Difficulty: Intermediate | Prerequisites: utility functions, MRS, indifference curves (see companion notes).
This builds on the utility function and MRS material. Once you know the shape of a consumer's indifference curves, the next question is: given a limited budget, which bundle will they choose? The answer depends on the type of utility function. Cobb-Douglas gives interior (tangency) solutions, perfect complements give kink solutions, and perfect substitutes give corner solutions. The price consumption curve then traces how the optimal bundle changes as one good's price varies, which is the foundation for deriving individual demand curves.
The budget constraint shows what a consumer can afford. The optimal bundle is where the consumer reaches the highest indifference curve that still touches the budget line. For Cobb-Douglas, set MRS = price ratio and solve with the budget equation. For perfect complements, use the fixed-ratio condition plus the budget. For perfect substitutes, compare MRS to the price ratio and pick a corner. The price consumption curve connects optimal bundles as one price changes.
Budget constraint
The set of all bundles a consumer can afford given their income and the prices of goods. The equation is PxX + PyY = M (or with inequality, PxX + PyY <= M). Think of it as the boundary of your shopping possibilities.
Optimal bundle
The bundle that maximises the consumer's utility subject to their budget constraint. This is the best the consumer can do given what they can afford.
Tangency condition
The condition MRS = Px/Py, which holds at an interior optimum for smooth, convex preferences (e.g. Cobb-Douglas). In simple terms, the indifference curve just touches the budget line, matching its slope.
Interior solution
An optimum where the consumer buys positive amounts of both goods. Typical of Cobb-Douglas preferences.
Corner solution
An optimum where the consumer spends their entire budget on just one good, buying zero of the other. Typical of perfect substitutes when MRS differs from the price ratio.
Price consumption curve (PCC)
The curve connecting the optimal bundles as the price of one good changes, holding income and the other good's price fixed. It traces how the consumer's choice shifts and is the basis for deriving the individual demand curve.
Income (M)
The total amount the consumer has available to spend on goods X and Y. Sometimes called the budget or wealth.
The budget equation
Px * X + Py * Y = M
Rearranged: Y = M/Py - (Px/Py) * X
This is a straight line with Y-intercept M/Py, X-intercept M/Px, and slope -(Px/Py)
Graphing the budget constraint
Plot the Y-intercept (0, M/Py) and the X-intercept (M/Px, 0), then connect with a straight line
The slope of the budget line = -Px/Py (the negative of the price ratio)
The feasible set is everything on or below this line (assuming the consumer can spend less than M)
What shifts the budget line
A change in income M shifts the line parallel (outward if M rises, inward if M falls)
A change in Px pivots the line around the Y-intercept (the Y-intercept stays fixed because M/Py is unchanged)
A change in Py pivots the line around the X-intercept
Method: tangency + budget (two equations, two unknowns)
Step 1: Set MRS = Px/Py. For U = X^a * Y^b, this gives (a/b)(Y/X) = Px/Py
Step 2: Solve for Y in terms of X (or vice versa)
Step 3: Substitute into the budget constraint PxX + PyY = M and solve
Shortcut for Cobb-Douglas
For U = X^a * Y^b, the consumer always spends a fraction a/(a+b) of income on X and b/(a+b) on Y. So:
X* = [a/(a+b)] * M/Px
Y* = [b/(a+b)] * M/Py
This shortcut works because Cobb-Douglas utility always produces constant expenditure shares.
Worked example: U(X,Y) = X^(1/4) * Y^(3/4), M = $60, Px = $5, Py = $3
Expenditure shares: a = 1/4, b = 3/4, so a/(a+b) = 1/4, b/(a+b) = 3/4
Spending on X = (1/4)(60) = $15. So X* = 15/5 = 3
Spending on Y = (3/4)(60) = $45. So Y* = 45/3 = 15
Optimal bundle: (3, 15)
Check: 5(3) + 3(15) = 15 + 45 = 60. Budget is exhausted.
Graphing the solution
Budget line: Y-intercept = 60/3 = 20, X-intercept = 60/5 = 12
Plot the line from (0, 20) to (12, 0)
The optimal bundle (3, 15) lies on this line, with an indifference curve tangent to the budget line at that point
Method: fixed ratio + budget (two equations, two unknowns)
Step 1: The consumer always buys at the kink, where aX = bY. Solve for Y = (a/b)X
Step 2: Substitute into the budget constraint PxX + PyY = M
Step 3: Solve for X, then Y
The tangency condition does not apply here because the indifference curve has a kink, not a smooth tangent point.
