Difficulty: Intermediate | Prerequisites: Basic calculus (partial derivatives, chain rule), introductory microeconomics (supply and demand, budget constraints).
Consumer theory is the backbone of intermediate microeconomics. It models how a rational individual allocates a limited budget across goods to maximise satisfaction, and it provides the formal machinery behind every demand curve you have seen since introductory economics. If you missed the early weeks on budget constraints and indifference curves, revisit those before diving in here, because the optimisation techniques below assume you already know what a budget line and a utility function represent.
This set of notes covers constrained optimisation via Lagrange, Marshallian demand functions, the Slutsky decomposition of price effects into substitution and income components, the distinction between normal and inferior goods, and the expenditure function. Together these tools let you predict how a consumer's choices shift when prices or income change, and they form the analytical core of most ECO 300 midterms.
A consumer with a Cobb-Douglas utility function splits income in fixed proportions across goods. You find those proportions using the Lagrange method, then decompose any price change into a substitution effect (movement along an indifference curve) and an income effect (shift to a new curve) via the Slutsky equation. The expenditure function tells you the minimum cost of reaching a given utility level.
Utility function
A mathematical representation of a consumer's preferences over bundles of goods. It assigns a number to each bundle such that preferred bundles receive higher numbers.
In simple terms, it is a scoring system: higher score means the consumer likes that combination more.
Cobb-Douglas utility function
A specific functional form U(X, Y) = X^a · Y^b where a and b are positive constants. It produces smooth, convex indifference curves and guarantees interior solutions.
Think of it as the "well-behaved" utility function that shows up on most exams because the maths works out cleanly.
Marshallian (ordinary) demand function
The quantity of a good a consumer chooses as a function of prices and income, derived from utility maximisation subject to the budget constraint. Written X*(Px, Py, I).
In simple terms, it answers: "Given these prices and this income, how much of each good does the consumer buy?"
Lagrange method (method of Lagrange multipliers)
A technique for solving constrained optimisation problems. You form a Lagrangian L = objective function + λ(constraint), take partial derivatives, set them to zero, and solve the resulting system.
Think of it as the standard recipe for maximising utility when you have a budget to respect.
Lagrange multiplier (λ)
The shadow price of the constraint. It tells you how much the maximum utility would increase if income rose by one unit.
In simple terms, λ is the marginal utility of an extra pound (or dollar) of income.
Slutsky decomposition (Slutsky equation)
A formula that breaks the total effect of a price change on demand into two pieces: the substitution effect (holding utility constant) and the income effect (the change in purchasing power).
Think of it as the "why did demand change?" decomposition: part is because relative prices shifted, part is because the consumer feels richer or poorer.
Substitution effect
The change in quantity demanded due solely to the change in relative prices, holding utility constant. It always works in the opposite direction to the price change (the law of demand for compensated demand).
Income effect
The change in quantity demanded due to the change in real purchasing power caused by a price change. For a normal good, a price increase reduces real income and reduces demand further. For an inferior good, the income effect works in the opposite direction.
Normal good
A good whose demand increases when income increases, holding prices constant. Formally, ∂X*/∂I > 0.
In simple terms, you buy more of it when you get a pay rise.
Inferior good
A good whose demand decreases when income increases. Formally, ∂X*/∂I < 0.
Expenditure function E(Px, Py, U)
The minimum income required to achieve a given utility level U at prices Px and Py. It is the "dual" of the indirect utility function and is derived by substituting Hicksian demands back into the budget constraint.
In simple terms, it answers: "What is the cheapest way to be this happy?"
The consumer's problem is to maximise U(X, Y) = X^0.5 · Y^0.5 subject to the budget constraint Px·X + Py·Y = I.
Step 1: Form the Lagrangian
L = X^0.5 · Y^0.5 + λ(I - Px·X - Py·Y)
Step 2: Take partial derivatives and set them to zero
∂L/∂X = 0.5 · X^(-0.5) · Y^0.5 - λPx = 0
∂L/∂Y = 0.5 · X^0.5 · Y^(-0.5) - λPy = 0
∂L/∂λ = I - Px·X - Py·Y = 0
Step 3: Solve for the optimality condition
Divide the first FOC by the second to eliminate λ:
(Y / X) = Px / Py, which gives Y = (Px / Py) · X
This is the tangency condition: MRS = price ratio.
