question archive (2+ 5=+ 1 + 2 +2) Consider an insurance market with insurance ?rms being competitive and risk neutral (Zero expected pro?ts in equilibrium) and a risk averse customer with a von-Neumann Morgenstern utility function 110:) = )8"
(2+ 5=+ 1 + 2 +2) Consider an insurance market with insurance ?rms being competitive and risk neutral (Zero expected pro?ts in equilibrium) and a risk averse customer with a von-Neumann Morgenstern utility function 110:) = )8"
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(2+ 5=+ 1 + 2 +2) Consider an insurance market with insurance ?rms being competitive and risk neutral (Zero expected pro?ts in equilibrium) and a risk averse customer with a von-Neumann Morgenstern utility function 110:) = )8". The customer is bcm with a wealth of $150. There is a probability of an accident p=0.3 in which case the customer will suffer a damage of $50. An insurance contract is ((11,(12), where (x1 = premium paid in the no accident state, and a2= net payout received in case of accident
(i) What is the customers utility without insurance?
(ii) If all Zero pro?t insurance contracts are offered, write down the customer's utility maximization problem. What contract would she choose? What will be the premium? What is the paycut?|
(iii) Is the customer happier with insurance? Show why or why not.
(iv) If there was another customer with p =0.5, (a high-risk customer) what contract will she choose? (assume everything else for this new customer is the same as the old one, and the customers are identi?able). Show your work.
(v) Suppose both contracts (for part (ii) and (M) are being offered by insurance companies, and all customers are free to choose whatever contract they prefer. Which one will the high type customer choose? Show your work.
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Answer:
1)
Damage payoff = 30%*-50+ 70%*0 = -15
Net Wealth = 150 - 15 = 135
Utility = 135^(1/3) = 5.13
2)
Let alpha 1 be x and alpha 2 be y
Wealth = 150 - x + 30%*(y - 50) = 135 - x + 0.3y
Maximize
U = (135 - x + 0.3y)^(1/3)
For zero profit,
x = 30%y
y = 50 [Since the customer has Morgenstern Utility function ]
x = 30%*50 = 15
Premium Paid = $15, Payout = $50
iii) The customer is equally content with the policy as the Utiliy remains same
iv)
With Probability = 50%
For zero profit,
Premium = Expected value of Pay out.
Since the customer has Morgenstern Utility function, therefore the customer will choose for highest payout.
Premium = 50%*50 + 50%*0
=> Premium = $25
Pay out = $50
v)
If the customer is free to choose any contract the customer with high probability will choose the contract in (ii) as it maximises its utility
Wealth = 150 - 15 + (50%*(50-50) + 50%*0) = 135
Compared to iv where wealth = 150 - 25 = 125
