Expected Value

Background§

Expected value represents the average outcome of a random variable if an experiment or a process were repeated many times. It serves as a measure in probability and statistics to envisage the long-term result of random variables associated with different probabilities.

Historical Context§

The concept originated in the field of probability theory and has been employed extensively in economics, finance, insurance, and other domains. The formal calculation methods for expected values were developed in the 17th century by mathematicians such as Blaise Pascal and Pierre de Fermat, who collaborated on problems pertaining to gambling and games of chance.

Definitions and Concepts§

The expected value $E[f(X)]$ of a discrete random variable $X$ with an associated probability distribution $P(x)$ is defined mathematically as:

$$E[f(X)] = \sum_{x} f(x) P(x)$$

Where:

• $X$ is a discrete random variable.
• $P(x)$ is the probability that $X$ takes the value $x$.
• $f(X)$ is a function of $X$.

For continuous random variables, the expected value is given by:

$$E[f(X)] = \int_{-\infty}^{\infty} f(x) p(x) , dx$$

Where $p(x)$ is the probability density function of $X$.

Major Analytical Frameworks§

Classical Economics§

In classical economics, expected value is used in various contexts, such as determining the expected profit from investments or the expected outcomes of different economic policies.

Neoclassical Economics§

Neoclassical economists use the concept of expected value in the analysis of consumer behavior, particularly in the probabilistic maximization of utility functions.

Keynesian Economics§

Keynesian models might utilize expected value in their treatment of investment decisions under uncertainty, where entrepreneurs base their behavior on expected future returns.

Marxian Economics§

Marxian economics does not typically integrate probabilistic approaches directly, but the concept can be applied to expected surplus values and profit calculations.

Institutional Economics§

In institutional economics, expected value can be related to the probabilistic outcomes of different rules, regulations, and institutional behaviors.

Behavioral Economics§

Behavioral economics scrutinizes the deviations from expected value theory evident in actual human behavior, addressing phenomena such as bias in probability judgments and decision-making anomalies.

Post-Keynesian Economics§

Post-Keynesian economists might employ expected value in scenarios involving uncertainty, economic forecasting, and behavioral unpredictability.

Austrian Economics§

Austrian economics often questions whether the formal mathematical expectations aligned with expected value adequately capture the real-world complexities they stress in human action.

Development Economics§

In development economics, expected value can be used to evaluate the probable outcomes of development policies, aid distributions, and intervention programs.

Monetarism§

Monetarist analyses incorporate expected value in regards to inflation expectations, money supply outcomes, and policy efficacy under stochastic frameworks.

Comparative Analysis§

While different strands of economic thought utilize the expected value differently, the shared essence is its role in decision-making under uncertainty. The divergent applications showcase the eclectic methodologies accommodated by the concept within economic theory and practice.

Case Studies§

• Investment Decisions: Use of expected value to predict returns on various financial assets.
• Insurance: Calculation of premium rates based on the expected value of potential payouts.
• Public Policy: Evaluating the expected benefits of implementing or discontinuing certain policies.

Suggested Books for Further Studies§

1. “Probability, Statistics, and Mathematics: A Process Approach” by Cohen and Nagel
2. “An Introduction to Probability Theory and Its Applications” by William Feller
3. “Probability and Statistics” by Morris H. DeGroot and Mark J. Schervish

Related Terms with Definitions§

• Random Variable: A variable that takes on different values based on outcomes of a random phenomenon.
• Probability Distribution: A function that provides the probabilities of occurrence of various possible outcomes.
• Variance: A measure of the spread between numbers in a data set, indicating how far the numbers lie from the expected value.
• Standard Deviation: The square root of the variance, representing the dispersion of a dataset relative to its mean.

By encapsulating these diverse applications and theoretical frameworks, the concept of expected value endures as a cornerstone of economic analysis, affording a crucial tool for decision-making under uncertainty.