Identity Elements

In math, there’s the concept of an identity element. Basically, it’s an object that, when used in an operation on another object, leaves the latter object unchanged. In subtraction and addition, for example, the identity element is 0, as adding 0 to anything leaves does not alter the original number. In multiplication it’s 1, as  x*1=x for all x, and with matrices it would be the identity matrix. In logic (and I hadn’t thought about this before doing a little research) an identity element is “truth” with the “and” operation, and “falsity” with the “or” operation, which is really interesting.

Identity elements can even be nontrivial. For example, a matrix that rotates a vector 360 degrees about the origin in R^2 is an identity element even though the general transformation that it performs (a rotation) can substantially change a vector. To be clear, a rotation of 360 degrees would just turn out to be the identity matrix:

but I think being an identity element is a property an object has, rather than the object somehow being identical to the identity element. Both the rotation matrix and the “standard” identity matrix have a similar form that contains certain properties, but they are distinct. The rotation matrix describes a rotation of x degrees about the origin, while the identity matrix as it stands has no geometric interpretation.

There are tons of identity elements spanning all of mathematics and I don’t know much about them. Yet, the idea is interesting to me. Intuitively, most operations have some type of effect on an object. It seems like applying an operation in a substantive sense to any thing is guaranteed to engender some type of change, or else you haven’t really applied an operation. Yet, if we generalize the idea of identity elements beyond math and into day-to-day living, they’re much more common than you’d think.

Let’s think of a non-mathematical identity element in a personal context as an object that invokes no change in an individual (person) when used with an operation. For example, a bowl of cereal when paired with the operation “eating” can be an identity element, as eating a bowl of cereal may satiate hunger, but doesn’t alter the individual in a substantial way. However, the object “God” when paired with the operation “contemplating” in most cases does not qualify as an identity element. Seriously reflecting on the divine is likely to change a person.

I’ll be the first to say generalizing from numbers to people is a terrible move. People are messy and complicated and don’t have any of the nice clean properties that numbers have. Is the element “work” when coupled with the operation “going to” count as an identity element? Heading to the office on the average day doesn’t change you, but the accumulation of each day over many years will certainly make you different. It’s a classic example of the sorites problem, but I’m going to table the issue for now. Even if we don’t have a clear idea of when a person is different in an important way from who they were a little bit ago, we can still draw parallels between identity elements in math and elements that might have a similar property when applied to people. Broadly speaking, some element operation combos leave people the same more often than not and others don’t.

Acknowledging the existence of identity-type elements that affect you (the person) is interesting. It means you’re aware that some actions elements and operations have the potential to change you in a certain way, and it’s incumbent on you to identify which changes are desirable and seek out the corresponding action. I’d hope most people like themselves and are comfortable with who they are, but I think an essential part of something important and true about living well is deliberately trying to be a better person. I also believe that “being a better person” doesn’t mean cultivating a specific set of skills like kindness or empathy that are exercised when needed, but is a result of fundamental changes that happen to you and not necessarily what you can and can’t do. If you believe this, then non-identity element-operation pairs are essential. They’re how you realize change.

I see this idea (if correct) as a tool of aspiration. If we’re seriously committed to “being a better person,” then we better take stock of all of the operations and elements around us and see whether they are nudging us in a direction we want to go. It forces us to consider what things will change us, and choose things that will. How will majoring in statistics vs engineering change you? Anthropology vs English? Living in one area vs another? How would you feel if some of these are identity elements after all, and leave you the same?


Circle Games

Interesting take on recent behavior by the women’s national team

The making of a youtube radical – A great piece. Living with a couple of college-aged boys that consume a lot of youtube, it’s easy to see how they can get sucked in to radical content.

Opportunity Atlas – This came out a bit ago, but I think it’s worth linking to for people who haven’t seen it. Raj Chetty and his team linked census and tax data over thirty years to identify how well people growing up in certain neighborhoods did later in life. Give it a look. Check out your neighborhood.

Weekly links

Instead of posting groups of links as I see fit, I’m going to experiment with compiling everything and making one post during the weekend. — highly recommended. I love finding blogs written by thoughtful people working in private industry. —This is just silly. — not entirely convincing due to limited scope of data and lack of control for “regional inflation” (scare quotes because I am making up the term. It may or may not be a legitimate economic concept). — The episode gets away from student debt towards the end when Dubner is just questioning Daniels on his political career, but it is still a good episode. Income share agreements look promising as a way to improve how people pay for college. Yet, they are not without detractors. No matter your stance on the degree to which private industry should get involved, it’s hard to argue against ISAs being a better form of financing college rather than debt.

Reader Response

A friend of mine penned a response to my post calling for a dramatic expansion of universities to put a dent in inequality. It’s reproduced below with his permission.


