Question

Use Utility Theory to explain why someone would insure his/her house, providing a numerical example and illustrating it graphically.

Answer 1

People’s preferences toward risk greatly differ. Most individuals generally prefer the less risky situation (that is, the situation with less variability in outcomes or rewards). In other words, most individuals seek to minimise risk and are called risk averter or risk averse.

However, some individuals prefer risk and are therefore called risk-seekers or risk lovers. Some other individuals are indifferent toward risk and are called risk-neutral. But it is important to note that these different preferences toward risk depend on whether for an individual marginal utility of money diminishes or increases or remains constant.

As shall be explained below, for a risk averse individual marginal utility of money diminishes as he has more money, while for a risk-seeker marginal utility of money increases as money with him increases. In case of risk-neutral individual marginal utility of money remains constant as he has more 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 10,000 + 0.5 x 30,000 = Rs. 20,000). Given that the probability of success or failure as a salesman is 0.5, the expected utility of the new job is given by,

E (U) = 0.5 U (10,000) + 0.5 U (30,000)

= 0.5 x 45 + 0.5 x 75

= 22.5 + 37.5

= 60.0

Thus with the present job with a fixed salary of Rs. 15,000 with no uncertainty is 55 whereas the expected utility of the new job or salesman on commission basis is 60. Though the individuals is risk-averse as revealed by the nature of his utility function of money income, but since the expected utility of the risky job is greater than the utility of the present job with a certain income he will choose the risky job.