Uncertainty

Choosing when outcomes are risky

When consumption is uncertain, we rank gambles by expected utility: the probability-weighted average of the utility of each outcome. If wealth is \(W_1\) with probability \(\pi\) and \(W_2\) with probability \(1-\pi\), \[ EU = \pi\,u(W_1) + (1-\pi)\,u(W_2). \] Compare that to the expected value of the gamble, \(EV = \pi W_1 + (1-\pi)W_2\) — the average money, not utility.

The shape of \(u\) is everything. A concave utility function — diminishing marginal utility of wealth — means a dollar lost hurts more than a dollar gained helps. Such a person is risk averse, and for them \[ u(EV) > EU, \] i.e. a sure \(EV\) beats the gamble. Two numbers make this concrete:

  • Certainty equivalent (CE): the guaranteed wealth that gives the same utility as the gamble, \(u(\text{CE}) = EU\). A risk-averse person has \(\text{CE} < EV\).
  • Risk premium: \(EV - \text{CE}\), the amount they’d pay to swap the gamble for its expected value with certainty. It’s also roughly what an insurer can charge.

The sim uses \(u(W)=\sqrt{W}\) (concave), so \(\text{CE} = EU^2\).

Try it. Set the two outcomes and the probability. The curve is \(u(W)=\sqrt{W}\); the straight chord is the gamble. \(EU\) sits on the chord above \(EV\); drop to the curve to read the certainty equivalent. The gap \(EV - \text{CE}\) is the risk premium.

What to notice

  • The gamble’s expected utility EU sits on the straight chord; the sure-thing utility \(u(EV)\) sits on the curve above it. Concavity is exactly why the curve is higher — risk aversion.
  • The certainty equivalent is always to the left of EV for a risk-averse person: they’d accept less guaranteed wealth to escape the risk.
  • The gap EV − CE is the risk premium — the most they’d pay for insurance. Push the two outcomes farther apart (more risk) and the premium grows; shrink the spread and it vanishes.