Consumer Theory: Preferences, Utility Functions, and Indifference Curves

Consumer Theory: Preferences, Utility Functions, and Indifference Curves

Introduction to Consumer Theory

Consumer theory is a foundational topic in microeconomics that analyzes how individuals make choices given their preferences and constraints. It provides the basis for understanding demand, market behavior, and welfare analysis.

• Consumer Choice Model: Examines how consumers allocate resources among different goods to maximize their satisfaction.

• Goods and Bundles: A bundle refers to a specific combination of quantities of goods consumed.

• Preferences: The ranking of bundles according to the consumer's satisfaction.

Types of Goods and Consumption Sets

Goods can be classified based on their consumption possibilities and the mathematical representation of bundles.

• Consumption Set: The set of all possible bundles a consumer can choose from.

• Boundary Bundle: A bundle where at least one good is consumed at its minimum (often zero).

• Interior Bundle: A bundle where all goods are consumed in positive amounts.

Examples of Consumption Sets

• Discrete Goods: Bundles are represented by integer values (e.g., number of cars).

• Continuous Goods: Bundles can take any non-negative real value (e.g., liters of water).

Preference Relations

Preferences are formalized using relations that compare bundles.

• "Better than" Relation (≻): Bundle x is strictly preferred to bundle y.

• "Equally preferred" Relation (∼): Bundle x is considered just as good as bundle y.

• "At least as good as" Relation (≽): Bundle x is at least as good as bundle y.

Completeness and Rationality of Preferences

Rational preferences are assumed to be complete (any two bundles can be compared) and transitive (if x ≽ y and y ≽ z, then x ≽ z).

• Complete Preferences: For any two bundles, the consumer can state a preference or indifference.

• Transitive Preferences: Ensures consistency in choices.

Utility Functions

Utility functions assign a numerical value to each bundle, representing the consumer's level of satisfaction.

• Ordinal Utility: Only the ranking of bundles matters, not the actual numbers.

• Cardinal Utility: The magnitude of utility differences is meaningful (less common in microeconomics).

Common Utility Functions

Utility Model	Utility Function (all parameters > 0)
Linear (Perfect Substitutes)	$U = a x_1 + b x_2$
Cobb-Douglas	$U = x_1^a x_2^b$
Leontief (Perfect Complements)	$U = ext{min}(a x_1, b x_2)$
CES (Constant Elasticity of Substitution)	$U = [a x_1^r + b x_2^r]^{1/r}$

Indifference Curves

An indifference curve represents all bundles that provide the same level of utility to the consumer.

• Properties: Indifference curves are downward sloping (for normal goods), do not cross, and are convex to the origin (for strictly convex preferences).

• Boundary vs. Interior Bundles: The shape and slope of indifference curves can change at boundaries.

Marginal Rate of Substitution (MRS)

The MRS measures the rate at which a consumer is willing to substitute one good for another while maintaining the same utility.

• Formula: $MRS = -\frac{MU_1}{MU_2}$, where $MU_1$ and $MU_2$ are the marginal utilities of goods 1 and 2.

• Interpretation: The absolute value of the slope of the indifference curve at a given point.

Classification of Goods

Goods can be classified based on how their consumption affects utility.

Type of Good	Utility Function	MRS
Good Good	$U = x_1 + x_2$	Negative
Bad Good	$U = x_1 - x_2$	Positive
Neutral Good	$U = x_1$	Zero

Examples and Applications

• Perfect Substitutes: $U = a x_1 + b x_2$; indifference curves are straight lines.

• Perfect Complements: $U = ext{min}(a x_1, b x_2)$; indifference curves are L-shaped.

• Cobb-Douglas: $U = x_1^a x_2^b$; indifference curves are smooth and convex.

• CES Utility: $U = [a x_1^r + b x_2^r]^{1/r}$; allows for varying degrees of substitutability.

Practice Problems and Solutions

The notes include a variety of practice problems involving utility functions, indifference curves, and the classification of goods. Solutions demonstrate how to compute MRS, interpret utility functions, and analyze consumer preferences.

• Example: Given $U = x_1^2 + x_2^2$, find the indifference curve for a fixed utility level $u_0$.

• Solution: $x_1^2 + x_2^2 = u_0$; this is the equation of a circle in the $x_1$-$x_2$ plane.

Summary Table: Utility Models and Properties

Model	Utility Function	MRS	Indifference Curve Shape
Perfect Substitutes	$U = a x_1 + b x_2$	Constant	Straight line
Perfect Complements	$U = ext{min}(a x_1, b x_2)$	Undefined at corner	L-shaped
Cobb-Douglas	$U = x_1^a x_2^b$	Decreasing	Convex
CES	$U = [a x_1^r + b x_2^r]^{1/r}$	Varies	Convex

Additional info:

• Notes include graphical illustrations of indifference curves, utility surfaces, and boundary/interior bundles.

• Practice problems cover both theoretical and applied aspects of consumer theory, including real-world applications (e.g., broadband service characteristics).

• Advanced topics such as the Young Economist's Algorithm for plotting indifference curves and the interpretation of MRS in various contexts are discussed.