Free study notes · Module EC3304
Mathematical Methods
for Economics
A free set of study notes in the mathematics that underpins economic theory: linear algebra and multivariate calculus, written for economics students who need the results to actually do something, from comparative statics to constrained optimisation.
Every topic is explained in plain English with the key definitions, results and derivations, worked examples throughout, and a companion set of fully solved practice problems. Maths is rendered cleanly so you can read it on any device.
Multivariate calculus
Multivariate calculus
study notes
The calculus you need for optimisation and comparative statics: implicit functions and the Jacobian, curvature through the Hessian, polynomial approximation with Taylor's theorem, and the integration tools used for expectations and present values.
The implicit function theorem
When an equation secretly defines a function, the Jacobian condition, and how the theorem delivers the MRTS and comparative-statics derivatives.
Read the notes → CurvatureThe Hessian, concavity & convexity
Second-order partials, Young's theorem, and the leading-principal-minor test that decides whether a function is concave, convex, or neither.
Read the notes → ApproximationMean value theorem & Taylor series
The mean value theorem, univariate and multivariate Taylor expansions, and linearising a whole system, the basis of log-linearisation.
Read the notes → AggregationIntegration for economics
The fundamental theorem, integration by parts and substitution, and Leibniz's rule for differentiating under the integral sign, with worked examples.
Read the notes → Worked solutionsPractice problems & solutions
Five multi-part problems spanning linear algebra and calculus, each with a complete step-by-step solution, from determinants to Kuhn–Tucker conditions.
Work the problems →Linear algebra
Linear algebra
study notes
Vectors and matrices, linear systems and determinants, rank, inverses and eigenvalues, the language in which multivariate calculus, econometrics and dynamic systems are all written.
Vectors & Euclidean spaces
n-vectors, inner products, norms and orthogonality, hyperplanes as budget constraints, and linear independence, span and basis.
Read the notes → SystemsMatrices & linear systems
Matrix algebra, rank via Gaussian elimination, determinants and the inverse, solving Ax = b, and Cramer's rule, with an IS-LM application.
Read the notes → Spectral theoryEigenvalues & diagonalisation
The characteristic equation, trace and determinant properties, diagonalisation and symmetric-matrix results, with stability applications.
Read the notes → DefinitenessQuadratic forms & definiteness
Quadratic forms and their symmetric matrices, definiteness by eigenvalues and principal minors, and the link to second-order conditions.
Read the notes → Worked examplesLinear algebra worked examples
A parametric linear system, matrix inversion and 3×3 eigenvalues, worked step by step as exam-style practice with full solutions.
Work the examples →The practice problems above also include fully worked linear-algebra questions, solving a linear system via the determinant and inverse, and analysing a linear system of differential equations with eigenvalues and eigenvectors. See Practice problems & solutions.
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