Expected Utility vs Expected Profit

In micro, one person's attitude to risk and one firm's choice were studied apart. In macro, those attitudes add up and move the whole economy.

People dislike shaky outcomes, so they buy insurance. Given the same average, they pick the sure thing. Firms read only average profit and take the riskier project. Splitting it among many shareholders shrinks the shakiness. The formulas below set the two yardsticks side by side. Everything here is risk, where the odds are known.

For example, say a coin flip decides your 💰 allowance. Heads pays 10,000 won, tails pays nothing. The average is 5,000 won, yet you might prefer a sure 4,500. The 500 you give up out of caution is just like an insurance premium.

EU=pWhi+(1−p)WloEU = p\sqrt{W_{hi}} + (1-p)\sqrt{W_{lo}}
E[π]=pπhi+(1−p)πloE[\pi] = p\pi_{hi} + (1-p)\pi_{lo}
σXˉ=σN\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{N}}

Part of these learning paths

Examples & Exercises

Concept check

One coin flip makes your wealth 160160 or 4040 (fifty-fifty).

For a person with utility u(x)=xu(x) = \sqrt{x}, what is the gamble's certainty equivalent, and what is the risk premium?

Wealth 160160 or 4040, fifty-fifty — the dots in the picture are exactly this case.
  1. Certainty equivalent 100100, risk premium 00
  2. Certainty equivalent 9090, risk premium 1010
  3. Certainty equivalent 110110, risk premium −10-10
  4. Certainty equivalent 9090, risk premium 6060
Show explanation

Answer: Certainty equivalent 9090, risk premium 1010

Expected utility is EU=0.5160+0.540≈9.49EU = 0.5\sqrt{160} + 0.5\sqrt{40} \approx 9.49, so the certainty equivalent is EU2=90EU^2 = 90 and the risk premium is expected wealth 100100 minus that, 1010. A zero premium would mean risk neutrality, and a certainty equivalent above expected wealth would mean risk loving. The 6060 is not the risk premium — it is the fair insurance premium when the gamble is read as insurance.

A firm must choose between a safe project (profit 4040 for sure) and a risky one (profit 8080 or 2020, fifty-fifty).

If a shareholder instead holds 1616 independent copies of the risky project, what is the spread of the average profit, and the chance the average falls below the safe profit 4040?

Sixteen copies — the dot in the lower panel sits at σ/16\sigma / \sqrt{16}.
Show proof

  1. The firm looks only at expected profit: the risky project gives 0.5⋅80+0.5⋅20=50>400.5 \cdot 80 + 0.5 \cdot 20 = 50 > 40, so it takes the risky one — even though it can come out at 2020. The yardstick is the average of profit, not of utility.

  2. One project's spread is σ=30\sigma = 30.

    The average of NN independent copies has spread σ/N\sigma / \sqrt{N}, so with 1616 copies it shrinks to 30/4=7.530 / 4 = 7.5.

    The average falls below 4040 only when at most 55 of the 1616 come out high — probability about 10.5%10.5\% (with a single copy it is 50%50\%).

  3. So the shareholder barely feels the risk: the expected profit 5050 survives while the danger evaporates. That is why a household tied to one undiversifiable job and a diversified shareholder face the same gamble with opposite attitudes.

9. Risk and Uncertainty