April 16, 2020

Written by: Ruhi Pungaliya

Edited by: Hetal Adhia

MaTH.CoMM was a 7-day online workshop that created a bridge between Mathematics, and Economics and Finance for around 85 pondering minds through the ages of 11 to 60+ year olds from 3rd to 9th April, 2020. This bridge was beautifully created by Mr. Hariharan R. who is a doctoral student at Michigan State University and also a co-founder of Raising A Mathematician Foundation.

The workshop was not only diverse on the age front but also on the background of the participants which ranged from middle-school and college students on one extreme to working professionals and engineering PhD students on the other. Also, the participants joined from three countries with majority of participants from India. Within India, participants belonged to eleven different states highlighting the geographical diversity of the participants in the workshop. One challenge in a workshop involving participants from diverse background was to bring all the participants on the same page. To ensure this, an online classroom was created, and pre-reading materials were shared. The course was designed to explain the math behind the concepts in economics and finance. The motivation for learning concepts like calculus and statistics was derived from economics and finance.

The workshop began by discussing cost functions and elasticity of demand using coordinate geometry, compound interest using Geometric Progressions. Later, discussions involved the use of binomial theorem and the derivation of ‘e’ which eventually brought us to calculus. The math concepts were further used to discuss revenue functions. A simple problem on profit maximization was discussed to understand pricing behavior under monopoly, duopoly, oligopoly, and perfect competition as a limiting case. As the course progressed, participants explored more on Statistics and Data Analytics.At first, they were skeptical of these intimidating jargons but as they continued their journey, they seemed to enjoy the discussions. Initial discussions involved financial mathematics, for example, understanding the difference between compounding and discounting. Later, measures of dispersion were discussed using examples from sports. Finally, regression analysis was introduced by using the data on COVID-19. This was followed by discussion on the need for multiple regression by using the example on relation of wages and years of education and ability. The participants agreed that explaining the relationship between two variables requires one to think beyond those two variables. All the mathematical concepts learnt during the workshop were applied in regression analysis.

All the sessions were interactive, and everyone was encouraged to ask questions as and when they had them. The participants also had help sessions, run by the volunteers of Raising a Mathematician Foundation (Aadityan Ganesh and Sundarraman, both undergrad students from Chennai Mathematical Institute), before the start of every session to obtain clarifications on topics discussed till date. The participants were given problem sets every day to track their understanding of the concepts covered during the workshop and identify the gaps on which they can focus their efforts. The learning appetite of the participants was amazing to say the least. Participants wanted rigorous explanations for concepts and did not accept things at face value. Day 7 ended with feedback given by various participants which included from school students and college students to a vice principal of high school.

Finally,the workshop ended with closing remarks from Mr. Vinay Nair (Secretary, RAM Foundation) and the instructor which ended a unique and fantastic online workshop.

**Day 1**

Day 1 started with cost functions and introduction to fixed cost and variable cost, and their graphical representation. Next a very important and useful topic of micro-economics, elasticity of demand was discussed using basic Coordinate geometry! Which was followed by a healthy debate on shares, profits and business strategies of companies. There was a discussion on information asymmetry and works of George Akerloff and Michael Spence. Later, the analysis shifted to non-linear functions and the need for calculus was discussed which eventually led to the need for understanding the concept of limits. The session ended with an introduction to limits.

**Day 2**

Before the start of Day 2 session, there was a help session conducted by the volunteers who patiently solved the doubts of the participants who needed more clarity. As the session progressed, the discussion on elasticity of demand was continued using linear demand functions. Then the topic of Compound Interest was discussed by connecting it to Geometric Progressions and gradually the limiting case of continuous compounding was explored using binomial theorem. Along the way, we stumbled across the constant ‘e’. In fact, we derived using continuous compounding and this laid the foundation for diving deeper into calculus.The discussion progressed to increasing and decreasing functions, and minima and maxima. The lecture ended with describing the ‘change in change’ i.e. the second-degree derivative. At the end of the session, some interesting questions came about non-linear relationships.

We leveraged on the derivation of to derive all the results in calculus. These derivations were shared through online classroom outside the workshop.

After Day 2, the participants were given a problem set to derive a few results in calculus and some applications of concepts from economics discussed in the past two days of the workshops.The solutions of these practice problems were written elegantly and rigorously by one of the volunteers.

**Day 3**

Day 3 of MaTH.CoMM started with another help session in which the solutions to the problem setfrom day 2were discussed and doubts were clarified on the derivation of the constant ‘e’.

