Consumer Optimization and Demand

What happens when your tastes meet your budget? Every day you choose what to buy and how much to save. Behind those choices is a balancing act: you want many things, but you have limited money. This chapter builds a clear model of that balancing act. It shows how a rational consumer picks the best combination of goods with a limited income, and how those choices change when prices or income change.

The Big Picture#

Imagine a consumer with clear preferences (captured by a utility function) and a fixed amount of money. Their goal is a constrained optimization problem: choose the combination of goods that gives the highest satisfaction without overspending. Solving this gives demand functions—for any prices and income, they tell us exactly what the consumer will buy. But that’s only half the story. By looking at the problem from the opposite side—finding the cheapest way to reach a certain level of satisfaction—we get a second, compensating view of demand (holding satisfaction constant). Together, these two views let us split a price change into two separate effects: the substitution effect (you change the mix because relative prices changed) and the income effect (your real purchasing power changed). Understanding this split helps us predict when a price hike reduces demand—and when, in rare cases, it might not.

The Budget Set and the Consumer’s Problem#

We begin by listing the consumer’s options. Suppose there are two goods—the ideas work for many goods. The consumer has money (income) to spend. Goods cost and each. A consumption bundle is a pair showing how much of each good they buy. The budget constraint says total spending cannot exceed income:

The budget set is all bundles that satisfy this rule. Usually we assume the consumer spends everything—leftover money gives no happiness—so the constraint is an equality: . This is the budget line, a straight line with slope .

Preferences are captured by a utility function . The numbers only rank bundles; bigger numbers mean more satisfaction. We assume “more is better” (monotonicity), so having more of a good always raises utility. We also assume indifference curves (sets of bundles giving equal utility) are smooth and bend inward (convex).

The consumer’s problem is to maximize subject to the budget constraint. In mathematical form:

We solve this with calculus. The solution conditions lead to a tangency rule:

The left side is the marginal rate of substitution (MRS)—how much of good 2 you would give up to get one more unit of good 1 and stay equally happy. The right side is the market trade‑off (the price ratio). At the best bundle, your personal trade‑off equals the market trade‑off. Combining this with the budget equation gives the exact quantities and .

Constrained optimization: The process of finding the best choice (highest utility) within the limits set by prices and income.

📝 Section Recap: The consumer’s problem is to maximize utility given the budget constraint. The optimal bundle satisfies two conditions: the MRS equals the price ratio, and the budget is fully spent.

Marshallian Demand Functions#

When we solve the problem for all possible prices and income, we get functions that give the best quantities to buy. These are called Marshallian (or ordinary) demand functions:

They tell us the best choice for any situation.

Marshallian demand: The function that gives the utility‑maximizing bundle of goods for given prices and income .

Marshallian demands have a few important properties that reflect the underlying optimization.

Homogeneity of degree zero. If all prices and income are scaled by the same factor , the budget set doesn’t change: . Since the problem is identical, the best bundle stays the same:

In plain terms: pure inflation (all prices and income doubling) doesn’t change what you buy.

Adding‑up (Walras’ law). By construction, the demands use up all the income. So for any and ,

This is the adding‑up condition. It just means the budget constraint holds exactly.

Example: Cobb‑Douglas preferences.
Suppose with . The tangency condition gives , so . Using the budget , we solve:

Check homogeneity: doubling both prices and leaves unchanged. Adding‑up: . These demands are straightforward, and we’ll revisit them in the dual view.

Marshallian demands are what we can observe—when prices or income change, these functions tell us the new chosen bundle.

📝 Section Recap: Marshallian demands give the optimal consumption bundle as functions of prices and income. They are homogeneous of degree zero (unaffected by proportional changes in all prices and income) and always satisfy the budget constraint exactly.

Expenditure Minimization and Hicksian Demand#

There is a mirror‑image problem that is really useful. Instead of maximizing utility for a given budget, ask: what is the cheapest way to reach a particular level of utility ? This is expenditure minimization:

At the best choice, the utility constraint holds with equality (you wouldn’t spend more than needed). The solution gives Hicksian (or compensated) demand functions:

Hicksian demand: The cheapest bundle that achieves a target utility level at given prices. It is “compensated” because we imagine adjusting income to keep utility constant.

Plugging these optimal quantities back into the cost gives the expenditure function:

This function shows the minimum amount of money needed to reach utility at those prices.

A striking result connects the expenditure function to Hicksian demand:

Shephard’s lemma: The partial derivative of the expenditure function with respect to a good’s price equals the Hicksian demand for that good:

Think of it this way: if the price of a good rises a tiny bit, the minimum cost to keep the same utility rises by exactly the amount you were buying of that good, because you would not change your bundle much right away.

Duality link. The two problems are closely connected. The indirect utility function gives the maximum utility from income . Then:

In words: ordinary demand at income is exactly the compensated demand at the utility level that can just afford. And compensated demand at utility is the ordinary demand you would choose if your income were exactly the minimum needed to reach .

Example, continued. For , solving the expenditure minimization problem gives

and

Differentiate with respect to and you get —a concrete Shephard’s lemma example.

