Mathematical Economics

Overview

Treat this unit as a toolkit with a fixed workflow for optimisation problems. First, write the objective and the constraints exactly, naming every variable and its domain. Second, form the Lagrangian or the Kuhn-Tucker conditions as the constraints require. Third, solve the first-order conditions. Fourth, check second-order conditions or concavity to confirm the nature of the critical point. Fifth, interpret the multipliers as shadow prices: the value of relaxing a constraint by one unit. Skipping the fourth or fifth step is the most common way to lose marks on an otherwise correct solution, because examinations test understanding, not just algebra. The same workflow, with time added, governs dynamic problems: find the steady state, linearise around it, and examine eigenvalues or roots for stability.

Sets, functions, continuity, sequences, series

A set is a collection of objects, and set operations of union, intersection, difference and complement organise economic domains such as feasible bundles, choice sets and strategy sets. A function assigns each element of its domain exactly one element of its codomain; in economics, demand functions, production functions and utility functions are the central examples. A function is injective if distinct inputs give distinct outputs, surjective if every codomain element is reached, and bijective if both hold, in which case an inverse function exists. Composite functions apply one function after another, and inverse functions undo them, which is how demand functions are inverted to give inverse demand. A sequence is an ordered list of numbers indexed by natural numbers, and a series is the sum of sequence terms. Convergence asks whether sequence terms approach a limit: a sequence converges if terms get arbitrarily close to a limit and stay there, and a series converges if its partial sums converge. Continuity formalises the absence of jumps: a function is continuous at a point if its limit there equals its value there, and continuous on an interval if it is continuous at every point of the interval. Continuity matters because it underpins the intermediate value theorem, which guarantees solutions to equations, and the extreme value theorem, which guarantees that a continuous function on a compact set attains a maximum and a minimum, the existence result behind most optimisation claims. Convex and concave sets and functions organise optimisation: a set is convex if the line segment between any two of its points lies in the set, and a function is concave if its graph lies above its chords, which makes first-order conditions sufficient for a global maximum.

Differential calculus and applications

The derivative of a function of one variable is the limit of the ratio of change in the function to change in the variable, and economically it is the marginal quantity: marginal utility is the derivative of utility, marginal cost the derivative of cost, marginal product the derivative of production. Rules of differentiation, including product, quotient and chain rules, let complex economic functions be differentiated systematically. Higher derivatives describe curvature: a positive second derivative means the function is convex and marginal quantities are rising, while a negative second derivative means concavity and diminishing marginal quantities. Points where the first derivative is zero are critical points, classified by the second derivative test or by sign changes of the first derivative.

With several variables, partial derivatives hold all but one variable fixed, so the partial derivative of a production function with respect to labour is marginal product of labour. The gradient collects all partial derivatives into a vector pointing in the direction of steepest increase, and the total differential sums partial effects weighted by changes in each variable, giving the linear approximation to a change in the function. The chain rule for several variables tracks indirect effects, for example how a change in an input price affects output through the optimal input choice as well as directly. Homogeneous functions scale in a regular way: a function homogeneous of degree one doubles when all inputs double, which is constant returns to scale, and Euler theorem states that such a function equals the sum of each variable multiplied by its partial derivative, the adding-up result behind factor payment exhaustion. Implicit differentiation finds derivatives of functions defined implicitly by equations, such as demand functions defined by first-order conditions, and comparative statics differentiates equilibrium conditions to find how endogenous variables respond to parameter changes, with signs often determined by second-order conditions and curvature assumptions.

Linear algebra: matrices and vector spaces

A matrix is a rectangular array of numbers, and matrix algebra organises systems of linear equations that arise whenever several markets or sectors interact. Matrix addition and scalar multiplication are element-wise, while matrix multiplication composes linear transformations and is not commutative: order matters. The transpose swaps rows and columns, the identity matrix leaves any conformable matrix unchanged under multiplication, and the inverse of a square matrix, when it exists, undoes multiplication by that matrix. A system of linear equations has a unique solution exactly when its coefficient matrix is non-singular, meaning its determinant is non-zero and its inverse exists. Determinants measure this: a zero determinant signals linear dependence among rows or columns and either no solution or infinitely many solutions, depending on the right-hand side. Cramer rule expresses solutions as ratios of determinants and is useful for small systems and for comparative statics, where it shows how a solution changes with a parameter.

Rank is the number of linearly independent rows or columns and determines how many independent equations a system contains. A vector space is a set closed under addition and scalar multiplication, spanned by basis vectors, with dimension equal to the number of vectors in any basis. Eigenvalues and eigenvectors solve the equation where a matrix applied to a vector merely scales it: the scaling factor is the eigenvalue and the vector is the eigenvector. Eigenvalues govern stability of dynamic systems and the definiteness of quadratic forms used in second-order conditions: a symmetric matrix is positive definite if all eigenvalues are positive, negative definite if all are negative, and indefinite otherwise. Quadratic forms appear as second-order terms in Taylor expansions, so their definiteness decides whether a critical point is a minimum, maximum or saddle point. Input-output analysis, linear programming and dynamic stability all reduce, at their computational core, to matrix operations, which is why fluency with inverses, determinants and eigenvalues pays off across the whole syllabus.

