Difficulty: Intermediate | Prerequisites: utility maximisation basics, MRS, budget constraints, Lagrangian method.
Not every utility function behaves like Cobb-Douglas. Some consumers view goods as interchangeable (perfect substitutes), some insist on fixed proportions (perfect complements), and some have preferences that produce corner solutions rather than interior tangencies. This set of notes walks through the four main special cases you will encounter in intermediate micro, each with a worked numerical example using P_x = 4, P_y = 6, and M = 116. Knowing how to recognise which type you are facing and which solution method applies is the core skill being tested.
Perfect substitutes lead to corner solutions (all spending on the cheaper good per unit of utility). Perfect complements use a fixed-ratio condition instead of tangency. Quasi-linear utility has a clean interior solution for the nonlinear good, with the remainder spent on the linear good. Convex preferences (like U = x^2 + 2y^2) also yield interior solutions via the Lagrangian.
Perfect substitutes
Goods that a consumer is willing to trade at a constant rate regardless of how much of each they have. Utility takes the form U = ax + by. Indifference curves are straight lines. Think of it as two brands of the same thing, like generic and branded paracetamol: the consumer just wants the most pain relief per pound.
Perfect complements (Leontief preferences)
Goods that are consumed in fixed proportions. Utility takes the form U = min(ax, by). Indifference curves are L-shaped. In simple terms, this means having more of one good without more of the other adds zero utility, like left shoes without right shoes.
Corner solution
An optimal bundle where the consumer spends all income on one good and zero on the other. This happens when the MRS does not equal the price ratio at any interior point, typically with perfect substitutes. Think of it as going all-in on whichever good gives more bang for the buck.
Interior solution
An optimal bundle where the consumer buys positive amounts of both goods. The tangency condition holds, and the solution sits in the interior of the budget line rather than at an endpoint.
Quasi-linear utility
Utility of the form U = f(x) + y, where f is nonlinear (often a square root or log) and y enters linearly. The demand for x depends only on its own price and not on income (above a threshold), while all remaining income goes to y. In simple terms, this means one good has diminishing marginal returns while the other serves as a "money" good.
Convex preferences
Preferences where averages of bundles are weakly preferred to extremes. If the utility function is strictly quasi-concave, the consumer prefers mixtures. This is the standard assumption that gives unique interior solutions. Think of it as preferring a balanced diet over eating only rice or only chicken.
All four problems below use the same parameters: P_x = 4, P_y = 6, M = 116.
MU_x = 2, MU_y = 1.
Check the tangency condition: MU_x / P_x vs MU_y / P_y gives 2/4 = 0.5 vs 1/6 = 0.167.
Since MU_x / P_x > MU_y / P_y, good x delivers more utility per pound at every quantity. The consumer spends everything on x.
Corner solution: x* = M / P_x = 116 / 4 = 29, y* = 0.
The decision rule for perfect substitutes is straightforward: compare MU_x / P_x with MU_y / P_y. Whichever is larger gets all the spending. If they are equal, any bundle on the budget line is optimal.
With Leontief preferences, the consumer always picks the kink of the L-shaped indifference curve, where the two arguments of the min function are equal: 2x = 3y.
This gives the fixed ratio: x = 3y/2.
Substitute into the budget constraint: (3y/2) * 4 + y * 6 = 116.
6y + 6y = 116
12y = 116
y = 116/12, which rounds to approximately 9.67*, and x = 3(9.67)/2, which rounds to approximately 14.5*.
In the problem set's integer treatment: y* = 9 (or the rounded value used), x* = 15 (adjusted to satisfy 2x = 3y exactly, giving 2(15) = 30 and 3(10) = 30, so x* = 15, y* = 10 if we use y = 116/12 more carefully).
The key step: always set the arguments of the min function equal and then use the budget constraint.
Set up the Lagrangian: L = x^2 + 2y^2 + lambda * (M - P_x * x - P_y * y).
First-order conditions:
dL/dx = 2x - lambda * P_x = 0, so lambda = 2x / P_x
dL/dy = 4y - lambda * P_y = 0, so lambda = 4y / P_y
dL/d(lambda) = M - P_x * x - P_y * y = 0
Set the two expressions for lambda equal: 2x / P_x = 4y / P_y.
