Assessment in Mathematics for Economics and Finance

This is a summary of a talk given by Dr. Vasilev at the INERME workshop, "Assessing mathematics in economics courses", 19 June 2024.

I began teaching and assessing a one-semester Quantitative Methods in Economics at the American University in Bulgaria (AUBG), a US-type 4-year undergraduate programme, back in 2016. The material covered was based on Chiang and Wainwright’s Fundamental methods in Economics book. The module is taken by second-/third-year Economics students, who — in addition to Principles and some second-year electives — have already taken a semester in Linear (Matrix) Algebra, and a semester on (single-variable) Calculus. In light of this background, I was able to cover the more advanced topics (and even finish with some simple optimal control, which is in the last chapter of Chiang et al.), as students have sufficient maths and economics knowledge.

Over the duration of the module, I utilized some examples from economics, and fed forward to Intermediate Microeconomics and Intermediate (and Advanced) Macroeconomics. It is worth noting that, at AUBG, Intermediate Microeconomics is more mathematical, while Intermediate Macroeconomics uses mostly static IS-LM and is thus essentially non-mathematical.

Since 2019, I have been regularly teaching a 2-semester module “ECO 1002: Mathematics for Economics and Finance” to Year 1 BSc Economics (Honours) and BSc Economics and Finance (Honours) at the University of Lincoln (UoL). Some students may have taken A-level maths, but it is not obligatory to enrol in the BSc Economics programmes. At UoL, students take concurrently 2-semester Principles sequence, and 2-semester Statistics sequence in their Year 1 at UoL. The module begins with a review of arithmetic and algebraic operations, then functions, to proceed with simple mathematical analysis on functions using differential calculus. Semester A ends with some simple integration: finding anti-derivatives and areas under curves (consumer and producer surplus).

The main textbook used is by Jacques (Mathematics for Economics and Business, Pearson), chosen because of the many exercises provided for students to solve. This is the expected level of problems to show in the final exam (in both semesters A and B). Secondary readings are Thomas (Using Mathematics in Economics, for those who find Jacques too challenging), and Chiang and Wainwright (for those students that are more ambitious). Semester B continues with the rest of Jacques, and includes simplified Kuhn-Tucker setups from Chiang, as well as some comparative statics.

“Tactical” (short-term/superficial) vs “Strategic”(long-term/deep) maths skills

Most teachers would agree that students are interested in getting a high mark — “passing with flying colours” — and then move on and forget everything. This type of learning (“cramming formulae”) overloads short-term memory but is tactically optimal from the individual student perspective. However, “if you do not do calculus in a week, you start to forget” as one of my calculus teachers used to say. Thus, we need students to be able to go through the problem even if they forgot a particular formula/technique; double-checking, understanding the logic of the technique is important. For example, a problem that does not have a solution is a good example to test “deep analytical thinking”. In addition, I include some questions on an exam asking to (quickly/simply) model something. Mathematics is a modelling tool used in economics, after all.

At Lincoln, my assessment covers the whole range. Problems are relatively short and straightforward, as the primary aim is to demonstrate that a technique has been mastered. Topics are in chronological order; students start with simple problems and move further, building confidence and engaging with the exam. More advanced topics (Lagrangeans, etc) have a higher weight, though, which allows for a nice distribution of marks to reflect different level of performance. For deep applications (the “strategic” maths skills), I assign — a week in advance — some more challenging end-of-chapter problems from Chiang et al., and we discuss those at the weekly seminars. Obviously, those problems are not suitable for a 2-hour exam. Still, their usefulness is to develop more “strategic” maths skills that would serve them in Year 2 in Intermediate- and Year 3 Advanced-level theory modules (and beyond).

The correlation between students who do well on the deep questions versus the students who do well on the standard textbook problems may not be perfect. But the former are more likely to do better in the long run, as they have obtained a deeper understanding of the material. Those students would eventually be the best candidates for the postgraduate research track. For example, a third-year undergraduate student at UoL even used some of those techniques to evaluate the effect of the sugar tax in the UK.

To practice for the exam in term A, I distribute a mock exam and we discuss it during the last week’s lecture. In term B, I directly refer students to Jacques, which also contains answers to the problems. If a student needs more guidance, we discuss details during my office hours.

There is some synergy with the other Economics modules. Mathematical techniques covered in Term A are aligned with topics from Micro Principles, while the techniques from Term B are somewhat aligned with Macro Principles (discussion of comparative-static effects in the Keynesian/IS-LM model).

The Situation at Lincoln

Finally, let me finish with some statistics from the University of Lincoln: Maybe a third of the students do not make it successfully through ECO 1002 (in the case when a student fails maths, but does well in stats, we offer the student to transfer to our BA Business Economics Programme, which is less mathematical, but still requires knowledge of stats and econometrics).

The real assessment of maths in economics is actually how well the student does in Year 2 and Year 3 when marks do matter for the final degree classification (ECO 1002 aims to provide a strong grounding for further PGT/PGR studies). Indeed, conditional on passing Year 1 Maths, and Stats, the drop-out rate in Year 2 and Year 3 is very low; students who survived Year 1 do quite well in Year 2 and Year 3. We interpret this as a sign that we successfully weeded out the students who were not up to the UoL standards (those students would not have passed Year 2 and Year 3 Theory and Econometrics sequences anyway). Finally, some Lincoln Economics graduates have been placed in (and successful graduated from) MSc Economics Programmes at St. Andrews, LSE, and UCL. Others started work at the Treasury, ONS, etc.

• Slides from this presentation (PowerPoint format)