Static optimisation problems

Static optimisation finds the best choice at a point in time, without dynamics. An unconstrained maximum of a function of one variable requires a zero first derivative and a negative second derivative at the optimum; for several variables, all partial derivatives must vanish and the Hessian matrix of second partial derivatives must be negative definite for a maximum and positive definite for a minimum. Concavity of the objective function makes these conditions sufficient for a global optimum, which is why concave utility and profit functions are so convenient. With equality constraints, the Lagrange method introduces a multiplier for each constraint and maximises the Lagrangian, the objective plus multipliers multiplied by constraint slacks. First-order conditions equate marginal trade-offs with price-like ratios, and the multiplier measures the shadow price of the constraint: the increase in the optimal objective value from relaxing the constraint by one unit. In consumer theory the multiplier is marginal utility of income; in cost minimisation it is marginal cost of output. Inequality constraints require Kuhn-Tucker conditions, which generalise Lagrange to problems with non-negativity and inequality restrictions. Each inequality constraint has a multiplier that is non-negative, and complementary slackness requires that either the constraint binds with equality or its multiplier is zero, never both with slack and a positive shadow price. This structure captures corner solutions that Lagrange alone misses, such as a consumer who buys zero of some good or a firm that uses zero of some input. The envelope theorem is a powerful shortcut: the derivative of the optimised value function with respect to a parameter equals the partial derivative of the Lagrangian with respect to that parameter, holding choices fixed, because induced changes in choices have no first-order effect at the optimum. This gives Roy identity, Shephard lemma and Hotelling lemma as special cases, and it explains why demand and supply responses can be read off value functions.

Input-output model and linear programming

The input-output model describes an economy as interdependent sectors, each needing inputs from other sectors to produce its output. Technical coefficients give the input of one sector needed per unit of output of another, collected in a matrix. Total output must cover intermediate demand from all sectors plus final demand from households, government and exports, giving a linear system whose solution expresses total output as the Leontief inverse multiplied by final demand. The Leontief inverse captures direct plus all rounds of indirect requirements: an increase in final demand for one sector raises output in its suppliers, which raises output in their suppliers, and so on, with the inverse summing the infinite geometric series of rounds provided the system is productive. The Hawkins-Simon conditions guarantee a non-negative solution for any non-negative final demand, and the model extends to employment, imports and pollution by attaching coefficients to the output solution.

Linear programming optimises a linear objective subject to linear constraints and is the computational counterpart of the same allocation logic. Its feasible region is a convex polyhedron, and the fundamental theorem says an optimum, if it exists, occurs at a vertex, which is why the simplex method moves from vertex to vertex improving the objective until no improvement is possible. Duality pairs each primal problem with a dual whose variables are shadow prices of primal constraints: primal maximisation becomes dual minimisation, and strong duality says optimal primal and dual values coincide, so dual variables value scarce resources exactly. Complementary slackness links the two optima: a primal variable is positive only if its dual constraint binds, and a dual variable is positive only if its primal constraint binds. Sensitivity analysis asks how the optimum changes with objective coefficients or resource availability, which is answered by dual values within ranges where the optimal basis is unchanged. Linear programming assumes linearity, divisibility and certainty; integer programming drops divisibility when choices are indivisible, at much higher computational cost.

Difference and differential equations

Difference equations describe discrete-time dynamics, where a variable next period depends on its current and past values; differential equations describe continuous-time dynamics, where the rate of change of a variable depends on its current value. First-order linear difference equations have a steady state where the variable is constant over time, found by equating next period value to current value, and a general solution combining that steady state with a term involving the coefficient raised to the power of time. Stability depends on the absolute value of that coefficient: below one, deviations decay and the system converges; above one, they explode; equal to one, they persist. Second-order equations have characteristic roots whose magnitudes govern stability and whose complex values produce oscillations. Economic examples include cobweb models, where supply responds to past prices with a lag, multiplier-accelerator interaction, and capital accumulation with discrete investment periods. First-order linear differential equations are solved with an integrating factor or by separation for the simplest forms, and their solutions likewise combine a steady state with an exponential term whose exponent sign determines stability: a negative exponent gives convergence, a positive exponent gives divergence. Systems of differential equations are analysed through their Jacobian matrix evaluated at a steady state: eigenvalues with negative real parts imply local stability, positive real parts imply instability, and mixed signs imply a saddle point with a stable manifold along which convergence is possible. Phase diagrams plot the loci where each variable is stationary and use arrows to show dynamics in each region, a qualitative tool that often answers stability questions without explicit solutions. Nonlinear systems are linearised around steady states for local analysis, but global behaviour can differ, with limit cycles and multiple equilibria possible.

Optimisation conditions

Stability in dynamic models

Matrix concepts at a glance