Difficulty: Intermediate | Prerequisites: Basic calculus (partial derivatives), budget constraints, indifference curves.
Cobb-Douglas utility functions (like U = T^a B^b) produce smooth, well-behaved indifference curves that always yield interior solutions. The optimal bundle is found by setting MRS equal to the price ratio and substituting into the budget constraint. A handy shortcut: with Cobb-Douglas preferences, each good receives a fixed share of income determined by its exponent.
Consumer choice theory sits at the heart of intermediate microeconomics. The question is deceptively simple: given a budget and a set of preferences, what bundle of goods will a rational consumer choose? Cobb-Douglas preferences are the workhorse case because they produce clean algebra and illustrate every core concept (MRS, tangency, demand functions, Engel curves) without the edge-case complications of perfect substitutes or complements. If you are comfortable solving the Cobb-Douglas optimisation problem, you have the template for most consumer theory questions on this exam.
Cobb-Douglas utility function
A utility function of the form U(X, Y) = X^a Y^b, where a and b are positive constants representing the consumer's relative preference for each good. In simple terms, it says "I like both goods, and the exponents tell you how much I lean toward each one."
Marginal Rate of Substitution (MRS)
The rate at which a consumer is willing to trade one good for another while staying on the same indifference curve. Formally, MRS = MU_X / MU_Y. Think of it as the consumer's internal exchange rate between the two goods.
Budget constraint
The set of all bundles a consumer can afford given prices and income: P_X · X + P_Y · Y = m. It is a straight line in (X, Y) space with slope equal to the negative price ratio.
Tangency condition
At the optimal interior bundle, the slope of the indifference curve equals the slope of the budget line: MRS = P_X / P_Y. This is the first of two equations you solve (the budget constraint is the second).
Engel curve
A function showing how the optimal quantity of a good changes as income changes, holding prices fixed. For Cobb-Douglas preferences, the Engel curve is a straight line through the origin because each good receives a constant share of income.
Normal good
A good whose quantity demanded increases when income rises. Both goods in a Cobb-Douglas utility function are always normal goods.
Budget share
The fraction of income spent on a particular good. For U = X^a Y^b, the budget share on X is a/(a + b) and on Y is b/(a + b). This is the constant-budget-share property unique to Cobb-Douglas preferences.
For U(T, B) = T^0.25 B^0.75:
MU_T = 0.25 T^(-0.75) B^(0.75)
MU_B = 0.75 T^(0.25) B^(-0.25)
MRS = MU_T / MU_B = (0.25 / 0.75) · (B / T) = B / 3T
The general pattern: for U = X^a Y^b, MRS = (a/b) · (Y/X). The exponents give you the ratio of coefficients, the variables flip.
Two equations, two unknowns:
Tangency condition: MRS = P_T / P_B, so B / 3T = 4 / 8 = 0.5, giving B = 1.5T
Budget constraint: 4T + 8B = 32
Substitute B = 1.5T into the budget constraint: 4T + 8(1.5T) = 32, so 16T = 32, giving T = 2 and B = 3.
The optimal bundle is (T, B) = (2, 3).
For U = T^0.25 B^0.75, the exponents sum to 1, so:
Share of income on T = 0.25 / (0.25 + 0.75) = 0.25, so spending on T = 0.25 × 32 = 8, and T = 8 / 4 = 2
Share of income on B = 0.75, so spending on B = 0.75 × 32 = 24, and B = 24 / 8 = 3
Same answer, less algebra. This shortcut works for any Cobb-Douglas utility function where the exponents are treated as weights.
The Engel curve expresses optimal quantity as a function of income, holding prices fixed.
From the tangency condition, B = 1.5T, so T = 2B/3. Substitute into the general budget constraint P_T · T + P_B · B = m:
4(2B/3) + 8B = m, which simplifies to (8B/3) + 8B = m, so (8B + 24B)/3 = m, giving 32B = 3m.
Result: B(m) = 3m / 32.
This is a linear function through the origin, confirming that Burritos are a normal good with a constant income elasticity of 1. The Engel curve for any Cobb-Douglas good is always a straight line through the origin.
If income rises from 32 to 48, the new optimal Burritos quantity is B(48) = 3(48)/32 = 144/32 = 4.5. Because the Engel curve is linear, a 50% income increase produces a 50% increase in quantity demanded.
