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IAI Actuarial Core Principles · Economic Modelling

Rational Choice Theory and Utility Explained

Rational choice theory says a person ranks options by preference and picks the best one. Utility is a number that represents that ranking. Under uncertainty, you compare expected utility: E[U(X)] = Σ p × U(x). To solve questions, compute each option's expected utility, then compare. Check risk attitude using the curvature of U.

What this chapter covers

This chapter is the base of CM2 Economic Modelling. It asks how a person chooses between options when resources are limited. You start with preferences and the axioms that make them consistent. You then represent them with a utility function, and extend the idea to risky outcomes with expected utility theory.

The second half applies the theory. Risk aversion explains why people buy insurance and why they accept a lower expected wealth for certainty. You learn to measure risk aversion, find certainty equivalents and risk premiums, and work out how much premium a person would pay. The chapter ends with behavioural critiques. These show where real people break the axioms, and how prospect theory describes what they do instead.

The chapter links to the rest of the paper. Measures of investment risk use utility to rank portfolios. Asset valuation and option theory use the idea that investors prefer more to less and dislike risk. If you understand utility well, those later chapters feel like applications, not new theory.

This chapter gives you ideas that the whole paper relies on, and it supports both multiple-choice and written questions. MCQs often test definitions, the sign of U′ and U″, and quick certainty equivalent calculations. Written questions ask you to derive, compute and interpret. Marks go to clear working and correct reasoning, so this is a chapter where careful practice pays off directly. It also helps you in the investment risk and valuation chapters, so time spent here is not wasted.

Rational choice theory and utility: topics in the order to study them

  1. 1Rational Choice Theory BasicsYou need the idea of preferences, completeness and transitivity before any utility function makes sense.
  2. 2Utility Functions and PreferencesOnce preferences are clear, you learn how a utility function represents them and what properties it must have.
  3. 3Expected Utility TheoryThis extends utility to uncertain outcomes and gives you the main calculation tool of the chapter.
  4. 4Risk Aversion and Insurance DemandThis applies expected utility to real decisions, using the curvature of U to explain insurance and risk premiums.
  5. 5Behavioural Critiques and Prospect TheoryStudy this last, because you can only judge the critiques once you know exactly what the standard theory assumes.

How to prepare Rational choice theory and utility

Aim to understand each assumption and then practise a small set of calculations until they are routine. Short daily sessions work well, even on a phone.

  1. Write the axioms of rational choice in your own words, and give one example of what breaks each.
  2. Learn what makes a utility function valid: more is preferred to less (U′ > 0), and the ranking matters, not the actual numbers.
  3. Practise expected utility: compute E[U(X)] = Σ p × U(x) for lotteries, using U(x) = ln x and U(x) = √x as standard examples.
  4. Learn the link between curvature and risk attitude: U″ < 0 is risk averse, U″ = 0 is risk neutral, U″ > 0 is risk seeking.
  5. Solve certainty equivalent and risk premium questions: find c with U(c) = E[U(X)], then risk premium = E[X] − c.
  6. Work through an insurance problem: compare expected utility with and without cover, and find the maximum premium the person would pay.
  7. List the standard behavioural critiques and the main features of prospect theory, then write a short answer on each from memory.

Common mistakes in Rational choice theory and utility

  • Applying the utility function to expected wealth instead of averaging the utilities.

    Fix: Always compute U at each outcome first, then weight by probabilities. Use the gap between the two to show risk attitude.

  • Stating that a risk averse person has U″ > 0, or mixing up the sign of the second derivative.

    Fix: Link it to diminishing marginal utility: each extra rupee adds less, so U′ falls and U″ < 0.

  • Giving the certainty equivalent as a utility value, not a wealth amount.

    Fix: Invert the utility function. For U = √x, c = (E[U])². For U = ln x, c = exp(E[U]).

  • Treating utility numbers as measurable quantities that can be compared across people.

    Fix: State that utility is a way of representing preferences. In expected utility theory, only positive linear transformations keep the same choices.

  • Describing prospect theory as just another form of risk aversion.

    Fix: Name all the features: reference dependence, loss aversion, diminishing sensitivity and probability weighting. Explain which standard axiom each one challenges.

Last-day revision: Rational choice theory and utility

  • Rational choice needs complete and transitive preferences.
  • Utility ranks options; only the order matters for certainty, not the size of the numbers.
  • More is preferred to less means U′ > 0.
  • Expected utility: E[U(X)] = Σ p × U(x).
  • Risk averse: U″ < 0, so U(E[X]) > E[U(X)] (Jensen's inequality).
  • Risk neutral: U″ = 0. Risk seeking: U″ > 0.
  • Certainty equivalent c solves U(c) = E[U(X)].
  • Risk premium = E[X] − c for a risk averse person.
  • A risk averse person with fair-priced cover would choose full insurance; they will pay more than the expected loss.
  • Prospect theory uses gains and losses from a reference point, not final wealth.
  • Loss aversion: losses hurt more than equal gains please.
  • Prospect theory uses probability weighting, so small probabilities are overweighted.

Rational choice theory and utility practice questions

Rational choice theory and utility in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Rational choice theory and utility: frequently asked questions

What is the difference between utility and expected utility?

Utility measures how much a person values a certain outcome, such as a given level of wealth. Expected utility is the probability-weighted average of the utilities of the possible outcomes of a risky choice. You use it to compare gambles.

How do I tell if a utility function shows risk aversion?

Check the second derivative. If U″(x) < 0 over the range of wealth, the function is concave and the person is risk averse. You can also compare U(E[X]) with E[U(X)]; for a risk averse person the first is larger.

Why would a risk averse person buy insurance?

Insurance replaces an uncertain loss with a certain smaller outcome. A risk averse person values certainty, so their expected utility with cover can be higher even when the premium exceeds the expected loss. The extra they will pay is linked to their risk premium.

How is prospect theory different from expected utility theory?

Expected utility theory judges final wealth and weights outcomes by their true probabilities. Prospect theory judges gains and losses from a reference point, treats losses as more painful than equal gains, and distorts probabilities. It aims to describe what people actually do.