Rational choice theory and utility: formula sheet

Key formulas

Completeness

For any A, B: A ≻ B, or B ≻ A, or A ~ B

The agent can always compare two options. Not being able to rank them violates completeness.

Transitivity

If A ≽ B and B ≽ C, then A ≽ C

A cycle such as A ≻ B, B ≻ C, C ≻ A breaks transitivity.

Non-satiation

If x contains more of a good than y and no less of any other, then x ≻ y

With divisible goods, this means the agent chooses a point on the budget line rather than inside it.

Rational choice rule

Choose x* in the feasible set F such that x* ≽ x for all x in F

Best available option, not the best option in general.

Utility representation

A ≽ B if and only if U(A) ≥ U(B)

Possible when preferences are complete and transitive (and, for infinite sets, suitably continuous).

Preference representation

A ≻ B ⇔ U(A) > U(B); A ~ B ⇔ U(A) = U(B)

Holds for complete, transitive preferences. Under ordinal utility any increasing transformation of U works.

Cardinal transformation

V(W) = a + b U(W), with b > 0

Gives the same choices under expected utility. Taking a non-linear transformation can change them.

Non-satiation

U′(W) > 0

More wealth is always preferred.

Diminishing marginal utility

U″(W) < 0

Utility is concave. This corresponds to risk aversion.

Marginal utility of a good

MU_x = ∂U(x, y) ÷ ∂x

Extra utility from one more unit of x, holding y fixed.

Slope of indifference curve

dy/dx = − MU_x ÷ MU_y

The absolute value is the marginal rate of substitution.

Expected utility

E[U(W)] = Σ p(i) × U(W(i))

Probabilities must sum to 1. Apply U to each outcome before weighting.

Expected wealth

E[W] = Σ p(i) × W(i)

Used for comparison with U(E[W]). Do not confuse it with E[U(W)].

Certainty equivalent

U(CE) = E[U(W)], so CE = U⁻¹(E[U(W)])

The certain wealth that gives the same utility as the gamble.

Risk premium

Risk premium = E[W] − CE

Positive for a risk-averse person. The most you would pay to avoid the risk.

Risk-aversion test

U'(W) > 0 and U''(W) < 0 means risk averse; U'' = 0 risk neutral; U'' > 0 risk seeking

Jensen's inequality: for concave U, E[U(W)] ≤ U(E[W]).

Invariance of utility

V(W) = a + b U(W), b > 0, gives the same choices

Utility is ordinal over gambles up to a positive linear transformation.

von Neumann-Morgenstern axioms

Completeness, transitivity, continuity, independence

Together they imply preferences can be represented by expected utility.

Expected utility

E[U(W)] = Σ p_i × U(w_i)

Use the probabilities of each wealth outcome. Wealth is final wealth, not just the gain or loss.

Risk attitude by curvature

U''(w) < 0: risk-averse; U''(w) = 0: risk-neutral; U''(w) > 0: risk-seeking

Assume U'(w) > 0 (more wealth is preferred).

Jensen's inequality for risk aversion

E[U(W)] < U(E[W]) for concave U and non-constant W

Reverses for convex U. Equality holds for linear U.

Certainty equivalent

U(CE) = E[U(W)]

Solve by inverting U. For U = ln w, CE = exp(E[ln W]).

Risk premium

π = E[W] − CE

Positive for risk-averse, zero for risk-neutral, negative for risk-seeking.

Maximum insurance premium

U(w − P) = E[U(w − X)], so P_max = w − CE = E[X] + π

X is the random loss and w is initial wealth. P_max exceeds E[X] for a risk-averse person.

Absolute risk aversion (Arrow-Pratt)

A(w) = −U''(w) ÷ U'(w)

Larger A means more risk-averse at wealth w.

Relative risk aversion

R(w) = w × A(w) = −w U''(w) ÷ U'(w)

Measures aversion to proportional risk.

Approximate risk premium

π ≈ ½ × A(w) × Var(W)

An approximation for small risks around mean wealth w. Not exact.

Expected utility

E[U(X)] = Σ p_i × U(x_i)

The rational benchmark. Choose the option with the higher expected utility. Independence axiom is the one the Allais paradox breaks.

Prospect theory value

V = Σ w(p_i) × v(x_i − reference point)

v is defined on gains and losses, not final wealth. w(p) is a decision weight, not the probability itself.

Typical value function form

v(x) = x^α for x ≥ 0; v(x) = −λ(−x)^β for x < 0, with λ > 1

λ is the loss aversion coefficient. Parameters are illustrative and given in the question if needed.

Loss aversion

|v(−x)| > v(x) for x > 0

A loss of a given size feels larger than a gain of the same size.

Independence axiom

If A ≻ B then p·A + (1−p)·C ≻ p·B + (1−p)·C

Mixing both options with the same third option should not reverse the ranking. Allais-type choices violate this.

Quick revision

• 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.

Common mistakes

• Confusing completeness with transitivity. Fix: Completeness is about every pair being comparable. Transitivity is about chains of comparisons being consistent.
• Choosing the agent's favourite option without checking affordability. Fix: Always filter to the feasible set first, then choose the best option inside it.
• Saying utility of 20 is twice the utility of 10 under ordinal utility. Fix: Under ordinal utility only the order matters. Make ratio or difference statements only for cardinal utility.
• Confusing diminishing marginal utility with falling total utility. Fix: Total utility still rises. U′ > 0 but U′ is falling, so U″ < 0.
• Averaging the wealth first and then applying U, getting U(E[W]) instead of E[U(W)]. Fix: Always apply U to each outcome first, then weight by probability. Write the Σ line explicitly.
• Reporting the certainty equivalent as E[U(W)]. Fix: Invert U to convert back to wealth. Check the answer lies between the lowest and highest outcomes.
• Taking U(E[W]) as the expected utility. Fix: Compute U for each outcome first, then weight by probability. Remember E[U(W)] ≠ U(E[W]) unless U is linear.
• Confusing certainty equivalent with risk premium. Fix: CE is a level of sure wealth. The risk premium is E[W] − CE, the amount given up. Always label which one you have found.
• Mixing up the Allais and Ellsberg paradoxes. Fix: Allais is about known probabilities and the certainty effect (independence axiom). Ellsberg is about unknown probabilities and ambiguity aversion.
• Saying prospect theory is the same as expected utility with a different utility function. Fix: State the three differences: reference point, separate treatment of gains and losses with loss aversion, and decision weights replacing probabilities.

Exam tips

• In written answers, name each assumption (completeness, transitivity, consistency, non-satiation) and say what it means in one line.
• For MCQs with preference lists, draw arrows and look for cycles before reading the options.
• Always mention constraints when describing choice. Marks are often lost by describing only preferences.
• When asked for criticisms, link each to an assumption and mention behavioural economics as the main response.
• Use standard notation: ≻ for strict preference, ~ for indifference, ≽ for weak preference.
• Always state the assumptions (complete, transitive, non-satiation) before using a utility function. Examiners give marks for them.
• When asked for the difference between ordinal and cardinal utility, mention what each allows: ranking only, versus meaningful differences and expected utility.
• Show both derivatives and their signs in written answers. Then explain in a sentence what the signs mean.