2.1 Utility functions and preference relations
7 min read•july 30, 2024
Utility functions and preference relations are key concepts in understanding how people make decisions. They help us model and analyze choices by assigning numerical values to outcomes, reflecting what individuals prefer.
These tools are crucial in economics and decision theory. They allow us to predict behavior, compare options, and explore how people trade off different goods or outcomes. Understanding these concepts is essential for grasping the foundations of rational choice theory.
Utility functions and preferences
Representing preferences with utility functions
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- Utility functions are mathematical representations that assign a numerical value to each possible outcome or bundle of goods, reflecting the preferences of an individual or decision-maker
- The utility value assigned to an outcome or bundle represents the level of satisfaction, happiness, or desirability associated with that outcome or bundle
- Utility functions capture the relative ordering of preferences, where higher utility values indicate more preferred outcomes or bundles compared to those with lower utility values
- For example, if bundle A has a utility value of 10 and bundle B has a utility value of 5, then the decision-maker prefers bundle A to bundle B
- The specific functional form of a utility function can vary depending on the context and assumptions made about the decision-maker's preferences, such as or diminishing marginal utility
- Common functional forms include linear, logarithmic, and exponential utility functions
- Utility functions are used to analyze decision-making under uncertainty, as they allow for the comparison and ranking of different outcomes based on their
- Expected utility is calculated by multiplying the utility value of each possible outcome by its probability and summing these products
Applications and limitations of utility functions
- Utility functions have wide-ranging applications in economics, finance, and decision theory, helping to model and predict choice behavior in various contexts (consumer choice, investment decisions, and strategic interactions)
- Utility functions can be derived from observed choice data using techniques such as revealed preference analysis or stated preference surveys
- Limitations of utility functions include the assumption of stable and well-defined preferences, the difficulty in measuring utility directly, and the potential for violations of rationality axioms ( and independence)
- Utility functions may not capture all aspects of decision-making, such as emotions, social norms, or bounded rationality
- Interpersonal comparisons of utility are challenging, as utility functions are subjective and may not be directly comparable across individuals
Preference relation properties
Completeness and transitivity
- Preference relations are binary relations that describe how an individual or decision-maker compares and ranks different alternatives or bundles of goods
- is a property of preference relations, which states that for any two alternatives, A and B, the decision-maker can always specify whether they prefer A to B, prefer B to A, or are between A and B
- Formally, for any A and B in the set of alternatives X, either A ≿ B, B ≿ A, or A ∼ B, where ≿ denotes "at least as good as" and ∼ denotes "indifferent to"
- Transitivity is another property of preference relations, which states that if a decision-maker prefers alternative A to B and prefers B to C, then they must also prefer A to C
- Formally, for any A, B, and C in the set of alternatives X, if A ≿ B and B ≿ C, then A ≿ C
- Preference relations that satisfy completeness and transitivity are called rational preference relations, as they ensure consistent and logical decision-making
- Violations of completeness or transitivity can lead to inconsistent or cyclic preferences, which may result in irrational or paradoxical decision-making behavior (Condorcet paradox or intransitive dice)
Other properties of preference relations
- Reflexivity is a property of preference relations, which states that an alternative is always at least as good as itself
- Formally, for any A in the set of alternatives X, A ≿ A
- Anti-symmetry is a property of preference relations, which states that if A is preferred to B, then B cannot be preferred to A, unless A and B are indifferent
- Formally, for any A and B in the set of alternatives X, if A ≿ B and B ≿ A, then A ∼ B
- Continuity is a property of preference relations, which states that if A is preferred to B, then alternatives sufficiently close to A are also preferred to B
- This property ensures that small changes in the attributes of alternatives do not lead to abrupt changes in preferences
- is a property of preference relations, which states that if bundle A contains at least as much of every good as bundle B, then A is at least as preferred as B
- This property reflects the idea that "more is better" for goods that are considered desirable
Indifference curves and preferences
Graphical representation of preferences
- Indifference curves are graphical representations of bundles of goods that provide the same level of utility or satisfaction to a decision-maker
- Each point on an represents a combination of two goods (or attributes) that the decision-maker is equally satisfied with, meaning they are indifferent between any two points on the same curve
