Consumer Theory, Utility Maximisation, and Demand -- ECON 300, Midterm Review
Source: ECON 300 Midterm Examination, Autumn 2016 (University of Washington, Prof. Greg Ellis)
Tags: consumer theory, utility maximisation, Lagrangian optimisation, Marshallian demand, Slutsky decomposition, income effect, substitution effect, expenditure function, Cobb-Douglas, intermediate microeconomics
Difficulty: Intermediate. Prerequisites: basic calculus (partial derivatives, constrained optimisation), familiarity with budget constraints and indifference curves.
Big Picture
Consumer theory is the backbone of microeconomics. It asks a deceptively simple question: given limited income and a set of prices, how does a rational consumer choose what to buy? This topic sits at the heart of ECON 300 and connects forward to market demand, welfare economics, and general equilibrium. You need to be comfortable setting up and solving constrained optimisation problems, interpreting the resulting demand functions, and decomposing price changes into their component parts. If you can do all of that fluently, most of the midterm's 30-point lead question is within reach.
Consumers maximise utility subject to a budget constraint. The Lagrangian method finds the optimal bundle, yielding Marshallian demand functions that depend on prices and income. When a price changes, the total effect on quantity demanded splits into a substitution effect (movement along the indifference curve) and an income effect (shift to a new indifference curve), per the Slutsky decomposition. The expenditure function flips the problem: it finds the minimum spending needed to reach 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 for how happy different combinations of goods make you.
Cobb-Douglas utility function
A specific functional form, U(X, Y) = X^a · Y^b, where a and b are positive constants. It is popular in textbook problems because it yields clean, closed-form demand functions. Think of it as the "well-behaved" workhorse of consumer theory.
Budget constraint
The set of all bundles a consumer can afford given their income and the prices of goods: Px·X + Py·Y = I. In simple terms, it is the line that says "you can spend up to this much, no more."
Lagrangian method (method of Lagrange)
A technique for solving constrained optimisation problems. You form a new function, L = U(X,Y) + λ(I - Px·X - Py·Y), take partial derivatives, set them to zero, and solve. Think of it as a systematic recipe for finding the best bundle you can afford.
Lagrange multiplier (λ)
The shadow price of the constraint. It tells you how much your maximum utility would increase if your income rose by one unit. In simple terms, it measures the marginal value of an extra pound (or dollar) of budget.
Marshallian (ordinary) demand function
The quantity of a good demanded as a function of prices and income: X*(Px, Py, I). It is what you get after solving the utility maximisation problem. Think of it as the consumer's shopping list, expressed as a formula.
Slutsky decomposition
A method for breaking the total effect of a price change on quantity demanded into two parts: the substitution effect (holding utility constant) and the income effect (the change in purchasing power). In simple terms, it answers "how much of the change is because the good got relatively more expensive, and how much is because you feel poorer?"
Substitution effect
The change in quantity demanded due solely to the change in relative prices, holding utility constant. The consumer slides along the same indifference curve. Always negative for a price increase (you buy less of the good that got relatively pricier).
Income effect
The change in quantity demanded due to the change in real purchasing power caused by the price change. A price increase makes you effectively poorer, which shifts you to a lower indifference curve.
Normal good
A good whose quantity demanded increases when income rises (and vice versa). Mathematically, ∂X*/∂I > 0. Most goods you can think of are normal goods.
Inferior good
A good whose quantity demanded decreases when income rises. Mathematically, ∂X*/∂I < 0. Instant noodles are the classic textbook example.
Expenditure function E(Px, Py, U)
The minimum amount of money needed to achieve a given utility level U at prices Px and Py. It is the dual of the utility maximisation problem. Think of it as answering: "What is the cheapest way to be this happy?"
The consumer wants to maximise U(X, Y) = X^0.5 · Y^0.5 subject to the budget constraint Px·X + Py·Y = I.
This is a Cobb-Douglas utility function with exponents a = b = 0.5.
Form the Lagrangian: L = X^0.5 · Y^0.5 + λ(I - Px·X - Py·Y)
Take partial derivatives and set each 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
Divide the first FOC by the second to eliminate λ. This gives the tangency condition: Y/X = Px/Py, which rearranges to Y = (Px/Py)·X.
Substitute the tangency condition into the budget constraint: Px·X + Py·(Px/Py)·X = I, so 2Px·X = I.
