Chapter 13 · Test 3
Constrained Optimization for Economics
Lagrangians, tangency conditions, and constrained second-order tests.
What you will learn
- Solve a utility-maximization problem
- Recognize the Lagrangian tangency condition
- Minimize cost subject to an output target
Worked example
Solve a utility-maximization problem: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Solve a utility-maximization problem
A consumer maximizes U(x, y) = √(xy) subject to 2x + y = 12. Find x* + y*.
For equal Cobb-Douglas exponents, spend half of income on each good.
In plain English: When the two Cobb-Douglas exponents are equal, the optimal allocation assigns one half of the budget to each good.
Bold blue terms in this symbol guide include definitions. Hover, focus, or tap a term. Each row also shows how to read the notation aloud.
- x, y
- quantities of two goodsHow to read: x and y
- U(x, y)
- Utility: A numerical representation of a consumer's preferences used to rank bundles of goods or outcomes.How to read: utility as a function of quantities x and y
- 2, 1
- the Price: The amount a buyer pays per unit of a good, service, or asset. When buyers and sellers face different prices, specify the price paid or the price received. of x and yHow to read: the prices of goods x and y, two and one respectively
- 12
- incomeHow to read: income of twelve
Solution
- The exponents on x and y are both 1/2, so each good receives one-half of income.
- Spend 6 on x. Since x costs 2, x* = 6/2 = 3.
- Spend 6 on y. Since y costs 1, y* = 6/1 = 6.
- Check the budget: 2(3) + 6 = 12.
- Therefore x* + y* = 3 + 6 = 9.
Logic and solving tips
- Cobb-Douglas expenditure shares follow exponents; quantities also depend on prices.
- Cobb-Douglas exponents determine spending shares; divide each expenditure share by its own price to get quantity.
Where economists use it
A consumer-demand analyst uses a Cobb-Douglas model to predict how a household divides a fixed budget between goods when income or prices change.
Key idea: Cobb-Douglas expenditure shares follow exponents; quantities also depend on prices.
Applied case study
Calibrating housing and food budget shares
Scenario
Economists use Cobb-Douglas utility as a transparent benchmark in which exponents determine expenditure shares. BLS reported 2024 average annual spending of $26,266 on housing and $10,169 on food per consumer unit.
Problem
Let U(H,F) = H^αF^(1-α), M = 36,435, and α = 26,266/36,435. Maximize utility subject to H + F = M. Find H* and F*.
H* = αM F* = (1 - α)M α = 26,266/36,435 ≈ 0.720900
In plain English: The same unknown values must satisfy every equation in the system at once, so the equations are solved jointly.
Worked solution
- Combine the two observed categories: M = 26,266 + 10,169 = 36,435.
- Compute α = 26,266/36,435 ≈ 0.720900.
- Cobb-Douglas demand gives H* = αM = 26,266.
- Then F* = (1 - α)M = 10,169, and the budget is exhausted.
Economic interpretation: A two-good Cobb-Douglas benchmark calibrated to observed shares reproduces the BLS housing and food expenditures by construction.
Source: U.S. Bureau of Labor Statistics, Consumer Expenditures in 2024 The spending amounts are survey estimates. The two-good budget and Cobb-Douglas utility are a calibration exercise, not evidence that BLS estimated household preferences.
Reasoning habits that help
- Cobb-Douglas expenditure shares follow exponents; quantities also depend on prices.
- At an interior consumer optimum, willingness to trade equals the rate at which the market permits trade.
- A binding production requirement can reduce a constrained problem to one variable through substitution.
Where economists use these methods
- A consumer-demand analyst uses a Cobb-Douglas model to predict how a household divides a fixed budget between goods when income or prices change.
- A household's interior optimum occurs where its willingness to trade one good for another matches the market price ratio, determining the chosen consumption bundle.
- A producer chooses the least-cost mix of labor and capital capable of meeting a required output target, then uses the result for budgeting and pricing.
Chapter vocabulary
Study 27 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Constrained optimization
Maximizing or minimizing an objective while restricting choices to values that satisfy one or more constraints.
Example: A consumer maximizes utility subject to a budget constraint.
Textbook reference: Chapter 13, Section Chapter introduction, printed page 503Functional constraintg(x) = 0
An equation or inequality involving the choice variables that restricts which combinations are feasible.
How to read: constraint g of x equals zero
Textbook reference: Chapter 13, Section Chapter introduction, printed page 503Objective function
The function whose value the decision maker seeks to maximize or minimize.
Example: Utility, profit, cost, and expenditure can serve as objectives.
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed pages 504-505Feasible point
A choice vector that satisfies every constraint in the problem.
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed pages 504-505Constraint curveg(x_1, x_2) = 0
For a two-variable equality constraint, the curve containing every feasible input pair.
How to read: g of x one and x two equals zero
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed pages 504-505Tangency conditionf_1/f_2 = g_1/g_2
At a regular interior constrained optimum, the objective's level curve and the constraint curve have the same slope.
