W )+p I U(W d) (1 p I )U(W α I 1 ) + pi U(W d α I 1 + αi 2 ) (1 p I )U(W α J 1 ) + pi U(W d α J 1 + αj 2 ) 17 / 19

18 1.c Lemma 1 We can ignore the (IR) constraint of type p H. That is, a pair of contracts is a solution to the problem (1) if and only if it is a solution to the problem derived from (1) by dropping the (IR) constraint for player p H. Lemma 2 In the optimal contract, we have (1 p L )U(W α L 1)+p L U(W d α L 1+α L 2) = (1 p L )U(W )+p L U(W d) Lemma 3 In the optimal contract, α H 2 = d. 18 / 19

19 1.c Summary: optimal contract cannot extract all surplus from all types of agents it provides partial insurance for the type p L the type that values insurance the most, gets complete insurance. 19 / 19