Difficulty: Intermediate | Prerequisites: Marginal utility, MRS, budget constraints, tangency condition for utility maximisation.
Big picture: The Price-Consumption Curve (PCC) is the counterpart to the Income-Consumption Curve. Where the ICC varies income and holds prices fixed, the PCC varies the price of one good and holds income and the other price fixed. The PCC is the bridge between indifference-curve analysis and the ordinary demand curve you will use for the rest of the course. Understanding how to move from one to the other is a core exam skill.
TL;DR: For Cobb-Douglas preferences like U = X²Y, the tangency condition (MRS = price ratio) plus the budget constraint pin down the optimal bundle at each price. Plot those bundles in (X, Y) space and you have the PCC. Read off the X-values against the prices that produced them and you have the demand curve for X.
Price-Consumption Curve (PCC)
The set of optimal bundles in (X, Y) space as the price of one good changes, holding income and the other price constant. Think of it as: "What path does the consumer's chosen point trace out as good X gets cheaper or more expensive?"
Cobb-Douglas utility function
A utility function of the form U(X, Y) = X^a · Y^b. It produces smooth, convex indifference curves, and the consumer always spends a fixed fraction of income on each good. For U = X²Y, the consumer spends 2/3 of income on X and 1/3 on Y.
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. Calculated as MU_X / MU_Y. For U = X²Y: MRS = 2XY / X² = 2Y / X.
Tangency condition
At the optimum, MRS = P_X / P_Y. This is the "bang for your buck" condition: the last pound spent on X and the last pound spent on Y must yield the same marginal utility.
Ordinary demand curve
A graph with the price of a good on the vertical axis and quantity demanded on the horizontal axis, holding income and other prices constant. Derived directly from the PCC by reading off the price and quantity pairs.
Budget constraint rotation
When P_X changes while P_Y and M stay fixed, the budget line pivots around the vertical intercept (M / P_Y). A lower P_X fans the line outward (the horizontal intercept M / P_X moves right); a higher P_X fans it inward.
Given: U(X, Y) = X²Y, so MU_X = 2XY and MU_Y = X². Income M = 240, P_Y = 1. The price of X varies: P_X = 1, 2, 5.
MRS = MU_X / MU_Y = 2XY / X² = 2Y / X.
Set MRS = P_X / P_Y:
2Y / X = P_X / 1
2Y = P_X · X
Y = P_X · X / 2
This is the key tangency relationship. It says that at the optimum, Y is proportional to P_X times X.
P_X · X + P_Y · Y = 240 becomes P_X · X + (P_X · X / 2) = 240, which simplifies to (3/2) · P_X · X = 240, so X = 160 / P_X.
Then Y = P_X · X / 2 = P_X · (160 / P_X) / 2 = 80.
Notice: Y = 80 regardless of P_X. The consumer always spends exactly 1/3 of income on Y (that is, 80 = 240/3), which is the Cobb-Douglas spending-share result.
P_X = 1: X = 160, Y = 80. Bundle is (160, 80). Budget intercepts: horizontal = 240, vertical = 240.
P_X = 2: X = 80, Y = 80. Bundle is (80, 80). Budget intercepts: horizontal = 120, vertical = 240.
P_X = 5: X = 32, Y = 80. Bundle is (32, 80). Budget intercepts: horizontal = 48, vertical = 240.
All three optimal bundles have Y = 80. The PCC is therefore a horizontal line at Y = 80, stretching from X = 32 to X = 160. This is a special feature of this utility function: because the exponent on Y is 1, the income share spent on Y is always 1/3, so Y = M / (3 · P_Y) = 80 regardless of P_X.
The graph shows three budget lines fanning out from the common vertical intercept at (0, 240). Each budget line is tangent to an indifference curve at the corresponding bundle. The PCC connects the bundles horizontally at Y = 80.
Sometimes the exam gives you a pre-drawn PCC graph rather than a utility function. The graph shows budget lines fanning from a common intercept, with optimal bundles marked along the PCC. Your job is to extract price-quantity pairs for the demand curve.
A consumer has M = 80 to spend on Juice (Y-axis) and Snacks (X-axis). The graph shows three optimal bundles on the PCC, each tangent to a different budget line. The Y-intercept for all budget lines is 20, so P_Y = M / 20 = 80 / 20 = 4.
From the graph, the three bundles and their associated budget-line X-intercepts are:
Bundle (10, 15): the budget line passes through (10, 15) and has Y-intercept 20. Using the budget constraint: P_X(10) + 4(15) = 80, so 10 · P_X = 20, giving P_X = 2.
Bundle (7, 15): 7 · P_X + 4(15) = 80, so 7 · P_X = 20, giving P_X = 2.86 (approximately 20/7).
Bundle (5, 15): 5 · P_X + 4(15) = 80, so 5 · P_X = 20, giving P_X = 4.
The demand schedule for Snacks:
Price of Snacks | Quantity Demanded |
|---|---|
2.00 | 10 |
2.86 | 7 |
4.00 | 5 |
Plotting Price on the vertical axis and Quantity on the horizontal axis gives the downward-sloping demand curve.
