Fiche de révision : Consumer Choice and Utility Theory

📋 Course Outline

1. Consumer Utility Maximization
2. Marginal Utility and Demand
3. Utility Measurement and Indifference
4. Budget Constraints and Optimization
5. Law of Demand and Elasticity
6. Price Elasticity of Demand
7. Types of Goods and Income Effects
8. Substitutes and Complements
9. Rational Decision-Making Assumptions
10. Expected Utility Theory

📖 1. Consumer Utility Maximization

🔑 Key Concepts & Definitions

• Utility Function (von Neumann & Morgenstern, 1944): A mathematical representation of a consumer’s preferences, assigning a real number to each option such that higher numbers indicate more preferred options, enabling analysis of choice behavior.

• Completeness and Transitivity (Arrow, 1951): Assumptions that preferences are complete (for any two options, the consumer can state a preference or indifference) and transitive (if A is preferred to B, and B to C, then A is preferred to C), ensuring preferences can be represented by a utility function.

• Expected Utility Theory (Bernoulli, 1738): A model where consumers maximize the expected utility of uncertain outcomes, capturing the idea that utility, not monetary value, guides decision-making under risk.

• Willingness-to-Pay (WTP): The maximum amount a consumer is willing to pay for an additional unit of a good, interpreted as the marginal utility of that good in quasilinear utility models.

• Marginal Utility of a Good (𝑉𝑚 𝑞): The additional utility obtained from consuming one more unit of a good, calculated as the derivative of the utility function with respect to quantity, often decreasing as quantity increases.

• Quasilinear Utility (Samuelson, 1947): A utility form where total utility is linear in money (or a monetary equivalent), simplifying analysis of consumer choice and marginal willingness to pay.

📝 Essential Points

• Consumers aim to maximize their utility subject to their budget constraint, which equates total expenditure to income (𝑡 + 𝑝𝑞 = 𝐼).

• The first-order condition for utility maximization in the quasilinear case is when marginal utility equals the price (𝑉′(𝑞) = 𝑝), indicating the consumer should stop purchasing when the marginal utility of an additional unit equals its price.

• Preferences are assumed to be complete and transitive, which allows representation via a utility function (𝑈), simplifying analysis of consumer choices.

• Expected utility models incorporate risk preferences, with utility derived from the probability-weighted outcomes, explaining behaviors like risk aversion.

• The marginal utility diminishes with additional units, reflecting the law of diminishing marginal utility, which influences consumption decisions and demand curves.

💡 Key Takeaway

Consumer utility maximization involves balancing the marginal utility of goods against their prices, underpinned by assumptions of rational preferences that can be represented by a utility function, enabling prediction of demand behavior even under uncertainty.

📖 2. Marginal Utility and Demand

🔑 Key Concepts & Definitions

• Marginal Utility (MU): The additional utility gained from consuming one more unit of a good or service. Mathematically, in the continuous case, MU is the derivative of the utility function with respect to quantity, denoted as $U_m(x) = U'(x)$. Economists generally assume MU decreases as consumption increases, reflecting diminishing marginal utility (KEYNES, 1936).

• Expected Utility (EU): A model where individuals evaluate risky prospects by calculating the expected utility, which is the probability-weighted average of utilities across outcomes. Originates from BERNOULLI (1738), emphasizing that utility, not monetary value, guides decision-making under risk.

• Willingness-to-Pay (WTP): The maximum amount a consumer is willing to pay for an additional unit of a good without decreasing overall utility. In the quasilinear utility case, WTP equals the marginal utility of the good, $V_m(q)$.

• Quasilinear Utility: A utility function of the form $U(q, t) = V(q) + t$, where $V(q)$ is utility from the good and $t$ is money. This simplifies analysis by making marginal utility of the good independent of income, allowing WTP to be directly interpreted as the maximum price a consumer is willing to pay (see example with baguettes).

• Marginal Rate of Substitution (MRS): The rate at which a consumer is willing to substitute one good for another while maintaining the same utility level, equal to the ratio of marginal utilities, $\frac{MU_1}{MU_2}$, and equal to the price ratio at the optimum.

📝 Essential Points

• Consumers aim to maximize their utility by choosing consumption levels where marginal utility equals the price of the good ($MU = p$). This condition ensures optimal allocation of resources (utility maximization).

