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Semester 1

Advanced Mathematical Economics (ECNM10085)







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Visiting students must have an equivalent of at least 4 semester-long Economics courses at grade B or above for entry to this course. This MUST INCLUDE courses in Intermediate Macroeconomics (with calculus); Intermediate Microeconomics (with calculus); and Probability and Statistics. If macroeconomics and microeconomics courses are not calculus-based, then, in addition, Calculus (or Mathematics for Economics) is required at grade B or above.

Course Summary

This course is about the advanced mathematical tools that are used in economics research. Each mathematical topic is explored in the context of an important economic problem.

Course Description

The topics covered vary from year to year. An example curriculum would be the following mathematics concepts illustrated in the context of general equilibrium theory: Naive Set Theory. This is the language of mathematics, and is widely used by economists. This is important for making precise hypotheses, such as "in every equilibrium, real wages increase over time", and for verifying these hypotheses with logically sound proofs. The main concepts are: sets, functions, logical connectives, quantifiers, countability, induction, proof by contradiction. * Real Analysis and Metric Spaces. This branch of mathematics focuses on continuity and nearness (topology) while putting geometric concepts like distance and angles into the background. These ideas are useful for determining whether an optimal decision is possible, whether an equilibrium of an economy exists, and determining when optimal decisions change drastically when circumstances change. The main concepts are: open sets, continuity, limits, interior, boundary, closure, function spaces, sup metric, Cauchy sequences, connected spaces, complete spaces, compact spaces, Bolzano-Weierstrass theorem, Banach fixed point theorem, Brouwer fixed point theorem. * Convex Analysis. This branch of geometry focuses on comparing extreme points and intermediate points that lie between extremes. These tools are useful for determining whether there is one or several optimal decisions in a particular situation, and determining in which direction optimal choices move when circumstances change. Convex analysis is related to the economic notions of increasing marginal cost and decreasing marginal benefit. The main concepts are: convex sets, convex and concave functions, quasi-convex and quasi-concave functions, supporting hyperplane theorem, separating hyperplane theorem. * Dynamic Programming. This branch of mathematics is about breaking up a complicated optimisation problem involving many decisions into many simple optimisation problems involving few decisions. For example, a lifetime of choices can be broken up into simple choices made day-by-day. The main concepts are: value functions, Bellman equations, Bellman operators. * Envelope Theorem. This is a calculus formula for calculating marginal values, such marginal benefit of saving money. The main concepts are: differentiable support functions, the Benveniste-Scheinkman theorem.

Assessment Information

Written Exam 90%, Coursework 10%, Practical Exam 0%

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