Automobile Insurance and Driver Ability: Contract Choice as a Screening Mechanism

Transcription

1 Automobile Insurance and Driver Ability: Contract Choice as a Screening Mechanism Lisa Posey Penn State University Paul Thistle University of Nevada Las Vegas

2 Compensation for Victims Two approaches to compensating victims of auto accidents Tort (3 rd party): Victim sues injurer under tort law Economic and general damages Injurer indemnified for damages by his/her insurer No fault (1 st party): Victim recovers from own insurer Economic damages only Pure: cannot sue for general damages Limited tort: can sue for general damages if economic damages exceed threshold 26 US states adopted some variation of no-fault/limited tort in early 1970s KY, NJ, PA allow drivers to choose between tort and no-fault

3 Contract Choice Is choice between contract forms a screening mechanism? Most analyses of adverse selection take contract form as given Use differences in contract terms (price, quantity) to screen E.g., Rothschild/Stiglitz We look at form of contract as potential screening mechanism

4 Basic Model Drivers are high or low risk, p H > p L Proportions are l Hs and 1 l Ls Three possible states of the world Cause accident, occurs with probability p i Are accident victim, occurs with probability p 0 = lp H + (1-l)p L No accident, occurs with probability 1 p i p 0 Proportion of drivers with no-fault is p, with tort 1 p Proportions are endogenous

5 Basic Model Economic damages To B when A is at fault, x ~ f(x) To A when B is at fault, y ~ h(y) To A when A is at fault, z ~ k(z) General damages assumed proportional to economic, g All drivers have PIP up to limit I Economic damages above I paid by at-fault driver s policy No-fault: cannot sue for general damages unless economic damages T For simplicity, all policies provide full coverage for liability to others.

6 Basic Model Drivers are risk averse expected utility maximizers U = utility of wealth W = exogenous initial wealth Drivers are not judgment-proof Fault determined without error Cost of adjusting/defending claims against 3 rd party is proportional, g, to damages

7 PA and NJ If driver is victim then Both full tort and no-fault collect economic damages, y, up to I from own insurer Both collect economic damages above I from at-fault driver Full tort can always collect general damages No-fault cannot collect general damages if y < T Can collect economic and general damages if y T

8 PA and NJ Actuarially fair premium is the same for both contracts a i = p i (1+g){E(z z I)K(I) +(1-p)[E(x-I x>i)[1-f(i)] + ge(x)] - pge(x x<t)f(t)} + p 0 E(y y<i)h(i) First three terms are expected claims when A is at fault Expected economic loss to A Expected liability to B when B has no-fault coverage Expected liabilty to B when B has full tort coverage Fourth term is expected indemnification to A when B is at fault

9 PA and NJ Expected utility depends on type, contract form Full tort: V i (a) = (1-p i -p 0 )U(w-a) + p i E[U(W-a-gz)] + p 0 U(W-a) No-fault V i (a) = (1-p i -p 0 )U(w-a) + p i E[U(W-a-gz)] + p 0 {E[U(W-a-gy) y<t]h(t) + U(W-a)(1-H(T))}

10 PA and NJ Expected utility is higher under full tort V i (a) V i (a) > 0 Difference is an increasing, convex function of a

11 PA and NJ Proposition 1: There is a separating equilibrium where Hs choose, pay the fair premium a H (l) Ls choose, pay the premium a L S that satisfies self selection V H FH (a H (l)) = V H (a LS ) Insurers earn zero expected profit on high risks and positive expected profits on low risks The separating equilibrium exists if, and only if, the break-even premium for the pooling full tort contract, a, is greater than a L S and the break-even premium for the pooling no-fault contract, a, is greater than a, where a satisfies V L (a ) = V L (a LS )

12 PA and NJ v V L (a LS ) V L V H (a H (l)) V L V H V H a a H (l) a L S a

13 PA and NJ Proposition 2: There is a full-tort pooling equilibrium Proposition 3: There is a no-fault pooling equilibrium Proposition 4: The equilibria are mutually exclusive

14 KY In NJ and PA, if full tort driver A injures no-fault driver B, then B s economic damages must exceed T before B can sue for general damages In KY, B can always sue A for general damages (T = 0) Premium now depend on driver type and contract form Expected utility with no-fault now depends on distribution of drivers across contract forms, p

15 KY Full tort a i = p i (1+g){E(z z<i)k(i) + E(x-I x>i)[1-f(i)] + ge(x)} + p 0 E(y y<i)h(i) V i (a) = (1-p i -p 0 )U(W-a) + p i E(U(W-a-gz) + p 0 U(W-a) No-fault a i = p i (1+g){E(z z<i)k(i) + E(x-I x>i)[1-f(i)] + ge(x) - pge(x x<t)f(t)} + p 0 E(y y<i)h(i) V i (a; p) = (1-p i -p 0 )U(W-a) + p i E(U(W-a-gz) + p 0 {U(W-a) [pe H (U(W-a) U(W-a-gy) y<t)]h(t)}

16 KY Expected utility is higher under full tort V i (a) V i (a; p) > 0 for p > 0 Difference is an increasing, convex function of a

17 KY Proposition 5: There is a separating equilibrium where Hs choose, pay the fair premium a H (l) Ls choose, pay the premium a L S that satisfies self selection V H (a H (l); l) = V H (a LS ) Insurers earn zero expected profit on high risks and positive expected profits on low risks The separating equilibrium exists if, and only if, the break-even premium for the low risk full tort contract, a L, is no greater than a L S and the break-even premium for the pooling no-fault contract, a, is greater than a, where a satisfies V L (a ; l) = V L (a LS )

18 KY v KY V L (a LS ) PA/NJ V L (a LS ) KY V H (a H (l); l) V H (a H (l)) PA/NJ V L V L (., l) V L KY V H a a H (l) a S L KY a H (l) PA/NJ & KY a L S PA/NJ V H V H (., l) a KY

19 KY Under separating equilibrium in Proposition 5, both high risks and low risks are better off than under the PA/NJ separating equilibrium. Proposition 6: There is a no fault pooling equilibrium Proposition 7: A full tort pooling equilibrium does not exist

20 Conclusion PA-NJ and KY allow drivers to choose between full tort and nofault auto insurance PA-NJ and KY no-fault are slightly different Can allowing drivers to choose contract be a screening mechanism in markets with adverse selection? Yes, there are separating equilibria in both PA-NY and KY systems Empirical tests for adverse selection are ambiguous May be able to use contract choice as an additional dimension to test empirically