Solve for the Marshallian demands:
X* = I / (2Px)
Y* = I / (2Py)
The consumer spends exactly half of income on each good. This equal-split property holds for any Cobb-Douglas function U = X^a · Y^b: the share spent on X is a/(a+b), and on Y is b/(a+b). With a = b = 0.5, both shares are 1/2.
Demand for each good is inversely proportional to its own price and directly proportional to income.
Demand for X does not depend on Py (and vice versa). This independence of cross-prices is a special feature of Cobb-Douglas preferences, not a general result.
Differentiate X* = I/(2Px) with respect to Px: dX*/dPx = -I/(2Px^2).
This is the total effect of a small price increase on quantity demanded. It is always negative, confirming the law of demand for this utility function.
The total effect of a change in Px on demand for X decomposes as:
dX*/dPx = (∂Xh/∂Px) - X* · (∂X*/∂I)
where the first term is the substitution effect (the derivative of the Hicksian/compensated demand) and the second term is the income effect.
From the Marshallian demand X* = I/(2Px), the income derivative is: ∂X*/∂I = 1/(2Px).
The income effect of the price change is: -X* · (∂X*/∂I) = -(I/(2Px)) · (1/(2Px)) = -I/(4Px^2).
Total effect = -I/(2Px^2).
Income effect = -I/(4Px^2).
Substitution effect = Total effect - Income effect = -I/(2Px^2) - (-I/(4Px^2)) = -I/(4Px^2).
For this utility function, the substitution effect and the income effect are equal in magnitude. Each accounts for exactly half of the total effect.
X is a normal good if ∂X*/∂I > 0.
We have ∂X*/∂I = 1/(2Px), which is always positive (prices are positive).
Therefore X is a normal good. When income rises, the consumer buys more of X.
Equivalently, the income effect of a price increase reinforces the substitution effect (both are negative), which is the hallmark of a normal good.
The expenditure function E(Px, Py, U) answers: what is the minimum income needed to reach utility level U at prices Px and Py? It is the solution to the expenditure minimisation problem, which is the dual of utility maximisation.
Start from the indirect utility function. Substitute the Marshallian demands into the utility function: V = (I/(2Px))^0.5 · (I/(2Py))^0.5 = I / (2·(Px·Py)^0.5).
The expenditure function inverts V: set V = U and solve for I. That gives E = 2U·(Px·Py)^0.5.
Alternatively, you can derive this directly by minimising Px·X + Py·Y subject to X^0.5 · Y^0.5 = U using a Lagrangian. The tangency condition is the same (Y/X = Px/Py), and substituting into the utility constraint yields the same result.
E is increasing in Px, Py, and U. Higher prices or a higher target utility level means you need more money.
E is homogeneous of degree 1 in prices. If all prices double, the minimum cost doubles.
E is concave in prices. This reflects the consumer's ability to substitute away from goods that become relatively more expensive.
Shephard's lemma: ∂E/∂Px = Xh (the Hicksian demand for X). This connects the expenditure function directly to the compensated demand functions.
Utility function: U(X, Y) = X^0.5 · Y^0.5
Budget constraint: Px·X + Py·Y = I
Lagrangian: L = X^0.5 · Y^0.5 + λ(I - Px·X - Py·Y)
Tangency condition (MRS = price ratio): Y/X = Px/Py
Marshallian demand for X: X* = I / (2Px)
Marshallian demand for Y: Y* = I / (2Py)
Total effect of price change: dX*/dPx = -I / (2Px^2)
Income effect: -I / (4Px^2)
Substitution effect: -I / (4Px^2)
Slutsky equation: dX*/dPx = (∂Xh/∂Px) - X*·(∂X*/∂I)
Indirect utility function: V(Px, Py, I) = I / (2·(Px·Py)^0.5)
Expenditure function: E(Px, Py, U) = 2U·(Px·Py)^0.5
The Slutsky decomposition is not just exam material. It is the analytical engine behind real policy questions. When a government raises a petrol tax, the substitution effect captures consumers switching to public transport or smaller cars, while the income effect captures the fact that households simply have less to spend. Policymakers who ignore the income effect (or assume it is small) will underestimate the impact on lower-income households, which is why compensating transfers sometimes accompany energy price reforms.
Expenditure functions underpin cost-of-living indices. The question "how much more does a household need to spend to be as well off as before prices rose?" is precisely the expenditure function evaluated at the new prices.
