Definition
Cardinal utility refers to a type of utility function that remains consistent under positive affine transformations. If we have a utility function \( U \), a positive affine transformation applied to it results in another utility function \( U’ \) given by \( U’ = a + bU \) where \( b > 0 \). Both \( U \) and \( U’ \) represent the same set of preferences, indicating the measure of cardinal utility.
Historical Context
The concept of cardinal utility dates back to early economic theories when economists tried to quantify satisfaction or happiness that consumers derive from goods and services. Unlike ordinal utility, which ranks preferences, cardinal utility attempts to measure the degree of preference.
Types/Categories
- Cardinal Utility Models: Models that quantify the utility in numerical terms.
- Expected Utility Function: A specific form where utility is expected over outcomes, often used in decision theory under uncertainty.
Key Events
- 19th Century: Emergence of the utility theory.
- 20th Century: Refinement by economists such as von Neumann and Morgenstern who formalized the Expected Utility Theory.
Detailed Explanation
Cardinal utility is often contrasted with ordinal utility. Ordinal utility merely ranks preferences but does not assign a specific measure to the intensity of those preferences. Cardinal utility, however, attempts to measure the intensity, allowing for comparisons in terms of “how much” one option is preferred over another.
Mathematical Formulation
Given a utility function \( U \), a positive affine transformation can be defined as:
Mermaid Diagram
graph TD;
U[Utility Function U] --> |Positive Affine Transformation| U'[Utility Function U']
U'[Utility Function U'] --> |Represents Same Preferences| U[Utility Function U]
Importance and Applicability
Understanding cardinal utility is crucial in various fields:
- Economics: Helps in evaluating the welfare changes.
- Finance: Applied in portfolio theory to assess risk preferences.
- Decision Theory: Used to model decisions under uncertainty.
Examples
- Expected Utility: In gambling, if a person derives a utility of 10 from winning $100 and 2 from losing $50, a utility function can represent this numerically to make decisions based on expected outcomes.
Considerations
- Cardinal utility requires assuming that utility can be measured and compared across different levels of goods or services.
- It is less commonly accepted than ordinal utility because it assumes a specific form of utility comparison.
Related Terms
- Ordinal Utility: Represents the order of preferences without measuring the degree.
- Interpersonal Comparisons: Comparing utility levels across different individuals, which is complex with cardinal utility.
Comparisons
| Feature | Cardinal Utility | Ordinal Utility |
|---|---|---|
| Measurement | Quantitative | Qualitative |
| Affine Transformation | Allowed | Not Considered |
| Complexity | Higher | Lower |
| Common Use | Less Common | More Common |
Interesting Facts
- The theory of cardinal utility has been foundational in developing risk and uncertainty models in finance.
Inspirational Stories
Von Neumann and Morgenstern’s work on the Expected Utility Theory revolutionized economics by providing a way to handle uncertainty quantitatively.
Famous Quotes
“Utility is not a measure of any empirical reality; it is merely a way of organizing preferences.” - Vilfredo Pareto
Proverbs and Clichés
- “Happiness cannot be measured.” - Reflects the critique of cardinal utility’s attempt to quantify satisfaction.
Expressions, Jargon, and Slang
- Utility Maximization: The process of making choices that result in the highest utility.
- Risk Aversion: Preference for a sure outcome over a gamble with higher or equal expected value.
FAQs
What is the primary difference between cardinal and ordinal utility?
Why is cardinal utility less commonly used?
References
- Von Neumann, J., & Morgenstern, O. (1944). “Theory of Games and Economic Behavior.”
- Pareto, V. (1909). “Manual of Political Economy.”
Summary
Cardinal utility is a concept in economics that attempts to quantify the satisfaction or happiness derived from consumption. It allows for positive affine transformations, preserving the order of preferences. While less commonly used than ordinal utility, it has profound applications in risk assessment, decision theory, and welfare economics. Understanding its implications, assumptions, and limitations is essential for deep insights into consumer behavior and economic modeling.
