Chapter 12 · Test 3
Multivariable Optimization for Economics
Stationary points, Hessian tests, and direct variable restrictions.
What you will learn
- Find a multivariable optimum
- Apply the two-variable Hessian test
- Check feasibility of an unconstrained optimum
Worked example
Find a multivariable optimum: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Find a multivariable optimum
Maximize f(x, y) = −(x − 2)² − (y + 1)² + 10. At the maximizer, what is x* + y*?
At an interior optimum: fₓ = 0 and fᵧ = 0
In plain English: At an interior optimum, neither a small change in x nor a small change in y can improve the objective to first order.
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
- the two choice variablesHow to read: x and y
- x*, y*
- the maximizing valuesHow to read: the maximizing values of x and y
- f(x, y)
- the Objective function: The function an optimization problem seeks to maximize or minimize, such as utility, profit, or cost.How to read: f of x and y
Solution
- Compute fₓ = −2(x − 2). Set it to zero: x* = 2.
- Compute fᵧ = −2(y + 1). Set it to zero: y* = −1.
- The Hessian is diagonal with entries −2 and −2, so it is negative definite.
- Therefore the point is a strict maximum and x* + y* = 2 + (−1) = 1.
Logic and solving tips
- For a strictly concave objective, the stationary point is the unique global maximum.
- Solve all first-order conditions together, then use the Hessian or visible curvature to classify the point.
Where economists use it
A firm solves several first-order conditions together to choose the combination of labor, capital, and output that maximizes profit.
Key idea: For a strictly concave objective, the stationary point is the unique global maximum.
Applied case study
A central bank's stationary policy target
Scenario
Economists solve simultaneous first-order conditions when several outcomes enter one objective. Federal Reserve researchers modeled Riksbank policy with a quadratic loss, a 2% inflation target, and an estimated output-gap weight near 1.1.
Problem
In a one-period simplification, minimize L(π,y) = (π - 2)² + 1.1y². Find the stationary inflation rate π and output gap y.
L(π,y) = (π - 2)² + 1.1y² L_π = 2(π - 2) L_y = 2.2y
In plain English: The same unknown values must satisfy every equation in the system at once, so the equations are solved jointly.
Worked solution
- Set L_π = 0: 2(π - 2) = 0, so π* = 2.
- Set L_y = 0: 2.2y = 0, so y* = 0.
- Both squared terms are nonnegative and equal zero at (2,0).
- Therefore (π*,y*) = (2,0) is the global minimum of the simplified loss.
Economic interpretation: The simplified policy loss is minimized when inflation is at its 2% target and the output gap is closed.
Source: Federal Reserve Board, Optimal Monetary Policy in an Operational Medium-Sized DSGE Model The policy setting, quadratic structure, 2% target, and approximate 1.1 weight are sourced. The static two-term objective omits expectations, dynamics, and interest-rate smoothing for teaching.
Reasoning habits that help
- For a strictly concave objective, the stationary point is the unique global maximum.
- The Hessian determinant and the sign of a leading second derivative distinguish local maxima, minima, and saddle points.
- A direct restriction changes the solution only when the unconstrained optimum is infeasible.
Where economists use these methods
- A firm solves several first-order conditions together to choose the combination of labor, capital, and output that maximizes profit.
- After locating a candidate combination of two inputs, an economist uses the Hessian to verify whether it is a local profit maximum, cost minimum, or saddle point.
- A planner compares an unconstrained optimum with nonnegativity, budget, and capacity limits so the recommended allocation is actually feasible.
Chapter vocabulary
Study 20 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Optimization
Choosing an allowed input vector that maximizes or minimizes an objective function.
Example: A firm chooses inputs to minimize the cost of a target output and then chooses output to maximize profit.
Textbook reference: Chapter 12, Section Chapter introduction, printed page 473Unconstrained optimization
Optimization in which every point in the stated domain is available and no additional restriction limits the choice variables.
Textbook reference: Chapter 12, Section Chapter introduction, printed pages 473-474Extreme value
A maximum or minimum value attained by a function over the relevant domain.
