Certainty Equivalent

The certainty equivalent of a random variable in political economy is the amount of guaranteed income that an individual perceives as equally desirable as a risky prospect. In other words, it's the sure amount that yields the same level of satisfaction - or expected utility - as a lottery, where the individual faces different potential outcomes depending on the result of a random variable (i.e., the state of the world).

U(Mx) = E(U)

Here's a simple example. Suppose a person flips a coin. If it lands on tails, they earn 5; if it lands on heads, they earn 3. The random variable can take on two outcomes - M₁ and M₂ (heads or tails) - each with a probability of p₁ = 0.5 and p₂ = 0.5. The utility function is U = f(x²). To calculate the expected utility, we proceed as follows:

E(U) = p₁ (x₁²) + p₂ (x₂²)
E(U) = 0.5 (25) + 0.5 (9)
E(U) = 12.5 + 4.5
E(U) = 17.0

The expected utility is 17. To find the certainty equivalent, we need to invert the utility function and solve for the original value of x. Since U = f(x²), we take the square root of 17, giving us approximately 4.12.

U = f(x²) = 17 when x = 4.12

This value - 4.12 - is the certainty equivalent of the lottery, meaning it's the guaranteed amount the individual considers equally attractive. Put differently, the person is indifferent between receiving a sure gain of 4.12 (which gives a utility of 17) and taking part in the lottery, where the possible outcomes are 3 or 5 (with an expected utility of 17), since both choices deliver the same level of utility.