a.) Given the Von Neumann-Morgenstern utility function of an individual, U (W) =W ½ , where W stands for amount of money. Comment upon attitude towards risk of such an individual with the help of a diagram.

b) Now suppose this individual possesses a building worth ₹1600. If the building catches fire, its value falls to ₹ 400. Let the probability of building catching fire be ¼. On the basis of the given information, find out whether the individual would be willing to pay a risk premium of ₹ 76 to the insurance company in order to eliminate the risk associated with the factory building.

a). The Von Neumann-Morgenstern utility function is a commonly used model for describing an individual's preferences for different outcomes. In this particular case, the utility function is given as U(W) = W^(1/2), where W represents the amount of money.

To explain the attitude toward risk we will consider a single composite commodity, namely, money income. An individual’s money income represents the market basket of goods that he can buy. It is assumed that the individual knows the probabilities of making or gaining money income in different situations. But the outcomes or payoffs are measured in terms of utility rather than rupees.

In Fig. 17.3 we have drawn a curve OU showing utility function of money income of an individual who is risk-averse. It will be seen from this figure that the slope of total utility function OL; decreases as the money income of the individual increases. Note that we measure money income on the X-axis and utility on the Y-axis.

It will be seen from Fig. 17.3 that as money income of the individual increases from 10 to 20 thousand rupees, his total utility increases from 45 units to 65 (that is, by 20 units) and when his money increases from 20 thousand to 30 thousand rupees, his total utility increases from 65 to 75 units (that is, by 10 units).

Thus in this concave utility function depicted in Fig. 17.3 marginal utility of money of an individual decreases as his money income decreases and therefore it represents the case of risk-averse individual. Suppose the individual is currently employed on a fixed monthly salary basis of Rs. 15000.

There is no uncertainty about the income from this present job on a the fixed salary basis and hence no risk. Now, suppose that the individual is considering to join a new job of a salesman on a commission basis. This new job involves risk because his income in this case is not certain.

This is because if he proves to be a successful salesman his income may increase to Rs. 30 thousand per month but if he does not happen to be a good salesman his income may go down to Rs. 10 thousand per month. Suppose in this new job there is 50-50, chance of either earning Rs. 30 thousands or Rs. 10 thousands (that is, each has a probability of 0.5). When there is uncertainty, the individual does not know the actual utility from taking a particular action.

But given the probabilities of alternative outcomes, we can calculate the expected utility. Whether the individual will choose the new risky job or retain the present salaried job with a certain income can be known by comparing the expected utility from the new risky job with the utility of the current job. It will be seen from the utility function curve OU in Fig. 17.3 that the utility of money income of Rs. 15,000 with certainty is 55.

Further, in case of new risky job if he is proved to be a successful salesman and his income increases to Rs. 30 thousands, his utility from Rs. 30 thousands is 75, and if he fails as a good salesman, his income falls to Rs 10 thousands which yields him utility of Rs. 45. (Note that in the new risky job, the expected income is 20,000 which is given by E(X) = 0.5 x __________ __________ _____ ______ ____ _____ ________ ________ __________ ____ _________ ______.
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