Textbook reference: Chapter 12, Section 12.1 First-Order Conditions, printed page 474Stationary valuenabla f(x*) = 0
A function value at a point where every first-order partial derivative is zero.
How to read: the gradient of f at x star equals zero
Textbook reference: Chapter 12, Section 12.1 First-Order Conditions, printed page 474Saddle point
A stationary point where the function increases in some directions and decreases in others, so it is neither a local maximum nor a local minimum.
Textbook reference: Chapter 12, Section 12.1 First-Order Conditions, printed pages 474-475First-order conditionf_i(x*) = 0 for every i
A derivative condition that an interior differentiable optimum must satisfy, commonly requiring every partial derivative to equal zero.
How to read: each first partial derivative at x star equals zero
Textbook reference: Chapter 12, Section 12.1 First-Order Conditions, printed pages 474, 476Multiproduct monopoly
A single firm that chooses quantities or prices for several related products while accounting for cross-demand and joint-cost effects.
Example: The firm solves all product first-order conditions together because changing one product can change another product's demand.
Textbook reference: Chapter 12, Section 12.1 First-Order Conditions, printed pages 479, 481Cournot duopoly
A two-firm oligopoly model in which each firm chooses output to maximize profit while treating its rival's output as given.
Textbook reference: Chapter 12, Section 12.1 First-Order Conditions, printed pages 481-482Reaction function (best-response function)
A rule giving one firm's profit-maximizing action for every possible action of its rival.
Example: The intersection of the two firms' reaction functions gives the Cournot equilibrium.
Textbook reference: Chapter 12, Section 12.1 First-Order Conditions, printed page 482Second-order condition
A curvature condition, usually stated through the Hessian or second-order differential, that classifies a stationary point as a maximum, minimum, or neither.
Textbook reference: Chapter 12, Section 12.2 Second-Order Conditions, printed pages 484, 490Local maximum
A point whose function value is at least as large as values at all sufficiently nearby feasible points.
Textbook reference: Chapter 12, Section 12.2 Second-Order Conditions, printed pages 484, 487Local minimum
A point whose function value is no larger than values at all sufficiently nearby feasible points.
Textbook reference: Chapter 12, Section 12.2 Second-Order Conditions, printed pages 484, 487Global maximum
A point whose function value is at least as large as the value at every point in the full feasible domain.
Textbook reference: Chapter 12, Section 12.2 Second-Order Conditions, printed pages 486-487Global minimum
A point whose function value is no larger than the value at every point in the full feasible domain.
Textbook reference: Chapter 12, Section 12.2 Second-Order Conditions, printed pages 486-487Interval constrainta_i <= x_i <= b_i
A restriction requiring a choice variable to remain between a lower and an upper bound.
How to read: x sub i lies between a sub i and b sub i
Example: A capacity limit can restrict a plant's output to 0 <= q <= q_bar.
Textbook reference: Chapter 12, Section 12.3 Direct Restrictions on Variables, printed pages 491, 493Interior solution
An optimum strictly inside all relevant bounds, where the usual zero-derivative condition applies.
Textbook reference: Chapter 12, Section 12.3 Direct Restrictions on Variables, printed pages 492, 494Boundary solution
An optimum at a lower or upper bound, where the derivative may be nonzero because motion in the improving direction is infeasible.
Textbook reference: Chapter 12, Section 12.3 Direct Restrictions on Variables, printed pages 492, 495Binding constraint
A restriction that is reached at the optimum and prevents the decision maker from choosing the unconstrained preferred point.
Example: An import quota is binding when the profit-maximizing unrestricted import quantity would exceed the quota.
Textbook reference: Chapter 12, Section 12.3 Direct Restrictions on Variables, printed pages 493, 498Shadow price
The marginal increase in the optimized objective from a small relaxation of a binding constraint.
Example: A quota shadow price measures the extra profit the firm could earn from permission to sell one more unit.
Textbook reference: Chapter 12, Section 12.3 Direct Restrictions on Variables, printed page 498Output quota
An upper bound on the amount a firm may produce, sell, or import in a market.
Textbook reference: Chapter 12, Section 12.3 Direct Restrictions on Variables, printed pages 497-498End-of-chapter problems
Practice all 5 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