Chapter 11 · Test 3
Multivariable Calculus for Economics
Partial derivatives, total differentials, Hessians, and curvature.
What you will learn
- Evaluate a partial derivative
- Use a total differential
- Calculate a cross-partial derivative
Worked example
Evaluate a partial derivative: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Evaluate a partial derivative
Let f(x, y) = x²y + 3y². Find fₓ(2, 1).
fₓ = ∂f/∂x
In plain English: The partial derivative f sub x measures how f changes with x while the other inputs are held fixed.
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 input variablesHow to read: x and y
- fₓ
- the Partial derivative: The derivative with respect to one input while all other inputs are temporarily held constant. of f with respect to x, holding y fixedHow to read: f sub x
Solution
- Differentiate x²y with respect to x: 2xy.
- The term 3y² is constant with respect to x, so its partial derivative is 0.
- Thus fₓ(x, y) = 2xy.
- Evaluate: fₓ(2, 1) = 2(2)(1) = 4.
Logic and solving tips
- A partial derivative changes one input while temporarily holding the other inputs constant.
- While differentiating with respect to x, temporarily treat every y as a constant coefficient.
Where economists use it
A production economist changes labor while holding capital fixed to estimate labor's marginal product and decide whether another worker adds enough output.
Key idea: A partial derivative changes one input while temporarily holding the other inputs constant.
Applied case study
Marginal labor product in output forecasting
Scenario
Economists use partial derivatives to isolate one input's marginal effect. CBO assesses Cobb-Douglas and CES production functions for macroeconomic forecasting, so consider a simplified capital-and-labor output index.
Problem
If Y = K^0.3L^0.7, what is the marginal product of labor at K = 100 and L = 100?
Y = K^0.3L^0.7 ∂Y/∂L = 0.7K^0.3L^-0.3
In plain English: The same unknown values must satisfy every equation in the system at once, so the equations are solved jointly.
Worked solution
- Hold K fixed and differentiate L^0.7.
- This gives ∂Y/∂L = 0.7K^0.3L^-0.3.
- Substitute K = L = 100: ∂Y/∂L = 0.7(100^0.3)(100^-0.3).
- The powers cancel, so ∂Y/∂L = 0.7.
Economic interpretation: Near this input combination, one additional unit of labor raises the output index by about 0.7, holding capital fixed.
Source: Congressional Budget Office, An Assessment of CES and Cobb-Douglas Production Functions CBO's production-function application is sourced. The exponents and input values are teaching assumptions, not CBO estimates for a specific year.
Reasoning habits that help
- A partial derivative changes one input while temporarily holding the other inputs constant.
- The total differential adds the first-order effects of all changing inputs.
- Cross partials measure how one marginal effect changes as another input changes.
Where economists use these methods
- A production economist changes labor while holding capital fixed to estimate labor's marginal product and decide whether another worker adds enough output.
- A forecaster combines small changes in labor, capital, and productivity to approximate their joint effect on output without recomputing the entire model.
- A firm examines a cross partial to learn whether more machinery raises workers' marginal productivity, which helps determine whether capital and labor are complements or substitutes.
Chapter vocabulary
Study 42 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Function of n variablesy = f(x_1, x_2, ..., x_n) = f(x)
A rule that assigns one output value to each permitted n-component input vector.
How to read: y equals f of x one through x n
Example: A production function can map labor, capital, land, and energy into a level of output.
Textbook reference: Chapter 11, Section 11.1 Partial Differentiation, printed page 393Continuity in ℝⁿ
A function is continuous at a point when inputs approaching that point along any path produce outputs approaching the function's value there.
Example: For two inputs, checking only horizontal and vertical paths is not enough because infinitely many approach paths exist.
Textbook reference: Chapter 11, Section 11.1 Partial Differentiation, printed pages 393-394Partial derivativepartial f / partial x_i = f_i(x)
The rate at which a multivariable function changes when one input changes and all other inputs are held fixed.
How to read: the partial derivative of f with respect to x sub i
Example: The partial derivative of revenue with respect to one product's sales holds sales of the other products fixed.
Textbook reference: Chapter 11, Section 11.1 Partial Differentiation, printed pages 394, 396Additively separable functionf(x) = g_1(x_1) + ... + g_n(x_n)
A multivariable function that can be written as a sum of one-variable functions, one for each input.
How to read: f of x equals g one of x one plus through g n of x n
Example: Because each term uses only one variable, every cross-partial derivative is zero.
Textbook reference: Chapter 11, Section 11.1 Partial Differentiation, printed pages 398-399Marginal productMP_i = partial y / partial x_i
The change in output per small increase in one input while other input levels are held fixed.
How to read: the marginal product of input i
Example: The marginal product of labor measures the extra output associated with a little more labor at the current capital stock.
Textbook reference: Chapter 11, Section 11.1 Partial Differentiation, printed pages 399-400Cobb-Douglas production functiony = A product_i x_i^alpha_i
A multiplicative production function in which exponents govern input responsiveness and, together, returns to scale.
