Learning Outcomes:
On completion of this model, students should be comfortable with the application of differentiation of non-linear functions, the operation of matrices and be able to apply models to constrained application problems involving multiple variables. In particular, students should be able to relate the relatively abstract methods taught in the course to applied economic problems they may encounter in further courses.
Indicative Module Content:
The textbook for this course is Chiang and Wainwright, Fundamental Methods of Mathematical Economics, Fourth Edition. This course primarily covers chapters 4-13. A brief revision of the content and concepts in chapters 1-3 will be given within the opening lecture. Whether the course will cover chapters 8 and 13 will depend on how the class progresses.
The course opens by considering static analysis of markets in equilibrium, including the mathematical definitions of supply and demand functions and the value of price and quantity traded at market equilibrium. As part of this, it considers solutions to simple linear and non-linear equations. The course expands on the simple case of one commodity to consider multiple commodities and multiple equilibria. It uses matrix algebra to find solutions to linear models.
Chapter 4 considers the fundamentals of matrices and matrix operations, including addition, multiplication and vector operations. Further topics include the identity, transpose and inverse matrices.
Chapter 5 expands on matrices, including consideration of matrix algebra. This includes understanding the conditions for non-singularity of a matrix including matrix rank and determinants. The various methods for finding the inverse matrix will be covered in chapter 5, including expansion by alien cofactors and Cramer's Rule. Application of matrices to economic models will be considered with a series of macro and micro examples.
Chapter 6 will consider comparative statics, considering the change between two equilibrium states in a model. This includes the fundamentals of differentiation of linear and non-linear functions. Chapter 6 will also include the concepts of limits and how they can be evaluated. It will go on to consider limit theorem and how they apply to various functions. The continuity and differentiability of functions will be considered, supported by simple graphical examples.
Chapter 7 addresses the fundamentals of differentiation and their application to comparative statics. This includes the various rules of differentiation and their application to different functions. Chapter 7 is limited to partial differentiation.
Chapter 8 considers the broader topics of total differentiation and the more general approach this enables. This includes differentials, the rules of differentials and total differentials. This will also include derivatives of implicit functions and extension to the simultaneous case.
Chapter 9 introduces the broad concept of optimisation. It considers non-linear functions and how analysis of their first and second derivatives inform our understanding of optimisation. Applications will include assessment of risk preferences, profit and total cost functions. Chapter 9 also includes the Maclaurin and Taylor Series Expansions.
Chapter 10 considers exponential and logarithmic functions and their application in economic models. Consideration includes the choice of the base of the logarithm. It also includes differentiation and optimisation of these functions and some common economic applications.
Chapter 11 extends optimisation to consider optimisation over multiple variables. This include first and second order conditions in the two variable case. It considers quadratic forms of second order total differentials. Chapter ll also considers extension to three variable cases as examined using matric algebra and the n-variable extensions. The formal examination of convexity and concavity is also consider.
Chapter 12 further extends analysis of optimisation to consider constrained optimisation. Specific topics including finding stationary values on functions to be optimised using Lagrangian and total differential approaches. Optimisation will be further extended to consider multiple variable and multiple constraint cases using matrix approaches including Jacobians and bordered Hessians. There is also further consideration of concavity and convexity in the n-variable case. Applications include utility maximisation and the Cobb-Douglas production functions.
Chapter 13 considers further extensions of the methods introduced earlier. This included non-linear programming and Khun-Tucker conditions and associated constraint qualifications. Applications including rationing and optimal pricing.