Difficulty: Intermediate | Prerequisites: Budget constraints, indifference curves, MRS, utility maximisation basics, perfect complements (see companion notes).
Cobb-Douglas utility is the workhorse of intermediate microeconomics. It produces smooth, convex indifference curves and well-behaved demand functions, making it the go-to setting for testing whether students can apply the tangency condition, trace out the price consumption curve (PCC), and decompose a price change into substitution and income effects (the Slutsky or Hicks decomposition). This set of notes covers all of that, plus how to read demand curves off a price-consumption graph. Together with the perfect-complements notes, these two documents cover the main consumer-theory problems you will face on exams.
For Cobb-Douglas utility, set MRS = price ratio and combine with the budget constraint to find the optimal bundle. The consumer always spends a fixed fraction of income on each good. When a price changes, the total effect on quantity demanded splits into a substitution effect (movement along the original indifference curve) and an income effect (shift to a new indifference curve).
Cobb-Douglas utility function
A utility function of the form U(X, Y) = X^a Y^b, where a and b are positive constants. The key property is that the consumer always spends a fixed share of income on each good, determined by the exponents.
Marginal rate of substitution (MRS)
The rate at which the consumer is willing to trade Y for X while staying on the same indifference curve. For U = X^a Y^b, MRS = (aY)/(bX). Think of it as: how much Y would you give up for one more unit of X, staying equally happy?
Tangency condition
At the optimal bundle, the slope of the indifference curve equals the slope of the budget line: MRS = Px/Py. This is the condition you set equal and solve, alongside the budget constraint.
Price consumption curve (PCC)
The line connecting all optimal bundles as the price of one good changes, holding income and the other price constant. For Cobb-Douglas preferences, the PCC for changes in Px is a horizontal line (Y stays constant) because the consumer always spends the same dollar amount on Y.
Substitution effect
The change in quantity demanded due solely to the change in relative prices, holding utility constant. In simple terms, this isolates "the good got relatively cheaper, so I want more of it" from any purchasing-power change.
Income effect
The change in quantity demanded due to the change in real purchasing power caused by the price change. When a price falls, you are effectively richer, and this part measures how much more (or less) you buy because of that.
Slutsky / Hicks decomposition
The method of splitting the total effect of a price change into the substitution effect and the income effect. Hicks holds utility constant (finds the compensated income); Slutsky holds purchasing power constant (finds the bundle you could just afford at new prices with old income). ECON 302 typically uses the Hicks version.
Compensating variation
The amount of money that must be taken away from (or given to) the consumer after a price change to return them to their original utility level. In simple terms, it answers: how much income adjustment would make the price change feel neutral?
The method has two equations and two unknowns:
Tangency condition: Set MRS = Px/Py. For U = X^a Y^b, this gives (aY)/(bX) = Px/Py.
Budget constraint: PxX + PyY = M.
Solve these simultaneously. Rearrange the tangency condition to express Y in terms of X (or vice versa), then substitute into the budget constraint.
A useful shortcut: for U = X^a Y^b, the consumer always spends fraction a/(a+b) of income on X and fraction b/(a+b) on Y. So:
X* = [a/(a+b)] x M/Px
Y* = [b/(a+b)] x M/Py
U(X, Y) = X^2 Y, so a = 2, b = 1. Py = 1, M = 360.
The spending-share shortcut gives: spend 2/3 of income on X, and 1/3 on Y.
Y* = (1/3)(360)/1 = 120 for every value of Px. This is why the PCC is horizontal at Y = 120.
X* = (2/3)(360)/Px = 240/Px.
For the three price levels:
Px = 1: X = 240, Y = 120. Budget line from (0, 360) to (360, 0).
Px = 2: X = 120, Y = 120. Budget line from (0, 360) to (180, 0).
Px = 5: X = 48, Y = 120. Budget line from (0, 360) to (72, 0).
All optimal bundles share Y = 120. The PCC is the horizontal line Y = 120, and as Px rises, the consumer slides leftward along it.
The indifference curve through each bundle satisfies X^2 Y = k, where k is the utility at that bundle. For instance, at Px = 1: k = 240^2 x 120 = 6,912,000.
