Consumers and Incentives

Introduction

This chapter explores the incentives that motivate consumer behavior, focusing on how consumers decide how much of each good and service to buy. The analysis covers the buyer’s problem, the derivation of the demand curve, the measurement of consumer surplus, and the consumers' responsiveness to changes in prices and income (elasticities).

• Key Terms: Budget set, budget constraint, consumer surplus, elasticity, price elasticity of demand, arc elasticity, indifference curve, utility, income effect, substitution effect, normal good, inferior good, cross-price elasticity, income elasticity.

The Buyer’s Problem

Defining the Buyer’s Problem

The buyer’s problem is to choose the quantities of goods and services that maximize the consumer’s benefit (utility), given their preferences, the prices of goods, and their income.

• Mathematical Formulation:

  • Maximize utility:

  • Subject to the budget constraint:

  • Where is utility, and re quantities of goods 1 and 2, and e their prices, and is income.

• Utility: A measure of satisfaction or benefit derived from consuming goods and services, shaped by individual preferences.

Budget Set and Budget Constraint

• Budget Set: All combinations of goods that a consumer can afford:

• Budget Constraint: The set of bundles that exhaust the budget:

• Opportunity Cost: The value of the next best alternative forgone. For good 1, the opportunity cost is units of good 2.

• Graphical Representation: The budget constraint is a straight line in space, with slope and intercepts.

Changes in the Budget Constraint

• Price Changes: A change in the price of a good rotates the budget constraint around the intercept of the other good.

• Income Changes: A change in income shifts the budget constraint outward (increase) or inward (decrease) in a parallel fashion.

• Example If, the intercepts are $60 (horizontal, good b), and the slope is.

Indifference Curves and Utility

Indifference Curves

Indifference curves represent all combinations of two goods that provide the same level of utility to the consumer.

• Properties:

  • Indifference curves never cross.

  • Higher indifference curves represent higher utility.

  • The slope of an indifference curve is the marginal rate of substitution between the two goods.

• Interpretation: If two bundles are on the same indifference curve, the consumer is indifferent between them.

Solving the Buyer’s Problem

Optimization Rule for Buyers

The optimal consumption bundle is found where the highest attainable indifference curve is tangent to the budget constraint.

• Mathematical Condition: where the marginal benefit of good

• Interpretation: The consumer allocates spending so that the marginal benefit per dollar is equalized across goods, and the budget is exhausted.

• Graphical Solution: The tangency point between the budget constraint and an indifference curve determines the optimal bundle.

Decomposing the Effects of a Price Change

• Substitution Effect: The change in quantity demanded due to a change in relative prices, holding utility constant.

• Income Effect: The change in quantity demanded due to the change in purchasing power from a price change.

• Total Effect: The sum of the substitution and income effects.

• Example: If the price of good 2 falls, the consumer substitutes away from good 1 to good 2 (substitution effect) and can afford more of both goods (income effect).

Tabular Approach to the Buyer’s Problem

Consumers can also solve the optimization problem by comparing marginal benefit per dollar for each good and allocating their budget accordingly.

Example: With income p_{milk} = p_{cookie} = $2, the optimal bundle is 2 milk and 8 cookies, where marginal benefit per dollar is equalized, and the budget is exhausted.

From the Buyer’s Problem to the Demand Curve

Solving the buyer’s problem at different prices for a good traces out the individual demand curve. The demand curve shows the quantity demanded at each possible price, holding preferences, other prices, and income constant.

Consumer Surplus

Definition and Measurement

• Consumer Surplus (CS): The difference between a consumer’s willingness to pay (marginal benefit) and the price actually paid.

• Formula: for an individual unit.

• Market Consumer Surplus: The area between the demand curve and the market price, up to the quantity purchased.

• Graphical Calculation: For a linear demand curve, where the base is the quantity and the height is the difference between the highest willingness to pay and the market price.

• Example: If the market price of bananas is CS = for that unit. For the market, if the highest WTP is.

Elasticities

Definition of Elasticity

• Elasticity: Measures the percentage change in one variable in response to a percentage change in another variable.

• General Formula:

• Percentage Change:

Price Elasticity of Demand

• Definition: Measures how responsive quantity demanded is to a change in price.

• Formula:

• Law of Demand: Price elasticity of demand is always nonpositive (usually negative).

Point and Arc Elasticities

• Point Elasticity:

• Arc Elasticity (Midpoint Formula):

• Arc elasticities are preferred for larger changes and are invariant to the direction of the change.

Interpreting Elasticity Values

• |ε| > 1: Demand is elastic (buyers are highly responsive to price changes).

• |ε| = 1: Demand is unit elastic (proportional response).

• 0 < |ε| < 1: Demand is inelastic (buyers are not very responsive).

• |ε| = ∞: Perfectly elastic demand (horizontal demand curve).

• ε = 0: Perfectly inelastic demand (vertical demand curve).

Elasticity and Revenue

• Revenue:

• Elastic Demand: Price increase decreases revenue; price decrease increases revenue.

• Inelastic Demand: Price increase increases revenue; price decrease decreases revenue.

• Unit Elastic: Revenue is unchanged by price changes.

• Perfectly Elastic: Any price increase eliminates all sales.

• Perfectly Inelastic: Revenue changes only with price, as quantity is fixed.

Determinants of Price Elasticity of Demand

• Availability of Substitutes: More substitutes make demand more elastic.

• Budget Share: Goods that take up a larger share of the budget have more elastic demand.

• Time Horizon: Demand is more elastic in the long run as consumers adjust.

Cross-Price Elasticity of Demand

• Definition: Measures how the quantity demanded of one good responds to a change in the price of another good.

• Formula:

• Interpretation: (substitutes), (complements), (unrelated goods).

Income Elasticity of Demand

• Definition: Measures how quantity demanded responds to changes in income.

• Formula:

• Interpretation:

  • : Inferior good

  • : Normal good

  • : Necessity

  • : Luxury

Summary Table: Types of Elasticity

Summary

• Consumers maximize utility subject to their budget constraint.

• The optimal bundle is where the marginal benefit per dollar is equalized across goods, and the budget is exhausted.

• The demand curve is derived by solving the buyer’s problem at all price levels.

• Consumer surplus measures the benefit to consumers from market transactions.

• Elasticities quantify how responsive consumers are to changes in price, income, and prices of related goods.