Revision#

ECON2125/6012 Lecture 12 Fedor Iskhakov

Announcements & Reminders

  • Tutorials this week: Q&A over the whole course + practice questions

  • Exam: Monday 06 November, 2023 from 9:00. 15 minutes reading time + 3 hours of work time. Centrally Invigilated Examination.

  • Locations:

    • Copland G39

    • Haydon-Allen G40

    • Moran G007

    • Moran G008

Tip

Plan for this lecture

  1. Main lessons in this course

  2. Second half: Q&A

Review#

Most importantly, make sure to fully understand and remember:

  • definitions

  • facts

  • all named facts and definitions are absolutely essential

Tip

Each good proof starts with definitions

Fundamentals#

  1. Sets 📖

    • language and symbols

    • operations with sets

    • Cartesian product 📖

    • Cardinality 📖

    • Bounded sets and epsilon-balls 📖

    • Open 📖 and closed 📖 sets

    • Compact sets 📖

    • Convex sets 📖

  2. Sequences and convergence

  3. Functions 📖

    • Domain, co-domain, range

    • Image, pre-image

    • Onto (surjections), one-to-one (injections)

    • Bijections, inverse function

    • Continuity of functions 📖

    • Convexity of functions 📖

    • Derivative 📖 and Taylor series 📖

    • Vector-valued functions

    • Partial deriavative 📖, total derivative 📖, gradient 📖

    • Hessian of a function 📖

  4. Correspondences 📖

    • classification, properties of values (smth-valued)

    • upper and lower hemi-continuity 📖

  5. Linear algebra 📖

    • Operations on vectors, inner product

    • Linear independence

    • Bases and dimension

    • Linear maps

    • Matrices and linear equations

    • Algebraic operations for matrices

    • Column space, rank, inverses

    • Determinants 📖, 📖

    • Eigenvalues and Eigenvectors

    • Quadratic Forms

Optimization#

  • General formulation of optimization problems 📖

  • Existence of optima

    • infima, suprema 📖

    • Weierstrass extreme value theorem 📖

  1. Univariate 📖 and bivariate 📖 case

    • Minimizers and maximizers

    • Stationary points

    • Solution algorithm: enumerations of stationary and boundary points 📖

  2. Multivariate unconstrained case

    • First order conditions 📖

    • Necessary second oder conditions with semi-definiteness 📖

    • Sufficient second oder conditions with strict definiteness 📖

    • Second order conditions in R2 case 📖

  3. Multivariate equality constrained case

    • Lagrange method 📖

    • Lagrangian and Lagrange multipliers 📖

    • Constraint qualification assumption

    • Necessary SOC with semi-definiteness 📖

    • Sufficient second oder conditions with strict definiteness 📖

    • Detecting definiteness on a linear constraint set, bordered Hessian 📖

  4. Multivariate inequality constrained case

    • Karush-Kuhn-Tucker conditions 📖

    • complementary slackness, binding vs. non-binding constraints

Convexity#

  • strict ⟹ uniqueness 📖

  • necessary FOCs → sufficient for global optima 📖

  • Detection of weak/strict convexity/concavity

    • Univariate case: second derivative test 📖

    • Multivariate case: Hessian based criterion 📖

Parametric#

  1. The maximum theorem

    • The maximum theorem 📖

    • The maximum theorem under convexity 📖

    • budget correspondence

  2. Envelope theorems

    • For unconstrained problems 📖

    • For constrained problems 📖

    • Lagrange multiplyers as shadow prices

Dynamic optimization#

  • General formulation of dynamic optimization problem 📖

  • Classification of dynamic optimization problems 📖

  • Bellman principle of optimality and dynamic programming

  • Finite and infinite horizon problems

  • Bellman equation 📖

  • Backwards induction algorithm 📖

  • Bellman operator 📖

  • Value function iteration algorithm 📖

  • Contraction mappings 📖, Banah theorem 📖, Blackwell condition 📖