Explain the concept of positive monotonic transformation and how it relates [10 marks] to ordinality.
The correct answer and explanation is:
Correct Answer (10 Marks):
A positive monotonic transformation is a mathematical transformation of a utility function that preserves the order (ranking) of preferences but not necessarily the magnitude of utility. It is a function f(u)f(u) such that f′(u)>0f'(u) > 0 for all uu, meaning it is strictly increasing. This concept is directly related to ordinal utility, which only requires the ranking of preferences, not their numerical differences. Thus, any utility function that is a positive monotonic transformation of another represents the same ordinal preferences.
Explanation (300 Words):
In microeconomics, particularly in consumer theory, utility functions are used to represent a consumer’s preferences. There are two key types of utility: ordinal and cardinal. Ordinal utility focuses on the order or ranking of preferences—what is preferred over what—without measuring how much more one bundle is preferred than another.
A positive monotonic transformation is any function that preserves the order of numbers. Mathematically, if u(x)u(x) is a utility function, and f(u)f(u) is a positive monotonic transformation (where ff is strictly increasing), then:
- If u(x1)>u(x2)u(x_1) > u(x_2), it implies f(u(x1))>f(u(x2))f(u(x_1)) > f(u(x_2))
This means that the transformed utility function f(u(x))f(u(x)) ranks goods in the same way as the original function u(x)u(x). Even though the actual numerical values may differ, the preference ordering remains unchanged.
This concept is crucial when discussing ordinal utility, where the numbers assigned to preferences do not matter—only their order does. For instance, if a consumer prefers bundle A over B, and B over C, any positive monotonic transformation of the utility function representing these preferences must preserve this order:
u(A)>u(B)>u(C)⇒f(u(A))>f(u(B))>f(u(C))u(A) > u(B) > u(C) \Rightarrow f(u(A)) > f(u(B)) > f(u(C))
In contrast, cardinal utility, which involves measuring the intensity of preferences, is not preserved under general monotonic transformations unless they are linear. Therefore, positive monotonic transformations are fully compatible with ordinal utility theory and reflect the idea that utility is not about the amount but the order of satisfaction.