Editing Risk aversion (section)

==Utility of money==
In [[expected utility hypothesis|expected utility]] theory, an agent has a utility function ''u''(''c'') where ''c'' represents the value that he might receive in money or goods (in the above example ''c'' could be $0 or $40 or $100).

The utility function ''u''(''c'') is defined only [[up to]] positive [[affine transformation]] – in other words, a constant could be added to the value of ''u''(''c'') for all ''c'', and/or ''u''(''c'') could be multiplied by a positive constant factor, without affecting the conclusions.

An agent is risk-averse if and only if the utility function is [[concave function|concave]]. For instance ''u''(0) could be 0, ''u''(100) might be 10, ''u''(40) might be 5, and for comparison ''u''(50) might be 6.

The expected utility of the above bet (with a 50% chance of receiving 100 and a 50% chance of receiving 0) is

:<math>E(u)=\frac{u(0)+u(100)}{2}</math>

and if the person has the utility function with ''u''(0)=0, ''u''(40)=5, and ''u''(100)=10 then the expected utility of the bet equals 5, which is the same as the known utility of the amount 40. Hence the certainty equivalent is 40.

The risk premium is ($50 minus $40)=$10, or in proportional terms

:<math>\frac{\$50-\$40}{\$40}</math>

or 25% (where $50 is the expected value of the risky bet: (<math>\tfrac {1}{2} 0 + \tfrac{1}{2} 100</math>). This risk premium means that the person would be willing to sacrifice as much as $10 in expected value in order to achieve perfect certainty about how much money will be received. In other words, the person would be indifferent between the bet and a guarantee of $40, and would prefer anything over $40 to the bet.

In the case of a wealthier individual, the risk of losing $100 would be less significant, and for such small amounts his utility function would be likely to be almost linear. For instance, if u(0) = 0 and u(100) = 10, then u(40) might be 4.02 and u(50) might be 5.01.

The utility function for perceived gains has two key properties: an upward slope, and concavity. (i) The upward slope implies that the person feels that more is better: a larger amount received yields greater utility, and for risky bets the person would prefer a bet which is [[stochastic dominance#First-order|first-order stochastically dominant]] over an alternative bet (that is, if the probability mass of the second bet is pushed to the right to form the first bet, then the first bet is preferred). (ii) The concavity of the utility function implies that the person is risk averse: a sure amount would always be preferred over a risky bet having the same expected value; moreover, for risky bets the person would prefer a bet which is a [[mean-preserving spread|mean-preserving contraction]] of an alternative bet (that is, if some of the probability mass of the first bet is spread out without altering the mean to form the second bet, then the first bet is preferred).