Worked example: U(X,Y) = MIN(4X, Y), M = $63, Px = $3, Py = $1
At the kink: 4X = Y
Budget: 3X + 1Y = 63
Substitute Y = 4X: 3X + 4X = 63, so 7X = 63, giving X* = 9
Then Y* = 4(9) = 36
Optimal bundle: (9, 36)
Check: 3(9) + 1(36) = 27 + 36 = 63. Budget is exhausted.
Utility: MIN(4(9), 36) = MIN(36, 36) = 36. The consumer is at the kink.
Graphing the solution
Budget line: Y-intercept = 63/1 = 63, X-intercept = 63/3 = 21
Plot the line from (0, 63) to (21, 0)
Draw the ray Y = 4X from the origin (the line of kinks)
The optimal bundle (9, 36) is where the budget line meets the ray
Draw the L-shaped indifference curve with its kink at (9, 36)
Method: compare MRS to price ratio
For U = aX + bY, MRS = a/b everywhere. Compare this to Px/Py:
If MRS > Px/Py (i.e. a/b > Px/Py): the consumer values X relatively more than the market does. Buy only X. Corner solution: X* = M/Px, Y* = 0.
If MRS < Px/Py (i.e. a/b < Px/Py): the consumer values Y relatively more. Buy only Y. Corner solution: X* = 0, Y* = M/Py.
If MRS = Px/Py: the consumer is indifferent along the entire budget line. Any bundle on the budget line is optimal.
Worked example: U(X,Y) = 4X + Y, M = $24, Px = $6, Py = $2
MRS = 4/1 = 4
Price ratio Px/Py = 6/2 = 3
Since MRS = 4 > Px/Py = 3, the consumer values X more than the market price requires. They should buy only X.
But wait: let us double-check by computing utility per dollar. Utility per dollar from X = MUx/Px = 4/6 = 2/3. Utility per dollar from Y = MUy/Py = 1/2. Since 2/3 > 1/2, X gives more bang per buck.
Corner solution: X* = 24/6 = 4, Y* = 0
Optimal bundle: (4, 0)
Utility: 4(4) + 0 = 16
Graphing the solution
Budget line: Y-intercept = 24/2 = 12, X-intercept = 24/6 = 4
The budget line goes from (0, 12) to (4, 0), with slope -Px/Py = -3
The indifference curves are lines with slope -MRS = -4 (steeper than the budget line)
The highest indifference curve that touches the budget line touches it at the X-intercept corner: (4, 0)
What it is
The price consumption curve connects all the optimal bundles as the price of one good (say Px) changes, holding income M and the other price Py constant. Each point on the curve is the solution to a separate optimisation problem with a different budget constraint.
How to construct it
Solve for the optimal bundle at each given price
Plot all optimal bundles on the same graph
Connect them: the resulting curve is the PCC
Worked example: U(X,Y) = MIN(X, 2Y), M = $48, Py = $2
At the kink: X = 2Y, so Y = X/2.
Solve for each Px:
When Px = $1:
Budget: 1X + 2Y = 48
Substitute Y = X/2: X + 2(X/2) = X + X = 2X = 48
X* = 24, Y* = 12
Optimal bundle: (24, 12)
When Px = $2:
Budget: 2X + 2Y = 48
Substitute Y = X/2: 2X + 2(X/2) = 2X + X = 3X = 48
X* = 16, Y* = 8
Optimal bundle: (16, 8)
When Px = $3:
Budget: 3X + 2Y = 48
Substitute Y = X/2: 3X + 2(X/2) = 3X + X = 4X = 48
X* = 12, Y* = 6
Optimal bundle: (12, 6)
Graphing the PCC
Draw all three budget lines on the same graph. They all share the Y-intercept at (0, 24) because M/Py = 48/2 = 24 in each case. The X-intercepts differ: 48/1 = 48, 48/2 = 24, 48/3 = 16.
Mark each optimal bundle on its budget line: (24, 12), (16, 8), (12, 6). Draw the L-shaped indifference curve through each.
