Difficulty: Intermediate | Prerequisites: MRS and indifference curves (see companion study notes), basic algebra, partial derivatives.
Once you know how to describe a consumer's preferences (utility functions, MRS, indifference curves), the next question is: what does the consumer actually buy? That depends on their income and prices, which define the budget constraint. The optimal bundle is where the consumer reaches the highest indifference curve they can afford. This section covers how to find that bundle for each utility type, and how the optimal bundle changes when a price changes, which traces out the price consumption curve (PCC). These mechanics are the foundation for deriving demand curves, which is where the course heads next.
The budget constraint is Px * X + Py * Y = M. For Cobb-Douglas, solve by setting MRS = Px/Py (tangency) and substituting into the budget constraint. For perfect complements, set the kink condition (the arguments of the MIN equal) and substitute into the budget constraint. For perfect substitutes, compare MRS to Px/Py to determine whether the consumer buys all X, all Y, or any mix along the budget line. The price consumption curve connects optimal bundles as one price varies.
Budget constraint
The set of all bundles the consumer can afford: Px * X + Py * Y = M, where M is income. Think of it as: the line that shows every combination of X and Y you can buy if you spend all your money.
Optimal bundle (utility-maximising bundle)
The bundle on the budget constraint that gives the highest utility. In simple terms, the best the consumer can do given their income and prices.
Tangency condition
For smooth, convex preferences (like Cobb-Douglas), the optimal interior bundle occurs where the indifference curve is tangent to the budget line: MRS = Px/Py.
Interior solution
An optimal bundle where the consumer buys positive amounts of both goods. This is the typical result for Cobb-Douglas preferences.
Corner solution
An optimal bundle where the consumer spends all income on just one good. This commonly occurs with perfect substitutes when MRS does not equal the price ratio.
Price consumption curve (PCC)
The locus of optimal bundles traced out as the price of one good changes, holding income and the other price constant. Think of it as: the path the consumer's chosen bundle follows as one good gets cheaper or more expensive.
Demand function
The relationship between the price of a good and the quantity demanded, derived from the optimal bundle. For Cobb-Douglas U = X^a * Y^b, the demand for X is X* = [a/(a+b)] * M/Px.
Equation: Px * X + Py * Y = M.
Slope of the budget line: -Px/Py.
X-intercept: M/Px (spend everything on X). Y-intercept: M/Py (spend everything on Y).
When Px changes, the X-intercept shifts but the Y-intercept stays fixed. This is what generates the PCC.
The general demand formulas for Cobb-Douglas are:
X* = [a / (a + b)] * (M / Px)
Y* = [b / (a + b)] * (M / Py)
The consumer spends a fixed share of income on each good: fraction a/(a+b) on X, and b/(a+b) on Y.
Worked example: U = X^(1/4) * Y^(3/4), M = 60, Px = 5, Py = 3.
X* = (0.25 / 1.00) * (60 / 5) = 0.25 * 12 = 3.
Y* = (0.75 / 1.00) * (60 / 3) = 0.75 * 20 = 15.
Optimal bundle: (3, 15). Check budget: 5(3) + 3(15) = 15 + 45 = 60. Correct.
Step 1: Set the kink condition aX = bY, which gives Y = (a/b)X.
Step 2: Substitute into the budget constraint Px * X + Py * (a/b)X = M and solve for X.
Worked example: U = MIN(4X, Y), M = 63, Px = 3, Py = 1.
Kink condition: 4X = Y.
Budget: 3X + 1(4X) = 63, so 7X = 63, giving X* = 9.
Y* = 4(9) = 36. Optimal bundle: (9, 36).
Compare MRS = a/b to the price ratio Px/Py:
If a/b > Px/Py: the consumer values X more (per pound spent) than the market charges. Buy only X. X* = M/Px, Y* = 0.
If a/b < Px/Py: the consumer values X less than what the market charges. Buy only Y. X* = 0, Y* = M/Py.
If a/b = Px/Py: any bundle on the budget line is optimal.
Worked example: U = 4X + Y, M = 24, Px = 6, Py = 2.
MRS = 4/1 = 4. Price ratio = 6/2 = 3.
Since 4 > 3, the consumer values X relatively more than the market. Buy only X.
X* = 24/6 = 4, Y* = 0. Optimal bundle: (4, 0).
Verification: U(4,0) = 16. Compare to all-Y corner: U(0,12) = 12. And interior point (2,6): U = 8 + 6 = 14. The all-X corner wins.
To trace the PCC, solve for the optimal bundle at several different values of Px (or Py), holding the other price and income fixed, then connect the optimal bundles.
Worked example: U = MIN(X, 2Y), M = 48, Py = 2, Px varies.
Kink condition: X = 2Y, so Y = X/2.
Budget: Px * X + 2(X/2) = 48, giving Px * X + X = 48, so X(Px + 1) = 48.
X* = 48 / (Px + 1), Y* = 24 / (Px + 1).
Results at each price:
Px = 1: X* = 24, Y* = 12.
Px = 2: X* = 16, Y* = 8.
Px = 3: X* = 12, Y* = 6.
The PCC connects (24, 12), (16, 8), and (12, 6). Notice all these bundles lie along the line Y = X/2 (the kink line), which is always the case for perfect complements.