While expanding access to large public institutions (such as UCLA) is certainly one method of improving social mobility in a society, the sheer size of such institutions unfortunately inhibits many of the extremely positive attributes that smaller, typically private institutions have. There are two primary reasons for this.

First, social capital – in the form of close bonds to alumni, professors, and peers – is much harder to achieve at large public institutions, partly because the number of alumni, professors, and students leads to a diminished sense of individual ownership and investment in the institution; large public universities don’t need your involvement to succeed since there are so many other successful alumni, whereas a small college can only succeed if every alumni is engaged and giving back, forcing greater effort on both the institution’s part and on the part of its constituents. This fosters a greater number of connections in general and, more importantly, a greater number of close connections, which have a strong tendency of leading to explicit financial benefits – internships, jobs, and investments – as well as benefits that simply improve quality of life, such as meaningful friendships and relationships. Individualized attention in the form of mentorship, guidance, and connection is incredibly difficult to achieve on a large institutional scale (i.e. auditorium-size classes) but much more doable on a small scale – this is also the reason why large conferences tend to be less impactful than intimate retreats.

Second, homogeneous cultures and rigid operating procedures typical of larger institutions limit innovation. Large public institutions have many more moving parts and immense oversight given the brand names they carry (and the huge amount of public funding they receive), meaning that they are unable to move quickly when student needs rapidly change.

Instead of expanding our large public institutions, states might find it more advantageous to fund small, highly-specialized public institutions with independent cultures and operating procedures. These small institutions could operate in a consortium model with other small public institutions nearby (i.e. the Claremont model or the Babson/Olin/Wellesley model), sharing resources while maintaining institutional independence and nimbleness. Public university systems could thus reap the benefits typically found at exclusive liberal arts institutions while maintaining a high level of scale and accessibility.


I have to admit, the consortium model did not cross my mind when I wrote my piece. The five C’s and Babson/Olin/Wellesley are all thriving institutions, so it’s worth examining whether their construction allows us to get the best of both educational access and quality.

More admissions foolery

UCLA conducted an internal report on quid-pro-quo athletic admissions way back in 2014.

It looks a lot like what happened last March. Underqualified athletes, large sums of money, a lack of transparency in the athletic admissions department. Hell, even Rick Singer was implicated. He was quoted in the report trying to claim money does not influence the application process.

The only difference this time is that the donations found their way into university accounts as opposed to sham charities or the pockets of coaches. This is why nobody received more than a slap on the wrist and no laws were broken.

Colonial Massachusetts education facts of the day

All of the following is from David Labaree’s Someone Has to Fail which I am working through now. All emphasis is added.

Only in new England was there a systematic effort by colonial governments to establish schools for all (white) members of a community. Boston established a public Latin school in 1635, only fifteen years after the Mayflower, and Harvard College was chartered in 1636. In 1647, the Massachusetts Bay Colony passed a law requiring that towns of a certain size should establish a primary school and that larger towns should also establish a grammar school. Other colonies in New England gradually followed suit by requiring the public provision of schooling in local communities.


These efforts to provide education in the American colonies had a significant impact on the literacy of the population, especially in New England. […] male literacy in New England reached 60 percent in 1660, 70 percent in 1710, 85 percent in 1760, and 90 percent in 1790. In contrast, male literacy among non-slaves in other colonies remained constant at around 67 percent throughout the 1700s. This was slightly higher than estimates for male literacy in England during the same period, which hovered around 60 percent, but markedly lower than New England, which may have been the first region in the world to achieve something approaching universal literacy in its white male population. [1]


[…] the major factor that promoted schooling in New England during this period was the intensity of the community’s commitment to the Protestant religion, especially the Puritan version that characterized the original English immigrants to the region.


At the core of the Protestant faith —especially the Calvinist version— was the belief that worshipers had a direct connection to God, which, in contrast with the Catholic belief, was not mediated by the church and its priesthood. As a result, the faithful could not afford to be left illiterate, which would make them dependent on a literate clergy to interpret and transmit the gospel. Instead they needed direct access to the word of God in order to maintain their faith, and this required learning how to read. Therefore, at the heart of the push for schooling in colonial America was a profoundly conservative vision of education’s mission: to preserve piety and maintain the faith.

For all of the criticism that religion endures in some circles, it was directly responsible for the emphasis on education that set the stage for America to become a wealthy industrialized nation.

Louis Vuitton Education

In light of the college admissions scandal, I’ve been thinking about the roles we expect our educational institutions to fulfill. We are repulsed by the those who sought to lie, bribe, and cheat their way into schools because we see education as a tool of social mobility. Typical American success stories often look like an immigrant/child of immigrants working their way through the educational system during their youth to land at an elite college with a ticket to the upper-middle class. Think Ben Carson or Shaan Patel.

Yet, why should we reasonably expect top-tier private universities to give economic opportunities to those that need them the most? It is true they have a moral/civic obligation to do so, but it does not fall exactly in line with their demonstrated goals. Big name schools want to retain prestige and power while doing what is necessary to avoid scrutiny.

