Jensen's Inequality and Precautionary Saving

Jensen's inequality: for a concave function f, E[f(X)] <= f(E[X]). In plain English, the expected utility of a random outcome is less than the utility of the expected outcome. The gap is the cost of uncertainty -- and the size of that gap is exactly what makes you willing to pay for insurance, accept a lower return for safety, and save more when the future is uncertain.

Worked example with round fictional numbers. Suppose your utility of wealth is the square root function, and you have $100M of wealth. A fair coin flip would pay you +$36M or -$36M with equal probability. Expected wealth after the flip is still $100M, but expected utility is 0.5 sqrt(64) + 0.5 sqrt(136) = 0.5 8 + 0.5 11.66 = 9.83. That is LESS than sqrt(100) = 10, the utility of certain wealth. The wedge -- 10 minus 9.83 -- is the certainty equivalent gap. You would rather have around $96.6M for sure than the fair gamble. That difference is what you would pay to avoid the uncertainty.

Precautionary saving rises with TWO things: the variance of expected income AND the degree of concavity (technically, the third derivative of utility, called prudence). A household facing 20 percent income variance with high prudence saves dramatically more than the same household facing 5 percent variance. This is why portfolio cash buffers should grow when career or business income becomes lumpier -- not because returns on cash improved, but because the wedge between expected utility and utility of expected wealth widened.

Jensen's inequality is also why the risk-free rate is structurally lower than the expected return on risky assets. Concave-utility investors will pay a premium for certainty -- that premium is the equity risk premium. Without Jensen, there is no reason a riskless Treasury yield should sit below the expected return on stocks; with Jensen, that gap is mathematically required.