Intro to Mathematical Economics Unit 9 Review: Probability Theory & Expected Utility

Key Concepts

• Probability theory provides a mathematical framework for quantifying and analyzing uncertainty
• Random variables assign numerical values to outcomes of random experiments
• Probability distributions describe the likelihood of different values of a random variable
• Expected value represents the average outcome of a random variable over many repetitions
• Variance measures the spread or dispersion of a random variable around its expected value
• Utility theory models decision-making preferences and risk attitudes of individuals
• Von Neumann-Morgenstern utility theorem establishes a framework for rational decision-making under uncertainty
• Applications of probability and utility theory in economics include game theory, finance, and insurance

Probability Basics

• Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1
  • 0 represents an impossible event, while 1 represents a certain event
• Sample space is the set of all possible outcomes of a random experiment
• Events are subsets of the sample space, representing specific outcomes or combinations of outcomes
• Probability of an event A is denoted as P(A) and can be calculated using the following methods:
  • Classical approach: P(A) = (number of favorable outcomes) / (total number of possible outcomes)
  • Empirical approach: P(A) = (frequency of event A) / (total number of trials)
• Conditional probability P(A|B) measures the probability of event A occurring given that event B has occurred
• Independent events have probabilities that do not depend on the occurrence of other events
  • For independent events A and B, P(A and B) = P(A) × P(B)

Random Variables and Distributions

• A random variable is a function that assigns a numerical value to each outcome in a sample space
• Discrete random variables take on a countable number of distinct values (integers)
  • Examples include the number of defective items in a production batch or the number of customers in a queue
• Continuous random variables can take on any value within a specified range (real numbers)
  • Examples include the weight of a randomly selected product or the time until a machine fails
• Probability mass function (PMF) describes the probability distribution of a discrete random variable
  • PMF gives the probability of each possible value of the random variable
• Probability density function (PDF) describes the probability distribution of a continuous random variable
  • PDF gives the relative likelihood of the random variable taking on a specific value
• Cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a given value
  • CDF is defined for both discrete and continuous random variables

Expected Value and Variance

• Expected value (or mean) of a random variable X, denoted as E(X), is the weighted average of all possible values
  • For discrete random variables: $E(X) = \sum_{x} x \cdot P(X = x)$
  • For continuous random variables: $E(X) = \int_{-\infty}^{\infty} x \cdot f(x) dx$
• Linearity of expectation states that for random variables X and Y: $E(X + Y) = E(X) + E(Y)$
• Variance, denoted as Var(X) or $\sigma^2$, measures the average squared deviation from the mean
  • $Var(X) = E[(X - E(X))^2] = E(X^2) - [E(X)]^2$
• Standard deviation $\sigma$ is the square root of the variance and has the same units as the random variable
• Chebyshev's inequality provides a bound on the probability of a random variable deviating from its mean
  • For any k > 0: $P(|X - E(X)| \geq k\sigma) \leq \frac{1}{k^2}$

Utility Theory

• Utility is a measure of satisfaction or preference that an individual derives from consuming goods or services
• Cardinal utility assigns numerical values to represent the level of satisfaction
  • Allows for interpersonal comparisons of utility
• Ordinal utility ranks preferences without assigning specific numerical values
  • Only the relative ordering of preferences matters
• Utility functions U(x) map the consumption of goods or wealth levels to utility values
  • Monotonicity: more is preferred to less, so utility functions are non-decreasing
• Marginal utility is the additional satisfaction gained from consuming one more unit of a good
  • Diminishing marginal utility: the additional satisfaction decreases as more units are consumed
• Risk attitudes describe an individual's preferences for uncertain outcomes
  • Risk-averse individuals prefer a certain outcome to a risky one with the same expected value
  • Risk-neutral individuals are indifferent between a certain outcome and a risky one with the same expected value
  • Risk-seeking individuals prefer a risky outcome to a certain one with the same expected value

Decision Making Under Uncertainty

• Von Neumann-Morgenstern (VNM) utility theorem provides a foundation for decision-making under uncertainty
  • Axioms: completeness, transitivity, continuity, and independence
• Expected utility theory states that a rational decision-maker chooses the option with the highest expected utility
  • Expected utility of an option: $EU = \sum_{i} p_i \cdot U(x_i)$, where $p_i$ is the probability of outcome $x_i$
• Certainty equivalent is the guaranteed amount that provides the same utility as a risky option
• Risk premium is the difference between the expected value of a risky option and its certainty equivalent
  • Measures the amount an individual is willing to pay to avoid risk
• Allais paradox demonstrates violations of the independence axiom in the VNM utility theorem
  • Highlights the limitations of expected utility theory in describing actual human behavior

Applications in Economics

• Game theory models strategic interactions between decision-makers
  • Players' payoffs are represented by utility functions
  • Nash equilibrium: a set of strategies where no player can improve their expected utility by unilaterally changing strategy
• Portfolio theory in finance uses probability and utility to optimize investment decisions
  • Mean-variance analysis balances the expected return and risk (variance) of a portfolio
  • Efficient frontier represents the set of portfolios with the highest expected return for a given level of risk
• Insurance markets rely on probability theory to calculate premiums and manage risk
  • Law of large numbers ensures the long-run stability of insurance companies
  • Adverse selection and moral hazard are key challenges in insurance markets
• Auction theory applies probability and utility to design and analyze auction mechanisms
  • Different auction formats (first-price, second-price, all-pay) lead to different bidding strategies and outcomes

Common Pitfalls and Misconceptions

• Gambler's fallacy: believing that past events influence the probability of future independent events
  • Example: thinking that a coin is "due" for heads after a series of tails
• Confusion between correlation and causation
  • Correlation does not imply a causal relationship between variables
• Neglecting base rates when evaluating conditional probabilities
  • Base rate fallacy: focusing on specific information and ignoring the underlying probability of an event
• Misinterpreting the meaning of expected value
  • Expected value is a long-run average, not a guaranteed outcome
• Assuming that all individuals have the same risk preferences
  • Risk attitudes can vary significantly between individuals and contexts
• Overreliance on expected utility theory to describe real-world decision-making
  • Behavioral economics highlights systematic deviations from the predictions of expected utility theory