This corresponds to what I will call the “Louis Vuitton” theory of higher education (stolen from Malcolm Gladwell in a conversation with Tyler Cowen).

Restricting supply is a surefire way to increase the price/value of a product if you’re a monopoly, or participate in what economists call “monopolistic competition,” which is what happens if you sell differentiated products of the same type. Louis Vuitton could sell many more bags than it already does, but then it would charge lower prices and be in the “commodity bad business” as Gladwell quips. Any monopolist knows she will make more money by reducing the accessibility of the product and waiting for the price to rise. This benefits consumers as well, if they can get their hands on a bag. Gladwell, and others, surely understands that part of the thrill of owning a luxury item is knowing few others have it. It’s the reason fashion companies charge exorbitant amounts for their products and destroy unsold merchandise.

I’m going to pick on Harvard. It’s safe to assume it wants to remain the best, most prestigious university in the world. Yet, prestige is inversely proportional to access. The reason why a Harvard undergraduate degree is so valuable is due in part because 96% of students who apply to get one do not. Any increase in the number of degrees awarded would decrease their value. Imagine if Harvard enrolled as many undergraduates as Ohio State or the University of Florida. 45,000 other students in your graduating class would surely put downward pressure on the perceived prestige of your degree.

Harvard, and other elite private universities, are “in the luxury handbag business, not the education business” according to Gladwell. This is the reason why enrollments are small and the price to get in —whether you’re paying indirectly by living in a good school district, paying private school tuition, hiring tutors, or bribing— is high. On the flipside, this confers huge benefits in the form of better future earnings prospects and increased social status to those that can finesse their way into a hyper-selective institution.

When you recognize some colleges behave like fashion houses, it seems downright irrational to expect elite private institutions to provide social mobility for Americans on the scale we desire. Even if they do commit to increasing the proportion of low-income undergraduates under threat of having their federal funding pulled, increasing percentages can only get you so far. There are still no incentives for these schools to dramatically increase their access.

It’s not like Harvard faces non-brand related barriers to expanding its educational reach, either. UCLA has an undergraduate enrollment of 32,000 on a campus of 419 acres. Harvard’s 6,500 undergraduates roam 567 acres, meaning space is definitely not an issue (UCLA is actually building even more dorms outside my window right now). Money is another restraint, to be sure, but I’d be skeptical of any financial excuses from a university that has a $40 billion endowment and just closed a $9.6 billion fundraising campaign. Yet, Harvard retains the status quo for the same reason LV doesn’t open a surplus store. You have to protect the brand.

Suppose Harvard increases the percentage of undergraduates from the bottom 20% of the income distribution to 25%, from where it currently stands at 4.5%. This is a jump from 292 to 1625 students from the bottom quintile of income. These numbers may seem impressive until you consider UCLA already educates 2656 students in the same economic bracket without having to undergo any significant demographic change in their undergraduate population.

Given, UCLA and other public universities are about five times as large their private elite counterparts, but this is exactly my point. I’ve written in the past about the educational limitations of massive public universities —and I stand by those views— but it appears institutions like those making up the University of California or the SUNY/CUNY system are the ones we should be paying attention to if we want the average American to be able to exercise some type of economic mobility. They do not fall prey to Louis Vuitton incentives and seem to understand they are providing a public good which entails occasional dings to their prestige. It’s never sexy to build bridges or roads or educate a poor student body, but nobody will argue these activities don’t serve vital long-term economic functions.

Public institutions actually have a chance of touching enough students to make anything resembling a dent in income inequality, and they’re good at it. In data compiled by the NY Times, 8 out of the top 10 colleges nationally with the highest social mobility index are public (Vaugh college, what I linked to, is private, but scroll down to the social mobility index row and click “all colleges” on the right hand side to see the top 10). Students going to these universities aren’t discussing heady political theory in an Ivy League seminar room, but the economic expected value of the average low-income individual applying to the City College of New York is much higher than submitting to Brown or Dartmouth.  

Dylan Mathews, writing for Vox, seems to recognize what a school that’s serious about social mobility would look like when he throws out that Harvard should double, or nearly triple its undergraduate enrollment to move the needle on poverty. The implicit statement is that a high percentage of low-income students is not enough, but an independently large number of such students must be graduating every year. Yet, some well-placed italics in the preceding clause, “if [Harvard] really wanted to expand the school’s impact on poverty and mobility,” indicate that he understands the Louis Vuitton mentality in higher education more than most. Mathews’ statement is a throwaway bit in an article about making a bad system transparent, but not better, by auctioning off spots to Ivies and the like. Princeton could accept these legal or illegal admission payments in ExxonMobil stock or bitcoin for all I care. Large public universities are the real engine of social mobility in the America. They are the way to realize the visions of opportunity we irrationally vest in private institutions.