- Indifference curves are downward sloping, as a decision-maker must be given more of one good to compensate for a reduction in the other good while maintaining the same level of utility
- This negative slope reflects the trade-off between the two goods and the decision-maker's willingness to substitute one good for the other
- Indifference curves cannot intersect, as it would violate the transitivity property of preferences. If two indifference curves intersect, it implies that the decision-maker is indifferent between two bundles that are on different curves, which is inconsistent
Properties and interpretation of indifference curves
- The slope of an indifference curve at any point represents the marginal rate of substitution (MRS), which measures the amount of one good a decision-maker is willing to give up to obtain one more unit of the other good while maintaining the same level of utility
- MRS = - (ΔY / ΔX), where ΔY is the change in the quantity of good Y and ΔX is the change in the quantity of good X
- Higher indifference curves represent higher levels of utility, as they contain bundles that are preferred to those on lower indifference curves
- The shape of indifference curves can provide information about the decision-maker's preferences, such as whether goods are perfect (straight line), perfect (L-shaped), or exhibit diminishing marginal rates of substitution (convex to the origin)
- Indifference maps, which consist of a set of indifference curves, can be used to represent a decision-maker's complete preference ordering over all possible bundles of goods
- The optimal choice for a decision-maker is the bundle that lies on the highest attainable indifference curve, given their budget constraint or other limitations
Ordinal vs cardinal utility
Ordinal utility and preference ordering
- is a concept where the utility of outcomes or bundles is ranked or ordered based on the decision-maker's preferences without assigning specific numeric values to the utility levels
- With ordinal utility, the focus is on the relative ordering of preferences rather than the absolute magnitude of utility differences between outcomes or bundles
- For example, if a decision-maker prefers bundle A to bundle B and bundle B to bundle C, we can assign ordinal utility values of 3, 2, and 1 to bundles A, B, and C, respectively
- Ordinal utility functions preserve the preference ordering but are unique up to monotonic transformations, meaning that any strictly increasing function applied to an ordinal utility function will result in an equivalent representation of preferences
- For instance, if U(A) = 3, U(B) = 2, and U(C) = 1 represent the ordinal utilities of bundles A, B, and C, then V(A) = 9, V(B) = 4, and V(C) = 1 also represent the same preference ordering
Cardinal utility and utility differences
- , on the other hand, assigns specific numeric values to the utility of outcomes or bundles, allowing for the measurement and comparison of utility differences
- Cardinal utility functions are unique up to positive affine transformations, which means that the utility values can be scaled by a positive constant and/or shifted by a constant without changing the underlying preferences
- For example, if U(A) = 10, U(B) = 5, and U(C) = 0 represent the cardinal utilities of bundles A, B, and C, then V(A) = 20, V(B) = 10, and V(C) = 0 also represent the same preferences and utility differences
- The choice between ordinal and cardinal utility depends on the assumptions made about the measurability and comparability of utility across individuals or decision-makers
- In many applications, ordinal utility is sufficient for analyzing preferences and decision-making, while cardinal utility is necessary when considering issues such as social welfare, interpersonal utility comparisons, or decision-making under uncertainty
- For instance, ordinal utility is often used in consumer theory to model preferences and demand, while cardinal utility is used in game theory to analyze strategic interactions and payoffs
Key Terms to Review (16)
Borda Count: Borda Count is a voting method that assigns scores to candidates based on their rank in voters' preferences, with the goal of determining a consensus choice. In this system, voters rank candidates in order of preference, and points are allocated according to their position on each voter's list, ultimately summing these points to identify the winner. This method reflects the idea that broader consensus can yield better decisions, as it takes into account not only the most preferred option but also the secondary choices of voters.
Cardinal Utility: Cardinal utility is a concept in economics that assigns a numerical value to the satisfaction or pleasure derived from consuming goods and services. This approach allows for the comparison of utility levels across different choices, enabling individuals to make decisions based on quantifiable measures of their preferences. Cardinal utility is often contrasted with ordinal utility, which only ranks preferences without assigning specific values.
Complements: Complements are goods that are typically consumed together, meaning that the demand for one good increases when the price of the other good decreases. This relationship highlights the interconnectedness of consumer preferences and utility, as a change in the availability or pricing of one complementary good can significantly impact the consumption patterns of another. Understanding complements is essential for analyzing how consumers allocate their resources and how utility functions can reflect these preferences.
