Interpersonal Comparisons: Understanding Utility Comparisons between Individuals

Interpersonal Comparisons involve comparing the welfare levels of different individuals using utility functions. This article explores the historical context, types, key events, and significance of interpersonal comparisons, along with detailed explanations, mathematical models, and examples.

Interpersonal comparisons involve comparing the welfare or utility of one individual with that of another. This concept is essential in welfare economics, social choice theory, and public policy for assessing and comparing individuals’ well-being.

Historical Context

The concept of utility can be traced back to early utilitarian philosophers such as Jeremy Bentham and John Stuart Mill, who believed in maximizing collective happiness. However, the formal methods for making interpersonal comparisons of utility developed significantly in the 20th century, especially through the works of economists like Vilfredo Pareto and John Hicks.

Types and Categories

Ordinal Utility

Ordinal utility ranks preferences without specifying the magnitude of differences. It involves an ordinal utility function which can be subject to any monotonic increasing transformation.

Cardinal Utility

Cardinal utility assigns numerical values to preferences, allowing comparisons of utility magnitudes. It can be subject to affine transformations where the form of comparability varies:

• Cardinal Unit Comparability: The same multiplication constant (b) but different additive constants (a).
• Cardinal Full Comparability: Both multiplicative and additive constants are the same across individuals.

Key Events

• 1920s: Introduction of the concept of utility by economists such as Pareto.
• 1950s: Kenneth Arrow’s Impossibility Theorem highlighting the challenges in aggregating individual utilities into a collective social welfare function.

Detailed Explanations

Mathematical Models

Ordinal Utility

For two consumers 1 and 2 with utility functions \(U^1\) and \(U^2\):

$$ U^1 = f(U^1), \quad U^2 = f(U^2) $$

The transformation \(f\) preserves the ranking between consumers.

Cardinal Utility

The transformation for cardinal utility varies:

• Cardinal Unit Comparability:

$$ U^1 = a_1 + bU^1, \quad U^2 = a_2 + bU^2 $$

• Cardinal Full Comparability:

$$ U^h = a + bU^h \quad \text{for all consumers} $$

Utility Comparisons and Welfare Ranking

Interpersonal comparisons determine if utility information can rank welfare levels consistently. This consistency is preserved under restricted transformations, leading to comparable utility functions across individuals.

Charts and Diagrams

Here is a basic diagram representing the utility transformations:

    graph TD;
	    A[U^1] --> B[f(U^1)];
	    C[U^2] --> D[f(U^2)];
	    E[U^1] --> F[a1 + bU^1];
	    G[U^2] --> H[a2 + bU^2];

Importance and Applicability

Interpersonal comparisons are crucial for:

• Evaluating public policies.
• Measuring economic inequality.
• Developing fair social welfare functions.

Examples

Consider two consumers with initial utilities of 50 and 60. Under cardinal unit comparability with \(a_1 = 10, a_2 = 20, b = 1\):

$$ U^1 = 10 + 1(50) = 60, \quad U^2 = 20 + 1(60) = 80 $$

The utility ranking is maintained.

Considerations

• Different transformations can change utility rankings if comparability constraints are not enforced.
• Accurate interpersonal comparisons require reliable utility measures and transformations.

Related Terms and Definitions

• Arrow’s Impossibility Theorem: Demonstrates the difficulty in creating a social welfare function that reflects individuals’ preferences.
• Social Welfare Function: Aggregates individual utilities into a collective measure of social welfare.

Comparisons

• Ordinal vs. Cardinal Utility: Ordinal utility only ranks preferences while cardinal utility measures the magnitude of preferences.
• Non-comparability vs. Comparability: Non-comparability allows different transformations for each individual, complicating welfare ranking.

Interesting Facts

• Early economic theories did not differentiate between ordinal and cardinal utility, leading to less precise welfare analyses.

Inspirational Stories

Amartya Sen’s work on welfare economics and capabilities highlighted the limitations of traditional utility-based comparisons and introduced more holistic approaches to evaluating well-being.

Famous Quotes

“The greatest happiness of the greatest number is the foundation of morals and legislation.” - Jeremy Bentham

Proverbs and Clichés

“One man’s trash is another man’s treasure.” - Reflects subjective nature of utility.

Expressions, Jargon, and Slang

• Utility Monster: A hypothetical being used in arguments about distribution of welfare.
• Pareto Efficiency: An allocation where no individual can be made better off without making someone else worse off.

FAQs

What is the main challenge in interpersonal comparisons of utility?

Ensuring transformations that preserve welfare rankings across individuals.

Can utility functions be both ordinal and cardinal?

No, they are treated separately based on the type of transformation they undergo.

References

1. Arrow, K. J. (1951). Social Choice and Individual Values. Yale University Press.
2. Sen, A. (1970). Collective Choice and Social Welfare. Holden-Day.
3. Bentham, J. (1789). An Introduction to the Principles of Morals and Legislation.

Summary

Interpersonal comparisons involve assessing and comparing individuals’ utility or welfare levels. Through transformations that preserve welfare rankings, economists can make meaningful comparisons to guide public policy and social welfare functions. Understanding both ordinal and cardinal utility is critical for effective welfare economics.

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This article provides a thorough exploration of interpersonal comparisons, incorporating historical context, types, mathematical models, and applications, aimed at educating readers on the complexities and importance of utility comparisons in economics and social sciences.