Step 4: Substitute back into the budget constraint
Px·X + Py · (Px / Py) · X = I
2Px·X = I
X* = I / (2Px)
Substituting back: Y* = I / (2Py)
The key insight with Cobb-Douglas: the exponents determine the income shares. Because the exponents here are equal (both 0.5), the consumer spends exactly half of income on each good.
From the Lagrange solution above, the Marshallian demand functions for this utility function are:
X*(Px, Py, I) = I / (2Px)
Y*(Px, Py, I) = I / (2Py)
Properties to note:
Demand is inversely proportional to own price. Double Px, and the consumer buys half as much X.
Demand is proportional to income. Double I, and the consumer buys twice as much of each good.
No cross-price effects. The demand for X does not depend on Py, and vice versa. This is a special feature of Cobb-Douglas utility with exponents that sum to 1; it does not hold in general.
Homogeneous of degree zero. If you double all prices and income simultaneously, demanded quantities do not change. This reflects the absence of money illusion.
When Px rises by a small amount dPx, the total change in demand for X is:
dX*/dPx = -I / (2Px²)
This is simply the derivative of X* = I/(2Px) with respect to Px.
The Slutsky equation decomposes this total effect:
Total effect = Substitution effect + Income effect
dX/dPx (total) = dX/dPx |_U=const - X* · (dX/dI)
To find the income effect using the Slutsky decomposition:
Income effect = -X* · (∂X*/∂I)
∂X*/∂I = 1/(2Px), so:
Income effect = -(I / 2Px) · (1 / 2Px) = -I / (4Px²)
To find the substitution effect (by subtraction):
Substitution effect = Total effect - Income effect
= -I/(2Px²) - (-I/(4Px²))
= -I/(2Px²) + I/(4Px²)
= -I/(4Px²)
Both effects are negative when the price rises, confirming that the substitution effect and income effect reinforce each other. This is always the case for a normal good.
A good is normal if demand rises with income. Formally, check the sign of ∂X*/∂I.
For the utility function U = X^0.5 · Y^0.5:
X* = I / (2Px), so ∂X*/∂I = 1/(2Px) > 0
Y* = I / (2Py), so ∂Y*/∂I = 1/(2Py) > 0
Both derivatives are strictly positive, so both X and Y are normal goods.
Why this matters for the Slutsky decomposition: for a normal good, the income effect reinforces the substitution effect when the own price rises. Demand unambiguously falls. For an inferior good, the income effect partially offsets the substitution effect, and in the extreme case of a Giffen good, the income effect dominates and demand rises when price rises.
The expenditure function E(Px, Py, U) gives the minimum income needed to achieve utility level U at prices Px and Py.
Derivation steps:
Start from the Marshallian demands and the indirect utility function.
With X* = I/(2Px) and Y* = I/(2Py), substitute into U:
V(Px, Py, I) = (I/2Px)^0.5 · (I/2Py)^0.5 = I / (2 · Px^0.5 · Py^0.5)
Now invert this: set V = U and solve for I:
U = I / (2 · Px^0.5 · Py^0.5)
I = 2U · Px^0.5 · Py^0.5
Therefore: E(Px, Py, U) = 2U · Px^0.5 · Py^0.5
Properties of the expenditure function:
Increasing in U (reaching higher utility costs more).
Increasing in prices (if goods become more expensive, so does reaching a given utility level).
Homogeneous of degree 1 in prices (doubling all prices doubles the expenditure needed).
Concave in prices.
By Shephard's lemma, ∂E/∂Px = Hicksian demand for X.
