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Economic Modelling · Rational choice theory and utility

Utility Functions and Preferences: Ordinal vs Cardinal Utility

Updated 11 October 2026 · Fact-checked

A utility function assigns a number to each outcome so that higher numbers mean more preferred outcomes. Under ordinal utility only the ranking matters. Under cardinal utility the size of differences also matters. Marginal utility is U′(W), the extra utility from a little more wealth. Diminishing marginal utility means U″(W) < 0.

Understand Utility Functions and Preferences

Economists start with preferences: how a person ranks outcomes. You do not need numbers to say you prefer ₹5,00,000 to ₹2,00,000. A utility function is a tool that turns this ranking into numbers so we can use maths on it.

For a rational person, preferences must be complete (any two outcomes can be compared) and transitive (if A is preferred to B and B to C, then A is preferred to C). The standard CM2 assumption is also non-satiation: more wealth is always preferred to less. Then a utility function U exists with U(A) > U(B) exactly when A is preferred to B.

Ordinal utility says only the order matters. Any increasing transformation of U represents the same preferences. So the statement 'utility of 10 is twice utility of 5' has no meaning. Cardinal utility says the numbers carry more information, so the size of differences matters. This is needed for choices under uncertainty, where we take expected values of utility. Cardinal utility is unique only up to a positive linear transformation, a + bU with b > 0.

Marginal utility is the change in utility from one extra unit of wealth, written U′(W). Non-satiation means U′(W) > 0. Diminishing marginal utility means each extra rupee adds less than the one before, so U″(W) < 0 and U is concave. A person with this shape is risk averse. This link is the bridge to expected utility theory and insurance demand.

An indifference curve joins combinations of goods or outcomes that give the same utility. Along one curve the person is indifferent. Curves further from the origin give higher utility. Standard curves slope downward, do not cross, and are convex to the origin. In the risk and return setting, the axes are often return and risk, and the curves slope upward for a risk-averse investor.

Key rules to remember

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.

How to solve Utility Functions and Preferences questions

Use this method for most questions on utility and preferences, whether MCQ or written.

  1. 1Identify what is asked: ranking, ordinal or cardinal comment, marginal utility, or indifference.
  2. 2State the assumptions: complete, transitive, non-satiation, and diminishing marginal utility if relevant.
  3. 3Write the utility function and differentiate: U′(W) for the first derivative, U″(W) for the second.
  4. 4Read the signs: U′ > 0 means more is preferred; U″ < 0 means diminishing marginal utility and concavity.
  5. 5For ranking, compute U for each outcome and compare the numbers. Do not interpret the size of the gaps unless utility is cardinal.
  6. 6For indifference, set U equal to a constant and solve for one variable in terms of the other, or use the MRS formula.
  7. 7Conclude in words, linking the result back to the economic meaning.

Quickest way: Sign check on U′ and U″

When to use it: Use for MCQs that ask whether a function shows non-satiation, diminishing marginal utility or risk aversion.

  1. Differentiate once. If the result is positive for all relevant W, more is preferred.
  2. Differentiate again. If negative, marginal utility diminishes and the person is risk averse.
  3. If a question compares two functions, check whether one is an increasing transformation of the other. If yes, ordinal preferences are the same.
  4. Under expected utility, only linear transformations a + bU (b > 0) keep choices the same.

Common mistakes in Utility Functions and Preferences

  • Saying utility of 20 is twice the utility of 10 under ordinal utility.

    Numbers look like measurements.

    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.

    Both involve something decreasing.

    Fix: Total utility still rises. U′ > 0 but U′ is falling, so U″ < 0.

  • Thinking any increasing transformation keeps expected utility choices unchanged.

    It is true for ordinal ranking of certain outcomes, so students over-extend it.

    Fix: For choices under uncertainty only a + bU with b > 0 is safe. A non-linear transformation can change the ranking of gambles.

  • Drawing indifference curves that cross.

    Students sketch quickly without checking transitivity.

    Fix: Crossing would break transitivity or non-satiation. State this reason if asked.

  • Dropping the minus sign in the MRS.

    The slope is negative but MRS is quoted as a positive number.

    Fix: Write dy/dx = − MU_x ÷ MU_y, then say MRS is the absolute value.

  • Treating concave utility as a statement about the amount of wealth.

    Students link risk aversion to being poor.

    Fix: Concavity describes the shape of preferences, not the level of wealth.

Worked examples

Example 1

An investor has utility U(W) = √W. (a) Show that she prefers more wealth and has diminishing marginal utility. (b) Compare the utility gain from raising wealth from ₹1,00,000 to ₹2,00,000 with the gain from ₹2,00,000 to ₹3,00,000. Use units of ₹1,000 for W.

Show the solution
  1. Use W in thousands: the wealth levels are 100, 200 and 300.
  2. U′(W) = 1/(2√W) > 0 for all W > 0, so more wealth is preferred.
  3. U″(W) = −1/(4 W^(3/2)) < 0, so marginal utility diminishes.
  4. U(100) = 10. U(200) = √200 = 14.142. U(300) = √300 = 17.321.
  5. First gain = 14.142 − 10 = 4.142. Second gain = 17.321 − 14.142 = 3.179.
  6. The second gain is smaller, which matches diminishing marginal utility.

Answer: U′ > 0 and U″ < 0. The gain from the first ₹1,00,000 rise is about 4.14 and from the second about 3.18, so the second extra ₹1,00,000 adds less utility.

Example 2

A consumer has utility U(x, y) = x·y. (a) Find the marginal utilities. (b) Find the slope of the indifference curve at (x, y) = (4, 6). (c) Find y on the same curve when x = 8.

Show the solution
  1. MU_x = ∂U/∂x = y and MU_y = ∂U/∂y = x.
  2. Slope dy/dx = − MU_x ÷ MU_y = − y ÷ x.
  3. At (4, 6): slope = − 6 ÷ 4 = −1.5.
  4. Utility at (4, 6) is 4 × 6 = 24.
  5. On the curve, x·y = 24. When x = 8, y = 24 ÷ 8 = 3.

Answer: MU_x = y, MU_y = x. The slope at (4, 6) is −1.5. At x = 8, y = 3.

Exam tips

  • 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.
  • In Paper B or any spreadsheet question, compute U for each outcome in a column and check the differences before commenting on diminishing marginal utility.

Practice questions from Rational choice theory and utility

Utility Functions and Preferences: frequently asked questions

What is the difference between ordinal and cardinal utility?

Ordinal utility only ranks outcomes, so any increasing transformation represents the same preferences. Cardinal utility gives meaning to the size of differences and is needed for expected utility. It is unique up to a + bU with b > 0.

What does diminishing marginal utility mean?

Each extra unit of wealth adds less utility than the previous one. Mathematically U′(W) > 0 and U″(W) < 0. The utility function is concave.

Why do indifference curves not cross?

If two curves crossed, the crossing point would lie on both, so bundles on different curves would be equally preferred. That breaks transitivity and non-satiation, so rational preferences cannot have crossing curves.

Is utility measured in real units?

No. Utility is a modelling tool, not a physical measure. Its job is to represent preferences so we can analyse choices.