120 economics and mathematics definitions
Mathematics for economics glossary
Build fluency in the language used by economists. These concise definitions cover the algebra, calculus, linear algebra, optimization, and economic-modeling terms used throughout the Mastery Lab.
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B
- Binding constraint
- An inequality constraint that holds with equality at a candidate and therefore has zero slack; binding alone does not imply a nonzero shadow price.
C
- Chain rule
- A derivative rule for compositions: D(g∘f)(x) = Dg(f(x))Df(x). In one variable, multiply the outer derivative by the inner derivative.
- Characteristic polynomial
- The polynomial det(A − λI); its roots are the eigenvalues of A.
- Closed form
- An informal, context-dependent term for a finite expression built from an agreed set of operations and functions rather than an unevaluated or numerical procedure.
- Closed interval
- An interval that includes both endpoints, written in the form [a, b].
- Cobb-Douglas
- A multiplicative function A∏xᵢ^αᵢ used for production or preferences. Production exponents are input elasticities; normalized preference exponents determine expenditure shares.
- Cobb-Douglas demand
- Consumer demand derived from Cobb-Douglas preferences; at an interior solution, each good receives a fixed share of expenditure.
- Cobb-Douglas exponent
- An exponent αᵢ in a Cobb-Douglas function. Its interpretation depends on the model, such as an input elasticity or a normalized expenditure-share parameter.
- Coefficient matrix
- The matrix containing the coefficients that multiply the unknowns in a linear system.
- Common ratio
- The fixed multiplier that takes one term of a geometric sequence to the next.
- Comparative statics
- The study of how an equilibrium or optimum changes when an exogenous parameter changes.
- Complementarity
- A positive interaction where increasing one choice or input raises the marginal payoff or productivity of another, represented locally by a positive cross partial.
- Complementary goods
- Two goods for which an increase in the price of one reduces demand for the other, holding other relevant factors fixed.
- Compounding
- Growth in which each period's return is added to the base on which later returns are calculated.
- Concavity
- On a convex domain, f is concave when f(tx + (1 − t)y) ≥ tf(x) + (1 − t)f(y) for all x, y and 0 ≤ t ≤ 1.
- Conformable
- Describes matrices whose dimensions are compatible for a requested operation, such as multiplication.
- Constant elasticity
- A property of a relationship whose elasticity has the same value throughout the relevant domain.
- Constraint
- A restriction that a choice must satisfy; choices satisfying every constraint form the feasible set.
- Continuity
- A function is continuous at a point when its limit there exists and equals the function's value.
- Continuity condition
- A restriction requiring specified pieces or values of a function to satisfy the equal-limit and function-value conditions for continuity.
- Convergence
- A sequence converges when its terms approach and remain arbitrarily close to a limit; a series converges when its sequence of partial sums converges.
- Convex combination
- A weighted sum of points with nonnegative weights that add to one. It lies in their convex hull and remains feasible when the feasible set is convex.
- Cost minimization
- Choosing the least-cost combination of inputs that can achieve a required output level.
- Cramer's rule
- A determinant-based formula for solving a square linear system when its coefficient determinant is nonzero.
- Cross partial
- A mixed second derivative showing how the marginal effect of one input changes with another input.
- Curvature
- Geometrically, how sharply a curve bends. In optimization, the term often informally means second-order behavior described by a second derivative or Hessian.
D
- Definiteness
- A classification of the signs taken by a quadratic form over nonzero vectors, including definite, semidefinite, and indefinite cases.
- Demand
- The relationship between a good's price and the quantity buyers are willing and able to purchase at each price, holding other relevant factors fixed.
- Derivative
- The local linear rate of change of a function; f′(a) = lim[h→0] (f(a + h) − f(a))/h when this limit exists.
- Determinant
- A number calculated from a square matrix that indicates, among other things, whether the matrix is invertible.
- Differentiation
- The process of finding a derivative from its limit definition or by applying valid derivative rules.
- Discontinuity
- A point where a function is not continuous, for example because of a hole, jump, or unbounded behavior.
- Discount factor
- The multiplier used to convert a future payment into present value; it is smaller than one when the discount rate is positive.
- Discounting
- Converting a future amount into today's value by reversing compound growth at a chosen rate.
- Dot product
- The scalar obtained by multiplying corresponding vector entries and adding those products.
E
- Eigenvalue
- A scalar λ for which Av = λv for some nonzero vector v.