Completeness: Completeness refers to a property of preference relations where every pair of alternatives can be compared by a decision-maker, meaning that for any two options, one is either preferred over the other, or they are viewed as equally preferred. This concept is crucial in understanding how individuals evaluate choices and form utility functions, ensuring that preferences are well-defined and consistent across all potential options.
Convexity: Convexity refers to the property of a set or function where any line segment connecting two points within the set or function lies entirely within that set or function. This concept is essential in understanding preferences and utility functions, as it implies that individuals prefer averages over extremes, suggesting a level of consistency in their choices.
Expected Utility: Expected utility is a concept used to quantify the preferences of an individual or decision-maker when faced with uncertain outcomes. It combines the probability of different outcomes occurring with the utility or satisfaction derived from those outcomes, allowing individuals to make rational decisions under uncertainty. This idea connects deeply with how strategies are evaluated and chosen, especially when considering risk and preferences in competitive environments.
Indifference Curve: An indifference curve represents a graph showing different combinations of two goods that provide the same level of utility to a consumer. This concept helps illustrate how consumers make choices based on their preferences while maintaining the same level of satisfaction, highlighting the trade-offs between different goods in the context of utility functions and preference relations.
Indifferent: Indifferent refers to a situation where a person has no preference between two or more options, meaning they derive the same level of satisfaction or utility from each choice. This concept is crucial in understanding utility functions and preference relations, as it illustrates how individuals evaluate different alternatives without a distinct inclination towards one over the others.
Monotonicity: Monotonicity refers to a property of utility functions where preferences are consistent and non-decreasing with respect to the consumption of goods. In other words, if a consumer prefers bundle A over bundle B, then they will prefer any bundle that contains more of at least one good in A, without decreasing the amount of any other good. This concept ensures that higher consumption levels yield equal or greater satisfaction, which plays a crucial role in understanding how individuals make choices based on their preferences.
Ordinal Utility: Ordinal utility is a concept in economics that ranks preferences based on the order of satisfaction derived from different goods or services, rather than assigning specific numerical values to those levels of satisfaction. It focuses on the relative ranking of choices and allows individuals to express preferences without needing to quantify how much more one option is preferred over another. This approach emphasizes that while we can say one bundle of goods is preferred to another, the exact degree of preference is not necessary for understanding consumer choices.
Pareto efficiency: Pareto efficiency refers to a situation in which resources are allocated in such a way that no individual can be made better off without making someone else worse off. It is a key concept in understanding optimal resource allocation and plays a significant role in various strategic interactions, showing how individuals or groups can reach outcomes where any change would harm at least one party involved.
Risk Aversion: Risk aversion is the preference for a sure outcome over a gamble with higher or equal expected value. This concept is crucial in understanding how individuals make choices under uncertainty and impacts various decision-making processes, especially when faced with potential losses. Recognizing that risk-averse individuals prefer safer options can help explain their behaviors in economic settings, as well as in game theory and experimental studies.
Strictly preferred: Strictly preferred refers to a situation in which an individual has a clear preference for one option over another, indicating that the preferred option provides a higher level of satisfaction or utility. This concept is fundamental in understanding how individuals make choices based on their preferences, where strictly preferred outcomes are consistently chosen over alternatives. It highlights the idea of comparative satisfaction that plays a crucial role in utility functions and preference relations.
Substitutes: Substitutes are goods or services that can replace each other in consumption, meaning that an increase in the price of one can lead to an increase in the quantity demanded of the other. This concept is crucial when analyzing utility functions and preference relations, as it helps to understand consumer behavior and how changes in price affect choices between different products. Understanding substitutes allows for a deeper insight into how consumers derive utility from various combinations of goods.
Transitivity: Transitivity refers to a property of preference relations where if an individual prefers option A over option B and prefers option B over option C, then they must also prefer option A over option C. This concept is crucial for establishing consistent and rational choice behavior in decision-making, linking it closely to utility functions, which represent how different choices lead to varying levels of satisfaction.
Utility maximization: Utility maximization is the concept in economics and decision theory that refers to individuals making choices that lead to the highest level of satisfaction or utility possible, given their preferences and constraints. This principle implies that people act rationally to achieve the greatest benefit from their available resources, balancing trade-offs between different options. It plays a crucial role in strategic decision-making and helps to understand how utility functions and preference relations drive consumer behavior.