How to read: output equals productivity A times the product of each input raised to its exponent
Textbook reference: Chapter 11, Section 11.1 Partial Differentiation, printed pages 400, 403Constant elasticity of substitution (CES) production functiony = A[delta x_1^-r + (1-delta)x_2^-r]^(-1/r)
A production function designed so the elasticity of substitution between inputs is constant along its isoquants.
How to read: the CES production function
Textbook reference: Chapter 11, Section 11.1 Partial Differentiation, printed pages 403-404Multivariable chain ruledy/dt = sum_i (partial f/partial x_i)(dx_i/dt)
A rule that adds every channel through which an underlying variable changes a multivariable function.
How to read: the total derivative of y with respect to t
Example: Output growth can be separated into the effect of capital accumulation and the direct effect of technical change.
Textbook reference: Chapter 11, Section 11.1 Partial Differentiation, printed pages 404-405Second-order partial derivativef_ij = partial^2 f / (partial x_i partial x_j)
The derivative of a first-order partial derivative with respect to one of the inputs.
How to read: f sub i j
Textbook reference: Chapter 11, Section 11.2 Second-Order Partial Derivatives, printed pages 407-408Gradient vectornabla f = [f_1, ..., f_n]^T
The vector containing every first-order partial derivative of a scalar-valued function.
How to read: the gradient of f
Example: The gradient reports the function's local rate of change in each coordinate direction.
Textbook reference: Chapter 11, Section 11.2 Second-Order Partial Derivatives, printed pages 408-409Hessian matrixH = nabla^2 f = [f_ij]
The square matrix containing all second-order partial derivatives of a scalar-valued function.
How to read: the Hessian of f
Example: The Hessian summarizes local curvature and is used to classify optimization candidates.
Textbook reference: Chapter 11, Section 11.2 Second-Order Partial Derivatives, printed pages 409-410Cross-partial derivativef_ij, i != j
A second-order partial derivative involving two different variables, measuring how one variable changes another variable's marginal effect.
How to read: the cross-partial f sub i j
Example: A positive labor-capital cross-partial means more capital raises labor's marginal product.
Textbook reference: Chapter 11, Section 11.2 Second-Order Partial Derivatives, printed page 411Young's theoremf_ij = f_ji
Under the required continuity conditions, changing the order of differentiation does not change a cross-partial derivative.
How to read: f sub i j equals f sub j i
Textbook reference: Chapter 11, Section 11.2 Second-Order Partial Derivatives, printed pages 411-412First-order total differentialdy = sum_i f_i dx_i
The linear approximation to the total change in a function produced by small changes in all of its inputs.
How to read: d y equals the sum of f sub i times d x sub i
Example: It approximates the effect on output when labor and capital change together.
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed pages 415, 417Implicit functionF(x, y) = 0
A functional relationship specified by an equation linking variables without isolating the dependent variable.
How to read: F of x and y equals zero
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed pages 418, 420Implicit differentiationdy/dx = -F_x/F_y
Finding derivatives from an implicit equation by differentiating the equation itself rather than first solving explicitly for the dependent variable.
How to read: d y by d x equals negative F sub x over F sub y
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed pages 419, 423Implicit function theorem
A theorem giving local conditions under which an equation such as F(x, y) = 0 defines y as a differentiable function of x.
Example: In the two-variable case, a key condition is F_y != 0 at the point being studied.
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed pages 420, 423Level set{x : f(x) = y_bar}
The set of all input combinations that produce one fixed value of a function.
How to read: the set of x such that f of x equals y bar
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed pages 423-424Level curve
A level set for a two-input function drawn as a curve in input space.
Example: Its slope shows how one input must change to hold the function value constant when the other input changes.
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed pages 423-424Isoquantf(x_1, x_2) = y_bar
A production-function level curve showing all input combinations that produce the same output.
How to read: output held at y bar
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed pages 425-426Marginal rate of technical substitution (MRTS)MRTS_12 = f_1/f_2 = -dx_2/dx_1
The amount of one input that can replace a small amount of another while output remains fixed, expressed as the negative isoquant slope.
How to read: the marginal rate of technical substitution of input one for input two
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed pages 425-426Indifference curveu(x_1, x_2) = u_bar
A utility-function level curve containing consumption bundles assigned the same utility level.
How to read: utility held at u bar
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed pages 430-431Positive monotonic transformationv(x) = T(u(x)), T'(u) > 0
A strictly increasing transformation of a utility representation that preserves every ranking and therefore preserves its indifference curves.
How to read: v is an increasing transformation of u
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed pages 431-432Marginal rate of substitution (MRS)MRS_12 = u_1/u_2 = -dx_2/dx_1
The amount of one good a consumer can give up for a small increase in another while remaining on the same indifference curve.
How to read: the marginal rate of substitution of good one for good two
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed page 433Perfect substitutes
Goods or inputs that can replace one another at a constant rate, producing straight-line level curves.
Example: For u = x_1 + x_2, one unit of either good replaces exactly one unit of the other.
Textbook reference: Chapter 11, Section 11.3 The First-Order Total Differential, printed page 434Second-order total differentiald^2y = dx^T H dx
The quadratic form that combines the Hessian with a vector of input changes to measure second-order curvature in any direction.
