Questions: Utility Functions and Preference Representation

Question 1

A consumer's preferences are represented by u(x₁, x₂) = x₁ · x₂. A colleague proposes switching to v(x₁, x₂) = ln(x₁) + ln(x₂) instead. What is the correct conclusion?

• A. v represents different preferences because ln(x₁) + ln(x₂) ≠ x₁ · x₂ for most bundles
• B. v is preferable because logarithms are easier to differentiate and optimize
• C. v represents exactly the same preferences, because ln(x₁ · x₂) = ln(x₁) + ln(x₂), making v a monotonic transformation of u
• D. v represents different preferences because the utility values differ — for example, u(2,2) = 4 but v(2,2) ≈ 1.39

Answer: (C)

Since ln is strictly increasing, v = ln(u) is a monotonic transformation. Any monotonic transformation of a utility function represents exactly the same preferences: if u(A) > u(B), then ln(u(A)) > ln(u(B)). The indifference curves are identical; only the numerical labels change. Options A and D both make the mistake of treating utility values as having absolute meaning. The scale changes; the underlying preference ordering does not.

Question 2

Suppose u(A) = 100 and u(B) = 25 under some utility function. Which of the following is a valid inference?

• A. Bundle A gives four times as much satisfaction as bundle B
• B. The consumer would trade four units of B for one unit of A at current prices
• C. Bundle A is preferred to bundle B, but the ratio 100/25 = 4 carries no behavioral meaning
• D. The consumer is indifferent between A and any bundle with utility value between 25 and 100

Answer: (C)

Utility is ordinal: the numbers encode ranking only, not magnitude. u(A) > u(B) tells us A is preferred to B — full stop. The ratio 4 is meaningless: applying the transformation u → √u gives values 10 and 5 (ratio 2); applying u → u² gives 10,000 and 625 (ratio 16). All three functions represent identical preferences. Statements about 'four times as much satisfaction' presuppose cardinal utility — that differences and ratios between utility values have meaning — which they do not in standard consumer theory.

Question 3

Multiplying a utility function by a positive constant produces a new utility function that represents different preferences.

• T. True
• F. False

Answer: False

Multiplying by a positive constant is a monotonic transformation — it preserves the ordering of every pair of bundles. If u(A) > u(B), then 3u(A) > 3u(B). The ranking of all bundles is unchanged, so the indifference curves are identical and the new function represents the same preferences. Only transformations that reverse the ordering (like multiplying by -1) would change which bundles are preferred.

Question 4

If two consumers have different utility functions, they is expected to have different underlying preferences.

• T. True
• F. False

Answer: False

The same preference ordering can be represented by infinitely many utility functions — any monotonic transformation of a valid utility function is equally valid. Two consumers might use u₁(x) = x₁x₂ and u₂(x) = x₁²x₂², which look very different but represent identical preferences (u₂ = u₁², a monotonic transformation). Behavior depends on indifference curves — not utility values — and monotonic transformations leave indifference curves unchanged.

Question 5

Why is it meaningless to say 'Bundle A gives me 10 utility units and Bundle B gives me 5, so A is twice as good as B'? What can legitimately be concluded from these numbers?

Model answer: Utility is ordinal: the numbers only encode the ranking of bundles, not the intensity of preferences. Any monotonic transformation — squaring, taking the log, multiplying by a constant — gives different numerical values while representing the same preferences. Under u → u², A has 100 units and B has 25 units (now 'four times as good'). The only valid conclusion from u(A) = 10 and u(B) = 5 is that A is preferred to B. Ratios and differences between utility values are arbitrary artifacts of which utility function was chosen.

Cardinal utility — where differences and ratios between utility values have meaning — requires additional assumptions beyond standard consumer theory (such as expected utility over lotteries). In the basic consumer model, only the ordinal ranking matters. The practical implication: any demand analysis is invariant to monotonic transformations. Two researchers using different utility functions for the same preferences make identical predictions about consumer behavior, because they trace the same indifference curves.