Utility type | Optimisation method | Demand for X | Demand for Y |
|---|---|---|---|
Cobb-Douglas X^a * Y^b | Tangency: MRS = Px/Py | [a/(a+b)] * M/Px | [b/(a+b)] * M/Py |
Perfect complements MIN(aX, bY) | Kink: aX = bY + budget | M / (Px + Py * a/b) | (a/b) * M / (Px + Py * a/b) |
Perfect substitutes aX + bY | Compare a/b to Px/Py | M/Px if a/b > Px/Py; 0 otherwise | M/Py if a/b < Px/Py; 0 otherwise |
Budget constraint: Px * X + Py * Y = M
Budget line slope: -Px / Py
X-intercept: M / Px
Y-intercept: M / Py
Budget constraints are the most directly observable part of consumer theory. Every time you decide between two purchases given a limited bank balance, you are solving this problem intuitively. The corner-solution logic for perfect substitutes explains why most people buy only one brand of a commodity good (whichever gives more utility per pound). The fixed-proportions logic for complements explains why a drop in the price of printers increases demand for ink cartridges. The PCC is the intermediate step between consumer theory and the demand curve you see in every introductory economics class.
Students often set MRS = Px/Py for perfect complements. This does not work because the MIN function has no well-defined MRS at the kink. Use the kink condition instead.
For perfect substitutes, students sometimes solve using the tangency condition MRS = Px/Py and find an interior solution. Unless MRS happens to equal Px/Py exactly, the solution is a corner, not interior.
Students forget to verify corner solutions for perfect substitutes. Always check utility at both corners (all X vs all Y) and confirm your answer gives higher utility.
When graphing the budget constraint, students sometimes draw it with the wrong slope sign or mix up which intercept moves when a price changes. Remember: only the intercept of the good whose price changed moves; the other intercept stays fixed.
For the PCC, students sometimes connect the wrong points. The PCC connects optimal bundles, not just any points on different budget lines.
Expect at least one question asking you to find the optimal bundle for each utility type. Know the method cold for all three.
Graphing the budget constraint with correct intercepts and labelling the optimal bundle on it is a standard exam task.
The PCC question often asks you to solve for optimal bundles at three different prices, graph all three budget lines, mark each optimal bundle, and connect them with the PCC.
For perfect substitutes, examiners test whether you can correctly identify corner vs interior solutions. Show your comparison of MRS to Px/Py.
True or false: For Cobb-Douglas preferences, the consumer always spends the same fraction of income on each good regardless of prices.
Fill in the blank: For U = MIN(2X, 5Y), the kink condition is ___.
True or false: If U = 3X + 7Y and Px/Py = 2, the consumer buys only Y.
Fill in the blank: The slope of the budget line is ___.
True or false: The PCC for perfect complements always lies along the kink line.
Answers: 1. True (the fixed-share property of Cobb-Douglas). 2. 2X = 5Y. 3. True (MRS = 3/7 < 2 = Px/Py, so Y is the better deal). 4. -Px/Py. 5. True (optimal bundles for complements always sit at the kink).
Q: U(X,Y) = X^(1/3) * Y^(2/3). The consumer has income M = 90, Px = 5, Py = 10. Find the optimal bundle.
A: X* = (1/3)(90/5) = 6. Y* = (2/3)(90/10) = 6. Optimal bundle: (6, 6). Check: 5(6) + 10(6) = 30 + 60 = 90.
Q: U(X,Y) = MIN(3X, Y). M = 100, Px = 4, Py = 2. Find the optimal bundle.
A: Kink condition: 3X = Y. Budget: 4X + 2(3X) = 100, so 10X = 100, X* = 10, Y* = 30.
Q: U(X,Y) = 2X + 3Y. M = 36, Px = 4, Py = 6. Should the consumer buy X, Y, or a mix?
A: MRS = 2/3. Px/Py = 4/6 = 2/3. Since MRS = Px/Py, any bundle on the budget line is optimal. The consumer is indifferent between all affordable combinations.
Q: For U(X,Y) = MIN(X, 3Y), M = 60, Py = 5, find the optimal X and Y when Px = 2, Px = 4, and Px = 6. Describe the PCC.
A: Kink: X = 3Y. Budget: Px * X + 5(X/3) = 60, so X(Px + 5/3) = 60.
Px = 2: X = 60 / (2 + 5/3) = 60 / (11/3) = 180/11 = 16.36, Y = 5.45.
Px = 4: X = 60 / (4 + 5/3) = 60 / (17/3) = 180/17 = 10.59, Y = 3.53.
Px = 6: X = 60 / (6 + 5/3) = 60 / (23/3) = 180/23 = 7.83, Y = 2.61.
The PCC lies along the kink line Y = X/3, moving closer to the origin as Px rises.
Q: Explain why a consumer with perfect-substitute preferences almost never buys both goods.
A: With a constant MRS, one good always delivers more utility per unit of spending than the other, unless MRS exactly equals the price ratio. The consumer maximises utility by spending everything on whichever good is the better deal, producing a corner solution.
The optimal bundle is the starting point for deriving individual demand curves: hold income and the other price fixed, vary Px, and plot X* against Px. This is exactly what the PCC does in (X, Y) space before you rotate it into (X, Px) space. The income consumption curve (ICC) does the same thing but varies M instead of a price, and leads to Engel curves. These tools also connect to the Slutsky decomposition later in the course, where a price change is broken into substitution and income effects.
In welfare economics, the optimal bundle determines how much consumer surplus a price change creates or destroys. In general equilibrium, two consumers' budget constraints and optimal bundles together pin down the equilibrium in an Edgeworth box.
budget constraint, budget line, optimal bundle, utility maximisation, tangency condition, corner solution, interior solution, price consumption curve, PCC, demand function, Cobb-Douglas demand, perfect complements optimisation, Leontief demand, perfect substitutes corner solution, income consumption curve, ICC, Engel curve, Slutsky decomposition, ECON 302, intermediate microeconomics, consumer choice, constrained optimisation