Students often think the substitution effect can be positive for a price increase. It cannot. The substitution effect of a price increase is always negative (you move away from the good that became relatively more expensive). The income effect is the part that can go either way.
Students often confuse the Slutsky and Hicks decomposition. The Slutsky version compensates income to keep the original bundle affordable. The Hicks version compensates to keep utility constant. In ECON 300, you are typically asked for the Hicks version unless told otherwise.
Students often forget that Cobb-Douglas demand for X is independent of Py. They waste time trying to find cross-price effects that are zero by construction. If you are working with Cobb-Douglas, each good's demand depends only on its own price and income.
Students sometimes claim a good is normal or inferior without computing ∂X*/∂I. Always show the derivative and confirm its sign.
⚠️ The Lagrangian setup and derivation of Marshallian demands is the single highest-value skill for this midterm. It appeared as question 1(a), worth the largest share of points. Be able to do it quickly and without errors.
⚠️ Know the Slutsky decomposition cold. You will be asked to identify the total effect, then decompose it. Show each step.
⚠️ The "Is X normal?" question is a quick-marks opportunity. Compute ∂X*/∂I, state its sign, and conclude. Do not waffle.
⚠️ The expenditure function derivation can be done two ways (inversion of indirect utility, or direct minimisation). Know at least one. Inversion is typically faster for Cobb-Douglas.
⚠️ Express answers in terms of Px, Py, and I as the question requests. Leaving λ in a final answer when not asked for it loses marks.
True or False: For a Cobb-Douglas utility function with equal exponents, the consumer spends the same fraction of income on each good.
Fill in the blank: The Marshallian demand for X when U = X^0.5 · Y^0.5 is X* = ______.
True or False: The substitution effect of a price increase can be positive.
Fill in the blank: A good is classified as normal if ∂X*/∂I is ______.
True or False: The expenditure function is decreasing in prices.
Answers: 1. True. 2. I/(2Px). 3. False. 4. Positive. 5. False (it is increasing in prices).
Q: Set up the Lagrangian for maximising U(X,Y) = X^0.5 · Y^0.5 subject to Px·X + Py·Y = I. What are the three first-order conditions?
A: L = X^0.5 · Y^0.5 + λ(I - Px·X - Py·Y). The FOCs are: (1) 0.5·X^(-0.5)·Y^0.5 = λPx, (2) 0.5·X^0.5·Y^(-0.5) = λPy, (3) Px·X + Py·Y = I.
Q: Derive the Marshallian demand functions for X and Y.
A: Divide FOC(1) by FOC(2) to get Y/X = Px/Py. Substitute into the budget constraint: Px·X + Px·X = I, so X* = I/(2Px). Similarly, Y* = I/(2Py).
Q: What is the total effect of a small increase in Px on the demand for X? Break it into substitution and income effects.
A: Total effect: dX*/dPx = -I/(2Px^2). Substitution effect: -I/(4Px^2). Income effect: -I/(4Px^2). Both are negative and equal in magnitude.
Q: Prove that X is a normal good for this utility function.
A: ∂X*/∂I = 1/(2Px) > 0 for all Px > 0. Since demand for X increases with income, X is a normal good.
Q: Derive the expenditure function E(Px, Py, U).
A: The indirect utility function is V = I/(2·(Px·Py)^0.5). Set V = U and solve for I: E = 2U·(Px·Py)^0.5.
This material connects directly to market demand (Topic 2 in these notes): individual Marshallian demands are aggregated horizontally to build the market demand curve. The expenditure function connects forward to welfare analysis, where compensating variation and equivalent variation use E(Px, Py, U) to measure the monetary impact of a price change. The Slutsky decomposition reappears when you study labour supply (leisure as a good whose "price" is the wage) and intertemporal choice (consumption today versus tomorrow).
Consumer theory, utility maximisation, utility maximization, constrained optimisation, Lagrange multiplier, Lagrangian method, first-order conditions, FOC, tangency condition, MRS equals price ratio, marginal rate of substitution, Marshallian demand, ordinary demand, uncompensated demand, Hicksian demand, compensated demand, Slutsky equation, Slutsky decomposition, income effect, substitution effect, normal good, inferior good, Giffen good, Cobb-Douglas, indirect utility function, expenditure function, duality, Shephard's lemma, Roy's identity, budget constraint, ECON 300, intermediate microeconomics