The 3rd session started with some properties of ‘e’, some basics of logarithms and calculus. Using continuous compounding, the so-calledfamous Rule of 72 was derived. Rule of 72 is used to determine the number of years it takes for an investment to double. Calculus was used to explain a problem on profit maximization and price determination in markets ranging from monopoly, duopoly, oligopoly, and the limiting case of perfect competition was examined. The session ended with deriving a relation between Marginal revenue, Average revenue and elasticity of demand using calculus.

**Day 4**

The Day 4 help session was heated as the participants were questioning every little detail which showed their enthusiasm and their mentality of not taking anything at face value.

The session started by reviewing the relationship derived on Day 3 of the workshop and the discussion on linear demand functions and derivation of relationship between average revenue, marginal revenue, and elasticity of demand was discussed. What the instructor showed us next could not be described less than a miracle. What seemed like a notorious economic problem was converted into a simple geometry problem! Using basic geometry like congruence and similarity, the relationship between average revenue, marginal revenue, and elasticity was established. That really opened my eyes (literally). We ended the economics part of the workshop by deriving some results in logarithms and economics.

After three and a half days, we started the Statistics and Data Analytics part of the workshop. At first, the participants were skeptical of these intimidating jargons but as they continued their journey, the concepts seemed less intimidating. Initial discussions were based on financial mathematics, on the difference between compounding and discounting. Later, the discussion was about measures of dispersion and the concept of variance and standard deviation was discussed using examples from Cricket.

**Day 5**

In the help session there were a lot of new claims and theories which were being formed. The discussion never stopped, and the participants were requested to discuss their ideas offline. On the fifth day of the workshop, the concept of variance was explored further. A few results involving variance were derived and why variance is computed as a deviation from mean was explained. Naturally, the discussion flowed into covariance and correlation. Next, we used the Cauchy-Schwarz inequality to derive the limits for correlation. Using this result we further included variance to create more results. At the end of the session, the participants had 7 bonus results along with the 4 main ones. The after-class discussion of Day 5 was on the money multiplier of the country, arbitrage, inflation, interest-rate parity, etc.

**Day 6**

The session of the second last day of the 7-day online workshop started with a quick recap of the result derived the previous day. Next another very important question for investors was answered… “If I want to invest in 2 companies with same expected returns. How should I invest/allocate my money such that the portfolio risk is minimized?” The answer involved a simple application of calculus to minimize a function. The participants were encouraged to explore the result for more than 2 stocks.

Next, the topic of Regression was introduced using COVID-19 data as a motivation to determine the approximate growth rate in the number of infected cases. Further, a famous example from labor economics i.e., the relation between wages and number of years of education was discussed to motivate the need for multiple regression. The participants acknowledged that understanding the relationship between two variables involves analyses that goes beyond those two variables and requires institutional knowledge that goes beyond mathematics and statistics. We used some calculus and basic summation to derive the line of best fit.

In the after-class discussions, some participants requested the instructor to take actual values and data to create regressions. There was a question asked about the current pandemic situation “Whether it is better to let the economy suffer due to lockdown or allow the lockdown”. The participants were provided with a link to podcast by Prof. Luigi Zingales on his economic analysis of lockdown.

**Day 7**

This was the final day of MaTH.CoMM 2020. The help session was in full swing. Everybody was actively participating, and the instructor and volunteers were clarifying the doubts in multiple ways using diverse examples. Thesession was a continuation of regressions and the result on obtaining line of best fit was reviewed. Further, the participants were encouraged to prove some algebraic results w.r.t regression. This was further leveraged to examine the necessary conditions required to obtain regression coefficients.

COVID-19 data was used to explore regressions and explain the utility of statistics in understanding real-world problems using a data-driven approach. A few participants were inspired by the discussion that took place on Day 6 and explored various concepts using real-world data. For example, one participant created some prediction models in python using stock-market data. Another participant created an excel sheet on Gordon’s dividend growth model while another participant had collected and analysed data on COVID-19. The data had many variables that are useful in cross-sectional variation in the number of infected cases.

Time flew by very fast and before we could realise it was time to say goodbye. Day 7 ended with feedback given by various participants- from school and college students to a vice principal of high school.

Finally, Mr. Vinay (Secretary of RAM Foundation) and the instructor gave closing remarks and that was the end of Math.COMM 2020, an unique and fantastic online workshop.

Ruhi Pungaliya

Grade X, Participant of MaTH.CoMM 2020

Pune, Maharashtra, India

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