• Marginal utility is assumed to diminish with increased consumption, which underpins the downward-sloping demand curve (law of demand).

• The concept of expected utility explains decision-making under risk, accounting for the non-linear valuation of money (BERNOULLI, 1738; KAHNEMAN and TVERSKY, 1979).

• In quasilinear utility, the maximum price a consumer is willing to pay for an additional unit (WTP) equals the marginal utility $V_m(q)$, facilitating the analysis of consumer behavior and demand.

• The law of demand emerges naturally from decreasing marginal utility: as price drops, consumers are willing to buy more because the additional utility from extra units exceeds the cost.

💡 Key Takeaway

Marginal utility explains how consumers decide how much of a good to consume; as they consume more, the additional satisfaction decreases, shaping the downward-sloping demand curve and guiding optimal purchasing decisions.

📖 3. Utility Measurement and Indifference

🔑 Key Concepts & Definitions

• Expected Utility (EU) (Bernoulli, 1738): A model where individuals evaluate risky prospects by the expected value of their utility, not monetary value, accounting for diminishing marginal utility of money.
• Completeness Assumption: The idea that consumers can always compare two options and state a preference or indifference, ensuring preferences are well-defined.
• Transitivity Assumption: The principle that if a consumer prefers option A to B and B to C, then they prefer A to C, ensuring consistency in preferences.
• Utility Function (U): A mathematical representation of preferences, assigning real numbers to options such that more preferred options have higher utility.
• Marginal Utility (𝑈𝑚(𝑥)): The additional utility gained from consuming one more unit of a good or service, often decreasing with increased consumption.
• Quasilinear Utility: A utility form where total utility is expressed as 𝑈(𝑞, 𝑡) = 𝑉(𝑞) + 𝑡, allowing the marginal utility of a good to be interpreted as willingness-to-pay (WTP).

📝 Essential Points

• Utility measurement relies on the assumption that preferences are complete and transitive, which guarantees they can be represented by a utility function (Theorem).
• Expected utility theory (Bernoulli, 1738) explains how individuals evaluate risky choices by considering the utility of outcomes, not just monetary amounts, addressing the non-linear valuation of money.
• Indifference between a certain amount 𝑥 and a risky prospect (e.g., 50% chance of €10,000) is formalized as 𝑈(𝑥) = 0.5 𝑈(10,000).
• Marginal utility decreases as consumption increases, reflecting diminishing returns and influencing consumer choices and demand.
• Quasilinear utility simplifies analysis by separating the utility of a good from money, making the marginal utility of the good equal to the willingness-to-pay, 𝑉𝑚(𝑞).
• Market models such as the Capital Asset Pricing Model and pricing kernels are based on expected utility with concave utility functions, emphasizing risk aversion.

💡 Key Takeaway

Utility measurement, grounded in expected utility theory and assumptions of preference consistency, provides a framework for understanding consumer choices under risk, highlighting how diminishing marginal utility influences demand and decision-making.

📖 4. Budget Constraints and Optimization

🔑 Key Concepts & Definitions

• Budget Constraint: The limit on the consumption choices of a consumer, defined by their income and the prices of goods, expressed as 𝑡 + 𝑝𝑞 = 𝐼, where 𝑡 is money, 𝑝 is price, 𝑞 is quantity, and 𝐼 is income. It represents all affordable combinations of goods and services (see section 3. Utility Measurement and Indifference).

• Utility Maximization: The decision process where consumers choose the combination of goods that provides the highest utility within their budget constraint, achieved when the marginal utility per dollar spent on each good is equalized (see section 3. Utility Measurement and Indifference).

• Marginal Utility per Dollar (𝑈𝑚/𝑝): The additional utility gained from spending one more dollar on a good, calculated as the marginal utility of the good divided by its price. Consumers allocate their budget so that 𝑈𝑚/𝑝 is equal across all goods (see section 3. Utility Measurement and Indifference).

• First-Order Condition for Utility Maximization: The condition where the consumer's optimal choice occurs when the marginal utility per dollar is equalized across all goods, i.e., 𝑈′(𝑞)/𝑝 = constant, ensuring no further utility gain from reallocating spending (see section 3. Utility Measurement and Indifference).