Connect the three optimal bundles: this is the price consumption curve. Notice all three bundles lie along the line Y = X/2 (the kink ray), because perfect complements always consume at the kink regardless of prices.
Students often try to use the tangency condition (MRS = Px/Py) for perfect complements. This does not work because the indifference curve has a kink, not a smooth point of tangency. Use the fixed-ratio condition instead.
For perfect substitutes, students sometimes assume the answer is always a corner. That is usually correct, but in the special case where MRS = Px/Py exactly, every bundle on the budget line is equally good.
When comparing MRS to Px/Py for perfect substitutes, students sometimes get the comparison backwards. Remember: if MRS > Px/Py, buy X (the consumer's valuation of X relative to Y exceeds the market's, so X is the bargain). If MRS < Px/Py, buy Y.
Students forget to label the kink ray on their graph for perfect complements problems. The kink ray is essential because it shows why the optimal bundle is where it is.
Solving for optimal bundles under all three utility types is a staple of intermediate micro exams. Expect at least one of each.
The Cobb-Douglas expenditure share shortcut (spend a/(a+b) of income on X) saves a lot of algebra. Know it.
For perfect complements, always start with the kink condition and substitute into the budget. This is a mechanical two-step process; getting comfortable with it guarantees points.
Accurate graphing matters: the budget line must have correct intercepts, the optimal bundle must sit on the budget line, and the indifference curve must be tangent (Cobb-Douglas) or have its kink at the optimal point (perfect complements) or touch the corner (perfect substitutes).
Price consumption curve questions test whether you can repeat the optimisation at multiple prices and draw everything on one graph. Label each budget line and each optimal bundle clearly.
True or False: For Cobb-Douglas utility U = X^(2/5) * Y^(3/5), the consumer spends 40% of income on X.
Fill in the blank: For U(X,Y) = MIN(3X, Y), the optimal consumption ratio is Y = ____ X.
True or False: If MRS = 2 and Px/Py = 3, a consumer with perfect substitute preferences should spend all income on Y.
Fill in the blank: A budget constraint with M = 100, Px = 5, Py = 10 has X-intercept ____ and Y-intercept ____.
True or False: The price consumption curve for perfect complements always lies along the kink ray.
Answers: 1. True (a/(a+b) = 2/5 = 40%). 2. Y = 3X. 3. True (MRS < Px/Py, so Y gives better value). 4. X-intercept = 20, Y-intercept = 10. 5. True.
Q: U(X,Y) = X^(1/3) * Y^(2/3), M = $90, Px = $5, Py = $2. Find the optimal bundle.
A: Expenditure on X = (1/3)(90) = $30, so X* = 30/5 = 6. Expenditure on Y = (2/3)(90) = $60, so Y* = 60/2 = 30. Optimal bundle: (6, 30).
Q: U(X,Y) = MIN(2X, 5Y), M = $70, Px = $4, Py = $2. Find the optimal bundle.
A: At the kink, 2X = 5Y, so Y = (2/5)X. Budget: 4X + 2(2/5)X = 4X + (4/5)X = (24/5)X = 70. X* = 350/24 = 14.58 (or 175/12). Y* = (2/5)(175/12) = 350/60 = 35/6 = 5.83. Optimal bundle: (175/12, 35/6).
Q: U(X,Y) = 3X + 2Y, M = $30, Px = $5, Py = $4. Where is the optimal bundle?
A: MRS = 3/2 = 1.5. Px/Py = 5/4 = 1.25. Since MRS > Px/Py, the consumer buys only X. X* = 30/5 = 6, Y* = 0. Optimal bundle: (6, 0).
Q: For U(X,Y) = MIN(X, 2Y), with M = $48, Py = $2, explain what happens to the optimal X as Px rises from $1 to $2 to $3.*
A: X* falls from 24 to 16 to 12. As Px rises, the budget line pivots inward (steeper), and the point where it hits the kink ray Y = X/2 moves closer to the origin. The consumer buys less of both goods, but always in the fixed ratio X = 2Y.
The optimal bundle is the foundation for deriving individual demand curves: solve for X* as a function of Px, and you have the demand function for X. The price consumption curve is the graphical version of that same relationship. Income and substitution effects (Slutsky decomposition) build directly on this material by asking what happens when a price change is broken into two parts. Welfare measures such as consumer surplus also rest on the demand curve, which in turn rests on the optimisation you learn here.
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