Difficulty: Intermediate | Prerequisites: Budget constraints, indifference curves, partial derivatives.
Cobb-Douglas preferences are the workhorse of intermediate microeconomics. Nearly every consumer-theory exam leans on them because the algebra is clean and the results generalise well. This set of notes walks through the full method: finding optimal bundles via the tangency condition, tracing how those bundles shift with income (Engel curves), and deriving demand as a function of price. If you can do these steps fluently, you can handle most utility-maximisation questions the course will throw at you.
For a Cobb-Douglas utility function U = x^a · y^b, the consumer's optimal bundle is found by setting the marginal rate of substitution equal to the price ratio, then solving simultaneously with the budget constraint. The exponents directly determine the share of income spent on each good, making Engel curves linear and demand curves hyperbolic. Both goods are always normal goods under Cobb-Douglas preferences.
Cobb-Douglas utility function
A utility function of the form U(x, y) = x^a · y^b, where a and b are positive constants (often summing to 1). In simple terms, it describes a consumer who always wants some of both goods and is willing to trade between them at a smoothly changing rate.
Marginal utility (MU)
The additional satisfaction a consumer gains from one more unit of a good, holding everything else constant. Think of it as the "payoff" from the next taco or the next burrito.
Marginal rate of substitution (MRS)
The ratio MU_x / MU_y, measuring how many units of good y the consumer would willingly give up for one more unit of good x while staying equally happy. In simple terms, it is the consumer's personal exchange rate between the two goods.
Tangency condition (slope condition)
The optimality rule stating MRS = P_x / P_y. At the best affordable bundle, the consumer's willingness to trade matches the market's terms of trade. This gives the first equation you need to solve for the optimum.
Budget constraint
The equation P_x · x + P_y · y = m, capturing all bundles the consumer can just afford. This is the second equation in your system of two unknowns.
Engel curve
A function showing how the quantity demanded of a good changes as income changes, holding prices fixed. For Cobb-Douglas, Engel curves are straight lines through the origin.
Normal good
A good whose quantity demanded rises when income rises. Under Cobb-Douglas preferences, both goods are always normal.
Demand curve
A function showing how the quantity demanded of a good changes as its own price changes, holding income and other prices fixed. For Cobb-Douglas, demand curves are downward-sloping hyperbolas (quantity = constant / price).
The approach is the same every time: two equations, two unknowns.
Step 1: Compute marginal utilities. Differentiate U(x, y) with respect to each good. For U(T, B) = T^0.25 · B^0.75, you get MU_T = 0.25 T^(−0.75) B^(0.75) and MU_B = 0.75 T^(0.25) B^(−0.25).
Step 2: Form the MRS. Divide MU_T by MU_B. The exponents and powers simplify. Here: MRS = B / (3T).
Step 3: Set MRS = price ratio. This is the tangency (slope) condition. With P_T = 4 and P_B = 8: B/(3T) = 4/8 = 1/2, which rearranges to B = 1.5T.
Step 4: Substitute into the budget constraint. Plug B = 1.5T into 4T + 8B = 32, solve for T, then back-substitute for B. Result: T* = 2, B* = 3.
For U = x^a · y^b, the consumer always spends a fraction a/(a+b) of income on good x and b/(a+b) on good y. With a = 0.25 and b = 0.75, that is 25% of income on tacos and 75% on burritos. You can verify: 4 × 2 = 8 (25% of 32) and 8 × 3 = 24 (75% of 32). This shortcut is worth memorising for speed on exams.
When income rises from $32 to $48, prices and preferences are unchanged, so the slope condition B = 1.5T still holds. Only the budget constraint shifts outward. Substituting into 4T + 8B = 48 gives T* = 3, B* = 4.5. Both goods increase with income, confirming they are normal goods.
The Engel curve expresses quantity as a function of income, holding prices fixed.
Keep the slope condition (B = 1.5T, so T = 2B/3).
Write the budget with prices plugged in but income left as m: 4T + 8B = m.
Substitute and solve for B: 4(2B/3) + 8B = m, so (32B/3) = m, giving B(m) = 3m/32.
This is a straight line through the origin with slope 32/3 when graphed with B on the horizontal axis and m on the vertical axis.
The demand curve expresses quantity as a function of the good's own price, holding income and the other price fixed.
Go back to the MRS = price ratio step, but do not plug in the price of the good you are finding demand for. Here, leave P_B as a variable: B/(3T) = 4/P_B, giving T = B·P_B / 12.
Write the budget with P_B left as a variable and m = 32: 4T + P_B · B = 32.
Substitute and solve: 4(B·P_B/12) + P_B · B = 32, so (4P_B · B)/3 = 32, giving B(P_B) = 24/P_B.
This is a rectangular hyperbola: as the price of burritos rises, quantity demanded falls, and total spending on burritos stays constant at $24 regardless of price.
