Question

Assume that an individual is risk-averse and has a utility function regarding income that can
is described mathematically as U =√Y. The benefits of increased income are positive, but declining
marginal utility.
A person with that benefit function knows that there is a ten (10) percent risk of being ill. He is in
average sick every ten days. The person receives 1,000 dollar per day when he works, which means
that the expected salary is 0.90 * 1000 per day.
a) How high an insurance premium is the individual willing to pay to get a guaranteed one
salary amount (of 1000 dollar minus the premium), even days with sick leave?
b) Within which interval can the insurance premium end up, given that all individuals have the same
utility function and average sick leave? (Tip: it depends on the competition
between the insurance companies!)

Answer 1

Diminishing marginal utility of income and wealth suggests that as income increases, individuals gain a correspondingly smaller increase in satisfaction and happiness.

In layman’s terms – “more money may not make you happy”

Alfred Marshall popularised concepts of diminishing marginal utility in his Principles of Economics (1890)

“The additional benefit a person derives from a given increase of his stock of a thing diminishes with every increase in the stock that he already has”

– Alfred Marshall, Principles of Economics

Utility means satisfaction, usefulness, happiness gained. Utility could be measured by the amount you are willing to spend on a good.

Example of why increasing income leads to diminishing returns

Marginal utility of first £100

If you have zero income and then gain £100 a week. This £100 will improve your living standards significantly. With this £100 you will be able to pay for the basic necessity of life – food, drink, shelter and heating. Without this basic £100 a week, life would be tough.

Marginal utility of income increasing from £500 to £600 (6th £100)

However, if you already gain £500 a week, an extra £100 has a proportionately smaller increase in utility. You may be able to eat out at restaurants more often, but it doesn’t significantly affect your standard of living and happiness. At £500 a week, you can afford most things you need. But, most people would be happy to gain an extra £100 to spend on luxuries like going out.

Marginal utility of income increasing from £10,000 to £10,100

If you are earning £10,000 a week – you would hardly notice an extra £100 a week. You may not even have the time or ability to spend it; this extra income is liable to be just saved. Therefore, we say the marginal utility of an extra £100 at this income level is very limited.

Therefore as income increases, the extra marginal benefit to individuals declines.

Diminishing marginal utility of wealth

An increase in wealth from £10 to £20 leads to a large increase in utility (3 utils to 8 utils)

However, an increase in wealth from £70 to £80 leads to a correspondingly small increase in utility (30 to 31).

This concave graph shows a diminishing marginal utility of money and a justification for why people may exhibit risk aversion for the potentially large losses with small probabilities.