U(X, Y) = XY. Px = 9, Py = 1, M = 36.
Part (a): Original optimal bundle.
Spending shares are 1/2 and 1/2 (since a = b = 1).
X* = (1/2)(36)/9 = 2
Y* = (1/2)(36)/1 = 18
Part (b): Utility at the original bundle.
U = 2 x 18 = 36.
Part (c): New bundle after Px falls to 4.
X* = (1/2)(36)/4 = 4.5
Y* = (1/2)(36)/1 = 18
Total effect on X: from 2 to 4.5, an increase of 2.5 units.
Part (d): Compensated income (Hicks).
Find the income M' such that, at the new prices (Px = 4, Py = 1), the consumer reaches the original utility level of 36.
At new prices, X = M'/(2 x 4) = M'/8, and Y = M'/(2 x 1) = M'/2.
Utility = (M'/8)(M'/2) = M'^2 /16 = 36, so M'^2 = 576, giving M' = 24.
Part (e): Decomposing the effects.
At compensated income M' = 24 with Px = 4, Py = 1:
X_comp = 24/8 = 3
Y_comp = 24/2 = 12
Substitution effect on X: from 2 to 3, an increase of 1 unit. This is the part due purely to the relative price change.
Income effect on X: from 3 to 4.5, an increase of 1.5 units. This is the part due to the increase in real purchasing power.
Check: 1 + 1.5 = 2.5 = total effect. Correct.
A price-consumption graph shows several budget constraints sharing a common vertical intercept (income/Py is fixed) but with different slopes (different Px values). Each budget line is tangent to an indifference curve at the optimal bundle.
To extract the demand curve for the good on the horizontal axis:
Find each budget line's slope. The vertical intercept is M/Py. The horizontal intercept is M/Px. So Px = M / (horizontal intercept).
Read off the optimal quantity from each tangency point.
Pair them up: each (Px, X*) pair is a point on the demand curve.
Worked Example from the graph (M = 240, vertical axis = Juice, horizontal axis = Snacks):
Budget line 1: vertical intercept 20, horizontal intercept 10. Py = 240/20 = 12. Px = 240/10 = 24. Optimal Snacks = 5. So (P_snacks = 24, Q = 5).
Budget line 2: vertical intercept 20, horizontal intercept 20. Px = 240/20 = 12. Optimal Snacks = 7. So (P_snacks = 12, Q = 7).
Budget line 3: vertical intercept 20, horizontal intercept 40. Px = 240/40 = 6. Optimal Snacks = 10. So (P_snacks = 6, Q = 10).
These three points trace out a downward-sloping demand curve for Snacks.
For U(X, Y) = X^a Y^b:
MRS = \frac{aY}{bX}\text{Tangency: } \frac{aY}{bX} = \frac{P_X}{P_Y}X^* = \frac{a}{a+b} \cdot \frac{M}{P_X} \qquad Y^* = \frac{b}{a+b} \cdot \frac{M}{P_Y}\text{Hicks compensated income: solve } U(X^*(M', P'), Y^*(M', P')) = \bar{U}\text{Substitution effect} = X^*(M', P_{\text{new}}) - X^*(M, P_{\text{old}})\text{Income effect} = X^*(M, P_{\text{new}}) - X^*(M', P_{\text{new}})The spending-share result is the single most useful formula for Cobb-Douglas. It tells you that the consumer always spends exactly a/(a+b) of income on X and b/(a+b) on Y, regardless of prices. Prices affect the quantity purchased but not the dollar amount spent.
Students often confuse the MRS formula. For U = X^a Y^b, MRS = aY/(bX), not aX/(bY). The exponent on X goes with Y in the numerator because MRS = MUx/MUy and MUx = aX^(a-1) Y^b.
A frequent error in the Hicks decomposition is using the original income instead of the compensated income to find the substitution effect. The substitution effect holds utility constant, not income.
Students sometimes compute the income effect as the total effect minus the substitution effect rather than solving for it directly. While the arithmetic works, it hides conceptual understanding and is more error-prone when the algebra has gone wrong earlier.
