Consumer Theory I: Utility, Preferences, and Choice

Consumer Theory I

Introduction

Consumer theory examines how individuals make choices about what goods and services to purchase, given their preferences and budget constraints. These decisions share similarities with the producer's optimization problem, particularly in the use of tangency conditions, but here the focus is on the consumer's budget line. This topic covers the consumer's problem, utility and demand functions, income and substitution effects, and the Slutsky and Hicks substitution effects.

• Consumer problem: How consumers maximize utility given their budget.

• Utility and demand functions: Mathematical representation of preferences and demand.

• Income and substitution effects: How changes in income and prices affect demand.

• Slutsky substitution effect: Decomposition of price effects holding purchasing power constant.

• Hicks substitution effect: Decomposition of price effects holding utility constant.

Utility and Choice

The central consumer problem is to maximize a utility function by choosing the quantities of two goods to consume, subject to a limited income or budget constraint.

• Utility function: Represents the satisfaction or happiness a consumer derives from consuming bundles of goods.

• Budget constraint: The consumer cannot spend more than their available income.

Utility Functions

A utility function is a way of representing or describing consumer preferences. It allows us to determine which bundle of goods is preferred over another. The actual number of utility units ("utils") is not important; what matters is the ranking of bundles.

• Ordinal utility: Only the order of preferences matters, not the magnitude.

• Indifference curve (IC): A curve representing all bundles that provide the same level of utility.

• Utility is constant along a given IC; higher ICs represent higher utility levels.

• Bundles with more total amounts of goods correspond to higher indifference curves.

• ICs are typically convex to the origin, reflecting diminishing marginal rates of substitution.

Examples of utility functions:

• Cobb-Douglas utility function: $U(x_1, x_2) = x_1^a x_2^b$

• Linear utility function (perfect substitutes): $U(x_1, x_2) = a x_1 + b x_2$

• Quasilinear utility function: $U(x_1, x_2) = f(x_1) + x_2$

• Perfect complements utility function: $U(x_1, x_2) = ext{min}igig(a x_1, b x_2igig)$

Constructing a Utility Function from Indifference Curves

Indifference curves can be used to construct a utility function by assigning utility levels to each curve. The distance from the origin measures the amount of goods in each bundle.

• Each indifference curve corresponds to a different utility level.

• Higher curves represent higher utility.

Perfect Substitutes

Perfect substitutes are goods for which the consumer is willing to substitute one for the other at a constant rate. The utility function for perfect substitutes is linear.

• Utility function: $U(x_1, x_2) = a x_1 + b x_2$

• Indifference curves are straight lines with a constant slope.

• Example: If a consumer is willing to trade 1 unit of good 1 for 1 unit of good 2, then $a = b = 1$.

Quasilinear Preferences

Quasilinear preferences are characterized by indifference curves that are vertically shifted versions of one another. The utility function is linear in one good and (possibly) non-linear in the other.

• Utility function: $U(x_1, x_2) = f(x_1) + x_2$

• Indifference curves are parallel shifts in the direction of the linear good.

• Quasilinear utility is not always realistic but is analytically convenient.

Perfect Complements

Perfect complements are goods that are always consumed together in fixed proportions. The utility function for perfect complements is based on the minimum amount of each good.

• Utility function: $U(x_1, x_2) = ext{min}(a x_1, b x_2)$

• Indifference curves are L-shaped, reflecting the fixed ratio of consumption.

• Example: Left and right shoes, or sugar and tea.

Marginal Utility (MU)

Marginal utility is the additional satisfaction gained from consuming one more unit of a good. It is calculated as the partial derivative of the utility function with respect to the good in question.

• Marginal utility of good 1: $MU_1 = \frac{\partial U}{\partial x_1}$

• Marginal utility of good 2: $MU_2 = \frac{\partial U}{\partial x_2}$

• Marginal utilities are typically positive: increased consumption increases utility.

Marginal Rate of Substitution (MRS)

The marginal rate of substitution (MRS) is the rate at which a consumer is willing to trade one good for another while maintaining the same level of utility. It is the negative slope of the indifference curve.

• Formula: $MRS = -\frac{MU_1}{MU_2}$

• MRS diminishes as the consumer substitutes one good for another.

Optimal Choice

The consumer's optimal choice is the bundle of goods that maximizes utility subject to the budget constraint. This occurs at the point of tangency between the highest attainable indifference curve and the budget line.

• Budget constraint: $p_1 x_1 + p_2 x_2 = m$

• Optimality condition: $MRS = \frac{p_1}{p_2}$

• All income is spent at the optimal bundle.

Demand Functions

Given a utility function and budget constraint, the demand function for each good can be derived. The demand function shows how the quantity demanded of a good depends on prices and income.

• Demand function for good 1: $x_1^* = f(p_1, p_2, m)$

• Demand function for good 2: $x_2^* = g(p_1, p_2, m)$

Comparative Statics: Changes in Income and Prices

Comparative statics analyze how demand changes in response to changes in income and prices.

• Normal goods: Demand increases as income increases.

• Inferior goods: Demand decreases as income increases.

• Income offer curve: Shows how optimal consumption changes as income changes.

• Price-consumption curve: Shows how optimal consumption changes as the price of a good changes.

Income and Substitution Effects

When the price of a good changes, the effect on demand can be decomposed into the substitution effect and the income effect.

• Substitution effect: Change in consumption due to a change in relative prices, holding utility constant.

• Income effect: Change in consumption due to a change in purchasing power, holding relative prices constant.

Slutsky Substitution Effect

The Slutsky substitution effect isolates the change in demand due to a change in price, holding the original bundle just affordable by adjusting income.

• Slutsky equation (discrete change): $x_1(p_1', m') - x_1(p_1, m) = [x_1(p_1', m') - x_1(p_1', m)] + [x_1(p_1', m) - x_1(p_1, m)]$

• The first term is the substitution effect; the second is the income effect.

Hicks Substitution Effect

The Hicks substitution effect isolates the change in demand due to a change in price, holding utility constant (compensated demand).

• Consumer is compensated to keep utility constant as price changes.

• Indifference curves are used to illustrate the effect.

Signs and Sizes of Substitution and Income Effects

The direction and magnitude of substitution and income effects depend on the type of good.

• Normal good: Both substitution and income effects are positive when price falls.

• Inferior good: Substitution effect is positive, income effect is negative when price falls.

• Giffen good: The negative income effect outweighs the positive substitution effect, leading to an upward-sloping demand curve.

Type of Good	Substitution Effect	Income Effect	Total Effect
Normal Good	Positive	Positive	Positive
Inferior Good	Positive	Negative	Depends on relative size
Giffen Good	Positive	Negative (larger)	Negative

Example: If the price of good 1 falls and both goods are normal, the consumer substitutes towards good 1 (substitution effect) and increases consumption due to higher purchasing power (income effect). If good 1 is inferior, the substitution effect is still positive, but the income effect is negative, partially offsetting the total effect.

Additional info: Some diagrams referenced in the notes (e.g., indifference curves, budget lines) are standard in microeconomics and can be found in most textbooks. The Slutsky and Hicks effects are typically illustrated with shifts in budget lines and indifference curves.