Formula | Expression |
|---|---|
Utility function | U(X, Y) = X^a Y^b |
MRS (general Cobb-Douglas) | (a/b) · (Y/X) |
Tangency condition | MRS = P_X / P_Y |
Budget constraint | P_X · X + P_Y · Y = m |
Optimal X (budget-share shortcut) | X = [a / (a + b)] · m / P_X |
Optimal Y (budget-share shortcut) | Y = [b / (a + b)] · m / P_Y |
Engel curve for Y | Y(m) = [b / (a + b)] · m / P_Y |
The budget-share property of Cobb-Douglas preferences is why economists use this functional form to model household spending patterns where categories like food, housing and transport each absorb a roughly stable percentage of income. The algebra mirrors how a household allocates a pay rise across its regular spending categories.
Students often think MRS = MU_X / MU_Y gives a constant number. It does not. MRS depends on the current quantities of X and Y, so it changes along the indifference curve. Only at the optimum does it equal the price ratio.
Students sometimes set MRS equal to P_Y / P_X instead of P_X / P_Y. The rule: MRS of X for Y = P_X / P_Y. Match the good in the numerator of MRS to the good in the numerator of the price ratio.
A common error is forgetting that Cobb-Douglas always yields an interior solution (positive amounts of both goods). If your answer has the consumer buying zero of one good, check your algebra.
Some students confuse the Engel curve with the demand curve. The Engel curve varies income with prices fixed. The demand curve varies one price with income fixed.
⚠️ Deriving MRS and setting it equal to the price ratio is tested in nearly every exam. Practise until the steps are automatic.
⚠️ The budget-share shortcut can save significant time on multiple-choice questions where you only need the final answer.
⚠️ Engel curve derivation is a common 5-mark question. Make sure you can go from the tangency condition to B(m) = 3m/32 in general terms, not just for m = 32.
⚠️ Questions that change income (e.g. from 32 to 48) are testing whether you can apply the Engel curve function you derived.
True or False: For a Cobb-Douglas utility function, MRS is constant along an indifference curve. (False)
Fill in the blank: For U = X^0.4 Y^0.6, the share of income spent on X is ____. (0.4, or 40%)
True or False: The Engel curve for a Cobb-Douglas good is a straight line through the origin. (True)
Fill in the blank: At the optimal bundle, MRS equals ____. (the price ratio, P_X / P_Y)
True or False: A Cobb-Douglas consumer might optimally buy zero of one good. (False)
Q: For U(T, B) = T^0.25 B^0.75 with P_T = 4, P_B = 8, m = 32, what is the MRS?
A: MRS = B / 3T. At the optimum (T = 2, B = 3), MRS = 3 / 6 = 0.5, which equals P_T / P_B = 4/8.
Q: Using the same utility function, derive B(m) when P_T = 4 and P_B = 8.
A: From the tangency condition B = 1.5T, so T = 2B/3. Substituting into 4T + 8B = m gives 4(2B/3) + 8B = m, so 32B/3 = m, and B(m) = 3m/32.
Q: If Tommy's income rises to $48, what is his new optimal bundle?
A: T = 0.25 × 48 / 4 = 3; B = 0.75 × 48 / 8 = 4.5. The bundle is (3, 4.5).
Q: What is the MRS for a general Cobb-Douglas U = X^a Y^b?
A: MRS = (a/b) · (Y/X).
Q: Why does a Cobb-Douglas consumer never choose a corner solution?
A: Because both marginal utilities approach infinity as either good approaches zero, the consumer always benefits from consuming at least a small amount of each good.
The Cobb-Douglas optimisation method (tangency + budget constraint) is the same two-equation approach used in producer cost minimisation, where MRTS replaces MRS and the isocost line replaces the budget line. If you can solve one, you can solve the other.
Engel curves connect directly to income elasticity of demand and the classification of goods as normal, inferior or luxury. For Cobb-Douglas, all goods are normal with unit income elasticity.
The contrast between Cobb-Douglas (interior solutions) and perfect substitutes or complements (corner or kink solutions) is a recurring exam theme. Understanding why Cobb-Douglas always produces an interior solution helps you recognise when a problem requires a different approach.
Cobb-Douglas, utility maximisation, consumer choice, MRS, marginal rate of substitution, budget constraint, tangency condition, optimal bundle, Engel curve, income expansion path, budget share, constant budget shares, normal good, interior solution, demand function, ECN 312, intermediate microeconomics, ASU, consumer theory, indifference curve, price ratio