U(T,B) = T^{0.25} \cdot B^{0.75}MU_T = 0.25\, T^{-0.75}\, B^{0.75} \qquad MU_B = 0.75\, T^{0.25}\, B^{-0.25}MRS_{T,B} = \frac{MU_T}{MU_B} = \frac{B}{3T}\text{Tangency condition: } \frac{B}{3T} = \frac{P_T}{P_B}\text{Budget constraint: } P_T \cdot T + P_B \cdot B = m\text{Engel curve for burritos: } B(m) = \frac{3m}{32}\text{Demand curve for burritos: } B(P_B) = \frac{24}{P_B}The general Cobb-Douglas demand result: for U = x^a · y^b, demand for good y is y* = [b/(a+b)] · m / P_y.
Students often plug in the price of burritos when deriving the demand curve for burritos. The whole point of the demand curve is to leave that price as a variable. Only plug in the prices of goods you are not solving for.
Students sometimes set MRS equal to P_B / P_T instead of P_T / P_B. The price ratio matches the order of the MRS: MU of the good on the x-axis divided by MU of the good on the y-axis equals P_x / P_y.
When deriving the Engel curve, a common error is plugging in both prices and income. You plug in prices but leave income as m, because the Engel curve shows how quantity changes with income.
Students confuse the Engel curve with the demand curve. The Engel curve varies income while holding prices fixed. The demand curve varies one price while holding income and the other price fixed.
The tangency method (MRS = price ratio + budget constraint) is the single most-tested technique in intermediate micro. Know it cold.
Exams frequently ask you to derive Engel and demand curves in the same question, varying what you leave as a variable and what you plug in. The decision rule is simple: leave the thing the curve is "a function of" as a variable, plug in everything else.
Cobb-Douglas demand curves always take the form quantity = constant / price. If your answer does not look like this, check your algebra.
The constant expenditure share property (spending on each good is a fixed fraction of income) is a distinctive feature of Cobb-Douglas. Exam questions sometimes test whether you recognise it.
Engel curves through the origin with a positive slope confirm normal goods. If an exam asks you to classify a good as normal or inferior, compute the Engel curve and check the sign.
True or false: For Cobb-Douglas preferences, the MRS depends on the ratio of quantities consumed, not on prices. (True.)
Fill in the blank: The tangency condition sets ______ equal to the price ratio. (The MRS.)
True or false: Under Cobb-Douglas, if income doubles, the optimal quantity of each good also doubles. (True.)
Fill in the blank: The Engel curve for burritos in this problem is B(m) = ______. (3m/32.)
True or false: The demand curve B(P_B) = 24/P_B implies that total spending on burritos changes as P_B changes. (False. Total spending is always $24.)
Q: Suppose U(T, B) = T^0.25 · B^0.75, P_T = 4, P_B = 8, m = 32. What is the optimal bundle?
A: Set MRS = B/(3T) equal to the price ratio 4/8 = 1/2, giving B = 1.5T. Substitute into 4T + 8B = 32 to get T* = 2, B* = 3.
Q: If income rises to $48 with the same preferences and prices, what is the new optimal bundle?
A: The slope condition B = 1.5T is unchanged. Substituting into 4T + 8(1.5T) = 48 gives T* = 3, B* = 4.5.
Q: Derive the Engel curve for burritos.
A: Rewrite the slope condition as T = 2B/3. Substitute into 4T + 8B = m: 4(2B/3) + 8B = m, so 32B/3 = m, giving B(m) = 3m/32.
Q: Derive the demand curve for burritos, using m = 32.
A: Leave P_B as a variable in the tangency condition: B/(3T) = 4/P_B, so T = B·P_B/12. Substitute into 4T + P_B·B = 32: (4P_B·B)/3 = 32, giving B(P_B) = 24/P_B.
Q: What fraction of income does this consumer spend on burritos?
A: The exponent on B is 0.75 and the exponents sum to 1, so 75% of income is spent on burritos regardless of prices or income level. At m = 32, spending on burritos = 8 × 3 = 24 = 0.75 × 32.
This connects to perfect complements (Leontief) preferences because the same two-equation method applies, but the first equation comes from the kink condition rather than the tangency condition. Knowing both cases side by side clarifies when to use MRS = price ratio and when to use the fixed-proportions condition.
The Engel curve and demand curve derivations reappear when you study income and substitution effects (Slutsky decomposition). Being comfortable moving between "hold price fixed, vary income" and "hold income fixed, vary price" is essential groundwork for that decomposition.
Cobb-Douglas preferences also connect to producer theory later in the course: Cobb-Douglas production functions have the same mathematical structure, and the cost-minimisation problem mirrors utility maximisation with inputs replacing goods.
Cobb-Douglas, utility maximisation, optimal bundle, tangency condition, MRS equals price ratio, marginal rate of substitution, budget constraint, Engel curve, demand curve, normal good, income expansion path, constant expenditure shares, ECN 312, intermediate microeconomics, consumer choice, utility function exponents, rectangular hyperbola demand