📝 Section Recap: The expenditure minimization problem asks for the cheapest way to reach a given utility. Its solution defines Hicksian (compensated) demands and the expenditure function. Shephard’s lemma gives a direct derivative link, and duality ties the two demand concepts together.

The Slutsky Equation: Breaking Down Price Changes#

Now we have two demand concepts: Marshallian (keeping income constant) and Hicksian (keeping utility constant). When the price of, say, good 1 rises, two things happen at once:

Substitution effect: Good 1 becomes relatively more expensive, so you substitute away from it toward other goods, even if we adjust your income to keep your original utility constant.
Income effect: The price rise makes you effectively poorer (your money buys less). This change in real purchasing power further changes your demand for all goods, depending on whether they are normal or inferior.

The Slutsky equation splits the total change in ordinary demand into these two parts. For a small change in on the demand for good :

where is the original utility level.

The left side is the total price effect. On the right, the first term is the substitution effect (the change in compensated demand with utility constant). The second term is the income effect: the change in demand because the price change acts like a change in real income, multiplied by the amount of good originally consumed.

Slutsky equation: The identity that splits the effect of a price change on Marshallian demand into a substitution effect (holding utility constant) and an income effect (accounting for lost purchasing power).

Own‑price effect. For , we have

The substitution effect is always non‑positive (a compensated price rise cannot make you buy more of that good). This is the law of demand for compensated goods. Why? The expenditure function is concave in prices, and its second derivative .

Fundamental law of demand (compensated): If you compensate a consumer to keep utility constant, a rise in a good’s own price will never increase the quantity demanded of that good.

The substitution matrix. The set of all compensated price derivatives has two key properties:

Symmetry: . This comes from Shephard’s lemma and the fact that mixed partial derivatives of the expenditure function are equal.
Negative semidefiniteness: The matrix of is negative semidefinite, which means own‑substitution effects are and compensated demand curves cannot slope upward.

Symmetry of the substitution matrix: The effect of a change in the price of good on the compensated demand for good equals the effect of a price change of good on the compensated demand for good .

These are logical consequences of optimization, not assumptions.

Giffen goods. Usually demand curves slope down: a higher price reduces quantity demanded. The Slutsky equation shows this can fail if the income effect is strongly positive (the good is inferior) and large enough to outweigh the substitution effect. If , we have a Giffen good—demand rises when price rises because the income effect dominates. The classic story is the Irish potato: when potato prices rose, poor households cut back on meat and bought even more potatoes just to survive. Giffen goods are rare and require a very strong inferior good with a large budget share, but they show the power of the Slutsky decomposition.

In most real‑world cases, the substitution effect dominates, and demand slopes downward—the familiar pattern you see every day.

📝 Section Recap: The Slutsky equation separates a price change into a pure substitution effect (utility‑constant) and an income effect (real‑income change). Compensated demand always obeys the law of demand: own‑price substitution effects are negative or zero. Giffen goods are the rare case where a positive income effect overwhelms the substitution effect, making demand rise with price.

Summary#

We have a complete model of consumer choice. Utility maximization gives us Marshallian demand: what a consumer will buy given prices and income. Expenditure minimization gives us Hicksian demand and the expenditure function, which have useful properties like symmetry. The Slutsky equation links them—it splits any price change into a substitution effect and an income effect. This helps us predict exactly how people adjust when prices or income change.

Key idea	What it means (plain English)	Why it matters
Budget constraint	The rule that total spending can’t exceed income: .	It shows which combinations of goods you can afford.
Utility maximization	Picking the affordable bundle that gives the most happiness.	It gives us the Marshallian demand functions we use to predict behavior.
Marshallian demand	The function showing the best bundle for any prices and income.	We can observe it in principle; it’s the workhorse of demand analysis.
Homogeneity of degree zero	Multiplying all prices and income by the same number doesn’t change the best choice.	Pure inflation doesn’t alter real decisions.
Adding‑up (Walras’ law)	The total spent on demanded goods always equals income: .	It ensures the numbers are consistent with the budget.
Expenditure minimization	Finding the lowest‑cost way to reach a certain happiness level.	Introduces compensated demand and the expenditure function.
Hicksian (compensated) demand	The cheapest bundle that gives a particular happiness level at given prices.	Keeps utility constant, so it shows pure substitution responses.
Expenditure function	: the minimum money needed to reach utility .	A building block for welfare analysis and the Slutsky split.
Shephard’s lemma	: a small price rise increases the minimum cost by the amount you buy.	Links the math of the expenditure function directly to demand.
Slutsky equation	.	Splits a price effect into substitution and income effects—essential for understanding demand slopes.
Substitution effect	Change in demand because a good became relatively more or less expensive, keeping happiness the same.	Always non‑positive for own‑price changes; the core of the law of demand.
Income effect	Change in demand because the price change made you feel richer or poorer.	Explains why inferior goods can sometimes break the “law of demand.”
Symmetry of substitution matrix	: cross‑price effects are mirrored.	A neat theoretical restriction that follows from optimization.
Negative semidefiniteness	Own‑substitution effects never positive; compensated demand curves don’t slope up.	Guarantees the law of demand for compensated goods.
Giffen good	A good where a higher price makes people buy more, because the income effect overpowers the substitution effect.	A rare but important exception that highlights the power of the Slutsky split.