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Intermediate Microeconomics Unit 3: Utility Maximization and Demand

Putting preferences and the budget together gives the consumer's optimal choice; solving that problem for general prices and income produces demand functions, which are the basis for every market model.

Covers: Tangency condition and the Lagrangian, Cobb-Douglas demand, Substitutes, complements and corner solutions, Quasilinear demand, Engel curves and income elasticity, Price elasticity and offer curves, Revealed preference

Study guide

A consumer picks the best bundle she can afford. With smooth, convex preferences and both goods bought, the optimum is a tangency: the indifference curve and the budget line have the same slope, so MRS = p₁/p₂, and the bundle is on the budget line. Equivalently, the marginal utility per dollar is the same for both goods, MUₓ/p₁ = MU_y/p₂. The Lagrangian method writes ℒ = u(x, y) + λ(m − p₁x − p₂y) and sets its partial derivatives to zero. The multiplier λ is the marginal utility of income: how much utility one more dollar would add. If MUₓ/p₁ is larger than MU_y/p₂, the consumer can do better by shifting spending toward x.

Solving the problem for any prices and income yields demand functions. For Cobb-Douglas u = x^a y^b, the consumer spends the fraction a/(a + b) of income on x, so x = [a/(a + b)]m/p₁. Shares are constant, own-price elasticity is −1, income elasticity is 1, and the other good's price does not affect x. For perfect complements min{ax, by}, the consumer buys at the corner ax = by, so demand is found by plugging that ratio into the budget. For perfect substitutes the consumer compares MRS with p₁/p₂ and usually buys only one good, a corner solution.

Quasilinear utility u = v(x) + y gives demand for x that depends only on prices, x solving v′(x) = p₁/p₂, as long as income is large enough to buy that amount. Extra income goes entirely to y. If income is too low, the tangency would require negative y, and the consumer instead spends everything on x. Always check whether a tangency is feasible; corner solutions also arise with perfect substitutes, non-convex preferences and kinked budget lines, where you compare the best point on each segment.

Demand functions are described by how they respond to their arguments. An Engel curve plots x against income; normal goods have positive income effects, inferior goods negative ones, and luxuries have income elasticity above 1. The price offer curve traces optimal bundles as one price changes, and the demand curve plots x against its own price. Revealed preference works backward from choices: if A was chosen when B was affordable, A is revealed preferred to B. The weak axiom says B can then never be revealed preferred to A, a testable restriction on data that any utility maximizer must satisfy.

Key ideas

• An interior optimum needs MRS = p₁/p₂ and the budget to hold with equality.
• Equivalently, MUₓ/p₁ = MU_y/p₂; λ is the marginal utility of income.
• Cobb-Douglas demand x = [a/(a + b)]m/p₁ has fixed budget shares, unit elasticities and zero cross-price effects.
• Perfect complements are bought at the kink; perfect substitutes usually give a corner solution.
• Quasilinear demand for x does not depend on income unless the consumer is at a corner.
• Engel curves, offer curves and demand curves show how optimal choices change with income and prices.
• Revealed preference tests whether choices are consistent with some utility function.

Formulas and rules

Common mistakes

• Using Cobb-Douglas exponents as shares without dividing by their sum.
• Applying the tangency rule to perfect substitutes or complements, where it does not hold.
• Reporting a quasilinear tangency that requires negative spending on the other good.
• Calling a good with income elasticity between 0 and 1 inferior; it is a normal necessity.
• Checking WARP with only one set of prices instead of pricing both bundles in both situations.

Before the test, make sure you can

• Solve a two-good utility maximization problem with the tangency condition or a Lagrangian.
• Derive demand functions for Cobb-Douglas, perfect complements, perfect substitutes and quasilinear utility.
• Identify corner solutions and check whether a tangency is feasible.
• Calculate income, own-price and cross-price elasticities from a demand function.
• Classify goods as normal, inferior, luxury or necessity from data or a demand function.
• Interpret λ as the marginal utility of income.
• Test a set of observed choices for consistency with WARP.

Words to know

Optimal bundle

The most preferred bundle on or inside the budget line.

Tangency condition

At an interior optimum, MRS equals the price ratio p₁/p₂.

Equal marginal principle

At an optimum the marginal utility per dollar is the same for every good bought.

Lagrange multiplier λ

In the consumer problem, the marginal utility of income at the optimum.

Corner solution

An optimum where the consumer buys none of at least one good.

Demand function

Optimal quantity of a good as a function of prices and income.

Engel curve

Quantity demanded plotted against income, with prices held fixed.

Income offer curve

The path of optimal bundles as income changes with prices fixed.

Price offer curve

The path of optimal bundles as one price changes with income fixed.

Normal good

A good whose demand rises with income.

Inferior good

A good whose demand falls as income rises.

Luxury

A normal good with income elasticity greater than 1.

Income elasticity

Percentage change in quantity demanded divided by the percentage change in income.

Own-price elasticity

Percentage change in quantity demanded divided by the percentage change in the good's own price.

Gross substitutes

Goods for which a rise in one price raises demand for the other.

Revealed preference

Inference that a chosen bundle is preferred to any other bundle that was affordable at the time.

Weak axiom of revealed preference

If A is directly revealed preferred to B, then B is never directly revealed preferred to A.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.