2x / 4 = 4y / 6
12x = 16y, so 3x = 4y, giving x = 4y/3.
Substitute into the budget constraint: (4y/3) * 4 + 6y = 116.
16y/3 + 6y = 116
16y + 18y = 348
34y = 348
y = 348/34, approximately 10.24*.
The problem set rounds to y = 10, x = 4(10)/3, approximately 13.3**.
Note: U = x^2 + 2y^2 is convex (not concave), which means the Lagrangian FOCs find a minimum of utility on the budget line, not a maximum. The actual maximum is a corner solution. This is a subtle but important point that some problem sets gloss over. Check the second-order conditions or compare utility at interior and corner points.
Set up the Lagrangian: L = 8 * sqrt(x) + y + lambda * (M - P_x * x - P_y * y).
First-order conditions:
dL/dx = 4 / sqrt(x) - lambda * P_x = 0, so lambda = 4 / (sqrt(x) * P_x)
dL/dy = 1 - lambda * P_y = 0, so lambda = 1 / P_y
Budget: P_x * x + P_y * y = M
Set lambda expressions equal: 4 / (sqrt(x) * P_x) = 1 / P_y.
4 * P_y = sqrt(x) * P_x
With P_x = 4, P_y = 6: 4 * 6 = sqrt(x) * 4, so 24 = 4 * sqrt(x), giving sqrt(x) = 6.
x = 36.*
Substitute into the budget constraint: 36 * 4 + 6y = 116, so 144 + 6y = 116.
This gives 6y = -28, which is negative. This signals that the consumer cannot afford the interior optimum for x and still buy any y.
When this happens, the consumer sets y = 0 and spends everything on x: x* = 116/4 = 29.
However, the problem set solution appears to treat M = 116 as large enough for an interior solution (possibly different parameter values were intended). The method itself is clear: solve for x from the tangency condition, check whether the remaining income covers a non-negative y, and if not, go to a corner.
The hallmark of quasi-linear utility: demand for x does not depend on income (above a threshold), because all extra income goes to y.
Perfect substitutes (U = ax + by)
Compare MU_x / P_x with MU_y / P_y. If a/P_x > b/P_y, buy only x. If a/P_x < b/P_y, buy only y. If equal, any combination on the budget line works.
Perfect complements (U = min(ax, by))
Set ax = by to find the consumption ratio. Substitute into the budget constraint to find the quantities. There is no tangency condition because indifference curves are not smooth at the kink.
Quasi-linear (U = f(x) + y)
From f'(x) / P_x = 1 / P_y, solve for x. This gives x independent of income. Then y = (M - P_x * x) / P_y. If y would be negative, set y = 0 and x = M / P_x.
General interior solution (Lagrangian)
Set up L = U(x,y) + lambda * (M - P_x * x - P_y * y). FOCs give two expressions for lambda. Set them equal (tangency), substitute into budget constraint, solve. Always check second-order conditions for non-standard utility functions.
Students often apply the tangency condition to perfect substitutes. Since the MRS is constant (the indifference curves are straight lines), the MRS either exceeds, falls short of, or equals the price ratio everywhere. There is no interior tangency to find unless MRS equals the price ratio exactly.
With perfect complements, students sometimes try to differentiate the min function. The min function is not differentiable at the kink. Use the equality condition ax = by instead.
Students forget to check whether the quasi-linear interior solution is feasible. If solving for x from the tangency condition leaves negative income for y, the answer is a corner solution with y = 0.
When U is convex rather than concave (e.g. U = x^2 + 2y^2), the Lagrangian FOCs find a utility minimum on the budget line, not a maximum. The true optimum is at a corner. This catches students who apply the Lagrangian mechanically without checking concavity.
Recognising the utility function type is the first step on any exam problem. Get this wrong and you will apply the wrong method.
Perfect substitutes and perfect complements are tested frequently because the solution method differs fundamentally from the standard Lagrangian. Expect at least one on every exam.
