# Understanding Social Welfare Comparisons without Math Anxiety

### by Carly Trachtman, Agricultural and Resource Economics

#### Teaching Effectiveness Award Essay, 2020

A key concept taught at the beginning of any development economics course is the social welfare function, as many poverty and inequality measures are examples of such functions. Specifically, a social welfare function is a function that takes as inputs the incomes of all the individuals in a society and outputs a single value, such that these values can be used to compare overall welfare levels between societies and over time. We also teach students about mathematical properties that social welfare functions should ideally satisfy. One such property, for example, is that when any number of a society’s people become richer, social welfare should not decrease. The first time I taught this material, I noticed that students really struggled with comprehension, as thinking mathematically about social welfare metrics as “functions” in the abstract felt unfamiliar. Additionally, economics students, many of whom are used to interacting with math in a strictly computational manner, expressed great unease at verifying whether social welfare functions satisfied theoretically desirable mathematical properties.

Hence, the next time I taught this course, before introducing the concept of a social welfare function formally, I had students break into small groups of three or four, and presented the class with a few different four-person “societies,” with information about the income level of each member of each society. Students then were tasked with ranking these societies from highest overall welfare level to lowest overall welfare level, and to be able to justify their rankings. I emphasized that there were not any right or wrong answers, and briefly visited each group during their deliberations to make sure they understood the exercise. A representative from each group then reported their ranking order to the class, which I recorded on the whiteboard. I then gave a lecture that introduced the concepts of social welfare functions and their properties, and afterwards we shifted our focus back to the groups’ rankings. As a class, we went through each property, and discussed which rankings on the board satisfied or violated the property.

In some sense, the initial task of ranking societies was very similar to having each group define their own social welfare function. However, importantly the exercise was devoid of mathematical context; no algebra was explicitly involved. All of the groups actively deliberated amongst themselves in completing the task, without the hesitation that I had seen in my previous class when engaging with the topic. Additionally, by essentially making up their own welfare functions, they were able to conceptualize the notion of quantifying social welfare in a less abstract, more direct way. Moreover, in addition to considering common examples of algebraic functions where these properties do and do not hold in the lecture, we also used the examples of the “functions” created by students themselves. As they had already spent time thinking about an ideal welfare ranking, students had an active stake in wanting to understand whether it satisfied properties that economists thought such a function should. Hence students could think more conceptually about how these properties could be violated, what such a violation would look like in practice, and why these properties might be desirable, all without necessarily thinking in terms of mathematical notation. Once they understood the general concepts, then mathematical notation was further covered in future lectures and problem sets.

I assessed the effectiveness of this method by comparing student attitudes regarding this concept and examination scores on related questions between the first time I taught this course (and students did not complete the ranking task) and the second time (when they did). Notably, the second time, students displayed a higher level of engagement with the material in lecture and less general unease in office hours. They performed better on associated problem sets and exam questions (which involved mathematical notation), too.