Difficulty: Intermediate | Prerequisites: MRS and indifference curves (see companion notes), basic algebra
This is where consumer theory becomes practical. Once you know a consumer's preferences (utility function) and their constraints (income and prices), you can solve for the bundle they will choose. The method for finding the optimal bundle depends entirely on the type of utility function, so recognising whether you are dealing with Cobb-Douglas, perfect complements, or perfect substitutes is the first step every time. The price consumption curve extends this by showing how the optimal bundle shifts when one price changes.
The budget constraint shows what a consumer can afford. The optimal bundle is where the consumer reaches the highest indifference curve while staying on or inside the budget line. For Cobb-Douglas, this is a tangency point. For perfect complements, it is on the kink line. For perfect substitutes, it is usually a corner solution. The price consumption curve traces how the optimal bundle moves as one good's price changes.
Budget constraint (budget line)
The set of all bundles (X, Y) a consumer can just afford given their income M and prices Px, Py. The equation is Px * X + Py * Y = M. Graphically, it is a straight line with slope -Px/Py, X-intercept M/Px, and Y-intercept M/Py. Think of it as the boundary of the consumer's shopping possibilities.
Feasible set (budget set)
All bundles on or below the budget line. Everything the consumer can afford, including bundles that do not use up all their income.
Optimal bundle (consumer's choice)
The bundle on the budget constraint that maximises the consumer's utility. This is the point the consumer will choose if they are rational. The method for finding it depends on the utility function type.
Tangency condition
For Cobb-Douglas (and other "well-behaved" preferences), the optimal interior bundle is where MRS = Px/Py. The indifference curve is tangent to the budget line at this point. This condition does not apply to perfect complements or perfect substitutes.
Corner solution
An optimal bundle where the consumer spends all their income on just one good (X = 0 or Y = 0). This happens with perfect substitutes when MRS does not equal the price ratio. The consumer picks whichever good gives more utility per pound spent.
Price consumption curve (PCC)
The set of optimal bundles traced out as the price of one good changes, holding income and the other good's price fixed. It shows how demand for both goods responds to a price change.
Method: tangency + budget constraint (two equations, two unknowns)
Step 1: Set MRS = Px/Py. Since MRS = (a/b)(Y/X), this gives (a/b)(Y/X) = Px/Py.
Step 2: Solve for Y in terms of X (or vice versa) from the tangency condition.
Step 3: Substitute into the budget constraint Px * X + Py * Y = M and solve.
Shortcut: For U = X^a * Y^b, the consumer always spends fraction a/(a+b) of income on X and b/(a+b) on Y. So X* = [a/(a+b)] * M/Px and Y* = [b/(a+b)] * M/Py.
Worked example (HW1 Q2a):
U(X,Y) = X^(1/3) * Y^(2/3), M = 18, Px = 3, Py = 2.
Spending shares: a/(a+b) = (1/3)/1 = 1/3 on X, 2/3 on Y.
X* = (1/3)(18)/3 = 6/3 = 2.
Y* = (2/3)(18)/2 = 12/2 = 6.
Optimal bundle: (2, 6). Budget line: X-intercept = 18/3 = 6, Y-intercept = 18/2 = 9.
Method: kink condition + budget constraint
Step 1: At the optimum, the consumer is always at the kink, where aX = bY. Spending money on extra X beyond the kink (or extra Y) is wasted because utility is determined by whichever argument is smaller.
Step 2: From aX = bY, get Y = (a/b)X.
Step 3: Substitute into Px * X + Py * Y = M and solve for X, then Y.
Worked example (HW1 Q2b):
U(X,Y) = MIN(6X, Y), M = 63, Px = 3, Py = 1.
Kink condition: 6X = Y.
Budget: 3X + 1(Y) = 63. Substitute Y = 6X: 3X + 6X = 63, so 9X = 63, X* = 7.
Y* = 6(7) = 42. Optimal bundle: (7, 42).
Budget line: X-intercept = 63/3 = 21, Y-intercept = 63/1 = 63.
Method: compare MRS to price ratio
MRS = a/b (constant). Compare to Px/Py.
If a/b > Px/Py: the consumer values X (relative to Y) more than the market does. Buy only X. Optimal bundle: X* = M/Px, Y* = 0.
If a/b < Px/Py: the consumer values X less than the market does. Buy only Y. Optimal bundle: X* = 0, Y* = M/Py.
If a/b = Px/Py: the consumer is indifferent along the entire budget line. Any bundle on the budget line is optimal.
Worked example (HW1 Q2c):
U(X,Y) = 2X + Y, M = 18, Px = 6, Py = 2.
MRS = 2/1 = 2. Price ratio = Px/Py = 6/2 = 3.
Since MRS (2) < price ratio (3), X is too expensive relative to how much the consumer values it. Buy only Y.
Optimal bundle: X* = 0, Y* = 18/2 = 9. This is a corner solution at (0, 9).
Budget constraint
Px * X + Py * Y = M
Slope of budget line = -Px/Py
X-intercept = M/Px (spend everything on X)
Y-intercept = M/Py (spend everything on Y)
Cobb-Douglas optimality (U = X^a * Y^b)
Tangency: (a/b)(Y/X) = Px/Py
Shortcut: X* = [a/(a+b)] * M/Px, Y* = [b/(a+b)] * M/Py
Perfect complements optimality (U = MIN(aX, bY))
Kink condition: aX = bY, i.e. Y = (a/b)X
Substitute into budget: Px * X + Py * (a/b) * X = M
X* = M / [Px + Py * (a/b)], Y* = (a/b) * X*
Perfect substitutes optimality (U = aX + bY)
If a/b > Px/Py: X* = M/Px, Y* = 0
If a/b < Px/Py: X* = 0, Y* = M/Py
If a/b = Px/Py: any bundle on the budget line
The price consumption curve (PCC) shows how the optimal bundle changes as the price of one good varies, with income and the other price held constant. You find it by solving for the optimal bundle at each price and then connecting the dots.
