EC3314: Mathematical Economics
AY2013/2014, Semester 1, Lecturer: Aditya Goenka
Course Coverage:
1. Linear Algebra
2. Convexity & Separation Theorems
3. Concave & Convex Functions
4. Unconstrained Optimization, Inverse & Implicit Function Theorems
5. Equality Constraints: Lagrangian Multiplier Method
6. Inequality Constraints: Karush-Kuhn-Tucker Conditions
7. Dynamic Programming
8. Correspondence, Continuity and Fixed Point Theorems
This is basically the simplified version of EC5104R. No kidding. No other modules in undergraduate economics will one encounter such mathematical module. This module provides significant rigour in mathematics that I have not seen out of Department of Mathematics, and illustrates how these mathematical theorems is used to obtain results that make economic sense. However, the recent version taught by Prof. Quah placed greater emphasis on proofs, which is similar to MA2108 and MA3110.
The first topic of the module is linear algebra. If you have went to the Academics page, you would have seen the C at the end of MA1101R. Second time going through this topic, I was very determined to perfect this segment and the results did not disappoint me. This part covers basic linear algebra, dealing with matrices, vectors and determinant. Some important things to know are linear independence, basis, rank and normalization. Cramer's Rule is a remarkable result, but I personally think it's to complicated if you don't have prior linear algebra background.
Next up is convexity, it's a rather simple concept but the non-mathematically inclined will faced some difficulties at first. Separation Theorems will talk about convex sets and hyperplanes, which are very abstract concepts. If you have done EC3101, then you can visualize the Edgeworth Box as a 3D space with price as the hyperplane separating the upper contour sets of utility functions. This then leads to the proving of Walrasian Equilibrium First & Second Theorems of Welfare Economics, which are the generalization of the pure exchange economy in EC3101. Then you have Farkas Lemma which proves the absence of arbitrage in financial markets, followed by definition concave/convex function and quasiconcave/quasiconvex functions using Hessian matrix which are important assumptions in many economic optimization.
The third part of the module focus on multivariable optimization of economic problems. In unconstrained segment, you will learn to prove the existence of a maxima, minima or saddle point using necessary and sufficient conditions. Other than the usual first order conditions, you will need to always check for second order conditions using Hessian matrix and principal minors, or alternatively, Bordered Hessian matrix. Then the Lagrangian Multiplier Method is introduced, but it's usually not a problem since it's covered in EC2104. Next will be the Karush-Kuhn-Tucker(KKT) for inequality constraints which is necessary if we no longer rely upon the usual assumptions in other modules. The Lagrangian-KKT question in the final exam proved to be a killer, but I was saved by real analysis knowledge I learn from MA2108 in the previous semester. The last section of optimization deals with dynamic programming, which is also called discrete optimal control theory. In finite-horizon, backward induction can be used, but the Bellman Equation becomes important in infinite-horizon optimization. This topic is the most tedious mathematical problem you will encounter in level-3000 EC-coded modules. This, however, proved to be very useful in understanding intertemporal optimization in higher level modules, especially EC4102.
The last segment is an introduction to slightly more advanced real analysis, with topics including correspondence, continuity, compactness and convergence which are fundamental to proving Banach, Brouwer and Kakutani Fixed Point Theorems, which in turn proves the existence of Nash Equilibria. This is the most abstract past of the whole module, but it's covered very briefly and Prof Goenka don't usually give too difficult things for finals, so you can treat it as enrichment unless you intend to do postgraduate research.
This module is a tough one, but exams are often very easy and application oriented with the exception of the last topic. So, with enough practice, it's not too difficult for those who are interested and not easily daunted by general equations.
Workload: Moderate
Difficulty: Difficult
Grade: A
Course Coverage:
1. Linear Algebra
2. Convexity & Separation Theorems
3. Concave & Convex Functions
4. Unconstrained Optimization, Inverse & Implicit Function Theorems
5. Equality Constraints: Lagrangian Multiplier Method
6. Inequality Constraints: Karush-Kuhn-Tucker Conditions
7. Dynamic Programming
8. Correspondence, Continuity and Fixed Point Theorems
This is basically the simplified version of EC5104R. No kidding. No other modules in undergraduate economics will one encounter such mathematical module. This module provides significant rigour in mathematics that I have not seen out of Department of Mathematics, and illustrates how these mathematical theorems is used to obtain results that make economic sense. However, the recent version taught by Prof. Quah placed greater emphasis on proofs, which is similar to MA2108 and MA3110.
The first topic of the module is linear algebra. If you have went to the Academics page, you would have seen the C at the end of MA1101R. Second time going through this topic, I was very determined to perfect this segment and the results did not disappoint me. This part covers basic linear algebra, dealing with matrices, vectors and determinant. Some important things to know are linear independence, basis, rank and normalization. Cramer's Rule is a remarkable result, but I personally think it's to complicated if you don't have prior linear algebra background.
Next up is convexity, it's a rather simple concept but the non-mathematically inclined will faced some difficulties at first. Separation Theorems will talk about convex sets and hyperplanes, which are very abstract concepts. If you have done EC3101, then you can visualize the Edgeworth Box as a 3D space with price as the hyperplane separating the upper contour sets of utility functions. This then leads to the proving of Walrasian Equilibrium First & Second Theorems of Welfare Economics, which are the generalization of the pure exchange economy in EC3101. Then you have Farkas Lemma which proves the absence of arbitrage in financial markets, followed by definition concave/convex function and quasiconcave/quasiconvex functions using Hessian matrix which are important assumptions in many economic optimization.
The third part of the module focus on multivariable optimization of economic problems. In unconstrained segment, you will learn to prove the existence of a maxima, minima or saddle point using necessary and sufficient conditions. Other than the usual first order conditions, you will need to always check for second order conditions using Hessian matrix and principal minors, or alternatively, Bordered Hessian matrix. Then the Lagrangian Multiplier Method is introduced, but it's usually not a problem since it's covered in EC2104. Next will be the Karush-Kuhn-Tucker(KKT) for inequality constraints which is necessary if we no longer rely upon the usual assumptions in other modules. The Lagrangian-KKT question in the final exam proved to be a killer, but I was saved by real analysis knowledge I learn from MA2108 in the previous semester. The last section of optimization deals with dynamic programming, which is also called discrete optimal control theory. In finite-horizon, backward induction can be used, but the Bellman Equation becomes important in infinite-horizon optimization. This topic is the most tedious mathematical problem you will encounter in level-3000 EC-coded modules. This, however, proved to be very useful in understanding intertemporal optimization in higher level modules, especially EC4102.
The last segment is an introduction to slightly more advanced real analysis, with topics including correspondence, continuity, compactness and convergence which are fundamental to proving Banach, Brouwer and Kakutani Fixed Point Theorems, which in turn proves the existence of Nash Equilibria. This is the most abstract past of the whole module, but it's covered very briefly and Prof Goenka don't usually give too difficult things for finals, so you can treat it as enrichment unless you intend to do postgraduate research.
This module is a tough one, but exams are often very easy and application oriented with the exception of the last topic. So, with enough practice, it's not too difficult for those who are interested and not easily daunted by general equations.
Workload: Moderate
Difficulty: Difficult
Grade: A