Insurance + Hidden Action Problem

Question 1

What is the Exp. utility of taking High Care?

Answer 1

EU = p U(Wt) + (1 - p) U(Wnt) - ß

Question 2

What is the Exp. utility of taking Low Care?

Answer 2

EU’ = p’ U(Wt) + (1 - p’) U(Wnt)

Question 3

What are the 2 Fair Odds Lines?

Answer 3

E(π) = 0 - Individ. chooses High Care
E(π') = 0 - Individ. chooses Low Care

Question 4

What does Contract J offer?

Answer 4

Wfi in Both States - Full Insurance
EU = U(Wfi) - ß
EU’ = U(Wfi)

Question 5

Why is Contract J not Incentive Compatible for High Care?

Answer 5

Incentive to Cheat and take Low Care after accepting contract –> E(π) < 0 - Losses

Question 6

Why is a contract at Fair Odds on High Care (J) not offered??

Answer 6

Both pass through same point –> EU’ = EU + ß
- At J - Certain Wealth regardless of Level of Care
– BUT Low Care gains ß more Utility
=> EU’ > EU
If they could- would locate at J + choose L.C => Exp. Losses

Question 7

Which I.C is steeper?

Answer 7

EU is Steeper than EU’ - due to p’ > p

Question 8

Would Indiv. take Insurance along E(π’) = 0 at Fair Odds (K)?

Answer 8

Can get Higher Utility from Partial Insurance w/ High Care priced at p’ - Tangent to E(π’) = 0 (Point L)
- Cut Full Insurance Line NE of K
As long as ß is Low enough - U(Wl) > U(Wfi’)

Question 9

Why is L better than K even w/ High Care?

Answer 9

Higher Fully Insured level of Wealth

• Higher Utility than at K - would’ve taken L.C + gained small ß
• Small Loss of ß < Gain by Lower Prob. of Theft

Question 10

Can L be Equilibrium + why?

Answer 10

L can NOT be Equilibrium
- H.C but charged a price at p’ > p - Greater than Fair Odds
– Partial Coverage at L –> Supernormal Profits
- Competed away by other companies - end up on E(π) = 0 - priced at p
BUT F.I at J can’t be Eq. –> Eq. at Partial but almost F.I - Point D

Question 11

Why is D Incentive Compatible?

Answer 11

Insured are Indifferent between:
- Incurring cost of care w/ Lower p –> Higher Exp. Wealth
- Saving cost of care c/ Higher p’ –> Lower Exp. Wealth
They choose H.C due to ‘epsilon-truthfulness’

Question 12

What happens to efficiency + welfare under Hidden Action?

Answer 12

Insurers - Indifferent between contracts on Zero Profit Lines
Insured- Worse off at D than J + Worse off at K than J
Pareto Sub-Optimal Outcome - Insured suffer - can’t commit to not cheating despite being in their interest

Question 13

What is Combinational Insurance Policy?

Answer 13

Insured can achieve Full Insurance w/ Several Partial Contracts
- Insurers often refuse to Co-Insure