How to read: the second-order total differential of y
Textbook reference: Chapter 11, Section 11.4 Curvature Properties: Concavity and Convexity, printed pages 436, 438Concave function
A function whose graph never falls below the line segment joining any two points on the graph.
Example: For a twice differentiable function, a negative-semidefinite Hessian is equivalent to concavity.
Textbook reference: Chapter 11, Section 11.4 Curvature Properties: Concavity and Convexity, printed pages 436, 450Strictly concave function
A concave function whose value at every nontrivial mixture of distinct inputs is strictly above the same mixture of their function values.
Example: A strictly concave profit function has at most one unconstrained maximizer.
Textbook reference: Chapter 11, Section 11.4 Curvature Properties: Concavity and Convexity, printed pages 436, 450Convex function
A function whose graph never rises above the line segment joining any two points on the graph.
Example: For a twice differentiable function, a positive-semidefinite Hessian is equivalent to convexity.
Textbook reference: Chapter 11, Section 11.4 Curvature Properties: Concavity and Convexity, printed pages 436, 450Strictly convex function
A convex function whose value at every nontrivial mixture of distinct inputs is strictly below the same mixture of their function values.
Example: A strictly convex cost objective has at most one minimizer over a convex feasible set.
Textbook reference: Chapter 11, Section 11.4 Curvature Properties: Concavity and Convexity, printed pages 436, 450Quasiconcave function
A function whose upper contour sets are convex, so mixtures of two inputs are never assigned less than the lower endpoint value.
Example: Utility and production functions are often assumed quasiconcave because this yields convex-to-the-origin level curves.
Textbook reference: Chapter 11, Section 11.5 More Properties of Functions with Economic Applications, printed pages 451, 456Quasiconvex function
A function whose lower contour sets are convex, so mixtures of two inputs are never assigned more than the higher endpoint value.
Textbook reference: Chapter 11, Section 11.5 More Properties of Functions with Economic Applications, printed pages 451, 456Bordered Hessian
A matrix formed by adding the first derivatives as a border around a Hessian, used to test quasiconcavity, quasiconvexity, and constrained optima.
Textbook reference: Chapter 11, Section 11.5 More Properties of Functions with Economic Applications, printed pages 454, 456Homogeneous functionf(sx) = s^k f(x)
A function for which scaling every input by the same positive factor scales output by that factor raised to a fixed degree.
How to read: f of s x equals s to degree k times f of x
Textbook reference: Chapter 11, Section 11.5 More Properties of Functions with Economic Applications, printed pages 457, 459Increasing returns to scalek > 1 for a degree-k homogeneous production function
A production property in which scaling all inputs by a common factor increases output by a larger proportion.
How to read: k is greater than one for a homogeneous production function of degree k
Textbook reference: Chapter 11, Section 11.5 More Properties of Functions with Economic Applications, printed pages 457, 459Constant returns to scalek = 1 for a degree-k homogeneous production function
A production property in which scaling all inputs by a common factor scales output in the same proportion.
How to read: k equals one for a homogeneous production function of degree k
Textbook reference: Chapter 11, Section 11.5 More Properties of Functions with Economic Applications, printed pages 457, 459Decreasing returns to scale0 < k < 1 for a degree-k homogeneous production function
A production property in which scaling all inputs by a common factor increases output by a smaller proportion.
How to read: k is between zero and one for a homogeneous production function of degree k
Textbook reference: Chapter 11, Section 11.5 More Properties of Functions with Economic Applications, printed pages 457, 459Euler's theoremsum_i x_i f_i(x) = kf(x)
For a differentiable function homogeneous of degree k, the sum of each input times its marginal effect equals k times the function value.
How to read: the input-weighted marginal products sum to k times output
Textbook reference: Chapter 11, Section 11.5 More Properties of Functions with Economic Applications, printed pages 459-460Elasticity of substitutionsigma = d ln(x_2/x_1) / d ln(f_1/f_2)
The percentage responsiveness of an input ratio to a percentage change in the marginal rate of technical substitution.
How to read: sigma equals the log change in the input ratio divided by the log change in the MRTS
Textbook reference: Chapter 11, Section 11.5 More Properties of Functions with Economic Applications, printed pages 460, 463Taylor series expansion
A local representation of a function using its value and derivatives at a reference point plus a remainder term.
Example: Economists use a second-order Taylor expansion to approximate welfare, profit, or utility changes near an equilibrium.
Textbook reference: Chapter 11, Section 11.6 Taylor Series Expansion, printed pages 464, 467Remainder formula
A Taylor formula that states the approximation error using a higher-order derivative evaluated at an intermediate point.
Textbook reference: Chapter 11, Section 11.6 Taylor Series Expansion, printed pages 465, 467Tangent hyperplaneT(x) = f(x^0) + nabla f(x^0)^T(x - x^0)
The first-order linear approximation to a multivariable function at a point.
How to read: the tangent approximation T of x
Textbook reference: Chapter 11, Section 11.6 Taylor Series Expansion, printed page 469End-of-chapter problems
Practice all 10 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