• Quasilinear Utility: A utility function of the form 𝑈(𝑞, 𝑡) = 𝑉(𝑞) + 𝑡, where 𝑉(𝑞) is utility from good consumption and 𝑡 is money. It simplifies analysis by allowing the marginal utility of money to be constant, making the willingness-to-pay (WTP) for additional units directly interpretable as marginal utility (see section 3. Utility Measurement and Indifference).

📝 Essential Points

• Consumers maximize utility subject to their budget constraint, which limits their consumption options based on income and prices.

• The optimal consumption bundle occurs where the marginal utility per dollar spent on each good is equalized, i.e., 𝑈′(𝑞₁)/𝑝₁ = 𝑈′(𝑞₂)/𝑝₂, known as the equimarginal principle.

• The budget line shifts with changes in income or prices: an increase in income shifts it outward, while a price change causes a rotation along the line.

• In the quasilinear utility case, the consumer's choice simplifies to maximizing the utility from the good, with money serving as a numeraire, and the willingness-to-pay (WTP) for an additional unit of a good is equal to its marginal utility.

• The first-order condition for utility maximization ensures that consumers do not reallocate their spending when the marginal utility per dollar is equalized across all goods.

• When prices change, consumers respond by adjusting quantities, moving along the demand curve; shifts in demand occur when income or prices of related goods change (see sections 4 and 5).

💡 Key Takeaway

Consumers optimize their utility by allocating their budget so that the marginal utility per dollar is equalized across all goods, with their choices constrained by income and prices; this principle underpins demand behavior and market equilibrium.

📖 5. Law of Demand and Elasticity

🔑 Key Concepts & Definitions

• Price Elasticity of Demand (𝜀): (Source: general economic theory) Measures the responsiveness of quantity demanded to a change in price, calculated as the percentage change in quantity divided by the percentage change in price.
  Formula: 𝜀 = (% change in quantity demanded) / (% change in price).
  Interpretation: If 𝜀 > 1, demand is elastic; if 𝜀 < 1, demand is inelastic; if 𝜀 = 1, demand is unit elastic.

• Marginal Utility (𝑈ₘ(𝑥)): (Source: utility theory) The additional utility gained from consuming one more unit of a good or service, mathematically the derivative of utility with respect to quantity (𝑈′(𝑥)).
  Key point: Marginal utility generally decreases as consumption increases (diminishing marginal utility).

• Law of Demand: (Source: classical microeconomics) States that, ceteris paribus, there is an inverse relationship between the price of a good and the quantity demanded, primarily due to decreasing marginal utility.

• Willingness-to-Pay (WTP): (Source: quasilinear utility models) The maximum amount a consumer is willing to pay for an additional unit of a good without decreasing their overall utility, often interpreted as the marginal utility of that good.

• Elasticity and Revenue Relationship: (Source: demand elasticity) When demand is elastic (𝜀 > 1), lowering prices increases total revenue; when demand is inelastic (𝜀 < 1), lowering prices decreases total revenue; at 𝜀 = 1, revenue is maximized.

📝 Essential Points

• The individual demand curve is derived from the consumer’s marginal utility, illustrating how much a consumer is willing to buy at different prices (see the law of demand).
• The market demand curve is obtained by horizontally summing individual demand curves, assuming identical consumers or aggregating diverse preferences.
• Price elasticity of demand (𝜀) quantifies how sensitive demand is to price changes, guiding firms’ pricing strategies to optimize revenue.
• The law of demand results from the principle of diminishing marginal utility: as price decreases, consumers are willing to buy more because the additional utility from extra units exceeds the marginal cost.
• Changes in the price of substitutes or complements cause shifts in demand curves (see section 7), affecting elasticity and total revenue.
• Revenue maximization occurs where the price elasticity of demand equals 1 (unit elasticity), as shown by the first-order condition for profit maximization.

💡 Key Takeaway

The law of demand reflects the inverse relationship between price and quantity demanded driven by diminishing marginal utility, and elasticity measures how demand responds to price changes, crucial for understanding revenue and market behavior.