Quasi-linear utility is a favourite for policy questions (e.g. taxation, subsidies) because the demand for the nonlinear good does not shift with income, simplifying welfare analysis.
Convex utility functions (concave indifference curves) are a trap. If the examiner gives you U = x^2 + 2y^2, they may be testing whether you recognise that the Lagrangian gives a minimum, not a maximum.
Lambda interpretation: for the quasi-linear case, lambda = 1/P_y at the optimum. This means the marginal utility of income is simply the inverse of the price of the linear good.
True or false: For U = 3x + 5y with P_x = 6 and P_y = 10, the consumer buys equal amounts of both goods. (False: MU_x/P_x = 3/6 = 0.5 and MU_y/P_y = 5/10 = 0.5, so the consumer is indifferent and any bundle on the budget line is optimal.)
Fill in the blank: For U = min(4x, 7y), the binding constraint at the optimum is ______ = ______ . (4x = 7y)
True or false: Quasi-linear demand for the nonlinear good depends on income. (False: it depends only on prices, provided income is large enough for an interior solution.)
True or false: If U = x^2 + y^2, the Lagrangian FOCs give the utility-maximising bundle. (False: U is convex, so the FOCs give a minimum. The maximum is at a corner.)
Fill in the blank: For perfect substitutes, the consumer buys only good x when MU_x / P_x ______ MU_y / P_y. (is greater than)
Q: Given U(x, y) = 2x + y, P_x = 4, P_y = 6, M = 116, find the optimal bundle.
A: MU_x / P_x = 2/4 = 0.5. MU_y / P_y = 1/6 = 0.167. Since 0.5 > 0.167, the consumer spends everything on x. x* = 116/4 = 29, y* = 0.
Q: Given U(x, y) = min(2x, 3y), P_x = 4, P_y = 6, M = 116, find the optimal bundle.
A: Set 2x = 3y, giving x = 3y/2. Substitute into 4x + 6y = 116: 4(3y/2) + 6y = 6y + 6y = 12y = 116, so y = 116/12. Then x = 3(116/12)/2 = 348/24. Simplifying: y is approximately 9.67 and x is approximately 14.5.
Q: For U(x, y) = 8 * sqrt(x) + y, explain why demand for x does not depend on income.
A: The tangency condition gives 4 / (sqrt(x) * P_x) = 1 / P_y. Income M does not appear in this equation, so x* depends only on prices. All extra income flows to y.
Q: If U(x, y) = x^2 + 2y^2, why might the Lagrangian method give a misleading answer?
A: Because U is convex. The second-order sufficient conditions for a maximum require the bordered Hessian to satisfy certain sign conditions, and here they fail. The Lagrangian FOCs identify a point where utility is minimised on the budget line. The true maximum is at one of the corners: either (M/P_x, 0) or (0, M/P_y), whichever gives higher utility.
Q: How do you decide between a corner solution and an interior solution for perfect substitutes?
A: Compare MU_x / P_x with MU_y / P_y. If one strictly exceeds the other, the consumer goes to the corner with the higher ratio. Only when the ratios are exactly equal is any combination along the budget line optimal.
Special utility functions reappear throughout intermediate micro. Perfect complements model situations like left and right shoes, or hardware and software bundles: anywhere fixed proportions matter. Perfect substitutes matter in trade theory (countries specialising in whichever good has comparative advantage) and in industrial organisation (homogeneous product markets).
Quasi-linear preferences are the workhorse of partial equilibrium welfare analysis. Because income effects are zero for the nonlinear good, consumer surplus is an exact measure of welfare change, which makes quasi-linear utility the go-to assumption for tax and subsidy problems.
The convexity trap connects to the broader theme of second-order conditions. In producer theory, you will see analogous issues with non-concave production functions.
Perfect substitutes, perfect complements, Leontief preferences, min function utility, corner solution, interior solution, quasi-linear utility, quasi-linear demand, income effect zero, convex preferences, concave utility, second-order conditions, bordered Hessian, Lagrangian method, budget constraint, utility maximisation, intermediate microeconomics, ECON, University of Washington.