Identify P_Y from the common vertical intercept: P_Y = M / (Y-intercept).
For each optimal bundle (X*, Y*), back out P_X from the budget constraint: P_X = (M - P_Y · Y*) / X*.
Pair each P_X with its X* to get a point on the demand curve.
Cobb-Douglas demand (general form for U = X^a · Y^b):
X^* = \frac{a}{a+b} \cdot \frac{M}{P_X}, \qquad Y^* = \frac{b}{a+b} \cdot \frac{M}{P_Y}For U = X²Y (a = 2, b = 1): the consumer spends 2/3 of income on X and 1/3 on Y.
Tangency condition:
\frac{MU_X}{MU_Y} = \frac{P_X}{P_Y}Demand curve (derived from the PCC):
From the Cobb-Douglas demand, X = (2/3) · M / P_X = 160 / P_X when M = 240. This is the equation of the ordinary demand curve for X.
Extracting P_X from a graph:
P_X = \frac{M - P_Y \cdot Y^*}{X^*}The PCC and the ordinary demand curve are the workhorses of applied microeconomics. Firms use demand curves derived this way to set prices, forecast revenue, and measure price elasticity. Governments use them to predict how a tax (which raises the effective price) will change consumption. The Cobb-Douglas spending-share result, where a fixed fraction of income goes to each good, is a surprisingly decent approximation for broad categories like food, housing, and entertainment in household budget data.
Students often forget that the PCC varies price while holding income constant, and the ICC varies income while holding prices constant. Mixing these up leads to drawing the wrong graph entirely.
When deriving demand from a graph, students sometimes read the budget-line intercept as the quantity demanded. The intercept is M / P_X, the maximum the consumer could buy if they spent everything on X. The actual quantity demanded is the X-coordinate of the optimal bundle.
For Cobb-Douglas, students sometimes think the spending share depends on prices. It does not. The share spent on each good depends only on the exponents in the utility function.
Students forget to check that the PCC bundles are consistent with the budget constraint. A quick plug-in check (P_X · X + P_Y · Y = M?) catches arithmetic errors.
⚠️ Deriving the demand curve from the PCC is a very common exam question, both algebraically and graphically. Be comfortable going in both directions.
⚠️ For U = X^a · Y^b, memorise the spending-share result: fraction on X = a/(a+b), fraction on Y = b/(a+b). It saves time and catches errors.
⚠️ On graph questions, always label the intercepts, the optimal bundles, the indifference curves, and the PCC itself. Partial marks come from correct labelling even if a calculation is off.
True or False: For Cobb-Douglas preferences, the fraction of income spent on each good depends on prices.
Fill in the blank: The PCC traces optimal bundles as ______ changes, holding income and the other price constant.
True or False: If U = X²Y and M = 240, P_Y = 1, the consumer always buys Y = 80 regardless of P_X.
Fill in the blank: The vertical intercept of the budget line is M / ______.
True or False: The demand curve for X is derived from the PCC by pairing each P_X with the corresponding optimal X.
Answers: 1. False (it depends only on exponents). 2. The price of one good. 3. True. 4. P_Y. 5. True.
Q: A consumer has U(X, Y) = X³Y², M = 500, P_Y = 2. What fraction of income is spent on X? How much X does the consumer buy when P_X = 5?
A: Fraction on X = 3/(3+2) = 3/5. Spending on X = (3/5)(500) = 300. Quantity: X = 300/5 = 60.
Q: You are given a PCC graph. All three optimal bundles have Y = 50, with X-values of 20, 40, and 100. Income is 200 and P_Y = 3. Find the three prices of X.
A: For each bundle, P_X = (M - P_Y · Y) / X = (200 - 150) / X = 50 / X. So P_X = 2.50 (when X = 20), 1.25 (when X = 40), 0.50 (when X = 100).
Q: Explain why the PCC for U = X²Y with P_Y = 1 is a horizontal line.
A: The spending share on Y is b/(a+b) = 1/3. So spending on Y = M/3 = 240/3 = 80. Since P_Y = 1, the consumer always buys Y = 80. Varying P_X changes only X, so the PCC is horizontal at Y = 80.
Q: A consumer has M = 80 to spend on Juice and Snacks. From a PCC graph, the optimal bundle at one price is (Snacks = 7, Juice = 15), and P_Juice = 4. What is the price of Snacks?
A: P_Snacks = (80 - 4 · 15) / 7 = (80 - 60) / 7 = 20/7 = 2.86 (approximately).
The PCC feeds directly into the demand curve, which you will use throughout the rest of the course when studying market equilibrium, consumer surplus, welfare analysis, and taxation. Understanding the PCC also prepares you for the Slutsky decomposition (income and substitution effects), since the same price changes that trace out the PCC can be broken into those two components.
For Cobb-Douglas, the constant spending-share result connects to the concept of unit income elasticity and the homothetic property. If you understand why Y stays constant in the PCC example above, you understand homotheticity, which matters for welfare comparisons and index-number theory later.
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