📖 6. Price Elasticity of Demand

🔑 Key Concepts & Definitions

• Price Elasticity of Demand (𝜀) (source): The percentage change in quantity demanded resulting from a 1% change in price. It measures consumer responsiveness to price changes.
  $\varepsilon = \frac{\%\ \text{change in quantity demanded}}{\%\ \text{change in price}}$

• Elastic Demand (𝜀 > 1) (source): Demand where the percentage change in quantity demanded exceeds the percentage change in price, indicating high sensitivity.
  Implication: Price reductions lead to proportionally larger increases in quantity demanded, raising total revenue.

• Inelastic Demand (𝜀 < 1) (source): Demand where the percentage change in quantity demanded is less than the percentage change in price, indicating low sensitivity.
  Implication: Price reductions decrease total revenue because the increase in quantity demanded is proportionally smaller.

• Unit Elastic Demand (𝜀 = 1) (source): The percentage change in quantity demanded equals the percentage change in price, maximizing total revenue at this point.

• Revenue Decision Rule (elasticity threshold) (source):

  • If 𝜀 > 1, lowering price increases revenue.
  • If 𝜀 < 1, lowering price decreases revenue.
  • If 𝜀 = 1, revenue is maximized.

📝 Essential Points

• Demand curve: The slope of the demand curve relates to elasticity; flatter curves tend to be more elastic, steeper curves more inelastic.
• Elasticity calculation: For a demand function $Q_P$, elasticity at a point is given by:
  $\varepsilon = \frac{dQ}{dP} \times \frac{P}{Q}$
• Market demand: The individual demand curves are summed horizontally to derive market demand, which influences overall elasticity.
• Revenue maximization: The point where elasticity equals 1 (unit elastic demand) corresponds to maximum total revenue, as shown by the first-order condition for revenue maximization ($dR/dQ=0$).
• Elasticity and pricing strategies: Understanding whether demand is elastic or inelastic guides firms in setting prices to optimize revenue (see the baguette example).
• Elasticity of demand for substitutes and complements: Demand for substitutes tends to be more elastic, while demand for complements tends to be more inelastic; changes in their prices shift the demand curve accordingly.

💡 Key Takeaway

Price elasticity of demand determines how consumers respond to price changes, guiding firms on optimal pricing strategies to maximize revenue based on whether demand is elastic, inelastic, or unit elastic.

📖 7. Types of Goods and Income Effects

🔑 Key Concepts & Definitions

• Normal Good: A good for which demand increases as consumer income rises, shifting the demand curve to the right (see section 44). Authors (source content) indicate that demand for normal goods responds positively to income changes.

• Inferior Good: A good for which demand decreases as consumer income increases, shifting the demand curve to the left (see section 43). Authors note that demand for inferior goods is inversely related to income.

• Substitute Goods: Goods that can replace each other; an increase in the price of one leads to an increase in demand for the other (see section 45). Authors state that demand for a good increases when the price of its substitute increases, shifting demand to the right.

• Complementary Goods: Goods that are used together; an increase in the price of one causes a decrease in demand for the other (see section 46). Authors explain that demand for a good decreases when the price of its complement increases, shifting demand to the left.

• Income Effect: The change in quantity demanded resulting from a change in consumer income, affecting demand for normal and inferior goods (see sections 41-43). Authors highlight that the income effect causes demand shifts based on income changes.

• Price Effect: The change in quantity demanded caused by a change in the good's own price, represented as movement along the demand curve (see section 48). Authors clarify that this does not shift the demand curve but causes a change in quantity demanded.

📝 Essential Points

• Demand shifts occur due to changes in income or the prices of related goods. For normal goods, demand increases with income; for inferior goods, demand decreases (sections 41-43).

• Substitutes and complements influence demand through cross-price effects: demand for substitutes rises when their prices increase, while demand for complements falls when their prices rise (sections 45-46).

• Impact of price changes on demand depends on the nature of the good: a price increase causes a movement along the demand curve, not a shift; however, changes in the price of related goods (substitutes or complements) shift the entire demand curve (sections 47-48).

• Market demand is obtained by summing individual demands horizontally (section 31), and the law of demand states that lower prices lead to higher quantities demanded (section 33).

• Elasticity determines how demand responds to price changes, influencing revenue outcomes when prices are adjusted (section 34-39). Elastic demand (ε > 1) means quantity responds strongly to price changes, affecting revenue positively or negatively depending on the direction.

