Question

Ann is risk-averse with a Bernoulli utility function u(w) = 100 + w^1/2 where w is her wealth in dollars. Ann’s current wealth is one million dollars, including her small boat valued at $180, 000. She estimates that with 10% probability the boat will sink and lose its full value; with 15% probability there will be damages and the boat will lose half its value, and with 25% probability the boat will lose a quarter of its value; otherwise, the boat will lose no value.

c) Bob is facing the exact same situation as Ann and is also risk averse but with a Bernoulli utility u(w) = w^3/4 . What is the maximum amount of money that Bob would pay to fully insure his boat? (1 mark) Would you say that Bob is more than, less than, or equally risk-averse as Ann? Explain your answer.

Answer 1

First, finding out for Ann,Ann's expected utility without insurance= 0.1 (100+820,000^0.5) + 0.15(100+910,000^0.5)+0.25(100+955,000^0.5)+0.5(100+1,000,000^0.5)= 1077.955

So finding out her equivalent wealth

1077.955=100+w^0.5

w=977.955²=956,395.953

Therefore Ann would be willing to pay upto 1,000,000-956,395.953=43,604.047$ to insure her boat.

Similarly finding out for Bob

Bob's expected utility= 0.1*820,000^0.75+0.15*910,000^0.75+0.25*955,000^0.75+0.5*1,000,000^0.75=30,593.189

Calculated his equivalent wealth

30,593.189=w^0.75

w=30,593.189^4/3=956,826.056

Therefore Bob would be willing to pay upto 1,000,000-956,826.056=43,173.94$ to insure his boat

Since Bob is willing to pay slightly less for insuring his boat, I would say Bob is less risk averse than Ann(Although both are willing to pay almost the same which is close to 43,500$ so it could be a bit ambiguous)

Hope it helps. Do ask for any clarifications required.