Introduction
This answer presents short notes on four important microeconomic concepts: von Neumann-Morgenstern (vNM) expected utility theory, Slutsky’s theorem, Arrow-Pratt measure of risk aversion, and the Bergson-Samuelson social welfare function. These are key tools in decision theory, consumer behavior, risk analysis, and welfare economics respectively.
(a) vNM Expected Utility Theory
The von Neumann-Morgenstern (vNM) expected utility theory is a model of how rational individuals make decisions under uncertainty. It was developed by mathematician John von Neumann and economist Oskar Morgenstern.
Main idea: When faced with risky choices or lotteries (where outcomes depend on probabilities), people choose the option with the highest expected utility, not necessarily the highest monetary value.
Key assumptions:
- Completeness: Individuals can rank all possible outcomes.
- Transitivity: Preferences are logically consistent.
- Independence: If A is preferred to B, then a lottery between A and C is preferred to a lottery between B and C.
- Continuity: Preferences are smooth and can be expressed in probabilities.
Expected Utility: E(U) = Σ pi × U(xi)
Here, pi is the probability of outcome xi, and U(xi) is the utility from that outcome.
This theory is widely used in economics, finance, and insurance for analyzing choices involving risk.
(b) Slutsky’s Theorem
Slutsky’s theorem explains how a change in the price of a good affects the quantity demanded, separating it into two effects:
- Substitution effect: The change in demand due to the good becoming relatively cheaper or costlier compared to other goods, holding utility constant.
- Income effect: The change in demand due to the change in real income or purchasing power caused by the price change.
Slutsky Equation:
Total Effect = Substitution Effect + Income Effect
It is usually expressed in mathematical form as:
∂x/∂p = ∂xh/∂p – ∂x/∂I × x
Where:
- x = demand
- p = price
- I = income
- h = Hicksian (compensated) demand
Importance: Slutsky’s theorem helps understand how consumers react to price changes and is a vital concept in consumer demand theory.
(c) Arrow-Pratt Measure of Risk Averseness
This is a mathematical way to measure how risk-averse an individual is. The measure was developed by Kenneth Arrow and John W. Pratt.
Key idea: Risk aversion depends on the curvature of the utility function. A more curved (concave) utility function shows greater risk aversion.
Formula:
A(x) = -U”(x) / U'(x)
Where:
- U(x) = utility function of wealth x
- U'(x) = first derivative (marginal utility)
- U”(x) = second derivative (how fast marginal utility is changing)
Interpretation:
- Higher A(x) → greater risk aversion
- Zero A(x) → risk neutrality
- Negative A(x) → risk loving
It is used in financial economics to understand investment behavior under risk.
(d) Bergson-Samuelson Social Welfare Function
The Bergson-Samuelson social welfare function is a mathematical representation of society’s preferences over different allocations of resources or utilities of individuals.
Developed by: Abram Bergson and Paul Samuelson
Form: W = W(U1, U2, …, Un)
Where W is the level of social welfare, and Ui is the utility of the i-th individual.
Importance:
- Provides a framework to compare different states of the economy in terms of welfare.
- Can incorporate ethical views or weights (e.g., give more importance to poor people’s utility).
- Useful in determining Pareto optimal points or designing redistribution policies.
Limitations: It requires interpersonal comparison of utility, which is often debated in welfare economics.
Conclusion
These four concepts represent essential tools in microeconomic analysis. vNM expected utility theory models decision-making under uncertainty; Slutsky’s theorem explains how price changes affect demand; Arrow-Pratt provides a method to measure risk aversion; and the Bergson-Samuelson welfare function helps analyze collective social preferences. Together, they provide deep insights into consumer behavior, risk, and welfare economics.