💡 Key Takeaway

Demand for goods is influenced by income and the prices of related goods, with normal and inferior goods responding differently to income changes, and substitutes and complements affecting demand through cross-price effects; understanding these relationships is essential for predicting market responses to price and income shifts.

📖 8. Substitutes and Complements

🔑 Key Concepts & Definitions

• Substitutes: Goods or services that can replace each other in consumption; an increase in the price of one leads to an increase in demand for the other (see impact of substitute goods, source content).
• Complements: Goods or services consumed together; an increase in the price of one decreases the demand for the other (see impact of complementary goods, source content).
• Cross-Price Effect: The change in demand for one good resulting from a change in the price of another; positive for substitutes, negative for complements (source content).
• Income Effect (related to goods type): The change in demand for a good resulting from a change in consumer income; normal goods see demand rise with income, inferior goods see demand fall (source content).
• Demand Shift vs. Movement Along Curve: A change in the price of a good causes a movement along the demand curve; a change in the price of a related good (substitute or complement) shifts the entire demand curve (source content).
• Elasticity of Substitutes and Complements: Measures how sensitive demand for a good is to changes in the price of related goods; high elasticity indicates significant demand response (source content).

📝 Essential Points

• The demand for a substitute increases when the price of its counterpart rises, shifting the demand curve to the right (source content).
• The demand for a complement decreases when the price of its counterpart rises, shifting the demand curve to the left (source content).
• Changes in the price of the good itself result in movement along the demand curve, whereas changes in the price of substitutes or complements cause shifts in the entire demand curve (source content).
• The cross-price elasticity of demand quantifies the responsiveness of demand for one good to price changes in another; elasticity > 0 indicates substitutes, elasticity < 0 indicates complements (source content).
• The impact of substitutes and complements on demand is crucial for pricing strategies, market competition, and understanding consumer behavior (source content).
• The theoretical foundation assumes that consumers respond rationally to price changes in related goods, aligning with the law of demand and elasticity principles (source content).

💡 Key Takeaway

Substitutes and complements significantly influence demand patterns; understanding their relationships and elasticity helps predict consumer responses to price changes and optimize market strategies.

📖 9. Rational Decision-Making Assumptions

🔑 Key Concepts & Definitions

• Completeness Assumption: The idea that consumers can always compare any two options and determine which they prefer or if they are indifferent, ensuring preferences are well-defined (see section 10).
• Transitivity Assumption: The principle that if a consumer prefers option A over B and B over C, then they must prefer A over C, ensuring consistency in preferences (see section 10).
• Expected Utility Theory: A model originating from Bernoulli (1738), which posits that individuals evaluate risky choices based on the expected utility rather than expected monetary value, capturing the idea that utility is a nonlinear function of money.
• Homo Economicus: The assumption that individuals act rationally to maximize their utility, with clearly defined and consistent preferences, acting in their best interests (see section 51).
• Source Theory: An extension of expected utility that models how individuals transform probabilities into subjective weights, as discussed by Abdellaoui et al. (2011) and Baillon et al. (2025), addressing ambiguity and subjective perceptions of uncertainty.

📝 Essential Points

• Rational decision-making in economics relies on the completeness and transitivity of preferences, which allow preferences to be represented by a utility function (theorem).
• The expected utility model explains decision-making under risk, emphasizing that individuals value outcomes based on utility, which often exhibits diminishing marginal utility (Bernoulli, 1738).
• The homo economicus assumption simplifies analysis by presuming individuals consistently maximize their utility, acting in their best interest based on well-defined preferences.
• Source Theory expands expected utility by incorporating how individuals subjectively transform probabilities, capturing behaviors like ambiguity aversion (Abdellaoui et al., 2011; Baillon et al., 2025).
• These assumptions underpin many economic models, including financial market theories, and are critical for predicting consumer behavior under risk and uncertainty.

💡 Key Takeaway

Rational decision-making assumes consumers have consistent, well-defined preferences that can be represented by a utility function, and they evaluate risky choices based on expected utility, though real behavior may deviate due to subjective perceptions of uncertainty.

📖 10. Expected Utility Theory

🔑 Key Concepts & Definitions

• Expected Utility (EU) (Bernoulli, 1738): A model where individuals evaluate risky prospects by calculating the probability-weighted average of their utility outcomes, rather than their monetary outcomes. It explains why people value money non-linearly and make consistent choices under risk.