When reading demand curves from a PCC graph, students often forget to compute the price. The price is not labelled on the graph; you must calculate it from the budget line intercepts and the given income.
For Cobb-Douglas, the PCC for changes in Px is always horizontal. Students who do not use the spending-share shortcut sometimes get confused about why Y does not change when Px changes.
⚠️ The Slutsky decomposition is one of the highest-value exam topics. Most exam questions give you a utility function, prices, and income, then ask you to decompose a price change. The worked example in the core content section above is the template.
⚠️ Graphing questions for the PCC require correctly drawn budget lines that share a common intercept on the axis of the good whose price is not changing. Label all intercepts, optimal bundles, and the PCC itself.
⚠️ For Cobb-Douglas, the substitution effect and income effect always go in the same direction for a normal good. If a price falls, both effects increase demand. You should state this explicitly in your answer.
⚠️ Questions that give you a graph and ask you to extract a demand curve are testing whether you can compute the price from budget-line intercepts. Show the calculation, not just the answer.
True or False: For U = X^3 Y^2, the consumer spends 60% of income on X.
Answer: True. a/(a+b) = 3/5 = 60%.
Fill in the blank: The PCC for changes in Px with Cobb-Douglas utility is a ______ line.
Answer: horizontal (at Y* = [b/(a+b)] x M/Py).
True or False: The substitution effect of a price decrease always increases demand for the good whose price fell.
Answer: True. The substitution effect always moves demand in the opposite direction to the price change.
Fill in the blank: To find the Hicks compensated income, you solve for the income that achieves the ______ utility level at the ______ prices.
Answer: original; new.
True or False: When reading a demand curve from a PCC graph, the price of the good is directly labelled on the graph.
Answer: False. You must calculate it from the budget-line intercepts and the given income.
Q: A consumer has U(X, Y) = X^3 Y. Px = 6, Py = 2, M = 240. What is the optimal bundle?
A: Spending shares: 3/4 on X, 1/4 on Y. X* = (3/4)(240)/6 = 30. Y* = (1/4)(240)/2 = 30.
Q: Same consumer as above. Px falls to 3. What is the new bundle, and what is the total effect on X?
A: X* = (3/4)(240)/3 = 60. Y* = (1/4)(240)/2 = 30. Total effect on X: 60 - 30 = +30.
Q: For U(X, Y) = XY with Px = 9, Py = 1, M = 36, decompose the effect of Px falling to 4 into substitution and income effects.
A: Original bundle: X = 2, Y = 18. U = 36. New bundle: X = 4.5, Y = 18. Compensated income M' = 24 (solve (M'/8)(M'/2) = 36). At M' = 24 with new prices: X = 3, Y = 12. Substitution effect = 3 - 2 = 1. Income effect = 4.5 - 3 = 1.5. Total = 2.5.
Q: A PCC graph shows budget lines with a common vertical intercept of 20 (Juice). Income is $240. The horizontal intercepts for Snacks are 10, 20, and 40, with optimal Snacks quantities of 5, 7, and 10 respectively. What three points lie on the demand curve for Snacks?
A: P_snacks = 240/horizontal intercept. So the three points are (P = 24, Q = 5), (P = 12, Q = 7), (P = 6, Q = 10).
Q: Explain why both goods are normal goods under Cobb-Douglas utility.
A: The demand functions X* = [a/(a+b)] x M/Px and Y* = [b/(a+b)] x M/Py are both increasing in M. As income rises, the consumer buys more of both goods. No Cobb-Douglas utility function can produce an inferior good.
The Slutsky decomposition connects to welfare economics and consumer surplus. The compensating variation you compute in the decomposition is the same concept used later in welfare analysis to measure the cost of a price change to consumers.
Cobb-Douglas demand functions are also the foundation for deriving individual and market demand curves, which come up in the next unit on market demand and elasticity.
Cobb-Douglas utility, marginal rate of substitution, MRS, tangency condition, price consumption curve, PCC, substitution effect, income effect, Slutsky decomposition, Hicks decomposition, compensating variation, demand curve derivation, normal goods, consumer choice, budget constraint, intermediate microeconomics, ECON 302, Penn State, utility maximisation, spending shares