U(X,Y) = MIN(3X, Y). M = 60, Py = 1. Find the optimal bundle for Px = 1, 2, and 3.
Step 1: Kink condition. At the optimum, 3X = Y (since U = MIN(3X, Y), the consumer wants 3X = Y).
Step 2: Solve at each price.
Px = 1: Budget is X + Y = 60. Substitute Y = 3X: X + 3X = 60, 4X = 60, X* = 15, Y* = 45. Budget line: X-intercept = 60, Y-intercept = 60.
Px = 2: Budget is 2X + Y = 60. Substitute Y = 3X: 2X + 3X = 60, 5X = 60, X* = 12, Y* = 36. Budget line: X-intercept = 30, Y-intercept = 60.
Px = 3: Budget is 3X + Y = 60. Substitute Y = 3X: 3X + 3X = 60, 6X = 60, X* = 10, Y* = 30. Budget line: X-intercept = 20, Y-intercept = 60.
Step 3: Graph. Plot all three budget lines (they share the same Y-intercept of 60 but have different X-intercepts). Mark each optimal bundle: (15, 45), (12, 36), (10, 30). The PCC is the line connecting these optimal bundles. Notice all three sit on the line Y = 3X, which is the kink line for this utility function.
Key insight: For perfect complements, the PCC always lies along the kink line Y = (a/b)X, because the optimal bundle is always at the kink regardless of prices. As Px rises, the consumer simply moves to a lower point along that same line.
Students often try to use the tangency condition (MRS = Px/Py) for perfect complements. This does not work. The MRS is undefined at the kink, so there is no tangency. Use the kink condition aX = bY instead.
With perfect substitutes, students sometimes try to find an interior tangency. Unless MRS happens to equal Px/Py exactly, the optimum is at a corner, not in the interior.
A frequent graphing error is drawing the budget line with the wrong slope. The slope is -Px/Py (negative), not -Py/Px. X is on the horizontal axis, so the price of the horizontal-axis good goes in the numerator.
Students sometimes forget that the optimal bundle must be on the budget line, not just below it. A consumer who does not spend all their income can always do better by buying more of something (assuming non-satiation).
Budget constraint problems are the workhorse of intermediate micro exams. Nearly every subsequent topic (income and substitution effects, labour supply, intertemporal choice) uses the same optimisation framework with different labels on the axes.
You will be asked to graph budget constraints with correct intercepts and slope. Label everything.
For Cobb-Douglas, know the spending-share shortcut cold. It saves time and reduces algebra errors.
For perfect complements, the kink-condition-plus-budget method appears regularly. Practice it until it is automatic.
For perfect substitutes, the key test is comparing MRS to Px/Py. Know the three cases and which corner to pick.
Price consumption curve questions require you to solve the optimisation problem multiple times (once per price). Organise your work clearly.
True or false: For Cobb-Douglas preferences, the optimal bundle is always in the interior of the budget line (the consumer buys some of both goods). (True.)
Fill in the blank: If U = X^(2/5) * Y^(3/5), the consumer spends ______ of income on X. (Answer: 2/5, or 40%.)
True or false: For perfect substitutes with MRS > Px/Py, the consumer buys only Y. (False: the consumer buys only X, because X gives more utility per dollar than Y does.)
Fill in the blank: The slope of the budget line is ______. (Answer: -Px/Py.)
True or false: For U = MIN(4X, 2Y), the optimal bundle always lies on the line Y = 2X. (True: the kink condition 4X = 2Y gives Y = 2X.)
Q: U(X,Y) = X^(1/2) * Y^(1/2). M = 100, Px = 5, Py = 10. Find the optimal bundle.
A: Spending shares: 1/2 on each good. X* = (1/2)(100)/5 = 10. Y* = (1/2)(100)/10 = 5. Optimal bundle: (10, 5).
Q: U(X,Y) = MIN(2X, 3Y). M = 50, Px = 4, Py = 2. Find the optimal bundle.
A: Kink condition: 2X = 3Y, so Y = (2/3)X. Budget: 4X + 2(2/3)X = 50, so 4X + (4/3)X = 50, (16/3)X = 50, X* = 150/16 = 9.375. Y* = (2/3)(9.375) = 6.25. Optimal bundle: (9.375, 6.25).
Q: U(X,Y) = 5X + 2Y. M = 40, Px = 10, Py = 5. Find the optimal bundle.
A: MRS = 5/2 = 2.5. Price ratio = 10/5 = 2. Since MRS (2.5) > price ratio (2), X gives more bang for the buck. Buy only X: X* = 40/10 = 4, Y* = 0. Optimal bundle: (4, 0).
Q: For the perfect complements problem in Q3 of HW1, why does the Y-intercept of the budget line stay the same at 60 for all three prices of X?
A: The Y-intercept equals M/Py = 60/1 = 60. Since neither income (M = 60) nor the price of Y (Py = 1) changes across the three scenarios, the Y-intercept is always 60. Only the X-intercept (M/Px) changes as Px changes.
The tangency condition MRS = Px/Py reappears in nearly every optimisation problem in intermediate micro, including labour-leisure choice (where the "goods" are consumption and leisure) and intertemporal choice (where the "goods" are consumption today versus consumption tomorrow). The corner-solution logic for perfect substitutes is the same logic that determines specialisation in trade models.
The price consumption curve is the building block for deriving individual demand curves. Each point on the PCC gives one point on the demand curve for good X: the price of X and the quantity of X chosen at that price.
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