- Eigenvector
- A nonzero vector v satisfying Av = λv for some eigenvalue λ; the line it spans is invariant under A.
- Elasticity
- The ratio of the percentage change in one variable to the percentage change in another, holding relevant factors fixed.
- Elimination
- A method for solving simultaneous equations by combining them so that one unknown cancels out.
- Endogenous solution
- A solution value determined inside a model as a function of its exogenous variables and parameters.
- Endogenous variable
- A variable whose value is determined inside the model by its equations and assumptions.
- Envelope theorem
- Under suitable optimality and differentiability conditions, a result that measures a value function's parameter response using the objective's direct effect at an optimizer.
- Equilibrium
- A solution or state in which a model's relevant conditions are mutually satisfied and there is no internal tendency to change.
- Equilibrium price
- The price at which quantity demanded equals quantity supplied.
- Equilibrium quantity
- The common quantity demanded and supplied at the equilibrium price.
- Expenditure share
- The fraction of a consumer's total budget spent on a particular good.
F
- Feasibility
- Whether a choice satisfies every constraint, or whether a problem has at least one such choice.
- Feasible
- Describes a choice that satisfies every constraint, or a problem that has at least one feasible choice.
- First-order condition
- A condition involving first derivatives that an optimum candidate must satisfy. For a differentiable unconstrained interior optimum, the gradient equals zero.
- Flow
- A quantity measured over an interval of time, such as income per year or investment during a quarter.
G
- Geometric sequence
- A sequence in which each term is obtained by multiplying the previous term by the same common ratio.
- Geometric series
- A sum of terms from a geometric sequence, each related to the previous term by a common ratio.
H
- Hessian
- For a twice partially differentiable scalar function, the Hessian is the matrix whose entries are its second partial derivatives.
I
- Implicit differentiation
- Differentiating a relation such as F(x, y) = 0 with the chain rule while treating y as a function of x, then solving for y′ when permitted.
- Indefinite
- A quadratic form is indefinite when it takes both positive and negative values on nonzero vectors.
- Indeterminate form
- A limit form such as 0/0 whose form alone does not determine the limit, so further analysis is required.
- Infeasible
- Describes a choice that violates at least one constraint, or a problem with no feasible choice.
- Infinite geometric series
- A series Σ arⁿ that converges to a/(1 − r) when |r| < 1 and otherwise diverges, apart from the trivial case a = 0.
- Inflow
- A flow that increases a related stock during the measurement interval.
- Intercept
- A point where a graph meets a coordinate axis. For y = mx + b, (0, b) is the y-intercept; an x-intercept satisfies f(x) = 0.
- Interior consumer optimum
- A consumer optimum at which every chosen good has a positive quantity, even though the budget constraint can still bind.
- Interior optimum
- An optimum in the interior, or appropriate relative interior, of the feasible set rather than on its boundary.
- Inverse matrix
- A matrix A⁻¹ that satisfies AA⁻¹ = A⁻¹A = I and reverses the effect of A.
- Invertible
- Describes a square matrix that has an inverse, so its associated linear transformation can be uniquely undone.
J
- Jump discontinuity
- A discontinuity where the finite left-hand and right-hand limits both exist but are unequal.
L
- Lagrange multiplier
- A coefficient attached to a constraint in a Lagrangian; under suitable conditions, its optimal value measures the effect of marginally relaxing that constraint.
- Lagrangian
- A function formed by combining an objective function with its constraints, each weighted by a Lagrange multiplier.
M
- Marginal
- The additional change associated with one more unit, or with a very small change, in an economic variable.
- Marginal cost
- The increase in total cost from a small increase in output; in a differentiable model, MC(q) = dC/dq.
- Marginal revenue
- The increase in total revenue from a small increase in quantity sold; in a differentiable model, MR(q) = dR/dq.
- Marginal utility
- The additional utility associated with a small increase in consumption of a good.
- Market clearing
- A condition in which quantity demanded equals quantity supplied, leaving no excess demand or excess supply.
- Market equilibrium
- A market outcome at which buyers' planned purchases equal sellers' planned sales, so the market clears.
- Market-clearing condition
- An equation or restriction imposing equality between quantity demanded and quantity supplied.
- Matrix-vector product
- The vector produced by taking the dot product of each matrix row with the input vector.
N
- Negative definite
- A quadratic form q(x) = xᵀAx is negative definite when q(x) < 0 for every nonzero x.
- Negative semidefinite
- A quadratic form q(x) = xᵀAx is negative semidefinite when q(x) ≤ 0 for every x.