U(X,Y) = X^{0.5} \cdot Y^{0.5}X^* = \frac{I}{2P_X}, \quad Y^* = \frac{I}{2P_Y}\frac{dX^*}{dP_X} = -\frac{I}{2P_X^2}\text{Slutsky: } \frac{\partial X}{\partial P_X}\bigg|_{\text{total}} = \frac{\partial X}{\partial P_X}\bigg|_{U=\bar{U}} - X^* \cdot \frac{\partial X^*}{\partial I}\text{Income effect} = -\frac{I}{4P_X^2}, \quad \text{Substitution effect} = -\frac{I}{4P_X^2}V(P_X, P_Y, I) = \frac{I}{2 \cdot P_X^{0.5} \cdot P_Y^{0.5}}E(P_X, P_Y, U) = 2U \cdot P_X^{0.5} \cdot P_Y^{0.5}The Cobb-Douglas model with equal exponents predicts that a household spends a fixed fraction of income on each category, regardless of price changes. This roughly matches observed spending patterns for broad categories like food vs non-food in developing economies, where Engel's law describes the income share spent on food declining as income rises (which would require unequal exponents). The expenditure function is used in welfare economics to calculate compensating and equivalent variation, the monetary measures of how much a price change helps or hurts a consumer.
Students often confuse the Marshallian (ordinary) demand with the Hicksian (compensated) demand. Marshallian holds income constant; Hicksian holds utility constant. The Slutsky equation connects the two.
A common error is to assume that the substitution effect can be positive (i.e., that the consumer buys more of a good when it becomes relatively more expensive). It cannot. The substitution effect is always negative for a price increase.
Students sometimes think that the income effect of a price change involves an actual change in nominal income. It does not. The consumer's money income stays the same; what changes is their real purchasing power.
Another mistake is to assume all Cobb-Douglas functions produce the same demand structure. The income shares depend on the exponents: for U = X^a · Y^b, the share spent on X is a/(a+b). Equal exponents are a special case.
⚠️ Lagrange optimisation is the workhorse method for this course. Expect to be asked to derive demand functions from scratch for a given utility function.
⚠️ The Slutsky decomposition is heavily tested. You need to be able to compute the total effect, identify the income and substitution effects separately, and explain their signs.
⚠️ Know how to determine whether a good is normal or inferior from the demand function. The test is the sign of ∂X*/∂I.
⚠️ The expenditure function derivation (inverting the indirect utility function) is a common exam question. Practice going from Marshallian demands to indirect utility to expenditure function fluently.
True or false: For a Cobb-Douglas utility function with equal exponents, the consumer always spends exactly half of income on each good.
True.
Fill in the blank: The Lagrange multiplier λ represents the marginal utility of ________.
Income.
True or false: The substitution effect of a price increase can be positive.
False. It is always negative (consumer substitutes away from the more expensive good).
Fill in the blank: A good is classified as normal if ∂X*/∂I is ________.
Positive.
True or false: The expenditure function is decreasing in prices.
False. It is increasing in prices.
Q: Given U(X,Y) = X^0.5 · Y^0.5, prices Px and Py, and income I, what are the Marshallian demand functions?
A: X* = I/(2Px) and Y* = I/(2Py). Each good receives half of income.
Q: What is the total effect on X of a small increase in Px?*
A: dX*/dPx = -I/(2Px²). Demand falls.
Q: Using the Slutsky decomposition, what is the income effect of the price change in the previous question?
A: Income effect = -X* · (∂X*/∂I) = -(I/2Px) · (1/2Px) = -I/(4Px²).
Q: How do you show that X is a normal good?
A: Compute ∂X*/∂I = 1/(2Px) > 0. Since the derivative is positive, demand for X rises with income, confirming X is normal.
Q: Derive the expenditure function for this consumer.
A: First find the indirect utility function V = I/(2·Px^0.5·Py^0.5). Invert to get E(Px, Py, U) = 2U·Px^0.5·Py^0.5.
This material connects directly to the theory of the firm, where producers maximise profit subject to cost constraints using the same Lagrange techniques. The expenditure function reappears in welfare economics when measuring compensating variation (CV) and equivalent variation (EV), the dollar measures of welfare change. The Slutsky decomposition is the foundation for understanding Giffen goods and the law of demand more generally.
Consumer theory, utility maximisation, Cobb-Douglas, Lagrange multiplier, constrained optimisation, Marshallian demand, ordinary demand, Hicksian demand, compensated demand, Slutsky equation, Slutsky decomposition, substitution effect, income effect, normal good, inferior good, Giffen good, expenditure function, indirect utility function, duality, Shephard's lemma, budget constraint, MRS, marginal rate of substitution, ECO 300, intermediate microeconomics