Attitudes toward Risktaking:

The rankings were different in each case and they obviously depend on the shape of the utility function over certain sums of money (u(Y)). Note also that lottery 1 is the least risky and 3 the riskiest, because the probability of winning the prize is highest and lowest, respectively, in these cases. In case 1, the consumer is indifferent between taking a risk, or not, and is said to be risk−neutral; in case 2 the consumer always prefer the less risky alternative, and is said to be risk−averse; in case 3 the consumer actually prefers the riskiest alternative and is therefore called a “risk lover”. Figure below shows case 2.

Two important areas of application of the theory of choice under uncertainty are to portfolio problems in financial theory and to the demand for insurance contracts. In financial theory it is an important question to derive prices of uncertain (or risky) asset relative to risk-free assets. In general, if most investors are risk-averse, risky assets will be sold at a lower price than risk-free ones (given that they pay out the same expected sum in the future), or, equivalently, investors demand a risk−premium to invest in such assets.

Example:

Let’s assume that we introduce a fourth lottery (L₀), where you always win 50, hence it is a risk-free alternative. The utility of this lottery (for the case of a risk-averse consumer) is: U²(L₀) = √50 = 7.071. How much extra (in expected value) does this consumer demand in order to accept lottery L₁ over L₀? In the example above L1 gave the utility: U²(L₁) = 5. If L₁ is to be as attractive as L₀ we must change the probability of getting the high outcome (π^∗) such that,

U² (L₁) = π^∗ . √100 + (1 − π^∗) .√0 = 7.071,

with the solution, π^∗ = 0.7071, and an expected value of, 0.7071 x 100 +(1 − 0.7071) x 0 = 70.71. The difference, 70.71 − 50 = 20.71, is called a compensatory risk premium.

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