- Nonsingular
- Describes a square matrix with a nonzero determinant and therefore a unique inverse.
O
- Objective function
- The function an optimization problem seeks to maximize or minimize, such as utility, profit, or cost.
- One-sided limit
- The value a function approaches from only the left or only the right of a point.
- Outflow
- A flow that decreases a related stock during the measurement interval.
P
- Parameter
- A value treated as fixed while a model is solved, although it may be changed to compare different scenarios.
- Parameterized
- Represented using one or more parameters whose values index different members of a family of models, functions, or solutions.
- Partial derivative
- The derivative with respect to one input while all other inputs are temporarily held constant.
- Perpetuity
- A stream of payments that continues indefinitely, commonly valued as a convergent discounted series.
- Point elasticity
- Elasticity measured locally at a point; for y = f(x), it is (dy/dx)(x/y) when the derivative and ratio are defined.
- Positive definite
- A quadratic form q(x) = xᵀAx is positive definite when q(x) > 0 for every nonzero x.
- Positive semidefinite
- A quadratic form q(x) = xᵀAx is positive semidefinite when q(x) ≥ 0 for every x.
- Present value
- The value today of a future payment or stream of payments after discounting.
- Present-value equation
- An equation that discounts one or more future cash flows to a common valuation date so amounts at different dates can be compared.
- 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.
- Price elasticity of demand
- The percentage change in quantity demanded divided by the percentage change in its own price, holding other demand factors fixed.
- Production
- The process of transforming inputs such as labor and capital into goods or services.
- Production function
- A relationship showing the maximum output obtainable from specified inputs under a given technology.
- Profit
- Total revenue minus total cost. Economic profit subtracts both explicit costs and implicit opportunity costs.
Q
- Quadratic form
- An expression such as xᵀAx whose variables all appear in terms of total degree two.
- Quantity demanded
- The amount of a good or service buyers are willing and able to purchase at a particular price, holding other relevant factors fixed.
- Quantity supplied
- The amount of a good or service sellers are willing and able to offer at a particular price, holding other relevant factors fixed.
R
- Reduced form
- A system or equation expressing endogenous variables directly as functions of exogenous or predetermined variables, parameters, and disturbances.
- Reduced-form derivative
- A derivative of a reduced-form solution showing how an endogenous variable changes with an exogenous variable or parameter.
- Removable discontinuity
- A discontinuity where the finite limit exists but the function is undefined there or has a different value.
S
- Saddle point
- A stationary point that behaves like a maximum in some directions and a minimum in others.
- Second derivative test
- A procedure using second derivatives or a Hessian to classify a stationary point; in semidefinite cases the test can be inconclusive.
- Second partial derivative
- A second-order partial derivative, pure when both differentiations use the same variable and mixed when they use different variables.
- Second-order condition
- A necessary or sufficient condition using second derivatives or Hessians to assess a local optimum; semidefinite cases can be inconclusive.
- Set complement
- All elements in the universal set that are not members of the specified set.
- Set intersection
- The set containing exactly the elements that belong to every set being compared.
- Stationary point
- A point where all relevant first derivatives are zero; it is a candidate for a maximum, minimum, or saddle point.
- Stock
- A quantity measured at a particular point in time, such as wealth, debt, capital, or inventory.
- Stock-flow identity
- An accounting relationship linking the change in a stock between two dates to the flows and other changes during that interval.
- Strict concavity
- On a convex domain, strict concavity means the concavity inequality is strict for distinct points and interior weights; a maximizer, if one exists, is unique.
- Substitutability
- The extent to which more of one input or choice can replace less of another while maintaining output or utility.
- Substitute goods
- Two goods for which an increase in the price of one raises demand for the other, holding other relevant factors fixed.
- Supply
- The relationship between a good's price and the quantity sellers are willing and able to offer at each price, holding other relevant factors fixed.
T
- Tangency condition
- At a differentiable interior optimum where the slopes exist, the objective's level curve has the same slope as the binding constraint.
- Tax incidence
- How the economic burden of a tax is divided among buyers, sellers, workers, or other groups.
- Total differential
- A first-order approximation that combines the effects of small changes in all of a function's inputs.
- Transpose
- A matrix formed by swapping rows with columns, so entry (i,j) becomes entry (j,i).
U
- Universal set
- The complete collection of elements under consideration in a particular problem.
- Utility
- A numerical representation of a consumer's preferences used to rank bundles of goods or outcomes.
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