The figure below shows the total utility that Raj, a first-year degree student, would receive from different levels of annual income. Assume at the moment that his annual income (from an allowance from his parents and some part-time work in a burger bar) is £4000. Spending this rationally gives him a total utility of 500 ‘utils’.
Assume that he is offered the chance to gamble the whole £4000 on the single toss of a coin. If he wins then he doubles his money and has an income of £8000. If he loses then he has to pay £4000 and so has no income left.
(a) If he takes the gamble, what will be his utility this year if he wins?
(b) If he takes the gamble, what will be his utility this year if he loses?................................................................................
(c) What would be his expected utility from the gamble?
(d) Why is it likely that he will not take the gamble, and thus be risk averse? ......................
Assume now that Raj does not take the gamble but is still worried because there is a 16 per cent chance that his £4000 will be stolen. An insurance company offers him a full insurance policy against this loss for a premium of £800.
(e) What is the expected value of taking the gamble (i.e. of not taking out the insurance)?
(f) What is his expected utility from taking the gamble (i.e. of not taking out the insurance)?
(g) What is his expected utility if he purchases the insurance (i.e. pays the £800 premium)?
(h) Would he buy the insurance policy for a premium of £800? Explain
(i) Would he buy the insurance policy for a premium of £750? Explain.
(j) Assume the insurance has many customers in exactly the same position as Raj (i.e. facing a 16 per cent chance that their income of £4000 will be stolen). How much will it pay out on average in claims per customer? Can it make a profit if it charges a premium of £750?
(k) Assume now that when customers like Raj take out the insurance policy they take less care with their money and the chances of it being stolen increase from 16 per cent to 25 per cent. Explain any problems this might cause the insurance company.