• Utility Function (U): A mathematical representation of individual preferences, assigning real numbers to options such that more preferred options have higher utility. It captures the subjective value of different consumption bundles or outcomes.

• Completeness and Transitivity (Preferences): Assumptions ensuring that individuals can always compare options (completeness) and that their preferences are consistent across choices (transitivity). These conditions guarantee preferences can be represented by a utility function (see section 2).

• Concave Utility Function: A utility function that exhibits diminishing marginal utility, meaning each additional unit of a good provides less additional utility. This reflects risk aversion and is central to expected utility theory.

• Willingness-to-Pay (WTP): The maximum amount an individual is willing to pay for an additional unit of a good or outcome without decreasing their overall utility, often interpreted as the marginal utility of that good (see quasilinear utility).

📝 Essential Points

• Expected Utility Theory (Bernoulli, 1738) posits that individuals evaluate risky choices by calculating the expected utility, which accounts for their subjective valuation of outcomes, not just monetary gains.

• Utility functions are used to represent preferences, with the completeness and transitivity assumptions ensuring these preferences can be mapped onto a utility scale (see section 2). This allows for consistent decision-making under risk.

• Risk preferences are captured by the shape of the utility function: concavity indicates risk aversion, linearity indicates risk neutrality, and convexity indicates risk seeking.

• Expected utility helps explain behaviors such as insurance purchase, gambling, and investment choices, where individuals weigh potential outcomes by their probabilities and subjective utilities.

• Bernoulli's paradox demonstrates that people do not value money linearly; instead, they value utility, which can be modeled with functions like the logarithm, preventing infinite expected utility in certain gambles.

• Source theory (see sources by Abdellaoui et al., 2011; Baillon et al., 2025) extends expected utility by incorporating ambiguity and subjective probability transformations, providing a richer framework for decision-making under uncertainty.

💡 Key Takeaway

Expected Utility Theory provides a foundational model for understanding how individuals make rational choices under risk by evaluating the subjective utility of outcomes, rather than their monetary values, ensuring consistent and risk-sensitive decision-making.

📊 Synthesis Tables

⚠️ Common Pitfalls & Confusions

1. Confusing expected utility with expected monetary value; utility accounts for risk preferences, not just amounts.
2. Assuming marginal utility remains constant; it actually diminishes with increased consumption.
3. Misinterpreting Willingness-to-Pay as the same as price; in quasilinear utility, it equals marginal utility, but not necessarily in other forms.
4. Overlooking the completeness and transitivity assumptions, which are essential for utility representation.
5. Mistaking the law of demand as solely price-driven; it fundamentally stems from diminishing marginal utility.
6. Ignoring risk aversion effects in expected utility models; risk-averse consumers have concave utility functions.
7. Assuming indifference curves are always linear; they are typically convex due to diminishing MRS.
8. Confusing utility maximization with profit maximization; utility maximization involves preferences, not monetary profit.
9. Overgeneralizing quasilinear utility; it simplifies analysis but may not reflect real preferences when income effects are significant.
10. Misunderstanding expected utility theory as only relevant under risk; it also underpins general consumer choice modeling.

✅ Exam Checklist

• Know von Neumann & Morgenstern's definition of the utility function and its role in representing preferences.
• Understand Arrow's assumptions of completeness and transitivity and their importance for utility representation.
• Be able to explain Bernoulli's expected utility theory and its application to decision-making under risk.
• Recognize the concept of Willingness-to-Pay (WTP) as the maximum price a consumer is willing to pay, especially in quasilinear utility models.
• Derive the condition for utility maximization where marginal utility equals price (MU = p).
• Describe how diminishing marginal utility leads to the downward-sloping demand curve.
• Understand the concept of the Marginal Rate of Substitution (MRS) and its equality to the price ratio at the consumer's optimum.
• Know the form and implications of quasilinear utility functions and how they simplify WTP and demand analysis.
• Explain the role of expected utility in modeling choices under risk, including risk aversion and concave utility functions.
• Be able to interpret indifference curves and their convexity due to diminishing MRS.
• Recognize the law of demand as a consequence of decreasing marginal utility.
• Master the differences between utility, expected utility, and monetary value in consumer decision-making.