How to read: the objective's marginal trade-off equals the constraint's marginal trade-off
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed pages 504, 507Lagrange multiplierlambda
An auxiliary variable that converts a constrained problem into stationarity conditions and, at a regular optimum, measures the marginal value of relaxing its constraint.
How to read: lambda
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed pages 506-507Lagrange function (Lagrangean)L(x, lambda) = f(x) + lambda g(x)
The objective function plus each constraint multiplied by its Lagrange multiplier.
How to read: L equals f plus lambda times g
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed pages 506-507Lagrange method
A procedure that forms the Lagrange function and solves its first-order conditions for the choices and multipliers.
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed pages 506-507Saddle point of the Lagrangean
A stationary point that has the optimizing curvature in the choice variables while behaving in the opposite direction with respect to the multiplier.
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed page 506Dual problem
A closely related optimization problem that reverses which outcome is fixed and which objective is optimized.
Example: Utility maximization fixes income and maximizes utility, while its expenditure-minimization dual fixes utility and minimizes spending.
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed pages 513-514Expenditure minimization
Choosing the least-cost consumption bundle that reaches a specified utility level at given prices.
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed pages 513-514Compensated demand
Demand obtained from expenditure minimization, showing quantities as functions of prices while utility is held fixed.
Example: After a price change, income is adjusted just enough to keep the consumer on the initial indifference curve.
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed pages 513-514Uncompensated demand
Ordinary demand obtained from utility maximization, showing quantities as functions of prices and fixed money income.
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed pages 513-514Compensated-price effect
The change in demand following a price change when expenditure is adjusted so utility remains fixed.
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed page 514Uncompensated-price effect
The change in ordinary demand following a price change when money income is held fixed.
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed page 514Expenditure functione(p, u_bar)
The minimum spending required to achieve a specified utility level at given prices.
How to read: minimum expenditure at prices p and utility u bar
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed page 514Shadow value of a constraint
The rate at which the optimized objective changes when the constraint constant is relaxed slightly, equal to the appropriately signed optimal multiplier.
Example: In cost minimization, the multiplier on the output requirement equals marginal cost under the chapter's sign convention.
Textbook reference: Chapter 13, Section 13.1 Constrained Problems and Approaches to Solutions, printed page 515Bordered Hessian for a constrained problem
The Hessian of the Lagrange function bordered by constraint gradients, whose minors provide sufficient second-order tests for a constrained optimum.
Textbook reference: Chapter 13, Section 13.2 Second-Order Conditions for Constrained Optimization, printed pages 517, 519Feasible setX
The complete set of choice vectors satisfying every restriction in an optimization problem.
How to read: the feasible set X
Textbook reference: Chapter 13, Section 13.3 Existence, Uniqueness, and Characterization of Solutions, printed pages 520, 522Weierstrass's theorem
A continuous function on a nonempty, closed, and bounded feasible set attains both a maximum and a minimum.
Textbook reference: Chapter 13, Section 13.3 Existence, Uniqueness, and Characterization of Solutions, printed pages 520, 522Nonempty feasible set
A feasible set containing at least one point, which is necessary for any constrained solution to exist.
Textbook reference: Chapter 13, Section 13.3 Existence, Uniqueness, and Characterization of Solutions, printed page 521Bounded set
A set contained within some finite distance, so one cannot move indefinitely far while remaining in the set.
Textbook reference: Chapter 13, Section 13.3 Existence, Uniqueness, and Characterization of Solutions, printed pages 521-522Closed set
A set that contains its boundary and limit points, preventing a best value from being approached but never reached solely because the boundary is excluded.
Textbook reference: Chapter 13, Section 13.3 Existence, Uniqueness, and Characterization of Solutions, printed page 522Constraint qualification
A regularity condition on constraint gradients that ensures suitable Lagrange multipliers can be solved for at an optimum.
Textbook reference: Chapter 13, Section 13.3 Existence, Uniqueness, and Characterization of Solutions, printed pages 522-523Global optimality
The property that a candidate is at least as good as every feasible point, not merely those in a small neighborhood.
Example: With a quasiconcave objective and a convex feasible set, suitable local maximum conditions can imply a global maximum.
Textbook reference: Chapter 13, Section 13.3 Existence, Uniqueness, and Characterization of Solutions, printed pages 523, 525Unique optimum
An optimization problem's single best feasible point rather than one of several equally good solutions.
Example: Strict quasiconcavity can rule out multiple maximizers on a convex feasible set.
Textbook reference: Chapter 13, Section 13.3 Existence, Uniqueness, and Characterization of Solutions, printed pages 524-525End-of-chapter problems
Practice all 4 Chapter Review Exercises
The index covers every numbered Review Exercise in this chapter of Mathematics for Economics, 3rd edition. Each reference opens the textbook exercise itself, including every listed subpart, with MFE hints, a symbol guide, a worked explanation, progress tracking, and similar-problem practice.
Practice end-of-chapter problems